19 Feb 2015 Graviton-Photon Scattering

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arXiv:1410.4148v2 [gr-qc] 19 Feb 2015 ∗ PTt418 IHES/P/14/32 IPhT/t14/148, uhrodcosscin o olm-iepotential. c Coulomb-like scattering a from for forward differing sections the behavior, cross that peculiar Rutherford very show a We evalu has to limit. photoproduction used spin-1 o be mass betwe case can difference zero the the limit discuss In massless and scattering the methods. graviton-photon how amplitude show helicity we of scatterin scattering Compton use gravitational the and gravita by QED various system evaluate the to both amplitudes examine We QED Abelian of terms euetefaueta h rvttoa opo scatterin Compton gravitational the that feature the use We a il orItrainlAaeyadDsoeyCenter Discovery and Academy International Bohr Niels NS R 36 -19 i-u-vte France Gif-sur-Yvette, F-91191 2306, URA CNRS, h il orIsiue lgmvj1,DK-2100 17, Blegamsvej Institute, Bohr Niels The E,Ih,F911GfsrYet,Fac,and France, Gif-sur-Yvette, F-91191 IPhT, CEA, ...Bjerrum-Bohr N.E.J. d ntttdsHautes des Institut rvtnPoo Scattering Graviton-Photon uoi PlantéLudovic -14 ue-u-vte France Bures-sur-Yvette, F-91440 c b ntttd hsqeThéorique Physique de Institut eateto Physics-LGRT of Department nvriyo Massachusetts of University mes,M 10 USA 01003 MA Amherst, oehgnØ Denmark Ø, Copenhagen e ray2,2015 20, February Abstract c ireVanhove Pierre , a 1 ar .Holstein R. Barry , tdsScientifiques, Etudes ´ npoo neatosadthe and interactions photon en h tnadTosnand Thomson standard the inlCmtnprocesses. Compton tional mltd atrzsin factorizes amplitude g c,d osscinfrgraviton for section ross rvttoa Compton gravitational f t h rs eto for section cross the ate rmamsiespin-1 massive a from g ∗ b , 1 Introduction

The treatment of electromagnetic interactions in quantum mechanics is well known and the discussion of electromagnetic effects via photon exchange is a staple of the graduate curricu- lum. In particular photon exchange between charged particles can be shown to give rise to the Coulomb potential as well as to various higher order effects such as the spin-orbit and Darwin interactions [1]. The fact that the photon carries spin-1 means that the electromagnetic current is a four-vector and manipulations involving such vector quantities are familiar to most physicists. In a similar fashion, graviton exchange between a pair of masses can be shown to generate the gravitational potential as well as various higher order effects, but in this case the fact that the graviton is a spin-2 particle means that gravitational “currents” are second rank tensors and the graviton propagator is a tensor of rank four. The resultant proliferation of indices is one reason why this quantum mechanical discussion of graviton exchange effects is not generally treated in introductory texts [2]. Recently, by the use of string-inspired methods, it has been demonstrated that the gravitational interaction factorizes in such a way that a gravitational amplitude can be written as the product of two more familiar vector amplitudes [3–7]. This factorization property, totally obscure at the level of the action, is a fundamental properties of gravity and has deep consequences at the loop amplitude level, since many gravitational amplitudes can be constructed by an appropriate product of gauge theory integrand numerators [8]. This feature has triggered a good deal of new results in extended supergravity [9–20], but quite remarkably these techniques can be applied as well to pure gravity [7, 21]. One remarkable property of amplitudes with emission of one or two gravitons is their factorization in terms of Abelian QED amplitudes [7, 22]. This factorization property has the important consequence that the low-energy limit of the gravitational Compton amplitude for graviton photoproduction is directly connected to the low-energy theorem for QED Compton amplitudes [7]. In a previous paper [22] this property was used to evaluate processes such as graviton photo- 1 production and gravitational Compton scattering for both spin-0 and spin- 2 systems by simply evaluating the corresponding electromagnetic amplitude for Compton scattering. This simplification permits the treatment of gravitational effects without long tedious computations, since they are now no more difficult than corresponding electromagnetic calculations. The simplicity offered through factorization has important consequences for the computations of long-range corrections to interaction potentials containing loops of intermediate photons- or gravitons [23–26]. In this paper we extend such considerations to electromagnetic and gravitational interactions of spin-1 systems. These calculations are useful not only as a generalization of our previous results but also, since the photon carries spin one, such methods can be used to consider the case of photon-graviton scattering, although there are subtleties in this case due to gauge invariance. In all the cases under study, we show that the low-energy limit of the differential cross section has a universal behavior independent of the spin of the matter field on which photon or graviton is scattered. We demonstrate that this is a consequence of the well-known universal low-energy behavior in quantum electrodynamics (QED) and the squaring relations between

1 gravitational and electromagnetic processes. The forward diﬀerential cross section for the Compton scattering of photons on a massive target has the well-known constant behavior of the Thomson cross-section

Comp 2 dσlab,S α lim = 2 , (1.1) θL→0 dΩ 2 m while the small-angle limit of gravitational Compton scattering of gravitons on a massive target has the expected behavior due to a 1/r long-range potential of a Rutherford-like cross section

g−Comp 2 2 dσlab,S 16 G m lim = 4 . (1.2) θL→0 dΩ θL We explain in section 6 why this formula reproduces the small-angle limit of the classical cross section for light bending in a Schwarzschild background. The forward limit of the graviton photoproduction cross section has the rather unique behavior photo dσlab,S 4 Gα lim = 2 . (1.3) θL→0 dΩ θL This limit is independent not only of the spin S but as well of the mass m of the target. The small-angle dependence is typical of an effective 1/r2 potential. We provide an explanation for this in section 6. It may be very difficult to detect a single graviton [27] but photons are easily detected so it is would be interesting to be able to use the graviton photoproduction process to provide an indirect detection of a graviton. The cross section in Eq. (1.3) is suppressed by a power of Newton’s constant G but, being independent of the mass m of the target, one can discriminate this effect from that of Compton scattering. In the next section then we quickly review the electromagnetic interaction and derive the spin-1 couplings. In section 3, we analyze the Compton scattering of a spin-1 particle. The corresponding gravitational couplings are derived in section 4 and the graviton photoproduction and gravitational Compton scattering reactions are calculated via both direct and factorization methods. In section 5 we discuss photon-graviton scattering and the subtleties associated with gauge invariance. In section 6 we consider the forward small-angle limit of the various scattering cross sections derived in the previous section. We show that Compton, graviton photoproduction and the gravitational Compton scattering have very different behavior. We summarize our findings in a brief concluding section.

2 Brief Review of Electromagnetism

In this section we present a quick review of the electromagnetic and gravitational interactions and the results given in our previous work. The electromagnetic interaction of a system may be found by using the minimal substitution i∂ iD = i∂ eA in the free particle Lagrangian, µ → µ µ − µ where Aµ is the photon ﬁeld. In this way the Klein-Gordon Lagrangian S=0 = ∂ φ† ∂µφ m2 φ†φ , (2.1) L0 µ − 2 which describes a free charged spinless ﬁeld, becomes

S=0 =(∂ ieA ) φ†(∂µ + ieAµ) φ + m2 φ†φ , (2.2) L µ − µ after this substitution. The corresponding interaction Lagrangian can then be identiﬁed as

S=0 = ieA φ†←→µφ + e2AµA φ†φ , (2.3) Lint µ ∇ µ where C←→D := C ~ D (~ D)C. (2.4) ∇ ∇ − ∇ 1 Similarly, for spin- 2 , the free Dirac Lagrangian

1 S= 2 = ψ¯ (i m) ψ , (2.5) L0 6∇ − becomes 1 S= 2 = ψ¯ (i e A m) ψ , (2.6) L 6∇ − 6 − and the interaction Lagrangian is found to be

1 S= 2 = e ψ¯ A ψ . (2.7) Lint − 6 The resulting single-photon vertices are then

p V (1)µ p = i e (p + p )µ , (2.8) f em iiS=0 − f i for spin-0 and (1)µ µ pf Vem pi 1 = i e u¯(pf )γ u(pi) , (2.9) S= 2 − 1 for spin- 2 , and in the case of spin 0 there exists also a two-photon (”seagull”) vertex

(2)µν 2 µν pf Vem pi S=0 =2 i e η . (2.10)

The photon propagator in Feynman gauge is iηαβ Dαβ(q)= − . (2.11) f q2 + iǫ

The consequences of these Lagrangians were explored in ref. [22] and in the present paper we extend our considerations to the case of spin-1, for which the free Lagrangian has the Proca form 1 S=1 = B† Bµν + m2 B† Bµ , (2.12) L0 −2 µν µ µ where Bµ is a spin one ﬁeld subject to the constraint ∂ Bµ = 0 and Bµν is the antisymmetric tensor Bµν = ∂µBν ∂ν Bµ . (2.13) −

3 The minimal substitution then leads to the interaction Lagrangian

S=1 = i e AµBν† η ←→ η ←→ Bα e2AµAνBα†Bβ(η η η η ) , (2.14) Lint να ∇ µ − αµ ∇ ν − µν αβ − µα νβ and the one, two photon vertices

p , ǫ V (1)µ p , ǫ = i e ǫ∗ (p + p )µηαβ ηβµpfα ηαµpiβ ǫ , f B em i A S=1 − Bβ f i − − Aα (2)µν 2 ∗ αβ µν αµ βν αν βµ pf , ǫB V em pi, ǫA = i e ǫBβ 2η η η η η η ǫAα . (2.15) S=1 − − However, Eq. (2.15) is not the correct result for a fundamental spin-1 particle such as the charged W -boson. Because the W arises in a gauge theory, there exists an additional W - photon interaction, leading to an “extra” contribution to the single photon vertex

p , ǫ δV (1)µ p , ǫ = i e ǫ∗ ηαµ(p p )β ηβµ(p p )α) ǫ . (2.16) f B em i AiS=1 Bβ i − f − i − f Aα The meaning of this term can be seen by using the mass-shell Proca constraints p ǫ = p ǫ = i · A f · B 0 to write the total on-shell single photon vertex as

p , ǫ (V + δV )µ p , ǫ = i e ǫ∗ (p + p )µη 2ηαµ(p p )β f B em em i A S=1 − Bβ f i αβ − i − f βµ α + 2η (pi pf ) ǫAα , (2.17) − wherein we observe that the coefficient of the term ηαµ(p p )β + ηβµ(p p )α has been − i − f i − f modified from unity to two. Since the rest frame spin operator can be identified via1

B†B B†B = i ǫ f S i , (2.19) i j − j i − ijk k the corresponding piece of the nonrelativistic interaction Lagrang ian becomes e = g f S~ i ~ A,~ (2.20) Lint − 2m · ∇ × where g is the gyromagnetic ratio and we have included a factor 2m which accounts for the normalization condition of the spin one ﬁeld. Thus the “extra” interaction required by a gauge theory changes the g-factor from its Belinfante value of unity [28] to its universal value of two, as originally proposed by Weinberg and more recently buttressed by a number of arguments [29,30]. Use of g = 2 is required (as shown in [31]) in order to assure the validity of the factorization result of gravitational amplitudes in terms of QED amplitudes, as used below.

1Equivalently, one can use the relativistic identity

1 i 1 ǫ∗ q ǫ ǫ q ǫ∗ ǫ pβqγ Sδ p p ǫ∗ qǫ q Bµ A Aµ B = q2 µβγδ i ( f + i)µ B A (2.18) · − · 1 2 m − 2m · · − m δ i δστζ ∗ where S = 2m ǫ ǫBσǫAτ (pf + pi)ζ is the spin four-vector.

4 3 Compton Scattering

The vertices given in the previous section can now be used to evaluate the Compton scattering amplitude for a spin-1 system having charge e and mass m by summing the contributions of the three diagrams shown in Figure 1, yielding

∗ ∗ ǫi piǫ pf ǫi pf ǫ pi AmpComp = e2 2ǫ ǫ∗ · f · · f · ǫ ǫ∗ S=1 A · B p k − p k − i · f " i · i i · f ∗ ǫ p ǫ p ǫ p ǫ pi g ǫ [ǫ∗ ,k ] ǫ∗ i · i i · f ǫ [ǫ ,k ] ǫ∗ f · f f · − A · f f · B p k − p k − A · i i · B p k − p k i · i i · f i · i i · f 1 1 g2 ǫ [ǫ ,k ] [ǫ∗ ,k ] ǫ∗ ǫ [ǫ∗ ,k ] [ǫ ,k ]ǫ∗ − 2p k A · i i · f f · B − 2p k A · f f · i i B i · i i · f (g 2)2 1 − ǫ [ǫ ,k ] p ǫ∗ [ǫ∗ ,k ] p − m2 2p k A · i i · i B · f f · f i · i 1 ǫ [ǫ∗ ,k ] p ǫ∗ [ǫ ,k ] p , (3.1) − 2p k A · f f · i B · i i · i i · f # with the momentum conservation condition pi + ki = pf + kf and where we have deﬁned

S [Q, R] T := S QT R S RT Q. · · · · − · · We can verify the gauge invariance of the above form by noting that this amplitude can be written in the equivalent form

e2 AmpComp = 2ǫ∗ ǫ (p F F p ) S=1 p k p k B · A i · i · f · i i · i i · f " + g [(ǫ∗ F ǫ )(p F p )+(ǫ∗ F ǫ )(p F p )] B · f · A i · i · f B · i · A i · f · f g2 [p k (ǫ∗ F F ǫ ) p k (ǫ∗ F F ǫ )] − 2 i · f B · f · i · A − i · i B · i · f · A 2 (g 2) ∗ ∗ − 2 [(ǫB Ff pf )(pi Fi ǫA) (ǫB Fi pi)(pi Ff ǫA)] , − 2m · · · · − · · · · # (3.2) where F µν = ǫµkν ǫν kµ and F µν = ǫ∗µkν ǫ∗ν kµ. Since F are obviously invariant under the i i i − i i f f f − f f i,f substitutions ǫi,f ǫi,f + λki,f , i =1, 2, it is clear that Eq. (3.1) satisﬁes the gauge invariance strictures → ∗µ ν Comp µ ν Comp ǫf ki Ampµν,S=1 = kf ǫi Ampµν,S=1 =0 . (3.3) Henceforth in this manuscript we shall assume the g-factor of the spin-1 system to have its “natural” value g = 2, since it is in this case that the high-energy properties of the scattering are well controlled and the factorization methods of gravity amplitudes are valid [29, 30].

5 (a) (b) (c)

Figure 1: Diagrams relevant to Compton scattering.

In order to make the transition to gravity in the next section, it is useful to utilize the helicity formalism [32], whereby we evaluate the matrix elements of the Compton amplitude between initial and final spin-1 and photon states having definite helicity, where helicity is defined as the projection of the particle spin along the momentum direction. We shall work initially in the center of mass frame. For a photon incident with four-momentum kiµ = pCM(1, zˆ) we choose the polarization vectors,

λi λi ǫ = xˆ + iλiyˆ , λi = , (3.4) i −√2 ± while for an outgoing photon with kfµ = pCM(1, cos θCMzˆ + sin θCMxˆ) we use polarizations

λf λf ǫ = cos θCMxˆ + iλf yˆ sin θCMzˆ , λf = . (3.5) f −√2 − ± We can deﬁne corresponding helicity states for the spin-1 system. In this case the initial and ﬁnal four-momenta are pi = (ECM, pCMzˆ) and pf = (ECM, pCM(cos θCMzˆ + sin θCMxˆ)) and there are transverse polarization four-vectors− − xˆ iyˆ ǫ± = 0, − ± , Aµ ∓ √2 cos θ xˆ iyˆ + sin θ zˆ ǫ± = 0, − CM ± CM , (3.6) Bµ ∓ √2 while the longitudinal mode has polarization four-vectors 1 ǫ0 = p , E zˆ , Aµ m CM − CM 1 ǫ0 = p , E (cos θ zˆ + sin θ xˆ) , (3.7) Bµ m CM − CM CM CM In terms of the usual invariant kinematic variables

2 2 2 s = p + k , t = k k , u = p k , i i i − f i − f 6 we identify

s m2 p = − , CM 2√s s + m2 E = , CM 2√s 1 1 1 (s m2)2 + st 2 m4 su 2 cos θ = − = − , 2 CM s m2 s m2 −1 − 1 st 2 sin θ = − . (3.8) 2 CM s m2 − The invariant cross section for unpolarized Compton scattering is then given by

dσComp 1 1 1 2 S=1 = B1(ab; cd) . (3.9) dt 16π(s m2)2 3 2 − a,b=−,0,+ c,d=−,+ X X where 1 Comp B (ab; cd)= pf , b; kf ,d AmpS=1 pi, a; ki,c , (3.10) is the Compton amplitude for scattering of a photon with four-mome ntum ki, helicity a from a spin-1 target having four-momentum pi, helicity c to a photon with four-momentum kf , helicity d and target with four-momentum pf , helicity b. The helicity amplitudes can be calculated straightforwardly. There exist 32 22 = 36 such amplitudes but, since helicity reverses under × spatial inversion, parity invariance of the electromagnetic interaction requires that2

B1(ab; cd) = B1( a b; c d) . − − − −

Also, since helicity is unchanged under time reversal, but initial and ﬁnal states are inter- changed, time reversal invariance of the electromagnetic interaction requires that

B1(ab; cd) = B1(ba; dc) .

Consequently there exist only twelve independent helicity amplitudes. Using Eq. (3.1) we can calculate the various helicity amplitudes in the center of mass frame and then write these results

2Note that we require only that the magnitudes of the helicity amplitudes related by parity and/or time reversal be the same. There could exist unobservable phases.

7 in terms of invariants using Eq. (3.8), yielding (s m2)2 + m2t 2 B1(++; ++) = B1( ; ) =2e2 − , −− −− (s m2)3(u m2) − − (m4 su)2 B1(++; ) = B1( ;++) =2e2 − , −− −− (s m2)3(u m2) − − m4t2 B1(+ ;+ ) = B1( +; +) =2e2 , − − − − (s m2)3(u m2) − − s2t2 B1(+ ; +) = B1( +;+ ) =2e2 , − − − − (s m2)3(u m2) − − B1(++;+ ) = B1( ; +) = B1(++; +) = B1( ;+ ) , − −− − − −− − m2t(m4 su) = 2e2 − , (s m2)3( u m2) − − B1(+ ;++) = B1( +; ) = B1( +;++) = B1(+ ; ) , − − −− − − −− m2t(m4 su) = 2e2 − . (3.11) (s m2)3( u m2) − − and B1(0+; ++) = B1(0 ; ) = B1(+0; ++) = B1( 0; ) , − −− − −− 2 2 2 √2m tm +( s m ) t(m 4 su) = 2e2 − − − , (s m2)3(u m2) − −p B1(0+;+ ) = B1(0 ; +) = B1(+0; +) = B1( 0;+ ) , − − − − − − 4 2 √2mst t(m su) = 2e − − , (s m2)3(u m2) − p − B1(0+; +) = B1(0 ;+ ) = B1(+0;+ ) = B1( 0; +) , − − − − − − √ 3 4 2 2m t t(m su) = 2e − − , (s m2)3(u m2) − p − B1(0+; ) = B1(0 ;++) = B1(+0; ) = B1( 0;++) , −− − 3−− − √2m t (m4 su) 2 = 2 e2 − − , (s m2)3t(u m2) − − 2tm2 +(s m2)2 (m4 su) B1(00; ++) = B1(00; ) =2e2 − − , −− (s m2)3(u m2) − − m2t((s m2)2 +2st 1 1 2 − B (00;+ ) = B (00; +) =2e 2 3 2 . (3.12) − − (s m ) (u m ) − − Substitution into Eq. (3.9) then yields the invariant cross section for unpolarized Compton scattering from a spin-1 target dσComp e4 S=1 = (m4 su+t2) 3(m4 su)+t2 +t2(t m2)(t 3m2) , (3.13) dt 12π(s m2)4(u m2)2 − − − − − − h i 8 which can be compared with the corresponding results for unpolarized Compton scattering 1 from spin-0 and spin- 2 targets found in ref. [22]— dσComp e4 S=0 = (m4 su)2 + m4t2 , dt 4π(s m2)4(u m2)2 − − − Comp 4 4 4 2 2 2 2 1 dσS= e (m su) 2(m su)+ t + m t (2m t) 2 = − − − . (3.14) dt h 8π(s m2)4(u m2)2 i − − Usually such results are written in the laboratory frame, wherein the target is at rest, by use of the relations s m2 = 2mω , u m2 = 2mω , − i − − f θ θ m4 su = 4m2ω ω cos2 L , m2t = 4m2ω ω sin2 L , (3.15) − i f 2 − i f 2 and dt d 2ω2(1 cos θ ) ω2 = i − L = f . (3.16) dΩ 2πd cos θ −1+ ωi (1 cos θ ) π L m − L Introducing the ﬁne structure constant α = e2/4π, we ﬁnd then

Comp dσ α2 ω4 θ θ ω θ 2 lab,S=1 = f cos4 L + sin4 L 1+2 i sin2 L dΩ m2 ω4 2 2 m 2 i " 16ω2 θ ω θ 32ω4 θ + i sin4 L 1+2 i sin2 L + i sin8 L , 3m2 2 m 2 3m4 2 Comp dσ 1 2 3 2 lab,S= α ω θ θ ω θ ω θ 2 = f cos4 L + sin4 L 1+2 i sin2 L +2 i sin4 L , dΩ m2 ω3 2 2 m 2 m2 2 i Comp dσ α2 ω2 θ θ lab,S=0 = f cos4 L + sin4 L . (3.17) dΩ m2 ω2 2 2 i We observe that the nonrelativistic laboratory cross section has an identical form for any spin

Comp NR dσ α2 θ θ ω lab,S = cos4 L + sin4 L 1+ i , (3.18) dΩ 2m2 2 2 O m which follows from the universal form of the Compton amplitude for scattering from a spin-S target in the low-energy ( ω m) limit, which in turn arises from the universal form of the Compton amplitude for scattering≪ from a spin-S target in the low-energy limit— S, M ; ǫ AmpComp S, M ; ǫ =2e2 ǫ∗ ǫ δ + ..., (3.19) f f S i i ω≪m f · i Mi,Mf and obtains in an eﬀective ﬁeld theory approach to Compton scattering [33].3

3 That the seagull contribution dominates the nonrelativistic cross section is clear from the feature that ∗ 2 ǫf pǫi p ω 2 ∗ Amp 2e · · Amp =2e ǫ ǫi. (3.20) Born ∼ p k ∼ m × seagull f · · 9 4 Gravitational Interactions

In the previous section we discussed the treatment the familiar electromagnetic interaction, using Compton scattering on a spin-1 target as an example. In this section we show how the gravitational interaction can be evaluated via methods parallel to those used in the electromagnetic case. An important diﬀerence is that while in the electromagnetic case we have the simple interaction Lagrangian = eA J µ , (4.1) Lint − µ where J µ is the electromagnetic current matrix element, for gravity we have κ = hµν T µν . (4.2) Lint − 2

Here the ﬁeld tensor hµν is deﬁned in terms of the metric via

gµν = ηµν + κhµν , (4.3) where κ is given in terms of Newton’s constant by κ2 = 32πG. The Einstein-Hilbert action is 2 = d4x √ g , (4.4) SEinstein−Hilbert − κ2 R Z where 1 1 √ g = detg = exp trlogg =1+ ηµνh + ..., (4.5) − − 2 2 µν p λ µν is the square root of the determinant of the metric and := R µλν g is the Ricci scalar Rµ curvature obtained by contracting the Riemann tensor R νρσ with the metric tensor. The energy-momentum tensor is deﬁned in terms of the matter Lagrangian via

2 δ√ g T = − Lmat . (4.6) µν √ g δgµν − The prescription Eq. (4.6) yields the forms

T S=0 = ∂ φ†∂ φ + ∂ φ†∂ φ g (∂ φ†∂λφ m2φ†φ) , (4.7) µν µ ν ν µ − µν λ − for a scalar ﬁeld and

1 S= 2 1 1 i Tµν = ψ¯[ γ i←→ + γ i←→ g ( ←→/ m)]ψ , (4.8) 4 µ ∇ ν 4 ν ∇ µ − µν 2 ∇ −

1 for spin- 2 , where we have deﬁned

ψi¯ ←→ ψ := ψi¯ ψ (i ψ¯)ψ . (4.9) ∇ µ ∇µ − ∇µ

10 αβ pf pf µν

pi pi µν

(a) (b)

Figure 2: (a) The one-graviton and (b) two-graviton emission vertices from either a scalar, spinor or vector particle.

αβ k k q − γδ µν q

Figure 3: The three graviton vertex

The one graviton emission vertices of ﬁgure 2(a) can now be identiﬁed as κ p V (1)µν p = i pµpν + pν pµ ηµν p p m2 , (4.10) f grav i S=0 − 2 f i f i − f · i − for spin-0,

(1)µν κ 1 µ ν 1 ν µ pf Vgrav pi 1 = i u¯(pf ) γ pf + pi + γ pf + pi u(pi) , (4.11) S= 2 − 2 4 4 1 for spin- 2 , and κ p , ǫ V (1)µν p , ǫ = i ǫ∗ ǫ pµpν + pνpµ ǫ∗ p pµǫν + ǫµ pν f B grav i A S=1 − 2 B · A i f i f − B · i f A A f ǫ p pνǫ∗µ + pµǫ∗ν + p p m2 ǫµ ǫ∗ν + ǫν ǫ∗µ − A · f i B i B f · i − A B A B ηµν p p m2 ǫ∗ ǫ ǫ∗ p ǫ p , (4.12) − i · f − B · A − B · i A · f for spin-1. There also exist two-graviton (seagull) vertices shown in ﬁgure 2(b), which can be found by expanding the stress-energy tensor to second order in hµν . In the case of spin-0 1 p V (2)µν,αβ p = iκ2 Iµν, Iξ (pζ pρ + pρ pζ ) ηµνIρζ,αβ + ηαβIρζ,µνpρ pζ f grav i S=0 ρξ ζ,αβ f i f i − 2 f i 1 1 Iµν;αβ ηµν ηαβ p p m2 , (4.13) − 2 − 2 f · i − 11 where 1 I = η η + η η . (4.14) αβ,γδ 2 αγ βδ αδ βγ 1 For spin- 2

(2)µν,αβ 2 1 µν α β β α pf Vgrav pi 1 = iκ u¯(pf ) η γ (pf + pi) + γ (pf + pi) S= 2 16 αβ µ h ν ν µ + η γ (pf + pi) + γ (pf + pi) 3 i + (p + p ) γ Iξφ,µνI ǫ,αβ + Iξφ,αβ I ǫ,µν 16 f i ǫ ξ φ φ i + ǫρσηλγ γ Iµν,η Iαβ,σζ p Iαβ,η Iµν,σζ p u(p ) , 16 λ 5 ζ fρ − ζ iρ i (4.15) while for spin-1

κ2 p , ǫ ; k V (2) µν,ρσ p , ǫ ; k = i f B f grav i A i S=1 − 4 + p p η (p p m2) η η + η η η η iβ fα − αβ i · f − µρ νσ µσ νρ − µν ρσ + η η p p + p p η p p η p p µρ αβ iν fσ iσ fν −αν iβ fσ − βν iσ fα η p p η p p +(p p m2 η η + η η − βσ iν fα − ασ iβ fν i · f − αν βσ ασ βν + η η p p + p p η p p η p p µσ αβ iν fρ iρ fν − αν iβ fρ − βν iρ fα η p p η p p +(p p m2 η η + η η − βρ iν fα − αρ iβ fν i · f − αν βρ αρ βν + η η p p + p p η p p η p p νρ αβ iµ fσ iσ fµ − αµ iβ fσ− βµ iσ fα η p p η p p +(p p m2 η η + η η − βσ iµ fα − ασ iβ fµ i · f − αµ βσ ασ βµ + η η p p + p p η p p η p p νσ αβ iµ fρ iρ fµ − αµ iβ fρ − βµ iρ fα η p p η p p +(p p m2 η η + η η − βρiµ fα − αρ iβ fµ i · f − αµ βρ αρ βµ η η p p + p p η p p η p p − µν αβ iρ fσ iσ fρ − αρ iβ fσ − βρ iσ fα η p p η p p + p p m2 η η + η η − βσ iρ fα − ασ iβ fρ i · f − αρ βσ βρ ασ η η p p + p p η p p η p p − ρσ αβ iµ fν iν fµ − αµ iβ fν − βµ iν fα η p p η p p +(p p m2 η η + η η − βν iµ fα − αν iβ fµ i · f − αµ βν βµ αν + η p η p η p η p αρ iµ − αµ iρ βσ fν − βµ fσ + η p η p η p η p ασ iν − αν iσ βρ fµ − βµ fρ + η p η p η p η p ασ iµ − αµ iσ βρ fν − βν fρ + η p η p η p η p ǫα (ǫβ )∗ . (4.16) αρ iν − αν iρ βσ fµ − βµ fσ A B

12 Finally, we require the triple graviton vertex of ﬁgure 3 i κ 1 3 τ µν (k, q)= I η η kµkν +(k q)µ(k q)ν + qµqν ηµν q2 αβ,γδ − 2 αβ,γδ − 2 αβ γδ − − − 2 + 2q q Iλσ, Iµν, + Iλσ, Iµν, Iλµ, Iσν, Iσν, Iλµ, λ σ αβ γδ γδ αβ − αβ γδ − αβ γδ µ h λν, λν, ν λµ, λµ, i + qλq ηαβI γδ + ηγδI αβ)+ qλq (ηαβI γδ + ηγδI αβ h q2 η Iµν, + η Iµν, ) ηµνqλqσ(η I + η I − αβ γδ γδ αβ − αβ γδ,λσ γδ αβ,λσ i + 2qλ Iσν, I (k q)µ +Iσµ, I (k q)ν Iσν, I kµ Iσµ, I kν γδ αβ,λσ − γδ αβ,λσ − − αβ γδ,λσ − αβ γδ,λσ h2 σµ, ν ν σµ, µν λ ρσ, ρσ, + q I αβIγδ,σ + Iαβ,σ I γδ + η q qσ Iαβ,λρI γδ + Iγδ,λρI αβ 1 1 i + k2 +(k q)2 Iσµ, I ν + Iσν, I µ ηµν I η η ) − αβ γδ,σ αβ γδ,σ − 2 αβ,γδ − 2 αβ γδ h k2η Iµν, +(k q)2η Iµν, . (4.17) − αβ γδ − γδ αβ i We work in harmonic (de Donder) gauge which satisﬁes, in lowest order, 1 ∂µh = ∂ h , (4.18) µν 2 ν with h = trhµν , (4.19) and in which the graviton propagator has the form i 1 D (q)= (η η + η η η η ) . (4.20) αβ;γδ q2 + iǫ 2 αγ βδ αδ βγ − αβ γδ

Then just as the (massless) photon is described in terms of a spin-1 polarization vector ǫµ which can have projection (helicity) either plus- or minus-1 along the momentum direction, the (massless) graviton is a spin-2 particle which can have the projection (helicity) either plus- or minus-2 along the momentum direction. Since hµν is a symmetric tensor, it can be described in terms of a direct product of unit spin polarization vectors—

(2) + + helicity = +2 : hµν = ǫµ ǫν , helicity = 2 : h(−2) = ǫ−ǫ− , (4.21) − µν µ ν and just as in electromagnetism, there is a gauge condition—in this case Eq. (4.18)—which must be satisﬁed. Note that the helicity states given in Eq. (4.21) are consistent with the gauge requirement, since

µν + + µν − − µ ± η ǫµ ǫν = η ǫµ ǫν =0, and k ǫµ =0 . (4.22) With this background we can now examine reactions involving gravitons, as discussed in the next section.

13 (a) (b)

Figure 4: Diagrams relevant to graviton photoproduction.

4.1 Graviton photoproduction We ﬁrst use the above results to discuss the problem of graviton photoproduction on a target of spin-1—γ + S g + S—for which the four diagrams we need are shown in Figure 4. → The electromagnetic and gravitational vertices needed for the Born terms and photon pole diagrams—Figures 4a, 4b, and 4d—have been given above. For the photon pole diagram we require the graviton-photon coupling, which is found from the electromagnetic energy-momentum tensor [34] 1 T = F F α + g F F αβ , (4.23) µν − µα ν 4 µν αβ and yields the photon-graviton vertex4 κ k , ǫ V (γ)µν k , ǫ = i ǫ∗ ǫ kµkν + kνkµ ǫ∗ k kµǫν + ǫµkν f f grav i i 2 f · i i f i f − f · i f i i f ǫ kh kνǫ∗µ + kµǫ∗ν +k k ǫµǫ∗ν + ǫν ǫ∗µ − i · f i f i f f · i i f i f ηµν k k ǫ∗ ǫ ǫ∗ k ǫ k . (4.24) − f · i f · i − f · i i · f i Finally, we need the seagull vertex which arises from the feature that the energy-momentum tensor depends on pi,pf and therefore yields a contact interaction when the minimal substitution is made, yielding the spin-1 seagull amplitude shown in Figure 4c. i p , ǫ ; k , ǫ ǫ T p , ǫ ; k , ǫ = κ e ǫ∗ (p + p ) ǫ∗ ǫ ǫ∗ ǫ f B f f f i A i i seagull 2 f · f i f · i B · A ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ǫB ǫi ǫf pf ǫf ǫA ǫB pi ǫf ǫi ǫf ǫA ǫA h ǫi ǫf pi ǫf ǫB − · · · − · · · − · · · ǫ p ǫ∗ ǫ ǫ∗ ǫ∗ ǫ∗ ǫ ǫ (p + p ) ǫ∗ ǫ∗ . (4.25) − A · f f · i f · B − f · A i · f i f · B i 4Note that this form agrees with the previously derived form for the massive graviton-spin-1 energy- momentum tensor—Eq. (4.12)—in the m 0 limit. → 14 The individual contributions from the four diagrams in Figure 4 are given in Appendix A and have a rather complex form. However, when added together we ﬁnda much simpler result—the full graviton photoproduction amplitude is found to be proportional to the already calculated Compton amplitude for spin-1—Eq. (3.1)—times a universal factor. That is, p ; k , ǫ ǫ T p ; k , ǫ = H ǫ∗ ǫ T αβ (S = 1) , (4.26) f f f f i i i × fα iβ Compton where ∗ ∗ κ p F p κ ǫ pf kf pi ǫ pi kf pf H = f · f · i = f · · − f · · , (4.27) 2e k k 2e k k i · f i · f ∗ αβ and ǫfαǫiβTCompton(S) is the Compton scattering amplitude for particles of spin-S calculated in the previous section. The gravitational and electromagnetic gauge invariance of Eq. (4.26) is obvious, since it follows directly from the gauge invariance already shown for the Comp- ton amplitude together with the explicit gauge invariance of the factor H. The validity of Eq. (4.26) allows the calculation of the cross section by helicity methods since the graviton photoproduction helicity amplitudes are given by C1(ab; cd)= H B1(ab; cd) , (4.28) × where B1(ab; cd) are the Compton helicity amplitudes found in the previous section. We can then evaluate the invariant photoproduction cross section using dσphoto 1 1 1 S=1 = C1(ab; cd) 2 , (4.29) dt 2 2 3 2 16π s m a=−,0,+ c=−,+ − X X yielding photo 2 2 4 dσS=1 e κ (m su) 4 2 4 2 = 4− 2 (m su + t ) 3(m su)+ t dt −96πt s m2 u m2 − − − − h + t2(t m2)(t 3m 2) . (4.30) − − i Since 1 κ m4 su 2 H = − , (4.31) | | e 2t − the laboratory value of the factor H is 2 2 2 1 2 κ m cos 2 θL Hlab = 2 2 1 , (4.32) | | 2e sin 2 θL the corresponding laboratory cross section is dσphoto dσComp lab,S=1 = H 2 lab,S=1 dΩ lab dt ω 4 θ θ θ θ ω θ 2 = Gα f cos2 L ctn2 L cos2 L + sin2 L 1+2 i sin2 L ω4 2 2 2 2 m 2 i " 16ω2 θ ω θ 32ω4 θ + i sin2 L 1+2 i sin2 L + i sin6 L . (4.33) 3m2 2 m 2 3m4 2 15 2 The factor Hlab can be thought of as “dressing” the photon into a graviton. We see that just as in Compton| | scattering the low-energy laboratory cross section has a universal form, which is valid for a target of arbitrary spin

photo dσ θ θ θ θ ω lab,S = Gα cos2 L ctn2 L cos2 L + sin2 L 1+ i . (4.34) dΩ 2 2 2 2 O m In this case the universality can be understood from the feature that at low energy the leading contribution to the graviton photoproduction amplitude comes not from the seagull, as in Compton scattering, but rather from the photon pole term,

∗ ∗ ǫf ǫi ǫf ki µ Ampγ−pole κ · · ki pf ; S, Mf Jµ pi; S, Mi . (4.35) ω−→≪m 2k k × f · i

The leading piece of the electromagnetic current has the universal low-energy structure

e p p p ; S, M J p ; S, M = p + p δ 1+ f − i , (4.36) f f µ i i 2m f i µ Mf ,Mi O m where we have divided by the factor 2m to account for the normalization of the target particle. Since ki (pf + pi) 2mω, we ﬁnd the universal low-energy amplitude · −→ω→0 ∗ ∗ ǫ ǫi ǫ ki AmpNR = κeω f · f · , (4.37) γ−pole 2k k f · i whereby the resulting helicity amplitudes have the form

θL 1 1+cos θL cos 2 2 θL sin θL = θ cos ++ = , κ e 2 1−cos θL sin L 2 AmpNR = 2 −− (4.38) γ−pole  θL 2√2 1 1−cos θL cos 2 2 θL  2 sin θL 1−cos θ = θ sin 2 + = + . L sin 2 − − Squaring and averaging, summing over initial, ﬁnal spins we ﬁnd

photo dσ θ θ θ θ lab,S Gα cos2 L ctn2 L cos2 L + sin2 L . (4.39) dΩ −→ω→0 2 2 2 2 as found above—cf. Eq. (4.34). The power of the factorization theorem is obvious and, as we shall see in the next section, allows the straightforward evaluation of even more complex reactions such as gravitational Compton scattering.

16 (a) (b)

Figure 5: Diagrams relevant for gravitational Compton scattering.

4.2 Gravitational Compton Scattering In the previous section we observed some of the power of the factorization theorem in the context of graviton photoproduction on a spin-1 target in that we only needed to calculate the simpler Compton scattering process rather than to consider the full gravitational interaction. In this section we tackle a more challenging example, that of gravitational Compton scattering— g +S g +S—from a spin-1 target, for which there exist the four diagrams shown in Figure 5. → The contributions from the four individual diagrams can now be calculated and are quoted in Appendix A. Each of the four diagrams has a rather complex form. However, when added together the result simpliﬁes enormously. Deﬁning the kinematic factor

κ2 p k p k κ4 (s m2)(u m2) Y = i · i i · f = − − , (4.40) 8e4 k k 16e4 t i · f the sum of the four diagrams is found to be given by

pf , ǫB; kf , ǫf ǫf Ampgrav pi, ǫA; ki, ǫiǫi S=1

= Y ] pf , ǫB; ki, ǫf Ampem pi, ǫA; ki, ǫi S=1 pf ; ki, ǫf Ampem pi; ki, ǫi S=0 , × × i (4.41) where ∗ ∗ ǫi pi ǫ pf ǫi pf ǫ pi p ; k , ǫ Amp p ; k , ǫ =2e2 · f · · f · ǫ∗ ǫ , (4.42) h f i f | em| i i iiS=0 p k − p k − f · i i · i i · f is the Compton amplitude for a spinless target. 1 In ref. [22] the identity Eq. (4.41) was veriﬁed for simpler cases of spin-0 and spin- 2 . This relation is a consequence of the general relations between gravity and gauge theory tree-level amplitudes derived from string theory as explained in [26]. Here we have shown its validity

17 for the much more complex case of spin-1 scattering. The corresponding cross section can be calculated by helicity methods using the identity

D1(ab; cd)= Y B1(ab; cd) A0(cd) , (4.43) × × where B1(ab; cd) is the spin-1 Compton helicity amplitude calculated in section 2 while

m4 su A0(++) = 2e2 − , s m2) u m2) − − m2t A0(+ ) = 2e2 − , (4.44) − s m2) u m2) − − are the helicity amplitudes for spin zero Compton scattering. Using Eq. (4.41) the invariant cross section for unpolarized spin-1 gravitational Compton scattering

dσg−Comp 1 1 1 S=1 = D1(ab; cd) 2 , (4.45) dt 2 2 3 2 16π s m a=−,0,+ c=−,+ − X X is found to be

dσg−Comp κ4 S=1 = (m4 su)2 3(m4 su)+ t2)(m4 su + t2) dt 768π s m2 4 u m2 2t2 − − − − − + m4t4(3m2 t)(m2 t) . (4.46) − − This form can be compared with the corresponding unpolarized gravitational Compton cross sections found in ref. [22]

g−Comp 4 3 4 2 6 4 2 dσ 1 4 S= κ m su 2(m su)+ t + m t 2m t 2 = − − − , dt 512π t2 s m2 4 u m2 2 − − g−Comp 4 dσS=0 κ 4 4 8 4 = 4 2 (m su) + m t . (4.47) dΩ 256π2 s m2 u m2 t2 − − − h i

18 The corresponding laboratory frame cross sections are

g−Comp dσ ω4 θ θ θ ω θ 2 lab,S=1 = G2m2 f ctn4 L cos4 L + sin4 L 1+2 i sin2 L dΩ ω4 2 2 2 m 2 i " 16 ω2 θ θ ω θ + i cos6 L + sin6 L 1+2 i sin2 L 3 m2 2 2 m 2 16 ω4 θ θ θ + i sin2 L cos4 L + sin4 L , 3 m4 2 2 2 g−Comp dσ 1 3 lab,S= ω θ θ θ ω θ θ θ 2 = G2m2 f ctn4 L cos4 L + sin4 L +2 i ctn2 L cos6 L + sin6 L dΩ ω3 2 2 2 m 2 2 2 i ω2 θ θ + 2 i cos6 L + sin6 L , m2 2 2 g−Comp dσ ω2 θ θ θ lab,S=0 = G2m2 f ctn4 L cos4 L + sin4 L . (4.48) dΩ ω2 2 2 2 i We observe that the low-energy laboratory cross section has the universal form for any spin

g−Comp dσ θ θ θ ω lab,S = G2m2 ctn4 L cos4 L + sin4 L + i . (4.49) dΩ 2 2 2 O m It is interesting to note that the “dressing” factor for the leading (++) helicity Compton amplitude— 2 4 2 2 2 θl ++ κ m su lab κ m cos 2 Y A = 2 − 2 2 θ , (4.50) 2e t −→ 2e sin 2 − —is simply the square of the photoproduction dressing factor H, as might intuitively be expected since now both photons must be dressed in going from the Compton to the gravitational Compton cross section.5 In this case the universality of the nonrelativistic cross section follows from the leading contribution arising from the graviton pole term

κ ∗ 2 µ ν µ ν κ Ampg−pole ǫf ǫi k kf + ki ki pf ; S, Mf Tµν pi; S, Mi . (4.52) ω−→≪m 4k k · f 2 f · i Here the matrix element of the energy-momentum tensor has the universal low-energy structure κ κ p p p ; S, M T p ; S, M = p p + p p δ 1+ f − i , (4.53) 2 f f µν i i 4m fµ iν fν iµ Mf ,Mi O m 5In the case of + helicity the “dressing” factor is − 2 − κ Y A+ = m2 . (4.51) 2e2 so that the nonleading contributions will have diﬀerent dressing factors.

19 where we have divided by the factor 2m to account for the normalization of the target particle. We ﬁnd then the universal form for the leading graviton pole amplitude

2 κ ∗ 2 Ampg−pole ǫf ǫi pi kf pf kf + pi ki pf ki δMf ,Mi . (4.54) non−→−rel 8mk k · · · · · f · i Since p k mω the corresponding helicity amplitudes become · ω−→≪m

2 4 θL 1+cos θL cos 2 = θ ++ = , sin2 L NR  2 1−cos θL 2 −− Ampg−pole =4πGm 2 (4.55)  1−cos θ 4 θL  L sin 2  = 2 θL + = + . 2 1−cos θL sin 2 − −  Squaring and averaging, summing over initial, ﬁnal spins we ﬁnd

g−Comp dσ θ θ θ lab,S G2 m2 ctn4 L cos4 L + sin4 L , (4.56) dΩ −→ω→0 2 2 2 as found in Eq. (4.49) above.

5 Graviton-Photon Scattering

In the previous sections we have generalized the results of reference [22] to the case of a massive spin-1 target. Here we show how these techniques can be used to calculate the cross section for photon-graviton scattering. In the Compton scattering calculation we assumed that the spin-1 target had charge e. However, the photon couplings to the graviton are identical to those of a graviton coupled to a charged spin-1 system in the massless limit, and one might assume then that, since the results of the gravitational Compton scattering are independent of charge, the graviton-photon cross section can be calculated by simply taking the m 0 limit of the → graviton-spin-1 cross section. Of course, the laboratory cross section no longer makes sense since the photon cannot be brought to rest, but the invariant cross section is well deﬁned in this limit— dσg−Comp 4πG2 3s2u2 4t2su + t4 S=1 − 2 2 , (5.1) dt m−→→0 3s t and it might be naively assumed that Eq. (5.1) is the graviton-photon scattering cross section. However, this is not the case and the resolution of this problem involves some interesting physics. We begin by noting that in the massless limit the only nonvanishing helicity amplitudes are s2 D1(++; ++) = D1( ; ) =8πG , m=0 −− −− m=0 t u2 D1( ;++) = D1(++; ) =8πG , −− m=0 −− m=0 t su D1(00; ++) = D1(00; ) =8πG , (5.2) m=0 −− m=0 t 20 which lead to the cross section g−Comp dσ 1 1 1 2 S=1 = D1(ab; cd) dt 16πs2 3 2 a=+,0,− c=+,− X X 1 1 s4 u4 s2u2 = (8πG)2 2 + + 16πs2 3 2 × × t2 t2 t2 4π s4·+ u4 + s2u2 = G2 , (5.3) 3 s2t2 in agreement with Eq. (5.1). However, this result demonstrates the problem. We know that in Coulomb gauge the photon has only two transverse degrees of freedom, corresponding to positive and negative helicity—there exists no longitudinal degree of freedom. Thus the correct photon-graviton cross section is actually dσ 1 1 1 gγ = D1(ab; cd) 2 dt 16πs2 3 2 a=+,− c=+,− X X 1 1 s4 u4 s4 + u4 = 8πG 2 2 + =2πG2 , (5.4) 16πs2 2 2 × × t2 t2 s2t2 · which agrees with the value calculated via conventional methods by Skobelev [35]. Alternatively, since in the center of mass frame dt ω = CM , (5.5) dΩ π we can write the center of mass graviton-photon cross section in the form

dσ 1 + cos8 θCM CM =2 G2 ω2 2 , (5.6) dΩ CM 4 θCM sin 2 ! again in agreement with the value given by Skobelev [35]. So what has gone wrong here? Ordinarily in the massless limit of a spin-1 system, the longitudinal mode decouples because the zero helicity spin-1 polarization vector becomes

2 0 1 m 1 m ǫµ p, p + + ... zˆ = pµ + 0, zˆ + ... (5.7) m−→→0 m 2p m 2p However, the term proportional to pµ vanishes when contracted with a conserved current by m gauge invariance while the term in 2p vanishes in the massless limit. That the spin-1 Compton scattering amplitude becomes gauge invariant for the spin-1 particles in the massless limit can be seen from the fact that the Compton amplitude can be written as

2 Comp e AmpS Tr FiFf FAFB + Tr FiFAFf FB + Tr FiFAFBFf =1 m−→→0 p q p q i · i i · f 1 Tr F F Tr F F + Tr F F Tr F F + Tr F F Tr F F , − 4 i f A B i A f B i B f A (5.8)

21 which can be checked by a bit of algebra. Equivalently, one can verify that the massless spin-1 amplitude vanishes if one replaces either ǫAµ by piµ or ǫBµ by pfµ. However, what happens when we have two longitudinal spin-1 particles is that the product of longitudinal polarization vectors is proportional to 1/m2, while the correction term to the four-momentum p is (m2) so that the product is nonvanishing in the massless limit. That is why the multipole µ O D(00; ++)m=0 = D(00; )m=0 is nonzero. One can deal with this problem by simply omitting the longitudinal degree−− of freedom explicitly, as we did above, but this seems a rather crude way to proceed. Should not this behavior arise naturally? The problem here is that as long as the mass of the spin-1 particle remains finite everything is fine. However, when the spin-1 particle becomes massless the theory becomes undefined. This can be seen from the neutral spin-1 (Proca) Lagrangian, which has the form 1 1 1 1 1 = F F µν + m2A Aµ = ∂ A ∂µAν ∂ A ∂ν Aµ + m2A Aµ . (5.9) L −4 µν 2 µ −2 µ ν − µ ν 2 µ The classical equation of motion then becomes

µ 2 ∂ Fµν + m Aν =0 . (5.10) Taking the divergence of Eq. (5.10) we ﬁnd

2 ν m ∂ Aν =0 , (5.11)

2 ν 2 which yields the constraint m ∂ Aν = 0. Then provided that m = 0 we have the stricture ν 6 ∂ Aν = 0, which is the condition that changes the number of degrees of freedom from four to three, as required for a spin-1 particle. However, in the massless limit, this is no longer the case. Another way to see this is to integrate by parts, whereby Eq. (5.9) can be written in the form 1 1 = A µν A , with µν = ηµν ✷ ∂µ∂ν . (5.12) Lm=0 2 µO ν O − −1 In particle physics the photon propagator is given by the inverse of this operator— µν —which is deﬁned via µν −1 = δµ [1]. However, the operator µν does not have an inverse,O since it O Oνα α O has a zero eigenvalue, as can be seen by operating on a quantity of the form ∂ν Λ(x) where Λ(x) is an arbitrary scalar function. The solution to this problem is well known. The Lagrangian must be altered by adding a gauge ﬁxing term

1 λ 2 1 F ∂ Aµ , (5.13) Lm=0 −→ −4 µν − 2 µ where λ is an arbitrary constant. We now have µν = ηµν ✷ (1 λ)∂µ∂ν which does possess O − − an inverse— −1 = 1 η 1−λ ∂µ∂ν . It is this gauge fixing term, which is required in the Oµν ✷ µν − λ ✷ massless limit, and which eliminates the longitudinal degree of freedom. This degree of freedom acts like simple scalar field (spin-0 particle) and must be subtracted from the massless limit of the spin-1 result. Indeed, from ref. [22] we see that the massless limit of the ++ graviton scattering from a spin-0 target becomes su D0(++) = (2e2)2 Y =8πG , (5.14) × t 22 while the + helicity amplitude vanishes. This scalar amplitude is identical to the ampli- − tude D1(00; ++) and eliminates the longitudinal degree of freedom when subtracted from the massless spin-1 limit. An alternative way to obtain this result is to use the Stueckelberg form of the spin-1 La- grangian, which involves coupling a new spin-0 field B [36] 1 m2 1 1 1 = F F µν + A + ∂ B Aµ + ∂µB ∂ Aµ + mB ∂ Aν + mB . (5.15) LS −4 µν 2 µ m µ m − 2 µ ν As long as m = 0 the fields A and B are coupled. However, if we take the massless limit 6 µ Eq. (5.15) becomes

1 µν 1 µ ν 1 µ S FµνF ∂µA ∂ν A + ∂µB∂ B, (5.16) L m−→→0 −4 − 2 2 and represents the sum of two independent massless ﬁelds—a spin-1 component Aµ with the Lagrangian (in Feynman gauge λ = 1) 1 1 1 1 = F F µν ∂ Aµ∂ Aν = Aµ✷A , (5.17) LS −4 µν − 2 µ ν −2 µ for which we do have an inverse and an independent spin-0 component having the Lagrangian 1 0 = ∂ B∂µB. (5.18) LS 2 µ It is the scattering due to the spin-1 component which is physical and leads to the graviton- photon scattering amplitude, while the spin-0 component is unphysical and generates the longitudinal component of the massless limit of the graviton-spin-1 scattering. As a ﬁnal comment we note that the graviton-graviton scattering amplitude can be obtained by dressing the product of two massless spin-1 Compton amplitudes [4]—

tot pf , ǫBǫB; kf , ǫf ǫf Ampgrav piǫAǫA; ki, ǫiǫi m=0,S=2 Comp = Y pf , ǫB; kf , ǫf Ampem pi, ǫA; kiǫi m=0,S=1 ×h | | i p , ǫ ; k , ǫ AmpComp p , ǫ ; k ǫ . (5.19) ×h f B f f | em | i A i iim=0,S=1 Then for the helicity amplitudes we have

2 1 2 E (++; ++)m=0 = Y B (++; ++)m=0 , (5.20) where E2(++; ++) is the graviton-graviton ++; ++ helicity amplitude while B1(++; ++) is the corresponding spin-1 Compton helicity amplitude. Thus we ﬁnd

κ2 su s 2 s3 E2(++; ++) = 2e2 =8πG , (5.21) m=0 16e4 t × u ut which agrees with the result calculated via conventional methods [37]. In this case there exist non zero helicity amplitudes related by crossing symmetry. However, we defer detailed discussion of this result to a future communication.

23 6 The forward cross section

The forward limit, i.e., θL 0, of the laboratory frame, Compton cross sections evaluated in section 3 has a universal structure→ independent of the spin S of the massive target

Comp 2 dσlab,S α lim = 2 , (6.1) θL→0 dΩ 2m reproducing the Thomson scattering cross section. For graviton photoproduction, the small angle limit is very diﬀerent, since the forward scattering cross section is divergent—the small angle limit of the graviton photoproduction of section 4.1 is given by

photo dσlab,S 4Gα lim = 2 , (6.2) θL→0 dΩ θL and arises from the photon pole in figure 4(d). Notice that this behavior differs from the familiar 1/θ4 small-angle Rutherford cross section for scattering in a Coulomb-like potential. This divergence of the forward cross section indicates that a long range force is involved but with an effective 1/r2 potential. This effective potential arising from the γ-pole in figure 4(d), is the Fourier transform with respect to the momentum transfer q = kf ki of the low-energy limit given in Eq. (4.37). Because of the linear dependence in the momen− ta in the numerator one obtains d3~q 1 1 ei~q·~r = , (6.3) (2π)3 ~q 2π2r2 Z | | and this leads to the peculiar forward scattering behavior of the cross section. Another con- trasting feature of graviton photoproduction is the independence of the forward cross section on the mass m of the target. The small angle limit of the gravitational Compton cross section derived in section 4.2 is given by

g−Comp 2 2 dσlab,S 16G m lim = 4 . (6.4) θL→0 dΩ θL The limit is, of course, independent of the spin S of the matter ﬁeld. Finally, the photon- graviton cross section derived in section 5, has the forward scattering dependence

2 2 dσCM 32G ωCM lim = 4 . (6.5) θCM→ 0 dΩ θCM The behaviors in Eq. (6.4) and (6.5) are due to the graviton pole in ﬁgure 5(d), and are typical of the small-angle behavior of Rutherford scattering in a Coulomb potential. The classical bending of the geodesic for a massless particle in a Schwarzschild metric produced by a point-like mass m is given by b = 4Gm/θ + O(1) [38], where b is the classical

24 impact parameter. The associated classical cross section is

dσclassical b db 16G2m2 = + O(θ−3) , (6.6) dΩ sin θ dθ ≃ θ4

matching the expression in Eq. (6.4). The diagram in figure 5(d) describes the gravitational interaction between a massive particle of spin-S and a graviton. In the forward scattering limit the remaining diagrams of figure 5 have vanishing contributions. Since this limit is independent of the spin of the particles interacting gravitationally, the expression in Eq. (6.4) describes the forward gravitational scattering cross section of any massless particle on the target of mass m and explains the match with the classical formula given above. Eq. (6.5) can be interpreted in a similar way, as the bending of a geodesic in a geometry curved by the energy density with an effective Schwarzschild radius of √2 GωCM determined by the center-of-mass energy [39]. However, the effect is fantastically small since the cross section in Eq. (6.5) is of order ℓ4 /(λ2 θ4 ) where ℓ2 =¯hG/c3 1.6210−35 m is the Planck length, and P CM P ∼ λ the wavelength of the photon.

7 Conclusion

1 In ref. [22] it was demonstrated that the gravitational interactions of a charged spin-0 or spin- 2 particle are greatly simplified by use of the recently discovered factorization theorem, which asserts that the gravitational amplitudes must be identical to corresponding electromagnetic amplitudes multiplied by universal kinematic factors. In the present work we demonstrated that the same simplification applies when the target particle carries spin-1. Specifically, we evaluated the graviton photoproduction and graviton Compton scattering amplitudes explicitly using direct and factorized techniques and showed that they are identical. However, the factorization methods are enormously simpler and allow the use of familiar electromagnetic calculational methods, eliminating the need for the use of less familiar and more cumbersome tensor quantities. We also studied the massless limit of the spin-1 system and showed how the use of factorization permits a relatively simple calculation of graviton-photon scattering. Fi- nally, we discussed a subtlety in this graviton-photon calculation having to do with the feature that the spin-1 system must change from three to two degrees of freedom when m 0 and studied why the zero mass limit of the spin-1 gravitational Compton scattering amplitude→ does not correspond to that for photon scattering. We noted that graviton-graviton scattering is also simply obtained by taking the product of Compton amplitudes dressed by the appropriate kinematic factor. We discussed the main feature of the forward cross section for each process studied in this paper. Both the Compton and the gravitational Compton scattering have the expected behavior, while graviton photoproduction has a different shape that could in principle lead to an interesting new experimental signature of a graviton scattering on matter. An extension of the present discussion at loop order and implications for the photoproduction of gravitons from stars [40, 41] will be given elsewhere.

25 ACKNOWLEDGMENTS

We would like to thank Thibault Damour, Poul H. Damgaard, John Donoghue and Gabriele Veneziano for comments and discussions. The research of P.V. was supported by the ANR grant reference QST ANR 12 BS05 003 01, and the CNRS grants PICS number 6076 and 6430.

APPENDIX A

Here we give the detailed contributions from each of the four diagrams contributing to graviton photoproduction and to gravitational Compton scattering. In the case of graviton photoproduction—Figure 4—we have the four pieces

Graviton photoproduction: spin-1

κe Born a : Amp (S =1)= ǫ p ǫ∗ ǫ ǫ∗ p ǫ∗ p ǫ∗ k ǫ∗ p ǫ∗ ǫ − a p k i · i B · A f · f f · f − B · f f · f f · A i · i ǫ p ǫ∗ p ǫ∗ ǫ∗ + p k ǫ∗ ǫ ǫ∗ ǫ∗ − A · f f · f f · B f · f f · A f · B + ǫ ǫ ǫ∗ k ǫ∗ p ǫ∗ p ǫ∗ k ǫ∗ p ǫ∗ k p k ǫ∗ p ǫ∗ ǫ∗ A · i B · i f · f f · f − B · f f · f f · i − f · i f · f f · B + p k ǫ∗ k ǫ∗ ǫ∗ f · f f · i f · B ǫ k ǫ∗ ǫ ǫ∗ p ǫ∗ p ǫ∗ k ǫ∗ p ǫ∗ ǫ ǫ p ǫ∗ p ǫ∗ ǫ∗ − A · i B · i f · f f · f − B · f f · f f · i − i · f f · f f · B + p k ǫ∗ ǫ ǫ∗ ǫ∗ f · f f · i f · B ǫ∗ ǫ∗ ǫ ǫ ǫ∗ p p k . (7.1) − B · f A · i f · f i · i

κe Born b : Amp (S =1)= ǫ p ǫ ǫ∗ ǫ∗ p ǫ∗ p ǫ∗ p ǫ∗ p ǫ∗ ǫ − b −p k i · f A · B f · i f · i − B · i f · i f · A i · f + ǫ k ǫ∗ p ǫ∗ ǫ∗ p k ǫ∗ ǫ ǫ∗ ǫ∗ A · f f · i f · B − i · f f · A f · B + ǫ∗ k ǫ ǫ ǫ∗ p ǫ∗ p ǫ p ǫ∗ p ǫ∗ ǫ + ǫ k ǫ∗ p ǫ∗ ǫ B · i A · i f · i f · i − i · i f · i f· A A · f f · i f · i p k ǫ∗ ǫ ǫ∗ ǫ − i · f f · A f · i + ǫ ǫ∗ ǫ k ǫ∗ p ǫ∗ p p k ǫ∗ p ǫ∗ ǫ + ǫ k ǫ∗ p ǫ∗ k i · B A · i f · i f · i − i · i f · i f · A A · f f · i f · i p k ǫ∗ ǫ ǫ∗ k − i · f f · A f · i ǫ ǫ∗ ǫ∗ p ǫ∗ ǫ p k . − A · f f · i B · i i · f (7.2)

26 Seagull c : Amp (S =1)= κ e ǫ∗ ǫ (ǫ∗ ǫ ǫ∗ (p + p ) ǫ p ǫ∗ ǫ∗ ǫ∗ p ǫ ǫ∗ ) − c f · i B · A f · f i − A · f B · f − B · i A · f ǫ∗ ǫ∗ ǫ ǫ ǫ∗ p ǫ ǫ∗ ǫ∗ ǫ ǫ∗ p + ǫ∗ ǫ ǫ∗ ǫ∗ ǫ (p + p ) , (7.3) − B · f A · i f · i − A · f B · i f · f f · A f · B i · f i and ﬁnally, the photon pole contribution e κ γ pole d : Amp (S =1)= − − d −2k k f · i ǫ∗ ǫ ǫ∗ (p + p )(k k ǫ∗ ǫ ǫ∗ k ǫ k ) × B · A f · f i f · i f · i − f · i i · f + ǫ∗ k (ǫ∗ ǫ k (p + p ) ǫ∗ k ǫ (p + p )) f · i f · i i · i f − f · i i · f i 2ǫ∗ p ǫ∗ ǫ (k k ǫ∗ ǫ ǫ∗ k ǫ k ) − B · i f · A f · i f · i − f · i i · f + ǫ∗ k (ǫ∗ ǫ ǫ k ǫ∗ k ǫ ǫ ) f · i f · i A · i − f · i i · A 2ǫ p ǫ∗ ǫ∗ (k k ǫ∗ ǫ ǫ∗ k ǫ k ) − A · f f · B f · i f · i − f · i i · f + ǫ∗ k (ǫ∗ ǫ ǫ∗ k ǫ∗ k ǫ ǫ∗ ) . (7.4) f · i f · i B · i − f · i i · B In the case of gravitational Compton scattering—Figure 5—we have the four contributions

Gravitational Compton Scattering: spin-1

1 Born a : Amp (S =1)= κ2 (ǫ p )2(ǫ∗ p )2ǫ ǫ∗ − a 2p k i · i f · f A · B i · i (ǫ∗ p )2ǫ p (ǫ k ǫ∗ ǫ + ǫ ǫ ǫ∗ p ) − f · f i · i A · i B · i A · i B · i (ǫ p )2ǫ∗ p (ǫ∗ ǫ∗ ǫ p + ǫ∗ k ǫ ǫ∗ ) − i · i f · f B · f A · f B · f A · f + ǫ p ǫ∗ p ǫ p ǫ k ǫ∗ ǫ∗ + ǫ p ǫ∗ p ǫ∗ p ǫ ǫ ǫ∗ k i · i f · f i · f A · i B · f i · i f · f f · i A · i B · f + (ǫ∗ p )2ǫ∗ ǫ ǫ ǫ p k +(ǫ p )2ǫ∗ ǫ∗ ǫ ǫ∗ p k f · f B · i A · i i · i i · i B · f A · f f · f + ǫ p ǫ∗ p (ǫ k ǫ∗ k ǫ ǫ∗ + ǫ∗ ǫ∗ ǫ ǫ p p ) i · i f · f A · i B · f i · f B · f A · i i · f ǫ p ǫ∗ p ǫ∗ ǫ∗ ǫ ǫ p k ǫ∗ p ǫ p ǫ ǫ ǫ∗ ǫ∗ p k − i · i f · i B · f A · i f · f − f · f i · f A · i B · f i · i ǫ p ǫ k ǫ∗ ǫ∗ ǫ∗ ǫ p k ǫ∗ p ǫ∗ k ǫ ǫ ǫ ǫ∗ p k − i · i A · i B · f f · i f · f − f · f B · f A · i i · f i · i + ǫ ǫ ǫ∗ ǫ∗ p k p k ǫ ǫ∗ m2ǫ∗ ǫ∗ ǫ ǫ ǫ∗ p ǫ p . A · i B · f i · i f · f i · f − B · f A · i f · f i · i (7.5)

27 1 Born b : Amp (S =1)= κ2 ǫ∗ p 2 ǫ p 2ǫ ǫ∗ − b − 2p k f · i i · f A · B i f 2 · + ǫ p ǫ∗ p ǫ k ǫ∗ ǫ∗ ǫ ǫ∗ ǫ∗ p i · f f · i A · f B · f − A · f B · i ∗ 2 ∗ ∗ + ǫf pi ǫi pf ǫB kiǫA ǫi ǫB ǫi ǫA pf · · · · − · · ǫ∗ p ǫ p ǫ∗ p ǫ k ǫ∗ ǫ ǫ∗ p ǫ p ǫ p ǫ ǫ∗ ǫ∗ k − f · i i · f f · f A · f B · i − f · i i · f i · i A · f B · i 2 2 ǫ p ǫ∗ ǫ∗ ǫ ǫ∗ p k ǫ∗ p ǫ∗ ǫ ǫ ǫ p k − i · f B · f A · f i · f − f · i B · i A · i f · i + ǫ∗ p ǫ p ǫ k ǫ∗ k ǫ ǫ∗ + ǫ∗ ǫ ǫ ǫ∗ p p f · i i · f A · f B · i i · f B · i A · f i · f + ǫ∗ p ǫ p ǫ∗ ǫ ǫ ǫ∗ p k + ǫ p ǫ∗ p ǫ ǫ∗ ǫ∗ ǫ p k f · i i · i B · i A · f f · i i · f f · f A · f B · i i · f ǫ∗ p ǫ k ǫ∗ ǫ ǫ ǫ∗ p k ǫ p ǫ∗ k ǫ ǫ∗ ǫ∗ ǫ p k − f · i A · f B · i i · f f · i − i · f B · i A · f f · i i · f + ǫ ǫ∗ ǫ∗ ǫ p k p k ǫ ǫ∗ m2ǫ∗ ǫ ǫ ǫ∗ ǫ p ǫ∗ p . A · f B · i i · f f · i i · f − B · i A · f i · f f · i (7.6)

κ2 Seagull c : Amp (S =1)= ǫ ǫ∗ 2 m2 p p ǫ ǫ∗ − c − 4 i · f − i · f A · B + ǫ p ǫ∗ p ǫ ǫ∗ 2 + ǫ p ǫ∗ p 2ǫ ǫ∗ ǫ ǫ∗ 2ǫ ǫ ǫ∗ ǫ A · f B · i i · f i · i f · f i · f A · B − A · 2 B · 1 + ǫ p ǫ∗ p 2ǫ ǫ∗ ǫ ǫ∗ 2ǫ ǫ ǫ∗ ǫ∗ i · f f · i i · f A · B − A · i B · f + 2ǫ p ǫ p ǫ ǫ∗ ǫ∗ ǫ∗ +2ǫ∗ p ǫ∗ p ǫ ǫ ǫ∗ ǫ i · i 1 · f A · f B · f f · f f · i A · i B · i 2ǫ p ǫ ǫ∗ ǫ p ǫ∗ ǫ∗ 2ǫ∗ p ǫ ǫ∗ ǫ ǫ ǫ∗ p − i · i i · f A · f B · f − f · f i · f A · i f · i 2ǫ p ǫ ǫ∗ ǫ ǫ∗ ǫ∗ p 2ǫ∗ p ǫ ǫ∗ ǫ∗ ǫ ǫ p − i · f i · f A · f B · i − f · i i · f B · i A · f 2 m2 p p ǫ ǫ∗ ǫ ǫ ǫ∗ ǫ∗ + ǫ ǫ∗ ǫ∗ ǫ , (7.7) − − f · i i · f A · i B · f A · f B · i

28 and ﬁnally the (lengthy) graviton pole contribution is

2 κ 2 g pole d : Amp (S =1)= ǫ∗ ǫ ǫ ǫ∗ 4k p p k +4k p k p − − d −16 k k B · A i · f i · i f · i f · i f · f i · f h h 2 2 p k p k + p k p k +6p p k k +4 ǫ k ǫ∗ p ǫ∗ p − i · i f · f f · i i · f i · f i · f i · f f · f f · i 2 i h + ǫ∗ k ǫ p ǫ p + ǫ k ǫ∗ k ǫ p ǫ∗ p + ǫ p ǫ∗ p f · i i · i i · f i · f f · i i · i f · f i · f f · i i 4ǫ ǫ∗ ǫ k ǫ∗ p p k + ǫ∗ p k p + ǫ∗ k ǫ p p k + ǫ p p k − i · f i · f f · i f · f f · f f · i f · i i · i f · i i · f i · i h i 4k k ǫ ǫ∗ ǫ p ǫ∗ p + ǫ p ǫ∗ p 4p p ǫ ǫ∗ ǫ k ǫ∗ k − i · f i · f i · i f · f i · f f · i − i · f i · f i · f f · i 2 i p p ǫ∗ ǫ ǫ∗ p ǫ p 10 ǫ ǫ∗ k k +4ǫ ǫ∗ ǫ k ǫ∗ k − i · f B · A − B · i A · f i · f i · f i · f i · f f · i 2 h 2 4 ǫ ǫ∗ k k 8ǫ ǫ∗ ǫ k ǫ∗ k + p p m2 ǫ ǫ∗ 4ǫ k ǫ∗ k − i · f i · f − i · f i · f f · i i · f − i · f A · i B · i ∗ ∗ ∗ ∗ + 4ǫA kf ǫ kf 2 ǫA ki ǫ kf + ǫAi kfǫ ki +6ǫhǫA ki kf · B · − · B · · B · B · · 2 ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ + 4 ǫi kf ǫA ǫ ǫ ǫ + ǫ ki ǫA ǫi ǫ ǫi + ǫi kf ǫ kf ǫA ǫi ǫ ǫ · · f B · f f · · B · · f · · B · f h + ǫ ǫ∗ ǫ∗ ǫ 4ǫ ǫ∗ ǫ k ǫ ǫ∗ ǫ∗ k + ǫ∗ ǫ∗ ǫ k A · f B · i − i · f i · f A · f B · f B · f A · f ∗ ∗ ∗ ∗ ∗ ∗ ∗ + ǫ ki ǫA ǫiǫi ki + ǫ h ǫi ǫA ki + ki kf ǫA ǫi ǫ ǫ + ǫ ǫi ǫA ǫ f · · B · B · · · · B · f B · · f 2 + ǫ ǫ∗ǫ k ǫ∗ k 2ǫ p ǫ∗ ǫ 2ǫ∗ k p k +2ǫ∗ k p k A · B i · f f · i − A · f f · i B · i i · i B · f i · f ii h h 2 + 3ǫ∗ p k k ǫ∗ k p k +ǫ∗ k p k +2 ǫ k ǫ∗ ǫ∗ ǫ∗ p B · i i · f − B · i i · f B · f i · i i · f B · f f · i + 2 ǫ∗ k 2ǫ∗ ǫ ǫ p +2ǫ k ǫ∗ k ǫ∗ ǫ ǫ∗i p +ǫ p ǫ∗ ǫ∗ f · i B · i i · i i · f f · i B · i f · i i · i B · f 2ǫ ǫ∗ ǫ k ǫ∗ ǫ∗ p k + ǫ∗ p ǫ∗ k + ǫ∗ k ǫ∗ ǫ p k+ ǫ∗ k ǫ p − i · f i · f B · f i · f f · i B · f f · i B · i i · i B · i i · i h i 2k k ǫ ǫ∗ ǫ∗ ǫ ǫ∗ p + ǫ∗ ǫ∗ ǫ p 2ǫ∗ p ǫ ǫ∗ ǫ k ǫ∗ k − i · f i · f B · i f · i B · f i · i − B · i i · f i · f f · i 2 i 2ǫ∗ p ǫ∗ ǫ 2ǫ k p k +2ǫ k p k +3ǫ p k k − B · i f · i A · i f · i A · f f · f A · f i · f h h 2 2 ǫ k p k + ǫ k p k ) +2 ǫ k ǫ ǫ∗ ǫ∗ p +2 ǫ∗ k ǫ ǫ ǫ p − A · i f · f A · f f · f i · f A · f f · f f · i A · i i · f i + 2ǫ k ǫ∗ k ǫ ǫ ǫ∗ p + ǫ p ǫ ǫ∗ 2ǫ ǫ∗ ǫ k ǫ ǫ∗ p k i · f f · i A · i f · f i · f A · f − i · f i · f A · f f · f h + ǫ∗ p ǫ k + ǫ∗ k ǫ ǫ p k + ǫ k ǫ p f · f A · f f · i A · i f · i A · i i · f i 2k k ǫ ǫ∗ ǫ ǫ ǫ∗ p + ǫ ǫ∗ ǫ p 2ǫ p ǫ ǫ∗ ǫ k ǫ∗ k . − i · f i · f A · i f · f A · f i · f − A · f i · f i · f f · i i (7.8)

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