Universality of ultra-relativistic gravitational scattering

We discuss the ultra-relativistic gravitational scattering of two massive particles at two-loop (3PM) level. We find that in this limit the real part of the eikonal, determining the deflection angle, is universal for gravitational theories in the two derivative approximation. This means that, regardless of the number of supersymmetries or the nature of the probes, the result connects smoothly with the massless case discussed since the late eighties by Amati, Ciafaloni and Veneziano. We analyse the problem both by using the analyticity and crossing properties of the scattering amplitudes and, in the case of the maximally supersymmetric theory, by explicit evaluation of the 4-point 2-loop amplitude using the results for the integrals in the full soft region. The first approach shows that the observable we are interested in is determined by the inelastic tree-level amplitude describing the emission of a graviton in the high-energy double-Regge limit, which is the origin of the universality property mentioned above. The second approach strongly suggests that the inclusion of the whole soft region is a necessary (and possibly sufficient) ingredient for recovering ultra relativistic finiteness and universality at the 3PM level. We conjecture that this universality persists at all orders in the PM expansion.


Introduction
At high energy and large impact parameter, gravitational scattering is dominated by the exchange of the highest-spin massless states in the theory [1,2,3,4,5].The focus of this work is on standard gravitational field theories in the two derivative approximation, so the highest spin particle is the graviton.The observable we study is the elastic 2 → 2 amplitude in the ultra-relativistic regime, where the centre-of-mass energy E cm = √ s is much larger than any other energy scale in the process (which goes under the name of Regge limit).Notice that this implies that the impact parameter b, which is related by Fourier transform to the total momentum exchanged in the scattering, is much larger than the gravitational length scale1 R D−3 ∼ G N √ s, defined in analogy with the Schwarzschild radius.Since the process is dominated by the exchange of gravitons, it is natural to expect a universal result for the high energy gravitational scattering.It is well known that this is the case to first order in R/b ≪ 1 where the result is captured by the leading eikonal phase δ 0 irrespectively of whether one works with strings or point particles, in higher dimensions, or in the presence of supersymmetry [2,3].It is then natural to ask whether such universality persists at higher orders in R/b, at least in the class of theories mentioned above 2 .The first non-trivial contribution to the ultra-relativistic result appears at sub-subleading order in the small R/b limit which captures the 3PM (post-Minkowskian) correction.This contribution can be encoded in terms of a correction δ 2 to the full eikonal.A first novelty with respect to the leading result is that δ 2 has both a real and an imaginary part, so e 2iδ is not just a phase.The imaginary part is related to non-conservative processes, such as the bremsstrahlung emission of massless particles, while the real part captures the conservative dynamics.A first result for δ 2 was obtained in [6] for the scattering in pure general relativity (GR) of two massless scalars representing, classically, the collision of two Aichelburg-Sexl shock-waves.The analysis of [7] suggested that the same result for Re(δ 2 ) should hold also for supergravity theories and this was explicitly verified in [8] for the case of the maximally supersymmetric theory.Explicit comparison between the GR and the N = 8 results shows that Im(δ 2 ) is not universal (although a crucial log(s)-enhanced term is), which is expected since the massless spectrum of the two theories is different.The universality of the 3PM conservative dynamics in the massless case has been confirmed and extended [9] to theories with different amounts of supersymmetry.
A similar, but slightly different setup is to consider the scattering of two scalar particles of masses m 1 and m 2 .There has been considerable interest, recently, in the use of new scattering amplitude techniques for extracting information about the conservative scattering process in the case s ∼ m 2 i [10,11,12,13,14,15,16,17,18]: the aim is to pin down the relevant ingredients needed for describing the inspiral phase of blackhole mergers.The importance of such a result for computing the templates for actual gravitational-wave experiments have been stressed, in particular by Damour [19].
The first result at 3PM level was derived a little while ago in [12,14] and was later confirmed in [20,21], while the case of maximally supersymmetric gravity was obtained recently in [22].The ultra-relativistic limit of these massive results is qualitatively different from the massless case mentioned above, as the leading term of Re(δ 2 ) has a log(s/(m 1 m 2 )) enhancement with respect to the one obtained in [6].From the amplitude perspective the origin of this log(s/(m 1 m 2 ))-enhanced contribution lies in a particular scalar integral which appears in exactly the same way in both the N = 0 and the N = 8 cases, hence the proposal of [22] that there is a separate universality class for the logarithmically-enhanced term of the massive scattering which is different from the massless case where there is no log(s)-enhancement.
This unexpected result of [12,14] has provoked quite a lot of discussion since, taken at face value, it would lead to a divergent deflection angle in either the massless or the ultrarelativistic limit.Since gravity is known [23] to be free of mass/collinear divergences the authors of [12,14] have immediately pointed out that their result only holds for sufficiently small values of the ratio q m where q ∼ b is the momentum transfer in the perturbative amplitude.They have also given [14] a one-loop example (involving a non-classical term) of how the m → 0 and the q → 0 limits can be quite different.On the other hand the above-mentioned divergence is also present at finite m for s → ∞ and would persist even if, for m > q, one would replace the log(s/(m 1 m 2 )) by a log(s/q 2 ).There is also some tension with expectations based on the "self-force" approach to PM dynamics.On these different grounds Damour suggested that log(s/(m 1 m 2 )) enhancement Re(δ 2 ) cannot be present in the ultra-relativistic limit.He proposed in [24] a modification of the result of [12,14] with a smooth massless limit that however differs (even in sign!) from the one of [6].Finally, a check [25,26] proposed in [24] for distinguishing the two alternatives at 6PN order has contradicted his original proposal while it is consistent with [12,14].Other alternatives, still at variance with [6], have been proposed in a later version of [24].
As mentioned above, in this short note we will reassess the ultra-relativistic limit s ≫ m 2 i and ask whether one recovers the massless shock-wave result for the real part of the eikonal δ at 3PM level.We will address this issue by using two complementary techniques.The first approach follows the argument of [6] where the 3PM result was derived by exploiting the analyticity and crossing properties of the scattering amplitudes among scalars.These properties imply a dispersion relation connecting the leading energy behaviour of the imaginary and real parts of the 2 → 2 amplitude.Then, in this limit, the conservative 3PM dynamics can be derived from a particular unitarity cut of the two-loop amplitude, thus replacing the full-fledged 2-loop calculation by a phase-space integral of a product of two tree-level amplitudes.As an aside, notice that this provides an explicit check that the 3PM conservative dynamics is entirely determined by classical on-shell data.The second approach follows the direct loop amplitude derivation of [12,14,22], where we just reconsider the calculation of the integrals without relying on the "potential" approximation.We use the eikonal approach to extract the classical contribution, instead of the effective field theory comparison of [10], and we work with the full result of the integrals in the "soft region" (i.e. in the limit of small momentum transfer).Both the approach based on analyticity/crossing and the one using the explicit loop amplitudes yield the same Re(δ 2 ) which also agrees with the massless result of [6,8,9], thus suggesting that the origin of the different ultra-relativistic behaviour of [12,14,20,22,21] lies in the use of the "potential" approximation in evaluating the loop integrals.When including the full contribution of the soft region, the ultra-relativistic limit at 3PM order is universal regardless of the mass of the external states.We conjecture that this is actually true to all orders in the PM expansion, since the relevant contributions are described by diagrams where the external states emit only gravitons and the two highly boosted lines representing them are connected through a tree-level GR amplitude [27].Both ingredients are universal in the class of theories we focus on.
The paper is organized as follows.In Section 2 we revisit the analyticity/crossing of [6] deriving on general grounds a relation between the ultra-relativistic limit of Re(δ 2 ) and the leading high energy behaviour of the imaginary part of the 2-loop amplitude.The analysis holds for arbitrary values of the masses as long as s ≫ m 2 i and also shows that the ultra-relativistic limit of Re(δ 2 ) can be universal only if Im(δ 2 ) is enhanced by a single power of log(s).In Section 3, we again follow [6] and evaluate the 2 → 3 tree-level amplitude relevant for the 3-particle cut determining the imaginary part of the 2-loop amplitude.Since we are interested in the ultra-relativistic case, we focus on a double Regge limit of the tree-level amplitude and show that, in this regime, it is universal and equal to the massless result of [6,28].This implies that the ultra-relativistic limit of Im(δ 2 ) is enhanced by a single factor of log(s) and that the ultra-relativistic limit of Re(δ 2 ) is also universal and has no enhancement at all.In Section 4 we follow the approach of [12,14] and in particular focus on the massive N = 8 case studied in [22].We provide the results for the integrals necessary to calculate the 3PM massive eikonal in N = 8 supergravity in the ultra-relativistic limit.As mentioned, we consider the full "soft" region result and show that the universal ultra-relativistic result for Re(δ 2 ) of Section 3 is recovered.We also provide the result for Re(δ 2 ) and the 3PM deflection angle for the generic values of s/m 2 i so as to facilitate the comparison with [22] even if we leave the discussion of the derivation to another work [29].In Appendix A we provide some further details on the consequences of analyticity and crossing used in the main text.In Appendix B we collect all the results we need for the high-energy scalar integrals in the soft region.

The analyticity/crossing argument revisited
In this section we review, improve and extend the arguments given in [6] for connecting the leading high-energy expression of Re(δ 2 ) to the inelastic contribution to the imaginary part of the two loop scattering amplitude.The latter is the convolution of two on-shell tree amplitudes and thus an easier object to deal with.For definiteness we will consider the case of the elastic gravitational scattering of two non-identical scalar particles of mass m 1 , m 2 .Crossing symmetry under the exchange of the two Mandelstam variables s and u simplifies considerably our analysis although we believe that the final results generalize to elastic processes that lack exact s ↔ u symmetry.
The essential ingredients of the argument presented in [6] are: i) real-analyticity of the scattering amplitude A(s * , t) = A * (s, t) as a function of the complex variable s at t ≤ 0, where −t = q 2 is the exchanged momentum (squared); 2 ); iii) some information about its high-energy asymptotic behaviour at each loop order, a property that fixes the number of subtractions needed in order to write a convergent dispersion relation for A, see Appendix A.
We will also make use of the exponentiation in impact-parameter space needed for recovering a classical limit.We will be helped by an amusing mathematical analogy with high-energy hadronic (QCD) scattering whereby the elastic hadronic amplitude A Had (s, t) is believed to behave, at high energy, as s log p s with some (not necessarily integer) p.An important quantity discussed in that context [30], [31] (see [32] for a recent review) is the ratio Re A Had (s,0) Im A Had (s,0) .Such a ratio (extended to the case of non-vanishing t) plays an important role in [6] as we shall see hereafter.Note that constraints based on analyticity and crossing, being linear, apply at each loop order, at each order in ǫ, and also separately to different terms in the high-energy expansion (like in the hadronic case where the Pomeron may be accompanied by subleading Regge pole contributions).Since it is not easy to find a selfcontained account of this methodology in the literature, and we are not constrained here by the Froissart bound, we will sketch for completeness the basic argument in Appendix A and refer to the above literature for further details.
We start by recalling the expression for the high energy tree-level amplitude A 0 (s, t) and the corresponding eikonal phase 2δ 0 for generic D = 4 − 2ǫ keeping only the first relevant terms in an expansion in inverse powers of s.One finds [33,13]: where Σ ≡ 2 (m 2 1 + m 2 2 ) and correspondingly 3 : (2.2) The kinematics is discussed in more detail before (3.4).In our conventions the eikonal phase is 2δ and is given by the Fourier transform of 1 4EcmP A(s, t), with ) is the product of the centre-of-mass energy and momentum.Following [13], we denote by M (s, b) the Fourier transform of 1 4EcmP M (s, t) for a generic function M (s, t).
The high-energy behaviour of Ã0 gives the asymptotic form of the leading eikonal phase: (2.3) In the Regge limit the full amplitude is encoded in the expression [8] where δ = δ 0 +δ 1 +δ 2 +• • • is the classical eikonal and ∆ encodes the quantum corrections.
At one-loop we know that A must include a leading imaginary contribution, growing like s 3 (without extra logs) and responsible for the start of the exponentiation of 2δ 0 .This indeed comes out of the box plus crossed box contribution in the form [13]: (2.5) which therefore fixes F 1 modulo analytic terms as t → 0. As explained in Appendix A analyticity and crossing symmetry imply the following form for the amplitude itself: as confirmed by explicit calculations [13].
It is known [6] that, in order to compute the classical two loop contribution to the eikonal, one has to take into account the first quantum contribution at one-loop level up to O(ǫ).It is also known from explicit calculations [34,13,35,8] that such a contribution contains two powers of s and no log s in its imaginary part 5 .Using Eq. (A.3) with p = 0 from Appendix A we get: In [6] this second structure was omitted since, in the specific case discussed there, there was no correction to the leading imaginary part appearing in Eq. (2.6) for D = 4.This was also shown [13] to be the case for non-vanishing 6 m 1,2 .However in other processes, in D = 4, or in supergravity theories, the structure Eq. (2.7) is also present.We shall see that, provided a certain exponent takes a particular value, A 1 actually drops out in the final expression for Re(δ 2 ), in agreement with several explicit calculations [8,9].
We finally turn to the two-loop amplitude A 2 .We know that it should have a leading O(s 4 ) real term in order to reproduce the correct third-order term in the exponentiation of δ 0 .Analyticity and crossing symmetry now fixes the amplitude to be of the form (see again Appendix A): As a consequence F 2 (t, m 2 i ) is known (up to analytic terms at t = 0).Note that the terms O(s 2 ) in Eq. (2.8) do not contribute to the classical phase shift.In order to have a classical contribution in the ultra relativistic (or massless) limit we need a term in the amplitude proportional to s 3 (up to logarithms) 7 contributing, in general, to both the real and the imaginary parts of A 2 .
Let us parametrize the latter in the form We may then use Eq.(A.4) of Appendix A to get 8 : Re

.10)
A highly non-trivial test of Eq. (2.8) and Eq.(2.10) is provided by the explicit results of [34].These are reported, for instance, in Eq. (B.1) of [8] where both the structure of Eq. (2.8) and the one of Eq. (2.9) are present.The process discussed there is not necessarily s − u symmetric but becomes so (at the order we consider) by choosing the external states appropriately 9 .After doing so all the real sub-leading terms (of O(sq 2 )) are matched either by the leading ones on the first line of (B.1) through Eq. (2.8), or by the logarithmically-enhanced imaginary terms through Eq. (2.10), or both.Nothing constrains instead the non logarithmically-enhanced imaginary terms at this level since their real counterparts would be sub-sub-leading in s. 7 We should mention that in the massive case other structures emerge both at the one and at the twoloop level.At one-loop the O(s 2 ) contribution actually contains a classical piece of the form (with D = 4 for concreteness) Re A At two loops we can get classical contributions from an amplitude going like s 2 m 2 log q 2 while an amplitude behaving like s 3 (m/q) log q 2 should also be present in order to accommodate the δ 0 δ 1 interference.Fortunately, all these terms arrange among themselves and do not interfere with the rest of our argument. 8In order to keep track of the first sub-leading correction in m 2 /s, as we did until now, we should include such corrections also in (2.9) and (2.10).We leave this point to our forthcoming paper [29]. 9This can be done by choosing an axion and a dilaton as incoming state, which implies that Â(0) = (1 + t s ) + . . . in the notations of [8].
We finally use exponentiation in impact parameter space to argue that: where we used Eq.(2.8) including the contribution 2s 3 tF 2 to 2 Re(δ 2 ).Analogously, where the term 4δ 0 Re(∆ 1 ) represents the full elastic contribution to the s-channel discontinuity of the amplitude while 2 Im(δ 2 ) represents the inelastic (3 particle) contribution.We are interested in extracting Re(δ 2 ) from Eqs. (2.11) and (2.12).In particular we would like to connect Re(δ 2 ) to Im(δ 2 ), an easier quantity to evaluate.Since F 1 (t, M 2 i ) and F 2 (t, m 2 i ) are both known in terms of δ 0 (see below) the only obstacle for obtaining the sought for relation is to eliminate the non-universal G 1 (t, m 2 i ) contribution appearing in Re(∆ 1 ) and Im(∆ 1 ).Using Eq. (2.10) a straightforward calculation gives: where in the last equation we have used Eq.(2.5) and Eq.(2.8) to express (tF 1 ) and (tF 2 ) in terms of δ 0 , and Eq.(2.7) to express G1 in terms of Im ∆ 1 .We thus notice that only for p = 1 is Re(δ 2 ) given entirely in terms of Im δ 2 and of δ 0 .For p = 1, Re(δ 2 ) will also depend on G1 i.e. on Im(∆ 1 ) which is non universal.We shall see in Section 3 that Im(δ 2 ) is indeed universal at high energy, and that Eq. (2.9) holds with p = 1.As a result, also Re(δ 2 ) is universal in the high-energy limit and given by:

High energy limit of the 3-particle cut
In this section we focus on the 3-particle cut contribution to A 2 as depicted in Fig. 1.Since we are interested in the ultra-relativistic limit, the amplitude is dominated by the graviton exchange whose contribution is universal.Thus for the sake of simplicity we focus on a gravity theory that can be obtained by taking the double copy of gauge theory The lines in bold represent energetic massive states, while the others represent massless states.The process depicted inside the dashed bubbles should not be interpreted as a specific Feynman diagram contribution, but just as a visual aid to recall the definition of the kinematic variables q i .We are interested in the full 2 → 3 tree level process, see (3.1) as an explicit example.
amplitudes or equivalently the field theory limit of string amplitudes.The presence of extra fields, and the dilaton in particular, becomes irrelevant in the limit s ≫ m 2 i .In order to obtain an explicit result we consider a 5-point amplitude in bosonic string theory where the external states p 1 and k 1 have a Kaluza-Klein momentum along one compact direction and p 2 and k 2 along another so they describe massive scalars in the uncompact dimensions 10 .By taking the field theory limit we obtain the following result for the 2 → 3 process in the left part of Fig. 1, where we introduced the variables The overall normalization was fixed by considering the leading high energy term for the Weinberg soft limit k → 0, where (3.1) has to reduce to the leading 2 → 2 amplitude times a universal factor.One can then calculate the contribution to the imaginary part of A 2 from the 3-particle cut by the usual phase space integral Here we are interested in the ultra-relativistic limit, so it is possible to approximate (3.2) in the double Regge limit11 where s = −(p 1 + p 2 ) 2 and s i = −(k + k i ) 2 .In this regime it is convenient to write the kinematic variables in terms of the (D − 2) space-like vectors orthogonal to the direction where the energetic states are boosted (which we take to be x D−1 ).By working in the Breit frame and taking light-cone variables for the time and longitudinal direction (p 0 + p D−1 , p, p 0 − p D−1 ), we have , m 1 e −y 1 ) ; p 2 = (m 2 e y 2 , q 2 , m 2 e −y 2 ) where y i are the rapidities of the external particle and m 2 1,2 = m 2 1,2 + q 2 4 .The intermediate states with momentum k 1 , k 2 , k (all incoming) are given by where In the double Regge limit, one can use approximations such as q 2 i ∼ q 2 i and (3.1) reduces to the result of [6] and Eq.(4.10) of [28] even if in those papers the external states are massless.This is not surprising since we are keeping the masses m 2 i fixed as the Mandelstam variables s, s i become large, however there is a point that deserves a further comment.By always keeping the leading order when rewriting the D-dimensional kinematics in terms of the transverse, we obtain that the ultra-relativistic limit of Eq. (3.2) agrees with the result of [6] Im × [( q 1 q 4 )( q 2 q 3 ) + ( q 1 q 2 )( q 3 q 4 ) − ( q 1 q 3 )( q 2 q 4 )] 2 q 2 1 q 2 2 q 2 3 q 2 4 , where we used the delta function in (3.2) in order to perform the integrals over the rapidities12 y ′ 1,2 and over the spatial components k.This result has a milder than naively expected IR behaviour since in (3.2) there are no terms diverging as q −2 1 q −2 2 (or as q −2 3 q −2 4 ) the the relevant q i are small.The cancellation of such contribution ensures that there are no 1/ǫ 2 contributions in the massless case and, crucially for us, no log 2 (s)-enhanced terms, i.e. no contribution with p = 2 in (2.9).This can be seen as follows: terms where the integration over the q i are factorised can produce 1/ǫ 2 contribution if each integration is independently IR divergent 13 , however in this case, one should keep subleading corrections to the approximation q 2 i ∼ q 2 i by including terms that break the factorisation such as q 2 1 = q 2 1 + A( q 1 + q 2 ) 2 + . . .and q 2 2 = q 2 2 + B( q 1 + q 2 ) 2 + . ..; then AB ∼ (m 1 m 2 /s) 2 acts as regulator in the deep IR region producing a contribution proportional to ǫ −1 log(m 1 m 2 /s) instead of 1/ǫ 2 .Since terms diverging as q −2 1 q −2 2 are absent in (3.1) then this mechanism does not apply and the only possibility to generate a factor of log(s) is from the integration over y.
Thus starting from (3.6) it is possible to follow the derivation of [6] and obtain This result is nothing else than 2 Im(δ 2 ) and we can use it in the general result (2.14).
First one can check that, as in [6] for the massless case, IR divergences cancel in Eq. (2.14) and notice that for this cancellation to happen it is crucial that (3.7) implies that p = 1 in (2.9).Then the finite term provides finite, smooth, and universal result for the highenergy limit of Re(δ 2 ) in D = 4: in agreement with [6,8,9].
4 Direct calculation of Re δ 2 In order to corroborate with an explicit example the general results obtained in the previous sections, in this section we provide an explicit expression for the four-point two-loop amplitude in N = 8 supergravity for the scattering of two scalar particles with masses m 1 and m 2 .This case has been already studied at one loop in [37] and at two loops in [22].We follow closely the procedure of [22] where the two-loop amplitude in N = 8 massive supergravity is written in terms of a set of basic scalar integrals I T , where the subscript T ∈ {III, III, IX, IX} indicates the diagram's topology, Here we follow the notation of Eq. (3.16) of [22] and, for the sake of simplicity, we have taken the angle appearing in that reference specifying the Kaluza-Klein reduction to be φ = π 2 ; in addition we have chosen the incoming particles to be an axion and a dilaton to give an amplitude that is symmetric under the exchange of s with u.Finally we neglected all the integral structures that are subleading in the high energy regime we are interested in or, equivalently, in limit of small momentum transfer.
The integrals I H and I H were computed in an ǫ expansion for arbitrary kinematics in [38], while the remaining integrals in (4.1) were studied in [22]: they adapted the differential equation approach, adopted in particular in [39] for the double box integral14 , by implementing the soft limit |t| ≪ s, m 2 1 , m 2 2 from the beginning and then further simplified the problem by calculating the boundary conditions of the relevant differential equation in the "potential" approximation.Here instead we do not take this extra approximation and we provide the result valid in the full soft region even if we focus on the ultra-relativistic (s ≫ m 2 i ) case.We collect in Appendix B the results for all scalar integrals needed and leave a detailed discussion of the generic kinematics to a followup paper [29].
When we add all the contributions we see that all log(s) in the real part of the amplitude cancel and one is left only with a log(s) in the imaginary part.In conclusion the complete ultra-relativistic amplitude is Before proceeding further, let us notice that this amplitude perfectly satisfies some of the general properties discussed in Sect. 2. In fact, the ratio between the leading and the real part of the subleading term at the order 1 ǫ 2 is equal to s 2t in agreement with the first line of Eq. (2.8), while, at the order 1 ǫ , the first two terms of the square bracket satisfy Eq. (2.10) for p = 1.This is the consequence of a non-trivial cancellation since the individual integrals contain higher log p (s) contributions, see for instance the contribution coming from I H in (B.15).There are also further non-trivial cancellations that ensure that the leading term at large distance, proportional to (q 2 ) −1+2ǫ , takes a particularly simple form in line with the expectation of the eikonal exponentiation (2.4).This is more easily seen in impact parameter space where we get In order to compute the new contribution to the eikonal we must first subtract the contribution of the lower eikonal δ 0 and ∆ 1 that are equal to15 Using them we can write the real part of the amplitude in Eq. (4.3) as follows: The last term in this equation is the ultra-relativistic limit of Re(δ 2 ) and is immediate to check that in D = 4 one recovers the universal result (3.8) obtained in the previous section by generalising the approach of [6].
For completeness, let us also present the real part of the eikonal to third post-Minkowskian order for generic s = m 2 1 + m 2 2 + 2m 1 m 2 σ (with σ ≥ 1) [29], Furthermore, the corresponding 3PM contribution to the scattering angle as a function of the angular momentum J reads Let us conclude with a few comments on the 3PM result (4.7) 16 .The first term in the first line is entirely determined by χ 1PM , while the first term in the second line is due to integrals I H and I H , see (B.15).Together they reproduce exactly Eq. (6.41) of [22].The second term in the second line is a contribution coming from the full soft-region analysis of the crossed double-box integrals.In the ultra-relativistic limit σ ≫ 1 the leading term O(σ 4 ) in each term in the round parenthesis in the second line cancels and only the first line survives reproducing the universal and finite ultra-relativistic result which was the main focus of this paper.Thanks to analyticity/crossing argument in Section 2, this cancellation is a consequence of the cancellation in the imaginary part mentioned below Eq. (4.2).Notice that the functional form of (4.7) matches exactly Eq. (3.65) of [24] (where of course the Schwarzschild contribution is substituted by the probe-limit relevant to this supersymmetric case) and so it is possible to define a function C(σ) also for the N = 8 case.It is interesting to look at the opposite limit σ → 1 which is relevant to the PN (Post-Newtonian) limit: the terms already present in Eq. (6.41) of [22] have a standard nPN expansion where n is integer.The new terms yield contributions only at half-integer PN orders, starting at 1.5PN, and so they do not modify the integer PN data.However, these half-integer PN terms, usually associated with dissipative phenomena, are somehow unexpected at order G 3 N , so it would certainly be interesting to repeat the same analysis for pure GR, instead of N = 8 supergravity case.If a similar pattern appears also in GR, as it seems reasonable since the same integrals I T appear in all cases, that would of course be the best setup where to investigate these issues in more detail.

A Asymptotic ρ ≡ Re A
Im A from analyticity and crossing Consider a crossing-symmetric (under s ↔ u), real-analytic amplitude A(s * , t) = A * (s, t) as a function of the complex variable s at t ≤ 0 and assume that, as s → ∞, Since real-analyticity and crossing symmetry imply Im A(−s + i0) = − Im A(s + i0), we can write an (n + 1)-times subtracted dispersion relation in the form: where s 0 = (m 1 + m 2 ) 2 is the s-channel threshold, Q 2m (s, t) is a polynomial of degree 2m, P denotes the principal part, and the integer m is defined by n = 2m for n even or by n = 2m + 1 for n odd 17 .Because of Eq. (A.1) the integral in Eq. (A.2) converges.From Eq. (A.2) we can compute ρ distinguishing the two cases.
For n even we get: while for n odd: where we have used the high-energy limit of the principal-part integrals appearing in Eq. (A.3) and Eq.(A.4).These are explicitly known in terms of special functions.An exceptional case (which we encounter for the amplitude A 1 of Eq. (2.6)) is the one of Eq. (A.4) with p = 0.In that case ρ is suppressed by a full power of s but the leading term comes from picking up the first subleading n-even correction in Eq. (A.1), which is fixed by crossing symmetry, and from applying to it Eq.(A.3).The last case needed, the one of A

B Summary of the soft integrals
In this appendix, we collect the expressions for the scalar integrals that enter the N = 8 two-loop amplitude.We focus on the contributions that are non-analytic in q 2 , which emerge from the soft region in the small-q expansion.In order to exhibit the relevant expressions for each topology T ∈ {III, III, IX, IX}, let us denote Here the dots stand for terms that are subleading in ǫ or in q, and we omit terms proportional to 1 q which are not needed to extract δ 2 from the amplitude but only to check the δ 0 δ 1 term arising from the exponentiation.Let instead trivial since the amplitude is purely real and crossing symmetry forces the combination s 4 + 2ts 3 + O(t 2 s 2 ).