weinberg QFT vol 2

The Quantum Theory of Fields Volume II Modern Applications Steven Weinberg University of Texas at Austin InThe Quantum Theory of Fields Nobel Laureate Steven Weinberg com- bines his exceptional physical insight with his gift for clear exposition to provide a self-contained, comprehensive, and up-to-date introduction to quantum field theory . Volume II gives an up-to-date and self-contained account of the methods of quantum field theory, and how they have led to an under- standing of the weak, strong, and electromagnetic interactions of the elementary particles . The presentation of modern mathematical methods is throughout interwoven with accounts of the problems of elementary particle physics and condensed matter physics to which they have been applied . Topics are included that are not usually found in books on quantum field theory, such as the Batalin-Vilkovisky formalism and its application to renormalization and anomalies in gauge theories ; the back- ground field method ; the effective field theory approach to symmetry breaking ; critical phenomena ; and superconductivity . The book contains original material, and is peppered with examples and insights from the author's experience as a leader of elementary particle physics . Problems are included at the end of each chapter . This will be an invaluable reference work for all physicists and mathe- maticians who use quantum field theory, as well as a textbook appropriate to graduate courses on quantum field theory . TheQuantumTheory o fFields Volume II Modern Applications Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 I R P 40 West 20th Street, New York, NY 100 1 1-421 1, USA 10 Stamford Road, Oakleigh, Melbourne 316 6, Australi a Cambridge University Press 199 6 First published 199 6 Printed in the United States of Americ a Acatalogue record for this hook isavailable from the British Librar y Library of Congress cataloguing in publication data availabl e Volume I ISB N 0 521 55041 7 hardback Volume iI ISBN 0521 55002 5 hardbac k Set of two volumes ISBN 0521 58555 4 hardbac k TAG Content s Sections marked with an asterisk are somewhat out of the book's main line of development and may be omitted in a first reading . PREFACE TO VOLUME I I NOTATIO N 15 NON-A BELIAN GAUGE T HEOR IES 15.1 Gauge Invariancexvii xx 1 2 Gauge transformations ❑Structure constants ❑Jacobi identity ❑Adjoint repre- sentation ❑Yang-Mills theory ❑Covariant derivatives ❑Field strength tensor ❑Finite gauge transformations ❑Analogy with general relativit y 15.2 Gange Theory Lagrangians and Simple Lie Groups 7 Gauge field Lagrangian ❑Metric ❑Antisymmetric structure constants ❑Simple, semisimple, and U(1 )Lie algebras ❑Structure ❑#' gauge algebra ❑Compact algebras ❑Coupling constant s 15.3 Field Equations and Conservation Laws 12 Conserved currents ❑Covariantly conserved currents ❑Inhornogeneous field equations ❑Homogeneous field equations ❑Analogy with energy-momentum tensor ❑Symmetry generator s 15.4 Quantization 14 Primary and secondary first-class constraints ❑Axial gauge ❑Gribov ambiguity ❑Canonical variables ❑Hamiltonian ❑Reintroduction ❑f A° ❑Covariant action ❑Gauge invariance of the measur e 15.5 T heDe Witt-Fa ddeev-Po pov Met hod 19 Generalization ❑f axial gauge results ❑Independence ❑f gauge fixing functionals ❑Generalized Feynman gauge ❑Form of vertice s 15.6 Ghosts 2 4 Determinant as path integral❑Ghostandantighost fie lds❑Feynmanrules for ghosts ❑Modifiedaction ❑Powercounting a ndrenormalizability vii V ill Content s 15.7 BRST Symmetry 2 7 Auxiliary field h, ❑BRST transformation ❑Nilpotence ❑Invariance of new action ❑BRST-cohomology ❑Independence ❑f gauge fixing ❑Application to electrodynamics ❑BRST-quantization ❑Geometric interpretatio n 15.8 Gene ralizations of BRST Symmetry' 36 De Witt notation ❑General Faddeev-Popov-De Witt theorem ❑BRST transfor- mations ❑New action ❑Slavnov operator ❑Field-dependent structure constants ❑Generalized Jacobi identity ❑Invariance of new action ❑Independence ❑#' gauge fixing ❑Beyond quadratic ghost actions ❑BRST quantization ❑BRST cohomology ❑Anti-BRST symmetr y 15.9 The Batalin-Vilkovisky Formalism` 42 Open gauge algebras ❑Antifields ❑Master equation ❑Minimal fields and trivial pairs ❑BRST-transformations with antifields ❑Antibrackets ❑Anticanonical transformations ❑Gauge fixing ❑Quantum master equatio n Appendix A A Theorem Regarding Lie Algebras 5 0 AppendixBThe Ca rtonCata log Problems References54 58 59 16 EXTERNAL FIELD METHODS 6 3 16.1 T he Quantum Ef fective Act ion 63 Currents ❑Generating functional for all graphs ❑Generating functional for connected graphs ❑Legendre transformation ❑Generating functional for one- particle-irreducible graphs ❑Quantum-corrected field equations ❑Summing tree graph s 16.2 Ca lculation of the Ef fective Potential 6 8 Effective potential for constant fields ❑One loop calculation ❑Divergences El Renormalization ❑Fermin loop s 16.3 Energy Interpretation 72 Adiabatic perturbation ❑Effective potential as minimum energy ❑Convexity ❑ Instability between local minima ❑Linear interpolatio n 16.4 Symmetries of the Effective Action 7 5 Symmetry a nd r enorrnalizatio n❑Slavnov-Tay lor identities❑Linear lyrealized symmetries❑Fermionic fie lds an dcurrents Problem s 7 8 References 78 Contents ix 17 RENORMALIZATION OF GAUGE THEORIES 80 17.1 The Zinn-Justin Equation 80 Slavnov-Taylor identities for BRST symmetry ❑External fields K„(x) ❑An- tibracket s 17.2 Renormalization :Dire ct Anal ysis 8 2 Recursive argument ❑BRST-symmetry condition ❑n infinities ❑Linearity in Kn(x) ❑New BRST symmetry ❑Cancellation ❑#' infinities ❑Renormalization constants ❑Nonlinear gauge condition s 17.3 Renormalization :General Gauge Theorie s' 91 Are `non-renorrnalizable' gauge theories renormalizable? ❑Structural constraints ❑Anticanonical change of variables ❑Recursive argument ❑Cohomology theorem s 17.4 B ackg roundFieldGauge 95 New gauge fixing functions ❑True and formal gauge invariance ❑Renormaliza- tion constant s 17.5 A One-Loop Calculation in Backgronnd Field Gauge 10 0 One-loop effective action ❑Determinants ❑Algebraic calculation for constant background fields ❑Renormalization ❑f gauge fields and couplings ❑Interpre- tation of infinitie s Problems References109 110 18 RENORMALIZATION G ROUPMET HOD S 11 1 18.1 W heredo theLargeLogarithms Co me From? 112 Singularities at zero mass ❑`Infrared safe' amplitudes and rates ❑Jets ❑Zero mass singularities from renormalization ❑Renormalized ❑peratar s 18.2 The SlidingScale 11 9 well-Mann-Law renormalization ❑Renormalization group equation ❑One- loop calculations ❑Application to 04 theory ❑Field renormalization factors C3 Application to quantum electrodynamics ❑Effective fine structure constant ❑ Field-dependent renormalized couplings ❑Vacuum instabilit y 18.3 Va rieties ofAsymptoti cBeha vior 130 Singularities at finite energy ❑Continued growth ❑Fixed point at finite coupling ❑Asymptotic freedom ❑Lattice quantization ❑Triviality ❑Universal coefficients in the beta function x Content s 18.4 Multiple Coupl ings and Mass Effects 139 Behavio r neara fixe dpoint ❑Invariant eigen values ❑Non renarmalizable t heor ies ❑Finite dimensional c ritical surfaces ❑Mass renormalization at zero mass❑ Renormalization g roup equ ations for masses 18.5 C ritical Ph enomena* 14 5 Low wave numbers ❑Relevant, irrelevant, and marginal couplings ❑Phase transitions and critical surfaces ❑Critical temperature ❑Behavior of correlation length ❑Critical exponent ❑4 - edimensions ❑Wilson-Fisher fixed point ❑ Comparison with experiment ❑Universality classe s 18.6 Minimal Subtract ion 14 8 Definition ❑f renormalized coupling ❑Calculation of beta function ❑Applica- tion to electrodynamics ❑Modified minimal subtraction ❑Non-renormalizable interaction s 18.7 Quantum Chromod ynamic s 152 Quark colors and flavors ❑Calculation of beta function 11Asymptotic freedom ❑ Quark and gluon trapping C7 Jets ❑e+-e- annihilation into hadrons ❑Accidental symmetries ❑Non-renormalizable corrections ❑Behavior of gauge coupling ❑ Experimental results for g, and A 18.8 Improved Perturbation Theory' 157 Leading logarithms ❑Coefficients of logarithm s Problem s References158 159 19 SPONTANEOUSLY BROKEN GLOBAL SYMMETRIES 16 3 19.1 D egenerate Vacua 163 Degenerate minima ❑#' effective potential ❑Broken symmetry ❑r symmetric super- positions? ❑Large systems n Factorization at large distances ❑Diagonalization ❑f vacuum expectation values ❑Cluster decompositio n 19.2 Gold stone Bo sons 16 7 Broken global symmetries imply massless bosons ❑Proof using effective potential ❑Proof using current algebra ❑F factors and vacuum expectation values ❑ Interactions of soft Goldstone boson s 19.3 Spontaneously Broken Approximate S ymmetries 1 77 Pseudo-Goldstone bosons ❑Tadpoles ❑Vacuum alignment ❑Mass matrix ❑ Positivity Content s 19.4 Pions as Gold stoneBosonsxi 182 SU(2) x SU(2) c hiral sym metry ❑f qu antum chramodynamics❑Breakdownto isospin ❑Vector a nd axia l-vectorweakcurrents❑Pion decay a mplitude❑Axial form factors ❑f nucleon❑Goldberger -Treimanrelation ❑Vacuum alignment❑ Quark and pion masses ❑Soft pioninteractions ❑Historica lnote 19.5 Ef fective Fie ldTheories: Pions an dNucleons 19 2 Current algebra for two soft pions ❑Current algebra justification for effective Lagrangian ❑6-model ❑Transformation to derivative coupling ❑Nonlinear realization of SU(2) xSU(2) ❑Effective Lagrangian for soft pions ❑Direct justification ❑f effective Lagrangian ❑General effective Lagrangian for pions ❑ Power counting ❑Pion-pion scattering for massless pions ❑Identification ❑f F-factor ❑Pion mass terms in effective Lagrangian ❑Pion-pion scattering for real pions ❑Pion pion scattering lengths ❑Pfion-nucleon effective Lagrangian E, Covariant derivatives ❑gA:~1❑Power counting with nucleons 0Pion-nucleon scattering lengths ❑a-terms ❑Isospin violation ❑Adler-Weisberger sum rul e 19.6 Effective F ield Theor ies: Ge neral Brok enSymmetries 21 1 Transformation to derivative coupling ❑Goldstone bosons and right carets ❑ Symmetric spaces ❑Carton decomposition n Nonlinear transformation rules ❑ Uniqueness ❑Covariant derivatives ❑Symmetry breaking terms ❑Application to quark mass terms ❑Power counting ❑Order parameter s 19.7 Effective Field Theories :SU(3) xSU(3) 225 SU(3) multiplets a ndmatrices❑Goldstonebosons of b roken SU(3) x SU(3) ❑ Quarkmass t erms❑Pseudoscalar meson masses ❑Electromagnetic co rrections ❑Quarkmassratios❑Higherterms in Lagrangian❑Nucleon mass s hifts 19.8 Anomalou sTermsinEffectiveField Th eories' 23 4 Wess -Zumino--Witten term ❑Five-dimensional form ❑Integer coupling ❑ Uniqueness and de Rham cohomolog y 19.9 Unbroken Symmetries 238 Persistent mass conjecture ❑Vafa-Witten proof ❑Small non-degenerate quark masse s 19.10 T he U(1) Pro blem Chiral U(1)symmetry ❑Implications for pseudoscalar masse s Problems243 246 References 247 xii Content s 20 OPERATOR PRODUCT EXPANSIONS 25 2 20.1 T he Expansion : Description an d Derivation 253 Statement of expansion ❑Dominance of simple operators ❑Path-integral deriva- tion 20.2Momentum Flaw' 255 02 contribution for two large momenta ❑Renormalized operators ❑Integral equation for coefficient function ❑0z contribution for many large moment a 20.3 Renormalization Group Equations for Coefficient Functions 263 Derivation and solution ❑Behavior for fixed points ❑Behavior for asymptotic freedo m 20.4 Symmetry Propert iesof Coefficient Functions 265 Invarianceunder spontan eouslybrokensymmetr ies 20.5 Spectral Function Sum Roles 266 Spectral functions defined o First, second, and third sum rules 0Application to chiral SU(N) x SU (N) o Comparison with experimen t 20.6 Deep Inelastic Scattering 2 72 Form factors WI and W2 DDeep inelastic differential cross section 0Bjorken scaling o Parton model oCallan-Grass relation ❑Sum rules ❑Form factors Tl and T2 ❑Relation between Tr and Wr ❑Symmetric tensor operators ❑Twist ElOperators of minimum twist o Calculation of coefficient functions 0Sum rules for parton distribution functions ❑Altarelli-Parisi differential equations o Logarithmic corrections to Bjorken scalin g 20.7 Reaormaloas" 283 Borel summation of perturbation theory ❑Instanton and renormalon obstruc- tions o Instantons in massless 04 theory o Renormalons in quantum chromody- namic s Appendix Momentum F low: The Ge neralCase Problems References288 292 293 21 SPONTANEOUS LYBRO KEN GAUGE SYMMET RIES 29 5 21.1 Unitarity Ga uge 295 Elimination of Goldstone bosons 0 Vector boson masses ❑Unbroken symmetries and massless vector bosons 0 Complex representations oVector field propagator ❑Continuity for vanishing gauge couplings Contents xiii 21.2Renormalizable ~-G auges 3[} 0 Gauge fixing function ❑Gauge-fixed Lagrangian ❑Propagator s 21.3 T he Electroweak T heory 305 Lepton-number preserving symmetries ❑S U(2) x U(1) ElW±, Z0, and photons ❑Mixing angle ❑Lepton-vector boson couplings ❑W i and Z° masses ❑Muon decay ❑Effective fine structure constant ❑Discovery of neutral currents ❑Quark currents ❑Cabibbo angle ❑c quark ❑Third generation ❑Kobayashi-Maskawa matrix ❑Discovery of W± and Z° 0 Precise experimental tests o Accidental symmetries o Nonrenormalizable corrections ❑Lepton nonconservation and neutrino masses ❑Baryon nonconservation and proton deca y 21.4 D ynam icallyBroken LocalSymmetri es* 31 8 Fictitious gauge fields ❑Construction of Lagrangian ❑Power counting ❑Gen- eral mass formula ❑Example :SU(2) xSU(2) ❑Custodial SU(2) xSU(2) ❑ Technicolo r 21.5 Electroweak-Strong Unification 327 Simple gauge groups o Relations among gauge couplings o Renormalization group flow ❑Mixing angle and unification mass ❑Baryon and lepton noncon- servatio n 21.6 S npercondactivityr 332 U(l) broken to Z2 ❑Goldstone mode ❑Effective Lagrangian ❑Conservation of charge ❑Meissner effect o Penetration depth o Critical field El Flux quan- tization ❑Zero resistance o ac Josephson effect ❑Laodau-Ginsburg theory o Correlation length ❑Vortex lines ❑U(1) restoration ❑Stability ❑Type I and II superconductors ❑Critical fields for vortices ❑Behavior near vortex center ❑ Effective theory for electrons near Fermi surface ❑Power counting ❑Introduc- tion of pair field ❑Effective action ❑Gap equation o Renormalization group equations ❑Conditions for superconductivit y Appendix Ge neralUnitarity Gaug e Probl ems References352 353 354 22 ANOMALIES 359 22.1 The noDecay Problem 359 Rate for 7c°--*2y❑Naive estimate ❑Suppression by chiral symmetry ❑ Comparison with experimen t 22.2 Transformation of the Measure : The Abelian Anomaly 3 62 Chiral and non-chiral transformations ❑Anomaly function ❑Chern-Pontryagin density ❑Nonconservation of current ❑Conservation of gauge-non-invariant xis Content s current ❑Calculation of no --+2,' o Euclidean calculation ❑Atiyah-Singer index theore m 22.3 Direct Calculation of Anomalies : The General Case 37 0 Fermion non-conservi ng currents❑Triangle graph calcu lation❑Shiftvectors ❑Symmetric anomaly❑Bardeenform ❑Adler-Bard een th eore m❑Massive fermions ❑Another approach❑Global anomalie s 22.4 Anomaly-Free Gauge Theories 383 Gauge anomalies must vanish ❑Real and pseudoreal representations ❑Safe groups ❑Anomaly cancellation in standard model ❑Gravitational anomalies ❑ Hypercharge assignments n Another U(1 ) 22.5 Ma ssless Bound S tates' 389 Composite quarks and leptons? ❑Unbroken chiral symmetries ❑`t Honft anomaly matching conditions o Anomaly matching for unbroken chiral SU(n) X 5U(n) with SU(N) gauge group ❑The case N = 3 ❑ ❑Chiral S U (3) x SU(3) must be broken ❑'t Hooft decoupling condition ❑Persistent mass conditio n 22.6 C onsistency Conditions 396 Wess-Zumino conditions ❑BRST cohomology ❑Derivation of symmetric anomaly ❑Descent equations ❑Solution of equations ❑Schwinger terms ❑ Anomalies in Zinn-Justin equation ❑Antibracket cohomology ❑Algebraic prof of anomaly absence for safe group s 22.7 Anomal iesand Gold stone Bo sons 40 8 Anomalymatch ing❑Solution of a nomalous Slav nov-Taylo r identities❑Unique- ness ❑Anomalo us Go ldstone boson i nteractio ns❑The caseSU(3) x SU(3) ❑ Derivationof W ens- Z umino-Witte n int eraction❑Eval uationof integercoeffi- cient ❑Generalization Probl ems References416 417 23 EXTENDED F IELDCONFIGURATIONS 42 1 23.1 Th e Uses o f Topology 422 Topological classifications ❑Homotopy ❑Skyrmions ❑Derrick's theorem ❑ Domain boundaries ❑Bagamol'nyi inequality ❑Cosmological problems ❑In- stantons ❑Monopoles and vortex lines ❑Symmetry restoratio n 23.2 Homotopy Groups 430 Multiplication rule for rcl (~~} ❑Associativity F1 Inverses ❑7[,(SI) 1:1Topological conservation laws ❑Multiplication rule for71k(,,#) ❑Winding number Content s 23.3 Monopole sxV 436 S U(2 )/ U(1) model ❑Winding number ❑Electromagnetic field ❑Magnetic monopole moment ❑Kronecker index ❑`t Hooft-Polyakov monopole ❑Another Bogomol'nyi inequality ❑BPS monopole ❑Dirac gauge ❑Charge quantization ❑GI(H' x U (l))monopoles o Cosmological problems ElMonopole-particle interactions o G/H monopoles with G not simply connected ❑Irrelevance of field conten t 23.4 Th e Cartan-Manrer Integral In variant 445 Definition of the invariant ❑Independence of coordinate system ❑Topological invariance ❑Additivity o Integral invariant for S 1 ~--* U(l) El Butt's theorem ❑ Integral invariant for S3 ~--> SU(2 ) 23.5Iastaatoa s 45 0 Evaluation of Carton-Maurer invariant ❑Chern-Pontryagin density o One more Bogomol'nyi inequality ❑v = 1 solution ❑General winding number ❑Solution of U[1] problem ❑Baryon and lepton non-conservation by electroweak instantons ❑Minkowskian approach n Barrier penetration ❑Thermal fluctuation s 23.6 T he Theta A ngle 455 Cluster decomposition ❑Superposition of winding numbers ❑P and CP non- conservation o Complex Permian masses ❑Suppression of P and CP non- conservation by small quark masses ❑Neutron electric dipole moment ❑Peccei- Quinn symmetry ❑Axions ❑Axion mass ❑Ammon interaction s 23.7 Quantum Fluctuations around Extended Field Configurations 462 Fluctuations in general ❑Collective parameters ❑Determinental factor ❑Cou- pling constant dependence ❑Counting collective parameter s 23.8Vacuum D ecay 464 False and true vacua ❑Bounce solutions ❑Four dimensional rotational invari- ance ❑Sign of action ❑Decay rate per volume El Thin wall approximatio n Appendix A Euclidean Path Int egrals AppendixB AList of H omotop yGroup s Probl ems References AUTHOR INDEX468 472 473 474 478 SUBJECT INDEX 484 xvi Content s OUTLINE OF VOLUME I 1 HISTORICAL INTRODUCTIO N 2 RELATIVISTIC QUANTUM MECHANICS 3SCATTERING THEOR Y 4 THE CLUSTER DECOMPOSITION PRINCIPLE 5 QUANTUM FIELDS AND ANTIPARTICLE S 6THE FEY NMAN RULE S 7 THE CANONICAL FORMALISM 8 ELECTRODYNAMIC S 9 PATH-INTEGRAL METHOD S 10 NON-PERTURBATIVE METHOD S 11 ONE-LOOP RADIATIVE CORRECTIONS IN QUANTUM ELEC"I'RO- DYNAMIC S 12 GENERAL RENORMALIZATION THEORY 13 INFRARED EFFECT S 14 BOUND STATES INEXTERNAL FIELDS Preface To Vo lumeII This volume describes the advances in the quantum theory of fields that have led to an understanding of the electroweak and strong interactions of the elementary particles . These interactions have all turned out to be governed by principles of gauge invariance, so we start here in chapters 15-17 with gauge theories, generalizing the familiar gauge invariance of electrodynamics to non-Abelian Lie groups . Some of the most dramatic aspects of gauge theories appear at high energy, and are best studied by the methods of the renormalization group . These methods are introduced in Chapter 1$, and applied to quantum chromodynamics, the modern non-Abelian gauge theory of strong in- teractions, and also to critical phenomena in condensed matter physics . Chapter 19 deals with general spontaneously broken global symmetries, and their application to the broken approximate SU(2) xS U(2) and SU(3) x SU(3) symmetries of quantum chromodynamics . Both the renor- malization group method and broken symmetries find some of their most interesting applications in the context of operator product expansions, discussed in Chapter 20 . The key to the understanding of the electroweak interactions is the spontaneous breaking of gauge symmetries, which are explored in Chap- ter 21 and applied to superconductivity as well as to the electroweak interactions . Quite apart from spontaneous symmetry breaking is the possibility of symmetry breaking by quantum-mechanical effects known as anomalies . Anomalies and various of their physical implications are presented in Chapter 22 . This volume concludes with a discussion in Chapter 23 of extended field configurations, which can arise either as new ingredients in physical states, such as skyrmions, monopoles, or vortex lines, or as non-perturbative quantum corrections to path integrals, where anomalies play a crucial role . It would not be possible to provide a coherent account of these de- velopments if they were presented in a historical order . I have chosen instead to describe the material of this book in an order that seems to me to work best pedagogically - I introduce each topic at a point wher e xvii xviii Prefac e the motivation as well as the mathematics can be understood with the least possible reference to material in subsequent chapters, even where logic might suggest a somewhat different order . For instance, instead of having one long chapter to introduce non-Abelian gauge theories, this material is split between Chapters 15 and 17, because Chapter 15 provides a motivation for the external field formalism introduced in chapter 15, and this formalism is necessary for the work of chapter 17 . In the course of this presentation, the reader will be introduced to various formal devices, including BRST invariance, the quantum effec- tive action, and homotopy theory . The Batalin-Vilkovisky formalism is presented as an optional side track . It is introduced in Chapter 15 as a compact way of formulating gauge theories, whether based on open or closed symmetry algebras, and then used in Chapter 17 to study the cancellation of infinities in `non-renormalizable' gauge theories, including general relativity, and in Chapter 22 to show that certain gauge theo- ries are anomaly-free to all orders of perturbation theory . The effective field theory approach is extensively used in this volume, especially in applications to theories with broken symmetry, including the theory of superconductivity .Ihave struggled throughout for the greatest possible clarity of presentation, taking time to show detailed calculations where I thought it might help the reader, and dropping topics that could not be clearly explained in the space available . The guiding aim of both Volumes I and Ii of this book is to explain to the reader why quantum field theory takes the form it does, and why in this form it does such a good job of describing the real world . Volume I outlined the foundations of the quantum theory of fields, emphasizing the reasons why nature is described at accessible energies by effective quantum field theories, and in particular by gauge theories .(Alist of chapters of Volume I is given at the end of the table of contents of this volume .) The present volume takes quantum field theory and gauge invariance as its starting points, and concentrates on their implications . This volume should be accessible to readers who have some familiarity with the fundamentals of quantum field theory . It is not assumed that the reader is familiar with Volume I(though it wouldn't hurt) . Aspects of group theory and topology are explained where they are introduced . Some of the formal methods described in this volume (such as BRST invariance and the renormalization group) have important applications in speculative theories that involve supersymmetry or superstrings . I am enthusiastic about the future prospects of these theories, but I have not included them in this book, because it seems to me that they require a whole book to themselves . (Perhaps supersymmetry and supergravity will be the subjects of a Volume III .) I have excluded some other interesting topics here, such as finite temperature field theory, lattice gauge calcula- Preface xix Lions and the large Nc approximation, because they were not needed to provide either motivation or mathematical techniques for the rest of the book, and the book was long enough . The great volume of the literature on quantum field theory and its applications makes it impossible for me to read or quote all relevant articles . I have tried to supply citations to the classic papers on each topic, as well as to papers that describe further developments of material covered here, and to references that present detailed calculations, data, or proofs referred to in the text . As before, the mere absence of a citation should not be interpreted as a claim that the material presented is original, but some of it is . In my experience this volume provides enough material for none-year course for graduate students on advanced topics in quantum field theory, or on elementary particle physics . Selected parts of Volumes I and II would be suitable as the basis of a compressed one-year course on both the foundations and the modern applications of quantum field theory . I have supplied problems for each chapter . Some of these problems aim simply at providing exercise in the use of techniques described in the chapter ; others are intended to suggest extensions of the results of the chapter to a wider class of theories . Imust acknowledge my special intellectual debt to colleagues at the University of Texas, notably Luis Boya, Phil Candelas, Bryce and Cecile De Witt, Willy Fischler, Joaquim Gomis, and Vadim Kaplunovsky, and especially Jacques Distler . Also, Luis Alvarez-Gaume, Sidney Coleman, John Dixon, Tony Duncan, aiirg Frohlich, Arthur Jaffe, Marc Henneaux, Roman Jackiw, Joe Polchinski, Michael Tinkham, Cumrun Vafa ., Don Weingarten, Edward Witten and Bruno Zumino gave valuable help with special topics . Jonathan Evans read through the manuscript of this volume, and made many valuable suggestions . Thanks are due to Alyce Wilson, who prepared the illustrations and typed the LATEX input files until I learned how to do it, to Terry Riley for finding countless books and articles, and to Jan Duffy for many helps . I am grateful to Maureen Storey and Alison Woollatt of Cambridge University Press for working to ready this book for publication, and especially to my editor, Rufus Neal, for his continued friendly good advice . STEVEN WEINBER G Austin, Texas December, 1995 Natatio n Latin indices i, j, k, and so on generally run over the three spatial coordi- nate labels, usually taken as 1, 2, 3 . Where specifically indicated, they run over values 1, 2, 3, 4, with x4 -== i t. Greek indices y, v, etc . from the middle of the Greek alphabet generally run over the four spacetime coordinate labels 1, 2, 3, 0,with x° the time coordinate . Greek indices a, fl, etc. from the beginning of the Greek alphabet generally run over the generators of a symmetry algebra . Repeated indices are generally summed, unless otherwise indicated . The spacetime metric q,,,v is diagonal, with elements q11= q22 -X33 = 1>q0a = -1 . The d'Alembertian is defined as ❑-q),''a21axpaxv=❑2 -02/at2,where ❑2is the Laplacian02/OxIOxt. The `Levi-Civita tensor' euvRa is defined as the totally antisymmetric quantity with E0123 = +1 . Spatial three-vectors are indicated by letters in boldface . Three-vectors in isospin space are indicated by arrows . A hat over any vector indicates the corresponding unit vector : Thus, V = V 'VI. A dot over any quantity denotes the time-derivative of that quantity . Dirac matrices yuare defined sothat yuyV +yvyu = 2qu, .Also , y5 = iYoYiY2Y3,and #=iYo-=Y¢ The step function 9(s) has the value +1 for s >0and 0for s c D . Notation xxi The complex con jugate ,transpose ,and Hermitian ad joint of a matrix or vector A are denoted A*, AT,and A t= A`7 ', respectively .The Hermitian adjo int of an operator 0is denoted fi t,except where an asterisk is used to emphas ize that a vector or matrix of operators is not transposed . +H.c.or +c.c,at the end of an expression indicates the add ition of the Hermitian adjoint or complex conjugate of the foregoing terms . A bar on a Dirac spinor u i sdefined b yu=utfl.The anti field of a field xin the Batalin- Vilkovisky formalism is denoted xtrather than x*to distinguish it from the ordinary complex conjugate or the antiparticle field . Units are usually used with h and the speed of light taken to be unity . Throughout -e is the rationalized charge of the electron, so that the fine structure constant is a = e2/47z ^_~ 1/137 . Numbers in parenthesis at the end of quoted numerical data give the uncertainty in the last digits of the quoted figure . Where not otherwise indicated, experimental data are taken from `Review of Particle Properties,' Phys . Rev .D50, 1173 (1994) . 15 Non-AbelianGauge T heories The quantum field theories that have proved successful in describing the real world are all non-Abelian gauge theories, theories based on principles of gauge invariance more general than the simple U(1)gauge invariance of quantum electrodynamics . These theories share with electrodynamics the attractive feature, outlined at the end of Section 8 .1, that the existence and some of the properties of the gauge fields follow from a principle of invariance under local gauge transformations . In electrodynamics, fields ip,(x) of charge en undergo the gauge transformation y)n(x) -- + exp(ienA(x)}Y)n(x) with arbitrary A(x) . Since a A)n(x)does not transform like y),(x), we must introduce a field AFz(x) with the gauge transformation property A,,(x) --- ),A,,(x)+a,n(x), and use it to construct a gauge-covariant derivative a uVn(x)-ienA ,(x)yyn(x ), which transforms just like tpn(x) and can therefore be used with y}n(x) to construct a gauge-invariant Lagrangian . In a similar way, the existence and some of the properties of the gravitational field g uy(x) in general relativity follow from a symmetry principle, under general coordinate transformations . Given these distinguished precedents, it was natural that local gauge invariance should be extended to invariance under local non-Abelian gauge transformations . In the original 1954 work of Yang and Mills, the non-Abelian gauge group was taken to be the S ZT (2)group of isotopic spin rotations, and the vector fields analogous to the photon field were interpreted as the fields of strongly-interacting vector mesons of isotopic spin unity . This proposal immediately encountered the obstacle that these vector mesons would have to have zero mass, like photons, and it seemed that any such particles would already have been detected . Another problem was that, like all strong-interaction theories at that time, there was nothing tha t Of course, both local gauge invariance and general covariance can be realized in a trivial way, by taking .A,(.x) and g,,,(x) to be non-dynamical c-number functions that simply characterize a choice of phase or coordinate system, respectively . These symmetries become physically significant when we treat A,(x) and g,,,(x) as dynamical fields, over which we integrate in calculating 5'-matrix elements . 2 15 Non-Abelian Gauge Theorie s could be done with it ; it seemed that the large coupling constant of the theory would preclude any use of perturbation theory . Gauge theories were soon generalized to arbitrary non-Abelian gauge groups,2 and their quantization continued to be studied mathematically, notably by Feynman,3 Faddeev and Popov,' and De Witt,5 in part as a warming up exercise for the harder problem of quantizing general relativity . They showed that the naive Feynman rules obtained by simply inspecting the Lagrangian need to be supplemented byadditional `ghost' loops . However, the physical relevance of these theories did not begin to be understood until the late 1960s . It eventually turned out that all of the observed interactions of elementary particles are generated by vector fields associated with local gauge symmetries ; the corresponding spin 1 particles are either very heavy, as a result of a spontaneous breakdown of the gauge symmetry, or are `trapped', as a result of the rise of the coupling constant at long distances . These matters will be the subjects respectively of Chapters 21 and 18 . In this chapter we shall explore the formulation of the non-Abelian gauge theories, and the derivation of their Feynman rules . 15.1GaugeInvaria nce We assume that the Lagrangian of our theory is invariant under a set of infinitesimal transformations on the matter fields ip/(x ) bwe(x)=ie"(x)(ta)em1Prn(x), (15.1.1) with some set of independent constant matrices** tIX, and with real in- finitesimal parameters e'(x) which (as for gauge transformations in elec- trodynamics) are allowed to depend on position in spacetime . We assume that these symmetry transformations are the infinitesimal part of a Lie group ; as shown in Section 2 .2, this requires that the t,,, obey commutation relations [t,X, tfi] = i C 7l~ty , (15.1.2) where Gya flare a set of real constants, known as the structure constants of the group . The antisymmetry of the commutator immediately tells u s In this book we shall generally label symmetry generators with letters a, P, etc . from the beginning of the Greek alphabet, in order to keep these labels distinct from the indices fie, v, etc . from the middle of the Greek alphabet that arc used to label spacetime coordinates . Later, in dealing with broken symmetries, we will often use letters a, b, etc. from the beginning of the Latin alphabet to label generators of spontaneously broken symmetries, and letters i, j, etc . from the middle of the Latin alphabet to label generators of unbroken symmetries . 15.1 Gauge .Invar iance that the structure co nstants are simi larly antisymmetric : Cyan = -C'#x. Also, from the Jacobi identit y 0=l[txat#], td+1[ty,ga],t #]+[[tp.tyl,ta] we see that the Cs sat isfy the further constrain t O=C100cEbY + C5Y~~,60+C5fl2Ce5a3 (15.1.3) (15.1.4) (15.1.5) Any set of co nstants Cy,,# thatsatisfy Eqs . (15 .1.3) an d(15.1.5)define at least o ne set of matrices tA,, : (t'd'a),y - -i C flya a (15.1.6) that satisfy the commutation relations (15.1.2)with structure constants CYa#: [tA,", tAP] = i CyIX#tAy ( 15.1.7) This is known as the `adjoint' (or `regular') representation of the Lie algebra with structure constants Cx #y. For example, in the original Yang-Mills theory, the matter fields were the doublet consisting of proton and neutron fields ippand yi n VnPP and th e tx wit ha=1, 2, 3 were t he isos pinmatrice s 1 4 1 1 4 -i ~ 1 0 t1 ~ 1 4 ~2 ~ 2t 00201 These satisfy the commutation relations (15.1.2)with C,ia~= E'yags where as usual is +1 or -1 if y, cc, Pis an even or an odd permutation of 1, 2, 3, respectively, and vanishes otherwise . We recognize this as the same as the Lie algebra (2 .4.18) of the three-dimensional rotation group ; the matrices to here furnish what we recognize as the spin 1 /2 representation of this Lie algebra . The matrices (15.1.6)of the adjoint representation are here (in a basis with rows and columns labelled 1,2,3) : 0 0 to = 00 0i0 -r 0t2 =0 0 -10 i 0 0 0 00-i0 t3 = i 00 0 0 0 This is the spin 1 representation of the Lie algebra of the rotation group . Now consider what is needed to make the Lagrangian invariant unde r the transformations (15.1.1).If there were no derivatives acting on the 4 15 Non-Abelian Gauge Theorie s fields, the task would be easy -any function of the matter fields that was invariant under the transformation (15.1.1)with FIX constant would also be invariant with elarbitrary real functions of the spacetime coordinates . This is not the case if the Lagrangian involves derivatives of the fields (as it must), because with position-dependent functions Fx(x), the derivatives of the matter fields do not transform like the fields themselves . Differentiating Eq.(15.1.1)gives 6(a,w,(x))= iF~(x)( t.)~ y"(auwm(x)) + i(apEa(x)) m Wm(x) .(15.1-8) To make the Lagrangian invariant, we need a field A",,, whose transfor- mation rule involves a term a.Ea, which can be used to cancel the second term in Eq . (15 .1.8). Since this field carries an a-index, we would expect it also to undergo a matrix transformation like Eq .(15.1.1),but with t'X replaced with the adjoint representation matrices (15.1.6).Let us therefore tentatively take the transformation relation of these new `gauge' fields a s or, using Eq . (15 .1.), This allows us to construct a `covariant derivative' :f (Dp y)(x)}r= (x)- i A ~u(x)(t#)e mv)m(x) (15 .1.9) (15.1.10) As planned, the term a,d in the transformation of A fi,in the second term of Eq .(15. 1.10)cancels the term proportional to a udin the transformation of the first term, leaving us wit h cS~D~'1f1}r = i ea (tIX)e'au~)yn - t C #yjEaAyF~(~~),,mWm +AY,(0( m ltoclm ' Wn or, using Eq .(15.1.2), (15.1.11) so that Nv transforms just like w itself . We also need to worry about derivatives of the gauge field . In order to eliminate the term O„audin the transformation of 0,,A#,U, we antisym- metrize with respect to µ and v, just as in electrodynamics . However, we still have terms in the transformation of avA#IU - a .Aflyproportional to first derivatives of E(x), arising from the second term in Eq .(15.1.9).The easiest way to construct a `covariant curl', F7 VP in whose transformatio n t As discussed in the next section, in writing Eq . (15 .1.1 d)we are tacitly supposing that any coupling-constant factors like the electric charge are included in the tfl, and hence also in the structure constants . 15.1Gauge Invariance 5 rule all such derivatives of F(x) cancel, is to consider the commutator of two covariant derivatives acting on a matter field y p- ([Dr . D,]1P)r =_"i(t]')lmFyvuwm where(15.1.12) (15.1.13) Eq.(15.1.12) makes it obvious that Fyvu must transform just like a matter field that happens to belong to the adjoins representation : SFP yt, - ie °`(tAa)fl-/Fy„),= cxCfljIXFY,,M. (15.1.14) The reader may check by direct calculation (using the relation (15.1.5)) that the quantity PVp defined in (15.1.13) actually has the simple trans- formation rule (15 .1. 14). For some purposes, it is useful to know that these infinitesimal gauge transformations can be upgraded to finite transformations . A group element can be parameterized by a set of real functions AI(x) so that it acts on a general matter field V((x) through the matrix transformatio n Vie(X) `YYA (x} = [exp (itA'(x))]??l(x) . We want the covariant derivative to transform in the same way : (~~ -- i txA"pA}VA =exp(itaA°`)(a1,-itxA'x)y3 ,(15.1.15) (15.1.16) so we must impose on A"Pthe transformation rule A~-4Al~, with 0Pexp(i t# A #)- it#exp(i txA°` )A~A= -~-i exp(i txna)t,~A # or in other word s taA"pA= exp(i t#A#) tA°`,, exp(-itfiAfl) - i P,, exp(it#Afl)] exp(-it#Afl) . (15.1.17) Eqs.(15.1.15) and (15.1.17) reduce to the previous transformation rules (15.1.1)and (15.1.9)in the limit where AII(x) is an infinitesimal Ea(x) . From Eq .(15.1.17), we can see that by a suitable choice of Afl(x), i t is always possible to make A",uA(x) vanish at any one point, say x = z . (Simply take Ax(z) to vanish, and OAa(x)/axA = -A" u(x)at x = z .) Also, it is always possible to choose Afl(x) so that any one spacetime component of A°` mA(x)vanishes for all a everywhere in at least a finite domain around any given point . For instance, to make A°`3 A(x)vanish, we must solve the set of ordinary first-order differential equations for the parameters Afl(x )a3 exp(it#V) = -i exp(i tflfl#) to A '3 , (15 .1.1$) which always have a solution in at least a finite domain around any ordinary point . 6 15Non-Abelian gauge theorie s However, in general it is not possible to choose 11°`(x) to make all four components Aapn(x) vanish in a finite region . For this purpose, we would have to be able to satisfy the partial differential equation s a.exp(rt~nfl)= -i exp(i tfl AP) toA°Gp, (15.1.19) which cannot be solved unless certain integrability conditions are satisfied . In particular, if A°`,uA(x) vanishes everywhere then so does F"~~vVx}, but since the field strength transforms homogeneously, F"uvA(x) can vanish only if PJt,,(x) does . A gauge field A",,{x} is called a `pure gauge' field if there exists a gauge transformation which makes it vanish everywhere . It is not difficult to show that the condition that PUv should vanish everywhere is not only necessary but also sufficient for Aa,,(x) to be expressible in any simply connected region as a pure gauge field .6 There is a deep analogy between the construction here of objects that transform simply under gauge transformations and the construction in general relativity of objects that transform covariantly under general co- ordinate transformations . Just as we use the gauge field to construct covariant derivatives Duyp( of matter fields with the same gauge trans- formation properties as the matter fields themselves, so we use the affine connection I-'i,,,Ax) to construct covariant derivatives of tensors TPd-,a...: 9... which are themselves tensors . Also, from the derivatives of the gauge field we constructed a field strength FI)I„ with the gauge transformation property of a matter field belonging to the adjoins representation of the gauge group ; correspondingly, from the derivatives of the affine connection we may construct a quantity : ar~ ar'~~ ~v _UK + rn r I-rn r~~v~ _ ~ axe axV ~v ~~ ~~ v~ which transforms as a tensor, the R iemann -Christoffel curvature tensor . The commutator of two gauge-covar iant der ivatives Dyand D,,may be expressed in terms of the field-strength tensor P~,v ; similarly ,the commutator of two co variant deri vativeswith respect to x'andXKmay be expressed in terms of the curvature : The necessary and sufficient condition for the existence of a gauge in which the gauge field vanishes in a finite simply connected region is the vanishing of the field-strength tensor, and the necessary and sufficient condition for the existence of a coordinate system in which the affine connection vanishes 15.2Gauge Theory Lagrangians and Simple die groups 7 in a finite simply connected region is the vanish ing of the Riemann - Christoffel curvature tensor . The analogy breaks down in one important respect :in general relativity the a ffine connection is itself constructed from first de rivativesof the metric tensor ,while in gauge theor ies the gauge fields are not expressed in terms of any more fundamental fields . 15.2 Gauge T heory Lagra ngians andSimple Lie Grou ps The transformation rules of the gauge-field tensor F°Cjj,, and the matter fields ip and their gauge-covariant derivatives do not involve the deriva- tives of the transformation parameters E°L(x), so if the Lagrangian is con- structed solely from these ingredients, and if it is invariant under global transformations with e' constant, then it is invariant under gauge trans- formations with general position-dependent Fa(x) . We therefore assume that the Lagrangian satisfies these conditions : that is , Y(V', D,,, ip,j}vr}~,W,..., P11, I }pf'apv...} with the invaria nce co ndition: w/ O('Uwe) a~ aLP a(DvD, .'P",) ~y aLP+ 0Dp~fl'Uy CPYaI}pF'IVP+ 0 .(15.2.1) (15.2.2) On the other hand, the Lagrangian may not depend on the gauge field it- self, except insofar as it appears in v and in gauge-covariant derivatives D.In particular, a mass term - ? rn~'0f #Aa,,A#Pis ruled out . We shall concentrate now on the terms in the Lagrangian that depend only on F .Just as in electrodynamics, for any massless particle of unit spin the Lagrangian must contain a free-particle term quadratic in 0PA"v - Ov Aar, and gauge invariance then dictates that this free-particle term should appear as part of a term quadratic in the field-strength tensor F .Lorentz invariance and parity conservation dictate its form a s LPA=- ? gx#FapvF#Pv (15.2.3) with a constant matr ix gad. If we do not assume pa rity (or CP or T ) conservat ion,then we may also include in the Lagrangian a ter m Yt A=-2Bad fPvPff P,,ti,F#p6 with another constant matrix O.This term is actually a der ivative, and therefore does not affect the field equations or the Feynman rules . Such a 8 15 Non-Abelian Gauge Theorie s term would, however, have non-perturbative quantum mechanical effects, tobediscussed in Section 23 .6. Before going on to consider the properties of the matrix gad, it is worth drawing attention to the fact that it is not possible to introduce a kinematic term for the gauge field Axjx} without also including interactions, the terms in Eq .(15.2.3)arising from the quadratic part of the field strength Fay„ defined by Eq .(15.1.13).This is one more respect in which non- Abelian gauge theories resemble general relativity, where the kinematic part of the Lagrangian for the gravitational field is contained in the Einstein-Hilbert Lagrangian density -~R/8nG, which also contains self-interactions of the field . The reasons in both cases are similar : the gravitational field interacts with itself because it interacts with anything that carries energy and momentum, and the gauge field interacts with itself because it interacts with anything that transforms according to a non-trivial representation (in this case the adjoins representation) of the gauge group . This is in contrast to the case of electrodynamics, where the photon does not carry electric charge, the quantum number with which it interacts, and it is consequently possible to introduce a kinematic term -'F,,,F""' for the electromagnetic field that does not entail interactions . The numerical matrix gad may be taken symmetric, and must be taken real to give a real Lagrangian . In order for this term to satisfy the gauge-invariance requirement (15.2.2),we must have for all S gjOFttmv CflY6FYP V=0. In order for this to be true without having to impose any functional relations among the Fs, the matrix gad must satisfy the condition : gaflCO76 = - gyflC1IX6.(15.2.4) There is one more important condition on the matrix g ,,#. Just as in quantum electrodynamics, the rules of canonical quantization and the positivity properties of the quantum mechanical scalar product require that the matrix ga~ in the Lagrangian (15.2.3)must be positive-definite . (That is, gxflu°LuP is positive for all real u, and vanishes for some real u only if u" = 0 for all a .) This is analogous to the requirement that in the kinematic Lagrangian - 2 ZaM0a1`o- Im2 02 for a real scalar field 0, the constant Z must be positive-definite . These requirements on the matrix ga #have far reaching implications . They form one of a set of three equivalent conditions : a: There exists a real symmetric positive-definite matrix gad that satisfies the invariance condition (15 .2.4). b:There is a basis for the Lie algebra (that is, a set of generators ia = with 91a real non-singular matrix) for which the structure 15.2Gauge Theory Lagrangians and Simple Lie Groups 9 constants C "y are antisymmetric not only in the lower indices (3 and y but in all three indices a, ftand y . (In this basis it is convenient to drop the distinction between upper and lower indices a, fl, etc ., and write Cady in place of C 'Py c: The Lie algebra is the direct sum of commuting compact simple and U(l) subalgebras .* Appendix A of this chapter presents a proof of the equivalence of the conditions a, b, and C Before going on to discuss the physical implications of this result, it will be useful to say a bit more about the condition of compactness . We will not use this here, but a compact Lie algebra consists of the generators of a compact Lie group : one for which the invariant volume of the group is finite . For instance, the rotation group is compact ; the Lorentz group is not . As a simple example of a simple Lie algebra that is not compact, consider the commutation relation s Iti,X23=-a t3 , [t 2, t3l=it~~ [0, t iI=i t2. The structure constant here is real, but not completely antisymmetric ; its non-vanishing components ar e X312 = _"C321 = _"1, C'23 = -C132 = 1,C231-= - C213 = 1. The metric given by Eq .(15.A.10)is here diagonal, with elements : 911-X 22= " 'g33 = -2. This is not a positive matrix, so the Lie algebra is not compact . It is in fact the Lie algebra of the non-compact group D(2,1 ), the Lorentz group in two space and one time dimensions . Some definitions : A subalgebra ff of a Lie algebra W is a linear space, spanned by certain real linear combinations t, _ Y,atx of the generato rs to of W, such that ,r is itself a Lie algebra, in the sense that the commutators of the t, with each other are of the form [t„ t3] = ick1 tR, A suba lgebra _~C is ca lled invariant if the commutator of any element of the whole algebra 5 with any e lement of the suba lgebra A" is in the subalgebra -ye, A simple Lie algebra is one wit hout invariant subalgebras . A U(l) subalgebra of 5 is one with just a sing le generator that commutes with all generators of the whole algebra W . A semi-simple Lie a lgebra is one that has no invariant Abelian subaEgebras, i .e., invariant subalgebras whose generators al lcommute with each other . Semi-simple Lie algebras are direct sums of simple (but not U(1)) Lie algebras . A simple or semi-simple Lie algebra is said to be compact if the matrix Tr {tAa tAfl I__Cy "6 C6 #"'is positive-definite . The meaning and importance of the properties of simplicity and compactness w ill be discussed furt her be low. In saying that a Lie a lgebra Wis a d irect sum of subgroups 'it is meant that it is possib le to find a basis for W with generators t„Q for which the structure constants take the for m /r ( n)r C na mb -~~m ~m n~ ab , where C('} ',6 is the structure constant of the subalgebra .fin. 10 15Non-Abelian gauge Theorie s Two sets of generators that differ by a real non-singular linear transfor- mation are considered to span the same Lie algebra, and generate the same group . This is not true for complex linear transformations of generators . In particular, any simple Lie algebra can be put into a compact form by a change of phase of the generators in a suitable basis . For instance, for the Lie algebra of the above example, it is only necessary to define new generators ti = iti, t2 -- its, t3 = t3, for which the commutation relations are K' 61 =i t" The structure constant is now real and totally antisymmetric : Ca bc =Eabc. Here gig = 2SRb, and the algebra is compact . We recognize this, of course, as the familiar algebra of the compact group 0(3)of rotations in three dimensions . To see that this is always possible for any simple Lie algebra, note that the matrix gib defined by Eq .(15.A.10) is real, symmetric, and non-singular, so that by a real orthogonal transformation it may be put in a diagonal form with non-zero elements along the main diagonal . It is then only necessary to multiply all the generators that correspond in this basis to the negative diagonal elements of gabby factors i . We note without proof that the finite-dimensional representations of compact Lie groups are all unitary, and the finite-dimensional repre- sentations of compact Lie algebras are correspondingly all Hermitian . Furthermore, it is easy to see that the only Lie algebras that can have any non-trivial representation by independent finite-dimensional Hermi- tian matrices t,, are direct sums of U{1} and compact simple Lie algebras . To show this, we may simply defin e g(X# = T ry t'Xtfl } Thismatrix is obvio uslypositive-definite, because gx#uxufl = Tr€(Uxta)21 is posi tive for any rea lu' a nd vanis hes on ly if u 't~ = 0, whichis no t possi bleunless a llu" va nish becausethe t" are ass umedindependent. Furthermorethis gad sat isfies theinvaria nce con dition (15 .2.4), as ca n be seen bymult iplying the commu tationrelation(15.1.2) wi th t6and t aking thetrace ; this give s i Cya #Tr €tytS j= Tr €[ta, to ]t6} = Tr €tStat fl- t#t,t6}, which is obviously antisymmetric in fland S . Having verified a, we can rely on the above theorem to infer condition c, so that the Lie algebra must be a direct sum of compact simple and U(1) subalgebras . Let's now return to the physics of gauge theories . In this section we have inferred the existence of a positive symmetric real matrix gx P, that satisfies the invariance condition (15.2.4), from the necessity of constructing a suitable kinematic term in the Lagrangian for the gauge field, and in 15.2Gauge Theory Lagrangians and Simple die Groups it Appen dix A of this chapterwehave s hown t hatthis resu ltis equivalent to a co ndition on theLie alge bra,thatitis a directsum of com pact simp le andU(1) subalgebras . For o ur purposes, t he im portantthing abo utthis result is that the sim ple Lie algebras are a llof cer tain limited types and dimensionalities. For i nstance, itis easy to see that theseisno simp leLie algebra wi th lessthan t hree genera tors, because in one or twodimensions there can be no no n-zero tota lly antisymme tric s tructure co nstants w ith three indices. Wi th t hree ge nerators, a ninvaria ntsubalgebra ca nbe avoi ded bytakingC312, Cz 31, and C12 3allnon-zero .In the basis in w hich the structure cons tantis rea land totally an tisymmetric, t hereis obv iously only one poss ibilit y: Here c is an arbitrary non-zero real constant, which can be eliminated by a change of scale of the generators, t, --+t,/c, so the Lie algebra i s [ta, tfi]=i E'~flyty. This may be recognized as the Lie algebra of the three-dimensional rotation group 0(3), and also of the group SU(2) of unitary unimodular matrices in two dimensions, and was used as a basis for the original non-Abelian gauge theory of bang and Mills . Continuing in the same way, it can be shown that there are no simple Lie algebras with 4, S, 6,or 7 generators, one with 8generators, and so on . Mathematicians (notably Killing and E . Cartan) have been able to catalog all simple Lie algebras . The compact forms of the simple Lie algebras form several infinite classes of algebras of the `classical' Lie groups -the unitary unimodular, unitary orthogonal, and unitary symplectic groups -plus just five exceptional Lie algebras . This catalog is presented in Appendix B of this chapter . It is also shown in Appendix A that under the equivalent conditions a, b, or c, the metric takes the for m gmu,rab ` gm2 6mra6ub [~-S .~.S} with real gam, where m and n label thesimple or U(l) subalgebras, and a and b label the individual generators of these subalgebras . We can eliminate the constants g.2 by a resealing of thegauge field s Ama __*Ama = gm 1 A ma (15.2.6) but then in order to keep the same formulas (15 .1.10) and (15 .1.13) for D,,W and Fay'', we must also redefine the matrices tx and the structure constants tma --*ima = gmtmn -) (15.2.7) C CUh ~ Cab=gmc Ch. (15.2.8) 12 15Non-Abelian Gauge Theorie s Thatis, we can always define the sca le of the gauge fie lds (now dro pping the tildes) so that gm i nEq. (15 .2.5) is unity : gad = d ap, (Z 5 .2.9) butthen th e tra nsformation ma trices ta a nd th e structure co nstants Cxfly contain an unknown mu ltiplicative factor g„, for eac hsimple or U{ 1} subalgebra . These factors are t hecoupling constants of the gauge theory . Alternatively,it is sometimes more conve nientto adoptsome fixe d th ough arbi trary norma lizationforthe t, an dstruct ure cons tants wit hineach simple or U{ 1} subalgebra, i nwhichcase t he coup ling consta nts appear inthe gauge -field L agra ngian(15.2.3) as t he fac tors gm z inEq. (15 .2.5). 15.3 Fie ldEquations andConservation Law s Using Eq. (15 .2.9)for the matr ix gad inEq. (15 .2.3),the fullLagra ngian densityis (15.3.1) where in t he absence of ga uge fie ldsYM {y),ay)} wo uld bethe `ma tter' Lagra ngiandensity. We cou ld, in principle, include adependence of YM on Fx,,,, as we llashighercova riantderivativesD,,Dmzp, D).Fapv, etc .,but we exc ludethese non-reno rmalizable terms here for the same reaso nas in electrodynamics : asdiscusse d in Section 12.3, such terms wo uld be highly suppresse dat or dinary e nergies by negative powers of some very large mass . For t hisreaso nthe stan dardmodelofthe weak, e lectromagnetic andstronginterac tionshas a Lagrang ian of t he ge neralform{15.3.1). The equations of mo tionofthe gauge fie ldarehere ~ O°P = -r~~`Fay" -aY ~ ~(OAv) Ma v _ -F ,°'Ca#A#,,-iOYM~D ta V3 vw and so yU V where f," is the current : OYM vu The current fa'is conserved in the ordinary sens e Ovfa v= 0,(15.3.2) (15.3.3) (15.3-4) 15.3 Field Equations and Conservation Laws 13 as can be seen ei ther from t he Euler-Lagrange eq uations for W an d theinvar iance e quivalent(15.2.2) or, more easi ly,directly from the fie ld equations (15 .3.2). ThederivativesinEqs. (15 .3.2) a nd(15.3.4) are o rdinaryderivatives, not the gauge-covaria ntderivativesDv, so the ga uge invariance of these equations is somew hat obscure.Itcan be ma demanifestby rewri ting Eq. (15.3.2)interms of t he gauge-cova riant deriva tive of the fieldstrength t-A 'UV Then Eq . (15.3.2) read s where.Iavis the currentof the ma tter fie lds alone ice ODvY3 taw{15.3.5} (15.3.6) (15.3.7) This is gauge-covariant, if YM is gauge-invariant . Also, by operating on Eq.(15.3.6)with Dv, using the commutation relatio n [Dv,pjFIP"=-i(tAl )aflFy,,uF#P6 = -C ja#FYvNF #Pa, we see t hat.Ia'' sa tisfies aga uge- covar iantconservatio n law DvJaV = 0, (15 .3.8) ratherthanthe ordinary co nserva tionlaw (15 .3.4) obeyed by the f ull current Kati . Also,itis stra ightforwar d(using E q. (15 .1.5})toderive the identities : (15-3.9) which hold whether or not the gauge fields satisfy the field equations . These results serve to underscore the profound analogy mentioned i n Section 15 .1 between non-Abelian gauge theories and general relativity . In general relativity there is a matter energy-momentum tensor Tv ., analogous to .ILL, which satisfies a generally covariant conservation law Tv,u;,, = 0, and stands on the right-hand side of the Einstein field equations in their generally covariant form, Rv~ - 26VR = -SrrGTv,s . However, T'',Uis not conserved in the ordinary sense : O,, PP does not vanish . On the other hand, moving the non-linear terms on the left-hand side of the Einstein equation to the right-hand side gives a field equation8 RvY- ~6v~,R = -Br~G~`',aLINEAR 14 15 Non-Abelian Gauge Theorie s where zv,, is the non-tenso r v Tv' ` + Br rG Rye-21 6vuR) NONLINEA R analogous to Like z",,is conserved in the ordinary sens e 0,,zvP=0 and may be regarded as the current of energy and momentum : P'U=f -co, d3x . It contains a purely gravitational term, because gravitational fields carry energy and momentum ; without this term, zy,, could not be conserved . Similarly, f' contains a gauge-field term (the first term on the right in Eq . {15 .3.3}) because for non-Abelian groups (those with Cx ,=)60)the gauge fields carry the quantum numbers with which they interact . Because fav is conserved in the ordinary sense, it can be regarded as the current of these quantum numbers, with the symmetry generators given by the time-independent quantities Ta =f,Oi3x. (15.3-10) (Also, the homogeneous equations (15.3.9)involve covariant derivatives, just as do the Bianchi identities of general relativity .) In contrast, none of these complications arises in quantum electrodynamics, because photons do not carry the quantum number, electric charge, with which the y interact. 15.4 Quantizatio n Wenowprocee d to quantize t he gauge theoriesdescribe dintheprevious two sec tions. TheLagrangian density is takeninthe form (15 .3.1): Y = -4Fx uvFa~`v+ Y1~ r(V,DmW ) with F'a,uv = OpAocv-OyAcc,u +Ccc#YA#p`4Yti D'UV~ aluw - itaAa ,,W(15.4.1) We ca nnotimme diatelyquantize t his theory by setting comm utators e qual to i times t he correspo ndingPoisso nbrackets . The problemis one of constraints. Intheterminology of Dirac,describedinSection7.6, there is a primary constraint that15.4Quantization n~oT c~~~ = 0O(O&Att ) a nda seco ndary co nstraintprovide d bythe fieldequat ionfor A ° OP +=OuFa"" + FY *)CYa~~#,u+Jao O(Oy Ax0) OAa015 (15.4.2) (15.4.3) where II aOY1 0(OoAk) =Fad is t he `momen tum' conjuga teto ~la,~, withk ru nnin g over the va lues 1, 2, 3 . The Poisson bracke ts of IIxo an d OkII,,k+IIYkCyx #A#k+.I,,,° vanis h(because thelatterquantityisindependent of AIX O), so these are first c lass co nstraints, whichcanno tbedealtwith b y replacingPoissonbrackets wit hDirac bracke ts. As in the case of electrodynamics, we deal w iththese co nstraintsby choosing a ga uge. The Co ulomb gauge adoptedfor e lectro dynam ics wou ld lead to pa infulcomplica tions here,* so instea dwe wi llwork in whatis knownas axial gauge, basedonthe condition A0= 0. (15 .4.4) The canonical variables of the gauge field are then Aaa, with i now running over the values Iand 2, together with their canonical conjugate s ~: =OY_ -Fad _ OUA~i - OiAxO + C(x#yAfloAyi . { 15 .4.5} O{OoAai} The fie ld Axe is not a nindependent canonicalvariable, but rat heris defined in terms of the ot her va riablesby the co nstraint(15.4.3). To see this, note that t he `electric' fie ldstrengths FPO ar e 0 = H, ~'cc~3 03Aa sothe constraint (15 .4.3) rea ds - {03}2Ao= ai llx, + IZ,liCya#A#i +.IIX(),(15.4.6) {15.4.7} which ca neasilybe solved (w ith reaso nable boundary conditions) to give Axe as a f unctiona lofIIyi , Api, an d.I,(). (We a reusing a s ummatio n In addition to purely algebraic complications, Coulomb gauge (like many other gauges) has a problem known as the Gribov ambiguiry :9 even with the condition that Ax vanishes at spatial infinity, for each solution of the Coulomb gauge condition ❑ AIX - 0 there are other solutions that differ by finite gauge transformations . The Gribov ambiguity will not bother us here, because we quantize in axial gauge where it is absent, and we shall use other gauges like Lorentz gauge only to generate a perturbation series . 16 1 5 N on-Abelian Gauge Theorie s convention, with indi ces i, j, e tc. summe dover t he va luesZand 2 .) It shouldbenoted thatthe canonicalconjuga tetothe matter fie ldV3c is OY OY M O(Oozp,,) 0{Dou~e}(15.4-8) so the time component of the matter current can be expressed in terms of the canonical variables of the matter fields alone Joc°= -ze y, } (toc)lmWm = -z 7 tC(ta)lmWm. (15.4.9) Hence Eq . (15 .4.7) defines A,,° at a given time as a functional of th e canonicalvariablesIIyi, Aft, roc,andWiz at the same t ime. Now t hatwehave i dentified t he canonicalvariables in can procee d tothe constr uctionof a Hamiltonian. Tf density is Af=rIaiOOAai + 7reOOtpl - Ythis gauge, we Hamiltonia n = II,i(F'x0i+OaAuo- Ca#vA#oAyi)+7rcO0Wc Fa0iFa0i + 2 FaijFocij + Z F ~i3F'cci3 - 2 F(x03F'ao3 - YM . (15 .4.10) Using Eqs. (15 .4.4) a nd(15.4.6),this is ~,Y='YeM+ Tlat (OiAao -Ca#YA#oAyt)+2 II,,trl,i + 2 FaijFaij + 2 03Aai O3Axi - Z 0:iAa0 O3Aa 0 where ff M is the ma tterHam iltonian density: AfM= 7tC0OY3,,-Y M .(15.4-11) {15.4.12} Followingthe genera lrulesderive din Sec tion 9 .2, we ca n nowusethis Hamiltonian density to ca lculate ma trix elemen ts as pa thintegrals over Apt, IIIXa, W/, an drr,,, w ith we ighting fac tor exp(iI), w here I = Jd4x[uo0A ai+rtcOpW(--*'+Fterms], {15 .4.13} in w hichthe `F terms' serve o nlyto suppl y the correc t imaginary infin ites- imalterms in propaga tor denominators . (See Sect ion9.2.) We note t hat Eqs. (15.4.7) an d(15.4.9) give AO as a functiona lofthe canonica lvariab les, linear in II ,,i an dter.Inspectionof E q. (15 .4-11) s hows t hen(assuming YM tobenomore than qu adratic inDuW) that theintegrand of the complete act ion(15.4.13) is no more t hanquadraticinI"I,i andrya. We could therefore carry o utthepath int egralover these cano nical`mome nta' by theusualrules of Ga ussian integra tion. Thetrouble wi th thisproce- dure is t hat the coefficie nts of the terms in E q. (15 .4.13) of seco ndorder 1 5.4Quantization 1 7 inII,,i are func tions of the f lea, so the Ga ussian in tegra lwouldyieldan awkwardfield-dependentdeterm inant factor . Also, the whole forma lism at this point loo ks hopelessly non-Lo rentz-i nvaria nt. Instead of procee dingin thi s way, we w ill apply a trick likethatused in the pat h int egra lformu lation of e lectrodynamicsinSect ion9.6. Note thatif for a mome nt we think of A Aas an in dependentvariab le, then the actio n(15.4.13) is ev idently qua dratic in f laa, wi ththe coefficient of the seco nd-order termAar{x}A #a{y} equalto the fie ld-independent k ernel (03)264(X-y). As we saw inthe a ppendix to C hapter 9, theintegra l of sucha Ga ussianover AA(x) is, u p to a co nstant fac tor, equa l tothe value of the i ntegrandatthe stationary ` point' of the arg umen tofthe exponential.But the varia tiona l derivative of the action here is M aI)r002Aao 6Aap OAuO so the sta tionary `poin t' ofthe act ion is the solutionofthe constraint equa- tion (15 .4.7).Hence, ins tead of using for A~o t he solution of Eq . (15 .4.7), we can just as we lltreat it as a n independent var iable of integrat ion. With Aar now regar dedas a nindependent var iable,the Ha miltonian f d3x is ev idently quadratic inIIai, w iththe coefficient of the seco nd- orderterm Ilat(x) II#j(y) give n by the fie ld-independentkernelZ64(x - y)6y . Assuming that the same is true for the matter var iable n,,,, we ca n evaluate pat hintegrals ove rn( and II,,i up to a cons tant fac tor by s imply setting n~ a nd II xi at the stat ionary ` points' of t he act ioncorrespondin g to Eq. (15.4.1}: 51 O'~f'M0 = ~oWr' - ,i-OiAuO+~uflyAfl OAyi =FxOi-rlxiO= bri_OOAui r l xi Inserting these back into Eq . (15.4-13) give s I = Jd4x[yM+ Z Fxoi Fooi - 2 Fcca j F cci j - Z 03Aai O3Axi +Z(03Aa0)2 = JdxY, (15.4-14) whereYisthe Lag rangian(15.3.1) withwhichwe started! Inother wor ds, we are to dopath integrals over Wc{x} and all ./our components of A,,,,(x), with a manifestly covariant weighting factor ex p{ii}given by Eqs . (15 .4-14) and ( 15.3.1),butwith th e axial-ga uge co nditionenforce d byinserting a 18 1 5Non-Abelian Gauge Theorie s factor (Aa3{x} ) x,a As long as OA, CAB    are gauge-invariant, we hav e TtOA(9B ...j)VACUU M ~ ~ ~ dy)I(X) 11 d A,,u(x) C',x IX 1lUIx x(9AOB...expfil + F terms 116 {A,,3{x}} , (15.4. 16) x,oc with Lorentz- and gauge-invariant action I given by Eq . (15 .4.14). For future reference, we note that the volume element fj a,x dAm{x} for the integration over gauge fields in (15 .4.16) is gauge-invariant, in the sense that dAAaN{x} =11dAau(x), (15.4.17) a,11,x a, ,U,x where A A,,(x) is the result of acting on A .. (x)with a gauge transformation having transformation parameters A,,,{x} . It will be enough to show that this is true for transformations near the identity, say with infinitesimal transformation parameters .SIX{x} . In this case , A~ ~ = A~ + [~~`tix + Cagy A~~.,; so the volume elements are related b y dA,~a,,(x)= I]et{ X}rl dAa,,(x) cc,,u,x a, Y,x where S'is the `matrix ' XIXyx,fivY-6~~ a~e (x) 4`x-y) 6~ I6a ►~+cal~y~.y('x)J. #V(y)  The determinant of Xis unity to fi rstorderin;,,,becausethetrace C,,,, vanis hes. In this chapter we shall assume that the volume element lj n,x dWn{x} for the integration over matter fields is also gauge-invariant . There are important subtleties here, to which we shall return in Chapter 22, but as shown there this assumption turns out to be valid in our present non-Abelian gauge theories of strong and electroweak interactions . 15.5 The De W itt-Faddeev -Popov Method 19 15.5 The De Witt-Faddeev-Popov Metho d Our formula (15 .4.16) for the path integral was derived in a gauge that is convenient for canonical quantization, but the Feynman rules that would be derived from this formula would hide the underlying rotational and Lorentz invariance of the theory . In order to derive manifestly Lorentz- invariant Feynman rules, we need to change the gauge . We first note that Eq . (15.4.16) is (up to an unimportant constant factor) a special case of a general class of functional integrals, of the form : fjdOn(X) W [01 B[f[01]Det,~F [01 n,x where 0n(x) are a set of gauge and matter fields ; ]j,,, do,{x} is a volume element ; and 1[0] is a functional of the 0,z{x}, satisfying the gauge- invariance condition : i:,_x n, .C where OAjx} is the result of o perating on 0 wi tha ga uge transfo rmation having parameters %,,{x} . (Usually w henthis is sa tisfiedboth th e functional I andthe vo lume eleme ntare se paratelyinvaria nt,but E q. (15 .5.2)is allweneed here .) A lso, f IX [O; x] is a non-gauge-invariant `ga uge-fix ing functional' of t hese fie lds that a lsodepends onx anda;B [f] is some numer icalfunctional define dfor genera lfunc tions f ,,(x) of x a nda; and .Fis the `matrix' .FIXI#Y6fX 10A ; X](15.5-3) 4(Y)a.=a (In accordance with our usual notation for functionals of functions or of functionals, B [f [ 0]]is understood to depend on the values taken by f',[O ; x] for all values of the undisplayed variables a and x, with the displayed variable, the function 0,{x}, held fixed .)Eq.(15.5.1)does not represent the widest possible generalization of Eq . (15 .4.16); we will see in Section 15 .7 that there is a further generalization that is needed for some purposes . We start here with Eq . (15 .5.1) because it will help to motivate the formalism of Section 15 .7, and it is adequate for dealing with non-Abelian gauge theories in the most convenient gauges . We now must check that the path integral (15 .4.16) is in fact a special case of Eq . {15.5.1}. In Eq . (15 .4.16) the fields 0&} consist of both A,,,{x} and matter fields y3/{x}, an d fx[fl, V); x] = A x3fix}, BLfa =II6(f,~(x)) x,a(15.5.4) (15.5.5) 20 15Non-Abelian Gauge Theorie s I:q[fl,y)]= exp fi I + F terms 1OAOB ... lld0 n(x)=fldwe( x) rldAP(x) n,x l,x la'AIX(15.5.6) (1s.s.7) (We are now dropping the distinction between upper and lower indices oc,P,--.)Comparison of Eq . (15 .4.16) with Eqs . (t 5 .5.E )-(t 5 .5.3)shows that these path integrals are indeed the same, aside from the factor Det .F[0] . For the particular gauge-fixing functional (15 .5.4), this factor is field-independent : if AIX{x} = 0, then the change in Ax(x) under a gauge transformation with parameters ~ .a(x) i s .q~,a(x)=03~a(x)=Id4Y ~IX(Y) 0364(x- Y), so that here Eq .(15.5.3)is the field-independent 'matrix ' The determinant in Eq . (15 .5.1) is therefore also field-independent in this gauge . As discussed in chapter 9, field-independent factors in the functional integral affect only the vacuum-fluctuation part of expectation values and S-matrix elements, and so are irrelevant to the calculation of the connected parts of the S-matrix . The point of recognizing the functional integral (15 .4.16) for non- Abelian gauge theories as a special case of the general path integral (15.5. 1} is that in this form we may freely change the gauge . Specifically, we have a theorem, that the integral (15.5.1)isactually independent (within broad limits) of the gauge-fixingfunctional fa [ 0; x], and depends on the choice of thefunctional B [f ]only through an irrelevant constant factor . Proof :Replace the integration variable 0everywhere in Eq . (15 .5.1) with a new integration variable OA, with A°`{x} any arbitrary (but fixed) set of gauge transformation parameters : J =f~dOnn(x) ~~ [OA] B[f[4)A]] Det~[OA] ra,x(15.5.8) (This step is a mathematical tri viality, like chang ing an integral f x,f(x)dx to read f f {y}dy, and does not yet make use of our assumpt ions regard- ing gauge invariance .)Now use the assumed gauge invariance (15 .5.2)of the measure II dotimes the functional 5[0] to rewrite th is as n,x5[01B [f[OA a]I]etJWr[can] . (15.5.9) j = J[lld4)11x ]) Since Aa(x) was ar bitrary, the lef t-hand sidehere ca nnot dependon it. Integra ting over A°`{x} wi thsome su itable we ight-functionalp[A](to be 15.5 The De Witt-Faddeet~-Popov Metho d chosen below) thus give s .~ J[ndAx)] P[A] fj dOn(x)INIOI C101a a,x n, x where C fjdAa(x)p [n]B[.f[OA l]Det.FLOA1 a,x Now ,Eq. (15 .5.3)gives ,Fax,flyLOA]=6f a [(OA; xl 6 10 (Y) ~),=O21 (15.5.10) (15.5.11) We are assuming that these transformations form a group ;that is ,we may write the result of performing the gauge transformation with parameters Aa(x)followed by the gauge transformation with parameters P(x) as the action of a single `product' gauge transformation with parameter s (OA =Ox(na,) (15.5.13) Using the chain rule of partial (functional) differentiation, we have the n ax,fly[OA]= ffax,,z[0,A]MYz,y[A]d4Z, where Ay (Z) A=A 5 A"(z) and ,Ryzpy[Al5 Ay(z;A,A) 6 )fl (Y) ~ A=O It follows that(15.5.15) (15.5.16) Det,flOn l= Det f[0,Al Det R[A]. (15 .5.17) We note that Det f[0, A] is nothing but the Jacobian of the transforma- tion of integration variables from the A'(x) to (for a fixed 0)thef,' [OA ;X1 Hence, if we choose the weight-function p(A) as p(A) = I 'Det R[A] then C f[iidAa(x) Det f[0,n]B[f[A]] f[fl df7 (x)j B [f C x,x(15.5.18) (15.5.19) 22 15 Non-Abelian Gauge Theorie s which is clearly independent of 0. (Eq .(15.5.18) may be recognized by the reader as giving the invariant (Haar) measure on the space of group parameters .) We have then at las t Cf [FI,,XdO,(x)1Cq1O1 J[H.dAa(x)]fi[n](15.5.20) This is clearly independent of our choice of fJO ; x], which has been reduced to a mere variable of integration, and it depends on B[ f ] only through the constant C, as was to be proved . Before proceeding with the applications of this theorem, we should pause to note a tricky point in the derivation . The integrals in the numerator and denominator of Eq .(15.5.20) are both ill-defined for the same reason . Since ~#[O] is assumed to be gauge-invariant, its integral over 0cannot possibly converge ; the integrand is constant along all `orbits,' obtained by sending 0into OAwith all possible 4°L(x) . Likewise, the integrand in the denominator is divergent, because p(A )IIdA is nothing but the usual invariant volume element for integrating over the group, and this is also constant along `orbits' A --+n(n,A). This divergence can be eliminated in both the numerator and denominator of Eq . (15 .5,4 ) by formulating the theory on a finite spacetime lattice, in which case the volume of the gauge group is just the volume of the global Lie group itself times the number of lattice sites . Because the gauge-fixing factor B[f ] eliminates this divergence in the original definition (15 .5.1) of the left-hand side of Eq . (15 .5.20), we may presume that, as the number of lattice sites goes to infinity, it cancels between the numerator and denominator of the right-hand side of Eq .(15.5.20). Now to the point . We have seen that the vacuum expectation value (15.4.16) in axial gauge is given by a functional integral of the general form (15.5.1).Armed with the above theorem, we conclude then tha t X L'A (nB...exp{i1frldw/(x) rl dAua(x) C,x a, P,x + £term sf.B[f[A,ya]]Det.~[A,iE)](15.5.21) for (almost) any choice of fx [A, V); x]and B [f ] .We are now therefore free to use Eq . (15 .5.21) to derive the Feynman rules in a more convenient gauge . The path integrals that we understand how to calculate are of Gaussians times palynoinials, so we will generally tak e BLf] =exp Jd4Xj-~, (Xy-OW (15 .5.22) 15.5TheDe Witt-Faddeev-Popov Method 23 with arbitrary real parameter ~ . With this choice, the effect of the factor B in Eq .(15.5.21)is just to add a term to the effective Lagrangia n The simplest Lorentz-invariant choice of the gauge-fixing function f,is the same as in electrodynamics fl= r7JUA~. (15.5.24) The bare gauge-field propagator can then be calculated just as in quantum electrodynamics . The free-vector-boson part of the effective action can be written IOA = - ~ d4x {(iAv - 0,,A,,,,)(O1`AaV - OvAxt ) +%(r~~~Ax~}(r7vAxU} + F term s 1f d4X 9ocpx,pv y Aof11(x)Ap v (y) where 2 x~~x,~ivy - ~ uv.t6 4(X -Y) cox0Y)_ 2 l - ~ 64(x - y) +e terms rxµryy, _ ~2 7r)-4'd4 P [11t"(P2- ie) - 1 - - p 'Up~,eip(x-y ) Taking the reciprocal of the matrix in square brackets, we find the prop- agator : tAxJJ V (x,Y) = (9) )xqixx VY -(27r)-4 1 d4p [;ipv+ (~ - 1}E,`py eip ~(x _- ,y) (15.5.25)P p t6 This is a generalization of both Landau and Feynman gauges, which are recovered by taking ~ = 0 and ~ = 1, respectively . For ~ -~ 0, the functional (15.5.22) oscillates very rapidly except near fx=0, so this functional acts like a delta-function imposing the Landau gauge condition O,Al` =0,leading naturally to a propagator satisfying the corresponding condition OF'Da1,,p„ = Q . For non-zero values of ~ the functional B[f] does not pick out gauge fields satisfying any specific gauge condition on the field A,,,,but it is common to refer to the propagator (15.5.25) as being in a `generalized Feynman gauge' or `generalized ~-gauge' . It is often a good 24 15Non-A belianGauge Theorie s strategy to calculate physical amplitudes with ~ left arbitrary, and then at the end of the calculation check that the results are ~-independent . With one qualification, the Feynman rules are now obvious : the contri- butions of vertices are to be read off from the interaction terms in the orig- inal Lagrangian ', with gauge-field propagators given by Eq .(15.5.25), and matter-field propagators calculated as before . To be specific, the trilinear interaction term i n - ~ Caflv(OuAav -OuAam)A fluAvv corresponds to a vertex to which are attached three vector boson lines . If these lines carry (incoming) momenta p, q, k and Lorentz and gauge-field indices pa, vfl, py, then according to the momentum-space Feynman rules, the contribution of such a vertex to the integrand i s i(27L)464(p+ q+k )~-i Caflj] [P!A pAPv + qAt 1vu-'-qut1yA+kuy1Av-kvnAuI (15.5.26) Also,the A 4interaction term in ', v-4Ceap Cc y6 AmpApvAy PAa corresponds to a vertex to which are attached four vector boson lines . If these lines carry (incoming) momenta p, q, k, (, and Lorentz and gauge indices pa, vii, py, and aS,then the contribution of such a vertex to the integrand i s z(27L)464(p+q+ k + OX f _ ...'GapCe~,S~lypqvr~ - ~F Uff~vP )L -CfaY CC6fl01A(F11 pv - qP v717p)-Cca6Cefl'j(t7'U vqpa qppflav)] (15.5.27) (Recall that the structure constants C ,pY contain coupling constant factors, so the factors (15 .5.26)and (15.5.27) are respectively of first and second order in coupling constants .) The one complication in the Feynman rules with which we have not yet dealt is the presence in Eq . (15 .5,21) of the factor Det ~F, which for general gauges is nota constant . We now turn to a consideration of this factor . 15.6 Gho sts We now consider the effect of the factor Det ~F in Eq . (15 .5.22)on the Feynman rules for a non-Abelian gauge theory . In order to be able to treat this effect as a modification of the Feynman rules, recall that as shown in 15.6Ghosts 25 Section 9 .5, the determinant of any matrix may be expressed as a path integra l Det _~Foc J[lldo(x)] [[I dr,~a (x)exp(iIG H}, oc,x x, x where(15.6.1) UGH fd4x d4Y ~~ ( x)Cofl (Y)-~~ax,flY. (15.6.2) Here co,,*, and (o,, are a set of independent anticommuting classical variables, and the constant of proportionality is field-independent . (We have to choose the cc)a and c .)L,* field variables to be fermionic in order to reproduce the factor Det _~F ; had we chosen these field variables to be bosonic, the path integral (15.6.1)would have been proportional to (Det -fl-1 .) The fields cc~a and cc~a are not necessarily related by complex conjugation ; indeed, in Section 15 .7 we shall see that for some purposes we need to assume that co* and co, are independent real variables . The whole effect of the factor Det F is the same as that of including IGH (cv, a)*) in the full effective action, and integrating over `fields' oi and w* . That is, for arbitrary gauge-fixing functionals ,f,(x) , (TtCA ...J)vOC f ridy)n(x) 11dAxu(x) n,x a,U,x x[ndw(x)do ;(x)]exp (IIMOD[4,,A,W,w*]) CAA... (15 .6.3) 1,X where IM Op is a modified actio n jMC]D = fd4x Y-1 ,fa.fa + UGH (15.6.4) The fields wa and wx are Lorentz scalars (at least in covariant gauges) but satisfy Fermi statistics . The connection between spin and statistics is not really violated here, because there are no particles described by these fields that can appear in initial or final states . For that reason, cva and r,)a are called the fields of `ghost' and `antighost' particles . Inspection of Eq. (15 .6.2)shows that the action respects the conservation of a quantity known as `ghost number,' equal to +1 for to,,, -1 for wa, and zero for all other fields . The Feynman rules for the ghosts are simplest in the case in which the `matrix' F may be expressed a s ~F='FO+ ~F, , (15.6.5) where FO is field-independent and of zeroth order in coupling constants, while ~F, is field-dependent and proportional to one or more coupling 26 15Non-Abelian Gauge Theorie s constant factors . In this case, the ghost propagator is jus t A,xp (x, y)_ 1) ,xx,p.v(15.6.6) and the ghost vertices are to be read off from the interaction ter m UGHfd4x d4Y ~~(~)too(.Y)(,F i }ux, flY. (15.6.7) For instance, in the generalized ~-gauge discussed in the previous sec- tion, we have f2 = all A M:x and for infinitesimal gauge parameters :c,Eq.(15.1.9) gives : so that ~. rS aAAa~(x) ax,QY6afl (y)~.=0 64(X y) + C"fl OX, 64( X This is of the form (15.6.5),withAl. (~F05CX'fly = -Cifty [Au(X)j4(X _ Y)] r~.xu(15.6.8) (15.6.9) (15.6.10) (15.6.11) From Eqs . (l 5 .6.6)and ( 15.6.14), we see that the ghost propagator i s A,p(x,y) = 6,# (271)-4f d4p (p' - ie)-' e'P"x-Y) , (15.6.12) so in this gauge the ghosts behave like spinless fermions of zero mass, transforming according to the adjoint representation of the gauge group . Using Eqs .(15. G.7)and (15.6. 11) and integrating by parts, we find that the ghost interaction term in the action is no w IGH = fdx C ,,,, x~~Aµ r~)fl. (15.6.13)r~ This interaction corresponds to vertices to which are attached one outgoing ghost line, one incoming ghost line, and one vector boson line . If these lines carry (incoming) momenta p, q, k respectively and gauge group indices oc,fl, y respectively, and the gauge field carries a vector index p,then the contribution of such a vertex to the integrand is given by the momentum- space Feynman rules a s i(2zc)4CS4(p + q + k)x ip ,,Cpr. (15 .6.14) 15.7 BRST Symmetry 27 The ghosts propagate around loops, with single vector boson lines attached at each vertex along the loops, and with an extra minus sign supplied for each loop as is usual for fermionic field variables . The extra minus sign for ghost loops suggests that each ghost field r~),,, together with the associated antighost field cu,*, represents something like a negative degree of freedom . These negative degrees of freedom are necessary because in using covariant gauge field propagators we are really over-counting ; the physical degrees of freedom are the compo- nents of Ax(x), less the parameters Ax(x) needed to describe a gauge transformation . In summary, the modified action (15 .6.4) may be written in generalized ~-gauge as IMC]D - .~d4x YMC7D (15.6.15) with a modified Lagrangian density : YMOD = Y1vf - 4FaUF aFiv - ~ (aJUA~)(OyAx) -Opc)~P...+Cxflr (c~ycvx } AY (j)p (15.6.16) It is important that this Lagrangian is renormalizable (if the matter La- grangian YM is), in the elementary sense that its terms involve products of fields and their derivatives of total dimensionality (in powers of mass) four or less . (The kinematic term - 0JU(0x 01'w, in Eq, (15.6.16) fixes the dimensionality of the fields cc) and cv* to be mass to the power unity, just like ordinary scalar and gauge fields .) However, there is more to renormalizability than power counting ; it is necessary also that there be a counterterm to absorb every divergence . In the next section we shall consider a remarkable symmetry that will be used in Section 17.2to show that non-Abelian gauge theories are indeed renormalizable in this sense, and that can even take the place of the Faddeev-Popov-De Witt approach that we have been following . 15.7 BRSTSymmetr y Although the Faddeev -Popov -De Witt method described in the previous two sections makes the Lorentz invariance of the theory manifest ,it still rests on a choice of gauge, and hence naturally it hides the underlying gauge invariance of the theory .This is a serious problem in trying to prove the renormalizability of the theory -- gauge invariance restricts the form of the terms in the Lagrangian that are available as counterterms to absorb ultraviolet divergences, but once we choose a gauge ,how do we 28 15 Non-Abelian GaugeTheories know that gauge invariance still restricts the ways that the infinities can appear ? Remarkably, however, even after we choose a gauge, the path integral still does have a symmetry related to gauge invariance . This symmetry was discovered by Becchi, Rouet, and Stora,' ❑(and independently by Tyutin,11) in 1975, several years after the work of Faddeev and Popov and De Witt, and is known in honor of its discoverers as BRST symmetry . This symmetry will be presented more-or-less as it was originally discovered, as a by-product of the method of Faddeev, Popov, and De Witt, but as we shall see it can also be regarded as a replacement for the Faddeev- Popov-De Witt approach . We have seen in Eqs .(15.6.3)and (15 .6.4) that the Feynman rules for a non-Abelian gauge theory may be obtained from a path integral over matter, gauge, and ghost fields, with a modified action, which we may write IM[JD =JEFF +IGH =JC14xYM[JD YlvroD=Y-2~fafx +wa*Ox where we have now introduced the quantit y A,X(x) =fd4YFax,#Y[A, W]t-oQ(Y). This is for the choice 4x,fj(15.7.1) B[f] acexp -2~fd(15.7.2) (15.7.3) (15.7.4) of the gauge-fixing functional in Eq .(15.5.21).For our present purposes, it will be helpful to rewrite BU]]as a Fourier integral : B [f f 11 dha(x)exp i2fha ha exp [if d4 x, faha a,x(15.7.5) We must now do our path integrals over the field ha (often known as a `Nakanishi-Lautrup' field"') as well as over matter, gauge, ghost and antighost fields, with a new modified actio n INEw =f d4x ~Y+ o)xAx + h jIX + Z~h ahIX). (15.7.6) This modified action is not gauge-invariant indeed, it had better not be, if we are to be able to use it in path integrals . However, it is invariant under a `BRST' symmetry transformation, parameterized by an 15.7BRST Symmetry 2 9 infinitesimal constant 0that anticomrrtutes with c.),,c.),*,,and all fermionic matter fields. For a given 0,the BRST transformation i s bow = it"80 ]aY>> csecox = -ah a bea)y_ - ~ a C aflY ~fl~v rSohIX = 0.(15.7.7) (15.7.8) (15.7.9) (15.7.11) (Recall that in fermionic path integrals, there is no connection between co, and w a, so that Eq .(15.7.9)does not need to be the adjoint of Eq . (15 .7.1).} Because hx is BRST-invariant, we could if we like replace the Gaussian factor exp(2 i~ f h~h ,)in Eq .(15.7.5)with an arbitrary smooth functional of h,, yielding an arbitrary functional B [ f ], without affecting the BRST invariance of the action . However, for the purposes of diagrammatic calculation and renormalization it will help to leave B[f] as a Gaussian . In checking the invariance of the action (15.7.1),it will be very useful first to note that the transformation (15.7.7)-(15 .7.11) is nilpotent ; that is, if F is any functional of W, A, w, rte', and h, and we define sF b y rSOF =_OsF then* rSe(sF )=0 or equivalently s(sF) = d.(15.7.12) (15.7.14) It is straightforward to verify this nilpotence when 60acts on a single field . First, acting on a matter field , (SgSZf] =etabB(COM - - 2tCal~Yta BCc}fl(c)yZp - t at#Cc}aBC~~g W - IlCaflYtIXBfiJp[.)Yw+ tat flBLoawfl W The product cc3acvp in the second term on the right is antisymmetric in a and fl, so we can replace t«tp in this term with 2 [tx, t #], and this term thus cancels the first term : SSW=0. (15.7.15) In the original work on BRST symmetry the functional B[f] was left in the form (15.7.4), so that h, was replaced in Eq .(15.7.9)with --f ,,/~, and the BRST transfor- mation was only nilpotent when acting on functions of co, and the gauge and matter fields, but not of col*. 30 15Non- Abelian Gauge Theorie s Next, acting on a gauge field, we hav e bgSAaP = 60 DI►a)(Y = 006 Ct)a+Ca#,cSeApjrwy + CxpvAa~~owv = 0( - 'C'XflYaj'(0jflw'j) + C'XflY(aj'Cqfl)W Y +Cafl.jCp6eA6uwfwY -iCja^C-,6FAfl uw6(0E) 1CxprCjSE146,uo)Fo)Q -1 zCaQiCY6,AQutobCDC~ The first two terms of the final expression cancel because Cxpris antisym- metric in #and y, and the third and fourth terms cancel because of the Jacobi identity (15.1.5),so Eqs.(15.7.9)and (15.7.11) show immediately tha t ssc.)*=0 and SSha = 0. Finally, gscc'x=- ztafl76 e(WflWT ) - ae (C :xpCp6 eW60) EWv +CxQ'jCvaeWfl(r)(W) El I J But w# commutes with w6wE, so this too vanishe s ssW~x =0.(15.7.16) (15.7.17) (15.7.18) (15.7.19) Now consider a product of two fields 01and 02,either or both of which may be y3,A,r.o,r.o*,or h,not necessarily at the same point in spacet ime. Then 6001 02) =O(SO002 +010002) -0[(,301)02 ± 01S02I where the sign ± is plus if 01is bosonic, minus if 01is fermionic . That is , 0102 )=(sO002 ±OI502 Since as we have seen rSg(soj) = 60 (42) = 0, the effect of a BRST transformation on s(OI 02) is 15.7 BRST Symmetry 31 But so always has statistics opposite to 0, so moving H to the left in the first term on the right-hand side introduces a sign factor + 600 1 02)= 0 T(SO 0002) ± (SO 1)(S02)] = 0 - Continuing in this way, we see that BRST transformations are nilpotent acting on any product of fields at arbitrary spacetime points : Any functional F[O] can be written as a sum of multiple integrals of such products with c-number coefficients, so likewis e 50sF[o]= Hss F [o] = 0 . (15 .7.20) This completes the proof of the nilpotency of the BRST transformation . Now let us return to the verification of the BRST invariance of th e action (15.7.6). First note that for any functional of matter and gauge fields alone, the BRST transformation is just a gauge transformation with infinitesimal gauge parameter Aa(x) = 00,),(X) . ( 15.7.21) Therefore the first term in Eq . (15 .7.6) is automatically BRST-invariant : 50 /d4x .=o . (15.7.22) To calculate the effect of a BRST transformation on the rest of the action (15.7.6),note that its effect on the gauge-fixing function is just the gauge transformation (15.7.21), so J d~ .(Y) ) .=o or in terms of the quantity (15.7.3) (15.7.23) (Note that 3w-is a bosonic quantity, so there is no sign change in moving 0to the left here .)Also recall that rSyr .)IX==-QIt IXand 69hx=0. Therefore the terms in the integrand of the `new' action (15.7.6)other than Ymay be written VJx0a + hxfa + 2 ~ hahx = S ~C~]a f x + Z ~Cc]~hx) or in other words jNEW - .~ d4x Y+ s T(t 5.7.Z4) 32 1 5Non-Abelian Gauge Theorie s where T = jd4x(cof+ i~wah,} (15.7.26) The nilpotence of the BRST transformation tells us immediately that the term sT as well as f d4x yis BRST-invariant . In a sense the converse of this result also applies : we shall see in Section 17.2 that a renormalizable Lagrangian that obeys BRST invariance and the other symmetries of the Lagrangian (15 .7.25) must take the form of Eq .(15.7.25), aside from changes in the values of various constant coefficients . But this is not enough to establish the renormalizability of these theories . BRST symmetry transformations act non-linearly on the fields, and in this case there is no simple connection between the symmetries of the Lagrangian and the symmetries of matrix elements and Greens functions . Using the external field methods developed in the next chapter, it will be shown in Section 17 .2 that the ultraviolet divergent terms in Feynman amplitudes (though not the finite parts) do obey a sort of renormalized BRST invariance, which allows the proof of renormalizability to be completed . Eq.(15.7.25)shows that the physical content of any gauge theory is contained in the kernel of the BRST operator (that is, in a general BRST- invariant term f d4x LP + sT), modulo terms in the image of the BRST transformation (that is, terms of the form sT) . The kernel modulo the image of any nilpotent transformation is said to form the cohomology of the transformation . There is another sense in which the physical content of a gauge theory may be identified with the cohomology of the BRST operator .12 It is a fundamental physical requirement that matrix elements between physical states should be independent of our choice of the gauge- fixing function f,,,or in other words, of the functional IFin Eq .(15.7.25). The change in any matrix element ~oc jfl} due to a change cSrN in T i s r~OC16INEW Ifl1= r ~1186T Ifl) - (15.7.27) (We use a tilde here to distinguish this arbitrary change in the gauge-fixing function from a BRST transformation or a gauge transformation .) We can introduce a fermionic BRST `charge' Q, defined so that for any field operator (D, or in other words, IQ,(DlT = zs(D , (15.7.28) the sign being - or + according as (Dis bosonic or fermionic . The nilpotence of the BRST transformation then gives 15.7BRST Symmetry 33 For this to be satisfied for all operators (D, it is necessary for Q2either to vanish or be proportional to the unit operator . But Q2cannot be proportional to the unit operator, since it has anon-vanishing ghost quantum number", so it must vanish : Q2= 0. From Eqs .(15.7.27) and (15.7.28), we have(t 5.7.29) (15.7.30) In order for this to vanish for all changes 3T in T, it is necessary tha t ~1IQ = Q W7= 0. (15.7.31) Thus physical states are in the kernel of the nilpotent operator Q . Two physical states that differ only by a state vector in the image of Q, that is, of form Q j  }, evidently have the same matrix element with all other physical states, and are therefore physically equivalent . Hence independent physical states correspond to states in the kernel of Q, modulo the image of Q -that is, they correspond to the cohomology of Q . To see how this works in practice, let us consider the simple example of pure electradynamicO Taking the gauge-fixing function as f = O.A~` and integrating over the auxiliary field h, the BRST transformation (15.7.8)- (15.7.10)is her e s AP=agc), s ccr*=a AAYI~ , S (O= 0. (15.7.32) We expand the fields in normal rnodest t Ag(x) = (27r)-3/ ' co(x) = ( 27c )-3/2dip [al'(p) d` + ag* (p) e-P" f V -2p () d3 2 0[c(p)ex + c* (P)e-ip'xl ~ 15.7.33} d 3p [b(p)e 'p x + b-(p) Cgyp 2 Recall that the ghost quantum number is defined as +1 for w ,, -1 for cox, and 0for all gauge and matter fields . f Eqs .(15.6.11) and (15.6.7)show that because the structure constants vanish in electro- dynamics, the ghosts here are not coupled to other fields . Nevertheless, electrodynamics provides a good example of the use of BRST symmetry in identifying physical states . Indeed, in analyzing the physicality conditions on `in' and `nut' states we ignore inter- actions, so for this purpose a non-Abelian gauge theory is treated like several copies of quantum electrodynamics . tf Just as (D"(x) is not to be thought of as the Hermitian adjoint of 0)(x), b' and c' are not the adjnints of c and b . But since Aµ(x) is Hermitian, co(x) is Hermitian if Q is . 34 15Non-Abelian Gauge Theorie s Mat ching coeffi cients of e±ip-" onbothsides of E q. (15 .7.28) yie lds = ppa~(P)/~ , (15 .7.34) Consi der any s tate ~ yp} sa tisfyingthe p hysica lity con dition(15.7.31): (15.7.35) The states ~ e, W} = e,,a*(p)jya } with one additional photon then satisfy the physicality condition Q e, y a }= 0 if e~ pP = 0. Also, the state ~ T}' b* (p)lyl) satisfies QIW~~= P~`d~ (pYT)~~, (15.7.36) sole + ap, ya} =fie, ya} + ~ocQ jyi)', and is therefore physically equivalent to fie,W}. From this we conclude that e ,"is physically equivalent to e l` + ocp l', which is the usual `gauge-invariance' condition on photon polarization vectors . On the other hand , so b* ltp} does not satisfy the physicality condition (15.7.31) Also, for any e,, with ep =~0, so c` ~ V~} is BRST-exact, and hence equivalent to zero . Thus the physi cal filbert space isfree of ghosts and antighosts . To maintain Lorentz invariance, we must interpret all four components of all(p) as annihilation operators, in the sense tha t 0 = a ,, (P)I0}. (15 .7.37) where Ais the BRST-invariant vacuum state . But the canonical commu- tation relations derived from the BRST-invariant action (say, with 1) give [a, (P), av(p')]- _ qpv5'( P-P') (15.7.3$) corresponding to the propagator in Feynman gauge . This violates the usual positivity rules of quantum mechanics, because Eqs .(15.7.37) and (15.7.38)yield " ~Qdo(P)aa(p')10}=-~010}. (15.7.39) Nevertheless we can rest assured that all amplitudes among physical states satisfy the usual positivity conditions, because these states satisfy Eq.(15.7.31), and for such states the transition amplitudes are the same 15.7BRST Symmetry 35 as they would be in a more physical gauge l ike Coulomb or axial gauge, where there is no problem of positivity or unita rity. The Faddeev-Popov-De Witt formalism described sofar necessarily yields an action that is bilinear in the ghost fields co,and r .,)a. This is adequate for renormalizable Yang-Mills theories with the gauge-fixing function ~'a = 0PAa, but not in more general cases . For instance, as we shall see in Section 17 .2,in other gauges renormalizable Yang -Mills theories need c ,)*co*cvcv terms in the action Lagrangian density to se rve ascounterterms for the ultraviolet divergences in loop graphs with four external ghost lines . Fortunately the Faddee v-Popov-De Witt formalism represents only one way of generating a class of equ ivalent Lagrangians that yield the same unitaryS-matrix . The BRST formalism provides a more general approach, that dispenses altogether with the Faddeev-Popov-De Witt formal ism. In this approach, one takes the action tobe the most general local functional of matter, gauge, cerA, car*Aand hAfields with ghost number zero that is invariant under the BRST transformation (15.7.7)-(15 .7.11) and under any other global symmetries of the theo ry.(For renormalizable theories one would also limit the Lagrangian density to operators of dimensionality four or less, but this restriction plays norole in the following discussion .) In the next section we shall prove, in a context more general than Yang - Mills theories ,that the most general action of this sort is the sum of a functional of the matter and gauge fields (collecti vely called 0)alone , plus a term given by the action of the BRST operator s on an arbitrary functional'' of ghost number -1 : jNEw[0~o},w`, fit] = 10 [01+ sT[0, co, r .o",h] , (15.7.40) as for instance in the Faddeev-Popov-De Witt action (15.7.25), but with sTnow not necessarily bilinear in ghost and ant ighost fields . By the same argument as before, the S-matrix elements for states that are annihilated by the BRST generator Q are independent of the choice ofTin Eq . (15 .7.40),soif there is any choice of Tfor which the ghosts decouple, then the ghosts decouple in general . In Yang -Mills theories, such a Tis provided by quantization of the theory in axial gauge, so in such theories ghosts decouple forarbitrary choices of the functional T[O, cv, cv *, h], not just those choices like (15.7.25) that are generated by the Faddee v-Popov De Witt formalism . We can go further, and free ourselves of all dependence on canonical quantization in Lorentz-non-invariant gauges like axial gauge . Again, take the action to be the most general functional of gauge, matter, coA, r.,)*A and hAfields with ghost number zero, that is invariant under the BRST transformation (15.7.7)-(15 .7.11) and under any other global symmetr ies of the theory, including Lorentz invariance . From the BRST invariance 36 15Non-Abelian Gauge Theorie s of the action we can infer the existence of a conserved nilpotent BRST generator Q . With the ghost and antighost fields treated as Herm itian, Q is also Herm itian. The space of phys ical states is defined as above as cons isting of states annihilated by Q, with two states treated as equivalent if their di fference is Q acting on another state . It has been shown that for Yang-Mills theories this space is free of ghosts and antighosts and has a positive-defin ite norm, and that the S-matr ix in this space isunitary .13a This procedure is known as BRS T quantization .It has been extended to theories with other local symmetries, such as general rela tivity and string theories . Unfortunately, it seems so far to be necessary to give separate proofs in each case that the BRST-cohomology is ghost-free and that the S-mat rix act ing in this space is unitary . The key point in these proofs is that, for each negative-norm degree of freedom, such as the time components of the gauge fields in Yang -Mills theories, there is one local symmetry that allows this degree of freedom to be transformed away . Although we shall not use it here, there is a beautiful geometric interpretation14 of the ghosts and the BRST symmetry that should be mentioned . The gauge fields Al'may be written as one-forms Aa - Aa,,dx1`, where dx" are a set of anticommuting c-numbers . (See Section 5 .$.)This can be combined with the ghost to compose cone-form P/x= Aa + wa in an extended space . Also, the ordinary exterior derivative d = dxu alax" may be combined with the BRST operator s to form an exterior derivative .9- d + s in this space, which is nilpotent because s2 = d2= sd + ds = 0 . The next chapter will introduce external field methods, which will be used along with the BRST symmetry in Chapter 17 to complete the proof of the renormalizability of non-Abelian gauge theories . 15.8 Ge neralizatio ns of BRST Sy mmetry* The BRST symmetry described in the previous section has a useful gener- alization to the quantization of a wide class of theories, including general relativity and string theories . In all these cases, we deal with an action I [ 0] and measure [d o] = jjr d or that are invariant under a set of infinitesimal transformations 0 r -->or+CAOAor. ( 15.8.1) This section ties somewhat out of the book's main line of development, and may be omitted in a first reading . 15.8Generalizations ofBRST Symmetry 3 7 This is an abbreviated "De Witt' notation, with randAincluding spacetime coordinates as well as discrete labels, and sums including integrals over these coordinates . For instance, for the gauge transformation (15.1.9), the index A consists of a group index oc and a spacetirne coordinate x, with 60x - e'(x), while the index r consists of a vector index yas well as a group index cc and a spacetirne coordinate x, with 0E" - A',,(x) ; in the notation of Eq .(1581), the variation 6AO" in the transformation (15.1.9) reads IX=M064(x- 6#Yoll Y)+Cry010, Y x 64(x "" y)r~x►~ As in the special case of Yang -Mills theor ies discussed in the previous section, BRST invariance can be used as a substitute for the Faddeev- Popov-De W itt formulation of these theories, one that is applicable even where the Faddeev-Popov-De Witt approach fails .Nevertheless, in order to motivate the introduction of BRST invariance, we shall begin here with theFaddeev-Popov-De Witt formulation of theories with general local symmetries, and then go onto consider further generalizations . By following the same arguments used to derive Eq .(15.5.21), we obtain the general Faddeev-Popov-De Witt theorem : ~j[dq] ei1"] y L01-j[dcb] e" «}BCf L0l lDet(5AfB [01) V [01,(15.8.2)n where V [ 0] is an arbitrary functional of 0'that is invariant under the gauge transformations (15.$.1);fA[0]are a set of gauge-fixing functionals *` of the car, chosen so that the `matrix' rSA f B [0]has a non-vanishing deter- minant, and B[f] is a more-or-less arbitrary functional of the fA (as for instance jIA rS(fA)}.The constant flis the volume of the gauge group, and the constant C is defined (as in Eq .(15.5-19)) by C =_ f [df] B[f] . (15.8.3) As we have seen, in gauge theories the importance of Eq . (1 S . $.2)is that it tells us that the integral on the right-hand side is independent of the choice of the gauge-fixing functionals fA,and depends on BU] only through the constant G . Where some meaning can be given to the usually infinite group volume fl,as in gauge theories on a finite spacetim e We use the same letters A, B, etc. to label the fA as the gauge variations 6A,in order to emphasize that there must be as many gauge-fixing functionals as there are independent gauge transformations . However, in some cases like string theory it is natural to use gauge-fixing functinnals f'° for which, although the index a runs over `as many' values as the index A on the gauge variations 6A, the values taken by these indices are quite different . No change is needed in the present formalism as long as we can define fA= cAaf°, with c Aa field-independent and non-singular . 38 15Non-Abelian Gauge Theories lattice, Eq .(15.8.2)can also have value as a formula for the integral on the left-hand side . To define a nilpotent BRST transformation, we must first express the functional B {f ] as a Fourier transfor m BLf] = f [dhl exp(ihAfA)94 [h] , (1 S.$.4) where [dh] = rjA dhA .Also, the determinant can be expressed as an integral over fermionic c-number fieldst(,)*AandOjA: Det(5AfB[0])Oc/[dw'] [dw] expOB , (15.8.5) where [day *] = rjAdr.,)"Aand [d (o] = rjAdr.,)A,and as usual "cc'means proportional with field-independent factors . Inserting these in Eq .(15.8-2) gives a general formula for the gauge-fixed path integra l [do]exp(ii[0] ) BCf[0]J Det (6A.faL01) V[01 oc I [d 01 [dh l[dr.o"] [dr.,)]exp(INEW[0, h, w, w "I }2 [h] V [0] , ( 15.$.6) where INEW is the new total action : As mentioned in Section 15 .6, we can think of the ghost fields as compensation for the fact that we are integrating over all 01, including those Vthat differ only by gauge transformations (15.8.1).Because ghosts are fermions, loops of ghost lines carry extra minus signs that allow these loops to compensate for the integration over gauge-equivalent Os. But for this to work, there must be just as many ghost fields w Aas there are independent gauge transformations . That is, since the c ,)A are independent, the gauge transformations (1 5.$.1)must all be independent . This is the case for gauge transformations in Yang-Mills theory and coordinate transformations in general relativity, but not always . The classic example of a theory with non-independent gauge transformations is the theory of p-form gauge fields, described in Section 8.8.A p-form A (an antisymmetric tensor of rank p )undergoes a gauge transformation A --> A + do, where 0is a (p - 1)-form, and d ois its exterior derivative (the antisymmetrized derivative) . Because d is nilpotent, for p ~ 2 we can shift 0by an amount dyewithout changing the gauge transformation, so there is a sort of invariance under gauge transformations of gaug e It is common instring theories and elsewhe reto find the ghost fields w*' and CO' writtenbA (or bA) a nd c', respectively . 15.8Generalizatio ns of BRS T Symmetry 3 9 transformations, in which the transformation parameters are the (p - 2)- forms y p. In such cases we must compensate for introducing too many ghosts by also introducing `ghosts of ghosts :15For p L> 3 we need to compensate further by introducing `ghosts of ghosts of ghosts,' and so on . In what follows we shall assume that the gauge transformations (15.8.1) are all independent, so that the ghost fields wA (and the antighosts (O*A) are all we need . Although the original symmetry (1 S . $.1)has been eliminated by the insertion of the non-gauge-invariant functional B[f],the new total action has an exact symmetry under the infinitesimal BRST transformation s where x is any of the or, (0A,(0A", or hA ;0is an infinitesimal anticom- muting c-number ; and s is the Slavnov operato r s=V)A6Aor b or- ~C tJBL~C f ABC5 -hA 6 *A (ZS.8.9) or 6~A rSw In Eq .(15.8.9)the subscript L denotes left differentiation, defined so that if6F = 6x G, then 6LF16x = G, and f ABC is the structure constanttt appearing in the commutation relatio n L5B, 5 cl=.fABC 6A . (15.8.10) The f ABc are field-independent in non-Abelian gauge theories and in string theories, though not always, but the BRST formalism is not limited to this case . A straightforward calculation give s s 2_ 2WA09B ~5AO'6LOEr)-6BOS6L~~A~r)_f'CAE 6C Or6L 60's bos I 60r -1WBCUe(t)D[IEBCIADE +6D(]" 6L.fAEC 6~(15.8.11)60r 6C~ Hence the condition that the BRST transformation be nilpotent is equiv- alent to the commutation relation (15.8.10), together with a consistency condition f E[SC f AD}E +6[DOr(oLfABCI/o)-_0 9 ( 15.8.12) where the brackets in subscripts indicate antisymmetrization with respect to the enclosed indices B, C, and D . Eq .(15.8.12) may be derived from the commutation relation (15.8.10) in the same way as the usual Jacob i For instance, for a gauge transformation acting on a matter field y .~(x) we have S~yw(x) = it pW(x)$4(x - Y), and sn6#yrS_~zy~(x) - -tytflya(x)64(x - y)64(x - z) . Hence in this case we have fI"`fl,7Z= Cxfl 764(x-Y)64(x - Z) . 40 15Non-Abelian Gauge Theorie s identity, and takes the place of the field-dependent structure constants . To show that the transformation note (recalling that 0anticommute{ rewrittenJacobi identity for symmetries wit h (15.$.$} is a symmetry of I NEW, we with ca*A} that Eq . (15.$.7) may b e The term I[01 is BRST-invariant ,because on the fieldsora BRST transformation is just a gauge transformat ion(15.8.1)with F Areplaced with Br .')A, wh ich commutes with all ca r. The term S (C,)*Af q)is BRST- invariant because BRST transformations are n ilpotent . For several reasons we may need to consider a wider class of actions than those that can be constructed by the Faddeev-Popov-De Witt ap- proach, by simply requi ring that the action is invariant under the BRST transformation (15.8.8).As a step toward show ing that such an action yields physically sensible results, we shall now prove the general result (al- ready used in the prev ioussection) that the most general BRST-invariant functional of ghost number zero is the sum of a functional of the 0alone, plus a term given by the action of the BRST operator s on an arbitrary functional Tof ghost number -1 : INEw L~~ to,to%hl = jo [ 01+ s'I'[ 0, c), w*, h] (15 .8.14) as for instance in the Faddeev-Popov -De Witt action (t5.8.13).In brief, the BRST cohomology consists of gauge-invariant functionals I [ 0] of the fields 0'alone . To prove Eq .(15.8.14), we note that the BRST transformation (15.8.8)- (15.8.9)does not change the total number of h Aand ,)*A fields, so if we expand I in a series of terms INthat contain definite total numbers Nof hAand ,)*Afields, then there can be no cancellations in sI between terms with di fferent N, so each term must be separately BRST-invariant : sIN=O . Wenextintroduce wha t is cal ledaHedge operator : t - (0*A 6 AA It is straightforward to check the anticommutation relatio n ls~ tt=_~*A CSL _hAf~ ~ ~~*A W(15.8.15) (15,8 .16) X15.$.17) Applying the operator ~s, tj to IN and using Eq .(15.$.15)then gives 15.8 Generalizations of BAST Symmetry 41 so each IN except for Io is $RST-exact, in the sense that it may be written as the operator s acting on some other functional . The complete functional I may therefore be written in the form Io + sT, wit h tINT (15.8.19) N=1N The term Io is by definition independent of co*A and hA, and since we assume it has zero ghost number it must also be independent of coA, as was to be proved . To show the invariance of physical matrix elements under changes in the definition of the gauge-fixing functional T, we define a fermionic `charge' Q, such that the change under a BRST transformation of any operator (Dis be (D -= i [D Q, (D]= iB [Q, (b]T (15.8.20) with the top or bottom sign in [x,AT = xy + yx according as [b is bosonic or fermionic . Just as in the previous section, the nilpotence of the BRST transformation then tells us that Q2 = Q . Matrix elements of gauge-invariant operators between physical states will be independent of the choice of Y' if and only if the physical states lac~ and (#Isatisf y so that physically distinguishable physical states are again in one-to-one correspondence with elements of the cohornology of Q . The general BRST- invariant action (15.8.3)will therefore yield physically sensible results for any gauge-fixing functional Y' if we can find some T,like axial gauge in Yang-Mills theories, in which the ghosts do not interact with other fields . If this ghost-free choice of T is inconvenient for actual calculation, as for instance axial gauge is inconvenient because it violates Lorentz invariance, we can adopt any gauge-fixing Y' we like, and still be confident that there is a unitary S-matrix with no ghosts in initial or final states . This approach works well in string theories, where so-called light-cone quantization takes the place of axial gauge . But in other theories like general relativity there is no way of choosing a coordinate system in which the ghosts decouple . Such theories may be dealt with by the BRST-quantization method described at the end of the previous section, using BRST invariance to prove that the S-matrix in a physical ghost-free Hilbert space is unitary . The discovery17 of invariance under an "anti-BRST' symmetry" showed that, despite appearances, there is a similarity between the roles of cr)A and co*A , which remains somewhat mysterious . 42 15 Non-Abelian Gauge Theorie s 15.9 Th e Batalin-Vilkovisky Formalism * This section will desc ribe a powerful formalism ,widely known as the Batalin-Vilkoviskyi9method . It is developed in the Lagrang ianframe- work, but has its roots in the earl ier Batalin-Fradk in-Vilkov isky formalism ,20 which had been derived in the Hamiltonian framework .(The two schemes have been proved perturbatively equivalent .21) As we shall see in Section 17 .1,the same formal machinery had been developed even earlier by Zinn-Just in22 in order to deal with the renormal ization of gauge theories . There are at least three areas where this formalism has proved invaluable : (i) Up to this point, we have considered only irreducible symmetries with an algebra that closes in the sense of Eq .(15,8.10). In some theories, such as supergravity (without auxiliary fields)23 the algebra is open : it closes only when the field equations are satisfied, so that terms appear in Eq. (15.8.10)proportional to R16 xn.Similar terms will also then appear in the consistency conditions (15.8.12).Eq.(15.8.11) then shows that in such theories s2 will not vanish, but rather will equal a linear combination of the derivatives R16xn .As we will see in this section, the Batalin- Vilkovisky method allows us to deal with very general gauge theories, including those with open or reducible gauge symmetry algebras . (ii) As mentioned above, essential aspects of the Batalin-Vilkovisky for- malism were originally developed by Zinn-Justin in order to prove the renormalizability of gauge theories . The crucial point, to be explained in Section 17 .1, is that although the sum of all one-particle-irreducible diagrams in a background field does not obey the BRST symmetries of the original action, it does share one of the key properties of the action, known as the master equation . (iii) The Batalin-Vilkovisky method provides a convenient way of analyz- ing the possible violations of symmetries of the action by quantum effects . It is used for this purpose in Section 22 .6. The starting point of the Batalin-Vilkovisky formalism is the introduc- tion of what are called "antifields,' one for each field in the theory . We letxn run over all the fields 0',r)A, cv`A, and hA, and for each xnwe introduce an external antifield** ,fin, with the same Bose or Fermi statistic s This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . The symbol t is used here in place of the more usual + in ❑rder to emphasize that it has nothing whatever to do with complex conjugation or charge conjugation . In particular, the antighost field (t)'A is notthe same as the antifield OJAof0)A. 15.9The BatuZin- Vilkoui .skyFormalism 43 and opposite ghost number as that of the BRST-transformed field sx ". That is, xn has the opposite statistics to fin, and a ghost number equal to --gh(xn )- 1, where gh(xn )is the ghost number of xn . In the simplest cases, including Yang-Mills theories and quantum gravity, the original gauge-invariant action 1[0] is supplemented with a term coupling the antifields Xn to the Fyn, giving an actio n This satisfies what is called the master equatio n 0 =6RS BLS Ix~5xn(15.9.1) (15.9.2) with `R' and `L' here denoting right- and left-differentiation . To check this, note that the terms in Eq.(15.9,2) of zeroth order in antifields just yield the condition of gauge invarianc e (Sor) ILI =O)A 6,1 [0] , 6o while the terms linear in the antifields O~ provide the condition of nilpo- tence 5L(Sxn)= ,2xn0=(sf1) 6xm(is.9.4) The xn are external fields, and must be given suitable values before we use S [x, xt] to calculate the S-matrix . For this purpose, we introduce an arbitrary fermionic functional T[X] with ghost number -1, and set t n = ~ Y' [ x] ~6xn Then Eq . (15 .9.1)becomes(15.9.5) (15.9.6) Comparison w ith Eq . (15 .8.14) shows that th isis the same as the gauge- fixed action I NEW [Z].Thus, using the same arguments as in the prev ious section, physical matrix elements are not affected by small changes in T. The act ion(15.8.7)constructed by the Faddeev --Popov -De W itt method corresponds to the cho iceT=-(o"AfA,for wh ichO~ = cc~' '~6fA16r~", cot= Q, and cvAt=-- f,q . So far ,nothing new ha sbeen accomplished .The first new point is that the master equat ion(15.9.2)can be used for more general theor ies t it is not necessary to distinguish between left- and right-differentiation here, because elttlerZn Or ST'dZnmust be bosonic . 44 15Non-Abelian Gauge Theorie s by letting S[x, xt] be a non-linear functional of the antifields Xt.(For reducible theories we must also include ghosts for ghosts among the ,fin, as discussed in the previous section, along with their antifields .) As above, we take the statistics of x~ to be opposite to that of ,fin, and its ghost number equal to -- gh(xn)- 1, and we require S [ X, xT] to be a bosonic operator of ghost number zero . Because co*A and hA have linear BRST transformations they are unaffected by the complications affecting the other xn (in this connection, see Section 16 .4), so they and their antifields enter in the action S [x, x f] in the same way as in Eq . (15 .9.1).That is , S=Smin hACAA~ (15.9.7) where ~~, cvA, and co? are the antifields of 0',cc~A, and co,"Awith ghost numbers - 1, -2, and 0, respectively, and Smin [0, CO, Ot, cvt] is some bosonic functional of ghost number zero . The last term in (15 .9.7) has no effect on the master equation, so Srninsatisfies the master equation by itself .tt Because Sm ;n has ghost number zero, its expansion in powers of antifields must take the for m + ? t.~ACt)B f rsAB L 01 O ~01 + ~A~B ~C f rDABC OrWD + 20) A COBWC CODfEFAacDL0]0)~~F +... (15.9.$) The term in the master equation (15.9.2)of zeroth order in antifields (and hence first order in c) A} yield s Q = f A [O] ~~101(15.9.9)0r which is just the statement that 1 [ 0] is invariant under the transformatio n with FA arbitrary infinitesimals . The term in the master equation propor- tional to on the right and cvAcoB on the left yield s o= fA L0]6A 10 ].fB101 6fA10] ao + ~ b[~]fl [01+ fC AB[01 f SC [01 (15.9.11) which, when the field equations 61160 = 0are satisfied, becomes the commutation relation (with structure constant f CAB [ 0]} for the transfor- mation (15.9.10).The other term in the master equation linear in antifield s tt The fields 0', co", 0~, wA are sometimes called minimal variables, while fields like rn'A and hA which together with their antifields enter bilinearly as in Eq .(15.9.7),are called trivial pairs . 15.9The Batali n-Yilkavisky Formalis m is proportional to coA wB CO Con the left and cot on the right ; it yields45 0 = ,t'~A [~]bf'ac~L01 6o fE[AB [01 f' c]EL01+fAac L01 5 O«,(15.9.12 where square brackets in the subscripts indicate antisymmetrization with respect to the enclosed indices A, B, and C . When the field equa- tions are satisfied this becomes the generalized Jacobi identity (15.8.12). Eq. (15 .9.1Z)is necessary for the consistency of the symmetry condition (15.9.9)(assuming that the fAfurnish a complete set of gauge symmetries), while Eq .(15.9.12) is necessary for the consistency of the commutation relations (15.9.11).Note that the terms in Eqs .(15.9.11) and (15 .9.12)that arise from the terms in Snin quadratic in antifields are proportional to 6I [0]15x, so they vanish when the field equations are satisfied, and in this sense are characteristic of open symmetry algebras . Terms in the master equation of second or higher order in antifields involve terms in Sm ;n that are of third and/or higher order in antifields . These provide consistency conditions for Eqs .(15.9.11) and (15.9.12), consistency conditions for these consistency conditions, and so on . It is a virtue of the Batalin-Vilkovisky formalism that all these consistency conditions are incorporated in the one master equation . The master equation may be reinterpreted as a statement of invariance of S under a generalized BRST transformation . In order to see this, and for future purposes, it is useful to introduce a formal device known as the antibracket .Returning now to our previous notation, the antibracket of two general functionals F [x, x t]and G [x, Z4] is defined b y (F,G)-5RF5LG 5RF5LG 6xn6x~- ~xn ~ xn(15.9.13) Note that right- and left-functional derivatives of a bosonic functional like S with respect to a bosonic or fermionic field variable are equal to each other or negatives of each other, respectively . Since just one of either xn or ,fin is always fermionic and the other bosonic, it follows that for the antibracket (S, S )the second term on the right-hand side in Eq .(15.9.13) changes sign if we reverse left- and right-differentiatio n 5RS5LS5LS 6RS 6RS5LS ~ x~ ~ xn 5x~ Jxn 6xn 6xn (The last step is allowed because, since one of the factors here is bosonic, their order is immaterial .) We see that the second term on the right in Eq. (15 .9.13)for(S,S) is the negative of the first . The master equation (15,9 .2)may therefore be written as the requirement that the antibracket of S with itself vanishes : (S,S)=fl. (15 .9.14) 46 15 Non-Abelian Gauge Theorie s This is a non-trivial requirement, because the antibracket has the general symmetry property (15.9.15) the sign being +1 where F and G are both bosonic, and -1 otherwise . In particular, (F, F) automatically vanishes if F is fermionic, but not if F is bosonic . The generalized BRST transformation is defined b y ~ xn ~~s= 5exn_ -B5xB(S, fin ), 8(S'x~) ,(15.9.16) (15.9.17) where B is a fermionic infinitesimal constant . (When S is of the form (15.9.1).,the transformation of Xis the same as the original BRST trans- formation 60x'= Bsxn.)To evaluate the effect of this transformation on general functionals, we note that the antibracket acts as a derivative, in the sense that (15.9.18) where the sign is -1 if G is fermionic and F is bosonic, and +1 otherwise . Hence if G and H are arbitrary functionals of x and xf with SaG,.-B(S, G )and 50H = -B(S, H), then 50(GH)_-B (S, G)H -GO (S, H) = B[(S, G)H ± G(S, H)], where the sign is + or - if G is bosonic or fermionic . Taking F in Eq.(15.9.18)equal to the bosonic functional S, we see then tha t Together with Eqs .(15.9.16) and (15.9. 17),this shows that for any func- tional F formed as a sum of products of fields and antifield s .. 5oF = -B(S, F). (15.9.19) The master equation (15 .9.14) may be interpreted as the statement that these generalized BRST transformations leave S invarian t A (15.9.20) Like the original BRST transformation, this symmetry transformation is nilpotent . To see this, we use the Jacobi identity for the antibracket : ±(F,(G,H)) + cyclic permutations = 0 , (15.9.21) where the sign in the first term is - if F and H are bosonic and + otherwise, with corresponding signs for the other two cyclic permutations 15.9 The Batulin-Yirka visky Formalism 4 7 of F, G, and H . Taking F - G = S, Eq .(15.9.21) become s 0 = +(S, (S,H)) T (H, (S, S )} + (S, (H, S) =T2(S, (S, H)) T (H, (S, S )}, where the sign is - or + if H is bosonic or fermionic . The master equation (15.9.14) then yields the nilpotency conditio n Because of this symmetry, the solution of the master equation is not unique . For instance, Eq .(15.9.22) shows that for any given solution S we can find another solution given by the infinitesimal transformatio n S' = S + (IF, S), (15 .9.23) with IF an infinitesimal functional of Xand xt that is arbitrary, except that it must be fermionic and have ghost number -1in order that S' should be bosonic and have ghost number zero . In particular, taking 6F to be a fermionic functional ET of xn alone give s S, S + F5T[X] IRS =S X1Xf+C IT] (15 .9.24) x5xn 5x These infinitesimal transformations may be trivially integrated, and show that the master equation is still satisfied if we shift the antifields to new variables Z~' x n- 5'T'/5,~' ~ The transformation (15 .9.23) is a special case of what are usually called canonical transformations, and will here be called `anticanonical transformations' to distinguish them from the canonical transformations of Chapter 7 . An anticanonical transformation is any transformation of fields and antifields, finite or infinitesimal, that leaves unchanged the fundamental antibracket relations : In (xn,rn)= (xn, x~ o . (15.9.25) For instance, consider the infinitesimal anticanonical transformation gen- erated by an infinitesimal fermion generator IF, under which any bosonic or fermionic functional G is transformed int o G--+G' =G+(IF,G). (15.9.26) It is easy to show that this does not change the fundamental antibrackets (15.9.25).For this purpose, note that the antibracket (G,H)of two functionals G and H is transformed into (G', H'), which to first order in infinitesimals i s Using the Jacobi identity {15 .9.21}, this i s (G', Hr' )=(G,H)±(IF, (G, H}} , (15 .9.7) 48 15 Non-Abelian Gauge Theorie s with the sign + if G and H are both bosonic, and -otherwise . In particular, if (G, H)is a c-number then it is unchanged by an anticanonical transformation . (This is another way of seeing that the transformation (15.9.23)leaves the master equation unchanged .) The fields and antifields have c-number antibrackets (15.9.25), so the same must be true for the transformed fields and antifields . To calculate the S-matrix we must give some definite value to the antifields . As in the simple case of closed gauge algebras where S is of the linear form (15.9.5),we can do this by taking the antifields in the form (15.9.5);that is, we calculate the S-matrix using the `gauge-fixed' actio n 6x where 'I'[x] is a fermionic functional of ghost number -1 . According to the remarks following Eq . {15 .9.4}, this is the same as taking the canonically transformed antifields xVequal to zero . The gauge-fixed action is invariant under a BRST transformation that acts on fields X"alone : 5OXn =USX" , wher e To check this, note thatSxn =6Xn x~- 6T16 x (oiS[x,xt ] SIB ~ x~ - SR Sfix, ~~ 6LS[x,x~~ 5 Xn 5xn )X~=6T16X +(5RS[Z,A 52 T[X1 6LS t ] ~x~b xm6 x~6xnx*=6T16 x The first term equation, and and n .(15.9.29) on the right-hand side vanishes as a result of the master the second because the summand is antisymrnetrict in m For closed algebras with S of the form (15 .9.1), the transformation (15.9.29)is the original BRST transformation 6OXn==6sxn.But for general open algebras, the transformation (15.9.29)is not the same as the original BRST transformation, and in general is not even nilpotent unless the field equations are satisfied . Instead, from the terms in the master equation o f For z" and x' both bosonic this is because SS 16Z" and SS/SZmanticommute . For ❑ne of x" and x' fermionic and the other bosonic, this is because the right- and left-derivatives of S with respect to whichever Z is fermionic have opposite signs . The terms with x" and ;gym both fermionic are antisyrnmetric because for these terms SL'~' /Sx"'Sx" is anlisyrnrnetric . 15.9 The Batalin -Yilko visky Formalis m first orderin the shiftedantifieldsxn~ = xh - 5Y~[,~]~~~n, we fi nd S2xm = + 6L6Rj x * ] ) 6ITIxI (5Z~5 xnxt=6T16x ~x~49 (15.9.30) with the s ign - or +ifx'is boson ic or fermion ic,respectively . Again , we see that the dependence on the field equations character istic of open gauge algebras is associated w ith terms in S quadratic in the antifields . Until now we have been conside ring only the formulat ion of a class ical field theory based on an open or closed gauge algebra . Now we must consider how quantum mechanical calculations are done in such theo - ries. Physical matrix elements may be calculated by funct ional integrals weighted w ith exp(il y[x] },where as expla ined above, I p[x]is obta ined from S[ X, xt] by settingXT = 5T[X]l6Xn,orin other words fin '= D. We wish to evaluate the effect of a change in T[x] on these matr ix elements . First consider the vacuum -vacuum amplitud e ZT= 111fix] e xP(rjT[xl}. Linder a shift 5Y' [x] in Y' [ X], this changes by an amount(15.9.31) 5Z=J[fl dX] eXP(iIT [XD5RS[Z,A 5(5T[Z]) t 6nX (15.9.32) Integrating by parts in field space, this become s ~Z=fElldX] exP(ijTLxI} X ERsLx,A 6 LITLx] 5x~6xn wherei0S [x, XT] xt=6T1 b x 6R 6L A 6x~bxn(15.9.34) We see that the condit ion for the Tindependence of the vacuum -vacuum amplitude is in general not the master equation (15.9.2),but rather what is called the quantum master equat ion (S,S)- 2i0S = 0 atxt = 5T/5x ( 15.9.35) In cgs units a factor 1 1h would accompany each factor of S [ X, XT], so the second term in Eq .(15.9.35) would have a coefficient -2ih in place of --2i. Thus whenever the quantum master equation (15.9.35) is satisfied, the term in S of zeroth order in h satisfies the original master equation (t5.9.E). Usually it is easy to construct an action that satisfies the classica l5T[x], (15 .9.33) 50 15Non-Abelian Gauge Theorie s master equation, so that the first term in Eq .(I5.9.35)vanishes, and the question is then whether the second term also vanishes . The case where the quantum master equation is not satisfied by a local action is considered in Chapter 22, on anomalies . Assuming the quantum master equation (15.9.35) to be satisfied, the change in the vacuum expectation value of an operator O [x]due to a change 5Y' in T is 6x ~ ~~~ (5Rsxxt) )6Z dX] exp(JT [Z]hn6T[X] xn ).Zt=6T/6X (15.9.36) The coefficient of the exponential in the integrand in Eq .(15.9.36)is just sU[X]. We see that expectation values of operators that are invariant*t under the generalized BRST transformation (15.9.16)-(15 .9.17) are unaf- fected by a change in the gauge-fixing fermion T . Corresponding results hold for vacuum expectation values of two or more operators . Appendix A A Theorem Regarding Lie Algebra s In this appendix we consider a general Lie algebra 1, with generators t,, and structure constants C" p~j, and prove the equivalence of three condi- tions : a: There exists a real symmetric positive-definite matrix g,,fl that satisfies the invariance conditio n gaflCflY6- _`grfi&A  (15 .A.1) (This is the condition (15 .2.4) shown in Section 15.2to be necessary on physical grounds .) b: There is a basis for the Lie algebra (that is, a set of generators T . = Yaptp, with 9 a real non-singular matrix) for which the structure constants C 'T are antisymmetric not only in the lower indices # and y but in all three indices oc, fl and y . c: The Lie algebra V is the direct sum of commuting compact simple and U(1) subalgebras gym . For open gauge theories there are generally no operators other than constants that are invariant under the transformations (15.9.16)-(15.9.17), Instead one should consider operators DQ, x~} that are invariant under the nilpotent `quantum BRST operator' 6, defined by a0 = (0, S) - FAD_Where 0depends only on x,this condition reduces to Eq.(15.9.36).If 6D = 0 then the expectation value of 0(T, ST/b x) is unaffected by a small change in the gauge-fixing fermion T24 Appendix A A Theorem Regarding Lie Algebras 51 We shall prove the equivalence of statements a, b, and c by showing that a implies b,bimplies c, and c implies a . As a by-product, we shall also show that if these conditions are satisfied, then it is possible to choose the generators 5§ as t„,,,, with m . labelling the simple or U(1) subalgebr a to which tma belongs and a labelling the individual generators in this subalgebra, and with the matrix 9,,,,,,,b that satisfies Eq . (15 .A.1) taking the form gma,nb -"'gm'5mn5ub (15.A.2) where the g .2 are arbitrary real positive constants . First, let us assume a, the existence of a real symmetric positive-definite g,flsatisfying the invariance condition (15 .A.1). We may then define new generators ia = (g-i,Z)a,a~,a , (15.A.3) in which the existence of a real inverse square-root matrix g-1/2is guar- anteed by the positive-definiteness of gIXa . These satisfy a Lie algebr a rt~' td= i COv77 , (15.A.4) where Cy0=(g-11Z)aa'(g-112)Qa'(g+112)yv'C~'oc,R,(15.A.5) (In this basis it is convenient to drop the distinction between upper and lower indices a, fl, etc., and write C,a in place of CYIX~ .} Then Eq .(15.A.1) tells us that Cay 6is antisymmetric in a and yas well as in yand 6, and hence is totally antisymmetric, verifying b . Next let us assume b, the existence of a basis for the Lie algebra for which the structure constants are totally antisymmetric . In this basis, the matrices (tA,)fly - -iC pyx of the adjoint representation are imaginary and antisymmetric, and hence Hermitian . According to a general theorem about Hermitian matrices,` the tA ,,are then either irreducible or totally reducible . If a se tofmatrices H, is not irreduc ible, the n bydefin ition there must be a set of vectors u, that spa na subspace (o therthanthe whole space) t hat isleft inva riant by theHa: thatis, fo rall a an dn,Hain=E(C ,),,,u,,.Inthis case we ca nadopt abasis consis ting of t he vec tors un together with a set ak that span the space o rthogo nalto all the gyn .IftheHa are Hermitia nthen(u,,,H.vk) = E,(C .)mn(u,,,,vk) = 0, so t he space spannedbythe ilk is also invaria nt under theHa:Hank= ~ ~(Dx}{kZ~f.In this basis the matrices Hx are simulta neous ly re duced to a block-diagonal form : Cx 0Ha =0 D , Contin uing inthis way, we ca ncom pletely re ducethe ma trices H.to a block-diagonal formwithirreducib le mat rices in th eblocks . 52 15 Non-Abelian Gauge Theorie s By an irreducible set of N x N matrices tA, is meant a set for which there exists no subspace of dimensionality <N that is left invariant by all the iA,, -that is, no set of less than Nnon-zeta vectors (ur)Qfor which (tAa)#y(ur)y is for each a and r a linear combination of vectors (us)a . Since the matrices (tA ,,)#y are proportional to the structure constants in this basis, this is equivalent to the statement that there is no set of linear combinations ~°l r ~u~ }yi`~ y that is closed under commutation with all the tAa, that is, for which rtAa, 37-r] is for each a and r a linear combination of the g-S. Such a set of matrices 9-r would furnish the generators of an invariant subalgebra of the full Lie algebra ; the absence of such a set means that the Lie algebra is simple . By a totally reducible set of matrices tAa, is meant one that by a suitable choice of basis may be written as block-diagonal supermatrice s (i AN)ma,nb=LtA(m)alabbmn , (15.A.6) where the submatrices tA(m) ,are either irreducible or vanish .*' Adopting this basis also for the Lie algebra itself, the structure constants are the n Ctc,ma,nb =Z(iA1c)ma,nb = Z (tAQn) ec)ab5mn. (15.A.7) But since this is totally antisymmetric, and proportional to 5mn, it must also be proportional to ben and be .: C11c,m a,nb = b"n6&1►nscab(15.A.8) In other wards, for any representation t(m)a = t„, of the Lie algebra in this basis, we have cab(15.A.9) with C cabreal and totally antisymmetric in the indices a, b ,c. The fact that it ispossible to construct a ba sisin wh ich the generators fall into set st('), with the commutators of the generators in one set with each other given by a linear combination of generators in that set ,and with all members of one set commuting with all members of any other set, is what we mean when we say that the Lie algebra is a direct sum of the subalgebras t(m )For each m ., the set of matrices of the ad joint representation t (m)'is either irreducible or zero, corresponding to a subalgebra that is e ither simple or consistsof so-called U(1) generators ,which commute with all generators of the whole algebra . We have thus shown that the most general Lie algebra with totally antisymmetric real structure constants is a direct sum of one or more Here m a nd n lab eltheblocks alo ngthe ma in di agonal, anda andb label rows a nd colu mns withintheseblocks. Also, m is notsummed, and the range of the indices a, b, etc.ingeneral depends on in. Appendix A A Theorem Regarding Lie Algebras 5 3 simple and/or U(1) Lie algebras . Furthermore, the simple subalgebras are compact, in the sense that each matrix -C(m)C(m) is positive-definite, because for any real vector ti,, -C ~dCbd~ua ub = EGd [EauaCa( Med']' is a sum of positive quantities, which cannot vanish unless Ua = 0, since for ua *0 the condition that Ea u aC~ ~ = 0 would imply that Ea ua~'(m)Q is itself an invariant Abelian subalgebra, in contradiction with the fact that the ttm7aform a simple Lie algebra . This completes the proof of c . Finally, let us assume c, that we have a Lie algebra that is the direct sum of a set of simple or U(1) Lie algebras ; that is, in some basis tam)= Yma,ata with real non-singular 9, we hav e where each subalgebra t(m) is either simple or commutes with everything . We furthermore assume that the simple subalgebras are compact, in the sense that the matrices 9~b) - C('n)C adC( m) dbc ( 15.A.10) are positive-definite . To construct a real positive-definite matrix gas satis- fying Eq .(15.A.1), in the basis of generators tma = tu"~}, we tak e gma,nb f ag}6rnra, ( 15.A.11) where gab)is taken as the matrix (15.A.10)when t("') is a simple subalgebra, and is taken as an arbitrary real symmetric positive-definite matrix when t(m}is a direct sum of one or more U(1) subalgebras . The matrix ~15 .A.11) is obviously real, symmetric, and positive-definite because each gabs is . To check the requirement (15.A.1),recall the Jacobi identity for the structure constants : C(M)cad C (m)dbe+C(m)cbd C (m)d ea + C(m )ced C( m)dab = 0(15.A.12) and contract with Orn)efc.After renaming the summation indices in the third term so that c--+d--).e--~,cand using Eq .(15.A.10), the result for the simple subalgebras may be written : gdf) C(m)d ab=C(m)C udC(m)dbeC(m)ecf-C(m)CfdC(m)dbeC(m)ec a The important point is that this shows that the left-hand side is antisym- metric in a and , f 9df)C(m) dab =-9da)C(m)dfb  (15.A.13) The same result holds trivially for the U(1) subalgebras, where the struc- ture constants vanish . The symmetry condition (15 .A.1) follows immedi- ately from Eq . (15 .A.13), thus completing the proof of a . This concludes our proof of the equivalence of statements a, b, and c . 54 15Non-Abelian Gauge Theorie s Now we come back to the matrix g,,fl . With totally antisymmetric structure constants, the invariance condition (15 .2.4) may be expressed as the statement that this matrix commutes with all the matrices in the adjoint representation of the Lie algebr a We have seen that all (tAcan be put in block-diagonal form, with irreducible (or zero) submatrices along the main diagonal . A well-known theorem25 tells us that then ga flmust also be block-diagonal, with blocks of the same size and position as in the tA,,and with the submatrix in each block proportional to the unit matrix . (Where two of the submatrices in thet`41,are equivalent, in the sense that they can be related by a similarity transformation, it may be necessary to make a suitable change of basis in order to bring the submatrices of gad into a form proportional to unit matrices .) In the notation of Eq .(15.A.11), the metric is then given by Eq.(15.A.2). AppendixBThe Carto nCata log We present here without proof the complete catalog of simple Lie algebras, worked out in its final form by E .Cartan.26These will be presented here in their `compact ' form - that is ,with generators that can be faithfully represented by finite-dimen sional Hermitian matrice s.The Lie algebras will be labelled with a subscript n ~1 indicating their `rank' -the number of independent commuting linear combinations of generators . An:This is the algebra of the special unitary group S U(n +1),the group of all unitary (Ut= U -1) unimodular ( Det U = 1) matrices in n +1 dimensions . Any such matrix that is infinitesimally close to the identity may be expre ssed as U=1+iH with infinitesimal H satisfying the condition s Ht=H, TrH=O , soAnis the Lie algebra of all Hermitian traceless matrices in n +1 dimen- sions . Any set of commuting Hermitian matrices may be simultaneously diaganalized, and the maximum number of independent diagonal traceless matrices in n+1 dimensions is n, so this is the rank of A.Any Hermitian matrix in n + 1 dimensions has (n + 1)2 independent real parameters (n + 1 real numbers on the main diagonal, and non +1)/2 complex numbers above the main diagonal, equal to the complex conjugates of those below Appendix B The Cartan Catalog 55 it), and one of these is eliminated by the tracelessness condition, so the dimensionality of A nis d(An) _ (n + 1)2 - 1=n(n + 2) . Allof the An are simple . B,: This is the algebra of the unitary orthogonal group 0(2n + 1), consisting of all unitary (61fi = U- '} and orthogonal BUT = U -') and hence real matrices in 2n + 1 dimensions . (The restriction to unitary matrices is sometimes indicated by calling the group Ua(2n + 1).)Any such matrix 0(that is infinitesimally close to the identity may be expressed as where A is an infinitesimal matrix satisfying the condition s A` =-A=AT . (It would make no difference here if we restricted ourselves to the subgroup SO(2n+ 1),for which U is subject to the further condition Det r,= 1, since any orthogonal matrix G that is close to the identity would have Det 60 = 1 anyway .) Any set of commuting imaginary antisymmetric (2n + l)- dimensional matrices may (bya common orthogonal transformation) be put in the form of a supermatri x ~l0-Z aZ 0- 2 0an0-2 where 0-2 is the usual 2 x 2 matrix 0 -i0-Z i 00 0 and at,    ,can are real . The rank of Bn is thus evidently n . An imaginary antisymmetric matrix is completely specified by the imaginary numbers above the real diagonal, so its dimensionality i s d(Bn)_(2n+~)(2n)= n(2n + l). All of the Bnare simple . There is an alternative definition of O(N) that will help in understanding the motivation for the next large set of simple Lie algebras . Instead of defining O(N) as the group of N-dimensional real matrices that satisfy the orthogonality condition (()T Co =1, it can just as well be defined as the 56 15Non-Abelian Gauge Theorie s group of N-dimensional real matrices .&that satisfy the conditio n .&Ty..# = y, where 9is an arbitrary positive-definite real symmetric matrix . This is because any such Ymay always be expressed as Y= WTM with M some real non-singular matrix, so there is a similarity transformation that takes any matrix ..&satisfying the above condition into a real orthogonal matrix, XMR-1 . Thus we may let Ybe various different real symmetric positive-definite matrices without changing the group . C,,: This is the algebra of the unitary symplectic group USp(2n), the group of unitary matrices . that leave an antisymmetric non-singular matrix c/ invariant : Tom.. =d, SjT = -'dDetd * U. (Note that in d dimensions, Det W = Det sVT = (-)dDetsl, so if d is odd then Det c/ must vanish . Thus there is no USp(d) unless dis even .} Any such antisymmetric non-singular (perhaps complex) matrix 'Q/may be written in the standard for m d='4Tdo' where Ris unitary and do is the supermatrix : 0 1 `~0 -10 (Alternatively, we could take si/o as ablack-diagonal supermatrix, with cis along the main diagonal .) Hence USp(2n) may be described as the group of unitary matrices 9satisfyin g yT'dpy=C/p since by a unitary transformation any such .SPcan be transformed into a unitary matrix .W= that satisfies the previous conditio n Tom.= d . Any such matrix Ythat is infinitesimally far from the identity may be written '5' = l+i~*, where *'is an infinitesimal matrix satisfying the condition s The most general 2n-dimensional matrix ~*'satisfying the se conditions Appendix B The Cartan Catalo g may be written as a supermatri x ~~ ['Q1 ~ -q' -'Cv*' where the n-dimensional complex submatrices c/, -4satisf y 'Wt--'W,_VT=_V.57 A maximal set of commuting generators have %4 diagonal and -4zero, and so are of the form cat 0 an-~1 0 -d n with al,   , an real . The rank of Cn is thus evidently n . The dimensionality ofC,is the number n 2 of independent real parameters in the Hermitian matrix %4, plus the number 2n(n + 1)/2 of independent real parameters in the complex symmetric matrix - 4 d(Cn)=n2 + 2 x non + 1)/2 ==n(2n + 1 ). All of the Cn algebras are simple . D,:This is the algebra of the unitary orthogonal group 0(2n), consisting of all unitary orthogonal matrices in 2n dimensions . The discussion of Bn can be carried over to Dn, except that here any set of commuting generators can be put in the for m a1CF2 0a2CF2 0 an CF2 so the rank is still n . Also, the dimensionality here i s 2 All of the Dn are simple except for D1,which is the Abelian algebra with just one generator, and D2,which is the direct sum B1 + B1 . Exceptional Lie Algebras : In addition to the above classical Lie algebras, there are just five special cases, the algebras G2(d =14);F4(d = 52); E6(d = 78);E7(d = 133) ; E8(d = 248) . 58 15Non-Abelian Gauge Theorie s Not all of the classical Lie algebras are really different . There are just four isomorphism s A1=B1=Ci, Cz= BZ, A3 = D3. These correspond to isomorphisms among Lie groups of which these are the algebras . However, isomorphisms like B1 = Al, B2=C2,and D3= A3do not mean that SO(3) is isomorphic to SU(2), or that SO(5) is isomorphic to USp(4), or that SO(6) is isomorphic to 4U(4) . Instead, SU(2), USp(4), and SU(4) are the simply connected covering groups of S0(3), S0(5), and 50(6) .(Covering groups are discussed in Chapter 2.) Nevertheless, the isomorphisms of the Lie algebras make it especially easy to construct the double-valued fundamental spinor representations of Sa (3), SO(S), and 5 O(6); they are just the defining representations ofSU(2), USp(4), and SU(4), respectively . Also SO(4) is isomorphic toSO(3) x 5O(3), so its double-valued spinor representation is just the defining representation of SU(2) xSU(2) .For d ~ 7 the double-valued spinor representations of SO(d) must be constructed by other means . The simplest technique uses Clifford algebras, discussed in Section 5 .4. Problem s 1. Derive the Bianchi identit y D,u F av;, +DvFcc;,v + DrFajjv=0. 2.Suppose we use generalized Coulomb gauge in a non-Abelian gauge theory ,taking the gauge -fixing function a s f, =❑A.Derive th e _ ghost Lagrangian .What is the ghost propagator? (Take B[f ] exp(-i f d4x fo:fa/2~)) 3. Suppose that in electrodynamics we use a gauge-fixing function f' = aMAu + cA AA#,with t' an arbitrary constant . Derive the ghost Lagrangian . (Take B [f ] = exp(-i f d4x f Ja12~).}What is the ghost propagator ? 4. Show that there is no simple Lie algebra with just four generators . 5. Show that if a field W~{x} belonging to a representation of a gauge group with generator matrices tx varies along a path x" = x/1(z) according to the differential equation u d~(-c)= ita~(-c)AIX~t(x(z)} ~ dz z References 59 (with both We(x) and Aa,,(x) classical c-number fields) then the change in W around any small closed path Yaround a point Xuis propor- tional to t,xya(X)FaElv (X)xP(z) d d~dz.L Find the constant of proportionality . Use this result to show tha t if Fa ,v (x) vanishes everywhere then it is possible by a gauge trans- formation to make Aa,(x) vanish throughout at least a finite region . (Hint : Follow the analogous argument in electrodynamics, or in general relativity, as in Ref .6.) 6. Applying path-integral methods to a general non-Abelian gaug e theory, calculate the propagator of the gauge field A ,,(x)if we choos e a gauge-fixing functional f IX = npAa, with nyarbitrary constants . (Take B [f ]=exp(-i f d4x f IX f a /2~}.}What is the ghost Lagrangian ? What is the ghost propagator? What is the ghost interaction vertex ? 7. Suppose we adopt BRST invariance instead of gauge invarianc e as a fundamental physical principle . Derive the most general La - grangian density constructed from sums of products of fields an d field deri vatives of dimensionality (mass )dwith d c 4, constructe d from A aji, cUa,cU~,*,h,,, and/or their derivatives, that is invariant (up to total derivatives) under Lorentz transformation s,ghost numbe r phase transformations ((aa --)- eioco,,coa~ e -in(oa), global gaug e transformations ( e, constant), and BRST invariance . 8.Show that the antibracket satisfies the symmetry condition (15.9.15) and the Jacobi identity (15.9.21). 9. Show that if a functional 0satisfies the condition (0,S)= i0S and the action S satisfies the quantum master equation then the quantum average (0)is independent of the gauge-fixing functional T . Reference s 1. C. N. Yang and R . L. Mills, Phys .Rev. 96, 191 (1954) . O. Klein ha d come close to a formulation of the S U(2) Yang-Mills theory in a tal k at a 1938 conference at Warsaw, reported in New Theories in Physic s (International Institute of Intellectual Cooperation, Paris, 1939) .For a critical discussion of Klein's theory, see D.J. Gross, 'Oscar Klei n and Gauge Theory,' talk at the 1994 Oscar Klein symposium a t Stockholm, Princeton preprint PUPT-1508 . 60 1 5Non-Abelian Gauge Th eories 2.R. Utiyama,Whys.Rev.101,1597 (1956) . 3.R.P.Feynman ,ActaPhys.Polonica 24,697 (1963). 4.L.D.Faddee vand V .N. Popov, Phys.Lett.25B, 29(1967) . 5.B.S.De Witt ,Phys.Rev.162,1195 ,1239(1967). 6. The proof is easily adapted from the standard proof of the corre - sponding result in general relativity that, for the existence of a co - ordinate transformation to a flat metric in a finite simply connecte d region, it is necessary and sufficient that the Riemann-Christoffe l curvature tensor should vanish . See, for example, S . Weinberg, Gra v- itation and Cosmology (Wiley, New York, 1972) :Sections 6 .3 an d 6.4. 7. The proof that a (with g,# =5,Q) implies band that bimplies c was given by M . Gell-Mann and S . L. Glashow, Ann.Phys .(NY) 15, 437(1961) . 8. See, for example, S . Weinberg, ibid ., Section 7 .6. 9. V. Gribov, Nucl .Phys .B139, 1 (1978) . Also see R . Jackiw, I . Muzinich, and G . Rebbi, Whys . Rev.D17, 1576 (1978) ; R. Jackiw , inNew Frontiers in H igh Energy Physics, eds . B. Kursunoglu, A . Perlmutter, and L . Scott (Plenum, New York, 1978) ; N. Christ an d T. D. Lee, Whys .Rev.D22, 939 (1980) ; R. Jackiw, in Current Algebr a and Anomalies (World Scientific, Singapore, 1985) : footnote 50 . 10. C. Becchi, A . Rouet, and R . Stara, Comm . Math . Whys .42, 127 (1975) ;in Renormalization Theory, eds. G. Vela and A .S.Wightman (Reidel, Dordrecht, 1976) ;Ann .Phys .98, 287 (1976) . 11. I. V. Tyutin, Lebedev Institute preprint N39 (1975) . lla. N. Nakanishi, Frog .Theor . Phys .35, 1111 (1966) ;B. Lautrup, Mat . Fys.Medd . Kon . Dan.Vid.-Sel. Medd .35, 29 (19 67). 12. The original source of this argument is unknown to me . I learned it from J.Polchinski . 13. S. N. Gupta, Proc .Whys . Soc . 63, 681 (1950) ;64, 850 (1951) ;K. Bleuler, Helv . Whys . Acta 3, 567 (1950) ;K. Bleuler and W . Heider, Progr . Theor .Whys . 5, 6 00 (1950). 13a.G.Curci and R . Ferrari, Nuovo Cimento 35, 273 (1976) ;T. Kugo and I. Ojima, 'rag .Theor . Whys .60, 1 869 (1978) ;frog .Theor . Whys . Suppl . 66, 1 (1979 ). Also see Ref .11a for the case of electrodynamics . References 61 14.J.Thierr y-Mieg ,J.Math.Whys.21, 2834(1980);R.Star a,inProgress in Gauge Theorie s: Proceedings of a Sympo sium at Garge se,1983, eds. G. 'tHooft, A.Jaffe,H.Lehm ann,P.K.Mitt en,I.M.Singer, and R.Stora (Plenum ,New York , 1984) :p.543;L.Ban osand P . Cotta-Ramusino, Commun . Math .Phys.87,589(1983);J.Manes,R. Stara, and B . Zumino, Commun .Math .Phys.102,157 (1985) ;A.S. Schwarz,Commun .Mat h.Whys.155, 249(1993) ;P.M.Lavrov,P.Yu Moshhin a nd A .A.Reshetnyak,Tomsk preprint hep-th /9507104;P. M. Lavrov, Tomsk pr eprint h ep-th/9507105. 15.W.Siegel ,Phys.Lett.93B,170(1980);T.Kimur a,Prog . Theor.Whys. 64, 357 (1980);65,338 (1981) . 16. Cr. Barnich, F . Brandt, and M . Henneaux, Gammon .Math .Whys .174, 57 (1995), Section 14 , 17.Cr. Curci and R .Ferrari,Nuovo Cimento 32A, 151(1976) ; I. Ojima, Prog .Theor.Whys.Supp .64,625 (1980) ;L.Baulieuand J .Thierr y- Mieg ,Nucl.Whys. B197,477 (1982) . 18.The anti-BRST transformation was extended to general local sym- metries by L . Alvarez-Craume and L . Baulieu, Nucr. Whys . B212, 255 (1983) .It is induced by the operator : s = co*A 6Aor h - 1(1)*e09.c A 6L *BCA 6 6o,. 2 fBC ~~~~ - CO h f BC6hA 6L It is straightforward to show that this transformation is nilpotent : sz= 0. Also, the BRST and anti-BRST transformations anticommute . Fur- thermore, there is a cohomology theorem (unpublished) for the anti-BRST transformations, like that for the BRST transformations : the most general functional Iofor,cr)A,CO*A, and hA that satisfies the anti-BRST invariance condition sl = 0and has ghost number zero is of the for m (Just use the Hodge operator t - c-)A 516hA in place of BRST symmetry can be an aid in enumerating possible Lagrangians and Greens functions, but 1 do not know of where it is indispensable .t.) Anti- terms i n any cas e 19.I.A.Batalinand G . A. Vilkovisky, Whys.Lett.B102 , 27 (1981) ; IVucl. Phys.B234,106(1984) ;J. Ma th.Whys.26, 172 (1985) .Alsosee B . 62 15 Non-Abelian Gauge Theorie s L. Voronov and I . V. Tyutin, Theor . Math .Whys . 50, 218, 628 (1982) . For a lucid review, see J . Gamic, J . Paris and S . Samuel, Whys . Rep . 259, 1 (1995 ). 20. E. S. Fradkin and G . A. Vilkovisky, Whys .Lett .B55, 224 (1975) ; CERN report TH2332 (1977) ;1. A. Batalin and G . A. Vilkovisky , Phys .Lett .B69, 309 (1977) ;I. S. Fradkin and T . E. Fradkina, Phys . Lett .B72, 334 (1977) . Also see M . Henneaux and C . Teitelboim , Quantization ofGauge Systems (Princeton University Press, Princeton , 1992) . 21. I. A. Batalin and I . V. Tyutin, Whys .Lett.B356, 373 (1995) . 22. J. Zinn-Justin, in Trends in Elementary Particle Theory - International Summer Institute on Theoretical Physics in Bonn 1974 (Springer- Verlag, Berlin, 1975) : p. Z. 23. D. Z. Freedman, P . van Nieuwenhuizen, an dS. Ferrara, Whys . Rev ., D13, 3214 (1976) ; R. E. Kallosh, ,1V ucr. Whys .B141 , 141 (t97$) ; B. de Wit and J . W. van Holten, Whys .Lett .,B79, 389 (1979) ; P. van Nieuwenhuizen Phys . Rep .,68, 189 (1981) . 24. M. Henneaux and C . Teitelboim, Ref . 20: Section 18 .1.4. 25. E. P. aligner, Group Theory (Academic Press, New York, 1959) : pp. 76-8 . 26. The original reference is E . Cartan, Sur laStructure desGroupesde Transformations Finis et Contin ue, (Paris, 1894 ; 2nd edn 1933) .For a textbook treatment, see, for example, R . Gilmore, LieGroups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974) : Chapter 9 . 16 Externa lFieldMet hods It is often useful to consider quantum field theories in the presence of a classical external field . One reason is that in many physical situations, there really is an external field present, such as a classical electromagnetic or gravitational field, or a scalar field with a non-vanishing vacuum expectation value . (As we shall see in Chapter 19, such scalar fields can play an important role in the spontaneous breakdown of symmetries of the Lagrangian .) But even where there is no actual external field present in a problem, some calculations are greatly facilitated by considering physical amplitudes in the presence of a fictitious external field . This chapter will show that it is possible to take all multiloop effects into account by summing `tree' graphs whose vertices and propagators are taken from a quantum effective action, which is nothing but the one- particle-irreducible connected vacuum-vacuum amplitude in the presence of an external field . It will turn out in the next chapter that this provides an especially handy way both of completing the proof of the renormalizabilty of non-Abelian gauge theories begun in Chapter 15, and of calculating the charge renormalization factors that we need in order to establish the crucial property of asymptotic freedom in quantum chromodynamics . 16.1 The Quantum Effective Actio n Consider a quantum field theory with action 1 [ 0], and suppose we `turn on' a set of classical currents .],(x) coupled to the fields or(X) of the theory . The complete vacuum vacuum amplitude in the presence of these currents is the n Z [J] _(VAC, autoVAC, in)j =JHdos(Y)exp(ii[cal+if d4x 0 r(x) Jr(X)+EterTT15 s,y (1G.1.1) Al 64 16Exter nalField Method s (The fields 0"(x) need not be scalars . They might even be fermionic, though until Section 16 .4 we shall not bother to keep track of the signs that would appear in that case .) The Feynman rules for calculating Z[J] are just the same as for calculating the vacuum-vacuum amplitude Z [0]in the absence of the external current, except that the Feynman diagrams now contain vertices of a new kind, to which a single OT-line is attached . Such a vertex labelled with a coordinate x contributes a `coupling' factor iJ,(x) to the integrand of the position-space Feynman amplitude . Equivalently, we could say that in the expansion of Z[J]in powers of J,the coefficient of the term proportional to i .lr(x )iJs(y )   is just the sum of diagrams with external lines (including propagators) corresponding to the fields 0r(x),0'(y), etc . In particular, the first derivative gives the vacuum matrix element of the quantum mechanical operator V(x) corresponding too''(x): S r [oir z[J] = iJ~dor(x) c~r(Y) exp trI [c~] + e terms} Jr (Y) J_ 0 r,x = i VAC, outpr(x}VAC, ln)j_o . (1 .1.2) We sometimes work with functianals Z[J] defined by (16.1.1)where the Or(x) are not elementary fields (that is, fields appearing in the action) but products of such fields . Where Or(X) is a product of N elementary fields, the new vertices in the Feynrnan rules for Z[J] have Nlines attached . Some of the results of this chapter (including Eq .(16.1.2))apply in this case, but where Feynman diagrams are involved, it will be tacitly assumed that the 0r(x) are elementary . Now, Z [J ]is given by the sum of all vacuum-vacuum amplitudes in the presence of the current J, including disconnected as well as connected diagrams, but not counting as different those diagrams that differ only by a permutation of vertices in the same or different connected subdia- grams .Ageneral diagram that consists of Nconnected components will contribute to Z[J] a term equal to the product of the contributions of these components, divided by the number N! of permutations of vertices that merely permute all the vertices in one connected component with all the vertices in another! Hence, the sum of all graphs i s ,X-I N=ON. The contribution of a Feynman diagram with Nconnected components containing n3 , n2, ' ' ' nN vertices will be proportional to a factor 1 1(nl +  '  nN } I from the Dyson expansion, and a factor (n1 +   - nN WN !equal to the number of permutations of these vertices, counting as identical those permutations that merely permute all the vertices in one component with all the vertices in another . 16.1 The Quantum Effective Action 65 where iW [J] is the sum of all connected vacuum -vacuum amplitudes, again not counting as different those d iagram sthat differ only by a permutation of vertices . Formany purposes ,it will be useful to goone step further ,and in place ofW(J) work with the sum of all connected one-particle-irreducible graphs . (A one-particle-irreducible graph is one that cannot be disconnected by cutting through any one internal line.)We can gi vea formal expre ssion forthis sum as follows . First, define or (x) as the vacuum expectation value of the operator V(x) in the presence of the current J : or (VAC, autj~r(x )IVAC, in)jc~i(x}(VAC, autIVAC, in) j or in terms of the sum of connected graphsi 6Z [J](16.1.4)Z[J] b.I,.(X) .Ir(x)(16.1.5) This formula can be inverted . Define JOr(x) as the current for which (16.1.4) has a prescribed value o'' (x )o f(x)=or(X)ifJr(X) _.1o,.(X). The quantum effective actio n1 r[O] is defined (as a functional of 0, not J) by the Legendre transformatio n r[o]_ -fd4xor(x) JO,(x)+W[JO]. (16.1.6) We will soon show that r[o] is the sum of all connected one-particle- irreducible graphs in the presence of the current J O. However, let us first take a look at another aspect of its physical significance . Note that the variational derivative of Flo] i s 6r [01 6 05 (Y) or,using Eq . (16 .1.5),dux 0r(X ) W(y)- JOsW 6 01 (Y)(16.1.7) Thus r[o] is the `effective action', in the sense that the possible values for the external fields 0`(y)in the absence of a current Jare given by the stationary `paints' of F : 6r101-0 for J=O . (16.1.8) 6 0s(y) 66 16 External Field Method s This may be compared with the classical field equations, which just require that the actual action 1[0] be stationary . Hence Eq . (16.1.$)may be regarded as the equation of motion for the external field 0, taking quantum corrections into account . Not only does F[O] provide the quantum-corrected field equations ; it is an effective action in the sense that iW [J] may be calculated as a sum of connected tree graphs for the vacuum-vacuum amplitude, with vertices calculated as if the action were T'[ 0] instead of 1[0]. By a tree graph is meant one that becomes disconnected if we cut any internal line . All the effects of loop diagrams are taken into account by using r[O] in place of IL0] To see this, 2 let us consider the quantity Wr [J, g] that we would get instead of W [ .1], if we used an action g-1F[O] in place of 1[0] exp{iWr [J, g]} rjdor(x) n,x +e terms ,exP i g _l [T'[0]+ f d4x0r(x)Jr(x)] (16.1.9) with arbitrary constant g. The propagator here is the inverse of the coefficient of the term in g 1I'(O) quadratic in 0, and is hence proportional tog, while all vertices make a contribution proportional to leg, so a graph with Vvertices (including those produced by the current J)and I internal lines (including those attached to the J vertices) is proportional to gj-V . For any connected graph, the number of loops is L = I- V +1,so the L-loop term in Wr- [J, g] has the g dependenc e (WIT[J,g]) cLg op5L-1 z lo Equivalently, we may write (at least formally ) Wr Lj,g] =EgL-1 W~~ ) [j] L=0(16.1.10) (1G.1.1t) where (as can be seen by setting g = 1),the quantity W~L} [J] is the L-loop contribution to the connected vacuum amplitude W [J, 1] that we would obtain if we used F[O] (without a factor g)in place of the action 1 [ 0]. Now, we are specially interested here in the sum of tree graphs, those without loops, calculated with vertices and propagators calculated as if the action were F[O] instead of 1[0].In our present notation, this is W(y)[J].In order to isolate the L = 0term in Eq .(16. 1.11), consider the limit g --+0. In this limit, the path integral (16.1.9)is dominated by the point of stationary phase , exp fiWr[J, g] Ioc exp {jgl [r[ Of]+j d4xOr(x)J,(x )] (16.1.12) 16.1The Quantum Effective Action 6 7 where, because of its definition as the field produce dby the current J, the field Oj is the stationary point of the exponent, in the sense tha t 6or(x) O_Oi(1G.1..13) The proportionality factor in Eq . (16.1.12) is in general a functional of J , but it is a power series in g starting with terms of order g° . Hence taking the logarithm of both sides and isolating the terms of order g -1gives Wr°)LJl= r10.11+ fd4X 0; (X)Jr (x). (16.1.14) By setting 0=Ojin Eq . (16 .1.6), we see that the right-hand side of Eq. (16 .1.14) is just YV [J] Wr7[J]=W [.I ] (1G.1.15) To recapitulate, this says that W [J1 may be calculated by using r [o] in place of 1[0] (the subscript r) and keeping only tree (0-loop) graphs : iW [J] z--fCONNECTED TREEn,x rf or (X) Jr (x) d 4X: (16.1.16) Now, any connected graph for iW [J] can be regarded as a tree, whos e vertices consist of one-particle-irreducible subgraphs . Thus in order for Eq.(16.1.16) to be correct, ir[k] must be the sum of all one-particle- irreducible connected graphs with arbitrary numbers of external lines, each external line corresponding to a factor 0rather than a propagator or wave function . For this reason, the coefficients in an expansion of Flo] in powers of fields and their derivatives around some fixed field coo may be regarded as renormalrzed coupling constants, with the renarmalizatian `point' specified by coo rather than by some set of momenta . Equivalently, ir[oo] for some fixed field oo(x) may be expressed as the sum of one-particle-irreducible graphs for the vacuum-vacuum amplitude, calculated with a shifted action I [ o+0o] i r1001- 1PI CON N fcL I t ,l][lldx ])exPfiILo`}-00]~n,x(1G.1.17) This is because any place where 00appears in any of the vertices or propagators within the one-particle-irreducible graphs in Eq.(16.1.E 7)is also a place where an external 0-line could be attached . (The restriction to one-particle-irreducible graphs plays an essential role in Eq .(16.1,17) ; without this restriction we could shift the variable of integration, yielding 68 16External Field Method s an integral that would be manifestly independent of coo .} In place of Eq. (16 .1.17), it is often convenient to writ e explirloolI = f rj dor(x)expIil[0+ co l} ~ ( 16.1.1$) 1P1nX in which we evaluate the path integral including all graphs, connected or not, in which each connected component is one-particle-irreducible . This formalism provides a simple method of summing tree graphs . As one example, consider the relation between the complete two-point func- tion 0r",Sy and its one-particle-irreducible part IIrx,sy From Eqs .(16.1.5) and (16.1.7),we find ❑rx,sy = rlrx,syS2W[J] 6jr(x) 64y ) blr [ol 6 orW 6 0' (Y)60J(x) _6-TS(Y) , 5JOr(x) 6 05(Y)(1G.1.19) It follows imm ediatelythatthe `matrices' ❑and II arerelated by _ -II-1. (16.1.21) This is the counterpart of the familiar relation (10.3.15)between propaga- tors and self-energy parts, with the extra term q2 +m2 in the denominator in Eq . (10.3.15)representing the zeroth-order term in the one-particle- irreducible two-point function . I6.2 Ca lculationof the Effect ive Pote ntial To see how the formalism of the previous section works in practice, consider a simple example, the renormalizable theory of a single real scalar field O(x), with actio n I 101 = - f d4x [~ + z0,00~'o+ z'n202 + za g0 4] (16.2.1) (We are here including a `cosmological constant' - Ain the Lagrangian density, for reasons that will become apparent) . Suppose for simplicity that we wish to calculate T" [coo] for a position-independent field 0o(x) = coo. Then every term in T'[0o] will contain a factor of the volume of 16.2Calculation of the Effective Potentia l i69 Figure 16 .1. Feynman diagrams with zero, one, or two loops for the quantum effective action of the theory of a neutral scalar field 0with interaction 04 . spacetime Y/1-4=fd4X =64(p- p)(27c)4(16.2.2) arising from the momentum conservation delta-function . We will therefore write, for 00constant (1f.2.3) where V(00 } is an ordinary function, known as the effective potential . In this section we will calculate the effective potential to one-loop order . This was originally done by Coleman and E . Weinberg3 in a study of spontaneous symmetry breaking, the subject of Chapters 19 and 21 . The results were also used by them in one of the early applications of the renormalization group, to be described in Section 18 .2. Shifting 0by co . othe action in Eq . (16 .1.18) is ' 1 L0+ col = - *~4 ~A+ zm20 o + zag~bo1 -[M200 + sg duo]f d4x0 d4X[!a 2(00)02] d4X [ Ig 0003 + _Lg 04 (16.2.4) where y2 is the field-dependent mas s It2(0 o)_MI+ Z goo (16.x.5) Note that there now appear new interactions proportional to 0(which have no effect on one-particle-irreducible graphs) and also 03, as well as terms with the same structure as those in the original action . The Feynman diagrams for T [coo] with up to two loops are shown in Figure 16 .1. The zero-loop term in the vacuum-vacuum amplitude is just given by the constant term in 1[0 + rho] r I'(o loop) [ 001 2M2o2 0+24o Q (16.2.6) -) - There are limitations on the applicability of perturbation theory for m2 c 0, discussed in the following section . 70 16 External Field Method s The one-loop term is given b y exp(ir'(1loop)1001) rl do(x) exp~ - ii f d4 x[a pOa ~'O +~2(0 o)02] X +E te rms Y(16.2.7) We learned how to calculate such integrals in Chapter 9;the result is given by Eq . (9.A.1$) iI`(1 loop) = In Det i K-"'_ - 1 Tr In(W ) 2 R where here -a2 Kx'Y~ C~1 x'~ C~1 yi(16.2.8) (1 G.2.9) As usual, to calculate such traces it is helpful to diagonalize the `matrix' K by passing to momentum space : 4 4J"'2n Z (27r)2 = ( p2 + µ2 (0 o)---ie)64( P - q). (16.2.10) The loga rithm ofthis diagonal matrix is just the d iagonal matrix with logarithms along the main diagonal : Inr K9P,4 and its trace is then= Ini (P2+P(Oo)- ie) 64(P - q) (16.2.11)171 iI'(1 loop) [00] = - 2 d4p Ini 71RP d4p in'(P2+P 2( 0 0 )(1.2.12)(2 27c }4 f n Putting together Eqs . (16.2.6)and (16.2.12), we have the effective potential to one-loop orde r V(0 o)= a. + ~ "~2o 0+24o Q+ ~f (~2(00)) (16 .2.13) where JG12} - -i 2(27r)4 f d4p In~i [P2 + F~2 - ie] (16 .2.14) It is painfully obvious that this formula for the effective potential contains ultraviolet divergences . Fortunately, these are naturally absorbed into a renormalization of the parameters of the theory . Although the 16.2 Calculation of the Effective Potential 71 integral (16 .2.14) is divergent, simple power-counting shows that it is made convergent by differentiating three times with respect to ,u2 (27r) 4 Once again, the -iE term tells us that we must rotate the p ocontou r counterclockwise ,so that p° = ip4, with p 4running from - coto -+-oo: X»>(y)= 1 21L)4J00 (2712k3dk 1 (k2+µi)3 327c2~U2 Integrating thrice, we have the n 41n ~.~(,u2)= ~G470+ f1+Bpz+Cµ 4 The constants A, B, C are not determined by this method of calculation, which is hardly a serious problem since they are obviously infinite anyway . We eliminate these constants by defining `renormalized' values for .Z, m2, and g: AR A +A+ Bm2 +Cm4 MR = m2 + gB + 2gm2 C , get=g+6g2C . Our final result for the potential to one-loop order is the n V(00) = AR + 1 mRo a+gxo~ +94(o0)In~2(oo) ~ ( 16.2.15)2 2 4 G47r where ,u(oo ) is the field-dependent mass defined by Eq .(16.2.5),which to this order can be calculated using MR and gR in place of m and g: ~~(o) = MR+ 7gx~~ . Similar results hold if the theory contains a complex spin ~ fermion field ip(x) that interacts with the scalar 0. For instance, if this interaction Hamiltonian density has the simple form G ofpip , then the mass M(oo) of the fermion in the presence of a constant scalar background field 00is of the form 11't(oo)= M(O) + GHQ. It is easy to see that in this case the potential (16.2.1)then receives an additional term : 2 24 64ir 2 M4(oo)In M2(oo)(16.2.16)32,g2 72 16 External Field Method s The numerical coefficient in the new term is twice that of the µ41n µ2term in Eq .(16.2.15), because of the two spin states of the fermion described by V, while the sign is opposite because (as shown in Chapter 9)fermionic path integrals of Gaussians give a result proportional to the determinant of the matrix coefficient in the exponent, while bosonic path integrals of Gaussians give a result proportional to the inverse of this determinant . 16.3 E nergy I nterpretatio n The effective action F[O] and potential V[4)] have an important interpre- tation in terms of the energy and energy density respectively .4 To see this, suppose that we turn on a current .In(x, t) that rises smoothly from zero at t = -oo to a finite value fn(x), and remains at that value for a long time T, after which it drops smoothly again to zero at t = +oo . The effect of this perturbation is to convert the vacuum to a state with a definite energy E [f] (a functional of f"(x)), in which it remains for a time T, after which it returns to the vacuum . However, although the `out' vacuum is the same physical state as the `in' vacuum, the state vectors differ by the phase exp(---iE [ f] T) accumulated during the time T : ~VAC ,outsVAC ,in} i= exp(-iE [,f] T). Comparing with Eqs . (16 .1.1) and (16.1.3),this give s W [J] _ -E [ J] T(16.3.1) (16.3.2) To see the connection between this energy and the effective action, suppose we seek the state no that minimizes the energy expectation valu e (n,Hn)(16.3.3) subject to the condition that the quantum fields (Dn(x, t) have the time- independent expectation value 0n(x ) (n, n ) It is convenient also to impose the condition that n is normalized(1G.3.4) (16.3.5) To minimize the expectation value (1 6.3.3) subject to the constraints (1 G.3.4) and (16.3.5),we use the method of Lagrange multipliers, and 1 6.3 Energy Interpretatio n instead minimize the quantit y (Q, HQ) - a(fl,Q) - I d3 X #'(X) (n, (D,(x)n) with no constraints on 0. This give s HO =OLQ+ Id3Xfln(x)(D, (x)Q . Both a and #n(x) are to be chosen in such a way as to satisfy the constraints (1 G .3.4) and (1 b.3.5) and therefore depend functionally on the prescribed expectation value 0n(x) . Now, we have said that in the presence of a currentfn(x), the Hamil- tonian H - f d3x fn(x)(Dn(x) has an eigenvalue E[ ,f] with a normalized eigenvectar Tt . Furthermore since slowl y on73 (1G.3.G) X16.3.7) (16.3.8) turnin g this current converts the vacuum into this energy eigenstate, we ca n presume that E [ f] is the lowest energy eigenstate in the presence of this current . Therefore Eqs . (16 .3-4), (16 .3.5), and (16.3-7) are satisfied by E VO ],(16.3.9) (16.3.10) (16.3.11) where fo(x) is the current for which (D(x) has an expectation value O(x) in the stateTlf. Setting f=f0in Eq .(16.3.8)and taking the scalar product with Tf" , the minimum energy of states in which the fields (Dn are constrained to have the expectation values 0,is seen to b e (H)n =E [JO] +Jd3xf(x) 0n(x). (16 .3.12) Recalling Eq .(16.3-2) and the assumed form of J(x), this i s ~~~5~ = 1 ~ - W L~~ + dux Jn(x)On(x) ]-= -1 r'~~~  (6 .3.13)T f T As noted in the previous section, if the field 4i(x) has a constant value ~i over a large spacetime volume `V4 = ' V3T, then we may write the effective action in terms of an effective potential Y(4) } r[0a = - V3 T V (O) In this case Eq . (16.3.13)tells us that the energy density is(16.3.14) (1G.3.15) 74 1 6External Field M ethod s This is the main result :V(O) is the minimum of the expectation value of the energy density for all states constrained by the condition that the scalar fields (D,,have expectation values On . One consequence is that in the absence of external currents the vacuum state will relax to a state in which the potential V(O) is not only stationary, which is required by the field equations (1 6.1.8), but also a minimum . This result helps to resolve a problem in the interpretation of the quantum effective potential . The Euclidean version of the path-integral formalism (described in Appendix A of Chapter 23)makes it manifest that the two-point function ❑is positive (in the matrix sense), so according to Eq. (1 G .1.2)the same is true of -II = ❑-1. Together with Eq .(16.2.3),this implies that for a single scalar field the effective potential V(O) must have a positive (or zero) second derivative with respect to 0. More generally, the effective potential must be convex :5 V(201 +G -2)02) :~~AM01 }+(1-2) Y(02) for ~:::-n: 2~I. But inspection of Eq .(16.2.13)shows that for m2 < 0 and g > 0 the zero-loop approximation to the effective potential for the scalar field theory with action (16 .2.1)has a negative-definite second derivative when 0is between the two minima of the effective potential at± ~mlvg This contradiction arises because the derivation of perturbation theory implicitly relies on the existence of a stable vacuum, but when V"(0) <0, the field 4) is at a value of ~ where V(~)- J0~is a maximum rather than a minimum, which means that the vacuum state in the presence of the current Jo is unstable . So what is the true effective potential for this scalar field theory when m2< 0 and 4) lies between the two minima of the potential? The result of this section is that we must find the state of minimum energy in which the expectation value of the operator (Dequals 0.As long as 0is between the two minima of the potential, we can give (Dan expectation value 4) bytaking the state as a suitable linear combination of the two states where 0is at the minima 0-+ 61mlj/g . The energy in this state is equal to the energy at the minima, so this state clearly minimizes the energy . (Interference terms here vanish in the limit of infinite volume, for reasons explained in Section 19.1.)Thus the effective potential between the two minima of the potential is a constant, satisfying the requirement that it have a nonpositive second derivative . The same argument shows that in more general theories where the potential has two local minima of unequal energy, the potential between these minima is linear . 16.4 Symmetries of the Effective Action 75 16.4 Symmetries of the Effective Actio n In some but not all cases, the symmetries of the action I [4i] are auto- matically also symmetries of the effective action r[o] . For instance, in the example of Section 16 .2, the action (16.2.1)has a symmetry under the discrete transformation 0 --+ -0. Thus it follows from their definition that Z [J] and W [J] are even under the corresponding reflection J--+-J.Eq. (16.1.5)then shows that 0_i = and hence J_0=-JO,so Eq .(16.1.6) shows that I'[ o] is even under - 0. This is borne out by the one-loop result (1 6.2.15). The fermion-loop contribution in (16.2.16) also exhibits the symmetry under 0--+-4) in the special case where M(O) =0, because in this case the action is invariant under the combined transformation ~--+-~~ W --+ Ystp We encounter problems in establishing the renormalizability of a theory, unless we can show that the symmetries that we impose on the action also apply to the effective action . For instance, in the example above, if f [0]were assumed to be even in 0but r[o] turned out not to be, then the coefficients of the terms in IF proportional to f d4x 0and f d4x4)3 would be divergent, but the symmetry of the action would not allow us to introduce counterterms to absorb these infinities . With this motivation, let's now turn to the important class of symmetries generated by infinitesimal transformation s x"(x) --), x '(x)+,EFn[x;x] , ( 16.4.1) where P is a function of x" that depends functionally on x ". (For instance, Fn [x ; x]may be an ordinary function of the x"and their derivatives at the point x .) We are now using the symbol xn rather than 4)" to denote the different types of fields, to emphasize that these include not only ordinary gauge and matter fields (which in the next chapter will be denoted 01( x)}, but all other fields appearing in the gauge-fixed action, including ghost fields . We repeat that these xII(x) may be of any type, not necessarily scalars . We assume that both the action and the measure are invariant under the symmetry transformation (16 .4.1): d V(x) + e P [x ;x] fjdxn(x). (16 .4.3) n,x n, x (It is actually sufficient for only the product (fln,x dx "(x)}exp(il) to be invariant, but where this is true usually Eqs . (16 .4.2) and (16 .4.3) both apply .) Replacing the integration variables in Eq . (16.1.1) with x'(x) + 76 1 6External Field Method s ,F F" [x ;X], we have t hen [Pd(Zn(X)+ eF'[x;X])] x exp i I [x +E F] + ijd4x(xn(x)+ E F" [x ; x])Jn(x) _ [fldx'x ]) exp i I [x] +ijd¢x (f(x)+eF{x ;1)Jn(x) } n,x f n,x xexpiI [x]+ iJd4x xn(x) Jn(x) and hence fd4y ~F '(y)}jJn(y)_0, (16.4.4) where ~ )jdenotes the quantum average in the presence of the current Jn(x), Z[.r]~Fn(y)},(lldxx ))F"[y;x]n,x x exp i I [x] + i ` d4x xn(x)Jn(x)(16.4.5)J normalized so that ~1} j= 1. But recall that Jn(y) is given in terms of the effective action r[x] by Eq .(16.1.7) Jn,x(Y)-- ~ r[x] 67Cn(Y) Therefore Eq . (1 G .4.4) may be written a s o = day ~Fn(y)},,6~ ~[x] bx v)(16.4.6) In other words , F[X] is invariant under the infinitesimal transformatio n (16.4.7) Such symmetry conditions are known as Slavnov--Taylor identities .7 Is this the symmetry transformation with which we started? It is for one very important class of infinitesimal symmetry transformations : those that are linear .For such symmetries Fis 16.4Symmetries of the Effective Action 7 7 (In the most common case sn(x) vanishes and tnm(x, y)is a constant matrix times 64(x - y).)For any linear F, we hav e ~F"(x)}, = 5n(x)+ ftnm(x,Y)~xm(Y)) id4y But for any fixed x,Jx is defined as the value of the current Jthat makes ~Xm(y)}jequal to x"i(y),50 ~P(X)),Ix= S"(X)+ Jtnm(x,Y)X'n(Y) d4Y = Fn [X; X]. (16.4-9) Hence Eq . (16 .4.6) requires that r[X] be invariant under all the functional linear transformations xn--,x"+EFn that leave I [x] and the measure invariant . We occasionally have to deal with symmetry transformations that are not linear . One important example is provided by the BRST transfor- mation discussed in Section 15 .7. For non-linear transformations, the symmetry transformation (16 .4.7) under which the effective action is in- variant is not generally the same as the assumed symmetry transformation (16.4.1) that leaves the original action invariant, because the average of a non-linear functional of fields is not generally the same as the functional of the average fields . Indeed, the form of ~F} Jx as a functional of x de- pends in general on the dynamics of the system, and is usually non-local . This complication will be dealt with in the next chapter by the method of antibrackets . Up to now, we have tacitly been assuming that the fields xnand the corresponding transformation functions ❑"and currents Jn are all bosonic . We will need to take note of the sign factors that appear when some of these are fermionic, as in particular for supersymmetry or BRST transformations, where E is fermionic and xnand ❑n have opposite statistics . With currents inserted to the right of fields, as in Eqs .(16.1.1) and (16 .4.5), Eqs .(16.1.5)and (1 6.1.7) hold in the for m 6R W [J] _ m ~Jm(Y) xa [Y~ 6Lr[x] with the subscripts Rand L indicating that the derivative is to the right orleft.Inconsequence, the Slavnov-Taylor identity should be written x(Y)(16.4-10) X16.4.11) act from y (16 .4.6) (16.4.12) 78 1 6External Field Method s Problem s 1. Consider a theory of real pseudoscalars O(x) and complex Dirac fields W(x),with masses Mand m respectively, and interaction g'y~ysipo . Evaluate the effective potential for ~ constant, V = 0 , to one-loop order . 2. Derive general formulas for63Wp]16j'(X)6j','(y)6J((z)and J¢W [J]l~Jn(x)bJ„,(y)dJ,,(z)6.Ik(w)in terms of the variational deriv- atives of r[o] with respect to ~ . Show which Feynman diagrams correspond to each term in these formulas . 3. Calculate the effective potential to one-loop order for the theory of a neutral scalar field 0with interaction Lagrangian density g03/6in six spacetime dimensions . 4. Suppose that the action I [01is invariant under a.f Hite matrix trans- formation 0n(x) --*E„a Mn„a On(x).Under which transformation of the currents is W [J] then invariant? Use this result to derive a symmetry property of F[4)] . Refere nces 1. The effective action r[o] was introduced by J . Goldstone, A . Salam , and S . Weinberg, Phys . Rev .127, 965 (1962), who defined it pertur - batively, as a sum over one-particle-irreducible connected diagrams . The non-perturbative definition (16 .1.6) was first given by G . Jana - Lasinia, Nuovo Cimento 3 4,1790 (1964) . 2. S. Coleman, Aspects of Symmetry (Cambridge University Press, Cam- bridge, 1985) :pp. 135-6 . 3. S. Colem anandE.Weinbe rg,Phys. Rev.D7,1888 (1973) . 4. K. Syrnanzik, Comm . Math . Phys .16, 48 (1970) ; S. Coleman, Aspects ofSymmetry (Cambridge University Press, Cambridge, 1985) : pp 139-42 . 5. K. Symanzik, Ref . 4; J. Iliapaulos, C . Itzykson, and A . Martin, Rev. Mod . Phys .47,165 (1975) . G.Y. Fujimoto , L.O'Raifeartaigh, andG. Parravicini, Nucl.Phys.B212 , 26$ (1983) ;R.W.Haymakerand J.Perez-Mercader,Phys.Rev.D27, 1948 (1983) ;C. M.Bender and F . Cooper, 1V ucd.Phys.B224 , 403 References 79 (1983); M.Hindmarsh a nd D . Johnston, J. Math . Phys .A19,141 (1986); V.Branchina, P. Castorina, an d D. Zappala,Phys.Rev.D41, 1948 (1990) ; K. Cahi ll,Phys. Rev.D52, 4704 (1995) . 7. A. A. Slavnav, Theor . Math .Phys .10, 152 (1972) [English transla- tion:Theor .andMath .Phys .10, 99 (1972)] ; J. C. Taylor, IVucl . Phys . B33, 436 (1971) . 17 Renormalization o f Gauge Theorie s We now return to gauge theories, and use the external field formalism described in the previous chapter to study the renormalizability of these theories and to carry out an important calculation . 17.1 T he Zinn-JustinEquation In this section the BRST symmetry described in Section 15 .7 will be used to demonstrate a fundamental property of the quantum effective action, first derived by Zinn-Justin .1 According to the general rules outlined in Section 16 .4, the BRST invariance of the action I [X] imposes on the effective action r[x] the conditio n fct4x ~0"(x)}jx6L~[x1_0,(17.1.1)6x(x) where the change in xn(x) under a BRST transformation with infinitesimal fermionic parameter B is boxn{x} = BOn(x) 17 .1.2) and ~ - -} here denotes a vacuum expectation value taken in the presence of a current J.that makes the vacuum expectation value of the operator fields X"(x) equal to the c-number functions xn(x) . The implied sum over n runs over all of the fields in the BRST formalism ; that is, over co ., co* , and h, as well as the gauge and matter fields that in Section 15 .8 we have collectively called 01. Because ❑n(x) is quadratic in the fields when x "is a gauge or matter field or co,, Eq .(17.1.1)does not in general tell us that the effective action is invariant under the same BRST transformations as the action itself . To handle this complication, we employ a trick that proves useful in dealing with any sort of nilpotent symmetry transformation . First, we introduce a set of c-number external fields Kn(x), and define a ne w 80 17.1 The Zinn-Justin Equatio n effective action b y r[x,K]- W [Jx,x,K]-fd4xx'(x)JxK n(X) ,81 (17.1.3) where the connected vacuum persistence amplitude W is here calculated with the gauge-fixed action* I + f dux Ong „ eiWVK][Hdf(x)] exp(iI+iJdux anon + iJduxxnJn)(17.1.4) n,x and Jx Kis the current required to give the fields the expectation values x in the presence of the external fields K ; hw[JA bJn(X) J=J x,k(17.1.5) (The Kn must have the same fermionic or bosonic statistics as 4 ', which is opposite to that of x" .)Since the BRST transformation is nilpotent, the quantities ❑n{x} are BRST-invariant, so in the same way as in Section 16.4 we can show that the new effective action r[x,K] satisfies a BRST- invariance condition : Id4x ❑nx6Lr[x'K]=0 ~ ~ ~~Jxk ~nx ~x~)(17.1.6) where ~ -)jK denotes a vacuum expectation value calculated in the presence of the current Jand the external fields K : f[u1.xdx'~(X)]d[x]exp (ii+a f d4x A"K, +if d4x xnJn ~d fix] ~~~f[Fin,xdxn(x)]exp(iI+i f dux AnKn+ifd4x xnJ, ) It is convenient to express the expectation value of ❑nas a variational derivative of the effective action . Taking the right variational derivative of Eq .(17.1.3)with respect to K give s 6K,rx bKnx jd4YZyn(Y)IRJXK m(Y) bKn(x)6Rj'V[J, K] 6RJxK m (Y) bJm(y) ~j_j, .K bKn(X) Using Eq .(17.1.5)we see that the last two terms cancel, and using th e Here Iis the action I NEr,,, mpdified as described in Section 15 .7 to depend on the ghost and antighost fields co, and co,,* and on the auxiliary field h„ with the subscript `NEW' dropped from now on . 82 17Renormalization of Gauge Theorie s definitions (17 .1.4) and (17 .1.7) gives us then our desired relatio n 6Rr[x,K] 6Kn(x)6R W [J, K] W n(X)J-JxX~On(x)~ .,x,,,x . (17.U The BRST symmetry condition (17 .1.6) may now be written as a simpl condition, the Zinn-Justin equation, involving the effective action alone : J ~~ ~ ~ x n~) x~ ) As remarked after Ey . (15.9.3), the interchange of fields and antifields (oi in this case xnand Kn) results simply in a change of sign of the left-hanc side of Eq .(17.1.9),so this can be written a s o, where the antibracket is calculated here with Ian in place of the antifield of x►'; ' 6 n (x) 6K x 6K(x) 6nx This is formally the same as the Batalin-Vilkovisky `master equation' discussed in Section 15 .9, but appears here as a constraint on the quantum effective action I' [x,K ]rather than on the fundamental action S [X, x t] The Zinn-Justin equation (17.1.10)will be used in the next two sections to show how to renormalize gauge theories, and in Section 22.6 to study anomalies in these theories . 17.2 Renorma lizatian : Direct A nalysis The simplest non-Abelian gauge theories are renormalizable in the `Dyson' sense that the operators in the Lagrangian density all have dimension- ality (in powers of mass) four or less . As we saw in Chapter 1 .2, this guarantees that the infinities in the quantum effective action only appear in terms that could be cancelled by the counterterms in interactions of dimensionality four or less . But there is more to renormalizability than this. The Lagrangian density is constrained by gauge invariance and other symmetries . For a theory to be renormalizable, it is necessary that the infinities in the quantum effective action satisfy the same constraints, up to possible renormalizations of the fields . The effective action I'[Z, K] is a complicated functional of both x and K, about which the symmetry condition (17 .1.9) says complicated things, but fortunately matters are much simpler for the infinite terms in F . We will write the action S [X, K] = I [X] +fd4xdnKn as the sum of a term SR [X, K ] 1 7.2Renormalization : Direct Analysis 83 in which masses and coupling constants are set equal to their renormalized values, plus a correction S,,[x, K], which contains the counterterms that we intend to cancel the infinities from loop graphs . Both SR and S . must be taken to have the symmetries of the original action S [x, K], so the question is whether the infinite parts of the higher-order contributions toFshare the same symmetries, so that they can be cancelled by the counterterms in Ste . We may expand F in a series of terms FN arising from diagrams with just Nloops, plus contributions from graphs with N - M loops (where 1 c McN)involving various counterterms in S ,, [x, K] that will be used to cancel infinities in graphs with a total of Mloops : r[x,K] = E rN[x,K]  (17 .2.1) N=O The symmetry condition (17.1.10) then reads,* for each N, N (rN', rN -N')= 0. (17 _2.2) N'=b In the sum (17.2.1)the leading term is just I'4 [x, K] = SR [ X, K], which of course is finite . Suppose that, for all M c N- 1, all infinities arising from M-loop graphs have been cancelled by counterterms in 5 ,.Then infinities can appear in Eq .(17.2.2) only in the N'= 0 and N' = N terms, which are equal, and the infinite part of this condition tells us that the infinite part FN,,, ofI,N is subject to the condition tha t (SR, rN,oc) = 0 . (17.2.3) This is a symmetry principle generated by SR, just like that described by Eqs.(15.9.16) and (15 .9.17). Note in particular that the transformation X H ( SR,X)acts on the external fields K, as well as the fields x ". Up to this point, we have used none of the special properties of a renormalizable Yang-Mills theory . Now note that, according to the gen- eral power-counting rules of renormali2ation theory, with all infinities cancelled in subgraphs of FN, the infinite part I'N,,,--[x, K] OfI'N [x, K] can only be a sum of products of fields (including K) and their derivatives o f Such order-by-order relations may be derived formally byrepeating the reasoning of Section 16.1.-When the action SR is replaced with g 'SR, the cont ribution of a connected L-loop graph with Iinternal lines and Vvertices is multiplied by a factor 9v-r = 9L-i If the counterterms in S te,associated with N-loop diagrams are also provided with factors g N,then I' Lis the value forg = 1 ofthe term in IFoforder gj-'. Eq.(17.2.2) then follows by requiring Eq. (17.1.10)to hold in each order in g . In cgs units the action has the same dimensions as h, and so appears in the path integral multiplied with a factor 1 lh, so we can also use hasa loop-counting parameter in place ❑f g. 84 17 Iienarmalizatian of Gauge Theorie s dimensionality (in powers of mass) four or less . Finally, the arguments of Section 16 .4 show that I-' [x,K] and hence also rN, ,,,[x, K] are invariant under all of the linearly realized symmetry transformations under which the action is invariant . (As described below, these are : Lorentz trans- formations, global gauge transf'ormations, antighost translations, and the ghost phase transformations associated with ghost number conservation . Ofcourse, the auxiliary fields K,, must be assigned suitable transforma- tion properties under these symmetry transformations .) These conditions together with Eq .(17.2.3)will suffice to tell us all we need to know about the structure of FN,,,-- [x,K] To implement these conditions, we need to know the dimensionalities of the external fields Kn . If a field xn has dimensionality do(that is, d ,, powers of mass), then ❑" correspondingly has dimensionality do +1 (as can be seen by inspection of the BRST transformation rules (15.7.7}- {15.7.11}}, so in order for f d4 xKnOn to be dimensionless, Kn must have dimensionality 3-dn . The fields A°` ,",co',and cv"* all have dimensionalities do = +1, so the corresponding K, all have dimensionalities +2 . (We do not introduce any external field corresponding to h', because this field is BRST-invariant .) Any spin t /2 matter fields W~ have dimensionalities 3 /2, and the corresponding & thus also have dimensionalities 3 /2. Thus a dimension four quantity like FN,,,-- [x, K] is at most quadratic in any of the K,,. Furthermore terms that are of second order in the K,, cannot involve any other fields, except that a term of second order in the Kn for spin 1/2 matter fields may involve at most one additional field of dimensionality unity . We can now use ghost number conservation to show that in fact FN,,,, [Z, K] does not contain any terms of second order in the K,, . For this purpose, we also need the ghost quantum numbers of the Kn . If xnhas ghost quantum number y,,, then 0 "has ghost quantum number yn + 1, so Kn must be assigned ghost quantum number -yn - 1. The ghost quantum numbers of the fields A`,,", Wt, cva , and c)2* are respectively 0, 0, +1, and -1, so the corresponding external fields Kn have ghost quantum number s and 0, respectively . This rules out any terms in rN,,,,fix, K] that are of second order in the fin, with the one possible exception of a term of second order in the external fields Ka associated with cva *(and involving no other fields) . However, these last terms are also forbidden, for a different reason . The BRST transformation of o)a* is linear in the fields, with Ax* ~ _h a so here(17.x.4) bKIX 17.2Renormalization :Direct Analysis 85 is independent of K,*. It follows that FN,,,--[Z, K] is linear in K',*, and depends on K°`'" only through a term - f d4x Ida h' .(Both K* and h' are bosonic, so their order is immaterial .) In particular, for N > 0,FN,,,-- [Z, K] is independent of Ka* . We have seen that FN,,,-- [x, K] is at most linear in all of the K, We will write it as FN,- [ 7C, K] = rlv,~c [Za d] +/d4x .[x ;x]Kn(x) . (17.2.5) We also recall that SR has the K dependence : SR [Z,K] = SR [X] +jct4x On[x; x]& (x). The terms in (17.2.2)of zeroth and first order in K therefore give* ` bx(x) x(x) and /d4x6xn(x) bxn(x)= 0 (17.2.6) = 0 , (17.2.7) respectively . These relations may be made more perspicuous by introduc- ing the quantities r{N}[x]= SR [x ]+Cr~v,x[x,01, and(17.2.8) (17.2.9) with e infinitesimal . Then (17.2.6)(together with the BRST invariance of SR }just says that I'N)[x]is invariant under the transformatio n xn(x)__*xn(x)+B0~)ni} , (17.2.10) while Eq .(17.2.7)(together with the nilpotence of the original BRST transformation) tells us that this transformation is nilpotent . We must now consider what form this nilpotent transformation may take. As already mentioned, FN,,, consists only of terms of dimensionality four or less, so 9N and hence ❑N)n(x) have at most the dimensionalit y "The second terms in Eqs .(17.2-6) and (17.2.7)have been put into the form shown here by recalling that, because x" and K,, have opposite statistics, for any bosonic functionals A and B, dxA dLB dLA SRB FRB dtA 6xn dKn 6xn dKn d Knbxn 86 17Renormalization of Gauge Theorie s of the original BRST transformation function On(x) . Also, !YN and hence also ❑N)'~(x) must have the same Lorentz transformation properties and ghost quantum numbers as ❑n(x). Hence the most general form of the transformation (17.2.10)is W-r W+iOCV a TaW, A~y --->Aal,+ B LB,fiaAo_)fl+ Dx#r AbuwY f..t)x -r Cva- ~ a Ea#''CU#CU,1 where Ta is some matrix acting on the spinor fields, and Bx fl, D,flj, and Ex #y are constants, with Ea #;,antisymmetric in fland y . Also, the transformations of cox and h, are linear, and are therefore unchanged : -Oh,, ha - -+ha Ctfa - *co* Next we impose the condition of nilpotence . The most important requirement is that E,,#jc,)#cvy should be invariant . This yields the require- ment that Exfly .E#s,wacv,coy should vanish, so that the part of Ea fl1Ep6F that is totally antisymmetric in 3,F, y vanishes : £a#,IE#ae + E xflFE#va + Ex#6EflFy= 0. But this just tells us that Easy is the structure constant of some Lie algebra &. Because Easy goes to the structure constant C ,#Y of the original gauge Lie algebra .4 for F-+0,(fmust be the same as 4, and the structure constants Ex p.,can differ from the original C~ #,,only by a multiplicative factor : Ex#Y=.YC~x#Y (This is for simple gauge groups ; in the general case we would have a separate factor I'for each simple subgroup .) Next we turn to the condition that the transformation (17 .2.10) be nilpotent when acting on the gauge fields . The requirement that Bx~ a,,cUfl+ Da#yA#,oybe invariant tells us tha t DxflvD#SE- D a#FDfiay= EflErDaafl = YCflE ;Da6# and B~%#EaYa = Da#SBpy , The requ irement o fglobalgauge i nvariancerules out any no n-trivialsimilar ity trans - format ionintherelationbetween 1:',flyandCzar . 17.2 Renormalization :Direct Analysi s The first condition has the unique solutiont t D~P'i= YC'Rv87 The second condition tells us that the matrix Bx# commutes with the adjoint representation of the gauge group, and hence (since we have chosen the structure constants totally antisymmetric) must be proportional to a Kronecker delta, with a coefficient we shall call YA' ; Bad =Y-41-~ afl. Finally, the condition that the transformation (17 .2.10)be nilpotent when acting on the fermion fields (if any) requires that co'Ty) is invariant . This tells us tha t so Ta differs from the generator t,,, in the original Lagrangian only by a factor Tx=Ytx. We have thus seen that, apart from the appearance of the new constants 9and V-, the transformation (17 .2.10) is just the BRST transformation with which we started : AaII-} A all + YO [s'012(ga +Ca#rA#,Lr~)-J (i)a~ Lv 'X- zY 0 Q~fl "rcoo co ye rUa ~ r,Oa -Ohs , hx ha .(17.2.11) (17.2.12) (17.2.13) (17.2.14) (17.2.15) Now we must use this symmetry to constrain the structure of the corrected action (17.2.8).Since this contains only the original renormalized action plus the infinite part of the 11T-loop contribution, it must be the integral of a Lagrangian density r" d4x N N(17.2.1d) with , .t~N~ a local function of fields and field derivat ives of d imension- ality (inpowers of mass) no greater than 4 . Furthermore, as we found in Section 16 .4,Al'(') must beinvariant under all the symmetries o fthe The matrix (D .Jx# - Df ,#IY satisfies the commutation relations of the gauge Lie algebra, [Dr, DE] - C #f,,DR .But the only representation of a simple Lie algebra with the same dimensionality and .sad transformation properties as the adjoint representation of ~dis the adjoinl representation itself . 88 17Renarmalization of Gauge Theorie s original Lagrangian that act linearly on the fields . To identify these sym- metries, recall that in generalized ~-gauge, the `new' Lagrangian density in Eq .(15.7.6)takes a form given by replacing the term -(0PAa)(aVAIX )/2~ in Eq .(15.6.16) with the terms hj,,~+Z~hah, in Eq .(15.7.6): ANEW = YM - AF,yFapv- ~~(~ aauCr3~ +Cxl~v ~~uc~a } Ay a -~?~haha. (17 .2.17) Inspection of this formula reveals the following linear symmetries : (1)Lorentzinvarian ce. (2) G lobalgauge invariance - that is, invariance under the transformation s 6wl(X)= Z ea(ta)emWm(X), 6AP,(x) Cfl,,e'A7,(x), 6C~~ (X) 6hfl(X)= C flyaeahy(x),(17.2.18) (17.2.19) (17.2.20) (17.2.21) (17.2.22) withcons tantparame ters e" . (3) A ntigho st tran slation invarian ce - that is, invariance under the trans- formation (1)IX(x) --* c-), (x) + c,~ , (17 .2.23) with arbitrary constant parameters cx . (4) G host numberconservation-that is, the conservation of a ghost number equal to +1 for c ),-1 for co*, and 0 for all other fields . We shall now proceed to work out the structure of the most general Lagrangian density that is renormalizable, in the sense that it consists only of terms of dimensionality +4 or less, that has these linearly acting sym- metries, and that is invariant under the modified BRST transformations (17.2.11)-(17 .2.15). From Eq . (17 .x. 17) we may conclude that the fields AP, co, co*, and h,, have the dimensionalities +1, +1, +1, and +2, respectively (in powers of mass) . Note also that ghost number conservation requires that CO and c)` come in pairs, while antighost translation invariance dictates that w* always appears as a derivative . Each pair of w andeUco* fields adds +3 to the dimensionality, so renormalizability rules out any term with more than one such pair . With one such pair we can have at most one more derivative or one additional gauge field, and Lorentz invariance dictates that we must have one or the other . The only renormalizable allowed interactions involving ghost fields are then linear combinations of terms of the form 014 avaMo# or awx Ay cv# . Next, let us consider the terms that involve the field h, and possibly other fields but not r .) or w* . This field has dimensionality +2, so 17.E Renormalization : Direct Analysis 89 renormalizability and Lorentz invariance allows this field to appear only multiplied with another h por0,,A~orA~AMy. Finally, the Lagrangian will contain renormalizable terms involving only the matter and gauge fields . We will call the sum of these terms 9'v,A . Putting this all together and using global gauge invariance, the most general renormalizable interaction allowed by the assumed symmetries (aside from BRST invariance) takes the form : Y NO=Yy,A+ z~~ha~a +ChxOuA~- efl',hA~Av u -Zaja ~~~~~auC'a) - d flv(Oc~a)cvflfty ~ (17 .2.24) where ~', Z ,,,, c, d ,,,#y, and ex#y are unknown constants, with no constraints except obvious symmetry properties such as global gauge invariance and e.pY = e,,yp .(As mentioned earlier, we are assuming for simplicity that the gauge group is simple, but the extension to a direct sum of simple and U(1) gauge groups would be trivial ; for instance, instead of one term proportional to hahIX we would have a sum of such terms, one for each simple subgroup of the gauge group .) Now we impose BRST invariance . The cancellation of terms in 8 ,T(,~ proportional to 009 haaucv , tells us tha t c =Z"/Y_N'. (17.2.25) The cancellation of terms in ~ Y('} proportional to B JAwp Ay (or Oa~~~r .~#aPcoy)requires tha t dIXflY= - (Zco'JV -)CaflY (17.2.26) The terms in 6Y{N) proportional to Oa~~a(O #~yASautomatically then cancel by virtue of the Jacobi identity for the structure constants . The cancellation of terms in 6Y(Iv' } proportional to Bha O'u(v#Ay(or BhaA~Ay~~g) yield s Finally, the effect of the infinitesimal transformation (17.2.10)on matter and gauge fields is the same as a local gauge transformation with gauge parameters ea = Y_41-0 c_)a and gauge couplings renormalized by a factor 11X (that is, with to and Ca#,i replaced with to = ta I,.' and Cady C,#ylX) ,so the cancellation of terms in 6YN)with only one factor of wa and no factors h,, or co,* simply tells us that the Lagrangian YW,Afor these fields is gauge-in variant, with a renormalized gauge coupling . We In a theory with scalar fields, we could also have renormalizable terms with hx multiplied with one or two scalar fields . Such terms cause no trouble, but for brevity they will not be considered here . 90 17 Renormalization of Gauge Theorie s conclude then that the most general renormalizable Lagrangian density allowed by our assumed symmetry principles i s LPN)=--ZAFU'tVFai,v -Z 4,~}~y`[a p- itxAx u]W+2~'hah a A~-Zw(aPcoo)( a~U)+Z(1)Oa#,,(auco *)w# AVju (17.2.2$) where the tilde on F,1," indicates that the field strength is to be calculated using the renormalized structure constant Cady CX#y/S'.But apart from the appearance of a number of new constant coefficients, this is the same Lagrangian with which we started . The new constants in this Lagrangian (including the gauge coupling constant) may be freely shifted by adjusting the Nth-order terms in the corresponding constants in the original unrenormalized Lagrangian . In particular, we can adjust these terms to make FN(6)=SR,in which case FN,,,, = 0, completing the proof . In the above proof we made important use of the accidental invariance of the gauge-fixed LagrangJan (17.2.17) under the antighost translation transformation (17.2.23).This symmetry would not be present for gauge- fixing functionals other than f ,=0YAa, which are not spacetime deriva- tives . One frequently cited example which preserves Lorentz invariance and global gauge invariance is 0PAa + a«# ;A~Aju, where aIX#v is a constant matrix, symmetric in #and y, which transforms as a tensor under global gauge transformations . (Such constant tensors exist for all S U (N) groups with N ~ 3.)Another more important example is the background gauge-fixing functional to be introduced in Section 17 .4. The absence of antighost translation invariance does not affect our argument that Nis a Lorentz- and global gauge-invariant local function of fields and field derivatives of dimensionality no greater than 4, that is invariant under the renormalized BRST transformation (17.2.1l)-(17 .2.15). But without antighost translation invariance there are new terms in Y{N) that satisfy these conditions . Since the transformation (0 .2.11)-(17 .2.15) is nilpotent, we can construct such terms as 5'F, where the transformation (17.2.1l)-(17 .2.15) is written as x"--*Xn + Bs'xn, and Fis an arbitrary Lorentz- and global gauge-invariant function of ghost number -1 . One such term i s aaflYs/(w,,*Afl,A~) =-axfly[hxA#uAy+2Yc_),,*(X0,wfl + C#SFAsMcva)Au1 . This causes no trouble : it is just a renormalized version of the usual ghost and gauge-fixing terms arising from terms a,,fl7A~A~~ in the gauge-fixing Functional f,:,. But there is also another possible term of the for m b1#7 '~r~~~[.tly~ _ -~?a #~[2h.ww y + 2 JCyde~a~~~S~e1 9 l 17.3Renormalizatio n:General Gauge Theories 91 where bIXay is a constant, antisymrnetric in a and P, which transforms as a tensor under global gauge transformations . (Such tensors exist for any Lie group ; for instance, we could take bx flY to be proportional to C,,,fly.)But the Faddeev-Popov-De Witt method cannot yield four-ghost interactions in the Lagrangian, so there are no counterterms available to absorb ultraviolet divergences in this term . This is not just a technical obstacle to proving renormalizability ; for gauge-fixing functionals like f0( = OPAa + a~flyA IAy~, one-loop graphs actually do yield divergences in four-ghost amplitudes that cannot be cancelled by counterterms in the Faddeev-Popov-De Witt Lagrangian . Aside from avoiding gauge-fixing functionals other thanOUA)', the only solution to this problem seems to be the one mentioned in Section 15.7. We must give up the Faddeev-Popov-De Witt approach, and instead take the action from the beginning as the most general renormalizable function of the gauge, matter, ghost, and auxiliary fields that is invariant under BRST and the other symmetries of the theory . According to the arguments of Section 15 .8, the action can be written in the form 10 + 5 y with I ❑ghost-free, and the S-matrix is independent of 1P, so we can justify this procedure by quantizing the gauge theory in axial gauge, where ghosts decouple, and then taking Y to be anything we like . In particular, we can include s(w`w` w)terms in the action that can serve as counterterms to the divergences in four-ghost vertices . 17.3 Renormali zation :General Gau geTheories* The proof of the renormalizability of non-Abelian gauge t heories in the previous sec tionreliedon a `b rute force' a nalysis of t he poss ible terms inthe ac tionof d imensio nality four o r less.Butas we saw i nChap ter 12,thislimitationon t hedimensiona lity of terms in the ac tioncan be at best a goo dapproximation. The successf ul renorma lizablequantum fie ld theor iesthatareused todescribe the st rong, weak, a nd elec tromag netic interactions are a lmost cer tainly effec tive fie ld th eories, accompa nied w ith terms of d imensionality d >4;these terms are norma llynotobserve d becausethey a re suppressed by4- d powers of some very large mass, perhaps of or der 10I'- 101 $GeV . Grav itatio ntoo ca n bedescri bed by an effective fie ldtheory, inwhichtheLagra ngiandens ity co ntainsnotonly the Ei nstein-H ilbert term - ,,[g_R116rt G, but also a llscalars co nstructed from fo ur or more derivatives of the gravitationa lfield. We need to This section lies somewhat ❑ut of the book's main line of development, and may be omitted in a first reading . 92 17 He normalizatio n of Gauge Theorie s show that ga ugetheories of this so rt, whichare no trenor malizableinthe power-cou nting sense, a re none theless renor malizable inthemodern sense thattheultraviolet divergences a re gover ned bythe gauge symme triesin sucha way thatthereis a co untertermavailableto cancelevery infinity .3 For this pu rpose, let us returntothe actio nS [X,xt]introduced in Section15,9, taken as a f unctionof in dependent fieldsxn(including ga uge andmatterfields 0' a ndghost fie lds wA as we llasthe non-m inima lfields COA*andhA), together w ith theirantifields Xt.Intheorieslikequantum gravity or Ya ng-Mi llstheories,thatarebasedona closedgauge algebra withstructure co nstants f CAB, this ac tionis cons trainedtobe of the form S = t[0]+(O A f A[0] 0 ~ + wA~B .~CAB[01 w~ - hA wA ~ (17.3.1) where t[0]is invariant underthe infi nitesima lgauge transformatio ns 0 r --* or + 0f A[0].(As inSect ion 15.9, the indices r, A, e tc,include a space time coo rdinate, over w hich we integrate insums ove r these indices .) We s hall not limit ourse lves here to act ions of this form, but we shallsuppose that thelocalsymmetries of thetheory are impose d by requiringthatthe act ionmustobey some `s tructura lconstraints' on its antifield dependence, of w hichEq. (17 .3. 1) prov idesjust one example .The structural co nstraints are ass umedtobe linear, inthe sense that ifthey are sa tisfiedfor S + 91 a ndforS + 92, thenfor a ny constants al, a2 they are sa tisfiedforS + a lYl + 92Y2 . We a lsoimposethequantum m aster equation(15.9.35) (S,S)- 2ihAS =0, (17 .3.2) with ❑S defined by Eq.(15.9.34), and the factor h now made explicit as a `loop-counting' parameter, in the sense described in the footnote of the previous section . The action is taken as a power series in h S =SR+hSI+h2Sz+ ,(17.3.3) where SR is an action of the same general form as S, but with all coupling parameters replaced with finite renormalized values, and the SN are a set of infinite counterterms . The action S is supposed to satisfy the quantum master equation (17 .3.2) for all h, so SR satisfies the classical master equation (SR,SR) _ 0, while the counterterms satisfy(17.3.4) N-I ( SR, SN) - - ~ ~ (SfSNf) +W N-I (17.3.5) M=I 17.3 Renormalization : General Gauge Theories 93 The cou nterterms SN are not by t hemselves suff icient to canceltheultra- violet divergences in loop grap hs. As a generalization of the conventional renorma lization of fiel ds, we a lso have to introduce a set of re norma lized fields andantifields,define din terms ofthe or iginalfields and a ntifields by an arb itrary anticanonica l transformation. An infinitesimal a nticanonical transformation maybedefined in terms of a n infinitesimalgenerating functional 6F by Eq. (15.9.26), so a seq uence of an ticanon ical transforma- tions G(t) --* G(t + 5 t) = G(t) + (F(t )5t,G(t)) (whereG( t) is any fu nctional of fiel ds andantifields, a ndF(t) is the genera ting functional) leadsto a finite ca nonical transforma tionG--+G - G(1), w ith dG(t) = (F(t), G(t)}dt If F(t) is given by a power ser iesG(O) = GG. (17.3.6) F(t) = gtF j + h2tIF2 + - - ' thenEqs. (17.3.3), (17 .3.6), a nd (17 .3.7) y ielda transforme daction(17.3.7) sR+hIsI+(FI,SR)] +h2[s2+(FI, SO+(F2, sR)+ 2(Ft, (Fr, SR))] +... (17.3.8) The quest ion is whet her we ca nuse w hatfreedom we have to choose the FN an dSN so as to cancelall infinities ar ising fromloop grap hs. As alrea dynoted,the first term SR in Eq. (17 .3.3)is automatically finite. Supposethat by ca ncellation of infin ities wi thSM an dFM for M< N ithas been possible to e liminate a ll infini ties in th e ter msI'M of or der hm i n th equantum effect ive ac tionwithM< N .As we saw in the prev ious sec tion,the Zinn-Justinequation (derivedhereby setting xn = Kn +6''16x~`)tellsus inthis case t hat theinfinite par tFN,,,,, ofthe term in t he qua ntum ef fective ac tionof or der h-N sat isfies the condition (SR.rrr,ou ) = 0 . (17.3.4) Thefieldandantifieldvariables xn a ndx~ are re lated tothe varia blesxnand Knby ananticanonical transfo rmation, wh ichpreserves all an tibrackets, sothe antibracketinEq. (17.3.9) may be ca lculated in terms of xnand xn instead of xnandKn. The condition ( 17.3.5) satisf iedby the counterterm SN is no t the same asthe conditi on (17 .3.9) sa tisfied byI-'N,~. However, g iven any S~ t hat satisfies Eq . (17 .3.5), we ca nfinda class of ot her so lutions SN=SN+SN, (17.3.1) where SN is arbitrary, except fo r the conditionthatSR + SN like SR+ SN satisfies the sa me symmetry con ditions as SR, a ndthat (SR,Sir)_0 (17.3.11) 94 17 Renormalization of Gauge Theorie s so as not to invalidate Eq .(17,3.5). The infinite part of the Nth-order term in the quantum effective action may therefore be writte n where XN consists of terms from loop graphs, as well as from the term SIN and various terms in Fthat involve S Mand FA4 for M< N .For instance, forN= 2 Eq . (17 .3.8) give s X2= S~ + 2(Fl, St) + (Ft, (FI, SR)) + two-loop terms involving only S R + one-loop terms involving S R,S1and Fl . For our purposes the only thing we need to know about X Nis that it does not involve SN or FN, and that it is invariant under any linearly realized global symmetries of S R. Now, because (SR, SR)=0,the operation F ~--*(SR,F)is nilpotent ; for all F, (5R,(SR,F)) = 0 . (17.3.13) Hence it follows fr omEqs. (17 .3.9)and (1 7.3.11)-(17.3.13)that (SRXN,c :)_0. (17 .3.14) Any term inXN,,,, ofthe form (S R, Y) may be ca ncelled in Eq. (17.3.12) by choos ing FN,,, eq ual to Y . Th usthe space of possible re maininginfinite termsin XN, ,,,that need to be ca ncelled by the counterterm SN ~, co nsist ofthose f unctionals X thatsatisfy (SR,X) = 0, coun ting as equ ivalent functionalsthatdiffer o nlybyterms of the form (SR, Y)- Inother wor ds, theinfinitiesin I 'Nthatneed tobe cancelled bythe counterterms SN belong to t he cnhnmoingy of t he mapping X ~-4 (SR, X)- The poss ible form of the co unterterrn SN islimited bytherequiremen t thatSR + SN mustsatisfy w hateverstructuralconstraints areimposedon the act ionS.Thus we ca ncompletethe proof of reno rmalizabilityif we canshow t hat the coho mology of t he ma pping X~--*(SR,X) consists only of fu nctionals that sat isfy this structura lconstraint. Inthe case of quantum gravi ty coup ledtothe Ya ng-M ills fie lds of a semisimp le gauge sy mmetry,the symmetries of the theo ry are impleme nted bythe structuralcons traint(17.3.1).Inthis case t hereis atheore mthat states tha t the co homology of the mapp ing X ~-4(SR, X) (o nthe space of local fu nctiona ts, rather than space time-dependent fu nctions, of g host numbe rzero) consists" of func tionals A [O]thatare invariantunderthe Strictly speaking, this is valid if one requires that the constant coefficients in Sdo not take special values for which S would be invariant under a larger group ❑f local symmetries . For instance, this excludes the case S = 0. 17.4 Background Field Gauge 95 gauge transforma tion0"-3 or + EAf A[0] with structure co nstantsf CAB . Any Nt h-orderinfinity of this sort maybe ca ncelledwitha co unterterm SN of the same form, so a lthough thesetheories are not renorma lizab le in the co nvent ional power-counting sense, they are renormalizable in th e sensethatall infinities ca n be eliminated by a c hoice of pa rame tersin th e original bare act ion 1[0] a ndby a su itable reno rmalizationof fie lds an d antifields . In other theories the co homology of the map X ~-4 (SR, X) con tains additional terms. Th isdoes no tnecessarily re quire a weaken ing of the struct uralconstra ints,because the additional termsin the cohomology may notcorrespo nd to ac tual ultraviolet divergences. Fo rinsta nce,in gauge theories wi thU(1) fac tors the co homology con tains terms4 cor respond ing to a redefinition of the ac tionofthe U( 1) gauge symmetry on t he various fields of thetheory, w hichifinfinite wou ld req uireusto weake nthe structuralconstraintsbyleavingthenormalizatio nofthetransforma tion functions f A[0] inEq. (17 .3.1) ar bitrary.But thisinfinity is forbidd enby the same sof t-photon theorems that tel lusthat ra tios of U(1) cou plings to va rious fields (like the rat ios of t he various lepto ncharges i n quantum electrodynamics) a re unaffected by radiative corrections . (See Sec tion 10.4.) W here t he extra terms in the co homology do contai nultravio let diverge nces, it isnecessa ry to weake n the structuralcons traint im posed o n S in o rderthatthere s hould be a cou nterterm for eve ry poss ible ultravio let divergence.It isnotknown w hether this wi ll always be possi ble;ifnot, some theo ries may have to be rejec tedbeca use of their unremovable ultraviolet diverge nces. 17.4 Background Field Gaug e We next turn to a method of calculation that explicitly preserves a sort of gauge invariance, and that therefore proves extremely conve- nient, especially in one-loop calculations . We consider the effective action 17[A, W, w,co*] as a functional° of classical external gauge, matter, ghost and antighost fields :A,,(x), y3,,(x), cox(x), r~),,*(x) . Even though ghosts and antighosts never appear in initial or final states, we are considering back- ground ghost and antighost fields as well as gauge and matter fields in order to deal with parts of diagrams that have external ghost or antighost lines . We are now returning to the specific choice of the gauge-fixing functional B[f] as the Gaussian (15 .7.4), and we are integrating out the auxiliary field h ,, so that the gauge-fixing term in the modified Lagrangian is just -fj,1 2~. 96 17 Renormalization of Gauge Theorie s Asdescribed in Section 16.1,r[A,y), w, w* ]isthe sum of co nnectedone- particle-irreducible graphs for the vac uum-vacuu mamplitude, ca lculated inatheory inwhich thequantum fie lds A', W', w', co'* over w hichwe integrate are rep laced in t he act ion wi th shiftedfields A + A', tP + V, co + w', w "+ w the pa th integralbeing taken over p rimedfields withthe unprimed fieldsheld fixed . We are free to c hoose t he gauge-fixing function fa(x) p retty muchany way we like; instead of o urprevious choicefa = aitA~(or0~[Aa+Ate]) we s hall now take5 .fa=01aAa + CaflYAfluAy (17.4.1) The reason for this choice is that it makes the gauge-fixing term f,,f a invariant under a formal transformation, in which the background field A~ transforms as a gauge field, while the quantum field Aa transforms homogeneously, like an ordinary matter field that happens to belong to the adjoint representation of the gauge grou p 6A~ - -Ca flv ~~ Ay(17.4.2) (17.4.3) Thetransformationproperties of f, can be seen mos teasily by wri ting it as a new so rtof cova riant de rivative fl= ~A?Pa where for any field 0,,in the adjoint representation(17.4.4) (17.4.5) We see that under the transformation (17 .4.2),(17.4.3), the function (17 .4.1) transforms just like A ': 5.fac =-CaflY'FflfY sothe termf a f,,,inthe mod ifiedLagrang ian isinvariant(17.4.6) (17.4.7) Also, the original Lagrangian density Ydepends on AandA'only through the sum A+A', which under the combined transformation (17.4.2), (17 .4.3) undergoes an ordinary gauge transformatio n 6(A~ + A ~)= Otte, - C x~yfQ(A Y+AY~. If we transfo rmthebackground and quantum ma tterfieldsby JP= 1 tx Ex y) 61p'=ito'FaY), thenalso(17.4.8) (17.4.9) (17.4.1d) (17.4.11) 17.4Background Field Gauge 9 7 The original Lagrangian Yis invariant under the original gauge trans- formations (17.4.8),(17.4.11), and only depends on A + A' and ~ ) +tp', so it is also invariant under the new formal transformations (17 .4.2), (17 .4.3), (17.4.9), (17 .4.10). It will be useful to make this invariance property more explicit, by writing Yin terms of the background-covariant derivative D.In general, we have (0,av av ~- L C aF~ QY[Q6i Ql~ Y Yv + YM (tp+ Y) I~ay(W +W')-ita(Axy +A'Ocy)(ip +W')) 2 - -" 4 ~Fa uv +I)jAa;v -DvAa~ + C~ flYA~~Ayti' } (1P+ Y)f, D,(tp -~- tp') - i txA' (tp + tp')) where, as in Eq . (17 .4.5), DJUAav = aPA~v + Cxa~4a1~Ay V alltp- i tocA a ytp and Fa ,v is the background field strengt h Fayv=-=ayAorv -Ov A ali +CIXfljAgyAjv .(17.4.12) (17.4.13) (17.4.14) (17.4.15) (The squareinthe first term of Yisintended to imp ly obvio usindex contractions.) C learly Y isinvaria nt underthe new transforma tions (17.4.2), (17 .4.3), (17 .4.9), ( 17.4.1Q), because it involves A,Y o nly inthe field strengthFy, a nd in backgroundcova riantderiva tivesbyof `matter' fields Aly,1P' and W. This new transfor mationshould be caref ullydistinguishedfrom a true gaugetransforma tion. Sucha tra nsforma tion ca n haveno effec ton A or y, w hich are j ustpresc ribed c lassicalbackground fie lds, and induces a n ordinary ga uge transfor mation o nA + A' andV, + tp', so 6TRLTEAa =PEx- C a#y,Ffl(AP+ Ay 1 = DPCa- C aflYEflAY and TRUE IP =0, 5TRUE Y1f =itaEa(IP+IPI)(17.4.16) (17.4.17) (17.4.18) (17.4.19) Of course,thisisthe same as the fo rmaltransformatio ns (17 .4.2), (17 .4.3 ), (17.4.9), (17 .4.10) i n its effectonA + A' and W + W ', a ndtherefore a lso leaves t he orig inal Lagrang ianYinvariant.However, fo rour new c hoice 98 17 He normalizatio n of Gauge Theorie s (17.4.1) of f,the term f a f,,,does notdepend only on A + A', and is not invariant under (17 .4.16) and (17 .4.17). Instead , TRUE f x CafljeflAy } (17 .4.20) with D~ given by (17 .4.5). Finally, let us consider the ghost Lagrangian in this new gauge . The quantity (15 .7.3) in the ghost action is given in general by just replacing Ex with the ghost field wx + wx in 6TRVEf ❑a= I]"[b'(wi + cva )- C aflY(rofl + w~)AY ] (17 .4.21) The ghost Lagrang ian in Eq. (15,6.2) is therefo re UGH = (w IX + wIX*)D,[b1L(w ,+(t)')- C 2~~(wo+ r~ ')AYuJ or, integrating by parts,(17.4.22.) ~C~~ _ -~ D~(wa + c.oa }} + wa} - C flY(UJfl+UJ~ )A~z).(17.4.23) This is manifestly invariant under the joint transformations (17 .4.2), (17.4.3), supplemented now with transformations on w and w' : 6wx=-CaflvEflw'1, cox=1 and likewise 6(,0a=-CaQY'Fp)y 60)a= -CaQYeaG)y(17.4.24) (17.4.25) (17.4.26) (17.4.27) We see that th e fo rmalcom bined tr ansforma tion (17 .4.2), (17 .4.3), (17.4.9), ( 17.4. 10), a nd (17 .4.24)-( 17.4.27) leave inva riant the comp lete Lagrangian in t he mo difiedaction (15 .6.4): We are integrating over A', y)', o)', and crag' with a measure that is pre- sumed to be invariant under the simple matrix transformations (17 .4.3), (17.4.10), (17 .4.25), and (17.4.27), so the effective action F [A, V), CO, (t)*] is invariant under the remaining transformations (17 .4.2), (17 .4.9), (17 .4.24), and (17.4.26). In other words, it is gauge-invariant in the same sense as the original action I [A, y), w, o)*] . This formal gauge invariance sets powerful constraints on the infinities that can occur in the effective action . The ultraviolet divergences in r appear in the coefficients of terms whose dimensionalities are [mass]` with d c 4, but here these terms are invariant under the background gauge transformations (t7 .4.2), (17 .4.9), (17 .4.24), and (17 .4.26). For instance, in 17.4Backgro undField Gauge 99 a gauge theory based on a simple gauge group ,with spin 1 /2 fermions belonging to anirreducible representation of this group, the only such terms are of the form IF,-=Jdux (17.4.29) 4 'UV -mLrn1~y)- Lr~ (f) 1A , (17.4.30) where here F,,,,v, Dfty), D,w,,and DPwx are constructed entirely from background fields :** Fauv = OiAAxv --OvAau+ cafl-,AfluArv , (17.4.31) (17.4.32) buw,=auwa + C L1flY~~pwa~ (17 .4.33) D~wx =a r~~~ + CxflYAflp{''a  (17 .4.34) Dimensional analysis leads us to expect that the constants LA, Lw, Lrn, and L(, are logarithmically divergent . To deal with these infinities, we note that the Lagrangian (17 .4.12) contains a purely classical piec e CLASS = - 4 Fx ;iv Fa v - ~p(Yiub,+~)tp- lbucoOc~(!)11wa) (17 .4.35) obtained bydroppi ng allterms inYMaD that involve thequantum fie lds All V ', w', r01*.Wedefine renorma lized fields tR au = 1+ LA A R = 1-+ L W Rwa = 1-+L r~wcc(17.4.36) (17.4.37) (17.4.38) (17.4.39) so that the sum of the terms (17 .4.30) and (17 .4.35) takes the for m CLASS + Yx) _ - gFu VFay V- ~ ~ ~~~R IPR A ' U where m Ris the renormalized mas s MR=m(l ~-Lm)l(l +Ly))(17.4.40) (17.4.41) The condition o fa simple gauge group i nsures that t here is just a single kinetictermfor Aand for w, proportional to FaPFa" and D~cc~a respectively, while the condition that 2l) transforms irreduciblyinsures that there is just a single kinetictermand a singlemassterm for W . It wo uldbe easy totreat more ge neral possib ilities, a t the cos t of a s light compl ication innotat ion. We are also i mplicitly using the conse rvation of ghos tnumber. 100 and17Renormali zation ofGauge Theorie s R R R R RAR D~waa~wa + Cfly``~aywR De w*R aw*R+ C R A~iRw*R ya A a oc~iY y Y(17.4.43) (17.4.44) The renormalized structure constants and group generators here are jus t cR Y=(1 + LA) -112Cafly to = ( 1+ LA ) -1/2 t o(17.4.46) (17.4.47) Because we assumed that the Lie algebra here is simple, the structure constant Cx#y and group generator t,, are fixed by the group structure except for a single common factor, the unrenormalized gauge coupling constant g . Eqs . (17 .4.46) and (17 .4.47) thus simply tell us that the gauge coupling constant gRfactor in CR and tR is renormalized b y gR- g(1+LA)-1/2. (17 .4.48) This resu ltexhibits the pa rticularvirtue of thebackgrou ndfield ga uge. In a genera lgauge we wo uld encounter ind epen dent renormalization factors for the gauge field a ndthe gauge co upling constant, andwe wo uld have to ca lculate two separate amp litudes (say, t he vacuum po larizatio n and t hree-ga uge-fieldvertex function) i norder to be able to sor t these out.Inbackgroun dfieldgauge, background fie ldgaugeinvar iance ties these two renormalizationstogetherby req uiringthat theinfiniteterms inthe effective Lagra ngian involvethe fie ldstreng th inits or iginalform (17.4.31), a nd so we ca ncalculate the charge reno rmalizationfactorby studying justone gauge fie ldamplitu de. 17.5 A One-Loop Ca lculation in Backgro und FieldGauge As an exe rcise, we are now goi ng to ca lculatethe one-loop re norma lization factorforthe gauge coup ling co nstantina ge neral non-Abeliangauge theory. As we will see i nthenext chapter, t his provi des anessen tialinput inso-ca lled `re norma lizationgroup' ca lculations of phys icalprocesses a t highenergy ;the resu lts we o btain he re will beusedtheretodemo nstrate the asymptot ic free dom of non-Abeliangauge theories. The met hod to be emp loyed here is somew hat novel . Us ually, one considersthe effective act ion in a s pacetime-dependent backgrou ndgauge field, and calculatesthetermsquadra ticinthis fiel d, extracting a fac tor (qyAIXv- gvA (xp}2 (w here q isthe gauge fieldfour-momen tum), and only 17.5 A One-Loop Calculation inBackground Field Gauge 101 then isolating the logarithm ic divergence by sett ing q = 0in the coefficient of this factor .Instead we shall follow the much simpler course of taking the gauge field to be spacet ime-independent from the beginn ing. In this case the terms in the effective action that are quadrat ic or cub icin the gauge field of course vanish ,but there isa non-vanishing quartic term that is ultraviolet divergent, and wh ich can be used to calculate the coupl ing constant renormalization factor (1+LA)-1I'.In this way our one-loop calculation becomes a matter of simple matr ix algebra . Note that this procedure can only work in background field gauge ;otherw ise there would be independent logarithmic divergences in the parts of the effective act ion that are quart ic and quadratic in the background gauge field . With th is mot ivation ,we turn to the calcination of the one-loop effective action in a background field for wh ich A ,,,,is constant and T= c.)=of-- 0. For such a background field ,the full modified Lagrang ianis* YMoD = Y + Yf + YGx, a(f,,, IUav v ce pj#YRFC Y►' 1(17.5.1) (17.5.2) (17.5.3) (17.5.4) One-loop graphs for vacuum-vacuum amplitudes are calculated from the part of the action that is quadratic in the quantum fields At, W% co', cc)'* over which one integrates . Keeping only such quadratic terms, we hav e YQuAn = - 4(DpA~v V` DvAau)2- 2 FavCaf~~rA~~Ay v + m)y3~ - ~ (D~Ax )2 - (DAct~a }(D~`wa} . (17 .5.5) The corresponding action may be put in the general quadratic form : IQUAD °Jd4x SQUA D - ? f dux day A~ (x)A~(Y) ~ ~U,y#v - f d4xd4.v ~ ~(x) W~(.v)gk,y( d4x d4y wa' (x)c.),(Y)-qxwa ,y#3'(17.5.6) See Eqs . (17 .4.12), (17 .4.4), and (17 .4.23), We are here specializing to the case of matter fields forming a multiplet ❑f shin ? fermions . The squares in Y and Yf include ❑bviaus index contractions . 1 02 With17 Renormalization of Gauge Theorie s _9 xx~,y~gv ~~v~ - rya ~a + ~~~~a~Sl~1 ~ ~ -6y~3 ~a + Cye~4AE~,1y1)641x -y) XA y - ~ ~byaa r~xv + CyaxAsV(x)~ ~ - 6,fl~ Oy~+ C,1E #Aejj(Y)}64(x -y) +F Y,U v('x)Cyaflb41X -y) + 1 ~ - 6,iaa~ + ~ya~Ab~(x}~ ~ - ~,~ a v + C yEflAE.V(y}~64(x - y) ~ r~ X M (17.5.7) A'Ye, Oy kc(17.5.8) _qa x~,y fl--rya OxA +CY~,14sz(x)~ ay).+ CyE#A E(.~)64(x - y) (17.5.9) (The minus signs in front of Ol axand 0/0ydrop out when we integrate by parts .) The one-loop contribution to the effective action is given (as in Section 16.2) b y exp (iF1 100P[A]) oc f P j(fl dA')(fl dWf)(fl dqp')(fj dco')(fj dry*'} x exp (iIQv ADLA'sy1%Y)'aco', o)'*;A] aC(Det gA )-1/'(Det _9W)+f(Det 9`'}+t . ( 17.5.10) (The exponents -1/2 and +1 appear because A' is a real boson field, while w',fp-', co', and cr )`* are distinct fermionic fields .) The calculation of such determinants is generally not an easy task . However, it becomes much simpler in the case of constant external fields, where the 9s can be diagonalized by passing to momentum space . Let's therefore now consider the case of constant background field AIX ,U. For non-Abelian gauge theories such a constant field cannot be removed by a gauge transformation, as shown by the non-zero values of various gauge-covariant fields FaEiv = CxflyAfluAvv , DAFa,ul,= C aFxCa#yAe.~A#uAvv(17.5.1Z) (17.5.12) and so on . Lorentz and gauge invariance tell us how to express the part of r[A ]of a given dimensionality as the integral of a finite number of local functions of Fx ,,,, D ),Fa,,, etc .; the coefficients of the terms in this expression can be inferred by comparing the contribution that these terms 17.5 A One-Loop Calculation in Background Field Gauge 103 make to r[A] for constant background field Aar with the results of a perturbative expansion . We transform each of the `matrices'9A,_9,v, and -91to a momentum basis by the usual normalized Fourier transfor m r a'X e_iq.x d4yeip.Y q--,p~,.-J(27r)2 f(27z)2x.., y.... With A constant, this gives(17.5.13) (17.5.14) where -denotes d iscrete ind ices, and the aare finite q-dependent matrices Gel -(-igv6~lx+ AavCja,x) (iq~5~,,# +'4,,uCvfl) +Fy1jv Gio +(-iqp5y(x +A6Y CY6x)(igvbvg+AEV CYFfl)l~ +eterm s , (17 .5.15) ~VW = 0 4- ita4 + m} V+ Eterm s , (17.5.16) a ~ W = H q),bya+ Aaa, Cyax)(iqAby#+A F'Cyffl) +F terms , (17.5.17) with FyF,,, given by Eq .(17.5.11).From Eq .(17.5.1 d)we have the n ir(1loop) [A]12InDet!2A +inDet-91'+In Det -9w _ - 2Tr 1ngA+Tr In 91P+ Tr In_9w _64(P_P)Id4q +trIn 1#'(q)] -2trIn A(q)+tr In .R`P(q) (17.5.18) We denote traces by `tr' instead of `Tr' in the last line of Eq .(17.5.18) (and from now on in this section) to indicate that these are the usual traces of finite matrices rather than of integral operators . Since we are aiming here at a calculation of the infinite factor LA multiplying FF terms in the effective action, let's isolate the term in (17.5.18) that is of fourth order in the background field A . For this purpose, it is convenient to divide each into terms n containing n= 0, 1, or 2 factors of A . = o + 1+ 2. (17.5.19) It is then elementary algebra to show that the term in (17-5 .18) of fourth 104 17Renormalization of Gauge Theorie s i ~~it r r ~ r ~ i♦ r i Figure 17 .1. One-loop Feynman diagrams for the term in the quantum effective action that is quartic in a constant background gauge field A.Here solid lines represent internal gauge, ghost, or matter lines ; dashed lines indicate factors of A.These three diagrams correspond to the three terms in Eq .(17.5.20). order in Axy i s [tr In ] Aa = tr {-~[,#-',#2]2 +[~'# 0 1 ~]2_ 41.2- ¢ ['#4 1.x]4 r ( 17.5.20) (To see this, insert factors e and e2 multiplying /#1 and 4#2 in Eq . (17.5.19), differentiate tr In four times with respect to e, divide by 4! and set e=0.)The01 factors here are just the usual propagators ; for ~ = 1,these are au,flv Leo(q)]~~=Li4+ m]V(17.5.21) (17.5.22) (17.5.23) Indeed, the three terms in Eq . (17.5.20)just correspond to the three Feyn- man diagrams shown in Figure 17 .1; the present method of calculation saves us from having to think about signs and combinatoric factors . For the A loop, Eq .(17.5.15) gives for ~ = 1 : where dAis the matri x forwhichWda~S/,,,q/11 - ~ YS/vlafl+FYpv Cyocfl5 iCa~yAy,~ ~'q1A a~lpJafl=-C aSyCbflF(AY~'4ep- Ayp`4eA) - - (CaS jCSpe+Cac6eC6py) Ay~Aep +CaaflCbEyAyAAEp= C x6#F6 pA. 17.5 A One-Loop Calculation in Background Field Gaug e The integrals have the structur e f4 f105 jd4qqMqvqPqf(q2)= 241Tuy,7Pcr+7YP,7va+7utT,7V,olJd4q (q')'f (q'). We then find, for~=1 : d4q tr_#A (q)-' ,#A(q)i2 4J tr['q/1'dA,dq' W +4_1'1"Cya#C6~pFy,uvF61"° d4q trlg(q)-iii(q)J2w o(q)-i ~~ (q)J = 4J d4 q tr1a(R')-i ii(q)j4 = 3 Jtr[2~2~~,~ '~~,~ + ~ A 11 where Jis the divergent integra l "f- Jd4q[q2-i,]-2, (17.5.24) whose significance is discussed below . Putting this together in Eq .(17.5.20), we hav e Jd4q~trIn'~A(R')~~¢ - 3 Jtr1.4IS/ zsr- W~ - S/,71 -2-1 CY ,x#Caa#Fyuv Fav Both terms are actually of the same form, and when combined yiel d Jd4q[trIn'/ffA(R')]A4 3JCyaf~ Cbaf~~`yF~v Fa v . (17.5.25) Now jumping ahead to the ghost loop, we see from Eq .(17.5.17) that We then fin d d4qtr-Wo'(q) (q)j2(17.5.26) d4q tr [)(q)_1 i~q)1 0 2 ~q)--J 4 d4qtr[.~ (R')-'~i(q)j =3 J + 106 17 Renormalization of ' Gauge Theorie s Thus for the ghost loop the integral of the quantity (17.5.20)is Jd4q [ Tr In M~'lR~llA4 = 6j q - qI izJC1aflCaaflF~yvF~v Finally, the vertices in the matter loop ar e so here there is only one term in Eq .(17.5.20)(17 .5.28) rd4R' [Tr In ` P(q)IA4 = -'Jcoq Tr [(i4 + m)-1ta-Aa]4 We are interested in the ultraviolet-divergent part of this integral, so we may drop the mass (which is negligible for large qv) and writ e j d4q[Tr In "(q&4 = -1 T'r ftt#t rtal AauA#vAyNAa u xId4q1'r~YYµqYvqY" 4Y° W--ie)8 T*atfltYtajAx.AflvA1PA6.96 X T' r{Zy~,Y',AY vY'YPyl'' + Y,~y~yr~y1 yAyp'yI yal,l (17.5.29) where Jis the same divergent integral as in Eq .(17.5.24). To calculate the traces of Dirac matrices, we use the anticommutation relations of these matrices to writ e Trf 2yAY17A YvYqY'°YqY' + Y alYqYVYI Y'°Y qYd1 = 8Tr~y'y°ypy'j - 4Tr ~y''ylypy's - 4Tr{y~`ypy"y'j _ - 64t711P,7v7+32Y7PvY~P(7-+32Y,,`TY7vp. Eq.(17.5.29)then give s Jd4q[TrIn " (q)]4= 3J Tr~ [tom,t#a Lty, 0 ~A«jj~ 1#v A~A ~ j, '/11V 6 Y Using Eqs . (17.5,25 ),(17.5.28), and (17.5.30)in Eq .(17.5,18)gives at las t 2n J 61 1~C+'aflC6oI 3 Trf tYt6I (17.5.31) 17.5 A One-Loop Calculation in Background Field Gauge 1 07 where we have expressed the momentum-space delta-function in (17 .5.14) as J(17.5.32) It is important that the result turns out to depend on A ,monly through the field strength (17.5.11), as required by background gauge invariance . Let us now (for the first time in this section) use the assumed simplicity of the gauge group and irreducibility of the matter field multiplet . In this case CyaflCt5afl=g1Cl r ya Tr~t .,t,~j= g'C2675,(17.5.33) (17.5.34) where g is the common gauge coupling constant appearing as a factor in Cy,#and ty, and Cl and C 2are numerical constants that characterize the gauge group and the representation of this group provided by the matter multiplet . For instance, in the original Yang-Mills theory the gauge group isSU(2) (or equivalently SO(3)) and the structure constants ar e C;'afl-9EaflY with a, fl, and y running over the values 1, 2, 3 . Comparing with Eq.(17.5.33), we see that here Cl = 2 . Also in this theory the matter field forms a doublet with t, given by g /2 times the usual Pauli matrix SIX, s o C2=:1/2. Somewhat more generally, for the group SU(N) with n f fermions in the defining representation, with a conventional normalization of generators we have** Cl = N , C2 = n f -/2. (17.5.35) Returning now to the general case ,Eqs.(17.5.33) and (17.5.34) give rA4loop) = [tg24 d4JCFyFiv~~~v1Cl-1C2] ( 17.5.36)2~)J 112 3 That is, the infinite constant LA in Eq, (17 .4.30) is 11 1LA (2n)4 (12 C1 3C~ (17.5.37) For SU(3)the r x are taken a sg/2 times the Geld-Mann ma tricesAaused in Sect ion 19 .7, sothat CIX#;, _(g/2) f'x#,, 108 17 Renormalization of Gauge Theorie s It remains to say a word about the interpretation of the divergent integral J. First, before we try to integrate over the three-momentum q, we can rotate the contour of integration of q4 in Eq . (17 .5.24) to the imaginary axis ; as usual, the -le is the denominator forces us to rotate counterclockwise, so that q4 = iq4, with q4 running from -oa to +ao . The integral is then facZn2g3dq 4 (Z7.5.3$) q where q is now the magnitude of the Eucl idean four-vector (ql, q2,q3,q4). To go further ,we ev identl yneed some method of regulating the integral . The simplest way of deal ing w ith the ultrav iolet divergence is just to cut off the integral for q above a scale A.However ,we also need a lower cut-off to deal with the infrared divergence . This is provided by the phys ics of the s ituat ion. If the momenta of the four vector part icles isnot zero ,then a momentum flows through the internal lines of the d iagrams, provid ing an infrared cut -off at the scale ,uof the se momenta .Similarly , if we evaluate the fourth variational derivative of r [A] with respect to A not at A=0but at a finite A,then the propagators of the internal l ines do not blow up at zero momenta ,and we have an infrared cut-off at a scale p --'gA.Either way , _ftakes the for m 27r2ifA dq= 2n 2 i In(A) q P and hence g 11 1 (A) 4 2rc212 ~f 3C2 In +O(g )LA= - ) Eq. (17 .4.48) then gives the renormalized coupling a s ~n gR=g1 +4n2In 1Z(ci_c2) +O(g4) 1(17.5.39) (17.5.40) (17.5.41) We note that while in quantum electrodynamics the radiative corrections discussed in Section 11 .2 decrease the physical coupling gR relative to the bare coupling g, in non-Abelian gauge theories they increase the physical coupling over the bare coupling, provided that the fermion multiplet is small enough so that C2<11 C,/4. The importance of this point will be explored in Chapter 18. Alternatively, we can deal with the ultraviolet divergence by the methods of dimensional regularization, discussed in Section 11 .2. Here, in place of Eq.(17.5.39), we write Ik27rzqa-tdq (qp)2 Problems 109 where d is a complex dimensionality, allowed to approach 4 at the end of the calculation, and yis an infrared cut-off, again taken of the order of the external momenta (or of the background fields times g).As long as d is complex with Re d <d and p2>a, this has the finite valu e j=_in2 ~_1~d-4A'S lI1 Z-Z 7 1 Analytically continuing to d -> 4, this i s .~ --~ -w-2i n2[ d1 4 + In P+... where    denotes finite p-independent terms . Here we have(17.5.43) g2(111 1 4~A ~n~ 12 Cl ~C2 ~-4 +ln ~c+ ... +D(g ) and so 2 get = g 1- 4n2 Ci -3C2(d4+InP +...+0(g4) (12 ( 17.5.44) Note that the ultraviolet divergence here takes a different form, but the dependence on the infrared cut-off pis the same . Eq. (17.5.44) will provide an important input to our discussion of asymptotic freedom in Section 18.7. Problems 1. Carry out the proof of renormalizability given in Section 17 .2, in- cluding elementary scalar fields in the Lagrangian . 2. Carry out the quantization of a non-Abelian gauge theory in back- ground field gauge, using the B SST method of quantization discussed at the end of Section 17 .2. 3. Derive the relation (17 .5.44) between the renormalized and unrenor- malized gauge couplings by calculating the terms in 170 l°°P} that are quadratic in a spacetime-dependent gauge field . 4. Calculate the one-loop relation between the renormalized and un- renormalized gauge couplings in a gauge theory containing elemen- tary scalar fields . 1 10 17 Renormalization of Gauge Theories References 1.J. Zinn- Justin, in Trends in Elementary Particle Theory Interna- tional Summer Institute on Theoretical Physics in Bonn 1974 Springer-Verlag, Berlin, 1975) . 2. Before the advent of BRST symmetry, proofs of the renormalizabilit y of non-Abelian gauge theories were based directly on the Slavnov - Taylor identities for gauge transformations ; see B . W, Lee and J . Zinn-Justin, Phys . Rev .D5, 3121, 3137 (1972) ;Phys . Rev .D7, 1049 (1972) ; G. 't Hooft and M . Veltman, Nucl .Phys .B50, 31 8 (1972) ; B. W. Lee, Phys .Rev.D9, 933 (1974) . The original proof o f renormalizability based on BRST symmetry was given by C . Becchi , A. Rouet, and R . Stora, Commu n.Math . Phys .42, 127 (1975) ; in Renormalization Theory --- Proceedings of the International School oj ' Mathematical Physics at Erice, A ugust1975, eds. G. Velo and A .S. Wightman (D . Reidel, Dordrecht, 1976) : pp. 269-97, 299-343 . Th e proof given here follows the general outline of J. Zinn-Justin, Ref . 1; B. W. Lee, in Methods in Field Theory, eds. R . Balian and J . Zinn-Justin (North-Holland, Amsterdam, 1976) : pp. 79-139 . 3. The point of view and presentation here is based on that of J.Gomi s and S . Weinberg, Kyoto-Texas preprint RIMS-1 036/UTTG-18-9 5 (1995), to be published in Nuclear Physics B . For earlier use of thes e methods, see B . L. Voronov and I . V. Tyutin, Theor . Math . Phys .50, 218 (1982) ;52, 628 (1982) ;B. L. Voronov, P . M. Lavrov, and I . V. Tyutin, Sov .J. Nucl . Fhys .36, 292 (1982); P. M. Lavrov and I . V. Tyutin Sav .J. Nucl .Fhys . 41, 1049 (1985) ;D. Anselmi, Class . and Quint . Grav . 11, 2181 (1994) ; 12, 319 (1995) ;M. Harada, T . Kugo , and K . Yamawaki, Prog . Theor . Phys . 91,801 (1994) . 4. G.Barnich and M .Henn eaux,Fhys.Rev.Lett.72, 1588 (1994) ; G. Barnich, F.Brandt ,and M .Henne aux,Phys . Rev .51,R143 (1995) ; Common .Math.Phys.174,57,93 (1995) ;Nucl .Phys.B455 , 357 (1995) . 5. The background field gauge was introduced by B . S. De Witt, Phys . Rev.162, 1195, 1239 (1967) .For the treatment of multi-loop effects , see G 't Hooft, in Functional an dProbabilistic Methods in Quantu m Field Theory : Proceedings of the 12th Karpacz Winter School of The - oretical Physics (Acta Universitatis Wratislavensis no . 38, 1975) ;B. S. De Witt, in Quantum Gravity II, eds . C. Isham, R . Penrose, and D . Sciama (Oxford University Press, oxford, 1982) ;L. F. Abbott, Nucl . Phys .B185, 189 (1981) . 18 Renormalization Group Method s The method of the renormalization group was originally introduced by Gell-Mann and Lowy as a means of dealing with the failure of perturbation theory at very high energies in quantum electrodynamics . An n-loop contribution to an amplitude involving momenta of order q, such as the vacuum polarization II ,U„(q), is found to contain up to n factors of ln(g2/m2 )as well as a factor an, so perturbation theory will break down when a Iln(g2/me)1 is large, even though the fine structure constant a is small . Even in a massless theory like a non-Abelian gauge theory we must introduce some scale pto specify a renormalization point at which the renormalized coupling constants are to be defined, and in this case we encounter logarithms ln(E/M), so that perturbation theory may break down if E >yor E CC y, even if the coupling constant is small . Fortunately, there is a modified version of perturbation theory that can often be used in such cases . The key idea of this approach consists in the introduction of coupling constants g .defined at a sliding renormalization scale y-that is, a scale that is not related to particle masses in any fixed way . By then choosing yto be of the same order of magnitude as the energy E that is typical of the process in question, the factors ln(E/y) are rendered harmless . We can then do perturbation theory as long as gu remains small . In particular, given the coupling constants defined at scale It,we can use perturbation theory to calculate physical amplitudes at an energy y+dy, and use these to calculate the coupling constants defined at a renormalization scale y+dy. By integrating the resulting differential equation we can then relate the coupling constants at the scale of interest to the coupling constants as conventionally defined . (The name `renarmalizatian group' arose originally because one is concerned here with equations that describe how the appearance of a theory changes under a redefinition of the renormalized coupling constants, but it really has nothing to do with group theory .) The method of the renormalization group can also provide qualitative guidance regarding asymptotic behavior at very high or (in massless theories) at very low energy, even where th e 111 112 18 Renormalization Group Method s coupling constants at the scale of interest are too large to allow the use of perturbation theory . Although the method of the renormali2ation group arose originally in connection with changes in the prescription used to define renormalized coupling constants, it has come to have a wider meaning . When we replace bare couplings and fields with renormalized couplings and fields defined in terms of matrix elements evaluated at a characteristic energy scale P, the integrals over virtual momenta will be effectively cut off at energy and momentum scales of order y. Thus as we change M, we are in effect changing the scope of the degrees of freedom taken into account in our calculations . The lesson of the renormalization group, that in order to avoid large logarithms we should take yto be of the order of the energy E typical of the process being studied, is a special case of a broader principle, that in order to do calculations at a given energy we should first get rid of the degrees of freedom of much higher energy . There are various other ways to accomplish this . As we saw in Section 1 2.4, in the approach to the renormalization group pioneered by Wilson 2 one introduces a finite explicit cut-off accompanied by a change in the parameters of the theory designed to keep physical quantities cut-off- independent . This approach requires introduction ❑f an infinite number of interaction types, all those allowed by the symmetries of the theory, and is therefore not particularly convenient in dealing with theories that are actually renormalizable, like quantum electrodynamics (although, as discussed in Section 12 .3, quantum electrodynamics is today regarded as only a very good approximation to anon-renarmalizable theory in which the higher-dimensional interactions are suppressed by negative powers of some very large mass .) Where the cut-off is imposed by quantizing a gauge theory on a finite spacetime lattice, the Wilson approach has the advantage that calculations can be done while maintaining manifest gauge invariance (the volume of the gauge group equaling the volume of the global symmetry group times the number of lattice sites), but it has the disadvantage of not maintaining manifest Lorentz or rotational invariance . In any case much of the formalism of the renarrnalizatian group remains the same whatever approach is used to eliminate the high-energy degrees of freedom . 18.1Where do the Large Logarithms Come From ? Let us first consider how large logarithms can arise at very high energies . Consider a physical amplitude or cross section orother rate parameter I-'(E, x, g, m ), that depends onanover-all energy scale E, on various angles and energy ratios collectively called x, on various dimensionless coupling 18.1 Where do the Large Logarithms Come From? 113 constants collectively called g, and on various masses collectively called m. If IF has dimensionality [mass]' (as, for instance, a cross section would have D = -2) then simple dimensional analysis tells us tha t I'(E,x,g,m) = EDy(Lx .g.mE We mi ghtexpect that in the lim itE --+ oa s uchanamplitude would beh ave asasimple powe r But this is not what is found . Instead, in perturbation theory calculations the factor EDis found to be accompanied by powers of ln(E/m), which invalidate this simple power-law behavior . Clearly, powers of in(E/m) can enter as E --+oo with fixed m only if the amplitude r at fixed energy E becomes singular as m - -+0. There are two classes of such mass singularities, one which is simply eliminated by calculating the right sort of amplitude or rate constant, the other of which requires a change in our renormalization procedure . Zero-mass singularities of the first sort arise from a confluence of poles of propagators on the mass shells of the corresponding particles . For instance, suppose that a Feynman diagram has an incoming line with total four-momentum p ", attached at a vertex to internal lines of mass m2, M2, .--, mn. According to the arguments of Chapter 1 0, the correspond- ing Feynman diagram will have a cut running along the negative real p2 axis, from p2 = -(mI + ...+mn)Z to -co . This does not lead to singularities if the external line is for a stable particle, with a mass M <mt + ...+Mn, because then p2 = -M2is off the cut . However, when M,mt,... , Mn all go to zero, the value of - p2on the mass shell and the branch point at the tip of this cut move together to join at p2 = 0,producing a singularity . This suggests that we can avoid the infrared divergences at m = 0 by simply staying off the mass shell, as, for instance, by letting p2for all external lines go to +oa along with all energy variables . We would then have to employ dispersion relations or some other technique of analytic continuation to use the results for the behavior of Feynrnan amplitudes in this limit to tell us anything about S-matrix elements . Often this continuation is unnecessary because we are interested not in on-shell S-matrix elements but rather in the matrix elements of currents carrying momenta q unrelated to any masses . For instance, the vacuum polarization function 7r(q2)defined in Section ZO .S is free of zero-mass singularities of the first sort except for q2 c 0 . Another approach to the elimination of mass singularities of the first type is suggested by the observation that zero-mass singularities typically occur if we try to calculate a cross section that becomes unmeasurable 114 18Renormalization Group method s in the limit m --+ 0. For instance, in quantum electrodynamics the cross section for any process involving definite numbers of electrons and photons becomes infrared-divergent in the limit me --), 0 , even if we sum over unlimited numbers of soft photons, because for me --*0 it is impossible to distinguish an electron from a jet of electrons, positrons, and photons with total charge -e, all moving in the same direction at the same speed . As shown in Chapter 13, such infrared divergences can be cured by considering only suitably integrated cross sections, which would be measurable for m, --+ 0 . For instance, instead of trying to calculate the cross section for a specific Compton scattering process, we would calculate the cross section for the scattering of a jet with total charge -e with another of total charge zero into two other such jets, plus soft photons . Such inclusive rates or cross sections, which remain finite when all masses vanish, are known as `infrared safe .' Our troubles are not over . Even where we avoid infrared divergences by integrating over cross sections or staying off the mass shell, the resulting integrated cross sections ❑r off-shell amplitudes for energies E contain mass singularities of a second type, leading to factors ln(E/m) that in- validate the naive power-law behavior suggested by dimensional analysis . The reason can be traced to the fact that renormalized coupling constants are conventionally defined in terms of amplitudes that become infrared- divergent when all masses vanish . For instance, consider the theory of a real scalar field with Lagrangian densit y 2 2 2 4 To one-loop order, the invariant elastic scattering amplitude for a scatter- ing process with initial four-momenta pt, P2, and final four-momenta pi, p'Z, is given by Eq . (12 .2.24) a s 2 'nz ~ - ~ 3 27c2 ~y dx In m2 - sx(1 --- x ) A2 Az +ln m2 - tx(1 - x) +~n mz ux(1 - x) - 3 + ~(g~~, (1$.1.3) where s, t, and u are the Mandelstam variable s s=-(Pi + P2)Z , t=-(P1-Pi )2u = -(P1- p2)2 and A is an ultraviolet cut-off . This has no zero-mass singularities as long as we keep s, t, and u away from the positive real axis, and in particular if they all go to -oa (which violates the mass shell condition s +t+u = 4m2 .) Ofcourse, the amplitude depends on the cut-off Aas well as on m, so even though there are no zero-mass singularities, we do not find the result A constant that would be found on the basis of naive scaling arguments 18.1 Where do the Large Logarithms Come From? 11 5 in the limit as s ,t, and u goto--co. The dependence onthe cut-off can be buried by renormalization ;we replace the bare coupling g with a renormalized coupling g R, defined as the value of A at some convenient renormalization paint .For i nstance ,we might tak e gx=A(s=t=u= 0) z z =9-3gInA 37r m Then (1$, Z .3)become s ~ sx(1 x)Z ~ ~ ~x + g3Z~2 ~ dx In 1 -m +1n 1 - tx(1 Z x) + In I-ux(1 Z X) +Q(g~)M m(18.1.4) (18.1.5 (We can freely replace g2 with gR in the second term, because the difference is only of order g3.) This is free of ultraviolet divergences, but it now has a singularity at m - 0, even where s, t, and u are all kept negative . In consequence, where s, t, and u all go to -oo, we again find an asymptotic behavior in disagreement with expectations based on naive scalin g 2 A--* 9 x+3~~2 InM2+Inm2 ~- 1n (-)Z - G (18.1.6)M2 (Much the same happens with any other `natural' definition of the renor- malized coupling ; for instance, we might define gR as the value of A at the on-shell symmetrical point s = t = u = 4m2 /3, and would recover the same asymptotic behavior as in Eq . (1$ .1.6), except that -6would be replaced some other numerical constant .) It is clear that the zero-mass singularity that is encountered when A is expressed in terms of gR arises entirely from the In m2 term in the formula (18 .1.4) for the renormalized coupling gR in terms of the bare coupling g . There are in addition other zero-mass singularities that are encountered when we calculate matrix elements of operators (such as off-shell Feynman amplitudes) rather than integrals of cross sections . These are due to the necessity of renormalizing these operators as well as coupling constants . For instance, suppose that in the scalar field theory with Lagrangian (18.1.2), we wish to calculate some matrix element (#J({1(p) j ay of the operator OP)° fd4xe1q2(x) . (18.1.7) In Feynman diagram terms, this corresponds to inserting a vertex in which two internal 0-lines come together, and through which flows a total four-momentum p in diagrams for the transition a --+ fl . (See Figure 18.1.) 1 16 18Renormalization Group Method s Figure 18.1.Momentum-space Feynman diagrams for the matrix element of the operator f d4x exp(-ip.X) r¢2(x) in the theory of an elementary scalar field O(x) . The cross-hatched disk represents the sum of diagrams with the indicated external lines . Apart from the pair of external lines that meet in a r¢ 1vertex, the other external lines attached to the disk represent the particles in the initial and final states between which the matrix element is evaluated . Figure 1$ .2. A class of Feynman diagrams for the matrix element of the operator f d4x exp(-ip  x)02(x) that exhibit an ultraviolet divergence . The notation is the same as in Figure 18 .1. Ultraviolet divergences arise from a class of diagrams in which this new vertex is part of a subdiagram that is connected to the rest of the graph by just two O-lines . (See Figure 1$ .2.)Dimensional analysis shows that the subgraph would be convergent if connected to the rest of the diagram by more than two O-lines, because the interaction 02 has dimensionality +2, and hence a subdiagram in which this vertex is connected to the rest of the graph by n >2 lines has dimensionality 4 - 2 - n <0.The divergent part of this subdiagram is just a logarithmically divergent constant, so 18.1 Where do the Large Logarithms Come From? > + >C ~>117 Figure 18 .3. The divergent part of the diagram in Figure 18.2,to one-loop order . the matrix elements of 02 can be made finite' by multiplying 02 with a suitable divergent constant ZO z . To order g2, the relevant subdiagram is given by the diagrams of Figure 18.3,and hence contributes to matrix elements of (1(p) a divergent facto r 2 4 F(p) = 1 + [-i(27r)4g]_i (ar)~ [k2+M2 -t][(Pk)2 M2 -te (18.1.8) Combining denominators, rotating the k ointegration contour, and impos- ing an ultraviolet cut-off Agives a result, for A--} o0 ~ n 2 F(p)= 1 - 3Z~c2 JOAIn ~2 + 2~ 1 _ ~ -1+Q(g2} .(18.1.9)p ( ) This has no zero-mass singularity (as long as we keep p2positive), bu t of course it does depend on the cut-off . This logarithmic divergence is eliminated by defining a renormalized 02 operator (18.1.10) with NW) chosen so that N W}F(p)has some definite finite value at some definite renormalization point . For instance, we could define the renormalized 02 operator so tha t 11T(02 )F(O) =1, inwhichcase N(0 Z) =1+ 3~~c2 In+Q(g2) .(18.1.11) (18.1.1Z) The matrix elements of the renormalized operator (02)R then contain a In making this argument it was assumed that any divergences arising from subsub- diagrams containing the 02 vertex which are attached to the rest of the subdiagram by just two 0-lines are eliminated in the same way . 118 factor18Renormalization Group Methods , FR(p) _= N «2}F(P)=--1 + 3~ 2 2dxin 1 +P'X(m2 x )+ 0(g)0 (18.1.13) This is finite for all p2 > 0 and m2 > 0, but it now contains an infrared singularity for m --*0, corresponding to large logarithms in the asymptotic behavior when p2 --*+oa. Of course in order to eliminate the cut-off in higher-order calculations we would have to introduce a renormalized coupling constant as well as a renormalized 02 ❑peratar, and we would encounter logarithms arising from both sources . Similar renormalization factors are needed for any sort of operator, not just 02(x) . In particular, taking a matrix element of one of the elementary fields Vof a theory introduces ultraviolet divergences that arise from radiative corrections to the corresponding propagator . As we saw in Chapter 12, these infinities can be cancelled by working with a renormalized field 1PR WR = ~ (Y)y, (18 .1.14) with IV (W)chosen to make the matrix element ❑f 1PR between acne-particle state and the vacuum the same as for a conventionally normalized field in the absence of interactions . This is related to the usual Z-factor of renormalization theory by Z(v) = i MY)1-z(18.1.15) For instance, let's recall our earlier results for the renormalization of the photon field in spinor quantum electrodynamics .** In this case, the renormalized electromagnetic field A~ is conventionally written in terms of the `bare' field A~ as APZ-1~2APR= 3 a with Z3 given by Eq . (11 .2.21) a s e2 n~ Z 3=--1 122In m2 +Q(e4)(18.1.16) This has a zero-mass singularity, which will affect the asymptotic behav- ior of matrix elements of the renormalized photon field . In particular , In the scalar field theory with 04 interaction used as an example above, the lowest- order terms in N(O) arise from a two-loop graph, so it will not be convenient to use this theory to illustrate the calculation of the N-factors . 18.2TheSlidingScale 119 Eq.(11.2.22) gives the self-energy function of the renormalized electro- magnetic field as r 7c(q2)e2 21r2 Ji dx x( 1- x)In 1 +qz x~m2X) +p(e~). (18 .1.17) 0 Thishas a sing ularity at m = 0, a ndsohas large logar ithms in its asymptotic behavior : for q2--*+00 7ER(q)e +Q(e4) . (1$.1.18) 2~27[26In m2 1 8 Electrodynamics has the special feature that the constant Z3 that ap- pears in the renormalization of the electromagnetic field also appears in the renormalization of the electric charge : eR = Z~/2eBA~tE , (18.1.19) but this is not generally the case . Renormalization group techniques were first applied in quantum electrodynamics, but the scalar field theory discussed here gives a more typical illustration of these methods, with separate renorrnalizations for fields and couplings . 1$.2The S lidin g Scal e We have seen in the previous section that the large logarithms that appear at high energy in suitably integrated cross sections or off-shell Feynman amplitudes can be traced to the prescription used to define renormalized coupling constants and operators . The central idea of the renormalization group method is to change this prescription . Suppose we find some way of defining a new kind of renormalized coupling constant g( p), that depends on a sliding energy scale P, but that (at least for M>m)has no dependence on the scale m of the masses of the theory . Then suitably integrated cross sections or other infrared-safe rate parameters may be expressed as functions of g .and Itinstead of gR . By dimensional analysis such functions may be written a s D M ) 11'(E, x> g~~~ m~ ~) = E I'1, x, 914 .) EE(18.2.1 (Our notation here is the same as in Section 18.1;in particular, x stands for all dimensionless angles, energy ratios, etc, on which I' may depend .) Since yis a completely arbitrary renormalization scale, we can choose y= E, in which case Eq .(18.2. 1)reads DM F(E, x, g ,,, m,/c)= E I' 1> x> 9E, E (18 .2.2) a 1 120 18Renormalization Group Method s This now has no zero-mass singularities because g Edoes not depend on m for m CC E, so there are no large logarithms, and we can use perturbation theory to calculate IFin terms of gE as long as gE itself remains sufficiently small . In particular, in any finite order of perturbation theory I' has the asymptotic behavior, for E >in, i'(E, x, g ., m, ~.c))EDI' (1, x, g E,0,1) (18 .2.3) (Non-perturbative corrections are considered in Section 18 .4.) It remains to calculate gE . For instance, in the scalar field theory with Lagrangian (18.1.2),we may define g ,,in terms of the value of the scattering amplitude at a renormalization point s = t = u = -Y2 : guA(s=t=u- _Y Z) 2 Jo z z _ + 0(g) ~18.~.4)- ~ ~ 3 27c2dx{lnM2A2 X) or, in terms of the conventional renormalized coupling (18 .1.4), 2 Jo')+ 0(gl)M2(18.2.5) But this formula is reliable only if the correction term is smaller than gR ; that is, only if 19R ln(p/m)l < 1 .If this were the case for y~E, then we would not need the methods of the renormalization group ; ordinary perturbation theory would be good enough . Instead of using formulas like (18.2.5)directly for large y, we must instead proceed in stages : gu may be calculated in terms ❑f gR as long as y/m is not much larger than unity ; then gu, may be calculated in terms of g ,,as long as lc'/yis not much larger than unity ; and so on, up to gE . Instead of discrete stages, this may also be done continuously . Dimensional analysis tells us that the relation between g .,and g .takes the form g~l= G( g,aM'/y,m/y) (18.2.G) Differentiating with respect to y' and then setting It'=yyields the differential equatio n whered m Y~~g~ ~u,Y (9p, _M) [-'G(g,,, z, m1ki) Z-2(18.2.7) (1$.Z.$) There are no zero-mass singularities here, so for y > m the differential 18.2The Sliding Scal e equation becomes simply d121 (18.2.9) which is often known as the Callan-Symarczik equation . We are to cal- culate gE by integrating the differential equation (18.2.9),with an initial value gmat some scale y = M,chosen in practice large enough so that for M~M we can neglect the masses m compared with ji, but small enough so that large logarithms ln(M/m) do not prevent us from using perturbation theory to calculate gmin terms of the conventional renormalized coupling constant gR . The solution may be formally writte n ln(E/M )= fX9 E dg/~3(g ) (18.2.10) as long as fl(g) does not vanish between gmand gE . The results of the previous paragraph do not rely on perturbation theory, but we usually need to use perturbation theory to calculate the functions G and fl. As an example, suppose we calculate g gin the scalar field theory with interaction g0¢/24, renormalizing by expressing g in terms of g ,,,rather than gR . Following the same procedure that led to Eq.(18.2.5),this give s 3g2I mZ + ~Zx(1 - x) . gr~'=gu- 32~ ~10dx InMz + ~(1zx 1 -x + O(g~~)} Then Eq .(18.2.8)gives }. m =+ 3g~'dx y2x(i - x)+ a l ~(g, )~ 16 7c2Jo mZ +pZx(1- x )~gPIt Fgry >m, this is 1671(18.2.11) 2 (18.2.12) In next order, the beta-function for P > m is3 ~(g,) = g U[3ZG~z17gu2 3 1Gnz +.. If we are content with the one-loop approximation the calculation of fl(g) can be done even more easily . In order to avoid large radiative corrections in matrix elements at energies of order p, we must write the bare coupling g in terms of a finite renormalized coupling gu, a s g-g ~+B(gu)1n~ +-~- . (18 .2.13) 122 18 Renormalization Group Method s For instance, from Eq . (1$ .1.3) we could have immediately read off that the In Aterms in g have coefficien t 3 )4 21[ _j 2 B(g)2[-i(271, 9(7E)427E2i = 13~ 2 (18.2.14) The unrenormalized coupling is of course independent of ji, so to lowest order d and so to lowest order (18.2.15) With B(g) given by Eq . (18 .2.14), this agrees with our previous result (18.2.12) forP(g) in the 04 scalar field theory . Instead of using a simple ultraviolet cut-off, which can interfere with gauge invariance, it is often more convenient to deal with ultraviolet divergences by use of dimensional regularizat2vn . For a spacetime dimen- sionality d <4, we find in place of ln(A/y) the convergent integra l kd-4dk k=4_d d-- -= >~ 44 - d-ln ji Thus instead of eliminating the cut-off dependence by writing the un- renormalized coupling constant as in (1$.2.13), we instead writ e with the same function B(gu)as before . Thus in order to calculate fl(gu), all we need to do is to pick out the coefficient of the singular factor 1/(4 - d) in the renormalized coupl2ng . This argument is extended to all orders of perturbation theory in Section 18 .6. As long as g ,,is sufficiently small in the scalar field theory with La- grangian (18.1.2),the solution of Eqs .(18.2.9)and (1 8.2.12)can be well approximated by 167c2~ (1$ .2.179P ~ 31n(y/1V 1)' where M is an integration constant . This expression illustrates a common aspect of renormalization group calculations, that dimensionless couplings like gR become replaced with parameters like M that have the dimension- ality of mass . The value of M may be related to gR by comparing the solution (18.2.17) with the behavior of the coupling for values of 11that are large enough to allow us to use approximations based on p > in,but 18.2The Sliding Scale small enough so that 19R ln(p/rri)J <<1, where (18.2.5)gives In this way, we findz ~ 9P^_~ gR + 'g~ ZIn1671 M 167E2M ' M ex p 39R so that Eq .(18.2.17) may be put in a more conventional for m -3 g ,u; 9R 1-~" In E t67rzM123 (18.2.18) (18.2.19) To repeat, this is valid provided gu is small, even if #gR In(y/m) I is of order unity, so it represents a significant improvement over the perturbative result (18.2.18). Of course, the condition that g ,,should be small will become violated when gR ln(p/m) is sufficiently close to the critical value 167r 2 /3. But at least Eq . (18 .2. 0)makes the unequivocal prediction that gE becomes large enough to invalidate perturbation theory at some energy E below the critical value (18.2.19). In calculating off -shell matrix elements of operators instead of integrated cross sections, we also need to take into account the N-factors that appear in the definition of the renormalized operators whose matrix elements are finite . We saw in the previous section that if these N-factors are defined in a conventional way (say, so that the correction factors produced by the divergent subgraphs are cancelled when the operator carries zero four- momentum, or is a field on its mass shell) then the formula for the N- factor involves zero-mass singularities as in Eq .(18.1.12) or Eq .(18.1-16), resulting in large logarithms at energies E :] m . The cure is to define renormalization constants NO)) at a sliding scale ji, so that in matrix elements of the renormalized operato r Uµ =IV~~ )U (18.2.21) the correction factor produced by divergent subgraphs containing operator (9are cancelled at a renormalization point characterized by four-momenta of order p. If MR is a matrix element of operators that are conventionally renvrrnalized, and M is one in which the operators are renormalized as in Eq.(18.2.21), then for any y MR (N/N)] mM(E, x, g,,, m.,p). (18.2.22) We can again use dimensional analysis (assuming M has dimensionality 124 18 Renormalization Group Method s D) and set p_E, to write this a s MR= ED (N(C)IN E(o))M 1 , x,gE,E ,1 (18.2.23) Thus to find the high energy behavior of an off-shell amplitude MR, we need to know how N ,,,varies with the renvrrnali2ation scale ji . For any two renormalization scales pandp', the renormalized operators N( 0)0and N(')0both have finite matrix elements, so the ratio N~~')/N~m) must be cut-off independent . On dimensional grounds, this ratio must take the form NteIN('~" = G(")(gu, y'/ju, m/y)P Differentiating with respect to p' and then setting y'= p gives wher e The solution isdp JZ=1 NE)acexpm d Y9'U'(18.2.24) (18.2.25) (18.2.26) (18.2.27) This is a useful result because the introduction of the sliding scale prevents the appearance of zero-mass singularities in 1V~ O)and N~~ ), and hence also in G{ "} and y{ "}. Hence as long as g uis small, there are no large logarithms that prevent the application of perturbation theory to calculate y{ "}. Also, fory > m, y(O(g,m/y) has a smooth limi t As an example, consider the operator(9 = 02 in the scalar field theory with interaction g,04/24 . Instead of renormalizing it so that the correction factor (18.1.9)is cancelled at p2= 0, we cancel it at a sliding scale p 2 =jjZ, by introducing a new renormalized 0Z operator N~ 02)~Z, wit h N~o2) - F {02 }(~Z)-l= 1 + g z dx[ln ~' -- 1327E0 m+ x 1 x ~ ( ) + a (g). 18.2TheSlidingScale Then the function (18 .2.24) for this operator i s z m N 1M2+ Zx 1- x ~(gp,)(0') i + 32 712dxInm2 'Zx 1 M- X) + a(g~}.125 (We can use g ,,here instead of g or gu, because the difference only affects the terms of order g~ or higher .) From Eq . (18.2.26), we have the n zm or for P >m,9'u JoIYZx1- x) dx+a ~167r2 „~z+z (~z}x1x ~ () 167E2 A(18.2.29) Another good example is provided by the N-factor associated with the renormalization of the electromagnetic field in quantum electrodynamics . Recall that the photon propagator can be made finite for all momenta by evaluating it for renormalized electromagnetic fields, or equivalently by multiplying the propagator of the unrenormalized field by 23 1 Op6(R) = Z3 10pA}. (18.2.30) Eq.(10.5,17) shows that this renormalized propagator may be writte n ❑plg(R) = LRZ -iE~[1¢ -n(qZ] + qpq¢-terms. (18.2.31)} Suppose we instead define a renormalized field N~A)Ap,whose propagator has a term proportional to r~p ,/ [q2 - iF] with a coefficient that is equal to unity at a sliding renormalization scale q z = jjZ . For this purpose, we must clearly tak e Using Eq . (11 .2.22), the function (18 .2.24) is the n G( A'(gu,P '1P ,M 1P) _ e2 i~ dx x(1 = 1 -4~2U + a(e~ } and so Eq .(18.2.26)gives7E(P'2)1/2 ~ - n(/1z) -x}lnM2 +P~x(1-x ) m2+ ~ x(1 - x) (18.2.33) (e2 Ix2(l- x )ZPZ 4) .`~}(e~~, M/1) = -2~Z~dxmz +~zx(1- x)+ a~e~( 18.2.34) 126 18 Renormalization Group Method s As promised, this has a smooth limit for p>m z Y(A)(ep)- y(A)(e'w0)= -` 12n2+ a(e~) . (18.2.35) As already mentioned, electrodynamics is a special case, because the renormalization constant Z3 3 /2 in the definition of the renormalized elec- tromagnetic field is just the reciprocal of the constant used to define the renormalized electric charge of the electron : ex = Z31/2 e. The natural definition of the renormalized electric charge at a sliding scale yis the n eIU_Nua)-le-Z3 11ZIIT~a)-ieR~ (18 .2.36) so that ep times the field N(A)AP renormalized at scale p is independent ofp. From Eq .(1$.2.25)we see that the function fl(e) which gives the y dependence of eY according to Eq .(18.2.9)fory > m has the valu e fl(e)=-eY(A y(e)=l~n2+ a(e5). (18.2.37) Using an earlier calculation3a of the fourth-order term in the vacuum polarization function 7E(q2), Gell-Mann and Low were able also to give the term in fl(e) of next order in e: 3 5 27E 47E In other words, the electric charge at a sliding scale y satisfies the renor- malization group equation 3d ~~ ~~7}.dy e 12~c2 + 647E2+ a~eA This shows ,for small ep,that euincreaseswith increa singy. We also need an initial condition . This is prov ided b y value of the convent ionally renormalized charge e R=Z~ ~Ze, oc- ex /47c = 1 /137.036....Eqs. (18 .2.32) and (18.2.36) yi eldthe know n for whic h ex/eu -Z31ZN~A) - 1 =1- eRZ 1 G ~x x( 1-- x )1n [1+2 ]x(1~ x~ + a(e R}.E~ m (18.2.40) We need to match this with the solution of Eq .(18.2.39) at a value of P which is large enough to justify the approximation P >mein(18.2.39)but small enough so that the logarithm in Eq . (18 .2.40) is still small enough compared with 47E 2 /eR to justify the use of perturbation theory . (For 18.2 The Sliding Scale 12 7 instance, we might take yto be of order 100 MeV .) For such values of ji, Eq. (18 .2.40) gives eP^-teR + eR 2 Iny-5(18.2.41)l2n me 6 On the other hand, the solution of Eq .(18.2.39)for eY small (keeping only the leading term on the right-hand side) is i-1/2 e.= constant -n 1Z Comparing Eqs . (18 .2.41) and (18 .2.42) gives the solutio n -1/2 e.- eR [i_jk(in5)1 n m e6(18.2.42) (18.2.43) UnlikeEq. (18 .2.41), Eq. (18 .2.43) i svalid a slong a s e2167[2 i s small , whether or not (eR/6n2)In (P/me)is small . For instance ,we have already seen in Sect ion 11 .3 that the lead ing fourth-order radiative correct iontothe magnet ic moment of the muon can be obtained by multiply ing the second-order (Schwinger) term (11.3.16) by the vacuum pola rization function are(O)atk2~ m~ ,which accord ing to Eq .(18.2.40)is the same (to th isorder) as us ing ems/47E in place of o cin the Schwinger term . Another example :Exper iment sat high energ yelectron -positron collid- ers such a sLEP at C ERN orSLC at SLAG now study ph ysical proces ses at energies of the order of the mass of the Z Oparticle , or 91 GeV . Eq.(18.2.43) shows that at these energies ,radiative correct ions in pure quantum electrodynam ics should be calculated using a value for the fine structure constant wh ichisnot oc = 1/137.036 but rathe r e2(91 Ge V) oc 1 47E 1- 2(11.25)a/37E 134.6(18.2.44) This is for a theory in which electrons are the only charged particles with masses below mz . In the real world there are many such particle types, and the effective fine structure constant4 at mz is (128.87± 0.12)-1 . The sliding scale at which the coupling parameters of a theory are cal- culated may be the value of an external field rather than the momentum of an external line . In one of the early applications of the renorrnaliaa- tion group method, Coleman and E . Weinberg4° considered the effective potential V(O) for a spacetime-independent external scalar field '0. In the simple case where the field interacts with itself alone, their one-loop result 128 18Renormalization GroupMethod s is given by Eq . (16 .2.15). It is particularly interesting to consider this potential in the case where the renormalized mass mx vanishes, where to one-loop order we may set ji Z(o) in the last term equal to gx02 /2, so that Eq. (16.2.15)here reads (with g a slightly redefined coupling) : 04+ ~2,0 4In ,02 VW =AR + 904+92,04 (18.2.45)24 2567E2 This looks like at first sight as if for g >0 the potential becomes less than JZR for very small 0, so that the point 0= 0 is a local maximum instead of a minimum, but for such small values of 0the third term is larger than the second, and the perturbation theory is obviously untrustworthy . Also, we would like to be able to argue that g must be positive in order for the potential to be bounded below at large fields, but Eq.(18245) shows that however small g is, there is some sufficiently large 0where perturbation theory breaks down, and hence Eq . (18 .2.45) cannot be trusted to tell us whether the potential is bounded below at large fields . We can do much better by using a coupling constant defined at a sliding scale yof field strength . Suppose we define a coupling g ,,by the condition that 24 If we had used g .as the coupling parameter from the beginning, then in place of Eq . (18 .2.45) we would have obtaine d 2o4 2 (18.x.47) V(4)) =2R + g4 04+2'~Z In (W)5b n which obviously satisfies Eq . (18.2.46).' The renormalization group equa- tion for g~ can be obtained from the condition that this effective potentia l -Eq .(18.2.47 differs from what we would get from Eq . (18 .2.45) by simply using Eq. (18 .2.46) to express g in terms of gu, in that the last term is proportional to g 'A rather than g2, The difference is of higher order in g, but can become significant if V(O) is evaluated at 0very different from u, where large logarithms can compensate for powers of coupling . If we use g ,as the coupling parameter from the beginning and take pto be of order 0then no such large logarithms occur, and the approximation Eq. (18 .2.47) is valid as long as g,, remains small . 18.2The Sliding Scale 12 9 is independent of y,** u dglu_3gP(18.2.48)dp 16~c~ (Terms involving the derivative of gu are dropped here, because they are of higher order in g ,,, and are hence negligible as long as gu is sufficiently small .) It is not a coincidence that this takes the same form as the renormalization group equation (18.2.9),(18.2.12), where ji was a renormalization momentum, because as we shall see in the next section the first two terms in the renormalization group equation are always independent of the way that we define the sliding scale . The solution of this equation is given by Eq . (18 .2.17), in general with a different integration constant M . Hence by taking p in Eq, (18 .2.17), we now have V(0)_2R -32nZo4 . (18 .2.49)31n(0Z/ M2) 327E2~i~.2.so~ 90~ 31n(02/11~12) This result should be used with some care, because it is only valid where the coupling constant go is small . The problem is not just that Eq. (18 .2.49) loses its validity for 0 near M ; we also cannot integrate the renormalization group equation through the singularity at '0 = M, so a knowledge of go on one side of this singularity tells us nothing about its behavior on the other side . 1 f go is found to have a small positive value for some 0() then from Eq. (1 8.2.5a) we know that M >1001, so go remains small and Eq . (18.2.49) is valid for 101 < 1001 . This shows that the point 0 = 0 is a local minimum of V(O), contrary to what Eq . (18 .2.45) might have led us to suppose . This means that the vacuum which is invariant under the symmetry transformation 0 --+ -0 is stable in this model, aside from the possibility of quantum mechanical barrier penetration . On the other hand, we do not know that Eq . (18 .2.49) becomes valid for 101 sufficiently large compared with M . The coupling might remain too large to use perturbation theory for all 101 > M, and even if it becomes s mallfor some 101 > M, the potential might be given for such 0 by Eq . (1$.2. 9) with a renormalizatio n To eliminate all cut-off dependence, 0should be written in terms of a renormalized field 44,, -- N ,0,0, which gives V(O 1,) aydependence arising from the ydependence of the renormalization constant N O,. This point is ignored here, because in the scalar field theory with interaction ac 0¢ the lowest-order graphs contributing to the Pdependence ofN,have two loops, so that to the order of the calculations presented here we can take N '0= 1. 130 18 Renormalization Group Method s scale M' >0, producing a second singularity . Thus in this case we cannot conclude that V(O) --+-oa for ICI -' oo . Similarly, if g ois found to have a small negative value for some Oo then we know that M <0(fl, so g oremains small and Eq . (18.2.49) is valid for 101 > IduoI. In this case we cannot conclude anything about the behavior of the potential for #c M, but here we can use Eq . (18.2.49) to see that VW --- * -oa for 101-> oo, ruling out the possibility of any stable vacuum . Because we are here considering the limit 101 --+ oo, this conclusion holds also for scalar fields of mass m >0, provided m cc 1,0()I. This is why it is necessary to assume that the 04 coupling (renormali2ed at any scale much larger than the scalar mass) is positive . 18.3 Varieties of Asymptotic Behavio r The renormalization group method provides useful insight into the types of possible asymptotic behavior that are encountered in quantum field theories, even in cases where the running coupling g ,,does not remain small enough to allow the use of perturbation theory . We will distinguish four different ways that g ,,,may behave for y --+oo, that correspond to four different shapes of the function fl(g) in theories with a single coupling constant . In the next section we will take up the case of theories with several independent couplings . Let's first recall the results for fl(g) obtained for the two examples considered in the previous section . One of these is the scalar field theory with interaction g,0¢/24, for which the beta-function for small g i s 218g3 fl(g) = 6~g nz _ 3 (i6nz~ +0(g4) . i The other is quantum electrodynamics . Instead of writing the beta- function here in the form (18.2.38), we will emphasize the similarities between this theory and the scalar field theory by writin g ag- e with fl(g.) now understood to be ydgu/d p,so that for small g(18.3.2) fl(g) = 2e12, 64, =9Z +9Z +0(g4) .(18.3.3)67E 32n Note that in both cases the physically allowed coupling constants fall in the range g ~ 0, where fl(g) ~0for small g . In electrodynamics this is simply 18.3Varieties of ' Asymptotic Behavior 13 1 (a) P(g) fib) ~d} Figure 18 .4. Schematic representation of four possible forms of the function di(g) . For such forms of fl{g}, the running coupling g ,, would : (a) approach infinity at a finite value of µ ; (b) continue to grow as It increases ; (c) approach a finite limit g .for It --)~oo;(d)approach zero for It--,cc. because the reality of the Lagrangian requires e to be real . In the scalar field theory, as we saw at the end of the previous section, it is necessary to have g >0in order to have any stable vacuum state . However, there are other examples that have fl(g) c0for g _>0. For instance, we can consider a scalar field theory with interaction Hamiltonian -g04/24, with g taken positive . This may be unphysical, but stability problems will not bother us as long as we stick to perturbation theory . Eq .(18.2.9)shows that if we redefine g -> -g the beta-function undergoes the change fl(g) --)'-fl(-g), so our previous result that fl(g) = 3g2J167T2+0(g)for an interaction g04/24 now gives NO =-3g2/167E2+ O(g)(18.3.4) for an interaction Hamiltonian density -g04/24, Ofgreater relevance to physics, we shall see in Section 18.7that non-Abelian gauge theories with not too many spinor fields have flc 0 for small positive gauge coupling constants . In what follows we will always define the coupling g so that g ~ 0, but we shall consider the cases of fl(g) either positive or negative for small g . Now let us turn to our list of possibilities . (See Figure 18 .4.) 132 18Renormalization Group Method s (a) Si ngularity at Fi nite E nergy Suppose that fl (g) > 0 for small positive g (as is the case for Eqs . (1$ .3.1 ) and (18 .3.3)), and that fl(g) remains positive and continues to rise suffi- ciently rapidly with increasing g, so that the integral f o° dg/fl (g) converges : fdgJc oa. 18 .3.5) fl(g) Then g ,,will move steadily away from g ,,= 0, and Eq .(18.2.10) shows that gE must become infinite at a finite value of E: E~=yexp f9dg ~ ~(g)(18.3.6) where p is any renormalization scale with y»in.We saw an example of this phenomenon in the previous section ; if the lowest-order formula fl(g) = 3g2/16n2for the beta-function in the scalar field theory were taken as exact for all values of g, then the running coupling (18.2.17) would become infinite at the energy (1$.2.19). Similarly, if the lowest order formulafl(g)= g2/d7E2(with g - e2)for the beta-function in spinor quantum electrodynamics were taken as exact for all g, then the energy (18.3.6)at which gE and eE become infinite would be (1$.3.7) Using Eq . (18 .2.43), we may ex pressthisinterms of the co nventional renormalized charge : Em^ me exp67E2 5 e2 + 6 + O(eR} = e646.6me (18.3.8) R Of course, the approximation that fl(g) = gZ /6n2will break down before this energy is reached, so all we can say with confidence is that eE will become large enough to invalidate perturbation theory at some energy E below Ems . (b) Co ntinuedGrow th Now suppose that in some theory fl(g) remains positive definite for g -> oo, but rises slowly enough (or decreases) so that f o° dglfl(g) is divergent . The coupling constant gE then continues to increase as E -> ao, but becomes infinite only for E = oa . Furthermore, the leading term in the asymptotic behavior of gE for E -> oo is independent of the conventionally renormalized coupling . For instance, if fl(g) behaves for large g like bgkl with b >0 and k <1, then the solution of Eq . (18 .2.9) is EZI(Z-k) gp . (18.3.9) gE = 1+(l- k)h 9N- 'In It 18.3 Varieties of Asymptotic Behavior 133 If gi,is sma llfor some µ (say, of o rderm) t hen the growth of gE is seenonly at energies w hich are exponentiall ylargecomparedwith thi s Y. However, inthe ext reme high e nergylimit the co uplin g grows acco rding to gE---)'[(1- k)hInE]11(1-k) a limiting behavior independent of g ,,!(1$.3.10) (c) Fixe dPoint at F inite C ouplin g Suppose next that fl(g) remains positive-definite for 0 <g<g., but drops to zero at g = g* and is negative thereafter . Then Eq . (18 .2.9) dictates that as y increases g~ will increase for g ,, < g. and decrease for gP>g*, in either case approaching the fixed point g . for µ -> oc . If the zero of fl(g) at g* is simple, then in the neighborhood of this point we have fl (g)-> a(g* -g )for g -> g * with a >0. The solution of Eq .(18.2.9)is then(18.3.11) (18.3.12) (The behavior of type (b)described above may be regarded as the special case where the fixed point g .is at infinity .) Also, y(g) for a general operator Cmay be expected to behave smoothly near g + Y(g) = Ag .) + C(g, - g) + 0 ((g* - 9)2). (18 .3.13) (We are here dropping the label U on y and c .) Hence in matrix elements of this (and perhaps other) operators, we encounter a factor (see Eq .(18.2.27)) PTE'ccexP[_fEy(gp)i] The product of the factors E- Y(9)can be lumped together with the factor EDin Eq .(1$.2.23), with the result that the whole matrix element goes a s MROCED', (1$.3.15) where the dimensionality D .is calculated adding an `anomalous dimen- sion' -y (g*)to the actual dimensionality of each operator appearing in the matrix element . (d) Asymptotic Freedo m In the examples that have been discussed so far, fl(g) was positive for small positive g, so that g,L is driven away from g = 0 as µ increases . Suppose that for some other theory fl(g) is negative for small positive g . 134 Then1 8Renormalization Group Method s ~(~) --'-hgn, where b >0. Here n is the order of the lowest-order diagrams that contribute to fl(g), and hence is always an integer greater than unity . (In the theories used as examples here, n = 2 .) The solution of Eq .(18.2.9) here is 9E = g ,1+b(n - 1)g~i-i InE /11 ForE --).ao, this has a limit independent of g ~, gp~--r [b (n -1) In El -1/0-1) .(18.3.17) (18.3.18) Since this gives a vanishing g EforE --+ oa, we can trust perturbation theory in this limit, provided only that gE for some finite E is within the region around g = 0where g and fl(g) have opposite signs . The anomalous dimensions y (Oof various operators U are expected to have the weak-coupling behavior (dropping the label 0} Y(g) -'egm , (18 .3.19) where m is the order of the lowest-order diagrams that contribute to the renormalization of the operator, and c is a real constant that can be positive or negative . The asymptotic behavior at high energies of the factor introduced into matrix elements by renormalization of this operator is then JEPTt c exp - Y(g")d1t rE exp[_C J[h(n-1) In y ]d1t oCexp -c [b(n - 1)](in E)'(1-m/(n - 1)) except that for m = n --1 NE1DC(InE)-rlh(n-Z )(18.3.20) We see that in the case of asymptotic freedom there are no corrections to those effective dimensionalities that determine the powers of energy appearing in the asymptotic behavior of the matrix element, but this asymptotic behavior is instead modified by powers of In E . For a toy model that exhibits asymptotic freedom, we can use the scalar field theory with interaction Hamiltonian density -g o4/24, with g taken 18.3Varieties of ' Asymptotic Behavio r positive . Eq. (18 .3.4) here gives the parameters of Eq .(1$.3.16)as: h=3/167c' , n=2 , so Eq .(18.3.17) gives,for E -> ao, 31nE-1 9E161r2135 (1$.3.22) X1$.3.23) Also, the operator 02 in this theory has an anomalous dimension given by Eq . (18 .2.29) as 1671 Hence Eq .(18.3.19) applies here, wit h c=- 1/162, m=1 .(1$.3.24) (18.3.25) Each 02 operator in a matrix element therefore contributes a factor given by Eq .(18.3.21) as PTE 1 0-- (In E )'I'. (18 .3.26) The scalar field 0itself in this theory has y(g) oc g2, and hence m = 2, so each 0operator in a matrix element therefore contributes a factor given byEq.(1$.3.20)as PTE'oc 1+O(ln'E) - (18.3.27) Wewillseeanother,morephysical,example ofasymptoticfreedomwhen we ta ke upquantum ch romody namics inSection18.7. In all cases where gE can be extended to infinite energy, its behavior in this limit turns out to be independent of the renormalized coupling gR. However, this does not necessarily mean that the theory involves no arbitrary dimensionless parameters . In all cases, in order to describe how gE approaches its limit for E -> oo, we need to specify a free parameter ~ . with the dimensions of energy . For case (b),Eq.(18.3.10)may be writte n For case (c), Eq .(18.3.12)may be written A), Finally, for case (d), Eq.(18.3.18) may be written 136 18Renormalization Group Method s Such theories in general do have a free dimensionless parameter : the ratio of;to the mass in . Coupling constants like eR that are renormalized at scales tied to m may be expressed as functions of m/~ . It is only when all masses in a theory vanish that we can say that the theory has no free dimensionless parameters . Of the four types of asymptotic behavior described here, types (a) and (b)lead to the apparently unphysical behavior that the running coupling gE becomes infinite, either at a finite energy (case (a)) or for E --).ao (case (b)).This does not in itself mean disaster : we have to look at how the coupling is defined . For instance, if g ,,drops smoothly from a finite value gm at p= m to zero for µ -> oo, and we define a new coupling gy = 9,u/[l -gyl92m],then g,, becomes infinite at µ =gym, but this is just an artifact of this particular choice of coupling parameter . However, the conventional renormalized couplings g ,,in both the 04 scalar field theory and quantum electrodynamics have been defined here in terms of the values of matrix elements at energies of order µ . Specifically, g ,,in 04 scalar field theory is defined as the invariant Feynman amplitude A for scalar-scalar scattering at s = t = u = µ2 , where A is supposed to be analytic . Also, g ,,= e~ in spinor electrodynamics is given b y 1 e~leR=Z3'I~T~A)2_ ~1-R(µ2(18.3.28) An infinity in e~ at a point M,,,would therefore produce a pole or other singularity inthe renormalized photon propagator at a positive value of p2, that is, at p 2=P~„where the propagator is supposed to be analytic . Thus, with g .as defined here, the type of asymptotic behavior described in case (a) isruled out physically . How then do our various quantum field theories behave? Years ago, Landau 4bargued that in quantum electrodynamics the increasing powers of 1n (EIM)encountered at each order of perturbation theory would add up to give singula rities (so-called `Landau ghosts' )at finite values of E . In modern terms, Landau could be said to have discovered possibility (a) above, but he did not give any argument against cases (b)or(c). Nevertheless, there is today a widespread view that interacting quantum field theories that are not asymptotically free ,like quantum electrodynam- ics or the scalar field theory with 04interaction, are not mathematically consistent . In quantum electrodynamics there is some evidence against case (c), the existence of a finite fixed point e .~. Such a fixed point would only be possible 4c if non-perturbative effects changed the qualitat ive nature of the operator product expansion ,the sub ject of Chapter 20, or if there were a non-perturbative renormalizat ion of the triangle anomaly discussed in Chapter 22.But even if case (c) i s indeed ruled out in quan- 18.3Varieties of Asymptotic Behavior 13 7 turn electrodynamics, there is still the possibility of case (b),a fixed point at infinite coupling . Most of the evidence against the consistency of interacting non- asymptotically free quantum field theories comes from the study of the scalar field theory in four spacetime dimensions with 04 interaction, quan- tized on a finite spacetime lattice . There are rigorous theorems4d to the effect that this theory (with arbitrary dependence of the parameters of the theory on lattice spacing) does not have an interacting continuum theory as its limit for zero lattice spacing unless the theory is asymptot- ically free, which of course is contrary to what is found for this theory in perturbation theory . This argument also seems inconclusive . It is true that if there were a consistent continuum scalar field theory that is not asymptotically free, then it would be possible to construct a lattice theory by integrating out the values of the scalar field at all points except on a spacetime lattice . But this would not he the lattice theory thatis considered in thesetheorems .It would be a lattice theory with every possible coupling allowed by symmetry principles -not just a term proportional to 04, but also terms proportional to06,03❑O, etc, with coefficients having a dependence on the cut-off (the inverse lattice spacing) governed by the Wilson renormalization group equations discussed in Section 12 .4. No one has proved anything about the continuum limit of such a theory . If it is really true that there is no interacting continuum scalar field theory in the limit of zero lattice spacing, then we must encounter some obstruction when we try to solve the Wilson renormalization group equa- tions, which for weak renormalized couplings would have to be at very small lattice spacings . Such a theory would appear like an interacting continuum field theory unless examined at very short distances . The renormalized coupling constant in this approximate continuum theory presumably has a singularity at finite energy, as in case (a) above, so that it too breaks down at short distances . (But for strong couplings there is no direct connection between the forms of the Wilson and Dell-Mann-Low renormalization group equations, so the existence of a singularity in the bare couplings at finite lattice spacing does not necessarily imply a singu- larity in the renormalized coupling constants at a finite renormalization scale .) Theories of this sort are sometimes called trivial, either because, under various assumptions about the bare couplings of the theory quantized on a lattice, the continuum limit turns out to be a free field theory, or because the only way to make a continuum theory of type (a) physically satisfactory at all energies is to adopt the solution g .=0of the renormalization group equation (18.2.9).Even if a field theory is trivial in either sense, there is no reason not to include it as part of a realistic theory of physical phenomena . The existence of an obstruction to the solution of the Wilson 138 18Renormalization Group Method s renormalization group equations for a field theory at very small lattice spacings is not important if in the real world there are other fields that must also be taken into account at such short distances . Similarly, the fact that a given quantum field theory has unphysical singularities at some large energy E,,,is not a physical problem if the theory in question is only a low-energy approximation to a larger theory, an approximation valid only at energies far below Em . In particular, long before we reach the energies near (18.3.8)where quantum electrodynamics could be expected to become singular, it becomes necessary to take even gravitation into account, and no one knows how to calculate the effects of strong gravitational forces at such energies . Despite these reassuring remarks, it is possible that in order to avoid unphysical singularities, all our separate quantum field theories like spinor quantum electrodynamics will eventually have to be integrated into an asymptotically free theory . Fortunately, the question of whether a theory is asymptotically free for some finite range of coupling constants can be settled by perturbative calculations : iffl(g) is negative as g --).0+, then the theory is asymptotically free for all renormalized couplings g ,,lying between zero and the first zero of fl(g). In this connection, it is worth noting that although the detailed form of fl(g) depends on the gauge and on precisely how the running coupling is defined, the first two terms in the power series for fl(g) do not . Suppose we have two definitions g ,,,and g .of the running coupling, perhaps employing different definitions of the renormalization scale Por different gauges . Since both g .and g ,,are dimensionless and cut-off independent, there is no way that g ,,for µ ]] m can depend on anything but g ,, We then have NO Pdµ dg p and so No = ddg NO. (18.3.29) As long as we are sticking to the same definition of the unrenormalized coupling, all renormalized couplings are equal in lowest order, so the power series for g in terms of g may be writte n g(g)= g +a g2 + O (g3) 1 8.4 Multiple Couplings and Mass Effects 139 or, equivalently, g=g - age+ O(g3). Thederiva tiveis dg Also, for the couplings we have been considering here (including g = e2 ) the power series for fl(g) takes the for m or in terms of g fl(k) =bg' + (b'-tab)g3 +0(g4) . From (18.3.29), we have the n ~(g) = [1 + 2a g2+ O(g)] [bg2+(b' -tab}g3 +0(g4)] = hg2 + b'g3 + a(g4). (18.3.30) We see that the first two terms in the power series for ~ in terms of g have the same coefficients as in the power series for Pin terms of g .However, this is definitely not the case for the higher-order terms . In fact, it is always possible to choose the function k(g) so that all terms in P(k) of higher than third order in g vanish, so we can describe the asymptotic behavior of REfor E -> oo by inspection of the first two terms in the perturbation series for fl(g). But this is of little value, since we would need to carry our calculations to all orders to determine how g depends on g, and without this we cannot use our knowledge of the asymptotic behavior of g to say anything about the asymptotic behavior of g, or of physical quantities . The same argument that led to Eq .(18.3.30) shows that, at small coupling, the Wilson renormalization group equation for the bare coupling constant as a function of lattice spacing for inverse lattice spacings greater than particle masses is the same as the Gell-Mann-Low renormalization group equation for the renormalized coupling constant as a function of renormalization scale for scales greater than particle masses . Hence if a continuum theory is asymptotically free, then there will be no obstruction in passing to the continuum limit of the theory quantized on a lattice . 18.4 M ultiple Cou plings andMass Effect s Up to now, we have considered theories with only one dimensionless coupling g . It is easy to extend the formalism to incorporate several such 140 18Renarmarizatiort Group Method s couplings g~ : for each g el, we have a renormalization group equation that for µ >m takes the form Pd gPW=KWO11 (18.4.1) with each fl~ depending in general on all the gs . There are now many more possibilities for the asymptotic behavior of the g "(µ)as µ --+oo; in a given theory we may have some trajectories in g-space that go off to infinity, at finite or infinite values of µ, other trajectories that approach fixed points, and yet other trajectories that approach closed curves known as `lima cycles' . To get a taste of some of the various possibilities, let's consider the behavior of g "(y) near a fixed point . Eq. (18 .4.1) has a fixed-point solution g ,(µ)= g* i f In the neighborhood of this point, Eq . (18 .4.1) become s ItaV(P) -g"Idµ where M is the matrixM'k[9h(P) -9! k M ek = afl'~(g). agk9=g The solution can be expanded in eigenvectors of this matri x m(18.4.2) (18.4.3) (18.4.4) (18.4.5) where Ym is a eigenvector of Mwith eigenvalue ~,,(normalized in any convenient way) : M/k Fm kAm YmI k(18.4.6) and the cm are a set of expansion coefficients .* (The summation convention is suspended in this section .) Eq. (18.4.5) shows that the coupling constants approach the fixed point asy-> oo if and only if cm = 0 for all eigenvectors with A,,>0. (For simplicity we are assuming here that none of the eigenvalues vanish .) Thu s We are assuming here that the eigenvectors V,,, form a complete set . This is not always so, but it is the generic case ; the eigenvectors of a finite matrix M will form a complete set if all of the roots of the secular equation Det (M - A1) _ 0are different . A matrix whose eigenvectors do not form a complete set can be regarded as a limiting case of a matrix with a complete set of eigenveclvrs when some of its eigenvalues become degenerate . 18.4Multiple Couplings and Mass Effects 141 in general the trajectories that are attracted to the fixed point lie on an N_-dimensional surface, where N_ is the number of negative eigenvalues of M ; the tangents to this surface at g .are the eigenvectors with negative eigenvalues . Trajectories that are not on this surface may approach close to the fixed point, but are eventually repelled, predominantly in the direction of eigenvectors with the largest positive eigenvalues .Of course, if all eigenvalues are negative then there is a finite region around the fixed point within which all trajectories converge on this point . Since the eigenvalues :~, are evidently important in learning the asymp- totic behavior of trajectories that approach a fixed point, it is useful to note that these eigenvalues are independent of the definition of the couplings . Suppose we introduce a new set of couplings g ", defined as functions of the gs . These satisfy renormalization group equation s d r►: so09Mg=&) V (g)= 1:aaggm($ ) #M(g) (1$.4.7) (That is,#transforms as a contra variant vector in coupling-constant space .)Differentiating ,we hav e M ag ag m M agagkagrnagk At a fixed point g . matrix equatio n whereMS=SM , k~~Ec=_ 0~ ogkk _k (g ) Sck- ak gg_g.(18.4.8) (18.4.9) (18.4.1Q) As long as the transformation g --+g is non-singular, Eq .(18.4.8)is a similarity transformation, and hence M and Mhave the same eigenvalues 4- The renormalization group formalism may be extended to non- renormalizable as well as renormalizable theories . As explained in Sec- tion 12.3, the infinities in non-renormalizable theories are eliminated by a suitable renormalization of coupling constants and masses, just as i nthe first term on the right vanishes ,so this gives the 142 18 Renormalization Group Method s renormalizable theories ; the only difference is that in non-renormalizable theories the Lagrangian must be supposed to contain all possible interac- tions allowed by the symmetries of the theory . If gB is the unrenormalized coupling constant multiplying an operator of dimensionality D,,,in the Lagrangian (that is, a product of fields and spacetime derivatives of fields whose dimensionality in powers of mass or energy is D,,~, then g~ will have a dimensionality 0 ,, = 4 - D,,. We may then re-express the bare couplings in terms of a set of dimensionless renormalized couplings gf (P) and scut-off A, through relations of the general for m gB=Y°C9'(p )+ b"0,m gk(y)g(p )In(A) + a(g(µ)) k,m ~ with the dimensionless numerical coefficients b"kmand similar coefficients in higher-order terms chosen to cancel the cut-off dependence of physical quantities . (In some theories the leading term might be trilinear or even higher order in couplings ; the modifications that would be needed here are obvious .) From the requirement that gB be cut-off-independent, we obtain the renormalization group equation (18 .4.1), wit h k,m Non-renormalizable interactions are those with D,,>4, or At < 0, so as long as the g '(p)all remain sufficiently small we expect the non- renormalizable renormalized couplings to have positive fl"and hence to grow with It, but no one knows what happens when the couplings become large enough to invalidate perturbation theory . However, as explained in the next section, even theories with infinite number of independent parameters commonly have fixed points g* at which the number N_ of negative eigenv alues of the matrix ( 18.4.6) is finite, just as it is at zero coupling . (In particular, often N_ =1.)Where N_ =~0, the fixed point lies on an N_-dimensional critical surface, consisting of trajectories that are attracted into the fixed point as ,u --+ac. A non- renormalizable theory with coupling parameters on such a critical surface, although not of course asymptotically free, is said to be asymptotically safe,-5 because the renormalized couplings remain finite for large values ofp. The condition of asymptotic safety in such a theory would play the role that used to be associated with the principle of renormalizability, of eliminating all but a finite number of free parameters, the coordinates of the critical surface . In a renormalizable theory, all physical quantities are made cut-off- independent by adjusting the cut-off dependence of a finite number of bare couplings . These bare couplings may be expressed in terms of an 18.4 Multiple Couplings and Mass Effects 143 equal number of it-dependent renormalized couplings, and the condition that the bare couplings are it-independent yields renormalization group equations relating only these renormalized couplings . From the broader point of view which allows non-renormalizable as well as renormalizable couplings, a renormalizable theory just corresponds to a finite-dimensional invaria nt surface in the infinite-dimensional space of all renormalizable and non-renormalizable theories ; that is, it is a surface for which fl'(g)at any point g on the surface is tangent to the surface at that point . So far in this section we have tacitly assumed that p > m, so that we could neglect the dependence of fi"on m/y . However, this is not necessary ; we can if we like treat a mass as just another coupling pararneter .6 That is, all renormalized couplings can be defined as before in terms of various Greens functions at off-mass-shell momenta of order it, but now evaluated with all bare masses zero . The dimensionless renormalized mass parameters for Dirac fields V)or scalar fields 0may be defined as (18.4.13 (18.4.14) where Nt ""7(A/y)are the dimensionless constants which, when multiplied into corresponding operators G, cancel the infinities in the matrix elements of these operators, also evaluated with all bare masses zero . (See Section 1$.1.)These new renormalized masses and couplings have no direct physical significance, but the true physical masses and all physical matrix elements can be expressed in terms of them . These matrix elements take the form of sums of matrix elements for zero bare mass, with any number of insertions of the renormalized mass operators IIT W}02and Nt~' Oy~tp, times the corresponding renormalized mass parameters . In this renormalization scheme the beta-functions for the various cou- plings are obviously mass-independent, and the beta-functions for the mass parameters are proportional to these parameters, with coefficients that depend on all the various couplings ; using Eq .(18.2.25), we hav e ditMV (Y ) For instance, we noted in Section 18.2that in the scalar field theory with Lagrangian ( 1$.1.2), the mass operator 02 has anomalous dimension (18.2.29) forin= 0, so her e ~ 671 144 18 Renormalization Group Method s Also, Eq . (1 1 .4.3) shows that the effect of higher-order corrections to the electron propagator is to replace the electron mass by me - E' (p, m ,), so the effect of these corrections on matrix elements of the operator y~ elpe between one-electron states of four-momentum p yis to multiply them by a factor 1aE*(P,rne)F(p) =I- erne me=o The renormalization constant Na'y'for the operator 1Detp,is there fore equal toF-1(p), evaluated with p 2equal to some renormalizat ion scale, say +,u2. According to Eq . (11 .4.8),to one-loop order this i s am,mp=0,p2=Y2 4rt2 fi(A2 =1-2~)4J~ dx In 1 + P2 ( I (27t)4 e2 A2 ~1-4n2In2-1x) (18.4.18) where n is an ultraviolet cut-off,** and we take the limit A > p .The anomalous dimension of the operator fpetpeis therefore given by (18.2.25) as 2 (w~v)~ In N(ww) +Oe4Y = ~dµ 2 ~2 (~ ~ so Eq . (18 .4.15) here reads z(18.4.19) The same formula holds in general gauge theories, with e2 replaced with the value of )]"(ta)2 for the particular species of fermion in question . The important difference between the m(y) and the other renormalized parameters of the theory is of course that bare masses have positive dimensionality, so as long as the couplings remain small the m(P) all decrease in magnitude . Our previous assumption that masses may be neglected as y --+ ao is justified if in fact m(y) does vanish for it --+ ao, However this is only known to be the case in asymptotically free theories, where the couplings all do remain small for ,u --+ao; in all other cases this assumption is just an educated guess . This notation is different from that of Eq_ (11.4.8), where the ultraviolet cut-off was called p. 18.5CritWal Phenomena 145 18.5 Critical Phenamena * For some purposes we may be interested in the limit of very low rather than high energies or wave numbers . The arguments of Section 18.2can be repeated to study this limit, except that here we must examine the case Y ---~ 0 rather than ,u --+ao. This limit is of course simplest if there are no masses in the theory, as, for instance, in quantum electrodynamics with a symmetry under the chiral transformation ~ ) --+ y$W which forbids an electron mass . In this particular case the only renormalizable coupling eAY~Dy,,tp as well as all non-renormalizable couplings have fil > 0 for sufficiently small couplings, so all trajectories in at least a finite region around the origin are attracted into the point g "= 0 as p--+0. The same considerations may be applied even to theories with very small but non-zero masses if we include these masses among the coupling parameters of the theory, as described in the previous section . The coefficient ❑in Eq . (18 .4. 12) is positive for a mass parameter, so in this case the trajectories can never reach the point g = 0, but they may come close if the masses are small . Of course, even if we can regard some degree of freedom like the electron field as having zero or very small mass, in the real world there are many other degrees of freedom whose masses are not small . The renormalization group should properly be applied not to the true theory that encompasses all these heavy degrees of freedom, but to an `effective' field theory, in which only massless or nearly massless degrees of freedom appear explicitly, with interactions that include the effects of internal heavy particle lines . (We shall have more to say about effective field theories in Chapter 19.) The low wave number limit is of particular interest in the study of critical phenomena, such as long-range correlations at or near a second- order phase transition (a smooth phase transition, with no latent heat) in condensed matter . Because we are interested in the limit y --).0, the important eigenvectors of the matrix (18 .4.4) are those with eigenvalues Ac 0, which are called relevant . The eigenvectors with ;t -= 0 and A > 0 are called marginal and irrele vant,respectively . Suppose that there is anon-trivial fixed point g* with just one negative eigenvalue Ao, perhaps corresponding approximately to a mass operator . The set of trajectories of g '~(,u) that are attracted into this fixed point for ,u---~ 0 therefore forms a critical surface of codimension one ; that is, a surface defined by a single condition on the couplings, the condition tha t This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 146 18 Renormalization GroupMethod s for g --+g., the tangents g' - g* have no components in the direction of the eigenvector with negative eigenvalue . There is a phase transition as the physical value of the couplings at any fixed characteristic scale approaches this surface . Because the critical surface has codimension one, the phase transition can be reached by adjusting any one parameter on which the couplings depend, such as the pressure or temperature . The fact that a wide variety of substances do exhibit phase transitions of this sort shows that it is common to encounter fixed points for which the matrix (18 .4.4) has a single negative eigenvalue, as already mentioned in the previous SeGt10I1. To be specific, as the temperature T approaches its critical value T, we expect the coefficient c() of the growing term in Eq . (18 .4.5) to become proportional to T- T, because there is no reason why it should be singular or why it should vanish faster than this . Hence for 0 and then T --+T, the couplings go a s (T-Tc)VagyA 0 , (1$.5.1) where Ao is the only negative eigen value at g *,and V o~is the correspond- ing eigen vector .**Applying our renormalization group arguments to wave numbers instead of energies ,theN-point funct ion (the Nth partial deriva- tive of the e ffectiveaction with respect to a field 0of dimens ionality [wave number]DO)at a small characteristic wave number scale x has the for ' where 7,p(g) is the anomalous dimension associated with the field 0, and d is the spacetime dimensionality, or in classical statistical mechanics the spatial dimensionality . It is convenient to rewrite this in the equivalent form rN(K) --~(T _TL )-,a-N[Do+^~O(g=)l14GN(rc(T- T0)'/`0) . (18.5.3) This shows for one thing that the correlation length ~ (the characteristic length that determines the scale over which the Fourier transform of FN varies) increases as T approaches T, lik e ~ cc (T - T(.)-' (18.5.4 where visaconventionally defined positive ` critical e xponent ',given b y Eq. (18 .5.3)as v = -1 /~o. (18.5.5) Other contributions to the couplings will go as (I' -- 7o)°pA1 with :ti>a_ Thus Eq_ ( 1$_5.1)is valid here provided T - To does not go to zero as fast as P"-'a. The function F,,, also depends on dimensionless angles and ratios of wave num- bers . Note that r'N has `naive' dimensionality d -- 1VD Obecause Sd(Erc)r must be dimensionless . 18.5 Critical Phenomena 147 Also, the zero-field effective action I-'o (or in statistical physics, the free energy) must be x-independent because it corresponds to graphs with no external lines . It follows that Eq .(18.5.3)here becomes for T--+Tc: I-'p - Fp cc (T - 7',)"d , (1$ .5.6) where the constant Fo is the effective action or free energy due to the heavy degrees of freedom which have been integrated out . Thus the exponent v also governs the behavior of the part of the free energy which is not analytic in temperature for T near T , In1972 Wilson and Fisher7used an expansion in powers of d - 4 both to show that the theory of a scalar field actually fits the above description, and also to carry out an approximate calculation of critical exponents such as v . Consider a theory with a single `light' degree of freedom, a scalar field 0, such as the magnetization in a ferromagnet, with a symmetry under 0 --+ -0that rules out interactions odd in 0. In addition to the `mass' term -g202 /2,the Lagrangian density of the effective field theory will contain interactions -g404/4?, -g606 /5!,etc. The dimensionality of the field 0in powers of wave number is (d - 2)/2 (so that f ddx(o(p}2 should be dimensionless) so the dimensionalities of the couplings 92, g4, 96,etc. in d dimensions are +2, 4 - d, 6-2d, etc . For the fixed point at zero coupling in three dimensions, there are two relevant couplings, 92 and g4, but this conclusion is changed by interactions at non-trivial fixed points . Let's examine the surface in coupling constant space in which only 92 and g4 are non-zero, and take g4 to be small .tt Eq . (18 .2.12) gives 9(g4) = 3g4 /15n2 +0(g4)for d = 4, and Eq . (18 .4.1 2) tells us that for d = 4 -- E dimensions we must add to this a term -E94, SO z . F~ g4(F~)= - Eg4(F~) +16~c2~ +0(9'(9) ) F~ Also, Eq . (18 .4.17) give s Therefore for small e there is a non-trivial fixed point a t 1Gn2E 94* = 3 g2*=o.0.5.7} (18.5.8) (18.5.9) These are the only rcnarmalizable couplings for 3 ~ d c 4, so for such dthis is an invariant surface . Note that we do not include the coefficient of (OO)2 among the couplings here, because this is a redundant coupling, in the sense described in Section 7.7. 148 18 Renormalizatio nGroup Method s The matrix (18 .4.4) at this fixed point is diagonal, with eigenvalue s 3 IN = M444 4+ 87z2 A2=X1 22= - 2 + 94*2 + O(g4*)-2 + + Q(E2}, 1Cr~ 3(18.5.10) (18.5.11) From Eq . (15.5.10)we see that the coupling g4 is actually irrelevant, so that there is just one relevant coupling here, signalling the presence of a second-order phase transition . From Eq . (18 .5.11) we see that the anomalous exponent (18.5.5)is 1 1 E (x$.5.12) For the physical value e=Ithe first two terms give v^-' 0.58.Three-loop calculation sgagive thi scritical e xponent to order e 3as z v = ~ + 1~E 7E +162 - 0.01904E3(18.5.13) which for E = Igives v = 0 .61. In the calculation presented here nothing was assumed about the system under study except that there is a second-order phase transition, near which the only long-wavelength degree of freedom is a single scalar field . There are a number of different physical systems that fit this description, such as the spontaneous appearance of magnetization (represented here by0)in ferromagnetic and antiferromagnetic materials, and also second- order phase transitions between liquids and gases and in binary fluids . All of these systems are therefore expected to have the same value of v . This is confirmed by experiment, which gives a value8 v = 0 .63 ± 0 .04, in good agreement with the three-loop result (1$.5.13), and in fair agreement even with the one-loop result (18.5.12).It is fortunate though still somewhat mysterious that an expansion in powers of Ishould work so well . More generally, all the systems that are described by the same set of long-wavelength degrees of freedom near their second-order phase transitions are said to belong to the same universality class .All the critical exponents are the same for all systems in a given universality class . 18.E Mi nimal S ubtraction We saw in Section 11 .2 that dimensional regularization provides a partic- ularly convenient method for calculating radiative corrections in quantum electrodynamics, because it preserves the conservation laws associated with gauge invariance . For the same reason, dimensional regularization 18.6Minimal Subtraction 149 also turns out to provide a very convenient alternative definition for the sliding scale of the renormalization group in general gauge theories . In calculations using dimensional regularization, ultraviolet divergences arise as poles in physical amplitudes when the spacetime dimensionality d approaches the physical value d = 4 . (For an example, see Eq . (11 .2.13).) To cancel these poles, the bare coupling constants gB(d)(including masses) must themselves have such poles, with residues fixed by the condition that physical amplitudes be regular as d -,4. These bare couplings in general have non-zero dimensionalities A,,(d)that depend on the spacetime dimensionality d, so it is convenient to consider the dimensionless quantity ge(d}~ where ,u is a sliding scale with the dimensions of energy or mass . This rescaled bare coupling may be expressed as a sum of terms proportional to positive-definite powers v of 11(d - 4), with coefficients bV fixed by the requirement of cancellation of singularities as d ---~4 in physical amplitudes, plus a remainder that is analytic in d at d = 4 . This remainder is identified as the dimensionless renormalized coupling constant V(y, d), so 00 v=t We are free to give the bare couplings any d dependence we like, as long as the singularities at d = 4 in physical amplitudes are cancelled ; we shall remove this ambiguity by requiring that gf(,v, d )be analytic in dnot only at d = 4, but for all d . To calculate the renormalization group equation satisfied by g "(,u, d), first differentiate Eq .(18.6.1)with respect to y -Y(d)g~ +j:(d- 4)-vbv( g)_fl`(ga d)+j: 1:bvm(g) fln(g, d)(d-4)-v v=Z v=1 r+T (18.6.2) where bfVm(g)-a yn~v~~~Og and as before ~ g'(,~,d) d), d) F~(1$.6.3) (18.6.4) Note that flf is a function of all of the g '(p, d)and also of d, but it cannot depend separately on ,u because, with rescaled masses included among the dimensionless coupling parameters, there are no other dirnensionful parameters besides ,u . As we have seen, the dimensionalities A,,(d)are always linear functions 150 18 Renormalization Group Method s of d, which we shall now write a s We rewrite the left-hand side of Eq .(18.5.2) as -p~,g'(d - 4)- [Atgt + b'(g)ptI -)rj j:(d - 4)-y v=t(18.6.5) pt b` V V+I(g) + Alb'(g)] The highest power of d in the analytic part here is of first order, so the same must be true on the right-hand side of Eq . (1$ .6.2), and therefore fl(g, d)must be linear in d: Equating terms of first and zeroth order in (18.6.2)gives the n aow _ -P tg and, more importantly , Nl(g) --At gt' -b1(g)p(b1 rrr(g) p nr~'n V=in](18.6.6) (x$.6.7) (18.6.8) It is noteworthy that the beta-function depends only on the coefficients of the s imple pole in the bare couplings . In fact ,these coefficients also determ ine the coefficients of all the higher poles ;equating the pole terms on the right and left of Eq .(18.6.2)yields the recursion relatio n ~~,Vv+1(g)-Y: pmgm bv+1m(g) = -4tbe'W- b v m($)#m($) (1$.6.9) ►n r n For instance, in order for fddx F,{vFJ"to be dimensionless, any gauge field Al`must have dimensions (in powers of mass) (d - 2 )/2,and since gRAPmust have the same dimensions as 010x~`, gB must have dimensions (4 - d )12,so that for gauge couplings ❑= 0 and p = -1/2.Eq. (1$ .6.$) gives then for a gauge theory with a single coupling constant : (18.6.10) In particular, Eq . (11 .2.20) shows that in quantum electrodynamics in one-loop order the bare electric charge has a pole at d --+4 wit h 1 eB=Z.1 1/2~e-e3 ~ 12r~2d-4 Setting b1(e) = ---e '112n2 inEq. (1$ .6.10) gives the n 3 in agreement with the previous result ( 1$.2.37). 18.6Minimal Subtraction 151 The coupling constants g '(y)introduced in this section a resaid to be defined by minimal subtraction .Thereis a slightly different scheme that is somewhat more convenient . The simple poles (d - 4 )-a typically arise from functions (4n)d/1-2I,(2---d12) (as in Eq .(11.2.13)} which for d--+4 have the limi t (471)z-d/z r(2- ~~ 2 -Id 12_ y + In 47c , (18.5.13) where y is the Euler constant, y =x .5772157 . It is therefore convenient to make the replacement everywhere in Eq . (18 .x.1) 1 ---*I+ 2 - Z In 47c (18 .6.14)d 4 d 4 With this prescription, the coupling constants are said to be defined by modified minimal subtraction . One of the distinguishing characteristics of the definition of couplings by minimal subtraction (or modified minimal subtraction) is that, since no factors of an ultraviolet cut-off ever appear in any calculation, loop diagrams have poles at d = 4 that correspond only to logarithmic ultra- violet divergences, not divergences that are linear, quadratic, etc . Hence a residue function bi(g )can contain a term of ordergagUgc   only if at d = 4 the dimensionality of g~ equals the total dimensionality of the couplings g~, g~, g~, etc .: and it follows then from (18.6.8)that the same is true of fl'~(g). In particu- lar, in a theory with no superrenormalizable couplings (like a gauge theory with massless spinors and no scalars, such as quantum electrodynamics with vanishing electron mass) all couplings have 0 /c0,sothe renormal- ization group eq uation s for the renormalizable couplings (with ❑(=0)are unaffected by the presence of any non-renormalizable interactions .5Also, in such a theory the beta-functions for the non-re norm alizable couplings are polynomials of finite order in the non-renormalizable couplings, with each coefficient in each polynomial given by an infinite series of powers of the renormalizable couplings . For instance, in the theory of photons and massless electrons (assuming invariance under y ) --+ y5y)and P), there are no nonrenormalizable interactions of dimensionality +5, and several interactions of dimensionality +6(four-fermion interactions as well as the purely photonic interaction F.❑Fl")with couplings f jof dimensionality -2. The beta-function for f' iis of the form 1:.jbij(e)fj,with the coefficients bije} given by a power series in e . 152 18Renormalization Group Method s 18.7 Q uantumChromodynamic s quantum chromodynamics is the modern theory of strong interactions . It is a non-Abelian gauge theory, based on the gauge group SU(3) .In addition to the gauge fields, quantum chromodynamics involves fields of spin Z particles known as quarks . There are quarks of six types, or `flavors', the u, c, and t quarks having charge 2e/3, and the d, s, and b quarks having charge -e13 . Quarks of each flavor come in three `colors' which furnish the defining representation 3 of the SU (3)gauge group .* Baryons like the protons and neutron may be approximately regarded as color-neutral bound states of three quarks, totally antisymmetric in quark colors, while mesons like the rho meson behave approximately like color-neutral bound states of quarks and antiquarks .11 In the approximation where the quark masses may be regarded as negli- gible compared with the energies of interest, the inversion of Eq . (17.5.44) shows that the bare coupling constant in a general gauge theory has a pole at spacetime dimensionality d --+4 with residue given b y 3 gB, 4~212(c1 - 3C2 +O(g) 1d4 where C 1and C 2are defined by Eqs .(17.5.33)and (17.5.34).That is, in the notation of the previous section , 3 gl(g) =47c2(-ci -1 Cz +O(g). X18 .7.1}3 Using this in Eq . (l$ .6.10)give s 3(11C1-1 C2) + 0(95) . (18 .7.2)3 For an S U(3) theory with nf massless quarks in the defining representation 3 of SU(3), Eq.(17.5.35)gives C1= 3, C 2= nf/2. ( 18.7.3) Because we are taking the quarks as massless here, this formula may be applied only in the effective field theory obtained by integrating out all quarks heavier than the typical energy E under consideration, so that nf is the number of quark flavors with masses much less than E . With thi s Before the final formulation of quantum chramodynamics several authors had spec- ulated that there might be three varieties of quarks of each flavar,10 both in order to account for the rate of decays like 7c°-~y + y(see Sections 22 .1 and 22 .2) and to introduce an additional degree of freedom that would explain how the wave func- tion of fermianic quarks in a baryon could be symmetric in spin, space, and flavor coordinates .ll 18.7Quantum Chromodynamic s understanding, Eqs .(18.7.2)and (18.7.3)yield 3 T-I I-,nf)+ow) - (4 6153 (18.7.4) We see that the theory is asymptotically free as long as the reareno mo re than 16quark flavors with masses below the energy scale of inte rest. Since in fact there seem to be onl y six quark fla vorsof an ymass,the theory of strong interactions based on the gauge group S U (3)is asymptotically free . It was the 1973 discovery of asymptotic f reedom in non-Abelian gauge theo riesof thi s sort b yGro ssand Wilczek12and Pol itzer13 that convinced theo retical physicists that this is the correct theory of strong interactions . Their calculation immediately expla ined the puzzling result of a famous 1968 expe riment l4at SLAG on deep- inelastic electron -nucleon scatter ing, that strong interactions seem to get weaker at high energ ies." (This expe riment will be discussed further in Section 2 0.6.)But the h istorical importance of the discovery of asymptotic freedom in Yang -Mills theor ies is not just that it explained an old experimental result ;it for the first time opened up the prospect of doing reliable perturbative calculations of strong interaction processes, at least at high energy . Asymptot ic freedom was soon found to have another important impli- cation . At first after the discovery of asymptotic freedom it was widely assumed that the gauge bosons in arealistic Yang-Mills theory of strong interactions would have to be quite heavy ,to explain why these strongly- interacting bosons had not been discovered long before . Following the precedent of the theory of weak and electromagnetic interactions (dis- cussed in Chapter 21), it was supposed that the masses of the gauge bosons arose from a spontaneou sbreakdown of the color SU(3) gauge group ,triggered by the vacuum expectation values of scalar fields in a non- trivial representation of this group . But these strongly interacting scalars would contribute pos itive terms to fl(g), wh ich could destroy asymptotic freedom . Even worse ,inatheory w ith strongly interacting scalar fields , radiative cor rections involving weak interactions would introduce large violations of various symmetries like charge conjugation invariance and flavor conse rvation which ,as we shall see,would not be v iolated with- out the scal ars."Then it was suggested to drop the strongly-interactin g Zee15 and perhaps other theorists had already understood that this experimental result could be understood in a theory with abets-function that becomes negative for small positive coupling, but calculations of P(g) in all renormalizable field theories except non-Abelian gauge theories gave P(g) >Q. On the other hand, by 1972 't Hooft had developed techniques for calculating fl(g) in Yang-Mills theories, and in June 1972 he announced at a conference on gauge theory at Marseilles16 that fl(g) <0,but he waited to publish this result and work out its implications while he was doing other things, so his result did not attract much attention . 154 18 Renormalization Group Method s scalars, and accept the consequence that the gluons, the SU(3) gauge bosons, have zero rnass .18The decrease of the strong coupling constant at high energy or short distance of course implies an increase at low energy or large distance, and it was suggested that this might explain why massless gluons and quarks had not been detected . According to this hypothes is, only color-neutral particles like baryons or mesons will ever appear in 150Iat10I1.19This is unfortunately still a hypothesis rather than a theo rem, but after two decades the reseems to be little doubt that it is cor rect. Even though quarks cannot mate rialize as free particles, they a rein a sense observed as jets produced in high energy collision processes . For instance, in many events in electron -positron annihilation, the final state consists of two narrowly collimated h adron jets, with a dist ribution in the angle 0between the colliding lepton momenta and the jet directions (inthe center-of-mass system) given by 1+sin2o,just as expected from the tree graph for electron -positron annihilation into quark-antiquark final states . 20This can be understood 21 in terms of the general analysis of infra red divergences in Section 13 .4. At extremely high energies we would expect the rate for a physical process to be g iven by lowest-order perturbation theory, provided it is `infrared-safe,' in the sense of not becoming infra red divergent when all masses are taken to zero . The total rate for electron-positron annihilation into hadrons is infrared-safe, since we sum over all hadronic final states .(We are igno ring higher- order electromagnetic effects here .)Therefore we can rely on perturbation theory, which immediately tells us that the ratio R of this rate to the rate for e ++e- y++p-is R = 3 Eql3~, where the sum runs over all quark flavors, Q qis their charge in units of e, and the factor 3 is the number of colors .(For instance, in the wide energy range between mh r: 4.5 GeV and mt :~ 180 GeV, R -3(2(2/3)2 + 3(-1/3)2) = 113 .}On the other hand, the rate for electron -positron annihilation into some definite state of quarks and gluons is not infrared-safe, and its rate therefore cannot be calculated in perturbation theory at all; in fact, it is zero . In between these two ext remes is the rate for electron -positron annihilation into a definite number of jets, each jet carrying a definite total momentum and charge, together with a set of unobserved hadrons with limited total energy outside the jets . As discussed in Section 13 .4, this rate is infrared-safe . It can therefo rebe calculated at high energy in the tree approximation of perturbation theory, identifying jets (inthis order of perturbation theory) with the outgoing quarks, antiquarks, and gluons . We can even calculate the rate forthree jet events, arising from tree diagrams in which a gluon is emitted from the outgoing quark or ant iquark ,and use the compa rison of the results with expe riment to measure the value of as(P).22But we cannot use perturbation theory to predict the d istribution of momenta within a jet, because such a di fferential rate is not infra red-safe . Similar remarks 18.7Quantum Chromodynamics 155 apply to the production of jets in deep inelastic lepton-hadron collisions, to be discussed in Section 20 .6, but the presence of hadrons in the initial state makes the analysis more complicated . Following the same reasoning as in Section 12 .5, with no scalar fields the most general renormalizable Lagrangian for quantum chromodynamics can be put in the for m 4F~vFxPv-)7,Vnl?-ig Aatx + MnJvna (18.7.5) n where A llis the color gauge vector potential ; Fa v is the color gauge- covariant field strength tensor ; g is the strong coupling constant ; t, are a complete set of generators of color SU(3) in the 3 representation (that is, Hermitian traceless 3 x 3 matrices with rows and columns labelled by the three quark colors), normalized so that Tr (txtp ) =~6a#; and the subscript n labels quark flavors, with quark color indices suppressed . Just as we found for electrodynamics in Section 12 .5, this Lagrangian has important accidental symmetries : it conserves space parity,t charge conjugation parity, and the numbers of quarks of each flavor (minus the number of the corresponding antiquarks), including the long-established `strangeness' quantum number, which counts the numbers of `s' quarks . Thus quantum chromodynamics immediately explained the mysterious fact that the strong interactions respect various symmetries that are not symmetries of all interactions . This argument also makes it clear why, as mentioned earlier, in this theory the weak interactions do not introduce large violations of parity, charge conjugation, strangeness, etc . Since all renormalizable interactions among quarks and gluons conserve these symmetries, at energies E much less than the masses mW of the particles that carry the weak interactions these symmetries could be violated only by non-renormalizable terms in the effective field theory, such as i P-yafPW interactions, which as discussed in Section 12 .3 would be suppressed by negative powers of mW as well as by the coupling constants of the weak interactions . Of course, it is possible that the quarks and gluons exhibit some new kind of strong interaction at an energy scale A' much larger than the scale n characteristic of quantum chromodynamics . For instance, as discussed in Section 22 .5, the quarks might be bound states of more fundamen- tal fermions, which interact with gauge fields whose asymptotically free couplings become strong at energies of order A', trapping them into the quarks . In that case the effective Lagrangian density for quarks at en- t Although it was not known in 1973, we shall see in Section 23 .6 that non-perturbalive effects can violate parity in quantum chramodynamics . Various ways of avoiding strong parity violation have been suggested, but it is not yet clear which is correct . 156 18Renormalization Group Method s ergies E < A' would contain non-renormalizable interactions such as qWi~ y),which are supp ressed only by powers of E/11.'.These interact ions could show up ,not only in small violations of symmet ries like parity and quark flavor conservation at ordinary energ ies, but also in departures 23 from the quantitative predictions of quantum chromodynamics at energies approaching X. Now let us cons ider the behaviour of the coupling constant of quantum chromodynam icsin greater detail .In lowest order ,the renormalization group equat ionis given by Eq .(18.7.4)as 3 P~dµgW=- 47E2} 4 -bra f The solution i s ~ ~ as(P) =g2 W ~ 127E 4n (33 - 2r~f)In(~/A)(18.7.6) (18.7.7) where A .is an integration constant . This formula exhibits a characte ristic property of theo ries of massless (or, for the quarks, approximately mass- less) particles : in such theories one of the dimensionless couplings in the Lagrangian is exchanged for a free dimensionful parameter . Eq .(18.7.7) involves no f ree dimensionless parameters, but it does in volve one f ree parameter with the dimensions of mass, the integration constant A . These calculations have been car ried to th ree-loop order .The renor- malization group equations to this order a re24 3 1~d~c~ g(l~)=-flQ 167r ~ 128ic4 81921c 6 where #n are the numerica lcoefficients : Igo = 11 - 32nf, fli= 51 --- 3n f , 2 = 2857 - 5033nf-325 z ~ 9 27nf The solution i s g2 aS~F~~ = ~F~)47c 471 flQIn(.c2/n,2)2#11n[In (P2 / n.2)3 fro In(jj21A2)(18.7.8) (18.7.9) (18.7.1Q) (18.7.11 ) fl O+ 4 2#~ 2 2 5 2((In[Irl(2/A2)1 - z +2- 4 )](18 .7.12) Idal n(p1A) f~i 18.8Improved Perturbation Theory 157 It should be recalled that nf in the above results is the number of quark flavors with masses below the energies of interest . In each energy range between any two successive quark masses we have a different value of n f, and also a different A, chosen to make g (p) continuous at each quark mass . In particular, experiments on the deep inelastic scattering of electrons typically involve energies above only the first four quark flavors (u, d, s, and c),so here we must take n f = 4 . On the other hand, experiments at electron-positron colliders like PEP, PETRA, TRISTRAN, and LEP are at energies well above the fifth (b)quark mass, so in these experiments we must take nf = 5 . But these results may be expressed in terms of those for n f= 4 by matching the solutions of the renormalization group equations at the b quark mass . In this way it is found25 (using the modified minimum subtraction prescription in calculating #2) that the strong coupling extrapolated to mz = 91 .2 GeV is a,(mZ) - gs (mZ)/4n _x .118 ± 0.006, corresponding to A ~ 250 MeV for energies ywith mh CC p < mt, where nf . = 5 . A more recent study 26 of hadronic production in e+-e- annihilation at the Z resonance has given a directly measured value a,(mi ) = 0 .1200 ± 0.0025, with a theoretical uncertainty of + 0.007$, corresponding to A= 253±96° MeV . 18.8Improved Perturbation Theory ' The ground-breaking papery of Gell-Mann and Low was in large part directed to the problem of `improving' perturbation theory -that is, of us- ing the ideas of the renormalization group and the results of perturbation theory to a given order to say something about the next order of pertur- bation theory . To illustrate this, let's return to the specific case studied by Gell-Mann and Low : vacuum polarization in quantum electrodynamics . Recall that the renormalized electric charge e . at a sliding scale yis given by Eq .(l 8.2.3)in terms of the bare charge e& a s eµ=N~A)- 1e8 where IV(A} is the constant which, when multiplied into the unrenormal- ized electromagnetic field, gives a field renormalized at scale µ . (See Eq.(18.2.21).)Thus we can define a renormalized (and hence cut-off- independent) complete photon propagator A'PCr (q,p, ej,) in terms of the complete propagator ❑'BPcr(q, eB) of the unrenormalized field a s Alp6(q,µ, e.) = N~A)1OsP~(q,eB) (18.8.2) This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 158 18 Renormalization Group Method s in such a way that the function e~~~, ff (q,y, eu) is both independent of y, because it equals e~0~PA,eB}, and independent of the cut-off, because both e~ and ❑~,(q,p, e,)are renormalized quantities . (We are not ex- plicitly displaying cut-off dependence here .) But Lorentz invariance and dimensional analysis tell us that this function must take the form : e~A~,T(q,µ,eU)_qP°d(q2Il~2' eu) + ~,pqaterms . (18.8.3) Since Eq .(18.8.3)is tt-independent, we can set p = V~_q_2 = q here, so tha t d(gZlPZ,eu) = d (1,eq) . (18.8.4) Now let us see what this tells us about the structure of the perturbation series for d(q2/tc2, eu) .The beta-function for e has the expansion : The renormalization group equation for e . then has a power series solutio n ~ 2 e~ = e~ - b le~ Inq- b2e~ In q (bib2ln2 2 + b31nq2 e~ + ... P If we also expand d : d(1, e) = e2+ die4 + d2e6+ d3e8 + ... then d(q2/µ2, e,,) = d (l, eq) = e~ - (bl1n_(/l)e_ b 2In Y ~ z z -2(b2b2ln2 ~ +(b3 - bld2)In ~ - d3 e~ + ....(18.8.6) (18.8.7) z ~ -d2 )e~ (18.8.8) Note that the leading powers of 1n(q2/tt2) in each order of d(q2/P2, eu) are respectively n, 1, 1, 2,3,....Also, if we calculate d(q2/P2, eu) to order e6~ and thus determine b1 and b2,we can immediately write down the coefficient of the leading logarithm in order e~, as - zb lb2. None of this would be easy to infer without using the method of the renormalization group . Problem s 1. Consider an SU(N) gauge theory with a scalar field in the defining representation of SU(N) . Calculate the beta-function for the gauge References 159 coupling to one-loop order, including the contribution of a scalar loop . (Recommendation-, Use the background field gauge, with a constant background field .) 2. Suppose that the beta-function fl(g) for a theory with positive cou - pling constant g has a simple zero at g = g ., where fl(g) a(g . - g ) with a >0. What is the asymptotic behavior of the correction t o the leading term oc E- Y6(90 in the factor IVE-Iassociated with th e inclusion of an operator 6 in a vacuum expectation value . 3. Show that in a theory with P(g) =bg2 + b'g3 +b"g4+   , it is possible by a redefinition of the coupling constant to make the coefficient b" anything we want . 4. Calculate the effective electric charge that should be used in studying processes at energy 1 00GeV, taking account of all known charged quarks and leptons with masses below 1 00 GeV . 5. Calculate the asymptotic behavior for large four-momentum of the electron propagator in quantum electrodynamics . (You may use the one-loop value of Z2 calculated elsewhere, for instance in Section 11.4.) 6. Calculate the anomalous exponent v to first order in the expansion ine= 4 - d for an 0(N)-invariant theory of scalar fields 0n(x) with n = 1,    , N belonging to the vector representation of 0 (N), and an interaction 4g(En 0n)2. References 1. M. Gell-Mann and F . E. Low, Phys . Rev .95, 130 0(1954) . Th e freedom of choosing the definition of renormalized coupling con - stants had been discussed a little earlier by E . C. G. Stueckelber g and A . Peterman, Helv .Phys .Acta 26,499 (1953) (who also intro - duced the unfortunate term `renormalization group') but without a n explanation of its relevance to the calculation of physical processe s at very high energy or very low energy . After the work of Dell-Man n and Low renormalization group methods were further developed b y N.N. Bogoliubov and D . V. Shirkov, Introduction tothe Theory o f Quantized Fields (Interscience, New York, 1959) : Chapter V111, an d references quoted therein . Interest in renormalization group method s in particle physics was revived in 1970by C .G.Callan, Phys . Rev . D2, 1541 (1970) ;K. Symanzik, Common . Math . Phys . 18, 227 (1970) ; 160 18Renormalization Group Method s C.G.Callan, S . Coleman, and R . Jackiw, Ann, of 'Phys . (New York), 59, 42 (1970) . 2. K.G.Wilson, Phy s.Rev.B4, 3174, 3184 (1971 );Rev. Mod . Ph ys .47, 773(1975 ). 3.J. C. Collins,Phys.Rev.D14,1213 (1974) . 3a. R. Jost and J . M. Luttinger, Help . Phys . Acta 23, 201 (1950) . 4. This is the value quoted by the Particle Data Group in 1994 . Mor e recent calculations have been summarized by G.Altarelli, CER N preprint CERN-TH-95 /203, to be published in Proceedings of th e Workshop on Physics at DA (D1VE, April 1995 . These more recen t values of a-l(mz) range from 128.89to 129 .08. 4a. S. Coleman an dE. We inberg, Phys.Rev.D7, 1888 (1973) . 4b.L.D.Landau, in Niels Bohr and the Development of Physics (Perga- mon Press, New York, 1955) : p. 52; and ear lier works cited therein . 4c. S. L. Adler, C . G. Callan, D . J. Gross, and R . Jackiw, Phy s.Rev.D6, 2982 (1972) ;M. Baker and K . Johnson, Physics 9 6A, 120 (1979) . 4d. For a discussion and references, see J . Glimm and A.Jaffe, Q uan- tum Physics -- A Functional Integral Point oj'View, Second Editio n (Springer-Verlag, New York, 1987) :Section 21 .6; R. Fernandez, J . Frohlich, and A . D. Sokal, Random Walks, Critical Phenomena, an d Triviality i n Qu antumField Theory (Springer-Verlag, Berlin, 1992) : Chapter 15 . 5. S. Weinberg, in General Relativity, eds . S. W. Hawking and W . Israel, eds. (Cambridge University Press, Cambridge, 1979) : p. 790. 6. S. Weinberg, Phys .Rev. D8, 3497 (1973) . 7. The original calculation is by K . G. Wilson and M . E. Fisher, Phys . Rev. Lett .28, 240 (1972) ;K.G.Wilson, Phys . Rev . Lett .28, 548 (1972) .For reviews, see K .G.Wilson and J . Kogut, Phys . Rep . 12C, No . 2 (1974) ; M. E. Fisher, Rev. Mod .Phys . 46, 597 (1974) ; E. Brezin, J . C. Le Ciuillou, and J . Zinn-Justin, in Phase Transitions and Critical Phenomena, eds. C. Domb and M . S. Green (Academic Press, London, 1975) . 8. See, e . g., P. M. Chaikin and T . C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 1995) : p. 231. References 161 8a. E.Brezin, J . C.Le Guillou,J. Zinn-Justin, a nd B . G. Nickel, PYays. Lett.44A, 227 ( 1973);K. G. WilsonandJ. Kogut, Ref. 7, Sec tionS. 9. G. 't Hooft, Nucl . PYays .B61, 455 (1973) ; Nucd . 'days .B82, 444 (1973) . The derivation presented here is a somewhat simplified version of 't Haft's . 10. 0.W.Greenberg,Pays.Rev. Lett .,13,598 (1964) ;M.Y.Han and Y . 1`+Tambu ,Phys .Rev.139,B1006(1965) ;W.A.Bard een,H.Fritzsch, and M .Dell-Mann ,inScale and Conformal Invariance in Hadron Physics, e d.R. Gatti (Wiley,NewYork, 1973) . 11.M. Gell-Mann ,Pays.Lett.8, 214(1964); G.Zweig, CERN pr eprint TH401 (1964) . 12.D.J.Gross a nd F . Wilczek, Plays.Rev. Lett .,30,1343 (1973) . 13. H. D. Pulitzer, Plays . Rev .,30, 1346 (1973 ) 14.E. D.Bloometal.,Plays.Rev.Lett.23,930(1969) ;M.Breidenbach et al.,Plyys.Rev.Lett.23,933 (1969) ;J.L.Friedman and H .W. Kendall ,Annual Reviews ofNuclea rScience 22,203(1972) . 15. A.Zee,unpubli shed. 16. G. 't Haft ,unpubli shed. 17. S.Weinberg,Plays. Rev .D8, 605(1973) . 18.D.J. Gross a nd F .Wilczek,Whys.Rev.D8,3633(1973) ;S.Weinberg, Plays.Rev.Lett.31,494(1973) . 19. A simi lar idea had been suggested before the discovery of asymptotic freedom by H . Fritzsch, M . Gell-Mann, and H . Leutwyler, Phys.Lett. 47B, 365 (1973) . 20. G. Hanson et ad ., Phys.Rev. Lett .35,1609 (1975) ;R. F. Schwitters , inProceedings of the International Conference on Lepton and Pho - tonInteractions at High Energy at Stanford, 1975, ed . W . T. Kir k (Stanford Linear Accelerator Center, Stanford, 1975) :p. 5; G. Han - son, Stanford Linear Accelerator Center Report SLAG-PLEB-1 S 1 4 (1976), unpublished . 21. G. Sterman and S.Weinb erg,Pays.Rev.Lett.39,1436(1977) 22. J. Ellis, M . K. Gaillard, and G . G. Ross, 11Tucd . Phys .B111 , 253 (1976) . 23. E.Eichten,K.Lane ,and M .Peskin,Phys. Rev . Lett .50, 811 (1983) . 162 18 Renormalization Group Method s 24. For a review, see I . Hinchliffe, in `Review of Particle Properties,' Plays .Rev.D50, 1177(1994) : Section 25 . 25. G.Altarelli, in Proceedings of the Rencontres deHanoi, CERN preprint CERN-PPE/94-71 (1994) . 26. K. Abe et al.(SLD collaboration), Plays . Rev .D51, 962 (1995) . A summary of earlier data on Z° decay into hadrons by M . Shifman, Minnesota preprint hep-ph/9501222 (1995), gave a value a,(mz) _ D.125 ± 0.005, corresponding to A :500 MeV . 19 Spontaneously Broken global symmetrie s Much of the physics of this century has been built on principles of symmetry : first the spacetime symmetries of Einstein's 1905 special theory of relativity, and then internal symmetries, such as the approximate SU(2) isospin symmetry of the 1930s . It was therefore exciting when in the 1960s it was discovered that there are more internal symmetries than could be guessed by inspection of the spectrum of elementary particles . There are exact or approximate symmetries of the underlying theory that are `spontaneously broken,' in the sense that they are not realized as symmetry trans formations of the physical states of the theory, and in particular do not leave the vacuum state invariant . The breakthrough was the discovery of a broken approximate global 5U(2) x SU(2) symmetry of the strong interactions, which will be discussed in detail in Section 19 .3. This was soon followed by the discovery of an exact but spontaneously broken local 5' U(2 )xU(1) symmetry of the weak and electromagnetic interactions, which will be taken up along with more general broken local symmetries in Chapter 21 . In this chapter we shall begin with a general discussion of broken global symmetries, and then move on to physical examples . 19,1 D egenerate Vac ua We do no t have tolookfar for exam ples of s pontaneous symme try breaking . Co nsidera chair.The equations gove rningthe atoms of the chair are ro tationally symmetric, but a sol utionofthese eq uations, the actualchair, has a definite orientat ioninspace . Here we wi ll be conce rned notso m uch wi th th ebreaking of symme tries by objects like chai rs, but ratherwith t he symmet rybreaking in the gro und state of a ny rea listic quantum fie ldtheory,the vacuum . A sponta neouslybrokensymme try infield theory is a lways assoc iated withadegene racy of vac uum sta tes. For i nstance, co nsider a symmetry transformatio nof the act ion, andofthe meas ureusedin integrating ove r IA 11 164 19Spontaneously Broken Global Symmetrie s fields, that acts linearly on a set of scalar fields din (x ) M (The can need not be elementary fields ; they can be composite objects, like yaly'nw .) As we saw in Section 16 .4, the quanturn effective action I'[o] will then have the same symmetry r pi =r [LO]  (19 .1.2) For the vacuum the expectation value of O(x) must be at a minimum of the vacuum energy -rp], say at O(x) =~ (a constant) . But if L~ *~, then this vacuum is not unique ; -r[O] has the same value at 0= L~ as it does at Vii . In the simple special case where the symmetry transformation (19.1.1) is a reflection, o ~ - o, if -r(O) has a minimum at a non-zero value 0of0, then it has two minima, at 0and -~, each corresponding to a state of broken symmetry . We are not yet ready to conclude that in such cases the symmetry is broken, because we have not yet ruled out the possibility that the true vacuum is a linear superposition of vacuum states in which dim has various expectation values, which would respect the assumed symmetry . For instance, in a theory with a symmetry 0 --+ -0, even if F(O) has a minimum for some non-zero value ~ of 0, how do we know that the true vacuum is one of the states VAC, ± )for which (Dhas expectation values and - ~, and not some linear combination like JVAC, +) + IVAC, -)that would respect the symmetry under 0 -~ -0? The assumed symmetry under the transformation 0~ - 0tells us that the vacuum matrix elements of the Hamiltonian ar e VAC, + IHIVAC, + ) = VAC, - IHIVAC, -)-a (with a real) an d VAC, +IHIVAC, - )= VAC, -JH IVAC, + )=b (with b real), so the eigenstates of the Hamiltonian are IVAC, +) + VAC, -), with energies a +JbI. These energy eigenstates are invariant (or invariant up to a sign) under the symmetry 0 --~, -0, In fact, the same issue also arises for chairs . The quantum mechanical ground state of an isolated chair is actually rotationally invariant ; it is a state with zero angular momentum quantum numbers, and hence with no definite orientation in space . Spontaneous symmetry breaking actually occurs only for idealized sys- tems that are infinitely large . The appearance of broken symmetry for a chair arises because it has a macroscopic moment of inertia I, so that its ground state is part of a tower of rotationally excited states whose 19.1Degenerate Vacua 16 5 energies are se parate dby o nlytiny amo unts, of o rder h2/ I. This g ives the state vec torof the chairanexquisite se nsitivi tyto externalperturb a- tions; eve nvery wea kexternal fie lds willshift the energy by muchmore than the energydifference of these rotationallevels.Inconsequence, any rotationally asymmetric ex ternalfieldwillcausethe gro und state orany otherstate of t he chair wi thdefinite angularmomentum numbersrapidly todevelop compone nts wi thotherangularmomentum q uantumnumbers. The states of the c hairthatarerelatively stable wi threspect to sma ll externalperturbations a renot those wi thdefinite angular mome ntum quantumnumbers,butrather those w ithadefinite orie ntation,in which the rota tionalsymmet ry of theunderlying theory is broken. Forthe vac uum also,thepossibility of spo ntaneous symmetry breaking is agai nrelated tothelarge size of the system, spec ifically to thelarge volume of space .In th e above exa mple of a reflec tionsymme try,the off- diagona lmatrix element b of t he Hami ltonianinvo lves a nintegrationover fieldconfigurations thattunnelfromthe minimum a t0 = 0 to t he one at 0 = -0, so itis smallerthan the diago nalmatrix e lementa by a barrier penetrationfactorthatfora spatia lvolume V is of the fo rm ex p(-C -r), where C is a posi tive cons tant*depending on the microscopic parameters of the theory . The two e nergy eige nstates VAC, +) ± VAC, -) a rethus essentiallydegenerate for a ny mac rosco pic vo lume, a ndso are strongly mixe d by any perturbation thatis anodd f unctionalof 0 . Eve n if sucha perturbation H' is very weak, i tsdiagonalelements VAC,+I H'IVAC,± ) will differ by muc hmore thanthe ex ponentially suppressedoff-diagonal elemen ts of ei ther H ortheperturb ation.Thusthe vac uum eigenstates ofthe pe rturb ed H amiltonianwill b e ve ry close to eitherone of the broken symme try states ~ VAC, ±) w hich diagonalizetheperturbation, andnottothe invarian tstates VAC, +) ± VAC, ---) . Whichone of the states IVAC, ± )is the true vac uum fo rvery sma ll perturbations? T his depends onthe perturbation,but sincethese two sta tes arerelated by a symme try transformation of the orig inal Hamiltonian,it doesn'tmatter; if the pertu rbation is sufficiently sma ll,no observe rwill be abletotell the difference. The vanishing of ma trix e lementsbetweenvacuum states wi thdifferent fieldexpectatio nvaluesbecomes exac tina space of infi nite volume .' For infinite vo lume, a ge neralvacuum s tate Jv) may bedefined as a s tate with For instance, by analogy with the classic wave mechanical problem of barrier penetration, for a Lagrangian density of the form -zr7,~Or7uO-V(O)1we have C = f ~ 2V(O) dO . We will not bother to calculate the off-diagonal matrix el- ement h here, because we shall soon give a general argument that shows that it vanishes for infinite volume . 166 19Spontaneously Broken Global Symmetrie s zero momentum P~V)=0 (19.1.3) for which this is a discrete momentum eigenvalue . (This excludes single- particle or multiparticle states, for which the momentum value zero is always part of a continuum of momentum values in a space of infinite volume .) In general there may be a number of such states . They can usually be expanded in a discrete set, and our notation will treat them as if they were discrete . They will be chosen to be orthonorma l ~utv) = 6uL . (19 .1.4) Any matrix element of a product of local Hermitian operators at equal times between these states may be expressed as a sum over states : ~uIA(x) B(D)jv) = j>j A(D)jw) ~w jB(a)~u) W +1d~~j> ~A(a)~N, P)~N,PIB(a)Ev)e-ip x,(19.1.5) where ~N,p)are a set of orthonormalized continuum states of defin ite three-momentum p that together with the jv)span the whole physical Hilbert space . (Here N may include continuous as well as discrete labels . Also, we are dropping time arguments .)We assume without proof that because the HIV,p)belong to the continuous spectrum of the momentum operator P, the dependence of matrix elements on p is smooth enough (that is, Lebesgue integrable) to allow the use of the Riemann Lebesgue theorem, 2so that the integral over p vanishes as x j --+ oo . In this limit, we have then ~ulA(x)B(a)1v5 )E~ulA(a )lw5(wlB(a)lv). (19.1.6)1xl-x w Likewise, ~uIB(a)A(x)IvS ) E ~ulB(a)jwS ~wIA( a)lu7 (9.1.7)ixi-->00w But causality tells us that the equal-time commutator [A(x), B(0)] vanishes for x =~ 0 (see Section 5 .1), so the matrix elements (19.1.6)and (19.1.7)are equal, and thus the Hermitian matrices ~u jA(D)jv~,(uB(0) jv), etc., must all commute with one another . It follows that they can all be simultaneously diagonalized . Changing if necessary to this basis, we have then for every Hermitian local operator A(x) of the theor y ~ufA(a )lu)=6uU QU (19.1.8) withat, a real number,the expectationvalue of A in the state ~u). So forinfinite volume any Hami ltonianconstructedfrom local ope rators will 19.2Goldstone Bosons 167 have vanishing matrix elements between the different vacua Jv) . In the absence of off-diagonal terms in the Hamiltonian, any two Iv)s connected by a symmetry operation will be degenerate . Asymmetry- breaking per- turbation built out of such local operators will be diagonal in the same basis, and will therefore yield a ground state that is one of the Iv)s, rather than a linear combination of them . It is reassuring that the vacuum states Jv) which are stable against small field-dependent perturbations are also vacuum states in which the cluster decomposition condition (see Chapter 4) is satisfied . This principle requires that for the physical vacuum state JVAC ) ~VACJA(x)B(0)JVAC) )(VAQA(x)VAC) ~VAC JB(0) JVAC ). (19 .1.9)~~00 This condition is satisfied if we take the vacuum state JVAQto be any one of the states Jv) in the basis defined by Eq .(19.1.8),but not if we take it to be a general linear combination of several of the Iv)s . 19.2 Go ldstone Bosons We now specia lizetothe case of a spo ntaneously brokencontinuous symme try.Inthis case there is a theorem, t hat (wit hone impor tant exception, to be co nsidered in Chapter21)the spec trum of p hysical particles m ustcontainone pa rticle of ze ro mass a ndspinfor each broken symme try. Suchparticles, k nownas Go ldstone boso ns (or Nambu- Goldstone bosons) were fi rstencountered in specific models by Goldstone' and N ambu4; two ge neralproofs of theirexistence we re thengiven by Goldstone, Salam, a ndmyse lf.5 This sec tion wil l present b othofthese proofs, a nd t hengo o n to co nsider theproperties of the Go ldstone boso ns. Supposethat the actionand m easureareinvaria nt un dera co ntinu - ous symme try,underwhich a se tof Hermitian sca larfields 0n(x) (ei ther elementary o rcomposite) are s ubjecte dtothelinear infinitesima ltrans- formatio ns OnW -,OnW + i cT,t►lmorn(x), (19.2.1) in withitem a fi nitereal matrix. As we fo und inSection 16 .4, the e ffective actionisthen also i nvariantunderthis transformatio n 1: 1 mo Jtnmom(.aC)d4aC =0.(19.2.2)nr n,m0l We shall specialize to the case of a translationally invariant theory with constant fields din, where as we saw in Section 16 .2, the effective action 168 takes the form19Spontaneously Broken Global Symmetries (19.2.3) where~Visthe spacetime vo lume a ndV(O) is k nownasthe ef fective potential. Eq. (19.2.2) may then be written r3V(0)tnmOm = 0 n,m0on(19.2.4) We will use this symmetry requirement in a form obtained by differenti- ating again with respect to 01: tn"+E tnmom = 0. (19 .2.5)a0n4l nom Now specialize to the case where On is at the minimum of V(O), that is, at the vacuum expectation value din . Since y(O)is stationary at its minimum, the first term in Eq .(19.2.5)vanishes, s o 02 V(O ) n,mOOn~~/0.4(19.2.6) The ge neralresults of Sec tion 16.1 show t hatthe seco nd d erivative inEq. (19.2.6) is j ust the sum of a llconnected o ne-particle-irreduci ble momen tum-space Feyn man diagrams wi thexternallines labelledn ande andcarrying zero fou r-mome ntum. As s hownat the endof Sect ion 16 .1, itis related to the reciprocalof the mome ntum-space pro pagator by 00n ~0e, soEq. (19.2.6) gives Ong(0)tnm ~m = 0. nm(19.2.7) (19.2.8) Thus, if the symme tryis broken, so t hatEm thrn*n is non-zero, then this is aneigenvectorof❑n1(D) w itheigenvalue zero. The existence of suc h aneigenvector mea ns that❑n~(q) has a pole a tq2 = D . Therankof the residue of t he pole at q2 = 0 is e qual to t he dime nsionality of the space of the vectors tai, wi th trunn ing ove rall the generators of co ntinu ous symmetries of thetheory .Roughly spea king, there is o ne mass lessboson forevery in dependent br okensymme try. In the c lassic examp le of a b rokensymmetry, the Lagrangianinvolves a set of Nrealscalar fiel ds din, a nd t akes the for m 1 .1& PI)z2 OpOn&On - - 1:OnOn -g- (E OnO2n 4 n(19.2.9) 19.2Goldstone Bosons 169 This is invarian t und er the group O(N), co nsisting of rotations of t he N-vector wit hcomponents din . Forconstantfields the effective po tential in th etree ap proximation is give nsimplyby mi nusthenon-deriva tive termsin th eLagrangi an density 2 1: O MOn + E O MOn VM~2 n 4 n(19.2.10) As usual we suppose that g is positive-definite . (Otherwise the minimum, if any, of Y(O) lies outside the range of validity of perturbation theory .) If'z is also positive, the minimum of V( O)is at the point 0=0,which is invariant under O(N) .On the other hand, for .2 c 0 the minimum is at points din at which n The mass matrix in the tree approximation is the n -02V(O ) Mn2M 0000m 26nrn + $5nm + 2g~n~ m 2$ On Om. This has one eigenvector with a non-zero eigenvalue : rn2 = 2 g E2121 n(19.2.11) (19.2.13) and N - 1 eige nvectorsperpendicular to ~ wit heigenvalue zero. The reason for t he appearance of o nly N - 1 Gol dstone boso nsis just that D(IV) is b roken down toD(1V - 1) ( the subgroupof O(N) t hatleaves ~ invariant), a ndsothe number of in dependent br okensymmetries is the dime nsionality of O(N) m inus thedimens ionality of D(1V - 1), or 1N(N -1) - i (N - 1) (1V - 2) = 1V - 1. (19 .2.14)2 2 Here is another proof of the existence of Goldstone bosons, one that makes no use of the effective action formalism . As we saw in Chapter 7, any continuous symmetry of the action leads to the existence of a conserved current J J` aa"(X) _0 r3xp(19.2.15) witha charge Q thatindu cesthe associate dsymme trytransformation Q=f d3X jO(X, 0) ,(19.2.16) 170 19 Spontaneously Broken Global Symmetrie s [L', 0ra(X)] - - T~ tnm Om0 in(19.2.17) Operator relations like Eqs .(19.2.15)-(19 .2.17) are unaffected by sponta- neous symmetry breaking, which is manifested in the properties of physical states . Now, consider the vacuum expectation value of the commutator of the current and field . Summing over intermediate states, this i s LaAW, ~nWI )VAC= (2n)-3 /d4 P[p(p) eP"y-x? where, using translation invariance,k(p) eiP (y-x)J -PN) ,(19.2.19) (27i)-3iP n(P)= E~ VAC ~J~(0)IN~ ~N10 n(a)IVAC)34(p ~27r)-3i Pn(P)= E~ VAC IOn(O)IN) ~Na~( a)IVAC)64(P - P N)(19.2.20) We usually take Pas well as din to be Hermitian operators, in which case Eqs.(19.2.19)and (19 .2.20)are complex conjugate s Pn(P) = -Pn*(p) but this will not be ass umed here. Lorentz invariance tells us that p and p must take the form s Pn(P) = P'Pn(r`P2)0(PI)'(19.2.21) (19.2.22) (19.2.23) (The factor B(p°), which is +1 for p° >0and zero otherwise, is required by the fact that p Nis the four-momentum of a physical state .) This give s ([JA(), On(x)l)VAC 01 - f d1t2 [p2+(y - xfAz) ~Y +pn(l12)A+(x- Y ;It2)] where❑+ is the fam iliarfunction 0+(z; ,12)=(27c)-3 Jr d4PO( PO) 6 (P2 + 112)e'Pz(19.2.24) Asremarkedin Chapter 5, Lo rentz invaria nce allows ❑+(z;yz)to depend only onz2, A2, a nd0(?), a nd fo rzz > 0 o nly onz2 a ndp2.Hence ❑+(x - y ; y2)and❑+(y-- x;M2)are equalforx - y space like, a nd so i n this cas e ([JA( .1)On(X)l)Vac y),dm2 IP42) +On(P)]A+(y - x; M2). (19.2.26) 19.2 Goldstone Boson s But for x - y spacelike all commutators must vanish, s o and therefore Eq . (19 .2.24) gives for general x and y171 (19.2.27) kA(Y),)VAC ~d+,2Pn(F~~~[A+(Y-~~ F~2) -A+ (x- Y,Ft2) I- y(19.2.28) ~Where Eq .(19.2.21) applies, Eq .(19.2.27)also shows that pn(p) isreal. Now let us use the fact that J '(y)is conserved . Applying the derivativ e 0/0y~to both sides of Eq .(19.2.28), and using the familiar equatio n we find, for all x and y(19.2.29) 0 =Idµ' µ'pn(µ) [D+(Y-X,µ2) - A+(x - Y,µ2)) , ( 19.2.30) and so (since ❑+(x - y )isnoteven for x - y timelike or lightlike ) Y2 Pn(,U 2) = 0 . (19.2-31) Normally we would conclude from this that p,'(1t2) vanishes for all µ' . However, this is not possible in the case of broken symmetry . Set A= 0 and x0 = YO=tin Eq .(19.2.28): ( IJO(Y, t), On (X,t)DVAC =2i(27r )-' I dp2Mit2) xJ'd4PV~ ~+1,2 ip-(y-X)6(p2 + t,2) =W(Y - x )JdE12 Pn(Fs2). Integrating and using Eqs . (19 .2.16)and (19 .2.17)gives 1tnm VAC = i Pn(Yz m Eqs.(19.2.31) and (19.2.32)can be reconciled only i f Pn(P?)trtm(om( O) ~VAC M(19.2.33) (This is real for Hermitian fields 0,,because in this case Eq . (19.2.1)requires tn,to be imaginary .) Thus as long as the symmetry is broken, Pn ([t) cannot vanish, but rather consists entirely of a term proportional to 6(µz) . Such a term can obviously only arise in a theory that has massless particles, because otherwise the spectrum of center-of-mass squared energies -p~ would not extend down to zero . Furthermore a delta-function S(µ) can only arise from single particle states of zero mass ; multiparticle states 172 19Spontaneously Broken Global Symmetrie s would contribute a continuum extending down to µz = 0 . The state 0n(0)IVAC} is rotationally invariant, so (IV~iO,(0)VAC )must vanish for any state Nof non-zero helicity . Also (VACJJ OIIIT~ vanishes for any state N that has different intrinsic parity or (unbroken) internal quantum numbers from JO.We conclude then that a broken symmetry with tn„z(Oh(0}yvAC =k0 requires theexistence of a massless particle of spin zero and the same parity and internal quantum numbers as AThese are our Goldstone bosons . The above argument breaks down when the spontaneously broken symmetry is a local rather than a global symmetry . Either we choose a Lorentz-invariant gauge, such as the Landau gauge with a,,Ay= 0, in which case as shown in Section 15 .7 the positivity assumptions of quantum mechanics are violated, or we adopt a gauge like the axial gauge with A~ - 0, in which case the ordinary rules of quantum mechanics apply but manifest Lorentz invariance is lost . As we shall see in Chapter 21, this exception is not just a technicality ; spontaneously broken local symmetries do not lead to Goldstone bosons . It will be useful to look in a little more detail at how the coefficient of the delta-function in p N(,u2)is related to the properties of the Goldstone boson . For a spin zero boson B of four-momentum q 11, Lorentz invariance requires the matrix element of the current between the vacuum and one- particle states to take the for m (VACJ .I'(x)j B~ = iFpBetpy 'x (27c)3/227B(19.2.34) where PB is the momentum of B, p 0B_ IpB I and F is a constant coefficient with the dimensions of energy . (This is consistent with current conservation because psa p~ = 0.) Also, the matrix element of the scalar fiel d between a single-particle state and the vacuum is of the for m (BIOn(AO ~ _Zne`Pe-v /:;:;;;-(19.2.35) where 2n is a dimensionless constant . From Eqs . (19 .2.34) and (19 .2.35), we have the n (27r)-'iPn( -P2 )P1O( PO ).1 `~-'N( vAC~ .r'{o}IB~ ~BIOO)P6 4( p- PB ) = 5(P° - ~P~)(27c)-'(2p°)-ipAF2n so MY 2)=Fzn6(,u'} . (19.2.36) 1 9.2Goldstone Boson s Comparing with Eq .(19.2.33), this give s iFZn = -- T, tnm (Om(0)VAC173 (19.2.37) More generally, we may have several broken symmetries with generators t,, and currents .Ia, which we can take independent in the sense that no linear combination of the to is unbroken . For each of these, there is a Goldstone boson IBp ), and we define Z,,, and Fabby FdeipB- x (VACJ .I~(x)jBb~ `~~p~ , (27c)3/2 26 AlOn(.y)IO~ Eq.(19.2.37) applies for each a, sofane"iPB'Y Pn)31'` 2P a t1: FahZbn - - X : [tulnm(Om( O)~VAC b m(19.2.39} (19.2.40) For instance, in the O(N) example discussed earlier, we can adapt our basis so that the vacuum expectation value points in the one-directio n Om = (Om(OVAC = v41 - ( 19.2.41) The IV-1 broken symmetry generators ta(with a = 2    N)can be defined as those for infin itesimal rotations in the 1 -a plane . With a convenient choice of normalization, these have the non-zero element s [talIa = - [tulul = l (19.2.42) (with no summation over a) . The unbroken a (11T - 1)symmetry, under which the N- 1 Goldstone bosons transform according to the vector representation, tells us tha t dab = 6a6F Then Eq .(19.2.40) requiresZul = ❑, F2--v. (19.2.44) It is conventional to adopt the field renormalization prescription that Z= 1, so F=v. Thus F is a measure of the strength of the symmetry breaking . As we shall now see, the parameter 1/F determines the strength with which the Goldstone bosons interact with each other and with other particles . A broken symmetry tells us more about the Goldstone bosons than just that they have zero mass ; it also tightly constrains their interactions at low energy . To see this in the simplest case, consider the matrix element o fZab= 2bRb. (19 .2.43) 174 19Spontaneously Broken Global Symmetrie s 0 a Figure 19.1.Feynman diagrams for pole terms in the matrix element of a symmetry current Jl'(x) between general states a and (3, due to a Goldstone boson internal line, denoted n . the current .It(x) associated with a broken symmetry . Between arbitrary states a, ~ (PIP(x)j~x) =eiq-x (pjy(0)11~ ,(19.2.45) with qy° Px -P~  ( 19.2.46) We know that P`(x)has a non-zero matrix element between the vacuum and a one-Goldstone-boson state ~B,q~,given by Eq .(19.2.34).Itfollows by the usual rules of polology (see chapter 10)that the matrix element (19.2.45) has a pole at q2__+0, with * where i (271)464(pa -- p#- q)Nf#a1{271}312(2p°)1/2 is the S-matrix element for emitting a Goldstone boson of four-momentum q in the transition a Let us therefore write iFqA #a M P where N~1is defined as the non-pole contributions to the matrix elemen t of the current . From Eq . (19 .2.45), we see that the conservation la w Where the Goldstone boson corresponds to an elementary field, Eq . (19 .2.47) may be obtained by inspection of the class of Feynman diagrams shown in Figure 19 .1. The factor i(27r )¢ in the S-matrix element for a -- ), P+ B is cancelled by the factor -i(21c)-4 associated with the B propagator . The general rules of polology tell us that the same holds even where the Goldstone boson is a composite particle . 1 9.2Goldstone Bosons 1 75 q q (a) (1)} Figure 19.2. Feynman diagrams for pole terms in the matrix element of a symmetry current Jy(x) between general states a and fl, due to internal lines close to their mass shell . Solid lines indicate `hard' external particle ; wavy lines indicate insertions of the current P(x) ; the cross-hatched disk represents the sum of diagrams with the indicated external lines . 0AP1 = 0 for the current P` requires (19 .2.48) to vanish when contracted with qu, and so MP =FquNfla(19.2.49) One immediate consequence is that unless IV',has a pole at q -- -+0, the matrix element Mfl,, for emitting a Goldstone boson in a transition a vanishes as q ---+0. This is called an "Adler zero .'6 In fact, it often happens that IV "does have a pole at q = 0 . This is because the vertex for the current PI(x) might be attached to an external line of the process a ~ fl. (See Figure 19.2.)For instance, if a four- momentum qu is carried away by a current Pinserted into an outgoing or incoming particle line of four-momentum p and mass m, then the internal line that connects this vertex to the rest of the diagram will carry a four-momentum p P+ qAor p "- qA respectively, and its propagator will 176 19Spontaneously Broken Global Symmetrie s therefore contribute to 1V~aa facto r [(p ± q )'+m2] -' = [f 2p  q + q2] -1 ---+ ±2p 19' (19 .2.50) For a fixed d irection of q ,the factor 1 / Iqlfrom Eq .(19.2.50) cancels the factor Iqlin Eq . (19.2.49), yielding a finite (though q-direction-dependent) result in the limit ~q~ --+ 0. On the other h and,the contribution to NI',, of d iagrams in which the current Pis attached to an internal line of t 9e process a - 4# has no singularity for Iql ---+ 0,and therefore is cancelled bythefactor ~ q~in Eq . (19 .2.49).Thus Eq . (19 .2.49) can be interpreted to mean that theamplitude for emitting a very soft Goldstone boson in a process a ~#can be calculat edfiratra graphs in whi chthe Goldstone boson isemitted only from the e xternal lines of the process ,with thevertex for emitting the Goldstone boson given b yapplying Eq .(19.2.49)to the transition between single-particle states . The effe ctive action formalism can be used to derive interesting results for the interactions of the Goldstone bosons with each other and w ith other scalars . For this purpose ,we note that if we define a set of renormalized Goldstone boson fields Kafor whic h e-iPB xbab 32,~~ V then Eq .(19.2.39) tells us tha t On(X) _ Y~Zan 7 ca(x)+... , (19.2.52) where ` ...' indicates fields that do not create Goldstone bosons . But Eq. (19 .2.40) gives Zan = EbFab'(itb0)n. Thus the amplitude for any reaction among Nzero-four-momentum Goldstone bosons 7c,,l,... ,7caN, is the same as would be calculated in the tree approximation from the effective interactio n with ONV(0) I 1 ! ga1...aNFalbl..FRntbnt(itb10)nj.. (ZtbNO)YiN (19.2.54) Eq. (19 .2.4) at 0 then implies that the sum of all "tadpole' graphs for a single Goldstone boson line disappearing into the vacuum vanishes, and Eq . (19 .2.6) tells us that the amplitudes vanish for Goldstone bosons to make transitions at zero four-momentum into any other scalars . To go beyond these results, we can continue to differentiate with respect to 1 9.3Spontaneously Broken Approximate Symmetries 17 7 the scalar fields . For instance, the derivative of Eq .(19.2.5)gives for any symmetry generator t : 1:a' V(O) ~'v(O) tn~+~nOOnOd~~a,V(O) tnm +F- 00"OO mOO" n,ktnkOk = a .(19.2.55) Taking t as one of the broken symmetry generators tu, setting contracting with (tbo)rn(t,~)z, and using Eq .(19.2.6)give s I:a3v~~} nm~(ta~Mtb4m(tckl= 0, (19.2.56) so the sum of all graphs with three external zero-four-momentum Gold- stone boson lines vanishes . In particular, this means that in general processes, to leading order in small Goldstone boson energies, low energy Goldstone bosons are not emitted from external low energy Goldstone boson lines . 19.3 S pontaneouslyBrokenApproximate Symme tries In the previous section we dealt with exact symmetries of the action that do not leave the vacuum invariant, and are said to be spontaneously broken . We shall now consider the effect of adding small symmetry-breaking terms to the action in such a theory . Such spontaneously broken approximate symmetries are important in the theory of strong interactions, and in some areas of condensed matter physics . As we shall see, the spontaneous breakdown of an approximate symmetry does not lead to the appearance of massless Goldstone bosons, but of low-mass spinless particles, often called pseudo- Goldstone bosons .7 We continue here to treat translationally invariant theories, in which the effective action is expressed as in Eq .(19.2.8} in terms of an effective potential V(O) that depends on a set of spacetime-independent scalar field expectation values din . For an action that obeys some set of approximate continuous symmetries with generators t, the effective potential may be writte n where Va{ O} satisfies the invariance condition * (ta~nrn orn = 0 nm ~~n(19.3.1) (19.3.2) We denote general symmetry generators as t ,,tfl, etc., in contrast with the independent broken symmetry generators, for which the subscript a takes values a, b, etc . 178 19Spontaneously Broken Global Symmetrie s and VI(O) is a small correction due to the symmetry breaking in the action . Suppose that this perturbation shifts the minimum of the potential from pia, the minimum of Va (O), to~= Oa + 01, where iii is small, of first order in the symmetry-breaking perturbation . The equilibrium condition for the vacuum is then 001, ~0=00+0 1 The zeroth-order term on the left is just [ 0Y0(O)10On]which vanishes because 00is defined as the minimum of Va(O) . Thus the first-order terms must also vanish, and so : a, Vowdim +yyz 00n4m 0_00~ V, (0) =o.00n 0=0a(19.3.4) Eq.(19.2.6)holds here with Vreplaced with the invariant term Vo and replaced with 00: a'`Va(O) nd(ta)rIr001= 0. (19.3.5) Hence multiplying Eq . (19 .3.4) with (t,O4)n and summing over n give s (t,XOO)n ~ ~~~)= o. on0=0o(19.3.6) Recalling the interpretation of V(O) as the generating function for one- particle-irreducible graphs, and noting that in the absence of the pertur- bation Vi the Goldstone components of On are those in the direction of taOo (see Eq .(19.2.33)), the left-hand side of Eq .(19.3.6)is proportional to the sum of all `tadpole' graphs, in which a pseudo-Golds tone boson disappears into the vacuum . Eq .(7 9.3.6)may thus be paraphrased as the condition that, to first order in VI, pseudo-Goldstone bosons have no tadpoles . The moral of this calculation is that if we do not start with a zeroth- order vacuum expectation value that satisfies Eq .(19.3.6),then even a small perturbation will produce a large change in Vii, invalidating the expansion of0around 00. Fortunately, for compact Lie groups it is always possible to choose 00to satisfy Eq . (19 .3.6). To see this, note that the invariance of the potential V4( O)under a group of real linear transformations 0 --+ LO implies that if 0.is one minimum of the potential, then so is LO .. For continuous groups of transformations, we can always parameterize the transformations as L(O), in such a way tha t a0a 19.3Spontaneously Broken Approximate Symmetries 179 where IV ,,bis a non-singular matrix depending on the group parameters O.(Recall that in a real representation, it is its rather than to that is real.) Now consider the function Vj{L(U)O* } . Where the group is compact, L(0)0 . maps out a compact manifold as 0runs over the group volume, and as long as VI(O) is continuous, it must have a minimum on any such compact surface, say at a point L(B .)O.. The derivative of Vj(L(0)O .) with respect to Bx i s 0V,(L(0)0.)_~ nrV,(r~)N~#(0)(it#L(B)~i.),j. (19.3.8)00200nO=L(e)0. This must vanish at the minimum 8 ., and since IITab is non-singular, this implies that aV,(0)o (tpL(B.)O*)n. (19 .3.9) But then Eq . (19 .3.6)is satisfied if we make the choice Oa = L(O.)O.. Eq.(19.3.6)is known as a vacuum alignment condition,' because it gen- erally has the effect of forcing the direction of the symmetry breaking by the vacuum into some sort of alignment with the symmetry-breaking terms in the Hamiltonian . For instance, consider the case of SO(N) spon- taneously broken to SO(N -1),introduced in the previous section . In the absence of any symmetry-breaking perturbation, there is no way to tell which SO(N -1)subgroup is left unbroken ; if the dynamics of the theory leads to a ground state that is invariant under the SO(N -1) subgroup of SO(N) that leaves some N-vector Oon invariant, then by performing an SO(N) rotation we can find a ground state that is in- variant under the SO(N -1)subgroup that leaves any other 11T-vector invariant . If we add a perturbation that transforms under SO(N) like, say, the component En uO,t of an SIT-vector On (not necessarily consisting of elementary scalars), then the Hamiltonian is invariant under a spe- cific SO(N -1)subgroup of SO(N), consisting of rotations that leave the vector u invariant . Without a vacuum alignment condition, we might think that the remaining exact symmetry is SO(N - 2), consisting of those rotations that leave invariant both u and the vector pia character- izing the vacuum symmetry . But with VI(O) = E n unpin, the condition (19.3.6)tells us that at the true vacuum, E,,(txOo),,u, = 0 for all SO(N) generators t, . The S0 (11T) generators t . span the space of all antisym- metric NxNmatrices, so this condition requires that 00must be in the same direction as u, and so the unbroken symmetry is S O(N -- 1),not SO(N - 2) . According to the general results of Section 16 .1, the mass matrix 1Vfab 180 19Spontaneously Broken Global Symmetrie s of the pseudo-Goldstone bosons is given to first order b y M2 ab =Y,ZdnZbmMnaomaon 0=00+01(19.3.10) where Zan is the field renormalization constant defined by Eq .(19.2.39). Since the mass matrix (19.3.10)vanishes in zeroth order, the first-order terms give a,VowM~b=~7,Zun2bm Mn0zvl{0}Ole + 0=0flaomaon(19.3.11) where 2u„ is here given by the zeroth-order approximation to (19 .2.40): Zan - ~ Fabl(400) M b(19.3.12) To calculate the mass matrix (19.3-11), we take t in Eq .(19.2.55) to be one of the broken symmetry generators ta, set 0=00, and contract with {tbOp}m Oll: a3Vow0 = E 0l~4taOO)n( tbOO) m nm( d(o) 1: 0- VOW (tu~jl)n(tb~ja )m+ 02 V nm a0n4m0_00n~NnaO~0-00(tutbOa)nOiz' The second term on the right-hand side vanishes according to Eq .(19.2.6), while the third may be rewritten using Eq . (19 .3.4), leaving us wit h Ea, Vow nme 0010o meo n 0-00OIAtaM nNO Q)m =0VI(O)(tatb OO) n eon 0=duo (19.3.13) Using this in Eq .(19.3.11) then yields a formula for the pseudo-Goldstone boson mass matrix in terms of V I: McdE Fca'Fdbl (taOa)n(tbOa)rn02 V 1(0) abaomaon0=0 o +(tatbOO)n vl(0) . eon0=ran(19.3.14) For this to be a sensible mass matrix, it had better be positive . To see that it is, it is convenient to rewrite this result in terms of derivatives with respect to the group parameters O.Differentiating Eq .(19.3.8)with respect to Bg,setting 0= B.,and using Eq .(19.3.9)and O4 = L([]*)O . 19.3Spontaneously Broken Approximate Symmetrie s gives ma6=ENaa, lay) Nb~(a.)Fa'Fbd' cda/3O2VI (L(8)0*) 00000fl 0=0.181 (19.3.15) The matrix on the right is positive, because 0* is the minimum of the function Vi(L(B)0* ). This formula has a somewhat more familiar version, in terms of the vacuum expectation value of a double commutator of symmetry generators with the symmetry-breaking perturbation . Suppose that the symmetry- breaking perturbation H1 in the Hamiltonian is a linear combinatio n HI= Eun(Dn ( 19.3.16) of operators On (not necessarily elementary scalar fields), which furnish a representation of the symmetry group with generators ta, in the sense tha t [T, On I =-(ta) nmOm , ( 19.3.17) where ?'x are the quantum mechanical generators of the symmetry group . According to the results of Section 1 6.3,the symmetry-breaking part of the potential is Vi(O)=(Hi~(O)-O_Y:unOn , (19.3.18) n the subscript in the middle expression indicating that the expectation value is to be taken in the state of minimum energy in which (Dn has expectation value 0,The vacuum alignment condition (19.3.6)then read s 0 un(ta04)n or using Eq .(19.3.17) 0= (Vot, H&o , (19.3.19) the subscript 0now indicating that the expectation value is to be taken in the vacuum state, in which (Dn has expectation value q5on. Also, Eq. (19.3.14) gives the mass matrix here a s Mc'd - - X:Fca1Fdb1 E Un(ttb 4o )n ab n and,using Eq. (19 .3.17), thisis M r d=_EFca1Fdb1 ff7'a~[T6,H1]]~o. (19.3.20) ab This is symmetric in c and d . To see this, note that the Jacobi identity and group commutation relations may be used to write the difference of Eq .(19.3.20) and the same with c and d interchanged as a linear 182 19Spontaneously Broken Global Symmetrie s combination of terms ffT,, HI]~0, which vanish according to the vacuum alignment condition (19.3.19).The mass matrix (19 .3.20) is also posi- tive, because the point 0 = 0 is at the minimum of the vacuum energy (exg(-iD ,,Tu)Hl exp (iB,TR) y~ for rotated vacuum states exp(iB,, T,,) Ia~. 19.4 Pions as Goldstone Boson s The classic example of a broken symmetry in elementary particle physics is the approximate symmetry of strong interactions known as chiral SU(2) xSU(2) .According to our present understanding, this symme- try arises because there are two quark fields, uand d,that happen to have relatively small masses . (An estimate is given in Section 19.7.)In the approximation that the uand dare massless, the Lagrangian (18.7.5)of quantum chromodynamics i s Y = ---RyPD, u - d yiD A d---- , (19 .4.1) where DPis a col or-gauge-covariant derivative (see Eq. (15.1.10})and ` ' ' refers to terms involving only gluon fields and/or other quark flavors, but not u or d . This Lagrangian is invariant under the transformation s ~ (19 .4.2) ~ ~ exp~i B~' ' t + i Y 5E~A.i ) where t 'is the three-vector* of isospin matrice s 1 d 1 1 0 -i 1 I 0 ~~ - 2 -I d t2 2 i 0 t3 2 0 - 1 and 6Vand BAare independent real three-vectors ." This Lie algebra may be written in terms of two commuting SU(2) subalgebras that act respectively only on the left- and right-handed parts of the quark fields, with generators tL= 1( 1+ y5}t , tR = -0 - y 5}t (19 .4.3)2 We use arrows for three-vectors in isotopic spin space to distinguish them from ordinary three-vectors, which will continue to be indicated by boldface letters , The Lagrangian (19 .4.1) has this symmetry because y5y~y~ =-qiY5Y1` =+V)Y~'yS. The Lagrangian also has two other continuous internal symmetries . One is baryon conservation, the invariance under a common phase transformation of the u and d quark fields . This is unbroken and commutes with the other symmetries, so it does not affect our discussion in this section . The other symmetry is invariance under multiplication of the quark doublet with cxp(icx75) . As discussed in Section 23 .5, this U(1) symmetry is strongly intrinsically broken by non-perturbative effects associated with instantons . 19.4Pions asGoldstone Boson s satisfying the commutation relation s ItLi,tLj] =t E'ijktLk , [tRi,tRj I= 1 E 'i jk tlZk [tLi,tRj] =d.183 (19.4.4) (19.4.5) (19.4.6) The underlying symmetry group is therefore identified as SU(2) xSU(2) . It has another obvious SU(2) subgroup, consisting of ordinary isospin transformations with 0A= d, and generator s t=tL+tR . (19 .4.7) The algebra of SU(2) xSU(2) may be written in terms of t and another triplet of generators : x=t L-tR=Y 5t with commutation relations [ti, t.iJ = iei jk tk [ti, xjJ = iFijk Xk 1xr,X.1] =tEi fk tk(19.4.8) (19.4.9) (19.4.10) (19.4.11) We will see that the SU(2) xSU(2) symmetry is spontaneously broken, while its isotopic spin subgroup generated by the t 'is an ordinary unbroken (though approximate) symmetry . By Noether's method (see, e .g., Section 7.3)we may derive from the Lagrangian (19 .4.1) the conserved vector and axial-vector current s VI= ig y"`tq Au=igY"Y5t4~ (19 .4.12) ~ Y 0 it VP0~1AP==0 where q is the quark doublet, 4 u=d(19.4.13) (19.4.14) Their associated charges are the generators respectively of isospin and of the remaining symmetries 3X - 0T=fd V (19.4.15) X _ fd3xA° . ( 19.4.16) The currents (19 .4.12) are normalized so that the quantum operators T and X satisfy the same commutation relations as the matrices t and x : [Ti,Til= 1 Fi jkTk, (19.4.17) 184 19 Spo ntaneo uslyBroken Global Symmetrie s [ Ti, X.1j- ZE 'ijkXk [Xi,x1j = i EijkTk(19.4.18) (19.4.19) Acting on the quark fields, these operators induce the transformation (19.4.2), in the sense that [1'.q, = -t q , (19 .4.20) ~X, q~ _ -x q  (19 .4.21) This symmetry if exact and unbroken would require any one-hadron state jh~ to be degenerate with another state X~hy of opposite parity and equal spin, baryon number, and strangeness ." No such parity doubling is seen in the hadron spectrum, so we are forced to conclude that if the chiral symmetry SU(2) xSU(2) is a good approximation at all, then it must be spontaneously broken to its isotopic spin SU(2) subgroup . In this case the operator X takes a one-hadron state jh~ into a hadron h plus a massless pseudoscalar Goldstone boson, so there is no need for parity doubling of the hadron spectrum . The question of whether quantum chromodynamics actually exhibits such a pattern of symmetry breaking involves all the complications of strong interaction dynamics . As we shall see in Section 19 .9, there are general grounds for believing that the isospin S U(2) is not spontaneously broken in quantum chromodynamics, but it is much more difficult to show that the chiral part of S U(2) xSU(2) is spontaneously broken . (But according to an argument given in Section 22 .5, the S U(3) xS U(3) symmetry of quantum chromodynamics with three massless quark flavors must be spontaneously broken .) It was something of a breakthrough in the 1960s to realize that one does not have to have a detailed understanding of the mechanism of the breaking of chiral symmetry ; we derive the most interesting consequences of this symmetry breaking by simply assumi ng that S U(2) x SU(2) is spontaneously broken to S U(2) . The uand dquarks have small but nonzero masses, so the S U(2) x SU(2) symmetry is not exact . A broken approximate chiral symmetry entails the existence of an approximately massless Goldstone boson with the same quantum numbers as the broken symmetry generator k : it must be a state of negative parity, zero spin, unit isospin, and zero baryon numbe r t one way to satisfy this condition is for the hadron to have zero mass, with two states ~±} of helicity ±~ and equal spin, baryon number, and strangeness ; the states I+}+~-} and 1+)-I-)would then have opposite parity . It is not true that an unbroken chiral symmetry necessarily implies a zero nucleon mass, unless we make further assumptions about the matrix elements of the axial-vector current . But as we shall see in Section 22,5, an exact unbroken chiral symmetry would in fact require that some baryons be massless . 19.4Pions asGoldstone Bosons 185 and strangeness .In fact the lightest of all hadrons is the pion, which has precisely these quantum numbers, so we are led to identify the pion as the Goldstone boson associated with the spontaneous breaking of approximate chiral symmetry . As we shall see below, it is m~ rather than m,, that is proportional to a linear combination of mu and md, and m2 /MN- 0.022 is very small, so the consequences we derive from spontaneously broken SU(2) x SU(2) should be reasonably accurate . In exploring the consequences of chiral symmetry for pion interactions, it is very useful to note that although the chiral symmetry of the strong interactions does not depend in any way on the existence of weak in- teractions, the symmetry currents Y" and At' happen to be the currents entering into strangeness-conserving semileptonic weak interactions like nuclear beta decay . As we shall see in Section 21 .3, the standard model of electroweak interactions requires that the effective Lagrangian for these interactions at low energy must take the form : YWkiGw k ~ v+ +M1+ y5}v1+H.c.(19.4.22) ~r where e runs over the renormalized fields of the three charged leptons e, pand z ; veruns over the renormalized fields of the associated neutrinos ; and V+and A+are the charge changing current s V~ = Vl ± i Y2 , At = Al ± i A~ . (19.4.23) The constanttt GWk may be measured from the rates of beta transitions between states of zero spin within the same isotopic multiplet, such as the decays n+--+ no + e+ + Ve and 140__+14N* + e+ + Ve. The momentum transfer in these transitions is very small, so parity conservation (in strong interactions) and rotational invariance tell us that only the matrix elements offd3x V 0 = ?',- iT2 enter in the S-matrix elements for these decays . This operator has matrix elements between states in a given isospin multiplet that are just known Clebsch-Gordan coefficients, so from the rates for these `d --+ 0' processes we may calculate a value9 for the coupling inEq.(19.4.22): GWk ^ I .149 59(3$) x 10-5GeV -2 . On the other hand, in the process of pion decay, ic+ --- +Ic+ + v,,, the only current matrix element we need is the matrix element of A ~_between cone-pion state and the vacuum (VACJAi1'(x)J7iJy =rF,,$; jpEnetP~'x 2(2ir)3/2 Apo(19.4.4) GAis rela ted to the conventionalFermi coup ling consta ntGF and t he Ca bibbo angle Qcby GWk - Gr cos BC ; see Sec tion21.3. 186 19Spontaneously Broken Global S ymmet ries which is completely known except for the factor F,, . The rate for pion decay turns out to b e I'(rc --* p+ v) =G2F2m2!m2 -M22 wk F,2, µ[n ~ 167cm39(19.4.25) From the known rate for ic+ -4 Ic+ + v , F= (2 .6033(24) x 10-$ s)-1, and the above value of Gwk, one finds tha t F,-184 MeV . (19 .4.26) Now let us consider the matrix element of AP between one-nucleon states . This is of interest in its own right, and as discussed in Section 19 .2, it provides information we need to calculate the emission of low energy pions in collisions of nucleons . Following the same reasoning as used for the electromagnetic current in Section 10 .6, one finds that Lorentz invariance and parity conservation require this matrix element to take the form# (pJA+(x)ln1= (2n) -3e=qx xiip [- iyµyJ (42)+q"Y5g(42) + i 4v [YP, YvIYSh(q2)]un , (19 .4.27) where q = pn - pp .In the approximation in which the SU(2) xSU(2) symmetry is exact, the conservation of current requires tha t q,,(PJA°+(x)any=0. Using the defining equations for the Dirac spinors up and u, , iip(i ~p+mN )-(r ~n+mN )Un =d , we see that qpupHYF`Y51Un = -2m NUpY5Un and so Eq . (19 .4.28) requires that(19.4.28) (19.4.29) Ifg(q2) had no singularity at q2= 0, then (19 .4.29) would require that either MN=0,which is certainly not true, or that f(a)= 0, which is not true either . In fact, the quantity f (0)is measured in low energy nuclea r It is common to encounter a pion decay constant f ,, which in terms of the F, used here is variously defined as F ,, FIi2- , or F,/2 . The currents At in the standard model have charge conjugation properties that make the coefficient h(q l) vanish . It is possible that there are `second-class' tcrms10 in the weak currents with opposite charge conjugation properties that give a non-vanishing h(q2 ), but there is no evidence for such terms . As we shall see, keeping the h(qZ) term here has no effect on the inferences drawn from chiral symmetry . 19.4Pions as Goldstone Boso ns 18 7 betadecays, dike neutron decay, w here it is us ually cal ledgA;it is fo und tohave t he value f(O) = gA = 1.2573(28) . (19 .4.30) The fact that neither MN nor f (0) = gA is small requires that in the limit of exact SU(2) xSU(2) symmetry, g(q2) must have a poke as q2 - +0: 2mNgA2 (19 .4.31) 4 Such a pole is naturally provided by the massless pion that would be required by the spontaneous breakdown of an exact SU(2) xSU(2) sym- metry . Suppose that the pion coupes to the one-nucleon state as if the interaction Lagrangian wereq -2iG ,N~cNYStN . Eq . (19 .4.24) tells us that the matrix element of the current between a one-pion state and . the vac- uum is the same as if there were a term in AF'(x) of the form FD/2 . Therefore in the limit q2 - + 0the matrix element (19 .4.27) has the pol e [()3 ]tq x [i(2ir2GN1 PA(x )jn~ ~ up~ry5 )un I (Zrc)q'[tqYF12]J Comparing with Eq . (19 .4.27), we see that one-pion exchange gives the function &2} a pole &2}_-+GgNF7r q2 (19 .4.32) for q2 --+0. Putting together Eqs . (19 .4.31) and (19 .4.32), we fin d GnN =2mNgA (19.4.33) F This is the famous Goldberger-Treiman relation ." It works reasonably well ; taking MN= (mp + mn)/2 = 938 .9 MeV, gA = 1 .257, and Fn = 184 MeV gives G ,N 1 2.7, in fair agreement with the value" G, 1N = 13.5 measured in various ways (including the effects of the one-pion poke in nucleon-nuclear scattering and the one-nucleon pole in pion-nucleon scattering .) In the real world the pion is not massless and the S U(2)x S U (2) symmetry is not exact (even before being spontaneously broken .) This circumstance may be analyzed using the general formalism presented i n This is the conventional definition of the pseudoscalar pion-nucleon coupling G . N. The factor 2 is introduced here to cancel the 1/2 in the isospin matrices . The textbook" value is C ,,Nl4rc=14.3, or G ,h = 13 .4. More recently, a high precision study13 of neutron-proton charge-exchange scattering at 162 MeV has given a value GnN /4rc =1 .4.6 ± 0 .3, or GnN - 13 .5. 1 88 19 Spontaneously Broken Global Symmetrie s the previous section . The Lagrangian (19 .4.1) yields a symmetry-breaking term in the Hamiltonia n Hl -= muuu + m ddd = (mu + m d)(D4+(mu - m d)[I~3 (19.4.34) where 04~(uu+ dd) , ~3 = z(uu - dd) . (19.4.35) The operators 04 and ~3 are spatial scalars and, as this notation is meant to suggest, they transform under S U(2)x S U (2)as components of independent chiral four-vectors (DI(X (D+ = agYJt4 (D=qtq ,(D4=zq~ (19.4.36) (D4 = - zi 4YA (19.4.37) These are chiral four-vectors, in the sense tha t [i.~n J= I X'(DJ~1:0- nm~ 0tm rn 0) M in(19.4.38) (19.4.39) where 9-and Xare Hermitian 4 x 4 matrices that furnish the four-vector representation of the algebra of S U(2)x S U (2)- 544 ): (9-0 br = - i Eabc ~ (9-a)b4=(9-u)4h = (-'7_rx)44 = 0 (19 .4.40) (Ya)b4 = - (?Ta)4b = -i 6ab, (Xc)ab = (Xc)44 = 0. (19 .4.41) This notation makes it easy to see that the vacuum alignment conditions for generators Ti, T2, X1, X2, and X3 respectively take the for m We have been assuming that in the absence of uand dquark masses, the symmetry S U(2) xSU(2) is spontaneously broken in such a way as to preserve unbroken the SU(2) symmetry generated by T as well as parity, in which case ((Dn yQ points in the four-direction and (~n y o= 0 vanishes, so that the conditions (19 .4.42) are satisfied . But with mu =and = 0,we can find other symmetry-breaking solutions with other definitions of parity by subjecting this one to an arbitrary SU(2) xSU(2) transformation . Thus in the absence of u and d quark masses, there would be no way to tell in what direction ((Dn ~~ = 0 should point, or which SU(2) subgroup of S U(2) x S U (2) is left unbroken, though in all cases ((Dn ~o = 0.The vacuum alignment condition (19 .4.42) tells us that, with the symmetry broken intrinsically by the perturbation (19 .4.34) and spontaneously in such a way that (cbn ~~ = 0,the vacuum must `line up" in such a way 19.4Pions as Goldstone Bosons 189 that ((D+ yo = 0points in the four-direction, so that the unbroken SU(2) symmetry is ordinary isaspin . This formalism may be used to calculate the pion mass . From Eqs . (19.4.39)-(19 .4.41) we fin d Wa, [Xb,(D41 I = auh ~4 [Xa, [Xh,~311= (D a43. 19.4.43) Also, isospin invariance tells us that the symmetry-breaking parameter Fah introduced in Section 19 .2 is proportional to bab, with a proportionality factor that according to (19 .4.24) is just F,/2, s o Fab = babFz/2. Eq.(19.3.20) thus gives the pion mass matrix a s Ma=6"hmzab R where m~ = 4(mu + md )((D 4)~IF~(19.4.44) (19.4.45) (19.4.46) It is striking that the masses of the charged and neutral pions turn out to be equal, even though we have made no assumpt ion about the ratio of mu and m j. We shall see below that th is ratio is not near unity ;isospin is a good quantum number not because the uand dquark masses are nearly equal, but because they are small .Also, as prom ised, we see that it is the square of the p ion mass that is proportional to the quark masses, so the quark masses should be qu itesmall .(See Sect ion 19 .7 for an estimate .) The observed p ion mass d ifference arises not from the u -d quark mass difference, but from electromagnetism . Indeed, the p ionisospin multiplet is the only one whose mass d ifference has been successfully calculated on the basis of one-photon exchange alone .14 Of course, even with quark masses taken into account, Eq .(19.4.24) can still be used to define F,; its divergence give s F,,6~m2 eiP~ x (VAC JOPA~(x)I~j~ = ~2(27c)3/2 2po(19.4.47) Instead of assuming that r71, Allvanishes, we can now assume that it is small, of order mom, except where a pion pole compensates for the smallness of mn . According to Eq . (19 .4.47, the behavior of matrix elements of a,'A near aone -pion pole is the same as if 0A~ were Fm~ times a properly renormalized pion field . For instance, the one-nucleon matrix element of auA+ should be 1~ ~"r 4(pl01,A~+(d)1n1-2 (27T)4(q2 + ~ri~[i22GNup (iy5)Un11 (19.4.48) 190 1 9Spontaneously Broken Global Symmetrie s so in terms of the form-factors in (19 .4.27) q+ m, :(19.4.49) This is expected to be valid when q2 is of order mom, and not only in the limit q2 _--+-rra2,because the pion pole dominates matrix elements of a,,Au for all such small q2 . Also, for such q2, in place of Eq . (19 .4.32) we hav e GIN F 7r q2+ M71(19.4.50) From Eqs . (19 .4.49) and (19 .4.50) we find that for q2 of order m2 g, J'(q2)-_G7rNFn /2MN - It should be no surprise that this function is roughly constant over the range of q2 from zero to of order mom, because it has no one-pion pole, and there is nothing else that could give it a substantial variation in such a small range of q2 . This constant value is approximately f {0} = gA, and Eq. (19 .4.51) thus again yields the Goldberger-Treiman relation . We can now use the results of Section 19 .2 to calculate the amplitude for emission of a single low energy pion in an arbitrary process a --+ P. We found that the amplitude is to be calculated from Feynman diagrams in which the pion is emitted only from the external lines of the process, and Eq. (19 .2.49) shows that these contributions are to be calculated as if the pion field interaction were 0uai.ANIF,in which the subscript N indicates that we are to drop the one-pion pole term in matrix elements of the axial-vector current . From Eq . (19 .4.27) (and using isospin invariance) we conclude that for soft pions emitted from a nucleon line this interaction is effectively --- F~ r~~ li~rNiP YStN . Using the free-particle Dirac equation, we the mass shell (that is, at the one-nucleon equivalent to the pseudoscalar interaction provides yet another demonstration of the (1g.4.33). ***can see that for nucleons on poles in Figure 19.2)this is -2iMNgA rc ' 1V 75NI F,,.This Goldberger-Treiman relatio n Our discussion here has not paralleled the historical development of these ideas . In fact, the course of historical development was chronologi- cally almost exactly opposite to the line of argument presented here . Bro- ken symmetries in particle physics started with the Goldberger-Treiman relation (19.4.33),which was derivedll in 1957 on the basis of a dynamical 19.4 Pions asGoldstone Bosons 191 calculation of pion decay . In order to explain the surprising success of this very approximate calculation, several thearists 15 introduced the idea of a `partial conservation of the axial-vector current' (PCAC), the idea that although the axial-vector current is not conserved (as shown by the fact that pions do decay) its divergence 0 uA+ is proportional to the pion field. In itself, this assumption is meaningless we saw in Chapter 10 that any field with a non-vanishing matrix element between the vacuum and one-pion states may be regarded as a pion field . Although it was not clear at the time, what was really being assumed was that the divergence of the axial-vector current is small, of order rra2,except where a pion pole gives it a large matrix element . The problem was greatly clarified by a 1960 paper of Nambu,l6 who pointed out that the axial-vector current could be regarded as exactly conserved in the limit of zero pion mass, in which case the Goldberger-Treiman relation could be derived as we have done here . In this and a subsequent paper with Jana-Lasinio,l7 Nambu recognized that the appearance of this massless or nearly massless pion was a symptom of a broken exact or approximate symmetry . With other collaborators, 18 Nambu also showed how to calculate the rates of emis- sion of a single soft pion in various processes . Subsequently Goldstone3 remarked that broken symmetries always entail massless bosons, and this was proved in 1962 by Goldstone, Salam, and myself,5 using the arguments presented here in Section 19 .2. None of this early work on pions as Goldstone bosons depended on any specific assumptions about the nature of the broken symmetry group ; for instance it might have been a direct product of three commuting U(1) groups, whose generators form an isotopic spin vector, or it might have been the non-compact group SO(3,1), for which a minus sign appears on the right-hand side of the commutation relation (9 .4.19). The nature of the broken symmetry group became important only with the consideration of processes involving more than one pion, starting with the Adler- Weisberger sum rule in 1965,19 whose success showed that the broken symmetry is indeed S U(2) x S U(2) . (Such processes are discussed in the following section .) The identification of the symmetry group S U(2) x S U(2) led to a shift of emphasis,20 toward the implications of this symmetry within strong interaction physics, and away from an earlier concentration21 on the currents themselves . All of this work was done without a specific theory of the strong interactions . One of the reasons for the rapid acceptance of quantum chromodynamics in 1973 as the correct theory of strong interactions was that it explained the S U(2) x S U(2) symmetry as a simple consequence of the smallness of the u and d quark masses . 192 19 Spontaneously Broken Global Symmetrie s 19.5 E ffective Fie ldTheories : Pions a ndNucleons In Section 19.2 we learned how to calculate the amplitude for emission of a single low energy Goldstone boson B in a transition a --* P +Bby applying the condition of current conservation to the matrix element of the symmetry current between the states a and fl. In this calculation we never had to use any information about the details of the broken symmetry group ; current conservation was all we needed . A new element enters if we wish to calculate the matrix element for the emission and/or absorption of two Goldstone bosons, as for instance in a Goldstone boson scattering process . Here we must apply the condition of current conservation to a matrix element like (PIT~al'(xl),J2 Z(X2)J la) where the states a and #contain the other particles besides the two Goldstone bosons participating in the reaction . But when we let the divergence operator 0/0xll act on this matrix element we encounter a non-zero contribution from the derivative of the functions 0(x° - x°) and B(x~ - x~ )that appear in the definition of the time-ordered product 7'~  ~ . This contribution is equal to the matrix element of an equal- time commutator S(x~ - x~}[J~ (xl),J~Z(x2)],whose value depends on the commutation relations of the group algebra . This makes such multi- Goldstone-boson processes specially interesting, because they can be used to decide experimentally on the nature of the broken symmetry group, in a way that is not possible for processes involving just a single Goldstone boson . Because of the appearance of such current commutators, this approach is known as the method of current algebra'21 The current algebra method was used in early calculations of multi- Goldstone-boson amplitudes .22 Unfortunately, such calculations are ted- ious, especially when three or more Goldstone bosons are involved, and it was also difficult to see how to deal with symmetries like the chiral sym- metry of quantum chromodynamics that are not exact . For this reason a different and more physical calculational technique was introduced, 23 based on the use of effective Lagrangians : We simply calculate the Gold- stone boson amplitudes by the methods of perturbation theory, using some Lagrangian for the Goldstone bosons and the other particles in the states a and Pthat obeys the assumed broken symmetry . Originally the justification for the effective Lagrangian procedure was based on current algebra . By using current algebra one could see that the amplitude for emission of a set of low energy Goldstone bosons in a process a --+ # + B1 +B2 +    is fixed once one knows the equal- time commutation relations of the currents associated with the broken symmetries as well as the matrix element for the process a ~ ~B and the 19.5Effective Field Theories :Pions and Nucleons 19 3 matrix elements of the currents between various one-particle states . We know that a Lagrangian that respects the broken symmetry will allow the construction of conserved currents with the appropriate equal-time commutators by Noether's technique, so if we simply calculate the low energy Goldstone boson amplitudes with such a Lagrangian and insert the correct values for Mpx and the one-particle matrix elements of the currents, we must get the same answer as provided by current algebra . In the case of interactions among Goldstone bosons alone, the states a and Pmay both be taken as the vacuum, and we do not need any extra information beyond the matrix element F between a Goldstone boson state and the vacuum . In the first example of this sort, 23the starting point was the Lagrangian of the Q-model .24 Restricting our attention for the moment to the bosonic sector of this model, its Lagrangian is the 5O(4)-invariant one used as an example in Section 19.2: 2 2 4 where n is understood to be summed over the values 1, 2, 3, 4, with an isovector pseudoscalar field and 04 an isoscalar scalar field . The immediate problem faced by any sort of effective Lagrangian is that in order to use it to calculate scattering amplitudes, we must either include all Feynman diagrams of all orders of perturbation theory, or else find some rationale for dropping higher-order diagrams . We can find no such rationale with Lagrangians like (19.5.1),in which the broken symmetry is realized through linear transformations on the various fields . Fortu- nately any such Lagrangian can be recast in a form that allows the use of Feynman diagrams to generate an expansion for scattering amplitudes in powers of Goldstone boson energies . To do this, we perform a symmetry transformation at every point in spacetime that eliminates the fields cor- responding to the Goldstone bosons of the theory . The Goldstone boson degrees of freedom reappear in the transformed theory as the parameters of this symmetry transformation . However, since the Lagrangian is in- variant under spacetime-independent symmetry transformations, it cannot have any dependence on the new Goldstone boson fields when they are constant, and so every term in the Lagrangian that involves these new Goldstone boson fields must contain at least one space time derivative of the field . These derivatives introduce factors of the Goldstone boson en- ergy when we calculate S-matrix elements for Goldstone boson reactions, and as we shall see we can use the Lagrangian to construct a series for the S-matrix elements in powers of these energies . For instance, to recast the Lagrangian (19.5.1)in a useful form, we write the four-vector 0,as a chiral rotation R acting on afoot-vector ~0, 0, 0, (7) 194 19Spontaneously Broken Global Symmetrie s whose first three components (the Goldstone part of 0,,)vanish : On(X) = Rn4 (x)a(x) with Rn„,(x) an orthogonal mat rix andthereforeRT(x} R (x)=1 a(x) = ~n(x)2 . n The Lag rangia n(19.5.1) then becomes 4 2 ~Y:(&40,19 + aa,Rn4) n=1(19.5.2) (19.5.3) (19.5.4) (19.5.5)2 4 Because R is an orthogonal matrix, the OyaOf`aterm is R-independent, and the cross-term vanishe s Rna-1~n so Y becomes~ Rn4au Rn 4 =1 2 au Rn4=0 n n a 2 4 2 2n~](19.5.6) If'A2is negative then a has a non-vanishing vac uum ex pectation va lue, given in lowes torder bythe position of t he minimum of t he sumof the lasttwoterms, ata= Inplace of the fie ldvariables fin, o urvaria bles now a re or - I and whatever ot her varia bles are neede dto parameterize the rotation R . For instance, theseparameters cou ld be sim ply chosen as theRao themse lves (where a, ta,    f rom now a re isovecto rindices running ove rthe va lues 1,2,3), wi th R 44 given bythe con dition t hatRis ort hogonal. Adifferent parame terizatio nwillgive sim plerfinal results, and was historically the first tobeusedforthesepurposes.It is to define and tak e Ra4- sothatOaQ-04 + 2(a21---~ ~RIQ=Raa2(a~ I+(19.5.7) Rib = bab - ~~~~~ (19 .5.8) 1 + C 2 a y 041(T=R44 =+2~(19.5.9) 19.5 Effective Field Theories :Pions and Nucleon s Then the Lagrangian (19.5.6)become s ~ ajtd~~`a - 2 Q2D~  D~` - ~ Zaz _ 4 as where a'U Diz 1+z195 (19.5.10) (19.5.11) Whatever parameterization we use, it is clear that the fields describe particles of zero mass, whose interactions all involve field derivatives . These are (up to a normalization) our new pion fields . Despite appearances, this Lagrangian is still invariant under SQ(4), but with SO(4) now realized non-linearly . Under an isospin transformation with infinitesimal parameters 0,the field simply rotates like an ordinary isovector, and a is an isoscalar : 6C=Bx C, ba=0, (19.5.12) so the Lagra ngian(19.5.10) is ma nifestly isos pin-invaria nt. On the other hand,underthebroken symmet rytransfo rmations parameterize d by an infinitesimalvecto rF, the origina lfieldstransform accor dingto From Eq . (19.5.7), we then find(19.5.13) (19.5.14) The Lagrangian (19.5.10)is invariant under the broken symmetry transfor- mation (19 .5.14) because DYundergoes a linear (though field-dependent) isospin rotation :25 6DII=2(( x0 x DY (19.5.15) and Eq .(19.5.10) is isospin-invariant . Because of the transformation rule (19.5.15) , DYis often called the covariant derivative of the pion field . The transformation rules (19.5.12) and (19 .5.14) specify what is called a non-linear realization of the group S U (2)X S U (2).25 The general theory of non-linear realizations of Lie groups is given in the next section ; we shall show there that, up to field redefinitions, the transformation rules (19.5.12) and (l 9 .5.14) provide the most general realization of SU(2) xSU(2) in which the isospin SU(2) subgroup is realized linearly o n We see that each interaction of these new pion fields is accompanied with a spacetime derivative, so that the effective coupling is weak for low pion energies . (This remark will be made more precise below .) Therefore for sufficiently small pion energies we may use this Lagrangian in the tree approximation to reproduce the soft-pion theorems of current algebra . For 196 19 Spontaneously Broken Global Symmetries this purpose, it is only necessary that the Lagrangian be SQ(4)-invariant . But since the a field is an SD(4) scalar, it plays no role in maintaining the SO(4) invariance of the Lagrangian, and may be simply discarded .* Of course, this procedure changes the physical content of the theory, but it does not change the amplitudes given by soft-pion theorems . The Lagrangian ( 19.5.10)then simplifies t o 2 ' Pa YFD1D1Fa~~ O PC(19.5.16)2 2 (1 + 2) 2 where F = 2 (a) = (As we shall soon see, this F is the same as the constant F,, discussed in the previous section .) For many purposes it is more convenient to work with a conventionally normalized pion fiel d A-F for which the Lagrangian (19.5.16) reads(19.5.17) 2 1U c2 F~ ~(19.5.18)(+/) The factor 1/F acts as a coupling parameter that accompanies the in- teraction of each additional pion . Eq .(19.5.18) describes what is often called the `non-linear a-model,' for the special case of SU(2) xSU(2) spontaneously broken to SU(2) . An important point : to derive Eq .(19.5.18) it was not really necessary to start with the `linear a-model' Lagrangian (19.5.1).Indeed, we did not need to start with any specific theory . Eq .(19.5.18) can be used simply because it is invariant under the SO(4) transformation (19.5.12), (19.5.14), and current algebra tells us that this is all we need to get the right results for low energy pion reaction amplitudes . Some years after the introduction of effective Lagrangians for soft pi- ons, there emerged a different justification for the effective field theory technique,26one that does not rely on current algebra and allows calcu- lations that are not limited to the limit of vanishingly small Goldstone boson energies . It is based on the realization (not yet formally embodied in a theorem) that when we calculate a physical amplitude from Feynman diagrams using the most general Lagrangian that involves the relevant degrees of freedom and satisfies the assumed symmetries of the theory, we are simply constructing the most general amplitude that is consistent with general principles of relativity, quantum mechanics, and the assumed symmetries . This was the point of view underlying Volume I of this book . In the present context, the `relevant' degrees of freedom are the Goldston e Alternatively, we can pass to the limit where .ifandAgo to infinity together, keeping the expectation value of rr constant . 19.5 Effective Field Theories :Pions andNucleons 197 bosons themselves, together with the particles in the states a and Pand any other particle states that can be produced from them by interactions with low energy Goldstone bosons . By invoking this justification for effective field theories, we are freed from any need to wrestle with the complications of current algebra . More importantly, the modern effective field theory approach yields results that take us beyond the extreme low energy limit and allows a systematic treatment of any intrinsic symmetry breaking . According to this approach, in order to calculate pion interaction amplitudes to any desired order in pion energies, we must use the most general Lagrangian involving a pion field that transforms according to the rules (19.5.12) and (19 .5.14): F2 D. ~~` -C4(DYD`}2 - c4 (DY  D, . } ~D~1 Dv } - .... ~ 19 .5.19}2 ~ 4 4 The terms indicated by ... will contain higher powers of the covariant derivative D Y, or higher covariant derivatives, whose general structure is described in the next section . The coefficients c4 and c4 are dimensionless, and all higher terms have coefficients with the dimensionality of negative powers of mass . Consider a general process involving arbitrary numbers of incoming and outgoing pions . We suppose that their energies and momenta are all at most of some order Q, which is small compared with a typical quantum chramadynamics energy scale (say, the nucleon or p mass) . Even though Lagrangians like (19.5.19) are not renormalizable in the usual sense, we saw in Section 12.3 that such Lagrangians can yield finite results as long as they contain all possible terms allowed by symmetries, for then there will be a counterterm available to cancel every infinity . If we define the renormalized values of the constants F2 , c4, c4, ...by specifying the values of various Goldstone boson scattering amplitudes at energies of order Q, then the integrals in momentum-space Feynman diagrams will be dominated by contributions from virtual momenta which are also of order Q (because renormalization makes them finite, and there is no other possible effective cut-off in the theory .) We can then develop perturbation theory as a power series expansion in Q .a Each derivative and hence each D.in each interaction vertex contributes one factor of Q to the order of magnitude of the diagram ; each internal pion propagator contributes a factor Q-2; and each integration volume d4q associated with the loops of the diagram contributes a factor Q4;so a general connected diagram makes a contribution of order Qy,wher e v Ytdi- 21 + 4L. (19 .5.20) 198 19Spontaneously Broken Global Symmetrie s Here d iis the number of derivatives in an interaction of type i, V iis the number of interaction vertices of type i in the diagram, I is the number of internal pion lines, and L is the number of loops . These quantities are related by the familiar topological identity (see Eq . (4.4.7)}: so we can eliminate I and writ e v Vi (di- 2) + 2L + 2 .(19.5.21) (19.5.22) The point of this is that each term here is positive ; every interaction in Eq.(19.5.19)has at least two derivatives, and of course L ~ 0. Therefore the leading term of each process is of order Q 2, and arises solely from tr ee graphs (that is, L = 0)constructed solely from the term in Eq .(19.5.19) with only two derivatives (that is, Vi = 0 for diz~ 2) : F2a7ray~i-`~2 = - 2 DF tD~` = -2(l+ ~ 2'F2)2  (19.5.23) Forinstance, the invaria ntamplitude M t hatappears in thepion-pion scatte ring S-ma trix element S = i (27c)464(PA+pa -PC-PD)M(2n)-6(16EAE,,EcED)-1/2 (19 .5.24) is given to this order b y Mab cd2)= 4F-2 ISab6,d(-P .~ ' Pa - Pc ' P~~ ) + S -,Ohd(PA ' PC+ PB  PD) + ~~ d&( P.a' PD + Pa ' PC) (19.5.25) where a, ta, c, dare the isovector indices of the pions A, B, C, D,respectively . (The effect of the finite pion mass will be taken up a little later in this section .) Using Eq .(19.5.25) as the leading term does not depend on any assumption of the smallness of the coupling constant Ain the original Lagrangian (19.5.1),or even on the validity of this formula for the Lagrangian, but only on the assumed smallness of the typical pion energy Q . The next term in the amplitude for any Goldstone boson reaction will be of order Q4, and arises both from one-loop graphs involving only the Lagrangian (19.5.16), and also from tree graphs constructed solely from the interaction (19.5.16) plus a single vertex arising from the d = 4 terms 19.5Effective Field Theories : Pions and Nucleon s in Eq .(19.5.19): M ab~~) --ba b4cd - 1252In(- s)-12(U2 - s2+3t2)In(-t)F 2n 12 ~ 121r2 (t2-S 2 + 3u 2) In( -u)+,~~2 (s2+ t2 + U)In A2199 (19.5.26) )+ c rossedterms ,, - ~ c4s2- 4c4~t2+u2 where `crossed terms' de notes te rms given bythe interchanges B ~ C a nd B+-*D, ands, t, a ndu are the Mandelstam va riables s = - '(P.a + pa)2,t = -(PA - Pc)2,U__(PA -PD)2 The dependence of this result on the cut-off A can be eliminated by a redefinition of the constants c4 and c' 4. The renormalized couplings ar e c4R= c4 -2In n2 (19.5.27)37c217 2 In (19.5.28) C4R=C43- 42 ~c~c where Pis an arbitrary renormalization scale of order Q inserted in order to make the logarithms well-defined . In terms of these renormalized couplings, the amplitude (19 .5.26)takes the for m m abc~)_bab 4~d _ 12s21n(-S) -1 ~ ~~2- s2+ 3 t2} In2n it 12a~ (Y') 1 ~.2 _ S2 + 3~2) 2In ;u) t 27c ~ + crossed terms .2CaRs2- 1caR (t2 +U2) 4 (19.5.29) This sort of calculation can be carried to arbitrary orders in Q, always with the result that in each order we encounter a finite number of new couplings whose renormalization serves to eliminate the cut-off depen- dence of physical amplitudes . Notice that the ratio of the leading v = 2 terms and the v = 4 corrections is of order Q2/B7E2 F2, so this is likely to be a useful expansion as long as the pion energies are all much less than an amount of order 2 7EF. This perturbative expansion may also be used to relate the ubiquitous parameter F to the measured pion decay amplitude F.Recalling the transformation rule (19 .5.14) for the axial-vector current is given b y UY ~~~ju 0 200 and so19Spontaneously Broken Global Symmetries 0(~JU0 (19.5.30) (This A~ is the Noether current of the symmetry generated by 2x, which on the nucleon doublet is represented by 2Y5t = y5i .} After integrating out all other fields, and using Eq .(19.5.17)to express in terms of the canonically normalized pion field A, we fin d AP=--Fa~`~(I -R ~~F~ ~ ( 1+ n21F2)NE(7~EOPA) (19.5.31) Using our perturbat ive expans ionin powers of the pion energy to calculate (VAQAP~7* we see that in lowest order the pion decay ampl itude here i s Fir= F. (19.5.32) Furthermore Lorentz invariance and Eq .(19.5.22)tell us that the higher- order corrections must be proportional to powers of p2/F,2,which vanishes for a massless pion, so Eq .(19.5.32) is actually exact in the limit m,, - +0. We may therefore guess that our perturbation expansion will be useful for pion energies that are less than an amount of order 27cFn= 1200 MeV . In order to make contact with experiment, we must deal with the fact that the pion mass is not zero . On the mass shell it is not possible for the time component of a pion four-momentum to be less than m, so in counting powers of the typical pion energy and momentum Q, we must regard m, as being of order Q . But we saw in the previous section that mn is proportional to a linear combination of quark masses, so our formula (19.5.22)for the order Qv of a given Feynman diagram should be rewritten to read v Vi (di+2mi-2) +2L+2, (19.5.33) where mi is the number of factors of quark masses in the interaction of type i . The interactions involving quark masses may be distinguished by their transformation properties under SU(2) xSU(2), or equivalently, under SO(4) .Eq. (19 .4.34) shows that the terms of first order in quark masses transform as a linear combination of two scalars, the fourth component of a chiral four-vector 4), ,with coefficient m u+ md, and the third component of a different chiral four-vector din , with coefficient mu-Md.Thus the terms ❑Y+ and ❑Y- in the effective Lagrangian that are of first order in mu + and and mu -and must have the chiral transformation properties of the fourth and third components of chiral four-vectors, respectively, and of course be Lorentz scalars . One obvious candidate for a chiral 1 9.5Effective Field Theories :Pions and Nucleons 201 four-vector whose fourth component is a scalar is the can with which we started in this section . According to Eq .(19.5.9),its fourth component is just (y{1 -1}/{1 + 2 } . The factor a may be dropped, as it is a chiral scalar, and therefore has no effect on the chiral transformation properties of this quantity . We can fix the normalization of this term by requiring that the coefficient of n2 = F'2-2 be -~m~12, so that, apart from an additive constant, 2 1 + 2 2 1 + n2/F'2(19.5.34) We shall see in the next section that this is the unique scalar function of the pion field without derivatives that transforms as the fourth component of a chiral four-vector . On the other hand, there is no scalar function of the pion field without derivatives that transforms as the third component of a chiral four-vector because such a function, if the third component of a chiral four-vector, would have to be odd in the pion field, and therefore pseudoscalar rather than scalar . Thus (19 .5.34) is the only interaction with di=0 and mi =Lit is striking that the isospin violating difference in the u and d quark masses has not only no effect on the pion masses, as we saw in the previous section, but also has no effect on any non-derivative multipion interaction . ❑Ve are now in a position to do a realistic calculation of the leading v = 2 terms in the pion-pion scattering amplitude . According to Eq .(19.5.33), these terms arise only from tree graphs constructed from the d = 2, in= 0 pion interaction in Eq .(19.5.23)or from the d = 0, m = 1 pion interaction in Eq . (19 .5.34). To this order, the invariant amplitude M defined by Eq. (19 .5.24) is no w M ~bcd)= 4F~2[~ab5cc(s_1n ~}+bacbbd(t-m~)+bad6br(u~.'yyt~) J In particular, at threshold s = 4m2 , t = u = 0, s o Nf(a'~~.a}threshold) = 4m~F,~ '[36abbc a -bac~~a - baabbc] 4mnF,~ 2 [7 N fRb) cd -2Mab)cd](19.5.35) (19.5.36) where 0°7 and Nf(2) are the appropriate tensors representing two-pion states with isospin T = 0or T = 2 : M(Q) = ~ babbrd ub,rd - 3(19.5.37) M(2) = 1(6acbbd+bRdb6c - 3 6ab&d) , (19 .5.38)ab,cd z normalized so that TrM(T) = 2T + 1 . This result is usually expressed in terms of the scattering lengths . According to Sections 3 .6 and 3 .7, the 202 19 Spontaneously Broken Global Symmetrie s scattering length aT for two pions in a state of isospin T is given27by 1/327Ern,, times the coefficient ofM(T)in Eq .(19.5.36): a° Sn F~ =0.16 rn,-1 a2- ~ 4nF2 = - 0.046 m,n The pion scattering lengths are difficult to measure, but careful study of processes liken+N --+n+n+N and K --*ar+n+e-}-v has given the results28 (0.26±0 .05)rn-1 and (-0.028± 0.012)rn,-,i for ao and a2, respectively, which are consistent with these theoretical values . Corrections of higher order in rnn/2n F,seem to improve the agreement . This formalism can be extended to describe the interactions of pions with nucleons or other particles, The easiest approach is to suppose that we add a term involving the nucleon doublet field N to the Lagrangian with which we started : YN= - N ( ~+g[04 +Zit0Ys])N, (19.5.39) where i is the isospin matrix vector for isospin 1/2 (that is, half the Pauli spinor z .) This is invariant under chiral S U(2)x SU(2)transformations with 60 2~04, 6N = -2iy5E t11T.604 =-29 - 0 (19 .5.40) (1.5.41) Now in eliminating the non-derivative pion couplings we must express the nucleon field Nas the 50 4) rotation R in the representation (19 .5.41) acting upon a new nucleon field IV ~ +T(19.5.42) with given again by Eq .(19.5.7).With this transformation the non- derivative term in Eq .(19.5.39) now depends only on F V and o IV[04 +21t. 0,15]1V = dNN . On the other hand, the derivative term involves derivatives of the matrix in Eq . (19.5.41), yielding a nucleonic Lagrangian : PN=-N 4+ ga+ 2i tGX 2~)2iy5t  ~N 1+~ orin terms of the ca nonicallynorma lized pion fie ld(19.5.17):(19.5.43) Zit(nxen) 2iY5tA ~'~ -N[e+gu +F~ [1 +A 2/F~]+F,r[1+n2/F2] N.(19.5.44} 19.5 Effective Field Theories : Pions and Nucleons 203 Since u has a non-vanishing vacuum expectation value, we see that the nucleon here has a non-zero rest mass, a result that would be prohibited if the symmetry under the transformation (19 .5.40), (19 .5.41) were unbroken . By its construction, the Lagrangian (19 .5.44) is chiral-invariant . This can also be seen directly, using the previously obtained chiral transformation properties of a and n, and now also noting that under the chiral transfor- mation (19 .5.40), (19 .5.41), the new nucleon field defined by Eq . (19 .5.42) transforms as bN=2it -[('xE]N . (19 .5.45) That is, under a chiral transformation 11T simply undergoes the same isospin rotation (19.5.15) as the quantity D ,,, but of course in the T = 1/2 representation . The isospin rotation parameter x E is spacetirne- dependent, so derivatives ofN do not have the same chiral transformation property, but it is straightforward to check that the combination of the first and third terms in Eq . (19 .5.43) behaves as a chiral-covariant derivative : that is, 6_9~N=2at [CXE]_qf1N , (19.5.46) where ~piv-_[0,+2i t '(CXOttoiv- (19.5.47)1+z Thus the Lagrangian (19 .5.43 (and hence (19 .5.44)) is obviously chiral- invariant, because it is isospin-invariant, and constructed solely from the ingredients N._9MN,a,and b.,all of which transform under chiral transformations with the same isospin rotation . Again, the specific Lagrangian with which we started here is not im- portant . As in the case of the pure pion theory considered earlier, the important thing is the chiral invariance of the Lagrangian . For chiral invariance, the Lagrangian must conserve isospin, and be constructed only from the ingredients N, _9^ and D ,,(together with higher covariant derivatives of these objects) . The most general chiral-invariant Lagrangian that is bilinear in the new nucleon field and involves no more than one derivative therefore takes the for m YN,o = -N -'~+MN +2agAY5t  ~ N (19.5.48) or in terms of the pion field (19.5.17): Zit  jai x fin } Fn [1 + ~i I/F~ ]+ZigA Ys t ' h F,~[1+n2/F~J N(19.5.49) Note that we have inserted an arbitrary constant gA in the last term in Eq .(19.5.48) because this term is chiral symmetric by itself, and so 204 19Spontaneously Broken Global Symmetrie s chiral symmetry cannot dictate its coefficient . (This is in contrast with the third term in Eq . (19 .5.43), whose coefficient is fixed by the condition that it and the first term together give a chiral invariant .)We can check that the constant gA inserted here is indeed the axial-vector coupling of beta decay by constructing the extra term in the axial-vector current that arises from the nucleonic Lagrangian (19.5.48).Alternatively, integrating by parts and using the Dirac equation, we find that the pion-nucleon interaction -ZigA NYSt  (tai)IV/F ,is equivalent on the nucleon mass shell to an interaction -4imNgaNY5t ' nNJF , corresponding to apion-nucleon coupling constant GnN = 2mN 9AI Fn , which is just the Goldberger -Treiman relation (19.4.33). Incidentally ,if we had u sed the Lagrangian (19.5.44) in this calculation , we would have obtained the Goldberger-Treiman relation with g A= 1. However ,this re sult is in fact an artifact of the particular form (19 .5.39) of the interaction with which we started .We might have included a non-renormali zable derivative coupling term* ` ~N = agN t (19.5.50) It is straightforward to show that in terms of the transformed fields n-, N, and this takes the form 2r7,, Since o- has a non-vanishing vacuum expectation value ,this makes a contribution to the pion -nucleon coupling con stant and to g Aproportional to g'.Hence the value g A = 1 isnotdictated by the broken SU(2) xSU(2) symmetry alone ;by including the interaction (19.5.50) and adjusting g ', we can make g Aanyth ing we like . Now let us con sider how to use this Lagrangian to calculate amplitudes for reactions in volving both pions and nucleon s.We will have to give special attention to the nucleon propagators, since a nucleon can never be a `soft' particle like a pion . A nucleon line that enters a diagram with an on-shell four-momentum p of order rn .N,and then from interactions wit h The left- and right-handed parts of the nucleon doublet transform according to the representations (Z 0) and (d„) of SU(2) x 5U(2), respectively . The bilinear N Y,7sN is thus a sum of terms quadratic in (2, 0)or ( 0, 2)terms, so it transforms like a direct sum of the (1,0),(0,0), and (O, i) representations . In Eq .(19.5.50) we have coupled the (t,0)+(0,1)terms in NruySN to the antisymmetric tensor formed from the S U(2) x S U(2) four-vectors can and 0,, 0n. lts invariance under the transformation (19.5.40), (19 .5.41) can of course be checked directly . 19.5Effective Field Theories : Pions and Nucleons 205 soft pions absorbs a net four-momentum q with components much less than mN, will have a propagator : -i(~+4)+MN )-i i+MN19.5.51 (P+q)2 + rn~ quo 2p  q ~) (The neglected terms may be taken into account by including higher derivative terms in the nucleon Lagrangian .) Suppose again that all external pion four-momenta have components at most of order Q and define all renormalized couplings at renormalization points of order Q, so that integrals converge in such a way that internal pion lines also have four-momenta Q . Then Eq . (19 .5.51) shows that internal nucleon lines make a contribution of order 1 1Q.A general Feynman diagram for such a process will make a contribution to the invariant amplitude of order Q', where now v iii(di + 2rnr ) - 21,- IN+ 4L . (19 .5.52) Here Vi is the number of vertices associated with interactions of type i, di is the number of derivatives in each such interaction, mi is the number of quark mass factors in each interaction, I, and IN are the numbers of internal pion and nucleon lines, respectively, and L is the number of loops in the graph . We use the familiar topological relations for connected graphs : L=1 ,+1N-~V i+1 (19.5.53) and 21N-}-E1V Vrni, (19.5.5) where Mi is the number of nucleon fields in interactions of type i, and E N is the number of external nucleon lines . Eliminating the quantities I Nand I,, this give s v Vi (dj + 2Yl'1i-}- 2-2+2L-Ems- }-2. (19.5.55) The important point here is that the coefficient di +2mi+ '-z ni -2 in the first term is always po sitive or zero .Wehave already seen that d i-}-2mj?2 forthe purely pionic interactions with rte - 0,and inspection of (19.5.49) shows that d i~ I for the pion -nucleon interaction swith M i= 2 and rni= 0. Any interaction with n i= 2 and m ;~ 1 or r~ i~ 4 clearly also has di~ 2. Hence the leading terms for Q <2nF,,are tree graphs (that is, 206 19Spontaneously Broken Global Symmetrie s .~ Figure 19.3.Feynman diagrams used with an effective chiral Langrangian to calculate the scattering of a soft pion from a nucleon . Dashed lines are pions ; solid lines are nucleons . L = 0)for which all interactions hav e di+2mi+~---2 =0 . The interactions that satisfy this condition are just those shown explicitly in Eqs .(19.5.23), (19.5.34), and (19 .5.49), plus possible interactions with di=0andna=4 . 01,ra~IT}(1~1raN} (19.5.56) where ra and ra are any matrices in spin and isospin space that yield Lorentz, space inversion, and isospin-invariant four-fermion interactions . These last interactions are important for multinucleon processes,29which will not be considered here . Let us now apply this method to pion-nucleon scattering . For pion energies roughly of order m, the pion-nucleon scattering amplitude is given by the Feynman graphs of Figure 19 .3, all of which make contribu- tions of order rn, to the invariant amplitude . However, at threshold the leading contribution is from Figure 19 .3(c), the others being suppressed by an extra factor of M,/M N. This is because at threshold in the rest frame the incoming and outgoing pion four-momenta q, q' and nucleon four-momenta p, p' are given by m~q=q'=(0,0,0, mn~ N P= MNP, (19 .5.57) Thus the invariant amplitude at threshold from either Figure 19 .3(a) or 1 9.5Effective Field Theories : Pions and Nucleon s 19.3(b) is propor tionalto (p±q)2+ rnN m2 uYS(-rj(1 ± rnnl MN) + m N)YSu (+2rn,,rnN - rn~ ~ TM7r2 ± M, u~i ~~1 ± Mr/MN) + mN}u m2rn~v = 2rnN±M n207 where u is a Dirac spinor with uu = 1 . In contrast, the diagram of Figure 19 .3(c) gives a contribution to M of order rn, We may write this amplitude as a 2 x 2 matrix in the nucleon isospin indices : -2i tcCabrr , where q' and q are the final and initial pion four-momenta and band a are the corresponding isovector indices . Using Eq .(19.5.57) and the momentum-space Dirac equation (i i+ rnN)u= 0, this become s r~~,~~ -4amn -4m, , I Mba= 2 tc 6a bc = ~,2 t'I~t~ ba F. a(19.5.58) where [t~'7]ba = -i cba, is the pion isovector matrix . The matrix t  tthas eigenvalues in states of total isospin T equal to 2 [ T(T + 1)--- 2 - 4 ], so in the two isospin states, T = 1/2 and T = 3 /2, the invariant amplitudes are30 Nf2/2 = 4m, ,IF~ , M3/2 = ~--2m ,IF~, These results are usually expressed in terms of the scattering lengths, which are defined as the invariant amplitudes divided by 4n(1 +MnIrnN): Mn a112 =n F,~2(] + m nlrnN)= 0.15 rn n ~ 2nF~(1 + M,,/MN)(19.5.59) a32 ="In--- -0 .075 rn~ 1 (19.5.60) These are in reasonable agreement with the experimental values12 a112 = (0.173 ± 0.003) m~ I and a3/2 = (-0.101 ± 0.004)m~ i. (The results (19.5.59) and (19.5.60) are only supposed to be valid to lowest order in M,,/MN, but we retain the factor 1 + M7,/MN, because it arises just from the definition of the scattering lengths .) The corrections to the scattering lengths of next order in m, arise from several sources . There are the Born diagram graphs of Figures 19.3(a) and 19.3(b), which are nominally of leading order but as we have 208 19 Spontaneously Broken Global Symmetrie s seen at threshold are suppressed by an extra factor of rn I/mN . There are additional tree graphs containing a vertex with two derivatives .Of special interest are tree graphs containing a vertex with no derivatives that arises from asymmetry- breaking interaction proportional to in.These interactions must have the chiral and spacetime transformation properties of the operators in Eq . (19 .4.34): the fourth and third components of two different chiral four-vectors (Dn and W..There are two obvious candidates for such operators that are bilinear in the nucleon fields : a cD4 term , ~2 N11T= 1 -~ NN-4i 11Ty5t11T, and a c D3term , Nt3N = NNVrt 1+~~i(_C3 1 V,1~V ~+2 Chiral symmetries act on j V_ ordinary isospin rotations, so they cannot mix up scalar and pseudoscalar nucleon bilinears . Thus there are really two independent (D4 operators : 2 -~ ~ NN ~+C2 and -}- y'~i'St11TC2 and two independent (D3 operators : andNt~V-23 2Nt -~ N i 2NYsIW 1+~(19.5.b1) (19.5.62) (19.5.63) (19.5.64 We shall show in the next section that these are the only operators with the transformation properties of the terms in Eq .(19.4.34) that are bilinear in the nucleon fields .The operators (19.5.62) and (19.5.64) evidently provide isospin-conserving and isospin-violating corrections to the Goldberger -Treiman formula for the p ion-nucleon couplings . The other two operators , (19 .5.61) and (19.5.63), contribute directly to both the nucleon mass and to low energy pion -nucleon scattering .From their contribution to the nucleon ma ss,wesee that these latter terms 19.5Effective Field Theories :Pions and Nucleons 209 enter into the effective Lagrangian in the form (now replacing with the conventionally normalized pion field) : 6rnP+6mn 1 - R2/~~F~NNE eff -2 I +7'21F72 -(brnp - 6mn)Nt~V -- F2 1 ~~ Fz Nt  rN X19 .5.65} ,r+/,r where JrnF and gym„ are the contributions of the quark mass terms (19 .4.34) to the proton and neutron masses . This makes a contribution to the pion- nucleon scattering amplitude (again written as a matrix in the isospin space of the nucleon) : W ha.-2[bYIZp+ S riln]dab +2 [b mp -b m~] ta~3b + tb~3a 19.5.66 F2F2 ( } } The first term in Eq .(19.5.65) is often known as the `o-term' . The second term is an isospin violating correction to the o-term, which shows up only in processes involving neutral pions, such as the charge-exchange processes n+ + n--+ no + p and n- + p --+no + n . Even though it is not possible to measure the nucleon mass shifts bmp,n directly, we shall see in Section 19.7 that an S U(3) symmetry allows their difference to be calculated from hyperon mass differences . This yields the result 6mP - 6m,=-2.5 MeV . Unfortunately, we still do not have a firm theoretical estimate of the coefficient bmp + 6m, of the first term in Eq .(19.5.65). Eq.(1.4.34) shows that in quantum chrornodynamics the coefficients 6rnp + 6rnn and 6mp-6mn in Eq . (19.5.66)are respectively proportional to m% + rnd and m& - in.We shall see in Section 19 .7 that mu and and are not at all degenerate, so these coefficients are roughly of the same order of magnitude ; the isospin-violating corrections to the Q-term are not much smaller than the Q-term itself . We see again that the reason that isospin conservation is such a good approximation in hadronic physics is not that the u and d quark masses are nearly equal, but just that they are small . The use here of Lagrangians like (19.5.19)or (19 .5.49) is one example of the method of effective field theories, already introduced in Section 12 .3. Similar techniques have been employed to deal with meson and baryon interactions including strange particles (see Section 19 .7), with quark and lepton interactions at energies below the scale of electroweak symmetry breaking (see Section 21.3),and even with superconductivity (see Section 21.6).In all these cases effective field theories provide the most convenient method for working out the consequences of symmetries and the general principles underlying quantum field theory . ~~~ 210 19 Spontaneously Broken Global Symmetries The low energy theorems provided by broken symmetries when com- bined with dispersion relations yield useful sum rules . Let's see how this works for chiral symmetry, ignoring the small u and d quark masses and the pion mass . Consider the forward scattering of a massless pion with isovector index a and four-momentum q on a nucleon of four-momentum p, yielding a pion with isovector index b.The scattering amplitude Nf, defined by S fi =-2n i b4(Pf - Pi)Nf fi, is given to first order in q by the Feynman diagrams of Figure 19 .3 as -2niN11 (fin)6(2qa) X u[( .(2m)42AY5tb~ Fn + i (2n)'2gAY5ta F7r tGEabc~ u .F27r-i -i (i+4)+MN (27r)l (P +q)2 +M2i(2n~2g aY5ta ~ F,r 4 -i - i(j-4)+mN-i(2n)42gAY5tb~ (19.5.67) To this order in q, the nucleon propagator in the first two terms inside the square brackets may by approximated b y (p±q)2+M2±2p q Eq.(19.5.67) can then be further simplified, using the relations 4 4 = q2 = 0, uyl`u = -ip  glrnN, and [t a, tb] = iF abctc. Also, in the laboratory frame (which for small q is the same as the center-of-mass frame) we have p  q = -mrv OL), where co - q° is the pion energy in the nucleon rest frame . Finally, the conventional forward scattering amplitude (whose absolute square is the differential cross section for forward scattering) is given by Eq. (3.6.9)asf - -4n2wM .Putting this all together, we find the forward scattering amplitude at pion energy co is given for co < MN by .fbu(CO) ~ -i F2 0-9A2)Fabcte. (19.5.68) n In particular, for n+-proton scattering, we must contract Eq . (1 .5.68)with vbva,where v is the normalized isovector (1, i, 0 )/,[2, and set t3 = +z. The forward low energy n+-proton scattering amplitude is then (19.5.69) Now, the dispersion relation for the forward scattering of a massless n+ on a proton is given by Eq . (10 .8.24) (with subtraction polynomial 19.6 ffective Field Theories :General Broken Symmetries 21 1 P(E) ac E) as io) 4n 2j0 E-w E+w where R is a possible subtraction constant ,and c oand Eare pion energies in the proton rest frame . Comparison of this dispersion relation with the low energy limit (19.5.69) shows that R = 0,and tha t 2 gA= 1 + F" 00 Zn[an+p (E)-a,~-p(E)] dE /E. (19.5.71) Thisisthe celebratedAdler-We isbergersum rule, 19 for the case of exact chiral sy mmetry, withmn = 0 . A s imilarsumrule maybe derived fo r the scatte ring of p ions onanybaryon o r meson, includingthe pion itself.30a 19.6 Effective Fie ldTheories ; Genera lBroken Sy mmetrie s The techniques described in the previous section for constructing effective Lagrangians for the case of SU(2) xSU(2) broken to SU(2) were soon generalized31 to the case of a general group G broken to an arbitrary subgroup H . Consider a quantum field theory whose Lagrangian is invariant under an arbitrary compact Lie group G of ordinary linear space time-independent transformations g of the fields Vn(x) : Vn(x))719r1mVm(x) - (19 .6.1) M We suppose that this symmetry group is spontaneously broken to some subgroup H =G of symmetry transformations that leave all vacuum expectation values invariant : for h cH, hnrra(y)rra(x)IVAC =(y)>z(x)I VAC (19.6.2) M (Not all of the yes need b escalars, but of course only the scalars will have non-vanishing vacuum expectation values .) In the example at the beginning of the previous section, G was the group 5O(4), the fields V, furnished a four-vector representation of this group, and their vacuum expectation values broke SO(4) down to the subgroup H = SO(3) of rotations that leave the four-axis invariant . In the general case the fields v,, may furnish a reducible rather than an irreducible representation of G, as for instance became the case when we introduced the nucleon fields in the previous section . 212 19 Spontaneously Broken Global Symmetrie s We next express a general `point' vn in field space as a G transformation acting on fields ~j~ ,from which the massless Goldstone mode has been eliminated : in For instance, in the previous section y was the SO(4) rotation we called R, and the condition satisfied by ~n was that the first three components of the four-vector should vanish . In order to formulate this condition more generally, let us suppose that we are working with a real representation of G . (There is no loss of gen- erality here, because we are not insisting on an irreducible representation, so if we start with a set of fields forming a complex representation of G we can always take the V~n to be the real and imaginary parts of these complex fields .) As shown in Section 19 .2, the massless eigenvectors of the mass matrix are just the independent linear combinations of the vectors Em[ta]nm(TWONAG, where to are the generators of G . (The reality of the representation means that ita is real, and the fact that G is compact means that we can choose the representation to be unitary and hence orthogonal, so that the to are imaginary and antisymmetric .) The condition that yin does not contain Goldstone modes may thus be formulated a s NX)[ ta1nrra(Vm(O)) VAC = 0. ( 19.6.4) rim The number of independent conditions here is the dimensionality of the group G minus the dimensionality of the subgroup H whose generators annihilate (Tm{O}yVAC As in the previous section, the Goldstone bosons whose fields are eliminated in this way will reappear in the dim(G) - dim(H) fields needed to parameterize the transformation ynm(x) . In general theq)n include all the heavy fields of the theory, including those (like nucleons in the previous section) that have different spacetime symmetry properties from the Goldstone bosons . It is necessary to show that we may always choose the transformation Y.(x) so as to satisfy Eq . (19.6.4). For this purpose, consider the quantit y VYr(g)=E Vngnm (Vm (a)) VAC ~ (19 .6.5) nm where gruns ove rthe whole group Ginthereal o rthogona l representation furnished by the yin. Thisis obviously a cont inuous realfunction of g, an d since the g roup is compactVy,(g) is also a boundedfunction of g . At eac h spacetime po intx,,,,(,)(g) thereforereaches a max imumvalue fo rsome groupelement, wh ichwe s hallcally(x). At g = y(x), V v(x)(g)mustbe stationary with r espectto arbitrary variations ing.Butand infinitesimal 19.6Effective Field Theories : General Broken Symmetries 21 3 shift in a group element g may always be written as a linear combinatio n a with real infinitesimal coefficients ca that may depend on g . Hence the condition that Y,,,(x)(g) be stationary with respect to the variation bg at g = y(x) read s 0 =6VY,(x)((x)) f-aY:yPn(x)^nr(x) ti~(Wm( ❑)~VAC , nor = a~eorY:[71(x)]tnWn(x)tim(Vm(0)~VAC a nyraI This must be satisfied for all variations, and thus for all c,,so we see that Eq. (19 .6.4) is satisfied for ~(x) = y-l(x)y(x), as was to be shown . In the purely bosonic theory with Lagrangian (19 .4.1), ~n was the four-vector field ( 0,0,0, u). This points in the direction of the vacuum expectation value (V ny,but this is not always the case . Incidentally, it is not necessary in dealing with complex fields to for- mulate the condition on flx) explicitly in terms of its real and imaginary parts . If a set of fields X(x) transform according to some complex represen- tation of G with Hermitian generators Tx, then in the real representatio n V(X)= Re X(x) Imx(x) the generators are -Im T' -Re T'itIX = (~e T°` -Im T a The condition (19 .6.4) may then be expressed directly in terms of Tx and X(x) as 1m(x(x)' 7'a (x(O)}SAC} _ 0. Because the Lagrangian is assumed to be only invariant under spacetime-independent G transformations, it will be found after the trans- formation (19.6.3)to depend on y(x) as well as on ip(x ), though always with at least one derivative in each y-dependent term . As already re- marked, the spacetime-dependent parameters needed to specify y(x) will play the role of Goldstone boson fields . We now have to consider how to parameterize y(x) . It must be recognized from the outset that the choice of y in Eq . (19.6.3) is generally not unique . Because (Vm(O)yVACis H-invariant in the sense of Eq.(19.6.2), the quantity Vy,(g )defined by Eq .(19.6.5)is invariant under 214 19Spontaneously Broken Global Symmetrie s right multiplication of g by any element h of the unbroken subgroup H : Vw(g)-VyJgh} for h C=H. (19.6.6) It follows that if y maximizes V,,(g),then so does yh, so that condition (19.6.4) is satisfied for ~j~ =h-1 y-1 y~ as well as tp = y- lya. Hence y is only defined so far up to right multiplication b yan element of H . (For instance in the example of the previous section, the four-dimensional rotation R could have been multiplied on the right with any rotation acting only on the first three components of four-vectors .) We may think of two group elements y jand yZ as equivalent if yl = yZh with h eH. This is an equivalence relation because it is reflexive (if yl is equivalent to y2 then Y2 is equivalent to yl ), symmetric (y is equivalent to itself), and transitive (if yl is equivalent to Y2 and yZ is equivalent to Y3 then yl is equivalent to y3) . The elements of the group G can therefore be sorted into disjoint `equivalence classes', each consisting of elements y that differ only by right multiplication by an element of H . These are known as the right cosecs of G with respect to H . What we need is a parameterization of the space (known as GBH )of right cosets . For this purpose we need only choose one representative group element from each right cost . For the SO(4) symmetry broken to 5 O(3)of the previous section, it was convenient to choose these representative elements as the rotations R parameterized by the three-vector There is another choice that is available for any compact group G broken to any subgroup H. Let us first adapt our notation for the generators of G to the pattern of symmetry breaking . The complete set of independent generators of H will be called ti . According to Eq . (19 .6.2)these satisf y Y:(ti)n»a(VmyVAC = '0. (19.6.7) M Since H is a subgroup, the t ; form a subalgebr a k We will take the x,, to be the other independent generators of G, in any basis with totally antisymmetric structure constants . (Such a basis always exists for compact groups ; see Section 15 .2.)Since x, does not appear on the right-hand side of Eq .(19.6.$), the structure constants Crjaall vanish, and since they are totally antisymmetric, it follows that C=pl = Q, s o b However, it is not necessarily the case that commutators of xs with each other are linear combinations of ts ; this depends on the nature of G and H, and also on how H is embedded in G . (Where C ab,vanishes, the coset 19.6Effective Field Theories :General Broken Symmetrie s space GBH is known as a symmetric space) .In general we may writ e Ixu,xG]- x ~ Cabita + x Y: Cu6cxc215 (19.6.10) Any set of generators with commutation relations of the form (19 .6.$)- (19.6.10)is called a Cartan decompos itionof the Liealgebra . An example is provided by the ch iral S U (2)x SU(Z) Lie algebra (19.4-9)-(19 .4.11), for which the C abcdid happen to vanish . Because t jand x aspan the Lie algebra of G, any finite element of G may be expressed in the for m g = ex piY:~axa exp i O iti (9.6.11) where ~a and H ; are a set of real parameters . But the transformation y(x) in Eq .(19.6.3)is only defined so far up to right-multiplication with an element of H, so we may standardize our definition of y by taking it in the form Y(x) = exp~a(x)xR (19.6.1 Z) The ~a(x) maybeidentified(apa rt from normalization) w iththe Goldstone boson fie lds. In what fo llows we wi llsimply assu methat re presentative elements have been c hosen f romeach rightcosec, and exp ressed as con tinuous func tions y(~) of so me parameters ~R, Eq . (19.6.12) in gene ralprov ides one ex plicit example,but we w ill notlimit ourse lves to t his para meterization. Now, suppose we use Eq . (19 .6.3)toreplace a llfields V(x) inthe Lagrangian wi thy(x)y~(x). Thederivatives of t he fields are given by OuYW = ~~'(x)lauflx) + (Y i(x)auY(x)R(x)]. (19.6.13) Therefore when we express the Lagrangian in terms of rprather than V, the Goldstone boson fields appear through the dependence of (x) apy(x) on ~a(x) and its derivatives . Any variation of a group element like y(x) may be written as the group element times a linear combination of the generators of the group . In our case, we may write this as a where DQ,u and Eiu take the form s D,,,(x) = 1:Dab(~W) O,WX) a(19.b.14) (19.b.15) 216 19 Spontaneously Broken Global Symmetrie s Ej,u (x) = Y:Eib(~(X))Oy4 W - (19.6.16) The Goldstone boson fields will thus enter the Lagrangian through the appearance of the quantities Da,,(x) and Eil,(x) (and their derivatives) . Note that for exact broken symmetries every interaction of the ~s must involve at least one derivative, so mass terms m~~~,,~h cannot appear in the Lagrangian, and all interactions vanish when and Goldstone boson four-momentum vanishes . Even where we do not know the details of the underlying Lagrangian, we can learn a great deal about the way that the quantities DuYand Ear appear in the transformed Lagrangian from their transformation properties . Under an arbitrary element g of the group G, the original field Ttransforms according to Eq .(19.6.1): V(x}--* O x} = g V(x} = g *(x)}ip(x). (19.6.17) Now, gy (~)is an element of G, so it must be in the same right coset as some y (~'), and may therefore be written in the for m g,~ (~ (x)) = 7(~'(x)) h (~(x),g) (19.6.18) where h is some element of the unbroken subgroup H . [)sing this in Eq. (19 .6.17), we find that V'(x) is of the form (19 .6.3) W,(X) = Y (v (X)) rp, (X ) with ~'(x) defined by Eq .(19.6.1$), and qV(x) ==h(~ (X), g) 1~ (X) -(19.6.19) (19.6.20) In the example at the beginning of the previous section, qa consisted of a single isoscalar a, and since this was invariant under the unbroken isospin subgroup, it was also chiral invariant . More generally, we find here that rp(x)is not necessarily invariant under general G transformations, but that its G transformation depends only on its transformation under the unbroken subgroup H . We saw this in the previous section when we introduced the nucleon fields ; under a general infinitesimal broken chiral symmetry, the fields N transformed according to the field-dependent isospin rotation (19 .5.45). The transformation rules and q7 specified by Eqs . (19.6.1 S) and (19 .6.20)make no reference to the particular linear G transformation properties of the original fields yin . Indeed, we did not need to start with a Lagrangian that was invariant under linear G transformations in order to deduce these transformation rules . Given and set of fields on which a group G acts linearly or non-linearly, with a subgroup H that leaves one special set of field values (their vacuum expectation values) invariant, 19.6Effective Field Theories : General Broken Symmetries 217 we can always express these fields (in at least a finite neighborhood of the special field values) in terms of a set ~ ,,and i p-, which transform under G according to the standard realization ~ ,, --* ~a,~ --* i~' defined by Eqs .(19.6.1$)and (19.6.20).The essential uniqueness of this realization was first proved25 for SU(2) xSU(2) broken to SU(2), and then for general Lie groups broken to any subgroup .31 For an easier argument, we may note that for any set of fields that transforms according to a non-linear realization of an arbitrary compact Lie group G, it is always possible to define functions of these fields (and perhaps additional fields*) that transform linearly under G .32Starting from this linear realization, the arguments of this section allow us to construct fields ~ and fpwith the transformation rules (19.b.1 S)and (19 .6.20). Although the transformation of ~(x) and flx} is complicated for general geG, it is much simpler when g is itself a member h of the unbroken subgroup H . It is usual and always possible to choose the Goldstone boson fields ~a(x) to transform according to some linear representation gab(h)of the unbroken subgroup H : hy(~)h-1=y(9(h)~) . (19.6.21) For instance, in the parameterization of the cost space S O(4)/SO(3)used in the previous section, the Goldstone boson fields ~ ,formed an isospin three-vector . For the more generally applicable parameterization based on Eq .(19.6.12), the commutation relations (19.6.9)show that the xa transform linearly under H : hxy,h-1 =Y:-9ab (h)xa (19.6.22) a from which Eq .(19.6.21)follows immediately . Comparing Eq .(19.6.21) with Eq .(19.6.2$), we see that for g = h eH, a rpn(x)= hnm~m(x) . (19 .6.24) M In other words, ~a and inn transform under H according to the represen- tations -9ab(h)and hnm itself, respectively . We must now consider how to construct the most general Lagrangian that is invariant under the transformations (19 .b.1 S)and (19 .6.20). The transformation ~ --*~' for general g cG is non-linear and rather compli- cated, and rules out the appearance of ~Q(x} in the Lagrangian except i n For instance, the polar and azimuthal angles B and ip furnish a non-linear realization of 50 3) ; by adding an additional variable r we can construct quantities xl, xz, and x3 that transform linearly under 503 ). 218 19 Spontaneously Broken Global Symmetrie s quantities like D,,,(x) and Ezt,(x) . Fortunately, these quantities obey very simple transformation rules . Differentiate Eq .(19.6.1$)with respect to xu and multiply on the left with its inverse . This give s h-1 (x), g) y (~'(x)) a, [y(~'(x)) h(~(x),g) =h-'(~(x),g) [y-'(~'(x))O,y(~'(x))] h(~(x) ) + h-i r~(x), g)Ouh(~(x)s g)1 and s o Y-; (~'(x)} O 1Y('(x)) = h((x), g)[Y_1(~(x), g)O,Y (~(x), g) ] h^i(~(x), g) -[3,4h((x),g)] h -1(~(x), g). (19.6.25) Eq.(19.6.12)gives the quantity 7_10p7 as a linear combinations of xs and ts. Also, the second term on the right-hand side of Eq .(19.6.25) is a linear combination of is alone . Since the xa and ti are all independent, the coefficients of each generator on both sides of Eq .(19.6.25)must all be equal : g= h(~ g)xaDau xaD~ Y:a u JA ( ih-i(~a$) +i [Ogh(~, g)]0(~, g) 9 or in more detail Da'p(X) = 1 : gab (h((x),g) ) x) = ESi ;(h(4(x),g)) Ej~(x) ->rib (h((x),g)) ~, 4(x)a(19.b.2b) (19.6.27) (19.6.28) where Dag and Ei.are defined by Eqs . (19.6.15) an d(19.6.16), an d IaPh(~(x),g)j h-1 (~(x),g) ~_-iE Afib(~(X))U,~b(X) ib ht1 h-1 tai j (h)ti I9.6Effective Field Theories : General Broken Symmetries 219 while -9ab isdefine d by Eq . (19.6.22). We see t hat the quant ities D,,,(x) a re `covarian t derivatives' of the Go ldstoneboson fie lds, transfor mingunder a genera lelementgcG muc h like the fie lds i~ :bothare subjected to t he H transfo rmationh(~ (x), g), though in different representations. For instance, inthe SO(4) theory of the prev ious section, the cova riant derivative of the pion fie ld was t he quan tity + ~ Z }, w hich is tra nsformedunderan infinitesimalchiral transformation by an isospin rotation (19 .5.15). Onthe o therhand,Ej,z(x)transfo rmsinhomogeneous ly, much like a gauge fie ld.The ext raterm in Eq . (19 .6.28) a llows a ca ncellationofthe inhomogeneous term in derivatives of i~ .Differentiating Eq . (19 .6.20), we have 0ufpl(x)=h(~(x),g) [0, fp (x)+h-1(~(x),g) 0,h(~(x),g)i~ (x)] . Combining this withEq. (19 .6.27) g ives (-9MVX)) = h(~(x),g) -9, fp(x) where is a cova riant derivative of theheavy particle fields :(19.6.29) (Eq.(19.5.47)provides an example of such a covariant derivative in the case of SO(4) broken to SO(3) .)Any Lagrangian that is invariant under the unbroken subgroup H and is constructed from i~, 9,S) , andDa,will thus also be invariant under the full group G . Renormalizability is not an issue here, so we can also consider quantities involving more than one spacetime derivative . One such is -9v9,i~, obtained by just repeating the operation (19,6.30 )0~u + i ~ t~ &~u y. (19.6.31) This transforms just like !?P T (gV-9,ufp)' =h(~, g)-qv-9mfp . (19.6.32) Another covariant quantity can be constructed from the covariant deriva- tive of the covariant Goldstone boson derivative (19.6.15), which is defined just like (19 .6.30), but with ti replaced with the corresponding matrix in the representation of H furnished by the x a ~2vDa,u=OvDcay+ECiabEivD6p b which transforms just like DRP a(19.6.34) 220 19 Spontaneously Broken Global Symmetrie s Also,by antisy rnmetrizing thederivative of Eit,we can eli minate the inhomogeneous te rm in (19.6.27).This y ields a `c urvature ' Ri,uv - OvEag - OuEiv - iCij kEjpEkv. 19.6.35) This is not really new ; it is easy to see that the antisymmetric covariant derivative of the 'matter' field y~ is proportional to this curvatur e 1-9v,emu]ip' tjRipvip'. (19.6.36) Of course, we can continue this process and construct yet more covariants by taking higher and higher covariant derivatives of 9µy~ and D,,,,. It is useful to note that the quantities Du,and Erg may be calculated once and for all for given groups G and H and a given parameterization of the cosec space GBH, and do not depend on the assumed transformation properties of the fields yin or the specific matrices xa, ti used to represent the Lie algebra of the broken symmetry group . For instance, in the exponential parameterization (19.6.12), the first few terms in the power series for these covariants are easily calculated from Eqs .(19.6.12) and (19.6.14) as Dal, =am~a+z Cabc~h0u~c be + ~E Y:Ccde Cbea + bed e + OWOUO I Eip=z E Cubt ~a~ lk + abCcdiCbia 4Uu ~d (19.b.37) CRCCI CF]dT ~LI 4O l~C abed (19.b.38) All we need to know in order to construct the most general Lagrangian involving the Goldstone boson fields ~a and a set of heavy particle fields is the pattern of spontaneous symmetry breaking, of Gto H ,and the transformation of the heavy particle fields under H, not G : the Lagrangian is taken to be the most general H-invariant function of ~,~,uq), Day, and higher covariant derivatives . Now consider the case where the symmetry under G is not only sponta- neously broken, but not even exact to begin with . Suppose that there is a term ❑Yin the initial Lagrangian that is not invariant under the group G of linear transformations V--*gy, but transforms as a linear combination of the components of some representation (reducible or irreducible) of G . That is, we take AY== E CACA (19.6.39) A 1 9.6 Effective FieldTheories :GeneralBroken Symmetries 221 where CAtransforms under G according to some representation D [g]A B CA E D [C]AB aBB(19.6.40) If we now replace the fields Vwith the set ~a,y~,this term in the Lagrangian will still take the form (19.6.39), but Eq . (19 .6.40) now read s CA[.f.(~,9), DCh(~,g)1q)]=EDC9IABOB ( 19.6.41) B where fa (~, g)is the result of applying the transformation g to ~ ,,, defined by Eq . (19 .6.18): g,j (~) =7 (f (~, g)) h(~, g) . (19.6.42) As we shall see, it is easy to find sets of operators satisfying (19 .6.41), and the solution is unique up to a specification of certain H-invariant functions of the V) . First, consider the case ~a = 0,with g In this case gy(~ ) so (0' ~tj5 h(0, Applying this in Eq . (19 .6.41), we find (now dropping the prime) : CA EDMOIAB OB [0,iP-] B(19.6.43) This gives the operators CA [~a, y~] for all ~ ,,if we know them for ~ ,,=0. Now take ~u = U, with g an element h of the unbroken subgroup H . Here we have gY(~ )= h, s o Applying this in Eq . (19 .6.41), we find her e (9 A10, hfpl = 1 : D[h]AB OB [0, q1] B(19.6.44) In other words ,theCAAfor ~ = 0 must have the same tran sformation under linear H tran sformation sas are found in the representation (19 .6.44)of G. (However, there is nothing in this that fixes the normalization of the different irreducible H repr esentation sfound in the G repre sentation (19.6.40); some of them ma yeven be absent altogether .) Finall y,we note that an yset of doperators satisf ying(19.6.44) allows the con struction using Eq . (19 .6.43) of operators sati sfying the general G-transformation rules (19 .6.40).Using Eq s.(19.6.43)and (19.6.44), the 222 19Spontaneously Broken Global Symmetries left-hand side of Eq . (19 .6.41) become s CA[fa(~, g),h(~,g)ip-] ED (f 9))IAB OB [0,h(~, g)rp] B EICY(f(,g))]ABD[h(, 9)]BCac [0,BC E D [7 (f(~, g)) h(~,9)]AB OB [0, B 1: D[97(~)]AB aB10,B 1:D[91AB aB[~' fp] B as was to be shown . As an example, consider the group SO(4) spontaneously broken to SO(3) as in the previous section, and suppose we want to construct operators that transform according to the four-vector representation of SO(4) out of the Goldstone boson field ~(x) and the other fields ~(x) . According to (9 .6.43), these must take the for m 0 [-*, i p-] = R -)( where R is the SO(4) rotation defined by Eqs . (19.5.8) and (19 .5.9): 2Ca 1_C2 Raa= 1+C 2 x44= ~ +C2 The condition that R is an orthogonal matrix is satisfied if we choose its other components as 2Ca4 2~a Thus any scalar (asopposed topseudoscalar) operators that transform as the fourth and third components of chiral four-vectors must appear in the effective Lagrangian in the form s + 2C ._+[O'~] + (1_~2+ (12) +C2 and 2C3 C 0-0,r 03 3 P] I + 213 ) (D 4[0, fP-] I+C2)~+2 The only condition on the operators (D+ [4, y~], (D4[0, y~], (D-[0,fl,and (D4[0,y~] is that they should transform under Lorentz and isospin trans- formations respectively as a pseudoscalar isovector, a scalar isoscalar, a 19.6Effective Field Theories : General Broken Symm etries 22 3 scalar isovector, and a pseudoscalar isoscalar, but they do not have to be related in any way . For processes involving only pions we use an effective Lagrangian that involves no fields except the pion field ~, so the operators (D+ [0, y~], (D- [0, y~], and (D4 [0,fp-] must all vanish, while c D4 [D, ~] is just a numerical constant . We see as promised in the previous section that the only non-derivative chiral symmetry breaking operator in the effective Lagrangian is proportional to (1 - ~ 2)/(1 + C2), and hence to the operator (19.5.34). In order to include symmetry-breaking operators bilinear in the nucleon field, parity and isospin conservation require that we tak e (D+ [0, f P-] OC 75-fV (D+ [01 f P' ~ ~-[4, y~]oc ~ITt1V , 04[o,.q]ocNYsPT. Thusasclaimed in the previous section the onl ynon-deri vativesymmetry- breaking term sinvolving nucleon bilinears are of the form (19.5.6t)- (19.5-64) . The phenomenological Lagrangian ma ybe used to calculate the ampli- tudes for processes involving particle sofsmall three-momentum in much thesame wa yas was done for the chiral theor yof pion sand nucleon sin theprevious section .In counting powers of a characteristic small momen- tum Q,Gold stone bo son internal line salwa yscontribute factor soforder Q-2.Also,an internal line forone of the hea vyparticles (ofany spin) described b ythe fields ~pthat has absorbed a net four-momentum q from the Gold stone boson field willhave a propagato r 1V(p +q) N(p) (P+4)2+ M2 2 P'4 (where Nis some polynomial, depending on the particle spin and field type, and M is the particle mass) so it will contribute a factor of order Q-1. The same argument that led to (19.5.55) applies here, and gives the number of powers of Q in a given Feynman diagram a s v =EVi(di+hd2 - 2 )-}- 2L - Eh + 2 , (19 .6.45) a where VI is the number of vertices of type i, di is the number of derivatives (or for approximate broken symmetries, derivatives or Goldstone boson masses) at a vertex of type i, h iis the number of heavy particle lines at a vertex of type i, L is the number of loops, and Eh is the number of external lines of heavy particles . In general the coefficient d i+ hj/2 - 2 is non-negative for all interactions allowed by chiral symmetries . The interactions among Goldstone bosons alone always have d ; ~ 2, because they must either be at least bilinear in the covariant derivative (19.6.15), or proportional to symmetry-breaking parameters (like quark masses in the previous section) which are of order of the squares of the Goldstone boson 224 19Spontaneously Broken Global Symmetrie s masses . Interactions between Goldstone bosons and heavy particles must involve the covariant derivatives (19 .6.15)or (19 .6.30), and so must have di ~ 1 as well as hi ~ 2 . The only interactions for which d i+ht/2 <2 would be trilinear interactions among the heavy particles, which we exclude because we are only considering graphs in which all heavy hadrons remain non-relativistic . With di + X112 ~ 2 for all interactions, the leading graphs are those constructed solely from interactions with di + h ;12 = 2 and no loops . Corrections are again provided by interactions with more derivatives and/or more heavy-particle fields, and/or one or more loops . In the absence of intrinsic symmetry breaking, the part of the La- grangian that involves only the Goldstone boson fields has a unique di = 2 term YGB1=-2EF h DagDu (19.6.46)h1, ah with ab some real positive -definite matrix . For (19 .6.12)and a large class of other parameterizations of the carets, y(~ )approaches 1 +i~"xa for ~Q --~ 0, so Eq . (19 .6.14) shows that the linear term in Du,is just aFt~a. The canonically normalized Goldstone boson fields na with a kinematic Lagrangian -2 Ea OpnuOyn,, are the n na = EFa64- (19.6.47) b In the SO(4) theory of the previous section, the generators xa and the fields ~a transform according to an irreducible representation of the un- broken subgroup H = Sn(3 ), so in this case F b = F~jab, with F, a single constant that characterizes the energy scale of the spontaneous symmetry breakdown . In the general case we can choose the generators (without changing the structure constants) so that FQ~, is diagonal, with diagonal elements all equal within each irreducible representation of H, but otherwise independent . We have seen how to construct theories of Goldstone boson fields by starting with fields belonging to linear representations of the broken symmetry group, such as the SU(2) xSU(2) four-nectar 0,(x) with which we started in Section 19.5. At very low energy the only significant degrees of freedom are the Goldstone bosons, so we generally throw away the non- Goldstone parts of the field, such as the field u(x) given by Eq . (19 .5.4). But there are circumstances in which it is necessary to return to full linear representations of the broken symmetry group, in which the Goldstone boson fields are just one part . This happens when, by varying the temperature or turning on an 19.7 Effective Field Theories : SU(3 )xSU(3) 225 external field of some sort, a system is brought close to a second-order transition, in which it smoothly goes from broken to unbroken symmetry . On one side of this transition the symmetry is broken, so we have massless Goldstone bosons, and various other massive excitations, not generally forming complete multiplets that would furnish linear representations of the broken symmetry group . On the other side of the transition the symmetry is unbroken, so here we have complete linear multiplets, not generally massless .If this transition is continuous, then very near the phase transition the Goldstone boson must form part of a complete linear multiplet of nearly massless excitations . Barring accidents or other symmetries, this multiplet will form an irreducible representation of the broken symmetry group . This irreducible multiplet of fields, which become massless only just at the phase transition, is known as the order parameter . This is a more precise definition of order parameters than usual . Often one speaks of an order parameter as any set of fields whose expectation values break the symmetry, but this is too vague -there is no one set of fields whose expectation values can be blamed for a broken symmetry . For instance, the S U(2) x SU(2) symmetry of quantum chramadynamics with massless u and d quarks is broken by the expectation value of iiu +dd, which is the fourth component of a chiral four-vector, but quartic and higher powers of quark fields also have non-vanishing vacuum expectation values, and these belong to other representations of S U(2) x SU(2) .In contrast, at a smooth phase transition at which SU(2) x SU(2) becomes unbroken the Goldstone boson becomes a member of just one massless representation of S U(2)x SU(2), and this provides an unambiguous definition of the order parameter . It is important to identify the correct order parameters, both in order to calculate the critical exponents discussed in Section 18.5, and to deal with configurations like vortex lines or magnetic monopoles, where, as we shall see in Sections 21 .6 and 23 .3, there is a singularity at which the broken symmetry becomes unbroken . It is often assumed that the order parameter for the chiral S U(2)x SU(2)symmetry of quantum chromodynamics is a chiral four-vector, but this is not known to be the case .32a 19.7 Effective F ieldTheories: SU(3) x SU(3 ) The approximate S U(2)isospin symmetry of nuclear physics was extended in the early 1960s by Gell-Mann33 and Ne'eman34 to an even less exact SU(3) symmetry, which grouped the known baryons and mesons in various irreducible representations : an octet of 1/2+ baryons p, n, 11a, E± ,O, -, -°a, an octet of 0- mesons K+ ❑, n~,a,ti °, K-' ❑,an octet of 1 -mesons K*+, ❑5 p±,0, c~, I~* -,0,and a decuplet of 3 /2+ baryons ❑++, +, o,-, F*+'Q'-~ 226 19 Spontaneously Broken Global Symmetries 2*❑,-, 0^ . (The Iand 0were not discovered until later .) After the successes of the chiral S U(2) x SU(2) symmetry in the mid-1960s, it became natural to suppose that the strong interactions also respect an approximate SU(3)x SU(3)symmetry, which like S U(2)x S U(2 )is spontaneously broken to its diagonal subgroup, the SU(3) of Gell-Mann and Ne'eman . Then after the advent of quantum chromodynamics it became clear that this symmetry arises because there are not two but three fairly light quarks ; the u and d, and a third quark s which like d has charge -1/3 . In this case the SU(3) xSU(3) symmetry consists of independent SU(3) transformations (analogous to Eq . (19 .4.2)) on the left- and right-handed parts of the u, d, s quark fields : u u d --+e)Cp[i(O'iL + as AaYS) d a s where A,,are a complete set of traceless Hermitian matrices : 0 1 0 100 0 0 0o0 1a,4 ao o loo 0 00 A7=0 0 -i oi 00-i0 a,2= i0 0 0 00 0 0 -i A5= o a 0 i0 0 to A8=73o1 o0111 a3= 0 0 0 A6=0 0 0 0 -20_1 0(19-7.1) 0 a , 0 o00i 10 (19.7.x) normalized so that Tr (AaiLb) = Z 6ab. The generators of SU(3) xSU(3) are thus represented on the quark fields by the generators ta = Aa of the unbroken S U(3) symmetry and the broken symmetry generators xa = ~ "75. Ta define the Goldstone boson fields and work out their transformation properties, we note that in the representation furnished by the quark fields, any 5 U(3)xSU(3) transformation may be written as exp(-iys A times a transformation exp(i EaOaAa) belonging to the unbroken SU(3) subgroup of SU(3) xSU(3) .(The minus sign accompanying ~ is inserted for convenience in later comparing components of this field with the pion fields introduced in Section 19 .5.) Hence in this representation, each right coset of SU(3) x SU(3)ISU(3)is represented by the matrix of the form y(~)= exp(-i75 Ea ~a Aa)that it contains, and aside from normalization, the parameters ~a of these right cosets may be taken as our Goldstone boson fields . According to Eq .(19.6.18), the transformation rule of these 19.7EffectiveFieldTheories : SU(3) x SU(3 ) fieldsisdictatedby exp zT[0a ~,a+~a ~aYS}expiYSE ~ a[x}it,a u a227 = ex p-P151:~Q(x)~,aexpiE9a(x)Aa ( 19.7.3) a a with 9,,(x) some function of 9V , 9A, and ~ (x) . Also, according to Eq.(19.6.3),the Goldstone-free quark fields q(x) are here defined b y u(x) q(x)= d(x) =exp -i 75E ~a(x)AaRW s(x)a and have the transformation rule given by Eq . (19 .6.24)- 4'(x) = exp i Oa(x)Aaq(x).(19.7.4) (19.7.5) We could proceed to introduce covariant derivatives (19.6.15) and (19.6.30), and use them to construct chiral-invariant Lagrangian densities, but for chiral symmetries there is a simpler approach available . Note that the parts of Eq. (19 .7.3)that are proportional to (1 + ys )and (1 - YS )read expiE9~.~a ex p-iE ~« (x)a u a a a a and exp iE9~ ~aexp i ~ ~ a(x)a R a a =CXp i E~a(x)A, exp i EOa(x)Aa a u where(19.7.6) (19-7 .7) By multiplying Eq .(19.7.7)on the right by the inverse of Eq . (19 .7.6), we find the simple transformation rul e U'(x) = exp(i>IaO )~U(X) exp(_ia )Oa (19.7.8) a a 228 19 Spontaneously Broken Global Symmetries where U(x) is the unitary unimodular matri x U(x) = ex p(2iE~,(x)Aa (19.7.9) I n other words, U(x) transforms according to the (j,3) representation of SU(3)x SU(3). This is still a non-linear realization of S U(3)x SU(3) because the components of U(x) are not independent ; they are subject to the nonlinear constraints UTU = Iand Det U=1. The unique (S U(3)x S U(3)}-invariant term in the Goldstone boson Lagrangian that is of second order in spacetime derivatives i s Y2deriv = -~~F2 Tr {[~~~1 [~ '`U~I, with F2 a positive constant to be determined . We can express the ~a in terms of conventionally normalized pseudoscalar meson fields by writin g ~A',~a,11 Fa~na + ~a n K-n+ T2 T6 FCUK+ K❑-,,lr2-B 0❑F 3 (19.7.11) so that the kinematic term in Eq .(19.7.10)is of the usual for m skin =~2OPn0O F1 n4-0 Yn+atin-oPK+Opk -OPK❑Ot`k❑-z0Pt~Opno. To determine the constant F, we note by comparing Eq . (19 .7.4) with Eq. (19 .5.42), that for ~ infinitesimal, the components ~ ;, ~2, and ~3 are the same as the Goldstone boson fields C1,C2, and C3 introduced in Section 19.5. Then, comparing (19.7.11) with (19.5.17) and (19.5.32), we see that the constant F defined by Eq .(19.7.10) is the same as the F ,=184 MeV introduced in Section 19 .4. The SU(3) xSU(3) symmetry of quantum chromodynamics is broken by the quark mass term, which in terms of the quark field q(x) defined by Eq. (19.7.4) is ama ss~ -4 Mqq= -q e-' ~~sBl ~'~~q e-i,1'2Y5B1FR (19 .7.12) where mu Mq0 00 md 00 0 ms(19.7.13) Eq. (t9 .7.1) contains a purely bosonic part, obtained by replacing the quark bilinear with its vacuum expectation value, given by the unbroken 19.7 Effective Field Theories :SU(3) xSU(3 ) SU(3) an d parity symmetries as * 'Z Z<4n7S4m >0= 0 <gnqrn>0= - vb,m The Goldstone boson mass term in the Lagrangian is then229 (19.7.14) 2Vv[ F~ Tr fB, JB, M, ~} = ~ F2 4mu no+ ~ ° +4(mu +and}n+ n Fn +4(mu +ms)K+K +4md 1 n ❑+ 8+4(md + ms)K ❑ka + 3ms(0) 2 From this, we can read off that35 2 24v m+ =moo = Cmu +m d] F2 2 4v x+ = 2 Cmu +ms s F 24vmKo = F2 [md+- Ind 204a4m,-+-and+-mu mY F2 302 (19.7-15) and there is also a term that mixes the n ❑and q❑ 2 4 u ~3 The same results can also be obtained by operator methods . theorem allows us to construct generators Ta and tea, for which(19.7.17) Noet her's [Ta,4)=-Aaq, lXa~ R ]= -75Aa4  (9 .7.18) We can write the Goldstone boson fields in a real basis as na, with n± = (nl ± i.2)1,/r2, n0 = n3, K± = (74 ± ing)I,/r2, K° == (75 + i7r7)/,/'2 ,f<0= (n5 -in WJ2 ,and rj❑= n$ .The SU(3) symmetry that survives the spontaneous symmetry breaking tells us that the F-matri x The fact that the conservation laws of parity, charge, and strangeness respected bythe vacuum are the same as those respected by the quark mass matrix is a result of the vacuum alignment condition discussed in Section 19 .3. Likewise, if the quark ficids are defined so that the quark masses are all positive, then with v> 0 the minus sign in Eq.(19.7.3) is required by the condition that the true vacuum be at a minimum rather than a maximum of the vacuum energy for vacuum states rotated by SU(3) xSU(3) transformations, for as we shall see, it is this sign that will yield positive masses for the pseudo-Goldstone boson octet . 230 19Spontaneously Broken Global Symmetrie s for these bosons takes the form Fah = F, ,bab The SU(3) xSU(3) sym- metry is also intrinsically broken by the mass term in the Hamiltonian Ht = muuU +mddd + msss = qM qq, where M qis the quark mass matrix (19.7.13).The mass matrix of the pseudo-Goldstone bosons is given here by(19.3.20) as M~b =_Fn2(4{Aa,fAh,Mqs} 4)0. (19.7.19) Since this is already of first order in quark masses, to this order we may use the unbroken SU(3) relations ( uu)a = (dd)a = ( ss)a=_-a, and find the results (19.7.16) and (19 .7.17), as before . The masses mK+ = 493 .65 MeV and mKo = 497 .7 MeV of the kaon doublet are quite close, and considerably larger than the pion mass, so we can see from (19 .7,1 b )that the mu and and must be considerably smaller than m, . In calculating quantities that are sensitive to mu and md, like the kaon mass difference or the pion masses, we should also take into account another small correction, the effect of electromagnetism . The electromagnetic current is J° = iegy yQq, where Q is a diagonal matrix with elements 2 /3, -1/3, and -1 /3. Its commutators with the SU(3) xSU(3) generators are 2 2 We see that Pcommutes with X3, X6, X 7,and X$as well as T3, T6, T7 , and T8,so the electromagnetic part of the Hamiltonian is invariant under the S U(2)x S U(2 )xU(1) xU(1) subgroup with these generators . Therefore in the limit of zero quark masses, electromagnetic effects give no mass to the neutral pseudo-Goldstone bosons35 n4, K0, K0, and rj❑,the Goldstone bosons associated with the spontaneous breakdown of the symmetries with generators X3, X6, X7and X8 . Also, in the zero-quark-mass limit there is an unbroken SU(2) symmetry generated by T6, T7 , and -,~13_T $ - T3, under which the K+ and n+ transform as a doublet, so that for zero quark masses the electromagnetic corrections to the K+ a nd n+ masses are equal .36 Since the quark mass terms and electromagnetic corrections are all small, it is reasonable to treat these effects as additive corrections to the effective quark Hamiltonian . We see then that with electromagnetism taken into account, the mass formulas (19 .7.16)should be corrected to read37 mK❑=m~a = 4v(md + m,)/F2 m~± = 4v(md+m~)/F~ + ❑, (19.7.24) , moo = 4v(mj +- m,,)/F,2 , m~ = 4v(mu+`and + -4ms)/3F~ 19.7Effec-tiveField Theories :SU(3) X SU(3) 231 where ❑is the common electromagnet iccorrection to the K+ and R + squared masses . These formulas impose one linear relation onthe five pseudo-Goldstone masses, a version of the Dell-Mann -Okuba relation :** 3m~ + 2m~+ - m~~, = 2m~+ + 2mh U . (19.7.21) Taking the kaon and pion masses from experiment, this relation yields an qmass of 566 MeV, in comparison with the experimental value of 547 MeV . The small discrepancy is usually attributed to the mixing of the q state with a heavier pseudoscalar particle, the q' at 958 MeV . From the mass formulas (19.7.20)we may derive formulas for the quark mass ratios in terms of the pion and kwon masses :37 mu mko+m~+ - mx+ ms moo + m~+--m~+2 2 2 2Mu 2m~o -moo - m,t+ + m K+ M,5= m~o+M2+_m~+ (19.7.22) Using the mass values m,+ = 139 .57 MeV, mRo = 134 .974 MeV, mK+ = 493.65MeV and moo = 497 .7 MeV gives the ratios md/ms = 0.050 and rnu/ms = 0.027. Thus the ratio of the d and u quark masses is closer to 2 than 1 . (A 1996 calculation38a using various other pieces of information as well as pseudoscalar meson masses has given values m d1ms = 0.053 ±0 .002 and mu/ms = 0.029 ± 0.003.) None of this gives definite values for the individual quark masses . Indeed, these masses are not well defined until we define a renormalization prescription for the quark bilinears . Often this prescription is fashioned so that rns equals the mass difference between isomultiplets differing by one unit of strangeness in some S U (3) multiplet . For instance, the lightest vector meson octet -consists of an isotopic doublet K* at 89 2 MeV, interpreted as a bound state of an s antiquark and a u ord; its antidaublet, consisting of an s and a Ti or d ; and a T = 1 pat 77 0 This relation was originally derived" on the basis of the approximate S U(3) symmetry of Gell-Mann and Ne'ernan generated by the TR, ignoring mass differences within isotopic multiplets . From this derivation, one cannot tell whether this relation should be applied to the pseudoscalar meson masses themselves, or as here to their squares . Applying this relation to the pseudoscalar masses themselves would not in fact work very well ; it gives an qmass of 613 MeV . The fact that the Gell-Mann-Okubo relation works well for the squares of the meson masses, but not for their first powers, supports the interpretation of these particles as Goldstone bosons of a spontaneously broken approximate SU(3) xS U(3) symmetry . Gell-Mann and Okubo also used the approximate SU(3) symmetry generated by the Tu to derive relations among the masses of other particles, such as (ignoring isospin violating effects) the relation 2mN + 2m .~= 3mA + m yamong the masses of the lightest baryon octet . For such rnultiplets, the average mass is so much larger than the mass differences within the multiplet that it makes little difference whether one applies the relation to the masses or their squares . 232 19Spontaneously Broken Global Symmetrie s MeV and a T = 0o) at 783 MeV, both interpreted as bound states of a u or dand a u or d.If we wish to attribute the difference between the K *mass and the average of the pand cv masses to the relatively large s quark mass, then we must renormalize the quark bilinear so that m,s - 2 (mu + m d)= mx  - 2 (m~, + m(~, ) = 120 MeV, yielding rns = 125 MeV . With the above quark mass ratios, this gives rn d=6.0MeV and mu = 3 .3 MeV . However, these estimates of mass values are much less reliable than the values of the mass ratios given in Eq .(19.7.22).Often ms is estimated381i as 18 0MeV rather than 1 25 MeV . The mass term (19.7.12) includes meson-meson interactions . Using Eqs. (19 .7-14), (19.7.11), and (19.7.9), we may write the purely bosonic part of this term in the Lagrangian a s Ymass,basonic = 2 vTr~Nlq(Ut + U) j (19.7.23) The form of this term could have been deduced from general symmetry considerations, which also allow us to find the allowed terms of higher order in Ilrl,.Suppose we invent an external 3 x 3 field x, and replace the mass term (19 .7.12) in the underlying quantum chromodynamics La- grangian with the x-quark coupling ter m Yx = -q [2M+ ys )x + 120 - ys )Xflq , (19.7.24) which becomes the same as Eq .(19.7.12) when we make the replacements x = X f=Mq.The point of this procedure is that the Lagrangian (19 .7.24) becomes formally invariant under SU(3) xSU(3) if we give Z the formal transformation rul e x ---> ex p(iiliO )ORx ex p1: AaBL a (Z ) -(19.7.25) Thus we can work out the allowed bosonic terms involving quark masses by w riting the most general SU(3 ) xSU(3) Lagrangian (up to some given order in de rivativesand Mq) involving U and x,requiring also invariance under the pa rity transformatio n U(x,t)+-*Ut(-x,t), and then making the replacement s x=xf=Mq . For instance, the interaction Tr (Ufx + Ux f)is invariant under S U (3 )x SU(3) and parity, and becomes of the same form as Eq .(19.7.23) when we give x the values (19.7-27) . Using this technique, Gasser and Leutwyler39 have given the complete effective Lagrangian for the pseudoscalar octet of fourth order in moment ax<-->xf , (19 .7.26) 19.7Effective Field Theories :SU(3) xSU(3) 233 or meson masses (with quark masses counted as being of second order in meson masses) a s Y4 = L1 Tr I 0,,UfOyU)2 +L2 Tr ~ O,UO,Uf ITr JOITOvUf I +L3 Tr ~ OuUapUf avUavUf I + L4 Tr ~ OiLUat`UtITrfwu + 01 +L5 Tr~a, U agUf (NIgU + U tMq)}+ L6 [Trl Mq(U + Ut)}]2 +L7[Tr ~( Uf-U)M q1] 2 +L8Trj((UM')l +(UtMq)2)I, (19 .7.28) where Li, ... ,L8are constants to be determined by comparison with experiment . The complete effective Lagrangian up to fourth order in meson masses and momenta i s yeR =Yz +Y4, (19 .7.29) where Y2 is the sum of the terms (19 .7.10)and (19.7.23): Y2 - - ,~F2Tr ~O~U 0pU~'~ + ~ vTr1llrlqV + U} j (19.7.30) Electroweak interactions may be included by replacing the derivatives 0.with suitable gauge-covariant derivatives Du, and adding a few additional terms to Y,ff. Following the same power-counting arguments as in Sections 19 .5 and 19.61to calculate S-matrix elements to fourth order in meson masses and momenta we must include both tree graphs to all orders in Y2 and up to first order in Y4, and also one-loop graphs constructed from Y2 alone . Gasser and Leutwyler and others have carried out a comprehensive study of meson dynamics (and associated electroweak interactions) using this effective Lagrangian .4° The quark mass term (19.7.12)naturally affects other SU(3) multiplets . For any multiplet other than the pseudoscalar octet this can be treated as a first-order perturbation, and therefore gives a shift in the mass matrix of a generic multiplet ji} equal to (19.7.31) 6rni~=~z~4Mg4l.1) and SU(3) may be used 41to relate the various matrix elements( i0 rqslk In this way, it is straightforward to show tha t imp - 6mn= ME -MI (19 .7.32) MU - md ms Using th is together with Eq .(19.7.22) givesthe quark mass contribution to the nucleon mass splitting bmP-6rnn.- -2 .5 MeV . This result may be used in Eqs .(19.5.65) and (19 .5.66)to calculate the leading violations of 234 19Spontaneously Broken Global Symmetrie s isospin symmetry in low energy pion-nucleon interactions . Of course, as defined here imp - 6mnis not the full proton-neutron mass difference, which receives an important contribution also from photon emission and absorption . Because the neutron is electrically neutral, this electromag- netic term is almost certainly positive, in agreement with the fact that the observed proton-neutron mass difference is -1 .3 MeV, leaving +1 .2 MeV to be accounted for by electromagnetism . Unfortunately, accurate calculation of this electromagnetic mass difference has proved difficult . 19.5 Anomalous Te rms inEffective F ield Th earies* In implementing the effective field theory program described in the previ- ous three sections, it is necessary that the action should contain all possible terms allowed by the assumed symmetries of the theory . The methods de- scribed in these sections allow us to identify all manifestly invariant terms, either by using the general formalism of covariant derivatives described in Section 19,6, or for chiral symmetries by using a linearly transforming field subject to non-linear constraints, like the U(x) of Section 19 .7. But it is possible that there may be other terms in the action of the effective field theory that are anomalous in the sense of being given by integrals of four-dimensional Lagrangian densities that are not invariant, but whose variation under the broken symmetry is a spacetime derivative, preserving the invariance of the corresponding term in the action . As we shall see in Section 22.7, such a term was discovered for S U(3 )x SU(3) by Wess and Zumino,42 in a study of `anomalies' due to quark loop graphs . However, the structure of this term can be understood without knowing anything about the underlying theory of quarks and gluons . The easiest way to describe the Wess-Zumino term is by an extension of spacetime to five dimensions, introduced for this purpose by Witten .43 As long as we require the fields in an effective field theory to approach a common limit as x1l -> oa in any direction, we can think of spacetime as having the topology of a sphere S4, with the point at infinity included as an ordinary point . As remarked in Section 19 .6, when a group G is broken to a subgroup H, the possible values of the Goldstone boson fields ~,,at any spacetime point may be regarded as defining a point in the coset space G/H (the space of elements of G, with two elements identified if they differ by multiplication on the right by elements of H), so a set of functions ~,(x) represents a mapping of the spacetime S4 sphere into GBH .Depending on the topology of GBH, it may be possible smoothly t o This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 19.8Anomalous Terms in Effective Field Theories 235 deform any four-sphere in G/H to a point ; that is, it may be possible to extend any ~a(x) to a continuous function ~ a(x;s)defined for 0 ~ s ~ 1, for which ~a(x ;0) = ~,,(x) and ~a(x ;1) is any fixed point (say, ~u = 0) on the original sphere . Where this is true, it is expressed mathematically as the statement that the homotopy group 7E4(GIH) is trivial . (For a discussion of homotopy groups, see Section 23 .2.) It is known that this is the case for SU(N) x SU(N) spontaneously broken to SU(N), where the coset space SU(N) x S U(1V )/S U( N)has the topology of SU(N) itself .` Hence in the case of physical interest, where G = SU(3 ) x SU(3) and H = SU(3), we may extend the Goldstone boson field, or equivalently U(x), (see Eq . (19 .7.9)) to a unitary unimodular matrix U(y) defined in a five-dimensional ball B5 with coordinates x" and s, whose surface is the four-dimensional sphere of spacetirne . Now consider the following function formed from U(y)- W(Y)iijk2mT'r~J-1OU U _1OU U -1OU U -1OU U -1OU = -240 R2 ~71~y~ Wyk ~y~ OyM (19.8.1) where indices i, j, etc . run over the five coordinate directions for the coordinates x" and s . (The phase and numerical coefficient are chosen for later convenience .) This is manifestly invariant under the chiral transformations (19.7.8).Also, because ~~jk2m is a tensor density, the integral of w(y) over the five-ball is manifestly independent (up to a sign) of the choice of five-dimensional coordinates y '. Furthermore, this integral depends only on the values taken by U(y) on the ball's surface ; that is, in spacetime . To check this last point, note that when we make an infinitesimal variation 5U(y) in U(y )in the interior of the ball, w(y ) changes by a derivative : -48~2rjkrm 0TrU-tOU U-iau U_1 OU U-1 OU U-1jU aym cry Oy ay Oy (19.8.2) (for this calculation, see Section 23 .4), so a change in U(y) that does not affect its value in spacetime also does not affect the integral of (19.8.1) over the five-ball B5 whose surface is spacetime . We can therefore include this integral as a term in the action : Iwzw [U]= n dsvW(y) (19 .8.3)$5 with n a coefficient that so far is arbitrary . This is shown by the fact that we can use 5U(N) elements U(x) to represent the right cosecs of SU(N) in 5 U(N) x 5U(N) . 236 19Spontaneously Broken Global Symmetrie s This term may be written as the four-dimensional integral of a La- grangian density, but not of an (SU(3 )x SU(3)}-invariant Lagrangian density . Using Eq .(19.7.9),the leading term in c)(x) in the limit of small meson fields i s ~ ~8-\/2ijklmOB OB OB O $OB (1g .$.4} Tr ~~i ~~~ Wyk ~yr gy m ~r)x ~5~2Fn 6 where B is the matrix (19 .7.11) of Goldstone boson fields . It follows then from Gauss's theorem tha t 18~'~6P vp~~d4x T r BOBOB OB r~B+0B6 wzw~U~-15~2F~~ Js40xE,fix"0x~'ax, F~n (19.8.5) Although chiral invariant, this cannot be written as the integral of a chiral-invariant density over spacetime, because any chiral invariant den- sity would have to be constructed out of the first and higher covariant derivatives of the Goldstone boson fields, and so, when expanded in pow- ers of the Goldstone boson fields, such an invariant density would begin with a term involving only derivatives of B, not B itself . As Witten noted, the inclusion of this term in the effective action resolved what otherwise would have been a conflict between the SU(3) x SU(3) effective field theory and experiment . Because no ~~'P' terms appear in the effective Lagrangians (19 .7.28)and (19.7.30) (or in higher- order terms of this sort) parity conservation imposes the requirement that these terms are even in the Goldstone boson fields, ruling out processes like K + K -> 37r . Not only is there no symmetry in the underlying theory of quantum chromodynamics that would account for such a selection rul e there is even experimental evidence against it, for as Witten pointed out, the 0meson is observed to decay both into K + Kand 3 7c final states . Eq.(19.8.5)shows that this unwanted selection rule is removed by the Wess-Zumino-Witten term in the action . Remarkably, the coefficient of the Weis-Zumina-Witten term is not a freely adjustable parameter . As Witten showed,43 this is because, although this term is not changed by smooth deformations of the function U(y ) in B5 that do not affect U(x) on the spacetime boundary S4, the Wess- Zumino-Witten term can be changed by a discontinuous change in U(y) that leaves U(x) unchanged on the boundary . We may think of the five- ball B5 as half of a five-sphere SS, with the spacetime S4 as the border between B5 and the other half B.(Think of SS as analogous to the surface of the Earth, with the spacetime S4 as the equator and B5 and BS as the northern and southern hemispheres .) Because S4 is also the boundary of the other half of S 5, we could just as well have written a 19.8Anomalous Terms in Effective Field Theories 237 Wes -Zumina -Witten term a s IwZw[U]=-n1Bd5Y(O(Y) (19.8.6) with a minus sign inserted because the boundary of BS is the four-sphere with opposite orientation .It is not possible to require that the terms (19.8.3) and (19 .8.6 should be equal for arbitrary Goldstone boson fields without setting n -- 0, but in order that the weighting factor exp(iI) in path integrals should be unaffected by the difference between the terms (19.8.3) and (19 .$.6) , it is only necessary to require that this difference should be 27E times an integer . That is , jwzw [U1- Iwzw [U] = n fdsyco(y) = 27cxinteger . (18 .8.7) S5 With the normalization factors we have inserted in the definit ion(19.8.1) ofw(y),the integral ofo)(y)over any five-sphere turns out to have the value44 2 7r. It follows that thecoefficientnmust be an integer. The example of the We ss-Zum ino--Witten term raises the question whether there may be other anomalous terms in the action, not nece ssarily related to quark loops, that also are invariant under SU(3) x SU(3), despite not being four-dimensional integra l sof (S U(3 ) xS U(3 ))-invariant Lagrang ian dens ities. Fortunately, the answer is no .It has been shown45 that for a general group G broken to an arb itrary subgroup H (with 7E4(GIH)=0),any term F[ fl in the action of the Goldstone boson fields ~a(x)may always be written as the integral of a G-invariant five-form 0 over a five-ball B5 whose boundary is the spacetime four-sphe reS4: fdSy6ijk[►~~~~04 O ~rO~dO ~e Dahrde (~(y)) (19.8.8)B5 Y Y Y Y Y In order that this should be independent of the particular way that ~a(x) is extended into the interior of the five-ball, it is necessary that 0should be exact, in the sense that there it is the external derivative of a four-for m Qahrde(O = (010~[a)yhcde~(O (19.5.9) (with square brackets as usual indicating antisymmetrization with respect to the enclosed indices), so tha t O~c~d F [fl d4x El"~ 16`3 Y~2hcd(~ ~- (19.8.10)Js Oxg Oxv OxP Oxf f 4 It follows that Qis also closed ; that is, it has a vanishing exterior derivativ e (010~U')Quhcde](~)=0. (19.8.11) Where the four-form Yab,d(~) is also G-invariant, the functional (19 .8.10) is just one of the ordinary manifestly G-invariant terms in the action, 238 19 Spontaneously Broken Global Symmetrie s discussed in the previous three sections .The anomalous terms in the action arise from the possibility that, although every term in SZ(y) is G-invariant and the exterior derivative of afour-farm, some terms may not be the exterior derivatives of G-invariant four-forms . Thus the new terms in the action may be identified with the closed G-invariant five- forms that are independent, in the sense that no real linear combination of them is the exterior derivative of a G-invariant four-form . These are known in mathematics as the generators of the deRham cohomorogy group HS(G/H ;R). (The group multiplication rule here is just simple addition .) The de Rham cohomology groups have been calculated for manifolds of various topologies . 46 In particular, H5(SU(N ) xSU(N)I SU(N );R} has a single generator, given by Eq . (19 .8.1). Thus without knowing anything about the underlying theory of quarks and gluons, we can learn everything about the anomalous terms in the Goldstone boson action, with the single exception of the value of the integer n . We will see in Section 22 .7 that in SU(1V,) gauge theories this integer equals the number N, of colors, which in quantum chromodynamics is n = 3 . 19.9 Unbroken Sy mmetries We have seen how the properties of Goldstone bosons and their low energy interactions may be deduced from an assumption that the theory is invariant (or approximately invariant) under a group G spontaneously broken to a subgroup H . But in applying these methods to the cases where G is S U(2) x S U(2 )or S U(3 )x S U(3 ), we had to take the pattern of sym- metry breaking, of SU(2) xSU(2) orSU(3) xSU(3) to their non-chiral SU(2) orSU(3) subgroups, from experiment . 1 n Section 2 2.5 we shall show that the SU(3) xSU(3) symmetry for massless u, d, and s quarks must in fact be spontaneously broken in quantum chromodynamics, but it is more difficult to show on the basis of quantum chromodynamics that the SU(2) xSU(2) symmetry with only u and d massless is also spontaneously broken! On the other hand, there is an intuitive argument that their non-chiral SU(2) or SU(3 )subgroups are not broken, based on a conjecture known as the persistent mass condition48 ,which states that composite particles will not be massless if the particles of which they ar e Weingarten47 has used lattice methods to show that whether or not chiral symmetry is broken, the lightest particle in quantum chromodynamics with massless u and d quarks must have the quantum numbers of the pion . As we will see in Section 22 .5, the existence of anomalies due to fennion loops in quantum chromodynarniGS together with the assumption of quark trapping requires that some hadron bemassless, so it follows that the pion is massless, which strongly suggests that chiral symmetry is spontaneously broken . 19.9 Unbroken Symmetries 239 composed are massive . Non-chiral symmetries like isospin conservation are not violated if we give the quarks equal masses, and then if they were spontaneouly broken we would have the massless Goldstone bosons formed as composites of massive quarks, in contradiction with the per- sistent mass condition . In what follows we shall present a proof by Vafa and Witten49 that in gauge theories like quantum chramodynamics those non-chiral symmetries that are not violated by quark masses can not be spontaneously broken . This result is of more than academic interest ; as we shall see in Section X1 .4, it is possible that the spontaneous breakdown of electroweak gauge symmetries is described by a `technicalar' theory similar to quantum chromodynamics, and in testing this idea it is important to know what symmetries are left unbroken in this theory . Consider a gauge theory like quantum chromodynamics, with a number of fermion "flavors' in identical representations of the gauge group . If all fermions have masses, then the theory will be invariant under all unitary global non-chiral transformations on the fermion flavors that commute with the fermion mass matrix . For instance, if ni fermions are degenerate with a common mass ml, n2 fermians are degenerate with some other common mass m2, and so on, then this global symmetry group is U(ni) x U(n2) X .... (As a special case, if there are no degeneracies we have a global symmetry under U(1) xU(1) x...; an example is the conservation of baryon number, strangeness, etc . in quantum chromodynamics .)These symmetries cannot be spontaneously broken . To prove this, let us consider a general Greens function for r fermion and antifermion fields ." In the path-integral formalism this is t ~T{iki(X1) ...Turk,(x,)Y ~,iI(yi)...Tirrr(vr)}SvAC~ ~f = zf[dA][dTJ['^.y)t]V) fUi kl( X I ) ..}1)urkrlXY1`i~L'j 1 1lJ 1 ~ ... '(~]4;rjrIyY J X exp(iIgaugeA] + i IDirac IT, W ~ ; A] )> (19 .9.1) where k j...kr and 11... lr are Dirac spin indices, ul ... u, and vt ... yr are fla- In the original work of Vafa and Witten,49 they first proved the absence of symmetry breaking in the case r = Iwith x = y, then observed that the absence of symmetry breaking in this vacuum expectation value did not rule out the possibility of a spontaneous symmetry breakdown occurring in other Greens functions, and so went on to different methods of proof . } As we saw in Section 15.5, both numerator and denominator are proportional to the infinite volume of the gauge group, which cancels in the ratio (19.91) .The presence of this infinite factor detracts from the rigor of the following arguments, but if we were to remove it byintroducing ghosts then some of the steps below that depend on the positivity of the action would raise difficulties . One way to deal with this problem is to replace the spacetime continuum with a finite lattice of points, in which case the gauge group has a finite volume, and no gauge fixing or ghosts are needed . 240 19Spontaneously Broken Global Symmetrie s gar indices, iga„ge [A] is the action for a pure gauge theory, Dirac [y)ay)t, A] is the action for the Dirac fields in the presence of a gauge field Ax (x ), and Z is the vacuum-vacuum amplitud e Z - f[dA][thr][thrt] exP (ilgauge[A] + ilDirac [T,Wf;A]}  (19.9.2) It will be necessary here to work in a Euclidean spacetime, with x4 = X4= ix°, y4 = Y4= iy0, and A4 = A4a = iAa, all real . (See Appendix A of Chapter 23.)In this case, the Dirac action i s IDirac [Y),Wt; A]= ifd3xjdx4 W, 19+ M] Ya , (19.9.3) where M is the fermian mass matrix, and Pis the Euclidean gauge- covariant derivative contracted with the Euclidean Dirac matrice s 4 - ~(Gi - it'Aia)yi =i(19.9.4) where a susual y4=V.Because the act ion isquadratic in fermian fields, we can explic itly perform the integral over these field s ..Turkr(xr)`'~, 2, (y1)...Tzrir(yr)j)VAc ~T{iiki(X1) . =If[dA] Det (1?+M) exp (ilgauge [ Al (]~ tX+M) xlul kl, y j vir1... ~~ + M)xrlurkr,Yr Urit ±permutations (19.9.5) where `± permutations' indicates that we must sum over all r! permuta- tions, of the Yfields, with a minus sign for odd permutations, an d Z=f[c4] Det (1?+ llrl)exp (ilgaugerA]} - ( 7g.9.6) The expression (19 .9.5)is manifestly invariant under any unitary transfor- mation on flavor indices that commutes with the mass matrix M .It does not matter what non-perturbative effects are produced by the functional integral over gauge fields and ghosts ; these fields are inert under the symmetries in question here, so that the remaining functional integral in Eq.(19.9.5)cannot break these symmetries .ft But for this argument to be convincing, we must show that the expression (19.9.5)is well defined . ~t It is more difficult to show that P, C,and T are not spontaneously broken in quantum chromodynamics with massive quarks, because these symmetries act non-trivially on the gauge fields . Vafa and Witten50 have applied the methods of Ref . 49 to show that P is not spontaneously broken in quantum chromodynamics . 19.9Unbroken Symmetries 24 1 This is not merely an academic question of mathematical rigor . As we saw in Section 19 .1, the sign that a symmetry is not spontaneously broken is not just that the ground state is invariant under the symmetry transformations, but also that the symmetry of the ground state is stable under small perturbations . If we break the symmetries of M by adding a small perturbation 6M ,and if the expression (7 9 .9.5)becomes singular as 6M -+0,then factors of 6M in symmetry-breaking matrix elements may be cancelled by the singularities in the matrix elements for 6M =0. Precisely this happens for chiral symmetries, which arise in a symmetry limit where some of the eigenvalues of Mvanish . In this case, as we approach the symmetry limit of zero mass, the factors of mass in the numerators of symmetry-breaking expectation values are cancelled by factors of mass in the denominators of the propagators (9+M) -1 wherever ,Phas zero eigenvalues . For this reason, the arguments here will apply only to the non-chiral symmetries of theories with mass matrices M whose eigenvalues are all non-zero . To set a bound on the matrix element (19.9.5)in this case, note first that the differential operator (19 .9.4) is here antihermitian, so as long as the hermitian matrix M has no null eigenvalues, 9+Mhas a well-defined inverse . It is still necessary to show that the remaining integration over the gauge fields in Eq .(19.9.5),including that in Z, does not make this expression singular when M satisfies the conditions for a symmetry, for instance that n of its eigenvalues are equal for a U(n) symmetry . As we shall see, this will insure that when a fermion mass matrix Althat is invariant under some global symmetry transformation is perturbed by adding a small term 6M that breaks this symmetry, the change in the expectation value (19 .9.4) under this symmetry transformation vanishes in the limit 6M-+ 0 . It is difficult to show that the coordinate-space Greens functions Eq.(19.9.5)are non-singular, so let us consider instead the matrix el- ement for smeared field s Tu[f]=Jd4xfkt(x .)PUk(x .). (19.9.7) wherefk(x) are arbitrary smooth square-integrable functions . In these terms, Eq .(19.9.5)read s KT{kPui [f,]... TUr [.ft']T', [g 1]... T' 19r]J)VAC [dA]Det(9+M) exp (iIgauge [A] ) Z 7 X[lY" + M/,~11U1 ~$1Z1...1J]~+ MI.~r fir .$rLr~'- permutationsI (19.9.8) 242 where19Spontaneously Broken Global Symmetrie s w +111}~ ~ g V/d4x .fd4y f"(x)(9+M)xuk, yUi g 1(Y). (19.9.9) In this basis, the fermion propagators for a given gauge field are not only well defined, but bounded uniformly in Aa(x) :f w+M) fu,gvi(19 .9.10) where m ,are the eigenvalues of M, and for convenience we are now normaliz ing the functions f 'i(x)and g j(y)sothat Jf(x .)f1(x .)d4x . =./g(y)gj(v)d4v =1. (19.9.11) (The summation convention is suspended here .) Furthermore, the weight function for the average over gauge fields is positive :# Det (P+M)exp(ilgauge[A]) = expJd3xjdx4 x Let ~(iD)2+11rf2].4 4 Fjz [=I j=I (19.9.12) With a positive weight function, the average of any function is bounded by the bound of that function . Using the bound (19 .9.10) in Eq .(19.9.8), ITo see this, we may expand in the eigenvectors of M MuXVU-maa'Cu~~u ~u 5ctfi and, adapting a trick of Vafa and Witten ,4y write Eq. (19 .9.9)as + "}fu KvCu*C afd4x Jd4Y.f kt(x) (JO+ m«)xk, y z gi(Y) ('x ±t.u'ca Jexp~-~ma~7) Jd4x~d4Y fkt(x) tCxP(+z9))xk, )lgt(Y) ~ where f is the sign of ma . The matrix c°' cv and the operator [exp(Tz P))k.!are both unitary, s o ucajd4xJd4y fkt(x) (exp(Tzjp ))xa,Yig t( Y) c Jd4xi f( x)~~ Id4 YIg( Y) I2=1- Using this inside the integral yields Eq .(19.9.10). In the Permian determinant, we use the fact that Det ( P+M)= Det y5 ( P+M)y5 19.10 T heU(1) Proble m the matrix element is bounded by243 r (T{'ji1i...T:firUr]'I't, [gt ]... 'Pt[9r] } 1 VAC ~ r! ~ ~ m a Thus there are no singu larities as the symmetry-breaking terms 6M in M go to zero that could cancel factors of 6M.To see this more explicitly, we can use the same methods to show that if we perturb M by a small symmetry-breaking term 6N f, then the term in Eq . (19 .9.8) of first order in6M is bounded b y b(7'I'~u,[1i ]...furVrj'~'v,[gI]...TUr[gr]} )VA C r r!E ~0 M) nbI a,b ma mbr-I Imal(19-9.14) so this vanishes when 6M ---} 0 . In the real world none of the quark masses are zero or degenerate, so it follows immediately from the above arguments that the U(1)symmetries like conservation of baryon number, strangeness, etc ., are neither sponta- neously nor intrinsically broken . The other strong-interaction symmetries like isospin or SU(3) are more problematic . These symmetries would be completely unbroken if the u and dquarks, or u, d,and s quarks, had equal non-zero masses, but as we saw in Section 19 .7, these masses are not at all degenerate ; the isospin and SU(3) symmetries arise because the quark masses are small, not equal . These symmetries do remain unbroken if we give two or three quarks equal masses and let the masses become arbitrarily small, so isospin or S U(3) will be good symmetries in any pro- cess that is not sensitive to the small quark masses . But not all processes are insensitive to these masses, because for two or three massless quarks, the pion or the pseudoscalar meson octet to which it belongs becomes massless . This is not a problem for isospin, because as we saw in Section 19.5, the pion triplet remains degenerate to first order in quark masses even though m,, * Md .For S U(3) the quark mass differences produce first- order mass differences among the pion, kaon, and eta, so that processes dominated by one-meson poles canshow large departures from SU(3) . 19.10 Th e U(1) Probl em The success of quantum chromodynamics in explaining the pattern of strong-interaction symmetries seemed at first to be marred by one failure . As we saw in Section 19.5, the broken SU(2) xSU(2) symmetry of the strong interactions is a natural consequence of the smallness of the u and 244 19 Spontaneously Broken Global Symmetrie s d quark masses . But the Lagrangian of quantum chromodynamics with small u and d quark masses also has another chiral symmetry ,"theUMA symmetry under transformation s u--+exp(iYsB)u , d --+exp(i750)d . (19.10.1) Such a symmetry if unbroken would, like SU(2) xSU(2), impose a parity doubling on the hadron spectrum, but no such parity doubling is observed . On the other hand, a broken U(1)A symmetry would imply the existence of an isoscalar Q- Goldstone boson with a mass comparable to that of the pion, and this Goldstone boson is also not observed . It is true that the qmeson is an isoscalar 0- boson, but it is considerably heavier than the pion, and as we saw in Section 19 .7, it is well understood as one of the Goldstone bosons of SU(3) x SU(3 ). With the s as well as the u and d quarks regarded as relatively light, the Lagrangian of quantum chromodynamics would have a U(1) chiral symmetry in addition to S U(3)x SU(3), under the transformation s u -} exp(iyAu, d --+exp(iy5E)ct , s -} exp(i 750) s. (19.10.2) The spontaneous breakdown of such a symmetry would require the exist- ence of two isoscalar 0- mesons : one the q, and the other with a mass comparable to that of the pion . This prediction can be made more explicit .52If we include the transfor- mation (19 .7 0.2)among the spontaneously broken symmetries of massless quantum chromodynamics, then in the presence of quark masses we en- counter a term (19.7.12): amass Mqq=-qCiNflj 581F,M. e-iN(2v5BIF,, where again mu 0 0 Mq =0 Md 0 00 ms(19.10.4) but now B includes a Goldstone boson field Cfor the broken symmetr y B n K- F"+0 IFS oIT+K+ 3" as (0 0 C 19.1 0 The U(1) Problem 245 with F~ the unknown coupling of the U(1 )A Goldstone boson to the cor- responding current (and the factor _- .13- inserted for future convenience .) We can again use the unbroken S U(3) symmetry to write the vacuum ex- pectation value of the quark bilinear in the form (19 .7.14), and expanding in powers of the boson fields, we find a Goldstone boson mass term in the Lagrangian of the for m - F2 TrfB,JBaMqjj - - F 4mu ~no +I no + F~~ n _,[2~F~ +4(mu + md)n+rc- + 4(mu + ms)K+fC- z +4m,d -In0+ + F"C y-'F~ z 3 _v[ F~ (19.1.6) The charged and strange meson masses are the same as before, but now the neutral non-strange mesons have a mass matri x mU~-rrt Urfa = 8vmu~~ 2,/3_ F72rMu-Md 2~'3_F2Ir mu+m+4m~ 6F,rMu-Td .Nf6-F,rF~ mu+md-2ms 3,12-F71 F~ Mu-mod Mu+md -ems m u+m +m s ,~6_F7rF~ 3-v'2-F7rF~ 3FC with rows and columns listed in the order n°, r7°, and C. In the limit where mu and m dvanish, this has two eigenvectors with eigenvalue zero : Ua----0 01o ub= F n F'+ ZF 12- F C In this orthonormal basis, the mass-squared matrix to first order in mu and m dis Z T 24v (mu + md mug '=-' Zda ~~}~a =F2(1 9.10.9) 246 19 Spontaneously Broken Global Symmetrie s z T m212v (m,~ +md) mbb ~b~Oub F,2 + 2F'C 2 2/3-v(m u -tea )mob= mha 2F,F,2 +2F2 C The effect of the off-diagonal term (19 .10.11)is to decrease the product of eigenvalues by a negligible fractional amount (mu - md)2/G4(mu +and)Z, while of course leaving the sum of the eigenvalues unchanged, so the eigenvalues are given to a good approximation by the diagonal elements (19.10.9)and (19.10.10).Comparing Eq . (19 .10.9)with Eq .(19.7.16), we see that the particle corresponding to the eigenvector uu is the n" . The other particle, corresponding to ub, has a mas s Mb~ gbh-~/ aMjr F~ ~ ~I,m 7r ~ FF7 -2F Thus a broken U(1}A symmetry would require a neutral pseudoscalar Goldstone boson with mass less than -,,/'3-m,,, in addition to the pion itself . It hardly needs to be said that no such strongly-interacting particle exists .* We shall see in Section 23 .5 that this problem was eventually solved by the discovery of non-perturbative effects that violate the extra U(1)Asymmetry . Problem s 1. Apply the general theory of broken global symmetries to the cas e where an S4(3 )symmetry group (with generators tl, t2, t3) is sponta - neously broken to its SO(2) subgroup (with generator t3) . How do th e Goldstone boson fields transform under infinitesimal SO(3) transfor - mations? (Use the exponential parameterization of the caret spac e SQ(3 )/S4(2 ).} Evaluate the covariant derivative D,,, of the Gold - stone boson field, and of a general field with a non-vanishing valu e q for the unbroken symmetry generator t3 . What is the most gen - eral SQ(3)-invariant Lagrangian involving Goldstone bosons alon e with no more than two derivatives? Use this Lagrangian to cal - culate the terms in the invariant amplitude for elastic scattering o f Note that taking FS >F.would give the extra neutral scalar a very light mass, but its interactions would be so weak that it might have escaped detection . It seems unlikely that a very large ratio of Fn to F could arise in quantum chromodynamics, but something like this happens in theories with extra field variables that have been proposed to avoid parity violation by instantons ; see Section 23 .6. References 247 the Goldstone bosons to lowest order in their energy . What is the most general SO(3)-invariant Lagrangian with two non-Goldstone field factors and at most one derivative? What is the most general function of the Goldstone boson field alone (with no derivatives) that transforms like the three-component of an SO(3) three-vector . Add this term to the Lagrangian, with a coefficient chosen to give the Goldstone boson a mass m, and recalculate the lowest order amplitude for Goldstone boson scattering . 2. Consider a theory with an SU(N) global symmetry, spontaneousl y broken to S U(IV - 1).Suppose we add a small symmetry-breakin g perturbation, belonging to the defining representation N of S U(N). Taking vacuum alignment into account, what symmetry group i s completely unbroken? What about the case where the symmetry - breaking perturbation belongs to the adjoint representation o f S U(N) ? 3. Calculate the pion-pion scattering amplitude to one-loop order, tak- ing account of a finite pion mass . 4. Derive the `Adler sum rule,' the analog for the case of pion-pion scattering of the Adler -Weisberger sum rule . 5. Calculate the SU(2) xSU(2) transformat ion properties of the pion field ~ ,,in the case where the corsets of SU(2) x SU(2)IS U(2 )are parameterized as exp(i~ axa). 6. Use Eq .(19.7.3 1)andS U(3) symmetry to derive the relation (19 .7.32). References 1. S.Coleman, `Secret symmetry :an introduction to spontaneous sym- metry breakdown and gauge fields,' in Aspects ofSymmetry :Selected Erice Lectures ofSidney Coleman (Cambridge University Press, Cam- bridge, 1985) . 2. E.C. Titchmarsh, Introduction to the Theory af'Fourier Integrals (Oxford University Press, Oxford, 1937) :Section 1 .4. 3. J.Goldstone .Nuovo Cimento 9,154(1961) . 4. Y. Nambu, Pays . Rev . Lett. 4,3$0 (1960) . 5. J.Goldstone, A. Salam, and S. Weinberg, Phys . Rev . 127,965(1962) . 248 19 Spontaneously Broken Global Symmetrie s G. S. Adder, Pays . Rev .137, B1022 (1965) . 7. S.Weinberg,Pays.Rev.Lett.,29, 169$(1972) . 8. R. Dasher, Pays .Rev.183, 1245 (1969) . For corrections to thi s relation, see J . F. Donoghue, B . R. Holstein, and D . Wyler, Phys . Rev.D47, 20$9 (1993) ; J. Bijnens, Pays .Lett.B306, 343 (1993) ; K. Maltman and D . Kotchan, Mod .Phys .Lett.,AS, 2457 (1990) ; R. Urech, N ucl. Phys .B433, 234 (1995) ; R. Baur and R . Urech, Zurich - Karlsruhe preprint ZU-TH 22/95, TTP95-31, hep-ph/9508393 (1995) . 9. For a review, see I . S. Towner et al.,chalk River preprint nucl- th/9507005 (1995) . 10. S. We inberg,Pays.Rev.112,1375(1958) . 11. M. L. Goldberger and S . Treiman, Pays .Rev.111, 354 (1966) . 12. T. Ericson and W . Weise, Pions and Nuclei (Clarendon Press, Oxford, 1988) . 13. T. Ericson et al.,CERN preprint CERN-TH/95-50 (1995) . 14.T.Das,G. S.Guralnik, V.S.Mathur ,F. E. Low, and J. E.Young , Phys.Rev.Lett.18,759(1967) . 15. J. Bernstein, S . Fubini, M . well-Mann, and W . Thirring, Nuovo Cimento 17, 757 (1960) ; M. Dell-Mann and M . Levy, Ref . 24; K-C . Chou, Soviet Physics JETP 12, 492 (1961) . 16. Y. Nambu, Phys . Re v.Lett .4,3$0 (1960) . 17. Y. Nambu and G . Jona-Lasinio, Pays . Rev .122, 345 (1961) . 18.Y. Na mbu andD.Lurie,Phys.Rev.125,1429 (1962) ; Y. Na mbu and E . Shrauner,Phys.Rev.128, $62(1962). 19. S.L. Adler, Pays .Rev. Lett .14, 1051 (1965) ; Phys . Rev .140, B73 6 (1965) ; W. I. Weisberger, Phys .Rev. Lett .14, 1047 (1965) ; Phys . Rev.143, 1302 (1965) . The effect of a finite pion mass was include d in a different way in Ref . 27, by combining the current a lgebr a results for the pion-nucleon scattering lengths with the sum rule fo r the scattering lengths derived from the forward scattering dispersio n relations by M . L. Goldberger, H . Miyazawa, and R . Oehme, Phys . Rev.99, 986 (1955) ; M. L. Goldberger, in Dispersion Relations an d Elementary Particles (Wiley, New York, 1960) : p. 146 . references 249 20. S. Weinberg, `Current Algebra - Rapparteur's Report' in Proceedings of the International Carafererace on High-Energy Physics, Vienna, 1968 (CERN, Geneva, 1968) : p. 253 . 21. M. well-Mann, Physics 1, 63 (1964) . 22. S. Weinberg, Pays .Rev. Lett .16, 879 (1966) . 23. S. Weinberg, P hys.Rev. Lett .18, 188 (1967) . 24. M. well-Mann and M . Levy, Nuovo Cimento 16, 705 (1960) . 25. S. Weinberg, P hys.Rev.166, 1568 (1968) . 26. S. Weinberg, Physica 96A, 327 (1979) . 27. These scattering lengths were first calculated, using current algebra techniques, by S . Weinberg, Pays . Rev .Lett .17, 616 (1966) . 28.For a review, see J.F. Donoghue, E . Golowich, and B . R. Hal - stein, Dynamics of the Standard Model (Cambridge University Press , Cambridge, 1992) :Section VI-4 . Measurements of the strong - interaction energy shift and width of the 1s state in 71-p atom s have yielded results a(n-P -> 7r-p) = 3a 11Z + 3a3/Z = a .0$85(9)mom 1 and a(rr-p-+rr°ra)= I,12-(a3- a,) =-0.136(10)m-' ; see D. Sigg et al., preprint ETHZ-1 PP PR-95-4, July 1995, to be published in Pays . Rev.Lett. 29. S. Weinberg, Phys .Lett .B251, 288 (1990) ;NucZ. Phys .B363,3 (1991) ; Phys .Lett .B295, 114 (1992) . C. Qrdonez and U . van Kolck , Phys .Lett .B291,459 (1992) ; C. Ordanez, L. Ray, a nd U. van Kolck , Phys .Rev. Lett .72, 1982 (1994) ; U. van Kolck, Phys . Rev .,C49, 2932 (1994) ; U. Wan Kolck, I . Friar, and T. Goldman, to appea r in Phys .Lett.B. This approach to nuclear forces is summarized i n G. Ordone2, L . Ray, and U . van Kolck, nucl-th/95113$0, submitte d to Pays .Rev. C; J. Friar, Few-Body Systems SuppZ .99, 1 (1996) . For application of these techniques to re lated nuc lear processes, se e T.-S. Park, D .-P. Min, and M . Rho, Phys .Rep.233, 341 (1993) ; nucl-th/9505017 ; S. R. Beane, C . Y. Lee, and U . van Kolck, Phys . Rev.,C52, 2915 (1995) ; T. Cohen, J . Friar, G . Miller, and U . van Kolck, nucl-th/9512036 ; D. B. Kaplan, M . Savage, and M . Wise , nucl-th/9605002 . 30. S. Weinberg, Ref . 27 . The pion -nucleon scattering lengths were independently calculated by Y .Tomozawa, Nuovo Cimento 46A, 707 (1966) . 250 19Spontaneously Broken Global Symmetries 30a. S. L. Adler, Phys . Rev .140,B736 (1965) . 31. S. Coleman, J. Wess, and B . Zumina, Pays . Rev . 177, 2239 ( 1969); C.G. Callan, S . Coleman, J. Wess, and B . Zumino, Pays . Rev . 177, 2247 (1969) . 32.R. S.Palais,J. Math . Mech.6, 673 (1957) ; G.D. Mostow, Annals qf Math.65,432 (1957) . 32a. For a discussion of other possibilities, see S . Weinberg, Physica Scripta 21, 773 (1980) . 33. M. Gell-Mann, Cal . Tech . Synchotron Laboratory Report CTSL-20 (1961), unpublished . This is reproduced along with other articles on SU(3) symmetry in M . Gell-Mann and Y . Ne'eman, The Eightfold Way (Benjamin, New York . 1964) . 34. Y. Ne'eman, NucZ. Phys .26, 222 (1961) . 35. M. well-Mann, R . J. Oakes, and B . Renner, Pays .Rev.175, 2195 (196$); S. Glashow and S . Weinberg, Phys .Rev. Lett .20, 224 (19 6$). 36. R. Dashen, Phys . Rev .183, 1245 (1969) .Corrections were considered by P . Langacker and H . Pagels, Pays . Rev . D8, 4620 (1973) . 37.S. Weinberg, contribution to a festschrift for I . I. Rabi, Trans .N.Y. Acad .Sci. 38, 185(1977) . 38.M. Cell-Mann, Ref . 33; S. Okubo, Prog . Theor . Pays .,27, 949 ( 1962). 38a. H. Leutwyler, Bern-CERN preprint GERM-TH/96-44, hep-ph/- 9602366, to be published (199 6). 38b.J. F. Donoghue, E . Golowich, and B . R. Holstein, Ref . 28. 39.J.Gasserand H .Leutwyler,Nucl.Pays. B250, 465 (1985) . Also see J.Gasserand H .Leutwyler,Ann.Phys.158,142(1984) . 40. For surveys, see H . Leutwyler, in Proceedings of the XXVI Interna - tional Conference on Hig h Energy Nuclear Physics, Dallas, 1992, ed . J. Sanford (American Institute of Physics, New York, 1993) :185; U.G. Meissner, Rep .Prog .Phys . 56, 903 (1993) ;A. Pith, Valen - cia preprint FTUV /95-4, February 1995, to be published in Report s on Progress in Physics ; J. Bijnens, G . Esker, and J . Gasser, in The Daphne PhysicsHandbook, Vol. 1, eds . L. Maiani, G . Panchen, an d N. Paver (INFN, Frascati, 1995) :Chapters 3 and 3 .1; G. Esker , preprint hep-ph/9501357, to be published in Progress in Particle an d Nuclear Physics, Vol . 35 (Pergamon Press, oxford) . References 251 41, S . Co leman and S .Glashow, Phys. Rev . Lett .6,423 (1961) ;S. Qkubo ,Phys.Lett.4,14 (19 63). 42.J.Wensand B . Zumino,Phys. Lett .37B, 95 (1971) . 43. E.Witt en, Nucl.Phys.B223 ,422 (1983) . 44.R.Battand R .Seely, Comm .Math .Phys.62, 235 (1978) . 45.E.D'HokerandS.Weinberg,Phys . Rev .D50,R6050 (1994) . 46. Encyclopedic Dictionary of Mathematics, eds. S . Iyanaga and Y . Kawada (MIT Press, Cambridge, 1980) ; W . Greub, S . Halperin , and R . Vanstone, Connections, Curvature and Cohomology, Vol III , (Academic Press, New York, 1976) ;A. Barel, Ann .Math .(2)57,115 (1953) ;also in A . Bore1, Collected Papers, Vol I (Springer Verlag , Berlin, 1983) . 47. D. Weingarten, Phys . Rev . Lett .51, 1830 (1983) . 48.J. Preskill and S . Weinberg, Phys . Rev .D24 , 1059 (1981) . This wa s a modified version of an earlier argument by 't Hooft, who ha d considered the limit of large quark masses ; see G . 't Hooft, lectur e given at the Cargese Summer Institute, 1979, in Recent Development s in Gauge Theories, eds. G 't Hooft et al.(Plenum, New York, 1980) , reprinted in Dynamical Gauge Symmetry Breaking, eds. E. Farhi an d R.Jackiw (World Scientific, Singapore, 19$2), and in G . 't Hooft , Under the Spell of the Gauge Principle (World Scientific, Singapore , 1994) . Also see Section 22 .5. 49. C. Vafa and E . Witten, Nucl .Phys .B234, 173 (084) . Also see C . Vafa and E . Witten, Commun . Muth . Phys .95, 257 (1984) . 50. C. Vafa and E . Witten, Phys . Rev . Lett .53, 535 (1984) . 51. S. L. Glashaw, in Hadrons and Their Interactions, ed. A. Zichich i (Academic Press, New York, 1968) ; S. L. Glashow, R . Jackiw, an d S-S. Shei, Phys . Rev .187, 1916 (1969) ; M. Gell-Mann, in Proc . Thir d Topical Conf . on Part . Phys .,eds. W . A. Simonds and S . F. Tua n (Western Periodicals, Los Angeles, 1970) ; in Elementary Particl e Physics (Springer-Verlag, Bonn, 1972) ;Acta Phys . Austriaca Suppl . IX, 1972 (1972) ; H. Fritzsch and M . Gell-Mann, in Proceedings o f theXVIInternational Conference onHigh Energy Physics, eds. J. D. Jackson and A . Roberts (Fermi National Accelerator Laboratory , Batavia, IL, 1972) ;Phys . Lett .47B, 365 (1973) . 52.S.Weinberg,Phys.Rev.D11, 3583 (1975) . 20 Operator ProductExpansions We often find ourselves needing to know how an amplitude behaves when the four-momentum brought in by one operator and out by another goes to infinity, with all other external lines held at fixed four-momenta . For instance, we will see in Section 20.6that the total cross section for the scattering of an electron by an initial hadron H, with arbitrary hadrons in the final state, is given by unitarity as a linear combination (with known coefficients) of the components of the amplitud e fd4x e-" (HjJ1'(x)J'(0)jH) , where k is the four-momentum transferred from the electron to the hadrons, and P(x) is the electromagnetic current . In the case of deep inelastic electric scattering the momentum k that is carried in by one current operator and out by the other is allowed to go to infinity . Sim- ilarly, in studying the high momentum limit of various propagators and deriving corresponding spectral function sum rules in Section 20 .5, we shall encounter the high momentum limit of similar Fourier transforms, but with the one-hadron state JH ) replaced with the vacuum . If an operator product such as JP(x)J'(0) were analytic in xP, then its Fourier transform would decrease exponentially as the Fourier variable k goes to infinity . The leading terms in the high momentum limit of the Fourier transform arise from the singularities of the operator product as the spacetime arguments approach one another . The study of such oper- ator products was initiated in 1969 by Wilson,' originally as an attempt to formulate a substitute for conventional quantum field theory . As has happened earlier (for instance with dispersion relations and Feynman's diagrammatic rules) the effort to bypass quantum field theory led to valu- able general results, but results that can best be understood as general properties of quantum field theory . The operator product expansion will be stated in Section 20 .1. The standard proof of this expansion was given in perturbation theory in 1970 by Zimmerman . 2 In Section 20 .1 we shall offer a non-perturbative and simpler though less rigorous derivation based on the path-integra l 1,Ir1, 20.1 TheExpansion : Description and Derivation 253 formulation of field theory . Section 20 .2 will present a different perspective on the operator product expansion, in terms of the flow of large momenta through Feynman graphs, which will lead us to a perturbative proof . There are several aspects of the operator product expansion that make it particularly useful for drawing consequences from theories like quantum chromodynamics . One such property, discussed in Section 20 .3, is that the functions describing the singularities in this expansion have a momentum dependence governed by renormalization group expansions, so that in asymptotically free theories they can be calculated at large momenta using perturbation theory . Another aspect, shown in Section 20 .4, is that these functions exhibit the full symmetry of the underlying theory, unaffected by possible spontaneous symmetry breaking . Applications will be considered in Sections 20.5 and 20 .6. 20.1The Ex pansion:Descriptionand D erivation Wilson1 hypothesized that the singular part as x --} y of the product A(x)B(y) of two operators is given by a sum over other local operator s A(x)B(y)EFcB(X - Y) C(1') , (20 .1.1) C. where FCB(x - y )are singular c-number functions . Dimensional analysis suggests that FHB ~x - y )behaves for x --} y like the power dc- dA - dB of x - y, where dois the dimensionality of the operator 0in powers of mass or momentum . Since do increases as we add more fields or derivatives to an operator 0,the strength of the singularity of FCAB(x - y )decreases for operators C of increasing complexity . The remarkable thing about the operator product expansion is that it is an operator relation ; that is, in applying it to any matrix element (#jA{x}B(y)ja), we get the same functions F~B~x - y)for all states day and jfl~. It is the decrease of the singularity in Eq . (20 .1.1) with operators C(y) of increasing complexity that makes this expansion useful in drawing conclusions about the behavior of the product A(x)B(y) for x --+ y .The simple power-counting argument above is modified by renormalization effects ; the expansion (20 .1.1) must be formulated in terms of operators renormalized at some scale p, and then yappears along with x - y in the coefficient function F~ AB (X - y } . We shall see in Section 20 .3 that in asymptotically free theories F~B~x --- y )does behave like the power dc- dA - dB of x - y suggested by dimensional analysis only up to a power of In (x - y)2.Even in more general theories, it is plausible that the singularities associated with various operators C(y) will decrease with the complexity of the operators . 254 20 Operator Product Expansion s The corresponding statement in momentum space is that fork --} oo, d4xQ-kxA(x)B(~)__*Y : VCB(k)e(0) C and correspondingl y fd4x .e-ik'x7'jA(x)B(O)J --}Y:UCB(k)C(0), C(20.t.2) (20.1.3) where V(jB (k)and U~B (k )are functions of k ythat for large k decrease increasingly rapidly for more and more complicated terms in the series . We are going to derive a generalized version of the Wilson expansion, in which the momenta carried by any number of operators go to infinity together . For this purpose, let us consider a Greens function for local operators A1(xj), A2(x2), etc . whose arguments approach a point x, as well as other local operators Bjyl }, B Ay2), etc . with fixed arguments : ff1do,(z) r, zQi(xi) a2(X2)... hi(Yt)b2(Yz)... eXPOIL01) (20.1.4) where the lower-case letters a and h indicate replacement of the field operators in the As and Bs with the c-number fields 0. Now surround the point x with a ball B(R) of radius R which is much larger than the separations among the XI, x2, etc . but much smaller than the separations between x, yl, yz, etc. Since the action is local, it may be written a s I=I d4Z y(Z) +d4zy(Z). (20.1.5)EB(x) fOB(x) Eq. (20 .1.4)maythen be pu t intheform Z4BtRy,~ X~ fj do,~(z)al(xl)a2(x2). zCB(x), ,I..N exp i JzVB(R)~(z) ..exp (if .~(z) EB{x}(20.1.6) in which the path integral over the fields inside the ball is constrained by the boundary condition that the fields merge smoothly at the ball's surface with the fields outside the ball . Aside from this boundary condition, the path integral over the fields inside the ball is completely unaffected by the behavior of the field outside the ball, so the integral over the fields inside 20.2 Momentum Flow 255 the ball may be expressed in terms of the values and derivatives of the fields on the surface of the ball, which in turn may be expressed in terms of the fields and their derivatives extrapolated from outside the ball to the interior point x . If we express this integral as a series in products* o(x) of the c-number fields and their derivatives extrapolated to x, then the coefficients can only be functions Uo'A2'"'(xl - x, xz - x, ...)of the coordinate differences . Since the points y i, y2, etc . are all far outside the ball B(R), the exclusion of this ball in the action for the fields outside the ball has no effect in the limit R --} 0, so in this limit Eq .(20.1.6)become s (7'IAI(xi), A2(x2),...Bj.YZ},$2(Y2)...1}0---+ x bl(.vi)b2(Y2) ... exp (ifY(z)[n(ici;]) e',z ?CU3(xl - x, x2 - x, ...} O(x) d 0 (20.1.7) for x1, x2, etc . all approaching x, where n(x) is the quantum-mechanical Heisenberg-picture operator corresponding to o(x) . In particular, by Fourier transforming with respect to the y variables and multiplying with appropriate coefficient functions, this yield s (#l~'~A~~~~}, Az~~z}, ...Ia)--~ ~U nI,Az,...(X1-X, xz - X,...) (P lo~}ja} 0 (20.1.8) for arbitrary states ja~ and (#I . Because this applies for arbitrary states, it is the operator product expansion in a generalized version : ~'jAj(xi)> A2(x 2), U~"A"--(XI -X,x2-X,...)O(x).(20.1.9) 0 20.2 Mome ntum Flow# We shall now consider the simplest example of the operator product expansion : the asymptotic behavior of an (n+2)-point Feynman amplitud e In order for the coefficients in this series to be finite, these products must be renor- malized by multiplication with suitable infinite factors . This will be made clearer in the derivation presented in the next section . This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 256 20 Operator Product Expansion s in the theory of a single real scalar field O(x) with mass m and interaction Lagrangian density - 'go', when a large four-momentum k flows in one line and out another, with all other external lines held at fixed momentum . This will lead us to a perturbative proof of the leading term in the operator product expansion in this case, but our real purpose here will be to gain some insight into the way that the flow of large momenta through Feynman diagrams leads to this expansion . An appendix to this chapter will discuss the extension of these results to the general case . Let us define r (k; p,  p,) as the sum of all connected graphs for the n-particle scattering amplitude, whose external lines carry incoming momenta k, p - k, and outgoing momenta p i   pn, where p - pl +P2-{- ..-{--pn . (It will be convenient to specify further that r(k ; p,... p,) includes propagators for the external lines with momenta kand p - k, but not for those with the fixed momenta pl   -pn.) We wish to show that in any finite order of perturbation theory, for k --} oo , r(k;pi...Pn)--} v0z(k)FOz(Pi.. pn)+0(k-5), (20.2.1) where U0z(k)is a sum of terms of order" k-4 , that is independent of pl ' ' ' pn and of n, and F4,z(pl    pn )is the amplitude for n 0lines with insertion of a single 02 vertex, times a suitable renormalization constant Z02 to make it finite . Because F0z(pl    pn )is a matrix element of the renormalized operator (02 )R Z0 2 0 2( 0 ) , Eq. (20 .2.1) corresponds to the statement that the leading term in the operator product expansion for k --*oo W r dax Q 'k xTj0 R(x)0R(0)jc ---3 U 0z(k)(02 (0)}R. (20.2.2).1 In assessing the asymptotic behavior of Feynman amplitudes, we must take account of the fact that there are parts of the range of integration in momentum space where some of the internal lines carry momenta of the same order as the external line momenta that are going to infinity, while other internal lines do not . The contribution to I' (k; pl    p,)from the part of the region of integration where the lines in some subgraph Y have momenta of order k has an asymptotic behavior of order Oy, where Dy Throughout this chapter, whenever it is said that an amplitude is of order 0, it should be understood that for k "= xnP where x .--*oo with W a fixed generic four-vector, the amplitude approaches rcA times a sum of powers of In K . t The subscript C on the time-ordered product here indicates that we are including only connected graphs . We have dropped the superscript cfoon Viz, because in this section we are concerned only with the operator product expansion for two 0fields . These fields are themselves renormalized, because we implicitly include counterterms along with radiative corrections for the propagators of the lines carrying momenta k and p-k, 20.2Momentum Flow 2 57 is the dimensionality of the subgraph SP, calculated according to the rules of Section 12 .1. IfYhas m external lines that connect it to the rest of the graph as well as the two external lines with momenta kand p -- k, then from Eq .(12.1.8)we have D y= 4 - 2 - m - 4 = -2 - m . (The term - 4 arises from the two propagators of the lines with momenta kand p - k, which we have specified are to be included in r (k;pl...pn).) Thus the asymptotic behavior of IF is dominated by the part of the momentum-space integral where the large momentum k flows either through the whole graph or through some sub graph, whichever has the smallest number ofexte rnal Iine.s.3 For n = 0 this is always the whole graph ; that is, the dominant part of the integral comes from the part of the region of integration where every line carries a momentum of order k, giving an asymptotic behavior of order k- 2. In this case the only operator in the operator product expansion contributing to this matrix element is the unit operator, C = 1 . This term is excluded here for n >0 because for the present we are limiting ourselves to connected graphs . For n = 2 the dominant contribution comes both from the whole graph, and from subgraphs in which the two external lines with momenta k and p - k are connected to the other two external lines by a bridge consisting of two internal lines,tt giving an asymptotic behavior of order k-4 . For n ~ 4 the dominant contribution comes only from subgraphs in which the two external lines with momenta k and p -k are connected to the n >2 other external lines by a bridge consisting of two internal lines, again giving an asymptotic behavior of order k-4 . The analysis of the cases n = 2 and n ~ 4 is complicated by the fact that a general graph may contain several of these two-particle bridges . Let us first consider the case n = 2 . We define i(k, k', p )as the sum of all graphs contributing to r(k ;pl,pz )(with pl = k', p 2= p - k' )that are two-particle-irreducible, in the sense that the two external lines with incoming momenta kand p - k cannot be disconnected from the two external lines with momenta k' and p - k' by cutting through any pair of internal lines . Then r(k ; k', p - k' )- i(k, k', p )consists of graphs that can be disconnected in this way, and may therefore be written (see Figure 20.1): r(k; kl,p-k') -1(k, k',p)=fd4k"~(~~k ",P)r(k» ;k',p- k).(20.2.3) (Liker(k;k',p -k'),the k ernel I (k,k',p)include spropagator sfor the l ines The possibility m - 1 is excluded because the symmetry of this theory under 0 -~ -fir rules out graphs or subgraphs with odd numbers of external lines . The possibility in - 0is excluded because r is defined to arise only from connected graphs . 258 20 Operator Product Expansion s k k kk, k k, r ~~ p-k ~~-kr P -k p- k + P p-7~ p- k r I I r Figure 20 .1. Diagrammatic representation of the integral equation (20.2.3). The cross-hatched disks marked F represent the sum of all connected Feynman diagrams with the indicated external lines, while the cross-hatched disks marked I, which are divided by a vertical line, represent the sum of all connected diagrams for which the lines on the left cannot be separated from those on the right by cutting through any pair of external lines . + :D> < Figure 20 .2. T ree and one-loop g raphs for t hekernel I(k, k', p) i n the t heory o f a sca lar fieldwithinteraction (D4 . with momenta kand p - k, but, to avoid double counting, not for the lines with momenta k' and p - k' .) The Feynman diagrams for I (k, k', p ) to order g2 are shown in Figure 20 .2. To evaluate the behavior of the right-hand side of Eq .(20.2.3),let's first consider the asymptotic behavior of the kernel I(k, k`, p) when k --} 00 with k' and p fixed . This is dominated by the region of momentum space in which all internal lines carry momenta of order k, which makes a contribution of order k-4 , because in any other region the subgraph consisting of internal lines that carry momenta of order k would have more than four external lines, and hence would make a contribution that decreases faster than k-4 . It follows that differentiating i (k, k', p )with respect to k' or p would reduce the asymptotic behavior of this kernel by a factor of k -l. Hence for k - -+oo with k' and p fixed, we hav e l(k,k ',P)~I,-x,(k) where i ,,(k) isafunction onl yofk,of order k -4(20.2.4) Unfortunately, we cannot simply replace I (k, k', p) in Eq .(20.2.3)with this asymptotic limit, because however large k may become, the integral will receive a large contribution from values of k' of order k . To deal with this complication, we shall now employ a trick, based on mathematical induction . In lowest order r (k; k',p-k')is given by a single vertex with 20.2 Momentum Flow two attached bare propagators ig (27r)4(k2+m2)((p-k)2 +m2)(lowes torder) ,259 for which it is easy to verify an asymptotic behavior of form Eq . (20 .2.1). Let us therefore assume that Eq . (20 .2.1) for n = 2 holds up to some given order Nin g -that is, that up to this order, the asymptotic behavior for k --} oo takes the for m r(k;k', p - k)--3,U0Z (k)F0z(k',P - k ') +0 (k-5), (20.2.5) and try to verify this behavior to the next order . In order to eliminate the contribution to the integral in Eq .(20.2.3)from values of k' of order k, we rewrite Eq .(20.2.3)as r(k; k', p- k)_I (k, ki,P) +fd4kh' j(k,V,p)[Fk" ;k',p -k')- U02(k")Fez(kf,p-kl)] +F0z(k', p - k' )JAll I(k,k", P)U0z(k"). (20.2.6) Since i (k, k', p)is at least of first order, we may use Eq .(20.2.5)in the right-hand side of Eq .(20.2.6).Hence in the second term on the right- hand side, the part of the region of integration where k" is of order k gives a contribution that vanishes like k-4+4-1 , and may therefore be neglected compared with the part where k" remains finite, which yields the convergent integra l ,(k) f d k11 [F(V ; k , p - k') -U02(V)FO2(k', p - k1)] Further, since the dominant part of fd4k" i(k,k", p )Utz(k") comes from the region of integration where every internal line of the graphs for I(k, k", p)carries a momentum of order k, differentiating this integral with respect to p would lower its asymptotic behavior by a factor of order k-1, so asymptotically i(k, k', p) may be replaced in this integral with I(k, k')- I (k, k', 0). Therefore for k --} oo, Eq . (20 .2.6)become s r(k>k',p-k') --31Fez (k' ,p- kf) fdak"I (k,k») U4,2(kll ) +Ioo(k) 1 +fd4k" [F(k" ;k',p -k') - U,2 (kff)F,z (kf, p -k')].(20.2.7) Let us therefore define U02(k) and Fez (k', p - k') to order N + 1 in g in 260 20 Operator Product Expansion s terms of these functions in lower orders of perturbation theory, b y U0z(k)= Clr,(k) + Fez(k',p -k')C-id4 k'1(k,k')Utz(k'), (20.2.8) x 1 +fd4k" [Fk' ;k',p - k') -U02 (0)F02(k%P - k' )] I, (20 .2.9) where C is a constant that can be chosen as we like . With these definitions, Eq.(20.2.5)follows from Eq .(20.2.7). It will be convenient to choose the constant C so that F~ (k',p - k')has the value unity at some renormalization point k' = k(p) and p = p(p), where k(p) and p(p) are standard four-momenta of order p. The n C = 1 +fd4k"r (k";k(u),p(p) - k (y))-fd4k" U 02(kff) . (20.2.10) Using Eq .(20.2.5),we see that the divergences in the two integrals in Eq. (20 .2.10) cancel . Eq.(20.2.9)may now be writte n F4,z(k'a P - k') = Z 4,2 {1+fd4k"F(k" ;k'p_k')} where Z02 =~1 +f d 4kttr(k// - k(p),p(p) - k (p))](20.2.11) (20.2.12) We can think of Zoz as the renormalization constant for the composite operator 02 , defined in such a way that the operator Z4,2 02 has the finite two-particle matrix element Fez (k, p - k), with the value unity for k = k( p) and p= p(p) . It is not particularly convenient to calculate U 0z(k)or F 0z(k,p - k) by using Eqs .(20.2.8)and (20 .2.11). Rather, it is simpler to calculate r(k; k', p - k' ), and read off Utz (k) or Fez (k, p - k)by comparison with Eq.(20.2.5).By multiplying the function (12.2.26) with the product of propagators for the lines with momenta k and p - k, we see that, to one-loop orde r 9 1 >C 1 327t2 l0 -}-+In m2 +4x(I m2-txi-i I-i dx In(m2+4x .(i.- x} /12/3 m2 - sx(l - x)[-i(27r)4g] - x) y 2/3 m2+4x(1 -X)112/3 1 - x) +In m2 -. ux(l - x )+... (0.2.13) 20.2 Momentum Flo w where s, t, and u are the Mandelstam variables261 yis the renormalization scale and g the corresponding renormalized coupling, defined as the Feynman amplitude at s = t = u = -4y 2 /3. This has the asymptotic behavio r X1 -S 32n~zg (27r)4(k2)2 dx In'n2+ 4x1 - x)tt2/3 rrt2+p2x(1 - x)Jo' (m2+41 - x)p2/3+2InMz + k 2x(1- x)+... To order g2, this agrees with Eq .(20.2.5)if we tak e U02(k)- ig (2n)a(kz) z X1-gjdxIn m+4x(1 )µ2/ 3 ( )16n2 a m2 + k2x1--X+...(20.2.14) (2Q.2.1 S) and g ! (m2+4x1 - x)µ2/3F02(k,p-k )= 1- 32n2fdx InM2+2xl-X~....(20.2.16)p( ) We have here chosen the renormalization point k(y), p(p) for the operator 02 to be related to that for the coupling constant g in such a way that p(µ)2 = 4P2 /3, so that Fez (k, p - k)= 1 when p2= 4P2 /3. Now consider the case where the number n of external lines with fixed momenta is greater than two . In accordance with our earlier discussion, in the limit k --+oc), the leading graphs for T (k; pl    pn )are those in which the two external lines carrying momenta kand p - k can be disconnected from the n external lines with fixed momenta by cutting through a pair of internal lines : r(k; P1... p►~)--+jA' I (k ak', p)r(k';Pz... Pn). (20.2.17) As before, we cannot simply use the asymptotic limit of the kernel I (k, k', p) for k ~ oo on the right-hand side, because this integral receives important contributions from k' of order k . We deal with this complication by 262 20 Operator Product Expansion s rewriting Eq .(20.2.17) in the for m r(k;P1...p,) --+Jd4k'I (k, k', p)[r(k';P1...Pn)-Utfiz(k')F'02(P1... pn) ] +F02(pl ... P.) f d4k'I(k', k, p)U02(k1) , (20.2.18) where by mathematical induction, we suppose that up to some given order N r(k;Pi... Pn) --+U0 z(k)F0z(Pi...Pn) , (20.2.19) with correction terms of order 1/ 0. We can now use the limit (20 .2.4) together with Eq . (20 .2.8) to rewrite Eq .(20.2.18)as 17(k;PI... Pn) ---~I,,o (k)f d4k' [I-'(k';PS... pn) - U02(k')F02(PJ... pn)] +F02(pl ...Pn)IU02(k) -Cl,,o(k)] This agrees with Eq . (20.2.19)to order N+ 1, provided we tak e GF,62 ( Pi ..Pn) = f d4k'[F(k' ;p...Pn) - U02 (k')F'02 (p 1... pn)J or, using Eqs . (20.2.10) and (20 .2.12), F0Z(P1... Pfl) = Zp zfd4k'r (k';p1... p1l). (20.2.20) This just says that F pz(pl    p ,,)is the matrix element of the renormalized operator Zpz02, so that Eq . (20,2 .1) corresponds to the operator product formula (20 .2.2). Note in particular that Upz(k)is the same coefficient function whatever value n or the momenta p1...pnmay take, as was to be shown . Strictly speaking, the (02)R operator is not the leading term in the expansion of the product of two Os . There is also the operator unity, which has lower dimensionality than (02 )R, but was excluded from Eq . (20 .2.2) because (as indicated by the subscript C)we are excluding disconnected graphs . As mentioned earlier, the graphs for F(k) with n = 0(and p~ = 0) are dominated for k --~.oo by the region of integration where all internal lines carry momenta of order k, and U1(k) is the contribution of thi s region . Higher-dimensional operators and more general theories will be con- sidered in an appendix to this chapter . 20.3 Renormalization Group Equations for Coefficient Functions 263 20.3 Renormalization G roupEquations for Coeffic ient Function s As mentioned earlier, one of the aspects of the operator product expansion that makes it so useful is that the momentum dependence of the coefficient functions is governed by renormalization group equations . This is because they arise from limiting values of sums of Feynman graphs (such as i,,,(k) in Section 20 .2) in which all relevant momenta are going to infinity together, so that masses may be set equal to zero without introducing singularities . Nevertheless the operator product coefficient functions do not obey simple scaling laws, because of renormalization effects : the functions are multiplied with scale-dependent renormalization constants, and depend on scale-dependent renormalized couplings . Consider the operator product expansion for a Greens function Tel,(k, k', p)in which the incoming momenta (collectively labelled k, with sum p)of a set Iof lines all go to infinity together, with the set Z' of remaining lines having fixed outgoing momenta (collectively labelled k', with sum p): T(j,(k, k', p) Ut'o(k)FGr~,(k', p). (20.3.1) The function F O,e,(k',p)is the mat rix element of a renormalized operator OR = ETl zC,~~(()', so its coefficient _Ukk)in the product of fields corre- sponding to the lines tis proportional to z ~ ,,, as well as to z (,,(, the direct product of all the renormalization matrix factors for the field (or composite) operators in the set Z'. Henc e ~-U(, Yc~' U~, ,;,Y~~~,~ + I~ (g) $U~. ~ 2 .3.2 F~ ~ „ where e'r a P oil and for simplicity we assume a single renormalizable coupling g ,,, defined as the value of some Feynman amplitude at a renormalization point with momenta of order p, with pdg~aldp = fl(g.).In order to be able to use dimensional analysis, we multiply all operators by powers of Pin such a way that they become dimensionless . This has the effect that the components of the Z and y matrices are also dimensionless, with the values in the limit of zero coupling given b y y,q' --+ 6Ce, N'(1') , Y C.C,, --> 6Cro'N'(O), (20.3.4) where N(O) is the dimensionality of the operator U, and 111' (11)is the total dimensionality of the set /of fields (the sum of s +1 for each field, where 264 20 Operator Product Expansion s s =0for scalar and massless gauge fields, s = 1/2 for Dirac fields, etc .) Also, dimensional analysis tells us that for k "= xnP with & fixed, the amplitude can depend on Konly through the ratio x/ p, aside from a factor rc4-4n(6 arising from the integrals used in defining the Fourier transform . The solution of Eq .(20.3.2)is then of the for m dU~(rcn)= 1C4-4n(l)~[i{ exp(IK ~Y(gp)) X 11, ~~(gK,n) i x M exp(JK dµ Y(gm)) (20.3.5) P where M denotes the `g-ordered' product, that is, each term in the expan- sion of the exponentials rearranged so that the factors are in the order of decreasing pfrom left to right . This gives especially simple results where g .approaches a fixed point g. for p -+ oo. The contribution of large ps to M{exp(f " y(gu)dy1µ)} is then a matrix factor rcy(g') that because of the p-ordering appears on the left.Thus Eq .(20.3.5)become s U~(rcn)= rC4-4n( ~) ErKr{~.} I~ ~wetk`IK_'~(g-) 1 0,,C ~ ~L1) (20 .3-6) where 16'is either a constant or a sum of powers of In rc, depending on the rate at which 9K approaches g .. Asymptotically free theories like quantum chromodynamics are a special case of particular physical interest . Here the fixed point is at g .=0, and, according to Eq . (2 .3.4), near this fixed point the y matrices go a s Also, if we write the renormalization group equation for the coupling in the form d z a (20 .3.8) then fxdyg~ ~ - 8 72In g~ + constant .b Using this in Eq .(20.3.5)yields the asymptotic behavio r -s~2~~h (g~)U~(Kn)K4-4n(f)+2V(~}-N(~7) 1:[(g2)- (20.3.9) where 16., is a constant matrix, equal to Wle"',(O, n) times constant factors that are not calculable in perturbation theory because they come from the 20.4 Symmetry Properties of Coefficient Functions 26 5 parts of the integrals in Eq .(20.3.5)where guis not small . For K - ;Oo, the behavior of the coupling constant is g2 --+87E2/b In rc, so Eq .(20.3.9) may also be writte n U~(rcn)-;K4-4n(1)+N(?)-N( O)~[(In K~ s~2C/h M~~', InK)-s712c1h (20.3.10) where is another constant matrix . The condition for Eqs .(20.3.9)and (20.3.10)to be valid is that gx /8n2 < 1,but it is not necessary for In K to be so large that only one eigen vector of the c mat rix contributes to the asymptot ic behavior . Eq.(2d.3.1 d)will be used to study deep inelastic scattering in Section 2 0.6. 20.4 Sy mmetry Properties of Coefficient F unctions The usefulness of the operator product expansion is greatly enhanced by the fact that the coefficient functions exhibit the full symmetry of the und- erlying theory, even where part or all of that symmetry is spontaneously broken .4 To prove this, we consider the operator product expansion for a product of renormalized operators O(i(x) that transform linearly under some symmetry with conserved current .J"(x), in the sense tha t [.IO(x, t), (),i(Ya t)]=-63 (x _" Y) ti~(PAy,t} , (20 .4.1) where t ijis a constant matrix . We can write the operator product ex- pansion as the statement that, as x1,    x, approach x together (with XI- x,    x„ -- x all having fixed ratios ) T{01i,(xi)...01ijxn}Jla)-; ~ UI,...an(xl _X'...xn - x )(9 I00)*. (20.4.2) Now suppose that the symmetry with current .II'is spontaneously broken, with the corresponding Goldstone boson 7r satisfyin g (VAC I.Iu(0)17)Fps (2n).3/22p°(20.4.3) Then, as we saw in Section 19 .2, the matrix element of the operator product between states with an additional low energy Goldstone boson i s 1 (2n)3/z 2p° F x fd4xa (flIT~011(xl }... din(xn)J°I(x)IW , (20.4.4)axp 266 20 Operator Product expansion s because only the Goldstone pole term survives in the integral on the right-hand side of Eq . (20 .4.4). Using Eq . (20 .4.1) and the conservation of the current, this can be put in the form 1 (2n)3/2 2poF r=1jr Now apply the operator product expansion to both sides of this formula . In the limit as xj    x„ all approach x together, we fin d Utt,..in (x1 - X'...JLn -JL)(flI0(i(x)ITLOL)_1 (2n)3/2IFn x1: E Ettr.]r vdl... jr...I~ (x1 - x,.x.-x) CIlx/I a~(20 .4.6) r=1jr i But as a special case of Eq . (20 .4.5), we hav e (27)3~2 1 Etij(# 27jrFi(20.4.7) Since all this holds for arbitrary states (#Iand la~, the coefficients of (#j®rj( x)Ca)on both sides of Eq .(20.4.6)must be equal, s o n 0_ -)7tij Uji---gin(x1_x'...X,,-x)+I: Et1rlrU jl...jr...in (X1 _x,...xn_x ) i r=1 Jr (20.4.8) This can be restated as the condition that Ua I an (xl - x,    xn - x) is invariant under the symmetry generated by t, with the action of this symmetry on the lower index contragredient to its action on the upper indices, in the sense that the matrix t is replaced with -tT . This is the same relation that would be expected if the symmetry generated by .I11 were not spontaneously broken . 20.5 S pectralFunctionSumRule s Spectral function sum rules are constraints on the spectral functions of various currents .5 We will start here with a set of currents Ja that are arbitrary, except for being Lorentz four-vectors, and then later consider more special examples . To define their spectral functions, we use Lorentz 20.5 Spectral Function Sum Rule s invariance to writ e i64(P -P er)WACI.Ixp (0)1 N) (VAC IJ~ (0)1 N}~_(2n)-3B(P° ) XL(q,v-ePvIP2)Pacfl(P) + PpPvPafl(P2)] in analogy with Eqs . (19 .2.19) and (19 .2.20)or Eq . (10 .7.4). Fourier transform and using the completeness of the states be writte n VaW.I~(a)}SAC =fdp2267 (20.5.1) Taking the CND, this ca n X [tjjuvpG1(tt1) -(p(l)(tl) +p(11(tt)1tt2apa" A+(x ; ji) where ❑+(x; µ2) is the function defined in Eq .(5.2.7): )A+(x; P2) =_ 1Jd4po(pO)6(p2+P2 )e'P-x(20.5.2) X20.5.3) Assuming the currents to have been chosen as Hermitian operators ,it is immediately apparent from Eq .(20.5.1)that p~~(~.c2}and p~~(p 2)are posi- tive Hermitian matrice s.Also,taking xyto be spacelike (for which ❑+(x) iseven) and using translation invariance and causality in Eq .(20.5.2),we see that p ~~0c2} and p a~0c2} are also symmetric . Now, for x --+ 0 with x2 > 0, the function ❑+(x; jc) goes a s 1A+(x;/12 )-+4n2x2+ 8n2[In(7PN[x2) 2I(20.5.4) where y is the Euler constant . The first few terms in the vacuum expecta- tion value of the expansion of .I,~,(x).I~(O)are thu s (A'(X) J v(0)IVAC1qlUv- 27r2 ~(X2)2 qPV 4n2x2,I. dFt2 Pas ~~~ )~~ +27r2(x2)2 +O(lnx2).4XJUXV 2 ( 2} 2 2 x231 f d~(P(O)(P ) x# + Pafl (P )IF S dtt2 (p(°)(i2)2 + PaS(F12 )} (20.5.5) Hence if some linear combination J:aflcad(J&u(x}.I~(0)}SAC of the two- point functions has a singularity as x -+ 0 which can be shown to be 268 2() Operator Product Expansion s weaker than of order 1 /x4, we hav e co(p dttI (P111(p) + P(11(tt1)/P2) =0 while if its singularity is also weaker than 11x 2 ,then andcapfdp2 pang(µ2) =0 afl 2 (O)V)P2 oCafl dpN(20.5.6) (2Q.5.7) (20.5.8) Eqs.(20.5.6), ( 20.5.7),and (20-5 .8)are known respectively as the first , second ,and third spectral function sum rule s. Let us see how this works out in the case of greatest interest ,where the J,"(x) are conserved currents in a theory like quantum chromodynamics . The conservation of current tells us that (20.5.1)vanishes when contracted with p,,, so p ad(-p2)must be proportional to 6(p2),and therefore au- tomaticall ysatisfies the third spectral function sum rule (20.5.8)for any cxfl.With 'p(0)(-p1)or6(-p2),it can recei ve contribut ionsonly from the terms IBa) in the sum over states in (20.5.1) consisting of a single massless particle Bd of zero spin, which in pract ice means a Goldstone boson . For such one-particle states, Lorentz invar iance give s (VACJ.I"(d)JBa) =i Fcx a PB (2n)3/2 V'P' B Using the relation 6(p° - Cpt)/2p° =B(p°)6(-p2),we see tha t px~(-p2) = 6(-p2)FaaF~u. el(2a.S.9) (20.5.10) In contrast, pM(--p2)is non-vanishing only for -p2 > 0. To be more specific, consider a renormalizable asymptotically free gauge theory with a number N of massless (or nearly massless) spin 1/2 fermions belonging to the same representation of the gauge group . Quantum chromodynamics fits this description, with N= 3 if we neglect the masses of the u, d, and s quarks, and with N = 2 if only the u and dare taken to be massless . As we have seen in Chapter 19, such a theory has an SU(N) x SU(N) global symmetry* under which the left- and right-hande d There is also a U(1) symmetry which is vectorial, in the sense that it acts the same way on the left- and right-handed parts of the light Permian fields . This is just the conservation of light quark number, and will not concern us here . The axial U(1) 20.5 Spectral Function Sum Rules 2 69 parts of the light fermion fields transform under the representations (N,1) and (1,N),respectively, where ` N' and `1' denote the defining and identity representations of SU(N), respectively . The currents of the left- and right-handed S U (N} symmetries ar e JLa(x) =-iq)(xW`(1 + Y s)AaW(x),JRaW = -iv- W Y"(1 - Y s)AaV (X) (Z0.5.11) where the ARform a complete set of Hermit ian tracele ssmatri cesacting on the `flavor' index that distinguishes the Nlight quarks ,as in Eq .(19.7.2)for N= 3. These currents have dimensionality (inpowers of mass) +3, so the coeffi cient of an operator Cof dimen sionality d (®r)in the expan sion (2 0.3.6) of the product of two currents is expected (apart from logarithms) to have a singularity of order x -5+d(6)as the separation xof their arguments goesto zero .Thus,if the expan sion of a linear combination of product s of currents contains the operator unity, then the corresponding linear combination of spectral functions goes as x- 6,and therefore in general satisfies neither the first nor the second spectral function sum rule ;if the lowest-dimensional operator in the vacuum expectat ion value of this expan sion i sa fermion bilinear with zero or one deri vative s,then the corresponding linear combination of spectral functions goes as x-3or x- 2, and therefore satis fies the first spectral function sum rule but generally not the second ;and if the lowest-dimensional operator in the vacuum expectation value of thi sexpansion i sa fermion bilinear with two or more derivatives or a fermion quadrilinear then the corresponding linear combination of spectral functions is less singular than x -',and therefore satisfies both the first and second spectral function sum rules . In order to tell which operators appear in the expansion of the product of two currents ,we need to classify the S U(N)x SU(N } representations contained in the product ,and to tell whether these operators have non- vanishing vacuum expectation values ,we have to ask which of them are invariant under the subgroup of SU(N) xS U(N) that i snotspontaneou sly broken . To answer these questions ,we note that the currents JL,,(x)and .IRa(x)transform under SU(N) xSU(N) respectively according to the (A,1)and (1,A)representation sof S U(N) xSU(N), where A and 1 are the ad joint and identity representations of SU(N) . Also ,we shall assume that the subgroup of S U(N) x SU(N) that is not spontaneously broken is the vectorial SU(N)Vwhose currents are .I~Q(x) +JRu(x),as is the case in quantum chromodynamics and (aswe sawin Section 19.9)a wide range of other theori es.We al so as sume that parity conservation is not spontaneously broken .These unbroke n symmetry of the Lagrangian, which acts differently on the left- and right-handed parts ❑f the light fermion fields, is broken by the quantum effects discussed in Section 23 .5. 270 20 Operator Product Expansion s symmetries have the consequence that 111(P2)](20.5.12)pL1a Lb(P2) = pR1~,Rh (y2)= 6Rh [p,2) +P A and Pi1a,Rb00 ~ = PR~ ,Lb( F~2}= 6a h[p3)(2)- PAl)(P2 )] (20.5.13) where 6abp(y)(P2)and 6abpA1)(P2)are the spectral functions defined by (20.5.2)for the current s ,]ya = -z fpY"AawI JAL = - iV)yEiyS AaY), (20.5.14) with generators Aa and ~uyS, respectively . Als o FLab = -F'Rab = 6abF where Eq .(20.5.9)here read s (VAC1.IA"d(0)jB h) so that (20.5.10)readsiF6ahPB (2n)3/22pB(20.5.15) (20.5.16) to) z_ (0) 2 (o) 2 fob z 2 z PLa Lb~~) - PRa rzb (P) = _LaRb(P)=-PRa Lb~~ } = F ~~~ )~a b (20.5.17) It will be convenient to consider separately the products of currents of like and unlike chirality . Like Chirality: The products JLa(X) .ILh(0) and .IRa(x)JRh(0) transform re- spectively as the (A xA, 1) and ( 1, A xA ) representations of S U (N)x S U(N ), where A and 1 are the adjoint and identity representations, respectively . For any group, A x A contains the identity representation, so the unit op- erator appears in the expansions of these products, with equal coefficients proportional to 6ab. Hence only the traceless parts of the operator product can satisfy spectral function sum rules . But as we have seen, the spectral functions have no traceless part, so the like-chirality spectral functions cannot satisfy any sum rules . Unlike Chirality: The products JLu(x)JRh(0) and JRa(x)JL'b(O) both transform as the ( A, A ) representation of SU(N) xSU(N) .The unit operator (and operators like Fau,. Fx") are of course SU(N) xSU(N) singlets, and therefore cannot appear in the expansion of these products . Non-derivative fermion bilinears like rya transform according to the (N, N) and (N, N') representations of S U(N) x SU(N), so they also cannot appear in the expansions of the products of currents with unlike chirality . The only gauge- and Lorentz-invariant fermion bilinears with a single derivative involve the gauge-covariant derivative operator yADu acting on V, which the field equations tell us vanishes . Thus these spectral functions satisfy 20.5 Spectral Function Sum Rules 271 both the first and second spectral function sum rules, which here rea d fdui[p)(2)-P~i111(1t)] /P2=F2 and Jdui[p9(2 ) l V- pA~(Y2}~ = 0 .(20.5.18) In the original work on spectral function sum rules,5the S U(2) x SU(2) spectral functions were assumed to be sharply peaked at values of µ that for p(1)(1c2) could be assumed to be mR = 770 MeV, and for pAl(p2)was taken to be at an unknown mass mA . That is , (11(p) g2b(P2 _ M2) PA(F12) ga6 (u-, ` WM2) Eqs.(20.5.18) and (20.5.19)then read° * 2 29 P9A= F ~m2 z P M A and ~P =9 A Eliminating the unknown gA, this gives a formul a 2 2t 1)-1 gP--F7r W2 -W2 P A Originally in 1967 this result was used together with a formula6 g~ _ 2F,2,m~ (whose justification was unclear but which agreed with experimen- tal measurements of the rate of the decay p -} e+ +e-) to derive the result MA=-~2mp The status of a possible ai resonance with the right quantum numbers to couple to the axial-vector current (that is, 1+ with T = 1 and C(a°) = +1) at around a mass _~2_mp remained unclear for many years, but a resonance with these quantum numbers is now reasonably well established at a mass 1230 MeV = 1.6mR. It seems preferable today to take the ratio mA/mR as an input, using either the value _~2_ suggested by certain models or the experimental value 1.6, and use it to predict gR . Taking the S U(2) generators Aa to be the Pauli matrices, as in Eq . (19.7.2), we have F=F ,=284MeV . 272 20 Operator Product Expansion s Since 1967 not only g Rbut the whole vector spectral function pV)(u2} of the 5 U(3)xSU(3) current in quantum chromodynamics has been cal- culated with prec ision for a w ide range of energies from the measured cross section for the proce ss e+ +e-} y --* hadrons ,using the fact that the electromagnetic current is a linear combination of SU(3) cur- rents . The axial-vector spectral funct ionpA~)(u2)of the S U(3) xS U(3) currents can in principle be measured in the process v +e -}hadrons , since the charged components of the currents (20.5.14) are the same as the hadron currents to which the leptons are coupled . However ,although antineutrino --elect ronscattering ha sbeen studied e xperimentally ,the low rate of these reactions precludes the use of colliding beams ,so that the electron target is essentially at rest . In order to reach typical hadronic energies of ,say, 3 GeV in the center-of-mass frame ,it would be necessary to have neutrino energies of (3GeV )2 / 2me - 1 0 TeV in the laboratory frame .Intense beams of neut rinoswith energies this high willnot be a vail- able for many years ,if ever . Fortunately, it has become possible to study the spectral functions in the process z --*v+hadrons, but the hadron energies here are strictly bounded by m T= 1.7 GeV .It is also possible to use effective ch iral Lagrangians to calculate the spectral functions at small u2, and to use quantum chromodynamics to calculate py)- pA~)at large u2, where it is quite small .A careful 1993 analysis by Donoghue and Golowich8 of all these inputs shows that the spectral function integrals are indeed dominated by the p and a lresonances ,and gives results that are consistent with the first and second spectral function sum rules . 20.6Deep Inelastic Scatteri ng The renormalization group coupled with the operator product expansion found one of its most important applications in the analysis of deep inelastic lepton-nucleon scattering . We will first review the early phen- omenological models for these reactions, and then see how the operator product expansion justifies these models and provides corrections to them . Consider a process in which an electron of four-momentum k collides with a nucleon N of four-momentum p, yielding a electron of four- momentum k' and a general unobserved hadron state H, over which we sum. To calculate the spin-averaged inclusive cross section, we shall need to know the quantit y (MN IP°n~)W~`v (q,P) -264(PH-p- R) (HIP`(0)IN~ (H Jv(O)IN)* gN H (20.6.1) where .I9is the electromagnetic current (divided here by a factor e) and 20.6 Deep Inelastic Scattering 27 3 q = k - k' is the momentum transfer from the electrons to the hadrons . Lorentz invariance tells us that WlV(q, p)must be a linear combination of p1pv, p 1qv, q l'pv, q ~"qV, and qW, with coefficients that can only depend on the two independent scalar functions of q and p :q2 and v - -q 'pImN. Current conservation requires that q .W"' = q,, WP' = 0, so Wmust take the form` (qPqv -q1'v)Wj(v,q2) +mi2(pp-~ ~ q P)(Pv- ~~ qv )W2(v, q2). (20.6.2) N Also, Eq .(20.6.1)shows that W" =Wye, so W1 and W2 are real, and WPv is a positive matrix, so Wl and W2 are positive . The differential cross section in the nucleon rest frame i s d2,g dS2dv_ "'(d92 MATT(20.6.3) where d 92= sin 0Adois the solid angle into which the electron is scattered, and (da/dQ)MaTT is the differential cross section for relativistic elastic scattering by a point spinless particl e dd e4 cosZ(B/2) dSlMAT T 4E~sin4(B/2)(20.6.4) with Ee = -k  p ImNthe incident electron energy in the nucleon rest frame . One might expect that for fixed values of -pH = -q2 + 2mNv + mN , the differential cross section should fall off very rapidly as q2 -} oo, because it should be proportional to the square of the form-factor for the transition from the nucleon to whatever particles or resonance have mass near -pH .It was therefore somewhat of a surprise that, two years after the 19 66opening of the Stanford Linear Accelerator Center, a SLAG-MIT collaboration headed by Friedman, Kendall, and Taylor9 discovered that in fact v W2(q2, v) is roughly constant in q2 for fixed values ofco= 2mNV lq2>1. (To be specific, W2(q2, v)for the proton was fitted to the curve v Wz (q2 , v) ^ 0 .35 - 0 .004r.) for EE = 10,13.5, and 1 6GeV, and 0= 6° and 10° . These experiments were insensitive to W1, because tangy ( 10°/2) = 7 .6 x 10-3 .} Note that in this limit -per -} (co-1)qZ -} oo, which is why this is called `deep-inelastic' scattering . At about the same time Bjorkenla was using current algebra to argue that W2 (q2, v) and Wi (q2, V) satisfy scaling laws : for q2 and v going t o The same formalism can be used for other deep-inelastic lepton scattering processes, such as v,, + p->p+ H, except that since parity is not conserved in these processes there is an additional term in W P(q, p)proportional to e,"P' pq,. For simplicity, we will limit ourselves here to deep-inelastic electron scattering . 274 20 Operator Product Expansions infinity together , vW2 (v,R2)__),F2(CO)~ W1(v,q2) --> F1 (w) , (20.6.5) where again o) - 2m Nvlq2. A more intuitive explanation was given a little later by Feynman .11 He supposed that in deep-inelastic scattering on a highly relativistic nucleon of momentum p, the nucleon behaves as if it consists of `partons' of various types labelled i, each with a probabilit y a(x)dx of having momentum between x pand (x + dx)p . Then for each i , fdx JF z(x) = 1 . (20.6.6) The condition that the nucleon has total momentum pyields the additional sum rule i 1.f>$i(x)xdx=(20.6.7) For elastic scattering of an electron (with me neglected) from a parton of four-momentum xp, we have x2mN = - (q +.gyp)2 =-q2 --- 2vm Nx + x2mN, so v = q2/2mNx . The inelastic cross section in this model is thus** 2d3a d2 a z'dx(1+2 tan20vq~ --dS~dv Al 2m2 X2 2 b 2M XMATT i Q n1 ~ (20.6.8) where Qiis the charge of the ith type of parton in units of e . (The tan2(0/2) term here is appropriate for a `Dirac' parton, with magnetic moment eQil2mNx .) Comparing this with Eq .(20.6.3)gives N'2(v,q2)~~i10 W1 (v, q2)=1 2 ~ ~~ Jadx.Fj(x)bv - 2m_q) Nx dxFj(x)~qx2 ~ v -Q2 VC0 (20.6.9) fi]V ~ 2mNx ~ 2mNW~~''a R ) This may be derived from the formulas (8.7.7)and (8.7.38) for Compton scattering on a spin 1/2 particle of four-momentum p and mass m in a general Lorentz frame . For this purpose, it is useful to note that in the gauge used to derive these formulas, for which the initial and final real polarization vectors e,, and e~ tsatisfy ez = e'z = 2, e  p = e'  p = e - k = e' -k' = 0, the polarization sum gives 20.6 Deep Inelastic Scatterin g This agrees with the Bjorken scaling rules (20.6.5), wit h 1 ~ 1 F2(w)_(0 and Fj((o) = (col2mN)F2(cv).275 (20.6.11) X20.6.12) Eq.(20.6.12)was originally derived by Callan and Gross . 12 It agrees with experiment within about 1 0-15%. On the assumption that the proton and neutron respectively consist of two uand one dor one uand two dquarks, plus any number of neutral partons, Eqs .(24.6.11)and (20.6.6)yield the sum rule for W 2 do)F2(CO)c~Q~1 1. 2/3 n(20.6.13) The integral receives a large contribution from large co, where F 2is difficult to measure . If we also assume that the total momentum of the nucleon is equally shared among three quarks (and no neutral partons) then in place of Eq .(20.6.7)we have the stronger relation f ,F;(x)xdx =1/3 for each quark ,which with Eq . (20.6.11)yield s F~(03) (02EQ ~ 2/9 nT This integral is easier to measure, and turns out to disagree badly with the sum rule, showing that a large fraction of the nucleon momentum is carried by the neutral partons . None of the above phenomenology depends on any specific field theory . It was the operator product expansion that finally provided a way of ap- plying an underlying field theory to deep-inelastic scattering . In particular, the operator product expansion made it clear that one needed an asymp- totically free field theory to explain the scaling assumptions (20.6.5)(and to calculate the corrections to scaling) . This asymptotically free theory was ultimately provided by quantum chromodynamics . To apply the operator product expansion to deep-inelastic scattering, first Fourier transform Eq .(20.6.1).Using translation invariance and the completeness of the hadron states JHy, this give s (MN 1Pnr)W"'(q , p)1 2(2n)4GNd4z e-"q`~1V ~ .I`'(z)P(0)E,Ny .(20.6.14) The asymptotic behavior of W""(q,p)as q -} oo is therefore related to the singularity of the operator product at x -} 0 . Feynman diagram calculations of the coefficient functions in operator product expansions refer directly not to the expansion for matrix elements 276 20 Operator Product Expansion s like those appearing in Eq . (20.6.14) for WW, but rather to matrix elements of the two-point Greens functio n (mNIPN) T,"v(q,P) =2 ~fd4z e-t`~,z(NIT {.Iv(z),~~`(Q)lI~~(2a~)4IN (20.6.15) We can express this in terms of the structure functions for ?'PV, defined by the analog to Eq .(20.6.2): (qPqV_01") T, (v, q2) +(pp_p-q qP)(pv_ p * q qv) T2(v, q2) , (20 .6.16) Nq2 q The connection between the T, . (v, q2) and Wr(v,q2) (with r = 1,2) is provided1-~ by the dispersion relationst for fixed q2 : I tWr (v, q2) 2 2 dv+2ari Ex v' -V(20.6.17) with the denominator in the integrand interpreted as a principal value function . The functions Wr(v, q2) vanish except for v >q212mN, so the dispersion relations may be rewritte n T, (v, q z) = 1 2Wr(-v,qz)+1 2 Wr(va R2~ 2a~i ,Zdv' W, .(v', q2) I +v- I ) . ~-v (20.6.t8) 4 ~2mN V +v We can categorize the operators that contribute to the operator product expansion for Tr according to the irreducible representation of the Lorentz group to which they belong . The only Lorentz-covariant functions of a single four-vector p 1lwith p2 =-mN fixed are proportional to the symmetric tensors p "I   pis, so the only operators that can contribute to the spin-averaged nucleon expectation value are symmetric traceless tensors ;t' , the subscript i distinguishing any different operators with this tensor structure . (For reasons that will become clear, we are using the same label i to distinguish operators that we used in the parton model t o t For q2 = 0, these may be derived exactly as in the derivation of the dispersion relation (10.8.16) for forward photon scattering . The derivation for fixed q2 * 0is more difficult . 20.6 Deep Inelastic Scattering 27 7 label types of partons .) These operators have matrix elements of the for m E(N ~as~~, ---"s a z s2 IN~ = ( mNIPN )[pPl ... pE U-Ntraces] (0(5:~ , (20.6.19) where the (Usi~ are constant coefficients . Such an operator makes a contribution to Ttl`'(q,p) proportional to s factors of the four-vector p, and hence makes a contribution to Tl(v,q2) and T2(v,q2) proportional to vs and v1-2 , respectively . (We are dropping terms involving p2, because such terms will be suppressed by factors of mN~g2 or mN~p  q .)If we ignore logarithmic corrections, then in an asymptotically free theory the q2dependence of the coefficients is of the forms(q2)(-4+6-d(s,i)-s)12and (q2)(--4+b-d(s,i)-s+Z)/2' respectively, where d(s, i)is the dimensionality of the operator a,i .tt Using v oc qZro, we see that the contributions of an operator asa to the structure functions are asymptotically of the form s T1,siocV'(R2)(2-d(s,i)-s)12oCWs(q2)(2--r(sj))12 and vT2,,t cc ys-I(q2)(4-d(s,t)-S)/z cc ca s-1 (g2)(z-its, :7)12 where r(s,i)is the `twist' of the operator aye, defined as14 z(s,i)=d(s,i)- s.(20.6.20) (20.6.21) (20.6.22) We see that the dominant terms in Tj and v T2 for q2 --*oa with fixed w are contributed by operators of minimum twist . Also, Eq .(20.6.18) shows there are no terms in Tr(v,q2)of odd order in v, so the only operators G', j that can contribute here are those with even s . The symmetric traceless tensors of rank s with minimum dimensionality are the operators =his- 2/ s! )fpf'Y~►,~D~z...Des}Ylf and FIX.,~,,DP3...Des Fa P21(20.6.23) (20.6.24) where flabels quark flavors ; D.is the gauge-covariant derivative ; and the brackets indicate a sum over permutations and subtraction of trace terms for the enclosed spacetime indices . (The symbol <-4 indicates half the difference of the derivative acting to the right and left . We take this difference of derivatives, because their sum would vanish in an y t}The -4 in the exponents arises from the integral over z in Eq .(20.6.14), and the+6 in these exponents is the dimensional ity of the two electric current operators . The terms -s/2 and -(s- 2)/2 serve to compensate for the powers of q 1` already in v sand vs-2, respectively . 278 20 Operator Product Expansion s matrix between states of equal four-momenta .) These operators have dimensionality 3 + (s --1)= 4 + (s - 2) = 2 + s, so they all have twist z = 2 . Thus an infinite number of operators contribute for q2 __*oo, all of which give contributions to Tj and v T2 that depend on w but only logarithmically on q2 . Asymptotic freedom thus confirms Bjorken scaling, but only up to logarithmic corrections . Keeping only operators of twist two, we see that (as anticipated in our notation) for each s there is indeed one operator for each parton type, with i running over quark types f (lumping quarks and antiquarks together) and also a value i -- 0 for the gluon . Now let us consider the logarithmic corrections . The asymptotic behav- ior of the coefficient functions in an asymptotically free theory is governed by Eq .(20.3.9). As discussed in Section 10 .4, when we ignore electro- magnetic radiative corrections no renormalization factors are needed for the electromagnetic current, because these currents are conserved . Thus the matrix c&, in Eq .(20.3.9)vanishes . The matrix c(,r,e,, has no elements connecting operators of different Lorentz transformation type, so that in the notation we are now using, cSj,s ;j, = 6 .ssiciii(s) . Thus Eq .(20.3.9)takes the form 9q) Isij i 1 2) Ws-2/g2 Si j(20.6.25) 87cZC(s)fib v T2(v,q I ~(gq "s j where Vand 2 are constants appearing in the coefficient function for the operator 6('.st in the operator product expansion of the product of two electric currents, 1,2 is a particular value of q2 at which we choose to define the coefficient functions, b is the constant in the one-loop renormalization group equation (20.3.8)for the strong coupling constant g ., and (asiy is the constant coefficient in the matrix elements (20.6.19). The operator product expansion coefficients are independent of the particular process in question and unaffected by quark trapping, so we can calculate the coefficients s lsjand *st by considering a fictitious simpler process, the scattering of an electron by a free quark of flavor f.The renormalized operators G, jmay conveniently be defined so that the one- quark matrix elements of the operators (20.6.23) and (20 .6.24) are given by the tree approximation : I Uflesall'orl) = YlI T I'UI U) p P2 ... p ju, I6fff 5ff fl (.f1 a'~Asa~.f11 ,(7")= 0.(20.b.27) (20.b.28) Averaging Eq .(20.6.27)over d' = a"and comparing the result with 20.6 Deep Inelastic Scatterin g (20.6.19) yields279 shas f~ .f' m PP'pPz...pis = b f(-1 i-T-r -i + m~' ~~ PlP112...PP, Pa .f.f2s! 2Po so that (for even s) (0'sf4'= 6.f.f'Irn.f and (6"s0f,= 0.(20.6.29) (2Q.6.30) With this definition of the operators &,Sj, at a renormalization momen- tum scale t2 which is large enough for the operator product expansion functions to be calculated using the tree approximation, these functions can be derived from the tree-approximation formulas for the electron- quark scattering cross section . Retracing the parton-model derivations of Eqs.(20.6.9)and (24.6.10), we see that Wr for an electron scattering on a quark of flavor fis given in the tree approximation by e vW2,.f= Qn Nlmf} Q~~~~ -1) Wl,f = Q ~6(w - 1}/2mf(20.6.3 t) (20.6.32) Inserting these results in the dispersion relations (20.6.18) yields, for c~ )=~1, Q~ MN 1 Tl,.f=z 2nim f mf w - ~ ~z ~, ~ vT2,f =2 Zari cap -I(20.6.33) (20.6.34) results that may also be obtained directly from the Feynman graphs for electron -quark scattering .Comparing the coe fficients of cv 'in Eqs .(20.6.33) and (20.6.34) with those in Eqs .(20.6.25) and (20.6.26)at the renormal- ization poin tq2=t2and us ing Eqs . (20.6.29) and (20.6.30), wefind that ~ 2 .Si = ~ psi = im~~t (20.6.35) with the gluon charge Qa of course taken as zero . To repeat, even though these values have been derived for electron-quark scattering, S/,z and 25i are factors in the operator product expansion of two electric currents, and therefore do not depend on the state in which we calculate the expectatio n The factor (mN /m f ~ is inserted in Eq . (27 .6.3 ] } because v is defined as -q -plmN instead of -q  p 1mf,while it is W2 /M Nrather than Wz1rn f that appears in Eq .(20.6.2). Eq. [27 .6.32} then follows from Eq . (20 .6.10). The quanlily co may be written in a mass-independent way, as a) = -2q -plqz. 2$0 20 Operator Product Expansion s value of this operator product . So we can use Eq . (20.6.35) in Eqs .(20.6.25) and (20.6.26), and fin d x S7c2 c(s)/ h 27r(9q ((Isj~ Sly xYlZN nV ~'2~v~~~~ --> ~ ~ CUs-1~ I[(g/g)tS2C(5)/h] ; ~~sj y sly J(20.6.36) (20.6.37) Now let us at last return to the structure functions W,(v, q2).We note that the coefficient of ws = (2mNV lq2)5 in Tr(v, q2) is given by Eq .(20.6.18) as -2 Z(~w co-1-N 2~~ R 2 ~ -~ itfW ~q 2yn 27I2 2lN 4 ~ ~ 2mrrdV fv W r(vq Comparing with Eqs .(20.6.36)and (20.6.37), we fin d ( (Dq2 2) I Q r[(g/g)812C(S)/h] ccI-Sdryw W12mN,, -}2J: rj i.T ~~as;~ d(oa)-Sv W22mN,q --+ MN ~Q i2[(g/g)8Th2C(S)/b] (61sJ). (20.6.39) The Wrsatisfying these equations may evidently be expressed in a manner similar to the equations (2 0.6.9)and (24.6.1D)of the Parton mode l zo)q W1 2m'q2 N wq2 vW2 'q2 2MN__42MN Q2,~Fp1(o,q2)(20.6.40) where fit is a Parton distribution function, now defined by the moment equations ~ j~ [~X X1 (x,q2)- ~(g fg)87cZc(.s)1h (a~(20.6.42) zJ In particular, we see that asymptotic freedom implies not only a corrected version of Bjorken scaling, but also the Callan-Gross relation (20 .6.12) between Wl and W2 . There is an elegant reformulation of the moment equations (20 .6.42) due to Altarelli and Parisi" that has become widely used in studies of deep-inelastic scattering . Note that Eqs . (20 .6.42) together with the 20.6 Deep Inelastic Scattering 281 renormalization group equation (20.3.8)imply the differential equation s q2 d dx.xs-1~Fi(x, q2)=-g~ c lj(s) dx x'fl~ t (xaqZ) dR 4 ~ fo (20.6.43) These equations, together with the `initial' condition at the renormalization point 11 A xs-l,~Fa(x,(Z) = z (6sr) (20.6.44) have a unique solution, so they can be used instead of the moment equations . Now, Eq.(20.6.43) is satisfied by the solution of the differential equation for ~F i d g Z dy(X)qZ d Z ~ i(x,~"2)= 2-Pig `~.i(Y,qz)q 47r Y Y where the matrix function PIj(z)is subject to the condition s Ii Zs-1Ptj(2) d2 = -47C2ci.J(s)(20.6.45) (20.6.46) (The factor 471 2 is conventional .) The matrix cjj(s) was calculated in quantum chromodynamics by Georgi and Politzer16and Gross and Wilczek .l7They assumed that there are N flavors of quarks that are sufficiently light to be treated as massless, while quarks of all other flavors are treated as very heavy and integrated out, so that they can be ignored except for their effect on the strong coupling constant . Their results in the basis provided by the operators (20.6.23) and {20 .6.24} ar e 1 1 1C°°(s) 2n2 C112 ~ s(s - 1)1 (s-+-1)(s -+- 2 ) 1 1 2CfO(s)n2C2~s+2 + sus +1)(s-}- 2) Cof(S)= C3 ~ - + - 1 )Sn s + s (s1) s $n sus +1) t_z t(20.6.48) (20.5.50) where 0 and fdenote the operators (20 .6.24) and (20.6.23), respectively ; the constants C1 and C2 are defined by Eqs .(17.5.33) and (17.5.34) (with only a single quark flavor included in the trace in Eq .(17.5.34)) ;Nis the+~1 +NCzr_2 t 3 (20.6.47) 282 20 Operator Product Expansion s number of quark flavors ; and C3 is defined (using the notation of Section 17.4) by txtot - C3 92 1. (2 4.6.51) In the realistic case of an S U(3) gauge group with quarks in its defining representation 3, these numbers ar e Ci= 3, C2 C34(20.6.52) 331 These rather complicated results become simpler when expressed in term of the Altarelli-Parisi functions . It is straightforward to check that Eq. (20 .6.46) is satisfied i f X2 P f-":::5ff[4( 1+ 3 (1-x)+) 1PAD=x2-x +2 4 2 PQfT3 x~2+x Poa6 1-x+x (l-x )-+-x(20.6.53) (20.6.54) (20.6.55) x + It - x) - 3 rS( 1- x)(1x)+ 1 2 where in an integral over x up to x = 1, 1 /(1 -- x)+ is defined b y f(x} _fW-f(1) (1 - x)+ 1 - x(2Q.b.57) For each s there is an obvious (N- l)-fold degenerate eigenvalue of the matrix c(s), equal to the coefficient of b' in Eq .(20.6.50): G(S,idjoint) =1C31-2--4 7E2 s(s + 1) t =zt with eigenoperators given by all independent linear combinations of the operators (20.5.23)with coefficients af satisfying Efa f= 0, which belong to the adjoint representation of the unbroken global SU(N) symmetry group of quantum chromodynamics with N quark flavors . In addition for each s there are two eigenoperators belonging to the singlet representation ofSU(N), given by linear combinations of the operator (20 .6.24) and the sum over J'of the operators (20.6.23). These eigenoperators and the corresponding eigenvalues can be found by diagonalizing the 2 x 2 matrix : cos N cr~r~(s~(s}singlet = f ~ (20-6 .59)c fo(s) c (s, adjoint) 20.7 Renormalon s For s = 2 this matrix takes the for m IvCZ/ 6nz c(2)singlet C2/37r2NC316nz C3'37z283 This has one zero eigenvalue, corresponding to the linear combination of024 and EfCz fthat is equal to the energy-momentum tensor, which like Pis not renormalized . The other eigenvalue of the matrix (20.6.36) is given by its trace, NC2/6ir2 + C3/37r3 . Now, the minimum of the eigenvalues of c ij(s)for a given s must be at least as large as the minimum of the eigenvalues of cjj(s') for any s' <s, because otherwise for sufficiently large q2the integral (20 .6.42) would eventually become larger for s than for s', in contradiction with the fact that this integral is a strictly decreasing function of s . Because the minimum eigenvalue for s = 2 is zero, we can conclude that all of the other eigenvalues for s>2 are positive . In fact, they are all positive-definite, because there are no unrenormalized operators for s >2. Hence strict Bjorken scaling occurs only in the extreme limit where g~ ~ D, where only the contribution of the energy- momentum tensor survives . The prediction of violations of strict Bjorken scaling have been confirmed experimentally by exhaustive studies of deep- inelastic electron-nucleon and muon-nucleon scattering . 20.7 Renormalans* Since the beginning of quantum field theory theorists have wondered whether the perturbation series for physical matrix elements converges, and if it does not, then what can be done about it? Very early in the modern period ❑ysonlS observed that the number of diagrams of nth order typically grows as n!, which suggested that the perturbation series has zero radius of convergence . There is a well-known technique known as a Borel transformationl9 for improving the convergence of a power series whose nth order term grows as n!, either to make the series converge, or at least to improve the behavior of the series so that it can be used as an asymptotic expansion for a larger range of coupling constants . For a given serie s F(g) = E.fngnn we consider the related serie s B(2) - Efn2nln! n(za.7.i) (20.7.2) This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 284 20 Operator Product Expansion s Iffngrows no faster than n! then B(z) will generally have at least a finite radius of convergence . The question is, can we recover the original series (20.7.1) from the resummed series (20 .7.2)? Using the familiar formul a 00 exp(-z/g) zn dz = n! gn+1fr? we see that, at least for mally, gF(g) = of exp(-z fig) B(z) d2. (20.7.3) Singularities of B(z) anywhere in the complex plane limit the radius of convergence of the series ( 24.7.2), but this is not an insuperable problem if these singularities are off the positive real axis . To calculate F (g)using Eq.(20.7.3) we need B(z) only for real positive values of z less than or of order g, which can be obtained from the power series (20.7.1)if the singularities of B(z) in the complex plane are all at distances from the origin much greater than g . Even if a few poles zl, 22, etc . have moduli of order g or less, we can calculate B(z) out to values of z of order g by using the power series for (z - 21) (z - 22) ...B(2), but for this purpose we have to know where the poles are . Singularities of B(z) on the positive real axis are much worse, for they invalidate Eq .(20.7.3). The contour in this integral may be distorted to avoid singularities on the positive real axis, but then we have an ambiguity : do we distort the contour above or below the singularity ? This section will show that some of the singularities of the Borel transform B(z) are associated with solutions of the classical field equations known as irastarataras, while other singularities, known as renormadans, are associated with terms in the operator product expansion . In quantum chromodynamics it is the renormalons that obstruct the use of the Borel transformation to sum the perturbation series . It was Lipatov20 who in 1976 showed that some of the singularities of the Borel transform B(z) are associated with the existence of classical solutions of the field equations . Consider a function F(g) defined by a Euclidean path integral : F(g) - /[d4] exp (I [0,gl) (2U.7.4) (The use of Euclidean path integration is discussed in Appendix A of Chapter 23.)The coefficients in the power series (2 0.7.1)are given b y fn =1 2niI [do]Jdg g-n-1exp (1[0,g]) 1 27ri f [do ]dgexp ( I[0,g] - ( n-{-1) In g) , (2U.7.5) 20.7 Renormalons 285 where fdenotes the integral counterclockwise over a closed curve in the complex g plane surrounding the point g = U . For very large n, it is reasonable to suppose that the integral is dominated by the point fin, gn where the argument of the exponential in the last line of Eq . (20 .7.5)is stationary in both 0and g: 6110,gnu=a,500 0=0n 0I[On, g] n -}- 1 09 ~ 9~96gn(2a.7.6) (24.7.7) For instance, suppose that I [ O, g] is the action for a massless scalar fiel d 1 L~,1fg1 = - ~ Jcd4x - g 04 d4x , (20.7-9)24j the sum running over the Euclidean coordinate directions 1, 2, 3, 4 . The n the field equation (20.7.6)reads ❑On= 6gn 0n We will see that gn is negative, so the solution ha s On(X)=(-gn)-i~z&} with Z(x) the g-independent solution of the equatio n 11x= - 6x3 . The condition (20.7.7)tells us tha t 1 n -+-- /dx ~n= 241 gn or in terms of the rescaled field (20.7.10) 1 d4x4 At this stationarypointthe action (20 .7.8)becomes(2a.7.9) (2Q.7.1Q) (24.7.11) (2.7.12) I [On, gn] =-1 - = /c2i~id4x24/dx24 286 20 Operator Product Expansion s Evaluating E q. (20 .7.5) at the stat ionarypointthengives for n - >co" -n- n 11:zgnn-iexp (Ion5 g0) = (n + 1)n+1-e 24fx4d4x ) n!_1/x4d4x) (20.7.14) The leading singularity in B(z) is therefore a pole at z -- zj, wher e Z 1 1 24fx4 d4x . (20.7.15) Because this is negative, it does not prevent us from carrying out the integration in Eq .(20.7.3).To calculate the pole position (20.7.15), we note that the field equation (20.7.11) has the solutio n 4,~J-a(20.7.1G)rz+az where r=(xixi)112, and a is an arbitrary parameter . This solution of the field equation is an elementary example of the `instanton' solutions to be discussed in Section 23 .5. (These are called instantons because instead of being concentrated along a worldline they are concentrated near a point in spacetime - in this case, the origin .) Fortunately the pole position does not depend on a : . qr2dryzi= - 96jT2 a4W + a2)4_ -16n2. (20 .7.17) We see that the perturbation series for B(z) can be used in Eq .(24.7.3)if g<16n2 . If g l16n2is of the order or greater than unity, we may still be able to calculate B(z) from the perturbation series for (z +167r 2)B(z) . We will see in Section 23 .5 that there are instanton solutions in non- Abelian gauge theories like quantum chromodynamics, but these also yield relatively harmless singularities of B(z) on the negative real axis . The real problem in quantum chromodynamics is with a different class of singularities, known as renorrnalons .2 l These were first discovered through the realization22 that single 2n-th order diagrams, like that shown in Figure 2 0.3, can make individual contributions that grow like n!, and hence according to Eq .(20.7.2)may lead to additional singularities in B (z). In this particular case the singularity is known as an infrared renormalon , Here the symbol 'A-,'should be interpreted to mean `asymptotically equal up to factors of constants and powers of n .' These factors arise from the factor 12-gn in Stirling's formula for (n + 1)!, from the ratio of (n+ 1}! and n!, and from the integral over the fluctuations of g and O(x) around the stationary point . Since we are not going to calculate the factors from the last source, there is no point in keeping track of the factors from the first two sources either . 20.7 Renormalons 287 Figure 2 0.3. One of a class of N-loop Feynman diagrams that grow as NL Solid lines are fermians ; wavy lines are gauge bosons . as it arises from virtual momenta that are much smaller than those used to define the running quantum chromodynamics coupling &} . Fortunately, it is possible to use the operator product expansion to locate the infrared renormalons, without having to look at individual Feynman diagrams . For a simple but important example, consider the sum II U'(q) of all vacuum diagrams in quantum chromodynamics with insertions of four- vector currents J11 and .I~ carrying a four-momentum q into and out of the diagram . (I11v(q) gives the hadronic contribution to the electroweak vacuum polarization, and its imaginary part yields the cross section for e+-e- and electron-antineutrino annihilation into hadrons .) As we saw in Section 20 .5, this receives contributions that go as q2 (the Fourier transform of xT6) from the operator 1, contributions that go as q-2from the operator Fx V FjxjjU, contributions that go as q-4 from four-fermion operators, and so on . (For instance, the contribution of the operator FIX VFa,Zy arises from diagrams like Figure 20 .3, with a momentum much less than q flowing through the chain of bubbles .) Dimensional analysis tells us that these momentum-dependent factors must be accompanied with vacuum matrix elements proportional to fl°, A4, A6, etc . But if we calculate Feynman diagrams using a running coupling defined at a scale P > A, where the coupling constants are small, then according to Eq. (1$ .7.7) A2=~2 eXp 12n T (~~.7.1$~((33- 2ra f )as(~) where n f is the number of quark flavors with mass much less than U. Operators of dimension d >0 would make a contribution in the operator product expansion with the coupling-constant dependenc e Ad QCexp end (20.7.19)((33-f )~Xs(P) 288 20 Operator Product Expansion s In quantum chromodynamics, perturbation theory yields a series in powers of as - g 2/4n rather than g, so we would write Eqs .(20.7.1)and (20.7.3) in terms of as F(a,)anS (20-7 .20) n aSF(as) = 10exp(-2/as) B(2) dz . (0.7.21) The presence of terms in II"" with coupling-constant dependence (20.7.19) indicates that B(z) must have singularities (not necessarily poles) a t 67rd(20-7 .22) (33 ,f) These are on the positive real axis, and make the integral (24.7.21) am- biguous . Thus Buret summation cannot be used to deal with low-energy effects in quantum chromodynamics . The fact that diagrams with small virtual momenta impede the use of perturbation theory is of course nothing new . As we saw in Section 20.2, the whole point of the operator product expansion is to separate the parts of Feynman diagrams where every line carries a large momentum, which in asymptotically free theories can be calculated using perturbation theory, from the contribution of the parts of Feynman diagrams through which small momenta flow, which cannot be calculated perturbatively . Appendix Mome ntum Flow ; The General Cas e In this appendix we shall consider the asymptotic behavior of an amplitude in a general renormalizable quantum field theory when the momenta of any set of two or more external lines become large, taking into account operators containing arbitrary numbers of field factors and derivatives, with dimensionality up to some limit N . To deal with this problem we shall have to introduce a more compact notation than that of Sections 20.1 or 20 .2. A letter /.",/.', etc . will denote a set of external lines i of specified types entering or leaving a Feynman diagram or part of a Feynman diagram . A letter k, k', etc . will denote the set of four-momenta ki of such lines, subject to the condition that their sum has some fixed value p . The amplitude T //,(k, k', p)is the sum of all graphs with a set ~,of incoming lines carrying momenta k and a set ~' of outgoing lines carrying momenta k', including the final bare propagators for the set (but not ('. As shown by the power-counting theorem3 quoted in Section 12 .1, the part of the region of integration where the momenta of order k flow only through a subgraph with external lines ~ and e.1""makes a contribution to Appendix Momentum Flow : The General Case 289 Fee,(k, k', p )of order 01e"7, where d(~, e") is the dimensionality (in powers of mass or momentum) of this subgraph : iEe,e" ic e (The last term in Eq .(20.A.1)arises from the propagators for the lines in the set /.} It will be convenient to rewrite Eq .(20.A.1)as where ra{O is the number of lines in the set (, and N( e.)is the total dimensionality of the fields for these lines : iEe ic e Here 3 ; is the `spin' of a line of type i, in the sense used in Section 12 .3: the dimensionality of the field of type i is 1 + si, and the bare propagator of such a field goes as k-2+2Si . (For scalars and gauge bosons, sI = 0; for spin 1/2, s i= 1/2 .). By taking account of the asymptotic behavior associated with these subgraphs, we wish to show that as k --+cc (all components going to infinity together, in generic directions), with k' and p fixed, the asymptotic behavior of r q,(k, k', p) in each order of perturbation theory is of the form (om) ]Fee, ( k, k', p) Ue(k)Fa,p (kIaP) + o k4-4n{ t,}+N (e)-1V(20.A.4) where the sum runs over operators C 9with dimensionality N ((9} c N ; the function Ul(k)is of order k 4+N(1)-4n(e }-Nt67; and o (k')denotes terms that vanish faster (byat least one factor 11k) than V. Toisolate the contribution of operators of dimensionality ~ N, we shall define an `N-ir reducible' amplitude I',(k, k', p )as the sum of all graphs for I -'~?,(k, k', p )in which the lines in the set ~ cannot be disconnected from the l ines in the set I' by cutting through any set ~" of inte rnal lines with IV(if")c N . Since the difference F- INconsists of graphs that ca nbe disconnected in th is way, it may be w ritten a s (N) I~~,(k,k,P) dk" I e»(k,V, P)re,,e,(k~~~k%P),(20.A.5) where N} denotes the sum over sets /"of particle lines with N V"} c N, and , f dk" is the integral over the components of the four-momenta in the set subject to the constraint that the sum of these momenta is p . The asymptotic behavior of the kernel IN„(k, k", p )is much simpler than that of (k, k", p ). For k --+cc with k" and p fixed it is dominated 290 20 Operator Product Expansion s by the part of the region of integration where every internal line carries a momentum of order k, which gives an asymptotic behavior kd{,1,e"}, because the terms where only some subregion carries such large momenta would have to be connected to the rest of the graph by a bridge ~"' of lines with N(("') > N ~ N (("),and this would be smaller as k --+co by at least a factor 0(e" }-N- 1Thus differentiating I N,(k, k", p )with respect to any component of k" or p reduces its asymptotic behavior to order k But differentiating d times lowers the asymptotic behavior to kdt1,(")-d only if d c N - N ((")+ 1, because for higher derivatives we get a larger contribution from the region of integration where the subgraph carrying momenta of order k is connected to the rest of the graph by a bridge of lines 1'" with total dimensionality greater than N, with the derivatives all acting on lines carrying momenta of order k" or p . Therefore in order to take into account the contribution of operators containing derivatives of fields, we may write the asymptotic behavior of INas IA,(k, V a P)IN"(k)Piny (V,p)+o (kti')_N+N(") {20.A.6} where P p»v{k", p)are a complete set of homogeneous polynomials of order dv in the n((") momenta k" and p, and I ~„v(k) are functions only of k, of order kd(e/")-das k --+co. We cannot immediately use Eq .(20.A.6)in Eq .(20.A.5), because the integral over k" receives important contributions from the region where some of the k" are of order k . To deal with this, we use mathematical induction : we assume that Eq . (20 .A.4) holds up to some given order in perturbation theory, and use it on the right-hand side of Eq .(20.A.5)to calculate the asymptotic behavior of IT to the next order of perturbation theory . We rewrite Eq .(24.A.5)in the for m r,,,,,(k, k' ,p) = I~'~', (ka k',p) (N)f (N} p (N) +Fa,~~~k~, P~ d~'tI~1, (k, kit, p) U"" (V) (20.A.7) ~":N~~~~~:c9N According to Eq . (20 .A.4), the quantity in square brackets in the second term on the right-hand side of Eq . (20.A.7)vanishes for k" --*oa faster than (k'')4-4n{ 1"}+N{.~"}--N,so the product of this factor with a polynomial PPAk"} of order vc N - 11T(~'' )vanishes faster than (k")4-4nv"}, and therefore has a finite integral over the 4(n(e ."") --1 )independent components Appendix Momentum Flow : The General Cas e ofV.*Hence we may use Eq . (20.A.6)in this term, and fin d Fl/,(k,k',p) --* IN ee"V(k)Idk ffP4~,, v(V, p) l"v:d,+N(r')<N (N)I~~~ (k) PI,v(k', p) (N}291 (N) (N ) + F~~,~~~k', p} dk''IN" (k, V, p) U~ (V)a (20.A.8) the correction being smaller than the terms shown by a factor Ilk. (Of course, the first term on the right-hand side of Eq .(20.A.8)is present only ifN(I')cN.) Now, for each value of /,'and vwith dv + N(e,) cN,there is an op- erator 0(with field factors corresponding to the lines in /,and with dv derivatives, such that in zeroth-order perturbation theory the vertex func- tion, with incoming momentum pcarried by the operator ®rand outgoing momenta k carried by external lines (, is the polynomial Plv(k, p ).Then the corresponding complete vertex function for a renormalized operator r rf Fc,l(k, p)_J:Zr.,G~~{¼,~ PIv,,(k,p)+fdk' Pl,,,ve,(k', p) rlo,I(k'~ kaP) (20.A.9) where and vr, label the types of fields and spacetime derivatives in the operator U . We see then that Eq . (20.A.4) is satisfied, wit h (N) ~~(k) r leN -1, (k)Z 014fdk,lUCee(k,~)Pe,(k„) (N) This overlooks the possibility that even though power-counting indicates the conver- gence of the integral in the second term on the right-hand side of Eq . (20.A,7) over the region where all k" go to infinity together, for n((") ~ 3 subintegrations may diverge . For this reason the arguments given in this appendix do not constitute a proof of the operator product expansion except for the simple case treated in Section 20 .2, where we consider the momenta of just two external lines to go to infinity, and we look for the terms in the power series expansion associated with operators that are quadratic in the fields . 292 20 Operator Product Expansion s in which we now use the abbreviation s (The p dependence of is dropped for the same reason as in section 20.2: whether or not k" is comparable with k, p is negligible here compared with k.)We will define the normalization constant Zei,r, so that at a renormalization point k (µ),p(p) the function Fe,l(k(y), p(µ)} has the same value 6ll,PC,(k(µ), p(µ)} that it would have in the absence of interactions : x ~161,,lP?v6'(k(µ),Am)) +Idk' P101 vf,? (k', p(p)) r1,11~ [k', k (y),p(p)) (20.A.12) For F = 0 this ha sa solution Z = 1 which i sunique (because thepoly- nomials for a given set of lines are supposed to be linearly independent), so by continuity Eq .(20.A.12) will have a unique solution for coupling constants in some finite range .Eq. (20.A.10)therefore provides a recur- sive de finition of the coefficient funct ions U ;(k)appearing in the general operator product expansion (2 0.A.4). Prob lems 1.Consider a theory of a fermion field W interacting with a scalar fiel d 0with interactions of the form f p-q) and 04. List the operators tha t appear in the operator product expansion of ip-(x)W(O) with coefficien t functions that (judging from perturbation theory) are singular an d non-vanishing for x --*0. Describe how you would calculate thes e coefficient functions to one-loop order . 2. Consider quantum chromodynamics with Nmassless quarks, and define the spectral functions of the scalar and pseudoscalar quark bilinears b y E~VAC IV)(O)AaV (O)IVAC } ~VAC lq~(a)A#V( a)VAC }* N (2n)-30(P')Pa~(_P2) Y> ACKPA Y~ ~o~V[0~VAC}NACITK0)Y5AflW(0 )I VAC} afl References 293 where Aa are a complete set of traceless Hermitian NxNmatrices, normalized so that Tr = 2 6q. What spectral function sum rules are satisfied by linear combinations of p5(µz) and p ~(µl) ? 3. Derive Eq .(20.6.8)in the Parton model from the formulas (8.7.7) and (8.7.38) for Compton scattering . 4. List the gauge-invariant symmetric traceless tensors of twist four in quantum chromodynamics . 5. In the massless scalar field theory with interaction -go4/24(with g>0),where in the complex plane would you expect the function (20.7.3)to have renormalon singularities ? Reference s 1.K. Wilson,Phys.Rev.179,1499 (1969 ) 2. W. Zimmerman, in Lectures on Elementary Particles and Q uantum Field Theory - I97 0Brandeis University Summer Institute in Theo- retical Physics, eds . S. Deser, H . Pendleton, and M . Grisaru, (MIT Press, Cambridge, 1970) . 3. S.Weinberg,Phys.Rev.118, 838 (1960) . 4. C. Bernard, A . Duncan, J . LoSecco, and S . Weinberg, Phys . Rev . D12, 792 (1975) . 5. S. Weinberg, Pays .Rev. Lett .18, 507 (1967). For the general case, see Ref . 4. 6.K.Kawarabayas hiand M . Suzuki ,Phys.Rev.Lett.16, 255 (1966) ; Riazudd in and Fa yyazuddin ,Phys.Rev.147, 1071 (1966) . 7. M. Ademollo, G . Veneziano, and S . Weinberg, Phys . Rev . Lett . 22, 83 (1969) . 8.J. F. Donoghue and E . Golowich, Pays . Rev . D49, 1513 (1994) . 9. This result was reported at the 1968 `Rochester' conference at Vienna, and published in E . D. Bloom et al., Pays . Rev . Lett .23,930 (1969) ; M. Breidenbach et al., Phys . Rev . Lett .23, 935 (1969) . 10. J.D. Bjorken, Phys.Rev.179,1547 (1969) . 11. R. P. Feynman, Phys .Rev. Lett .23, 1415 (1969) ;Photon-Hadron Interactions (Benjamin, New York, 1972). 294 20 Operator Product Expansion s 12. C. G. Callan and D . J. Gross, Phys . Rev . Lett .22, 156 (1969) .The reader should be warned that the symbol ca as used by Callan and Gross is 2/co in the notation used here . 13. D. Gross and S . Treiman, P hys.Rev.D4, 1059 (1971) . 14.N. Christ,B.Hasslacher, andA.H.Muell er,Phys.Rev.,D6, 35 43 (1972) . 15. G. Altarelli and G. Parisi, Nucl .Phys . B126, 298 (1972) . 16. H. Georgi and H .D.Politzer, Phys . Rev . D9, 416 (1974) . 17. D. J. Gross and F . Wilczek, Phys .Rev. D9, 98 0(1974) . 18.F. J. Dyson, Phys . Rev . 85, 631(1952) . 19. See, e .g., G .N.Hardy, Divergent Series (Oxford University Press, oxford, 1949) . 20. L. N. Lipatov, Leningrad Nuclear Physics Institute report, 1976 (unpublished) ;Proceedings of the XVIII International Conference on High Energy Physics at Tbilisi, 1976 . 21. G. 't Hooft, in The Whys ofSubnuclear Physics - Proceedings of the 1977 price Summer School, ed. A. Zichichi (Plenum, New York, 1978) . For a collection of articles on renormalons and high orders of perturbation theory, see Large Order Behavior of Perturbation Theory, eds. J. C. Le Guillou and J . Zinn-Justin (North-Holland, Amsterdam, 1990) . For further work on infrared renormalons, see A. H. Mueller, Phys .Lett .B 30 8, 355 (1993) ;Nucl. Phys .B250, 327 (1995) ; A. Duncan and S . Pernice, Pays . Rev .D51, 1956 (1995), and articles quoted therein . 22. B. Lautrup, Phys . Lett . 76B,109 (1977) . 21 Spont aneouslyBrok en Gauge Sy mmetries The 1961 theorem' that broken symmetry implies massless spin zero Goldstone bosons was seen at first as a serious obstacle to the search for broken symmetries in nature . A few years later several authors noted an exception to this theorem, already mentioned in Section 19 .2: the Goldstone bosons are absent where the broken symmetry is local, rather than global .2 Instead, these degrees of freedom show up as the helicity zero states of the vector particles associated with the broken local symmetries, which thereby acquire a mass . This phenomenon, now generally known as the Higgs mechanism, was not at first applied in any sort of realistic theory, perhaps because by the mid- 19G0s it had become clear that the pion is a Goldstone boson of a spontaneously broken approximate symmetry, and attention therefore shifted away from the effort to avoid Goldstone bosons . But soon after, spontaneously broken local symmetries turned out to provide the natural framework for understanding the weak and electromagnetic interactions of the elementary particles .3 21.1Unitarity Ga uge We saw in Chapter 19 that in a theory with a global s ymmetry group G that isspontaneously broken to a subgroup H ,there is a massle ss`Gold stone' boson forevery independent broken symmetry, in the sense that the mass matrix Mn,of real spinless fields 0,(x)hasa zero eigen value w ith eigen vector for each independent broken symmetry generator to of G . (We are here considering the case where these Goldstone bosons are included among the elementary spinless particles represented by scalar or pseudoscalar fields 0nappea ring in the Lagrangian ; the more general case w ill be taken up inSection 21 .4.)We also saw that we could rotate away these Goldstone modes ,bysubjecting the fields to a G transformation y-1(x) On(x} y wnl(x}om(x} 295 296 21Spontaneously Broken Gauge Symm etries such that the new fields are orthogonal to the Goldstone direction s = E ~n(X)( ta)nmVm a (21.1.2) nm where v, is the vacuum expectation value, vn - (0n(0)}vAC . After rotating the fields so that they satisfy Eq . (21 .1.2), the Goldstone boson fields then reemerged as the spacetime-dependent parameters in y(x) . The point of this procedure was that since the Lagrangian was invariant under the transformation (21 .1.1) with constant y(x), all dependence on y(x) dropped out except where y(x) is acted on by derivatives . On the other hand, if the Lagrangian is invariant not only under constant G transformations but also under G transformations that depend on spacetime position, then the transformation (21 .1.1)is a true symmetry of the theory, and all dependence on y(x) drops out of the Lagrangian, so that we can simply replace 0n(x) everywhere with ~M(x) . This is a choice of gauge, fixed by imposing the condition (21 .1.2) on ~(x) rather than by imposing conditions on the gauge fields themselves . (For instance, in the electrodynamics of a charged scalar field 0 cal + 02,we can choose a gauge like Lorentz or Coulomb gauge by imposing conditions a,AP = 0 or D' A = 0, but we can also choose a gauge by imposing a condition on 0, as for instance that 0be real, or in other words by rotating the two-vector ~Reo, Imol into the 1-direction .) The gauge defined by Eq . (21 .1.2) is called u nitarity gauge,3abecause in this gauge it will be obvious that the theory does not have any degrees of freedom with negative probability, like timelike gauge bosons . More generally, the unitarity gauge makes manifest the menu of physical particles of the theory . Eq. (21 .1.2) shows that there are no Goldstone boson fields in unitarity gauge . Since the theory is gauge-invariant this means that there are no physical Goldstone bosons, whatever gauge we choose . What about the vector bosons? If the can are elementary canonically normalized scalar fields, the Lagrangian will contain a term 2 YO=- 21 E (0'(~n - Z1:trIXZrn Aa (21.1.3} n i1t, a where t 'runs over all the generators of the gauge group G . (From now on we shall drop tildes, it being understood henceforth in this section that we are already in unitarity gauge .) We are assuming that the symmetry G is broken by the vacuum expectation value vn of 0,,, so in order to see th e Here we are working with a real reducible or irreducible representation of the symmetry algebra, for which the matrices its are real . The transition to a complex representation is described below . 21.1Unrtarrty Gaug e nature of the particle spectrum, we define a shifted field 0' On =Vn + can297 (21.1.4) (It is sometimes convenient to take v„ in Eqs . (21 .1.2) and (21 .1.4) as the vacuum expectation value in the tree approximation . In order to generate a useful perturbation theory it is only necessary that vn should agree with the true vacuum expectation value in lowest order .) Expanding Eq . (21 .1.3) to second order in 0' and A, we have : -ro ,puAD_ - 2U0nna 2tomAaut?n in,IX(21.1.5) Using Eq .(21.1.2),we see that the 0'-Across term vanishes ,yield ing YO,QUAI] =-21: amo n0'`on - 2E ,aflAau A~ , (21.1.6) where 2 (~ 1:torn t~~ ~~ V,1. nm?'(21.1.7) Combining this with the quadrat ic terms in the Yang -Mills Lagrangian -4FaPvFa'',we see that the vector particles have a mass matrix yafl. In our notation the generators tn~are proportional to gauge coupling constants , so Eq . (21 .1.7)yield svector boson ma sses that are also proport ional to these coupling constants . Let's take a look at some of the algebra ic properties of E .c2,fl. Since t nm is is imaginary and antisymmetric (and hence Hermitian) the matrix µ~fl real,symmetric ,and positive .Also , if a certain real line ar combination of generators & cx tais unbroken ,then ECcx(to(}nmVM =0' a,m in which case Eq .(21.1.7)shows that, as noted by Kibble,2 a~ c~ = 0.(21.1.8) (21.1.9) That is, we still have a massless gauge boson for every unbroken gauge symmetry . The converse is also true ; from Eq .(21.1.7)we see that for arbitrary real constants c a Eµ2flcacfl_ aflCa  itn mtlm ~0, n a,m(21.1.10) and this can vanish only if c, satisfies Eq .(21.1.8). 298 21 Spontaneously Broken Gauge Symmet ries In particular, if there is just one unbroken symmetry Y :, eatx, then the general gauge field can be written (21.1.11) where ` - - -' denotes a linear combination of the gauge fields of definite non- zero mass, and ca is the coefficient of to in the single unbroken generator q1 q Gofta (21.1.12) so that the field AP will appear with zero coefficient in the mass term --2 &,µaOA~A~ fl.In order for the coefficient of -~(aPAv - avAP)2inthe kinetic gauge Lagrangian -~ &(OyAIXV - 0VA,,p)2 to have the canonical value 1, the ca must also be normalize d c2 IX(2.1.1.13) Note that q is the charge to which AP is coupled, in the sense tha t (21.1.14) where `  ' again denotes terms involving massive gauge fields . We shall use these general results in studying the electroweak theory in Section 21.3. These results have been derived here for scalar fields that form a real representation of the gauge group, but they can be straightforwardly converted to a form appropriate for complex representations . We saw in Section 19.6that a complex scalar field X(x) that transforms according to a representation of the gauge group with Hermitian generators T"' may be written as a set of real fields ReX(x) ImX(X) which furnish a real representation of the gauge group with generator s --~-Im?'a - Re?''i t°L = RE ? 'a -Im T '(21.1.16) Inserting Eqs .(21.1.15) and (21.1.16) in Eq .(21.1.7)gives the vector boson mass matrix (21.1.7)as = Re ((X)AC , TaTfl(x)VAC} = Z((X)TAC I€TaI Tfl I~xIVAC)µ~fl (2.1.17) Now let's take a closer look at the vector field propagator . Including the quadratic term in the Yang-Mills Lagrangian, the part of the total 2I.1 U nitarity Gauge 299 Lagrangian that is quadratic in A i s - a ~(0,jAcev - OvAa.Z)2 - zI:~a~ Aoc AAA = Z 1 :Aa''_9~X,,,flA(O) A/ + total derivatives , (2 1.1.1$) afl where Juafl ?1v'Z (2 1.1.19) Suppose for simplicity that all gauge symmetries are broken, so that µa has no zero eigenvalues . According to the general rules described in Section 9 .4, the gauge-field propagator in momentum space i s Dav,#A(k) =-(_9-t),xv,fl)#k) = [( k2+ µ2)-1 (nvA + Y-2kv k j)j~,. (21 .1.20) Because of the ky kti term, the propagator has an asymptotic behavior A(k) - Q(k°), which does not allow us to use the usual power-counting arguments to prove renormalizability . Fortunately, as we shall see in the next section, there is another gauge in which renormalizability is obvious, at the price of obscuring the particle content of the theory . It is important to note that although Goldstone bosons suddenly reap- pear in the physical spectrum in the limit of zero gauge couplings, physical matrix elements are perfectly continuous in this limit . This is because in unitarity gauge the gauge bosons do not entirely decouple for zero gauge couplings . Consider the matrix element for a scattering process A + B --*C + D, where Aand C belong to some representation of the gauge algebra, and B and D belong to some different representation of the gauge algebra . The S-matrix element for this process receives a contribution from vector boson exchang e Scp,BD =Z(27t)464( PA + PB - PC - PD )(C IJNaIA)❑av,fla,(k)~DIJNaB ), (21.1.21) where k = pA - PC = PD - PB, and .INS is the current to which the gauge bosons are coupled, with the subscript Nto remind us that in this gauge we omit the Goldstone boson pole term in this current . This current is proportional to gauge coupling constants, so the only terms in Eq. (21 .1.21) that survive in the limit of zero gauge couplings are the ones involving the matrix µ-2 , which becomes singular in this limit . Hence for 300 21 Spontaneously Broken Gauge Symmetrie s zero gauge couplings, the S-matrix element i s SCD,BD -+ i(2rz)464(PA + pB - PC - P D)kvkA(y') ocfi(CIJIV~A )k~ ~Dj .IN~JB} (21.1.22) The gauge coupling constants in the currents are cancelled by the gauge coupling constants in the matrix µ2 . Now let us compare this result with the contribution of Goldstone boson exchange . For vanishing gauge couplings there is a set of Goldstone bosons B, associated with generator t ,,(assuming for simplicity that all gauge symmetries are spontaneously broken) with the kinematic term in the Lagrangian given by - Z ~ A0ar~~`Ofl En Za nZfln, where ~y,~ is the component of the Goldstone boson B,, in the spinless field 0,defined by Eq .(19.2.39).In order that the Goldstone boson fields be canonically normalized, we must therefore hav e ZcxnZ#n = 6xfi. (21.1.23) n According to Eq . (19 .2.49), the Goldstone boson Ba couples to the currents .INS with a coupling constant Fad , where Ffl is the coupling of the Goldstone boson associated with generator tx to the current .IN'defined by Eq .(19.2.38).Thus the exchange of Goldstone bosons would give a scattering matrix elemen t ScD,aD= i(2n)464(PA+pB - Pc - P D)kvkjay1F-y1~G JJNaJA} 1 F2 ~DJJN~IB}. (21.1.24) But Eqs . (19.2.40), (21.1.7),and(21.1.23)give the vector boson mass matrix as a~ _ Y:FavZvnFfl6Z6n -Fav Fflv , (21 .1.25) n so in the limit of vanishing gauge boson coupling, the gauge boson exchange matrix element (21 .1.22) is the same as the Goldstone boson exchange matrix element (21 .1.24). This argument can be reversed ; the requirement of continuity at zero gauge coupling can be used to derive a formula for the gauge boson masses even in the case where all other couplings are strong . 21.2 Renormalizable ~-Ga uges In 1971 't Hooft4 showed that path integrals in spontaneously broken gauge theories can be calculated in a gauge in which the vector boson 21.2 Renormaliza$[e ~-Ganges 301 propagators vanish for momentum k --*ao as k-2, so that these theories satisfy the power-counting test for renormalizability described in Section 12.2. Here we shall describe a larger class of renormalizable gauges, parameterized by an arbitrary constant ~, that was introduced a little later by Fujikawa, Lee, and 5anda .5 In general gauges the kinematic term (21.1.3)in the Lagrangian for the scalar fields of a theory contains a cross ter m i auo~ n ta nmAa 1`vm nmoc where v . is the vacuum expectation value of 0 ., and 0n is the shifted field defined by Eq . (21 .1.4). In unitarity gauge this term vanished as a consequence of the gauge condition (21 .1.2). We will adopt a different approach here, similar to that of Sections 15.5and 15 .6. A functional S[f ] is introduced into the path integral, wit h B [f]= exp i(2~(21.2.1) This is equivalent to adding a gauge-fixing term in the Lagrangia n Ygf = -1 ~~fa fu (21.2.2) Instead of taking f,= aAa as in Section 15.5,we shall now take the gauge-fixing function a s .fa=aPA~ - x ~(ta)nmon Vm a (21.2.3) which is designed so that the above cross term in (21.1.3)is cancelled by the cross term in Eq . (21 .2.2). Unitarily gauge is now a special case ; for ~--),oo, the gauge-fixing functional (21 .2.1) is infinitely sharply peaked at a0' that satisfies the unitarily gauge condition (21 .1.2). Another special case is provided by the limit ~ --*0; here the gauge-fixing functional is peaked at a gauge field that satisfies the Landau gauge condition a uAa = O . We also include in the Lagrangian a quartic polynomial -P(O), subject to the gauge-invariance conditio n OP(O) (ta0nmOm = 0 .on(21.2.4) We must of course also include in the Lagrangian a gauge-field ter m Y,q = - 1 4~ F yvFa,v , (21.2.5) 302 21 Spontaneously Broken Gauge Symmetrie s The total Lagrangian density of gauge fields and scalars is the n _ - ~EFaVFccmv+ ~ ~(t"O)n(tflO)nA~Al~u - ~ E(auAa) (avAa) 4 a naffs ~ ~ X - 1 ~0,U0 napo n + ~2IXniii~(t"V)n(taV)mOnOm P~ -P(O) + i~ ayOn(ta)nmO;,zA~ + total derivatives . (21.2.6) cxnm As we saw in Section 15 .6, the introduction of a gauge-fixing functional B[f] requires also the introduction of a ghost field c),(x), with Lagrangian depending on the gauge-transformation properties of fx. Under a general gauge transformation (with an arbitrary function c,(x) ) 6Aa=- ECa~ye~ Ayy + fl'i On = t ~ Fx(tx)nmO m we have(21.2.7) (21.2.8) ~.fx=El E (F,Ay,`)+~E(tav)nFfl(tflO)n. (21.2.9) PT na According to the general results of Section 15 .6, this yields a ghost Lagrangia n 03a❑wa -~--~ ~Cx#Ya~(o)#A,} +~E( txv)nCt3#(t#O) n nfl Finally, if the theory involves spin Z fermivns there will also be a general renormalizable ter m where 40 is the matrix representation of the generators of the gauge group for the fermions (including coupling-constant factors), and m and Fn are constant matrices (in general, linear combinations of terms proportional to Dirac matrices 1 and T5)satisfying the gauge-invariance condition s [tP, Y4m0] =0 [tx~'}aY4r.] +(21.2.12) EIrromn y 4rm = 0. (21.2.13) m (The factor y4 - iyQ arises from the definition ip - WTy4. It is relevant only if taV) involves terms proportional to y5 .) The general theorem proved in Sections 15.5and 15 .6 guarantees that the S-matrix calculated from a 21.2 Renormalizable ~-Gacuges 3 03 Lagrangian given by the sum of Eqs .(21.2.6),(21.2.10), and (21 .2.11) is independent of the choice of the parameter ~ appearing in the gauge-fixing function (21.2.3), and so for any ~ will give the same results as the choice = oo corresponding to unitarity gauge . To derive the propagators for all these fields, we need the part of the Lagrangian that is quadratic in fields : SQUAD = -4E (a" Aav-a''Aa'') (a .A.„ - a,,Axu) 12 1 1 2 aA PAf~E, -~~E(aMAa) (aAa v) - ~ ~ (aito r~) 0110n) - 1 ~ 2 MmOnO m n nm P (X UP + total derivatives , where y IXflis the vector boson mass matrix (21.1.7): n and Mnm and m are new scalar and fermion mass matrices : ~2_02p( d~) -urnaonaomO=U YYZ = 1 '11[ + t'nt1n.E( taV)n(taV)m2a(21.2.14) (21.2.15) (21.2.16) We see from Eq . (21 .2.14) that the ghosts have gauge-dependent masses, equal to J times the corresponding vector boson masses . These expressions give the particle masses in the zeroth order of per- turbation theory . To this order, the vacuum expectation value v, is just the location of the minimum of the polynomial `potential' P(O ) aon4=v Also, as we saw in Section 19 .2, it follows from Eqs . (21 .2.4) and (21.2.17) that a2p(o) (tIXv)m=0 inaonaomO=u for all a . It follows then that in place of the Goldstone modes with mass zero, the scalar boson mass-squared matrix in Eq .(21.2.16)has eigenvalues 304 21 Spontaneously Broken Gauge Symmetrie s equal to 5 times the non-zero vector boson masses . That is, if /ca flha an eigenvector cfl with eigenvalue y2 , then >:#c#t#v is an eigenvector c M2 with an eigenvalue ~µ2 : M nm c#tPUP aflC#(tuv)n = ~~~ catav (21 .2.19 m n The other eigenvectors of 1V1n„= are then orthogonal to all of these, any hence to all t ,v, so these eigenvectors and the corresponding eigenvalue are just the same as for the matrix {a2p( 0/00n4 m)O=z.We see that fv the unitarily gauge value ~ -} oo, the Goldstone bosons are so heavy tha they drop out of the theory, while the other boson masses are as usual . The propagators are calculated by the usual rules : if the free-partich part of the Lagrangian takes the form (after integration byparts ; -6fg(a)~ for a complex field ~ or- 2 for a real field C, then the propagator of this field is -q-'Pik} . This gives the propagators : A (k)_ ~ 12(nuv y (1 ~ I)k~ kU (21.2.20) ~ + p k+~~ Onm(k) = (k2 +M 2) nrn+(taV)n(t(iv)m(k2)-Z(k2 +~P2) (21.2.21) (21.2.22) (21.2.23) The poles in Eq . (21 .2.20) at unphysical mass squares proportional to ~ are cancelled by the poles at the same masses in Eq . (21 .2.21). Note that now for finite ~ all propagators have the same asymptotic behaviors as in the unbroken symmetry case, as required for renormalizability . In particular, the k ,,kV term in the vector boson propagator no longer presents any problem for renormalizability, because it is accompanied with an extra factor (k2+y2~)-1 . We can even drop this term by choosing Feynman gauge, with ~ = 1 . It is only in the unitarity gauge case where ~ - )~oo that this factor fails to give the propagator the asymptotic behavior needed for renormalizability . Even with well-behaved propagators, it is still necessary to show that the ultraviolet divergences in these theories are constrained by the broken gauge symmetries in such a way that every infinity can be cancelled by the renormalization of a field or a parameter in the Lagrangian . This can be done by the same techniques as in Sections 17 .2 and 17 .3, but treating the vacuum expectation values of the spinless fields as external fields rather than fixed quantities that break the symmetries .~~ 22.3The Electroweak Theor y 21.3 The Electroweak Theorycos The most important application of spontaneously broken gauge theories has been to the theory of weak and electromagnetic interactivns .3 Weak interactions at low energies are well described by an effective Lagrangian given by a sum of products of vector (including axial-vector) currents, as in Eq . (19 .4.22). This suggests that these interactions may like electro- magnetism be described by some sort of gauge theory . In order to insure the separate conservation of electronic-type and muonic-type leptons and baryons (or quarks) we may guess that the known electronic-type and muonic-type leptons and the quarks all form separate representations of the gauge group . With this assumption, there are only a few possibilities for the structure of the gauge group . Let's first consider the electronic-type lepton fields . As far as we know, these consist only of the left- and right-handed parts of the electron field e: andapurelyleft-handedelectron-neutrino fie ldvex, Y5VeL =vPL(21.3.1) (21.3.2) The fields in any representation of the gauge group must all have the same Lorentz-transformation properties, so the representations of the gauge group here divide* into a left-handed doublet(v,L, eL) and a right-handed singlet eR . The largest possible gauge group is the n S U (2)LXUMzXU(1}R, under which the fields transform a s Ve)=i[- .- (Ve eCt+-FOL +-eRtR] e (21.3.3) where the generators ar e = 4~ 1+Y5}0 1 t tLOC 0 + y5}i o o1-i 1 0(21.3-4)o o - i ),(0 tR OC 0 -75) 3(21.3.5) (21.3.6) If we allow gauge couplings that change electron-type lepton number, then it is possible to include the left-handed field e~ along with vEL and PL in a representation of the gauge group . This was the basis for an early SO(3) variant' of the electroweak theory, which has since been ruled out by experiment . 306 21Spontaneously Broken Gauge Symmetrie s with g a constant to be chosen later . It will be convenient instead of tL and tR to consider the generator s 1+ y$ 1 0 1 -75 y-g 4 D1+ 2 and 1 ~-Ys 1 0 1-75 ne ~~~ 2 0 1 + 2(21.3.7) (21.3.8) where g' and g" are constants like g to be chosen later . The generator y appears along with t3 in a linear combination that plays a special role in physics ; it is the electric charg e q.= e ~ _~ -et3- ~ y . (2 1.3.9) g g Also, ne is the electron-type lepton number . We want to include both charge-changing weak interactions (like beta decay) and electromagnetism in our theory, so we will assume that there are gauge fields Al` and B" coupled to t and y . In addition, we may or may not want to include a gauge field coupled to the one remaining independent linear combination of tL and tR, which can be taken as the electron-type lepton number (21.3.8).There are very stringent limits7 on the long-range forces that would be produced by a massless gauge field coupled to nE, so in order to include in our theory a gauge field coupled to n, with a strength g" comparable to the weak and electromagnetic interactions, we would have to assume that this gauge symmetry is spontaneously broken ." However, there is no experimental evidence for the weak interaction that would be produced by such a gauge coupling (and plenty of evidence by now against it) so we shall simply exclude ne from the generators of the gauge group . The gauge group is the n with generators 1,ygiven by Eqs . (21 .3.4) and (21.3.7)respectively . The coupling constants g and g' are to be adjusted so that the gauge fields AP and B ucoupled to these generators are canonically normalized . The most general gauge-invariant and renormalizable Lagrangian that involves jus t Note that this is possible without violating the global conservation law of electronic lepton conservation . We would have to assume that the Lagrangian is invariant under both a global phase transformation acting only on electron-type lepton fields, and also a local phase transformation that acts on electron-type lepton fields as well as on some scalar field that does not interact with leptons . The vacuum expectation value of this scalar would break the local symmetry, giving the gauge boson coupled to ne a mass, without breaking the global symmetry . 21.3TheElectroweak Theor y these gauge fields and electronic leptons is then307 Y Y Y ~ L GYM +ye = -4 (auAV-0,,A~ + gA~,xAv}- 4 (a u Bu -aV$u)2 a-?(0- i'AtL - iAy y. (21-3 .11) (We here use the fact that the structure constants of S U(2)L and U (l) are Cask = -i g Ft jk and zero, respectively .) Of course, of the four gauge fields coupled to t and y, only one linear combination, the electromagnetic field A A, is actually massless . We therefore must assume that SU(2)L X U(1)is spontaneously broken to a subgroup U (1)gym, with generator given by the charge (21.3.7).The details of the symmetry-breaking mechanism will be considered a little later . However, whatever this mechanism may be, we know that the canonically normalized vector fields corresponding to particles of spin one and definite mass consist of one field of charge +e with mass ma y another of charge -e and the same mass : WP*(Al - iAll)(21.3.12) (21.3.13) and two electrically neutral fields of mass mz and zero respectively, given by orthonormal linear combinations of A~ and S ~ ZIA_ cos 0 A~ + s in0BP, A" sin d A 3 + cos B B~`, or equivalently A~ = cos 0Z,` - sin d AP, R~` = sin 0V+ cos F] A J`(21.3.14) (1.3.15) X21.3.16) (21.3.17) According to the general result (21 .1.11)-(21.1.12),the generator of the unbroken symmetry ,which is here electromagnetic gauge invariance, is given by a linear combination of generators in which the coeffi cients a re the same as the coefficients of the corresponding massless field in the expansion of the canonically normalized gauge fields coupled tothese generators . Inspecting Eqs .(21.3.16) and (21.3.17) shows tha t Comparing this with Eq .(21.3.9)gives then(21.3.18) , ' g = -e/ sin O9 = -e/ cvs O . (21.3.19) 308 21 Spontaneously Broken Gauge Symmetrie s The complete lepton -gauge boson coupling can be expressed in terms of the coupl ings g and g': tY1- - ve e ~e - vee *(tiL+it2L) t t sin 0 +y Cos 0 ve + ~ ( 3L ~vs B + ysin Q) + ~(- ~L}1( e y5 9 (Ve f'~j (1 +1 5 g eYV(1+)ve ~ T 2~`2 1Vrg~ T~+75) (9 2-g12 } (lY5 )+ 2 2 ~e ) 1- +gZ 215 )e-e(~,Ae) To complete the theory, we must now make some assumption about the mechanism of symmetry breaking . We want this mechanism to give masses not only to the Wt and ZQ, but to the electron as well . Now, the only way that this is possible in a renvrmalizable weakly-coupled theory is to have a scalar field coupled without derivatives to IR and IL (and also ?L and ~R ).Then SU (2)L X U(1) invariance requires that the scalar be an S U(2)L doublet like h, but with a shifted value of y and hence of q. We thus assume a `Yukawa' couplin g Y'Pe=-Ge v~e eR + H .C. 21 .3.21) )L where ( 0+,0°)is a doublet, on which the SU(2) xU(1) generators ar e represented by the matrices : 2 1o i0 0 - 1 yW 91/2 1 001 so that the charge matrix i s 9g o0(21.3.23) It is possible that there are other scalar multiplets in the theory, but for the moment let's suppose that this is the only one . We must add a gauge-invariant term involving scalar and gauge fields to the Lagrangian . The most general form consistent with S U (2)xU(1) 21.3 The Electroweak Theory 309 gauge invariance, Lorentz invariance, and renormalizability is : 1'~a~ - iA~ - t[~l _ i B 2 ~ yt~~~~l~ _ ~~(21.3.25)2 4 where A>0, and 00 (21 .3.26) For µ2c D, there is a tree-approximation vacuum expectation value at the stationary point of the Lagrangia n We can always perform an SU(2) X U(1) gauge transformation to a unitarily gauge, in which 0+ = 0 and 0Q is Hermitian, with a positive vacuum expectation value . (This is why we normalized the complex doublet 0so that an unconventional factor-1appears in the kinetic term in Eq .(21.3.25);Re0° is the only physical scalar field, and Eq .(21.3.25) makes this a canonically normalized field .) In unitarily gauge the vacuum expectation values of the components of 0are The scalar Lagrangian (2 1 .3. 5)then yields a vector meson mass ter m 21 (0))(0)12 = -, (g _ ___g, B --I (A-,, - PO) + B,,y -- A 'u T -'u) ( 0) 22 2 v 4 S We see that as expected, the photon mass is zero, while the W± and Z Q have the masses 2 2 Also, from Eqs .(21.3.21) and (21 .3.28) we see that the electron is given a lowest-order mass me =Gev. (21.3.31) It is difficult to study reactions among electron-type leptons atone, though by now there are data on scattering processes tike vP+e---+ve+e . For high precision data we have to consider reactions that also involve at least involve muonic-type leptons, such as the well-studied process of muon decay, µ+--~e++Ve+vP.It is trivial to extend the above model to include muon-type leptons -just add to the Lagrangian terms Y. 310 21 Spontaneously Broken Gauge Symmetrie s and Yo., like the last terms in Eqs .(21.3.11) and (21.3.21), with the fields e and Ve replaced with the muon and muon-neutrino fields it-and v ., and with Ge replaced with G.= Ge (m./m,.).Inspection of (21.3.20) and the corresponding term with e and ve replaced with itand v,, shows that W exchange between low energy, e-type and it-type leptons produces the effective interactio n g ' 1 m yt,~'Y~~+ Ys v evEtYA1 ~ y5 ~ + H .c. (2 1.3.32) 2 This may be compared with the interaction of the effective `V - A' theory which is known to give a good description of muon deca y G~(~7" ( 1+Ys) Ve)(v'YA 0 + ys)µ}+H.c. (21.3.33) Here GF is the conventional Fermi coupling constant, known from the muon decay rate to have the value G F=1.16639(2)x 1()^5GeV-2 . Com- paring these two expressions, we fin d g2/rnY1, =4~Gv. (21 .3.34) This allows an immediate determination of the vacuum expectation value v, given by Eq .(21.3.30) as 2m WV . ^ 247 GeV (213.35) = 1 1/2 2114GF Also ,Eq.(21.3.31)shows that Gehas the very small value 0.511 Me V ~e =2.07 x10-6 247GeV(21.3.36) From Eq .(21.3.30)we see that mz >mw. We cannot use Eq .(21.3.30)to determine the actual values of mz and mw without knowing something about g and g' . Using Eqs .(21.3.30) and (21.3.19), we can express mz and myy in terms of the electroweak mixing angle 0: ev 37 .3 Ge VM W 21 sin 01 Isin D l ev 74 .6 GeV mz ~ 21 sin 011cos01 = I sin 20 1 These are the original results obtained in Ref . 3. Of course, there are radiative corrections of all sorts, most of which depend on details of the theory that have not yet been specified in this section . But there is one particularly large radiative correction that can be readily calculated with- out further information . The above values for myy and mz were calculated using the conventionally defined electronic charge for e . However, as explained in Section 18 .2, this is not precisely the appropriate value to use 21.3 The Electroweak Theory 311 in calculations of processes at energies E >me;we should instead use the electric charge eF, defined at a sliding scale pcomparable to the energies of interest . For Pof the order of 90 GeV the effective fine structure constant e~/4~ is about 1/129 (and quite insensitive to the precise value of P),so the above values for mw and mz should be multiplied with V1371129, giving mW38.4 GeV(21.3.37)1sinB ~ mz =76.9 Ge V 1sin 201 (21 .3.38) Whatever the value of 0, these masses are too large for there to have been any hope of detecting the W or Z in the 1960s or early 1970s . Exper- imental evidence for the electroweak theory had to come instead from the discovery of the new class of weak interactions predicted by the theory, the neutral current processes produced by ZO exchange .9 The first observation of a neutral current process was the 1973 bubble chamber detection of the purely leptonic process of v .-e- elastic scattering .10 Although these processes are easy to deal with theoretically, the frequency of events is relatively low, because the cross section is proportional to the square of the center-of-mass energy . It was years before the purely leptonic neutral current reactions could be used to give a reasonably precise value for the parameter sin 2 B . By 1994, the study of purely leptonic neutral current processes like v,, + e- --* v,, + e- and v,,+ e- -~ vt, + e- had yielded the value 0 .222 ± 0 .011, which would give mw = 81 .5 GeV and mz = 92 .5 GeV . Even before the discovery of neutral currents, the electroweak theory had been extended to the weak and electromagnetic interactions of hadrons with each other and with leptons . By the mid-1960s, it had become understood that weak interaction processes in which charge is exchanged between leptons and hadrons are well described at low energy by the effective Lagrangia n GF[ey(1 + 75)Ve +pYA(1-}-Y5)vp ] Jz + H.c. , (21 .3.39) where Pis an hadronic current . Within the quark model, the commutation and conservation properties of Pallowed it to be identified with the quar k The cross sectio nisproportional to Gr, so inorder to havethe d imensions of cncrgy Z, itmust also beproportional to so me energy s quare d. Where the center-of-mass ene rgy is m uchlarger than the e lectron m ass, it is t he only energy that ca nappear in thi s formula. 312 21 Spontaneously Broken GaugeSymm etries current P= uY'~(1 + Ys )dcosB,+uy~(1+y5 )s sin B,. (21.3.40) Here u, d, s are the fields of the up, down, and strange quarks, and O.is another angle, known as the Cabibbo angle .ll Experiments on processes like 014 -~N14*+e++vP and K+ ---- *7i° + e+ + ve confirm that GF has very nearly the same value as that measured in the purely leptonic process bi+--+v+e++V.and give for H ,the value 12 sin B ,= 0.220 +0.003. We naturally conclude that the quarks provide another S U(2) xU(1) double t (1-}- Y5u -2 = 2 d cos H ,+s sin B, as well as right-handed singlets, with y values adjusted to give the quark charges 2 e13and --e13.By itself this would lead to a serious difficulty . The Z° boson interacts with the quark neutral curren t -27N t3Lcos0+ y sin0).2 -)'y"(t3Lsec 0+q tan 0)_9, (21 .3.42) 2 2 with the sum running over all quark doublets 2 like (21 .3.41). The charge matrix qis diagonal in quark flavors, but if (21 .3.41) were the only quark doublet then the term involving the matrix t3L would contain cross terms proportional to 3y"(1 + y 5)d and dy f`(1 + y5)s, leading to effective Z exchange interactions like s+d*-->d+sands +d+-->u++y- with the strength of ordinary first-order weak interactions . Such effects would lead to rates for processes like K°-K° oscillations and K° --+M++ M- many orders of magnitude greater than observed . Also, even without neutral current terms in the Lagrangian, the one-loop diagrams involving the interaction (21.3,39 )with the charged current (21 .3.40) would lead to an effective interaction s + d --+d+swhich is smaller than an ordinary first-order weak interaction only by a factor of order a /27r, leading to a rate for KO--KO oscillations that is still much too large . In order to avoid this last difficulty, it was proposed13 that there is another term in J'` ; in modern notation, Z!Y-~(l+75) [-d sin ac+ s cos OJ, {21 .3.43} where c is a fourth quark, like u with charge 2e13.Adding (21 .3.43) to (21.3.40), the charged current may be writte n J~=(u cos B,- c' sin H)Y A(1+ 75) d + (u sin 0,+ccos H ,)YA(1 + Ys )s. The only reason that the interactions of the W with this current do not conserve strangeness is that the c and u have different masses, leading to transitions between u cos B,--- c sin Band u sin B,+c cos B,But this means that the loop diagrams for the effective interaction s+d-).d+3are 21.3 The Electroweak Theory 313 suppressed by additional factors (since mu CC m ,)of m ~ /my~,,bringing the rate for K° -K° oscillations into agreement with experiment . It was subsequently noted14 that this also solves the problem of the strangeness changing Z Ointeractions . In the context of the S U(2) xU(1) gauge theory the combination -dLsinB,+ SL cos0,cannot be a singlet , but must be part of another double t 1+75 C(21.3.44)2 -d sinp,+scos B , Including this doublet in the weak neutral current (21.3.42), the strangeness non-conserving terms proportional to s yu(1+ y5) d and d yu(1+y5)s cancel , removin g the problem of excessive Zexchange contributions to processes like KO-KO oscillations and K o--~y++y Particles containing the c quark in ac-cbound state were di scovered 1 5in 19 74,and indicated a mass m . 1.5GeV, ffi This completed two generations of quarks and leptons : a(u, d) quark doublet mixed with a (c, s) quark doublet ,together with two lepton doublets (vE,e)and (v ,,Y). The first sign of a third generation was the discovery of a third charged lepton ,16ther.Later a fifth quark t ype,the b , wasdiscovered, 17 w ith charge - e13 and a mass of about 4 .5 GeV .A sixth, the twith charge 2e13, then became theoretically necessary ,and after a long interval it too was discavered, l$with a mass quoted in 1995 as 181 ±12 GeV .ly Today thehadronic current Pin(21.3.39) is expressed a s u d J~ = C Y~~1 +Y 5}V s t b(21.3.45) where V is an incompletely known 3 x 3 unitary matrix, known as the Kobayashi-Maskawa matrix ." In the SU(2) x U(1) gauge theory, this means that there are three quark doublets : + Ys U 2 V udd+YuSs +Yubh + Y5 C 2 V cdd+Vcss +Vcbb 1 + Y 5 2 Vtd d+Vtss +Vthb(21.3.46) (Z1.3.47) (21.3.48) fifi We do not observe quarks in isolation, so their masses are not precisely defined . The mass of the c quark quoted here is roughly half the mass of the .T-y)particle, interpreted as a cc bound state . The band t quarks are so heavy that their masses can be taken from the masses of the hadrons containing them with little ambiguity . 314 21Spontaneously Broken Gauge Symmetrie s It is important to recognize that this is just what we should naturally expect on general grounds for three quark doublets . The most general renormalizable (S U(3) xSU(2) xU(1 )}-invariant interactions of the scalar doublets 0n with the quarks must in general take the for m W. ijn -EH ijnUiL DiL UiL DiLon,U1R On off. DjR +H.c. , n(21.3.49) where Uiand D iwith i = 1, 2, 3 are three independent quark fields of charge 2e13 and -e/3, respectively, L and R denote the left- and right- handed parts of the quark fields, and G ~ and H ~ are unknown constants . The vacuum expectation values of the neutral scalars then produce a quark mass term -~ Lm°DAR-}- H .C., Ym - -~ UiLM' U jR ~ ~ where MU j =~G" (O nO)vAc MRM(0 n)VA n n(21.3.50} (21.3.51) The matrices m f~ and m° are not constrained in any way, and in particular may be complex and non-diagonal, in which case parity- and flavor-non- conserving terms appear in gym . But we can introduce new quark fields UR = AR UR, UL = AL UL, D~ =~IRDR, DL = ALDL, where the As are 3 x 3 matrices constrained only by the condition that they must be unitary in order to preserve the form of the kinematic term (19 .4.1). Then the mass term (21 .3.50)takes the same form when rewritten in terms of the primed quark fields, but with the matrices mu and m° replaced wit h MU' = AL mUARf mD' = A°m°ARf. (21.3.5) Now it is a general theorem that for any matrix m, it is always possible to choose unitary matrices Aand B such that Amy is real and diagonal . (Use the polar decomposition theorem to write m -= H U, where H is Hermitian and U is unitary, and choose A = S tand B _ UfiS, where S is the unitary matrix that diagonalizes H .) We can therefore choose the As so that mom' and m°i are real and diagonal, in which case the quark fields u,c, t, d, s, and b are to be identified with the components of UL+QTRandDL + D. The weak doublets are now written a s ( AiU-1Ui)jQiL=(AD-'D 1-1DL)i 2 1.3The Electroweak Theory 315 but we can just as well take the doublets as linear combinations AL QL that have charge 2e/3quarks of definite mass u, c, t as their top component, in which case these doublets take the form (21 .3.46)-(21 .3.48), wit h V = ALq°-a (21.3.53) Within 90% confidence limits, the best present (1995) values for the absolute values of the elements of the Kobayashi-- Maskawa matrix areM ° 0.9745to0.9757 0.219 to 0 .224 0.002 to 0 .005 0.2:1s8 to 0 .224 0 .9736 to 0 .9750 0 .036 to 0 .046 0.004 to 0.014 0.034 to 0 .046 0.9989 to 0.9993 with rows labelled u, c,and t ,and columns labelled d ,s,and b . If there were only twoquark doublet sformed from u ,d, c, ands quarks, it would be possible to choose the phases of the quark fields sothat all V al are real , so that the V matrix is orthogonal ,and the doublets (21.3.46)and (21.3.47) (with bomitted) take the form (21.3.41) and (2 1.3.44),respecti vely. In this case the gauge interactions would automatically conserve T and CP. The great importance of the third generation is that it is no longer always possible to choose quark phases so that the V matrix is real, and therefore the gauge interactions can violate T and CP conservation .But for unknown rea sons the element sVub,Vb,Vtd,and V t,that connect the third generation with the first two are all quite small ,so the physics of the first two generations is hardly affected by the presence of the third , which explains in a more-or-less natural waywhythe Cabibbo a ssumption (21.3.40) works so well and why the violation of T and CP conservation issoweak .T and CP conservat ion can also be violated by scalar boson interactions if there are two ormore scalar doublets ;'Oh here the violation of T and CP conservation is expected to be weak because the scalar doublets couple weaklyto light quark s.It isstill unknown which of these mechanisms is respons ible for the observed violation of T and CP conservation in K~ decay ,discussed in Section 3 .3. Neutral current processes i nvolving hadrans, such as neutrin o-nucleon deep- inelastic scattering ,were discovered in 1 973,21shortly after the de- tection of the purely leptonic process v p+e--*v,, +e.Becau se of the much greater mass of the target particle here ,it became possible before long to observe large numbers of events ,and use them to confirm the electroweak theory and measure its parameters . Additional information on lepton -hadron neutral current interactions came from the ob servation of parity violation in atomic physics .By 1983 all direct measurements o f Adjust the phases of d and s so that Vw and V, ..are real . Unitarity then requires that V,,d and V, .s have the same phase, which can be eliminating by adjusting the phase of c. 316 21 Spontaneously Broken Gauge Symm etries sin 20had become consistent, and gave a combined value sing 0 = 0.23, yielding the predictions m w=80.1GeV and mz = 91 .4 GeV . Then in 1983 the W was discovered, with the Z following soon after . 22 Their measured masses are now (in 1995 ) MW= 80.410±0.180 GeV23 , m Z= 91 .1857 +0.0022 GeV24 , in satisfactory agreement with the predictions of the electroweak theory . The very great accuracy of the measurement of the Z mass, whic h has been achieved by tuning the energy of e+- e-collisions to the Z resonance at LEP (CERN's Large Electron Positron collider) and the SLC (Stanford Linear Collider), has changed the way that electroweak data is analyzed . Instead of comparing predictions of W and Z masses with observed values, the Z mass is taken as an experimental input, along with the Fermi coupling constant GF = 1.16639(2) x 10 -1GeV-2 taken from the rate of moon decay (including radiative corrections to order a}, and the fine structure constant a(mZ) _ (128.87±0.12)-1, extrapolated from low energy measurements as described in Section 18 .2. In this way sin2 0becomes a derived quantity ; if defined by Eq .(21.3.38), it takes the value sin20= 0 .2312 ± 0 .003 . With these inputs, the electroweak theory can be used to make predictions of other quantities like mw with sufficient precision that it becomes necessary to take electroweak radiative corrections into account .21 In one-loop order these radiative corrections involve the masses of the t quark and scalar ('Higgs') boson, and thus can be used to estimate these masses . For instance, before the top quark was discovered the agreement between theory and experiment set bounds on these radiative corrections which implied a top quark mass in the range 13 0-200GeV,"' in agreement with the value subsequently found experimentally . The W mass is predicted (in 1994) to be 80.29GeV, with an uncertainty of ±0 .02 GeV from uncertainties in the inputs mz, GF, and a(mz ), and an uncertainty of ±0 .11 GeV from the range of possible values of mt and mH,ggS . One 1995 study27 concludes that mHiggs < 225 GeV . The precise measurement of mw expected at the LEP 2 electron-positron collider at CERN will allow a useful estimate of mx ;ggs The most general renormalizable Lagrangian with the field content and S U(3) x SU(2) xU(1) gauge symmetries of the electroweak theory automatically respects baryon and lepton conservation . This is obviously true for the gauge interactions and bare mass terms, because the quarks , antiquarks ,leptons, and antileptons all belong to distinct representations ofSU(3) x SU(2) xU(1) .With scalars all belonging to 5 U(3)neutral SU(2) doublets w ithU(1) quan tum number ±1/2,the only renormalizable interactions of scalars with fermions and /or antifermions are with quark - 2 1.3The Electroweak Theory 31 7 antiquark and lepton-antilepton pairs, which of course conserve baryon and lepton number . (In much the same way, one can see that the charged hadronic currents with which leptons interact are necessarily linear combinations of the currents associated with the spontaneously broken SU(3) xSU(3) symmetry described in Section 19.7,as assumed without knowing the explanation in the original work on this broken symmetry .) These results depend critically on the assumption that the standard model is renormalizable . But as we have repeatedly emphasized, the renormalizable Lagrangian of the standard model is expected to be accom- panied with non-renormalizable terms of dimensionality d >4, suppressed by 4 - d powers of some very large mass M . The leading corrections to the predictions of the renormalizable standard model come from terms with the smallest possible dimensionality greater than four . The only Lorentz-invariant terms of dimensionality five that can be constructed out of the fermion and other fields of the standard model are at most bilinear in fermion fields and also contain either two scalars, or one scalar and one gauge-invariant derivative, or no scalars and two gauge-invariant derivatives (including their commutator, a field strength tensor) . Color S U(3) invariance requires that the fermion fields in such an interaction appear in either a quark-antiquark bilinear or a pair of lepton and/or antilepton fields, all of which operators conserve baryon number . There are a great number of such terms, but to violate lepton number conservation they must involve a product of two lepton fields or of their conjugates . The left-handed lepton doublets (tLi, vi) and right- handed charged lepton singlets tRi (with i = e, M, or z) have U(1) quantum numbers 1/2 and +1, respectively, while the scalar doublet (or doublets) (0+, 0o)have U(1) quantum number -1/2, so we can construct UM - invariant interactions of dimensionality five out of two left-handed lepton doublets and two scalar doublets . With only a single type of scalar doublet, there is just one such term that satisfies S U(2) and Lorentz invariance :27a ij where i and j are lepton flavor indices, and c denotes the charge conjugate field . At energies below the electroweak breaking scale, this yields an effective interaction ,fijvavj{00~2 (2 1.3.55) ij We expect fijto be of order 11M, perhaps multiplied with small coupling constants, so this gives lepton number non-conserving neutrino masses at most of arder211 (300 GeV)21M .We shall see in Section 21 .5 that M 31$ 21 Spontaneously Broken Gauge Symmet ries isexpected to be of order 10t 1-10t$GeV ,so we would expect neutrino masses in the range 10- 4-10-1eV,or less if suppressed by small coupling constants . Such masses are too small for direct measurement ,but there is no reason for the neut rino mass mat rix to be diagonal, soneutrino masses might be detected in oscillations of one neutrino type into another over long flight path s. A similar analysis shows that there are interactions of dimensionality six that violate both baryon and lepton number conservation ,involving three quark field sand one lepton field.27cSuch interaction swould ha ve coupling constants of order M -2,and would lead to processes like proton decay ,with rates proportional to M -4. 21.4 D ynamicall yBrok en Local Symmetries* Our discussion of spontaneously broken local symmetries has so far been entirely within the context of perturbation theory . To some extent, this limitation is inevitable . Whereas for spontaneously broken global symmetries it is possible to prove exact theorems about the existence and interactions of massless Goldstone bosons, the spontaneous breakdown of a local symmetry does not lead to any such precise consequences . Even the existence of massive vector bosons is not really a general theorem ; for sufficiently strong gauge coupling these particles decay so rapidly that they lose their identity as distinct resonances of definite spin j = 1 . On the other hand, if the gauge couplings like e or g or g' are sufficiently small then the theory with a spontaneously broken local symmetry must be very close to one with a spontaneously broken global symmetry, about which exact theorems can be proved . It is therefore possible to derive useful approximate results for such gauge theories, even if the other non- gauge couplings are verystrong .One example is provided by the standard SU(2) xU(1) electroweak theory with a large scalar self-coupling :~ (and hence a large scalar mass ; see Eq .(21.3.27}).Amore intriguing possibility is that the breakdown of electroweak symmetry is due to strong forces associated with some new gauge group acting on a set of new fermions . We will here consider the results that can be obtained for all such theories, without reference to the specific mechanism for spontaneous symmetry breaking .2g We assume that in the limit of zero gauge couplings, our theory is invariant under some group G of global symmetries, spontaneously broke n This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 21.4 Dyn amicallyBroken Local Symm etries 31 9 to a subgroup H . As discussed in Section 19.5,in this case the theory can be written in terms of a set of Goldstone boson fields ~a, plus other matter fields ~, whose G-transformation properties are such that the Lagrangian is G-invariant if it is H-invariant and constructed solely from y~ and covariant derivatives Day, D,,~etc., given by Eqs . (19 .6.14), (19.6.30), etc. We now `turn on' the gauge couplings . The gauge group 21 is of course required to be a subgroup ~#=G of the group G of all symmetries of the theory, and when G is spontaneously broken to H, 5must be spontaneously broken to a subgroup _*', equal to the intersection of ~# with H . The generators .37-x of the gauge group ~#may be expressed as linear combinations of the generators TAof the full group G : J~= ~e,.ATA(21.4.1) A with coefficients ems, the gauge couplings, that are taken very small . The index A runs over the labels i, aof the unbroken symmetry generators tt and the broken symmetry generators xu,(We are here taking the generators TA to be conventionally normalized ; that is, they are represented by matrices with elements of order unity . In particular, in contrast to the Via, the structure constants of the xQ and ti do not include factors of coupling constants .) In the underlying theory in which G invariance is linearly realized, we introduce the coupling of gauge fields to other fields ipby replacing ordinary derivatives with gauge-covariant derivative s 1P 1 :TAAAu~> > (21.4.2) A where AAA e~,4x,u . (21.4.3) The resulting theory is then invariant under formal local transformations, under which the fields transform according t o 1P~ gyp , (21 .4.4) TA AAA, g 1:TA AAA g-1 - i (vug)g-I, (21.4-5) A A where g(x) is an arbitrary space time-dependent element of the group G . This is a purely formal invariance, because the gauge couplings in general actually break G, as shown by the fact that the transformation (21 .4.5) does not in general preserve the form of the linear combination (21 .4.3). Nevertheless, we can temporarily forget about Eq . (21 .4.3), treating AA as an unconstrained classical external field, and analyze the structure 320 21 Spontaneously Broken Gauge Symmetrie s of the Lagrangian for the matter fields and their interaction with the gauge fields by requiring that it be invariant under local transformations (21.4.4), (21 .4.5). In this way we will insure not only that the Lagrangian is invariant under the true local symmetry subgroup 5(and for eIXA--+0 under the larger global symmetry group G),but also that the currents, the variational derivatives of the matter action with respect to 'W", will have the correct transformation under the broken global symmetry group G . Later we will restrict AAto the form (21 .4.3), and treat the field Ofas a quantum field, supplying a suitable kinematic term in the Lagrangian for this field . In order to explore the implications of the spontaneous breakdown of the invariance group Gto its subgroup H, we will proceed as we did in Section 19 .b, First, replace ipand Awith new fields y~, A: AA CAB(7AB B(21.4.6) X21.4.7) where y (~)is the standard G transformation that eliminates the Goldstone boson degrees of freedom in ip, and D(g) is the representation of G furnished by the gauge fields : g TA g-1 DBA(g) TB. ( 1.4.8) B These Goldstone degrees of freedom reappear in the spacetime-dependent parameters ~R on which y (~)depends .By the same calculations as in Section 19 .6, for local as well as global transformations the transformation rule (21 .4.4) translates into the transformation s a__*~u=fR(~'g) y~ ---*~'= h(~, g) Cp where h and fare defined b y g Y(~ ) =Y(f (~, g ))h(~,g)(21.4.9) (1.4.10) (1.4.11) with h in the unbroken subgroup H . We also need to work out the transformation rule for AA.Recall that, according to Eq . (21 .4.5), under these local transformations the linear combinations of gauge fields in Eq.(21.4.2) transform a s TAAA,-~~TAAAµ = g A AT'AAA y- r g-1 a pgg-I A Mult iplying on the lef t and right by and y(~') respecti vely,and 21.4 Dynamicall y Broken Local Symmetrie s using Eqs . (21 .4.7), (21 .4.$), and (21 .4.11), we can write this a s i1:TAAA , = h (~,g)i1:T'AAA~~ +Y 1~~)LgYlau 9IY~~)h-1(~,g) A A321 (21.4.12) To see how to cancel the inhomogeneous g -l a,g term, we note that by differentiating Eq . (21 .4.11) and multiplying on the left with its inverse, we hav e So to cancel the inhomogeneous term, we must subtract Eq . (Z 1.4.12) from Eq. (21 .4.13): A IT- A ..."[h(,g)] h-1(~, g).We therefore de fine new gauge-covariant quantit ies-9and ~ 'by t'a luxa+Z~'i,ti =7 - t 1: T 'A,!, q, a A with transformation propertie s where(21.4.14) (21.4.15) (21.4.1) -91apxQ = h(~ 'g)E-gapxah- g), (21.4.17) a a (21.4.18) just as in Eqs .(19.6.26) and (19.6.27). We can use 6'to construct fully covariant derivatives of matter fields (21.4.19) as well as higher derivatives like 9v9uzj~, etc . Because of the inhomoge- neous term in Eq . (21 .4.12), we cannot freely introduce AAA or covariant derivatives like (21 .4.19) of IaU into the Lagrangian . However, it is 322 21Spontaneously Broken Gauge Symmetrie s easy to construct a `curl' that transforms covariantly under both local transformations and global G transformations . It is FApv=EDAB(Y-l(O) altABvayAB y-1:CBCDACuADy CD (21.4.Z0) This transforms under formal local G transformations a s FAQ„ --~ FApv ,C AB(h(~, g)) FBpv. (21 .4-21) B The Lagrangian is therefare invariant under formal local G tran sformations if it is constructed as a n arbitrary function of y~, -9 ap, Dp~j~, FA pV, and higher covariant derivatives, that satisfies global H invariance . Now let us return to reality, and treat AAas a quantum field of the restricted form (21 .4.3). Eqs . (21 .4.15) and (21 .4.20) now becom e i -gap xQ + ~ a s iptt a i y-~0'Uy(O- r1:TADAB(Y-V)}ea$_<VxM ,4 Ba and Fapv DABV1(o) ea8~FXp v , Ba where J~,7pp v7~_-apdfl v -avd#p-1: W flYb _dYp.~bV Y6(21.4.22) (21.4.23) (21.4.24) with W #y6the structure constant of the gauge group, related to the structure constant of G by CDfl We include in the Lagrangian as the kinematic term for this field the usual Yang-Mills term Y.1:0 J~ ' aUJv9pV (21.4-26) in which, by linear transformations of the slx,and correspondingly of thee1A, we have adjusted the coefficient of to be just 6,,#. The linear term in is just ~ pslIXV - so Eq . (21 .4.26) has the effect of making &/xp a canonically normalized vector field . The effective Lagrangian density is therefore to be taken as a function of ~p, ~pip-, and 9,,p that is invariant under global H transformations, plus possible terms 21.4 Dynamically Broken Local Symmetries 323 that conserve Ibut not G, with coefficients proportional to two or more factors of the e ,,A. Now let us see what sort of perturbation theory we can construct out of these ingredients . We know that the gauge bosons become massless in the limit e,,A --+0,where they decouple from the matter fields that experience the spontaneous symmetry breakdown . Let us therefore ten- tatively anticipate that their mass for small e,,A is of order eM, where e is a typical value of ea,q (the generators TA being normalized to have structure constants of order unity) and M is an energy scale typical of the dynamics that leads to spontaneous symmetry breaking . We shall consider here a general Feynman diagram involving gauge and Goldstone bosons of energy or momentum Q ~ eM, and with all particles of higher energy or momentum and all heavier matter particles buried in corrections to the coupling constants in an effective field theory . Our perturbation theory will be an expansion in powers of e and Q/ M. Following the same analysis as in Sections 19 .4-6, the total number of powers of e and/or QOM in any such diagram is v ViA + ei - 2} + 2L + 2, (21.4.27) where Vj is the number of vertices of type i ; di and ei are the number of derivatives and factors of e,,A,respectively, in an interaction of type a ; and L is the number of loops . With the constraint (21 .4.3), a field AA,, orAAA contributes one factor of e . Inspection of Eq . (21 .4.15) shows that each Goldstone boson covariant derivative contributes +1 to di+ ej, and inspection of Eqs . (21 .4.19) and (21 .4.15) shows that each additional covariant derivative of ~,,,Ucontributes another +1 to d i+ ei. All allowed terms in the Lagrangian have di+ e; ? Z, so the dominant contributions are those from tree graphs (L -- 0)constructed entirely from interactions with d i+ et = 2 . The only such interactions are the Goldstone boson kinematic term ab the Yang-Mills term (21,4 .26), and possible symmetry-breaking non- derivative terms of second order in the e A. To see the physical significance of the field ~a, note that the linear term in_9a, is pay1LiN=ap~a e,Q.dx,, (21.4.29) As shown in the appendix to this chapter ,we may always choose a 324 21Spontaneously Broken Gauge Symmetrie s `unitarity gauge' in which, for all a , Fa2 b~ueah = 0 ab(21.4.30) which makes the cross term in Eq . (21 .4.28) vanish . To clarify the sig- nificance of this condition, note that in the special case where all broken symmetries are gauge symmetries, any xQ can be written as a linear combination of gauge generators and unbroken generator s xu - ~ Caa,~7a+ Cant a j and therefore-~ Cacx ~ eabxb + exi ti + Iccxitf caxeab =da b IX Contract ing Eq .(21.4.30) with cap,, we see then that ~ a = 0 ; there are no Goldstone bosons at all in this gauge . More generally, Eq .(21.4.30) leaves us with just those Goldstone bosons that do not correspond to gauge symmetries .Some of these are associated with elements of G that are broken by the gauge interactions, and therefore have masses of second order in gauge couplings ; these are called pseudo-Goldstone bosons . With ~ chosen to satisfy the unitarity gauge condition (21.4.30), the quadratic part of the Lagrangian (21.4.28) is simpl y (Y~)QuAD Z1:F eap~aa~~6- 2E ~, (21.4.31) ab a fl where 2 2 ah(21.4.32) This has two important implications . First, we note that ~a may be expressed in terms of a canonically orthonormalized field tea, a s Fabl 7Th b(21.4.33) where FQb is the positive square root of the positive matrix F.This shows that Fabl are the factors analogous to F .-Z that accompany the emission and absorption of low energy Goldstone bosons . Second, since s/,u has been defined to be a canonically normalized vector field, Eq. (21.4.31) shows that ,uaflis the square of the vector boson mass matrix . Eq. (21 .4.32) is a universal formula for the vector boson mass matrix, valid to second order in the gauge couplings but to all orders in all other interactions . By using Eq . (21 .4.1) in Eq .(21.1.7), it is easy to see that our 21.4 Dynamically Broken Local Symmetries 325 previous result Eq .(21.1.7)is a special case of Eq . (21 .4.32), with Fj, - - E( xa)nm(xb)ndU,,Ud ramd Eq. (21 .4.32) may also be understood on the basis of the continuity arguments outlined (in a somewhat different notation) at the end of Section 21 .1. It guarantees that the effects of gauge boson exchange that survive in the limit of zero gauge coupling are the same as the effects that would have been produced by Goldstone boson exchange if there were no gauge coupling . In general, we cannot calculate the Fah matrix, but we do know that it must be invariant under the unbroken subgroup H, in the sense tha t z 2 CsbdFdr +CicdFbdl_0. d This condition allows us to put useful constraints on the gauge boson masses (21 .4.3). As an example, consider the case of the electroweak gauge group SU(2) xU(1), spontaneously broken to the U(l) of electromagnetism . The three broken symmetry generators Xa can be taken as the three generators of S U(2) (called t1, t2, 0 in Section 21 .3), without the coupling constant factor g, and the one unbroken symmetry generator t can be taken as the charge q, without the factor e . That is, the x, and t are represented on the lepton doublets by the matrice s X= 4 (1 + Y5} ~ ~),( 0 i 00 t=-01-i), (10 0 U -1 ' The gauge generators are then given b y .~=gX, .may =9'(X3---0. (21.4.34) That is, the non-zero coefficients eaR of x,, in the gauge generator !i-IX ar e ell =e22 -X33 -9, ey3=g1 Also, since t subjects the three-vector x~ to a rotation around the three- axis, this unbroken symmetry requires the matrix FQb to have the non-zero components 2 z 2 z z X11 - F22 - FC F33 F N According to Eq . (21 .4.32), the mass squared matrix of the gauge bosons then has the non-vanishing element s X11 -X 22 - g2FC 2 2 ~X33 = 9~FN /13Y -ggI FN Pyy -g'2 FN 326 21Spontaneously Broken Gauge Symmetrie s Its eigenvalues ar e Mw = gZF~ , M z =(g2+g 2)FN MA=0.(21.4.35) To go further, we need a relation between FC and FN . This is provided if the theory is invariant in the limit g - g' =0under a global symmetry group G larger than SU(2) xU(1), which breaks spontaneously to a subgroup H, that includes three-dimensional rotations under which x rotates as a three-vector . Such an unbroken symmetry would require F 2ab to be proportional to SRb, and henc e FC=FN . Any such symmetry is known as a `custodial' symmet ry. It has the consequence that m z /mwis given in terms of the gauge coupling constants by the successful formula discussed in Section 21.3 ML/MW = V 1 + 1gZ = 1/sin B. (21.4.36) For instance, in the absence of the gauge couplings, the Lagrangian (21.3.28) for the scalar doublet c kin the simplest version of the SU( 2)x U(1) electroweak theory may be written 2 2 2 4(OnOn )2, where Of ° Im O+ , 02 = Re o+ , 03 = Im Oa 04 = ReOa This is automatically invariant under an `accidental' SO(4) - SU(2) x S U(2) global symmetry group, which is spontaneously broken by the vacuum expectation value of ReO° down to an approximate unbroken SO(3) custodial subgroup .29The result (21 .4.36) applies even if there is more than one scalar doublet, because even though in this case the mass and interaction terms in the scalar Lagrangian will not in general respect the custodial symmetry, it is only the kinematic term that enters in the derivation, and this always has the full SO(4) symmetry . Custodial symmetries can be found in other theories . Consider for in- stance a theory with no scalar fields, but with new extra-strong vector gauge interactions,30called technicolor interactions, that act on a new S U(2)x U(1) doublet (Ur, D, .)of `techniquarks' U.and D, ., with r a tech- nicolor index . As long as the left- and right-handed parts of both Ur and D ,all transform in the same way under the technicolor gauge group, the Lagrangian will be invariant in the limit of vanishing electroweak couplings under the group S U(2) x S U (2)of independent S U(2) transfor- mations on the left- and right-handed techniquark doublets . According to the arguments described in Section 19.9,the subgroup S U(2)v consisting 21.5Electroweak-Strong Unificatio n 327 of simultaneous S U(2) transformations on both the left- and right-handed techniquark doublets will not be spontaneously broken . It is reasonable to suppose that the technicolor interactions will produce a spontaneous breakdown of SU(2) xSU(2) toS U(2)v, just as color interactions led to a spontaneous breakdown of the chiral SU(2) xSU(2) symmetry of quantum chromodynamics (for vanishing u and d quark masses) to its isospin subgroup . Under the unbroken S U( 2) y symmetry the electroweak generator x or 9-rotates as a three-vector, leading again to the relation FC = FN, and the consequent successful prediction of the relation between the W and Z masses . The technicolor idea is attractive, because it provides a natural mech- anism for breaking the electroweak symmetry at a characteristic scale which is very much smaller than what is often supposed to be the fun- damental scale of physics, generally assumed (as in string theories) to be of the order of the Planck mass, or about 1 018GeV . It is only necessary to suppose that just below the fundamental scale there is an unbroken gauge group consisting of the S U(3) x S U( 2)x U (1) of the strong and electroweak interactions, plus a technicolor gauge group, all with compa- rable small coupling constants . If the technicolor gauge interactions are asymptotically free the technicolor coupling like the QCD color coupling will increase slowly with decreasing energy, becoming strong at an energy much less than the fundamental scale . This energy where the technicolor coupling becomes strong would set the scale for the parameters Fab that appear in our formula, Eq . (21 .4.32), for the gauge boson masses, and would therefore presumably be of the order of 30 0GeV . Since the increase of coupling with decreasing energy is logarithmic, a moderate difference in the beta-functions for technicolor and color can easily give rise to the three orders of magnitude difference between the energy scales where color and technicolor forces become strong . Unfortunately, although technicolor provides a very attractive picture of the spontaneous breaking of SU(B) x U(1), it does not by itself offer a mechanism for giving masses to the quarks and leptons . For this reason it has been suggested to add additional `extended technicolor' gauge interac- tions with transformations that link quarks and techniquarks .31 Such the- ones have potential problems with flavor-changing neutral current weak interactions, and though these problems may be surmounted, the added complications reduce their attractiveness . The question of elementary weakly coupled scalars versus dynamical symmetry breaking remains open . 21.5 Electroweak-Strong Unificatio n We saw in Section 15 .2 that a gauge theory will have an independent coupling constant for each simple or U(l) subgroup of the gauge group . 328 2 1 Spontaneously Broken GaugeSymmetries Thus the electroweak theory which is based on the gauge group S U(2) x U(1), has two independent couplings, g and g' .In order to reduce the number of free parameters it was suggested32 that the SU(2) X U(1) gauge group might be embedded in a simple SU(3) gauge group, which would give g' = g/,13-, but this has been ruled out experimenta lly. After the advent of quantum chromodynamics, theorists confronted a gauge group S U(3 )x ,S U(Z) x U(1), which has three independent gauge coupling constants : the coupling gS of quantum chromodynamics, and the coup lings g and g' of the electroweak interactions . To reduce these to just a single free parameter, it was proposed to embed S U(3) x S U(Z) x U(1) in various simple Lie groups :* S U(4) x S U(4 ),33 S U(5),3 4 OT ,Sn(I0).35 Such models are often known as grand-unified theories . Fortunately, the consequences of these and a large class of other models for the ratios of the S U(3) x S U(2) x U(1) coupling constants are inde- pendent of the details of the individual models .36 This class of models is characterized by the fact that the observed generations of quarks and leptons are the only fermions in the models, or at least the only fermions that are not neutral under SU(3) X SU(2) X U(1) . As shown in Section 15 .2, for any simple compact Lie group there is a conventional choice of generators Ttt with total ly antisymmetric structure constants, which in each reducible or irreducible representation D satisfy the normalization condition : TrITIXTfll = N D6x#. (21.5.1) We are assuming that all left-handed fermions form ng generations : Ve vu vz ... ~ L ~ L Z IL Z~R PR"CR ... u dL UR dRC ~5 )L~ L... ~ Q TR... YR bR.... The S U(3 )generator ~ g, .~3 has eigenvalues : + 1g,for the red quark doublets and the white antiquark singlets ; - 2gs for the white quark doublets and the red antiguark singlets ; and zero for all other left-hande d The group S U(4) x S U(4) is made simple by inclusion of a discrete symmetry operator that interchanges the two S U(4)s . 22.5Electroweak-Strong Unificatio n fermions; soits squarehas thetrace Tr( 29sA3)2 =4ng X ( ids)2 + 4ng X (- 2g5)2 = 2~ aggS329 (21.5.0 The SU(2) generator t3 has eigenvalues : 2g for the red, white, and blue quarks of charge 3 and the neutrinos ; - 2g for the red, white, and blue quarks of charge - 3 and the charged leptons ; and zero for all other fermions, so its square has the trac e Tr(t3 )2 =[3ng +ng] X [( '2g)2 +( -Zg) 2]=2ngg2 . (21.5.3) Finally, the U(1) generator y = t3 - q has eigenvalues :I g'for neutrinos and charged leptons ; -g' for charged antileptons ; - bg' for quarks ; 3g' for antiquarks of charge - 3 ; and - 3 g' for antiquarks of charge + 3, so its square has the trac e Try= 2ng( 2g')2 _ ,a .2 Tng g6 3 (21.5.4) Eq. (21 .5.1)require sthe trace s (21.5.2)-(21.5.4)to be all equal , so in th is class of models the embedding of SU(3) xSU(2) xU(1) in a simple Lie group imposes the coupling constant relation s gs == g2 = ~g12  (2 1.5.5) Now, Eq . (21 .5.5) is in gross disagreement with the observed values of the coupling constants . The ratio g'2/g2= 3 implies an electroweak mixing angle with sine B - g'2/(g2 +g'2)_3, while the experimental value is sine B = 0.231. Even worse, the strong coupling g~ is of course much larger than g2 or gi2 . The solution36 to this problem is that coupling-constant relations like Eq. (1 .5.5)apply only to the couplings measured at an energy scale comparable to the typical rnass47 M of the gauge bosons that become massive in the spontaneous breakdown of the simple gauge group to S U(3) x S U(2) x U(1) .If the energy Eat which the couplings is measured is very much less than M, then there will be large radiative corrections proportional to ln(1VI /E ) . As emphasized in Chapter 19,there are no large logarithms in the relation between couplings measured at nearby energies pand y- d,u, so by integrating this relation from M down to Ewe can calculate the couplings at energies ECC M without encountering large logarithms . For this to be done it is only necessary that the couplings should stay small over this whole range . For the S U (3 )xS U(2) xU(1) couplings with gag 330 21 Spontaneously Broken Gauge Symmetrie s ferznion generations, Eq . (18 .7.2)gives* * d gs (µ) 11 ngd~gs(F~)- - 4~c2 4 3 d g3(µ)11 n g F~d~~~F~~ = 47c2 6 3 d g'3 01} 5ng f,g W 2 (--) -(Z1.5.b) (21.5.7) (21.5.8) The solutions of these equations ar e 1-2 - ~Z 11- ~~ 9In M (21 .5.9) g~ s(y)~5 ~ ~ ~ Y 1 1 i(22 4ngIn(M) (21.5.10) 1 1 1 _20"x In M (21.5.11)12 Also, Eq .(21.5.5)should be interpreted to mean tha t g; (1VI) =g2(M)= 3gi2(1Vl). (21 .5.1) We can therefore eliminate the couplings (21.5.12)and the number of generations by subtracting Eq .(21.5.10) from Eq .(21.5.9): 21 _ g2~4~2 InM (21 .5-13) gs _ (Y and by subtracting 3/5 of Eq .(21.5.11) from Eq .(21.5.10): 1 - ~~it2 In M(21-5 .14) ~j i27 ( ) Taking the ratio of these two equations gives a formula for sin2 9 9t2/(g2+g,a): singB =1 +~2(m ~z) gs(z )(21.5.15) The second term in the brackets in Eqs . (21 .5.6) and (21.5.7)is given by Eqs . (l $ .7.2) and (18.7.3)as -n/6, where n farc the numbers of fermions in the defining representation of the respective SU(N) gauge groups . But this was calculated under the assumption that left- and right-handed fermions are in the same representation of the gauge group . If we count only left-handed ferrniflns (and antifermions), then the second term in the brackets in Eqs . (21 .5.6) and (21.5.7)should be -of/12 .For SU(3) there are two left-handed quark triplets and two left-handed antiquark triplets per generation, so ny = Ong, while for S U(2) there are three left-handed quark doublets and one left-handed lepton doublet per generation, so again of - 4ng_ For U(1), the beta- function is g'/24n2times the sum of the squares ❑f the U(1) charges for left-handed fermions and antifermions (compare Eq .(18.2.38)), which according to Eq.(21.5.4) is (g'/24n2) X (ion9912/3). 21.5 Electroweak-Strong Unification 331 In this formula we have set pequal to the typical energy of the processes used to measure sin2o,that is, p.; m Z. This has the advantage that we w ill be using the renormalization group equations (21.5.6)-(21 .5.8)only above mz, where they are not strongly affected by the spontaneous breakdown of S U(2 )xU(1) .Eqs.(21.5.13) and (21.5.14) can also be combined to give a formula for the unification scale M : M 4n2 In -----mZ 11e28e2(MZ) 3 g5(Mz)(21.5.16) where again, to avoid the effects of electroweak symmetry breaking on the renormalization group equations, we have taken pas of order mz . We saw in Section 18.2that the value of e(p) atu~ mz is given by e(mz)2/4n = (1 2$.$7 ± 0 .12)-1 . This is the charge defined in the conventional (Gell-Mann-Low) manner, in terms of the vacuum po- larization . For purposes of comparison with gs, g', and g, it is bet- ter to use the coupling defined (as in Section 1$ .6)by modified min- imum subtraction :e(m f)2/4n=(127 .9+0.1)-l .The greatest uncer- tainty in Eqs .(21.5.13) and (21 .5.14) is in the value of gs(mZ ). As discussed in Section 1$,7, extrapolation of gs from lower energy data gives g~(mZ)/4n = 0 .118 ~ +- 0,006, while direct measurement from the rate of Z° decay into hadrons gives gs (ml )/4n = 0 .120 +0.0025 . For g; (mz)/4n = 0.118 and e(mz)2/4n = 1/128, Eqs .(21.5.15) and (21.5.16) give sing B=0.203 and M 1.1 x 101 GeV . As mentioned in Section 21.3, there is no reason to expect that baryon and lepton number should be conserved by the suppressed non- renormalizable terms in the effective Lagrangian that describes physics at ordinary energies, so we may expect the presence of an ( SU(3)x SU(2)x U(1)}-conserving four fermion (three quarks and a lepton) interaction, with a coefficient that on dimensional grounds would be of order M-2 . The proton lifetime was first estimated on this basis, and found to be of order1032years .36 Such baryon- and lepton-non-conserving four-fermion interactions are produced in models like those of Refs . 33-35 from the ex- change of gauge bosons with masses of order M . More generally, once the standard model explained why baryon- and lepton-non-conserving pro- cesses are naturally suppressed, there ceased to be any reason to believe that either baryon or lepton number are exactly conserved . We have seen that the prediction (21.5.15) comes quite close to the measured value 0 .23 of sin 2 B, but the accuracy of the measurements and calculations has become good enough to make clear that they are not in precise agreement . The extra particles in supersymmetric theories appear to remove this discrepancy, and lead to a value of M ~_-2 x1016GeV, about an order of magnitude larger .36a It is very interesting that this M is not so very different from the energy 1018 GeV at which the gravitational 332 21 Spontaneously Broken Gauge Symmetrie s interactions become strong . The larger value of M also has the effect of increasing the proton lifetime, which is proportional to M4 . 21.6Superconductivity* Superconductivity is quite different from the elementary particle phenom- ena that chiefly concern us in this book, but it is worth some consideration here, both as the earliest realistic example of a spontaneously broken gauge symmetry, and also as an exceptionally enlightening example of the power of effective field theories and of the use of topological arguments in field theory . A superconductor is simply a material in which electromagnetic gauge invariance is spontaneously broken .** Detailed dynamical theories are needed to explain why and at what temperatures this symmetry breaking occurs, but they are not needed to derive the most striking aspects of superconductivity : exclusion of magnetic fields, flux quantization, zero resistivity, and alternating currents at a gap between superconductors held at different voltages . As we shall see here, these consequences of broken gauge invariance can be worked out, in a manner somewhat like our treatment of soft pions, solely on the basis of general properties of the Goldstone mode .39 The action for any system will be invariant under gauge transformations, which in cgs units take the for m Ap(x) --} Au(x) + aun(x), (21.6.1) Wn(x)--~ exp (iqnA(x)/h)lpn(x), (21.6.2) where A(x) is arbitrary, and qn is the electric charge destroyed by field yin. All charges are assumed to be integral multiples of the electron charge -e, so this group is compact : the phases Aand A+ 27rhle are regarded as identical . This symmetry group is assumed to be broken in a superconductor by non-vanishing expectation values of operators carrying charge -2e (such as products of two electron fields), so ther e This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . This is not the way that most experts have historically thought about superconduc- tivity . Early phenomenological theories were known to violate electromagnelic gauge invariance, but this was regarded as more annoying than enlightening . Broken sym- metry is never mentioned in the seminal paper by Bardeen, Cooper, and Schrieffer37 that first gave us a microscopic theory of superconductivity, Andersan38 subsequently stressed the important role of broken symmetry in superconductors, but even today most textbooks explain superconduclivity in terms of detailed dynamical models, with broken symmetry rarely mentioned . 21.6Superconductivity 333 is an unbroken Z2 subgroup, consisting of gauge transformations with A= 0 and A= nhle . We introduce a Goldstone boson field O(x) by writing all charged fields as y),(x) = exp (iqi(x)/h)qn(x) . The field O(x) parameterizes the coset space U(1)/Z2, and so is given the gauge transformation propert y O(x) --} O(x) +A(x) . Because O(x) parameterizes U(1)/Z2 rather than U(1), we must identify O(x) and O(x) + nh1e .All the ~n(x) are gauge-invariant, and so when integrated out leave the Lagrangian as a gauge-invariant functional of 0and A Palone . It follows that the Lagrangian for the Goldstone and electromagnetic fields may be written as(21.6.3) (21.6.4) where L5 is an imperfectly known functional . The electric current and charge density here are J (X)~Ls 6A(x) 6Ls BL S 6A°(x)T6~(x)(21.6.6) .I°(x) = The equations of motion for the Goldstone boson field are the n r76L` _ 6LS = ❑BLS r7t ~c~s(} 60(x) 6A(x)(21.6.7) (21.6.8) which in light of Eqs . (21.6.6)and (21.6.7)is equivalent to the conservation law of electric charge DJ+~ JO =O . (21 .6.9) Now let us see how this formalism explains the remarkable properties of superconductors . About Ls we will only need to assume that the system is stable in the absence of Goldstone or external electromagnetic fields, so that the energy is at least at a local minimum at A.= r7.0, with non-vanishing second derivatives with respect to A.-amo. One immediate consequence is that, deep in a large superconductor where boundary conditions are unimportant, the electromagnetic field is 334 21 Spontaneously Broken Gauge Symmetrie s a pure gauge : AP=0110, (21.6.10) so that in particular the magnetic field must vanish . This is known as the Meissner effect .It is possible to be a little more quantitative about what we mean by `deep in a large superconductor .' Since the energy is a minimum when Eq .(21.6.10) is satisfied, for small values of JA - Vol it must be of order JA - ❑o 12 L3/A2, where Ais some length depending on the nature of the material, and L3 is the superconductor volume .Ifa magnetic field of order B penetrated the superconductor, then we would have JA-❑ol of order BL, so the energy cost in allowing the magnetic field into the superconductor would be of order B2L~/A2.On the other hand, the energy cost in expelling a magnetic field B from a volume L3 is of order B2 V.Hence a weak magnetic field will be expelled from a superconductor if $2LS/a2>B2L3, or in other words if L » )y. For this reason Ais known as the penetration depth of the superconductor . The same energetic considerations tell us that for any superconducting material, there is a critical magnetic field, above which superconductivity is extinguished . The existence of superconductivity at zero magnetic field means that the material in its normal state has an energy per unit volume which is higher than that in the superconducting state, say by an amount ❑. When a superconductor with linear dimensions much larger than ). is placed in a magnetic field B, the magnetic field is expelled from most of the material, at an energy cost per volume of B2/2 . Hence it is energetically favorable for the material to be in the superconducting state if and only if the magnetic field is below a critical valu e Bc= 2 ❑ (21.6.11) (This is for uniform superconductors . As we shall see below, for cer- tain kinds of superconductor it is possible to maintain superconductivity throughout most of the material for magnetic fields in a finite range above B, by the formation of narrow vortex lines with normal metal at their cores .) A magnetic field B <B, will penetrate a superconductor to a depth a, but this does not extinguish superconductivity in this layer ; indeed, as shown by the field equation V x B=J, it is in this surface layer of the superconductor that electric currents can flow . Now consider a thick superconducting wire, with thickness much larger than A, bent into a closed ring . We can draw a closed contour W running deep inside the wire, along which JA - ❑ol must vanish . This does not mean that either A or 0 vanishes on this contour, but we do know that in going around the ring 0 must return to an equivalent value, and can therefore only change by an amount nirh/e, with n a positive or negative integer or zero . It follows then by Stokes's theorem that the magnetic flux 21.6 Superconductivity 335 through the area Wsurrounded by Wis subject to the fl ux quantization rule (1.6.12) /Bds=Adx=V'dx= . The electric current that maintains the magnetic flux (21.6.12) flows in a layer of thickness Ajust below the surface of the superconducting wire . The quantization of flux shows that this current cannot decay smoothly, but only in jumps at which the flux (21.6.12) drops by multiples of nh/e, so there can be no ordinary electrical resistance in the superconductor . The absence of resistance in a superconductor can be shown in a more general context than closed rings, by considering time-dependent effects in superconductors . Note that Eq .(21.6.7)can be interpreted as the statement that -J° is the canonical conjugate to 0. The Hamiltonian HS is thus to be regarded as a functional of 0and .I° rather than 0and with the time dependence of 0given by the Hamiltonian equatio n ~(x) =~Hs (21.6.13) 6( J (x)} Now, the `voltage' V(x) at any point is just the change in energy density per change in the charge density at that point, so Eq .(21.6.13) gives the time dependence of the Goldstone boson field a s ~(x)=-V(x ). (21.6.14) It follows that a piece of superconducting wire that carries a steady current, with time-independent fields, must have zero voltage difference between its ends, because otherwise O(x) would have a time-dependent gradient . A zero voltage difference at finite current is what we mean by zero resistance . Now consider a gap between two pieces of superconducting material . In the absence of any gradients along the surface of the gap, or any vector potential, gauge invariance requires LS to depend only on the difference ❑Obetween the Goldstone boson fields in the two superconductors : Ljunction = Q/F(0o), (21.6.15) where Q/is the area of the junction . Furthermore, we can shift 0in either superconductor by any integer multiple of nh/ewithout physical effect, so the function F must be periodi c F(AO) = F(OO+nhnle) . (21.6.16) fThis function was calculated by Josephson,' who found it to be proportional to cas(2e4O/h), but this is an approximate result, while the periodicity is exact . 336 21Spontaneously Broken Gauge Symmetrie s A current flows through such a gap, which can be calculated by considering the junction in the presence of a vector potential A . Gauge invariance then tells us that in place of ❑O, the function F must depend o n the integral being taken over a line joining the two superconductors . Eq.(21.6.6)then shows that the current density i s .T=6Ljunctio n 6A= _ nF.'(AAO) where nis the unit vector perpendicular to the gap . We can now take away the vector potential, and find the curren t If we now suppose that the two superconductors are maintained at uniform voltages, with a voltage difference ❑V, then, according to Eq . (21 .6.14), the difference in the Goldstone boson fields will have the time dependenc e ❑O = -tA V + constant . (1.6.18) Using this in Eq . (21 .6.17)and recalling Eq . (21 .6.16), we see that the current oscillates at a frequenc y v = elAV11nh . (21.6.19) This is the ac Josephson effect .'It is possible to measure frequencies and voltages with great accuracy, so this effect provides a very accurate method of measuring the constant e1h. As pointed out at the end of Section 19 .6, the description of a system with broken symmetry in terms of Goldstone modes alone becomes in- adequate when a system is brought close to the point where the broken symmetry becomes unbroken . Under these circumstances the Goldstone mode is accompanied with other modes that have nearly zero frequency in the long-wavelength limit, which together with the Goldstone mode forms a linear representation, usually irreducible, of the symmetry group, known as the order parameter . It is plausible to assume that a uni- form superconductor in slowly varying external fields is described by a local order parameter, because any non-locality would be characterized by microscopic distance scales (such as the mean electron-electron spac- ing) that are much smaller than the scales of distances over which the electromagnetic and Goldstone boson fields are assumed to be varying . For superconductivity there is no doubt about the nature of this order parameter . The only non-trivial irreducible linear representation of the group U(1) is a real two-vector yin, or equivalently a Goldstone mode 0 21.6Superconductivit y and modulus field p, wit h W1+ iY'2= p exp (2ie/h) =W.337 The coefficient of icy in the exponential must be 2elh in order that a gauge transformation with A = nh/e (and with no smaller A)should leave Wn invariant . For a nearly uniform time-independent system close to the symmetry-breaking transition, the order parameter is small and slowly varying in space, and so the Lagrangian in an external vector potential A may be approximated (from now on using natural units with h = 1)by d3x - 2 (VW" - tie tr~ ►~A~V ►~~ 2 + ~~~ Wnwn - where t is the Hermitian U(1)generato r t=~ 0 i(1.6.21) and g must be taken positive to give a bounded Hamiltonian . This is the Ginzburg-Landau theory of superconductivity .41It has been derived by Gor'kov 42 from the microscopic theory of superconductivity presented below, in the case of a short-range potential and a temperature close to the critical temperature at which the material loses its superconductivity . In terms of p and 0, Eq.(21.6.21) become s + - g p4-(Op)2] . (21 .6.22)LS^, ~ d3 x [_2e2p2 (v-A) 2 The field equations a re the n ❑2 p = -map + 9P3+ 4e2P(0o-A)2. (21,6 .24) The U(1 )symmetry is broken if these field equations are satisfied for p* 0, which in a field-free homogeneous material will be the case if m2>D, in which case ptakes the value (p)= m /,,,,fg-.The penetration depth ~was defined earlier as the inverse square root of the coefficient of - 2(Oo - A)2, so here i _ J94e2(Pge m This is the distance that according to Eq .(21.6.23)characterizes variations in the magnetic field . On the other hand, variations in the modulus p are characterized by a distance scale known as the correlation length, given 338 21 Spo ntaneou slyBroken Gauge Symmetrie s according to Eq . (21 .6.24) bytt ~ =1/m,~'2_ . (21 .6.26) Also, the superconducting state with p = ~p )has an energy per unit volume that is less than that of the normal state with p = 0 by an amoun t A=2Im2(P)I -4(P)4=M414g. (21.6.27) Eliminating the parameters m and g from Eqs . (21 .6.5)-(21 .6.27), we find one important approximate relation among the observable quantities A, and A: A^' 1  (21.6.28)8e2a,2~2 The modular field becomes important in the dynamics of superconduct- ing vortex lines . These arise when a superconductor of a certain type is placed in a magnetic field that is strong enough so that it is energetically favorable for tubes of magnetic flux known as vortex lines to penetrate the material .43 (The conditions for the appearance of vortex lines are dis- cussed below .) By drawing a closed curve,6 around the tube at a distance much larger than the penetration depth, where the magnetic field vanishes, and repeating the argument contained in Eq . (21 .6.12), we see that the magnetic flux through the area W surrounded by W must be equal to the change of 0 around the curve, and hence equal to an integer multiple of the flux quantum n/e, just as for the flux through a thick superconducting ring. Where this flux is not zero, there must be a line within each tube along which electromagnetic gauge invariance is not broken . To see this, note that if we shrink the curve (e into the region of high magnetic field it becomes no longer true that ❑O = A, but the change of 0 around the curve must remain an integer multiple of n/e, and so by continuity cannot change . Thus we must eventually encounter a line (conceivably of finite thickness) along which p vanishes, so that 0 becomes ill-defined . (This is an elementary example of the sort of topological reasoning we shall use in Chapter 23 .) Near this line we must take both p and 0 into account as dynamical variables . The quantization of magnetic flux shows that a superconducting vortex line of minimum flux nee is stable . A vortex line of higher flux cannot simply disappear, but magnetic flux quantization alone does not prevent it from breaking up into vortex lines of smaller flux . Bogomol' I1y143' has shown that vortex lines of flux nn/ewith n >1 are unstable against breakup into n vortex lines of flux ir/e if and only if av > ~ . The factor 2is included along with mbecause at p the derivati ve of the function -mp+gp3in Eq .(21_6_2 4) is 2m2. 21.6Superconductivity 339 For this and other reasons it is convenient to divide superconducting materials into two classes :type I superconductors (most pure metals, except niobium) have ~ >~., while type II superconductors (niobium and most alloys) have A>~. The corresponding distinction in the electroweak standard model is between theories where the scalar mass (analogous to 1/~)is less than or greater than the W and Z masses (analogous to 1 1A). From the definitions of the correlation length ~ and penetration depth A, it follows that the modular parameter will rise from zero at the central line of a vortex to its equilibrium value (p)in a distance of the order of the correlation length ~, while the magnetic field will decay with distance from the central line in a distance of the order of the penetration depth A.Hence a vortex solution in a type Isuperconductor with ~ >Awould consist of a thin inner cylinder of nearly normal metal, within which the magnetic field drops to zero, surrounded by a much thicker outer cylinder within which the modular parameter rises to its asymptotic value (p). In contrast, a vortex in a type 11superconductor with A» ~ consists of a thin inner cylinder of constant magnetic field, within which the modular parameter rises to its asymptotic value (p), surrounded by a much thicker outer cylinder of superconducting material within which the magnetic field falls to zero . Vortex solutions exist for both types of superconductor and for any magnetic field, but as we shall now see, it is only in type II superconductors and in a finite range of magnetic fields that vortex lines are energetically favored . Because each vortex line has a cross-sectional area of order 7C~2 within which the material is in its normal state or nearly so, the extra energy per volume required to create these vortex lines is of order A/'7C~20, where A' is the number of vortex lines per area . This vortex density is limited by the condition that N' < I/7r~2, since otherwise the cylinders of normal metal would overlap, and the material would be considered to be in its normal state . The magnetic field must be expelled from a fraction 1 - .N'nA2 of the material if A' < 1 /nA2 , and from the whole material if A"> j/ 7c ;,2, So the energy per volume of the vortex state, relative to the superconducting state in the absence of magnetic fields, i s WY 5 :t~ 2x .Vn~2❑+ ZB 1dc n :~~(21.6.29) (Numerical factors like z and itare kept here to remind the reader of the origin of these expressions, but should not be taken literally .) For comparison, the energy per volume of the normal metal exceeds that in the superconducting state by WN =+0, and the energy per volume required to expel all magnetic fields from a superconductor is W s= B2/2 . We can decide which state is present at a given magnetic field by checking which ofWS, WN, or WV is smallest . 340 21 Spontaneously Broken Gauge Symmetrie s In type I superconductors we must distinguish between magnetic fields less or greater than the critical field B ,- 20. Recalling that *' < 1/n~2, and here ~ > )., we also have AV<1/n~2. Hence for B<B, Eq.(21.6.29) gives WV >Z ,B 2 + ,N'n(~2 -- ~ .2)❑>WS, so there can be no vortex lines . Also, for such fields WN >Ws, so the material is superconducting . On the other hand, for B >B, Eq .(21.6.29) implies that WV>❑[1+.N*n(~2- a2)]>WN, so here again there are no vortex lines . Also for such fields Ws>War, so the material is in its normal state . In type II superconductors we need to distinguish between magnetic fields in three ranges : B<B,1, B,1<B<Be,, and B>8,2,where Bc1 and 8,2 are a pair of critical fields, of orde r As we have seen, for type II superconductors the only stable vortex lines are those with the minimum flux nee, so in a magnetic field Bwe should put the number of vortices per area A"-in Eq . (21.6.9)equal to eB/ir. For B <B,, we can use Eq . (21 .6.8)to show that eB1n<11n~.~. The coefficient of the vortex density in Eq . (21 .6. 9)is therefore the positive quantity n~2 ❑-1B2),2, so here WV>Ws, and there are no vortex lines for B <B,1. Also, for such fields War >WS,so the material is entirely superconducting . For B >Be,, Eq . (21 .6.28)shows that the vortex density ~N' = eBliT is greater than 1/nA2, so in the vortex state the magnetic field completely penetrates the superconductor, and the energy per volume is given by Eqs .(21.6.29) and (21,6,2$) a s WV.=4N'n~2❑= eB ~2❑.-t~(BIB22)WN.~(BL,1IB)WS. Hence for Bc1 <B<Bc2 we have WV <War and Wv<Ws, so the material is in its vortex state . For B >B,2 we still have WV <WS but now WN <WV, so the vortices disappear and all superconductivity is extinguished . The ability of type II superconductors with a >~ to carry magnetic fields much higher than the critical value B, ;t~,/A--deduce d The derivation of flux quantization given above for an isolated vortex line is not fully applicable here . As we shall see, the separation of the vortex lines for B >B,1 is less than the penetration depth A, so it is not possible to find a contour W on which A -Obi = 0by simply drawing a circle around the vortex line at a distance from the vortex much greater than A . Instead, it is necessary to appeal to considerations of continuity . (M . Tinkham, private communication .) Suppose we draw an arbitrary continuous curve between the centers of any two vortex lines . As shown below, on this curve the vector A - ❑0 will be very large close to either vortex, but pointing to opposite sides of the curve, Thus there is at least one point on each such curve where A - Vo = 0.Because A - V0 is gauge-invariant, it must be canlinuaus, so there is a closed contour W around each vortex line where A - ❑O - D . 21.6 Superconductivity 34 1 earlier is important in technological applications of superconductivity, including magnets in high energy accelerators . The Ginzburg-Landau theory holds only where the material is near the transition between its normal and superconducting states, so let us apply it near the center of a vortex line, where p drops to zero . Close to the center of a vortex line we can ignore its curvature, and assume cylindrical symmetry . The field A - orp is taken to have only an azimutha l component : (A - VOL =A(r), so that the magnetic field has only an axial component : BZ =(❑x (A-❑O)) Z =A'(r) + A(r)fir(21.6.30) (21.6.31) while p is a function only of r . The structure of the vortex line is thus governed by the pair of coupled differential equations : ,,t(p) ~ (P(r)_ p Z 2~(p)(21.6.32) 4e2p(r)AZ(r ).(21.4.33) When r is small compared with both the correlation length ~ and the penetration depth Awe can neglect the terms in Eqs . (21.6.32)and(21.6.33) proportional to 1g2 and 1/A2, and find the simplified equation s AY+ A'ir- Alr 2= 0 , Eq. (21 .6.34) has the general solutio n Br C A(r) _ ~ + ter(21.4.34) (21.6.35) (21_6.36) where B is a constant that according to Eq . (21.4.31) is the magnetic field along the vortex line, and C is a real constant that is so far arbitrary . Using this in Eq . (21 .6.35)shows that for r --+0, the solution for p(r) is a linear combination of rlt'I and r-ICI, The order parameter p exp(2ieo) must be a smooth function of position, so we can conclude that ICIis a positive integer f, with p oc r( and 0=±1'rpl2e + constant for r --1,0. Note that this solution for 0is consistent with Eq .(21.6.36) with C = ±~ ; the azimuthal component of OO approaches ±f12er, while analyticity requires the azimuthal component of A to vanish as r --~-0. By a non-singular gauge transformation we can arrange that 0=±~,rpl2eeverywhere, so by the same reasoning as in Eq . (21 .6.12)the magnetic flux carried by a single vortex line in a superconductor much larger than the penetration depth is ±7re'/e . We see not only that, as expected, the order parameter 342 21 Sponta neously Broken Gauge Symmetrie s vanishes at the center of the vortex line, but that it vanishes with a power of r equal to the magnitude of the magnetic flux in units of 7r/e . This solution for 0obeys the `quantization' condition that 0(2n) - 0(0) is an integral multiple of 7r/e, which implies the quantization of magnetic flux. In the theory of Goldstone boson and electromagnetic fields based on Eq.(21.6.5)this condition on 0has to be imposed by hand on the solution of the field equations, while the Ginzburg-Landau equations `know' about this condition, because these equations are based on an appropriate choice of order parameter . Although the most dramatic properties of superconductors can be de- rived directly from the assumption that electromagnetic gauge invariance is spontaneously broken, a microscopic theory of superconductivity is needed to understand how and when this occurs . The derivation of the microscopic superconductivity theory of Bardeen, Cooper, and Schrieffer37 has been recast44,11 in the language of effective field theory, using power- counting methods similar to those that we have used here in Sections 19.5 and 21 .4. For this purpose, suppose we integrate out the degrees of freedom associated with the ions in a superconductor,#t leaving only an effective interaction among electrons . For simplicity, we shall work at zero temperature, and at first assume no external field, so that the La- grangian is invariant under translations and the time reversal operation T. We shall also assume spin-independent forces, so that the Lagrangian is invariant under SU(2) transformations that act on spin indices alone, but we shall not need to assume invariance under rotations acting on momenta . Electrons are then characterized by a momentum p and spin index s - ± 2, and are described by annihilation and creation operators a(p, s,t)and a~( p,s,t),with a Lagrangian of for m r+E(p) a(p, s,1) L = - ~ Jdap at(p, s, t} - ia at +E fdpi d'P`d3 p3 `~3 P4Vs'S25154(PI~Pza pay P4)NJ52S3 4 xat(P~,s1,t)a'(P202 , r} a( p3,83, t)a(p4aS4,t}63(pz + P2-P3-P4) + ... (21 .6.37) Strictly speaking, it is not possible to integrate out all degrees of freedom except electrons, because the phonon is a Goldstone boson whose frequency vanishes for very large wavelengths, like a massless particle in relativistic theories . However those effects of phonon inlerxctions that cannot be represented as effective electron-electron interactions are suppressed by inverse factors of the ion masses . 21.6Superconductivity 343 where ` . -.' represents terms with six or more creatio nand annihilation operators, and E(p) is the e lectron energy minus the chemical potential . For free electrons, E(p) =p2/2me - E F where EF is the energy of electrons at the Fe rmi surface .Interactions will inevitably change this function, but it is natura lto expect that since E(p) vanishes for free electrons at momenta pon the sphere I pI = 2meEF, in the presence of interactions it will still vanish on some closed Fermi surface Y E(p) = 0 for pon 9 . (21 .6.38) For reasons that will become apparent, we shall integrate out all electrons except those within a thin shell of thickness K around the Fermi surface .a (Later we will be able to remove the cut-ofd K by introducing a renormal- ized electron-electron potential .) The remaining degrees of freedom are then electrons with momenta of the for m p=k+n(k)e' , (21 .6.39) where k is on the Fermi surface 9, n(k) is the unit vector normal to the surface at k, and O c 1~ K. For such momenta , E(p) - VF(k)to, where VF(k) = n(k)(vE(p))p=k(21.6.40) (2L6 .41) The electron propagator in wave number -frequency space is the n 1 0)-- vF(k)e+if(21.6.42) Now let us consider how a general connected matrix element scales with K as K --+ 0 . We have an integral over frequencies for every loop, and a propagator for every internal line, so the integral of the product of propagators over all frequencies will have an e dependence e"L-I where L is the number of loops and I is the number of internal lines . To count the number of I integrals, it is important to note that for generic momenta the delta-function in the interaction term in Eq . (21 .6.37) constrains the ks, not the ts . (For instance, if momentum conservation constrains the momenta pi and p2 of two electron lines to have total momentum P=~ 0, then the integral over pj runs over the intersection of two closed shells of thickness K, one centered on P and the other on zero . The intersection of these shells is a closed ring of thickness x, so we have to integrat e For forces that are not spin-independent, there are two Fermi surfaces, one for each cigenvalue of the matrix E S,,s{p},and we integrate over all electrons in each spin eigcnsta tc except those within a thin shell around the corresponding Fermi surface . 344 21Spontaneously Broken Gauge Symmetrie s over one k-component which gives position around the ring, and two t"s which give position within the cross section of the ring .) Thus there are I integrals over ~s, and since the integrand goes as the matrix element will vary as NIacx--L (21.6.43) The number of loops is related to the number of internal lines and the numbers Vi of vertices of type i by the familiar relatio n L =I-~V, +1 . Also ,the number of internal lines is related to the numbers of vertices and the number E of external lines by another familiar relatio n 21 + E niVi , (21.4.45) where ni is the number of electron operators in the interaction of type i . Eliminating I from Eqs . (21 .6.44) and (21 .4.45) give s 2 2(21.6.46) Terms in the action with ni = 2 just serve to change the function E( p), and hence to shift the Fermi surface . True interactions have eta > 2, and so it appears from Eqs . (21 .6.43) and (21 .6.46) that they yield terms in the matrix element that make relatively negligible contributions for x --*0. In the language of Section 18 .5, this would mean that all interactions are irrelevant operators . This is why electrons near the Fermi surface in normal metals behave pretty much like free particles . There is, however, an exception to this conclusion . If a pair of electron lines disappears into the vacuum, then translation invariance requires the momenta of these two lines to be equal in magnitude and opposite in direction, so if one momentum is near the Fermi surface, then the other is also . (Time reversal invariance requires that E(p) is even in p, so even though the Fermi surface is generally not spherical, it is still true that if E(p) = 0 then E(- p) = 0 .) The integral over the momentum p of one of these lines is then over a shell of thickness K, or in other words over two ks and one / .For each interaction involving two such lines, we have one rather than two integrals over ~s, so that instead of reducing the order of magnitude of the matrix element by a factor ,c as indicated in Eqs. (21.6.43) and (21 .6.46), such an interaction has no effect on the order of magnitude of the matrix element . In other words, interactions involving four electron operators become marginal rather than irrelevant if they act between electron lines that eventually disappear into the vacuum . 21.6Superconductivity 345 To see the consequences of this exception, it is convenient to use a trick, known as the Hubbard-Stratonovich transf o'rmati on,45that was mentioned briefly in Section 10 .7. We are now going to include slowly varying external electromagnetic fields, so it will be convenient to work in coordinate space . Gauge invariance tells us that the Lagrangian (2 1.6.37)then become s L = - ~ dux ips (x, t) - iata- Aa(x, r) + E ~ - X + A(x, t) } vPsNt} +E.Id'xi d 3x2d'x d'X4 Vsl sZ si Sa(xl ,x2, x3 ,x4) 6152 s3 s q X IPs,(Xly t) V . 2(xZy t~ Ws3 (x3,t} ~s4(X4, 0, where(21.6.47) W,(x, t) = (2 7c)-3,2 fdip exp(i p x)a(p, s, t) . (21.6.48) We are now dropping terms with more than four electron operators, because they are all irrelevant . To this Lagrangian we add the ter m ❑L -- 4 d 3xld3X2[3x3d3X4 VS,S23234(xi,x2, x3, x4 ) 315 32Sq  [Tt , (xz, x 1,t)- ~s (x 1,t)~} 2 (Xz, t)1  [Ts3sa(X 3y X4,0-Ws3(X3,t) IPs4(X4,0J` (21.6.49) and do path integrals over the new `pair' field 'I'(x, s, x', .s', t) as well as the electron field w . This is allowed because ❑Lis quadratic in the pair field, with a field-independent coefficient of the second-order term, so, according to the appendix of Chapter 9, the integration over 'I'SS,(x, x', t) in path integrals just has the effect of setting TSSr(x, x', t) equal to the stationary point 'I'„,(x, x', t) = y~S(x, t) y~,,(x', t) of the Lagrangian, at which DL = 0 . This additional term is chosen so that the terms in L + 0 Lthat are quartic in electron fields cancel, leaving only terms quadratic in these fields : L+AL } }rs(x . t} _ - ~ d`~x ~s (x, t) - i ~t - Aa(x, t) + E ~ - N + A(x, t) S _1 4 [3xi d3 x2d`~x3d3 x4VSO52.S354x1IX 2 9 X3, x 4} Sl52S3 S4 X Vsl (XI, t) Wt (X2, t) Ts3salx3~ x4~ t) + TssI lxZy Ala tlY)s3lx3, 0 Ws4(x4, t l -T'SZSI(X21) xlIt) T5354 (x3,x4,t) (21.6.50) 346 21 SpontaneousrV Broken G augeSymmetrie s Now to the point . We have shown that the interaction V is irrelevant except where it acts on a pair of electron lines that disappear into the vacuum . When we calculate the quantum effective action r[T] in the presence of an external, slowly varying pair field LI's,,(x, x', t), this means that we drop all diagrams except those that become disconnected when we slice through any internal lI' line . But the general definition of the quantum effective action requires that we drop all diagrams that do become disconnected when we slice through any internal line .Therefore we must not include anyinternal lI'lines at all .Because the electron field enters quadratically in Eq .(21.6.50), the only graphs that survive when we integrate out the electron field are a one-vertex graph arising from the last term in the square brackets in Eq .(21.6.50), plus cone-loop graph given, by the same reasoning as in Section 16 .2, by the logarithm of the determinant of the coefficient of the terms quadratic in electron fields : r f r'[T] =1 4 EJ[fit J[ 3.x1(~`~aCZ (~`~aC3 C ~`~x4K';]sa s3 .cQ(x1Ix2, x3, x4) slsi s3 sq x T5251(X2, X 1 ,t)Ts3s4(X3a X4, 0 2 (InDot B _A T whereAandBare the 'matrices' : Bx,s'r,,xsr = -As's(x', x, t )6(t' - t) and ❑is the gap function :+ constant . X+A(x, t))63(Xi(21.6.51) I I:2f d3yd3y, x,y', y)i's,s(x',x,t) 6~Q(21.6.52) (21.6.53) (21.6.54) (Here and below, we ignore an additive constant in r arising from the modes that have been integrated out .) This is one problem in which the effective action can be calculated without any assumption about the smallness of the interaction in the Lagrangian . In order to simplify our further discussion, let us now specialize to the case of a spin-singlet pair fiel d T+-(x', x, t) = -T-+(x, x, t) = lI'(x', x, t) , (21.6.55) where subscripts + and - stand for spin indices +1/2 and -1/2, 21.6Superconductivity 347 respectively .aa Using invariance under rotations of spin indices, the com- ponents of the potential needed here are the n V+-+-( x',x, t} - V+__+( x',x, t} _ - V_++_( x',x,t)+ V_+-+( x',x,t) 2V(xI X1 t) - (21.6-56 ) Then Eq . (21 .6.54) give s The quantum effective action (21.6.51) is no w QT] =JdtJd3x1d3xZd3x3d3x4V(X1, X2, X3, X 4) XV(X Z,xl,t) T 1x3, X4, 0 -iInDet-~T where(21.6.57) (21.6.58) (21.6.59) Aa(x,t)+E ~ - N+ A(x ,r)}63(x' ---x)6(t'-t), at and(21.6.60) (21.b.b1) (21.4.62) Let's first use these results to consider the translationally invariant ease, with no external electromagnetic fields . Then the pair and gap fields can be expressed as Fourier transform s ( x) J ❑(x'x)_ (2 7r)-3 JLl3I) dap(x'-x)❑(P)a and the electron-electron potential appears here in the for m Id3xId3 x2d3 x3d3 x4eip'(xl-x2)eap(x3-x4)jj (xl,,xZa X3, X 4)(21.6.63) (21.6.64) (21.6.b5) In liquid He3 the non-vanishing components of the pair field form a spin triplet, with components T, - T~ ~, YO = /2-"I'+_ = 12-'Y-+, T-1 =T_ 348 21 Spontaneously Broken Gauge Symmetrie s where *_4 is the spacetime volum e f_4=fd3x fdt 1. (21.6.66) By going over to wave number-frequency space, the `matrices' (21 .6.60) and (21.6.61) become diagonal, and the calculation of the determinant becomes trivial . The effective potential V [T] (not to be confused with the electron-electron potential) was defined in Section 16.1as minus the effective action per spacetime volume z + J dcv c~-~p I n1 -- W2 I E(P)I (21 .6.67)(27r)4 (p)te with A(p) = -f d3P'V(p, p') T(p') . (21.6.68) (The ie term can be inferred from the Feynman rules for the electron loop graphs, and as shown in Section 9 .2, it ultimately arises from the conditions on the electron field at t --+±(Da.} Wick rotating, integrating over r .i, and expressing lI' in terms of A,this become s V[D] = - J'd'p d' p'❑* (p') V-1( p',P)O(P) - I -3P [ E2( P) + IM( P) ~ -- E( P),. (21 .6.69)2~cId(} As we saw in Section 14 .1, the field equations are just the condition that the effective action is stationary, which in our case yields the famous gap equation for the equilibrium gap function Ao(p) : r5V [O] 6A (p) ~ A=At ) d3P, V-'(p, p')Do( P') -I Ao(p) 2(2n)3 z 2E (P) + IDa(p)I or, in a more familiar form , Da(P) ~ - ~ d3pV(P,P~~AO W) (21 .6.70)2(27c)- A/ E ~ 2(P.) + lDa( P.)1z All along, all integrals over momenta have been implicitly understood to be limited to momenta of the form (21 .6.39) within a thin shell of 21.6Supercond uctivity 349 thickness K around the Fermi surface . The effective potential (21.6.69) is thus V [A] =-x2 ~ d2k d2k'❑*(k')V-'(k', k )0(k) '~ (k) + IO (k)IZ - ~ v, (k)] .(21.6.7 1)3 JaLd2k[2 t(Z7c) The effective potential is now to be understood as a functional only of th e gap function ❑on the Fermi surface . Since K is arbitrary, the potential V(k', k)must be given a K-dependence such that V[A] is K-independent . Most applications44 of renormalization group methods to superconductivity have followed the Wilson approach outlined in Section 12 .4, deriving a differential equation for the dependence of V(k', k)on rc, and studying the behavior of its solutions for 1c-+0. For the sake of flexibility it is useful to note that one can just as well adopt the approach of Dell-Mann and Low, and introduce a renormalized electron-electron potential44 a V~ f ( k',k)62V[0](21.6.72) where y is a sliding renormalization scale, like that introduced in Section 1$.2. By expressing the original electron--electron potential in terms of V ', Eq. (21 .6.71)become s V [A] _ - ~ d2 k dZk' ❑"(k')V,-'(k', k}0(k) d2k[V/~2 V2[(k)+ J0(k)~ z - I~vF(k)fV L (27r)3 JA(k)12 it2IA(k)12 ~(~2VF~k~ + ~Z~1~~ + 4~~~~'r(k) + jc2)3/Z 21.6.73 The integral over /now converges if we take the cutoff x to infinity, and yields V[A] = - Ld2k d2k'❑"(k') V,-'(k', k)A(k ) + 13d 2kI0(k)i2 [In -11(21.6.74)2 2n fr ~vk ()F() , The condition that V [A] be stationary at ❑= 0a yields the gap equation in the more useful for m Ao(k) = 1~ V (k, k')~v'(k')Da(k')InJA°k')(21.b.75)2(2~)L,2kft 350 21Spontaneously Broke nGauge Symmetrie s Using this in Eq . (21 .6.74) shows that the energy density of the supercon- ducting state is less than that of the normal state by an amoun t z L2(27T)3vk~( ) Of course the y dependence of the electron-electron potential V~ mus t be chosen to satisfy a renormalization group equation that insures that the effective potential (21 .4.74) is independent of the arbitrary renormalization scale u : d,u 2(27r)3 VF(k) or, equiva lently, It~ V~~( k',k} = 1 d2k" V k'k") v. 1 (k")V,~(k",k).d~ 2(2 7c)3fu ~ (21.6.77) This can be usefully rewritten in terms of the Hermitian kerne l K2(2ic)-VF f /2( k') V~ 1/2 ( k) Vu(k', k) (21 .4.78) as zd~Kit = Kit (21.6.79) The eigenvectors u,,(k) of K p(k, k' )are then independent of p, while the eigenvalues take the form 1 /ln(An/µ), where the An are integration constants, like the A of quantum chromodynamics . The potential therefore takes the for m V(k', k) = 2(2n)3 v~ 2 ( k) vF 2( k') E un(k)u;~(k') ln(An/ft) n(21.6.80) where we have chosen the eigenvectors to be orthonormal, so that the completeness relation takes the for m n The effective potential (21 .6.74) may then be written as(21.6.81) V[❑]~dzk dzk,E❑"(k')~u;~(k,)A(k) un(k) In I0(k)l- 1( } A 2 21r~ ~vkt~ z A nnF( } F( (21.6.82) For a material with local rotational invariance the Fermi surface is a sphere and the eigenvectors un(k) are spherical harmonic functions of the coordinates on this sphere, but Eq . (21 .4.82)holds without any assumption of rotational invariance . 21.6Superconductivity 35 1 We can now use this formalism to get an idea of when superconductivity occurs . The logarithm in Eq . (L1 .5.82)is large and negative for very small ❑, and large and positive for very large ❑, so as the overall scale of A increases from zero to infinity, VIA] drops from zero to negative values and then rises to infinity . It therefore always has a minimum at a non- vanishing value of Afor any electron-electron potential . But this result is subject to an important qualification . When we take the cut-off rc to infinity, the integral in Eq .(21.6.73) is effectively cut off at 1,of order IAI1aaF, while the Fermi surface has a radius of order k F, where 8nkF/3(2n)3 equals the electron number density . Since we have been assuming that the electrons taken into account here are in a thin shell around the Fermi surface, this derivation is only valid if A l lCC wD, where COD is the Debye frequency kFVF . In particular, in the case of rotational invariance and a direction-independent gap function, the effective potential reaches a local minimum for a gap function of the order of 115 w,,, so the symmetry is spontaneously broken as long as 115 Wave CC WD . Another way of stating this result is that for s-wave superconductivity, the s-wave projection of the electron---electron potential (21 .6.80)must be attractive if renormalized at a scale y & COD, But itdoes not matter how strong this renormadizect attractive potential is . This feature of superconductivity, that a Goldstone boson forms for an attractive potential however weak the potential may be, is a consequence of the existence of a Fermi surface, which enhances long-range effects . In quantum field theories without elementary spinless fields like those of Section 21 .5, we would not normally expect a spontaneous symmetry breakdown in empty space unless the interactions are sufficiently strong . Now let us return to the case of an external electromagnetic field . As usual, we can introduce the Goldstone boson field O( x,t) by writing each charged field of the theory, which here is just the gap field 0(x, x', t) or equivalently the pair field T(x, x', t), as a gauge transformation with gauge parameter O(x, t) acting on the corresponding gauge-invariant field, distinguished with a tilde : T(x,x',t)= exp ( - io( x,t)}T( x,x',t)exp(- io(x',t)). (21 .6.83) The effective action is then given by using Eq . (2 .6.83)inEq. (21 .6.59). By a gauge transformation, we can then remove the 0dependence in Eq. (21 .6.83), provided we replace Ay(x) in Eq . (21 .6.60)with A',(x)- 0,0(x) . When the material is not only superconducting but far from the transition between normal and superconducting states, we may also integrate out the gauge invariant degrees of freedom associated with tI'(x, x', t), which simply means that we replace it with its equilibrium value TQ(x, x', t) . The part of the effective action that depends on the 352 21 Spontaneously Broken Gauge Symmetrie s Goldstone and external electromagnetic fields is the n r[,0,A] = ro=Q[A] - i InDetsy ~4 T+ iInDet ~ _~T, (21.6.84) where now } + eAo(x, t) -- ec~(x, t) + E ~ - M + eA(x, t) - e Vo(x, t) iat (21.6.85) Quantitative properties of the superconductor such as the penetration depth can be read off46 from the expansion of Eq . (21 .6.84) in powers of AQ(x,t)-~(x,t) and A(x, t) - VO{x, t) . Appendix General U nitarit y Gaug e In this appendix we shall show that in general spontaneously broken gauge theories it is always possible to adopt a `unitarity' gauge in which the Goldstone boson fields satisfy Eq . (21 .4.30): F b ~ aeab =0. (21.A.1) Rb Using the exponential parameterization for all groups, we note first that any element of G, in at least a finite neighborhood of the identity, may be put in the for m g=exp --i Ef~. 3ra exp iEO axaexp iEp itd a a i with Oa subject to the linear constraint that for all a Fz ub Oa eab = 0. ab(21.A.2) (21.A.3} This is easy to see when g is infinitesimally close to the identity . Any such g may be written 9- 1 +t 0oaxa+Z ~i ti a with 0~ and PQ infinitesimal . Equivalently , g + i~ Oaxa+i~ ~C~t~--i~B xJa,(21.A.4) (21.A.5) Problem s where 0,is an arbitrary infinitesimal, an d Oa(O) =Oa + oaexa353 (21.A.6) (21.A.7) For any given 40,we can choose B ,,to minimize the positive quantit y EFb0,040 }. ab(21.A.8) At this minimum, the quantity (21.A.8)is stationary with respect to variations of Ba, so Oa{0}satisfies Eq .(21.A.3).For 4), µ, and 0infinitesimal, Eq. (21 .A.5)is the same as Eq . (2Z .A.2), so we see that the set of all gs of the form (21.A.2)(with 0,,satisfying Eq .(21.A.3)} includes all gs infinitesimally close to the identity . It follows then from continuity that the same is true of all gs in at least some finite neighborhood of the identity . Next, consider the particular group elemen t g= y(~)= eXp i ~ ~ax a and write it in the form (21.A.2 )Y(~ ) = exp(21.A.9) exp i (21.A.10) with subject to Eq .(21.A.3).This just says that the gauge transfor- mation exp (i E, Oa(~) .57_,) transforms ~a int o Now dropping the prime, we have thus succeeded in constructing a gauge in which ~u satisfies Eq .(21.A.1),as was to be shown . Probl ems 1. Calculate the effective ghost Lagrangian in a `generalized unitarit y gauge,' with S[f] given as usual by Eq .(15.5.22), but now wher e f(Jx) = i0n(x)(ta)nm(0y„(0))vAC, with 0,real scalar fields, and t , imaginary antisymmetric matrices representing the Lie algebra o f the gauge group . What is the ghost propagator? Is this part of th e Lagrangian renormalizable ? 2. What would be the effect in the SU(2) xU(1) electroweak theory if the gauge symmetry were broken by the vacuum expectation value 354 2 1 Spontaneously Broken Gauge Symmetrie s of a field 03 belonging to a real triplet 0 = (,0+,0❑,0-)instead of the usual complex doublet (00,0-)? 3. Consider the standard electroweak theory, with a single scalar doub- let. In one-loop order, calculate the effect of Z° and neutral scalar exchange on the anomalous magnetic moment of the muon . 4. What is the lowest-order magnetic moment of the W+ andZQ particles in the standard electroweak theory` ? 5. What would be the effect of the discovery of a fourth generation of quarks and leptons on the predictions of the unified theories of strong and electroweak interactions discussed in Section 2 1.5? 6.Suppose that several fields with incommensurate values of the electric charge had nonvanishing vacuum expectation values in a superconductor . What effect would this have on the superconductors discussed in Section 2 1.6? Referencesproperties o f 1. J. Goldstone .Nuovo Cimento 9, 154(1961) ;J. Goldstone, A . Salarn, and S . Weinberg, Phys . Rev .127, 965 (1962) . 2. P. W. Higgs, Phys . Lett .12, 132 (1964) ;Phys.Rev. Lett .13, 508 (1964) ;Phys . Rev.145, 1156 (1966) ; F. Englert and R. grout, Phys . Rev. Lett .13, 321 (1964) ; G. S. Guralnik, C .R. Hagen, and T . W. B. Kibble, Fhys . Rev . Lett .13, 585 (1964) ; T. W.B. Kibble, Phys . Rev.155, 1554 (1967) ; S. Weinberg, Phys .Rev. Lett .18, 507 (1967) : footnote 7 . 3. S. Weinberg, Phys .Rev. Lett .19, 1264 (1967) .A. Salam, in Ele- mentary Particle Physics, ed . N. Svartholm (A1mgWist and Wiksells, Stockho1m,1968) : p. 367 . 3a. The Feynman rules for general gauge theories in unitarity gauge were given by S . Weinberg, Phys . Rev, D7,1068 (1973) . 4. G. 't Hooft, Nucl . Phys .B35,167 (1 971); B. W. Lee, Phys .Rev.D5, 823(1972) . 5. K. Fujikawa , B . W. Lee, and A . Sanda, Phys . Rev. D6, 2923 (1972) . 5a.B. W. Lee and J. Zinn-Justin, Phys . Rev .D5, 3121, 3137 (1972) ; Phys. Rev .D7, 1049 (1972) ; G. 't Hooft and M . Veltman, Nucl . Phys . B5U1 318 (1972) ;B. W. Lee, Phys .Rev.D9, 933 (1974) . Reference s 6. H. Georgi and S . L. Glashow, Phys .Rev. Lett .2$,1494 (1972) . 7.T.D. Lee and C . N. Yang, Phys . Rev .98,101 (1955) .355 8. Mode ls with an incomplete SU(2) x U(1) symmetry were proposed by S . L. Glashow, N uci. Phys .22, 579 (1961) ; A. Salam and J. Ward, Phys . Lett .13, 168 (1964) . Also see J . Schwinger, Ann.Fhys .(N.Y) 2, 407 (1957) . 9. Speculations about possible neutral currents go back to work of G . Gamow and E . Teller, Phys .Rev. 51, 288 (1937) ;N. Kemmer, Phys. Rev.52, 906(1937) ;G. Wentzel, Hely .Phys .Acta 10,108(1937) ;S. Bludman, N uovoCimento 9, 433(1958) ;J. Leiter-Lope, Nucd . Fhys . S, 234 (1958) . 10. Neutral currents were fir st obse rved in a single v.e scattering event atCERN, by F . J. Hasert et al.Aachen -brussels -CERN -Ecole Polytechnique -MilanJ Qrsay-London collaboration), Phys.Lett . 46, 121 (1973) . 11. N. Cabibbo, Phys . Rev . Lett .10, 531 (1963) . 12. I. S. Towner, E . Hagberg, J . C. Hardy, V . T. Koslowsky, and G . Savard, Chalk River preprint nucl-th/9507005 (1995) . 13. S.L. Glashow,J.Iliopoulos, a nd L ,Maiani,Whys.Rev.D2,1285 (1970) . 14. S. Weinberg, Phys . Rev . Lett .27, 1688 (1971) ;Phys .Rev.5,1413 (1972) . 15.J.J. Aubert et ad .,Fhys.Rev. Lett .33, 1404 (1974) ; J. E. Augustin et al., Ph ys.Rev. Lett .33, 1406 (1974) . 16. M. L.Perl et al.,Fhys.Rev.Lett.35, 1489 (1975) . 17. S. W.Herb et al.,Whys.Rev. Lett .39, 252 (1977) . 18. F. Abe et al.,Fhys.Rev. Lett .74, 2626 (1995) ; S. Abac hi etal.,Fhys. Rev. Lett .74, 2632 (1995) . 19. The values given by the CDF and D ocollaborations at Fermila b are respectively 176 ± 8±10GeV (F . Abe et al .,Ref.18.)and 199±z~ ±22 GeV (S. Abachi et al.,Ref, 18), with the first an d second errors statistical and systematic, respectively . These result s are interpreted to give a combined value of 181 ± 12 GeV by J . Ellis , G. L. Fogli, and E . Lisi, CERN-BARI preprint hep-ph/9507424 . 20. M.Kobayash iand K .Maskawa, Frog. Theo r.Phys.49, 282 (1972) . 356 2 1 Spontaneously Broken Gauge Symmetrie s 20a. F.J. Gilman, K .Kleinknecht, an dB.Renk, Carneg ie Me llon-Mainz preprint CMU-HUP95-19-DOE-ER/40682- 107 (1995), to be pub- lished in t he 1995 Review of Particle Properties . 20b. S. Weinberg, Phys . Rev . Lett .37, 657 (1976) .For earlier models in which scalar fields are responsible for CP and T non-conservation see T . D. Lee, Phys . Rev .D8, 1226 (1973) ;Phys .Rep.9C, 143 (1974) . 21. F. J. Hasert et al.,Phys .Lett .46B, 138 (1973) ;P. Musset, Jour . dePhysiq ue 11/12, T34 (1973) .Neutral current events were see n at about the same time by the Harvard-Pennsylvania-Wisconsin - Fermilab group at Fermilab, but publication of their paper wa s delayed, so they took the opportunity to rebuild their detector, an d at first did not find the same signal . Evidence for neutral current s was published by this group in A . Benvenuti et al ., Phys .Rev. Lett . 32,800 (1974) . 22. G. Arnison et al.Fhys .Lett.122B, 103 (1983) ;126B, 398 (1983) ; 1298, 273 (1983) ;134B, 469 (1984) ;147B, 241 (1984) . 23. F. Abe et al .(CDF collaboration), Phys .Rev. Lett . 75, 11 (1 995) . The Wmass is measured here in observations of the decays W--*y+ v and W -),e + v . 24. CE RN report LEPE,WWG/95-01, unpublished (1995) . 25. M. Green and M . Veltman, Nucd .Phys .B169, 137 1980) ;V. A. Novikov, L . B. Okun', and M . I. Vysotsky, Nucl .Phys . B397, 3 5 (1992) .For later studies, see P . Langacker, in Precision Tests of th e Standard Model (World Scientific, Singapore, 1994) ; P. Bamert, C . P. Burgess, and I . Maksymyk, McGill-Neuchatal-Texas preprint hep - ph/9505339 (1995) ;W. Hollik, Karlsruhe preprint hep-ph/950740 6 (1995) . 26. S. Fanchotti, B . Kniehl, and A . Sirlin, Phys . Rev .D48, 3 07(1973), and references cited therein . 27.J. Ellis, G . L. Fog li, and E . Lisi, Ref. 19. 27a. S. Weinberg, Phys . Rev . Lett .43, 1566 (1979) . 27b. For instance, in the so-called see-saw mechanism, a neutrino mas s of this order would be produced by exchange of a heavy neutra l lepton of mass M ; see M . Dell-Mann, P . Ramond, and R . Slansky , in Supergravity, ed . P. van Nieuwenhuizen and D . Freedman (Nort h Holland, Amsterdam, 1979) : p. 315 ; T. Yanagida, Frog . Theor .Phys . B135, 66 (1978) . References 357 27c. S. Weinberg, Ref . 27a ; F. Wilczek and A . Zee, Phys . Rev . Lett .43, 1571 (1979) . Also see S . Weinberg, Phys .Rev.D22, 1694 (1980) . 28. This general formalism was described by S . Weinberg, Phys .Rev. D13, 974 (1976) .For earlier work on dynamical symmetry breakin g in gauge theories, see R . Jackiw and K . Johnson, Phys . Rev. D8 , 2386 (1973) ;J. M. Cornwall and R . E. Norton, Fhys .Rev.D8, 333 8 (1973) . 29. This interpretation of the results of the standard model is due to L . Susskind, Ref . 30. 30. S. Weinberg, ref . 28 and Phys . Rev .D19, 1277 (1979) ; L. Susskind, Phys . Rev .D19 , 2619 (1979) . The term `technicolor' is due to Susskind . 31. S. Dimopoulos and L . Susskind, Nuc[ .Phys .B15, 237 (1979) ; E. Eichten and K . Lane, Phys . Lett .90B, 125 (1980) ; S. Weinberg, unpublished work quoted by Eichten and Lane . For a review, see E . Farhi and L Susskind, Phys .Rep.74, 27 7 (1981 ) . 32. S. Weinberg, Phys . Rev . D5, 19 62(1972) . 33. J. Pati and A . Salam, Phys . Rev.Lett.31, 275 (1973) . 34. H. Georgi and S . L. Glashow, Phys .Rev.Lett.32, 438 (1974) . 35. H. Georgi, in Particles and Fields 1974, ed . C. Carlson (Amer . Inst. of Physics, New York, 1975). 36. H. Georgi, H . R. Quinn, and S . Weinberg, Phys . Rev . Lett .33, 451 (1974) . 36a. S. Dimopoulos and H . Georgi, Nucl . Phys .B193, 15 0 (1981) ;J. Ells, S . Kelley, and D . V. Nanopoulos, Phys . Lett .B260, 131 (1991 ); U. Amaldi, W . de Boer, and H . Furstmann, Phys .Lett .B260, 447 (1991) .For more recent analyses of the data, see P . Langacker and N. Polonsky, Phys . Rev. D47, 4028 (1993) ;D49, 1454 (1994) ; L. J. Hall and U . Sarid, Phys .Rev. Lett .70, 2673(1993) . 37. J. Bardeen, L . N. Cooper, and J . R. Schrieffer, Phys .Rev.108, 1175 (1957) . 38. P. M. Anderson Phys .Rev, 130, 439 (1963) . 39. S. Weinberg, Prog . Theor . Phys . Suppl .No. 86, 43 (1986) . 40. B.D. Josephson, Phys . Lett .1, 25 (1962) . 358 21 Spontaneously Broken Gauge Symmetrie s 41. V. L. Ginzburg and L . D. Landau, JETP (USSR) 20,1064 {195 0}. 42. L. P. Gor'kov, Soviet Fhys .JETP 9,1364 (1959) . 43. A. A. Abrikosov, Soviet Phys . JETP 5, 1174 (1957) . 43a. E. B. Bogomol'nyi, Sov .J.Nucd .Phys .24, 449 (1976) .For numerical calculations, see E . B. Bogomol'nyi and A . I. Vainshtein, Sov. J.Nucd . Phys .23, 588 (1976) . 44. G. Benfatto and G . Gallavotti, J . Scat .Phys .59, 541 (1990) ;Phys . Rev.42, 9967 (1990) ;J. Feldman and E . Trubowitz, Hely .Phys .Actcr 63,157(1990) ;64,213 ( 1991) ;65,679 (1992) ;R. Shankar, Physic s A177, 53 0(1991) ;Rev. Mod . Phys . 66, 129 (1993) ;J. Polchinski , inRecent Developments inParticle Theory, Proceedings of the 1992 TASI, eds. J. Harvey and J . Polchinski (World Scientific, Singapore , 1993) . 44a. S. Weinberg, N ucd.Phys .B413, 567 (1994) . 45. R. L. Stratonovich, Sov .Phys . Dokl .2, 416 (1957) ; J. Hubbard, Phys . Rev. Lett . 3,77 (1959) . 46. A. A. Abrikosov, L . P. Gor'kov, I . Ye. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics, end edn ., transl . by D . E. Brown (Pergamon Press, oxford, 1965) . 47. In order to give a precise meaning to the mass Al it is necessary to take into account both two-loop corrections to the renormalization group equations (21 .5.6)-(21 .5.8), and one-loop `threshold correc- tions' to condition (21 .5.5). See S . Weinberg, Phys . Lett .BZB, 387 (1979) . 22 Anoma lies There are subtleties in the implications of symmetries in quantum field theory that have no counterpart in classical theories . Even in renormaliz- able theories, the infinities in quantum field theory require that some sort of regulator or cut-off be used in actual calculations . The regulator may ❑iolate symmetries of the theory, and even when this regulator is removed at the end of the calculation it may leave traces of this symmetry violation . This problem first emerged in trying to understand the decay rate of the neutral pion, in the form of an anomaly that violates a global symmetry of the strong interactions . Anomalies can also violate gauge symmetries, but in this case the theory becomes inconsistent, so that the condition of anomaly cancellation may be used as a constraint on physical gauge theories . The importance of anomalies will become even more apparent in the next chapter, where we shall study the non-perturbative effects of anomalies in the presence of topologically non-trivial field configurations . 22.1 The n° Decay Prob lem By the mid-1960s the picture of the pion as a Goldstone boson associated with a spontaneously broken SU(2) (DSU(2) symmetry of the strong interactions had scored a number of successes, outlined here in Chapter 19. However, this picture also had a few outstanding failures . The most disturbing had to do with the rate of the dominant decay mode of the neutral pion, no --+2y. It was the solution of this problem that led to the discovery of symmetry-breaking anomalies . After integrating out all heavy and trapped particles, we would expect the effective Lagrangian for 7r° --+2y to be given by the unique gauge- and Lorentz-invariant term with no more than two derivatives : Y'9Ty = ~~ oe~`~P;,FjsV Fp),, ( 22.1.1} where g is an unknown constant with the dimensions [mass]- 1. Th e 359 360 22 Anornadie s methods of Section 3 .4 can then be used to calculate the rate for no --+27 M32 r(,~Q ~2y)~ ~~ One might naively expect g to be of orde r e2 g '& Sn2Fn(22.1.2) (22.1.3) where F, L-~ 190MeV is used as a typical strong interaction mass scale, and a factor l/Src2 is inserted because the graphs responsible for no --+27 would have to include at least one loop . For instance, in 1949, using the pre-QCD theory of pions and nucleons with interaction Lagrangian iG7rN7C  N2ty5N, Steinberger' calculated that the contribution to g from triangle graphs with a single proton loop i s e2G7rNgo = (22 .1.4)321r2 MN This is numerically not very different from Eq .(22.1.3), because the Goldberger-Treiman relation (see Section 19 .4) gives G IN =2MNgAI F em. This estimate of the amplitude for no --*2y leaves out the special constraints imposed by the S U(2) (D SU(2) symmetry . The electromag- netic interaction violates most of this symmetry, but at least formally it has no effect on the U(1) x U (1) subgroup generated by the electrically neutral generators of S U(2) (DS U(2) . Acting on quarks, the infinitesimal pseudoscalar element of this subgroup has the effect : bu = i Ey5u, 6d=- i1EY5d , (22.1.5) so that the e lectric current is invariant : S 3 uy1`u - ~ c~ypc~=0. (22.1.6) (This argument was made by Sutherland and Veltman before the advent of quantum chromodynamics, with the proton and neutron in place of the u and d quarks .) Because the no is the Goldstone boson associated with this symmetry, a non-derivative interaction like (22 .1.1) can arise only from the breaking of this symmetry by quark masses, and so must be proportional to m~ oc mu + Md.With this in mind, we would expect the constant g to be suppressed2,3by an extra factor m~~mN 2 2e Mn g (22.1.7)Sn2F,~(MN (In place of mN we might have used the chiral-symmetry-breaking scale 27[Fg= 1200 MeV discussed in Section 19 .3, with little change in our results) . There are also chiral-invariant effective nQyy interactions involving 22.1 The a° Decay Problem 361 the derivative of the nofield . Lorentz invariance requires that these involve at least two additional derivatives . Using the homogeneous Maxwell equation a,FA.-OvFyA=-OAF,,,and integrating by parts, we see that there is only one independent chiral-invariant coupling with just two extra derivatives ,given by inserting a d'Alembertian acting on the nofield in Eq. {22 .1.1 }. On the pion mass shell this is just the same as an interaction (22.1.1)with an extra factor rn~ ,leading to the same estimate (22.1.7).It was sometimes said that chiral symmetry would forbid the decay no --+ 2y, but itismore accurate to say that chiral symmetr ywould make the rate for this process go as m ~instead of rn ~for rn, , ---> 0. The difficulty is that the observed rate for no --+ 2yis much larger than would be expected from Eq . (22 .1.7),and is in fact much closer to what would be inferred from the naive result Eq .(22.1.3).To be specific, from Eqs. (22.1.7) and (22.1.2)we would expect a decay r ate M712 r'(na--+ 2y)= 3 ~2 4 1.9X 10I3 5 -I(22.1.8)4~ F ,EmN while using Eq .(22.1.3)in place of Eq . (22.1.7)would giv e M392 r(,g°--+ 2y) z Z =4.4x106s-'. (22.1.9)47r- Fn The observed rate is r(7r° --+ 2y) =(1.19 ± 0.08) x 1016s-1, which is in satisfactory agreement with the naive rough estimate (22.1.9),and almost three orders of magnitude larger than the `improved' result (22.1.8)!One is forced to the conclusion that something anomalous here is invalidating the chiral symmetry that led us to introduce the additional factor mn /mN in g. Similar problems arise in trying to understand the rates of some other processes, such as q°--*y + y . In 19 69 the source of this anomaly was traced by Bell and lackiw4 to the violation of chiral symmetry by the regulator that is needed in order to derive consequences of the conservation of the neutral axial vector current for one-loop Feynman diagrams . Their result was confirmed, generalized, and extended to higher orders by Adler5.It was subsequently realized in 1979 by Fujikawa 6that in the path-integral formulation of field theory the chiral-symmetry-breaking anomaly enters only in the measure used to define the path integral over fermion fields . As we shall see in the next section, this approach makes it simple to evaluate the amplitude for 71°--+ 2y produced by this anomaly to all orders of perturbation theory . After that, we shall return to the direct calculation of anomalies in more general theories, and discuss various applications . 362 22 Anomalie s 22.2 Transformation of t he Measure : The Abe lian Ano maly We now turn to a calculation of anomalies of the sort relevant to 7° de- cay. For this purpose we adopt Fujikawa's interpretation6 of the anomaly as a symptom of the impossibility of defining a suitably invariant mea- sure for integrations over fermionic field variables . Fujikawa's analysis of this problem was based on the use of path integrals in Euclidean space, and on an expansion of the fermionic variables of integration in eigenfunctions of the gauge-invariant Dirac operator, which is Hermitian in four-dimensional Euclidean space . Here we shall first present a less rigorous derivation, based on the familiar path integrals in Minkowski space, which will allow us to obtain the correct answer with a minimum of work . The Euclidean approach will be taken up briefly at the end of this section, and used to derive a famous index theorem . We will start by evaluating the anomaly in the transformation of the measure under an arbitrary local matrix transformation W(x) --+U(x)Y)(x) of a column y)n(x) of massless complex spin 1/2 fermion fields that interact non-chirally with a set of gauge fields Aa(x) (such as the u and dquark fields interacting with the electromagnetic vector potential AII(x) in the problem of calculating the rate for no --* 2y) .These are fermionic variables, so the measure is transformed not with the determinant of the transformation matrix, but with its inverse : [dW]LdFpl--* (Det,"11Deter)-'[dW][dqa], (22.2.1) where Vxn~yrn_U(X)nm54(X-y)(22.2.2) Wxrt,ym = 174U(X)' Y4]nm64(x -Y)(22.2.3) and y4 - iy° is the matrix used to define y~ =Y)TY4 . The indices n, m run over flavor labels and Dirac spin indices . The reader may wonder at this point why we bother to include factors of Y4 in Eq .(22.2.3),since they just amount to a unitary transformation that should not affect the determinants . The answer is that in order for calculations to be meaningful we will find it necessary to regulate the sum over fermion modes in calculating propagators and determinants, and we shall find that the Y4 factors do affect the regulated determinants . Whether or not we include the Y4 factors thus depends on the method of regularization that is used to make these hand-waving manipulations meaningful . We include the factors of y4 because we wish to regulate in such a way as to preserve Lorentz invariance, and in the cases of interest here it is 74 U(x)ty4 rather than U(x)t that transforms as a scalar . First, let us consider the case where U(x) is a unitary non-chiral trans- 22.2 Transformation of 'the Measure : The Abelia n Anomal y formation363 (22.2.4) witht an o rdinary Hermitian matrix (not involving 75, butnotnecessar ily traceless) an da(x) a narbitraryreal f unctionof x .Inthis case J& is pseudounitary : *0&= 1 , (22.2.5) so the measure is invariant under this sort of transformation . I n particular, the symmetry under the gauge group itself, where t is one of the non-chiral generators tom, is not spoiled by any anomalies . Next, consider a local chiral transformation, wit h witht aga inanordina ryHermitianmatrix and m(x) realfunctionof x.Inthis case OW ispseudo-Hermitian: *=v. (22.2.7) Themeasure is notinvariant und erthe chiraltransforma tion;rather, we have Ldp}][dqi]--+(DetW)-ZLdy)][dp]. (22.2.8) Now let us specialize to the case of an infinitesimal local chiral trans- formation . Taking a(x) to be infinitesimal in Eq .(22.2.6),we have her e [V - 1]nx,rrty = i *X)[ YS t]nm 64(x - .Y). (22 .2.9) Using the identity DetM = exp Tr In M and the limiting formula ln(1 + x)--+x for x --+0, the measure now has the transformation propert y [d V)] [dtp] --* exp i fd4x oc(x).~I(x) [dy) ] [dfp-], whereslis the anomaly function: (22.2.11) with `Tr' here denoting a trace over both Dirac and species indices . The measure [dW] [d fp]appears in the path integral weighted with a factor expli f d4xy(X)l,so the factor exp fi f d4x a(x).(V(x)} in the transforma- tion rule (22 .2.10) for the measure has the same effect as if the Lagrangian density Y(x) were not invariant under these transformations, but instead Y(x) --+Y(x) + a(x)s1(x). Hence when we use an effective Lagrangian with the fermions integrated out, to take account of the anomaly we mus t(22.2.b) agai nanarbitrary 364 22 Anomalie s be sure to include a non-invariant term so tha t Yeff(x) --> Yeff (x) + a(x)'C/(x) . (22.2.12) It now remains to calculate the anomaly function Qi(x) . At first sight it does not look like we can get any definite result for the anomaly . The delta-function is infinite, but the trace vanishes . To do better, we must introduce a regulator to make 64(x - x) meaningful . We can do this in a gauge-invariant manner by inserting a differential operator f(_ p2/M2)acting on the delta-function before taking its argument to zero : sl{x}=-2 [Tr{YStf(-,V2/M2)}54(x-Y) 1y~x . (22.2.13) HereDxis theDiracdifferentialoperatorinthe prese nce o fthe gauge fieldA~(x) (Dx),u_a~ - it aAa~,(x).(22 .2.14)ax Also, M is some large mass, eventually to be taken to infinity, and f (s) is a smooth function, subject only to the condition* that as s goes from 0 to oo, f(s) must drop smoothly from 1 to 0 : sf'(s) =0 a ts= U ands=00 . (22 .2.16) Note that we do not take the regulator function to be a function of ~, because we want to preserve gauge invariance, and we do not take it to be a function of DF1D,,, because we need it to regulate not only the determinant but also the fermion propagator .9T1. To evaluate the expression (22.2.13), we use the Fourier representation of the delta-function, and write the anomaly function a s sit(x) = --2 d4k f (2g)4y=x --2fK Tr tf [i k+ 'VX] 2/ A12) (27r) 4 (Aderivative with respect to x gives zero when acting on the extreme right in the second expression, but not when acting on Aa,,(x) .} Resealing the momentum k ilby a factor M, this i s .n~(x)- -2 1I4 J4d k Tr (27c)4{y5t1( [f k+.pxlMl2~~ (22.2.17) For instance, we might take f (s) - e $2 , as done originally by Fujikawa, or f (s) 1/(1 +s) . 22.2TranSformatio nof the Measure : The AbelianAnomaly 365 The argument of the cut-off function may be writte n Vx Z z i k  Dx (',X 2 i~+~ .k - M.- M (22.2.15) Eq.(22.2.17) receives contributions in the limit M oa only from terms in the expansion of f(-[i ]C+ p x/NI]Z)that have no more than four factors of 1 /111,and also only from terms that contain at least four Dirac gamma matrices, because otherwise the trace over Dirac indices vanishes . This leaves us with only the terms of second order in9x Pk 2n 4f"(kZ)Tr{y5t 94} , ( 22.2.19) which are now independent of the regulator mass M . To evaluate the integral over k, we perform the same sort of rotation of the k ocontour of integration as in evaluating Feynman diagrams, so that in effect k° is replaced with ik4, with k4 running from -oo to +oa . (This is a step that can really only be justified by working with Euclidean path integrals from the beginning .) The integral is then in effec t Jd4kf"(kZ)= i 2nZrc3 d~c f"{KZ} . (22.2.20)a By repeated integration by parts using Eq .(22.2.16)and then Eq .(22.2.15), this i s f d4kf"(k2)= ins fC* dssf"(s) _ -i Y[Z f00 ds f' '(s)=ins .(22.2.21)Ja To calculate the trace, we writ e 9x = f(Dx)"a(Dx)U! Jju, YVI+ 4 L(Dx)ga(Dx)'']17u, 7V1 = Dx - 4 its Fa"v [Yy,Yv] (22 .2.22) The only term in .9x that contributes to the trace over Dirac indices is the one involving a product of four Dirac matrices, which give s try fY5 [Y 'U'YVI[Yp"/,T ] } = 16i ElOPQ ~ X22 .2.23) where `try' denotes a trace over Dirac indices only, and as usual EUvPTis the totally antisymmetric tensor with e0123=+1. Using Eqs . (22 .2.21)- (22.2.23)in Eq . (22 .2.19)then gives the anomaly function a s (x) _-1 2 EF 4 vperF~~"'(x)F~ ~ cT ( x)tr€txt#tj, (22 .2.24)16~ with `tr' here denoting a trace only over indices labelling the various fermion species . In the special case in which t is the unit matrix, the quantity (22.2.24) is known as the Chern -Pont ryagin density . This result can be expressed in terms of the current associated with the anomalous symmetry . If we assume for simplicity that the action itself 366 22 Anomalie s is invariant under the symmetry transformation V(X) ---> Y)(x) + ity5 1Vy(x) with constant infinitesima lparameter oc, then as discussed in Section 7.3, the change in the action when we make such a transformation with a spacetime-dependent parameter a(x) may be written as H = fd4X.I5 (x)r~F~a~x}, where J5 (x)is the current that becomes conserved when the field operators satisfy the dynamical equations that make the action statio nary with respect to arbitrary variations in the fie lds. When we make the change of variab les 6V)(x) = itYSoc (x)V)(x), the change in the path integral over the fe rmionic fields i s J/[d][di5] ear = i ~ dux j[d][dfJi][d(x)x(x ) y) + J5(x)0.oc(x)]ear (22.2.25) But this is a mere change of variables, and so for arbitrary a(x) it cannot affect the path integral . Therefore for arbitrary gauge field s ~aPJ5 ~A=~ - -16 7cz EJ`°~'aFlu"F~5tr {tx t fl tj (22.2.26) where for any operator 60,(OAis the quantum average of a in a fixed background field AII(x) ~a}A = f [dip]Ldq)l eira f[d~~ [di p-1 ear(22.2.27) Incidentally, it is possible to rewrite Eq .(22.2.26) as a conservation condition . Consider the special case where tr {t xt#t} is proportional to bafl tr i [a t flt}= Nb,fl . Define a c urrentknownas the Che rn-Simons class : GA 2EE`4AY A`4YP + 3CaftyA,,, Aflj`4YP = EuvAP[AYV Fy:P-~ ~.&AavAflAA,P which satisfies the identit y r~~G2EFyFFy~p Eq.(22.2.30) allows us to rewrite Eq .(22.2.26) as the conditio n where KA(J5)A +81z2GA(22.2.30) (22.2.31) (22.2.32) 22.2 Transformation of the Measure : The Abelian Anomaly 367 However, we cannot use the conservation of the current K uto argue, as we did in the previous section, that the process no --* 2 y is suppressed, because in making that argument we assumed not only the chiral symmetry associated with axial-vector current conservation but also electromagnetic gauge invariance, and as we see from Eq .(22.2.29), the current K" though conserved is not gauge-invariant . Our derivation of the formula (22 .2.24) for the anomaly shows that if we had evaluated the anomaly function using a differential operatorf(_02/M2) in place of f(-p2/M2)in Eq .(22.2.13), we would have obtained a vanishing anomaly function . In fact, with this regulator procedure the axial-vector current is KA, not J.The trouble with this procedure, as mentioned above, is that the regulator operator is no longer gauge-invariant, resulting in the presence of a non-gauge-invariant term in KA . There is no regulator procedure for fermion propagators as well as determinants that is both gauge- and chiral-invariant . We may now return to the problem that gave anomalies their start, and use our results to calculate the actual rate of the process n° --+ 2y. The symmetry of interest here is generated by charge-neutral chiral transformations of the light quark fields : 6u = i oc75U , bd = -i ocYSd. (22 .2.33) This symmetry is anomaly-free in pure quantum chromodynamics, because theuand d belong to the same representation of the color gauge group, so that their contributions to the gluon -gluon terms in the anomaly of the symmetry (22.2.33) cancel . On the other hand, in the presence of an electromagnetic field A ft(x), this symmetry has an anomal y SJI( x)1= -16~Z E~„ pTF~`v(x)FPu(x)tr{q2T3 where qis the quark charge matrix and Ta is the diagonal 2 x 2 matrix with elements +1 for nand -1 for d . If we assume as usual that there are N,u quarks of charge 2e/3 and an equal number of d quarks of charge -e/3, the trace is ()2(+1 )trI c~2 z3}=1V +(e)2 z so the anomaly f unction here is z -4$`~Z EA„P6 F "" (x) FP" (x) . (22.2.34) We m ust theninclude in the effective Lagrangia nterms thatunder the chiraltransformation(22.2.33) g iveit thetransformation (22 .2.12), t hatis, z byeff (x)= oc~~x} _ - $`~2 EAV~,~ F~"' (x)F~'~(x)oc . (22.2.35) 368 22Anomalie s Under the transformation (22.2.33) the pion field transforms a s 6'g° = ocFr, (22.2.3b) where F,, = 184 M eVis thepion decay a mplitudeintroduced in Chap- ter 19. (Thenormalization co nventionfor th is constant is fixe dby our definitionofthe symmetry ge nerato ras 75 z3 = 275t3.}It follows t hat we must inc lude i n the effective Lagra ngianaterm n4(x),~/w N,~ Z 'FLV P,F4"(x) F"' (x) nO(x). (22.2.37)F,r 48~ F,r Comparing this w iththe ge neralformul a (22 .1.1) fo rthe effective La- grangian fo rthe decay no ~ 2y, we see t hat the constant g in Eq . (22.2.1) musthavethe value NeZ9 48~cZF,r (22 .2.38) (This shows that our previous crude order-of-magnitude estimate (22.1.3} was too large by just a factor 6 1N,.)The rate (22 .1.2) for pion decay is thus predicted to be N~12M'3 =(Nc)2 144n3F72r 3x 1.11 x 1016s-1. (22.2.39) The observed rate is r(n° --+ 2y) _(1.19± 0.08) x1016s-1, in good agreement with the theoretical result (22.2.39) if and only if N,= 3. The success of this calculation was one of the first pieces of firm evidence that there are three colors of quarks . Remarkably, as we have seen in the previous section, shortly after the discovery of the noSteinberger calculated g from a single proton- loop diagram, and obtained the result g = e2 G/32n2 MN, where G is the pseudoscalar pion-nucleon coupling constant . This result would precisely agree with Eq .(22.2.38) forN,,= 3 if we used the Goldberger-Treiman relation with gA = 1 to set G = 2rnNIF,The correct result is larger than the Steinberger result by a factor gA = 1 .56. The reason that Steinberger got nearly the right result is because the answer is determined by the triangle anomaly, which is proportional to tr {q2 T3} . For one proton, this trace has the value e2, which is the same as the quark value found above for three colors of quarks . As mentioned earlier, a more rigorous derivation of the anomaly can be given by using path integrals in Euclidean spacetime . (The use of Euclidean path integrals is briefly discussed in Appendix A of Chapter 23.)We introduce a Euclidean fourth coordinate x4 = ix° =-ixa, and correspondingly a4 = -i aa,74= iy a , and A4a = i~1~ . The space time 22.2 Transformation of theMeasure : The Abelian Anomaly 369 volume is then expressed as d4x = -i(d4x )E,where (d4x)E is the Euclidean volume element (d4x)E = dxldx2dx3dx4 .In Euclidean spacetime the fields W(x) and ip(x) must be regarded as entirely independent, with local chiral transformations defined by SAP(x) = ioc (x}tY51P(x},bfP(x)=--iatx K5(x}ty5 . The transformation of the measure is then again given by Eq . (22 .2.10), with the anomaly function !%I(x) given by Eq . (22 .2.11). Introducing a regulator function as before, this again gives the formula (22.12.13) for .!V(x) . One great advantage of the Euclidean approach is that with x4 and A4,,real, the Dirac operator i .P in Eq .(22.2.13)is Hermitian : iP=li0i + taAla] Ya a (22.2.40) with i, j, etc . here summed over the values 1, 2, 3, 4 . It therefore has orthonormal spinor eigenfunctions cp,jx}: tP(ph=Ax(pK J(x)E cpK(x)tcpK(x)= 5 ,a,',(22.2.41) (22.2.42) with real eigenvalues aL, ,. We also are assuming, as throughout this section, that t commutes with i .P, so that we can choose the c o,,so that tcpk = tKcpK . These eigenfunctions satisfy a completeness relatio n where `1' is the 4 x 4 unit matrix . Therefore the anomaly function may now be written as the limit of a manifestly convergent sum : -4(x) _ -21imm,xTr ~y5 t f(_p2/Ml) 1:rp,~~x)rpx~x) } tK(Ax/M2) (22 .2.44) (p(x)y5qc(x)) . _ -21imm-rte .~ In justthe same way t hat we derived fo rmula (22 .2.24) for the anomaly function, we can s howhere that 'PI(X)= 16?LZ E~kl Fiji Fkl fltI'{tatfit} , 22 .2.45) where e~kl is the totally antisymmetric tensor with E~Z34 = +1 . (The difference of sign in Eqs . (22 .2.24) and (22 .2.45) arises because Eq . (22 .2.45) is missing two factors of i as compared with Eq . (22 .2.24): one factor i from Eq. (22 .2.23), becaus e tryfY5[Yi,Yj11Yk, Y.i1j=16e~kl together with another factor i from the replacement of d4k with (d4k )Ein Eq. (22 .2.20).} 370 22 Anomalie s Now, given any eigenfunction cph(x) of i pand t with eigenvalue A,00, there is another normalized eigenfunction (p,-(x) with eigenvalues A,= -A,and t,, given by c0,-(X) =y5cp,(x) . (Recall that in the notation used throughout this book, Y5 is the Hermitian matrix y5= -iyzy2y3y0 = y1727374) Furthermore (p(X)QK(X))=(4_(x)pK_(x))so the terms K and K- cancel in the sum in Eq . (22 .2.44). This leaves just a sum over the eigenfunctions with AK= 0 . These are not generally paired ; rather, since Y5 anticommutes with i p, they can be chosen as simultaneous orthornormal eigenfunctions cpu, cps, of i.P with eigenvalue zero and ofy5with eigenvalues +1 and -1, respectively . ipqu =0, ipcpz;=0)Y5(pu = (Pu, ys9v = -(Po Using the fact that f (0) = 1, Eq . (2.2.44) then become s .rI(x)=-2Etu(cpu(x)(Pu(x)) - to ((Pu(x)(P"(x)} U(22.2.46) (22.2.47) Now, since the cpuand cpvare normalized as in Eq . (22 .2.42), the integral of Eq . (22 .2.47) gives (d4X)E,d(X)=-2[E tu-Etv u v with the sums over u and v running over left- and right-handed zero modes of the operator i p, respectively . In particular, for the case where t is the unit matrix, by using Eq . (22 .2.45) this may be expressed as a relation between a functional of the gauge field and the numbers of the zero modes of the Dirac operator with definite chiralities : 32~c2.~( d4x), E~klFaiiFfik?tr[tt] _ n+- n- ~ (22.2.49) where here n+ are the numbers of zero modes of Pthat have eigenvalues ±1 for y5 . This is the celebrated Atiyah-Singer index thearem .6aAmong other things, it shows that under variations in the gauge field the integral on the left-hand side of Eq . (22 .2.49) cannot change smoothly, but only by integers, so this integral can only depend on the topology of the gauge field. Its dependence on this topology will be described in Section 23 .5. 22.3 Direct Ca lculation of Ano malies: The General Cas e We saw in Section 22 .2 how to use the elegant Fujikawa approach to calculate anomalies of chiral symmetries in gauge theories like quantum 22.3 Direct Calculation of Anomalies : The General Case 371 chromodynamics, where the gauge interactions are non-chiral and fermion number is conserved . This method can also be used to deal with more general problems, but there it becomes less.7In this section we will treat the anomaly by direct calculation, as it was originally done . This will provide a useful additional perspective on the anomalies, and will incidentally allow us to discuss anomalies in general theories with very little extra trouble . To deal with the general case, we will unite all left-handed fermion fields (including antifermions, where that distinction is meaningful) in a single column Z . For instance, if T is a column containing all quark and lepton (as opposed to antiquark and antilepton) fields, the n Z= [ z(1+75)v 20 + 75*22.3.1)1. 1 = [ where 16is the matrix defined in Section 5 .4 by. ~~. 'P16-1le7ll which is needed in order that all components of x should belong to the same (1/2,0) representation of the Lorentz group . Under an infinitesimal fermion number (e.g., baryon number or baryon number minus lepton number) conserving gauge transformation : 6W = i0a[ (l + 75)tL + M - 75)tRYY (22.3.2) this column undergoes the transformatio n where6Z_ic,pax, tx0 t~ 0 a(22.3.3) X22.3.4) We shall not limit ourselves here to theories that conserve fermion number, so the 7'x will now be any Hermitian representations of the gauge algebra, not necessarily of the block-diagonal form (22 .3.4). For the moment we will consider only massless fermions, returning to the effect of fermion masses a little later . We shall be concerned here with the one-loop three-point functio n rx~Y(x,y,z) =_ (Tjjx(x),.~~(v}, ,]P(z)j) VAC a (22.3.5) where jIX is the fermionic current, calculated in terms of free fields : (22.3.6) The two Feynman diagrams of Figure 22 .1 give 372 .7~(Y) J~"(Z)22Anomalies jp(7-) i; (Y) Figure 22 .1. Two triangle diagrams for the anomaly in the current ja~(x) . Solid lines are fermions ; wavy lines indicate fictitious gauge fields, coupled to the currents . r'xflY(x,y, z )_ -iTr [ S(X- y)7'flyvPLS (y - z)' T'Yy~'PLS (2 -- x )Txy~`PL] -iTr [S(x - z )7'yyPPLS (z - y )T#y''PLS (y - x )7'xy"PL] (22.3.7) where PL is the projection operator on the left-handed fermion field s PL =(I+ yI2 and S(x) is the propagator of a massless fermion field(2.3.8) S(x)=ifd'P 2i~~ e 'p'x. (22.3.9)(27E)4F (For further comments on this use of the Feynman rules ,see the end of this section .) Collecting factors ,Eq. (22.3.7)now read s ra~~(x,Y,Z)= 12d4k1d4k2et(ki+k2)'x eik i'yeikz-zfd4p(Z7z) X tTT'1+ ~ 7v + 0_Y p + T2+~ ,~ y1 + ~'5 (P - kl + a)Z - a e(p + a)2 - i e(p + k2 +a)2- ae 2 x tr [TP7',TIXl +tr~(p- kz +b)2- iEY(p+b)2-- ie7(p+ ki+ h)Z -icy 2 x tr [ ~, T#Tx] , (22.3-10) where `tr' here denotes a trace over either Dirac or group indices, de- pending on its argument . We have introduced the arbitrary constant four-vectors a and bbecause, although the expression (22 .3.10) is con- vergent and hence independent of the labelling of the momenta carried 22.3 Direct Calculation of Anomalies : The General Case 373 by internal lines, the evaluation of 0Itra~~ involves the manipulation of divergent integrals that do depend on this labelling . We will see that the freedom to choose al` and Vcorresponds to a freedom to move the anomaly in these integrals from one current to another, but does not allow its to eliminate the anomaly altogether . In taking the divergence of Eq . (22.3.10), we use the identitie s and fin d a ra"~Y(x'Y,Z) =~1 12 C~4kId4k2e-i{kl+k2}~xeiklyeikzz OC~ 4Px~(27E)12 I xtr[Tp7'Y 7' ,:,] trIT# T pTx +tr ITY~~T'XItr ~_ 9i+0V tr +0n I(P+a)2--ie~ tr -9z+0 1(P-k2+b)2-~ -tr [TYTfl T xjtr +0yv(P+a)Z- ic 2 ~ + 9z+0 v 1 + Ys (P + k2 + a)2- iey 2 ~+ 0 1+p vY5 ie~(P +b)2- iey 2 (p+kl +b )2-iey2] (22.3.11) At this point it is convenient to separate the three-point function into terms symmetric and antisymmetric in the group indices, by writin g and where D ,,P,is the totally symmetr ic quantit y ? tC[IT aITfljTi and the coefficient of the structure constant Cxfl,1is defined b y The terms that are antisymmetric in group indices are in general non- zero, but they do not represent any breakdown of the symmetry . Just as in the derivation of the Ward identity in Section 10 .4, in a formal calculation of the divergence of the matrix element (22.3.5)we encounter contributions from the time-derivatives of the theta-functions in the time- 374 22 Anomalie s ordered product ,equal t o ax~r~~y(x, Y, Z) _ - iCa#6 64(x- Y ) (.76v (Y).7~~Z})VASformal -iCay56 4lx - ZA(jfl,(Y)j6P(Z) )VAC.(22.3.13) It isnot hard to see that the antisymmetric terms in Eq .(22.3.11)just reproduce Eq .(22.3.13).The anomaly is contained in the symmetric part of Eq .(22.3.11): [/_F(xvz)] _ ~~ 12 DIX #,, d4k1 d4kZ e i{k1+kz} -x eiky yeikz-z anom X d4P Itr0 ~v ~+ 0~P+75 - kl + a)Z - i~ (p + a)Z - i~ 2 (p -tr ~ +0p ~ + ~2+ 0 ,, ~+75 (P+a)2---icy (P+k2+Q)Z-iF72 kZ+0~ P ~ +0 v + Y5+ tr(P- kz+b)Z-iE'~ (P+b)2-ie~ 2 -tr (p + b)Z - i e~(P + ki + b)Z - ieY 2 (22 .3.14) Grouping together the first and fourth traces, and the third and second traces, this may be put in the for m a[r ;(x Y,z) = ! D u#Y dk1 Cr4k2 ek eklyeik2z Y 27i f2 J (27E) anom {yKyVyAyPl +Y5 +tr 171"yPyAy'' 1 Z ~~'5 1IK~.(b- a --~- k2,a,a+kZ)(22.3.15) where IK,Akaca d) = fd4P[fr;.a(P + k, c, d) - f x~(P,c,d)] , (22.3.16) (P + c )x(P + d ),~ .~h~~P~ c, d} °[(p+ ~)Z - i el[(p+d)2- ie](22.3.17) To evaluate these integrals consider the expansion of the function fKA(p + k,C,d)in powers of k: n=O n! The zeroth-order term fKA(p, c, d )clearly cancels in Eq .(22.3.16).All other terms in Eq .(22.3.16) are integrals of derivatives with respect to p, and 22.3Direct Calculation of Anomalies :The General Case 375 hence after Wick rotation may be written as surface integrals over a large three-sphere, say of radius P . The nth derivative of fthen yields the surface integral of a function that behaves asp-2-("-I)'while the area of the three-sphere of radius Pgoes asp3,so the only terms that can contribute for P -- +co are those with n =1 and n = 2 : 4OfhA (P,c, d) + 2k~,~y Z .f(P,v, d)(22..3.18} I(k, c, d ) _V d pONfd4pOP~a P P A straightforward calculation then give s ~,,A(k, c, d )='inZ{2k2cK+2kdA -kAd,,-k,;,ct-q,Ak'(k+c+d)].(22.3.19) We must now separately consider the terms in the traces in Eq .(22.3.15) arising from the 1 and the yS in the projection matrix Z(1 +y5 ). The terms arising from the 1 involve tr[y"y`'yY'], which is symmetric in K and A, and also in v and p, so the integrals here appear in the combinatio n . +IKa,(b - a- kz, a,a+kz) + Ia,,,(b- a - k2, a, a +k2) Using Eq .(22.3.19), it is not difficult to see that this vanishes if and only if we choose the arbitrary constant vectors so tha t a = - b. (22.3.20) Furthermore, this is a choice that avoids the non-chiral anomaly for all three currents, because in (r~ 10yv)ra~~(x, y, z ), a and bwould be replaced with a' = k2+ a and Y= -k2 +b, while in (O/Pzp)r'vp(x, y, z), a and b would be replaced with a" = kl + a and V=-kI+b,so that taking a = - balso insures that a' _ -b' and a" = -b" . We are left with the term in the trace involving75. This is totally antisymmetnc trIY"Yv YYpY51=-4ae"vAp(22.3.21) where iF"v4 is the totally antisymmetric tensor with X0123= 1. Using this in Eq .(22.3.15) with b=-a give s a[T(xvz)] - D dk d4k elz )x anom = ( 2 7 z ) 1 2a~?'I1 2 x eik4-y eik2Z it2exVZFaK ski+k2)A . (22 .3.22) We could eliminate the anomaly (22.3.22) in the current Ja (x) by taking a oc kl + k2.Although this is possible, it would not eliminate the anomaly altogether ; it would appear in (0/0 yv}ra~P(aC, y, z )or (a1a zp)ravP~aC, y, z} . The symmetry of the problem indicates that the anomaly will 6e absent in(a/ray" ) rIXsp(x, y, z) if and only if (k2+a)-(-k2 +b)oc kl, or in other 376 22 Anomalie s words a + k2 oc kl, and will be absent in (a /azp)rtt,'p(x, y, z) if and only if (-kl + a)- (kl + b)or- k2, or in other words a - k joc k2 . It is possible to choose am to satisfy any two of these conditions, so that the anomaly can be removed from any two of the currents, but for non-parallel kl and k2 it is evidently impossible to simultaneously satisfy all three conditions ---- a oc kl + k2 and a + k -,oc kl and a -- kl oc k2 . (For instance, the first two conditions would imply that a =---kl - k2, in contradiction with the third condition .) We therefore conclude that although we have some freedom in deciding which current exhibits the anomaly, the non-vanishing of Da #~j shows definitely that there is an anomaly in at least one of the currents JIX1(x), J` (y ),or JP(z) .This is one of the chief results of these calculations, and widbe exploited in the next section as a source of constraints on the matter content of gauge theories . We have seen that the evaluation of the anomalies depends on the choice we make of the shift vector al` . Unfortunately, there is no one choice that is uniformly satisfactory, so we have to choose all in accordance with the special features of the problem at hand . In one class of problems of great importance, Ja (x) is the current of a global symmetry, while J~ (y)and Jp(z) are currents of gauge symmetries, that is, currents to which gauge fields are coupled . (We dealt with such a problem in the previous section .) In such cases, we must choose all so that the anomaly is solely in J, ,I'(x), not in J v(y)or Jp (z). As we have seen, this requires that a +k2 oc kl and a - kl oc k2, which leads to the unique result that a = kl -- kZ . (22.3.23) Adopting this value of a m, the anomaly (22.3.22)is a r"P(xY,Z) = 1 D d4k d4k ei(k I+k2)x e~ Iy eik2-z1 2 [~xy4Y ano m'' r2~ 12 IX~Y1 x 47c2E"v)Pklkk2J _ 1 ~E~cvxp064(y-x)064(Z -- x) (22.3.24)4nZ ~`~ Y eykOzA We note in passing that a result like this can only arise from a theory that involves massless particles .7a Otherwise, we would expect the Fourier transform of r ""p(x, y, z )to have a power series expansion around zero momentum . The only terms in such an expansion that could possibly lead to a current divergence of the form (22 .3.24) are pseudotensors of first order in moment a [r(ki,k2)] =_Jd4yd4zeikl-ye-ik2-z[F(O,V,z) ]anom anoi n =ey''p6{A,JkIt7 + Ba# ;k2a~ 22.3 Direct Calculation ofAnomalies :The General Cas e where Aa #y and Bair are constants . According to Eq . (22 .3.24), ski + k2),, [F(k1 .k2)]anom 4~ZD~`(~~'Kv7pklKk2; so AaflY-$xpY=47E2 D"#Y377 But the symmetry of the amplitude among the three currents requires tha t AaflYkja + Bx#yk2o- = -AaYflk2a - BocYflklg = -Ayafl(-klQ - k2 er)-~~apk2 Q so, equating the coefficients of kl ,and k2 ,, Taking the difference of these equations give s Since this is proportional to D ,#y it must be totally symmetric, so Bx#,j and hence also Aafl,jare totally symmetric . But then the conditions for the symmetry of the three-point function read A = -B =B - A, which implies that A ,,flY = BIX#Y = 0, in contradiction to Eq . (22 .3.24). By the same reasoning, it is not possible to cancel the anomaly by adding a local term to the interaction whose contribution to the divergence of the current Jx cancels that given by Eq . (22 .3.24). Returning now to Eq . (22 .3.24), we can express this result in terms of the vacuum expectation value of the current JIX in the presence of gauge fields coupled to the currents J~ and J.The triangle graphs make a contribution to the current in the presence of the gauge field s (, ~x}} ❑ 12fd4y d4z r2"flY(x, y, z)A~(y)A"p(z) Using Eq . (22 .3.24), this has the anomalous divergence . p1UjaI(x))p1anom = - 2 ~x~Y~~ v~pOKA~(x)[~;,AP(x)8~(22.3.25) (22.3.26) There are additional diagrams, shown in Figure 22 .2, that also exhibit anomalies . Gauge invariance requires that the diagrams of Figure 22 .2 should add up to give the gauge-invariant resul t [(OjJa(X))] anom =1 -TILZDxflvEKvaPF13ti(x)FAp(x). (22 .3.27) As a check, consider a fermion-conserving theory, with generators 7'a of the form (22 .3.4). The constant D ,#Y in the anomaly (22.3.26)is here given by ztr[Itx atP1 tL'] atr[4tIX,t~jt~i(22.3.28) 378 22 A nomalies Figure 22 .2. One-loop diagrams for the anomaly in a current indicated by the dashed line . Solid lines are fermions ; wavy lines are gauge fields with which they interact . More specifically, in Section 22 .2 we calculated the divergence of an axial- vector current JS , with tL = -tR = t, due to gauge-field interactions with vector currents J~ and JP(called Ja and J~ in Section 22 .2) with tL - tRtfl, and likewise for t,, . Hence in this case DIX flY is replaced with~-~-trt, t oItY] = trt #, tYIt], and Eq .(22.3.27) become s 1(PPJ5 (x)}]anom = - 322 tr[lt#, t ., It] F~ (x) F~ P(x), (22.3.29) in agreement with Eq . (22.2.26). Where no gauge fields are coupled to any of the currents .I,,I',(x), JI(y), or,Tp(z), the choice of the shift vector al' is a matter of convenience . When some but not all currents are associated with symmetries that are spontaneously broken, we keep the unbroken symmetries manifest if we choose aP so that there are no anomalies in the currents corresponding to the unbroken symmetries . (This will be important in Section 22.7.) For instance, in quantum chromodynamics and similar theories, where the generators Ta of global symmetries like chiral S U(3) x SU(3) take the form (22 .3.4), all currents are either vector, with to = t~, corresponding to unbroken symmetries, or axial-vector, with t~ = -ta, corresponding to broken symmetries . We see from Eq . (22.3.28)that in this case the only triangle graphs with anomalies are those with one axial-vector and two vector currents, or three axial-vector currents . 1 n the case of one axial-vector and two vector currents, we choose al' so that there is no anomaly that interferes with the conservation of the vector currents . Thus, if ,T,,II(x) is the axial-vector current and J I(y)and Jp(z) the two vector currents, then as in Eq .(22.3.23) we must take ag = k- k, so that the anomaly is given by Eq . (22 .3.24). On the other hand, for three axial-vector currents, there is no reason to require that any one of them should be free of anomalies . Instead, it is natural to give aP a value that respects the symmetry among the three currents . Guided by Lorentz invariance, suppose we try a = akl + flk2, where a and #are constants . 2 2.3Direct Calculation of Anomalies : The General Case 379 Then symmetry would require that the momentum of eachinternal line should be p plus a times the momentum flowing out at the end of the line plus #times the momentum flowing out at the beg inning of the line. That is ,if we requ ire that a = akl + #kZ, then symmetry requires also that a- k, = -a(kl +k+2)+ #kland a - k z=akz-#(kl+ kz). These three relations are satisfied for non-parallel k land k2 if and only if a = ---#=1/3, so that a=3(kl - k2) . (22..3.30) Using this in Eq .(22.3.22) and comparing with Eq . (22.3.23), we see that the anomaly in the axial-vector current in a Feynman amplitude for three axial-vector currents is one-third what it would be for one axial-vector and two vector currents . The divergence of the current contains additional anomalies from the graphs of Figure 22 .2. Gauge invariance is no guide here, because even the triangle graph does not yield conserved currents . The total anomaly has been calculated by Bardeen for the chiral SU(3) xSU(3) symmetry of the strong interactions, with momentum shift vectors chosen so that vector currents are conserved and graphs with all axial currents are symmetric among these currents . Although this S U(3) x SU(3) symmetry (which acts on quark flavors rather than colors) is not gauged in quantum chromodynamics, it is convenient to express the anomaly as a failure of gauge invariance in a functional r[V, A] of fictitious weakly coupled gauge fields : an octet of vector fields VP(x) and an octet of axial-vector fields A~~x} . We also introduce infinitesimal gauge-transformation operators ' ~~a 6 6 6a(x)=-- (~x~ ~ ~a~~x~ -fabcVbu(x)~ ~c~~x}- fabcAby(x)6 Acu (x) (22.3-31 ) and iYaW = -a-6 - ,fabc VbM~~W 6 - .!abc AbMW 6 aOx"6Aa,u(aC) ~~,,u(x) 6 VCu(x) where fQb,are the StJ(3 )structure constants . As mentioned above, the labelling of internal line momenta is chosen so that the vector current is not anomalous : The operators V,,and yahere are what in Ref .8were called XQ and Y,. This switch in notation is made to maintain consistency with the notation of Chapter 19,where broken symmetry generators are consistently labelled X,, . 380 22 Anomalie s but then an anomaly appears in the axial-vector current s i ~a r[V'A]~ 32~Ze~lp~Tr~.a [~~v spa + 3A~ .vApa - ZA,aAvApA Q + t (A A vVpa + A~uVp,7Av + VpdAuAv ) ~ where A,,,are the SU(3) matrices given by Eq .(19.7.2),and Vp= z AaVjea Am= 2tia Apa VIN=ONVv-OvVu- 1[Vu,Vv~ - Z [AuiAv!(22.3.34) (22.3.35) (22.3.37) The factor 1/3 accompanying the second term on the right-hand side in Eq. (22 .3.34) has already been explained as a consequence of the different choice of aP in the AV V and AAA graphs . In Section 22 .6 we will describe consistency conditions that allow the cubic and quartic terms in Eq. (22 .3.34) to be calculated from the quadratic terms . For anomalies involving symmetries that are all spontaneously broken, there is no reason to choose al' in a way that distinguishes among the different currents, and instead it is natural to label internal fermion lines in a way that is symmetric among the attached gauge boson lines . As we have seen, this means that the triangle graph is to be calculated with the choice a = 3 (kl -- k2).This gives a triangle anomaly one-third of the value given by (22.3.26). When square and pentagon graphs are included, this result becomes 8a 1 [(D~JL~~anom- -24~~e'~v7pTr T, [0,A, r~a.AP-~ir~-AvA;~Ap + ziA,;OvA,~Ap~iA,~A,,r~;~ApI (22 .3.38) where Au = A PT, We will not derive this result here, because we have already seen that in such cases the terms quadratic in A.are given by one-third the corresponding terms in Eq .(22.3.26), and in Section 22 .6 we shall be able to use consistency conditions to derive the remaining terms in Eq .(22.3.38) from the quadratic terms . Now we must consider possible corrections to these results . A careful derivation of the anomaly to all orders of perturbation theory was given by Adler and Bardeen ; what follows is intended to give only the gist of their analysis . The argument that led to Eq . (22.3.22)may be repeated in any order of perturbation theory, and shows that the anomaly in general arises from 22.3 D irect Cal culationof'Anomal ies:The General Case 38 1 momentum-space integrals that can be reexpressed as surface terms . In consequence, as we saw here in Eq .(22.3.18), the only graphs for the current divergence that contribute to the anomaly are those for which the integral over the momentum circulating in the fermion loop has dimensionality (in powers of momentum) zero or greater . Interactions of the fermions in the loop with virtual gauge bosons would reduce the dimensionality of the integral over the fermion loop momentum enough to eliminate the anomaly, so the anomaly receives no contribution from such radiative corrections . (It is true that the integral over the momenta of the virtual gauge bosons as well as the fermion loop would have non-negative dimensionality, but the gauge boson propagators can be regulated, as the fermion propagator cannot, without interfering with the chiral symmetry in question .) The anomaly is affected by interactions of the gauge bosons attached to the fermion loop with other gauge bosons and fermion loops, but these just serve to renormalize operators like E "O Fn(x)~'~P(x). By the same argument, any fermian mass that respects the symmetries in question (if this were possible) would not change the anomaly, because extracting factors of this mass would lower the dimensionality of the momentum-space integral . The last remark of the previous paragraph raises the question of whether we can calculate the anomaly without knowing all the fermians in a theory, heavy fermians as well as light or massless ones . Yes, we can ; we shall now show that no f ermion that is allowed by a given symmetry to have a mass can contribute to the anomaly for that symmetry .In the general class of theories considered here, a mass term in the Lagrangian density would take the form mass = - 1:ZrrnErrrr' Mn n1Zrr1n,+ H .C. , (22.3.39) nmuu ' where d is the two-component spinor index of the (~, Q)representation of the Lorentz group, eQ ,,is the antisymmetric matrix with 617'-l needed for Lorentz invariance, and M is a symmetric mass matrix .** Now , in order for Ymass to respect gauge invariance, the mass matrix mus t In fermion-number-conserving theories where xhas the form (22.3.1),Mis related to the usual mass matrix m by M~ 1 0m2MT0 With all inversion phases equal to unity, parity, charge conjugation, and time-reversal invariance respectively have the further consequences that m is Hermitian, symmetric, or real . 382 satisfy22 Anomalie s T,TM=MT'. (22.3.4) The index n may be replaced with an index r labelling different irreducible representations of the gauge group, and an index s labelling components within each irreducible representation, so tha t `TOrs,r's' =5rr'(TX1r})s,s' and we writ e Eq. (22 .3.40) then becomes(22.3.41) (22.3.42) (22.3.43) (This is not summed over r or r' .) Schur's lemma10 tells us that whenever the matrices of a pair of irreducible representations are related by such an equation, the matrix that relates them is either zero or non-singular . (See Section 5.5.)Thus, either (i)M(r,r') =0,or else (ii) -T,(r)z and T,( r) are related by a similarity transformation, and likewise for T# and Ty. In the latter case the contributions to the anomaly constant (22 .3.12) from fermions belonging to the individual irreducible representations r and r' are related by D(r) _-D (r) , (22.3.4.4)'fly xpY so the anomaly either vanishes (for r =r') or cancels between the two representations (for r :~r').The anomalies in a given set of symmetries therefore are unaffected by the possible presence of fermians with a mass allowed by these symmetries . There is a fine point in the use of the Feynman rules to calculate the three-point function (22.3.7),to which we now return . In using the conventional fermion propagator (22.3.9), we have in effect doubled the number of fermion fields ; in additional to the purely left-handed fields &} given by Eq .(22.3.1),the propagator (22.3.9)includes right-handed modes (unrelated to the fields (l. -75)tpof fermion-number-conserving theories) that do not interact with the gauge fields . Combining these non-interacting right-handed fields with the interacting left-handed fields in a single spinor T, the fermian Lagrangian density is -TPT, where P is now 1+ ys,~_ ~ --- i¢ raTa 2 (22 .3.45) 22.4 Anomaly-Free Gauge Theories 383 The one-loop vacuum functional for spin Z fields interacting with real or fictitious gauge fields is just Det P. The failure of gauge invariance in this determinant can be blamed on the fact that the operator (22 .3.45) is the sum of two terms P = 0 (1 - 2Y5 + P(1+7 52 (22 .3.46) of which only the second is gauge-invariant . There is another way of looking at anamalies,101 which provides a framework for calculations of a large class of anomalies, of coordinate as well as gauge symmetries in spaces of any dimensionality . Instead of working with the operator (22 .3.45), which has a well-defined determinant, one deals instead with just the second term in Eq . (22 .3.46) ~+ ys PL =P2 (22 .3.47) This is perfectly gauge-invariant, but does not have a well-defined deter- minant, because this operator does not map the space of fermion fields of one handedness into itself, but rather into the space of fermian fields of the other handedness . One can try to define a gauge-invariant vac- uum functional Det PLby writing differential equations for Det PL in the space of gauge fields, modulo gauge transformations, but then one may encounter obstructions . There are local obstructions due to violations of the necessary integrability conditions, which correspond to the anomalies we have been discussing . Even where these local obstructions are absent, in cases where the infinite-dimensional space of gauge-field configurations, modulo gauge transformations, is not simply connected, topological ob- structions can prevent the definition of single-valued functionals on this space . Such a global anomaly was found by Witten'0b for the gauge group SU(2) (which is free of local anomalies) with an odd number of massless left-handed fermions in SU(2) doublets . 22.4 Anomaly-Free Gauge Theorie s We have calculated the effect of anomalies on the conservation of a general current R. Where this current is itself coupled to a gauge field, gauge invariance requires that the anomaly be absent . We saw in the previous section that the anomaly is proportional to the completely symmetric constant factor Da#, defined by Eq .(22.3.12), so for gauge currents we must havell 384 22 Anomalie s where Ta is the representation of the gauge algebra on the set of all left- handed fermion and antifermion fields, and `tr' again denotes a sum over these ferrnion and antifermian species . This condition may be satisfied for any gauge group if the fermian fields furnish a suitable reducible or irreducible representation of the group . In addition, there are some gauge groups for which Eq . (22 .4.1) is satisfied for fermions in any representation of the group . 12 (The Batalin-Vilkovisky formalism is used in Section 22 .6 to give a purely algebraic proof that for such gauge groups the anomaly is absent to all orders of perturbation theory .) The condition (22 .4.1 ) is obviously satisfied if the left-handed fermion (and antifermian) fields furnish a representation of the gauge algebra that is equivalent to its complex conjugate, in the sense tha t OTa)#= S (iTa)S-1 or equ ivalently (since we a lways take Ta Hermitian ) Ta = -STa5-1 . (22 .4.2) (Inserting Eq.(22.4.2) in Eq .(22.3.12) gives Dx #y = -Da #.,.)Such a representation may be either real, in which case it is possible by a similarity transformation 7'a = R 7' xR-1 to convert the representation to a form in which T,,' is imaginary and antisymmetric, or pseudoreal, in which case this is impossible . (For instance, the three-dimensional irreducible representation of SU(2) is real, while the two-dimensional representation is pseudoreal .) There is no anomaly for gauge algebras that have only real or pseudoreal representations, namely13 S O(2n+1) (including SU(2) SO(3 )}, S0 (4n)for n ~ 2, USp(2n) for n >_3,G2,F4,E7, and fig, and all of their direct sums . A few other algebras also have only representations for which Da#,1vanishes, even though some representations are neither real nor pseudareal .l2 These are SO(4n +2) (except for 50 2) = U(1) and SO(6) =SU(4)) and E6, and their direct sums with each other and the above algebras . Anomalies are thus only possible for gauge algebras that include SU(n) (for n ~ 3)orU(1) factors . As it happens, these are among the most important gauge algebras in today's physics . The standard model is based on the gauge group SU(3) xSU(2) xU(1), so we must rely on cancellations among the quarks and leptons to make the theory free of anomalies . Table 22 .1 gives a classification of the left-handed spinor fields of the first generation of the standard model according to the representations they furnish of the color S U(3) group and the electroweak SU(2) group, and the value of the U(1) quantum number y/g' = t3/g - q1e . We can now check whether Dx p-,vanishes when T,,, T#, and T ., run over all of the generators of S U (3) x S U (2)xU(1) .We need only consider those combinations of generators for which the product of T,7'p, and T,1 22.4 Anomaly-Free Gauge Theories 385 Table 22 .1. First-generation left-handed fermion and antifermion fields of the standard model . Fermions Ud )L uR d~ ve e eRS U(3) 3 3 3 t L 1S U(2) 2 1 1 2 IUMLY1g'] -1/6 +2/3 -1/3 1/2 -1 is neutral under S U(3) x S U(2) x U(1), since D ,,#y obviously vanishes for all the others . We can make invariants out of either zero, two, or three SU(3) generators (because 8 x 8and 8x8x8both contain singlets), zero, two, or three S U(2 )generators, and any number of U(I) generators, so we only need to check the following cases : SU(3 )-SU (3)-SU(3 ):Here Daffy vanishes because the left-handed fermions furnish a representation 3 + 3 + 3 + 3 + I+I+IofSU(3), which is real . SU(3)-SU(3)-U(1) :Here the anomaly is proportional t o 1 1 2 Ylg6-6+3-3=0 . 33 SU(2 )-S U(2 )-SU(2 ): There is noanomaly here because SU(2) only has real or pseudoreal representations . SU(2) --SU(2) -U(1) :Here the anomaly is proportional t o YIgt =3 --~+~ =0 . doublets 386 22 Anomalie s U(1)-U( 1)-U( 1):Here the anomaly is proportional t o 13 X:(Y1g')3= 66~ 3 1313 3 3 2 We see that all anomalies cancel for the gauge symmetries of the standard made1 .13a This result can be neatly understood by noting that S U(3)x S U (2) x U(1) may be embedded in SOS 10) .14 All of the repre- sentations of SO(10) are anomaly-free, so the same property is inherited by any reducible representation of SU(3) xSU(2) xU(1) that furnishes a complete representation of SO (10). As it happens, the left-handed fields of a single generation of quarks, leptons, antiquarks, and antileptons plus one additional (SU(3) x SU(2) x U(1))-singlet forms a complete 16-dimensional representation of SO (10) (the fundamental spinor representation), so for this set of left-handed fermians there are no S U (3)x S U (2)xU(1) anoma- lies. The singlet would not contribute to such anomalies anyway, so there are no anomalies in the gauge symmetries of the standard model . There is one more anomaly that needs to be evaluated . All species of fermians interact with gravitation in the same way . Calculation of the fermian loop graph for the expectation value of the current xTy9x (where x is a column of all left-handed fermion and antifermion fields) in the presence of an external gravitational field yields an anamaly15 au(xTyYx} proportional to trf TJ'F'"'P'T Rey K;,R,6 " In particular, to avoid a gravitational violation of a gauge symmetry like (22.3.3),the generators must satisf y trJT,J = D . (22 .4.3) Like the pure gauge anomaly, this vanishes for gauge generators satisfying Eq. (22 .4.2), so this anomaly receives no contribution from fermions that form real or pseudoreal representations of the gauge group, and it can therefore be calculated taking into account only those fermians whose masses arise from a breaking of the gauge symmetry . Also, this condition is automatically satisfied by the generators of any simple subalgebras like SU(2) orSU(3) of the gauge algebra . (trfT,,,J is just a number which commutes with all the Tp, so if it is non-zero then the algebra is not simple .) Hence we only need to check that Eq . (22 .4.3) is satisfied by the U(1) generators of the gauge algebra . In the standard model, the sum of the values of the weak hypercharge y for all left-handed fermions i s ylgr=6(-I) +3(2)+3(-l) +2(l) +(-l)=056 3 3 2 so there are no gravitational anomalies in the standard model currents . 22.4 Anomaly-Free Gauge Theories 387 The requirement of vanishing of anomalies may be used as a guide in formulating realistic theories . For instance, the values of the weak hy- percharges yfar various SU(3) xSU(2) multiplets were originally taken from experiment, but one may wonder why these weak hypercharges (and the corresponding average electric charges in each multiplet) take the ob- served values . To answer this question, suppose we assign arbitrary weak hypercharges a,b, c, d,and e to the multiplets (uL, dL), uR, dR, (VL, EL ), and eR, respectively . The conditions for anomaly cancellation tell us that : SU(3)-SU(3)-U(1) : Ty=2Q+b+c- o 3,3 SU(2)-SU(2)-U(1) : y=3a +d=Q ; doublet s U(1)-U(1)-U(1): y3 = 6U3 + 3b3+3C3 + 2d3+e3 = D ; gravitongravito nU(1) : Ey = 6a+ 3 b+ 3c+2d+ e=0. Aside from the possibility of interchanging UR and dR, these equations have only two solutions, which we may call U(1) and U(1) ' U(1) U(1)'b/a = -4, c/a = 2, d/a = -3 , eta = 6 , b= -c, a =c=d=e= 0. Furthermore, these solutions are exclusive ; we cannot suppose that both U(].)and U(1)' are local symmetries, because then we would encounter a U (]. )'-U (]. )'-U (1)anomaly proportional to (-4) + (+2) :~ 0 and a U(1)'-U(1)-U(1) anomaly proportional to (-4)2 - (+2)2 :~D. The U(1) generator is just the weak hypercharge of the standard electroweak theory (with the overall constant factor a absorbed into the definition of g'), while the U(1)' symmetry resembles nothing observed in nature . This little calculation provides a rational explanation of the assignment of y values, or equivalently of electric charges, in the standard model, and it shows that if all gauge anomalies must cancel within a single generation of quarks and leptons, then it is not possible to couple a gauge boson to any other U (1) quantum number, in addition to weak hypercharge . On the other hand, although it is reasonable to guess that the SU(3 )x SU(2) x U(].)gauge bosons of the standard model may couple only to the known quarks and leptons (both to explain why no other fermions have 388 22 Anomalie s been discovered and to preserve the beautiful cancellation of anomalies in the standard model), there may be other U(1)' gauge bosons that couple to other undetected (S U(3)x S U (2)x U(1))-neutral fermions as well as to the known quarks and leptons . Suppose we denote the U(1)' quantum numbers y' of the multiplets (uL, dL ), uR, dx, (vL, e L), and eR as a', b', c',d, and e', respectively . Since we don't know anything about possible (SU(3)xS U(2) x U(1))-neutral fermians, the requirement of cancellation of the U(1)'-U(1)'-U(1)' and graviton graviton-U(1)' anomalies does not help us to constrain a', h', c', d', or e' . The remaining conditions of anomaly cancellation tell us that : 3,3 doublet s UM -UMI-001 : yyi2 = 6a "+ 3(-4 )hi2 + 3(2 )c'2+ 2(-3)d'2 + (6)e'2 = 0 . The general solution has y' a linear combination of y and a quantum number B - L (where B and L are the conventional baryon and lepton numbers) that takes values 1/3, -1/3, -1/3, -1, and +1for the multiplets (UL,dL), uR, dR, K, ed, and eR, respectively . If B - L is a local symmetry, with a coupling that is not many orders of magnitude smaller than e, then it must be spontaneously broken, since ordinary bodies have macroscopic values of B-L. To avoid conflict with observations of neutral currents, the characteristic scale Fof the symmetry breaking would have to be larger than that of the electroweak interactions, but not necessarily many orders of magnitude larger . Thus a neutral vector boson somewhat heavier than the ZO and coupled to B - L seems like the most plausible addition to the standard model . This has all been for a single generation of the standard model . With three generations there are many more anomaly-free symmetries . One class of symmetries that are not broken by anomalies or (as far as we know) by anything else consists of the differences in the numbers of leptons of the various flavors . Together with B - L, this will be important in Section 23 .5 in classifying the baryon- and lepton-non-conserving processes produced by anomalies . 22.5 Massless Bound States 38 9 22.5 Mass less Bound States * It is sometimes speculated that quarks and leptons may be bound states of more fundamental particles .Ifthese hypothetical fundamental par- ticles had asymptotically free gauge interactions, like those in quantum chromodynamics, then we would expect them to be trapped, which would explain why they are not observed . However, there is a difficulty with this picture . Na internal structure of the quarks and leptons has ever been observed, so the characteristic energy scale A' (analogous to the A: 200 MeV of quantum chromodynamics) of these gauge interactions must be very high . For instance, as remarked in Section 12 .3, the agree- ment between theory and experiment for the magnetic moment of the muon indicates that A' >3 TeV . But, aside from Goldstone bosons, we would normally expect the bound states in such a theory to have masses of order A', or perhaps 2nA', just as in quantum chromodynamics the proton mass is of order 2nAQCD . This expectation is of course in sharp contradiction with the fact the observed masses of the quarks and leptons are much less than A' . To put this problem another way, if the leptons and quarks are bound states, then why is their size (as measured by anomalous magnetic moments, etc .) so much smaller than their Compton wavelengths ? One way to answer this question is to suppose that, unlike quantum chromodynamics, this theory has unbroken chiral symmetries that keep the quarks and leptons massless, aside from small corrections from other interactions . In general, a chiral symmetry is any symmetry for which the massless elementary fields of some given helicity (including the complex conjugates of the fields of opposite helicity) furnish a complex represen- tation . By operating on the vacuum with products of the elementary fields we can construct other states of definite helicity that also furnish complex representations of these symmetries . If any of these states are actual composite particles then they must be massless, because all helic- ity components of the state of a massive particle must furnish the same representation of any symmetry that commutes with rotations, so a given helicity component of a particle together with the antiparticle of the op- posite helicity component would together furnish a real representation . Of course, it may not be so easy to tell which of the states constructed in this way correspond to actual composite massless particles, but if they do, their masslessness is natural, because in order for some massless particles of a given helicity that belong to a complex representation of the symmetr y This section lies somewhat out of the book's main line of development, and may be omitted in a first reading . 390 22 Anomalie s group to become massive as we change some parameter of the theory, their symmetry properties would have to change discontinuously from a complex to a real representation . Although this reasoning shows that there are theories in which it is natural to have massless or very light composite particles, it does not give any indication of when this actually happens . The question is an interesting one, quite apart from the problem of understanding quarks and leptons as possible composite particles . 't Hoaft 16 has suggested a powerful way of answering this question, based on considerations of anomalies . Briefly, if the underlying theory has global chiral symmetries (not broken by gauge anomalies, and not spontaneously broken) consisting of transformations on the elementary left-handed spin z fermions (and antifermions) x with symmetry generators Ta,Tfl, etc., and if the anomaly constant tr[I T,,, Tfl I Ty] of these global symmetries is non-zero, then the spectrum of bound states must include spin z massless particles on whose left-handed states the same symmetries induce transformations with generators I a, 17-#, etc ., with the same anomaly constan t tr[{9_,,JfljJj= tr[ ITa,Tfl I T y]. (22.5.1) 't Hooft's argument was as follows . Imagine that some weakly- interacting gauge bosons are coupled to generators T , Tp , etc . of the global symmetries of the underlying theory . Suppose also that although some of the coefficients D,#,, - tr[t Ta, T~I Ty] are non-zero, this anomaly is cancelled by anomalies due to other "spectator' massless fermions, which do not feel the strong forces that trap the constituents of the composite particles . Physical processes at energies much less than the characteristic energy scale A' of the trapping interactions will be described by an effective Lagrangian in which the trapped fermians do not appear . If the symme- tries with generators Tx, T #, etc. are not spontaneously broken, then there are no Goldstone bosons, so the only particles in this effective Lagrangian will be the weakly coupled gauge bosons and spectator fermians, plus any massless bound states of the trapped ferrnions and strongly interacting gauge bosons . The consistency of the effective field theory requires that it must be anomaly-free, but the spectator fermions by assumption have an anomaly constant equal to -D ,#1, so there must be massless bound states, to provide an anomaly constant equal and opposite to this, and hence equal to that of the original trapped fermians ."* Note that this argumen t 1 t remains to show that these particles must have spin ? . We are assuming that there are no Goldstone bosons here, and that other spinless particles would not naturally be massless . The elementary gauge bosons of the theory are assumed neutral under the anomalous symmetry transformations, so they could not contribute to the anomaly . Composite particles of spin j ~ Iare ruled out by a different argumcnt .ls° This is 22.5Massless Bound States 39 1 works no matter how weak are the gauge interactions w ith generators T , T#, etc., so these gauge bosons and the untrapped fermions do not have to be real particles to reach the conclusion that the massless spin 1 bound states reproduce theanomalies of the trapped elementary spin 11ferrnions of whichtheyarecompo sed. As a s imple example, suppose that the underly ing theory contains n`fla- vors' of massless fermians, each of wh ich has both left- and right-handed parts in the defin ing representation N of an asymptotically free S U( IV) gauge gr oup.We require Nto be odd so that there can be untrapped SU(N)-neutral ferm iaric bound states .Just asin quantum chromodynam- ics,this theory has an automatic global S U L(n)x S UR(n )xUv(1)symmetry, with the left- and right-handed massless fermion srespecti velyin its (n ,1) and(1,n)representat ions, both hav ing equal values for the Uy(1)quantum number, wh ich can be taken as un ity. There are non-vanishing anomaly constants in the underlying theory fortheS U(rt)L-,SU(ra)L-U(1)Yand S U(n )R-S U(n)R-U(1)Ycurrent t riplets ,which have the value s DaL,hL ,Q = DuR,hR ,Q=N6 ab, where a, b, etc . label the 5U(n) generators Au, normalized so that in the defining n-component representation tr{A,,Ah} =16,,h.For n >2 there are also non-vanishing anomaly constants for the SU(n)L-S U(n )L-S U(n)L and SU(n)R-S U(n)R-S U(n)R currents, equal t o DaL,bL,cL =DaR,bR, cR = Ntr[{ava, Agl,Ad . We suppose here that the SUL(n) x SUR(n) xU(1) symmetry is not spontaneously broken . Because of trapping, the only fermionic bound states in the physical spectrum will be contained in representations of this symmetry that can be formed from mL and MR elementary fermions of helicity + Z and - ~, respectively, and mL and mR of their antiparticles, with mL+mR - mL -- mR = kN , (22.5.2) where k is any positive or negative odd integer . In consequence, the only irreducible representations (r, s)of SU(n)L x S U R(n) encountered are those for which r is in the direct product of mL of the defining representations o f a theory in which the anomalous currents can be constructed as Lorentz four-vector functions of the elementary spin 12 fields, and in order to contribute to the anomaly these currents would have to have non-vanishing matrix elements between any massless composite particles of spin j = 1, which would ❑ivtate Lorentz invariance . Massless composite particles of spin j>3/2 are ruled out because in this theory it is also possible to construct a conserved energy-momentum tensor, which would have to have non-vanishing matrix elements between these massless composite particles, which for j Z 3 /2 would also violate Lorentz invariance . 392 22Anomalies SU(n) and rraL❑f their complex conjugates, while s is in the direct product ofrrtRof the defining representations of 5U(n) and mR of their complex conjugates, and the U(1)y quantum number is k117, with k, ML, mR, mL, and mR subject to Eq . (22.5.2). Let p( r, s, k)be the number of times an irreducible representation (r,s) of SUL(n) X SUR(n) with iT(1)vquantum number kN appears among the helicity + 2 bound states . Eq .(22,5.1) then reads p(r, s, k )ds tr(r) [ f.T,,, = Ntr[fAa,AbI Ael , (2 .5.3) r,s,k Ar, s, k)dsk tr(r)If J_a,J_bJ1= tr[{ Aa,Abl] (22.5.4) r,s,k where tr(r) denotes the trace in the irreducible representation r ❑f SU(n), and ds is the dimensionality of representation s of S U(N) .The only other constraint ❑n the p( r, s,k)is that they must be positive integers . The complex conjugate (r, s, -k )of any representation (r ,s, k) of SU(2)L xSU(2)R x U(1)vhas values of the traces tr ('')[{-Ta,9big C] and k tr(r) [{~ ~, ~b~] opposite to those of the representation (r, s, k), so Eqs. (2..5.3) and (22 .5.4) only restrict the values o f Recall that in the notation we have been using here, these traces are over all massless bound states of helicity + ', including the antiparticles of the massless bound states of helicity - ~, which transform according to the complex conjugate representations . The complex conjugate of any representation ❑f S U (2)LxSU(2)R X U (1)Vhas values of tr(r) [{ 37-a, TnIg-C ] and ktr(r) [{opposite to those of that representation . Thus we can sum in (22.5.3)and (22 .5.4) only over the representations with U(1 )v quantum number kN >0 Et(r, s,k)dstr(r)[{9-a,9-h~-Tel =Ntr[{)~a,Ab!)J , (22 .5.5) r,s,k>Q /'(r, s, k)dsktr(r)[{J a, J b l]=tr[{Aa>AbJ1 (22.5.6) r,s,k>0 with e(r, s,k)equal to the number of times an irreducible representation (r, s)of S UL (n)XSUR(n) with U(1 )y quantum number kN>0appears among the helicity + ~ bound states, minus the number of times the same representation appears among the helicity - ~ bound states .(If parity is unbroken then a representation r, s, k must occur among the helicity - Z bound states as often as the representation s, r, k occurs among the helicity + 2 bound states, so in this case t(r, s,k) = -e(s, r,k).) First, consider the case of n = 2 flavors . There is no way ❑f coupling three three-vectors together symmetrically to make an ,S U(2)-invariant, 22.5Massless Bound S tates 3 93 so both sides of (22.5.5)vanish automatically, leaving only the condition (22.5.6).The defining two-component representation of SU(2) is contained in the product of any odd number of these two-component representations, so we can always find a solution ❑f(22.5.6)with e(r,s,k) = 0 for all SU(2)L xSU(2)R x U (I)Vrepresentations with r non-trivial except the one in which r is the defining representation, s is the trivial representation, and k = 1, for which we take Z'= 1. Unfortunately, this solution is far from unique -there is an infinite number of ways of reproducing the anomalies of the underlying theory . For general n and Nit is usual to find many solutions of Eqs . (22.5.5) and ( 22.5.6), but there are some cases where no solutions can be found . In such theories we can reach the conclusion that the S U (n)L x 5 U (n)RX U(1 }y symmetry must be partly or completely spontaneously broken . This has particularly interesting implications for quantum chromodynamics, so let us now specialize to the case of an SU(3 )gauge group . To be concrete, we will focus on the representations, with k = 1 and mL = mR = 0, that can be formed from just three elementary fermions and no antifermions . These are : (a) r is the symmetric third-rank 5U(n) tensor ; s is the trivial representa- tion. (b) r is the antisymmetric third-rank 5U(n) tensor ; s is the trivial repre- sentation . (c) r is the third-rank SU(n) tensor of mixed symmetry ; s is the trivia l representation . (d)ris the symmetric second-rank S U(n) tensor ; s is the S U(n) vector . (e) r is the antisymmetric second-rank S U(n) tensor ; s is the S U(n) vector . (f) r is the SU(n) vector ; s is the symmetric second-rank SU(n) tensor . (g) r is the 5 U(n) vector ; s is the antisymmetric second-rank S U(n) tensor . (h) r is the trivial representation ; s is the symmetric third-rank 5U(n) tensor . (i) r is the trivial representation ; s is the antisymmetric third-rank SU(n ) tensor . {j) r is the trivial representation ; s is the third-rank S U(n) tensor ❑f mixed symmetry . fi All SU(N) vectors and tensors here are understood to be contravariant . 394 22 Anomalie s For n> 2, Eqs. (22 .5.5)and (22.5.6)here read ff and(n +3)fin +6Yu + ~ (n -3)(n -6)eb +(n2 -g}~C + n(n+4)ed +n(n - 4)e,+ ~ n2(n+1)ef + ~ n2(n - 1 )fig= 3 (22.5.7) (n+2)fin +3)ea + ~ (n - z)(n -3)~h +(n2-3)Z'c +n(n+2)Zd +n(n - 2 )/'e+ ~ non+1)~ f + ~ non - 1 )fig = 1. (22.5.8) There is no problem in satisfying Eq .(22.5.7),but note that if n is a multiple of three then for all values of the es, each term on the left- hand side of Eq .(22.5.8)is also a multiple of three, which makes it impossible to satisfy this condition . We conclude in particular that the S U(3)LXS U(3)RXUG}ysymmetry of quantum chromodynamics with three flavors of massless quarks must bespontaneously broken . This result is not limited to the representations (a)-(j), but applies to any representations ofSU(3)L xSU(3)R XU(1} Vthat can be formed from color-neutral combinations of quarks and antiquarks . Aside from special cases such as an S U(3) gauge group with n = 3 elementary fermion SU(3)-triplets, the 't Hooft anomaly matching condi- tion is not very restrictive ; in general it allows a wide variety of massless bound states when chiral symmetry is unbroken . 't Hooft also proposed adecoupling condition, which requires that when one or more of the ele- mentary fermion flavors become very heavy there should be no unbroken chiral symmetries that prevent composite particles that contain heavy el- ementary fermions from acquiring masses . For instance, in the case ❑f an SU(3) color gauge group, if we give one of the n quark types a large mass then those three-fermion bound states that contain a single massive quark will furnish representations (r',s')of the group S Urn -1)x SU(n-1)that are eithe r (v) r' is the symmetric second rank SU(n - 1)tensor ; s' is the trivial representation , (w) r' is the antisymmetric second rank SU(n - 1)tensor ; s' is the trivial representation , (x) r'and s 'are S U(n) vectors , It in R ef 16, 't Hooftassumed that parity is no tspontaneous ly broken, so he gave t hese formula s fo r th e cas ewhere ea = -e h, ea = - fir,t, = -ej, ea --If,and ee--fig . As we see here, t hemain c onclusion does not depen d on p arity conservation. 22.5Massless Bound States 395 (y)r' is the trivial representat ion,s' is the symmetric second rank SU(n-1 ) tensor , (z) r' is the trivial representation, s' is the antisymmetric second rank S U(n - 1)tensor . In order for the three-fermion mass state to acquire a large mass, it is necessary that P(r',s')= 0, where Z'(r',s')is the number of times that an irreducible representation fir', s'} occurs among the helicity + ~ bound states minus the number of times the same representation appears among the helicity - 2 bound states . By inspecting the list of three-fermion representations (a)-O} of 5U(n) x 5 U(n) to see what representations of .5 Urn -1)x 5 U(n - 1)they yield, we see then that the 't Hooft decoupling condition requires that 0_e'v=(a+ ec+ed 0_ew= 6 + ec + fe 0_tx tf + tg +td + te (22.5.9) Unfortunately, in most cases there are still an infinite number of solutions, though there are no solutions in which the es are n-independent integers . The decoupling condition seems quite plausible, but its use by 't Hooft was questioned17on the ground that, as one or more fermion masses increase, one usually encounters phase transitions that make the mass spectrum different from what it would be for small fermion masses . There is a stronger condition, known as the persistent mass condition, which requires that when one or more ❑f the elementary fermion flavors acquire any mass, there should be no unbroken chiral symmetries that prevent composite particles that contain these massive elementary fermions from acquiring some masses ." If valid, the persistent mass condition would lead to the same consequences, such as Eqs .(22.5.9),that were described by 't Hooft, with no chance of being invalidated by phase transitions . It is easy to construct non-realistic models in which the persistent mass condition is violated, such as theories with spontaneously broken non-chiral symmetries, which lead to massless Goldstone bosons formed as composites of massive fermions ." (These models also exhibit phase transitions with increasing fermion masses, which invalidate 't Hooft's conclusions from the decoupling condition .) But the work of Vafa an d t In the parity conserving case studied by 't Hooft, the fourth and fifth equations are identical to the first and second, while the third is empty . 396 22 Anomalie s Witten discussed in Section 19 .9 shows that, in a variety of more realistic QCD-like theories, non-chiral symmetries cannot be spontaneously bro- ken, so this should not be taken as a serious objection to the persistent mass condition . 22.6 Consistency Condition s The numerical coefficient appearing in the anomaly for any symmetry depends on the matter content of the theory . On the other hand, the form ❑f the anomaly is largely independent of the details of the theory, because it is governed by consistency conditions, first presented in 1971 by Wess and Zumino .18 Even when we are interested in anomalies in global symmetry currents, in order to derive the consistency conditions it is convenient to imagine that all symmetry currents are coupled to gauge fields, which for non- Abelian symmetries also couple to each other, in such a way that these symmetries become local symmetries of the Lagrangian density . We can always return to the case of global symmetry by letting the corresponding gauge coupling constants become infinitesimal . Apart from anomalies, the effective action I'[A ]in a background gauge field Aa,,(x) will in this formalism be invariant under infinitesimal transformations Apu(y)-* A#,,(y)+ i f d4X fa(x) T,(x)A&(y) on the gauge field, where in order to reproduce the transformation (15.1.9)we must tak e iJI-'(x) =-a. ~ - G~flvAfl~(x)6(22.6.1)Ox u Taking anomalies into account, 'Y-a(x)no longer annihilates F[A], but rather Ta(x)T[A] = Ga [x ; Al , (22.6.2) where G . [x; A] represents the effect of the anomaly . Eq.(22.6.2)may also be written as a formula for the covariant divergence of the expectation value ❑f the current : Du (.Ta (x)) =-iGx [x;A], (22 .6.3) where 6a,(X) and Duis the gauge-covar iant derivative (15. 1.10), taken here in the ad joint representat ion w ith (t p)ya = -iC xfly. 22.6 Consistency Conditions 39 7 The Wess-Zumino consistency conditions follow from the commutation relations [J a(x)' g-fl(Y)]= iC,,flY64(x- y)~7-y(x). (22.6.5) From Eqs .(22.6.2)and(22.6.3),we derive the general consistency conditio n ~y-a(x)G# [y; A]--17-#(x)Ga[y;A]= aC a# Y6 4(x-y)G,~ [y ; A].(22.6.6) These cons istency cond itions were or iginally der ived by Wens and Zu- mino for the chiral ,5 U(3 )x 5U(3 )symmetry of the strong interact ions, a special case of phys ical as well as histor icalimportance .Here the generators J a(x)acting on the gauge fields consist of the even parity generator s ~~(,,(x) defined by Eq . (22 .3.31),and the odd par ity gene rators Xa(x)defined by ( 22.3.32).M*They sat isfy the commutat ionrelation s ['Na(x)aUb(y) 1= i64(X- y)fabcVc(x) , PWA x~, -Tb(y)1=i64( x - y )fabc y,( x ) V a(x), -T h(y)l = z64(X - y)fabcWc(x ) where }'~b ,are the S U(3) structure constants . Since the SU(3) subgroup generated by the ~Ja is not spontaneously broken , itis conven ient to treat the integration over fermion momenta in such a way as to preserve invariance under gauge transformations generated by N,,, so tha t leaving us with the non-zero anomal y The non-trivial consistency conditions are the n 'ya(X)Gh(y)= i64(X - WahcGc(X) and The first of these simply says that G,,(x) transforms like an octet under ordinary S U(3) transformations . The second condition imposes ❑ther strong constraints on Gu(x ). The reader may check that this condition i s The factor -i was inserted in Eq . (22.b.1)in order to provide the conventional factor +i accompanying the structure constant Cx fly. in this commutation relation . A reminder : we are using a basis for the Lie algebra in which the structure constants are totally antisymmetric, and so we do not distinguish between upper and lower gauge indices . Another reminder : as mentioned in Section 22 .3, the and ON',, used here are what were called Y,, and XA in Ref 8. 398 22 Anomalie s satisfied by the Bardeen formula (22 .3.34) for G« (x). We will not go into this here, but will instead use as an illustration a general gauge theory with all currents treated symmetrically . In Section 22.3 we quoted the formula (22.3.38) for the anomaly in this case, but did not derive the terms in this formula of higher than second ❑rder in the gauge fields . Here we shall show that these terms are dictated by the consistency conditions (22.6.6). For this purpose, and also to allow for further generalizations, it is very convenient to reformulate the set of Wess-Zumino consistency conditions as a condition of invariance under the BRST transformations described in Section 15 .7. Let us introduce a ghost field co,,, and define the nilpotent BRST operator s for a general gauge theory b y sAa1U=allwa+ C,~ #,jA#ua'v SCCla_ - 2 Caflnjfl3pfl3Y(22.6.7) (22.6.8) it being understood that s satisfies the distributive rule s(AB) =(sA)B ± A(sB), the sign being negative where A is a fermionic quantity like (t),, and otherwise positive . In place of the anomaly function Ga[x ;A], we shall work with a functiona l G [cv,A]= fw(x)G[x ;A] d4x , Then (keeping in mind that coa is fermionic ) sG[C r],.4];C,#y/d4x w# (x)CC]Y(X)Ga[x;A](22.6.9) -1d4Xcvx(x)fd4v[i3wstv)+ c~y 6Ayu(y)a)a(y)~a[x;A] y #,u(y) d4xId4yWM~x~ ~~~Y~ [- 2C a# Yb 4(x-y)Gy[x;A] Because the ghost fields anticommute with each other, this can be writte n sG[co,A] d4xfd4v COAX)wOW X12Cxfi Y6 4(X y)Gy[xaA]+,~T#( y)Qx[X;A]-~Fa(X)Gfl(y)] We see that the consistency condition (22.6.6)will hold if and only if G[cv, A] is BRST-invariant sG[cu,A] = 0 forallghost field s cva(x) .(22.6.10) Now consider the possibility that the anomaly G [A;cca]could be written 22.6Consistency Conditions 399 as the BRST operator s acting on a local functional F[A] : G [A;cry]= sF [A ]. (22 .6.11) (Note that the functional F is necessarily independent ❑f the ghost field, because the operator s adds one ghost field factor, and the anomaly functional G is already linear in the ghost field .) The BRST operator satisfies s2 = 0, so such an anomaly would satisfy the consistency condition sG = 0 . If F [A] is a local functional' of the gauge field, then it could be subtracted from the action, thus cancelling the anomaly . The same is true of any term in the anomaly that can be written as the BRST operator s acting on a local functional ; such a term satisfies the consistency condition by itself, and can be cancelled by adding a local term to the action . The possible anomalies that interest us are thus the local functionals G[o) ;A] ❑f ghost number unity that satisfy the consistency condition (22.6.10), modulo terms that can be expressed as s acting on some local functional of ghost number zero . In accordance with the usual terminology for nilpotent ❑perators, the equivalence classes of such functionals form what is called the cohomorogy of the s operator at ghost number unity . We can also express this in terms of the local densities themselves . We can write the anomaly (or any term in the anomaly) as G = f d4x !§(x), where I(x) is a power series in the gauge and ghost fields and their derivatives at the spacetime point x . The condition SG = 0then is equivalent to the statement tha t sq(x)=O'U'P( x) (22 .6.12) for some function ~P(x) of fields and field derivatives . Likewise, the terms in9that can be cancelled by adding local terms to the action are those that are of the form ,sue up to possible derivatives . Thus the anomalies that interest us are the local functions ❑f ghost number unity that satisfy the consistency condition (22.6.12), modulo terms that can be expressed as s acting on some local functional of ghost number zero, modulo possible derivatives . This is known as the cohomology of s at ghost number unity in the space of local functions, modulo derivatives, and denoted H'(sld) . Algebraic methods have been used to work out the cohomology ❑f the BRST operator s, and thereby deduce the form ❑f the anomaly in general gauge theories-19 This approach leaves unknown only constant coefficients that depend on the matter content of the theory and need to be calculated by the methods of Sections 22 .2 or 22.3. Since we have already calculated the terms in the anomaly of second ❑rder in the gauge fields for general gauge theories, including their constant coefficients, here we will use th e ~ By a local functional is meant the integral of a local function, that is, a function of fields and field derivatives at a given point . 400 22 Anomalie s consistency condition (22.6.12) to calculate the terms of higher order in the fields . We saw in Section 22 .3 that when all currents are treated symmetrically, the terms of second order in the gauge fields are one-third the expression (22.3,26). This is an operator of dimensionality four (in units of mass), while the Wess-Zumino consistency condition (22.6.6)relates only oper- ators of the same dimensionality, so to satisfy this condition we should add only terms of higher order in the gauge field that have the same dimensionality . We therefore seek a solution of the consistency conditions in the (not necessarily unique) for m Ga= i[ (ap ,Tocanam -24n2 fKvAp Tr Tx [ a,.4❑aAAp+ iC1 aKA,,A~Ap +ic'2AKau.4jAp +ic3AhAVaAAPc4A,,AVAAAp1(22.6.13} where A ,,= AIX ',T,and the ci are constants to be determined . In order to save a great deal ❑f effort, it will be convenient to rewrite this in the language of differential forms . (See Section 8.8.)We introduce a set of c-number parameters dxt' that are taken to anticpmmute with themselves and all fermionic fields, such as the ghost field co, The dx` then also anticommute with the BRST operator s . Because dx'1dx°dx ~dxP is totally antisymmetric, it may be written a s dx'dx''dX),dxp=FKv4d4X , d4X = dXQdX'd.x2dX3. (22.6.14) We also introduce the exterior derivativ e d = dxu ax u which since derivatives commute is nilpotent as well as anticommuting with s : d2= 0, Finall y, we intr oduce the anti commut ing qu antitie s co= if13a T,. In this notation, Eq .(22.6.13) reads [(dA)2+cl(dA)A2 G[cr),A] =247r2 TrW + c2A(dA)A + c3A2(dA)+ c4A411(22.6.17) In order to implement the consistency condition (22.6.10), we note 22.6 Consistency Conditions 40 1 that the BRST transformation rules (22 .6.7) and (22.6.8)may be written as sco=cod(22.6.1 S) (22.6.19) Now, the BRST transformation of the last term in Eq .(22.6.17) is given by sTr [wA4] = Tr [CO2A4-a )IA, w}A3 +coAfA,co}A 2 -wA2fA,co}A+ coA3 fA,w}A1 + codcvA3 terms _ -Tr[CO2A4]+ codcoA3 terms . There is no other contribution to sG proportional to Tr [a2A4], so the consistency condition (22 .6.10)can only be satisfied if c4 = a . With c4 = 0, a straightforward calculation give s sG = 24 2 Tr -(dA)2cv2 + c odcoA dA - dc ocodAA -Aco dA dco - ~ .oA dw d A +cl [wdAdo)A - co dA A dw] +c2 1co dcc7 dA A --- c oA dA dcol +c3 1co dr.o A dA - coAdw dAl -c, I- coA dco A2 + c odcoA3 + co dAA2cO I -c2 [wA? dcoA - coA dco A2 + rUA dA dw] -c3 I- coA 3dw+roA2dw A +c.oA2dAcol We do not need to assume that the integrand vanishes, but only that it is the derivative of some local function, so that its integral vanishes . This condition must be satisfied separately for the terms involving two derivatives and those involving one derivative, since no cancellation can occur between terms with different numbers of derivatives . It is not hard to see that the terms in the integrand involving just one derivative are of the form d.Fif we take cl = -c2 = +c3 = c .The remaining terms are a total derivative if c = -1/2, thus justifying the previously quoted result 402 22 Anomalie s (22.3.3$). This result is often expressed more compactly a s G[co' A]241r2 1 242Tr cod1AdA - 2A 3~ JTr d1AF-~- A3~, where F is the matrix-valued field-strength two-for m F = 2 itaF ,~,,dx'~ dxv = dA - A2 .(22.6.20) (Z2.6.21) The anomaly does not have to be put in the form (22.6.13), so Eq .(22.6.20) is not the unique result for G[c), A].Results quoted in the next section show that, for any subgroup H of G for which Tr {t1{t1, tk II_0for all generators of H, it is possible to add local terms to the action in such a way that the anomaly Gi vanishes when ti is any generator of H . There is an elegant algebraic tool, known as the Stora-Zumino descent equatiaras,18afor constructing a solution of the consistency conditions . It is just as easy to describe this method in a spacetime of any even dimen- sionality as in four spacetime dimensions, so we will take the spacetime dimensionality to be 2n . To start, one must imagine that at least two additional variables have been added to the 2n coordinates of space and time, in order to give meaning to the (2n + 2)-form TrF"+I . Note tha t dF = -d(A 2) =-(dA)A + A(dA ) =[A,F], so that Tr Fn+i is closed : dTr Fn+l = (n + 1)Tr f(dF)Fnj = Tr f[A,Fn+1] } = 0(22.6.22) (22.6.23) As long as the extended spacetime is simply connected ,Poincare 's theorem then tells us that Tr fFn+l Iis exact ,in the sense that there is a (2n+1)-form S22r+1 (known as the Chern -Simons form), for whic h Tr {F2n+21 = dS22n+i. (22 .6.24) Further, Tr IFn+l } is manifestly gauge-invariant and depends only on the gauge field, so it is BRST-invariant : sTrfFn+1J = 0. (22 .6.25) The 'differentials' dxY a reunderstood to anticommute with fer mionic fie lds like the g host fie ldcoo, so t he operator d anticommutes wi th the opera tor sdefined by Eqs . (22 .6.7) a nd(22.6.8): sd+ds=0. Because s is ni lpotent, it follows t hen thats nZn+i is a lso closed : des S22n+ 1 ) = -sTr ~Fn+ 'j = 0. 22.6 Consistency Conditions 403 Again using Poincare's theorem, this means that there must be a 2n-form S22n, of first order in co,,, for whic h s02n+ 1 = ffi2n (22.6.26) Furthermore, d(s S22n) = -s202,+1 = 0, so there is a lso a (2n - 1)-form 522n_1, of seco ndorderin th e ghos tfield,for w hich Sn2n = d ~2n- 1(22.6.27) It follows than that the integral of S22n over the 2n dimensions of space and time is BRST-invariant : s S~i2n=0,spacctim c even though S2 21n is not itself BRST-invariant . We can thus find a candidate f L21,, for the anomaly functional G[co, A] by integrating the two first-order differexitial equations dS 22n+1 =Tr {F3 jand ffi2n = s 522,+1 . General (not unique) solutions of these equations ar e ri S~2n+1 = (n + 1 )JdtTrfA Fr0(22.b.29) J= ra(ra + 1 ) cat (1 - t)Tr ~cv d(AF1 )l (22.6.30) where Ft - tF + (t - t2)A2 . Evaluation of the integral (22.6.30) shows that Eq. (22.6.20) gives a result for G[w, A] proportional to f S24 in the case of four spacetime dimensions . We can continue this descent, and derive other useful results . In particular, it follows from Eq . (22.6.27)and the nilpotence of s that d(sS22n_1 ) = 0, so Poincare's theorem tells us that sS22n_1 is of the form dS232n_2, so the integral of j22 2n-1 over the 2n --- 1 coordinates of space is BRST-invariant : s ~n2,-,= Q. (22.6.31) "ace Such BRST-invariant functionals of second order in the ghost fields are candidateslgb for so-called Schwinger terms .lg'Schwinger terms of the sort that concern us here arise as anomalous terms 5,,,p(x, y) in the equal-time commutation relations of the time components of two symmetry currents : J.(X,0,J~ (Y1 t)]=iC'fl'j,'O(X, t)b'"-'(x - y) + Sxo(x, y,t) (22.6,32) (All operators in this paragraph are taken at the same time t, which will henceforth not be shown explicitly .) From the antisymmetry of the commutator we have Sx#(x, y) = -S#,,(y, x), so all information about 404 22 Anomalie s Sad (x, y) is contained in the functiona l S [W] = Jd21xd2fl_1y a (x)a# (Y)S(x, Y)- ( 22.6.33) Note that S ,#(x,y)depends in general on various matter and gauge fields, so S [ w] generally depends on these fields as well as on the ghost field coa(x) . Taking the commutator of Eq .(22.6.32) with a third current JO(z), contracting with claa(x) co#(y) c.),,(z),integrating over x, y, and z, and using the Jacobi identities, we fin d 0=Jd21xd2 1ydzri-Iz wax ) (')~ ( y) coo, (z) x[iCx#a rya(z, x)6an-l(x - Y) +[j y(2),Safi (x, Y)] I On functionals of gauge and matter fields like S,,fl(x, y),the action of the BRST operator s is the same as that of a gauge transformation with transformation parameter co,,, s o s 5"fl(X, Y)= i ~ ~ don-i z 01), (z)J~°(z),Says(Xa Y)). Recalling Eq .(22.6.8), we find that the functional (22.6.33) is BRST- invariant sS [w] = 0. (22 .6.34) Also, by adding terms to the currents we can change S [r .o] by terms of the form s7'[r o], so the set of possible Schwinger terms that could notbe removed by adding terms to the currents is given by the cohomology of the BRST operator s at ghost number two -that is, by BRST-invariant functionals S of second order in the ghost field that are not themselves of the form s T.Eq. (22 .6.31) shows that f 0 2,~_~ is a candidate for such a functional . The analysis of anomalies given so far in this section is strictly applicable only to anomalies in one-loop order . It is true that a theory with one- loop anomalies in currents to which quantum gauge fields are coupled is inconsistent, and therefore does not need to be studied in higher orders . But the converse does not hold ; if a theory with quantum gauge fields is anomaly-free in one-loop order, we still need to show that anomalies are absent in higher orders . Also, there is nothing inconsistent in theories with anomalies in global symmetries, such as the chiral symmetries of quantum chromodynamics, and for these we need to know whether higher-order corrections affect the anomalies . Since BRST transformations act non-linearly on the fields, even in the absence of anomalies we would not necessarily expect the quantum 22.6Consistency Conditions 405 effective action F[w, A] to be BRST-invariant beyond the one-loop ap- proximation . As we saw in Section 17 .1, beyond this approximation we need to consider functionals not only of gauge and ghost fields but also of their antifields . (The introduction of antifields is sometimes important even in one-loop order for another reason : a local functional of fields alone that satisfies the Wess-Zumino consistency conditions, and that is not expressible as the BRST operator acting on a local functional of fields alone, will not be a candidate anomaly if it canbe expressed as the antibracket of the action with a local functional of fields and antifields, because in this case the anomaly can be cancelled by subtracting this term from the action . This corresponds to a change in the action of the fields combined with a change in the gauge symmetry obeyed by this action .) The study of anomalies with antifields included in the action turns out also to have a cohomological formulation .X21 The analysis of this problem is based on the Zinn-Justin version (17.1.10) of what Batalin and Vilkovisky called the master equation . In the absence of anomalies (r, r) =o, so that in one-loop order (S, Tt) = o, where S is the zeroth- order action and rl is the one-loop contribution to the quantum effective action . In the presence of anomalies we have instea d (S, r I)= G1 , (22 .6.35) where G1 is some functional of fields and antifields which, since S and T1 have ghost number zero, must have ghost number unity . The action is assumed to satisfy the classical master equation (S, S )=0, so the antibracket operation in Eq . (22 .6.35) is nilpotent, and therefore (S, G I) = 0. But if G1 = (S, Fl )for some local functional Fl of ghost number zero then we can cancel the anomaly to one-loop order by subtracting the term Fl (which is treated as a quantum correction of order h) from the action and hence from Ft . (Of course, G1 = (S, 171), but in the presence of massless particles T1 is not a local functional .) Thus the candidate anomalies are those local functionals G ,of ghost number unity that are closed, in the sense that they satisfy (S, G1 )=0, but that are not exact, that is, that cannot be expressed as G1 = (S, Fly for local functionals Fl of ghost number unity . In other words, the candidate anomalies correspond to the cohomology of the antibracket operation X 1--4(S, X)at ghost number unity on the space of local functionals of the fields and antifields . This is just like the result found earlier in this section, but with the antibracket (S,    )replacing the BRST operator s . If we set the antifields equal to zero in Eq . (22.6.35)and recall that (bT1/bx")xt=0 = sxn, we find that the condition (22.6.35)in fact yields the condition sF1 = 0, which as we have seen is equivalent to the requirement that rl satisfies the Wess-Zumino consistency conditions . But there is a sense in which the analysis based on Eq . (22.6.35)can be extended to higher orders . 406 22 Anomalie s To see this, suppose that it is found that the antibracket operation Xt--►(S, X )has an empty cohomology at ghost number unity in the space of local functionals, and that we redefine the action as described above to make G1 = (S, T1) = 0. An anomaly that violates the master equation (F, F) = 0in two-loop order is represented by a function G2 for whic h 07, r,) +2(s,r2) =G2, But since(S, Ti ) = 0 a nd(S,S) = 0, any such G2 wo uldsatisfy ( S,G2) =Q. Onthe ass umptionofan e mpty cohomo logy, thismeansthatitcan be expresse das G2 =(S, F2) wit hF2 a localfunctional of ghost number zero, soin this or der the anomaly ca n be ca ncelled by subtracting F2 from t he action. This arg umentcan be extended to allorders. Suppose we have ca ncelled the anomalies in the mas ter eq uation up to or derN-1, sothat M M- 1 0= Gm (FL, IFM-L) = 2 (S' FM) + 1 :(TL, 11-M-L) for all M<N.The Nth-order term in the antibracket (T, F) is likewis e N-1 GN= 2 (S,FN J+ M=t(rMe FN-M) so, using the Jacobi identity (15.9.21)(in which for three bosonic operators all signs are -) and the above formula for (S, TM ), we find : N-1 N-1M-1 (SI GN -2 1 :((S, FM), rN-Al) = E E M=] M=1 L= 1 This can be written in a more symmetric wa y N-2 N -2N-2 (SGN)=->2>2>2 Mt- lM2=1M3=1((FL ,FM-L} ,rN-M } 5N,M1 +Mz+M3((F1,, (F .FM) ) Since the ranges of Ml,M2, and M3 are the same, we can write the double antibracket in this sum as a sum over the 3!permutations over these indices, which vanishes according to the Jacobi identity (15.9.21), leading to the conclusion that (S, Gam ) =0. If as assumed the cohomology is empty then this implies that there is a local functional F Nfor which GN = (S, FN), so by subtracting FN from the action the anomaly can be cancelled in order N,as was to be shown . Using purely algebraic methods Barnich, Brandt, and Henneaux22 have succeeded in showing that for Yang-Mills theories (in four spacetime dimensions) based on semisimple gauge groups, the cohomology of the 22.6Consistency Conditions 407 antibracket operation X ~-4(S, X) (atghost number unity on the space of local functionals) cons ists entirely of a linear combination of terms of the form (22.6.20), one for each simple subgroup of the gauge group , with unknown coe fficient s.TT This show swithout an yreference to the matter content of the theory that in one-loop order ,where the anomaly G1automatically satisfies (S ,GI) =0,the anomaly for a semi-simple gauge group must be a linear combination of terms of the of the form (22.6.20),with only the constant coefficient for each simple subgroup left to be determined b ydetailed calculations that take into account the matter content of the theor y.Further ,wesaw in Section 22 .4 that there are gauge groups for which the trace in Eq .(22.6.20) vanishes automatically for any menu of fermion fields . (These are the se rnisirnple gauge groups with no S U(n) factors with n>3.) In such cases the theorem of Ref . 22 shows that the cohomology of the antibracket operation X ~--> (S ,X)at ghost number unity is zero . As we have seen ,this means that in such theories there is no anomaly in any order of perturbation theory . There i sanother sense in which the anomal yisrelated to the cohomol- ogy of the antibracket operation .21In deriv ing the Slavno v-Taylor identity (16.4.6),we assumed that the measure dx'(x)is invariant under the symmetry transformation in question . In Section 15 .9 the Zinn-Justin equation was derived from the Sla vnov-Taylor identity for the symmetry transformation x'->xn + DDS /bxn,so the derivat ion of the Zinn-Justin equation given in Section 15 .9 breaks down unless rjnxdxn(x)is invariant under this transformation ,or in other words unless AS=0, where ❑is the operator (15 .9.34). Where ❑S=~0,itstill may be possible to save the Zinn-Justin equation by adding local functionals to S that violate the classical master equation (S, S) =0in such a way as to cancel the effect of the non-invariance of the measure . It turns out that the cond ition for this cancellation is nothing but the quantum master equation (15.9.35).To construct an action S that satisfies this equation ,we start with a zeroth- order action Sothat satisfie sthe classical master equation ,(So,SO)= 0, and add quantum corrections .In the ca se in which the cohomology of the operation X -> (S o,X)(at ghost number unity in the space of local functionals) is empt y, the same proof that wa sused above to show the absence in this case of anomalies to all orders can be used to show that a local functional can be added to S osothat the quantum master equation is satisfied to all orders . It is not necessary to specify the representation of the gauge algebra in which the trace inEq. (22 .6.20)is to be calculated, because this trace is the same up to a constant coefficient for all representations of a simple Lie algebra,"° and the constant coefficient is not determined anyway by this cohomology theorem . 408 22 Anomalie s 22.7 A nom alies andGoldstoneBosons In the same 1971 paper in which they introduced the consistency con- ditions, Ness and Zumino18 also noted that the possibility of anomalies has important consequences for the interactions of Goldstone bosons . To understand their point, it is helpful to apply the `anomaly matching' ar- gument developed by 't Hooftlb in 1979, which has already been used in Section 22 .5. Consider a broken global symmetry group G that is realized linearly in some underlying theory of trapped massless fermions, as for instance the global chiral S U (3) x S U(3) symmetry in quantum chromodynamics with three massless quarks . In this underlying theory introduce fictitious gauge fields, so that aside from possible anomalies the global symmetry G becomes local . In general this local symmetry will be broken by anomalies, since in the real world the symmetry is purely global, and there is no reason why a global symmetry should admit an extension to an anomaly-free local symmetry . However these anomalies can be cancelled by adding suitable massless spectator fermions . As long as the gauge couplings introduced in this way are sufficiently small, and the spectator fermions have only these very weak gauge interactions, the dynamics of the theory will not be substantially changed by these modifications . Next consider the effective field theory that describes physics at low energy, where the trapped fermions are unobservable . The only degrees of freedom in this theory will be the massless particles : the fictitious gauge bosons and spectator fermions, and a set of Goldstone bosons with fields ~ ,,, one for each independent broken symmetry . Since the underlying theory had been made gauge-invariant and anomaly-free, the same must be true of the effective field theory . But the spectator fermions produce an anomaly which had previously cancelled the anomaly due to the trapped massless fermions in the underlying theory, so in order for them to cancel the anomaly due to the Goldstone bosons, the gauged effective field theory of the Goldstone bosons must have an anomaly .for the fictitious local symmetries which is equal to that produced by the trapped fermiaras in theunderlying theory .That is, in place of Eq .(22.6.2), the effective action r[~, A] of the fictitious gauge fields and Goldstone bosons is subject to the conditio n 9-0(x) rJ ~a A]= Gfl[x; A] , (22.7.1) where G p[x; A] is the anomaly function of the underlying theory, in which there are no Goldstone bosons, and .T#is a generator of the gauge group G, now acting on both gauge and Goldstone boson fields . (The index 9on9 -flruns over values i labelling a complete set of independent generators °JI of the unbroken symmetry subgroup H, together with values 22.7Anomalies and Goldstone Bosons 409 a labelling a set of independent broken symmetry generators Xa; there is one Goldstone boson field ~a for each Ya .} Of course, Eq .(22.7.1)may also be used to study the interactions of Goldstone bosons with real weakly coupled gauge fields . For instance, where the underlying theory includes electroweak gauge bosons coupled to the quarks, we would identify some of what we have called `fictitious' gauge fields with the electraweak gauge fields . In such cases some of the `spectator' fermions must also be real, in order to cancel the anomalies produced by loops of trapped fermions in the gauge symmetries of the real weakly coupled gauge fields, as for instance the leptons cancel the electroweak anomalies produced by the quarks . Now let us turn to the implications of Eq .(22.7.1).To calculate the gauge symmetry generator J fl(x) in this context, note that under a general group transformation g = expt-i f C#(x),T#(x)d4x ), the Goldstone boson fields ~a(x) transform into fields ~a(x) given by Eq .(19.6.18), and the gauge fields A~(x) transform into gauge-transformed fields A'~(x), so tha t (x)= ~,A (x)+j,-~(x). (22 .7.2) Here l ~ (x) acts on the gauge fields and is given by Eq . X22 .6.1 ZJ ~ ~JC~ =- f~~~ [~A ~- CflYAYF ,(x) CAA ~ s (22.7.3) x ~S ,(x) a~(x ) where Cady are the totally antisymmetric structure constants of the gaug e group G, while T~(x) acts on the Goldstone fields and is given by the infinitesimal limit of Eq .(19.6.17), which (for the exponential parameteri- zation y(~) = exp(i~,,X,,)) reads :* TflexpRaWX a) =-J-fl~(x) exp (i~a(x)Xa)+ exp (i&(x)X,,) D#a(x)Yi (22.7.4) Here To are matrices representing the generators of G in any represen- tation ; they are divided into sets Xaand Yz, which represent the broken and unbroken symmetry generators, respectively . Also the 0#j(x) are ~-dependent functions whose form will not concern us here . About the anomaly functions G #[x; A], we will assume only the consis- tency conditions (22 .6.6) g-a(x)G# LY;A] - 9-#(x)Gx [Y;A]= iCa#v64(x- Y)Gy LY; Al (22.7.5) The minus sign in the first term on the right-hand side of Eq . (22 .7.4) and the factor -i on the left-hand side of Eq .(22.7.3)appear because it is exp[-i f A~ (x).ag(x)] that induces a gauge transformation (15.1.17) with gauge parameter A pas can be seen by requiring the 9-#(x) to satisfy the commutation relations (22.6.5). 410 22 Anomalie s and the absence of anomalies in the unbroken symmetries (22.7.6) As mentioned in the previous section, as long as the trace Tr [7't{7' j,Tk}] vanishes for the generators of unbroken symmetry subgroup (as it does for the non-chiral generators of SU(3) xSU(3)), it is always possible to add a local functional to the action so that Eq .(22.7.6)is satisfied . Under the assumptions (22.7.5)and (22.7.6), it is always possible to find a solution of the anomalous Slavnov-Taylor identities (22.7.1): 1 dtf~b(y)Gb[y;A_t~I d4y (22.7-7)0 where [A_t&)],, is the result of acting on A .- TpA,,flwith a gauge transformation (15.1.17)having Aa = -tea and Ai=0: [A-t~ (x)]~ = exp (-itXa~a (x))A,,(x) exp (itXu~ ,(x)) -i [01,exp (-itX«~ a(x)}Iexp (i ffp~a(x) ). (22 .7.8 In contrast with the case in which unbroken symmetries have anomalies, Eqs.(22.7.7)and (22.7.8)define a local (though complicated) functional of gauge and Goldstone boson fields . Any other solution of Eq .(22.7.1)will differ from this one by an anomaly-free functional . Here is an outline of the proof 24 that the action (22.7.7)satisfies Eq.(22.7.1).Instead of working with the local generator J ~ (x), it is convenient to introduce an arbitrary function rj#(x), and defin e Cnl =Id4xqfl(x)j7-fl(x) To evaluate .Y[] Wx},we introduce the matrix(22.7.9) rj_t&) = exp (-iX(x)t) exp(iXa~a(x)t) 1q --t&~~PT# (22.7.10) where n(x) - nfl(x) 7'# . The n a q-t&) =Axa~AXOI-t~(X)] +i(~Y- [,I] &(X))X a so that a 111-1~(x)Ib +iCub ~u (x)[n-cjx)I,,. (22.7.11)at To evaluate J"[rj] Gb[y, A],we apply 9-[rl]to the gauge field ,and after a straightforward calculation find tha t ~jq]CA-tjx)],,= (g-A[n-c~IAM (x)}A-.A_,~ '(22.7.12) 22.7 Anomalies and Goldstone Bosons 41 1 so, using the consistency condition (22.7.5), J~[?j]G h[,}l; A_t~ ] =Jd4x[q-t~( x)]y(r:(x) Gh U';A])A--,A-j g fd4X lq-t~(X)Ia(g-A(Y) G.[x;A]A->A_t ~ +iCy bU[q-ts(Y)jyG. Lv; A_r~] . (22.7.13) The structure constant terms in Eqs .(22.7.11)and (22.7.13)cancel, yieldin g J [q] r[~,A] = ~dt d 4.Y~at[[t/_t]b] G 6LJ'; A-r~ ~.lo f t ilq-t~(Y)Ia (37-[fl G.[y; A]) -(22.7.14) Another straightforward calculation shows tha t o-[A_t~ {x} ],,= i~'iA(~)Au (x))AAA-r&(22.7.15)at so the terms in the integrand in Eq . (22 .7.14) add up to a t-derivativ e t J[qI rj~,A] = - dtday ~t{fr/_t'nh Gb[Y;A-c~] a t=i r=o At t = 1 we hav e q-~(X) = exp(-ix"~"(X)) [q(x) + 9-[q]] exp(ix'~"(X))- From Eq . (22 .7.4)we see that this is a linear comb ination of the generators of the unbroken symmetry subgroup H,so the coeffic ient [q_Jy }]bof any broken symmetry generator Xbvanishes .Also ,Eqs. (22.7.1 U)and (22.7.8) show immediately that at t = 0 ,q_t~(y)=q(y) and [A_t~(y)],s = A ,(y),so Eq.(22.7.16) yield s = Jd4v [qa(.v)]b Gb[v;Aa]=f d4.vqb(.v)Gb 1Y ; A],(22.7.17) which with Eq . (22.7.6)is equivalent to the desired result (22.7.1),as was to be proved . The solution (22.7.7)of Eq .(22.7.1)is not unique, but it is the unique 412 solution that vanishes for ~22Anomalie s 0.To see this, note tha t exp [ - if q# (x)9-#(x)d4 x]T[~, A ]= r[~f~~A'] , (22.7.1$) where the primes indicate a gauge transformation with transformation parameter qp. It is convenient to represent the exponential here a s a so Eqs .(22.7.1)and (22.7.18) yiel d JO1-1W., Alldtexp(zt)z, dt exp [_-itftip(x)T~ (x} d 4x ~ Jt/h(v)Gb[v ;A1d4 v In particular, if we take and qt = 0, then ~a = 0, in which case by assumption Eq .(22.7.19)has a vanishing right-hand side, and therefore yields the formula : 1 r[~,A]=-i dtexp[itJ(x) ~ ~(x) ~x] ~~(.v)Gb[.v;A]d4.vo The funct ional operator exp[it f ~ ,,(x)te a(x} d4x ] in Eq .(22.7.20)simply produces a gauge transformation (15 .1.17) with gauge parameter A fl(x)= -t~jx), so that Eq .(22.7.20)may be written as in Eq .(22.7.7). Eq. (2.7.7)may be applied to study the electroweak interactions of the octet of pseudoscalar Goldstone bosons, but it has important implications for the interactions of the Goldstone bosons themselves, in the absence of real gauge fields. In the case where A =0, Eq .(22.7.8)becomes a `pure gauge' field [A-t jx)J,u=-i 10,, exp ( -dffa~a (x))]exp(tffa~a(x)) = -i[O'V(t~(X))]t where V(t~(X)) _=exp( -iffa~a(x))- With Aa(x)= 0, Eq .(22.7.1)gives so the result of using Eq .(22.7.21) in Eq .(22.7.7)is a G-invariant local functional of the Goldstone boson field ~,Jx}, though, as we shall see, it is not in general the integral over spacetime of a G-invariant function of ~a(x) and its derivatives . 22.7 Anomalies and Goldstone Bosons 41 3 The simplest example is the case of a completely broken symmetry group . Here the condition (22.7.6)is empty, and we can use the symmetric form (22.3.38) for the anomaly : GQ[x; A]~~4~2E"v'~P Tr 7'R[0xAv (aC) O~A~[aC] - 2 iOxA„(x)A~{x}Ap{x } + 2 iAx (x)0,AA(x)A,, (X) 2iA,~(x)Av(x)6AA~(x)] , (22 .7.24) where now T',, is the specific representation of the group generator fur- nished by the left-handed fermions (including antifermions, where the dis- tinction is relevant) of the theory . In this case when we use Eq .(22.7.21) in Eq . (22 .7.24), we find that the only terms in the trace in (22 .7.24) that survive when contracted with e"4 are then all proportional t o Tr~T"(O"V)V-I(O,V)V-I(o ;V)V-1(00V)V-II, with coefficients -1 , + ? , - and + ?, respecti vely,so Eq .(22.7.7) becomes here 248ic JdtTr TG [~V(t(y))]a XV-I(t~(Y)) [O'V(t~(Y))] V-I(t~(Y)) [0,V(t~w) ] XV_1(t~(.Y))[ap1l(t~(Y)}] V-f (t~(Y}) (22.7.25) As promised, this is not the integral of an invariant function of fields and field derivatives . For instance, for small Goldstone boson fields Eq. (22.7.25) become s r[~,0]= - 1 2 CxvzpTr€T'aT'bTc Td T'e240ir +0 W) .d4Y~caOr:4Ov~c OZ~d ap e Any function of the covariant derivatives of Goldstone boson fields would have a term of lowest order in the fields which would be simply a product of partial derivatives of the Goldstone fields, and therefore could not have a lowest-order term of the form (22.7.26). For an example of greater practical importance, consider the 5' U(3 )x 5' U(3)chiral symmetry of quantum chromodynamics with massless u, d, and s quarks, spontaneously broken to the diagonal SU(3) subgroup of Dell-Mann and Ne'eman . In order to use the results of this section, we must label internal momenta in the fermion loop integral so that the vector currents of the diagonal 5' U(3)subgroup are anomaly-free, in which case 414 22 Anomalie s the anomaly takes the Bardeen form (2 2.3.34): GAV ' A]=i n 167L2 Tr to[V, Vpd + 3A~vAoa - 2 A~AvANAf f + 3 i(A~AV Vp~ + Au VpQAv +VpQAmAv) ] where V, u, Au, V,uV, and A 4vare defined by Eqs .(22.3.35)-(22 .3.37);t, is here half the Dell-Mann matrix ~,,, given by Eq .(19.7.2);and n is the number of quark types `colors') of each flavor . (We are now using a lower case t for the matrices representing the group generators, because in this trace we sum only over the left-handed quarks, not the left-handed antiquarks .) For a pure gauge field like (22.7.21)the field strengths V,,,vand A,,v vanish, so that using Eq .(22.7.27)in Eq .(22.7.7)yields the anomalous effective Goldstone boson actio n r'i duxTr[~a0] = -2n 2jdtf (22-7 .28) where `A' now denotes the axial-vector as opposed to the vector gauge field. To find [A_tj, we use Eq . (22 .7.21), which here read s IV t~(x)1 p +YS [A-t~ (x)],u=10J,exp ~ - itYSta~u (X)1J exp(ity5ta~a (x)). Multiplying with {1 +Y5}/2and {1- Y5}/2 and taking the difference yields the axial-vector term Pexp ~ - it~ at« exp (tata) [A-c~(x) ]u = - Zi a +Zi~exp (itata)] eX~] ~ -Etata I 1 l = Zi exp(itta)U-1(t(x)) [U(t(x))] expl where U(t) - exp (2ftaat) . Using this in Eq .(22.7.28)yields the effective anomalous actio n rRI 01 =n -DZ 2EJI vp6,~dt J~d4xTx {ataU_1(t(X)) [0,U( t~~)}] u-,(t~(X))[OVU(t~(X))]U-I(t~(X))[O,U(t~(X))]u-,(t~(X) ) [0,U(t~w)] I - (22 .7-32)Kata), 22.7 Anomalies and Goldstone Bosons 415 As noted by Witten,25 this may be expressed in a convenient five- dimensional form . We take t as our fifth coordinate, and define ~a (x, t} ta(x) . Then Eq. (22 .7.32) become s M'01 = -i n 240712Cijkrmfdz Tr U-I((z}l l [U ( `(Z)l1] Xu-,(~(Z))[aiu(~(Z))]u-,(~(Z))[O,U(~(Z))]u-,(~(Z) ) X[OIU(~(Z))]U-I(~(z))[OnU(~(z))] (22-7 .33) where i, j, etc . run over the values 1, 2, 3, 0, 5, with Zi = Xi for i = 1, 2, 3, 0 and z5 = t, and the integral is over the region 0 c z5 c 1 . (An extra factor 115 appears in Eq .(22.7.33) to take account of the fact that any one of the five indices i, j, k, 1, or m may take the value 5.)Since ~Q(z) takes a fixed value zero for z5 = 0 and all values of the other components zY of zt, we can identify these values of z ias a single point, and consider the region of integration in Eq .(22.7.33) as a five-dimensional ball, with the four-dimensional boundary z5 = 1 taken as ordinary spacetime . Eq .(22.7.33) is thus a special case of the "Wess Zumino-Witten' action given by Eqs .(19.8.1)and (19.8.3),which was also proportional to an integer n ; the only difference is that n is now identified as the number of colors . We saw in Section 19 .8 that the integral (19.8.3) depends only on the values of ~u(z} on the spacetime boundary of the five-ball, so in deriving Eq .(22.7.33) we have shown that Eq .(19.8.2) applies for any continuation of ~u(x) into the interior of the five-ball, not just for ~Jx, t) = t~(x ). More generally ,consider an arb itrary gauge group G that is sponta- neously broken to a subgroup H that by itself is anomaly-free - that is, for which the D-symbol (2Z.3.12) vanishes for any three generators of H . Chu, Ho ,and Zumino 26 have shown that we can add a local function B [A]to the action , in such a way that the currents of the subgroup H are anomaly free, even when the gauge fields of the broken symmetries are taken into account . This functional i s 1 dux Tr ~ EAR, Avg (FP° + F~ ") + A`A~A~A~B ~A, 4$1c2 E~`V p`~ A~A' 'APA6 +~AhA' 'A~A¢I, (22.7.34) where again AP.- T'xA~ and Fu" ~ T~Fa", while A~ = T'iA~' and F~ T'if;"are the terms in A Pand FW in the algebra of the unbroken symmetry 416 22 Anomalie s subgroup H . The total anomaly is the n G~ [x ;A]= Gp[x;A] + ~T p(x) B [A] (22.7.35) (where G fl[x;A]is the symmetrized anomaly (22.3.38)) and satisfies the desired condition Ga [x; A]= 0. (22.7.36) The anomalous part of the effective action for Goldstone bosons and gauge fields is given by using Eq .(22.7.35) in place of G #in Eq .(22.7.7). In this way, Chu et a1 .26 found an anomalous effective actio n r'[~,A]= r[~,A] - B [A_ ~]+ B LAS, (22.7.37) where T[~, A] is the previously derived effective action (22.7.7),and A_~ (x ) is obtained by setting t = 1 in Eq .(22.7.8).In particular, where the gauge field vanishes A_~ is the pure gauge fiel d [A_~(x)],u =-ileftV (~ (X)) IV-' (~ (X))~ V (~(x)) =exp(-iXa~,,(x)) 9 so here the Goldstone boson action i s rv>oi=maOi- 1 2 Fpypa Jd4x Tr AA hA h4 -AA u ,Ap ,Aa ,+2All~ h ElvAP~ y~ A6 (22.7.38) This result is not unique ; in particular, in parity-conserving theories like quantum chromodynamics we can add additional local terms to the effective action to cancel any parity-non-conserving terms in Eq .(22.7.38). Problem s 1. Calculate the rate of the decay process q--o. T + y, to leading order in ms, with m,, md = 0. 2. Consider a chiral S U(3) symmetry under which the left-handed part s of the spin2fields of a ferm ion-number-conserv ing theory form N defining representations 3 of 5 'U(3),while the right-handed parts ar e all singlets . Evaluate the anomaly in the SU(3)symmetry .Wha t is the anomaly if we add Mfermion fields whose left-handed part s are singlets, and whose right-handed parts transform as symmet ric tracele ss second-rank SU(3)tensor s? References 417 3. Find a solution of the 't Hooft anomaly matching conditions (22.5.5) and (22.5.6)for the case of n = 4 flavors . Find a solution for n = 2 flavors other than the one given in the text . 4. Derive the Zinn-Justin equation from the quantum master equation, without assuming that ❑S = 0 . Reference s 1. J. Steinberger, Phys .Rev. 76, 1180 (1949) ; Also see R . J. Finkelstein , Phys . Rev . 72, 415 (1949) ; H. Fukuda and Y . Miyamoto, Prog . Theor .Pays .4,347(1949) ; J. Schwinger, Phys .Rev. 82, 664 (1951) ; L. Rosenberg, Phys . Rev .129, 2756 (1963) . A rough estimate of thi s decay rate was made by S . Sakata and Y . Tanikawa, Phys .Rev. 57, 548 (1940), before the experimental discovery of the 7r° meson . 2. D. G. Sutherland, N ucl.Pays .B2, 433 (1967) 3.M. Veltman, Proc . Roy .Soc.A301, 107 (1967) . 4. J. S. Bell and R . Jackiw, Novo Cimento 60A, 47 (1969) . Also seeR. Jackiw, in Lectures on Current Algebra and its Applications, (Princeton Press, Princeton, 1972) . 5. S. Ad ler,Phys . Rev .177, 2426 (1969) . Also see S . Ad ler, in Lectures on Elementary Particles and Quant umField Theory - 1970 Brandeis University Summer Institute in Theoretical Physics, eds. S. Deser, M. Grisaru, and H .Pendleton (MIT Press, Cambridge, MA, 1970) : Volume I . 6. K. Fujikawa, Phys . Rev . Lett .42, 1195 (1979) . 6a. M. F. Atiyah and I . M. Singer, Proc .Nat. Acad .Sci.81, 2597 (1984) . 7. L. Alvarez-Gaume and P . Ginsparg, Ann .Phys .161, 423 (1985) . 7a. For a more detailed argument, see Y . Frishman, A . Schwimmer, T . Banks, and S . Yankielowicz, N ucl.Phys .B177, 157 (1981) . 8. W. A. Bardeen, Pays . Rev .184, 1848 (1969) . 8a. See, e .g., B. Zumino, Y-S . Wu, and A . Zee, Nucl .Phys .B239, 477 (1984) . 9. S. L. Adler and W . A. Bardeen, Pays .Rev.182, 1517 (1969) . 418 22 Anomalie s 10. See, e .g., H . Georgi, Lie Algebras in Particle Physics (Benjamin- Cummings, Reading, MA, 1982) :pp. 15, 198.The original reference is I. Schur, Sitz. Preuss . Akad .,p. 406 (1905). I0a. M. F. Atiyah and I . M. Singer, Ref . 6a ; B. Zumino, in Relativity , Groups, and Topology II, eds. B. S. De Witt and R . Story (North - Holland, Amsterdam, 1984) ; L. Alvarez-Gaume and P . Ginsparg , Ref. 7; W. Bardeen and B . Zumino, N ucl.Phys .B244, 421 (1984) ; R. Stara, in Progress in Gauge Field Theory, eds. G . 't Hooft e t al. (Plenum, New York, 1984) : 543 ; 0. Alvarez, I . Singer, and B . Zumino, Comm . Math .Phys 96, 409 (1984) ; L. Alvarez-Gaume an d P. Ginsparg, N ucl.Phys .B262, 439 (1985) ; J. Manes, R . Story, an d B. Zumino, Commun . Math.Pays .102, 157 (1985) ; B. Zumino, IVucl . Pays .B253, 477 (1985) ; J. M. Bismut and D . S. Freed, Commun . Math . Phys .106, 159 (1986) ;107, 103 (1986) ; D. S. Freed, Common . Math . Phis .107, 483 (1986) . 10b. E. Witten, Phys . Lett .117 B , 324 (1982) . 11. D. J. Gross and R .Jackiw, P hys.Rev.96, 477 (1969) . 12.H. Georgi and S . L. Glashow, Phys .Rev.D6, 429 (1972) . 13. M.L. Mehta, J.Math .Phys .7,18 4 (1966) ; M.L. Mehta and P . K. Srivastava, J. Math . Pays .7,1833 (1966) . 13a. The cancellation of anomalies in the four-quark version of the stan - dard electroweak theory was shown by C . Bouchiat, J . Iliopoulos , and Ph . Meyer, Phys . Lett 38B, 519 (1972) ;S. Weinberg, in Funda - menta lInteractions in Physics and Astrophysics, eds . G. Iverson et al . (Plenum Press, New York, 1973) :p. 157 . 14. H. Georgi, in Particles and fields 1974, ed. C. Carlson Amer . Inst. of Physics, New York, 1975) . 15. R. Delbourgo and A . Salam, Phis .Lett .40B, 381 (1972) ; T. Eguch i and P . Freund, Pays . Rev . Lett .37, 1251 (1976) . Also see N . K. Nielsen, M . T. Grisaru, R . Romer, and P . van Nieuwenhuizen, Nucl . Pays .B140, 477 (1978) ; M. J. Perry, Nucl . Phys .B143, 114 (1978) ; S. W. Hawking and C . Pope, Nucl .Phys .B146, 381 (1978) ; S. M. Christensen and M . J. Duff, Phys . Lett .76B, 571 (1978) ; R. Critchley , Pays . Lett .78B, 410 (1978) ; A. J. Hanson and H . Romer, Phys . Lett . 80B, 58 (1978) . For a general review, see T . Eguchi, P . B. Gilkey , and A . J. Hanson, Phys . Rep .66, 213 (1980) . In 4n + 2 dimension s there is also an anomaly in the divergence of the energy momentu m tensor ; see L . Alvarez-Gaurne and E . Witten, Nuc. Phys .B234, 26 4 (1984) . References 419 16. G. 't Hooft, lecture given at the Cargese Summer Institute, 1979 , in Recent Developments in Gauge Theories, eds. G 't Hooft et al . (Plenum, New York, 1980), reprinted in Dynamical Gauge Symmetr y Breaking, eds . E. Farhi and R . Jackiw (World Scientific, Singapore , 1982), and in G . 't Hooft, Under the spell of the Gauge Pri nciple (World Scientific, Singapore, 1994) . Also see S . Dimopoulos, S . Raby, and L . Susskind, Nucl .Phys .B173, 208 (1980) ; S. Colema n and E . Witten,, Pays . Rev . Lett .45, 1000 (1980) ; Y. Frishman, A . Schwimmer, T . Banks, and S . Yankielowicz, Nucl .Phys .B177, 15 7 (1981) ; A. Zee, Univ . of Pennsylvania report, 1980 (unpublished) ; R. Barbieri, L . Maiani, and R . Petronzio, Phys .Lett .9613, 63 (1980) ; G. Farrar, Phys .Lett.96B, 273 (1980) ; R. Chanda and P . Roy Phys . Lett .99B, 453 (1981) . 16a. S. Weinberg and E . Witten, Phys . Lett .9613, 59(1980). 17.J.Preskill and S. We inber g,Pays. Rev .D24,1059 (1 981). 18.J.Wens and B . Zumin o,Phys.Lett.37B,95(1971) . 18a. R. Stara, in Progress in Gauge FieldTheory, eds. G 't Hooft et al . (Plenum, New York, 1984) :543 ; B. lumina, in Relativity, Groups an d Topology II, eds. B. S. De Witt and R . Story (Elsevier, Amsterdam , 1984) :1293 ; J. Manes, R . Stara, and B . Zumino, Commu n.Math . Phys .102, 157 (1985) . 18b. J. Schwinger, Phys .Rev. Lett . 3,296(1959) . 18c.B. Zumin o,Nucl.Pays.B253 ,477 (1985) . 19. J. A. Dixon, unpublished preprints (1976-9) ;Common .Math . Pays . 139. 495 (1991) ;F. Brandt, N . Dragon, and M . Kreuter, Nuc2 . Phys . 13332, 224 (1990); M. Dubois-Violette, M . Henneaux, M . Talon, an d C. M. Viallet, Phys .Lett .B289, 361 (1992) ;M. Dubois-Violette, M . Talon, and C . M. Viallet, Phys .Lett .B158, 231 (1985) ;Commun . Math .Phys . 102, 105(1985) ;J. Manes and B . Zumino, in 5upersym - metry and its Applications :Superstrings, Anomalies, and Supergravity , eds. G. W. Gibbons, S . W. Hawking, and P . K. Townsend (Cambridg e University Press, Cambridge, 1986} . 19a.See,e.g.,H.Georgi,Ref.10: Eq. (XXV11.9). 20. R. Stara, in New Directions in Quantum Field Theory and Statistical Mechanics -Lectures at the 1976 Cargese Summer School, eds. M . Levy and P . Metter (Plenum, New York, 1977) . 21. J. A.Dixon, Ref.19. 420 22Anomalie s 22. G. Barnich and M . Henneaux, Pays . Rev . Lett .72, 1588 (1994) ; G. Barnich, F . Brandt, and M . Henneaux, Common . Math .Phys .174, 57, 93 (1995) . 23.W. Troost, P . van Nieuwenhuizen, and A . Van Proeyen, Nucl . Pays . B333, 727 (1990) . 24. 1 learned about this proof from B . Zumino, private communication . 25. E. Witten, IV ucl.Phys .B223, 422 (1983) . 26. C-S . Chu, P-M . Ho, and B . Zumino, Berkeley preprint LBL-38276, UCB-PTH-96/05, hep-th/9602093, to be published (1996) . 23 Exte ndedFieldConfiguration s Most of this book has been devoted to applications of quantum field the- ory that can at least be described in perturbation theory, whether or not the perturbation series actually works well numerically . In using perturba- tion theory, we expand the action around the usual spacetime-independent vacuum values of the fields, keeping the leading quadratic term in the expo- nential exp(il ), and treating all terms of higher order in the fields as small corrections . Starting in the mid-1970s, there has been a growing interest in effects that arise because there are extended spacetime-dependent field configurations, such as those known as instantons,1 that are also stationary `points' of the action . In principle, we must include these configurations in path integrals and sum over fluctuations around them . (In Section 20 .7 we have already seen an example of an instanton configuration, applied in a different context .) Although such non-perturbative contributions are often highly suppressed, they are large in quantum chromodynamics, and produce interesting exotic effects in the standard electroweak theory . There are also extended field configurations that occur, not only as correction terms in path integrals for processes involving ordinary par- ticles, but also as possible components of actual physical states . These configurations include some that are particle-like, such as as magnetic monopoles2 and skyrmions,3 which are concentrated around a point in space or, equivalently, around a world line in spacetime . There are also string-like configurations,4 similar to the vortex lines in superconductors discussed in Section 21 .6, which are concentrated around a line in space or, equivalently, around a world sheet in spacetime . Then there are config- urations that are sheet-like, like the domain walls between spatial regions in which discrete symmetries are broken in different ways . In contrast, the instantons mentioned above are event-dike, concentrated about a point in spacetime, and therefore never appear as components of actual physical states . Some extended field configurations are stabilized because of boundary conditions that are imposed by the nature of the problem in which they appear . An example is the `bounce' solution, which appears in th e 421 422 23Extended Field Configuration s analysis of vacuum decay,6 and will be discussed in Section 23.8. Other configurations are stable because they carry a quantum number whose conservation forbids any possible decay mode .7 In this chapter we shall mostly be concerned with extended field con- figurations that are stabilized by their topology . In analyzing all such configurations we use the same topological tools, chiefly homotopy theory, so we shall begin by considering all topologically stabilized configurations together, in a space or spacetime of arbitrary dimensionality d . 23.1The Uses of Topolo gy It often happens that the space of all possible field configurations may be given a non-trivial topology by the condition that some functional S of the various fields is finite . In classical field theory S is the potential energy (or in some cases the potential energy per unit area or per unit length) ; no finite perturbation can produce a configuration where this is infinite . In classical statistical mechanics S is the Hamiltonian, and in quantum field theory formulated in Euclidean spacetime S is the Euclidean action or is proportional to it . (Euclidean path integrals and some of their applications are discussed in Appendix A of this chapter .) We construct a perturbation theory by starting with some equilibrium field configuration for which the Euclidean action or Hamiltonian is finite, and then integrating over fluctuations which leave it finite . Two field configurations are said to be topologically equivalent if it is possible to deform one of them continuously into the other without passing through forbidden configurations with S infinite . This is evidently an equivalence relation (in the sense of being reflexive, symmetric, and transitive), and therefore divides the set of all field configurations into equivalence classes, each consisting of configurations of the same topology . For example, if S is the potential energy (the Hamiltonian for time- independent fields) in d space dimensions, then topologically different field configurations are forbidden by an infinite energy barrier from being transformed into one another . In particular, extended configurations with a different topology from the usual spatially uniform vacuum fields cannot spread out to become spatially uniform . The topological classification is also useful when we are looking for a local minimum of S . If we can find a configuration that minimizes S for all configurations of a given topological type, then that field configuration must be at least a local minimum of S for all configurations of any type, since no small variation of the fields can change their topological type . Such a configuration is therefore a solution of the field equations, which are equivalent to the condition that S is stationary . This sort of problem 23.1 The Uses of Topology 423 comes up not only in stability problems, where S is the Hamiltonian, but also in finding field configurations around which we may expand the field variables in path integrals in Euclideanized d-dimensional spacetime . Here S is the negative of the Euclidean action I, and we must look for a local minimum so that the leading term in the expansion around this configuration will be a quadratic free-field action with second-order terms of the right sign . Here are some examples : (a) Skyrmions, etc . Consider the real Goldstone boson fields iru associated with the spontaneous breakdown of a continuous global symmetry group G to a subgroup H . As we saw in Chapter 19, the potential energy for these Goldstone bosons in a Euclidean space of dimensionality d >2 will take the form S~nl=fddxZ1:gab(n)OinuOinb+... ( 23.1.1) ab where guy, is a positive-definite matrix, and`+...' denotes possible terms ofhigher or der in the deriva tives of it . Alternatively, Eq. (23 .1.1) ca n beregarded as mi nusthe a ction fora Goldstone boson fie ldin a d- dimensionalEuclideanspacetime . Fieldconfigurat ions of f inite S must have Ot 7ra(x) van ishing at i nfinity fasterthan I xI^d/2 (where xj - x;xj), sothatnR(x)mustapproac ha constantnQoo as x--+ oo with a rema inder va nishing faster thanIx~(2-d)/2 . The Go ldstone boson fie lds na a tanypoint form a homogeneous s pace, t he coset s pace G/ H, for w hichit ispossible to transform a ny one fie ldvalue to any othe rby a t ransformationofG, so by a g lobal G tr ansformation itis always poss ible to arrange t hat the asym ptotic limit na. ta kes any specific value, say nay, = 0 .The fie ldn,,{x} thus re presents amapping of the who le d-dimensionalspace, wi th the s phere r = co takenas a si ngle point,intothemanifold G BHofall field values. Now, a d- dimensionalEuclideanspace wit h the (d - I)-dimensional spherica lsurface at infinity identifie das a s inglepoint istopologica lly the same as Sd, the d- dimensionalsphere (that is, the su rface of a (d + I)- dimensionalball), i n the se nsethateither ca n be con tinuous lymapped intothe other. The fie lds n( x) that a pproac hzero as x-> oo maytherefore be classifie daccor dingto thetopologicallydistinctmappin gs of S dinto the ma nifoldG/Hof the fie ldvariables, for w hichthepoint at infinity ismapped into zero . The set o fclasses of suc htopo logicallydistinct mappings Sd w ith o ne po intof Sdmappedinto a f ixedpoint of,/# isknow nas nd (,&), the dthhomotopy group ofthemanifold&.These homotopy grou ps wil lbediscusse d(and t heir grou pstructure ex plaine d inthenext sect ion, andalistof homo tapy grou ps for var ious manifolds 424 23 Extended Field Co nfiguration s is given in Appendix B of this chapter . For the present it will be enough to mention that, although when the manifold &is a linear space the homotopy group 774 ) is trivial (in the sense that any field configuration n(x) that approaches a constant value as x --+oo can be continuously deformed into one in which the field takes that value everywhere), the manifold &- = GBH of the Goldstone boson fields often has a non-trivial homotopy group . In the cases relevant to quantum chromodynamics, of S U(2) 0 SU(2)broken to S U(2) or of S U(3)0SU(3) broken to S U(3), the manifold GBH is the same as S U(2) or S U(3), respectively, for which according to Appendix B the homotopy groups n3 (H) are non-trivial . The topologically non-trivial fields at local minima of the potential energy with d = 3 are known as skyrmtons .3 Baryons like the proton may in some respects be regarded as skyrmion solutions in a pure meson theory . The functional (23.1.1)does not have skyrmion stationary points, unless terms involving higher powers of O;na are included in the integrand . In the absence of such terms, any topologically non-trivial field configurations will have a continuum of values for S, extending down to a lower bound S = 0 at which n becomes singular, so topology cannot stabilize such configurations . This is generally known as Derrick's theorem .9To prove this theorem, note that for any field configuration n,(x) we may introduce another configuration with the same topology , n~ (x) = n ,(x/R ) withRanarbitrary rea l positive sca le fac tor.Thenfortheterms s hown explicitlyinEq. (23.1.1) S[nR]=Rd-zs [n] For d >2 this is a decreasing function of R as R --->0, so there is a continuum of values of S [nR ] extending down to a value S = 0 . Furthermore S [7r] >0, because S [7r] can only vanish if n(x) is constant, which is not possible because we assumed that inis topologically non- trivial . Hence this lower bound is attained only at R = 0, at which n~ (x) becomes singular . Goldstone boson field configurations can be stabilized by adding higher derivative terms to S . For instance, if we take S [n] =T [n] + D [n] wit h T[7r] =JddXZj:g,b(n)OinaOinb?O ab D[7r] =IddX .fah,d(n}onQ ❑nb❑n, '❑nd ~~!0, thenD [n R] = Rd-4D [n],while as before T [nR]=Rd-2T[7r], so S [nR] reaches a minim umata finiteR if 2<d C 4, andin p articular for the phys icallyinterest ing case d = 3 . 23.1The Uses of Topology 425 The pro blemwith th etheory of skyrmio ns is not tha twehave to includehigher-derivative ter mslike D [7r] inthe ac tion. As discusse d in Section 19 .5, we expectsuch term sinthe actionofany effectivefield theory of Go ldstone boso ns.Theproblemis that thereisno ratio nale for exc luding an i nfinitenumber of o therhigher-derivative terms, a llof whic hare ge nerically of t he same or der of magn itude forconfigurations thatare sta bilized by abalance between terms w ith diff erentnumbers of derivatives, w hich makes rea listic ca lculationsimposs ible. (b) Domain boundaries . When a discrete symmetry is broke n, we have thepossibilitythatthe symmetry is bro kenin differentways in different domains, separa ted bydoma in b oundariesinwhichthe vacuum fie lds make atransitionfrom one m inimumof thepotentialto ano ther. For instance, co nsider a flat doma in boundaryin th e y-z plane, a ndsuppose that the energyper unitarea is give nby S 101 £d x where O(x} is a real scalar field that is assumed to depend only on th e distance x along the direction normal to the boundary, and V(O) is a potential satisfying the reflection symmetry 0-} - 0, with minima only at field values ±~ . For convenience, we will adjust an additive constant in V(O) so that the minimum value of V(O) is zero, in which case V(O) 0, and V(O) = 0 only at 0 = ±~. To keep S finite, it is necessary for c~ to approach either +~or - 0as x --*oo, and also to approach either + 0or --~as x --*-oo. We then have four topologically distinct configurations, in two of which 0approaches the same limits as x -} ±oo, so that the configuration can be smoothly deformed into the vacuum configurations with O(x) constant everywhere, and in two of which 0approaches opposite limits as x --*±oa, which are topologically stable . Here we are classifying field configurations according to no(G), where n~ (,&) for any manifold A/ is conventionally defined as the set of connected components of -&,a G is the symmetry group, which in our case is the Z2 group of reflections This is a good place to introduce a trick due to Bagomol'nyi'O that will prove useful in Sections 23 .3 and 23 .5 in dealing with the more complicated cases of monopoles and instantons . Rewrite Eq .(23.1.2)in the for m S101 =2 foo dxdoT Vr2_V _(o)2 dx±000 ,~ ~ ;~[_lV'f2 . (23.1.3) The integra linthe seco nd term o n the rig ht-hand side of Eq . (23 .1.3) can be regar dedas a ` topologicalcharge,' w hich depends only on the values taken bythe fie ldatx -> ±Ga .For co nfigurat ions t hat ap proach the same limit as x -> +oo a ndx -> -Ga, this integra lvanishes, a nd theminimum 426 23 Extended Field Configuration s value of S is zero, reac hedfor co nstantfields. For a fie ldO(x) that takes different values a tx = ±oo, we ca nchoose t he ± s ign in Eq. (23 .1.3)to yieldalower boun d S101 (23.1.4)~Lv2vndf . This bound is reached when the first term in Eq .(23.1.3)vanishes, or i n otherwords, when foO(x)dfx ± 2V + x° , (23.1.5)(.f) where xO is a n integrat ionconstant, whichevidently givestheposition of the ce nter of thedomai n boundary . Note thatDerr ick's theorem is no obstacleto a so lution here, because fo r domai nboundaries d = 1, so for a resca ledfieldO(xl R) the integrals of thetwo ter ms inthe i ntegra ndin Eq. (23 .1.2) go as R-1 a nd R +i, res pectively. Eq. (23 .1.5) cou ldhave been o btained m ore direct ly, by deriving a seco nd-orderdiffere ntialequation for O(x) fro mthe co nditionthat Eq. (23 .1.2)must be stationary under sma llvariatio nsin O(x), a ndthen using thisdifferential equation to showthat thequantity2(dc~/dx)2-V(c~) is co nstantinx.The adva ntage of t hederivatio n basedon th e for mula (23.1.3) is thatitshows imme diately that the so lution (23 .1.5)is stable agai nstsmall perturbations that maintain th eflatness of theboundary, aside fro mthe `zero-mo de' associa tedwithchanges i n theboundary lo- cation xO .By adding a term 2 (dc~ ldy)2+ Z (dry/dz)2inthe integran d of E q. (23 .1.2), we can see thatthis sol utionis also sta ble agai nst any perturb ationSo(x, y, z), providedSo(x, y, z) --+0forx --+ boa wi th fi xed y andz. Ifthere are discre te spontaneous ly brokensymmetries then domain boundaries wou ldhave for medwhenthese symme tries became broken in the ear ly universe .Ifthe do mainboundaries d idnot disappear t hey wou ldproduce gross distortions of the obse rvedisotropy an dhomo- geneity of thepresentuniverse.5Wedonot nowknow o fany ofany exact discrete symmet ries exceptCPT , orof a ny spo ntaneous ly bro ken approxima teorexact discretesymmetries, so for thepresent th is is nota problem. (c)Instanton s,etc. Now consider a gauge theory, wit h S [A]=a f ddx F xijFxi j,(23.1.6) where FxiJ is the usual field strength tensor, and we take d ~ 4 . This can either be regarded as the action for quantum gauge fields in a Euclidean 23.1The Uses of Topology 427 d-dimensional spacetime, or the potential energy for classical gauge fields in temporal gauge, with A~= 0, in (d + I)-dimensional spacetime . In order for S [A] to be finite, Fxij(x) must vanish as x -} oa . This can be achieved by having A,,t(x) vanish sufficiently rapidly as x -> 00, but even for d ~ 4 it is also possible for 5 [A] to be finite for a field A,i{x} that vanishes as slowly as 111x , as long as the field approaches a pure gauge as jxj - -+co itaAcci (x)-*g-1( x) ~ig ~x) a (23 .1.7) where g (x) is a direction-dependent element of the gauge group G . Fur- thermore A ai(x) is unaffected if we replace g(x) with g og(X^) for any fixed group element g o eG, so by choosing g o= g- 1 (x2) we can arrange that g(xl)=Ifor any one direction x j. Each gauge field of finite S [ A] therefore defines a mapping from the unit sphereIx~ = I to the group manifold, with the pointAl mapped into the unit element of G .(In the case in which the gauge field vanishes faster than 1 /jxjas fix '-} oa, this mapping takes all points on the unit sphere into the identity element of the gauge group .) The set of classes of such topologically dist inct mappings S d_1~--*G, with one point of S d_i mapped into a fixed element of G, is known as n d_1(G), the (d - 1)th homotopy group of the group manifold .As indicated in Appendix B of this chapter, 73(G)is non-trivial for any semisimple Lie group G . The topologically non-t rivial stationary points of 5 [A] for d =4 are known as instantons .1Their importance in quantum chromodynamics is discussed in Sections 23.5 and 23 .6. In order for S[A] to be stationary at a field A(x), it is necessary that A(x) should satisfy the field equation s OjFIXi j = 0. (23.1.8) A simple scaling argument again limits the values of the dimensionality d where we can hope to find a topologically non-trivial local minimum of S [A] . Define AI(x) - A(x /R)/R.Then S [AR]=Rd-asLAS, so for d =~4 there can be no topologically non-trivial stationary point of S [A] unless S [A] = 0 . But if S [A] = 0 then Frj= 0 everywhere, so by a gauge transformation we can make Aat also vanish everywhere . As we shall see in Section 23.5, for d = 4 it is possible to find instanton solutions where S [A] (here identified as -I[A]) is stationary, with F "iJ not equal to zero except at infinity . The scaling argument above shows that if A(x) is such an instanton solution then so is A(x/R)/R, but this degeneracy is removed by quantum corrections . 428 23 Extended Field Configuration s (d)Monopole s,vortex lines ,etc.Now consi der a t heory of gauge fie lds together wi thscalarsthatfurnishalinear re presentatio nof the gauge grou p, with s [O,A]=IddX iY:$ab(O)DiOraDiOb + abdFaijFaij + U(O) a (23 .1.9) where gab(O) is a positive-definite matrix (usually 0-independent), U(O )is bounded below, and shifted by a constant term so that its minimum value is zero, and Fats and Di are the usual field strength and gauge-covariant derivative . We require that U(O)is a scalar and gab(O) is a tensor under transformations in the gauge group G . Again, Eq . (23 .1.9) gives either the action for a quantum field theory in Euclidean d-dimensional spacetime, or the potential energy for a classical field theory in temporal gauge in (d + 1)-dimensional spacetime . For S [0, A] to be finite it is necessary for U(O(x)) to vanish as x --* oo . The set of cis at which U(O) vanishes is invariant under G and may be discrete or continuous . In case (b) above we have dealt with an example where this set is discrete . Let us now consider the broken symmetry case where the zeros of U(O )form a continuous manifold 0 consisting of fields related by transformations g c G . In this case each 0(x) may be obtained by a transformation y(x) E G acting on the value O(xl) of the field in any one direction xl . We may therefore consider the field O(x) to define a mapping Sd_1 ~ G/H into a coset space G/H ; in other words, into the group G with elements gl and 92 identified if they differ only by right multiplication with some element h of the subgroup H =G that leaves 4i) invariant, that is, if gl = g2h . In particular, the point xl is mapped into the subgroup H, in order that y(xl) acting on 0(xl) should yield 0(xl ) itself . The fields that approach values on the manifold &0 as x -> oo may therefore be classified according to the topologically distinct mappings of Sd_i into GBH that map the point xl into the fixed `unit' element Hof GBH . The set of classes of such topologically distinct mappings Sd_1 r- -*GBH with one point of Sd_1 mapped into a fixed element of GBH is known as nd-t (GIH), the(d - 1)th homotopy group of the manifold G/H . In this case Oao(x) goes as 1/~xj for x --+ oo . In order for S[ O] to be finite Dick must vanish faster than IxI-d/2 for x -> oo, so it is necessary for itaAa ;(x) to approach y-'(x)Oiy(x) faster than IxI-d/2 for x -> oo . This is a pure gauge field, so the field strength tensor Faij( X) vanishes faster than IxI-d/2-1, which is fast enough to make f dux FIXt jFtjconverge . Derrick's theorem does not apply for the gauge theory defined by Eq. (23 .1.9), but it is interesting to see where the same reasoning takes us. For any given fields O(x) and A(x), again define ^ x} - O(x/R),and now also A'(x) _= A(x/R)/R . The three terms in the Hamiltonian (23 .1.9) 23.1 The Uses of Topology 429 now have t he sca lingproperties T [OR, AR] = Rd-2T[ 0, A] , K [AR]= R1-4K [A],V [OR] = Rd V [O], where T [0,A]-7f ddX E ah gab (O)DiOaDiOb,K[A] = '.I'd d x FajFij~ and V[01 =fdux U(O ).Now, for d>4,S [OR, AR] has no minimum at any finite value of H, so there is no stable configuration with non-trivial topology . For 0 <d<4 there is no difficulty in finding a finite value of R at which S [OR, Ate] is a minimum . In the physically interesting case d = 3, topologically non-trivial field configurations are classified according to the homotopy group n2(G/H), which is non-trivial for a simply connected group G (such as S U (2)) broken to an H containing the U(1) of electromagnetism . The topologically non- trivial classical field configurations with d = 3 are known as magnetic mnnopoles .2 As we shall see in Section 23 .3, their magnetic pole strength is quantized, the different values corresponding to different elements of n2(G/H) . For d = 2, topologically non-trivial configurations correspond to ele- ments of nl (G/H), which is non-trivial when G is a non-simply connected group like U(1)orSO(3), broken either completely or to a discrete sub- group . The topologically non-trivial classical field configurations with d = 2 are the cross sections of vortex lines .One example is provided by superconductivity, where G = U(1) is spontaneously broken to H = Z2 . We have seen in Section 21 .6 that vortex lines occur in type II super- conductors for magnetic field strengths in a certain range, and that the magnetic flux carried by a vortex line is quantized, the different values of the flux corresponding to different elements of ni (U(1)/2z) . Vortex lines can also occur in relativistic quantum field theories,4 and may be produced in symmetry-breaking transitions in the early universe, in which case they are known as cosmic strings-11 Monopoles and vortex lines share a remarkable feature that can be deduced on purely topological grounds . In both cases the forms of the Goldstone boson fields n,,(x) on large spheres (SIfor vortex lines, S 2for monopoles) surrounding the configurations are twisted, in such a way that they cannot smoothly be deformed into constants . In particular, it is impossible smoothly to reduce the radii of these spheres to zero without encountering some sort of singularity, because a non-singular field na(x) on a sphere would have to become a constant as the radius of the sphere shrinks to zero . The singularity in both cases occurs in a core (a line or perhaps a tube for vortex lines, a point or perhaps a ball for monopoles) within which the group G is no longer broken, so that the system is no longer described by Goldstone boson fields, but by an order parameter that transforms linearly under G . For d = 4, the function S [OR, A ']of R can only have a minimum 430 23 Extended Field Configuration s at some finite R if T [O, A] = V [O] = 0, which would require that O(x) everywhere takes a value at which U(O) = 0. Assuming that these values form a continuum related by transformations in the gauge group G, by a gauge transformation they may be made constants, O(x) = O4, Then in this gauge the condition T [0, A]= 0 implies that A,,(x) = 0 for all broken symmetries, for which to =~0. Both T[O,A] and V[O] are stationary at such a field configuration, so in order for S [0,A]to be stationary K [0,A] must also be stationary, which means that the non-zero gauge fields Aj,' (which belong to the subgroup H c:G that is unbroken by 0()) must satisfy the Euclidean Yang-Mills field equation s 01AFjyv = 0 . (23.1.10) This case therefore reduces to case (c), but with the gauge group G replaced with its unbroken subgroup H . 23.2 Homotop y Group s We learned in the previous section to classify field configurations, at which the Hamiltonian or other functionals are finite, in correspondence with the elements of appropriate homotopy groups . But we have not yet explained in what sense the the homotopy groups are groups, nor have we given any physical significance to the group structure . As we shall see, there is a natural definition of the multiplication rule for elements of homotopy groups, according to which two extended configurations of fields forming a manifold in d dimensions, that belong to different elements cl and c2 of nd(,&), can only fuse continuously to form a configuration belonging to the element elxC2of n~(,&) . We will begin by defining the first homotopy group nl ( ) of an arbi- trary manifold -&,alknown as the fundamental group of the manifodd . As we have seen, the existence of a non-trivial ni(GBH) for some coset space GBH is the condition for the topological stability of a vortex line in three dimensions (or a monopole in two dimensions) . After considering nl(,#), we will then move on to more general homotopy groups . A connected manifold &is said to be multiply connected if there is some closed curve of points p(z) on the manifold, parameterized by a single variable z with 0 c z c 1 and p(O) = p(l), which cannot be contracted to a point by a continuous deformation . Since on a connected manifold we can always continuously deform any such closed curve so that any one point on the curve is anywhere we like on the manifold, we may restrict our attention only to curves for which p(O) = p(l) = p o, where po is any fixed point of the manifold, known as the base point .Two such closed curves p1(z) and P2(z) are said to be homotopically equivalent 23.2 Homotopy Groups 43 1 if they can be deformed into each other, that is, if there is a continuous function p (z,t)for0c t c 1, wit h P(z, 0)=Pi (Z) , P(z,1) = P2 (Z) PLO, t} = p 0,0= Po The relation of homotopic equivalence is an equivalence relation in the sense of being symmetric, reflexive, and transitive, so it divides the space of closed curves on the manifold into equivalence classes : two closed curves are in the same class if and only if they are homotoptcally equivalent . The set of these equivalence classes is known as the first homotopy group of the manifold, nI ( ,). To define the multiplication rule for nI (}, choose a standard curve p[z; c] that starts and ends at the base point po for each equivalence class c in n j(.). For any two equivalence classes cI and c 2, define the `product' Clx c2 as the equivalence class containing the curve p[z,Cl,c2] that starts at po, follows p[z, cI ]back to po,and then follows p[z, c2} back again to po Formally, we take p [2z, cl} 0 S z c Z PLZac19 C21 - PL 2Z --- 1, C21 ~ 2 Z C I We must now show that multiplication defined in this way satisfies the conditions for a group . First, let us check that this multiplication is associative . For this purpose, note that (Clx c2 )xC3is the equivalence class containing a curve p[z, cz Xc2,C31 that goes along the standard curve p[z, elx c2] from the base point and back again, and then along the standard curve p[z, c 31from the base point and back again, while Clx(C2xC3)is the equivalence class containing a curve p[z, cI, c2X c3] that goes along the standard curve p[z, el} from the base point and back again, and then along the standard curve p[z, c2 x c3] from the base point and back again . By definition, the curve p[z, Clx c2] may be deformed into a curve that goes along p[z, cI} from the base point and back again and then along p[z, c 2]from the base point and back again, while p[z,c2 x c3 1 may be deformed into a curve that goes along p[z, c2] from the base point and back again and then along p[z, c 3] from the base point and back again . Hence both curves p[z, cI XC2, c3] and p[z, Cl,c2 x C3] may be deformed into the curve that goes along p[z, cj] from the base point and back again, then along p[z, cZ] from the base point and back again, and finally along p[z, c3 1from the base point and back again, and hence they may be deformed into each other, showing tha t (Cl ?C C2 )XC3 = Cl?C(C2XC3). The unit element e of nl (~} is defined as the equivalence class con- 432 23 Extended Field Configuration s taining the curve p[z, e ]= po that stays at the base point . To check that e x c ==c,note tha t p[z, e, c ] 1Pd p[2z- 1,elOC Z C Z 2 C Z C I But this curve can be continuously deformed into p[z, c] by taking 0_PO 0 C z C t /2 P(z,p[(2z - t)I(2 - t ), C} t/2 C zC 1 ' which is the same as p[z, e, c] for t = 1 and the same as p[z, c] for t = 0 . The product e x c is the equivalence class containing p[z, e, c ],which we now see is the same as the equivalence class containing p[z, c], which is just c.The proof that c x e = c is similar . The `inverse' c-I of the equivalence class c is the equivalence class that contains a curve p- I[z, c] which goes around the same path as the standard curve p[z, c] but in the opposite direction ; that is , P-1 ~~, C] =P[I - zy C]  This is not necessarily the same as the `standard' curve p[z, c°-I}, but by the definition of the equivalence class c1 . the two curves may be deformed into each other . To check that c-Ixc= e, note that by deforming p[z, c- into p'[z, c], the curve p[z, c i , c] may be deformed int o z C-1C]-}p[I-2Z,c] 0C Z C Z But this curve can be continuously deformed into p[z, e] = p oby takin g P(z' 0 = p Litz + 1- 2t, c]OCZC 2ZC ~ which is the same as p[z, c- 1, c]for t = Iand the same as p[z, e] for t = 0 . The product cixc is the equivalence class containing p[z, c-l, c],which we now see is the same as the equivalence class containing p[ z, el, which is just e . The proof that c x c-' = e is similar . The existence of a unit element and inverses shows that these equivalence classes form a group . The classic example of a manifold Wwith a non-trivial first homotopy group is the circle itself, ,#= SI . This may be parameterized by an angle Bwith 0= 0 and B =inn (with n any positive or negative integer) identified as the same point . The homotopy groups consist of classes of functions O(z) for 0 c z c 1 that begin at some base point, 0(0) = e0, and end at the same base point, 0(1) = B () +inn. Two such functions may be continuously deformed into each other if and only if they have the same value of n, so nl(Sz) consists of a denumerably infinite number of classes cn labelled by the positive or negative integer n . Furthermore, the `product' 23.2Homotopy Groups 433 of two c lasses cn a ndc,, co nsists of c urves t hatgo n t imes aro und the circle from thebase po intand b ackagain, andthenmtimes arou ndthe circle from thebase po intand back aga in, so here m ultiplication is just addition : CnX Cm = Cn+ m andhence(23.2.1) (23.2.2) where Z is the group of addition of positive and negative integers . As an immediate physical application, we note that when a gauge group SO(2) is completely spontaneously broken, the coset space is just SO(2) itself, which has the topology of a circle, so in this case there is an infinite number of types of topologically stable vortex line, characterized by a positive or negative integer n . For instance, this is the case in a type II superconductor, where as we saw in Section 21 .6the U(I)of electromagnetic gauge invariance is spontaneously broken to a discrete subgroup Z2 . More generally, all spheres Sd with d >1 are simply connected, which means that they have a trivial first homotopy group, a statement conven- tionally expressed as nl (Sd) = 0 for d >1. (23.2.3) Only a few of the more familiar Lie groups are multiply connected : 7c1(G)_Z G = U(k) Z2 G = SO(k ) 0 G = Spin(k ) 0 G = S U(k ) 0 G = USp(2k ) 0 G_Ga, F4,Es, E 7, E8k~1 k~3 k2:3 k~2 k~l(23.2.4) Here Z2 is the group with two elements 1 and -1, with group multi- plication defined as ordinary multiplication, and Spin(n) is the simply connected covering group of SO(n) . (As we saw in Section 2.7,Spin(3) is the same as SU(2) .)Also, for a direct product of two manifolds Wand .', the fundamental group i s 7T 1(. x d) = R r(,#) X 71z(-W) - (23 .2.5) We can appreciate the physical significance of the group structure of nI (. } by asking what happens when two distant parallel vortex lines in three dimensions are brought together . When the vortex lines are sufficiently far apart their fields do not interact, so the configuration can be described by specifying the classes c'and c"in7cI(GIH)to which 434 23 Extended Field Configuration s each belongs . The class c to which the whole configuration belongs is determined by the behavior of the fields on a very large circle surrounding both vortex lines . By a continuous deformation we can distort this eery large circle into two large circles, one surrounding each vortex line, which intersect at a point midway between them . As we go around this closed curve in two-dimensional space we first trace out a curve in G/H which consists of one of the closed curves in the class c', and then one of the closed curves in the class c", just as in the definition of the product of the classes . We conclude then that the whole configuration is in a class c = c' x c", and so the two vortex lines can only fuse together to form a vortex line of this class . In particular, they can only annihilate if C~~ _e~-i For instance, when 7rI (G/H) =Z (as in superconductivity, where G/H -- SO(2)/Z2), in three dimensions two vortex lines of classes cn and cm can fuse together to form a vortex line of class cn+m, so they can only annihilate if n = -m . On the other hand, when 7EI(GIH)= Z2 (as when G = SO(IV) with N _> 3 and H is trivial or discrete) there is only one kind of vortex line in three dimensions, corresponding to the element -1 of Z2, and since (-1 )2 = 1, any two can annihilate . Now let us consider the general homotopy group 71k(.). This is much like 7E1(.), except that instead of considering mappings of the circle SI into a manifold ., we consider mappings of the k-sphere Sk (the surface of a (k +I)-dimensional ball) into , W,agaiwith one point of 5k always mapped into the same `base point' p oof. Two such mappings are equivalent if one can be continuously deformed into the other, keeping the same point of Sk always mapped into the base point . The kth homotopy group nk( ) has elements consisting of equivalence classes of these mappings . It is often convenient to picture the d-sphere Sd as the interior of a d-dimensional hypercube, with all points on the boundary identified as a single point . For instance, we have already seen that the circle 5 ,can be treated as the interval 0 c Bc 2n, with points 0= 0 and 0= 2n identified . Similarly we can make a map of an S 2like the earth's surface by cutting out the south pole and spreading out the resulting sheet on the unit square 0 c z ,c 1, 0 c z2 c I.Continuous mappings of this square into . must take all points on the boundary into the same point of ,, because all points on the boundary are the same point, the south pole . In general, two mappings p(zI,    zd ) and p'(zI,    z d)of Sd into . are homotopically equivalent if one can be continuously deformed into the other, while keeping p on the boundary of the hypercube equal to the base point pa. As before, for each equivalence class c we choose a standard mapping p(zl, - -  z d; c). The product of cZ and c2 is defined as the equivalence class 23.2 Homotopy Groups 435 that contains the mapping p(2ZI, Z2,...Zd;C0 0 C Z 1C P(Zl,Z2 ,...Zdy C1, C2) = p(2ZI - Z2a...Zd;C2) 2 <Z,C~ (23.2.6) The unit element e is defined as the equivalence class that contains the mapping with p = pa for all z, and the inverse C' of c is defined as the equivalence class that contains the mapping wit h P-I[zI, zz,... Irv;Cl= P[l- zI, z2, ... zN;Cl. (23 .2.7) In the same way a s for nI,it can be shown that thi smultiplication i s associative, and that e x c = cx e = cand c f1x c= c x c- I= e. All nn(.) for n ~ 2 are Abelian . (There are manifolds ,. for which 7rl()is non-Abelian , such a sthe plane with two or more point s removed .) In any case in which n k(..#) = Z, there must be a one-to-one mapping of the k-sphere 5 kinto a k-s phere S k in-0, which correspond sto the element `one 'of Z (not the unit element, which is zero) .The element v ofZwith v = 2,3,...corresponds to the mapping of S kinto the same k-sphere S kin which covers Sk v times,with the Jacobian of the transformation S k-> Skpositive. The element v of Zwith v = - 1,-2.... corresponds to the mapping S k-> Sk,which covers S k JvJtimes, with the Jacobian of the transformation S k-}Sknegative . For in stance ,wesawintheprevious s ection that magnetic monopole s arise when a simply connected group G i sbroken to the U(1) of electro- magnetism . In this case the appendix shows tha t n2(G/U(1)} = ni (U(1 )} = Z , (23 .2.8) so a magnetic monopole carries an integer-valued quantum number v, which as shown in Section 23 .3 is proportional to the magnetic charge . This quantum number gives the number of times that a two-sphere of large radius surrounding the monopole is mapped into a two-sphere in the manifold G/ U(1)of Goldstone boson fields (with the relative orientation of the two two-spheres being the same or opposite for v positive or negative, respectively) and is therefore known as the winding number . The structure of the group Z shows that this quantum number is conserved, in the sense that a monopole of quantum number v can fuse with a monopole of quantum number v' only to form a monopole with quantum number v +v'. If the unbroken subgroup is SO(n) with n ~ 3, then according to Appendix B of this chapte r nAG/SO(n)}=7cI(SO(n)} = Z2 . (23 .2.9) In this case there is just one sort of `monopole,' corresponding to the element -1 of Z2, which can annihilate only in pairs . It is important to 436 23 Extended Field Configuration s distinguish between this case and that in which SO(n) is replaced with its simply connected covering group Spin(n), for which there are no monoples . We will come back to this point at the end of the next section . Another example : we saw in the previous section that the skyrmions in quantum chromodynamics with n light quarks correspond to elements of X3(5 U(n)}, which according to the appendix is Z . Thus these skyrmions carry a conserved integer-valued quantum number v, which perhaps may be identified with baryon number . Similarly, recall from the previous section that the instantons in a gauge theory based on a simple gauge group G correspond to the elements of nAG}, which according to the appendix is Z, so these instantons like skyrmions carry an integer-valued quantum number v, also known as the winding number . In Section 23.5 we shall see how to express this quantum number as a local functional of the gauge field . 23.3 Mo nopoles As a detailed example of a topologically non-tr ivial field configuration, weshall now consider the monopole of 't Hooft and Polyako v,2and its general izations . We saw in Section 23 .1 that when a simply connected gauge group G is spontaneously broken to the U(1) of electromagnetism , the configurations of finite energy are classified according to the elements of the group 712(GIU(1)}= 7EI (U(1 )} = Z. (The case of non-simply con- nected Lie groups is con sidered at the end of this section .)According to the physical interpretation of homotopy groups discussed in Section 23 .2, this mean sthat these configurations have a conserved additive quantum number . But we st ill need to show that any of these stationary con- figurations actually exist ,and to give a physical interpretation of their topological quantum numbers . As an illustrative example, consider a theory (like the Georgi -Glashow electroweak model1 2)in wh ich an SU(2) gauge group is spontaneously broken b ythe vacuum expectat ionvalue o fanSU(2) triplet of scalar fields 0,(It is explained at the end of th is section why inthis case we say that the gauge group i sSU(2)rather than SO(3) .} The Lagrangian den sity for the scalars and gauge fields in Minkowskian spacet ime is taken a s 4F►zjjvF~tv -ZDFj0nD ~on- V(OnOn) , (23.3.1) where Fn,uV = OpAn u - rev Ay :11 +e enmlAm F ,A,va (23 .3.2) D -0 and the function V(OnOn}is assumed to be positive, with the value zero 23.3Monopoles 437 at a non-vanishing value ~ O)(taken positive) of (In much work on monopoles Vis taken to be the quartic polynomial A (OnOn --- WY , with A>0, but we shall not make this assumption here .) Eq .(23.3.1) describes a theory with a spacetime-independent vacuum solution with Any = 0, in which a vacuum expectation value can with 0,0, = ~0)2 breaks S U(2)to its U(1)subgroup, which can be identified with the gauge group of electrodynamics . Instead we here shall seek topologically non-trivial inhomogeneous but time-independent classical solutions in temporal gauge, with An = 0 but An ~ 0 . The Lagrangian density in this case is the negative of the potential energy density ,*, given b y ,X,,=aFnij+i~~i~n) 2+ V~~~ ~ (23.3.4) with squares including obvious index contractions . Because each term in (23.3.4) is positive, the integral of each term must separately converge in a configuration of finite energy . In particular, for the integral of V(OnOn) to converge, the vector On must have the fixed length ~ O} at infinity, so each configuration of finite energy defines a smooth mapping of a large two-sphere Y surrounding the monopole configuration into the two-sphere of the 2 0,with 0,0,= (0 )As x runs over .51, can may run over the sphere 0,0, = ~0)2any integer number Nof times, either with Jacobian Det (Ox/r7O) positive, in which case we say that the winding number is N,or with Jacobean negative, in which case the winding number is -N . To see what the winding number has to do with the magnetic monopole moment, we must first consider what in this theory is observed as `the' magnetic field . Whatever the field configuration, we can introduce a gauge in which the scalar field can points in some definite direction, say the three-direction, in any given finite region, so that in this region the gauge field associated with the unbroken U(1) subgroup of SU(2) is A3 i. 't Hooft2 found a gauge-invariant tensor which reduces to the usual electromagnetic field strength tensor OFzA3,, -r7,,A31z in this gauge : A, I A A, A ,~FFlv = F'nFev On - e 6nmlOnDFtO►nDv0l(23.3.5) where c~~ =can/ c~mc~m To check that is the ordinary elec-.. tromagnettc field strength tensor in a gauge with constant can (and for later purposes) we use Eqs . (23 .3.2) and (23.3.3)and the identity 'Fubc,Fude =60 ce-60 cdto write F,,, in the form13 r r A A ~v = aF~lOnAnv ) - a vl0,,A,,,) - ~ Enm [0n OU0m OV 01 . (23 .3.6) Thus in a gauge in which On is a fixed unit vector in the three-direction, 438 23 Extended Field Configuration s wehave as pro mised ,uv=OuA3v - OvA3u . The magnetic monopole moment g of any localized field configuration is defined as 1/47E times the magnetic flux through a large closed surface 9around the configuration : 4Trg = ~~'a .ikL,jjd2s k (23 .3.7) The first two terms in Eq . (23 .3.6) for 3W'ijare derivatives, and therefore do not contribute to the integral (23.3.7),so tha t 1 f,. , . 2 98ne6'jk~nrnlJ~on O iom0joldSk. (23.3.8) This has the important property of being a topological invariant- inte- grating by parts where necessary, we can see that the change in g under,. an infinitesimal variation 60„ in 0n is 3 f A .. ~g S7te~ajk~'nml JfOnOaOmajOt L1~2'Sk .`. +' . A A But because 0 is a unit vector, 60 as well as r7iO andaj0 are all in the plane perpendicular to 0, s o ~nml6 0na i0ma .j0I= 0 , and therefore r5g = 0.The quantity (23.3.8)is related to the topological invariant known as the Kronecker index . Because g is a surface integral, it is additive : for any two distant localized configurations, the surface 9'used to calculate g may be taken to be a pair of spheres, one surrounding each configuration, connected by a thin neck between them, so the value of g for the whole system will be the sum of the values of gZ, 92 for the individual localized configurations . Furthermore, since g is a topological invariant, we will have g = 91+ 92 for any field configuration that is formed by a smooth fusion of two configurations with magnetic monopole moments g ,and 92- It follows that g must be proportional to the winding number . Arafune, Freund, and Goebel13 verified this and calculated the coefficient of proportionality by using the formula (23.3.8)for general winding number . Here we will simply calculate the coefficient by studying the 't Hooft-Polyakov monopole,2 in which the fields have unit winding number . As we saw in the previous section, the `identity' (as opposed to the unit) element Jof7c2(SU( 2)/U(1)}, which corresponds to the element `one' of Z, consists of configurations in which the two-sphere 9at infinity is mapped once (with positive Jacobian) into the sphere described b y As a representative of this homotopy class, we may take a configuration 23.3Monopole s in which at infinity this configuration let 0 = 101 ,02,031and ansatz :439 can is in the same direction as x . To construct us impose a symmetry under joint rotations on x, as well as parity conservation, and make th e On=Gn ~o)Fir)~ en~rxrG(r) Ani= .er(23.3.9) (23.3.10) There is an important similarity between this field configuration and that for a vortex line in a superconductor . The solution 0=~-~-~rpl2e for the Goldstone boson field in a vortex line found in Section 21 .6 shows that, although gauge invariance and rotational invariance are spontaneously broken, the vortex line solution is invariant under the combination of a global gauge transformation for which 0 --+ 0 + A and a rigid rotation rp -> rp ± 2eA/,~,.Similarly, a monopole solution of the form (23 .3.9)- (23.3.10)is not invariant under rotations or gauge transformations, but it is invariant under the combination of a rigid three-dimensional spatial rotation and an equal global 50(3) gauge transformation . As already mentioned, in order for the integral of V(0n0n) to converge it is necessary for c~nc~n to approach (0)2 as r -> oo, so in this limit F(r) -> 1. To derive the limiting behavior of G(r ), note that the covariant derivative of the scalar field i s Den ~ 0) [( I -G(r)) (6n i - xr~xi) F( r)+ xnxiF'(r)r so the scalar term in the Hamiltonian density i s lDi~r~ l2 F2`1- C~i)2 F~2 r2 + 2(23.3.11) For this to have a finite integral it is necessary also that G(r) --+1 and F'(r) -> 0 as r -> cc . Finally, the field strength (23 .3.2)is Fnii=eek [_.!G'(r)( & r, ,--,Ai ) - ~ ~~G( r~ - G~( r'~} xnx : sothe Ya ng-Mi llsterm in theHamiltonian densi tyis a( Fnrj)2=2eGi2 (2G - GZ)2 r2 + 2r4(23.3.13) (23.3.14) This has an integral that converges at large distances as long as G'(r) vanishes sufficiently rapidly as r -> oo . We can use these results to calculate the magnetic monopole moment of this configuration . Eq.(23.3.13), together with the limits G(oo) = 1 and 440 23 Extended Field Configuration s G'(oo) = 0, shows that for r --+oo, we hav e ^ iFnijEijkXkXn er2 Since DaOn vanishes rapidly as r -> oo, the magnetic part strength tensor in this gauge is given for r -> co by the Eq.(23.3.5),so at large distances the magnetic field become s A ^ Xi $i = 2 ~'i jk~ jk ~ 2 Ei jkOnFn jk 1 erg(23.3.15) of the field first term i n Thus this configuration hasmagnetic monopole moment g = Ile. According to the general argument above, the magnetic moment of a configuration with winding number v, corresponding to the element v of Z, is the n gy =vie . (23.3.17) It requires a detailed numerical calculation to find the stable config- uration that minimizes the integral of the Hamiltonian density (23 .3.4). However there is a limiting case where an analytic solution is available . To see this, it is useful first to derive a general lower bound due to Bogomol'nyi10 on the monopole energy for a given magnetic monopole moment g . Note that (23 .3.4) may be writte n 2 _1Y°' q (Fnij+ Fijk DkOPI) ± 21FijkFnij DkOn+V(OnOn) (23.3.18) Using the Bianchi identity (15.3.9), the second term may be writte n ± 2 QjkFni1 DkOn = ± ?E ijkDk(FnijOrt) = ± 2 E'tjkak(FnijOn) so its integral is given by the magnetic monopole moment g ± ~~' i.ik fd3x FnajDkOn=~-~-~O) fBdA = -4~~O)g Since every other term in W'is positive, we have a general lower bound on the energy of a configurat ion with magnetic monopole moment g E ^/d3xr ~ 47r~O)IgI. (23.3.19) For g = ~-~-Ile, this gives an energy E ~ 4-r(O)l e,which for small coupling constant e is much greater than the corrections due to quantum fluctuations ,which are at most of order ~O).This i swhy we can take such a class ical con figuration seriously a sthe leading term in a perturbat ion expansion . Now ,itis tempting to try to minimize the energy for a given magnetic monopole moment by setting the first term in Eq . (23 .3.18)equal to zero, so that Fnij = ±FtjkDk On , (23 .3.20) 23.3 Monopoles 44 1 but in general this does not lead to a configuration at which the energy is stationary . The condition that the energy should be stationary with respect to variations in the scalar field is the field equatio n DkDk On = 20nV'(OnOn), (23.3.21) while Eq .(23.3.20)together with the Bianchi identity (15.3.9)would imply that DkDkran = 0 . This argument suggests that in the special case where V(OnOn)is very small, it will be possible nearly to reach the lower bound (23.3.19), and hence minimize the energy for a given magnetic monopole moment, by imposing the condition (23.3.20).(Where V is the quartic polynomial).(Onon - (0)2)2, the assumption that V is small means that ,~CC e2, as in a type I superconductor .) Such stable configurations were studied in this way by Bogomol'nyi1° . They had been found earlier by Prasad and Sommerfeld14 without direct use of Eck . (23 .3.20), and are usually called BPS monopoles . The Bogomol'nyi condition (23.3.20) provides first-order differential equations for F(r) and G(r), which are much easier to solve than the second-order field equations derived directly from the condition that the energy should be stationary . Using Eqs .(23.3.11) and (23.3.13), the terms inEq. (23 .3.20) proportional to eijk [Ski - xkx, ,] and Fi jkXkXn respectively yield the differential equations (23.3.22) (23.3.23) With the boundary condition that F(r) --+1 and G(r) --+1 as r --+ ao, these equations have the solutio n F=coth 1 ' G=t -- p P P Binh p(23.3.24) where p - e~ O}r.Note in particular that the field On given by Eqs . (23.3.9) and (23 .3.24) vanishes for r --+ 0, so as remarked in Section 23 .1, the S U(2) symmetry is restored at the center of the monopole . Let's now return to the case of a potential V of arbitrary strength . The 't Hooft-Polyakov monopole is stable, because are no configurations of smaller topological quantum number into which it could decay . Config- urations of higher magnetic monopole moment are generally unstable? There are also interesting configurations with both magnetic monopole moments and electric charge, known as dyons . 15 There is another way of understanding the value Ile of the 't Hooft- Polyakov magnetic monopole moment, which goes back to the original work of Dirac on magnetic monopoles .16 As mentioned earlier, instead of the gauge we have been using, we can make a gauge transformation on --+ Rn,,,(x)om that rotates On COpoint in a fixed direction V, for instance 442 23 Extended Field Configuration s the three-direction . Then the field strength $nk=z Fi jk Fni j is transformed into Rn~B,,.k, which approaches vnackler2 for r ~ ao, so here we do not have to project this on a local unbroken symmetry direction . The price we need to pay for these conveniences is that the gauge transformation is singular ; the rotation that takes a vector in the x direction into some fixed direction v i s R(x;v) =1- ~1-vx}vvT+v(x- (xv) v) T +( x- (x-v) v) v-" ( x- (x_ v}v) ( x- (xv) v) - + (23 .3.25)lxv which is singular at x = -v . This Ris not unique ; for instance, we could perform the rotation R(x ; -v} that tapes x into the -v direction, followed by a fixed rotation of 180° around some axis perpendicular to v, but this would become singular at x = +v . To avoid the singularities, we have to adopt different gauges in different regions ; for instance, with v in the three-direction, we can use a gauge for 0 < 0 < (gyp that is singular at 0 = it and a gauge for 00 c B <it that is singular at 0 = 0, where 0a is an arbitrary angle with 0 < B0 < it, often taken as 7c/2 . Everywhere except at 0 = 0 and 0 = it the magnetic fie ld will be given at large distances by B - -+gx/r2 , where g is the magnetic monopole strength . This can be written as the curl ❑xAof a vector potential whose only non-vanishing component is in the azimuthal rp direction . For 0 < 0 < Fop we must take A,, = g(1 - cos 0)/r sin 0, which is only singular at 0 = 7r, while for 00 < 0 <it we must take A,,= -g (1 + cos O) firsin 0, which is only singular at 0 = D . The difference ❑A between these two vector potentials is a gradient ❑A with A = 2grp, which of course does not affect the magnetic field for 0 < 0 < it, but could affect the dynamics of charged fields . A gauge transformation with A = 2grp will change a field of charge q by a factor exp(2iggrp), which is not single-valued unless 2qg is an integer . This is the Dirac quantization condition ; the existence of any magnetic monopole with monopole moment g would require all electric charges to be integer multiples of (2g)-1 . For the 't Hooft-Polyakov monopole this condition is automatically satisfied, because here g = 11e, and all charges in the Georgi-Glashow model are integer multiples of e/2 . The Georgi-Glashow model was ruled out as a theory of weak and elec- tromagnetic interactions by the discovery of neutral currents, but magnetic monopoles are expected to occur in other theories, where a simply con- nected group G is spontaneously broken not to U(1), but to some subgroup H' x U(1), where H' is simply connected . (According to Appendix B of this chapter, for simply connected groups G we have n2(G/H) = nl(H), which for H = H' x U(1) equals 7r1( H')Xnl(U(1)} = n1(U(1)} =Z.) There are no monopo les produced in the spontaneous breaking of the gauge group 23.3 Monopoles 443 SU(2) x U(1) of the standard electroweak theory, which is not simply connected . (About this, more below .) But we do find monopoles when the simply connected gauge group G of theories of unified strong and electroweak interactions, such as SU(4) xSU(4) or SU(5) or Spiny 10 ), is spontaneously broken to the gauge group SU(3) x SU(2) x U(1) of the standard model . (See Section 21 .5.) The monopoles in this case are expected to have a mass larger by an inverse square gauge coupling con- stant than the vecto rboson masses M  10 15_ 1016 GeV produced by this symmetry breaking . Such monopoles would have been produced when the universe underwent a phase transition in which G was spontaneously broken to SU(3) x SU(2) X U(1), at a temperature T of order M. This poses a problem for some cosmological models .17 The scalar fields before this phase transition would have necessarily been uncorrelated at distances larger than the horizon distance, the furthest distance that light could have travelled since the initial singu larity . At an early time t in standard cosmological theories18 the horizon distance is of order t.= (G NT4)-112 (where GN ^f (1019 GeV)-2 is Newton's constant), so the number density of monopoles produced at this time would have been of order t-3 .:: (G N1Vf4)3/2, which is smaller than the photon density M3 at T ~t : M by a factor of order (GN M2)3/2 . For M .~: 1015 GeV this factor is of order 10-5 .If monopoles did not find each other to annihilate, then this ratio would remain roughly constant to the present, but with at least 1D9 microwave background photons per nucleon today, this wou ld give at least 103 monopoles per nucleon, in gross disagreement with what is observed . This potential paradox was one of the factors leading to inflationary cosmological models,19 in which there was a period of exponential expansion, which if it occurred before the monopoles were produced would have greatly extended the horizon, and if it occurred after the production of monopoles (but before a period of reheating) would have greatly diluted the monopole density . The discovery of monopoles of any sort would create opportunities for the observation of remarkable phenomena, including the existence of fermion-monopole configurations of fractional fermion number,'° and the violation of baryon conservation in fermion-monopole scattering .21 In the above discussion we have considered only monopoles associated with the spontaneous breakdown of a simply-connected gauge group G . This raises a question . For every Lie group G, whether simply connected or not, there is a simply-connected group G with the same Lie algebra, known as its covering group . (For examples, see Section 2.7.)Any non- simply connected group has fewer representations than its covering group (for instance, the doubly-connected groups SO(n) have only scalar, vector, 444 23Extended Field Configuration s and tensor representations, while their covering groups Spin(n) also have spinor representations) . If a theory does not happen to involve fields that belong to the extra representations of the covering group, then are we are free to consider the gauge group of the theory to be either the non- simply connected group G or its covering group G? In particular, does the menu of possible monopoles depend on whether the theory contains only fields transforming as representations of a non-simply connected group G, or additional fields that would only furnish representations of its covering group G? For instance, in the original Georgi-Glashow model12 the only scalar fields belonged to a representation of S O(3), the three- vector, but there were fermions that belonged to spinor representations of the covering group Spin(3) =SU(2) .Would the allowed values of the magnetic monopole moment change if we added scalar fields belonging to the spinor representations of SU(2)? Would it change if we removed the fermions ? The answer is that the menu of possible monopoles does not depend on whether we say that the gauge group is a non-simply connected group G or its covering group G, and therefore is unaffected when we add or remove fields belonging to representations of G that are not representations of G . As we saw in Section 23 .1, in general the topologically stable monopole- like configurations are classified according to the elements of 7r2(G/H) . According to a result quoted in Appendix B of this chapter, this homotopy group consists of those elements of n1(H)that correspond to the trivial element of n1(G) when H is embedded in G . But if we replace G with its covering group G, we also replace H with a different subgroup H', because some of the loops in H do not return to the base point when H is embedded in G . These are just the loops that do not become trivial when H is embedded in G, so 712(G/H )= 7r1(H'), and thus as far as monopoles are concerned we could just as well say that the gauge group is G rather than G . For instance, as far as the scalar fields are concerned, the gauge group of the Georgi-Glashow model might be considered to be the doubly- connected SO(3) rather than its simply-connected covering group SU(2) . The unbroken subgroup is then SQ(Z ), in which we identify transforma- tions that differ by a 360° rotation . Thus 7r,(SO(2)} includes loops that extend from the unit element to a 360° rotation, which would not be loops when SO(3) is embedded in SU(2) .But7E2(Sn(3 )/SQ(2 )}is not the same as 7r,(Sa(2)), but rather excludes the loops that are hornotopically non-trivial when SO(3) is embedded in SU(2), which are just the loops that extend from the unit element to a 360° rotation, so 7E2(SO(3)/SQ(2 )} is the U(1) subgroup of SU(2), just as if the gauge group were taken to beSU(2) from the beginning . It is convenient always to consider the gauge group G associated with 23.4 The Cartan-Maurer Integral Invariant 44 5 any semi-simple gauge algebra to be the simply-connected covering group, so that we can use the simple result that nZ(G IH) - n1(H) . As we have just seen, the connectivity properties of H are then fixed by its embedding in G, or more precisely, by the embedding of the Lie algebra of H in the Lie algebra of G . For instance, an SU(3) gauge algebra might be spontaneously broken either into an SU(2) sub algebra, under which the defining representation of S U(3) transforms as a doublet plus a singlet, or into an SO(3) subalgebra, under which the defining representation of SU(3) transforms as a three-vector . In the first case we do not have the option of considering the unbroken subgroup to be SO(3) ; since nl(S U(2 )) = 0, there are no monopoles here . In the second case the unbroken gauge group must be regarded as Sa(3), not SII(Z), so the theory does have monopole-like configurations, classified according to the elements of n, (S O(3)} = Z2 . The nature of the unbroken subalgebra of H and its embedding in the gauge algebra G may be dynamically affected by the variety of field types that we introduce in the Lagrangian, but once the algebra of H and its embedding in the algebra of G are fixed, the menu of monopoles is otherwise entirely unaffected by the variety of fields in the theory . In particular, the argument that led to the Dirac quantization condition shows that, in any theory in which a Lie algebra G is spontaneously broken to a subalgebra including the electric charge operator, the allowed magnetic monopole moments are integer multiples of the reciprocal of the smallest electric charge that appears in the representations of the covering group of G, whether or not there is actually any particle that carries that charge in the theory . If the algebra of G itself contains a U(l) generator, then we must consider the covering group of this U(1), which is the non-compact group of translations along the real line .If this U(1) generator appears as a term in the electric charge operator, as in the standard electroweak theory, then there is no minimum electric charge in the representations of the covering group, and hence no monopoles . 23.4 The Cartan -Maurer Integral Invarian t In understanding the topology of various compact manifolds, it is a great help that there is often a topologically invariant quantity that can be written as an integral over the manifold . This will beimportant in our discussion of instantons in the next section, and it has already been used in studying Wess-Zumino-Witten terms in Sections 19 .8 and 22.7. Consider a mapping of an arbitrary compact manifold Yof odd dimen- sionality dwith coordinates01, BZ, .- Bdinto a manifold of matrices g(B1,02'. Od)with Lletg =~D. (For the applications that concern us here, 446 23 Extended Field Configuration s .SP is usually a sphere S d, and the g are the elements of a Lie group G in some representation .) We define a functional of g(O), known as the Cartan-Maurer form : j 191 =JdO1d02 ...dod Etitz...ad XTr{g_1(Oj~ g (0 )~ g- ( 0) (23.4.1)r3B~1 a o~2 aOEd ' where ftji2 td isthetotallyantisymmetric quantity with612"'d= 1. From the fact that€ 1 2--id = -(-1 )dC2Z... idil, we see that J[g] vanishes where .S' is even-dimensional, so we will restrict ourselves here to the case where d is odd .The usefulness of this quantity arises from its several remarkable properties . First, this integral is independent of the coordinate system used to parameterize the manifold Y . This follows rather obviously from the fact that ei0=...idis a contravariant tensor density, in the sense tha t 0e4aeiZ00=d (00 ) Second, the integral (23 .4.1) is also invariant under small deformations of the mapping Y~--* Using the properties of the trace, we see that under an infinitesimal change g --+g + Sg of the function g(B ), the change in each factor g- 'OglaDi in (23 .4.1) makes the same contribution to the change in ._flg] 6J[g ] =d IdB 1 d02... d0dEili2---ia ae0e Now, the last factor in the trace is(g-'(O) acid 6(g1o )~g(0} g 1(0) 6g(O) g, (0)ag(O)+gI(O)069(0) aoid 00Ed Mid =g-i(y)aoi d(gog-10) g(O) When we integrate by parts, the derivative a/ae'dgives no contribution when acting on the partial derivatives Og(0)100 in because E'i02...tdis anti- symmetric . The remaining d - 1 terms where 0/00 id acts on g- 1 (0) are all equal except for an alternating sign, so since there are an even number of them they add up to zero . Finally, let us specialize to the case where Yis the sphere S d. Because .fi[g] is invariant under small variations of g(0 ), it can be regarded as a function .~(c) only of the homotopy class c to which g(O) belongs . The 23.4 The GartanMaurer Integral Invariant 44 7 integrals Y,_(c) for strictly speaking, exp~J(c)~-) furnish a representation of the homotopy group Rd(), in the sense tha t ,f(Ca Xc'b) = ~Jr(Ca)+ -0- (c:b). (23.4.2) (If ga(0) and gb (H) are elements of the homotopy classes c ,,and cb respec- tively, then the homotopy class c . Xcb consists of mappings homotopically equivalent t o gab(O)2 $6(201-1,02...Od)zC01C1 The part of the integral J[gAover the hemispheres with 0 <02 c 1/2 and 1/2 ~ 01c 1 can be done by changing variables to 01 = 20t and Hi = 2B1 - 1, respectively, yielding the terms J(c,,) and J(Cb)in (23 .4.2),) In particular, this tells us that the homotopy classes e, c, c xc, etc ., and r-1, c -l x c-1, etc ., have J'(c') = n -0(c). (23.4.3) IfJ(c):~0 for some c, then these invariants are all different, so the classes c 'are all different, and therefore form a subgroup Z of 7rd(.,#). This goes far to explain the great difference between the sizes of the homotopy groups for odd and even dimensions shown in Appendix B of this chapter . For instance, 7rj(U(1 )} =Z,7E3(G) = Z for all simple Lie groups G, and 715(5 U(n}) -= Z for all n ? 3, while for all Lie groups G, R2(G) = 0 and ff4(G) is finite . As a simple example where J(c) :~0, consider the homotopy group RJ(U(1)), which is the same as the group rrl (SI) used as an example at the beginning of the previous section . Any mapping Si F-+ U(1) may be characterized by v, the number of times that the phase of the U(1) element goes counterclockwise around Si minus the number of times it goes around Si in the opposite direction, as the coordinate 0goes around S1,two mappings being hornotopically equivalent if and only if they have the same v . The with class contains the mapping g,, (B) = exp(2iv7rO) with 0 ~0:S:1, for which i J1[9v] = indF] exp(-2iv8)dexp(2ivO) = 2iv ~d0 thus verifying that nj(U(1 )} = Z . As an aid to calculating J(g) in less simple cases, suppose that we can continuously deform the manifold .into a Lie group H of dimension- ality d . The result of performing the H transformation with parameters 0followed by the H transformation with parameters rp is an H transfor- mation, say with parameters 0'(0,(p}. In terms of a matrix representation 448 g(B), this reads23Extended Field Configuration s g((P)g(o)=g(a'(O' (A Differentiating with respect to 0' with rp fixed and multiplying on the left with the inverse of this equation give s 001i 9 00i 9 001 i The integrand of J[g] at a point B' is therefore 1 (of ) 00'j1 ao'.i2 001 .ia =nit 00) etlaZ'..edTr'(0ag(~~ - V)j g(o)...1(0 Now, every Lie group Hhasa metric yij(O)(not nece ssarily unique )which is form-invariant in the sense tha t yi.l(01) For instance, we can takeask0e1, 001i001iyk~(e~ 1yz;(a)_-Tr{g_1(o)g_1(o) 00; For any choice of y ij(O), the determinant of Eq.(23.4.4) give s Det00 Det y(0'} 00' Det y(B j(23.4.4) (23.4.5) By replacing the coordinates 0inEq.(23.4.1) with 0',we find {g_1(01)~)g_1(01 ~~~2...9-1(0)Did 1XI]et ~ddofhet y(E]'). (23 .4.6) Since the parameters rp of the second H transformation are arbitrary, we may regard B and 0'as independent variables, and evaluate the right-hand side of (23 .4.6) at any value of 0, say 0'= 0. It will be convenient to normalize the generators tj and coordinates 9i so that for 0 - -+ 0, g(0)--*1+2iOtti. In this case, Eq . (23 .4.6) read s J1[g]=(2i)dezla2...~1 TrItal tie ...tidI1 d do, etY(D)(23.4.7) I]ety( O'). (23.4.8) 23.4 The Cartan-Maurer Integral Invariant 44 9 We will be especially interested in the case d = 3 . Bott22 has shown that for any simple Lie group G, all continuous mappings S3 ~ G may be continuously deformed into mappings of S3 into a `standard' SU(2) sub- group of G . (Where G - = S U(n),this standard S U(2 )subgroup is the one that acts only on the first two components of the defining representation of SU(n) . Not all SU(2) subgroups of SU(n) are equivalent to this one .) As remarked in Section 2 .7, in a 2 x 2 representation the general element ofSU(2) may be writte n g(O)_04 + t03 02 + t0l= EJ 4 + 2i8-t,-02 + A B4- iO.3 1 D -i 1 1 t2 2 0 ~3 2 d(23.4.9) where as usua l 1 D 1 tj=( ) 2 1 00 -1 and 04 and 0are real, with (B4)2 = 1 - B2 . (Note that Eq . (23 .4.9) is consistent with the normalization convention (23 .4.7).) A straightforward calculation gives the metric (23 .4.5) as OiOj _02 so that I]et y(O) - 1o2 Hence E q. (23 .4.8) here reads d' OJ[g] = -81e'jk Tr~titjtkjf 1B Using 4tit j = 6i j + 2idp'teand Tr ~ tftkj = z S~k , we see tha t $FijkTrIti tjtkj= 2iEtjk6ijk = 12i .(23.4.10) (23.4-11) Also, for the `identity' mapping g2, the integral here runs twice over the interior of the unit ball (because 04 can be positive or negative), and gives d302 t47rrZ dr-22 1-02JO - r 2 For the class c of mappings homotopic to gj, we have the n J(c) = 24-K2 and so J(cv } = 247r 2 v .(23.4.1) (23.4.13) The integer v is known as the winding number .This result is for a representation and normalization conventions for which the standard 450 23 Extended Field Configuration s SU(2) sub algebra has generators to with structure constants Eijk and Tr (tt t j)= z Sad . More generally, if [t i, tj] = igE ijk tk and Tr (ti t j)= 2 ~1Tg2 6,j, then J(c'')= 247r2Nv. (23 .4.14) The results (23 .4.13) or (23 .4.14) show incidentally that for every simple Lie group, 7rAG} contains Z . As listed in Appendix B to this chapter, 7r3(G) = Z for all simple Lie groups . Thus the homotopy class of g(O) for any simple Lie group is entirely determined by its hornotopy class when the group is deformed into its standard SU(2) subgroup . 23.5Instantons As we saw in Section 23.1, the topologically non-trivial solutions of a pure gauge theory with a simple gauge group G in d = 4 Euclidean spacetime dimensions correspond to the elements of the homotopy group 7r3(G) = Z . These are four-dimensional field configurations, known as instantons, that (for reasons discussed in the next section) must be included along with their fluctuations in path integrals . After Belavin, Polyakov, Schwarz, and Tyupkin' demonstrated the existence of these solutions, it Hooft23 showed that the inclusion of these configurations in path integrals solved theU(1) problem outlined in Section 19 .10. Here we shall first discuss the instantons themselves, and then consider their role in path integrals . According to Eq. (23 .1.7),in order for a topologically non-trivial gauge field to have finite action, the gauge fields must approach a pure gauge at r -4oa (where here r - xixi, with i summed over the values 1, 2, 3, 4) : iAj(x )--+g-1(x)~3ig(x) , (23.5.1) where Ai .= t,,A,,z and g(ac) is a direction-dependent element of the gauge group G . The topological invariant discussed in the previous section may therefore be written in terms of the asymptotic behavior of the gauge fiel d j 191 _fdel dB2 d O3F"bc ae aeC 11mr,., r3 dO'd02dO3rah`OXzaXaO~kTrjAjAjAk}(23.5.2).l aeaa0b a0c where O" with a = 1, 2, 3 are any three parameters used to specify the direction of the unit four-vector 3t . This surface integral can be evaluated using Gauss's theorem . In analogy with the current (22.2.29)in Minkowski 23.5Instanton s spacetime, we can define a current in Euclidean spacetim e GI=Jajk[AYiFj.ik - 3CaflYAai Afl.iAyk] whosedivergenceis 1E0dGd=- zCijkdFaijF alcd-451 (23.5.3) (23.5.4) (Here F ~klis the totally antisymmetric tensor with J 1~34 - 1.)We are using a representation of the gauge group with totally antisymmetric structure constants, so that Tr ~tatg} = 2N 6ag, (23 .5.5) with N a constant that depends on the representation in which we calculate the trace in Eq .(23.5.2).Hence Eq . (23 .5.4) may be writte n GI- (2/N )FE Iijk Tr [AFJk + (2i/3 )A= A jAkI For r --), oa , the field strength F kjvanishes ,so GI --), (4i/3N )eEIijk TrIAiA jAkI Hence Eq .(23.5.2)gives 'f [g] -=- -(3NI 4)Jd4x01 GI =-(3N/8) ~Ekdfd4xFaijFxkr(23.5.6) (23.5.7) (23.5.8) Thus in order to demonstrate the existence of topologically non-trivial field configurations, we have to show that there are configurations for which the Chern-Pontryagin density F ski Fa ijFake has a non-vanishing integral . For this purpose, it is very useful to take advantage of what is known as the Bogorraol'nyi inequality .10From the fact tha t 0C ~ (Fjj + Z~ jklFczkl )d4x (with squares indicating obvious index contractions), we hav e S[A]? F~krIFaijFakid4'x=If [g] II3N where S [A ]is (up to a factor) the Euclidean actio n S[A]= 41Fai jFoci jLl~4x.(23.5.9) (23.5.10) The lower bound (23.5.9)is evidently reached if and only if the gauge field is self-dual or anti-self-dual, in the sense tha t Fay j = _ ± Z EI~Jkt Fcckl . (23 .5.11) Hence any solution of the first-order equation (23.5.11) is a minimum of S [A] for gauge fields of winding number unity, and hence also a solution of the second-order Yang-Mills field equation . 452 23 Extended Field Configuration s Belavin et a[ .1 found a solution of Eq . (23 .5.11)of the for m riAi(x) = rz + R2 91Imaigo) , (23.5.12) where R is an arbitrary scale factor, and gl (x) is an element of an SU(2) subgroup of the gauge group, with x4+2ix t gl~x} = r and l 0 1 1 0 -a 1 1=- t 12 ~~ 2t3 02 i 0 20-1(23.5.13) (23.5. 14) It is clear that th is solution has the asymptot ic behav ior(23.5.1),with g(x) the same as the `identity' mapping (23 .4.9), so th is solut ion belongs to the homotopy class of the identity map, and therefore as we saw in the previous section has winding number v =1. From Eqs . (3 .4.14)and (23.5.9) (which is here an equality) we have then (23.5.15) From Eqs .(23.5.10) and (23 .5.11)(with positive sign), we also hav e C7 EkdfFaajFakid4x = 64 n 2 (23.5.16) This solution is not unique, because it can be translated or subjected to a gauge transformation, but aside from these degrees of freedom, there are no other solutions of the field equations with winding number unity .14 Because we have found field configurations with v = 1, we know that there are also field configurations with any integer v . For instance, solutions with v a positive integer .+' can be constructed by superimposing X solutions with v = Iwith centers so far apart that at these distances the non-linearities of the field equations become unimportant . A solution with v = -1 negative can be found by replacing gl in Eq .(23.5.12) with gl and solutions with v a negative integer - S' can be found by superimposing Xof these at large separations . For general winding number, Eqs .(23.5.15) and (23 .5.16) becom e s[A]= sn2lvl, e7 EkdIFaij F1tl d4x = 647r 2 v(23.5.17) (23.5.18) These results are for a gauge field normalized as in Eqs .(23.5.11)- (23.5.13).With this normalization, the action I [A] is not -S[A ], but (23.5.19) 23.5 Instantons 45 3 where g is the conventional coupling constant . If we had used our usual convention of including a factor g in the generators and structure constant, then the action I [A ]would be the same as --S [A ], but with AIX , and Fx,,v carrying a factor 11gwe would have S [A] = 8n2/g2instead of Eq. (23 .5.14), so in this case again the action would be -8rc2/g2,but now in place of Eq . (23 .5.14) we would hav e F~jklfFxijFxktd4x = 647C2v/g2. (23.5.20) We shall see in the next section that in path integrals we must sum the effects of instantons of all winding numbers . The contribution of configurations of winding number v *0to Euclidean path integrals is suppressed by a factor exp(I[A]) = exp(-8tvln2/g2) . In Section 23 .7 we will see that the coefficient of this exponential is a negative power -n of g: in quantum chromodynamics, n = 12 . The function g-n exp(--8f vJn2/g2) and all its derivatives with respect to g vanish at g = 0, so such contribu- tions are non-perturbative -they will never be encountered in any order of perturbation theory . This does not necessarily mean that these contributions are small . As we saw in Chapter 18, in quantum chromodynamics the coupling g is not a fixed dimensionless parameter, but a function of a sliding energy scale, and becomes large at low energies . The effective energy scale to use in the coupling in Eq .(23.5.19) is determined by quantum fluctuations, the subject of Section 23 .7, but on dimensional grounds it cannot be very different from 11R, where R is the instanton size in Eq .(23.5.12).The instanton size is not fixed, but must be integrated over, with some weight function that depends on the process under consideration . In quantum chromodynamics the running coupling constant g ,,is given for large ,u by Eq.(18.7.7)as z87r2 ,u flo ln(p/A ) where (3 0 = 11- fin f /3andA.= 25 0MeV is the quantum chromodynami c scale factor . The factor exp(-Sn2/g211R) is therefore, for small instantons , exp ~ - $7E2/g 121R ~ _(RA)° . We cannot calculate this factor for large instantons, with RA ~~ 1, but it is clear that in this case there is no suppression of instanton effects . Despite initial hopes, the discovery of instantons has not led to much improvement in our ability to do quantitative calculations in quantum chromodynamics . On the other hand, as we shall now see, it has pro- duced spectacular qualitative changes in our understanding of quantum chromodynamics and other gauge theories . 454 23 Extended Field Configuration s The mere fact that there are solutions of the field equations for which the integral (23 .5.15) does not vanish is enough to provide a solution of the U(1 )problem discussed in Section 19.10.Under global U(1) transformations y )-+exp(iy59)W, the measure for integration over quark fields undergoes a change given by Eq . (22 .2.10): [dW][di-p]--),exp icc jd(xi4x} [dV)] [d~v] , (23.5.21) where (with the matrix t here equal to unity) the anomaly function ~Ql(x) is given by Eq . (22 .2.45): ,-V/(x)= 1~2- F~kr F= jxFk!fltr txt fl. (23.5.22) The existence of this anomaly would not in itself solve the U(1) problem, because the quantity (23.5.22) is a total derivative, and so would have a zero integral for a non-singular gauge field that vanishes sufficiently rapidly at infinity . The instanton solution vanishes only as 11r, and yields a non-vanishing value for this integral, given by Eqs .(23.5.22), (23 .5.18), and (23.5.5)as fd4xa(x) = 2Nv, (23 .5.23) showing that the anomaly does violate the U(1 )chiral symmetry . As we saw in Section 22 .4, the currents of baryon and lepton number also contain anomalies due to the interaction of quarks and leptons with the SU(2) xU(1) gauge fields of the standard model . Instanton configurations of the SU(2) gauge field therefore produce violations of baryon and lepton conservation .25 As noted in Section 22 .4, there are various currents whose conservation is not violated by anomalies or anything else, such as baryon number minus lepton number, and the differences of electron, muon, and tau lepton numbers, so these will be conserved in any baryon- and lepton-non-conserving process . For instance, the decay of a proton or deuteron is forbidden, but the decay He3 -4 e+ + P+ + vx is allowed . The amplitudes for these effects are suppressed by the same factor exp(-$1vtn2/g2 )as before, but now with g the ,S U(2)coupling constant e/ sin B, evaluated not at a sliding scale but at the natural scale for electroweak processes, roughly of order mz . Taking e2 /4n = 1/129 (see Section 18.2)and sing B =x .23, the suppression factor for Jvj = 1 is exp(-373) . Hey decay into three antileptons is not likely to be observed . *** There is another approach26that illuminates some aspects of instantons . In temporal gauge, with Aa4(x,X4) =0,the gauge field for x4 - )'±oo is 23.6The Theta Angl e expected to approach time-independent pure gauges for x4 --)'±oo iA(x,x4) itIX AIX(x,x4)`g+ (x)Ogt(x)455 (23.5.24) where g±(x) are group elements in the representation generated by t ,,,. Assum ing that A a(x, Xq) vanishes for x --), oa, the group elements g±(x) must approach constants g tfor x --+ oo, so the three-spaces at x 4--~co may be regarded as three-spheres ,with the point at infinity regarded as an ordinary po int.Following the same argument that led to Eq . (23 .4.13), we then hav e Eiik Jd3XTr{1(x) Oig+(x)g±1(x)Vjg±(x)g±1(x) akg±(x) = 24n2n+ (23.5.25) where n± are integers . The integral over the boundary of four-space in Eq.(23.5.2)may be regarded as the integral over the `plane' x4 = + 00 minus the integral over the `plane' x4 - -oo, so from (23.5.8)and (23.5.18) (with N = 1) we have v = n+ - n_ . (23 .5.26} The exponential factor exp(-8Cvtn2/g2) from Eq .(23.5.19) may therefore be regarded as the amplitude for a transition from a configuration with spatial winding number n_ at x4 --;'-oa to one of spatial winding number rt+ at x4 - ►+oa. The exponential form of this factor for v =#0reflects the fact that this is a tunneling process ; no continuous sequence of pure gauge fields can take us from a configuration of one spatial winding number to a configuration with a different spatial winding number . The interpretation of the factor exp(-8 C v Cn2/g2 )as a tunneling am- plitude suggests that baryon- and lepton- no n-con serving processes may proceed rapidly at temperatures above about 1 TeV, where instead of having to tunnel through the barrier, thermal fluctuations can take the vacuum over the barrier .27 This process may be of cosmological impor- tance, but it is still subject to the selection rules mentioned above : thermal fluctuations will not change the density of baryon number minus lepton number, or the differences of the densities of the three varieties of lepton number . 23.E The Theta Angl e We have seen that there are configurations of arbitrary integer winding number, but how do we know that these configurations must be included in the path integral? To keep an open mind, suppose we add up configura- tions with arbitrary weight factors ,f (v) for each winding number, leaving 456 23 Extended Field Configuration s open the possibility that some or most of these weight factors might van- ish. The expectation value of a local observable 0located within a large Euclidean spacetime volume Qis the n EVf(v) fv [do]exp(In [01) cc101r23.b.}1 - ( Ev.~(v) fv [do] exp(Ic[41] ) where 0stands for all the fields of the theory ; the subscript v on the integrals over 0indicates that we are to include only configurations of winding number v ; and In [ 0] is the integral of the Lagrangian density over the spacetime volume 0. Now suppose that Qis divided into two very large volumes 01and 02, with Ccin the volume 5 11. The integral over all fields with winding number v may be written as an integral over all fields with winding number vi in volume Q1 and winding number v2 in volume Q2, with vl and v2 summed over all values with vl + v2 = v, and so to a good approximation Eq .(23.6.1)become s Ev1,y2f (v1 + v 2)fv, [do] exp(in, ~~~) Cc [0] fv, [do] exP(In2101) Ev,,v2f(vl+ V2)fv1 [do] exp (In, [01) Jv,[do] exP (I02[0l ~ (23.6.2) But then for general weight factors the average is not the same as if we omitted the volume 0.2, in contradiction with our general ideas about cluster decomposition . (See Chapter 4 .) In order for the factors involving the volume 02 to cancel in this ratio, we must hav e This will be the case if and only if f(v) is of the for m f(v)= exp(iOv) , (23 .6.3) where 0is a free parameter . Thus in particular we cannot arbitrarily discard all configurations with non-zero winding number, because then an instanton of winding number v in one region would have to be balanced with an instanton of winding number -v in some other region, making it impossible to calculate expectation values without considering what is happening far from the location of the operators being measured . The factor f(v) may be put in a more familiar form . According to Eq.(23.5.18), with gauge fields normalized so that in the standard 5' U(2) subgroup the structure constants are-Fift, the winding number may be written as an integral : 1V==64n2 ,I(d4x)E ~~kd Focij F ofkr  ( 23.6.4) 23.6The Theta Angle 457 This can be expressed in terms of a Minkowskian path integral ; since (d4x),E=id4X,Fa34 =-iFa3 0; and -F1230= -1, Eq. (23 .6.4) may be writte n 1 4 Apa fdx e~cFa~~.Fx~~ u = - 64n2 Thus inclusion of the weighting factor (23.6.3)is therefore equivalent to adding a term 0 KApU ~B b4~c~F FUKJ xPU to the Lagrangian density . But as mentioned at the beginning of Section 15.2, we might have included such a term in the Lagrangian of any non-Abelian gauge theory anyway, with arbitrary real 0. The inclusion of a term (23 .6.6)in the Lagrangian density would violate P and CP conservation . We could of course simply set 0= 0, but this would invalidate what in Section 18 .7 was scored as one of the successes of quantum chromodynamics : it made it automatic for P and CP to be conserved by the strong interactions even though they are evidently violated by the weak interactions . To assess the physical consequences of including the term ( 23.6.6)in the Lagrangian, consider the effect of redefinition of all the fermion field s Vf--;'exp(iy59f)Wf , ( 23.6.7)(23.x.5) (23.6.6) where f is a flavor index, and ccf are a set of real phases . According to Eqs. (22 .2.10) and (22 .2.24), the effect on the measure for path integrals over fermion fields i s Ldy~jLdVl---),exp 32n~f d4x epvp6F." FP,' E af [dTj [dq) ] s(23.6.8) (We are using generators normalized so that Tr (txt g)= 6afl/2.) Comparing this with Eq .(23.6.6)shows that this is equivalent to shifting 0 by 0 - - - > 0 + 2 c . 1. (23.6.9) J, The redefinition of the fermion fields will also change the mass terms in the Lagrangian density . In order to take account of masses involving y5, let us write the fermion mass term in the Lagrangian as (23.6.10) YM= - 2 ~ ~~ f fPf (1 +75)Ws -12E~.f~3s(1-75M f s with mass parameters ~ f which , if complex ,would violate P and CP conservation . Then the rede finition (3 .6.7) changes these parameter sby f--+exp(2icc f) f . (23.6.11) 458 23 Extended Field Configuration s A mere change of path integration variables cannot have any physical effect, so observable quantities cannot depend separately on 0or the phases of the mass parameters iff, but only on the combinatio n t' In particular, we can always define the fermion fields so that B = 0, but at the price of perhaps introducing P and CP violating phases in the mass parameters . This discussion shows that if any of the quark masses were to vanish, then the theta angle would have no effect and there would be no P or CP non-conservation in quantum chromodynamics . The analysis of quark mass ratios in Section 19 .7 indicates that the quarks all have non-zero masses, though it is sometimes suggested that the inclusion of terms of second order in rra5in this analysis might allow rrauto vanish .28 But there is no question that the u and d quarks are quite light, which leads to some suppression of the effects of a non-zero theta angle . We saw in Section 19.4 that mu and mdare roughly of order rran /mom (where MN is used here as a typical quantum chromodynamic mass scale), so we might expect the effects of the theta angle to be suppressed by four factors of m, but this is not quite correct . With P and CP not conserved there would be a non-zero amplitude for the n° to disappear into the vacuum, proportional to m~, but the `tadpole' graphs with pion lines ending in such vacuum transition vertices would be enhanced by a factor mn 2 from the pion propagator, giving a net effect proportional to m~, not mom . In particular, if we define fermion fields so that all iff are real, then a non-zero theta angle would produce a P- and T-non-conserving neutron electric dipole moment proportional to 101and m~, and hence on dimensional grounds of order29 do 1p1erra~/rra~-:t10-16CO1e cm . (23 .6.13) The neutron electric dipole moment is known to be less than about 10-25 e Cm, so 101 < 10 -4. In order to explain in a natural way why 0is so small, Peccei and Quinn30 proposed a theory in which 0became in effect a dynamical variable, which could relax to a minimum of the effective potential, at which P and CP would be conserved . Their idea was taken up by Wilczek and myself,31 who noted that it would require the existence of a light spinless particle, the axion . The axion that appeared in the original Peccei-Quinn model has been ruled out by experiment, but there are more general possibilities,32 in which the axion couples too weakly to ordinary matter to have been observed . The common feature of all versions of the axion theory is that there 23.6The Theta Angle 459 is some U(1) symmetry that is spontaneously broken at energies much higher than those associated with quantum chromodynamics, and is also broken by an anomaly involving the gluon fields . According to the general formalism of Chapters 19 and 22, the low energy effective field theory will contain a Goldstone boson field 0, so that, under the symmetry transformation 0 --). 0+ Foe, (23.6.14) the effective Lagrangian undergoes the transformatio n ~ ~eff eAf~va F11v FPc (23.6.15) xF.~4Tr?Ep where A is a dimensionless constant of order unity, characterizing the anomaly, and FO is a constant of the order of the energy scale at which the symmetry is spontaneously broken (with 0defined to be canonically normalized .) Then the terms in the effective Lagrangian involving 0are Y'P~ ~~~ ~~u~ ~ ~~vp~FavFa6 + ... ~ (23 .6.16)64n where M = FoIA and ` - ' denotes possible interactions involving deriva- tives of ~ . Comparing Eqs .(23.6.1b)and (23.6-6), we see that, for a constant 0, all observables will be functions not of 0and 0 sepa- rately, but only of 0+ OIM (This is with fermion fields defined to make the mass parameters fall real ; otherwise observables will depend on )-Y:fArg.,W f + 01M .} If everything in the theory but the theta term 23.6.6)and the 0interaction in Eq . (23.6.16)conserves P or CP, then the ,ffective potential will be even in 0+01M, so it will have a stationary point at 0+ O/M = 0,preserving the conservation of both P and CP . In he real world P and CP are not exact, but the only observed violations ire in the weak interactions, and would shift the expectation value of 0 only a small way33 from -Mo. Even without specifying the underlying theory, by using effective field heOry techniques it is possible to say a fair amount about the general roperties of the axion . The most general Lagrangian for the axion field ~, now including the theta term and all interactions with u and d quarks p to order I / M, is of the for m 1 0401"0 + ' - [0+ 01 ,,, FW FPcz 6_472M i Mfu Ou01757'U -~f-d 0u0d7571' d, (23.6.17) here fu and fdare dimensionless coupling constants, expected to be of der unity . As shown by Eq .(23.6.8), the redefinition (23.6.7)of the park fields has the effect of subjecting the quantity O(x)/M + 0in the 460 23 Extended Field Configuration s second term of the Lagrangian (23 .6.17)to the replacemen t OW+0--),OX)+ 0 + 2 [au(x) +UdWl -M M(23.6.18) By choosing at =-(8+011Vf)cf /2 with constant coefficients cfsatisfying cu+c d= 1, we eliminate the term in the Lagrangian involving 'F'U,,p0-FIlvFf6, and change the mass term in the low energy quantum chromodynamic Lagrangian density t o YM = -MU u expI- acu (B +O/M)751u - m dd exp icAB + 01M)Y51 d . (23.6.19) In addition, from the kinematic part of the quark Lagrangian we pick up a derivative interaction ter m 2iCu(uY"75)u00'M+ 2iC~j(C1~~111y5)Cl~ [~C]' M, (23.6.20) so that fuand fd in Eq . (23 .6.17)are replaced with ft,=fu -- cu /2and fd'=f d- Cd' 2. To derive an effective Lagrangian for low-energy pions and axions, we follow the procedure of Section 19 .5 (but now with u and d quarks in place of protons and neutrons), and make the replacements : dd--;-u cos(no/F,,) , dy5d -> -iv sin( no/F,,) , (23 .6.21) idyuY5__*2 F~0~ ~0+...a where v is the constant v=(uu) = (dd},and `  'denotes terms that do not have the one -pion pole .From Eqs .(23.6.19)and ( 23.6.20) wefind the effective pion -axion Lagrangia n 2 2 (2M ) (3 r ~ -muv cas F ~ mdv cos(70+ ~ , ( 23.6.22) where 0' is the difference between the axion field and its expectation valu e Mp. (23.6.23) Since cu and Cd are arbitrary except for the condition that Cu + Cd=1, we are free to eliminate the OyoaunO cross term by taking cu=; + f u-fdand cd= 2 + fd - fu, so that f u' = fd. The quadratic part of the Lagrangian (23.6.22) is U ~ ( ] squad = -2 Op7 0OP7 O-2aPor ayof - 2 ~1MU"~(23.6.24) 23.6 The Theta Angl e where ~2 (M u + M d) ~' /F~ (-MuC~ + mdcd) v/F~ ~ ❑ (-Mu~'u + MaCa) vIF,,M (muC~+mdc~} U~~Irf~461 (23.6.25) ForM ~~ F, one eige nvalue of Mo2 is (mu + m d) vIF,2,, which acco rding to Eq. (19 .7.20) is the 7i4 s quared mass. The othe reigenvalueis thenthe axio nmass 2 vMdMu ~a - M2ma + m uFa77MdMu 2- M2 (Md+M,,)2 11~ yytI, (23.6.26) For the quark m ass ratio derived in Section19.7, th is gives ma = 13 MeV/M(GeV) . We can also use this formalism to say something about the interactions of ax ions with hadrons . The eigen vector of M~ with eigen value m~ has a component along the original 7r❑direction equal to (mucu - rrt dcd)F,,I(rrau+ MOM .As mentioned earlier, because of the one-pion pole this is the dominant axion -hadron coupling . We see that the ratio of the axion and pion production interaction amplitudes will typically be of order F,1M. The fact that axions are not observed in such collisions ind icates that M > 3 TeV, in contradict ion with the original expectation 3"' that the anomalous U(1) symmetry is spontaneously broken by the same scalar vacuum expectation values of order 0.3 TeV that break the electroweak SU(2 )x U(1) symmetry . It is possible to explain why axions are not found in reactor or accelerator expe riments by taking Mas an independent parameter,32 much larger than the e lectroweak break ing scale, but there are still astrophysical limitations . Limits on the rate of cooling of red giant stars gi ve34M > 107GeV, while observations of the supernova SN1987A indicate31that M >la'❑GeV .*Cosmological arguments suggest 36an upper bound M<1012GeV, leaving an open but narrow window of allowed axion parameters . 'For M >147GeV the axion mass would be less than about IeV, so that stars are hot enough to produce axions_ The ratio of the axion and nodecay rates into two photons is expected to be of order (F,,IM)' times a phase space ratio of order (m,,/m,)', o r , M) I'{7c°~y+Y } - (M)( Hence for M >10' GeV the axion lifetime is expected to be longer than about 1a24 s, which is ample time for the axion to travel even cosmological distances before decaying . 462 23.723Extended Field Configuration s Quantum F luctuatio ns aro und Exte ndedFieldConfigurat ions Topologically non-trivia lfour- dimensionalfieldconfig urations such as in- stantons provide only zeroth-o rdercontributions to pat h integra ls. Now wemust co nsiderthe effect of quantum flu ctuations aro und these config- urations. To start in a very genera lway, co nsider a set of fie lds, co llectively called fi(x), whose dyna micsisdescribed by the E uclideanaction 1[0], a nd suppose we have a se tof co nfigurations 0v,, at wh ichI [0]is statio nary, where v labels the topo logica ltype of the config uration, and u represe nts a set of co ntinuous col lective pa rameters on wh ich t he conf iguration depen ds. For instance, for ins tantons v is the wi ndin g nu mber, an intege r, anduincludes the posit ion and sca le as wel las its `d irection'inthe gauge group. Euclidean pat hintegra lsmaythenbe wr itten f[do]exp (I[1)0 = ~:Jdu ~[ do~]exp(I[cbv,u + 0' l} Cc~(23.7.1) V where the su bscript on theintegral ove rthe fluctuation0'indicates that we a re to integrate on ly over f luctuations that do not e ntailchanges in the co llectiveparameters, a nd0 stan ds forany p roduct of localfunctions of fie ldoperators.BecauseI[0]is stationary at its expa nsionto secondorderin the fl uctuations takes the for m I10v,u + 01]--'Iv --- Z 1d4xd4yKxI,ym(v, u)0i(x )0' (y) , (23.7.2) where l and m here include spin and species indices, and I . = I[o,,,u] is a function of v alone because the action is supposed to be stationary at all field configurations 0v,u . The integral over the fluctuation 0' in Eq.(23.7.1)will then yield a sum of terms from contractions of the fields in (9, times (for real bosonic fields) an over-all factor [Det K(v,u)]-1/', it being understood that K now acts only in the subspace of fluctuations that do not entail changes in the collective parameters u . This factor can be written as a product over the eigenvectors ~,,(v, u) of the `matrix' K(v, u) : (Det K(v, u)}1/2 n(23.7.3) the prime here indicating that we are to exclude the `zero modes', the zero eigenvalues of K, for which the eigenfunctions correspond to changes in the collective parameters . These remarks allow us to refine the results given in Section 23.5 for the coupling-constant dependence of instanton contributions with different winding numbers . Suppose we define the fields of our theory in such 23.7 Quantum Fluctuations around extended Field Configurations 4 63 a way that the act ion ta kes the for m I [0, g]= g-2 I, [0]whereI,[0] is indepen dent of co upling consta nts. (For instance, as discussedat the e nd of Sect ion 15.2, in Yang--M ilts theories we wo uld use as a ga uge field the canon icallynormalizedgauge fie ldtimes a facto rg.)Thenalleigenvalues ofKare propo rtionalto g-2, a ndthe factor (23 .7.3)is proportionalto g to a power e qualtothenumberofnon-zero eigenval ues of K . This powe r is of co urse infin ite,but it can be writtenas the numberof all e igenvalues of K, w hich is alsoinfinitebutisindependent of v, minus the n umb erX(V) of zero modes, e qualto the n umb erof co llective pa rameters, wh ichis finite anddepen dentonv. We co nclude the nthat asi de fromfactors that do notdepen don v, the co ntribution of fluctuations a roundconfigurations of topo logica ltype v is a facto rwith a coupling-constant dependence that is givenin terms of the numb erA"(v) of co llective pa rameters by g-.~tv} (23.7.4) This est imateisbasedon the app roximation (23 .7.2), w hichcorrespo nds to a o ne-loop app roximation, so when h igher-order terms are ta keninto acco unt the facto r(23.7.4) w ill be multipliedwith a power series ing, whose detailsdependon t he ope rators C appearing in t he pa thinte- gral. Let's see how this applies to instantons . The configuration with v = 0 of course has no collective parameters, and so its contribution to path integrals isjust a power series in g . The configuration with v = 1 has four collective parameters giving the spacetime location of the instanton, one collective parameter giving the scale of the instanton, and a number Nlof collective parameters corresponding to the rotat ions and /or global gauge group transformations that do not leave the instanton invariant, so (now including the factor exp(I v )given by Eq . (3 .5.19)}, the couplin g- constant dependence for v = 1 instantons is g-5-N' exp (-87E 2/g2). For the v =1instanton (23.5.12) i n an SU(2) Yang -Mills theory, there are three independent rotations and three independent S U(2) transformations, but since the instanton is invariant under three combined rotations and S U( 2) transformations, we have MIT I= 3, so fluctuations around the v = I instanton yield a coupling-constant dependence g T8exp( -87E2/g2). In an S U(3) Yang-Mills theory we have three independent rotat ions and eight independent S U(3) transformations, but again the instanton is invariant under three independent combined rotations and S U(3) transformations in the standard S U(2) subgroup (say, acting on the first two components of the defining representation of S U(3)), and now also inva riant under one additional 5 U(3) transformation that (like hypercharge) commutes with all generators of the standard S U(2) subgroup .Hence 11TH = 3 +8 - 3 - 1= 7, and the v = Iinstanton yields a factor proportional to g-12exp(-8 7i2 /g2}. 464 23 Extended Field Configuration s Now suppose that the action I also contains a ter m d4xId4 Y W1 (x) -'*'px,my(v, u)TM (Y ) involving independent fermion fields Tand v. If none of these field s appear in ( 9then the integral over these fields yields a factor Det -*'(v, u), which vanishes if -V'(v, u) has any zero modes . This result can be simply understood in terms of the rules for integration over fermionic parameters . By expanding yp(x) and fp(x) in the eigenmodes of Y, we can write the integral over V(x) and V(x) as an integral over the coefficients in these expansions . The coefficients of the zero modes do not appear in the quadratic approximation to the action, so for each such fermionic zero mode we have an integral over a fermionic parameter that does not appear in the integrand, which vanishes according to the general rules of Section 9.5. The only terms in an integral over fermionic variables that do not vanish are those for which the integrand contains a single factor of each integrated variable . Hence the integral over fermionic fields in Eq .(23.7.1) will not vanish only if there is a single fermionic field in ( 9for each zero mode of Y. For instantons there are fermionic zero modes whose numbers and chiralities are governed by index theorems, like the Atiyah- Singer index theorem derived in Section 22 .2, so only certain processes are allowed for a given winding number . It was on this basis that 't Hooft25 showed that the baryon- and lepton-nonconserving effective interaction produced by v = 1 instantons in the electroweak standard model must involve just one of each lepton flavor . 23.8 Vacuum Deca y A vacu umstate is sta bleif its sca larfieldexpectatio nvalues are at a trueminimum of the effective pote ntial.Butif the sca lar fie ldexpectat ion values are at a localminimum that ishigher tha nthe true mi nimum thenthis vacu um wi ll bemetasta ble. A metasta ble `fa lse' vac uum state correspon ding to a local minimum willdecay into t he sta ble `true' vac uum correspo ndingtothe trueminimum by a process of barrierpenet ration, analogousto nuclearalpha decay o rsponta neous fission . This is not a process tha tcan be observe d in ourlaboratories, bu t ithas pres um- ably occ urredseveraltimes inthehistory of t heuniverse as vario us symmetries have become spontaneo uslybroken, so itisimpo rtant tobe able to calc ulate the rate of s uchfalse vac uumdecay . As we shall now see, t his calculation involves co nsideration of yet a nother exten dedfield configuration .6 Letus conce ntrate on the co mponent 0 of the sca lar fie ld mul tiplet that 23.8Vacuum Decay 465 acquires anexpectat ion va lue (0) in the t rue vac uum. Fo r instance,in the t heory of broken c hiralsymmetrydiscussed in Section 19.5, 0 wo uld be the fou rth co mponent of a ch iralfour-vector .It willturn out that, wherebarrierpenet ration is st rongly suppressed, t he otherscala rfields (including t hose of a ny Go ldstone bosons) donot affect the do minant suppression fac tor in thedecay rate . For definiteness, we will take t he Lagrangiandensity in the fo rm Y = -- ~a~00~0-V~~}. X23 .8.1} We ass ume that t helowest-o rdereffect ive potentia lV(¢,) has a t rue minimum at 0 = (0) a nd alocal minimum at 0 = 0, a ndwe a djust an additive co nstantin the Lagra ngiandensity so t hat V(O) = 0, i nwhich case Y( (O)) E 0 . We wa nt to ca lculate the rate at w hich t he false vac uum state withscalarfieldexpectatio nvalue zero willdecay into the true vac uum state with sca lar fie ldexpectat ionvalue (0) . Theresults of Appen dix A of t his chapte r(Eqs . (23.A.6), (23 .A.21), an d (23.A.23)) s how t hat the e nergy Eo of t he false vac uum state in which t he scalar fie ldvacuum expectation va lue vanishes is given by EO=-lim 1In ~',x' T exp S [0;TJ) do(x, t} x,t where S[0; T] is the E uclideanactio n derive dfromEq. (23 .8. 1) +T1a S [O ;7'] _Jd3xJ_dtT/a(23.8.2) ()2 Z ~0) + 1(O~i)z +V(O)(23.8.3)Ot 2 and the integral in Eq .(23.8.2)is over all fields O(x, t) satisfying the conditions O(x, T/2) =O(x, -TIC) = 0. (23 .8.4) The energy (23.8.2)is complex ; its imaginary part will give us the decay rate. To calculate the functional integral in Eq . (23.$.2), we look for a stationary `point' of the Euclidean action S [ 0, T]. The fields at which (23.8.3)is stationary satisfy the field equation s 60 Ot2dpi(3.8.5) subjectto the boundary con ditions (23 .8.4).Because of t heseboundary conditions, sucha solution is known as a bounce , We sha ll lookfor these bounce so lutionsbymakingthe ansa tz that O(x,t)isinvariantunder rotations aro unda pointxO, toinfour dimens ions: O(x, t) = 0(P) whe re p = fi x-xo}2 + (t - to~ . (23 .8.6) 466 23 Extended Field Configuration s There are so lutions t hat are notrotationa llyinvariantin four di mens ions, but these othe rsolutions have h igher va lues37 of S, an dsobecome negligible forlarge T . Using Eq. (23.8.6)in Eq . (23.8.5) yie lds the ordinary differentialequation d a~ + 3do=V(O )dP Pdp(23.8.7) Strictly speaking, a so lutionof this form is consiste nt wit htheboundary conditions (23 .8.4) on ly whenT is very large co mparedwith the char- acter istic time assoc iatedwithV(p), inwhich case we ca ntake T to be infinitein Eq. (23 .8.4), w hichthen becomes the con ditionthat O(p) must vanish when p --+oo. Also O( x, t)mustbe an a nalytic f unctionofx near x= 0 at a lltincluding at t = to, so O(p) is apowerseriesinp2forp--* 0, and in particulardoldp= 0 at p = D .With these con ditions, Eq . (23 .8.7) is the equation of motionof a part icle of unit mass wit h`position' 0 at `time'p,movingunder the influence of a potentia l-V(O) a nd a visco us force-(31p)doldp, that t ravels from rest at some fi nite in itialvalue 00 of 0 at p = 0,and just reaches 0 = 0 at p-* a7, losing itsinitial`energy' -V(o ❑) > 0 to v iscosity along t he way .The E uclidean act ion (23 .8.2) fo r such a so lution is BJ2~2p=~dp a1(4) 2 2dp+ V(O ) (23.8.8) Itis crucialtodetermine the sign of B . For t his pu rpose,38 we use the same device that was usedto p rove Derrick's t heorem inSection 23 .1, andcons ider the act ion(23.8.8) for a modifiedfieldOR(u) - O(pl R).By rescaling the variable of integratio n, we f ind 00 S[CR]= fo2712p3dp ~2 do 2 +R4V(O)2 (dp)(23.8.9) IfO(p) is the solution of Eq .(23.8.7)then the action must be stationary with respect to any variation in 0, so that dS [OR] /dR must vanish at R = 1, and therefor e 10 100 It follows t hen that3dP(4)2 =-4 P3dP V(O)f P z ~ 2 pddo>a. B= Z fo p3d(23.8.1) We will co mebacklater to a nexplicit approximate so lutionfor O(p), but first let's co nsiderhow s uchsolutions are to beused. 23.8 Vac uum Decay 467 We have here not just a single one-bounce configuration at which the Euclidean action is stationary, but a continuum, characterized by the collective coordinates x ❑and to . According to the results of Section 23.7, we must integrate over these parameters, which, since B is independent of xo and to, yields factors of ~Vand T in a box of spatial volume -V. The contribution of all one-bounce configurations to the functional integral in Eq. (23 .8.2) is given in one-loop order b y ,VTA ex p(-B). (23 .8.12) The coeff icientA is p roportiona l* to a p roductj~'n ~n1/2, w here t he Zn are the e igenva lues of t he kernelb2S[O]/60(x,t)aO( x', t'), a nd the prime indicates t hat we are to om it the zero eigenva lues, w hich co rrespo ndto changes in t he collective coo rdinates x4 andto. Us ing Eq . (23 .8.10), we cansee that t he seco nd d erivative of E q. (23 .8.9) with respect to R is negative at R=1, so t hereis atleast one (an d in fact j ust one39) negative eigenva lue, an d hence to t his orderA is imaginary. We w ill not attempt to calculateAhere, but will j ust note that Aunlike exp(-B) does not have adramaticdependence on t he pa rameters of t he theory, so that we ca n estimate it on dimensiona lgrounds to be rough ly of o rder iM-4, w here M is so me characteristic mass sca le of the theo ry. For la rge T an dV we can fi ndadditionalstationa ry config urationsby superimposing a ny nu mberN of these bounce co nfigurations, yielding a contributionthat is the Nth power of the quantity (23 .8.12), divided by N!to take acco unt of the fact t hatinintegrating ove rN of t he xO an dto we a re summing ove rconfigurations t hat on ly differ by per mutatio ns of the N identicalbounces. Summi ng over N t hen g ives the expo nentialof the quantity (23 .8.12), so the e nergy (23 .8.2) is j ust thequantity (23 .8.12) divided by -T E❑_ -Y,"A exp(-B) . (23.8.13) Since A is of order iM-4, the decay rate per volume of the false vacuum is thus of order I-"I'V ;:~M-4 exp(-S) . (23 .8.14) Note that this is adecay rate per vo lume,because the decay does not occ ur by a c hange inthe sca larfieldsimultaneo usly eve rywhere inspace, but by the occ urrence of bubbl es of t rue vac uum i n a fa lse vac uum b ackground. " Where a continuous symmetry G is spontaneously broken to a subgroup H, there are additional collective coordinates giving the orientation of H within G . The integration over these parameters yields an additional factor in A equal to the volume of the coset space GBH . 468 23 Extended Field Configuration s Theresult(3,8 .14)is chiefly usef ul in the case w here B is large, so thatbarrierpenet ration is st rongly supp ressed, and we ca nestimate the suppressio nfacto ras simply exp(-B) . Fort unately, themost nat ural circumstanceinwhich B islarge is one in w hichit is poss ible to ca lculate Binclosed fo rm. Thisisthe case in whichthe energy V((O)) = -e of the true vac uum i s only slightlybelow t he zero e nergy of the fa lse vac uum, but V(O) is posit ive an dnot sma ll between 0 = 0 and ¢, = (0) .To minim ize the Euclideanaction (23 .8.3) in t his case, we must take 0 to be near(0) w ithin a four-dimensional ballwith a largeradius R, at which 0 drops to ze ro withina shellof thickness g ivenby somelength L ' :M-1 characterist ic of the potentia lin the limit e --* 0 . (Th is is so metimes ca lled the `th inwall app roximatio n,'but perhaps a better te rmwould b e thebig bubbl e approximation .) The act ion(23.8.3) inthis app roximationis S(R) ^'-;7r'R4~' +27r'R3y , (23 .8.15) where 9 is a s urface te nsion, equalto the she llcontribution to the act ion perarea. The seco ndtermontheleft-han dside of Eq . (23 .8.7)becomes negligible forp;:~R, so th isis now essent ially a one-dimensional pro blem. We can t herefore take t he surface tension f rom E q. (23 .1.4), wh ich fo ra solution of the fie ldequations is an eq uality, a nd inpresent notation reads .V 2V (f ) df (23.8.1)J4 The act ion (23 .8.15) is statio nary at a ra dius R^J 3~I-e , (3.8.17) so the action at its stationary point has the valu e 277r2Y4 B ^ (23.8.18)2.0 Note that B islarge fo rsmalle, so in this case t he decay rate of the fa lse vacuu m is strongly suppresse d. After t hebarrierhasbeen penet ratedthe bubble of t rue vacuu mwillgrow at t he spee dof light, colliding eve ntually with ot her bubb les,untilall space is in the state of lowest e nergy. Appendix A E uclidean Path Integra ls This appen dix will outline the use of E uclideanpath int egralsinquant um fieldtheory. As mentio ned in Section9.1,itispossible to for mulate quantumfieldtheoryina four-dimensional E uclideanspaceti me.Instea d of going i nto the non-trivialanalytic cont inuation neededto ca lculate S-matrix e lements inthis app roach,here we s hallillustrate the use of Appendix A Euclidean Path Integrals 469 Euclideanpath integralsby ad dressing a p roblem for w hichthey a re naturally suited. We co nsidera set of Hermitian cano nicalvariablesQuand P Q, with commutation relations [QQ,Pb]= iSab, (23.A.1) (23.A.2) Inquantum field theory the index a is un derstoo d, as i nSection9.1, to co nsist of a spat ialposition x and anydiscrete Loren tz and spec ies indices m, an dthe Kronecker deltain Eq . (23 .A.1) is understoo das bxm, yn = 63( x -y)b, We define eigenstates of t heQa Qalq) = qaIq), normalizedso that u(23.A.3) (23.A.4) and likewise fo reigenstates 1p) of t heP,. The p roblem we cons ider is the calculationof the matrix element F(q',q;T) =(R'~ exp (- H [Q, P]T)IRS, (23.A.5) where H is the Hamiltonian, and Tis an arbitrary positive constant . One application is to the study of ground-state energies . If the smallest eigenvalue of H is E0,with eigenvector 10), then for T --> o o so E0=-lim T--,,In F(q',q;T) T(23.A.6) Also, we can calculate the partition function of statistical mechanics from the trace : Z(I3} = Tr exP(-#H) = f[fJdq] F(q, q ; fl)~ (23.A.7) a where 1 /#is the temperature . To derive a path integral formula for F(q', q ;T) we define Euclidean time dependent operator s Qa(t)=eHtQae-Nt9 Pa (t)_exrPae-xiI (23.A.8) andcorrespon dingleft- a nd right-eigenstate s IR,t)-exp(Ht) Iq), (q,tI =(qI exp (-Ht} , (23.A.9) 470 such tha t and23Extended Field Configuration s IPAt)=exp(Ht)IP) , (p,tj=API exp(- Ht), (23.A.1a) Qa(t)lq,t}= qalq,t)~ Pa(t) IP,0=nnlP,0,(q, tiQa(t)=qu(q, tl, (Pa tlPa(t} = Pa (P, rI-(23.A.11) (Z3.A.12) One differencebetwee nthis and the usual Mink owsk ian fo rmalism is that the `time' evo lution of the operators is gove rned by a non-unitary s imilarity transfo rmation (23 .A.8), so t here is no simplerelation between the right- eigenstates (q, tjofQ(t) and the Hermitian adjoint of the left-cigenstates Iq, t), except at t = 0 . Inthislanguage, t hedefinition (23 .A.5) of F(q', q ;T)mayberewritte n Let us first convention haveF(q',R;T)=(qt, 7'I2 Iq, -T/2) . (23 .A.13) calculate (q', t + dtlq, t )for infinitesimaldt. Adopti ng the thatH(Q, P) is writte nwithallQs to t he left of a llPs, w e (q',t+dtlq,t)= (q', tlexp(-H(q',P)dt) Iq, ty. Letus expa nd~q, t)ina complete set of e igenstates of t heP,, (t) operators. FromEq. (23 .A.1) we have as usual (q,t~P,r)_ HCXp(iPuxu) 27ia so(P,t~R, t )_Rexp(-ipuxQ ) 27ia (q',t+ dtI q,t)=fi adPaexp i 1:Pu(qa - ~ 'a)-H(~'',P)dt 27ii aa (The summation convention is suspended here .) As in Section 9.1,we divide the time interval from -T/2 to T/2 into a large number of very small intervals and insert a sum over Q eigenstates for each interval . Defining functions q(t) and p(t) that interpolate between the values of the Qand Peigenvalues at each interval, we obtain the path integral expression for F in its most general for m F(q', q ;T)=. icl(- T/2)=q, q(T12)~q'fjdqa(t))f,dpaW 271a,t a, t r xexpT/2 Jdi[iEqu(t)Pa(t) - H (q(t),p(t))(23.A.14) T/z Q Appendix A Euclidean Path In tegrals 47 1 To calculate the partition function (23.A.7)we would integrate over the ps and qs subject only to the condition that q(t) is periodic with period equal to the inverse temperature P Z (fl) _i~q(#12)=q(-#12)J ( 1 jd 2~t}} x exp dt[i qa~t}pa(t} - H (q(t), p(t )(L'a ) I )(23.A.15) Eqs. (23 .A.14) and (23.A.15) look a little odd, with one term in the exponent real, and the other imaginary . These formulas begin to look more familiar after we do the path integral over the p,,(t)s . This integral is trivial in the important class of theories where H(q, p )is quadratic in the Ps : H(q, p) =1 2YAab(q)PuPh + ~ B a(q)Pa+C(q) a,h a(23.A.16) As shown in the appendix to Chapter 9, the integral over the ps in Eq. (23 .A.14) yield s F(q'aqaT)=Jq T z - ~ 2 -,(fJdqa(t)) ( Det [2i2rci(q)])11 2 T12 xexp ~ T2dt[iEqa(t)pu(t} - H (q(t),p(t))](23.A.17) ~ a where 4(q) is the `matrix' (23.A.18) and p(t) is the stationary `point' of the argument of the exponential in Eq. (23 .A.17) - that is, the solution of the equatio n 6H (4(t),p) i4aW_ 6Pa{23.A.19} P=P(t ) The factor ihere should not be surp rising,because Eq .(23.A.19)is the same equation that is satis fied by the non-Hermitian operators (23.A.8): iOa(t) =i1H, Qa(t)](Q(t), (o) 5Pa (t) For a Hamiltonian in the general quadratic form (23 .A.16), the solution of Eq .(23.A.19) is pa [A-'(q)] ", (iqb - Bb(q)) (23.A.20) 472 23Extended Field Configuration s and so Eq .(23.A.17)takes the for m F(q', q ; T) = j1dq,(t)) (Det [2i7EQI(q)])-1,2 exp(-S [q]) R(0)R,q(T)=q'a,t (23.A.21) where S [q] i sthe actio n T/2 S[q] = Clot 1 ~ A ab(q)4a4h +iEAt~b(q)Ba (q)4b LT/2 ab a b -~ Aah(q)Ba(q)Bh(q)+C(q) (23.A.22) ab In the special (but common) case where Ba(q) =0, this simplifies t o T12 S [4]=LT/2dt (23.A.23) ab Thus in this case the `Lagrangian' appearing in the path integral is equal to what the Hamiltonian would be in Minkowskian spacetime when the ps are expressed in terms of qs and 4s . Appendix B A List of Hamatapy Group s This appendix presents a list of homotopy groups4° for various manifolds . Here Z denotes the group of the integers, with group composition defined by addition, so that zero is the unit element . Also, Zn is the group of the integers modulo n . The trivial group, consisting of the element 0, is denoted 0. Homotopy groups for direct products of manifolds may be obtained from the homotopy groups of the manifolds themselves by the product rule : Spheres Zn(Sm )= 0 for nC m Rn(Sn)=Z 7En+ t(Sn} = Z2 except TCAS 2} _ 0 ; 7C3 (S2)= Z 7En+2(Sri)= Z2 except R3(SO = 0 7En+3(Sn)=Z24 except 7c4(SI)= 0;7c5(S2)_Z2 ;7E6(S3)= Z12, 7t7(S4)=Z>CZ12 arn(SI) = 0 except 7cI(Sl) = Z Problem s Lie Gro up Ma nifolds RI(G) _ R2(G) =0 R3(G} = ZZ G = U(1 ) Z2 G = SO(n) (n ~ 3) 0 other simple compact connected Lie groups G any compact connected Lie grou p G any compact connected simple Lie group473 Z2 XZ2G=Sn(4),Spin(4 ) 7r4(G)= Z 2 G = USp(2n), SU(2), SO(3),Spin(5), SO(5) 0 G=SU(n) (n 7 3), SO(n) (n 7 6), G2, F4, E n 7E4n+2(USp(2n )} = Z(2n+2),neven7T2n(SU(n)} = Znt22(2n+2) nodd Batt P eriodicityTheorems Forn -,,~(k-1)/4,k _>2, Z k = 3, 7 (mod $ ) 7ck(USp(2n )} = Z2 k = 4,5 (mod 8 ) 0 k =0, 1,2,6 (mad 8 ) Fornzk+2,k ~!!2, 2 7ck(SO(n))= Z 2 Forn _>(k+1)/2,k ~!!2,k=3,7 (mod 8) k=O,1(mod 8) k=2,4,5,b(mod S ) 7ES U n2 k odd~ ( ~ }} _0k eve n Coset Space s For a nyLie gro up G a nd anyLie subgroupH=G, 7c2(G/H)=kerI 7cl(H)~-4 7c 1(G)I. That is, 7E2(GIH)is the subgroup of 7EI(H) that maps into the trivial element of 7rt(G) when H is embedded in G . As a special case , 7c2(GIH)-7cI(H) for 7ci(G) = 0. Problem s 1. What sort of term would need to be added to the action for Goldstone boson fields in four-dimensional Euclidean spacetime to make it 474 23 Extended Field Configuration s possible to have topologically non-trivial field configurations at which the action is stationary ? 2. Consider a theory of scalar fields in six space dimensions in which the chiral symmetry SU(2) xSU(2) of the Lagrangian is sponta- neously broken to the S U(2) of isospin . Suppose that enough higher derivative terms are added to the Lagrangian so that skyrmions are stabilized . What sort of conservation law do these skyrmions obey? (Hint : Note that, as shown in Section 2.7, SU(2 )is topologically the same as S 3.) 3. Show that a ll7r,(. ) for n > 1and ar bitrary manifolds are Abelian. 4. What is the coupling-constant dependence of the contribution of instantons of unit winding number in an SU(4) gauge theory ? 5. Derive a formula for the axion mass in the case in which mz{, m d, and ms are all small . 6. Consider the theory of a real scalar field 0with Lagrangian densit y where rn2 and a are positive, and g is very small . Calculate the exponential suppression factor exp(-B) in the rate of decay of the metastable vacuum state in terms of m2, A, and g . References 1. A. A. Belavin, A . M. Polyakov, A . S. Schwarz and Yu . S. Tyupkin, Phys . Lett .59B, 85 (1975) . 2. G. 't Hooft, Nucl. Phys .B79, 276 (1974) ; A. M.Polyakov, JETP . Letters 20,194 (1974) . 3. T. H. R. Skyrme, Proc . Roy . Soc . London A260, 127 (1961) . 4. K. Nielsen and P . Oleson, Nuel . Phys .B61, 45 (1973) . 5. Ya . B. Zel'dovich, I . Yu . Kobzarev, and L . B. Okun', Sn v. Phys . JETP 40,2 (1975) . 6. S. Coleman, Phys . Rev . I] 15, 2929 (1977) ;C. G. Callan and S . Coleman, Phys . Rev .D16, 1762 (1977) . The discussion in Section 23.8is based on these references . For earlier work on the formation of regions of broken symmetry, see T . D. Lee and Cr . C. Wick, Phys . References 475 Rev.D9, 2291 (1974) ; M. B. Voloshin, I . Yu .Kobzarev, and L . B. Okun', Sov .J. Nucl . Phys .20, 644 (1975) . 7. S. Coleman, Nucl .Phys . B262, 263 (1985) ;T. D. Lee and Y . Pang, Phys . Rep .221, 251(1992), and other references by Lee and collaborators quoted therein . 8. For an extremely lucid review of these topics and references to much of the original literature, see S . Coleman, Aspects of Symmetry (Cambridge University Press, Cambridge, 1985) : Chapters 6 and 7 . A good, more recent review is given by E . J. Weinberg, Ann . Rev . Nucl.Part .Sci.42, 177 (1992) . 9. G. H. Derrick, J. Math .Phys. 5, 1252 (1964) . 10. E. B. Bogomal'nyi, Sav. J. Nucl .Pays . 24, 449 (1976) . 11. Homotopy theory has been used to study symmetry breaking i n the early universe by T . W. B. Kibble, J. Phys . A 9,1387 (1976) ; A. Vilenkin and E . P. S. Shellard, Cosmic Strings and Other Topo - logical Defects (Cambridge University Press, Cambridge, 1994), an d references quoted therein . 12. H. Georgi and S . L. Glashow, Pays . Rev . Lett . 28, 1494 (1972) . 13.J.Arafune, P . G. O. Freund, and C . J. Goebel, J. Math . Phys .16, 433(1975) . 14. M. K. Prasad and C . M. Sommerfeld, Phys . Rev . Lett .35, 7 60 (1975) . 15. B. Julia and A . Zee, Phys . Rev .D11, 27 6(1974) ; D13, 819 (1976) . 16. P. A. M. Dirac, Proc . Roy . Soc . London A133, 6 0(1934) ;Phys . Rev . 74, 817 (1948) . 17. Ya . B. Zel'dovich and M . Yu . Khlopov, Phys . Lett .79B, 239 (1978) ; J. Preskill, Pays . Rev . Lett .43, 1365 (1979) . For a review, see J . Preskill, Annual Rev . Nucl . Part . Science 34,461 (1984) . 18. See, e . g., S. Weinberg, Gravitation and Cosmology Wiley, New York, 1972) :Chapter 15 . 19. A. Guth, Phys .Rev.D23 , 347 (1981 ) ; A. D. Linde, Phys . Lett .108B, 389 (1982) ; A. Albrecht and P . Steinhardt, Phys .Rev. Lett .48,1220 (1982) . 20. R. Jackiw and C . Rebbi, Phys .Rev.D13, 3398 (1976) . 476 23 Extended Field Configuration s 21. V. A. Rubakov, JETP Lett .33, 644 (1991) ; IITucl. Phys .B212 , 391 (1982) ; C. G. Callan, Phys .Rev.D25 , 2141 (1982) ; Phys . Rev .D26 , 2058 (1982) ; PTued.Phys .B212 , 391 (1982) . 22. R. Bott, Bull. Soc . Math . France 84, 251 (1956) . 23. G. 't Hooft, Phys . Rev .D14, 3432 (1976) ;Phys . Rep .142, 357 (1986) . 24. M. Atiyah and R . Ward, Commun . Math . Phys .55, 117 (1977) . 25. G. 't Hooft, Phys . Rev .Lett.37, 8 (1976) . Instead of the identificatio n of anomaly-free conserved currents used here, 't Hoaft's derivatio n of the selection rules was based on an analysis of the zero modes o f the Dirac operator (see Section 23.7),which gives the change in eac h quantum number for an instanton of a given winding number . 26. C. G. Callan, R . F. Dashen, and D . J. Gross, P hys. Lett . 63B, 334 (1976) ; R. Jackiw and C . Rebbi, Pays . Rev . Lett . 37, 172 (1976) . 27. V. A.Kuzmin, V . A. Rubakov, and M . E. Shaposhnikov, Phys . Lett . 155B, 36 (1985) . The transition is dominated by field configuration s known as sphalerons ; see N . S. Manton, P hys. Rev . D28, 201 9 (1983) ; F. R. Klinkhammer and N . S. Manton, Phys .Rev.D30, 221 2 (1984) ; R. F. Dashen, B . Hasslacher, and A . Neveu, Phys . Rev .I]10 , 4138 (1974) . 28. H. Georgi and I . McArthur, unpublished (1981) ;D. B. Kaplan and A. V. Manohar, Pays . Rev .Lett .56, 2004 (198 6); K. Choi, Nucl . Phys .B383, 58 (1992 ). 29. For a more detailed calculation, see V . Baluni, Phys . Rev .D19, 2227 (1978) ; R. J. Crewther . P. Di Vecchia, G . Veneziano, and E . Witten, Fhys . Lett .88B, 123(1979) . 30. R. D. Peccei and H . Quinn, Phys . Rey . Lett .38, 1440 (1977) ;Phys . Rev.D16, 1791 (1977) . 31. S. Weinbe rg,Phys . Rev . Lett .40, 223 (1978) ; F Wilczek, Phys . Rev . Lett .40, 279 (1978) . 32. J. E. Kim, Pays . Rev . Lett .43, 1 03(1979) ;M. A. Shifman, A . I. Vainshtein, and V . I. Zakharov, Nucl .Phys .B166,493 (1980) ;A. R. Zhitnitsky, Sav. J. Nucl . Phys .31, 2 60 (1980) ;M. Dine, W . Fischler, and M . Srednicki, Phys . Lett .104B, 199 (1981 .). 33. S. Weinberg, in Neutrinos '78, ed. E. C. Fowler Purdue, Lafayette, 1978) . References 477 34. D. A. Dicus, E . W. Kolb, V .I. Teplitz, and R . V. Wagoner, Phys . Rev . Il18, 1829 (1978) ;D22, 839 (1980) . 35. For reviews, see M . S. Turner, Phys . dept .197, 67 (1990) ; G. G. Raffelt, Phys . Rept .198, 1 (1990) . 36. J.PreskilL M . B. Wise, and F . VLjilczek, Pays .Lett .120B , 127 (1983) ; L. F. Abbott and P . Sikivie, Phys . Lett .120B , 127 (1983) ; M. Dine and W . Fischler, Phys .Lett .120B, 137 (1983) . For reviews, see J . E. Kim, Phys .Rep.150, 1 ( 1987) ; M. S. Turner, Phys . Rep .197, 6S (1990) . 37. S. Coleman, V . Glaser, and A . Martin, Canamun . Math .Fhys .58,211 (1978) . 38. S. Coleman, Ref .8,Section 6 .2. 39. S. Coleman, Ref .8,Section 6 .5. 40. A. Actor, Rev. Mod . Phys .51, 461 (1979 );Encyclopedic Dictio nary ofMathematics (MIT Press, Cambridge, 1980) :Appendix A . AuthorIndex Where page numbers are given in italics, they refer to publications cited in lists of references . Abachi, S. 355 Abbot, L.110, 477 Abe, F .355, 356 Abe, K .162 Abrikosov, A . A.358 Actor, A . 477 Ademollo, M .293 Adler, S .160,248,250,361, 38 0, 417 Albrecht, A .475 Altarelli, G .160,162, 28 0, 294 Alvarez, O .418 Alvarez-Gaume, L. 61, 417, 418 Amaldi, U .357 Anderson, P . M. 332, 357 Anselmi, D .110 Arafune, J. 438, 475 Arnison, G . 356 Atiyah, M . F.417, 418, 476 Aubert, J . J.355 Augustine, J . E.355 Baker, M .160 Balian, R .110 Baluni, V . 476 Bamert, P . 356 Banks, T .417, 419 Barbieri, R . 419 Bardeen, W . A.161, 332, 342, 357, 379, 380, 417, 418 Barnich, G . 61, 110, 406, 4 20 Batalin, Y . A.61, 62, 405 Baulieu, L . 61 Baur, R .248 Beane, S . R.249 Becchi, C . 28, 60,110Belavin, A. A. 450, 4521 474 Bell,J. S. 361, 4 17 Bender, C. M, 78 Benfatt o,G.358 Benvenu ti, A. 356 Bernard, C. 293 Bernstein, J.248 Bijnens,J. 248, 25 0 Bismut,J. M.418 Bjorken, J. D. 273, 293 Bleuler, K.60 Bloom,E.D.161, 243 Bludm an, S . 355 Bogoliubov, N . N. 159 Bogomol'nyi, E .B. 338, 358, 425, 440, 441,475 Bonora,L. 61 Borel, A .251 Bott,R. 251, 449, 476 Bouchiat, C.418 Branchina, V. 79 Brandt, F. 61,1M 406, 419,420 Breidenbach,M.161,293 Brezin, E.160, 16 1 Brout,R.354 Brown,D. E. 358 Burgess, C .P.356 Cabibbo, N . 355 Cahill, K . 79 Callan, C . C.159,160, 250, 275, 294, 474, 47 6 Carlson, C . 357, 418 Cartan, E .11,26,62 Castorina, P . 79 Author Index Chaikin, P . M.160 Chanda, R . 419 Choi, K .476 Chou, K .-C. 248 Christ, N . 60, 294 Christensen, S . M. 418 Chu, C-S . 416, 42 0 Cohen, T . 249 Coleman, S . 69, 78, 127,160, 247, 250, 251, 4 19, 474, 475, 47 7 Collins, J . C. 160 Cooper, F . 78 Cooper, L . N. 332, 342, 357 Cornwall, J . M.357 Cotta-Ramusino, P . 61 Crewther, R . J. 476 Critchley, R . 418 Curci, G . 60, 6 1 D'Hoker, E .251 Das, T .248 Dashen, R . F.248, 250, 476 de Boer, W .357 de Wit, B . 61 De Witt, B . S. 2, 2$, 6 0, 110,418, 419 Delbouro, R .418 Derrick, G . H. 475 Deser, S .293, 417 Di Vecchia, P . 476 Dicus, D . A. 477 Dimopoulos, S .357, 419 Dine, M . 476, 477 Dirac, P . A. M. 14, 475 Dixon, J . A. 419 Domb, C .160 Donoghue, J. F.248, 249, 250, 272, 293 Dragon, N . 419 Dubois-Violette, M . 419 Duff, M . J. 418 Duncan, A . 293, 294 Dyson, F . J. 283, 294 Dzyaloshinskii, I . Ye. 358 Esker, G . 250 Eguchi, T . 418 Eichten, E . 16I, 35 7 Einstein, A .163Ellis, J .161, 355, 356, 357 Englert, F . 354 Ericson, T . 248479 Faddeev, L . D. 2,28, 60 Fanchotti, S . 356 Farhi, E .251, 357, 419 Farrar, G . 419 Fayyazuddin 293 Feldman, J .358 Fernandez, R . 160 Ferrara, S . 62 Ferrari, R . 60, 61 Feynman, R . P. 2,60, 274, 293 Finkelstein, R.J. 417 Fischler, W . 476, 47 7 Fisher, M . E. 147, 160 Fogli, G . L. 355, 356 Fowler, E . C. 476 Fradkin, E . S. 61 Fradkina, T . E.61 Freed, D . S.418 Freedman, D . Z. 62, 356 Freund, P . G. D. 418,438, 475 Friar, J .249 Friedman, J . L. 161, 273 Frishman, Y .417, 419 Fritzsch, H .161 Frohlich, J . 160 Fubini, S .248 Fujikawa, K . 301, 354, 361, 362, 417 Fujimoto, Y . 78 Fukuda, H .417 Forstmann, H .357 Gaillard, M . K. 161 Gallavotti, G. 358 Gasser, J . 232, 233, 250 Gamow, G . 355 Gatto, R .161 GeII-Mann, M . 60,111, 126, 157, 159, 161,225, 22 6, 231, 248, 249, 250, 349, 356, 413 Georgi, H .281, 294, 355, 357, 418, 419, 475, 47 6 Gibbons, G . W. 419 Gilkey, P . B. 418 480 Author Index Gilman, F . J.356 Gilmore, R . 62 Ginsparg, P . 417, 418 Ginzburg, V . L. 358 Glaser, V . 477 Glashow, S . L. 6[), 25 0,251, 355, 357, 418, 47 5 Cxlimm, J .160 Goebel, C . J. 438, 475 Goldberger, M . L.248 Goldman, T . 249 Goldstone, J . 78, 1 67, 19 1, 247, 354 Golowich, E .249,250,272,293 Gomis, J . 62,110 Gor'kov, L . P. 337, 358 Green, M .356 Green, M . S.160 Greenberg, O . W. 161 Greub, W . 251 Gribov, V .60 Grisaru, M .293, 417, 4 18 Gross, D . J. 59, 153, 160, 161, 275, 28 1, 294, 418, 476 Gupta, S . N. 60 Guralnik, G . S.248, 354 Guth, A . 475 Hagberg, E . 355 Hagen, C . R.354 Hall, L . J. 357 Halperin, S . 251 Han, M . Y. 161 Hanson, A .J.418 Hanson, G . 161 Harada, M .110 Hardy, G. N. 294 Hardy, J . C.355 Harvey, J . 358 Hasert, F . J.355, 356 Hasslacher, B . 294, 476 Hawking, S . W.160, 418, 4 19 Haymaker, R . W. 78 Heitler, W . 60 Henneaux, M .61,62,1I0, 406, 419, 420 Herb, S . W.355Higgs, P.W. 354 Hinchtiffe,I. 162 Hindm arsh,M. 79 Ho,P-M .420 Hollik, W. 356 Holstein,B.R.248, 249, 250 Hubb ard, J. 358 Iliopoulis, J . 78, 355, 418 Isham, C .110 Israel, W . 160 Itzykson, C . 78 Iverson, G.418 Iyanaga, S . 251 Jackiw, R . 60, 160,251, 357, 361, 417, 418, 419,475, 97 6 Jaffe, A . 61, I60 Johnson, K . 160,357 Johnston, D . 79 Jona-Lasinio, C . 78, 191, 248 Josephson, B . D. 335, 357 Jost, R .160 Julia, B . 475 Kallosh, R . E. 62 Kaplan, D . B. 249, 47 6 Kawarabayashi, K .293 Kawada, Y .251 Kelley, S . 357 Kemmer, N . 355 Kendall, H . W.161, 273 Khlopov, M . Yu .475 Kibble, T . W. B.354, 475 Killing, W .11 Kim, J . E. 476, 477 Kimura, T . 61 Kirk, W . T.161 Klein, O . 59 Klinkhammer, F . R. 476 Kleinknecht, K . 356 Knieh], B. 356 Kobayashi, M . 355 Kobzarev, I . Yu . 474, 475 Kogut, J.161 Kolb, E . W. 477 Koslowsky, V . T.355 Author Inde x Kotchan, D .248 Kreuter, M . 419 Kugo, T .60, 110 Kursunoglu, B .60 Kuzmin, V . A. 476 Landau, L . D. 136, 160, 358 Lane, K .161, 357 Langacker, P .250, 356, 3 57 Lautrup, B . 60,294 Lavrov, P . M. 61, 110 Le Guillou, J.C. 160, 161, 294 Lee, B . W.110, 301, 354 Lee, C . Y.249 Lee, T . D. 60,355, 356, 474, 475 Lehmann, H . 61 Leites-Lopes, J .355 Leutwyler, H . 161, 232, 233, 250 Levy, M .248,249, 41 9 Lipatov, L . N, 284, 294 Lisi, E .355, 356 LoSecco, J .293 Low, F . E. 111,126,157, 159,248, 349 Lubensky, T . C.160 Lurie, D .248 Luttinger, J . M. 160 Maiani, L .250,355, 419 Maksymyk, I . 356 Maltman, K .248 Manohar, A . V. 476 Maties,3 .b1,418,419 Manton, N . S. 476 Martin, A . 78, 477 Maskawa, K .355 Mathur, V . S.248 McArthur, 1 . 476 Mehta, M . L.418 Meissner, U . G.250 Meyer, Ph . 418 Miller, G .249 Mills, R . L. 1,l l,59 Min, D .-P. 249 Mitter, P . K. 61, 419 Miyamoto, Y . 417 Miyazawa, H . 248 Moshhin, P . Yu 61Mostow, G . D.250 Mueller, A . H. 294 Musset, P . 356 Muzinich, I. 60481 Nakanishi, N . 60 Nambu, Y . 161, 167,191,247,248 Nanopoulos, D . V. 357 Ne'cman, Y . 225, 226, 231, 250, 413 Neveu, A . 476 Nickel, B .G.161 Nielsen, H . 474 Nielsen, N . K.418 Norton, R . E. 356 Novikov, V . A. 356 O'Raifeartaigh, L . 78 Oakes, R . J.250 Oehme, R .248 Ojima, I .60,61 Okubo, S . 231, 250, 25 1 Okun', L. 356, 474, 475 Qjeson, P . 474 Ordonez, C . 249 Pagels, H .250 Palais, R . S.250 Pancheri, G. 250 Pang, Y . 474 Parravicini, G.78 Paris J .62 Parisi, G . 280, 294 Park, T .-S.249 Pati, J . 357 Paver, N .250 Peccei, R . D. 458, 4 76 Pendleton, H .293, 417 Penrose, R .110 Perez-Mercader, J . 78 Purl, M . L.355 Perlmutter, A . 60 Pernice, S . 294 Perry, M . J.418 Peskin, M . 161 Peterman, A .159 Petronzio, R .419 Pich, A .250 482 Polchinski,J.60, 35 8 Politzer,H.D.153, 161, 281, 294 Polonsky,N. 357 Polyakov, A . M. 436, 450,474 Pope, C.418 Popov, V .N. 2, 28, 60 Prasad,M.K.441,475 Preskill, J.251,419,475,477 Quinn, H . R. 357, 4 5$,476 Rabi, I . I.250 Raby, S . 419 Raffelt, G . G. 477 Ramond, P . 356 Ray, L .249 Rebbi, C . 60,475, 47b Rink, B. 3S6 Renner, B .250 Reshetnyak, A . A.61 Rho, M .249 Riazuddin 293 Romer, R .418 Rosenberg, L . 417 Rout, A . 28, 6 0,110 Roy, P .419 Rubako, V . A. 476Author Index Sakata, S .417 Salam, A . 78, 167, 19 1, X47, 354, 355 Samuel, S .62 Sanda, A . 301, 354 Sanford, J .250 Sarid, U . 357 Savage, M .244 Savard, G .355 Schrieffer, J . R. 332, 342, 3 57 Schur, I .418 Schwarz, A . S. 61, 450, 474 Schwimmer, A .417, 419 Schwinger, J.355, 417, 419 Schwitters, R . F.161 Sciama, D . Y10 Scott, L .60 Seely, R .251 Shankar, R . 358 Shaposhnikov, M . E. 4765hellard, E . P. S. 475 Shifman, M . A.162, 47 6 Shirkov, D . V. 159 Shrauner, E .248 Siegel, W .61 Sigg, D .249 Sikivie, P .477 Singer, I . M.61,417, 418 Sirlin, A . 356 Skyrme, T . H. R. 474 Slansky, R . 356 Slavnov, A . A. 79 Sokal, A . D.160 Sommerfield, C . M. 441, 475 Srednicki, M . 476 Srivastava, P . K. 418 Steinberger, J . 360, 417 Steinhardt, P . 475 Sterman, G .161 Stora, R . 28, 6 0,61,110, 418, 419 Stratonovich, R . L. 358 Stueckelberg, E . C. G.159 Susskind, L . 357, 41 9 Sutherland, D . G. 360, 417 Suzuki, M .293 Svartholm, N .354 Symanzik, K . 78, 159 't Hooft, G . (T, 110, 153, 1 51,251, 294, 300, 35 4, 390, 394, 395, 408, 418 Talon, M .419 Tanikawa, Y . 417 Taylor, J . C. 79, 273 Teitelboim, C. 62 Teller, E .355 Teplitz, V . I. 477 Thierry-Mieg, J .61 Thirring, W . 248 Tinkham, M . 340 Titchmarsh, E . C. 247 Tomozawa, Y .249 Towner, I .S.248, 35 5 Townsend, P . K. 419 Treiman, S .248,294 Troost, W. 420 Trubowitz, E .258 Author Inde x Turner, M . S. 477 Tyupkin, Yu . S. 450, 474 Tyutin, I . V. 28, 6 0,62, 1 10 Urech, R .248 Utiyarna, R .60 Vafa, C . 239, 240, 251, 39 5 Vainshtcin, A . I.358, 476 van Holten, J . W. 62 van Kolck, U .249 van Nieuwenhuizen, P . 62, 35 6, 413, 420 Van Proeyen, A 420 Vanstone, R .251 Velo, G .60,110 Veltman, M .110, 354, 356, 3 60, 417 Veneziano, G .293, 47 6 Viallet, C . M. 419 Vilenkin, A . 475 Vilkovisky, G . A. bT, 62, 405 Voloshin, M . B. 475 Voronov, B . L. 62,110 Vysotsky, M . I. 356 Wagoner, R . V. 477 Ward, J.355 Ward, R .476 Weinberg, E . 69, 78, 127,475 Weinberg, S . 60, 78, 110, 160, 161, 167, 191,247, 248, 249, 250, 251, 2 93. 354, 355, 356, 357, 358, 418, 4 19, 458, 475, 476 Weingarten, D .238, 251 Weisberger, W . I.248483 Weise, W .248 Wentzel, G .355 Wess, J . 234, 250,251, 396, 397, 408, 419 Wick, G . C. 474 Wightman, A .S.60,110 aligner, E . P.62 Wilczek, F .15 3, 161,281, 294, 357, 458, 476, 47 7 Wilson, K . G. 112, 147, 160, 161, 252, 293 Wise, M . B.249, 47 7 Witten, E . 234, 23 6, 239, 240, 251, 383, 396, 415, 418, 4 19, 420, 476 Wu, Y-S . 417 Wyler, D . 248 Yamawaki ,K. ,i 10 Yanagida ,T. 356 Yang ,C.N.1,11,59, 355 Yankielowic z, S.417,419 Youn g,J.E.248 Zakharov, V . I. 476 Zappala, D . 79 Zee, A .153,161,357, 417, 419,475 Zel'dovich, Ya .B. 474, 475 Zhitnitsky, A . 476 Zichichi, A .294 Zimmerman, W . 252, 293 Zinn-Justin, J.42,62,80,110,160, 1 61, 294, 35 4 Zumino, B . 61, 234, LSD, 251, 3 96, 397, 408, 4 17, 418, 419,420 Zweig, G .161 Subject Ind ex al meson, 27 1 accidental symmetries, 155 in quantum chromodynamics, 155 in standard model, 317-18, 32 6 adjoint representation, defined, 3 Adler-Bell-Jackiw anomaly, see anoma- lies in symmetries Adler zero, 175 Adler-Bardeen theorem, 38 0-1 Adler-Weisberger sum rule, 1 91, 210- 11,24 8 affine connection, 6r7 Altarelli-Parisi equations, 280-2 anomalies in symmetries, 42, 359-416, 454, 45 9 anomalous dimensions, 133 anomaly constant D~ p.,, defined, 373 antibrackets, 45-7, 82 ; also see Batalin- Vilkovisky formalism, anticanoni- car transformation s anti-BRST transformations, 41, 62 anticanonical transformations, 47, 93 antifields, 42-3, 92, 40 5 antighosts, seeghost and antighost fields antighost translation invariance, 88 asymptotic freedom, 133-6, 153 Atiyah-Singer index theorem, 37 0. 464 axial coupling constant 9A, 187, 203.4 axial gauge, 5, 15 -18, 3 5, 41 axions, 458-6 1 b quark, 154, 313-15 background field gauge , 95-100 Bardeen formula, 379 -80,398,414Bardeen-Cooper-Schrieffer theory, 342- 52 barrier penetration, see tunneling baryon number, 239, 436, 443, 454-5, 464 baryons, 2 25-6;also see nucleons base point, defined, 430 Batalin-Vilkovisky formalism, 42-50, 91-5,404- 7 beta-function defined, 12 1 dependence on coupling definition, 138-9,14 1 for electrodynamics, 12 6r7, 150 for minimum subtraction, 15 0 for multiple couplings, 140 2 for quantum chromodynamics, 152- 3 for scalar field theory, 121, 129 also see renormalization group Bianchi identity (for gauge fields), 13 , 58 Bjorken scaling, 273-5, 280, 283 Bogomol'nyi inequalities, 425, 440, 451 Borel transform, defined, 28 3 Bott periodicity theorems, 473 bounce, 421-2, 465-468 bound states, 238-9, 3$9- 96 BPS monopoles, 44 1 BRST quantization, 35-- 6, 41, 91 BRST symmetry, 271, 46, 81, 398- 404 bubble formation, 467- 8 cquark,152, 312-1 5 Ain A Subject Index Cabibbo angle, 185, 312-13 Callan-Gross relation, 275, 280 Callan-Symanzik equation, 121 ; als o see renormalization group canonical transformations (in Batalin- Vilkovisky formalism), see anticanon- ical transformation s Cartan decomposition, 215 Cartan-Maurer integral invariant, 445- 450 charge conjugation invariance, 155, 240 Chern--Pontryagin density, 365, 451 Chern -Simons form, 366, 40 2 chiral symmetry, 182-5, 19 1;also see S U(2) x S U(2), S U(3 ) XS U(3 ) cluster decomposition principle, 1 67, 456 cohomology, antibracket, 94-5, 405-7 anti-BRST, 6 1 BRST, 32-3, 41, 39 9 de Rham, 23 8 collective parameters, 46,4, 467 color, defined, 1 52 compact Lie algebras, defined, 9 compact Lie groups, 10 consistency conditions, seeWess-Zumino consistency condition s constraints, 15 convexity of effective potential, 74 correlation length, 337- $ cosets, defined, 214 cosmic strings, 42 9 cosmology, 426, 443, 455, 461, 464 Coulomb gauge, 15 coupling constant s electroweak, 30 5-10, 325-6, 32 9-30, 464 for non-Abelian gauge fields, 11-12, 100, 108-9, 29 7 for scalar fields, 120 covariant derivatives in gauge theories, 4 in general relativity, 6 in spontaneous symmetry breaking, 195, 219, 321485 covering groups, 58, 443 -4 GP non-conservation, 7 -8, 315, 457-61 critical exponents, phenomena, 145-$ currents , axial-vector, 185-90,199 in gauge theories, 12-13 in superconductors, 323, 33 6 of anomalous symmetr ies, 365 -7,377- 8,38 0 of broken symmetries, 169-70 of semileptonic weak interactions, 311- 13 used to define effective action, 63-5 vector (ofisospin )185 curvature, see R iemann-Christoffel cur- vature tenso r custodial symmetry, 326 cut-off, 108,112, 364-5 d quarks, see u and d quarks Debye frequency, 351 deep inelastic scattering, 153, 252, 272- 83 operator, defined, 49 ❑+-function, defined, 170 Derrick's theorem, 424, 428--9, 466 descent equations, 402- 4 De Witt notation, 3 7 De Witt-Faddeev-Popov method, see Faddeev-Popov-De Witt method differential forms 36, 38-9, 400 dimensional regularization, 108-9,122 , 148-9 ;also see minimum subtrac- tion Dirac quantization conditions, 441-2, 445 direct sums, defined, 9, 52 domain boundaries, 421, 425- 6 effective action, seequantum effective actio n effective field theory, 14 5 general broken global symmetry, 211- 25, 42 5 pions and nucleons, 192-211 superconductivity, 342 - 52 also see Wes-Zumino-Witten terms 486 Subject Inde x effective potential, 6 8-74, 127-3 0,168, 347-5 0 eightfold way, seeS U(3) x S U(3 ) electric dipole moment of neutron, 458 electrodynamics, 1,33-4, 108, 118- 19, 125-7,150,157-8 electron, 3051 t, 38 5 e Q+ annihilation, 127, 154, 27 2, 273, 287, 31 6 also see deep-inelastic scattering, elec- trodynamic s electroweak mixing angle 0, 3 07-$, 310- IL330 electroweak theory, 3 05-18, 3 25-7, 454, 464 electroweak-strong unification, 327-3 2 energy-momentum tensor, 13-14, 283, 391 qmeson, 22 5-6, 229-31, 2445, 361 Euclidean path integrals, 240, 362, 358- 70, 451, 468-47 2 exceptional Lie groups and algebras, 11,57, 38 4 extended technicolor, 32 7 exterior derivative, defined, 36, 400 external fields Kn, 8 0-1 F parameter for broken symmetry, de- fined, 17 2 Faddeev--Popov-De Witt method, 2, 19 24,4 3 Fermi coupling constant, 185, 316 Fermi surface, 343 ferromagnetism, 14 7 Feynman gauge, 2 3 field strength tensor F,, defined, 5 fixed point, 133, 145- 8 flavor, defined, 152 flux quantization, 335, 338, 340, 342, 429 free energy, 147 Fujikawa-Lee-Sanda gauge, 301-4 fundamental group, 43 0 gap function ,348-9 Gasser -Leutwyler parameters, 232 -3 gauge fix ing functional ,19,43,48,301gauge invariance, 2 7, 3 19, 332 Dell-Mann-Low function, see beta-function general relativity, see gravitation generalized Feynman (~) gauge, 23-4 generators (of symmetries), 14 Georgi-Glashow electroweak model, 3 05, 436, 442, 44 4 ghost and antighost fields, 24- 7, 38, 302- 4 ghost number, 25,43, 88 ghosts of ghosts, 39 CIIM mechanism, 3 12 Ginzburg-Landau theory, 337, 341 gluons, 154-6, 27 8 Goldberger Treiman . relation, 187, 190- 1,204,36 0 Goldstone bosons, 167-9 1, 215-25, 295- 6,40$-16 ;also see mesons, pions, skyrmion s grand unification, seeelectroweak-strong unificatio n gravitation, 1, 6r8, 13-14, 36, 43, 91, 386 Gribov ambiguity, 1 5 Haar measure, 2 2 hadrons, see mesons, baryons Higgs boson, 316 Higgs mechanism, defined, 29 5 Hodge operator, 40, 62 homotopic equivalence, defined, 430-1 homotopy groups, 235, 423-5, 4 27-36, 444-7,472- 3 Hubbard-Strat«novich transformation, 345 hypercharge y, defined, 30 6 Infrared divergences, 113 14, 154 infrared-safe, defined, 11 4 instantons, 286, 421, 426-7, 436, 450- 64 irreducible, defined, 51-2 irrelevant interactions, 145, 148, 344 isotopic spin, 3, 163, 182-3, 24 3 J/y)particle, 3 13 Jacobi identity, 3, 39, 45, 46 Subject Index jets, 114, 154-5 Josephson junction, 335- 6 K mesons, 228-31, 236, 312-13 Kobayashi-Maskawa matrix, 313 Kronecker index, 43 8 KSFR formula, 271 Amatrices, defined, 22 6 A parameter in quantum chromody- namics, 155-7 Landau (or Lorentz) gauge, 23, 301 Landau ghost, 132, 13 6 lattice approximation, 22, 37-8, 112, 136-9,238,23 9 Legendre transformation, 6 5, 81 lepton number, 3 06, 317-18, 46 4 Lie algebras, see compact Lie algebras, semisimple Lie algebras, simple Lie algebras, U(1) Lie algebra s Lie groups, see compact Lie groups, exceptional Lie groups, orthogonal groups, symplectic groups, unitary group s magnetic monopoles, 225, 421, 429, 435-4 5 Mandelstam variables, defined, 114 marginal interactions, 14 5 master equation, 43, 45, 82 ; also see quantum master equation measur e for integration over fermion fields, 361-70,45 4 for integration over gauge fields, 18 Meissner effect, 33 4 mesons, 225- 34, 236, 243-6, 271 ;also seepions metric for gauge groups, 7-12, 50-3 for Goldstone boson fields, 423 for Lie groups, 44 8 minimal variables, 4 4 minimum subtraction, 122,146-51, 157 modified minimum subtraction, see min- imum subtractio n Mott cross section, defined, 273muon ,309-11,389487 Nakanishi -Lautrup field ,28 Nambu -Goldstone bosons, seeGold- stone boson s neutral currents, 311 13, 315, 327 neutrinos, 3 05-11, 38 5 v-e and v- ereactions ,272, 311 masses and oscillations, 317-18 neutron, see elect ric dipole moment o f neutron, nucleo n non-Abelian gauge theor ies, 1-58, 86 - 91,152-7, 42 6r30, 436 ;also see electroweak theory, quantum chro- modynam ics, electroweak -strong unificatio n non-linear realizations, 1 95,215-2 4 nucleon as skyrmion, 43 6 axial form factors, 18 6-7,190 interaction with pions, 202- 9 masses 209, 233- 4 a) meson, 232 one-particle-irreducible graphs, 65-7 open gauge algebras, 42, 44, 48-50 operator product expansions, 252 83, 287-9 2 order parameters, 224-5, 33 6r7, 429 orthogonal groups, 11, 55-8, 384, 473 14Q decay, 185, 31 2 p-form fields, see differential forms parity conservation, 155, 240, 3 15, 416, 457-6 1 partial conservation of axial current (PCA Q,191 partition function, 469, 471 Parton model, 274-5 Peccei-Quinn symmetry, 458-61 penetration depth, 334, 337, 35 2 persistent currents, 335- 6 persistent mass condition, 238-9, 395- 6 phase transitions, 148 0meson, 236 488 pions as Goldstone bosons, 182-211 coupling to nucleon Gn, 187 7c± decay, 185-6, 18 9 7c° decay, 359-61, 368 mass, 189, 201Subject Inde x scattering or, nucleons, 2 06r9 scattering on pions, 198-2 02, 249 tadpoles, 45 8 power counting, 197-8, 200, 2 05-6, 203-4, 323, 34 4 proton decay, 318 , pseudo-Goldstone bosons, 177-82, 324 pseudo-real representations, defined, 384 pure gauge fields, 6, 334, 412, 427, 428, 449,45 5 quantum chromodynamics, 152-7, 181, 192, 225, 238, 239, 287--8, 421, 453 ; also see quarks, gluons, asymptotic freedom, deep inelastic scatterin g quantum effective action, 63-8, 75-7, 164,167,346-7 , 352 quantum electrodynamics, see electro- dynamic s quantum master equation, 49, 92, 407 quarks, 152-7, 194, 225-7, 275, 312- 15 masses, 231-2, 243, 4 57-61 also see anomalie s real and pseudoreal representations, 384 reducible gauge algebras, 38- 9 relevant interactions, 145, 148 renormalization at sliding scale, 111-12, 119-30 in general theories, 91-5, 141- 2 in `renarmalizable' gauge theories, 82- 91 of electron-electron potential, 349 of fields, 99-10 0, 118- 19 of gauge coupling constants, 100, tU8-g, 119 of general operators, 115-18, 1 23, 255,260,29 1 of masses, 99, 143^ 4 of pion self-interaction coupling con- stants, 199of scalar coupling constants, 114-15 renormalization group, 111-5 8, 263- 5, 329-32, 349--50, 453 ;also see anomalous dimensions, asymptotic freedom, asymptotic safety, beta function, critical phenomena, fixed point, Landau ghos t renormalons, 283-8 p mesons, 232, 271 Riemann-Christoffel curvature tensor , 6,386 Riemann-Lebesgue theorem, 166 right cosets, defined, 21 4 running coupling constant, see cou- pling constants, beta-functio n s quark, 152, 226, 244,312-15 Schwinger terms, 4 03-4 second class currents, 18 6 6 model, 193-6 uterm, 209 simple Lie algebras, defined, 9 skyrmions ,42t,423-6 Slavnov operator ,defined, 39 Slavno v-Taylor identities ,76r7, 8 1,407 SO(10 )(or Sp in(10)} symmetry, 328, 386 spectral function sum rules, 26 6r72 sphalerons, 47 6 Spin(n) group, 436 ,444 spontaneous symmetry breaking 63 approx imate symmetries, 177-82 dynamical symmetry breaking, 3 t 8- 27 global symmetries, 163 -246, 265 -6 in superconductivity, 33 2 local symmetr ies,2, 29 5-352 standard model, 384-88 ;also seeelec- troweak theory, quantum chromo- dynamic s standard SU(2) subgroup, 449 Stora -Zumino descent equations ,see descent equation s strangeness, 155, 239 string theor ies, 36 ,37,38, 39, 41 Subject Index structure constants, 2-3, 8-11, 39,50- 3,89 structure functions, 272-80 subalgebras, defined, 9 superconductivity, 225, 332-52 supergravity, 42 supersymmetry, 33 1 surface tension, 426, 46 8 SU(2) x U(1) symmetry, see electroweak theor y S U(2) xS U(2) symmetry, 182-5, 191, 271,360- 1 SU(3) and S U(3) xS U(3) symmetry, 225-38,243,273 SU(4) xSU(4) symmetry, 32 8 SU(S )symmetry, 3 28 symmetric space, defined, 215 symplectic groups, 9, 56-8, 384, 47 3 t quark, 152, 154, 313- 16 z lepton, 272, 313 technicolor, 239, 326 - 7 temperature effects, 146-7, 455, 469, 471 temporal gauge, 427-8, 437, 454 theta angle, 455-6 1 thin wall approximation, 468 threshold corrections, 3 58 time reversal invariance, 240, 315 topology, 422-30 ;also see homotopy, cohomolog y totally reducible, defined, 52 trapping, 2, 1 53-5 tree graphs, defined, 66 trivial pairs, 44 triviality, 137- 8 tunneling, 165, 455, 464, 468 twist, 277489 u and d quarks, 152, 182, 226, 244, 275, 312-15,367,39 5 U(l)Lie algebras, defined, 9 U(1) problem, 243-6, 454 unbroken symmetries, 238-43 unitarity gauge, 295-30 1, 324, 352-3, 473 unitary groups and algebras, 11, 54-5, 58, 384, 433, 47 3 universality classes, 14 8 vacuum alignment, 179, 1$1, 188-9 vacuum decay, 12930, 421-2, 464-8 vacuum degeneracy, 163- 7 vortex lines, 225, 338- 42, 421, 42 9 W± particles, 307-1 1, 316 weak interactions, 18 5 Wens-Zumino consistency conditions, 396--407, 409, 411 Wess-Zumino-Witten terms, 234-8, 413- 15 Wilson-Fisher expansion, 147-8 winding number, 435-6, 449, 452-3, 455- 7 Yang -Mill stheory , see non-Abel ian gaug e theories Yukawa interacti ons,30$-10 Z and 2„ groups, defined, 433, 472 Z Oparticle, 127, 157, 307-11, 31 6 zero mass singularities, seeinfrared di- vergence s zero modes, 426 . 462, 464, 467, 476 Zinn-Justin equation, 42, 80-2,93, 405-5