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This is the final manuscript (October 1995) of Georg G. Raffelt's monograph, published by the University of Chicago Press in 1996. It uses stellar energy-loss arguments to constrain new particles, covering stellar evolution, white dwarfs, neutron stars, globular clusters, axion processes, neutrino properties and oscillations, solar neutrinos, and supernova 1987A. It is a book by another author, kept in the particle physics folder.

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Georg G. Raﬀelt Stars as Laboratories for Fundamental Physics The Astrophysics of Neutrinos, Axions, and Other Weakly Interacting Particles Final Manuscript, October 1995 Published by the University of Chicago Press, 1996 Georg G. Raﬀelt is a staﬀ researcher at the Max-Planck-Institut f¨ ur Physik (MPP) in Munich, Germany, and a member of the Sonderforschungsbereich Astroteilchen- physik (Special Research Center Astroparticle Physics) at the Technische Univer- sit¨ at M¨ unchen. He is co-editor of the journal Astroparticle Physics . He studied physics at the University of Munich and the University of California at Berkeley, received a Master of Arts from the latter, and a Diplom and in 1986 a doctorate from the former. Before taking up his position at the MPP in 1990 he has completed several years of postdoctoral research at the Astronomy Department in Berkeley, the Institute for Geophysics and Planetary Physics in Livermore, and the Institute for Advanced Study in Princeton. To be published by the University of Chicago Press, Chicago. c⃝University of Chicago Press (1996) Table of Contents Preface (xiv) Acknowledgments (xxii) 1. The Energy-Loss Argument 1.1 Introduction (1) 1.2 Equations of Stellar Structure (5) 1. Hydrostatic Equilibrium (5) ⋄2. Generic Cases of Stellar Structure (7) ⋄ 3. Energy Conservation (10) ⋄4. Energy Transfer (11) ⋄5. Gravitational Settling (13) 1.3 Impact of Novel Particles (14) 1. Energy Loss (14) ⋄2. Application to the Sun (16) ⋄3. Radiative Energy Transfer (17) ⋄4. Opacity Contribution of Arbitrary Bosons (18) ⋄5. Solar Bound on Massive Pseudoscalars (20) 1.4 General Lesson (21) 2. Anomalous Stellar Energy Losses Bounded by Observations 2.1 Stages of Stellar Evolution (23) 1. The Main Sequence (23) ⋄2. Becoming a Red Giant (28) ⋄3. Helium Ignition (33) ⋄4. The Horizontal Branch (34) ⋄5. From Asymptotic Giants to White Dwarfs (35) ⋄6. Type I Supernovae (36) ⋄7. Intermediate-Mass Stars (37) ⋄8. Massive Stars and Type II Supernovae (37) ⋄9. Variable Stars (39) 2.2 White-Dwarf Cooling (42) 1. Theoretical and Observed White-Dwarfs Properties (42) ⋄2. Cooling Theory (45) ⋄3. Neutrino Cooling (47) ⋄4. Cooling by Boson Emission (50) ⋄5. Period Decrease of Variable White Dwarfs (52) 2.3 Neutron Stars (54) 1. Late-Time Cooling (54) ⋄2. X-Ray Observations (56) ⋄3. Nonstandard Cooling and Heating Eﬀects (58) ⋄4. Cooling by Particle Emission (59) v vi Table of Contents 2.4 Globular-Cluster Stars (60) 1. Observables in the Color-Magnitude Diagram (60) ⋄2. Theoretical Re- lations (65) ⋄3. Observational Results (70) ⋄4. Interpretation of the Observational Results (74) ⋄5. An Alternate Analysis (76) ⋄6. Systematic Uncertainties (78) 2.5 Particle Bounds from Globular-Cluster Stars (79) 1. Helium-Burning Lifetime (79) ⋄2. Helium Ignition (83) 2.6 Summary (87) 3. Particles Interacting with Electrons and Baryons 3.1 Introduction (89) 3.2 Compton Process (91) 1. Vector Bosons (91) ⋄2. Scalars (93) ⋄3. Pseudoscalars (94) ⋄4. Neutrino Pairs (95) ⋄5. Energy-Loss Rates (96) ⋄6. Applying the Energy-Loss Argument (98) 3.3 Pair Annihilation (99) 3.4 Free-Bound and Bound-Free Transitions (100) 3.5 Bremsstrahlung (101) 1. Nondegenerate, Nonrelativistic Medium (101) ⋄2. High Degeneracy: Pseudoscalars (103) ⋄3. High Degeneracy: Neutrino Pairs (105) ⋄4. Neutron- Star Crust (106) ⋄5. Applying the Energy-Loss Argument (107) 3.6 Astrophysical Bounds on Yukawa and Gauge Couplings (109) 1. Pseudoscalars (Axions) (109) ⋄2. Energy Loss by Scalar and Vector Bosons (110) ⋄3. Long-Range Forces (112) ⋄4. Leptonic and Baryonic Gauge Interactions (114) 3.7 Graviton Emission from Stars (116) 4. Processes in a Nuclear Medium 4.1 Introduction (117) 4.2 Axionic Bremsstrahlung Process (119) 1. Matrix Element for NN→NNa (119) ⋄2. Energy-Loss Rate (120) ⋄ 3. Nondegenerate Limit (121) ⋄4. Degenerate Limit (123) ⋄5. Bremsstrah- lung Emission of Scalars (124) ⋄6. Mixture of Protons and Neutrons (125) 4.3 Neutrino Pair Emission (126) 1. Structure Function (126) ⋄2. Bremsstrahlung Emission of Neutrino Pairs (130) 4.4 Axion Opacity (131) Table of Contents vii 4.5 Neutrino Opacity (132) 1. Elastic Scattering (132) ⋄2. Pair Absorption (132) ⋄3. Inelastic Scattering (133) 4.6 Structure Functions (136) 1. Formal Deﬁnition (136) ⋄2. Nonrelativistic Limit (138) ⋄3. The f-Sum Rule (139) ⋄4. Long-Wavelength Properties (141) ⋄5. Axion Emission in the Classical Limit (143) ⋄6. Classical vs. Quantum Result (146) ⋄ 7. High-Density Behavior (147) 4.7 Eﬀective Nucleon Mass and Coupling (151) 4.8 The URCA Processes (152) 4.9 Novel Phases of Nuclear Matter (155) 1. Pion Condensate (155) ⋄2. Quark Matter (157) ⋄3. Bubble Phase (159) 4.10 Emission of Right-Handed Dirac Neutrinos (160). 5. Two-Photon Coupling of Low-Mass Bosons 5.1 Electromagnetic Coupling of Pseudoscalars (165) 5.2 Primakoﬀ Process in Stars (168) 1. Screened Cross Section and Emission Rate (168) ⋄2. Plasmon Decay and Coalescence (170) ⋄3. Axion Emission from Electromagnetic Plasma Fluctuations (172) ⋄4. Solar Axion Spectrum (175) ⋄5. Globular-Cluster Bound on ga (176) 5.3 Search for Cosmic Axions (176) 5.4 Axion-Photon Oscillations (179) 1. Mixing Equations (179) ⋄2. Solar Axions (181) ⋄3. Shining Light through Walls (182) ⋄4. Vacuum Birefringence (183). 5.5 Astrophysical Magnetic Fields (185) 1. Transitions in Magnetic Fields of Stars (185) ⋄2. Birefringence in a Pulsar Magnetosphere (186) ⋄3. Conversion of Stellar Arions in the Galactic Field (188) ⋄4. Polarimetry of Distant Radio Sources (190) ⋄5. Temperature Fluctuations in the Cosmic Microwave Background (191). 5.6 Summary of Constraints on ga (191) 6. Particle Dispersion and Decays in Media 6.1 Introduction (193) 6.2 Particle Dispersion in Media (196) 1. Refractive Index and Forward Scattering (196) ⋄2. Particle Momentum and Velocity (198) ⋄3. Wave-Function Renormalization (200) viii Table of Contents 6.3 Photon Dispersion (202) 1. Maxwell’s Equations (202) ⋄2. Linear Response of the Medium (204) ⋄ 3. Isotropic Polarization Tensor in the Lorentz Gauge (206) ⋄4. Lowest- Order QED Calculation of Π (209) ⋄5. Dispersion Relations (212) ⋄ 6. Renormalization Constants ZT;L(217) 6.4 Screening Eﬀects (219) 1. Debye Screening (219) ⋄2. Correlations and Static Structure Factor (222) ⋄3. Strongly Coupled Plasma (223) ⋄4. Screened Coulomb Scattering (224) 6.5 Plasmon Decay in Neutrinos (227) 1. Millicharged Neutrinos (227) ⋄2. Neutrino Dipole Moments (228) ⋄ 3. Standard-Model Couplings (229) ⋄4. Summary of Decay Rates (231) ⋄ 5. Energy-Loss Rates (232) ⋄6. Astrophysical Bounds on Neutrino Electro- magnetic Properties (234) 6.6 Neutrino Form Factors in Media (237) 6.7 Neutrino Refraction (241) 1. Neutrino Refractive Index (241) ⋄2. Higher-Order Eﬀects (245) ⋄3. The Sun a Neutrino Lens? (247) 6.8 Majoron Decay (248) 7. Nonstandard Neutrinos 7.1 Neutrino Masses (251) 1. The Fermion Mass Problem (251) ⋄2. Dirac and Majorana Masses (253) ⋄ 3. Kinematical Mass Bounds (254) ⋄4. Neutrinoless Double-Beta Decay (257) ⋄5. Cosmological Mass Bounds (258). 7.2 Neutrino Mixing and Decay (260) 1. Flavor Mixing (260) ⋄2. Standard-Model Decays of Mixed Neutrinos (263) 7.3 Neutrino Electromagnetic Form Factors (267) 1. Overview (267) ⋄2. Single-Photon Coupling (268) ⋄3. Two-Photon Coupling (271) 7.4 Electromagnetic Processes (272) 7.5 Limits on Neutrino Dipole Moments (275) 1. Scattering Experiments (275) ⋄2. Spin-Flip Scattering in Supernovae (277) ⋄3. Spin-Flip Scattering in the Early Universe (277) ⋄4. Search for Radiative Neutrino Decays (278) ⋄5. Plasmon Decay in Stars (279). Table of Contents ix 8. Neutrino Oscillations 8.1 Introduction (280) 8.2 Vacuum Oscillations (282) 1. Equation of Motion for Mixed Neutrinos (282) ⋄2. Two-Flavor Oscilla- tions (284) ⋄3. Distribution of Sources and Energies (287) ⋄4. Experimental Oscillation Searches (289) ⋄5. Atmospheric Neutrinos (290) 8.3 Oscillations in Media (293) 1. Dispersion Relation for Mixed Neutrinos (293) ⋄2. Oscillations in Homo- geneous Media (296) ⋄3. Inhomogeneous Medium: Adiabatic Limit (297) ⋄4. Inhomogeneous Medium: Analytic Results (299) ⋄5. The Triangle and the Bathtub (301) ⋄6. Neutrino Oscillations without Vacuum Mixing (303) 8.4 Spin and Spin-Flavor Oscillations (304) 1. Vacuum Spin Precession (304) ⋄2. Spin Precession in a Medium (306) ⋄ 3. Spin-Flavor Precession (306) ⋄4. Twisting Magnetic Fields (308) 9. Oscillations of Trapped Neutrinos 9.1 Introduction (310) 9.2 Kinetic Equation for Oscillations and Collisions (313) 1. Stodolsky’s Formula (313) ⋄2. Matrix of Densities (315) ⋄3. Free Evolution: Flavor Oscillations (316) ⋄4. Interaction with a Background Medium (317) 9.3 Neutral-Current Interactions (320) 1. Hamiltonian (320) ⋄2. Neutrino Refraction (321) ⋄3. Kinetic Terms (322) ⋄4. Recovering Stodolsky’s Formula (324) ⋄5. Weak-Damping Limit (325) ⋄6. Small Mixing Angle (328) ⋄7. Flavor Conversion by Neutral Cur- rents? (328) 9.4 Charged-Current Interactions (329) 1. Hamiltonian (329) ⋄2. Kinetic Terms (329) ⋄3. Weak-Damping Limit (330) 9.5 Flavor Conversion in a SN Core (332) 1. Rate Equation (332) ⋄2. Neutrino Interaction Rates (334) ⋄3. Time Scale for Flavor Conversion (335) 9.6 Sterile Neutrinos and SN 1987A (338) 10. Solar Neutrinos 10.1 Introduction (341) 10.2 Calculated Neutrino Spectrum (347) 1. Individual Sources (347) ⋄2. Standard Solar Models (351) ⋄3. Uncer- tainties of Standard Neutrino Predictions (353) x Table of Contents 10.3 Observations (357) 1. Absorption Reactions for Radiochemical Experiments (357) ⋄2. Chlo- rine Detector (Homestake) (360) ⋄3. Gallium Detectors (SAGE and GALLEX) (362) ⋄4. Water Cherenkov Detector (Kamiokande) (365) ⋄ 5. Summary (372) 10.4 Time Variations (372) 1. Day-Night Eﬀect (372) ⋄2. Seasonal Variation (372) ⋄3. Correlation with Solar Cycle at Homestake (373) ⋄4. Summary (376) 10.5 Neutrino Flux Deﬁcits (377) 1. Boron Flux (377) ⋄2. Beryllium Flux (378) ⋄3. An Astrophysical Solution? (379) 10.6 Neutrino Oscillations (380) 1. Which Data to Use? (380) ⋄2. Vacuum Oscillations (381) ⋄3. Resonant Oscillations (MSW Eﬀect) (384) 10.7 Spin and Spin-Flavor Oscillations (387) 10.8 Neutrino Decay (389) 10.9 Future Experiments (390) 1. Superkamiokande (390) ⋄2. Sudbury Neutrino Observatory (SNO) (392) ⋄3. BOREXINO (393) ⋄4. Homestake Iodine Detector (394) ⋄5. Sum- mary (394) 11. Supernova Neutrinos 11.1 Stellar Collapse and Supernova Explosions (395) 1. Stellar Collapse (395) ⋄2. Deleptonization and Cooling (399) ⋄3. Su- pernova Explosions (401) ⋄4. Nucleosynthesis (405) 11.2 Predicted Neutrino Signal (407) 1. Overall Features (407) ⋄2. Energies and Spectra (408) ⋄3. Time Evolution of the Neutrino Signal (411) 11.3 SN 1987A Neutrino Observations (414) 1. Supernova 1987A (414) ⋄2. Neutrino Observations (415) ⋄3. Analysis of the Pulse (423) ⋄4. Neutrino Mass and Pulse Duration (426) ⋄5. Anomalies in the Signal? (427) 11.4 Neutrino Oscillations (430) 1. Overview (430) ⋄2. Prompt eBurst (432) ⋄3. Cooling-Phase e’s (434) ⋄4. Shock Revival (436) ⋄5. R-Process Nucleosynthesis (437) ⋄ 6. A Caveat (442) 11.5 Neutrino Propulsion of Neutron Stars (443) 11.6 Future Supernovae (445) Table of Contents xi 12. Radiative Particle Decays from Distant Sources 12.1 Preliminaries (449) 12.2 Laboratory Experiments (451) 1. Spectrum of Decay Photons (451) ⋄2. Electron Neutrinos from Reac- tors (453) ⋄3. Heavy Neutrinos from Reactors (455) ⋄4. Neutrinos from a Beam Stop (456) 12.3 Particles from the Sun (458) 1. Electron Neutrinos (458) ⋄2. Heavy Neutrino Admixtures (461) ⋄3. New Particles (462) 12.4 Supernova 1987A (462) 1. Decay Photons from Low-Mass Neutrinos (462) ⋄2. SMM Observa- tions (465) ⋄3. Radiative Decay Limit: Low-Mass Neutrinos (467) ⋄ 4. Decay Photons from High-Mass Neutrinos (469) ⋄5. Radiative Decay Limits: High-Mass Neutrinos (472) ⋄6. Summary of →′ Limits (477) ⋄7. Limit on →ee+e−(480) ⋄8. Heavy, Sterile Neutrinos (482) ⋄ 9. Axions (483) ⋄10. Supernova Energetics (483) 12.5 Galactic Supernovae and →ee+e(484) 1. Bounds on the Positron Flux (484) ⋄2. Can the Tau Neutrino Be Heavy? (485) 12.6 Neutrinos from All Stars (486) 12.7 Cosmological Bounds (488) 1. Neutrinos (488) ⋄2. Axions (491) 13. What Have We Learned from SN 1987A? 13.1 Introduction (493) 13.2 Basic Characteristics of the Neutrino Burst (494) 1. Fluence (494) ⋄2. Energy Distribution (495) ⋄3. Prompt eBurst (496) ⋄4. Nonobservation of a -Ray Burst (497) 13.3 Dispersion Eﬀects (497) 1. Photons vs. Antineutrinos (497) ⋄2. Neutrinos vs. Antineutrinos (498) ⋄3. Intrinsic Dispersion of the e-Pulse (499) 13.4 Duration of Neutrino Emission (501) 1. A General Argument (501) ⋄2. Analytic Criterion in the Free-Streaming Limit (504) ⋄3. Trapping Limit (506) 13.5 Axions (508) 1. Numerical Studies (508) ⋄2. Impact of Multiple-Scattering Eﬀects (511) 13.6 How Many Neutrino Flavors? (513) xii Table of Contents 13.7 Neutrino Opacity (514) 13.8 Right-Handed Neutrinos (516) 1. Dirac Mass (516) ⋄2. Right-Handed Currents (519) ⋄3. Magnetic Dipole Moments (521) ⋄4. Millicharges (522) ⋄5. Charge Radius (523) 14. Axions 14.1 The Strong CP-Problem (524) 14.2 The Peccei-Quinn Mechanism (521) 1. Generic Features (526) ⋄2. Axions as Nambu-Goldstone Bosons (528) ⋄3. Pseudoscalar vs. Derivative Interaction (531) ⋄4. The Onslaught of Quantum Gravity (532) 14.3 Fine Points of Axion Properties (534) 1. The Most Common Axion Models (534) ⋄2. Axion Mass and Coupling to Photons (535) ⋄3. Model-Dependent Axion-Fermion Coupling (536) 14.4 Astrophysical Axion Bounds (538) 14.5 Cosmological Limits (540) 15. Miscellaneous Exotica 15.1 Constancy of Fermi’s Constant (545) 15.2 Constancy of Newton’s Constant (546) 1. Present-Day Constraints from Celestial Mechanics (546) ⋄2. Big-Bang Nucleosynthesis (547) ⋄3. Properties of the Sun (549) ⋄4. White Dwarfs (551) ⋄5. Globular Clusters (551) 15.3 Test of the Equivalence Principle (554) 15.4 Photon Mass and Charge (555) 15.5 Free Quarks (556) 15.6 Supersymmetric Particles (557) 15.7 Majorons (558) 1. Particle-Physics and Cosmological Motivations (558) ⋄2. Majorons and Stars (562) 15.8 Millicharged Particles (564) 16. Neutrinos: The Bottom Line 16.1 Standard Neutrinos (568) 16.2 Minimally Extended Standard Model (570) 1. Cosmological Mass Limit for All Flavors (570) ⋄2. Oscillations of Solar Neutrinos (571) ⋄3. Oscillation of Supernova Neutrinos (572) ⋄4. Elec- tromagnetic Properties (573) Table of Contents xiii 16.3 New Interactions (574) 1. Majorana Masses (574) ⋄2. “Heavy” Neutrinos and Fast Decays (574) ⋄3. Electromagnetic Properties (576) ⋄4. Summary (579) Appendices A. Units and Dimensions (580) B. Neutrino Coupling Constants (583) C. Numerical Neutrino Energy-Loss Rates (585) 1. Plasma Process (585) ⋄2. Photoneutrino and Pair-Annihilation Pro- cess (586) ⋄3. Bremsstrahlung (588) ⋄4. Total Emission Rate (589) D. Characteristics of Stellar Plasmas (591) D.1 Normal Matter (591) 1. Temperatures and Densities (591) ⋄2. Relativistic Conditions for Elec- trons (593) ⋄3. Electron Degeneracy (593) ⋄4. Plasma Frequency (596) ⋄ 5. Screening Scale (596) ⋄6. Summary (598) D.2 Nuclear Matter (600) 1. The Ideal p n e eGas (600) ⋄2. Kinetic and Chemical Equilibrium (600) ⋄3. Cold Nuclear Matter (601) ⋄4. Hot Nuclear Matter (602) References (606) Acronyms (642) Symbols (644) Subject Index (649) Preface Ever since Newton proposed that the moon on its orbit follows the same laws of motion as an apple falling from a tree, the heavens have been a favorite laboratory to test the fundamental laws of physics, notably classical mechanics and Newton’s and Einstein’s theories of gravity. This tradition carries on—the 1993 physics Nobel prize was awarded to R. A. Hulse and J. H. Taylor for their 1974 discovery of the binary pulsar PSR 1913+16 whose measured orbital decay they later used to identify gravitational wave emission. However, the scope of physical laws necessary to understand the phenomena observed in the super- lunar sphere has expanded far beyond these traditional ﬁelds. Today, astrophysics has become a vast playing ground for applications of the laws of microscopic physics, in particular the properties of elementary particles and their interactions. This book is about how stars can be used as laboratories to probe fundamental interactions. Apart from a few arguments relating to grav- itational physics and the nature of space and time (Is Newton’s constant constant? Do all relativistic particles move with the same limiting ve- locity? Are there novel long-range interactions?), most of the discussion focusses on the properties and nongravitational interactions of elemen- tary particles. There are three predominant methods for the use of stars as particle- physics laboratories. First, stars are natural sources for photons and neutrinos which can be detected on Earth. Neutrinos are now routinely measured from the Sun, and have been measured once from a collapsing star (SN 1987A). Because these particles literally travel over astronom- ical distances before reaching the detector one can study modiﬁcations of the measured signal which can be attributed to propagation and dis- persion eﬀects, including neutrino ﬂavor oscillations or axion-photon oscillations in intervening magnetic ﬁelds. It is well known that the discrepancy between the calculated and measured solar neutrino spec- tra is the most robust, yet preliminary current indication for neutrino oscillations and thus for nonvanishing neutrino masses. xiv Preface xv Second, particles from distant sources may decay, and there may be photons or even measurable neutrinos among the decay products. The absence of solar x- and -rays yields a limit on neutrino radia- tive decays which is as “safe” as a laboratory limit, yet nine orders of magnitude more restrictive. An even more restrictive limit obtains from the absence of -rays in conjunction with the SN 1987A neutri- nos which allows one to conclude, for example, that even must obey the cosmological limit of m∼<30 eV unless one invents new invisible decay channels. Third, the emission of weakly interacting particles causes a direct energy-loss channel from the interior of stars. For neutrinos, this eﬀect has been routinely included in stellar evolution calculations. If new low- mass elementary particles were to exist such as axions or other Nambu- Goldstone bosons, or if neutrinos had novel interactions with the stellar medium such as one mediated by a putative neutrino magnetic dipole moment, then stars might lose energy too fast. A comparison with the observed stellar properties allows one to derive restrictive limits on the operation of a new energy-loss or energy-transfer mechanism and thus to constrain the proposed novel particle interactions. While these and related arguments as well their application and re- sults are extensively covered here, I have not written on several topics that might be expected to be represented in a book on the connection between particle physics and stars. Neutron stars have been speculated to consist of quark matter so that in principle they are a laboratory to study a quark-gluon plasma. As I am not familiar enough with the liter- ature on this interesting topic I refer the reader to the review by Alcock and Olinto (1988) as well as to the more recent proceedings of two top- ical conferences (Madsen and Haensel 1992; Vassiliadis et al. 1995). I have also dodged some important issues in the three-way relation- ship between cosmology, stars, and particle physics. If axions are not the dark matter of the universe, it is likely ﬁlled with a “background sea” of hypothetical weakly interacting massive particles (WIMPs) such as the lightest supersymmetric particles. Moreover, there may be ex- otic particles left over from the hot early universe such as magnetic monopoles which are predicted to exist in the framework of typical grand uniﬁed theories (GUTs). Some of the monopoles or WIMPs would be captured and accumulate in the interior of stars. GUT monopoles are predicted to catalyze nucleon decay (Rubakov-Callan- eﬀect), providing stars with a novel energy source. This possibility can be constrained by analogous methods to those presented here which limit anomalous energy losses. The resulting constraints on the pres- xvi Preface ence of GUT monopoles in the universe have been reviewed, for exam- ple, in the cosmology book of Kolb and Turner (1990). Because nothing of substance has changed, a new review did not seem warranted. WIMPs trapped in stars would contribute to the heat transfer be- cause their mean free path can be so large that they may be orbit- ing almost freely in the star’s gravitational potential well, with only occasional collisions with the background medium. Originally it was thought that this eﬀect could reduce the central solar temperature enough to solve the solar neutrino problem, and to better an alleged discrepancy between observed and predicted solar p-mode frequencies. With the new solar neutrino data it has become clear, however, that a reduction of the central temperature alone cannot solve the problem. Worse, solving the “old solar neutrino problem” by the WIMP mecha- nism now seems to cause a discrepancy with the observed solar p-mode frequencies (Christensen-Dalsgaard 1992). In addition, a signiﬁcant ef- fect requires relatively large scattering cross sections and thus rather contrived particle-physics models. Very restrictive direct laboratory constraints exist for the presence of these “cosmions” in the galaxy. Given this status I was not motivated to review the topic in detail. The annihilation of dark-matter WIMPs captured in the Sun or Earth produces high-energy neutrinos which are measurable in terres- trial detectors such as Kamiokande or the future Superkamiokande, NESTOR, DUMAND, and AMANDA Cherenkov detectors. This in- direct approach to search for dark matter may well turn into a serious competitor for the new generation of direct laboratory search experi- ments that are currently being mounted. This material is extensively covered in a forthcoming review Supersymmetric Dark Matter by Jung- man, Kamionkowski, and Griest (1995); there is no need for me to duplicate the eﬀort of these experts. The topics covered in my book revolve around the impact of low- mass or massless particles on stars or the direct detection of this ra- diation. The highest energies encountered are a few 100 MeV (in the interior of a SN core) which is extremely small on the high-energy scales of typical particle accelerator experiments. Therefore, stars as labora- tories for fundamental physics help to push the low-energy frontier of particle physics and as such complement the eﬀorts of nonaccelerator particle experiments. Their main thrust is directed at the search for nonstandard neutrino properties, but there are other fascinating top- ics which include the measurement of parity-violating phenomena in atoms, the search for neutron or electron electric dipole moments which would violate CP, the search for neutron-antineutron oscillations, the Preface xvii search for proton decay, or the search for particle dark matter in the galaxy by direct and indirect detection experiments (Rich, Lloyd Owen, and Spiro 1987). It is fascinating that the IMB and Kamiokande water Cherenkov detectors which had been built to search for proton decay ended up seeing supernova (SN) neutrinos instead. The Fr´ ejus detec- tor, instead of seeing proton decay, has set important limits on the oscillation of atmospheric neutrinos. Kamiokande has turned into a major solar neutrino observatory and dark-matter search experiment. The forthcoming Superkamiokande and SNO detectors will continue and expand these missions, may detect a future galactic SN, and may still ﬁnd proton decay. ⋄ All currently known phenomena of elementary particle physics are either perfectly well accounted for by its standard model, or are not explained at all. The former category relates to the electroweak and strong gauge interactions which have been spectacularly successful at describing microscopic processes up to the energies currently available at accelerators. On the other side are, for example, the mass spectrum of the fundamental fermions (quarks and leptons), the source for CP violation, or the relationship between the three families (why three?). There must be “physics beyond the standard model!” In the standard model, neutrinos have been assigned the most min- imal properties compatible with experimental data: zero mass, zero charge, zero dipole moments, zero decay rate, zero almost everything. Any deviation from this simple picture is a sensitive probe for physics beyond the standard model—thus the enthusiasm to search for neu- trino masses and mixings, notably in oscillation experiments, but also for neutrino electromagnetic properties, decays, and other eﬀects. In astrophysics, even “minimal neutrinos” play a major role for the en- ergy loss of stars as they can escape unscathed from the interior once produced. Moreover, in spite of their weak interaction there are two astrophysical sites where they actually reach thermal equilibrium: the early universe up to about the nucleosynthesis epoch and in a SN core for a few seconds after collapse. Neutrinos thus play a dominant role in the cosmic and SN dynamical and thermal evolution—little wonder that these environments are important neutrino laboratories. Nonstandard neutrino properties such as small Majorana masses or magnetic dipole moments would be low-energy manifestations of novel physics at short distances. Another spectacular interloper of high- energy physics in the low-energy world would be a Nambu-Goldstone xviii Preface boson of a new symmetry broken at some large energy scale. Such a particle would be massless or nearly massless. The most widely dis- cussed example is the “invisible axion” which has been postulated as an explanation for the observed absence of a neutron electric dipole moment which in QCD ought to be about as large as its magnetic one. The axion also doubles as a particle candidate for the dark matter of the universe; two beautiful experiments to search for galactic axions are about to go on line in Livermore (California) and Kyoto (Japan). Be- cause of their weak interactions, the role of axions and similar particles in stellar evolution is closely related to that of neutrinos. ⋄ The complete or near masslessness of the particles studied here (neu- trinos, photons, axions, etc.) opens up a rich phenomenology in its own right. One intriguing issue is the production and propagation of these objects in a hot and dense medium which modiﬁes their dispersion rela- tions in subtle but signiﬁcant ways. One of the most important neutrino production processes in stars is the “photon decay” →which is enabled by the modiﬁed photon dispersion relation in a medium, and by an eﬀective neutrino-photon coupling mediated by the ambient elec- trons. Another example is the process of resonant neutrino oscillations (MSW eﬀect) which is instrumental at explaining the measured solar neutrino spectrum, and may imply vast modiﬁcations of SN physics, notably the occurrence of r-process nucleosynthesis. The MSW eﬀect depends on a ﬂavor-birefringent term of the neutrino dispersion rela- tion in a medium. In a SN, neutrinos themselves make an important contribution to the “background medium” so that their oscillations be- come a nonlinear phenomenon. In the deep interior of a SN core where neutrinos are trapped, a kinetic treatment of the interplay between os- cillations and collisions requires a fascinating “nonabelian Boltzmann collision equation.” One may think that low-energy particle physics in a dense and hot medium requires the tools of ﬁeld theory at ﬁnite temperature and density (FTD) which has taken a stormy development over the past ﬁfteen years. In practice, its contribution to particle astrophysics has been minor. The derivation of dispersion relations in a medium can be understood in kinetic theory from forward scattering on the medium constituents—dispersion is a lowest-order phenomenon. The imaginary part of a particle self-energy in the medium is physically related to its emission and absorption rate which in FTD ﬁeld theory is given by Preface xix cuts of higher-order graphs. These methods have been applied only in a few cases of direct interest to particle astrophysics where the re- sults seem to agree with those from kinetic theory in the limits which have been relevant in practice. A systematic formulation of astrophys- ically relevant particle dispersion, emission, and absorption processes in the framework of FTD ﬁeld theory is a project for another author. Tanguy Altherr had begun to take up this challenge with his collabo- rators in a series of papers. However, his premature death in a tragic climbing accident on 14 July 1994 has put an abrupt end to this line of research. My presentation of dispersion, emission, and absorption processes is based entirely on the old-fashioned tools of kinetic theory where, say, an axion emission rate is given by an integral of a squared matrix ele- ment over the thermally occupied phase space of the reaction partners. Usually this approach is not problematic, but it does require some tin- kering when it comes to the problem of electromagnetic screening or other collective eﬀects. I would not be surprised if subtle but important collective eﬀects had been overlooked in some cases. A simple kinetic approach is not adequate in the hot nuclear medium characteristic for a young SN core. However, in practice quantities like the neutrino opacities and axion emissivities have been calculated as if the medium constituents were freely propagating particles. At least for the dominant axial-vector current interactions this approach is not consistent as one needs to assume that the spin-ﬂuctuation rate is small compared with typical thermal energies while a naive calculation yields a result much larger than T. Realistically, it probably saturates at O(T), independently of details of the assumed interaction potential. This conjecture appears to be supported by our recent “calibration” of SN opacities from the SN 1987A neutrino signal. However, a calculation of either neutrino opacities or axion emissivities on the basis of ﬁrst principles is not available at the present time because FTD eﬀects as well as nuclear-physics complications dominate the problem. ⋄ It has been challenging to hammer the multifarious and intertwined aspects of my topic into the linear shape required by the nature of a book. Chapters 1 −6 are mainly devoted to the stellar energy-loss argument. It is introduced in Chapter 1 where its general aspects are developed on the basis of the stellar structure equations. Chapter 2 establishes the observational limits on those stellar evolutionary time xx Preface scales that have been used for the purposes of particle astrophysics. For each case the salient applications are summarized. Chapters 3 −6 deal with the interaction of “radiation” (neutrinos, axions, other low- mass bosons) with the main constituents of stellar plasmas (photons, electrons, nucleons). In these chapters all information is pulled together that pertains to the given interaction channel, even if it is not directly related to the energy-loss argument. For example, in Chapter 5 the limits on the electromagnetic coupling of pseudoscalars with photons are summarized; the stellar energy-loss ones are the most restrictive which justiﬁes this arrangement. Chapter 6 develops the topic of particle dispersion in media. The medium-induced photon dispersion relation allows for the plasma pro- cess →which yields the best limit on neutrino dipole moments by virtue of the energy-loss argument applied to globular-cluster stars. The neutrino dispersion relation is needed for the following discussion of neutrino oscillations, establishing a link between the energy-loss ar- gument and the dispersion arguments of the following chapters. Dispersion and propagation eﬀects are particularly important for massive neutrinos with ﬂavor mixing. To this end the phenomenology of massive, mixed neutrinos is introduced in Chapter 7. Vacuum and matter-induced ﬂavor oscillations as well as magnetically induced spin oscillations are taken up in Chapter 8. If neutrinos are in thermal equi- librium as in a young SN core or the early universe, neutrino oscillations require a diﬀerent theoretical treatment (Chapter 9). Chapters 10 −13 are devoted to astrophysical sources where neu- trinos have been measured, i.e. the Sun and supernovae (for the lat- ter only the SN 1987A signal exists). Neutrino oscillations, notably of the matter-induced variety, play a prominent role in Chapter 10 (solar neutrinos) and Chapter 11 (SN neutrinos). Radiative particle decays, especially of neutrinos, are studied in Chapter 12 where the Sun and SN 1987A ﬁgure prominently as sources. In Chapter 13 the particle-physics results from SN 1987A are summarized, including the ones related to the energy-loss and other arguments. Chapters 14 −16 give particle-speciﬁc summaries. While axions play a big role throughout this text, only the structure of the interaction Hamiltonian with photons, electrons, and nucleons is needed. Thus everything said about axions applies to any pseudoscalar low-mass bo- son for which they serve as a generic example. In Chapter 14 these results are interpreted in terms of axion-speciﬁc models which relate their properties to those of the neutral pion and thus establish a nearly unique relationship between their mass and interaction strength. Often- Preface xxi quoted “astrophysical bounds on the axion mass” are really transformed bounds on their interaction strength. This chapter also summarizes recent developments of the putative cosmological role of axions. Chap- ter 15 takes up a variety of hypotheses which can be tested by the methods developed in this book. Finally, Chapter 16 is an attempt at a bottom line of what we have learned about neutrino properties in the astrophysical laboratory. ⋄ Parts of my presentation are devoted to theoretical and calcula- tional ﬁne points of particle dispersion and emission eﬀects in media. I ﬁnd some of these issues quite intriguing in their own right. Still, the main goal has been to provide an up-to-date overview of what we know about elementary particles and their interactions on the basis of estab- lished stellar properties and on the basis of measured or experimentally constrained stellar particle ﬂuxes. All those whose lives are spent searching for truth are well aware that the glimpses they catch of it are necessarily ﬂeet- ing, glittering for an instant only to make way for new and still more dazzling insights. The scholar’s work, in marked contrast to that of the artist, is inevitably provisional. He knows this and rejoices in it, for the rapid obsolescence of his books is the very proof of the progress of scholarship. (Henri Pirenne, 1862 −1935) In spite of this bittersweet insight, and in spite of some inevitable errors of omission and commission, I hope that my book will be of some use to researchers, scholars, and students interested in the connection between fundamental physics and stars. Munich, May 1995. Acknowledgments While writing this book I have beneﬁtted from the help and encourage- ment of many friends and colleagues. In particular, I need to mention Hans-Thomas Janka, Lothar Luh, G¨ unter Sigl, Pierre Sikivie, Thomas Strobel, and Achim Weiss who read various parts of the manuscript, and the referees Josh Frieman and J. Craig Wheeler who read all or most of it. Their comments helped in no small measure to improve the manuscript and to eliminate some errors. Hans-Thomas, in particular, has spared no eﬀort at educating me on the latest developments in the area of supernova physics. Several chapters were written during a visit at the Center for Particle Astrophysics in Berkeley—I gratefully ac- knowledge the ﬁne hospitality of Bernard Sadoulet and his staﬀ. Stay- ing for that period as a guest in Edward Janelli’s house made a huge diﬀerence. During the entire writing process Greg Castillo provided encouragement and a large supply of Cuban and Puerto Rican music CDs which have helped to keep my spirits up. At the University of Chicago Press, I am indebted to Vicki Jennings, Penelope Kaiserlian, Stacia Kozlowski, and Eleanore Law for their expert handling of all editorial and practical matters that had to be taken care of to trans- form my manuscript into a book. David Schramm as the series editor originally solicited this opus (“just expand your Physics Report a little bit”). It hasn’t quite worked that way, but now that I’m ﬁnished I am grateful that David persuaded me to take up this project. xxii Chapter 1 The Energy-Loss Argument Weakly interacting, low-mass particles such as neutrinos or axions con- tribute to the energy loss or energy transfer in stars. The impact of an anomalous energy-loss mechanism is discussed qualitatively and in terms of homology relations between standard and perturbed stellar models. The example of massive pseudoscalar particles is used to il- lustrate the impact of a new energy-loss and a new radiative-transfer mechanism on the Sun. 1.1 Introduction More than half a century ago, Gamow and Schoenberg (1940, 1941) ushered in the advent of particle astrophysics when they speculated that neutrinos may play an important role in stellar evolution, particularly in the collapse of evolved stars. Such a hypothesis was quite bold for the time because neutrinos, which had been proposed by Pauli in 1930, were not directly detected until 1954. That their existence was far from being an established belief when Gamow and Schoenberg wrote their papers is illustrated by Bethe’s (1939) complete silence about them in his seminal paper on the solar nuclear fusion chains. Even after the existence of neutrinos had been established they seemed to interact only by βreactions of the sort e−+ (A, Z)→ (A, Z−1) + νeor (A, Z−1)→(A, Z) +e−+νe, the so-called URCA reactions which Gamow and Schoenberg had in mind, or by fusion pro- cesses like pp→de+νe. The URCA reactions and related processes become important only at very high temperatures or densities because of their energy threshold. While the Sun emits two neutrinos for every helium nucleus fused from hydrogen, the energy loss in neutrinos is only a few percent of the total luminosity and thus plays a minor role. 1 2 Chapter 1 Still, neutrinos can be important in normal stars. This became clear when in 1958 Feynman and Gell-Mann as well as Sudarshan and Marshak proposed the universal V−Ainteraction law which implied a direct neutrino-electron interaction with the strength of the Fermi constant. Pontecorvo (1959) realized almost immediately that this in- teraction would allow for the bremsstrahlung radiation of neutrino pairs by electrons, and that the absence of a threshold renders this process an important energy loss mechanism for stars. Of course, their typical energies will correspond to the temperature of the plasma (about 1 keV in the Sun) while neutrinos from nuclear reactions have MeV energies. On the basis of the bremsstrahlung process, Gandel’man and Pinaev (1959) calculated the approximate conditions for which neutrino losses would “outshine” the photon luminosity of stars. For the Sun, thermal neutrino emission is found to be irrelevant. Subsequently, the neutrino emissivity was calculated by many au- thors. It was quickly realized that the dominant emission processes from a normal stellar plasma are the photoneutrino process γe−→e−νν, the bremsstrahlung process e−+(A, Z)→(A, Z)+e−+νν, and the plasma process (“photon decay”) γ→ννwhich is possible because photons have an eﬀective mass in the medium. These and related reactions will be discussed at length in Chapters 3 −6. In the late 1960s Stothers and his collaborators1established that the observed paucity of red supergiants is best explained by the fast rate with which these carbon-burning stars spend their nuclear fuel due to neutrino emission. Therefore, the existence and approximate magnitude of the direct electron-neutrino coupling was at least ten- tatively established by astrophysical methods several years before its experimental measurement in 1976. While standard neutrino physics today is an integral part of stellar evolution and supernova theory, they could have novel couplings to the plasma, for example by a magnetic dipole moment. Then they could be emitted more eﬃciently than is possible with standard interactions. Moreover, new concepts of particle physics have emerged that could be equally important despite the relatively low energies available in stellar interiors. In various extensions of the standard model, the spontaneous breakdown of a symmetry of the Lagrangian of the fundamental inter- actions by some large vacuum expectation value of a new ﬁeld leads to the prediction of massless or nearly massless particles, the Nambu- Goldstone bosons of the broken symmetry. The most widely discussed 1For a summary see Stothers (1970, 1972). The Energy-Loss Argument 3 example is the axion (Chapter 14) which arises as the Nambu-Goldstone boson of the Peccei-Quinn symmetry which explains the puzzling ab- sence of a neutron electric dipole moment, i.e. it explains CP conserva- tion in strong interactions. The production of axions in stars, like that of neutrinos, is not impeded by threshold eﬀects. Clearly, the emission of novel weakly interacting particles or the emission of neutrinos with novel properties would have a strong impact on the evolution and properties of stars. Sato and Sato (1975) were the ﬁrst to use this “energy-loss argument” to derive bounds on the coupling strength of a putative low-mass Higgs particle. Following this lead, the argument has been applied to a great variety of particle- physics hypotheses, and to a great variety of stars. In the remainder of this chapter the impact of a novel energy-loss mechanism on stars will be discussed in simple terms. The main mes- sage will be that the emission of weakly interacting particles usually leads to a modiﬁcation of evolutionary time scales. By losing energy in a new channel, the star eﬀectively burns or cools faster and thus shines for a shorter time. In the case of low-mass red giants, however, particle emission leads to a delay of helium ignition and thus to an extension of the red-giant phase. Either way, what needs to be observationally established is the duration of those phases of stellar evolution which are most sensitive to a novel energy-loss mechanism. In Chapter 2 such evolutionary phases will be identiﬁed, and the observational evidence for their duration will be discussed. In the end one will be able to state simple criteria for the allowed rate of energy loss from plasmas at certain temperatures and densities. One may be tempted to think that one should consider the hottest and densest possible stars because no doubt the emission of weakly in- teracting particles is most eﬃcient there. However, this emission com- petes with standard neutrinos whose production is also more eﬃcient in hotter and denser objects. Because neutrinos are thermally emitted in pairs their emission rates involve favorable phase-space factors which lead to a temperature dependence which is steeper than that for the emission of, say, axions. Thus, for a given axion coupling strength the relative importance of axion emission is greater for lower temperatures. Of course, the temperatures must not be so low that neither neutrino nor axion emission is important at all relative to the photon luminos- ity. Consequently, the best objects to use are those where neutrinos just begin to have an observational impact on stellar observables. Exam- ples are low-mass red giants, horizontal-branch stars, white dwarfs, and old neutron stars. They all have masses of around 1 M⊙(solar mass). 4 Chapter 1 For low-mass stars plenty of detailed observational data exist, allowing one to establish signiﬁcant limits on possible deviations from standard evolutionary time scales. From the astrophysical perspective, the energy-loss argument re- quires establishing evolutionary time scales and other observables that are sensitive to a novel “energy sink.” This problem will be taken up in Chapter 2. From the particle-physics perspective, one needs to cal- culate the energy-loss rate of a plasma at a given temperature, density, and chemical composition into a proposed channel. For this purpose it is not necessary to know any of the underlying physics that leads to the hypothesis of nonstandard neutrino properties or the prediction of new particles such as axions. All one needs is the structure of the inter- action Hamiltonian with the constituents of stellar plasmas (electrons, nucleons, nuclei, and photons). For example, it is enough to know that an axion is a low-mass particle with a pseudoscalar interaction with electrons to calculate the axion emission rate. As such “axion” stands for any particle with a similar interaction structure. The motivation for “real axions” and their detailed model-dependent properties are not discussed until Chapter 14. In Chapters 3 −5 and partly in Chapter 6 various interaction structures will be explored and constraints on the overall coupling strengths will be derived or summarized. Also, they will be put into the context of evidence from sources other than the stellar energy-loss argument. Putative novel particles almost inevitably must be very weakly in- teracting or else they would have been seen in laboratory experiments. Still, one is sometimes motivated to speculate about particles which may be so strongly interacting that they cannot freely escape from stars; their mean free path may be shorter than the stellar radius. It is often incorrectly stated that such particles would be astrophysically allowed. Nothing could be further from the truth. Particles which are “trapped,” i.e. which cannot freely stream out once produced, con- tribute to the radiative transfer of energy in competition with photons. Radiative energy transfer occurs because particles produced in a hot region get absorbed in a neighboring somewhat cooler region. This mechanism is more eﬃcient if the mean free path is larger because then ever more distant regions with larger temperature diﬀerences are ther- mally coupled. Therefore, particles which interact more weakly than photons would dominate the radiative energy transfer and thus have a tremendous impact on the structure of stars. Very roughly speaking, then, novel particles must be either more strongly interacting than photons, or more weakly interacting than neu- The Energy-Loss Argument 5 trinos, to be harmless in stars. Of course, depending on whether the particles are bosons or fermions, and depending on the details of their interaction structure, this statement must be reﬁned. Still, the impact of novel particles on stellar structure and evolution is maximized when their mean free path is of order the geometric dimension of the system . The energy-transfer argument is equally powerful as the energy-loss ar- gument. The only reason why it has not been elaborated much in the literature is because there is usually little motivation for considering “strongly” interacting novel particles. When it comes to the evolution of a supernova (SN) core after col- lapse even neutrinos are trapped. Such a newborn neutron star is so hot ( Tof order 30 MeV) and dense ( ρexceeding nuclear density of 3×1014g cm−3) that neutrinos take several seconds to diﬀuse to the surface. Particles like axions can then compete in spite of the extreme conditions because they freely stream out if their coupling is weak enough. The energy-loss argument can be applied because the neu- trino cooling time scale has been established by the SN 1987A neutrino observations. While the energy-loss argument in this case is fundamen- tally no diﬀerent from, say, white-dwarf cooling, the detailed reasoning is closely intertwined with the issue of neutrino physics in supernovae, and with the details of the SN 1987A neutrino observations. There- fore, it is taken up only in Chapter 13. The groundwork concerning the interactions of neutrinos and axions with nucleons, however, is laid in Chapter 4 within the series of chapters devoted to various modes of particle interactions with the constituents of stellar plasmas. 1.2 Equations of Stellar Structure 1.2.1 Hydrostatic Equilibrium To understand the impact of a novel energy-loss mechanism on the evolution of stars one must understand the basic physical principles that govern stellar structure. While a number of simplifying assumptions need to be made, the theory of stellar structure and evolution has been extremely successful at modelling stars with a vast range of properties. For a more detailed account than is possible here the reader is referred to the textbook literature, e.g. Kippenhahn and Weigert (1990). One usually assumes spherical symmetry and thus excludes the ef- fects of rotation, magnetic ﬁelds, tidal eﬀects from a binary companion, and large-scale convective currents. While any of those eﬀects can be important in special cases, none of them appears to have a noticeable 6 Chapter 1 impact on the overall picture of stellar structure and evolution, with the possible exception of supernova physics where large-scale convective overturns may be crucial for the explosion mechanism (Chapter 11). Second, one usually assumes hydrostatic equilibrium, i.e. one ignores the macroscopic kinetic energy of the stellar medium. This approxima- tion is inadequate for a study of stellar pulsation where the inertia of the material is obviously important, and also inadequate for “hydrody- namic events” such as a supernova explosion (Sect. 2.1.8), and perhaps the helium ﬂash (Sect. 2.1.3). For most purposes, however, the changes of the stellar structure are so slow that neglecting the kinetic energy is an excellent approximation. Therefore, as a ﬁrst equation one uses the condition of hydrostatic equilibrium that a spherical shell of the stellar material is held in place by the opposing forces of gravity and pressure, dp dr=−GNMrρ r2. (1.1) Here, GNis Newton’s constant, pandρare the pressure and mass den- sity at the radial position r, andMr= 4π∫r 0dr′ρ r′2is the integrated mass up to the radius r. With apologies to astrophysicists I will usually employ natural units where ¯ h=c=kB= 1. In Appendix A conversion factors are given between various units of mass, energy, inverse length and time, tem- perature, and so forth. Newton’s constant is then GN=m−2 Plwith the Planck mass mPl= 1.221×1019GeV = 2 .177×10−5g. Stellar masses are always denoted with the letter Mto avoid confusion with an absolute bolometric brightness which is traditionally denoted by M. In general, the pressure is given in terms of the density, temperature, and chemical composition by virtue of an equation of state. For a classical monatomic gas p=2 3uwith uthe density of internal energy. One may multiply Eq. (1.1) on both sides with 4 πr3and integrate from the center ( r= 0) to the surface ( r=R). The r.h.s. gives the total gravitational energy while the l.h.s. yields −12π∫R 0dr p r2after a partial integration with the boundary condition p= 0 at the surface. With p=2 3uthis is−2Uwith Uthe total internal energy of the star. Because for a monatomic gas Uis the sum of the kinetic energies of the atoms one ﬁnds that on average for every atom ⟨Ekin⟩=−1 2⟨Egrav⟩. (1.2) This is the virial theorem which is the most important tool to under- stand the behavior of self-gravitating systems. The Energy-Loss Argument 7 As a simple example for the beauty and power of the virial theo- rem one may estimate the solar central temperature from its mass and radius. The material is dominated by protons which have a gravita- tional potential energy of order −GNM⊙mp/R⊙=−2.14 keV where M⊙= 1.99×1033g is the solar mass, R⊙= 6.96×1010cm the solar radius, and mpthe proton mass. The average kinetic energy of a proton is equal to3 2T(remember, kBhas been set equal to unity), yielding an approximate value for the solar internal temperature of T=1 32.14 keV = 0 .8×107K. This is to be compared with 1 .56×107K found for the central temperature of a typical solar model. This exam- ple illustrates that the basic properties of stars can be understood from simple physical principles. 1.2.2 Generic Cases of Stellar Structure a) Normal Stars There are two main sources of pressure relevant in stars, thermal pres- sure and degeneracy pressure. The third possibility, radiation pressure, never dominates except perhaps in the most massive stars. The pres- sure provided by a species of particles is proportional to their density, to their momentum which is reﬂected on an imagined piston and thus exerts a force, and to their velocity which tells us the number of hits on the piston per unit time. In a nondegenerate nonrelativistic medium a typical particle velocity and momentum is proportional to T1=2so that p∝(ρ/µ)Twith ρthe mass density and µthe mean molecu- lar weight of the medium constituents. For nonrelativistic degenerate electrons the density is ne=p3 F/3π2(Fermi momentum pF), a typical momentum is pF, and the velocity is pF/me, yielding a pressure which is proportional to p5 For to n5=3 eand thus to ρ5=3. The two main pressure sources determine two generic forms of be- havior of overall stellar models, namely normal stars such as our Sun which is dominated by thermal pressure, and degenerate stars such as white dwarfs which are dominated by degeneracy pressure. These two cases follow a very diﬀerent logic. A normal star is understood most easily if one imagines how it ini- tially forms from a dispersed but gravitationally bound gas cloud. It continuously loses energy because photons are produced in collisions between, say, electrons and protons. The radiation carries away energy which must go at the expense of the total energy of the system. If it is roughly in an equilibrium conﬁguration, the virial theorem Eq. (1.2) in- 8 Chapter 1 forms us that a decrease of ⟨Ekin+Egrav⟩causes the gravitational energy to become more negative, corresponding to a more tightly bound and thus more compact system. At the same time the average kinetic energy goes up which corresponds to an increased temperature if the system can be considered to be locally in thermal equilibrium. Therefore, as the system loses energy it contracts and heats up. Self-gravitating sys- tems have a negative speciﬁc heat! As a protostar contracts it becomes more opaque and soon the en- ergy loss is limited by the speed of energy transfer from the inner parts to the surface, i.e. essentially by the photon diﬀusion speed. Before an internal energy source for stars was known it was thought that gravita- tional energy provided for their luminosity. Stellar lifetimes seemed to be given by the “Kelvin-Helmholtz time scale” for thermal relaxation which is ﬁxed by the speed of energy transfer. The total reservoir of gravitational energy of the Sun is estimated by GNM2 ⊙/2R⊙and its luminosity is L⊙= 3.85×1026W = 3 .85×1033erg/s so that the thermal relaxation scale is about τKH≈1 2GNM2 ⊙R−1 ⊙L−1 ⊙= 1.6×107yr. Pressed by thermodynamic theory and the authority of Lord Kelvin, geologists tried to adjust the age of the Earth to this short time scale against sound evidence to the contrary. At the beginning of our century the discovery of radioactivity and thus of nuclear processes revealed that stars had another source of energy and consequently could live much longer than indicated by τKH. A further contraction is thus intercepted by the onset of hydrogen burning which commences when the temperature is high enough for protons to penetrate each other’s electrostatic repulsive potential. The hydrogen-burning reactions are more fully discussed in Chapter 10; the bottom line is that four protons and two electrons combine to form a helium nucleus ( αparticle), releasing 26 .73 MeV of energy. A few percent are immediately lost in the form of two neutrinos which must emerge to balance the electron lepton number, but most of the en- ergy is available as heat. Because the nuclear reaction rates have a steep temperature dependence, a further contraction and heating of the star leads to much more nuclear energy generation, which quickly increases the average Ekinof the nuclei and thus leads to an expansion and cooling by the same virial-theorem logic that led to contraction and heating when energy was lost. Thus a stable conﬁguration of “thermal equilibrium” is reached where the energy lost is exactly balanced by that produced from nuclear reactions. Stars as fusion reactors are per- fectly regulated by the “negative speciﬁc heat” of a self-gravitating system! The Energy-Loss Argument 9 The most salient feature of a normal stellar conﬁguration is the in- terplay of its negative speciﬁc heat and nuclear energy generation. Con- versely, if the pressure were dominated by electron degeneracy it would be nearly independent of the temperature. Then this self-regulation would not function because heating would not lead to expansion. Thus, stable nuclear burning and the dominance of thermal pressure go insep- arably hand in hand. Another salient feature of such a conﬁguration is the inevitability of its ﬁnal demise because it lives on a ﬁnite supply of nuclear fuel. b) Degenerate Stars Everything is diﬀerent for a conﬁguration dominated by degeneracy pressure. Above all, it has a positive heat capacity so that a loss of energy no longer implies contraction and heating. The star actu- ally cools. This is what happens to a brown dwarf which is a star so small ( M<0.08M⊙) that it did not reach the critical condi- tions to ignite hydrogen: it becomes a degenerate gas ball which slowly “browns out.” The relationship between radius and mass is inverted. A normal star is geometrically larger if it has a larger mass; very crudely R∝ M. When mass is added to a degenerate conﬁguration it becomes geometrically smaller as the reduced size squeezes the electron Fermi sea into higher momentum states, providing for increased pressure to balance the increased gravitational force. The l.h.s. of Eq. (1.1) can be approximated as p/Rwhere p∝ρ5=3is some average pressure. Because ρ≈ M /R3one ﬁnds p/R∝ M5=3/R6while the r.h.s. of Eq. (1.1) is proportional to Mρ/R2and thus to M2/R5. Therefore, a degenerate conﬁguration is characterized by R∝ M−1=3. Increasing the mass beyond a certain limit causes the radius to shrink so much that the electrons become relativistic. Then they move with a velocity ﬁxed at c(or 1 in natural units), causing the pressure to vary only as p4 Forρ4=3. In this case adding mass no longer leads to a suﬃcient pressure increase to balance for the extra weight. Beyond this “Chandrasekhar limit,” which is about 1 .4M⊙for a chemical composi- tion with Ye=1 2(number of electrons per baryon), no stable degenerate conﬁguration exists. In summary, the salient features of a degenerate conﬁguration are the inverse mass-radius relationship R∝ M−1=3, the Chandrasekhar limit, the absence of nuclear burning, and the positive speciﬁc heat which allows the conﬁguration to cool when it loses energy. 10 Chapter 1 c) Giant Stars A real star can be a hybrid conﬁguration with a degenerate core and a nondegenerate envelope with nuclear burning at the bottom of the envelope, i.e. the surface of the core. The core then follows the logic of a degenerate conﬁguration. The envelope follows the self-regulating logic of a normal star except that it is no longer dominated by its self- gravity but rather by the gravitational force exerted by the compact core. Amazingly, the envelopes of such stars tend to expand to huge dimensions. It does not seem possible to explain in a straightforward way why stars become giants2except that “the equations say so.” Low- mass red giants will play a major role in Chapter 2—a further discussion of their fascinating story is deferred until then. 1.2.3 Energy Conservation Returning to the basic principles that govern stellar structure, energy conservation yields another of the stellar structure equations. If the lo- cal sources and sinks of energy balance against the energy ﬂow through the surface of a spherical mass shell one ﬁnds dLr/dr= 4πr2ϵρ, (1.3) where Lris the net ﬂux of energy through a spherical shell of radius r while ϵis the eﬀective rate of local energy production (units erg g−1s−1). It is a sum ϵ=ϵnuc+ϵgrav−ϵ−ϵx, (1.4) where ϵnucis the rate by which nuclear energy is liberated, ϵis the energy-loss rate by standard neutrino production, and ϵxis for novel particles or nonstandard neutrinos with, say, large magnetic dipole mo- ments. Further, ϵgrav=cpT(∇ad˙p/p−˙T/T) (1.5) is the local energy gain when Tandpchange because of expansion or contraction of the star. This term is the dominant heat source in a red-giant core which contracts because of the mass gained by hydrogen shell burning. Conversely, it is a sink which absorbs energy when helium ignites in such a star: the would-be explosion is dissipated by expansion 2For recent attempts see Faulkner and Swenson (1988), Eggleton and Cannon (1991), and Renzini et al. (1992). The Energy-Loss Argument 11 against the force of gravity. In Eq. (1.5) cpis the heat capacity at constant pressure. Further, the quantity ∇ad≡(∂lnT/∂lnp)s, taken at constant entropy density s, is the “adiabatic temperature gradient.” It is not really a gradient. It is a thermodynamic quantity characteristic of the medium at the local conditions of ρ,p,T, and the chemical composition. The calculation of ϵxfor a number of hypotheses concerning the existence of novel particles or novel properties of neutrinos will be a major aspect of this book. Even for a given interaction law between the particles and the medium constituents this is not always a straight- forward exercise because the presence of the ambient medium can have a signiﬁcant impact on the microscopic reactions. This is equally true for the nuclear energy generation rates. Nuclear reactions are slow in stars because the low temperature allows only few nuclei to penetrate each other’s Coulomb barriers. Therefore, screening eﬀects are important, and the nuclear cross sections need to be known at energies so low that they cannot be measured directly in the laboratory. Much of the debate concerning the solar neutrino problem revolves around the proper extrapolation of certain nuclear cross sections to solar thermal energies. 1.2.4 Energy Transfer The transfer of energy is driven by the radial temperature gradient. In the absence of convection heat is carried by photons and electrons mov- ing between regions of diﬀerent temperature, i.e. by radiative transfer and by conduction. In this case the relationship between the energy ﬂux and the temperature gradient is Lr=−4πr2 3κρd(aT4) dr, (1.6) where aT4is the energy stored in the radiation ﬁeld ( a=π2/15 in natural units) and κis the opacity (units cm2/g). It is given by a sum κ−1=κ−1 +κ−1 c+κ−1 x, (1.7) where κ is the radiative opacity, κcthe contribution from conduction by electrons, and κxwas included for a possible contribution from novel particles. The quantity ( κ ρ)−1=⟨λ ⟩Ris the “Rosseland average” of the photon mean free path—its precise deﬁnition will be given in Sect. 1.3.4. 12 Chapter 1 One of the main diﬃculties at calculating the opacity is that heavier elements, notably iron, are only partially ionized for typical conditions. Resonant transitions of electrons between diﬀerent bound states are very important, an eﬀect which causes stellar models to be rather sen- sitive to the amount of “metals” (elements heavier than helium). The construction of an opacity table is a major eﬀort as it requires includ- ing huge numbers of electronic energy levels. Widely used were the Los Alamos and the Livermore Laboratory opacity tables. Recently, the Livermore tables were systematically overhauled (Igle- sias, Rogers, and Wilson 1990; Iglesias and Rogers 1991a,b), resulting in the new OPAL tables which since have become the standard in stel- lar evolution calculations. The main diﬀerences to the previous tables are at moderate temperatures so that no substantial changes in the deep interior of stellar structures have occurred. However, envelope phenomena are aﬀected, notably convection near the surface and stel- lar pulsations. A number of previous discrepancies between theory and observations in this area have now disappeared. If the equation of state, the energy generation rate, and the opacity are known one can construct a stellar model for an assumed compo- sition proﬁle by solving the stellar structure equations with suitable boundary conditions. (It is not entirely trivial to deﬁne surface bound- ary conditions because the star, strictly speaking, extends to inﬁnity. A crude approach is to take T= 0 and p= 0 at the photosphere.) It may turn out, however, that in some locations this procedure yields a temperature gradient which is so steep that the material becomes unstable to convection—it “boils.” An adequate treatment of convection is one of the main problems of stellar evolution theory. A simpliﬁcation occurs because convection is extremely eﬀective at transporting energy and so the temperature gradient will adjust itself to a value very close to the “adiabatic gra- dient” which marks the onset of the instability. At this almost ﬁxed temperature gradient a nearly arbitrary energy ﬂux can be carried by the medium. This approximation tends to be justiﬁed for regions in the deep interior of stars while the “superadiabatic convection” found near the surface requires a substantial reﬁnement. One usually applies the “mixing length theory” which contains one free parameter, the ratio be- tween the convective mixing length and the pressure scale height. This parameter is empirically ﬁxed by adjusting the radius of a calculated solar model to the observed value. Main-sequence stars like our Sun with M∼<M⊙have a radiative interior with a convective surface which penetrates deeper with decreas- The Energy-Loss Argument 13 ingM; stars with M∼<0.25M⊙are fully convective. For M∼>M⊙ the outer regions are radiative while the core is convective out to an ever increasing mass fraction of the star with increasing M. A star with M near 1M⊙is very special in that it is radiative almost throughout; the Sun is thought to have only a relatively minor convective surface layer. Besides transporting energy, convection also moves matter and thus aﬀects the composition proﬁle of a star. This is seen, for example, in the upper panels of Fig. 2.4 where the hydrogen depletion of a solar model (which is radiative) is a function of the local nuclear burning rates while for the convective helium core of a horizontal-branch (HB) star the helium depletion reaches to much larger radii than nuclear burning. The long lifetimes of HB stars cannot be understood without the convective supply of fuel to the nuclear furnace at the center. The Sun, on the other hand, will complete its main-sequence evolution when hydrogen is depleted at the center, corresponding to about a 10% global depletion only. The extent of convective regions can change during the course of stellar evolution. They can leave behind composition discontinuities which are a memory of a previous conﬁguration. For example, on the lower red-giant branch (RGB) the convective envelope reaches so deep that it penetrates into the region of variable hydrogen content caused by nuclear burning. Later, the convective envelope retreats from the ad- vancing hydrogen-burning shell which encounters a discontinuity in the hydrogen proﬁle. This causes a brief “hesitation” on the RGB ascent and thus a “bump” in the distribution of stars in the color-magnitude diagram of globular clusters on the lower RGB. This bump has been identiﬁed in several clusters (Figs. 2.18 and 2.19); its location is in good agreement with theoretical expectations (Fusi Pecci et al. 1990). 1.2.5 Gravitational Settling The composition proﬁle of a star can also change by diﬀusion, and no- tably by gravitational settling of the heavier elements. This eﬀect was ignored in most evolution calculations because the time scales are very large. Still, the settling of helium will displace hydrogen from the cen- ter of a hydrogen-burning star and thus accelerate the depletion and main-sequence turnoﬀ. The gravitational settling of metals will lead to an opacity increase in the central regions. Helium settling reduces the inferred globular cluster ages by 1 Gyr or more which is about a 10% eﬀect (Proﬃtt and Michaud 1991; Chaboyer et al. 1992). Be- cause the inferred globular cluster ages are larger than the expansion 14 Chapter 1 age of the universe, this “cosmic age problem” is slightly alleviated by gravitational settling. Helium and metal settling was recently included in solar models (Bahcall and Pinsonneault 1992, 1995; Kovetz and Shaviv 1994; Proﬃtt 1994). It increased the predicted neutrino ﬂuxes on the 10 −30% level, depending on the speciﬁc treatment of gravitational settling and on the neutrino source reaction. These modiﬁcations are not huge, but surely not entirely negligible. They go in the direction of aggravating the solar neutrino problem. 1.3 Impact of Novel Particles 1.3.1 Energy Loss One of the most interesting possibilities to use stars as particle-physics laboratories is to study the backreaction of the novel energy-loss rate ϵximplied by the existence of new low-mass particles such as axions, or by nonstandard neutrino properties such as magnetic dipole moments. The impact on degenerate stars such as white dwarfs is rather obvi- ous: the new energy-loss rate accelerates the cooling. Therefore, the observationally established cooling speed allows one to constrain this process or to detect evidence for it. The impact of a novel energy-loss mechanism on a nondegenerate star like the Sun is less obvious. According to the virial theorem one expects that the loss of energy leads to heating and contraction up to a point where the temperature has risen enough that increased nuclear burning provides for the extra energy loss. Because of the steep temper- ature dependence of ϵnucone expects that the overall stellar structure changes very little in response to ϵx—the main impact is to accelerate the consumption of nuclear fuel and thus the completion of hydrogen burning. This argument was cast into a quantitative form by Frieman, Di- mopoulos, and Turner (1987). They asked how a given equilibrium structure of a star would change in response to turning on a new energy- loss rate ϵx. The main simplifying assumption is that the perturbed conﬁguration is obtained by a homology transformation, so that “the distance between any two points is altered in the same way as the ra- dius of the conﬁguration.” Thus, if the new radius of the star is given byR′=yRwith a dimensionless scaling factor y, then every point in the star is mapped to a new position r′=yr. The mass interior to the new radius is identical with that interior to the old location, The Energy-Loss Argument 15 M′(r′) =M(r), and the chemical composition at r′is the same as that at r. The density is transformed by ρ′(r′) =y−3ρ(r), and from Eq. (1.1) one ﬁnds that the pressure scales as p′(r′) =y−4p(r). The equation of state for a nondegenerate, low-mass star is approximately given by the ideal-gas law where p∝ρT/µ , where µis the average molecular weight of the electrons and nuclei. Since µ′(r′) =µ(r) by assumption, the temperature is found to scale as T′(r′) =y−1T(r), and the temperature gradient as dT′(r′)/dr′=y−2dT(r)/dr. The assumption that the star reacts to new particle emission by a homologous contraction imposes restrictions on the constitutive rela- tions for the eﬀective energy generation rate and the opacity. In par- ticular, for a chemically homogeneous star one needs to assume that ϵ∝ρnTand κ∝ρsTp. (1.8) For the opacity, Frieman et al. took the Kramers law with s= 1 andp=−3.5 which is found to be a reasonable interpolation for- mula throughout most lower main-sequence interiors. Hence, the local energy ﬂux scales as L′(r′) =y−1=2L(r). (1.9) The hydrogen-burning rate ϵnucalso has the required form with n= 1, and for the ppchain ν= 4−6; it dominates in the Sun and in stars with lower mass. It is assumed that the new energy-loss rate ϵxfollows the same proportionality; the standard neutrino losses ϵare ignored because they are small on the lower main sequence. If the star is not in a phase of major structural readjustment one may also ignore ϵgrav in Eq. (1.4) so that in Eq. (1.3) ϵ= (1−δx)ϵnuc, (1.10) where δx<1 is a number which depends on the interaction strength of the new particles. From Eq. (1.3) one concludes that L′(r′) =y−(3+)(1−δx)L(r), (1.11) leading to y= (1−δx)2=(2+5). (1.12) Assuming δx≪1, Frieman et al. then found for the fractional changes of the stellar radius, luminosity, and interior temperature, δR R=−2δx 2ν+ 5,δL L=δx 2ν+ 5,δT T=2δx 2ν+ 5. (1.13) Therefore, the star contracts, becomes hotter, and the surface photon luminosity increases—it overcompensates for the new losses. Moreover, 16 Chapter 1 even if the luminosity Lxin “exotics” is as large as the photon lumi- nosity ( δx=1 2) the overall changes in the stellar structure remain mod- erate. The predominant eﬀect is an increased consumption of nuclear fuel at an almost unchanged stellar structure, leading to a decreased duration of the hydrogen-burning phase of δτ/τ≈ −δx. (1.14) The standard Sun is halfway through its main-sequence evolution so that a conservative constraint is δx<1 2. In general, the exotic losses do not have the same temperature and density dependence as the nuclear burning rate, implying a breakdown of the homology condition. However, to lowest order these results will remain valid if one interprets δxas a suitable average over the en- tire star, δx=Lx/(Lx+L ), (1.15) with the photon luminosity L and that in exotics Lx. To lowest order Lxcan be computed from an unperturbed stellar model. For a convective structure ( M∼<0.25M⊙main-sequence stars) Frieman et al. found by a similar treatment δR R=−2δx 2ν+ 11,δL L=−5δx 2ν+ 11,δT T=2δx 2ν+ 11.(1.16) These stars also contract, and the internal temperature increases, but the surface luminosity decreases. 1.3.2 Application to the Sun For the Sun, the radius and luminosity are very well measured and so one may think that small deviations δRandδLfrom a standard model were detectable. This is not so, however, because a solar model isdened to produce the observed radius and luminosity at an age of 4.5 Gyr. The unknown presolar helium abundance Yinitial is chosen to reproduce the present-day luminosity, and the one free parameter of the mixing-length theory relevant for superadiabatic convection is calibrated by the solar radius. In a numerical study Raﬀelt and Dearborn (1987) implemented ax- ion losses by the Primakoﬀ process in a 1 M⊙stellar model, metal- licity Z= 0.02, which was evolved to 4 .5 Gyr with diﬀerent amounts of initial helium and diﬀerent axion coupling strengths. Details of the emission rate as a function of temperature and density are studied in The Energy-Loss Argument 17 Table 1.1. Initial helium abundance for solar models with axion losses. g10Yinitial δx Xc 0 0.274 0.00 0.362 10 0.266 0.16 0.307 15 0.256 0.32 0.292 20 0.241 0.51 0.245 25 0.224 0.65 0.151 Sect. 5.2. For the present discussion the axion losses represent some generic energy-loss mechanism with a rate proportional to the square of the axion-photon coupling strength ga . Without exotic losses a presolar helium abundance of Yinitial = 0.274 was needed to reproduce the present-day Sun. For several values of g10≡ga /10−10GeV−1Raﬀelt and Dearborn found the initial helium values given in Tab. 1.1 necessary to produce the present-day luminos- ity. The values for δxin Tab. 1.1 are deﬁned as in Eq. (1.15) with Lx the axion luminosity of the (perturbed) present-day solar model which hasL =L⊙. Also, the central hydrogen abundance Xcof the present- day model is given. For g10= 30, corresponding to δx≈0.75, no present-day Sun could be constructed for any value of Yinitial. The primordial helium abundance is thought to be about 23%, and the presolar abundance is certainly larger. Still, a value of δxless than about 0 .5 is hard to exclude on the basis of this calculation. Therefore, the approximate solar constraint remains δx∼<1 2orLx∼<L⊙as found from the analytic treatment in the previous section. One may be able to obtain an interesting limit by considering the oscillation frequencies of the solar pressure modes. Because of the excel- lent agreement between standard solar models and the observed p-mode frequencies there is little leeway for a modiﬁed solar structure and com- position. This method has been used to constrain a hypothetical time variation of Newton’s constant (Sect. 15.2.3). 1.3.3 Radiative Energy Transfer If novel particles are so weakly interacting that they escape freely from the star once produced their role is that of a local energy sink. Neu- trinos are of that nature, except in supernova cores where they are “trapped” for several seconds. One could imagine new particles with 18 Chapter 1 such large interactions that they are even trapped, say, in the Sun. Because their mean free path (mfp) is now less than the geometric di- mension of the star, they remove energy from one region and deposit it at an approximate distance of one mfp. This is precisely the mechanism of radiative energy transfer: the particles now contribute to the opac- ity. In a transition region where the mfp is on the order of the stellar radius this mode of energy transfer couples distant regions and thus cannot be described in the form of the diﬀerential equation (1.6). In this case the diﬀerence between an energy-loss and an energy-transfer mechanism is blurred. Equation (1.6) is justiﬁed when the mfp is less than the temper- ature scale height ( dlnT/dr )−1. For radiative transfer ( κρ)−1is an average mfp. Therefore, Eq. (1.6) informs us that the energy ﬂux car- ried through a sphere of radius ris a product of a numerical factor, the area, the photon mfp, the number density of photons, and the temper- ature gradient. Radiative transfer is more eﬃcient for a larger mfp, i.e. for a more weakly interacting particle! If a second photon existed with a coupling strength α′instead of α=1 137, it would contribute more to the energy transfer for α′< α. The observed properties of the Sun and other stars conﬁrm that the standard opacities certainly cannot be wrong by more than a factor of a few. Therefore, the new photon must interact about as strongly as the standard one to be in agreement with the properties of stars, or it must interact so weakly that it freely escapes and the integrated volume emission L ′is less than the photon luminosity L . 1.3.4 Opacity Contribution of Arbitrary Bosons In order to implement the energy-transfer argument one needs a prop- erly deﬁned expression for the Rosseland opacity contribution of arbi- trary bosons. Following the textbook derivation of the photon radiative opacity such a deﬁnition was provided by Carlson and Salati (1989) as well as Raﬀelt and Starkman (1989). For a suﬃciently short mean free path ℓthe radiation ﬁeld of the new bosons is taken to be locally isotropic. The local energy ﬂux is then found to be F!=−1 3β!ℓ!∇B!where the index ωindicates that a quan- tity refers to the boson energy ω. Moreover, B!andF!are understood to be “speciﬁc,” i.e. an energy density and ﬂux per unit energy interval. The velocity is β!= [1−(m/ω)2]1=2for a boson with mass m. (Recall that natural units with ¯ h=c=kB= 1 are always used.) In local thermal equilibrium the energy density for massive bosons correspond- The Energy-Loss Argument 19 ing to the phase-space element d3pisg ω(e!=T−1)−1d3p/(2π)3where gis the number of polarization degrees of freedom and ( e!=T−1)−1 is the thermal boson occupation number. With an angular integration d3pbecomes 4 πp2dpwhere p=|p|. Using p= (ω2−m2)1=2one ﬁnds p dp=ω dω so that B!=g 2π2ω2(ω2−m2)1=2 e!=T−1. (1.17) The total energy ﬂux is found by integrating over all frequencies, F=−1 3∇T∫∞ mdω β !ℓ!∂TB!, (1.18) where∇B!=∂TB!∇Twas used with ∂T=∂/∂T . For photons, one usually writes F=−(3κ ρ)−1∇aT4where aT4 is the total energy density in photons ( a=π2/15). Together with Eq. (1.18) this deﬁnes the photon Rosseland mean opacity κ . For other bosons one deﬁnes a corresponding quantity, 1 κxρ≡1 4aT3∫∞ mdω ℓ !β!∂TB!. (1.19) The “exotic” opacity thus deﬁned appears in the stellar structure equa- tion in the way indicated by Eq. (1.7). The production and absorption of bosons involves a Bose stimu- lation factor. This eﬀect is taken account of by including a factor (1−e−!=T) under the integral in Eq. (1.19). The “absorptive” opacity thus derived is usually referred to as the reduced opacity κ∗which is the quantity relevant for energy transfer. In practice, it is not very diﬀerent from κasωis typically 3 Tfor massless bosons. In the large-mass limit ( m≫T) the reduction factor may be ignored entirely and one ﬁnds to lowest order, 1 κxρ=g15 4π4(m T)3 e−m=T∫∞ 0dy ℓ !(y)y e−y, (1.20) where y≡β2 !m/2Twas used so that the energy of a nonrelativistic boson is given by ω=m+yT. 20 Chapter 1 1.3.5 Solar Bound on Massive Pseudoscalars In order to illustrate the energy-loss and energy-transfer argument for a boson with arbitrary mass I use the example of pseudoscalars χwhich couple to electrons by a “ﬁne-structure constant” αx. This is the only case in the literature where both arguments have been applied without assuming that the particle mass mxis small relative to T. The new bosons can be produced and absorbed by a variety of reactions. For purposes of illustration I focus on the Compton-type process γe↔eχ. For mx≪Tthe energy-loss rate per unit mass will be given in Eq. (3.23) in a more general context. For arbitrary masses it was derived by Raﬀelt and Starkman (1989). The result is ϵx=160ααx πYeT6 mum4 eF(mx/T), (1.21) where F(z) = 1 for mx≪TandF(z) = (20√ 2)−1z9=2e−zforT≪ mx≪me. Further, Yeis the number of electrons per baryon and muis the atomic mass unit. (It was used that approximately ne/ρ=Ye/mu with the electron density ne.) ForT≪mx≪methe Compton absorption rate is found to be Γx= 4παα xm2 xm−4 ene, leading to an opacity contribution of the pseudo- Fig. 1.1. Eﬀects of massive pseudoscalar particles on the Sun (interior tem- perature about 1 keV). Above the dashed line they contribute to the radia- tive energy transfer, below they escape freely and drain the Sun of energy. The shaded area is excluded by this simple argument. (Adapted from Raﬀelt and Starkman 1989.) The Energy-Loss Argument 21 scalars of κx=2(2π)9=2ααx 45YeT5=2emx=T m1=2 xm4 e = 4.4×10−3cm2g−1×αxYem−0:5 keVT2:5 keVemx=T, (1.22) where mkeV=mx/keV and TkeV=T/keV. The observed properties of the Sun then allow one to exclude a large range of parameters in the mx-αx-plane. The energy-loss rate integrated over the entire Sun must not exceed L⊙. Moreover, κ−1 xmust not exceed the standard photon contribution κ−1 ≈1 g/cm2, apart from perhaps a factor of order unity. Taking a typical solar interior temperature of 1 keV, these requirements exclude the shaded region in Fig. 1.1. The dashed line marks the parameters where the mfp is of order the solar radius. 1.4 General Lesson What have we learned? Weakly interacting particles, if they are not trapped in stars, carry away energy. For a degenerate object this en- ergy loss leads to additional cooling, for a burning star it leads to an accelerated consumption of nuclear fuel. New particles would cause sig- niﬁcant eﬀects only if they could compete with neutrinos which already carry away energy directly from the interior. If new particles are trapped because of a short mfp, they contribute to the energy transfer. They dominate unless their mfp is shorter than that of photons. Such particles probably do not exist or else they would have been found in laboratory experiments. This justiﬁes the usual focus on the energy-loss argument or “free streaming” limit. Still, the impact of new low-mass particles is maximized for an mfp of order the stellar radius, a fact which is often not appreciated in the literature. If the particles were so heavy that they could not be produced by thermal processes in a stellar plasma they would be allowed for any cou- pling strength. How heavy is heavy? The average energy of blackbody photons is about 3 T. Therefore, if mx∼<3Tthe particle production will not be signiﬁcantly suppressed. However, the solar example of Fig. 1.1 illustrates that for a suﬃciently large coupling strength even a particle with a mass of 30 Tcould have a signiﬁcant impact. While it can be produced only by plasma constituents high up in the tails of the thermal distributions, the Boltzmann suppression factor e−mx=T 22 Chapter 1 can be compensated by a strong coupling. For particles with a diﬀerent interaction structure the equivalent of Fig. 1.1 would look qualitatively similar, albeit diﬀerent in detail. Particles are not harmless in stars just because they are trapped, or just because their mass exceeds a typical temperature. However, in practice new particles are usually thought to be very weakly interacting and either essentially massless on the scales of stellar temperatures, or else very massive on those scales. In this book I will focus on low-mass particles with very weak interactions so that the energy-loss argument will be in the foreground of the discussion. The opposite case of weakly interacting massive particles (WIMPs) is of great interest in the framework of a hypothesis which holds that such objects are the dark matter of the universe. (Another dark matter candidate are axions which fall into the low-mass category covered in this book.) WIMP masses would be in the 10 GeV range and above so that they cannot be produced in stars, but they can be captured from the galactic background. They would contribute to the energy transfer by conduction. Hence their role in stars is similar to that of electrons, except for their large mfp which allows them to contribute signiﬁcantly even if their concentration is low. I do not treat this subject because it requires entirely diﬀerent formal tools from those relevant for low- mass particles. The fascinating story of WIMPs in stars will be told comprehensively within a forthcoming review paper Supersymmetric Dark Matter by Jungman, Kamionkowski, and Griest (1995). Chapter 2 Anomalous Stellar Energy Losses Bounded by Observations After a description of the main phases of stellar evolution, a review is given of the observations that have been used to constrain novel stellar energy losses. The main arguments involve the cooling speed of white dwarfs and old neutron stars, the delay of helium ignition in low-mass red giants, and the helium-burning lifetime of horizontal-branch stars. The latter two arguments, which are based on observations of globular- cluster stars, are cast into the form of an easy-to-use analytic criterion that allows for a straightforward application in many diﬀerent cases. The cooling speed of nascent neutron stars is discussed in Chapter 11 in the context of supernova neutrinos. 2.1 Stages of Stellar Evolution 2.1.1 The Main Sequence To identify those observables of stellar structure and evolution that can be used to constrain or perhaps discover a novel energy-loss mechanism it is necessary to understand how stars live and die. In Chapter 1 the basics of the theory of stellar structure have already been discussed. It is now time to give an overview of the stages of stellar evolution, and how they manifest themselves in the observable properties of stars. Because stellar evolution is an old subject, the literature is vast. I will give reference only to a few recent original papers that I need to support speciﬁc points. Otherwise, detailed quantitative accounts of 23 24 Chapter 2 stellar structure and evolution and their observational consequences are found in the textbook and review literature.3 How do stars form in the ﬁrst place? While a detailed understanding of this process is still elusive, for the present discussion it is enough to know that gravitationally bound clouds of gas ultimately fragment and condense because of their continuous energy loss by electromagnetic radiation. As discussed in Sect. 1.2.2, the negative speciﬁc heat of a self-gravitating system enforces its contraction and heating. For the fragmentation process a variety of parameters may be important such as overall pressure, angular momentum, or magnetic ﬁelds. Because the formation process of stars is poorly understood the initial mass function (IMF) is not accounted for theoretically. Typically, the number of stars per mass interval may be given by Salpeter’s IMF, dN/dM ∝ (M/M⊙)−1:35, which means that there are a lot more small than large stars. However, the overall range of stellar masses is quite limited. The largest stars have masses of up to 100 M⊙; a value beyond this limit does not seem to allow for a stable conﬁguration. At the small mass end, stars with M<0.08M⊙never become hot enough to ignite hydrogen; such “brown dwarfs” have never been unambiguously detected. It has been speculated that they could make up the dark matter of spiral galaxies. A search in our galaxy by the gravitational microlensing technique has yielded ﬁrst candidates (Alcock et al. 1993, 1995; Aubourg et al. 1993, 1995). The ﬁrst stars consist of a mixture of X≈0.75 (mass fraction of hydrogen) and Y≈0.25 (mass fraction of helium), the material left over from the big bang of the universe. Subsequent generations also contain a small amount of “metals” which in astronomers’ language is anything heavier than helium; our Sun has a metallicity Z≈0.02 (mass fraction of metals). These heavier elements were bred by nuclear fusion in earlier generations of stars which returned some of their mass to the interstellar 3An excellent starting point for the nonexpert is Shu (1982). The classics are Chandrasekhar (1939), Schwarzschild (1958), Clayton (1968), and Cox and Giuli (1968). A recent general textbook is Kippenhahn and Weigert (1990). Recent mono- graphs specializing on the Sun in general are Stix (1989), and on solar neutrinos Bahcall (1989). The theory of stellar pulsation is covered in Cox (1980). A classic on the physics of compact stars is Shapiro and Teukolsky (1983). A recent monograph on neutron stars is Lipunov (1992) and on pulsars M´ esz´ aros (1992). Recent collec- tions of papers or conference proceedings are available on the physics of red giants (Iben and Renzini 1981), white dwarfs (Barstow 1993), pulsating stars (Schmidt 1989), neutron stars (Pines, Tamagaki, and Tsuruta 1992), and supernovae (Brown 1988; Petschek 1990). Many excellent reviews are found in the Annual Review of Astronomy and Astrophysics . Anomalous Stellar Energy Losses Bounded by Observations 25 medium. Mass loss in advanced stages of stellar evolution can take on the benign form of a “stellar wind,” or it can occur in gigantic “supernova explosions.” The end state of stellar evolution must be complete disruption, or a compact object (white dwarf, neutron star, or black hole), because a normal star, supported by thermal pressure, can exist only as long as its nuclear fuel lasts (Sect. 1.2.2). The masses of degenerate stars are constrained by their Chandrasekhar limit of about 1.4M⊙for white dwarfs, and a similar value for neutron stars. Most massive stars seem to return enough mass to the interstellar medium that their typical end states are degenerate stars, not black holes. The primordial material evolves chemically because it is cycled through one generation of stars after another. Today, the continuing birth and death of stars takes place mostly in the disks of spiral galaxies. Such galaxies also have a population of old halo stars and of globular clusters, about 150 in our Milky Way, each of which is a gravitationally bound system of about 106stars (Fig. 2.1). Most of the globular clusters are in the galactic halo, far away from the disk. The gravitational escape velocity for stars or gas from a cluster is rather small, about 10 km s−1. A supernova explosion, on the other hand, ejects its material with typical velocities of several 103km s−1. A single supernova is enough to sweep a globular cluster clean of all Fig. 2.1. Globular cluster M3. (Image courtesy of Palomar/Caltech.) 26 Chapter 2 Fig. 2.2. Zero-age main sequence in the Hertzsprung-Russell diagram, with a composition of 68.5% hydrogen, 29.4% helium, and 2.1% metals. (Adapted from Kippenhahn and Weigert 1990.) gas; then no further star formation is possible.4Therefore, these sys- tems are very clean laboratories for studies of stellar evolution as they contain an early generation of stars (typical metallicities 10−3−10−4) which in a given cluster are all coeval and have almost equal chemical compositions. To a ﬁrst approximation the stars in a globular cluster diﬀer only in one parameter—their initial mass. The virial theorem explains (Sect. 1.2.2) that the negative speciﬁc heat of a protostellar cloud leads to contraction and heating until it has reached a thermal equilibrium conﬁguration where the nuclear burning rate exactly balances its overall luminosity. At this point the conﬁgu- ration is determined entirely by its mass, apart from a small inﬂuence of the metal content. Therefore, after the initial contraction hydrogen- burning stars of diﬀerent mass form the “zero-age main sequence” in a Hertzsprung-Russell diagram where the eﬀective surface temperature is plotted on the horizontal axis and the stellar luminosity on the verti- cal axis (Fig. 2.2). As hydrogen is consumed small adjustments of the conﬁguration occur. However, essentially a star remains at its initial main-sequence location for most of its life. 4The escape velocity from our galaxy is about 500 km s−1. However, in a galactic spiral arm the interstellar medium is dense enough to dissipate a SN explosion which is only able to blow a hole into the interstellar material. Anomalous Stellar Energy Losses Bounded by Observations 27 A 1M⊙star lives about 10 Gyr (1 Gyr = 109yr) on the main sequence; our Sun is thought to have completed about half of this episode. Heavier stars burn brighter (crudely L∝ M3) and thus live shorter lives. Because the universe is 10 −20 Gyr old, stars with M∼<0.7−0.9M⊙have not yet completed their hydrogen-burning phase, even if they formed shortly after the big bang. Because glob- ular clusters formed very early one expects a main-sequence turnoﬀ Fig. 2.3. Color magnitude diagram for the globular cluster M3 according to Buonanno et al. (1986), based on the photometric data of 10,637 stars. Following Renzini and Fusi Pecci (1988) the following classiﬁcation has been adopted for the evolutionary phases. MS (main sequence): core hydrogen burning. BS (blue stragglers). TO (main-sequence turnoﬀ): central hydro- gen is exhausted. SGB (subgiant branch): hydrogen burning in a thick shell. RGB (red-giant branch): hydrogen burning in a thin shell with a growing core until helium ignites. HB (horizontal branch): helium burning in the core and hydrogen burning in a shell. AGB (asymptotic giant branch): he- lium and hydrogen shell burning. P-AGB (post-asymptotic giant branch): ﬁnal evolution from the AGB to the white-dwarf stage. (Original of the ﬁgure courtesy of A. Renzini.) 28 Chapter 2 near this mass, depending somewhat on their metallicity. Indeed, the color-magnitude diagram5of the cluster M3 (Fig. 2.3) clearly shows this eﬀect. The location of the turnoﬀ (TO) point in such a diagram can be converted to the mass and thus to the age of a star which is just about to complete hydrogen burning. Globular clusters, therefore, allow one to establish a signiﬁcant lower limit for the age of the universe. 2.1.2 Becoming a Red Giant A main-sequence star naturally is hottest and densest at the center where hydrogen is depleted fastest. (See Fig. 2.4 for the overall struc- ture of a typical solar model which represents a low-mass star about halfway through its main-sequence evolution.) After all of the hy- drogen is consumed at the center, the further evolution depends on the total mass—I begin with the fascinating story of low-mass stars (M∼<2M⊙). After central hydrogen exhaustion a new conﬁguration establishes itself where the “helium ashes” which have accumulated at the cen- ter form an ever denser core surrounded by a hydrogen burning shell (Fig. 2.5). At the same time the material above the core begins to ex- pand which implies that the stellar surface becomes larger and thus the surface temperature lower according to the Stefan-Boltzmann law. Put another way, the star becomes redder, and ultimately a “red giant.” In the color-magnitude diagram of the globular cluster M3 (Fig. 2.3), the collection of stars which begins to form a contracting core and an expanding envelope is marked as the subgiant branch (SGB). 5The brightness of a star is photometrically measured on the basis of the lumi- nosity in a wavelength band deﬁned by a ﬁlter with a well-deﬁned spectral response. There exist many diﬀerent color systems, corresponding to many diﬀerent ﬁlters. In Fig. 2.3 the visual brightness is deﬁned by V=−2:5 logLV+const :withLVthe lu- minosity measured with the “visual” ﬁlter, centered in the yellow-green waveband, and a similar deﬁnition applies to the brightness Bin the blue. The “color” B−V is a measure of the surface temperature with lower temperatures (redder color) to the right. For technical deﬁnitions see Allen (1963). Note that the downward turn of the horizontal branch in the blue is determined by the Vﬁlter; bolometrically the HB is truly horizontal. An absolute bolometric brightness is given by Mbol= 4:74−2:5 log(L=L⊙). It is a dimensionless number, but often the unit mag (magnitude) is appended for clarity. The visual brightness in Fig. 2.3 is given in apparent magnitudes which de- pend on the distance of the cluster. However, because of the logarithmic brightness deﬁnition a diﬀerent distance leads only to a vertical shift of the entire picture; the color and relative brightnesses remain unchanged. Anomalous Stellar Energy Losses Bounded by Observations 29 Fig. 2.4. Left panels: A typical solar model (Bahcall 1989) as an ex- ample for a low-mass star which is halfway through its hydrogen-burning (main-squence) phase. Right panels: Horizontal-branch star (central helium burning, shell hydrogen burning) with a metallicity Z= 0:004 after about 2:5×107yr which is about a quarter of the HB lifetime. (Model from the calculations of Dearborn et al. 1990.) 30 Chapter 2 Fig. 2.5. Main evolutionary phases of low-mass stars. The envelope and core dimensions depend on the location on the RGB, HB, or AGB, respectively. The given radii are only meant to give a crude orientation. Anomalous Stellar Energy Losses Bounded by Observations 31 As hydrogen burning continues it dumps more and more helium on the core which at ﬁrst supports itself by thermal pressure. Soon it becomes so dense, however, that the electrons become degenerate, in- verting the mass-radius relationship to R∝ M−1=3(Sect. 1.2.2). Thus, as the helium core mass Mcgrows, the core radius Rcshrinks. The gravitational potential Φ cat the edge of the core is determined entirely by the core because the envelope contributes little due to its large exten- sion so that Φ c≈ −GNMc/Rc∝ M4=3 c. Because the hydrogen-burning shell above the core still supports itself by thermal pressure, the tem- perature near the core edge is determined by Φ c∝ M4=3 c. The growing core mass causes the hydrogen burning shell to become ever hotter! Because of the steep Tdependence of the hydrogen burning rates, the growing core causes the star to become ever brighter (Fig. 2.6). In this Fig. 2.6. Evolutionary track of a 0 :8M⊙star ( Z= 0:004) from zero age to the asymptotic giant branch. The evolutionary phases are as in Fig. 2.3. (Calculated with Dearborn’s evolution code.) 32 Chapter 2 shell-burning phase the luminosity is determined almost entirely by the core mass . On the MS it was determined by the total mass, a parameter which now hardly matters. The red-giant branch (RGB) ascension is a fast process compared with the main-sequence (MS) evolution (Fig. 2.6). Therefore, the red giants in the observational color-magnitude diagram of Fig. 2.3 diﬀer only by very small amounts of their initial mass. The stars beyond the MS have nearly identical properties while those on the MS diﬀer by their total mass. Therefore, the evolved stars in Fig. 2.3 essentially trace out the evolutionary path of a single star with an approximate mass corresponding to the MS turnoﬀ (TO). For example, the stars on the RGB in Fig. 2.3 essentially constitute snapshots of the single-star evolutionary track shown in the upper panel of Fig. 2.6. A red giant is a star with a compact energy source at the center and a large convective envelope. In the present case we have a degenerate helium core with shell hydrogen burning, but other conﬁgurations are possible. The core is a star unto itself with near to negligible feed- back from the envelope. The envelope, on the other side, is strongly inﬂuenced by the nontrivial central boundary conditions provided by the core. Convective conﬁgurations with a ﬁxed mass and a prescribed luminosity from a central point source occupy the “Hayashi line” in the Hertzsprung-Russell diagram. A low-mass red giant ascends the Hayashi line corresponding to its envelope mass. The inﬂation of a star to red-giant dimensions is a remarkable phe- nomenon which deﬁes an intuitive explanation, except that the stellar structure equations allow for such solutions. Another conceivable struc- ture is a much smaller envelope with radiative rather than convective energy transfer—upper MS stars are of that nature. Prescribing a core with a given luminosity one can imagine these two extreme conﬁgura- tions. Indeed, bright stars ( L∼>102L⊙) are usually found either near the MS or near the RGB which together form a V-shaped pattern in the Hertzsprung-Russell diagram (Fig. 2.9). The empty space in the V is known as the “Hertzsprung gap”—the few stars found there are thought to be on the move from one arm of the V to the other, i.e. from a radiative to a convective envelope structure. Massive stars af- ter completing hydrogen burning move almost horizontally across the Hertzsprung gap and inﬂate to red-giant dimensions. Remarkably, they can deﬂate and move horizontally back, executing a “blue loop.” Anomalous Stellar Energy Losses Bounded by Observations 33 2.1.3 Helium Ignition The core of a red giant reaches its limiting mass when it has become so hot and dense that helium ignites. Because the nucleus8Be which consist of two αparticles (He nuclei) is not stable, He burning proceeds directly to carbon, 3 α→12C (“triple- αreaction”), via an intermediate 8Be state. Because it is essentially a three-body reaction its rate de- pends sensitively on ρandT. In a red-giant core helium ignites when Mc≈0.5M⊙with central conditions of ρ≈106g cm−3andT≈108K where the triple- αenergy generation rate per unit mass, ϵ3, varies ap- proximately as ρ2T40. This steep temperature dependence allows one to speak of a sharp ignition point even though there is some helium burning at any temperature and density. The helium core of a red giant is like a powder keg waiting for a spark. When the critical temperature is reached where ϵ3exceeds the neutrino losses a nuclear runaway occurs. Because the pressure is mainly due to degenerate electrons the energy production at ﬁrst does not lead to structural changes. Therefore, the rise in temperature is unchecked and feeds positively on the energy generation rate. As this process continues the core expands nearly explosively to a point where it becomes nondegenerate and the familiar self-regulation by the gravitational negative speciﬁc heat kicks in (Chapter 1). The “explosion energy” is absorbed by the work necessary to expand the core from about 106g cm−3to about 104g cm−3. The core temperature of the ﬁnal conﬁguration remains at about 108K because of the steep temperature dependence of ϵ3which allows only for a narrow range of stationary burning conditions. The ﬁnal conﬁguration with a helium-burning core and a hydrogen- burning shell (Fig. 2.5) is known as a horizontal-branch (HB) star, a term which is justiﬁed by the location of these objects in the color- magnitude diagram Fig. 2.3. Note that the overall luminosity has de- creased by the process of helium ignition (Fig. 2.6) because of the core expansion which lowers the gravitational potential at the core edge and thus the temperature in the hydrogen-burning shell which continues to be regulated by the core mass and radius. The total luminosity of an HB star is given to about 1 /3 by He burning and 2 /3 by hydrogen shell burning (Fig. 2.4, right panels). Because helium ignition is an almost explosive process on dynamical time scales it is known as the “helium ﬂash.” For the same reason, a realistic numerical treatment does not seem to exist (for a review, see Iben and Renzini 1984). There are two main problems. First, normal 34 Chapter 2 stellar-evolution codes use hydrostatic stellar structure equations which ignore kinetic energies which are small in stationary phases or phases of slow expansion or contraction (Sect. 1.2.1). However, in a phase of fast expansion kinetic energies can be large. It is not known whether the helium ﬂash is a hydrostatic or a hydrodynamic event. Secondly, convection undoubtedly plays a large role at transferring energy from the ignition point. Note that helium ignites oﬀ-center because neutrino losses cause a temperature dip at the center. A fundamental theory of convection does not exist; it is likely that one would have to per- form a three-dimensional hydrodynamic calculation to develop a full understanding of the helium ﬂash. 2.1.4 The Horizontal Branch After helium ignition the overall luminosity has dropped as explained above, and the surface has shrunk considerably, leading to a substan- tially hotter (bluer) conﬁguration. Precisely how blue an HB star be- comes depends on the size of its envelope and thus on its total mass. A substantial amount of mass loss occurs on the RGB where the pho- tosphere of the star is so inﬂated that it is only weakly gravitationally bound; a red giant may typically lose about 0 .2M⊙of its envelope before helium ignites. Presumably, the amount of mass loss is not ex- actly the same from star to star; a small spread of order 0 .03M⊙is enough to explain the wide range of surface radii and thus colors found for these objects which form a horizontal branch in the color-magnitude diagram.6The downward turn of the HB in Fig. 2.3 is an artifact of the ﬁlter which determines the visual brightness—the bolometric brightness is the same. It is ﬁxed by the core properties. In practice, the HB morphology found in diﬀerent clusters is com- plicated. Some clusters have widely spread HBs such as that shown in Fig. 2.3 while others have stars only near the red or blue end, and gaps and blue tails occur. One important parameter is the metallicity of the envelope which determines the opacity to which the envelope structure is very sensitive. However, one ﬁnds clusters with the same chemical composition but diﬀerent HB morphologies, a phenomenon which has prompted a search for the “second parameter.” Many suggestions have been made by some and refuted by others. One candidate second pa- rameter which appears to remain viable is the cluster age, or rather, the MS mass of the stars observed today on the HB. The envelope mass 6For recent synthetic HBs see Lee, Demarque, and Zinn (1990, 1994). Anomalous Stellar Energy Losses Bounded by Observations 35 remaining for an HB star after mass loss on the RGB will still depend on its initial value, rendering this a rather natural possibility. How- ever, it implies that the globular clusters of our galaxy did not form at the same time—an age spread of several Gyr is required with the older clusters predominantly at smaller galactocentric distances (Lee, Demarque, and Zinn 1994 and references therein). Apart from some anomalous cases such an age spread appears to be enough to explain the second parameter phenomenon. Returning to the inner structure of an HB star, it evolves quietly at an almost ﬁxed total luminosity (Fig. 2.6). The core constitutes essentially a “helium main-sequence star.” Because its inner core is convective it dredges helium into the nuclear furnace at the very center, leaving a sharp composition discontinuity at the edge of the convective region (Fig. 2.4). After some12C has been built up,16O also forms. At the end of the HB phase, the helium core has developed an inner core consisting of carbon and oxygen. 2.1.5 From Asymptotic Giants to White Dwarfs After the exhaustion of helium at the center a degenerate carbon- oxygen (CO) core forms with helium shell burning and continuing hy- drogen shell burning (Fig. 2.5). Again, the star grows progressively brighter and inﬂates. Put another way, it becomes very similar to a star which ﬁrst ascended the RGB: it ascends its Hayashi line for a second time. The track in the Hertzsprung-Russell diagram asymp- totically approaches that from the ﬁrst ascent (“asymptotic giants”). The upper RGB can be observationally diﬃcult to distinguish from the asymptotic giant branch (AGB) even though they are reasonably well separated in Fig. 2.3. In low-mass stars carbon and oxygen never ignite. The shell sources extinguish when most of the helium and hydrogen has been consumed so that the star has lost its entire envelope either by hydrogen burning or by further mass loss on the AGB. The remaining degenerate CO star continues to radiate the heat stored in its interior. At ﬁrst these stars are rather hot, but geometrically very small with a typical ra- dius of 104km. Their small surface area restricts their luminosity in spite of the high temperature and so they are referred to as “white dwarfs.” Because they are supported by electron degeneracy pressure, their remaining evolution is cooling by neutrino emission from the in- terior and by photon emission from the surface until they disappear from visibility. 36 Chapter 2 Fig. 2.7. Ring Nebula in Lyra (M57), a planetary nebula. (Image courtesy of Palomar/Caltech.) For stars which start out suﬃciently large the mass loss on the AGB can be so dramatic that one may speak of the ejection of the entire envelope. It forms a large shell of gas which is illuminated by its central star, the newborn white dwarf. Such systems are known as “planetary nebulae” (Fig. 2.7). 2.1.6 Type I Supernovae A white dwarf can make a spectacular reappearance if it is in a binary system. If the other member expands because it is in an earlier evolu- tionary phase it can transfer mass, allowing for renewed nuclear burning on the white-dwarf surface. This adds mass to the CO conﬁguration which ultimately becomes so hot and dense that these elements ignite, leading to a (subsonic) deﬂagration or (supersonic) detonation front which sweeps through the star and disrupts it entirely. This course of events is the standard scenario for type I supernova explosions which are among the most energetic events known in the universe. They must be carefully distinguished from type II supernovae which are related to the collapse of evolved, massive stars. Anomalous Stellar Energy Losses Bounded by Observations 37 2.1.7 Intermediate-Mass Stars The evolutionary scenario described so far applies to stars with a mass below 2 −3M⊙. For larger masses the core conditions evolve contin- uously to the ignition of helium which is then a quiet process. For a given chemical composition the transition between the two scenarios is a sharp function of the total mass. Because red giants are very bright they dominate the total luminosity of an old stellar population so that this transition aﬀects its integrated brightness in a discontinuous way. Some authors have used the concept of an “RGB phase transition” to describe this phenomenon (Sweigart, Greggio, and Renzini 1990, and references therein; Bica et al. 1991). Stars with masses of up to 6 −8M⊙are expected to end up as CO white dwarfs. Their evolution on the AGB can involve many interesting phenomena such as “thermal pulses”—for a key to the literature see Iben and Renzini (1984). Intermediate-mass stars have thus far played little role for the purposes of particle astrophysics, and so their evolution does not warrant further elaboration in the present context. 2.1.8 Massive Stars and Type II Supernovae The course of evolution for massive stars ( M>6−8M⊙) is qualita- tively diﬀerent because they ignite carbon and oxygen in their core, allowing them to evolve further. This is possible because even after mass loss they are left with enough mass that their CO core grows toward the Chandrasekhar limit. Near that point the density is high enough to ignite carbon which causes heating and thus temporarily re- lieves the electron degeneracy. Next, the ashes of carbon burning (Ne, Mg, O, Si) form a degenerate core which ultimately ignites Ne burning, and so forth. Ultimately, the star has produced a degenerate iron core, surrounded by half a dozen “onion rings” of diﬀerent burning shells. The game is over when the iron core reaches its Chandrasekhar limit because no more nuclear energy can be released by fusion. The tem- perature is at 0 .8×1010K = 0 .7 MeV, the density at 3 ×109g cm−3, and there are about Ye= 0.42 electrons per baryon. Further contraction leads to a negative feedback on the pressure as photons begin to dis- sociate iron, a process which consumes energy. Electrons are absorbed and converted to neutrinos which escape, lowering the electron Fermi momentum and thus the pressure. Therefore, the core becomes un- stable and collapses, a process which is intercepted only when nuclear density (3 ×1014g cm−3) is reached where the equation of state stiﬀens. 38 Chapter 2 Fig. 2.8. The Crab nebula, remnant of the supernova of A.D. 1054. (Image courtesy of the European Southern Observatory.) At this point a shock wave forms at the edge of the core and moves outward. The implosion can be said to be reﬂected and thus turned into an explosion. In practice, it is diﬃcult to account for the subsequent evolution as the shock wave tends to dissipate its energy by dissociating iron. Currently it is thought that a revival of the stalled shock is needed and occurs by neutrinos depositing their energy in the “hot bubble” below the shock. This region has a low density yet high temperature, and thus a high entropy per baryon by common astrophysical standards. Within about 0 .3 s after collapse the shock has moved outward and ejects the entire overburden of the mantle and envelope. This course of events is the scenario of a type II supernova (SN) explosion. What remains is an expanding nebula such as the Crab (Fig. 2.8) which is the remnant of the SN of A.D. 1054, and a central neutron star (radius about 10 km, mass about 1 M⊙) which often appears in the form of a pulsar, a pulsating source of radiation in some or all electromagnetic wave bands. The pulsed emission is explained by a complicated interplay between the fast rotation and strong magnetic ﬁelds (up to 1012−1013G) of these objects. Returning to the moment after collapse of the iron core, it is so dense (nuclear density and above) and hot (temperature of several 10 MeV), Anomalous Stellar Energy Losses Bounded by Observations 39 that even neutrinos are trapped. Therefore, energy and lepton num- ber are lost approximately on a neutrino diﬀusion time scale of several seconds. The neutrinos from stellar collapse were observed for the ﬁrst and only time when the star Sanduleak −69 202 in the Large Magel- lanic Cloud (a small satellite galaxy of the Milky Way) collapsed. The subsequent explosion was the legendary SN 1987A. After the exhaustion of hydrogen, massive stars move almost hori- zontally across the Hertzsprung-Russell diagram until they reach their Hayashi line, i.e. until they have become red supergiants. However, sub- sequently they can loop horizontally back into the blue; the progenitor of SN 1987A was such a blue supergiant. The SN rate in a spiral galaxy like our own is thought to be about one in a few decades, pessimistically one in a century. Because many of the ones occurring far away in our galaxy will be obscured by the dust and gas in the disk, one has to be extremely lucky to witness such an event in one’s lifetime. The visible galactic SNe previous to 1987A were Tycho’s and Kepler’s in close succession about 400 years ago. Both of them may have been of type I—no pulsar has been found in their remnants.7Of course, in the future it may become possible to detect optically invisible galactic SNe by means of neutrino detectors like the ones which registered the neutrinos from SN 1987A. 2.1.9 Variable Stars Stars are held in equilibrium by the pull of gravity which is opposed by the pressure of the stellar matter; its inertia allows the system to oscillate around this equilibrium position. If the adiabatic relationship between pressure and density variations is written in the form δP/P = γ δρ/ρ , the fundamental oscillation period Pof a self-gravitating ho- mogeneous sphere (density ρ) is found to be8P−1= [(γ−4 3)GNρ/π]1=2. This yields the period-mean density relationship P(ρ)1=2≈const. which is often written in the form P(ρ/ρ⊙)1=2=Qwith the average solar density ρ⊙= 1.41 g cm−3and the pulsation “constant” Q. It is in the range 0 .5−3 h, depending on the adiabiatic coeﬃcient γand the 7Observationally, type I and II SNe are distinguished by the absence of hydrogen spectral lines in the former which is explained by their progenitor being an accret- ing white dwarf which explodes after carbon ignition. Therefore, it is diﬃcult to establish the type of a historical SN unless a pulsar is detected as in the Crab. 8For < 4=3 the star is dynamically unstable. We have already encountered this magic number for the relationship between pressure and density of a degenerate relativistic electron gas where it led to Chandrasekhar’s limiting white-dwarf mass. 40 Chapter 2 detailed structure of the star. When these oscillations are damped, stars are found in their equi- librium conﬁgurations with constant color and brightness. However, it is possible that a small deviation from equilibrium is ampliﬁed, leading to a growing oscillation amplitude. Throughout a star there can be re- gions which try to excite oscillations, and others which damp them. If the driving mechanism is strong enough the star is found in a continuing oscillation which manifests itself in an oscillating lightcurve. Of particular interest are the Cepheid-type variables. Their oscilla- tions are excited by the “ κmechanism” where the driving force is the heat valve provided by the opacity of the stellar matter near the sur- face. There, hydrogen and helium are only partially ionized so that a temperature increase leads to increased ionization which increases the opacity because the Thomson cross section on free electrons is much larger than the Rayleigh one on neutral atoms. Therefore, the temper- ature increase caused by compression makes it more diﬃcult for energy to escape: the valve shuts. In most regions of the star the opposite hap- pens. Increased temperature makes it easier for energy to leak out from the compressed region, the valve opens, and oscillations are damped. Because the operation of the κmechanism is determined by condi- tions near the stellar surface which is characterized by its temperature and luminosity, Cepheid-type variables are found on a certain locus of color and brightness. This “instability strip” extends throughout the Hertzsprung-Russell diagram (hatched band in Fig. 2.9); it applies to stars of radically diﬀerent internal structure. Variable stars are found wherever the instability strip overlaps with an actual stellar population. At the bright end of the strip which is close and nearly parallel to the red-giant Hayashi line one ﬁnds the classical Cepheids ( δCepheids) with luminosities 300 −30,000 L⊙and periods 1 −50 days. The brightness variation can be up to 1 mag (visual), or a luminosity which changes by up to a factor of 3. Classical Cepheids are massive stars which cross the otherwise unpopulated Hertzsprung gap from the main-sequence to the red-giant region, and red giants and supergiants which execute blue loops and so temporarily move to the blue of the Hayashi line. The linear relationship between period and brightness of Cepheids as well as their large intrinsic luminosities are the key for their prominent role as standard candles and thus as astronomical distance indicators. The crossing of the instability strip with the HB is the domain of the RR Lyrae stars. Their luminosities are around 50 −100L⊙, their periods 1 .5−24 h. Their color coincides with the MS turnoﬀ in globular clusters (Fig. 2.3). RR Lyrae stars play an important role in the ongoing Anomalous Stellar Energy Losses Bounded by Observations 41 Fig. 2.9. Schematic Hertzsprung-Russell diagram of the main types of stars. The hatched band is the instability strip, the locus of Cepheid-type variables. The “Hertzsprung gap” between the main sequence and the red-giant region is populated with stars crossing between these branches, and by red giants and supergiants which execute blue loops. debate about the age of globular clusters. Toward fainter luminosities the instability strip crosses the MS, a region where the δScuti stars and dwarf Cepheids (periods of 1 −3 h) are located. Fainter still, the only remaining population of stars are white dwarfs. At the crossover of their locus with the instability strip the ZZ Ceti variables are found; they have periods of a few minutes. Because the oscillation period depends on the luminosity, an observed secular change of the period of a ZZ Ceti star can serve as a sensitive diagnostic for a decrease of its luminosity and thus for its cooling speed. 42 Chapter 2 2.2 White-Dwarf Cooling 2.2.1 Theoretical and Observed White-Dwarf Properties After this general survey of how stars evolve it is time to study individ- ual aspects in more detail, and notably, how stellar evolution is aﬀected by the emission of weakly interacting particles. I begin with the con- ceptually most transparent case of stars for which the loss of energy simply accelerates their cooling, i.e. white dwarfs and neutron stars. The former represent the ﬁnal state of the evolution of stars with initial masses of up to several M⊙, perhaps up to 8 M⊙(Sect. 2.1). For reviews of the theory and observed properties see Hubbard (1978), Liebert (1980), Shapiro and Teukolsky (1983), Weidemann (1990), and D’Antona and Mazzitelli (1990). Because white dwarfs (WDs) are sup- ported by electron degeneracy pressure the hydrostatic and thermal properties are largely decoupled. In Sect. 1.2.2 we had encountered their inverted mass-radius relationship; in a polytropic approximation of the WD structure one ﬁnds quantitatively (Shapiro and Teukol- sky 1983) R= 10,500 km (0 .6M⊙/M)1=3(2/µe)5=3. (2.1) Here,Mis the stellar mass and µe=ρ m−1 un−1 e=Y−1 ethe “mean molecular weight of the electrons” with ρthe mass density, muthe atomic mass unit, nethe electron density, and Yethe number of elec- trons per baryon. WDs do not contain any hydrogen in their interior—it would immediately ignite—so that µe= 2. Typically they consist of carbon and oxygen, the end products of helium burning in the core of the progenitor star. The central density of a polytropic model is ρc= 1.46×106g cm−3(M/0.6M⊙)2(µe/2)5, (2.2) assuming nonrelativistic electrons. If the mass is so large (the radius so small) that the electrons become relativistic there exists no stable conﬁguration, i.e. the masses of WDs must lie below the Chandrasekhar limit (Shapiro and Teukolsky 1983) MCh= 1.457M⊙(2/µe)2. (2.3) Observationally the WD mass distribution is strongly peaked near M= 0.6M⊙(Weidemann and Koester 1984) so that a nonrelativistic treat- ment of the electrons is justiﬁed. The low mass of observed WDs is understood by the large rate of mass loss during the red giant and Anomalous Stellar Energy Losses Bounded by Observations 43 asymptotic giant evolution which can amount to an ejection of the en- tire envelope and thus to the formation of a planetary nebula (Fig. 2.7). A theoretical evolutionary track for a 3 M⊙star from the MS to the WD stage was calculated, e.g. by Mazzitelli and D’Antona (1986). The central stars of planetary nebulae are identiﬁed with nascent WDs. The rate of WD formation inferred from the luminosity function discussed below agrees within a factor of about 2 with the observed formation rate of planetary nebulae, which means that both quantities agree to within their statistical and systematic uncertainties. The hottest and brightest WDs have luminosities of L≈0.5L⊙ while the faintest ones are observed at L≈0.5×10−4L⊙. Thus, be- cause of their small surface area WDs are intrinsically faint (see the Hertzsprung-Russell diagram Fig. 2.9). This implies that they can be observed only in the solar neighborhood, for bright WDs out to about 100 pc (300 lyr). Because their vertical scale height in the galactic disk is about 250 pc (Fleming, Liebert, and Green 1986) the observed WDs homogeneously ﬁll a spherical volume around the Sun. One may then express the observed number of WDs in terms of a volume density; it is of order 10−2pc−3. The observed luminosity function (the space density of WDs per brightness interval) is shown in Fig. 2.10 and listed in Tab. 2.1 according to Fleming, Liebert, and Green (1986) and Liebert, Dahn, and Monet (1988). The operation of a novel cooling mechanism can be constrained by three important features which characterize the luminosity function: its slope, which signiﬁes the form of the cooling law, its amplitude, which characterizes the cooling time and WD birthrate, and its sudden break at log( L/L ⊙)≈ −4.7, which characterizes the beginning of WD formation. Even the oldest WDs have not had time to cool to lower luminosities. From this break one can infer an age for the galactic disk of 8 −10.5 Gyr (Winget et al. 1987; Liebert, Dahn, and Monet 1988; Iben and Laughlin 1989; Wood 1992) while Hernanz et al. (1994) ﬁnd 9 .5−12 Gyr on the basis of their cooling calculations which include crystallization eﬀects. Because of the WD mass-radius relationship the surface tempera- ture and luminosity are uniquely related for a given WD mass. There- fore, instead of the luminosity function one may consider the temper- ature distribution which, in principle, is independent of uncertain WD distance determinations. Fleming, Liebert, and Green (1986) gave the distribution of their sample of hot WDs in several temperature bins (Tab. 2.2). Numerical cooling calculations of Blinnikov and Dunina- Barkovskaya (1994) found good agreement with this distribution, as- 44 Chapter 2 Table 2.1. Observed WD luminosity function according to Fleming, Liebert, and Green (1986) and Liebert, Dahn, and Monet (1988). For the hot and bright degenerates (upper part of the table) a large fraction of their spectrum lies in the ultraviolet, causing a large discrepancy between the absolute visual magnitude MVand the absolute bolometric magnitude Mbol. For the hot dwarfs the bins orginally had been chosen on the MVscale with a width of 0:5 mag, centered on the half-magnitudes; the listed Mbolis the mean in these intervals. Mean Mean dN/dM bol MV Mbol log(L/L ⊙) [pc−3mag−1] log( dN/dM bol) 9.5 5 .50 −0.31 1 .22×10−6−5.91 (+0 .18,−0.31) 10.0 6 .88 −0.86 1 .01×10−5−5.00 (+0 .14,−0.21) 10.5 7 .84 −1.25 2 .16×10−5−4.67 (+0 .13,−0.18) 11.0 8 .92 −1.68 9 .56×10−5−4.02 (+0 .12,−0.16) 11.5 10 .12 −2.16 1 .21×10−4−3.92 (+0 .11,−0.15) 12.0 11 .24 −2.61 1 .51×10−4−3.82 (+0 .11,−0.16) 12.5 11 .98 −2.90 2 .92×10−4−3.54 (+0 .11,−0.16) 13.0 12 .55 −3.13 6 .07×10−4−3.22 (+0 .20,−0.39) 13.50 −3.51 0 .89×10−3−3.05 (+0 .14,−0.21) 14.50 −3.91 1 .34×10−3−2.87 (+0 .14,−0.20) 15.50 −4.31 0 .24×10−3−3.62 (+0 .18,−0.31) Table 2.2. Temperature distribution of hot WDs according to Fleming, Liebert, and Green (1986). Teﬀ[103K] Fraction of WDs 40−80 (7 .06±1.27)×10−3 20−40 0 .235±0.026 12−20 0 .759±0.083 suming that the WD mass function was peaked around 0 .7M⊙which is somewhat larger than the canonical value of 0 .6M⊙. For the present purpose, however, the smallness of this diﬀerence is taken as a conﬁr- mation of the standard WD cooling theory. Anomalous Stellar Energy Losses Bounded by Observations 45 Fig. 2.10. Observed WD luminosity function as in Tab. 2.1. The dot- ted line represents Mestel’s cooling law with a constant WD birthrate of B= 10−3pc−3Gyr−1. The dashed line is from the numerical cooling curve of a 0 :6M⊙WD (Koester and Sch¨ onberner 1986), including neutrino losses and assuming the same constant birthrate. 2.2.2 Cooling Theory A WD has no nuclear energy sources and so it shines on its residual ther- mal energy: the evolution of a WD must be viewed as a cooling process (Mestel 1952). Because electron conduction is an eﬃcient mechanism of energy transfer the interior can be viewed, to a ﬁrst approximation, as an isothermal heat bath with a total amount of thermal energy U. Because the nondegenerate surface layers have a large “thermal resis- tance,” they insulate the hot interior from the cold surrounding space, throttling the energy loss L by photon radiation. Of course, WDs can also lose energy by neutrino volume emission L, and by novel particle emission Lx. Hence, WD cooling is governed by the equation dU/dt =−(L +L+Lx). (2.4) This simple picture ignores the possibility of residual hydrogen burn- ing near the surface, a possibly important luminosity source for young WDs (e.g. Castellani, Degl’Innocenti, and Romaniello 1994; Iben and Tutukov 1984). I will get back to this problem below. 46 Chapter 2 In order to translate this equation into the observable luminosity function I assume a constant WD birthrate Bso that the total number density of degenerates is N=B tgal(age of the galactic disk tgal). Taking the above values N≈10−2pc−3andtgal≈9 Gyr one ﬁnds B≈10−3pc−3Gyr−1. Because the number density of WDs in a given magnitude interval dMbolis proportional to the time interval dtit takes to cool through this magnitude range one readily obtains dN dMbol=Bdt dMbol=−BdU/dM bol L +L+Lx. (2.5) The photon luminosity is L = 78.7L⊙10−2Mbol=5in terms of the bolo- metric magnitude, equivalent to log( L /L⊙) = (4 .74−Mbol)/2.5.L is related to the internal temperature Tby the thermal conductance of the surface layers so that one may derive a function T(L ). The quanti- tiesU,L, and Lxare given in terms of Tso that they can be expressed in terms of L and hence of Mbol. In hot WDs the thermal energy is largely stored in the nuclei which form a nearly classical Boltzmann gas. At low Tthe ideal-gas law breaks down and eventually the nuclei arrange themselves in a lattice. The internal energy is then a more complicated function of temperature. The heat capacity per nucleon, which is3 2for an ideal gas, rises to 3 near the Debye temperature Θ Dand then drops approximately as (16π4/5) (T/ΘD)3to zero (Shapiro and Teukolsky 1983). However, the observed WDs have a relatively small ρbecause of their small mass around 0 .6M⊙so that even the oldest WDs have not yet crystallized. Therefore, as a reasonable ﬁrst approximation the internal energy is U=C Twith the ideal-gas heat capacity for the entire star of C=3 2M mu∑ jXj Aj= 3.95×10−2L⊙Gyr 107KM M⊙∑ jXj Aj, (2.6) where Xjis the mass fraction of the element j, atomic mass Aj, and mu= 1.661×10−24g is the atomic mass unit. The thermal conductance of the surface layers is calculated by solv- ing the stellar structure equations. Using a Kramers opacity law, κ=κ0ρ T−7=2, one ﬁnds (van Horn 1971; Shapiro and Teukolsky 1983) L= 1.7×10−3L⊙M M⊙(T 107K)7=2 ≡K T7=2, (2.7) where Tis the internal temperature. The observed WD luminosities of Tab. 2.1 vary between 0 .5×10−4and 0 .5L⊙, corresponding to a range 0.4−6×107K of internal temperatures. Anomalous Stellar Energy Losses Bounded by Observations 47 With these results the luminosity function is found to be dN dMbol=4 ln(10) 35BC(L /K)2=7 L +L+Lx. (2.8) With B3≡B/10−3pc−3Gyr−1this is numerically dN dMbol=B32.2×10−4pc−3mag−1 ×10−4Mbol=35L⊙ 78.7L⊙10−2Mbol=5+L+Lx(M M⊙)5=7∑ jXj Aj.(2.9) If one ignores LandLxthis is dN dMbol=B32.9×10−6pc−3mag−1102Mbol=7(M M⊙)5=7∑ jXj Aj. (2.10) Taking M= 0.6M⊙and an equal mixture of12C and16O one ﬁnds log(dN/dM bol) =2 7Mbol−6.84 + log( B3), (2.11) a behavior known as Mestel’s cooling law. For B3= 1 this function is shown as a dotted line in Fig. 2.10. Detailed cooling curves and luminosity functions have been calculated, for example, by Lamb and van Horn (1975), Shaviv and Kovetz (1976), Iben and Tutukov (1984), Koester and Sch¨ onberner (1986), Winget et al. (1987), Iben and Laugh- lin (1989), Segretain et al. (1994), and Hernanz et al. (1994). From Fig. 2.10 it is evident, however, that Mestel’s cooling law pro- vides a surprisingly good representation for intermediate luminosities where it is most appropriate. At the bright end, the luminosity func- tion is slightly depressed, providing evidence for neutrino cooling. It rapidly falls oﬀ at the faint end, presumably indicating the beginning of WD formation as discussed above. 2.2.3 Neutrino Cooling For the hottest WDs volume neutrino emission is more important than surface photon cooling. The photon luminosity of Eq. (2.7) can be expressed as an eﬀective energy-loss rate per unit mass of the star, ϵ = L /M= 3.3×10−3erg g−1s−1T3:5 7with T7=T/107K. For the upper relevant temperature range, neutrinos are emitted mostly by the plasma 48 Chapter 2 process γ→νeνewhich is studied in detail in Chapter 6; numerical emission rates are discussed in Appendix C. The neutrino energy-loss rate as a function of temperature is shown in Fig. C.6; for a WD the short-dashed curve (2 ρ/µ e= 106g cm−3) is most appropriate. Neutrino cooling causes a depression of the WD luminosity func- tion at the bright end. The dashed line in Fig. 2.10 represents a nu- merical cooling calculation for a 0 .6M⊙WD which included neutrinos (Koester and Sch¨ onberner 1986); the “neutrino dip” is clearly visible. The photon decay γ→ννis a neutrino process made possible by the medium-induced photon dispersion relation and by the medium- induced eﬀective photon-neutrino coupling (Chapter 6). As such this process is not observable in the laboratory, although there is no doubt about its reality. Still, it is encouraging to see it so plainly in the WD luminosity function. If neutrino emission were much stronger than standard, the neu- trino dip would be correspondingly deeper. Stothers (1970) used the observation of several bright WDs in the Hyades cluster to constrain the eﬃciency of neutrino cooling; at that time the existence and magnitude of a direct neutrino-electron interaction had not yet been experimen- tally established. Stothers found that an emission rate 300 times larger than standard could be conservatively excluded. Recently, Blinnikov and Dunina-Barkovskaya (1994) have studied this subject in detail. Their motivation was the possible existence of a neutrino magnetic dipole moment which would enhance the eﬀec- tive neutrino-photon coupling and thus the eﬃciency of the plasma process. For the present general discussion it is enough to think of their study as an arbitrary variation of the neutrino emissivity, even though the magnetic-moment induced plasma emission rate has a dif- ferent density dependence—see Eq. (6.94) for details. For the perti- nent conditions the energy-loss rate induced by the assumed dipole moment µis roughly 0 .06µ2 12times the standard one where µ12≡ µ/10−12µB(Bohr magneton µB). Blinnikov and Dunina-Barkovskaya (1994) calculated the luminosity function for 0 .6M⊙WDs for the stan- dard case ( µ12= 0, dashed line in Fig. 2.11) and for a roughly 25-fold increased rate of neutrino cooling ( µ12= 20, dotted line in Fig. 2.11). The birthrate of WDs was assumed constant and adjusted to opti- mize the agreement with the observations. For µ12= 0 they needed B= 0.62×10−3pc−3Gyr−1while for µ12= 20 the best ﬁt was achieved for 0.67 in these units. Blinnikov and Dunina-Barkovskaya (1994) pointed out that a par- ticularly sensitive observable for the neutrino dip is the temperature dis- Anomalous Stellar Energy Losses Bounded by Observations 49 Fig. 2.11. Luminosity function for 0 :6M⊙WDs for two values of 12(Blin- nikov and Dunina-Barkovskaya 1994) compared with the observations quoted in the upper part of Tab. 2.1 (Fleming, Liebert, and Green 1986). Fig. 2.12. Relative number of 0 :6 and 0 :8M⊙WDs in the two hot tempera- ture bins of Tab. 2.2 as a function of the anomalous neutrino cooling implied by a magnetic dipole moment (Blinnikov and Dunina-Barkovskaya 1994). The number of WDs in the temperature range 12,000 −40,000 K is normal- ized to unity. The shaded bands correspond to the observations (Tab. 2.2). 50 Chapter 2 tribution of hot WDs derived by Fleming, Liebert, and Green (1986)— see Tab. 2.2. They calculated cooling sequences for M= 0.6M⊙and 0.8M⊙with varying amounts of nonstandard cooling. The relative number of WDs in the two hot temperature bins of Tab. 2.2 are shown as a function of µin Fig. 2.12. These results seem to indicate that a signiﬁcantly enhanced rate of neutrino emission can be conservatively excluded. However, this view may be challenged if one includes the possibility of residual hydro- gen burning near the WD surface which could mask neutrino cooling because it would ﬁll in some of the “neutrino dip” in the luminosity function. Because it is not known how much hydrogen is retained by a WD after the planetary nebula phase one has an adjustable parame- ter to provide a desired amount of heating (Castellani, Degl’Innocenti, and Romaniello 1994). However, preliminary investigations seem to in- dicate that even when residual hydrogen burning is included the impact ofµis masked only in one of the temperature bins used by Blinnikov and Dunina-Barkovskaya (1994) so that a signiﬁcant deformation of the luminosity function appears to remain (Blinnikov and Degl’Innocenti 1995, private communication). 2.2.4 Cooling by Boson Emission Standard or exotic neutrino emission from WDs (or neutron stars) has the important property that it switches oﬀ quickly as the star cools because of the steep temperature dependence of the emission rates. Therefore, neutrinos cause a dip at the hot end of the luminosity func- tion while older WDs are left unaﬀected, even for signiﬁcantly enhanced neutrino cooling (Fig. 2.11). One may construct other cases, however, where this is diﬀerent. One example is when a putative low-mass bo- son is emitted in place of a neutrino pair, say, in the bremsstrahlung process e+ (Z, A)→(Z, A) +e+νν. The reduced ﬁnal-state phase space then reduces the steepness of the temperature dependence of the energy-loss rate. The possible existence of such particles is motivated by theories involving spontaneously broken global symmetries. The most widely discussed example is the axion which will be studied in some detail in Chapter 14. For the present discussion all that matters is the temperature variation of an assumed energy-loss rate. The bremsstrahlung rate for pseudoscalar bosons will be calculated in Chapter 3. For the highly degenerate limit the result is given in Eq. (3.33) where α′=g2/4πis the relevant “ﬁne-structure constant.” Because this rate depends on the density only weakly through a factor Anomalous Stellar Energy Losses Bounded by Observations 51 Fwhich includes Coulomb screening by ion correlations and electron relativistic corrections one may easily “integrate” over the entire star. For an assumed equal mixture of carbon and oxygen one ﬁnds for the luminosity in pseudoscalar “exotica” Lx=α262.0×10−3L⊙(M/M⊙)⟨F⟩T4 7, (2.12) where α26=α′/10−26andT7=T/107K (internal temperature T). Further, ⟨F⟩ ≈1.0 within a few 10% (Sect. 3.5.2). This energy-loss rate varies with internal temperature almost as the surface photon luminosity of Eq. (2.7). If Lxdominates in Mes- tel’s cooling law Eq. (2.11) the slope10 35Mbolis replaced with the al- most identical value12 35Mbol. Thus, for a given WD birthrate the main impact of pseudoscalars is to reduce the amplitude of the luminosity function. Conversely, the inferred birthrate is larger by about a factor 1 +Lx/L ≈1 +α26. Because the formation rate of planetary nebulae, the progenitors of WDs, agrees with the standard inferred birthrate to within a factor of about 2, one ﬁnds α26∼<1.9 The observed break of the luminosity function at the faint end (Fig. 2.10) has been interpreted as the beginning of WD formation. If boson cooling were dominant the faintest luminosities would have been reached in a shorter amount of time, reducing the inferred age of the galactic disk; standard cooling implies tgal= 8−12 Gyr. Even the solar system is 4 .5 Gyr old and so a reduction of tgalby more than a factor of 2 is ruled out. This implies Lx< L so that α26<1.0 as before. The galactic age constraint appears to be much more reliable than the one based on the formation rate of planetary nebulae. The former could be avoided if the observations reported by Liebert, Dahn, and Monet (1988) were crudely incomplete at the faint end of the luminos- ity function, i.e. if many faint WDs had been overlooked so that the break in the luminosity function of Fig. 2.10 were a major observational selection eﬀect. Barring this remote possibility the limit on α26is con- servative. Even for a much smaller value of α26, the inferred value for tgalis reduced by an approximate factor (1 + α26)−1which may still be signiﬁcant. Wang (1992) calculated numerical 1 M⊙WD sequences with vary- ing amounts of pseudoscalar cooling while Blinnikov and Dunina-Bar- kovskaya (1994) performed a more detailed study for 0 .6M⊙WDs. 9In the original derivation (Raﬀelt 1986b) ion correlations were ignored, leading to⟨F⟩ ≈3 and to the limit 26∼<0:3. 52 Chapter 2 They calculated the distribution for the temperature bins of Tab. 2.2 in analogy to the discussion of neutrino dipole moments in the previous section. Their results are shown in Fig. 2.13 as a function of α26where I have corrected from ⟨F⟩= 3, which they used in order to compare with Raﬀelt (1986b), to the more appropriate value ⟨F⟩= 1. Fig. 2.13. Relative number of 0 :6M⊙WDs in the hot and intermediate temperature bins of Tab. 2.2 as a function of the “ﬁne-structure constant” 26=′=10−26of pseudoscalar bosons (Blinnikov and Dunina-Barkovskaya 1994). I have corrected from ⟨F⟩= 3 to the more appropriate value 1. The number of WDs in the temperature range 12,000 −40,000 K is normalized to unity. The shaded bands correspond to the observations (Tab. 2.2). This temperature method is now relatively insensitive because axion cooling leaves the shape of the luminosity function nearly unchanged, except that the neutrino dip is washed out for strong cooling. In detail, however, important changes occur. It is noteworthy that boson cool- ing has a signiﬁcant impact on the shape of the luminosity function even for α26<1. It appears that the possibility of residual hydrogen burning in young WDs would make the present argument more con- servative because it has the same eﬀect as boson emission, namely to reduce or wash out the neutrino dip in the luminosity function. 2.2.5 Period Decrease of Variable White Dwarfs The luminosity function allows one to determine the WD cooling speed because one is looking at a large WD ensemble of diﬀerent age. Recently it has become possible for the ﬁrst time to measure WD cooling directly for a single object by virtue of the period change of a ZZ Ceti star. Anomalous Stellar Energy Losses Bounded by Observations 53 White dwarfs have residual atmospheres which may be hydrogen rich (DA white dwarfs) or helium rich (DB); the DA stars are about four times more frequent. The surface layers of DA white dwarfs are not fundamentally diﬀerent from those of main-sequence or giant stars and so for appropriate conditions one expects Cepheid-type pulsations. Indeed, where the faint continuation of the Cepheid instability strip intersects with the locus of WDs in the Hertzsprung-Russell diagram (Fig. 2.9) one ﬁnds the DA variables (DAV), also known as ZZ Ceti stars after their prototype example. They have pulsation periods of a few minutes. The oscillation period depends on the temperature of the layer which exhibits the κinstability (Sect. 2.1.9) and thus excites the pul- sations, and also on the radius of the star. Therefore, the slowing down of the period Pis a direct measure of the temperature decrease and thus of the WD cooling speed. Standard pulsation theory yields ˙P/P≈ −a˙T/T +b˙R/R where the dimensionless constants aandb are of order unity (Winget, Hansen, and van Horn 1983). Because a WD has an almost ﬁxed radius, the second term may be ignored. The time scale of cooling and of the period change are then related by T/˙T=−a P/ ˙P. ZZ Ceti stars have surface temperatures in the neigh- borhood of 13,000 K where the cooling time scale is of order 1 Gyr. For a period of a few minutes one is talking of a period decrease ˙P=O(10−14s s−1), not an easy quantity to measure. After upper limits on ˙Phad been established over the years for a number of cases, Kepler et al. (1991) succeeded at a measurement for the DAV star G117–B15A (Tab. 2.3) using the Whole Earth Telescope which allows for nearly 24 h a day coverage of a given object. A variety of model calculations give ˙P= 2−5×10−15s s−1, some- what smaller than the measured value, i.e. the star appears to cool Table 2.3. Properties of the DAV star G117–B15A. Surface temperature Teﬀ 13,200 K Luminosity log(L/L ⊙)−2.3 Bolometric brightness Mbol 10.49 mag Mass M (0.49±0.03)M⊙ Pulsation period P (215.197,387 ±0.000,001) s Period change ˙P (12±4)×10−15s s−1 P/˙P (0.57±0.17) Gyr 54 Chapter 2 faster than predicted by these models (Kepler et al. 1991). It should be noted, however, that part of the observed period change can be at- tributed to a Doppler shift if G117–B15A is a physical binary with its proper-motion companion G117–B15B. Isern, Hernanz, and Garc´ ıa-Berro (1992) speculated that an addi- tional cooling agent may be operating and notably that this star is cooled by axion emission. For a ﬁducial WD model ( M= 0.5M⊙, internal temperature T= 1.8×107K) the cooling time scale is found to be T/˙T= 1.0 Gyr while P/˙P≈1.4 Gyr. Therefore, Isern et al. found that Lx= 0.5−2.6L was needed to account for the observed ˙P, although other models needed little or no axion cooling. For their ﬁducial model the required axion cooling yielded a coupling constant α26= 0.2−0.8. This interpretation is speculative, of course, but appar- ently not in conﬂict with any other constraints on the electron coupling of pseudoscalars (Sect. 3.6.1). The most conservative interpretation of the ˙Pmeasurement of the star G117–B15A is that it agrees with theoretical calculations within the observational and model uncertainties. Therefore, it provides in- dependent evidence that the WD cooling speed is known to within a factor of O(1) so that any novel cooling agent is constrained to be less eﬃcient than O(L ). 2.3 Neutron Stars 2.3.1 Late-Time Cooling Neutron stars are born when the degenerate iron core of an evolved massive star becomes unstable and collapses to nuclear densities, an implosion which is partly reﬂected at the core bounce and leads to a type II supernova explosion (Sect. 2.1.8). These events and the ﬁrst few seconds of neutron star cooling are discussed more fully in Chapter 11. For a few seconds the star emits most of its binding energy in the form of MeV neutrinos which were observed from SN 1987A. Afterward, the temperature at the neutrino sphere (the analogue of the photosphere in ordinary stars) has dropped so much that the detectors are no longer sensitive to the neutrino ﬂux although the star continues to cool by surface neutrino emission. After 10 −100 yr the internal temperature has dropped to about 109K≈100 keV where the neutron star becomes entirely transpar- ent to neutrinos and continues to cool by neutrino volume emission. After about 105yr it reaches an inner temperature of about 2 ×108K, Anomalous Stellar Energy Losses Bounded by Observations 55 a point at which photon emission from the surface becomes the dom- inant form of cooling. In Fig. 2.14 the central temperature, surface temperature, neutrino luminosity, and photon luminosity are shown as functions of age according to a numerical calculation of Nomoto and Tsuruta (1987). Fig. 2.14. Cooling of a neutron star with baryon mass 1 :4M⊙(gravitational mass 1 :3M⊙) according to Nomoto and Tsuruta (1987). The solid line is for an equation of state of intermediate stiﬀness (model FP), the dotted line for a stiﬀ model (PS). All “nonstandard” eﬀects were ignored such as nucleon superﬂuidity, a meson condensate, magnetic ﬁelds, and so forth. Temperatures and luminosities are local, ignoring the gravitational redshift. 56 Chapter 2 2.3.2 X-Ray Observations Because the surface temperature of neutron stars is in the keV regime they can be observed only by x-ray satellites such as the Einstein Obser- vatory which was launched in 1979, and more recently by EXOSAT and Table 2.4. Selected x-ray observations of SN remnants and pulsars.a Age [yr] Dist. Remnant (in 1995) [kpc] PulsarbT∞ eﬀ[106K]c d 3C58 (814) (2.6) — 2.2±0.2 E 1 Crab 941 1.7–2 0531+21 <1.6 R 2 (r, o, x, γ) RCW 103 1500 ±500 2 — 2.15±0.15 E 1 <1.2 R 3 MSH15–52 1850 ±250 4.2 (r, x) detected E 1 Vela X ∼12,000 0.5 0833–45 0 .95±0.15 E 1 (r, o, x, γ) 1 .6±0.2 R 4 Monogem ∼110,000 (0.5) 0656+14 0 .90±0.04 R 5 Ring (?) (r, x) (0.30±0.05) — ∼340,000 0.15–0.4 Geminga 0 .52±0.10 R 6 (x,γ) — ∼540,000 0.5–1.5 1055–52 0 .65±0.15 X 7 (r, x, γ) 0 .75±0.06 R 8 aAdapted from Tsuruta (1986) and updated. bPulsed radiation: r = radio, o = optical, x = x-rays, = -rays. cE = Einstein, R = ROSAT, X = EXOSAT observation. dReferences: 1. See the review by Tsuruta (1986) for references to the original literature. 2. Becker and Aschenbach (1995). 3. Becker et al. (1992). 4.¨Ogelman, Finley, and Zimmermann (1993). 5. Thompson et al. (1991). Finley, ¨Ogelman, and Kizilo˘ glu (1992). 6. Halpern and Ruderman (1993). 7. Brinkmann and ¨Ogelman (1987). 8.¨Ogelman and Finley (1993). Anderson et al. (1993). Anomalous Stellar Energy Losses Bounded by Observations 57 ROSAT. X-rays have been observed from a number of compact sources in supernova remnants and from several isolated pulsars. However, with the limited spectral resolution of these instruments it is diﬃcult to extract the actual thermal surface emission because there can be signiﬁcant nonthermal x-ray emission from the magnetosphere. There are three candidates in the 103yr age category (3C58, the Crab pulsar, RCW 103) from which x-rays have been observed by the Einstein Observatory (Tab. 2.4). However, ROSAT did not detect a compact source in RCW 103; only an upper ﬂux limit has been re- ported (Tab. 2.4). The Crab pulsar is x-ray bright mostly from non- thermal emission so that ROSAT could only establish an upper limit on its surface temperature. The remaining Einstein source 3C58 probably should also be interpreted as an upper limit on thermal surface emis- sion. These upper limits are in agreement with the standard cooling curves of Nomoto and Tsuruta (1987) shown in Fig. 2.15. The Einstein data point for the Vela pulsar at an age of about 104yr is somewhat low, a fact which has given rise to the speculation that “exotic” cooling eﬀects may be operating such as neutrino emission by virtue of a meson condensate. However, a blackbody spectral ﬁt to the ROSAT observations yields a much higher temperature (Tab. 2.4, Fig. 2.15. Surface temperature (observed at inﬁnity) of a neutron star with baryon mass 1 :4M⊙(gravitational mass 1 :3M⊙) according to Nomoto and Tsuruta (1987). The solid line is for an equation of state of intermediate stiﬀness (model FP), the dotted line for a stiﬀ (PS), and the dashed line for a soft model (BPS). All “nonstandard” eﬀects were ignored. The measure- ments of Tab. 2.4 are also shown. 58 Chapter 2 open circle in Fig. 2.15). Such a high temperature is not compatible with the total x-ray luminosity unless the radius of the neutron star is very small (3 −4 km). Either way, it may be premature to reach deﬁnite conclusions regarding neutron star cooling on the basis of Vela. At still larger ages, ROSAT measurements of the surface temper- ature of the pulsars PSR 0656+14, PSR 1055–52, and Geminga have been reported which lie close to the theoretical standard cooling curves. However, the inferred temperature values depend sensitively on the as- sumed circumstellar atmospheric models which can modify the spec- trum and thus lead to an erroneous temperature assignment. Still, the old isolated pulsars give one a ﬁrst realistic observational handle at the issue of neutron star cooling. 2.3.3 Nonstandard Cooling and Heating Eﬀects The so-called “standard” neutron star cooling scenario should be called a “reference” or “minimal” scenario because there are many eﬀects that will alter the cooling history; no doubt at least some of them will be in operation in some or all neutron stars. For reviews see Tsuruta (1986, 1992), for recent numerical cooling curves including various nonstan- dard eﬀects see Umeda, Tsuruta, and Nomoto (1994). The occurrence of nucleon superﬂuidity slows the neutrino emission by the URCA process (Sect. 4.8). However, the cooling curves including superﬂuidity (Nomoto and Tsuruta 1987) do not seem to diﬀer signif- icantly from the reference curves at ages above a few hundred years unless extreme assumptions are made. When nucleon superﬂuidity is important, ννbremsstrahlung emis- sion by electrons in the crust dominates. However, electron band- structure eﬀects may suppress this process, and the crust mass may be smaller than previously thought (Pethick and Thorsson 1994). There- fore, the cooling may be slowed even further. Slowed cooling may also occur by a number of heating eﬀects (accretion, polar cap heating, vor- tex creep, and others), although such eﬀects become important only for relatively old neutron stars ( t∼>104yr). Of course, heating eﬀects related to accretion will not be important in isolated pulsars which thus are preferred laboratories to study neutron star cooling. The cooling is accelerated if the equation of state provides enough protons to allow for the direct URCA process (Sect. 4.8). In this case the surface temperature drops catastrophically at an age of about 100 yr (Page and Applegate 1992; Lattimer et al. 1994) until superﬂuidity sets in which essentially stops neutrino cooling from the core. The temper- Anomalous Stellar Energy Losses Bounded by Observations 59 ature then stays almost constant for a long time until photon cooling from the surface begins to dominate. In this scenario the cooling curve depends sensitively on the “on switch” set by the occurrence of the di- rect URCA process anywhere in the star, and by the “oﬀ switch” from superﬂuidity. As the occurrence of these eﬀects depends on ﬁne points of the equation of state as well as on the density and thus the stellar mass there may not be a universal cooling curve for all neutron stars. Another eﬀect which would accelerate cooling is the occurrence of a meson condensate (Sect. 4.9.1) because of the increased eﬃciency of neutrino emission. Again, this eﬀect depends sensitively on the equa- tion of state and thus on the density and the stellar mass. The most recent numerical study of neutron-star cooling with a pion condensate was performed by Umeda, Nomoto, and Tsuruta (1994). 2.3.4 Cooling by Particle Emission The emission of novel particles would also accelerate the cooling of neutron stars. Iwamoto (1984) considered axion emission e+ (Z, A)→ (Z, A) +e+ain the crust and found unacceptably fast cooling unless α′∼<10−25where α′is the axionic ﬁne-structure constant (Chapter 3). However, this result is quite uncertain, notably in view of the above Pethick and Thorsson (1994) band-structure suppression of the brems- strahlung rate. Moreover, in view of the white-dwarf and globular- cluster bounds of α′∼<10−26it appears that crust cooling by axions is not important in neutron stars. In the interior of neutron stars, axions can be emitted by the neu- tron bremsstrahlung process nn→nna(Sect. 4.2). With a numerical implementation of Iwamoto’s (1984) bremsstrahlung rate Tsuruta and Nomoto (1987) found a limit gan∼<10−10for the axion-neutron Yukawa coupling, based on a comparison with the 103yr old sources. In their calculation the eﬀect of superﬂuidity apparently was not included which would diminish the bound. Conversely, including protons in a regime where neither protons nor neutrons are superﬂuid would increase the emission rate (Sect. 4.2.6). Of course, the more recent ROSAT results suggest that thermal surface emission has not been observed for any of the 103yr old sources anyway. Most recently, Iwamoto et al. (1995) considered the plasma decay process γ→ννin the crust under the assumption of a large neutrino magnetic dipole moment. They found that for µof order 10−10µBa signiﬁcant eﬀect would obtain, but that a value as large as 5 ×10−7µB would be consistent with current data. In view of the globular-cluster 60 Chapter 2 limit of µB∼<3×10−12µB(Sect. 6.5.6), I interpret these results to mean that a neutrino magnetic dipole moment leaves neutron-star cooling unaﬀected—one less nonstandard eﬀect to worry about. The rough agreement between the reference cooling curves and the data points in Fig. 2.15, notably for the old pulsars, suggests that “non- standard” eﬀects cannot be much more eﬃcient than standard neutrino cooling unless all of the old isolated pulsar surface temperatures have been incorrectly assigned—not a likely scenario. Therefore, it is clear that these and future observations of cooling neutron stars will be piv- otal as laboratories to study novel phenomena such as the occurrence of nonstandard phases of nuclear matter. However, at the present time it is not clear if the emission of weakly interacting particles such as axions could still have an interesting impact on neutron star cooling. At the present time it appears easier to make the reverse statement that such cooling eﬀects likely are not important in view of the restrictive limits on the interaction strength of axions or nonstandard neutrinos set by other astrophysical objects. Therefore, at present it appears that for the more narrowly deﬁned purposes of particle physics the role of old neutron stars as laboratories is less useful than had been thought in the early works discussed above. It also appears that novel weakly interacting particles usually would have a more dramatic impact on the ﬁrst few seconds of Kelvin-Helmholtz cooling of a protoneutron star (Chapter 11) than they do on the cooling of old pulsars. 2.4 Globular-Cluster Stars 2.4.1 Observables in the Color-Magnitude Diagram Globular clusters are gravitationally bound associations of typically 106stars (Fig. 2.1); the clusters themselves (about 150 in our galaxy) form an approximately spherical galactic halo. The metallicity is in the range Z= 10−4−10−2; it is usually the same for all stars in one cluster. The low metallicity is one indicator for their great age—like isolated halo stars they belong to the Population II which formed early from a relatively “uncontaminated” hydrogen-helium mixture left over from the big bang of the universe. Stars found in the galactic disk belong to the later Population I which continue to form even today from the interstellar gas. Clusters of disk stars are usually less populous and less tightly bound—the so-called “open clusters.” Anomalous Stellar Energy Losses Bounded by Observations 61 Fig. 2.16. Observables in the color-magnitude diagram of a typical globular cluster (here M15 after Sandage 1986). Depending on the metallicity, the red-giant “bump” can also appear below the HB brightness. In the color-magnitude diagram of a globular cluster (Fig. 2.3) the stars arrange themselves in a characteristic pattern which is schemati- cally shown in Fig. 2.16. Each branch corresponds to a certain evo- lutionary phase as discussed in Sect. 2.1. Typically, all stars in a given cluster are coeval and have a ﬁxed chemical composition; they diﬀer only in their mass. After formation, they all began their evo- lution on the zero-age main-sequence as in Fig. 2.2. The more mas- sive stars evolve faster, become red giants and explode as supernovae (M∼>8M⊙), or become white dwarfs ( M∼<8M⊙). Recently, many pulsars, the remnants of type II supernovae and thus of massive stars, have indeed been found in globular clusters. The upper main sequence (MS) is depleted of stars down to a limiting value below which they have not had time to complete hydrogen burning. Therefore, the stel- lar mass corresponding to the MS turnoﬀ (TO) is a precise measure of the cluster age. A typical value is 0 .7−0.9M⊙, depending on the precise age and metallicity. The color-magnitude diagram of a globular cluster represents an “isochrone” of a stellar population. It shows the locus of coeval stars with diﬀerent initial masses. It is to be distinguished from the evolu- tionary track of a single star (Fig. 2.6) which shows a star of a ﬁxed mass at diﬀerent ages. However, the evolution beyond the MS is very 62 Chapter 2 fast. Therefore, the stars along the red-giant branch (RGB), horizon- tal branch (HB), and asymptotic giant branch (AGB) in a globular cluster have almost identical initial masses whence for these phases a single-star track is practically identical with an isochrone. On the other hand, the TO region requires the construction of detailed theoretical isochrones to compare theory and observations and thus to determine the ages of globular clusters. In order to associate a certain stellar mass with the TO in a cluster one needs to know the absolute brightness of the stars at the TO, i.e. one needs to know the precise distance. All else being equal, the inferred age varies with the TO luminosity as ∂log(age) /∂logLTO=−0.85 or ∂log(age) /∂V TO= 0.34 (Iben and Renzini 1984). Therefore, a 0 .1 mag error in VTOleads to an 8% uncertainty in the inferred cluster age. Put another way, because L∝(distance)2a 10% uncertainty in cluster distances leads to an 18% age uncertainty. This is the main problem with the age determination of globular clusters. A particularly useful method to measure the distance is to use RR Lyrae stars as standard candles. As discussed in Sect. 2.1, their luminosity is determined almost entirely by their core mass (apart from a dependence on chemical composition), which in turn is ﬁxed by he- lium ignition on the RGB which, again, depends only on the chemical composition and not on the red-giant envelope mass. Therefore, the brightness of the HB is nearly independent of stellar mass. Conse- quently, the brightness diﬀerence ∆ VTO HBbetween the HB and the TO is a distance-independent measure of the TO mass and thus of the clus- ter age. Moreover, because the color of RR Lyrae stars coincides with that of the TO region it is not necessary to convert from the measured brightness with a certain ﬁlter (e.g. visual brightness V) to a bolomet- ric brightness, i.e. there is no need for a bolometric correction (BC). Also, RR Lyrae stars are bright and easily identiﬁed because of their pulsations. Therefore, ∆ VTO HBis one of the most important observables in the color-magnitude diagram of globular clusters (Iben and Renzini 1984; Sandage 1986). As an example the recent ∆ VTO HBdeterminations of Buonanno, Corsi, and Fusi Pecci (1989) in 19 globular cluster are shown in Fig. 2.17 as a function of metallicity; the logarithmic metallicity measure [Fe/H] is deﬁned in Eq. (2.15). The best linear ﬁt is ∆VTO HB= (3.54±0.13)−(0.008±0.078) [Fe /H], (2.13) so that the HB brightness varies with metallicity almost exactly as the TO brightness. The measured points are in agreement with a Gaussian Anomalous Stellar Energy Losses Bounded by Observations 63 Fig. 2.17. Brightness diﬀerence between main-sequence turnoﬀ (TO) and horizontal branch (HB) in 19 galactic globular clusters according to Buon- anno, Corsi, and Fusi Pecci (1989). The shaded band indicates the 1 sta- tistical error of the best ﬁt Eq. (2.13). distribution about the mean. This result illustrates the level of precision that presently can be achieved at determining ∆ VTO HB. In principle, the color of the TO also speciﬁes the location on the MS and thus the TO mass and cluster age. In practice, the color of a star is theoretically less well determined than its luminosity because it depends on the treatment of the photosphere and on the surface area and thus on the radius which is partly ﬁxed by the treatment of convection. Still, the TO color is a useful measure for the relative ages between clusters of identical metallicities. Notably, the distance- independent color diﬀerence ∆( B−V) between the TO and the base of the RGB (Fig. 2.16) has been used to establish an age diﬀerence of about (3 ±1) Gyr between the clusters NGC 288 and 362 (VandenBerg, Bolte, and Stetson 1990; Sarajedini and Demarque 1990). In order to constrain the operation of a novel energy-loss mechanism the brightness of the RGB tip is particularly useful because particle emission (neutrinos, axions) from a red-giant core delays helium igni- tion. This delay allows the stars to develop a more massive core and thus to turn brighter before they become HB stars. Again, the inferred luminosity of the brightest star on the RGB depends on the distance whence the most useful observable is the distance-independent bright- ness diﬀerence between the RGB tip and the HB. However, because the color of RR Lyrae stars and red giants is very diﬀerent one must 64 Chapter 2 convert to an absolute bolometric brightness diﬀerence ∆ Mtip HBrather than using the visual brightness diﬀerence ∆ Vtip HB. Fig. 2.18. Evolutionary speed on the RGB for a model with M= 0:8M⊙, metallicity Z= 10−4, and initial helium abundance Y= 0:240 (Raﬀelt and Weiss 1992). The dashed line is the tangent near the bright end. Fig. 2.19. Luminosity function of the RGB of the globular cluster NGC 2808 which has [Fe =H] =−1:37 (Fusi Pecci et al. 1990). Anomalous Stellar Energy Losses Bounded by Observations 65 A very important measure of the speed of evolution of a single star along the RGB, HB, and AGB is the relative number of stars found on these branches in a given cluster. Because for these advanced evolution- ary phases a single-star track is essentially an isochrone, these number ratios Rgive us directly the relative amounts of time spent on these branches (“ R-method”). For example, a novel energy-loss mechanism that operates mostly in the nondegenerate core of an HB star would shorten the helium-burning lifetime. This possibility is constrained by the relative number of stars found on the HB and RGB where the RGB in this context refers to that part which is brighter than the HB. When a star ascends the RGB, its hydrogen-burning shell encoun- ters at some point a discontinuity in the composition proﬁle left behind by a previous deep penetration of the envelope convection into the region of varying hydrogen content caused by nuclear burning. The as- cent is brieﬂy interrupted and the star stays at a ﬁxed luminosity for a brief period of time. Afterward, the hydrogen shell works itself through a constant composition proﬁle which was prepared by the convective envelope. Therefore, at a brightness near the HB one expects to ﬁnd a “bump” in the number of stars on the RGB, i.e. in the distribution ∂N/∂M bolwhich essentially corresponds to dt/dM bolfor a single-star evolutionary track (Fig. 2.18). The bump was recently identiﬁed in a number of clusters; for a particularly beautiful example see Fig. 2.19. It has been suggested to use it as a standard candle to calibrate the RR Lyrae brightness-metallicity relation (Fusi Pecci et al. 1990). An important observable is the absolute brightness of RR Lyrae stars which are not members of globular clusters. No reason is known why these ﬁeld stars for a given metallicity should be any diﬀerent from those found in a cluster and so brightness determinations of nearby RR Lyrae stars provide important information about their luminosity calibration. 2.4.2 Theoretical Relations In order to test the standard stellar-evolution picture against the ob- servables introduced in Sect. 2.4.1 they need to be related to stellar properties such as mass and chemical composition. In the past, exten- sive grids of stellar evolutionary sequences have been calculated and have been used to derive analytic approximations for the connection between various stellar parameters; as a canonical standard I use the evolutionary HB and RG sequences of Sweigart and Gross (1976, 1978). 66 Chapter 2 a) Composition Parameters The chemical composition is characterized by the helium content and the metallicity. The convective envelope on the RGB at some point reaches down into the region of a variable composition proﬁle caused by nuclear burning, causing a certain amount of processed material (helium) to be dredged up. Therefore, the helium mass fraction Yenv of the envelope is slightly larger than the value Yat formation. The amount of dredge-up is approximately given by (Sweigart, Renzini, and Tornamb´ e 1987) ∆Ys≡Yenv−Y≈0.0136 + 0 .0055Z13, (2.14) where the metallicity parameter Z13is deﬁned in Eq. (2.16) below. However, because the exact amount of convective dredge-up is some- what model dependent, and because measurements of12C/13C ratios in metal-poor ﬁeld red giants seem to indicate that the dredge-up of processed material is more eﬃcient than predicted by standard calcula- tions (Sneden, Pilachewski, and VandenBerg 1986) it is best to use Yenv rather than Yas an independent parameter to characterize red giants and HB stars.10 Gravitational settling of helium throughout the MS evolution has the opposite eﬀect of reducing the envelope abundance relative to the initial homogeneous value. In recent evolutionary sequences which were calculated to estimate the eﬀect of helium diﬀusion on the inferred globular-cluster ages a decrease between ∆ Ys=−0.009 and −0.015 was found (Proﬃtt and Michaud 1991; Chaboyer et al. 1992). Therefore, the eﬀect of MS gravitational settling and that of RG convective dredge- up appear to cancel each other more or less so that Yenvappears to be much closer to the initial value Ythan had been thought previously. The metallicity is usually characterized by the mass fraction Zof elements heavier than helium. Because iron is most important for the opacities one often uses the abundance of iron relative to hydrogen as a metallicity measure. It is characterized by the quantity [Fe /H] which is the logarithmic abundance of iron over hydrogen relative to the solar value. If the solar metallicity is taken to be Z⊙= 0.02 so that 10In their calculation of RG sequences, Sweigart and Gross (1978) used the symbol Yto denote the MS helium abundance; the envelope abundance near the helium ﬂash can be inferred from their tabulation of ∆ Ysvalues for each sequence. In their 1976 study of HB sequences, they used the symbol Yto denote the envelope abundance which is here consistently called Yenv. Anomalous Stellar Energy Losses Bounded by Observations 67 logZ⊙=−1.7 one ﬁnds logZ= [Fe /H] + log Z⊙= [Fe /H]−1.7. (2.15) Because globular-cluster metallicities cover a range of Z= 10−4to 10−2 a typical average value is log Z=−3 or [Fe /H] =−1.3. The primordial helium abundance is thought to be about 23%. Therefore, it is useful to employ the “reduced” composition parameters Y23≡Yenv−0.23, Z13≡logZ+ 3 = [Fe /H] + 1 .3, (2.16) which are zero for a typical globular cluster. b) Core Mass at Helium Ignition One of the most important quantities to be aﬀected by a novel energy- loss mechanism is the core mass at the helium ﬂash because helium ignition is an extremely sensitive function of the temperature. Based on the Sweigart and Gross (1978) models Raﬀelt (1990b) has derived the analytic approximation Mc= 0.500−0.22Y23−0.011Z13−0.021M7+δMc, (2.17) where all stellar masses are understood in units of the solar mass M⊙. Here,Mcis the core mass at helium ignition and M7≡ M − 0.7 is the “reduced total mass.” I have increased Mcby 0.004 relative to the original calculation to account for the corrected plasma neutrino emission rate (Haft, Raﬀelt, and Weiss 1994). Within about 0 .003M⊙ Raﬀelt and Weiss (1992) found the same expression (when corrected for the plasma rates) except for a slightly shallower metallicity dependence. Recently, Sweigart (1994) has reviewed the core-mass calculations at the helium ﬂash. All workers seem to agree within a few 10−3M⊙ except for Mazzitelli (1989) who found core masses larger by some 0.020M⊙. Sweigart (1994) claims that this disagreement cannot be attributed to the algorithm adopted to accomplish the “shell shifting” of the numerical grid which represents the star on a computer. Because of substantial mass loss on the RGB the meaning of the total mass Min this equation is not entirely obvious. If there were enough time to relax to equilibrium one would think that it is the in- stantaneous mass at the helium ﬂash. Indeed, Raﬀelt and Weiss (1992) found in an evolutionary sequence with mass loss that the end mass de- termined Mc. However, in this calculation the mass loss was stopped 68 Chapter 2 sometime before helium ignition and so this ﬁnding is not surprising. Castellani and Castellani (1993) studied RG sequences with mass loss in more detail and found the surprising result that the core mass at helium ignition was determined by the initial stellar mass while the en- velope structure followed the instantaneous envelope mass. Apparently, in these calculations the core retained memory of a previous conﬁgu- ration. The total amount of mass loss on the RGB may be of order 0.2M⊙, causing a maximum discrepancy between the two scenarios of about 0 .005M⊙in the expected Mc. In Eq. (2.17) a deviation δMcwas explicitly included which rep- resents nonstandard changes of Mc. The core-mass increase δMcis the main quantitity to be constrained by observations. It should be thought of as a function of the parameters which govern the physics which causes the delay of helium ignition such as coupling constants of particles which contribute to the energy loss, or a more benign param- eter such as the angular frequency of core rotation. c) Brightness at Helium Ignition Next, the brightness at helium ignition is needed, identical with the brightness at the tip of the RGB. In the Sweigart and Gross (1978) calculations, both the luminosity and the core mass at the helium ﬂash are functions of Z,Yenv, andM. For the present purposes, however, the core mass at helium ignition must be viewed as another free parameter which is controlled, for example, by the amount of energy loss by novel particle emission. In order to determine how the luminosity at helium ignition varies with Mcif all other parameters are held ﬁxed Raﬀelt (1990b) considered ∂logL/∂Mcfor a grid of Sweigart and Gross tracks near the ﬂash. An interpolation yields logLtip= 3.328 + 0 .68Y23+ 0.129Z13+ 0.007M7+ 4.7Mc, (2.18) with Ltipthe luminosity at the RGB tip in units of L⊙. Ignoring the dependence on the total mass here and in Eq. (2.17) one ﬁnds for the absolute bolometric brightness of the RGB tip11 Mtip=−3.58 + 0 .89Y23−0.19Z13−11.8δMc, (2.19) slightly diﬀerent from the results of Raﬀelt (1990b) who used the mass of RR Lyrae stars for the total mass in Eq. (2.18). 11Recall that the absolute bolometric brightness is given by M= 4:74−2:5 logL forLin units of L⊙,Min magnitudes. Anomalous Stellar Energy Losses Bounded by Observations 69 d) Brightness of RR Lyrae Stars Theoretically, details of the evolution of stars on the HB are diﬃcult to account for. Notably, they may move in and out of the RR Lyrae insta- bility strip so that the stars found there cannot be trivially associated with a speciﬁc age after the beginning of helium burning. Therefore, it is easiest to use the absolute bolometric brightness of zero-age HB stars with a mass chosen such that they fall into the RR Lyrae strip (Buzzoni et al. 1983; Raﬀelt 1990b) MRR= 0.66−3.5Y23+ 0.16Z13−∆RR−7.3δMc, (2.20) where ∆ RRis an unknown amount of deviation between real RR Lyrae stars and the zero-age HB models of Sweigart and Gross (1976) that served to derive this relation. It is expected that ∆ RRis a positive number of order 0 .1 mag, i.e. on average RR Lyrae stars are thought to be somewhat brighter than zero-age HB star models. This conclusion is supported by Sandage’s (1990a) investigation of the vertical height of the HB by means of the pulsational properties of RR Lyrae stars. Sandage found an intrinsic width between 0 .2 mag for the most metal-poor and about 0 .4 mag for the most metal-rich clusters, i.e. an average deviation between 0.1 and 0.2 mag between zero-age and average HB stars. Lee, Demarque, and Zinn (1990) have constructed synthetic HBs for a range of metallicities and helium content on the basis of new evolutionary sequences. For a MS helium content of 0.20, which in their calculation amounts to Yenv= 0.22, they found (see also Lee 1990) MRR= 0.70 + 0 .22Z13. (2.21) Comparing this with Eq. (2.20) at Y23=−0.01 and δMc= 0 one ﬁnds ∆ RR≈0. This is not in contradiction with the brightening of RR Lyrae stars relative to zero age, it only means that there is a slight oﬀset relative to the analytic representation Eq. (2.20) derived from the Sweigart and Gross (1976) calculations. ∆ RRshall always refer to the brightness diﬀerence of real RR Lyrae stars relative to Eq. (2.20), it does not refer to an oﬀset relative to real zero-age HB stars. e) Brightness Diﬀerence between HB and RGB Tip The main observable to constrain a deviation from the standard core mass at the helium ﬂash is the brightness diﬀerence between the HB 70 Chapter 2 (at the RR Lyrae strip) and the RGB tip for which one ﬁnds with Eqs. (2.19) and (2.20) ∆Mtip HB≡MRR−Mtip = 4.24−4.4Y23+ 0.35Z13−∆RR+ 4.5δMc (2.22) which is deﬁned such that it is a positive number. f) Ratio of HB/RGB Stars The relative duration of the HB vs. RGB phase is given by the number ratio of the stars on these branches where the RGB is deﬁned as that part which is brighter than the HB. The HB lifetime is found to be (Buzzoni et al. 1983; Raﬀelt 1990b) log(tHB/yr) = 8 .01 + 0 .37Y23+ 0.06Z13−2.9δMc. (2.23) The RGB lifetime cannot be expressed easily in terms of a simple linear formula because of the RGB bump discussed earlier. A simple approx- imation to the lifetime ratio R=tHB/tRGB is (Buzzoni et al. 1983; Raﬀelt 1990b) logR= 0.105 + 2 .29Y23+ 0.029Z13+ 0.33 ∆ RR−0.70δMc. (2.24) This quantity is particularly sensitive to the helium content of the stars and almost independent of the core mass at helium ignition. 2.4.3 Observational Results a) Brightness at the RGB Tip The brightness at the tip of the RGB can be estimated by the brightest red giant in a given globular cluster. A homogeneous set of observations of the brightest RGs in 33 globular clusters are those of Cohen, Frogel, and Persson (1978), Da Costa, Frogel, and Cohen (1981), Cohen and Frogel (1982) and Frogel, Persson, and Cohen (1981, 1983). According to Frogel, Cohen, and Persson (1983) only in 26 of the 33 clusters the brightest giant was likely observed; for those cases the bolometric brightness diﬀerence between the brightest RGs and the HB are shown as a function of metallicity in Fig. 2.20. Also shown is a linear ﬁt ∆Mtip HB= 4.06 + 0 .38Z13. The observational errors are thought to be less than about 0 .05 mag. Anomalous Stellar Energy Losses Bounded by Observations 71 Fig. 2.20. Bolometric brightness diﬀerence between the horizontal branch and the brightest red giant in 26 globular clusters from the observations referenced in the text. However, because there are relatively few stars near the tip on the RGB (on average about 10 mag−1in the observed clusters), the bright- est RG is on average about 0 .1 mag below the actual RGB tip. Raﬀelt (1990b) estimated the richness of the RGB near the tip for each cluster on the basis of the ﬁrst few brightest stars provided by the observa- tions and thus estimated the expected brightness diﬀerence between the brightest RG and the tip. This yields a linear regression ∆Mtip HB= (4.19±0.03) + (0 .41±0.06)Z13, (2.25) about 0 .13 mag brighter than the ﬁt shown in Fig. 2.20. The slope of Eq. (2.25) agrees very well with the theoretical expectation Eq. (2.22), provided that ∆ RRdoes not introduce a large modiﬁcation. More recent observations are those of Da Costa and Armandroﬀ (1990) who also found excellent agreement between the theoretical slope of the brightness of the RGB tip luminosity as a function of metallicity. Because the coeﬃcient of the metallicity dependence agrees well with the predicted value one may restrict a further comparison be- tween theory and observation to ∆ Mtip HBat a given metallicity for which it is best to use the average value [Fe /H] =−1.48 or Z13=−0.18 of the globular clusters used in Fig. 2.20. Inserting these values into Eqs. (2.22) and (2.25) and adding the errors quadratically one ﬁnds 4.4Y23+ ∆ RR−4.5δMc= 0.06±0.03. (2.26) If ∆ RR= 0.2 mag this result implies that the envelope helium abun- 72 Chapter 2 dance, which presumably is very close to the primordial value, has to be 0.20 to achieve perfect agreement, or else the core mass at helium ignition has to be 0 .030M⊙larger than given by the standard the- ory. Alternatively, the zero-age HB models may represent the bright- ness of RR Lyrae stars better than anticipated so that ∆ RR≈0 in which case there is perfect agreement with the standard values Yenv≈Yprimordial ≈0.23 and δMc= 0. b) Ratio of HB/RGB Stars Buzzoni et al. (1983) have determined the number of stars on the HB (including RR Lyrae stars),12NHB, and on the RGB brighter than the HB,NRGB, in 15 globular clusters. The resulting values for the number ratio R=NHB/NRGBis shown in Fig. 2.21 as a function of metallicity. The individual errors are found by assuming standard deviations of N1=2for the number counts and adding the errors of NHBandNRGB quadratically (Raﬀelt 1990b). The data are ﬁt by a linear regression logR= (0.162±0.016) + (0 .065±0.032)Z13, (2.27) also shown in Fig. 2.21. Fig. 2.21. Number ratio of HB/RGB stars for 15 globular clusters according to Buzzoni et al. (1983). Comparing this result with the theoretical prediction of Eq. (2.24) one ﬁnds that the metallicity dependence essentially agrees within the 12Buzzoni et al. (1983) call this quantity NHB+RR . Anomalous Stellar Energy Losses Bounded by Observations 73 stated uncertainty; it is very shallow anyway. Therefore, one compares at the average metallicity of the 15 clusters, [Fe /H] =−1.54 or Z13= −0.24, adds the errors quadratically, and ﬁnds Y23+ 0.14 ∆ RR−0.31δMc= 0.021±0.008. (2.28) With ∆ RR≈0.2 mag and δMc≈0 this conﬁrms a primordial helium abundance of around 23%. c) RR Lyrae Absolute Brightness The RR Lyrae absolute brightness as well as the precise variation of MRRwith metallicity is the single most discussed issue about globular- cluster color-magnitude diagrams because the distance and thus the age determination depends critically on this quantity. For example, when using ∆ VTO HBmeasurements like the ones shown in Fig. 2.17, the inferred relative ages of globular clusters depend crucially on the slope aofMRR=a[Fe/H] +b. One possibility to determine ais the use of the pulsation frequencies of these variable stars. The result is about 0.35, almost twice as large as that obtained by theoretical zero-age HB models or by synthetic HBs, an issue known as the Sandage period shift eﬀect (Sandage 1990 and references therein; Iben and Renzini 1984; Renzini and Fusi Pecci 1988). While this issue is crucial for a relative age determination of globular clusters, it is of relatively minor importance for the present discussion where the zero point bfor an intermediate metallicity is the most crucial quantity. Probably the most direct determination of MRRis to use nearby ﬁeld RR Lyrae stars for which, in principle, a distance determination by parallax measurements is possible. Barnes and Hawley (1986) applied the method of statistical parallaxes to a sample of 142 stars and found a mean absolute visual brightness of (0 .68±0.14) mag. Assuming a bolometric correction for RR Lyrae stars of −0.06 mag this leads to ⟨MRR⟩= (0.62±0.14) mag . (2.29) The average metallicity of this sample is probably [Fe /H]≈ −1.4. The Baade-Wesselink method applied to a total of 25 ﬁeld RR Lyrae stars, and using a bolometric correction of −0.06, leads to MRR= 0.72 + 0 .19Z13 (2.30) (Sandage and Cacciari 1990, and references therein). 74 Chapter 2 Most recently, the brightness of RR Lyrae stars in the Large Mag- ellanic Cloud was measured; it has a distance which is thought to be well determined by other methods. Walker (1992) found MRR= 0.48 + 0 .15Z13, (2.31) if the same bolometric correction −0.06 is assumed. In summary, a reasonably conservative estimate of the absolute RR Lyrae bolometric brightness is MRR= (0.60±0.15) + 0 .17Z13. (2.32) A comparison with Eq. (2.20) yields 3.5Y23+ ∆ RR+ 7.3δMc= 0.06±0.15. (2.33) This is in good agreement with the standard values Yenv= 0.23,δMc= 0, and ∆ RR≈0.1 mag. 2.4.4 Interpretation of the Observational Results In order to interpret the observational results it is ﬁrst assumed that there is no anomalous core-mass increase. However, considering the uncertainties entering the calculation of Mcsuch as the precise value of the relevant total stellar mass, uncertainties in the electron conductive opacities, etc., it appears that a plausible range of uncertainty is δMc= ±0.010M⊙even in the absence of any novel phenomena. Adopting this uncertainty and adding it quadratically to the previous uncertainties, the three observables from Eq. (2.26), (2.28), and (2.33) yield Yenv= (0.244±0.012)−0.23 ∆ RR from ∆ Mtip HB, Yenv= (0.251±0.009)−0.14 ∆ RR from R, Yenv= (0.247±0.048)−0.29 ∆ RR from MRR. (2.34) The primordial helium abundance likely is in the range 22 −24%; the envelope abundance in globular clusters is probably slightly larger, de- pending on details of gravitational settling on the MS and convective dredge-up on the RGB. Therefore, with ∆ RRbetween 0 and 0 .2 mag these results are perfectly consistent. In order to constrain YenvandδMcsimultaneously it is assumed that ∆ RR= 0.1±0.1. Adding this error quadratically in Eq. (2.26), Anomalous Stellar Energy Losses Bounded by Observations 75 (2.28), and (2.33) one ﬁnds Yenv−1.0δMc= 0.239±0.024 from ∆ Mtip HB, Yenv−0.3δMc= 0.237±0.016 from R, Yenv+ 2.1δMc= 0.247±0.043 from MRR. (2.35) Bands of allowed values for YenvandδMcare shown in Fig. 2.22. From Fig. 2.22 one reads oﬀ that approximately Yenv= 0.24±0.02 and |δMc|<0.025M⊙. (2.36) Therefore, the core mass at the helium ﬂash is tightly constrained to lie within 5% of its standard value. This result allows one to constrain the operation of a novel energy- loss mechanism in the core of a red giant, i.e. at densities of around 106g cm−3and at a temperature of about 108K. In some cases, how- ever, an energy-loss mechanism is suppressed by degeneracy eﬀects in a RG core, while it may be fairly eﬃcient in the helium-burning core of an HB star ( ρ≈104g cm−3,T≈108K) which is essentially nonde- generate. In this case δMc≈0 while the energy loss from the HB core Fig. 2.22. Allowed values for the envelope helium abundance of evolved globular-cluster stars and of an anomalous core-mass excess at helium igni- tion. The limits were derived from the observed brightness diﬀerence ∆ Mtip HB between the HB and the RGB tip, from the “ R-method” (number counts on the HB vs. RGB), and from the brightness determination of nearby ﬁeld RR Lyrae stars ( MRR) by statistical parallaxes and the Baade-Wesselink method as well as the brightness of RR Lyrae stars in the LMC. 76 Chapter 2 may be substantial and thus would shorten the helium-burning lifetime as the nuclear fuel would be consumed faster. The HB/RGB number ratio Rindicates that the acceleration of the HB evolution must not be too extreme. From Eq. (2.24) together with the observational result Eq. (2.27) one ﬁnds δlogR= 0.01±0.06, (2.37) if one adopts Yenv= 0.24±0.02, ∆ RR= 0.1±0.1, and δMc= ±0.010M⊙. Therefore, Rcannot be smaller by much more than about 10% of its standard value. Put another way, the helium-burning life- time of low-mass stars is determined within about 10% from the ratio of HB/RGB stars in globular clusters. With lesser statistical signiﬁcance this result is corroborated by number counts of clump giants in open clusters which consist of more recently formed stars in the galactic disk (Population I). Clump giants in open clusters correspond to HB stars in globular clusters; instead of forming a horizontal branch they are concentrated in a clump near the base of the RGB. Cannon (1970) compared the number of clump giants in the open cluster M67 with the number of stars per luminos- ity interval near the MS turnoﬀ and found tHe≈1.5×108yr, with a large statistical uncertainty, however, because there were only 5 clump giants. (Open clusters tend to be much less populous than globular ones.) Tinsley and Gunn (1976) derived tHe= (1.27±0.29)×108yr from low-mass giants of the old galactic disk population. These results are in full agreement with Eq. (2.23). 2.4.5 An Alternate Analysis The above discussion of the globular-cluster limits is a somewhat up- dated version of my own previous work (Raﬀelt 1990b). Very recently, Catelan, de Freitas Pacheco, and Horvath (1995) have provided an in- dependent new and extended analysis. While they closely follow the line of reasoning of Raﬀelt (1990b) they have changed numerous de- tails. Of the 26 globular clusters which enter the ∆ Mtip HBargument, and of the 15 clusters which enter the R-method, they have discarded several with an extreme HB morphology. For the absolute brightness of RR Lyrae stars they have employed Walker’s (1992) values which depend on a precise knowledge of the LMC distance. For the bright- ness diﬀerence between zero-age HB stars and RR Lyrae stars they use ∆RR= 0.31 + 0 .10 [Fe /H] = 0 .18 + 0 .10Z13. Anomalous Stellar Energy Losses Bounded by Observations 77 These authors have introduced a fourth observable, the so-called mass to light ratio Aof RR Lyrae stars. This observable has also been used by Castellani and Degl’Innocenti (1993) to constrain a possible core-mass excess. It amounts to a determination of the core mass of an RR Lyrae star on the basis of its luminosity and pulsation period. Going through similar steps as in the previous sections, Catelan, de Freitas Pacheco, and Horvath (1995) then found the results Yenv−0.9δMc= 0.207±0.039 from ∆ Mtip HB, Yenv−0.3δMc= 0.218±0.018 from R, Yenv+ 2.1δMc= 0.250±0.049 from MRR, Yenv+ 1.5δMc= 0.238±0.008 from A. (2.38) Bands of allowed values for YenvandδMcare shown in Fig. 2.23 which is analogous to Fig. 2.22. From this analysis one infers a best-ﬁt value for the envelope he- lium abundance which is somewhat low. The primordial abundance probably is not lower than 22%. The only possibility to reduce the envelope abundance from this level is by gravitational settling which is counteracted by convective dredge-up, and perhaps by other eﬀects that might eject helium into the envelope from the core. Therefore, Fig. 2.23. Allowed values for the envelope helium abundance of evolved globular-cluster stars and of an anomalous core-mass excess at helium igni- tion according to the analysis of Catelan, de Freitas Pacheco, and Horvath (1995). For comparison see Fig. 2.22. 78 Chapter 2 Yenv>0.22 is probably a conservative estimate. This takes us back to the conclusion that δMc≈0 within at least 0 .025M⊙. Again, the core mass at helium ignition is found to agree with the standard result to better than 5%. 2.4.6 Systematic Uncertainties The globular-cluster argument yields some of the most restrictive lim- its on novel modes of energy loss. Therefore, it is important to under- stand some of the systematic uncertainties that enter the nominal limit δMc∼<0.025M⊙. Core rotation has often been quoted as an eﬀect to change Mc. However, as it would actually delay helium ignition it cannot be invoked to compensate for anomalous cooling eﬀects. In addition, if fast core rotation were an important eﬀect one would expect it to vary from star to star, causing a random broadening of the distribution of ∆ Mtip HB, an eﬀect not indicated by the observations. The scatter of ∆ Mtip HBis completely within observational errors and within the scatter caused by the eﬀect that the brightest red giant is not exactly at the tip of the RGB in a given cluster. Further discussions of the rotational impact onMcare found in Catelan, de Freitas Pacheco, and Horvath (1995). The uncertainty of the conductive opacities are relatively large as stressed by Catelan, de Freitas Pacheco, and Horvath (1995). However, conceivable modiﬁcations of Mcdo not seem to exceed the 0 .010M⊙ level even with extreme assumptions. One of the main theoretical weaknesses of the helium-ignition ar- gument is that the helium ﬂash has never been properly calculated. Because helium ignites oﬀ-center one expects that convection plays a major role in the process of heating the entire core and its expansion. One may worry that in the process of the ﬂash, parts of the core are ejected into the stellar envelope, reducing its post-ﬂash size. However, if signiﬁcant amounts of helium were ejected, the inferred Yenvwould be changed dramatically. Even the ejection of 0 .010M⊙of helium would increase Yenvby 0.03 if one assumes an envelope mass of 0 .3M⊙and thus would brighten RR Lyrae stars by 0 .12 mag. Within the stated limit of δMc<0.025M⊙, mass ejection from the core is a dramatic eﬀect that would be hard to hide in the data. One signiﬁcant systematic uncertainty arises from the relative abun- dance of metals among each other which is usually ﬁxed by the solar mixture (Ross and Aller 1976). The assumption of a Ross-Aller mix- ture for metal-poor systems like globular-cluster stars has been called Anomalous Stellar Energy Losses Bounded by Observations 79 into question in recent years (e.g. Wheeler, Sneden, and Truran 1989). It is thought that in these systems the “ αelements” (mostly oxygen) are enhanced relative to iron. RG sequences calculated by Raﬀelt and Weiss (1992) with somewhat extreme αenhancements indicated that the red-giant core mass and luminosity at the helium ﬂash are only moderately changed. Most of the change that did occur was due to the reduction of Fe at constant Zbecause of the αenhancement. Put another way, if Z13= [Fe /H] + 1 .3 is used as the deﬁning equation for the above “reduced metallicity” the eﬀect of αenhancements appear to be rather minimal. Catelan, de Freitas Pacheco, and Horvath (1995) have taken the point of view that for enhanced αelements one should rescale the metallicity according to a recipe given by Chieﬃ, Straniero, and Salaris (1991). Put another way, the metallicity parameter of Eq. (2.16) that was used in the previous sections should be redeﬁned as Z13≡[Fe/H] + 1 .3 + log(0 .579f+ 0.421). (2.39) Here, fis the enhancement factor of the abundance of αelements relative to the solar value. For f= 3 one ﬁnds that Z13must be oﬀset by +0 .33. Catelan, de Freitas Pacheco, and Horvath (1995) went through their analysis with this modiﬁcation, causing the allowed bands of Fig. 2.23 to be slightly shifted relative to each other. However, the overall change is small, well within the stated upper limit on δMc. In summary, no eﬀect has been discussed in the literature that would cause the predicted core mass at helium ignition to deviate from its standard value beyond the adopted limit of ±0.025M⊙. Therefore, any new energy-loss mechanism that would cause a signiﬁcantly larger core- mass excess would have to be compensated by a hitherto unidentiﬁed other novel eﬀect. 2.5 Particle Bounds from Globular-Cluster Stars 2.5.1 Helium-Burning Lifetime A particularly simple argument to constrain the properties of novel particles arises from the observed duration of helium burning of low- mass stars, i.e. form the lifetime of stars on the horizontal branch (HB). In Sect. 2.4.4 it was argued that the number ratio Rof stars on the HB vs. RGB in globular clusters agreed with standard predictions to within 10%. Therefore, the helium-burning lifetime tHeagrees with standard predictions to within this limit. A less signiﬁcant conﬁrmation arises 80 Chapter 2 from the number of clump giants in open clusters and from the old galactic disk population. In Sect. 1.3.1 it was shown that the main impact of a nonstandard energy-loss rate on a star is an acceleration of the nuclear fuel con- sumption while the overall stellar structure remains nearly unchanged. The temperature dependence of the helium-burning (triple- αreaction) energy generation rate ϵ3∝T40is much steeper than the case of hy- drogen burning discussed in Sect. 1.3.1 and so the adjustment of the stellar structure is even more negligible. With L3the standard helium- burning luminosity of the core of an HB star and Lxthe nonstandard energy-loss rate integrated over the core, tHewill be reduced by an ap- proximate factor L3/(Lx+L3). Demanding a reduction by less than 10% translates into a requirement Lx∼<0.1L3. Because this con- straint is relatively tight one may compute both LxandL3from an unperturbed model. If the same novel cooling mechanism has delayed the helium ﬂash and has thus led to an increased core mass only helps to accelerate the HB evolution. Therefore, it is conservative to ignore a possible core-mass increase. The standard value for L3is around 20 L⊙; see Fig. 2.4 for the properties of a typical HB star. Because the core mass is about 0 .5M⊙ the core-averaged energy generation rate is ⟨ϵ3⟩ ≈80 erg g−1s−1. Then a nonstandard energy-loss rate is constrained by ⟨ϵx⟩∼<10 erg g−1s−1. (2.40) Previously, this limit had been stated as 100 erg g−1s−1, overly con- servative because it was not based on the observed HB/RGB number ratios in globular clusters. However, in practice Eq. (2.40) does not improve the constraints on a novel energy-loss rate by a factor of 10 because the appropriate average density and temperature are somewhat below the canonical values of ρ= 104g cm−3andT= 108K. For a simple estimate the energy-loss rate may be calculated for average conditions of the core. Typically, ϵxwill depend on some small power of the density ρ, and a somewhat larger power of the temperature T. For the HB star model of Fig. 2.4 the core-averaged values ⟨ρn⟩and ⟨Tn⟩are shown in Fig. 2.24 as a function of n. The dependence on n is relatively mild so that the ﬁnal result is not sensitive to ﬁne points of the averaging procedure. In order to test the analytic criterion Eq. (2.40) in a concrete exam- ple consider axion losses by the Primakoﬀ process. The energy-loss rate will be derived in Sect. 5.2.1. It is found to be proportional to T7/ρ and to a coupling constant g10=ga /10−10GeV−1. For a typical HB Anomalous Stellar Energy Losses Bounded by Observations 81 Fig. 2.24. Average values of nandTnfor the HB star model of Fig. 2.4 where 4==104g cm−3andT8=T=108K. core I ﬁnd ⟨T7 8/ρ4⟩ ≈0.3 which leads to ϵ≈g2 1030 erg g−1s−1. Thus, forg10= 1 one concludes that the helium-burning lifetime should be reduced by a factor 80 /(80 + 30) = 0 .7. This result may be compared with numerical evolution sequences for 1.3M⊙stars with an initial helium abundance of 25% and a metal- licity of Z= 0.02 (Raﬀelt and Dearborn 1987). The helium-burning lifetime was found to be 1 .2×108yr which was modiﬁed to 0 .7×108yr forg10= 1. This means that the axion losses on the HB led to a tHe reduction by a factor 0.6, in good agreement with the analytic estimate. This comparison corroborates that the analytic criterion represents the claimed impact of a novel energy loss on the helium-burning lifetime with a reasonable precision. Equation (2.40) may now be applied to many diﬀerent cases. An overview over the most salient results is given in Tab. 2.5 where the original references are given, and the sections of this book are indi- cated where a more detailed discussion can be found. Apart from the listed examples, the argument has also been applied to low-mass super- symmetric particles (Fukugita and Sakai 1982; Bouquet and Vayonakis 1982; Ellis and Olive 1983). However, it is now thought that super- symmetric particles, if they exist, are probably not light enough to be produced in stars whence the interest in this case has waned. 82 Chapter 2 Table 2.5. Constraints on low-mass particles from the observed duration of helium burning in globular-cluster stars. Details are discussed in the indicated sections of this book. Particle property Dominant process Constrainta[Ref.] Sect. Yukawa coupling of Bremsstrahlung gS<1.3×10−14[1] scalar (vector) boson e+4He→4He + e+ϕ g V<0.9×10−143.5.5 ϕto electrons Yukawa coupling Compton gS<4.3×10−11[1] to baryons γ+4He→4He + ϕ g V<3.0×10−113.6.2 Photoproduction of Photoproduction σ∼<3×10−50cm2[2] X◦boson γ+4He→4He + X◦ Yukawa coupling of Compton g <4.5×10−13[3] pseudoscalar boson γ+e→e+a 3.2.6 ato electrons Yukawa coupling of Compton g <3.2×10−13[4] “paraphoton” γ′γ+e→e+γ′3.2.6 γγcoupling of Primakoﬀ ga <0.6×10−10[5] pseudoscalar boson γ+4He→4He + a GeV−15.2.5 Neutrino dipole Plasmon decay µeﬀ<1×10−11µB [6] moment γ→νν 6.5.6 aAs derived in this book; may diﬀer from the quoted references. References: 1. Grifols and Mass´ o (1986); Grifols, Mass´ o, and Peris (1989). 2. van der Velde (1989); Raﬀelt (1988b). 3. Dicus et al. (1978, 1980); Georgi, Glashow, and Nussinov (1981); Barroso and Branco (1982); Fukugita, Watamura, and Yoshimura (1982a,b); Pantziris and Kang (1986); Raﬀelt (1986a). 4. Hoﬀmann (1987). 5. Raﬀelt (1986a); Raﬀelt and Dearborn (1987). 6. Sutherland et al. (1976); Fukugita and Yazaki (1987); Raﬀelt and Dearborn (1988); Raﬀelt, Dearborn, and Silk (1989). Anomalous Stellar Energy Losses Bounded by Observations 83 2.5.2 Helium Ignition Another powerful constraint arises from the agreement between the predicted and observationally inferred core mass at the helium ﬂash (Sect. 2.4.4). An energy-loss mechanism which is eﬃcient in a degen- erate medium ( ρ≈106g cm−3,T≈108K) can delay helium ignition. To establish the core-mass increase as a function of nonstandard par- ticle parameters one needs to evolve red giants numerically to the he- lium ﬂash. However, for a simple analytic estimate one observes that the core mass of a red giant grows by hydrogen shell burning. Because it is a degenerate conﬁguration its radius shrinks and so the core releases a large amount of gravitational binding energy which amounts to an average energy source ⟨ϵgrav⟩. If a novel energy-loss rate ⟨ϵx⟩is of the same order then helium ignition will be delayed. In order to estimate ⟨ϵgrav⟩one may treat the red-giant core as a low-mass white dwarf. Its total energy, i.e. gravitational potential energy plus kinetic energy of the degenerate electrons, is found to be (Chandrasekhar 1939) E=−3 7GNM2 R. (2.41) The radius of a low-mass white dwarf (nonrelativistic electrons!) may be expressed as R=R∗(M⊙/M)1=3withR∗= 8800 km so that ⟨ϵgrav⟩=−˙E M=GNM⊙ R∗(M M⊙)1=3˙M M⊙. (2.42) From the numerical sequences of Sweigart and Gross (1978) one ﬁnds that near the helium ﬂash M ≈ 0.5M⊙and ˙M ≈ 0.8×10−15M⊙s−1 so that ⟨ϵgrav⟩ ≈100 erg g−1s−1. Therefore, one must require ⟨ϵx⟩ ≪ 100 erg g−1s−1in order to prevent the helium ﬂash from being delayed. In order to sharpen this criterion one may use results from Sweigart and Gross (1978) and Raﬀelt and Weiss (1992) who studied numerically the delay of the helium ﬂash by varying the standard neutrino losses with a numerical factor Fwhere F= 1 represents the standard case. The results are shown in Fig. 2.25. Note that for F<1 the standard neutrino losses are decreased so that helium ignites earlier, causing δMc<0. It is also interesting that for F= 0 helium naturally ignites at the center of the core while for F>1 the ignition point moves further and further toward the edge (Fig. 2.26). This behavior is 84 Chapter 2 Fig. 2.25. Change of the red-giant core mass at helium ignition, Mc, as a function of a factor Fwhich multiplies the standard neutrino energy-loss rate. Triangles: Metallicity Z= 10−3(Sweigart and Gross 1978). Squares: Z= 10−4(Raﬀelt and Weiss 1992). Fig. 2.26. Variation of the red-giant core mass at helium ignition, Mc, as a function of Fas in Fig. 2.25 for Z= 10−4,Y= 0:22, and M= 0:8M⊙. Also shown is the mass coordinate Migwhere helium ignites (Raﬀelt and Weiss 1992). Anomalous Stellar Energy Losses Bounded by Observations 85 understood from the relatively steep density dependence of the plasma neutrino emission rate which is most eﬃcient at the center. In order to prevent the core mass from exceeding its standard value by more than 5% ( δMc<0.025M⊙) one must require F∼<3 or ⟨ϵx⟩∼<2⟨ϵ⟩. According to the Sweigart and Gross (1978) red-giant se- quences the neutrino luminosity of the core at helium ignition is approx- imately 1 L⊙so that ⟨ϵ⟩ ≈4 erg g−1s−1. Therefore, an approximate analytic criterion to constrain a nonstandard energy loss is ϵx∼<10 erg g−1s−1, (2.43) where ϵxis to be evaluated in a helium plasma at the average density of the core of about 2 ×105g cm−3(the central density is about 106g cm−3) and at the almost constant temperature of T= 108K. The standard neutrino plasma emission rate evaluated at 2 ×105g cm−3and 108K is 3 erg g−1s−1, in good agreement with the above average neutrino luminosity of the Sweigart and Gross models. Table 2.6. Increase of the core mass at helium ignition because of the emis- sion of pseudoscalars (Raﬀelt and Weiss 1995). α′[10−26]δMc[M⊙] 0.0 0.000 0.5 0.022 1.0 0.036 2.0 0.056 A simple application of Eq. (2.43) is the case of bremsstrahlung emission of pseudoscalars e+4He→4He + e+a. For a degenerate medium the emission rate is given in Eq. (3.33); for T= 108K it is approximately α′2×1027erg g−1s−1with the “axion ﬁne-structure constant” α′. Then, Eq. (2.43) yields α′∼<0.5×10−26. (2.44) The same case was treated numerically by Raﬀelt and Weiss (1995) who implemented the energy-loss rate Eq. (3.33) with varying values ofα′in several red-giant evolutionary sequences.13They found the 13In a previous numerical treatment by Dearborn, Schramm, and Steigman (1986) the correct emission rate had not yet been available and so they overestimated the energy-loss rate by as much as a factor of 10 at the center of a red-giant core. 86 Chapter 2 Fig. 2.27. Increase of the core mass of a red giant at helium ignition due to the emission of pseudoscalars according to Tab. 2.6 (Raﬀelt and Weiss 1995). core-mass increases given in Tab. 2.6 and shown in Fig. 2.27. The re- quirement that the core not exceed its standard value by more than 5% reproduces the analytic bound. This example nicely corroborates the surprising precision of the simple criterion Eq. (2.43). Another important case where a detailed numerical study is avail- able is the emission of neutrinos by the plasma process γ→ννwhen they have nonstandard magnetic dipole moments µ. The emission rates are derived in Sect. 6.5.5 and the simple criterion Eq. (2.43) is applied in Sect. 6.5.6. It yields a limit µ12∼<2 where µ12=µ/10−12µB with the Bohr magneton µB=e/2me. The numerical variation of the core mass with µis shown in Fig. 2.28 according to Raﬀelt and Weiss (1992). An analytic approximation is δMc= 0.025M⊙[ (µ2 12+ 1)1=2−1−0.17µ3=2 12] . (2.45) The requirement δMc∼<0.025M⊙then translates into µ∼<3×10−12µB, (2.46) a result which, again, is almost identical with the analytic treatment, supporting the power of the simple criterion stated in Eq. (2.43). Anomalous Stellar Energy Losses Bounded by Observations 87 Fig. 2.28. Increase of the core mass of a red giant at helium ignition as a function of an assumed neutrino dipole moment according to Raﬀelt and Weiss (1992) with a total stellar mass 0 :8M⊙and an initial helium abun- dance of Y= 0:22 or 0 :24 (the core-mass increase is found to be the same). The triangles refer to the metallicity Z= 10−3, the squares to 10−4. The open circles are the corresponding results of Castellani and Degl’Innocenti (1992) with the same stellar mass, Y= 0:23, and Z= 2×10−4. The solid line is the analytic ﬁt Eq. (2.45). 2.6 Summary What is the bottom line after studying the impact of an anomalous energy-loss rate on a variety of stellar-evolution phases? While the Sun remains an interesting object from a pedagogical point of view, its main use is that of a distant particle source for terrestrial experimentation, notably to study neutrino properties (Chapters 10 and 12). Supernovae and their collapsed cores (newborn neutron stars) remain very impor- tant; they will be discussed in Chapters 11 −13. At the present time the cooling of old neutron stars appears to be a useful laboratory to study conventional phenomena (Does the direct URCA process occur? Are there meson condensates or other exotic phases? What is the role of magnetic ﬁelds?). The emission of weakly interacting particles other than standard neutrinos appears to play a lesser role in view of other astrophysical limits on their interaction strength. Then, apart from the supernova arguments to be discussed later, the most useful and reliable observables to constrain the operation of a 88 Chapter 2 nonstandard energy-loss mechanism in stars are the white-dwarf lumi- nosity function, the helium-burning lifetime of horizontal-branch stars, and the nondelay of helium ignition in low-mass red giants as observed by the brightness of the tip of the red-giant branch in globular clus- ters. It was possible to condense the latter two arguments into two exceedingly simple criteria, namely that an anomalous energy-loss rate in the cores of HB stars as well as in red-giant cores before helium ignition must not exceed about 10 erg g−1s−1. The emission rate is to be calculated at the pertinent plasma conditions, i.e. at an approx- imate temperature of 108K = 8 .6 keV, an electron concentration of Ye= 0.5, and an average density of about 0 .6×104g cm−3(HB stars) or 2×105g cm−3(red giants). The former case corresponds to roughly nondegenerate conditions, the latter case to degenerate ones so that it depends on the density dependence of the emission rates which of these cases will yield a more restrictive limit. A red-giant core is essentially a 0 .5M⊙helium white dwarf. The observed “real” white dwarfs have typical masses of about 0 .6M⊙; they are thought to consist mostly of carbon and oxygen. Both the helium-ignition argument and the white-dwarf luminosity function al- low one to constrain a novel energy-loss mechanism roughly on the level of standard neutrino emission. Therefore, it is no surprise that bounds derived from both arguments tend to be very similar. There may be other objects or phenomena in the universe that mea- sure novel particle-physics hypotheses even more sensitively than the cases discussed here. They still need to make their way into the particle astrophysics literature. Chapter 3 Particles Interacting with Electrons and Baryons The stellar energy-loss argument is applied to weakly interacting par- ticles which couple to electrons and baryons. The emission from a nor- mal stellar plasma can proceed by a variety of reactions, for example the Compton process γe−→e−χwhereχstands for a single particle (axion, paraphoton, etc.) or a neutrino pair νν. Other examples of practical interest are electron bremsstrahlung e−(Z,A)→(Z,A)e−χ ore−e−→e−e−χ, electron free-bound transitions, and pair annihila- tione−e+→γχore−e+→νν. The corresponding energy-loss rates of stellar plasmas are studied for temperatures and densities which are of interest for stellar-evolution calculations. For particles coupled to baryons some of the same processes apply in a normal plasma if one sub- stitutes a proton or a helium nucleus for the electron. Reactions speciﬁc to a nuclear medium are deferred to Chapter 4. In addition to draining stars of energy, scalar or vector bosons would mediate long-range forces. Leptonic long-range forces would be screened by the cosmic neutrino background. Thermal graviton emission from stars is mentioned. 3.1 Introduction If weakly interacting particles couple directly to electrons, they can be produced thermally in stellar plasmas without nuclear processes. For neutrinos, a direct electron coupling was ﬁrst contemplated after the universal V−Atheory for their interactions had been proposed in 1958. It was realized immediately that such a coupling would allow for thermal pair production by bremsstrahlung e−(Z,A)→(Z,A)e−νν 89 90 Chapter 3 (Pontecorvo 1959; Gandel’man and Pinaev 1959) or by photoproduc- tionγe−→e−νν(Ritus 1961; Chiu and Stabler 1961). The set of processes important for normal stars was completed by Adams, Rud- erman, and Woo (1963) who discovered the plasma process γ→νν. Neutrino emission by these reactions is now a standard aspect of stellar- evolution theory. If other weakly interacting particles were to exist which couple di- rectly to electrons they could essentially play the same role and thus add to the energy loss of stars. The main speculation to be followed up in this chapter is the possible existence of weakly interacting bosons that would couple to electrons. Among standard particles the only low- mass bosons are photons and probably gravitons. The former dominate the radiative energy transfer in stars, the latter are so weakly interact- ing that their thermal emission is negligible (Sect. 3.7). Why worry about others? Such a motivation arises from several sources. Low-mass bosons could mediate long-range forces between electrically neutral bodies for which gravity is the only standard interaction. It is an interesting end in itself to set the best possible bounds on possible other forces which might arise from the exchange of novel scalar or vector bosons. Their existence seemed indicated for some time in the context of the “ﬁfth-force” episode alluded to in Sect. 3.6.3 below. It is also possible that baryon or lepton number play the role of physical charges similar to the electric charge, and that a new gauge interaction is associated with them. The baryonic or leptonic photons arising from this hypoth- esis are intriguing candidates for weakly interacting low-mass bosons (Sect. 3.6.4). It will turn out, however, that typically massless bosons which mediate long-range forces are best constrained by experiments which test the equivalence principle of general relativity. Put another way, to a high degree of accuracy gravity is found to be the only long- range interaction between neutral bodies. Long-range leptonic forces can be screened by the cosmic neutrino background. In this case the stellar energy-loss argument remains of importance to limit their possible strength (Sect. 3.6.4). The remaining category of interesting new bosons are those which couple to the spin of fermions and thus do not mediate a long-range force between unpolarized bodies. In the simplest case their CP-con- serving coupling would be of a pseudoscalar nature. Such particles arise naturally as Nambu-Goldstone bosons in scenarios where a global chiral U(1) symmetry is spontaneously broken at some large energy scale. The most widely discussed example is the Peccei-Quinn symmetry that was Particles Interacting with Electrons and Baryons 91 proposed as an explanation of CP conservation in strong interactions. It leads to the prediction of axions which will be discussed in some detail in Chapter 14. The most restrictive limits on their coupling strength arise from the stellar energy-loss argument, and it cannot be excluded that in fact they play an important role in the evolution of some stars (Sect. 2.2.5). No wonder that axions have played a primary role in studies concerning the impact of new weakly interacting particles on stellar evolution. The main focus of this chapter is an application of the stellar energy- loss argument to weakly interacting bosons which couple to electrons by a variety of interaction structures. The relevant processes are entirely analogous to those which emit neutrino pairs except for the plasma pro- cess which requires a two-body ﬁnal state. Therefore, I will presently study photo and bremsstrahlung production of weakly interacting par- ticles, including standard neutrinos. One may consider the same processes with protons substituted for electrons. For neutrino emission this variation is of no interest because the rate is much smaller. It is signiﬁcant for low-mass bosons which couple only to baryons. The energy-loss argument will be systematically applied, yielding restrictive limits on the possible Yukawa and gauge couplings of novel bosons to electrons and baryons. For very low-mass scalar or vector bosons these limits are discussed in the context of those arising from the absence of novel long-range interactions. 3.2 Compton Process 3.2.1 Vector Bosons The simplest process for the emission of weakly interacting particles from the hot and dense interior of a star is the Compton process where a photon from the heat bath interacts with an electron and is thus con- verted into a neutrino pair, an axion, or some other boson (Fig. 3.1). These processes are analogous to the usual Compton scattering of pho- tons. Therefore, I begin with this well-known case which is based on the standard electron-photon interaction Lint=ieψeγµψeAµwith the elec- tron charge e, the electron Dirac ﬁeld ψe, and the photon ﬁeld A. With the ﬁne structure constant α=e2/4π≈1/137, the electron mass me, andσ0≡πα2/m2 ethe total Compton cross section is (e.g. Itzykson and 92 Chapter 3 Fig. 3.1. Compton processes for photon scattering as well as for axion and neutrino pair production (“photoneutrino process”). In each case there is another amplitude with the vertices interchanged. Zuber 1980) σ=σ0[16 (ˆs−1)2+ˆs+ 1 ˆs2+2 (ˆs2−6 ˆs−3) (ˆs−1)3ln(ˆs)] . (3.1) Here, ˆs≡s/m2 ewith√sthe CM (center of mass) energy.14This cross section is shown in Fig. 3.2 as a function of the CM photon energy ω. Forω≪methe CM frame is the electron rest frame, ˆ s→1, and one recovers the Thomson cross section σ=8 3σ0. Forω≫meone has ˆs≫1 and soσ= (σ0/ˆs) [2 ln(ˆs) + 1] = (πα2/ω2) [ln(2ω) +1 4] because in this limit s= (2ω)2. The standard Compton cross section can also be used to study the photoproduction of novel low-mass vector particles which couple to electrons in the same way as photons except that the ﬁne-structure constant must be replaced by the new coupling α′. Then σ0≡παα′/m2 e withα′≡g2/4π, (3.2) a deﬁnition that pertains to all bosons which couple to electrons with a dimensionless Yukawa or gauge coupling g. If novel vector bosons such as “paraphotons” exist (Holdom 1986) they likely couple to electrons by virtue of an induced magnetic moment rather than by a tree-level gauge coupling (Hoﬀmann 1987), Lint= (g/4me)ψeσµνψeFµν, (3.3) wheregis a dimensionless eﬀective coupling constant and Fthe para- 14The square of the CM energy is s= (P+K)2with P= (E;p) and K= (!;k) the four-vectors of the initial-state electron and photon, respectively. The CM frame is deﬁned by p=−kso that s= (E+!)2= [(m2 e+!2)1=2+!]2with !the initial- state photon energy in the CM frame. In the frame where the target electron is at rest ( p= 0) one ﬁnds s= 2!me+m2 ewhere now !is the photon energy in the electron frame. Thus, with !the photon energy in the respective frames, ! me={ 1 2(ˆs−1)=√ ˆsin the CM frame, 1 2(ˆs−1) in the electron rest frame. Particles Interacting with Electrons and Baryons 93 photon ﬁeld tensor. In the nonrelativistic limit this yields the same to- tal cross section as the interaction with pseudoscalars Eq. (3.6) below. This is seen if one compares the matrix elements between two electron statesiandffor the two cases. For paraphotons (momentum k, polar- ization vector ϵ) it isg⟨f|eik·r(k×ϵ)·σ|i⟩while for pseudoscalars it is g⟨f|eik·rk·σ|i⟩. After an angular average the two expressions are the same. Of course, one must account for the two paraphoton polarization states by an extra factor of 2. Fig. 3.2. Total cross section for the Compton process with a ﬁnal-state vector, scalar, or pseudoscalar boson according to Eqs. (3.1), (3.5), and (3.9), respectively, with 0deﬁned in Eq. (3.2) and !the CM initial photon energy. 3.2.2 Scalars Grifols and Mass´ o (1986) studied the stellar emission of scalars ϕwhich couple according to Lint=gψeψeϕ. (3.4) Integrating their diﬀerential cross section I ﬁnd σ=σ0[−16 (ˆs−1)2+1−3ˆs 2ˆs2+(ˆs+ 3)2 (ˆs−1)3ln(ˆs)] (3.5) shown in Fig. 3.2 (dashed line). For small and large photon energies this is half the cross section for massless vector bosons which have two polarization degrees of freedom. For intermediate energies the two results are not related by a simple factor. 94 Chapter 3 3.2.3 Pseudoscalars Next, turn to the photoproduction of low-mass pseudoscalars ϕwhich couple to electrons by the interaction Lint= (1/2f)ψeγµγ5ψe∂µϕorLint=−igψeγ5ψeϕ, (3.6) wherefis an energy scale and ga dimensionless coupling constant. Both interaction laws yield the same Compton cross section with the identiﬁcation g=me/f(see the discussion in Sect. 14.2.3). If a pseudoscalar (frequency ω, wavevector k) is emitted in a transi- tion between the nonrelativistic electron states |i⟩and|f⟩, the matrix element is Mpseudoscalar =g 2me1√ 2ω⟨ feik·rσ·ki⟩ . (3.7) This is to be compared with the corresponding matrix element for pho- ton transitions, Mphoton =−e 2me1√ 2ω⟨ feik·r[2ϵ·p+σ·(k×ϵ)]i⟩ , (3.8) with the photon polarization vector ϵand the electron momentum oper- atorp. Therefore, transitions involving pseudoscalars closely compare with photonic M1 transitions, a fact that was used to scale nuclear or atomic photon transition rates to those involving axions (Donelly et al. 1978; Dimopoulos, Starkman, and Lynn 1986a,b). The relativistic Compton cross section for massive pseudoscalars was ﬁrst worked out by Mikaelian (1978). The most general discussion of the matrix element was provided by Brodsky et al. (1986) and by Chanda, Nieves, and Pal (1988) who also included an eﬀective photon mass relevant for a stellar plasma. For the present purpose it is enough to consider massless photons and pseudoscalars. With σ0as deﬁned in Eq. (3.2) one ﬁnds σ=σ0(ln(ˆs) ˆs−1−3ˆs−1 2ˆs2) , (3.9) shown in Fig. 3.2 (dotted line). Forω≫meone ﬁndsσ= (παα′/2ω2) [ln(2ω)−3 4], similar to the scalar case. For ω≪me, however, σ=4 3σ0(ω/m e)2, (3.10) so that the cross section is suppressed at low energies. This reduction is related to the M1 nature of the transition. Particles Interacting with Electrons and Baryons 95 3.2.4 Neutrino Pairs The photoneutrino process was ﬁrst studied by Ritus (1961) and by Chiu and Stabler (1961). The eﬀective neutral-current Hamiltonian is Hint=GF√ 2ψeγµ(CV−CAγ5)ψeψνγµ(1−γ5)ψν, (3.11) whereGFis the Fermi constant, and the dimensionless couplings CV andCAare given in Appendix B. Of course, in the early sixties neutral currents were not known—one used Fierz-transformed charged currents which gave CV=CA= 1. For general CV’s andCA’s the cross section was ﬁrst calculated by Dicus (1972) who found15(Fig. 3.3) σ=σ0[ (C2 V+C2 A) ˆσ+−(C2 V−C2 A) ˆσ−] , ˆσ+=49 12+5 (13 ˆs−7) (ˆs−1)2+15−117 ˆs−55 ˆs3 12 ˆs2 +25−28 ˆs−27 ˆs2−2 ˆs3+ 2 ˆs4 (ˆs−1)3ln(ˆs), ˆσ−=−39 +120 ˆs (ˆs−1)2−8 ˆs−1 ˆs2+12 (2 + 2 ˆs+ 5 ˆs2+ ˆs3) (ˆs−1)3ln(ˆs), σ0=αG2 Fm2 e 9 (4π)2. (3.12) In the nonrelativistic (NR) limit this is (CM photon energy ω) σNR=σ06 35(C2 V+ 5C2 A) (ˆs−1)4 =σ096 35(C2 V+ 5C2 A) (ω/m e)4. (3.13) In the extreme relativistic (ER) limit it is σER=σ0(C2 V+C2 A) 2ˆs[ ln(ˆs)−55 24] =σ0(C2 V+C2 A) 16 (ω/m e)2[ ln(2ω/m e)−55 48], (3.14) whereσ−does not contribute. 15With C2 V=C2 A= 1 this result agrees with that of Ritus (1961) while Chiu and Stabler (1961) appear to have an extra factor 2ˆ s=(ˆs+1). In the nonrelativistic limit with ˆ s→1 this deviation makes no diﬀerence while in the extreme relativistic limit their result is a factor of 2 larger than that of Ritus (1961) and Dicus (1972). 96 Chapter 3 Fig. 3.3. Dimensionless total cross sections ˆ +and ˆ−for the photoneutrino process e−→e−according to Eq. (3.12) with !the initial photon energy in the CM frame. 3.2.5 Energy-Loss Rates The Compton-type processes are typically important when the elec- trons are nondegenerate (otherwise bremsstrahlung dominates) and nonrelativistic (otherwise e+e−annihilation dominates). In these lim- its one may use the cross sections without Pauli blocking corrections. Because the recoil of the target electron is neglected, the energy ωof a photon impinging on an electron is identical with the energy carried away by the new boson or neutrino pair. Therefore, the energy-loss rate per unit volume is a simple integral over the initial-state photon phase space, weighted with their Bose-Einstein occupation numbers, Q=ne∫2d3k (2π)3σω eω/T−1, (3.15) whereneis the number density of electrons, Tthe temperature, and the factor 2 is for two photon polarization states. In this expression the photon “plasma mass” ωPhas been neglected. IfωP∼>3T, corresponding to a typical thermal photon energy, the pho- ton dispersion relation would have to be included properly in both the phase-space integration and in the cross section calculation. However, these are insigniﬁcant ﬁne points for the cases to be studied below. In the nonrelativistic limit it is easy to estimate a suppression fac- torFdegby electron degeneracy. If recoil eﬀects can be neglected, the Particles Interacting with Electrons and Baryons 97 initial- and ﬁnal-state electrons have the same momentum, reducing the calculation to an average of the Pauli blocking factor over all electrons Fdeg=1 ne∫2d3p (2π)31 e(E−µ)/T+ 1( 1−1 e(E−µ)/T+ 1) , (3.16) whereµis the electron chemical potential and E2=m2 e+p2. Then, Fdeg=1 neπ2∫∞ mepEdEex (ex+ 1)2, (3.17) wherex≡(E−µ)/T. For degenerate conditions the integrand is strongly peaked near x= 0 so that one may replace pandEwith the valuespFandEFat the Fermi surface ( x= 0), and one may extend the lower limit of integration to −∞. The integral then yields Tso that Fdeg= 3EFT/p2 F, (3.18) wherene=p3 F/3π2was used. Returning to the nonrelativistic, nondegenerate limit note that the cross sections are of the form σ=σ∗(ω/m e)pso that Q=σ∗neTp+4 π2mp e∫∞ 0dxxp+3 ex−1=(p+ 3)!ζp+4 π2σ∗neTp+4 mp e.(3.19) Here,ζn=ζ(n) is the Riemann zeta function which shall be set equal to unity.16In a medium of mass density ρthe electron density is ne= Yeρ/m uwhereYeis the electron number fraction per baryon and mu the atomic mass unit. Therefore, the energy-loss rate per unit mass is ϵ=(p+ 3)! π2Yeσ∗Tp+4 mump e. (3.20) The average energy of the photons which are converted into weakly interacting particles is ⟨ω⟩= (p+ 3)T. (3.21) Forp= 0 one recovers ⟨ω⟩= 3Tfor the average energy of blackbody photons.17 16n→1 rather quickly with increasing n; for example 4=4=90≈1:082. Therefore, the error is small if one takes n= 1 which corresponds to using a Maxwell-Boltzmann rather than a Bose-Einstein distribution. They diﬀer at small !where an eﬀective photon mass in the plasma should be taken into account anyway. Therefore, at the crude level of accuracy where one uses massless photons nothing is gained by using the Bose-Einstein distribution. 17With a Bose-Einstein distribution it is ⟨!⟩=T= 34=3=4=303≈2:70. 98 Chapter 3 Beginning with the case of scalars, the low-energy cross section is constant at σ∗=4 3παα′/m2 e, leading to ϵscalar =8αα′ πYeT4 mum2 e=α′5.7×1029erg g−1s−1YeT4 8, (3.22) whereT8=T/108K. Vector bosons carry an extra factor of 2 for their polarization states. Turning to pseudoscalars, the low-energy cross section was found to beσ∗=4 3(παα′/m2 e) (ω/m e)2, i.e.p= 2, so that ϵpseudo =160αα′ πYeT6 mum4 e=α′3.3×1027erg g−1s−1YeT6 8.(3.23) The average energy is ⟨ω⟩ ≈5Tor⟨ω⟩/me≈0.08T8. Finally, turn to neutrino pair production for which the NR cross section was given in Eq. (3.13). The energy-loss rate is ϵνν= (C2 V+ 5C2 A)96α π4G2 Fm6 e muYe(T me)8 = (C2 V+ 5C2 A) 0.166 erg g−1s−1YeT8 8. (3.24) Becausep= 4 for this process, ⟨ω⟩ ≈7T. Relativistic corrections be- come important at rather low temperatures. For example, at T= 108K the true emission rate is about 25% smaller than given by Eq. (3.24). 3.2.6 Applying the Energy-Loss Argument After the derivation of the energy-loss rates it is now a simple matter to apply the energy-loss argument. In Chapter 2 it was shown that the most restrictive limits obtain from the properties of globular-cluster stars; two simple criteria were derived in Sect. 2.5 which amount to the requirement that a novel energy-loss rate must not exceed 10 erg g−1s−1 for the typical conditions encountered in the core of a horizontal-branch star, and in the core of a red giant just before helium ignition which both haveT≈108K. The Compton process is suppressed by degeneracy eﬀects in a dense plasma as discussed above—see Eq. (3.18). Therefore, at a ﬁxed tem- perature the emissivity per unit mass decreases with increasing density. Because a red-giant core is nearly two orders of magnitude denser than the core of an HB star it is enough to apply the argument to the latter case. Particles Interacting with Electrons and Baryons 99 The energy-loss rates for scalars and pseudoscalars in Eqs. (3.22) and (3.23) are independent of density, and proportional to T4andT6, respectively. The averages over the core of a typical HB star are ⟨T4 8⟩= 0.40 and ⟨T6 8⟩= 0.37 (Fig. 2.24). With Ye= 0.5 appropriate for helium, carbon, and oxygen the 10 erg g−1s−1limit yields α′∼<{0.9×10−28scalar, 1.6×10−26pseudoscalar.(3.25) These bounds apply to bosons with a mass below a few times the tem- perature,m∼<20−30 keV. For larger masses the limits are signiﬁcantly degraded because only the high-energy tail of the blackbody photons can produce the particles. For massive pseudoscalars this eﬀect was explicitly studied in Sect. 1.3.5 in the context of solar limits. For vector bosons which interact by means of a Yukawa coupling, the same limits to α′apply except that they are more restrictive by a factor of 2 because of the two polarization states which increases the emission rate. For vector bosons which couple by means of a “magnetic moment” as the “paraphotons” in Eq. (3.3), the bound on gis the same as for pseudoscalars apart from an extra factor of 2 in the emission rate from the two polarization states. 3.3 Pair Annihilation Electron-positron pair annihilation can produce new bosons by the “crossed” version of the Compton amplitude while the conversion into neutrino pairs does not require the participation of a photon (Fig. 3.4). For pseudoscalars the cross section for e+e−→γais (Mikaelian 1978) σ=2παα′ s−4m2 eln[s 4m2 e( 1 +√ 1−4m2 e/s)] . (3.26) Fore+e−→ννit is (’t Hooft 1971; Dicus 1972) σ=G2 F 12π(C2 V+C2 A) (s−m2 e) + 3 (C2 V−C2 A)m2 e√ 1−4m2 e/s. (3.27) Because pair annihilation requires the presence of positrons it is impor- tant only for relativistic plasmas. In this limit σ∝s−1ln(s) for the production of bosons (pseu- doscalar, scalar, vector) and σ∝sfor neutrino pairs. Therefore, the 100 Chapter 3 Fig. 3.4. Pair annihilation processes for the production of neutrino pairs or new bosons where a second amplitude with the vertices interchanged is not shown. importance of stellar energy-loss rates into new scalars relative to neu- trino pairs decreases with increasing temperature and density. The impact of new bosons on stellar evolution relative to neutrinos is then expected to be most pronounced for low-mass stars where other pro- cesses such as photoproduction dominate. Consequently, the pair pro- cess has not played any signiﬁcant role at constraining the interactions of new bosons. 3.4 Free-Bound and Bound-Free Transitions Photons, new bosons, or neutrino pairs can be emitted in transitions where a free electron is captured by an ion to form a bound state. For the case of axions this eﬀect was dubbed “axio-recombination” (Dimopoulos et al. 1986). In the Sun, it contributes about 4% of the total axion ﬂux which is mostly from bremsstrahlung. The energy-loss rate scales as T3/2, bremsstrahlung as T5/2, and Compton emission as T6. Thus, axio-recombination is of importance in low-mass stars which have low internal temperatures; for main-sequence stars with M∼< 0.2M⊙it would be the dominant axion emission process. However, given the limits on the coupling of pseudoscalars to electrons from other arguments, no observable eﬀects can be expected. Of some practical interest is the inverse process where an axion un- binds an atomic electron, the “axio-electric eﬀect” (Dimopoulos, Stark- man, and Lynn 1986a,b). It serves to constrain the solar ﬂux of axions or other pseudoscalars which could produce keV electrons in a Ge spec- trometer designed to search for double- βdecay (Avignone et al. 1987). Unfortunately, the resulting bound of α′∼<10−21is not very restrictive. Pseudoscalars saturating this limit would be a major energy drain of the Sun and thus not compatible with its observed properties. For neutrinos, free-bound transitions were ﬁrst discussed by Pinaev (1963). Recently, Kohyama et al. (1993) studied the corresponding Particles Interacting with Electrons and Baryons 101 stellar energy-loss rate in detail and found that it dominates the other neutrino emission processes only in such regions of temperature and density where the overall neutrino luminosity is very small. Therefore, free-bound transitions do not seem to be of practical importance as a stellar energy-loss mechanism. 3.5 Bremsstrahlung 3.5.1 Nondegenerate, Nonrelativistic Medium The last emission process to be discussed is bremsstrahlung (Fig. 3.5) where an electron emits a boson or a neutrino pair when scattering oﬀ the Coulomb ﬁeld of a nucleus. Conceptually, bremsstrahlung is closely related to the Compton process (Fig. 3.1) because in both cases the electron interacts with electromagnetic ﬁeld ﬂuctuations of the am- bient medium which have nonvanishing power for all wavenumbers and frequencies. The Compton process corresponds to wavevectors which satisfy the photon dispersion relation so that a real (on-shell) excita- tion is absorbed. However, in a typical stellar plasma there is more power in the “oﬀ-shell” electromagnetic ﬁeld ﬂuctuations associated with the charged particles.18Moreover, degeneracy eﬀects do not sup- press bremsstrahlung at high densities, in contrast with the Compton process. Fig. 3.5. Bremsstrahlung emission of bosons or neutrino pairs by an electron which scatters from the Coulomb ﬁeld of a nucleus of charge Ze. In each case a second amplitude with the vertices interchanged is not shown. For neutrino pairs bremsstrahlung dominates over other processes only in the highly degenerate regime (Appendix C). For pseudoscalars, 18In a typical stellar plasma there are many more charged particles than black- body photons. In the solar center, for example, the electron density is 6 ×1025cm3 while for photons at a temperature of 1 :3 keV it is 2 (3)T3=2= 6×1022cm3. 102 Chapter 3 however, it is important even in environments which are approximately nondegenerate and so I begin with this simple case. The calculation amounts to a straightforward evaluation of the ma- trix element corresponding to the amplitude of Fig. 3.5 and an inte- gration over the Maxwell-Boltzmann distributions of the electrons. In order to account for screening eﬀects the Coulomb propagator is mod- iﬁed according to Eq. (6.72), |q|−4→[q2(q2+k2 S)]−1wherekSis a screening wave number (Sect. 6.4). For the emission of pseudoscalars one then ﬁnds for the energy-loss rate per unit volume to lowest order ink2 S(Krauss, Moody, and Wilczek 1984; Raﬀelt 1986a) Q=128 45√πα2α′ me(T me)5/2 ne ×∑ jnj[ Z2 j√ 2( 1−5 8k2 S meT) +Zj( 1−5 4k2 S meT)] . (3.28) Here, the sum is extended over all nuclear species with charges Zjeand number densities nj; note thatne=∑ jZjnj. The term quadratic in Zj corresponds to electron-nucleus collisions while the linear term is from electron-electron scattering which yields a nonnegligible contribution under nondegenerate conditions.19 Raﬀelt (1986a) incorrectly used Eq. (6.61) as a modiﬁcation of the Coulomb propagator, a procedure which enhances the terms propor- tional tok2 Sby a factor of 2. Either way, screening is never an impor- tant eﬀect. The screening scale is k2 S=k2 D+k2 ibecause both electrons and nuclei (ions) contribute. Then k2 S meT=4πα meT2( ne+∑ jnjZ2 j) , (3.29) which is about 0 .12 at the center of the Sun and 0 .17 in the cores of horizontal-branch (HB) stars. Ignoring screening eﬀects, the energy-loss rate per unit mass is ϵ=α′5.9×1022erg g−1s−1T2.5 8Yeρ∑ jYj( Z2 j+Zj/√ 2) ,(3.30) whereT8=T/108K,ρis in g cm−3, andYj=Xj/Ajis the number fraction of nuclear species jrelative to baryons while Xjis the mass fraction,Ajthe mass number. 19Note that ee→ee vanishes to lowest order because two particles of equal mass moving under the inﬂuence of their Coulomb interaction do not produce a time-varying electric dipole moment because their center of mass and “center of charge” coincide. However, the emission of pseudoscalars corresponds to M1 rather than E1 transitions and so it is not suppressed. Particles Interacting with Electrons and Baryons 103 Grifols, Mass´ o, and Peris (1989) have worked out the bremsstrah- lung rate for a scalar boson; in this case e−e−collisions can be ignored relative to electron-nucleus scattering. They found in the nonrelativis- tic and nondegenerate limit, ignoring screening eﬀects which are small, ϵ=2α2α′ π3/2neT1/2 mum3/2 ea∑ jXjZ2 j Aj =α′2.8×1026erg g−1s−1T0.5 8Yeρ∑ jXjZ2 j Aj, (3.31) whereais an angular integral which numerically is found to be 8.36, muis the atomic mass unit, and ρis in g cm−3. For low-mass vector bosons the same result pertains with an extra factor of 2 for the two polarization states. 3.5.2 High Degeneracy: Pseudoscalars Bremsstrahlung is a particularly important eﬀect under conditions of degeneracy. In this case one neglects e−e−collisions entirely which are suppressed by degeneracy relative to the the electron-nucleus process. If the target nuclei are taken to be inﬁnitely heavy, Raﬀelt (1990) found for the volume emissivity of pseudoscalars (“axions”) Q=4α2α′ π2∑ jZ2 jnj∫∞ medE1f1∫E1 medE2(1−f2)∫dΩ2 4π∫dΩa 4π ×|p1||p2|ω2 q2(q2+κ2)[ 2ω2P1P2−m2 e+ (P2−P1)K (P1K)(P2K)+ 2−P1K P2K−P2K P1K] , (3.32) whereP1is the four-vector of the incoming, P2of the outgoing electron, f1,2are the electron Fermi-Dirac occupation numbers at energies E1,2, temperature T, and chemical potential µ. Further,Kis the four-vector of the outgoing pseudoscalar (energy ω=E1−E2, direction Ω a,afor axion), q=p1−p2−kis the momentum transfer to the nucleus, and kSa screening scale. The Coulomb propagator was modiﬁed according to Eq. (6.72). As long as the plasma is not strongly coupled one may usekS=kiwhile the electrons do not contribute because kTF≪kiin a strongly degenerate plasma. If the electrons are very degenerate, the energy integrals can be done analytically. Moreover, all electron momenta are close to the Fermi surface, |p1| ≈ |p2| ≈pFwithpFthe Fermi momentum deﬁned 104 Chapter 3 byne=p3 F/3π2. This also implies that |q|2≈ |p1−p2|2≈2p2 F(1−c12) wherec12is the cosine of the angle between p1andp2. With these approximations and the velocity at the Fermi surface βF≡pF/EF= pF/(m2 e+p2 F)1/2one ﬁnds Q=π2α2α′ 15T4 m2 e(∑ jnjZ2 j) F, (3.33) where F=∫dΩ2 4π∫dΩa 4π(1−β2 F) [2 (1−c12)−(c1a−c2a)2] (1−c1aβF) (1−c2aβF) (1−c12)(1−c12+κ2) (3.34) withκ2≡k2 S/2p2 F. For a single species of nuclei with charge Zeand atomic weight A the energy-loss rate per unit mass is ϵ=π2α2α′ 15Z2 AT4 mum2 eF =α′1.08×1027erg g−1s−1Z2 AT4 8F, (3.35) where again T8=T/108K. Because Fis of order unity for all condi- tions, the bremsstrahlung rate mostly depends on the temperature and chemical composition, and ϵis not suppressed at high density. Expanding Eq. (3.34) in powers of βFfor nonrelativistic or partially relativistic electrons one ﬁnds F=2 3ln(2 +κ2 κ2) +[2 + 5κ2 15ln(2 +κ2 κ2) −2 3] β2 F+O(β4 F). (3.36) Therefore, in contrast to the nondegenerate calculation this expression would diverge in the absence of screening. Another approximation can be made if one observes that Coulomb scattering is mostly forward, i.e. the main contribution to the integral is fromc12≈1 which implies c1a≈c2a. Withc2a=c1aonly in the denominator one obtains F=2 3ln(2 +κ2 κ2) +[2 + 3κ2 6ln(2 +κ2 κ2) −1] f(βF) (3.37) with f(βF) =3−2β2 F β2 F−3 (1−β2 F) 2β3 Fln(1 +βF 1−βF) . (3.38) The function f(βF) is 0 atβF= 0 and rises monotonically to 1 for Particles Interacting with Electrons and Baryons 105 βF→1. Hence in the relativistic limit F=2 +κ2 2ln(2 +κ2 κ2) −1, (3.39) somewhat diﬀerent from what Iwamoto (1984) found who used the Thomas-Fermi wave number as a screening scale. For a strongly coupled, degenerate plasma typical for white dwarfs the factorFwas calculated numerically by Nakagawa, Kohyama, and Itoh (1987) and Nakagawa et al. (1988) who also gave analytic approxi- mation formulae for the axion emission rate, applicable to nonrelativis- tic and relativistic conditions. For a12C plasma with densities in the range 104−106g cm−3and temperatures of 106−107K it is found that F= 1.0 within a few tens of percent. Therefore, for simple estimates this value is a satisfactory approximation. Altherr, Petitgirard, and del R´ ıo Gaztelurrutia (1994) have calcu- lated the bremsstrahlung process with the methods of ﬁnite tempera- ture and density (FTD) ﬁeld theory. The main point is that one com- putes directly the interaction of the electrons with the electromagnetic ﬁeld ﬂuctuations which are induced by the ambient charged particles. The result for the emission rate is similar to the one derived above. 3.5.3 High Degeneracy: Neutrino Pairs Neutrino pair bremsstrahlung (Fig. 3.5) was the ﬁrst nonnuclear neu- trino emission process ever proposed (Pontecorvo 1959; Gandel’man and Pinaev 1959). A detailed calculation of the energy-loss rate in a degenerate medium, relativistic and nonrelativistic, was performed by Festa and Ruderman (1969) while conditions of partial degeneracy were studied by Cazzola, de Zotti, and Saggion (1971). The Festa and Ruderman calculation was extended by Dicus et al. (1976) to include neutral-current interactions. After a calculation very similar to the one presented above for pseudoscalars the emission rate for neutrino pairs is Q=2πα2 189G2 FT6(∑ jnjZ2 j)[ 1 2(C2 V+C2 A)F++1 2(C2 V−C2 A)F−] . (3.40) With a single species of nuclei (charge Z, atomic mass A) the energy- loss rate per unit mass is ϵ= 0.144 erg g−1s−1(Z2/A)T6 8[...], (3.41) whereT8=T/108K and the square bracket is from Eq. (3.40). The temperature dependence is steeper than for axions by two powers. 106 Chapter 3 The factors F+andF−are of order unity. Dicus et al. (1976) de- rived analytic expressions in terms of a screening scale and the Fermi velocity of the electrons. In fact, because F−is always much smaller thanF+andC2 V−C2 Ais much smaller than C2 V+C2 A, the “minus” term may be neglected entirely. Therefore, the Dicus et al. (1976) re- sult is identical with that of Festa and Ruderman (1969). Either one is correct only within a factor of order unity because Eq. (6.61) was used as a screening prescription with the Thomas-Fermi wave number as a screening scale. However, in a degenerate medium electrons never dominate screening. The most important eﬀect is from the ion corre- lations which, in a weakly coupled plasma (Γ ∼<1), can be included by Eq. (6.72) with the Debye scale kiof the ions as a screening scale. While it is easy to replace kTFwithkiin these results, the modiﬁcation of the Coulomb propagator according to Eq. (6.72) cannot be implemented without redoing the entire calculation. A systematic approach to include ion correlations (i.e. screening eﬀects) was pioneered by Flowers (1973, 1974) who showed clearly how to separate the ion correlation eﬀects in the form of a dynamic structure factor from the matrix element of the electrons and neutrinos. This approach also allows one to include lattice vibrations when the ions form a crystal in a strongly coupled plasma. In a series of papers Itoh and Kohyama (1983), Itoh et al. (1984a,b), and Munakata, Kohyama, and Itoh (1987) followed this approach and calculated the emission rate for all conditions and chemical compositions. As an estimate, good to within a factor of order unity, one may use F+= 1 andF−= 0. Moreover, inspired by the axion results one can guess a simple expression which can be tested against the numerical rates of Itoh and Kohyama (1983). I ﬁnd that F−= 0 and F+≈ln(2 +κ2 κ2) +κ2 2 +κ2(3.42) is a reasonable ﬁt even for strongly coupled conditions (Appendix C). 3.5.4 Neutron-Star Crust The degenerate bremsstrahlung emission of ννpairs is relevant in such diverse environments as the cores of low-mass red giants, white dwarfs, and in neutron-star crusts. Pethick and Thorsson (1994) noted that in the latter case the medium is so dense that band-structure eﬀects of the electrons become important. The band separations can be up Particles Interacting with Electrons and Baryons 107 toO(1 MeV), suppressing electron scattering and bremsstrahlung pro- cesses for temperatures below this scale ( T∼<5×109K). Bremsstrah- lung ofννpairs from the crust was thought to dominate neutron-star cooling for some conditions while Pethick and Thorsson (1994) now ﬁnd that it may never be important. These ﬁndings also diminish Iwamoto’s (1984) axion bound based on the bremsstrahlung emission from neutron-star crusts. 3.5.5 Applying the Energy-Loss Argument One may now easily derive astrophysical limits on the Yukawa couplings of scalars and pseudoscalars, in full analogy to Sect. 3.2.6 where the Compton emission rates were used. I begin with the same case that was considered there, namely the restriction ϵ∼<10 erg g−1s−1in the cores of horizontal-branch stars. For nondegenerate conditions the emission rates are Eq. (3.30) and Eq. (3.31), respectively. They are proportional toρT0.5(pseudoscalar) and ρT2.5(scalar). With ⟨ρ4⟩= 0.64,⟨T0.5 8⟩= 0.82,⟨T2.5 8⟩= 0.48, and a chemical composition of pure helium one ﬁnds α′∼<{1.4×10−29scalar, 1.6×10−25pseudoscalar.(3.43) Comparing these limits with those from the Compton process Eq. (3.25) reveals that for scalars the present bremsstrahlung limit is more restric- tive, for pseudoscalars the Compton one. Because bremsstrahlung is not suppressed in a degenerate plasma, one can also apply the helium-ignition argument of Sect. 2.5.2 which again requires ϵ∼<10 erg g−1s−1at the same T≈108K, however at a density of around 106g cm−3. For these conditions the plasma is degen- erate but weakly coupled (Appendix D) so that Debye screening should be an appropriate procedure. Therefore, for the emission of pseu- doscalars the emission rate Eq. (3.35) with Ffrom Eq. (3.36) should be a reasonable approximation. The screening scale is dominated by the ions,kS=ki= 222 keV, the Fermi momentum is pF= 409 keV so thatκ2= 0.15 whileβF= 0.77, yielding F≈1.8. In Fig. 3.6 the energy-loss rate of a helium plasma at T= 108K is plotted as a function of density, including the Compton process, and the degenerate (D) and nondegenerate (ND) bremsstrahlung rates. This ﬁgure clariﬁes that bremsstrahlung, of course, is suppressed by degeneracy eﬀects relative to the ND rates, but it is not a signiﬁcantly decreasing function of density. In this ﬁgure a simple interpolation (solid line) between the regimes of high and low degeneracy is shown 108 Chapter 3 Fig. 3.6. Energy-loss rate of a helium plasma at T= 108K as a function of density for pseudoscalars with the coupling ′= 10−26. The Compton rate is given in Eq. (3.23), suppressed with the factor Fdegof Eq. (3.18) at high density. The nondegenerate (ND) and degenerate (D) bremsstrahlung rates are from Eqs. (3.28) and (3.35), respectively. The solid line is the interpolation formula of Raﬀelt and Weiss (1995). that was used in a numerical study of axion emission from red giants by Raﬀelt and Weiss (1995); details of how it was constructed can be found there. AtT= 108K the compound rate (solid line) coincidentally is almost independent of density. The requirement ϵ∼<10 erg g−1s−1then yields a constraint α′∼<10−26for any density whence the helium-burning life- time of HB stars as well as the core mass at helium ignition yield an almost identical constraint. In detail one needs to apply the helium- ignition argument at an average core density of 2 ×105g/cm3(the cen- tral density is about 106g/cm3) and at the almost constant tempera- ture ofT= 108K. The degenerate emission rate is then approximately α′2×1027erg g−1s−1so that α′∼<0.5×10−26. (3.44) This is the most restrictive available bound on the Yukawa coupling of pseudoscalars to electrons. Particles Interacting with Electrons and Baryons 109 The same case was treated numerically by Raﬀelt and Weiss (1995) who implemented the compound energy-loss rate of Fig. 3.6 with vary- ing values of α′in several red-giant evolutionary sequences. The re- sults of this work have been discussed in Sect. 2.5.2 where it was used as one justiﬁcation for the simple 10 erg g−1s−1energy-loss constraint that was derived there. This detailed numerical study yielded the same limit Eq. (3.44) on a pseudoscalar Yukawa coupling to electrons. 3.6 Astrophysical Bounds on Yukawa and Gauge Couplings 3.6.1 Pseudoscalars (Axions) The main concern of this chapter was novel bosons which interact with electrons by a dimensionless Yukawa coupling. It will become clear shortly that these results can be easily translated into limits on bary- onic couplings as well. It may be useful to pull together the main results found so far, and discuss them in the context of other sources of information on the same quantities. The best studied case of boson couplings to electrons is that of pseudoscalars because the existence of such particles is motivated by their role as Nambu-Goldstone bosons of a spontaneously broken chiral symmetry of the fundamental interactions. Within this class, axions (Chapter 14) have been most widely discussed; they usually serve as a generic example for low-mass pseudoscalars. For vector bosons which couple by means of a “magnetic moment” (Eq. 3.3) such as “para- photons” the same limits apply apart from an extra factor of 2 in the emission rate from the two polarization states of these particles. The simplest constraint on the Yukawa coupling gaof pseudoscalars (axions) to electrons ( αa=g2 a/4π) arises from the argument that the age of the Sun precludes any novel energy-loss mechanism to be more eﬃcient than the surface photon luminosity (Sect. 1.3.2). The relevant emission processes are the Compton reaction with the energy-loss rate given in Eq. (3.23), and bremsstrahlung with electrons scattering on electrons, protons, and helium nuclei; the emission rate was given in Eq. (3.28). An integration over a typical solar model yields an axion luminosity (Raﬀelt 1986a) La=αa6.0×1021L⊙, (3.45) with about 25% from the Compton process, 25% from eebremsstrah- lung, and 50% from bremsstrahlung by electrons scattering on nuclei. 110 Chapter 3 The requirement La∼<L⊙yields a constraint αa∼<1.7×10−22. A much more restrictive limit arises from the white-dwarf luminos- ity function as axion emission would accelerate white-dwarf cooling. This argument was studied in detail in Sect. 2.2.4 as a generic case for the use of the white-dwarf luminosity function; the resulting con- straint is given in Tab. 3.1. The cooling speed of white dwarfs was also established from a measurement of the period decrease of the DA variable (ZZ Ceti) star G117–B15A which thus yields a similar limit. However, the period decrease of this star may be slightly faster than can be attributed to standard cooling processes; it has been speculated that axions with a coupling strength of about αa≈0.5×10−26could be responsible (Sect. 2.2.5). The limit Eq. (3.44) that was derived in the previous section from the helium-ignition argument in globular clusters is of a similar magnitude, but not restrictive enough to exclude this hypothesis. All of these constraints apply to low-mass bosons. The most restric- tive one is based on the helium ignition argument with T≈108K = 8.6 keV. Therefore, these constraints apply if m∼<10 keV. However, it would be incorrect to think that for larger masses there was no con- straint. There is one, but it is degraded because threshold eﬀects limit the particle production to the high-energy tails of the thermal distri- butions of the plasma constituents. For massive pseudoscalars, this question has been studied in Sect. 1.3.5 in the context of general solar particle constraints. For the more restrictive globular-cluster limits, such a detailed investigation does not exist in the literature. 3.6.2 Energy Loss by Scalar and Vector Bosons The couplings of low-mass scalar or vector particles ϕare easy to con- strain by the same methods. Because the energy-loss rates have been calculated only for nondegenerate conditions, only the arguments in- volving the solar age and the helium-burning lifetime can be employed. The latter yields a constraint on the ϕ-ecoupling of αϕe∼<1.4×10−29. (3.46) It is based on the bremsstrahlung process e+4He→4He +e+ϕas discussed above in Sect. 3.5.5. For vector bosons which couple by a current-current structure analogous to photons the same results apply except for a factor of two in the emission rate which improves the limit by a factor of 2. Particles Interacting with Electrons and Baryons 111Tab. 3.1. Astrophysical bounds on the Yukawa coupling gaof pseudoscalars (axions) to electrons. Upper limit Astrophysical Dominant emission Detailed αa=g2 a/4πobservable process discussion 1.7×10−22Solar age Bremsstrahlung (75%), Compton (25%) here e+X→X+e+a(X=e,p,4He) γ+e→e+a 1.0×10−26Galactic age inferred from break Bremsstrahlung (degenerate) Sect. 2.2.4 in white-dwarf luminosity function e+12C (16O)→12C (16O) +e+a similar Period decrease in DAV star G117–B15A same Sect. 2.2.5 1.6×10−26Helium-burning lifetime of Compton Sect. 3.2.6 horizontal-branch stars γ+e→e+a 0.5×10−26Core mass at helium ignition in Bremsstrahlung (degenerate) Sect. 2.5.2 low-mass red giants e+A→A+e+a Sect. 3.5.5 (A=4He,12C,16O) 112 Chapter 3 One may also consider the Yukawa coupling of scalar (vector) bosons to baryons (Grifols and Mass´ o 1986; Grifols, Mass´ o, and Peris 1989b). In this case one may use the Compton process γ+4He→4He +ϕon a helium nucleus for which the emission rate is given mutatis mutandis by the same formula as for nonrelativistic electrons. Assuming the same coupling to protons and neutrons the emission rate is coherently enhanced by a factor 42. Moreover, in Eq. (3.22) one must replace YewithYHe(number of4He nuclei per baryon) which is1 4for pure helium,meis to be replaced by mHe≈4mu, andα→4αto account for the coherent photon coupling. Pulling these factors together one ﬁndsϵ=αϕN0.7×1023erg g−1s−1for pure helium. Because the helium- burning lifetime argument limits this energy-loss rate to 10 erg g−1s−1 one ﬁnds αϕN∼<1.5×10−22(3.47) as a limit on the coupling of a scalar boson to a nucleon N. Again, for vector bosons this limit is a factor of 2 more restrictive. 3.6.3 Long-Range Forces Scalar or vector particles mediate long-range forces between macro- scopic bodies. For pseudoscalars this is not the case because their CP- conserving coupling to fermions has a pseudoscalar structure, i.e. in the nonrelativistic limit they couple to the fermion spin. Therefore, even if the mass of the new particles is very small or exactly zero, they do not mediate a long-range force between unpolarized bodies. The resid- ual force caused by the simultaneous exchange of two pseudoscalars is found to be extremely small (e.g. Grifols and Tortosa 1994). Consider a scalar of mass m=λ−1(Compton wave length λ) which couples to nucleons with a Yukawa strength g. It mediates an at- tractive force between two nucleons given in terms of the potential −(g2/4π)r−1e−r/λ. Two macroscopic test bodies of geometric dimen- sion much below λare then attracted by virtue of the total potential V(r) =−GNm1m2 r(1 +βe−r/λ), (3.48) whereGN=m−2 Plis Newton’s constant, mPl= 1.22×1019GeV is the Planck mass, m1andm2are the masses of the bodies, and β≡g2 4πm2 Pl u1u2. (3.49) Here,u1,2=m1,2/N1,2withN1,2the total number of nucleons in each body. Apart from small binding-energy eﬀects which are diﬀerent for Particles Interacting with Electrons and Baryons 113 diﬀerent materials u1≈u2≈muis the atomic mass unit, approxi- mately equal to a nucleon mass mN. For this force not to compete with gravity one needs g<O(10−19) orα=g2/4π<O(10−37) so that low-mass scalars, if they exist, need to have extremely feeble couplings to matter. Their smallness implies that such particles would not have any impact whatsoever on the energy loss of stars. Therefore, for boson masses so small that the Compton wave length λis macroscopic, the most restrictive limits on gobtain from analyzing the forces between macroscopic bodies, not from the energy-loss argu- ment. The existence of a composition-dependent “ﬁfth force” in nature with a strength of about 1% of gravity and a range λof a few hun- dred meters seemed indicated by a reanalysis of E¨ otv¨ os’s original data (Fischbach et al. 1986). Subsequently many experiments were carried out to search for this eﬀect, with no believable positive outcome (for a review see Fischbach and Talmadge 1992). However, these investi- gations produced extremely restrictive limits on β, depending on the assumed range λof the new force. The limits also depend on the pre- sumed coupling; if the new force couples to baryon number one ﬁnds thatβ∼<10−3forλin the cm range, or β∼<10−9forλof order the Earth-Sun distance and above. Thus, novel long-range forces must be much weaker than gravity. No bounds on βseem to exist for λbelow the cm range, i.e. for boson masses of order 10−3eV and above, apart from the stellar energy-loss argument. The eﬀect of a novel force with intermediate range on the equa- tions of stellar structure was discussed by Glass and Szamosi (1987, 1989). Solar models including such a force and the impact on the solar oscillation frequencies were discussed by Gilliland and D¨ appen (1987) and Kuhn (1988). For a force so weak or weaker than indi- cated by the laboratory limits, no observable consequences for stel- lar structure and evolution seem to obtain. Also, the impact of the new ﬁeld on the value of fundamental coupling constants even at com- pact objects such as neutron stars is far below any observable limit (Ellis et al. 1989). A signiﬁcant bound obtains from the orbital decay of the Hulse- Taylor binary pulsar. In order for the energy loss in the new scalars to remain below 1% of the gravitational wave emission the Yukawa cou- pling to baryons must satisfy g∼<3×10−19(Mohanty and Panda 1994). This translates into β∼<1 for scalar boson masses below the orbital pul- sar frequency of 2 π/P = 2.251×10−4s−1, i.e. forλ−1∼<1.5×10−19eV orλ∼>1.3×1014cm. This “ﬁfth-force limit,” however, is weaker than those derived by terrestrial laboratory methods. 114 Chapter 3 3.6.4 Leptonic and Baryonic Gauge Interactions The physical motivation for considering long-range interactions medi- ated by vector bosons, besides the ﬁfth-force episode, is the hypothesis that baryon number or lepton number could play the role of physical charges similar to the electric one (Lee and Yang 1955; Okun 1969). Their association with a gauge symmetry would provide one explana- tion for the strict conservation of baryon and lepton number which so far has been observed in nature. In the framework of this hypothesis one predicts the existence of baryonic or leptonic photons which couple to baryons or leptons by a charge eBoreL, respectively. The novel gauge bosons would be massless like the ordinary photon. Therefore, the limits established in the previous sections on the di- mensionless couplings of vector bosons can be readily restated as limits on the values of putative baryonic or leptonic charges. The energy-loss argument applied to helium-burning stars yields eL∼<1×10−14, eB∼<3×10−11, (3.50) according to Eqs. (3.43) and (3.47), respectively. Tests of the equiv- alence principle (i.e. of a composition-dependent ﬁfth force) on solar- system scales yield β∼<10−9(Sect. 3.6.3) so that eB∼<1×10−23. (3.51) Apparently this is the most restrictive limit on eBthat is currently available. One may be tempted to apply the limits from the equivalence prin- ciple also to a leptonic charge eL. However, in this case one has to worry about the fact that even neutrinos would carry leptonic charges. The universe is probably ﬁlled with a background neutrino sea in the same way as it is ﬁlled with a background of microwave photons. This neu- trino medium would constitute a leptonic plasma which screens sources of the leptonic force just as an electronic plasma screens electric charges (Zisman 1971; Goldman, Zisman, and Shaulov 1972; C ¸ift¸ ci, Sultansoi, and T¨ urk¨ oz 1994; Dolgov and Raﬀelt 1995). Debye screening will be studied in Sect. 6.4.1. For the screening wave number one ﬁnds the expression k2 S= 2 (eL/π)2∫∞ 0dpf pp(v+v−1), (3.52) wherep=|p|is the momentum (isotropy was assumed), v=p/E the velocity of the charged particles, and fpthe occupation number of Particles Interacting with Electrons and Baryons 115 mode p. The overall factor of 2 relative to Eq. (6.55) arises because neutrinos and antineutrinos contribute equally. The smallest kS(the largest radius over which leptonic ﬁelds remain unscreened) arises if the background neutrinos have such small masses that they are still relativistic today ( v= 1). The neutrino number density is given by nν+ν= 2∫fpd3p/(2π)3=∫∞ 0fpp2dp/π2. Therefore, in the relativistic limit one ﬁnds roughly k−1 S≈e−1 Ln−1/3 ν+ν≈e−1 L0.2 cm, (3.53) independently of details of the neutrino momentum distribution. In this estimate the predicted number density of background neutrinos in each ﬂavor of nν+ν≈100 cm−3was used. The largest conceivable kSwould obtain if neutrinos were nonrel- ativistic today, and if some of them were bound to the galaxy. The escape velocity from the galaxy is vesc≈500 km s−1so that the maxi- mum momentum of a gravitationally bound neutrino is pmax=mνvesc. Because neutrinos obey Fermi statistics, the largest conceivable galactic neutrino density is nmax≈p3 max≈m3 νv3 esc. A typical neutrino velocity is of order the galactic velocity dispersion, i.e. of order vesc. Therefore, from Eq. (3.52) one estimates k2 S≈e2 Ln2/3 maxv−1 esc≈e2 Lm2 νvesc. Because the largest cosmologically allowed neutrino mass is about 30 eV one ﬁnds that neutrino screening cannot operate on scales below e−1 L10−5cm. Therefore, the screening scale is reduced by no more than six orders of magnitude by the fact that cosmic neutrinos could be nonrelativis- tic today. The stellar energy-loss result Eq. (3.50) informs us that for rela- tivistic neutrinos kS∼>1013cm≈1 AU where 1 AU = 1 .5×1013cm (astronomical unit) is the distance to the Sun. If neutrinos were non- relativistic, leptonic forces could be screened over distances six orders of magnitude smaller, i.e. over about 100 km. However, it is proba- bly safe to assume that leptonic forces with a strength comparable to gravity would have been noticed in terrestrial experiments searching for a composition-dependent ﬁfth force. Therefore, even with neutrino screening it appears inconceivable that eLcould exceed about 10−19. In that case the screening scale would always exceed about 0 .1 AU so that terrestrial limits would easily apply. Thereby one could gain a few orders of magnitude in the limit, and so one could even use so- lar system constraints, taking one back to a result of order the bary- onic one Eq. (3.51). A similar conclusion was reached by Blinnikov et al. (1995). 116 Chapter 3 Finally, the matter of the galaxy would exert a leptonic force on neutrinos propagating, say, from a distant supernova to us. This eﬀect would cause an energy-dependent dispersion of the measurable neutrino burst (Sect. 13.3.3). However, when eL∼>10−20, which is necessary to cause an interesting eﬀect on supernova neutrinos, then the galactic leptonic charge is completely screened over the relevant length scales, even if the cosmic background neutrinos are relativistic. 3.7 Graviton Emission from Stars The one nonelectromagnetic long-range force that is actually known to exist is gravity. Whatever the ultimate quantum theory of gravity, there is little doubt that there will be quantized wave excitations, the gravitons. They would be massless spin-2 particles. In fact, gravity is the only possible force that can be mediated by a massless spin-2 boson because as a source it needs a conserved rank-2 tensor. The energy- momentum tensor, which acts as a source for the gravitational ﬁeld, is the only example. Classical gravitational waves are an inevitable consequence of Ein- stein’s theory of general relativity. The orbital decay of the binary pulsar PSR 1913+16 (Hulse and Taylor 1975) yields ﬁrm evidence for their emission (Taylor and Weisberg 1989; Damour and Taylor 1991). Gravitons can be produced in hot plasmas in analogy to axions or neutrino pairs; typical processes are bremsstrahlung e+p→e+p+g (gravitong) and the Primakoﬀ eﬀect (gravitons have a two-photon coupling). Because of their weak interaction gravitons can freely es- cape once produced in the interior of stars. Early calculations of the emission rates were summarized by Papini and Valluri (1977). More recent discussions include Sch¨ afer and Dehnen (1983), Gould (1985), and del Campo and Ford (1988). As the graviton coupling involves the inverse of the Planck mass (1 .2×1019GeV) the graviton luminosity of stars is inevitably small. For the Sun it is about 1015erg s−1≈10−19L⊙, much too small to be of any observational relevance. The same conclu- sion holds for other stars. Therefore, gravity itself illustrates that the mediation of long-range forces is a far more important eﬀect of low-mass bosons than their thermal emission from stellar plasmas. Unless, of course, they only couple to the fermion spins rather than to a “charge.” Pseudoscalars such as axions are in that category. Chapter 4 Processes in a Nuclear Medium The interaction rates of neutrinos and axions with nucleons in a nuclear medium are studied with a focus on neutral-current processes such as bremsstahlung emission of axions NN→NNa and of neutrino pairs NN→NNννas well as neutrino scattering. A severe problem with the perturbative rate calculations at high density is discussed which has a strong impact on axion emissivities and the dominant axial-vector contribution to the neutrino opacities. 4.1 Introduction New particles which couple to nucleons are emitted from ordinary stars by analogous processes to those discussed in Chapter 3 for electrons. For example, axions can be produced by the Compton process γp→pa; the previous results can be easily adapted to such reactions. Presently I will focus on processes involving neutrinos or axions that are speciﬁc to a nuclear medium, i.e. to supernova (SN) cores or neutron stars. The main focus of the literature which deals with microscopic pro- cesses in a nuclear medium was inspired by the problem of late-time neutron-star cooling. The recent progress of x-ray astronomy has led to reasonably safe ROSAT identiﬁcations of thermal surface emission from a number of old pulsars (Sect. 2.3). Together with the spin-down age of these objects one can begin to test neutron-star cooling sce- narios, notably those that involve novel phases of nuclear matter such as superﬂuidity, meson condensates, quark matter, etc. (Shapiro and Teukolsky 1983; Tsuruta 1992). 117 118 Chapter 4 From the perspective of particle physics, however, a more interesting nuclear environment is a young neutron star for the ﬁrst few seconds after the progenitor collapsed. The nuclear medium here is so hot that it is essentially nondegenerate, and neutrinos are trapped. Therefore, the production of even more weakly interacting particles such as axions or right-handed neutrinos can compete with neutrino energy transfer which is essentially a diﬀusion process. For a quantitative understand- ing of the emissivities of the new particles, but also for the conventional transport of energy and lepton number by neutrinos, a knowledge of the microscopic interaction rates is needed. The neutrino opacities that went into standard SN collapse and ex- plosion calculations as well as the particle emissivities that went into the derivation of, say, axion bounds from SN 1987A were based on the assumption that the hot nuclear medium can be treated as an ideal Boltzmann gas of free particles, except for degeneracy eﬀects that are easy to include. It turns out, however, that the approximations made are internally inconsistent, notably for the dominant processes which involve couplings to the nucleon spin (axial-vector current interactions). In order for the “naive” neutral-current neutrino opacities to be correct one needs to assume that the nucleon spins do not ﬂuctuate too fast on a time scale set by the temperature. A naive perturbative calculation, however, yields a spin-ﬂuctuation rate which is much larger than this limit. This large rate went into the axion emissivities. Therefore, the existing studies of SN axion bounds are based on microscopic interac- tion rates which simultaneously make use of the opposite limits of a spin-ﬂuctuation rate very small and very large compared with T. Beyond the ideal-gas approximation there is virtually no literature on the microscopic interaction rates in a hot nuclear medium, presum- ably because of the historical focus on old neutron stars, and presum- ably because degenerate nuclear matter is more reminiscent of actual nuclei. Thus there is a dearth of reliable microscopic input physics for conventional studies of SN evolution, and for variations involving novel particle-physics ideas. Most of this chapter focusses on the dominant axial-vector current interactions of neutrinos and axions with nucleons in a dense and hot medium. Most of the material is based on a series of papers which I have co-authored (Raﬀelt and Seckel 1991, 1995; Keil, Janka, and Raﬀelt 1995; Keil et al. 1995) and as such does not represent a com- munity consensus. On the other hand, I am not aware of a signiﬁcant controversy. Rather, it appears that very little serious interest has been taken in the diﬃcult question of weakly interacting particles in- Processes in a Nuclear Medium 119 teracting with a hot nuclear medium, even though these issues are of paramount importance for a proper quantitative understanding of SN physics where the interaction of neutrinos with the medium dominates the thermal and dynamical evolution. My discussion can only be a starting point for future work that may actually yield some answers to the questions raised. 4.2 Axionic Bremsstrahlung Process 4.2.1 Matrix Element for NN!NNa The simplest neutral-current process of the kind to be discussed in this chapter is bremsstrahlung emission of axions or other pseudoscalars (Fig. 4.1) because the single-particle axion phase space is particularly simple. It will turn out that the result thus derived can be applied to neutrino processes almost without modiﬁcation. The interaction Hamiltonian with nucleons is of the form Hint=−CN 2faψNγµγ5ψN∂µϕ, (4.1) wherefais an energy scale (the Peccei-Quinn scale for axions), CNwith N=norpis a dimensionless, model-dependent coupling constant of order unity, the ψNare the proton and nucleon Dirac ﬁelds, and ϕis the axion ﬁeld or any other pseudoscalar Nambu-Goldstone boson. Consider a single species of nonrelativistic nucleons interacting by a one-pion exchange (OPE) potential. The spin-summed squared matrix element is (Brinkmann and Turner 1988; Raﬀelt and Seckel 1995) ∑ spins|M|2=16 (4π)3α2 παa 3m2 N (k2 k2+m2 π)2 +(l2 l2+m2 π)2 +k2l2−3 (k·l)2 (k2+m2 π)(l2+m2 π)] .(4.2) Here,αa≡(CNmN/fa)2/4πandαπ≡(f2mN/mπ)2/4π≈15 withf≈ 1 are the axion-nucleon and pion-nucleon “ﬁne-structure constants,” respectively. Further, k=p2−p4andl=p2−p3withpithe momenta of the nucleons Nias in Fig. 4.1. In a thermal medium k2≈3mNTso that [ k2/(k2+m2 π)]2varies between 0.86 for T= 80 MeV and 0.37 for T= 10 MeV. Therefore, neglecting the pion mass causes only a moderate error in a SN core (Brinkmann and Turner 1988; Burrows, Ressell, and Turner 1990). 120 Chapter 4 Fig. 4.1. Feynman graph for nucleon-nucleon axion bremsstrahlung. There is a total of eight amplitudes, four with the axion attached to each nucleon line, and an exchange graph each with N3↔N4. Raﬀelt and Seckel (1995) showed that including mπcauses less than a 30% reduction of typical neutrino or axion rates for T >20 MeV. Be- cause this will be a minor error relative to the dominant uncertainties the term in square brackets is approximated as [3 −(ˆk·ˆl)2]. The remaining ( ˆk·ˆl)2term is inconvenient without yielding any signiﬁcant insights. In a degenerate medium it averages to zero in expressions such as the axion emission rate while in a nondegenerate medium it can be as large as about 1.31 (Raﬀelt and Seckel 1995), leading to an almost 50% reduction of the emissivity. Still, for the present discussion I will neglect this term and use ∑ spins|M|2= 16 (4π)3α2 παam−2 N. (4.3) While this may seem somewhat arbitrary, it must be stressed that us- ing an OPE potential to model the nucleon interactions in a nuclear medium is in itself an approximation of uncertain precision. For the present discussion a factor of order unity will not change any of the conclusions. 4.2.2 Energy-Loss Rate The axionic volume energy-loss rate of a medium is the usual phase- space integral, Qa=∫d3ka 2ωa(2π)3ωa∫4∏ i=1d3pi 2Ei(2π)3f1f2(1−f3)(1−f4) ×(2π)4δ4(P1+P2−P3−P4−Ka)1 4∑ spins|M|2,(4.4) whereP1,2are the four-momenta of the initial-state nucleons, P3,4are for the ﬁnal states, and Kais for the axion. The factor1 4is a statistics Processes in a Nuclear Medium 121 factor to compensate for double counting of identical fermions in the initial and ﬁnal state. The occupation numbers f1,2and the Pauli blocking factors (1 −f3,4) are for the nucleons while the axions are assumed to escape freely so that a Bose stimulation factor as well as backreactions (axion absorption) can be neglected. In the nonrelativistic limit the nucleon mass mNis much larger than all other energy scales such as the temperature or Fermi energies. The nucleon momenta are then much larger than the momentum carried by the radiation. A typical nonrelativistic nucleon kinetic energy is Ekin= p2/2mNso that a typical nucleon momentum is p= (2mNEkin)1/2. In a bremsstrahlung process, the radiation typically takes the energy Ekinwith it, less in a degenerate medium, so that a typical radiation momentum is ka=Ekin≪p. Therefore, one may ignore the radiation in the law of momentum conservation so that Eq. (4.4) is simpliﬁed according to δ4(P1+P2−P3−P4−Ka)→ →δ(E1+E2−E3−E4−ωa)δ3(p1+p2−p3−p4). (4.5) In this case, the second integral expression in Eq. (4.4), which “knows” about axions only by virtue of the energy-momentum transfer Kain theδfunction, is only a function of the axion energy ωa. Therefore, in terms of a dimensionless function s(x) of the dimen- sionless axion energy x=ωa/Tone may write the energy-loss rate in the form (baryon density nB) Qa=(CN 2fa)2 nBΓσ∫d3ka 2ωa(2π)3ωas(ωa/T)e−ωa/T =αanBΓσT3 4πm2 N∫∞ 0dxx2s(x)e−x. (4.6) Γσwill turn out to represent the approximate rate of change of a nucleon spin under the inﬂuence of collisions with other nucleons. 4.2.3 Nondegenerate Limit To ﬁnd Γ σands(x) turn ﬁrst to an evaluation of Eq. (4.4) in the nondegenerate limit. The initial-state nucleon occupation numbers f1,2 are given by the nonrelativistic Maxwell-Boltzmann distribution fp= (nB/2) (2π/m NT)3/2e−p2/2mNTso that∫2fpd3p/(2π)3=nBgives the nucleon (baryon) density where the factor 2 is for two spin states. Pauli blocking factors are omitted: (1 −f3,4)→1. 122 Chapter 4 Because the matrix element has been assumed to be a constant it can be pulled out of the integral which reduces to a phase-space volume. Nonrelativistically, d3pi/[2Ei(2π)3] =d3pi/[2mN(2π)3] while Ei=p2 i/2mNin the energy δfunction. The axion momentum is ignored according to Eq. (4.5). One uses CM momenta p1,2=p0±pandp3,4= p0±qwhere p0=1 2(p1+p2) =1 2(p3+p4) and deﬁnes u2≡p2/mNT andy≡v2≡q2/mNT. Then one ﬁnds explicitly Γσ= 4√πα2 πnBT1/2m−5/2 N, s(x) = 4∫ dudvu2v2e|x|−u2δ(u2−v2− |x|) =∫∞ 0dye−y( |x|y+y2)1/2≈√ 1 +|x|π/4. (4.7) The analytic form is accurate to better than 2.2% everywhere; it has the correct asymptotic behavior s(0) = 1 and s(|x|≫1) = (|x|π/4)1/2. We will see in Sect. 4.6.3 that for |x| ≫1 the nondegenerate s(x) must actually decrease , in conﬂict with this explicit calculation. This problem reveals a pathology of the OPE potential which is too singular at short distances. For the present discussion this is of no concern so that I stick to the explicit OPE result in order to facilitate comparison with the existing literature. To determine the total emission rate one uses the ﬁrst representation ofs(x) and performs∫dxﬁrst to remove the δfunction; the remaining integrals are easily done. Explicit results for sn≡∫∞ 0xns(x)e−xdxare given in Tab. 4.1. The normalized axion energy spectrum dNa/dx= xs(x)e−x/s1is shown in Fig. 4.2 (solid line). The average axion energy is⟨ωa⟩/T=s2/s1= 16/7. The total nondegenerate energy-loss rate is Table 4.1. sn=∫∞ 0xns(x)e−xdxfor nondegenerate (ND) and degenerate (D) conditions. n s n(ND)sn(D) 1 8/5 2 ζ3+ 6ζ5/π2 2 128/35 31 π4/315 3 256/21 24 ζ5+ (180/π2)ζ7 4 4096/77 82 π6/315 Processes in a Nuclear Medium 123 Fig. 4.2. Normalized axion spectrum x s(x)e−x=s1from nucleon-nucleon bremsstrahlung emission. The nondegenerate and degenerate functions s(x) are given in Eqs. (4.7) and (4.9), respectively, while s1(ND) and s1(D) are found in Tab. 4.1. explicitly QND a=128αaα2 π 35√πn2 BT7/2 m9/2 N, ϵND a=αa1.69×1035erg g−1s−1ρ15T3.5 MeV, (4.8) whereρ15=ρ/1015g cm−3,TMeV=T/MeV, andϵND a=QND a/ρis the energy-loss rate per unit mass. 4.2.4 Degenerate Limit Details of the nucleon phase space in the degenerate limit can be found in Friman and Maxwell’s (1979) calculation of ννemission. The axion energy-loss rate is expressed as in Eq. (4.6). One ﬁnds Γσ=4α2 π 3πpFT3 nBands(x) =(x2+ 4π2)|x| 4π2(1−e−|x|), (4.9) wherepFis the nucleon Fermi momentum. As in the nondegenerate case s(0) = 1 while s(|x|≫1) =|x|3/4π2. Values for sn=∫∞ 0xns(x)e−xdx are given in Tab. 4.1. The normalized axion spectrum is shown in Fig. 4.2 (dotted line), the average energy is ⟨ωa⟩/T=s2/s1≈3.16. 124 Chapter 4 The total energy-loss rate is QD a=αaα2 π31π2 945pFT6 m2 N, ϵD a=αa1.74×1031erg g−1s−1ρ−2/3 15T6 MeV, (4.10) (Iwamoto 1984; Brinkmann and Turner 1988). The degenerate and nondegenerate rates are best compared in terms of a parameter ξ≡p2 F/(2πm NT) which approaches η/πin the degen- erate limit (degeneracy parameter η), QD a QND a=31π4 1536√ 2ξ−5/2≈1.39ξ−5/2. (4.11) Therefore, they are equal for ξ≈1, or a degeneracy parameter of η≈3.5. This deﬁnes the dividing line between the regimes where these approximations can be reasonably used. In the degenerate limit the nucleon phase-space integrals can be done analytically with the inclusion of a nonzero mπ. Themπ= 0 rates must be supplemented with a factor20(Ishizuka and Yoshimura 1990) G(u) = 1−5u 6arctan(2 u) +u2 3(u2+ 4)+ +u2 6√ 2u2+ 4arctan(2√ 2u2+ 4 u2) ,(4.12) whereu=mπ/pF. For only one species of nucleons (as approximately in a neutron star) pF= 515 MeV ρ1/3 15so thatu= 0.26ρ−1/3 15 withρ15 the mass density in 1015g/cm3. For this case Gis shown as a function ofρin Fig. 4.3 (solid line). 4.2.5 Bremsstrahlung Emission of Scalars The previous results equally apply to pseudoscalars with a coupling igaNψNγ5ψNϕwithαa≡g2 aN/4πif a derivative pion-nucleon interac- tion is used (Sect. 14.2.3). However, for scalars which couple according 20Eq. (4.12) diﬀers from the corresponding result of Friman and Maxwell (1979) which is identical with that of Iwamoto (1984) who apparently did not take the third term of the matrix element Eq. (4.2) properly into account. Processes in a Nuclear Medium 125 Fig. 4.3. Correction of the degenerate bremsstrahlung rate for a nonzero pion mass. For pseudoscalars, G(mπ=pF) is given explicitly in Eq. (4.12). It is assumed that only one species of nucleons is present. togψNψNϕthe results are diﬀerent. The degenerate bremsstrahlung energy-loss rate was worked out by Ishizuka and Yoshimura (1990), Qscalar =α′α2 π44 153(T mN)4 p5 FGscalar(mπ/pF), (4.13) withα′≡g2/4π. The function Gscalar(u) is similar to Eq. (4.12); it is plotted in Fig. 4.3 (dashed line). 4.2.6 Mixture of Protons and Neutrons For a mixed medium of protons and neutrons one needs to consider the individual Yukawa couplings gan=CnmN/faandgap=CpmN/faas well as the isoscalar and isovector combinations g0,1=1 2(gan±gap) and the “ﬁne-structure constants” αj=g2 j/4πwithj=n,p,0,1. For equal couplingsαn=αp=α0whileα1= 0. In the nondegenerate limit, the main diﬀerence is that npscattering beneﬁts from the exchange of charged pions which couple more strongly by a factor√ 2. Depending on the chemical composition of the medium, the emission rate will be increased by up to a factor of 2. On the other hand, some reduction factors have been ignored such as the ( ˆk·ˆl)2term and the pion mass. Therefore, ignoring this enhancement essentially compensates for the previously introduced errors. In the degenerate limit the changes are more dramatic. The role of the Fermi momentum is played by pF→(3π2nB)1/3. It sets the 126 Chapter 4 scale for the proton and neutron Fermi momenta which are pn,p F= (3π2nB)1/3Y1/3 n,p. According to a result of Brinkmann and Turner (1988) the eﬀective coupling is then αa→αnY1/3 n+αpY1/3 p+ (28 3α0+20 3α1) (Y2/3 n+Y2/3 p)1/2 ×1 2√ 2( 2−|Y2/3 n−Y2/3 p| Y2/3 n+Y2/3 p) .(4.14) The third term is from npcollisions; it has the remarkable feature that it does not vanish as Yp→0. For equal couplings ( Cn=Cp) the variation of the emission rate is shown in Fig. 4.4 as a function of Yp= 1−Yn. For all proton concentrations the npcontribution dominates. Fig. 4.4. Variation of the degenerate nucleon-nucleon bremsstrahlung rate with Yp(proton number fraction) according to Eq. (4.14) with equal cou- plings so that n=p=0and1= 0. 4.3 Neutrino Pair Emission 4.3.1 Structure Function Neutrino pair emission in nucleon-nucleon collisions (Fig. 4.5) is anal- ogous to that of axions. Therefore, instead of embarking on a new calculation it is worth understanding their common features—the neu- trino rates can be obtained for free on the basis of the axion ones. This approach amounts to deﬁning the dynamical structure functions Processes in a Nuclear Medium 127 Fig. 4.5. Bremsstrahlung emission of neutrino pairs in nucleon-nucleon colli- sions. There is a total of eight amplitudes, four with the neutrinos attached to each nucleon line, and an exchange graph each with N3↔N4. of the medium, quantities which are of utmost importance to under- stand the general properties of the emission, absorption, and scattering rates independently of phase-space details of the neutrinos or axions. Contrary to the ννbremsstrahlung emission in electron-nucleus col- lisions (Sect. 3.5.3), nonrelativistically only the nucleon axial-vector coupling contributes in nucleon-nucleon collision (Friman and Maxwell 1979). This diﬀerence originates from the interaction potential of the colliding particles which involves a spin-dependent force between nucle- ons so that the spin ﬂuctuations caused by collisions are more dramatic than those of the velocity—see Sect. 4.6.5 below. Hence, the part of the interaction Hamiltonian relevant for neutrino pair bremsstrahlung is Hint=CN AGF√ 2ψNγµγ5ψNψνγµ(1−γ5)ψν, (4.15) whereCp A≈ −Cn A≈1 2; see Appendix B for a discussion of the appro- priate values in a nuclear medium. The interaction Hamiltonian Eq. (4.15) has the same structure as that for axions Eq. (4.1). The squared matrix elements are then of the general form ∑ spins|M|2=  (CN AGF/√ 2)2MµνNµνfor neutrinos, (CN/2fa)2MµνKµ aKν afor axions.(4.16) Here,Kais the axion four-momentum while Nµν= 8( Kµ 1Kν 2+Kµ 2Kν 1−K1·K2gµν−iϵαβµνK1αK2β) (4.17) with the neutrino and antineutrino four momenta K1andK2(Gaemers, Gandhi, and Lattimer 1989). NµνandKµ aKν aare the squared matrix elements of the neutrino and axion current, respectively. 128 Chapter 4 The matrix Mµνis the nuclear part of the squared matrix element. It is exactly the same for axion or neutrino interactions because in Eq. (4.16) the global coupling constants have been explicitly pulled out. Therefore, one may go one step further and perform the entire nucleon phase-space integration for both cases directly on Mµν, Sµν≡1 nB∫4∏ i=1d3pi 2Ei(2π)3f1f2(1−f3)(1−f4) ×(2π)4δ4(P1+P2−P3−P4+K)Mµν. (4.18) Here, thePiare the nucleon four momenta, K=−Kafor axion emis- sion, andK=−(K1+K2) for neutrino pairs. Thus, Kis the energy- momentum transfer from the radiation (axions or neutrino pairs) to the nucleons. Because Sµνknows about the radiation only through the energy-momentum δfunction it is only a function of K= (ω,k), apart from the temperature and chemical potentials of the medium. The slightly awkward deﬁnition of the sign of Kfollows the common deﬁnition of the structure function where a positive energy transfer ω refers to energy given to the medium. The energy-loss rates by ννor axion emission are then the phase- space integrals Qνν=(CN AGF√ 2)2 nB∫d3k1 2ω1(2π)3d3k2 2ω2(2π)3SµνNµν(ω1+ω2), Qa=(CN 2fa)2 nB∫d3ka 2ωa(2π)3SµνKµ aKν aωa. (4.19) Here, it was assumed that both axions and neutrinos can escape freely from the medium so that ﬁnal-state Pauli blocking or Bose stimulation factors can be ignored. In the nonrelativistic limit the nucleon current in Eqs. (4.1) and (4.15) reduces to χ†τiχwhereχis a nucleon two-spinor and τi(i= 1,2,3) are Pauli matrices representing the nucleon spin operator. Put another way, in the nonrelativistic limit the axial-vector current rep- resents the nucleon spin density. Therefore, it has only spatial com- ponents so that Sµν→Sij(i,j= 1,2,3). In order to construct the most general tensorial structure for Sijin an isotropic medium only δij is available. Recall that in the nonrelativistic limit Sµνdoes not know about the momentum transfer kbecause of Eq. (4.5). There is then no Processes in a Nuclear Medium 129 vector available from which a spatial tensor can be constructed. Thus, the structure function has the most general form Sij(ω) =Sσ(ω)δij. (4.20) For nonrelativistic nucleons all axion or axial-vector neutrino processes involve only one scalar function Sσ(ω) of the energy transfer. The contraction of δijwithKµ aKν afor axion emission yields k2 a= ω2 a. (The spatial Kronecker δshould be viewed as a Lorentz tensor with a zero in the 00 position.) Therefore, SµνKµ aKν a→ω2 aSσ(−ωa) in Eq. (4.19). For neutrino emission, K1·K2=ω1ω2−k1·k2and the contraction of δijwithgµνis−3. Therefore, the contraction of δijwithNµνyields 8ω1ω2(3−cosθ) whereθis the angle between the νandνmomenta. The neutrino phase-space integration will always average cos θto zero so that it may be dropped. Therefore, SµνNµν→ 24ω1ω2Sσ(−ω1−ω2) in Eq. (4.19). One may then immediately perform the integration over one of the neutrino energies and is left with an integration over the energy transfer. Thus, in the nonrelativistic limit the energy-loss rates Eq. (4.19) are of the form Qνν=(CN AGF√ 2)2nB 20π4∫∞ 0dωω6Sσ(−ω), Qa=(CN 2fa)2nB 4π2∫∞ 0dωω4Sσ(−ω). (4.21) The medium properties are embodied in a common function which may be expressed in the form Sσ(ω) =Γσ ω2s(ω/T)×{1 forω>0, eω/Tforω<0.(4.22) Because of the “detailed-balance relationship” between positive and negative energy transfers to be discussed more fully below, s(x) must be an even function. In the nondegenerate and degenerate limits Γ σ ands(x) have been determined above. They allow one to calculate the ννemission rate without any eﬀort. 130 Chapter 4 4.3.2 Bremsstrahlung Emission of Neutrino Pairs In order to calculate the neutrino pair emission rate explicitly turn ﬁrst to the nondegenerate limit for which one uses Γ σands(x) given in Eq. (4.7) and the integrals of Tab. 4.1. The average energy of a neutrino pair is ( s4/s3)T≈4.36Tand the total energy-loss rate is QND νν=2048 385π7/2C2 AG2 Fα2 πn2 BT11/2 m5/2 N, ϵND νν= 2.4×1017erg g−1s−1ρ15T5.5 MeV, (4.23) withρ15=ρ/1015g cm−3,TMeV=T/MeV, andϵND νν=QND νν/ρis the energy-loss rate per unit mass. For the degenerate rate one uses Eqs. (4.9). The average energy of a neutrino pair is ( s4/s3)T≈5.78Tand the total energy-loss rate is21 QD νν=41π 4725C2 AG2 Fα2 πpFT8, ϵD νν= 4.4×1013erg g−1s−1ρ−2/3 15T8 MeV. (4.24) One can correct for a nonvanishing value of the pion mass by virtue of Eq. (4.12). For a mixture of protons and neutrons the same remarks as in Sect. 4.2.6 apply. Apart from a small correction the neutrino cou- pling is isovector ( Cp A≈ −Cn A) so thatα0≈0 while the other α’s are approximately equal (Appendix B). For degenerate conditions, with a small modiﬁcation the dependence on the proton concentration is the same as that shown in Fig. 4.4. Again, the absolutely dominating con- tribution is from npcollisions unless protons are so rare that they are nondegenerate.22 21Friman and Maxwell’s (1979) total energy-loss rate is 2 =3 of the one found here. Apparently they did not include the crossterm in the squared matrix element, i.e. the third term in Eq. (4.2). 22This conclusion, based on the work of Brinkmann and Turner (1988), is in conﬂict with the results of Friman and Maxwell (1979). They found that Qwas proportional to the proton Fermi momentum which is relatively small in neutron- star matter. On the other hand, for small proton concentrations Brinkmann and Turner’s pFin Eq. (4.24) approaches the neutron Fermi momentum. I am in no position to decide between these conﬂicting results. Processes in a Nuclear Medium 131 4.4 Axion Opacity When axions or other pseudoscalar bosons interact so “strongly” that they are trapped in a young SN core, they still contribute to the ra- diative transfer of energy and can thus have an impact on the cooling speed. In order to include axions in a numerical evolution calculation one needs to determine the opacity of the medium to axions (Burrows, Ressell, and Turner 1990). The “reduced” Rosseland mean opacity relevant for radiative trans- port by relativistic bosons (Sect. 1.3.4) is deﬁned by 1 ρκa=15 4π2T3∫∞ 0dωℓ ω(1−e−ω/T)−1∂TBω(T), (4.25) whereρis the mass density of the medium, ℓωthe boson mean free path against absorption, and Bω(T) = (2π2)−1ω3(eω/T−1)−1is the boson spectral density for one spin degree of freedom. After ∂T=∂/∂T has been taken one ﬁnds 1 ρκa=15 8π4∫∞ 0dxℓ xx4e2x (ex−1)3, (4.26) wherex≡ω/T, and (1 −e−x)−1=ex(ex−1)−1was used. The axion opacity thus deﬁned is to be added to the photon opacity byκ−1 tot=κ−1 γ+κ−1 a. The energy ﬂux is then given by the usual ex- pressionF=−(3κtotρ)−1∇aγT4whereaγ=π2/15 gives the radiation density of photons . Burrows, Ressell, and Turner (1990) have deﬁned an axion opacity which is to be used in conjunction with the axion radi- ation density which is half as large because there is only one spin degree of freedom, i.e. aa=π2/30. Theirκ−1 ais twice that in Eq. (4.26) so that the energy ﬂux is the same. I prefer the present deﬁnition because it allows for the usual addition of all opacity contributions. It is easy to determine the axion absorption rate from the discussion in Sect. 4.3.1. Starting from Qain Eq. (4.21) one removes the phase- space integral∫∞ 0dω4πω2/(2π)3and a factor ωbecauseQawas an energy-loss rate. One includes a factor eω/Tto account for the detailed- balance relationship (see Eq. 4.43) so that altogether ℓ−1 ω=(CN 2fa)2nBΓσ Ts(x) 2x, (4.27) wherex=ω/T. Therefore, the reduced Rosseland mean opacity is κa=(CN 2fa)2Γσ Tm Nˆκ, (4.28) 132 Chapter 4 whereρ/n B≈mNwas used. The dimensionless opacity is ˆκ−1≡15 8π4∫∞ 0dxx4e2x (ex−1)32x s(x). (4.29) For the nondegenerate case s(x) was given in Eq. (4.7), for the degen- erate one in Eq. (4.9). Then one ﬁnds numerically ˆ κND= 0.46 and ˆκD= 1.53. The respective Γ σ’s were given in Eqs. (4.7) and (4.9). 4.5 Neutrino Opacity 4.5.1 Elastic Scattering The trapping of neutrinos in a hot SN core allows them to escape only by diﬀusion. This implies that they transport energy in a fashion similar to radiative transfer. Apart from proper spectral weights as in the Rosseland mean opacity, the main ﬁgure of merit that determines the transport eﬃciency is the neutrino mean free path (mfp). While the charged-current absorption of νe’s is very important for practical SN core-cooling calculations it has no direct bearing on the main issues of interest here. However, all neutrino ﬂavors interact by neutral-current interactions which allow for the scattering on nucleons. The inverse mean free path is then simply the cross section times the nucleon density. If there is only one species of nondegenerate nucleons one easily ﬁnds λ−1= (C2 V+ 3C2 A)G2 FnBω2 1/π, (4.30) whereω1is the energy of the incident neutrino, assumed to be much smaller than mNso that recoil eﬀects can be neglected. For a mixture of protons and neutrons one has to take a proper average with the coupling constants CVandCAfor protons and neutrons (Appendix B). Forνe’s which have a large chemical potential, a Pauli blocking factor must be included, and the same for nucleons if they are degenerate. 4.5.2 Pair Absorption Because the neutrinos are assumed to be trapped there is an ambient bath ofν’s andν’s which allows for neutrino absorption by the inverse bremsstrahlung process ννNN→NN, i.e. Fig. 4.5 read from right to left. The rate for this process is closely related to that for pair emission: Processes in a Nuclear Medium 133 it is based on the same matrix element with the neutrinos “crossed” into the initial state. Then one ﬁnds in analogy to Sect. 4.3.1 λ−1=(CAGF√ 2)2nB 2ω1∫d3k2 2ω2(2π)3f2SµνNµν =3C2 AG2 FnB 2π2∫∞ 0dω2ω2 2f2Sσ(ω1+ω2), (4.31) where 1 refers to the νfor which the mfp is being determined while 2 refers to a νfrom the thermal environment (occupation number f2). With the detailed-balance relationship Sσ(ω) =Sσ(−ω)eω/T(see Eq. 4.43) and writing Sσ(|ω|) = (Γ σ/ω2)s(ω/T) as before one ﬁnds λ−1= 3C2 AG2 FnBΓσT∫∞ 0dx2f2x2 2s(x1+x2) 2π2(x1+x2)2, (4.32) wherexi=ωi/T. Then one may use the previously determined Γ σand s(x) to ﬁnd the mfp for given conditions (degenerate or nondegenerate). The ratio between the inverse mfp’s from elastic scattering and pair absorption is, ignoring the contribution from C2 V, λ−1 pair λ−1 scat=Γσ T∫∞ 0dx2f2x2 2s(x1+x2) 2πx2 1(x1+x2)2. (4.33) An average with regard to a thermal x1distribution yields about 0 .02 for the dimensionless integral where Fermi-Dirac distributions with chemical potentials µν= 0 were used. The main ﬁgure of merit, how- ever, is Γ σ/T, the ratio of a typical spin-ﬂuctuation rate and the ambi- ent temperature. In the nondegenerate limit one ﬁnds with Eq. (4.7) γσ≡Γσ T= 4π1/2α2 πnB m5/2 NT1/2≈16ρ ρ0(30 MeV T)1/2 , (4.34) with the nuclear density ρ0= 3×1014g/cm3. 4.5.3 Inelastic Scattering It appears that for typical conditions of a young SN core, pair absorp- tion is almost as important as elastic scattering. However, even though the quantity γσis larger than unity, the pair-absorption rate has an unfavorable phase-space factor from the initial-state νso that the di- mensionless integral in Eq. (4.33) is a small number. This would not be the case for the inelastic scattering process νNN→NNν shown in 134 Chapter 4 Fig. 4.6. Inelastic neutrino-nucleon scattering. There is a total of eight amplitudes, four with the neutrinos attached to each nucleon line, and an exchange graph each with N3↔N4. Fig. 4.6. In this process, neutrinos can give or take energy even though recoil eﬀects for heavy nucleons are small in the elastic scattering pro- cessνN→Nν. This reaction is identical with pair absorption with the antineutrino line crossed into the ﬁnal state. The corresponding mfp is given by the same expression as for pair absorption, except that the index 2 now refers to the ﬁnal-state ν, the initial-state occupation number f2is to be replaced with a ﬁnal-state Pauli blocking factor (1 −f2), and the energy transfer is ω=ω2−ω1, λ−1=3C2 AG2 FnB 2π2∫∞ 0dω2ω2 2(1−f2)Sσ(ω1−ω2). (4.35) However, because the energy transfer can be zero, and because Sσ(ω)∝ ω−2, this expression diverges. The ω−2behavior seemed harmless before because it was moderated by powers of ωfrom the phase space of axions or neutrinos. The occurrence of this divergence could have been predicted without a calculation by inspecting Fig. 4.6. If one cuts the intermediate-state nucleon line, this graph falls into two sub-processes (nucleon-nucleon scattering and nucleon-neutrino scattering) which are each permitted by energy-momentum conservation, allowing the intermediate nucleon in the compound process to go “on-shell.” Therefore, the pole of the propagator which corresponds to real particles causes a divergence of the cross section. Physically, the divergence reﬂects a long-range inter- action which occurs because the intermediate nucleon can travel arbi- trarily far when it is on its mass shell. Still, the inelastic scattering process is an inevitable physical possi- bility. For nonzero energy transfers its diﬀerential rate is given by the unintegrated version of Eq. (4.35). For a vanishing energy transfer it is elastic and then its rate should be given by Eq. (4.30). Processes in a Nuclear Medium 135 However, what exactly does one mean by a vanishing energy trans- fer? The nucleons in the ambient medium constantly scatter with each other so that their individual energies are uncertain to within about 1/τcollwith a typical time between collision τcoll. Therefore, one would expect that the structure of Sσ(ω), which was calculated on the basis of free nucleons which interact only once, is smeared out over scales of order ∆ω≈τ−1 coll. In particular, this smearing-out eﬀect naturally regulates the low- ωbehavior of the structure function (Raﬀelt and Seckel 1991). Because Γ σ≈τ−1 collis a typical nucleon-nucleon collision rate, a sim- ple ansatz for a modiﬁed Sσ(ω) is a Lorentzian (Raﬀelt and Seckel 1991) Sσ(|ω|)→Γσ ω2+ Γ2 σ/4s(ω/T) =1 Tγσ x2+γ2 σ/4s(x), (4.36) withx=ω/T. Forγσ≪1 one has∫+∞ −∞dωS σ(ω) = 2π+O(γσ) for the modiﬁedSσ(ω). Thus, for γσ≪1 essentially Sσ(ω) = 2πδ(ω) so that the total “inelastic” scattering rate Eq. (4.35) reproduces the elastic scattering one. At the same time Sσ(ω) has wings which, for ω≫Γσ, give the correct inelastic scattering rate which is of order γσ. This simple ansatz can be expected to give a reasonable approx- imation to the true Sσ(ω) only in the limit γσ≪1. For practical applications in cooling calculations of young SN cores, however, one has to confront the opposite limit γσ≫1 causing the smearing-out ef- fect by multiple nucleon collisions to be a dominating feature of Sσ(ω) even forω≫T. This implies that for typical thermal energies of neutrinos and nucleons a reasonably clean separation between elastic and inelastic scattering processes is not logically possible—there is only one structure function Sσ(ω), broadly smeared out, which governs all axial-vector scattering, emission, and absorption processes. This observation has important ramiﬁcations not only for neutrino scattering, but also for the bremsstrahlung emission of axions and neu- trino pairs from a nuclear medium. Earlier it seemed that one did not have to worry about details of the behavior of Sσ(ω) nearω= 0 be- cause the low- ωpart was suppressed by axion or neutrino phase-space factors. However, since the notion of “low energy” presently means ω∼<Γσ, and because T≪Γσ,allrelevant energy transfers are low in this sense, and even the emission processes are dominated by multiple- scattering eﬀects. Consequences of this behavior are explored in more detail below after a formal introduction of the structure functions and their general properties. 136 Chapter 4 4.6 Structure Functions 4.6.1 Formal Deﬁnition To treat axion and neutrino pair emission and absorption on the same footing it became useful in Sect. 4.3.1 to deﬁne the quantity Sµνwhich was the nuclear part of the squared matrix element of nucleon-nucleon collisions, integrated over the nucleon phase space. The structure func- tion thus obtained embodied the medium properties relevant for diﬀer- ent processes without involving the radiation phase space. Clearly, this method is not limited to the bremsstrahlung process. For example, one could include the interaction of nucleons with thermal pions or a pion condensate, three-nucleon collisions, and so forth. Whatever the de- tails of the medium physics, in the end one will arrive at some function Sµν(ω,k) of the energy and momentum transfer which embodies all of its properties. High-density properties of the medium should also ap- pear in the structure function so that multiple-scattering modiﬁcations can be consistently applied to all relevant processes such as neutrino scattering, pair emission, axion emission, and others. A formal deﬁnition of the structure function without reference to speciﬁc processes begins with a neutral-current interaction Hamiltonian Hint= (gVVµ+gAAµ)Jµ, (4.37) wheregVandgAare (usually dimensionful) coupling constants. The radiation or “probe” is characterized by a current Jµwhich for axions is∂µϕ, for neutrino interactions ψνγµ(1−γ5)ψν, and for photons the electromagnetic vector potential. The medium is represented by the vector and axial-vector currents VµandAµ. If the probe couples only to one species Nof nucleons, Vµ=ψNγµψNandAµ=ψNγµγ5ψN. Next, one imagines that the medium properties are experimentally investigated with a neutrino beam with ﬁxed momentum k1which is directed at a bulk sample of the medium, and the distribution of ﬁnal- state momenta and energies are measured. The transition probability W(k1,k2) is proportional to ( g2 VSµν V+g2 ASµν A+gVgASµν V A)NµνwhereNµν was deﬁned in Eq. (4.17). A standard perturbative expansion (Sect. 9.3) yields for the dynamical structure functions Sµν V(ω,k) =1 nB∫+∞ −∞dteiωt⟨ Vµ(t,k)Vν(0,−k)⟩ , (4.38) and analogous expressions for Sµν Ain terms of ⟨AµAν⟩and forSµν V A involving ⟨VµAν+AµVν⟩. The expectation values are to be taken with respect to a thermal ensemble of medium states. Processes in a Nuclear Medium 137 In addition, one frequently uses the static structure functions which are functions of the momentum transfer alone. For example, Sµν V(k) =∫+∞ −∞dω 2πSµν V(ω,k) =1 nB⟨ Vµ(k)Vν(−k)⟩ . (4.39) It was used that∫+∞ −∞dωeiωt= 2πδ(t) so that∫dtin Eq. (4.38) is trivially done and yields the operators at equal times. In Eq. (4.39) V(k) isV(t,k) at an arbitrary time, for example t= 0. It only matters that bothV(k) andV(−k) are taken at equal times. In an isotropic medium the tensorial composition of the dynamical structure function can be obtained only from the energy-momentum transferKand the four-velocity Uof the medium; U= (1,0,0,0) in its rest frame. The general form of the vector term is (Kirzhnits, Losyakov, and Chechin 1990) Sµν V=S1,VUµUν+S2,V(UµUν−gµν) +S3,VKµKν+S4,V(KµUν+UµKν). (4.40) An analogous expression pertains to Sµν Awhile the mixed term is Sµν V A=iSV AϵµναβUαKβ (4.41) because of its transformation properties under parity. The functions Sℓ,VandSℓ,A(ℓ= 1,..., 4), andSV Adepend on medium properties and on the Lorentz scalars K2andU·Kthat can be constructed from UandK; the third possibility U2= 1 is a constant. Instead of K2and U·Kone may use the energy and momentum transfer ωandk=|k| measured in the medium rest frame. The structure functions are deﬁned for both positive and negative energy transfers because the medium can both give or take energy from a probe. Taking axion emission and absorption as an example, the rate of change of the occupation number of an axion ﬁeld mode kis given by ∂tfk=(CA 2fa)2nB 2ω[ (fk+ 1)Sµν A(−ω,k)−fkSµν A(ω,k)] KµKν. (4.42) If axions are trapped and reach thermal equilibrium, ∂tfk= 0 andfk= (eω/T−1)−1, a Bose-Einstein distribution. This implies the detailed- balance condition Sµν A(ω,k) =Sµν A(−ω,k)eω/T. (4.43) Recall that a positive energy transfer is energy given to the medium. 138 Chapter 4 4.6.2 Nonrelativistic Limit In a nuclear medium one is interested primarily in the nonrelativistic limit. The vector current is then dominated by V0=ρ=χ†χwhereχ is a nucleon two-spinor. Here, ρis the operator for the nucleon number density. The spatial component Vis suppressed by a nonrelativistic velocity factor v. The reverse applies to the axial-vector current where A0is suppressed; it is dominated by the spin density s=1 2χ†τχwhere τis a vector of Pauli matrices. In Eqs. (4.40) and (4.41) all terms arise from correlators such as ⟨A0Vi⟩which are suppressed by v2, except for the ﬁrst term of Sµν V which arises from ⟨V0V0⟩, and the second term of Sµν Awhich arises from⟨AiAi⟩. Therefore, the only unsuppressed components are S00 V(ω,k) =Sρ(ω,k) andSij A(ω,k) =Sσ(ω,k)δij, (4.44) where the density and spin-density dynamical structure functions are Sρ(ω,k) =1 nB∫+∞ −∞dteiωt⟨ ρ(t,k)ρ(0,−k)⟩ , Sσ(ω,k) =4 3nB∫+∞ −∞dteiωt⟨ s(t,k)·s(0,−k)⟩ (4.45) (Iwamoto and Pethick 1982). In order to determine the overall normalization in the nonrelativis- tic limit consider the static structure function for an ensemble of NB nucleons enclosed in a large volume V. At a given time the system is characterized by a wavefunction Ψ which depends on the locations ri of the nucleons. Then ρ(r)|Ψ⟩=∑NB i=1δ(r−ri)|Ψ⟩while the Fourier- transformed operator V−1∫d3reik·rρ(r) is ρ(k)|Ψ⟩=1 VNB∑ i=1eik·ri|Ψ⟩. (4.46) Therefore, a given conﬁguration of nucleons yields ⟨Ψ|ρ(k)ρ(−k)|Ψ⟩=1 VNB∑ i,j=1eik·rij=NB V+1 VNB∑ i;j=1 i̸=jeik·rij,(4.47) where rij≡ri−rj. When averaged over a thermal ensemble the second term will disappear if there are no spatial correlations, and nB=NB/V so thatSρ(k) = 1. Processes in a Nuclear Medium 139 For the static spin-density structure function the same steps can be performed with the inclusion of the spin operators1 2σifor the individual nucleons. This takes us to the equivalent of Eq. (4.47) ⟨Ψ|s(k)·s(−k)|Ψ⟩=1 VNB∑ i,j=1ek·rij1 4⟨Ψ|σi·σj|Ψ⟩= =1 VNB∑ i=11 4⟨Ψ|σ2 i|Ψ⟩+1 VNB∑ i;j=1 i̸=jeik·rij1 4⟨Ψ|σi·σj|Ψ⟩. (4.48) Noting that ⟨(1 2σi)2⟩=1 2(1 +1 2) =3 4the ﬁrst term is3 4NB/V=3 4nB. Therefore, in the absence of correlations one ﬁnds Sσ(k) = 1. Even in a noninteracting medium there exist anticorrelations be- tween degenerate nucleons. Standard manipulations yield in this case (e.g. Sawyer 1989) Sρ,σ(k) =1 nB∫2d3p (2π)3fp(1−fp+k), (4.49) with the Fermi-Dirac occupation number fpfor the nucleon mode p. For small temperatures this result can be expanded to yield Sρ,σ(k) = 3k/2pF+ 3Tm N/p2 F. 4.6.3 The f-Sum Rule The structure functions have a number of general properties, indepen- dently of details of the interactions of the medium constituents. We have already seen that they must obey the normalization condition ∫+∞ −∞dω 2πSρ(ω,k) = 1 +1 nB⟨NB∑ i;j=1 i̸=jcos(k·rij)⟩ , ∫+∞ −∞dω 2πSσ(ω,k) = 1 +4 3nB⟨NB∑ i;j=1 i̸=jσi·σjcos(k·rij)⟩ .(4.50) This can be referred to as a “sum rule” because the strength of Sρ,σis “summed” (integrated) over all frequencies ω. Usually we will assume that the correlation expressions on the r.h.s. are negligible. A more nontrivial sum rule obtains when a factor ωis included under the integral. The deﬁnition of the density structure function 140 Chapter 4 then yields ∫+∞ −∞dω 2πωSρ(ω,k) =∫+∞ −∞dω 2πω∫+∞ −∞dteiωt⟨ρ(t,k)ρ(0,−k)⟩ (4.51) and a similar expression for Sσ. Under the integral, a partial integration with suitable boundary conditions allows one to absorb the ωfactor, at the expense of ρ(t,k)→˙ρ(t,k). Because Heisenberg’s equation of motion informs us that i˙ρ= [ρ,H] withHthe complete Hamiltonian of the system one ﬁnds (Sigl 1995b) ∫+∞ −∞dω 2πωSρ(ω,k) =1 nB⟨[ ρ(k),H] ρ(−k)⟩ , ∫+∞ −∞dω 2πωSσ(ω,k) =4 3nB⟨[ σ(k),H] ·σ(−k)⟩ . (4.52) Here it was used, again, that∫dωeiωt= 2πδ(t) andρ(k)≡ρ(0,k) and σ(k)≡σ(0,k). In order to evaluate this sum rule more explicitly one must assume a speciﬁc form for the interaction Hamiltonian. In the simplest case of a medium consisting of only one species of nucleons one may assume thatHconsists of the kinetic energy for each nucleon, plus a general nonrelativistic interaction potential between all nucleon pairs which depends on the relative distance and the nucleon spins, i.e. H=NB∑ i=1p2 i 2mN+1 2NB∑ i;j=1 i̸=jV(rij,σi,σj), (4.53) where again rij≡ri−rj. One can then proceed to evaluate the commu- tators in Eq. (4.52). By virtue of the continuity equation for the particle number one can then show (Pines and Nozi` eres 1966; Sigl 1995b) ∫+∞ −∞dω 2πωSρ(ω,k) =k2 2mN, ∫+∞ −∞dω 2πωSσ(ω,k) =k2 2mN +4 3nB⟨NB∑ i;j=1 i̸=j[ σ(k),V(rij,σi,σj)] ·σ(−k)⟩ .(4.54) For the density structure function this exact relationship is known as the f-sum rule. Processes in a Nuclear Medium 141 For the spin-density structure function one can go one step further by expressing the most general nonrelativistic interaction potential as (Sigl 1995b) V(rij,σi,σj) =U0(rij)−US(rij)σi·σj −UT(rij)[ 3(σi·ˆrij)(σj·ˆrij)−σi·σj] , (4.55) whereU0is a spin-independent potential of the interparticle distance rij=|rij|,USis the scalar, and UTthe tensor part of the spin-dependent potential. Of these terms, only the tensor part does not conserve the total spin in nucleon-nucleon collisions and thus is the only part con- tributing to spin ﬂuctuations. With VS ij≡US(rij)σi·σjandVT ij≡ UT(rij) [3(σi·ˆrij)(σj·ˆrij)−σi·σj] one ﬁnds ∫+∞ −∞dω 2πωSσ(ω,k) =k2 2mN +4 3nB⟨NB∑ i;j=1 i̸=j[ 1−cos(k·rij)] VS ij+[ 1 +1 2cos(k·rij)] VT ij⟩ .(4.56) The f-sum of the spin-density structure function is thus closely related to the average spin-spin interaction energy in the medium. In order for the eigenvalues of the Hamiltonian to be bounded from below one must require that the potentials U0,S,T(r) are not more sin- gular than 1 /r2. In this case the r.h.s. of the spin-density f-sum rule exists as a nondivergent expression. In our representation Sσ(|ω|) = (Γσ/ω2)s(ω/T), the necessary existence of the l.h.s. of Eq. (4.56) im- plies thats(x) must be a decreasing function of xfor largex. This is not the case for the s(x) derived from the OPE potential, indicating that this interaction model is pathological in the sense that it is too singular. Indeed, it corresponds to a dipole potential and thus varies as 1/r3. Real nucleon-nucleon interaction potentials have a repulsive core and thus do not exhibit this pathology. 4.6.4 Long-Wavelength Properties In calculations involving the emission or scattering of neutrinos or ax- ions as in Sect. 4.3.1 one usually neglects the momentum transfer k because the medium constituents are so heavy that recoil eﬀects are 142 Chapter 4 small. In this “long-wavelength limit” one is only interested in the small- kstructure functions Sρ,σ(ω)≡lim k→0Sρ,σ(ω,k). (4.57) It should be stressed that this quantity is not identical with Sρ,σ(ω,0). For example, in Eq. (4.47) for k= 0 the interference term does not average to zero. The structure function becomes N2 B/Vand thus co- herently enhanced because the momentum transfer is so small that a target consisting of many particles in a volume Vcannot be resolved. The limit k→0 is understood such that |k|−1remains much smaller than the geometrical dimension V1/3of the system. In the long-wavelength limit the normalization Eq. (4.50) and the f-sum rule Eq. (4.56) yield for the spin-density structure function ∫+∞ −∞dω 2πSσ(ω) =∫∞ 0dω 2π(1 +e−ω/T)Sσ(ω) = 1 +4 3nB⟨NB∑ i;j=1 i̸=jσi·σj⟩ , ∫+∞ −∞dω 2πωSσ(ω) =∫∞ 0dω 2πω(1−e−ω/T)Sσ(ω) =4 3nB⟨NB∑ i;j=1 i̸=j3 2VT ij⟩ . (4.58) These relations will be of great use to develop a general understanding of the behavior of Sσ(ω) at high densities. The second column of expres- sions follows from the ﬁrst by detailed balance. Because Sσ(ω)≥0 it is evident that all of these expressions are always positive, independently of details of the medium interactions. In a noninteracting medium the operators ρ(t,k) and s(t,k) are constant so that Sρ,σ(ω) = 2πδ(ω), allowing for scattering (zero energy transfer), but not for the emission of radiation. This behavior is familiar from a gas of free particles which can serve as targets for collisons, but which cannot emit radiation because of energy-momentum constraints. In an interacting medium the density correlator retains this prop- erty because in the long-wavelength limit it depends on ρ(t,k→0) = V−1∫d3rρ(t,r) which remains constant. Therefore, even in an inter- acting medium one expects Sρ(ω) = 2πδ(ω), in agreement with the ﬁnding that the neutrino vector current does not contribute to brems- strahlung in the nonrelativistic limit relative to the axial-vector current (Friman and Maxwell 1979). The relevant quantity for the latter is V−1∫d3r s(t,r) =1 2∑NB i=1σi withσithe individual nucleon spins. If the evolution of diﬀerent spins Processes in a Nuclear Medium 143 is uncorrelated one may ignore the cross terms in the correlator. With the single-nucleon spin operator σone ﬁnds then Sσ(ω) =1 3∫+∞ −∞dteiωt⟨ σ(t)·σ(0)⟩ . (4.59) With Eq. (4.21) one obtains d˙Ia dω=(CN 2fa)2ω4 12π2∫+∞ −∞dte−iωt⟨ σ(t)·σ(0)⟩ (4.60) for the diﬀerential axion energy-loss rate (radiation power) per nucleon. Because of collisions with other nucleons, and because of a spin depen- dent interaction potential caused by pion exchange, the nucleon spins evolve nontrivially so that the correlator has nonvanishing power at ω̸= 0, allowing for axion emission. 4.6.5 Axion Emission in the Classical Limit The correlator representation Eq. (4.60) of the axion emission rate is extremely useful to develop a general understanding of its main proper- ties without embarking on a quantum-mechanical calculation. To this end the nucleon spin σis approximated by a classical variable, basically a little magnet which jiggles around under the impact of collisions with other nucleons. With an ergodic hypothesis about the spin trajectory on the unit sphere one may replace the ensemble average in Eq. (4.60) by a time average. The radiation intensity (time-integrated radiation power) emitted during a long (inﬁnite) time interval is then dIa dω=(CN 2fa)2ω2 12π2∫+∞ −∞dteiωt˙σ(t)2 , (4.61) where two powers of ωwere absorbed by a partial integration with suitable boundary conditions at t=±∞. In this form the energy-loss rate is closely related to a well-known expression for the electromagnetic radiation power from a charged par- ticle which moves on a trajectory r(t). The nonrelativistic limit of a standard result (Jackson 1975) is dIγ dω=2α 3π∫+∞ −∞dteiωta(t)2 , (4.62) where a(t) =¨r(t) is the particle’s acceleration on its trajectory. 144 Chapter 4 The same result can be found with the above methods applied to the spatial part of the vector current. Easier still, it can be obtained directly from Eq. (4.61). To this end note that photon emission by an electron involves nonrelativistically ( e/m e)p=evso that we must substitute ˙σ→˙v=a. The role of k(axions) is played by the polarization vector ϵ(photons) so that k2=ω2must be replaced by ϵ2= 1. With (CA/2fa)→e, usingα=e2/4π, and inserting a factor of 2 for two photon polarization states completes the translation. In order to understand the radiation spectrum consider a single “inﬁnitely hard” collision with ˙σ(t) = ∆ σδ(t). The radiation power is dIa dω=(CN 2fa)2ω2 12π2|∆σ|2. (4.63) For photons one obtains the familiar ﬂat bremsstrahlung spectrum dIγ/dω = (2α/3π)|∆v|2which is hardened, for axions, by the addi- tional factor ω2from their derivative coupling. In the form Eq. (4.63) the total amount of energy radiated in a sin- gle collision is inﬁnite. In practice, collisions are not arbitrarily hard, and the backreaction of the radiation process on the emitter must be included. This is not rigorously possible in a classical calculation, it re- quires a quantum-mechanical treatment. Therefore, a classical analysis is useful only for the soft part of the spectrum where backreactions can be ignored, i.e. for radiation frequencies far below the kinetic energy of the emitter. In a thermal environment, a classical treatment then appears reasonable for ω∼<T. Next, consider a large random sequence of nhard collisions with a spin trajectory ˙σ(t) =n∑ i=1∆σiδ(t−ti). (4.64) This yields the average radiation intensity per collision of d˙Ia dω=(CN 2fa)2ω2 12π2Γcoll⟨(∆σ)2⟩F(ω), (4.65) where Γ collis the average collision rate and ⟨(∆σ)2⟩is the average squared change of the spin in a collision. Further, F(ω) = 1 + limn→∞1 nn∑ i;j=1 i̸=j∆σi·∆σjcos[ω(ti−tj)] ⟨(∆σ)2⟩, (4.66) where the ﬁrst term (the “diagonal” part of the double sum) gives the total radiation power as an incoherent sum of individual collisions. The Processes in a Nuclear Medium 145 second term takes account of interference eﬀects between the radiation emitted in diﬀerent collisions. The interference term yields a suppression of the incoherent sum- mation because the ∆ σin subsequent collisions are anticorrelated. The spin is constrained to move on the unit sphere whence a kick in one direction is more likely than average followed by one in the opposite direction. A similar argument pertains to the velocity; a ∆ vin one di- rection is more likely than average to be followed by one in the opposite direction. The radiation spectrum with ω∼>Γcollwill remain unaﬀected while forω∼<Γcollit is suppressed. The low- ωsuppression of brems- strahlung is known as the Landau-Pomeranchuk-Migdal eﬀect (Landau and Pomeranchuk 1953a,b; Migdal 1956; Knoll and Voskresensky 1995 and references therein). The summation in Eq. (4.66) can be viewed as an integration over the relative time coordinate ∆ t=ti−tjwith a certain distribution functionf(∆t). For a random sequence of “kicks” one expects an ex- ponential distribution of the normalized form f(∆t) =1 4Γσe−Γ∆t/2 where Γ σis some inverse time-scale. This implies the Lorentzian shape (e.g. Knoll and Voskresensky 1995) F(ω) =ω2 ω2+ Γ2 σ/4. (4.67) A Lorentzian model is familiar, for example, from the collisional broad- ening of spectral lines. A comparison with Eq. (4.60) indicates that Γcoll⟨(∆σ)2⟩ ω2+ Γ2 σ/4=∫+∞ −∞dteiωt⟨ σ(t)·σ(0)⟩ . (4.68) An integral over dω/2πreveals a normalization ⟨σ2⟩so that Γσ=⟨(∆σ)2⟩ ⟨σ2⟩Γcoll. (4.69) Therefore, Γ σis identiﬁed with a collisional spin-ﬂuctuation rate. For nucleons interacting by an OPE potential one may estimate Γ σ without much eﬀort. The NN cross section is dimensionally α2 π/m2 N. A typical thermal nucleon velocity is v= (3T/m N)1/2yielding for a typical collision rate Γ coll=⟨vσNN⟩nB≈α2 πT1/2m−5/2 NnB. Because |∆σ| ≈ 1 in a collision, Γ σ≈Γcoll. This estimate agrees with the detailed result of Eq. (4.7) apart from a numerical factor 4√π. 146 Chapter 4 4.6.6 Classical vs. Quantum Result In order to make contact with the quantum calculation that led to the axion emission rate in Eq. (4.6) it is useful to juxtapose it with the classical result in terms of the spin-density structure function. The classical calculation led to Sσ(ω) =Γσ ω2×ω2 ω2+ Γ2 σ/4(4.70) while the quantum result is Sσ(ω) =Γσ ω2×s(ω/T)×{1 forω>0, eω/Tforω<0.(4.71) The function s(x) has the property s(0) = 1, it is even, and according to the f-sum rule must decrease for large x. In Fig. 4.7 the classical and quantum results are shown as dotted and dashed lines, respectively, where for the purpose of illustration s(x) = (1 +x2/4)−1/4has been assumed. Fig. 4.7. Classical, quantum, and compound spin-density structure func- tion in the nondegenerate limit according to Eq. (4.70) and Eq. (4.71) with Γσ=T= 0:2 and s(x) = (1 + x2=4)−1/4. This comparison highlights an important weakness of the classi- cal result: it does not obey the detailed-balance requirement Sσ(ω) = Sσ(−ω)eω/T. The classical correlators are invariant under time reﬂec- tiont→ −tand thus symmetric under ω→ −ω. Again, the classical result is adequate only for |ω|∼<T. Processes in a Nuclear Medium 147 Equation (4.71) also highlights an important weakness of the quan- tum result: It does not include the interference eﬀect from multiple collisions at |ω|∼<Γσbecause the calculation was done assuming indi- vidual, isolated collisions. The normalization condition Eq. (4.58) can then be satisﬁed only by accepting a pathological infrared behavior of Sσ(ω) like the one suggested by Sawyer (1995). The classical and the quantum results thus both violate fundamental requirements. The quantum calculation applies for |ω|∼>Γσwhile the classical one for |ω|∼<T. If Γ σ≪T(“dilute medium”) the regimes of validity overlap for Γ σ∼<|ω|∼<Tand the results agree beautifully. In this case the two calculations mutually conﬁrm and complement each other. The compound structure function, where the quantum result is multiplied with ω2/(ω2+Γ2 σ/4), fulﬁlls the detailed-balance requirement and approximately the normalization condition. Therefore, it probably is a good ﬁrst guess for the overall shape of Sσ(ω). 4.6.7 High-Density Behavior In a SN core one is typically in the limit Γ σ≫Tso that the regime of overlapping validity Γ σ∼<|ω|∼<Tbetween a classical and a per- turbative quantum calculation no longer exists. Rather, one is in the opposite situation where for T∼<|ω|∼<Γσneither approach appears directly justiﬁed. Because the structure function Sσ(ω) determines all axial-vector interaction rates in the long-wavelength limit, it is rather unclear what their high-density behavior might be. Still, one has important general information about Sσ(ω). The de- tailed-balance condition Sσ(ω) =Sσ(−ω)eω/Treveals that it is enough to specify Sσ(ω) for positive energy transfers (energy given to the medium). In the classical limit Sσ(|ω|) = Γ σ/ω2, while the f-sum rule informs us that the quantum version must fall oﬀ somewhat faster with large|ω|. Finally, if spin-spin correlations can be neglected, the nor- malization condition∫∞ 0dω(1 +e−ω/T)Sσ(ω) = 2πobtains. In order to illustrate the overall impact of the high-density behavior on axion or neutrino pair emission rates and on neutrino scattering rates, it is enough to take the classical limiting case for large |ω|, even though it does not have an integrable f-sum. Thus a simple ansatz is Sσ(|ω|) =Γσ ω2+ Γ2/4, (4.72) where Γ is to be determined by the normalization condition. In the dilute limit (Γ σ≪T) this implies Γ ≈Γσwhile in the dense limit (Γσ≫T) one ﬁnds Γ ≈Γσ/2. 148 Chapter 4 For axion bremsstrahlung, the energy-loss rate of the medium is given byϵa=Qa/ρ∝∫∞ 0dωω4Sσ(ω) so that with Eq. (4.72) one needs to evaluate ϵa∝Γσ∫∞ 0dωω4e−ω/T ω2+ Γ2/4(4.73) with Γ determined from the normalization condition. In the dilute limit (Γ ≈Γσ≪T) one may ignore Γ in the denominator, so that ϵa∝Γσ. This is indeed what one expects from a bremsstrahlung process for which the volume energy-loss rate is proportional to the density squared, and thus ϵaproportional to the density which appears in the spin-ﬂuctuation rate Γ σ. In the high-density limit (Γ ≈Γσ/2≫T) the denominator in Eq. (4.73) is dominated by Γ because the exponential factor suppresses the integrand for ω≫Tso that one expects ϵato be a decreasing function of Γ σ. In Fig. 4.8 the variation of ϵawith Γ σ/Tis shown (solid line), taking Eq. (4.72) for the spin-density structure function. The dashed line shows the “naive rate,” based on Γ σ/ω2which ignores multiple-scattering eﬀects, and which violates the normalization con- dition. Fig. 4.8 illustrates that even very basic and global properties ofSσ(ω) reveal an important modiﬁcation of the axion emission rate at high density: they saturate with an increasing spin-ﬂuctuation rate, Fig. 4.8. Schematic variation of the axion emission rate per nucleon with Γσ=T, taking Eq. (4.72) for the spin-density structure function. The dashed line is the naive rate without the inclusion of multiple-scattering eﬀects, i.e. it is based on Sσ(|!|) = Γ σ=!2. Processes in a Nuclear Medium 149 and may even decrease at large densities, although such large values for Γσmay never be reached in a nuclear medium as will become clear be- low. The high-density downturn of the axion emission rate can be in- terpreted in terms of the Landau-Pomeranchuk-Migdal eﬀect (Landau and Pomeranchuk 1953a,b; Feinberg and Pomeranchuk 1956; Migdal 1956) as pointed out by Raﬀelt and Seckel (1991). The main idea is that collisions interrupt the radiation process. The formation of a radi- ation quantum of frequency ωtakes about a time ω−1according to the uncertainty principle and so if collisions are more frequent than this time, the radiation process is suppressed. Classically, this eﬀect was demonstrated in the language of current correlators in Sect. 4.6.5. The impact of the high-density behavior of Sσ(ω) on the neutral- current neutrino opacity is crudely estimated by the inverse mean free path given in Eq. (4.35), averaged over a thermal energy spectrum of the initial neutrino. Moreover, all expressions become much simpler if one replaces the Fermi-Dirac occupation numbers with the Maxwell- Boltzmann expression e−ωi/T; the resulting error is small for nondegen- erate neutrinos. The relevant quantity is then ⟨ λ−1⟩ ∝∫∞ 0dω1∫∞ 0dω2ω2 1ω2 2e−ω1/TSσ(ω1−ω2). (4.74) One integral can be done explicitly, leaving one with an integral over the energy transfer alone. With Eq. (4.72) for the structure function one ﬁnds ⟨ λ−1⟩ ∝∫∞ 0dωΓσ(T2+Tω/2 +ω2/12)e−ω/T ω2+ Γ2/4. (4.75) This expression is constant for Γ σ≪Twhere Γ = Γ σand thus the structure function is essentially 2 πδ(ω). Indeed, the average scattering cross section (or mean free path) is not expected to depend on the density. For dense media (Γ σ≫T), however, the broadening of Sσ(ω) be- yond a delta function leads to a decreasing average scattering rate. This means that at a ﬁxed temperature the medium becomes more trans- parent to neutrinos with increasing density, even without the impact of degeneracy eﬀects. This behavior is shown in Fig. 4.9 in analogy to the axion emission rate Fig. 4.8. A decreasing cross section is intuitively understood if one recalls that the nucleon spin is typically ﬂipped in a collision with other nucle- ons because the interaction potential couples to the spin. Neutrino scat- tering with an energy transfer ωimplies that properties of the medium 150 Chapter 4 Fig. 4.9. Schematic variation of the neutrino scattering rate per nucleon (axial-vector interaction only) with Γ σ=T, taking Eq. (4.72) for the spin- density structure function. The dashed line is the naive rate without the inclusion of multiple-scattering eﬀects, i.e. it is based on Sσ(!) = 2 (!). that ﬂuctuate on faster time scales cannot be resolved. Therefore, if a nucleon spin ﬂips many times within the time scale ω−1, the probe “sees” a vanishing average spin and the scattering rate is reduced ac- cordingly. For the vector current, the relevant zeroth component does not ﬂuc- tuate so that this contribution to neutrino scattering is not suppressed (Sect. 4.6.4). This observation is intimately tied to the absence of bremsstrahlung emission of neutrino pairs by the vector current. Both of these eﬀects are summarized in the statement that the vector-current structure function Sρ(ω) is always given as232πδ(ω), allowing for scat- tering (energy transfer ω= 0), but not for bremsstrahlung. 23This statement is based on the assumption that there are no spatial correlations among the nucleons—possibly a poor approximation in a dense medium. Moreover, collective oscillations may occur so that it can be too simplistic to treat the medium as consisting of essentially free, individual nucleons (Iwamoto and Pethick 1982). Calculations of the long-wavelength structure factor by Sawyer (1988, 1989) in a speciﬁc nucleon interaction model showed a substantial suppression of S(0) and thus, of the neutrino mfp. Other related works are those of Haensel and Jerzak (1987) and of Horowitz and Wehrberger (1991a,b) who calculated the dynamic structure functions within certain interaction models. Unfortunately, these works do not shed much light on the high-density behavior of the quantities which are of prime interest to this book such as the axion emission rate from a hot SN core. Processes in a Nuclear Medium 151 The axion emission and the neutrino scattering rates begin to be suppressed at high density if Γ σexceeds a few T. The numerical value Eq. (4.34) that was derived in a simple OPE calculation indicates that such large Γ σmay be expected in a nuclear medium. Of course, one may well ask if a perturbative calculation of the spin-ﬂuctuation rate is adequate if such a calculation fails for the neutrino scattering rate. Nu- cleon spins ﬂuctuate because of their spin-dependent interaction with other nucleons and thus are themselves subject to spin averaging eﬀects. The question of the true value of Γ σin a nuclear medium (as opposed to the OPE-calculated one) can be addressed empirically by virtue of the SN 1987A neutrino signal and theoretically by the f-sum rule. Because the axial-vector current contribution to the standard neu- tral-current scattering rate dominates (the cross section is proportional toC2 V+ 3C2 A), it would be much easier for neutrinos to diﬀuse out of the hot SN core if the axial-current scattering rate were signiﬁcantly suppressed. The observed SN 1987A neutrino signal would have been much shorter than predicted in standard cooling calculations. The impact of reduced neutrino opacities on the SN 1987A signal has been studied by Keil, Janka, and Raﬀelt (1995); see Sect. 13.6. The main conclusion is that a suppression of the axial-vector current opacity by more than about a factor of 2 is not compatible with the SN 1987A signal duration. This result would indicate that the eﬀective Γ σnever becomes much larger than a few T. Theoretically, the f-sum rule Eq. (4.58) allows one to relate Γ σto the average spin-spin interaction energy in the medium. According to Sigl’s (1995b) estimate one concludes, again, that Γ σdoes not exceed a fewTin a nuclear medium. These results imply that even in a dense medium the axial-vector scattering rate is not suppressed as strongly as one may have expected on the basis of a naive estimate of Γ σ. It still remains impossible to cal- culate its exact magnitude from ﬁrst principles. Therefore, the neutral- current neutrino opacity remains an adjustable function of density for practical SN cooling calculations much as the equation of state. 4.7 Eﬀective Nucleon Mass and Coupling In a dense medium it is not necessarily possible to use the vacuum masses and coupling constants to determine interaction rates. The axial-vector neutrino couplings are probably suppressed somewhat (Ap- pendix B). For the pion-nucleon coupling, Turner, Kang, and Steigman 152 Chapter 4 (1989) argued on the basis of the nonlinear sigma model that the combi- nation of parameters α2 παa/m2 Nshould remain approximately constant. Mayle et al. (1989) similarly found that this parameter should remain somewhere in the range 0 .3−1.5 of its vacuum value. The nucleon eﬀective mass m∗ Ndeviates substantially from the vac- uum valuemN= 939 MeV. This shift has an impact on the kinematics of reactions and the nucleon phase-space distribution. A calculation of m∗ Nhas to rely on an eﬀective theory which describes the interaction of nucleons and mesons. A typical result from a self-consistent rela- tivistic Brueckner calculation including vacuum ﬂuctuations is shown in Fig. 4.10. For conditions relevant for a SN core, an eﬀective value as low asm∗ N/mN= 0.5 is conceivable. Fig. 4.10. Eﬀective nucleon mass in a nuclear medium according to a self- consistent relativistic Brueckner calculation (Horowitz and Serot 1987). Nu- clear density corresponds to about 0= 3×1014g=cm3. 4.8 The URCA Processes Neutral-current weak processes are of prime interest for the topics stud- ied in this book because they are closely related to “exotic” reactions involving new particles such as axions. For completeness, however, it must be mentioned that the charged-current reactions depicted in Fig. 4.11 dominate the neutrino energy-loss rate of old neutron stars, and also dominate the νeopacity in young SN cores. The processes n→peνe(neutron decay) and ep→nνeare usually called the URCA Processes in a Nuclear Medium 153 reactions after a casino in Rio de Janeiro. It must have been as easy to lose money there as it is to lose energy in reactions which produce a neutrino whether the electron is in the initial or ﬁnal state. Fig. 4.11. The URCA processes. For the modiﬁed versions, there are obvious other graphs with the leptons attached to other nucleon lines, and exchange amplitudes. However, in a neutron star the direct URCA processes can be highly suppressed by energy-momentum conservation. Because all participat- ing degenerate fermions are close to their Fermi surface one must re- quirepF,p+pF,e>pF,n. Because the proton and electron concentration is small in a neutron star, it was thought that this “triangle condition” could not be satisﬁed. However, the equilibrium proton concentration depends on details of the equation of state—even in a naive model with free fermions it is a sensitive function of the eﬀective nucleon mass (Ap- pendix D). Therefore, it is conceivable that the direct URCA processes actually do take place in neutron stars (Boguta 1981; Lattimer et al. 1991), in which case they provide a cooling mechanism much faster than all other proposed possibilities. If the triangle condition is fulﬁlled, the energy-loss rate is (Lattimer et al. 1991) QURCA =457π 10080G2 Fcos2θC(1 + 3C2 A)m2 NpF,eT6, (4.76) whereθC≈0.24 is the Cabbibo angle and CAis the charged-current axial-vector constant which is −1.26 in vacuum while in nuclear matter it is suppressed somewhat (Appendix B). 154 Chapter 4 If the triangle condition is not satisﬁed the URCA processes re- quire bystander particles to absorb momentum, leading to the “modi- ﬁed URCA process” shown in Fig. 4.11 (Chiu and Salpeter 1964). Of course, as below in Sect. 4.9.1, the missing momentum can be provided by pions or a pion condensate (Bahcall and Wolf 1965a,b). An explicit result for the modiﬁed URCA rate in the OPE model for the nucleon interactions is (Friman and Maxwell 1979) Qmod.URCA =11513π 120960α2 πG2 Fcos2θCC2 ApF,eT8, (4.77) ignoring factors of order unity to account for a nonzero mπand nucleon correlations. As in the ννprocesses, only the axial-vector coupling contributes. Some details of the phase-space integration can be found in Shapiro and Teukolsky (1983).24 The modiﬁed URCA reactions (Fig. 4.11) are closely related to νν bremsstrahlung and the inelastic scattering process νNN→NNν dis- cussed earlier. It is interesting that the rate for the epn→nnν epro- cess does not diverge, in contrast with νNN→NNν where multiple- scattering eﬀects had to be invoked to obtain a sensible result. The divergence was due to the intermediate nucleon going on-shell for a vanishing energy transfer. In epn→nnν ethe electron has the energy EF,e, the neutrino T, and because EF,e≫T(degenerate electrons!), the minimum energy transfer to the leptons is EF,eand thus never zero. Of course, this is the reason why the modiﬁed URCA reaction was in- voked in the ﬁrst place: if the intermediate nucleon line is cut, the two sub-processes are suppressed by energy-momentum constraints. The direct and modiﬁed URCA process should be expressed in terms of a common structure function applicable to charged-current processes. One may expect that at high density, spin and isospin ﬂuctuations may suppress these reactions in analogy to the neutral-current processes discussed in the previous section. However, the URCA processes have not been discussed in the literature from this particular perspective. The equivalent of the modiﬁed URCA process for quark matter was calculated by Iwamoto (1980, 1982), more recently by Goyal and Anand (1990), and numerically by Ghosh, Phatak, and Sahu (1994) who claim that Iwamoto’s phase-space approximations can lead to substantial er- rors in the emission rate. 24Shapiro and Teukolsky’s phase-space volume involves a factor p3 F;ebecause in their treatment of the nuclear matrix element there is no cancellation of p2 F;ewith 1=!2from the nucleon propagator which occurs in a bremsstrahlung calculation. Processes in a Nuclear Medium 155 4.9 Novel Phases of Nuclear Matter 4.9.1 Pion Condensate Free particles cannot radiate because of energy-momentum constraints which can be overcome by exchanging momentum with “bystander” particles (bremsstrahlung). No bystander is required if the momentum is taken up by the pion ﬁeld directly which in Fig. 4.1 only mediated the nucleon interaction (Bahcall and Wolf 1965a,b). This possibility is par- ticularly important if a pion condensate develops so that the medium is characterized by a macroscopic, classical pion ﬁeld. The pion dis- persion relation can be such that the lowest energy state involves a nonvanishing momentum kπ(Baym 1973; Kunihiro et al. 1993; Migdal et al. 1990). Nucleons can exchange pions with the condensate and thereby pick up a momentum kπwhich then allows for the radiation of axions or other particles (Fig. 4.12). Fig. 4.12. Axion emission by nucleons scattering oﬀ a pion condensate. There is another amplitude with the axion attached to N1. The pion condensate causes a periodic potential for the nucleons which thus must be described as quasi-particles or Bloch states with a main momentum component pand admixtures p±kπ. Because an eigenstate of energy is no longer an eigenstate of momentum, these quasi-particles can emit radiation without violating energy-momentum conservation. Therefore, the process of Fig. 4.12 can be equally de- scribed as a decayfN1→fN2a. The rate for this reaction was calculated by Muto, Tatsumi, and Iwamoto (1994) who found that the dominant contribution was from a π◦condensate. It forms a periodic potential Asin(kπ·r) whereAis a dimensionless amplitude which is small compared to unity for a weakly developed condensate. The Bloch states are to lowest order in A fN± p=N± p∓Aκ0 2(N± p+k Ep+k−Ep+N± p−k Ep−k−Ep) , (4.78) 156 Chapter 4 whereN± pis a plane-wave nucleon state with energy Epand spin ori- entation ±relative to kπ, andκ0is a coupling constant. In the nonrelativistic limit the axion energy-loss rate corresponding to the decayfN1→fN2ais written as Qa=∫d3k 2ω(2π)3ω∫d3p1 (2π)3∫d3p2 (2π)3f1(1−f2) ×2πδ(E1−E2−ω)∑ spins|M|2, (4.79) where kis the axion momentum, p1,2the nucleon momenta, and f1,2 their occupation numbers. The matrix element, averaged over axion emission angles (the condensate is not isotropic!) is found to be ⟨∑ spins|M|2⟩ =(C0−egAC1 2fa)2 4 3A2κ2 0 ×(2π)3[ δ3(∆p+kπ) +δ3(∆p−kπ)] , (4.80) where ∆ p=p1−p2−k. The isoscalar and isovector axion coupling constants are C0=1 2(Cp+Cn) andC1=1 2(Cp−Cn) in terms of their couplings to protons and neutrons. The isovector current carries a renormalized axial-vector coupling constant for which Muto, Tatsumi, and Iwamoto (1994) found egA≈0.43×1.26 = 0.54. The phase-space integration was carried out by neglecting the axion momentum in δ3(∆p±kπ) and taking the degenerate limit. Then, Qa=π 45(C0−egAC1 2fa)2A2κ2 0m2 NT4 |kπ|, (4.81) an emission rate with a relatively soft temperature dependence. For a numerical estimate Muto, Tatsumi, and Iwamoto (1994) found that in cold neutron-star matter near the critical density (about 2.1 times nuclear) kπ≈410 MeV, the neutron and proton Fermi momenta are 410 and 150 MeV, respectively (corresponding to Yp= 0.04), and the coupling strength to the condensate is κ0≈105 MeV. Therefore, Qa=αaA2π2 45κ2 0T4 |kπ|=αaA21.03×1044erg cm−3s−1T4 9,(4.82) whereαa≡[2mN(C0−egAC1)/2fa]2/4πandT9≡T/109K. Processes in a Nuclear Medium 157 Instead of an axion one may also emit a neutrino pair in Fig. 4.12. The corresponding energy-loss rate was worked out by Muto and Tat- sumi (1988) who found Qνν=2π 945eC2 AG2 Fm2 NA2κ2 0T6 |kπ|. (4.83) Here,eCAis the renormalized neutron neutral-current coupling constant; in vacuum CA=−1.26/2. A somewhat diﬀerent emission rate was found by Senatorov and Voskresensky (1987). A process related to that shown in Fig. 4.12 is pionic Compton scatteringπN→Nainvolving thermal pions. The equivalent of the axion emission rate was calculated by Turner (1992) and by Raﬀelt and Seckel (1995). This process could be of some importance in the hot nondegenerate medium of a SN core. However, a thermal population of pions yields a rate which is always less important than bremsstrahlung. For a charged pion or kaon condensate, the equivalent of the mod- iﬁed URCA process can occur by exchanging a meson with the con- densate rather than with a bystander nucleon. Recent calculations of such processes include Senatorov and Voskresensky (1987), Muto and Tatsumi (1988), and Thorsson et al. (1995). 4.9.2 Quark Matter It has been speculated that a “neutron” star may actually undergo a phase transition where the nucleons dissolve in favor of a quark-gluon plasma. Notably, it is possible that the true ground state of nuclear matter consists of “strange quark matter” with about equal numbers of up, down, and strange quarks. Such a system can emit axions by virtue of quark-quark bremsstrahlung (Fig. 4.13). Fig. 4.13. Axion emission by quark-quark bremsstrahlung. There is a total of eight amplitudes, four with the axion attached to each quark line, and an exchange graph each with q3↔q4. 158 Chapter 4 The gluon in Fig. 4.13 couples to the quarks with the strong ﬁne- structure constant αsand the gluon propagator involves an eﬀective mass for which Anand, Goyal, and Iha (1990) used m2 g= (6αs/π)p2 F wherepFis the Fermi momentum of the quark sea. The axion cou- pling to quarks is the usual derivative form ( Cq/2fa)ψqγµγ5ψq∂µϕwith q=u,d,s . After a cumbersome but straightforward calculation Anand et al. found for the energy-loss rate in the degenerate limit Qa=62π2α2 s 945pFT6 (2fa)2×{ C2 q(I1+ 2I2) forqq→qqa, (C2 q+C2 q′)I1forqq′→qq′a,(4.84) where the angular integrals are I1≡1 2π2∫π 0dθ∫2π 0dϕs5 θ(1 +c4 ϕ) (s2 θs2 ϕ+κ2)2, I2≡1 2π2∫π 0dθ∫2π 0dϕs5 θ (s2 θs2 ϕ+κ2)(s2 θc2 ϕ+κ2), (4.85) wheresθ≡sin(θ/2),sϕ≡sin(ϕ/2),cϕ≡cos(ϕ/2), andκ2≡m2 g/2p2 F= 3αs/2π. Numerically, I1= 12.3, 4.23, and 2.25 for αs= 0.2, 0.4, and 0.6, whileI2= 2.34, 1.30, and 0.86, respectively. The total emission rate involves three processes with equal quarks (qq=uu,dd,ss), and three with diﬀerent ones ( qq′=ud,us,ds) so thatQais the prefactor of Eq. (4.84) times ( C2 u+C2 d+C2 s)(3I1+ 2I2), to be compared with Eq. (4.10) for degenerate neutron matter. The ratio between the rates is Qqq a Qnn a=C2 u+C2 d+C2 s C2 npF,q pF,nα2 s α2 π2π(3I1+ 2I2) G(mπ/pF,n), (4.86) where the function G(u) was given in Eq. (4.12). The ratio of coupling constants is model dependent (Sect. 14.3.3), the Fermi momenta are approximately equal for equal densities. For a typical value αs= 0.4 one hasα2 s/α2 π= 0.6×10−3while the last factor is about 140 with G= 0.7 (Fig. 4.3) so that the last two factors together are about 0.08. Hence, the emission rate from quark matter is much smaller than that from neutron matter because απis so large relative toαs. In the inevitable presence of protons in a neutron star, thenpprocess will be even more important than nn(Fig. 4.4), further enhancing the emission rate of nuclear matter relative to quark matter. For neutrino pairs the ratio of the emission rates is about the same as for axions and so quark matter is much less eﬀective at ννemission Processes in a Nuclear Medium 159 than nuclear matter (Burrows 1979; Anand, Goyal, and Iha 1990). This observation has little eﬀect on the neutrino cooling rate of a quark star because the URCA processes, which are based on charged-current reactions, dominate for both nuclear or quark matter. For axions, however, it may seem that the emission rate is much suppressed relative to nuclear matter. This is certainly true if “ax- ion” stands for any generic pseudoscalar Nambu-Goldstone boson. The QCD axion, however, which was introduced to solve the CP problem of strong interactions (Chapter 14) necessarily has a two-gluon cou- pling which allows for the gluonic Primakoﬀ eﬀect (Fig. 4.14) which is analogous to the photon Primakoﬀ eﬀect discussed in Sect. 5.2. Fig. 4.14. Axion emission by the gluon Primakoﬀ eﬀect. Altherr (1991) has calculated the emissivity of a hot quark-gluon plasma and found that it was similar to that of a nuclear medium at the same temperature and density. Ellis and Salati (1990) found a much smaller emission rate because they included only the gluonic plasmon decay process gT→gL+γ, much in analogy to the corre- sponding photonic process discussed in Sect. 5.2.2. This decay process, however, is only part of the axion emissivity by the gluon ﬁeld ﬂuctu- ations. For highly degenerate quark matter corresponding to old neutron stars the emission rate has not be calculated as far as I know. 4.9.3 Bubble Phase In a hot lepton-rich neutron star the nuclear medium in the density regime1 2ρ0toρ0(nuclear matter density ρ0= 3×1014g cm−3) may form a “bubble phase” with regions of low density embedded in the medium. Leinson (1993) has estimated the bremsstrahlung rate for the neutral-current reactions N+ bubble →bubble +N+νν. Naturally, he found that only the axial-vector current contributes. Also, it is easy to translate his results into an axion emission rate. Apparently, the bubble phase can be important for early neutron-star cooling. 160 Chapter 4 4.10 Emission of Right-Handed Dirac Neutrinos So far in this chapter neutrinos were assumed to be massless parti- cles which interact only by the standard left-handed weak current. It is possible, however, that neutrinos have a Dirac mass in which case any reaction with a ﬁnal-state (anti)neutrino produces both positive and negative helicity states. Typically, the “wrong-helicity” states will emerge in a fraction ( mν/2Eν)2of all cases because of the mismatch between chirality (eigenstates of γ5) and helicity. For Eν≫mνthe wrong-helicity states correspond approximately to right-handed chiral- ity states and so their interaction-rate with the ambient medium is weaker by an approximate factor ( mν/2Eν)2. This implies that for a suﬃciently small mass they would not be trapped in a SN core and thus carry away energy in an “invisible” channel, allowing one to set constraints on a Dirac neutrino mass from the observed neutrino signal of SN 1987A (Sect. 13.8.1). Here, the relationship between the pro- duction rate of left- and right-handed neutrinos is explored in some detail because the simple scaling with ( mν/2Eν)2is not correct in all cases. When a neutrino interacts with a medium the transition probability from a state with four-momentum K1= (ω1,k1) to one with K2= (ω2,k2) is written as W(K1,K2). The function Wis deﬁned for both positive and negative energies. The emission probability for a pair ν(K1)ν(K2) is thenW(−K1,K2), the absorption probability for a pair isW(K1,−K2). The collisional rate of change of the occupation number fk1of a neutrino ﬁeld mode k1is then given by dfk1 dt coll=∫d3k2 (2π)3[ WK2,K1fk2(1−fk1)−WK1,K2fk1(1−fk2) +W−K2,K1(1−fk1)(1−fk2)−WK1,−K2fk1fk2] ,(4.87) where the variables of Wwere written as subscripts. The ﬁrst term corresponds to neutrino scatterings into the mode k1from all other modes, the second term is scattering out of mode k1into all other modes, the third term is pair production with a ﬁnal-state neutrino k1, and the fourth term is pair absorption of a neutrino of momentum k1 and an antineutrino of any momentum. fkis the occupation number for the antineutrino mode k; (1−fk) or (1−fk) represent Pauli blocking factors. This collision integral only includes (eﬀective) neutral-current processes while charged-current reactions were ignored. Processes in a Nuclear Medium 161 According to the discussion of Sect. 4.6 the transition probability is given as W(K1,K2) =G2 F 8nB 2ω12ω2SµνNµν, (4.88) whereSµνis an eﬀective structure function given in terms of the vector, axial-vector and mixed structure function of the medium deﬁned in Eq. (4.38), and nBis the baryon density. The tensor Nµνis the squared matrix element of the neutrino current; it was given in Eq. (4.17) in terms ofK1andK2. In an isotropic medium the tensorial composition of the dynamical structure function can be expressed in terms of the energy-momentum transfer Kand the four-velocity Uof the medium; U= (1,0,0,0) in its rest frame. Thus, in analogy to Eqs. (4.40) and (4.41) one may write Sµν=S1UµUν+S2(UµUν−gµν) +S3KµKν +S4(KµUν+UµKν) +iS5ϵµναβUαKβ,(4.89) where the functions Sℓ(ℓ= 1,..., 5) depend on medium properties and on the energy-momentum transfer K= (ω,k). The transition probability is then found to be (Raﬀelt and Seckel 1995) W(K1,K2) =G2 FnB 4[ (1 + cosθ)S1+ (3−cosθ)S2 −2(1−cosθ)(ω1+ω2)S5] , (4.90) whereθis the neutrino scattering angle. The terms proportional to S3 andS4vanish identically. Next, one may consider processes involving massive Dirac neutri- nos with speciﬁed helicities. Gaemers, Gandhi, and Lattimer (1989) showed that in this case the same expressions apply with Nµνas con- structed in Eq. (4.17) if one substitutes Ki→1 2(Ki±mνSi),i= 1 or 2, where the plus sign refers to ν, the minus sign to ν, andSiis the covariant spin vector. For relativistic neutrinos one may consider a noncovariant lowest-order expansion in terms of mν. In this limit Ki remains unchanged for left-handed states while Ki= (ωi,ki)→fKi= (mν/2ωi)2(ωi,−ki) (4.91) for right-handed ones. After this substitution has been performed all further eﬀects of mνare of higher order so that one may neglect mν everywhere except in the global “spin-ﬂip factor.” 162 Chapter 4 The dispersion relation of neutrinos in a SN diﬀers markedly from the vacuum form; in the core the “eﬀective mν” is several 10 keV. How- ever,mνin Eq. (4.91) is the vacuum mass which couples left-handed to right-handed states and thus leads to spin ﬂip while the medium- induced “mass” only aﬀects the dispersion relation of left-handed states. This view is supported by a detailed study of Pantaleone (1991). Of course, for nonrelativistic neutrinos the situation is more complicated because an approximate identiﬁcation of helicity with chirality is not possible. Next consider a speciﬁc process which involves a left- and a right- handed neutrino such as “spin-ﬂip scattering” νL(KL)→νR(KR). Then, one needs to construct NµνfromK1= (ωL,kL) andK2= (mν/2ωR)2(ωR,−kR). The contraction with Sµνleads to (Raﬀelt and Seckel 1995) fW(KL,KR) =G2 FnB 4(mν 2ωR)2[ (1−cosθ)S1+ (3 + cosθ)S2 + 4ω2 R(1−cosθ)S3−4ωR(1−cosθ)S4+ 2(ωR−ωL)(1 + cosθ)S5] . (4.92) Following Gaemers, Gandhi, and Lattimer (1989) it must be empha- sized that this expression diﬀers in more than the factor ( mν/2ωR)2 from the nonﬂip case Eq. (4.90). This diﬀerence is due to the changed angular momentum budget of reactions with spin-ﬂipped neutrinos. This angular-momentum diﬀerence between spin-ﬂip and no-ﬂip processes is nicely illustrated by virtue of the pion decay process π◦→ νν. If both ﬁnal-state neutrinos are left-handed, i.e. if νhas nega- tive andνpositive helicity, this decay is forbidden by angular momen- tum conservation. This is seen most easily if one recalls that it is the pion current ∂µπ◦that interacts with the left-handed neutrino current. The squared matrix element of the pion current is thus proportional toKµ π◦Kν π◦. For a pion decaying at rest only the 00-component con- tributes. Contraction with Nµνleads to identically zero if one recalls that for the ﬁnal-state neutrinos ω1=ω2andk1=−k2. For the spin- ﬂip process one must use the reversed momentum instead so that in Nµνone must use ω1=ω2andk1=k2, leading to a nonvanishing contribution. The decay of thermal pions is an important process for populating the right-handed states of massive Dirac neutrinos in the early universe (Lam and Ng 1991). Contrary to a discussion by Na- tale (1991), however, pion decays do not seem to provide a particularly strong contribution in SN cores (Raﬀelt and Seckel 1991). Processes in a Nuclear Medium 163 Right-handed neutrinos can be produced by the interaction with all medium constituents. For a SN core, the production rate from the interaction with charged leptons such as e+e−→νLνRorνLe−→e−νR was explicitly studied by P´ erez and Gandhi (1990) and by Lam and Ng (1992). Still, in a SN core the dominant contribution to the interaction between neutrinos and the medium is due to the nucleons. Therefore, one may simplify the expression Eq. (4.92) by applying the same approximations that were used earlier in Sect. 4.6 in the con- text of purely left-handed neutrinos. Notably, one may use the nonrel- ativistic and long-wavelength limits which reveal, again, that only the terms proportional to S1andS2contribute, and that these structure functions can be taken to be functions of the energy transfer ωalone.25 Moreover, the neutrino phase-space integration will always average the cosθterm to zero. Then, the spin-ﬂip reaction rate is indeed simply the nonﬂip rate times the “spin-ﬂip factor” ( mµ/2Eν)2. Right-handed neutrinos can be produced by spin-ﬂip scattering of left-handed ones νL→νR, or by the emission of pairs νLνRorνRνL. In a SN core, left-handed neutrinos are trapped while right-handed ones can freely escape whence the quantity of interest is the energy- loss rate of the medium in terms of right-handed states. The total energy-loss rate in νRis thenQR=Qscat+Qpairwhere “scat” refers to spin-ﬂip scattering and “pair” to the pair-emission process. The two contributions are Qscat=∫d3kL (2π)3d3kR (2π)3fWKL,KRfkLωR, Qpair=∫d3kL (2π)3d3kR (2π)3fW−KL,KR(1−fkL)ωR. (4.93) An analogous expression pertains to the emission of νR; if the trapped left-handed neutrinos are nondegenerate this contribution has the same magnitude as that for the emission of νR. Next, consider the limit where the transition probabilityfWcan be represented in terms of a single structure function26S(ω) =S1(ω) + 3S2(ω). Further, the possibility of neutrino degeneracy is ignored which allows one to approximate the left-handed neutrino Fermi-Dirac distri- bution by a Maxwell-Boltzmann one, ( eω/T+ 1)−1→e−ω/T. Then the 25In the notation of Sect. 4.6 and for a single species of nucleons one has in this limit S1=C2 VSandS2=C2 AS. 26In the notation of Sect. 4.6 this is S=C2 VS+ 3C2 ASin a medium consisting of a single species of nucleons. 164 Chapter 4 energy-loss rates are Qscat=G2 Fm2 νnBT4 4π4∫∞ 0dω[(ω T)2 + 6ω T+ 12] S(ω), Qpair=G2 Fm2 νnBT4 4π4∫∞ 0dω1 12(ω T)4 S(ω), (4.94) where the detailed-balance condition S(−ω) =S(ω)eω/Twas used. These rates include a factor of 2 for the emission of νRandνR. As noted by Raﬀelt and Seckel (1995), unless the functional form ofS(ω) is very bizarre one has Qscat≫Qpair. This is easily under- stood if one observes that in a scattering process the energy available for the ﬁnal-state νRis that of the initial-state νLplus energy that is contributed by the medium. For pair emission, the energy for the ﬁnal-stateνLandνRboth have to be provided by the medium. There- fore, the spin-ﬂip scattering process is favored by the neutrino phase space.27In a SN core the production of right-handed Dirac neutri- nos is given essentially by the standard scattering rate times the fac- tor (mν/2Eν)2. 27This conclusion is in contrast to a suggestion that the pair-emission rates could be relatively important in a SN core (Turner 1992). This statement was not based on a self-consistent treatment of the medium structure functions, but rather on a perturbative treatment of the processes involved. The conﬂict between Turner’s statement and our ﬁnding, which essentially was based on phase-space considerations, highlights the inadequacy of naive perturbative results in a nuclear medium. Chapter 5 Two-Photon Coupling of Low-Mass Bosons Well-known particles such as neutral pions, or hypothetical ones such as axions or gravitons, each have a two-photon interaction vertex. It allows for radiative decays of the form π◦→2γas well as for the Primakoﬀ conversion π◦↔γin the presence of external electric or magnetic ﬁelds. The Primakoﬀ conversion of photons into gravitons in putative cosmic magnetic ﬁelds could cause temperature ﬂuctuations of the cosmic microwave background radiation. In stars, the Primakoﬀ conversion of photons leads to the production of gravitons and axions, although the graviton luminosity is always negligible. The Primakoﬀ conversion of axions into photons serves as the basis for a detection scheme for galactic axions, and was used in several laboratory axion search experiments. The modiﬁed Maxwell equations in the presence of very low-mass pseudoscalars would substantially modify pulsar elec- trodynamics. These physical phenomena are explored and current lab- oratory and astrophysical limits on the axion-photon coupling are re- viewed. 5.1 Electromagnetic Coupling of Pseudoscalars The idea that symmetries of the fundamental interactions can be bro- ken by the “vacuum” or ground state of the ﬁelds plays an important role in modern particle physics. The breakdown of a global symme- try by the vacuum expectation value of a Higgs-like ﬁeld leads to the existence of massless particles, the “Nambu-Goldstone bosons” of the broken symmetry. One well-known example are the pions which are 165 166 Chapter 5 the Nambu-Goldstone bosons of an approximate SU(2) symmetry of the system of nucleons and pions in the sigma model (e.g. Itzykson and Zuber 1983). Another example is the hypothetical axion which is the Nambu-Goldstone boson of the Peccei-Quinn chiral U(1) symmetry that would solve the CP problem of strong interactions (Chapter 14). Both pions and axions carry a small mass because the underlying sym- metry is not exact at low energies; they are sometimes called “pseudo Nambu-Goldstone bosons.” An example for a true Nambu-Goldstone boson is the hypothetical majoron which arises from the spontaneous breakdown of a symmetry by a Higgs ﬁeld which would give the neu- trinos Majorana masses (Sect. 15.7). In these examples the Nambu-Goldstone bosons are pseudoscalars because the underlying symmetry is chiral. Unless the CP symmetry is violated, their possible coupling to photons must be of the form Lint=−1 4ga FeFa=ga E·Ba, (5.1) whereais the pseudoscalar ﬁeld; the axion will serve as a generic example. Further, ga is a constant with the dimension (energy)−1,F is the electromagnetic ﬁeld strength tensor,eFits dual, and EandBare the electric and magnetic ﬁelds, respectively. Because Eis a polar and Ban axial vector, E·Bis a pseudoscalar under a CP transformation and so Lintremains invariant. New scalar particles ϕalmost inevitably would also couple to pho- tons with a CP conserving structure Lint∝1 4FFϕ=1 2(E2+B2)ϕ. Therefore, everything that will be said about axions applies mutatis mutandis to scalar particles as well. Gravitons would also have a two-photon vertex. Much of what is said about axions also applies to them, except that their weak cou- plings render most of the arguments irrelevant—an exception will be mentioned in Sect. 5.5.5. Returning to the case of pseudoscalar Nambu-Goldstone bosons, by the very construction of the underlying models they interact with certain fermions ψby a pseudoscalar coupling of the form Lint=igψγ5ψa, (5.2) whereg=m/f is a Yukawa coupling given in terms of the fermion massmand an energy scale frelated to the vacuum expectation value of the Higgs ﬁeld which breaks the underlying symmetry—for axions the Peccei-Quinn scale fa(Chapter 14), for pions the pion decay con- stantf= 93 MeV. In the latter example the relevant fermions are Two-Photon Coupling of Low-Mass Bosons 167 the nucleons, for axions they are the quarks, possibly the charged lep- tons, and perhaps some exotic heavy quark state not contained in the standard model. For majorons, such couplings exist to neutrinos, and possibly to other fermions. An interaction of the form Eq. (5.2) with charged fermions auto- matically leads to an electromagnetic coupling of the form Eq. (5.1) because of the triangle amplitude shown in Fig. 5.1. For one fermion of chargeeand massman explicit evaluation leads to the relationship (e.g. Itzykson and Zuber 1983) ga =α πg m=α πf, (5.3) wheremwas taken to be much larger than the axion and photon en- ergies. Remarkably, because g=m/f this coupling does not depend on the fermion mass, but only on the scale fof symmetry breaking. In general, one must sum over all possible fermions, taking account of the appropriate charges which are fractional for quarks, and also of the proper pseudoscalar coupling to the individual fermions which may vary fromm/f by model-dependent factors of order unity. Fig. 5.1. Triangle loop for the coupling of a pseudoscalar a(axion) to two photons. For massive pseudoscalars the two-photon coupling allows for a de- caya→2γwith a width Γa→2 =g2 a m3 a/64π. (5.4) For pions in the sigma model, the only charged fermion is the proton. Theng◦ =α/πf and Γ ◦→2 =α2m3 /64π3f2 = 7.6 eV, in close agreement with the experimental value. (For subtleties of interpretation of this result in the context of current algebra see the standard ﬁeld theory literature, e.g. Itzykson and Zuber 1983). 168 Chapter 5 The main point for the present discussion is that pseudoscalar mass- less or low-mass bosons are a natural consequence of certain extensions of the standard model, and that these particles couple to photons ac- cording to Eq. (5.1) with a strength ga =α πfaCa , (5.5) wherefais the energy scale of symmetry breaking and Ca is a model- dependent factor of order unity. (For axions, model-dependent details of the couplings are discussed in Chapter 14.) In the following I will explore a variety of consequences arising from this interaction. 5.2 Primakoﬀ Process in Stars 5.2.1 Screened Cross Section and Emission Rate The two-photon coupling of pions or other pseudoscalars allows for the conversion a↔γin an external electric or magnetic ﬁeld by virtue of the amplitude shown in Fig. 5.2. This process was ﬁrst proposed by Primakoﬀ (1951) to study the π◦-γ-coupling which is experimentally diﬃcult to measure in free decays π◦→2γ. In stars, this process allows for the production of low-mass pseudoscalars in the electric ﬁelds of nuclei and electrons (Dicus et al. 1978). Fig. 5.2. Primakoﬀ conversion between axions or other pseudoscalars and photons in an external electromagnetic ﬁeld. The Primakoﬀ process turns out to be important for nonrelativistic conditions where T≪meso that both electrons and nuclei can be treated as “heavy” relative to typical energies of the ambient photons. Therefore, ignoring recoil eﬀects one ﬁnds for the diﬀerential cross sec- tion in this limit (target charge Ze) dσ →a dΩ=g2 a Z2α 8π|k ×ka|2 q4, (5.6) where q=k −kais the momentum transfer; the axion and photon energies are the same. Two-Photon Coupling of Low-Mass Bosons 169 This cross section exhibits the usual forward divergence from the long-range Coulomb interaction. For ma̸= 0 it is cut oﬀ in vacuum by the minimum necessary momentum transfer qmin=m2 a/2ω(ma≪ω); the total cross section is then σ →a=Z2g2 a [1 2ln(2ω/m a)−1 4]. In a plasma, the long-range Coulomb potential is cut oﬀ by screening eﬀects; according to Sect. 6.4 the diﬀerential cross section is modiﬁed with a factor q2/(k2 S+q2). In a nondegenerate medium the screening scale is given by the Debye-H¨ uckel formula k2 S=4πα TnB( Ye+∑ jZ2 jYj) , (5.7) wherenB=ρ/m u(atomic mass unit mu) is the baryon density while YeandYjare the number fractions per baryon of the electrons and various nuclear species j. With this modiﬁcation the total scattering cross section is easily calculated (Raﬀelt 1986a). Summing over all targets one may derive an expression for the transition rate (“decay rate”) of a photon of frequency ωinto an axion of the same energy, Γ →a=g2 a Tk2 S 32π[( 1 +k2 S 4ω2) ln( 1 +4ω2 k2 S) −1] , (5.8) where the plasma “mass” of the initial-state photon and the axion mass were neglected relative to the energy ω. In the limit ω≪kSthis ex- pression expands as Γ →a=g2 a ω2T/16πwhich is entirely independent of the density and chemical composition. For a stellar plasma, however, this approximation is usually not justiﬁed. Ignoring the plasma frequency for the initial-state photons, the energy-loss rate per unit volume is Q=∫2d3k (2π)3Γ →aω e!=T−1=g2 a T7 4πF(κ2), (5.9) whereκ≡kS/2Tand F(κ2) =κ2 2π2∫∞ 0dx[ (x2+κ2) ln( 1 +x2 κ2) −x2]x ex−1,(5.10) withx=ω/T. This function is shown in Fig. 5.3. In a standard solar modelκ2≈12 throughout the Sun with a variation of less than 15%. In the core of an HB star with ρ= 104g/cm3andT= 108K it isκ2≈2.5. One ﬁndsF= 0.98 and 1.84 for κ2= 2.5 and 12, respectively. 170 Chapter 5 Fig. 5.3. Function F(2) according to Eq. (5.10). 5.2.2 Plasmon Decay and Coalescence Several authors have used the plasma frequency ωPinstead of the Debye-H¨ uckel wave number as a screening scale. In a nondegenerate plasmak2 S/ω2 P≈me/T(Sect. 6.3) and so they overestimated the emis- sion rate. Another source of confusion are statements that the decay γ→γawas another axion emission process enabled by the 2 γcoupling. This issue can be easily clariﬁed, but one needs to draw heavily on the discussion of photon dispersion of Sect. 6.3. A medium allows for both transverse and longitudinal electromag- netic excitations. In a nondegenerate and nonrelativistic plasma the dispersion relation of the former is ω2 T−k2 T=ω2 P(plasma frequency ωP) while the latter oscillate essentially with a ﬁxed frequency ωL=ωP, independently of their wave number. Therefore, the plasmon decay processγT→γLaas well as the plasmon coalescence γTγL→aare indeed kinematically possible for kL>ω P. It is easy to calculate the inverse lifetime of transverse excitations against these processes Γ T→a=∫d3kL 2ωL(2π)3d3ka 2ωa(2π)31 2ωTZTZL|M|2(2π)4 ×[δ4(KT+KL−Ka) e!L=T−1+δ4(KT−KL−Ka) 1−e−!L=T] ,(5.11) whereZT;Lare the vertex renormalization factors. Here, the ﬁrst term in square brackets corresponds to coalescence and so it involves a Bose- Einstein occupation number for the initial-state γL. The second term Two-Photon Coupling of Low-Mass Bosons 171 is from decay; it involves a Bose stimulation factor [1+( e!L=T−1)−1] = (1−e−!L=T)−1for the ﬁnal-state γL. For the axions, a stimulation factor is not included because they are assumed to escape immediately. Because a longitudinal excitation has no magnetic ﬁeld, the elec- tric ﬁeld in the matrix element corresponding to Eq. (5.1) must be associated with a longitudinal, the magnetic ﬁeld with a transverse ex- citation. In general B=∇ ×AandE=−∇A0−∂tAwith the vector potentialA. For a propagating mode A∝e−i(!t−k·x)ϵwhereϵis a polarization vector. This implies that B∝kT×ϵTwhere ϵTis a po- larization vector transverse to kTwith ϵ2 T= 1, and E∝ωLϵL−kLϵ0 L whereϵL= (k2 L,ωLkL)k−1 L(ω2 L−k2 L)−1=2according to Eq. (6.27) so that E∝ˆkL(ω2 L−k2 L)1=2. Then one ﬁnds from the usual Feynman rules |M|2=g2 a (ω2 L−k2 L)|ˆkL·(kT×ϵT)|2. (5.12) Note that |ˆkL·(kT×ϵT)|2=|(ˆkL×kT)·ϵT|2. Averaging over the two transverse polarization states yields1 2|ˆkL×kT|2. If one writes ZL=eZLω2 L/(ω2 L−k2 L) as in Sect. 6.3 and performs the d3kLintegration in Eq. (5.11) one ﬁnds Γ T→a=g2 a 16 (2π)2∫ dΩadωaZTeZLωaωL ωT|ˆkL×kT|2 ×[δ(ωT+ωL−ωa) e!L=T−1+δ(ωT−ωL−ωa) 1−e−!L=T] . (5.13) In a nondegenerate, nonrelativistic plasma typically ωP≪T. Because ωL=ωPone may expand the exponentials so that ( e!L=T−1)−1→T/ω L and also (1 −e−!L=T)−1→T/ω L. IfωT=O(T)≫ωPone may ignore ωLin theδfunctions which are then trivial to integrate, Γ T→a=g2 a T 8 (2π)2∫ dΩaZTeZL|ˆkL×kT|2. (5.14) In this limit ahas the same energy as γTwhileγLhas only provided momentum. To ﬁnish up, note that in the nondegenerate, nonrelativistic limit ZT=eZL= 1. Because ωP≪TandωT=O(T) we haveωP≪ωT so thatkT≈ωT. Moreover, kL=kT−kayielding |ˆkL×kT|2= |ka×kT|2/|ka−kT|2=ω2 T(1−z2)/2(1−z) =1 2ω2 T(1+z) wherezis the 172 Chapter 5 cosine of the angle between kaandkT. The angular integration then averageszto zero and leaves us with Γ T→a=g2 a T 16π. (5.15) Collective longitudinal oscillations only exist for kL∼<kD(Debye screen- ing scale). Because a momentum transfer |kT−ka|=O(ωT) is required, this result applies only for ωT∼<kD. This conversion rate agrees with the low-ωexpansion of the Primakoﬀ result Eq. (5.8). It must be stressed that Eq. (5.15) is not an additional contribution to the conversion rate, it is the same result derived in a diﬀerent fashion. Here, longitudinal plasmons in the static limit were used as the external electric ﬁeld in which the Primakoﬀ eﬀect takes place. Before, the static limit was taken from the start; the collective behavior of the electron motion was reﬂected in the screening of the Coulomb potential. These two paths of performing the calculation in the end extracted the same information from the electromagnetic polarization tensor which deﬁnes both screening eﬀects and the dispersion behavior of electromagnetic excitations. 5.2.3 Axion Emission from Electromagnetic Plasma Fluctuations These simple calculations of the axion emission rate apply only in the classical (nondegenerate, nonrelativistic) limit. Even though this is the most relevant case from a practical perspective it is worth mentioning how one proceeds for a more general evaluation. To this end note that the 2γinteraction Eq. (5.1) corresponds to a source term for the axion wave equation, (+m2 a)a=ga E·B, (5.16) where =∂∂=∂2 t−∇2. Axions are then emitted by the E·Bﬂuc- tuations caused by the presence of thermal electromagnetic radiation as well as the collective and random motion of charged particles. The Primakoﬀ calculation in Sect. 5.2.1 used the (screened) electric ﬁeld of charged particles and the magnetic ﬁeld of (transverse) electro- magnetic radiation (photons) as a source. Actually, one can include the magnetic ﬁeld of moving charges for this purpose. Then axions are emitted in the collision of two particles (Fig. 5.4), a process sometimes referred to as the “electro Primakoﬀ eﬀect.” Unsurprisingly, the emis- sion rate is much smaller because the magnetic ﬁeld associated with Two-Photon Coupling of Low-Mass Bosons 173 Fig. 5.4. Example for the electro Primakoﬀ eﬀect. nonrelativistic moving particles is small. (For an explicit calculation see Raﬀelt 1986a.) Put another way, the dominant contribution to magnetic ﬁeld ﬂuctuations in a nonrelativistic plasma is from photons, not from moving charges. In order to illustrate the relationship between ﬁeld ﬂuctuations and axion emission consider the amplitude for the conversion of a classical transverse electromagnetic wave into a classical axion wave in the pres- ence of an electric ﬁeld conﬁguration E(x). One ﬁnds from Eq. (5.16) f(Ω) =ga 4π(ϵ×k)·∫ d3xe−iq·xE(x) =ga 4π(ϵ×k)·E(q), (5.17) where kandϵare the wave and polarization vector of the incident wave, respectively, qis the “momentum” transfer to the axion, and E(q) is a Fourier component of E(x). The diﬀerential transition rate isdΓ/dΩ =|f(Ω)|2so that dΓ →a dΩ=g2 a (4π)2(ϵ×k)i(ϵ×k)jEi(−q)Ej(q), (5.18) where it was used that E(x) is real so that E∗(q) =E(−q). IfE(x) is a random ﬁeld conﬁguration one needs to take an ensemble average so that the transition rate is proportional to ⟨EiEj⟩q≡ ⟨Ei(−q)Ej(q)⟩. The electric and magnetic ﬁeld ﬂuctuations of a plasma are inti- mately related to the medium response functions to electric and mag- netic ﬁelds, i.e. to the polarization tensor. For a plasma at tempera- tureTone can show on general grounds (Sitenko 1967) ⟨EiEj⟩q=∫+∞ −∞dω 2π2 e!=T−1 ×[qiqj q2ImϵL |ϵL|2+( δij−qiqj q2)ImϵT |ϵT−q2/ω2|2] ,(5.19) whereϵL;T(ω,q) are the longitudinal and transverse dielectric permit- tivities of the medium (Sect. 6.3.3). A quantity such as Im ϵL/|ϵL|2is known as a spectral density—here of the longitudinal ﬂuctuations. 174 Chapter 5 One ﬁnds explicitly ⟨EiEj⟩q=ˆqiˆqjT/(1 +q2/k2 S) in the classi- cal limit (Sitenko 1967) where k2 Sis the Debye-H¨ uckel wave number of Eq. (5.7). With this result one easily reproduces the Primakoﬀ transi- tion rate Γ →a(Raﬀelt 1988a). The language of spectral densities for the electromagnetic ﬁeld ﬂuc- tuations forms the starting point for a quantum calculation of the axion emission rate in the framework of thermal ﬁeld theory. This program was carried out in a series of papers by Altherr (1990, 1991), Altherr and Kraemmer (1992), and Altherr, Petitgirard, and del R´ ıo Gaztelurrutia (1994). Naturally, in the classical limit they reproduced the Primakoﬀ transition rate Γ T→aof Eq. (5.8). In the degenerate or relativistic limit their results cannot be rep- resented in terms of simple analytic formulae. The most important astrophysical environment to be used for extracting bounds on ga are low-mass stars before and after helium ignition with a core temper- ature of about 108K (Sect. 5.2.5). Altherr, Petitgirard, and del R´ ıo Gaztelurrutia (1994) gave numerical results for the energy-loss rate for this temperature as a function of density shown in Fig. 5.5 (solid line). The dashed line is the classical limit Eq. (5.9); it agrees well with the general result in the low-density (nondegenerate) limit. In the degen- Fig. 5.5. Energy-loss rate of a helium plasma at T= 108K by axion emission with ga = 10−10GeV−1. The solid line is from transverse-longitudinal ﬂuctuations; the dashed line is the corresponding classical limit. The dotted line is from transverse-transverse ﬂuctuations, i.e. in the axion source term ga E·Bboth ﬁelds are from transverse ﬂuctuations. (Adapted from Altherr, Petitgirard, and del R´ ıo Gaztelurrutia 1994.) Two-Photon Coupling of Low-Mass Bosons 175 erate limit, the emission rate drops precipitously, an important feature which will be taken advantage of in Sect. 5.2.5 below. 5.2.4 Solar Axion Spectrum As a ﬁrst practical application it is easy to calculate the expected ﬂux of axions at Earth from the Primakoﬀ conversion in the Sun where the classical approximation is well justiﬁed. To this end van Bibber et al. (1989) have integrated Eq. (5.9) over a standard solar model which yields an axion luminosity La=g2 101.7×10−3L⊙, (5.20) withL⊙the solar luminosity and g10≡ga ×1010GeV. (Recalling that ga = (α/πf a)Ca this corresponds to fa/Ca = 2.3×107GeV.) The diﬀerential ﬂux at Earth is well approximated by the formula dFa dωa=g2 104.02×1010cm−2s−1keV−1(ωa/keV)3 e!a=1:08 keV−1(5.21) which is shown in Fig. 5.6. The average axion energy is ⟨ωa⟩= 4.2 keV. The total ﬂux at Earth is Fa=g2 103.54×1011cm−2s−1. The “standard Sun” is about halfway through its main-sequence evolution. Therefore, the solar axion luminosity must not exceed its Fig. 5.6. Axion ﬂux at Earth according to Eq. (5.25) from the Primakoﬀ conversion of photons in the Sun. 176 Chapter 5 photon luminosity; otherwise its nuclear fuel would have been spent before reaching an age of 4 .5×109yr. This requirement yields a bound ga ∼<2.4×10−9GeV−1. (5.22) Detailed solar evolution calculations of Raﬀelt and Dearborn (1987) showed that this bound was ﬁrm, but also that it could not be im- proved easily (Sect. 1.3.2). The present-day properties of the Sun could be obtained by a suitable adjustment of the unknown presolar helium abundance. 5.2.5 Globular-Cluster Bound on ga Armed with the Primakoﬀ emission rate Eq. (5.9) it is an easy task to derive a bound on ga from the energy-loss argument applied to globular-cluster stars (Sect. 2.5). We need to require that at T≈108K the axionic energy-loss rate is below 10 erg g−1s−1for a density of about 0.6×104g cm−3, corresponding to a classical plasma, and for about 2×105g cm−3, corresponding to degeneracy. From Fig. 5.5 it is evi- dent that the emission rate is a steeply falling function of density when degeneracy eﬀects become important. Obviously, the more restrictive limit is found from the low-density case which is based on the helium- burning lifetime of HB stars (Sect. 2.5.1). In order to calculate the average energy-loss rate of the core of an HB star one needs ⟨T7/ρ⟩if in Eq. (5.9) one uses a constant κ2= 2.5 orF= 1.0. For a typical HB-star model (Fig. 1.4) one ﬁnds ⟨T7 8/ρ4⟩ ≈0.3 whereT8=T/108K andρ4=ρ/104g cm−3. There- fore,⟨ϵa⟩ ≈g2 1030 erg g−1s−1so that the criterion Eq. (2.40) yields a constraint ga ∼<0.6×10−10GeV−1orfa/Ca ∼>4×107GeV.(5.23) The temperature 108K corresponds to 8 .6 keV; a typical photon energy is 3T≈25 keV. Therefore, this bound applies to pseudoscalars with a massma∼<30 keV while for larger masses it would be degraded. 5.3 Search for Cosmic Axions One of the most interesting ramiﬁcations of the electromagnetic cou- pling of pseudoscalars is the possibility to search for dark-matter axions. It is brieﬂy explained in Chapter 14 that axions would be produced in the early universe by a nonthermal mechanism which excites classical Two-Photon Coupling of Low-Mass Bosons 177 axion ﬁeld oscillations, i.e. a highly degenerate axion Bose condensate that would play the role of “cold dark matter,” and notably provide the unseen mass necessary to explain the rotation curves of spiral galax- ies such as our own (e.g. Kolb and Turner 1990). Because there are uncertainties with regard to details of the primordial axion production mechanism the exact value of the relevant axion mass is not known. However, typically it is of order 10−5eV so that it is a reasonable spec- ulation that the mass of our galaxy is dominated by very low-mass bosons. As these particles are bound to the galaxy they must be non- relativistic; a typical velocity dispersion corresponding to the galactic gravitational potential is around 10−3in units of the speed of light. Sikivie (1983) proposed to search for galactic axions by means of a Primakoﬀ-like method. The a→γconversion of nonrelativistic axions in theµeV mass range produces photons in the microwave (GHz) range. Therefore, the idea is to place a microwave cavity in a strong magnetic ﬁeld and wait for cavity modes to be excited by the axion ﬁeld. In this context one may view the electromagnetic modes of the cavity and the free axion ﬁeld modes as oscillators which are coupled by virtue of the interaction Eq. (5.1) where Bis the external static ﬁeld while Eis from an electromagnetic cavity mode. Then, power is transferred from the axion ﬁeld to the cavity excitations by virtue of the oscillator beats induced by the coupling; detailed calculations of the conversion rate were performed by Sikivie (1985) and Krauss et al. (1985). It is worth noting that with a mass of 10−5eV and a velocity of 10−3a typical axion momentum is 10−8eV which corresponds to a wave length of about 20 m. Thus on laboratory scales the axion ﬁeld is homogeneous. This does not apply to free microwaves—their en- ergy and momentum are the same ( ω =|k |); for 10−5eV their wave length is 2 cm. The role of the resonant cavity is to overcome this mo- mentum mismatch: on resonance the fundamental cavity frequency is degenerate with nonrelativistic axions of a certain mass for which the energy transfer is maximized. In a search experiment the cavity must be stepped through a range of resonant frequencies which deﬁnes the range of axion masses to which a given experimental setup is sensitive. Two pilot experiments of this sort were completed several years ago (Wuensch et al. 1989; Hagmann et al. 1990). Assuming an axionic dark- matter density at the Earth of 5 ×10−25g cm−3= 300 MeV cm−3allowed these groups to exclude the range of masses and coupling constants shown in Fig. 5.7. The solid line indicates the relationship between ga andmain axion models where E/N = 8/3 orξ= 1 in Eq. (14.24), i.e. wherega = (ma/µeV) (0.69×1016GeV)−1. 178 Chapter 5 Most excitingly, the galactic axion search is going to be taken up again with two new experimental setups. The one in Livermore (Cali- fornia) has an increased detection volume and magnetic ﬁeld ( B2V= 14 T2m3), and a reﬁned microwave detection method (van Bibber et al. 1992, 1994). Within a running time of two or three years it will be pos- sible to explore the axion mass range 1 .3−13µeV down to a coupling strengthga which is only a factor of about 2.5 shy of the “axion line” in Fig. 5.7. If axions interact with photons somewhat stronger than indicated by this line, or if the local dark-matter density is somewhat larger than assumed in Fig. 5.7, one may already be able to detect axions in this round of measurements. The second experiment (Kyoto, Japan) will use Rydberg atoms in a novel scheme to detect single microwave quanta (Matsuki et al. 1995). Because of the intrinsic low noise of this detector one can go to lower physical temperatures, thereby reducing thermal noise, and thus allow- ing one to use smaller cavities. This in turn permits one to search for larger axion masses than is possible with the Livermore-type large cav- ities. Together, the two experiments can probably cover two decades of axion masses, between about 10−6and 10−4eV. Fig. 5.7. Results of the galactic axion search experiments of the Rochester- Brookhaven-Fermilab (RBF) collaboration (Wuensch et al. 1989) and of the University of Florida (UF) experiment (Hagmann et al. 1990). The hatched areas are excluded, assuming a local dark-matter axion density of 5×10−25g cm−3= 300 MeV cm−3. The “axion line” is the relationship be- tween axion mass and coupling strength for = 1 or E=N = 8=3 according to Eq. (14.24). Two-Photon Coupling of Low-Mass Bosons 179 5.4 Axion-Photon Oscillations 5.4.1 Mixing Equations The “axion haloscope” discussed in the previous section was based on the Primakoﬀ conversion between axions and photons in a macroscopic magnetic or electric ﬁeld. The same idea can be applied to other axion ﬂuxes such as that expected from the Sun (“axion helioscope,” Sikivie 1983). Typical energies of solar axions are in the keV range (Sect. 5.2.4) so that a macroscopic laboratory magnetic ﬁeld is entirely homogeneous on the scale of the axion wavelength. In this case the conversion process is best formulated in a way analo- gous to neutrino oscillations (Anselm 1988; Raﬀelt and Stodolsky 1988). This approach may seem surprising as the axion has spin zero while the photon is a spin-1 particle. States of diﬀerent spin-parity can mix, however, if the mixing agent (here the external magnetic ﬁeld) matches the missing quantum numbers. Therefore, only a transverse magnetic or electric ﬁeld can mix a photon with an axion; a longitudinal ﬁeld respects azimuthal symmetry whence it cannot mediate transitions be- tween states of diﬀerent angular momentum components in the ﬁeld direction. The starting point for the magnetically induced mixing between axions and photons is the classical equation of motion for the system of electromagnetic ﬁelds and axions in the presence of the interaction Eq. (5.1). In terms of the electromagnetic ﬁeld-strength tensor Fand its dualeFone ﬁnds, apart from the constraint ∂eF= 0, ∂F=J+ga eF∂a, (+m2 a)a=−1 4ga FeF, (5.24) whereJis the electromagnetic current density. In a physical situation with a strong external ﬁeld plus radiation one may approximate ga eF∂a→ga eF ext∂abecause a term ga eF rad∂a is of second order in the weak radiation ﬁelds. Moreover, if only an ex- ternal magnetic ﬁeld is present, the wave equation for the time-varying part of the vector potential Aand for the axion ﬁeld are A=ga BT∂ta, (−m2 a)a=−ga BT·∂tA, (5.25) where BTis the transverse external magnetic ﬁeld. If one specializes to a wave of frequency ωpropagating in the z-direction, and denoting the 180 Chapter 5 components of Aparallel and perpendicular to BTwithA∥andA⊥, respectively, one ﬁnds (Raﬀelt and Stodolsky 1988)  ω2+∂2 z+ 2ω2 n⊥−1nR 0 nRn∥−1ga BT/2ω 0ga BT/2ω−m2 a/2ω2   A⊥ A∥ a = 0, (5.26) where the oﬀ-diagonal terms were made real by a suitable global trans- formation of the ﬁelds. Further, a photon index of refraction was in- cluded because in practice one never has a perfect vacuum. The refractive index is generally diﬀerent for the two linear polar- ization states parallel and perpendicular to BT(Cotton-Mouton eﬀect). Also, there may be mixing between the A⊥andA∥ﬁelds, i.e. the plane of polarization may rotate in optically active media, an eﬀect charac- terized bynR. In general, any medium becomes optically active if there is a magnetic ﬁeld component along the direction of propagation (Fara- day eﬀect). Therefore, in general the refractive indices n⊥;∥depend on the transverse, the index nRon the longitudinal magnetic ﬁeld. Equation (5.26) is made linear by an approach that will be discussed in more detail for neutrinos in Sect. 8.2. For propagation in the positive z-direction and for very relativistic axions and photons one may expand (ω2+∂2 z) = (ω+i∂z)(ω−i∂z)→2ω(ω−i∂z). Then one obtains the usual “Schr¨ odinger equation”  ω+ ∆⊥∆R 0 ∆R∆∥∆a 0 ∆ a ∆a +i∂z  A⊥ A∥ a = 0, (5.27) where ∆ ∥;⊥= (n∥;⊥−1)ω, ∆ R=nRω, ∆ a=−m2 a/2ω, and ∆ a = 1 2ga BT. If one ignores a possible optical activity or Faraday eﬀect ( nR= 0), the lower part of this equation represents a 2 ×2 mixing problem. The matrix is made diagonal by a rotation about an angle 1 2tan 2θ=∆a ∆∥−∆a=ga BTω (n∥−1)2ω2+m2 a. (5.28) In analogy to neutrino oscillations (Sect. 8.2.2) the probability for an axion to convert into a photon after travelling a distance ℓin a trans- verse magnetic ﬁeld is prob(a→γ) = sin2(2θ) sin2(1 2∆oscℓ), (5.29) where ∆2 osc= (∆ ∥−∆a)2+ ∆2 a so that the oscillation length is ℓosc= 2π/∆osc. Two-Photon Coupling of Low-Mass Bosons 181 By adjusting the gas pressure within the magnetic ﬁeld volume one can make the photon and axion degenerate and thus enhance the tran- sition rate (van Bibber et al. 1989). This applies, in particular, to solar axions which have keV energies so that the corresponding pho- ton dispersion relation in low- Zgases is “particle-like” with the plasma frequency being the eﬀective mass. If there is a gradient of the gas density, for example near a star, or if the gas density and magnetic ﬁeld strength change in time as in the expanding universe, suitable conditions allow for resonant axion-photon conversions in the spirit of the neutrino MSW eﬀect (Yoshimura 1988; Yanagida and Yoshimura 1988). For the magnetic conversion of pseudoscalars in the galactic mag- netic ﬁeld one must worry about density ﬂuctuations of the interstellar medium which can be of order the medium density itself. In this case Eq. (5.29) is no longer valid because it was based on the assumption of spatial homogeneity of all relevant quantities. Carlson and Garretson (1994) have derived an expression for the conversion rate in a medium with large random density variations. They found that it can be sig- niﬁcantly suppressed relative to the naive result. 5.4.2 Solar Axions An axion helioscope experiment was performed by Lazarus et al. (1992) who used a vacuum pipe of 6′′diameter which was placed in the bore of a dipole magnet of 72′′length (1.80 m); the ﬁeld strength was 2 .2 T. The helioscope was oriented so that its long axis pointed along the azimuth of the setting Sun. This provided a time window of approximately 15 min every day during which the line of sight through the vacuum region pointed directly to the Sun. As a detector they used an x-ray proportional chamber at the end of the pipe. Data were taken on several days with He at diﬀerent pressures in the pipe. At 1 atm helium provides a plasma mass of about 0 .3 keV to x-rays. For vacuum, a 3 σbound ofga <3.6×10−9GeV−1forma< 0.050 eV was found. For a helium pressure of 55 Torr the limit was 3.9 in the same units, applicable to 0 .050< m a/eV<0.086, and for 100 Torr it was 3.4, applicable to 0 .086< m a/eV<0.110. These bounds assume an axion ﬂux as given by Eq. (5.21) which in turn assumes an unperturbed Sun. Unfortunately, this assumption is not consistent because the present-day age of the Sun already requires the bound Eq. (5.22). 182 Chapter 5 However, an ongoing experimental project at the Institute for Nu- clear Physics in Novosibirsk may be able to improve the helioscope signiﬁcantly. The conversion magnet has been gimballed so that it can track the Sun, providing much longer exposure times. First results can be expected for late 1995—see Vorobyov and Kolokolov (1995) for a status report. Another possibility would be to use the straight sections of the beam pipe of the LEP accelerator at CERN as an axion helioscope. Hoogeveen and Stuart (1992) have calculated the times and dates of alignment with the Sun. They proposed an experimental setup that might allow one to reach a sensitivity in ga down to 4 ×10−10GeV−1, which would be very impressive, but still far from the globular-cluster bound Eq. (5.23). Finally, Paschos and Zioutas (1994) proposed to use a single crystal as a detector where the Primakoﬀ conversion of solar axions is coher- ently enhanced over the electric ﬁelds of many atoms. Put another way, one would expect a strong enhancement via Bragg scattering. Even with this improvement, however, it does not seem possible to beat the bound from globular-cluster stars. 5.4.3 Shining Light through Walls Instead of using the solar axion ﬂux one can make one’s own by shining a laser beam through a long transverse magnetic ﬁeld region where it develops an axion component. Then the laser beam is blocked while the weakly interacting axions traverse the obstacle. In a second magnet they are back-converted into photons so that one “shines light through walls” (Anselm 1985; Gasperini 1987; van Bibber et al. 1987). Instead of a freely propagating beam one may use resonant cavities on either side of the wall which are coupled by the axion ﬁeld (Hoogeveen and Ziegenhagen 1991). Another possibility to improve the sensitivity is to use squeezed light (Hoogeveen 1990). An actual experiment was performed by Ruoso et al. (1992) who used two superconducting magnets of length 440 cm each with a ﬁeld strength of 3 .7 T. The light beam was trapped in a resonant cavity in the ﬁrst magnet, allowing for about 200 traversals; the incident laser power was 1 .5 W. At the end of the second magnet photons were searched for by a photomultiplier. For an axion mass ma∼<10−3eV an upper bound ga <0.7×10−6GeV−1was found. Two-Photon Coupling of Low-Mass Bosons 183 5.4.4 Vacuum Birefringence Besides the conversion between photons and axions there is a more subtle eﬀect that can serve to search for the two-photon vertex of low- mass pseudoscalars. In an external transverse EorBﬁeld the mix- ing between the A∥component with aleads to a backreaction on A∥ (Fig. 5.8b) which amounts to a retardation of its phase. Put another way, the ∥and⊥polarization states have diﬀerent refractive indices in vacuum with an EorBﬁeld (vacuum Cotton-Mouton eﬀect). There- fore, if one shines a light beam which is linearly polarized at 45◦relative to a transverse Bﬁeld, the beam will develop a small degree of elliptic polarization (Maiani, Petronzio, and Zavattini 1986). A vacuum Cotton-Mouton eﬀect is expected even in the absence of axions from the QED amplitude shown in Fig. 5.8a; in the nonforward direction it describes Delbr¨ uck scattering on a charged particle. More generally, an electron loop mediates an eﬀective γγinteraction which for low energies can be described by the Euler-Heisenberg Lagrangian, L =2α2 45m4 e[ (E2−B2)2+ 7(E·B)2] (5.30) (Heisenberg and Euler 1936; see also Itzykson and Zuber 1983). Fig. 5.8. Vacuum birefringence in the presence of external ﬁelds. (a) QED contribution according to the Euler-Heisenberg interaction. (b) aor ◦oscillations in an external EorBﬁeld. (c) Photon birefringence in an external axion ﬁeld (axionic domain walls, cosmic axion ﬁeld). (d) Axion- mediated contribution in a strong E·Bﬁeld, e.g. near a pulsar. 184 Chapter 5 The vacuum refractive index in an external magnetic ﬁeld which results from this interaction was studied by a number of authors— see Tsai and Erber (1975, 1976) for references to the early literature. Adler (1971) provided a comprehensive study and derived the correct expression for the related photon splitting rate γ→γγin an external ﬁeld.28The refractive indices for the ∥and⊥polarization states are29 n∥= 1 + 72α2 45B2 m4 eandn⊥= 1 + 42α2 45B2 m4 e, (5.31) where (2α2/45)B2/m4 e= 1.32×10−32(B/Gauss)2. (5.32) Note that 1 Gauss corresponds to 10−4Tesla, and to 1 .95×10−2eV2in natural units (Appendix A). Clearly one needs very strong magnetic ﬁelds for vacuum birefrin- gence eﬀects to become important. So far, no positive experimental measurement exists. A proposal to measure the acquired elliptic polar- ization of a laser beam was put forth by Iacopini and Zavattini (1979). More recently Cantatore et al. (1991) proposed to use polarized light scattered oﬀ an electron beam, a method which allows one to obtain polarized GeV photons. As the relative phase shift is ( n∥−n⊥)ωℓfor a distance of travel ℓin the magnetic ﬁeld, high-energy photons show a much stronger eﬀect for otherwise equal conditions. An experiment to search for the axion contribution of Fig. 5.8b was recently performed (Semertzidis et al. 1990; Cameron et al. 1993). Note that there are twoaxion-induced eﬀects on a laser beam trapped in an optical cavity. One is the birefringence eﬀect analogous to the QED eﬀect which leads to a small amount of elliptical polarization. Another is the loss of ∥photons into the axion channel which depletes the amplitude of the ∥mode relative to the ⊥one, which in turn leads to a rotation of the plane of polarization. Both eﬀects are of the same order in the coupling constant; experimentally, the rotation eﬀect led to 28The photon-splitting box graph with one external ﬁeld and three real photons attached to an electron loop does not contribute. The lowest-order amplitude is with the external ﬁeld attached three times, and three real photons (hexagon dia- gram). For references to the early literature and a discussion of the astrophysical implications of the photon-splitting process see Baring (1991). 29∥;⊥refer to the electric ﬁeld of the wave relative to the external transverse B ﬁeld while Adler (1971) refers with ∥;⊥to the magnetic ﬁeld of the wave. Note also that I use rationalized units where =e2=4= 1=137 while in the literature on photon refraction unrationalized units with =e2= 1=137 are often employed. Two-Photon Coupling of Low-Mass Bosons 185 a more restrictive limit of ga <3.6×10−7GeV−1forma∼<7×10−4eV. For a larger mass an a-γoscillation pattern develops on the length scale of the optical cavity, leading to an “oscillating limit” as a function of ma. In this regime the ellipticity measurement was superior. New experimental eﬀorts in the birefringence category include a proposal by Cooper and Stedman (1995) to use ring lasers. A laser experiment which is actually in the process of being built is PVLAS (Bakalov et al. 1994) which will be able to improve previous laboratory limits onga by a factor of 40, i.e. it is expected to be sensitive in the regime ga ∼>1×10−8GeV−1as long asma∼<10−3eV. While such strong couplings are astrophysically excluded it is intriguing that this experiment should be able to detect for the ﬁrst time the standard QED birefringence eﬀect of Fig. 5.8a. 5.5 Astrophysical Magnetic Fields 5.5.1 Transitions in Magnetic Fields of Stars Certain stars have very strong magnetic ﬁelds. For example, neutron stars frequently have ﬁelds of 1012−1013G (e.g. M´ esz´ aros 1992), and even white dwarfs can have ﬁelds of up to 109G. Therefore, one may think that axions produced in the hot interior of neutron stars at a tem- perature of, say, 50 keV would convert to γ-rays in the magnetosphere (Morris 1986). However, the vacuum refractive term suppresses the conversion rate because the photon momentum for a given frequency isk =n∥;⊥ω > ω with the refractive indices Eq. (5.31) while for the axionska=ω−m2 a/2ω <ω . Therefore, in the presence of a magnetic ﬁeld axions and photons are less degenerate so that it is more diﬃcult for them to oscillate into each other. In principle, the refractive index can be cancelled by the presence of a plasma where the photon forward scattering on electrons induces a negativen−1, i.e. something like a photon eﬀective mass. In the aligned rotator model for a magnetized neutron star a self-consistent solution of the Maxwell equations with currents requires the presence of an electron density of about ne= 7×1010cm−3B12P−1 swhereB12is the magnetic ﬁeld along the rotation axis in units of 1012G andPsis the pulsar period in seconds (Goldreich and Julian 1969). The corresponding plasma frequency is ω2 P= 4παn e/m e= 0.97×10−10eV2B12P−1 s. This implies k−ω=−ω2 P/2ω=−5×10−14eVB12P−1 sω−1 keVwithωkeV=ω/keV, to be compared with ∆ ∥= (n∥−1)ω= 0.92×10−4eVB2 12ωkeVwhich is much larger, allowing one to ignore the plasma term. 186 Chapter 5 The oﬀ-diagonal term in the mixing matrix Eq. (5.27) is ∆ a = 1 2ga B= 0.98×10−9eVg10B12withg10=ga /(10−10GeV−1). Ignoring mathe oscillation length is 2 π(∆2 ∥+ ∆2 a )−1=2≈1.3 cmB−2 12ω−1 keVwhile the geometric dimension of the dipole ﬁeld is of order the stellar radius, i.e. of order 10 km. Therefore, many oscillations occur within the mag- netosphere, and the average transition probability between photons and axions isθ2with the mixing angle θ≈∆a /∆∥= 1.1×10−5g10B−1 12ω−1 keV. Therefore, the transition rate is very small. The vacuum refractive index is larger than unity, the plasma contri- bution less than unity, and so near the stellar surface a crossover must occur where axions and photons are degenerate. However, the length scales do not work out to have a resonant MSW-type transition (Raﬀelt and Stodolsky 1988; Yoshimura 1988). At a pulsar, reducing Bincreases the mixing angle and thus the transition rate while the oscillation length becomes larger. When ℓosc far exceeds the geometric dimension Rof the stellar magnetosphere, and when the mixing angle is small, one can expand the sine functions in Eq. (5.29) so that the transition probability is ( θ∆oscR)2≈(∆a R)2. This transition rate scales with B2as expected and becomes smaller for smallerB. Therefore, the optimal situation is when the magnetic ﬁeld strength and geometric dimensions are matched such that R≈ℓosc. This condition is approximately met in magnetic white dwarfs with, say,B= 109G,ω= 10 eV, and R= 103km. In these systems one may even expect a resonant level crossing if they have a dilute atmosphere with an appropriate scale height (Raﬀelt and Stodolsky 1988; Gnedin and Krasnikov 1992). However, a fortuitous combination of particle and white-dwarf parameters is required, and, even then, observable eﬀects apparently have not been proposed. Most recently, Carlson and Tseng (1995) have performed a study of the conversion of very low-mass pseudoscalars in the magnetic ﬁeld of sunspots. They ﬁnd that for certain parameters the x-ray ﬂux from the conversion process could be observable in solar x-ray telescopes such as SXT and Yohkoh. 5.5.2 Birefringence in a Pulsar Magnetosphere In the previous section it was shown that near a pulsar the QED vac- uum Cotton-Mouton eﬀect (Fig. 5.7a) induces a sizeable amount of birefringence between the photon states which are linearly polarized parallel or perpendicular to the transverse component of the magnetic ﬁeld. Recently, Mohanty and Nayak (1993) showed that in addition Two-Photon Coupling of Low-Mass Bosons 187 there can be a strong and potentially observable circular birefringence eﬀect along the polar direction if massless pseudoscalars exist. The masslessness of the pseudoscalars is crucial for this scenario and so ax- ions (Chapter 14) which generically must have a mass do not fulﬁll this requirement. Therefore, I use “arions” as a generic example which are like axions in all respects except that they are true Nambu-Goldstone bosons of a global chiral U(1) symmetry and thus strictly massless (Anselm and Uraltsev 1982a,b; Anselm 1982). The main idea of the pulsar birefringence scenario is that in the oblique rotator model a strong E·Bdensity exists in the pulsar mag- netosphere which serves as a source for the arion ﬁeld. Therefore, a pulsar would be surrounded by a strong classical arion ﬁeld density which constitutes an optically active “medium,” causing a time delay between the two circular polarization states of the pulsed radio emis- sion from the polar cap region. As a Feynman graph, this situation is represented by Fig. 5.8d. In detail, Mohanty and Nayak (1993) considered the oblique ro- tator model where the pulsar magnetic dipole axis is tilted with re- gard to its rotation axis by an angle α. The instantaneous rotating magnetic dipole ﬁeld is B= (B0R3/r3) [3ˆr(ˆr·ˆµ)−ˆµ], whereB0 is the magnetic ﬁeld strength at the poles of the pulsar surface (ra- diusR) and ˆµis the instantaneous magnetic dipole direction with the angleαrelative to the angular velocity vector Ω. The time av- erage of the electric ﬁeld which is induced by the rotating magnetic dipole, and which matches the boundary condition that the electric ﬁeld component parallel to the pulsar surface vanishes, is found to be ⟨E⟩=B0R5Ω cosαr−4[3 (sin2θ−2 3)ˆr−2 sinθcosθˆθ] whereθis the polar angle relative to the rotation axis. The time-averaged value for the pseudoscalar ﬁeld density is ⟨E·B⟩=−B2 0ΩR8r−7cosαcos3θ. (5.33) It appears as a source for the arion ﬁeld on the r.h.s. of Eq. (5.24). Tak- ing account of the relativistic space-time metric outside of the pulsar, Mohanty and Nayak (1993) found for the resulting arion ﬁeld a=−ga 2 575B2 0R8Ω cosα (GNM)3cosθ r2+O(r−3), (5.34) whereGNis Newton’s constant and Mthe pulsar mass. The en- tire magnetosphere contributes coherently to this result. If the pseu- doscalars had a mass, only the density E·Bwithin a distance of about 188 Chapter 5 m−1 awould eﬀectively act as a source for the local aﬁeld and so it would be much smaller. An inhomogeneous pseudoscalar ﬁeld conﬁguration represents an optically active medium (Fig. 5.8c) as was noted, for example, in the context of axionic domain wall conﬁgurations (Sikivie 1984). To low- est order the dispersion relation for left- and right-handed circularly polarized light is (e.g. Harari and Sikivie 1992) k=ω±1 2ga ˆk· ∇a, (5.35) so that the momentum is shifted by a frequency-independent amount. The corresponding refractive index is n= 1±1 2ga ω−1ˆk· ∇a. Taking account of the relativistic metric in the strong gravitational ﬁeld of a pulsar, Mohanty and Nayak (1993) then found for the time de- lay between circularly polarized waves which propagate approximately along the polar axis δt=2 5(2 575)2 g4 a B4 0R11Ω2cos4α ω2(GNM)6. (5.36) For the pulsar PSR 1937+21 a polarimetric analysis yields a time delay 0.37±0.67µs and thus a 1 σupper limit of δt < 1µs (Klein and Thorsett 1990). For typical pulsar parameters this allowed Mo- hanty and Nayak (1993) to place a limit on the arion-photon coupling ofga ∼<2×10−11GeV−1. 5.5.3 Conversion of Stellar Arions in the Galactic Field Stars are powerful sources for pseudoscalars which can be produced in the hot interior by the Primakoﬀ process, i.e. by the conversion γ→a in the electric ﬁelds of the charged medium constituents. Outside of the star, the pseudoscalars can be converted back to photons in the galactic magnetic ﬁeld so that stars would appear to be sources of x- orγ-rays, depending on the characteristic energy of the stellar core (Carlson 1995). In the galaxy, photons propagate with an eﬀective mass given by the plasma frequency ωPwhich for typical electron densities of order 0.1 cm−3is of order 10−11eV. As discussed in Sect. 5.4.1 the pseu- doscalar to photon conversion process is an oscillation phenomenon with a mixing angle given by Eq. (5.28); in the present context it is 1 2tan 2θ=ga BTω m2 a−ω2 P. (5.37) A typical galactic ﬁeld strength is 1 µG, a typical energy at most of or- der 100 MeV for axions from supernovae. With ga <0.6×10−10GeV−1 Two-Photon Coupling of Low-Mass Bosons 189 one ﬁndsga BTω∼<10−19eV2. For the allowed range of axion masses this mixing angle is too small to yield a signiﬁcant conversion eﬀect. Therefore, this entire line of argument is only relevant for massless (or at least very low-mass) pseudoscalars which again shall be referred to as “arions.” Carlson (1995) considered the star α-Ori (Betelgeuse), a red supergiant about 100 pc away from us. He estimated its arion luminosity from the Primakoﬀ process, and compared the expected x-ray ﬂux with data from the HEAO-1 satellite. As a result, a new limit ofga ∼<2.5×10−11GeV−1emerged which is more restrictive than the above bound from globular-cluster stars. Carlson’s argument yields an even more restrictive limit if applied to SN 1987A. One may estimate the arion luminosity of the SN core on the basis of the Primakoﬀ process. If arions couple to quarks or elec- trons, the luminosity can only be higher because existing axion limits already indicate that arions cannot be trapped by these couplings. In order to evaluate Eq. (5.9) an average temperature of 30 MeV and an average density of 3 ×1014g cm−3with a proton fraction of 0 .3 is used (Sect. 13.4.2). The Debye screening scale by the protons is then found to be 36 MeV so that κ2= 1.41 in Eq. (5.10) leading to F= 0.72. Therefore, the average energy loss rate is about g2 101.4×1016erg g−1s−1. Taking a core mass of 1 M⊙= 2×1033g and a duration of 3 s one ex- pects about g2 101050erg to be emitted in arions which is about 10−3g2 10 of the energy emitted in each neutrino ﬂavor. Typical arion energies are 3 T≈100 MeV so that Eq. (5.37) together withωP≈10−11eV in the interstellar medium reveals that mixing is nearly maximal for the relevant circumstances. Therefore, the oscil- lation length is given by ℓosc= 4π/g a BTwhich is about 40 kpc for BT= 1µG andga = 10−10GeV−1. Therefore, ℓoscfar exceeds the relevant magnetic ﬁeld region which is of order 1 kpc as discussed in Sect. 13.3.3b. The conversion rate is then prob(a→γ) = (1 2ga BTℓ)2 = 2.3×10−2g2 10(BTℓ/µG kpc)2, (5.38) whereℓis the eﬀective conversion region (distance to source or distance within magnetic ﬁeld region), and g10≡ga /10−10GeV−1. With Eq. (5.38) and an eﬀective magnetic conversion region of ℓ= 1 kpc the expected energy showing up as γ-rays at Earth corresponds to about 10−5g4 10relative to the energy in one neutrino species. In Sect. 12.4.3 the radiative decays of low-mass neutrinos from SN 1987A was discussed. On the basis of the SMM data it was found that less 190 Chapter 5 than about 10−9of a given neutrino species may show up in the form of decay photons (Fig. 12.9) if the spectral distribution is taken to be characterized by T≈30 MeV. Therefore, one ﬁnds a limit of g4 10∼< 10−4orga ∼<10−11GeV−1, applicable if the particle mass is below about 10−10eV. This is more restrictive than Carlson’s original limit, and of the same order as the PSR 1937+21 birefringence limit quoted after Eq. (5.36). 5.5.4 Polarimetry of Distant Radio Sources Nambu-Goldstone bosons aare by deﬁnition the result of a sponta- neously broken global symmetry. The cosmic evolution from a very hot initial phase begins with the unbroken symmetry; as the universe expands and cools a phase transition will occur where the ﬁeld respon- sible for the spontaneous breakdown must ﬁnd its new minimum. As this process occurs independently in each causally connected region of the universe at that time, the universe today will be characterized by diﬀerent orientations of the ground state, i.e. by diﬀerent values of a classical background aﬁeld. If no inﬂation occurred in the universe after the phase transition, and if the Nambu-Goldstone bosons remain truly massless (in contrast with axions), the background ﬁeld will not have relaxed to a common ground state everywhere. In this scenario a radio signal from a distant source travels through regions with diﬀerent values of the classical aﬁeld, and thus through regions of gradients ∇awhich act as an optically active medium accord- ing to Eq. (5.35). Therefore, linearly polarized light will experience a random rotation of its plane of polarization. This eﬀect is also expected from the Faraday rotation caused by intervening magnetic ﬁelds which induce optical activity in the cosmic background plasma. As this eﬀect is frequency dependent it can be removed by observing a given object at diﬀerent wavelengths. The eﬀect induced by pseudoscalars, on the other hand, is independent of frequency. A systematic correlation between the geometric shape of distant radio sources and the linear polarization of the emitted radiation has been observed. This correlation proves that no random rotation of the plane of polarization occurs over cosmic distances, except for the Faraday eﬀect which can be removed from the data. Therefore, the maximum allowed coupling strength of photons to a random cosmic Nambu-Goldstone ﬁeld can be constrained. Harari and Sikivie (1992) found thatCa ∼<50 in Eq. (5.5), independently of the symmetry break- Two-Photon Coupling of Low-Mass Bosons 191 ing scalefa. This bound supersedes Sikivie’s (1988) previous scenario where he tried to explain the polarization features of certain sources by the conversion of cosmic-string-produced Nambu-Goldstone bosons to photons in cosmic magnetic ﬁelds. This scenario would have re- quiredCa ≈105. 5.5.5 Temperature Fluctuations in the Cosmic Microwave Background In the presence of large-scale magnetic ﬁelds in the universe, photons of the cosmic microwave background radiation (CMBR) could convert into arions. The angular variations of the CMBR temperature have been measured by the COBE satellite and other instruments to be extremely small; a typical value is δT/T≈10−5. Therefore, the conversion process must not have been very eﬃcient between the surface of last scattering and us. This argument has been studied in detail by Chen (1995) for photon- graviton conversion which is a very similar eﬀect due to the two-photon coupling vertex which the massless gravitons must have. The coupling constant involves the inverse Planck mass. Therefore, one may also expect interesting eﬀects for hypothetical arions which could couple to photons more strongly than gravitons do. 5.6 Summary of Constraints on gaγ The astrophysical and experimental bounds on the photon coupling of arbitrary pseudoscalars are summarized in Fig. 5.9. “Haloscope” refers to the search for galactic axions discussed in Sect. 5.3 and so these constraints (Fig. 5.7) apply only if the pseudoscalars are the dark matter in our galaxy. The dotted line is the search regime for the ongoing experiment mentioned in Sect. 5.3. “Helioscope” refers to the search for solar axion to x-ray conversion (Sect. 5.4.2). It is shown as a dashed line because it is not self-consistent in that it assumes an unperturbed Sun—the area enclosed by the dashed line is already excluded by the solar age. “Telescope” refers to the search for decay photons from the cosmic axion background (Sect. 12.7.2, Fig. 12.23). It is assumed that the pseudoscalars were in thermal equilibrium in the early universe. “Laser” refers to the birefringence and shining-light-through-walls experiments discussed above. The most restrictive such limit is from the rotation of the plane of polarization of a laser beam trapped in 192 Chapter 5 an optical cavity in a strong transverse magnetic ﬁeld (Sect. 5.4.4). The dotted line marks roughly the expected range of sensitivity of the PVLAS experiment (Sect. 5.4.4). The solar limit (Eq. 5.22) is based on the Primakoﬀ energy loss and the requirement that axions must not exceed the photon luminosity; otherwise the Sun could not have reached its present-day age. The HB-star limit (Eq. 5.23) comes from the requirement that these objects do not spend their nuclear fuel so fast that their observable number in globular clusters is reduced by more than a factor of ≈2. The “axion line” refers to models where E/N = 8/3 orξ= 1 in Eq. (14.24). For very low-mass bosons ( ma∼<10−10eV) the SN 1987A ﬂux of pseudoscalars would be eﬃciently converted into γ-rays, leading to a limitga ∼<10−11GeV−1(Sect. 5.5.3). Fig. 5.9. Bounds on the photon coupling ga as a function of mafor arbitrary pseudoscalars; see the text for details. (Adapted from Cameron et al. 1993.) Chapter 6 Particle Dispersion and Decays in Media Dispersion eﬀects in media have a signiﬁcant impact on the propaga- tion of some low-mass particles (photons, neutrinos) while others are left unaﬀected (axions and other Nambu-Goldstone bosons). The rela- tionship between forward scattering and refraction is derived, and the dispersion relations for photons and neutrinos are thoroughly studied. Modiﬁed particle dispersion relations allow certain decay processes to occur in media that cannot occur in vacuum, notably the photon de- cayγ→ννwhich dominates the neutrino emissivity in a wide range of temperatures and densities (“plasma process”). Other examples are the neutrino and majoron decay ν→νχandχ→νν, respectively. The rates for such processes are derived. The plasma process allows one to derive the most restrictive limits on neutrino magnetic dipole moments. Screening eﬀects in reactions involving Coulomb scattering, and neutrino electromagnetic form factors in media are discussed. 6.1 Introduction Particles are the quantized excitations of certain ﬁelds—photons of the electromagnetic ﬁeld, electrons of the electron ﬁeld, and so forth. It is usually convenient to expand these ﬁelds in plane waves character- ized by frequencies ωand wave vectors k; the excitations of these modes then exhibit a temporal and spatial behavior proportional to e−i(ωt−k·x). The frequency for a given wave number is determined by the disper- sion relation. Because ( ω,k) is a four-vector, and because of Lorentz invariance, in vacuum the quantity ω2−k2=m2is the same for all 193 194 Chapter 6 frequencies; mhas the usual interpretation of a particle mass. One con- sequence of this covariant dispersion relation is that decays of the sort 1→2+3 are only possible if m1>m 2+m3so that in the rest frame of particle 1 there is enough energy available to produce the ﬁnal states. In media the dispersion relations are generally modiﬁed by the co- herent interactions with the “background.” In the simplest case a parti- cle acquires a medium-induced eﬀective mass. For example, photons in a nonrelativistic plasma acquire a dispersion relation ω2=ω2 P+k2with the plasma frequency given by ω2 P= 4παn e/me(electron density ne). ForωP>2mνthis implies that the decay γ→ννbecomes kinemati- cally possible and occurs in stars because the ambient electrons mediate an eﬀective neutrino-photon interaction (Adams, Ruderman, and Woo 1963). In fact, this “plasma process” is the dominant neutrino source in a wide range of temperatures and densities which covers, for example, white dwarfs and red-giant stars (Appendices C and D). Neutrinos may have nonstandard electromagnetic couplings, no- tably magnetic dipole moments, which would enhance the plasma pro- cess and thus the cooling of stars (Bernstein, Ruderman, and Feinberg 1963). Observational constraints on anomalous cooling rates derived from white dwarfs and globular-cluster stars then provide the most restrictive limits on neutrino electromagnetic couplings (Sect. 6.5.6). Within the standard model all fermions are fundamentally massless; they acquire an eﬀective mass by their interaction with the vacuum expectation value Φ 0of a scalar Higgs ﬁeld (Sect. 8.1.1). Therefore, even vacuum masses can be interpreted as “refractive” phenomena. Because the scalar Φ 0is Lorentz invariant the dispersion relation thus induced is of the standard form E2=m2+p2. “Normal” media, however, single out a preferred Lorentz frame, usually causing E(p) to be a more complicated function than ( m2+p2)1/2. Notably, the dispersion relation can be such that the four-momen- tumP= (E,p) is “space-like,” P2=E2−p2<0, which amounts to a “negative mass-square” P2=m2 eﬀ<0. There is nothing wrong with such “tachyons” because the speed of signal propagation safely remains below the speed of light (Sect. 6.2.2). The dispersion relation in isotropic media is often expressed as k=|k|=nωin terms of a refractive index n. Space-like excitations correspond to n>1; examples are photons in water or air. In this case the well-known decay process e→eγis kinematically allowed for suﬃciently fast moving electrons (Cherenkov radiation). The dispersion relation can also depend on the spin polarization of the radiation. In “optically active” media, the left- and right-handed Particle Dispersion and Decays in Media 195 circular photon polarizations acquire diﬀerent refractive indices. In this sense all media are optically active for neutrinos where only the left- handed states interact while the right-handed ones are “sterile.” For Majorana neutrinos the helicity-plus states are equivalent to ν’s which acquire an opposite energy shift from νso that there is an energy gap betweenν(p) andν(p). Therefore, in a medium the majoron decays ν→νχorχ→ννbecome possible where the majoron χis a massless particle (Sect. 6.8). The interaction of νµandντwith a normal medium is diﬀerent from that of νebecause of a charged-current νe-e−scattering ampli- tude. Therefore, normal media are “ﬂavor birefringent” in the sense that the medium induces diﬀerent dispersion relations for neutrinos of diﬀerent ﬂavors. The importance of this eﬀect for neutrino oscilla- tions, which eﬀectively measure relative phases in the propagation of diﬀerent-ﬂavored neutrinos, cannot be overstated. It must be stressed that usually allparticles acquire nontrivial dis- persion relations in media although it depends on the detailed circum- stances whether or not the refractive eﬀect is signiﬁcant. For example, until recently one found statements in the literature that in a suﬃciently dense medium where ωP>2mephotons were damped by electromag- netic pair production γ→e+e−. However, this is incorrect because the charged leptons also acquire a medium-induced eﬀective mass which is so large that this decay never occurs (Braaten 1991). On the other hand, the above majoron decay is only possible because the majorons χare Nambu-Goldstone bosons and thus remain massless even in a medium, at least to lowest order (Sect. 6.8). Besides modifying the dispersion relation of particles it is also possi- ble that the presence of the medium allows for entirely new excitations. The best known example is the longitudinal polarization state of the electromagnetic ﬁeld which exists in a plasma in addition to the usual states with transverse polarization. These “plasmons” were ﬁrst dis- cussed by Langmuir (1926). Another example from electromagnetism are the “plasminos,” spin-1 2excitations of a plasma that were discussed for the ﬁrst time only very recently (Klimov 1981; Weldon 1982b, 1989; Pisarski 1989; Braaten 1992). For many purposes such (quantized) col- lective modes play the same role as the usual particles. For example, in a medium both photons and plasmons can decay into neutrinos and thus contribute to the plasma process of neutrino emission. In the present chapter I will follow up these questions in detail. While the dispersion relations and couplings of particles in media are formally best dealt with in terms of ﬁeld theory at ﬁnite temperature 196 Chapter 6 and density, most of the results relevant for particle physics in stars predate the development of this formalism; they were based on the old-fashioned tools of kinetic theory. Indeed, for simple issues of dis- persion or collective eﬀects a kinetic approach seems often physically more transparent while yielding identical results. At any rate, the fol- lowing discussion is based entirely on kinetic theory. 6.2 Particle Dispersion in Media 6.2.1 Refractive Index and Forward Scattering How does one go about to calculate the all-important dispersion rela- tion for a given particle in a medium with known properties? Usually it is enough to follow the elementary approach of calculating the for- ward scattering amplitude of the relevant ﬁeld excitations with the con- stituents of the background medium, an approach which has the added advantage of physical transparency over a more formal procedure.30 To begin, consider a scalar ﬁeld Φ which may be viewed as repre- senting one of the photon or electron polarization states. If a plane wave excitation of that ﬁeld with a frequency ωand a wave vector kin- teracts with a scatterer at location r= 0 an additional spherical wave will be created. The asymptotic form of the original plus scattered wave is Φ(r,t)∝e−iωt( eik·r+f(ω,θ)eikr r) , (6.1) wherek=|k|,r=|r|, andfis the scattering amplitude. It was as- sumed that it has no azimuthal dependence, something that will always apply on average for a collection of randomly oriented scatterers. The diﬀerential scattering cross section is dσ/d Ω =|f(ω,θ)|2. If there is a collection of scattering centers randomly distributed in space, all of the individual scattered waves will interfere. However, because of the random location of the scatterers, constructive and de- structive interference terms will average to zero. Thus the total cross section of the ensemble is the (incoherent) sum of the individual ones. In the forward direction, however, the scattered waves add up co- herently with each other and with the parent wave, leading to a phase shift and thus to refraction. This is seen if one considers a plane wave in thez-direction incident on an inﬁnitesimally thin slab (thickness δa) at 30The derivation below follows closely the exposition of Sakurai (1967). Particle Dispersion and Decays in Media 197 z= 0 which contains nscattering centers per unit volume, and which is inﬁnite in the x- andy-directions. At a distance zfrom the slab, large compared with k−1, the asymptotic form of the parent plus scattered wave is, ignoring the temporal variation e−iωt, Φ(z)∝eiωz+nδa∫∞ 0eik(ρ2+z2)1/2 (ρ2+z2)1/2f(ω,θ) 2πρdρ, (6.2) whereρ≡(x2+y2)1/2andθ= arctan(ρ/z). Moreover, it was assumed that in vacuum the wave propagates relativistically so that k=ω. The integral in Eq. (6.2) is ill deﬁned because the integrand os- cillates with a ﬁnite amplitude even for large values of ρ. It is made convergent by substituting k→k+iκwithκ >0 an inﬁnitely small real parameter. Integration by parts then yields Φ(z)∝eiωz[ 1 +i2πnδa ωf0(ω)] , (6.3) where a term of order ( ωz)−1was neglected which becomes small for largez. Here,f0(ω)≡f(ω,0) is the forward scattering amplitude . Turn next to a slab of ﬁnite thickness a. The phase change of the transmitted wave is obtained by compounding inﬁnitesimal ones with δa=a/jand taking the limit j→ ∞ , lim j→∞[ 1 +i2πna jωf0(ω)]j =ei(2πn/ω )f0a. (6.4) Inserting this result in Eq. (6.3) reveals that over a distance ain the medium the wave accumulates a phase einrefrωawhere nrefr= 1 +2π ω2nf0(ω) (6.5) is recognized as the index of refraction. If the relativistic approximation |nrefr−1| ≪1 is not valid one must treat the wave self-consistently in the medium and distinguish carefully between frequency and wavenumber. In this case one ﬁnds (Foldy 1945) n2 refr= 1 +4π k2nf0(k), (6.6) where the forward scattering amplitude must be calculated taking the modiﬁed dispersion relation into account. For the propagation of a ﬁeld with several spin or ﬂavor components the same result applies if one remembers that “forward scattering” not 198 Chapter 6 only refers to scattering in the forward direction, but that all properties of the wave and the scatterer are left unchanged. If the medium parti- cles have a distribution of momenta, spins, etc. the forward scattering amplitude must be averaged over those quantities, and diﬀerent species of medium particles must be summed over. For a practical calculation it helps to recall that dσ/d Ω =|f(θ)|2 so that |f0|is the square root of the forward diﬀerential cross section. For example, the Thomson cross section for photons interacting with nonrelativistic electrons is dσ/d Ω = (α/m e)2|ϵ·ϵ′|2with the polariza- tion vectors ϵandϵ′of the initial- and ﬁnal-state photon. Forward scattering implies |ϵ·ϵ′|2= 1 so that |f0|=α/m e. The dispersion rela- tion is then ω2=k2+ω2 Pwith the plasma frequency ω2 P= 4παn e/me. Of course, the absolute sign of f0has to be derived from some other information—for photon dispersion see Sect. 6.3. The forward scattering amplitude and the refractive index are gen- erally complex numbers. Physically it is evident that in a medium the intensity of a beam is depleted as e−z/ℓ. The mean free path is given by ℓ−1=σnvwhereσis the total scattering cross section, nis the number density of scatterers, and vis the velocity of propagation. Thus the amplitude of a plane wave varies as eikz−z/2ℓ. Moreover, the derivation of the refractive index indicates that the amplitude varies according to einrefrωz, yieldingk= Renrefrωand (2ℓ)−1= Imnrefrω. For relativistic propagation ( v= 1) the last equation implies σ(ω) = (4π/ω) Imf0(ω), a relationship known as the optical theorem. For the applications discussed in this book speciﬁc interaction mod- els between the propagating particles and the medium will be assumed so that it is usually straightforward to calculate the dispersion relation according to Eq. (6.5). One should keep in mind, however, that nrefras a function of ωhas a number of general properties, independently of the interaction model. For example, its real and imaginary part are con- nected by the Kramers-Kronig relations (Sakurai 1967; Jackson 1975). 6.2.2 Particle Momentum and Velocity The four-vector ( ω,k) which governs the spatial and temporal behavior of a plane wave can be time-like ( ω2−k2>0) as for massive particles in vacuum, it can be light-like ( ω2−k2= 0) as for photons in vacuum, or it can be space-like ( ω2−k2<0) as for visible light in water or air. Because the quantized excitations of such ﬁeld modes are interpreted as particles, E= ¯hωis the particle’s energy. (I have temporarily re- stored ¯heven though it is 1 in natural units.) Similarly one may be Particle Dispersion and Decays in Media 199 tempted to interpret p= ¯hkas the particle’s momentum. In vacuum a particle’s velocity is p/E, a quantity which exceeds the speed of light for space-like excitations. Occasionally one reads in the literature that for this reason only those branches of a particle dispersion relation were physical where |p|< E. Such statements are incorrect, however, and the underlying concern about tachyonic propagation is unfounded. The quantity p/Ehas no general physical relevance. Two signiﬁ- cant velocity deﬁnitions are the phase velocity and the group velocity of a wave (Jackson 1975). The former is the speed with which the crest of a plane wave propagates, i.e. it is given by the condition ωt−kz= 0 or vphase =ω/k=n−1 refr. For a massive particle in vacuum vphase>1. How- ever, the phase velocity can drop below the speed of light in a medium. When this occurs for electromagnetic excitations in a plasma, electrons can “surf” in the wave which thus transfers energy at a rate propor- tional to the ﬁne structure constant α(Landau 1946), an eﬀect known as Landau damping. As long as vphase>1 the photon propagation is damped only by Thomson scattering which is an eﬀect of order α2. The group velocity vgroup =dω/dk is the speed with which a wave packet or pulse propagates. In terms of the refractive index it is v−1 group =nrefr(ω) +ωdn refr/dω (6.7) (Jackson 1975). For a massive particle in vacuum with ω2= (k2+m2)1/2 it isvgroup =k/ω< 1, and also in a medium normally vgroup<1. Near a resonance it can happen that vgroup>1, but there is still no reason for alarm. The fast variation of nrefr(ω) as well as the presence of a large imaginary part near a resonance imply that the issue of signal propagation is much more complicated than indicated by the simple approximations which enter the deﬁnition of the group velocity. For a detailed discussion of electromagnetic signal propagation in dispersive media see Jackson (1975). Evidently a naive interpretation of ¯ hkas a particle momentum can be quite misleading. Another example relates to the diﬃculty of sepa- rating the momentum ﬂow of a (light) beam in a medium into one part carried by the wave and one carried by the medium. There was a long- standing dispute in the literature with famous researchers on diﬀerent sides of an argument that was eventually resolved by Peierls (1976); see also Gordon (1973). Experimentally, it was addressed by shining a laser beam vertically through a water-air interface and measuring the deformation of the surface due to the force which must occur because of a photon’s change of momentum between the two media (Ashkin and Dziedzic 1973). 200 Chapter 6 The problem of the physical momentum ﬂow associated with a wave will be of no concern to the issues addressed in this book. In microscopic reactions the quantity which appears in the law of “energy-momentum conservation” is the wave vector. For example, in the plasma process γ→ννthe momenta of the outgoing neutrinos must balance against the wave vector of the decaying electromagnetic excitation. In this book dispersion eﬀects will be important only for pulse propagation from distant sources, for particle oscillation eﬀects, and for energy- momentum conservation in microscopic reactions. In these cases the naive interpretation of ¯ hkas a particle’s momentum is safe. For the remainder of this book the wave number (or pseudomomentum) and the momentum of a ﬁeld excitation will not be distinguished. 6.2.3 Wave-Function Renormalization In particle reactions the main impact of medium-induced modiﬁcations of the dispersion relations is on the kinematics, notably if a threshold condition is involved. One is thus tempted to proceed with the usual Feynman rules and take account of the dispersion relations only in the phase-space integration, notably in the law of energy-momentum conservation. In most practical cases this approach causes no problems, although an exception are interactions involving longitudinal plasmons (Sect. 6.3). Therefore, one should be aware that the matrix element also must be modiﬁed because of the subtle issue of what one means with a “particle” in a medium. After a spatial Fourier transform the equations of motion for the Fourier components ϕkof a free ﬁeld are those of a harmonic oscillator. Interpreting the amplitude ϕkand its velocity ˙ϕkas conjugate variables, the canonical quantization procedure leads to quantized energy levels ¯hωk, whereωkis the classical frequency of the mode kaccording to its dispersion relation. Conversely, a quantized excitation with energy ¯ hωk has a certain ﬁeld strength which determines its coupling strength to a source, e.g. the coupling strength of a photon to an electron. In a medium, the energy associated with a frequency ωkis still ¯hωk. However, because of the presence of interaction energy between ϕand the medium, the ﬁeld strength associated with a quantized excitation is modiﬁed. For example, photons with a given frequency couple with a diﬀerent strength to electrons in a medium than they do in vacuum. This modiﬁcation can be lumped into a “renormalization factor”√ Z of the coupling strength of external photon lines in a Feynman graph. Particle Dispersion and Decays in Media 201 In order to determine this factor from the dispersion relation con- sider a scalar ﬁeld ϕin the presence of a medium which induces a re- fractive index. This means that the Klein-Gordon equation in Fourier space, including a source term ρ, is of the form [−K2+ Π(K)]ϕ(K) =gρ(K), (6.8) whereK= (ω,k) is a four-vector in Fourier space and gis a coupling constant. Π( K) is the “self-energy” which includes a possible vacuum massm2and medium-induced contributions which are calculated from the forward scattering amplitude. The homogeneous equation with ρ= 0 has nonvanishing solutions only for K2= Π(K) which deﬁnes the dispersion relation ω2 k−k2= Π(ωk,k). (6.9) This equation determines implicitly the frequency ωkrelated to a wave number kfor a freely propagating mode. A problem with Eq. (6.8) is the general dependence of Π on ω andkwhich implies “dispersion,” i.e. in coordinate space it is not a simple second-order diﬀerential equation. Otherwise the equation of motion for a single ﬁeld mode ϕkwould be (∂2 t+k2+ Π k)ϕk=gρk. Apart from the source term this is a simple harmonic oscillator with frequencyω2 k=k2+ Π k. The canonical quantization procedure then leads to quantized excitations with energy ¯ hωk—the usual “particles.” In a medium one follows this procedure in an approximate sense by expanding Π k(ω)≡Π(ω,k) to lowest order around ωk, Πk(ω) = Π k(ωk) + Π′ k(ωk)(ω−ωk), (6.10) where Π′ k(ω)≡∂ωΠk(ω). To this order the Klein-Gordon equation is [ −ω2+ω2 k+ Π′ k(ωk)(ω−ωk)] ϕk(ω) =gρk(ω), (6.11) where the dispersion relation Eq. (6.9) was used. To ﬁrst order in ω−ωk one may use 2 ω= 2ωk=ω+ωkwhich allows one to write Z−1(ω2−ω2 k)ϕk(ω) =gρk(ω), (6.12) where Z−1≡2ωk−Π′ k(ωk) 2ωk= 1−∂Π(ω,k) ∂ω2 ω2−k2=Π(ω,k). (6.13) 202 Chapter 6 BecauseZis a constant for a ﬁxed kthe approximate equation of motion EE405 corresponds to a Hamiltonian H=H0+Hint=1 2Z−1(˙ϕ2 k+ω2 kϕ2 k) +gϕkρk. (6.14) The free-ﬁeld term is of the standard harmonic-oscillator form if one substitutes ϕk=√ Zeϕk, i.e. free particles are excitations of the ﬁeld eϕkwhich has a renormalized amplitude relative to ϕk. In terms of the renormalized ﬁeld the interaction Hamiltonian is now of the form Hint=√ Zgeϕkρkwhich means that particles in the medium interact with an external source with a modiﬁed strength√ Zg. There- fore, in Feynman graphs one must include one factor of√ Zfor each external line of the ϕﬁeld. Equivalently, the squared matrix element involves a factor Zfor each external ϕparticle. For relativistic modes where |ω2 k−k2| ≪ω2 kthe modiﬁcation is inevitably small, |Z−1| ≪1. For (longitudinal) plasmons, however, the dispersion relation in a nonrelativistic plasma is approximately ω=ωP with the plasma frequency ωP, i.e. they are far away from the light cone, and then Zis a nonnegligible correction (Sect. 6.3). In the original calculation of the plasma decay process γ→ννan incorrect Zwas used for the longitudinal excitations (Adams, Ruderman, and Woo 1963). The correct factor was derived by Zaidi (1965). 6.3 Photon Dispersion 6.3.1 Maxwell’s Equations For the astrophysical applications relevant to this book the photon re- fractive index in a fully ionized plasma consisting of nuclei and electrons will be needed. On the quantum level, this system is entirely described by quantum electrodynamics (QED). It is sometimes referred to as a QED plasma—in contrast with a quark-gluon plasma which is described by quantum chromodynamics (QCD). The calculation of the refractive index amounts to an evaluation of the forward scattering amplitude of photons on electrons, a simple task except for the complications from the statistical averaging over the electrons which are partially or fully relativistic and exhibit any degree of degeneracy. Recently Braaten and Segel (1993) have found an astonishing simpliﬁcation of this daunting problem (Sect. 6.3.4). A more conceptual complication is the occurrence of a third photon degree of freedom in a medium (Langmuir 1926), sometimes referred to Particle Dispersion and Decays in Media 203 as Langmuir waves or plasmons.31Still, one should not think of photons as becoming literally massive like a massive vector boson which also carries three polarization states. A photon mass is prohibited by gauge invariance which remains intact. However, the medium singles out an inertial frame and thus breaks Lorentz invariance, an eﬀect which is ultimately responsible for the possibility of a third polarization state. It is useful, then, to begin with some general aspects of photon prop- agation in a medium which are unrelated to speciﬁc assumptions about the medium constituents. Notably, begin with the classical Maxwell equations for the electric and magnetic ﬁelds ∇ ·E=ρ,∇ ×B−˙E=J, ∇ ·B= 0, ∇ ×E+˙B= 0. (6.15) An additional condition is that the electric charge density ρand current density Jobey the continuity equation ∂·J= ˙ρ− ∇ · J= 0, (6.16) whereJ= (ρ,J) and∂= (∂t,∇). Covariantly, Maxwell’s equations are ∂µFµν=Jν, ϵµνρσ∂µFρσ= 0, (6.17) whereFµνis the antisymmetric ﬁeld-strength tensor with the nonvan- ishing components F0i=−Fi0=−Ei, andFij=−Fji=−ϵijkBk. Applying∂µto the inhomogeneous equation and observing that Fµν is antisymmetric and ∂µ∂νsymmetric under µ↔νreveals that for consistency Jmust obey the continuity equation. An equivalent formulation arises from expressing the ﬁeld strengths in terms of a four-potential A= (Φ,A) by virtue of Fµν=∂µAν−∂νAµ, (6.18) which amounts to E=−∇Φ−˙AandB=∇×A. This representation is enabled by the homogeneous set of Maxwell equations which are then automatically satisﬁed. The inhomogeneous set now takes the form A−∂(∂·A) =J, (6.19) where =∂·∂=∂µ∂µ=∂2 t− ∇2. 31They are sometimes called “longitudinal plasmons” in contrast to “transverse plasmons.” In this nomenclature the term “plasmon” refers to any excitation of the electromagnetic ﬁeld in a medium while “photon” refers to an excitation in vacuum. 204 Chapter 6 The ﬁeld-strength tensor contains six independent degrees of free- dom, the EandBﬁelds. The redundancy imposed by the constraint of the homogeneous equations was removed by introducing the vector po- tential. There still remains one redundant degree of freedom related to a constraint imposed by current conservation. The Maxwell equations remain invariant under a “gauge transformation” A→A−∂αwhereα is an arbitrary scalar function. The modiﬁed Ayields the same ﬁelds EandBwhich are the physically measurable quantities. The relationship to current conservation is easiest recognized if one recalls that Maxwell’s equations can be derived from a Lagrangian −1 4F2−J·AwhereF2=FµνFµν. A gauge transformation introduces an additional term J·∂αwhich is identical to a total divergence ∂·(αJ) if∂·J= 0 and thus leaves the Euler-Langrange equations unchanged. Indeed, current conservation is a necessary and suﬃcient condition for the gauge invariance of the theory (Itzykson and Zuber 1983). A judicious choice of gauge can simplify the equations enormously. Two important possibilities are the Lorentz gauge and the Coulomb, transverse, or radiation gauge, based on the conditions ∂·A= 0, Lorentz gauge, ∇ ·A= 0, Coulomb gauge. (6.20) Maxwell’s equations are then found to be (Jackson 1975) Φ =ρ, A=J, Lorentz gauge, −∇2Φ =ρ, A=JT, Coulomb gauge, (6.21) where JTis the transverse part of Jcharacterized by ∇ ·JT= 0. In the absence of sources ( ρ= 0 and J= 0) the potential Φ vanishes in the Coulomb gauge while Aobeys a wave equation. A Fourier trans- formation leads to ( −k2+ω2)A= 0 whence the propagating modes have the dispersion relation k2=ω2corresponding to massless parti- cles. Because of the transversality condition k·A= 0 there are only two polarizations, the usual transverse electromagnetic waves. They are characterized by an electric ﬁeld Etransverse to kand a magnetic ﬁeld of the same magnitude transverse to both. 6.3.2 Linear Response of the Medium Maxwell’s equations allow one to calculate the electromagnetic ﬁelds in the presence of prescribed external currents. However, the charged Particle Dispersion and Decays in Media 205 particles which constitute the currents move themselves under the in- ﬂuence of electromagnetic ﬁelds. Therefore, the interaction between ﬁelds and currents must be calculated self-consistently. If the ﬁelds are suﬃciently weak one may assume that the reaction of the currents to the ﬁelds can be described as a linear response. (For a general review of linear-response theory in electromagnetism see Kirzhnits 1987.) In general this statement cannot be made locally in the sense that the currents at space-time point ( t,x) were only linear functions of A(t,x). Within the restrictions imposed by causality the relation- ship between ﬁelds and currents is nonlocal; for example, a solution of Maxwell’s equations with prescribed currents requires integrations over the sources in space and time. After a Fourier transformation, however, the assumption of a linear response can be stated as Jµ ind=−ΠµνAν. (6.22) The polarization tensor Π( K) withK= (ω,k) is a function of the medium properties. Besides the induced current there may be an externally prescribed oneJextwhich is unrelated to the response of the microscopic medium constituents to the ﬁelds; the total current is J=Jind+Jext. Maxwell’s equations (6.19) are then in Fourier space (−K2gµν+KµKν+ Πµν)Aν=Jµ ext. (6.23) Invariance under a gauge transformation Aν→Aν+Kναrequires that ΠµνKν= 0. Because the external and total currents are conserved the induced current is conserved as well, leading to K·Jind= 0 or KµΠµν= 0. Altogether KµΠµν= ΠµνKν= 0 (6.24) which is an important general property of the polarization tensor. Considering the Maxwell equations in Coulomb gauge in the ab- sence of external currents, the transversality of Astill implies that it provides only two wave polarization states, albeit with modiﬁed disper- sion relations due to the presence of Π. With regard to the Φ equation note that in an isotropic medium the induced charge density ρindmust be a spatial scalar and so can depend only on Φ and the combination k·A= 0 which is the only available scalar linear in A. Therefore, the homogeneous equation for Φ is (k2+ Π00)Φ = 0. (6.25) Because Π00is a function of ωandkthis is a wave equation with the dispersion relation k2+ Π00(ω,k) = 0. The electric ﬁeld associated 206 Chapter 6 with this third polarization degree of freedom is proportional to kΦ, along the direction of propagation—hence the term longitudinal exci- tation. There is no magnetic ﬁeld associated with it. Physically, it corresponds to a density wave of the electrons much like a sound wave. Obviously this mode requires being carried by a medium, as opposed to the transverse waves which propagate in vacuum as well. The wave equation Eq. (6.23) corresponds to a Langrangian density in Fourier space which involves a new term −VwithV=1 2AµΠµνAν which plays the role of a medium-induced potential energy for the ﬁeldA. In vacuum Πµνcan be constructed only from gµνandKµKν which both violate the gauge condition Π K= 0. Notably, this forbids a photon mass term which would have to be of the form Πµν mass=m2gµν. In a medium an inertial frame is singled out, allowing one to construct Π from the medium four-velocity Uand to ﬁnd a structure which obeys the gauge constraint. Strictly speaking, however, the medium does not induce an eﬀective-mass term which would be of the form m2 eﬀgµνand which remains forbidden by gauge invariance. 6.3.3 Isotropic Polarization Tensor in the Lorentz Gauge The Coulomb gauge is well suited to treat radiation in vacuum because the propagating modes are neatly separated from the scalar potential, and the gauge component is easily identiﬁed with the longitudinal part ofA. In a medium, however, the diﬀerent appearance of Φ and Ain their respective wave equations is cumbersome. The Maxwell equations in Lorentz gauge are symmetric between Φ and Awhich allows one to treat all polarization states on the same footing. In order to construct the most general Π( K) for an isotropic medium it is useful to deﬁne four basis vectors for Minkowski space which are adapted to the symmetry of the medium as well as to the Lorentz condition (Weldon 1982a; Haft 1993). For that purpose one may use the preferred directions in Minkowski space, namely Kand the four- velocity of the medium Uwhich is (1,0) in its inertial frame. Moreover, the notation ωandkis used for the frequency and wave vector of K in the medium frame; they are covariantly given by ω=U·Kand k2=k2= (U·K)2−K2. In Lorentz gauge the physical Aﬁelds obey K·A= 0. Therefore, one deﬁnes a basis vector for the gauge degree of freedom by eg≡K/√ K2. (6.26) Next, one chooses a vector which is longitudinal relative to the spatial Particle Dispersion and Decays in Media 207 part ofKand which obeys K·eL= 0, eL≡ωK−K2U k√ K2=(k2,ωk) k√ K2, (6.27) where the second expression refers to the medium rest frame. There remain two directions orthogonal to egandeL, or equivalently, to K andU. Ifkis taken to point in the z-direction two possible choices are the unit vectors exandey, respectively. However, in order to retain the azimuthal symmetry around the kdirection the “circular polarization vectors” e±= (ex±iey)/√ 2 are needed. Then e±≡(0,e±) (6.28) in the rest frame of the medium; a suitable covariant formulation is also possible. The basis vectors obey e∗ ±=e∓whileeg,Lare real for K2>0 (time-like) and e∗ g,L=−eg,LforK2<0 (space-like). They are normalized according to e∗ ±·e±=e∓·e±=−1 ande∗ L·eL=∓1 and e∗ g·eg=±1, depending on K2being time- or space-like. Evidently, eg andeLswitch properties between a time- and space-like K. This choice of basis vectors is only possible if K2̸= 0. IfKis light-like one must make some other choice, for example (1 ,0) and (0,ˆk) in the rest frame of the medium. Because the goal is to describe electromagnetic excitations in a medium, and because usually K2̸= 0 for such waves, this is no serious limitation. It will turn out that the dispersion relation of (longitudinal) plasmons crosses the light-cone, i.e. there is a wave number for which ω2−k2= 0. The degeneracy of eg witheLat this single point will cause no trouble. The most general polarization tensor compatible with the gauge condition Eq. (6.24) must be constructed from e±andeLalone. Az- imuthal symmetry about the kdirection requires that they occur only in the scalar combinations eµ ae∗ν a. Therefore, one deﬁnes the projection operators on the basis vectors Pµν a≡ −eµ ae∗ν a, (a=±,L). (6.29) The most general polarization tensor is then given as Πµν=∑ a=±,LπaPµν a, (6.30) where the πaare functions of the Lorentz scalars K2andU·Kor equivalently of ωandkin the medium frame. They represent the medium response to circularly and longitudinally polarized A’s. 208 Chapter 6 The homogeneous Maxwell equations in Lorentz gauge in an iso- tropic medium then have the most general form ( −K2gµν+∑ a=±,LπaPµν a) Aν= 0. (6.31) The metric tensor is g=Pg+PL+P++P−wherePµν g=eµ ge∗ν g. Hence one obtains decoupled wave equations [ −ω2+k2+πa(ω,k)]Aa= 0 for the physical degrees of freedom Aa=PaAwitha=±,L. The corre- sponding dispersion relation is −ω2+k2+πa(ω,k) = 0. (6.32) It yields the frequency ωkfor modes with a given polarization and wave number. The so-called eﬀective mass is then m2 eﬀ=πa(ωk,k). This expression is diﬀerent for diﬀerent polarizations and wave numbers, and may even be negative. Generally, an isotropic medium is characterized by three diﬀerent response functions because the left- and right-handed circular polariza- tion states may experience diﬀerent indices of refraction (Nieves and Pal 1989a,b). Such optically active media are not symmetric under a parity transformation. For example, a sugar solution changes under a spatial reﬂection because the sugar molecules have a deﬁnite handedness. If the medium and all relevant interactions are even under parity the circular polarization states have the same refractive index. Then one needs to distinguish only between transverse and longitudinal modes; one deﬁnes πT≡π+=π−andPT=P++P−which projects on the plane transverse to KandUin Minkowski space. In macroscopic electrodynamics the medium eﬀects are frequently stated in the form of response functions to applied electric and mag- netic ﬁelds instead of a response to A. The displacement induced by an applied electric ﬁeld is D=ϵEwithϵthe dielectric permittivity. Simi- larly, the magnetic ﬁeld is H=µ−1Bfor an applied magnetic induction whereµis the magnetic permeability. For time-varying and/or inho- mogeneous ﬁelds these relationships are understood in Fourier space where the response functions depend on ωandk. The magnetic ﬁeld Hand the transverse part of D, characterized byk·DT= 0, do not have independent meaning (Kirzhnits 1987). Therefore, among other possibilities one may choose H=B,DT= ϵTET, and DL=ϵLEL. In this case ϵL≡ϵis the longitudinal and ϵT≡ϵL+ (1−µ−1)k2/ω2the transverse dielectric permittivity. Particle Dispersion and Decays in Media 209 The relationship to the transverse and longitudinal components of the polarization tensor is (Weldon 1982a) ϵL= 1−πL/(ω2−k2) and ϵT= 1−πT/ω2. (6.33) This yields the well-known dispersion relations (Sitenko 1967) ϵL(ω,k) = 0 and ω2ϵT(ω,k) =k2(6.34) for the longitudinal and transverse modes. Calculations of the polarization tensor from the forward scattering amplitudes on microscopic medium constituents are usually performed in a cartesian basis and thus yield an expression for Πµν. The lon- gitudinal and transverse components are projected out by virtue of πL=e∗µ LΠµνeν LandπT=e∗µ ±Πµνeν ±, or explicitly in the medium frame (Weldon 1982a) πL= (1−ω2/k2) Π00andπT=1 2(Tr Π−πL), (6.35) with Tr Π = gµνΠµν. Recall that the dispersion relations are given by ω2−k2=πT,L(ω,k). Thus the frequency ω(k) and the “eﬀective mass” of a given mode are generally complicated functions of k, notably in a medium involving bound electrons where various resonances occur. It can be shown on general grounds (Jackson 1975), however, that for frequencies far above all resonances the transverse mode has a particle-like dispersion relation ω2−k2=m2 TwheremTis the “transverse photon mass” which is a constant independent of the wave number or frequency. 6.3.4 Lowest-Order QED Calculation of Π On the level of quantum electrodynamics (QED) the potential V= 1 2AµΠµνAνwhich modiﬁes the free Lagrangian is interpreted as the self-energy of the photons in the medium. As a Feynman graph, it cor- responds to an insertion of Πµνinto a photon line of four-momentum K and thus corresponds to forward scattering on the medium constituents (Fig. 6.1), entirely analogous to the interpretation of the refractive in- dex in terms of a forward scattering amplitude in Sect. 6.2.1. One then concludes that Πµν(K) is the truncated matrix element for the forward scattering of a photon with momentum K, i.e. it is the matrix ele- ment of the medium constituents alone, uncontracted with the photon polarization vectors ϵµandϵν. In general, the calculation of Π requires the methods of ﬁeld the- ory at ﬁnite temperature and density. To lowest order, however, this 210 Chapter 6 Fig. 6.1. Polarization tensor as photon self-energy insertion. formalism is not required because the only contribution is from lowest- order forward scattering on charged particles. Moreover, because the scattering amplitude involves nonrelativistically the inverse mass of the targets one may limit one’s attention to the electrons. Then one takes the standard (truncated) Compton-scattering ma- trix element (e.g. Bjorken and Drell 1964; Itzykson and Zuber 1983) and takes an average over the Fermi-Dirac distributions of the elec- trons. To lowest order in α=e2/4πthis yields (Altherr and Kraemmer 1992; Braaten and Segel 1993) Πµν(K) = 16πα∫d3p 2E(2π)3(1 e(E−µ)/T+ 1+1 e(E+µ)/T+ 1) ×(P·K)2gµν+K2PµPν−P·K(KµPν+KνPµ) (P·K)2−1 4(K2)2, (6.36) whereP= (E,p) andE= (p2+m2 e)1/2, apart from refractive eﬀects for the electrons and positrons. The phase-space distributions represent electrons and positrons at temperature Tand chemical potential µ. Over the years, the phase-space integration has been performed in various limits (Silin 1960; Tsytovich 1961; Jancovici 1962; Klimov 1982; Weldon 1982a; Altherr, Petitgirard, and del R´ ıo Gaztelurrutia 1993). The most comprehensive analytic result is that of Braaten and Segel (1993) which contains all previous cases in the appropriate limits. The main simpliﬁcation occurs from neglecting the ( K2)2term in the denominator of Eq. (6.36). For light-like K’s this is exactly cor- rect, and in the nonrelativistic limit where meis much larger than all other energy scales the approximation is also trivially justiﬁed. In the relativistic limit it is only justiﬁed if one is interested in Π( K) near the light cone ( ω=k) in Fourier space. In the relativistic limit both trans- verse and longitudinal excitations have dispersion relations which are approximately ( ω2−k2)1/2≈eToreEFin the nondegenerate and de- generate limits, respectively. As detailed by Braaten and Segel (1993), this deviation from masslessness is small enough to justify the approxi- mation if one aims at the dispersion relations. Including1 4(K2)2yields anO(α2) correction—it can be ignored in an O(α) result. Particle Dispersion and Decays in Media 211 In a higher-order calculation one has to include the proper e±disper- sion relations which imply that electromagnetic excitations are never damped by γ→e+e−decay (Braaten 1991), in contrast with state- ments found in the previous literature. Dropping the ( K2)2term in the denominator of Eq. (6.36) prevents Π from developing an imaginary part from this decay, even with the vacuum e±dispersion relations. Therefore, the “approximate” integral actually provides a better repre- sentation of the O(α) dispersion relations than the “exact” one. With the approximation K2= 0 in the denominator of Eq. (6.36) the angular integral is trivial.32With Eq. (6.35) one ﬁnds πL=4α πω2−k2 k2∫∞ 0dpf pp2 E[ω kvlog(ω+kv ω−kv) −ω2−k2 ω2−k2v2−1] , πT=4α πω2−k2 k2∫∞ 0dpf pp2 E[ω2 ω2−k2−ω 2kvlog(ω+kv ω−kv)] ,(6.37) wherev=p/E is thee±velocity and fprepresents the sum of their phase-space distributions. The remaining integration can be done analytically in the classical, degenerate, and relativistic limits where one ﬁnds πL=ω2 P[ 1−G(v2 ∗k2/ω2)] +v2 ∗k2−k2, πT=ω2 P[ 1 +1 2G(v2 ∗k2/ω2)] . (6.38) Here,v∗is a “typical” electron velocity deﬁned by v∗≡ω1/ωP. (6.39) The plasma frequency ωPand the frequency ω1are ω2 P≡4α π∫∞ 0dpf pp(v−1 3v3), ω2 1≡4α π∫∞ 0dpf pp(5 3v3−v5). (6.40) The function G(Fig. 6.2) is deﬁned by G(x) =3 x[ 1−2x 3−1−x 2√xlog(1 +√x 1−√x)] = 6∞∑ n=1xn (2n+ 1)(2n+ 3). (6.41) Note thatG(0) = 0,G(1) = 1, and G′(1) =∞. 32In the degenerate limit, Jancovici (1962) has calculated analytically the full integral without the K2= 0 approximation. 212 Chapter 6 Fig. 6.2. Function G(x) according to Eq. (6.41). The most astonishing observation of Braaten and Segel (1993) is that Eq. (6.38) is a good approximation for allconditions, not only for the limiting cases for which it was derived. As the approximation is much better than 1%, which is the approximate accuracy of an O(α) result, these representations can be taken to be exact to this order. 6.3.5 Dispersion Relations In order to determine the photon dispersion relation for speciﬁc condi- tions one must determine ωPandv∗corresponding to the temperature Tand chemical potential µof the electrons. In Fig. 6.3 contours for v∗andγ≡ωP/Tare shown in the T-ρ-plane of a plasma. Analytic limiting cases are (Braaten and Segel 1993) v∗=  (5T/m e)1/2Classical, vF Degenerate, 1 Relativistic,(6.42) ω2 P=  4παn e me( 1−5 2T me) Classical, 4παn e EF=4α 3πp2 FvF Degenerate, 4α 3π( µ2+1 3π2T2) Relativistic,(6.43) wherevF=pF/EFis the velocity at the Fermi surface, “classical” refers to the nondegenerate and nonrelativistic limit, and “relativistic” is for any degree of degeneracy. Particle Dispersion and Decays in Media 213 Fig. 6.3. Contours for v∗andγ=ωP/Tas deﬁned in Eqs. (6.39) and (6.40) where Yeis the number of electrons per baryon. Next, with Eq. (6.38) one must solve the transcendental equations πT,L(ω,k) =ω2−k2which are explicitly ω2−k2=ω2 P[ 1 +1 2G(v2 ∗k2/ω2)] Transverse, ω2−v2 ∗k2=ω2 P[ 1−G(v2 ∗k2/ω2)] Longitudinal. (6.44) In the classical limit this is to lowest order in T/m e ω2=k2+ω2 P( 1 +k2 ω2T me) Transverse, ω2=ω2 P( 1 + 3k2 ω2T me) Longitudinal. (6.45) For small temperatures the longitudinal modes oscillate with an al- most ﬁxed frequency, independently of momentum, while the transverse modes behave almost like massive particles (Fig. 6.4). The general result Eq. (6.38) and the behavior of the function G(x) reveal that for transverse excitations ω2−k2can vary only between ω2 P and3 2ω2 P. Also,ω2−k2>0 so thatK2is always time-like. For k≫ωP the transverse dispersion relation approaches that of a massive particle with a ﬁxed mass mT, the “transverse photon mass”. With Eq. (6.38) and because k/ω→1 fork≫ωPone ﬁndsm2 T=ω2 P[1 +1 2G(v2 ∗)], or 214 Chapter 6 Fig. 6.4. Electromagnetic dispersion relations in the classical limit according to Eq. (6.45) for v∗= (5T/m e)1/2= 0.2. The shaded area indicates the “width” of ω(k) in the longitudinal case due to Landau damping. withω=kdirectly from Eq. (6.37) m2 T=4α π∫∞ 0dpf pp2 E. (6.46) Limiting cases are m2 T ω2 P=  1 Classical, 3 2v2 F[ 1−1−v2 F 2vFlog(1 +vF 1−vF)] Degenerate, 3 2Relativistic.(6.47) In Fig. 6.5 ( ω2−k2) is shown for several values of v∗as a function ofk. It is quite apparent how the transverse mass is asymptotically approached. The dispersion relation for longitudinal modes is more interesting in several regards. First, according to Eq. (6.44) the oscillation frequency is only a function of v∗kand so the natural scale for kisωP/v∗. In Fig. 6.6 I show ω2−v2 ∗k2as a function of v∗k. Particle Dispersion and Decays in Media 215 Fig. 6.5. Dispersion relation for transverse modes according to Eq. (6.44). Fig. 6.6. Dispersion relation for longitudinal modes according to Eq. (6.44). Second, for v∗<1 there is always a wave number k1whereω(k) “crosses the light cone” ( ω/k= 1), k2 1=4α π∫∞ 0dpf pp2 E[1 vlog(1 +v 1−v) −1] =ω2 P3 v2 ∗[1 2v∗log(1 +v∗ 1−v∗) −1] . (6.48) The second identity (Braaten and Segel 1993) applies at the same level of approximation as πT,Lin Eq. (6.38). Some analytic limiting cases 216 Chapter 6 are k2 1 ω2 P=  1 + 3T/m e Classical, 3 v2 F[1 2vFlog(1 +vF 1−vF) −1] Degenerate, ∞ Relativistic.(6.49) Then, fork>k 1the four-momentum is space-like, ω2−k2<0. As discussed in Sect. 6.2.2 there is nothing wrong with a space-like four-momentum of an excitation. In media with electron resonances such as water or air even (transverse) photons exhibit this behavior which allows kinematically for their Cherenkov emission e→eγor absorption γe→e. In a plasma, transverse excitations are always time-like and thus cannot be Cherenkov absorbed. Their lowest-order damping mechanism is Thomson scattering γe→eγwhich is not in- cluded because it is an O(α2) eﬀect. Longitudinal excitations with k >k 1, in contrast, can and will be Cherenkov absorbed by the ambi- ent electrons, leading to an O(α) damping rate. It corresponds to an imaginary part of the dispersion relation (an imaginary part of πL). In the expression Eq. (6.36) this damping eﬀect corresponds to a vanishing denominator, essentially to P·K= 0, which occurs when the intermediate electron in Compton scattering “goes on-shell.” Evidently, P·K=Eω−p·kcan never vanish for k<ω while fork>ω there are always some electrons, even in a nonrelativistic plasma, which satisfy this condition. When the phase velocity ω/kbecomes of the order of a typical thermal velocity the number of electrons which match the Cherenkov condition becomes large, and then the damping of plasmons becomes strong. Because v∗measures a typical electron velocity this occurs for k∼>ω/v∗(Fig. 6.4). Therefore, while nothing dramatic happens where the dispersion relation crosses the light cone, it ﬁzzles out near the “electron cone.” For k∼>ω/v∗there are no organized oscillations of the electrons—longitudinal modes no longer exist. This damping mechanism of plasma waves was ﬁrst discussed by Landau (1946) and is named after him. A calculation in terms of Che- renkov absorption was performed by Tsytovich (1961). In the classical limit the Landau damping rate (the imaginary part of the frequency) is ΓL ωP=√π 8(kD k)3 e−k2 D 2k2=√π(5 2)3/2(ωP v∗k)3 e−5 2ω2 P v2∗k2,(6.50) wherekD= 4παn e/Tis the Debye screening scale. (Note that ω2 P/k2 D= T/m e=v2 ∗/5.) For a given wave number a plasmon must be viewed Particle Dispersion and Decays in Media 217 as a resonance with a ﬁnite width Γ L. The approximate uncertainty ±ΓL(k) of the energy ω(k) is shown in Fig. 6.4 as a shaded area. The damping rate Eq. (6.50) does not show a threshold eﬀect at k=ωbecause it was calculated nonrelativistically so that the high- energy tail of the electron distribution contains particles with velocities exceeding the speed of light. Tsytovich (1961) calculated a relativis- tic result with the correct threshold behavior. However, because the damping rate is exceedingly small for k≪kD, the correction is very small ifωP≪kD, equivalent to T≪me. For the degenerate case, explicit expressions for the imaginary parts of πT,L(ω,k) were derived by Altherr, Petitgirard, and del R´ ıo Gaztelurrutia (1993). The general expressions Eq. (6.38) for the real parts of πLand πTwere derived without the need to assume that Kwas time-like. Therefore, they should also apply “below the light cone,” even though Braaten and Segel (1993) conﬁned their discussion to the region k<ω . As expected, these expressions break down for k>ω/v ∗where Landau damping becomes strong. 6.3.6 Renormalization Constants ZT;L Armed with the dispersion relation one may determine the vertex renor- malization constants ZT,Lrelevant for the coupling of external photons or plasmons to an electron in the medium (Sect. 6.2.3), Z−1 T,L= 1−∂πT,L(ω,k) ∂ω2 ω2−k2=πT,L(ω,k). (6.51) With the same approximations as before, Braaten and Segel (1993) found an analytic representation accurate to O(α), ZT=2ω2(ω2−v2 ∗k2) ω2[3ω2 P−2 (ω2−k2)] + (ω2+k2)(ω2−v2 ∗k2), ZL=2 (ω2−v2 ∗k2) 3ω2 P−(ω2−v2 ∗k2)ω2 ω2−k2. (6.52) In each case ωandkare “on shell,” i.e. they are related by the disper- sion relation relevant for the T and L case, respectively. Inspection of Eq. (6.52) reveals that ZTis always very close to unity, as expected for excitations with only a small deviation from a massive- particle dispersion relation. The contours in Fig. 6.7 conﬁrm that ZT never deviates from unity by more than a few percent. 218 Chapter 6 Fig. 6.7. Contours for the vertex renormalization factor ZTfor transverse electromagnetic excitations in a medium according to Eq. (6.52). Fig. 6.8. Modiﬁed vertex renormalization factoreZLfor longitudinal electro- magnetic excitations in a medium according to Eq. (6.53). The longitudinal case is more complicated. ZLis a product of two factorseZLandω2/(ω2−k2) where the former, eZL≡2 (ω2−v2 ∗k2) 3ω2 P−(ω2−v2 ∗k2), (6.53) is a function of the variable v∗kalone because for plasmons ωis a function of v∗kalone. The functioneZL(v∗k) is shown in Fig. 6.8; for v∗k≫ωPit quickly drops to zero. However, in a relativistic plasma Particle Dispersion and Decays in Media 219 withv∗= 1 the complete factor is ZL= 2ω2/[3ω2 P−(ω2−k2)] and thus rises quickly with kbecauseω≈k. As discussed in the previous section, the dispersion relation crosses the light cone at k=k1, a point at which ZLdiverges and changes sign. The sign change is compensated by the change of the polarization vector eL(Eq. 6.27) at the light cone where it becomes imaginary. Because in the squared matrix element a factor e∗µ Leν Lappears, and because this expression changes sign at the light cone, the expression ZLe∗µ Leν L remains positive. As for the divergence, it is harmless in reactions of the sort γL→νν (plasmon decay), e→eγL(Cherenkov emission), γLe→e(Cherenkov absorption), and γTγL→a(plasmon coalescence into axions) which are of interest in this book. These reactions involving three particles are constrained by their phase space to either time-like excitations (plas- mon decay), or to space-like ones (Cherenkov and coalescence process). Therefore, the threshold behavior of the phase space moderates the divergence in these cases. 6.4 Screening Eﬀects 6.4.1 Debye Screening Scattering processes in the Coulomb ﬁeld of charged particles such as Rutherford scattering, bremsstrahlung, or the Primakoﬀ eﬀect typically lead to cross sections which diverge in the forward direction because of the long-range nature of the electrostatic interaction. In a plasma this divergence is moderated by screening eﬀects which thus are crucial for a calculation of the cross sections or energy-loss rates. Screening eﬀects are revealed by turning to the static limit of Max- well’s equations in a medium, [ k2+πL(0,k)] Φ(k) =ρ(k), [ k2+πT(0,k)] A(k) =J(k), (6.54) where the current must be transverse in both Coulomb and Lorentz gauge as∂tρ= 0 in the static limit. Notably, the equation for Φ in vacuum is the Fourier transform of Poisson’s equation and thus gives rise to a 1/rCoulomb potential if the source is point-like, ρ(r) =eδ(r). In a QED plasma πL,T(ω,k) are given by the integrals Eq. (6.37). In the static limit ( ω= 0) one ﬁnds πT(0,k) = 0 because all terms in the integrand involve factors of ω. Therefore, stationary currents 220 Chapter 6 (∂tJ= 0) are not screened. The magnetic ﬁeld associated with a sta- tionary current is the same at a distance whether or not the plasma is present. Not so for the electric ﬁeld associated with a charge. In the static limit one ﬁnds πL(0,k) =4α π∫∞ 0dpf pp(v+v−1). (6.55) Because this expression does not depend on kit can be identiﬁed with the square of a ﬁxed wave number kS, leading to Poisson’s equation in the form ( k2+k2 S) Φ(k) =ρ(k). (6.56) Because for a point-like source this gives a Yukawa potential Φ(r)∝r−1→r−1e−kSr(6.57) electric charges are screened for distances exceeding about k−1 S. Evaluating Eq. (6.55) explicitly in the classical limit reproduces the well-known Debye screening scale (Debye and H¨ uckel 1923; for a text- book discussion see Landau and Lifshitz 1958) k2 S=k2 D=4παn e T=me Tω2 P. (6.58) At this point one recognizes that k2 Dis independent of the electron mass, in contrast with the plasma frequency ω2 P. Therefore, it is no longer justiﬁed to ignore the ions or nuclei; they contribute little to dispersion because of their reduced Thomson scattering amplitude, but they contribute equally to screening. Therefore, one ﬁnds k2 S=k2 D+k2 i with k2 i=4πα T∑ jnjZ2 j, (6.59) where the sum is over all species jwith charge Zje. Comparing Eq. (6.55) with the corresponding expression for the plasma frequency Eq. (6.40) reveals that in the relativistic limit ( v= 1) k2 D= 3ω2 P. It is clear that apart from a numerical factor they must be the same because a relativistic plasma has only one natural scale, namely a typical electron energy. Particle Dispersion and Decays in Media 221 In the limit of degenerate electrons the integral is also easily solved and leads to the familiar Thomas-Fermi wave number33(Jancovici 1962) k2 S=k2 TF=4α πEFpF=3ω2 P v2 F. (6.60) However,k2 Dalways exceeds k2 TFso that in a medium of degenerate electrons and nondegenerate ions the main screening eﬀect is from the latter. Recall that the Fermi momentum is related to the electron density byne=p3 F/3π2and the Fermi energy is EF= (p2 F+m2 e)1/2. To compare the Thomas-Fermi with the Debye scale take the non- relativistic limit ( kTF/kD)2=3 2T/(EF−me). This is much less than 1 or the medium would not be degenerate whence kTF≪kD. There- fore, if the electrons are degenerate and the ions nondegenerate, a test charge is mostly screened by the polarization of the ion “ﬂuid” because the electrons form a “stiﬀ” background. Unfortunately, one often ﬁnds calculations in the literature which include screening by the electrons (screening scale kTF) but ignore the ions. The resulting error need not be large because the screening scale typically appears logarithmically in the ﬁnal answer (see below). Screening eﬀects in Coulomb processes are often found to be imple- mented by a modiﬁed Coulomb propagator 1 |q|4→1 (q2+k2 S)2, (6.61) where qis the momentum transfer carried by the intermediate photon. This substitution arises if one considers Coulomb scattering from a Yukawa-like charge distribution. It corresponds to the substitution Eq. (6.57), i.e. to a single charge with an exponential screening cloud. This picture is appropriate if the Coulomb scattering process itself is so slow that the charged particles move around and rearrange themselves so much that the probe, indeed, sees an average screening cloud. In the opposite limit, a given probe sees a certain conﬁguration, a diﬀerent probe a diﬀerent one, etc., and one has to average over all of these possibilities. In this case one needs to square the matrix element ﬁrst, and then take an average over diﬀerent medium conﬁgurations. For Eq. (6.61) one averages ﬁrst, obtains an average scattering ampli- tude or matrix element, and squares afterward. 33For a textbook derivation from a Thomas-Fermi model see Shapiro and Teukol- sky (1983). Note that they work in the nonrelativistic limit: their EF=p2 F/2me. 222 Chapter 6 It depends on the physical circumstances which procedure is a better approximation. If one considers bremsstrahlung processes with degen- erate electrons scattering oﬀ nondegenerate nuclei, the crossing time of an electron of a region the size k−1 Sis short compared to the crossing time of nuclei. Hence, the latter can be viewed as static, the probe sees one conﬁguration at a time, and one certainly should use the “square ﬁrst” procedure instead of Eq. (6.61) to account for screening. This is achieved by the following consideration of correlation eﬀects. 6.4.2 Correlations and Static Structure Factor The screening of electric ﬁelds in a plasma is closely related to correla- tions of the positions and motions of the charged particles. If a negative test charge is known to be in a certain position, the probability of ﬁnd- ing an electron in the immediate neighborhood is less than average, while the probability of ﬁnding a nucleus is larger than average. It is this polarization of the surrounding plasma which screens a charge. Take one particle of a given species to be the origin of a coordinate system, and take their average number density to be n. The electro- static repulsion of the test charge causes a deviation of the surrounding charges from the average density by an amount S(r) =δ3(r) +nh(r), (6.62) whereh(r) measures the particle correlations. They vanish in an ideal Boltzmann gas: h(r) = 0. The Fourier transform S(q) =∫ d3rS(r)e−iq·r(6.63) is the static structure factor of the electron distribution. In the absence of correlations ( h= 0) one has trivially S(q) = 1. In order to make contact with Debye screening consider the Yukawa potential of Eq. (6.57) which represents a charge density ρ(r) =δ3(r)−k2 S 4πe−kSr r. (6.64) The volume integral of ρ(r) vanishes, giving zero total charge, i.e. com- plete screening at inﬁnity. If one imagines that only one species of charged particles is mobile on a uniform background of the opposite charge, then Eq. (6.64) implies correlations between the mobile species ofnh(r) =−(k2 S/4πr)e−kSr. As expected, Debye screening corresponds Particle Dispersion and Decays in Media 223 to spatial anticorrelations of like-charged particles. Fourier transform- ing Eq. (6.64) yields the important result S(q) =q2 q2+k2 S(6.65) for the structure factor. The assumption of one mobile species of particles on a uniform background corresponds to the model of a “one-component plasma.” It is approximately realized in the interior of hot white dwarfs or the cores of red-giant stars where the degenerate electrons form a “stiﬀ” background of negative charge in which the nondegenerate ions move. In a nondegenerate situation, however, there are at least two mo- bile species, ions of charge Zeand electrons. For this two-component plasma Salpeter (1960) derived the structure functions See(q) =q2+Zk2 D q2+ (1 +Z)k2 D, Sii(q) =q2+k2 D q2+ (1 +Z)k2 D, Sei(q) =k2 D q2+ (1 +Z)k2 D. (6.66) The Fourier transform of the screening cloud around an electron is S(q) =See(q)−ZSei(q) =q2 q2+k2 S, (6.67) withk2 S=k2 D+k2 i= (1 +Z)k2 D. (Note that for only one species of ions k2 i=Zk2 D.) Hence one reproduces a screened charge distribution which causes a Yukawa potential. However, the small- qbehavior of SeeorSii is very diﬀerent: See(0) =Z/(1 +Z) in a two-component plasma while See(0) = 0 for only one component. 6.4.3 Strongly Coupled Plasma For low temperatures, the screening will not be of Yukawa type and the structure factor will deviate from the simple Debye formula. A plasma can be considered cold if the average Coulomb interaction en- ergy between ions is much larger than typical thermal energies. To quantify this measure, one introduces the “ion-sphere radius” aiby virtue ofn−1 i= 4πa3 i/3 whereniis the number density of the mobile 224 Chapter 6 particle species. Hence, a measure for the Coulomb interaction energy isZ2α/a i, assuming the ions have charge Ze. One usually introduces the parameter Γ≡Z2α aiT=(kiai)2 3(6.68) as a measure for how strongly the plasma is coupled, where k2 i= 4πZ2α/T. For Γ ≪1 it is weakly coupled and approaches an ideal Boltzmann gas. The Debye structure factor of a one-component plasma can be writ- ten as SD(q) =|aiq|2 |aiq|2+ 3Γ. (6.69) This result applies even for large Γ if |aiq| ≪1. For Γ ≫1, the plasma is strongly coupled, and for Γ ∼>178 the ions will arrange themselves in a body centered cubic lattice (Slattery, Doolen, and DeWitt 1980, 1982). In Fig. 6.9 I show SandSDas functions of aiq=|aiq|for Γ = 2, 10 and 100 where Swas numerically determined (Hansen 1973; Galam and Hansen 1976). The emerging periodicity for a strongly coupled plasma is quite apparent. It is also clear that for Γ ∼<1 the Debye formula gives a fair representation of the structure factor while for a strongly coupled plasma it is completely misleading. The interior of white dwarfs is in the regime of large Γ, and old white dwarfs are believed to crystallize. (See Appendix D for an overview over the conditions relevant for stellar plasmas.) 6.4.4 Screened Coulomb Scattering Armed with these insights one may turn to the issue of Coulomb scatter- ing processes in a plasma. In the limit of nonrelativistic and essentially static sources for the electric ﬁelds the relevant quantity entering the matrix element is the Fourier component ρ(q) of the charge distribu- tionρ(r) where qis the momentum transferred by the Coulomb ﬁeld to the sources. The squared matrix element thus involves the quantity ρ(q)ρ∗(q) which isρ(q)ρ(−q) becauseρ(r) is real. Taking a statistical average over all possible conﬁgurations of the charge distribution leads to a rate proportional to S(q) =⟨ρ(q)ρ(−q)⟩. (6.70) This is the static structure factor introduced earlier as a measure of the correlation between the charged particles of the medium. Particle Dispersion and Decays in Media 225 Fig. 6.9. Static structure factor for a one-component plasma according to the numerical calculations of Hansen (1973) and Galam and Hansen (1976). The dashed lines correspond to the Debye structure factor Eq. (6.69). 226 Chapter 6 Without correlations, the squared matrix element involves |q|−4 from the Coulomb propagator. In order to account for screening ef- fects one should substitute |q|−4→ |q|−4S(q) (6.71) which implies 1 |q|4→1 q2(q2+k2 S)(6.72) in the weak-screening limit (Debye screening). The diﬀerence in a scattering cross section implied by Eq. (6.72) relative to (6.61) is easily illustrated. Observe that a cross section involving a Coulomb divergence is typically of the form ∫+1 −1dx(1−x)f(x) (1−x)2, (6.73) wherexis the cosine of the scattering angle of the probe. Here, f(x) is a slowly varying function which embodies the details of the scattering or bremsstrahlung process. If this function is taken to be a constant, the two screening prescriptions amount to the two integrals ∫+1 −1dx1 (1−x+κ2)= log(2 +κ2 κ2) , ∫+1 −1dx(1−x) (1−x+κ2)2= log(2 +κ2 κ2) −2 2 +κ2, (6.74) whereκ2≡k2 S/2p2is the screening scale expressed in units of the initial- state momentum of the probe. Usually, it far exceeds the screening scale whenceκ2≪1. Then Eq. (6.72) yields a cross section proportional to log(4p2/k2 S) while Eq. (6.61) gives [log(4 p2/k2 S)−1]. Thus, if one is only interested in a rough estimate, either screening prescription and any reasonable screening scale yield about the same result. For an accurate calculation, however, one needs to identify the dominant source of screening (for example, the nondegenerate ions in a degenerate plasma and not the electrons), and the appropriate moderation of the Coulomb propagator, usually Eq. (6.72). Particle Dispersion and Decays in Media 227 6.5 Plasmon Decay in Neutrinos34 6.5.1 Millicharged Neutrinos Transverse and longitudinal electromagnetic excitations in a plasma are both kinematically able to decay into neutrino pairs (Fig. 6.10) of suﬃciently small mass, namely 2 mν< K2whereKis the plasmon35 four-momentum. In the following, the neutrinos are always taken to be massless relative to the plasma frequency and so K2>0 is required which restricts longitudinal excitations to k < k 1, the wave number where their dispersion relation crosses the light cone. Fig. 6.10. Plasmon decay in neutrinos. In addition, a ν-γ-interaction is required which does not exist in the standard model. Still, plasmon decays occur because the medium itself mediates an eﬀective coupling as will become clear below. As an easy start, however, consider the hypothesis that neutrinos carry small electric charges (“millicharges”). Interestingly, this possibility is not excluded by the structure of the standard model and has received some recent attention in the literature (Sect. 7.3.2). With a neutrino millicharge eνthe interaction with the electromag- netic vector potential Ais the standard expression Lint=−ieνψνγαψνAα. (6.75) The spin-summed squared matrix element is of the form ∑ spins|M|2=MαβPαPβ(6.76) where explicitly Mαβ= 4e2 νZ(gαβ+ 2ϵ∗ αϵβ). (6.77) Here,Zis the renormalization constant (Sect. 6.2.3 and 6.3.6), Pand Pare theνandνfour-momenta, and ϵis the plasmon polarization vector for which one uses the basis vectors of Eqs. (6.27) and (6.28). 34I closely follow Haft (1993). 35In this section the term “plasmon” refers to both transverse and longitudinal electromagnetic excitations in a medium. 228 Chapter 6 The decay width of a plasmon with four-momentum K= (ω,k) in the medium frame, and with a deﬁnite polarization is Γ =∫d3p 2Ep(2π)3d3p 2Ep(2π)3(2π)4δ4(K−P−P)1 2ω∑ spins|M|2. (6.78) Because of Eq. (6.76) one may use Lenard’s (1953) formula ∫d3p 2Epd3p 2EpPαPβδ4(K−P−P) =π 24( K2gαβ+ 2KαKβ) . (6.79) Withαν≡e2 ν/4πthis leads to Γ =ανZ(ω2−k2)/3ω, (6.80) where the normalization ϵ∗ αϵα=−1 for transverse and time-like lon- gitudinal plasmons was used as well as ϵ·K= 0. Γ applies to both transverse and longitudinal plasmons with the appropriate ZT,L. For a chosen three-momentum k=|k|the quantities Z,ω, andK2=ω2−k2 are all functions of kby virtue of the dispersion relation K2=πT,L(K). In the classical limit transverse plasmons propagate like massive particles with K2=ω2 PandZT= 1. Then Γ T=1 3ανω3 P(ωP/ω) where the last factor is recognized as a Lorentz time-dilation factor. For a general dispersion relation which is not Lorentz covariant it makes little sense, of course, to express the decay rate in the plasmon frame. An example is the classical limit for the longitudinal mode for which to zeroth order in T/m ethe frequency is ω=ωP,ZL=ω2/K2, and then ΓL=1 3ανωPwith the restriction k<ω P. 6.5.2 Neutrino Dipole Moments Another direct coupling between neutrinos and photons arises if the for- mer have electric or magnetic dipole or transition moments (Sects. 7.2.2 and 7.3.2) Lint=1 2∑ a,b( µabψaσµνψb+ϵabψaσµνγ5ψb) Fµν. (6.81) Here,Fis the electromagnetic ﬁeld tensor, σµν=γµγν−γνγµ, andψa witha=ν1,2,3orνe,µ,τare the neutrino ﬁelds. Particle Dispersion and Decays in Media 229 The squared matrix element is of the form Eq. (6.76); for a magnetic dipole coupling µof a single ﬂavor one ﬁnds Mαβ= 4µ2Z(2KαKβ−2K2ϵ∗ αϵβ−K2gαβ). (6.82) This leads to a decay rate Γ = (µ2/24π)Z(ω2−k2)2/ω, (6.83) applicable to either plasmon polarization with the appropriate Zand dispersion relation. In the presence of electric and magnetic transition moments one obtains the same result with µ2→∑ a,b( |µab|2+|ϵab|2) . (6.84) There is no interference term between the electric and magnetic cou- plings. The presence of transition moments allows for plasmon decays of the sort γ→νaνbwith diﬀerent ﬂavors a̸=b, doubling the ﬁnal states which may be νaνborνbνa. 6.5.3 Standard-Model Couplings In a medium there is an eﬀective coupling between electromagnetic ﬁelds and neutrinos mediated by the ambient electrons. For the purpose of a plasmon-decay calculation it can be visualized with the Feynman graph Fig. 6.11 which corresponds to the “photoneutrino process” or Compton production of neutrino pairs (Sect. 3.2.4). However, the ﬁnal- state electron can have the same four-momentum as the initial state which amounts to forward scattering of the electrons. In this case the energy-momentum transfer to the electron vanishes, allowing for a coherent superposition of these amplitudes from all electrons. Fig. 6.11. Compton production of neutrino pairs (nonzero momentum trans- fer to electrons) or plasmon decay (electron forward scattering). 230 Chapter 6 With the dimensionless coupling constants CVandCAgiven in Ap- pendix B the neutral-current ν-e-interaction is Lint=−GF√ 2ψeγα(CV−CAγ5)ψeψνγα(1−γ5)ψν. (6.85) The vector-current has the same structure that pertains to the elec- tron interaction with photons, Lint=−ieψeγαψeAα. Therefore, af- ter performing a thermal average over the electron forward scattering amplitudes the plasmon decay is represented by the Feynman graph Fig. 6.12 which is identical with Fig. 6.1 with one photon line replaced by a neutrino pair. As far as the electrons are concerned, photon for- ward scattering γ→γis the same as the conversion γ→νν. Fig. 6.12. Photon-neutrino coupling by the photon polarization tensor; the second photon line in Fig. 6.1 was replaced by a neutrino pair. Put another way, in a plasma the propagation of an electromagnetic excitation is accompanied by an organized oscillation of the electrons. This is particularly obvious for longitudinal modes which arethe col- lective oscillation of the electron gas, but it also applies to transverse modes. The collective motion is the medium response Jind= ΠAto an electromagnetic excitation, a response which is characterized by the polarization tensor. The coherent electron oscillations serve as sources for the neutrino current whence they emit neutrino pairs. The electron collective motion is an oscillation of their location or density while their spins remain unaﬀected apart from relativistic cor- rections. Because the axial-vector current represents the electron spin density, and because to lowest order no collective spin motion is ex- pected as a response to an electromagnetic excitation, the axial-vector neutrino coupling to electrons will contribute very little to plasmon de- cay. This remains true in a relativistic plasma. For example, Koyama, Itoh, and Munakata (1986) found numerically that the axial vector con- tributes less than 0.01% to neutrino emission by the plasma process for all conditions of astrophysical interest. A detailed study of the axial response function can be found in Braaten and Segel (1993). For the Particle Dispersion and Decays in Media 231 present discussion I will not worry any further about the axial-vector contribution. The matrix element for the interaction between neutrinos and pho- tons can then be read from the eﬀective vertex iCVGF e√ 2AαΠαβψνγα(1−γ5)ψν. (6.86) The electric charge in the denominator removes one such factor con- tained in Π which was calculated for photon forward scattering. In the matrix element, Π is to be taken at the four-momentum Kof the photon. Besides plasmon decay γ→νν, this interaction also allows for processes such as Cherenkov absorption γν→νor emission ν→γν. For the decay of a plasmon with a polarization vector ϵand four- momentum Kthe squared matrix element has the form Eq. (6.76) with Mαβ= 8G2 F 2e2π2 T,L(gαβ+ 2ϵ∗ αϵβ). (6.87) Because the plasmon is a propagating mode it obeys its dispersion relation, i.e. πT,L=ω2−k2. The decay rate is then Γ =C2 VG2 F 48π2αZT,L(ω2−k2)3 ω. (6.88) This result was ﬁrst derived by Adams, Ruderman, and Woo (1963), the correct Zfor the longitudinal case was ﬁrst derived by Zaidi (1965). This equation is understood “on shell” where ωdepends on kthrough the dispersion relation ω2−k2=πT,L(ω,k). 6.5.4 Summary of Decay Rates In order to express the decay rates in a compact form, recall that on shell the scale for K2is set by the plasma frequency ωP. Therefore, it is useful to deﬁne ˆπT,L(k)≡πT,L(ωk,k) ω2 P, (6.89) whereωkis the frequency related to kas a solution of the dispersion equationω2−k2=π(ω,k). Recall that 1 ≤ˆπT<3 2while 0 ≤ˆπL≤1 for a time-like K2(k<k 1). Atk=k1the L dispersion relation crosses the light cone. 232 Chapter 6 The decay rates of Eqs. (6.80), (6.83) and (6.88) of a plasmon with three-momentum kare then expressed as Γk=4π 3Zk ωk×  ανω2 Pˆπk 4πMillicharge, µ2 2(ω2 Pˆπk 4π)2 Dipole Moment, C2 VG2 F α(ω2 Pˆπk 4π)3 Standard Model,(6.90) where forZkand ˆπkthe T or L value appropriate for the chosen polar- ization must be used. 6.5.5 Energy-Loss Rates It is now an easy task to calculate stellar energy-loss rates for the plasma process. An integration over the Bose-Einstein distributions of the transverse and longitudinal plasmons yields for the energy-loss rate per unit volume QT=2 2π2∫∞ 0dkk2ΓTω eω/T−1, QL=1 2π2∫k1 0dkk2ΓLω eω/T−1. (6.91) InQTthe factor 2 counts two polarization states. In QLthe integration can be extended only to the wave number k1where the L dispersion relation crosses the light cone—for k > k 1decays are kinematically forbidden. In either case Γ T,Landωare functions of k, the latter given by the dispersion relation. For the speciﬁc neutrino interaction models discussed in the previ- ous section one obtains with the decay rates of Eq. (6.90) Q=8ζ3 3πT3×  ανω2 P 4πQ1 Millicharge, µ2 2(ω2 P 4π)2 Q2 Dipole Moment, C2 VG2 F α(ω2 P 4π)3 Q3 Standard Model,(6.92) whereζ3≈1.202 refers to the Riemann Zeta function. The dimension- less emission rates Qnfor the three cases are each a sum of a transverse Particle Dispersion and Decays in Media 233 and longitudinal term, Qn=QL,n+QT,nwhere QL,n=1 4ζ3T3∫k1 0dkk2ZLˆπn L eω/T−1=1 4ζ3T3∫k1 0dkk2ω2 ω2 PeZLˆπn−1 L eω/T−1, QT,n=1 2ζ3T3∫∞ 0dkk2ZTˆπn T eω/T−1. (6.93) The second equation for QL,nrelies on the deﬁnition Eq. (6.53), i.e. ZL=eZLω2/(ω2−k2), andω2−k2=πLwas used. The normalization factors were chosen such that QT,n= 1 if the plasmons are treated as eﬀectively massless particles for the phase-space integration. Then ZT= ˆπT= 1 which is a reasonable approximation in a nondegenerate, nonrelativistic plasma. In that limit to lowest orderk1=ωP,eZL= 1, andπL=ω2 P−k2. Therefore, in this limit QL,n≪QT,n. In fact, the longitudinal emission rate is of comparable importance to the transverse one only in a narrow range of parameters of astrophysical interest (Haft, Raﬀelt, and Weiss 1994). These simple approximations, however, are not adequate for most of the conditions where the plasma process is important. In Appendix C the numerical neutrino emission rates are discussed; a comparison be- tween Fig. C.1 and Fig. 6.3 reveals that the plasma process is important for 0.3∼<ωP/T∼<30, i.e. transverse plasmons can be anything from relativistic to entirely nonrelativistic. For a practical stellar evolution calculation one may use the analytic approximation formula for the plasma process discussed in Appendix C, based on the representation of the dispersion relations of Sect. 6.3. The main issue at stake in this book, however, is nonstandard neu- trino emission from the direct electromagnetic couplings discussed in Sect. 6.5. Instead of constructing new numerical emission rate formulae one uses the existing ones for the standard-model (SM) couplings and scales them to the novel cases. Numerically, one ﬁnds Qcharge QSM=ανα(4π)2 C2 VG2 Fω4 PQ1 Q3= 0.664e2 14(10 keV ωP)4Q1 Q3, Qdipole QSM=µ2α2π C2 VG2 Fω2 PQ2 Q3= 0.318µ2 12(10 keV ωP)2Q2 Q3, (6.94) wheree14=eν/10−14eandµ12=µ/10−12µBwithµB=e/2me. Contours for Q1/Q3andQ2/Q3are shown in Fig. 6.13 according to Haft, Raﬀelt, and Weiss (1994). Replacing these ratios by unity in a practical stellar evolution calculation introduces only a small error. 234 Chapter 6 Fig. 6.13. Contours of Q1/Q3andQ2/Q3deﬁned by Eq. (6.93) in the plane deﬁned by the plasma frequency ωPand a “typical” electron velocity v∗ discussed in Sect. 6.3. See Fig. 6.3 for contours of v∗andωPin the T-ρ- plane. (Adapted from Haft, Raﬀelt, and Weiss 1994.) 6.5.6 Astrophysical Bounds on Neutrino Electromagnetic Properties The plasma decay process is the most important neutrino emission process for a large range of temperatures and densities. Moreover, it has an observable impact on the cooling of hot white dwarfs, and on the core mass at helium ignition in low-mass red giants. For a neutrino millicharge eν∼>10−14eand a dipole moment µ∼>10−12µBthe nonstandard plasmon decay rates in Eq. (6.94) begin to compete with the standard one if the plasma frequency is around 10 keV. Therefore, neutrino millicharges or dipole moments of this magnitude will have observable eﬀects on these stars and thus can be excluded. Particle Dispersion and Decays in Media 235 In Sect. 2.2.3 it has been discussed that neutrino emission cools hot white dwarfs so fast that there is a clear depression of the white- dwarf luminosity function at the hot and bright end relative to the simple Mestel law which takes only surface photon emission into ac- count. This depression would be enhanced by additional cooling caused by dipole moments or millicharges, allowing one to derive a limit of aboutµ12<10, although even for µ12≈3 a nonnegligible eﬀect is apparent. A more restrictive and probably more reliable limit can be derived from the properties of globular-cluster stars (Raﬀelt 1990b). To this end one may use the simple criteria derived in Sect. 2.5 which state that a novel energy-loss rate is constrained by ⟨ϵx⟩∼<10 erg g−1s−1for the average core-conditions of a horizontal branch star, and for those of a low-mass red giant before the helium ﬂash. In both cases T≈108K. For this temperature, the plasma loss rates are shown in Fig. 6.14 as a function of density. The anomalous rates were obtained from the standard one according to Eq. (6.94), taking Q1/Q3=Q2/Q3= 1 and using the zero-temperature plasma frequency given in Eq. (D.12) as a function of density. The ﬁrst criterion of Sect. 2.5, based on the helium-burning lifetime of HB stars, requires calculating the energy loss rate at an average density which is below 104g cm−3. Therefore, the medium is so dilute Fig. 6.14. Neutrino energy-loss rate in helium at T= 108K. Solid line: Total standard rate. Long dashes: Standard plasma rate. Short dashes: Plasma rate induced by a dipole moment µν= 2×10−12µB.Dots: Plasma rate induced by a neutrino “millicharge” eν= 10−14e. 236 Chapter 6 that one may employ the simple analytic form Eq. (6.92) of the emission rate withQn= 1. The core of HB stars consists at ﬁrst of helium, later also of carbon and oxygen, for all of which Ye= 0.5. Then, ϵx= 1 erg g−1s−1×T3 8×  5.0e2 14 Millicharge, 0.098µ2 12ρ4Dipole Moment, 0.0127ρ2 4 Standard Model,(6.95) whereT8=T/108K andρ4=ρ/104g cm−3. The core averages for a typical HB star are ⟨T3 8⟩= 0.44,⟨T3 8ρ4⟩= 0.47 and ⟨T3 8ρ2 4⟩= 0.57. The requirement ⟨ϵx⟩<10 erg g−1s−1then gives the limits eν∼<2×10−14e andµν∼<14×10−12µB. (6.96) Of course, for such large dipole moments the core would grow far be- yond its standard value before helium ignites, causing an additional acceleration of the HB lifetime. In fact, this indirect impact on the HB lifetime would be the dominant eﬀect as shown, for example, by the numerical calculations of Raﬀelt, Dearborn, and Silk (1989). From Fig. 6.14 it is clear that the dipole-induced emission rate is larger for the conditions of the second criterion, based on the helium- ignition argument where ⟨ρ⟩ ≈2×105g cm−3. According to Eq. (D.12) the relevant plasma frequency is ωP= 8.6 keV so that Qcharge/QSM≈ 1.2e2 14andQdipole/QSM≈0.4µ2 12in Eq. (6.94). The average total emis- sion rate is then given by the standard rate times Fν= 1 +Qj/QSM wherejstands for “charge” or “dipole.” In order to prevent the core mass at helium ignition from exceeding its standard value by more than 5% one must require Fν<3. Then one ﬁnds eν∼<1.3×10−14e andµν∼<2×10−12µB. For the dipole case, a detailed numerical imple- mentation yielded µν∼<3×10−12µB(Sect. 2.5.2), nearly identical with this simple analytic estimate. The limit on the charge could also be slightly degraded and so I adopt eν∼<2×10−14e andµν∼<3×10−12µB (6.97) as the ﬁnal globular-cluster limits. Particle Dispersion and Decays in Media 237 6.6 Neutrino Form Factors in Media In a medium, neutrinos can interact with photons using electrons or other charged particles as go-betweens. The basic idea is to consider the Compton process of Fig. 6.11 with the initial- and ﬁnal-state electrons in the same state, i.e. forward scattering for the electrons. Then one may sum over all electrons of the medium. This coherent superposition of the amplitudes from all electrons was used in the previous section to calculate the standard-model plasmon decay rate. There, only on-shell (propagating) photons were considered. In general one may consider other cases, for example electromagnetic scattering by the exchange of a space-like photon, or the behavior of neutrinos in an external electric or magnetic ﬁeld. The neutrino electromagnetic form factors in vacuum will be studied in Sect. 7.3.2. They can be classiﬁed as a charge radius, an anapole moment, and an electric and a magnetic dipole moment. They are induced by intermediate (virtual) charged particles such as charged leptons orWbosons. In the present case the form factors are induced by the real particles of the ambient heat bath. The eﬀective Lagrangian Eq. (7.19) is fundamentally Lorentz covariant, a fact which reduces the number of possible form factors to four. While in a medium the couplings may also be written in a nominally Lorentz covariant form, the medium singles out an inertial frame, leading to more complicated structures. This is analogous to dispersion which is simple in vacuum (a mass term is the only possibility) while in a medium the dispersion relations can be excruciatingly complicated. Limiting the couplings to the ones mediated by electrons and pro- tons, the induced photon coupling to the neutrino is proportional to Aµ because these fermions couple by the usual eψγµψAµinteraction. The neutrinos couple by their standard eﬀective neutral-current interaction (GF/√ 2)ψνγα(1−γ5)ψνψeγα(CV−CAγ5)ψewith the weak coupling constantsCVandCAgiven in Appendix B. Therefore, after summing over all intermediate electron states the eﬀective neutrino-photon in- teraction may be written in the form Leﬀ=−√ 2GFψνγα1 2(1−γ5)ψνΛαβAβ, (6.98) where Λαβis a matrix which depends on the medium properties and on the energy-momentum transfer, i.e. the energy momentum Kof the photon line. It consists of a symmetric piece Λαβ Vwhich is proportional toCV, and an antisymmetric piece Λαβ Awhich is proportional to CA. 238 Chapter 6 Explicit expressions in terms of the electron phase-space integrals were derived, e.g. by D’Olivo, Nieves, and Pal (1989) and by Altherr and Salati (1994), Λαβ V= 4eCV∫d3p 2E(2π)3[ fe−(p) +fe+(p)] ×(P·K)2gαβ+K2PαPβ−P·K(KαPβ+KαPβ) (P·K)2−1 4(K2)2, Λαβ A= 2ieC Aϵαβµν∫d3p 2E(2π)3[ fe−(p)−fe+(p)] ×K2PµKν (P·K)2−1 4(K2)2, (6.99) wherefe±(p) is the electron and positron phase-space distribution with P= (E,p) the electron or positron four-momentum. Instead of a neutrino pair, another photon can be thought of as being coupled to the electron line in Fig. 6.11 or 6.12, a process which represents photon forward scattering. Therefore, apart from overall coupling constants Λαβ Vis identical with the electronic contribution to the photon polarization tensor Παβstudied earlier in this chapter Λαβ V= (CV/e) Παβ. (6.100) In an isotropic plasma, Παβis characterized by the two medium charac- teristicsπT(ω,k) andπL(ω,k) which are functions of the photon four- momentum K= (ω,k) withk=|k|. For the antisymmetric piece, in an isotropic medium the phase-space integration averages the spatial part ofPαto zero so that Λαβ A= 2ieC Aϵαβµ0Kµa(ω,k) (6.101) in terms of a single medium characteristic a. The single most important application of the eﬀective neutrino elec- tromagnetic coupling is the photon decay process γ→ννthat was studied in the previous section. It turns out that Λαβ Acontributes very little to the decay process so that Λαβ V, or rather the polarization tensor Παβdetermines all aspects of the plasma process. Another possible process is the Cherenkov emission of photons by neutrinos. Of course, because in a plasma transverse photons acquire an “eﬀective mass,” i.e. their dispersion relation is time-like ( ω2−k2>0), Particle Dispersion and Decays in Media 239 this process is kinematically forbidden. However, longitudinal elec- tromagnetic excitations (plasmons), which exist only in the medium, propagate such that for some momenta ω2−k2<0 (space-like four- momentum), allowing for Cherenkov emission. One ﬁnds statements in the literature that the neutrino energy transfer to the medium by this process exceeded the transfer by (incoherent) ν-escattering (e.g. Oraevski˘ ı and Semikoz 1984; Oraevski˘ ı, Semikoz, and Smorodinski˘ ı 1986; Semikoz 1987a). This is in conﬂict with the discussion of Kirzh- nits, Losyakov, and Chechin (1990) who found on general grounds that the energy loss of a neutrino propagating in a stable medium was al- ways bounded from above by the collisional energy loss, apart from a factor of order unity. Granting this, the Cherenkov process does not seem to be of great practical importance. If neutrinos have masses and mix, decays of the form ν2→ν1γare possible in vacuum, and can be kinematically possible in a medium if the photon “eﬀective mass” does not exceed the neutrino mass diﬀer- encem2−m1. If kinematically allowed, this decay receives a contri- bution from the medium-induced coupling which may far exceed the vacuum decay rate. Explicit calculations were performed by a num- ber of authors36who unfortunately ignored the kinematic constraint imposed by the photon dispersion relation. This is not a reasonable approximation in view of the relatively small neutrino masses that re- main of practical interest. Further, in order to judge the importance of the medium-induced decay it is not relevant to compare with the vacuum decay rate, but rather one should compare with the collisional transition rate ν2e→eν1(mediated by photon exchange) which is the process with which the coherent reaction directly competes. The photon decay as well as the Cherenkov process and the medium- induced neutrino decay all have in common that the neutrino couples to an electromagnetic ﬁeld which is a freely propagating wave, obeying the dispersion relation in the medium which is ω2−k2=πT,L(ω,k) for transverse and longitudinal excitations, respectively. However, one may also consider the eﬀect of a static external electric or magnetic ﬁeld.37To this end, one must take the static limit ω→0 of the vertex functions Λαβ V,A(ω,k). For an external static electric ﬁeld the only nonvanishing component of the vector potential AµisA0. Then 36D’Olivo, Nieves, and Pal (1990); Kuo and Pantaleone (1990); Giunti, Kim, and Lam (1991). 37This issue has been investigated in many works, e.g. Oraevski˘ ı and Semikoz (1985, 1987), Semikoz (1987a,b), Nieves and Pal (1989c, 1994), Semikoz and Smorodinski˘ ı (1988, 1989), and D’Olivo, Nieves, and Pal (1989). 240 Chapter 6 Λαβ Adoes not contribute so that only the Λ00 Vcomponent remains of interest. In the static limit Π00is simply given by πL(0,k) which in turn can be identiﬁed with the square of the screening scale k2 Sin a medium (Sect. 6.4.1). This implies that the neutrino interacts with the external electric ﬁeld as if it had a charge eν=−(CV/e)√ 2GFk2 S. (6.102) In a classical (nondegenerate, nonrelativistic) hydrogen plasma the screening scale is given by the Debye scale through k2 S= 2k2 D= 2e2ne/T with the electron density neso that the induced neutrino charge is eν= CVe2√ 2GFne/T. This induced charge is explained by the medium polarization caused by the weak force exerted by the presence of the neutrino. While this induced charge is conceptually very interesting it does not seem to have any immediate practical consequences. Notably, it is not the relevant quantity for the interaction with a static magnetic ﬁeld, i.e. one may not infer that neutrinos move on curved paths in magnetic ﬁelds. The presence of a “neutrino charge” was derived for the interaction with a static electric ﬁeld! The relevant form factor for the interaction with a static magnetic ﬁeld is identiﬁed by noting that now only the spatial components of Aµare nonzero. In the static limit only the component Π00of the polarization tensor survives when contracted with Aµ. Then there is no contribution from Λαβ Sfor the neutrino interaction with a magnetic ﬁeld. The contribution from Λαβ A can be interpreted as a “normal” or “Dirac magnetic moment” induced by the medium (Semikoz 1987a; D’Olivo, Nieves, and Pal 1989) µν=−eCA√ 2GF4π∫∞ 0dp[ fe−(p)−fe+(p)] . (6.103) In the limit of a classical plasma this is µν= (eν/2me)(2CA/CV) where eνis the induced electric charge of Eq. (6.102). This induced Dirac magnetic moment is to be compared with the electron’s Dirac moment e/2me, not with an anomalous moment. The former arises from the eψeγµψeAµcoupling, the latter is described by 1 2µeψeσµνψeFµν. This means that the induced dipole moment does not lead to neutrino spin precession—it only couples to left-handed states. It entails an energy diﬀerence between neutrinos moving in opposite directions along a magnetic ﬁeld. The transverse part of the ﬁeld has no impact on the neutrino—there is no spin precession, and no curvature of the trajectory. (These conclusions pertain to the limit of weak magnetic ﬁelds. For strong ﬁelds the modiﬁcation of the electron Particle Dispersion and Decays in Media 241 wavefunctions, i.e. Landau levels rather than plane waves, would have to be used for a self-consistent treatment of the photon polarization tensor and thus, for the neutrino coupling to a magnetic ﬁeld. For a ﬁrst discussion see Oraevski˘ ı and Semikoz 1991.) In summary, on the basis of the existing literature it appears that the medium-induced electromagnetic form factors of neutrinos are of practical importance only for the photon decay process that was dis- cussed in the previous section. 6.7 Neutrino Refraction 6.7.1 Neutrino Refractive Index When neutrinos propagate in a medium they will experience a shift of their energy, similar to photon refraction, due to their coherent inter- action with the medium constituents (Wolfenstein 1978). The neutrino refractive index can be calculated in the same way as that for any other particle which propagates in a medium, namely on the basis of the forward scattering amplitudes as discussed in Sect. 6.2.1. As one needs only forward scattering, and as the relevant medium constituents are protons, neutrons, electrons, and possibly other neutrinos, only the Feynman graphs of Fig. 6.15 need to be considered.38 In most situations of practical interest the energies of the neutrinos and of the medium particles are much smaller than the WandZmass (80.2 and 91.2 GeV) so that the energy and momentum transferred by the gauge bosons is always much less than their mass.39This justiﬁes to expand their propagators (energy-momentum transfer Q) as Dµν(Q) =gµν m2 Z,W+Q2gµν−QµQν m4 Z,W+... (6.104) and keep only the ﬁrst term. (The second term is needed if the con- tribution of the ﬁrst one cancels as in a CP symmetric medium—see 38In the formalism of ﬁnite temperature and density (FTD) ﬁeld theory the am- plitudes may be written in a more compact form so that the relevant Feynman graphs reduce to a tadpole and a bubble graph (N¨ otzold and Raﬀelt 1988; Nieves 1989; Pal and Pham 1989). Apart from a more compact notation, however, the FTD formalism leads to the same expressions as the “pedestrian” approach chosen here. 39See however Learned and Pakvasa (1995) as well as Domokos and Kovesi- Domokos (1995) for a discussion of the oscillations of very high-energy cosmic neu- trinos for which this approximation is not adequate. 242 Chapter 6 Fig. 6.15. Amplitudes contributing to forward scattering: (a) Neutral- current scattering for any νorνon any forfas target. (b) νe-νescattering. (c)νe-echarged-current scattering. (d) νe-e+charged-current scattering. (e) Eﬀective four-fermion vertex in the low-energy limit. Sect. 6.7.2.) This amounts to reducing the weak interaction to the low-energy Fermi eﬀective Hamiltonian represented by graph (e) in Fig. 6.15. For the neutral-current processes (a) and (b) it is explicitly Hint=GF√ 2ψfγµ(CV−CAγ5)ψfψνℓγµ(1−γ5)ψνℓ, (6.105) whereψνℓis a neutrino ﬁeld ( ℓ=e,µ,τ ) whileψfrepresents fermi- ons of the medium ( f=e,p,n , or even neutrinos νℓ′). Here,GF= 1.166×10−5GeV−2is the Fermi constant. The relevant values of the vector and axial-vector weak charges CVandCAare given in Ap- pendix B. In the low-energy limit the charged-current reactions (c) and (d) can also be represented as an eﬀective neutral-current interac- tion of the same form with CV=CA= 1. It is now straightforward to work out the forward scattering ampli- tudes. The axial-vector piece represents the spin of fand so it averages to zero if the medium is unpolarized. Then one ﬁnds for the refractive index of a neutrino (upper sign) or antineutrino (lower sign) with en- ergyω nrefr−1 =∓C′ VGFnf−nf ω√ 2, (6.106) wherenfandnfare the number densities of fermions fand antifer- mionsf, respectively. The eﬀective weak coupling constants C′ Vare identical with the CVgiven in Appendix B except for neutrinos as Particle Dispersion and Decays in Media 243 medium particles40which are left-handed and thus polarized. There- fore, (1 −γ5)ψν= 2ψνandC′ V= 2CV. Because we are dealing with forward scattering where recoil eﬀects do not occur, the contributions from free or bound nucleons are the same. Therefore, Eq. (6.106) allows one to determine the refractive index of any normal medium. Because electric neutrality implies an excess density of electrons over positrons which balances against the protons, their neutral-current contributions cancel. (A possible excep- tion is aπ−condensate that may exist in neutron stars.) An excess ofνeoverνeappears to occur only in a young supernova core where neutrinos have a large chemical potential for the ﬁrst few seconds after collapse. All told, the dispersion relation for unmixed neutrinos, valid even in the nonrelativistic limit (Chang and Zia 1988), can be written in terms of a potential energy as (ω−V)2=k2+m2, (6.107) whereV=−(nrefr−1)ω. For all practical cases V=±√ 2GFnB×  (−1 2Yn+Ye+ 2Yνe) forνe, (−1 2Yn+Yνe) forνµ,τ,(6.108) (upper sign ν, lower sign ν). Here,nBis the baryon density and Yf≡nf−nf nB(6.109) are the particle number fractions commonly used in astrophysics. Nu- merically, √ 2GFnB= 0.762×10−13eVρ g cm−3(6.110) with the mass density ρ. A remark concerning the absolute sign of Vis in order. The relative signs between the diﬀerent CV’s can be worked out easily from the weak interaction structure of the standard model. Also, the relative sign of the eﬀective neutral-current amplitudes which follow from Z◦ andWexchange follows directly, for example, from the FTD approach (N¨ otzold and Raﬀelt 1988). Thus to ﬁx the overall sign it is enough 40In a supernova core or in the early universe it is not possible to distinguish between a “test neutrino” and a “medium neutrino.” There, one has to study the nonlinear evolution of the entire ensemble self-consistently (Sect. 9.3.2). 244 Chapter 6 to understand the absolute sign of neutrino-neutrino scattering. It is a case where identically “charged” fermions scatter by the exchange of a vector boson. The structure of this process is analogous to Coulomb scattering of like-charged particles which experience a repulsive force. (The exchange of a spin-0 or spin-2 boson leads to an attractive force.) Thus a neutrino of given momentum in a region of space ﬁlled with other neutrinos will have a positive potential energy Vin addition to its kinetic energy. The correct absolute sign was ﬁrst pointed out by Langacker, Leveille, and Sheiman (1983). The deviation from relativistic propagation described by Eq. (6.107) can be expressed as an eﬀective refractive index which includes the vacuum neutrino mass, nrefr=[( 1−V ω)2 −m2 ω2]1/2 → 1−V ω−m2 2ω2(6.111) (relativistic limit). Therefore, the eﬀect of a medium can be expressed as an eﬀective mass m2 eﬀ=m2+ 2ωV. (6.112) A numerical comparison with the vacuum mass is achieved by ( 2ω√ 2GFnB)1/2= 3.91×10−4eV(ρ g cm−3)1/2(ω MeV)1/2 . (6.113) Of course, the term “eﬀective mass” is a misnomer because meﬀdepends on the energy ω, andm2 eﬀcan be negative depending on the vacuum mass, the medium composition, the ﬂavor of the neutrino, and whether it isνorν. The phase velocity vphase =ω/k=n−1 refrcan be larger or less than the speed of light, depending on those parameters. However, the group velocityvgroup =dω/dk remains at its vacuum value (1 + m2/k2)−1/2 for a given momentum kbecauseωis shifted by a constant amount V, independently of k. Because normal media contain about equal numbers of protons and neutronsYn≈Ye≈1 2and soVνe≈1 4√ 2GFnB≈ −Vνµ,τ. Therefore, νe andνµ,τare shifted by almost exactly opposite amounts. An exception is the proton-rich material of normal stars which initially contain about 75% hydrogen. Another exception is the neutron-rich matter of neutron stars where even Vνeis negative. Particle Dispersion and Decays in Media 245 The absolute shift of the neutrino “masses” is rather negligible be- cause we are dealing with highly relativistic particles. Even in this limit, however, the diﬀerence between the dispersion relation of diﬀerent ﬂa- vors is important for oscillation eﬀects. Hence the most noteworthy medium eﬀect is its ﬂavor birefringence :νeandνµ,τacquire diﬀerent eﬀective masses because of the charged-current contribution from νe-e scattering. The diﬀerence of their potentials is Vνe−Vνµ,τ=√ 2GFnL withnL=YLnBthe lepton-number density where the number frac- tion of leptons is YL=Ye+Yνe. Of course, neutrinos as a background medium contribute only in a young supernova core. 6.7.2 Higher-Order Eﬀects In the early universe one has nearly equal densities of particles and an- tiparticles with an asymmetry of about 10−9, leading to a near cancel- lation of the refractive terms Eq. (6.106). One may think that the next most important contribution is from ν-γscattering, a process closely related to the ν→ν′γγdecay brieﬂy discussed in Sect. 7.2.2. If one approximates the weak interactions by an eﬀective four-fermion cou- pling the relevant amplitude is given by the graph Fig. 6.16 which on dimensional grounds should be of order αGF. However, electromag- netic gauge invariance together with the left-handedness of the weak interaction implies that it vanishes identically (Gell-Mann 1961). For massive neutrinos the amplitude is proportional to αGFmν, but even in this case it vanishes in the forward direction (Langacker and Liu 1992). Fig. 6.16. Neutrino-photon scattering with an eﬀective four-fermion weak interaction and a charged lepton ℓin the loop. This amplitude vanishes entirely for massless neutrinos, and for massive ones it still vanishes in the forward direction. For the lowest-order ν-γcontribution to the refractive index one must then use the full gauge-boson propagator and include all one-loop amplitudes required by the standard model. Such calculations were performed by Levine (1966) and by Cung and Yoshimura (1975) who found that the scattering amplitude was proportional to αGFs/m2 W (center of mass energy√s). Recently this problem was revisited by 246 Chapter 6 Dicus and Repko (1993) who worked out explicitly the matrix elements and cross sections. From their results one can extract the forward scattering amplitude which leads to a refractive index for νℓ(ℓ=e,µ,τ ) in a photon bath, nrefr−1 =α 4πGF m2 W[ 1 +4 3ln(m2 W m2 ℓ)] ⟨Eγ⟩nγ, (6.114) where⟨...⟩means an average. Numerically, the term in square brackets is 32.9 forℓ=eand thus not small, but with 4 πin the denominator the whole expression is still of order αGF/m2 W, i.e. of order41G2 F. (See also Nieves, Pal, and Unger 1983; Nieves 1987; Langacker and Liu 1992.) Because the ν-γterm is so small a larger refractive index arises if one includes the second term in the expansion Eq. (6.104) of the gauge-boson propagators. Of course, in graphs (a) and (c) of Fig. 6.15 forward scattering implies Q= 0 so that only the ﬁrst term contributes, except when f=νwhere the exchange graph has Q̸= 0. This case and graphs (b) and (d) yield a second-order contribution (N¨ otzold and Raﬀelt 1988). For a neutrino of ﬂavor ℓwhich is either e,µ, orτit is nrefr−1 =8√ 2GF 3m2 Z( ⟨Eνℓ⟩nνℓ+⟨Eνℓ⟩nνℓ) +8√ 2GF 3m2 W( ⟨Eℓ−⟩nℓ−+⟨Eℓ+⟩nℓ+) . (6.115) In this case the contributions from background fermions and antifermi- ons add with the same sign, and the global sign remains the same for νandνas test particles. In practice, only an electron-positron back- ground is of relevance in the early universe so that ντ, for example, only feels a second-order contribution from other ντ’s andντ’s. These results are of order G2 F/αand thus they are the dominant contribution in a CP-symmetric plasma. One-loop corrections to the amplitudes of Fig. 6.15 yield other higher-order terms which are of order G2 Flike Eq. (6.114). They are still interesting because the loops involve charged leptons with a mass depending on their ﬂavor. Therefore, the universality of the eﬀective neutral-current interaction is broken on this level, leading to diﬀerent refractive indices for diﬀerent νℓ. Betweenνeandνµorντthe medium is already birefringent to lowest order from νe-echarged-current inter- actions. Between νµandντthe one-loop correction dominates. Assum- 41Note that m2 Z= cos2ΘWsin2ΘW√ 2GF/πα andm2 W= sin2ΘW√ 2GF/πα. Particle Dispersion and Decays in Media 247 ing electric neutrality it is (Botella, Lim, and Marciano 1987; see also Semikoz 1992 and Horvat 1993) Vντ−Vνµ=3G2 Fm2 τ 2π2nB[ ln(m2 W m2 τ) −1 +Yn 3] , (6.116) with a sign change for Vντ−Vνµ. The shift of the “eﬀective mass” m2 eﬀ is numerically 2ω(Vντ−Vνµ) =( 2.06×10−6eV)2ρ g cm−3ω MeV6.61 +Yn/3 7, (6.117) much smaller than the corresponding diﬀerence between νeandνµ,τ. 6.7.3 The Sun a Neutrino Lens? The most important consequence of neutrino refraction in media is its impact on neutrino oscillations because diﬀerent ﬂavors experience a diﬀerent index of refraction. In optics, the most notable consequence of refraction is the possibility to deﬂect light and thus to use lenses and other optical instruments. In principle, the same is also possible for neutrinos. The Sun, for example, could act as a gigantic neutrino lens. One may easily calculate the deﬂection caused by a given body. If sis a unit vector along the direction of a propagating wave, and if s is a coordinate along the beam, the deﬂection is given by (Sommerfeld 1958) |ds/ds|=n−1 refr|s× ∇nrefr|, (6.118) wherenrefris the refractive index. One may equally write |dα/ds|=n−1 refr|∇⊥nrefr|, (6.119) whereαis the angle relative to the local tangential vector, i.e. dα/ds is the local curvature of the beam, and ∇⊥is the transverse gradient. The total angle of deﬂection is |∆α|=∫+∞ −∞ds|∇⊥nrefr| (6.120) if the curvature is small which is the case for |nrefr−1| ≪1. If the beam hits a spherically symmetric body (radius R) at an impact parameter b < R its angle against the radial direction at a radiusrfrom the center is sin β=b/rso that ∇⊥nrefr= (b/r)∂rnrefr. 248 Chapter 6 Moreover, if sis measured from the point of closest approach one has s= (r2−b2)1/2and sods= (r2−b2)−1/2rdr. Altogether one ﬁnds ∆α=−2∫R bdrb∂rnrefr√ r2−b2. (6.121) The refraction by the surface of the body was ignored: it is assumed that it is “soft” with ∂rnrefr= 0 atr=R. The absolute sign was chosen such that ∆ αis positive if the spherical body acts as a “focussing lens.” This is the case, for example, for light passing through the atmosphere of the Earth where ∂rnrefr<0 and so the Sun near the horizon appears “lifted” (Sommerfeld 1958). With the results for the neutrino refractive index of Sect. 6.7.1 one has for a beam of energy ω ∆α= 2√ 2GFnB,c ω∫R bdrb∂r[nB(−1 2Yn+Ye)] nB,c√ r2−b2, (6.122) wherenB,cis the baryon density at the center of the lens. This ex- pression applies to νewhile forνµorντthe termYeis absent, and for antineutrinos the overall sign changes. For the Sun with a central density of about 150 g cm−3the overall coeﬃcient is 2 .3×10−18(10 MeV/ω). The integral expression is dimen- sionless and thus of order unity. Therefore, the focal length of the Sun as a neutrino lens is of order 1018R⊙(solar radius) for 10 MeV neutrinos, or about the radius of the visible universe! 6.8 Majoron Decay As a ﬁrst application for the neutrino dispersion relation I consider the interaction of neutrinos with the hypothetical majorons (Sect. 15.7). These particles are Nambu-Goldstone bosons of a symmetry which is spontaneously broken by a Higgs ﬁelds which gives the neutrinos Ma- jorana masses. For the present purposes it is enough to specify a pseu- doscalar interaction Lint=ihψνγ5ψνχ (6.123) between the neutrinos of a given family and the massless majorons. Here,ψνis a Majorana neutrino ﬁeld while his a dimensionless Yukawa coupling constant. Majorana neutrinos are fermions with only two degrees of freedom. They correspond to a helicity-minus νand a helicity-plus ν. In fact, if Particle Dispersion and Decays in Media 249 they are massless there is no operational distinction between a Majo- rana neutrino and the two active degrees of freedom of a Dirac neutrino. Therefore, according to Eq. (6.107) the dispersion relation for the he- licity±states of a Majorana neutrino is E±= (m2+p2)1/2∓V, (6.124) where the medium-induced potential Vwas given in Eq. (6.108). For νeit isV=√ 2GF(ne−1 2nn) with the electron and neutron densities neandnn, respectively. This dispersion relation implies that the medium is “optically ac- tive” with regard to the neutrino helicities, just as some media are birefringent with regard to the photon circular polarization. In the optical case the left-right symmetry (parity) is broken by the medium constituents which must have a deﬁnite handedness; sugar molecules are a well-known example. In the neutrino case parity is broken by the structure of the interaction; the medium itself is unpolarized. Because there is an energy diﬀerence between the Majorana helicity statesν±for a given momentum, decays ν−→ν+χare kinematically allowed. For relativistic neutrinos the squared matrix element is found to be|M|2= 4h2P1·P2with the four-momenta P1,2of the initial and ﬁnal neutrino state. The diﬀerential decay rate is then dΓ =4h2 2E1d3p2 2E2(2π)3d3k 2ω(2π)3P1·P2(2π)4δ4(P1−P2−K) (6.125) with the majoron four-momentum K. Integrating out the d3kvariable removes the momentum δfunction. The remaining diﬀerential decay is dΓ dE2=αχ∫+1 −1dxP1·P2p2 2 E1E2ωδ(E1−E2−ω), (6.126) wherex= cosθfor the angle between p1andp2andαχ≡h2/4π is the majoron “ﬁne-structure constant.” In the δfunction one must useω=k=|p1−p2|= (p2 1+p2 2−2p1p2x)1/2. With∫dxδ[f(x)] = |df/dx|−1=ω/p 1p2and with energy-momentum conservation which yieldsP1−P2=Kand thusP1·P2=1 2(P2 1+P2 2) one ﬁnds dΓ dE2=αχ(E2 1+E2 2−p2 1−p2 2)p2 2E1E2p1. (6.127) 250 Chapter 6 If one ignores the vacuum mass relative to Vone hasp1,2=E1,2∓V so that to lowest order in V dΓ dE2=αχV(E1−E2) E2 1. (6.128) Therefore, the ﬁnal-state neutrino spectrum has a triangular shape whereE2varies between 0 and E1. The integrated decay rate is Γ =1 2αχV (6.129) as ﬁrst shown by Berezhiani and Vysotsky (1987). Various subleties were covered in the detailed discussion by Giunti et al. (1992). Interestingly, under the same circumstances the decay χ→ν+ν−is equally possible (Rothstein, Babu, and Seckel 1993) and proceeds with the same rate Eq. (6.129)—see Berezhiani and Rossi (1994). In these calculations it was assumed that the majoron is massless as it behooves a Nambu-Goldstone boson. Does this remain true in a medium? The majorons could have a Yukawa coupling gto electrons in which case one would expect on dimensional grounds that they de- velop a medium-induced “mass” of order g(ne/me)1/2in analogy to the photon plasma mass. Indeed, if one works with a pseudoscalar cou- pling analogous to Eq. (6.123) one ﬁnds such a result. Even if they did not couple to electrons, in a supernova core there is a background of neutrinos to which majorons couple by assumption. However, a pseudoscalar coupling is not appropriate for a Nambu- Goldstone boson as it is not invariant under a shift χ→χ+χ0. The pseudoscalar expression is only the lowest-order expansion of an expo- nential coupling which respects the symmetry. Equivalently, a deriva- tive coupling of the sort (1 /2f)ψγµγ5ψ∂µχcan be used which satisﬁes the symmetry explicitly (Sect. 14.2.3). Either way one ﬁnds that the forward scattering amplitude between Nambu-Goldstone bosons and fermions vanishes—there is no refractive index. The same conclusion was reached by Flynn and Randall (1988) on more general grounds. For a suitable choice of parameters the medium-induced decay of electron neutrinos can deplete the solar neutrino ﬂux before it leaves the Sun. However, it is doubtful if this eﬀect could explain all current solar neutrino measurements (Sect. 10.8). For h∼>10−6a radical mod- iﬁcation of the neutrino signal from a supernova collapse is expected (Sect. 15.7.2). Chapter 7 Nonstandard Neutrinos The phenomenological consequences of nonvanishing neutrino masses and mixings and of electromagnetic couplings are explored. The decay channels and electromagnetic properties of mixed neutrinos are dis- cussed. Experimental, astrophysical, and cosmological limits on neu- trino masses, decays, and electromagnetic properties are summarized. 7.1 Neutrino Masses 7.1.1 The Fermion Mass Problem In the physics of elementary particles one currently knows of two cat- egories of apparently fundamental ﬁelds: the spin-1 2quarks and lep- tons on the one-hand side, and the spin-1 gauge bosons on the other. The former constitute “matter” while the latter mediate the electro- magnetic, weak, and strong forces. The gauge-theory description of the interactions among these particles is renowned for its elegance and stunning in its success at accounting for all relevant measurements. At the same time it has many entirely loose ends. Perhaps the most puz- zling problem is that of fermion masses and the related issue of the threefold replication of families: The electron and neutrino as well as the up and down quarks (which make up protons and neutrons) each come in two additional “ﬂavors” or families which seem to diﬀer from the ﬁrst one only in their masses (Fig. 7.1). The standard model of particle physics holds that all fermions and gauge bosons are fundamentally massless. The gauge symmetry for- bids a fundamental mass for the latter while the masslessness of the former is indicated by the handedness of the weak interaction: only left-handed (l.h.) fermions feel this force while the right-handed (r.h.) 251 252 Chapter 7 Fig. 7.1. Mass spectrum of elementary fermions according to the Review of Particle Properties (Particle Data Group 1994). For the top quark see CDF Collaboration (1995) and D0 Collaboration (1995). For neutrinos the experimental upper limits of Tab. 7.2 are shown. ones are “sterile.” Handedness, however, is not a Lorentz-invariant concept because a particle with a spin opposite to its momentum (l.h. or negative helicity) is r.h. when viewed by a suﬃciently fast-moving observer. Only massless particles cannot be “overtaken” because they move with the speed of light and so they can be classiﬁed into l.h. and r.h. states without reference to a speciﬁc Lorentz frame. Any deviation from relativistic propagation is thought to be a “re- fractive” eﬀect, much as a photon acquires a nontrivial dispersion in a medium. Here, the “medium” is the spin-0 Higgs ﬁeld Φ, a hypothetical third category of fundamental objects, which is believed to take on the nonzero classical value Φ 0≈246 GeV in vacuum due to self-interactions Nonstandard Neutrinos 253 (“spontaneous symmetry breaking”). Fermion ﬁelds ψinteract with the Higgs ﬁeld by virtue of a Lagrangian gΦψψwheregis a dimensionless (Yukawa) coupling constant. In vacuum, this coupling leads to an in- teraction term gΦ0ψψwhich has the form of a standard Dirac mass termmψψ. Diﬀerent fermion masses thus arise from diﬀerent Yukawa couplings. Φ 0does not depend on the Lorentz frame because of the scalar nature of the Higgs ﬁeld and so gΦ0ψψis the same in all frames, unlike a refractive photon “mass” in a medium. It is not known whether the Higgs mechanism is the true source for the masses of the fundamental fermions. Experimentally, the Higgs particle (excitations of the Higgs ﬁeld) has not yet been discovered, while theoretically the fermion masses are the least appealing aspect of the standard model because they require a host of ad hoc coupling constants which must be experimentally determined. 7.1.2 Dirac and Majorana Masses Neutrinos break the pattern of Fig. 7.1 in that they are much lighter than the other members of a given family, a discrepancy which is most severe for the third family where cosmologically mντ∼<30 eV, eight and ten orders of magnitude less than mτandmt, respectively! Moreover, neutrinos are diﬀerent in that their r.h. chirality states are sterile be- cause of the handedness of the weak interaction. The r.h. states interact with the rest of the world only by gravity and by a possible Yukawa coupling to the Higgs ﬁeld. It is frequently assumed that neutrinos do not couple to the Higgs ﬁeld, and that the r.h. components do not even exist, assumptions which are part of the particle-physics standard model. In this case there are only two neutrino states for a given family as opposed to four states for the charged leptons. Actually, one may interpret the two components of such a neutrino as the spin states of a Majorana fermion which is de- ﬁned to be its own antiparticle. Fermions with four distinct states are known as Dirac fermions . Naturally, a Majorana fermion cannot carry a charge as that would allow one to distinguish it from its antiparticle. A magnetic or electric dipole moment is equally forbidden: its orienta- tion relative to the spin is reversed for antiparticles. For example, the neutron cannot be a Majorana fermion among other reasons because it carries a magnetic moment. Because the r.h. neutrino components of a given family, if they exist, are sterile anyway, there is no practical distinction between massless Dirac and Majorana neutrinos except in a situation where gravitational 254 Chapter 7 interactions dominate. For example, even otherwise sterile neutrinos should be thermally emitted from black holes which are thought to emit blackbody radiation of all physical ﬁelds. Equally, they would have been produced in the very early universe when quantum gravitational eﬀects dominate. However, their present-day cosmic density, like that of primordial gravitons, would be very dilute relative to microwave background photons. If neutrinos were Majorana particles they could still have a mass even though it could not arise from the usual Higgs ﬁeld which in- duces Dirac masses. However, a Majorana mass could arise from the interaction with a second Higgs ﬁeld which also develops a vacuum ex- pectation value. Then the smallness of the neutrino masses could be due to a small vacuum value of the new Higgs ﬁeld while the Yukawa couplings would not need to be anomalously small. It is also possible that the r.h. components of the neutrinos do ex- ist, but are themselves (sterile) Majorana fermions with large masses. It should be noted that any Dirac fermion (four components) can be viewed as a combination of two Majorana fermions (two components each) with degenerate masses. Certain variations of such models (“see- saw mass models”) predict for the light, interacting neutrinos m1:m2:m3=m2 e:m2 µ:m2 τorm2 u:m2 c:m2 t. (7.1) The smallness of the neutrino masses is then a suppression eﬀect by the large mass scale of the heavy sterile state whose mass would arise, for example, at the grand uniﬁcation scale of 1015−1016GeV. For an elementary introduction to the most common models for neu- trino masses see, for example, Mohapatra and Pal (1991). The dizzying variety of such models alone attests to the fact that even very ba- sic questions about the nature of neutrinos remain unanswered. While some mass schemes like the see-saw relationship Eq. (7.1) are intriguing, they have no predictive power because there are many other possibili- ties. Therefore, it is best to remain open to all possibilities which are not excluded by experimental or astrophysical arguments. 7.1.3 Kinematical Mass Bounds Unsurprisingly, much experimental eﬀort goes into attempts to measure or narrow down the range of possible neutrino masses, an area where astrophysics and cosmology have made their most renowned contribu- tions to particle physics. Direct laboratory experiments rely on the Nonstandard Neutrinos 255 kinematical impact of a mass on certain reactions such as nuclear de- cays of the form ( A,Z)→(A,Z+ 1)e−νewhere the continuous energy spectrum of the electrons originally revealed the emission of another particle that carried away the remainder of the available energy. The minimum amount of energy taken by the neutrino is the equivalent of its mass so that the upper endpoint of the electron spectrum is a sen- sitive measure for mνe. Actually, the most sensitive probe is the shape of the electron spectrum just below its endpoint, not the value of the endpoint itself. The best constraints are based on the tritium decay 3H→3He+e−νewith a maximum amount of kinetic energy for the electron of Q= 18.6 keV. This unusually small Q-value ensures that a large fraction of the electron counts appear near the endpoint (Boehm and Vogel 1987; Winter 1991). In Tab. 7.1 the results from several recent experiments are sum- marized which had been motivated by the Moscow claim of 17 eV < mνe<40 eV (Boris et al. 1987). This range is clearly incompatible with the more recent data which, however, ﬁnd negative mass-squares. This means that the endpoint spectra tend to be slightly deformed in the opposite direction from what a neutrino mass would do. This eﬀect is particularly striking and signiﬁcant for the Livermore exper- iment where it is nearly impossible to blame it on a statistical ﬂuc- tuation. Therefore, at the present time one cannot escape the con- clusion that this experimental technique suﬀers from some unrecog- nized systematic eﬀect. This problem must be resolved before it will become possible to extract a reliable bound on mνealthough it ap- pears unlikely that an mνein excess of 10 eV could be hidden by what- Table 7.1. Summary of tritium decay experiments. Experiment m2 νe±σstat±σsyst Reference [eV2] Los Alamos −147±68±41 Robertson et al. (1991) Tokyo −65±85±65 Kawakami et al. (1991) Z¨ urich −24±48±61 Holzschuh et al. (1992) Mainz −39±34±15 Weinheimer et al. (1993) Livermore −130±20±15 Stoeﬄ and Decman (1994) Troitsk −18±6 Belesev et al. (1994)a aSee Otten (1995) for a published description. 256 Chapter 7 ever eﬀect causes the “wrong” deformation of the end-point spectrum. In spite of apparent systematic problems, the Troitsk group (Belesev et al. 1994) claims a limit of mνe<4.5 eV at 95% CL; see also Ot- ten (1995). One has attempted to determine mνµby measuring the muon mo- mentum from the decay of stopped pions, π+→µ+νµ, leading to m2 νµ= m2 π++m2 µ−2mπ+(m2 µ+p2 µ)1/2. A recentpµmeasurement (Daum et al. 1991) implied a negative squared mass of m2 νµ=−(0.154±0.045) MeV2, probably due to large systematic uncertainties in the determination of mπ+. Hence, it seemed that the often-quoted bound of mνµ<0.27 MeV did not apply. An older experiment studied the in-ﬂight decay of pi- ons with a result m2 νµ=−(0.14±0.20) MeV2, largely independent of the pion mass (Anderhub et al. 1982). This implies a 90% CL upper limit ofmνµ<0.50 MeV. Most recently, the mass of the negative pion was reconsidered by Jeckelmann, Goudsmit, and Leisi (1994). Their previous experiment allows for two mass assignments, mπ= 139.56782±0.00037 MeV or 139 .56995±0.00035 MeV. The larger value had previously been rejected on the basis of evidence which now appears questionable. Together with a new pµmeasurement (Assamagan et al. 1994) one ﬁnds m2 νµ=−0.148±0.024 MeV2or−0.022±0.023 MeV2. The ﬁrst value is negative by 6.2 standard deviations and thus may be rejected as unphysical. The second solution is compatible with zero and gives a 90% CL upper limit of mνµ<0.16 MeV. Forντthe best bounds also come from limits on missing energy in certain reactions, the only form in which ντhas ever been “ob- served.” The ARGUS Collaboration (1988, 1992) studied the decay τ−→3π−2π+ντwith a total of 20 events with good energy deter- minations for all ﬁve pions, leading to mντ<31 MeV at 95% CL. A similar experiment by the CLEO Collaboration (1993) based on a much larger data sample gave mντ<32.6 MeV at 95% CL. Most re- cently, the ALEPH Collaboration (1995) at CERN has reported a new Table 7.2. Experimental neutrino mass limits. Flavor Limit CL Reference νe (5 eV) — See Tab. 7.1 νµ 0.16 MeV 90% Assamagan et al. (1994) ντ 23.8 MeV 95% ALEPH Collaboration (1995) Nonstandard Neutrinos 257 95% CL mass limit of 23 .8 MeV on the basis of 25 events of the form τ→5πντand 5ππ◦ντwhereπstands for a charged pion. Another kinematical method to be discussed in Sect. 11.3.4 uses the neutrino pulse dispersion from a distant supernova (SN). For νethe observed neutrinos from SN 1987A gave mνe∼<20 eV, less restrictive than the tritium experiments. However, if the neutrino pulse from a future galactic SN will be detected one may be able to probe even a ντ mass down to the cosmologically interesting 30 eV range (Sect. 11.6)! For Dirac neutrinos there is another essentially kinematical con- straint from the SN 1987A neutrino observations. The sterile νDirac components can be produced in scattering processes by helicity ﬂips. In a supernova core this eﬀect leads to an anomalous energy drain, limiting a Dirac mass to be less than a few 10 keV (Sect. 13.8.1). 7.1.4 Neutrinoless Double-Beta Decay If neutrinos have Majorana masses, lepton number is not conserved as one cannot associate a conserved “charge” with a Majorana particle. One observable consequence would be the occurrence of neutrinoless nuclear decay modes of the form ( A,Z)→(A,Z+2) 2e−which would violate lepton number by two units. There are several isotopes which can decay only by the simultaneous conversion of two neutrons. Re- cently it has become possible to observe the electron spectra from the standard two-neutrino mode ( A,Z)→(A,Z+2) 2e−2νe; for a recent review see Moe (1995). The decay76Ge→76Se 2e−2νe, for example, is found to have a half-life of (1 .43±0.04stat±0.13syst)×1021yr (Beck 1993). The age of the universe, by comparison, is about 1010yr. In the 0νdecay mode, loosely speaking, one of the emitted Majorana neutrinos would be reabsorbed as an antineutrino with an amplitude proportional to mνe,Majorana and thus a rate proportional to m2 νe,Majorana . In a measurement of the combined energy spectrum of both electrons the 0νmode would show up as a peak at the endpoint. The best current upper bound is from the Heidelberg-Moscow76Ge experiment which yields mνe,Majorana<0.65 eV (Balysh et al. 1995), a number which will likely improve to 0 .2 eV over the next few years. This nominal limit must be relaxed by as much as a factor of 2 −3 for the uncertainty in the nuclear matrix elements which are needed to translate an experimental limit on the neutrinoless decay rate into a mass limit. With neutrino mixing (Sect. 7.2) the other ﬂavors also contribute so that the bound is really on the quantity ⟨mν⟩ ≡∑ jλj|Uej|2mjwhereλj is a CP phase equal to ±1, and the sum is to be extended over all two- 258 Chapter 7 component Majorana neutrinos. In this language a four-component Dirac neutrino consists of two degenerate two-component Majorana ones withλ= +1 and −1 so that their contributions cancel exactly, reproducing the absence of lepton number violation for Dirac neutrinos. If one takes the largest cosmologically allowed value mντ= 30 eV and the largest experimentally allowed mixing amplitude (Sect. 8.2.4) of |Ue,3| ≈0.16 one may have a contribution as large as 0 .8 eV fromντ. 7.1.5 Cosmological Mass Bounds Cosmology arguably yields the most important neutrino mass bounds (Kolb and Turner 1990; B¨ orner 1992). In the framework of the big-bang scenario of the early universe one expects about as many “blackbody neutrinos” in the universe as there are cosmic microwave photons. In detail, the cosmic energy density in massive neutrinos is found to be ρν=3 11nγ3∑ i=1mi, (7.2) withnγthe present-day density of microwave background photons and mithe neutrino masses. In units of the cosmic critical density this is Ωνh2=3∑ i=1mi 93 eV, (7.3) wherehis the Hubble constant in units of 100 km s−1Mpc−1. The observed age of the universe together with the measured expansion rate yields Ω h2∼<0.4 so that for any of the known families mν∼<30 eV. (7.4) If one of the neutrinos had a mass near this bound it would be the main component of the long-sought dark matter of the universe. Certain scenarios of structure formation currently favor “hot plus cold dark matter” where neutrinos with mνe+mνµ+mντ≈5 eV play a sub-dominant dynamical role but help to shape the required spectrum of primordial density perturbations (Pogosyan and Starobinsky 1995 and references therein). Preferably, the three neutrino masses should be degenerate rather than one dominating ﬂavor. If neutrinos were unstable and if they decayed so early that their decay products were suﬃciently redshifted by the expansion of the uni- verse, the cosmological mass bound can be violated without running into direct conﬂict with observations. The excluded range of masses Nonstandard Neutrinos 259 Fig. 7.2. Cosmological bounds on neutrino masses and lifetimes as described in the text. The experimental limits are shown above the main panel. If the dominant decay channel is the majoron mode →′the BBN-excluded range extends between the dashed lines. The dotted line is ν|Ue3|2for standard-model decays 3→1according to Eq. (7.17). and lifetimes according to Dicus, Kolb, and Teplitz (1977)42is shown in Fig. 7.2 as a shaded area marked “Mass Density.” Decaying neutrinos would cause a second cosmic epoch of radiation domination, suppressing the growth of density ﬂuctuations and thus the formation of structure (Steigman and Turner 1985; Krauss 1991; Bond and Efstathiou 1991). Somewhat schematically, the area above the shaded band in Fig. 7.2 marked “Structure Formation” is excluded by this more model-dependent argument. For masses and lifetimes on this band, neutrinos would actually have the beneﬁcial eﬀect of modify- ing the primordial spectrum of density ﬂuctuations such as to avoid the problem of too much small-scale power in cold dark matter universes (Bardeen, Bond, and Efstathiou 1987; Bond and Efstathiou 1991; Do- delson, Gyuk, and Turner 1994; White, Gelmini, and Silk 1995). 42Note that the corresponding limits discussed in the book by Kolb and Turner (1990) are somewhat less restrictive because their treatment does not seem to be entirely self-consistent (G. Gelmini, private communication). 260 Chapter 7 The expansion rate and thus the energy density of the universe are well “measured” at the epoch of nucleosynthesis ( T≈0.3 MeV) by the primordial light-element abundances (Yang et al. 1984). This big- bang nucleosynthesis (BBN) argument has been used to constrain the number of light neutrino families to Nν∼<3.4 (Yang et al. 1984; Olive et al. 1990). Even though the measured Z◦decay width has established Nν= 3 (Particle Data Group 1994) the BBN bound remains of interest as amass limit because massive neutrinos contribute more than a mass- less one to the expansion rate at BBN. For a lifetime exceeding about 100 s this argument excludes 500 keV ∼<mν∼<35 MeV (Kolb et al. 1991; Dolgov and Rothstein 1993; Kawasaki et al. 1994), with even more re- strictive limits for Dirac neutrinos (Fuller and Malaney 1991; Enqvist and Uibo 1993; Dolgov, Kainulainen, and Rothstein 1995). In Fig. 7.2 the region thus excluded is hatched and marked “BBN.” Kawasaki et al. (1994) have considered the majoron mode ν→ν′χ (Sect. 15.7) as a speciﬁc model for the neutrino decay. Including the energy density of the scalar χthey ﬁnd even more restrictive limits which exclude the region between the dashed lines in Fig. 7.2. 7.2 Neutrino Mixing and Decay 7.2.1 Flavor Mixing One of the most mysterious features of the particle zoo is the threefold repetition of families (or “ﬂavors”) shown in Fig. 7.1. The fermions in each column have been arranged in a sequence of increasing mass which appears to be the only signiﬁcant diﬀerence between them. There is no indication for higher sequential families; the masses of their neutrinos would have to exceed1 2mZ= 46 GeV according to the CERN and SLAC measurements of the Z◦decay width (Particle Data Group 1994). If the origin of masses is indeed the interaction with the vacuum Higgs ﬁeld, the only diﬀerence between the fermions of a given column in Fig. 7.1 is their Yukawa coupling to Φ. If the only diﬀerence between, say, an electron and a muon is the vacuum refraction, any superposition between them is an equally legit- imate charged lepton except for the practical diﬃculty of preparing it experimentally. When such a mixed state propagates, the two compo- nents acquire diﬀerent phases along the beam exactly like two photon helicities in an optically active medium, leading to a rotation of the plane of polarization. Of course, now this “polarization” is understood in the abstract ﬂavor space rather than in coordinate space. Nonstandard Neutrinos 261 In three-dimensional ﬂavor space one is free to choose any superpo- sition of states as a basis. It is convenient and common practice to use the mass eigenstates (vacuum propagation eigenstates) for each column of Fig. 7.1. Thus by deﬁnition the electron is the charged lepton with the smallest mass eigenvalue, the muon the second, and the tau the heaviest, and similarly for the quarks. All fermions interact by virtue of the weak force and thus couple to theW±andZ◦gauge bosons, the quarks and charged leptons in ad- dition couple to photons, while only the quarks interact by the strong force and thus couple to gluons. The W±(charged current) interac- tion has the important property of changing, for example, a charged lepton into a neutrino as in the reaction p+e−→n+νe(Fig. 7.3). If the initial charged lepton was an electron (the lightest charged lepton mass eigenstate), the outgoing neutrino state is deﬁned to be an “elec- tron neutrino” or νewhich in general will be a certain superposition of neutrino mass eigenstates. Fig. 7.3. Typical charged-current reaction. This phenomenon of Cabbibo mixing is well established among the quarks. For example, in the process of Fig. 7.3 the transition among the quarks is between uand cosθCd+ sinθCswhere cosθC= 0.975 refers to the Cabbibo angle. Kinematics permitting, the ﬁnal-state hadron will sometimes be udswhich constitutes the Λ particle with a mass of 1.116 GeV compared with 0 .934 GeV for the neutron ( udd). The superposition of quark states into which utransforms by a charged-current interaction is commonly denoted by d′, charm couples tos′, and top to b′while the unprimed states refer to the ﬁrst, second, and third mass eigenstates in the d-column of Fig. 7.1. Ignoring the third family one has (d′ s′) =(cosθCsinθC −sinθCcosθC)(d s) . (7.5) It is only by convention that the mixing is applied to the d-column of the quarks rather than the u-column or both. 262 Chapter 7 Including the third generation, the mixing is induced by the three- dimensional Cabbibo-Kobayashi-Maskawa (CKM) matrix V. After re- moving all unphysical phases by an appropriate redeﬁnition of the quark ﬁelds this unitary matrix is given in terms of four signiﬁcant parameters. The Particle Data Group (1994) recommends a standard parametriza- tion in terms of three two-family mixing angles θij<π/ 2 and one phase 0≤δ <2π. WithCij≡cosθijandSij≡sinθijthis standard form is (Fritzsch and Plankl 1987) V= 1 0 0 0C23S23 0−S23C23  C13 0S13e−iδ 0 1 0 −S13eiδ0C13  C12S120 −S12C120 0 0 1  = C12C13 S12C13 S13e−iδ −C23S12−C12S23S13eiδC12C23−S12S23S13eiδC13S23 S12S23−C12C23S13eiδ−C12S23−C23S12S13eiδC13C23  ≈ 1 S12S13e−iδ −S12 1S23 S12S23−S13eiδ−S23 1 . (7.6) Experimentally one has the 90% CL ranges (Particle Data Group 1994) 0.218<S 12<0.224, 0.032<S 23<0.048, 0.002<S 13<0.005. (7.7) The approximation in Eq. (7.6) is justiﬁed by the small mixing angles. They and the CP-violating phase δ= 3.3×10−3(Wolfenstein 1986) are measured parameters of the standard model which, like the fermion masses, are not theoretically accounted for at the present time. If neutrinos have masses one naturally expects that they follow a similar scheme and so the “weak interaction eigenstates” νℓ(ℓ=e,µ,τ ) are thought to be given as linear superpositions of the mass eigenstates νiby virtue of νℓ=3∑ i=1Uℓiνi, (7.8) where the unitary matrix Uplays the role of the CKM matrix. Unless otherwise stated ν1will always refer to the dominant mass admixture ofνeand so forth. It seems plausible that m1<m 2<m 3, a hierarchy that is often assumed. Nonstandard Neutrinos 263 Flavor mixing is the only possibility for members of one family (one row in Fig. 7.1) to transform into those of a diﬀerent family. This phe- nomenon is known as the absence of avor-changing neutral currents ; it means that the Z◦coupling to quarks and leptons, like the photon coupling, leaves a given superposition of fermions unaltered. For ex- ample, muons decay only by the ﬂavor-conserving mode µ−→νµe−νe (Fig. 7.4); the experimental upper limits on the branching ratios for µ−→e−γandµ−→e−e+e−are 5×10−11and 1.0×10−12, respectively. In the absence of neutrino masses and mixing the individual lepton ﬂa- vor numbers are conserved: a lepton can be transformed only into its partner of the same family, or it can be created or annihilated together with an antilepton of the same family. Fig. 7.4. Allowed and forbidded decays. When neutrinos have masses and mixings, ﬂavor-violating lepton decays become possible, but their rate would be so small that their experimentally observed absence does not yield interesting constraints on neutrino parameters. Because neutrino masses must be very small if they exist at all, the most signiﬁcant observable eﬀect is that of neutrino oscillations. 7.2.2 Standard-Model Decays of Mixed Neutrinos For massive neutrinos it is kinematically possible to decay according toν→ν′γ,ν→ν′γγ, orν→ν′ν′′ν′′. In the absence of mixing, of course, all of these modes are forbidden. Even in the presence of mixing, Fig. 7.5. Allowed and forbidden tree-level decays of mixed neutrinos. 264 Chapter 7 however, the three-neutrino channel remains forbidden by the absence of ﬂavor-changing neutral currents because the ﬂavor content of νand ν′in Fig. 7.5 must remain unaltered by the Z◦vertex. Therefore, in the standard model low-mass neutrinos can decay only by higher-order (radiative) amplitudes. “Heavy” neutrinos νhwithmh>2me≈1 MeV may decay at tree- level through the channel νh→νee+e−at a rate 1 τe+e=|Ueh|2G2 F 3 (4π)3m5 hΦ(mh) =|Ueh|23.5×10−5s−1m5 MeVΦ(mh), (7.9) whereUehis the mixing amplitude between νhandνe,GFis the Fermi constant, and mMeV≡mh/MeV. The phase-space factor is (Shrock 1981) Φ(mh) = (1 −4a)1/2(1−14a−2a2−12a3) + 24a2(1−a2) ln1 + (1 −4a)1/2 1−(1−4a)1/2(7.10) witha≡m2 e/m2 h; it is shown in Fig. 7.6. Fig. 7.6. Phase space factor for h→ee+e−according to Eq. (7.10). The one- and two-photon decay modes arise in the standard model with mixed neutrinos from the amplitudes shown in Fig. 7.7. Turn ﬁrst to the one-photon decay νi→νjγwith the neutrino masses mi>m j. Nonstandard Neutrinos 265 Fig. 7.7. Feynman graphs for neutrino radiative decays. There are other similar graphs with the photon lines attached to the intermediate Wboson. In general the matrix element can be thought of as arising from an eﬀective interaction Lagrangian of the form Lint=1 2ψiσαβ(µij+ϵijγ5)ψjFαβ+ h.c., (7.11) whereFαβis the electromagnetic ﬁeld tensor, ψiandψjare the neutrino ﬁelds, and µijandϵijare magnetic and electric transition moments which are usually expressed in units of Bohr magnetons µB=e/2me. The decay rate is 1 τγ=|µij|2+|ϵij|2 8π(m2 i−m2 j mi)3 = 5.308 s−1(µeﬀ µB)2 δ3 mm3 eV, (7.12) whereµ2 eﬀ≡ |µij|2+|ϵij|2,meV≡mi/eV, andδm≡(m2 i−m2 j)/m2 i. An explicit evaluation of the one-photon amplitude of Fig. 7.7 yields for Dirac neutrinos (Pal and Wolfenstein 1982) µD ij ϵD ij} =e√ 2GF (4π)2(mi±mj)∑ ℓ=e,µ,τUℓjU∗ ℓif(rℓ). (7.13) For Majorana neutrinos one has instead µM ij= 2µD ijandϵM ij= 0 or µM ij= 0 andϵM ij= 2ϵD ij, depending on the relative CP phase of νiandνj. In Eq. (7.13) rℓ≡(mℓ/mW)2where the charged-lepton masses are me= 0.511 MeV,mµ= 105.7 MeV, and mτ= 1.784 GeV while the W±gauge boson mass is mW= 80.2 GeV. Thus for all charged leptons rℓ≪1; in this limit f(rℓ)→ −3 2+3 4rℓ. (7.14) If one inserts the leading term −3 2into the sum in Eq. (7.13) one ﬁnds that its contribution vanishes because the unitarity of Uimplies that its rows or columns represent orthogonal vectors. Because the ﬁrst nonzero 266 Chapter 7 contribution is from3 4rℓ, the transition moments are suppressed by (mℓ/mW)2, an eﬀect known as GIM cancellation after Glashow, Il- iopoulos, and Maiani (1970). Explicitly, the transition moments are µD ij/µB ϵD ij/µB} =3GFme√ 2 (4π)2(mi±mj)(mτ mW)2∑ ℓ=e,µ,τUℓjU∗ ℓi(mℓ mτ)2 = 3.96×10−23mi±mj 1 eV∑ ℓ=e,µ,τUℓjU∗ ℓi(mℓ mτ)2 .(7.15) These small numbers imply that neutrino radiative decays are exceed- ingly slow in the standard model. Dirac neutrinos would have static or diagonal ( i=j) magnetic dipole moments while the electric dipole moments vanish according to Eq. (7.13). Their presence would require CP-violating interactions. Majorana neutrinos, of course, cannot have any diagonal electromag- netic moments. For µD iithe leading term of Eq. (7.14) in Eq. (7.13) does not vanish because the unitarity of Uimplies that the sum equals unity fori=j. Therefore, µD ii µB=6√ 2GFme (4π)2mi= 3.20×10−19meV, (7.16) much larger than the transition moments because it is not GIM sup- pressed. The two-photon decay rate νi→νjγγis of higher order and thus may be expected to be smaller by a factor of α/4π. However, it is not GIM suppressed so that it is of interest for a certain range of neu- trino masses (Nieves 1983; Ghosh 1984). Essentially, the result in- volves another factor α/4πrelative to the one-photon rate, and f(rℓ) in Eq. (7.13) is replaced by ( mi/mℓ)2. As an example consider the diﬀerent decay modes for ν3→ν1, assuming that m3≫m1and that the mixing angles are small so that ν3≈ντandν1≈νe. Then one has explicitly 1 τ≈ |Ue3|2G2 Fm5 3 3 (4π)3×  Φ(m3), ν3→ν1e+e−, 27 8α 4π(mτ mW)4 , ν 3→ν1γ, 1 180(α 4π)2(m3 me)4 , ν 3→ν1γγ,(7.17) where Φ(mh) was given in Eq. (7.10) and shown in Fig. 7.6. The γγ decay dominates in a small range of m3just below 2 me. Nonstandard Neutrinos 267 The quantity τ|Ue3|2is shown as a dotted line in Fig. 7.2. Even without nonstandard physics the decay rate is fast on cosmological scales ifm3∼>2me. Because experimentally m3may be as large as 24 MeV the cosmological mass bound of 30 eV does not automatically apply. Experimental limits on Ue3together with the BBN mass bound, however, exclude a heavy standard ν3. Even without reference to BBN it can be excluded on the basis of the SN 1987A neutrino radiative lifetime limits (Sect. 12.5.2). 7.3 Neutrino Electromagnetic Form Factors 7.3.1 Overview When Wolfgang Pauli in 1930 ﬁrst postulated the existence of neutri- nos he speculated that they might interact like a magnetic dipole of a certain moment µν. If that were the case they could be measured by their ionizing power when they move through a medium; this ionizing power was ﬁrst calculated by Bethe (1935). Nahmias (1935) measured the event rates in a Geiger-M¨ uller counter in the presence and ab- sence of a radioactive source and interpreted his null result as a limit µν∼<2×10−4µB(Bohr magneton µB=e/2me) on the neutrino dipole moment. He concluded that “since this limit is already smaller than a nuclear magneton, it seems probable that the neutrino has no moment at all.” Subsequent attempts to measure ever smaller neutrino dipole moments have consistently failed. The main diﬀerence between then and now is the advanced the- oretical understanding of neutrino interactions in the context of the standard model of electroweak gauge interactions. A magnetic dipole interaction couples l.h. with r.h. states so that the latter would not be strictly sterile. This would be in conﬂict with the standard model where neutrinos interact only by their l.h. coupling to WandZgauge bosons. Thus neutrino dipole moments must vanish identically because weak interactions violate parity maximally. This picture changes when neutrinos have masses because even the r.h. components of a Dirac neutrino are then not strictly sterile as they couple to the Higgs ﬁeld—or else they would not have a mass. Indeed, an explicit calculation in the standard model with neutrino masses gave a magnetic dipole moment µν= 3.20×10−19µB(mν/eV) (Eq. 7.16). If neutrinos mix, they also obtain transition magnetic and electric moments. However, they are even smaller because of the GIM suppression eﬀect—see Eq. (7.15). 268 Chapter 7 Much larger values would obtain with direct r.h. neutrino interac- tions. For example, in left-right symmetric models there exist heavier gauge bosons which mediate r.h. interactions; parity violation would occur because of the mass diﬀerence between the l.h. and r.h. gauge bosons. For a neutrino νℓ(ﬂavorℓ=e,µorτ) the dipole moment in such models is (Kim 1976; Marciano and Sanda 1977; B´ eg, Marciano, and Ruderman 1978) µν=eGF 2√ 2π2[ mℓ( 1−m2 W1 m2 W2) sin 2ζ+3 4mνℓ( 1 +m2 W1 m2 W2)] , (7.18) whereζis the left-right mixing angle between the gauge bosons WL andWR;W1,2are their mass eigenstates. Because of the smallness of the mass-induced standard dipole mo- ments any evidence for neutrino electromagnetic interactions would rep- resent evidence for interactions beyond the standard model. Therefore, the quest for neutrino electromagnetic interactions is more radical than that for masses and mixings. In a dense medium even standard massless neutrinos interact with photons by an eﬀective coupling which is mediated by the ambient elec- trons. This coupling can be expressed in terms of an eﬀective neutrino charge radius. Presently I focus on neutrino interactions in vacuum, leaving a discussion of their properties in media to Sect. 6.6. 7.3.2 Single-Photon Coupling There are many possible extensions of the standard model which would give sizeable neutrino dipole and transition moments by some novel r.h. interaction. For the purposes of this book the underlying new physics is of no concern; all we need is a generic representation of its observable eﬀects in terms of neutrino electromagnetic form factors. The most general interaction structure of a fermion ﬁeld ψwith the electromagnetic ﬁeld can be expressed as an eﬀective Lagrangian Lint=−F1ψγµψAµ−G1ψγµγ5ψ∂µFµν −1 2ψσµν(F2+G2γ5)ψFµν, (7.19) whereAµis the electromagnetic vector potential and Fµνthe ﬁeld strength tensor. The interpretation of the coupling constants is that of an electric charge for F1, an anapole moment for G1, a magnetic Nonstandard Neutrinos 269 dipole moment for F2, and an electric dipole moment for G2. In the matrix element derived from this Lagrangian these couplings should be viewed as form factors which are functions of Q2whereQis the energy-momentum transfer to the fermion, i.e. the energy momentum of the photon line attached to the fermion current. The interpretation of a charge etc. then pertains to the Q2→0 limit. It is usually assumed that neutrinos are electrically neutral, i.e. that F1(0) = 0 because electric charge quantization implies that elementary particles carry only charges in multiples of1 3ewhereeis the electron charge. In recent discussions of electric charge quantization43it was stressed, however, that the standard model of electroweak interactions without grand uniﬁcation requirements does allow neutrinos to carry small electric charges. Their possible magnitude is thus an experimental issue; existing limits are reviewed in Sect. 15.8. Because these limits are very restrictive, i.e. because neutrino electric charges must be very small, it appears likely that electric charge is quantized after all so that neutrino electric charges vanish identically. Even if neutrinos are electrically neutral, as shall be assumed hence- forth, they can virtually dissociate into charged particles and so they will have a form factor F1(Q2) which does not vanish for Q2̸= 0. One may visualize the neutral object as a superposition of two charge dis- tributions of opposite sign with diﬀerent spatial extensions. In terms of a power series expansion of F1(Q2) one usually deﬁnes the charge radius by virtue of ⟨r2⟩= 6∂F1(Q2) e∂Q2 Q2=0(7.20) where ⟨r2⟩may be both positive or negative. For neutrinos, the interpretation of the charge radius as an ob- servable quantity is a rather subtle issue as it is probed by “oﬀ-shell” photons (Q2̸= 0), i.e. by intermediate photons in processes such as scattering by photon exchange. Because the form factor is proportional toQ2such scattering processes do not exhibit a Coulomb divergence. The charge radius induces a short-range or contact interaction simi- lar to processes involving Z◦exchange. Therefore, the charge radius represents a correction to the standard tree-level electroweak scatter- ing amplitude between neutrinos and charged particles. This tree-level 43Babu and Mohapatra 1990; Babu and Volkas 1992; Takasugi and Tanaka 1992; Foot, Lew, and Volkas 1993; Foot 1994. References to earlier works are given in these papers. 270 Chapter 7 amplitude will receive radiative corrections from a variety of diagrams, including photon exchange, which must be considered simultaneously so that it is not at all obvious that one can extract a ﬁnite, gauge- invariant, observable quantity that can be physically interpreted as a charge radius.44 The charge radius, even if properly deﬁned, represents only a cor- rection to the tree-level weak interaction and as such it is best studied in precision accelerator experiments. In the astrophysical context, weak interaction rates involving standard l.h. neutrinos cannot be measured with the level of precision required to test for small deviations from the standard model. Indeed, a recent compilation (Salati 1994) reveals that experimental bounds on ⟨r2⟩are more sensitive than astrophysical limits, except perhaps for ντfor which experimental data are scarce and so its standard-model neutral-current interactions are not well tested. The matrix element for the anapole interaction in the Lorentz gauge is proportional to Q2. Therefore, it vanishes in the limit Q2→0, i.e. for real photons coupled to the neutrino current. The role of the anapole form factor G1is thus very similar to a charge radius: it represents a correction to the standard tree-level weak interaction and as such does not seem to be of astrophysical interest. The form factors F2andG2are of much greater importance be- cause they may obtain nonvanishing values even in the Q2→0 limit. Henceforth I shall refer to µ≡F2(0) as a magnetic dipole moment, toϵ≡iG2(0) as an electric dipole moment, respectively. This iden- tiﬁcation is understood if one derives the Dirac equation of motion i∂tψ=Hψfor a neutrino ﬁeld ψ(massm) in the presence of an exter- nal, weak, slowly varying electromagnetic ﬁeld Fµν. From Eq. (7.19) one ﬁnds for the Hamiltonian H=−i·∇+β[ m−(µ+iϵγ5)(i·E+·B)] , (7.21) where1 2σµνFµν=i·E+·Bwas used. In the Dirac representation one has =(0 0) , β =(I0 0−I) , =(0 0) ,(7.22) where is a vector of Pauli matrices while Iis the 2 ×2 unit matrix. For a neutrino at rest the Dirac spinor is characterized by its large 44For recent discussions of these matters see Lucio, Rosado, and Zepeda (1985), Auriemma, Srivastava, and Widom (1987), Degrassi, Sirlin, and Marciano (1989), Musolf and Holstein (1991), and G ongora-T. and Stuart (1992). Nonstandard Neutrinos 271 component, a Pauli two-spinor ϕ, which is then found to evolve as i∂tϕ= (µB0+ϵE0)·ϕ, (7.23) where the index 0 refers to quantities in the neutrino rest frame. A neutrino polarized in or opposite to the ﬁeld direction has the energy ±µB0or±ϵE0, respectively, so that µandϵare indeed magnetic and electric dipole moments, respectively. The electromagnetic form factors obey certain constraints for Dirac and Majorana neutrinos; detailed discussions were provided by a num- ber of authors.45For Dirac neutrinos, all form factors must be real relative to each other (no relative phases) if CP invariance holds. For the diagonal case (coupling to one neutrino species) all form factors must be real, and CP invariance implies that the electric dipole moment must vanish. For Majorana neutrinos, a magnetic transition moment (F2) must be imaginary, an electric transition moment ( G2) real. If CP invariance holds, in addition one of them must vanish, i.e. there is either a transition electric, or a transition magnetic moment, but not both. Majorana neutrinos cannot have diagonal electric nor magnetic moments, nor can they have a charge or charge radius; they may have an anapole form factor. 7.3.3 Two-Photon Coupling Discussions of neutrino electromagnetic form factors are usually re- stricted to the eﬀective neutrino coupling to an electromagnetic wave or static ﬁeld. However, a two-photon coupling of neutrinos is also pos- sible and of some interest. Historically, it was thought for some time that the process γγ→ννcould be of great importance for the emis- sion of neutrinos from stars until it was shown by Gell-Mann (1961) that the amplitude for this process vanishes identically if neutrinos have only l.h. local interactions with electrons. Several authors dis- cussed theγγ→ννprocess when neutrinos are massive, or when they have more general interaction structures (Halprin 1975; Fischbach et al. 1976, 1977; Natale, Pleitez, and Tacla 1987; Gregores et al. 1995). How- ever, there does not seem to be a plausible scenario where this process would be of serious astrophysical interest. In the standard model, there is an eﬀective two-photon coupling to neutrinos because the interaction is not local; rather, it is medi- ated by ﬁnite-mass gauge bosons. Early calculations of the eﬀective 45For example Nieves (1982), Kayser (1982), Shrock (1982), and Li and Wilczek (1982). 272 Chapter 7 coupling in gauge theories were performed by Levine (1966) and Cung and Yoshimura (1975); for more recent discussions in the framework of the standard model see Nieves, Pal, and Unger (1983), Nieves (1987), Dodelson and Feinberg (1991), Liu (1991), Langacker and Liu (1992), Kuznetsov and Mikheev (1993), and Dicus and Repko (1993). This two-photon coupling leads to a higher-order contribution to the neu- trino refractive index in a bath of photons (Sect. 6.7.2). However, this and other consequences do not seem to be important in any astrophys- ical or laboratory setting of practical interest. For neutrinos with masses and mixings there is a decay amplitude ν→ν′γγwhich can dominate for a small range of neutrino masses belowmν= 2meas discussed by Nieves (1983) and Ghosh (1984). Explicit results were given in Sect. 7.2.2 above. One of the photons may represent an external electric or magnetic ﬁeld, i.e. one may consider the neutrino decay ν→ν′γin the pres- ence of a strong external ﬁeld which would modify the propagators of the intermediate charged leptons. For very strong ﬁelds it appears that a substantial decay rate can obtain (Gvozdev, Mikheev, and Vas- silevskaya 1992a,b, 1993, 1994a,b). From this literature it does not seem to become entirely clear under which if any circumstances these results might be of practical interest for, say, the decay of supernova or neutron-star neutrinos. 7.4 Electromagnetic Processes The presence of electromagnetic dipole and transition moments implies that neutrinos couple directly to the electromagnetic ﬁeld, allowing for a variety of nonstandard processes (Fig. 7.8). Most obviously, neutrinos can scatter on electrons by photon exchange. The ν-escattering cross section (electrons at rest in the laboratory frame) was given, e.g. by Vogel and Engel (1989) dσ dT=G2 Fme 2π[ (CV+CA)2+ (CV−CA)2( 1−T Eν)2 + (C2 A−C2 V)meT E2 ν] +αµ2 ν[1 T−1 Eν] , (7.24) whereCVandCAare the weak coupling constants given in Appendix B, Tis the electron recoil energy with the limits 0 ≤T≤2E2 ν/(2Eν+me), andµνis the neutrino dipole moment. Nonstandard Neutrinos 273 Fig. 7.8. Important processes involving direct neutrino electromagnetic cou- plings, notably magnetic dipole moments and magnetic or electric transition moments. Because an anomalous neutrino charge radius causes a contact in- teraction it would manifest itself by the modiﬁcation CV→CV+ 1 3√ 2πα⟨r2⟩/GF, i.e. there would be a small correction to the overall cross section. A dipole moment, on the other side, modiﬁes the energy spectrum of the recoil electrons, notably at low energies, because of its forward-peaked nature from the Coulomb divergence. If neutrinos had electric dipole moments, or electric or magnetic transition moments, these quantities would also contribute to the scat- tering cross section. Therefore, the quantity measured in electron recoil experiments involving relativistic neutrinos is (e.g. Raﬀelt 1989) µ2 ν=∑ j=νe,νµ,ντ|µij−ϵij|2, (7.25) whereirefers to the initial-state neutrino ﬂavor. Therefore, in principle there is a possibility of destructive interference between the magnetic and electric transition moments of Dirac neutrinos. (Majorana neutri- nos have only magnetic or electric transition moments, but not both if CP is conserved.) 274 Chapter 7 A related process to scattering by photon exchange is the spin- precession in a macroscopic magnetic or electric ﬁeld into r.h. states of the same or another ﬂavor, i.e. electromagnetic oscillation into “wrong- helicity” states. Such eﬀects will be studied in Sect. 8.4 in the general context of neutrino oscillations. In principle, this process can be impor- tant at modifying the measurable l.h. solar neutrino ﬂux as discussed in Sect. 10.7 in the context of the solar neutrino problem. Spin oscilla- tions can also be important in supernovae where strong magnetic ﬁelds exist, although a detailed understanding remains elusive at the present time (Sect. 11.4). The most interesting process caused by dipole moments is the pho- ton decay into neutrino pairs, γ→νν, which is enabled in media where the photon dispersion relation is such that ω2−k2>0. It is the most interesting process because it occurs even in the absence of dipole mo- ments due to a medium-induced neutrino-photon coupling (Chapter 6). This is the dominant standard neutrino emission process from stars for a wide range of temperatures and densities. Details of both the stan- dard and the dipole-induced “plasma process” depend on complicated ﬁne points of the photon dispersion relation in media that were taken up in Chapter 6. The salient features, however, can be understood in an approxima- tion where photons in a medium (“transverse plasmons”) are treated as particles with an eﬀective mass equal to the plasma frequency ωPwhich in a nonrelativistic medium is ω2 P= 4παn e/mewith the electron den- sityneand electron mass me. The decay rate of these electromagnetic excitation in their own rest frame is then Γγ=µ2 νω3 P 24πwithµ2 ν=∑ i,j( |µij|2+|ϵij|2) , (7.26) while in the frame of the medium where the photon has the energy ωa Lorentz factor ωP/ωmust be included. The sum includes all ﬁnal-state neutrino ﬂavors with mi≪ωP; otherwise phase-space modiﬁcations occur, and even a complete suppression of the decay by a neutrino mass threshold. In contrast with Eq. (7.25) relevant for the scattering rate, no destructive interference eﬀects between magnetic and electric dipole amplitudes occur. Transition moments would allow for the radiative decay νi→νjγ. Again, because a neutrino charge radius or anapole moment vanish in theQ2→0 limit relevant for free photons, radiative neutrino decays are most generally characterized by their magnetic and electric transition Nonstandard Neutrinos 275 moments. The decay rate is found to be Γνi=µ2 νm3 i 8πwithµ2 ν=∑ j( |µij|2+|ϵij|2) . (7.27) The sum is extended over all νjwithmj≪mi, otherwise phase-space corrections as given in Eq. (7.12) must be included. There is no destruc- tive interference between electric and magnetic amplitudes. Because of the long decay path available in the astronomical environment, the search for decay photons from known astrophysical neutrino sources is by far the most eﬃcient method to set limits on radiative decays. This method and its results are studied in detail in Chapter 12. Neutrino radiative decays, the plasma process, and scatterings by photon exchange all depend on the same electromagnetic form factors. Therefore, a limit on the radiative decay time can be expressed as a limit on neutrino transition moments, and a limit on a transition moment from, say, a scattering experiment can be translated into a limit on a radiative decay time. In much of the literature this simple connection has not been made; the search for neutrino decays and that for dipole moments were strangely dealt with as separate and unrelated eﬀorts. There are a number of interesting neutrino electromagnetic pro- cesses which I will not discuss in any detail because they either yield very small rates or have not so far led to any new results. Among them are the bremsstrahlung emission of photons in neutrino electron colli- sionsνe→νeγ(Mour˜ ao, Bento, and Kerimov 1990; Bernab´ eu et al. 1994). Another possibility is nuclear excitation due to neutrino dipole moments (Dodd, Papageorgiu, and Ranfone 1991; Sehgal and Weber 1992) or the disintegration of deuterons (Akhmedov and Berezin 1992). Also, Cherenkov radiation (Grimus and Neufeld 1993) or transition ra- diation (Sakuda 1994) of neutrinos with magnetic moments have been studied in the literature. 7.5 Limits on Neutrino Dipole Moments 7.5.1 Scattering Experiments In principle, there are a number of possibilities to search for direct neutrino electromagnetic couplings in the laboratory. In practice, the only method that has so far yielded signiﬁcant limits is a study of the neutrino-electron scattering cross section that would receive a forward- peaked contribution from a dipole moment. 276 Chapter 7 The best limits on µνeare based on the use of reactor neutrinos as a source. The measurement of the νe-escattering cross section by Reines, Gurr, and Sobel (1976) was interpreted by Kyuldjiev (1984) to yield a bound of µνe<1.5×10−10µB. Since then, the reactor νe spectrum has been much better understood. Vogel and Engel (1989) stressed that a literal interpretation of the old results by Reines, Gurr, and Sobel (1976) would actually yield evidence for a dipole moment of about 2 −4×10−10µB. There is, however, a more recent limit of µνe<2.4×10−10µB (7.28) from the Kurchatov Institute (Vidyakin et al. 1992). When interpreting this bound as a limit on the νe-ντtransition mo- ment, recall that for an experimentally allowed ντmass in the 10 MeV regime this limit would not apply as reactor neutrinos have relatively low energies (below about 10 MeV). Therefore, it remains useful to con- sider the bounds from the neutrino beam at LAMPF with a νeendpoint energy of 52 .8 MeV µνe<10.8×10−10µB, µνµ<7.4×10−10µB (7.29) (Krakauer et al. 1990). A similar limit of µνµ<8.5×10−10µBwas obtained by Ahrens et al. (1990). An improvement of the bound on µνeby an order of magnitude or more can be expected from a new reactor experiment currently in preparation (“MUNU experiment,” Broggini et al. 1990). However, results will not become available before a few years from now. The transition moments between ντand other sequential neutrinos are bounded by the above experiments involving initial-state νe’s or νµ’s or their antiparticles. The diagonal ντmagnetic dipole moment, however, is much less constrained because no strong ντsources are avail- able in the laboratory; the ντ-ecross section has never been measured. However, the calculated ντﬂux produced in a proton beam dump from the decay of Dsmesons can be used to derive an upper limit on the ντ-ecross section which can be translated into a bound µντ<5.4×10−7µB, (7.30) (Cooper-Sarkar et al. 1992). Of course, in this range the ντelectromag- netic cross section would far exceed a typical weak interaction one. Nonstandard Neutrinos 277 7.5.2 Spin-Flip Scattering in Supernovae Neutrino scattering by photon exchange can also be important in astro- physical settings, notably if neutrinos are Dirac particles. The magnetic or electric dipole coupling is such that it ﬂips the helicity of relativistic neutrinos, i.e. the ﬁnal state is r.h. for an initial l.h. neutrino. This spin ﬂip is of no importance in experiments where the electron recoil is measured, but it can have dramatic consequences in supernovae where l.h. neutrinos are trapped by the standard weak interactions. The spin- ﬂip scattering by a electromagnetic dipole interaction would produce “wrong-helicity” states that could freely escape unless they scattered again electromagnetically. The SN 1987A neutrino signal indicates that this anomalous cooling channel cannot have been overly eﬀective, yield- ing a constraint of around µν∼<3×10−12µBon all Dirac diagonal or transition moments in the sense of Eq. (7.25); see Sect. 13.8.3 for a more detailed discussion. One should keep in mind, however, that for dipole moments in this range the spin precession in the strong macroscopic magnetic ﬁelds that are believed to exist in and near SN cores could also cause signiﬁcant left-right transitions. Notably, the back conversion of r.h. neutrinos could cause a transfer of energy between widely separated regions of the SN core, and might even help at the explosion (Sect. 13.8.3). The role of relatively large neutrino dipole moments in SN physics has not been elaborated in enough depth to arrive at reliable regions of parameters that are ruled out or ruled in by SN physics and the SN 1987A neutrino signal. 7.5.3 Spin-Flip Scattering in the Early Universe Neutrino spin-ﬂip scattering has important consequences in the early universe as it can bring some or all of the “wrong-helicity” Dirac neu- trino degrees of freedom into thermal equilibrium. The usual big bang nucleosynthesis (BBN) argument previously mentioned in Sect. 7.1.5 allows one to exclude this possibility because even one additional ther- mally excited neutrino degree of freedom appears to be forbidden by the spectacular agreement between the predicted and observed primordial light-element abundances. This argument was ﬁrst advanced by Morgan (1981a,b). Unfortu- nately, he used an unrealistically small cutoﬀ for the Coulomb diver- gence of the spin-ﬂip scattering cross section, leading to an overestimate of the eﬃciency by which r.h. Dirac neutrinos can be brought into ther- 278 Chapter 7 mal equilibrium. A reasonable cutoﬀ by the Debye screening scale was used by Fukugita and Yazaki (1987) who found µν∼<0.5×10−10µB, (7.31) about a factor of 3.5 looser than Morgan’s original constraint. This limit can be avoided for the diagonal ντmagnetic moment which would contribute to the ντντ→e−e+annihilation process. If theντalso had a mass in the 10 MeV regime the spin-ﬂip excitation of the r.h. degrees of freedom would be compensated by the annihilation depletion of the ντandντpopulation before nucleosynthesis (Giudice 1990). A detailed analysis (Kawano et al. 1992) reveals that a µντlarger than about 0 .7×10−8µBis allowed in the mass range between a few and about 30 MeV. Like in a SN core, there could exist large magnetic ﬁelds in the early universe that would allow for left-right transitions by the mag- netic dipole induced spin precession (Sect. 8.4); for early discussions of this possibility see Lynn (1981) and Shapiro and Wasserman (1981). Arguments of this sort naturally depend on assumptions concerning the primordial magnetic ﬁeld distribution; such ﬁelds may be required as seeds for the dynamo mechanism to create present-day galactic mag- netic ﬁelds. A quantitative kinetic understanding of the process of populating the r.h. neutrinos requires a simultaneous treatment of the neutrino spin-precession and scattering much along the lines of Chap- ter 9 where ﬂavor oscillations are studied in an environment where neu- trinos scatter frequently. The most recent investigation of primordial neutrino magnetic oscillations is Enqvist, Rez, and Semikoz (1995); see their work for references to the previous literature. In certain plausi- ble scenarios of primordial magnetic ﬁeld distributions neutrino Dirac dipole moments as small as 10−20µBseem to be in conﬂict with BBN. 7.5.4 Search for Radiative Neutrino Decays The search for radiative decays of reactor, beam, solar, supernova, and cosmic neutrinos will be discussed at length in Chapter 12. For electron neutrinos, the eﬀective transition moment in the sense of Eq. (7.27) will be found to be limited by µν∼<  0.9×10−1µB(eV/mν)2Reactors, 0.5×10−5µB(eV/mν)2Sun, 1.5×10−8µB(eV/mν)2SN 1987A, 3×10−10µB(eV/mν)2.3Cosmic background,(7.32) Nonstandard Neutrinos 279 wheremνis the mass of the decaying parent neutrino. Of these limits, all except the cosmic one are based on measured neutrino ﬂuxes. For nonelectron neutrinos, the cosmic and SN 1987A limits apply equally because these sources emit neutrinos of all ﬂavors. The SN 1987A constraints apply in this form only for mν∼<40 eV, and for total lifetimes which in the laboratory exceed the transit time between the SN and Earth. The cosmological limit assumes that the ra- diative channel dominates and as such it requires mν∼<30 eV; otherwise the universe would be “overclosed” by neutrinos. Because mνe∼<5 eV these conditions are plausibly satisﬁed for νe. Forνµandντone may contemplate masses in excess of 30 eV if one simultaneously contem- plates novel interactions which allow for invisible fast decays. There- fore, radiative decay limits for νµandντmust be derived as a func- tion of the assumed mass and of the assumed total decay time. For SN 1987A this exercise will be performed in Sect. 12.4.5, the results are displayed in Fig. 12.17. For cosmic neutrinos, I do not know of a published comparable contour plot. 7.5.5 Plasmon Decay in Stars The last and most interesting constraint arises from the energy-loss argument applied to globular cluster stars. The neutrino emissivity by the plasma process γ→ννwould be too large unless µν∼<3×10−12µB. (7.33) This bound, which applies for mν∼<5 keV, has been derived in detail in Sect. 6.5.6. Chapter 8 Neutrino Oscillations The phenomenon of neutrino oscillations in vacuum and in media as well as in magnetic ﬁelds is studied. Experimental constraints on neu- trino mixing parameters are reviewed. 8.1 Introduction If neutrinos do not have novel interactions that allow them to decay fast then they must obey the cosmological mass limit of mν∼<30 eV. This is even true for ντalthough it could decay suﬃciently fast into thee+e−νechannel if it had a mass in the 10 MeV range. However, the absence of γrays from SN 1987A in conjunction with the neutrino signal (Sect. 12.5.2) and independently arguments of big-bang nucleosynthesis (Fig. 7.2) exclude this option. If neutrino masses are indeed so small then there is no hope for a direct experimental measurement at the present time, with the possible exception of mνewhich could still show up in tritium βdecay or neutrinoless ββdecay experiments as discussed in Chapter 7. Pontecorvo (1967) was the ﬁrst to realize that the existence of sev- eral neutrino ﬂavors (two were known at the time) allows even very small masses to become visible.46A “weak-interaction eigenstate” which is produced, say, in the neutron decay n→pe−νeis in gen- eral expected to be a mixture of neutrino mass eigenstates. The phe- nomenon of particle mixing (Sect. 7.2.1) is familiar from the quarks 46Pontecorvo’s (1957, 1958) original discussion referred to ν↔νoscillations in analogy to the experimentally observed case of K◦↔K◦. For a historical overview see Pontecorvo (1983). In this book I will not discuss ν↔νoscillations any further—see Akhmedov, Petcov, and Smirnov (1993) for a recent reexamination of “Pontecorvo’s original oscillations.” 280 Neutrino Oscillations 281 and hence does not appear to be an exotic assumption. Whatever the physical cause of particle masses, it seems unrelated to their gauge in- teractions! Expanding the neutrino state in plane waves, each mass eigenstate propagates as e−i(ωt−ki·x)where k2 i=ω2−m2 i. Therefore, the diﬀerent mass components develop phase diﬀerences, causing the original superposition which formed a νeto turn partially into other ﬂavors. Therefore, one can search for the disappearance of neutrinos of a given ﬂavor from a beam, or one can search for the appearance of “wrong-ﬂavored” states in a beam. The measured deﬁcit of solar νe’s (Chapter 10) has long been attributed to the oscillation phenomenon even though a deﬁnitive proof is still missing. Neutrino oscillations eﬀectively measure a phase diﬀerence between two components of a beam, much as the rotation of the plane of polar- ization of linearly polarized light represents a phase diﬀerence between the circularly polarized components of a beam in an optically active medium. This method is sensitive to small diﬀerences in the refractive index of the two components. For example, the Faraday rotation ef- fect can be used to measure very weak interstellar magnetic ﬁelds even though the interstellar medium is quite dilute. Both for neutrinos and photons, ﬁne points of the dispersion relation have a signiﬁcant impact on the oscillation eﬀect. Wolfenstein (1978) was the ﬁrst to recognize that the medium- induced modiﬁcation of the neutrino dispersion relations (Sect. 6.7.1) is not an academic aﬀair, but rather of immediate relevance for some neutrino oscillation experiments. In Mikheyev and Smirnov’s (1985) seminal paper it was shown that oscillations can be “resonant” when a beam passes through such a density gradient that the ﬂavor branches of the dispersion relation cross. This Mikheyev-Smirnov-Wolfenstein (MSW) eﬀect is very important in astrophysics because neutrinos are naturally produced in the interior of stars and stream through a density gradient into empty space. An adiabatic crossing of the dispersion relations has the eﬀect of interchanging the ﬂavor content of the neutrino ﬂux even if the mixing angle is very small. This eﬀect is one version of the oscillation solution of the solar neutrino problem. For suitable parameters it is also signiﬁ- cant in supernovae where diﬀerent-ﬂavored neutrinos are thought to be produced with diﬀerent energy spectra. The MSW eﬀect could swap the spectral characteristics of the neutrinos emerging from a newborn neutron star, allowing for a number of fascinating novel eﬀects. If neutrinos had large magnetic dipole moments they could spin- precess in magnetic ﬁelds. This eﬀect is completely analogous to ﬂavor 282 Chapter 8 oscillations, except that here the two helicity components rather than the ﬂavor components get transformed into each other. Again, this is a standard eﬀect familiar from the behavior of electrons in magnetic ﬁelds. In astrophysical bodies large magnetic ﬁelds exist, especially in supernovae, so that magnetic helicity oscillations are potentially inter- esting. However, much larger magnetic dipole moments are required than are predicted for standard massive neutrinos. Thus, ﬂavor oscil- lations have rightly received far more attention. Presently I will develop the theoretical tools for neutrino oscilla- tions, and summarize the current experimental situation. In Chapter 9 I will discuss the more complicated phenomena that obtain when os- cillating neutrinos are trapped in a supernova core. The story of solar neutrinos (Chapter 10) is inextricably intertwined with that of neu- trino oscillations, especially of the MSW variety. Finally, oscillations may also play a prominent role for supernova neutrinos and the inter- pretation of the SN 1987A signal (Chapter 11). 8.2 Vacuum Oscillations 8.2.1 Equation of Motion for Mixed Neutrinos In order to derive a formal equation for the oscillation of mixed neu- trinos I begin with the equation of motion of a Dirac spinor νiwhich describes the neutrino mass eigenstate i. It obeys the Dirac and thus the Klein-Gordon equation ( ∂2 t− ∇2+m2 i)νi= 0. One may readily combine all mass eigenstates in a single equation (∂2 t− ∇2+M2) Ψ = 0 , (8.1) where M2≡ m2 10 0 0m2 20 0 0 m2 3 and Ψ ≡ ν1 ν2 ν3 . (8.2) Eq. (8.1) may be written in any desired ﬂavor basis, notably in the basis of weak-interaction eigenstates to which one may transform by virtue of Eq. (7.8),  νe νµ ντ =U ν1 ν2 ν3 . (8.3) The mass matrix transforms according to M2→UM2U†and is no longer diagonal. (Note that U−1=U†because it is a unitary matrix.) Neutrino Oscillations 283 As usual one expands the neutrino ﬁelds in plane waves of the form Ψ(t,x) = Ψ k(t)eik·xfor which Eq. (8.1) is (∂2 t+k2+M2) Ψk(t) = 0 . (8.4) In general one cannot assume a temporal variation e−iωtbecause there are three diﬀerent branches of the dispersion relation with ω2 i=k2+m2 i. A mixed neutrino cannot simultaneously have a ﬁxed momentum and a ﬁxed energy! In practice one has always to do with very relativistic neutrinos for which k=|k| ≫mi. In this limit one may linearize Eq. (8.4) by virtue of∂2 t+k2= (i∂t+k)(−i∂t+k). For each mass eigenstate i∂t→ωi≈k and one needs to keep the exact expression only in the second factor where the diﬀerence between energy and momentum appears. Thus ∂2 t+k2≈2k(−i∂t+k), leading to the Schr¨ odinger-type equation i∂tΨk= Ω kΨkwhere Ω k≡( k+M2 2k) . (8.5) The vector Ψ originally consisted of neutrino Dirac spinors but it was reinterpreted as a vector of (positive-energy) probability amplitudes. For negative-energy states (antineutrinos) a global minus sign appears in Eq. (8.5). The Schr¨ odinger equation (8.5) describes a spatially homogeneous system with a nonstationary temporal evolution. In practice one usu- ally deals with the opposite situation, namely a stationary neutrino ﬂux such as that from a reactor or the Sun with a nontrivial spatial variation. Then it is useful to expand Ψ( t,x) in components of ﬁxed frequency Ψ ω(x)e−iωt, yielding (−ω2− ∇2+M2) Ψω(x) = 0 . (8.6) In the relativistic limit and restricting the spatial variation to the z-direction one obtains in full analogy to the previous case i∂zΨω=−KωΨωwhere Kω≡( ω−M2 2ω) . (8.7) This equation describes the spatial variation of a neutrino beam prop- agating in the positive z-direction with a ﬁxed energy ω. Ultimately one is not interested in amplitudes but in the observable probabilities |Ψℓ|2= Ψ∗ ℓΨℓwith ℓ=e,µ, orτ. One may derive an 284 Chapter 8 equation of motion for these quantities which is most compact in terms of a density matrix ρab= Ψ∗ bΨa. (8.8) Then i∂tρk= [Ω k, ρk] ori∂zρω=−[Kω, ρω] where [ A, B] =AB−BA is a commutator of matrices. Therefore, i∂tρ= (2k)−1[M2, ρ] or i∂zρ= (2ω)−1[M2, ρ], (8.9) where the indices ωorkhave been dropped. A beam evolves from the z= 0 state or density matrix as Ψω(z) =eiKzΨω(0) or ρω(z) =e−iKzρω(0)eiKz. (8.10) An analogous result applies to the case of temporal rather than spatial oscillations. In the weak-interaction basis eiKzwill be denoted by W, W(z)≡(eiKz)weak=U(eiKz)massU†. (8.11) If the neutrino is known to be a νeat the source ( z= 0) its probability for being measured as a νeat a distance z(“survival probability”) is |Wee(z)|2. One may be worried that the simple-minded derivation and interpre- tation of these results is problematic because the neutrino wave function is never directly observed. What is observed are the charged leptons absorbed or emitted in conjunction with the neutrino production and detection. However, if one performs a fully quantum-mechanical calcu- lation of the probability (or cross section) for the compound process of neutrino production, propagation, and absorption, the naive oscillation probability described by the elements of the Wmatrix factors out for all situations of practical interest, and notably in the relativistic limit (Giunti et al. 1993; Rich 1993). 8.2.2 Two-Flavor Oscillations The neutrino mixing matrix Ucan be parametrized exactly as the Cabbibo-Kobayashi-Maskawa (CKM) matrix in the quark sector in Eq. (7.6). If one of the three two-family mixing angles is much larger than the others (as for the quarks) one may study oscillations between the dominantly coupled families as a two-ﬂavor mixing problem. More- over, because so far all experiments—with the possible exception of solar and certain atmospheric neutrino observations—yield only upper limits on oscillation parameters one usually restricts the analysis to a Neutrino Oscillations 285 two-ﬂavor scenario. Of course, if one were to observe the appearance of a certain ﬂavor—rather than the disappearance of νeas for solar neutrinos or of νµin the atmospheric case—one would have to consider the possibility that they arise from sequential transitions of the sort νe→νµ→ντ. Note that in the bottom-left entry of the approximate expression for the CKM matrix Eq. (7.6) the term S12S23had to be kept because it is larger than the direct term S13. The neutrino mixing angles could show a similar hierarchy. With these caveats in mind we turn to the two-ﬂavor mixing case where Uhas the 2 ×2 Cabbibo form Eq. (7.5), U= cos θ I+isinθ σ2, (8.12) with the mixing angle θ, the 2×2 unit matrix I, and the Pauli matrix47 σ2. The mass matrix may be written in the form M2/2ω=b0−1 2B·, (8.13) where b0= (m2 1+m2 2)/4ω. In the weak-interaction basis B=2π ℓosc sin 2θ 0 cos 2θ , (8.14) a vector which is tilted with regard to the 3-axis by twice the mixing angle (Fig. 8.2). Further, ℓosc≡4π ω m2 2−m2 1(8.15) is the oscillation length . Its meaning will presently become clear. In this representation it is straightforward to work out the spatial behavior of a stationary neutrino beam. From K=ω−b0+1 2B· one ﬁnds W=ei(ω−b0)z[ cos(πz ℓosc) −isin(πz ℓosc) (−cos 2θsin 2θ sin 2θcos 2θ)] . (8.16) Assuming that the oscillations are among the ﬁrst two families, the appearance probability for a νµand the νesurvival probability are for an initial νe prob ( νe→νµ) =|Weµ|2= sin2(2θ) sin2(πz/ℓ osc), prob ( νe→νe) =|Wee|2= 1−prob ( νe→νµ). (8.17) The oscillation behavior is shown in Fig. 8.1. 47The Pauli matrices are σ1=( 0 1 1 0) ,σ2=( 0−i i0) , and σ3=( 1 0 0−1) . 286 Chapter 8 Fig. 8.1. Oscillation pattern for two-ﬂavor oscillations (neutrino energy ω). The ﬂavor oscillations described by Eq. (8.16) are fully analogous to the rotation of the plane of polarization in an optically active medium or to the spin precession in a magnetic ﬁeld. This analogy is brought out more directly if one starts with the equation of motion for the density matrix Eq. (8.9). Suppressing the index ωthe matrices can be expressed as ρ=1 2(1 +P·) and K=ω−b0+1 2B·, (8.18) and a similar representation for Ω where Bis expressed as a function ofkby virtue of ω→kto lowest order for relativistic neutrinos. The vector Pis a avor polarization vector . In the weak-interaction basis|Ψe|2=1 2(1 +P3) and|Ψµ|2=1 2(1−P3) give the probability for the neutrino to be measured as νeorνµ, respectively. P1andP2contain phase information and thus reveal the degree of coherence between the ﬂavor states. For a pure state |P|= 1 while in general |P|<1. For P= 0 one has a completely incoherent equal mixture of both ﬂavors. In optics, Pdescribes the degree of polarization of a light beam in the Poincar´ e sphere representation of the Stokes parameters (Poincar´ e 1892; Born and Wolf 1959). The equation of motion for the polarization vector in any ﬂavor basis is found to be (Stodolsky 1987; Kim, Kim, and Sze 1988) ∂zP=B×Por∂tP=B×P. (8.19) Here, Bplays the role of a “magnetic ﬁeld” and Pthat of a “spin vector.” The precession of Pfor an initial νewhere P(0) = (0 ,0,1) is shown in Fig. 8.2. Neutrino Oscillations 287 Fig. 8.2. Flavor oscillation as a “spin precession.” (After Stodolsky 1987.) 8.2.3 Distribution of Sources and Energies If the neutrino source region is not point-like relative to the oscilla- tion length, one has to average the appearance or survival probabilities accordingly. If the source locations z0are distributed according to a normalized function f(z0) the νµappearance probability is prob ( νe→νµ) = sin22θ∫ dz0f(z0) sin2π(z−z0) ℓosc. (8.20) For example, consider a Gaussian distribution f(z0) =e−z2 0/2s2/s√ 2π of size sfor which prob ( νe→νµ) =1 2sin22θ[ 1−e−2π2(s/ℓosc)2cos(2 πz/ℓ osc)] .(8.21) Fors=1 5ℓoscthis result is shown in Fig. 8.3. For s= 0 Eq. (8.21) is identical with Eq. (8.17) while for s≫ℓoscit is1 2sin22θwhich reﬂects that the beam is an incoherent mixture: the relative phases between diﬀerent ﬂavor components have been averaged to zero. No source is exactly monochromatic; usually the neutrino energies are broadly distributed. With a point source and a normalized distri- bution g(ω) one ﬁnds prob ( νe→νµ) = sin22θ∫ dω g(ω) sin2(m2 2−m2 1)z 4ω. (8.22) As an example let g(ω) such that ∆ = 2 π/ℓ oscfollows a Gaussian distribution e−(∆−∆0)2/2δ2/δ√ 2πof width δand with ∆ 0= 2π/ℓ 0. Then prob ( νe→νµ) =1 2sin22θ[ 1−e−δ2z2/2cos(2 πz/ℓ 0)] (8.23) which is shown in Fig. 8.4 for δ=1 10∆0. For δ= 0 Eq. (8.23) reproduces Eq. (8.17) while for z≫δ−1it approaches1 2sin22θ. 288 Chapter 8 Fig. 8.3. Oscillation pattern for a Gaussian source distribution with s= 1 5ℓoscaccording to Eq. (8.21). Fig. 8.4. Oscillation pattern for a mixture of neutrino energies with δ= 1 10π/ℓ0according to Eq. (8.23). These phenomena are well described in the picture of a precessing polarization vector which represents a density matrix and thus is de- signed to deal with incoherent or partially coherent beams. Notably, for a distribution of energies the polarization vector is P=∫dω g(ω)Pω. Because the components Pωprecess with diﬀerent frequencies about a common “magnetic ﬁeld” direction the component of Ptransverse to B disappears as the Pωapproach a uniform distribution on the precession cone in Fig. 8.2. The projection of PonB, however, is conserved so thatP→(P·ˆB)ˆBfort→ ∞ . If originally P= (0,0,1) for initial νe’s the geometry of Fig. 8.2 indicates that P3→cos22θfort→ ∞ and so prob( νe→νµ) =1 2sin22θ. Neutrino Oscillations 289 8.2.4 Experimental Oscillation Searches Because neutrino masses must be very small the oscillation length in- volves macroscopic scales. Numerically, it is ℓosc= 2.48 mEν 1 MeV1 eV2 ∆m2. (8.24) Therefore, the modulation of the ﬂavor content of a neutrino beam can occur on large, even astronomical length scales. There exists a large number of oscillation searches using terrestrial neutrino sources (reactors, accelerators); for detailed references see Par- ticle Data Group (1994). In Fig. 8.5 (curves a–f) I show the most restrictive limits on oscillations between the known neutrinos where Fig. 8.5. Experimental limits on neutrino masses and mixing angles. Re- actors, νedisappearance: (a) Bugey 4 (Achkar et al. 1995), superseding the G¨ osgen limits (Zacek et al. 1986); (b) Kurchatov Institute (Vidyakin et al. 1987, 1990, 1991). Accelerator experiments: (c) BNL Experiment 776, wide- band beam, νeandνeappearance (Borodovsky et al. 1992). (d) BNL Ex- periment 734, measurement of νe/νµratio (Ahrens et al. 1985). (e) Fermilab Experiment 531, ντappearance (Ushida et al. 1986); similar constraints were reported by the CHARM II Collaboration (1993). (f) CDHS Experiment, νµdisappearance (Dydak et al. 1984). (g) Anticipated range of sensitivity for the CHORUS and NOMAD experiments which are currently taking data at CERN (DiLella 1993; Winter 1995). 290 Chapter 8 the analysis was always based on the assumption that two-ﬂavor os- cillations dominate. The disappearance experiments, of course, also constrain oscillations into hypothetical sterile neutrinos. Even though the experimental results look very impressive, a glance on the CKM matrix Eq. (7.6) reveals that one could not yet have ex- pected to see oscillations in the νe↔ντorνµ↔ντchannel if the neutrino mixing angles are comparably small. It is very encouraging that the NOMAD and CHORUS experiments which are currently tak- ing data at CERN (DiLella 1993; Winter 1995) anticipate a range of sensitivity (curve gin Fig. 8.5) which is promising both in view of the possible cosmological role of a mνin the 10 eV range and the small mixing angles probed. Other future but less advanced projects for ter- restrial oscillation searches were reviewed by Schneps (1993, 1995). At the time of this writing the LSND Collaboration has reported a signature that is consistent with the occurrence of νµ→νeoscillations (Athanassopoulos et al. 1995). If this interpretation is correct, the corresponding ∆ m2would exceed about 1 eV2, while sin22θwould be a few 10−3. The status of this claim is controversial at the present time— see, e.g. Hill (1995). No doubt more data need to be taken before one can seriously begin to believe that neutrino oscillations have indeed been observed. 8.2.5 Atmospheric Neutrinos Besides reactors and accelerators, one may also use atmospheric neu- trinos as a source to search for oscillations. Primary cosmic ray pro- tons produce hadronic showers when interacting with atmospheric nu- clei ( A). Neutrinos are subsequently produced according to the simple scheme p+A→n+π/K +. . . π/K→µ+(µ−) +νµ(νµ) µ+(µ−)→e+(e−) +νe(νe) +νµ(νµ). (8.25) Therefore, one expects twice as many νµ’s as νe’s, and equally many neutrinos as antineutrinos of both ﬂavors. At a detector, the neu- trino ﬂux is approximately isotropic except at energies below about 1 GeV where geomagnetic eﬀects become important. Because the neu- trinos come from anywhere in the atmosphere, from directly overhead or from as far as the antipodes, oscillation lengths between about 10 Neutrino Oscillations 291 Fig. 8.6. Limits on neutrino masses and mixing angles from atmospheric neutrinos. (a) The shaded area is the range of masses and mixing angles required to explain the νe/νµanomaly at Kamiokande (Fukuda et al. 1994); the star marks the best-ﬁt value for the mixing parameters. The hatched areas are excluded by: (b) νe/νµratio at Fr´ ejus (Fr´ ejus Collaboration 1990, 1995; Daum 1994). (c) Absolute rate and (d) stopping fraction of upward go- ing muons at IMB (Becker-Szendy et al. 1992). Also shown are the excluded areas from the experimental limits of Fig. 8.5. and 13000 km are available.48The energy spectrum and absolute nor- malization of the ﬂux must be determined by calculations and thus is probably uncertain to within about ±30% while the νe/νµﬂavor ratio is likely known to within, say, ±5%. Several underground proton decay experiments have reported mea- surements of atmospheric neutrinos. The Fr´ ejus detector (an iron calorimeter) saw the expected νe/νµﬂavor ratio and thereby excluded the range of masses and mixing angles marked bin Fig. 8.6 for νe-νµ andνµ-ντoscillations (Fr´ ejus Collaboration 1990, 1995; Daum 1994). Instead of measuring the neutrinos directly one may also study the ﬂux of secondary muons produced by interactions in the rock surround- 48The eﬀect of matter must be included for νe-νatmospheric neutrino oscilla- tions. For a recent detailed analysis see Akhmedov, Lipari, and Lusignoli (1993). 292 Chapter 8 ing the detector. (Electrons from νeinteractions range out much faster in the rock and so one expects mostly muons from νµ’s.) This method is sensitive to the high-energy spectral regime of the atmospheric νµﬂux. Moreover, one may select upward going muons which are produced from νµ’s which traversed the entire Earth and thus have a large oscillation length available. The IMB detector excludes range cby this method (Becker-Szendy et al. 1992). Also, one may determine the fraction of muons stopped within the detector to those which exit, allowing one to constrain a spectral deformation caused by the energy dependence of the oscillation length. Range dis excluded by this method according to the IMB detector (Becker-Szendy et al. 1992). However, several detectors see a substantial deﬁcit of atmospheric νµ’s relative to νe’s, a ﬁnding usually expressed in terms of a “ratio of ratios,” i.e. the measured over the expected ratio of e-like over µ-like events (Fig. 8.7). While this procedure is justiﬁed because it is largely free of the uncertain absolute ﬂux normalization, one must be careful at interpreting the signiﬁcance of the ﬂux deﬁcit. The error of a mea- sured ratio does not follow a Gaussian distribution; a representation like Fig. 8.7 tends to overemphasize the signiﬁcance of the discrepancy (Fogli and Lisi 1995). Fig. 8.7. Measured ratio of the atmospheric νµ/νeﬂuxes relative to the expected value (“ratio of ratios”) in ﬁve detectors. Where two results are shown they refer to diﬀerent signatures or data samples. (See Goodman 1995 for references.) Neutrino Oscillations 293 Apparently, the anomaly observed at the Kamiokande water Che- renkov detector (Hirata et al. 1992; Fukuda et al. 1994) can be ex- plained in terms of oscillations for neutrino parameters in the shaded area in Fig. 8.6; the best-ﬁt value is indicated by a star. These results are a combined ﬁt for the sub-GeV and multi-GeV data as published by the Kamiokande collaboration (Fukuda et al. 1994). The oscilla- tion hypothesis appears to be buttressed by a zenith-angle variation of the eﬀect observed in Kamiokande’s multi-GeV data sample although the claimed signiﬁcance of this eﬀect has been critiqued, e.g. by Fogli and Lisi (1995) and by Saltzberg (1995). The required large mixing angle as well as the exclusion regions of the other experiments make it appear dubious that the anomaly is caused by oscillations. Still, it is a serious eﬀect that cannot be blamed easily on problems with the Kamiokande detector. Also, the reliability of some of the exclusion areas in Fig. 8.6 may be called into question, notably because of their dependence on absolute ﬂux normalizations. The intuition against a large νµ-ντmixing angle may be misguided. In the future, it will be possible to test the relevant regime of mixing parameters in long-baseline laboratory experiments (e.g. Schneps 1995). At the time of this writing, the possibility that the atmospheric neutrino anomaly may be revealing neutrino oscillations remains a lively-debated possibility. 8.3 Oscillations in Media 8.3.1 Dispersion Relation for Mixed Neutrinos The neutrino refractive index of a normal medium is extremely small and so its only potentially observable eﬀect occurs in neutrino oscilla- tions. The refractive index is diﬀerent for diﬀerent ﬂavors—the medium is “ﬂavor birefringent”—and so neutrinos from diﬀerent families which propagate with the same energy through the same medium acquire dif- ferent phases. If in addition these ﬂavors mix, the medium-induced phase shift between them shows up in the interference between the mixed states. Without a medium the phase diﬀerence between mixed states arises from their mass diﬀerence. Hence, medium eﬀects will be noticable only if the induced “eﬀective mass” is of the same order as the vacuum masses. Therefore, medium refraction is important for neutrino oscillations in certain situations because the vacuum masses are very small. 294 Chapter 8 The dispersion relation for a single ﬂavor in a medium was given by Eq. (6.107). For three ﬂavors which mix according to Eq. (8.3) the Klein-Gordon equation in Fourier space is    ω−GFnB√ 2 3Ye−1 0 0 0 Ye−1 0 0 0 Ye−1  2 −k2−U m2 10 0 0m2 20 0 0 m2 3 U†   νe νµ ντ = 0,(8.26) where a possible neutrino background was ignored and Yn= 1−Yp= 1−Yewas used. Also, the higher-order diﬀerence between νµandντwas ignored. This equation has nonzero solutions only if det {. . .}vanishes, a condition that gives us the dispersion relation for the three normal modes. For unmixed neutrinos where Uis the unit matrix one recovers Eq. (6.107) for each ﬂavor. For all practical cases the neutrinos are highly relativistic so that one may linearize this equation, (ω−k−M2 eﬀ/2k) Ψ = 0 (8.27) where it is easy to read the matrix M2 eﬀfrom Eq. (8.26). Because to lowest order ω=kone may equally use M2 eﬀ/2ωin order to derive the dispersion relation, depending on whether one wishes to write ωas a function of kor vice versa. For two-ﬂavor mixing between νeandνµorντ(mixing angle θ0) the eﬀective mass matrix may be written in the same form as in vacuum M2 eﬀ/2ω=b0−1 2B·, (8.28) where b0= (m2 1+m2 2)/4k+√ 2GFnB(Ye−1 2) and B=2π ℓosc sin 2θ 0 cos 2θ =m2 2−m2 1 2ω sin 2θ0 0 cos 2θ0 −√ 2GFne 0 0 1 . (8.29) This equation deﬁnes implicitly the mixing angle θas well as the os- cillation length ℓoscin the medium in terms of the masses, the vacuum mixing angle θ0, and the electron density ne. Neutrino Oscillations 295 Fig. 8.8. Mixing angle, oscillation length, and neutrino dispersion relation as a function of the electron density. The medium was taken to have equal numbers of protons and neutrons ( Ye=1 2), the ratio of neutrino masses was taken to be m1:m2= 1 : 2, and sin22θ0= 0.15. 296 Chapter 8 Explicitly one ﬁnds for the mixing angle and the oscillation length in the medium the following expressions, tan 2θ=sin 2θ0 cos 2θ0−ξ, sin 2θ=sin 2θ0 [sin22θ0+ (cos 2 θ0−ξ)2]1/2, (8.30) where ξ≡√ 2GFne2ω m2 2−m2 1= 1.53×10−7Yeρ g cm−3ω MeVeV2 m2 2−m2 1, (8.31) and ℓosc=4π ω m2 2−m2 1sin 2θ sin 2θ0. (8.32) Form2> m 1these functions are shown in Fig. 8.8; they exhibit a “resonance” for cos 2 θ0=ξ. The dispersion relation has two branches which in vacuum corre- spond to ω1,2= (k2−m2 1,2)1/2. In the relativistic limit they are ω1,2−k=m2 1+m2 2 4k+√ 2GFnB(Ye−1 2) ±m2 2−m2 1 4k[ sin22θ0+ (cos 2 θ0−ξ)2]1/2,(8.33) a result schematically shown in Fig. 8.8. The “resonance” of the mix- ing angle corresponds to the crossing point of the two branches of the dispersion relation. Of course, the levels do not truly cross, but rather show the usual “repulsion.” Because the medium eﬀect changes sign for antineutrinos, a reso- nance occurs between νeandνµifm2> m 1, while none occurs between νeandνµ. If the mass hierarchy is the other way round, a resonance occurs for νeandνµ, but not for νeandνµ. 8.3.2 Oscillations in Homogeneous Media In a homogeneous medium the treatment of neutrino oscillations is exactly as in vacuum except that one must use the eﬀective medium mixing angle and oscillation length given above. If the medium is suf- ﬁciently dilute it will not aﬀect the oscillations at all. This is the case Neutrino Oscillations 297 when the quantity ξof Eq. (8.31) is much smaller than unity. In the opposite limit one ﬁnds for the mixing angle in the medium sin 2θ=m2 2−m2 1√ 2GFne2ωsin 2θ0. (8.34) The oscillation length becomes ℓosc=2π√ 2GFne= 1.63×104kmg cm−3 Yeρ, (8.35) independent of the neutrino masses or energy. Typically the eﬀect of the medium is, therefore, to suppress the mixing angle and thus the possibility to observe oscillations. Of course, for normal materials with a density of a few g cm−3and neutrino masses in the eV range one needs TeV neutrino energies for the medium to be relevant at all. For a review of the impact of oscillations in a medium on neutrino experiments or the observation of atmospheric, solar, or supernova neutrinos see, for example, Kuo and Pantaleone (1989). 8.3.3 Inhomogeneous Medium: Adiabatic Limit In an inhomogeneous medium the oscillation problem is much more complicated. Recall that the spatial variation of a stationary neutrino beam in the z-direction is given by i∂zΨ =−KΨ according to Eq. (8.7) if one drops the index ω. The matrix K=ω−M2 eﬀ/2ωis now a function ofz. Formally, the solution is Ψ( z) =WΨ(0) with W=Sexp( i∫z 0K(z′)dz′) , (8.36) where Sis the space-ordering operator. An explicit solution is not available because the matrices K(z) generally do not commute for dif- ferent z. Of course, for a constant Kone recovers the previous result W=eiKz. In certain limits one may still ﬁnd simple solutions. The most in- teresting case of neutrinos moving through an inhomogeneous medium is the emission from stars, notably the Sun, where they are produced in a relatively high-density region and then escape into vacuum. The density at the center of the Sun is about 150 g cm−3, the solar radius 6.96×1010cm, yielding an extremely shallow density variation by ter- restrial standards! Therefore, consider the adiabatic limit where the density of the medium varies slowly over a distance ℓoscwhich is the characteristic length scale for the oscillation problem. 298 Chapter 8 This case is best understood for two-ﬂavor mixing if one studies the (temporal) evolution of the neutrino ﬂavor polarization vector P. Recall that it evolves according to the spin-precession formula ˙P= B×Pwhere the “magnetic ﬁeld” is now a function of time. If B varies slowly relative to the precession frequency the “spin” follows the magnetic ﬁeld in the sense that it moves on a precession cone which is “attached” to B. If the spin is oriented essentially along the magnetic ﬁeld direction it stays pinned to that direction. Therefore, it can be entirely reoriented by slowly turning the external magnet. For the case of neutrino oscillations this means that in the adia- batic limit a state can be entirely reoriented in ﬂavor space, i.e. an initial νecan be turned almost completely into a νµeven though the vacuum mixing angle may be small. Consider θ0≪1, an initial den- sity so large that the medium eﬀects dominate, begin with a νe, and letm1< m 2. This means that in Fig. 8.8 (lowest panel) begin on the upper branch of the dispersion relation far to the right of the crossover. Then let the neutrino propagate toward vacuum through an adiabatic density gradient. This implies that it stays on the up- per branch and ends up at ne= 0 (vacuum) as the mass eigenstate of the upper eigenvalue m2which corresponds approximately to a νµ. This behavior is known as “resonant neutrino oscillations” or MSW ef- fect after Mikheyev, Smirnov, and Wolfenstein. Mikheyev and Smirnov (1985) ﬁrst discovered this eﬀect when they studied the oscillation of solar neutrinos while Wolfenstein (1978) ﬁrst emphasized the impor- tance of refraction for neutrino oscillations. A simple interpretation of resonant oscillations in terms of an adiabatic “level crossing” was ﬁrst given by Bethe (1986). In order to quantify the adiabatic condition return to the picture of a spin precessing around a magnetic ﬁeld. For ﬂavor oscillations the precession frequency is 2 π/ℓ osc. The “magnetic ﬁeld” is tilted with an angle 2 θ(medium mixing angle) against the 3-direction (Fig. 8.2) and so its speed of angular motion is 2 dθ/dt . For spatial rather than temporal oscillations the adiabatic condition is |∇θ| ≪π/ℓ osc. (8.37) This translates into ∇θ=1 2ξ(sin22θ/sin 2θ0)∇lnnewhile ℓoscis given by Eq. (8.32) and ξby Eq. (8.31). Thus, the adiabatic condition is ξsin32θ|∇lnne| ≪sin22θ0|m2 2−m2 1|/2ω. (8.38) This condition must be satisﬁed along the entire trajectory. Neutrino Oscillations 299 If the neutrino crosses a density region such that a resonance oc- curs, this part of the trajectory yields the most restrictive adiabaticity requirement. On resonance ξ= cos 2 θ0and sin 2 θ= 1. In this case one deﬁnes an adiabaticity parameter γ≡m2 2−m2 1 2ωsin 2θ0tan 2θ0 |∇lnne|res, (8.39) where the denominator is to be evaluated at the resonance point. The adiabatic condition is γ≫1. It establishes a relationship between vac- uum mixing angles and neutrino masses for which resonant oscillations occur. 8.3.4 Inhomogeneous Medium: Analytic Results For practical problems, notably the oscillation of solar neutrinos on their way out of the Sun, one may easily solve the equation i∂zΨ = −KΨ numerically for prescribed proﬁles of the electron density and neutrino production rates. The main features of such calculations, how- ever, can be understood analytically because for certain simple density proﬁles one can ﬁnd analytic representations of Eq. (8.36) indepen- dently of the adiabatic approximation. Consider the situation where a νeis produced in a medium and subsequently escapes into vacuum through a monotonically decreasing density proﬁle. Because in the adiabatic limit the production and de- tection points are separated by many oscillation lengths, the oscillation pattern will be entirely washed out and one may use average probabil- ities for the ﬂavor content. Initially, the projection of the polarization vector on the “magnetic ﬁeld” direction is cos 2 θwhere θis the medium mixing angle at the production point. Because the component of Pin theBdirection is conserved when Bchanges adiabatically, the ﬁnal av- erage projection of Pon the 3-axis is cos 2 θ0cos 2θ. Then the survival probability is prob( νe→νe) =1 2(1 + cos 2 θ0cos 2θ). This result applies in the adiabatic limit whether or not a resonance occurs. Next, drop the adiabatic condition but assume that the production and detection points are many oscillation lengths away on opposite sides of a resonance so that the oscillation pattern remains washed out; then it remains suﬃcient to consider average probabilities. In this case it is useful to write prob( νe→νe) =1 2+ (1 2−p) cos 2 θ0cos 2θ, (8.40) where the correction pto the adiabatic approximation is the probability that the neutrino jumps from one branch of the dispersion relation to 300 Chapter 8 the other (Fig. 8.8, lowest panel) when it moves across the resonant density region. A linear density proﬁle near the resonance region, which is always a ﬁrst approximation, yields the “Landau-Zener probability” p=e−πγ/2(8.41) which was ﬁrst derived in 1932 for atomic level crossings. The adia- baticity parameter γwas deﬁned in Eq. (8.39); in the adiabatic limit γ≫1 one recovers p= 0. For a variety of other density proﬁles and without the assumption of a small mixing angle one ﬁnds a result of the form p=e−(πγ/2)F−e−(πγ/2)F′ 1−e−(πγ/2)F′, (8.42) where F′=F/sin2θ0andFis an expression characteristic for a given density proﬁle. For a linear proﬁle ne∝rone has F= 1 so that for a small mixing angle one recovers the Landau-Zener probability. The proﬁle ne∝r−1leads to F= cos22θ0/cos2θ0. Of particular interest is the exponential ne∝e−r/R 0which yields F= 1−tan2θ0. (8.43) All of these results are quoted after the review by Kuo and Pantaleone (1989) where other special cases and references to the original literature can be found. Going beyond the Landau-Zener approximation requires assuming one of the above speciﬁc forms for ne(r) for which analytic results ex- ist. Recently, Guzzo, Bellandi, and Aquino (1994) used a somewhat diﬀerent approach which is free of this limitation. They derived an approximate solution to the equivalent of Eq. (8.36) by the method of stationary phases for the space-ordered exponential. They found p=(1−γ′ 1 +γ′)2 sin2(θ0−θ) +2γ′ (1 +γ′)2[ cos2(θ0−θ) + cos2(θ0+θ)] , (8.44) where γ′≡πγ/16. It was assumed that a resonance occurs between the production point (mixing angle θ) and the detection point which is taken to lie in vacuum (mixing angle θ0). Eq. (8.44) can be used only in the nonadiabatic regime as it works only for γ <16/π. Neutrino Oscillations 301 8.3.5 The Triangle and the Bathtub The most important application of resonant neutrino oscillations is the possible reduction of the solar νeﬂux, an issue to be discussed more fully in Chapter 10. Here, I will use the above simple analytic results to discuss schematically the survival probability prob( νe→νe) of neutrinos produced in the Sun. To this end I use an exponential proﬁle for the electron density, ne=nce−r/R 0, which is a reasonable ﬁrst approximation with R0= R⊙/10.54 (Bahcall 1989). Then |∇lnne|=R−1 0is a quantity indepen- dent of location so that there is no need to evaluate it speciﬁcally on resonance. Thus |∇lnne|res= 3×10−15eV is independent of neutrino parameters. Therefore, the adiabaticity parameter is γ=1 6×103sin 2θ0tan 2θ0∆m2 meV2MeV Eν. (8.45) An electron density at the center of nc= 1.6×1026cm−3yields ξc= 40meV2 ∆m2Eν MeV. (8.46) The quantity ξwas deﬁned in Eq. (8.31). It is further assumed that all neutrinos are produced with a ﬁxed energy by a point-like source at the solar center. They will encounter a resonance on their way out if ξc>cos 2θ0. In this case one may calculate the “jump probability” paccording to Eq. (8.42) with Ffrom Eq. (8.43) for the exponential proﬁle. The survival probability is then given by Eq. (8.40) with the mixing angle at the solar center cos 2θ=cos 2θ0−ξc [(cos 2 θ0−ξc)2+ sin22θ0]1/2. (8.47) Ifξc<cos 2θ0no resonance is encountered; the vacuum parameters dominate throughout the Sun. In this case one may use p= 0. In the framework of these approximations it is straightforward to evaluate prob( νe→νe) as a function of ∆ m2and sin22θ0. In Fig. 8.9 contours for the survival probability are shown for Eν= 1 MeV. Be- cause the energy always appears in the combination ∆ m2/2Eνone may obtain an analogous plot for other energies by an appropriate vertical shift. This kind of plot represents the well-known MSW triangle. The dashed line marks the condition γ= 1 and thus divides the param- eter plane into a region where the oscillations are adiabatic and one 302 Chapter 8 where they are not. The dotted line marks ξc= cos 2 θ0and thus indi- cates for which neutrino parameters a resonance occurs on the way out of the Sun. Fig. 8.9. The MSW triangle for the simpliﬁed solar model discussed in the text with Eν= 1 MeV. Fig. 8.10. The MSW bathtub for the simpliﬁed solar model discussed in the text with ∆ m2= 3×10−5eV2and sin22θ0= 0.01. Neutrino Oscillations 303 It is also instructive to look at a vertical cut through this contour plot, except that it is more useful to represent it as a function of Eνfor a ﬁxed ∆ m2rather than the reverse. Both possibilities are equivalent because what appears is the combination ∆ m2/2Eν. The result is the MSW bathtub shown in Fig. 8.10 for ∆ m2= 3×10−5eV2and sin22θ0= 0.01 which is one set of parameters that might solve the solar neutrino problem (Chapter 10). The main point of Fig. 8.10 is that resonant neutrino oscillations modify the spectrum of observable νe’s diﬀerently for diﬀerent energies. Low-energy neutrinos remain entirely unaﬀected, intermediate-energy ones are strongly suppressed, and the eﬀect on high-energy states varies over a broad interval. There, the shape of the observable spectrum is modiﬁed relative to the source spectrum. 8.3.6 Neutrino Oscillations without Vacuum Mixing Neutrino oscillations in vacuum or in a medium arise because in the weak-interaction basis the matrix of eﬀective neutrino masses, M2 eﬀ, has oﬀ-diagonal elements which induce transitions between, say, an initial νeand a νµ. The eﬀect of the medium is to modify the diagonal elements of this matrix, possibly such that diﬀerent ﬂavor states become degenerate, causing the eﬀect of resonant oscillations. Conceivably, oscillations could occur even if M2 eﬀ= 0 in vacuum so that in the standard model there are no oﬀ-diagonal terms. The absence of oﬀ-diagonal medium contributions to M2 eﬀwithin the standard model reﬂects the absence of ﬂavor-changing neutral cur- rents (Sect. 7.2). It cannot be excluded experimentally, however, that for neutrinos such currents exist on some level, implying that a νe, for example, sometimes emerges as a νµfrom a collision with another parti- cle. In the forward direction this would cause an “oﬀ-diagonal refractive index” so that mixing and thus oscillations would be induced by the medium (Valle 1987; Fukugita and Yanagida 1988). Certain supersym- metric extensions of the standard model with R-parity breaking predict such eﬀects (Guzzo, Masiero, and Petcov 1991; Kapetanakis, Mayr, and Nilles 1992). Therefore, if neutrino oscillations indeed explain the solar neutrino problem (Chapter 10) this does not inevitably imply neutrino vacuum masses and mixings—it could also point to the existence of ﬂavor-changing neutral currents. An analytic description of two-ﬂavor oscillations for this case was given by Guzzo and Petcov (1991) while a detailed parameter study for the solution of the solar neutrino problem was provided by Barger, Phillips, and Whisnant (1991). 304 Chapter 8 8.4 Spin and Spin-Flavor Oscillations 8.4.1 Vacuum Spin Precession According to Sect. 7.2 neutrinos may have magnetic dipole moments. In particular, if neutrinos have masses they inevitably have small but nontrivial electromagnetic form factors. Novel interactions can induce large dipole moments even for massless neutrinos. In the presence of magnetic ﬁelds such a dipole moment leads to the familiar spin preces- sion which causes neutrinos to oscillate into opposite helicity states. If a state started out as a helicity-minus neutrino, the outgoing par- ticle is a certain superposition of both helicities. The “wrong-helicity” (right-handed) component does not interact by the standard weak in- teractions, diminishing the detectable neutrino ﬂux. As the Sun has relatively strong magnetic ﬁelds it is conceivable that magnetic spin oscillations could be responsible for the measured solar neutrino deﬁcit (Werntz 1970; Cisneros 1971)—see Sect. 10.7. Magnetic spin preces- sions could also aﬀect supernova neutrinos, and in the early universe it could help to bring the “wrong-helicity” Dirac neutrino degrees of free- dom into thermal equilibrium. Presently I will focus on some general aspects of neutrino magnetic spin oscillations rather than discussing speciﬁc astrophysical scenarios. In the rest frame of a neutrino the evolution of its spin operator S= 1 2(Pauli matrices ) is governed by the Hamiltonian H0= 2µB0·S; this follows directly from Eq. (7.23). Here, B0refers to the magnetic ﬁeld in the neutrino’s rest frame as opposed to Bin the laboratory frame. The equation of motion of the spin operator in the Heisenberg picture is then given by i∂tS= [S, H0] or ˙S= 2µB0×S, (8.48) which represents the usual precession with frequency 2 µB0around the magnetic ﬁeld direction. Equivalently, one may consider Eq. (7.23) for the neutrino spinor which has the form of a Schr¨ odinger equation. Usually one will be concerned with very relativistic neutrinos so that the magnetic ﬁeld B0in the rest frame must be obtained from the electric and magnetic ﬁelds in the laboratory frame EandBby virtue of the usual Lorentz transformation B0= (B·ˆv)ˆv+γ[ˆv×(B×ˆv) +E×v], (8.49) where vis the velocity of the neutrino, ˆvthe velocity unit vector, and γ= (1−v2)−1/2=Eν/mthe Lorentz factor with Eνthe neutrino en- ergy and mits mass. The spin precession as viewed from the laboratory Neutrino Oscillations 305 system, however, involves a time-dilation factor γ−1so that the Hamil- tonian for the spin evolution in the laboratory system is H=µγ−1B· or explicitly H=µ[ (m/ω)(B·ˆv)ˆv+ˆv×(B×ˆv) +E×v] ·. (8.50) If the spin is quantized along the direction of motion (helicity states) and if there is only a magnetic ﬁeld in the laboratory ( E= 0), the Hamiltonian is H=µB(γ−1cosθ e−iφsinθ e+iφsinθ−γ−1cosθ) , (8.51) with θthe magnetic ﬁeld direction relative to the neutrino direction of motion and φits azimuthal direction which may be chosen as φ= 0. The bottom line is that very relativistic neutrinos ( γ→ ∞ ) spin- precess like nonrelativistic ones, except that the relevant magnetic ﬁeld is the transverse component BT; the eﬀective laboratory Hamiltonian is H=µBT(0 1 1 0) . (8.52) Therefore, if one begins with left-handed neutrinos a complete preces- sion into right-handed ones will always occur (assuming a suﬃcient path length), independently of the direction between the laboratory magnetic ﬁeld and the neutrino direction of motion (Fujikawa and Shrock 1980). If there is only an electric ﬁeld in the laboratory frame, magnetic oscillations will nevertheless occur because Eq. (8.50) implies that the neutrino sees an eﬀective magnetic ﬁeld BT=E×v(Okun 1986). The precession thus takes place in the plane of Eandv. Contrary to ﬂavor oscillations, the oscillation length does not de- pend on the energy. Therefore, a broad energy spectrum would not cause a depolarization of the neutrino ﬂux from an astrophysical source (Sun, supernovae). Rather, neutrinos of all energies would spin-precess in step with each other. Because the precession frequency is 2 µBTthe neutrinos will have reversed their spin after a distance 1 2ℓosc=π 2µBT= 5.36×1013cm10−10µB µ1 G BT, (8.53) with µB=e/2methe Bohr magneton. With a solar BTin the kG regime the oscillation length is of order a solar radius R⊙= 6.96×1010cm. A 306 Chapter 8 substantial reduction of the left-handed solar neutrino ﬂux then re- quires magnetic dipole moments of order 10−10µB. Typical galactic magnetic ﬁelds are of order 10−6G with coherence scales of order kpc. Therefore, neutrinos from a galactic source such as a supernova could be “ﬂipped” before reaching Earth if their dipole moment is of order 10−12µB. These values are in the neighborhood of what is experimen- tally and observationally allowed so that there remains interest in the possibility that magnetic spin oscillations could have a signiﬁcant im- pact on the neutrino ﬂuxes from the Sun or from supernovae. 8.4.2 Spin Precession in a Medium The picture of magnetic spin oscillations thus developed is incomplete. If neutrinos propagate in a medium as in the Sun they are subject to medium-induced refractive eﬀects which shift the energy of left-handed states relative to right-handed ones which propagate as in vacuum be- cause they are sterile. Therefore, including medium eﬀects the eﬀective Hamiltonian for the evolution of the two helicity states of νeis H=(0 µBT µBT√ 2GF(ne−1 2nn)) , (8.54) where neandnnare the electron and neutron densities, respectively. Here, “spin up” refers to the direction of motion, i.e. to right-handed states, “spin down” to left-handed ones. The two helicity states are no longer degenerate, and the spin precession will no longer lead to a complete reversal of the spin. One may write H=b+µB·so that there is eﬀectively a longitudinal magnetic ﬁeld, i.e. parallel to the direction of motion or z-direction BL=GF(ne−1 2nn)√ 2µ= 66.6 kG10−10µB µρ(Ye−1 2Yn) g cm−3. (8.55) Here, ρis the mass density and Ye,nthe electron and neutron number per baryon. If BL≫BTthe precession is around a direction close to the direction of motion, and so the spin reversal will always be far from complete. Hence, the presence of the medium suppresses magnetic spin oscillations (Voloshin, Vysotski˘ ı, and Okun 1986). 8.4.3 Spin-Flavor Precession If neutrinos were Majorana particles they could not have magnetic dipole moments, but they could still possess transition moments be- tween diﬀerent ﬂavors. In this case the magnetic oscillation would be, Neutrino Oscillations 307 say, between a helicity-minus νeto a helicity-plus νµwhich, because of its assumed Majorana nature, is identical to a νµ. Because typically the νeandνµmasses will be diﬀerent they must be included in the phase evolution of the neutrinos. Thus, for relativistic states (momentum p) one arrives at a two-level equation of motion of the form i∂t(νe νµ) =(m2 νe/2p µB T µBT m2 ν/2p)(νe νµ) . (8.56) The transition magnetic moment µthus leads to simultaneous spin and ﬂavor oscillations (Schechter and Valle 1981). Moreover, neutrinos will be converted to antineutrinos (and the reverse). If such spin-ﬂavor oscillations occurred, say, in the Sun one might actually be able to measure a ﬂux of antineutrinos from that source. A transition magnetic moment implies that the individual ﬂavor lepton numbers are not conserved. Thus most likely there is also stan- dard ﬂavor mixing which allows for, say, νe↔νµoscillations. If the mass eigenstates are m1andm2, respectively, and if the vacuum mixing angle is θ, the four-level equation of motion in vacuum is i∂t νe νµ νe νµ = ∆mc2θ∆ms2θ 0 µBT ∆ms2θ−∆mc2θµBT 0 0 µBT ∆mc2θ∆ms2θ µBT 0 ∆ ms2θ−∆mc2θ  νe νµ νe νµ , (8.57) where ∆ m≡(m2 2−m2 1)/4p,c2θ≡cos 2θ, and s2θ≡sin 2θ. Any of νe, νµ,νe, and νµcan oscillate into any of the others. Including medium eﬀects further complicates this matrix because the energies of νeandνµare shifted by diﬀerent amounts, and each is shifted in the opposite directions from its antiparticle. With Vνe=√ 2GF(ne−1 2nn) and Vν=√ 2GF(−1 2nn) the matrix becomes  ∆mc2θ+Vνe ∆ms2θ 0 µBT ∆ms2θ−∆mc2θ+Vν µBT 0 0 µBT ∆mc2θ−Vνe ∆ms2θ µBT 0 ∆ms2θ−∆mc2θ−Vν .(8.58) If there are density gradients (solar neutrinos!) one may have resonant magnetic conversions between, say, νeandνµwhere the barrier to spin- precessions caused by the mass diﬀerence is compensated by matter eﬀects, i.e. one may have resonant spin-ﬂavor oscillations (Akhmedov 1988a,b; Barbieri and Fiorentini 1988; Lim and Marciano 1988). For Dirac neutrinos, transitions to antineutrinos are not possible and so one needs to consider oscillations separately in the four-level 308 Chapter 8 system ( νL e, νL µ, νR e, νR µ) and ( νL e,νL µ,νR e,νR µ). Moreover, one may have both diagonal and transition magnetic moments so that for neutrinos the r.h.s. of the equation of motion is  ∆mc2θ+Vνe ∆ms2θ µeeBTµµeBT ∆ms2θ−∆mc2θ+VνµeµBTµµµBT µeeBT µµeBT ∆mc2θ∆ms2θ µeµBT µµµBT ∆ms2θ−∆mc2θ  νL e νL µ νR e νR µ , (8.59) where the right-handed neutrinos do not experience a medium-induced energy shift. For certain values of the parameters the oscillations between two states may dominate which are then described by a certain 2 ×2 sub- matrix. Then the general treatment is much like the ﬂavor mixing problems studied earlier. In general, a much richer collection of solu- tions obtains—for a review see Pulido (1992). Electric transition moments can also play a role, and precessions in electric ﬁelds may be important. Majorana neutrinos have either electric or magnetic transition moments, but not both as long as CP remains conserved. Still, even electric transition moments would lead to spin-ﬂavor oscillations in macroscopic magnetic ﬁelds. 8.4.4 Twisting Magnetic Fields The problem of spin oscillations is further complicated by the possibility that the magnetic ﬁeld changes its direction along the neutrino trajec- tory, i.e. that it may be “twisting” (Aneziris and Schechter 1991). In this case the equation of motion of a two-level spin-precession problem is based on the Hamiltonian H=µ(BLBTe−iφ BTeiφ−BL) , (8.60) where the three parameters BL,BT, and φvary along the neutrino trajectory. The transverse ﬁeld strength BTis a physical magnetic ﬁeld while the longitudinal one BLis an eﬀective magnetic ﬁeld which represents the neutrino mass diﬀerence (spin-ﬂavor oscillations!) and refractive medium eﬀects. As long as BTmaintains its direction along the trajectory, the phase φcan be globally chosen to be zero. Ifφvaries along the trajectory so that the neutrino experiences it to be a function of time, it is useful to transform the equation of motion to a coordinate system which corotates with BTso that in the new frame one is back to the old situation of a ﬁxed direction for BT(Smirnov Neutrino Oscillations 309 1991). This transformation is achieved by ν′=Uνwhere νrepresents a two-level wave function and U=(e−iφ/20 0 eiφ/2) , (8.61) so that the spin-up and down states acquire opposite, time-dependent phases. However, because the transformation does not mix spin up with spin down, the transition rate between the two levels is the same in both coordinate systems. The equation of motion i∂tν′=H′ν′involves the Hamiltonian H′=µ(BLBT BT−BL) +1 2˙φ(−1 0 0 1) . (8.62) Put another way, the eﬀective longitudinal magnetic ﬁeld in the rotating frame is B′ L=BL−1 2˙φ/µwhile BTremains unchanged. |B′ L|can be less or larger than |BL|so that the transition rate between the two levels can be increased or decreased by a twist. BLmay even be cancelled entirely, enabling resonant oscillations. For a systematic discussion of the full four-level spin-ﬂavor problem see, for example, Akhmedov, Petcov, and Smirnov (1993). Chapter 9 Oscillations of Trapped Neutrinos In a supernova (SN) core or in the early universe neutrinos scatter on the background medium and on each other. Therefore, oscillations of mixed neutrinos are frequently interrupted, leading to ﬂavor equilib- rium if there is enough time. A Boltzmann-type kinetic equation is derived that accounts simultaneously for neutrino oscillations and col- lisions for arbitrary neutrino degeneracy. It is applied to a SN core, yielding an estimate of the time scale to reach ﬂavor equilibrium. On the basis of the SN 1987A neutrino observations a limit on the mixing of sequential neutrinos with a hypothetical sterile ﬂavor is derived. 9.1 Introduction So far oscillations were discussed in the context of “beam experiments” where neutrinos of a known ﬂavor are produced at a certain location, for example in a power reactor or in the Sun, then propagate over a distance, and are then detected with a device that allows one to dis- tinguish between diﬀerent ﬂavors. It was assumed that there were no interactions between the production and detection point, with the pos- sible exception of decays, allowing one to treat neutrino oscillations as a simple propagation phenomenon fully analogous to light propagation in an optically active medium. This approach is not adequate for the early universe and a young SN core. In both cases there are frequent neutrino collisions which aﬀect the free evolution of the phases. The impact of these collisions can be understood if one assumes for the purpose of illustration that one 310 Oscillations of Trapped Neutrinos 311 neutrino ﬂavor (say νe) scatters with a rate Γ while the other (say νµ) does not. An initial νewill begin to oscillate into νµ. The probability for ﬁnding it in one of the two ﬂavors evolves as previously discussed and as shown in Fig. 9.1 (dotted line). However, in each collision the momentum of the νecomponent of the superposition is changed, while theνµcomponent remains unaﬀected. Thus, after the collision the two ﬂavors are no longer in the same momentum state and so they can no longer interfere: each of them begins to evolve separately. This allows the remaining νeto develop a new coherent νµcomponent which is made incoherent in the next collision, and so forth. This process will come into equilibrium only when there are equal numbers of νe’s andνµ’s. This decoherence eﬀect is even more obvious when one includes the possibility of νeabsorption and production by charged-current reactions νen↔pe. Because of oscillations an initial νeis subsequently found to be aνµwith an average probability of1 2sin22θ(mixing angle θ) and as such cannot be absorbed, or only by the reaction νµn↔pµ if it has enough energy. The continuous emission and absorption of νe’s spins oﬀ a νµwith an average probability of1 2sin22θin each collision! Chemical relaxation of the neutrino ﬂavors will occur with an approximate rate1 2sin22θΓ where Γ is a typical weak interaction rate for the ambient physical conditions. An initial νepopulation turns into an equal mixture of νe’s andνµ’s as shown schematically in Fig. 9.1 (solid line). Fig. 9.1. Neutrino oscillations with collisions (solid line). In the absence of collisions and for a single momentum one obtains periodic oscillations (dotted line), while for a mixture of energies the oscillations are washed out by “dephasing” (dashed line). 312 Chapter 9 A certain “damping” of ﬂavor oscillations occurs even without colli- sions when the neutrinos are not monochromatic because then diﬀerent modes oscillate with diﬀerent frequencies. This “dephasing” eﬀect was shown in Fig. 8.4 and is repeated in Fig. 9.1 (dashed line). While the dephasing washes out the oscillation pattern, it does not lead to ﬂavor equilibrium: the probability for νeends at a constant of 1 −1 2sin22θ. For the rest of this chapter “damping of oscillations” never refers to this relatively trivial dephasing eﬀect. The interplay of collisions and oscillations leads to ﬂavor equilibrium between mixed neutrinos. In a SN core the concentration of electron lepton number is initially large so that the νeform a degenerate Fermi sea. The other ﬂavors νµandντare characterized by a thermal distri- bution at zero chemical potential. However, if they mix with νethey will achieve the same large chemical potential. In a SN core heat and lepton number are transported mostly by neutrinos; the eﬃciency of these processes depends crucially on the degree of neutrino degeneracy for each ﬂavor. Therefore, it is of great interest to determine the time it takes a non- νeﬂavor to equilibrate with νeunder the assumption of mixing (Maalampi and Peltoniemi 1991; Turner 1992; Pantaleone 1992a; Mukhopadhyaya and Gandhi 1992; Raﬀelt and Sigl 1993). Moreover, if νemixes with a sterile neutrino species, conversion into this inert state leads to the loss of energy and lepton number from the inner core of a SN. The observed SN 1987A neutrino signal may thus be used to constrain the allowed range of masses and mixing angles (Kainulainen, Maalampi, and Peltoniemi 1991; Raﬀelt and Sigl 1993; see also Shi and Sigl 1994). These applications are discussed in Sects. 9.5 and 9.6 below. A simple estimate of the rate of ﬂavor conversion and the emission rate of sterile neutrinos from a SN core requires not much beyond the ap- proximate rate1 2sin22θΓ. However, a proper kinetic treatment of the evolution of a neutrino ensemble under the simultaneous action of os- cillations and collisions is an interesting theoretical problem in its own right. Notably, it is far from obvious how to treat degenerate neutri- nos in a SN core because the diﬀerent ﬂavors will suﬀer diﬀerent Pauli blocking factors. Does this eﬀect break the coherence between mixed ﬂavors in neutral-current collisions? The bulk of this chapter is devoted to the derivation and discus- sion of a general kinetic equation for mixed neutrinos (Dolgov 1981; Rudzsky 1990; Raﬀelt, Sigl, and Stodolsky 1993; Sigl and Raﬀelt 1993). This equation provides a sound conceptual and quantitative framework for dealing with various aspects of coherent and incoherent neutrino Oscillations of Trapped Neutrinos 313 interactions with a medium, whether the neutrinos are degenerate or not. As a free spin-oﬀ it will provide a proper formalism to deal with the refractive eﬀects of neutrinos propagating in a bath of neutrinos, a problem that is of interest in the early universe where interactions among neutrinos produce the dominant medium eﬀect, and in the neu- trino ﬂow from a SN core where the density of neutrinos is larger than that of nonneutrino background particles (Sect. 11.4). Besides neutrino ﬂavor oscillations, magnetically induced spin or spin-ﬂavor oscillations are also of potential interest because in super- novae and the early universe strong magnetic ﬁelds are believed to exist. Spin relaxation (the process of populating the r.h. degrees of freedom by the simultaneous action of spin oscillations and collisions) is a very sim- ilar problem to that of achieving chemical equilibrium between diﬀerent ﬂavors which is discussed here. Therefore, similar kinetic methods can be applied (Enqvist, Rez, and Semikoz 1995 and references therein). 9.2 Kinetic Equation for Oscillations and Collisions 9.2.1 Stodolsky’s Formula The loss of coherence between mixed neutrinos in collisions cannot be properly understood on the amplitude level because it is not the am- plitudes, but only their relative coherence that is damped—the ﬂavor states “decohere,” they do not disappear. This is diﬀerent from the decay of mixed particles where one of the amplitudes can be viewed as decreasing exponentially so that the total number of particles is not conserved. A natural description of decoherence is achieved by a density matrix as in Eq. (8.8); for a two-ﬂavor mixing problem it was expressed in terms of a polarization vector Paccording to Eq. (8.18), ρ=1 2(1+P·). In the weak interaction basis its diagonal elements are the probabilities for measuring νin, say, the νeorνµstate, respectively, while the oﬀ-diagonal elements contain relative phase information. In a two-level system, the length of the polarization vector measures the degree of coherence: length 1 corresponds to a pure state, shorter P’s to some degree of incoherence, and length zero is the completely mixed or incoherent state (Stodolsky 1987). In this latter case ρis proportional to the unit matrix which is invariant under a transforma- tion of basis. This state of “chemical” or “ﬂavor equilibrium” has no oﬀ-diagonal elements in any basis. 314 Chapter 9 Even coherent or partially coherent density matrices can be diago- nalized in some basis. The interactions with the background medium will also be diagonal in some basis; for neutrinos, this is the weak inter- action basis. If the density matrix and the interactions are diagonal in the same basis, there is no decoherence eﬀect. For example, a density matrix diagonal in the weak interaction basis implies that there is no relative phase information between, say, a νeand aνµand so collisions which aﬀect νe’s andνµ’s separately have no impact on the density matrix. However, a density matrix which is diagonal in the mass basis, assumed to be diﬀerent from the weak interaction basis, will suﬀer a loss of coherence in the same medium. All told, the loss of coherence is given by a shrinking of the length ofP. More precisely, only the component PTis damped which repre- sents the part “transverse” to the interaction basis, i.e. which in the interaction basis represents the oﬀ-diagonal elements of ρ. Thus, in the presence of collisions the evolution of Pis given by (Stodolsky 1987) ˙P=V×P−DPT. (9.1) The ﬁrst part is the previous precession formula, except that here a temporal evolution is appropriate since one has in mind the evolution of a spatially homogeneous ensemble rather than the spatial pattern of a stationary beam. The “magnetic ﬁeld” Vis V=2π tosc sin 2θ 0 cos 2θ , (9.2) whereθis the mixing angle in the medium and toscthe oscillation period. They are given in terms of the neutrino masses and momen- tum, the vacuum mixing angle, and the medium density by Eqs. (8.30) and (8.32) where strictly speaking the neutrino energy is to be replaced by its momentum as we have turned to temporal rather than spatial oscillations. The damping parameter Dis determined by the scattering amplitudes on the background. The evolution described by Eq. (9.1) is a precession around the “magnetic ﬁeld” V, combined with a shrinking of the length of Pto zero. This ﬁnal state corresponds to ρ=1 2where both ﬂavors are equally populated, and with vanishing coherence between them. For D=t−1 oscand sin 2θ=1 2the evolution of the ﬂavors is shown in Fig. 9.1 (solid line). Ignoring the wiggles in this curve it is an exponential as can be seen by multiplying both sides of Eq. (9.1) with P/P2which Oscillations of Trapped Neutrinos 315 leads to ˙P/P =−D(PT/P)2withP=|P|andPT=|PT|. For a small collision rate relative to the oscillation frequency one may use the precession-averaged PTwhich is found by taking the transverse part of the projection of PonV. Elementary geometry in Fig. 8.2 yields ⟨PT⟩/P= cos 2θsin 2θso that ⟨˙P/P⟩=−Dcos22θsin22θ. If one neutrino interacts while the other is sterile (for example νµwith regard to charged-current absorption) Dis half the collision rate Γ of the active ﬂavor (Stodolsky 1987). For small mixing angles one recovers the previous intuitive relaxation rate1 2sin22θΓ. Equation (9.1) is based on a single-particle wave function picture of neutrino oscillations and thus it is applicable if eﬀects nonlinear in the neutrino density matrices can be ignored. It does not allow one to include the eﬀect of Pauli blocking of neutrino phase space, which is undoubtedly important in a SN core where the νeFermi sea is highly degenerate. In this case it is rather unclear what one is supposed to use for the damping parameter D. Ifνeandνµscatter with equal amplitudes, there is no damping at all because the collisions do not distinguish between ﬂavors, preserving the coherence between them. Does this remain true if νecollisions are Pauli blocked by their high Fermi sea while those of νµare not? Such and other related questions can be answered if one abandons a single-particle approach to neu- trino oscillations, i.e. if one moves to a ﬁeld-theoretic framework which includes many-body eﬀects from the start. 9.2.2 Matrix of Densities Which quantity is supposed to replace the previous single-particle den- sity matrix as a means to describe a possibly degenerate neutrino ensemble? For unmixed neutrinos the relevant observables are time- dependent occupation numbers fpfor a given mode pof the neu- trino ﬁeld. They are given as expectation values of number operators np=a† papwhereapis a destruction operator for a neutrino in mode p anda† pthe corresponding creation operator. The expectation value is with regard to the state |⟩of the entire ensemble. For several ﬂavors it is natural to generalize the fp’s to matrices ρp=ρ(p) of the form49 49Strictly speaking (p) is deﬁned by ⟨a† j(p)ai(p′)⟩= (2)3(3)(p−p′)ij(p) and similar for (p). Therefore, the expectation values in Eq. (9.3) diverge because they involve an inﬁnite factor (2 )3(3)(0) which is related to the inﬁnite quantization volume necessary for continuous momentum variables. In practice, this factor al- ways drops out of ﬁnal results so that one may eﬀectively set (2 )3(3)(0) equal to unity. 316 Chapter 9 (Dolgov 1981) ρij(p) =⟨ a† j(p)ai(p)⟩ andρij(p) =⟨ b† i(p)bj(p)⟩ , (9.3) whereai(p) anda† i(p) are the destruction and creation operators for neutrinos of ﬂavor iin mode pwhilebis for antineutrinos which other- wise are referred to by overbarred quantities. The reversed order of the ﬂavor indices in the deﬁnition of ρ(p) guarantees that both matrices transform in the same way under a unitary transformation in ﬂavor space. Also, for brevity ⟨|...|⟩is always written as ⟨...⟩. The diagonal elements of ρpandρpare the usual occupation num- bers while the oﬀ-diagonal ones represent relative phase information. In the nondegenerate limit, up to a normalization ρpplays the role of the previously deﬁned single-particle density matrix. Therefore, the ρp’s andρp’s are well suited to account simultaneously for oscillations and collisions. In fact, one can argue that a homogeneous neutrino ensemble is completely characterized by these “matrices of densities” (Sigl and Raﬀelt 1993). It remains to derive an equation of motion which in the appropriate limits should reduce to the previous preces- sion equation, to a Boltzmann collision equation, and to Stodolsky’s damping equation (9.1), respectively. 9.2.3 Free Evolution: Flavor Oscillations The creation and annihilation operators which appear in the deﬁnition ofρpare the time-dependent coeﬃcients of a spatial Fourier expansion of the neutrino ﬁeld [notation dp≡d3p/(2π)3] Ψ(t,x) =∫ dp[ ap(t)up+b† −p(t)v−p] eip·x. (9.4) More precisely, apis an annihilation operator for negative-helicity neu- trinos of momentum pwhileb† pis a creation operator for positive- helicity antineutrinos. The Dirac spinors upandvprefer to mass- less negative-helicity particles and positive-helicity antiparticles, re- spectively; the spinor normalization is taken to be unity. For nﬂavors, apandb† pare column vectors of components ai(p) andb† i(p), respec- tively. They satisfy the anticommutation relations {ai(p),a† j(p′)}= {bi(p),b† j(p′)}=δij(2π)3δ(3)(p−p′). In the massless limit and when only left-handed (l.h.) interactions are present one may ignore the right-handed (r.h.) ﬁeld entirely. How- ever, in order to include ﬂavor mixing one needs to introduce a n×n Oscillations of Trapped Neutrinos 317 mass matrix Mwhich is nondiagonal in the interaction basis. Even in this case the “wrong” helicity states will be ignored because in the ultrarelativistic regime spin-ﬂip reactions are suppressed by an approx- imate factor ( mν/2Eν)2≪1. In this limit lepton number violating eﬀects from possible Majorana masses are also ignored. In the absence of interactions Ψ satisﬁes the free Dirac equation, im- plyingap(t) =ap(0) exp( −iΩ0 pt) andbp(t) =bp(0) exp( −iΩ0 pt), where Ω0 p≡( p2+M2)1/2(9.5) is an×nmatrix of “vacuum oscillation frequencies.” In the mass basis it has only diagonal elements which are the energies Ei= (p2+m2 i)1/2. Therefore, one may use H0=∫ dpn∑ i,j=1[ a† i(p)Ω0 ij(p)aj(p) +b† j(p)Ω0 ij(p)bi(p)] (9.6) as a free neutrino Hamiltonian. In order to ﬁnd the evolution of ρpandρpone needs to study the equations of motion of the n×noperator matrices ˆρij(p,t)≡a† j(p,t)ai(p,t) and ˆρij(p,t)≡b† i(p,t)bj(p,t).(9.7) With a Hamiltonian Htheir evolution is given by Heisenberg’s equa- tion, i∂tˆρ= [ˆρ,H], (9.8) and similar for ˆρp. WithH=H0from Eq. (9.6) one ﬁnds i∂tˆρp= [Ω0 p,ˆρp] andi∂tˆρp=−[Ω0 p,ˆρp]. (9.9) Taking expectation values on both sides yields equations of motion forρpandρp. For two ﬂavors one may use the representation Ω0 p= ω0 p+1 2Vp·andρp=1 2fp(1 +Pp·). Then Eq. (9.9) leads to the precession formulas ˙Pp=Vp×Ppand˙Pp=−Vp×Pp. 9.2.4 Interaction with a Background Medium Interactions with a medium are introduced by virtue of a general in- teraction Hamiltonian Hint(B,Ψ) which is a functional of the neutrino ﬁeld Ψ and a set Bof background ﬁelds; speciﬁc cases will be discussed in Sects. 9.3 and 9.4 below. The equation of motion for ˆ ρpis found from 318 Chapter 9 Heisenberg’s equation with H=H0+Hint. Taking an expectation value with regard to the initial state yields ˙ρp(t) =−i[ Ω0 p,ρp(t)] +i⟨[ Hint(B(t),Ψ(t)),ˆρp(t)]⟩ , (9.10) and an analogous equation for ρp(t). These equations are exact, but they are not a closed set of diﬀerential equations for the ρpandρp. To this end one needs to perform a perturbative expansion. To ﬁrst order one may set the interacting ﬁelds B(t) and Ψ(t) on the r.h.s. of Eq. (9.10) equal to the free ﬁelds50B0(t) and Ψ 0(t). Under the assumption that the original state contained no correlations between the neutrinos and the background the expectation value factorizes into a medium part and a neutrino part. With Wick’s theorem and ignoring fast-varying terms such as b†b†it can be reduced to an expression which contains only ρp’s andρp’s. The result gives the forward-scattering or refractive eﬀect of the interaction. To include nonforward collisions one needs to go to second order in the perturbation expansion. At a given time ta general operator ξ(t) =ξ(B(t),Ψ(t)) which is a functional of Band Ψ is to ﬁrst order ξ(t) =ξ0(t) +i∫t 0dt′[ H0 int(t−t′),ξ0(t)] , (9.11) whereξ0andH0 intare functionals of the freely evolving ﬁelds B0(t) and Ψ 0(t). Applying this general iteration formula to the operator ξ= [Hint(B,Ψ),ˆρp] which appears on the r.h.s. of Eq. (9.10) one arrives at ˙ρp(t) =−i[ Ω0 p,ρp(t)] +i⟨[ H0 int(t),ˆρ0 p]⟩ −∫t 0dt′⟨[ H0 int(t−t′),[ H0 int(t),ˆρ0 p]]⟩ ,(9.12) and similar for ρp(t). The second term on the r.h.s. is the ﬁrst-order refractive part associated with forward scattering. The second-order term contains both forward- as well as nonforward-scattering eﬀects. 50These free operators are the solutions of the equations of motion in the absence ofHint. However, internal interactions of the medium such as nucleon-nucleon scattering are not excluded. Moreover, Ψ(0) = Ψ 0(0) etc. are taken as initial conditions for the interacting ﬁelds. Also, the mass term is ignored in the deﬁnition of Ψ 0; its eﬀect is included only in the ﬁrst term on the r.h.s. of Eq. (9.10), the “vacuum oscillation term.” Therefore, the free creation and annihilation operators vary as a0 j(p; t) =aj(p;0)e−iptetc. for all ﬂavors with p=|p|. This implies that the operators ˆ 0(p) and ˆ0(p), which are constructed from the free a’s and b’s, are time independent. Oscillations of Trapped Neutrinos 319 Because all operators on the r.h.s. of Eq. (9.12) are free the expec- tation values in the ﬁrst- and second-order term factorize between the neutrinos and the medium. This leads one to equations for ˙ ρp(t) and ˙ρp(t) which on the r.h.s. involve only ρp(t) andρp(t) as well as ⟨ˆρ0 p⟩ and⟨ˆρ0 p⟩besides expectation values of Boperators. The interactions described by Hintare taken as individual, isolated collisions where the neutrinos go from free states to free states as in ordinary scattering theory. The duration of one collision (the inverse of a typical energy transfer) is assumed to be small relative to the time scale over which the density matrices vary substantially, i.e. small relative to the oscillation time and the inverse collision frequency. Phys- ically this amounts to the restriction that the neutrino collision rate is small enough that multiple-scattering eﬀects can be ignored. Further, it is assumed that the medium is not changed much by the interactions with the neutrino ensemble, allowing one to neglect evolution equations for the medium variables which can thus be taken to be externally pre- scribed, usually by conditions of thermal equilibrium. If the medium is not stationary it is assumed that the time scale of variation is large compared to the duration of typical neutrino-medium collisions. One may then choose the time step of iteration tin Eq. (9.12) both small relative to the evolution time scale and large relative to the duration of one collision. Under these circumstances the time integral can be extended to inﬁnity while setting ρp(t) equal toρp(0) = ⟨ˆρ0 p⟩. This leads to ˙ρp(0) =−i[ Ω0 p,ρp(0)] +i⟨[ H0 int(0),ˆρ0 p]⟩ −1 2∫+∞ −∞dt⟨[ H0 int(t),[ H0 int(0),ˆρ0 p]]⟩ ,(9.13) and similar for ρp. Here,∫∞ 0dt⟨...⟩was replaced by1 2∫+∞ −∞dt⟨...⟩. The diﬀerence between these expressions corresponds to a principle- part integral which leads to a second-order correction to the refractive term which is ignored. In the form of Eq. (9.13) the time integral leads to energy conservation in individual collisions. An explicit evaluation of the r.h.s. of Eq. (9.13) for a given interac- tion model yields the desired set of diﬀerential equations for the ρp’s andρp’s at timet= 0. It will be valid at all times if the correlations built up by neutrino collisions are “forgotten” before the next colli- sion occurs. This assumption corresponds to “molecular chaos” in the derivation of the usual Boltzmann equation. 320 Chapter 9 9.3 Neutral-Current Interactions 9.3.1 Hamiltonian In order to make Eq. (9.13) explicit one must use a speciﬁc model for the interactions between neutrinos and the medium. To this end I begin with fermions which interact by virtue of an eﬀective neutral-current (NC) Hamiltonian, HNC=GF√ 2∑ a∫ d3xBµ a(x)Ψ(x)Gaγµ(1−γ5)Ψ(x). (9.14) Here,Bµ atypically is also a bilinear of the form ψaγµψaorψaγ5γµψa with a Dirac ﬁeld ψawhich describes fermions of the medium. It is assumed that all neutrino ﬂavors scatter on a given species ain the same way apart from overall factors which are given as a hermitian n×nmatrixGaof dimensionless coupling constants. In the absence of ﬂavor-changing neutral currents it is diagonal in the weak interaction basis. As a concrete example consider the νeandνµﬂavors in a medium of ultrarelativistic electrons which may be classiﬁed into a l.h. and a r.h. “species,” a=LorR, so thatBµ L,R=1 2ψeγµ(1∓γ5)ψe. With the standard-model couplings given in Sect. 6.7.1 one ﬁnds in the weak interaction basis GL= 2 sin2θW+σ3andGR= 2 sin2θW. For the calculations it is convenient to write Eq. (9.14) in momen- tum space, HNC=GF√ 2∑ a∫ dpdp′Bµ a(p−p′)ΨpGaγµ(1−γ5)Ψp′,(9.15) whereBµ a(∆) =∫d3xBµ a(x)e−i∆·xis the Fourier transform of Bµ a(x) and Ψ p=apup+b† −pv−pin terms of the annihilation and creation operators of Eq. (9.4). A special case of NC interactions are those among the neutrinos themselves with a Hamiltonian that is quartic in Ψ. In momentum space these “self-interactions” are given by HS=GF√ 2∫ dpdp′dqdq′(2π)3δ(3)(p+q−p′−q′) ×ΨqGSγµ(1−γ5)Ψq′ΨpGSγµ(1−γ5)Ψp′.(9.16) In the standard model with three sequential neutrino families GSis the 3×3 unit matrix. For the evolution of a normal and a hypothetical sterile ﬂavor one would have GS= diag(1,0). Oscillations of Trapped Neutrinos 321 9.3.2 Neutrino Refraction As a ﬁrst simple application one may recover neutrino refraction by a medium that was previously discussed in Sect. 6.7.1. An explicit evaluation of the second term of Eq. (9.13) with Hint=HNCyields i˙ρ= [Ω0 p,ρp] +∑ ana[Ga,ρp], (9.17) wherena≡ ⟨Bµ a⟩Pµ/P0wherePis the neutrino four momentum and thusP/P 0their four velocity. In an isotropic medium the spatial parts of⟨Bµ a⟩vanish so that nais the number density of fermions a. If the medium is unpolarized, axial currents do not contribute. In the standard model with the coupling constants of Appendix B one ﬁnds for an isotropic, unpolarized medium of protons, neutrons, and electrons, i˙ρp=[ (Ω0 p+√ 2GFNℓ),ρp] , −i˙ρp=[ (Ω0 p−√ 2GFNℓ),ρp] , (9.18) whereNℓ= diag(ne,0,0) in the ﬂavor basis (electron density ne). For two ﬂavors this is equivalent to the previous precession formula. No- tably, theνandνoscillation frequencies are shifted in opposite direc- tions relative to the vacuum energies. Neutrino-neutrino interactions make an additional contribution to the refractive energy shifts, i.e. to the ﬁrst-order term in Eq. (9.13). After the relevant contractions one ﬁnds51(Sigl and Raﬀelt 1993) ΩS p=√ 2GF∫ dq{ GS(ρq−ρq)GS+GSTr[ (ρq−ρq)GS]} . (9.19) ΩS pis given by the same formula with ρqandρqinterchanged. The trace expression implies the well-known result that neutrinos in a bath of their own ﬂavor experience twice the energy shift relative to a bath of another ﬂavor. The early universe is essentially matter-antimatter symmetric so that higher-order terms to the refractive index must be included as discussed in Sect. 6.7.2; see also Sigl and Raﬀelt (1993). In stars, 51If the neutrino ensemble is not isotropic one has to include a factor (1 −cospq) under the integral, where pqis the angle between pandq. 322 Chapter 9 neutrinos are important only in young SN cores where one may ignore antineutrinos. With the total neutrino matrix of densities ρ≡∫ dpρp, (9.20) and withGS= 1 in the standard model, the neutrino contribution to the refractive energy shift is ΩS p=√ 2GFρ. The trace term was dropped because it does not contribute to the commutator in the equa- tion of motion. The diagonal entries of ρare the neutrino densities. However, in the presence of mixing and oscillations ρalso has oﬀ- diagonal elements, i.e. there are “oﬀ-diagonal refractive indices” as ﬁrst realized by Pantaleone (1992b). In a SN core the complete ﬁrst-order equation of motion for ρpis then i˙ρp=[ Ω0 p,ρp] +√ 2GF[ (Nℓ+ρ),ρp] , (9.21) which is intrinsically nonlinear. Interestingly, if one integrates both sides overdpone obtains an equation for ρwhich is linear as the neu- trino term drops out from the commutator. Therefore, even though individual modes of the neutrino ﬁeld oscillate diﬀerently in the pres- ence of other neutrinos, the instantaneous rate of change of the overall ﬂavor polarization is as if they were absent. In a SN core the refractive eﬀects are dominated by nonneutrino particles, notably by electrons. However, above the neutrino sphere the ﬂow of neutrinos itself represents a particle density exceeding that of the background medium (Sect. 11.4). Also, the medium of the early universe is dominated by neutrinos so that self-interactions and the corresponding nonlinearities of the neutrino ﬂavor oscillations must be carefully included. For recent studies of primordial neutrino oscillations see Samuel (1993), Kostoleck´ y, Pantaleone, and Samuel (1993), and references to the earlier literature given there. 9.3.3 Kinetic Terms The refractive term (ﬁrst order) of Eq. (9.13) is just a sum over diﬀerent medium components, whereas the collision term (second order) in gen- eral contains interference terms between diﬀerent target species. How- ever, if they are uncorrelated, corresponding to ⟨Bµ aBν b⟩=⟨Bµ a⟩⟨Bν b⟩ fora̸=b, these interference terms only contribute to second-order forward-scattering eﬀects which are neglected. The collision term is then an incoherent sum over all target species so that in the following one may suppress the subscript afor simplicity. Oscillations of Trapped Neutrinos 323 After a lengthy but straightforward calculation one arrives at the NC collision term (Sigl and Raﬀelt 1993) ˙ρp,coll=1 2∫ dp′[ WP′,PGρp′G(1−ρp)−WP,P′ρpG(1−ρp′)G +W−P′,P(1−ρp)G(1−ρp′)G−WP,−P′ρpGρp′G+ h.c.] , (9.22) wherePandP′are neutrino four momenta with physical (positive) en- ergiesP0=|p|andP′ 0=|p′|. The nonnegative transition probabilities WK′,K=W(K′,K) are Wick contractions of medium operators of the form W(K′,K) =1 8G2 FSµν(K′−K)Nµν(K′,K), (9.23) whereKandK′correspond to neutrino four-momenta with K0andK′ 0 positive or negative. The “medium structure function” is Sµν(∆)≡∫+∞ −∞dtei∆0t⟨Bµ(t,∆)Bν(0,−∆)⟩, (9.24) where the energy transfer ∆ 0can be both positive and negative. In the ultrarelativistic limit the neutrino tensor can be written as Nµν=1 2(UµU′ν+U′µUν−U·U′gµν−iϵµναβUαU′ β), (9.25) whereU≡K/K0andU′≡K′/K0are the neutrino four velocities. Therefore,Nµνis an even function of KandK′. Note that the deﬁni- tion Eq. (9.25) diﬀers slightly from the corresponding Eq. (4.17). The ﬁrst two terms of the collision integral Eq. (9.22) are due to neutrino scattering oﬀ the medium. The positive term represents gains from scatterings νp′→νpwhile the negative one is from losses by the inverse reaction. The third and fourth expressions account for pair processes, i.e. the creation or absorption of νpνp′by the medium. The pair terms are found by direct calculation or from the scattering ones by “crossing,” P→ −Pandρp→(1−ρp). (9.26) For example, the reaction νpX→X′νp′transforms to X→X′νpνp′ under this operation where XandX′represent medium conﬁgurations. The collision integral for ρpis found by direct calculation or by applying the crossing operation Eq. (9.26) to all neutrinos and antineu- trinos appearing in Eq. (9.22). The neutrino gain terms then transform to the antineutrino loss terms and vice versa. 324 Chapter 9 Equation (9.22) and the corresponding result for ρpwere derived by Sigl and Raﬀelt (1993) whose exposition I have closely followed. In the nondegenerate limit where (1 −ρp)→1 it agrees with a kinetic equation of Dolgov (1981) and Barbieri and Dolgov (1991). Moreover, a similar equation was derived by Rudzsky (1990) which can be shown to be equivalent to Eq. (9.22) in the appropriate limits. The relatively complicated collision term that follows from the neu- trino-neutrino Hamiltonian Eq. (9.16) has been worked out by Sigl and Raﬀelt (1993). However, in a SN core the collisions of neutrinos with each other are negligible relative to interactions with nucleons and elec- trons. In the limit of a single neutrino ﬂavor, or several unmixed ﬂavors, the role ofρpis played by the usual occupation numbers fpwhile the matrixGis unity, or the unit matrix. Then Eq. (9.22) is ˙fp,coll=∫ dp′[ WP′,Pfp′(1−fp)−WP,P′fp(1−fp′) +W−P′,P(1−fp)(1−fp′)−WP,−P′fpfp′] (9.27) which is the usual Boltzmann collision integral. The main diﬀerence to Eq. (9.22) is the appearance there of “nonabelian Pauli blocking factors” which involve noncommuting matrices of neutrino occupation numbers and coupling constants. 9.3.4 Recovering Stodolsky’s Formula The damping of neutrino oscillations becomes particularly obvious in the limit where a typical energy transfer ∆ 0in a neutrino-medium in- teraction is small relative to the neutrino energies themselves. This would be the case for “heavy” and thus nonrelativistic background fermions. Then pair processes may be ignored and neutrinos change their direction of motion in a collision, but not the magnitude of their momentum, which also implies W(P,P′)≈W(P′,P). If the neu- trino ensemble is isotropic one then has ρp=ρp′under the integral in Eq. (9.22). For the matrix structure of the collision term this leaves 2GρpG−GGρ p−ρpGG=−[G,[G,ρp]] which puts the nature of the collision term as a double commutator in evidence—see also Eq. (9.13). One may deﬁne a total scattering rate for nondegenerate neutrinos of momentum pby virtue of Γp=∫ dp′W(P′,P). (9.28) Oscillations of Trapped Neutrinos 325 In the present limit this yields a collision “integral” ˙ρp,coll=−1 2Γp[G,[G,ρp]], (9.29) where terms nonlinear in ρphave disappeared even though the neutri- nos may still be degenerate. Eq. (9.29) is more transparent in the case of two-ﬂavor mixing where one may write ρp=1 2fp(1 +Pp·) andG=1 2(g0+G·). The total occupation number fpis conserved while the polarization vector is damped according to ˙Pp,coll=−1 2ΓpG×(G×Pp). (9.30) The r.h.s. is a vector transverse to G, allowing one to write ˙Pp,coll=−1 2Γp|G|2Pp,T. (9.31) Thus one naturally recovers Stodolsky’s damping term Eq. (9.1) with D=1 2Γp|G|2. Forνeandνµand if one writes G= diag(gνe,gν) in the weak interaction basis D=1 2Γp(gνe−gν)2. This representation reﬂects that the damping of neutrino oscillations depends on the dif- ference of the scattering amplitudes: Dis the square of the amplitude diﬀerence, not the diﬀerence of the squares. If one ﬂavor does not scatter at all, Dis half the scattering rate of the active ﬂavor. Collisions thus lead to chemical equilibrium as discussed in the in- troduction to this chapter and as shown in Fig. 9.1. However, the simple exponential damping represented by Stodolsky’s formula can be reproduced only in the limit of vanishing energy transfers in collisions, an assumption which amounts to separating the neutrino momentum degrees of freedom from the ﬂavor ones. In a more general case the evo- lution is more complicated. In particular, collisions usually lead to a transient ﬂavor polarization in an originally unpolarized ensemble if the momentum degrees of freedom were out of equilibrium. Still, the neu- trinos always move toward kinetic and chemical equilibrium under the action of the collision integral Eq. (9.22) in the sense that the properly deﬁned free energy never increases (Sigl and Raﬀelt 1993). 9.3.5 Weak-Damping Limit Even for two-ﬂavor mixing the general form of the collision integral Eq. (9.22) remains rather complicated. However, for the conditions of a SN core one may apply two approximations which signiﬁcantly simplify the problem. First, one is mostly concerned with the evolution 326 Chapter 9 ofνe’s because initially they have a large chemical potential and thus are far away from chemical equilibrium with the other ﬂavors. Their high degree of degeneracy implies that νe’s may be ignored and with them all pair processes. Therefore, the evolution of νe’s mixed with one other ﬂavor (standard or sterile) is given by ˙ρp=i[ρp,Ωp] +1 2∫ dp′[ WP′PGρp′G(1−ρp) −WPP′ρpG(1−ρp′)G+ h.c.] . (9.32) The matrix of oscillation frequencies includes vacuum and ﬁrst-order medium contributions. With the momentum-dependent oscillation pe- riodtoscit is Ωp= (2π/tosc)1 2vp·with vp≡(sp,0,cp). (9.33) Here, sp≡sin 2θpandcp≡cos 2θp (9.34) with the momentum-dependent mixing angle in the medium θp. The second approximation for the conditions of a SN core is the weak-damping limit or limit of fast oscillations. It is easy to show that for the relevant physical conditions 2 π/toscis typically much faster than the scattering rate. Therefore, it is justiﬁed to consider density matrices eρpaveraged over a period of oscillation. While the ρp’s are given by four real parameters which are functions of time, the eρp’s require only two, for example the occupation numbers of the two mixed ﬂavors. It is straightforward to show that in the weak interaction basis eρp=(fe p0 0fx p) +1 2tp(fe p−fx p)(0 1 1 0) , (9.35) wheretp≡tan 2θp=sp/cp,fe pis the occupation number of νe, not of electrons, and fx prefers to a standard or sterile ﬂavor νx. One way of looking at the weak-damping limit is that between col- lisions neutrinos are best described by “propagation eigenstates,” i.e. in a basis where the eρpare diagonal. Then the matrix of coupling con- stantsGis no longer diagonal and so ﬂavor conversion is understood as the result of “ﬂavor-changing neutral currents” where “ﬂavor” refers to the propagation eigenstates. However, because in general the eﬀective mixing angle is a function of the neutrino momentum one would have to use a diﬀerent basis for each momentum, an approach that compli- cates rather than simpliﬁes the equations. Therefore, it is easiest and Oscillations of Trapped Neutrinos 327 physically most transparent to work always in the weak interaction basis. In order to derive an equation of motion for eρpone evaluates the collision term in Eq. (9.32) by inserting eρp’s under the integral. Ex- panding the result in Pauli matrices leads to an expression of the form 1 2(ap+Ap·). In general, the polarization vector Approduced by the collision term is not parallel to vp= (sp,0,cp) because the collision term couples modes with diﬀerent mixing angles. However, the assumed fast oscillations average to zero the Apcomponent perpendicular to vp. Therefore, the r.h.s. of Eq. (9.32) is of the form1 2[ap+(vp·Ap) (vp·)]. With a matrix of coupling constants in the weak basis of G=(ge0 0gx) (9.36) the collision integral Eq. (9.32) becomes explicitly ˙fx p=1 4∫ dp′{ w(νx p→νx p′)[ (4−s2 p)g2 x+ (2−s2 p)tptp′gegx] +w(νe p→νx p′)[ −s2 pg2 x−s2 ptptp′gegx] +w(νx p→νe p′)[ s2 pg2 e−(2−s2 p)tptp′gegx] +w(νe p→νe p′)[ s2 pg2 e+s2 ptptp′gegx]} , (9.37) where w(νa p→νb p′)≡WP′Pfb p′(1−fa p)−WPP′fa p(1−fb p′) (9.38) (Raﬀelt and Sigl 1993). The equation for fe pis the same if one exchanges e↔xeverywhere. In the absence of mixing sp=tp= 0, leading to the usual collision integral for each species separately. Ifνxis neitherνµnorντbut rather some hypothetical sterile species its coupling constant is gx= 0 by deﬁnition. In this case the collision integral simpliﬁes to ˙fx p=1 4s2 pg2 e∫ dp′[ WP′Pfe p′(2−fe p−fx p) −WPP′(fe p+fx p)(1−fe p′)] .(9.39) If theνestay approximately in thermal equilibrium, detailed balance yields ˙fx p=1 4s2 pg2 e∫ dp′[ WP′Pfe p′(1−fx p)−WPP′fx p(1−fe p′)] .(9.40) If in addition the mixing angle is so small that the νxfreely escape one may setfx p= 0 on the r.h.s. so that the integral expression becomes∫dp′WP′Pfe p′. 328 Chapter 9 9.3.6 Small Mixing Angle In practice the mixing angle is usually small, allowing for substantial further simpliﬁcations. In this limit the approach to ﬂavor equilibrium is much slower than that to kinetic equilibrium for each ﬂavor separately (νxis taken to be one of the active ﬂavors νµorντ). Therefore, each ﬂavor is characterized by a Fermi-Dirac distribution so that it is enough to specify the total number density nνxrather than the occupation numbers of individual modes. Integrating Eq. (9.37) over all modes, using detailed balance to lowest order in s2 p, and withtp=spone ﬁnds for the evolution of the νxnumber density ˙nνx=1 4∫ dpdp′WPP′[ (gxsp−gesp′)2fe p(1−fx p′) −(gesp−gxsp′)2fx p(1−fe p′)] ,(9.41) and a similar equation for ˙ nνe. Together with the condition of βequi- librium,µn−µp=µe−µνe, that of charge neutrality, np=ne, and the conservation of the trapped lepton number, d(ne+nνe+nνx)/dt= 0, these equations represent diﬀerential equations for the chemical poten- tialsµνx(t) andµνe(t) if the temperature is ﬁxed. 9.3.7 Flavor Conversion by Neutral Currents? Next I turn to the conceptually interesting question whether ﬂavor conversion (or the damping of neutrino oscillations) is possible by NC collisions alone. Considering only standard ﬂavors the matrix of cou- pling constants Gis then proportional to the unit matrix. In this case Stodolsky’s damping formula in the form Eq. (9.29) gives ˙ ρp,coll= 0. This formula applies in the limit when the neutrino energies do not change in collisions (a medium of “heavy” fermions). If one lifts this restriction the situation is more complicated, but it simpliﬁes again for weak damping and a small mixing angle. Then one may apply Eq. (9.41) with ge=gx= 1, ˙nνx=1 4∫ dpdp′WPP′(sp−sp′)2[ fe p(1−fx p′)−fx p(1−fe p′)] .(9.42) If in a collision |p|=|p′|and thussp=sp′one recovers the previous result ˙nνx= 0. However, if the mixing angle is a function of the neutrino momentum, NC collisions do lead to ﬂavor conversion and thus to the damping of oscillations. Of course, if only true NC interactions existed, the mixing angle in the medium would be ﬁxed at its vacuum value and so no ﬂavor Oscillations of Trapped Neutrinos 329 conversion could occur. The deviation of θfrom its vacuum value in a medium is entirely from charged-current interactions with electrons, even though they may be written in an eﬀective NC form. The coherent neutrino energy shifts by an electron background are enough to allow true NC collisions with, say, neutrons to achieve ﬂavor equilibrium! 9.4 Charged-Current Interactions 9.4.1 Hamiltonian Besides neutrino scattering or pair processes one must also include charged-current (CC) reactions where neutrinos are absorbed or pro- duced by the medium (converted into or from charged leptons) such that the total lepton number of the neutrino ensemble changes by one unit. The corresponding interaction Hamiltonian can be written in the form HCC=GF√ 2∫ d3xΥ(x)Ψ(x) + h.c., (9.43) where the neutrino ﬁeld Ψ is, again, a column vector in ﬂavor space with the entries Ψ ℓ,ℓ=e,µ,τ in the standard model. Further, Υ is a row of Dirac operators representing the medium. In the interaction basis Υ ℓ carries the lepton number corresponding to the ﬂavor ℓ. For example, in a medium of nucleons and electrons the ﬁeld Υ ecorresponding to the electron lepton number can be written for standard-model couplings as Υe=γµ(1−γ5)ψeψnγµ(CV−CAγ5)ψp, (9.44) whereψp,ψn, andψeare the proton, neutron, and electron Dirac ﬁelds, respectively, while CV= 1 andCA= 1.26 are the dimensionless CC vector and axial-vector nucleon coupling constants. 9.4.2 Kinetic Terms One may now insert HCCinto Eq. (9.13) in order to derive the explicit CC collision integral for the evolution of ρpandρp. The operators Υℓviolate the lepton number Lℓcorresponding to ﬂavor ℓ. Therefore, ⟨Υℓ⟩= 0 at all times if the medium is in an eigenstate of Lℓ(ℓ=e,µ, τ, or additional exotic ﬂavors). This assumption implies that the CC interaction Eq. (9.43) does not contribute to refractive eﬀects given by the ﬁrst-order term in Eq. (9.13). 330 Chapter 9 In the second-order term Hintappears quadratic so that one obtains expressions like ⟨ΥℓΥk⟩. However, because the medium is assumed to be in an eigenstate of Lℓthey do not contribute for ℓ̸=k. Thus, in the ﬁnal result the contributions of diﬀerent ﬂavors can be added incoherently. The rates of production Pℓ ∆and absorption Aℓ ∆of aνℓare functions of the energy-momentum transfer ∆ to the medium, Pℓ ∆=1 2G2 F∫+∞ −∞dte−i∆0t⟨ Υℓ(∆,t)γµ∆µΥℓ(∆,0)⟩ , Aℓ ∆=1 2G2 F∫+∞ −∞dte−i∆0t⟨ Tr[ γµ∆µΥℓ(∆,0)Υℓ(∆,t)]⟩ .(9.45) These expressions are deﬁned for both positive and negative energy transfer ∆ 0because Pℓ −Pplays the role of an absorption rate for an- tineutrinos with physical ( P0>0) four momentum while Aℓ −Pplays that of a production rate. Put another way, Aℓ ∆andPℓ ∆represent the rate of absorption or production of lepton number of type ℓ, indepen- dently of the sign of ∆ 0. It is useful to deﬁne a ﬂavor matrix of production rates which in the weak basis has the form P∆≡1 2 Pe ∆0 0 0Pµ ∆0 0 0 Pτ ∆ . (9.46) An analogous deﬁnition pertains to A∆. Then one ﬁnds for the CC collision integrals (Sigl and Raﬀelt 1993) ˙ρp,CC={PP,(1−ρp)} − {A P,ρp}, ˙ρp,CC={A−P,(1−ρp)} − {P −P,ρp}, (9.47) where {·,·}is an anticommutator. The kinetic term for ρpis related to that forρpby the crossing relation Eq. (9.26). The r.h.s. of Eq. (9.47) for ρpis the diﬀerence between a gain and a loss term corresponding to the production or absorption of a νp. For a single ﬂavor they take on the familiar form PP(1−fp) andAPfpwhere (1−fp) is the usual Pauli blocking factor. 9.4.3 Weak-Damping Limit The meaning of Eq. (9.47) becomes more transparent if one makes var- ious approximations which are justiﬁed for the conditions of a SN core. As discussed in Sect. 9.3.5 one may ignore the antineutrino degrees Oscillations of Trapped Neutrinos 331 of freedom, and one may use the weak-damping limit where neutrino oscillations are much faster than their rates of collision or absorption. Then one ﬁnds for two ﬂavors (Raﬀelt and Sigl 1993) ˙fx p= (1−fx p)Px P−fx pAx P +1 4s2 p[ (2−fx p−fe p)(Pe P− Px P)−(fx p+fe p)(Ae P− Ax P)] , (9.48) wherefe pandfx pare the occupation numbers for νeandνxas in Sect. 9.3.5. The corresponding equation for νeis found by exchang- inge↔xeverywhere. Forνx=νµﬂavor conversion can build up a nonvanishing muon den- sity in a SN core because they are light enough to be produced initially when the electron chemical potential is on the order of 200 −300 MeV. Forνx=ντor some sterile ﬂavor, the direct production or absorption is not possible, Ax P=Px P= 0. This simpliﬁes Eq. (9.48) considerably, ˙fx p=1 4s2 p[ (2−fx p−fe p)Pe P−(fx p+fe p)Ae P] . (9.49) If neitherνenorντare occupied because, for example, the medium is transparent to neutrinos so that they escape after production, one has ˙fx p=1 2s2 pPe Pso that the production rate of νxis that ofνetimes 1 2sin22θpas one would have expected. In a SN core where normal neutrinos are trapped Eq. (9.49) is more complicated as backreaction and Pauli blocking eﬀects must be included. It becomes simple again if s2 p≪1, because then the produc- tion ofνxcauses only a small perturbation of βequilibrium. Therefore, one may use the detailed-balance condition (1 −fe p)Pe P−fe pAe P= 0. Inserting this into Eq. (9.49) leads to ˙fx p=1 4s2 p[ (1−fx p)Pe P−fx pAe P] , (9.50) so that now the νxfollow a Boltzmann collision equation with rates of gain and loss given by those of νetimes1 4sin22θp. If theνxare sterile they escape without building up so that their production rate is 1 4sin22θpthat ofνe. 332 Chapter 9 9.5 Flavor Conversion in a SN Core 9.5.1 Rate Equation As a ﬁrst application of the formalism developed in the previous sec- tions consider two-ﬂavor mixing between νeand another active neutrino speciesνx=νµorντ(vacuum mixing angle θ0). In a SN core immedi- ately after collapse electron lepton number is trapped and so the νe’s have a large chemical potential on the order of 200 MeV. The trapped energy and lepton number diﬀuses out of the SN core and is radiated away within a few seconds. Will νxachieve equilibrium with νeon this time scale and thus share the large chemical potential? For the µﬂa- vor this would also imply the production of muons by the subsequent charged-current absorption of νµso that the lepton number would be shared between e,µ,νe, andνµ. If the mixing angle in the medium were not small, ﬂavor conversion would occur about as fast as it takes to establish βequilibrium. In this case a detailed calculation is not necessary so that one may focus on the limit of small mixing angles. In addition the oscillations are fast which allows one to use Eqs. (9.41) and (9.50). Moreover, the medium properties are assumed to be isotropic so that the production and absorption rates PeandAeofνe’s depend only on their energy E. Also, the transition rate WPP′for the scattering of a neutrino with four momentum Pto one with P′may be replaced by an angular average which depends only on the energies EandE′. Altogether one ﬁnds a rate of change for the νxdensity of ˙nνx=1 4∫ dps2 p[ (1−fx p)Pe E−fx pAe E] +1 4∑ a∫ dpdp′Wa EE′[ (ga xsp−ga esp′)2fe p(1−fx p′) −(ga esp−ga xsp′)2fx p(1−fe p′)] ,(9.51) wherefe pandfx pare the occupation numbers of νeandνxwhich are given by Fermi-Dirac distributions because kinetic equilibrium was as- sumed for both ﬂavors. Also,1 4s2 p=1 4sin22θE=θ2 Efor small mixing angles. A summation over diﬀerent species aof medium fermions was re- stored;ga eandga xare dimensionless eﬀective NC coupling constants of νeandνxto fermion species a. Fora=norpthese constants are the same for all active neutrino species. Electrons as scattering targets are very relativistic so that they may be classiﬁed into a l.h. and a Oscillations of Trapped Neutrinos 333 r.h. “species.” The scattering with νeandνxis then described by the eﬀective NC Hamiltonians HL,R=GF 2√ 2ψeγµ(1∓γ5)ψeΨGL,Rγµ(1−γ5)Ψ, (9.52) where Ψ is again a neutrino column vector in ﬂavor space. Further, GL=(2 sin2θW+ 1 0 0 2 sin2θW−1) , GR=(2 sin2θW 0 0 2 sin2θW) , (9.53) where sin2θW≈1 4will be used. Hence the eﬀective NC coupling con- stants are diﬀerent for νeandνxinteracting with l.h. electrons while they are the same for r.h. ones. TheνeFermi sea is very degenerate. With regard to the neutrino distributions one may thus use the approximation T= 0 so that neu- trino occupation numbers are 1 below their Fermi surface, and 0 above. Because the chemical potential µνeof theνepopulation exceeds µνx, and because neutrinos can only down-scatter in the T= 0 limit, the term proportional to fx p(1−fe p′) vanishes. Moreover, the detailed-balance requirement (1 −fe p)Pe E=fe pAe Eimplies Pe E= 0 forE > µ νewhere fe p= 0. Then altogether ˙nνx=∫µe µxdE(θ2 EPe EE2 2π2+∫E µxdE′∑ aWa EE′(ga xθE−ga eθE′)2E2E′2 4π4) . (9.54) Because for degenerate neutrinos nν=µ3 ν/6π2Eq. (9.54) can be written as a diﬀerential equation for µνx. According to Eq. (8.30) the mixing angle in a medium which is dominated by protons, neutrons, and electrons is given by tan 2θE=sin 2θ0 cos 2θ0−E/E ρ, (9.55) where the density-dependent “resonance energy” is Eρ≡∆m2 2√ 2GFne, (9.56) with the electron density ne. ForE=µνeone ﬁnds E Eρ=(68 keV)2 ∆m2YeY1/3 νeρ4/3 14, (9.57) whereρ14is the density in units of 1014g cm−3and as usual Yjgives the abundance of species jrelative to baryons. The approximate parameter 334 Chapter 9 range where resonance is important is shown in Fig. 9.3 as diagonal shaded band. For conditions near resonance there is no need for a detailed cal- culation because then the mixing angle is large. Therefore, one may focus on the limiting cases where ∆ m2is either so small or so large that θ2 E≪1, |θE|=θ0×{1 ifEρ/E≫1 (large ∆m2), Eρ/E ifEρ/E≪1 (small ∆m2).(9.58) Thus, for large ∆ m2theνxproduction rate is ˙nνx=θ2 0∫µe µxdE(Pe EE2 2π2+∫E µxdE′∑ aWa EE′(ga x−ga e)2E2E′2 4π4) ,(9.59) while for small ∆ m2it is ˙nνx=(θ0∆m2 2√ 2GFne)2∫µe µxdE(Pe E 2π2+∫E µxdE′∑ aWa EE′(ga xE′−ga eE)2 4π4) . (9.60) 9.5.2 Neutrino Interaction Rates In order to evaluate these integrals one ﬁrst needs the production rate Pe Eofνe’s with energy Edue to the CC reaction p+e→n+νe. The leptons are taken to be completely degenerate, the nucleons to be completely nondegenerate. Because they are also nonrelativistic the absorption of an e−produces a νeof the same energy. Therefore, Pe E=σEnpwherenpis the proton density and σE= (C2 V+3C2 A)G2 FE2/π is the CC scattering cross section for electrons of energy E. Here, CV= 1 andCA= 1.26 are the usual vector and axial-vector weak couplings. Altogether one ﬁnds Pe E=C2 V+ 3C2 A πG2 FE2np. (9.61) In practice this rate is reduced by various factors. First, Pauli blocking of nucleons cannot be neglected entirely. Second, the degeneracy of electrons is not complete. Third, the axial-vector scattering rate may be suppressed in a medium at nuclear densities (Sect. 4.6.7). For NC scattering, nucleons may be neglected entirely. In Eq. (9.59) their contribution vanishes identically because ge=gx. In Eq. (9.60) it is suppressed because they are relatively heavy so that E′≈E. Oscillations of Trapped Neutrinos 335 For l.h. electrons the coupling strengths are diﬀerent, and they are relativistic so that recoil eﬀects are not small. However, they are degen- erate so that their contribution is expected to be smaller than the ep process. It is not entirely negligible, however, especially if the nucleon contribution is partly suppressed by many-body eﬀects. The transition rates were worked out in detail by Raﬀelt and Sigl (1993) for entirely degenerate leptons. They found WR EE′≈G2 Fµ2 e 3π(E−E′)E′ E2, WL EE′≈G2 Fµ2 e 3π(E−E′)E E′2, (9.62) a result which is the lowest-order term of an expansion in powers of E/µ e(electron chemical potential µe). 9.5.3 Time Scale for Flavor Conversion Given enough time the ντ’s will reach the same number density as the νe’s. Therefore, it is most practical to discuss the approach to chemical equilibrium in terms of a time scale τ−1≡ −d dtln(nνe−nνx nνe) . (9.63) Ifτwere a constant independent of nνx/nνethe diﬀerence between the number densities would be damped exponentially. Collecting the results of the previous section one then ﬁnds easily for the case of a “large” ∆ m2of Eq. (9.59) τ−1=θ2 03G2 F 5πnpµ2 νe[ (C2 V+ 3C2 A)Fp+µνe µeFe] , (9.64) whereFpandFegive the contributions of protons (CC process) and electrons (eﬀective NC process). They are functions of µνxwhich is parametrized by η≡µνx/µνe. (9.65) One ﬁnds (Fig. 9.2, left panel) Fp= (1−η5)/(1−η3), Fe= (5 6+1 2η+1 4η2+1 12η3) (1−η)3/(1−η3), (9.66) 336 Chapter 9 Fig. 9.2. Dependence of the ﬂavor relaxation rate on =νx=νefor the case of a “large” ∆ m2Eq. (9.66) and a “small” ∆ m2Eq. (9.69). Feis the contribution of electron targets ( e→e) while Fpis from protons (ep↔ne). and numerically 3G2 F 5πnpµ2 νe= 1.0×109s−1(Ypρ 1014g cm−3)5/3(µνe µe)2 , (9.67) which sets the time scale for ﬂavor conversion. For the case of a “small” ∆ m2one ﬁnds from Eq. (9.60) and from the production and scattering rates of the previous section τ−1=θ2 0(∆m2)2 8πnp[ (C2 V+ 3C2 A)Fp+µνe µeFe] , (9.68) whereFp= 1 and Fe=9 5η−1−137 40+3 4(1 +η4) logη+5 3η+η3−101 120η4−1 5η5 8 3(1−η3)(9.69) (Fig. 9.2, right panel). Numerically, (∆m2)2 8πnp= 1.4×102s−1(1014g cm−3 Ypρ)((∆m2)1/2 1 keV)4 . (9.70) This time scale is independent of Fermi’s constant because a factor G2 F from the scattering rate cancels against G−2 Ffromθ2 E. Oscillations of Trapped Neutrinos 337 Fig. 9.3. Contour plot for log( 2 0) with in seconds according to Eqs. (9.64) and (9.68), taking Fe= 0, Fp= 1, C2 V+3C2 A= 4, and νe=e= 1. (Adapted from Raﬀelt and Sigl 1993.) From Fig. 9.2 it is clear that eﬀective NC scattering on electrons slightly accelerates the initial rate of ﬂavor conversion, but it does not dramatically aﬀect the overall time scale for achieving equilibrium. This time scale is crudely estimated by ignoring Feentirely in Eqs. (9.64) and (9.68) and by setting Fp= 1 andC2 V+ 3C2 A= 4. With µνe≈µe one then ﬁnds results for τθ2 0shown as contours in Fig. 9.3. The diag- onal band refers to the resonance condition of Eq. (9.57); there ﬂavor equilibrium would be established on a time scale nearly independent of the vacuum mixing angle. However, the detailed behavior in this range of parameters has not been determined. Armed with these results it is straightforward to determine the range of masses and mixing angles where νxwould achieve ﬂavor equilibrium and thus would eﬀectively participate in βequilibrium ep↔nν. The initially trapped lepton number escapes within a few seconds. There- fore,τ∼<1 s is adopted as a criterion for νxto have any novel impact on SN cooling or deleptonization. The relevant density is about three times nuclear while Yp≈0.35 so thatYpρ= 3×1014g cm−3is adopted. Otherwise the same parameters are used as in Fig. 9.3. Then one ﬁnds sin22θ0∼>{0.02 (keV2/∆m2)2for ∆m2∼<(100 keV)2 2×10−10for ∆m2∼>(100 keV)2(9.71) as a requirement for νxto reach chemical equilibrium. This range of parameters is shown as a hatched region in Fig. 9.4. 338 Chapter 9 Fig. 9.4. In the hatched parameter range µorτwould achieve chemical equilibrium with ein a SN core within about one second after collapse. The cosmological mass limit of about 30 eV prevents νµorντfrom playing any novel role in the cooling or deleptonization of a SN core. However, only a very small mixing angle is required to achieve equi- librium if one of the neutrinos deﬁed either standard particle physics or standard cosmology and had a mass in the keV range or above. Of course, if the mass were of Dirac type the cooling eﬀect from the pro- duction of spin-ﬂipped neutrinos would be too large to be compatible with the SN 1987A neutrino signal, yielding a bound on Dirac neutrino masses in the 10 keV range (Sect. 13.8.1). It is in this context that ﬂa- vor conversion was ﬁrst discussed by Maalampi and Peltoniemi (1991), Turner (1992), and Pantaleone (1992a). 9.6 Sterile Neutrinos and SN 1987A If a hypothetical sterile neutrino νxexisted, it would be produced in the inner core of a SN by virtue of its assumed mixing with νe. Theνx would escape directly from the inner SN core, carrying away energy and lepton number. If this process occurred too fast the observed neutrino signal of SN 1987A would have been unduly shortened, allowing one to exclude a certain range of νxmasses and mixing angles with νe (Kainulainen, Maalampi, and Peltoniemi 1991). For small mixing angles one may use Eqs. (9.40) and (9.50) as a starting point for the rate of change of the νxoccupation numbers fx p. Oscillations of Trapped Neutrinos 339 Because the νxare assumed to escape freely one may set fx p= 0 on the r.h.s. of these equations. Moreover, for the νxcoupling constants one may use gx= 0 because it is sterile. For an isotropic medium and taking an angular average of the scattering rate as in the previous section one ﬁnds ˙fx p=1 4s2 p( Pe E+∑ a(ga e)2∫ dp′Wa E′Efe p′) . (9.72) A summation over diﬀerent target species for eﬀective NC scattering was restored. Taking the νe’s to be completely degenerate one has fe p= 1 forE=|p|<µ νeandfe p= 0 otherwise so that ˙fx p=1 4s2 p( Pe E+∑ a(ga e)2∫ dE′E′2Wa E′E 2π2) . (9.73) With this result and ˙nL=−∫µe 0dE˙fx p and ˙Q=−∫µe 0dE˙fx pE (9.74) the volume loss rates for lepton number and energy can be easily de- termined. Turn ﬁrst to the case ∆ m2∼>(100 keV)2so that one may use the vacuum mixing angle in a SN core. One must now include both the CC processep→nνxas well as the NC process νeN→Nνxwhich is not suppressed because for a sterile νxthere is no destructive interference eﬀect. For nondegenerate nucleons and using C2 V+ 3C2 A≈4 for both CC and NC processes one easily ﬁnds from the results of Sect. 9.5.2 ˙nL=−1 4sin22θ02G2 F 5π3nB(Yp+1 4)µ5 νe, (9.75) and the same for ˙Qwith1 5µ5 νe→1 6µ6 νe. Electrons as NC scattering targets may be neglected because of their degeneracy. Lepton number is thus lost at a rate ˙YL=−sin22θ0τ−1(Ye+1 4)Y5/3 νe, (9.76) where τ−1=3 5(36π)1/3G2 Fn5/3 B= 7.7×1010s−1ρ5/3 15. (9.77) Here,Yp=Ye,YL≡nL/nB, andYνnB=nν=µ3 ν/6π2was used, and ρ15isρin units of 1015g cm−3. 340 Chapter 9 BecauseYνeis aboutYL/4, lepton number and energy are lost at about a rate of sin22θ01010s−1. Because the SN 1987A signal lasted for several seconds, a conﬂict with these observations is avoided if sin22θ0∼<10−10(9.78) (Kainulainen, Maalampi, and Peltoniemi 1991; Raﬀelt and Sigl 1993). Ifmνx∼>1 MeV the assumed mixing with νeallows for decays νx→ νee−e+andνx→νee−e+γ. The resulting γsignal from the SN 1987A νxﬂux (Sect. 12.4.7) does not allow for a dramatic improvement of the bound Eq. (9.78). However, the decay argument does exclude the possibility of a mixing angle so large that even the “sterile” νxwould be trapped by virtue of its mixing with νe. The case of ∆ m2∼<(100 keV)2is complicated because neutrinos in a certain energy range below their Fermi surface encounter a νe-νxmixing resonance. This implies that during the SN infall phase a large amount of lepton number can be lost. The impact on the equation of state can be strong enough to prevent a subsequent explosion. Shi and Sigl (1994) found that this argument requires sin22θ0∼<10−8keV2/∆m2for ∆m2∼>(1 keV)2. They found additional constraints from the anoma- lous contribution to the cooling by νxemission. These are all limits on the mixing of a sterile neutrino with νe, the only case that has been studied in the literature. Historically, this is related to the now forgotten episode of the 17 keV neutrino which for some time seemed to exist and which could have been a sterile neutrino mixed with νe. However, similar limits can be derived for νµ-νxorντ-νx mixing. The main diﬀerence is that the nonelectron neutrinos would not normally obtain a chemical potential so that only a thermal νµand ντpopulation can be converted. A typical temperature is 30 MeV or more, the average energy of a thermal population of relativistic fermions is about 3T∼>100 MeV, so there will be a signiﬁcant thermal muon population ( mµ= 106 MeV). Thus sterile states can be produced in charged-current muon scatterings in analogy to the above discussion ofνxproduction involving electron scattering. The resulting limit on the mixing angle may be slightly weaker than in the νe-νxcase, but it will be of the same general order of magnitude.52Forντ-νxmixing the situation is diﬀerent in that there are no thermally excited τleptons. Still, any reaction that produces ντντpairs can also produce νxandνx particles. 52This remark is relevant in the context of recent speculations about the existence of a 34 MeV sterile neutrino (Barger, Phillips, and Sarkar 1995) as an explanation of an anomaly observed in the KARMEN experiment (KARMEN Collaboration 1995). Chapter 10 Solar Neutrinos The current theoretical and experimental status of the Sun as a neu- trino source is reviewed. Particle-physics interpretations of the appar- ent deﬁcit of measured solar neutrinos are discussed, with an emphasis on an explanation in terms of neutrino oscillations. 10.1 Introduction The Sun, like other hydrogen-burning stars, liberates nuclear binding energy by the fusion reaction 4p+ 2e−→4He + 2 νe+ 26.73 MeV (10.1) which proceeds through a number of diﬀerent reaction chains and cy- cles (Fig. 10.2). With a total luminosity of L⊙= 3.85×1033erg s−1= 2.4×1039MeV s−1, the Sun produces about 1 .8×1038s−1neutrinos, or at Earth (distance 1 .50×1013cm) a ﬂux of 6 .6×1010cm−2s−1. While this is about a hundred times less than the νeﬂux near a large nu- clear power reactor it is still a measurable ﬂux which can be used for experimentation just like the ﬂux from any man-made source. The most straightforward application of the solar neutrino ﬂux is a search for radiative decays by measurements of x- and γ-rays from the quiet Sun. Because of the long decay path relative to laboratory experiments one obtains a limit which is about 9 orders of magnitude more restrictive (Sect. 12.3.1). A more exciting application is a search for neutrino oscillations. In fact, the current measurements of the solar neutrino ﬂux are nei- ther compatible with theoretical predictions nor with each other (“solar neutrino problem”); all discrepancies disappear with the assumption of 341 342 Chapter 10 neutrino oscillations. At the present time, however, this interpretation is not established “beyond reasonable doubt”—a ﬁnal verdict can be expected from the new experiments currently in preparation. They may be able to discover a characteristic distortion of the neutrino spectrum, or they may actually measure the “wrong-ﬂavored” neutrinos that were produced by oscillations from the νeoriginating in the Sun. Contrary to the νeﬂux from a power reactor, the solar neutrino spectrum arises from a small number of speciﬁc reactions (Fig. 10.1). The main contribution (91%) is from the reaction p+p→d+e++νe with a maximum neutrino energy 0 .420 MeV (“ ppneutrinos”). Second at about 7% of the ﬂux are the “beryllium neutrinos” from the electron- capture reaction e−+7Be→7Li +νewith a ﬁxed energy 0 .862 MeV. Finally, very small fraction ( ≈10−4) are the “boron neutrinos” from 8B→8Be + e++νe. Still, they are of major importance because their large energies of up to 15 MeV allow for a less diﬃcult detection procedure than is required for the soft part of the spectrum. The ﬁrst solar neutrino experiment is based on the nuclear reaction νe+37Cl→37Ar + e−. With a threshold of 0 .814 MeV it picks up both beryllium and boron neutrinos, although the argon production rate is dominated by the latter. The target consists of about 4 ×105 liters (615 tons) of perchloroethylene (C 2Cl4) in a huge tank which is located in the Homestake Mine in South Dakota (U.S.A.), about 1.5 km underground for protection against the cosmic-ray background. After an exposure of a few months to the solar neutrino ﬂux a few argon atoms have been produced (about 0.4 atoms per day). They are chemically extracted and counted by their subsequent decays (half- life 35 .0 days). When this pioneering experiment ﬁrst produced data (Davis, Har- mer, and Hoﬀman 1968) there appeared a deﬁcit relative to the theoret- ically expected ﬂux, a discrepancy which has persisted ever since—the experiment is still taking data today! On the theoretical side, the ever reﬁned predictions of Bahcall and his collaborators (e.g. Bahcall 1989) were instrumental at establishing the notion that this discrepancy—a factor of around 3—was to be taken seriously. However, in spite of the acknowledged experimental care of Davis and his collaborators, doubt has always lingered about the reliability of the data because this de- tector has never been subject to an on-oﬀ test as the Sun is the only available neutrino source powerful enough to cause a detectable signal. On the solar side, the boron neutrino ﬂux depends crucially on the reaction rate p+7Be→8B +γwith a cross section that is relatively poorly known. Solar Neutrinos 343 Fig. 10.1. Solar neutrino ﬂux at Earth according to the Bahcall and Pinsonneault (1995) solar model. Upper panel: Continuum spectra in cm−2s−1MeV−1, line spectra in cm−2s−1. Solid lines are the sources of dominating experimental signiﬁcance. Above: Range of sensitivity of cur- rent and near-future solar neutrino experiments. Lower panel: Cumulative spectrum integrated from a given energy to inﬁnity. The situation changed radically when the Kamiokande detector, originally built to search for proton decay, began in 1987 to mea- sure the solar neutrino ﬂux by virtue of the elastic scattering reaction νe+e−→e−+νewhich is detected by the Cherenkov light emitted by the kicked electron. With a threshold of about 9 MeV (later 7 MeV) it is exclusively sensitive to the boron neutrino ﬂux. Because of its direc- 344 Chapter 10 tional sensitivity it is a true “neutrino telescope” and for the ﬁrst time established that indeed neutrinos are coming from the direction of the Sun. It is perplexing, however, that the measured ﬂux is less suppressed relative to solar-model predictions than that found in the Homestake experiment. This is the reverse from what would be expected on the grounds that37Cl is sensitive to boron andberyllium neutrinos. The situation changed yet again when the experiments SAGE (So- viet-American Gallium Experiment) and GALLEX began to produce data in 1990 and 1991, respectively. They are radiochemical experi- ments using the reaction νe+71Ga→71Ge + e−which has a threshold of 233 keV. Therefore, these experiments pick up the dominant ppneu- trino ﬂux which can be calculated from solar models with a precision of a few percent unless something is radically wrong with our understand- ing of the Sun. Therefore, a substantial deﬁcit of measured ppneutrinos would have been a “smoking gun” for the occurrence of neutrino os- cillations. While the ﬁrst few exposures of SAGE seemed to indicate a low ﬂux, the good statistical signiﬁcance of the data that have since been accumulated by both experiments indicate a ﬂux which is high enough so that no ppneutrinos are reported missing, but low enough to conﬁrm the existence of a signiﬁcant problem with the high-energy part of the spectrum (beryllium and boron neutrinos). With four experiments reporting data, which represent three diﬀer- ent spectral responses to the solar neutrino ﬂux, the current attention has largely shifted from a comparison between experiments and theoret- ical ﬂux predictions to a “model-independent analysis” which is based on the small number of possible source reactions each of which produces neutrinos of a well-deﬁned spectral shape. This sort of analysis cur- rently indicates a lack of consistency among the experiments which can be brought to perfect agreement if neutrinos are assumed to oscillate. Even though the attention has currently shifted away from theo- retical solar neutrino ﬂux predictions it should be noted that in recent years there has been much progress in a quantitative theoretical treat- ment of the Sun. Independently of the interest in the Sun as a neutrino source it serves as a laboratory to test the theory of stellar structure and evolution. Particularly striking advances have been made in the ﬁeld of helioseismology. There are two basic vibration patterns for the Sun, one where gravity represents the restoring force (g-modes), and normal “sound” or pressure (p) modes. The former are evanescent in the solar convection zone (depth about 0 .3R⊙from the surface) and have never been unambiguously observed. The oscillation period of the highest-frequency g-modes would be about 1 h. Solar Neutrinos 345 There exist vast amounts of data concerning p-modes (periods be- tween 2 min and 1 h) which can be measured from the Doppler shifts of spectral lines on the solar surface. The main point is that one can establish a relationship between the multipole order of the oscillation pattern and the frequency. Because diﬀerent vibration modes probe the sound speed at diﬀerent depths one can invert the results to derive an empirical proﬁle for the sound speed in the solar interior. The agree- ment with theoretical expectations for the square of the sound speed is better than about 0.3% except in the inner 0 .2R⊙which are not probed well by p-modes (Christensen-Dalsgaard, Proﬃtt, and Thomp- son 1993; Dziembowski et al. 1994). Such results make it very diﬃcult to contemplate the possibility that the Sun is radically diﬀerent from a standard structure. On the other hand, these results are not precise enough to reduce the uncertainty of solar neutrino predictions which arise from the uncertainty of the opacity coeﬃcients. The diﬀerences between the solar neutrino predictions of diﬀerent authors are minimal when identical input parameters are used. There- fore, the expected error resulting from solar modelling is very small, except that some key input parameters remain uncertain. The dom- inant source of uncertainty for the boron neutrino ﬂux is the cross section for the reaction p+7Be→8B +γwhich appears as a multi- plicative factor for the solar ﬂux prediction and thus is unrelated to solar modelling. A more astrophysical uncertainty are the opacity coeﬃcients. Al- though they are thought to be well known for the conditions in the deep solar interior, even a relatively small error translates into a non- negligible uncertainty of the boron ﬂux prediction because of its steep temperature dependence. Crudely, a 10% error of the opacity coeﬃ- cients translates into a 1% error of the central solar temperature and then into a 20% uncertainty of the boron ﬂux. Indeed, a measure- ment of the solar neutrino ﬂux was originally envisaged as a method to measure precisely the inner temperature of the Sun. The Sun as a neutrino source naturally has received much attention in the literature because of its outstanding potential to ﬁnally conﬁrm the existence of neutrino oscillations in nature. Because this book is not primarily on solar neutrinos I will limit my discussion to what I consider the most important features of this unique neutrino source, and the role it plays for particle physics. A lot of key material can be found in Bahcall’s (1989) book on solar neutrinos, in the more recent Physics Report on the solar interior by Turck-Chi` eze et al. (1993), and on the Sun in general in the book by Stix (1989). 346 Chapter 10 Fig. 10.2. Reaction chains PPI −PPIII and CNO tri-cycle. Nuclear reactions, including decays, are marked with a bullet ( •). Average (av) and maximum (max) energies in MeV are given for the neutrinos. For photons (wavy arrows) numbers in brackets refer to the total energy of a cascade; otherwise it is the energy of a monochromatic line. Solar Neutrinos 347 10.2 Calculated Neutrino Spectrum 10.2.1 Individual Sources A hydrogen-burning star like the Sun liberates nuclear binding energy by helium fusion from hydrogen which proceeds by virtue of the pp chains and the CNO cycle (Fig. 10.2). The energy-generation rate for the CNO cycle is a much steeper function of temperature whence it dominates in hot stars. For a 2% mass fraction of CN elements, typical for population I stars like the Sun, the crossover temperature is at about 1 .8×107K (Clayton 1968). The central temperature of the Sun is about 1 .56×107K and so the CNO reactions contribute only about 1% to the total energy budget. In fact, of the CNO tri-cycle shown in Fig. 10.2, in practice only the ﬁrst loop (the CN cycle) is of importance in the Sun as the branching rates into the second or even third loop are extremely low. In the Sun, the ppchains terminate in about 85% of all cases via PPI, i.e. by the fusion of two3He nuclei. In this case the only neutrino- producing reactions are pp(pp→de+νe) and pep ( pe−p→dνe), the latter occurring very rarely. In about 15% of all cases the termina- tion is via PPII where7Be is formed from4He +3He. Because of the small energy diﬀerence between the ground states of the7Be and7Li Table 10.1. Source reactions for solar neutrinos. ReactionQ(a) [MeV]Flux at Earthb [cm−2s−1]Uncertaintyb [ % ] pp→2He+νe 0.420 c 5 .9×1010+1 −1 pe−p→2Hνe 1.442 1.4×108+1 −2 3Hep→4Hee+νe18.77 c 1 .2×103Factor 6 7Bee−→7Liνe 0.862 4.6×109+6 −7 7Bee−→7Li∗νe 0.384 5.2×108+6 −7 8B→8Be∗e+νe≈15 c 6 .6×106+14 −17 13N→13Ce+νe 1.199 c 6 .2×108+17 −20 15O→15Ne+νe 1.732 c 5 .5×108+19 −22 17F→17Oe+νe 1.740 c 6 .5×106+15 −19 Total 6.5×1010+1 −1 aMaximum eenergy for continuum (c) sources. bBahcall and Pinsonneault (1995). 348 Chapter 10 nuclei, the usual weak decay by e+νeemission is not possible and so the conversion proceeds by electron capture, leading to the emission of an almost monochromatic neutrino. In about 10% of all cases the capture reaction goes to the ﬁrst excited state (478 keV) of7Li so that there are two neutrino lines. Instead of an electron,7Li very rarely captures a proton and forms 8B which subsequently decays into8Be, a nucleus unstable against spon- taneous ﬁssion into4He +4He. This PPIII termination occurs in about 0.02% of all cases, too rare to be of importance for nuclear energy gen- eration. Its importance arises entirely from the high energy of the8B neutrinos which are the ones least diﬃcult to measure. The neutrino-producing reactions are summarized in Tab. 10.1 with their maximum energies, and with the resulting neutrino ﬂux at Earth found in the solar model of Bahcall and Pinsonneault (1995). Apart from small screening and thermal broadening eﬀects, the spectral shape for each individual source is independent of details of the solar model. The pp,13N,15O, and17F reactions are allowed or superallowed weak transitions so that their spectra are dN/dE =A(Q+me−E)[ (Q+me−E)2−m2 e]1=2E2 F , (10.2) where Qis the maximum e+kinetic energy and also the maximum νeenergy, Ais a normalization constant, and Fis a function of Ee+ which takes the e+ﬁnal-state interactions into account. For the low- Z nuclei under consideration this correction is small for most of the neu- trino spectrum. With F= 1 the normalization constants are given in Tab. 10.2. In Fig. 10.3 the normalized spectra from the ppand the15O processes are shown where the ppspectrum was taken from the tabula- tion of Bahcall and Ulrich (1988). Using Eq. (10.2) instead would cause a change so small that it would be nearly hidden by the line width of the curve in Fig. 10.3. Table 10.2. Normalization of the spectrum Eq. (10.2) with F= 1. Source Q[MeV] A[MeV−5] pp 0.420 193.9 13N 1.199 3.144 15O 1.732 0.668 Solar Neutrinos 349 Fig. 10.3. Normalized spectra of neutrino source reactions in the Sun. It is much more diﬃcult to obtain the spectrum from8B decay be- cause the ﬁnal-state8Be∗nucleus is unstable against spontaneous ﬁssion into two αparticles. Even though several states of8Be contribute to the transition, it is dominated by the 2 .9 MeV excitation and so the neutrino spectrum can be determined with relatively little ambiguity by folding Eq. (10.2) with the experimental αspectrum (Kopysov and Kuzmin 1968). A more recent and more detailed analysis was per- formed by Bahcall and Holstein (1986). An analytic approximation to their tabulated spectrum is dN/dE = 8.52×10−6(15.1−E)2:75E2 , (10.3) where the neutrino energies are in MeV. This normalized spectrum is also shown in Fig. 10.3 where, again, the diﬀerence between the tabulated values and the analytic approximation would be hidden by the line width (maximum deviation less than 0.02 in units of the vertical axis in Fig. 10.3). Neutrinos from the hep reaction extend to the highest energies of all solar sources, but their overall ﬂux is very small and very uncertain because of large uncertainties in the low-energy3Hepcross section— see Bahcall and Pinsonneault (1992) for a detailed discussion. The tabulated spectrum (Bahcall and Ulrich 1988) can be represented by dN/dE = 2.33×10−5(18.8−E)1:80E1:92 , (10.4) where the quality of the ﬁt is equally good as that for the8B neutrinos. 350 Chapter 10 In principle, any of the neutrino-producing reactions in the Sun with a ﬁnal-state positron can also occur with an initial-state electron such as pe−p→d νeinstead of pp→de+νe. In most cases, however, the electron-capture process is strongly suppressed relative to positron emission because of an unfavorable phase-space factor, or equivalently because of the relatively small electron density. (For a general com- parison between these two reaction channels see Bahcall 1990.) The only exception is pep which yields a nonnegligible contribution because theQvalue of the ppreaction and thus the available positron phase space are rather small. Another exception are the beryllium neutri- nos because positron emission is inhibited entirely by the small energy diﬀerence between the ground states of7Be and7Li. The shapes of the neutrino spectra from the individual source re- actions are determined entirely by the matrix element and phase space of the microscopic reactions apart from small broadening eﬀects by the thermal motion of the reaction participants. As typical thermal en- ergies in the Sun are 1 keV the spectral modiﬁcation by such eﬀects is entirely negligible. (For a detailed discussion of this point see Bah- call 1991.) The high-energy part of the solar neutrino spectrum between about 2 −15 MeV is dominated by the boron neutrinos (Fig. 10.1), i.e. by a single source reaction. Therefore, this energy range is a clean example to test the spectral shape. The conﬁrmation of the expected spectral shape above 7 MeV by the Kamiokande II detector excludes Fig. 10.4. Normalized spectra of the thermally and Doppler broadened beryl- lium neutrino lines where E0is the corresponding laboratory energy (Bahcall 1994). The line shapes involve an integral over a solar model. Solar Neutrinos 351 a certain range of neutrino masses and mixing angles (Sect. 10.3.4). Of course, the Superkamiokande detector with its much improved sen- sitivity and lower threshold could still detect a deviation; this would be a clear indication for neutrino oscillations or other neutrino-related novel phenomena. For the neutrino lines from e−+7Be→7Li +νe(861.8 keV) and e−+7Be→7Li∗+νe(384.3 keV) thermal broadening eﬀects are of some interest because they dominate the line shape. A detailed discussion is found in Bahcall (1994) who calculated the observable spectra at Earth shown in Fig. 10.4 where E0is the energy of the transition in the laboratory. The peak of the lines is shifted to higher energies by 0.43 and 0 .19 keV, respectively, while the average energy is shifted by an even larger amount because of the asymmetric form. Perhaps in some next-century detector this line shape could be used to measure the central temperature of the Sun. 10.2.2 Standard Solar Models In order to calculate the expected solar neutrino ﬂux at Earth one needs to construct a model of the Sun. A “standard solar model” is obtained by solving the stellar structure equations discussed in Sect. 1.2 in several time steps to evolve it to the solar age of 4 .5×109yr. At this point it must produce the observed present-day luminosity of the Sun of 1L⊙= 3.85×1033erg s−1(it is about 30% brighter than a zero-age model). This agreement is enforced by tuning the unknown presolar helium abundance Yinitial to the required value which is usually found to be about 27%. Another present-day boundary condition is the measured solar ra- dius of 1 R⊙= 6.96×1010cm which is adjusted by the mixing-length parameter which enters the standard treatment of convection. In the Sun, the outer layers (depth about 0 .3R⊙) are found to be convective; in lower-mass main-sequence stars convection reaches deeper, in higher- mass ones it disappears entirely near the surface while the central region becomes convective. The Sun calibrates the mixing-length parameter which is then used in evolutionary calculations of other stars. In order to calculate a standard solar model one needs a variety of calculated or measured input information, notably the photon opaci- ties, the equation of state, nuclear cross sections with an appropriate screening prescription, diﬀusion coeﬃcients, a prescription to treat con- vection, the abundances of metals (elements heavier than helium), and the solar age, luminosity, and radius. Because of the large number 352 Chapter 10 of details that have to be minded it is sometimes diﬃcult to compare exactly the solar models of diﬀerent workers. However, wherever a detailed comparison has been performed, diﬀerent results for such out- put quantities as predicted neutrino ﬂuxes usually can be understood in terms of diﬀerent choices regarding some of the input physics (e.g. Bahcall and Pinsonneault 1992; Turck-Chi` eze and Lopes 1993). Thus, there is very little ambiguity in solar modelling for a common choice of input physics. Nonstandard solar models are ones where standard input parame- ters have been chosen outside the range of recognized uncertainties (for example the opacities), or which include entirely new physical eﬀects such as strong magnetic ﬁelds, fast rotation in the deep interior, non- standard nuclear reaction rates (involving free quarks, for example), energy transfer by a nonstandard mechanism (e.g. by trapped massive weakly interacting particles), and others. Some such possibilities can be excluded by the helioseismologically determined sound-speed proﬁle of the Sun. I will not discuss nonstandard solar models any further— for an overview see Bahcall (1989). Naturally, it is always possible that a new “nonstandard” eﬀect is recognized which could be important for the solar structure and neutrino ﬂuxes. One standard physical eﬀect that has made its way into standard solar models only recently is the gravitational settling of helium and metals. This eﬀect leads to a stronger concentration of helium in the central region than is caused by nuclear burning alone so that the present-day Sun is “more evolved.” Consequently, the central tem- perature is slightly higher, leading to an increased ﬂux of boron neu- trinos. The settling of metals leads to an increased opacity, causing a further increase of the temperature and of the neutrino ﬂuxes by a similar amount. Solar models with gravitational settling of helium signiﬁcantly improve the already good agreement with the helioseismo- logically inferred sound-speed proﬁle (Christensen-Dalsgaard, Proﬃtt, and Thompson 1993) while the settling of metals does not seem to have a strong additional impact on the p-mode frequencies (Proﬃtt 1994). Bahcall and Pinsonneault (1992) were the ﬁrst to include helium set- tling in a standard solar model; in their 1995 paper they included metal settling as well. They ﬁnd that the compound eﬀect is to change the8B ﬂux by +36%, the7Be ﬂux by +14%, and the ppﬂux by −1.7%. Sim- ilar changes (+31%, +13%, and −1.7%) were found by Proﬃtt (1994) while Kovetz and Shaviv (1994), who included only helium settling, obtained smaller eﬀects relative to the corresponding case of Bahcall and Pinsonneault (1992). Solar Neutrinos 353 While helium and metal diﬀusion increases the neutrino ﬂuxes, the changes are roughly within the claimed errors of previous standard solar model predictions. Many improvements of input physics over the years have left the neutrino ﬂux predictions of Bahcall and his collaborators surprisingly stable over 25 years (Bahcall 1989, 1995). Bahcall (1994) has compiled the central temperature predictions from a heterogeneous set of 12 standard solar models without diﬀusion calculated by diﬀerent authors since 1988. The temperature predictions are almost uniformly distributed on the interval 15 .40−15.72×106K, i.e. these authors agree with each other on the value 15 .56×106K within ±1%. Thus, in spite of diﬀerences in detail there exists a broad consensus on what one means with a standard solar model. Therefore, the neu- trino ﬂuxes of the Bahcall and Pinsonneault (1995) model with element diﬀusion (Tab. 10.1 and Fig. 10.1) can be taken to be representative. The general agreement on a standard solar model does not guarantee, of course, that there might not exist problems related to incorrect stan- dard assumptions or incorrect input parameters common to all such models. 10.2.3 Uncertainties of Standard Neutrino Predictions a) Opacities In order to compare the neutrino ﬂux predictions with the experimental measurements one needs to develop a sense for the reliability of the calculations. Naturally, it is impossible to quantify the probability for the operation of some hitherto unknown physical eﬀect that might spoil the predictions; an error analysis can only rely on the recognized uncertainties of standard input physics. Very detailed error analyses can be found for the standard solar models of Bahcall and Pinsonneault (1992) and of Turck-Chi` eze and Lopes (1993). As for solar modelling, the dominating uncertainty arises from the radiative opacities which largely determine the temperature proﬁle of the Sun. A reduction of the Rosseland mean opacity in the central region by 10% reduces the temperature by about 1%. There is not a one-to-one correspondence between the central solar temperature Tcand the neutrino ﬂuxes because it is not possible to adjust Tcand leave all else equal. Conversely, one must modify some input parameters and evolve a self-consistent solar model. Still, if one allows all input parameters to vary according to a distribution deter- mined by their measured or assumed uncertainties one ﬁnds a strong 354 Chapter 10 correlation between Tcand the neutrino ﬂuxes which allows one to un- derstand the impact of certain modiﬁcations of a solar model on the neutrino ﬂuxes via their impact on Tc. Bahcall (1989) found Neutrino Flux ∝  T−1:2 c forpp, T8 c for7Be, T18 c for8B.(10.5) It is noteworthy that the ppﬂux decreases with increasing Tcbecause of the constraint imposed by the solar luminosity. One concludes that a 1% uncertainty in Tctranslates roughly into a 20% uncertainty of the boron ﬂux. There are two sources of uncertainty for the opacity. First, for an assumed chemical composition, the actual opacity calculation which in- volves complicated details of atomic and plasma physics. Second, there is the uncertain metal content in the central region of the Sun ( Z/X ). Because iron retains several bound electrons even for the conditions at the solar center it contributes substantially to the Rosseland mean opacity; removing iron entirely would reduce it by 25 −30%. For a recent calculation and detailed discussion of solar opacities see Iglesias and Rogers (1991a). In view of the relatively small deviations between diﬀerent opacity calculations for the relevant conditions Turck- Chi` eze and Lopes (1993) as well as Bahcall and Pinsonneault (1992) agree that the radiative opacities are likely calculated with a precision of better than a few percent.53 The actual amount of heavy elements in the Sun, notably iron, is determined by spectroscopic measurements in the photosphere, and by the abundance in meteorites which are assumed to represent the preso- lar material. The results from both methods seem to agree essentially on a common value—see Bahcall and Pinsonneault (1995) for a sum- mary of the recent status. They believe that the uncertainty in the 8B ﬂux caused by the uncertainty of Z/X is about 8%. Of course, in the central regions of the Sun the metal abundance is also determined by gravitational settling as discussed above. The uncertainty of the8B ﬂux inherent in the treatment of gravitational settling is thought to be 8% as well. 53However, most recently Tsytovich et al. (1995) have claimed that relativistic corrections to the electron free-free opacity as well as a number of other hitherto ignored eﬀects reduce the standard total opacity by as much as 5%. Solar Neutrinos 355 Bahcall and Pinsonneault (1995) then ﬁnd that the opacity-related uncertainty of the8B neutrino ﬂux is about 12%. For the7Be ﬂux the errors are thought to be roughly half as large, in agreement with Eq. (10.5). However, as stressed by Bahcall and Pinsonneault (1995), the interpretation as an eﬀective 1 σerror is misleading as the main source of uncertainty is of a systematic and theoretical nature. Previ- ously, these authors had stated “theoretical 3 σerrors;” in that sense an opacity-related uncertainty of the8B ﬂux of about 30% was found in Bahcall and Pinsonneault (1992). Turck-Chi` eze and Lopes (1993) adopted 15% without a commitment to a speciﬁc number of sigmas. None of these errors can be interpreted in a strict statistical sense. Rather, they give one an idea of what the workers in that ﬁeld consider a plausible range of possibilities. b) Beryllium-Proton Reaction A dominating uncertainty for the important ﬂux of boron neutrinos arises from the cross section7Be + p→8B +γwhich plays no role whatsoever for the energy generation in the Sun because the PPIII termination of the ppchain is extremely rare. Therefore, a modiﬁcation of this cross section has no impact on the structure of the Sun and thus no other observable consequence but to modify the high-energy neutrino ﬂux. The cross section for this reaction is parametrized for low energies in the usual form with an astrophysical S-factor σ(E) =S(E)E−1e−2(E), (10.6) with the Sommerfeld parameter η=Z1Z2e2v−1. (10.7) Here, Z1;2eare the charges of the reaction partners, vtheir relative velocity, and Etheir CM kinetic energy. The S-factor is expected to be essentially constant at low energies unless there is a resonance near threshold. The six “classical” measurements of the7Be + p→8B +γreaction are referenced in Tab. 10.3. In the Sun, the most eﬀective energy range is around E= 20 keV, far in the tail of the thermal distributions of the reaction partners, but still far below the lowest laboratory energies of around 120 keV in the experiments of Kavanagh et al. (1969) and Filippone et al. (1983). Therefore, one must extrapolate the factor 356 Chapter 10 Table 10.3. Extrapolation of S17toE= 0 for measurements of7Bep→8B according to Johnson et al. (1992). Experiment S17(0) [eV b] Kavanagh et al. (1960) 15 ±6 Parker (1966, 1968) 27 ±4 Kavanagh et al. (1969) 25 .2±2.4 Vaughn et al. (1970) 19 .4±2.8 Wiezorek et al. (1977) 41 .5±9.3 Filippone et al. (1983) 20 .2±2.3 S17down to the astrophysically interesting regime. The most recent comprehensive reanalysis was performed by Johnson et al. (1992) who found the values listed in Tab. 10.3. They tend to be smaller by around 10% relative to previous extrapolations which did not take into account that for laboratory energies there is a contribution from d-waves in the entrance channel. Johnson et al. used two diﬀerent interaction models for the extrapolation which yielded identical results within 2%. This does not necessarily imply that the theoretical extrapolation is known with this precision; for example, Riisager and Jensen (1993) suggest much smaller values for S17(0). As stressed by Johnson et al. (1992) it is problematic to combine the results of Tab. 10.3 to a “world average” because they show systematic discrepancies. In fact, the data of the two “low-energy” experiments (Kavanagh et al. 1969; Filippone et al. 1983) agree very well with each other over a wide range of energies with a systematic oﬀset by a factor 1.34. A similar oﬀset exists between the “high-energy” data of Parker (1966, 1968) and Vaughn et al. (1970) with a factor 1 .42 (Gai 1995). Because it is unknown which of the experiments is right (if any) Johnson et al. (1992) combined the data according to a prescription adopted by the Particle Data Group (1994) for such cases. It amounts to the usual weighted average, but increasing the error by a certain fac- tor derived from the statistical signiﬁcance of the discrepancies. John- son et al. then arrive at a world average S17(0) = (22 .4±2.1) eV b . (10.8) Bahcall and Pinsonneault (1992, 1995) used this value, i.e. they used a 1σuncertainty of 9%. Turck-Chi` eze and Lopes (1993) used (22 .4± 1.3stat±3.0syst) eV b, i.e. they adopted an uncertainty of ±15%. Solar Neutrinos 357 Xu et al. (1994) have discussed a unique relationship between S17(0) and the nuclear vertex constant of the overlap wave function for the virtual decay8B→7Be+p; the nuclear vertex constant can be predicted from other nuclear data. These authors’ calculation of S17(E) agrees remarkably well with the data points of Filippone et al. (1983) at low energies, and with Vaughn et al. (1970) at higher energies; these are the experiments which gave a low S17(0). Xu et al. (1994) ﬁnd an even lower value of S17(0)≈17.6 eV b. On the other hand, Brown, Cs´ ot´ o, and Sherr (1995) studied the relationship between the Coulomb displacement energy for the A= 8, J= 2+,T= 1 state and S17. They found a high value S17(20 keV) = (26.5±2.0) eV b. TheS17(0) factor was recently determined from the Coulomb dis- sociation of8B, i.e. by the Primakoﬀ-type process8B→7Be + pin the electric ﬁeld of a208Pb nucleus (Motobayashi et al. 1994). These authors ﬁnd a very low preliminary value of S17(0) = (16 .7±3.2) eV b. Langanke and Shoppa (1994) think that the true value may be signif- icantly lower still if allowance is made for a possible E2 amplitude in the Primakoﬀ reaction. However, this possibility is heavily disputed by Gai and Bertulani (1995); see also the reply by Langanke and Shoppa (1995). All of these new results are somewhat preliminary at the present time. Independently of their ultimate status it is evident that the S17 factor must be considered the weakest link in the prediction of the boron neutrino ﬂux. 10.3 Observations 10.3.1 Absorption Reactions for Radiochemical Experiments The longest-running solar neutrino experiment is the Homestake chlo- rine detector which is based on a reaction proposed by Pontecorvo (1948) and Alvarez (1949), νe+37Cl→37Ar + e (threshold 0 .814 MeV). (10.9) The two other data-producing radiochemical experiments use gallium as a target according to the reaction νe+71Ga→71Ge + e (threshold 0 .233 MeV). (10.10) Because of its low threshold, the gallium experiments can pick up the solar ppneutrino ﬂux. The absorption cross sections as a function of 358 Chapter 10 neutrino energy are shown in Fig. 10.5 according to the tabulation of Bahcall and Ulrich (1988). Fig. 10.5. Neutrino absorption cross section on37Cl and71Ga according to Bahcall and Ulrich (1988). Because the absorption cross sections are steeply increasing func- tions of energy while the predicted solar neutrino spectrum (Fig. 10.1) steeply decreases, the low ﬂux of boron neutrinos yields the dominant contribution to the expected counting rate for chlorine, and a sizeable contribution to gallium. This is illustrated in Fig. 10.6 where the pre- dicted counting rates from solar neutrinos, integrated between energy Eand inﬁnity, are shown as a function of E. These plots correspond to the lower panel of Fig. 10.1 if the diﬀerential ﬂux is weighted with the relevant absorption cross section. It is customary to express the absorption rate per nucleus in “solar neutrino units” 1 SNU = 10−36s−1, (10.11) not to be confused with 1 SNu, the supernova unit, which quantiﬁes the rate of supernova occurrences in a galaxy. Because one measures a fully integrated ﬂux one only needs the absorption cross sections folded with the spectra of the individual source reactions. For37Cl and 71Ga they are given in Tab. 10.4. Multiplying the predicted ﬂuxes with these cross sections and applying a factor 1036gives the absorption rate in SNUs. Solar Neutrinos 359 Table 10.4. Spectrally averaged absorption cross sections on37Cl and71Ga for diﬀerent solar source reactions according to Bahcall (1989), except for 8B neutrinos on37Cl which is according to Garcia et al. (1991). The unit is 10−46cm2. pp pep7Be8B13N15O 37Cl 0 16 2.4 10,900 1.7 6.8 71Ga 11.8 215 73.2 24,300 61.8 116 Fig. 10.6. Absorption rate by37Cl and71Ga of the solar neutrino ﬂux at Earth, integrated from a given neutrino energy to inﬁnity. The predicted ﬂux is according to Bahcall and Pinsonneault (1995), the absorption cross sections according to Bahcall and Ulrich (1988)—see Fig. 10.5. 360 Chapter 10 10.3.2 Chlorine Detector (Homestake) Solar neutrino observations were pioneered by the chlorine detector of Davis (1964) which is located in the Homestake Gold Mine at Lead, South Dakota (U.S.A.). The target consists of about 615 tons of per- chloroethylene (C 2Cl4), a cleaning ﬂuid, from which argon is extracted every few months by an intricate chemical procedure. Because37Ar has a half-life of 35.0 days the amount of neutrino-produced37Ar be- gins to saturate after a couple of months whence a longer exposure time is not warranted. The extracted argon—only a few atoms in a small amount of carrier gas—is then viewed by a proportional counter which registers the Auger electrons which are ejected when37Ar de- cays by electron capture. The introduction in 1970 of an electronic system which analyzes the pulse rise time greatly enhanced the sensi- tivity. Therefore, usually only the results after 1970 are quoted, be- ginning with run 18. Except for a period from May 1985 to Octo- ber 1986 where the experiment was down due to successive electrical failures of the circulation pumps, data have been taken continuously since 1967. The known backgrounds for the experiment are cosmic-ray produced 37Ar atoms which correspond to (0 .29±0.08) SNU, and an average neu- tron background corresponding to (0 .13±0.13) SNU. Because these average backgrounds as well as the backgrounds of the proportional Fig. 10.7. Distribution of the counting rates of 99 runs (18 −117). The black bars represent the data shown in Fig. 10.7, the shaded histogram the expected distribution (adapted from Lande 1995). The absorption rate in SNU is obtained by multiplying the argon production rate by 5.31. Solar Neutrinos 361 Table 10.5. Predicted absorption rate (in SNU) by37Cl for diﬀerent solar source reactions. BP92 give “theoretical 3 ” uncertainties. Diﬀusion pp pep7Be8B13N15O Total BP95 He, metals 0.0 0.2 1.2 7.2 0.1 0.4 9 .3+1:2 −1:4 BP92 He 0.0 0.2 1.2 6.2 0.1 0.3 8 .0±3.0 BP92 — 0.0 0.2 1.2 5.5 0.1 0.2 7 .2±2.7 TL93 — 0.0 0.2 1.1 4.6 0.1 0.2 6 .4±1.4 BP92 = Bahcall and Pinsonneault (1992). BP95 = Bahcall and Pinsonneault (1995). TL93 = Turck-Chi` eze and Lopes (1993). Fig. 10.8. Solar neutrino ﬂux at the Homestake experiment ( e37Cl→ 37Are) for the past quarter century. The shaded band indicates the 1 uncertainty of the global best-ﬁt average neutrino ﬂux (Lande 1995). counters must be subtracted, the counting rate attributed to solar neu- trinos in a given run is sometimes found to be formally negative. In those cases, a zero counting rate is adopted. The distribution of count- ing rates for 99 runs (18 −117) is shown in Fig. 10.7. The global average given in Eq. (10.12) is based on a maximum-likelihood analysis which includes the background. It would not be correct to average the data points with individual background subtractions to obtain a global av- erage signal. 362 Chapter 10 The recognized systematic errors are 1.5% for the extraction ef- ﬁciency, 3% for the proportional counter eﬃciency, 3% for the cos- mic-ray background, 5% for the neutron background, and 2% for the proportional counter background, which amounts to a total of 7% or 0.18 SNU. The measured counting rate for the individual runs at Homestake is shown in Fig. 10.8. The current global best-ﬁt average for the solar neutrino ﬂux measurement, derived from runs 18 −124 is (Lande 1995) (2.55±0.17stat±0.18syst) SNU = (2 .55±0.25) SNU (10.12) where the errors were combined in quadrature. The average argon production rate per day in the detector is obtained by dividing the SNUs by 5.31 so that it is found to be 0 .4837Ar/day. Eq. (10.12) is to be compared with the predictions from diﬀerent solar models shown in Tab. 10.5. 10.3.3 Gallium Detectors (SAGE and GALLEX) The gallium experiments involve far more complicated chemical extrac- tion procedures for the neutrino-produced71Ge which is not a noble gas. The Soviet-American (now Russian-American) Gallium Experi- ment (SAGE) used at ﬁrst 27 tons, later 55 tons of metallic gallium while the European GALLEX collaboration uses 100 tons of an aque- ous gallium chloride solution, corresponding to 30.3 tons of gallium. The SAGE experiment is located in the Baksan Neutrino Observatory in Mount Andyrchi, Caucasus Mountains (Russia) while GALLEX is located in the Gran Sasso tunnel near Rome (Italy). SAGE has been taking data since January 1990, GALLEX since May 1991. The cur- rent results of SAGE were published by Abdurashitov et al. (1994) and Gavrin (1995), those of GALLEX by the GALLEX Collaboration (1994, 1995b) and Kirsten (1995). The half-life of71Ge is 11 .43 days so that one needs only about a three-week exposure, allowing for frequent extractions. SAGE has already accumulated a total of 21 analyzed runs, GALLEX a total of 39. In Fig. 10.9 the individual counting rates are shown as well as the global best-ﬁt averages and the distribution of counting rates in 25 SNU bins. For the SAGE data, formally negative rates after background subtraction are forced to zero. In both cases, recognized background signals of 7 −8 SNU were subtracted. The average counting Solar Neutrinos 363 rates attributed to solar neutrinos are found to be (in SNU) SAGE: 74 + 13 /−12stat+ 5/−7syst= 74±14, GALLEX: 77±9stat+ 4/−5syst= 77±10, (10.13) where systematic and statistical errors were added in quadrature. These results are to be compared with the predictions from standard solar models shown in Tab. 10.6. Fig. 10.9. Solar neutrino ﬂux at the gallium detectors ( e71Ga→71Gee). The shaded bands indicate a 1 uncertainty of the global best-ﬁt average neutrino ﬂuxes (statistical and systematic errors added in quadrature). In the SAGE data, formally negative ﬂuxes are forced to zero. In the right panels, the bin size is 20 SNU. 364 Chapter 10 Table 10.6. Predicted absorption rate (in SNU) by71Ga for diﬀerent solar source reactions. BP92 give “theoretical 3 ” uncertainties. Diﬀusion pp pep7Be8B13N15O Total BP95 He, metals 69.7 3.0 37.7 16.1 3.8 6.3 137+8 −7 BP92 He 70.8 3.1 35.8 13.8 3.0 4.9 131 .5+21 −17 BP92 — 71.3 3.1 32.9 12.3 2.7 4.3 127+19 −16 TL93 — 71.1 3.0 30.9 10.8 2.4 3.7 122 .5±7 BP92 = Bahcall and Pinsonneault (1992). BP95 = Bahcall and Pinsonneault (1995). TL93 = Turck-Chi` eze and Lopes (1993). The GALLEX experiment was subjected to an “on-oﬀ test” by virtue of a laboratory νesource strong enough to “outshine” the Sun in its local neutrino ﬂux (GALLEX collaboration 1995a). To this end a container with activated chromium was inserted in the center of the tank containing the target ﬂuid. The relevant isotope is51Cr which decays to51V with a half-life of 27 .71 d by electron capture. The νe spectrum consists of four monoenergetic lines of energies 426 keV (9%), 431 keV (1%), 746 keV (81%), and 751 keV (9%). The dominating line is very close to the solar beryllium line. An analysis of the ﬁrst seven exposures reveals a ratio between the measured and expected counting rate of 1 .04±0.12. At the present time, four further extractions are still being analyzed. Meanwhile, the source is being reactivated at the Silo´ e nuclear reactor in Grenoble (France) in order to perform further exposures. The GALLEX collaboration has interpreted the source experiment as a global test of their detector eﬃciency. Most recently, Hata and Haxton (1995) have advocated a somewhat diﬀerent view. They argue that the gallium absorption cross section for the 746 keV line is poorly known because of excited-state contributions which have not been di- rectly measured, and which are more uncertain than had been acknowl- edged in the previous literature. According to Hata and Haxton the source experiment should not be taken as measuring, say, the GALLEX extraction eﬃciency but rather as measuring the excited-state contri- butions to the absorption cross section for beryllium neutrinos. Solar Neutrinos 365 10.3.4 Water Cherenkov Detector (Kamiokande) A water Cherenkov detector like the one located in the Kamioka metal mine (Gifu prefecture, Japan) is a large body of water, surrounded by photomultipliers which register the Cherenkov light emitted by rel- ativistic charged particles. Solar neutrinos are detected by virtue of their elastic scattering on the electrons bound in the water molecules, ν+e→e+ν. (10.14) This process has no signiﬁcant threshold, although in practice the detec- tion of the electrons is background-limited to relatively large energies. The Kamiokande detector began its measurement of solar neutrinos in January 1987 with an eﬀective analysis threshold of about 9 MeV which was reduced to about 7 MeV in mid 1988. The much larger Superkamiokande detector, which is scheduled to begin data-taking in April 1996, will have a threshold of about 5 MeV. In contrast with the radiochemical detectors which are eﬀectively based on the charged-current reaction νe+n→p+e, the elastic scattering on electrons is sensitive to all (left-handed) neutrinos and antineutrinos. The cross section is given by the well-known formula (e.g. Commins and Bucksbaum 1983) dσe dy=G2 FmeE 2π[ A+B(1−y)2−C yme E] , (10.15) where GFis the Fermi constant and the coeﬃcients A,B, and Care tabulated in Tab. 10.7. Further, y≡E−E′ E=Te Eand 0 < y <2E 2E+me, (10.16) where EandE′ are the initial- and ﬁnal-state neutrino energies while Teis the kinetic energy of the ﬁnal-state electron. Table 10.7. Coeﬃcients in Eq. (10.15) for elastic neutrino electron scattering. Flavor A B C νe (CV+CA+ 2)2(CV−CA)2(CV+ 1)2−(CA+ 1)2 νe (CV−CA)2(CV+CA+ 2)2(CV+ 1)2−(CA+ 1)2 ν; (CV+CA)2(CV−CA)2C2 V−C2 A ν; (CV−CA)2(CV+CA)2C2 V−C2 A CV=−1 2+ sin2ΘW≈ −0:04, CA=−1 2. 366 Chapter 10 ForE>5 MeV, i.e. for energies above the detection threshold of Kamiokande and Superkamiokande, the total cross section is well approximated by σe≈G2 FmeE 2π(A+1 3B) = 9.5×10−44cm2E 10 MeV×  1 for νe, 1 2:4forνe, 1 6:2forν;, 1 7:1forν;. (10.17) Of course, for a ﬂux of νe’s such as that from a supernova collapse, the dominant signal in a water Cherenkov detector is from the charged- current process νe+p→n+e+(Chapter 11). Theνeelastic scattering cross section is strongly forward peaked forE≫me. One easily ﬁnds dσ dcosθ= 4me E(1 +me/E)2cosθ [(1 + me/E)2−cos2θ]2dσ dy, (10.18) where dσ/dy was given in Eq. (10.15) with y=2 (me/E) cos2θ (1 +me/E)2−cos2θ. (10.19) Here, θis the angle between the direction of motion of the ﬁnal-state electron relative to the incident neutrino; one ﬁnds 0 ≤cosθ≤1. For E= 5 MeV and 10 MeV this cross section is shown in Fig. 10.10, normalized to unity for θ= 0. The electron keeps the direction of the incident neutrino within θ∼<(2me/E)1=2. In a water Cherenkov detector, the direction of the charged par- ticles can be reconstructed from the ring of Cherenkov light hitting the photomultipliers. Therefore, the direction of the incident neutrino is known within an uncertainty which is determined by the angular distribution of Fig. 10.10, and by the random motion of the electron due to multiple scattering in the Coulomb ﬁelds of the medium con- stituents. At an electron energy of 10 MeV, the Kamiokande detec- tor has an angular resolution of about 28◦for the direction of the electron. For this energy, the electron keeps the neutrino direction within about 18◦so that the uncertainty is dominated by the electron Coulomb scattering. Apart from these eﬀects which blur the reconstruction of the in- cident neutrino direction, a water Cherenkov detector is an imaging Solar Neutrinos 367 Fig. 10.10. Diﬀerential scattering cross section for ee→eeaccording to Eq. (10.18), normalized to unity for = 0, where is the angle between the incident neutrino and ﬁnal-state electron. device and thus a true “neutrino telescope.” Thus, for the ﬁrst time the Kamiokande detector actually proved that neutrinos are coming from the direction of the Sun. The ﬁrst main publication of these results was by Hirata et al. (1991) from a data sample of 1040 live de- tector days, taken between 1987 and 1990; an update including data until July 1993 (a total of 1670 live detector days) was given by Suzuki (1995). The angular distribution of the registered electrons relative to the direction of the Sun is shown in Fig. 10.11. Even though there remains an isotropic background, probably from radioactive impurities in the water, the solar neutrino signal beautifully shows up in these measurements. In elastic neutrino-electron scattering, the energy distribution of the kicked electrons is nearly ﬂat so that the spectral shape of the incident neutrino ﬂux is only indirectly represented by the measured electron spectrum. In Fig. 10.12 the electron recoil spectrum is shown for an incident spectrum of8B neutrinos. A water Cherenkov detector can resolve the energy of the charged particles from the intensity of the measured light which for low-energy electrons is roughly proportional to the energy. Therefore, the electron recoil spectrum from the interac- tion with solar neutrinos can be resolved at Kamiokande. The measured shape relative to the theoretically expected one is shown in Fig. 10.13, arbitrarily normalized at E= 9.5 MeV (Suzuki 1995). Within statis- tical ﬂuctuations the agreement is perfect. 368 Chapter 10 Fig. 10.11. Angular distribution of the solar neutrino events at Kamiokande for 1670 live detector days (January 1987 −July 1993), according to Suzuki (1995). There remains an isotropic background, probably from radioactive impurities in the water. Fig. 10.12. Normalized spectra of8B neutrinos and of recoil electrons (kinetic energy Te) from the elastic scattering process ee→ee. Even though the Sun is thought to produce only νe’s one may spec- ulate that some of them are converted into νe’s on their way to Earth by spin-ﬂavor oscillations in magnetic ﬁelds (Sect. 8.4) or by matter- induced majoron decays (Sect. 6.8). The νe’s would cause a signal by the reaction νep→ne+with an isotropic angular distribution. The remaining measured isotropic background shown in Fig. 10.11 can thus Solar Neutrinos 369 Fig. 10.13. Measured over expected spectrum of recoil electrons from solar neutrinos at Kamiokande, arbitrarily normalized to unity at E= 9:5 MeV (Suzuki 1995). Fig. 10.14. Limit on a solar eﬂux relative to8B solar e’s from the isotropic background shown in Fig. 10.11. The data used for this result extend until August 1992 (Suzuki 1993). be used to constrain the νeﬂux. At diﬀerent energies a 90% CL up- per limit on the solar νeﬂux relative to8B solar neutrinos is shown in Fig. 10.14 according to Suzuki (1993). The best relative limit is atE= 13 MeV where the νeﬂux is less than 5.8% of8B solar νe’s at 90% CL. If the cause of the isotropic background could be reliably identiﬁed these upper limits could be improved accordingly. 370 Chapter 10 Table 10.8. Predicted ﬂux of8B neutrinos at Earth. For BP92, the uncer- tainty is a “theoretical 3 error.” Model Diﬀusion8B Flux [106cm−2s−1] BP95 He, metals 6.6+0:9 −1:1 BP92 He 5.7±2.4 BP92 — 5.1±2.2 TL93 — 4.4±1.1 BP95 = Bahcall and Pinsonneault (1995). BP92 = Bahcall and Pinsonneault (1992). TL93 = Turck-Chi` eze and Lopes (1993). Fig. 10.15. Measured8B solar neutrino ﬂux at Kamiokande (Suzuki 1993). The shaded band gives the average with 1 statistical and systematic errors added in quadrature. Next, one may determine the absolute ﬂux of solar8B neutrinos. As a function of time it is shown in Fig. 10.15. Assuming that the spectrum is indeed that of8B neutrinos, the measured ﬂux from the data of January 1987 until July 1993 is (units 106cm−2s−1) 2.89 + 0 .22/−0.21stat±0.35syst= 2.89±0.41 (10.20) where the errors were added in quadrature. This measurement is to be compared with the predictions shown in Tab. 10.8 Solar Neutrinos 371Table 10.9. Current and near-future solar neutrino experiments. Experiment OperationaDetection Thresh. Solar MeasurementcPrediction from until [MeV] ﬂux BP95 TL93 Homestake 1967 (?) Radiochemical 0.814 not pp2.55±0.25 9 .3+1:2 −1:46.4±1.4 νe37Cl→37Are SNU SAGE 1990 (?) Radiochemical 0.233 all 74 ±14 SNU 137+8 −7123±7 νe71Ga→71Gee 77±10 SNU ... ... GALLEX 1991 (?) ... ... ... Kamiokande II 1987 (1996) H 2O Cherenkov 78B 2 .89±0.41 6 .6+0:9 −1:14.4±1.1 νxe→e νx(b) ×106cm−2s−1 Superkamiok. (1996) (?) ... 5 ... — ... ... SNO (1996) (?) D 2O Cherenkov8B — ... ... νed→p p e (1.44) νxe→e νx(b) 5 νxd→p n ν x(b) 2.225 BOREXINO (1997) (?) Scintillator 0.257Be — 5.2+0:3 −0:4— νxe→e νx(b) ×109cm−2s−1 aIn brackets anticipated.bx=e,, or.cStatistical and systematic 1 errors added in quadrature. BP95 = Bahcall and Pinsonneault (1995), with helium and metal diﬀusion. TL93 = Turck-Chi` eze and Lopes (1993), no diﬀusion. 372 Chapter 10 10.3.5 Summary The main features and results of the solar neutrino experiments dis- cussed in this section, and of near-future experiments to be discussed in Sect. 10.9, are summarized in Tab. 10.9. More technical aspects can be found in the original papers quoted in this section and in previous papers by the referenced authors. Overviews of many experimental as- pects can be found in Bahcall (1989), Davis, Mann, and Wolfenstein (1989), and Koshiba (1992). The two theoretical predictions shown are somewhat extreme in that TL93 yields lowish neutrino ﬂuxes when compared with BP92 (no diﬀusion), while BP95 with the inclusion of helium and metal diﬀusion is presently at the upper end of what is being predicted on the basis of standard solar models. It is clear, of course, that including diﬀusion would also increase the TL93 ﬂuxes. 10.4 Time Variations 10.4.1 Day-Night Eﬀect It is commonly assumed that the Sun is in a stationary state so that the solar neutrino ﬂux should be constant in time. One may still ex- pect certain temporal variations of the measured ﬂux. Between day and night the line of sight between the Kamiokande detector and the Sun intersects with diﬀerent parts of the Earth. If neutrinos oscil- late, for certain masses and mixing angles the Earth’s matter would alter the oscillation pattern such that the counting rate at Kamiokande would be expected to vary between day and night. The day rate is found to be 0 .90±0.10stat±0.12systtimes the average, the night rate 1.04±0.10stat±0.12syst, i.e. there is no signiﬁcant diﬀerence (Suzuki 1995). The radiochemical detectors with their long exposure times can- not resolve a possible day-night diﬀerence. 10.4.2 Seasonal Variation Because of the ellipticity of the Earth’s orbit the distance to the Sun varies during the year from a minimum of 1 .471×1013cm in January to a maximum of 1 .521×1013cm in July, i.e. it varies by ±1.67% from its average during the year. Therefore, the solar neutrino ﬂux varies by±3.3% from average between January and July. This eﬀect is too small to be observed by any of the present-day experiments. This variation can be ampliﬁed if neutrinos oscillate. Notably, if the vacuum oscillation length of the monochromatic7Be neutrinos is Solar Neutrinos 373 of order the annual distance variation, the7Be ﬂux measured in the chlorine and gallium detectors could vary between zero and its predicted full rate. Moreover, at diﬀerent times of the year the neutrinos have to traverse on average a diﬀerent amount of terrestrial matter which could aﬀect neutrino oscillations and thus lead to an annual variation. There is no evidence for such an eﬀect in any of the detectors. GALLEX reports a counting rate of (82 ±15stat) SNU for October − March and (78 ±14stat) SNU for April −September, i.e. there is no sig- niﬁcant diﬀerence (GALLEX collaboration 1994). A semiannual variation could be caused by nonstandard neutrino in- teractions with the solar magnetic ﬁeld (Sect. 10.7). The solar equato- rial plane is at an angle of 7◦15′relative to the Earth’s orbital plane (the ecliptic). Around 7 June and 8 December, the solar core is viewed from Earth through the solar equator where the magnetic ﬁeld is thought to be weaker than at higher latitudes. The Kamiokande II data (1040 live detector days in 1987 −1990) were subdivided into three-months periods which include (rate Γ I) or exclude (Γ II) the intersection points, respectively. The relative diﬀerence in counting rate was found to be (ΓI−ΓII)/(ΓI+ Γ II) =−0.06±0.11stat±0.02syst, i.e. there was no in- dication for a time variation (Hirata et al. 1991). The Homestake data also do not show any evidence for a semiannual variation. 10.4.3 Correlation with Solar Cycle at Homestake The Sun shows a prominent magnetic activity cycle which is thought to be due to dynamo action within the convective surface layers, driven by the nonuniform rotation of the Sun (Stix 1989). One of the best- known manifestations of this activity is the cycle of sunspots, measured by their total number appearing on the solar disk.54A given solar cy- cle lasts for about 11 years from one minimum of sunspot number to the following; the ﬁrst recorded cycle begins with the minimum around A.D. 1755. Sunspots are caused by magnetic ﬂux tubes which break through the surface; they are thought to be manifestations of a sub- surface toroidal magnetic ﬁeld with opposite directions between the southern and northern hemisphere. In addition, the Sun has a poloidal (dipole) ﬁeld. The ﬁelds reverse polarity after 11 years so that the full magnetic cycle lasts 22 years. 54In the following, the “number of sunspots” refers to the international sunspot index according to the Z¨ urich system where both spots and spot groups are counted and averaged from the reports of many solar observatories. The sunspot index is regularly published in Solar Geophysical Data . 374 Chapter 10 The solar neutrino measurements of the chlorine experiment seem to anticorrelate with solar activity as ﬁrst suggested by Subramanian (1979) and Bazilevskaya, Stozhkov, and Charakhch’yan (1982). These latter authors also noted a correlation with the primary cosmic proton ﬂux which, however, is known to correlate with the solar magnetic cycle because the solar magnetic ﬁeld aﬀects the ﬂux of charged cosmic rays hitting the Earth. The cosmic-ray ﬂux is not expected to aﬀect the chlorine experiment directly; the average cosmic-ray induced37Ar production rate is about 0 .055 day−1, compared with an average solar neutrino signal of 0 .48 day−1. Figure 10.16 shows the temporal variation from 1970 −1989 of the Homestake argon production rate, the integral number of sunspots, and the neutron counts at the McMurdo station in Antarctica. The latter are indicative of the primary cosmic-ray proton ﬂux with energies above 0 .4 GeV. While a certain degree of anticorrelation between the number of sunspots and the argon production rate is plainly visible in Fig. 10.16, there are two important questions: How signiﬁcant is the eﬀect? If it is signiﬁcant, what are its physical origins? As for the latter question, the only plausible explanation55that has been put forth over the years is the hypothesis that neutrinos possess a small magnetic moment which allows them to spin-precess into sterile right-handed states in the presence of magnetic ﬁelds (Voloshin and Vysotski˘ ı 1986; Voloshin, Vysotski˘ ı, and Okun 1986a,b), and variations of this scheme which allow for simultaneous spin and ﬂavor oscillations (Sect. 10.7). In this case one would expect some degree of a semiannual variation as explained above, but none is found in the Homestake data. The issue of statistical signiﬁcance was addressed on the basis of the 1970−1989 data by Filippone and Vogel (1990), Bieber et al. (1990), and Bahcall and Press (1991). These latter authors argued for rank- ordering as a statistical method because of the likely nonlinear relation- ship (if any) with indicators of solar activity, and because of the possibly non-Gaussian nature of the experimental errors. They produced pairs of an observed argon production rate and the corresponding sunspot index. Next, the Homestake data and sunspot indices were replaced by the ordinal rank in their respective data sets, i.e. the smallest value has rank 1, the second-largest rank 2, and so forth. In order to discover correlations they applied two standard rank statistical tests, the Spear- 55The refractive term in the MSW conversion probability from convective currents could play a role and could be coupled to solar activity (Haxton and Zhang 1991). However, extreme conditions are required to explain the observed eﬀect. Solar Neutrinos 375 Fig. 10.16.37Ar production rate at Homestake, integral number of sunspots, and neutron counts at McMurdo (Antarctica) which are indicative of the primary cosmic-ray proton ﬂux above 0 :4 GeV; annual bins each. In the center panel, the number of the solar cycle is also indicated. (Adapted from Bieber et al. 1990.) man rank-order correlation coeﬃcient and Kendall’s tau (e.g. Press et al. 1986). They found a moderate statistical signiﬁcance level of 1.3% (Spearman) and 0.9% (Kendall) for the argon-production rate to be correlated with the sunspot index.56 Most recently, a correlation between the 1970 −1991 argon produc- tion rate and the solar activity cycle was investigated by Oakley et al. (1994). They used the Mt. Wilson 150-ft. tower magnetograms as an indicator for the solar magnetic surface ﬂux, both for the full solar disk 56The signiﬁcance level gives approximately the probability that a random shuf- ﬂing of the data gives an equally good or better (anti)correlation. A signiﬁcance level smaller than 1% is considered “highly signiﬁcant.” 376 Chapter 10 and for the central 14◦×14◦which is the region most signiﬁcant for the neutrino ﬂight path to us. The disk-centered magnetic-ﬁeld cycle lags full-disk indicators by about 1 y. Oakley et al. (1994) found a signiﬁ- cance level of 0.001% for an anticorrelation between the Homestake rate and the disk-centered magnetic ﬂux. Including the ﬂux from increas- ingly higher latitudes reduced the signiﬁcance level of the correlation. With only high-latitude magnetic information the correlation was lost. Another test of time variation is a comparison as in Fig. 10.7 be- tween the expected and measured distribution of argon production rates. A time variation would broaden the measured distribution (black histogram) relative to the expectation for a constant neutrino ﬂux (shaded histogram). A certain degree of broadening is certainly com- patible with Fig. 10.7, but a precise statistical analysis does not seem to be available at the present time. At any rate, the rank-ordering re- sult does not specify the amplitude of a time varying signal relative to a constant base rate while the width of the distribution in Fig. 10.7 would be sensitive mostly to this amplitude. Therefore, the two methods yield rather diﬀerent information concerning a possible time variation. The gallium experiments have not been running long enough to say much about a time variation on the time scale of several years. Also, there does not seem to be a signiﬁcant time variation in the Kamiokande data between 1987 and 1992 which covers a large fraction of the cur- rent solar cycle No. 22. Notably, there is no apparent (anti)correlation with sunspot number (Suzuki 1993). However, a correlation with disk- centered magnetic indicators may yield a diﬀerent result—according to the analysis of Oakley et al. (1994) the use of a disk-centered indicator with its inherent time-lag relative to full-disk spot counts is crucial for the strong anticorrelation at Homestake. The constancy of the Kamiokande rate, if conﬁrmed over a more extended period, severely limits a conjecture that the nuclear energy generating region in the Sun was not constant (e.g. Raychaudhuri 1971, 1986). Because the Kamiokande detector is mostly sensitive to the boron neutrinos with their extreme sensitivity to temperature, they would be expected to show the most extreme time variations if the solar cycle was caused by processes in the deep interior rather than by dynamo action in the convective surface layers. 10.4.4 Summary There is no indication for a day-night variation of the solar neutrino ﬂux at Kamiokande and none for a seasonal variation at Kamiokande, Solar Neutrinos 377 GALLEX, or Homestake. The Homestake argon production rate ap- pears to be strongly anticorrelated with solar disk-centered indicators of magnetic activity, with no evidence for a long-term variation at Kamiokande. However, the period covered there (1987 −1992) may be too short, and the Kamiokande rate has not been correlated with disk- centered indicators. No homogeneous analysis of both Kamiokande and Homestake data relative to solar activity exists at the present time. It is very diﬃcult to judge what the apparent Homestake anticorre- lation with solar activity means. Is there an unrecognized background which correlates with solar activity such as cosmic rays? Is there a long-term drift in some aspect of the experiment which then naturally correlates with other causally unrelated long-term varying phenomena such as solar activity? Is it all some statistical ﬂuke? Or is it an indica- tion of nonstandard neutrino properties, notably magnetic dipole mo- ments which spin precess in the solar magnetic ﬁeld into sterile states? This latter possibility will be elaborated further in Sect. 10.7. 10.5 Neutrino Flux Deﬁcits 10.5.1 Boron Flux Even if one ignores for the moment a possible time variation of the Homestake neutrino measurements there remain several “solar neutrino problems.” All of the solar neutrino ﬂux measurements discussed in Sect. 10.3 exhibit a deﬁcit relative to theoretical predictions. The sim- plest case to interpret is that of Kamiokande because it is sensitive only to the boron ﬂux. In this case the most uncertain input parameter is the cross section for the reaction p7Be→8Bγwhich enters the ﬂux pre- diction as a multiplicative factor, independently of other details of solar modelling. As discussed in Sect. 10.2.3, the astrophysical S-factor for this reaction depends on theoretical extrapolations to low energies of experimental data which themselves seem to exhibit relatively large sys- tematic uncertainties. Therefore, one may turn the argument around and consider the solar neutrino ﬂux measurement at Kamiokande as another determination of S17(0). According to Eq. (10.20) the measured ﬂux is 2 .89×(1±0.14) in units of 106cm−2s−1while the prediction is 4 .4×(1±0.21)×S17(0)/22.4 for TL93 (Turck-Chi` eze and Lopes 1993) where S17is understood in units of eV b. For the BP95 (Bahcall and Pinsonneault 1995) it is 6.6×(1±0.15)×S17(0)/22.4 where I have used an average symmetric 378 Chapter 10 error for simplicity. This yields S17(0)/eV b ={14.7±3.7 TL93, 9.8±2.0 BP95,(10.21) to be compared with the world average 22 .4±2.1 (Sect. 10.2.3). While the discrepancy is severe, one would still be hard-pressed to conclude with a reasonable degree of certainty that the solar neutrino problem is not just a nuclear physics problem. 10.5.2 Beryllium Flux The Kamiokande solar neutrino problem may be explained by a low S17(0) factor, perhaps in conspiration with lower-than-standard opaci- ties that cause the central solar temperature to be lower than expected. In this case the spectral shape would remain unchanged, leading to an unambiguous prediction for the neutrino signal that must be caused by 8B neutrinos in the chlorine and gallium detectors. This contribution may be subtracted from the Homestake and SAGE/GALLEX data, re- spectively, to obtain a measurement of the remaining neutrino sources (Tab. 10.10)—see Kwong and Rosen (1994). Moreover, the predicted pp and pep signals in these detectors do not seem to involve any signiﬁcant uncertainties so that they may be subtracted as well, eﬀectively leading to a measurement of the ﬂux of7Be and CNO-neutrinos (Tab. 10.10). These ﬂuxes are then found to be formally negative; they are consistent with zero within the experimental measurement errors. These remaining ﬂuxes inferred from the chlorine and gallium ex- periments are to be compared with the predictions for the7Be and CNO neutrinos. In Tab. 10.10 the predictions of BP95 are shown; those of TL93 and other authors are similar. Bahcall (1994b) has compiled the predictions for the7Be ﬂux from a heterogeneous set of 10 solar models by diﬀerent authors whose predictions agree to within ±10%—there is a broad consensus on the standard value of this ﬂux. Therefore, the errors of the predictions are much smaller than the experimental ones, leaving one with a very signiﬁcant discrepancy between the predicted and observed ﬂux both at Homestake and at SAGE/GALLEX, even if one were to ignore the CNO contributions entirely. In the analysis of Hata and Haxton (1995) which uses the GALLEX source experiment as a constraint on the gallium absorption cross sec- tion, it is found that at the 99% CL the measured7Be ﬂux is less than 0.3 of the BP95 prediction. This high signiﬁcance of the beryl- lium problem relies on the assumption that all solar neutrino experi- Solar Neutrinos 379 Table 10.10. Measured vs. predicted7Be and CNO neutrino ﬂuxes (in SNU). Homestake GALLEX/ (37Cl) SAGE (71Ga) Measurements: Total 2.55±0.25 77±10 8B (inferred from 3 .15±0.44 7±1 Kamiokande) pp + pep 0.20±0.01 74±1 (calculated) Remainder −0.8±0.5 −4±10 BP95 prediction:a 7Be 1.24 37.7 CNO 0.48 10.1 Total 1.72 47.8 aBahcall and Pinsonneault (1995). ments are correct within the acknowledged uncertainties. If one ignored Homestake the 99% CL limit would be at 0.5 of the BP95 prediction. Therefore, the “new solar neutrino problem” consists in the mea- sured absence of the beryllium neutrino ﬂux. This is a far more serious deﬁcit than that of the boron ﬂux which has been measured to be a signiﬁcant fraction of its expected value, and for which the prediction is very uncertain anyway. 10.5.3 An Astrophysical Solution? The beryllium problem is very signiﬁcant because its predicted ﬂux is closely related to that of the boron ﬂux. A glance at the solar reaction chains (Fig. 10.2) reveals that both ﬂuxes start with7Be, 7Be + e→7Li +νe, 7Be + p→8B→8Be + e++νe. (10.22) Therefore, the ﬂux ratio of7Be/8B neutrinos essentially measures the branching ratio between the electron and proton capture on7Be, ad- mittedly averaged over slightly diﬀerent regions of the solar core. The 380 Chapter 10 boron and beryllium ﬂuxes respond very diﬀerently to a modiﬁcation of the solar central temperature (Eq. 10.5). Because the boron ﬂux has been measured, even at a reduced strength, it is not possible to explain the missing beryllium neutrinos by a low solar temperature un- less one is willing to contemplate simultaneously an extremely reduced temperature and an extremely enhanced S17factor. Many authors57have recently investigated the question of the sig- niﬁcance of the solar neutrino problem, and if it can be solved by a plausible, or even implausible, combination of erroneous nuclear cross sections, solar opacities, and so forth. The consensus is that an as- trophysical solution is not possible, even if one ignores part of the ex- perimental data. There is no obvious uncertain input quantity into solar modelling or the ﬂux predictions that could be tuned to obtain the measured ﬂuxes. Therefore, something mysterious is wrong with the calculation of the solar neutrino ﬂuxes and/or the solar neutrino experiments. However, if our understanding of the solar neutrino source and of the detection experiments is not totally wrong, an attractive alternative is to contemplate the option that something happens to the neutrinos as they propagate from the solar core to us. 10.6 Neutrino Oscillations 10.6.1 Which Data to Use? The idea to test the hypothesis of neutrino oscillations by means of the solar neutrino ﬂux goes back to Pontecorvo (1957, 1958, 1967) and Maki, Nakagawa, and Sakata (1962). After the ﬁrst measurements in 1968 it was revived by Gribov and Pontecorvo (1969) as well as Bahcall and Frautschi (1969). If some of the νe’s produced in the Sun transformed into νe’s, into other sequential neutrinos ( ν,ν), or into hypothetical sterile states νs, the measurable ﬂux at Earth would be depleted. Even the Kamiokande detector which responds to νandν because of the possibility of neutral-current ν-escattering would show a reduced ﬂux because of the smaller cross section. The measured ﬂux 57These include Bludman, Kennedy, and Langacker (1992a,b); Bludman et al. (1993); Bahcall and Bethe (1993); Castellani, Degl’Innocenti, and Fiorentini (1993); Castellani et al. (1994b); Hata, Bludman, and Langacker (1994); Shi and Schramm (1994); Shi, Schramm, and Dearborn (1994); Kwong and Rosen (1994); Bahcall (1994b); Berezinski (1994); Degl’Innocenti, Fiorentini, and Lissia (1995); Parke (1995); Hata and Haxton (1995); Haxton (1995). Solar Neutrinos 381 deﬁcits in all detectors may give us the ﬁrst indication for neutrino oscillations and thus for nonvanishing neutrino masses and mixings. Because of the statistically high signiﬁcance of the anticorrelations with solar activity of the Homestake results it is not entirely obvious how to proceed with a quantitative test of the oscillation hypothesis which can cause only a day-night or semiannual time variation. In the present section a possible long-term variability of the solar neutrino ﬂux or its detection methods is ignored, i.e. the apparent anticorrela- tion with solar activity at Homestake is considered to be a statistical ﬂuctuation. In Sect. 10.7 a possible explanation in terms of neutrino interactions with the solar magnetic ﬁeld is considered. One may take the opposite point of view that the variability at Homestake is a real eﬀect, and that it is not related to neutrino mag- netic moments because that hypothesis requires fairly extreme values for the dipole moments and solar magnetic ﬁelds (Sect. 10.7). An in- terpretation of the data in terms of neutrino oscillations then becomes diﬃcult because one admits from the start that unknown physical ef- fects are either operating in the Sun, in the intervening space, or in the detectors. One could argue perhaps that the apparent variability of the Homestake data in itself was evidence that something was wrong with this experiment. In this case one could still test the hypothesis of neutrino oscillations under the assumption that the signal recorded at Homestake was spurious in which case one must discard the entire data set, not only parts of it as has sometimes been done. Ignoring the Homestake data does not solve the solar neutrino problem, but its signiﬁcance is reduced. Still, as no one has put forth a plausible hypothesis for a speciﬁc problem with the Homestake experiment it is arbitrary to discard the data. Admittedly, it is also arbitrary to consider the time variation spurious even though standard statistical methods seem to reveal a highly signiﬁcant anticorrelation with solar activity. 10.6.2 Vacuum Oscillations A formal treatment of vacuum neutrino oscillations was presented in Sect. 8.3. If the observer is many oscillation lengths away from the source, and if the neutrinos are produced with a broad spectrum of en- ergies from an extended source, one will observe an incoherent mixture of the mass eigenstates. If one considers two-ﬂavor oscillations νe↔νa with νa=ν,ν, or some hypothetical sterile ﬂavor, the νeﬂux is re- duced by a factor 1 −1 2sin22θ(vacuum mixing angle θ). Therefore, 382 Chapter 10 one needs a large mixing angle to achieve a substantial suppression of the measurable ﬂux. If one includes the possibility of three-ﬂavor os- cillations one can achieve, in principle, a maximum νereduction by a factor1 3. In this picture the νeﬂux is reduced by the same factor for all ener- gies, contrary to what is indicated by the observations. Therefore, one needs an oscillation length of order the Earth-Sun distance (astronom- ical unit 1 AU = 1 .496×10−13cm) because in this case the oscillation pattern is not completely smeared out. Then one may measure a diﬀer- ent ﬂux suppression for diﬀerent energies; the wiggles in the reduction factor as a function of neutrino energy can be resolved (Fig. 10.17). For two-ﬂavor oscillations the relationship Eq. (8.15) between neu- trino energy E, the mass-square diﬀerence ∆ m2 , and the oscillation length ℓosccan be written as ∆m2 = 1.66×10−10eV21 AU ℓoscE 10 MeV. (10.23) Therefore, the neutrino masses are very small if these “long-wavelength oscillations” are the explanation for the observed ﬂux deﬁcits. Fig. 10.17. Survival probability of solar e’s for ∆ m2 = 0:8×10−10eV2 and sin22= 0:8, parameters which reconcile all measurements with the standard solar ﬂux predictions. The solid line is for the minimum annual distance to the Sun of 1 :471×1013cm (January), the dashed line for the maximum distance 1 :521×1013cm (July). The right panel is an enlarged section around the line of beryllium neutrinos. Between January and July, their ﬂux is suppressed by very diﬀerent amounts. Solar Neutrinos 383 Theνesurvival probability as a function of energy is shown in Fig. 10.17 for ∆ m2 = 0.8×10−10eV2and sin22θ= 0.8. The Sun is treated as a point-like source because the extension of its neutrino- producing core is minute relative to the oscillation length of order the Earth-Sun distance. The solid line is for the minimum, the dashed line for the maximum annual distance to the Sun. The nearly monochro- matic beryllium neutrinos ( E= 0.862 MeV) easily retain their phase relationship because they are distributed in energy only over a width of about 2 keV (Fig. 10.4). Therefore, one would expect a pronounced annual ﬂux variation which is not observed, thereby reducing the al- lowed range of masses and mixing angles where long-wavelength oscil- lations could solve the solar neutrino problem. The analysis of Barger, Phillips, and Whisnant (1992) and more recently of Krastev and Petcov (1994) reveals that there remain a few small parameter pockets with sin22θaround 0 .8−1 and ∆ m2 around 0 .6−0.9×10−10eV2. Krastev and Petcov (1994) also conclude that oscillations into sterile neutrinos are slightly disfavored. While the spectral deformation of the8B neutrino spectrum is quite dramatic, the spectrum of recoil electrons smears out most of this eﬀect (Fig. 10.18). Notably, above the Kamiokande detection threshold of about 7 MeV the expected distortion is too small to be detected and Fig. 10.18. Spectrum of8B neutrinos and of recoil electrons. Dotted lines: Standard spectrum from Fig. 10.12. Solid lines: Surviving espectrum after oscillations (suppression factor from Fig. 10.17) and modiﬁed electron recoil spectrum. For oscillations into ;one would have to include their neutral- current scattering on electrons with a cross section about1 6that of e. 384 Chapter 10 so the spectral shape measured at Kamiokande (Fig. 10.13) does not further constrain the vacuum oscillation hypothesis. Barger, Phillips, and Whisnant (1992) as well as Krastev and Pet- cov (1994) gave precise conﬁdence contours in the sin22θ-∆m2 -plane where the solar neutrino problems are solved. However, some of the input information involves large systematic uncertainties so that it is well possible that, for example, a revised S17factor would shift the allowed regions beyond their stated conﬁdence contours—see, for ex- ample, Berezhiani and Rossi (1995). The main message is that there re- main parameters in the quoted range where vacuum oscillations (“just so oscillations”) could reconcile all solar ﬂux measurements with the standard ﬂux predictions, except for the apparent anticorrelation of the Homestake data with solar activity. 10.6.3 Resonant Oscillations (MSW Eﬀect) The solar neutrino problem has become tightly intertwined with the is- sue of neutrino oscillations thanks to the work of Mikheyev and Smirnov (1985) who showed that even for small mixing angles one can achieve a large rate of ﬂavor conversion because of medium-induced “resonant oscillations.” This “MSW eﬀect” was conceptually and quantitatively discussed in Sect. 8.3. For a practical application to the solar neutrino problem two main features of the MSW eﬀect are of great relevance. One is the bathtub- shaped suppression function of Fig. 8.10 which replaces a constant reduction factor (short-wavelength vacuum oscillations) or the wiggly shape of Fig. 10.17 (long-wavelength vacuum oscillations). It implies thatνe’s of intermediate energy can be reduced while low- and high- energy ones are left relatively unscathed. This is what appears to be in- dicated by the observations with the beryllium neutrinos more strongly suppressed than the ppor boron ones. Another important feature are the triangle-shaped suppression con- tours in the sin22θ-∆m2 -plane for a ﬁxed neutrino energy (Fig. 8.9). Each experiment produces its own triangular band of neutrino param- eters where its measured rate is reconciled with the standard ﬂux pre- diction. As the signal in diﬀerent detectors is dominated by neutrinos of diﬀerent energies, these triangles are vertically oﬀset relative to one another and so there remain only a few intersection points where all experimental results are accounted for. Therefore, as experiments with three diﬀerent spectral responses are now reporting data the MSW so- lution to the solar neutrino problem is very well constrained. Solar Neutrinos 385 The allowed ranges from the diﬀerent experiments as well as the combined allowed range are shown in Fig. 10.19 according to the anal- ysis of Hata and Haxton (1995). They used the Bahcall and Pinson- neault (1995) solar model with helium and metal diﬀusion and its un- certainties. The black areas represent the 95% CL allowed range if one includes all the experimental information quoted in Sect. 10.3, includ- ing the GALLEX source experiment. The center of the allowed area for the “Large-Angle Solution” is roughly sin22θ= 0.6, ∆m2 = 2×10−5eV2, (10.24) while for the “Nonadiabatic Solution” or “Small-Angle Solution” it is sin22θ= 0.6×10−2, ∆m2 = 0.6×10−5eV2. (10.25) Both cases amount essentially to the same neutrino mass-square diﬀer- ence. Other recent and very detailed analyses are by Fiorentini et al. (1994), Hata and Langacker (1994), and Gates, Krauss, and White (1995) who ﬁnd similar results. In detail, their conﬁdence contours diﬀer because they used other solar models or a diﬀerent statistical analysis. A certain region of parameters (dotted area in Fig. 10.19) can be excluded by the absence of a day/night diﬀerence in the Kamiokande counting rates which are expected for the matter-induced back oscilla- tions into νewhen the neutrino path intersects with part of the Earth at night. (Even though the detector is deep underground in a mine the overburden of material during daytime is negligible.) The excluded range of masses and mixing angles is independent of solar modelling as it is based only on the relative day/night counting rates. Except for the Kamiokande day/night exclusion region one should be careful at taking the conﬁdence contours in a plot like Fig. 10.19 too seriously. The solar model uncertainties are dominated by systematic eﬀects which cannot be quantiﬁed in a statistically objective sense. One of the main uncertain input parameters is the astrophysical S-factor for the reaction p7Be→8Bγ. Speculating that all or most of the missing boron neutrino ﬂux is accounted for by a low S17has the eﬀect of reducing the best-ﬁt sin22θof the small-angle solution by about a factor of 3 while the large-angle solution disappears, at least in the analysis of Krastev and Smirnov (1994). The best-ﬁt ∆ m2, however, 386 Chapter 10 Fig. 10.19. Allowed range of neutrino masses and mixing angles in the neu- trino experiments if the ﬂux deﬁcit relative to the Bahcall and Pinsonneault (1995) solar model is interpreted in terms of neutrino oscillations. The ex- perimental data include all summarized in Sect. 10.3. (Plot adapted from Hata and Haxton 1995.) is rather stable against variations of S17(Krastev and Smirnov 1994; Berezinsky, Fiorentini, and Lissia 1994). Another approach is to allow S17to ﬂoat freely when performing a maximum-likelihood analysis, i.e. to ﬁt it simultaneously with the neu- trino parameters from all solar neutrino experiments (Hata and Lan- gacker 1994). The best-ﬁt value is found to be 1 .43+0:65 −0:42times the stan- dard 22 .4 eV b. Of course, the 95% CL range for the best-ﬁt neutrino parameters is now much larger, allowing any sin22θbetween about 10−3and 0.8. Notably the range of allowed large-angle solutions is vastly increased. In summary, the hypothesis of neutrino oscillations can beautifully explain almost all experimental results to date. Only the curious an- ticorrelation of the Homestake data with solar activity remains unac- counted for. It is interpreted as a statistical ﬂuctuation. Solar Neutrinos 387 10.7 Spin and Spin-Flavor Oscillations If the anticorrelation between disk-centered magnetic indicators of so- lar activity and the event rate at Homestake is taken seriously, the only plausible explanation put forth to date is that of magnetically induced neutrino spin or spin-ﬂavor transitions that were discussed in Sect. 8.4. It was ﬁrst pointed out by Voloshin and Vysotski˘ ı (1986) that the vary- ing magnetic-ﬁeld strength in the solar convective surface layers could cause a time-varying depletion of the left-handed solar neutrino ﬂux. A reﬁned discussion was provided by Voloshin, Vysotski˘ ı, and Okun (1986a,b) after whom this mechanism is called the VVO solution to the solar neutrino problem. Strong magnetic ﬁelds may also exist in the nonconvective interior of the Sun. In principle, they could also cause magnetic oscillations and thus reduce the solar neutrino ﬂux (Werntz 1970; Cisneros 1971). However, the amount of reduction would not be related to the magnetic activity cycle which is conﬁned to the convective surface layers with an approximate depth of 0 .3R⊙≈2×1010cm = 200 ,000 km. The oscilla- tion length is given by Eq. (8.53) so that over this distance a complete spin reversal is achieved for µBT≈3×10−10µBkG, (10.26) where µB=e/2meis the Bohr magneton and BTis the magnetic- ﬁeld strength perpendicular to the neutrino trajectory. Because the magnetic ﬁeld is mainly toroidal, the condition of transversality is au- tomatically satisﬁed. Magnetic spin oscillations in vacuum do not have any energy depen- dence and so the solar neutrino ﬂux would be reduced by a common factor for the entire spectrum. However, a large conversion rate is only achieved if the spin states are nearly degenerate which is not the case in media where refractive eﬀects change the dispersion relation of left-handed neutrinos. In the context of spin-ﬂavor oscillations, degen- eracy can be achieved by a proper combination of medium refraction and mass diﬀerences (resonant spin-ﬂavor oscillations). In this case the diagonal elements of the neutrino oscillation Hamiltonian involve energy-dependent terms of the form ( m2 2−m2 1)/2E, causing a strong energy dependence of the conversion probability. Therefore, the mea- sured signals in the detectors as well as the time variation at Homestake and the absence of such a variation at Kamiokande can all be explained by a suitable choice of neutrino parameters and magnetic-ﬁeld proﬁles 388 Chapter 10 of the Sun.58The simple estimate Eq. (10.26) remains approximately valid. A neutrino mass-square diﬀerence m2 2−m2 1below about 10−5eV2 is required because of the small matter densities encountered in the con- vective layers. The bounds on neutrino dipole and transition moments discussed in Sect. 6.5.6 yield µ∼<3×10−12µBso that a magnetic-ﬁeld strength B∼>100 kG is required in the solar convection zone. Typical ﬁeld strengths measured in sunspots where the ﬂux breaks through the sur- face are of a few kG. While this may not be representative of the large- scale toroidal ﬁeld, several general arguments suggest that 10 kG is a generous upper limit (Shi et al. 1993). If there were much stronger ﬁelds they would have to be conﬁned to ﬂux ropes which would not be eﬀective at inducing neutrino spin oscillations. Therefore, the spin oscillation scenario would require anomalously large magnetic ﬁelds, in conﬂict with the arguments presented by Shi et al., or dipole moments in excess of what is allowed by the bounds of Sect. 6.5.6. Even smaller dipole moments than 3 ×10−12µBrequire novel neutrino interactions. The solar magnetic ﬁeld is believed to consist of two opposite ﬂux tori, separated by the solar equatorial plane. In the course of a year the line of sight from Earth to the solar core varies between ±7◦15′solar latitude and so the neutrinos measured here traverse a ﬁeld conﬁgu- ration which varies in the course of a year. The predicted semiannual ﬂux variation (Voloshin, Vysotski˘ ı, and Okun 1986a), however, has not been conﬁrmed by any of the experiments. In the spin-ﬂavor oscillation scenario involving Majorana neutrinos the Sun would be a source for antineutrinos, some of which could be νe’s by a combination of spin-ﬂavor ( νe→ν) and ﬂavor ( ν→νe) oscillations. However, the Kamiokande data already yield restrictive limits on a solar νeﬂux (Barbieri et al. 1991; see also Fig. 10.14). In summary, the magnetic spin oscillation scenario requires new neu- trino interactions to generate large magnetic dipole moments, and new astrophysics to allow for suﬃciently strong magnetic ﬁelds in the so- lar convection zone. Neither a semiannual variation of the ﬂux nor νe’s have been observed; each would have been a smoking gun for the occur- rence of this eﬀect. Therefore, one is led to disfavor the magnetic spin oscillation scenario. Then, of course, one is back to a statistical ﬂuctu- ation as an explanation for the time structure of the Homestake data. 58Recent detailed investigations were performed by Akhmedov, Lanza, and Petcov (1993, 1995), Krastev (1993), Nunokawa and Minakata (1993), Guzzo and Pulido (1993), and Pulido (1993, 1994). For a review of earlier works see Pulido 1992. Solar Neutrinos 389 10.8 Neutrino Decay A deﬁcit of solar neutrinos measured at Earth can be related to neutrino decay. However, the in-ﬂight decay of neutrinos does not provide the required deformation of the spectrum because the decay rate in the laboratory system involves a Lorentz factor m/Eso that low-energy neutrinos decay faster. If νeis a mixture of mass eigenstates only the heavier one decays. It is possible that the heavy admixture decays fast so that the spectrum is reduced by a constant factor. Even this extreme case does not provide a good ﬁt to the data. A detailed analysis of diﬀerent cases was performed by Acker and Pakvasa (1994) who found that the in-ﬂight decay solution was ruled out at the 98% CL, even when allowing for the solar model uncertainties.59 Neutrino decays can yield a solar νeﬂux which is, in principle, measurable. Such a ﬂux can be produced if neutrinos are Majorana particles, and if they couple to majorons χ(Sect. 15.7). Some fraction ofν→ν′+χdecays ﬂip the helicity of the neutrino so that the ν′is eﬀectively a ν′. Thus after an MSW conversion νe→ν;one could have decays ν;→νe+χ(Raghavan, He, and Pakvasa 1988). Even without oscillations one can have matter-induced decays of the form νe→νe+χas discussed in Sect. 6.8. Detailed predictions for the νeﬂux for this type of scenario were worked out by Berezhiani et al. (1992) and by Berezhiani, Moretti, and Rossi (1993). The Kamiokande detector has already produced limits on solar νe’s (Fig. 10.14), with much better limits to be expected from Superkamiokande. However, in view of other limits on the neutrino-majoron coupling Berezhiani, Moretti, and Rossi (1993) found that it seemed unrealistic to hope for a detectable solar νesignal. Malaney, Starkman, and Butler (1994) showed that in decays of the form ν→ν′+ boson, ﬁnal-state stimulation eﬀects (“neutrino lasing”) could enhance the decay rate. However, the best-motivated case is that of majoron decays which involve a γ5coupling. The shape of the resulting majoron spectrum is such that the crucial emission of low-momentum bosons and thus the lasing eﬀect is suppressed whence Acker and Pakvasa’s conclusions remain valid. For models involving scalar or vector bosons a detailed new analysis is required. At the present time it looks rather unconvincing that the solar neu- trino problem is related to some form of neutrino decays. 59See Acker and Pakvasa (1994) for references to earlier discussions of neutrino decay as a potential solution to the solar neutrino problem. 390 Chapter 10 10.9 Future Experiments 10.9.1 Superkamiokande If some form of neutrino oscillations are the explanation for the mea- sured solar ﬂux deﬁcits relative to standard predictions, how are we ever going to know for sure? One needs to measure a signature which is characteristic only for neutrino oscillations. The most convincing case would be a measurement of the “wrong-ﬂavored” neutrinos, i.e. theνorνappearance rather than the νedisappearance. Other clear signatures would be a deformation of the8B spectrum or a diurnal or seasonal ﬂux variation. The latter cases can be very well investigated with the Superkamio- kande detector which is scheduled to begin its operation in April of 1996. It is a water Cherenkov detector like Kamiokande, with about 20 times the ﬁducial volume. With about twice the relative coverage of the surface area with photocathodes and a detection threshold as low as 5 MeV it will count about 30 events/day from the solar boron neutrino ﬂux, as opposed to about 0.3 events/day at Kamiokande. Fig. 10.20. Expected signal at night relative to the average daytime sig- nal in a water Cherenkov detector as a function of the angle between the Sun and the detector nadir. The large-angle example is for sin22= 0:7 and ∆ m2 = 1×10−5eV2, the small-angle case for sin22= 0:01 and ∆m2 = 0:3×10−5eV2. Also shown are the existing Kamiokande measure- ments and the expected Superkamiokande error bars after 1 month and 1 year of running, respectively. (Adapted from Suzuki 1995.) Solar Neutrinos 391 The small- and large-angle MSW solutions suggested by Fig. 10.19 would cause a day/night variation of the solar neutrino signal as indi- cated in Fig. 10.20. The expected signal is shown in bins for the angle between the Sun and the nadir of the detector, i.e. in bins of the inter- section length of the neutrino ﬂight path with the Earth. Also shown are the current Kamiokande measurements, and the expected error bars after 1 month and 1 year of Superkamiokande running time, respec- tively. Shortly after Superkamiokande starts taking data one should be able to decide whether the large-angle MSW solution applies! The MSW solutions would also cause a spectral distortion of the re- coil electron spectrum from the primary boron neutrinos. The expected spectral shape relative to the standard one, arbitrarily normalized at an electron kinetic energy of Te= 10 MeV, is shown in Fig. 10.21 for sev- eral values of the assumed mixing angle. After several years of running, Superkamiokande should be able to identify clearly the small-angle so- lution if it applies. If Superkamiokande measures neither a spectral distortion nor a day/night eﬀect, the deﬁciency of the boron ﬂux probably would have Fig. 10.21. Spectral distortion of the recoil electrons from the primary boron neutrinos in a water Cherenkov detector. The ratio relative to the standard spectrum is arbitrarily normalized at an electron kinetic energy of Te= 10 MeV. For the large-angle example (solid line) the assumed mass-square diﬀerence is ∆ m2 = 2×10−5eV2, for the small-angle examples (broken lines) it is ∆ m2 = 0:6×10−5eV2. The anticipated error bars after 5 years of running Superkamiokande are also indicated for two energies. (Adapted from Krastev and Smirnov 1994.) 392 Chapter 10 to be attributed to a small astrophysical S17factor. In this case, the ex- planation for the deﬁciency of beryllium neutrinos could not be resolved by this detector. 10.9.2 Sudbury Neutrino Observatory (SNO) The Sudbury Neutrino Observatory (SNO), also scheduled to take up operation in 1996, is a heavy-water Cherenkov detector which is ex- pected to be able to measure the appearance of “wrong-ﬂavored” neu- trinos if the MSW eﬀect solves the solar neutrino problems (Sudbury Neutrino Observatory Collaboration 1987; Lesko et al. 1993). The SNO detector consists of 1000 tons of heavy water (D 2O) in a spherical acrylic vessel of 12 m diameter, immersed in an outer vessel of ultrapure light water (H 2O), surrounded by about 9600 photomultiplier tubes of 20 cm diameter each. The detector is located 2000 m underground in the Creighton mine, an operating Nickel mine, near Sudbury in Ontario (Canada). Neutrinos can be detected by three diﬀerent reactions in this detector: by electron elastic scattering ν+e→e+νand by the deuterium dissociation reactions νe+d→p+p+eandν+d→p+n+ν. The electron elastic scattering reaction is analogous to the Kamio- kande and Superkamiokande detectors: the recoiling electron is mea- sured by the detection of its Cherenkov light. The eﬀective detection threshold is expected to be at 5 MeV as in Superkamiokande. This reaction is sensitive to both νeandν;, albeit with a reduced cross section for the latter (Eq. 10.17). The charged-current deuterium dissociation νed→ppehas a thresh- old of 1 .44 MeV, i.e. the ﬁnal-state electron kinetic energy is essentially Te=E−1.44 MeV. The electron is detected by its Cherenkov light; the eﬀective energy resolution is about 20%. The angular distribution relative to the incident neutrino is given by 1 −1 3cos Θ. The cross section for this reaction is large. For an incident spectrum of boron neutrinos one expects 9 times more electron counts above 5 MeV than from electron elastic scattering; above 9 MeV even 13 times as many. One can search for neutrino oscillations by a spectral distortion of the electron spectrum, similar to Fig. 10.21 for Superkamiokande. In fact, the spectral distortion is more pronounced as it is not washed out by a broad ﬁnal-state distribution of electron energies. Also, one can search for a day/night eﬀect. The most important detection reaction, however, is the neutral- current deuteron disintegration νd→pnν which has the same cross section for all ﬂavors and so it measures the total (left-handed) neutrino Solar Neutrinos 393 ﬂux above its threshold of 2 .2 MeV, independently of the occurrence of oscillations. The main problem here is the measurement of the ﬁnal- state neutron by the detection of γrays from the subsequent neutron capture, or by a neutron detector array in the heavy water. Ultrapure water, acrylic, and other materials are needed to prevent an excessive radioactive background that would spoil this measurement. One an- ticipates to obtain the full unsuppressed8B ﬂux with a precision of about 1% after 5 years of operation. The measured ratio between the charged-current and neutral-current deuterium disintegration will give an immediate measure of the electron survival probability and thus of the occurrence of neutrino oscillations. If spin or spin-ﬂavor oscillations occur such that a sizeable νeﬂux is produced, it can be detected by the reaction νe+d→n+n+e+ which produces three detectable particles (Balantekin and Loreti 1992). Of course, a possible νeﬂux is already constrained by Kamiokande (Fig. 10.14), and can be detected or constrained by Superkamiokande. Suggestions for solar model independent methods of analyzing fu- ture solar neutrino data were made, for example, by Spiro and Vignaud (1990), Bilenky and Giunti (1993, 1994), and Castellani et al. (1994). 10.9.3 BOREXINO Superkamiokande and SNO are both limited to a measurement of the boron neutrino ﬂux because of their relatively high detection thresholds. If the boron ﬂux is partly or mostly suppressed by a low S17factor or a low central solar temperature instead of neutrino oscillations these experiments may have diﬃculties at identifying oscillations which would still be indicated by the missing beryllium neutrino ﬂux. Therefore, it is interesting that another experiment (BOREXINO60) is being prepared which would be sensitive dominantly to the beryllium neutrinos. The main detection reaction is elastic ν-escattering as in the light- water Cherenkov detectors. However, the kicked electron is detected by virtue of scintillation light rather than Cherenkov radiation, the former 60The name of this experiment is derived from BOREX (boron solar neutrino experiment), a proposed detector that was to use11B as a target (Raghavan, Pak- vasa, and Brown 1986; see also Bahcall 1989). For a practical implementation it was envisaged to use a boron loaded liquid scintillator (e.g. Raghavan 1990); be- cause of the relatively small size of this detector the Italian diminutive BOREXINO (baby BOREX) emerged. Ultimately, the idea of using a borated scintillator was dropped entirely, leaving boron only in the name of the experiment. Confusingly, then, BOREXINO is unrelated to a boron target, and also unrelated to the solar boron neutrinos because the experiment is designed to hunt the beryllium ones. 394 Chapter 10 yielding about 50 times more light at the relevant energies below about 1 MeV. Naturally, as a target one needs to use an appropriate scintilla- tor rather than water. The advantage of a lowered threshold is bought at the price of losing all directional information. However, because of the good energy resolution the monochromatic beryllium neutrinos at 862 keV should be clearly detectable as a distinct shoulder in the energy spectrum of the recoil electrons. Optimistically, a scintillation detector could have a threshold as low as E= 250 keV. The main challenge at implementing this method is to lower the radioactive contamination of the scintillator, its vessel, and the sur- rounding water bath to an unprecedented degree of purity. For exam- ple, the allowed mass fraction of238U of the scintillator is less than about 10−16g/g. The feasibility of this method is currently being stud- ied at the CTF (Counting Test Facility) experiment, located in the Gran Sasso underground laboratory. Assuming a positive outcome, BOREXINO would be built, consisting of 300 tons of scintillator, sur- rounded by 3000 tons of water. Optimistically, data taking with this facility could commence in 1997. 10.9.4 Homestake Iodine Detector Currently, a modular 100-ton iodine detector is under construction in the Homestake mine. It is similar to the Homestake chlorine detector, except that it uses the127I→127Xe transition to measure the νeﬂux. It has an eﬀective threshold of 0 .789 MeV, similar to the chlorine detector. However, for a standard solar neutrino ﬂux the detection rate should be about four times higher if the cross section calculations are correct. The currently built detector should take up operation in mid-1995. It may be expanded at a later time after running experience has been obtained, and after the neutrino cross sections have been measured (Bahcall et al. 1995; Engel, Krastev, and Lande 1995). 10.9.5 Summary The hypothesis of neutrino ﬂavor oscillations is strongly supported by the results of all existing solar neutrino experiments. With the new generation of detectors which will begin to take up operation in 1996 it looks plausible that nonstandard neutrino properties can be ﬁrmly established on the basis of the solar neutrino ﬂux before the mille- nium ends. Chapter 11 Supernova Neutrinos The general physical picture of stellar collapse and supernova (SN) explosions is described with an emphasis on the properties of the ob- servable neutrino burst from such events. The measurements of the neutrino burst from SN 1987A are reviewed. Its lessons for particle physics are deferred to Chapter 13 except for the issue of neutrino masses and mixings. Future possibilities to observe SN neutrinos are discussed. 11.1 Stellar Collapse and Supernova Explosions 11.1.1 Stellar Collapse A massive star ( M∼>8M⊙) inevitably becomes unstable at the end of its life. It collapses and ejects its outer mantle in a SN explosion as brieﬂy described in Sect. 2.1.8. Within fractions of a second the collapsing core forms a compact object at supranuclear density which radiates its gravitational binding energy Eb≈3×1053erg within a few seconds in the form of neutrinos. Gamow and Schoenberg (1940, 1941) were the ﬁrst to speculate that neutrino emission would be a major eﬀect in the collapse of a star. The only direct observation of neutrinos from such an event occurred on 23 February 1987 when the blue su- pergiant Sanduleak −69 202 in the Large Magellanic Cloud exploded in what became known as SN 1987A. The neutrinos, and possibly other low-mass particles, emitted from a collapsing star are the main topic of this chapter, with SN 1987A playing a primary role. Before an evolved massive star collapses, its core is a degenerate conﬁguration made up of iron-group elements. They cannot release nuclear energy by fusion as they are already the most tightly bound 395 396 Chapter 11 nuclei so that no further nuclear burning stage can be ignited. The precollapse inner “iron white dwarf” has a mass of about 1 :5M⊙, a central density of about 3 :7×109g cm−3, a central temperature of about 0:69 MeV, and a number fraction of electrons per baryon of Ye≈0:42 (Brown, Bethe, and Baym 1982). As the mass of this object grows and its radius shrinks it reaches its Chandrasekhar limit: relativistic electrons cannot support a self-gravitating body. In practice, a thermal pressure contribution cannot be neglected. The collapse is triggered when the temperature has become so high that the photodissociation of iron commences, +56Fe→13+ 4n, a reaction which consumes 124 :4 MeV of energy. This energy loss reduces the thermal contribution of the electron pressure. Therefore, compres- sion yields a lesser pressure increase than would occur in the absence of photodissociation. A star near its Chandrasekhar mass is close to a point where the increased gravitational pull caused by a small con- traction is no longer overcompensated by a large enough push from the corresponding pressure increase. Therefore, a small reduction of the adiabatic index Γ ≡(@lnp=@ln)sis enough to cause an insta- bility. Once the collapse has begun, pressure support is also lost by the capture of electrons on heavy nuclei which amounts to the reaction e−+p→n+e. It converts electrons to neutrinos which escape freely. The inner part of the core ( M ≈ 0:6M⊙) has Γ ≈4 3. Its collapse is homologous, i.e. it maintains its relative density proﬁle (Goldreich and Weber 1980). The collapse velocity is proportional to the radius with v=r= 400−700 s−1, yet it remains subsonic and so this part of the core is in good communication with itself. The nearly free fall of the outer part is supersonic. At a certain density the neutrinos will no longer be able to stream freely from the core. When their diﬀusion time exceeds a dynamical collapse time scale they will be trapped (Mazurek 1974, 1975, 1976; Sato 1975). Neutral-current scatterings on large nuclei are particularly eﬀective at trapping neutrinos because the cross section is coherently enhanced (Freedman 1974).61One ﬁnds a trapping density of around 1012g cm−3for 10 MeV neutrinos (Brown, Bethe, and Baym 1982). The neutrino trapping radius as a function of time is shown as a dotted line in Fig. 11.1. 61Neutrinos with 10 MeV energies cannot “resolve” the nucleus, causing it to act as a single scattering center. Because the neutral-current interaction with protons is reduced by a factor 1 −4 sin2ΘW(weak mixing angle Θ W) with sin2ΘW≈0:23 (Appendix B), the elastic scattering cross section of a nucleus scales with the square of the neutron number. Supernova Neutrinos 397 Fig. 11.1. Schematic picture of the core collapse of a massive star ( M∼> 8M⊙), of the formation of a neutron-star remnant, and the beginning of a SN explosion. There are four main phases numbered 1 −4 above the plot: 1. Collapse. 2. Prompt-shock propagation and break-out, release of prompt eburst. 3. Matter accretion and mantle cooling. 4. Kelvin-Helmholtz cooling of “protoneutron star.” The curves mark the time evolution of several characteristic radii: The stellar iron core ( RFe). The “neutrino sphere” ( R) with diﬀusive transport inside, free streaming outside. The “inner core” (Ric) which for t∼<0:1 s is the region of subsonic collapse, later it is the settled, compact inner region of the nascent neutron star. The SN shock wave ( Rshock) is formed at core bounce, stagnates for several 100 ms, and is revived by neutrino heating—it then propagates outward and ejects the stellar mantle. The shaded area is where most of the neutrino emission comes from; between this area and Rneutrinos still diﬀuse, but are no longer eﬃciently produced. (Adapted from Janka 1993.) Neutrino trapping has the eﬀect that the lepton number fraction YLis nearly conserved at the value Yewhich obtains at the time of trapping. However, electrons and electron neutrinos still interconvert (equilibrium), causing a degenerate esea to build up. The core of a collapsing star is the only known astrophysical site apart from the early universe where neutrinos are in thermal equilibrium. It is the only site where neutrinos occur in a degenerate Fermi sea as the early universe is thought to be essentially CP symmetric with equal numbers of neutrinos and antineutrinos to within one part in 109. When neutrino trapping becomes eﬀective, the lepton fraction per baryon is YL≈0:35, 398 Chapter 11 not much lower than the initial iron-core value, i.e. not much of the lepton number is lost during infall. The conditions of chemical and thermal equilibrium dictate that neutrinos take up only a relatively small part (about1 4) of the lepton number (Appendix D.2). A typical YLproﬁle after collapse is shown in Fig. 11.2 The collapse is intercepted when the inner core reaches nuclear den- sity ( 0≈3×1014g cm−3), a point where the equation of state stiﬀens. Because the inner core collapse is subsonic, the information about the central condition spreads throughout, i.e. the collapse of the entire ho- mologous core slows down. However, this information cannot propagate Fig. 11.2. Snapshots of the proﬁles of temperature Tand lepton number fraction YLin the collapsed core of a massive star. The indicated times are in seconds after collapse. The shaded arrows in the upper panel indicate the motion of the temperature maximum. For a soft equation of state the maximum temperature can be up to about 70 MeV. (Adapted from Burrows and Lattimer 1986.) Supernova Neutrinos 399 beyond the sonic point at the edge of the inner core which now encom- passes about 0 :8M⊙. As material continues to fall onto the inner core at supersonic velocities a shock wave builds up at the sonic point which is at the edge of the inner core, not at its center. As more material moves in, more and more energy is stored in this shock wave which almost immediately begins to propagate outward into the collapsing outer part of the iron core—see the thick solid line in Fig. 11.1. As- suming that enough energy is stored in the shock wave it will eventually eject the stellar mantle outside of what was the iron core. The rebound or “bounce” of the collapse turns the implosion of the core into an explosion of the outer star—a SN occurs. This “bounce and shock” scenario of SN explosions was ﬁrst pro- posed by Colgate and Johnson (1960) and then elaborated by a number of authors (see Brown, Bethe, and Baym 1982 and references therein). In practice, however, the story of SN explosions appears to be more complicated than this “prompt explosion scenario.” Neutrino losses and the dissociation of the iron material through which the shock wave propagates dissipate much of the shock’s energy so that in typical calcu- lations it stalls and eventually recollapses. It is currently believed that the energy deposition by neutrinos revives the shock wave, leading to the “delayed explosion scenario” detailed in Sect. 11.1.3 below. 11.1.2 Deleptonization and Cooling After core bounce and the formation of a shock wave the next dra- matic step in the evolution of the core is when the outward propagating shock breaks through the “neutrino sphere,” i.e. the shell within which neutrinos are trapped, most eﬀectively by the coherent scattering on heavy nuclei. As the passage of the shock wave dissociates these nu- clei, it is easier for neutrinos to escape. Moreover, the protons newly liberated from the iron nuclei allow for quick neutronization by virtue ofe−+p→n+e, causing a short eburst which is often called the “prompt eburst” or “deleptonization burst” (phase No. 2 in Figs. 11.1 and 11.3). However, the material which is quickly deleptonized encom- passes only a few tenths of a solar mass so that most of the leptons remain trapped in the inner core (Fig. 11.2). At this stage, the object below the shock has become a “protoneu- tron star.” It has a settled inner core within the radius where the shock wave ﬁrst formed and which consists of neutrons, protons, elec- trons, and neutrinos (lepton fraction YL≈0:35). The protoneutron star also has a bloated outer part which has lost a large fraction of 400 Chapter 11 Fig. 11.3. Schematic neutrino “lightcurves” during the phases of (1) core collapse, (2) shock propagation and shock breakout, (3) mantle cooling and accretion, and (4) Kelvin-Helmholtz cooling. (Adapted from Janka 1993.) its lepton number during the eburst at shock break-out. This outer part settles within the ﬁrst 0 :5−1 s after core bounce, emitting most of its energy in the form of neutrinos. Also, more material is accreted while the shock wave stalls. As much as a quarter of the expected total amount of energy in neutrinos is liberated during this phase (No. 3 in Figs. 11.1 and 11.3). Meanwhile, the stalled shock wave has managed to resume its out- ward motion and has begun to eject the overburden of matter. There- fore, the protoneutron star after about 0 :5−1 s can be viewed as a star unto itself with a radius of around 30 km which slowly contracts and cools by the emission of (anti)neutrinos of all ﬂavors, and at the same time deleptonizes by the loss of e’s. After 5 −10 s it has lost most of its lepton number, and slightly later most of its energy. This is the “Kelvin-Helmholtz cooling phase,” marked as No. 4 in Figs. 11.1 and 11.3. Afterward, the star has become a proper neutron star whose small lepton fraction is determined by the condition of a vanishing neutrino chemical potential (Appendix D.2), and whose further cooling history has been discussed in Sect. 2.3. Immediately after collapse the protoneutron star is relatively cold (see the t= 0 curve in Fig. 11.2). Half or more of the energy to be ra- diated later is actually stored in the degenerate electron Fermi sea with typical Fermi momenta of order 300 MeV. The corresponding degener- Supernova Neutrinos 401 ate neutrino sea has its Fermi surface at around 200 MeV. The lepton number proﬁle (lower panel of Fig. 11.2) has a step-like form, with the step moving inward very quickly when the shock breaks through the neutrino sphere. The quick recession of the lepton proﬁle during the ﬁrst 0 :5 s represents deleptonization during the mantle cooling phase. After this initial phase, however, the bloated outer part of the star has settled; it is more diﬃcult for neutrinos to escape from this compact object. Still, the steep gradient of lepton number drives an outward dif- fusion of neutrinos which move toward regions of lower Fermi momen- tum and thus, of lower degeneracy energy. Therefore, they “downscat- ter,” releasing most of the previous electron and neutrino degeneracy energy as heat. Hence near the edge of the lepton number step the medium is heated eﬃciently. In Fig. 11.2 it is plainly visible that the temperature maximum of the medium is always in the region of the steepest lepton number gradient.62Therefore, the medium ﬁrst heats near the core surface, and then the temperature maximum moves in- ward until it has reached the center. At this time the core is entirely deleptonized; it continues to cool, the temperature maximum at the center drops to obscurity. The neutrino radiation leaving the star has typical energies in the 10 MeV range, compared with a 200 MeV neutrino Fermi energy in the interior. Therefore, the loss of lepton number by itself is associated with relatively little energy. Put another way, only a small excess of e overeis needed to carry away the lepton number. The total energy is carried away in almost equal parts by each (anti)neutrino ﬂavor. 11.1.3 Supernova Explosions How do supernovae explode? For some time it was thought that the outer layers of the star were ejected by the momentum transferred from the outward neutrino ﬂow which is released after the core collapse (Col- gate and White 1966). This scenario had to be abandoned after the discovery of neutral-current neutrino interactions which trap these par- ticles so that they are released only relatively slowly. With the demise of the neutrino explosion scenario the earlier suggestion of Colgate and Johnson (1960) of a hydrodynamic shock wave driving the explosion be- came the standard, the so-called “prompt explosion scenario” or “direct mechanism.” It continued to malfunction, however, because in numer- ical calculations the shock tended to stall because of energy dissipation 62I thank David Seckel for explaining this point to me which is not usually stressed in the pertinent literature. 402 Chapter 11 associated with the dissociation of the remaining iron shell it had to work through before it could ﬂy. Recall that the shock wave forms relatively deep inside of the collapsing iron core of the progenitor star. A possible solution is the “delayed mechanism” (Wilson 1983; Bethe and Wilson 1985) where the shock wave lingers at a constant radius for a few 100 ms and then takes oﬀ again (Fig. 11.1), powered both by the accretion of material and by the energy deposition of the neutrino ﬂow. In this regard the dilute hot region ( of order 106−108g cm−3,Tof order 1 MeV) below the stalled shock plays a major role at absorbing neutrino energy.63Unfortunately, it is not certain that the neutrino ﬂux can deposit enough energy to revitalize the shock as the energy transfer is relatively ineﬃcient. However, if one adjusts the amount of neutrino energy transfer to a value above a rather well deﬁned threshold one obtains beautiful explosions (Fig. 11.4). It should be noted that the spectacular explosion of a SN—which at the peak of its lightcurve outshines an entire galaxy—is only a “dirt eﬀect” relative to the release of neutrino energy which equals the gravitational binding energy of the newborn neutron star of about GNM=R≈3×1053erg with M ≈ 1:4M⊙andR≈10 km. The total energy released in the kinetic energy of the ejecta and in electromagnetic radiation is a few641051erg, on the order of 1% of the total neutron-star binding energy. Therefore, on energetic grounds alone there is no problem at tapping the neutrino ﬂux for explosion energy. A variety of schemes are currently being discussed to achieve a suc- cessful shock revival. It is thought that convection may play a major role at transporting energy to the surface of the bloated protoneutron star which has formed after the break-out of the shock (e.g. Burrows and Lattimer 1988; Wilson and Mayle 1988; Mayle and Wilson 1993; Burrows and Fryxell 1992, 1993; Janka and M¨ uller 1993b). It may be that this mechanism can boost the eﬀective neutrino luminosity for a few hundred milliseconds, enough in some calculations to trigger an explosion (see however Bruenn and Mezzacappa 1994). At any rate, in the absence of a fundamental treatment of convection this method is essentially one way of parametrizing the initial amount of neutrino heating below the shock. 63Goodman, Dar, and Nussinov (1987) proposed that the pair-annihilation pro- cess→e+emight be the dominant mode of energy transfer. However, Cooper- stein, van den Horn, and Baron (1987) critizised the neutrino emission parameters of that study while Janka (1991) found that a proper treatment of the phase space renders this process less eﬃcient than had been originally thought. 64This unit is sometimes referred to as 1 foe, for “ten to the fiftyoneergs.” Supernova Neutrinos 403 Convection is also important at transporting energy within the di- lute region between the protoneutron star and the stalling shock wave (Herant, Benz, and Colgate 1992; Herant et al. 1994; Janka and M¨ uller 1993a, 1994, 1995a,b; Sato, Shimizu, and Yamada 1993; Burrows, Fig. 11.4. Unsuccessful (upper panel) and successful (lower panel) SN explo- sion. In each case, the location of several mass shells is shown as a function of time. The thick shaded line indicates the location of the shock. The only diﬀerence between the two cases is the adjusted neutrino luminosity from a central source; it was chosen as L= 2:10×1052erg s−1(upper panel) and L= 2:20×1052erg s−1(lower panel), respectively. In the upper case, which is just below threshold for a successful explosion, the shock displays an in- teresting oscillatory behavior. (Curves courtesy of H.-T. Janka, taken from Janka and M¨ uller 1993a.) 404 Chapter 11 Fig. 11.5. Entropy contours between neutron star and shock wave in a 2- dimensional calculation of a SN explosion. The entropy per nucleon is shown in contours at equal steps of 0 :5kBbetween 5 and 16 kB, and in steps of 1 kB between 16 and 23 kB. This snapshot represents model T2c of Janka and M¨ uller (1995b) at t= 377 ms after bounce. (Original of the ﬁgure courtesy of H.-T. Janka.) Supernova Neutrinos 405 Hayes, and Fryxell 1995). For the ﬁrst time 2- and 3-dimensional cal- culations have become possible. They reveal a large-scale convective overturn (Fig. 11.5) which helps at revitalizing the shock because it brings hot material from depths near the neutrino sphere quickly up to the region immediately behind the shock, and cooler material down to the neutrino sphere where it absorbs energy from the neutrino ﬂow. Successful explosions can be obtained for amounts of neutrino heating where 1-dimensional calculations did not succeed. The sharp transition between failed and successful explosions as a function of neutrino heat- ing that was found in 1-dimensional calculations (Fig. 11.4) is smoothed out, but neutrino heating still plays a pivotal role at obtaining a suc- cessful and suﬃciently energetic explosion. In summary, then, the current standard picture of SN explosions is a modiﬁcation and synthesis of the Colgate and Johnson (1960) shock- driven and the Colgate and White (1966) neutrino-driven explosions. At the present time there may still remain a quantitative problem at obtaining enough neutrino energy deposition behind the shock wave to guarantee a successful and suﬃciently energetic explosion. It remains to be seen if this scenario withstands the test of time, or if a novel ingredient will have to be invoked in the future. 11.1.4 Nucleosynthesis The universe began in a hot “big bang” which allowed for the formation of nuclei from the protons and neutrons originally present in thermal equilibrium with the ambient heat bath. The primordial abundances “froze out” at about 22 −24% helium, the rest hydrogen, and a small trace of other light elements such as lithium. The present-day distri- bution of elements was bred from this primeval mix mostly by nuclear processes in stars; they eject some of their mass at the end of their lives (Chapter 2), returning processed material to the interstellar medium from which new stars and planets are born. However, the normal stellar burning processes can produce elements only up to the iron group which have the largest binding energy per nucleon. Thus, the heavy elements must have been produced by dif- ferent processes at diﬀerent sites. It has long been thought that nu- clei with A∼>70 were predominantly made by neutron capture, no- tably the s- and r- (slow and rapid) processes (Burbidge et al. 1957; Cameron 1957; Clayton 1968; Meyer 1994). The site for the occur- rence of the r-process has remained elusive for the past three decades, although many diﬀerent suggestions have been made. The crux is that 406 Chapter 11 one needs to produce the heavy elements in the observed proportions, and with a total amount compatible with a plausible galactic history. It has long been held that the r-process elements were made in SN explosions; in this case one needs a yield of about 10−4M⊙of heavy elements per SN. Perhaps the ﬁrst realistic scenario that appears to meet these re- quirements is r-process nucleosynthesis in the hot bubble between a protoneutron star and the escaping shock wave in a core-collapse SN explosion at a time of a few seconds after core bounce (Woosley and Hoﬀmann 1992; Meyer et al. 1992; Woosley et al. 1994; Witti, Janka, and Takahashi 1994; Takahashi, Witti, and Janka 1994; Meyer 1995). The material in this region is very dilute because of the successful ex- plosion, yet very hot—around 109K or 100 keV in the region where the r-process is thought to occur. This hot bubble is not entirely empty because of a neutrino-driven stellar wind. Therefore, one is talking about a high-entropy environment (a few hundred kBper baryon), i.e. a large number of photons per baryon (a few ten). For such con- ditions the required neutron/proton ratio is achieved even for elec- tron fractions of Ye≈0:40 typical for the material outside of a col- lapsed SN core. This scenario appears to be qualitatively and quantitatively almost perfect except that the necessary combination of entropy, electron frac- tionYe, and expansion time scale do not seem to be quite born out by current numerical calculations. Whatever the explanation of this prob- lem, it is fascinating that both the occurrence of a successful and suﬃ- ciently energetic SN explosion as well as the occurrence of the r-process in the high-entropy environment of the “hot bubble” seem to depend crucially on the neutrino energy transfer which thus plays a dominant role in this scenario. One may expect that r-process nucleosynthesis will turn into a tool to calibrate the neutrino ﬂux from a nascent neutron star, and perhaps into a tool to study nonstandard neutrino proper- ties (for a ﬁrst example see Sect. 11.4.5). In eﬀect, the distribution and quantity of r-process elements gives us a measure of SN neutrino ﬂuxes, independent of direct observations! This is not unlike big-bang nucle- osynthesis where the primeval light-element abundances have been an extremely useful tool to study the properties of the primordial neutrino heat bath (Kolb and Turner 1990). At the present time, of course, a quantitative understanding of SN nucleosynthesis in conjunction with a quantitative understanding of SN explosions is a ﬁeld in its infancy—it remains to be seen if it grows up to be as beautiful as big-bang nucle- osynthesis. Supernova Neutrinos 407 11.2 Predicted Neutrino Signal 11.2.1 Overall Features One of the most important aspects of SN physics relevant to particle astrophysics is the immense ﬂux of neutrinos liberated after the core collapse. This ﬂux has been measured from SN 1987A, and with luck will be measured again from a galactic SN in the future. Therefore, it is important to understand the neutrino signal to be expected from this sort of event. On a crude level of approximation one can understand the main fea- tures of the overall neutrino signal on the basis of very simple physical principles. The overall amount of energy to be expected is given by the binding energy of the compact star that formed after collapse Eb≈3 5GNM2 R= 1:60×1053erg(M M⊙)2(10 km R) : (11.1) It is reasonable to expect the energy to be equipartitioned among the diﬀerent neutrino ﬂavors and so to expect about1 6Ebin each of the six standard (anti)neutrino degrees of freedom. (Here and in the following Newtonian physics is used; general relativistic corrections to energies, temperatures, etc. as viewed from a distant observer can be as large as several 10% due to gravitational redshifts.) Neutrinos are trapped in the interior of the high-density neutron star. Therefore, they are emitted from the relatively well deﬁned surface at a radius of 10 −20 km, depending on the mass and the nuclear equa- tion of state. As long as the material near the surface is nondegenerate it must support itself against the local gravitational ﬁeld by normal thermal pressure. One may apply the virial theorem (Chapter 1) which informs us that the average kinetic energy of a typical nucleon near the neutron-star surface must be half of its gravitational potential, i.e. 2⟨Ekin⟩ ≈GNMmN=R(nucleon mass mN). With a neutron-star mass ofM= 1:4M⊙and a radius R= 15 km one ﬁnds ⟨Ekin⟩ ≈25 MeV orT=2 3⟨Ekin⟩ ≈17 MeV. Therefore, thermal neutrinos emitted from the neutron-star surface are characterized by a temperature of order 10 MeV. The duration of neutrino emission is a multiple of the neutrino dif- fusion time scale over the dimension of the neutron-star radius, tdiﬀ≈R2=; (11.2) where is a typical mean free path. A typical neutral-current weak 408 Chapter 11 scattering cross section on nonrelativistic nucleons is given by ≈G2 FE2 == 1:7×10−42cm2(E=10 MeV)2: (11.3) Nuclear density ( 0≈3×1014g cm−3) corresponds to a nucleon density of about 1 :8×1038cm−3so that ≈300 cm for 30 MeV neutrinos. This yields a diﬀusion time scale tdiﬀ=O(1 s). In summary, one expects an energy of about 0 :5×1053erg to be emitted in each (anti)neutrino degree of freedom over a time scale of order 1 sec with typical energies of order several 10 MeV. 11.2.2 Energies and Spectra These global properties of the expected neutrino signal are broadly conﬁrmed by detailed numerical calculations of neutrino transport.65 However, there are a number of important “ﬁne points” to keep in mind. First, the nonelectron neutrino degrees of freedom ;and; have smaller opacities; their energies are too low for charged-current reactions of the sort +p→n++because of the large masses of the andleptons. These ﬂavors decouple at higher densities and tem- peratures than eandeand so they are emitted with higher average energies. Equally important, e’s have lower energies than e’s because the opacities are dominated by e+n→p+e−ande+p→n+e+, respectively, and because there are fewer protons than neutrons. Typ- ically one ﬁnds (Janka 1993) ⟨E⟩=  10−12 MeV for e, 14−17 MeV for e, 24−27 MeV for ;and;,(11.4) i.e. typically ⟨Ee⟩ ≈2 3⟨Ee⟩and⟨E⟩ ≈5 3⟨Ee⟩for the other ﬂavors. The number ﬂuxes of the nonelectron ﬂavors are smaller than those ofebecause the energy is found to be approximately equipartitioned between the ﬂavors: the total Ee+elies between1 3and1 2ofEb. Simi- larly, the number ﬂux of eis larger than that of e(the lepton number is carried away in e’s!) so that, again, the energy is approximately equipartitioned between eande. The total Eeis found to lie between 1 6and1 4ofEb. The SN 1987A observations were almost exclusively sen- sitive to the eﬂux. The total Eeinferred from these measurements 65See, e.g., Burrows and Lattimer (1986); Bruenn (1987); Mayle, Wilson, and Schramm (1987); Burrows (1988); Janka and Hillebrandt (1989a,b); Myra and Bludman (1989); Myra and Burrows (1990). For reviews see Cooperstein (1988) and Burrows (1990a,b). Supernova Neutrinos 409 must then be multiplied with a factor between 4 and 6 to obtain an estimate of Eb. Even though the emission of neutrinos is a quasithermal process their energies are not set at the “neutrino sphere” which is deﬁned to be the approximate shell from where they can escape without sub- stantial further diﬀusion. Of course, even the notion of a neutrino sphere is a crude concept because of the E2 dependence of the scatter- ing cross section on nonrelativistic nucleons which implies that there is a separate neutrino sphere for each energy group. The scattering with nucleons does not allow for much energy transfer apart from recoil ef- fects. What is relevant for determining the neutrino energies is their “energy sphere” where they last exchanged energy by the scattering on electrons, by pair processes, and by charged-current absorption. Nat- urally, this region lies interior to the neutrino sphere—see the shaded areas in Fig. 11.1 as opposed to the dotted line which represents the neutrino sphere. The concept of an “energy sphere” (where neutrinos last exchanged energy with the medium) and of a “transport sphere” (beyond which they can stream oﬀ without further scattering) helps to explain the apparent paradox that the spectrum is, say, twice as hard as that ofe, yet the same amount of energy is radiated. Both ﬂuxes originate from the same radius of about 15 km so that the Stefan-Boltzmann law ( L∝R2T4) would seem to indicate that the ﬂux should carry 16 times as much energy. However, the place to which the Stefan- Boltzmann law should be applied is the energy sphere, yet the neutrinos cannot escape from there because the ﬂow is impeded by neutral-current scattering on an overburden of nucleons. One may crudely think of the energy sphere being covered with a skin that does not allow the ra- diation to stream oﬀ except through some holes. Thus the eﬀectively radiating surface is smaller than 4 R2. (For a more technical elabora- tion of this argument see Janka 1995a.) Evidently, neutrino transport is a rather complicated problem, es- pecially in the transition region between diﬀusion and free escape. The most accurate numerical way to implement it would be a Monte Carlo integration of the Boltzmann collision equation (Janka and Hillebrandt 1989a,b). In practice, this is not possible because of the constraints imposed by the limited speed of present-day computers so that a va- riety of approximation methods are used to solve this problem. While there is broad agreement on the general features of the expected neu- trino signal, there remain diﬀerences between the predicted spectra and lightcurves of diﬀerent authors. 410 Chapter 11 Apart from the highly nontrivial problem of neutrino transport, the expected signal depends on a variety of physical assumptions such as the nuclear equation of state which determines the stellar equilib- rium conﬁguration and the amount of energy that is liberated, on the properties of the progenitor star (notably the iron core mass), on the duration of the accretion phase while the shock stalls, on the treatment of convection during the ﬁrst few 100 ms, and others. In the Monte Carlo integrations of Janka and Hillebrandt (1989a,b) the neutrino spectra at a given time are found to be reasonably well described by the Fermi-Dirac shape dL dE∝E3 1 +eE=T−; (11.5) where Tis an eﬀective neutrino temperature and an eﬀective degen- eracy parameter. This ansatz allows one to ﬁt the overall luminosity by a global normalization factor as well as the energy moments ⟨E⟩and ⟨E2 ⟩; ﬁner details of the spectrum are probably not warranted anyway. Throughout the emission process, edecreases from about 5 to 3, e from about 2.5 to 2, and ;;;from 2 to 0. This eﬀective degeneracy parameter is the same for ;and;, in contrast with a real chemical potential which changes sign between particles and antiparticles. Fig. 11.6. Normalized neutrino spectral distribution according to a Maxwell- Boltzmann distribution and a Fermi-Dirac distribution with an eﬀective de- generacy parameter = 2, typical for the Monte Carlo transport calcula- tions of Janka and Hillebrandt (1989a,b). The temperatures are T=1 3⟨E⟩ (Maxwell-Boltzmann) and T= 0:8321 3⟨E⟩(Fermi-Dirac with = 2). Supernova Neutrinos 411 Figure 11.6 shows a normalized Maxwell-Boltzmann spectrum and a Fermi-Dirac spectrum with = 2, both for the same average en- ergy⟨E⟩. This choice implies T=1 3⟨E⟩(Maxwell-Boltzmann) and T= 0:8321 3⟨E⟩(Fermi-Dirac with = 2). What is shown in each case is the normalized number spectrum, i.e. dN=dE ∝E2 e−E=T (Maxwell-Boltzmann) and E2 =(1 +eE=T−) (Fermi-Dirac). It is ap- parent that for a ﬁxed average energy the Fermi-Dirac spectrum is “pinched,” i.e. it is suppressed at low and high energies relative to the Maxwell-Boltzmann case. The most important diﬀerence between the two cases is the sup- pressed high-energy tail of the pinched spectra which causes a signif- icant reduction of neutrino absorption rate. This may be important for neutrino detection in terrestrial detectors as well as for neutrino- induced nuclear reactions in the SN mantle and envelope. However, for the sparse SN 1987A signal the diﬀerences would not have been overly dramatic. Therefore, in view of the many other uncertainties at predicting the spectrum most practical studies of neutrino emission and possible detector signals used simple Maxwell-Boltzmann spectra, or Fermi-Dirac spectra with a vanishing chemical potential which diﬀer from the former only in minor detail. 11.2.3 Time Evolution of the Neutrino Signal The schematic time evolution of the (anti)neutrino luminosities of the diﬀerent ﬂavors was shown in Fig. 11.3. The prompt eburst has rela- tively high energies ( ⟨Ee⟩ ≈15 MeV), but the total energy content of a few 1051erg renders it negligible relative to the integrated luminosity of the subsequent emission phases. The average energies and luminosities of the other ﬂavors rise during the ﬁrst few 100 ms while the shock stalls, matter is accreted, and the initially bloated outer region of the protoneutron star contracts. The temperature of the region near the edge of the lepton number step rises substantially during this epoch (Fig. 11.2). During the ﬁrst 0 :5 s somewhere between 10% and 25% of the total binding energy is radiated away; the remainder follows during the Kelvin-Helmholtz cooling phase of the settled star. Detailed parametric studies of the Kelvin-Helmholtz cooling phase were performed by Burrows (1988), and more recently by Keil and Janka (1995). In these works the expected SN 1987A detector signal was studied as a function of the assumed nuclear equation of state (EOS), the mass of the collapsed core at bounce, the amount of post- 412 Chapter 11 bounce accretion, and the temperature proﬁle of the core after col- lapse. As expected, a soft EOS leads to a large amount of binding energy and thus to large integrated neutrino luminosities; a large core mass or large postbounce accretion rate has a similar eﬀect. A soft EOS leads to relatively high temperatures during deleptonization, causing large neutrino opacities and thus long emission time scales. If the EOS is too soft, or the core mass too large, the ﬁnal conﬁguration is not stable and collapses, presumably to a black hole. This must not occur too early to avoid conﬂict with the duration of the observed SN 1987A Fig. 11.7. Luminosity and temperature of the eﬂux from the protoneutron star model 55 of Burrows (1988) which is based on a “stiﬀ equation of state,” an initial baryonic core mass of 1 :3M⊙, and an accretion of 0 :2M⊙within the ﬁrst 0 :5 s. The dotted line in the upper panel indicates a t−1behavior, in the lower panel it indicates e−t=4with ≈10 s. The neutrino spectral distribution was taken to be thermal. Supernova Neutrinos 413 signal (Sect. 11.3). However, in some cases studied by Keil and Janka (1995) with an EOS including hyperons this ﬁnal collapse occurs so late (at 8 s after bounce in one example) that black-hole formation is diﬃcult to exclude on the basis of the SN 1987A observations, notably as no pulsar has yet been found there. Needless to say, with so many parameters to play it is not diﬃcult to ﬁnd combinations of EOS, core mass, accretion rate, and initial temperatures which ﬁt the observed SN 1987A signal well within the statistical uncertainties of the observations. An example is model 55 of Burrows (1988) which is based on a “stiﬀ EOS,” an initial baryonic core mass of 1 :3M⊙, and an accretion of 0 :2M⊙within the ﬁrst 0 :5 s. The evolution of the eﬀective eluminosity and temperature is shown in Fig. 11.7. After about 1 s the decay of the temperature is ﬁt well by an exponential e−t=4with ≈10 s while the decay of the luminosity is poorly ﬁt by an exponential; it decays approximately as t−1after 1 s. For later reference, the time-integrated ﬂux (ﬂuence) of this model is shown in Fig. 11.8. It must be stressed that the “cooling behavior” (decrease of the av- erage eenergy) shown in Fig. 11.7 may not be generic at early times. Initially the star is quite bloated, and relatively cold. Therefore, the Fig. 11.8. Expected eﬂuence (time-integrated ﬂux) from SN 1987A, as- suming a distance of 50 kpc, and taking Burrows’ (1988) model 55 neutrino luminosity shown in Fig. 11.7. The solid line is for an assumed Maxwell- Boltzmann energy spectrum for a given average neutrino energy, the dashed line for a Fermi-Dirac spectrum with a degeneracy parameter = 2 as in Fig. 11.6. 414 Chapter 11 initial neutrino ﬂux may be characterized by a decreasing radius (de- creasing ﬂux), yet increasing temperature. For an overview of various model calculations see Burrows (1990b). One should be careful not to take details of the time evolution of the temperature and luminosity of any speciﬁc calculation too seriously. 11.3 SN 1987A Neutrino Observations 11.3.1 Supernova 1987A Shelton’s (1987) sighting of a supernova (SN 1987A) in the Large Mag- ellanic Cloud (LMC), a small satellite galaxy of the Milky Way at a distance from us of about 50 kpc (165,000 ly), marked the discovery of the closest visual SN since Kepler’s of 1604. It was close enough that several underground detectors which were operational at the time were able to measure the neutrino ﬂux from the core collapse of the progenitor star, the blue supergiant Sanduleak −69 202. The observed neutrinos were registered within a few seconds of 7:35:40 UT (universal Fig. 11.9. Early optical observations of SN 1987A according to the IAU Circulars, notably No. 4316 of February 24, 1987. The times of the IMB, Kamiokande II (KII) and Baksan (BST) neutrino observations (23:07:35) and of the Mont Blanc events (23:02:53) are also indicated. The solid line is the expected visual brightness, the dotted line the bolometric brightness according to model calculations. (Adapted, with permission, from Arnett et al. 1989, Annual Review of Astronomy and Astrophysics, Volume 27, c⃝1989, by Annual Reviews Inc.) Supernova Neutrinos 415 time) on 23 February 1987 while the ﬁrst evidence for optical brighten- ing was found at 10:38 UT on plates taken by McNaught (1987)—see Fig. 11.9. The main neutrino observations come from the Irvine-Michigan- Brookhaven (IMB) and the Kamiokande II water Cherenkov detectors, facilities originally built to search for proton decay, while a less sig- niﬁcant measurement is from the Baksan Scintillator Telescope (BST). A likely spurious observation is from the Mont Blanc Liquid Scintilla- tor Detector (LSD). It preceded the other observations by about 5 h, with no contemporaneous signal at Mont Blanc with the other signals, and no contemporaneous signal at the other detectors with the Mont Blanc event. The Mont Blanc detector was built to search for neutrinos from core collapse supernovae, except that it was optimized for galac- tic events within a distance of about 10 kpc. The neutrino output of a normal SN in the LMC could not have caused an observable signal at Mont Blanc; the reported events probably represent a background ﬂuctuation. Koshiba (1992) has given a lively account of the exciting and ini- tially somewhat confusing story of the neutrino measurements and their interpretation. Early summaries of the implications for astrophysics and particle physics of the neutrino and electromagnetic observations were written, for example, by Schramm (1987), Arnett et al. (1989), and Schramm and Truran (1990). A more recent review of SN 1987A is McCray (1993). A nontechnical overview was provided in a book by Murdin (1990). For the present purposes the bottom line is that SN 1987A broadly conﬁrmed our understanding of SN physics as outlined in Sect. 11.1. A remaining sore point is the lack of a pulsar observation in the SN remnant so that one may continue to speculate that a black hole has formed in the collapse. 11.3.2 Neutrino Observations In the IMB and Kamiokande water Cherenkov detectors neutrinos are measured by the Cherenkov light emitted by secondary charged par- ticles, e±for the relatively low-energy (anti)neutrinos emitted from a stellar collapse. The IMB detector is now defunct while Kamiokande continues to measure, for example, solar neutrinos until the much larger Superkamiokande detector will take up operation in 1996. In the Bak- san Scintillator Telescope (BST) one measures the scintillation light produced by charged secondary particles. 416 Chapter 11 The relevant neutrino interaction processes in water are elastic scat- tering on electrons (Sect. 10.3.4), and the charged-current reactions ep→n e+ande16O→16Fe−(Arafune and Fukugita 1987; Haxton 1987). The cross section for the epreaction is given by =G2 F cos2C( C2 V+ 3C2 A) peEe(1 +) = 9:4×10−44cm2(1 +)peEe=MeV2; (11.6) where GFis the Fermi constant and cos2C≈0:95 refers to the Cabibbo angle. The charged-current vector and axial-vector weak coupling con- stants are CV= 1 and CA= 1:26, and incorporates small corrections from recoil, Coulomb, radiative and weak magnetism corrections (Vo- gel 1984). Further, peandEerefers to the positron momentum and energy. Ignoring recoil eﬀects, the latter is Ee=E−mn+mp≈ E−1:3 MeV; the threshold is 1 :8 MeV because the minimum Eeisme. A general expression for the e16O cross section is much more com- plicated. A simple approximation, taking only the 2−state of the16F nucleus, is ≈1:1×10−44cm2(E=MeV−13)2(11.7) (Arafune and Fukugita 1987). Fig. 11.10. Total cross sections for the measurement of neutrinos in a water Cherenkov detector according to Eqs. (10.17), (11.6), and (11.7). The curves refer to the total cross section per water molecule so that a factor of 2 for protons and 10 for electrons is already included. Supernova Neutrinos 417 All of the relevant cross sections per water molecule are shown in Fig. 11.10 as a function of E. The curves incorporate a factor of 2 for proton targets (two per H 2O), and a factor of 10 for electrons (ten per H2O). Above its threshold, the e16O cross section rises very fast; it is then the dominant detection process for e’s. Still, the epreaction is the absolutely dominant mode of observing SN neutrinos. The BST detector is ﬁlled with an organic scintillator based on “white spirit” C nH2n+2with n≈9. The dominant detection reaction is alsoep→ne+. In addition, elastic scattering on electrons is possible, and the process e12C→12Ne−occurs for Ee∼>30 MeV. The trigger eﬃciencies relevant for the three detectors are shown as a function of the e±energy in Fig. 11.11. Analytic ﬁt formulae to these curves were given by Burrows (1988) for IMB and Kamiokande. IMB reports a dead time of 13% during the SN burst (Bratton et al. 1988); the IMB curve includes a factor 0.87 to account for this eﬀect. The ﬁducial volume of Kamiokande II relevant for the SN 1987A obser- vations was 2,140 tons, for IMB 6,800 tons, and for BST 200 tons. It corresponds to a target of 1 :43×1032protons at Kamiokande, 4 :6×1032 at IMB, and 1 :88×1031at BST. With the epcross section of Eq. (11.6) and the eﬃciency curves of Fig. 11.11 one may compute a prediction for the number of events Fig. 11.11. Trigger eﬃciency for electron (positron) detection at the Kamiokande (Hirata et al. 1988) and IMB (Bratton et al. 1988) water Che- renkov detectors, and the Baksan scintillator telescope (Alexeyev et al. 1988), relevant for the SN 1987A neutrino observations. In the IMB curve a factor 0.87 is included to account for their reported dead time of 13%. 418 Chapter 11 per energy interval due to the dominant epreaction in the detectors. For Kamiokande and IMB an example is shown in Fig. 11.12, based on Burrows’ (1988) model 55 ﬂux calculation which was tuned to ﬁt the data. The predicted ﬂuence at Earth per unit energy was shown in Fig. 11.8 where a distance of 50 kpc was adopted. The solid lines are for the case when the instantaneous espectra are assumed to be Maxwell-Boltzmann. Of course, the time-integrated ﬂux is then no longer thermal as it is a superposition of Maxwell-Boltzmann spectra at diﬀerent temperatures. The solid line leads to a total expectation of 13.1 events at Kamiokande and 6.3 at IMB. The dashed lines cor- respond to spectra which are instantaneously pinched (Fig. 11.6) with a degeneracy parameter = 2. Again, the time-integrated spectra are not necessarily pinched. This case leads to 11.6 events at Kamiokande and 3.9 at IMB. Fig. 11.12. Expected number of events per energy interval at Kamiokande and IMB from the SN 1987A eﬂux on the basis of the ep→ne+reaction. The ﬂux prediction is based on Burrows’ (1988) model 55 which was tuned to ﬁt the SN 1987A data. The distance is taken to be 50 kpc, and the detector eﬃciency curves of Fig. 11.8 are used. The dashed lines refer to neutrino spectra which are instantaneously “pinched” as in Fig. 11.6. Water Cherenkov detectors, as opposed to scintillation ones, are imaging devices in that they can resolve the direction of the electron (positron) because of the directionality of the emitted Cherenkov light. The main limitation is multiple Coulomb scattering of low-energy e± in the medium. A typical path length in water is only a few cm so that hard collisions are relatively unlikely. The rms angular deviation due Supernova Neutrinos 419 to multiple scattering varies from about 34◦atEe= 5 MeV to 22◦at 20 MeV (Hirata 1991). The direction of motion of the charged lepton relative to the inci- dent neutrino is well preserved in ecollisions (Sect. 10.3.4) so that the main limitation to a reconstruction of the primary neutrino direction is multiple Coulomb scattering. For the epreaction, the angular distribu- tion is isotropic, apart from a small (about 10%) backward asymmetry (e.g. Boehm and Vogel 1987). The e16O reaction yields a distribution approximately proportional to 1 −1 3cos Θ, i.e. it is also nearly isotropic with a backward bias (Haxton 1987). As a SN is expected to produce all (anti)neutrino ﬂavors in about equal numbers, the signal is dominated by the isotropic epreaction. Table 11.1. Neutrino burst at the Kamiokande detector (Hirata et al. 1988). The time is relative to the ﬁrst event at 7:35:35 ±0:01:00 UT, 23 Feb. 1987. The energy refers to the detected e±, not to the primary neutrino. Event Time Angle Energy [s] [degree] [MeV] 1 0.000 18 ±18 20 :0±2:9 2 0.107 40 ±27 13 :5±3:2 3 0.303 108 ±32 7 :5±2:0 4 0.324 70 ±30 9 :2±2:7 5 0.507 135 ±23 12 :8±2:9 6a0.686 68 ±77 6 :3±1:7 7 1.541 32 ±16 35 :4±8:0 8 1.728 30 ±18 21 :0±4:2 9 1.915 38 ±22 19 :8±3:2 10 9.219 122 ±30 8 :6±2:7 11 10.433 49 ±26 13 :0±2:6 12 12.439 91 ±39 8 :9±1:9 13a;b17.641 : : : 6:5±1:6 14a;b20.257 : : : 5:4±1:4 15a;b21.355 : : : 4:6±1:3 16a;b23.814 : : : 6:5±1:6 aUsually attributed to background. bQuoted after Loredo and Lamb (1995). 420 Chapter 11 Table 11.2. Neutrino burst at the IMB detector (Bratton et al. 1988). The time is relative to the ﬁrst event at 7:35:41.374 ±0:00:00.050 UT, 23 Feb. 1987. The energy refers to the detected e±, not to the primary neutrino. Event Time Angle Energy [s] [degree] [MeV] 1 0.000 80 ±10 38 ±7 2 0.412 44 ±15 37 ±7 3 0.650 56 ±20 28 ±6 4 1.141 65 ±20 39 ±7 5 1.562 33 ±15 36 ±9 6 2.684 52 ±10 36 ±6 7 5.010 42 ±20 19 ±5 8 5.582 104 ±20 22 ±5 Table 11.3. Neutrino burst at the Baksan detector (Alexeyev et al. 1987, 1988). The time is relative to the ﬁrst event at 7:36:06.571+02:000 −54:000UT, 23 Feb. 1987. The energy refers to the detected e±, not to the primary neutrino. Event Time Energy [s] [MeV] 0a0.000 17 :5±3:5 1 5.247 12 :0±2:4 2 5.682 18 :0±3:6 3 6.957 23 :3±4:7 4 12.934 17 :0±3:0 5 14.346 20 :1±4:0 aUsually attributed to background. The energy of the electron (positron) can be reconstructed from the total amount of Cherenkov or scintillation light emitted. For small energies it is roughly proportional to the number of photomultipliers hit in a given event. Because of the reaction threshold and recoil eﬀects, the energy of the primary neutrino in the ep→ne+reaction is about 2 MeV larger than the measured e+energy. For the rare ecollisions, Supernova Neutrinos 421 the ﬁnal-state electron energy distribution is broad so that one can infer only a lower limit to the energy. The measured events at the Kamiokande (Hirata et al. 1987, 1988), IMB (Bionta et al. 1987; Bratton et al. 1988), and BST (Alexeyev et al. 1987, 1988) detectors are summarized in Tabs. 11.1, 11.2, and 11.3. The absolute timing at IMB is accurate to within ±50 ms while at Kamiokande only to within ±1 min. At BST, the clock exhibited an erratic behavior which led to an uncertainty of +2 =−54 s. Within the timing uncertainties the three bursts are contemporaneous and may Fig. 11.13. SN 1987A neutrinos at Kamiokande, IMB, and Baksan. The energies refer to the secondary positrons, not the pimary neutrinos. In the shaded area the trigger eﬃciency is less than 30%. The detector clocks have unknown relative oﬀsets; in each case the ﬁrst event was shifted to t= 0. In Kamiokande and Baksan, the events marked with open circles are usually attributed to background. 422 Chapter 11 thus be simultaneously attributed to SN 1987A. The events are shown in the t-E-plane in Fig. 11.13 and for the Cherenkov detectors in the cos Θ- E-plane in Fig. 11.14 where Θ is the angle relative to the opposite direction of the SN, i.e. relative to the direction of the neutrino ﬂux. In principle, the bursts of events observed in the detectors could be due to rare background ﬂuctuations rather than due to SN 1987A. The Kamiokande group has performed a detailed analysis of this possibility by analyzing the multiplicity of events in 10 s time intervals, i.e. the number of chance events in an arbitrarily chosen 10 s time interval. They found a probability of about 0 :6×10−7that the observed burst is a random ﬂuctuation of a constant background. However, there are time-correlated backgrounds, notably the spal- lation of oxygen induced by primary muons which can cause clusters of events with large multiplicities; the Kamiokande group found one clus- Fig. 11.14. SN 1987A neutrinos at Kamiokande and IMB, excluding the ones which likely are due to background. Supernova Neutrinos 423 ter of 53 events! Therefore, any cluster following a high-energy muon would be very suspicious. Performing a cut on the data for this back- ground leaves no burst with multiplicity 3 or larger in several data sets of several hundred days each (Hirata 1991). Therefore, it is extremely unlikely that the event cluster 10 −12 in the Kamiokande data has been caused by background. The BST detector has a relatively large background rate. Event clusters of multiplicity 5 or more within 9 s occur about once per day. Thus the probability for such a background cluster to fall within a minute of the IMB and Kamiokande events is about 5 ×10−4. 11.3.3 Analysis of the Pulse Many authors have studied the distribution of energies and arrival times of the reported events. Probably the most signiﬁcant work is that of Loredo and Lamb (1989, 1995) who performed a maximum-likelihood analysis, carefully including the detector backgrounds and trigger ef- ﬁciencies. Loredo and Lamb also gave detailed references to previous works, and in some cases oﬀered a critique of the statistical method- ology employed there. Their more extensive 1995 analysis supersedes certain aspects of the earlier methodology and results. Because of the small number of neutrinos observed, a relatively crude parametrization of the time-varying source is enough. Among a variety of simple single-component emission parametrizations, Loredo and Lamb (1989, 1995) found that an exponential cooling model was preferred. It is characterized by a constant radius of the neutrino sphere, R, and a time-varying eﬀective temperature T(t) =T0e−t=4; (11.8) so that is the decay time scale of the luminosity which varies with the fourth power of the temperature according to the Stefan-Boltzmann law. It should be noted, however, that numerical cooling calculations do not yield exponential lightcurves. For example, the model shown in Fig. 11.7 displays an exponential decline of the eﬀective temperature, but a power-law decline of the neutrino luminosity. Other calculations even yield early heating and a constant temperature for some time (see Burrows 1990b for an overview). Of course, for the time-integrated spectrum the exponential cooling law is just another assumption concerning the overall spectral shape. For example, one easily ﬁnds that the average eenergy of the time- integrated spectrum is ⟨Ee⟩= 2:36T0if Fermi-Dirac distributions with 424 Chapter 11 = 0 are taken for the instantaneous spectra. The time-integrated spectrum of the exponential cooling model looks quite similar to the time-integrated spectrum shown in Fig. 11.8. Loredo and Lamb also used ≡(R=10 km) (50 kpc =D)g1=2as a ﬁt parameter where Dis the distance to SN 1987A and ga statistical weight factor which is unity if only left-handed, massless or low-mass neutrinos of the three sequential ﬂavors are emitted. The registration time of the ﬁrst neutrino in each detector is taken as a free parameter relative to the arrival time of the ﬁrst neutrinos. In the 1995 analy- sis, Loredo and Lamb included the Baksan signal without “event 0” which is attributed to background because it precedes the main bunch by 5 s. The following six parameters are then allowed to ﬂoat freely in order to achieve a maximum-likelihood result: T0,,,toﬀ(IMB), toﬀ(KII), andtoﬀ(BST). All best-ﬁt oﬀset times are found to be zero. The other best-ﬁt values are = 4:02,= 4:37 s, and T0= 3:81 MeV. This initial temperature of the exponential cooling model corresponds to an average neutrino energy of the time-integrated ﬂux of ⟨Ee⟩= 9:0 MeV. In Fig. 11.15, the 68% and 95% credible regions are shown in the T0-- plane where T0has been translated into ⟨Ee⟩which is of greater direct relevance. Fig. 11.15. Two-dimensional marginal distribution for the parameters and ⟨Ee⟩= 2:36T0of the exponential cooling model. (Curves courtesy of Tom Loredo, taken from Loredo and Lamb 1995.) Supernova Neutrinos 425 Given these parameters one infers that the number of expected Kamiokande events is 16.9 plus 5.6 background, 4.0 events at IMB, and 1.8 plus 1.0 background at Baksan. The inferred best-ﬁt neutrino- sphere radius is 40 :2 km, and the inferred total emitted eenergy is 0:84×1053erg which corresponds to a total binding energy of the neu- tron star of 5 :02×1053erg if exact equipartition of the energy among the neutrino ﬂavors is assumed. The inferred average neutrino energy, the luminosity-decay time scale, radius of the source, and total energy emitted all agree reasonably well with what one expects from a core collapse SN, even though ⟨Ee⟩ is somewhat low, the radius and inferred binding energy somewhat large. Of course, all of the inferred quantities carry large uncertainties because of the sparse data. Loredo and Lamb (1995) have also considered two-component cool- ing schemes where the neutrino signal is modelled to consist of Kelvin- Helmholtz cooling, plus a low-energy component which mimics the neu- trinos emitted by the accreting matter during the stalled-shock phase in the delayed-explosion scenario. With more parameters they nat- urally ﬁnd a better ﬁt to the data. More interestingly, the inferred Fig. 11.16. Two-dimensional marginal distribution for the parameters and ⟨Ee⟩= 2:13T0of the Kelvin-Helmholtz component of the Loredo and Lamb (1995) best-ﬁt two-component cooling model. The parameters andT0are those of a displaced power law as described in the text. (Curves courtesy of Tom Loredo, taken from Loredo and Lamb 1995.) 426 Chapter 11 neutrino-sphere radius of 18 km and the inferred total binding energy of 3:08×1053erg correspond much better to theoretical expectations. Perhaps this ﬁnding can be taken as a hint that the delayed-explosion scenario with a signiﬁcant matter accretion phase is favored by the SN 1987A data over a prompt-explosion picture. In Loredo and Lamb’s best-ﬁt two-component model the Kelvin- Helmholtz signal is described by a “displaced power law” cooling model with a neutrino sphere of ﬁxed radius Rand thermal neutrino emission with T(t) =T0=(1 +t=3). It turns out that the luminosity, which is proportional to T4, follows a surprisingly similar curve to the exponen- tiale−t=so that the parameter has practically the same meaning as before. ⟨Ee⟩for the time-integrated ﬂux is given by 2 :13T0, very similar to 2 :36T0for the exponential. In Fig. 11.16 the 68% and 95% credible regions are shown in the T0--plane. While the best-ﬁt value is not too diﬀerent from the single-component exponential model of Fig. 11.15, the 95% credible region is much larger, including the lowest values of the typical theoretical ⟨Ee⟩predictions quoted in Eq. (11.4). 11.3.4 Neutrino Mass and Pulse Duration Zatsepin (1968) was the ﬁrst to point out that the eburst expected from stellar collapse oﬀers a possibility to measure or constrain small neutrino masses. Because a neutrino with mass mtravels slower than the speed of light its arrival at Earth will be delayed by ∆t= 2:57 s(D 50 kpc)(10 MeV E)2(m 10 eV)2 : (11.9) Because the measured e’s from SN 1987A were registered within a few seconds and had energies in the 10 MeV range, the memass is limited to less than about 10 eV. A detailed study must proceed along the lines of the maximum- likelihood analysis of Loredo and Lamb (1989, 1995) quoted in the previous section where the detector background is included, and such parameters as the unknown oﬀset times between the detectors are left unconstrained. Including the possibility that some of the registered events are due to background is particularly important because the neutrino mass limit is very sensitive to the early low-energy events at Kamiokande which have a relatively high chance of being due to back- ground. Loredo and Lamb (1989) found a vanishing best-ﬁt neutrino mass and a 95% CL upper limit of me<23 eV. This bound is less restrictive than limits found by previous authors on the basis of less Supernova Neutrinos 427 thorough statistical analyses. In their 1995 paper, Loredo and Lamb have not studied neutrino mass limits which likely would change some- what because of corrections to their previous approach. An analysis by Kernan and Krauss (1995) on the basis of a similar method yields a limit 19 :6 eV at 95% CL. Apparently, the reduction of the limit is due to their inclusion of the 13% dead-time eﬀect in the IMB detector. The diﬀerence to the Loredo and Lamb (1989) limit illustrates that changing a relatively ﬁne point of the analysis procedure can signiﬁcantly change a so-called 95% CL limit. Therefore, instead of quoting a speciﬁc conﬁdence limit it is at present more realistic to state qualitatively that a violation of the mass limit me∼<20 eV (11.10) would have caused a signiﬁcant and perhaps intolerable modiﬁcation of the SN 1987A signal. This limit is weaker than the current bounds from the tritium decay endpoint spectrum (Sect. 7.1.3). Therefore, the above analysis can be turned around in the sense that the observed neutrino signal duration is probably representative of the duration of neutrino emission at the source. The observed long time scale of Kelvin-Helmholtz cooling which is indicated by the late IMB and Kamiokande events cannot be blamed on neutrino dispersion eﬀects. In this context it is interesting to observe that Loredo and Lamb (1989) also performed a maximum- likelihood analysis with meheld ﬁxed at their 95% CL upper limit 23 eV. The best-ﬁt time scale in an exponential cooling model changed from 4 :15 to 2 :96 s. Therefore, even assuming a large value for medid not allow one to contemplate a signiﬁcantly shorter Kelvin-Helmholtz cooling phase than implied by massless neutrinos. 11.3.5 Anomalies in the Signal? The distribution of the Kamiokande and IMB events shows a number of puzzling features. The least worrisome of them is a certain discrepancy between the neutrino energies observed in the two detectors which point to a harder spectrum at IMB. The maximum-likelihood analysis in the exponential cooling model of Loredo and Lamb (1989) was also per- formed for the two detectors separately. The 95% conﬁdence volumes projected on the T0-Eb-plane are shown in Koshiba (1992); a similar re- sult is found in Janka and Hillebrandt (1989b). There is enough overlap between the conﬁdence contours to allow for a joint analysis. Still, the best-ﬁt value for Kamiokande lies outside the 95% CL volume of IMB. 428 Chapter 11 Including the pinching eﬀect discussed in Sect. 11.2.2 would enhance the discrepancy between the signals in the two detectors. Either way, the IMB detector with its high energy threshold is mostly sensitive to the high-energy tail of the neutrino spectrum. Therefore, the IMB- inferred ⟨Ee⟩depends sensitively on the assumed spectral shape and is thus a poor indicator of the true average energies. A more conspicuous anomaly is the 7 :3 s gap between the ﬁrst 9 and last 3 events at Kamiokande. Ideas proposed to explain the alleged pulsed structure of the signal range from the occurrence of a phase transition in the nuclear medium (pions, quarks) to a secondary collapse to a black hole. It should be noted, however, that the gap is partially ﬁlled in by the IMB and Baksan data, thus arguing against a physical cause at the source. The random occurrence of a gap exceeding 7 s with three or more subsequent events can be as high as several percent, but naturally it is sensitive to the expected late-time signal (Lattimer and Yahil 1989). The most signiﬁcant and thus the most troubling anomaly is the remarkable deviation from isotropy of the events in both detectors, in conﬂict with the expected signature from ep→ne+which actually predicts a slight (about 10%) backward bias. LoSecco (1989) found a probability of about 1.5% that the combined Kamiokande and IMB data set was drawn from an isotropic distribution. Kie lczewska (1990) analyzed the expected signal from standard SN cooling calculations and found agreement only at the 0.8% CL with the measured angu- lar distribution. The combined set of IMB plus those Kamiokande data which are above the IMB threshold, i.e. the combined set of “high-energy” events is consistent with isotropy only at the 0.07% level (van der Velde 1989); the four relevant events at Kamiokande are all very forward. The IMB collaboration claims that their reconstruction of the event direction was not seriously impeded by the outage of about a quarter of their phototubes due to the failure of a high-voltage supply. They con- ducted a detailed calibration of their detector to investigate this point (Bratton et al. 1988). Therefore, one must accept that the forward- peaked angular distribution shown in Fig. 11.14 is not a problem of the detectors or the event reconstruction. A forward-peaked distribution is expected from eelastic scattering which has a much lower cross section than the epprocess (Fig. 11.8). One expects far less than one event due to escattering from the cool- ing phase, although the ﬁrst Kamiokande event has sometimes been interpreted as being a scattering event due to the prompt eburst. Supernova Neutrinos 429 The forward events in both detectors have relatively large energies compared with the isotropic ones at Kamiokande,66contrary to what would be expected from e→ewhere some of the energy is carried away by the secondary neutrino. Assuming a larger-than-standard ﬂux ofe’s with larger-than-standard energies (LoSecco 1989) does not solve the problem because above 30 −35 MeV the process e16O→16Fe− takes over (Fig. 11.8) which has a backward bias. Anomalously large ﬂuxes of ;or;are diﬃcult to arrange on energetic grounds—the binding energy of the neutron star is limited. Even allowing for extreme values of Eband extreme temperature diﬀer- ences between (anti)electron neutrinos and the other ﬂavors improves the agreement only marginally; one can achieve an agreement at the 5% CL with the observed angular distribution (Kie lczewska 1990). The simple problem with elastic escattering to explain the data is that this process is too strongly forward peaked, especially for high-energy neutrinos, hence it does not ﬁt the data very well either. This is espe- cially true for the IMB events which are selected for high energies by the detector threshold, and yet are very broadly distributed around the forward direction. A very speculative idea was put forth by van der Velde (1989) who proposed the existence of a new neutral boson X◦which could pro- duce photons when interacting with nucleons. These MeV photons would look very similar to charged particles in the detectors. In the forward direction, the X◦cross section on16O would be coherently en- hanced, causing the observed forward bias for high-energy events while low-energy ones would naturally follow a more isotropic distribution. However, the opposite process +4He→4He + X◦would then con- tribute to the energy-loss of horizontal-branch stars (Raﬀelt 1988b). The resulting bound on the interaction cross section (Tab. 2.5), valid at an energy of about 10 keV, excludes van der Velde’s scenario unless increases with energy at least as E2, a scaling which could bring it up to the requisite level for Ein the 10 MeV range, relevant for the SN detection. The Primakoﬀ conversion of axions or similar particles on oxygen is much too ineﬃcient in view of the restrictive limits on the axion-photon interaction strength. In summary, the angular and energy distributions of the IMB and Kamiokande events appear to indicate a low-energy isotropic and a 66The Kamiokande events have an obvious correlation between energy and di- rection. The application of Spearman’s rank-ordering test (e.g. Press et al. 1986) gives a conﬁdence level of 0.06% where event No. 6 was excluded as background. Therefore, the Kamiokande data alone show a fairly signiﬁcant “angular anomaly.” 430 Chapter 11 high-energy forward component; they are not ﬁt well by the assumed dominant detection process ep→ne+which is supposed to yield an isotropic positron distribution with no directional correlation with en- ergy. Elastic escattering, however, does not ﬁt the data well either because it is too forward peaked, and anyhow it is disfavored by a small cross section unless the ﬂux of ;or;was extremely high. However, because no plausible and/or viable nonstandard cause for the observed events has been proposed one has settled for the interpre- tation of a statistical ﬂuctuation for the apparent anomalies. After all, it is diﬃcult to imagine a small sample drawn from any distribution without some “anomalies” which are easy to overinterpret. Still, if a reasonable alternative to the standard interpretation of the signal were to come forth this topic would have to be reconsidered. 11.4 Neutrino Oscillations 11.4.1 Overview The expected neutrino signature from a stellar collapse and conversely the inferred protoneutron star properties from the SN 1987A neutrino signal both depend on the assumption that “nothing happens” to the neutrinos on their way to us. One simple modiﬁcation of the expected signal is a dispersion of the eburst caused by a nonvanishing mein the 10 eV range (Sect. 11.3.4). Dispersion eﬀects could also be caused by novel interactions with the galactic magnetic ﬁeld, dark matter, the neutrino background, or simply by decays. All of these scenarios require relatively exotic particle-physics assumptions which can be constrained by the SN 1987A signal (Chapter 13). The assumption of small neu- trino masses and mixings, however, ﬁts into the standard model with minimal extensions, and may already be implied by the solar neutrino observations (Sect. 10.6). Therefore, it is prudent not to ignore the possible impact of oscillations on SN neutrinos. The most obvious consequence is that the prompt eburst could os- cillate into another ﬂavor which then would be much harder to observe because of the reduced -ecross section for non- eﬂavors (Fig. 11.10). Notably, if the solar eﬂux is depleted by resonant oscillations one may expect the same in the SN mantle and envelope where a large range of densities and density gradients is available. It will turn out, however, that the small-angle MSW solution to the solar neutrino problem leaves an observable eburst (Sect. 11.4.2). Supernova Neutrinos 431 Another interesting possibility is a partial swap e↔;ande↔ ;by oscillations. Because the energy spectrum of the non- eﬂavors is much harder than that of eore, a number of interesting consequences obtain. First, the detected e’s could have larger average energies than expected. Conversely, the SN 1987A-implied emission temperature and neutron-star binding energy could be an overestimate of the true values. It will turn out that one seriously needs to worry about these eﬀects, for example, if the large-angle MSW solution or the vacuum solution to the solar neutrino problem obtain, or if the atmospheric neutrino anomaly is caused by oscillations (Sect. 11.4.3). Even more importantly, a swap e↔ore↔would cause a more eﬃcient energy transfer from the neutrino ﬂux to the matter behind the stalled shock after core bounce but before the ﬁnal explo- sion. As enhanced neutrino heating actually appears to be required to obtain successful and suﬃciently energetic explosions, neutrino oscil- lations may help to explode supernovae! This scenario works only if the spectral swap occurs inside of the stalled shock wave. In view of the relevant medium densities, resonant transitions obtain for neutrino masses in the cosmologically interesting range of 10 −100 eV. A mixing angle as small as sin22∼>3×10−8would be enough (Sect. 11.4.4). Hardening the espectrum by a swap with or, however, can suppress r-process nucleosynthesis just outside of the nascent neutron star a few seconds after core bounce. Normally the espectrum is harder than the espectrum, driving equilibrium in the hot bub- ble to the required neutron-rich phase. The oscillation scenario can cause the reverse. For this eﬀect the oscillations would need to oc- cur close to the protoneutron star surface and so again a relatively large neutrino mass-square diﬀerence is required which falls into the cosmologically interesting range. However, the required mixing angle is larger (sin22∼>10−5), leaving ample room for, say, small e- mixing angles where ’s with cosmologically relevant masses could help explode supernovae without disturbing r-process nucleosynthe- sis (Sect. 11.4.5). Finally, neutrino oscillations could allow the non- eﬂavors to par- ticipate in equilibrium in the inner core and thus build up their own degenerate Fermi seas. This possibility has been studied in Sect. 9.5 where it turned out that a signiﬁcant ﬂavor conversion obtains only for large neutrino masses (keV range and above). Such large masses are cosmologically forbidden unless neutrinos decay fast into invisible channels, a hypothesis that would require novel neutrino interactions beyond masses and mixings (Sect. 12.5.2). 432 Chapter 11 11.4.2 Prompt eBurst The prompt eburst from a core collapse SN can be detected, in princi- ple, by the forward-peaked signal from the elastic ee→eescattering in a water Cherenkov detector. In the Kamiokande SN 1987A observa- tions, the ﬁrst event could have been caused by the prompt eburst, but naturally one event contains no statistically signiﬁcant information. It could have been caused by ep→ne+and simply happen to point in the forward direction. Therefore, the main interest in the prompt e burst is the possibility that it could be observed from a future galactic SN by the Superkamiokande or SNO detectors which would yield sta- tistically signiﬁcant signatures. A possible oscillation of the eburst into other ﬂavors would reduce the number of forward events because of the reduced -escattering cross section of non- eﬂavors (Fig. 11.10). If the neutrino mass hierarchy is normal where the lightest mass eigenstate is the dominant eadmixture, the medium-induced neutrino refractive index in the stellar mantle and envelope can cause a “mass inversion” and thus level crossing between, say, eandin analogy to the solar MSW eﬀect. Therefore, one may expect resonant ﬂavor conversion of the prompt eburst in a collapsing star as shown by a number of authors;67I follow the analysis of N¨ otzold (1987). When the shock wave breaks through the neutrino sphere and lib- erates the prompt eburst, the overlaying part of the progenitor star has not yet noticed the collapse of its core so that the density proﬁle is given by that of the progenitor star. The electron density is reasonably well approximated by a simple power law for which N¨ otzold (1987) used ne≈1034cm−3r−3 7; (11.11) where r7≡r=107cm. Note that 107cm = 100 km is the approximate radius of the shell from where the e’s originate. According to the discussion in Sect. 6.7.1 the electron density causes an energy shift between eandorof ∆V=√ 2GFne= 1:3×10−3eVr−3 7. Com- paring this with the energy shift ∆ m2 =2pof neutrinos with momentum pone ﬁnds an eﬀective medium-induced eﬀect of ∆ m2 eﬀ= 2p∆V= 3×104eV2p10r−3 7where p10=p=10 MeV. However, because the prompt eburst itself constitutes a large local edensity, the neutrino-induced refractive index may be more impor- tant than the standard electron-induced contribution. The total num- 67Mikheev and Smirnov (1986), Arafune et al. (1987a,b), Lagage et al. (1987), Minakata et al. (1987), N¨ otzold (1987), Walker and Schramm (1987), Kuo and Pantaleone (1988), Minakata and Nunokawa (1988), and Rosen (1988). Supernova Neutrinos 433 ber of e’s in the burst is of order 1056, its duration of order 50 ms. Thus, while it passes it represents a edensity of about 1032cm−3r−2 7which exceeds the local electron density for r∼>109cm. However, the phase- space distribution of the neutrinos is locally far from isotropic and so the energy shift involves a factor ⟨1−cos Θ⟩where Θ is the angle be- tween the “test neutrino” and a “background neutrino;” the average is to be taken over all background neutrinos (Sect. 9.3.2). A typical angle between two neutrinos moving within the burst at the same location is given by the angle subtended by the neutrino sphere as viewed from the relevant radial position, i.e. Θ ≈R=rwith Rthe radius of the neu- trino sphere. For a large rone thus ﬁnds ⟨1−cos Θ⟩ ≈Θ2≈(R=r)2. Then, with R≈107cm the eﬀective neutrino density is approximately 1032cm−3r−4 7, a value which is always smaller than the electron density Eq. (11.11). Therefore, in the present context one may ignore neutrino- neutrino interactions. This will not be the case for the issue of r-process nucleosynthesis (Sect. 11.4.5). One may proceed to determine the MSW triangle as in Fig. 8.9 for solar neutrinos, except that there an exponential electron density proﬁle was used while now Eq. (11.11) pertains. N¨ otzold (1987) found a conversion probability in excess of 50% if ∆m2 sin32∼>4×10−9eV2E=10 MeV ; (11.12) assuming that ∆ m2 ∼<3×104eV2E=10 MeV so that a resonance can occur outside of the neutrino sphere. The region in the ∆ m2 -sin22- plane (mixing angle ) with a conversion probability exceeding 50% for E= 20 MeV is shown as a shaded area in Fig. 11.17, together with the MSW solutions to the solar neutrino problem. For orientation, the Kamiokande-allowed range for solar neutrinos is also indicated. The solar small-angle MSW solution would seem to have a small impact on the prompt eburst from a collapsing star. Thus, the ﬁrst Kamiokande SN 1987A event may still be interpreted as a prompt e, and one may well observe nearly the full eburst from a future SN. It is interesting that the MSW triangle for the prompt eburst reaches to relatively large neutrino masses. In the 3 −30 eV regime neu- trino masses would play an important cosmological role as dark matter and for the formation of structure in the universe. Such massive neu- trinos likely would mix with e. Unless the mixing angle is very small the appearance of an unoscillated prompt eburst from a stellar col- lapse would be in conﬂict with a cosmological role of massive neutrinos (Arafune et al. 1987b). 434 Chapter 11 Fig. 11.17. MSW triangle for the prompt eburst from a stellar collapse. In the shaded area the conversion probability exceeds 50% for E= 20 MeV, assuming the electron density proﬁle of Eq. (11.11). The MSW solutions to the solar neutrino problem and the Kamiokande allowed range for solar neutrinos are indicated (see Fig. 10.19). 11.4.3 Cooling-Phase e's Neutrino oscillations would cause a partial swap e↔;ande↔ ;so that the measured eﬂux at Earth could be a mixture of the original eandorsource spectra (Wolfenstein 1987). The energy spectra of the neutrinos emitted during the Kelvin-Helmholtz cooling phase are ﬂavor dependent (Eq. 11.4); typically, one ﬁnds ⟨E⟩= (1:3−1:7)×⟨Ee⟩. From the SN 1987A measurements one infers a low value of roughly ⟨Ee⟩ ≈10 MeV. While the lowest typical predictions are about 14 MeV, this discrepancy is not a serious problem. However, it is probably not tolerable that a signiﬁcant fraction of the observed events were due to oscillated ’s. One may contemplate an “inverted” mass hierarchy where the pre- dominant mass component of eis larger than that of, say, . In this case one would obtain resonant oscillations and thus a complete spectral swap in the shaded triangle of mixing parameters shown in Fig. 11.17. Even if ⟨E⟩is only 1 :3×⟨Ee⟩this would be in contradiction with the softeenergies observed from SN 1987A. Therefore, an inverted mass scheme looks excluded for a large range of masses and mixings. Supernova Neutrinos 435 A “normal” mass hierarchy prevents a level crossing among an- tineutrinos. Still, if the large-angle or the vacuum solution to the so- lar neutrino problem obtain, the mixing angle would be so large that the spectral swapping could still be uncomfortably large. Smirnov, Spergel, and Bahcall (1994) have considered in detail the probability for swapping the ewith the orspectrum. For vacuum oscilla- tions, relevant for small ∆ m2 , the fractional exchange of the spectra isp=1 2sin22(vacuum mixing angle ) with a maximum of p= 0:5, i.e. the observed spectrum could be as much as an equal mixture of the primary ones. In general, medium refractive eﬀects must be included, although for a large range of ∆ m2one may still take pto be indepen- dent of energy. Contours for pin the sin22-∆m2 -plane are shown in Fig. 11.18. In the shaded area (A) the Earth eﬀect is important so that the amount of conversion is energy dependent and diﬀers between the Fig. 11.18. Contours for the “swap fraction” pbetween the eand the orﬂuxes from the protoneutron star cooling phase. The matter eﬀect of the stellar envelope and of the Earth are included. In (A) the Earth eﬀect is important; in this area pis an average over neutrino energies while otherwise it does not depend on the energy. In (B) the exact contours depend on the detailed matter distribution of the stellar envelope. Black areas indicate the approximate mixing parameters which would explain the solar neutrino problem. (Adapted from Smirnov, Spergel, and Bahcall 1994.) 436 Chapter 11 two detectors. In this case the contours refer to an average value ⟨p⟩. In the shaded area (B) the contours are not independent of the detailed matter distribution in the stellar envelope. In Fig. 11.18 the large-angle MSW and the vacuum oscillation solu- tions of the solar neutrino problem are indicated. If either one of them is correct the measurable espectrum in a detector is a substantial mix- ture of diﬀerent-ﬂavor source spectra. Smirnov, Spergel, and Bahcall (1994) argued on the basis of a joint analysis between the SN 1987A signals at the Kamiokande and IMB detectors that p <0:17−0:27 at the 95% CL, depending on the assumed primary neutrino spectra. If pwere any larger, the expected spectra would be much harder than has been observed. This analysis excludes the vacuum oscillation solu- tion to the solar neutrino problem. Kernan and Krauss (1995) arrive at the opposite conclusion that all mixing angles are permitted by the SN 1987A signal, and that sin22= 0:45 is actually a favored value. 11.4.4 Shock Revival The “swap fraction” of the ewith the more energetic orspec- trum discussed in the previous section is always very small unless the mixing angle is very large because of the assumed normal mass hier- archy which prevents resonant conversions. By the same token a swap of the ewith the orspectrum will be resonant for certain mix- ing parameters and so it can be almost complete even for very small mixing angles. If the resonance is located between the neutrino sphere and the stalling shock wave after bounce, but before the ﬁnal explo- sion, the shock would be helped to rejuvenate because the higher-energy ;’s are more eﬃcient at transferring energy once they have converted intoe’s (Fuller et al. 1992). Of course, approximately equal luminosi- ties in all ﬂavors have been assumed. A typical density proﬁle for a SN model 0 :15 s after bounce is shown in Fig. 11.19; the step at a radius of about 400 km is due to the shock front. A resonance occurs if ∆ m2 =2E=√ 2GFne. On the right scale of the plot the quantity mres≡(√ 2GFne2E)1=2is shown for E= 10 MeV and Ye= 0:5. A resonance occurs inside of the shock wave only for neutrino masses in the cosmologically interesting regime of order 10 eV and above. Fuller et al. (1992) have performed a detailed numerical calculation of the additional heating eﬀect for (∆ m2 )1=2= 40 eV; they found a 60% increase of the energy of the shock wave. The conversion probability between neutrinos is large if the adia- baticity parameter deﬁned in Eq. (8.39) far exceeds unity. Accord- Supernova Neutrinos 437 Fig. 11.19. Typical density proﬁle for a SN model at 0 :15 s after the core bounce (Fuller et al. 1992). The right-hand scale is mres= (√ 2GFne2E)1=2 which indicates the resonance value for (∆ m2 )1=2forE= 10 MeV, assuming an electron number fraction of Ye= 0:5. ing to the Landau-Zener formula Eq. (8.41) the swap probability is 1−e− =2. It exceeds 86% if one requires >4=or sin22 >8 E ∆m2 |∇lnne|res; (11.13) where the vacuum mixing angle was assumed to be small. For (∆m2 )1=2= 40 eV the density scale height at the resonance region is |∇lnne|−1 res≈50 km so that sin22∼>10−8E=10 MeV : (11.14) Therefore, unless the mixing angle with eis very small a cosmologically interesting neutrino mass for, say, the may help to explode super- novae! According to the discussion in Sect. 11.4.2 this would imply that the prompt eburst would also oscillate. Then the ﬁrst Kamiokande event could not be associated with the prompt eburst. 11.4.5 R-Process Nucleosynthesis a) Basic Picture A partial swap of the ecooling ﬂux with the more energetic or ﬂux can prevent the synthesis of heavy nuclei by the r-process neutron 438 Chapter 11 capture. As discussed in Sect. 11.1.4, the hot bubble between the set- tled protoneutron star and the escaping shock wave at a few seconds after core bounce might be an ideal high-entropy environment for this process for which no other site is currently known that could reproduce the observed galactic heavy element abundance and isotope distribu- tion. Naturally, the r-process can only occur in a neutron-rich medium (Ye<1 2). The p=nratio in the hot bubble is governed by the reac- tions en↔pe−andep↔ne+. Because the neutrino number density is much larger than the ambient e+e−population the proton/neutron fraction is governed by the neutrino spectra and ﬂuxes. The system is driven to a neutron-rich phase because normally the e’s are more energetic than the e’s. They emerge from deeper and hotter regions of the star because their opacity is governed by the same reactions, and because the core is neutron rich, yielding a larger opacity for e. If an exchange e↔;occurs outside of the neutrino sphere the subsequent eﬂux is more energetic than the eﬂux which did not undergo a swap. (A normal mass hierarchy has been assumed.) Even a partial swap of a few 10% is enough to shift the medium to a proton- rich state, i.e. to Ye>1 2, to be compared with the standard values of 0:35−0:46. Therefore, the occurrence of such oscillations would be in conﬂict with r-process nucleosynthesis in supernovae (Qian et al. 1993). Fig. 11.20. Density proﬁle for a SN model at 6 s after the core bounce, typical for the “hot bubble phase” (Qian et al. 1993). The right-hand scale ismres= (√ 2GFne2E)1=2which indicates the resonance value for (∆ m2 )1=2 forE= 10 MeV, assuming Ye= 0:5. (Note that out to a few km above the neutrino sphere Ye≪0:5.) Supernova Neutrinos 439 The approximate parameter range for which this eﬀect is important can be estimated from the density proﬁle of a typical SN core a few seconds after bounce (Fig. 11.20). Again, one expects resonant conver- sions for the cosmologically interesting neutrino mass range as in the above shock revival scenario. However, the present eﬀect reaches down tomof about 3 eV; for smaller masses the oscillations would occur at radii too large to have an impact on nucleosynthesis. With regard to the required mixing angle there is an important dif- ference to the previous case because the neutron star has already settled so that the neutrino sphere is now at a radius of about 11 km rather than at 50 km. Therefore, the relevant length scales in Fig. 11.20 are reduced relative to Fig. 11.19 which corresponds to a postbounce but preexplosion conﬁguration. For example, at mres≈40 eV the density scale height is now |∇lnne|−1 res≈0:3 km, about two orders of magni- tude smaller than before so that the lower limit on sin22is about two orders of magnitude larger than it was for the shock revival sce- nario. From a more detailed analysis Qian et al. (1993) found the hatched area in Fig. 11.21 where Yewould be driven beyond 0 :5 and so this area would be in conﬂict with r-process nucleosynthesis in super- novae. Fig. 11.21. Mass diﬀerence and mixing angle of ewithorwhere a spec- tral swap would be eﬃcient enough to help explode supernovae (schemat- ically after Fuller et al. 1992), and where it would prevent r-process nu- cleosynthesis (schematically after Qian et al. 1993; Qian and Fuller 1994). 440 Chapter 11 b) Impact of Neutrino-Neutrino Interactions The discussion so far has been relatively simplistic because the role of neutrino-neutrino interactions has been ignored. During the hot-bubble phase the neutrino refractive index caused by other neutrinos is not nec- essarily negligible (Pantaleone 1995; Qian and Fuller 1995). In terms of the neutrino luminosity the number ﬂux and thus the density of neutri- nos of a given species at a radius ris given by n=L⟨E⟩−1(4r2)−1. Moreover, as in the discussion in Sect. 11.4.2 one must include an aver- age of the factor (1 −cos Θ) to account for the anisotropy of the neu- trino phase space distribution (angle Θ between test and background neutrino). At a given distance rthe neutron star (neutrino sphere radius R) subtends an angle given by sin Θ R=R=r. For a radially moving test neutrino one ﬁnds (for a more rigorous treatment see e.g. Qian and Fuller 1995) ⟨1−cos Θ⟩=∫1 cos Θ R(1−cos Θ) dcos Θ/∫1 cos Θ Rdcos Θ =1 2[ 1−√ 1−(R=r)2] ; (11.15) which for large rapproaches1 4(R=r)2. Therefore, the eﬀective neutrino density n⟨1−cos Θ⟩varies as r−4at large distances. The electrons and positrons cause a refractive energy shift between eand, say, of ∆V=√ 2GF(ne−ne+) while the eﬀect of the neutrinos is ∆V≈√ 2GF(ne−ne−n+n)⟨1−cos Θ⟩ ≈√ 2GF⟨1−cos Θ⟩ 4r2(Le ⟨Ee⟩−Le ⟨Ee⟩−L ⟨E⟩+L ⟨E⟩) : (11.16) The tau-ﬂavored neutrino contribution has not been included because it cancels exactly between and. The same is true for the mu- ﬂavored terms before oscillations have taken place. For a test-neutrino of momentum pone ﬁnds numerically 2p∆V≈420 eV2(10 km =r)2⟨1−cos Θ⟩ × ×p 1051erg s−1(Le ⟨Ee⟩−Le ⟨Ee⟩−L ⟨E⟩+L ⟨E⟩) : (11.17) A few seconds after bounce typical values might be ⟨Ee⟩= 11 MeV, ⟨Ee⟩= 16 MeV, and ⟨E⟩=⟨E⟩= 25 MeV, the luminosities can Supernova Neutrinos 441 be taken to be the same at 3 ×1051erg s−1each, and R= 11 km. For these conditions the estimated contributions to ∆ Vfrom electrons and neutrinos are shown in Fig. 11.22 as a function of radius. As long as no swap has occurred the neutrinos do not play a major role because the contribution of cancels exactly against . Further, the diﬀerence between eandeis smaller than the electron contribu- tion, and it has the same sign whence it simply causes a slightly larger eﬀective matter density. However, on resonance a relatively large number of e’s exchange ﬂavor with ’s so that the neutrino contribution changes sign. More- over, on resonance the vacuum ∆ m2 by deﬁnition cancels against the medium-induced contribution. Therefore, “switching on” the neutrino term shifts the resonance position for given vacuum mixing parameters. Also, in a self-consistent treatment one needs to consider the full non- Fig. 11.22. “Mass splitting” between eand(equivalently ) of mo- mentum p= 10 MeV caused by the regular medium (the electrons), and by diﬀerent neutrino ﬂavors according to Eq. (11.17). For the electrons the density proﬁle of Fig. 11.20 was used with Ye= 0:5; in a real SN core Yeis much lower near the neutrino sphere so that the thick line would increase less steeply toward the neutrino sphere than shown here. For the neutrinos, a luminosity in each degree of freedom of 3 ×1051erg s−1was assumed, the average energies were taken to be ⟨Ee⟩= 11 MeV, ⟨Ee⟩= 16 MeV, and ⟨E⟩=⟨E⟩= 25 MeV, and the neutrino-sphere radius is 11 km. The signs of the contributions relative to electrons are: + for eand,−for eand. The contribution of cancels exactly against unless a swap e↔has taken place. 442 Chapter 11 linear equations of motion for the neutrino density matrix which causes an “oﬀ-diagonal refractive index” as discussed in Sect. 9.3.2. Qian and Fuller (1995) have performed an approximately self-consistent analysis of this problem. All told, they found that the neutrino-neutrino interac- tions have a relatively small impact on the parameter space where ﬂavor conversion disturbs the r-process. Within the overall precision of these arguments and calculations, their ﬁnal exclusion plot is nearly identi- cal with the schematic picture shown in Fig. 11.21. Qian and Fuller’s ﬁndings are corroborated by a study performed by Sigl (1995a). c) Summary In summary, there remains a large range of mixing angles where neu- trino oscillations between eandorwith a cosmologically in- teresting mass could help to explode supernovae, and yet not disturb r-process nucleosynthesis. If one assumes a mass hierarchy with e dominated by the lightest, by the heaviest mass eigenstate, the cos- mologically relevant neutrino would be identiﬁed with . It is inter- esting that the relevant range of mixing angles, 3 ×10−4∼<∼<3×10−3, overlaps with the mixing angle among the ﬁrst and third family quarks which is in the range 0 :002−0:005 (Eq. 7.6). Therefore, a scenario where a massive plays a cosmologically important role, helps to ex- plode supernovae, and leaves r-process nucleosynthesis unscathed does not appear to be entirely far-fetched. This scenario leaves the possi- bility open that the MSW eﬀect solves the solar neutrino problem by e-oscillations. 11.4.6 A Caveat All existing discussions of neutrino ﬂavor oscillations in SNe were based on spherically symmetric, smooth density proﬁles. However, there can be signiﬁcant density variations, convection, turbulence, and so forth. Therefore, it is clear that many of my statements about medium- induced oscillation eﬀects are provisional. Further studies will be re- quired to develop a more complete picture of SNe and their neutrino oscillations as 3-dimensional events. A ﬁrst study of SN neutrino oscillations with an inhomogeneous den- sity proﬁle was recently performed by Loreti et al. (1995). They added a random density ﬁeld to the standard smooth proﬁle and studied the impact on neutrino oscillations. They found that the shock-revival sce- nario involving MSW oscillations can be signiﬁcantly aﬀected in the Supernova Neutrinos 443 sense that less additional energy is transferred because the stochastic density ﬁeld can prevent a complete swap of the neutrino spectra. On the other hand, for the r-process prevention a complete swap is not necessary. Therefore, the mixing parameters for which r-process nucle- osynthesis is prevented by oscillations is not signiﬁcantly changed, even if the amplitude of the stochastic density component is as large as 1%. 11.5 Neutrino Propulsion of Neutron Stars Shortly after the discovery of pulsars (neutron stars) it became clear that they tend to have the largest peculiar velocities of all stellar pop- ulations. Recent determinations of pulsar proper motions by radio- interferometric methods (Bailes et al. 1990; Fomalont et al. 1992; Har- rison, Lyne, and Anderson 1993) and by interstellar scintillation obser- vations (Cordes 1986) reveal typical speeds of a few 100 km s−1. Tak- ing into account selection eﬀects against high-velocity pulsars, Lyne and Lorimer (1994) argued that the mean pulsar velocity at birth was 450±90 km s−1. Associating certain pulsars and SN remnants would indicate velocities of up to 2000 km s−1(Frail and Kulkarni 1991; Car- aveo 1993; Stewart et al. 1993) while the interaction of PSR 2224+65 with its local environment produces a nebula which reveals a transverse velocity of at least 800 km s−1(Cordes, Romani, and Lundgren 1993). Therefore, the distribution of pulsar peculiar speeds appears to have a mean of 400 −500 km s−1with the largest measured values of 1000−2000 km s−1. Recall that the galactic rotation velocity is about 200 km s−1, the escape velocity about 500 km s−1. Therefore, the fastest pulsars will eventually escape from the galaxy. In most cases the mi- gration is away from the galactic disk in agreement with the picture that pulsars are born in the disk where massive stars can form from the interstellar gas. However, there seem to be a few puzzling cases of pulsars which move toward the disk, apparently having formed in the galactic halo. Massive stars probably do not form with much larger velocities than other stars and so pulsars are likely accelerated in conjunction with the SN collapse that produced them or during their early evolution. One suggestion for an acceleration mechanism holds that large neutron-star “kick velocities” are related to the breakup of close binaries, notably during the SN explosion of the second binary member (Gott, Gunn, and Ostriker 1970; Dewey and Cordes 1987; Bailes 1989; see also the review by Bhattacharya and van den Heuvel 1991). 444 Chapter 11 Another possibility is that the SN explosion itself is not spheri- cally symmetric and thus imparts a kick velocity on the neutron star (Shklovski˘ ı 1970). Indeed, as discussed in Sect. 11.1.3 SN explosions likely involve large-scale convective overturns below and above the neu- trino sphere which could lead to an explosion asymmetry of a few percent, enough to accelerate the compact core to a speed of order 100 km s−1, but not enough to account for the typically observed pulsar velocities (Janka and M¨ uller 1994). An interesting acceleration mechanism was proposed by Harrison and Tademaru (1975) who considered the rotation of an oblique mag- netic dipole which is oﬀ-center with regard to the rotating neutron star. The radiation of electromagnetic power is then asymmetric relative to the rotation axis and so a substantial accelerating force obtains, enough to cause velocities of several 100 km s−1. The velocity reached should not depend on the magnitude of the magnetic dipole moment while its direction should correlate with the pulsar rotation axis. These predic- tions do not seem to be borne out by the data sample of Anderson and Lyne (1983) although the more recent observations may be less disfavorable to the “electromagnetic rocket engine.” The correlation between peculiar velocity and pulsar magnetic moment may now be less convincing (Harrison, Lyne, and Anderson 1993; Itoh and Hiraki 1994). Another intriguing mechanism ﬁrst proposed by Chuga˘ ı (1984) relies on the asymmetric emission of neutrinos (“neutrino rocket engine”). Recall that the total amount of binding energy released in neutrinos is about 3 ×1053erg; because neutrinos are relativistic they carry the same amount of momentum. If the neutron-star mass is taken to be 1 M⊙, and if all neutrinos were emitted in one direction, a recoil velocity of 0:17c= 5×104km s−1would obtain. Thus an asymmetric emission of 1.5% would be enough to impart a kick velocity of 800 km s−1. Neutrino emission deviates naturally from spherical symmetry if large-scale convection obtains in the region of the neutrino sphere. Janka and M¨ uller (1994) believe that 500 km s−1is a generous upper limit on the kick velocity that can be achieved by this method. For a reliable estimate one needs to know the typical size of the convective cells as well the duration of the convective phase in the protoneutron star. To this end one needs to perform a fully 3-dimensional calculation. No such results are available at the present time. An asymmetric neutrino emission would also obtain in strong mag- netic ﬁelds because the opacity is directional for processes involving initial- or ﬁnal-state charged leptons, i.e. URCA processes of the type e−+p→n+eore++n→p+e. The rates for such processes in the Supernova Neutrinos 445 presence of magnetic ﬁelds were discussed by a number of authors.68 Because of the left-handedness of the weak interaction both neutrinos and antineutrinos would be emitted preferentially in the same direction singled out by the magnetic ﬁeld. Its eﬀects become substantial only for ﬁeld strengths near and above the critical strength m2 e=e= 4:4×1013G. In order to obtain neutron-star kick velocities of several 100 km s−1 it appears that magnetic ﬁelds several orders of magnitude larger are required which, however, may possibly exist in some SN cores after collapse. Also, the asymmetric emission of neutrinos may be aided by the formation of a pion condensate and perhaps other processes (Par- fenov 1988, 1989). Certain special ﬁeld conﬁgurations seem to allow for signiﬁcantly anisotropic neutrino emission (Bisnovatyi-Kogan and Janka 1995). It remains to be seen if suﬃciently anisotropic neutrino emission can be established as a generic property of a protoneutron star’s Kelvin- Helmholtz cooling phase. Meanwhile, the neutrino rocket engine re- mains a fascinating speculation for accelerating neutron stars. 11.6 Future Supernovae The neutrino observations from SN 1987A gave us a wealth of infor- mation in the sense that they conﬁrmed the broad picture of neutrino cooling of the compact object formed after collapse. The data were much too sparse, however, to distinguish between, say, diﬀerent equa- tions of state or diﬀerent assumptions concerning neutrino transport, or to detect or signiﬁcantly constrain neutrino masses and mixing parame- ters. Some worry is caused by the apparent anomalies of the SN 1987A data, notably the angular distribution of the secondary charged par- ticles. No doubt it would be extremely important to observe a SN neutrino signal with greater statistical signiﬁcance, or from a greater distance. What is the prospect for such an observation? SN neutrinos can be observed in a number of underground detec- tors which are operational now or in the near future, or which have only been proposed. An extensive overview was given by Burrows, Klein, and Gandhi (1992). Of the experiments which will become operational within the foreseeable future, the upcoming Superkamiokande water Cherenkov detector (Sect. 10.9) would yield by far the largest num- 68O’Connell and Matese (1969a,b); Matese and O’Connell (1969); Ivanov and Shul’man (1980, 1981); Dorofeev, Rodionov, and Ternov (1984); Loskutov (1984a,b); Cheng, Schramm, and Truran (1993). 446 Chapter 11 ber of events. Its ﬁducial mass for the detection of SN neutrinos is about 32,000 t, to be compared with 2,140 t for Kamiokande, i.e. it has a target mass about 15 times larger. Thus one may be able to recog- nize a neutrino signal from a SN perhaps as much as 4 times farther away than the Large Magellanic Cloud, which is out to about 200 kpc. However, the closest large galaxy is M31 (Andromeda) at a distance of about 700 kpc, allowing Superkamiokande (and all other near-future detectors) to observe SNe only in our own galaxy and in the Large and Small Magellanic Clouds. The rate at which SNe occur in our Galaxy as well as in the LMC is rather uncertain. From observations in other galaxies the rate of core-collapse SNe in the Milky Way is estimated to be about 7 :3h2 per century with hthe Hubble parameter in units of 100 km s−1Mpc−1 (van den Bergh and Tammann 1991). Thus, for a low hof order 0 :5 one may expect only about 2 such events per century. Roughly the same number was found in a more recent study by Tammann, L¨ oﬄer, and Schr¨ oder (1994). The record of historical SNe, on the other hand, suggests a signiﬁcantly larger number. Thus it is optimistic, but not entirely implausible, to hope for an observation within a decade of Superkamiokande running time. The rate for the LMC is thought to be about 0.5 per century (Tammann, L¨ oﬄer, and Schr¨ oder 1994)—one cannot reasonably expect another SN there within our lifetime. To reach beyond the limits of our own galaxy and the LMC one would need much more sensitive (much bigger) detectors. It would not be enough to go as far as Andromeda because this galaxy appears to have an anomalously low SN rate (van den Bergh and Tammann 1991). In order to achieve a SN rate of at least 1 per year one may need to use the Virgo cluster of galaxies at about 15 Mpc (300 times the distance to the LMC) although it may be enough to reach to the nearby starburst galaxies M82 and NGC 253 within about 4 Mpc which have a very high SN rate because of their high rate of star formation (Becklin 1990). However, a recent estimate of the SN rate for each of these galaxies is only about 1 per 10 years (van Buren and Greenhouse 1994). A novel detection scheme (Cline et al. 1990) that may allow one to build big enough detectors is based on the neutral-current reaction + (Z; N)→(Z; N−1) + n+which can have a much enhanced cross section in some nuclei due to collective eﬀects; one would detect the ﬁnal-state neutron. The necessary detector volume can be achieved by using natural deposits of minerals which contain the relevant target nuclei. Naturally, the main concern would be to reduce sources of background in order to isolate the feeble signal from a distant SN. Supernova Neutrinos 447 An even more ambitious goal would be to measure the cosmic e background ﬂux from all past SNe in the universe. For energies below around 10 MeV this ﬂux is swamped by many orders of magnitude by that from the nuclear power plants on Earth. Above a few 10 MeV the atmospheric neutrino ﬂux would dominate and so there is only a small window where the cosmic SN ﬂux might be detectable. However, even a moderately sized (200 tons) scintillation detector located on the moon would have a chance of measuring this ﬂux because these backgrounds do not exist there (Mann and Zhang 1990). Returning to Earth one may speculate about what could be learned if a galactic SN were indeed observed at Superkamiokande. For the pur- pose of argument a distance of 10 kpc (5 times closer than the LMC) is assumed. (Recall that the solar system is at a distance of about 8 kpc from the galactic center.) Then one expects at Superkamiokande about 4000 events from the reaction ep→ne+, compared with 270 at Kamiokande, and with 12 measured there from SN 1987A (Totsuka 1990). This would be enough to determine a statistically very signiﬁ- cant and very detailed “neutrino lightcurve.” Interestingly, one expects about 13 events within the ﬁrst few ms from the prompt eburst. Its presence would indicate that the e’s have not oscillated, say, into ’s as would be expected for a cosmolog- ically interesting mass (Sect. 11.4.2), thus excluding a large range of masses and mixing angles (Fig. 11.17). Moreover, the prompt burst could not have been dispersed by a neutrino mass and so a mebound of order 1 eV could be derived. A number of other conclusions tentatively reached for SN 1987A could be aﬃrmed (Sect. 13.2). The nonobserva- tion of the prompt burst, on the other hand, would be more diﬃcult to interpret as its absence could have a variety of causes ranging from neutrino oscillations to some ﬂaw in the standard picture of SN collapse. Theelightcurve which would last, say, 10 s would not allow one to extract interesting bounds on merelative to the ones already obtained from SN 1987A and from laboratory experiments. From Eq. (11.9) one concludes that for a distance D≈10 kpc one is sensitive to masses in the 100 eV range. In Superkamiokande one would expect to see about 40 events for each+and+from the elastic -escattering process. Because the ﬁnal-state electrons are strongly forward peaked one can separate them from the isotropic ep→ne+signal. Therefore, a ormass would manifest itself through late forward events. Seckel, Steigman, and Walker (1991) found that from the signal in water Cherenkov de- tectors one could be sensitive to a mass down to about 75 eV. Of 448 Chapter 11 course, masses so large and larger are excluded from cosmology—such neutrinos would have to be unstable. In order to obey the cosmologi- cal limits neutrinos emitted at a distance of 10 kpc would decay before reaching Earth if their mass exceeds a few 10 keV. Thus, the obser- vation of a galactic SN would allow one, at best, to probe the mass window 100 eV ∼<m∼<30 keV. A similar conclusion was reached by Acker, Pakvasa, and Raghavan (1990) who considered the signature in the proposed BOREX detector. It may be possible, however, to probe a somewhat smaller mass for, say, the down to the cosmologically interesting range of 30 eV if one takes advantage of all aspects of the observed neutrino signal (Krauss et al. 1992). Because the late part of the neutrino lightcurve is expected to be similar for eand the other ﬂavors one can hope to extract the behavior of the source from the esignal. Then one would be more sensitive to modiﬁcations of the lightcurve caused by dispersion eﬀects. However, in order to identify the ’s one would have to use energy cuts ( e’s and ’s have diﬀerent spectra!) in addition to angular cuts. In the analysis of Krauss et al. (1992) the possibility of e↔oscillations was not included which would weaken the range of accessible masses because of the modiﬁed energy spectra. Of course, the r-process nucleosynthesis argument of Sect. 11.4.5 would indicate that MSW oscillations did not take place late even if they took place early and rendered the prompt eburst unobservable. Still, a mass relevant for the dark matter content of the universe as well as for scenarios of structure formation may be much lower than 30 eV. Therefore, on the basis of current analyses the prospect of being able to recognize a cosmologically relevant orin the neutrino signal of a galactic SN appears relatively dim. A far more positive view was taken by Cline et al. (1994) who ar- gued that masses down to 15 eV may be accessible by the simultaneous operation of Superkamiokande and their previously proposed (Cline et al. 1990) Supernova Burst Observatory (SNBO) which is based on neutral-current reactions alone. This would obviate the need to sep- arate the charged-current en→pe+detection at Superkamiokande from the neutral-current reaction e→eby angular and energy cuts. One could use the Superkamiokande esignal to monitor the SN neu- trino lightcurve, notably its sharp onset, and relate it to the onset of the neutral-current events at SNBO which would be “washed out” for a cosmologically interesting mass of, say, the . One must hope that SNBO will become a real project in the near future. Chapter 12 Radiative Particle Decays from Distant Sources If neutrinos, axions, or other low-mass particles had radiative decay channels, the decay photons would appear as x- or -ray ﬂuxes from stellar sources where these particles can be produced by nuclear or plasma processes. This chapter is devoted to limits on such decays, including decays into charged leptons, that are based on observational limits on photon or positron ﬂuxes from stellar sources, notably the Sun and supernova 1987A. For comparison, laboratory and cosmological limits are also reviewed. 12.1 Preliminaries This book is largely about the properties of electrically neutral particles whose electromagnetic interactions are correspondingly weak. However, because they can virtually dissociate into charged states, they will still interact with photons through higher-order amplitudes. The focus of the present chapter is the possibility of radiative decays of the form →′ (neutrinos) or a→ (axions), but also →′e+e−and →′e+e− . Cowsik (1977) was the ﬁrst to recognize that the huge path lengths available in the astrophysical environment allow one to obtain much more restrictive limits on such decays than from labora- tory experiments. For example, the absence of single-photon counts in a detector near a ﬁssion reactor indicates a bound69 =me>22 s=eV 69In this chapter will always denote the partial neutrino decay time into radia- tion while totis the total decay time if hypothetical invisible channels are included. Then −1 =B −1 totwith the branching ratio B . 449 450 Chapter 12 on the process e→′ . The corresponding limit based on a compar- ison between the measured solar neutrino ﬂux and the measured limit on x- or -rays from the quiet Sun is 7 ×109s=eV, almost 9 orders of magnitude more restrictive. Moreover, this method is fully analogous to a laboratory experiment as it is based on a measured neutrino ﬂux and a measured upper limit photon ﬂux. To a lesser degree this re- mark also applies to the even better SN 1987A constraints which are applicable to all neutrino ﬂavors. In addition, less directly established particle ﬂuxes can be used such as those from the stars in the galactic bulge or from all hydrogen- burning stars or supernovae in the universe. Even more indirectly, one may study the impact of the radiative decay of the cosmic background sea of neutrinos or axions which are predicted to exist in the framework of the big-bang theory of the early universe. Usually, the bounds on →′ thus obtained are presented as lim- its on the radiative decay time . Even if one does not aim at an imme- diate theoretical interpretation, however, the signiﬁcance of is limited because it represents a combination of the ﬁnal-state phase-space vol- ume and the matrix element. The latter can be expressed in terms of an eﬀective transition moment eﬀas in Eq. (7.12) of Sect. 7.2.2. This moment characterizes the interaction strength independently of phase-space eﬀects, providing a much more direct link between the ex- perimental results and an underlying theory. A “heavy” neutrino hwith mh>2me≈1 MeV can decay into ee+e−. This channel is often included in the notion of “radiative” de- cays because relativistic charged leptons cause experimental signatures similar to rays. If this decay proceeds by virtue of a mixing amplitude Uehbetween handethe rate is given by Eq. (7.9). Therefore, it is characterized by |Ueh|in a phase-space independent way. Ifeis a mixture of diﬀerent mass eigenstates, any esource such as a power reactor or the Sun produces all components. If their mass diﬀerences are small one needs to consider in detail the phenomenon of neutrino oscillations as in Chapter 8. However, if the oscillation length is much smaller than the distance between the detector and the source, the neutrino ﬂux can be considered an incoherent mixture of all mass eigenstates; at a reactor this is the case for ∆ m2∼>1eV2. The ﬂux of “heavy” neutrinos from a esource is then given by Fh(E) =|Ueh|2(E)Fe(E): (12.1) The velocity (E) = (1 −m2 h=E2 )1=2enters from the phase space of nonrelativistic neutrinos in the production process. From esources Radiative Particle Decays 451 one can then derive limits on Uehfor any h, even a hypothetical sterile one, if mh∼>1 MeV because it is the same mixing amplitude that allows for its production in the source and for its h→ee−e+decay. In the following I will discuss radiative lifetime limits from diﬀerent sources approximately in the order of available decay paths, from labo- ratory experiments (a few meters) to the radius of the visible universe (about 1010light years). 12.2 Laboratory Experiments 12.2.1 Spectrum of Decay Photons Perhaps the simplest neutrino source to use for laboratory experiments is a nuclear power reactor which produces a strong eﬂux from the weak decays of the uranium and plutonium ﬁssion products. If a detector is placed at a certain distance from the reactor core one may assume that the local neutrino ﬂux F(E) is known (units cm−2s−1MeV−1). As a ﬁrst step one then needs to compute the expected ﬂux F (E ) of photons from the decay →′ . The result will also apply to station- ary stellar sources such as the Sun while for the short neutrino burst from SN 1987A one needs to derive a separate expression (Sect. 12.4). Because the decay is a dipole transition, the general form of the photon angular distribution in the rest frame of the parent neutrino is dN =dcos#=1 2(1−cos#); (12.2) where #is the angle between the spin polarization vector and the photon momentum. For Majorana neutrinos the decay is isotropic, independently of their polarization, and thus = 0. Similarly for axion decays, a→ , as these particles have no spin so that in their rest frame no spatial direction is favored. For polarized Dirac neutrinos the possible parameter range is −1≤≤1. The decay photon has an energy !=mm=2 in the rest frame of the parent neutrino—see Eq. (7.12). If it is emitted in a direction with regard to the laboratory direction of motion, the energy in the laboratory frame is E =!(E+pcos)=m. Hence, E =1 2mE(1 +cos); (12.3) where =p=E= (1−m2 =E2 )1=2is the neutrino velocity. For a left- handed parent neutrino the spin is polarized opposite to its momentum so that cos =−cos#anddN =dcos=1 2(1 + cos). For a very 452 Chapter 12 relativistic parent neutrino = 1 and so the normalized photon energy distribution corresponding to Eq. (12.2) is dN dE =1 mE( 1−+ 2E mE) ; 0< E < mE:(12.4) The decays of Majorana neutrinos or axions ( = 0) produce a box- shaped spectrum while for =±1 it is triangle shaped (Fig. 12.1). Fig. 12.1. Photon spectrum from the decay of a relativistic neutrino (en- ergyE) according to Eq. (12.4). A fraction ( m=E)(d = ) of neutrinos decay before reaching the detector if the laboratory lifetime is large compared with the decay path d . Here, is the rest-frame radiative decay time and E=m the time dilation factor. Integrating over the neutrino source spectrum then yields F (E ) =m d ∫∞ E =mdE m( 1−+ 2E mE)F(E) E2 :(12.5) Most sources emit either antineutrinos (for example e’s from a ﬁssion reactor) or neutrinos (for example e’s from the Sun). In these cases Eq. (12.5) gives us directly the expected ﬂux as a function of the as- sumed value for . However, there are some examples for simultaneous andsources. In those cases one needs to know which value of to use for if a certain value for has been assumed. Under very general assumptions the laws of particle physics are invariant under a simultaneous transformation which takes particles into antiparticles (charge conjugation C), reﬂects all spatial coordinates (parity transformation P), and inverts motions (time reversal T). In this case the CPT theorem states that the masses and total decay times of Radiative Particle Decays 453 corresponding particles and antiparticles are the same. However, the partial decay rates into speciﬁc channels need not be identical, and indeed, K◦andK◦show such CP-violating decays: A transformation under CP leads to a “mirror world” which is diﬀerent from the one we live in. Therefore, in the most general case one may only assume that andhave the same total lifetimes while their radiative decay rates may be diﬀerent. However, it is common to analyze the available data under the as- sumption of CP conservation for all neutrino interactions. For the radiative decay of a polarized in its rest frame, the CP-mirrored de- cay is one where a polarized with the same spin decays into ′and of reversed momenta.70Therefore, andof the same polarization are characterized by opposite values for in Eq. (12.2). In the source, however, the ’s and ’s are produced by weak interactions which vio- late parity maximally. If they are relativistic they both have negative (left-handed) chiralities which means that the ’s have negative and the ’s positive helicities. Thus, relative to their momentum left-handed ’s and ’s show the same distribution of decay photons so that in Eq. (12.5) one must use the same for both. 12.2.2 Electron Neutrinos from Reactors Fission reactors are superb neutrino sources. At a thermal power of 2800 MW, for example, one expects about 5 ×1020e=s. At a distance of 30 m this corresponds to a ﬂux of about 4 ×1012cm−2s−1, almost a hundred times larger than the solar eﬂux of about 6 :6×1010cm−2s−1. Reactor e’s emerge from many weak decays of the products of the neutron induced ﬁssion of235U and239Pu while the ﬁssion of238U and 241Pu contributes less than 10% to the total rate. The spectral distribu- tion can be inferred from a measurement of the corresponding spectra together with the reasonably well-known distribution of the end point energies of the ﬁssion products (von Feilitzsch et al. 1982; Schrecken- bach et al. 1985). The distribution of eenergies per ﬁssion of235U and 239Pu is shown in Fig. 12.2 with a total of about 6 e’s per ﬁssion. An early, relatively crude, but often-quoted (Particle Data Group 1994) limit on the decay e→′ was derived by Reines, Sobel, and Gurr (1974) on the basis of the upper limit ﬂux in a scintillation detec- tor near the Savannah River reactor (U.S.A.). A weaker but more reli- 70A parity transformation P inverts all polar vectors, e.g. momenta, currents, or electric ﬁelds, while it leaves axial vectors unchanged, e.g. angular momenta, magnetic moments, or magnetic ﬁelds. 454 Chapter 12 Fig. 12.2. Spectrum of reactor e’s per ﬁssion of239Pu and235U (von Feil- itzsch et al. 1982; Schreckenbach et al. 1985). The width of the lines gives the total error of the spectra. able bound was inferred by Vogel (1984) from data taken at the G¨ osgen reactor (Switzerland). The most recent analysis is, again, based on data taken with a scintillation counter at G¨ osgen (Oberauer, von Feil- itzsch, and M¨ ossbauer 1987). From a comparison of the “reactor on” with the “reactor oﬀ” photon counts for several energy channels in the MeV range and using the experimentally established neutrino spectrum (Fig. 12.2), these authors found the 68% CL lower limits on the era- diative decay times of =me>22 s=eV for =−1, 38 s =eV for = 0, and 59 s =eV for = +1. It was assumed that ′is massless so that m= 1 in Eq. (12.5). With Eq. (7.12) the =−1 constraint translates into a bound on the eﬀective electromagnetic transition moment of eﬀ<0:092Bm−2 eV; (12.6) not a very restrictive limit even if esaturates its upper mass bound of about 5 eV. Even this weak limit would cease to apply if eand′became nearly degenerate. Therefore, Bouchez et al. (1988) performed an ex- periment where they searched for optical decay photons at the Bugey reactor (France). They excluded a certain region in the parameter plane spanned by =meandm. Their greatest sensitivity was ap- proximately at m= 2×10−5where they found =me∼>0:04 s=eV. With Eq. (7.12) this is eﬀ∼<107B=m2 eVwhich, unfortunately, is ir- relevant as a constraint. Hence, for nearly degenerate neutrinos there Radiative Particle Decays 455 is no meaningful limit on eﬀfrom decay experiments. This example highlights the importance of separating the intrinsic coupling strength, or the magnitude of the matrix element, from phase-space eﬀects in the interpretation of such results. 12.2.3 Heavy Neutrinos from Reactors Considering the decay of einto a diﬀerent neutrino species implies entertaining the notion of the nonconservation of the electron lepton number, forcing one to contemplate the possibility of neutrino ﬂavor mixing as well. Therefore, reactors will be sources for other ﬂavors and notably of “heavy neutrinos” according to Eq. (12.1). The photon ﬂux from radiative hdecays can now be calculated as before, except that the dwelling time for nonrelativistic h’s in the decay volume is increased by −1so that the velocity cancels between this factor and Eq. (12.1). However, the photon spectrum shown in Fig. 12.1 must be modiﬁed for the decays of nonrelativistic neutrinos because < 1 in Eq. (12.3). Therefore, the overall expression for F (E ) becomes somewhat more involved. As long as mh∼<1 MeV one may treat the hﬂux as relativistic. Then the radiative decay limits are the same as for e, except that they are diminished by the reduced neutrino ﬂux. Thus, the bound Eq. (12.6) translates into eﬀ<0:92×10−13B|Ueh|−1m−2 MeV: (12.7) With mhup to an MeV this bound has a lot more teeth than the one on electron neutrinos. Formh>2me≈1 MeV the decays h→ee+e−will become kinematically possible and probably dominate. The scintillation coun- ters that were used to search for decay photons near a power reactor are equally sensitive to electrons and positrons—for many purposes relativistic charged particles may be treated almost on the same foot- ing as rays. Therefore, the same G¨ osgen data have been analyzed to constrain the mixing amplitude Ueh(Oberauer, von Feilitzsch, and M¨ ossbauer 1987; Oberauer 1992). Even more restrictive limits were obtained from data taken at the Rovno reactor (Fayons, Kopeykin, and Mikaelyan 1991), and most recently at the Bugey reactor (Hagner et al. 1995). Note that in this method the same mixing probability |Ueh|2appears in Eq. (12.1) to obtain the hﬂux from a esource, and in Eq. (7.9) to obtain the decay probability. Hence, the expected e+e− ﬂux is proportional to |Ueh|4. 456 Chapter 12 In Fig. 12.3 the excluded range of masses and mixing angles71is shown together with similar constraints from other neutrino sources. The reactor bounds are weaker than those from the Sun and SN 1987A, but they remain important because of the short decay path involved! If one accepts the big-bang nucleosynthesis bounds (Sect. 7.1.5) the total lifetime must be so short that it would not escape from the mantle of a SN before decaying, and perhaps not even from the Sun. 12.2.4 Neutrinos from a Beam Stop Another powerful laboratory source for both e’s and ’s is a beam stop where neutrinos are produced from the decay of stopped pions, +→+and the subsequent decay of stopped muons, +→e+e. In a recent experiment of this sort (Krakauer et al. 1991), the neu- trino intensity was 4 :3×1013=s with a total of 8 :52×1019. In these decays the has a ﬁxed energy of ( m2 −m2 )=2m= 29 :8 MeV while the other normalized spectra are (3 =Y4)(3Y−2E)E2 for and (12 =Y4)(Y−E)E2 forewith Y≡1 2m= 52 :8 MeV. For e, the 90% CL radiative lifetime limit as a function of the “anisotropy parameter” is =me>(15:9 + 9 :8+ 0:32) s=eV, somewhat less restrictive than the reactor results. Forandone obtains slightly diﬀerent limits because of the diﬀerent source spectra. Under the assumption of CP invariance the radiative lifetimes for andare the same. In this case the combined limit is =m>(36:3 + 21 :65+ 0:752) s=eV while the individual limits are about half this value. In terms of an eﬀective transition moment the most conservative case ( =−1) yields eﬀ<0:11Bm−2 eV: (12.8) For the admixture of other mass eigenstates this result may be trans- lated in a fashion analogous to the discussion of reactor neutrinos. Beam-stop neutrinos from meson decays may also be used to con- strain the e+e−decays of heavy admixtures. Because of the larger amount of available energy one may probe higher masses for hwhile the reactor bounds drop out above a few MeV because of the relatively soft spectrum. In Fig. 12.3 the most restrictive such constraints are summarized. 71It is customary to display |Ueh|2when constraining the mixing parameters of heavy, decaying neutrinos while one shows sin22ehfor light, oscillating neutrinos as in Chapter 8. For easier comparison I always use the mixing angle. Recall that |Ueh|= sin ehso that for small mixing angles sin22eh= 4|Ueh|2. Radiative Particle Decays 457 Fig. 12.3. Bounds on e-hmixing from the absence of h→ee+e−decays. Meson decays: (a,b) Leener-Rosier et al. (1986) and (c) Bryman et al. (1983). Reactor neutrinos: (d) Hagner et al. (1995). Absence of reactor neutrino oscillations: (e) Zacek et al. (1986). Absence of solar positrons: Toussaint and Wilczek (1981); see also Sect. 12.3.2. These results are based on esources which produce hby their mixing which also leads to the subsequent h→ee+e−decay. Of the known sequential neutrinos hcan be identiﬁed only with 3, the dom- inant mass component of with an allowed mass of up to 24 MeV. If one makes this identiﬁcation, stronger limits are obtained from di- rect≈3sources. One example is the beam stop at the Big Euro- pean Bubble Chamber (BEBC) where a strong ﬂux of charmed strange mesons72Dswas produced which subsequently can decay as Ds→ besides the dominant hadronic modes (WA66 Collaboration 1985). Ac- cording to Babu, Gould, and Rothstein (1994) who quote a private communication from the WA66 collaboration, a model-independent constraint from the BEBC experiment is =m3>0:15 s=MeV or eﬀ<1:1×103Bm−2 eV: (12.9) Another constraint is e+e=m3>0:18 s=MeV. As a constraint on the mixing amplitude, |Ue3|2<1:6×105m−6 MeV, it is weaker than those shown in Fig. 12.3. However, as it is based on a direct ﬂux it is valid even if the decays are not induced by mixing but by exotic intermediate states (Babu, Gould, and Rothstein 1993). 72TheDsused to be called Fas in the quoted reference. 458 Chapter 12 12.3 Particles from the Sun 12.3.1 Electron Neutrinos Like a terrestrial power reactor, the Sun is a proliﬁc neutrino source ex- cept that it emits e’s rather than e’s. The expected spectrum as well as the relevant measurements were discussed in Chapter 10. Suﬃce it to recall that the solar neutrino ﬂux is now experimentally estab- lished without a shred of doubt. There remain signiﬁcant discrepancies between the predicted and measured spectral shape of the spectrum which may be explained by neutrino oscillations. However, the solar neutrino problem is a ﬁne point in the context of the present discus- sion because the following results depend mostly on the low-energy ppﬂux. One may proceed exactly as in the previous section in order to trans- late the solar neutrino spectrum shown in Fig. 10.1 into an expected ﬂux of x- and -rays from the Sun. In Fig. 12.4 I show this ﬂux at Earth for =me= 10 s =eV (about the laboratory lifetime limit) and for the values ±1 for the “anisotropy parameter” . The spectrum as shown is based on the calculated neutrino ﬂux. The shoulders corresponding to other than the ppneutrinos likely would have to be reduced somewhat. The magnitude of the photon ﬂux is enormous because the decay path d is the entire distance to the Sun of 1 :5×1013cm = 500 s. The quiet Sun is a signiﬁcant source of soft x-rays from the quasi- thermal emission of the hot corona at T≈4:5×106K. The ﬂux mea- surements of Chodil et al. (1965) are marked as open diamonds in Fig. 12.4. The ﬂux of decay photons in Fig. 12.4 would outshine the solar corona by some 4 orders of magnitude! Moreover, the corona spectrum falls oﬀ sharply at larger energies. In the hard x- and soft -ray band very restrictive upper limits exist on the emission of the quiet Sun that are shown in Fig. 12.4. They are based on balloon- borne detectors ﬂown many years ago (Frost et al. 1966; Peterson et al. 1966). These upper limits are still far above the estimated albedo (ra- diation from cosmic rays hitting the surface of the Sun), leaving much room for improvement. Alas, the quiet Sun is not an object of great interest to -ray astronomers and so more recent measurements do not seem to exist. In order to respect these measured upper limit photon ﬂuxes one must shift the decay spectrum in Fig. 12.4 down by about 8 orders of magnitude (thin solid line in Fig. 12.4). This yields a lower radiative liftime limit for eof =me∼>7×109s=eV (Cowsik 1977; Raﬀelt 1985). Radiative Particle Decays 459 Fig. 12.4. Spectrum of photons from the solar neutrino decay e→′ for the indicated values of the anisotropy parameter . Measurements of the x-ray emission of the solar corona (open diamonds) according to Chodil et al. (1965). Upper limit x- and -ray ﬂuxes according to Frost et al. (1966) and Peterson et al. (1966). Estimated albedo according to Peterson et al. (1966). Thin solid line: Maximally allowed photon spectrum from neutrino decay. (Figure adapted from Raﬀelt 1985.) With Eq. (7.12) this translates into eﬀ<5×10−6Bm−2 eV: (12.10) It must be stressed, again, that the recent progress in solar neutrino astronomy has placed this result on the same footing as a terrestrial experiment—the magnitude of the solar neutrino ﬂux and its main spectral features are now experimentally established! 460 Chapter 12 This bound is diminished if eand′are nearly degenerate so that in Eq. (7.12) m≪1. The structure of the photon ﬂux as a function of min Eq. (12.5) is such that for m<1 the spectrum can be obtained, in a doubly logarithmic representation such as Fig. 12.4, by shifting it “to the left” and “upward” by the amount |logm|each, the shape itself remaining unchanged (Raﬀelt 1985). The excluded regime in the plane of =meandmis shown in Fig. 12.5 (left panel). Again, it is more appropriate to express these limits in terms of eﬀandmby virtue of Eq. (7.12), leading to Fig. 12.5 (right panel). As expected, for ﬁxed methe limits on eﬀquickly degrade with small m. Fig. 12.5. Excluded parameters for e→′ from the Sun for nearly degen- erate neutrino masses (adapted from Raﬀelt 1985). A limit on =mesimilar to the solar one can be obtained from the central bulge of the galaxy. Within 2 :5 kpc it contains a luminosity of about 2 ×1010L⊙and thus a neutrino luminosity similarly enhanced. With its distance of (8 :7±0:6) kpc it is about 2 ×109times farther away from us, leading to a much smaller local neutrino ﬂux than that from the Sun. However, the neutrino decay path is also 2 ×109times larger. Even though the neutrino ﬂux scales with the inverse of the distance squared from the source, the ﬂux of decay photons scales only with the inverse distance! Therefore, the local ﬂux of decay photons would be larger than the solar one by perhaps a factor of ten. The measured hard x- and soft -ray ﬂux from the central region of the galaxy is similar in magnitude to the upper limit solar ﬂux so that one obtains a similar constraint on radiative decays. Because no dramatic improvement is expected a detailed analysis is not warranted. However, a substantial improvement is achieved by considering all hydrogen-burning stars in the universe as a source (Sect. 12.6). Radiative Particle Decays 461 12.3.2 Heavy Neutrino Admixtures In full analogy to the case of reactor experiments the solar bound Eq. (12.10) can be reinterpreted as a limit on radiative decays of heavy eadmixtures. To this end one interprets meV=mh=eV and introduces the factor |Ueh|−1on the r.h.s. of Eq. (12.10). It must be stressed, how- ever, that this simple procedure is only applicable to mh∼<30 keV because the main part of the solar neutrino spectrum is relatively soft. In the remaining range of interest, 30 keV ∼<mh∼<1 MeV, the hﬂux is partly suppressed. Moreover, a large fraction of it will be nonrelativis- tic or only moderately relativistic so that the ﬂux of decay photons will have a nonnegligible angular divergence. The detectors used to derive the constraints shown in Fig. 12.4 had a limited forward aperture of about 0 :15 sr, relevant in the energy range 18 −185 keV (Peterson et al. 1966), and 1 sr, relevant in the energy range 163 −774 keV (Frost et al. 1966). Therefore, a certain part of the photon ﬂux that would have come from angles relatively far away from the Sun would have been cut out, weakening the bounds on radiative decays of h. Formh>2meone would, again, expect the decay h→ee+e− to dominate. For mhup to about 14 MeV one may use the8B neu- trinos from the Sun as a source spectrum. The ﬂux of interplanetary positrons from cosmic ray secondaries is measured to be approximately 10−4cm−2s−1sr−1MeV−1at kinetic energies of about 5 MeV. Inter- preting this ﬂux as an upper limit to possible decay positrons from solar neutrinos, Toussaint and Wilczek (1981) derived |Ueh|2∼<  2×10−4(mh= 2 MeV), 2×10−5(mh= 5 MeV), 3×10−6(mh= 10 MeV).(12.11) Because they used the theoretically expected rather than the experi- mentally measured solar8B neutrino ﬂux I have discounted their orig- inal numbers by a factor of 3. These bounds are included in Fig. 12.3; in the applicable mass range they are more restrictive than those from laboratory experiments. They are valid only if the hﬂux is not dimin- ished by invisible decay channels on its way between Sun and Earth, i.e. the total h(laboratory) lifetime must exceed about 500 s. Because in the mass range of a few MeV the time dilation factor for about 10 MeV neutrinos is not large, the bounds apply for total hlifetimes exceeding about 100 s. Such “long-lived” MeV-mass neutrinos are in conﬂict with the big-bang nucleosynthesis constraints shown in Fig. 7.2. 462 Chapter 12 12.3.3 New Particles Besides neutrinos, stars can also produce other weakly interacting par- ticles by both plasma and nuclear processes. With a temperature of about 1 :3 keV in the solar center the former reactions would produce a relatively soft spectrum and so I focus on nuclear reactions where MeV energies are available. If the new particle is a boson and if it couples to nucleons, it will substitute for a photon with certain relative rates r in reactions with ﬁnal-state -rays. A short glance at the nuclear reac- tion chains shown in Fig. 10.2 reveals that a particularly useful case is p+d→3He + with E = 5:5 MeV. This reaction occurs about 1 :87 times for every4He nucleus produced by fusion in the Sun and so it must occur about 1 :7×1038s−1. A certain fraction of the particles pro- duced will be reabsorbed or decay within the Sun. If their probability for escaping is pthe Sun emits r p1:7×1038s−1of the new objects. If the new particle is a scalar boson ait will have a decay channel a→2 . Because this decay is isotropic in a’s rest frame the spectrum of decay photons is box-shaped (Fig. 12.1) with an upper endpoint of 5:5 MeV. If a fraction qof the particles decays between the Sun and Earth the local ﬂux is r p q2:2×1010cm−2s−1MeV−1. The upper limit photon ﬂux shown in Fig. 12.4 at 5 :5 MeV is 0 :8×10−3cm−2s−1MeV−1 (Peterson et al. 1966). From there, Raﬀelt and Stodolsky (1982) found the general upper bound r p q < 4×10−14. They also calculated r,p, andqfor the speciﬁc case of “standard axions” and were able to derive a strong limit on the properties of this hypothetical particle. Together with many laboratory constraints (Particle Data Group 1994) standard axions are now entirely excluded, the main motivation to consider “in- visible axions” instead (Chapter 14). 12.4 Supernova 1987A 12.4.1 Decay Photons from Low-Mass Neutrinos The most signiﬁcant constraints on radiative particle decays from stellar sources can be derived on the basis of the neutrino burst from supernova (SN) 1987A. The neutrino observations and their interpretation were discussed in Chapter 11. For the present purpose it is enough to know that the neutrinos arrived in a short burst lasting a few seconds with a total emitted energy per ﬂavor of about 1 ×1053erg = 6 :2×1058MeV. Forethe measured spectral distribution is consistent with a thermal Radiative Particle Decays 463 emission at Te≈4 MeV so that the ﬂuence73of’s plus ’s per ﬂavor was about F= 1:4×1010cm−2(4 MeV =T); (12.12) taking ⟨E⟩= 3T. Approximately the same result is thought to apply toandwith about 1 :3−1:7 times the temperature, although for those ﬂavors there is no direct measurement. Because SN 1987A occurred in the Large Magellanic Cloud (LMC) at an approximate distance of dLMC= 50 kpc = 1 :5×1023cm an enor- mous decay path was available for the neutrinos from this measured source. A very restrictive upper limit photon ﬂux was provided by the gamma ray spectrometer on the solar maximum mission (SMM) satel- lite which was operational at the time of the neutrino signal and did not register any excess counts above the normal background. In or- der to use this result one needs to compute the expected signal from neutrino decay. Because one is dealing with a short neutrino burst the previous results for stationary sources do not apply directly: the time structure of the expected photon burst must be taken into account. This is easy when the mass of the parent neutrino is below about 40 eV; the pulse dispersion is then not much larger than the duration of the observed eburst. Because for low-mass neutrinos the decay photons have essentially the same time structure as the neutrino burst one considers the ﬂuence for a time interval of about 10 s around the ﬁrst neutrino arrival. The non- eﬂavors could be heavier if they violate the cosmological mass limit of a few 10 eV. Then the photon pulse will be correspondingly stretched, a case to be studied in Sect. 12.4.4 below. If the neutrino masses are not degenerate so that in Eq. (12.5) m= 1 the expected diﬀerential ﬂuence is F′ (E ) =Fm dLMC∫∞ E dE( 1−+ 2E E)Φ(E) E2 ; (12.13) where Φ (E)≡ F′ (E)=Fis a normalized spectrum (units MeV−1). For this expression CP conservation was assumed so that ’s and ’s are characterized by the same values of andas discussed in Sect. 12.2.1. 73With ﬂuence one means the time-integrated ﬂux. In this book I use the sym- bolFfor a diﬀerential particle ﬂux (cm−2s−1MeV−1), the symbol Ffor a ﬂuence (cm−2), and F′=dF=dEfor a “diﬀerential ﬂuence” which includes spectral infor- mation (cm−2MeV−1). 464 Chapter 12 Fig. 12.6. Expected ﬂuence of photons from the decay →′ of low-mass SN neutrinos according to Eq. (12.15) with m= = 10−15eV=s and the indicated neutrino temperatures. The shaded bands for each temperature are for the range −1≤≤+1 with the harder edge corresponding to = +1. Also shown are the upper limits from the GRS channels taken from the 10 s column of Tab. 12.1. Because details of the spectral form are not known it is easiest to use a Boltzmann distribution as a generic case, Φ(E) =E2 e−E=T 2T3 : (12.14) Then one ﬁnds explicitly74 F′ (E ) =Fm dLMC 2T2 [ (1−)e−"+ 2 " E 1(")] ; (12.15) where "≡E =T. In Fig. 12.6 this spectrum is shown for T= 4 and 8 MeV with m= = 10−15eV=s. The envelopes of the shaded bands in Fig. 12.6 correspond to =±1 where for each temperature the “harder” edge corresponds to = +1. The observational constraints give a limiting ﬂuence for certain energy bands. Thus one needs the expected ﬂuence for a given energy 74The exponential integral function is deﬁned as En(x) =∫∞ 1dt e−x t=tn. Note thatEn(∞) = 0 while for n >1En(0) = ( n−1)−1. Radiative Particle Decays 465 range ( E ;1; E ;2) for which one ﬁnds by integration of Eq. (12.15) F ;1;2=Fm dLMC 2T{ (1−)e−"+ 2[" E2(") +E3(")]}"1 "2; (12.16) where the expression in braces is meant to be taken as a diﬀerence between the two limits for ". For "1= 0 and "2=∞it is equal to 1, independently of , as it must because the angular distribution of photon emission leaves the total number of decay photons unchanged. 12.4.2 SMM Observations The gamma ray spectrometer (GRS) on the SMM satellite consists of seven NaI detectors surrounded on the sides by a CsI annulus and at the back by a CsI detector plate (Forrest et al. 1980). The three energy bands shown in Tab. 12.1 have been analyzed for -ray emission from SN 1987A (Chupp, Vestrand, and Reppin 1989; Oberauer et al. 1993). At the detection time of the ﬁrst neutrino event at IMB (7:35:41.37 UT) the GRS was observing the Sun. A time interval of 223 :232 s until it went into calibration mode was used to search for a photon signal above background in each energy band. The background was determined by analyzing the rates measured during an interval of 151 :6 s before the ﬁrst neutrino event. In Fig. 12.7 the recorded number of events per 2:048 s is shown in each band as a function of time. No excess counts were found in any of them and the distributions of the rates are in good agreement with a Gaussian shape. Because the GRS was observing the Sun, -rays associated with the neutrino burst would have hit the instrument almost exactly from the side and so they had to traverse about 2 :5 g cm2of spacecraft alu- minum before being recognized in one of the detectors. This eﬀect has been included to calculate the eﬀective detector areas which allow one to convert “counts” into a -ray ﬂuence. In Tab. 12.1 the corre- sponding 3 limits are shown for the time until 223 :232 s after the ﬁrst neutrino arrival (Oberauer et al. 1993). In order to constrain low-mass neutrinos, only a time interval of 10 s around the burst is of interest; the corresponding limits are also given (Chupp, Vestrand, and Rep- pin 1989). The ﬂuence limits (cm−2) can be expressed as limits on an average diﬀerential ﬂuence (cm−2MeV−1) by dividing with the width ∆E of a given channel. For the 10 s column they are shown as dotted histograms in Fig. 12.6 while for 223 :2 s they are shown in Fig. 12.13. 466 Chapter 12 Fig. 12.7. Event rates measured in the Gamma Ray Spectrometer (GRS) of the Solar Maximum Mission (SMM) satellite encompassing the observed neutrino burst of SN 1987A (the dashed line is for the ﬁrst neutrinos observed in the IMB detector). The time interval for each bin is 2 :048 s. The rates to the left of the dashed line are used to determine the background while the ones to the right would include photons from neutrino decay. (Figure from Oberauer et al. 1993 with permission.) Table 12.1. GRS 3 upper ﬂuence limits. Channel Energy Band Fluence Limit [cm−2] [MeV] (10 s)a(223:2 s)b 1 4:1−6:4 0.9 6.11 2 10−25 0.4 1.48 3 25−100 0.6 1.84 aChupp, Vestrand, and Reppin (1989) bOberauer et al. (1993) For the 10 s and 223 :2 s time intervals one can compute an average ﬂux limit (cm−2s−1) for each channel. Then one expects that for the longer time interval it is more restrictive by the ratio of (∆ t)1=2, i.e. by (10 s=223:2 s)1=2= 0:21. This expectation is approximately borne out by the data in Tab. 12.1, conﬁrming their consistency. Radiative Particle Decays 467 12.4.3 Radiative Decay Limit: Low-Mass Neutrinos We are now armed to derive a radiative lifetime limit for low-mass neutrinos ( m∼<40 eV) by comparing the expected ﬂuence according to Eqs. (12.12) and (12.16) for channels 1, 2, and 3 with the observational upper limits given in Tab. 12.1 for a 10 s time interval surrounding the observed SN 1987A eburst. Depending on the assumed neutrino spectral distribution which was parametrized by Tand the anisotropy parameter , diﬀerent channels give the most restrictive limits. These are shown in Fig. 12.8 as a function of Tfor= 0;±1. For normal neutrinos the relevant temperature range is between 4 and 8 MeV. In Fig. 12.8 a much larger range is shown because one may also consider the emission of sterile neutrinos or axions from the deep interior of a SN core where temperatures of several 10 MeV and Fermi energies of several 100 MeV are available (Sect. 12.4.6). For Dirac neutrinos the most conservative case is =−1 where for 4 MeV ∼<T∼<8 MeV the bound is approximately constant at =m>0:8×1015s=eV. Therefore, it applies equally to eand low- mass and. For Majorana neutrinos ( = 0) the limit is more restrictive by a factor of 2 −3, depending on the assumed T. With Eq. (7.12) the most conservative overall limit75(=−1) translates into eﬀ<1:5×10−8Bm−2 eV: (12.17) It applies if the total laboratory lifetime exceeds the time of ﬂight of 5:7×1012s from the LMC to us. With a typical E= 20 MeV one must require tot=m∼>3×105s=eV in the neutrino rest frame. If the total lifetime is shorter than this limit all neutrinos decay before they reach the Earth. Therefore, one can only derive a limit on the branching ratio B of the radiative channel. One easily ﬁnds that Eq. (12.13) is to be replaced by F′ (E ) =FB ∫∞ E dE(1−+ 2 E =E) Φ(E)=E:(12.18) Therefore, using the same Boltzmann source spectrum the photon spec- trum is slightly harder. Going through the same steps as before one ﬁnds the upper limits on B as a function of the assumed Tand 75Somewhat stronger constraints found in the literature were based on a less de- tailed analysis, notably with regard to the spectral dependence and the dependence on. Published results are =m>0:83×1015s=eV (von Feilitzsch and Oberauer 1988), 1 :7×1015(Kolb and Turner 1989), 6 :3×1015(Chupp, Vestrand, and Reppin 1989), and 2 :8×1015(Bludman 1992). 468 Chapter 12 Fig. 12.8. Lower limit on =mform∼<40 eV and tot=m∼>5×105s=eV (most neutrinos pass the Earth before decaying). Fig. 12.9. Upper limit on the radiative branching ratio B form∼<40 eV andtot=m∼<5×105s=eV (most neutrinos decay between SN 1987A and Earth). shown in Fig. 12.9. The most conservative case yields approximately B <3×10−10. As a bound on an eﬀective transition moment this is eﬀ<0:7×10−5Bm−2 eV(meV=s)1=2; (12.19) where s=tot=s. Of course, if totbecame so short that the neutrinos would decay while still within the envelope of the progenitor even this Radiative Particle Decays 469 weak limit would not apply. This is the case for lab∼<Renv≈100 s (envelope radius Renvof the progenitor star) and so with E≈20 MeV one needs to require tot=m∼>10−5s=eV. 12.4.4 Decay Photons from High-Mass Neutrinos For “high-mass” neutrinos with m∼>40 eV a calculation of the photon ﬂuence is more involved because of the dispersion of the neutrino burst and the corresponding delay of the decay photons. The parent neutrino travels with a velocity = (1−m2 =E2 )1=2≈1−m2 =2E2 so that it arrives at Earth with a time delay of about ( m2 =2E2 )dLMC relative to massless ones; here, dLMC = 50 kpc = 5 :1×1012s is our distance to the LMC where SN 1987A had occurred. With T≈6 MeV for oran average neutrino energy is 3 T≈20 MeV, spreading out the arrival times of massive neutrinos over an approximate interval of 10−2s (m=eV)2. As the radiative decay may occur anywhere between the LMC and here, the arrival times of the decay photons will be spread out by a similar amount even though the photons themselves travel with the speed of light. Thus for m∼<40 eV all decay photons fall within about a 10 s time window around the arrival time of the ﬁrst e; the results of Sect. 12.4.3 apply to this case. For m∼<200 eV they fall within the 223 :2 s interval for which GRS ﬂuence limits exist. Therefore, one may easily scale the previous limits to this case by using the 223 :2 s ﬂuence limits in Tab. 12.1 instead of the 10 s ones. Of course, for a given neutrino mass there would be an optimum time window for which ﬂuence limits could be derived on the basis of the original data. For larger masses only a certain portion of the photon pulse falls into the 223 :2 s window. In order to calculate this fraction, I follow Oberauer et al. (1993) and begin with the simple case where all neu- trinos are emitted at the same time with a ﬁxed energy E. The ra- diative decay occurs at a time tDafter emission and thus at a distance dD=tDfrom the source (neutrino velocity ), and the photon is emit- ted at an angle labrelative to the neutrino momentum (Fig. 12.10). It has to travel a distance d until it arrives here; elementary geom- etry yields d = [d2 LMC−d2 D(1−cos2lab)]1=2−dDcoslab. Therefore, relative to the ﬁrst (massless) neutrinos the photons are delayed by t=tD+d −dLMC. This delay has two sources: The parent moves with a speed less than that of light, and the photon is emitted at an angle so that a detour is taken from the LMC to us. For ultrarelativis- tic parents both eﬀects disappear as the relativistic transformations squeeze all laboratory emission angles into the forward direction. 470 Chapter 12 Fig. 12.10. Geometry of the radiative neutrino decay. The photons which arrive ﬁrst are the ones from decays near the source. Then the limit dD≪dLMCleads to d =dLMC−dDcoslaband t=tD(1−coslab). With cos lab= (+ cos )=(1 +cos) where is the angle of photon emission in the parent frame one ﬁnds t=tD 2(1 +x); (12.20) where =E=mis the neutrino Lorentz factor and x≡cos. There- fore, photons detected between tandt+dtresult from decays at tD= 2(1 + x)tduring an interval dtD= 2(1 + x)dt. (Recall that tis measured after the ﬁrst massless neutrinos arrived while tD is measured after emission at the source.) The number of parent neu- trinos diminishes in time as e−tD= totwith totthetotal decay time. Therefore, the number of photons traversing a spherical shell of radius dLMCper unit time is ˙N (t) = (1 +x) e− (1+x)t=tot; (12.21) where as before is the radiative decay time. Photons produced before the parent has left the envelope of the progenitor star (radius Renv) cannot be detected at Earth. If this absorption eﬀect is to be included, Eq. (12.21) will involve a step function76Θ(dD−Renv) with dD=tD= 2(1 + x)t. Put an- other way, photons emitted at an angle in the rest frame will ﬁrst arrive at a time t0=Renv[ 2(1 +x)]−1. The progenitor of SN 1987A has been unambiguously identiﬁed as the blue supergiant Sanduleak −69 202 (Schramm and Truran 1990). From its surface temperature (15,000 K), its luminosity (5 ×1038erg=s) and the distance to the LMC one can infer its radius to be Renv≈3×1012cm = 100 s. Using this 76The step function is deﬁned by Θ( z) = 0 for z <0 and Θ( z) = 1 for z >0. Radiative Particle Decays 471 value, t0is shown in Fig. 12.11 as a function of the neutrino velocity for several values of x= cos . The absorption eﬀect is negligible except for nonrelativistic neutrinos or for backward emission which, in the laboratory frame, corresponds to very soft photon energies. Fig. 12.11. Arrival time t0of ﬁrst decay photons from a parent neutrino with velocity , taking the envelope radius of the source to be Renv= 100 s. The curves are marked with the respective values of x= cos , the direction of photon emission in the neutrino rest frame. In the parent frame the photons (energy !) follow a normalized distribution f(!; x) which yields d3N (t; !; x ) = ˙N (t)f(!; x)dt d! dx with ˙N (t) from Eq. (12.21). The usual relativistic transformations lead to a laboratory photon energy of E = (1 +x)!. Transforming from d!dx tod!dE and integrating over the unobserved rest-frame energy !yields d2N dE dt=1 E ∫! !+d!f(!; x) !2e−E t=! totΘ( E t Renv−!) ; (12.22) where !±=E [ (1±)]−1andx= (E = !−1)=. The photon ﬂux at Earth is obtained by multiplication with the neutrino ﬂuence Fof Eq. (12.12) and integration over a suitable spectrum Φ (E) of neutrino energies. If the neutrinos are suﬃciently long-lived (the exact meaning of this is quantiﬁed below) the exponential can be ignored. If one also ignores the absorption eﬀect by the progenitor star ( Renv= 0), the 472 Chapter 12 time structure of Eq. (12.22) reduces to Θ( t), i.e. its spectral form is time independent except that it begins at t= 0. This is somewhat surprising because the energy of a photon in the laboratory frame is related to the angle of emission in the neutrino rest frame which in turn determines the “detour” taken from the source to us (Fig. 12.10). However, the ﬁrst photons come from decays immediately at the source and so any angle of emission leads to the same initial arrival time. It must be stressed that the expression Eq. (12.22) depends on the assumption of decays not too far from the source ( dD≪dLMC) and so only the head of the photon pulse is correctly described while its tail would require including decays even close to the Earth. Strictly speak- ing, the photon burst never ends because even if the parent neutrinos have passed the Earth, some photons will be received from backward emission. However, because one is interested in neutrino masses so large ( m∼>200 eV) that the photon burst is much longer than the GRS measurement window, it is enough to account for the head of the photon pulse. 12.4.5 Radiative Decay Limits: High-Mass Neutrinos As a ﬁrst explicit case for the distribution of photon energies and emis- sion angles I take the two-body decay →′ with a massless daugh- ter neutrino and with the dipole angular distribution of Eq. (12.2) with x= cos =−cos#for a left-handed parent. This amounts to f(!; x) =1 2(1 +x)(!−1 2m) (12.23) in Eq. (12.22). Because of the function it is trivial to integrate, d2N dE dt=1 m 2E p[ 1 +2E −E p] e−t=Θ(t−tenv);(12.24) where ∗≡m 2E tot; t env≡m2 2E pRenv: (12.25) Moreover, the ﬂux vanishes if the chosen value for E does not fall between1 2(E±p), or equivalently, unless E> E +m2 =4E ; (12.26) a condition on the minimum required neutrino energy. For the simplest case when the neutrinos are relativistic ( p=E), long-lived ( e−t== 1), and absorption eﬀects by the progenitor can be Radiative Particle Decays 473 ignored ( tenv= 0) the spectrum is shown in Fig. 12.12 for the anisotropy parameters = 0;±1. Note the diﬀerence to the triangular shape of Fig. 12.1 for a stationary source. High-energy photons are now en- hanced because lower-energy ones correspond to larger emission angles in the parent frame and so they take a larger “detour” from the source to us (Fig. 12.10). Hence, their ﬂux is spread out over a larger time interval even though photons of all energies begin to arrive at the same time if tenv= 0. Fig. 12.12. Photon spectrum from the decay of a short burst of relativistic neutrinos, energy E, according to Eq. (12.24) taking p=Eande−t== 1 (relativistic and long-lived parent), and ignoring absorption eﬀects by the progenitor ( tenv= 0). In order to compare with the GRS ﬂuence limits one needs to in- tegrate the expected ﬂux between t= 0 and t=tGRS = 223 :2 s. The Θ function in Eq. (12.24) is accounted for by using tenvas a lower limit of integration. Integrating also over the neutrino source spectrum FΦ(E) yields F′ =FtGRS m ∫∞ EmindEΦ2E p[ 1 +2E −E p] I; (12.27) where I≡e−tenv=−e−tGRS= tGRS=∗: (12.28) For suﬃciently long-lived parents ( ∗≫tGRS) the exponentials can be expanded and I= (tGRS−tenv)=tGRS. The lower limit of integra- tion is set by the condition Eq. (12.26) and by the requirement that tenv< t GRS, i.e. that I > 0. This condition may be expressed as p>(m2 =2E ) (Renv=tGRS). 474 Chapter 12 The two-body decay is mostly interesting for m∼<2me, a limit in which one may safely ignore all nonrelativistic corrections, including the progenitor absorption eﬀect. In this case I=1−e−tGRS= tGRS=∗; (12.29) which is unity for ∗∼>tGRS. With typical photon energies of 3 T≈ 20 MeV and tGRS= 223 :2 s this requirement translates into mtot∼> 1010eV s. In the relativistic limit and with I= 1 one can easily integrate Eq. (12.27) with the Boltzmann spectrum Eq. (12.14) and ﬁnds F′ =FtGRS m [ (1−)"+ (1 + )"2] e−"; (12.30) where "=E =T. This spectrum is shown in Fig. 12.13 for T= 4 and 8 MeV with m = 1018eV s. The envelopes of the shaded bands in Fig. 12.13 correspond to =±1 where for each temperature the “harder” edge corresponds to = +1. The expected ﬂuence for each GRS channel of Tab. 12.1 is found by integration. The GRS ﬂuence limits then yield the lower bounds on m shown in Fig. 12.14. Fig. 12.13. Expected ﬂuence of photons from the decay →′ of “high- mass” but relativistic SN neutrinos, 200 eV ∼<m∼<1 MeV, according to Eq. (12.30) with m = 1018eV s and the indicated neutrino temperatures. The shaded bands for each temperature are for the range −1≤≤+1 with the harder edge corresponding to = +1. Also shown are the upper limits from the GRS channels taken from the 223 :2 s column of Tab. 12.1. Radiative Particle Decays 475 Fig. 12.14. Lower limit on m for 200 eV ∼<m∼<1 MeV, assuming mtot∼>1010eV s. For Dirac neutrinos the most conservative case is =−1 and the temperature range relevant for andis between 6 and 8 MeV. This yields an approximately temperature independent bound of m > 7×1018eV s. For Majorana neutrinos ( = 0) the limit is about77 m >12×1018eV s. The most conservative case ( =−1) trans- lates with Eq. (7.12) into eﬀ<1:6×10−10Bm−1 eV; (12.31) assuming mtot∼>1010eV s. Note the diﬀerent dependence on meV relative to the low-mass result Eq. (12.17). Ifmtotviolates this condition because it is below 1010eV s means that the neutrinos decay so fast that the photon burst eﬀectively ends before the GRS integration time tGRSis over. Then the photon burst is again “short” even though the neutrino mass is large. In this case one can state a limit on B as in Sect. 12.4.3. If totis only slightly shorter so that mtot∼<4×108eV s, all photons arrive within tGRS= 10 s and one may directly apply Eq. (12.19), originally derived for small neutrino masses. For the narrow region where the photon burst ends between 10 and 223 :2 s this bound must be discounted by about a factor of 2 because of the less restrictive ﬂuence limits for the larger integration time. 77The result here is after Oberauer et al. (1993) which is similar to 6 ×1018eV s of Bludman (1992) but substantially more restrictive than 0 :84×1018eV s of Kolb and Turner (1989). These works all refer to the isotropic case ( = 0). 476 Chapter 12 Fig. 12.15. Nonrelativistic correction of the expected photon ﬂuence for the GRS channels of Tab. 12.1 according to Eq. (12.32) with Renv= 100 s, tGRS= 223 :2 s, and T= 6 MeV. So far the nonrelativistic corrections for a neutrino mass in the 10 MeV mass range have been ignored. As an example I consider the photon ﬂuence of Eq. (12.27) in the limit of a large ∗where the ex- ponentials in Eq. (12.28) can be expanded. The Boltzmann spectrum for nonrelativistic neutrinos should include an extra factor =p=E. Then for = 0 the ﬂuence is78 F′ (E ) =FtGRS m ∫∞ EmindEEe−E=T T3 ( E −m2 2pRenv tGRS) ;(12.32) with Emin= max  m 1 +(m 2E Renv tGRS)2 1=2 ;( E +m2 4E )  :(12.33) Relative to the massless case, the integral expression is suppressed if m∼>T. A straightforward numerical integration then yields the sup- pression of the expected ﬂuence for each GRS channel as shown in 78Strictly speaking, the ﬂuence of massive neutrinos must be calculated by de- termining their neutrino sphere which is diﬀerent from the massless case. This problem was recently tackled by Sigl and Turner (1995) by solving the Boltzmann collision equation by means of an approximation method known from calculations of particle freeze-out in the early universe. However, because only masses of up to 24 MeV are presently considered, a precise treatment of the neutrino spectrum changes the resulting limits only by a small amount. Radiative Particle Decays 477 Fig. 12.15. (The overall factor m−1 of Eq. 12.32 is not included, of course.) Up to neutrino masses of about 10 MeV one may essentially ignore the nonrelativistic corrections while for larger masses one has to worry about them. However, because this discussion applies to stan- dard neutrinos, the largest relevant mass is about 24 MeV and so the nonrelativistic corrections never overwhelm the result. 12.4.6 Summary of !′ Limits In order to summarize the decay limits I begin in Fig. 12.16 with the rel- evant regimes of mandtot. Above the upper dotted line the neutrinos live long enough so that most of them pass the Earth before decaying while below the lower dotted line they decay within the envelope of the progenitor star. In the areas 1 and 5 the photon burst is “short” (∆t ∼<10 s), in 2 and 4 it is “intermediate” (10 s ∼<∆t ∼<223:2 s), and in 3 it is “long” (223 :2 s∼<∆t ). The exact boundaries as well as the relevant constraints are summarized in Tab. 12.2. In Fig. 12.17 the limits on eﬀare summarized as a contour plot. If one restricts possible neutrino decays to the radiative channel one hastot= which depends only on eﬀandm. Then one may use directly the upper limits on eﬀgiven in Tab. 12.2 for the areas 1 −3, depending on the assumed mass. Put another way, the conditions on totare then automatically satisﬁed. These limits certainly apply to eas the eburst from SN 1987A has been measured. The ﬂuxes of the other ﬂavors were only theoretically implied. If they have only standard weak interactions they must have been emitted approximately with the same eﬃciency as e. Large dipole moments, however, imply large nonstandard interactions: The same electromagnetic interaction vertex that allows for radiative decays also allows for scattering on charged particles by photon exchange! For MeV energies, for example, the scattering cross section on electrons by regular weak interactions and that by photon exchange are the same foreﬀof order 10−10B. Hence in the lower left corner of Fig. 12.17 the neutrinos would interact much more strongly by photon exchange than by ordinary weak interactions, causing them to emerge from higher layers of the SN core than normally assumed. Their ﬂuence and eﬀective temperature is then much smaller than standard. Put another way, for eﬀ∼>10−10Bthe above constraints are not self-consistent (Hatsuda, Lim, and Yoshimura 1988). However, because large dipole moments can be constrained by other methods (Sect. 7.5.1) a detailed investigation of their impact on SN physics is not warranted. 478 Chapter 12 Fig. 12.16. Diﬀerent regimes of neutrino masses and total lifetimes referred to in the text. ∆ t is the duration of the burst of decay photons. The radiative lifetime limits in the areas 1 −6 are summarized in Tab. 12.2. Fig. 12.17. Upper limits on eﬀective neutrino transition moments from SN 1987A as given in Tab. 12.2. The contours are marked with log( eﬀ=B). In the shaded lower right region the neutrinos decay within the progenitor. Toward the lower left side the bounds are not self-consistent because large dipole moments induce large nonstandard scattering cross sections which enhance neutrino trapping. The discussion of the previous sections focussed on standard neu- trinos which are emitted approximately with the same eﬃciency and similar energies as e’s and e’s. It is possible, however, that these neu- Radiative Particle Decays 479 Table 12.2. Neutrino radiative lifetime limitsafrom SN 1987A. AreabBoundariescRadiative lifetime Transition momentd limitceﬀ=B 1 m<40 m−1 >0:8×10151:5×10−8m−2 3×105< totm−1 2 40 < m <200 m−1 >0:2×10150:8×10−8m−2 3×105< totm−1 3 200 < m <107 m>7×10181:6×10−10m−1 9×109< totm 4 m<107B <1:2×10−91:4×10−5m−3=2 −1=2 tot 4×108< totm<9×109 10−5< totm−1 <3×105 5 m<107B <3×10−100:7×10−5m−3=2 −1=2 tot totm<4×108 10−5< totm−1 <3×105 6 totm−1 <10−5B <0:01 — aFor the anisotropy parameter =−1. bNumbered as in Fig. 12.16. cNeutrino masses in eV, lifetimes in s. dUpper limit. trinos have Dirac masses and thus right-handed partners which could be emitted from the inner core of the SN by helicity-ﬂipping processes (Sect. 13.8). Moreover, entirely new particles could be produced and escape from there. The present bounds can be scaled to such cases if one calculates the total energy Ex;tot=fx1053erg emitted in the new xparticles, where 1×1053erg is the total energy that was used for a standard plus. Self-consistency requires fx<1, of course. In addition, one needs the average energy ⟨Ex⟩of the new objects which allows one to deﬁne an approximate equivalent temperature Tx=1 3⟨Ex⟩. Depending on the x mass and total lifetime one can then read the radiative lifetime limits directly from Figs. 12.8, 12.9, and 12.14, except that they must be relaxed by a factor fxfor the reduced ﬂuence. 480 Chapter 12 12.4.7 Limit on !ee+e− Neutrinos with a mass exceeding 2 mecan decay into ee+e−, a channel which probably dominates over ′ . Among the standard neutrinos, the role of the parent can be played only by (or rather 3) with its upper experimental mass limit of about 24 MeV. Within the standard model where the decay is due to ﬂavor mixing the rate is given by Eq. (7.9). In order to derive bounds on the e+e−channel from the GRS obser- vations, photons need to be produced. At ﬁrst one may think that the positrons would quickly annihilate so that a strong prompt ﬂux can be expected (Takahara and Sato 1987; Cowsik, Schramm, and H¨ oﬂich 1989). Following Mohapatra, Nussinov, and Zhang (1994), however, the gas density outside of the progenitor is too low, in spite of a substan- tial stellar wind during the progenitor’s supergiant evolution. Also, the annihilation of the charged leptons from the decay among each other is moderately eﬃcient only if the decays occur close to the source. Typ- ical galactic magnetic ﬁelds have a strength of about 3 G; they may well be larger in the Large Magellanic Cloud, and the circumstellar ﬁeld of the SN 1987A progenitor may have been larger still. The gyromag- netic radii for 5 MeV positrons is then less than 1010cm≪Renvso that one may think that the charged leptons were locally trapped (Cowsik, Schramm, and H¨ oﬂich 1989). However, the momentum carried by the ﬂux of the charged decay products is so large that those ﬁelds would have been swept away (Mohapatra, Nussinov, and Zhang 1994). Altogether it appears that the decay positrons will linger in inter- stellar space for a long time before meeting annihilation partners unless most decays occur immediately outside of the progenitor. Therefore, prompt photons are mostly produced by bremsstrahlung →ee+e− which is suppressed relative to the decay rate only by a factor of about =≈10−3(Dar and Dado 1987). Neutrinos with masses in the MeV range which are emitted at MeV temperatures are nearly nonrelativistic. Their rest frame is then ap- proximately equal to the laboratory frame, but they still move essen- tially with the speed of light. To escape from the progenitor before decaying their rest-frame lifetime totmust exceed a few 100 s. This also guarantees that the pulse of decay photons will outlast the GRS integration time: one is automatically in the region of a “long” photon burst which was “case 3” in Fig. 12.16. In fact, if one assumes that decays are induced by mixing, the decay rate Eq. (7.9) together with the laboratory bounds on Uehshown in Fig. 12.3 easily guarantees that they fulﬁll this requirement. Radiative Particle Decays 481 In order to estimate the expected photon ﬂux from Eq. (12.22) one needs to know the distribution of photon energies and emission angles in the frame of the parent neutrino. In the absence of a detailed calcu- lation I follow Oberauer et al. (1993) and assume approximate isotropy for the photon emission. The soft part of the spectrum dN =d!from a bremsstrahlung process is given by the rate of the primary process times ( =)!−1(Jackson 1975). Extending this behavior up to photon energies of1 2mI use 1 f(!; x) == e+e1 2!Θ(1 2m−!); (12.34) an expression which does not depend on xbecause of the assumed isotropy. ( = 1=137 is the ﬁne-structure constant, not the previous anisotropy parameter.) After dropping the exponential in Eq. (12.22) because the neutri- nos are long-lived, one integrates over a Boltzmann source spectrum for nonrelativistic neutrinos, integrates over the GRS energy channels, and compares the expected ﬂuence with the measured upper limits of Tab. 12.1. The resulting bound on e+ecorresponds with Eq. (7.9) directly to a limit on |Ue3|2. In Fig. 12.18 I show these bounds (trans- formed into bounds on the mixing angle) as a function of the assumed neutrino mass for T= 4 and 6 MeV. There remains a strong limit even for masses far exceeding the temperature because the exponential Fig. 12.18. SN 1987A limits on sin22e3= 4|Ue3|2with 3≈for T= 6 MeV (solid line) and 4 MeV (dashed line). Also shown are the corre- sponding laboratory and solar limits from Fig. 12.3. 482 Chapter 12 suppression of the ﬂux is compensated by the steep phase-space factor m5 in the expression for the decay rate. These bounds are far more restrictive than those from laboratory experiments (Fig. 12.3). The reason is that a SN explosion is a strong source; such a source is diﬃcult to make in the laboratory. This is the reason why the has never been directly measured by its charged- current conversion into . The strong SN 1987A bounds on |Ue3|2imply that a heavy with only standard-model interactions must be rather long-lived, in fact too long to be compatible with cosmological limits derived from big-bang nucleonsynthesis (Sect. 7.1.5). Those bounds together with the present results imply that a heavy cannot exist unless it has fast, invis- ible decays induced by interactions beyond the standard model. If this were the case it could well decay before leaving the SN progeni- tor. Therefore, the laboratory experiments with their short distance between source and decay volume remain important for anomalously short-lived neutrinos. 12.4.8 Heavy, Sterile Neutrinos The bounds on Ue3from reactors, beam stops, or the Sun were based on aeﬂux which partially converts into 3’s which subsequently decay. Because the SN emits about equal numbers of all ordinary neutrino ﬂavors, this approach is obsolete with regard to 3. However, one may still consider hypothetical sterile neutrinos which interact only by virtue of their mixing with e. By assumption these states would not interact through ordinary weak interactions and so they would not be trapped in the SN core. Hence the expected hﬂux would emerge from the deep interior rather than the surface of the core. In this case, however, the sterile neutrinos would carry away energy much more eﬃciently than the ordinary ones and so the requirement that enough energy was left for the observed e’s from SN 1987A al- ready gives one the approximate limit |Ueh|2∼<10−10(Sect. 9.6). If this limit is approximately saturated one expects that about as much energy is carried away by has by the ordinary ﬂavors. Taking account of the harder energies of neutrinos emitted from the SN core one still obtains about the same limit on |Ueh|2as on|Ue3|2before. Put an- other way, the GRS observations do not dramatically improve on the cooling argument of Sect. 9.6, although they range in the same general magnitude. Radiative Particle Decays 483 12.4.9 Axions Another hypothetical particle that could have been emitted abundantly from SN 1987A is the axion. The impact of the axionic energy loss is discussed in Sect. 13.5. If axions are more strongly interacting than a certain limit, implying that their mass is larger than a few eV, they are emitted from the surface of the SN core with a luminosity similar to that of neutrinos. Kolb and Turner (1989) found that the axion ﬂuence from SN 1987A would have been Fa≈6×1010cm−2m−12=11 eV with meV≡ma=eV at a temperature of Ta≈15 MeV m−4=11 eV. From the GRS ﬂuence limits, Kolb and Turner found that mamust be less than a few 10 eV, a bound which is less restrictive than, for example, the limit from globular cluster stars (Sect. 5.2.5). 12.4.10 Supernova Energetics To derive the various GRS limits one had to assume that the radiative decays occurred outside of the progenitor’s envelope and so neutrinos falling into the shaded area in Fig. 12.16 were not accessible to these arguments. However, in this case the stellar envelope itself serves as a “detector” as discussed by Falk and Schramm (1978) many years ago; see also Takahara and Sato (1986). Supernova observations in general, and those of SN 1987A in particular, indicate that of the approximately 3×1053erg of released gravitational binding energy only a small fraction on the order of one percent becomes directly visible in the form of the optical explosion as well as the kinetic energy of the ejecta. In contrast, even if only one of the neutrino species decayed radiatively within the progenitor, about 30% of the binding energy would light up! If the lifetime were so short that the parent neutrinos would never get far from the SN core one would not have to worry. Therefore, the critical range of decay times is between the core dimensions of about 30 km = 10−4s and the envelope radius of about 100 s. If the neutrinos had nonradiative decay modes, and if their total laboratory lifetime fell into this range, one could only conclude that B ∼<10−2. If they decayed only into radiation, and taking Eto be 10 MeV, the quantity =mcannot lie between about 10−11and 10−5s=eV (for heavy neutrinos includes the e+e−channel). For an eﬀective tran- sition moment eﬀ, an interval between about 102and 105Bm−2 eVis excluded, moderately interesting only for large masses. Then, however, the cosmological limits strongly suggest the presence of nonradiative decay channels. 484 Chapter 12 12.5 Galactic Supernovae and !ee+e 12.5.1 Bounds on the Positron Flux A core-collapse SN produces about 3 ×1057’s which may subsequently decay into ee+e−. What is the long-term fate of all these positrons? If the’s decay mostly outside of the galaxy it is well possible that the positrons will linger in intergalactic space “forever.” Those positrons produced within the galaxy, however, will be trapped by the magnetic ﬁelds (typical strength a few G) which render the galactic disk a mag- netic “bottle” for charged particles. The interstellar electron density is on the order 1 cm−3, leading to a positron lifetime against annihilation of order 105years. Moreover, elastic e+e−scattering (Bhabha scatter- ing) is very eﬃcient at slowing down relativistic positrons because of the perfect mass match which allows for an eﬃcient energy exchange in collisions. Therefore, most annihilations occur at rest, producing a sharp -ray feature at 511 keV. The galactic SN rate is a few per century while the decay positrons annihilate on a much longer time scale. Therefore, the galactic disk should contain a stationary positron population with a density deter- mined by the galactic SN rate and the lifetime. A comparison with the measured photon ﬂux at E = 511 keV of about 5 ×10−3cm−2s−1 then leads to a very restrictive limit on the e+e−decay channel (Dar, Goodman, and Nussinov 1987). In detail these authors used a galactic rate of two core-collapse SN per century, an e+lifetime against annihilation of 105yr, and a typical distance of the decays from Earth of 10 kpc. If most neutrinos decay within the galactic disk, these assumptions lead to an expected photon ﬂux of 400 cm−2s−1. Thus, it is enough that one in 105neutrinos decays within the galactic disk to outshine the measured ﬂux. This estimate is corroborated by the more recent work of Skibo, Ra- maty, and Leventhal (1992) who devised detailed models of the positron distribution in the galaxy in order to account for the 511 keV diﬀuse galactic line feature measured in the direction away from the galac- tic center. They found a total stationary positron annihilation rate in the galaxy of 0 :6−3×1043s−1, where the precise coeﬃcient depends on model assumptions. With two core-collapse SN per century the average galactic production rate is 2 ×1048s−1. Again, it is enough if one in 105injects an e+into the galaxy to account for the observations. If all positrons produced within about 1 kpc = 3 ×1021cm from the source (the scale height of the galactic disk) were magnetically trapped, Radiative Particle Decays 485 while those produced further away escaped into intergalactic space, a lifetime below 105×1 kpc≈1016s in the laboratory frame is excluded. Because SN neutrinos with MeV masses are nearly nonrelativistic the rest-frame lifetime is identical with the laboratory lifetime to within a factor of a few, excluding e+e∼<1015s. If the decays occur too close to the source the annihilation with electrons from the neutrino decay is of some importance (Mohapa- tra, Nussinov, and Zhang 1994). Therefore, decay times below about 104s cannot be excluded by the present argument (Dar, Goodman, and Nussinov 1987). Actually, the galactic positron ﬂux is thought to be associated with supernovae, albeit not from decay but rather from the +decays of certain nuclei which are synthesized in a SN explosion (Chan and Lingenfelter 1993). These authors performed a detailed analysis of the probability for positrons to escape without annihilation from the SN environment into the galaxy. 12.5.2 Can the Tau Neutrino Be Heavy? Armed with this result we can return to the question raised in Sect. 7.2.2 if awith a mass exceeding 2 meis compatible with the cosmologi- cal requirement that such particles and their decay products do not “overclose” the universe. It turns out that the SN constraints on →ee+e−presented in this chapter exclude this possibility so that either respects the cosmological mass limit of a few 10 eV or else it must have fast invisible decay channels which inevitably require inter- actions beyond the standard model. In Fig. 12.19 the available constraints on e+eare summarized. The SN 1987A bound from the absence of a prompt burst, the cosmological requirement, and the above limit from galactic positron annihilation together exclude the entire range of possible masses and lifetimes. The margins of overlap are so enormous that each of the arguments has several orders of magnitude to spare for unaccounted uncertainties. A heavy standard is also excluded on the basis of arguments in- volving big-bang nucleosynthesis (BBN). The usual limit on the num- ber of eﬀective neutrino degrees of freedom at nucleosynthesis alone is enough to reach this conclusion (Sect. 7.1.5). Moreover, charged leptons and secondary photons from the e+e−decay channel would de- stroy some of the synthesized nuclei (Lindley 1979, 1985; Krauss 1984; Kawasaki, Terasawa, and Sato 1986). The main virtue of the SN limits is, therefore, that no reference to BBN is required to exclude a heavy . 486 Chapter 12 Fig. 12.19. Excluded areas of the mass and lifetime if the standard-model decay →ee+e−is the only available channel. The laboratory results refer to the bounds on sin22e3of Fig. 12.3, translated into a limit on e+e by virtue of Eq. (7.9). The SN 1987A bound is that from Fig. 12.18 while the cosmological one is from Fig. 7.2. The excluded range indicated by the vertical arrow refers to the argument of Sect. 12.5.1. 12.6 Neutrinos from All Stars All stars in the universe contribute to a diﬀuse cosmic background ﬂux of MeV neutrinos. If they decayed radiatively they would produce a cosmic x- and -ray background which must not exceed the measured levels. Because the entire radius of the visible universe is available as a decay path, one can derive rather restrictive limits on (Cowsik 1977). In order to derive such limits I assume that neutrinos of energy E are produced with a constant rate ˙N(cm−3s−1). Assuming a zero- curvature model of the universe, Kolb and Turner (1989) found for the resulting isotropic ﬂux of decay photons d2F dE dΩ=m 1 49 21=25˙Nt2 U E3=2 E1=2 ; (12.35) where tUis the age of the universe. Moreover, it was assumed that in→′ the daughter neutrino is massless, and that the decays are isotropic in the parent frame (anisotropy parameter = 0). In a ﬂat universe one has tU=2 3H−1 0=h−12:05×1017s where H0= Radiative Particle Decays 487 h100 km s−1Mpc−1is the present-day Hubble expansion parameter and where observationally 0 :4∼<h∼<1. There exist numerous measurements of the diﬀuse cosmic x- and -radiation (for example Sch¨ onfelder, Graml, and Penningfeld 1980). Between a few 100 keV and a few 10 MeV the isotropic ﬂux is reasonably well approximated by d2F dE dΩ= 2×10−2cm−2s−1sr−1MeV−1(MeV E )2 : (12.36) Comparing this with Eq. (12.35), the most restrictive limit on is obtained for the highest possible photon energy, E =E. The re- quirement that the decay ﬂux does not exceed the measurements at this energy leads to the upper limit m ∼<1:6×10−40eV scm−3s−1 ˙Nh2; (12.37) which does not depend on the assumed value for Ebecause Eq. (12.35) and (12.36) both scale with E−2. This limit applies if the total neutrino lifetime totexceeds tU; otherwise only a limit on the branching ratio B can be found. The most proliﬁc stellar neutrino source in the universe are hy- drogen-burning stars which produce two e’s with MeV energies for every synthesized4He nucleus. Because most of the binding energy that can be liberated by nuclear fusion is set free when single nucleons are combined to form4He, most of the energy emitted by stars can be attributed to hydrogen burning. Therefore, it is easy to translate the optical luminosity density of the universe into an average rate of neutrino production. The average luminosity density of the universe in the blue ( B) spec- tral band is about h2:4×108L⊙;BMpc−3where L⊙;Bis the solar Blu- minosity. The Sun produces about 1 ×1038e=s and so one arrives at ˙Ne≈h2:4×1046Mpc−3s−1= 0:8×10−27cm−3s−1. (Of course, this es- timate is relatively crude in that the neutrino luminosity scales directly with the average bolometric luminosity of a stellar population, but not precisely with LB.) With Eq. (12.37) this leads to a constraint for e of =me∼>5×1012s=eV, or eﬀ∼<2×10−7Bm−2 eV. The (core-collapse) supernovae in the universe are also very promi- nent neutrino sources, and, more importantly, they are thought to pro- duce MeV neutrinos of all ﬂavors. Such SNe do not occur in ellipti- cal galaxies, and their present-day rate in spirals depends sensitively 488 Chapter 12 on the Hubble type, varying from 0 :2h2SNu for Sa spirals to about 5h2SNu for Sd (van den Bergh and Tammann 1991) where the su- pernova unit is deﬁned by 1 SNu ≡1 SN per century per 1010L⊙;B. Adopting 1 h2SNu as a representative value and about 5 ×1057neu- trinos plus antineutrinos of a given ﬂavor per SN yields for each ﬂa- vor ˙N≈h31:3×10−27cm−3s−1, a rate almost identical to that from hydrogen-burning stars.79The radiative lifetime limit is then also iden- tical, except that it applies to neutrinos of all ﬂavors. All of these bounds are weaker than those from SN 1987A. There- fore, decaying stellar neutrinos cannot actually contribute to the ob- served x- and -ray background. 12.7 Cosmological Bounds 12.7.1 Neutrinos Within the big-bang scenario a cosmic background sea of neutrinos is an inevitable consequence of the hot early universe. Its contribution to the cosmic energy density was already used in Sect. 7.1.5 to derive ex- tremely restrictive neutrino mass limits. If neutrinos decay radiatively, further constraints can be obtained. For one, the decay photons can show up directly as a diﬀuse, isotropic cosmic background radiation. If the decays occur before recombination, i.e. before the universe be- came transparent to radiation, but so late that the photons could not be thermalized entirely, they contribute to a spectral distortion of the cosmic microwave background radiation (CMBR). The resulting limits were discussed, for example, by Kolb and Turner (1990) who found that those areas of masses and lifetimes are excluded that are hatched in Fig. 12.20. It was assumed that neutrinos decay only radiatively. The contribution of the neutrinos and their decay products to the mass density of the universe leads to the constraints shown in Fig. 7.2 which are based on the present-day value of Ω h2and on the expansion 79Multiplying this rate with the age of the universe of about 3 ×1017s and the speed of light of 3 ×1010cm=s, and using h= 0:5 one ﬁnds an estimated present-day ﬂux at Earth of about 1 cm−2s−1. In a recent detailed study, Totani and Sato (1995) ﬁnd a ﬂux which is larger than this crude estimate by as much as a factor of 30. The Kamiokande II detector has set an upper limit on the cosmic background ﬂux of eof about 103cm−2s−1for eﬀective temperatures in the 3 −4 MeV range (Zhang et al. 1988). It is conceivable that this background will be measured by the Superkamiokande detector. Note that at the Kamiokande site the eﬂux from power reactors is roughly 1000 times larger than the background ﬂux, except that it falls oﬀ sharply beyond about 10 MeV. Radiative Particle Decays 489 Fig. 12.20. Cosmological limits on neutrino radiative lifetimes according to Kolb and Turner (1990). The radiative mode is assumed to be the only decay channel. The shaded area is excluded according to Sect. 7.1.5 (Fig. 7.2). rate at nucleosynthesis. These limits (shaded area in Fig. 12.20) are more general because they do not depend on the nature of the ﬁnal states in the decay. Kolb and Turner’s (1990) exclusion plot is somewhat schematic. Ressell and Turner (1990) performed a much more detailed analysis on the basis of the diﬀuse photon backgrounds in all wavebands. Probably the most interesting region is that of small neutrino masses and large lifetimes (the upper left corner of Fig. 12.20). The excluded range of eﬀective electromagnetic transition moments for m<30 eV is shown in Fig. 12.21. It may be useful to approximate the excluded range analytically by eﬀ∼<3×10−11B(eV=m)2:3(12.38) which is shown as a dashed line in Fig. 12.21. Using favored cosmological parameters (Ω h2≈0:3) neutrinos with m≈30 eV would be the dark matter of the universe. With smaller masses there would have to be another component, but neutrinos could 490 Chapter 12 Fig. 12.21. Limits on eﬀaccording to Ressell and Turner’s (1990) bounds on the radiative lifetime of long-lived neutrinos from the diﬀuse cosmic back- ground radiations. The dashed line corresponds to Eq. (12.38). Fig. 12.22. Limits on decaying neutrinos in clusters of galaxies. (a) A1413, A2218, and A2256 (Bershady, Ressell, and Turner 1991). (b) Coma and Virgo (Henry and Feldman 1981). (c) A665 (Davidsen et al. 1991). (d) Extragalactic background light (Overduin, Wesson, and Bowyer 1993). (e) Background light from decay of unclustered neutrinos (Ressell and Turner 1990). Radiative Particle Decays 491 still play a signiﬁcant dynamical role. Recently, such mixed dark matter scenarios have received much attention where m= 5 eV is a favored value. Such low-mass particles cannot cluster on galactic scales, but likely they would reside in clusters of galaxies. With radiative decays →′ and a total lifetime exceeding the age of the universe one then expects clusters of galaxies to be strong sources of optical or ultraviolet photons. Several limits are summarized in Fig. 12.22. A case has been made that radiatively decaying neutrino dark mat- ter is actually required to solve certain problems, notably the ioniza- tion of galactic hydrogen clouds (e.g. Melott and Sciama 1981; Melott, McKay, and Ralston 1988; Sciama 1990a,b; Sciama 1993a,b, 1995). The predictions are very speciﬁc: An energy of decay photons of E = (14:4±0:5) eV and thus a neutrino mass of m= (28 :9±1:1) eV with a radiative lifetime of = (2±1)×1023s which translates into eﬀ= (6:3±2)×10−15B. Such a large transition moment would require particle physics beyond the standard model. Nominally, this possibility is already excluded by the absence of a uv line from the cluster A665 (Davidsen et al. 1991). However, a bound from a single source is always subject to the uncertainty of unrecognized absorbing material in the line of sight or internal absorption. Moreover, the dark matter in the core of this cluster may be mainly baryonic (Sciama, Persic, and Salucci 1993; Melott et al. 1994). Bounds from the diﬀuse extragalactic background light are more reliable in this regard. While they marginally exclude Sciama’s neutrino (Overduin, Wesson, and Bowyer 1993) it is perhaps too early to pronounce it entirely dead. A decisive test will be performed with a future satellite experiment where the uv line from neutrinos decaying in the solar neighborhood deﬁnitely would have to show up if neutrinos were the bulk of the galactic dark matter (e.g. Sciama 1993b). 12.7.2 Axions The axion lifetime from a→2 is= 6:3×1024s (ma=eV)5=2where is a model-dependent number of order unity (Sect. 14.3.2). Therefore, axions with eV masses have radiative lifetimes in the neighborhood of the above neutrino limits whence they can be constrained by similar methods (Kephart and Weiler 1987). Moreover, such axions would con- tribute substantially to the mass density of the universe because they would have been in thermal equilibrium until relatively late.80Their 80In the early universe, axions are also produced by the relaxation of the coherent initial ﬁeld conﬁguration at the onset of the QCD phase transition. This process 492 Chapter 12 contribution to the cosmic mass density would be Ω ah2= 0:082ma=eV (Turner 1987; Ressell 1991) and so one can expect that clusters of galax- ies contain substantial amounts of axions even if they are not the main dark matter component. Observations of the diﬀuse extragalactic background radiation limit the axion mass to values below about 8 eV unless is very small (Ressell 1991). Overduin and Wesson (1993) found <0:43, 0.07, and 0.02 for ma=eV = 5 :3, 8.6, and 13, respectively (Fig. 12.23). Moreover, some axions would reside in the halo of our own galaxy so that their decays would light up the night sky. Its brightness yields a conservative bound of <(6 eV =ma)5(Ressell 1991). Fig. 12.23. Constraints on axion decays in galaxies and galaxy clusters; parametrizes the coupling to photons with = 1 corresponding to common axion models. (a) Line emission from clusters A2256 and A2218 (Bershady, Ressell, and Turner 1991; Ressell 1991). (b) Diﬀuse extragalactic background radiation according to Ressell (1991) and (c) Overduin and Wesson (1993). (d) Our galaxy (Ressell 1991). The most interesting limits arise from a search for axion decay lines from the intergalactic space in the clusters of galaxies A2256 and A2218. Bershady, Ressell, and Turner (1991) and Ressell (1991) found <0:16 0.078, 0.039, 0.032, 0.016, and 0.011 for ma=eV = 3 :5, 4.0, 4.5, 5.0, 6.0, and 7.5 respectively (Fig. 12.23). These limits are placed into the context of other constraints in Fig. 5.9. yields Ω ah2≈(10−5eV=ma)1:175—see Eq. (14.5). While the overall coeﬃcient of this expression is very uncertain it is clear that Ω a= 1 saturates for masomewhere between 1 eV and 1 meV. In this range axions never achieved thermal equilibrium. Chapter 13 What Have We Learned from SN 1987A? The lessons for particle physics from the SN 1987A neutrino burst are studied. First, neutrinos could have decayed or oscillated into other states on their way out of the SN core and to us. Second, propagation eﬀects could have caused a time delay between photons and neutrinos or between νe’s of diﬀerent energy. Third, nonstandard cooling agents could have shortened the neutrino burst below its observed duration. These arguments are applied to a variety of speciﬁc cases. 13.1 Introduction In Chapter 11 the neutrino observations from SN 1987A were dis- cussed and it was shown that they agree well with standard theoret- ical expectations from the core collapse and subsequent explosion of an evolved massive star. The signal displays several anomalies (time gap at Kamiokande, anisotropy in both detectors) which render it a less beautiful specimen of the expected signal characteristics than is sometimes stated in the literature. Still, in the absence of plausible alternatives one must accept that the Kamiokande II, IMB, and Bak- san event clusters observed at 7:35 UT on 27 February 1987 represent theνecomponent of the neutrino burst from the core collapse of the SN 1987A progenitor star rather than some other particle ﬂux, or some other reaction than the expected dominant νep→ne+process. Accepting this, there is a host of consequences concerning a variety of fundamental physics issues. The ﬁrst and simplest set of arguments is based on the fact that the νepulse and perhaps the prompt νeburst 493 494 Chapter 13 were observed, constraining various mechanisms that could have re- moved neutrinos from the beam such as decays. Equally important, the nonobservation of a γ-ray burst in coincidence with the neutrino burst constrains radiative decays of neutrinos and other particles (Sect. 12.4). More intricate arguments involve signal dispersion, either between photons and neutrinos, between νe’s andνe’s, or the intrinsic dispersion of theνeburst, constraining various eﬀects that could cause signal dispersion such as a nonzero neutrino mass or charge. Most importantly, the inferred cooling time scale of a few seconds of the newborn neutron star precludes an eﬃcient operation of a non- standard cooling agent and thus yields constraints on the emission of new particles from the SN core, notably of right-handed (r.h.) neutri- nos or axions. This line of reasoning is analogous to the “energy-loss argument” which for normal stars has been advanced in Chapter 2. 13.2 Basic Characteristics of the Neutrino Burst 13.2.1 Fluence The neutrinos from a SN are expected to consist of two major compo- nents: the prompt νeburst, and quasi-thermal emission of about equal total amounts of energy in (anti)neutrinos of all ﬂavors. The water Cherenkov detectors would register the νeburst by virtue of the reac- tionνe+e−→e−+νewhere the scattered electron is strongly forward peaked, while the cooling signal is registered by νe+p→n+e+with an essentially isotropic e+signal. Even though both detectors observed a forward peaked overall signal, it cannot be associated with νe-ecolli- sions (Sect. 11.3.5). Most or all of the events are interpreted as νe’s. The observation of a νeﬂuence (time-integrated ﬂux) roughly in agreement with what is expected from a stellar collapse precludes that these particles have decayed on their way from the SN to us, yielding a constraint on their lifetime of (Frieman, Haber, and Freese 1988) τe/me∼>6×105s/eV. (13.1) However, this simple result must be interpreted with care because mas- sive neutrinos are expected to mix. The heavy νeadmixtures could decay and may violate this bound. Theνe’s were not removed by excessive scattering on cosmic back- ground neutrinos, majorons, dark-matter particles etc., leading to con- straints on “secret interactions” (Kolb and Turner 1987). Take the What Have We Learned from SN 1987A? 495 scattering on cosmic background neutrinos as an example. The present- day density of primordial neutrinos is about 100 cm−3in each neu- trino and antineutrino ﬂavor. With a distance to the Large Magellanic Cloud of about 50 kpc = 1 .5×1023cm one has a column density be- tween SN 1987A and Earth of about 1025cm−2so that the νe-νcross section must be less than about 10−25cm2. If the cosmic background neutrinos are massless they have a temperature of about 1 .8 K and so⟨E⟩ ≈ 3T≈5×10−4eV. Because the measured SN neutrinos have a characteristic energy of 30 MeV the center of mass energy is√s≈200 eV. The cross-section bound is not particularly impressive compared with a standard weak cross section of order G2 Fs≈10−51cm2. How- ever,ννcross sections have never been directly measured and so the SN 1987A limit provides nontrivial information. As an example, neu- trinos could scatter by majoron exchange, or they could scatter di- rectly on a background of primordial majorons. Kolb and Turner then found a certain constraint on the neutrino-majoron Yukawa coupling (Sect. 15.7.2). As another example, the proposition that the solar neu- trino ﬂux could be substantially depleted by scatterings on cosmic back- ground particles (Slad’ 1983) is excluded. Other particles besides neutrinos may have been emitted from the SN and could have caused detectable events. Engel, Seckel, and Hayes (1990) have discussed the case of axions; they can be absorbed in water by oxygen nuclei, a16O→16O∗, which subsequently produce γrays by decays of the sort16O∗→16Oγ,16O∗→15Onγ, and16O∗→15Npγ. Theγrays would cause electromagnetic cascades and so they are de- tectable about as eﬃciently as e±. The axion emission was estimated by identifying their unit optical depth for a given interaction strength in a simpliﬁed model of the SN temperature and density proﬁle. More than 10 extra events would be expected at Kamiokande for an axion-nucleon Yukawa coupling in the range 1×10−6∼<gaN∼<1×10−3(13.2) which is thus excluded. In the middle of this interval, up to 300 ad- ditional events would have been expected. However, axions with cou- plings in this interval are also excluded by other methods (Sect. 14.4). 13.2.2 Energy Distribution The energy distribution of the events at the IMB and Kamiokande detectors broadly conﬁrms the expected quasi-thermal emission with 496 Chapter 13 a temperature of around 4 MeV. This precludes that a major swap by oscillations with the higher-energetic νorνﬂux has taken place. The impact of neutrino oscillations on the observable signal has been discussed in Sect. 11.4. Also,ν;orν;decays with ﬁnal-state νe’s would produce addi- tional higher-energy events. While the SN 1987A data are probably too sparse to extract signiﬁcant information on the presence or absence of this eﬀect, a future galactic SN would certainly allow one to exclude a certain range of masses and decay times or to detect this eﬀect (Soares and Wolfenstein 1989). The trapping of neutrinos in a SN core together with the condition ofβequilibrium inevitably implies that there is a large νechemical potential, leading to typical νeenergies of order 200 MeV. Moreover, the inner temperature during deleptonization reaches values of up to 40−70 MeV so that typical thermal (anti)neutrino energies of up to 100−200 MeV are available. Therefore, if neutrinos could escape di- rectly from the inner core they would cause high-energy events in the detectors which have not been observed. A mechanism to tap the inner-core heat bath directly is the pro- duction of r.h. neutrinos by a variety of possible eﬀects such as spin- ﬂip scattering by a Dirac mass term or a magnetic dipole moment (Sect. 13.8). R.h. states could not be detected directly because they are sterile with regard to standard l.h. weak interactions, a property which allows them to avoid the SN trapping. However, they could produce detectable l.h. states by decays (Dodelson, Scott, and Turner 1992) or by magnetic oscillations (N¨ otzold 1988; Barbieri and Mohapatra 1988). 13.2.3 Prompt eBurst The prompt νeburst can be seen in a water Cherenkov detector by the reactionνee→eνewhere the ﬁnal-state electron essentially preserves the direction of the incident neutrino. At Kamiokande, the directional- ity of the ﬁrst event81is consistent with the interpretation that it was caused by this reaction. However, the expected ﬂuence corresponds only to a fraction of an event and so the ﬁrst event may also be due to theνep→ne+reaction and point coincidentally in the forward di- rection. A random direction has about a 5% chance of being forward within 25◦which is approximately the uncertainty of the Kamiokande directional event reconstruction. 81In the ﬁrst publication of the Kamiokande group (Hirata et al. 1987) the second event was also reported forward; its most probable direction was later revised. What Have We Learned from SN 1987A? 497 Still, the observation of the prompt νeburst from a future galactic SN would allow for a number of interesting conclusions. For example, one could exclude or ﬁnd evidence for neutrino oscillations (Sect. 11.4). Signal dispersion caused by a neutrino mass or other eﬀects which are discussed below for the cooling-phase signal would be even more sig- niﬁcant for the prompt burst because of its short duration. For exam- ple, the cooling signal with a duration of about 10 s is sensitive to νe masses in the 10 eV regime. As the prompt burst is at least 100 times shorter one is sensitive to a factor of 10 smaller masses, i.e. to mein the eV range. The (anti)neutrino signal during the prompt burst phase allows one to decide if the SN consisted of antimatter rather than matter. In that case one would expect a prompt νeburst with a scattering cross section on electrons which is about a factor of 2.4 smaller (Eq. 10.17). Moreover,νe’s are dominantly absorbed by the isotropic νep→ne+ reaction and so the prompt burst would cause a substantial isotropic signal within the ﬁrst 50 ms. In a matter SN the cooling νe’s have larger energies than the νe’s; the reverse for antimatter. Therefore, the observable νesignal from the cooling phase would be reduced. In a detector like Kamiokande one would then expect 6 −20% of the total νesignal from the prompt burst, in contrast with at most 1% for a regular matter SN (Barnes, Weiler, and Pakvasa 1987). Of course, in the foreseeable future one can hope to acquire the relevant data only from a galactic SN which, no doubt, consists of matter. 13.2.4 Nonobservation of a -Ray Burst Noγrays in conjunction with the SN 1987A neutrino burst were ob- served by the solar maximum mission (SMM) satellite which was oper- ational at the relevant time. Therefore, one can derive some of the most restrictive limits on neutrino radiative decays as detailed in Sect. 12.4. 13.3 Dispersion Eﬀects 13.3.1 Photons vs. Antineutrinos The optical sighting of SN 1987A followed the detection of the νeburst by only a few hours (Fig. 11.7), a delay which is expected on the basis of the simple reasoning that some time must pass before the mantle of a SN “notices” the collapse of the inner core. Hence the two signals must have propagated through space with an almost identical velocity 498 Chapter 13 so that the speed of light and that of neutrinos are equal to within (Longo 1987; Stodolsky 1988) c−c c ∼<2×10−9, (13.3) assuming an uncertainty of ±3 h in the relative duration of the transit times from the LMC to us. This was interpreted as the most stringent test of special relativity to date in the sense that it proves with high precision the universality of a relativistic limiting velocity.82 This result can also be interpreted as testing the weak equivalence principle of general relativity (Krauss and Tremaine 1988). In the post- Newtonian approximation one predicts that a gravitational potential V(r) delays a light signal (Shapiro time delay) by an amount ∆t=−2∫A EV[r(t)]dt, (13.4) where the integral is taken along the trajectory r(t) of the beam between the points of emission (E) and absorption (A). This delay is the same for neutrinos and photons to within ∆t−∆t ∆t <0.7−4×10−3, (13.5) where the uncertainty reﬂects the uncertain modelling of the gravita- tional potential between Earth and SN 1987A.83This result has been used to constrain the parameters of a speciﬁc model of C- and P- violating gravitational forces (Almeida, Matsas, and Natale 1989), and to constrain the parameters of a class of nonmetric theories of gravity (Coley and Tremaine 1988). 13.3.2 Neutrinos vs. Antineutrinos Assuming that the ﬁrst event at Kamiokande represents the prompt νe burst one may also constrain the diﬀerence in transit time between νe andνeand thus conﬁrm the equivalence principle between matter and antimatter (LoSecco 1988; Pakvasa, Simmons, and Weiler 1989). Of course, in order to make such results reliable one would need to observe the prompt burst from a future SN with greater statistical signiﬁcance. 82For a recent laboratory experiment which addresses the Lorentz limiting veloc- ity, see Greene et al. (1991), and references there to earlier works. See also the book by Will (1993). 83See Will (1993) for a review of many other empirical tests of general relativity. What Have We Learned from SN 1987A? 499 13.3.3 Intrinsic Dispersion of the e-Pulse a) Neutrino Mass So far the transit time of diﬀerent particle species was compared un- der the assumption of a ﬁxed velocity each. However, the most likely eﬀect of signal propagation over large distances is dispersion due to an energy-dependent speed of propagation. The most widely discussed84 case is that of a nonzero neutrino mass (Zatsepin 1968). The main problem at extracting information about the signal dispersion is the unknown behavior of the source which must be modelled according to some theoretical assumptions. A particularly detailed discussion is that of Loredo and Lamb (1989) who found a mass limit of me<23 eV (Sect. 11.3.4). In a similar analysis which included the 13% dead-time eﬀect at IMB, Kernan and Krauss (1995) found me<20 eV. b) Neutrino Charge The absence of an energy-dependent dispersion of the neutrino pulse can be used to constrain other neutrino properties. A small electric chargeewould bend the neutrino path in the galactic magnetic ﬁeld, leading to a time delay of ∆t t=e2 (BTdB)2 6E2 , (13.6) whereBTis the transverse magnetic ﬁeld and dBthe path length within the ﬁeld. This leads to a constraint of e e∼<3×10−17(1µG BT)(1 kpc dB) (13.7) (Barbiellini and Cocconi 1987; Bahcall 1989). Note that a typical ﬁeld strength for the ordered magnetic ﬁeld in the galactic spiral arms is 2−3µG and that the path length of the neutrinos within the galactic disk is only of order 1 kpc because the LMC lies high above the disk (galactic latitude about 33◦). 84Limits on mefrom the SN 1987A data were derived, among others, by Abbott, de R´ ujula, and Walker (1988), Adams (1988), Arnett and Rosner (1987), Bah- call and Glashow (1987), Burrows and Lattimer (1987), Burrows (1988a), Chiu, Chan, and Kondo (1988), Cowsik (1988), Kolb, Stebbins, and Turner (1987a,b), Midorikawa, Terazawa, and Akama (1987), Sato and Suzuki (1987a,b), Spergel and Bahcall (1988), Loredo and Lamb (1989), and Kernan and Krauss (1995). 500 Chapter 13 c) Long-Range Forces Speculating further one may imagine some sort of neutrino “ﬁfth-force charge.” If electrons, protons, or dark-matter particles also carry such a charge the bending of the neutrino trajectory in the ﬁfth-force ﬁeld of the galaxy would lead to an energy-dependent time delay. This and re- lated arguments were advanced by a number of authors (Pakvasa, Sim- mons, and Weiler 1989; Grifols, Mass´ o, and Peris 1988, 1994; Fiorentini and Mezzorani 1989; Malaney, Starkman, and Tremaine 1995). The most plausible form for such a long-range interaction is one me- diated by a massless vector boson, i.e. a new gauge interaction, perhaps related to a novel leptonic charge (Sect. 3.6.4). In this case neutrinos and antineutrinos would carry opposite charges so that the cosmic neu- trino background would be essentially a neutral plasma with regard to the new interaction. The resulting screening eﬀects then invalidate the SN 1987A argument (Dolgov and Raﬀelt 1995). Screening eﬀects would not operate if the force were due to a spin-0 or spin-2 boson which always cause attractive forces. However, any force mediated by a massless spin-2 boson must couple to the energy- momentum tensor and thus is identical with gravity. The force medi- ated by a scalar boson between a static source and a relativistic neutrino is suppressed by a Lorentz factor. Therefore, even if scalar-mediated forces existed between macroscopic bodies, their eﬀect would be weak- ened for relativistic neutrinos. In summary, the SN 1987A signal does not seem to carry any simple information concerning putative nongravitational long-range forces. d) Fundamental Length Scale Fujiwara has proposed a quantum ﬁeld theory where the velocity of par- ticles increases with energy, leading to an energy-dependent advance of the arrival times by ∆ t/t=−1 2(ℓ0E)2. Here,ℓ0is a fundamental length scale. Whatever the merits of this theory, a value ℓ0∼<10−18cm would not be in conﬂict with the SN 1987A neutrino signal (Fujiwara 1989). e) Lorentz Addition of Velocities If relativistic particles (photons, massless neutrinos) are emitted by a moving source (velocity vS) their velocity c′in the laboratory frame should be equal to c(velocity in the frame of the source). The Galilean addition of velocities, on the other hand, would give c′=c+vS. In general one may assume that velocities add according to c′=c+KvS What Have We Learned from SN 1987A? 501 withK= 0 representing the Lorentzian, K= 1 the Galilean law of adding velocities. For photons, the most stringent laboratory bound is K ∼<10−4derived from the time of ﬂight of decay photons π◦→2γ from a pulsed π◦source (Alv¨ ager et al. 1964). A much more stringent constraint ( K ∼<2×10−9) obtains from an analysis of the photon signal from a pulsed x-ray source (Brecher 1977). The absence of dispersion of the νepulse can be used to derive con- straints on K(Atzmon and Nussinov 1994). The observed SN 1987A neutrinos were produced by microscopic processes involving nearly rel- ativistic nucleons, and subsequently scattered several times on such nucleons before leaving the star. If the last nucleon on which they scatter is considered the source with vS≈0.2ctheir laboratory speed c′ will be represented by a distribution of approximate width 0 .2K aroundcbecause of the random orientation and distribution in magni- tude ofvS. Thus one derives a bound K∼<10−11from the absence of a spread in arrival times exceeding about 10 s. Atzmon and Nussinov (1994) warn, however, that this simple argu- ment may be too naive as the motion through the progenitor’s enve- lope may cause the particles of the envelope to be the true source of the “neutrino waves” as there is a substantial amount of refraction be- tween the neutrino sphere and the stellar surface. If one follows Atzmon and Nussinov’s reasoning, there remains only a much weaker bound of K∼<10−5from the absence of an anomalous time delay between the neutrino signal and the optical sighting of the SN. 13.4 Duration of Neutrino Emission 13.4.1 General Argument The most intricate way to use SN 1987A as a laboratory arises from the observed duration of neutrino cooling. While the neutrino luminosity during the ﬁrst few 100 ms until the shock has been revived is largely powered by accretion and by the contraction and settling of the bloated outer core, the long tail is associated with cooling, i.e. emission from the neutrino sphere which is powered by energy originally stored deep in the inner core. If a direct cooling channel existed for that region, such as the emission of r.h. neutrinos or axions, the late cooling phase would be deprived of energy. Put another way, a novel cooling channel from the inner core would leave the schematic neutrino light curves of Fig. 11.3 more or less unchanged before about 1 s while the long Kelvin-Helmholtz cooling phase would be curtailed. 502 Chapter 13 Of course, this reasoning is identical with the energy-loss argument previously studied for normal stars in Chapters 1 and 2. The main diﬀerence is that neutrinos are trapped so that particles which interact more weakly can dominate the thermal evolution by volume emission. In normal stars, photons are trapped and neutrinos can dominate the energy loss by volume emission as, for example, in the early cooling of a white dwarf. This general argument is best illustrated with axion emission. These particles are pseudoscalars which for the purpose of this argument are taken to interact with neutrons and protons with a common Yukawa coupling strength gawhich is the only free parameter in the problem. For very small values of gaaxions will play no role, but with an increas- ing coupling strength their emission from the inner core by bremsstrah- lung processes, NN→NNa , will begin to compete with neutrino cool- ing. Of course, if gaexceeds some critical value axions will be trapped and emitted from an “axion sphere” at about unit optical depth. Be- yond some large coupling they will be trapped so eﬀectively that their contribution to the cooling of the SN core is, again, negligible and the neutrino signal assumes its standard duration. This general behavior is shown in Fig. 13.1 on the basis of the numerical cooling calculations85 of Burrows, Turner, and Brinkmann (1989) and Burrows, Ressell, and Turner (1990). These authors used the quantity ∆ t90%as a measure of the cooling time; it represents the time at which 90% of the expected number of events have arrived at a detector. ∆ t90%was calculated sep- arately for Kamiokande II and IMB; in Fig. 13.1 an average relative signal duration is shown, normalized to the value when axions are not important. It is apparent that a large range of gavalues can be excluded on the basis of the observed duration of the neutrino signal. One is here considering the time scale of neutrino emission at the source while the detectors register a pulse which conceivably could have been lengthened by dispersion eﬀects. However, in view of the recent laboratory limits of me∼<5 eV this is not a serious concern. If one contemplates nonstandard neutrinos, a relatively short emis- sion time scale at the source is compatible with the observations if ν’s 85In the free-streaming regime these calculations were based on axion emission rates which do not take the high-density multiple-scattering eﬀects into account that were discussed in Sect. 4.6.7. Therefore, the free-streaming part of Fig. 13.1 probably overestimates the import of axion emission. For the present purpose of discussing the general aspects of a novel cooling channel, however, this problem is of no concern. The axion case is the only one where numerical cooling calculations are available for both the volume-emission (free-streaming) and the trapping limit. What Have We Learned from SN 1987A? 503 Fig. 13.1. Relative duration of neutrino cooling of a SN core as a function of the axion-nucleon Yukawa coupling ga. In the free-streaming limit axions are emitted from the entire volume of the protoneutron star, in the trapping limit from the “axion sphere” at about unit optical depth. The solid line is according to the numerical cooling calculations (case B) of Burrows, Turner, and Brinkmann (1989) and Burrows, Ressell, and Turner (1990); the dotted line is an arbitrary completion of the curve to guide the eye. The signal duration is measured by the quantity ∆ t90%discussed in the text; an average for the IMB and Kamiokande detectors was taken. orν’s decay. The ﬁnal states could include νe’s which are detectable at IMB and Kamiokande so that one can obtain late-time events by a suitable combination of mass and lifetime. For a 17 keV Majorana neutrino a lifetime around 104s would allow one to explain the signal duration even with a short emission time scale at the source (Simpson 1991; see also Cline 1992). In the following it will be assumed that this is not the explanation of the observed signal duration. Another loophole is that some or all of the late-time events at Ka- miokande, which are separated from the main bunch by a 7 s gap, were caused by eﬀects other than core cooling. Recall that a similar problem exists with the x-ray observations of old neutrons stars (Sect. 2.3) where it is not always clear that one is observing blackbody surface emission from thermal cooling rather than magnetospherically produced x-rays. In the present case, one possibility is the fall-back of material onto the core, i.e. late-time accretion which could cause signiﬁcant neutrino emission. However, on the basis of an analytic estimate Janka (1995b) has argued that even with extreme assumptions this is not a likely explanation of the late events. 504 Chapter 13 While it is clear that a large range of gavalues can be excluded, the quantity ∆ t90%is a relatively crude measure of the length of the cooling phase. In principle, one should perform a maximum likelihood analysis for a given range of particle properties. Moreover, one would need to consider a variety of models for the protoneutron star where the equation of state (EOS), mass, accretion rate, neutrino opacities, and perhaps other parameters should be varied to optimize the agreement with the observed signal when a novel cooling mechanism operates. Another caveat applies to the “trapping regime” of the new par- ticles. One may expect that they play a signiﬁcant role during the infall phase and shock formation of a SN collapse, an issue that was addressed only by a small number of authors in the context of majoron bounds (Fuller, Mayle, and Wilson 1988) and bounds on neutrino dipole moments (N¨ otzold 1988). Hence, in general it is not obvious that pa- rameters allowed by the cooling argument on the trapping side would remain allowed if one took account of these eﬀects. Moreover, on the trapping side the novel particles interact about as strongly as neutrinos and so they could also cause a signal in the detectors. For axions, this argument rules out the values of gagiven in Eq. (13.2). 13.4.2 Analytic Criterion in the Free-Streaming Limit In order to estimate the impact of a novel cooling channel on the neu- trino signal it is obviously useful to evolve a protoneutron star numer- ically with the new physics included, and to calculate the expected neutrino signal for a varying strength of the new eﬀect. Considering the many uncertainties involved in this procedure one may well ask if it is not just as reliable to perform a simple analytic estimate. At about 1 s after core bounce the neutrino luminosity in all six (anti)neutrino degrees of freedom together is about 3 ×1052erg s−1. The mass of the object is around 1 .5M⊙= 3×1033g so that its average energy-loss rate is L/M ≈ 1×1019erg g−1s−1. A novel cooling agent would have to compete with this energy-loss rate in order to aﬀect the total cooling time scale signiﬁcantly. Therefore, the observed signal duration indicates that a novel energy-loss rate is bounded by ϵx∼<1019erg g−1s−1. (13.8) It is to be evaluated at typical core conditions, i.e. at a temperature of around 30 MeV and a density of around 3 ×1014g cm−3. The nuclear medium is then at the borderline between degeneracy and nondegener- acy while the electrons are highly degenerate. What Have We Learned from SN 1987A? 505 Fig. 13.2. Proﬁle of various pa- rameters for the protoneutron star model S2BH 0 of Keil, Janka, and Raﬀelt (1995), 1 s after core bounce. The degen- eracy parameters were approx- imated by N= (EF−mN)=T with E2 F=p2 F+m2 Nand with the eﬀective nucleon mass. 506 Chapter 13 The proﬁle of various parameters as a function of the mass coor- dinate is shown in Fig. 13.2 for model S2BH 0 of the cooling calcula- tions of Keil, Janka, and Raﬀelt (1995); it illustrates typical physical conditions encountered in the core of a protoneutron star during the Kelvin-Helmholtz phase. For this model, the average value of ( ρ/ρ0)n with the nuclear density ρ0= 3×1014g cm−3and of (T/30 MeV)nis shown in Fig. 13.3 as a function of n. Fig. 13.3. Average values for ( =0)nwith the nuclear density 0= 3×1014g cm−3and of ( T=30 MeV)nfor the protoneutron star model of Fig. 13.2. As an example one may apply this criterion to the bremsstrahlung energy-loss rate NN→NNa for the emission of some pseudoscalar bosona(axion) with a Yukawa coupling ga. The nondegenerate energy- loss rate Eq. (4.8) is ϵa=g2 a2×1039erg g−1s−1ρ15T3:5 30whereT30= T/30 MeV and ρ15=ρ/1015g cm−3. From Fig. 13.3 one ﬁnds that ⟨ρ15⟩ ≈ 0.4 and ⟨T3:5 30⟩ ≈ 1.4. The criterion Eq. (13.8) then yields ga∼<10−10, similar to what one would conclude from Fig. 13.1. Using the degenerate emission rate Eq. (4.10) yields an almost identical result. Therefore, a simple criterion like Eq. (13.1) is not a bad ﬁrst estimate for the import of a novel energy-loss rate. 13.4.3 Trapping Limit When the new particles (for example, axions) interact strongly enough, they will be emitted from a spherical shell where their optical depth is about unity rather than by volume emission. Again, one is concerned What Have We Learned from SN 1987A? 507 mostly with a time later than 0 .5−1 s where the outer core has set- tled and the shock has begun to escape. The density of the protoneu- tron star falls within a thin shell from supranuclear levels to nearly zero, causing the “photosphere” radius rxof the new particles to be essentially the radius R≈10 km of the settled compact star. With a “photosphere” temperature Txof the new objects their luminosity is 4πr2σT4 xwith the Stefan-Boltzmann constant σwhich isgπ2/120 in natural units with gthe eﬀective number of degrees of freedom (2 for photons). Therefore, one must demand that Tx∼<8 MeVg−1=4, (13.9) in order to stay below the total neutrino luminosity of 3 ×1052erg s−1. It is nontrivial, however, to determine the temperature Txwhich corresponds to about unit optical depth. Following the approach of Turner (1988) who carried this analysis through for axions one may assume a simple model for the run of temperature and density above the settled inner core. A simple power-law ansatz is ρ(r) =ρR(R/r)n with the density ρR= 1014g cm−3at a radius R≈10 km. A plausible ansatz for the temperature proﬁle is T(r) =TR[ρ(r)/ρR]1=3withTR (temperature at radius R) of around 10 MeV. From the opacity κas a function of density and temperature one may then calculate the optical depth asτ(rx) =∫∞ rxκρdr . From the condition τ(rx)≈2 3one can determine the “photosphere” radius rxand thus its temperature Tx. The opacity of axions for a medium of nondegenerate nucleons was given in Eq. (4.28). One may deﬁne τR≡κRρRRso thatκρR = τR(ρ/ρR)2(TR/T)1=2where Eq. (4.28) yields τR=g2 a3.4×1016. Then one ﬁnds for Turner’s model an optical depth τaat the axion-sphere temperature Ta τa=τR(11 6n−1) (Ta/TR)11=2−3=n. (13.10) Becausenis a relatively large number such as 3 −7 the criterion τa∼<2 3 yieldsτR∼>n(TR/Ta)6. With the requirement Ta∼<8 MeV and with TR≈20 MeV as taken by Turner one ﬁnds ga∼>2×10−7, not in bad agreement with what one would conclude from the numerical results shown in Fig. 13.1. Still, this argument is rather sensitive to the detailed model as- sumptions concerning the protoneutron star structure. Also, as axions contribute to the transfer of energy within the star, a self-consistent model must take this eﬀect into account. Moreover, for novel fermions such as r.h. neutrinos one must distinguish carefully between their neu- trino sphere (from where they can escape almost freely) and the deeper 508 Chapter 13 region where their energy ﬂux is set. Put another way, for fermions the concept of blackbody emission from a neutrino sphere is not adequate, making it impossible to apply the Stefan-Boltzmann in a simplistic way. The transport of r.h. neutrinos in the trapping limit is an equally com- plicated problem as that of l.h. ones! Therefore, a proper treatment of the trapping limit is generally a tricky subject; axions are the only case where it has been studied in some detail. Occasionally one may wish to construct a particle-physics model that avoids the SN limit. It would be incorrect to believe that this is achieved when the interaction strength has been tuned such that the mean free path is of order the neutron star radius. On the con- trary, when this condition obtains the impact on the cooling rate is maximized. This is analogous to the impact of novel particles on the structure and evolution of the Sun as depicted in Fig. 1.2; the cooling rate is maximized when the mfp corresponds to a typical geometric di- mension of the object. In the trapping regime a new particle is harmless only if it interacts about as strongly as the particles which provide the standard mode of energy transfer. 13.5 Axions 13.5.1 Numerical Studies The most-studied application of the SN cooling-time argument is that of invisible axions as these particles are well motivated (Chapter 14). Moreover, they have attracted much interest because they are one of the few particle-physics motivated candidates for the cosmic dark mat- ter. Early analytic studies in the free-streaming limit are Ellis and Olive (1987), Raﬀelt and Seckel (1988), and Turner (1988) who also discussed the trapping regime; his line of reasoning was presented in Sect. 13.4.3 above. Numerical studies in the free-streaming limit were performed by Mayle et al. (1988, 1989) and by Burrows, Turner, and Brinkmann (1989) while the trapping regime was numerically studied by Burrows, Ressell, and Turner (1990). The numerical studies by diﬀerent work- ers in the free-streaming limit used diﬀerent assumptions concerning the axion couplings, emission rates, and other aspects. In my pre- vious review (Raﬀelt 1990d) I have attempted to reduce the results of these works to a common and consistent set of assumptions; apart from relatively minor diﬀerences which could be blamed on diﬀerent input physics (e.g. softer equation of state and thus higher temperatures in the Mayle et al. papers) the results seemed reasonably consistent. A What Have We Learned from SN 1987A? 509 recent numerical study by Keil (1994) who used the same axion emis- sion rates as Burrows, Turner, and Brinkmann (1988) conﬁrmed their results. Here, I present the numerical studies of Burrows and his collabora- tors where axions were assumed to couple with equal strength to pro- tons and neutrons. The axial-vector coupling to nucleons is written in the form (C/2fa)ψγγ5ψ∂awith a model-dependent numerical factor C, the Peccei-Quinn energy scale fa, the nucleon Dirac ﬁeld ψ, and the axion ﬁelda. Under certain assumptions detailed in Sect. 14.2.3 it can be written in the pseudoscalar form −igaψγ5ψwherega=Cm N/fa is a dimensionless Yukawa coupling (nucleon mass mN); Burrows et al. usedC=1 2. All results will be discussed in terms of gaand as such they apply to any pseudoscalar particle which couples to nucleons accord- ingly. In Sect. 14.4 the available constraints on axions will be expressed in terms of the axion mass ma. In the free-streaming limit the energy loss by axions was imple- mented according to the numerical rates of Brinkmann and Turner (1988); limiting cases of these rates were discussed in Sect. 4.2. In the trapping regime, the transfer of energy by axions as well as axion cool- ing from an “axion sphere” was implemented by means of an eﬀective radiative opacity as discussed in Sect. 4.4. The protoneutron star mod- els are those of Burrows and Lattimer (1986) and of Burrows (1988b). In the latter study, cooling sequences were presented for diﬀerent equa- tions of state (EOS), and diﬀerent assumptions concerning the mass and early accretion rate of the stars. A ﬁducial case in these studies is model 55 with a “stiﬀ EOS,” an initial baryon mass of 1 .3M⊙, and an initial accretion of 0 .2M⊙. The compatibility of a given model with the SN 1987A observations should be tested by a maximum-likelihood analysis of the time and energy distributions of the events in both the IMB and Kamiokande II detectors. In practice, it is easier to consider a few simple observables. Burrows and his collaborators chose the total number of events NKII andNIMBin the two detectors as well as the signal duration deﬁned by the expected times tKIIandtIMBit takes to accrue 90% of the expected total number of events. As both detectors measured approximately 10 events each, the time of the last event probably is a reasonable estimate of tKIIandtIMB. Finally, Burrows et al. calculated the total energy carried away by neutrinos and axions. The run of these quantities with gais shown in Fig. 13.4. Re- call from Sect. 11.3.2 that the observed SN 1987A numbers of events areNIMB= 8 andNKII= 10−12, depending on whether event No. 6 510 Chapter 13 Fig. 13.4. Results from protoneutron star cooling sequences with axions. The free-streaming regime (small ga) is according to Burrows, Turner, and Brinkmann (1989), the trapping regime (large ga) according to Burrows, Ressell, and Turner (1990). For models A, B, and C (corresponding to models 57, 55, and 62 of Burrows 1988b) the amount of early accretion and the type of EOS (“stiﬀ” or “soft”) is indicated. The models were calculated until 20 s after collapse. What Have We Learned from SN 1987A? 511 was actually due to background (quite possible) and whether event No. 1 was due to the prompt νeburst (possible but not necessary). The last events were registered at 5 .6 s after the ﬁrst (IMB) and 12 .4 s (Kamiokande II). Recall also that the absolute timing between the two detectors is uncertain to within a minute although it seems plausible that in both cases the ﬁrst event essentially marks the arrival of the ﬁrst neutrinos. Finally, recall that the signal at Kamiokande II exhibits a peculiar 7 .3 s time gap before the last three events; event No. 9 was registered at 1 .9 s after the ﬁrst. Naturally, it is worrisome that the large Kamiokande time scale rests on the last three events, i.e. in order to take the Kamiokande pulse duration seriously one needs to appeal to a rare statistical ﬂuctuation. The total number of events observed is not very sensitive to the amount of axion cooling which has an impact mostly on the late-time neutrino signal. Interestingly, in the trapping regime (large ga) the number of events at IMB actually increases because the axionic energy transfer heats the neutrino sphere to higher temperatures. NIMBre- sponds sensitively to the neutrino spectrum because of the high thresh- old at IMB. However, for this reason it is a bad indicator for the actual neutrino ﬂux because the high-energy tail of the spectrum is relatively uncertain. For example, if it is described by a Fermi-Dirac function with a degeneracy parameter η= 2−3 rather than η= 0 reduces NIMB by about a factor of two (Fig. 11.11). No numerical results are available in the intermediate regime be- tween free streaming and trapping where the axion mean free path is of order the neutron star radius. In this range of coupling constants the impact of axions on the star is maximized. Moreover, a substantial modiﬁcation of the initial collapse phase obtains. As emphasized before, the most sensitive observable is the duration of the neutrino signal at the detectors. Therefore, nominally a range of coupling constants 1 ×10−10∼<ga∼<3×10−7is excluded. Within this range the observed neutrino signal likely would be shortened too much to be compatible with the observations. 13.5.2 Impact of Multiple-Scattering Eﬀects The results presented in the previous section were based on a naive perturbative calculation of the axion emission rate without taking the modiﬁcation of the spin-density structure function into account that must occur at high density as outlined in Sect. 4.6.7. The density de- pendence of the axion emission rate is encapsuled in the spin-ﬂuctuation 512 Chapter 13 rate Γ which, in the nondegenerate limit, was given in Eq. (4.7) on the basis of a perturbative one-pion exchange (OPE) calculation. For the protoneutron star model displayed in Fig. 13.2 the proﬁle of this Γ /T is shown in Fig. 13.5. In Fig. 4.8 the axion emission rate was shown as a function of Γ , revealing that for the conditions of interest one is in the neighborhood of the maximum of the solid curve. In a realistic nuclear medium, the true spin ﬂuctuation rate may be smaller than the OPE calculated value, taking one perhaps somewhat to the left of the maxi- mum. Therefore, the true axion emission rate corresponds to the naive one (dashed line in Fig. 4.8) at Γ /T≈3−5 which at temperatures around 30 MeV corresponds to around 20% nuclear density. Fig. 13.5. Proﬁle for the nondegenerate spin-ﬂuctuation rate Γ of Eq. (4.7) in the protoneutron star model S2BH 0 of Keil, Janka, and Raﬀelt (1995) shown in Fig. 13.2. Given the overall uncertainties involved in this discussion it is best to derive a plausible limit on gaby the analytic criterion Eq. (13.8). The relevant average temperature is about 30 MeV for which the maximum emission rate corresponds to the naive one at about 5 ×1013g cm−3. From Eq. (4.8) one ﬁnds an approximate axion energy-loss rate of g2 a1×1038erg g−1s−1. Then Eq. (13.8) indicates that one needs to re- quirega∼<3×10−10, about a factor of 3 less restrictive than the nomi- nal bound from the numerical calculations above. Altogether one may adopt 3×10−10∼<ga∼<3×10−7(13.11) as a range excluded by the SN 1987A cooling-time argument. What Have We Learned from SN 1987A? 513 13.6 How Many Neutrino Flavors? One may ask how the neutrino signal from a SN would be modiﬁed if there existed additional light sequential neutrino ﬂavors beyond νe, ν, andν. Of course, the Z◦decay width measured at CERN already reveals that there are exactly three sequential neutrino ﬂavors (Particle Data Group 1994); the same conclusion is reached from studies of big bang nucleosynthesis (e.g. Kolb and Turner 1990). Burrows, Ressell, and Turner (1990) calculated several protoneu- tron star cooling sequences, varying the number of ﬂavors from 3, the standard value, to 11. This increases the eﬃciency of energy transfer within the SN core and also allows for a more eﬃcient radiation from the neutrino sphere as there are more degrees of freedom. Thus one expects a shortened signal in the Kamiokande II and IMB detectors, as well as a reduced number of events because the available energy is shared between more neutrino degrees of freedom of which mostly the Fig. 13.6. Number of events NKIIandNIMBin the Kamiokande and IMB detectors as well as the signal duration tKIIandtIMB(in sec) as a function of the assumed number of neutrino ﬂavors (Burrows, Ressell, and Turner 1990). The signal duration is deﬁned as the time it takes to accrue 90% of the total expected number of events. 514 Chapter 13 νe’s are detected. These expectations are borne out by the numeri- cal results shown in Fig. 13.6. A doubling of the number of ﬂavors is probably excluded by the observed signal duration. 13.7 Neutrino Opacity The cooling time scale of a young SN core is determined by the neutrino opacities which in turn are dominated by the neutral-current scattering ν+N→N+νof neutrinos on nucleons. Apart from ﬁnal-state Pauli blocking eﬀects these opacities are given in terms of the scattering cross sectionσ= (G2 F/π) (C2 V+ 3C2 A)E2 where the neutral-current nucleon weak-coupling constants CV;Awere given in Appendix B. Therefore, the neutrino opacities are dominated by the axial-vector, i.e. the nucleon spin-dependent interaction. In Sect. 4.6.7 it was discussed that a naive application of perturbation theory in a nuclear medium likely is not appropriate because of the large spin ﬂuctuation rate implied by this method. It would indicate that the spin of a given nucleon ﬂuctuates so fast in a SN core that a neutrino would “see” on average a nearly vanishing contribution. This would lead to a decrease of a typical axial- vector scattering rate as estimated in Fig. 4.9. To test if such a suppression eﬀect is compatible with the SN 1987A neutrino signal, Keil, Janka, and Raﬀelt (1995) calculated a series of protoneutron star cooling sequences with modiﬁed neutrino opacities. To this end they substituted C2 A→FC2 Ain the numerical subroutine which evaluates the opacities where F= (1−a) +a 1 +b(13.12) with b=1 12(Γ T)2 ≈(ρ 3×1013g cm−3)210 MeV T. (13.13) Here,a= 1 represents full suppression while smaller values of aallow one to dial a lesser reduction of the opacities. The predicted neutrino signal at IMB and Kamiokande II was, again, characterized by the total number of expected events NIMBand NKIIas well as the signal durations tIMBandtKIIwhich represent the time at which 90% of the total number of expected events have been ac- crued. In Fig. 13.7 these quantities are shown as a function of awhere the suppression eﬀect was implemented for both, neutral- and charged- current axial-vector interactions. The results marked with open circles What Have We Learned from SN 1987A? 515 Fig. 13.7. Number of events in the Kamiokande and IMB detectors as well as the signal durations as a function of the assumed “opacity suppression parameter” deﬁned by Eq. (13.12). Filled circles refer to a suppression of both neutral- and charged-current axial-vector interactions while open circles refer to a suppression of neutral-current interactions only. (Adapted from Keil, Janka, and Raﬀelt 1995.) refer to a suppression of the neutral-current reactions alone. The modi- ﬁcation of the results between those cases is relatively minor, indicating that the neutral-current interactions represent the dominant opacity source for the overall cooling time scale. The increase of the counting rates at the two detectors with decreas- ing opacities is explained by the neutrino sphere moving to deeper and hotter layers, yielding larger neutrino energies. The detectors register mostlyνe’s so that the number of events is relatively sensitive to the charged-current opacity which aﬀects only the electron ﬂavor. NIMBis particularly sensitive to the νespectrum because of its high threshold. 516 Chapter 13 By the same token, NIMBis quite sensitive to spectral pinching, an ef- fect not included in these calculations where an equilibrium neutrino transport scheme was used. Therefore, the total number of events, notably at IMB, is a poor measure to characterize the neutrino signal. In the calculations of Keil, Janka, and Raﬀelt (1995) the mass of the initial neutron-star model as well as its temperature proﬁle and the equation of state were varied. While such modiﬁcations cause changes in the predicted signal durations and event counts, none of these param- eters appears likely to be able to compensate for an extreme suppres- sion of the neutrino opacities. The SN 1987A neutrino signal excludes a suppression eﬀect stronger than, say, a∼>0.5. These ﬁndings are in agreement with those of the previous section where the number of neutrino ﬂavors had been increased. Essentially that procedure amounted to increasing the eﬃciency of neutral-current energy transfer and so it is not very diﬀerent from the decreased opac- ities used here. Therefore, it appears that the “standard” opacities which ignore fast spin ﬂuctuations provide a reasonable representation of what is observed. This result appears to imply that the spin-ﬂuctuation rate Γdoes not exceed O(T) in a SN core—see Sect. 4.6.7 for a discussion of these matters. However, it is surprising that the best ﬁt is achieved by the naive opacities because the spin-ﬂuctuation eﬀect is only one reason to expect reduced axial-vector opacities. Other reasons include reduced eﬀective values for CAin a nuclear medium, and spin-spin correlations which tend to “pair” the spins and thus tend to reduce the opacities. The question of the appropriate neutrino opacities in a SN medium remains worrisome. However, with regard to particle bounds the eﬀect of reduced opaci- ties goes in the direction of making those constraints more conservative as a reduction of the opacities, like an anomalous energy loss, shortens the neutrino signal. 13.8 Right-Handed Neutrinos 13.8.1 Dirac Mass Right-handed neutrinos (helicity-minus neutrinos, helicity-plus anti- neutrinos) do not interact by the standard weak interactions and so they would not be trapped in the interior of a SN core. Therefore, the SN 1987A neutrino signal allows one to constrain any mechanism that What Have We Learned from SN 1987A? 517 could produce these “wrong-helicity” states.86The main possibilities are the existence of novel r.h. interactions which couple directly to r.h. neutrinos, the existence of neutrino magnetic or electric dipole moments which allow for left-right scatterings or magnetic oscillations, and the existence of neutrino Dirac masses. Of course, these possibilities are not necessarily distinct as the existence of r.h. currents or a Dirac mass would usually also induce magnetic dipole moments. Beginning with the assumption that neutrinos have a Dirac mass, the mismatch between chirality and helicity for massive fermions im- plies that in purely l.h. interactions a ﬁnal-state neutrino or antineu- trino sometimes has the “wrong” helicity and thus nearly r.h. chirality. Then it is essentially noninteracting and thus may escape almost freely. In principle, there are two production channels, the spin-ﬂip scattering of trapped l.h. states, and the production of pairs νLνRorνRνLby the medium. In Sect. 4.10 it was shown that the neutrino phase space favors the spin-ﬂip process by a large margin. Moreover, in a nonrel- ativistic medium the spin-ﬂip scattering rate was found to be simply the nonﬂip scattering rate times the factor ( m/2E)2. The energy- loss rate was then given by Qscatin Eq. (4.94) in terms of a dynamic structure function of the medium.87 In a dilute medium consisting of only one species of nucleons one hasS(ω) = (C2 V+ 3C2 A) 2πδ(ω), leading to an energy-loss rate of ϵR=3(C2 V+ 3C2 A)G2 Fm2 T4 2π3mN ≈0.7×1019erg g−1s−1(m 30 keV)2(T 30 MeV)4 , (13.14) whereC2 V+3C2 A≈1 was used (see Appendix B). Even though the axial- vector structure function is not a δfunction, Eq. (13.14) is probably a reasonable estimate because the results of the previous section indicate that the neutrino scattering rate in a dense medium is probably not 86Such bounds naturally can be avoided if one assumes that the r.h. neutrinos have other novel interactions which are strong enough to trap them eﬃciently in a SN core. Explicit models were constructed, for example, by Babu, Mohapatra, and Rothstein (1992) or Rajpoot (1993). These authors aimed at avoiding the SN 1987A bound on Dirac neutrino masses. 87Besides the neutrino spin-ﬂip rate due to the neutrino weak interactions with nucleons or other particles, there is also a spin-ﬂip scattering term in the gravita- tional ﬁeld of the entire neutron star (Choudhury, Hari Dass, and Murthy 1989). However, the resulting energy loss was found to be small except for low-energy neutrinos. 518 Chapter 13 too diﬀerent from that found in the dilute-medium limit. If one applies the analytic criterion Eq. (13.8) one ﬁnds m∼<30 keV (13.15) as a limit on a possible Dirac neutrino mass. This agrees with the bounds originally estimated by Raﬀelt and Seckel (1988) and Gaemers, Gandhi, and Lattimer (1989) while Grifols and Mass´ o (1990a) esti- mated a slightly more restrictive limit (14 keV). This sort of bound only applies if the mass is not so large that the “wrong-helicity” states interact strongly enough to be trapped themselves. This would occur for a mass beyond a few MeV. Therefore, a Dirac-mass νwith, say, m=O(10 MeV) is not excluded by this argument. In a numerical study Gandhi and Burrows (1990) implemented the spin-ﬂip energy-loss rate and calculated the expected event counts and signal durations at the IMB and Kamiokande II detectors. They found a bound almost identical with Eq. (13.15). A similar numerical study by Burrows, Gandhi, and Turner (1992) corroborated this result. An- other numerical study was performed by Mayle et al. (1993) who found a somewhat more restrictive limit of about 10 keV, essentially because their equation of state allows the core to heat up to much higher tem- peratures than are found in Burrows’ implementation with a stiﬀ EOS. In the Mayle et al. (1993) study, a more restrictive bound of around 3 keV was claimed if the pion-induced pair emission process π+N→ N+νR+νLwas included. This result is incorrect if one accepts the pre- dominance of the spin-ﬂip scattering over the pair-emission processes that was discussed in Sect. 4.10. It does not seem believable that the presence of pions would enhance the scattering cross section on nucle- ons. In the form implemented by Mayle et al. (1993), the pair emission rate and their neutrino opacities were not based on a common and consistent axial-vector dynamical structure function. A massiveνorνlikely would mix with νe. In this case the degen- erateνesea initially present in a SN core would partially convert into a degenerate νorνsea (Maalampi and Peltoniemi 1991; Turner 1992; Pantaleone 1992a). In Sect. 9.5 it was shown that the ﬂavor conversion would be very fast even for rather small mixing angles. In this case the spin-ﬂip scattering energy-loss rate involves initial-state neutrinos with much larger average energies than those of a nondegenerate distribu- tion that was used above. However, even though the initial energy-loss rate in r.h. neutrino is much larger than before, the degeneracy eﬀect disappears after the core has been deleptonized and so the late-time What Have We Learned from SN 1987A? 519 neutrino signal is not aﬀected as signiﬁcantly as one might have ex- pected. The Dirac mass limit becomes only slightly more restrictive if mixing is assumed (Burrows, Gandhi, and Turner 1992). Neutrinos with masses in the keV range must decay suﬃciently fast in order to avoid “overclosing” the universe. If r.h. Dirac-mass neu- trinos escape directly from the inner core of a SN they have energies far in excess of l.h. neutrinos emitted from the neutrino sphere. If their decay products involve sequential l.h. neutrinos or antineutrinos, these daughter states would have caused high-energy events at IMB or Kamiokande II, contrary to the observations. Dodelson, Frieman, and Turner (1992) found that this argument excludes the lifetime range 10−9s/keV∼<τ/m ∼<5×107s/keV, (13.16) for Dirac masses in the range 1 keV ∼<m∼<300 keV, assuming that the “visible” channel dominates. 13.8.2 Right-Handed Currents On some level r.h. weak gauge interactions may exist as, e.g. in left- right symmetric models where the gauge bosons which couple to r.h. currents would diﬀer from the standard ones only in their mass. In the low-energy limit relevant for processes in stars one may account for the novel couplings by a “r.h. Fermi constant” which is given as ϵGFwithϵ some small dimensionless number which may be diﬀerent for charged- and neutral-current processes. In left-right symmetric models one ﬁnds explicitly for charged-current reactions (Barbieri and Mohapatra 1989) ϵ2 CC=ζ2+ (mWL/mWR)4, (13.17) wheremWR;Lare the r.h. and l.h. charged gauge boson masses while ζ is the left-right mixing parameter. In order to constrain ϵCCone assumes the existence of r.h. νe’s so that the dominant energy-loss mechanism of a SN core is e+p→ n+νe;Rwhere the ﬁnal-state r.h. neutrino escapes freely. Initially, a substantial fraction of the thermal energy of a SN core is stored in the degenerate electron sea. Therefore, the time scale of cooling is estimated by the inverse scattering rate for e+p→n+νe;R. The usual charged-current weak scattering cross section involving nonrelativistic nucleons is G2 F(C2 V+3C2 A)E2 e/πwithC2 V+3C2 A≈4 in a nuclear medium (Appendix B). Using a proton density corresponding to nuclear matter at 1015g cm−3and using 100 MeV for a typical electron energy one ﬁnds 520 Chapter 13 a charged-current scattering rate of 0 .6×1010s−1or an approximate cooling time-scale by r.h. neutrinos of ϵ−2 CC2×10−10s. The requirement that this timescale exceeds a few seconds leads to the constraint ϵCC∼<10−5, (13.18) in agreement with a result of Barbieri and Mohapatra (1989) while Raf- felt and Seckel (1988) found a somewhat less restrictive limit of ϵCC∼< 3×10−5. Laboratory experiments yield a limit of order ϵCC∼<3×10−2 (e.g. Jodidio et al. 1986) which is much weaker but does not depend on the assumed existence of r.h. neutrinos. Mohapatra and Nussinov (1989) extended the SN 1987A bound to the case of r.h. Majorana neutrinos which mix with νe. In order to constrain r.h. neutral currents, equivalent to constraining the mass of putative r.h. Z◦gauge bosons, one considers the emission of r.h. neutrino pairs νRνR. The dominant emission process is by the nucleons of the medium; in a dilute medium it can be represented as the bremsstrahlung process NN→NNν RνR. Apart from a global scaling factorϵ2 NC, the bremsstrahlung energy-loss rate for a nondegenerate medium was given in Eq. (4.23). However, in a dense medium this rate probably saturates at around 10% nuclear density as in the case of axion emission (Sect. 4.6.7). Evaluating Eq. (4.23) at 10% nuclear density (ρ15= 0.03) and at T= 30 MeV, and applying the analytic criterion Eq. (13.8) one ﬁnds ϵNC∼<3×10−3. (13.19) This is less restrictive from what was found by Raﬀelt and Seckel (1988) or Barbieri and Mohapatra (1989). The translation of a limit on ϵNCinto one on a r.h. gauge boson mass depends on details of the couplings to quarks and leptons, and notably on the mixing angle between the new and the standard Zbosons. De- tailed analyses were presented by Grifols and Mass´ o (1990b), Grifols, Mass´ o, and Rizzo (1990), and Rizzo (1991). Because these authors did not consider multiple-scattering eﬀects and the resulting saturation of the bremsstrahlung process, their bounds on the Z′mass are somewhat too restrictive, perhaps by a factor of 2 or 3. Still, mZ′has to exceed at least 1 TeV, except for special choices of the mixing angle. The SN 1987A limits on r.h. neutral currents are weaker than those from big bang nucleosynthesis ( ϵNC∼<10−3) which are based on the requirement that r.h. neutrinos must not have come to thermal equi- librium after the QCD phase transition (at T∼<200 MeV) in the early universe (e.g. Olive, Schramm, and Steigman 1981; Ellis et al. 1986). What Have We Learned from SN 1987A? 521 13.8.3 Magnetic Dipole Moments Turning to neutrino magnetic dipole moments, the main production process of r.h. states would be spin-ﬂip scattering on charged parti- cles, notably on protons. The scattering cross section was discussed in Sect. 7.4. It involves the usual Coulomb divergence which in a medium is cut oﬀ by screening eﬀects. In a SN core, the electrons are initially very degenerate while the protons are essentially nondegenerate and so the main contribution to screening is from the protons. Accord- ing to Eq. (D.17) they are electrically weakly coupled at the prevailing temperatures and densities so that Debye screening should be an ap- proximately adequate prescription; the Debye scale for the protons is found to be around 30 MeV. As typical neutrino energies are somewhat larger but of the same order, the Coulomb logarithm is approximately unity. Thus the cross section is approximately αµ2 withµthe mag- netic dipole or transition moment. This is to be compared with the spin-ﬂip cross section from a Dirac mass which is approximately G2 Fm2 /4π. In Sect. 13.8.1 a Dirac mass bound of about 30 keV was derived which translates into µ∼<4×10−12µB (13.20) withµB=e/2methe Bohr magneton. This bound is similar to that derived by Barbieri and Mohapatra (1988), but less restrictive by about an order of magnitude than that claimed by Lattimer and Cooperstein (1988). See also Nussinov and Rephaeli (1987), Goldman et al. (1988), and Goyal, Dutta, and Choudhury (1995). Eq. (13.20) applies to all magnetic, electric, and transition moments of Dirac neutrinos. Numerically, it is similar to the bound Eq. (6.97) derived from the absence of excessive plasmon decay in globular cluster stars. However, because this latter result applies also to Majorana transition moments it is more general than the SN limit. The SN limit, on the other hand, applies to masses up to a few MeV while the globular cluster bound only for m∼<5 keV. The simple cooling argument that led to Eq. (13.20) is not necessar- ily the end of the story of neutrino magnetic dipole moments in SNe. If r.h. neutrinos were indeed produced in the inner core and escaped freely, they could rotate back into l.h. ones in the magnetic ﬁeld around the SN and in the galaxy. For a galactic magnetic ﬁeld of order 10−6Gauss, extended over, say, 1 kpc (the ﬁeld is conﬁned to the disk, the LMC lies far above the disk) this eﬀect would be important for µ∼>10−12µB. 522 Chapter 13 As the r.h. neutrinos escape from the inner core with much larger ener- gies than those from the neutrino sphere one would expect high-energy events in the Kamiokande and IMB detectors, contrary to the observa- tions. Therefore, one probably needs to require µ∼<10−12µBfor the diagonal dipole moments; spin-ﬂavor oscillations could be suppressed by the neutrino mass diﬀerences. As the spin precession is the same for all neutrino energies, this limit would not apply if the Earth happened to be in a node of the oscillation pattern between SN 1987A and us. Neutrino magnetic moments of order 10−12µBcould also aﬀect the infall phase of SNe. The spin-ﬂip scattering on nuclei would be coher- ently enhanced relative to protons. Therefore, neutrinos could escape in the r.h. channel for much longer so that eﬀectively trapping would set in much later than in the standard picture (N¨ otzold 1988). In and near the SN core there probably exist strong magnetic ﬁelds of order 1012Gauss or more which would induce spin-precessions be- tween r.h. and l.h. neutrinos. Therefore, the sterile states produced in the deep interior by spin-ﬂip scattering could back-convert into active ones near the neutrino sphere. Depending on details of the matter- induced neutrino energy shifts, the vacuum mass diﬀerences, and the magnetic ﬁeld strengths and conﬁgurations this conversion could take place inside or outside of the neutrino sphere. The observable neutrino signal could be aﬀected, but also the energy transfer within the SN core and outside of the neutrino sphere. Perhaps, a more eﬃcient transfer of energy to the stalled shock wave could help to explode SNe in the de- layed explosion scenario. Various aspects of these scenarios have been studied by Dar (1987), Nussinov and Rephaeli (1987), Goldman et al. (1988), Voloshin (1988), Okun (1988), Blinnikov and Okun (1988), and Athar, Peltoniemi, and Smirnov (1995). Clearly, Dirac magnetic or transition moments in the 10−12µBrange and below would aﬀect SN dynamics and the observable neutrino signal in interesting ways. However, because there are so many parameters and possible ﬁeld conﬁgurations, it is hard to develop a clear view of the excluded or desired neutrino properties. If compelling evidence for nonstandard neutrino electromagnetic properties in this range were to emerge, SN dynamics likely would have to be rethought from scratch. 13.8.4 Millicharges Within the particle physics standard model it is not entirely impos- sible that neutrinos have small electric charges (Sect. 15.8). In this case neutrinos would have to be Dirac fermions and so the r.h. states What Have We Learned from SN 1987A? 523 can be produced in pairs by their coupling to the electromagnetic ﬁeld. Obvious production processes are the plasmon decay γpl→νRνRand pair annihilation e+e−→νRνRwith an intermediate photon. An in- termediate photon can also be coupled to hadronic components of the medium. Mohapatra and Rothstein (1990) considered nucleon-nucleon bremsstrahlung, and notably an amplitude where the electromagnetic ﬁeld is coupled to an intermediate charged pion. One may well wonder, however, if this sort of naive perturbative bremsstrahlung calculation is adequate in a nuclear medium. A simple estimate of the plasmon decay process begins with the energy-loss rate Eq. (6.94). Because the electrons in a SN core are highly relativistic the plasma frequency is given by Eq. (6.43) as ω2 P= (4α/3π) (µ2 e+1 3π2T2) whereµeis the electron chemical potential. For a SN core ωP≈10 MeV is a reasonable estimate while the relevant temperature is about T= 30 MeV. Taking approximately Q1= 1 in Eq. (6.94) and applying the approximate criterion Eq. (13.8) one ﬁnds e∼<10−9e. (13.21) This is similar to Mohapatra and Rothstein’s (1990) result. Including thee+e−annihilation process would slightly improve this limit and extend it to somewhat larger masses. The bound Eq. (13.21) would apply to any millicharged particle which is not trapped in the SN core. Mohapatra and Rothstein (1990) estimated that for a charge in excess of about 10−7ethe particles would be suﬃciently trapped by scatterings oﬀ electrons to leave the SN cool- ing time scale essentially unaﬀected. If they are Dirac neutrinos with a mass in excess of a few MeV trapping by spin-ﬂip scattering would be- come important. Again, it is not obvious how strong the impact of such trapped particles would be during the infall phase of SN collapse, i.e. one should not infer that millicharged particles in the trapping regime would not have a strong impact on SN physics just because their impact on the Kelvin-Helmholtz cooling phase is small. 13.8.5 Charge Radius If r.h. neutrinos existed and had an eﬀective electromagnetic interaction by virtue of a charge radius, they would be produced in a SN core by the same processes as above where they were assumed to have a charge. On the basis of the e+e−annihilation process Grifols and Mass´ o (1989) found a limit of about 3 ×10−17cm on a r.h. charge radius. Chapter 14 Axions The idea of axions is introduced and their phenomenological properties are reviewed. The constraints on pseudoscalars that have been derived throughout this book are systematically applied to axions. 14.1 The Strong CP-Problem All fermions, with the possible exception of neutrinos, have magnetic dipole moments—see Tab. 14.1 for several important examples. The minimal electromagnetic coupling of charged spin-1 2fermions (charge q, massm) automatically yields a “Dirac moment” q/2m. Higher-order amplitudes lead to an additional “anomalous” contribution, which is the only one for neutral particles. For example, the magnetic moments of massive neutrinos calculated in the standard model were given in Eq. (7.16). The QED prediction of the electron anomalous magnetic moment is perhaps the most stunning quantitative success of theoretical physics. On the other hand, no particle electric dipole moment has ever been detected—see Tab. 14.1 for some upper limits. At ﬁrst this is quite satisfying because an electric dipole moment would allow one to distinguish between matter and antimatter in an absolute sense.88The electroweak and strong gauge interactions are CP-conserving, so the observed broad symmetry between particles and antiparticles appears natural. 88In the nonrelativistic limit the electric dipole operator dmust be proportional to the spin operator swhich is an axial vector. Because the electric ﬁeld is a polar vector the energy d·Ereverses sign under P and thus under CP whence its absolute sign yields an absolute distinction between fermions and antifermions. 524 Axions 525 Table 14.1. Particle magnetic and electric dipole moments. Fermion Magnetic MomentcElectric Momentd [10−26ecm] Protona2.792,847,3(86)µN 4000±6000 Neutrona−1.913,042,(75)µN <11e Electrona1.001,159,652,1(93)µB −0.3±0.8 Neutrinob∼<3×10−12µB ∼<6000 aParticle Data Group (1994). bAll ﬂavors with mν∼<5 keV (Sect. 6.5.6). cBohr magneton B=e=2me; nuclear magneton N=e=2mp. d1ecm = 5 :18×1010B= 0:951×1014N(Appendix A). e95% CL. This symmetry is not respected, however, by the generally complex Yukawa couplings to the Higgs ﬁeld which are thought to induce the fermion masses (Sects. 7.1 and 7.2). The resulting complex quark mass matrixMqcan be made real and diagonal by suitable transformations of the quark ﬁelds. This involves a global chiral phase transformation (angle Θ = arg det Mq), leading to a term in the QCD Lagrangian L= Θαs 8πGeG. (14.1) Here,αsis the ﬁne-structure constant of strong interactions and GeG≡ Gµν beGbµνwhereGµν bis the color ﬁeld strength tensor,eGbµν=1 2εµνρσGρσ b its dual, and the implied summation over brefers to the color degrees of freedom. Of course, det Mqand thus Θ would vanish if one of the quarks were exactly massless, but this does not seem to be the case. Under the combined action of charge conjugation (C) and a parity transformation (P) the Lagrangian Eq. (14.1) changes sign,89violating the CP invariance of QCD. It leads to a neutron electric dipole moment |dn| ≈ |Θ|(0.04−2.0)×10−15ecm (Baluni 1979; Crewther et al. 1979; see also Cheng 1988). This is |dn| ≈ | Θ|(0.004−0.2)µNin units of nuclear magnetons. Hence, for |Θ|of order unity one expects a neu- tron electric dipole moment almost as large as its magnetic one. The 89The structure of GeGisEcolor·Bcolor, i.e. the scalar product of a polar with an axial vector and so it is CP-odd. 526 Chapter 14 experimental limit (Tab. 14.1), however, indicates |Θ|∼<10−9, surpris- ingly small in view of the phase δ= 3.3×10−3which appears in the Cabbibo-Kobayashi-Maskawa matrix Eq. (7.6) and which explains the observed CP-violating eﬀects in the K◦-K◦system. Even worse, QCD alone produces a term like Eq. (14.1) because of the nontrivial topological structure of its ground state (Callan, Dashen, and Gross 1976; Jackiw and Rebbi 1976; t’Hooft 1976a,b). The coeﬃ- cient Θ QCDis a parameter characterizing the “Θ-vacuum.” It is mapped onto itself by a transformation Θ QCD→ΘQCD+ 2πso that diﬀerent ground states are characterized by values in the range 0 ≤ΘQCD<2π. The phase of the quark mass matrix and the QCD-vacuum together yield Θ≡ΘQCD+ arg detMqas a compound coeﬃcient for Eq. (14.1). The experimental bounds then translate into ΘQCD+ arg detMq∼<10−9. (14.2) The CP-Problem of strong interactions consists of the smallness of Θ which implies that the numbers Θ QCDand arg det Mqare either sepa- rately very small, or cancel each other with very high accuracy. How- ever, both are expected to be of order unity, or perhaps of order δin the case of arg det Mq, and completely unrelated to each other. 14.2 The Peccei-Quinn Mechanism 14.2.1 Generic Features An attempt to explain the smallness of Θ may be overambitious as long as we do not have an understanding of the origin of the Yukawa couplings that went into Mqand of the other seemingly arbitrary pa- rameters of the standard model. Still, whatever determines these “con- stants of nature,” the strong CP-problem can be elegantly explained by the existence of a new physical ﬁeld, the axion ﬁeld, which allows Θ to vanish dynamically (Peccei and Quinn 1977a,b; Weinberg 1978; Wilczek 1978). In this scheme, the CP-violating Lagrangian Eq. (14.1) is literally switched oﬀ by its own force. To this end the new ﬁeld a(x) must be a pseudoscalar which couples to gluons according to Eq. (14.1) with Θ replaced by −a/fa. The constant90fawith the dimension of an energy is the Peccei-Quinn scale 90In the literature one often ﬁnds fa=N,Fa=N,vPQ=Netc. for what I call fa. It was stressed, e.g. by Georgi, Kaplan, and Randall (1986) that a discussion of the generic properties of all axion models does not require a speciﬁcation of the model- dependent integer Nwhich can be conveniently absorbed in the deﬁnition of fa. Axions 527 oraxion decay constant . Axions must be fundamentally massless so that all observable eﬀects remain unchanged under a global shift a(x)→ a(x) +a0wherea0is a constant. (A mass term1 2m2 aa2would spoil this possibility.) This invariance allows one to absorb Θ in the deﬁnition of the axion ﬁeld. Including a kinetic term, Eq. (14.1) is replaced by L→ L a=1 2(∂µa)2−αs 8πfaaGeG. (14.3) It conserves CP because axions were assumed to be pseudoscalar (odd under CP), similar to neutral pions. Even though axions were constructed to be massless they acquire an eﬀective mass by their interaction with gluons. It induces transitions to qqstates and thus to neutral pions (Fig. 14.1) which means physically thataandπ◦mix with each other. Axions thereby pick up a small mass which is approximately given by (Bardeen and Tye 1978; Kandaswamy, Salomonson, and Schechter 1978) mafa≈mπfπ, (14.4) wheremπ= 135 MeV is the pion mass and fπ≈93 MeV its decay constant. This mass term implies that at low energies the axion La- grangian contains a potential V(a) which expands to lowest order as 1 2m2 aa2. Because of the invariance of Lwith respect to Θ →Θ + 2π the gluon-induced potential V(a) is a function periodic with 2 πfa. Fig. 14.1. Axion mixing with qqstates and thus ◦. The curly lines represent gluons, the solid lines quarks. The ground state of the axion ﬁeld is at the minimum of its po- tential ata= 0, explaining the absence of a neutron electric dipole moment. If one could produce a static nonvanishing axion ﬁeld a0in some region of space, neutrons there would exhibit an electric dipole moment corresponding to Θ =−a0/fa. The Lagrangian Eq. (14.3) is the minimal ingredient for any ax- ion model: the aGeGcoupling is their deﬁning feature as opposed to other pseudoscalar particles. Then axions inevitably acquire an eﬀec- tive mass at low energies. Thus the concept of a “massless axion” for some arbitrary pseudoscalar is a contradiction in terms. 528 Chapter 14 Because of the mixing with π◦, axions share not only their mass, but also their couplings to photons and nucleons with a strength reduced by aboutfπ/fa. Therefore, they generically couple to photons so that the general discussion of Chapter 5 applies directly except for those aspects which required massless pseudoscalars. The eﬀective axion mass is a low-energy phenomenon below Λ QCD≈ 200 MeV. Above this energy pions and other hadrons dissociate in favor of a quark-gluon plasma. Then a= 0 is no longer singled out so that any value in the interval 0 ≤a <2πfais physically equivalent. Because the universe is believed to begin with a hot and dense “big bang,” any initial value for ais equally plausible, or diﬀerent initial conditions in diﬀerent regions of space. As the universe expands and cools below Λ QCD, however, the axion ﬁeld must relax to its newly singled-out ground state at a= 0. This relaxation process produces a population of cosmic background axions which is, in units of the cosmic critical density (e.g. Kolb and Turner 1990), Ωah2≈(fa/1012GeV)1.175. (14.5) The exact value depends on details of the cosmic scenario and of the relaxation process. Modulo this uncertainty, values exceeding fa≈ 1012GeV are excluded as axions would overdominate the dynamics of the universe. With Eq. (14.4) this corresponds to ma∼<10−5eV; axions near this bound would be the cosmic dark matter. A search strategy for galactic axions in this mass range was discussed in Sect. 5.3. 14.2.2 Axions as Nambu-Goldstone Bosons The invariance of Lin Eq. (14.1) against transformations of the form Θ→Θ+2π, and the corresponding invariance of the axion Lagrangian against transformations a→a+ 2πfa, calls for a very simple interpre- tation of the axion ﬁeld as the phase of a new scalar ﬁeld. A transparent illustration is provided by the KSVZ axion model (Kim 1979; Shifman, Vainshtein, and Zakharov 1980) where one in- troduces a new complex scalar ﬁeld Φ which does not participate in the weak interactions, i.e. an SU(2) ×U(1) singlet. There is also a new massless fermion ﬁeld Ψ and one considers a Lagrangian with the usual kinetic terms, a potential Vfor the scalar ﬁeld, and an interaction term, L=( i 2Ψ∂µγµΨ + h.c.) +∂µΦ†∂µΦ−V(|Φ|) −h( ΨLΨRΦ + h.c.) . (14.6) Axions 529 The Yukawa coupling his chosen to be positive, and Ψ L≡1 2(1−γ5)Ψ and Ψ R≡1 2(1 +γ5)Ψ are the usual left- and right-handed projections. This Lagrangian is invariant under a chiral phase transformation of the form Φ→eiαΦ,ΨL→eiα/2ΨL,ΨR→e−iα/2ΨR, (14.7) where the left- and right-handed ﬁelds pick up opposite phases. This chiral symmetry is usually referred to as the Peccei-Quinn (PQ) sym- metry U PQ(1). The potential V(|Φ|) is chosen to be a “Mexican hat” with an abso- lute minimum at |Φ|=fPQ/√ 2 wherefPQis some large energy scale. The ground state is characterized by a nonvanishing vacuum expecta- tion value ⟨Φ⟩= (fPQ/√ 2)eiφwhereφis an arbitrary phase. It spon- taneously breaks the PQ symmetry because it is not invariant under a transformation of the type Eq. (14.7). One may then write Φ =fPQ+ρ√ 2eia/fPQ(14.8) in terms of two real ﬁelds ρandawhich represent the “radial” and “angular” excitations. The potential Vprovides a large mass for ρ, a ﬁeld which will be of no further interest for these low-energy considerations. Neglecting all terms involving ρthe Lagrangian Eq. (14.6) is L=( i 2Ψ∂µγµΨ + h.c.) +1 2(∂µa)2−mΨeiγ5a/fPQΨ, (14.9) wherem≡hfPQ/√ 2. The variation of the fermion ﬁelds under a PQ transformation is given by Eq. (14.7) while a→a+αfPQ. The invariance of Eq. (14.9) against such shifts is a manifestation of the UPQ(1) symmetry. It implies that arepresents a massless particle, the Nambu-Goldstone boson of the PQ symmetry . Expanding the last term in Eq. (14.9) in powers of a/fPQ, the zeroth- order term mΨΨ plays the role of an eﬀective fermion mass. Higher orders describe the interaction of awith Ψ, Lint=−im fPQaΨγ5Ψ +m 2f2 PQa2ΨΨ +... . (14.10) The dimensionless Yukawa coupling ga≡m/f PQis proportional to the fermion mass. The fermion Ψ is taken to be some exotic heavy quark with the usual strong interactions, i.e. an SU C(3) triplet. The lowest-order interaction 530 Chapter 14 ofawith gluons is then given by the triangle graph of Fig. 14.2. With the ﬁrst term of Eq. (14.10) it yields an eﬀective a-gluon interaction of LaG=−ga mαs 8πaGeG, (14.11) whereαs≡g2 s/4π. All external momenta were taken to be small relative to the mass mof the loop fermion. Fig. 14.2. Triangle loop diagram for the interaction of axions with gluons (strong coupling constant gs, axion-fermion Yukawa coupling ga). An anal- ogous graph pertains to the coupling of axions with photons if the fermion carries an electric charge which replaces gs. In more general models, several conventional or exotic quark ﬁelds Ψjmay participate in this scheme. The transformation of each ﬁeld under a U PQ(1) transformation is characterized by its PQ charge Xj, Ψj L→eiXjα/2Ψj L. (14.12) The totalaGeGinteraction is obtained as a sum over Eq. (14.11) for all Ψj. Becausegaj=Xjmj/fPQthe fermion masses drop out. With N≡∑ jXjandfa≡fPQ/N (14.13) one has then found the required coupling Eq. (14.3) which allows one to interpret aas the axion ﬁeld. The potential V(a) is periodic with 2 πfa= 2πfPQ/N. The interpre- tation ofaas the phase of Φ, on the other hand, implies a periodicity with 2πfPQso thatNmust be a nonzero integer. This requirement restricts the possible assignment of PQ charges to the quark ﬁelds. It also implies that there remain Ndiﬀerent equivalent ground states for the axion ﬁeld, each of which satisﬁes Θ = 0 and thus solves the CP problem. Axions 531 14.2.3 Pseudoscalar vs. Derivative Interaction There has been considerable confusion in the literature concerning the proper structure for the coupling of axions to fermions. The lowest- order term in the expansion Eq. (14.10) is a pseudoscalar interaction which is frequently used because of its simplicity. However, there is an inﬁnite series of terms which sometimes must be taken into account. For example, the axion scattering on fermions is second order in the ﬁrst term of Eq. (14.10), but ﬁrst order in the second whence both must be included. Such complications can be avoided if one redeﬁnes the fermion ﬁeld in Eq. (14.9) by a local transformation, ψL≡e−ia/2fPQΨL, ψ R≡eia/2fPQΨR. (14.14) The last term in Eq. (14.10) is then a simple mass term mψψ. The interaction between ψandanow arises from the kinetic Ψ term in Eq. (14.9), Lint=1 2fPQψγµγ5ψ∂µa. (14.15) This interaction is of derivative nature, and it is linear in awith no higher-order terms. The fermion part has the form of an axial-vector current, bringing out a useful similarity between neutrino and axion interactions. Pions play the role of Nambu-Goldstone bosons of a spontaneously broken U(2) L−Rsymmetry of QCD and so their interactions with nucle- ons should involve similar higher-order terms. The diﬀerence between derivative and “naive” pseudoscalar couplings should become apparent in bremsstrahlung processes of the type shown in Fig. 14.3 where two Nambu-Goldstone bosons are attached to one fermion line. A useful template is provided by p+p→p+p+π◦where existing data, in- deed, favor the derivative case (Choi, K. Kang, and Kim 1989; Turner, H.-S. Kang, and Steigman 1989). It should be noted in this context that the pion-nucleon interaction is described by the Lagrangian (Carena and Peccei 1989), Lint=gπNNγµγ5N·∂µ+f2 πNNγµN·∂µ×, (14.16) wheregπN≡f/m π≈1/mπandfπN≡1/2fπare the relevant coupling constants, is a vector of Pauli isospin matrices, is the isovector of the neutral and charged pion ﬁelds, and Nis the isodoublet of neutron and proton. Thus there appears an extra dimension-6 term compared 532 Chapter 14 Fig. 14.3. Nucleon-nucleon bremsstrahlung emission of axions or pions. with the axion example Eq. (14.15) where only one Nambu-Goldstone boson was present as opposed to the pion isotriplet. However, this ad- ditional term does not contribute to the bremsstrahlung process in the limit of nonrelativistic nucleons so that the above conclusion regarding the derivative coupling remains valid. In the process NN→NNa (Fig. 14.3) axions andpions appear so that again two Nambu-Goldstone bosons are attached to one fermion line. It is then necessary to use a derivative coupling for at least one of them (Raﬀelt and Seckel 1988). For other bremsstrahlung processes such ase−p→pe−a, where the particles interact through a virtual photon (a gauge boson) the pseudoscalar coupling causes no trouble. Also for the Compton process γe−→e−aone may use either the pseu- doscalar or the derivative axion coupling: both yield the same result. Because it is not always a priori obvious whether the pseudoscalar and derivative couplings yield the same result it is a safe strategy to use the derivative coupling in all calculations. 14.2.4 The Onslaught of Quantum Gravity A heavy critique was levied against the PQ mechanism by quantum- gravity inspired phenomenological considerations. The main idea is that generally the PQ symmetry, like any other global symmetry, will not be respected by gravity (Georgi, Hall, and Wise 1981). For exam- ple, a black hole can “swallow” any amount of PQ charge without a trace, while a swallowed electric charge remains visible by its Coulomb force. At energy scales exceeding the Planck mass mPl= 1.2×1019GeV quantum gravitational eﬀects are expected and so mPlis a phenomeno- logical cutoﬀ for any quantum theory which does not fundamentally include gravitation. In the “low-energy” world it should manifest itself by all sorts of eﬀective interactions which are not forbidden by a sym- metry and which likely involve inverse powers of the cutoﬀ scale mPl. Axions 533 Notably, the Higgs ﬁeld Φ which gives rise to the axion probably exhibits eﬀective interactions of dimension 2 m+n Vgrav(Φ) =geiδ(ΦΦ†)mΦn m2m+n−4 Pl, (14.17) wheregandδare real numbers. Because such interactions violate the PQ symmetry for n̸= 0 they induce an eﬀective potential for the axion after spontaneous symmetry breaking. The full potential is then of the form (Kamionkowski and March-Russell 1992) V(a) f2 a=m2 QCD[ 1−cos(a/fa)] +m2 grav[ 1−cos(δ+na/f a)] ,(14.18) wheremQCDis the usual QCD axion mass while gravity induces m2 grav=gm2 Pl(fa/√ 2mPl)2m+n−2. (14.19) Forgof order unity and for low values of mandnone needs a very smallfafor the QCD eﬀect to dominate. Therefore, unless gravity for some reason favors a minimum at the CP-conserving position for athe PQ scheme will be ruined entirely.91 Whatever the ultimate quantum theory of gravitation, no doubt it will be very special. Therefore, it is by no means obvious that the above arguments, which do not go far beyond a dimensional analysis, correctly represent the low-energy eﬀects of Planck-scale physics. Even then the PQ mechanism still works if the PQ global symmetry is an “automatic symmetry” of a gauge theory; in this case it is protected from the assault of quantum gravity. Such models can be constructed (Holman et al. 1992) and in fact may be quite generic (Barr 1994). Either way, in order for axions to solve the strong CP problem one must assume that the PQ scheme is not ruined by quantum gravity. This discussion illustrates an important feature of axion models, or any model involving a broken global symmetry and its Nambu- Goldstone boson. These particles are interlopers in the low-energy world—axions really belong to the high-energy world at the PQ scale. These roots make them susceptible to physics at large energy scales, at the Planck mass, for example. By the same token, if axions were ever detected, for example by the galactic axion search (Sect. 5.3), they would be one of the few messengers that we can ever hope to receive from a high-energy world which is otherwise inaccessible to experimen- tal enquiry. 91These issues were studied by Barr and Seckel (1992) and by Kamionkowski and March-Russell (1992). See also the earlier papers by Georgi, Hall, and Wise (1981), Lazarides, Panagiotakopoulos, and Shaﬁ (1986), and Dine and Seiberg (1986). 534 Chapter 14 14.3 Fine Points of Axion Properties 14.3.1 The Most Common Axion Models Axions generically mix with pions so that their mass and their couplings to photons and nucleons are crudely fπ/fatimes those of π◦. In detail, however, these properties depend on the speciﬁc implementation of the PQ mechanism. Therefore, it is useful to review brieﬂy the most common axion models which may serve as generic examples for an interpretation of the astrophysical evidence. In the standard model, the would-be Nambu-Goldstone boson from the spontaneous breakdown of SU(2) ×U(1) is interpreted as the third component of the neutral gauge boson Z◦, making it impossible for the scalar ﬁeld Φ of which axions are the phase to be the standard Higgs ﬁeld. Therefore, one needs to introduce two independent Higgs ﬁelds Φ 1 and Φ 2with vacuum expectation values f1/√ 2 andf2/√ 2 which must obey (f2 1+f2 2)1/2=fweak≡(√ 2GF)−1/2≈250 GeV. In this standard axion model (Peccei and Quinn 1977a,b; Weinberg 1978; Wilczek 1978) Φ1gives masses to the up- and Φ 2to the down-quarks and charged leptons. With x≡f1/f2and 3 families the axion decay constant is fa=fweak[3 (x+ 1/x)]−1∼<42 GeV. This and related “variant” models (Peccei, Wu, and Yanagida 1986; Krauss and Wilczek 1986), however, are ruled out by overwhelming experimental and astrophysical evidence; for reviews see Kim (1987), Cheng (1988), and Peccei (1989). Therefore, one is led to introduce an electroweak singlet Higgs ﬁeld with a vacuum expectation value fPQ/√ 2 which is not related to the weak scale. Taking fPQ≫fweak, the mass of the axion becomes very small, its interactions very weak. Such models are generically re- ferred to as invisible axion models . The ﬁrst of its kind was the KSVZ model (Kim 1979; Shifman, Vainshtein, and Zakharov 1980) discussed in Sect. 14.2.2. It is very simple because the PQ mechanism entirely decouples from the ordinary particles: at low energies, axions interact with matter and radiation only by virtue of their two-gluon coupling which is generic for the PQ scheme. The KSVZ model in its simplest form is determined by only one free parameter, fa=fPQ, although one may introduce N > 1 exotic quarks whence fa=fPQ/N. Also widely discussed is the DFSZ model introduced by Zhitnitski˘ ı (1980) and by Dine, Fischler, and Srednicki (1981). It is a hybrid be- tween the standard and KSVZ models in that it uses an electroweak singlet scalar ﬁeld Φ with a vacuum expectation value fPQ/√ 2andtwo electroweak doublet ﬁelds Φ 1and Φ 2. There is no need, however, for ex- Axions 535 otic heavy quarks: only the known fermions carry Peccei-Quinn charges. Therefore,Nis the number of standard families. Probably N= 3 so that the remaining free parameters of this model are fa=fPQ/Nand x=f1/f2which is often parametrized by x= cotβor equivalently by cos2β=x2/(x2+ 1). From a practical perspective, the main diﬀerence between the KSVZ and DFSZ models is that in the latter axions couple to charged leptons in addition to nucleons and photons. The former is an example for the category of hadronic axion models . BecausefPQ≫fweakin these models, one may attempt to identify fPQwith the grand uniﬁcation scale fGUT≈1016GeV (Wise, Georgi, and Glashow 1981; Nilles and Raby 1982). However, the cosmologi- cal boundfa∼<1012GeV disfavors the GUT assignment. There exist numerous other axion models, and many attempts to connect the PQ scale with other scales—for a review see Kim (1987). In the absence of a compelling model fashould be viewed as a free phenomenological parameter. 14.3.2 Axion Mass and Coupling to Photons The axion mass which arises from its mixing with π◦can be obtained with the methods of current algebra to be (Bardeen and Tye 1978; Kandaswamy, Salomonson, and Schechter 1978; Srednicki 1985; Georgi, Kaplan, and Randall 1986; Peccei, Bardeen, and Yanagida 1987) ma=fπmπ fa(z (1 +z+w)(1 +z))1/2 = 0.60 eV107GeV fa, (14.20) where the quark mass ratios are (Gasser and Leutwyler 1982) z≡mu/md= 0.568±0.042, w≡mu/ms= 0.0290±0.0043. (14.21) Aside from these uncertainties there are higher-order corrections to the current-algebra axion mass which have not been estimated in the literature. 536 Chapter 14 By their generic coupling to gluons, axions necessarily mix with pions and hence couple to photons according to Lint=−1 4gaγFµνeFµνa=gaγE·Ba, (14.22) whereFis the electromagnetic ﬁeld strength tensor andeFits dual. In models where the quarks and leptons which carry PQ charges also carry electric charges, there is a contribution from a triangle loop diagram as in Fig. 14.2, replacing gswith the electric charge Qjeof the lepton. It yields an axion-photon coupling proportional to E≡2∑ jXjQ2 jDj, (14.23) whereDj= 3 for color triplets (quarks) and 1 for color singlets (charged leptons). The total axion-photon coupling strength is then (Kaplan 1985; Srednicki 1985) gaγ=−α 2πfa3 4ξ=meV 0.69×1010GeVξ, (14.24) where ξ≡4 3(E N−2 34 +z+w 1 +z+w) =4 3(E N−1.92±0.08) (14.25) andmeV≡ma/eV. In the DFSZ or grand uniﬁed models one has for a given fam- ily of quarks and leptons E/N = 8/3. Neglecting wthis yields ξ≈ (8/3)z/(1 +z)≈1. However, one may equally consider models where E/N = 2 so that ξ= 0.1±0.1, i.e. the axion-photon coupling is strongly suppressed and may actually vanish (Kaplan 1985). 14.3.3 Model-Dependent Axion-Fermion Coupling The discussion in Sect. 14.2.3 implies that axions interact with a given fermionj(massmj) according to a pseudoscalar or a derivative axial- vector interaction, Lint=−iCjmj faΨjγ5ΨjaorCj 2faΨjγµγ5Ψj∂µa, (14.26) whereCjis an eﬀective PQ charge of order unity to be deﬁned below. Evidentlygaj≡Cjmj/faplays the role of a Yukawa coupling and αaj=g2 aj/4πthat of an “axionic ﬁne structure constant.” Numerically, gae=Ceme/fa=Ce0.85×10−10meV, gaN=CNmN/fa=CN1.56×10−7meV (14.27) for electrons and nucleons. Axions 537 Various axion models diﬀer in their assignment of PQ charges. How- ever, in all models N=∑ quarksXjis a nonzero integer. The assignment at high energies is not maintained in the low-energy sector because the spontaneous breakdown of the weak SU L(2)×UY(1) symmetry at fweak≈250 GeV mixes the axion with the would-be Nambu-Goldstone boson which becomes the longitudinal component of the Z◦gauge bo- son. Hence the PQ charges must be shifted such that the physical axion does not mix with the Z◦; these shifted values are denoted as X′ j. Also, below the QCD scale Λ QCD≈200 MeV free quarks do not exist, so one needs to consider the eﬀective coupling to nucleons which arises from the direct axion coupling to quarks and from the mixing with π◦andη, leading to PQ charges X′ pandX′ nfor protons and neutrons. The above eﬀective PQ charges are obtained by Cj≡X′ j/Nin order to absorb N in their deﬁnition just as it was absorbed in fa=fPQ/N. In the KSVZ model Ce= 0 at tree level (“hadronic axions”) al- though there are small radiatively induced couplings (Srednicki 1985). In the DFSZ model Ce= cos2β/N f, (14.28) whereNfis the number of families, probably 3. The nucleon interactions in general axion models were investigated by Kaplan (1985) and Srednicki (1985). They were revisited by Mayle et al. (1988, 1989), Cp= (Cu−η)∆u+ (Cd−ηz)∆d+ (Cs−ηw)∆s, Cn= (Cu−η)∆d+ (Cd−ηz)∆u+ (Cs−ηw)∆s, (14.29) whereη≡(1 +z+w)−1withzandwwere given in Eq. (14.21). For a given quark ﬂavor, q=u,d, ors, the interaction strength with protons depends on the proton spin content carried by this par- ticular quark ﬂavor, Sµ∆q≡ ⟨p|qγµγ5q|p⟩whereSµis the proton spin. Similar expressions pertain to the coupling with neutrons; the two sets of expressions are related by isospin invariance. Neutron and hyperon β-decays as well as polarized lepton scattering experiments on nucleons yield a consistent set of ∆ q’s (Ellis and Karliner 1995) ∆u= +0.85,∆d=−0.41,∆s=−0.08 (14.30) with an approximate uncertainty of ±0.03 each. In the DFSZ model, Cs=Cd=Ce,Cu+Cd= 1/Nf, andCu−Cd= −cos2β/N f, leading to Cu= sin2β/N fandCd=Cs=Ce= cos2β/N f. 538 Chapter 14 WithNf= 3 one ﬁnds Cp=−0.10−0.45 cos2β, Cn=−0.18 + 0.39 cos2β. (14.31) In the KSVZ model and in other hadronic axion models Cu=Cd= Cs= 0 which yields Cp=−0.39, Cn=−0.04. (14.32) In Fig. 14.4 these couplings are shown, for DFSZ axions as a function of cos2β. They are all uncertain to within about ±0.05, but even then CpandCnnever seem to vanish simultaneously. Fig. 14.4. Axion couplings to fermions according to Eqs. (14.28), (14.31), and (14.32). 14.4 Astrophysical Axion Bounds Because axions couple to nucleons, photons, and electrons it is easy to translate the bounds on such couplings derived in previous chapters of this book into bounds on the Peccei-Quinn scale or equivalently, on the axion mass. A possible modiﬁcation of the usual axion models by the quantum gravity eﬀects discussed in Sect. 14.2.4 is ignored for the present discussion.92 92Barr and Seckel (1992) studied astrophysical axion bounds when quantum grav- ity eﬀects are taken seriously. Axions 539 Bounds on the Yukawa coupling to electrons of various novel parti- cles were derived in Chapter 3; for pseudoscalars a summary was given in Tab. 3.1. The most restrictive limit was obtained from the delay of helium ignition in low-mass red giants that would be caused by exces- sive axion emission; in terms of the axion-electron Yukawa coupling it isgae∼<2.5×10−13. With Eq. (14.27) this translates into maCe∼<0.003 eV and fa/Ce∼>2×109GeV. (14.33) Axions which interact too strongly to escape freely from the interior of stars would still contribute to the transfer of energy. For the Sun, this issue was studied in Sect. 1.3.5. One easily ﬁnds that for Ce= 1 Fig. 1.2 excludes axion masses below about 50 keV. In hadronic axion models Ce= 0 at tree level and so no interesting bounds on maandfaobtain. In the DFSZ model, Cewas given in Eq. (14.28). Taking the number of families to be Nf= 3 one ﬁnds macos2β∼<0.01 eV and fa/cos2β∼>0.7×109GeV.(14.34) These limits depend on the parameter cos2βwhich, in principle, can be equal to 0. The axion-photon coupling is best constrained by the lifetime of horizontal-branch (HB) stars as outlined in Sect. 5.2.5. The limit Eq. (5.23) translates into maξ∼<0.4 eV and fa/ξ∼>1.5×107GeV, (14.35) whereξwas deﬁned in Eq. (14.25). In addition, approximately the mass range 4 −14 eV is excluded by the “telescope search” for a line from the radiative decay of cosmic axions (Fig. 12.23). The most restrictive limit on the axion-nucleon coupling arises from the duration of the neutrino signal of SN 1987A. The formally ex- cluded range for the axion-nucleon Yukawa coupling was speciﬁed in Eq. (13.11); as discussed in Sect. 13.5 it is fraught with uncertainties because no reliable calculation of the axion emission rate from a nuclear medium is available at the present time. In terms of the axion mass and axion decay constant the nominally excluded range is 0.002 eV ∼<CNma∼<2 eV, 3×106GeV∼<fa/CN∼<3×109GeV. (14.36) The case of large ma(smallfa) is the trapping regime where axions contribute to the energy transfer in a SN core, and where they are 540 Chapter 14 emitted from an “axion sphere” rather than the entire volume of the protoneutron star. These bounds were derived assuming equal couplings to protons and neutrons. However, a glance at Fig. 14.4 reveals that KSVZ axions es- sentially do not couple to neutrons while Cp≈ −0.36. For DFSZ axions the couplings vary with cos2β, although for cos2β≈0.5 about the same values as for KSVZ axions apply which are thus taken as generic. As- suming a proton fraction of about 0 .3 for the relevant regions of the SN core I estimate an eﬀective nucleon coupling of CN≈0.31/20.36≈0.2. Therefore, 0.01 eV ∼<ma∼<10 eV, 0.6×106GeV∼<fa∼<0.6×109GeV (14.37) are formally adopted as the SN 1987A excluded axion parameters. Axions on the “trapping side” of the SN argument can still be ex- cluded because they would have caused additional events in the IMB and Kamiokande water Cherenkov detectors. The excluded range of Eq. (13.2) translates into the approximate mass exclusion range of 20 eV−20 keV. These limits on the axion mass and decay constant are summarized in Fig. 14.5. The slanted end of the SN 1987A exclusion bar is a reminder of the potentially large uncertainty of this limit. Except for very special choices of model-dependent parameters axions with a mass above 0.01 eV are excluded. The high-mass end of the stellar exclusion bars in Fig. 14.5 has not been worked out in detail because of their overlap with laboratory limits. The globular cluster limits apply without modiﬁcation up to a mass of, say, 30 keV because the temperature in the cores of HB stars and red giants are about 10 keV. However, axions with masses in this range interact much more strongly than those at the low-mass end of the exclusion bar so that the Boltzmann suppression of the emission rate for a large mass is partly balanced by the increased coupling strength. 14.5 Cosmological Limits The stellar-evolution bounds on axions have received much attention because they push the Peccei-Quinn scale to such large values that axions appear to play a signiﬁcant cosmological role if they exist at all. Therefore, to place the stellar constraints into context it may be useful to close this chapter with a brief summary of the cosmological Axions 541 Fig. 14.5. Astrophysical and cosmological bounds on “invisible axions.” One globular-cluster limit (white exclusion bar) is based on the axion-electron coupling and thus applies only if axions are of the DFSZ type (cos2= 1 was used). For the coupling to photons = 1 was assumed. Slanted ends of exclusion bars indicate an estimated uncertainty of the bounds. The antic- ipated range of sensitivity of the Livermore and Kyoto search experiments are also indicated (Sect. 5.3). limits which I quote from my contribution to the axion session of the XVth Moriond Workshop Dark Matter in Cosmology, Clocks, and Tests of Fundamental Laws , (Villars-sur-Ollon, Switzerland, January 21–28, 1995). The proceedings of this meeting will provide many up-to-date accounts of diﬀerent aspects of the axion saga. If axions were suﬃciently strongly interacting ( fa∼<108GeV) they would have come into thermal equilibrium before the QCD phase tran- sition and so we would have a background sea of invisible axions in analogy to the one expected for neutrinos (Turner 1987). This pa- rameter range is excluded by the astrophysical arguments summarized in Fig. 14.5. Hence axions must be so weakly interacting that they have never come into thermal equilibrium. Still, the well-known mis- 542 Chapter 14 alignment mechanism will excite coherent oscillations of the axion ﬁeld (Abbott and Sikivie 1983; Dine and Fischler 1983; Preskill, Wise, and Wilczek 1983; Turner 1986). When the temperature of the universe falls below fathe axion ﬁeld settles somewhere in the brim of its Mex- ican hat potential. When the hat tilts at the QCD phase transition, corresponding to the appearance of a mass term for the axion, the ﬁeld begins to move and ﬁnally oscillates when the expansion rate of the uni- verse has become smaller than the axion mass. In units of the cosmic critical density one ﬁnds for the axionic mass density Ωah2≈0.23×10±0.6(fa/1012GeV)1.175Θ2 iF(Θi), (14.38) wherehis the present-day Hubble expansion parameter in units of 100 km s−1Mpc−1. The stated range reﬂects recognized uncertainties of the cosmic conditions at the QCD phase transition and uncertainties in the calculations of the temperature-dependent axion mass. The cosmic axion density thus depends on the initial misalignment angle Θ iwhich could have any value between 0 and π. The function F(Θi) encapsules anharmonic corrections to the axion potential for Θ ≫0. (For a recent analytic determination of Fsee Strobl and Weiler 1994.) The age of the universe indicates that Ω h2≈0.3, causing a problem with Ω = 1 models if his around 0.8 as indicated by recent measure- ments. For the present purpose I take Ω ah2= 0.3×2±1for axions which constitute the dark matter where the adopted uncertainty of a factor of 2 likely covers the whole range of plausible cosmological mod- els. Then, axions with ma=O(1µeV) are the cosmic dark matter if Θiis of order 1. Because the corresponding Peccei-Quinn scale of fa=O(1012GeV) is far below the GUT scale one may speculate that cosmic inﬂation, if it occurred at all, did not occur after the PQ phase transition. If it did not occur at all, or if it did occur before the PQ transition with Treheat>f a, the axion ﬁeld will start with a diﬀerent Θ iin each region which is causally connected at T≈faand so one has to average over all possibilities to obtain the present-day axion density. More importantly, because axions are the Nambu-Goldstone mode of a complex Higgs ﬁeld after the spontaneous breaking of a global U(1) symmetry, cosmic axion strings will form by the Kibble mechanism (Davis 1986). The motion of these global strings is damped primarily by the emission of axions rather than by gravitational waves. At the QCD phase transition, the U(1) symmetry is explicitly broken (axions acquire a mass) and so domain walls bounded by strings will form, get sliced up by the interaction with strings, and the entire string and domain wall system will quickly Axions 543 decay into axions. This complicated sequence of events leads to the production of the dominant contribution of cosmic axions. Most of them are produced near the QCD transition at T≈ΛQCD≈200 MeV. After they acquire a mass they are nonrelativistic or mildly relativistic so that they are quickly redshifted to nonrelativistic velocities. Thus, even the string and domain-wall produced axions form a cold dark matter component. In their recent treatment of axion radiation from global strings, Battye and Shellard (1994a,b) found that the dominant source of axion radiation are string loops rather than long strings, contrary to what was assumed in the previous works by Davis (1986) and Davis and Shellard (1989). At a given cosmic time tthe average loop creation size is parametrized as ⟨ℓ⟩=αtwhile the radiation power from loops isP=κµwithµthe renormalized string tension. The exact values of the parameters αandκare not known; the cosmic axion density is a function of the combination α/κ. Forα/κ < 1 the dependence of Ωah2onα/κ is found to be rather weak. Battye and Shellard favor α/κ≈0.1 for which Ω ah2= 18×10±0.6(fa/1012GeV)1.175, about an order of magnitude smaller than originally found by Davis (1986) and Davis and Shellard (1989). The overall uncertainty has the same source as in Eq. (14.38) above. With Ω ah2= 0.3×2±1the mass of dark-matter axions is found to be ma= 30−1000µeV; the cosmologically excluded range of axion masses is indicated in Fig. 14.5. These results are plagued with systematic uncertainties. Battye and Shellard (1994a,b) argue that the largest uncertainty was the impact of the backreaction of axion emission on the string network. (They believe that a current numerical study will allow them to pin down the parameter α/κto within, say, a factor of 2.) Further, Sikivie and his collaborators (Harari and Sikivie 1987; Hagmann and Sikivie 1991) have consistently argued that the motion of global strings was over- damped, leading to an axion spectrum emitted from strings or loops with a ﬂat frequency spectrum. In Battye and Shellard’s treatment, wavelengths corresponding to the loop size are strongly peaked, the motion is not overdamped. In Sikivie et al.’s picture, much more of the string-radiated energy goes into kinetic axion energy which is red- shifted so that ultimately there are fewer axions; it was argued that the cosmic axion density was then of order the misalignment contribu- tion. Therefore, following Sikivie et al. one would estimate the mass of dark-matter axions at about ma= 4−150µeV where the range reﬂects the same overall uncertainties that bedevil the Battye and Shellard estimate, or the misalignment contribution. 544 Chapter 14 While the cosmic axion bounds claimed by both groups of authors still diﬀer signiﬁcantly, the overall uncertainty within either scenario is larger than the mutual disagreement, i.e. the range of masses where axions could be the dark matter overlaps signiﬁcantly between the pre- dictions of the two groups (Fig. 14.5). Moreover, there remain diﬃcult to control uncertainties, for example, with the “dilute instanton gas” calculation of the temperature-dependent axion mass near the QCD phase transition. There may be other unaccounted systematic prob- lems which may increase the adopted uncertainty of the cosmic mass prediction which is represented in Fig. 14.5 by the slanted end of the cosmic exclusion bar. The astrophysical and cosmological limits on axions leave a narrow window of parameters where axions could still exist (Fig. 14.5); they would then be some or all of the dark matter of the universe. While the SN 1987A as well as the cosmological bound are each very uncertain, there is a recent trend toward an allowed range near ma=O(1 meV) where no current or proposed experimental eﬀort appears to be sensi- tive. Of course, the overall quantitative uncertainty of the predicted cosmic axion density is large, especially if one includes the possibility of late-time inﬂation after the Peccei-Quinn phase transition or late- time entropy production after the QCD phase transition. Therefore, the microwave cavity experiments (Sect. 5.3) in Livermore and Kyoto no doubt have a fair chance of detecting galactic axions if they are the dark matter. In September 1995, the Livermore search experiment has taken up operation (K. van Bibber, private communication). Chapter 15 Miscellaneous Exotica Stellar-evolution constraints on a variety of hypotheses are discussed and compared with limits from other sources. Speciﬁcally, a possible time variation of Fermi’s and Newton’s constant, the validity of the equivalence principle, a photon mass and charge, the existence of free quarks and supersymmetric particles, and the role of majorons and millicharged particles are considered. 15.1 Constancy of Fermi's Constant One of the basic physical assumptions commonly made in astrophysi- cal research is that the laws of nature are the same at diﬀerent places in the universe, and at earlier times here and elsewhere. Apparently this assumption has never been challenged seriously by any experiment or observation that would have indicated a spatial or temporal varia- tion of parameters such as particle masses or coupling constants. At the present time it is not known what ﬁxes the values of such “funda- mental numbers.” Therefore, the possibility that they vary in time or space cannot be a priori rejected. Notably, Dirac (1937, 1938) is often quoted for his speculation that the large value of some dimensionless numbers occurring in physics are related to variations of some physical constants on cosmological time scales. Whatever the merit of Dirac’s large numbers hypothesis, it remains an interesting task to isolate sim- ple observables that are sensitive to variations of certain “constants.” One instructive stellar-evolution example was discussed by Scherrer and Spergel (1993) who considered the constancy of Fermi’s constant GFwhich governs weak-interaction physics. The particle-physics stan- dard model gives G−1 F=p 2 Φ2 0in terms of the Higgs-ﬁeld vacuum expectation value Φ 0. Scherrer and Spergel noted that a nonconstant 545 546 Chapter 15 vacuum expectation value of a physical ﬁeld was more plausibly subject to variations than dimensionless constants such as gauge and Yukawa couplings. Type Ia supernovae are thought to represent the nuclear deﬂagration of a white dwarf pushed beyond its Chandrasekhar limit by accretion (e.g. Woosley and Weaver 1986b). Therefore, their light curves are very reproducible and indeed serve as standard candles in an attempt to improve measurements of the cosmic expansion rate. The shape of the lightcurve is determined by radioactive heating of the SN remnant by the decay of56Co; its lifetime is proportional to G−2 F. A change of order 10% in the slope of type Ia SN lightcurves in neighboring galaxies (Leibundgut et al. 1991) would be readily observable so that GFmust be constant within at least 5% over 30 Mpc distances. In addition, Scherrer and Spergel showed that the agreement be- tween the observed primordial light element abundances and big-bang nucleosynthesis calculations imply that GFlay within 1% and +9% of its standard value at this early epoch. A stringent constraint on the local93time variation of GFobtains from the analysis of ores from the Oklo uranium mine where a natu- ral ﬁssion reactor is thought to have operated about 2 Gyr ago (e.g. Maurette 1976). A shift of the ground state diﬀerence between150Sm and149Sm by more than 0 .02 eV is inconsistent with the isotope ra- tios at Oklo (Shlyakhter 1976, 1983). According to this author the contribution of the weak interaction to the nuclear binding energy is about 200 eV, excluding a change of GFby more than 0 .1% over the past 2 Gyr. 15.2 Constancy of Newton's Constant 15.2.1 Present-Day Constraints from Celestial Mechanics Another “constant of nature” that might vary in time is Newton’s con- stantGN. Indeed, there exist self-consistent alternative theories to general relativity which actually predict a temporal variation of GNon cosmological time scales—see Will (1993) for a summary of such the- ories and detailed references. A typical scale for the rate of change is the cosmic expansion parameter Hso that it is natural to write ˙GN/GN=σHwithσa dimensionless model-dependent number. Some 93The Earth moves with the galaxy and the local group relative to the cosmic microwave background (CMB). Taking 600 km s1for this peculiar velocity the Earth moves by about 1 :2 Mpc in 2 Gyr relative to a frame deﬁned by the CMB. Miscellaneous Exotica 547 Table 15.1. Bounds on the present-day ˙GN=GN. (Adapted from Will 1993.) Method ˙GN/GN References [10−12yr−1] Laser ranging (Moon) 010 M¨ uller et al. (1991) Radar ranging (Mars) 210 Shapiro (1990) Binary pulsar 1913+16 11 11 Damour and Taylor (1991) Spin-down PSR 0655+64 <55 Goldman (1990) recent discussions have also addressed the possibility of an oscillating GN(Hill, Steinhardt, and Turner 1990; Accetta and Steinhardt 1991) with a rate of change much faster than H. The following discussion does not generically address such extreme model assumptions. TheGNrate of change can be tested by a detailed study of the orbits of celestial bodies. Particularly precise data exist in the so- lar system from laser ranging of the moon and radar ranging of plan- ets, notably by the Viking landers on Mars. Very precise orbital data also exist beginning 1974 for the binary pulsar PSR 1913+16; among other things its orbital decay reveals the emission of gravitational ra- diation. A weaker but also less model-dependent bound can be de- rived from the spin-down rate of the pulsar PSR 0655+64. These present-day limits on ˙GN/GNare summarized in Tab. 15.1; altogether j˙GN/GNj<2010−12yr−1is probably a safe limit. The Hubble expan- sion parameter today is H0=h100 km s−1Mpc−1=h1.0210−10yr−1 (observationally 0 .4<h<1). Thus today jσj<0.2h−1. 15.2.2 Big-Bang Nucleosynthesis An interesting constraint on the value of GNin the early universe arises from the observed primordial light element abundances as ﬁrst dis- cussed by Barrow (1978).94In a Friedman-Robertson-Walker model of the universe the expansion rate is given by H2= (˙R/R)2=8 3GNρ in terms of the energy density ρwhich, during the epoch of nucleosyn- thesis, is dominated by radiation (photons, neutrinos). It is a standard 94Apparently there is an earlier discussion of this limit by G. Steigman in an unpublished essay for the 1976 Gravity Research Foundation Awards. Subse- quent reﬁnements include Rothman and Matzner (1982), Accetta, Krauss, and Romanelli (1990), Damour and Gundlach (1991), and Casas, Garc´ ıa-Bellido, and Quir´ os (1992). 548 Chapter 15 argument to constrain ρfrom the yield of4He and other light elements, and thus to constrain the eﬀective number of neutrino degrees of free- dom at nucleosynthesis (Yang et al. 1979, 1984; Olive et al. 1990; Walker et al. 1991). Because the number of low-mass sequential neutrino fam- ilies is now known to be 3, such constraints can be translated into constraints on the value of GNpertaining to the nucleosynthesis epoch. An extra neutrino species would add around 15% to ρ. It appears reasonably conservative to assume that big-bang nucleosynthesis does not allow for a deviation of the standard number of eﬀective neutrino degrees of freedom by more than 1 so that GNat that time must have been within about 15% of its present-day value. It is possible, however, that GNwas considerably smaller if this reduction was com- pensated by additional exotic degrees of freedom such as right-handed neutrinos which increase ρ. Moreover, it was assumed that Fermi’s constant had its present-day value at nucleosynthesis, contrary to the speculations discussed in Sect. 15.1. Still, barring fortuitous compen- sating eﬀects, nucleosynthesis excludes an O(1) deviation of GNfrom its standard value at nucleosynthesis. This sort of result can be compared with the present-day bounds of Sect. 15.2.1 only by assuming a speciﬁc functional form for GN(t) which is often taken to be GN(t) =GN(t0) (t0/t), (15.1) wheret0refers to the present epoch. Assuming that GNat nucleosyn- thesis was within 50% of its standard value one ﬁnds jβj<0.01 which would imply j˙GN/GNjtoday<10−12yr−1, at least a factor of ten below the present-day limits. One should keep in mind, however, that a power- law variation of GNis a relatively arbitrary assumption. For example, in scalar-tensor extensions of general relativity such as the Brans-Dicke theoryGNvaries as a power law during the matter-dominated epoch while it remains constant when radiation dominates. Either way, while the nucleosynthesis bounds are probably somewhat more restrictive than the celestial-mechanics ones it is interesting that the resulting bounds j˙GN/GNjtoday<11010−12yr−1are of the same general order of magnitude. They leave room for a considerable variation of GNover cosmic time scales. Miscellaneous Exotica 549 15.2.3 Properties of the Sun IfGNdid vary in time one would expect a modiﬁcation of the standard course of stellar evolution as ﬁrst stressed by Teller (1948). By means of a beautifully simple homology argument, Teller showed that the lu- minosity of the Sun is approximately proportional to G7 NM5withM the solar mass which was also allowed to vary in the spirit of Dirac’s (1937, 1938) large numbers hypothesis. Teller then proceeded to esti- mate the temperature on Earth in the past, taking a modiﬁcation of its orbit from the GNvariation into account. Depending on whether GN was larger or smaller in the past the average terrestrial surface tem- perature would have been larger or smaller. Teller estimated that if GNfell outside 10% of its standard value it would be unlikely that life on Earth could be sustained. Therefore, GNshould have remained constant to within this accuracy at least during the past 500 million years or more where life has been known to exist on Earth.95 Later, a similar homology argument based on the solar age was pre- sented by Gamow (1967). Detailed models of the Sun with a varying GNwere constructed by Pochoda and Schwarzschild (1964), Ezer and Cameron (1966), Roeder and Demarque (1966), Shaviv and Bahcall (1969), Chin and Stothers (1975, 1976), Demarque et al. (1994), and Guenther et al. (1995). The crux with constraining ˙GNfrom the Sun is that the presolar helium abundance Yinitial and the mixing-length pa- rameterαcan and must be tuned to reproduce the Sun’s present-day luminosity and radius. In Sect. 1.3.2 it became clear that even extreme anomalous energy-loss rates could be compensated by an adjustment ofYinitial; a similar eﬀect pertains to variable- GNsolar models. Even though the present-day central temperature, density, and helium abun- dance could diﬀer vastly from standard predictions, their main impact would be on the neutrino ﬂux which, however, is not a reliable probe of the solar central conditions as it may get modiﬁed by neutrino oscil- lations. At the present time the most sensitive probe of a variant internal solar structure is aﬀorded by the measured p-mode frequencies which agree well with standard predictions, especially when the gravitational settling of helium is taken into account. Demarque et al. (1994) have constructed solar models with a varying GNand then analyzed their p-mode spectra in comparison with the observations. They assumed a GNtime variation of the form Eq. (15.1) with t0= 15 Gyr for the age of 95Apparently there is more recent evidence for primitive lifeforms on Earth as early as 3 :5 Gyr ago (e.g. Gould 1994). 550 Chapter 15 Table 15.2. Characteristics of the Demarque et al. (1994) solar models with a varying GNaccording to Eq. (15.1). β α X initialXcTcρcRenv37Cl71Ga [%] [%] [106K] [g/cm3] [R⊙] [SNU] [SNU] 0.4 1.832 69.42 45.6 15.14 125.5 0.739 0.2 1.895 70.10 42.1 15.28 134.0 0.731 5.5 117 0.1 1.936 70.51 40.0 15.37 139.5 0.724 0.0 1.983 70.98 37.6 15.47 146.2 0.721 6.8 124 0.1 2.036 71.51 34.8 15.58 154.6 0.716 0.2 2.104 72.11 31.7 15.72 165.1 0.710 8.7 134 0.4 2.291 73.56 23.9 16.07 197.3 0.695 the universe. Some characterisitics of their solar models as a function of βare summarized in Tab. 15.2 where αis the mixing-length parameter, Xinitial the presolar hydrogen abundance, the quantitites with index c refer to central conditions of the present-day Sun, and Renvto the radius of its convective envelope. The last two columns are the predicted counting rates in the chlorine and gallium solar neutrino experiments under the assumption that there are no neutrino oscillations. The most important eﬀect of the GNvariation is a shift of the base of the convective envelope which is caused by the required change of α and the initial helium abundance Yinitial. It is this modiﬁcation of the convection zone which has the largest impact on the observable p-mode frequencies. However, an identical shift can be produced by other eﬀects such as modiﬁed opacities, equation of state, surface boundary condi- tions, degree of gravitational helium settling, and perhaps by magnetic ﬁelds. Therefore, only relatively crude limits can be extracted at the present time. Demarque et al. (1994) believe that jβj<0.4 is a rea- sonably conservative limit which probably can be improved by a factor of four within the next decade by more precise p-mode observations. With Eq. (15.1) the current limit corresponds to j˙GN/GNj<3010−12yr−1, (15.2) similar to the celestial-mechanics bounds of Tab. 15.1. The precise func- tional form Eq. (15.1) is not crucial for the solar bound as it probes GN only for the last 4 .5 Gyr of the assumed 15 Gyr cosmic age. Therefore, one could have equally assumed a linear form for GN(t). Miscellaneous Exotica 551 Most recently, a similar study was completed by Guenther et al. (1995) who focussed on the predicted g-mode spectrum. At the present time there is no generally accepted observation of solar g-modes (re- call that they are evanescent in the convection zone). If one were to take the claimed observations by Hill and Gu (1990) seriously, a bound jβj<0.05 would obtain. Therefore, if an unambiguous identiﬁcation of g-modes would emerge from a number of forthcoming observational projects, the Sun may yet provide one of the most restrictive limits on the constancy of Newton’s constant. 15.2.4 White Dwarfs A large impact of a time-varying gravitational constant can be expected on the oldest stars which “integrate” GN(t) into the more distant past than does the evolution of the Sun. One well understood case are white dwarfs, the faintest of which likely formed shortly after the birth of the galactic disk. Therefore, the age of the galactic disk implied by the fast drop of the white-dwarf luminosity function at the faint end (Sect. 2.2.1) depends on the GNevolution in the past. In an early study Vila (1976) concluded on the basis of the observa- tions then available that ˙GN/GNas large as 75 10−12yr−1was not ex- cluded. Garc´ ıa-Berro et al. (1995) constructed detailed luminosity func- tions under the assumption of a decreasing GN. For an assumed age of the galactic disk of 7 Gyr, which probably is a lower plausible limit, the best ﬁt for the faintest data point requires ˙GN/GN=1010−12yr−1 while the curves for 0 and 3010−12yr−1lie somewhat outside of the 1σerror bar of this all-important data point. Still, the white-dwarf luminosity function does not seem to yield signiﬁcant limits relative to the celestial-mechanics ones. 15.2.5 Globular Clusters The oldest stellar objects in the galaxy are globular-cluster stars which are thus expected to yield the most restrictive stellar-evolution limits on˙GN/GN. A color-magnitude diagram for an intermediate-aged galac- tic cluster was constructed by Roeder (1967) while detailed studies of globular clusters were performed by Prather (1976) and Degl’Innocenti et al. (1995). Roeder (1967) and Prather (1976) used a time variation for a speciﬁc Brans-Dicke cosmology where GNdecreases approximately as in Eq. (15.1) with β0.03 while Degl’Innocenti et al. (1995) con- sidered more generic cases of GN(t). 552 Chapter 15 The main impact of the assumed GNvariation is a change in the time it takes for a star to burn out hydrogen at its center and thus to leave the main sequence (MS). The subsequent fast evolution (ascend- ing the RGB, HB evolution, etc.) is determined by the present-day value ofGN. The stellar evolutionary tracks in the color-magnitude di- agram can look signiﬁcantly diﬀerent from the standard ones if the GNvariation was suﬃciently severe. However, within the range of possibilities left open by the above ˙GN/GNbounds, the present-day isochrone cannot be observationally distinguished from the standard case (Degl’Innocenti et al. 1995). Apparently, then, the only signiﬁ- cant consequence of a time-varying gravitational constant is that the true ageτof a globular cluster is diﬀerent from its apparent age τ∗ which is inferred from its color-magnitude diagram in the framework of a constant-gravity scenario. The change of the MS lifetime can be estimated by Teller’s (1948) homology relation L/G N. Based on a speciﬁc assumption for the opacity variation with temperature and density Teller found γ= 7 while a more appropriate value for low-metallicity globular-cluster stars isγ= 5.6 (Degl’Innocenti et al. 1995). Either way, one can easily show that the true ( τ) and apparent age ( τ∗) at the MS turnoﬀ are approximately related by τ∗=∫t0 t0−dt[ GN(t)/GN(t0)] (15.3) (Prather 1976; Degl’Innocenti et al. 1995). For all practical purposes this analytic result can be considered to be exact because it agrees with numerical calculations surprisingly well. Unless one wishes to probe the very early universe it is fairly generic to assume a linear GNvariation of the form GN(t) =[ 1 + Γ 0(tt0)] GN(t0), (15.4) where Γ 0=˙GN(t0)/GN(t0) is the present-day rate of change of Newton’s constant. Then one ﬁnds explicitly τ τ∗=γ1Γ0τ 1(1Γ0τ) 1=1(1γ1Γ0τ∗)1= 1 Γ0τ∗, (15.5) whereγ1γ+ 1. Given a present-day rate of change Γ 0one can thus determine the modiﬁcation of the globular-cluster age if a certain apparent age or a certain true age is assumed. The observed color-magnitude diagrams of globular clusters yield apparent ages τ∗in the range 14 to 18 Gyr. With these values one Miscellaneous Exotica 553 Fig. 15.1. Required present-day ˙GN=GNin order to achieve a true globular- cluster age , given that the apparent age is ∗. A linear GN(t) variation as in Eq. (15.4) was assumed. can relate a desired true age τto a required value for Γ 0(Fig. 15.1). Conversely, it is probably safe to assume that the true ages of globular clusters do not exceed 20 Gyr. Then Fig. 15.1 implies that today ˙GN/GN<710−12yr−1. (15.6) A lower age limit is less certain. Taking 8 Gyr one ﬁnds ˙GN/GN> 3510−12yr−1which is less certain, and also less interesting relative to the limits discussed in the previous sections. Fig. 15.2. Summary of limits on the present-day ˙GN=GN. The big-bang nucleosynthesis limit is not shown as it depends sensitively on the assumed GN(t) variation at early cosmic times. 554 Chapter 15 Actually, a certain reduction of the true globular-cluster ages rela- tive to their apparent ones would be a welcome cosmological eﬀect as they are, at best, marginally compatible with other cosmic age indica- tors. In view of the current limits on ˙GN/GNsummarized in Fig. 15.2 this possibility cannot be excluded at present. 15.3 Test of the Equivalence Principle The equivalence principle of Einstein’s general theory of relativity im- plies that the space-time trajectories of relativistic particles should be independent of internal degrees of freedom such as spin or ﬂavor, and independent of the type of particle under consideration (photons, neu- trinos). A number of astronomical observations allow one to test this prediction. Laboratory tests of various consequences of the equivalence principle are discussed in Will’s (1993) book. Nonsymmetric extensions of general relativity (e.g. Moﬀat 1991) predict that diﬀerent polarization components of electromagnetic waves propagate with diﬀerent phase velocities in gravitational ﬁelds. This birefringence eﬀect would lead to the depolarization of the Zeeman components of spectral lines emitted in magnetically active regions of the Sun. The absence of this depolarization eﬀect leads to signiﬁcant constraints on Moﬀat’s theory and others (Gabriel et al. 1991). In a similar approach one uses the diﬀerence of the Shapiro time delay between diﬀerent particles or between diﬀerent polarization states of a given particle which propagate through the same gravitational ﬁeld. In Sect. 13.3 the absence of an anomalous shift between the SN 1987A photon and neutrino arrival times gave limits on violations of the equivalence principle because both pulses moved through the same galactic gravitational potential. Also, one may search for diﬀerences in the arrival times of left- and right-handed polarized electromagnetic signals from distant pulsars (LoSecco et al. 1989). The best bound was obtained from an analysis of the pulse arrival times from PSR 1937+21 which is about 2 .5 kpc away from Earth. One may write the eﬀective gravitational potential in the formV(r) =V0(r) [1 +A1ˆr+A2v+A3ˆr(v)] where r,v, and represent the location, velocity, and spin of the particles (photons, neutrinos). The PSR 1937+21 data then yield a constraint jA1j< 410−12andjA2j<110−12(Klein and Thorsett 1990), apparently the most restrictive limits of their kind. Miscellaneous Exotica 555 A violation of the equivalence principle could also manifest itself by a relative shift of the energies of diﬀerent neutrino ﬂavors in a gravita- tional ﬁeld. For a given momentum pthe matrix of energies in ﬂavor space (relativistic limit) is E=p+M2/2p+ 2pϕ(r)(1 +F) whereM2 is the squared matrix of neutrino masses, ϕ(r) is the Newtonian gravi- tational potential, and Fis a matrix of dimensionless constants which parametrize the violation of the equivalence principle; in general rela- tivityF= 0. A nontrivial matrix Fcan lead to neutrino oscillations in analogy to the standard vacuum oscillations which are caused by the matrixM2(Gasperini 1988, 1989; Halprin and Leung 1991; Pantaleone, Halprin, and Leung 1993; Iida, Minakata, and Yasuda 1993; Minakata and Nunokawa 1995; Bahcall, Krastev, and Leung 1995). Values for Fijin the general 10−1410−17range could account for the solar neu- trino problem and perhaps could be probed with future long-baseline oscillation experiments. 15.4 Photon Mass and Charge Even though in classical electrodynamics gauge invariance implies that photons must be massless, quantum electrodynamics (QED) can be formulated consistently with the inclusion of a photon mass, and the limitm !0 takes the modiﬁed theory smoothly over to massless QED (St¨ uckelberg 1941). Therefore, the possibility of a small photon mass cannot be excluded theoretically; limits must be set by laboratory and astrophysical methods. A still up-to-date review of the laboratory lim- its was given by Goldhaber and Nieto (1971); the best is m <10−14eV from a test of Coulomb’s law (a photon mass would modify the inverse- square behavior). More recent experiments worked at low temperature (Ryan, Accetta, and Austin 1985; Chernikov et al. 1992); the resulting limits onm are relatively weak, however. An astrophysical limit may be set by the absence of an anomalous dispersion of photon signals from distant sources, notably the pulsed signal from radiopulsars. This method is limited by the presence of the ionized interstellar medium. It causes a dispersion relation for photons which mimics the eﬀect of a photon mass m =ωPwhere the plasma frequency is given by ω2 P= 4παn e/m e. With a typical electron density neof order 0.1 cm−3the photon plasma mass is of order 10−11eV so that a vacuum mass much smaller than this value cannot be probed. In Sect. 13.3.3 a limit on a hypothetical νecharge was derived from the absence of a dispersion of the SN 1987A neutrino pulse. The path of 556 Chapter 15 charged particles in the galactic magnetic ﬁeld would be curved, lead- ing to an energy-dependent time-delay (Barbiellini and Cocconi 1987). Because the same argument can be applied to photons, the signals from radio pulsars also allow one to set a limit on a putative photon electric charge (Cocconi 1988). However, the resulting dispersion eﬀect scales with photon frequency in the same way as the eﬀect caused by a photon mass or by the plasma eﬀect so that this method, again, is limited by the standard dispersion eﬀect (Raﬀelt 1994). One ﬁnds a bound on the photon charge of Q <10−29e. Returning to a hypothetical photon mass, its value can be extracted, in principle, from the spatial distribution of static magnetic ﬁelds of celestial bodies. The measured ﬁelds can be ﬁtted by an appropriate multipole expansion in which m is kept as a free parameter. The most restricitve limit of this sort was derived from Jupiter’s magnetic ﬁeld on the basis of the Pioneer-10 observations; Davis, Goldhaber, and Nieto (1975) found a limit m <0.610−15eV. The same method applied to the Earth’s magnetic ﬁeld yields an almost equivalent bound ofm <0.810−15eV (Fischbach et al. 1994). As detailed in a review by Barrows and Burman (1984) more restric- tive limits obtain from detailed considerations of astrophysical objects in which magnetic ﬁelds, and hence the Maxwellian form of electrody- namics, play a key role in maintaining equilibrium or creating long-lived stable structures. The most restrictive such limit of m <10−27eV is based on an argument by Chibisov (1976) concerning the magneto- gravitational equilibrium of the gas in the Small Magellanic Cloud which requires that the range of the interaction exceeds the charac- teristic ﬁeld scale of about 3 kpc. This limit, if correct, is surprisingly close to 10−33eV where the photon Compton wavelength would exceed the radius of the observable universe and thus would cease to have any observable consequences. 15.5 Free Quarks It is thought that quarks cannot exist as free particles; they always occur bound in hadrons which are neutral (“white”) with regard to the “color charge” of the strong interaction. In order to test this hypoth- esis of conﬁnement it remains an important task to search for single quarks. The observation of fractional charges in the experiment of LaRue, Phillips, and Fairbanks (1981) has never been conﬁrmed. How- ever, if their observations were caused by unconﬁned quarks bound to Miscellaneous Exotica 557 nuclei it would correspond to an abundance of one quarked nucleus (Q-nucleus) in 6 1017normal ones. A small abundance of Q-nuclei could signiﬁcantly alter the stel- lar thermonuclear reaction chains and among other eﬀects change the solar neutrino predictions (Boyd et al. 1983). Detailed nuclear reac- tion chains involving Q-nuclei were studied by Boyd et al. (1985). For strangelets (lumps of strange quark matter) trapped in stars the nuclear networks were investigated by Takahashi and Boyd (1988); the eﬀect of strangelets is similar to that of Q-nuclei. The Q-nuclear reactions were implemented in a stellar evolution code by Joseph (1984). Predictions for the solar neutrino ﬂux were worked out by Sur and Boyd (1985). With a Q-nuclear abundance of order 10−15the modiﬁed reaction chains would compete with the standard ones. In the Sun one could achieve a reduction of the high-energy solar neutrino ﬂux and thus solve the “old solar neutrino problem” (missing boron neutrinos). However, Sur and Boyd (1985) predict an increase of the low-energy ﬂux, corre- sponding to a substantially increased counting rate at the gallium solar neutrino experiments. As this contradicts the ﬁndings of SAGE and GALLEX (Sect. 10.3) one concludes that Q-nuclear burning is not the answer to the solar neutrino problem. Turning the SAGE/GALLEX observations around one concludes that in the Sun the abundance of Q-nuclei is below about 10−15. 15.6 Supersymmetric Particles Supersymmetric extensions of the particle-physics standard model are very popular, among other reasons because the lightest supersymmet- ric particle (LSP) could play the role of the cosmic dark matter. In these models, there is a fermionic partner to all standard bosons, and a bosonic partner to all standard fermions. The supersymmetric part- ners of the photon, the Z◦gauge boson, and the neutral Higgs bo- son (photino, Zino and Higgsino) would be Majorana fermions. They would be very much like Majorana neutrinos except that their interac- tion strength is not ﬁxed by the Fermi constant but rather depends on details of the supersymmetric models. If these “neutralinos” had low enough masses they would be pro- duced in the interior of stars by the same processes that create neu- trinos, except that the coupling strength has to be adjusted according to the particular model that one has in mind. Limits to an anomalous energy loss of stars yielded early constraints on supersymmetric models 558 Chapter 15 (Bouquet and Vayonakis 1982; Fukugita and Sakai 1982; Anand et al. 1984). The neutrino burst of SN 1987A yielded more interesting limits for the case of low-mass photinos (Ellis et al. 1988; Grifols, Mass´ o, and Peris 1989; Grifols and Mass´ o 1990b). To avoid that too much energy is carried away by photinos they inferred that squark masses in the approximate range 60 GeV to 2 .5 TeV were excluded. Low-mass neutralinos are disfavored by laboratory limits. More- over, if the LSP plays the role of cold dark matter its mass likely is above several 10 GeV. In this case the stellar energy-loss arguments would not yield any constraints as all supersymmetric particles would be too heavy to be emitted. Stars would still play an interesting role as they could trap the dark-matter particles. Their annihilation in the Sun or Earth would lead to a high-energy neutrino signal which has been constrained by the Kamiokande detector (Mori et al. 1992). It may well be found at the Cherenkov detectors Superkamiokande, NESTOR, DUMAND, or AMANDA and thus lead to the indirect discovery of par- ticle dark matter in the galaxy. These important issues are discussed at length in the forthcoming review Supersymmetric Dark Matter by Jungman, Kamionkowski, and Griest (1995). 15.7 Majorons 15.7.1 Particle-Physics and Cosmological Motivations Axions (Chapter 14) are one representative of a variety of Nambu- Goldstone bosons of spontaneously broken global symmetries that have appeared in the literature over the years. Another widely discussed ex- ample are the majorons ﬁrst introduced by Chicashige, Mohapatra, and Peccei (1981) as a scheme to generate small neutrino Majorana masses. An important variation by Gelmini and Roncadelli (1981) and Georgi, Glashow, and Nussinov (1981) led to a model where neutri- nos had small Majorana masses and coupled to the massless majoron (a pseudoscalar boson like the axion) with a relatively large Yukawa strength. The main phenomenological interest in this sort of conjec- ture lies in the intriguing possibility that neutrinos could have relatively strong interactions with the majorons and with each other by virtue of majoron exchange. As majorons would not necessarily show up in in- teractions with ordinary matter one could well speculate that neutrinos might have “secret interactions” which would be of relevance only in a neutrino-dominated environment such as the early universe, perhaps the present-day universe if neutrinos have a cosmologically signiﬁcant Miscellaneous Exotica 559 mass, and in supernovae. Another motivation to consider majoron models is the possibility to account for fast neutrino decays in order to avoid cosmological neutrino mass bounds. In the laboratory, majorons could show up in experiments searching for neutrinoless 2 βdecays. The main motivation for the introduction of majorons is the puz- zling smallness of neutrino masses (if they have nonvanishing masses at all) relative to other fermions. As outlined in Sect. 7.1, in the particle- physics standard model it is thought that all Dirac fermions acquire a mass by their interaction with a background Higgs ﬁeld which takes on a classical value (vacuum expectation value) Φ 0everywhere; neutrino masses could well arise in the same fashion, except that the Yukawa couplings to the Higgs ﬁeld would have to be extremely small. Alter- natively, one may speculate that neutrino masses are so small because they arise in a diﬀerent fashion. Notably, the known sequential neu- trinosνe,ν, andνcould well be Majorana fermions, i.e. their own antiparticles so that the νeis really equivalent to a helicity-plus νe. As long as neutrinos are massless this picture is equivalent to an inter- pretation where the standard left-handed neutrinos are the two active components of a four-component Dirac spinor while the two remain- ing sterile components would never have been observed because they do not interact. With a nonvanishing mass these interpretations are vastly diﬀerent because helicity ﬂips in collisions would allow one to produce the (almost) sterile “wrong-helicity” Dirac components. This possibility was exploited in Sect. 13.8.1 to set bounds on a neutrino Dirac mass from the SN 187A neutrino signal. For Majorana neu- trinos, a helicity-ﬂipping collision takes an active νeinto an active νe, thus violating lepton number by two units. Therefore, Majorana masses could not arise from the coupling to the standard Higgs ﬁeld which is lepton-number conserving. Majorana masses could arise, however, by interacting with a dif- ferent Higgs ﬁeld which would develop a vacuum expectation value by virtue of the usual spontaneous breakdown of a global symmetry. The resulting Nambu-Goldstone boson is the majoron. (Recall that the Nambu-Goldstone boson of the standard Higgs ﬁeld shows up as the third polarization degree of the massive Z◦gauge boson so that there is no massless Nambu-Goldstone degree of freedom in the standard model.) In the original model of Chicashige, Mohapatra, and Peccei (1981), the “singlet majoron model,” the new vacuum expectation value was considered to be much larger than the standard Φ 0250 GeV. Large masses would be given primarily to sterile neutrinos postulated to exist; the standard sequential neutrinos would obtain their small 560 Chapter 15 masses by a see-saw type mixing eﬀect with the heavy states. The ma- joron coupling to standard neutrinos would be extremely small in this model, leading to no interesting consequences besides small Majorana masses forνe,ν, andν. Gelmini and Roncadelli (1981) and Georgi, Glashow, and Nussinov (1981) suggested instead to do away with the unobserved heavy sterile neutrinos and give a small Majorana mass directly to the sequential neutrinos by the interaction with the new Higgs ﬁeld. The intriguing feature of this model is that it requires a very small vacuum expectation valuev, perhaps in the keV regime. As all couplings of the new Higgs ﬁeld to fermions scale with the inverse of v, the majoron would have a rather strong coupling to neutrinos. Among many fascinating phe- nomenological and astrophysical consequences (e.g. Georgi, Glashow, and Nussinov 1981; Gelmini, Nussinov, and Roncadelli 1982) this model predicted, however, that the new Higgs ﬁeld should contribute precisely the equivalent of two massless neutrino species to the Z◦decay width. The measurements of this width at CERN and SLAC in 1989 1990, however, correspond exactly to the known three neutrino ﬂavors (Parti- cle Data Group 1994), leaving no room for this “triplet majoron model.” Other “doublet majoron models” which would contribute one-half of an eﬀective neutrino species to the Z◦decay width are also excluded (for references see, e.g. Berezhiani, Smirnov, and Valle 1992). It is possible, however, to construct majoron models for Majorana neutrino masses which retain the original idea of Chicashige, Mohapatra, and Peccei (1981) and yet provide large majoron-neutrino couplings (e.g. Berezhi- ani, Smirnov, and Valle 1992 and references therein; see also Burgess and Cline 1994a; Kikuchi and Ma 1994, 1995). The main motivation for going out of one’s way to construct such models does not arise from particle theory but rather from experi- ments and astrophysics. In Sect. 7.1.4 it was outlined that those nuclei which decay predominantly by a double beta channel (emission of 2 e− and 2νe) can also decay in a neutrinoless mode if νehas a Majorana mass, allowing an emitted νeto be eﬀectively reabsorbed as a νe. In majoron models of Majorana neutrino masses there is a third decay channel where the intermediate νein the 0νmode radiates a majoron so that eﬀectively 2 e−plus one majoron χare emitted. The expected sum spectrum of the electron energies would be continuous as in the 2νmode, but with a diﬀerent spectral shape. Once in a while, ex- periments which search for the 0 νmode (a sharp endpoint peak of the 2e−sum spectrum) have reported a continuous spectral signature which allegedly could not be ascribed to the dominant 2 νmode or other Miscellaneous Exotica 561 backgrounds, although such claims have tended to disappear with the collection of more signiﬁcant data. If interpreted in terms of an up- per limit to the majoron- νeYukawa coupling, current experiments give aboutg<210−4(Beck et al. 1993 and references therein). At the present time there does not appear to exist a compelling signature for majorons in any of the experiments, although several of them seem to ﬁnd certain spectral anomalies—see, e.g. Burgess and Cline (1993, 1994b) for an overview and references. If the spectral anomalies were to represent the ﬁrst evidence of ma- joron emission in 2 βdecays, the Yukawa coupling would have to be near the 10−4level. The neutrino-majoron coupling is given as g=m/vin terms of the neutrino mass and the symmetry breaking scale v. With me<1 eV for a Majorana mass from measured limits on the 0 νde- cay mode one ﬁnds the requirement v<10 keV which is an extremely small scale of symmetry breaking. The majoron is the “angular degree of freedom” of a complex scalar ﬁeld; the “radial degree of freedom,” often referred to as the ρﬁeld, has a mass typically of order v. There- fore, a low-mass scalar particle beyond the majoron would appear in the low-energy sector of the theory. One severe limitation on such models is provided by big-bang nu- cleosynthesis which has been widely used to set limits on additional low-mass degrees of freedom which are thermally excited during the epoch of nucleosynthesis; an upper limit of about 0 .3 is often quoted as the maximum allowed extra contribution in units of eﬀective neutrino degrees of freedom (e.g. Walker et al. 1991). On the face of it, this limit excludes majoron models where the majoron (and possibly the ρ) interact suﬃciently strongly with neutrinos to reach thermal equilib- rium. In one recent study Chang and Choi (1994) found that one must requireg<10−5for the largest majoron Yukawa coupling to any neu- trino species in order to avoid thermalization of the majorons before nucleosynthesis. Such constraints rely on the assumption of 3 standard light neu- trino species being in thermal equilibrium at the epoch of nucleosyn- thesis. However, in majoron models the usual cosmological neutrino mass bound does not apply because of the possibility of fast decays of the typeν!ν′χ(majoronχ) which are induced by ﬂavor oﬀ-diagonal Yukawa couplings. Therefore, ν’s may have a mass of, say, a few MeV and may have disappeared by decays and annihilations before nu- cleosynthesis, thus making room for majorons; for detailed numerical studies see Kawasaki et al. (1994) and papers quoted there. Such heavy, short-lived ν’s may provide interesting eﬀects on scenarios of galaxy 562 Chapter 15 formation and thus could be a novel ingredient for cold dark matter cosmological models (Dodelson, Gyuk, and Turner 1994). Even if in the long run the 2 βexperiments do not yield any com- pelling evidence for majoron emission, one may consider a decaying- neutrino cosmology as a motivation in its own right for majoron mod- els. Such cosmologies may explain the discrepancy between the cosmic density ﬂuctuation spectrum inferred from the cosmic microwave back- ground and from galaxy correlations which persists in a purely cold dark matter cosmology (Bond and Efstathiou 1991; Dodelson, Gyuk, and Turner 1994; White, Gelmini, and Silk 1995). Another neutrino- related explanation of this discrepancy is a hot plus cold dark matter cosmology which involves neutrinos with a mass of a few eV. In summary, the simplest majoron model which implied large cou- plings to neutrinos (the Gelmini-Roncadelli model) is experimentally excluded although one can construct more complicated ones which re- tain sizeable neutrino-majoron couplings and yet are compatible with theZ◦decay width. The possibility of such models is entertained be- cause 2βdecay experiments may yet turn up compelling evidence for majoron decays, and because certain cosmological models of structure formation may be taken to suggest massive, decaying neutrinos. 15.7.2 Majorons and Stars Majorons could also have an impact on stellar evolution. Besides in- teracting with neutrinos, they typically also couple to other fermions, allowing one to apply the astrophysical bounds on pseudoscalars de- rived throughout this book to majoron models. Because of the possibility of fast decays ν!ν′χthe neutrino signal from distant sources, notably from the Sun or from SN 1987A, would be aﬀected. Such decays are not likely to be able to explain the so- lar neutrino problem as discussed in Sect. 10.8. It remains interesting, however, that the matter-induced νe-νeenergy splitting allows for de- caysνe!νeχ(Sect. 6.8). Should a solar νeﬂux show up in future measurements, it could be an indication for such decays. As for SN neutrinos, the decay of massive ν’s orν’s with ﬁnal- stateνe’s could modify the νesignal observed in a detector, notably the energy distribution and duration of the observed pulse (Soares and Wolfenstein 1989; Simpson 1991). The SN 1987A data have not been analyzed in detail with regard to this possibility, although the observed signal can be accounted for without invoking such eﬀects. It is not clear if one could derive signiﬁcant constraints on majoron models from Miscellaneous Exotica 563 SN 1987A on the basis of this decay argument. Aharonov, Avignone, and Nussinov (1988a) predicted for certain parameters a dramatic in- crease of the number of observable events from the prompt νeburst. One interesting SN 1987A limit is based on the interaction of the pulse of observed νe’s with the cosmic majoron background that would be expected to exist for majorons which thermalized in the early uni- verse. In order not to deplete the pulse too much by collisions, Kolb and Turner (1987) found a certain upper limit on the majoron-neutrino Yukawa coupling. Unfortunately, their bound was based on an in- correct cross section for the process νχ!νχ. They used a pseu- doscalar coupling of the form igψγ5ψχrather than a derivative cou- pling (1/2v)ψγγ5ψ∂χwhereg=m/vandvis the majoron sym- metry breaking scale. As discussed in Sect. 14.2.3, in processes which involve two Nambu-Goldstone bosons attached to one fermion it is mandatory to use the derivative coupling in order to obtain the correct interaction rate. The pseudoscalar coupling yields a scattering cross section (g4/64π)s−1times an expression of order unity which depends on the neutrino mass and s, the squared CM energy. Put another way, the cross section is ( m/v)4s−1times numerical factors. The derivative coupling, on the other hand, leads to a cross section m2 /v4times numer- ical factors (Choi and Santamaria 1990). Therefore, Kolb and Turner’s (1987) bound translates approximately into g(eV/m )1=2<310−4. It is not a bound on galone. Typical energies of neutrinos and other particles which prevail in the interior of a SN core are in the range of tens to hundreds of MeV. In majoron models with a symmetry breaking scale below this range, the symmetry may be restored in the interior of the star, and both com- ponents of the complex Higgs ﬁeld will be thermally excited. Quick deleptonization may be achieved by reactions involving the majoron ﬁeld, leading to a high-entropy collapse. Substantial majoron emission may shorten the SN 1987A νesignal too much, although decays of heav- ier neutrinos in ﬂight could, perhaps, provide the late events observed in the detectors. SN scenarios involving majorons were discussed by a number of authors.96There is little doubt that majoron models will have an important impact on SN physics for Yukawa couplings some- where in the range 10−610−3. From the available literature, however, 96Kolb, Tubbs, and Dicus (1982); Dicus, Kolb, and Tubbs (1983); Manohar (1987); Fuller, Mayle, and Wilson (1988); Aharonov, Avignone, and Nussinov (1988b, 1989); Choi et al. (1988); Grifols, Mass´ o, and Peris (1988); Konoplich and Khlopov (1988); Dicus et al. (1989); Berezhiani and Smirnov (1989); Choi and Santamaria (1990). 564 Chapter 15 the present author has not been able to develop a clear view of the precise range of parameters that can be ruled out or ruled in by the SN 1987A neutrino signal.97 15.8 Millicharged Particles It is commonly assumed that all particles have charges in multiples of 1 3e(electron charge); notably neutrinos are thought to be electrically neutral. While gauge invariance and anomaly cancellation constraints pose limits on the possible charge assignments in the standard model, electric charge quantization is not entirely assured. Two of the three neutrino species may carry small charges if one gives up the assumption that the three fermion families diﬀer only in the mass of their mem- bers (Takasugi and Tanaka 1992; Babu and Volkas 1992; Foot, Lew, and Volkas 1993). Moreover, if one allows for charge nonconservation, all neutrinos could have small charges (Babu and Mohapatra 1990; Maruno, Takasugi, and Tanaka 1991). Finally, the existence of novel particles with small electric charges is possible and actually motivated by certain models involving a “mirror sector” where the mirror symme- try is slightly broken (Holdom 1986; see also Davidson, Campbell, and Bailey 1991). Therefore, it is interesting to study the experimental, as- trophysical, and cosmological bounds on the existence of particles with small electric charge.98 The most severe constraints obtain for nonstandard charge assign- ments in the ﬁrst family of quarks and leptons. The most model- independent charge limit on νewas derived from the absence of an anomalous dispersion of the SN 1987A neutrino signal (Sect. 13.3.3) 97An incomplete list of issues that ought to be considered in a study of majorons in SNe are the following. If the symmetry is broken within the SN core, a derivative majoron coupling to neutrinos should be used instead of a pseudoscalar one (Choi and Santamaria 1990). For light neutrinos, the medium-induced dispersion rela- tion may dominate the cross section result. Besides the medium-induced processes !, the process !should be included (Sect. 6.8). The decay of heavy neutrinos outside of the SN could contribute to the measurable signal. The trapping of neutrinos and majorons due to reactions with each other should be properly un- derstood along the lines discussed by Dicus et al. (1989); it is dubious, for example, that a process like !really contributes to the majoron opacity. The eﬀect of majorons during the infall phase must be understood, especially the possibility of early deleptonization. 98Such studies were performed by Dobroliubov and Ignatiev (1990), Davidson, Campbell, and Bailey (1991), Babu and Volkas (1992), Mohapatra and Nussinov (1992), and Davidson and Peskin (1994). Miscellaneous Exotica 565 which led to (Barbiellini and Cocconi 1987; Bahcall 1989) ee<310−17e. (15.7) If electric charge conservation is assumed to hold in βprocesses such as neutron decay, one ﬁnds a more restrictive limit of ee<310−21e. (15.8) It is based on a limit for the neutron charge of en= (0.41.1)10−21e (Baumann et al. 1988) and on the neutrality of matter which was found to beep+ee= (0.80.8)10−21eassuming a vanishing neutron charge (Marinelli and Morpurgo 1984). The deﬂection of charged neutrinos in the toroidal magnetic ﬁeld in the solar convection zone would modify the observable ﬂux at Earth (Ignatiev and Joshi 1994, 1995). However, in view of the above limits an unrealistically large ﬁeld gradient is required to obtain signiﬁcant ﬂux modiﬁcations. Babu and Volkas (1992) derived a limit on the νelectric charge from the measured νecross section which would receive a contribution from photon exchange, e<10−9e. (15.9) Similar limits could be derived for νe. The following arguments apply to millicharged neutrinos or novel particles alike. They would appear as virtual states in higher-order am- plitudes. For example, they would contribute to the anomalous mag- netic moment of electrons and muons, and to the Lamb shift between the 2P 1=2and 2S 1=2states of the hydrogen atom. Of these quantitities, the Lamb shift gives the most restrictive limit (Davidson, Campbell, and Bailey 1991), ex<0.11em x/MeV, (15.10) whereexandmxare the charge and mass of the millicharged particle, respectively. This result applies to mx>1 keV. Davidson, Campbell, and Bailey (1991) have reviewed more restric- tive bounds from a host of accelerator experiments (Fig. 15.3). A simple astrophysical constraint is based on avoiding excessive en- ergy losses of stars which can produce millicharged particles by various reactions, most notably the plasma decay process. To avoid an unac- ceptable delay of helium ignition in low-mass red giants, and to avoid an 566 Chapter 15 Fig. 15.3. Summary on limits on the electric charge exand mass mxof generic millicharged particles which may be sequential neutrinos or novel particles. (Adapted from Davidson, Bailey, and Campbell 1991.) In order to avoid overclosing the universe, additional model-dependent parameter regions are excluded. The big-bang nucleosynthesis (BBN) excluded region is larger in some models. undue shortening of the lifetime of horizontal-branch stars, one needs to require (Sect. 6.5.6) ex<210−14e. (15.11) It applies for mx<ωP/2 withωPthe plasma frequency. It is larger for red giants before helium ignition than for HB stars because of the larger average density of 2 105g cm−3which corresponds to ωP8.6 keV. The white-dwarf luminosity function yields about the same limit. Ifexexceeds around 10−8ethe mean free path of the millicharged particles will be less than the physical size of a white dwarf or red- giant core. For larger exthe particles will be trapped and contribute to the transfer of energy. Their impact on stellar evolution will become negligible when exis so large that other forms of energy transfer (photon radiation, convection) are more important. Because the new particles act essentially as radiation their mean free path must be less than that of photons, or very crudely, their charge must be of order an electron charge. (The main opacity source is probably Coulomb scattering on charged particles.) Therefore, if their mass is below a few keV, even the properties of the Sun would imply that there is not an allowed range of largeexon the trapping side of the red-giant argument, except perhaps forexso large that it is excluded by experimental arguments. Miscellaneous Exotica 567 For a narrow range of charges, these limits can be extended to larger masses by the SN 1987A cooling argument (Sect. 13.8.4). Accordingly, 10−9e<ex<10−7e (15.12) is excluded for mxup to several MeV, perhaps up to 10 MeV. On the trapping side of this range (large ex) these particles surely would have an important impact on SN physics even though they cannot be excluded on the basis of the simple cooling argument. If millicharged particles reach thermal equilibrium in the early uni- verse before nucleosynthesis they contribute to the energy density and thus to the expansion rate. If they are one of the sequential neutrinos, this means that the right-handed degrees of freedom of that species are excited (they must be Dirac particles!), adding an eﬀective neu- trino degree of freedom. If they are nonneutrinos, even more energy is contributed, depending on their spin degrees of freedom. Even one additional eﬀective neutrino degree of freedom is excluded and so for any millicharged particle Davidson, Campbell, and Bailey (1991) found ex<310−9e (15.13) ifmx<1 MeV. In certain models where the millicharged particles are associated with a shadow sector, more stringent limits apply (Davidson and Peskin 1994). Additional regions in the mass-charge plane can be excluded by the requirement that the novel objects do not overclose the universe. However, these arguments depend on the annihilation cross section in the early universe so that one needs to know all of their interactions apart from the millicharge. It is hard to imagine novel particles which interact only by their small electric charge! For certain speciﬁc cases the excluded regime was derived by Davidson, Campbell, and Bailey (1991) and Davidson and Peskin (1994). All of these constraints leave the possibility open that ν’s have a mass in the 1 24 MeV range and a charge in the 10−510−3erange. Then they would annihilate suﬃciently fast before nucleosynthesis to actually reduce their eﬀective contribution to the expansion rate (Foot and Lew 1993). In the standard model with small neutrino charges, however, twosequential neutrino species must carry a millicharge of equal but opposite magnitude (Babu and Volkas 1992; Takasugi and Tanaka 1992). Because the large charges required for Foot and Lew’s scenario are excluded for νeandνone would need to require that only νcarries a relatively large charge, forcing one to espouse even more exotic particle-physics models. Chapter 16 Neutrinos: The Bottom Line Most of the particle-physics arguments discussed in this book are closely related to neutrino physics because these particles play an important role in stellar evolution whether or not they have nonstandard prop- erties. Besides a summary of some recent developments of standard- neutrino astrophysics, a synthesis is attempted of what stars as neutrino laboratories have taught us about these elusive objects, and what one might reasonable hope to learn in the foreseeable future. 16.1 Standard Neutrinos The main theme of this book has been an attempt to extract infor- mation about the properties of neutrinos and other weakly interact- ing particles from the established properties of stars. However, even standard-model neutrinos (massless, no mixing, no exotic properties) play a signiﬁcant role in astrophysics. There have been some recent de- velopments in “standard-neutrino astrophysics” which deserve mention in a summary. It is now thought that neutrinos play an active role in supernovae besides carrying away the binding energy of the newborn neutron star (Chapter 11). In the delayed-explosion scenario they are crucial to revive the stalled shock wave which is supposed to expel the stellar mantle and envelope. Moreover, they have a strong impact on r-process nucleosynthesis which is thought to occur in the high-entropy region above the neutron star a few seconds after collapse. For both purposes it is crucial to calculate the SN “neutrino lightcurve” for the ﬁrst seconds after collapse. Convection below the neutrino sphere and large-scale convective turnovers in the region between the neutron star and the shock wave are both important and need to be understood better on 568 Neutrinos: The Bottom Line 569 the basis of 2- and 3-dimensional hydrodynamic calculations which are only beginning to appear in the literature. In addition, however, the neutrino opacities must be calculated with greater reliability. In Chapter 4 the weaknesses of a naive calculation of the axial-vector opacities have been amply demonstrated. In my opinion, a far better understanding of the interaction rates of neutrinos with a hot nuclear medium is required before one can calculate a SN neutrino lightcurve (notably the duration of Kelvin-Helmholtz cooling) with a reasonable precision. Neutrinos presumably play a key role for the self-acceleration of neu- tron stars which are observed to have huge “kick velocities” (Sect. 11.5). The required anisotropic neutrino emission of 1 2% may be caused by temperature ﬂuctuations on the neutrino sphere (convection!), or by magnetic-ﬁeld induced anisotropies of the neutrino opacities. Of course, other phenomena may be responsible for anisotropic neutrino emission and for the kick velocities. For the ﬁrst time ever neutrinos have been observed from a collaps- ing star (SN 1987A), conﬁrming the expected signal behavior within the large uncertainties caused by the small number of observed events (Chapter 11). Neutrinos are routinely observed from the Sun—currently in four diﬀerent detectors with three diﬀerent spectral response characteristics (Chapter 10). At least three further detectors will soon take up opera- tions. The measured solar neutrino spectrum diﬀers signiﬁcantly from theoretical predictions. This “solar neutrino problem” has no obvious “astrophysical solution” that would involve plausible variations of in- put parameters such as nuclear cross sections or photon opacities. Still, the gallium detectors SAGE and GALLEX have for the ﬁrst time mea- sured the dominant low-energy ﬂux of ppneutrinos, conﬁrming that the Sun cannot be completely diﬀerent at its center from what had been thought. There is also some more benign news. The interest of some authors to apply the methods of ﬁnite temperature ﬁeld theory to astrophysical problems has led to a new formulation of the photon dispersion relation in a plasma (Sect. 6.3). It helped to correct some “fossilized errors” in the literature on plasma neutrino emission (see also Appendix C.1). It may well be worthwhile to scrutinize other standard aspects of stellar- evolution input physics that involve subtle dispersion or screening ef- fects. 570 Chapter 16 16.2 Minimally Extended Standard Model 16.2.1 Cosmological Mass Limit for All Flavors Neutrinos with nonstandard properties would have more radical impli- cations in astrophysics. A minimal and most plausible extension of the standard model is the possibility that they have masses and mixings like the other fermions. Taking this hypothesis in a literal sense means that neutrinos would need to have Dirac masses like the charged fer- mions. Therefore, one needs to postulate the existence of right-handed neutrinos and Yukawa couplings to the standard Higgs ﬁeld to generate masses and mixings. What do we know about neutrinos in the context of this Minimally Extended Standard Model? Perhaps the most dramatic lesson is that the mass of all sequen- tial neutrinos ( e,,) must be less than the cosmological limit of approximately 30 eV (Sect. 7.1.5). This conclusion is not to be taken for granted as massive neutrinos with mixings can decay by virtue of !′ , and by !ee+e−ifmexceeds about 2 me. While the standard-model radiative decays are too slow to avoid the cosmological limit, the e+e−channel can be fast on cosmological time scales if the mixing angle is not too small. This decay channel is an option only forwhich may have a mass of up to 24 MeV while the experimental mass limits on the other ﬂavors are below 0 :16 MeV. Such a “heavy” , however, can be excluded by several arguments. The stellar evolution one based on SN arguments was presented in Sect. 12.5.2. The bremsstrahlung emission of photons in !ee+e− would produce a -ray ﬂux in excess of the SMM limits for SN 1987A unless sin22e3<10−9. (For a detailed dependence of this limit on the assumed mass see Fig. 12.18.) In addition, the integrated positron ﬂux from all galactic supernovae over the past, say, 100,000 years would exceed the observed value unless the decays are very fast (near the SN) or very slow (outside of the galactic disk). Together, these limits leave no room for a heavy (Fig. 12.19). In addition, the mass range 0:5 MeV <m<35 MeV can be excluded on the basis of big bang nucleosynthesis arguments (Fig. 7.2) unless the neutrinos are shorter lived than permitted by the Minimally Extended Standard Model. Neutrino masses near the cosmological limit would be important for cosmology as they could contribute some or all of the dark matter of the universe. The latter option is disfavored by theories of structure formation. However, a subdominant “hot dark matter” contribution in the form of, say, 5 eV neutrinos might be cosmologically quite welcome. Neutrinos: The Bottom Line 571 Neutrino masses in this general range are also accessible to time- of-ﬂight measurements. The neutrino burst of SN 1987A has already provided a limit of me<20 eV (Sect. 11.3.4), a result which remains interesting in view of the confusing situation with the tritium de- cay endpoint experiments which seem to be plagued by systematic ef- fects which cause the appearance of a negative neutrino mass-square (Sect. 7.1.3). The observation of the prompt eburst from a future galactic SN would allow one to reduce this limit to a few eV. More- over, one may well be able to detect or constrain a ormass in the 1020 eV range, assuming the simultaneous operation of a water Cherenkov detector such as Superkamiokande and a neutral-current de- tector such as the proposed Supernova Burst Observatory (Sect. 11.6). 16.2.2 Oscillations of Solar Neutrinos Small neutrino masses or rather, small neutrino mass diﬀerences can have dramatic consequences if neutrinos also mix; this is expected in the present scenario. Neutrino oscillations then lead to the possibility that a diﬀerent neutrino ﬂavor is measured in a detector than was produced in the source. The observed characteristics of the solar neutrino ﬂux strongly suggest that neutrino oscillations may in fact be occurring. In terms of the mass diﬀerence and mixing angle between eandor there remain three solutions which account for the presently available data from the chlorine (Homestake), gallium (GALLEX and SAGE), and Cherenkov (Kamiokande) experiments (Tab. 16.1). In view of pos- sible systematic uncertainties concerning such quantities as the solar opacities and the p7Be cross section the values of the favored mixing angles can be somewhat diﬀerent from those shown in Tab. 16.1. How- ever, the required mass diﬀerences remain rather stable against large nonstandard modiﬁcations of the solar model. Table 16.1. Approximate neutrino parameters which explain all current solar neutrino observations in the framework of standard solar model assumptions. Solution ∆m2[eV2] sin22 Large-angle MSW 2 10−50.6 Nonadiabatic MSW 0 :610−50.006 Vacuum oscillations 0 :810−100:81 572 Chapter 16 The main aspect of the current situation is that there does not seem to be a simple “astrophysical solution” to reconcile the solar source spectrum with the measured ﬂuxes in experiments with three diﬀerent spectral response characteristics. Even allowing for large modiﬁcations of the p7Be cross section or the solar central temperature does not yield consistency unless one stretches the experimental uncertainties of the ﬂux measurements beyond reasonable limits. Still, a ﬁnal verdict on the question of solar neutrino oscillations can be expected only from the near-future experiments Superkamiokande, SNO, and BOREXINO as discussed in Chapter 10. 16.2.3 Oscillation of Supernova Neutrinos Naturally, neutrino oscillations would also aﬀect the characteristics of SN neutrinos which have been observed only from SN 1987A, although it is not unrealistic to hope for the observation of a galactic supernova at Superkamiokande or SNO within, say, a decade of operation. In wa- ter Cherenkov detectors, the main signal of SN neutrinos is thought to be from the ep!ne+reaction, although the angular characteristics of the SN 1987A observations do not square well with this assumption ex- cept that there is no convincing alternate interpretation (Sect. 11.3.5). The MSW solution in the Sun requires a “normal” mass hierarchy with ebeing dominated by the smaller mass eigenstate. In this case reso- nant oscillations do not occur among the ’s. Still, for large-angle vacuum oscillations such as those corresponding to the solar vacuum solution, the signal in the IMB and Kamiokande detectors would have been “hardened” by the partial swap of, say, the ewith the spectrum. It is not entirely obvious from the current literature if this eﬀect is ruled in or ruled out by the SN 1987A obser- vations (Sect. 11.4.3). If the mixing angle of ewith both andis small (sin22<0:1) there is no impact on the eSN signal. Still, the prompt eburst could be aﬀected even for small mixing angles because of the possibility of resonant oscillations (Sect. 11.4.2) and so the observation of a future galactic supernova could serve to measure this eﬀect. If one were to contemplate more general neutrino mass matrices with an “inversion” so that eis not dominated by the lowest-mass eigenstate, there could be resonant oscillations in the sector which would then lead to dramatic modiﬁcations of the SN signal in a large range of masses and mixing angles. This possibility has not been ex- plored much in the literature. Neutrinos: The Bottom Line 573 Apart from the detector signal, the oscillation of SN neutrinos can have important implications for SN physics itself because of the swap of, say, the ewith the spectrum which is much harder. For a mass diﬀerence corresponding to the cosmologically interesting range of a few to a few tens of eV, resonant oscillations could occur so close to the neu- trino sphere that the “crossover” point is within the stalling shock wave in the delayed-explosion scenario. The eﬀective hardening of the e spectrum would then enhance the neutrino energy transfer to the shock wave, thus helping to explode supernovae (Sect. 11.4.4). Conversely, a few seconds after collapse the same eﬀect would drive the hot wind proton rich which is driven from the surface of the compact remnant. This eﬀect would prevent the occurrence of r-process nucleosynthesis which requires a neutron-rich environment (Sect. 11.4.5). Interestingly, because the remnant is more compact at “late” times (few seconds after collapse), the adiabaticity condition can be met only for relatively large mixing angles. Thus, there is a plausible range of neutrino parameters where oscillations may help to explode supernovae, and still r-process nucleosynthesis may proceed undisturbed (Fig. 11.20). At any rate, it is impossible to ignore neutrino oscillations for SN physics if neutrino masses happen to lie in the cosmologically interesting range. 16.2.4 Electromagnetic Properties In the Minimally Extended Standard Model neutrinos have magnetic and electric diagonal and transition moments which are proportional to their assumed masses (Sect. 7.2.2). Because of the cosmological mass limit these quantities are so small that they do not seem to be important anywhere. However, one may toy with the idea that neutrinos actually carry small electric charges, a possibility that is not entirely excluded by the structure of the Standard Model if one gives up the notion that the second and third particle families are exact replicas of the ﬁrst except for the masses (Sect. 15.8). The possible magnitude of a echarge is limited by ee<310−17e from the absence of an anomalous dispersion of the SN 1987A neu- trino burst (Sect. 13.3.3). All neutrino charges are limited by e< 210−14efrom the absence of anomalous cooling of globular-cluster stars (Sect. 6.5.6). For andthe cosmological mass limit is crucial for this bound because their emission would be suppressed by threshold eﬀects if their mass exceeded the relevant plasma frequency of a few keV. The assumption of charge conservation in decay yields a more restrictive limit ee<310−21(Sect. 15.8). 574 Chapter 16 Either way, possible charges of all neutrinos must be so small that the quantization of charge is very accurately realized among the fermi- ons of the Standard Model. Therefore, charge quantization is probably exact so that the neutrino electric charges vanish exactly, those of the quarks are exactly1 3eand2 3e, respectively. 16.3 New Interactions 16.3.1 Majorana Masses Because of the cosmological limit all neutrino masses are found to be so small relative to those of the corresponding charged fermions (Fig. 7.1) that it is hard to maintain the pretense that neutrinos are essentially like the other fermions. In this sense the assumption of a Minimally Extended Standard Model is self-defeating. One reaction may be to return to the assumption of massless two-component neutrinos. It re- mains to be seen for how much longer this option remains viable in view of the expected progress in solar neutrino astronomy and, perhaps, in laboratory experiments (including atmospheric neutrino observations). They may soon yield unrefutable evidence for neutrino oscillations. Another reaction is to embrace the notion of neutrinos being very diﬀerent from the charged leptons with a possible wealth of novel and unexpected properties. The most benign assumption is to maintain the notion of only two neutrino components per family which are charac- terized by a Majorana mass term which may arise by new physics at some large energy scale. In this case all that was said about neutrino masses and oscillations in the previous section remains applicable. 16.3.2 \Heavy" Neutrinos and Fast Decays If one postulates novel neutrino interactions one may speculate about the possibility of neutrino decays which proceed faster than in the Min- imally Extended Standard Model. Besides accelerated radiative decays one may speculate about “fast invisible decays” of the form !′′′′′′ or!′where is some new boson such as the majoron (Sect. 15.7). In this case one can escape the cosmological mass limit. In fact, such fast-decaying neutrinos can be a welcome feature of theories for the formation of structure in the universe (Sect. 7.1.5). “Heavy” ’s or ’s are not in obvious conﬂict with normal stellar evolution even though their emission could now be suppressed by the mass threshold. Thermal production of neutrinos from a stellar plasma Neutrinos: The Bottom Line 575 is dominated by the plasmon decay process !for a large range of stellar conditions (Appendix C). This process occurs mostly by the vector-current coupling to electrons of the medium. Because the weak mixing angle has the special value sin2ΘW= 0:231 4the vector- current coupling of andto electrons nearly vanishes (Appendix B) and so !andplays near to no role anyway—in this sense stars are “blind” to the issue of heavy ormasses. However, assuming that the heavy neutrinos are Dirac particles one can limit their mass by a SN argument: The energy loss by right-handed states produced in spin-ﬂip collisions must not be too large, yielding a limit of m<30 keV (Sect. 13.8.1). In addition, these neutrinos have very high energies, typical of SN core temperatures, and so their decays might produce high-energy daughter e’s (100200 MeV) which were not observed from SN 1987A (Sect. 13.8.1). Even Majorana neutrinos would not be harmless for SN physics if they would mix suﬃciently strongly with e. In this case they would eﬀectively participate in equilibrium (Sect. 9.5) so that the spin- ﬂip scattering of, say, would eﬀectively lead to ’s and thus to deleptonization without the need of transporting lepton number to the stellar surface. In addition, even though heavy Majorana neutrinos would be emit- ted from the neutrino sphere so that their decays would not produce high-energy daughter products, the decays could still add to the de- tectable signal. It is not obvious from the existing literature which (if any) range of masses and decay times is ruled out or ruled in by the SN 1987A observations (Sect. 13.2.2). If the fast decays were due to some sort of majoron model, large “secret” neutrino-neutrino interactions are conceivable, possibly in con- junction with a small vacuum expectation value of a new Higgs ﬁeld so that the symmetry may be restored in a SN core. While a substantial body of literature exists on this sort of scenario (Sect. 15.7.2) I believe that in this context the story of SN physics would have to be rewritten more systematically than has been done so far. However, there appears to be little doubt that for neutrino-majoron Yukawa couplings in excess of about 10−5one would expect dramatic modiﬁcations of the transport of energy and lepton number. In summary, “heavy” neutrinos with fast invisible decays are a way to circumvent the cosmological mass limit, and may indeed be desirable in certain scenarios of cosmic structure formation. Depending on their detailed properties they could have a substantial impact on SN physics and the signal observable in a detector. It is diﬃcult, however, to 576 Chapter 16 state general constraints as one may easily postulate, for example, that the decays do not involve ﬁnal-state e’s, that even “wrong-helicity” Dirac neutrinos are trapped in a SN core by novel interactions, or that heavy Majorana ’s have only negligible mixings with e. Surely other loopholes could be found. 16.3.3 Electromagnetic Properties a) Spin and Spin-Flavor Oscillations If one contemplates neutrino interactions beyond the Minimally Ex- tended Standard Model, neutrino dipole and transition moments no longer need to be small. In particular, they do not need to be pro- portional to the neutrino masses; one example are left-right symmetric models where even massless neutrinos would have large dipole moments (Sect. 7.3.1). Dipole and transition moments can lead to spin or spin- ﬂavor oscillations in external magnetic ﬁelds, they allow for spin-ﬂip scattering on charged particles, for the plasmon decay !in stars, and for radiative decays !′ . One motivation for studying neutrino dipole moments is the ap- parent ﬂux variability of the solar neutrino signal in the Homestake detector. It anticorrelates with solar magnetic activity too closely to blame it comfortably on a statistical ﬂuke (Sect. 10.4.3). The only physical explanation put forth to date is that of Voloshin, Vysotski˘ ı, and Okun of a partial depletion of left-handed (measurable) neutrinos by spin or spin-ﬂavor oscillations. Unfortunately, the required value for B(neutrino dipole moment , magnetic ﬁeld Bin the solar convec- tion zone) exceeds by about two orders of magnitude what is allowed by typical models of the solar magnetic ﬁeld and by limits on . There- fore, this scenario appears to be in big trouble. Still, if one ignores the limits or speculates about large convection-zone magnetic ﬁelds one may ﬁt all currently available solar neutrino data (Sect. 10.7). Spin or spin-ﬂavor oscillations can be very important in and near the cores of supernovae where ﬁelds of order 1012G exist, and perhaps pockets with much larger ﬁelds. If neutrinos are Dirac particles so that their spin-ﬂipped (right-handed) states are sterile, the combination of spin-ﬂip scattering on charged particles and the magnetic spin oscilla- tion in large-scale magnetic ﬁelds can lead to nonlocal modes of energy transfer where energy can be deposited in one region that was depleted from a distant other region. Thus, energy transfer could no longer be treated with simple diﬀerential equations which involve local gradients Neutrinos: The Bottom Line 577 of temperature and lepton number. (Of course, a nonlocal energy trans- fer mechanism is already thought to be important for reviving the shock wave in the delayed-explosion scenario.) As far as I know, nothing more quantitative than back-of-the-envelope estimates of this scenario exist in the literature. Therefore, alleged bounds of order 10−12B(Bohr magneton B=e=2me) on Dirac-neutrino dipole moments probably have to be used with some reservation (Sect. 13.8.3). Conversely, such dipole moments may actually help to explode supernovae. With regard to the observable neutrino signal, spin and spin-ﬂavor oscillations both in the SN and in the galactic magnetic ﬁeld may cause vast modiﬁcations of the eﬂuxes and spectra observable in water Che- renkov detectors if neutrinos have dipole moments in the ballpark of 10−1210−14B. Spin and spin-ﬂavor oscillations can be very important in the early universe where strong magnetic ﬁelds may exist, and where a popu- lation of the r.h. degrees of freedom would accelerate the expansion rate of the universe. These issues are being investigated in the current literature; ﬁnal conclusions do not seem to be available at the present time. Still, it appears that this eﬀect may well be the most signiﬁcant impact of small Dirac neutrino dipole or transition moments anywhere in nature. b) Laboratory Limits Less problematic bounds on neutrino dipole moments arise from labo- ratory experiments where one studies the recoil spectrum of electrons in the reaction +e!e+′where ′can be the same or a diﬀerent ﬂavor (Sect. 7.5.1). A sensitivity down to, perhaps, as low as 10−11B can be expected from a current eﬀort involving reactor neutrinos as a source (MUNU experiment). Current limits on dipole or transition moments are about 2 10−10Bifeis involved, and about 7 10−10B ifis involved. For transition moments, these limits are subject to the assumption that there is no cancellation between a magnetic and an electric dipole scattering amplitude. c) Huge Dipole Moments or Millicharges In principle, the possibility of a large diagonal moment for remains open as it has not been possible to produce a strong source in the lab- oratory so that only extremely crude limits exist on the -e-scattering cross section. The globular-cluster bounds discussed below do not 578 Chapter 16 apply to a “heavy” so that one is confronted with a nontrivial al- lowed region in -mspace where even MeV masses become cos- mologically allowed because of the dipole-induced annihilation process !e+e−(in the ballpark of 10−7B). Of course, such large dipole moments must be caused by a fairly nontrivial arrangement of intermediate charged states and so one may wonder if a large mag- netic moment could be realistically the only manifestation of these new particles and/or interactions. Still, a large dipole moment and the correspondingly large an- nihilation cross section in the early universe is one possibility to toler- ate a large mass without the need for fast decays—such a particle could be entirely stable. Another similar possibility is that has a “huge millicharge” in the neighborhood of 10−510−3e. Such a scheme would require the violation of charge conservation as the possibility of neutrino charges within a simple extension of the Standard Model dis- cussed above always gives charges to two neutrino ﬂavors; for eor the required value is not tolerable (Sect. 15.8). For the issues of stellar evolution, the only conceivable consequence of such large electromagnetic interaction cross sections would be a reduced contribution to the energy transfer in SNe because of the reduced mean free path. In the study discussed in Sect. 13.6 one should have included the possibility of only two eﬀective ﬂavors! Still, there is little doubt that large cross sections could be accommodated in what one knows about SNe today. d) Astrophysical Bounds on Dipole and Transition Moments For all neutrinos with a mass below a few keV a very restrictive limit on dipole or transition magnetic or electric moments arises from the ab- sence of anomalous neutrino emission from the cores of evolved globular- cluster stars, notably of red-giant cores just before helium ignition (Sect. 6.5.6). One ﬁnds a limit <310−12Bwhich applies to Dirac and Majorana neutrinos, and which does not allow for a destructive interference between electric and magnetic amplitudes. Neutrino transition moments would reveal themselves by radiative decays. Because the decay rate involves a phase-space factor m3 this method is suitable only for large masses. In the cosmologically allowed range with m<30 eV, the only radiative limit which can compete with the globular-cluster bound is from the cosmic diﬀuse background radiations (Fig. 12.21). In fact, Sciama has proposed a scheme where a 28:9 eV neutrino with a radiative decay time corresponding to a tran- Neutrinos: The Bottom Line 579 sition moment of 0 :610−14Bplays a signiﬁcant cosmological role (Sect. 12.7.1). While this scenario is probably excluded it highlights the possibility of interesting cosmological eﬀects for radiatively decay- ing neutrinos in the range allowed by the globular-cluster bound. Radiative decay limits which are based on astronomical decay paths suﬀer from the uncertainty of other invisible decay channels which may compete with the radiative mode. Limits based on the cosmic back- ground radiations (Fig. 12.20) imply that in a large range of cosmolog- ically allowed neutrino masses and lifetimes the dominant decay chan- nel must be nonradiative. Therefore, one should use the cosmic back- ground radiations to derive limits on the branching ratio. One could then construct a contour plot of the excluded transition moments in them-plane. From the SN 1987A radiative lifetime limits I have constructed such a plot in Fig. 12.17. For large dipole moments in excess of, say, 10−10Bthese bounds are not self-consistent because neutrinos would be trapped too strongly by electromagnetic scatterings. How- ever, the laboratory limits exclude large transition moments. More- over, the globular-cluster bound yields more restrictive limits if mis less than a few keV. However, for relatively large neutrino masses, and for lifetimes not so short that the decays would have occurred within the progenitor star, SN 1987A yields the most restrictive limits on tran- sition moments. Naturally, one must assume that orwere actually emitted with about the standard ﬂuxes. Trapping eﬀects by additional new interactions could circumvent this assumption. 16.3.4 Summary Neutrinos with nonstandard interactions may well saturate the experi- mental mass limits, and may have a variety of novel properties. How- ever, it is nearly impossible to derive generic constraints on quanti- ties like magnetic transition moments without specifying an underlying particle physics model. Many constraints, notably those related to SN 1987A or to cosmology, can be circumvented by postulating suf- ﬁciently bizarre neutrino properties. Therefore, it is probably more important to know the arguments that can serve to learn something about neutrinos in astrophysics than it is to know a list of alleged lim- its. If a concrete conjecture turns up, or a speciﬁc theoretical model needs to be constrained, one can easily go through the list of arguments and check if they apply or not. Perhaps this book can be of help at this task. Appendix A Units and Dimensions In the astrophysical context, frequently occurring units of length are centimeters, (light) seconds, light years, and parsecs. Conversion fac- tors are given in Tab. A.1. For example, 1 pc = 3 :26 ly. Using both centimeters and (light) seconds as units of length implies a system of units where the speed of light cis dimensionless and equal to unity. In this book I always use natural units where Planck’s constant ¯ h and Boltzmann’s constant kBare also dimensionless and equal to unity. This implies that (length)1, (time)1, mass, energy, and temperature can all be measured in the same unit by virtue of x=ct,E=mc2, E= ¯h!,!= 2=t, and E=kBT. In Tab. A.2 conversion factors are given. For example, 1 K = 0 :862104eV or 1 erg = 0 :9481027s1. The most confusing aspect of natural units is that of an electromag- netic ﬁeld strength. The square of a ﬁeld strength is an energy density (erg=cm3) which, in natural units, is (energy)4or (length)4. Thus, an electric or magnetic ﬁeld may be measured, for example, in eV2or cm2. In natural units, electric charges are dimensionless numbers. However, there is a general ambiguity in the deﬁnition of charges and ﬁeld strengths because only their product (a force on a charged particle) is operationally deﬁned. All physical quantities stay the same if the charges are multiplied with an arbitrary number and the ﬁeld strengths are divided by it. However, the ﬁne-structure constant 1=137 is dimensionless in all systems of units, and its value does not depend on this arbitrary choice. If eis the charge of the electron one has =e2=4in the rationalized system of (natural) units which is always used in modern works on ﬁeld theory, and is used throughout this book. The energy density of an electromagnetic ﬁeld is then1 2(E2+B2). In the older literature and some texts on electromagnetism, unrationalized units are used where =e2and the energy density is ( E2+B2)=8. 580 Units and Dimensions 581 In the astrophysical literature the cgs system of units is very popular where magnetic ﬁelds are measured in Gauss (G). Confusingly, this system happens to be an unrationalized one. Field strengths given in Gauss can be translated into our rationalized natural units by virtue of 1 G!√ 1 erg=cm3 4= 1:953102eV2= 0:502108cm2;(A.1) where I have converted erg and cm1into eV according to Tab. A.2. The energy density of a magnetic ﬁeld of strength 1 G is, therefore, 1 2(1:953102eV2)2= 1:908104eV4= 3:979102erg cm3= (1=8) erg cm3. For a further discussion of electromagnetic units see Jackson (1975). It is sometimes useful to measure very strong magnetic ﬁelds in terms of a critical ﬁeld strength Bcritwhich is deﬁned by the condition that the quantum energy corresponding to the classical cyclotron fre- quency ¯ h(eB=m ec) of an electron equals its rest energy mec2so that in natural units Bcrit=m2 e=e: (A.2) Note that the Lorentz force on an electron in this ﬁeld is proportional toeBcritso that the electron charge cancels. Hence, Eq. (A.2) is the same in a rationalized or unrationalized system of units. In our ratio- nalized units e=p 4= 0:303 so that Bcrit= (0:511 MeV)2=0:303 = 0:8621012eV2which, with Eq. (A.1), corresponds to 4 :4131013G, in accordance to what is found in the literature (M´ esz´ aros 1992). Magnetic dipole moments of electrons and neutrinos are usually discussed in terms of Bohr magnetons Be=2me. For particle elec- tric dipole moments, on the other hand, one commonly uses 1 ecm as a unit. The conversion is achieved by 1 ecm = (2 mecm) ( e=2me) = 5:181010B. 582 Appendix ATab. A.1. Conversion factors between diﬀerent units of length. cm s ly pc cm 1 0:33410101:0610180:3251018 s 2:99810101 0:3171070:973108 ly 0:94610183:1561071 0.307 pc 3:0810181:0281083.26 1 Tab. A.2. Conversion factors in the system of natural units. s1cm1K eV amuaerg g s11 0:33410100:76410110:65810150:70710241:05510271:1731048 cm12:99810101 0.2289 1:9731052:11810143:16110170:3521037 K 1:31010114.369 1 0:8621040:92610131:38110161:5371037 eV 1:51910150:5071051:1601041 1:0741091:60210121:7831033 amu 1:41510240:47210141:08110130:9311091 1:4921031:6611024 erg 0:94810270:31610170:72410160:62410120:6701031 1:1131021 g 0:85210482:84310370:65110370:56110330:60210240:89910211 aAtomic mass unit. Appendix B Neutrino Coupling Constants Neutrinos can interact with other fermions and with each other by the exchange of WorZbosons. Because the astrophysical phenomena relevant for this book take place at very low energies compared with the WorZmass, one may always use an eﬀective four-fermion coupling which is parametrized in terms of the Fermi constant and the weak mixing angle GF= 1:16610−5GeV−2; sin2ΘW= 0:23250:0008: (B.1) The tree-level relationship of these quantities with the gauge-boson masses is p 2GF= m2 Wsin2ΘW= m2 Zsin2ΘWcos2ΘW; (B.2) wheremZ= 91:2 GeV and mW= 80:2 GeV. The eﬀective charged-current interaction between nucleons and lep- tons is written in the form Hint=GFp 2 p µ(CVCA 5) n ℓ µ(1 5) νℓ; (B.3) where the jare the proton, neutron, charged-lepton, and the corre- sponding neutrino ﬁeld. The vector-current coupling constant is CV= 1 while the axial-vector coupling for free nucleons is CA= 1:26. However, in large nuclei this value is suppressed somewhat, and the commonly used value for nuclear matter is CA= 1:0 (e.g. Castle and Towner 1990). This quantity would be relevant, for example, for reactions in super- nova cores and neutron stars. 583 584 Appendix B The eﬀective charged-current interaction between charged leptons and their own neutrinos, e.g. between e−ande, is written in the same form withCV=CA= 1. By virtue of a Fierz transformation it is brought into the form of a neutral current (see below). Neutral-current interactions between a neutrino and a fermion f are written in the form Hint=GFp 2 f µ(CVCA 5) f ν µ(1 5) ν: (B.4) Iffis the charged lepton corresponding to , there is a contribution withCV=CA= 1 from a Fierz-transformed charged current. The compound eﬀective CV’s andCA’s for various combinations of fand are given in Tab. B.1. (Note that the jCV,Ajfor neutral currents are typically1 2, a factor which is sometimes pulled out front so that the global coeﬃcient is GF=2p 2 while the couplings are then twice those of Tab. B.1.) For neutrinos interacting with neutrinos of the same ﬂavor a factor 2 for an exchange amplitude for identical fermions was included. TheCA’s for nucleons were thought to be given by isospin invariance to be1:26=2. However, because of the strange-quark contribution to the nucleon spin there is an isoscalar piece as well giving rise to the values shown in Tab. B.1—for a discussion and references to the original literature see Raﬀelt and Seckel (1995). Moreover, in a nuclear medium a certain suppression is expected to occur. In analogy to the charged- current couplings Raﬀelt and Seckel (1995) suggested the values Cp A 1:09=2 andCn A 0:91=2. Table B.1. Neutral-current couplings for the eﬀective Hamiltonian Eq. (B.4) in vacuum. Fermionf Neutrino CV CAC2 VC2 A Electron e +1 2+ 2 sin2ΘW +1 20.9312 0.25 µ,τ 1 2+ 2 sin2ΘW 1 20.0012 0.25 Proton e,µ,τ +1 22 sin2ΘW+1:37=2 0.0012 0.47 Neutron e,µ,τ 1 21:15=2 0.25 0.33 Neutrino (a)a +1 +1 1 1 b̸=a +1 2+1 20.25 0.25 Appendix C Numerical Neutrino Energy-Loss Rates In normal stars with densities below nuclear there are four main reac- tions that contribute to the energy loss by neutrino emission: pl! Plasma process, e!e Photoneutrino process, e+e! Pair annihilation, e(Z; A)! (Z; A)eBremsstrahlung.(C.1) Individual processes dominate in the regions of density and temperature indicated in Fig. C.1. In the following, various analytic ﬁt formulae for these neutrino emission rates are reviewed. C.1 Plasma Process Widely used formulae for the plasma process are those of Beaudet, Pet- rosian, and Salpeter (1967), Munakata, Kohyama, and Itoh (1985), and Schinder et al. (1987), which all agree with each other to better than 1% if the same eﬀective coupling constants are used. All of these rates are poor approximations for T<108Kwhich is relevant for low-mass stars because they were optimized for higher temperatures. Itoh et al. (1989) have attempted to improve the accuracy at low temperatures, and Blin- nikov and Dunina-Barkovskaya (1994) gave rates which were optimized for low-mass stars but fail for temperatures above about 108K. At high temperatures and densities, a poor approximation to the photon dis- persion relation was used in all of these works (Braaten 1991) whence none of these rates are satisfactory. A new ﬁt by Itoh et al. (1992) still contains islands in the -T-plane with errors of several 10%. 585 586 Appendix C Fig. C.1. Regions of density and temperature where the indicated neutrino emission processes contribute more than 90% of the total. µeis the electron “mean molecular weight,” i.e. roughly the number of baryons per electron. The bremsstrahlung contribution depends on the chemical composition. The solid lines are for helium, the dotted ones for iron which yields a larger bremsstrahlung rate. A detailed comparison of these formulae with the exact rates was performed by Haft, Raﬀelt, and Weiss (1994). They provided a new ﬁtting formula which approximates the analytic emission rate to within 5% in the entire regime where the plasma process dominates. C.2 Photoneutrino and Pair-Annihilation Process Beaudet, Petrosian, and Salpeter (1967) provided analytic approxi- mations for the photoneutrino and pair-annihilation processes. Dicus (1972) gave global correction factors to these rates to include neutral- current eﬀects. Schinder et al. (1987) numerically recalculated the emis- sion rates in the standard model and found good agreement with the BPS formulae together with the Dicus correction factors. They supple- mented the BPS rates for the temperature range 10101011K. An alternate set of approximation formulae was provided by Itoh et al. (1989) who improved on their previous work (Munakata, Koh- yama, and Itoh 1985). In Fig. C.2 I show the relative deviation between the Itoh et al. (1989) with the Schinder et al. (1987) rates. The total Numerical Neutrino Energy-Loss Rates 587 Fig. C.2. Deviation between the Schinder et al. (1987) and the Itoh et al. (1989) rates for the photoneutrino and pair-annihilation processes. Com- pared are the total emission rates where the plasma rate of Haft, Raﬀelt and Weiss (1994) and the bremsstrahlung rate (helium) of Itoh and Kohyama (1983) were used. The contours indicate were the individual processes dom- inate (Fig. C.1). energy-loss rates for the photo and pair process was calculated accord- ing to these authors, while in each case the plasma rate of Haft, Raﬀelt, and Weiss (1994) and the bremsstrahlung rate for helium of Itoh and Kohyama (1983) were taken. Therefore, deviations occur only in the range of temperatures and densities where the photo or pair process dominates. The largest deviations in the lower left corner of Fig. C.2 are around 25%. However, there the absolute magnitude of neutrino emission is very small (see below) so that the diﬀerence between the rates in this regime does not appear to be of much practical signiﬁcance. Another analytic approximation formula for the pair process was derived by Blinnikov and Rudzski˘ ı (1989). 588 Appendix C C.3 Bremsstrahlung Bremsstrahlung dominates for low temperatures and high densities where electrons are degenerate and the nuclei are strongly correlated. In a series of papers the emission rate was calculated by Itoh and Ko- hyama (1983), Itoh et al. (1984a,b), and Munakata, Kohyama, and Itoh (1987). For simple estimates one may use the approximate rate given in Eq. (11.40). In Fig. C.3 I display the error of this approximation for iron relative to the results of Itoh and Kohyama (1983). For orientation the contours of Fig. C.1 for iron are also shown, but only the brems- strahlung rates are compared. The simple approximation is not a bad ﬁt in the regions where bremsstrahlung could be of interest. For carbon the ﬁt is almost as good, but it is substantially worse for helium. Fig. C.3. Relative deviation between the bremsstrahlung rates of Itoh and Kohyama (1983) for iron and the simple approximation formula Eq. (11.40). The contours where diﬀerent processes dominate are for iron. Numerical Neutrino Energy-Loss Rates 589 C.4 Total Emission Rate The total neutrino energy-loss rate for helium is shown in Figs. C.4–C.6 where the photoneutrino and pair-annihilation rates are from Schinder et al. (1987), the plasma process from Haft, Raﬀelt, and Weiss (1994), and bremsstrahlung (helium) from Itoh and Kohyama (1983) Fig. C.4. Contour plot for the total neutrino energy-loss rate ϵper unit mass for helium. The thin contours are at intervals of a factor of 10 for ϵ. The regions where the individual processes dominate are also indicated. 590 Appendix C Fig. C.5. Neutrino energy-loss rate as a function of density. The thin lines are for temperatures 2, 3, 4, etc. times the value indicated on the corresponding thick line. Fig. C.6. Neutrino energy-loss rate as a function of temperature for the indicated values of 2 ρ/µ e. Appendix D Characteristics of Stellar Plasmas D.1 Normal Matter D.1.1 Temperatures and Densities The material encountered in stars is usually in a state of thermal equi- librium. In the absence of strong magnetic ﬁelds, the plasma is entirely characterized by its temperature T, mass density , and a set of chem- ical composition parameters X,Y,X12, etc. which determine the mass fractions of the elements1H,4He,12C, and so forth. The mass fraction of all elements heavier than helium (“metals”) is denoted by Z. The number density of a species with mass fraction Xj, atomic weight Aj, and charge Zjeis given by nj= (=m u)Xj=Aj; (D.1) where mu= 1:66×10−24g = 0 :932 GeV is the atomic mass unit.99The number density of electrons is ne=∑ jZjnj= mu∑ jXjZj Aj= emu; (D.2) where eis the “mean molecular weight” per electron, not to be con- fused with the electron chemical potential. (Strictly speaking ne= ne−ne+, the number density of electrons minus that of positrons.) 99The proton and neutron mass are 0 :9383 and 0 :9396 GeV, respectively. An exact translation between mass and number density thus requires taking nuclear binding energies into account whence the Ajare not exact integers. For the purposes of this book these diﬀerences are negligible. 591 592 Appendix D For all elements except hydrogen Zj=Aj≈1 2. Notably, this applies to helium and compounds of -particles such as12C and16O. Therefore, Ye≈−1 e≈X+1 2(Y+Z); (D.3) where Yeis the mean number of electrons per baryon. Here, the mass fraction Zof metals must not be confused with a nuclear charge. Some examples for typical conditions encountered in stars are shown in Fig. D.1, ignoring neutron stars (density around nuclear). Aside from the example of an evolved massive star, all conditions refer to the centers of stars. Except for the hydrogen main squence, the abscissa is essentially the physical density because for most chemical compositions Ye≈1 2. Horizontal-branch (HB) stars correspond essentially to the helium main sequence at 0 :5M⊙. Fig. D.1. Typical temperatures and densities encountered in stars, ignoring neutron stars. The hydrogen, helium, and carbon main sequence (H-MS, He-MS and C-MS) represent the conditions at the center of zero-age models; for selected cases their M=M⊙is indicated (adapted from Kippenhahn and Weigert 1990). The highly evolved 25 M⊙star is according to Woosely and Weaver (1986b) where the open circles are marked with the energy source of the diﬀerent burning shells (He-burning etc.). There is no nuclear burning at the center (Fe). Also shown is the evolution of the central conditions of a 0:8M⊙star from the hydrogen main sequence to the helium ﬂash (Haft, Raﬀelt, and Weiss 1994). The rear ends of the arrows mark the indicated values of the absolute surface brightness in magnitudes. Characteristics of Stellar Plasmas 593 D.1.2 Relativistic Conditions for Electrons The nuclei in normal stellar matter are always nonrelativistic; relativis- tic corrections begin to be important only in neutron stars. The elec- trons, on the other hand, tend to be at least partially relativistic. Even at the center of the Sun at a temperature of 1 :3 keV, a typical thermal electron velocity is about 9% of the speed of light. In Fig. D.2 contours for the thermal average ⟨v2⟩1=2are shown in the -T-plane. The loci of the stellar models of Fig. D.1 are also indicated. For low-mass stars, the electrons are mildly relativistic, although a nonrelativistic treatment is often enough as a ﬁrst approximation. Fig. D.2. Contours for ⟨v2⟩1=2, the average thermal velocity of electrons. The loci of the stellar models of Fig. D.1 are also indicated. D.1.3 Electron Degeneracy The phase-space occupation numbers of fermions in thermal equilib- rium are characterized by a Fermi-Dirac distribution fp=1 e(Ep−)=T+ 1; (D.4) where Epis the energy of the momentum mode p. If dispersion eﬀects can be ignored, E2 p=m2+p2with the fermion vacuum mass m. The (relativistic) chemical potential is denoted by ; for electrons it should not be confused with the mean molecular weight e. The distribution 594 Appendix D of antifermions is given by the same expression with → −. Then is implicitly given by the phase-space integral nf=∫2d3p (2)3(1 e(Ep−)=T+ 1−1 e(Ep+)=T+ 1) ; (D.5) where the second term represents antifermions and the factor 2 is for the two spin degrees of freedom. Again, the fermion density is understood to mean the density of fermions minus that of antifermions. At vanishing temperature, fpbecomes a step function Θ( −Ep). If > 0 so that nf>0, i.e. an excess of fermions over antifermions, there are no antifermions at all at T= 0. The fermion integral yields nf=p3 F=32; (D.6) where the Fermi momentum is deﬁned by 2 0=p2 F+m2with 0the zero-temperature chemical potential. The Fermi energy is deﬁned by E2 F=p2 F+m2 e, i.e. EF=0. Equation D.6 is taken as the deﬁnition of the Fermi momentum even atT > 0; it is a useful parameter to characterize the fermion density, whether or not they are degenerate. Numerically it is pF= 5:15 keV ( Ye)1=3(D.7) for electrons with the mass density in units of g cm−3. In general, Eq. (D.5) cannot be made explicit for ; it has to be solved numerically or by an approximation method. In the -T-plane, contours for the electron chemical potential are shown in Fig. D.3. Above the main plot, the electron density is characterized by pF. On the right side, the temperature is shown in units of keV. Recall that 107K = 0 :8621 keV (Appendix A). For nonrelativistic electrons the contours in Fig. D.3 are very sensi- tive to the exact value of . Therefore, in this regime the nonrelativistic chemical potential ˆ≡−m (D.8) is a more appropriate parameter. Often ˆ is referred to as thechem- ical potential. This can be very confusing when relativistic eﬀects are important. In terms of ˆ , the relativistic Fermi-Dirac distribution is fp=1 e(Ekin−^)=T+ 1; (D.9) with the kinetic energy Ekin=Ep−m→p2=2m(nonrelativistic limit). Characteristics of Stellar Plasmas 595 Fig. D.3. Contours for the electron chemical potential . The solid lines are marked with the relevant value for , the dotted lines with −me. Fig. D.4. Contours for the electron degeneracy parameter = (−me)=T. Also shown are the loci of the stellar models of Fig. D.1. 596 Appendix D A Fermi gas becomes degenerate when a typical thermal (kinetic) energy is on the order of ˆ . Therefore, the degeneracy parameter ≡ˆ=T = (−m)=T (D.10) is frequently used to characterize the fermions. They are degenerate forlarger than a few, and nondegenerate for < 0. For electrons, contours of in the -T-plane are shown in Fig. D.4. A comparison with the loci of the stellar models of Fig. D.1 reveals that electrons in stellar plasmas are often at least partially degenerate. D.1.4 Plasma Frequency Other characteristic properties of a plasma refer to the behavior of electromagnetic waves. The photon dispersion relation (Sect. 7.4) is characterized by the plasma frequency which at T= 0 is !2 0= 4 n e=EF= (4=3)p3 F=EF: (D.11) Nonrelativistically, it takes on the familiar form !2 0= 4 n e=me. Nu- merically it is !0= 28:7 eV(Ye)1=2 [1 + (1 :019×10−6Ye)2=3]1=4; (D.12) where the mass density is in units of g cm−3. D.1.5 Screening Scale An electric test charge will be screened by the polarization of the plasma. If the plasma is weakly coupled (see below) the screened Coulomb potential takes the form of a Yukawa potential r−1e−kSrwhere kSis the screening scale; its inverse is the screening radius. The plasma is polarized by the test charge because the positive constituents of the plasma are repelled while the negative ones are attracted, or the re- verse. Therefore, both electrons and ions contribute to screening. If both are nondegenerate the total contribution is k2 S=k2 D+k2 iwhere the electron contribution is known as the Debye scale, k2 D= 4n e=T= (4=3)p3 F=T: (D.13) The ions (charge Zje, atomic weight Aj) contribute k2 i=4 T∑ jZ2 jnj=4 T mu∑ jXjZ2 j Aj: (D.14) Characteristics of Stellar Plasmas 597 For only one species of ions with charge Zeone has k2 i=Z k2 D. Nu- merically, kD= 222 eV ( Ye=T 8)1=2; (D.15) where is in units of g cm−3andT8=T=108K. Contours in the -T- plane are shown in Fig. D.5. Fig. D.5. Contours for the Debye scale kDin keV. When the electrons are degenerate they cannot form a Debye-H¨ uckel cloud around a test charge. Rather, their distribution is characterized by a Thomas-Fermi model which results in the screening scale k2 TF= 4pFEF=. Because kTF≪kDthe electron screening can be neglected relative to the ions whence kS≈ki. Degenerate electrons form an essentially inert background of nega- tive charge in which the ions move, subject to their mutual Coulomb interaction. They can be treated as a weakly coupled Boltzmann gas as long as a typical thermal energy exceeds a typical Coulomb interaction energy. As a quantitative measure one uses the plasma parameter Γ =Z2=a iT; (D.16) where Zeis the nuclear charge and aithe ion-sphere radius deﬁned by n−1 i= 4a3 i=3 with the ion density ni. Numerically this is Γ = 1 :806×10−3T−1 8(Z52Ye)1=3; (D.17) 598 Appendix D with the mass density in units of g cm−3andT8=T=108K. Recall that for a single nuclear species Ye=Z=A (atomic weight A). The plasma is weakly coupled for Γ ∼<1, it is in the liquid metal phase for 1 ∼<Γ<178, and forms a body-centered cubic lattice for Γ>178 (Slattery, Doolen, and DeWitt 1980, 1982). Debye screening by the ions is appropriate for a weakly coupled plasma; otherwise the Debye approximation for the ion-ion correlations is misleading. Con- tours for Γ in the -T-plane are shown in Fig. D.6. For the purposes of this book, a strongly coupled plasma occurs only in the interior of white dwarfs. Fig. D.6. Contours for the plasma coupling parameter Γ. Also shown are the loci of the stellar models of Fig. D.1 which had to be shifted relative to each other according to the nuclear charge Zrelevant for each chemical composition. D.1.6 Summary The characteristic plasma properties for a number of typical astrophys- ical sites that are important in the main body of the book are summa- rized in Tab. D.1. Characteristics of Stellar Plasmas 599Table D.1. Plasma characteristics for some typical astrophysical sites. Center of Red-giant core standard Core of just before solar model HB stars helium ignition White dwarf Characteristic nondegenerate nondegenerate degenerate degenerate nonrelativistic nonrelativistic weakly coupled strongly coupled Temperature 1:55×107K ≈108K ≈108K 3×106−2×107K = 1:3 keV = 8:6 keV = 8:6 keV = 0:3−1:7 keV Density 156 g cm−3≈104g cm−3≈106g cm−31:8×106g cm−3(a) Composition X= 0:354He,12C,16O4He12C,16O Electron density 6 :3×1025cm−33:0×1027cm−33:0×1029cm−35:3×1029cm−3 Fermi momentum 24 :3 keV 88 keV 409 keV 495 keV Fermi energyb0:58 keV 7:6 keV 144 keV 200 keV Plasma frequency 0 :3 keV 2:0 keV 18 keV 23 keV Plasma coupling Γ = 0 :07 0.12 0.57 144−22 Debye screening electrons + ions electrons + ions ions — kS= (k2 D+k2 i)1=2kS= (k2 D+k2 i)1=2kS=ki (strong = 9:1 keV = 27 keV = 222 keV screening) aCenter of WD with M= 0:66M⊙.bNonrelativstic Fermi energy EF−me. 600 Appendix D D.2 Nuclear Matter D.2.1 The Ideal p n e eGas For the topics discussed in this book, the properties of hot nuclear matter in a young supernova core are of great interest. The relevant range of densities and temperatures is about 3 ×1012−3×1015g cm−3 and 3−100 MeV, respectively. The properties of matter at such con- ditions is determined by its equation of state which takes the nuclear interaction fully into account. However, in order to gain a rough un- derstanding of the behavior of the main constituents of the medium (protons, neutrons, electrons, and electron neutrinos) it is worthwhile to study a simple toy model where these particles are treated as ideal Fermi gases. To this end, neutrinos and electrons are treated as massless. Their dispersion relation is dominated by the interaction with the medium. Because they interact only by electroweak forces their “eﬀective mass” is always much smaller than their energies. D.2.2 Kinetic and Chemical Equilibrium The reaction e p↔n ewhich establishes equilibrium is fast com- pared to other relevant time scales. Therefore, the relative abundances ofn,p,e, and eare determined by the conditions of kinetic and chem- ical equilibrium. The physical condition of the medium is then deter- mined by the baryon density nB, the temperature T, and the condition of electric charge neutrality nB=nn+np Baryon density, np=ne Charge neutrality, (D.18) e+p=n+e equilibrium. Here, the jare the relativistic chemical potentials of the fermions which determine their number densities njaccording to the Fermi-Dirac distribution Eq. (D.5). Recall that njis the diﬀerence between fermions and antifermions of a given species. In addition, one of two extreme assumptions is made. In a young SN core the neutrinos are trapped so that the local lepton number is conserved. In this case the lepton fraction YLis the fourth required input parameter, YLnB=ne+ne Lepton conservation. (D.19) Characteristics of Stellar Plasmas 601 As a neutron star cools it becomes transparent to neutrinos. In this case their chemical potential vanishes which yields e= 0 Free neutrino escape (D.20) as the other extreme additional condition. D.2.3 Cold Nuclear Matter The limit T→0 relevant for old neutron stars is particularly simple because it allows one to express the Fermi-Dirac distributions as step- functions. One may express all Fermi momenta in units of the eﬀective nucleon mass, i.e., xj≡pj F=m∗ N. Then baryon conservation is x3 B=x3 p+x3 n; (D.21) where nj= (pj F)3=33was used, and xB≡(32nB)1=3=m∗ N= 0:2551=3 14(mN=m∗ N); (D.22) with 14the baryonic mass density in units of 1014g cm−3. Because the star is transparent to neutrinos one may use e= 0. Then the equation of -equilibrium becomes e+p−n= 0 or xp+ (1 + x2 p)1=2−(1 +x2 n)1=2= 0; (D.23) where xe=xpwas used from the condition of charge neutrality. This is easily solved to yield (Shapiro and Teukolsky 1983) x2 p=x4 n 4 (1 + x2 n): (D.24) This result may be expressed in terms of the usual composition param- eters YpandYnwhich give the number of protons and neutrons per baryon; Yp+Yn= 1. Then Yn;p= (xn;p=xB)3so that Yp=(xB 2)3(1−Yp)2 [1 +x2 B(1−Yp)2=3]3=2: (D.25) When xB≪1 this is Yp= (xB=2)3= 2:1×10−314(mN=m∗ N)3so that the proton fraction is small—hence the term “neutron star”—although the exact Ypfor a given density depends sensitively on the nucleon dispersion relation. For inﬁnite density ( xB→ ∞ ) a maximum of Yp=1 9is reached. 602 Appendix D D.2.4 Hot Nuclear Matter In a supernova core right after collapse the temperature is so high (sev- eral tens of MeV) that the nucleons are nearly nondegenerate. More- over, the neutrinos are trapped so that locally a ﬁxed value for YL determined by initial conditions is assumed. Most of the lepton num- ber will reside in electrons, causing the proton concentration Ypto be approximately equal to YL. A more accurate determination requires a numerical solution of Eqs. (D.18) and (D.19). In Figs. D.7 (a) −(c) the results of such an exercise are presented forYL= 0:3, which is a typical value for the material in a SN core just after collapse. The proton concentration Yp, the diﬀerence between the neutron and proton chemical potentials, and the degeneracy parameters for neutrons and protons are shown. In each case, a solid line refers to the assumption of an eﬀective nucleon mass as in Fig. 4.10 while a dotted line refers to the vacuum mass. As expected, the reduced eﬀective mass increases somewhat the medium’s degeneracy. Because n−p=e−e, the contours of Fig. D.7 (b) also give the diﬀerence between the surfaces of the electron and neutrino Fermi seas. Note that the leptons are much more degenerate than the nucleons because they are essentially massless. This remark does not apply to the upper-left corner of the plots where actually an excess of antineutrinos is enforced—there are more protons than leptons ( Yp> Y L)! Fig. D.7. (a) Contours for Ypin hot neutron-star matter with YL= 0:3. Solid lines for the eﬀective nucleon mass as in Fig. D.7, dotted lines for the vacuum mass. Characteristics of Stellar Plasmas 603 Fig. D.7. (b) n−p=e−e(in MeV). Fig. D.7. (c) Degeneracy parameter for protons and neutrons. 604 Appendix D Fig. D.8. Degeneracy suppression of neutrino scattering on “heavy” nucleons forYL= 0:3 and an eﬀective nucleon mass as in Fig. D.7. Fig. D.9. Ratio of the suppression factor of Fig. D.8 between the case with an eﬀective nucleon mass and with the vacuum one. When neutrinos scatter on a “heavy” nucleon recoil eﬀects can be neglected. The nucleon does not change its momentum so that the degeneracy suppression is given by a factor =2 nN∫d3p (2)3fp(1−fp); (D.26) where fpis a Fermi-Dirac occupation number. 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AGB asymptotic giant branch AU astronomical unit (distance to the Sun) BBN big-bang nucleosynthesis BC bolometric correction BP Bahcall and Pinsonneault BS blue straggler CC charged current CKM Cabbibo-Kobayashi-Maskawa CL conﬁdence level CM center of mass CMB cosmic microwave background CMBR cosmic microwave background radiation CNO carbon-nitrogen-oxygen CO carbon-oxygen D degenerate DAV DA Variable DFSZ Dine-Fishler-Srednicki-Zhitniski˘ ı EOS equation of state FTD ﬁnite temperature and density GIM Glashow-Iliopoulos-Maiani GUT grand uniﬁed theory HB horizontal branch IAU International Astronomical Union IMF initial mass function KII Kamiokande II (detector) pc parsec (3 :081018cm) KSVZ Kim-Shifman-Vainshtein-Zakharov 642 Acronyms 643 L longitudinal l.h. left-handed l.h.s. left-hand side (of an equation) ly light year LMC Large Magellanic Cloud LSP lightest supersymmetric particle mfp mean free path MS main sequence MSW Mikheyev-Smirnov-Wolfenstein NC neutral current ND nondegenerate T transverse TL Turck-Chi` eze and Lopes TO main-sequence turnoﬀ OPE one pion exchange PQ Peccei-Quinn QED quantum electrodynamics QCD quantum chromodynamics RG red giant RGB red-giant branch r.h. right-handed r.h.s. right-hand side (of an equation) SGB sub-giant branch SN supernova SNe supernovae SNU solar neutrino unit SNu supernova unit SNBO Supernova Burst Observatory UT universal time VVO Voloshyn-Vysotski˘ ı-Okun WD white dwarf Symbols Symbols which have only a local meaning in, say, one paragraph are not listed here. Four-momenta are usually denoted by uppercase italics such asK, three-momenta are boldface lowercase letters such as k, the modulus of a three-momentum is in lowercase italics such as k=jkj. Lorentz indices are usually given in Greek letters ( µ,ν,α,β, etc.), three-indices and ﬂavor indices in Latin letters ( i,j,k, etc.). Functions, Operators Besides the usual functions and operators, the following convention may be noteworthy. ∂x partial derivative ∂/∂x ln natural logarithm log logarithm base 10 Latin Symbols a axion, axion ﬁeld, annihilation operator, acceleration, radiation constant (a=π2/15 in natural units) ai ion-sphere radius A electromagnetic vector potential, atomic mass number B magnetic ﬁeld, operator for “background” medium B! speciﬁc energy density of a radiation ﬁeld c speed of light ( c= 1 in natural units) cj speed of propagation of particle j cp heat capacity at constant pressure CV;A vector and axial-vector weak-coupling constants Cj eﬀective Peccei-Quinn charge of fermion j d electric dipole moment d,D distance 644 Symbols 645 dp diﬀerential in phase space, dp=d3p/(2π)3 e electron, electric charge of electron, photon polarization four- vector ej charge of particle j E electric ﬁeld, energy of a particle EF Fermi energy f occupation numbers, dimensionless function, energy scale fPQ Peccei-Quinn scale fa axion decay constant f pion decay constant (93 MeV) fp occupation number of mode p F electromagnetic ﬁeld-strength tensor, number ﬂux of particles, dimensionless function of order unity F! speciﬁc energy ﬂux in a radiation ﬁeld F ﬂuence (time-integrated ﬂux) g dimensionless Yukawa coupling, gluon, graviton, number of spin degrees of freedom ga axion-photon coupling strength (units GeV−1) g10ga /10−10GeV−1 G color ﬁeld-strength tensor, matrix of coupling constants, dimen- sionless function GN Newton’s constant ( GN=m−2 Pl) GF Fermi’s constant ( GF= 1.16610−5GeV−2) H Hubble expansion parameter H0 Hubble expansion parameter today ( h100 km s−1Mpc−1) H Hamiltonian density k momentum of a particle, momentum transfer in a reaction kB Boltzmann’s constant ( kB= 1 in natural units) kD Debye screening scale of the electrons ki Debye screening scale of the ions kS screening scale kTF Thomas-Fermi screening scale l momentum transfer in a reaction ℓ mean free path, ﬂavor index L luminosity of a star L⊙ solar luminosity (3 .851033erg) L Lagrangian density m mass ma axion mass me electron mass (0 .511 MeV) m pion mass (135 MeV) 646 Symbols mN nucleon mass (938 MeV) mu atomic mass unit (931 MeV) mPl Planck mass (1 .2211019GeV) mWW-boson mass (80 .2 GeV) mZZ-boson mass (91 .2 GeV) M absolute magnitude of a star, mass matrix of quarks or neutrinos Mbolabsolute bolometric magnitude of a star M mass of a star, matrix element M⊙solar mass (1 .991033g) n particle density nB baryon density ne;p;n; electron, proton, neutron, neutrino density nrefr index of refraction N nucleon, number of degenerate Θ vacua in QCD N matrix element of neutrino current p proton, momentum of a particle, pressure, dimensionless power index pF Fermi momentum P four-momentum of a particle, stellar pulsation period q momentum transfer, charge of a particle Q energy loss or generation rate per unit volume r radial coordinate R radius of a star R⊙ solar radius (6 .961010cm) s entropy density, dimensionless power index, square of CM en- ergy, dimensionless spin-structure function S static or dynamical structure function t time T temperature T7T/107K T8T/108K T30T/30 MeV U total internal energy of a star, four-velocity v velocity, vacuum expectation value of a Higgs-like ﬁeld V potential energy, volume w up/strange quark mass ratio W W -boson WK;K′Transition probability from KtoK′. x dimensionless energy ( ω/T), spatial coordinate, dimensionless Fermi momentum ( pF/m) X mass fraction of hydrogen Symbols 647 Xj Peccei-Quinn charge of a fermion j Y mass fraction of helium Yj number of fermions jper baryon Ye mass fraction of helium in stellar envelope YL lepton number per baryon z up/down quark mass ratio, z-coordinate Z mass fraction of “metals,” nuclear charge Z,Z◦Z-boson Greek Symbols α ﬁne-structure constant ( e2/4π= 1/137), asymmetry parameter in angular distribution of decay photons α′ﬁne-structure constant for general bosons, α′=g2/4πwith the Yukawa coupling g αa; ﬁne-structure constant for axions, majorons αs strong ﬁne-structure constant α pionic ﬁne-structure constant, α= (f2mN/m)2/4π15 with f1.0 β velocity of a particle, parameter of axion models, parameter in nucleon-nucleon-axion bremsstrahlung rate γ photon, Dirac matrix, adiabatic index, dimensionless ﬂuctuation rateγ= Γ/T Γ rate for decay or ﬂuctuations, plasma coupling parameter Γ spin-ﬂuctuation rate in a nuclear medium δ δ function, Kronecker δ, diﬀerential quantity ϵ energy loss or generation rate per unit mass, electric dipole mo- ment, polarization vector ϵij neutrino transition electric moment η degeneracy parameter, η= (µm)/T θ mixing angle, scattering angle Θ Θ parameter of QCD ΘW weak mixing angle (sin2ΘW= 0.2325) κ opacity, dimensionless screening or momentum scale κ∗reduced opacity λ wave length, mean free path µ magnetic moment, chemical potential, muon, Lorentz index, mean molecular weight ˆµ nonrelativistic chemical potential, ˆ µ=µm µij neutrino transition magnetic moment 648 Symbols µB Bohr magneton ( e/2me) µN nuclear magneton ( e/2mp) µe electron mean molecular weight ( µeY−1 e), electron chemical potential ν neutrino, Lorentz index, dimensionless power index ξ dimensionless correction factor, dimensionless axion-photon cou- pling constant π π = 3.1415..., pion ρ mass density, density matrix ρ density matrix for antineutrinos ρ0 nuclear density (3 1014g cm−3) ρ14ρ/1014g cm−3 ρ15ρ/1015g cm−3 σ scattering cross section, Pauli matrix, spin operator τ lifetime of a particle, duration of a stellar evolution phase, Pauli matrix, optical depth Φ Higgs ﬁeld Φ0 vacuum expectation value of Higgs ﬁeld χ majoron, majoron ﬁeld ψ fermion ﬁeld Ψ column vector of fermion ﬁelds (several ﬂavors) ω energy of a particle, energy transfer in reactions ωP plasma frequency ω0 zero-temperature plasma frequency Ω cosmic density parameter, matrix of energies for mixed neutrinos Subject Index A -Ori 189 A665, A1413, A2218, A2256 (galaxy clusters) 490–92 adiabatic index 396 adiabatic oscillations !neutrino oscillations adiabatic temperature gradient 10–12 adiabaticity parameter 299 AGB (asymptotic giant branch) 27, 35, 61 aligned rotator 185 AMANDA xvi, 558 anapole moment 268–70 Andromeda 446 angular anomaly of SN 1987A neutrino signal 422, 428–30 antimatter supernova 497 arions 187–90 automatic symmetry 533 asymptotic giant branch 27, 35, 61 atmospheric neutrinos 290–3 atomic mass unit 582, 591 axio-electric eﬀect 100 axion !pseudoscalar boson bounds 191f, 539–41 cavity experiment 177f, 191f, 544 cosmic density 491f, 528, 541–44 cosmic thermal production 541 couplings electron 536f fermions general 530–2 gluon 159, 527, 530 nucleon 536–38 photon 167f, 176, 191f, 536 decay 167f, 191f decay constant 526f, 529, 535direct search decay photons of cosmic axions 491f galactic (cavity search) 177f, 191f, 544 laboratory 182, 184f, 191f solar 100, 181, 191f, 462 SN 1987A 483 mass 527, 535 mixing with pion 527f, 534 models DFSZ 535–40 KSVZ 528–30, 542–44 other 534–36 Nambu-Goldstone boson 190f, 528–30 quantum gravity 532f solution to strong CP problem 527f sphere 507 axio-recombination 100 B decay 255 decay 257f, 560f Baade-Wesselink method 73 Baksan scintillator telescope (BST) 414f, 417, 420 band-structure eﬀects 58, 106f baryonic force 114 beam-stop neutrinos 456f Betelgeuse 189 big-bang nucleosynthesis: bounds majorons 259f, 561f millicharged particles 522f, 566f neutrino Dirac dipole moment 277f mass 259f time variation Fermi’s constant 546 Newton’s constant 547f 649 650 Subject Index binary pulsar !PSR 1913+16 birefringence neutrino ﬂavor 195, 281 photon in media 180f, 194f, 208 vacuum 183–5, 187f, 190f, 554 Bloch state 155 blue loop 32, 39–41 bolometric correction 62 bolometric magnitude: deﬁnition 28 Boltzmann’s constant 580 Boltzmann collision equation (for mixed neutrinos) 323–31 BOREXINO 343, 371, 393f bounce and shock 399, 401–5 bound-free transition 100f branching ratio 449 Brans-Dicke theory 548 bremsstrahlung classical limit 143–47 electronic 101-9 nucleonic axial-vector vs. vector 127, 142 bubble phase 159 meson condensate 155–57 neutrino pairs 127–30 pseudoscalars 119–26, 143–45, 148f suppression at high density 148f quark matter 157f brown dwarfs 9, 24 bubble phase 159 bump on RGB 61, 64f C Cabbibo mixing 261, 285 Cabbibo-Kobayashi-Maskawa matrix 262 carbon-burning stars 2 cavity axion experiment 177f, 191f, 544 Cepheids 40f Chandrasekhar limit 9, 25, 37, 42, 396 charged current collision term (in neutrino oscillations) 331 coupling constants 583 Hamiltonian 329, 583 reactions 1, 58f, 152–54 charge quantization 564–67 chemical potential 593–96, 600fCherenkov eﬀect 194, 216, 238f detectors !IMB, Kamiokande, Superkamiokande chirality 160, 253 chiral symmetry 166, 529 chlorine detector (solar neutrinos) 342f, 357–62, 371 CHORUS experiment 289f CKM matrix 262 clump giants 76, 80 CM energy 92 CNO cycle 346f COBE satellite 191 coherent neutrino scattering 396, 446 collision equation for mixed neutrinos 323–31 color-magnitude diagram evolutionary track 31 globular cluster M3 27 globular-cluster observables 61 populations of stars 41 zero-age main sequence 26 Compton process 20f, 91–98, 108 conduction 11, 22, 78 convection description 12f helium dredge-up 66, 77 mixing length 12, 16, 351, 549f RGB bump 13, 61, 64f superadiabatic 12 supernova 402–5, 444 core red giant !red giant core mass supernova !supernova core correlations: Coulomb plasma 222–25 cosmic background ﬂuxes -rays 487 microwaves 191, 488, 546, 562 neutrinos core-collapse SNe 447, 487f hydrogen-burning stars 487 primordial !neutrinos: cosmic background photons 489f cosmic strings 191, 542–44 cosmic structure formation 258f cosmion xvi Cotton-Mouton eﬀect 180, 183 Subject Index 651 Coulomb gauge 204 Coulomb propagator in plasma 221f, 226 CP symmetry 241, 246, 262, 271, 453, 524–26 Crab Nebula 38, 56f critical ﬁeld strength 581 crossing relation 323 current algebra 167, 535 D Cepheids 40f Scuti stars 41 damping !neutrino oscillations: damping by collisions DA Variables 53 dark matter xv–xvi, xviii, 22, 24, 177f, 258f, 528, 542–44, 558 decoherence 311, 313f degeneracy parameter 124, 595f, 603 delayed explosion mechanism 399, 402–5, 426, 436f, 522 Delbr¨ uck scattering 183 deleptonization burst !prompt neutrino burst deleptonization: supernova core 339, 398, 401, 408 density matrix (neutrino ﬂavor) collision equation 319, 323–31 deﬁnition 284, 315–16 derivative coupling 119, 531f, 536, 563 detailed balance 129, 137, 146 DFSZ axions 535–40 dielectric permittivity 173, 208f diﬀusion elements !gravitational settling neutrinos !neutrino opacity dipole moment !neutrino dipole moments neutron electric 524–26 other particles 525 units 525, 581 Dirac matrices 270 Dirac neutrinos !neutrino mass, neutrinos: right-handed direct explosion mechanism 399, 401fdispersion !neutrino dispersion, photon dispersion general theory 193–202 double beta decay !decay DUMAND xvi, 558 dynamical structure function ! structure function E Einstein x-ray satellite 56–58, 189 electron charge 581 electro-Primakoﬀ eﬀect 172f energy conservation in stars 10f energy-loss argument analytic treatment 14–16 applied to HB stars 79–82, 98f, 107–9, 112, 176, 236 neutron stars 59f red giants 83–87, 107–9, 236 red supergiants 2 Sun 16f, 20f, 109f, 175f supernova core 501–23 white dwarfs 47–52 ZZ Ceti stars 52–54 introduction 3–5, 21f numerical treatment 16f, 47–52, 81, 83–87, 108f, 508–11, 513–16 energy-loss rate bremsstrahlung electronic 102–5 nucleonic 121–6, 128–30, 147–49, 156f Compton process 96–98 plasma process 232–36 Primakoﬀ process 169, 174 quark matter 157f URCA processes 153f energy-momentum conservation 121, 200 energy transfer !convection, opacity equation 11–13 particle bounds general argument 4f, 17f Sun 20f SN 1987A 503, 506–11 652 Subject Index energy transfer (cont'd) radiative 4, 17–19, 131 WIMPs 22 equation of state impact on SN neutrino signal 412f normal stars 6 nuclear 600 equivalence principle: tests 113, 498, 554f escape velocity galaxy 26 globular clusters 25 Euler-Heisenberg Lagrangian 183 EXOSAT 56f explosion mechanism !supernova: type II F Faraday eﬀect 180 Fermi constant 545f, 583 energy 594 momentum 594 Fermi-Dirac distribution 593–96 forward scattering 196–98 ﬁfth force 112–16, 500 ﬁne-structure constant axion-nucleon 119 electromagnetic 580 new bosons 92 pion-nucleon 119 ﬂavor-changing neutral current 263f, 303, 326 ﬂavor conversion !neutrino oscillations ﬂuence: deﬁnition 463 foe 402 form factor !neutrino form factor free-bound transition 100f Fr´ ejus experiment 291f FTD ﬁeld theory xviii–xix, 105, 174, 241 fundamental length scale 500 G G117–B15A 53fGALLEX 343f, 357–59, 362–64, 371, 373 gallium detectors 343f, 357–59, 362–64, 371 gauge bosons: new 114–16 invariance 204–6 General Relativity !gravitation giant star !red giant deﬁnition 10 GIM suppression 266f globular cluster !red giants, HB stars, RR Lyrae stars ages 62f, 66, 552–54 general properties 25f M3 25, 27 observables interpretation 74–79 in the color-magnitude diagram 60–65 observational results 70–74 theoretical relations 65–70 gluons 159, 525, 527, 530 g-mode !helioseismology gravitation Brans-Dicke theory 548 equivalence principle: tests 113, 498f, 554f graviton 116, 166, 191, 254 induced neutrino mixing 555 nonmetric theory 498 nonsymmetric extension 554 quantum 532f Shapiro time delay 498, 554 time-varying GN546–54 gravitational settling description 13f globular-cluster ages 13, 66 red-giant envelope helium abundance 66, 74, 77 solar neutrino ﬂux 14, 352, 354 graviton 116, 166, 191, 254 group velocity 199 GRS instrument 465f Subject Index 653 H Hayashi line 32, 35, 40 HB (horizontal branch) location in color-magnitude diagram 27, 41, 61 morphology 34 HB stars !RR Lyrae stars energy-loss argument 79–82, 98f, 107–9, 112, 176, 236 inner structure 29f, 80f, 592, 599 lifetime 70, 76, 79–82 overview 34f particle bounds: tabulation 82 helioscope 179, 181f helioseismology xvi, 17, 113, 344f, 549–51 helium abundance presolar 16f, 351, 549f primordial 24, 67 red-giant envelope 66f, 74–78 helium-burning stars !HB stars helium ﬂash !helium ignition helium ignition !red giant core mass brightness 63f, 68–72 description 33f delay by particle emission 63, 83–87, 236 oﬀ-center 34, 78, 83f Hertzsprung gap 32, 40 Hertzsprung-Russell diagram !color-magnitude diagram Higgs ﬁeld 3, 165f, 194, 252f, 528, 534, 545, 559 Homestake (solar neutrino detector) chlorine 342f, 357–62, 371, 373–76 iodine 343, 394 homologous models 14–16, 549, 552 horizontal branch !HB hot bubble 38, 397, 406 hot plus cold dark matter 258 Hulse-Taylor binary pulsar !PSR 1913+16 Hyades 48 hydrodynamic event 6 hydrogen-burning chains 341, 346f hydrostatic equilibrium 5–7I, J IMB detector atmospheric neutrinos 291f SN 1987A neutrinos 415–23 inﬂation 542 ions contribution to screening 220f, 596, 599 correlations 106, 222–25 ion-sphere radius 223f, 597 isochrone 61f Jupiter magnetic ﬁeld 556 K mechanism 40 Kamiokande (Cherenkov detector) atmospheric neutrinos 291f dark-matter search xvi neutrino detection eﬃciency 417 electron scattering 365–67, 416f proton absorption 416f proton decay xvii SN 1987A neutrinos 418–23 solar neutrinos 343f, 368–73, 383 Kelvin-Helmholtz cooling Sun 8 supernova core 397, 400, 407–13, 501–23 Kibble mechanism 542 kick velocity (neutron stars) 443–45 Klein-Gordon equation for mixed neutrinos 282–84, 294 Kramers-Kronig relation 198 KSVZ axions 528–30, 542–44 L Lamb shift 565f Landau damping 199, 214, 216f Landau-Pomeranchuk-Migdal eﬀect 145, 148f Landau-Zener approximation 300 Langmuir wave !plasmon: longitudinal Large Magellanic Cloud 39, 74, 414, 446 large-numbers hypothesis 545, 549 654 Subject Index left-right symmetric model 268 leptonic force 114–16, 500 lepton fraction in nuclear matter 600, 602–5 masses 252, 265 proﬁle in SN core 398 level crossing 295f, 298 linear-response theory 204–6, 208f long-range force !fth force, scalar particles, photon long-wavelength limit 121, 128f, 141–43 Lorentz addition of velocities 500f gauge 204 Lorentzian model for structure function 135, 144–47 LPM eﬀect 145, 148f M M1 transition 94, 102 M3 (globular cluster) 25, 27 magnetic dipole moment !neutrino dipole moments units 581 magnetic ﬁeld Earth 556 galactic 188–90, 484, 499 large-scale cosmic 191 Jupiter 556 neutron stars 185–88, 444f primordial 278 Small Magellanic Cloud 556 solar 186, 373–76, 387f, 554, 565 supernova 522 twisting 308f white dwarfs 185f magnetic oscillations !neutrino spin precession magnetic permeability 208f magnetogram 375 main sequence helium 35, 592 internal stellar conditions 592, 599 lifetime 27, 552 location in color-magnitude diagram 27, 41, 61 turnoﬀ 27, 61–63 zero-age 26Majorana neutrinos !neutrino mass majorons !pseudoscalar bosons decay 560f decay in media 195, 248–50, 259f interaction with supernova neutrinos 495 motivation 558–62 neutrino decay: limits 562–64 nucleosynthesis constraints 259f solar neutrino decay 389 supernova physics 562–64 mass fraction (of elements) 591f mass loss 25, 34, 36 mass-radius relationship 9, 42 Maxwell’s equations 202–4 mean molecular weight 591 meson condensate 59, 155–57, 243 metallicity -enhanced 78f globular cluster 26, 34, 60, 66f, 78f impact on opacity 12 Sun 24 Mexican hat 529, 542 Mikheyev-Smirnov-Wolfenstein eﬀect !neutrino oscillations: resonant millicharged particles 227f, 232–36, 564–67 misalignment mechanism 541f mixing angle in medium 294–7 mixing between particles !neutrino mixing, neutrino oscillations axion-photon 179–81 axion-pion 527f, 534 general theory 260–63 graviton-photon 191 gravitationally induced 555 mixing length 12, 16, 351, 549f molecular chaos 319 monopoles xv MSW bathtub 301–3 eﬀect!neutrino oscillations solution of solar neutrino problem 384–86 triangle 301–3, 386, 434 Subject Index 655 multiple scattering Cherenkov detector 366 nuclear medium 135, 143–51, 511f MUNU experiment 276 muon decay 263 N Nambu-Goldstone bosons 165f, 250, 529–31, 559f NESTOR xvi, 558 neutral current collision term in neutrino oscillations 323f coupling constants 584 ﬂavor-changing 263f, 303, 326 Hamiltonian 95, 119, 136, 320, 531, 584 neutrality of matter 565 neutrino absorption !neutrino opacity neutrino charge: electric 227f, 232–36, 240, 499, 522f, 564–67 neutrino cooling !supernova core red-giant core 83–85 white dwarfs 47–50 neutrino coupling constants: standard 583f neutrino cross section electrons 365f, 416 protons and nuclei 416 neutrino dark matter 258–60 neutrino decay !neutrino radiative decay fast invisible 248–50, 389, 485, 559 standard-model 263–67 neutrino dipole moments astrophysical impact and bounds early universe 277f HB stars 82, 236 neutron stars 59f red-giant core mass 86f, 236 supernova 277, 477–79, 483, 521f white dwarfs 48–50 electric vs. magnetic 265f, 268–71, 273–75, 525 experimental bounds !neutrino radiative decay scattering experiments 267, 275fneutrino dipole moments (cont'd) interaction structure 228, 265, 268–71 medium induced 240 processes plasmon decay 229, 232–36, 274 scattering 272f summary 272–75 radiative decay !neutrino radiative decay summary of limits 275–79 theoretical prediction Dirac vs. Majorana 265, 271, 451 left-right symmetric model 268 standard-model 265f structure of interaction 265, 268–71 neutrino dispersion Cherenkov eﬀect 238f deﬂection 247f density-matrix treatment 321f general theory 241–47 inhomogeneous medium 442f neutrino background 321f, 432f, 440–42 nonisotropic medium 321, 432f, 440–42 oﬀ-diagonal refractive index 322 relation for massive neutrinos in media 295f spin-ﬂip scattering 162 neutrino emission: standard !solar neutrino ux, supernova core astrophysical impact red-giant core mass 83f red supergiant 2 white-dwarf cooling 47–50 historical perspective 1f numerical rates 585–90 speciﬁc processes bremsstrahlung (electronic) 105-7, 588 bremsstrahlung (nucleonic) 127–30, 157, 159 Compton (photoneutrino) 95f, 98, 586f free-bound transition 100f pair annihilation 99f, 586f plasma 585f 656 Subject Index neutrino form factor: electromagnetic !neutrino dipole moment anapole moment 268–70 charge 227f, 232–36, 240, 268, 564–67 charge radius 269f, 523 general theory 267–72 in plasma 230f, 237–41 neutrino lasing 389 neutrino lightcurve !supernova core neutrino mass bounds decay 257f cosmological 258–60 experimental 252, 254–58 future supernovae 447f, 497 SN 1987A signal dispersion 426f SN 1987A signal duration 516–19 Dirac 253f, 516–19 eﬀective !neutrino dispersion inverted hierarchy 434 Majorana 253f, 257–59 neutrino mixing !neutrino oscillations bounds atmospheric neutrinos 291 decay experiments 456f, 461 oscillation experiments 289f r-process nucleosynthesis 437–42 SN 1987A -rays 481 SN 1987A prompt burst 432–34 SN 1987A signal duration 434–36 supernova core: ﬂavor conversion 338 decay rates and dipole moments 264–67 induced by ﬂavor-changing neutral currents 303 gravity 555 mass matrix 262, 282 neutrino opacity 132–35, 149–51, 329f, 334f, 408, 514–16 neutrino oscillations damping by collisions kinetic equation 327–29, 332–34 simple picture 310–12 spin relaxation 313 time scale in SN core 335–38 equations of motion 282–84neutrino oscillations (cont'd) experimental searches 289f historical introduction and overview 280–82 in media adiabatic limit 297–99 analytic results 299–300 homogeneous 296f mixing angle 294–7 neutrino background in supernovae 432f, 440–42 oscillation length 285f, 289, 294–97 primordial 322 resonant analytic results 299f description 281, 298 Landau-Zener approximation 300 level crossing 295f, 298 schematic model for the Sun 301–3 solution of solar neutrino problem 384–86 supernova core 333f supernova mantle 432–42 spin-precession picture 286f, 314f supernova cooling phase 434–36 explosion mechanism 436f overview 430f prompt burst 432–34, 496f, 498 r-process nucleosynthesis 437–42 Sun MSW solutions 384–86, 434f schematic model of MSW eﬀect 301–3 vacuum solution 381–84 survival probability general expression 284–86 with energy distribution 288 with source distribution 287f temporal vs. spatial 283f three-ﬂavor 284f two-ﬂavor 284–88 neutrino-photon scattering 245f neutrino radiative decay !neutrino dipole moment, neutrino two-photon coupling Subject Index 657 neutrino radiative decay (cont'd) bounds from beam stop 456f cosmic background neutrinos 489–91 diﬀuse ﬂux from all stars 487f reactors 453–55 solar positrons 457, 461 solar x-rays 459–61 SN 1987A -rays 467f, 474f, 477–79 photon spectrum stationary source 451–53 pulsed source 463–65, 469–73 standard-model predictions 264–67 summary of bounds 278f neutrino rocket engine 443–45 neutrino sphere 397, 407, 409 neutrino spin precession supernova magnetic ﬁeld 521f theoretical description equations of motion 304f in electric ﬁelds 305 in medium 306 oscillation length 305 spin-ﬂavor 306–9 time variation of solar neutrino ﬂux 374, 387f neutrino trapping 396 neutrino two-photon coupling 245, 265f, 271f neutrinos: cosmic background screening of leptonic force 114f, 500 mass density 258–60, 489–91 radiative decay 489–91 neutrinos: number of families 251, 260, 513 neutrinos: right-handed component of Dirac 253f e+edecay limits 479, 482 SN 1987A limit charge radius 523 dipole moment 521f Dirac mass 516–19 mixing with sequential ﬂavors 338–40 right-handed currents 519–20 spin-ﬂip production 160–64 neutrinos: solar !solar neutrino ux neutrinos: strongly interacting 558f, 563neutrinos: supernova !SN 1987A, supernova core neutron stars cooling 54–60 crust 58–60, 106f formation 38 kick velocities 443–45 internal conditions 601 magnetosphere 185–88 pseudoscalar ﬁeld around 187 x-ray observations 56–58 Newton’s constant value 6 time variation 546–54 NOMAD experiment 289f nonabelian Boltzmann collision equation 323f nuclear matter novel phases 155–59 properties 151f, 600–5 nucleon mass: eﬀective 151f, 602–5 nucleosynthesis !big-bang nucleosyn- thesis, supernova (type II) number fraction: deﬁnition 243 O oﬀ-diagonal refractive index 322 Oklo natural reactor 546 one-pion exchange potential 119, 122, 124f, 141 opacity -enhanced 78f conductive 11, 22, 78 deﬁnition 11 neutrino 132–35, 149–51, 329f, 334f, 408, 514–16 particle 11, 18f, 127–35 radiative 18f reduced 19 Rosseland average 11, 18f solar 21, 345, 353–55 tables 12 OPAL 12 open clusters 60, 76 OPE potential 119, 122, 124f, 141 optical activity !birefringence optical theorem 198 oscillation length 285f, 289, 294–97, 305 658 Subject Index oscillations (particles) !neutrino oscillations axion-photon 179–82, 185f, 188–90 graviton-photon 191 inhomogeneous media 181, 442f pion-photon 183 oscillations (stars) !helioseismology, stars: variable P pair annihilation 99f paraphoton 82, 92f, 99 Pauli matrices 286 Peccei-Quinn mechanism 526–28, 532f scale 166, 526f, 529, 535 peculiar velocity (neutron stars) 443–45 permittivity !dielectric permittivity phase velocity 199, 244 photon baryonic 114 charge limits 555f decay!plasma process leptonic 114–16 mass !plasma frequency eﬀective 208 vacuum 206, 555f transverse 209, 214 refraction !photon dispersion splitting 184 weakly interacting 18 photon dispersion approximate expressions 213f external ﬁelds (vacuum birefringence) 183–5, 187f, 190f, 554 general theory 203–19 mixing with axions 180f transverse mass 209, 214 photoproduction !Compton process X-particles in HB stars 82, 429 pinching of neutrino spectra 410f pion !one-pion exchange potential Compton scattering 157 condensate 59, 155–57, 243 coupling to nucleons 119, 151f, 531pion (cont'd) decay 162, 167f, 256 decay constant 166f, 527 mass 119, 124f, 527 mixing with axion 527f, 534 Nambu-Goldstone boson 165f Planck mass: deﬁnition 6 Planck’s constant 580 planetary nebulae 36, 43, 51 plasma conditions in stars 591–99 coupling parameter Γ 224, 597–99 crystallization 46, 598 ﬂuctuations 172–74 frequency 198, 211f, 596, 599 one-component 223–25 strongly coupled 223f two-component 223 plasma process !neutrino dipole moments astrophysical impact HB star lifetime 80–82 neutron-star crust 59f red-giant core mass 83–87 supernova core 523 general theory 227–36 gluonic 159 plasmino 195 plasmon coalescence 170–72 e+edecay 195, 211 longitudinal 170–72, 195, 202f, 213 decay!plasma process p-mode !helioseismology Poincar´ e sphere representation 286 polarimetry of radio sources 190f polarization tensor 205–12, 230f, 238 polarization vector neutrino ﬂavor deﬁnition 286f degree of coherence 313 individual modes 317 shrinking by collisions 314f transverse 314, 325 plasma excitations 206f positron ﬂux: interplanetary 461, 487f pp-chain 15, 341, 346f pressure sources in stars 7–10 Subject Index 659 Primakoﬀ process 116, 159, 168–75, 177, 357 prompt explosion mechanism 399, 401f neutrino burst 397, 399f, 432–34, 496f, 498 proton decay xvii fraction in nuclear matter 601f spin 537, 584 protoneutron star !supernova core pseudomomentum 200 pseudo Nambu-Goldstone bosons 166 pseudoscalar bosons !arions, axions, majorons astrophysical impact and bounds HB star lifetime 80–82, 98f, 176, 191f neutron-star cooling 59 red-giant core mass 85f, 107-9 SN 1987A 501–4, 508–12 Sun 16f, 20f, 176, 191f white-dwarf cooling 50–52 ZZ Ceti period decrease 54 bounds on couplings to photons 17, 81f, 176, 191f electrons 20f, 51f, 82, 85, 99f, 107–11 nucleons 512 conversion to photons 179–82, 185f, 188–90 derivative coupling 119, 531f, 563 emission processes bremsstrahlung (electronic) 102–5, 107f bremsstrahlung (nucleonic) 119–26, 143–45, 148f, 155f bremsstrahlung (quarks) 157f Compton 20, 94, 98, 108f energy spectrum 123, 175 pair annihilation 99f Primakoﬀ (gluonic) 159 Primakoﬀ (photonic) 168–75, 177 interaction Hamiltonian 119 long-range force 112 opacity contribution 18f, 21, 131f, 506–8 refractive index 250 solar direct search 100, 181, 191fPSR 0655+64 !547 PSR 1913+16 (Hulse-Taylor binary pulsar) xiv, 113, 116, 547 PSR 1937+21 !188, 190, 554 pulsar !neutron star pulse dispersion 554f pulsation constant 40 periods 40f period decrease 53f PVLAS experiment 185, 192 Q QCD: CP problem 524–26 Q-nuclei (quarked nuclei) 557 quark free 556f matter xv, 157–59, 557 masses 252, 525 mass ratios 535 mixing 261f quantum gravity 532f R radiative energy transfer !energy transfer, opacity random media: particle oscillations 181, 442f rank-ordering statistics 374f reactor: nuclear natural 546 neutrino radiative decay limits 454f neutrino spectrum 450, 453f, 455 red-giant branch!RGB core mass at helium ignition increase by particle emission 83–87 observational 75–79 theoretical 67f uncertainties 78f core rotation 78 properties and evolution 28–32, 592, 599 reduced opacity 19, 131 660 Subject Index refractive index !photon dispersion, neutrino dispersion deﬁnition 194 forward scattering 196–98 neutrino 242, 244 relativistic limiting velocity 497f renormalization: wave function 200, 217–19, 228, 233 repulsion of energy levels 296 resonant oscillations !neutrino oscillations axion-photon 181, 186 RGB (red-giant branch) brightest star 63f, 68–72 bump 61, 64f location in color-magnitude diagram 27, 41 phase transition 37 right-handed currents 268, 519–20 neutrinos !neutrinos: right-handed Ring Nebula in Lyra 36 R-method 65, 70, 72, 74–78 ROSAT 56–58 Rosseland average 11, 18f r-process nucleosynthesis 405–6, 437–42 RR Lyrae stars absolute brightness 62, 65, 69f, 73f ages of globular clusters 62 color 62 location in color-magnitude diagram 40f mass-to-light ratio 77 Rubakov-Callan eﬀect xv Rydberg atom 178 S S17factor 355–57, 377f, 386 SAGE 343f, 357–59, 362f, 371 Sanduleak 69 202!39, 414 scalar ﬁeld !scalar particles, Higgs eld around a neutron star 113 scalar particles bounds on coupling electrons 82, 99, 107 nucleons 82, 113scalar particles (cont'd) emission from binary pulsar 113 emission processes bremsstrahlung 103, 124f Compton 93, 98 long-range force 112f, 500 photon coupling 166 Sciama’s neutrino 491f screening eﬀects background neutrinos 114f Debye-H¨ uckel scale 220f, 596f, 599 general theory 219–26 in processes bremsstrahlung 102–6 Primakoﬀ 169 spin-ﬂip scattering 277f ion contribution 220f, 596f, 599 modiﬁcation of Coulomb propagator 221f, 226 Thomas-Fermi scale 221 secret neutrino interactions 558f, 563 see-saw mechanism 254 self-energy 201, 209 Shapiro time delay 498, 554 shining light through walls 182 shock wave 397, 399–405 Small Magellanic Cloud 446, 556 SMM satellite 465f SN!supernova SN 1987A neutrino pulse analysis 423–26 anomalies 427–30 measurements 419–23 optical lightcurve 414f progenitor 39, 414 SN 1987A bounds antimatter supernova 497 arion-photon conversion 189 axions direct detection 495 neutrino signal duration 501–4, 508–12 fundamental length scale 500 Lorentz addition of velocities 500f neutrino charge 499, 522f decay 494, 496, 519 mass 426f, 499 Subject Index 661 SN 1987A bounds (cont'd) neutrino (cont'd) neutrino-neutrino cross section 494f number of families 513 secret interactions 494f neutrino oscillations cooling phase 434–36 prompt burst 432–34 pseudoscalar boson couplings 501–4, 508–12 radiative particle decays 467f, 474f, 477–79, 483 relativistic limiting velocity 497f right-handed currents 519–20 right-handed neutrinos charge radius 523 dipole moment 521f Dirac mass 516–19 mixing with sequential neutrinos 338–40 secret neutrino interactions 563 supersymmetric particles 558 weak equivalence principle 498 SNBO 448 SNO 343, 371, 392f SNu (supernova unit) 488 SNU (solar neutrino unit) 358 solar axions 100, 181f, 191f solar maximum mission satellite 465f solar neutrino ﬂux antineutrino component from majoron decay 389 from spin-ﬂavor oscillations 388 limits from Kamiokande 369 counting rate prediction for detection Cherenkov 370 chlorine 359, 361 gallium 359, 364 future experiments 390–4 measurements Cherenkov 368–70 chlorine 360–62 gallium 362–64 modiﬁed by electrically charged neutrinos 565 gravitational settling 14, 352, 354 opacities 353–55 neutrino decay 389 neutrino-neutrino scattering 495solar neutrino ﬂux (cont'd) modiﬁed by (cont'd) resonant oscillations 301–3, 384–86, 434f temperature 354 WIMP energy transfer xvi strange quark matter 557 Q-nuclear burning 557 time-varying GN549–51 vacuum oscillations 381–84 radiative decay limits 458–62 source reactions beryllium 343, 347–50, 355, 378f boron 343, 347–49, 355–57, 368–70, 377f, 383 CNO 343, 347–49 electron capture vs. decay 350 hep 347–49 pep 347, 350 pp343, 347–49 time variation day-night 372, 385f, 390f semiannual 372f, 382f solar cycle 373–76, 387f solar neutrino problem introduction and historical overview 341–45, 380f ﬂux deﬁcits beryllium 378f beryllium/boron branching ratio 379f boron 377f ﬂux variation at Homestake 373–77 MSW solution 384–86 vacuum solution 381–84 VVO solution (magnetic oscillations) 387f Sommerfeld parameter 355 space-like excitations 194, 198f, 207, 216, 215f, 238f spectral density 173f speed of light 497f spin-ﬂavor oscillations !neutrino spin precession spin ﬂip 160–64, 277f, 304–9, 317, 516–23 spin-ﬂuctuation rate 118, 121-23, 127, 133, 144f spin relaxation 313 662 Subject Index spin-spin interaction potential 141, 151 starburst galaxies 446 stars ages 27, 35, 43, 62f, 552–54 formation 24–27 initial mass function 24 intermediate-mass 37 mass loss 25, 34, 36 mass range 24 massive 37–39 populations in color-magnitude diagram 41 variable 12, 39–41, 52–54 statistical parallaxes 73 Stefan-Boltzmann law 409 stellar collapse !supernova: type II stellar evolution bibliography 24 descriptive overview 23–41 evolutionary track 31 main phases 30 stellar oscillations !helioseismology, variable stars, ZZ Ceti stars stellar structure convective 12f equations 5–14 examples for models 29f generic cases 7–10 homologous models 14–16, 549, 552 long-range force: new 113 sterile neutrinos !neutrinos: right-handed Stodolsky’s formula 313–15, 324f Stokes parameters 286 strange quark matter xv, 157–59, 557 structure function Coulomb plasma 106, 222–25 dynamical classical 146f detailed balance 129, 137, 146 formal deﬁnition 136–43, 323 long-wavelength limit 121, 128f, 141–43 Lorentzian model 135, 144–47 nuclear medium 128f, 135, 146–51, 161–63 spin-density 138 static 137, 222–25 subgiant 27, 28sum rules 139–41 Sun !solar neutrino ux activity cycle 373–76, 387f axion spectrum 175 bounds on axion ﬂux 100, 181, 191f time-varying GN549–51 deﬂection of neutrinos 247f energy-loss argument 16f, 20f, 109f, 175f global properties central temperature 7, 353 convective layer 550 distance 341, 372 Kelvin-Helmholtz time scale 8 helium abundance 16f, 351, 549f luminosity 8, 341 magnetic ﬁeld 186, 373–76, 387f, 554, 565 mass 7 plasma properties 599 radius 7, 351 opacity 21, 345, 353–55 positron ﬂux limits 457, 461 spots 186, 373–76 standard model 29, 351–53 x- and -ray ﬂux 458–62 superﬂuidity in neutron stars 58f Superkamiokande (Cherenkov detector) 343, 371, 390–2 supernova !SN 1987A burst observatory 448 core collapse !supernova: type II energetics: particle bounds 483 future 445–48 Kepler’s 39 galactic positron ﬂux 484f rate 39, 446, 487f remnants 38f, 56–58 SN 1054 !38 Tycho’s 39 type I!36, 546 unit (SNu) 488 Subject Index 663 supernova core binding energy 407 characteristics of the medium 398, 505f, 512, 602–5 matter accretion 397, 400, 425f, 503 neutrino cooling !SN 1987A analytic emission models expected detector signal 418 schematic picture 397, 400, 407f spectral characteristics 408–11 time evolution 411–14 neutrino ﬂavor conversion 332–38 particle cooling axion emission 501–4, 508–12 general argument 501–8 numerical studies 508–11, 513–16 structure (numerical model) 505f, 512 supernova: type II !supernova core description 37–39 explosion mechanism 401–5, 436f, 522 neutrino oscillations cooling phase 434–36 explosion mechanism 436f overview 430f prompt burst 432–34 r-process nucleosynthesis 437–42 nucleosynthesis 405–6, 437–42 stellar collapse 395–99 supersymmetric particles dark matter xv–xvi, 22, 558 emission from stars 81, 557f supersymmetry: ﬂavor-changing neutral current 303 SXT satellite 186 T tau neutrino charge 567 dipole moment 278, 455, 457, 476f e+edecay: bounds galactic supernovae 484–86 reactors 455, 457 SN 1987A -rays 480–82 solar positrons 457, 461 mass 256, 258–60, 485f theoretical decay rate 264thermal broadening of beryllium line 350f thermal equilibrium in stars 8 Thomas-Fermi scale 221 transition moment !neutrino dipole moments transverse current 204, 219f transverse gauge 204 transverse part of the ﬂavor polarization vector 314, 325 transverse photon mass 209, 214 triangle condition 153f triangle loop 167f, 530, 536 triangle: MSW 301–3, 386, 434 triple- reaction 33, 80 tritium decay 255 twisting magnetic ﬁeld 308f two-photon coupling neutrinos 245, 265f, 271f various bosons 165–68 U, V units conversion factors 580–82 natural 6 rationalized 184, 580 URCA process 1, 58f, 152–54 vacuum birefringence 183–5, 187f, 190f, 554 vector bosons Compton process 91–93, 98 energy-loss bounds 82, 99, 110–12 long-range force 112–16, 500 virial theorem introduction 6f negative speciﬁc heat 7f VVO eﬀect 387 W weak damping limit (of neutrino oscillations) 326f, 330f weak decay spectrum 348 weak mixing angle 583 Weinberg angle 583 664 Subject Index white dwarfs bounds on neutrino dipole moments 48–50 pseudoscalar bosons 50–52 time-varying GN551 characteristics 9, 42–45, 599 cooling theory 45–47 formation 35f inferred galactic age 43, 51 location in color-magnitude diagram 40f luminosity function 43–45 magnetic 185f mass-radius relationship 9, 42 neutrino cooling 47–50 variable 41, 52–54 vs. red giant core 88Whole-Earth Telescope 53 WIMP xv, 22 X, Y, Z x-rays neutron stars 56–58 particle bounds SN 1987A arion-photon conversion 189 SN 1987A axion decays 483 SN 1987A radiative neutrino decays 467f, 474f, 477–79 solar x-rays 458–62 satellites 56–58, 186, 465f Yohkoh satellite 186 ZZ Ceti stars 41, 52–54