Gravitational dynamics at $O(G^6)$: perturbative gravitational scattering meets experimental mathematics

TL;DR

This work delivers a complete analytic evaluation of classical gravitational scattering at $O(G^6)$ (6PM) and $6\mathrm{PN}$ accuracy by importing multi-loop quantum-field-theory techniques to classical GR. It develops a two-pronged approach: high-precision numerical reconstruction and direct Harmonic Polylogarithm integration, to obtain analytic expressions for the nonlocal-in-time scattering contributions and the associated tail terms. The authors provide full analytic results for the 6PM nonlocal tail coefficient $A_2^h$, determine a crucial rational coefficient $D$, and derive the minimal time-localization factor $f(t)$ and the corresponding 6PN periastron-precession correction, thereby significantly improving the analytic understanding of binary dynamics. The work demonstrates a promising synergy between GR and QFT methods, with potential impact on precision gravitational-wave modeling and the analytic study of high-order classical scattering observables.

Abstract

A recently introduced approach to the gravitational dynamics of binary systems involves intricate integrals, linked to nonlocal-in-time interactions arising at the 5-loop level of classical gravitational scattering. We complete the analytical evaluation of classical gravitational scattering at the sixth order in Newton's constant, $G$, and at the sixth post-Newtonian accuracy. We use computing techniques developed for the evaluation of multi-loop Feynman integrals to obtain our results in two ways: high-precision arithmetic, yielding reconstructed analytic expressions, and direct integration {\it via} Harmonic Polylogarithms. The analytic expression of the tail contribution to the scattering involve transcendental constants up to weight four.