PHYSICAL REVIEW LETTERS 121, 171601 (2018) General Relativity from Scattering Amplitudes N. E. J. Bjerrum-Bohr,1 Poul H. Damgaard,1 Guido Festuccia,2 Ludovic Plant´e,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 3Minist`ere de l’economie´ et des finances, Direction g´en´erale des entreprises, 94200 Ivry-sur-Seine, France 4Institut de Physique Th´eorique, Universit´e 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.