General Relativity from Scattering Amplitudes

PHYSICAL REVIEW LETTERS 121, 171601 (2018)

N. E. J. Bjerrum-Bohr,1 Poul H. Damgaard,1 Guido Festuccia,2 Ludovic Planté,3 and Pierre Vanhove4,5 1Niels Bohr International Academy and Discovery Center, The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 2Department of Physics and Astronomy, Theoretical Physics, Ångströmlaboratoriet, Lägerhyddsvägen 1, Box 516 751 20 Uppsala, Sweden 3Ministère de l’economie´ et des finances, Direction générale des entreprises, 94200 Ivry-sur-Seine, France 4Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette Cedex, France 5National Research University Higher School of Economics, 123592 Moscow, Russian Federation (Received 21 June 2018; published 25 October 2018)

Weoutline the program to apply modern quantum field theory methods to calculate observables in classical general relativity through a truncation to classical terms of the multigraviton, two-body, on-shell scattering amplitudes between massive fields. Since only long-distance interactions corresponding to nonanalytic pieces need to be included, unitarity cuts provide substantial simplifications for both post- Newtonian and post-Minkowskian expansions. We illustrate this quantum field theoretic approach to classical general relativity by computing the interaction potentials to second order in the post-Newtonian expansion, as well as the scattering functions for two massive objects to second order in the post- Minkowskian expansion. We also derive an all-order exact result for gravitational light-by-light scattering.

DOI: 10.1103/PhysRevLett.121.171601

Today it is universally accepted that classical general simplification, and most of today’s amplitude computations relativity can be understood as the ℏ → 0 limit of a quantum for the Standard Model of particle physics would not have mechanical path integral with an action that, minimally, been possible without this method. In classical gravity, the includes the Einstein-Hilbert term. It describes gravitational long-distance terms we seek are precisely of such a interactions in terms of exchanges and interactions of spin-2 nonanalyticpffiffiffiffiffiffiffiffi kind, being functions of the dimensionless gravitons with themselves (and with matter) [1,2].The ratio m= −q2, where m is a massive probe, and qμ language of effective field theory encompasses this view- describes a suitably defined momentum transfer [4]. This point, and it shows that a long-distance quantum field leads to the proposal that these modern methods be used to theoretic description of gravity is well defined order by order compute post-Newtonian and post-Minkowskian perturba- inaderivativeexpansion[3,4]. Quantum mechanics thus tion theory of general relativity for astrophysical processes teaches us that we should expect classical general relativity to such as binary mergers. This has acquired new urgency due be augmented by higher-derivative terms. More remarkably, to the recent observations of gravitational waves emitted what would ordinarily be a quantum mechanical loop during such inspirals. expansion contains pieces at an arbitrarily high order that While the framework for classical general relativity as are entirely classical [5,6]. A subtle cancellation of factors of described above would involve all possible interaction ℏ is at work here. This leads to the radical conclusion that one terms in the Lagrangian, ordered according to a derivative can define classical general relativity perturbatively through expansion, one can always choose to retain only the ℏ the loop expansion. Then plays a role only at intermediary Einstein-Hilbert action. Quantum mechanically this is steps, a dimensional regulator that is unrelated to the classical inconsistent, but for the purpose of extracting only classical physics the path integral describes. results from that action, it is a perfectly valid truncation. For the loop expansion, central tools have been the This scheme relies on a separation of the long-distance unitarity methods [7] that reproduce those parts of loop (infrared) and short-distance (ultraviolet) contributions in “ ” amplitudes that are cut constructable, i.e., all nonanalytic the scattering amplitudes in quantum field theory. We will terms of the amplitudes. This amounts to an enormous follow that strategy here, but one may apply the same amplitude methods to actions that contain, already at the classical level, higher-derivative terms as well. In the Published by the American Physical Society under the terms of future, this may be used to put better observational bounds the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to on such new couplings. the author(s) and the published article’s title, journal citation, In Ref. [8], Damour proposed a new approach for and DOI. Funded by SCOAP3. converting classical scattering amplitudes into the

0031-9007=18=121(17)=171601(7) 171601-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 121, 171601 (2018) Z effective-one-body Hamiltonian of two gravitationally d3l⃗ i 1 1 i ¼ − : ð4Þ interacting bodies. In this Letter, we take a different route 3 4 ⃗2 ⃗ 2 32 j⃗j jl⃗j≪m ð2πÞ m l ðl þ qÞ m q and we show how scattering amplitude methods, which build on the probabilistic nature of quantum mechanics, may be used to derive classical results in gravity. We show This result can be obtained by performing the large mass how tree-level massless emission from massive classical expansion of the exact expression for the triangle integral as sources arises from quantum multiloop amplitudes, thus shown in the Supplemental Material [13]. providing an all-order argument extending the original In Eq. (4) we recognize the three-dimensional integral of observations in Ref. [6]. We apply this method to derive the two static sources localized at different positions, repre- scattering angle between two masses to second post- sented as shaded blobs, and emitting massless fields Minkowskian order using the eikonal method. We start with the Einstein-Hilbert action coupled to a scalar field ϕ: ð Þ Z 2 5 4 pffiffiffiffiffiffi 1 1 μν m 2 S ¼ d x −g R þ g ∂μϕ∂νϕ − ϕ : ð1Þ 16πG 2 2

Here R is the curvature and gμν is the metric, defined as the η Below we show how this allows us to recover the first post- sum of a flat Minkowskipffiffiffiffiffiffiffiffiffiffiffi component μν and a perturbation Newtonian correction to the Schwarzschild metric from κhμν with κ ≡ 32πG. It is coupled to the scalar stress- ≡∂ ϕ∂ ϕ − ðη 2Þð∂ρϕ∂ ϕ − 2ϕ2Þ quantum loops. We now explain how the classical part energy tensor Tμν μ ν μν= ρ m . emerges from higher-loop triangle graphs, starting with Scalar triangle integrals [9] are what reduces the one- two-loop triangles loop, two-graviton scattering amplitude to classical general relativity [4,10–12] in four dimensions. For the long- distance contributions these are the integrals that produce the tree-like structures one intuitively associates with classical general relativity. To see this, consider first the triangle integral of one massive and two massless propagators,

ð6Þ

⃗ In the large mass limit jlij ≪ m1 for i ¼ 1, 2, 3 and ðl þ Þ2 − 2 ≃ 2l ≃ 2 l0 ð2Þ approximating i p1 m1 ip1 m1 i , the integral reduces in that limit to ¼ð ⃗2Þ ¼ð − ⃗2Þ ≡ − 0 ¼ð0 ⃗Þ with p1 pE;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq= , p2 E; q= and q p1 p1 ;q , ¼ 2 þ ⃗2 4 and E m1 q = , and we work with the mostly I⊳⊳ð1Þðp1;qÞ negative metric ðþ−−−Þ. The curly lines are for massless Z Y3 4l 1 fields and the left solid line is for a particle of in- ¼ − d i 0 ð2πÞ4 l2 þ ϵ coming momentum p1, outgoing momentum p1, and mass i¼1 i i 2 ¼ 02 ¼ 2 P p1 p1 m1. ð2πÞ3δð3Þð 3 l⃗ þ ⃗Þ i¼1 i q 2πδðl0 þ l0 þ l0Þ In the large mass approximation, we focus on the region × 2 × 1 2 3 ⃗ 2 2 2 ðl1 þ qÞ þ iϵ jlj ≪ m1,andwehaveðlþp1Þ −m1 ¼l þ2lp1 ≃2m1l0; 1 1 1 1 therefore the integral reduces in that limit to þ × 0 0 0 0 Z 2m1l1 þ iϵ 2m1l2 − iϵ 2m1l3 þ iϵ 2m1l1 − iϵ 1 4l 1 1 1 d ð Þ 1 1 4 2 2 : 3 þ ð Þ 2m1 ð2πÞ l þ iϵ ðl þ qÞ þ iϵ l0 þ iϵ 0 0 : 7 2m1l3 þ iϵ 2m1l2 − iϵ

We perform the l0 integration by closing the contour of integration in the upper half-plane to get We note that

171601-2 PHYSICAL REVIEW LETTERS 121, 171601 (2018) 1 1 δðl0 þ l0 þ l0Þ A generalization of the identity [Eq. (8)] implies that the 1 2 3 0 0 2m1l þ iϵ 2m1l − iϵ sum of all the permutation of n massless propagators 1 2 1 1 1 1 connected to a massive scalar line results in the coupling þ þ of classical sources to multileg tree amplitudes [20,21]. The 2 l0 þ ϵ 2 l0 − ϵ 2 l0 þ ϵ 2 l0 − ϵ m1 3 i m1 1 i m1 3 i m1 2 i same conclusion applies to massive particles with spin as ¼ 0 þ OðϵÞ; ð8Þ we will demonstrate elsewhere. This analysis applies directly to the computation of an 0 so that only the l0 residue at 2m1l ¼iϵ contributes, off-shell quantity such as the metric itself. Consider the giving absorption of a graviton Z 3 ⃗ 3 ⃗ hp2jTμνjp1i i d l1 d l2 1 1 Z I⊳⊳ð1Þðp1;qÞ¼ 4 4 2 ð2πÞ3 ð2πÞ3 ⃗2 ⃗2 −i d l Pρσ;αβ Pκλ;γδ m1 l1 l2 ¼ 2 ð2πÞ4 l2 þ ϵðlþ Þ2 þ ϵ 1 1 m1 i q i × : ð9Þ ρσ κλ αβ;γδ ⃗ ⃗ 2 ⃗ 2 τ1 ðp1 −l;p1Þτ1 ðp1 −l;p2Þτ3μν ðl;l−qÞ ðl1 þ l2 þ q⃗Þ ðl1 þ q⃗Þ ð Þ × 2 2 ; 14 ðlþp1Þ −m1 þiϵ We now consider the large mass expansion of the graph where τ1 is the vertex for the coupling of one graviton to a scalar given in Ref. [[22], Eq. (72)], τ3 is the three graviton vertex given in Ref. [[22], Eq. (73)], and Pμν ρσ is the ð Þ ; 10 projection operator given in Ref [[22], Eq. (30)]. In the large m limit jqj=m ≪ 1 projects the integral on the 00- μν 2 μ ν component of the scalar vertex τ1 ðp1 − l;p1Þ ≃ iκm1δ0δ0. Focusing on the 00 component, we have in this limit [21] which, to leading order, reads Z 4l 3 3 3 d 2 ⃗2 ð Þ hp2jT00jp1i≃4iπGm q⃗− l I⊳⊳ðdÞ p1;q 1 ð2πÞ4 8 2 Z 1 Y3 4l 1 1 ¼ − d i 3 ð2πÞ4 l2 þ ϵ ×ðl2 þ ϵÞ½ðlþ Þ2 þ ϵ½ðlþ Þ2 − 2 þ ϵ i¼1 i i i q i p1 m1 i 3 ð3Þ 3κ2 3 × ð2πÞ δ ðl1 þ l2 þ l3 þ qÞ m ¼ jq⃗j; ð15Þ 1 1 128 2πδðl0 þ l0 þ l0Þ × 1 2 3 0 0 2m1l1 þ iϵ 2m1l2 − iϵ where we used the result of the previous section to evaluate 1 1 1 1 the triangle integral. This reproduces the classical first post- þ þ 0 0 0 0 ; Newtonian contribution to the 00-component of the 2m1l þ iϵ 2m1l − iϵ 2m1l þ iϵ 2m1l − iϵ 3 1 3 2 Schwarzschild metric evaluated in Ref. [22]. It also immedi- ð11Þ ately shows how to relate a conventional Feynman-diagram evaluation with the computation of Duff [23] who derived and evaluates to such tree-like structures from classical sources. Scalar interaction potentials.—For the classical terms Z 3 ⃗ 3 ⃗ 1 d l1 d l2 1 1 1 we need only the graviton cuts, and instead of computing I⊳⊳ð2Þðp1;qÞ¼ : 12 2 ð2πÞ3 ð2πÞ3 ⃗2 ⃗2 ⃗ ⃗ 2 classes of diagrams, we apply the unitarity method directly m1 l1 l2 ðl1 þ l2 þ q⃗Þ to get the on-shell scattering amplitudes initiated in ð12Þ Ref. [10] and further developed in [11,12]. We first consider the scattering of two scalars of masses m1 and Equations (9) and (12) are precisely the coupling of three m2, respectively. At one-loop order this entails a two- static sources to a massless tree amplitude graviton cut of a massive scalar four-point amplitude. We have shown that classical terms arise from topologies with loops solely entering as triangles that include the massive states. When we glue together the two on-shell scattering amplitudes, we thus discard all terms that do not correspond to such topologies. Rational terms are not needed, as they correspond to ultralocal terms of no relevance for the ð13Þ long-distance interaction potentials.

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We first recall the classical tree-level result from the one- unitarity method [10–12]. The previous analysis graviton exchange shows that the classical piece is contained in the triangle graphs

ð16Þ

2 02 2 where incoming momenta are p1 and p2 and p1 ¼p1 ¼m1, 2 02 2 0 p2 ¼ p2 ¼ m2 and the momentum transfer q ¼ p1 − p1 ¼ ð17Þ 0 −p2 þ p2. The two-graviton interaction is clearly a one-loop amplitude that can be constructed using the on-shell with, for the interaction between two massive scalars,

2 5 2 4 2 2 3 2 2 2 2 4 cðm1;m2Þ¼ðq Þ þðq Þ ð6p1 ⋅ p2 − 10m1Þþðq Þ ð12ðp1 ⋅ p2Þ − 60m1p1 ⋅ p2 − 2m1m2 þ 30m1Þ 2 2 2 2 4 4 2 6 − ðq Þ ð120m1ðp1 ⋅ p2Þ − 180m1p1 ⋅ p2 − 20m1m2 þ 20m1Þ 2 4 2 6 6 2 2 8 2 6 2 þ q ð360m1ðp1 ⋅ p2Þ − 120m1p1 ⋅ p2 − 4m1ðm1 þ 15m2ÞÞ þ 48m1m2 − 240m1ðp1 ⋅ p2Þ : ð18Þ

At leading order in q2, using the result of Eq. (4), the two ⃗2 ⃗2 ⃗4 ⃗4 ¼ p1 þ p2 − p1 − p2 gravitons exchange simplifies to just, in agreement with H 3 3 2m1 2m2 8m1 8m2 Ref. [[24], Eq. (3.26)] and Refs. [10,25], 2 ð þ Þ − Gm1m2 − G m1m2 m1 m2 6π2 2 r 2r2 M ¼ G ð þ Þ½5ð ⋅ Þ2 − 2 2þ ðj ⃗jÞ ð Þ 2 m1 m2 p1 p2 m1m2 O q : 19 3 ⃗2 3 ⃗2 7 ⃗ ⋅ ⃗ ð ⃗ ⋅ ⃗Þð ⃗ ⋅ ⃗Þ jq⃗j − Gm1m2 p1 þ p2 − p1 p2 − p1 r p2 r 2 2 2 ; 2r m1 m2 m1m2 m1m2r Note the systematics of this expansion. The Einstein ð20Þ metric is expanded perturbatively, and all physical momenta are provided at infinity. Contractions of momenta are performed with respect to the flat-space Minkowski which precisely leads to the celebrated Einstein-Infeld- metric only, and no reference is made to space-time Hoffmann equations of motion. It is crucial to correctly coordinates. This is a gauge invariant expression for the perform the subtraction of the iterated tree-level Born term classical scattering amplitude in a plane wave basis that is in order to achieve this. independent of coordinate choices (and gauge choices). To The post-Minkowskian expansion.—The scattering prob- derive a classical nonrelativistic potential, we need to lem of general relativity can be treated in a fully relativistic choose coordinates: we Fourier transform the gauge invari- manner, without a truncated expansion in velocities. To this ant momentum-space scattering amplitude. This introduces end, we consider here the full relativistic scattering ampli- coordinate dependence even in theories such as quantum tude and expand in Newton’s constant G only. For the scalar- electrodynamics. Moreover, just as in quantum electrody- scalar case we thus return to the complete classical one-loop namics, we must also be careful in keeping subleading result [Eq. (17)]. The conventional Born-expansion expres- terms of this Fourier transform and thus expand in q0 sion that is used to derive the quantum mechanical cross consistently. This forces us to keep velocity-dependent section is not appropriate here, even if we keep only the terms in the energy that are of the same order as the naively classical part of the amplitude. That expression for the cross defined static potential. One easily checks that the overall section is based on incoming plane waves, and will not sign of the amplitudes in Eqs. (16) and (19) are precisely match the corresponding classical cross section beyond the the ones required for an attractive force. leading tree-level term. In fact, even the classical cross The result of this procedure has been well documented section is unlikely to be of any interest observationally. So a elsewhere, starting with the pioneering observations of more meaningful approach is to use the classical scattering Iwasaki [5], and later reproduced in different coordinates in amplitude to compute the scattering angle of two masses Refs. [10,26]. Although we are unable to reproduce the colliding with a given impact parameter b. individual contributions in Ref. [[5], Eqs. (A.1.4)–(A.1.6)] We use the eikonal approach to derive the relationship our final result for the interaction energy is to this order: between small scattering angle θ and impact parameter b.

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Generalizing the analysis of Refs. [27,28] (please refer to contributed to the exponentiation of the tree-level ampli- the Supplemental Material [13] for some details on the tude [28,30]. The Fourier transform around two dimensions eikonal method to one-loop order) to the case of two scalars is computed using of masses m1 and m2, we focus on the high-energy regime s, t large, and t=s small. Note that in addition to expanding Z ðαþD D 2−D αþ Γ 2 d q ⃗ ð2πμÞ 2 D 2Þ in G, we are also expanding the full result [Eq. (17)]inq , μ2−D e−iq⃗⋅bjq⃗jα ¼ : ð2πÞD 4π b ð2−α−D and truncating already at next-to-leading order. We go to Γ 2Þ the center of mass frame and define p ≡ jp⃗1j¼jp⃗2j. The ð Þ impact parameter is defined by a two-dimensional vector b⃗ 26 in the plane of scattering orthogonal to p⃗1 ¼ −p⃗2, with b ≡ jb⃗j. In the eikonal limit we find the exponentiated The scattering functions then read relationship between the scattering amplitude Z ˆ 4 2 2 M − 2m1m2 1 ⃗ χ ð Þ¼2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − ðπμ Þ − γ ð⃗Þ ≡ 2 ⃗ −iq⃗⋅b ð⃗Þ ð Þ 1 b G log b E ; M b d qe M q ; 21 ˆ 4 2 2 d −2 M − 4m1m2 3π 2 þ and scattering function χðbÞ to be G m1 m2 ˆ 4 2 2 χ2ðbÞ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5M − 4m1m2Þ; ð27Þ ˆ 4 2 2 b 8 M − 4m1m2 ⃗ iχðb⃗Þ MðbÞ¼4pðE1 þ E2Þðe − 1Þ: ð22Þ

In order to compare with the first computation of post- where μ is a regularization scale. The scattering angle to Minkowskian scattering to order G2 [29], we introduce this order reads new kinematical variables M2 ≡ s, Mˆ 2 ≡ M2 − m2 − m2. 1 2 2 ¼ ˆ 4 2 2 We go to the center of mass frame where p θ 4GM M − 2m1m2 ð ˆ 4 − 4 2 2Þ 4 2 θ 2 sin ¼ M m1m2 = M . In terms of the scattering angle 2 b Mˆ 4 − 4m2m2 ≡ 2 ¼½ðˆ 4 − 4 2 2Þ 2ðθ 2Þ 2 1 2 we have t q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM m1m2 sin = =M , and ˆ 4 2 2 3π Gðm1 þ m2Þ 5M − 4m m 4 ð þ Þ¼2 ˆ 4 − 4 2 2 þ 1 2 : ð28Þ p E1 E2 M m1m2. Keeping, consistently, 16 ˆ 4 2 2 b M − 4m1m2 only the leading order in q2 of the one-loop amplitude [Eq. (21)], we obtain This result agrees with the expression found by Westpfahl −2M ∂ [29] who explicitly solved the Einstein equations to this 2 sinðθ=2Þ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½χ1ðbÞþχ2ðbÞ; ð23Þ order in G and in the same limit of small scattering angle. ˆ 4 2 2 ∂b M − 4m1m2 We find the present approach to be superior in efficiency, and very easily generalizable to higher orders in G. where χ1ðbÞ and χ2ðbÞ are the tree-level and one-loop Taking the massless limit m2 ¼ 0 and approximating scattering functions given respectively by the Fourier 2 sinðθ=2Þ ≃ θ, we recover the classical bending angle of 2 2 2 transform of the scattering amplitudes light θ ¼ð4Gm1=bÞþð15π=4ÞðG m1=b Þ, including its first nontrivial correction in G, in agreement with Z 2 1 d q⃗ ⃗ χ ð Þ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −iq⃗⋅bM ð⃗Þ ð Þ Ref. [[31], §101]. We have additionally computed the full i b 2 e i q : 24 4 2 2 ð2πÞ expression for the classical part of the scalar-fermion (spin 2 Mˆ − 4m m 1 2 1=2) amplitude up to and including one-loop order, but do 2 not display the results here for lack of space. We stress that At leading order in q the tree-level and one-loop ampli- the small-angle scattering formula derived above is based tudes in Eqs. (16) and (19) read on only a small amount of the information contained in the 8πG full one-loop scattering amplitude [Eq. (17)]. M ð⃗Þ¼ ð ˆ 4 − 2 2 2Þ Light-by-light scattering in general relativity.—Photon- 1 q ⃗2 M m1m2 ; q photon scattering is particularly interesting, as our analysis 3π2 2 G ˆ 4 2 2 will show how to derive an exact result in general relativity. M2ðq⃗Þ¼ ðm1 þ m2Þð5M − 4m m Þ; ð25Þ 2jq⃗j 1 2 As explained above, classical contributions from loop diagrams require the presence of massive triangles in the where higher order terms in q2 correspond in position loops. For photon-photon scattering, there are no such space to quantum corrections. Only the triangle con- contributions to any order in the expansion, and we tribution contribute to the one-loop scattering function conclude that photon-photon scattering in general relativity because the contributions from the boxes and cross-boxes is tree-level exact, as follows:

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Danish National Research Foundation (DNRF91). P. V. is partially supported by “Amplitudes” ANR-17- CE31-0001-01, and Laboratory of Mirror Symmetry National Research University Higher School of Economics, Russian Federation Government Grant No. agreement No. 14.641.31.0001. G. F. is supported by the ERC STG Grant No. 639220.

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