Graviton Physics

More Graviton Physics

Niels BJERRUM-BOHR, Barry HOLSTEIN, Ludovic PLANT´eand Pierre VANHOVE

Institut des Hautes Etudes´ Scientiﬁques 35, route de Chartres 91440 – Bures-sur-Yvette (France)

Septembre 2015

IHES/P/15/05 IPHT-t15/011/, IHES/P/15/05

More Graviton Physics

N.E.J. Bjerrum-Bohra, Barry R. Holsteinb, Ludovic Plantéc, Pierre Vanhovec,d a Niels Bohr International Academy and Discovery Center The Niels Bohr Institute, Blegamsvej 17, DK-2100 Copenhagen Ø, Denmark b Department of Physics-LGRT University of Massachusetts Amherst, MA 01003 USA c Institut de Physique Théorique CEA, IPhT, F-91191 Gif-sur-Yvette, France CNRS, URA 2306, F-91191 Gif-sur-Yvette, France d Institut des Hautes Etudes´ Scientifiques, F-91440 Bures-sur-Yvette, France

The interactions of gravitons with spin-1 matter are calculated in parallel with the well known photon case. It is shown that graviton scattering amplitudes can be factorized into a product of familiar electromagnetic forms, and cross sections for various reactions are straightforwardly evaluated using helicity methods. Universality relations are identiﬁed. Extrapolation to zero mass yields scattering amplitudes for photon-graviton and graviton-graviton scattering.

1. INTRODUCTION

The calculation of photon interactions with matter is part of any introductory (or advanced) course on quantum mechanics. Indeed the evaluation of the Compton scattering cross section is a standard exercise in relativistic quantum mechanics, since gauge invariance together with the masslessness of the photon allow the results to be presented in terms of relatively simple analytic forms [1]. Naively, one might expect a similar analysis to be applicable to the interactions of gravitons since, like photons, gravitons are massless and subject to a gauge invariance. Also, just as virtual photon exchange leads to a detailed understanding of electromagnetic interactions between charged systems, a careful treatment of virtual graviton exchange allows an understanding not just of Newtonian gravity, but also of spin-dependent phenomena—geodetic precession and Lense-Thirring frame dragging—associated with general relativity which have recently been verified by gravity probe B [2]. However, despite these parallels, examination of quantum mechanics texts reveals that (with one ex- ception [3]) the case of graviton interactions is not discussed in any detail. There are at least 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 more complex, involving symmetric and traceless second rank tensors rather than simple Lorentz four-vectors; ii) there exist fewer experimental results with which to confront the theoretical calculations. added text Funda- mental questions beyond the detection of quanta of gravitational fields have been exposed in [4]; iii) in order to guarantee gauge invariance one must include, in many processes, the contribution from a graviton pole term, involving a triple-graviton coupling. This vertex is a sixth rank tensor and contains a multitude of kinematic forms. added text: Hundred years after the classical theory of general relativity and Einstein1 argument for a quantization of gravity [5], we are still looking after experimental signature of quantum gravity effects. This paper present and extend recent works where elementary quantum gravity processes display new and very distinctive behaviour.

1 “Gleichwohl müßtendie Atome zufolge der inneratomischen Elektronenbewegung nicht nur elektromagnetische, sondern auch Gravita- tionsenergie ausstrahlen, wenn auch in winzigem Betrage. Da dies in Wahrheit in der Natur nicht zutreffen dürfte,so scheint es, daß die Quantentheorie nicht nur die Maxwellsche Elektrodynamik, sondern auch die neue Gravitationstheorie wird modifizieren müssen” [5]. 2

Recently, however, using powerful (string-based) techniques, which simplify conventional quantum ﬁeld theory calculations, it has been demonstrated that the scattering of gravitons from an elementary target of arbitrary spin factorizes [6], a feature that had been noted ten years previously by Choi et al. based on gauge theory arguments [7]. This factorization property, which is sometime concisely described by the phrase “gravity is the square of a gauge theory”, permits a relatively elementary evaluation of various graviton amplitudes and opens the possibility of studying gravitational processes in physics coursework. In a previous paper published in this journal [8] it was shown explicitly 1 how, for both spin-0 and spin- 2 targets, the use of factorization enables elementary calculation of both the graviton photoproduction,

γ + S g + S, → and gravitational Compton scattering,

g + S g + S, → reactions in terms of elementary photon reactions. This simpliﬁcation means that graviton interactions can now be discussed in a basic quantum mechanics course and opens the possibility of treating interesting cosmological applications. In the present paper we extend the work begun in [8] to the case of a spin-1 target and demonstrate and explain the origin of various universalities, i.e., results which are independent of target spin. In addition, by taking the limit of vanishing target mass we show how both graviton-photon and graviton-graviton scattering may be determined using elementary methods. In section 2 then, we generate the electromagnetic interactions of a spin one system. In section 3 we calculate 1 the ordinary Compton scattering cross section for a spin-1 target and compare with the analogous spin-0 and spin- 2 forms. In section 4 we examine graviton photoproduction and gravitational Compton scattering for a spin-1 target 1 and again compare with the analogous spin-0 and spin- 2 results. In section 5 we study the massless limit and show how both photon-graviton and graviton-graviton scattering can be evaluated, resolving a subtlety which arises in the derivation. Finally, our results are summarized in a brief concluding section 6. Two appendices contain formalism and calculational details.

2. SPIN ONE INTERACTIONS: A LIGHTNING REVIEW

We begin by generating the photon and graviton interactions of a spin-1 system. We generate the photon interaction by writing down the free Lagrangian for a scalar ﬁeld φ

S=0 † µ 2 † = ∂µφ ∂ φ m φ φ (1) L0 − and using the minimal substitution [9]

i∂µ iDµ i∂µ eAµ −→ ≡ − This procedure leads to the familiar interaction Lagrangian

S=0 † µ 2 ν † = iAµφ ←→∂ φ + e AµA ηµν φ φ (2) Lint − where e is the particle charge and Aµ is the photon ﬁeld and implies the one- and two-photon vertices

(1)µ µ < pf V pi > = ie(pf + pi) | em | (2)µν 2 µν < pf V pi > = ie η (3) | em | The corresponding charged massive spin-1 Lagrangian has the Proca form [10] 1 S=1 = B† Bµν + m2 B† Bµ , (4) L0 −2 µν µ µ µ µν where B is a spin one ﬁeld subject to the constraint ∂µB = 0 and B is the antisymmetric tensor

Bµν = ∂µBν ∂ν Bµ . (5) − 3

The minimal substitution then leads to the interaction Lagrangian

S=1 µ ν† α 2 µ ν α† β = i e A B ηνα←→∂ µ ηαµ←→∂ ν B e A A (ηµν ηαβ ηµαηνβ)B B , (6) Lint − − − and the one, two photon vertices

(1)µ ∗ µ αβ βµ α αµ β pf , B V pi, A = i e (pf + pi) η η p η p Aα , em S=1 − Bβ − f − i (2)µν 2 ∗ αβ µν αµ βν αν βµ pf , B V pi, A = i e 2η η η η η η Aα . (7) em S=1 Bβ − − However, Eq. (7) 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, the ﬁeld tensor is not given by Eq. (5) but rather is generated from the charged— q 1 (x iy)—component of 2 ±

Bµν = DµBν Dν Bµ ggaBµ Bν (8) − − × ± where gga is the gauge coupling. This modiﬁcation implies the existence of an additional W γ interaction, leading to an “extra” contribution to the single photon vertex

(1)µ ∗ αµ β βµ α pf , B δV pi, A S=1 = i e η (pi pf ) η (pi pf ) ) Aα . (9) em i Bβ − − −

The signiﬁcance of this term can be seen by using the mass-shell Proca constraints pi A = pf B = 0 to write the total on-shell single photon vertex as · ·

µ ∗ µ αβ βµ α pf , B (Vem + δVem) pi, A = i e (pf + pi) η 2η (pi pf ) Aα S=1 − Bβ − αµ β + 2η (pi pf ) , (10) − − αµ β βµ α wherein, comparing with Eq. 9 we observe that the coefficient of the term η (pi pf ) + η (pi pf ) has been modified from unity to two. Since the rest frame spin operator can be identified− via2− −

† † B Bj B Bi = i ijk f Sk i , (12) i − j − the corresponding piece of the nonrelativistic interaction Lagrangian becomes e int = g f S i ∇ A , (13) L − 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 [11] to its universal value of two, as originally proposed by Weinberg [12] and more recently buttressed by a number of additional arguments [13]. Henceforth in this manuscript then 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 properties of gravitational amplitudes are valid [14].

3. COMPTON SCATTERING

The vertices given in the previous section can now be used to evaluate the ordinary Compton scattering amplitude,

γ + S γ + S, →

2 Equivalently, one can use the relativistic identity ∗ ∗ 1 i β γ δ 1 ∗ Bµq · A − Aµq · B = 2 µβγδpi q S − (pf + pi)µB · qA · q (11) − q m 2m 1 m2 δ i δστζ ∗ where S = 2m BσAτ (pf + pi)ζ is the spin four-vector. 4

(a) (b) (c)

FIG. 1: Diagrams relevant to Compton scattering.

for a spin-1 system having charge e and mass m by summing the contributions of the three diagrams shown in Figure 1, yielding

( ∗ ∗ Comp 2 ∗ i pif pf i pf f pi ∗ AmpS=1 = 2e A B · · · · i f · pi ki − pi kf − · · · ∗ ∗ ∗ i pi i pf ∗ f pf f pi A [f , kf ] B · · A [i, ki] B · · − · · pi ki − pi kf − · · pi ki − pi kf · · · · ) 1 ∗ ∗ 1 ∗ ∗ A [i, ki] [f , kf ] B A [f , kf ] [i, ki]B (14) − pi ki · · · − pi kf · · · · with the momentum conservation condition pi + ki = pf + kf . It is then straightforward to verify that Eq. (14) satisﬁes the gauge invariance strictures

∗µ ν Comp µ ν Comp f ki Ampµν,S=1 = kf i Ampµν,S=1 = 0 . (15) In order to pave the way for the transition to gravity in the next section, it is useful to utilize the helicity formalism [15], wherein one evaluates 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 work initially in the center of mass frame and, for a photon incident with four-momentum µ ki = pCM(1, zˆ), we choose the polarization vectors

λi λi = xˆ + iλiyˆ , λi = , (16) i −√2 ± µ while for an outgoing photon with kf = pCM(1, cos θCMzˆ + sin θCMxˆ) we use polarizations

λf λf = cos θCMxˆ + iλf yˆ sin θCMzˆ , λf = . (17) f −√2 − ± We can deﬁne corresponding helicity states for the spin-1 system. In this case the initial and ﬁnal four-momenta µ µ are pi = (ECM, pCMzˆ) and pf = (ECM, pCM(cos θCMzˆ + sin θCMxˆ)) and there exist two transverse polarization four-vectors − − xˆ iyˆ ±µ = 0, ± − , A √2 cos θCMxˆ + iyˆ sin θCMzˆ ±µ = 0, ± ∓ , (18) B √2 in addition to the longitudinal mode with polarization four-vectors

0µ 1 = pCM, ECMzˆ , A m − 0µ 1 = pCM, ECM(cos θCMzˆ + sin θCMxˆ) , (19) B m − 5

In terms of the usual invariant kinematic (Mandelstam) variables

2 2 2 s = pi + ki , t = ki kf , u = pi kf , − − we identify

s m2 s + m2 pCM = − ,ECM = , 2√s 2√s 1 1 1 2 2 2 4 2 2 1 (s m ) + st m su 1 st cos θCM = − = − , sin θCM = − . (20) 2 s m2 s m2 2 s m2 − − − The invariant cross-section for unpolarized Compton scattering is then given by

Comp dσ 1 1 X 1 X 2 S=1 = B1(ab; cd) . (21) dt 16π(s m2)2 3 2 − a,b=−,0,+ c,d=−,+ where

1 Comp B (ab; cd) = pf , b; kf , d AmpS=1 pi, a; ki, c , (22) is the Compton amplitude for scattering of a photon with four-momentum 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 now be calculated straightforwardly. There exist 32 22 = 36 such amplitudes but, since helicity reverses under spatial inversion, parity invariance of the electromagnetic× interaction requires that3

1 1 B (ab; cd) = B ( a b; c d) . − − − − Also, since helicity is unchanged under time reversal, but initial and ﬁnal states are interchanged, T-invariance of the electromagnetic interaction requires that

1 1 B (ab; cd) = B (ba; dc) .

Consequently there exist only twelve independent helicity amplitudes. Using Eq. (14) we calculate the various helicity amplitudes in the center of mass frame and then write these results in terms of invariants using Eq. (20), yielding the forms quoted in Appendix A. Substitution of these helicity amplitudes Eq. (86) into Eq. (21) then yields the invariant cross-section for unpolarized Compton scattering from a charged spin-1 target

dσComp e4 h i S=1 = (m4 su + t2)3(m4 su) + t2 + t2(t m2)(t 3m2) , (23) dt 12π(s m2)4(u m2)2 − − − − − − which can be compared with the corresponding results for unpolarized Compton scattering from charged spin-0 and 1 spin- 2 targets found in ref. [8]. Often such results are written in the laboratory frame, wherein the target is at rest, by use of the relations

2 2 s m = 2mωi, u m = 2mωf , − − − 4 2 2 θL 2 2 2 θL m su = 4m ωiωf cos , m t = 4m ωiωf sin , (24) − 2 − 2 and

2 2 dt d 2ωi (1 cos θL) ωf = ωi − = . (25) dΩ 2πd cos θL −1 + (1 cos θL) π m −

3 Note 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. 6

Introducing the ﬁne structure constant α = e2/4π, we ﬁnd then

Comp 2 4 " 2 dσlab,S=1 α ωf 4 θL 4 θL ωi 2 θL = 2 4 cos + sin 1 + 2 sin dΩ m ωi 2 2 m 2 16ω2 θ ω θ 32ω4 θ + i sin4 L 1 + 2 i sin2 L + i sin8 L , 3m2 2 m 2 3m4 2 (26) Comparing with the corresponding forms found in [8], We observe that the nonrelativistic laboratory cross-section has an identical form for any spin

Comp NR dσ 2 lab,S α 4 θL 4 θL ωi = 2 cos + sin 1 + , (27) dΩ m 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, Comp 2 ∗ S, Mf ; f Amp S, Mi; i = 2e i δMi,M + ..., (28) S ωm f · f and obtains in an eﬀective ﬁeld theory approach to Compton scattering [16].4

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 µ int = eAµJ , (30) L − where J µ is the electromagnetic current matrix element, for gravity we have

κ µν int = hµν T . (31) L − 2

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

gµν = ηµν + κhµν , (32) where κ is given in terms of the Cavendish constant G by κ2 = 32πG. The Einstein-Hilbert action is Z 4 2 Einstein−Hilbert = d x √ g , (33) S − κ2 R where

p 1 1 µν √ g = detg = exp trlogg = 1 + η hµν + ..., (34) − − 2 2 λ µν is the square root of the determinant of the metric and := R µλν g is the Ricci scalar curvature obtained by µ R 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 mat Tµν = − L . (35) √ g δgµν −

4 That this seagull contribution dominates the non relativistic cross-section is clear from the feature that ∗ · p · p 2 f i ω 2 ∗ Amp ∼ 2e ∼ × Amp = 2e · i. (29) Born p · k m seagull f 7

(a) (b)

FIG. 2: Diagrams relevant to graviton photoproduction.

The various gravitational vertices can then be found and are given in Appendix B. We work in harmonic (de Donder) gauge which satisﬁes, in lowest order,

µ 1 ∂ hµν = ∂ν h , (36) 2 with

h = trhµν , (37) and in which the graviton propagator has the form i 1 Dαβ;γδ(q) = (ηαγ ηβδ + ηαδηβγ ηαβηγδ) . (38) q2 + i 2 −

Then just as the (massless) photon is described in terms of a spin-1 polarization vector µ 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 = µ ν , helicity = 2 : h(−2) = −− , (39) − µν µ ν and just as in electromagnetism, there is a gauge condition—in this case Eq. (36)—which must be satisﬁed. Note that the helicity states given in Eq. (39) are consistent with the gauge requirement since

µν + + µν − − µ ± η µ ν = η µ ν = 0, and k µ = 0 . (40) With this background under our belt, we can now examine speciﬁc graviton reactions.

4.1. Graviton Photoproduction

We ﬁrst use the above results to discuss the problem of graviton photoproduction on a spin-1 target—γ+S g+S— for which the relevant four diagrams are shown in Figure 2. The electromagnetic and gravitational vertices needed→ for the Born terms and photon pole diagrams—Figures 2a, 2b, and 2d—are given in Appendix B. For the photon pole diagram we require the graviton-photon coupling, Eq. (90)5

5 Note that this form agrees with the previously derived form for the massive graviton-spin-1 energy-momentum tensor—Eq. (87)—in the m → 0 limit. 8

The individual contributions from the four diagrams in Figure 2 are quoted in Appendix C and have a rather complex form. However, when combined, we ﬁnd a much simpler result. The full graviton photoproduction amplitude is found to be proportional to the already calculated Compton amplitude for spin-1—Eq. (14)—times a universal kinematic factor, i.e.,

4 X ∗ αβ pf ; kf , f f T pi; ki, i = Amp (Fig.4(i)) = H iβT (S = 1) , (41) × fα Compton i=1 where ∗ ∗ κ pf Ff pi κ f pf kf pi f pi kf pf H = · · = · · − · · , (42) 2e ki kf 2e ki kf · · ∗ αβ and fαiβTCompton(S) is the spin-1 Compton scattering amplitude calculated in the previous section. The gravitational and electromagnetic gauge invariance of Eq. (41) is obvious, since it follows directly from the gauge invariance already shown for the Compton amplitude together with the explicit gauge invariance of the factor H. The validity of Eq. (41) allows the straightforward calculation of the cross-section by helicity methods since the graviton photoproduction helicity amplitudes are given simply by

C1(ab; cd) = H B1(ab; cd) , (43) × where B1(ab; cd) are the Compton helicity amplitudes found in the previous section. We can then evaluate the invariant photoproduction cross-section using

photo dσS=1 1 1 X 1 X 1 2 = 2 C (ab; cd) , (44) dt 16πs m2 3 2 − a=−,0,+ c=−,+ yielding

photo 2 2 4 dσS=1 e κ (m su) h 4 2 4 2 = 4− 2 (m su + t ) 3(m su) + t dt −96πts m2 u m2 − − − −i + t2(t m2)(t 3m2) . (45) − − Since

1 κ m4 su 2 H = − , (46) | | e 2t − the laboratory value of the factor H is

2 2 2 1 2 κ m cos 2 θL Hlab = 2 2 1 , (47) | | 2e sin 2 θL and the corresponding laboratory cross-section is

photo Comp dσlab,S=1 2 dσlab,S=1 = Hlab dΩ dt " θ ω 4 θ θ θ ω θ 2 = Gα cos2 L f ctn2 L cos2 L + sin2 L 1 + 2 i sin2 L 2 ωi 2 2 2 m 2 16ω2 θ ω θ 32ω4 θ + i sin2 L 1 + 2 i sin2 L + i sin6 L . (48) 3m2 2 m 2 3m4 2

1 Comparing Eq. (48) with the spin-0 and spin- 2 cross sections found in [8], 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 . (49) dΩ 2 2 2 2 O m 9

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,

∗ ∗ f i f ki µ Ampγ−pole κ · · ki pf ; S, Mf Jµ pi; S, Mi . (50) ω−→m 2kf ki × · The leading piece of the electromagnetic current has the universal low-energy structure e pf pi pf ; S, Mf Jµ pi; S, Mi = pf + pi δM ,Mi 1 + − , (51) 2m µ f O m where we have divided by the factor 2m to account for the normalization of the target particle. Since ki (pf +pi) · ω−→→0 2mω, we ﬁnd the universal low-energy amplitude

∗ ∗ NR f i f ki Ampγ−pole = κ e ω · · , (52) 2kf ki · whereby the helicity amplitudes have the form

 θL 1 1+cos θL cos 2 2 θL  sin θL = θ cos ++ = ,  2 1−cos θL L 2 NR κ e sin 2 −− Ampγ−pole = (53) cos θL 2√2  1 1−cos θL 2 2 θL  sin θL = θ sin + = + . 2 1−cos θL sin 2 2 − − Squaring and averaging, summing over initial, ﬁnal spins we ﬁnd then

photo dσ θ θ θ θ ω lab,S G α cos2 L ctn2 L cos2 L + sin2 L 1 + i . (54) dΩ ω−→m 2 2 2 2 O m as determined above—cf. Eq. (49). Comparing the individual contributions from Appendix C with the simple forms above, the power of factorization is obvious and, as we shall see in the next section, permits the straightforward evaluation of even more complex reactions such as gravitational Compton scattering.

4.2. Gravitational Compton Scattering

In the previous section we observed the power of factorization 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 description. In this section we consider an even 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 3. → The contributions from the four individual diagrams can be calculated using the graviton vertices given in Appendix B and are quoted in Appendix C. Each of the four diagrams has a rather complex form. Again however, when added together the total simpliﬁes enormously. Deﬁning the kinematic factor

2 4 2 2 κ pi ki pi kf κ (s m )(u m ) Y = 4 · · = 4 − − , (55) 8e ki kf 16e t · the sum of the four diagrams is found to be given by

4 X pf , B; kf , f f Ampgrav pi, A; ki, ii S = Amp (Fig.5(i)) i=1

= Y ] pf , B; ki, f Amp pi, A; ki, i pf ; ki, f Amp pi; ki, i S=0 , × em S × em i (56) 10

(a) (b)

FIG. 3: Diagrams relevant for gravitational Compton scattering.

with S = 1, where ∗ ∗ 2 i pi f pf i pf f pi ∗ pf ; ki, f Ampem pi; ki, i S=0 = 2e · · · · f i , (57) h | | i pi ki − pi kf − · · · is the Compton amplitude for a spinless target. 1 In [8] the identity Eq. (56) was veriﬁed for simpler cases of S = 0 and S = 2 . This relation is a consequence of the general connections between gravity and gauge theory tree-level amplitudes derived using string-based methods as explained in [17]. Here we have demonstrated its validity for the much more complex case of spin-1 scattering. The corresponding cross-section can be calculated by helicity methods using

D1(ab; cd) = Y B1(ab; cd) A0(cd) , (58) × × where D1(ab; cd) is the spin-1 helicity amplitude for gravitational Compton scattering, B1(ab; cd) is the ordinary spin- 1 Compton helicity amplitude calculated in section 3, and A0(cd) are the helicity amplitudes for spin zero Compton scattering arising from Eq. (57) calculated in [8], as given in Appendix A. Using Eq. (56) the invariant cross-section for unpolarized spin-1 gravitational Compton scattering

g−Comp dσS=1 1 1 X 1 X 1 2 = 2 D (ab; cd) , (59) dt 16πs m2 3 2 − a=−,0,+ c=−,+ is found to be

g−Comp 4 dσS=1 κ 4 2 4 2 4 2 = 4 2 (m su) 3(m su) + t )(m su + t ) dt 768πs m2 u m2 t2 − − − − − + m4t4(3m2 t)(m2 t) . (60) − − and this form can be compared with the unpolarized gravitational Compton cross-sections found in [8]. The corresponding laboratory frame cross-section is

g−Comp 4 " 2 dσlab,S=1 2 2 ωf 4 θL 4 θL 4 θL ωi 2 θL = G m 4 ctn cos + sin 1 + 2 sin dΩ ωi 2 2 2 m 2 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 (61) 11

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 1 + i . (62) 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 = − , (63) 2e2 t −→ 2e2 sin2 θL − 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.6 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 f i kf kf + ki ki pf ; S, Mf Tµν pi; S, Mi . (65) ω−→m 4kf ki · 2 · Here the matrix element of the energy-momentum tensor has the universal low-energy structure

κ κ pf pi pf ; S, Mf Tµν pi; S, Mi = pfµpiν + pfν piµ δM ,Mi 1 + − , (66) 2 4m f O m 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 f i pi kf pf kf + pi ki pf ki δMf ,Mi . (67) non−→−rel 8m kf ki · · · · · · Since p k mω the helicity amplitudes become · ω−→m

2  4 θ 1+cos θL cos L  = 2 ++ = ,  sin2 θL NR  2 1−cos θL 2 −− Ampg−pole = 4πGm 2 (68) 4 θ 1−cos θL sin L  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 , (69) dΩ ω−→m 2 2 2 as found in Eq. (62) above.

5. GRAVITON-PHOTON SCATTERING

In the previous sections we have generalized the results of [8] to the case of a massive spin-1 target. Here we show how these spin-1 results 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, →

6 In the case of +− helicity the “dressing” factor is 2 +− κ 2 Y A = m . (64) 2e2 so that the non-leading contributions will have diﬀerent dressing factors. 12 the laboratory cross-section no longer makes sense since the photon cannot be brought to rest, but the invariant cross-section is well deﬁned

g−Comp 2 2 2 2 4 dσ 4π G 3s u 4t su + t 4πG2(s4 + u4 + s2u2) S=1 − = , (70) dt m−→→0 3s2t2 3s2t2 and it might be (naively) assumed that Eq. (70) 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

2 1 1 s D (++; ++)m=0 = D ( ; )m=0 = 8π G , −− −− t 2 1 1 u D ( ; ++)m=0 = D (++; )m=0 = 8π G , −− −− t 1 1 su D (00; ++)m=0 = D (00; )m=0 = 8π G , (71) −− t which lead to the cross-section g−Comp dσS=1 1 1 X 1 X 1 2 = D (ab; cd) dt 16πs2 3 2 a=+,0,− c=+,− 1 1 s4 u4 s2u2 = (8πG)2 2 + + 16πs2 3 2 × × t2 t2 t2 4π s4·+ u4 + s2u2 = G2 , (72) 3 s2t2 in agreement with Eq. (70). However, this result reveals 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 obtained by deleting the contribution from the D1(00; ++) and D1(00; ) multipoles −− dσgγ 1 1 X 1 X 1 2 = 2 D (ab; cd) dt 16πs 3 a=+,− 2 c=+,− 4 4 4 4 1 1 2 s u s + u = 8πG 2 + = 2π G2 , (73) 16πs2 2 2 × × t2 t2 s2t2 · which agrees with the value calculated via conventional methods by Skobelev [18]. Alternatively, since in the center of mass frame dt ω = CM , (74) 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 , (75) CM 4 θCM dΩ sin 2 again in agreement with the value given by Skobelev [18]. 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 µ p, p + + ... zˆ = pµ + 0, zˆ + ... (76) m−→→0 m 2p m 2p

However, the term proportional to pµ vanishes when contracted with a conserved current by gauge invariance while m the term in 2p vanishes in the massless limit. However, what takes place when two longitudinal spin-1 particles are present is that the product of longitudinal polarization vectors is proportional to 1/m2, while the correction term to 2 the four-momentum pµ is (m ) so that the product is nonvanishing in the massless limit and this is why the multipole O D(00; ++)m=0 = D(00; )m=0 is nonzero. One deals with this problem by simply omitting the longitudinal degree of freedom explicitly, as−− done above. 13

5.1. Extra Credit

Before leaving this section it is interesting to note that graviton-graviton scattering can be treated in a parallel fashion. That is, the graviton-graviton scattering amplitude can be obtained by dressing the product of two massless spin-1 Compton amplitudes [19]—

tot pf , BB; kf , f f Ampgrav piAA; ki, ii m=0,S=2 Comp = Y pf , B; kf , f Ampem pi, A; kii m=0,S=1 × h |Comp | i pf , B; kf , f Amp pi, A; kii m=0,S=1 . (77) ×h | em | i Then for the helicity amplitudes we have

2 1 2 E (++; ++)m=0 = Y B (++; ++)m=0 , (78) where E2(++; ++) is the graviton-graviton ++; ++ helicity amplitude while B1(++; ++) is the corresponding spin-1 Compton helicity amplitude. Thus we ﬁnd

2 2 3 2 κ su 2 s s E (++; ++)m=0 = 2e = 8π G , (79) 16e4 t × u ut which agrees with the result calculated via conventional methods [20].

6. THE FORWARD CROSS-SECTION

Before closing, we note some intriguing physics associated with the forward-scattering limit. In this limit, i.e., θL 0, in the laboratory frame, the Compton cross-section evaluated in section 3 has a universal structure independent→ of the spin S of the massive target

Comp 2 dσlab,S α lim = 2 , (80) θL→0 dΩ 2m reproducing the well-known Thomson scattering cross-section. For graviton photoproduction, however, 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 , (81) θL→0 dΩ θL and arises from the photon pole in ﬁgure 2(d). 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 . (82) θL→0 dΩ θL where again the limit is independent of the spin S of the matter ﬁeld. The behavior in Eqs. (82) is due to the graviton pole in ﬁgure 3(d), and is typical of the small-angle behavior of Rutherford scattering in a Coulomb potential.

7. CONCLUSION

1 In [8] it was demonstrated that the gravitational interaction of a charged spin-0 or spin- 2 particle is greatly simplified by use of factorization, which asserts that the gravitational amplitudes can be written as the product of corresponding electromagnetic amplitudes multiplied by a universal kinematic factor. In the present work we demonstrated that the same simplification applies when the target particle carries spin-1. Specifically, we evaluated 14 the graviton photoproduction and graviton Compton scattering amplitudes explicitly using direct and factorized techniques and demonstrated that they are identical. However, the factorization methods are enormously simpler, since they require only electromagnetic calculations and eliminate the need to employ less familiar and more cumbersome tensor quantities. As a result it is now straightforward to include graviton interactions in a quantum mechanics course in order to stimulate student interest and allowing access to various cosmological applications. We 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, discussing 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 we explained why the zero mass limit of the spin-1 gravitational Compton scattering amplitude does not correspond→ to that for photon scattering. The graviton-photon cross section may possess interesting implications for the attenuation of gravitational waves in the cosmos [23]. We also calculated the graviton-graviton scattering amplitude. Finally, we discussed the main feature of the forward cross-section for each process studied in this paper. Both the 4 Compton and the gravitational Compton scattering have the expected 1/θL behavior, while graviton photoproduction has a different shape that could in principle lead to an interesting new experimental signature of a graviton scattering 2 on matter— 1/θL. Again this result has potentially intriguing implications for the photoproduction of gravitons from stars [24,∼ 25].

Appendix A: Compton Scattering Helicity Amplitudes

For a spin-0 target we have

A0(cd) L(s, u)H0(cd) (83) ≡ where 2e2 L(s, u) = (s m2)(u m2) − − and

H0(++) = m4 su , − H0(+ ) = m2t , (84) − − For a spin-1 target we ﬁnd

B1(ab; cd) K(s, u)H1(ab; cd) (85) ≡ where 2e2 K(s, u) = (s m2)3(u m2) − − 15

p2 p2 ρσ

µν

p1 p1 µν

FIG. 4: (a) The one-graviton and (b) two-graviton emission vertices from a vector particle.

and

1 1 2 2 2 2 H (++; ++) = H ( ; ) = ((s m ) + m t) , −− −− − 1 1 4 2 H (++; ) = H ( ; ++) = (m su) , −− −− − 1 1 4 2 H (+ ; + ) = H ( +; +) = m t , − − − − 1 1 2 2 H (+ ; +) = H ( +; + ) = s t , − − − − 1 1 1 1 H (++; + ) = H ( ; +) = H (++; +) = H ( ; + ) , − −− − − −− − = m2t(m4 su) , 1 1 − 1 1 H (+ ; ++) = H ( +; ) = H ( +; ++) = H (+ ; ) , − − −− − − −− = m2t(m4 su) , 1 1 − 1 1 H (0+; ++) = H (0 ; ) = H (+0; ++) = H ( 0; ) , − −− p − −− = √2m(tm2 + (s m2)2) t(m4 su) , 1 1 − 1 − − 1 H (0+; + ) = H (0 ; +) = H (+0; +) = H ( 0; + ) , − −p− − − − = √2mst t(m4 su) , 1 1 − − 1 1 H (0+; +) = H (0 ; + ) = H (+0; + ) = H ( 0; +) , − −p − − − − = √2m3t t(m4 su) , 1 1 − − 1 1 H (0+; ) = H (0 ; ++) = H (+0; ) = H ( 0; ++) , −− p− −− − = √2m t(m4 su)3 , 1 1 − − 2 2 2 4 H (00; ++) = H (00; ) = (2tm + (s m ) )(m su) , −− − − 1 1 2 2 2 H (00; + ) = H (00; +) = m t((s m ) + 2st) . (86) − − − Appendix B: Gravitational Vertices

Using the formalism discussed in section 3, the needed gravitational couplings can be obtained. The spin 1-one single graviton emission vertex shown in Fig. 4(a) is (1)µν κ ∗ µ ν ν µ ∗ µ ν µ ν pf , B V pi, A = i A p p + p p pi p + p grav S=1 − 2 B · i f i f − B · f A A f ν ∗µ µ ∗ν 2 µ ∗ν ν ∗µ A pf pi B + pi B + pf pi m AB + AB − · · − µν 2 ∗ ∗ η pi pf m A pi A pf , (87) − · − B · − B · ·

There also exista a two-graviton (seagull) vertex shown in Fig. 4(b), which are found by expanding the stress-energy 16

αβ k k q − γδ µν q

FIG. 5: The three graviton vertex

tensor to second order in hµν .

2 (2) µν,ρσ κ pf , B; kf V pi, A; ki = i grav S=1 − 4 2 + piβpfα ηαβ(pi pf m ) ηµρηνσ + ηµσηνρ ηµν ηρσ − · − − + ηµρ ηαβ piν pfσ + piσpfν ηαν piβpfσ ηβν piσpfα − − 2 ηβσpiν pfα ηασpiβpfν + (pi pf m ηαν ηβσ + ηασηβν − − · − + ηµσ ηαβ piν pfρ + piρpfν ηαν piβpfρ ηβν piρpfα − − 2 ηβρpiν pfα ηαρpiβpfν + (pi pf m ηαν ηβρ + ηαρηβν − − · − + ηνρ ηαβ piµpfσ + piσpfµ ηαµpiβpfσ ηβµpiσpfα − − 2 ηβσpiµpfα ηασpiβpfµ + (pi pf m ηαµηβσ + ηασηβµ − − · − + ηνσ ηαβ piµpfρ + piρpfµ ηαµpiβpfρ ηβµpiρpfα − − 2 ηβρpiµpfα ηαρpiβpfµ + (pi pf m ηαµηβρ + ηαρηβµ − − · − ηµν ηαβ piρpfσ + piσpfρ ηαρpiβpfσ ηβρpiσpfα − − − 2 ηβσpiρpfα ηασpiβpfρ + pi pf m ηαρηβσ + ηβρηασ − − · − ηρσ ηαβ piµpfν + piν pfµ ηαµpiβpfν ηβµpiν pfα − − − 2 ηβν piµpfα ηαν piβpfµ + (pi pf m ηαµηβν + ηβµηαν − − · − + ηαρpiµ ηαµpiρ ηβσpfν ηβµpfσ − − + ηασpiν ηαν piσ ηβρpfµ ηβµpfρ − − + ηασpiµ ηαµpiσ ηβρpfν ηβν pfρ − − ) α β ∗ + ηαρpiν ηαν piρ ηβσpfµ ηβµpfσ ( ) . (88) − − A B

We also require the triple graviton vertex of Fig. 5, which was given in [8]. µν i κ 1 µ ν µ ν µ ν 3 µν 2 τ (k, q)= Iαβ,γδ ηαβηγδ k k + (k q) (k q) + q q η q αβ,γδ − 2 − 2 − − − 2 h λσ, µν, λσ, µν, λµ, σν, σν, λµ, i + 2qλqσ I αβI γδ + I γδI αβ I αβI γδ I αβI γδ − − h µ λν, λν, ν λµ, λµ, + qλq ηαβI γδ + ηγδI αβ) + qλq (ηαβI γδ + ηγδI αβ

2 µν, µν, µν λ σ i q ηαβI γδ + ηγδI αβ) η q q (ηαβIγδ,λσ + ηγδIαβ,λσ − − h λ σν, µ σµ, ν σν, µ σµ, ν + 2q I γδIαβ,λσ(k q) +I γδIαβ,λσ(k q) I αβIγδ,λσk I αβIγδ,λσk − − − − 2 σµ, ν ν σµ, µν λ ρσ, ρσ, i + q I αβIγδ,σ + Iαβ,σ I γδ + η q qσ Iαβ,λρI γδ + Iγδ,λρI αβ h 2 2 σµ, ν σν, µ 1 µν 1 + k + (k q) I αβIγδ,σ + I αβIγδ,σ η Iαβ,γδ ηαβηγδ) − − 2 − 2 2 µν, 2 µν, i k ηαβI γδ + (k q) ηγδI αβ . (89) − − 17 where 1 Iαβ,γδ = (ηαγ ηβδ + ηαδηβγ ) 2 The photon-graviton coupling was given in [8]

(γ)µν κ h ∗ µ ν ν µ ∗ µ ν µ ν kf , f V ki, i = i i k k + k k ki k + k grav 2 f · i f i f − f · f i i f ν ∗µ µ ∗ν µ ∗ν ν ∗µ i kf k + k + kf ki + − · i f i f · i f i f µν ∗ ∗ i η kf ki i kii kf . (90) − · f · − f · · 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 Fig. 2c.

i h ∗ ∗ ∗ pf , B; kf , f f T pi, A; ki, i = κ e (pf + pi) i A seagull 2 f · f · B · ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ i pf A pi i A A i pi − B · f · f · − B · f · f · − · f · f · B ∗ ∗ ∗ ∗ ∗ ∗ i A pf i A i (pf + pi) . (91) − · f · f · B − f · · f · B Appendix C: Graviton Scattering Amplitudes

In this section we list the independent contributions to the various graviton scattering amplitudes which must be added in order to produce the complete and gauge invariant amplitudes quoted in the text. We leave it to the (perspi- cacious) reader to perform the appropriate additions and to verify the equivalence of the factorized forms shown earlier.

Graviton Photoproduction: spin-1

For the case of graviton photoproduction, we ﬁnd the four contributions, cf. Fig. 2, κe ∗ ∗ ∗ ∗ ∗ ∗ Fig.2(a) : Ampa(S = 1) = i pi B A f pf f pf B kf f pf f A pi ki · · · · − · · · ∗ ∗· ∗ ∗ ∗ ∗ A pf pf + pf kf A − · f · f · B · f · f · B ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ + A i ki pf pf kf pf ki pf ki pf · B · f · f · − B · f · f · − · f · f · B ∗ ∗ ∗ + pf kf ki · f · f · B ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ A ki i pf pf kf pf i i pf pf − · B · f · f · − B · f · f · − · f · f · B ∗ ∗ ∗ + pf kf f i f B · · · ∗ ∗ ∗ A i pf pi ki . (92) − B · f · f · ·

κe ∗ ∗ ∗ ∗ ∗ ∗ Fig.2(b) : Ampb(S = 1) = i pf A B f pi f pi B pi f pi f A −pi kf · · · · − · · · ∗ ∗ ·∗ ∗ ∗ ∗ + A kf pi pi kf A · f · f · B − · f · f · B ∗ ∗ ∗ ∗ ∗ ∗ ∗ + ki A i pi pi i pi pi A + A kf pi i B · · f · f · − · f · f · · f · f · ∗ ∗ pi kf A i − · f · f · ∗ ∗ ∗ ∗ ∗ ∗ ∗ + i A ki pi pi pi ki pi A + A kf pi ki · B · f · f · − · f · f · · f · f · ∗ ∗ pi kf f A f ki − · · · ∗ ∗ ∗ A pi i pi kf . − · f f · B · · (93) 18 ∗ ∗ ∗ ∗ ∗ ∗ ∗ Fig.2(c) : Amp (S = 1) = κ e i( A (pf + pi) A pf pi A ) c f · B · f · − · B · f − B · · f ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ A i pi A i pf + A i (pf + pi) , (94) − B · f · f · − · f B · f · f · f · B · and ﬁnally, the photon pole contribution e κ Fig.2(d) : Ampd(S = 1) = −2kf ki · ∗ ∗ ∗ ∗ A (pf + pi)(kf ki i ki i kf ) × B · f · · f · − f · · ∗ ∗ ∗ + ki( i ki (pi + pf ) kii (pf + pi)) f · f · · − f · · ∗ ∗ ∗ ∗ 2 pi A (kf ki i ki i kf ) − B · f · · f · − f · · ∗ ∗ ∗ + ki ( i A ki ki i A) f · f · · − f · · ∗ ∗ ∗ ∗ 2A pf f B(kf ki f i f ki i kf ) − · · · · − · · ∗ ∗ ∗ ∗ ∗ + ki ( i ki ki i ) . (95) f · f · B · − f · · B Gravitational Compton Scattering: spin-1

In the case of gravitational Compton scattering—Figure 3—we have the four contributions

2 1 2 ∗ 2 ∗ Fig.3(a) : Ampa(S = 1) = κ (i pi) (f pf ) A B 2pi ki · · · ∗ 2 ∗· ∗ ( pf ) i pi(A ki i + A i pi) − f · · · B · · B · 2 ∗ ∗ ∗ ∗ ∗ (i pi) f pf (B f A pf + B kf A f ) − · ∗ · · ·∗ ∗ · ∗· ∗ ∗ + i pi pf i pf A ki + i pi pf pi A i kf · f · · · B · f · f · f · · B · ∗ 2 ∗ 2 ∗ ∗ ∗ + (f pf ) B i A i pi ki + (i pi) B f A f pf kf · ∗ · · ∗ · ∗ · ∗ ∗· · · + i pi f pf (A ki B kf i f + B f A i pi pf ) · ∗ · ∗ ·∗ · · ∗ · · · ∗ ∗ i pi f pi B f A i pf kf f pf i pf A i B f pi ki − · · ∗ · ∗ ∗ · · − ∗ · ∗ · · · ∗ · i pi A ki B f f i pf kf f pf B kf A i i f pi ki − · · · · · − · · · · · ∗ ∗ ∗ 2 ∗ ∗ ∗ + A i pi ki pf kf i m A i pf i pi . · B · f · · · f − B · f · f · · (96)

2 1 ∗ 2 2 ∗ Fig.3(b) : Ampb(S = 1) = κ f pi i pf A B − 2pi kf · · · · 2 ∗ ∗ ∗ ∗ ∗ + i pf pi A kf A pi · f · · B · f − · f B · ∗ 2 ∗ ∗ + f pi i pf B kiA i B i A pf ∗ · · ∗ · · ∗ − · ∗ · ∗ ∗ pi i pf pf A kf i pi i pf i pi A ki − f · · f · · B · − f · · · · f B · 2 ∗ ∗ ∗ ∗ 2 ∗ i pf A pi kf pi i A i pf ki − · B · f · f · − f · B · · · ∗ ∗ ∗ ∗ ∗ + f pi i pf A kf B ki i f + B i A f pi pf ∗ · · ∗ · ·∗ · · ∗ · · ∗ ∗ + f pi i pi B i A f pf ki + i pf f pf A f B i pi kf ∗ · · ∗· · ∗ · · ∗ · · ∗ ∗ · · f pi A kf B i i f pf ki i pf B ki A f f i pi kf − · · · · · − · · · · · ∗ ∗ ∗ 2 ∗ ∗ ∗ + A i pi kf pf ki i m i A i pf pi . · f B · · · · f − B · · f · f · (97) 19

2 κ ∗ 2 2 ∗ Fig.3(c) : Amp (S = 1) = i m pi pf A c − 4 · f − · · B ∗ ∗ 2 ∗ ∗ ∗ ∗ + A pf pi i + i pi pf 2i A 2A 2 1 · B · · f · f · · f · B − · B · ∗ ∗ ∗ ∗ ∗ + i pf f pi 2i f A B 2A i B f · · · ∗ ∗ · ∗ − ∗ · ∗ · ∗ + 2i pi 1 pf A f B f + 2f pf f pi A i B i · · ∗ · ∗ · ∗ ∗ · · ∗ · ∗ · 2i pi i f A pf B f 2f pf i f A i f pi − · · ∗ · ∗ ∗ · − ∗ · · ∗ ∗ · · 2i pf i f A f B pi 2f pi i f B i A pf − · · · · − · · · · 2 ∗ ∗ ∗ ∗ ∗ 2 m pf pi i A i + A i , (98) − − · · f · B · f · f B · and ﬁnally the (lengthy) graviton pole contribution is 2 κ ∗ h ∗ 2h 3(d) : Ampd(S = 1) = B A i f 4ki pi pf ki + 4kf pi kf pf −16 ki kf · · · · · · · i h 2 ∗ ∗ 2 pi ki pf kf + pf ki pi kf + 6pi pf ki kf + 4 i kf pf pi − · · · · · · · f · f · ∗ 2 ∗ ∗ ∗ i + ki i pi i pf + i kf ki i pi pf + i pf pi f · · · · f · · f · · f · ∗ h ∗ ∗ ∗ i 4i i kf pi pf kf + pf kf pi + ki i pi pf ki + i pf pi ki − · f · f · · f · · f · · · · · ∗ ∗ ∗ ∗ ∗ i 4ki kf i i pi pf + i pf pi 4pi pf i i kf ki − · · f · f · · f · − · · f · f · ∗ ∗ h ∗ 2 ∗ ∗ pi pf A pi A pf 10 i ki kf + 4i i kf ki − · B · − B · · · f · · f · f · ∗ 2 ∗ ∗ i 2h ∗ 2 ∗ 4 i ki kf 8i i kf ki + pi pf m i 4A ki ki − · f · − · f · f · · − · f · B · ∗ ∗ ∗ ∗ + 4A kf kf 2 A ki kf + A kf ki + 6 A ki kf · B · − · B · · B · B · · h 2 ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ + 4 i kf A + ki A i i + i kf kf A i · · f B · f f · · B · · f · · B · f ∗ ∗ i ∗ h ∗ ∗ ∗ ∗ + A i 4i i kf A kf + A kf · f B · − · f · · f B · B · f · ∗ ∗ ∗ ∗ ∗ ∗ ∗ + ki A i ki + i A ki + ki kf A i + i A f · · B · B · · · · B · f B · · f ∗ ∗ ii h ∗ 2h ∗ ∗ + A i kf ki 2A pf i 2 ki pi ki + 2 kf pi kf · B · f · − · f · B · · B · · ∗ ∗ ∗ i 2 ∗ ∗ ∗ + 3 pi ki kf ki pi kf + kf pi ki + 2 i kf pi B · · − B · · B · · · B · f f · ∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ + 2 ki i i pi + 2i kf ki i pi + i pi f · B · · · f · B · f · · B · f ∗ h ∗ ∗ ∗ ∗ ∗ ∗ ∗ i 2i i kf pi kf + pi kf + ki i pi ki + ki i pi − · f · B · f · f · B · f · B · · B · · ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ i 2ki kf i i pi + i pi 2 pi i i kf ki − · · f B · f · B · f · − B · · f · f · ∗ h ∗ 2h 2 pi i 2A ki pf ki + 2A kf pf kf + 3A pf ki kf − B · f · · · · · · · i 2 ∗ ∗ ∗ 2 A ki pf kf + A kf pf kf ) + 2 i kf A pf + 2 ki A i i pf − · · · · · · f f · f · · · ∗ ∗ ∗ ∗ h ∗ + 2i kf ki A i pf + i pf A 2i i kf A pf kf · f · · f · · · f − · f · · f · ∗ ∗ i + pf A kf + ki A i pf ki + A ki i pf f · · f · · · · · ∗ ∗ ∗ ∗ ∗ i 2ki kf i A i pf + A i pf 2A pf i i kf ki . − · · f · f · · f · − · · f · f · (99) Acknowledgement 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 20

01, and the CNRS grants PICS number 6076 and 6430.

[1] See, e.g., B.R. Holstein, Advanced Quantum Mechanics, Addison-Wesley, New York (1992). [2] C.W.F. Everitt et al., ”Gravity Probe B: Final Results of a Space Experiment to Test General Relativity,” Physical Review Letters 106, 221101, pp. 1-4 (2011). [3] M. Scadron, Advanced Quantum Theory and its Applications through Feynman Diagrams, Springer-Verlag, New York (1979). [4] F. Dyson, “Is a Graviton Detectable?,” Int. J. Mod. Phys. A 28 (2013) 1330041. [5] A. Einstein, “Approximative Integration of the Field Equations of Gravitation,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1916 (1916) 688. [6] A review of current work in this area involving also applications to higher order loop diagrams is given by Z. Bern, “Perturbative Quantum Gravity and its Relation to Gauge Theory,” Living Rev. Rel. 5, 5, pp. 1-57 (2002); see also H. Kawai, D.C. Lewellen, and S.H. Tye, “A Relation Between Tree Amplitudes of Closed and Open Strings,” Nucl Phys. B269, 1-23 (1986). [7] S.Y. Choi, J.S. Shim, and H.S. Song, “Factorization and Polarization in Linearized Gravity,” Phys. Rev. D51, 2751-69 (1995. [8] B.R. Holstein, ”Graviton Physics,” Am. J. Phys. 74, 1002-11 (2006). [9] J.D. Jackson, Classical Electrodynamics, wiley, New York (1970) shows that in the presence of interactions with an external vector potential Aµ = (φ, A) the relativistic Hamiltonian in the absence of Aµ p H = m2 + p2

is replaced by H − eφ = pm2 + (p − eA)2 Making the quantum mechanical substituations ∂ H → i p → −i∇ ∂t we find the minimal substitution given in the text. [10] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York (1964). [11] F.J. Belinfante, Phys. Rev. 92, 997-1001 (1953). [12] S. Weinberg, in Lectures on Elementary Particles and Quantum Field Theory, Proc. Summer Institute, Brandeis Univ. (1970), ed. S. Deser, MIT Press, Cambridge, MA (1970), Vol. 1. [13] B.R. Holstein, ”How Large is the ”Natural” Magnetic Moment,” Am. J. Phys. 74, 1104-11 (2006). [14] B.R. Holstein, ”Factorization in Graviton Scattering and the ”Natural” Value of the g factor,” Phys. Rev. D74, 085002 pp. 1-12 (2006). [15] M. Jacob and G.C. Wick, “On the General Theory of Collisions for Particles with Spin,” Ann. Phys. (NY) 7, 404-28 (1959). [16] B.R. Holstein, “Blue Skies and Effective Interactions,” Am. J. Phys. 67, 422-27 (1999). [17] N. E. J. Bjerrum-Bohr, J. F. Donoghue and P. Vanhove, “On-Shell Techniques and Universal Results in Quantum Gravity,” JHEP 2014-02, 111, pp. 1-28 (2014). [18] V.V. Skobelev, “Graviton-Photon Interaction”, Sov. Phys. J. 18, 62-65 (1975). [19] S.Y. Choi, J.S. Shim, and H.S. Song, “Factorization and Polarization in Linearized Gravity,” Phys. Rev. D51, 2751-69 (1995). [20] J.F. Donoghue and T. Torma, “Infrared Behavior of Graviton-Graviton Scattering,” Phys. Rev. D60, 024003 (1999). [21] L. D. Landau and E. M. Lifschits, The Classical Theory of Fields : Course of Theoretical Physics, Volume 2, Butterworth-Heinemann; 4 edition (December 31, 1975). [22] D. Amati, M. Ciafaloni and G. Veneziano, “Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions,” Int. J. Mod. Phys. A3,1615-61 (1988). [23] P. Jones and D. Singleton, “Gravitons to Photons – Attenuation of Gravitational Waves,” arXiv:1505.04843 [gr-qc]. [24] W. K. De Logi and A. R. Mickelson,“Electrogravitational Conversion Cross-Sections in Static Electromagnetic Fields,” Phys. Rev. D16, 2915-27 (1977). [25] G. Papini and S. R. Valluri, “Gravitons in Minkowski Space-Time. Interactions and Results of Astrophysical Interest,” Phys. Rept. 33, 51-125 (1977).