Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order

We present the amplitude for classical scattering of gravitationally interacting massive scalars at third post-Minkowskian order. Our approach harnesses powerful tools from the modern amplitudes program such as generalized unitarity and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. Adapting methods for integration and matching from effective field theory, we extract the conservative Hamiltonian for compact spinless binaries at third post-Minkowskian order. The resulting Hamiltonian is in complete agreement with corresponding terms in state-of-the-art expressions at fourth post-Newtonian order as well as the probe limit at all orders in velocity. We also derive the scattering angle at third post-Minkowskian order.

The past decade has also witnessed immense progress in the study of scattering amplitudes, where understanding mathematical structures within gauge theory and gravity has yielded new physical insights and efficient methods for calculation. In particular, the Bern-Carrasco-Johansson (BCJ) color-kinematics duality and associated double copy construction [14] allow multiloop gravitational amplitudes to be constructed from sums of products of gauge-theory quantities. This has yielded a variety of new results in supergravity (see Ref. [15] for recent results). The BCJ construction is intimately tied to the Kawai-Lewellen-Tye (KLT) relations [16], which relate tree amplitudes of closed and open strings.
In this paper, we apply modern amplitude methods to derive the classical scattering amplitude for two massive spinless particles at O(G 3 ) and to all orders in the velocity, i.e. at the third post-Minkowskian (3PM) order. We use generalized unitarity [17] to construct the corresponding two-loop integrand from tree amplitudes of gravitons and massive scalars, obtained straightforwardly from the double-copy construction. While the double copy introduces dilaton and antisymmetric tensor degrees of freedom [18] which are absent in pure Einstein gravity, we remove these unwanted states efficiently by restricting the state sums in unitarity cuts to gravitons alone. As we will show, we can calculate in strictly D = 4 dimensions for the classical dynamics, where spinor he-licity variables [19,20] dramatically simplify the required tree amplitudes. The viability of working in D = 4 offers optimism for extending our results to higher orders.
Afterwards, we integrate the two-loop integrand via a procedure adapted from EFT, in which energy integrals are evaluated in the potential region via residues before performing spatial integrations [21]. Using EFT matching [21,22] we then derive the 3PM conservative Hamiltonian for compact spinless binaries. We show that the 4PN terms in our Hamiltonian are, up to a coordinate transformation, physically equivalent to corresponding terms in state-of-the-art results. We also verify that our result agrees in the probe limit with the Hamiltonian for a test body orbiting a Schwarzschild black hole to 3PM order. Finally, we derive a compact expression for the 3PM scattering angle in terms of amplitude data.
Double copy and unitarity. Dynamics at 3PM order is encoded in the two-loop scattering amplitude for two massive, gravitationally interacting scalars. Our calculation begins with a construction of the corresponding two-loop integrand via generalized unitarity. Because we are interested in classical scattering, we need not assemble the full quantum-mechanical integrand. Rather, as emphasized in Refs. [21][22][23], the classical potential only receives contributions with a single on-shell matter line per loop and with no gravitons starting and ending on the same matter line. For this reason we focus solely on the unitarity cuts shown in Fig. 1.
We obtain the tree amplitudes in the unitarity cuts via two methods. In the first approach, we work in general D space-time dimensions. Exploiting color-kinematics duality [14], we derive gravitational amplitudes straightforwardly from simpler gauge-theory amplitudes by replacing color factors with corresponding kinematic factors. For the unitarity cuts of the classical limit of the two-loop scattering amplitude, the reference momenta that complicate projection onto graviton physical states can be eliminated, simplifying the calculation [24]. The primary purpose of our D-dimensional construction is to confirm FIG. 1. Unitarity cuts needed for the classical scattering amplitude. The shaded ovals represent tree amplitudes while the exposed lines depict on-shell states. The wiggly and straight lines denote gravitons and massive scalars, respectively. explicitly the completeness of our second method, where we work in strictly D = 4 so as to benefit from very simple expressions for gauge-theory amplitudes in terms of spinor helicity [19] variables. We then build the two corresponding gravitational amplitudes via the KLT relations [16]. At two loops, both approaches are efficient, but at higher loops, helicity amplitudes offer a much more compact starting point.
For concreteness, consider the first generalized unitarity cut in Fig. 1, which we refer to as C H-cut and is comprised of products of four three-point and one four-point amplitudes. Since four-point tree amplitudes are already very simple there is little computational advantage to imposing the on-shell conditions on matter lines. Thus, we replace the pairs of three-point amplitudes at the top and bottom of the cut with four-point amplitudes and then impose the matter cut conditions at the end. The resulting iterated two-particle cut is then where M 4 denotes the tree-level four-point amplitude for gravity minimally coupled to two massive scalars denoted here by legs 1 s , 2 s , 3 s , 4 s . In this cut, legs 1 s , 4 s have mass m 1 while legs 2 s , 3 s have mass m 2 . All momenta in each tree amplitude are taken to be outgoing. The sum runs over graviton states for legs 5, 6, 7, 8, where the minus signs on the labels indicate reversed momenta. The four-point gravity tree amplitudes needed in the cuts are obtained from gauge-theory ones via the fieldtheory limit of KLT relations [16], (1,2,4,3), (2) where the A 4 are tree-level color-ordered gauge-theory four-point amplitudes and s ij = (p i + p j ) 2 , working in mostly minus metric signature throughout. Strictly speaking, the KLT relations apply only to massless states. However, they can be applied here by interpreting the scalar masses, in the sense of dimensional reduction, as extra-dimensional momentum components. While we have not included coupling constants, these are easily restored at the end of the calculation by including an overall factor of (8πG) 3 , where G is Newton's constant.
In terms of the spinor-helicity conventions of Ref. [20], the independent tree-level gauge-theory amplitudes needed in Eq. (1) are where t ij = 2p i · p j and the ± denote gluon helicities. The dilaton and antisymmetric tensor states are removed from unitarity cuts by correlating the gluon helicities on both sides of the double copy. The unwanted states correspond to one gluon in the double copy of positive helicity and the other of negative helicity. An internal graviton state is obtained by taking the corresponding gluons in the KLT formula in Eq. (2) to be of the same helicity.
Using spinor evaluation techniques, it is straightforward to obtain a compact expression for the iterated twoparticle cut in Eq. (1) (e.g. see Ref. [25]). Imposing cuts on the matter lines, as indicated in the first unitarity cut of Fig. 1, further simplifies it and gives C H-cut . We find where we have defined The simplicity of this expression is a reflection of the double-copy structure: the same building blocks appear in the simpler corresponding gauge-theory cut.
The spurious double-pole in s 23 can be explicitly cancelled by adding terms proportional to the Gram determinant formed from the five independent momenta at two loops which vanishes in D = 4. In fact, the expression derived from the D-dimensional approach is automatically free of such spurious singularities. While these Gram determinants contribute quantum mechanically, we have checked explicitly that they vanish in the classical limit. This is not accidental-such terms are of the wrong form to generate the required log(s 23 ) needed to contribute to the classical 3PM amplitude (see Ref. [24] for details).
The remaining two independent generalized unitarity cuts in Fig. 2 are more complicated because they require five-point tree amplitudes with two massive scalar legs. The four-dimensional input gauge-theory amplitudes are simple to compute using modern methods (e.g. see Ref. [26]). For our D construction we obtain a BCJ representation, allowing us to express the gravity cuts directly in terms of local diagrams. The particular representation was chosen such that we can ignore the reference momenta when projecting the internal states into gravitons. Further details will be given elsewhere [24].
To facilitate integration, we merge the cuts into a single integrand whose cuts match those in Fig. 1. This is achieved using an ansatz in terms of eight independent diagrams with only cubic vertices displayed in Fig. 2. The diagrammatic numerators are polynomials of the appropriate dimension exhibiting the symmetries of the corresponding diagram. Their coefficients are then fixed via the method of maximal cuts [27], whereby cuts of the integrand are constrained to match the known ones. This approach is sufficient for the two-loop problem. Integration. Our method of integration follows Ref. [21]. For convenience, we give a short summary here, leaving details to Ref. [24]. Terms in the integrand take the form, where i labels each matter line, which has energy ω i , spatial momentum k i , and mass m i . The matter propagators can be factored into particle and antiparticle poles, ω i ± k 2 i + m 2 i . We then express the integrand as I = N × i 1 zi , i.e. in terms of the particle poles z i = ω i − k 2 i + m 2 i and an effective numerator N which absorbs the rest of the integrand.
Following the procedure outlined in Ref. [21], we first evaluate the energy integrals. At two loops, i.e. 3PM order, we integrate over two independent combinations of energies, ω and ω ′ , in the potential region. As we will prove in detail in Ref. [24], the result is where the sum runs over distinct pairings (i, j) of matter poles and z i = z j = 0 when (ω, ω ′ ) = (ω ij , ω ′ ij ). Here S ij is a calculable symmetry factor whose sign and magnitude depend on the topology of the cut graph. Note that the residue for an (i, j) pairing will vanish if there are no values of ω and ω ′ for which z i = z j = 0.
The resulting quantity I depends on two independent spatial loop momenta. To integrate over them we employ dimensional regularization to deal with ultraviolet divergences stemming from the renormalization of delta function contact interactions which do not contribute classically. Due to the localization on energy residues, I is a complicated, non-polynomial function of threedimensional invariants involving square roots. Nevertheless, we can series expand I in large m 1,2 , yielding polynomials of kinematic invariants which we can integrate at each order. After expanding, nearly all the spatial integrals are simple bubbles for which there are known analytic expressions [28]. The remaining integrals are evaluated via integration-by-parts identities [29].
For diagrams free from infrared (IR) singularities generated by iterations of lower-loop graviton exchanges, we have checked that our integrated results accord with several standard methods in the Feynman integral literature, including the Mellin-Barnes representation [28,30], numerical integration via sector decomposition [31], and differential equations [32] derived through integration-byparts reduction [29,33]. Since the classical contribution comes from certain residues on matter poles, the system of differential equations omits integrals without support on such residues. Amplitude and potential. We apply the integration procedure outlined above order by order in the large-mass expansion, i.e. in powers of velocity. Combining an explicit evaluation of the 3PM amplitude up to 7PN order with information on the pole structure of individual integrals and exact, manifestly relativistic analytic results for certain graph topologies, we conjecture a full, all orders in velocity expression for the 3PM amplitude (whose uniqueness will be discussed in Ref. [24]): where the log scale dependence is absorbed into a delta-function ultraviolet counterterm. Here we use center-of-mass coordinates where the incoming and outgoing particle momenta are ±p and ±(p − q), respectively. We emphasize that M 3 includes the nonrelativistic normalization factor, 1/4E 1 E 2 , where E 1,2 = p 2 + m 2 1,2 . We also define the total mass m = m 1 + m 2 , the symmetric mass ratio ν = m 1 m 2 /m 2 , the total energy E = E 1 + E 2 , the symmetric energy ratio ξ = E 1 E 2 /E 2 , the energymass ratio γ = E/m, and the relativistic kinematic invariant σ = p1·p2 m1m2 . Note that the arcsinh factor is actually proportional to the sum of particle rapidities, arctanh |p|/E 1,2 .
Eq. (8) only includes q-dependent terms which persist in the classical limit. In particular, the log q 2 term ultimately feeds into the conservative Hamiltonian through the Fourier transform log q 2 FT = − 1 2π|r| 3 . Meanwhile the remaining IR-divergent contributions, parameterized by F 1 = k1 in the notation described in Eq.(12) of Ref. [21], will cancel in the EFT matching.
The Hamiltonian is extracted from the amplitude via EFT methods developed in Refs. [21,22,34] (see Ref. [12] for another approach). Consider massive spinless particles interacting via the center-of-mass Hamiltonian H(p, r) = p 2 + m 2 1 + p 2 + m 2 2 + V (p, r), where r is the distance vector between particles and i labels PM orders. The above Hamiltonian is in a gauge in which terms involving p·r or time derivatives of p are absent. We then compute the scattering amplitude of massive scalars, comes from diagrams with two or fewer loops that depend on c 1 , c 2 , and c 3 . In Ref. [21], the coefficients c 1 and c 2 were extracted analytically to all orders in velocity. Inserting these into M where for convenience, the expressions for c 1 and c 2 in Ref. [21] are reproduced here with slightly different normalization and in our current notation. As emphasized in Ref. [21], the cancellation of IR divergences between M (EFT) 3 and M 3 depends critically on c 1 and c 2 and thus provides a nontrivial check of our calculation. Consistency checks. Our results pass several nontrivial albeit overlapping consistency checks (see Ref. [24] for details). First and foremost, we have verified that the 4PN terms in our Hamiltonian are equivalent to known results up to a canonical coordinate transformation, with ellipses denoting higher order corrections entering as a power series in G/|r|, p 2 , and (p · r) 2 /r 2 (for past treatments, see Ref. [35,36]). To derive this coordinate transformation we generate an ansatz for A, B, C, D and constrain it to preserve the Poisson brackets, i.e. {r, p} = {R, P } = 1 with all other brackets vanishing, in the spirit of Ref. [37]. We verify that within this space of canonical transformations exists a subspace which maps our Hamiltonian in Eq. (10) to the one in the literature, e.g. as summarized in Eq.(8.41) of Ref. [9], up to the intersection of 3PM and 4PN accuracy.
Second, applying the methods of Ref. [21] we have checked that the full-theory amplitude M 3 in Eq. (8) is identical to the amplitude M (EFT) 3 computed from the conservative Hamiltonian in Ref. [9] up to 4PN accuracy.
Third, we have extracted from our Hamiltonian the coordinate invariant energy of a circular orbit as a function of the period. Working at 2PN order-the highest order subsumed by 3PM which is relevant to a virialized system-we agree with known results [8].
Fourth, we have extracted from our Hamiltonian the 3PM-accurate classical scattering angle in the center-ofmass frame, given by where J = b|p| is the angular momentum, b is the impact parameter, and we have defined d 1 = mγξq 2 M ′ 1 /|p|, d 2 = mγξ|q|M ′ 2 , and d 3 = mγξ|p|M ′ 3 / log q 2 . The primed quantities denote the IR-finite parts of the nonrelativistically normalized amplitudes that enter the Hamiltonian coefficients as defined here and in Ref. [21], so and M ′ 3 is the log q 2 term in Eq. (8). Truncated to 4PN order, Eq. (12) agrees with known results [38].
Last but not least, in the probe limit m 1 ≪ m 2 , our result exactly coincides with the Hamiltonian for a point particle in a Schwarzschild background to O(G 3 ) and all orders in velocity, e.g. as given in Eq.(8) of Ref. [39]. Conclusions. We have presented the 3PM amplitude for classical scattering of gravitationally interacting massive spinless particles. From this amplitude we have extracted the corresponding conservative Hamiltonian for binary dynamics to 3PM order.
The 3PM Hamiltonian in Eqs. (9) and (10) will be employed in a forthcoming paper [40] to compute approximants for the binding energy of binary systems moving on circular orbits and assess their accuracy against numerical-relativity predictions. This is relevant for understanding the usefulness of PM calculations when building accurate waveform models for LIGO/Virgo data analysis.
Our paper leaves many avenues for future work, e.g. including obtaining higher orders in the PM expansion, incorporating spin [41], radiation [42], and finite-size effects, as well as connecting to other recent amplitude approaches [18,43] and the effective one-body formalism [3,11,12,44].
The simplicity of the 3PM amplitude in Eq. (8) and potential in Eq. (10) bodes well for future progress. Moreover, since the amplitude and EFT methods employed in this paper are far from exhausted, we believe that the results we have reported mark only the beginning.