The Sub-Leading Scattering Waveform from Amplitudes

We compute the next-to-leading order term in the scattering waveform of uncharged black holes in classical general relativity and of half-BPS black holes in $\mathcal{N}=8$ supergravity. We propose criteria, generalizing explicit calculations at next-to-leading order, for determining the terms in amplitudes that contribute to local observables. For general relativity, we construct the relevant classical integrand through generalized unitarity in two distinct ways, (1) in a heavy-particle effective theory and (2) in general relativity minimally-coupled to scalar fields. With a suitable prescription for the matter propagator in the former, we find agreement between the two methods, thus demonstrating the absence of interference of quantum and classically-singular contributions. The classical $\mathcal{N}=8$ integrand for massive scalar fields is constructed through dimensional reduction of the known five-point one-loop integrand. Our calculation exhibits novel features compared to conservative calculations and inclusive observables, such as the appearance of master integrals with intersecting matter lines and the appearance of a classical infrared divergence whose absence from classical observables requires a suitable definition of the retarded time.


Introduction
Much like accelerated charges in Maxwell's theory emit electromagnetic radiation, accelerated masses in gravitational theories emit gravitational radiation. Increasingly more precise waveform models, yielding the asymptotic space-time metric, play a critical role in the detection and analysis of gravitational wave signals from compact binaries [1]. Current waveform computations for binaries in bound orbits use effective-one-body methods [2][3][4][5][6] and numerical-relativity approaches [7,8], in addition to direct solutions of Einstein's equations sourced by the motion of the bodies in the post-Newtonian approximation [9,10], and computations in the effective-field theory approach pioneered by Goldberger and Rothstein Ref. [11][12][13][14][15], see [16,17] for reviews.
The post-Minkowskian (PM) approximation, keeping exact velocity dependence for each power of Newton's constant, is natural for bound binary systems on eccentric orbits and for unbound binaries [18]. Inclusive dissipative observables through O(G 3 ) and waveforms to leading order in Newton's constant have been discussed with traditional methods in e.g. [19][20][21][22][23][24][25] and amplitude and worldline methods in Ref. [26][27][28][29][30][31][32][33][34][35][36] and at O(G 4 ) in [37]. The properties of scattering waveforms -such as the absence of periodicity, short duration, and low amplitude -pose a challenge for current gravitational-wave detectors. They may, however, be interesting goals for future terrestrial and space-based detectors. From a theoretical standpoint, scattering waveforms are an important part of the program to leverage scattering amplitudes and quantum field theory methods for precision predictions of gravitational wave physics, complementing the effort to determine the conservative motion and inclusive dissipative observables. It is an important challenge for the future to identify the analytic continuation that connects them to bound-state waveforms, as it is the case for certain (parts of) inclusive observables such as the scattering angle vs. the periastron advance and the energy loss [38][39][40][41].
A lesson from the calculations of conservative effects is that amplitudes exhibit vast simplifications when expanded in the classical limit. An important feature of this expansion is that the leading term is not always the classical one. The first L terms at L loops are classically singular, or super-classical, and can be interpreted as iterations of lower-loop terms. Since the KMOC formalism contains terms bilinear in amplitudes, one may naively expect to find classical contributions that arise from the interference between superclassical terms in one factor and classically subleading (quantum) contributions from the other. Intuitively however such terms should not contribute to classical physics and we demonstrate explicitly that at one-loop order this is indeed the case. Assuming the exponential representation of the scattering matrix proposed in Refs. [60,61] we derive a necessary condition for a given term to be included in the classical amplitude.
The heavy-particle effective theory (HEFT) of Refs. [62,63] provides a strategy to isolate the classical amplitude, discarding the super-classical and quantum operators from the outset. It has been demonstrated in Ref. [63] that this approach indeed gives the correct four-point classical amplitudes through two-loop order. The use of this framework for the construction of the one-loop five-point amplitude with outgoing radiation requires that the i0 prescription of the uncut linearized matter propagators be clearly stated. For four-point amplitudes specifying a presciption is not required through two loops because the uncut matter propagators are always off-shell due to kinematic constraints. This is no longer the case in the presence of an outgoing graviton; sewing a one-loop five-point amplitude into a four-point higher-loop one suggests that a similar feature may appear at three loops for this multiplicity. In this paper, we find, by comparing the HEFT and direct calculation of the classical one-loop five-point amplitude in GR coupled to scalars, that the correct prescription at this order for these uncut matter propagators is that they are principalvalued. With this prescription, the HEFT construction reproduces the classical expansion of the full theory amplitude once the super-classical terms are subtracted from the latter.
We also compute the classical amplitude and scattering waveform for half-BPS black holes in N = 8 supergravity [64,65]. We use the existing d dimensional one-loop five-point amplitude of maximally supersymmetric Yang-Mills theory [66,67], the double copy [51] and dimensional reduction to construct the relevant parts of the four-massive-scalar-onegraviton amplitude in four dimensions. Their remarkable simplicity leads to a (relatively) compact integral representation for the waveform.
We encounter several interesting features, not present in the amplitudes-based approach to inclusive observables. One of the requirements of the classical limit is that the matter particles are always separated. In the four-point classical amplitudes relevant for inclussive observables, this requirement leads to the absence of diagrams with intersecting matter lines. As we will see in section 4, the five-point amplitude relevant for the scattering waveform does receive contributions from certain graphs with this topology. We will understand how their presence is consistent with the separation of the matter particles. Furthermore, we find that the presence of the outgoing graviton introduces a certain asymmetry between matter lines in contributing diagrams and enhances the importance of the i0 prescription. Last, but not least, the classical five-point amplitudes in both GR and N = 8 supergravity are infrared divergent. The structure of infrared divergences in quantum gravitational amplitudes was understood long ago in Ref. [68]. We remarkably find that the IR divergence of the classical amplitude is a pure phase that can be absorbed into the definition of the retarded time. This indicates that the asymptotic waveform contains no information about the lapsed time between the scattering event and observation.
We directly integrate the resulting master integrals and find the complete classical amplitude. The construction of the asymptotic spectral waveform requires a further Fourier transform to impact parameter space while that of the time-domain waveform also requires a Fourier transform of the outgoing graviton frequency.
The late-time properties of the waveform can be inferred directly from its integral representation. We find that, while there is an O(G 2 ) correction to the gravitational wave memory in GR, such a correction is absent in N = 8 supergravity. We will later demonstrate that the memory is proportional to the scattering angle at O(G 2 ). Thus, we trace the absence of memory in N = 8 supergravity to the absence of an O(G 2 ) correction to the scattering angle or, equivalently, to the absence of one-loop triangle integrals in this theory. For both N = 8 supergravity and GR, we analytically evaluate the frequency integral and all but one of the integral transforms to impact-parameter space. The integral was evaluated numerically. We leave a complete analytic evaluation of the spectral and time-domain waveform to future work.
Our paper is organized as follows. In section 2 we review the classical limit of amplitudes and the HEFT of Refs. [62,63]. In section 3 we review observable-based formalism for waveforms, demonstrate the cancellation of two-matter-particle-reducible (2MPR) contributions, and spell out the relation between the two-matter-particle-irreducible part of the amplitude and the waveform. In section 4, we obtain the HEFT prediction for the classical one-loop five-point amplitude and compare it with the result of direct calculation in GR coupled to massive scalar fields. In section 5, we construct the classical one-loop five-point amplitude in N = 8 supergravity using the double copy. In section 6 we discuss aspects of the integration of the relevant one-loop master integrals while leaving further details to appendix B. In section 8 we numerically compute the waveform for N = 8 supergravity and discuss our results. In section 9, we discuss our conclusions. Appendix A contains an argument that all super-classical and interference between super-classical and quantum terms (i.e. / contributions) terms correspond to 2MPR graphs and thus do not contribute to local classical scattering observables.

Note added:
While this paper was in preparation, we became aware of Refs. [69] and [70], which partly overlap with aspects of our analysis. We thank the authors for communicating and for sharing copies of their drafts prior to publication.

The classical limit and HEFT amplitudes
Integrands of scattering amplitudes simplify considerably in the classical limit [71][72][73]. It is therefore advantageous to take this limit as early as possible and weed out terms that do not contribute to classical observables from the outset.
One approach to this limit uses the correspondence principle, according to which classical physics emerges in the limit of large charges. Thus, considering the scattering of two massive spinless bodies, the classical regime emerges for masses much larger than the Planck mass and for orbital angular momenta much larger than unity (in natural units). This limit also corresponds to the inter-particle separation being much larger than the de Broglie wavelength. As the separation of particles is Fourier-conjugate to the change in the momentum of each particle, it follows that the momentum transfer q i is much smaller than the momenta of the two particles. If there is any massless radiation in the initial or final state, its momentum should be much smaller than the massive particles. This is implemented by the rescaling (q, k, ) → (λq, λk, λ ) , (2.1) where k and are, respectively, the momenta of external and internal gravitons, and expanding at small λ. This process is later called the soft expansion. Accounting for the classical nature of Newton's potential, classical L-loop four-scalar amplitudes depend on Newton's constant and λ as M cl. 4+ng,L-loop ∼ G L+1+ 1 2 ng λ −2+L , where n g is the number of external gravitons.
Another perspective on the classical limit was taken in Ref. [42] and involves a suitable restoration of the dependence on Planck's constant and an expansion at small . From this perspective, external momenta and masses pick up a factor of −1 , which is equivalent to messenger momenta picking up a factor of through a change of variables. Thus, Planck's constant effectively plays the role of the momentum transfer and its restoration in four dimensions, realizes the classical limit as a limit of small messenger momenta. With this scaling, the classical four-point amplitudes scale as M cl. 4,L-loop ∼ −3 , which can also be recovered from the correspondence principle perspective by identifying λ and and further rescaling G → G/λ, so that the contributions to the classical amplitude have the same scaling at all loop orders.
The generalization to amplitudes with four scalars and any number of gravitons is straightforward based on the observation that the emission of arbitrarily-many low-energy messengers is a classical process, so it should not involve additional quantum suppression. Thus, allowing for an arbitrary (even) number of scalars n φ and messengers n g , the amplitude scales as in four dimensions. Indeed, eq. (2.3) can be verified for the tree-level five-point amplitude of Ref. [74] and we will use it to extract the classical limit of the one-loop four-scalar-onegraviton amplitude. As for the four-point amplitude, the method of regions provides a systematic way of isolating the relevant contributions to the amplitude [75]. Interestingly, unlike the one-loop four-scalar amplitude, not all internal messengers need to be in the potential region. However, this fact is unsurprising as, through the unitarity method, the one-loop five-point amplitude is part of the three-loop four-point amplitude, which receives contributions from messenger momenta in the radiation region. The identification of the classical limit as a large-mass expansion accompanied by small messenger momenta establishes a connection with heavy-quark effective theory [76][77][78][79], first utilized for classical gravitational scattering in Ref. [80] and further developed in Ref. [81,82]. One approach to constructing the heavy-particle effective theory starts with the action of a scalar field coupled to gravity, Building on the assumption that the messenger momenta are soft, one considers a process in which scalar fields exchange some gravitons. Decomposing the soft part of the scalar momenta and redefining the fields so that they only depend on the soft momenta, leads to the new action where we have neglected the terms with a highly oscillatory phase e ±2imu·x . Denoting the momentum of χ as q, the propagator is 1 As one might expect from the propagator of a massive field, the leading term is O( −1 ), the next to leading term is O( 0 ), etc. Upon constructing scattering amplitudes from the Lagrangian in eq. (2.6) extended with the Einstein-Hilbert action, all vertices on a matter line must be symmetrized as a consequence of Bose symmetry. Therefore, all leading-order propagators are replaced by Dirac delta functions through the identity and its multi-propagator generalization [83]. This phenomenon can be visualized as where the red vertical line represents the cut massive propagator, and the two blobs connected by the cut commute (2.10) To leading order in the classical expansion, the exposed propagators of heavy scalar states can be treated as on-shell in the gravitational heavy-particle effective theory. At loop level, we also encounter propagators that form a principal value combination, (2.11) As we will see later, they often show up in classical amplitudes. We take a classical expansion of the full tree amplitudes to compute the HEFT tree amplitudes. The HEFT amplitudes are naturally organized as a sum of terms manifesting the factorization properties of Feynman diagrams with the on-shell matter propagator being linearized. Doubled (and perhaps higher powers of) linearized propagators, arising from the expansion in eq. (2.7), are also present. This is analogous to the structure obtained by directly expanding in the soft region. A more systematic way to construct the classical parts of tree-level HEFT two-scalar-graviton amplitudes without reference to a Lagrangian (though probably equivalent to one) that also exhibits double-copy properties was proposed in Ref. [62]. For example, the three and four-point gravitational Compton amplitudes, expanded up to the classical order, are given by 1 3 The three-point amplitude scales classically under eq. (2.2) as O( −1/2 ). The first term in the four-point amplitude exhibits super-classical O( −2 ) scaling, which contains the characteristic delta function that localizes it on a special momentum configuration, while the second term scales classically as O( −1 ). The classical part of the amplitude is referred to as the "HEFT amplitude".
At loop level, we will focus on amplitudes with four scalars (two distinct massive scalar lines). The classical expansion naturally decomposes amplitudes into two matter-particle reducible (2MPR) and two matter-particle irreducible (2MPI) contributions. We define the 2MPR contribution as follows: • The diagram becomes disconnected by cutting two matter propagators.
• The two cut matter propagators are both represented by δ(p i · ), and the residue on the cut follows from the factorization of the amplitude's integrand.
Being complementary to 2MPR, the 2MPI contributions include the following two classes of diagrams: 2. If the diagram becomes disconnected after cutting two matter propagators, then at least one of the matter lines exhibits a principal-value propagator. Consequently, this cut has zero residue.
Therefore, by construction, the 2MPR diagrams are given by the product of their 2MPI components on the support of the explicit delta functions that enforce the two-matter cuts.
Some simple examples of 2MPR and 2MPI diagrams are given in figure 1.
It was further demonstrated in Ref. [63] that, together with a suitable application of generalized unitarity, the classical HEFT tree amplitudes, such as eq. (2.12) and the second term of eq. (2.13), yield the classical part of the two-loop four-scalar amplitude, which can be identified with the radial action. As expected, the classical contributions are all 2MPI. More generally, the prescription of Ref. [63] manifests the 2MPR versus 2MPI classification by construction, such that the 2MPI contributions have the correct classical scaling while the 2MPR parts are super-classical. The only way to make the 2MPR diagrams have a classical contribution is to include local quantum contributions in the HEFT tree amplitudes. Such terms are, however, discarded from the outset; since they can appear in a complete calculation, one may wonder whether such / contributions can appear in classical scattering observables. By comparing the HEFT calculation of the five-point amplitude with the analogous result GR coupled with scalar fields we will see that such contributions are in fact absent. We postpone for appendix A a more general discussion of these points.
Finally, an essential aspect of using the HEFT tree amplitudes in a unitarity-based construction of loop-level amplitudes is fixing the i0 prescription for the uncut linearized matter propagators. We find that for the one-loop five-point problem considered in this work, we can treat all the uncut linearized matter propagators as principal-valued. We justify this approach by showing that, with this prescription, the HEFT result agrees with the classical part of the full quantum amplitude. It is beyond the scope of this work to identify a general i0 prescription for the uncut matter propagators when applying the HEFT construction at loop levels.

Waveforms in the observable-based formalism
We begin this section by briefly reviewing the observable-based approach of Kosower, Maybee, and O'Connell [42] (KMOC) to inclusive and local scattering observables. We then discuss certain cancellations present in this formalism, which can be made manifest through the use of the HEFT organization of amplitudes.
The KMOC formalism constructs quantum scattering observables, described by some operatorÔ, by comparing the expectation value ofÔ in the final and initial states of the scattering process: where ψ out and ψ in are the outgoing and incoming state respectively. Further using that the incoming and outgoing states are connected by the time-evolution operator whose matrix elements form the scattering matrix,Ŝ, andT is the transition matrix, eq. (3.1) becomes [42] Thus, using eq. (3.3), scattering observables are expressed in terms of (i.e. phase space integrals of) scattering amplitudes (i.e. matrix elements ofT ) and their cuts, dressed withÔ. The operatorsÔ may correspond to global (inclusive) observables such as the impulse of the matter particles or the radiated momentum [42], to local (exclusive) observables of the waveform of the outgoing radiation [43], or to a combination of both. The corresponding classical observables can be obtained in the appropriate classical limit. Impact parameter space provides one convenient means to taking this limit. The classical limit corresponds to an impact parameter significantly larger than the de Broglie wavelengths of the scattered particles and the horizon radii of black holes of masses equal to their masses; these, in turn, imply that the orbital angular momentum is large, making contact with Bohr's correspondence principle. To make use of this, and assuming that the initial state contains no incoming radiation, the initial two-particle state is built in terms of wave packets over the tensor product of a single-particle phase-space of measure dΦ[p]: The combination b 1 − b 2 is the impact parameter (i.e. the separation between the incoming particles), while b 1 + b 2 is the conjugate of the momentum of the center of mass. The wave packets φ i are sufficiently localized not to interfere with the classical limit conditions [42], i.e. their widths φ(p i ) obey horizon namely, it is much greater than the horizon size horizon of the black holes but much smaller than their separation.

A brief review of the waveform in the observable-based formalism
The waveform in the infinite future, obtained by measuring the linearized Riemann curvature tensor [43], is the focus of this paper. The relevant operator, written in terms of graviton creation and annihilation operators, is where the antisymmetrization has strength 2, and dΦ(k) is given by eq. (3.4) at zero mass, ε h (k) are polarization vectors normalized as In the second relation, we choose h = ± to represent the ±1 helicity state. The operator a hh (k) annihilates a graviton with polarization ε µν h = ε µ h ε ν h . We note that, even though this operator superficially depends on an arbitrary space-time point, x, the formulation of the measurement process through scattering theory in eq. (3.1) implicitly assumes that this point is both in the far future and at spatial infinity.
Assuming temporarily that the observable is measured at finite distance and there is no gravitational radiation in the infinite past, the waveform is given by [43,84] At large distances, |x 0 | → ∞ and |x| → ∞, the integral over the angular directions of the on-shell graviton momentum, k = ω(1, n k ) , with ω > 0 and n 2 k = 1 , (3.10) can be evaluated through various methods [43,84] and each of the exponentials in eq. (3.7) leads to a linear combination of advanced and retarded propagators while n k is localized to the spatial unit vector n ≡ x/|x| at the observation point x, In the infinite future, only the terms depending on the retarded time t = x 0 − |x| are relevant. We thus drop the second line of eq. (3.11). The waveform W µνρσ (t, n) for the curvature tensor and the spectral waveform, f µνρσ (ω, n), is given by A convenient presentation of the curvature tensor (and consequently of the gravitational waveform) is in terms of Newman-Penrose scalars [85]. They are constructed as projections of the curvature tensor on a complex basis of null vectors. Following Ref. [43] we choose these vectors to be where M · M * = −1 following eq. (3.6), and ζ is a gauge choice such that ζ · ε ± = 0 and L · N = L · ζ = 1. The independence of L on the frequency of the outgoing graviton makes eq. (3.13) a suitable basis both for the waveform and the spectral waveform. The Newman-Penrose scalars are defined by the independent contractions of the Weyl tensor with the vectors in eq. (3.13). The one with the slowest decay at large distances, typically denoted by Ψ 4 , describes the transverse radiation propagating along L [85], where the ellipsis stands for terms suppressed at large distance. Using the transversality and null property of M µ and eq. (3.11), we can write the spectral representation of Ψ ∞ 4 as For an asymptotically flat spacetime, outgoing radiation at large distances is described by linearized general relativity in transverse traceless gauge. Using that k · N = ω, the Newman-Penrose scalar Ψ 4 takes the form For k = (ω, 0, 0, ω), the negative-helicity polarization vector is ε − = − 1 √ 2 (0, 1, i, 0) and the + and × graviton polarizations are In general, the × and + polarization are defined with respect to the vector L µ pointing along the graviton momentum. They are related to the real and imaginary parts of the outgoing negative helicity polarization tensor, see section 8. The metric perturbation h µν is in transverse traceless gauge and is normalized in such a way that, at spatial infinity, it falls off as Therefore, we may directly identify the Fourier transform of the metric at infinity in terms of the frequency-space Newman-Penrose scalar: Thus, in frequency space, the standard gravitational-wave polarizations, h +,× , are given directly in terms of scattering amplitudes. Let us now discuss the matrix elements ψ in |Ŝ †â −− (k)Ŝ|ψ in and ψ in |Ŝ †â † ++ (k)Ŝ|ψ in , which defineJ andJ † . We will focus on the former and obtain the latter by complex conjugation. We first go to momentum space and consider the matrix element with an initial state with momenta p 1 and p 2 , We then expandŜ in terms of the transition matrixT , which leads to The first term is simply the 2-to-3 scattering amplitude , (3.22) where M contains implicitly the momentum-conserving delta function, and M is the reduced amplitude with this delta function stripped off. The second term gives the s-channel unitarity cuts of this virtual amplitude, where we have judiciously inserted a complete set of states betweenT † andâ hh (k), and X stands for graviton exchanges. This term can be graphically represented by We note that X implicitly contains both the integration over the graviton phase spaces and the sum over polarizations, together with symmetry factors for identical particles. The phase factor in eq. (3.20) can be reorganized using the momentum-conserving delta functions. We introduce q i = p i − p i , which are related to the momentum of the outgoing graviton as q 1 + q 2 = k. The phase factor thus becomes e iq 1 ·b 1 +iq 2 ·b 2 = e iq 1 ·(b 1 −b 2 )+ik·b 2 . The second term can be absorbed into the e ikx factor in eq. (3.7) by redefining the position x.
Since the b i are finite, this redefinition is irrelevant at large distances. We can thus choose the impact parameter to be b = b 1 −b 2 , which is the Fourier-conjugate of q 1 . 2 Consequently, it is equivalent to performing the Fourier transform in eq. (3.20) as Finally, the soft expansion in the classical limit, q i p i , introduces substantial simplifications. After introducingp i ≡ p i + (1/2)q i , we can rewrite the measure dΦ(p i ) as whereδ(x) ≡ 2πδ(x), and we have used the fact that external physical particles always have positive energy. We further identify the initial and final momentum space wave packets in this limit, φ(p i ) → φ(p i ) and φ * (p i ) → φ * (p i ). Accounting for these features of the classical limit, eq. (3.25) now takes its final form, where the matrix element p 1 − q 1 , p 2 − q 2 |Ŝ †â hh (k)Ŝ|p 1 p 2 should also be evaluated in the classical limit. The matrix element ψ in |Ŝ †â † hh (k)Ŝ|ψ in follows by complex conjugation. The reason we included an explicit factor of (−i) is to make B h be given by amplitudes directly, which are defined as matrix elements ofT , see eqs. (3.21) and (3.22). We will now discuss some important properties of these matrix elements.

On the structure of local (exclusive) observables
The matrix element p 1 p 2 |Ŝ †â hh (k)Ŝ|p 1 p 2 determining the spectral waveform is written out explicitly in terms of scattering amplitudes in eq. (3.21). As discussed in Refs. [29,86] the two different i0 prescriptions that appear in the last term in eq. (3.21) pose no difficulty to the evaluation of inclusive observables. For example, in the related KMOC calculation of inclusive observables [29,86], the terms bilinear in amplitudes were evaluated through reverse unitarity [87][88][89]. We may simply construct the five-point virtual amplitude M(p 1 p 2 → p 1 p 2 k h ) and then take its s-channel cut to find the second term in eq. (3.21). 3 However, such direct calculations can obscure additional simplifications as propagators with opposite i0 conventions can exhibit nontrivial cancellations. In particular, we argue that only cuts of 2MPI graphs contribute to classical local observables due to such cancellations.
At one loop, it is clear from eq. (3.21) that the 2MPR graphs are subtracted in the classical waveform. The second term in eq. (3.21), p 1 p 2 |T †â h (k)T |p 1 p 2 , evaluates to the 2MPR contribution plus one-loop 3-particle cuts, while the first diagram, being 2MPR and superclassical, manifestly cancels the corresponding contribution in p 1 p 2 k h |iT |p 1 p 2 , the first term of eq. (3.21). The diagrams in the bracket only have zero energy support, and they further subtract out certain s-channel cuts. We now demonstrate this feature for generic cases, namely, the classical waveform does not contain 2MPR diagrams. Consequently, only (cuts of) 2MPI diagrams contribute to the waveform at all loop orders. To see this, it is convenient to use the exponential representation of the classical Smatrix found in Refs. [60,61], which we now briefly review. Ref. [60] argued that, in the classical limit, the conservative 2-to-2 S-matrix has an exponential representation, S = e iIr (3.29) where the exponential is defined via its series expansion with the product being the integral over the two-matter-particle phase space. It was further argued in Ref. [61] that this exponential structure continues to hold as the exponent is promoted to an operator that has 2 + n → 2 + m matrix elements, where n and m are initial and final state graviton whereN is a Hermitian operator and the product of operators is defined by inserting the identity operator in the complete Hilbert space of states, where P 2,m is the identity operators in the (2 + m)-particle Hilbert space. We can perturbatively expandN in κ, where the subscript denotes the loop order, and the superscript signals the presence of graviton emission. We note that graviton emission also contributes to even orders of κ, which we suppress here for simplicity. We can perform a similar expansion forT and solve N i , the matrix element ofN i , order by order. For example, when restricted to two-particle initial and final states, we have [61], where the blobs represent virtual amplitudes, which are matrix elements ofT , and the cut propagator is integrated with measure dΦ[p]. Crucially, we now prove that the matrix elements ofN do not exhibit local s-channel cuts by induction. A term exhibits a local s-channel cut if the term contains an s-channel cut whose residue can be interpreted as a product of trees. For N 1 , this is apparent due to the explicit subtraction of the cut shown in eq. (3.33), and one can check this property order by order through direct computations. To see this for more generic cases, we carry out the expansion in eq. (3.30) up to a certain power of κ. Should anN i in this expansion contain an s-channel cut in addition to those explicitly present due to the insertion of the projectorsP 2,m , then it would contribute a termÂ aP2,mBb ∈N i . BothÂ a andB b here, which may contain graviton emissions, have a lower power of κ. They should have already been included in the respectiveN a andN b , otherwise it would be inconsistent with the symmetry factors coming from eq. (3.30). Therefore, we have proved thatN i is free of s-channel cuts. We now substitute eq. (3.30) into eq. (3.1), finding a sum of nested commutators, The above discussion is completely general, applying to the full quantum theory. This form of the S-matrix operator manifests all the s-channel cuts, which can only appear between twoN operators through the insertion of the phase-space projectorP 2,n . Restricting ourselves to the waveform operator, or more generally those that only measure properties of particular external states, one finds additional identities among the matrix elements in the classical limit when a two-massive-particle projector is inserted, This relation follows from reformulating eq. (2.10) in operator language, which states that the difference between the two matrix elements in eq. (3.35) is subleading in the momentum transfer q. It is crucial that we are considering an operatorÔ that does not participate in the integrals implied by the phase-space projector. As discussed above,N does not contain local s-channel cuts. We have therefore demonstrated the absence of two-massiveparticle cuts at all loop orders. 4 As a direct consequence of this result, only 2MPI diagrams contribute to the classical waveform in the KMOC formalism. This generalizes the intuitive picture that the iteration part of the scattering amplitudes (often superclassical) should not contribute. We will evaluate the 2MPI part of the classical five-point amplitude at one loop order in sections 4 to 6.

Infrared divergences of amplitudes and the classical waveform
Five-point (and in general n-point) gravitational amplitudes are typically IR divergent. In cross-section calculations, some divergences exponentiate to a harmless total phase while others must be removed by summing over final state radiation [91,92] or dressing the external states [93][94][95]. In observables that are both linear and bilinear in scattering amplitudes, such as the waveform, some of these divergences are in the 2MPR contributions and therefore cancel against the bilinear-in-T terms in eq. (3.22). It is important to understand the IR divergences of the surviving 2MPI diagrams on e.g. eqs. (3.12a) and (3.27) as these are relevant for classical observables. However, in the classical limit, we find the surviving 2MPI IR divergences correspond to a total phase that can be safely absorbed into a linear re-definition of t in eq. (3.12). Similar treatment was first discussed in Refs. [13,14] in the context of PN expansion.
Ref. [68] famously showed that the virtual IR divergences of gravitational amplitudes come from loop-momentum integration regions in which a graviton connecting the external particles with momenta p a and p b becomes soft. They factorize and exponentiate as [68] Figure 2: Typical contribution to the IR divergence. The soft graviton is shown in red.
where M(α → β) is the all-order α → β amplitude, and M 0 (α → β) is its counterpart without the virtual soft gravitons. In the exponent, the factor (p a ·p b ) 2 − 1 2 m 2 a m 2 b comes from the contraction of two stress-energy tensors with the numerator of the graviton propagator. The summation a,b runs over all the unordered pairs (a, b) of external particles and η a = ±1 depending on whether p a is outgoing or incoming. We follow Ref. [96] and dimensionally regularize J ab by using d = 4−2 , µ is the scale of dimensional regularization, and Λ is the cutoff | 2 | < Λ 2 that defines the virtual soft momenta. 5 RR: Some tweaks.
The IR-divergent integral J ab has both a real and an imaginary part [68]. One option to eliminate them and define IR-finite S-matrix elements is by choosing suitable asymptotic states [94,96,97]. While we will not pursue this approach here, it would be interesting to understand if it can be realized by judiciously choosing the wave packets φ(p i ) in eq. (3.4). Instead, we will show here that, in the classical limit, the 2MPI diagrams do not contribute to the real IR divergence. The remaining IR-divergent phase can be absorbed into the definition of the time variable of the waveform.
Following Ref. [68], let us consider the scattering of two massive particles (labeled as 1 and 2) with graviton emission in the final state. In the classical limit (that is, expanding in the soft region), all the matter propagators are linearized. The counting further implies that, for a connected amplitude, there can be at most one graviton (labeled as k) in the final state which can be relevant to classical obserables [98,99]. 6 In the following discussion, the "virtual soft graviton" is even softer than the soft region, namely, Λ |q| and |k|. Under this setup, the IR divergence receives contributions from the following three configurations shown in figure 2: (I) The virtual soft graviton starting and ending on the same particle: For such a configuration, the soft expansion implies that the corresponding loop integral is either scaleless or its propagators are linearly-dependent. The former has the topology of an external bubble, which integrates to zero in dimensional regularization. 7 The latter 5 These momenta should note be confused with the momenta in the soft region as defined in eq. (2.1). 6 We note that disconnected amplitudes at higher points may still contribute to KMOC-type observables on the support of zero graviton energy. However, as we will see later, such configurations are not relevant for the waveform. 7 A regulator, corrresponding e.g. to an external particle being slightly off shell, may be required to prevent matter propagators from being on shell and to allow the determination of contributions from graphs with the topology of bubbles on external lines. In the complete amplitude the potentially-singular propagator cancels out and the regulator can be removed explicitly. While the bubble integral is not scaleless in the presence of the regulator, it is so -and thus vanishes in dimensional regularization -after requires partial fractioning [29,100] and the resulting integrals are either scaleless or finite. Thus, this graviton configuration does not lead to an IR divergence in the classical limit.
(II) The virtual soft graviton connecting different massive particles: In the classical amplitudes, the incoming and outgoing momenta of the same massive particle are actually equal. This is because, in the full amplitude, their difference is quantum, and the soft expansion homogenizes the counting in each diagram. As a result, the exponent in eq. (3.36) becomes This term contributes a real IR divergence, but it belongs to the 2MPR part of the amplitude. Therefore, this configuration does not contribute to the IR divergence of the 2MPI amplitude. This also shows that in the 2 → 2 scatterings, the IR divergence is real and captured by 2MPR diagrams, while the 2MPI contributions are finite.
(III) The virtual soft graviton connecting a massive and a massless particle: For this configuration, the exponent of eq. (3.36) is given by The evaluation proceeds by first integrating over 0 , following [68,96]. For the first integral in eq. (3.39), we close the 0 contour from above, picking up the pole 0 = −| | + i0 and 0 = 1 ω k k k · + i0. For the second integral, we close the contour from below, picking up the pole 0 = | | − i0. The contribution from −| | + i0 and | | − i0 cancel each other, leaving only an imaginary IR divergence from the pole 0 = 1 where we have used the fact that p b · k > 0 for physical processes. If there are more external gravitons, then the internal soft graviton connecting two external gravitons may the regulator is removed, so we may take it to vanish from the outset.
also contribute to the IR divergence. However, as we have discussed before, amplitudes that are relevant to classical physics can only contain at most one external graviton [98,99]. Therefore, we have shown that for classical 2MPI amplitudes, the IR divergence is purely imaginary. For the four-scalar-one-graviton amplitude, it is a pure phase To leading order in Newton's constant M 0,2MPI,cl. (p 1 p 2 → p 1 p 2 k) is the tree-level five-point amplitude. We will indeed verify the one-loop part of this relation in section 7.
We can now assemble the waveform from eqs. (3.12a), (3.25) and (3.27) and understand the fate of the IR-divergent phase. Indeed, after the cancellation of the 2MPR part of the five-point amplitude, the remaining 2MPI part and the bilinear-in-T contribution to the matrix element in eqs. (3.25) and (3.27), and consequently the spectral waveform (3.12b), have the same IR-divergent phase. Using the solution for the on-shell condition in eq. (3.10) for the outgoing graviton, we can write its argument as which may be removed by defining the observation time as where t is the retarded time first defined in eq. (3.11). We thus conclude that the IRdivergent phase of the classical five-point amplitude can be removed by choosing a suitable origin of the observation time or, alternatively, focusing on observation-time intervals. IR divergences similar to those of amplitudes appear in the far-zone EFT; in the PN expansion they were discussed in Refs. [13,14] where they were also absorbed in the definition of the retarded time.
Interestingly, the dependence of the scattering amplitude with no soft-graviton contribution, M 0,2MPI,cl. (p 1 p 2 → p 1 p 2 k), is irrelevant after the soft divergence is absorbed in the definition of the time. Indeed, these O( ) terms contribute to the → 0 limit only if they are multiplied by some 1/ factor. Since all IR divergences are eliminated from the spectral waveform by eq. (3.43) and the classical amplitude is free of ultraviolet divergences, 8 it follows that O( ) can be ignored when evaluating the waveform f µνρσ (ω, n) or R µνρσ (x) in eq. (3.12).

Waveforms from amplitudes: summary and further comments
We will discuss the calculation of waveforms at leading order and next-to-leading order by applying this formalism to classical N = 8 supergravity and GR in section 8. To facilitate this application, we collect here the relevant formulae and further organize them, using the properties of amplitudes in the classical limit to streamline their connection to waveforms in the time domain.
The spectral waveform (or the frequency-space curvature tensor), the frequency-space Newman-Penrose scalar, and the frequency-space metric in a transverse-traceless gauge are given by eqs. (3.12b), (3.15) and (3.19). After the IR divergence is absorbed in the definition of time, they become where the superscript 0 indicates that the IR divergences have been absorbed in the definition of the observation time τ , see eq. (3.43). The term proportional to δ (ω) is a gauge degree of freedom, and we will ignore it in the following. The remaining δ(ω) term in eq. (3.46) can be present due to specific contributions that only have support on zero graviton energy and will, at most, lead to a time-independent background that the initial condition will fix. Additionally, such terms are irrelevant specifically for the evaluation of the asymptotic Newman-Penrose scalar Ψ ∞ 4 because of the additional factor of ω 2 in eq. (3.45). The time-domain observables follow from the Fourier-transform in eq. (3.12a), The matrix element (−i) ψ in |Ŝ †â hh (k)Ŝ|ψ in and its conjugate (+i) ψ in |Ŝ †â † hh (k)Ŝ|ψ in are then computed using eq. (3.27), where, as for frequency-domain observables, the superscript 0 indicates that the IR divergent phase has been removed from the matrix element and absorbed into the definition of the observation time. The gravitational memory of the observable F is defined as the difference between its initial and final values, which can be derived by integrating the derivative of F between τ = ±∞. Therefore, the memory is determined solely by the residue of F at zero frequency.
To evaluate frequency-domain observables, it is necessary to evaluate the q 1 and q 2 integrals in eq. (3.48); only one of them is nontrivial because of the momentum-conserving constraint The two explicit delta functions, as well as the phase factor, suggest that it is convenient to decompose the integration variable into components alongū 1 ,ū 2 , b, and a fourth vector orthogonal on these, as described in Ref. [43].
To evaluate classical time-domain observables, it is convenient first to evaluate the ω integral because it localizes parts of the remaining integrals. To expose this structure, we use properties of classical amplitudes -and thus of the matrix elements ψ in |Ŝ †â hh (k)Ŝ|ψ in -under the soft-region rescaling in eq. (2.1), where λ 2−L comes from the scaling of the L-loop classical amplitude, and λ d comes from the momentum conserving delta function δ (d) (q 1 + q 2 − k) implicit in the matrix element. Choosing λ = ω −1 and changing integration variables q i = ωq i we may therefore isolate the ω dependence of B to the Fourier phase and overall factors: wherek = k/ω = (1, n). The tildes on q i can now be dropped as they are dummy integration variables. Schematically, R L and I L are defined as where Re(M L loops µν ) and Im(M L loops µν ) are respectively the real and imaginary part of the L loops with the polarization vectors stripped off. In the following, we will simply refer R L and I L as the real and imaginary parts of the L-loop matrix elements. In the conjugate matrix element, we scale out (−ω) because Θ(−ω) localizes the integrals to the domain (−ω) > 0.
We can now explore the structure of eq. (3.47) given integrands of the form in eqs. (3.44) to (3.46). Since all IR divergences have been removed, we may set d = 4. The relevant integral to compute the time-domain observables at L-loop order is where the additional factor of ω 2n accounts for and generalizes such factors in eqs. (3.44) and (3.45). For now we keep the exponent n to be a real number. As we will see in sections 7 and 8, the amplitude contains a logarithmic dependence on ω; we may find the relevant Fourier transform by simply differentiating with respect to n. This logarithmic dependence on the outgoing-graviton frequency yields the so-called gravitational-wave tail first studied in [101,102] and represents the effect of the scattering of the leading-order gravitational wave off the gravitational field of the source. For the integer part of the exponent, depending on the parity of the loop order L the contribution to the waveform from the real or imaginary part of the matrix element (3.53) localize because: For example, this localization occurs at tree level for the real part of the matrix element (3.53) (L = 0 and n = 1), where R L=0 is just the tree-level five-point amplitude.
It also occurs at one loop for the imaginary part of the matrix element (3.53) (L = 1 and n = 1), where I L=1 is effectively the imaginary part of the 2MPI part one-loop five-point amplitude with subtracted IR divergences. In terms of J n,L , the time-domain Newman-Penrose scalar and waveform have very compact expressions, where we have assumed that the wavepackets for the massive states are highly localized.
Let us now spell out the ingredients necessary for the evaluation of the waveform h ∞ + + ih ∞ × at leading order, O(κ 3 ), and next-to-leading order, O(κ 5 ). At O(κ 3 ), the matrix element determining B h in eq. (3.48) is simply the tree-level 2 → 3 amplitude evaluated at k = (1, n), where we do not decorate the right-hand side with the "2MPI" designation because it is irrelevant at tree level. Thus we have R L=0 = M cl. 5,tree and I L=0 = 0 at the leading order. At O(κ 5 ) the matrix element determining B h in eq. (3.48) is the 2MPI part of the oneloop five-point amplitude in which the IR-divergent contribution of soft virtual gravitons has been removed per eq. (3.43), and theT -bilinear terms in eq. (3.3): where here is a single graviton whose polarization a is summed over. In the second line, we only keep terms at O(κ 5 ) order. It consists of a five-point classical amplitude M cl. * 5,tree , which contributes at O(κ 3 ), and the disconnected pieces of the six-point amplitude M disc.
6,tree , which contributes at O(κ 2 ). Diagrammatically, these terms are It is not difficult to see that such amplitudes are kinematically forbidden unless the two outgoing gravitons have zero energy. They would at most contribute to the δ(ω) terms in the metric (3.46), which corresponds to a time-independent background that can be subtracted. The factors of graviton energy ω in eq. (3.45) imply that such configurations do not contribute to the Newman-Penrose scalar and the spectral waveform. Thus for our calculation, we may neglect the second line of eq. (3.59) and write Upon the rescaling (3.51), it identifies R L=1 and I L=1 as the real and imaginary parts of M 0,2MPI,cl.

5,1 loop
Note that we have used the fact that ε * + = ε − . In the next two sections, we will evaluate the integrand and then the integrals of this one-loop classical amplitude. We collect them and discuss their properties in section 7. In section 8, we proceed to discuss waveform observables for N = 8 supergravity and GR.

The five-point classical integrand in minimally scalar-coupled GR
The integrand for the complete one-loop four-scalar-one-graviton amplitude in general relativity coupled to self-interacting scalars and in N = 0 supergravity was constructed in Ref. [103] through double-copy methods. In this section, we construct the corresponding one-loop HEFT amplitude and compare it with the 2MPI part of the classical limit of the full theory amplitude, which we find convenient to re-derive directly from generalized unitarity considerations. We isolate from the classical field-theory amplitude the 2MPR diagrams, in which both matter lines are cut. As discussed in the previous sections, these contributions are super-classical and cancel in the waveform. The remaining diagrams, which have exactly one matter line cut, reproduce the HEFT amplitude, thus demonstrating the absence of / contributions to the waveform. 9

Preliminaries
We begin by setting up the notation and variables for five-point kinematics. While we are ultimately interested in physical kinematics, with two incoming and three outgoing particles, it is convenient to take all momenta to be outgoing, To cleanly separate different orders in the soft expansion, it is convenient to introducep 1,2 variables that are respectively orthogonal on q 1,2 , the momentum transfer from particle 1 and 2 [74], The external graviton momentum k is related to the momentum transfers as k = q 1 + q 2 . The on-shell conditions expressed in terms of the shifted matter momenta arē We define barred four-velocityū 1 =p 1 /m 1 andū 2 =p 2 /m 2 such thatū 2 1 =ū 2 2 = 1. It is also convenient to define y =ū 1 ·ū 2 . We will also use normal four-velocity u 1 = p 1 /m 1 and u 2 = p 2 /m 2 , and define σ = u 1 · u 2 . The difference between y and σ is of the order O(q 2 ).
The classical four-scalar-one-graviton tree-level amplitude in GR was given in Ref. [74]. In the notation above and for real kinematics, it can be written as where ε(k) µν = ε(k) µ ε(k) ν is the graviton polarization tensor, and the conjugation indicates that it is an outgoing graviton. The coupling κ is related to Newton's constant through κ 2 = 32πG. One can easily verify that both ε * · P 12 and ε · Q 12 are gauge invariant. We will use this expression in section 7 to compare the IR divergences of the classical amplitude with the prediction of Weinberg's analysis (3.41) and in section 8 to construct the leading-order waveform. Here is a list of tree amplitudes that will be used in unitarity cut constructions, including the three-point amplitudes that are uniform in , In eq. (4.6), the full quantum amplitudes are given in the first entries, which we compute through the standard Feynman diagram approach. The amplitudes with a superscript "cl" are classical amplitudes, which are obtained from the full amplitudes through a soft expansion and keeping the terms with the classical scaling (2.3). These classical tree amplitudes will be used later to construct HEFT cuts. The sum over the physical graviton states is a common ingredient in both HEFT and full amplitude calculations, as it enters in the evaluation of generalized unitarity cuts. Figure 3: Topology of integrals given in eq. (4.11). where is the physical-state projector for a vector field, and r µ is an arbitrary null reference vector that should drop out of physical expressions. This sum simplifies considerably if the amplitudes being sewn obey generalized Ward identities, i.e., they obey the Ward identity for external leg i without using the transversality of the polarization vectors for any of the other external gravitons [73,104,105]. For such amplitudes, all terms proportional to the momentum of the sewn legs drop out so we can effectively use the much simplified (and manifestly covariant) graviton state sum Ref. [104] showed that, through simple manipulations, it is always possible to put scattering amplitudes into a form that obeys the generalized Ward identities. In fact, by being manifestly written in terms of linearized field strengths, HEFT amplitudes already obey such generalized Ward identities without any additional manipulations. We will use such amplitudes in our loop calculations. The expressions for the 2MPI amplitudes, both in HEFT and the full theory calculation, are naturally expressed in terms of scalar integrals of pentagon topology and with two linear propagators. One of our results, which is natural in the HEFT approach, is that one matter line is always cut; thus, all integrals will be of the special cases of where the delta function realized as We will omit the ± superscript if the linearized matter propagator is absent, i.e., when a 4 = 0. This is the generalization to the five-point case of the analogous feature present in four-point amplitudes [71][72][73]. The diagrammatic representation of the master integrals are shown in figure 3.  Figure 4: The spanning cuts of the five-point one-loop amplitude. All exposed lines are cut. The loop momentum 2 = − q 1 and 3 = + q 2 follow the clockwise direction.
Unlike the four-point case, the five-point classical amplitudes depend on master integrals with contact matter vertices, namely, a 1 = 0 in eq. (4.11). They come from the IBP reduction of integrands with higher topologies, and their coefficients contain non-local dependence on q 2 i through the Gram determinants generated by IBP. As a result, these contributions are relevant to the classical long-range interaction in the position space after a Fourier transform. 10

The five-point 2MPI HEFT integrand
We use generalized unitarity in d dimensions [44][45][46] to construct the 2MPI HEFT integrand from the spanning set of generalized cuts in figure 4. Unlike generalized unitarity in the full quantum theory, the cut matter line, denoted by a red vertical line, is permanently cut and dressed by a delta functionδ(2p i · ). In addition, the input tree amplitude for each blob contains only terms with classical scaling in the full tree amplitude. Such terms are free of the delta-function contributions that are super-classical, as reviewed in section 2. We can safely ignore the super-classical delta-function dependent terms because they correspond to 2MPR graphs, which are ultimately subtracted by the KMOC formalism. Finally, we focus on integrands where matter line 2 is cut as the integrands where matter line 1 is cut can be derived from a relabelling. We note that cuts with no cut matter lines, for example, propagator remains cut in the final expression. 11 These HEFT cuts are given by: 14) As first mentioned in section 2, we make all the uncut linear propagators symmetric in i0 and thus principal-valued in the HEFT cuts. The HEFT Cut 2 is technically divergent due to external bubble contributions such as Thus we need to construct the cut with an additional regulator, such as taking external scalars off-shell or not imposing momentum conservation. The regularization will break the generalized Ward identity. As a result, we need to use the full graviton state projector (4.8) to compute HEFT Cut 2. Since such external bubbles are scaleless after the regulator is removed, we can simply subtract them from HEFT Cut 2 to reach a finite result. We now merge the cuts in eq. (4.14) into an integrand we follow Ref. [106] and simply add them together and subtract the overlap shown in figure 5 and given by (4.16) Thus, the resulting one-loop HEFT 2MPI integrand is which we will later IBP-reduce to master integrals. The relative signs between the cut  Figure 6: The spanning cuts of the five-point one-loop amplitude in GR coupled to scalars. All exposed lines are cut. The loop momentum i follows the clockwise direction. Contributions captured solely by GR Cut 3 involve intersecting matter lines which do not contribute in the classical limit but whose possible appearance is related to the absence of four-scalar contact terms in the classical action. To construct the integrand we consider all relabelings of external legs.
contributions, C (1,H) and C (2,H) , and the overlap contribution, C (12,H) , are a consequence of the factors of i in the definition of matrix elements and of propagators.

The five-point classical integrand from the quantum integrand
In this section, we construct the classical limit of the four-scalar-one-graviton amplitude in GR coupled to two scalar fields to verify the completeness of the HEFT amplitude and to understand the fate of terms of / type that naturally appear in the classical expansion of cuts of the full theory. The spanning set of generalized unitarity cuts determines the (classical part) of the fivepoint amplitude in GR and is given in figure 6. The terms in the integrand determined solely by GR Cut 3 contain (1) 1PR mushroom graphs and (2) graphs with intersecting matter lines with numerators that are polynomial in external and loop momenta. The former is required by gauge invariance; the latter, while not having a Feynman vertex counterpart, may appear depending on choices made in the construction of the integrand. The integral corresponding to the latter does not depend on the momentum transfer (q 1 − q 2 ) and as such these contact terms contribute only δ(b) terms to the waveform and we could ignore it. The same integral also appears in the IBP reduction of terms determined by the first three cuts; their coefficients turn out to have a rather nontrivial dependence on the momentum transfer (q 1 −q 2 ) and thus contribute nontrivially to the waveform. We will find that for the integrand we construct, the contact terms with the topology of GR Cut 3 vanish identically.
In our construction, we will ignore 2MPI contributions to the classical amplitude which are not captured by these cuts, because the corresponding (master) integrals are scaleless and thus vanish in dimensional regularization. We use d-dimensional generalized unitarity [44][45][46] to construct the five-point integrand. In terms of tree amplitudes, the cuts in Figure 7: Topologies of the contact terms that contribute to the classical limit of the five-point amplitude before reduction to master integrals. The complete basis includes all inequivalent permutations of these diagrams.
where the i are defined according to figure 6. We use complete tree-level amplitudes of GR minimally-coupled to scalar fields that obey the generalized Ward identities [73,104,105], so the sum over the internal graviton physical states, labeled here by h 1,2,3 , is given by eq. (4.10). The resulting cuts reproduce those used in the construction of the quantum five-point integrand in Ref. [103] up to the contributions of four-scalar contact terms which do not contribute in the classical limit but are natural in the double copy construction used there. 12 Merging these cuts using the method of maximal cuts [49] while maintaining quadratic propagators for the matter fields leads to the relevant part of the one-loop integrand, which includes only graphs with three, four, and five propagators with at least one matter line in the loop. The relevant topologies are shown in figure 7. Diagrams with fewer propagators either do not have internal matter lines or intersecting matter lines, neither of which contribute to the classical amplitude.
(4. 19) In practice, we also convert top i variables defined in eq. (4.2) at this step, which will introduce additional q i dependence in the above expansion. The leading soft-region scaling of the five-point one-loop amplitude is super-classical, as expected from the existence of graphs with two-particle matter cuts. Direct inspection of contributing diagrams suggests that one of them, figure 7b, scales as q −3 while the diagrams in figures 7a and 7c to 7e scale as q −2 , and the remaining two triangle graphs exhibit classical scaling, q −1 , at leading order. After soft expanding to O(q −1 ), in which the diagrams with topology figure 7b must be expanded to second order, all matter propagators have linear dependence in loop momenta [107] and may be raised to a power higher than one. Upon soft expanding the integrand, the two matter propagators in the top matter line of figures 7b and 7d become linearly dependent because the momentum of the outgoing graviton is of the same order, O( ), as the loop momentum. Linear dependence of propagators prevents a direct IBP reduction. These integrands are first partial-fractioned, and the resulting terms are assigned to the box diagrams in which the graviton is attached to the left and right vertex on that matter line, A similar feature occurs in the calculation of radiative observables at two and higher loops [29] and in the calculation of the tail effect at three loops [100].
To expose cut matter propagators and make contact with the HEFT integrand we separate each diagram into its symmetric and antisymmetric parts with respect to permutations of vertices on the two matter lines. Using the identity in eq. (4.12), the symmetric part effectively cuts a matter propagator [83]. In the analogous four-point one-loop calculation, it suffices to carry out this procedure for only one matter line in each diagram; if the diagram has a second matter line, it gets cut by changing integration variables and using the same identity. Here, becausep 1 · q 2 = 0 =p 2 · q 1 , we need to actively decompose each diagram into symmetric and antisymmetric parts with respect to both matter lines. 13 The antisymmetric part can then be reinterpreted as the principal value of the matter propagator. Interestingly, the terms in which both matter propagators are replaced by their principal values cancel out for the classical amplitude and, as one might expect, the resulting classical integrand has at least one cut matter line. This cut, which we will shortly identify with the cut present in all HEFT diagrams, implies that at least one of the gravitons present in each diagram is in the potential region. 14 We note that the principalvalued propagators appear naturally from a first principle unitarity calculation without any assumptions. Since the principal-valued matter propagator has no pole (and no imaginary 13 We remove the i0 prescription in the linearized matter propagators that are independent of the loop momentum. They will at most contribute terms proportional to δ(ūi · k) ∼ δ(ω), which are irrelevant to the waveform computation as discussed in section 3.4.
14 For example, in an integral with the topology of figure 7f, the cut matter line implies that the left graviton is in the potential region while the right graviton could be in the radiation region. As we will see in the next section, the integral I1,0,1,0 described here indeed has both a real and an imaginary part. part), the cut of such propagators vanishes. Therefore, diagrams with one principal-valued matter propagator and one cut matter propagator can contribute to the term bilinear in the transfer matrix in the KMOC expression for the waveform only if a cut through a graviton propagator is allowed kinematically which, according to the discussion in section 3.2, is not the case here.

Integrand reduction
We have computed the integrand using generalized unitarity at the level of HEFT and the full quantum amplitude. We now reduce the integrand to a basis of master integrals with kinematic coefficients. We first expand the polarization tensor, ε µν = ε µ ε ν and ε 2 = 0, in a basis of external momenta: This decomposition, equivalent to Passarino-Veltman reduction [108], introduces spurious poles in the form of Gram determinants, G[p 1 ,p 2 , q 1 , q 2 ], which should cancel in the final integrated expression. The resulting separated and soft-expanded integrands are reduced to master integrals using FIRE [58,59]. We keep pentagon, box, triangle, and bubble integrals that do not vanish in the classical limit. While bubble integrals are independent of the momentum transfer, their coefficients, which are generated by the IBP reduction, can exhibit such a dependence and thus can contribute nontrivially in the classical limit.
Integrands with different numbers of cut propagators form distinct sectors under IBP reduction. Diagrams with two cut matter lines are the 2MPR contributions are not included (though of course computable) in the HEFT amplitude. Factorization of the one-loop amplitude of the full theory identifies these terms as the product of the classical limit of four-point and five-point tree amplitudes. Integrals with a single cut matter line, including mushroom graphs, are the same as in the reduction of the HEFT 2PMI amplitude (4.24) and we have verified that the coefficients are also the same. This demonstrates the complete cancellation of the / terms in the full-theory calculation; such terms appear at intermediate stages, with the numerator coming from quantum terms in one tree-level factor in a cut and the denominator from superclassical terms in another.
Ultimately, the classical 2MPI part of the amplitude becomes a linear combination of the master integrals where the integrals are defined in eq. (4.11). The master integral coefficients c i are lengthy and are included in the ancillary Mathematic-readable file. The symmetric combinations I + 1,1,0,1 + I − 1,1,0,1 and I + 1,1,1,1 + I − 1,1,1,1 correspond to replacing the linear matter propagators of these integrals with their principal value. We have checked with the authors of Ref. [69] and we find full agreement on the master integral coefficients for GR. The seven master integrals in the second and third line of eq. (4.22) have contributions from radiation region gravitons. They are thus complex and contain the radiation reaction as discussed in Ref. [70].

The classical five-point integrand in N = 8 supergravity
The N = 8 supergravity provides an important laboratory to explore properties of gravitational theories in a setting where amplitudes have somewhat simpler expressions. In this section, we construct the classical five-point amplitude in this theory, with the massive scalars being the lightest Kaluza-Klein modes for scalar modes of gravitons in the compact dimensions while all other particles are zero-compact-momentum modes, following [107].
We construct massless maximally-supersymmetric supergravity amplitude in generic dimensions via the double copy. The classical tree-level five-point amplitude was constructed in Ref. [74]. With the notation introduced in eq. (4.4), it is given by, It differs from that of GR result in eq. (4.4) only by the inclusion of the dilaton exchange here. At one loop, we start with the one-loop five-gluon BCJ numerators of maximallysupersymmetric Yang-Mills theory [66,67], which consist of only pentagon and box topology, shown in figure 8, where the scalar and vector t 8 tensors are defined as The traces entering the definition of t 8 are over the Lorentz indices and the linearized oneand two-particle field strengths, f µν i and f µν ij , are where k i and e i are respectively the massless momentum and polarization in higher dimensions. From eq. (5.2), we can get the numerators of maximal supergravity through the double copy N SUGRA = (N SYM ) 2 . We introduce four compact dimensions and use a dimensional reduction in which the compact momenta responsible for the scalar masses are in orthogonal directions, 15 k 1 = (p 1 , m 1 , 0, 0, 0) , k 2 = (p 2 , 0, m 2 , 0, 0) , Since the masses arise from higher-dimensional momenta, they obey conservation relations, i.e., they change signs with the orientation of the corresponding momentum. The on-shell condition k 2 1,2,3,4 = 0 for higher dimensional massless momenta thus lead to the massive onshell condition p 2 1 = p 2 4 = m 2 1 and p 2 2 = p 2 3 = m 2 2 . The kinematic configuration in eq. (5.6) gives the following reduction rules for Mandelstam variables, while · k i = · p i for i = 1, 2, 3, 4, and · k 5 = · k. The massive scalars are realized as the scalar graviton modes in the compact dimensions, e 1 = e 4 = (0, 0, 0, 1, 0) , e 2 = e 3 = (0, 0, 0, 0, 1) , e 5 = (ε, 0, 0, 0, 0) , (5.8) such that dot products involving polarization vectors are reduced by e 1 · e 4 = e 2 · e 3 = −1 and e i · e j = 0 otherwise, e i · k j = 0 for i = 1, 2, 3, 4 and arbitrary j ε · p j for i = 5 and j = 1, 2, 3, 4 .
The diagrams that survive as these relations are plugged into the one-loop integrand are the ones that are consistent with higher-dimensional momentum conservation at each vertex. Namely, we keep the diagrams in which the two matter lines connecting {p 1 , p 4 } and {p 2 , p 3 } do not cross each other. They are all captured by the spanning set of cuts in figure 6. As in the GR calculation outlined in section 4.3, we expand the resulting integrand in the soft limit. By using eq. (4.12), we can expose all the 2MPR diagrams. The remaining 2MPI diagrams have one cut matter propagator. Upon reduction to master integrals, the uncut linear matter propagators turn into principal-value propagators. The 2MPI part of the classical amplitude takes the same form as eq. While the coefficients are somewhat unwieldy, it is not difficult to verify that each of them is separately gauge-invariant, as they should be. Interestingly, in the classical limit, the N = 8 amplitude includes triangle and bubble integrals, unlike the quadratic-propagator counterpart [109]. Similar to the GR five-point amplitude, they contribute nontrivially to the waveform because their coefficients exhibit nontrivial dependence on the momentum transfer (q 1 − q 2 ).

Integrating in the Rest Frame
In this section, we evaluate, for physical kinematic configurations, the bubble, triangle, and pentagon master integrals eq. (4.22) that appear in GR and N = 8 supergravity one-loop five-point classical amplitudes. We also list the expressions for the box integrals and relegate the details of their evaluation to appendix B. We note that the results given in this section are in full agreement with Ref. [69]. For all master integrals, it is convenient to work in the rest frame of particle 2, in whichū 2 = (1, 0, 0, 0), and integrate outδ(2ū 2 · ) in the numerators. This projects out the temporal component of the loop momentum such that = (0, ). We are thus left with a Euclidean loop integral in 3 − 2 dimensions, expressed in terms of non-covariant quantities. We can uplift the result back into a generic frame by usinḡ where the left hand side comes from the components ofū 1 , q 1 and q 2 written in theū 2 rest frame,ū 1 = (ū 0 1 ,ū ū u 1 ), q 1 = (E q 1 , q q q 1 ) and q 2 = (0, q q q 2 ). In the physical region, we havē u 1 · q 2 > 0 andū 2 · q 1 > 0 because the outgoing graviton has positive energy. 16 We also have q 2 1 < 0 and q 2 2 < 0, because the momentum transfer of particle 1 and 2 is spatial-like. With these preparations, let us first discuss the bubble integral I 0,0,1,0 as the simplest example, In the classical amplitude, the bubble integral usually comes with a divergent coefficient 1 d−4 that encodes part of the IR divergence. We expand this combination up to the terms finite in , where we have defined for convenience,

4)
and γ E is the Euler constant. It is crucial to track the i0 prescription to determine the analytic continuation into the physical region of external kinematics. 17 We will mainly consider the results in d = 4 only, so we will omit the O( ) terms in the following.

Triangle integrals
Next, we consider the triangle integral I 1,1,0,0 . We first introduce the Feynman parameterization to combine the denominator while going to theū 2 rest frame. The integral is then straightforward to work out, This integral is purely real as expected because further cutting either 2 or ( + q 2 ) 2 leads to vanishing results due to on-shell three-point kinematics. In contrast, the other triangle master integral I 1,0,1,0 is complex. We start with the same Feynman parameterization, and the integration proceeds as before, We note that the argument of arcsin in the second line is greater than 1 in the physical region. The analytic continuation in +i0 prescription is By using eq. (6.1), we can uplift the result to a generic frame, where we have also used the identity arccosh √ x + 1 = arcsinh √ x. To compute the integrals with a linear matter propagator, we use a different parameterization to combine the denominator, Applying eq. (6.9) to I + 0,0,1,1 , we get where α = y(ū 2 ·q 1 ) y 2 −1 and β = − (ū 2 ·q 1 ) 2 (y 2 −1) 2 + i0. The integration over x is divergent at x → ∞, which encodes the IR divergence due to the linear matter propagator 2ū 1 · . However, in the classical amplitude, the linear matter propagator only appears in the principal-valued combination I + 0,0,1,1 + I − 0,0,1,1 . After including I − 0,0,1,1 , we find that the range of x gets truncated and we get a finite result, The +i0 prescription in β leads to the following analytic continuation of the arcsinh function in the physical region, Therefore, the final result of this triangle integral is We can apply the same technique to compute . (6.15) In particular, the principal value combination is given by where γ =ū 1 ·q 2 y 2 −1 .

Box and pentagon integrals
The box master integrals are all individually IR divergent. For I 1,1,1,0 , the IR divergence is due to the presence of a massless three-point vertex, The other box integrals, I ± 1,1,0,1 , I ± 1,0,1,1 and I ± 0,1,1,1 , all have an IR divergence due to the linear matter propagator 2ū 1 · . This IR divergence cancels in the classical amplitude because the linear matter propagator always appears as a principal value, On the other hand, I + 0,1,1,1 + I − 0,1,1,1 remains IR divergent due to the presence of the same massless three-point vertex as in I 1,1,1,0 , We leave the derivations of these integrals to appendix B. Scalar pentagon integrals with quadratic propagators in d = 4 − 2 dimensions can be written as a sum of box integrals and a six-dimensional pentagon integral [54,110,111]. We derive here an analogous decomposition for our pentagon integral with linear matter propagators one of which is cut, I ± 1,1,1,1 . We first decompose the loop momentum into its four-dimensional and extra-dimensional component, 2 = 2 4 +µ 2 , where 4 can be expressed as a linear combination of external kinematic data, 4 = α 1ū1 + α 2ū2 + α 3 q 1 + α 4 q 2 . The coefficients α i contain the dependence through scalar products v · 4 = v · with v ∈ {ū 1 ,ū 2 , q 1 , q 2 }. We then plug the above relation for 4 into the identity express all the v · in terms of inverse propagators, and perform the standard tensor reduction. This will lead to a linear relation that expresses the pentagon integral in terms of the box integrals in d = 4 − 2 and another pentagon integral I ±,d=6−2 1,1,1,1 in d = 6 − 2 , I ± 1,1,1,1 = β 1 I ± 1,1,0,1 + β 2 I ± 1,0,1,1 + β 3 I ± 0,1,1,1 + β 4 I 1,1,1,0 + β 5 I ±,d=6−2 where β i only depend on external Mandelstam variables. The pentagon integral I ±,d=6−2 1,1,1,1 comes from evaluating the µ 2 term contribution by the dimension shift relation [45], In the principal value combination I + 1,1,1,1 + I − 1,1,1,1 , this term is finite in d = 6 − 2 such that it contributes at most to O( ). The coefficients β i in eq. (6.21) are straightforward to compute. Here we just give the final result of the pentagon integral, where C 1 and C 2 are given by

The five-point amplitude and its infrared divergences
Having evaluated all the relevant master integrals, we can assemble the classical amplitudes and carry our various consistency checks. When writing down the classical amplitude we can also remove the distinction between {m i , p i , u i , σ = u 1 · u 2 } and {m i ,p i ,ū i , y =ū 1 ·ū 2 }, as they differ only by positive powers q 1 and q 2 , i.e. by terms with quantum scaling in the soft expansion. By inspecting the non-rational terms in the integrals evaluated in section 6 and appendix B it is straightforward to see that they are all linear combinations of the functions with rational coefficients. Here µ is the scale of dimensional regularization, and this µdependent logarithm is intimately related to IR divergence of amplitudes. In both GR and N = 8 supergravity classical two-scalar-one-graviton five-point ampli-tude has the general form

5,tree
where the coefficient functions A R,I i are rational combinations of momentum invariants and polarization vector ε. Their scaling in the soft limit is homogeneous and it is such that M 2MPI,cl.
where the dependence on the dimensional regularization scale µ has been replaced by cutoff defining the virtual IR gravitons Λ due to eq. (3.43). We have also explored the fate of spurious singularities. The Gram determinant G[p 1 ,p 2 , q 1 , q 2 ] arising from the decomposition (4.21) of the graviton polarization tensor into a basis of external momenta cancels in eq. (7.2) within each coefficient A R,I i with the help of four dimensional identities involving the vanishing Gram determinant G[p 1 ,p 2 , q 1 , q 2 , ε]. Other spurious singularities occur for kinematic configurations that set to zero denominator factors arising from the IBP reduction. They are solutions to the equations It is not difficult to check that when either of these relations is satisfied, the logarithmic functions in eq. (7.1) are no longer linearly-independent. Therefore, these singularities can cancel only in the complete expression, which they indeed do. While all four ∆'s appear in the GR amplitude, only ∆ 1 and ∆ 2 appear in the N = 8 amplitude.
In the real part of the amplitude, the rational coefficient has a very simple structure. For the GR amplitude, it reads where we set κ = 1 in the tree amplitude because the overall κ dependence has been pulled out. On the other hand, the square-root functions in eq. (7.1) originate only from the triangle integrals I 1,1,0,0 , I 1,0,1,0 and their up-down flip, while their coefficients are more complicated in GR. However, such master integrals are absent for N = 8 supergravity, see eq. (5.11), and thus A R,N =8 1,2,3,4 = 0. This makes the real part of the five-point N = 8 amplitude much simpler than eq. (7.3) implies. The complete real part is given by A R rat only, which is similar to its GR counterpart, We note that this is the complete R L=1 that is used to compute the waveform for N = 8 in eq. (3.55). Curiously, the σ-dependent factor in eq. (7.6) vanishes in the ultrarelativistic limit, σ → ∞, implying that in this limit only the imaginary part of the amplitude contributes to the waveform at O(G 2 ) in N = 8 supergravity according to eq. (3.55). Even though the real part of the N = 8 classical one-loop amplitude is proportional to the tree-level amplitude, the real part of the one-loop spectral function is different from the tree-level one because the ω-dependent distributional factor is now Θ(ω) − Θ(−ω) for L = 1, see eq. (3.55). Finally, we have verified that our one-loop amplitudes reproduce the expected soft limit under k → 0. At the leading order, the imaginary part vanishes, and the real part factorizes The leading soft limit for the 2MPI amplitude of N = 8 vanishes because at four points it is only contributed by the cut box, which is 2MPR. Meanwhile, the 2 → 2 amplitude in GR receives contributions from triangles, and it is given by where q is the momentum transfer for this 2 → 2 scattering. At the leading order, we can take q = q 1 = −q 2 , such that the soft factor S is given by The gravitational memory only receive contribution from the leading soft limit. Thus there is no memory effect for N = 8 at NLO, and we will explicitly compute the memory for GR in the next section.

Leading order and next-to-leading order waveforms
Having constructed the relevant part of the one-loop five-point amplitude, we now use the formulae summarized in section 3.4 to construct the leading order and next-to-leading order waveform observables focusing on N = 8 supergravity. The time-domain leading-order asymptotic metric was first discussed in Ref. [22] and more recently from the worldline QFT perspective in Ref. [35]. The last ingredient that we need are polarization vectors ε ± (k) for a massless particle with momentum k. We may use spinor-helicity notations for it; for numerical evaluation, however, it is more convenient to start with the special outgoing momentum k 0 and obtain the general angle-dependent polarization vector through a rotation: The third angle of a general rotation simply multiplies ε by a phase, so we will ignore it. This rotation also maps k 0 tok = (1, n k ) with n k a general unit vector. We stress that, as discussed before, the complex nature of these polarization tensors does not change the definition of R L and I L in eq. (3.53), because they are stripped off before taking the real and imaginary part. Indeed, we may decompose the outgoing polarization tensor ε −− = ε − ⊗ε − as and use separately the real polarizations ε + and ε × to define the real and imaginary parts of the matrix element B − and subsequently the waveforms. The asymptotic metric h ∞ µν is defined in eq. (3.18). We further parametrize it as µν + . . . (8.4) where M = m 1 + m 2 is the total mass. Hereĥ (1) µν andĥ (2) µν are respectively the reduced waveform at 1PM and 2PM order. Similar to the previous section, we will not distinguish barred and unbarred variables since their difference is quantum.
We will perform explicit calculations in the center-of-mass (COM) frame in which the black holes move along the z axis and b µ is along the x axis, 18 We will plot waveforms at various locations (θ, φ) at spatial infinity with different velocities v.

Gravitational memory
By plugging eqs. (3.46) to (3.48) into eq. (3.50), we can write the gravitational-wave memory as In the soft limit, we have seen explicitly in eq. (7.7) through one loop order, that the matrix elements become wherek = (1, n), and we have used Θ(0) = 1/2. The graviton soft theorem implies this result holds at all orders. A similar calculation can be applied to the conjugate matrix element. 19 Therefore, eq. (8.6) becomes The typical integral in eq. (8.8) is We can fix d = 4 and decompose q in terms of Ref. [43] We then integrate over the coefficients {z 1 , z 2 , z b , z v }. It is easy to see that the two delta functions fix z 1 = z 2 = 0. The integration over z v vanishes because the integrand is odd in z v . As a result, I µ must be proportional to b µ . ε µ − ε ν − S(k, q) µν can therefore be pulled out of the integral by replacing q µ → −i∂/∂b µ . Recalling that, up to two loops, the constrained Fourier-transform of the real part of the 2MPI four-point amplitude is expected to be the radial action [63,100], we find The action of ∂/∂b µ on the radial action is proportional to the scattering angle χ, where ν = m 1 m 2 /M 2 ,b µ = b µ / √ −b 2 and χ = −∂I r /∂J. The angular momentum J is given by J = p ∞ √ −b 2 , and p ∞ is the norm of the initial COM momentum, which is given by the prefactor in eq. (8.12). Therefore, the memory is also proportional to χ, Importantly, the angular dependence of the memory is completely encoded in the soft factor. We now plug in the 1PM and 2PM scattering angles for GR [112,113] and N = 8 19 We note that in this calculation, cancelling ω in k leads to an extra minus sign because k = (−ω, −ωn) and ω < 0. This sign will be absorbed by redefining q → −q to align the exponential factor e iq·b because the soft factor Sµν is odd in q. The final result differs from eq. (8.7) only by a complex conjugation on M 2MPI.cl.

4
. supergravity [107], where X = 1 for GR and X = 0 for N = 8 supergravity. The final result for gravitational memory is 15) whereĥ is defined in eq. (8.4). In particular, there are no memories for N = 8 at 2PM. We note that the above relation between memory and the soft factor was first predicted in Ref. [114].
With foresight on the integrals required by the evaluation of time-domain observables, it is convenient to follow the strategy of Ref. [43] and compute the J α 1 ,α 2 β 1 ,β 2 integrals in d = 4 by decomposing the integration variable q 1 along four orthogonal fixed vectors as in eq. (8.10). The integrals over z 1 , z 2 , and z b are localized due to the three delta functions in eq. (8.18). This leaves the integral over z v as the only nontrivial integral. The first six integrals in eq. (8.17) have been evaluated in Ref. [43]. Using this method, the last three integrals can be brought to a one-parameter integral over a finite range. The background at τ → −∞ is also subtracted. To demonstrate the final result at LO, we plot the evolution of the waveform at a particular location at the spatial infinity in figure 9. The GR and N = 8 waveforms have the same qualitative features. The difference is approximately an overall scale. Here and after, all the plots are made with m 1 = m 2 .

Next-to-leading order (NLO)
The NLO time-domain waveform follows from the L = 1 component of eq.
We first compute the contribution from M 0,2MPI, cl 5,1 loop , which can be written as where H R,I are obtained by combining eqs. (3.54), (3.55) and (3.62): (8.23) We note that H R and H I do not correspond to the real and imaginary part of the final waveform, because the polarization vectors in these expressions are still complex. The tail contribution will be computed later in this section.
As at leading order, we compute the integral over q 1 in d = 4 by decomposing this variable in the basis of 4d vectors in eq. (8.10). For both GR and N = 8 supergravity, it is easier to compute H I due to the presence of Thus we can use the three delta functions in H I to localize the integrals over the z 1 , z 2 and z b variables. The integrand is now an algebraic function of z v parametrized by τ . The resulting integral over z v is convergent and performed numerically. The background at τ → −∞ is also subtracted at the end. In contrast, there are only two delta functions in H R , which localize z 1 and z 2 . For N = 8 supergravity, the integrand is a rational function in z b and z v according to eq. (7.6). Therefore, we can integrate z b using the residue theorem, resulting in an algebraic function of z v , which will be integrated numerically. For N = 8 supergravity, the subtraction of background at τ → −∞ can be done before or after the final z v integral since it does not affect the convergence.
However, for GR, after localizing z 1 and z 2 , the integrand contains square roots according to eq. (7.3). For this case, we still first integrate over z b . After some rescaling, the integral has the following generic form, where f (z b ) is a rational function, and we have changed the variable to z b = w 2 −1 2w . Since the original z b integral is convergent, there are no poles at w = 0 and w = ∞ after the  change of variable. We can evaluate this integral by summing over all the residues, which leads to the final answer in eq. (8.25). Now if we subtract the background at τ = −∞, the final z v integral is convergent and we integrate it numerically. Unlike the previous cases, the background subtraction needs to be done before the z v integrals because the integral is divergent otherwise. Alternatively, if one keeps d = 4 − 2 , the integration and background subtraction can be done in any order, the difference being at most some local δ(|b|) terms in the waveform and memory. We plot the NLO waveforms in GR observed at several angles at the spatial infinity in figure 10. In this figure, we fix the COM velocity of the two black holes to be v = 1/5. Then, in figure 11, we fix the observation angle to be (θ, φ) = ( 7π 10 , 7π 5 ) and vary the velocities of the black holes. At low velocities, the waveforms have a much larger magnitude and are more spread out in the time domain; we may understand this intuitively by noting that at lower velocities, the two particles have a smaller minimal separation, so they experience larger accelerations and therefore radiate more. We also note that at lower velocities, we are closer to the boundary of the PM regime, which requires that Jv 2 be sufficiently large.
Finally, for comparison, we include in figures 12 and 13 the NLO waveforms in N = 8 supergravity. As expected, the N = 8 waveforms do not display any memory effect at NLO. At low velocities, for example, v = 1/20, the imaginary part exhibits a small residual    at τ → ∞; this is merely an artifact of our initial and final times not being infinite. We note that the waveform profile is spread over a larger time interval and that they have a larger amplitude than the GR waveform. A possible explanation is that fields of the N = 8 supergravity yield an additional net attractive force increasing the acceleration experienced by the two particles relative to GR. We remark that at certain angles, our GR waveform displays features which, for lack of a better name, we refer to as "kinks." For example, such kinks can be seen in figure 10 forĥ (2) + at (θ, φ) = ( 7π 10 , 7π 5 ) and ( 7π 10 , 2π 5 ). At these locations, several terms in the integrand become sharply peaked (but finite), and the resulting large values of the integral cancel over many orders of magnitude. Our numerical evaluation further suggests that more prominent kinks exist at some other angles, such as (θ, φ) = ( π 2 , 0), which corresponds to an observation direction parallel to b in the plane of the two incoming particles. It is unclear whether these features are merely numerical artifacts due to cancelation between large numbers, or they are physical. A definitive answer can be provided by an analytic evaluation of the waveform, which we leave for future work.
We now turn to the evaluation of the gravitational-wave tails. To this end we plug M tail of eq. (8.20) into I L of eq. (3.55). As discussed in sec. 3.4, to evaluate the Fourier transform to the time domain in the presence of the logarithmic dependence on ω we simply differentiate eq. (3.55) with respect to n. Thus, Since the integral is finite, we compute it directly in four dimensions. To this end we parametrize q 1 as in (8.10) and evaluate the z 1,2 integrals using the explicit δ-functions. Unlike eqs. (8.23) and (8.24), the logarithmic dependence on ω prevents the appearnce a third δ function. Since the integrand does not exhibit branch cuts in z v we evaluate this integral analytically using Cauchy's residue theorem. The last integral, over z b , is evaluated numerically.
To this end we also need to choose a value for the cutoff Λ defining the infrared soft virtual gravitons. 20 It is required to be well-inside the soft region defining the classical limit, Λ |q|. We may therefore set a dimensionless bound on the product of Λ and the impact parameter b, which is O(|q| −1 ) being the Fourier-conjugate of the momentum transfer. We will choose Λ|b| = 5 × 10 −10 . Since in the classical limit |q| and the frequency ω of the outgoing graviton are of the same order, we may also relate Λ and the lowest frequency accessible to a detector. The results for GR and N = 8 supergravity are plotted in figures 14 to 17, and we have chosen the same observation angles and velocities as before.    Figure 17: The tail contribution to NLO waveform in N = 8 supergravity on the COM velocities at the fixed angle θ = 7π/10 and φ = 7π/5, and IR cut off Λ|b b b| = 5 × 10 −10 .
We note the absence of a memory contribution from the gravitational-wave tail, in agreement with the vanishing soft limit of the imaginary part of the one-loop five-point amplitude, as well as the close similarities of the velocity dependence of the tail and M 0,2MPI, cl 5,1 loop to the two gravitational-wave polarizations at fixed angle. In both contributions a higher amplitude of the wave corresponds to lower-velocity scattering, in agreement with the intuition that for fixed impact parameter lower-velocity particles experience a larger acceleration. Moreover, the angular dependence of the tail and of M 0,2MPI, cl 5,1 loop contributions toĥ (2) + are also similar in shape and velocity dependence. In contrast, their contributionsĥ (2) × exhibit quite different, cf. e.g. the right panels of figures 11 and 15. While the gravitationalwave tail contributions may be changed somewhat by varying the cutoff Λ, a difference between (θ, φ) < (π/2, π) and (θ, φ) > (π/2, π) persists. It is tempting to speculate that such a difference might offer an observable signature distinguishing the gravitational-wave tail from the local-in-time effects.

Conclusions
The observable-based formalism [42,43] directly links scattering amplitudes and observables of unbound binary systems in the classical regime. This framework naturally includes both conservative and dissipative effects, providing a means to finding local observables such as the asymptotic gravitational waveform for a scattering event in addition to inclusive ones such as the impulse or the energy and angular momentum loss. With the possible exception of singular momentum configurations, they are governed by four and five-point amplitudes [98]. We examined in detail this connection for the scattering waveform and found that, beyond leading order, the exponential form of amplitudes proposed in Refs. [60,61] implies that only (the suitably-defined) two-matter-particle-irreducible components of amplitudes can contribute to this observable.
We computed these parts of the four-scalar-one-graviton amplitude at one-loop order in GR with minimally-coupled scalars and in N = 8 supergravity. We obtained the former using generalized unitarity, both by evaluating the classical part of the complete amplitude and by using a HEFT approach which effectively corresponds to truncating the tree amplitudes to classical order. Comparing the two results and requiring that they agree identifies a prescription for treating uncut matter propagators, that they be treated as principalvalued. We obtained the corresponding integrand in N = 8 supergravity, which describes the emission of gravitational radiation from the scattering of two half-BPS black holes, by using the double copy of the corresponding integrand in N = 4 super-Yang-Mills theory and dimensional reduction.
Upon reduction to master integrals, the five-point classical amplitudes exhibit novel features compared to their four-point counterparts. For example, the outgoing graviton momentum injects a scale in some of the scaleless (and thus vanishing) integrals in the fourpoint amplitude, implying that they can contribute to the five-point classical amplitude.
The one-loop master integrals are sufficiently simple so we evaluated them by direct integration for physical kinematics. 21 A distinguishing feature of the (2MPI part of the) classical five-point amplitude is that it exhibits an infrared divergent phase. The structure of this phase divergence is governed by Weinberg's classic result [68] and the classical nature of the amplitude; we argue that at this order -and perhaps to all orders in perturbation theory -it can be absorbed as a suitable shift in the definition of the retarded time. Consequently, the observation of the gravitational waveform is sensitive only to differences of retarded times rather than the absolute time of the process. This is consistent with the basic assumption of scattering theory that observations are carried out at infinite times. We also gave a precise relation between the gravitational wave memory and the soft limit of the corresponding scattering amplitude, along the lines of Ref. [114], relating the amplitude of the memory to the scattering angle to at least O(G 3 ). Interestingly, this implies that the gravitational-wave memory vanishes at O(G 2 ) for half-BPS black holes in N = 8 supergravity, which is confirmed by the explicit calculation.
Even though our results are analytic in frequency and momentum space and in timemomentum space, we resorted to numerical integration to evaluate the one-dimensional integral completing the transformation to time-impact parameter space. A complete analytic evaluation of the waveform, which we leave for future work, would involve obtaining an analytic expression for this last one-dimensional integral. The result would resolve certain numerical features that we encountered at special observation angles and would open the door to the evaluation of interesting inclusive and local observables, such as the energy and angular momentum loss at O(G 4 ) and the radiated energy spectrum E(ω) through traditional general relativity methods [115][116][117][118]. The classical one-loop five-point amplitude constructed here allows the computation of inclusive quantities at 4PM order through the KMOC formalism, along the lines of the 3PM calculations of Refs. [29,33,86].
While the next-to-leading order is the current state of the art for relativistic scattering waveforms, it is natural to think about higher orders. The experience with N -extended supergravity loop calculations [119][120][121][122][123][124] suggests that the construction of the relevant classical integrand through double copy and generalized unitarity should scale well to higher orders and the results will provide tests of the HEFT approach to classical amplitudes beyond those already available. We expect, however, that the evaluation of the resulting master integrals will benefit from advanced techniques such as differential equations and method of regions, as was done for the full quantum two-loop five-point amplitude of N = 8 supergravity in Ref. [125]. Going to higher order could also help us better understand radiation-reaction, which already appears at one-loop [70], and other physical phenomena that could appear.
The current approach to analytic and semi-analytic binary inspiral waveforms makes use of the effective one-body formalism [2,4], whose input is an off-shell binary Hamiltonian and radiation reaction forces. It would be interesting to identify features of gravitational waveforms that can be analytically continued between bound and unbound systems, in analogy with inclusive observables discussed in Refs. [38,39,41]. We may expect that a detailed understanding of the analytic structure of the waveforms, both in the bound and unbound case, will be important. we only need to remove two matter propagators and one integral measure instead, which leads to a = 4. Thus, if a = 8 -i.e. if the cut of the eight-scalar amplitude is connected -then it has classical scaling upon use of eq. (A.2) and the truncation to classical order is iteratively consistent. In contrast, if a = 4 -i.e. if the cut of the eight-scalar amplitude is disconnected -the cut is suppressed by 4 , which means that the disconnected components contain quantum contributions.
Repeating the calculation above for any pair of matter lines, it follows that any 2MPR contribution with a classical scaling requires quantum information about some amplitude contributing to it. Since the HEFT approach prescribes that tree amplitudes be truncated to classical orders, the scaling of 2MPR diagrams and cuts with classical vertices is superclassical, with one factor of −1 for each present two-matter-particle cut. On the other hand, if we explicitly include some quantum operators in the Lagrangian and keep higher-order terms in the HEFT tree amplitudes, we will get classical 2MPR contributions. However, they will be consistently subtracted in the KMOC formalism. Thus, only 2MPI cuts -and consequently only 2PMI diagrams -should contribute to classical scattering observables to any loop order.

B Evaluating box master integrals
In this appendix, we derive all the box master integrals listed in section 6.2. We will repeatedly use the bubble integral with a cut matter propagator and a generic propagator in d = 4 − 2 , where Q 1 is a generic vector.

(B.5)
The result agrees with eq. (6.17) in the main text.
B.2 I ± 1,1,0,1 We next consider the box integrals with one uncut matter propagator. We start with (B.6) 22 One can show this by using for the integral. The remaining x integral is convergent.
where we have used the eq. (6.9) to combine 2 and 2ū 1 · + i0. The IBP identity can reduce the double propagator . (B.7) The integral proportional to d − 4 contributes to O( ) order, for a reason similar to the discussion below eq. (B.3). Now using eq. (B.1) for the bubble integral in the second line of eq. (B.7), we get the IR divergence and the finite part of this integral, Here we can omit the i0 prescription in logū 1 ·q 2 −q 2 2 because the argument is positive. To compute I − 1,1,0,1 , we first redefine the loop momentum to flip the sign of i0, . We apply the same strategy to compute the last set of box integrals, . (B.12) The only difference here is that the combination I + 1,0,1,1 + I − 1,0,1,1 is now complex, with the imaginary part given by further cutting ( − q 1 ) 2 . Starting with I + 1,0,1,1 , we first apply eq. (6.9) to combine ( − q 1 ) 2 and 2ū 1 · + i0, followed by using an IBP identity to get, where I + a and I − b are defined as . (B.14) Similar to eq. (B.9), we can rewrite I − 1,0,1,1 as where I − a,b = I + a,b q 1 →−q 1 . Therefore, we have such that we only need to compute the combination I + a − I − a and I + b − I − b . If we use eq. (B.1) with d = 4 to integrate out the bubble in I ± a , we will find that the resulting x integral is divergent at x → ∞. However, by taking the difference I + a − I − a , we get a finite result since the range of x gets truncated. Interestingly, the result is proportional to the triangle integral (6.14), We first use the same Feynman parameterization as in eq. (B.2) to bring together the two massless propagators, . (B.20) The resultant double propagator ( − q 1 + xk 5 ) 4 is then reduced by the IBP relation, where G(x) is a quadratic polynomial in x, G(x) = x 2 (ū 1 · q 2 ) 2 + 2x(1 − x)y(ū 1 · q 2 )(ū 2 · q 1 ) + (1 − x) 2 (ū 2 · q 1 ) 2 .