Radiation Reaction from Soft Theorems
Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo, Gabriele Veneziano
aa r X i v : . [ h e p - t h ] J a n CERN-TH-2021-008NORDITA 2021-001QMUL-PH-21-03UUITP-03/21
Radiation Reaction from Soft Theorems
Paolo Di Vecchia a,b , Carlo Heissenberg b,c , Rodolfo Russo d ,Gabriele Veneziano e,f a The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark b NORDITA, KTH Royal Institute of Technology and Stockholm University,Roslagstullsbacken 23, SE-10691 Stockholm, Sweden c Department of Physics and Astronomy, Uppsala University,Box 516, SE-75120 Uppsala, Sweden d Queen Mary University of London, Mile End Road,E1 4NS London, United Kingdom e Theory Department, CERN, CH-1211 Geneva 23, Switzerland f Coll`ege de France, 11 place M. Berthelot, 75005 Paris, France
Abstract
Radiation reaction (RR) terms at the third post-Minkowskian (3PM) or-der have recently been found to be instrumental in restoring smooth continu-ity between the non-relativistic, relativistic, and ultra-relativistic (includingthe massless) regimes. Here we propose a new and intriguing connection be-tween RR and soft (bremsstrahlung) theorems which short-circuits the moreinvolved conventional loop computations. Although first noticed in the con-text of the maximally supersymmetric theory, unitarity and analyticity ar-guments support the general validity of this 3PM-order connection that weapply, in particular, to Einstein’s gravity and to its Jordan-Brans-Dicke ex-tension. In the former case we find full agreement with a recent result byDamour obtained through a very different reasoning.
Introduction
The gravitational scattering of classical objects at large impact parameter b is rel-evant for the study of the inspiral phase of black-hole binaries since it can be usedto determine the parameters of the Effective-One-Body description (see [1] and ref-erences therein). For this reason, gravitational scattering has been at the centreof renewed attention and has been recently investigated using a variety of tech-niques, including the use of quantum field theory (QFT) amplitudes to extract therelevant classical physics [2–23]. Here we will focus in particular on the eikonalapproach [24–27], where the classical gravitational dynamics is derived from stan-dard QFT amplitudes by focusing on the terms that exponentiate in the eikonalphase e iδ . The Post-Minkowskian (PM) expansions writes δ as a perturbative se-ries in the Newton constant G at large values of b and the state-of-the-art resultsdetermine the real part of the 3PM ( i.e. δ (or the closely re-lated scattering angle) and to some extent the imaginary part, both in standardGR [10, 13, 21, 28–30] and various supersymmetric generalisations [31–34].In this letter we expand on the approach discussed in [28, 34] where the relationbetween the real and the imaginary part of δ was used to derive the 3PM scat-tering angle in the ultrarelativistic limit and to show that it is a universal featureof all gravitational theories in the two derivative approximation. Furthermore, itwas shown in [34] for N = 8 supergravity that taking into account the full softregion in the loop integrals was crucial to obtain a smooth interpolation betweenthe behaviour of δ in the non-relativistic, i.e. Post-Newtonian (PN), regime andthe ultrarelativistic (or massless) one. The additional contributions coming fromthe full soft region had the feature of contributing half-integer terms in the PNexpansion and were therefore interpreted as radiation-reaction (RR) contributions.This connection was further confirmed in [30] by Damour, who used a linear re-sponse relation earlier derived in [35] to connect these new RR terms to the loss ofangular momentum in the collision. In this way the result of [34] was extended tothe case of General Relativity [30].In this paper we argue that there is actually a direct relation between the RRand the much studied soft-bremsstrahlung limits. We claim that the real part ofthe RR eikonal at 3PM (indicated by Re 2 δ ( rr )2 ) is simply related to the infrareddivergent contribution of its imaginary part (Im 2 δ ). This relation holds at allenergies and reads lim ǫ → Re 2 δ ( rr )2 = − lim ǫ → [ πǫ (Im 2 δ )] , (1.1)where, as usual, ǫ = − D is the dimensional regularisation parameter. On theother hand, there is a simple connection (see e.g. [36]) between the infrared di-vergent imaginary part of δ and the so-called zero-frequency limit [37] of thebremsstrahlung spectrum reading:lim ǫ → [ − ǫ (Im 2 δ )] = dE rad ~ dω ( ω → ⇒ lim ǫ → Re 2 δ ( rr )2 = π ~ dE rad dω ( ω → , (1.2)1o that, in the end, RR gets directly related to soft bremsstrahlung. We stress thatall (massless) particles can contribute to the r.h.s. of (1.2) and therefore to the RR.This result was first noticed in the N = 8 supergravity setup of [33, 38] byusing the results of [34, 39] where the full 3PM eikonal is derived by a direct com-putation of the 2-loop amplitude describing the scattering of two supersymmetricmassive particle. Here we give an interpretation of this connection and conjectureits general validity in gravity theories at the 3PM level (the first non trivial one)by reconstructing the infrared divergent part of Im 2 δ from the three-body discon-tinuity involving the two massive particles and a massless particle. The buildingblock is of course the 2 → N = 8 case, one needs to consider, in addition tothe graviton, the contributions of the relevant vectors and scalar fields (includingthe dilaton). Once all massless particles that can appear in the three-particle cut aretaken into account, one obtains (5.17) which, as already mentioned, satisfies (1.1).The basic idea underlying all cases is that the calculation of Im 2 δ from sewingtree-level, on shell, inelastic amplitudes is far simpler than the derivation of the fulltwo-loop elastic amplitude even when focusing on just the classical contributions.Both for GR and for N = 8, the infrared divergent piece of δ can be equivalentlyobtained exploiting the exponentiation of infrared divergences in momentum spacefor the elastic amplitude itself (details will be presented elsewhere). The argumentssupporting (1.1) appear to be valid within a large class of gravitational theories andso this equation provides a direct, general way to calculate the RR contributionsat the 3PM level. It remains to be seen whether this approach can be generalized,and in which form, beyond 3PM.The paper is organized as follows. In Sect. 2 we introduce our kinematical set-up for the relevant elastic (2 →
2) and inelastic (2 →
3) processes and discuss thestandard soft limit of the latter in momentum space. In Sect. 3 we present theempirical connection between Re 2 δ ( rr )2 and the IR divergent part of Im 2 δ in themaximally supersymmetric case. Using unitarity and analyticity of the scatteringamplitude, we provide arguments in favour of its general validity. We also outlinethe logic of the calculations that follow. In Sect. 4 we transform the soft-limit resultsof Sect. 2 to impact-parameter space in the large- b limit. In Sect. 5 we use these tocompute the divergent part of Im 2 δ and, through our connection, the RR termsin Re 2 δ . This is first done for the case of N = 8 supergravity, where we recoverthe result of [34], and then for Einstein’s gravity, reproducing the result of [30], andfor Jordan-Brans-Dicke theory. 2 Soft Amplitudes in Momentum Space
Let us start by better defining the processes under consideration. We shall beinterested in the scattering of two massive scalar particles in D = 4 − ǫ dimensions,with or without the additional emission of a soft massless quantum. For GR, wethus consider minimally coupled scalars with masses m , m in 4 − ǫ dimensions.For N = 8 supergravity, that can be obtained by compactifying six directions in ten-dimensional type II supergravity, we instead choose incoming Kaluza–Klein (KK)scalars whose (10 − ǫ )-dimensional momenta read as follows: P = ( p ; 0 , , , , , m ) , P = ( p ; 0 , , , , m sin φ, m cos φ ) , (2.1)where the last six entries refer to the compact KK directions and provide p , p with the desired effective masses m , m in 4 − ǫ dimensions. The angle φ thusdescribes the relative orientation between the KK momenta,We work in a centre-of-mass frame and for our purposes it is convenient to regardthe amplitudes as functions of ¯ p , encoding the classical momentum of the massiveparticles, the transferred momentum q (which is related to the impact parameterafter Fourier transform) and the emitted momentum k . We thus parametrise themomenta of the incoming states as follows, p = ( E , ~p ) = ¯ p − aq + ck , ¯ p = ( E , , . . . , , ¯ p ) ,p = ( E , − ~p ) = ¯ p + aq − ck , ¯ p = ( E , , . . . , , − ¯ p ) , (2.2)while the outgoing states are a soft particle of momentum k and massive stateswith momenta k = − ¯ p − (1 − a ) q − ck , k = − ¯ p + (1 − a ) q − (1 − c ) k . (2.3)We singled out the direction of the classical momentum ¯ p , while q is non-trivialonly along the 2 − ǫ space directions orthogonal to ¯ p i . In the elastic case of course k = 0 = c and we have a = 1 /
2. For the inelastic amplitudes one can fix a and c by imposing the on-shell conditions and using ¯ p i q = 0, but we will not need theirexplicit expression in what follows.We shall now collect the tree-level amplitudes that will enter our calculation ofIm 2 δ via unitarity, focusing for the most part on N = 8 and commenting alongthe way on small amendments that are needed to obtain the GR amplitudes.The simplest building block for our analysis of N = 8 supergravity is the elastictree-level amplitude A tree ≃ − πGm m ( σ − cos φ ) t , with σ = − p p m m , (2.4)where we retained only the terms with the pole at t = − q = 0, since we restrict ourattention to long-range effects. When φ = π , the KK momenta are along orthogonal We treat all vectors as formally ingoing. t = 0 corresponds to the exchange of thegraviton and of the dilaton that are coupled universally to all massive states withthe following three-point on-shell amplitudes in D = 4: A µν = − iκ ( p µj k νj + p νj k µj ) , A dil = − iκ √ m j , (2.5)with j = 1 , κ = √ πG . Using the vertices (2.5) and standard propagators,the graviton and the dilaton exchanges yield A grtree ≃ − πGm m (2 σ − t , A diltree ≃ − πGm m t . (2.6)Their sum reproduces (2.4) for φ = π . For generic φ , in addition to the couplingsmentioned above, we also need to consider massless vectors and scalars coming fromthe KK compactification of the ten dimensional graviton. We have a scalar and avector whose three-point amplitudes involving the massive fields are A µ = − iκm √ p − k ) µ , A = − iκ m ,A µ = − iκm √ p − k ) µ cos φ , A = − iκ m cos φ . (2.7)Including also the contribution of these states one can reproduce the tree-levelamplitude (2.4) for φ = 0, which provides a useful cross-check for the normalizationof the three-point amplitudes. The particle with mass m couples to another vectorand another scalar with a strength depending to the other component of the KKmomentum, B µ = − iκm √ p − k ) µ sin φ , B = − iκ m sin φ . (2.8)There is also an extra scalar related to the off-diagonal components of the inter-nal metric whose coupling is proportional to cos φ sin φ ; here we will not use thiscoupling as we will mainly focus on the cases φ = 0 and φ = π .Let us now move to the inelastic, 2 → k − for k →
0. It is given by the product of the elastic tree-level amplitude times a softfactor. For instance, the leading term for the emission of a soft graviton is [40]: A µν ≃ κ (cid:18) p µ p ν p k + k µ k ν k k + p µ p ν p k + k µ k ν k k (cid:19) A tree , (2.9)while in the case of the dilaton one finds [41] A dil ≃ − κ √ (cid:18) m p k + m k k + m p k + m k k (cid:19) A tree . (2.10)We now use (2.2) and (2.3) and keep the leading terms in the soft limit k → classical contributions, which are captured by the linearterms in the q → A µν ≃ κ (cid:20)(cid:18) ¯ p µ ¯ p ν (¯ p k ) − ¯ p µ ¯ p ν (¯ p k ) (cid:19) ( qk ) − ¯ p µ q ν + ¯ p ν q µ (¯ p k ) + ¯ p µ q ν + ¯ p ν q µ (¯ p k ) (cid:21) A tree (2.11)4or the graviton and A dil ≃ κ √ (cid:18) m ( qk )(¯ p k ) − m ( qk )(¯ p k ) (cid:19) A tree (2.12)for the dilaton.From now on we focus for simplicity on the case φ = π and so only the first lineof (2.7) is non-trivial; together with the contribution of (2.8) we need to considerthe emission of the two vectors and of the two scalars. For the soft amplitudes wefind: A µ ≃ κm √ (cid:18) ¯ p µ ( qk )(¯ p k ) − q µ ¯ p k (cid:19) A tree , B µ ≃ κm √ (cid:18) − ¯ p µ ( qk )(¯ p k ) + q µ ¯ p k (cid:19) A tree , (2.13) A ≃ κm ( qk )(¯ p k ) A tree , B ≃ κm ( qk )(¯ p k ) A tree . (2.14) In this section we briefly present our arguments for the validity, at two-loop leveland for generic gravity theories, of the relation (1.1). We leave a more detaileddiscussion to a longer paper [39].Our starting point is an empirical observation made in the context of a recentcalculation in N = 8 supergravity [34] whose set-up has been recalled in the previoussection. An interesting outcome of that calculation (made for cos φ = 0) was theidentification of a radiation-reaction contribution to the real part of the (two loop)eikonal phase, given byRe 2 δ ( rr )2 = 16 G m m σ ~ b ( σ − " σ + σ ( σ − σ − cosh − ( σ ) + O ( ǫ ) . (3.1)This contribution emerges from the inclusion of radiation modes in the loop integralsand gives rise to half-integer-PN corrections to the deflection angle.Considering the full massive N = 8 result [39], we then noticed a simple relationbetween the contribution in eq. (3.1) and two terms appearing in the imaginary partof the same eikonal phase so that, in the full expression for δ ( rr )2 , there are threeterms that appear in the following combination: (cid:20) iπ (cid:18) − ǫ + log( σ − (cid:19)(cid:21) Re 2 δ ( rr )2 . (3.2)The two imaginary contributions to 2 δ ( rr )2 that appear in (3.2) are an IR-singularterm, which captures the full contribution proportional to ǫ − , and a log( σ − σ = ± δ (let us recallthat Im 2 δ contains just the inelastic contribution to the cut [28]). Furthermore,using real-analyticity of the amplitude forces the log( σ −
1) to appear in δ aslog(1 − σ ) = log( σ − − iπ yielding precisely the analytic structure of (3.2).Combining these two observations, which are based purely on unitarity, analyticityand crossing symmetry, we are led to conjecture the validity of (1.1) independentlyof the specific theory under consideration.As anticipated, this relation opens the way to a much simpler calculation of RReffects since it trades the computation of Re 2 δ ( rr )2 to that of the IR-divergent partof Im 2 δ . In the following sections we will carry out this calculation both for thesupersymmetric case at hand, for pure gravity where we shall recover a recent resultby Damour [30], and for the scalar-tensor theory of Jordan-Brans-Dicke.For the purpose of computing the IR-divergent piece in Im 2 δ , one can focuson the leading O ( k − ) term in the soft expansion of the inelastic amplitudes givenin Sect 2. This allows us to factor out, for each specific theory, the correspondingelastic amplitude. Next, and in this order, one has to take the leading term in asmall- q expansion so as to get the sought-for classical contribution. In terms of theimpact parameter b which will be introduced in (4.1), the small- q limit is equivalentto an expansion for large values of b . Since the soft factor is linear in q (it goesto zero at zero scattering angle), and the tree amplitude has a q − singularity, theresult for the inelastic amplitude is (modulo ǫ dependence) of O ( b − ) and thus ofthe desired O ( b − ) in Im 2 δ . b -space We now start from the momentum space soft amplitudes given in Sect. 2 and go toimpact parameter space using for a generic amplitude the notation˜ A ( b ) = Z d − ǫ q (2 π ) − ǫ A ( q )4 m m √ σ − e ib · q . (4.1)We can now simply replace the factors of q j in the numerators of the various am-plitudes A by the derivative − i ∂∂b j and then perform the Fourier transform wherethe q -dependence appears only in A tree . Starting from the N = 8 elastic tree-levelamplitude with φ = π , given, up to analytic terms as q →
0, by A tree = 8 πβ ( σ ) m m q , β ( σ ) = 4 Gm m σ , (4.2)6he leading eikonal takes the form2 δ = − β ( σ ) Γ(1 − ǫ )( πb ) ǫ ǫ ~ √ σ − ⇒ − i ∂∂b j δ = 2 i Γ(1 − ǫ ) b j ( πb ) ǫ b ~ √ σ − β ( σ ) . (4.3)As clear from (2.6), one can move from N = 8 to the case of pure GR simply byreplacing the prefactor β ( σ ) by β GR ( σ ) = 2 Gm m (2 σ − . (4.4)We then obtain the following result for the classical part of the soft gravitonand soft dilaton amplitudes in impact parameter space˜ A µν ( σ, b, k ) ≃ i κβ ( σ )( πb ) ǫ b √ σ − × (cid:20) ( kb ) (cid:18) ¯ p µ ¯ p ν (¯ p k ) − ¯ p µ ¯ p ν (¯ p k ) (cid:19) − ¯ p µ b ν + ¯ p ν b µ (¯ p k ) + ¯ p µ b ν + ¯ p ν b µ (¯ p k ) (cid:21) , (4.5)˜ A dil ( σ, b, k ) ≃ i κβ ( σ )( πb ) ǫ p σ −
1) ( kb ) b (cid:20) m (¯ p k ) − m (¯ p k ) (cid:21) , (4.6)where we approximated the factor of Γ(1 − ǫ ) in (4.3) to 1 as we are interested inthe D → b ǫ .Having obtained Eqs. (4.5), (4.6) with the appropriate normalization, we followthe same procedure to go over to b -space for the other fields relevant to the N = 8analysis. For the two vectors we obtain˜ A µ ≃ i √ κm β ( σ )( πb ) ǫ b √ σ − (cid:20) ( kb )¯ p µ (¯ p k ) − b µ ¯ p k (cid:21) , (4.7)˜ B µ ≃ − i √ κm β ( σ )( πb ) ǫ b √ σ − (cid:20) ( kb )¯ p µ (¯ p k ) − b µ ¯ p k (cid:21) , (4.8)while for the two scalars we get˜ A ≃ i κm β ( σ )( πb ) ǫ b √ σ − kb )(¯ p k ) , ˜ B ≃ i κm β ( σ )( πb ) ǫ b √ σ − kb )(¯ p k ) . (4.9)Note that all our soft amplitudes are homogeneous functions of ω and b of degree − − ǫ , respectively. Motivated by the discussion of Sect. 3 and armed with the results of the Sect. 4,we now turn to the calculation of the infrared divergent part of Im 2 δ from the7hree-particle unitarity cut. Indeed the unitarity convolution in momentum spacediagonalizes in impact parameter space giving (see e.g. [42])2Im 2 δ = X i Z d D − ~k | ~k | (2 π ) D − | ˜ A i | , (5.1)where the sum is over each massless state in the theory under consideration. Forspin-one and spin-two particles this also includes a sum over helicities. Instead ofseparating different helicity contributions, we use the fact that all the 2 → i.e. η µν for the vectors and ( η µρ η νσ + η µσ η νρ − η µν η ρσ ) for the graviton.Equation (5.1) implies that β ( σ ) always factors out of the integral over ~k . Inspherical coordinates the latter splits into an integral over the modulus | ~k | and oneover the angles defined by the following parametrisation of the vector ~k : ~k = | ~k | (sin θ cos ϕ, sin θ sin ϕ, cos θ ) , ( kb ) = −| ~k | b sin θ cos ϕ , (5.2)that implies (¯ p k ) = | ~k | ( E − ¯ p cos θ ) , (¯ p k ) = | ~k | ( E + ¯ p cos θ ) , (5.3)where we have taken b in (5.2) along the x axis. It is clear that the integral over | ~k | = ~ ω in (5.1) factorises together with an ǫ -dependent power of b to give δ ∼ Z dωω ω − ǫ ( b ) − ǫ ∼ ( b ) − ǫ Z dωω ( ωb ) − ǫ (5.4)where the factor ( b ) − ǫ is precisely the one expected (also on dimensional grounds)to appear in δ . On the other hand, the integral over ω produces a ǫ divergence inthe particular combination: Z dωω ( ωb ) − ǫ = − ǫ ( ωb ) − ǫ = − ǫ + log ωb + O ( ǫ ) (5.5)where ωb is an appropriate upper limit on the classical dimensionless quantity ωb .To determine ωb one can argue as follows. By energy conservation: ~ ω = ∆ E + ∆ E (5.6)where ∆ E i is the energy loss for the i th particle. On the other hand, in order for thespatial components of the momentum transfers q i = − ( p i + k i ) to provide a classical We need to keep D = 4 − ǫ only for the integral over | ~k | while the integration over the angularvariables can be done for ǫ = 0, so that effectively d D − ~k = | ~k | − ǫ d | ~k | sin θ dθ dϕ . ~ /b ≪ | ~p i | . But then we can estimate (5.6)by using (for on-shell particles):∆ E i . | ~p i | E i | ∆ ~p i | ( i = 1 , . (5.7)Combining (5.6) and (5.7) we arrive at ωb . | ~p | E + | ~p | E . (5.8)Using now the following (centre-of-mass) expressions,¯ p ≃ | ~p | = m m √ σ − p m + m + 2 m m σ ,E = m m + σm p m + m + 2 m m σ , E = m m + σm p m + m + 2 m m σ , (5.9)we find: ωb ∼ √ σ − (cid:18) m m + σm + m m + σm (cid:19) = √ σ − O ( σ − . (5.10)Therefore, inserting this result in (5.5) and using the real-analyticity argument men-tioned in Sect. 3, precisely the combination appearing in (3.2) is indeed recovered.This is the essence of our argument for conjecturing (3.2) as a general connectionbetween RR and soft limits. The rest of this section provides examples and nontrivial tests of such a connection. N = 8 Supergravity
We evaluate separately the O ( ǫ − ) contribution to (5.1) for each massless state: thegraviton, the dilaton, two vectors and two scalars coupling to the particle of mass m and other two vectors and two scalars coupling to the particle of mass m . Wefirst start from the dilaton contribution. By using (4.6) in (5.1) we obtain(Im 2 δ ) dil ≃ κ β ( σ )4 b ( σ − Z d | ~k || ~k | − ǫ − π ) Z − dx π (1 − x ) (cid:20) m ( E − ¯ px ) − m ( E + ¯ px ) (cid:21) , (5.11)where x = cos θ . The extra factor of π sin θ = π (1 − x ) in the integrand followsfrom the integration over the angle ϕ . As already mentioned the integral over | ~k | factorises out of the whole integral and provides the sought for ǫ − factor. Finally,by using (5.9), we express everything in terms of σ introduced in (2.4). Then,using (5.9) in (5.11) and performing the integral over x , we obtain(Im 2 δ ) dil ( σ, b ) ≃ − ǫ Gβ ( σ ) π ~ b ( σ − " σ + 23 − σ ( σ − cosh − ( σ ) . (5.12)9ote that the final result depends on the masses only through σ even if the integranddepends on m , m and σ separately. The term with the factor of cosh − ( σ ) emergesfrom the cross-product of the square in (5.11), while the other terms yield onlyrational contributions in σ .For the graviton’s contribution, using (4.5) in (5.1), we obtain(Im 2 δ ) gr ( σ, b ) ≃ − κ β ( σ )2 b ( σ − (cid:18) − ǫ π ) (cid:19) π Z − dx × ( (cid:20) m ( E − ¯ px ) + m ( E + ¯ px ) − m m σ ( E − ¯ px )( E + ¯ px ) (cid:21) (5.13) − − x (cid:20) m ( E − ¯ px ) + m ( E + ¯ px ) − m m (2 σ − E − ¯ px ) ( E + ¯ px ) (cid:21) ) . The integral over x is again elementary . In terms of the variable σ we obtain:(Im 2 δ ) gr ( σ, b ) ≃ − ǫ Gβ ( σ ) π ~ b ( σ − " − σ − σ (3 − σ )( σ − cosh − ( σ ) . (5.14)Following the same procedure for the contribution of the two vectors in (2.13)we get (Im 2 δ ) vec ( σ, b ) ≃ − ǫ Gβ ( σ ) π ~ b ( σ − (cid:20)
83 ( σ − (cid:21) (5.15)and for the sum of the two scalars in (2.14) we obtain(Im 2 δ ) sca ( σ, b ) ≃ − ǫ Gβ ( σ ) π ~ b ( σ − (cid:20)
23 ( σ − (cid:21) . (5.16)In the last two types of contributions the soft particles are attached to the samemassive state, so there are no terms in the integrand with the structure appearingin the cross term of (5.11) and hence no factors of cosh − ( σ ) in the final result. Alsothe graviton and the dilaton results contain contributions of this type correspondingto the terms in the integrands which depend only on E or E . In the N = 8 setupthese contributions cancel when summing over all soft particles. Notice also thatthe static limit σ → N = 8 eikonal or deflectionangle is due to the vectors and the scalars in (2.7) and (2.8). Surprisingly, it turns out to be the same as the integral appearing in Eq. (4.4) of [30] and thusreproduces exactly the function I in (4.7) of that reference.
10y summing the contributions (5.12)–(5.17), we get the following result for theinfrared divergent part of the three-particle discontinuity in N = 8 supergravitywith φ = π (Im2 δ ) ≃ − Gβ ( σ ) π ~ b ǫ ( σ − " σ + σ ( σ − σ − cosh − ( σ ) (5.17)and we can check that it is consistent with (1.1) and (3.1).Further checks of the relation (1.1) could be performed by extending the sameanalysis to the case of cos φ = 0 or to supergravity theories with 0 < N < The calculation in pure GR follows exactly the same steps with only the contributionof the graviton and yields again the result in Eq. (5.14) just with the prefactor( β GR ( σ )) in place of β ( σ ). Then, assuming that Eq. (1.1) is also valid in GR, weget (Re 2 δ ( rr )2 ) GR ( σ, b ) = G ( β GR ( σ )) ~ b ( σ − " − σ − σ (3 − σ )( σ − cosh − ( σ ) (5.18)and, from it, we obtain the deflection angle( χ ( rr )3 ) GR = − ~ | ~p | ∂ Re2 δ ( rr )2 ∂b = G ( β GR ( σ )) | ~p | b ( σ − " − σ − σ (3 − σ )( σ − cosh − ( σ ) (5.19)which reproduces the one given in Eq. (6.6) of [30]. At the moment, the physicalreason for this agreement is unclear.The results obtained so far allow one to derive in a straightforward way the zero-frequency limit (ZFL) of the energy spectrum dE rad dω . Indeed, the energy spectrumis just the integrand of (5.1) for the graviton multiplied by an extra factor of ~ ω (see also [4]) so that, E rad = Z d D − k π ) D − ˜ A ∗ µν (cid:18) η µρ η νσ − η µν η ρσ (cid:19) ˜ A ρσ ≡ Z ∞ dω dE rad dω . (5.20)Since we computed only the k → dE rad dω ( ω →
0) = lim ǫ → [ − ~ ǫ (Im2 δ )] . (5.21)In the case of GR we can use (5.14) with ( β GR ( σ )) in place of β ( σ ) and reproduceEq. (2.11) of [43] (taken from [44]) by taking the static limit σ → dEdω ( ω →
0) = 32 G m m πb . (5.22)11ur result (5.21) should hold true at all values of σ , extending Smarr’s originalresult [37] to arbitrary kinematics (see [43]). Possibly, our approach can be extendedto compute the energy spectrum to sub and sub-sub leading order in ω and toreproduce, in particular cases, the results of [45], [36] and [46].On the other hand, our method looks inadequate to deal with the full spectrumand with the total energy loss . For instance, extrapolating the ZFL result (5.21) tothe upper limit given in (5.10) would reproduce, at large σ , the qualitative behaviourof Eq. (5.10) of [43]. But, as anticipated to be the case in [43], and discussed in [48]and [49], such a result needs to be amended, as in the ultra-relativistic/masslesslimit, at fixed G , it would violate energy conservation.Our connection between RR and soft limits readily applies to Jordan-Brans-Dicke (JBD) scalar-tensor theory. The coupling of the massless scalar to massiveparticles is very much like that of the dilaton except for a rescaling of the couplingby a function of the JBD parameter ω J (the coefficient of the JBD kinetic term): g JBD = 1 √ ω J + 3 g dil . (5.23)The string dilaton case is recovered for ω J = −
1. It is then straightforward tocalculate the RR in JBD theory. It amounts to inserting in (5.14) and in (5.12) theJBD β ( σ ) factor, β JBD ( σ ) = 4 Gm m (cid:18) σ − ω J + 12 ω J + 3 (cid:19) , (5.24)and to further multiplying the dilaton’s contribution of (5.12) by a factor (2 ω J +3) − . Thus the contribution to the radiation reaction part of the eikonal from theJBD scalar reads G ( β JBD ) (2 ω J + 3) − ~ b ( σ − " σ + 23 − σ ( σ − cosh − ( σ ) . (5.25)In the limit ω J → ∞ , this result vanishes leaving just the contribution of thegraviton and thus reproducing the GR result. Since the present lower limit on ω J is about 4 × the effect is unfortunately unobservable. Acknowledgements
We thank Enrico Herrmann, Julio Parra-Martinez, Michael Ruf and Mao Zeng forsharing with us a first draft of their paper [47] and for useful comments on ours.We also thank Zvi Bern, Emil Bjerrum-Bohr, Poul Henrik Damgaard, ThibaultDamour, Henrik Johansson, Rafael Porto and Ashoke Sen for valuable observationson a preliminary version of this letter. The research of RR is partially supported T. Damour kindly informed us that he has carried out the explicit check. Such a calculation has been recently tackled by a different approach in [47].
12y the UK Science and Technology Facilities Council (STFC) Consolidated GrantST/P000754/1 “String theory, gauge theory and duality”. The research of CH(PDV) is fully (partially) supported by the Knut and Alice Wallenberg Foundationunder grant KAW 2018.0116.
References [1] T. Damour, “Gravitational scattering, post-Minkowskian approximation andEffective One-Body theory,”
Phys. Rev.
D94 (2016) no. 10, 104015, arXiv:1609.00354 [gr-qc] .[2] W. D. Goldberger and I. Z. Rothstein, “An Effective field theory of gravity forextended objects,”
Phys. Rev. D (2006) 104029, arXiv:hep-th/0409156 .[3] S. Melville, S. G. Naculich, H. J. Schnitzer, and C. D. White, “Wilson lineapproach to gravity in the high energy limit,” Phys. Rev.
D89 (2014) no. 2, 025009, arXiv:1306.6019 [hep-th] .[4] W. D. Goldberger and A. K. Ridgway, “Radiation and the classical doublecopy for color charges,”
Phys. Rev. D (2017) no. 12, 125010, arXiv:1611.03493 [hep-th] .[5] A. Luna, S. Melville, S. G. Naculich, and C. D. White, “Next-to-softcorrections to high energy scattering in QCD and gravity,” JHEP (2017) 052, arXiv:1611.02172 [hep-th] .[6] A. Luna, I. Nicholson, D. O’Connell, and C. D. White, “Inelastic Black HoleScattering from Charged Scalar Amplitudes,” JHEP (2018) 044, arXiv:1711.03901 [hep-th] .[7] N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Plant´e, andP. Vanhove, “General Relativity from Scattering Amplitudes,” Phys. Rev. Lett. (2018) no. 17, 171601, arXiv:1806.04920 [hep-th] .[8] C. Cheung, I. Z. Rothstein, and M. P. Solon, “From Scattering Amplitudes toClassical Potentials in the Post-Minkowskian Expansion,”
Phys. Rev. Lett. (2018) no. 25, 251101, arXiv:1808.02489 [hep-th] .[9] D. A. Kosower, B. Maybee, and D. O’Connell, “Amplitudes, Observables, andClassical Scattering,”
JHEP (2019) 137, arXiv:1811.10950 [hep-th] .[10] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng,“Scattering Amplitudes and the Conservative Hamiltonian for BinarySystems at Third Post-Minkowskian Order,” Phys. Rev. Lett. (2019) no. 20, 201603, arXiv:1901.04424 [hep-th] .1311] A. Koemans Collado, P. Di Vecchia, and R. Russo, “Revisiting the secondpost-Minkowskian eikonal and the dynamics of binary black holes,”
Phys. Rev. D (2019) no. 6, 066028, arXiv:1904.02667 [hep-th] .[12] A. Cristofoli, N. Bjerrum-Bohr, P. H. Damgaard, and P. Vanhove,“Post-Minkowskian Hamiltonians in general relativity,”
Phys. Rev. D (2019) no. 8, 084040, arXiv:1906.01579 [hep-th] .[13] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng,“Black Hole Binary Dynamics from the Double Copy and Effective Theory,” arXiv:1908.01493 [hep-th] .[14] G. K¨alin and R. A. Porto, “From Boundary Data to Bound States,”
JHEP (2020) 072, arXiv:1910.03008 [hep-th] .[15] N. Bjerrum-Bohr, A. Cristofoli, and P. H. Damgaard, “Post-MinkowskianScattering Angle in Einstein Gravity,” JHEP (2020) 038, arXiv:1910.09366 [hep-th] .[16] G. K¨alin and R. A. Porto, “From boundary data to bound states. Part II.Scattering angle to dynamical invariants (with twist),” JHEP (2020) 120, arXiv:1911.09130 [hep-th] .[17] T. Damour, “Classical and quantum scattering in post-Minkowskian gravity,” Phys. Rev. D (2020) no. 2, 024060, arXiv:1912.02139 [gr-qc] .[18] A. Cristofoli, P. H. Damgaard, P. Di Vecchia, and C. Heissenberg,“Second-order Post-Minkowskian scattering in arbitrary dimensions,”
JHEP (2020) 122, arXiv:2003.10274 [hep-th] .[19] G. K¨alin and R. A. Porto, “Post-Minkowskian Effective Field Theory forConservative Binary Dynamics,” JHEP (2020) 106, arXiv:2006.01184 [hep-th] .[20] G. K¨alin, Z. Liu, and R. A. Porto, “Conservative Tidal Effects in CompactBinary Systems to Next-to-Leading Post-Minkowskian Order,” Phys. Rev. D (2020) 124025, arXiv:2008.06047 [hep-th] .[21] G. K¨alin, Z. Liu, and R. A. Porto, “Conservative Dynamics of BinarySystems to Third Post-Minkowskian Order from the Effective Field TheoryApproach,” arXiv:2007.04977 [hep-th] .[22] G. Mogull, J. Plefka, and J. Steinhoff, “Classical black hole scattering from aworldline quantum field theory,” arXiv:2010.02865 [hep-th] .[23] M. Accettulli Huber, A. Brandhuber, S. De Angelis, and G. Travaglini,“From amplitudes to gravitational radiation with cubic interactions and tidaleffects,” arXiv:2012.06548 [hep-th] .1424] D. Amati, M. Ciafaloni, and G. Veneziano, “Superstring Collisions atPlanckian Energies,”
Phys. Lett.
B197 (1987) 81.[25] D. Amati, M. Ciafaloni, and G. Veneziano, “Classical and Quantum GravityEffects from Planckian Energy Superstring Collisions,”
Int. J. Mod. Phys. A3 (1988) 1615–1661.[26] I. J. Muzinich and M. Soldate, “High-Energy Unitarity of Gravitation andStrings,” Phys. Rev.
D37 (1988) 359.[27] B. Sundborg, “High-energy asymptotics: the one loop string amplitude andresummation,”
Nucl. Phys.
B306 (1988) 545–566.[28] D. Amati, M. Ciafaloni, and G. Veneziano, “Higher Order GravitationalDeflection and Soft Bremsstrahlung in Planckian Energy SuperstringCollisions,”
Nucl. Phys.
B347 (1990) 550–580.[29] C. Cheung and M. P. Solon, “Classical gravitational scattering at O (G ) fromFeynman diagrams,” JHEP (2020) 144, arXiv:2003.08351 [hep-th] .[30] T. Damour, “Radiative contribution to classical gravitational scattering atthe third order in G ,” Phys. Rev. D (2020) no. 12, 124008, arXiv:2010.01641 [gr-qc] .[31] P. Di Vecchia, S. G. Naculich, R. Russo, G. Veneziano, and C. D. White, “Atale of two exponentiations in N = 8 supergravity at subleading level,” JHEP (2020) 173, arXiv:1911.11716 [hep-th] .[32] Z. Bern, H. Ita, J. Parra-Martinez, and M. S. Ruf, “Universality in theclassical limit of massless gravitational scattering,” Phys. Rev. Lett. (2020) no. 3, 031601, arXiv:2002.02459 [hep-th] .[33] J. Parra-Martinez, M. S. Ruf, and M. Zeng, “Extremal black hole scatteringat O ( G ): graviton dominance, eikonal exponentiation, and differentialequations,” arXiv:2005.04236 [hep-th] .[34] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, “Universality ofultra-relativistic gravitational scattering,” Phys. Lett. B (2020) 135924, arXiv:2008.12743 [hep-th] .[35] D. Bini and T. Damour, “Gravitational radiation reaction along generalorbits in the effective one-body formalism,”
Phys. Rev. D (2012) 124012, arXiv:1210.2834 [gr-qc] .[36] A. Addazi, M. Bianchi, and G. Veneziano, “Soft gravitational radiation fromultra-relativistic collisions at sub- and sub-sub-leading order,” JHEP (2019) 050, arXiv:1901.10986 [hep-th] .1537] L. Smarr, “Gravitational Radiation from Distant Encounters and fromHeadon Collisions of Black Holes: The Zero Frequency Limit,” Phys. Rev. D (1977) 2069–2077.[38] S. Caron-Huot and Z. Zahraee, “Integrability of Black Hole Orbits inMaximal Supergravity,” JHEP (2019) 179, arXiv:1810.04694 [hep-th] .[39] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano (in preparation).[40] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. (1965) B516–B524.[41] P. Di Vecchia, R. Marotta, M. Mojaza, and J. Nohle, “New soft theorems forthe gravity dilaton and the Nambu-Goldstone dilaton at subsubleadingorder,”
Phys. Rev. D (2016) no. 8, 085015, arXiv:1512.03316 [hep-th] .[42] D. Amati, M. Ciafaloni, and G. Veneziano, “Towards an S-matrix Descriptionof Gravitational Collapse,” JHEP (2008) 049, arXiv:0712.1209 [hep-th] .[43] S. J. Kovacs and K. S. Thorne, “The Generation of Gravitational Waves. 4.Bremsstrahlung,” Astrophys. J. (1978) 62–85.[44] R. Ruffini and J. A. Wheeler, “Relativistic cosmology and space platforms,”ESRO (1971), 45-174.[45] B. Sahoo and A. Sen, “Classical and Quantum Results on Logarithmic Termsin the Soft Theorem in Four Dimensions,” JHEP (2019) 086, arXiv:1808.03288 [hep-th] .[46] A. P. Saha, B. Sahoo, and A. Sen, “Proof of the classical soft gravitontheorem in D = 4,” JHEP (2020) 153, arXiv:1912.06413 [hep-th] .[47] E. Herrmann, J. Parra-Martinez, M. Ruf, and M. Zeng, “GravitationalBremsstrahlung from Reverse Unitarity,” to appear.[48] A. Gruzinov and G. Veneziano, “Gravitational Radiation from MasslessParticle Collisions,” Class. Quant. Grav. (2016) no. 12, 125012, arXiv:1409.4555 [gr-qc] .[49] M. Ciafaloni, D. Colferai, and G. Veneziano, “Infrared features ofgravitational scattering and radiation in the eikonal approach,” Phys. Rev.
D99 (2019) no. 6, 066008, arXiv:1812.08137 [hep-th]arXiv:1812.08137 [hep-th]