Graviton_physics

arXiv:gr-qc/0607045v1 11 Jul 2006Graviton Physics Barry R. Holstein Department of Physics—LGRT University of Massachusetts Amherst, MA 01003 February 3, 2008 Abstract The interactions of gravitons with matter are calculated in parallel with the familiar photon case. It is shown that graviton scat tering amplitudes can be factorized into a product of familiar elec tromagnetic forms, and cross sections for various reactions are straigh tforwardly evaluated using helicity methods. 1 1 Introduction The calculation of photon interactions with matter is a stap le in an intro- ductory (or advanced) quantum mechanics course. Indeed the evaluation of the Compton scattering cross section is a standard exercise in relativistic quantum mechanics, since gauge invariance together with th e masslessness of the photon allow the results to be presented in terms of sim ple analytic forms[1]. On the surface, a similar analysis should be applicable to th e interactions of gravitons. Indeed, like photons, such particles are mass less and subject to a gauge invariance, so that similar analytic results for gra viton cross sections can be expected. Also, just as virtual photon exchange leads to a detailed understanding of electromagnetic interactions between ch arged systems, a careful treatment of virtual graviton exchange allows an un derstanding not just of Newtonian gravity, but also of spin-dependent pheno mena associated with general relativity which are to be tested in the recentl y launched gravity probe B[2]. However, despite this obvious parallel, examin ation of quantum mechanics texts reveals that (with one exception[3]) the ca se of graviton interactions is not discussed in any detail. There are at lea st three reasons for this situation: i) the graviton is a spin-two particle, as opposed to the spin -one photon, so that the interaction forms are somewhat more complex, inv olving symmetric and traceless second rank tensors rather than sim ple Lorentz four-vectors; ii) there exist few experimental results with which to compa re the theo- retical calculations; iii) as we will see later in some processes, in order to guaran tee gauge invariance one must include, in addition to the usual Born an d seagull diagrams, the contribution from a graviton pole term, invol ving a triple- graviton coupling. This vertex is a sixth rank tensor and con tains a multitude of kinematic forms. However, recently, using powerful (string-based) techniq ues, which simplify conventional quantum field theory calculations, it has rece ntly been demon- strated that the elastic scattering of gravitons from an ele mentary target of arbitrary spin must factorize[4], a feature that had been no ted ten years pre- viously by Choi et al. based on gauge theory arguments[5]. Th is factorization 1 permits a relatively painless evaluation of the various gra viton amplitudes. Below we show how this factorization comes about and we evalu ate some rel- evant cross sections. Such calculations can be used as an int eresting auxiliary topic within an advanced quantum mechanics course In the next section we review the simple electromagnetic cas e and develop the corresponding gravitational formalism. In section 3 we give the factor- ization results and calculate the relevant cross sections, and our results are summarized in a concluding section 4. Two appendices contai n some of the formalism and calculational details. 2 Photon Interactions: a Lightning Review Before treating the case of gravitons it is useful to review t he case of photon interactions, since this familiar formalism can be used as a bridge to our understanding of the gravitational case. We begin by genera ting the pho- ton interaction Lagrangian, which is accomplished by writi ng down the free matter Lagrangian together with the minimal substitution[ 6] i∂µ−→iDµ≡i∂µ−eAµ whereeis the particle charge and Aµis the photon field. As examples, we discuss below the case of a scalar field and a spin 1/2 field, s ince these are familiar to most readers. Thus, for example, the Lagrang ian for a free charged Klein-Gordon field is known to be L=∂µφ†∂µφ+m2φ†φ (1) which becomes L= (∂µ−ieAµ)φ†(∂µ+ieAµ)φ+m2φ†φ (2) after the minimal substitution. The corresponding interac tion Lagrangian can then be identified— Lint=−ieAµ(∂µφ†φ−φ†∂µφ) +e2AµAµφ†φ (3) Similarly, for spin 1/2, the free Dirac Lagrangian L=¯ψ(i/negationslash∇−m)ψ (4) 2 (a) (b) (c) Figure 1: Diagrams relevant to Compton scattering. becomes L=¯ψ(i/negationslash∇−e/negationslashA−m)ψ (5) whereby the interaction Lagrangian is found to be L=−e¯ψ/negationslashAψ (6) The single-photon vertices are then <pf|Vµ em|pi>S=0=e(pf+pi)µ(7) for spin zero and <pf|Vµ em|pi>S=1 2=e¯u(pf)γµu(pi) (8) for spin 1/2—and the amplitudes for photon (Compton) scatte ring— cf. Fig- ure 1—can be calculated. In the case of spin zero, three diagrams are involved—two Born terms and a seagull—and the total amplitu de is AmpCompton (S= 0) = 2e2/bracketleftbigg2ǫi·piǫ∗ f·pf pi·ki−ǫi·pfǫ∗ f·pi pi·kf−ǫ∗ f·ǫi/bracketrightbigg (9) Note that all three diagrams must be included in order to sati sfy the stricture of gauge invariance, which requires that the ampl itude be unchanged under a gauge change ǫµ−→ǫµ+λkµ Indeed, we easily verify that under such a change for the inci dent photon δAmpCompton (S= 0) =λ2e2/bracketleftbiggpi·kiǫ∗ f·pf ki·pi−ki·pfǫf·pi pf·ki−ki·ǫ∗ f/bracketrightbigg =λ2e2ǫ∗ f·(pf−pi−ki) =λ2e2ǫ∗ f·kf= 0 (10) 3 In the case of spin 1/2, there exists no seagull diagram and on ly the two Born diagrams exist, yielding[7] AmpCompton (S=1 2) =e2¯u(pf)/bracketleftbigg/negationslashǫ∗ f(/negationslashpi+/negationslashki+m)/negationslashǫi 2pi·ki−/negationslashǫi(/negationslashpf−/negationslashki+m)/negationslashǫ∗ f 2pf·ki/bracketrightbigg u(pi) (11) Again, one can easily verify that this amplitude is gauge-in variant— δAmpCompton (S=1 2) =λe2¯u(pf)/bracketleftbigg/negationslashǫf∗(/negationslashpi+/negationslashki+m)/negationslashki 2pi·ki−/negationslashki(/negationslashpf−/negationslashki+m)/negationslashǫ∗ f 2pf·ki/bracketrightbigg u(pi) = ¯u(pf)/bracketleftbig /negationslashǫ∗ f−/negationslashǫ∗ f/bracketrightbig u(pi) = 0 (12) The corresponding cross sections can then be found via stand ard methods, as shown in many texts[7]. The results are usually presented in the labora- tory frame— pi= (m,/vector0)—wherein the incident and final photon energies are related by ωf=ωi 1 + 2ωi msin21 2θ(13) whereθis the scattering angle. For unpolarized scattering, we sum (average) over final (initial) spins via /summationdisplay λǫ∗µ λǫν λ=−ηµν,/summationdisplay su(p,s)i¯u(p,s)j=(/negationslashp+m)ij 2m(14) and the resulting cross sections are well known— dσlab(S= 0) dΩ=α2 m2(ωf ωi)21 2(1 + cos2θ) (15) and dσlab(S= 1/2) dΩ=α 2m2(ωf ωi)2[ωf ωi+ωi ωf−1 + cos2θ)] =α2 m2ω2 f ω2 i[1 2(1 + cos2θ)(1 + 2ωi msin21 2θ) + 2ω2 i m2sin41 2θ] (16) 4 2.1 Helicity Methods For use in the gravitational case it is useful to derive these results in an alternative fashion, using the so-called ”helicity formal ism,” wherein one de- composes the amplitude in terms of components of definite hel icity[8]. Here helicity is defined by the projection of the particle spin alo ng its momentum direction. In the case of a photon moving along the z-directi on, we choose states ǫλi i=−λi√ 2(ˆx+iλiˆy), λ i=± (17) while for a photon moving in the direction ˆkf= sinθˆx+ cosθˆz we use states ǫλf f=−λf√ 2(cosθˆx+iλfˆy−sinθˆz), λ f=± (18) Working in the center of mass frame we can then calculate the a mplitude for transitions between states of definite helicity. Using ǫ± i·pf=−ǫ∗± f·pi=∓p√ 2sinθ ǫ∗± f·ǫ± i=−1 2(1 + cosθ), ǫ∗± f·ǫ∓ i=−1 2(1−cosθ) we find for spin zero Compton scattering A++=A−−=−e2/parenleftbigg 1 + cosθ+p2sin2θ pi·kf/parenrightbigg A+−=A−+=−e2/parenleftbigg 1−cosθ−p2sin2θ pi·kf/parenrightbigg (19) While these results can be found by direct calculation, the p rocess can be simplified by realizing that under a parity transformation t he momentum reverses but the spin stays the same. Thus the helicity rever ses, so parity conservation assures the equality of Aa,bandA−a,−bwhile under time re- versal both spin and momentum change sign, as do initial and fi nal states, guaranteeing that helicity amplitudes are symmetric— Aa,b=Ab,a. 5 Using the standard definitions s= (pi+ki)2, t= (ki−kf)2, u= (pi−kf)2 it is easy to see from simple kinematical considerations tha t p2=(s−m2)2 4s,cos1 2θ=((s−m2)2+st)1 2 s−m2=(m4−su)1 2 s−m2 sin1 2θ=(−st)1 2 (s−m2)(20) We can write then[9] A++=A−−= 2e2(s−m2)2+st (s−m2)(u−m2) A+−=A−+= 2e2−m2t (s−m2)(u−m2)(21) (It is interesting that the general form of these amplitudes follows from simple kinematical constraints, as shown in ref. [10].) The cross s ection can now be written in terms of Lorentz invariants as dσ dt=1 16π(s−m2)21 2/summationdisplay i,j=±|Aij|2 = 4e4(m4−su)2+m4t2 16π(s−m2)4(u−m2)2(22) and can be evaluated in any desired frame. In particular, in t he laboratory frame we have s−m2= 2mωi, u−m2=−2mωf m4−su= 4m2ωiωfcos21 2θ, m2t=−4m2ωiωfsin21 2θ (23) Since dt dΩ=d 2πdcosθ/parenleftbigg −2ω2 i(1−cosθ) 1 +ωi m(1−cosθ)/parenrightbigg =ω2 f π(24) the laboratory cross section is found to have the form dσlab dΩ(S= 0) =dσ(S= 0) dtdt dΩ=α2 m2ω2 f ω2 i(cos41 2θ+ sin41 2θ) (25) 6 and, using the identity cos41 2θ+ sin41 2θ=1 2(1 + cos2θ) Eq. 25 is seen to be identical to Eq. 15 derived by conventiona l means. A corresponding analysis can be performed for spin 1/2. Work ing again in the center of mass frame and using helicity states for both photons and spinors, one can calculate the various amplitudes. In this c ase it is conve- nient to define the photon as the ”target” particle, so that th e corresponding polarization vectors are ǫλi i=−λi√ 2(−ˆx+iλiˆy) ǫλf∗ f=−λf√ 2(sinθˆx+ cosθˆz−iλfˆy) (26) for initial (final) state helicity λi(λf). Working in the center of mass, the corresponding helicity amplitudes can then be evaluated vi a Bsfλf;siλi= ¯u(pf,sf)[2ǫi·pi/negationslashǫ∗ f pi·ki−2ǫi·pf/negationslashǫ∗ f pi·kf+/negationslashǫ∗ f/negationslashki/negationslashǫi pi·ki−/negationslashǫi/negationslashki/negationslashǫ∗ f pi·kf]u(pi,si) (27) Useful identities in this evaluation are O1≡ /negationslashǫ∗ f/negationslashki/negationslashǫi=pλiλf 2/parenleftbigg A+ Σλi(A+ Σ) −λi(A+ Σ)−(A+ Σ)/parenrightbigg O2≡ /negationslashǫi/negationslashki/negationslashǫ∗ f=pλiλf 2/parenleftbigg A−Σ−λi(A−Σ) λi(A−Σ)−(A−Σ)/parenrightbigg O3≡ǫi·pf/negationslashǫ∗ f=psinθλiλf 2/parenleftbigg 0 Υ −Υ 0/parenrightbigg (28) where A=λiλf+ cosθ Σ = (cos θλi+λf)σz+λisinθσx−isinθσy Υ =−cosθσx+ sinθσz−iλfσy (29) We have then Bsfλf;siλi=E+m 2m/parenleftBig χ† f−2sfp E+mχ† f/parenrightBig ×/bracketleftbigg1 s−m2O1−1 u−m2(O2+O3)/bracketrightbigg /parenleftbiggχi 2sip E+mχi/parenrightbigg (30) 7 and, after a straightforward (but tedious) exercise, one fin ds the amplitudes[11] B1 21;1 21=B−1 2−1;−1 2−1=2√se2pcos1 2θ m2−u(−1 m+m ssin21 2θ) B1 21;1 2−1=B−1 21;−1 2−1=B1 2−1;1 21=B−1 2−1;−1 21=−2e2mp (m2−u)√ssin21 2θcos1 2θ B1 2−1;1 2−1=B−1 21;−1 21=−2√se2p m(m2−u)cos31 2θ B1 21;−1 21=−B−1 2−1;1 2−1=B−1 21;1 21=−B1 2−1;−1 2−1=−2e2p (m2−u)sin1 2θcos21 2θ B1 2−1;−1 21=−B−1 21;1 2−1=−2e2p m2−usin31 2θ B1 21;−1 2−1=−B−1 2−1;1 21=−2e2m2p s(m2−u)sin31 2θ (31) Summing (averaging) over final (initial) spin 1/2 states we d efine spin-averaged photon helicity quantities |B++|2 av=|B−−|2 av=2p2e4ssin2 1 2θ m2(m2−u)2/bracketleftbigg 1 + cos41 2θ+ sin41 2θm4 s2(1−2s m2)/bracketrightbigg |B+−|2 av=|B−+|2 av=2p2e4sin41 2θ (m2−u)2/bracketleftbigg 2m2 scos21 2θ+ (1 +m4 s2) sin21 2θ/bracketrightbigg (32) which, in terms of invariants, have the form |B++|2 av=|B−−|2 av=e4 2m2(u−m2)2(s−m2)2(m4−su)(2(m4−su) +t2) |B+−|2 av=|B−+|2 av=e4t2(2m2−t) 2(s−m2)2(u−m2)2(33) The laboratory frame cross section can be determined as befo re (note that this expression differs from Eq. 25 by the factor 4 m2, which is due to the different normalizations for fermion and boson states— m/Efor fermions and 1/2Efor bosons) dσ(S= 1/2) dΩ=4m2ω2 f 16π2(s−m2)2[|B++|2 av+|B+−|2 av] =α2 m2ω2 f ω2 i[1 2(1 + cos2θ)(1 + 2ωi msin21 2θ) + 2ω2 i m2sin41 2θ] (34) 8 which is seen to be identical to the previously form—Eq. 16. So far, all we have done is to derive the usual forms for Compto n cross sections by non-traditional means. However, in the next sec tion we shall see how the use of helicity methods allows the derivation of the c orresponding graviton cross sections in an equally straightforward fash ion. 3 Gravitation The theory of graviton interactions can be developed in dire ct analogy to that of electromagnetism. Some of the details are given in Ap pendix A. Here we shall be content with a brief outline. Just as the electromagnetic interaction can be written in te rms of the coupling of a vector current jµto the vector potential Aµwith a coupling constant given by the charge e Lint=−ejµaµ(35) the gravitational interaction can be described in terms of t he coupling of the energy-momentum tensor Tµνto the gravitational field hµνwith a coupling constantκ Lint=−1 2κTµνhµν(36) Here the field tensor is defined in terms of the metric via gµν=ηµν+κhµν (37) whileκis defined in terms of Newton’s constant via κ2= 32πG. The energy- momentum tensor is defined in terms of the free matter Lagrang ian via Tµν=2√−gδ√−gLint δgµν(38) where√−g=/radicalbig −detg= exp1 2trlogg (39) is the square root of the determinant of the metric. This pres cription yields the forms Tµν=∂µφ†∂νφ+∂νφ†∂µφ−gµν(∂µφ†∂µφ−m2φ†φ) (40) 9 for a scalar field and Tµν=¯ψ[1 4γµi← →∇ν+1 4γνi← →∇µ−gµν(i 2/negationslash← →∇−m)]ψ (41) for spin 1/2, where we have defined ¯ψi← →∇µψ≡¯ψi∇µψ−(i∇µ¯ψ)ψ (42) The matrix elements of Tµνcan now be read off as <pf|Tµν|pi>S=0=pfµpiν+pfνpiµ−ηµν(pf·pi−m2) (43) and <pf|Tµν|pi>S=1 2= ¯u(pf)[1 4γµ(pf+pi)µ+1 4γν(pf+pi)µ]u(pi) (44) We shall work in harmonic (deDonder) gauge which satisfies, i n lowest order, ∂µhµν=1 2∂νh (45) where h= trhµν (46) and yields a graviton propagator Dαβ;γδ(q) =i 2q2(ηαγηβδ+ηαδηβγ−ηαβηγδ) (47) Then just as the (massless) photon is described in terms of a s pin-one polar- ization vector ǫµwhich can have projection (helicity) either plus or minus on e along the momentum direction, the (massless) graviton is a s pin two particle which can have the projection (helicity) either plus or minu s two along the momentum direction. Since hµνis a symmetric tensor, it can be described in terms of a simple product of unit spin polarization vector s— helicity = +2 : h(2) µν=ǫ+ µǫ+ ν helicity =−2 :h(−2) µν=ǫ− µǫ− ν (48) and just as in electromagnetism, there is a gauge condition— in this case Eq. 45—which must be satisfied. Note that the helicity states giv en in Eq. 48 are consistent with the gauge requirement, since ηµνǫµǫν= 0,andkµǫµ= 0 (49) With this background we can now examine various interesting reactions in- volving gravitons, as we detail below. 10 3.1 Graviton Photoproduction Before dealing with our ultimate goal, which is the treatmen t of graviton Compton scattering, we first warm up with a simpler process—t hat of gravi- ton photoproduction γ+s→g+s, γ +f→g+f The relevant diagrams are shown in Figure 2 and include the Bo rn dia- grams accompanied by a seagull and by the photon pole. The exi stence of a seagull is required by the feature that the energy-momen tum tensor is momentum-dependent and therefore yields a contact interac tions when the minimal substitution is made, yielding the amplitudes <pf;kf,ǫfǫf|T|pi;ki,ǫi>seagull=κe/braceleftbigg−2ǫ∗ f·ǫiǫ∗ f·(pf+pi)S= 0 ¯u(pf)/negationslashǫ∗ fǫ∗ f·ǫiu(pi)S=1 2(50) For the photon pole diagram we require a new ingredient, the g raviton-photon coupling, which can be found from the expression for the phot on energy- momentum tensor[6] Tµν=−FµαFα ν+1 2gµνFαβFαβ(51) This yields the photon pole term <pf;kf,ǫfǫf|T||pi;kiǫi>γ−pole=e<p f|jα|pi>1 (pf−pi)2 ×κ 2[2ǫ∗α f(kf·kiǫ∗ f·ǫi−ǫ∗ f·kiǫi·kf) + 2ǫi·kf(ǫ∗ f·ǫikα f−ǫi·kfǫ∗α f)] (52) Adding the four diagrams together, we find (after considerab le but simple algebra— cf.Appendix B) a remarkably simple result <pf;kf,ǫfǫf|T|pi;ki,ǫi>=H×/parenleftBig ǫ∗ fαǫiβTαβ Compton (S)/parenrightBig (53) whereHis the factor H=κ 4eǫ∗ f·pfkf·pi−ǫ∗ f·pikf·pf ki·kf(54) andǫ∗ fαǫiβTαβ Compton (S) is the Compton scattering amplitude for particles of spin S calculated in the previous section. The gauge invaria nce of Eq. 53 11 (a) (b) (c) (d) Figure 2: Diagrams relevant to graviton photoproduction. is obvious, since it follows directly from the gauge invaria nce already shown for the corresponding photon amplitudes together with that of the factor H under ǫf→ǫf+λkf. Also we note that in Eq. 53 the factorization condition menti oned in the introduction is made manifest, and consequently the corres ponding cross sections can be obtained trivially. In principle, one can us e conventional techniques, but this is somewhat challenging in view of the t ensor structure of the graviton polarization vector. However, factorizati on means that he- licity amplitudes for graviton photoproduction are simple products of the corresponding photon amplitudes times the universal facto rH, and the cross sections are then given by the simple photon forms times the u niversal factor H2. In the CM frame we have ǫ∗ f·pi=−ǫ∗ f·kiand the factor Hassumes the form |H|=κ 4e|ǫ∗ f·kikf·pf ki·kf|=κ 4epsinθ√ 2s−m2 −t=κ 2e/bracketleftbiggm4−st −2t/bracketrightbigg1 2 (55) In the lab frame this takes the form |Hlab|2=κ2m2 8e2cos21 2θ sin2 1 2θ(56) 12 and the graviton photoproduction cross sections are found t o be dσ dΩ=Gαcos21 2θ ×/braceleftBigg (ωf ωi)2[ctn21 2θcos21 2θ+ sin21 2θ] S= 0 (ωf ωi)3[(ctn21 2θcos21 2θ+ sin2 1 2θ) +2ωi m(cos41 2θ+ sin4 1 2θ) + 2ω2 i m2sin2 1 2θ]S= 1/2 (57) The form of the latter has previously been given by Voronov.[ 12] 3.2 Graviton Compton Scattering Finally, we can proceed to our primary goal, which is the calc ulation of gravi- ton Compton scattering. In order to produce a gauge invarian t scattering amplitude in this case we require fourseparate contributions, as shown in Figure 3. Two of these diagrams are Born terms and can be writt en down straightforwardly. However, there are also seagull terms b oth for spin 0 and for spin 1/2, whose forms can be found in Appendix A. For the sc alar case we have <pf;kf,ǫfǫf|T|pi;kiǫiǫi>seagull =/parenleftBigκ 2/parenrightBig2/bracketleftbig −2ǫ∗ f·ǫi(ǫi·piǫ∗ f·pf+ǫi·pfǫ∗ f·pi) −1 2(ǫ∗ f·ǫi)2ki·kf/bracketrightbigg (58) while in the case of spin 1/2 <pf;kf,ǫfǫf|T|pi;kiǫiǫi>seagull =/parenleftBigκ 2/parenrightBig2 ¯u(pf) ×/bracketleftbigg3 16ǫ∗ f·ǫi(/negationslashǫiǫ∗ f·(pi+pf)+/negationslashǫ∗ fǫi·(pi+pf)) +i 16ǫ∗ f·ǫiǫρσηλγλγ5(ǫiηǫ∗ fσkfρ−ǫfηǫiσkiρ)/bracketrightbigg u(pi) (59) Despite the complex form of the various contributions, the fi nal form, which results (after considerable algebra- cf. Appendix B) upon summation of the various components, is remarkably simple: ǫfαǫfβMαβ;γδ gravǫiγǫiδ=F×/parenleftbig ǫ∗ fµǫiνTµν Compton (S= 0)/parenrightbig ×/parenleftBig ǫ∗ fαǫiβTαβ Compton (S)/parenrightBig (60) 13 (a) (b) (c) (d) Figure 3: Diagrams relevant for gravitational Compton scat tering. whereFis the universal factor F=κ2 8e4pi·kipi·kf ki·kf(61) and the Compton amplitudes are those calculated in section 2 . The gauge in- variance of this form is again obvious from the already-demo nstrated gauge invariance of the photon amplitudes and this is the factoriz ed form guar- anteed by general arguments[5]. Again, it is in principle po ssible but very challenging to evaluate the cross section by standard means , but the result follows directly by the use of helicity methods. From the for m of Eq. 60 it is clear that the helicity amplitudes for graviton scatteri ng have the simple form of a product of corresponding helicity amplitudes for s pinless and spin S Compton scattering. That is, for graviton scattering from a spinless target we have |C++|2=|C−−|2=F2|A++|4 |C+−|2=|C−+|2=F2|A+−|4(62) while for scattering from a spin 1/2 target, we find for the tar get-spin aver- aged helicity amplitudes |D++|2 av=|C−−|2 av=F2|A++|2|B++|2 av |C+−|2 av=|C−+|2 av=F2|A+−|2|B+−|2 av (63) 14 Here the factor F has the form F=κ2 8e4(s−m2)(u−m2) t(64) whose laboratory frame value is Flab=κ2m2 8e41 sin21 2θ(65) The corresponding laboratory cross sections are found then to be dσlab dΩ(S= 0) =ω2 f πF2 1 16π(s−m2)2(|C++|2+|C+−|2) =G2m2(ωf ωi)2[ctn41 2θcos41 2θ+ sin41 2θ] (66) for a spinless target and dσlab dΩ(S=1 2) =ω2 f πF2 1 16π(s−m2)2(|D++|2 av+|D+−|2 av) =G2m2(ωf ωi)3[ctn41 2θcos41 2θ+ sin41 2θ) + 2ωi m(ctn21 2θcos61 2θ+ sin61 2θ) + 2ω2 i m2(cos61 2θ+ sin61 2θ)] (67) for a spin 1/2 target. The latter form agrees with that given b y Voronov[12]. We have given the results for unpolarized scattering from an unpolarized target, but having the form of the helicity amplitudes means that we can also produce cross sections involving polarized photons or gravitons. That is, however, a subject for a different time and a different pape r. 4 Summary While the subject of photon interactions with charged parti cles is a standard one in any quantum mechanics course, the same is not true for t hat of gravi- ton interactions with masses despite the obvious parallels between these two topics. The origin of this disparity lies with the complicat ions associated with 15 the tensor structure of gravity and the inherent nonlineari ty of gravitational theory. We have argued above that this need not be the case. In deed in an earlier work we showed how the parallel between the exchan ge ofvirtual gravitons and photons could be used in order to understand th e phenomena of geodetic and Lense-Thirring precession in terms of the the r elated spin-orbit and spin-spin interactions in quantum electrodynamics[2] . In the present pa- per, we have shown how the treatment of graviton scattering p rocesses can benefit from use of this analogy. Of course, such amplitudes a re inherently more complex, in that they must involve tensor polarization vectors and the addition of somewhat complex photon or graviton pole diagra ms. However, it is remarkable that when all effects are added together, the resulting am- plitudes factorize into simple products of photon amplitud es times kinematic factors. Using helicity methods, this factorization prope rty then allows the relatively elementary calculation of cross sections since they involve simple products of the already known photon amplitudes times kinem atical factors. It is hoped that this remarkable result will allow introduct ion of graviton re- actions into the quantum mechanics cirriculum in at least pe rhaps a special topics presentation. In any case, the simplicity associate d with this result means that graviton interactions can be considered a topic w hich is no longer only associated with advanced research papers. Appendix A: Gravitational Formalism Here we present some of the basics of gravitational field theo ry. Details can be found in various references[13, 14]. The full gravitatio nal action is given by Sg=/integraldisplay d4x√−g/parenleftbigg1 16πGR+Lm/parenrightbigg (68) whereLmis the Lagrange density for matter and Ris the scalar curvature. Variation of Eq. 68 via gµν→ηµν+κhµν yields the Einstein equation Rµν−1 2gµνR=−8πGT µν (69) where the energy-momentum tensor Tµνis given by Tµν=2√−g∂ ∂gµν(√−gLm) (70) 16 We work in the weak field limit, with an expansion in powers of t he gravita- tional coupling G gµν≡ηµν+κh(1) µν+... gµν=ηµν−κh(1)µν+κ2h(1)µλh(1) λν+... (71) where here the superscript indicates the number of powers of Gwhich appear and indices are understood to be raised or lowered by ηµν. We shall also need the determinant which is given by √−g= exp1 2trlogg= 1 +1 2κh(1)+... (72) The corresponding curvatures are given by R(1) µν=κ 2/bracketleftBig ∂µ∂νh(1)+∂λ∂λh(1) µν−∂µ∂λh(1)λ ν−∂ν∂λh(1)λ µ/bracketrightBig R(1)=ηµνR(1) µν=κ/bracketleftbig ✷h(1)−∂µ∂νh(1)µν/bracketrightbig (73) In order to define the graviton propagator, we must make a gaug e choice and we shall work in harmonic (or deDonder) gauge— gµνΓλ µν= 0—which requires, to first order in the field expansion, 0 =∂βh(1) βα−1 2∂αh(1)(74) Using these results, the Einstein equation reads, in lowest order, ✷h(1) µν−1 2ηµν✷h(1)−∂µ/parenleftbigg ∂βh(1) βν−1 2∂νh(1)/parenrightbigg −∂ν/parenleftbigg ∂βh(1) βµ−1 2∂µh(1)/parenrightbigg =−16πGTmatt µν (75) which, using the gauge condition Eq. 74, can be written as ✷/parenleftbigg h(1) µν−1 2ηµνh(1)/parenrightbigg =−16πGTmatt µν (76) or in the equivalent form ✷h(1) µν=−16πG/parenleftbigg Tmatt µν−1 2ηµνTmatt/parenrightbigg (77) Gravitational Interactions: Spin 0 17 The coupling to matter via one-graviton and two-graviton ve rtices can be found by expanding the spin zero matter Lagrangian √−gLm=√−g/parenleftbigg1 2DµφgµνDνφ−1 2m2φ2/parenrightbigg (78) via √−gL(0) m=1 2(∂µφ∂µφ−m2φ2) √−gL(1) m=−κ 2h(1)µν/parenleftbigg ∂µφ∂νφ−1 2ηµν(∂αφ∂αφ−m2φ2)/parenrightbigg √−gL(2) m=κ2 2/parenleftbigg h(1)µλh(1)ν λ−1 2h(1)h(1)µν/parenrightbigg ∂µφ∂νφ −κ2 8/parenleftbigg h(1)αβh(1) αβ−1 2h(1)2/parenrightbigg (∂αφ∂αφ−m2φ2) (79) The one- and two-graviton vertices are then respectively ταβ(p,p′) =−iκ 2/parenleftbig pαp′ β+p′ αpβ−ηαβ(p·p′−m2)/parenrightbig ταβ,γδ(p,p′) =iκ2/bracketleftbig Iαβ,ρξIξ σ,γδ/parenleftbig pρp′σ+p′ρpσ/parenrightbig −1 2(ηαβIρσ,γδ+ηγδIρσ,αβ)p′ρpσ −1 2/parenleftbigg Iαβ,γδ−1 2ηαβηγδ/parenrightbigg/parenleftbig p·p′−m2/parenrightbig/bracketrightbigg (80) where we have defined Iαβ;γδ=1 2(ηαγηβδ+ηαδηβγ) 18 We also require the triple graviton vertex τµν αβ,γδ(k,q) whose form is τµν αβ,γδ(k,q) =iκ 2/braceleftbigg (Iαβ,γδ−1 2ηαβηγδ)/bracketleftbigg kµkν+ (k−q)µ(k−q)ν+qµqν−3 2ηµνq2/bracketrightbigg + 2qλqσ/bracketleftbig Iλσ, αβIµν, γδ+Iλσ, γδIµν, αβ−Iλµ, αβIσν, γδ−Iσν, αβIλµ, γδ/bracketrightbig + [qλqµ(ηαβIλν, γδ+ηγδIλν, αβ) +qλqν(ηαβIλµ, γδ+ηγδIλµ, αβ) −q2(ηαβIµν, γδ+ηγδIµν, αβ)−ηµνqλqσ(ηαβIγδ,λσ+ηγδIαβ,λσ)] + [2qλ(Iσν, αβIγδ,λσ(k−q)µ+Iσµ, αβIγδ,λσ(k−q)ν −Iσν, γδIαβ,λσkµ−Iσµ, γδIαβ,λσkν) +q2(Iσµ, αβIγδ,σν+Iαβ,σνIσµ, γδ) +ηµνqλqσ(Iαβ,λρIρσ, γδ+Iγδ,λρIρσ, αβ)] + [(k2+ (k−q)2)/parenleftbigg Iσµ, αβIγδ,σν+Iσν, αβIγδ,σµ−1 2ηµνPαβ,γδ/parenrightbigg −(k2ηγδIµν, αβ+ (k−q)2ηαβIµν, γδ)]/bracerightbig (81) Gravitational Interactions: Spin 1/2 For the case of spin 1/2 we require some additional formalism in order to extract the gravitational couplings. In this case the matte r Lagrangian reads √eLm=√e¯ψ(iγaeaµDµ−m)ψ (82) and involves the vierbein eaµwhich links global coordinates with those in a locally flat space[15, 16]. The vierbein is in some sense the “ square root” of the metric tensor gµνand satisfies the relations ea µeb νηab=gµν, ea µeaν=gµν eaµebµ=δa b, eaµeaν=gµν(83) The covariant derivative is defined via Dµψ=∂µψ+i 4σabωµab (84) where ωµab=1 2eaν(∂µebν−∂νebµ)−1 2ebν(∂µeaν−∂νeaµ) +1 2eaρebσ(∂σecρ−∂ρecσ)eµc(85) 19 The connection with the metric tensor can be made via the expa nsion ea µ=δa µ+κc(1)a µ+... (86) The inverse of this matrix is eaµ=δµ a−κc(1)µ a+κ2c(1)µ bc(1)b a+... (87) and we find gµν=ηµν+κc(1) µν+κc(1) νµ+... (88) For our purposes we shall use only the symmetric component of the c- matrices, since these are physical and can be connected to th e metric tensor. We find then c(1) µν→1 2(c(1) µν+c(1) νµ) =1 2h(1) µν We have dete= 1 +κc+...= 1 +κ 2h+... and, using these forms, the matter Lagrangian has the expans ion √eL(0) m=¯ψ(i 2γαδµ α← →∇µ−m)ψ √eL(1) m=−κ 2h(1)αβ¯ψiγα← →∇βψ−κ 2h(1)¯ψ(i 2/negationslash← →∇−m)ψ √eL(2) m=κ2 8h(1) αβh(1)αβ¯ψiγγ← →∇λψ+κ2 16(h(1))2¯ψiγγ← →∇γψ −κ2 8h(1)¯ψiγαhαλ← →∇λψ+3κ2 16h(1) δαh(1)αµ¯ψiγδ← →∇µψ +κ2 4h(1) αβh(1)αβ¯ψmψ−κ2 8(h(1))2¯ψmψ +iκ2 16h(1) δν(∂βh(1)ν α−∂αh(1)ν β)ǫαβδǫ¯ψγǫγ5ψ (89) 20 The corresponding one- and two-graviton vertices are found then to be ταβ(p,p′) =−iκ 2/bracketleftbigg1 4(γα(p+p′)β+γβ(p+p′)α)−1 2ηαβ(1 2(/negationslashp+/negationslashp′)−m)/bracketrightbigg ταβ,γδ(p,p′) =iκ2/braceleftbigg −1 2(1 2(/negationslashp+/negationslashp′)−m)Pαβ,γδ −1 16[ηαβ(γγ(p+p′)δ+γδ(p+p′)γ) +ηγδ(γα(p+p′)β+γβ(p+p′)α)] +3 16(p+p′)ǫγξ(Iξφ,αβIφ ǫ,γδ+Iξφ,γδIφ ǫ,αβ) +i 16ǫρσηλγλγ5(Iαβ,ηνIγδ,σνk′ ρ−Iγδ,ηνIαβ,σνkρ)/bracerightbigg (90) Appendix B: Graviton Scattering Amplitudes In this section we summarize the independent contributions to the various graviton scattering amplitudes which must be added in order to produce the complete amplitudes quoted in the text. We leave it to the (pe rspicacious) reader to perform the appropriate additions and to verify th e factorized forms shown earlier. Graviton Photoproduction: Spin 0 Born−a : Ampa= 4eκ(ǫ∗ f·pf)2ǫi·pi 2pi·ki Born−b : Ampb=−4eκ(ǫ∗ f·pi)2ǫi·pf 2pi·kf Seagull : Ampc=−2eκǫ∗ f·ǫiǫ∗ f·(pi+pf) γ−pole : Ampd=eκ ki·kf[ǫ∗ f·(pi+pf)(ki·kfǫ∗ f·ǫi−ǫ∗ f·kiǫi·kf) +ǫ∗ f·ki(ǫ∗ f·ǫiki·(pi+pf)−ǫ∗ f·kiǫi·(pi+pf)] (91) 21 Graviton Photoproduction: Spin 1/2 Born−a : Ampa=eκǫ∗ f·pf 2pi·ki¯u(pf)[/negationslashǫf∗(/negationslashpi+/negationslashki+m)/negationslashǫi]u(pi) Born−b : Ampb=−eκǫ∗ f·pi 2pi·kf¯u(pf)[/negationslashǫi(/negationslashpi−/negationslashkf+m)/negationslashǫ∗ f]u(pi) Seagull : Ampc=−eκ¯u(pf)/negationslashǫ∗ fu(pi) γ−pole : Ampd=eκ1 ki·kf¯u(pf)[/negationslashǫ∗ f(ki·kfǫ∗ f·ǫi−ǫ∗ f·kiǫi·kf) +/negationslashkfǫ∗ f·ǫiǫ∗ f·ki−/negationslashǫi(ǫ∗ f·ki)2]u(pi) (92) Graviton Scattering: Spin 0 Born−a : Ampa= 2κ2(ǫi·pi)2(ǫ∗ f·pf)2 pi·ki Born−b : Ampb=−2κ2(ǫ∗ f·pi)2(ǫi·pf)2 pi·kf Seagull : Ampc=κ2/bracketleftbigg ǫ∗ f·ǫi(ǫi·piǫ∗ f·pf+ǫi·pfǫ∗ f·pi)−1 2ki·kf(ǫ∗ f·ǫi)2/bracketrightbigg g−pole : Ampd=4κ2 ki·kf/bracketleftbig ǫ∗ f·pfǫ∗ f·pi(ǫi·(pi−pf))2+ǫi·piǫi·pf(ǫ∗ f·(pi+pf))2 +ǫi·(pi−pf)ǫ∗ f·(pf−pi)(ǫ∗ f·pfǫi·pi+ǫ∗ f·piǫi·pf) −ǫ∗ f·ǫi/parenleftbig ǫi·(pi−pf)ǫ∗ f·(pf−pi)(pi·pf−m2) +ki·kf(ǫ∗ f·pfǫi·pi+ǫ∗ f·piǫi·pf) +ǫi·(pi−pf)(ǫ∗ f·pfpi·kf+ǫ∗ f·pipf·kf) +ǫ∗ f·(pf−pi)(ǫi·pipf·ki+ǫi·pfpi·ki)/parenrightbig + (ǫ∗ f·ǫi)2/parenleftbigg pi·kipf·ki+pi·kfpf·kf−1 2(pi·kipf·kf+pi·kfpf·ki) +3 2ki·kf(pi·pf−m2)2/parenrightbigg/bracketrightbigg (93) 22 Graviton Scattering: Spin 1/2 Born−a : Ampa=κ2ǫ∗ f·pfǫi·pi 8pi·ki¯u(pf)[/negationslashǫf∗(/negationslashpi+/negationslashki+m)/negationslashǫi]u(pi) Born−b : Ampb=−κ2ǫ∗ f·piǫi·pf 8pi·kf¯u(pf)[/negationslashǫi(/negationslashpi−/negationslashkf+m)/negationslashǫ∗ f]u(pi) Seagull : Ampc=κ2¯u(pf)/bracketleftbigg3 16ǫ∗ f·ǫi(/negationslashǫiǫ∗ f·(pi+pf)+/negationslashǫ∗ fǫi·(pi+pf)) +i 8ǫ∗ f·ǫiǫρσηλγλγ5(ǫiηǫ∗ fσkfρ−ǫ∗ fηǫiσkiρ)/bracketrightbigg u(pi) g−pole : Ampd=κ2 ki·kf¯u(pf)/bracketleftbig (/negationslashǫiǫ∗ f·ki+/negationslashǫ∗ fǫi·kf)(ǫi·piǫ∗ f·pf−ǫi·pfǫ∗ f·pi) −(ǫ∗ f·ǫi)/parenleftbig ki·kf(/negationslashǫ∗ fǫi·kf+/negationslashǫiǫ∗ f·pi) +/negationslashki(ǫ∗ f·pfǫi·pi−ǫ∗ f·piǫi·pf) +pi·ki(/negationslashǫiǫ∗ f·ki+/negationslashǫ∗ fǫi·kf)/parenrightbig + (ǫ∗ f·ǫi)2/negationslashki(pi·ki−1 2ki·kf)/bracketrightbigg u(pi) (94) Acknowledgement This work was supported in part by the National Science Found ation under award PHY02-44801. 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