Local in Time Conservative Binary Dynamics at Fourth Post-Minkowskian Order

Leveraging scattering information to describe binary systems in generic orbits requires identifying local and nonlocal in time tail effects. We report here the derivation of the universal (nonspinning) local in time conservative dynamics at fourth post-Minkowskian order, i.e., O ð G 4 Þ . This is achtieved by computing the nonlocal-in-time contribution to the deflection angle, and removing it from the full conservative value in [C. Dlapa et al. , Phys. Rev. Lett. 128 , 161104 (2022).; C. Dlapa et al. , Phys. Rev. Lett. 130 , 101401 (2023).]. Unlike the total result, the integration problem involves two scales — velocity and mass ratio — and features multiple polylogarithms, complete elliptic and iterated elliptic integrals, notably in the mass ratio. We reconstruct the local radial action, center-of-mass momentum and Hamiltonian, as well as the exact logarithmic-dependent part(s), all valid for generic orbits. We incorporate the remaining nonlocal terms for ellipticlike motion to sixth post-Newtonian order. The combined Hamiltonian is in perfect agreement in the overlap with the post-Newtonian state of the art. The results presented here provide the most accurate description of gravitationally bound binaries harnessing scattering data to date, readily applicable to waveform modeling.

Introduction.Motivated by the impending era of high-precision gravitational-wave (GW) astronomy with observatories such as LISA [3], the Einstein Telescope [4] and the Cosmic Explorer [5], and the incredibly rich amount of information expected from compact binary sources [6][7][8][9][10][11], the (long dormant [12]) Post-Minkowskian (PM) expansion in general relativity-entailing a perturbative series in G (Newton's constant) but to all orders in the relative velocity-has experienced a resurgence in recent years, e.g.[1,2,.This is, in part, thanks to the repurposing of modern integration techniques from collider physics (see [42,44] and references therein), which have led to a plethora of new results.Notably, using worldline effective field theory (EFT) methodologies [47][48][49][50][51], the rapidly evolving state of the art includes the total relativistic impulse (yielding the scattering angle and emitted GW flux) of nonspinning [1,2] and spinning [39,40] bodies to O(G 4 ), akin of a 'three-loop' calculation in particle physics, as well as partial results in the conservative sector at 5PM [46].
The derivations in [1,2], together with a (Firsov-type [52,53]) resummation scheme [21,22], have led to an unprecedented agreement between analytic results and numerical simulations [54][55][56], paving the way to more accurate waveform models for hyperbolic encounters.However, due to nonlocal-in-time effects [57,58], unbound results cannot be used to describe generic ellipticlike motion (away from the large-eccentricity limit [1]).As shown in [23], the binding energy for quasi-circular orbits obtained from scattering results (via the "boundaryto-bound" (B2B) analytically continuation [21,22]) does not reproduce-other than logarithms-the known Post-Newtonian (PN) values [57][58][59][60][61] (see also [62]).Hence, to fully harness the power of scattering calculations, a separation between local-and nonlocal-in-time effects was thus imperative.In this letter we report the derivation of the universal (non-spinning) local-in-time con-servative dynamics of binary systems at O(G 4 ).This is obtained via a direct computation of the nonlocalin-time contribution to the scattering angle.Following [23,63], the calculation entails an integral over the energy spectrum times the logarithm of the center-of-mass GW frequency.To solve the integration problem, we implement the methodology of differential equations, already used in [1,2].However, unlike the total impulse, which obeys a simple (power-law) mass scaling [16,21], isolating (non)local effects in a gauge-invariant fashion entails dealing with two relevant scales: the velocity and mass ratio.Despite the complexity of the two-scale problem, we find that it can be factorized into solving two secondorder Picard-Fuchs (PF) equations.The nonlocal part of the angle features multiple polylogarithms (MPLs), complete elliptic integrals, and integrations thereof.We find agreement in the overlap with the 6PN values in [63].
We derive the local-in-time contribution to the conservative scattering angle by removing the (unbound) nonlocal terms from the total result in [1,2].The local radial action follows directly via the B2B map [21][22][23].Using the relations in [21,22], we reconstruct the universal local-in-time center-of-mass momentum and Hamiltonian in isotropic gauge, together with the complete logarithmic dependence, all applicable to generic motion.We also provide-for all practical purposes-results expanded to 30 orders in the (symmetric) mass ratio and all orders in the velocity (with an error beyond 30PN).To incorporate the remaining (non-logarithmic) nonlocal part of the bound dynamics, we adapt to our isotropic gauge the values obtained in [63] to 6PN order.The combined Hamiltonian at O(G 4 ) perfectly matches in the overlap with the state of the art in PN theory [62][63][64].The results presented here can be directly inputed onto waveforms models for gravitationally-bound eccentric orbits, potentially increasing their accuracy by incorporating an infinite tower of (local-in-time) velocity corrections.
(Non)local-in-time tail effects.The scattering of the emitted radiation off of the binary's gravitational potential, or "tail effect", enters in the 4PM conservative dynamics both through local-as well as nonlocal-in-time interactions [57][58][59][60][61].Because of this, although an effectively local description is possible to any order [1,2], the coefficients of the radial action (or Hamiltonian) depend on the type of motion, and therefore are not related via analytic continuation for generic orbits.Our strategy is to identify the local-and nonlocal-in-time parts of S r , the total radial action.Due to the structure of tail effects [23,57,58], the nonlocal-in-time tail terms can be shown to take the gauge-invariant form where ω ≡ +∞ −∞ dω 2π , E and dE dω are the total (incoming) energy and emitted GW spectrum in the center-ofmass frame.The 'renormalization scale' µ, which cancels against a similar term in the local-in-time part [1,34,58], can be arbitrarily chosen.The factor of 4e 2γE ( with γ E Euler's constant) follows the PN conventions [57,58].An explicit derivation of (1) in the context of the PM expansion can be found in [23], see also [63] for a discussion in the PN regime.
For unbound motion, the scattering angle is given by χ 2π = −∂ j I r , with I r ≡ Sr GM 2 ν and j ≡ J GM 2 ν the (reduced) radial action and angular momentum, and the total mass, mass ratio, and symmetric mass ratio, respectively.We split the PM coefficients of the deflection angle in impact parameter space as, where γ ≡ u 1 • u 2 (using the mostly negative metric convention), u 1,2 the incoming velocities, and Γ ≡ E/M = 1 + 2ν(γ − 1).(The reader should keep in mind that logarithms of the velocity may still appear in both coefficients.)In the remaining of this letter we choose µ ≡ 1/GM for the renormalization scale.
Integration.To solve for the master integrals, we derive differential equations in x and the mass ratio, q, where x is given by γ = 1 2 x + 1 x .We then adopt the strategy of an ǫ-(and ǫ-)regular basis [67], such that we can set ǫ = 0, and consider the expansion of the integrand, differential equations, and boundary constants, only to O(ǫ).The latter are determined via a small-q expansion, together with the techniques described in [42] (adapted to the new factors of ǫ).From this setting, it is then straightforward to find a solution of the differential equations through iterated integration.
For the parts containing MPLs, and similarly to the x variable, it is useful to rationalize the square root of the energy E m1 = 1 + 2γq + q 2 = (q + x) q + 1 x , by introducing a new variable, y, defined through Hence, we find the traditional harmonic polylogarithms with letters {x, 1 + x, 1 − x} [26,33], as well as MPLs which depend on the velocity and mass ratio via the new letters: {y, 1 + y, 1 − y, y − 1+x 1−x , y − 1−x 1+x , 1 + 2 1−x 1+x y + y 2 }.In addition to MPLs, the solution to the differential equations depend on another set of functions, through an a priori irreducible fourth-order PF equation, already at O(ǫ 0 ).However, a Baikov representation [68,69] of the maximal cut suggests a simpler Calabi-Yau two-fold as the relevant geometry.Indeed, in terms of the variables (qx, q x ), the differential equations can be solved, in the first and subsequently the second variable, via two equivalent second-order PF equations (per variable).The solution can then be written in terms of products of K's (such as the f 1 in (5) below) as well as the leading three derivatives w.r.t. the mass ratio.As in previous PM computations, e.g.[1,2,34], K(z) = , is the complete elliptic integral of the first kind.
After the leading order solution is known, it is then straightforward to obtain the O(ǫ) part.We find that it can be written in terms of (at most) two-fold iterated integrals, with elliptic kernels depending on the mass ratio, q, as the integration variable.The full set is given by: Remarkably, while individually this is not the case, the combination of complete elliptic integrals in f 1 has a simple power-series expansion in the PN limit (x → 1).Furthermore, the f i 's are real, and have (at most) simple poles in q.Let us point out, however, that a simplified version of the iterated integrals may still be possible.
In particular, upon assigning to K(z) a transcendental weight one, we notice that the iterated integrals would have up to weight four, in contrast to the MPL part with maximum weight two.Hence, we expect that either the naïve assignment is incorrect or an even simpler form exists.
We leave this open for future work.
Scattering angle.After solving for the master integrals and plugging them back into the integrand, we arrive at the radial action, and from there to the nonlocalin-time contribution to the deflection angle, χ b(nloc) and χ (4) log b(nloc) , at 4PM order.As anticipated in [34], the logarithmic part takes on a simple closed form, −2ν with χ 2ǫ introduced in [34], and the h 5,9,10 are polynomials depending only on γ, which also enter the nonlogarithmic part (see below).The latter, on the other hand, also involves a set of iterated integrals in the mass ratio.Despite its complexity, it is straightforward to construct a "self-force" (SF) expanded version, for which we find the generic form, valid to any nSF order, Due to the structure of the full solution, except for the h 1 , h 3 and h 4 carrying information from the (iterated) elliptic sector (h 1,4 ) and the new letters in the MPLs depending on the mass ratio (h 3 ), the remaining h i 's are SF-exact.We find the nSF coefficients may be split as where the h i (γ), are polynomials in γ only.
The ∆h i (γ, ν) vanish except when i = 1, 3, 4, for which they become polynomials both in γ and ν, up to O(ν n ).We provide in an ancillary file their values up to n = 30.The 30SF result (with an error beyond 30PN) is in perfect agreement in the overlap with the 6PN values in [63].
Let us emphasize that the definition of nonlocal-intime in [63,64] includes not only the expression in (1) (W 1 in [63]), but also an extra contribution (W 2 in [63]).
Due to the local-in-time (and gauge-dependent) nature of W 2 , we do not add it to (1).Therefore, (7) agrees (in the overlapping realm of validity) with the scattering angle obtained from the W 1 -only terms in Eq. (3.14) of [63].
After subtracting from the total conservative angle in [1,2], we arrive at the local-in-time counterpart, 1  where we used the fact that the log µb Γ cancels out in the total value.The result in ( 9) can now be used to describe generic bound orbits, as we discuss next.
Universal logarithms.Nonlocal-in-time tail effects also contribute with a log r term in the bound dynamics.Performing a small-eccentricity expansion of (1), and using Kepler's law (log Ω = − 3 2 log r + • • • , with Ω the 1PM orbital frequency), we find ĉlog 1 Although amenable to a conservative-like description of the relative dynamics, we keep the other (time-symmetric) radiationreaction corrections, i.e. "2rad" in [2], in the dissipative part.
where ξ is the energy flux at 3PM order [23,34].Similarly, consistently with (6).Hence, adding both terms, Ĥell(log and likewise, for the full logarithmic dependence of the bound Hamiltonian and center-of-mass momentum at 4PM.
Towards the complete bound dynamics.Putting together the local-in-time coefficient plus exact logarithmic part, the total bound Hamiltonian up to 4PM order may be written as where we have absorbed the factor of e 2γE that arises from (1) into the logarithm.The ĉ1|2|3(loc) are the known local-in-time PM coefficients up to 3PM order [18,20,25,26], and ĉ4(loc) is reported here for the first time.
To complete the knowledge of the bound dynamics, we are still missing the non-log r e 2γ E nonlocal-in-time contributions, ĉi(nloc) , which depend on the trajectory.These are more difficult to compute in a PM scheme, since they are often needed in the opposite limit of quasicircular orbits, thus entering at all PM orders!Yet, they can be readily obtained within the PN approximation by evaluating the radial action in (1) in a small-eccentricity expansion.Adapting the (W 1 -only) results in [63] to the isotropic gauge, we quote their values in the supplemental material to 6PN and eight order in the eccentricity.The combined Hamiltonian in (18) is in perfect agreement to O(G 4 p6 ) with the Ĥell 6PN(4PM) derived in [62] using the state of the art in PN theory, while at the same time it incorporates all-order-in-velocity corrections.Ready-touse expressions for the full results and 30SF-approximate are collected in ancillary files.

Conclusions. Novel integration techniques in combination with EFT methodologies have been extremely
successful in reaching the very state of the art in our understanding of scattering dynamics in general relativity, including conservative and dissipative effects [42,46].However, as illustrated in [23,62], although local-in-time and logarithms are universal, the full hyperbolic results fail to describe quasi-circular binaries.This is due to the presence of orbit-dependent (non-logarithmic) nonlocalin-time effects, which preclude a smooth analytic continuation via the B2B map [21,22].Hence, up until now, we were lacking a direct correspondence to generic bound motion, notably for the conservative sector.
We have computed the nonlocal-in-time contribution to the deflection angle, and removed it from the total conservative value in [1,2], thus yielding the local-in-time counterpart.We then derived the radial action, centerof-mass (isotropic-gauge) momentum and Hamiltonian, as well as the total logarithmic-dependent part(s), all applicable to generic motion.Upon adapting the (nonlogarithmic) nonlocal-in-time effects for elliptic-like orbits computed in the PN expansion [63], the combined total Hamiltonian becomes the most accurate description of gravitationally-bound binary systems obtained from PN/PM data to date, readily applicable to waveform modelling.Studies assessing the implications of our results towards constructing high-precision GW templates, as well as the derivation of a PM version of nonlocal-intime effects for bound orbits, are underway.

Supplemental Material
An accurate hybrid description of elliptic-like orbits can be constructed by incorporating into the Hamiltonian in (18) For the remaining terms, translating from the results in [63]  Upon PN expanding the total Hamiltonian in (19), we find agreement with the expression for Ĥell 6PN(4PM) in [62], in the overlapping realm of validity.
Let us conclude with a few remarks on the (non)local-in-time dynamics in (19).In addition to γ = u 1 • u 2 , the nonlocality in time introduces a new (mass-dependent) scale in the scattering problem, i.e. u 1(2) • u com , yielding [63]non-log r e 2γ E nonlocal-in-time (W 1 -only) contributions computed in PN theory to 6PN and O(e 8 ) in the small-eccentricity expansion in[63], yielding