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D 99, 024021 (2019)

Effective action of as the classical double copy of Yang-Mills theory

† ‡ Jan Plefka,1,* Jan Steinhoff,2, and Wadim Wormsbecher1, 1Institut für Physik und IRIS Adlershof, Humboldt-Universität zu Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany 2Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam D-14476, Germany

(Received 30 October 2018; published 14 January 2019)

We compute the classical effective action of color charges moving along worldlines by integrating out the Yang-Mills gauge field to next-to-leading order in the coupling. An adapted version of the Bern- Carrasco-Johansson (BCJ) double-copy construction known from scattering amplitudes is then applied to the Feynman integrands, yielding the prediction for the classical effective action of point in dilaton gravity. We check the validity of the result by independently constructing the effective action in dilaton gravity employing field redefinitions and gauge choices that greatly simplify the perturbative construction. Complete agreement is found at next-to-leading order. Finally, upon performing the post- Newtonian expansion of our result, we find agreement with the corresponding action of scalar- theories known from the literature. Our results represent a proof of concept for the classical double-copy construction of the gravitational effective action and provides another application of a BCJ-like double copy beyond scattering amplitudes.

DOI: 10.1103/PhysRevD.99.024021

I. INTRODUCTION conjectural at the loop level, the double-copy procedure is extremely efficient in generating integrands for gravita- There is a growing body of evidence for a fascinating tional theories at high-perturbative orders in theories with perturbative duality between Yang-Mills theory and quan- and without [6–14] and as such has passed tum gravity known as the double-copy construction or many nontrivial checks. The present record being at the color-kinematics duality due to Bern, Carrasco and four point five-loop level for maximal [15]. Johansson (BCJ) [1–3]. It provides a concrete prescription Moreover, an elaborate web of theories connected via for transforming scattering amplitudes in non-Abelian color-kinematics duality exists that includes cou- gauge theories into scattering amplitudes in gravitational plings and various numbers of [16–18]. theories upon replacing the non-Abelian color degrees of All these results point towards the double copy being a freedom by kinematical ones. In the simplest scenario pure generic property of gravity alluding at a hidden kinematical Yang-Mills theory double copies to dilaton gravity coupled algebra which has resisted discovery so far. to an also termed as N ¼ 0 supergravity (the In of these findings the natural question arises, massless sector of bosonic strings). This relation was whether the double copy generalizes beyond the realm of proven for tree-level amplitudes [2], where the double scattering amplitudes. In particular, does it also play a role copy is equivalent to the Kawai-Lewellen-Tye relations of in classical ? Here a number of encour- theory [4], a precursor of the BCJ duality. It is also aging results have been obtained: Schwarzschild, Kerr and manifest in the Cachazo-He-Yuan formulation of and Taub-NUT were shown to be double copies of tree-level amplitudes [5]. While remaining classical gauge-theory solutions [19,20], were further extended to certain classes of perturbative spacetimes [21], and a classical double-copy description of the *[email protected] † [email protected] gravitational radiation from accelerating black holes was ‡ [email protected] developed in Ref. [22]. Following up the last direction, a very interesting setting is that of gravitational radiation Published by the American Physical Society under the terms of produced by binary sources, not the least due to the the Creative Commons Attribution 4.0 International license. spectacular observations of gravitational waves and the Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, need for high precision predictions to generate waveform and DOI. Funded by SCOAP3. templates [23]. Pioneering work was done in [24–26]

2470-0010=2019=99(2)=024021(15) 024021-1 Published by the American Physical Society PLEFKA, STEINHOFF, and WORMSBECHER PHYS. REV. D 99, 024021 (2019) where perturbative classical solutions for binary color- start out with color charges moving along a worldline as in charged point coupled via Yang-Mills theory [27] [24–26,30]. This may be viewed as a of the were shown to double copy to their counterparts in dilaton- massive . gravity radiation (see also [28,29]). In these works certain Our paper is organized as follows: In Sec. II we give an color-kinematics replacement rules were employed which overview of the structure of a theory of classical color- appear somewhat distinct to the color factor/numerator charged massive point particles moving along worldlines, replacement rules familiar from scattering amplitudes. Also interacting via a Yang-Mills theory, in the first-order the question of a double-copy-respecting representation of formalism. In Sec. III we propose a double-copy prescrip- the perturbative gauge-theory solutions remained open. tion for the classical effective action. We compute the This situation was clarified very recently in the work of effective action of the theory introduced in Sec. II up to Shen [30], which pushed the perturbative approach of next-to-leading order in the gluon coupling. Along the solving the equations of motions via double copy to the way we introduce the notion of a trivalent representation next-to-leading order (NLO) in the for graphs involving worldlines and at the end of the section expansion. In this process a color-kinematics duality we perform the double copy. In Sec. IV we repeat the representation of the Yang-Mills-radiation solution was computation of the effective action in the weak-field/post- found, which parallels the color-kinematics duality rules of Minkowskian approximation for massive point particles BCJ [1] and replaces the nonstandard double-copy rules of interacting in dilaton-gravity theory and we compare the [24]. An alternative route for finding the graviton radiation resulting expression to the double-copy result. In Sec. V of binaries via the double copy was taken in [31], where a we reexpand our result for the post-Minkowskian dilaton- tree scattering amplitude in coupled to scalar gravity effective action in velocities, thus yielding the post- fields was double copied to establish the leading order Newtonian approximation. After solving the Feynman gravitational radiation emitted from the scattering of two integrals, we compare the resulting expression to known black holes (modeled by two massive scalar fields). results in the literature. Our conclusions are presented in In our present work, we generalize these approaches to Sec. VI. Finally, our conventions, the Feynman rules the double copy of gravitationally interacting binaries by and a discussion of self- can be found in the ascending from the level of equations of motion to the appendices. classical effective action. This approach makes direct contact to the post-Minkowskian (weak-field) and post- II. YANG-MILLS OF Newtonian (weak-field and slow-motion) expansions of COLOR-CHARGED MASSIVE PARTICLES the gravitational potential for which high-order results We consider the worldline action of a massive, color- exist in the literature, namely at the fourth post- charged (non-Abelian) point coupled to the Yang- Newtonian order (four loop) for nonspinning bodies: using a Mills gauge field Aμ; see appendix A for notation and a canonical formalism of general relativity [32], a Fokker conventions. The action for the classical colored point Lagrangian [33,34], and partial results within an effective charge (pc) is given by [24,25,27] field theory formalism [35]. The latter makes use of Z Z Feynman diagrammatic methods and sophisticated tools pffiffiffiffiffi ¼ − τ ¼ − τð 2 − ψ † μ ψÞðÞ for effective field theories in the context of classical gravity Spc d Lpc d m u iu Dμ 1 [36]. The question how the classical gravitational potential Z may be extracted from the quantum scattering amplitude of pffiffiffiffiffi 2 † μ a a massive scalars has in fact a long history starting in the ¼ − dτðm u − iψ ψ_ − gu Aμc Þ; ð2Þ 1970s [37–39]. Recent works have updated this by employ- ing modern unitarity methods of which is invariant under reparametrization of τ. Here uμ ¼ [40–42]. Damour recently proposed an alternative approach x_μ is the 4-velocity of the particle along the worldline xμðτÞ for converting scattering angles and amplitudes to the and ψðτÞ an associated fundamental vector carrying the effective one-body Hamiltonian [43,44]. Hence, in order color degrees of freedom of the particle—often called the to perform the construction of the interaction potential color wave function. Moreover using the double copy, one could study the scattering amplitude of a massive scalar field coupled to either Yang- ca ≔ ψ †Taψ ð3Þ Mills or dilaton-gravity theory and relate to a potential using one of the mentioned works. Representing the gauge- is the color charge carried by the particle. In this form, there theory result in a color-kinematics respecting fashion is only a single gluon coupling to the scalar worldline. should then yield the effective gravitational potential via As the gravity counterpart also has higher graviton cou- a suitably defined double copy. Here we follow a more plings to the worldline of a massive particle, this poses an direct approach, where the classical interaction potential is immediate obstacle to a double-copy relation. Also a the direct outcome of a path integral. For this purpose, we massive scalar field, for which a double copy of scattering

024021-2 EFFECTIVE ACTION OF DILATON GRAVITY AS THE … PHYS. REV. D 99, 024021 (2019) Z amplitudes exist [31], has linear and quadratic couplings to i i i eℏSeff;YM ¼ eℏSpc;free M ¼ const DAeℏScYM : ð10Þ . However, in a first-order formalism this situation is YM remedied. Defining canonical momenta The (divergent) normalization constant on the right-hand ∂ M ¼ 1 → 0 Lpc uμ a a 2 2 side is chosen such that YM for g and Spc;free is pμ ¼ ¼ m pffiffiffiffiffi − gAμc ; ⇒ m ¼ðp þ gAcÞ ; ð4Þ ∂ μ 2 þ ˜ → 0 u u given by Spc Spc for g . The perturbative expansion in M the coupling g of YM can always be brought into the we find the Hamiltonian and the action in the first-order schematic form formalism   Z Z X∞ X xI Y 2 i 4 CINI 2 2 M ¼ ð2gÞ n dτˆ d lI x ; H ¼ H þ λ½ðp þ gAcÞ − m ; YM iI pc can ℏ SIDI n¼0 I∈Γn iI ð4Þ ¼ μ − ¼ ψ †ψ_ ð Þ Hcan pμu Lpc i ; 5 ð11Þ Z n−1 μ where Γn is the set of N LO trivalent graphs with two S ¼ − dτðpμu − H Þ pc pc external worldlines of the colored point particles, C Z I denotes the color factor associated with graph I, i.e., μ † 2 μ a ¼ − τð μ − ψ ψ_ − λ½ þ 2 μ d p u i p gp Aac functions of the cðτÞ’s, NI the associated kinematic numerator factors and D the propagators 2 b b μ a 2 I þ g Aμc Aac − m Þ: ð6Þ appearing in the graph I. The xI is defined as the number of vertices (bulk and worldline) minus the number of H “ ” H Here, can is the canonical Hamiltonian and pc is the propagators in the trivalent diagram and lI is the number of “Dirac” Hamiltonian where the -shell constraint was bulk vertices. Note that the integral measure along the added using a Lagrange multiplier λðτÞ. Now we have both worldline dτ always comes with a corresponding Lagrange a single and a double gluon coupling to the worldline, multiplier λðτÞ in order to guarantee reparametrization similar to a massive scalar field. Note that λðτÞdτ is invariance. We collect this into a new density notation via reparametrization invariant. We now consider Yang-Mills gauge theory coupled to a dτˆi ≔ dτiλðτiÞ: ð12Þ set of two scalar particles in the worldline description. Notationally we separate the worldlines by placing a tilde The SI finally, denote the symmetry factor of the graph I in on all variables associated to one of the worldlines. the trivalent representation, i.e., the graph has to be drawn Therefore the full action for classical Yang-Mills (cYM) in such a way that it reproduces the color structure CI.We reads will explain our notion of a trivalent representation more precisely in the next section. ¼ þ þ þ ˜ ð Þ ScYM SYM Sgf Spc Spc; 7 The double-copy prescription that we are adopting here then amounts to representing the exponential of the where a gauge fixing part Sgf was added. With this, the effective action of two massive particles coupled to a gluon propagator reads weak-field (post-Minkowskian) expansion of dilaton gravity (dg) as ℏ a b hAμðxÞAν ðyÞi0 ¼ ημνδ Dðx − yÞ; ð8Þ   Z Z ab X∞ X x Y i 2 i I 4 NINI M ¼ ðiκÞ n dτˆ d lI x ; dg ℏ iI ¼0 ∈Γ SIDI where Dðx − yÞ is defined via n I n iI ð13Þ □Dðx − yÞ¼−δðx − yÞ: ð9Þ where κ is the gravitational coupling constant. After that, the effective action in dilaton gravity follows from III. EFFECTIVE ACTION OF COLORED MASSIVE i i ℏSeff;dg ¼ ℏSpc;free M ℏ → 0 PARTICLES AND ITS DOUBLE COPY e e dg. Only classical terms ( ) are retained in Seff;dg in our considerations. At higher order, A. Principle of the double-copy construction CI and NI are required to fulfill the usual BCJ color- The double-copy scheme to be applied here is to first kinematics duality (Jacobi relations). We conclude this a integrate out the gauge field Aμ in order to obtain the section by stressing that our double-copy prescription is effective action Seff;YM for two colored, massive particles proposed at the level of M and not at the level of the with Yang-Mills interactions, i.e., effective action.

024021-3 PLEFKA, STEINHOFF, and WORMSBECHER PHYS. REV. D 99, 024021 (2019)

B. The trivalent representation of the effective action At leading order we have the three graphs of colored particles to next-to-leading order We now compute the effective action for two color-charged massive point particles moving on their worldlines to next-to-leading order, Oðg4Þ. Furthermore, we will clarify what we mean by a trivalent represen- ð Þ tation of a diagram involving worldlines. We will use the 15 short-hand notation

ci ≔ cðτiÞ;pi ≔ pðτiÞ; Yn Dij ≔ DðxðτiÞ − xðτjÞÞ;dτˆ1…n ≔ dτiλðτiÞ: ð14Þ i¼1

Additionally, we will use the tilde notation for the “right” worldline in our diagrams. ð16Þ

ð17Þ

Since the last term vanishes in dimensional regularization, we neglect any graph involving a bubble, as shown above, or any other scaleless integral. Note that we do not mention the mirrored counterparts to every graph with an uneven number of untilded and tilded variables. They can be trivially obtained by replacing untilded and tilded and it is understood that we add them to our final results. At the next-to-leading order we encounter the graphs

ð18Þ

ð19Þ

where we used a delta function in the last step in order to introduce a dummy τ3 integration. This can be understood as pulling apart the gluon-gluon-worldline vertex into two gluon-worldline vertices, i.e., arriving at our definition of a trivalent representation necessary for our proposed double-copy prescription (13). The next graph is

ð20Þ

μ1μ2μ3 where V123 is the color independent part of the Yang-Mills three gluon vertex

024021-4 EFFECTIVE ACTION OF DILATON GRAVITY AS THE … PHYS. REV. D 99, 024021 (2019)

μ μ μ μ μ 1 2 3 μ1μ2 3 3 V123 ¼ η ð∂1 − ∂2 Þþcyclic; ð21Þ R ¼ 4 and G12˜ 3˜ d xD1xD2˜xD3˜x. We proceed in an analogous fashion with

ð22Þ

ð23Þ

ð24Þ

ð25Þ

We also have graphs that are not mirrored counterparts of C. The double copy of the gauge-theory above mentioned graphs, i.e., effective action In order to perform the double copy according to (11) we first need to investigate the independent ð Þ 26 color structures from the previous section. They are given by which we only include implicitly due to their very similar analytic form as previously computed graphs. In addition LO∶ ðc · c˜Þ; ðc · cÞ; ðc˜ · c˜Þ: we neglect graphs that represent quantum corrections, since they are suppressed in the classical effective action. In NLO∶ ðc · c˜Þ2; ðc · c˜Þðc˜ · c˜Þ; ðc˜ · cÞðc · cÞ; ðc · cÞ2; ðc˜ · c˜Þ2; particular we neglect the following type of graphs: abc abc abc abc f cacbcc;f cac˜bc˜c;f c˜acbcc;f c˜ac˜bc˜c; ð28Þ ð27Þ

yielding the backbone to write down the terms in (11).We where the grey blob denotes loop topologies. start with the H-graph

024021-5 PLEFKA, STEINHOFF, and WORMSBECHER PHYS. REV. D 99, 024021 (2019)

ð29Þ where ð33Þ ¼ð ˜ Þ −1 ¼ CH c1 · c2 ;DH D12˜ ; where SH ¼ 1;NH ¼ðp1 · p˜ 2Þ: ð30Þ −1 ¼ ¼ð ˜ Þð ˜ Þ ¼ 2 DV D13˜ D24˜ ;CV c1 · c3 c2 · c4 ;SV ; Next we have the single-worldline (self-interaction) ℏ δðτ1 − τ2Þ NV ¼ðp1 · p˜ 3Þðp2 · p˜ 4Þþ ðp˜ 3 · p˜ 4Þ D-graph 2i λ2 ℏ δðτ˜ − τ˜ Þ þ 3 4 ð Þ ð Þ ˜ p1 · p2 ; 34 2i λ4 and the Y-graph

ð31Þ where ð35Þ ¼ð Þ −1 ¼ CD c1 · c2 ;DD D12; where SD ¼ 2;ND ¼ðp1 · p2Þ: ð32Þ ¼ abc a ˜b ˜c −1 ¼ ¼ 2 CY f c1c2c3;DY G12˜ 3˜ ;SY ; Turning to next-to-leading order contributions, we have the 1 μνρ N ¼ − V p1μp˜ 2νp˜ 3ρ: ð36Þ V-graph Y 2 12˜ 3˜

Next there is the C-graph

ð37Þ

where

¼ð ˜ Þ𘠘 Þ −1 ¼ ¼ 2 CC c1 · c2 c3 · c4 ;DC D12˜ D3˜ 4˜ ;SC ; ℏ δðτ˜ − τ˜ Þ ¼ð ˜ Þð ˜ ˜ Þþ 2 4 ð ˜ Þ ð Þ NC p1 · p2 p3 · p4 ˜ p1 · p3 : 38 i λ4

Note that here we use the symmetry factor of the first topology of the C-graph. This is consistent with our double-copy prescription which states that the correct symmetry factor is the one attributed to the trivalent graph. Next in line is the B-graph

ð39Þ

024021-6 EFFECTIVE ACTION OF DILATON GRAVITY AS THE … PHYS. REV. D 99, 024021 (2019) where Again, the symmetry factor is given by the first topology of the B-graph. ¼ð˜ ˜ Þ𘠘 Þ −1 ¼ ¼ 8 The same constructions apply to the mirrored graphs and CB c1 · c2 c4 · c3 ;DB D1˜ 2˜ D3˜ 4˜ ;SB ; ℏ δðτ˜ − τ˜ Þ the graphs mentioned in (26). Performing the double copy ¼ð˜ ˜ Þð ˜ ˜ Þþ2 4 1 ð ˜ ˜ Þ ð Þ as discussed in Sec. III A yields our prediction for the NB p1 · p2 p3 · p4 ˜ p2 · p3 : 40 i λ4 exponential of the effective action in dilaton gravity

Z Z  i N N N N M¯ ¼ 1 − κ2 dτˆ H H þ dτˆ D D dg ℏ H S D D S D   Z H H Z D D Z  i 2 N N N N N N þ κ4 dτˆ V V þ dτˆ C C þ dτˆ B B ℏ V S D C S D B S D Z V V C C B B 4 i NYNY þ κ dτˆY þðmirroredÞ; ð41Þ ℏ SYDY M¯ where we introduced the bar notation for dg to point out that it is computed via our double-copy prescription. At this point it is necessary to comment on a subtle feature of the above result. An essential technicality in the previous construction was the notion of pulling apart the gluon-gluon-worldline vertex using a delta function, i.e., obtaining a trivalent representation. δð0Þ M¯ Doing so, our double-copy prescription introduces terms in dg. This is a potential hazard but since such terms are of Oðℏ0Þ , they are quantum corrections to Seff;dg and we neglect them for the time being. ℏ Another important property of Seff;dg is that all negative powers of exponentiate, which is a priori not obvious. We check this by taking the logarithm of our double-copy result, obtaining a perturbative expansion of the effective action, Z Z ℏ κ2 ¯ ¯ 2 2 2 S ¼ S þ log M ¼ S − κ dτˆ ˜ ðp1 · p˜ 2Þ D ˜ − dτˆ12ðp1 · p2Þ D12 eff;dg pc;free i dg pc;free 12 12 2 Z   κ4 1 2 þ τˆ μνρ ˜ ˜ 2 d 12˜ 3˜ 2 V12˜ 3˜ p1μp2νp3ρ G12˜ 3˜ Z κ4 þ τˆ ð ˜ Þð ˜ Þð ˜ ˜ Þ 2 d 12˜ 3˜ p1 · p2 p1 · p3 p3 · p2 D12˜ D13˜ Z 4 þ κ dτˆ12˜ 3˜ ðp1 · p˜ 3Þðp1 · p˜ 2Þðp˜ 3 · p˜ 2ÞD12˜ D2˜ 3˜ Z κ4 þ τˆ ð Þð Þð Þ 2 d 123 p1 · p2 p1 · p3 p2 · p3 D12D13 þðmirroredÞþOðℏÞ: ð42Þ

Z In the following we will check whether a direct construc- 2 4 pffiffiffiffiffiffi μ S ¼ − d x −g½R − 2∂μϕ∂ ϕ tion within dilaton gravity reproduces this prediction. dg κ2 þðGHY boundary termÞð43Þ IV. INTERACTION OF POINT MASSES IN DILATON GRAVITY Z 2 4 pffiffiffiffiffiffi μν ρ λ ¼ − d x −g½g ðΓ μλΓ νρ In this section we are performing an analogous analysis κ2 as in the previous section for a system of two massive ρ λ μ worldlines that are interacting in dilaton gravity. − Γ μνΓ ρλÞ − 2∂μϕ∂ ϕ; ð44Þ pffiffiffiffiffiffiffiffiffiffiffi κ ¼ −1 ¼ 32π A. Dilaton gravity where mPl G is the gravitational coupling 1 ϕ The action of dilaton gravity is given by (with Newton constant G and Planck mass mPl), is a real scalar field called the dilaton, R is the usual α αβ Ricci scalar, Γ μν ¼ð∂μgνβ þ ∂νgμβ − ∂βgμνÞg =2 is the 1 Christoffel connection, gμν is the metric, and g ¼ detðgμνÞ. Note that the Gibbons-Hawking-York (GHY) boundary term [45,46] is nonzero also in dimensional regularization for asymp- The worldline action of a point mass (pm) coupled to totically flat spacetimes, leading to (44). gravity and a dilaton is defined by

024021-7 PLEFKA, STEINHOFF, and WORMSBECHER PHYS. REV. D 99, 024021 (2019) Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ μ ν then performing the field redefinitions S ¼ −m dτe gμνu u ; ð45Þ pm   1 μ μ μ μ hμν → hμν − ημν h μ þ 2ϕ with the worldline x ðτÞ, the 4-velocity u ¼ x_ and ϕ, gμν 2 are evaluated at xμðτÞ. We again pass to the first-order  1 ρ 1 ρ σ 1 ρ version þ κ − hμνh ρ þ ημνh ρh σ þ hμρh ν − 2ϕhμν Z 2 8 2  ¼ − τð μ − λ½ −2ϕ μν − 2Þ ð Þ Spm d pμu e g pμpν m : 46 2 ρ þ 2ϕ ημν þ ϕhμνh ρ ; ð53Þ The full action reads 1 μ þ þ þ ˜ ð Þ ϕ → ϕ þ h μ; ð54Þ Sdg Sgf Spm Spm; 47 4 with a gauge fixing part Sgf specified below. Note that we and finally, adding the total derivative do not need an axion field here, which will be important Z  when adding to the point particles [26]. 0 ¼ ¼ 4 ∂ ðð∂ μκÞ ν Þ − ∂ ð μνð∂ κ ÞÞ STD d x μ νh h κ μ h κh ν   B. Weak-field expansion 1 μν σν 1 μλ κν þ κ ∂μðh ð∂νh ÞhσνÞ − ∂μðð∂νh Þhλκh The weak-field approximation is defined as the 4 4  expansion of the full metric around a flat Minkowski 1 1 background, i.e., − ∂ ð μνð∂ Þ ρλÞþ ∂ ð μ νλð∂ σÞÞ 4 μ h λhνρ h 4 μ h νh σhλ ;

gμνðxÞ¼ημν þ κhμνðxÞ; ð48Þ ð55Þ

μν μν μν 2 μλ ν 3 g ðxÞ¼η − κh ðxÞþκ h ðxÞhλ ðxÞþOðκ Þ; ð49Þ to the action. This procedure decouples the point masses on 2 the worldline from the dilaton, pffiffiffiffiffiffi μ κ μν μ 2 −g ¼ 1 þ κh μðxÞ − ðh ðxÞhμνðxÞ − ðh μðxÞÞ Þ Z    2 2 μ μν μν κ μ ρν 3 S ¼ − dτ pμu − λ η − κh þ h ρh pμpν þ Oðκ Þ; ð50Þ pm 2  where κ is our weak-field (or post-Minkowskian) pertur- − m2 þ Oðκ3Þ; ð56Þ bation parameter and hμν is the graviton. In principle we are ready to start the computation of the effective action by integrating out the graviton and dilaton field. Nevertheless, which means we only need to worry about the graviton- at this point one encounters large expressions during dependent part of the field action for the computation of the intermediate steps, e.g., the three-graviton vertex will have classical effective action Seff;dg at next-to-leading order. around 170 terms [47]. However, it is known that one can Furthermore we observe a considerable simplification in achieve a high simplification by performing field redefi- the three-graviton interaction, i.e., it is given by nitions and adding the appropriate gauge fixings [48]. Here, þ þ ð Þ we aim at a field redefinition that removes the coupling of Sdg Sgf STD 57 the dilaton to the worldline and also simplifies the three- Z  4 1 ρ μν κ μ ν ρσ graviton vertex to the square of the Yang-Mills one. Our ¼ d x ∂ρhμν∂ h þ ðhμν∂ ∂ hρσh procedure is given by: 2 4 σ μ ν ρ σ μ ρ ν ρ σ μν First, choosing the following gauge fixing terms to þ 2hμν∂ h ρ∂ h σ − hμν∂ hρ ∂ h σ − hρσ∂ hμν∂ h Oðκ2Þ,  Z ρμ σν 2 − ∂ρ∂σhμνh h Þ þ Oðκ ; ϕÞð58Þ 1 4 pffiffiffiffiffiffi μ S ¼ d x −gf fμ; ð51Þ gf κ2 Z    1 κ κ2 1 1 ¼ 4 ∂ ∂ρ μν þ μαγ νβδ μ ¼ Γμ νσ þ − ð∂ κλÞ μ − ð∂μ κλÞ d x 2 ρhμν h 4 3! V123V123h1μνh2αβh3γδ f νσg 2 4 κh hλ 4 h hκλ · þ Oðκ2; ϕÞ; ð59Þ κ μλ 3 μ κ λ 3 κ μ λ þð∂ h Þhκλ þ ð∂ h κÞh λ − ð∂ h κÞh λ 16 8 μαγ  where V is again the color independent part of the three 3 123 − ð∂λ κ Þ μ ð Þ gluon interaction defined by (21). We also introduced 8 h κ h λ ; 52 fiducial indices 1, 2, 3, on hμν to indicate on which hμν

024021-8 EFFECTIVE ACTION OF DILATON GRAVITY AS THE … PHYS. REV. D 99, 024021 (2019)

μ1μ2μ3 the partial derivatives in V123 are applied. The propagator Again the (divergent) normalization constant on the M ¼ 1 κ → 0 therefore reads right-hand side is chosen such that dg for þ ˜ κ → 0 ℏ and Spm;free is given by Spm Spm for . We also hhμνðxÞhρσðyÞi0 ¼ ημðρησÞνDðx − yÞ; ð60Þ neglect quantum corrections and keep all mirrored graphs i implicit. M where the round brackets on the indices indicate a sym- The relevant classical diagrams for dg to order metrization of unit weight, κ4 read 1 η η ¼ ðη η þ η η Þ ð Þ μðρ σÞν 2 μρ σν μσ ρν : 61 ð63Þ

C. Effective action of gravitating particles to next-to-leading order Similar to the Yang-Mills case in Sec. III B, we compute Seff;dg by integrating out the graviton and scalar fields, Z ð64Þ i i i ð þ þ þ˜ Þ ℏSeff;dg ¼ ℏSpm;free M ¼ D Dϕ ℏ Sdg Sgf Spm Spm e e dg const h e :

ð62Þ

ð65Þ

ð66Þ

ð67Þ

ð68Þ

ð69Þ

024021-9 PLEFKA, STEINHOFF, and WORMSBECHER PHYS. REV. D 99, 024021 (2019)

ð70Þ

ð71Þ

Summarizing, we find upon taking the logarithm that the effective action in dilaton gravity in a graphical notation reads

ð72Þ

Now that we have obtained all classical contributions from v2 κ2ðm þ m˜ Þ r ∼ ; ð74Þ integrating out the graviton and dilaton, we can compare c2 c232πr this result with that from the double copy (41). Going ˜ through the expression term by term we indeed find where r ¼jx − xj, vr is the relative velocity of the two particles, and we have restored the c. The Z weak-field approximation therefore implies small velocities ¯ ¼ þ τ ðψ †ψ_ þ ψ˜ †ψ˜_ Þ ð Þ −1 Seff;dg Seff;dg d i ; 73 in the case of bound binaries. It is convenient to use c as a formal post-Newtonian counting parameter, such that κ ¼ Oðc−1Þ as well as where the last term comes from the difference between S and S . However, the dynamics of the auxiliary μ 0 −1 pc;free pm;free ðu Þ¼ð1; vÞ¼ðOðc Þ; Oðc ÞÞ; ð75Þ field ψ is now decoupled and trivial such that it may be 0 −1 dropped. Therefore we conclude that our double-copy ðpμÞ¼ðE; −pÞ¼ðOðc Þ; Oðc ÞÞ; ð76Þ prescription (13) yields the correct classical effective action up to next-to-leading order in the weak-field expansion. 0 −1 λ ¼ Oðc Þ and ∂t ¼ Oðc Þ. The post-Newtonian expan- sion of the propagator then reads

V. POST-NEWTONIAN EVALUATION Z μ 4 −ikμx OF THE EFFECTIVE ACTION ð Þ¼ d k e D x 4 μ ð2πÞ kμk þ iϵ In this section, we evaluate the effective action in the Z   d3k eik·x ∂2 ∂4 post-Newtonian approximation. After adopting a gauge for ¼ − 1 − t þ t þ … δðtÞ: ð77Þ the worldline parameter, the equations of motion for the ð2πÞ3 k2 k2 k4 Lagrange multipliers λ and the p0 of the particles become algebraic and these variables can be eliminated Recall that the effective action is invariant under repar- from the action. This leads to an action in Hamiltonian ametrizations of the worldline parameters. We may therefore form, which agrees with previous results in scalar-tensor pick a gauge for them. In the post-Newtonian approximation, theory [49], that includes dilaton gravity as a special case. it is useful to fix τ ¼ t ¼ coordinate time. Then the effective The post-Newtonian approximation is a refinement of action is the sum of the free terms Spm;free and the connected the weak-field approximation to bound binaries. The third graphs shown in (72). Most of these graphs are actually Kepler law, or the virial theorem, then tells us that (with vanishing self-interactions, see Appendix B.Thisleavesus equality in the circular orbit case) with (stripping off the overall factors i=ℏ)

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ð78Þ

p2 κ2mm˜ p4 2κ4mm˜ ð2m þ m˜ Þ E ¼ m þ − − þ ð85Þ ð79Þ 2m 16πr 8m3 ð32πrÞ2   κ2 ˜ p2 p˜ 2 3ðp p˜Þ ðn pÞðn p˜ Þ − mm þ − · − · · 32πr m2 m˜ 2 mm˜ mm˜ þ Oðc−6Þ; ð86Þ   1 p2 κ2m˜ ð80Þ λ ¼ 1 − − þ Oðc−4Þ; ð87Þ 2m 2m2 16πr

and substituting these solutions into the action, we arrive at 2 where n ¼ðx − x˜Þ=r,andweused (recall κ ¼ 32πG) Z Z   ddk eik·x 1 Γðd=2 − αÞ x2 α−d=2 S ¼ dt½p · v þ p˜ · v˜ − H; ð88Þ ¼ : ð81Þ eff;dg ð2πÞd ðk2Þα ð4πÞd=2 ΓðαÞ 4 p2 p˜ 2 2Gmm˜ H ¼ m þ m˜ þ þ − Moreover in (78) we anticipated that λ ¼ const þ 2m 2m˜ r −2 Oðc Þ¼E, allowing us to drop time derivatives of λ, E. p4 p˜ 4 2 2 ˜ ð þ ˜ Þ − − þ G mm m m The effective action in the post-Newtonian approximation 3 3 2 8m 8m˜ r finally reads   ˜ p2 p˜ 2 3p p˜ ðn pÞðn p˜ Þ − Gmm þ − · − · ·  2 2 Z r m m˜ mm˜ mm˜ 2 2 2 S ¼ dt p · v − E þ λðE − m − p Þ −6 eff;dg þ Oðc Þ: ð89Þ þ p˜ · v˜ − E˜ þ λ˜ðE˜ 2 − m˜ 2 − p˜ 2Þ   Notice the extra factor of 2 in the Newtonian potential, κ2λλ˜E2E˜ 2 p · p˜ which is due to the dilaton. Throughout this derivation, þ 2 − 4 þ v · v˜ − ðn · vÞðn · v˜Þ 8πr EE˜ higher-order time derivatives (like accelerations) are  removed by inserting the respective equations of motion. κ4λλ˜2E2E˜ 4 κ4λλ˜ 2E˜ 2E4 þ þ þ Oðc−6Þ : ð82Þ This corresponds to a variable redefinition at the considered 2ð4πrÞ2 2ð4πrÞ2 order [50]. [In particular, this justifies the dropping of time derivatives of λ, E and the use of p ¼ mv þ Oðc−3Þ above.] Next, we vary the effective action with respect to λ and E Further note that H is the Hamiltonian describing the to arrive at motion of the binary system in dilaton gravity. We may compare our result to scalar-tensor theory [49]. 0 ¼ E2 − m2 − p2 Scalar-tensor theories (in the so called Einstein frame) are   based on the same field action as dilaton gravity (43). κ2λ˜E2E˜ 2 p · p˜ þ 2 − 4 þ v · v˜ − ðn · vÞðn · v˜Þ However, the source part (45) is more generic in scalar- 8πr EE˜ tensor theories, κ4 λ˜2 2 ˜ 4 þ 2λλ˜ 4 ˜ 2 Z þ E E E E þ Oð −6Þ ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 c ; 83 ¼ − τ μ ν ð4πÞ 2r Spm;st m d gμνu u   1 2 2 1 þ α ϕ þ ðα2 þ β Þϕ2 þ Oðκ3Þ ð Þ κ λλ˜EE˜ × 0 0 0 ; 90 0 ¼ −1 þ 2λE þ þ Oðc−4Þ; ð84Þ 2 2πr with the so-called sensitivities α0 and β0. Comparing to and similar for the tilded variables. Solving iteratively for λ, (45) we find that α0 ¼ 1 and β0 ¼ 0 for dilaton gravity. E, using also p ¼ mv þ Oðc−3Þ, The Hamiltonian for scalar-tensor gravity, following via a

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Legendre transform from the Lagrangian stated in Eq. (3.7) adaptation of the BCJ double copy to classical gravity. This of Ref. [49], reads is an important research direction due to the foreseeable improvements of observations over the p2 p˜ 2 ˜ ˜ p4 p˜ 4 ¼ þ ˜ þ þ − Gmm − − next decade(s), demanding ever more accurate predictions H m m 2 2 ˜ 8 3 8 ˜ 3 for the motion and radiation of compact astrophysical  m m r m m  Gm˜ m˜ 3p2 3p˜ 2 7p · p˜ ðn · pÞðn · p˜ Þ binaries. − þ − − An improvable part of our double-copy procedure is the r 2m2 2m˜ 2 2mm˜ 2mm˜   split of worldline vertices with more than one gluon, using 2 ˜2 2Gmα0m˜ α˜ 0 p p˜ G mm˜ a delta distribution of the worldline parameter. This leads to þ − þ ðm þ m˜ Þ r m m˜ 2r2 singular terms involving δð0Þ in the quantum corrections to 2 ˜ the effective action, which we boldly drop in our classical þ G mm ð α2β˜ þ ˜ α˜ 2β ÞþOð −6Þ ð Þ 2 m 0 0 m 0 0 c ; 91 consideration. While one cannot expect a consistent treat- 2r ment of all quantum corrections in an approach based on ˜ ¼ ð1 þ α α˜ Þ classical worldlines, addressing this problem might still where G G 0 0 . This Hamiltonian agrees with lead to a crucial improvement of our approach. An ours for α0 ¼ 1 and β0 ¼ 0, as expected. Note that for α ¼ 0 ¼ β interesting solution is to integrate out the auxiliary field 0 0 one obtains the result of general relativity. ψ along the worldline. Since its propagator is a step In closing let us compare the complexities of establishing function, the delta distributions used to split worldline the result above with the more traditional approach of vertices can be produced by derivatives in the numerators Ref. [49]. At this low order in perturbation theory it is hard acting on these worldline propagators. This will avoid the to claim a clear superiority of the double-copy approach. terms δð0Þ. However, after the double-copy step, this will We regard our work as a proof of concept, which we expect produce an auxiliary field ψ propagating along the world- to simplify calculations at higher orders dramatically, as it line in the dilaton gravity, too. But it appears possible to is the case in scattering amplitudes. But already at the next- surgically remove this unwanted feature from the result. to-leading order established here, our approach allows one Alternatively, one can possibly give the auxiliary field an to obtain the post-Minkowskian Feynman integrals in a additional property that one desires of the theory, like a simple way, where in a more traditional approach the three- spin. It is interesting to note that step-function propagators graviton vertex contains around 170 terms [47]. along the worldline basically introduce a time ordering. This time ordering is also present in the Wilson loop, which VI. CONCLUSIONS can be used to calculate the potential between a - In the present paper, we proposed an adaption of the BCJ antiquark pair. double copy [1–3] to the classical effective action of binary We may also compare our work to a direct calculation of systems and demonstrated its validity to next-to-leading radiation via equations of motion in [24–26,30], which is order in the weak-field/post-Minkowskian approximation. based on the same classical color charges (Yang-Mills) and This is another application for a BCJ-like double copy point-masses (dilaton gravity) as our work. If we descend beyond (quantum) scattering amplitudes, next to classical from the effective action to the equations of motion through solutions for the field equations [19–21] and weak-field a variation, then derivatives of the propagators may appear. approximations for the radiation from classical binary These derivatives are not part of the numerators which take point-sources [24–26,28–30]. The former application of part in the double copy. Adapting the double copy at the the double copy operates at the level of the gauge field level of equations of motion then faces the challenge of and metric which are gauge-covariant quantities, unlike identifying numerators and (derivatives of) propagators. gauge-invariant scattering amplitudes. Similarly, in the This problem was elegantly solved by Shen [30]: replacing present work, we formulated a double copy between the Yang-Mills numerators by a copy of the color factor effective actions, which are not manifestly gauge indepen- should lead to the corresponding result in bi-adjoint scalar dent either: the actions depend on gauge-dependent canoni- theory, thus providing a prescription to separate numerator cal momenta, and on the dilaton-gravity side the positions and propagator structures. Here we avoided this compli- are gauge dependent, too.2 This provides growing evidence cation by working at the action level. On the other hand, it that the BCJ double copy can be adapted to gauge-covariant is straightforward to calculate the (gauge independent) quantities, which, if true, would liberate it to a considerably radiation emitted by a binary system when working with – larger realm of applications. Furthermore, our work is an equations of motion [24 26,30]. While maybe less straightforward, extending our work to

2 radiation from binaries will be an interesting future The (conservative) effective actions encode the binding direction. Note that the calculation of classical binary (energy levels in a quantum mechanical analog), which are gauge independent. However, the variables (like momenta) interaction potentials presented here was an effective field that are used when the double copy is performed are not gauge theory approach similar to [36]. Following this line of invariant. research [51], the calculation of radiation aided by a double

024021-12 EFFECTIVE ACTION OF DILATON GRAVITY AS THE … PHYS. REV. D 99, 024021 (2019) copy (at the level of the action) could succeed. Past work also suggests that an extension of our work to spinning ðA5Þ point particles [26] or a projection to pure general relativity [14,31] will be possible. But the most important future extension of our present paper will be to demonstrate the double-copy prescription at higher orders. At next-to- μ1μ2μ3 where we used the abbreviation V123 from (21). next-to-leading order in the effective action, the BCJ μ Our definition for the Riemann tensor is R ναβ ¼ color-kinematics duality (Jacobi identity) will start to play μ μ μ λ μ λ ∂αΓ νβ − ∂βΓ να þ Γ λαΓ νβ − Γ λβΓ να, from which the a crucial role in identifying suitable color factors and α μ Ricci tensor Rμν ¼ R μαν and Ricci scalar R ¼ R μ are numeraors for the double copy. As for all work in the area of color-kinematics duality and double copy, it will be obtained through contractions. The Feynman rules for highly interesting to see if our proposal (or a modification dilaton gravity, for our choice of gauge and field redefi- thereof) survives at the next order. For the original BCJ nitions, follow straightforwardly from (59) and (56) as case, the answer to this was so far always positive.

ACKNOWLEDGMENTS ðA6Þ We wish to thank Alessandra Buonanno, Thibault Damour, Radu Roiban, Chia-Hsien Shen and Justin Vines for helpful discussions. The work of J. P. is supported ðA7Þ through funds of Humboldt-University Berlin in the frame- work of the German excellency initiative.

APPENDIX A: CONVENTIONS AND FEYNMAN RULES The signature of spacetime is −2 (“mostly minus”). We ðA8Þ use units such that the speed of light c ¼ 1 and use c−1 as a formal post-Newtonian counting parameter. ¼ TheR Yang-Mills action is the standard one, SYM 1 4 a 2 a b − 4 d xðFμνÞ , with the color convention trðT T Þ¼ ab a δ =2. The field strength Fμν is given by

a a a a a bca b c ½Dμ;Dν¼−igFμνT ⇒ Fμν ¼ ∂μAν − ∂νAμ þ gf AμAν; ðA1Þ ðA9Þ a a a b abc c where Dμ ¼ ∂μ − igAμT , ½T ;T ¼if T and the abc 1 μα νβ να μβ structure constants f are totally antisymmetric here. where Pμναβ ¼ 2 ðη η þ η η Þ. Working in Feynman gauge we have the coordinate space By comparing the Yang-Mills and dilaton-gravity Feynman rules of Yang-Mills theory, Feynman rules, it is clear that the double copy of many diagrams is trivial, since the vertices already show the double-copy structure, except for the vertex (A8).

ðA2Þ APPENDIX B: SELF-INTERACTIONS IN THE POST-NEWTONIAN APPROXIMATION ðA3Þ In this appendix, we wish to discuss the self-interactions, which vanish using the post-Newtonian expansion of the propagator (77). Using the gauge choice τ ¼ t, the simplest self-interaction at leading order reads

ðA4Þ

ðB1Þ

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Z 0 iκ2 d3kdtdt0 eik·½xðtÞ−xðt Þ on the light-cone. Now, the worldlines of massive particles ¼ − ½ ð Þ ð 0Þ2 can not intersect the light cones emanating from them, so 2ℏ ð2πÞ3 p t · p t k2   that these self-interaction should be zero. Similar argu- ∂ ∂ 0 ments apply to the other self-interactions at next-to-leading × 1 þ t t þ … δðt − t0Þ: ðB2Þ k2 order in the effective action

After performing one of the time integrations, the times 0 are identified t ¼ t0 and in the end eik·½xðtÞ−xðt Þ ¼ 1. The k-integral then turns into a scaleless one and vanishes in dimensional regularization. Hence one gets a vanish- ing result at each post-Newtonian order. This result is physically sensible. Indeed, a crucial property of the propagator (in Minkowski signature) is that its support is which all vanish in dimensional regularization.

[1] Z. Bern, J. J. M. Carrasco, and H. Johansson, New relations [12] G. Mogull and D. O’Connell, Overcoming obstacles to for gauge-theory amplitudes, Phys. Rev. D 78, 085011 colour-kinematics duality at two loops, J. High Energy (2008). Phys. 12 (2015) 1. [2] Z. Bern, T. Dennen, Y.-t. Huang, and M. Kiermaier, Gravity [13] S. He, R. Monteiro, and O. Schlotterer, String-inspired BCJ as the square of gauge theory, Phys. Rev. D 82, 065003 numerators for one-loop MHV amplitudes, J. High Energy (2010). Phys. 01 (2016) 171. [3] Z. Bern, J. J. M. Carrasco, and H. Johansson, Perturbative [14] H. Johansson and A. Ochirov, Pure via color- as a Double Copy of Gauge Theory, kinematics duality for fundamental matter, J. High Energy Phys. Rev. Lett. 105, 061602 (2010). Phys. 11 (2015) 046. [4] H. Kawai, D. C. Lewellen, and S. H. H. Tye, A relation [15] Z. Bern, J. J. Carrasco, W.-M. Chen, A. Edison, H. between tree amplitudes of closed and open strings, Johansson, J. Parra-Martinez, R. Roiban, and M. Zeng, Nucl. Phys. B269, 1 (1986). Ultraviolet properties of N ¼ 8 supergravity at five loops, [5] F. Cachazo, S. He, and E. Y. Yuan, Scattering equations and Phys. Rev. D 98, 086021 (2018). matrices: From Einstein to Yang-Mills, DBI and NLSM, [16] A. Anastasiou, L. Borsten, M. J. Duff, M. J. Hughes, A. J. High Energy Phys. 07 (2015) 149. Marrani, S. Nagy, and M. Zoccali, Twin supergravities from [6] Z. Bern, C. Boucher-Veronneau, and H. Johansson, N ≥ 4 Yang-Mills theory squared, Phys. Rev. D 96, 026013 supergravity amplitudes from gauge theory at one loop, (2017). Phys. Rev. D 84, 105035 (2011). [17] A. Anastasiou, L. Borsten, M. J. Duff, A. Marrani, S. Nagy, [7] Z. Bern, J. J. M. Carrasco, L. J. Dixon, H. Johansson, and M. Zoccali, Are all supergravity theories Yang-Mills and R. Roiban, Simplifying multiloop integrands and ultra- squared?, Nucl. Phys. B934, 606 (2018). violet divergences of gauge theory and gravity amplitudes, [18] C. Cheung, C.-H. Shen, and C. Wen, Unifying relations Phys. Rev. D 85, 105014 (2012). for scattering amplitudes, J. High Energy Phys. 02 (2018) [8] J. J. M. Carrasco, M. Chiodaroli, M. Gunaydin, and R. 095. Roiban, One-loop four-point amplitudes in pure and matter- [19] R. Monteiro, D. O’Connell, and C. D. White, Black holes coupled N ≤ 4 supergravity, J. High Energy Phys. 03 and the double copy, J. High Energy Phys. 12 (2014) 056. (2013) 056. [20] A. Luna, R. Monteiro, D. O’Connell, and C. D. White, The [9] Z. Bern, S. Davies, T. Dennen, Y.-t. Huang, and J. classical double copy for Taub NUT spacetime, Phys. Lett. Nohle, Color-kinematics duality for pure Yang-Mills and B 750, 272 (2015). gravity at one and two loops, Phys. Rev. D 92, 045041 [21] A. Luna, R. Monteiro, I. Nicholson, A. Ochirov, D. (2015). O’Connell, N. Westerberg, and C. D. White, Perturbative [10] M. Chiodaroli, M. Gunaydin, H. Johansson, and R. Roiban, spacetimes from Yang-Mills theory, J. High Energy Phys. Scattering amplitudes in N ¼ 2 Maxwell-Einstein and 04 (2017) 069. Yang-Mills/Einstein supergravity, J. High Energy Phys. [22] A. Luna, R. Monteiro, I. Nicholson, D. O’Connell, and C. 01 (2015) 081. D. White, The double copy: Bremsstrahlung and accelerat- [11] M. Chiodaroli, M. Gunaydin, H. Johansson, and R. Roiban, ing black holes, J. High Energy Phys. 06 (2016) 023. Complete Construction of Magical, Symmetric and Homo- [23] B. P. Abbott et al. (Virgo and LIGO Scientific Collabora- geneous N ¼ 2 Supergravities as Double Copies of Gauge tions), Observation of Gravitational Waves from a Binary Theories, Phys. Rev. Lett. 117, 011603 (2016). Merger, Phys. Rev. Lett. 116, 061102 (2016).

024021-14 EFFECTIVE ACTION OF DILATON GRAVITY AS THE … PHYS. REV. D 99, 024021 (2019)

[24] W. D. Goldberger and A. K. Ridgway, Radiation and the [37] Y. Iwasaki, Quantum theory of gravitation vs. classical classical double copy for color charges, Phys. Rev. D 95, theory—fourth-order potential, Prog. Theor. Phys. 46, 1587 125010 (2017). (1971). [25] W. D. Goldberger and A. K. Ridgway, Bound states and the [38] M. J. Duff, Quantum tree graphs and the Schwarzschild classical double copy, Phys. Rev. D 97, 085019 (2018). solution, Phys. Rev. D 7, 2317 (1973). [26] W. D. Goldberger, J. Li, and S. G. Prabhu, Spinning [39] B. R. Holstein and J. F. Donoghue, and particles, axion radiation, and the classical double copy, Quantum Loops, Phys. Rev. Lett. 93, 201602 (2004). Phys. Rev. D 97, 105018 (2018). [40] D. Neill and I. Z. Rothstein, Classical space-times from the [27] A. P. Balachandran, P. Salomonson, B.-S. Skagerstam, and S matrix, Nucl. Phys. B877, 177 (2013). J.-O. Winnberg, Classical description of particle interacting [41] N. E. J. Bjerrum-Bohr, J. F. Donoghue, and P. Vanhove, with non-Abelian gauge field, Phys. Rev. D 15, 2308 On-shell techniques and universal results in quantum (1977). gravity, J. High Energy Phys. 02 (2014) 111. [28] W. D. Goldberger, S. G. Prabhu, and J. O. Thompson, [42] N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Classical gluon and graviton radiation from the bi-adjoint Plante, and P. Vanhove, General Relativity from Scattering scalar double copy, Phys. Rev. D 96, 065009 (2017). Amplitudes, Phys. Rev. Lett. 121, 171601 (2018). [29] D. Chester, Radiative double copy for Einstein-Yang-Mills [43] T. Damour, Gravitational scattering, post-Minkowskian theory, Phys. Rev. D 97, 084025 (2018). approximation and effective one-body theory, Phys. Rev. [30] C.-H. Shen, Gravitational Radiation from color-kinematics D 94, 104015 (2016). duality, J. High Energy Phys. 11 (2018) 162. [44] T. Damour, High-energy gravitational scattering and the [31] A. Luna, I. Nicholson, D. O’Connell, and C. D. White, general relativistic two-body problem, Phys. Rev. D 97, Inelastic black hole scattering from charged scalar ampli- 044038 (2018). tudes, J. High Energy Phys. 03 (2018) 044. [45] G. W. Gibbons and S. W. Hawking, Action integrals and [32] T. Damour, P. Jaranowski, and G. Schäfer, Nonlocal-in-time partition functions in quantum gravity, Phys. Rev. D 15, action for the fourth post-Newtonian conservative dynamics 2752 (1977). of two-body systems, Phys. Rev. D 89, 064058 (2014). [46] J. W. York, Jr., Role of Conformal Three Geometry in [33] L. Bernard, L. Blanchet, A. Boh´e, G. Faye, and S. Marsat, the Dynamics of Gravitation, Phys. Rev. Lett. 28, 1082 Energy and periastron advance of compact binaries on (1972). circular orbits at the fourth post-Newtonian order, Phys. [47] B. S. DeWitt, Quantum theory of gravity. 3. Applications of Rev. D 95, 044026 (2017). the covariant theory, Phys. Rev. 162, 1239 (1967). [34] T. Marchand, L. Bernard, L. Blanchet, and G. Faye, [48] Z. Bern and A. K. Grant, Perturbative gravity from QCD Ambiguity-free completion of the equations of motion of amplitudes, Phys. Lett. B 457, 23 (1999). compact binary systems at the fourth post-Newtonian order, [49] T. Damour and G. Esposito-Farese, Tensor multiscalar Phys. Rev. D 97, 044023 (2018). theories of gravitation, Classical Quantum Gravity 9, [35] S. Foffa, P. Mastrolia, R. Sturani, and C. Sturm, Effective 2093 (1992). field theory approach to the gravitational two-body dynam- [50] T. Damour and G. Schäfer, Redefinition of position vari- ics, at fourth post-Newtonian order and quintic in the ables and the reduction of higher order Lagrangians, Newton constant, Phys. Rev. D 95, 104009 (2017). J. Math. Phys. (N.Y.) 32, 127 (1991). [36] W. D. Goldberger and I. Z. Rothstein, An effective field [51] W. D. Goldberger and A. Ross, Gravitational radiative theory of gravity for extended objects, Phys. Rev. D 73, corrections from effective field theory, Phys. Rev. D 81, 104029 (2006). 124015 (2010).

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