Inelastic Exponentiation and Classical Gravitational Scattering at One Loop

We calculate the inelastic $2\to3$ one-loop amplitude for the scattering of two point-like, spinless objects with generic masses involving the additional emission of a single graviton. We focus on the near-forward, or classical, limit. Our results include the leading and subleading orders in the soft-region expansion, which captures all non-analytic contributions in the transferred momentum and in the graviton's frequency. This allows us to check the first constraint arising from the inelastic exponentiation put forward in Refs. 2107.12891, 2112.07556, 2210.12118 and to calculate the $2\to3$ one-loop matrix element of the $N$-operator, linked to the $S$-matrix by $S = e^{iN}$, showing that it is real, classical and free of infrared divergences. We discuss how our results feature in the calculation of the $\mathcal O(G^3)$ corrections to the asymptotic waveform.

Recent years have witnessed renewed efforts in the study of two-body systems undergoing classical gravitational collisions, motivated by the ultimate objective of providing increasingly accurate waveform templates for gravitational wave detection [4,5].While at first sight counterintuitive, scattering-amplitude methods borrowed from collider physics [6][7][8][9][10][11][12] have proven to be powerful tools for describing such systems and providing state-of-the-art predictions in the Post-Minkowskian (PM) regime, when the two colliding objects are sufficiently far apart and interact weakly [13][14][15][16][17][18].Interactions between astrophysical black holes or neutron stars involved in such collisions are indeed classical, since their typical quantum wavelength is much smaller than the length scale associated to the gravitational curvature they induce, a statement that for black holes of mass M translates to GM 2 / ≫ 1 [16,19,20].This inequality, which is of course amply satisfied by such objects, signals, however, a breakdown of conventional perturbation theory, since the effective coupling to gravity is not small.Therefore scattering amplitudes, which are organized as a weak-coupling G-expansion, must actually be supplemented with a nonperturbative principle in order to correctly capture the classical limit.One such guiding principle, which is also familiar from the non-relativistic WKB approximation, is that in the classical limit the S-matrix should be dominated by the exponential e 2iδ of a large phase 2δ, which plays the role of a large action in units of .For the elastic 2 → 2 amplitude this resummation is known as the eikonal exponentiation and the eikonal phase, or the closely related radial action, has been employed to extract from the amplitude the deflection angle(s) for two-body collisions up to 4PM order [17][18][19][21][22][23][24][25][26][27][28][29][30][31][32][33].The nonperturbative nature of the problem manifests itself at each loop order via "superclassical" or "iteration" terms, contributions that scale with higher powers of the large ratio GM 2 / , or, for short, of −1 .The eikonal exponentiation dictates how such spurious terms should be subtracted, by matching with the power series expansion of the exponential, and fixes all ambiguities associated to possible remainders, providing a direct connection to the impulse and to the deflection angle via a saddle point approximation [27,[34][35][36][37][38][39].
However, by focusing on the elastic 2 → 2 amplitude, the conventional eikonal framework fails to capture possible subtractions associated to inelastic channels.For instance the infrared (IR) divergent imaginary part in the 3PM eikonal signals the fact that at O(G 3 ) an inelastic 3-particle channel involving the two massive states and a graviton opens up, and the standard eikonal exponentiation does not capture it.This problem has been studied and solved in [1][2][3]33], the basic idea being that, in a more comprehensive framework, the eikonal should be promoted to the exponential of i times a suitable Hermitian operator that is able to appropriately combine all needed channels.The approach of Ref. [1] is to apply this principle to the full S-matrix, writing S = e iN with N † = N and building N -matrix elements out of conventional scattering amplitudes, which are of course T -matrix elements with S = 1 + iT .
A complementary approach is provided by the formalism first introduced by Kosower, Maybee and O'Connell (KMOC) [40] and later developed in Refs.[2,[41][42][43][44][45][46].This framework is based on the principle that, after identifying a well-defined classical observable O associated to the collision, its expectation value in the final state dictated by the S-matrix, in|S † OS|in will be free of superclassical terms and thus possesses a well-defined classical limit.The state |in models the two incoming massive particles with given impact parameter(s) via an appropriate superposition of plane-wave states built with suitable wave-packets, whose details become immaterial after the cancellation of superclassical terms.
In this paper, we explore further the exponentiation in the classical limit and the connection between amplitudes and classical observables.We focus on the 2 → 3 amplitude for the scattering of two minimally-coupled massive scalars plus the emission of a single graviton in General Relativity, whose loop expansion reads A µν = A µν 0 + A µν 1 + • • • , or, pictorially, and discuss the calculation of its 1-loop part, A µν 1 , starting from the integrand provided in Ref. [47].We focus on the non-analytic terms in the near-forward limit, whereby the transferred momentum and the emitted graviton's momentum are simultaneously taken to be small, O( ), in comparison with the particles' masses, O( 0 ).To this end we apply the method of regions and restrict our attention to the soft region, in which the loop momentum assigned to the exchanged gravitons is also small, O( ).We employ dimensional regularization, letting ǫ = 4−D 2 , and express the result as a Laurent expansion around ǫ = 0.The calculation of A µν 1 in the soft region constitutes one of the main new results of this work, and represents a first step in generalizing the studies of graviton emissions during collisions of ultrarelativistic or massless objects [48][49][50] to the case of massive objects with generic velocities.The amplitude A µν 1 , as expected, involves both superclassical, O( −2 ), and classical, O( −1 ), contributions, for each of which we calculate both infrared (IR) divergent and finite terms.For the IR divergent pieces, we find complete agreement with the well-known exponential pattern [51] according to which IR divergences in a given one-loop amplitude are equal to a one-loop-exact divergent factor W times the tree-level amplitude with the same external states [27,52].
In the Weinberg limit, in which the emitted graviton's frequency becomes very small, k µ ∼ O(λ) with λ → 0, the one-loop amplitude A µν 1 must also exhibit O(λ −1 ) terms whose form is completely fixed by the leading soft graviton theorem [53] as the factor F µν = √ 8πG n p µ n p ν n /(p n •k) times the 2 → 2 one-loop amplitude without graviton emissions.This 1/λ pole is a frequency-space manifestation of the memory effect [54,55].Comparing with the results available from the literature [14,23,56], we find perfect agreeement with this prediction, reproducing in particular the terms arising from the 2PM deflection encoded in one-loop 2 → 2 "triangle" contributions, i.e. from the sub-leading eikonal phase 2δ 1 .Moreover, exploiting the conventional exponentiation of the 2 → 2 amplitude, this factorization allows us to check the inelastic exponentiation of Refs.[2,3] to leading order in the soft limit.Throughout the paper, we focus on emitted gravitons with positive frequencies, so that we do not include in our analysis terms with support localized at ω = 0 in frequency space, which are associated to static effects in time domain (see e.g.[57,58] for their concrete appearance in the tree-level expressions).The inclusion of such terms has been discussed in [59,60] and can be typically performed by means an appropriate dressing of the initial and final states with a modified Weinberg factor √ 8πG n p µ n p ν n /(p n •k − i0).After constructing the appropriate subtractions dictated by the N -matrix formalism [1], we calculate the 2 → 3, N -matrix element B µν 1 from the amplitude A µν 1 .In this way we

S-channel
Table 1: The s, s ′ , s 1 , s 2 channels and their scaling in the classical limit.
obtain a purely classical object, B µν 1 , which is also real and free of IR divergences.Indeed, by comparing the operator power series is easy to see that, at one loop, the operator exponentiation of [1] boils down to simply dropping the imaginary parts of the amplitude, i.e. to subtracting its unitarity cuts, 2 Im A µν 1 = (cuts).For the process under considerations, there are four distinct channels, which we depict in Table 1.
Two of them, often referred to as s and s ′ , involve cutting an intermediate state with two massive particles, and we shall call them collectively "S-channel".We find that the subtraction of the S-channel is actually enough to get rid of all superclassical terms.This is in accordance with Refs.[2,3], since this subtraction in momentum space is equivalent in b-space to the subtraction of 2iδ 0 Ãµν 0 .Indeed, since this cut contributes schematically via + i 2 (S-channel) to the amplitude and each of the two diagrams in the first line of Table 1 contributes as 2δ 0 Ãµν 0 in b-space, it is crucial to consider both diagrams in order to get the right combinatoric factor in front.
The remaining two cuts in Table 1, s 1 and s 2 , instead involve an intermediate state with a massive particle and a graviton, in which the latter re-scatters against the massive line in the gravitational analog of a Compton process.For this reason, we may call them collectively "C-channel".Albeit classical as far as the scaling is concerned, the C-channel involves an infrared divergence and its subtraction is crucial in order to make the resulting B µν 1 (real and) finite as ǫ → 0.
The amplitude A µν 1 , and the N -matrix element B µν 1 , encode the dynamical information needed in order to evaluate the O(G 3 ) corrections to asymptotic value of the metric fluctuation far away from the collision, which provide the next order in the PM expansion compared to the results of Refs.[57,58,[61][62][63][64].For this reason we investigate the construction of the associated KMOC kernel, i.e. the object whose Fourier transform from q-space to b-space provides the waveform in frequency domain.We find that this kernel is not simply given by iB µν 1 , as perhaps expected.Rather, it equals iB µν 1 minus 1 2 times the IR divergent Cchannel cuts.By its very nature, the associated IR pole in 1/ǫ can be exponentiated to a q-independent phase, amounting to a (divergent) shift of the origin of the observer's retarded time, at the price of introducing a logarithm involving an unspecified scale in the finite part.
Neither the phase nor this logarithm appear in the energy-momentum spectra, and could as such be considered "harmless".The appearance of ambiguous logarithms in the waveform, as a result of the long-range nature of the gravitational force, is a known issue [65,66] and is associated to an ambiguity in the definition of the asymptotic detector's retarded time induced by so-called tail or rescattering effects [67][68][69][70].We leave further investigations of this issue for future work, together with the calculation of the (b-space) waveform and with a comparison with subleading log λ-corrected soft theorems [71][72][73][74], which are also intimately related with long-range corrections to the asyptotic interactions.
The paper is organized as follows.In Section 2 we present our conventions for dealing with the external states of the scattering, illustrating a useful choice of variables and polarizations, while more details on them are available in Appendix A. In Section 3 we discuss the classical limit and how focusing on the soft region simplifies the integration.We list the corresponding 9 independent master integrals in Appendix B and in the file master integrals.m.Section 4 is devoted to illustrating our result for the amplitude, which is collected in computer-readable format in the file Results Ampl 5pt.nb, discussing the consistency checks offered by the exponentiation of IR divergences and from factorization in the soft limit.We also discuss the implications of unitarity and the subtraction of superclassical iterations, explicitly proving the leading constraint coming from the inelastic exponentiation.The tree-level amplitudes needed to perform such checks are presented in Appendix C. In Section 5, we discuss the calculation of the gravitational field, of the associated spectrum and of the asymptotic waveform, before presenting a summary of our conclusions and a prospect of possible future directions in Section 6.

Conventions:
We employ the mostly-plus signature, η µν = diag(−, +, +, +).All momenta are regarded as formally outgoing, so that p 3 , p 4 and k are the physical momenta of the final states of the scattering, while p 1 and p 2 are minus the physical momenta of the initial states.
Note added 1: While working on this project we became aware of independent progress by Refs.[75][76][77], whose scope partly overlaps with our analysis.These groups' work was also presented as a series of seminars [78][79][80] at the "QCD Meets Gravity 2022" conference.
Note added 2: In Ref. [81] it was pointed out that an extra term must be taken into account in the KMOC cuts contributing to the one-loop waveform discussed in Section 5. See Refs.[82,83] for its inclusion.
We consider the scattering of two massive objects with masses m 1 (depicted with a thick blue line) and m 2 (thick green line) and the emission of a graviton (thin red line), The figure in (2.1) is meant to help remembering the definition of the "transferred momenta" and does not represent an actual topology.Let us begin by discussing a useful choice of variables.

Physical variables
We let In this way, and (2.7) The momentum transfers q µ 1 and q µ 2 are not independent because of momentum conservation (2.3) and of the mass-shell condition k 2 = 0, which implies Five independent invariant Lorentz products can be taken as follows: (2.9) The variable y is the relative Lorentz factor of two observers with four-velocities u µ 1 , u µ 2 .Letting v denote the velocity of the former as seen from the rest frame of the latter (or vice-versa), 3), we see that ω 1 and ω 2 are the frequency of the graviton measured by these two observers, In order to simplify square roots that frequently appear in the calculations, it is convenient to define the following dimensionless variables with the inverse relations given by so that they obey the inequalities The limit x → 0 corresponds to the high-energy (ultrarelativistic) regime and x → 1 to the low-energy one.

Polarization tensor
It is convenient to contract the amplitude with an appropriate polarization tensor, which we can build as follows.We start from a vector We solve the transversality condition, by letting (2.17) We then define the polarization tensor This transverse, thanks to (2.17).It can be also made traceless and related to more standard choices of graviton polarizations as detailed in Appendix A. We introduce the symbol to denote the contracted amplitude.Defining gauge invariance requires that we can freely replace ε µ with εµ The new polarization vector (2.20) is independent of d + , which thus constitutes a free parameter that ought to drop out from the final expression.This serves as a very useful cross-check of the calculations.

Classical Limit and Integration
Let us spell out the decomposition of our amplitude in the classical or near-forward limit.In this regime the momentum transfers q 1 , q 2 are taken to be simultaneously small with respect to the masses of the incoming particles, or, equivalently, the masses are taken to be large with respect to the exchanged momenta.We can therefore use a common scaling parameter as a bookkeeping device for the associated power counting.We shall use for this bookkeeping purpose and define the scaling by We emphasize again that (3.1)only serves to keep track of powers of the transferred momenta and does not refer to the actual dependence of the amplitude on the Planck constant after restoring standard units [40].We shall also scale the integrated momentum ℓ associated to exchanged gravitons in the same way as the exchanged momenta q 1,2 , This enforces the expansion for the loop integrals in the soft region, which is the appropriate one to capture all the non-analytic dependence on q 1 , q 2 in the amplitude.From Eq. (3.1), it also follows that where c 1,2 and d 1,2 are the decomposition coefficients in (2.15).We follow the numbering of the 24 topologies, G j with j = 1, . . ., 24, associated to the integrand numerators given in Ref. [47], which are depicted in Table 2 and can be grouped into five families as in Table 3.As we shall discuss in the Subsection 3.1, all integrals belonging to the P , P ′ , M and M ′ families can be mapped to a collection of linearized pentagon integrals.The 5 integrals of the quantum family are manifestly associated to intermediate processes, like creation of black-hole-/anti-black-hole pairs, which ought be disregarded in the classical limit, and indeed the associated integrals vanish in the soft region.In this way, the 16 master integrals in Table 4 below suffice to decompose the integrand via Integration By Parts and to evaluate the resulting integrals relevant for our purposes.
Table 2: Topologies of the 24 numerators of Ref. [47].Color code: yellow = pentagon (P ), white = pentagon prime (P ′ ), red = mushroom (M ), orange = mushroom prime (M ′ ), gray = quantum topologies.Our conventions on the external states are summarized in Eq. 2.1.Each of the 16 numerators belonging the P , P ′ , M and M ′ families should be multiplied by the appropriate propagators dictated by its diagram, and summed over the 8 independent permutations

Family Topology
generated by the following ones, The trivial transformation (identity element).
σ 2 : The permutation interchanging the endpoints of the blue line in Eq. (2.1), sending (3.5) The permutation interchanging the endpoints of the green line in Eq. (2.1), sending u µ 2 → −u µ 2 and correspondingly σ 4 : Particle-interchange symmetry, which corresponds to replacing the blue line with the green one and vicevesa, and to interchanging all particle labels 1 ↔ 2.
Of course, these operations should be performed while leaving ε µ in Eq. (2.15) invariant.For this reason, after expanding it as in (2.15), one should compensate for the transformations of the basis vectors by also sending c 1 → −c 1 (resp.c 2 → −c 2 ) when performing σ 2 (resp.σ 3 ) and by sending c 1 ↔ c 2 , d 1 ↔ d 2 when performing σ 4 .Moreover, each diagram should only be summed over its nontrivial permutations.Equivalently, when summing over the whole P 8 , each diagram should be supplied with the appropriate symmetry factor s i accounting for the fact that a subset of the permutations may leave it invariant.Massless bubble diagrams G 20 , G 23 carry an additional factor of 1/2 due to the freedom of relabeling the loop momentum.
After evaluating the integrals in each family using the integration measure and summing over the allowed permutations, the last step is to multiply by the overall normalization factor N given by with µ an arbitrary energy scale introduced by dimensional regularization.All in all, we may summarize this construction as follows, (3.9)

Mapping to the pentagon family
In the limit (3.1), with an appropriate choice of loop momentum routing, all integrals in the pentagon family can be mapped to the following collection of integrals (3.10) In Eq. (3.10) and in the following, the −i0 prescription is left implicit for brevity.The family (3.10) is obtained from the conventional scalar pentagon with two massive lines (the momentum flows clockwise in the loop and, as in Eq. (2.1), the external momenta are all outgoing) by linearizing the two massive propagators and factoring out m1 , m2 : In our conventions, the sign of each propagator is fixed due to the −i0 prescription, e.g.
A basis of for the family of integrals (3.10), which determine all the others via Integration By Parts (IBP), can be obtained using LiteRed [84,85] and is given by the 16 elements in Table 4 (although 7 of them can be deduced from the remaining 9 by using σ 4 ).Using HyperInt [86] and dimensional shift identities [87][88][89], we have found the values of all such master integrals up to transcendental weight 2. We present our results in Appendix B in the Euclidean region, and discuss their analytic continuation to the physical one in the Subsection 3.2.It turns out that the pentagon prime, mushroom and mushroom prime families can be also mapped to the integrals (3.10) in the limit (3.1), by suitably decomposing the linearized propagators into partial fractions and applying symmetry transformations.Let us discuss this step in detail focusing on a prototypical integral for each family, with propagators raised I 0,0,1,0,1 I 0,0,1,1,0 I 0,1,0,0,1 I 1,0,0,1,0 Table 4: Topologies of the 16 master integrals for the pentagon family.Color code: lighter green = non-analytic in q 2 , darker green = analytic in q 2 .The appearance of the latter type of topologies where matter lines "touch" and which do not appear in Table 2 is induced by the IBP reduction, whose coefficients can be non-analytic in q 2 and thus induce long-range effects in position space.Such contributions would be scale-less in the 2 → 2 kinematics.
to the first power.Generalizing this procedure to any other positive power is straightforward.A typical integral of the pentagon prime family takes the form and decomposing the second and third propagator into partial fractions leads to (3.17)We can change integration variable in the second integral, letting ℓ → q 1 − ℓ, and obtain The two integrals do not simply cancel against each other, due to (3.14), (3.15), but we can map them to the family (3.10) by applying permutations σ 2 and σ 3 to the second one, For a typical integral of the mushroom family, we have (3.21) Noting that the first integral on the right-hand side is scaleless, and sending ℓ → q 1 − ℓ in the second one, we find Finally, a typical mushroom prime integral takes the form and leads to the following partial fractions Performing the permutation σ 3 in the first integral and sending ℓ → q 1 − q 2 − ℓ in the second one, Let us comment that, since one is ultimately summing over all allowed permutations (3.10) to build the full integrand from the 19 diagrams in Table 2, it is not strictly necessary to apply σ 2 σ 3 as in (3.19) and σ 3 as in (3.25).One can also first treat the two contributions to each equation as separate objects, perform the IBP reduction and mapping to the master integrals only for the appropriate permutation, and then perform all 8 permutations on the result.

Euclidean variables and analytic continuation
The amplitude is the boundary value of an analytic function which develops branch cuts when its variables are located in the physical region.For this reason it is convenient to introduce complex variables y E , ω 1E , ω 2E such that the physical region corresponds to setting with y, ω 1 , ω 2 the invariant products defined in Eq. (2.9).The new variables allow for manifestly real expressions of the master integrals (3.10) when they are taken in the Euclidean region, defined by We similarly define new rationalized variables according to so that The rationalized variables fall in the physical region when with x, w 1 , w 2 as in Eq. (2.12).The conditions (3.27) defining the Euclidean region instead translate to the following ones in terms of the rationalized variables, Mapping back to the physical variables discussed in Subsection 2.1 via (3.26), (3.30), one encounters branch singularities when the variables fall in the physical region.We have expressed our master integrals (see Appendix B) in terms of the analytic functions log x E , log(x E − 1) , log w 1E , log(w 1E ± 1) , log w 2E , log(w 2E ± 1) , and so that their expressions are manifestly real in the Euclidea region (3.31).We then perform the analytic continuation back to the physical region (3.30) by letting where we have kept all analytic continuations in principle arbitrary.We find that, consistently with the i0 prescriptions in (3.30) or equivalently by demanding consistency with the exponentiation of infrared divergences (see Subsection 4.1), The sign of the i0 prescription q I for x E matches the elastic calculation in [27] where for instance in the double box solution log x E → log x + iπ.The elementary choices made in (3.34) then resolve all other branch ambiguities upon reducing the analytic continuation of the dilogarithms to those of conventional logarithms via which holds whenever z doesn't belong to the positive real axis.
4 Structure of the Amplitude in the Classical Limit Looking at the integrand obtained by combining the diagrams in Table 2, we find the following structure in the limit (3.1), (3.2) (including the scaling of the measure element d 4 ℓ), where We expand each coefficient for small ǫ = 4−D 2 , defining The first (second) index within square brackets thus refers to the (resp.ǫ) scaling.
After inputting the values of the master integrals, we find that (for nonzero graviton frequencies) This cancellation is expected because it mirrors a similar one occurring for the tree-level amplitude A 0 , whose classical limit also naively goes like −3 (and it indeed involves terms localized at zero frequency at that order), while its actual scaling is −2 in our present conventions.Eq. (4.4) also serves a nontrivial check of the symmetry factors because it relies on s 3 = 2s 1 and on s 6 + s 24 = s 11 + s 14 .We also find that the coefficient of the double pole in ǫ vanishes, A For instance, both G 2 , G 4 and G 9 would naively diverge like 1/ǫ 2 to order O( −2 ), but thanks to the transversality condition (2.17) such divergences cancel between G 2 and G 9 , and separately in G 4 .This serves as a cross check that s 2 = s 9 .This cancellation is also expected on general grounds [51] as we shall discuss more in detail shortly.
Taking into account the vanishing of these coefficients, we find the following structure of the amplitude in the classical limit,

.6)
The functions A constitute the main results of the present work and are all provided in the ancillary files in attachment, where for convenience we collect their expressions after dividing by N 4 defined in (3.8).We have checked that our expressions for the classical terms, A , agree with the results of Ref. [? ] on numerical points, up to an overall sign.In its turn, this also ensures agreement wit the results of Ref. [? ].The remainder of this section is devoted to the discussion and illustration of Eq. (4.6).
We find that the coefficients of the ǫ −1 poles, A , are in complete agreement with the prediction obtained from the exponentiation of infrared divergences, which fixes them completely in terms of the tree-level five-point amplitude A 0 times a universal factor [51].The combinations A /N 4 have uniform transcendental weight 1, i.e. they are rational functions of the invariants x, w 1 , w 2 in (2.12), q 1 and q 2 times iπ.For all terms displayed in (4.6), we also find agreement with Weinberg's soft theorem [53], which dictates that, as the frequency of the graviton tends to zero, their most singular term must reduce to a universal factor times the one-loop four-point amplitude A , when written in terms of c 1 and c 2 , arise from the cuts of the amplitude exactly as predicted by unitarity.
The terms that do not multiply the imaginary unit in and all dependence on the logarithms and dilogarithms drops out in such terms, so that they reduce to rational functions of the invariants x, w 1 , w 2 in (2.12), q 1 and q 2 times π 2 .After this simplification, the structure of this piece is thus analogous to that of the O( −1 ) in the elastic 2 → 2 amplitude at one loop (Eq.(4.34) below).The terms that multiply the imaginary unit in A [−1,0] 1 /N 4 (and similarly the combination A [−2,0] 1 /N 4 ), instead, can have transcendental weight either 2, i.e. reduce to rational functions times the logarithms (4.7) times iπ, or 1, i.e. reduce to rational functions times iπ.Schematically, = π 2 Q(x, w 1 , w 2 , q 1 , q 2 ) + iπ j (log x j ) R j (x, w 1 , w 2 , q 1 , q 2 ) + S(x, w 1 , w 2 , q 1 , q 2 ) (4.9) where Q, R j and S are real, rational functions of the invariants and log x j are the logarithms (4.7).

Exponentiation of infrared divergences
Infrared divergences in gravity amplitudes follow a simple exponential patter first clarified by Weinberg [51] (see also [29,90]), with W α→β an infrared-divergent, one-loop-exact exponent whose expression in terms of the states α, β takes a universal form.Accordingly, the infrared divergences of any one-loop five-point amplitude are equal to w nm (4.11) times the tree-level five-point amplitude with the same external states.When the 5 particles are massive, letting (η n is +1 if n is outgoing and −1 if n is incoming) we have where η nm = +1 provided n = m and n and m are both outgoing or both incoming, and vanishes otherwise.Moreover w nn = m 2 n /2.When m 5 → 0 and p 5 → k, the function W is smooth and reduces to with Λ an arbitrary energy scale.One can explicitly check this by taking the limit in (4.12) and by using momentum conservation to show that a potentially dangerous log m 5 cancels out, leaving behind an arbitrary reference scale Λ in the logarithm in (4.14).All in all, this dictates the IR divergences of our amplitude, with A 0 = ε µ A µν 0 ε ν the tree-level five-point amplitude (C.2).In particular, (4.16) predicts that no double pole 1/ǫ 2 should occur and Eq.(4.5) is consistent with this prediction.
We checked that, expanding to leading and subleading order in , where Therefore, to this order in , the Weinberg exponent is purely imaginary.W [0] is a divergent phase arising from soft graviton exchanges between lines 1 − 2 and 3 − 4 while W [1] arises from those between the outgoing graviton line and lines 3, 4. The other ingredient is the tree-level five-point amplitude, which in the classical limit is given by A + O( 0 ) as in (C.13) [52,57,59,91,92].Note the lack of O( −1 ) corrections to this leading O( −2 ) result, as discussed below Eq (C.13).Consistently with these general facts, we checked that our result (4.6) obeys 4.2 Factorization in the soft limit the one-loop five-point amplitude A µν 1 must factorize according to the Weinberg soft graviton theorem [53] as times the one-loop four-point amplitude A , where F = ε µ F µν ε ν in analogy with (2.19).The limit (4.20) is best taken after performing the decomposition in Eq. (A.3) and following, introducing also the exchanged momentum whose decomposition reads The limit (4.20) should then be understood for fixed u 1 , u 2 and q µ ⊥ , so that and to leading order To leading order in this limit, the five-point kinematics (2.4) reduces to the four-point one introduced in Ref. [93], Both sides of the factorization (4.22) should be expanded in the near-forward limit (3.1) in order to be applied to our results.One finds the following near-forward limit for the Weinberg factor [19], with Note the absence of O( 1) corrections in (4.30).For the other ingredient of (4.22), the one-loop four-point amplitude, one has instead [14,23,56] for generic ǫ, while we will only need the O( −1 ) term A Eq. (4.22) then translates into the following relations which can be also expanded for small ǫ.Of course, the soft limit of the 1/ǫ terms, 1 also obeys the factorization of infrared divergences.For the O( −2 ) term, this dictates 0 , i.e. (see e.g.Eq. (3.52) of [29]) For the O( −1 ) terms in (4.36), the absence of a Weinberg pole ∼ 1/λ is also ensured by (4.19) because W [1] carries an extra power of λ.
Constraints on the soft limit of A 1 independent of the exponentiation of infrared divergences instead involve the finite parts, We have checked that our results are consistent with these constraints.It is instructive to see how they translate to impact-parameter space, by letting where the 2 → 2 amplitude (see (C.1), (C.12) for the tree level) obeys the eikonal exponentiation [23,56] Ã(4) Since we work to leading order in the soft limit, we can apply the same Fourier transform (4.39) to the 2 → 3 amplitude as well, finding that (4.35) translates to were we have used that FT[q α ( • ] and the relation between the impulse and the eikonal phase up to 2PM, Of course the soft theorem also holds for the tree-level amplitude, and one has Combining (4.41) and (4.43) provides a check of the leading inelastic exponentiation of the one-loop level amplitude in b-space, to first order in the soft limit whereby the "superclassical" term of the one-loop inelastic amplitude factorizes in terms of the elastic tree-level amplitude times the inelastic one in b-space.Moreover, the relations (4.41) can be seen as a manifestation order by order in G of the non-perturbative pattern discussed in [33,60] according to which the soft dressing governing the soft theorem/memory effect for processes with generic deflections can be obtained from the Weinberg factor (4.21) by replacing the perturbative momentum transfer q α with the classical impulse Q α given by (4.42).

Imaginary parts and unitarity
The "superclassical" contributions are purely imaginary and, by unitarity, they must correspond to appropriate intermediate states for the 2 → 3 process under consideration.We find that they are equal to the the sum of two processes where one "cuts" two intermediate massive states, which we may term "S-channel" by analogy with the situation at four points, The first process on the right-hand side is obtained by gluing together a tree-level 2 → 2 amplitude A 0 involving four massive states (C.1) and a tree-level 2 → 3 amplitude A 0 for the inelastic process under consideration (C.2).The second process formally corresponds to a 2 → 3 amplitude glued together with a partially disconnected 3 → 3 one, in which the graviton line simply "passes through" the second blob, so that in practice it corresponds to gluing together A (4) 0 and A 0 in the opposite order.Equivalently, it can be obtained from the first one by applying the permutation σ 2 σ 3 , which flips ω 1 , ω 2 and leaves y unaltered.
We have checked Eq. (4.45) in two ways.First, by leaving the signs of the analytic continuations arbitrary as in (3.34), i.e. without imposing (3.35), one sees that the left-hand side of (4.45) is only sensitive to q I , the sign of the analytic continuation of y, which is the invariant associated to propagation in the S-channel.More precisely, denoting by f (q I , q O , q A ) this generalized version of the amplitude obtained by leaving the three signs of the analytic continuations arbitrary, the left-hand side of (4.45) coincides with the S-channel discontinuity Second, we have explicitly constructed the integrand for the right-hand side of (4.45) by gluing together A 0 and A 0 (for completeness, we provide their explicit expressions in Appendix C) in the classical limit and performed the integration over phase space via reverse unitarity [41,42,[94][95][96][97].This amounts to treating the Lorentz-invariant phase space delta functions like formal propagators, performing the IBP reduction while dropping all integrals that do not possess the cut (4.45), and lastly substituting the master integrals with 2 times their imaginary parts obtained by applying Disc S as defined by (4.46) (i.e. the imaginary parts associated to their S-channel discontinuities).
The purely imaginary term A arise instead due to intermediate processes whereby one cuts a massive line and a graviton line.Since these cuts are built using the "Compton" amplitude (C.7) involving two massive states and two gravitons, together with the tree-level 2 → 3 amplitude A 0 , we may term these "C-channel" cuts, (4.47) Once again these processes can be also thought of as 2 → 3 amplitudes glued with partially disconnected 3 → 3 ones.We have checked (4.47) in the same two ways as for (4.45), both by calculating the discontinuity of the f (q I , q O , q A ) with respect to q O (and separately q A ), and by building the integrand for the cuts and evaluating it via reverse unitarity. 4ombining (4.45) with (4.47), we reconstruct the complete unitarity relation for the one-loop amplitude,

Removing superclassical iterations
We follow Ref. [1] and consider the operator N linked to the S-matrix by As usual, we define their matrix elements by stripping a momentum-conserving delta function, where A α→β are the conventional scattering amplitudes.From (4.49), one trivially obtains at tree level.Going to the next order and inserting a complete set of free intermediate states to resolve the terms involving T 2 , one finds In view of the unitarity relation (4.48), all imaginary parts of A µν 1 cancel out and one is left with the real and IR-finite result In fact, the subtractions in the first line of (4.52) are enough in order to remove all O( −2 ) "superclassical" terms, while as already discussed the ones in the second line are O( −1 ).Both types of subtractions involve imaginary infrared divergences, the ones in the first line being associated to W [0] , i.e. to soft-graviton exchanges between massive lines, and the ones in the second line being associated to W [1] , i.e. soft-graviton exchanges between the graviton and an outgoing massive line, as also suggested by the respective figures.Letting B 1 = ε µ B µν 1 ε ν , as already mentioned, B 1 /N 4 has uniform transcendental weight 2 and takes the form of π 2 multiplying a rational function of the invariants.
Let us conclude this section by commenting on the parity properties of B 1 under the transformation 5ω 1,2 → −ω We find that with B 1E (resp.B 1O ) even (odd) under (4.54).In particular, the odd piece is equal to an x-dependent function times iπ times the coefficient of the −1 ǫ −1 pole Time-reversal-odd terms in the finite real part thus arise from the analytic continuation of logarithms left behind by the 1/ǫ in the imaginary part.This mechanism is highly reminiscent of how radiation reaction enters the eikonal phase at two loops [19,27,33,60,98] (see also [80? ]).
5 Gravitational Field, Spectrum and Waveform Following Refs.[2,40,43], let us model the initial state of the collision by with in terms of wavepackets ϕ 1,2 and impact parameters b 1 , b 2 .We can then take the expectation value of the graviton field in the state obtained by applying the S matrix (4.49), We denote this expectation by ("c.c" stands for "complex conjugate") Defining the Fourier transform by the generalization of (4.39) to the 2 → 3 kinematics, and introducing a shorthand notation for the wavepacket average we find where (5.10) Notably, although B 0 and B 1 are real and finite, the two infrared divergent C-channel cuts have "reappeared" in the loop-level result for the KMOC kernel W µν 1 in (5.10).This comes about because, to this order, using S = e iN = 1 Inserting a complete set of states, we see that at one loop the term in|a µν (k)N 2 |in includes all cuts depicted in Table 1, while out|N a µν (k)N |out only contains the second S-channel cut.Thanks to the factor of − 1 2 , this leads to a cross-cancellation of the S-channel cuts in (5.11), which leaves behind the C-channel cuts as in (5.10). 6e may rewrite W µν 1 as follows7 where we have used that the infrared divergence is proportional to the tree-level result by (4.19) and (4.18).By exponentiating it, we may also rewrite W µν in the following way in terms of the infrared-finite object As usual, the factorization of an infrared-divergent scale has left behind the logarithm of an arbitrary scale, and one can verify that such log μ2 terms neatly combine with the leftover log q 2 1,2 terms in Im A µν[−1,0] 1 to reconstruct logarithms of dimensionless quantities.Let us now turn to the spectral rate, where omitting the µν indices stands for contraction according to Then it is clear that, although W µν in (5.13) has an IR-divergent phase, the spectrum is free from infrared divergences, since this overall phase cancels out, and retaining terms up to (5.17) In principle this cancellation could leave behind ambiguities associated to the log μ2 terms in (5.13), (5.14).To see why this is not the case, let us denote by the combination appearing in the log μ2 terms, which is insensitive to the Fourier transform (5.7).Terms of type B 0 E log μ2 in the imaginary part M 1 of the waveform kernel enter the spectral rate via In this way, we see that the log μ2 terms in (5.13) do not contribute to the spectral rate and a fortiori to the energy emission spectrum.
One then considers a detector with four-velocity t µ placed at a spatial distance r from the scattering event in the angular direction characterized by the null vector nµ , so that and takes the asymptotic limit r → ∞ , u, nµ fixed. (5.21) In this limit, the asymptotic field (5.9) takes the form [43,99,100] so that using (5.13) (5.23)In this way, the classical information extracted from the one-loop amplitude can be used to build the O(G 3 ) corrections to the asymptotic metric fluctuation √ 32πG h µν , and its IRdivergent phase can be formally reabsorbed via a constant shift of the detector's retarded time [65,66].

Conclusions and Outlook
In this paper we calculated the 2 → 3 amplitude for the collision of two massive scalars and the emission of a graviton.We focused on the near-forward regime, where the exchanged momenta are small, O( ), compared to the masses, and on the soft region in which the loop momentum associated to the exchanged gravitons is of the same order.This allowed us to perform the integration of the integrand first obtained in Ref. [47], calculating the result up to and including O( −1 ) and O(ǫ 0 ).The result passes nontrivial consistency checks.It displays the appropriate structure of IR divergences predicted by Ref. [51] as well as the correct factorization in the soft limit [53].After checking that the operator version of the eikonal exponentiation [1][2][3] indeed works as expected and produces a classical, real and finite matrix element for the "eikonal", or more precisely N -operator [1], we sketched the calculation of the asymptotic waveform and spectral emission rates.We derived an expression for such quantities, showed that the spectra are free of ambiguities, while the waveform itself is affected by an IR divergent phase or, once such an irrelevant phase is discarded, by the presence of the logarithm of an arbitrary scale.
The appearance of logarithms of arbitrary parameters left behind by infrared divergences in waveform calculations is a manifestation of the so-called "hereditary" or "tail" effects [17,[67][68][69]101] and can be ascribed to an arbitrariness in fixing the origin of the detector's retarded time [65,66].It is interesting that these features already appear in the one-loop five-point calculation performed here, whereas they only intervene at three loops in the fourpoint calculation [17,18,102,103].We leave further investigations of this point for future work.Another interesting issue to which we plan to return is the comparison with subleading log-corrected soft theorems [72][73][74], which are also intimately related to tail effects and to the long-range nature of the gravitational force in four spacetime dimensions.In analogy with the tree-level case, such checks will likely require to first obtain sufficient analytic control of the b-space expression of the waveform.More generally, but also in connection with the issue of infrared divergences, which here we removed by following the exponentiation [51], it will be interesting to investigate how our results fit within the broader program of the eikonal operator and to understand whether an improved operator formalism is actually able to directly provide an infrared finite answer, possibly fixing the associated scale ambiguity.Of course, for all such open issues, extremely valuable guidance will come from comparisons with the available PN results (see e.g.Ref. [70] and references therein).