Gravitational scattering at the seventh order in $G$: nonlocal contribution at the sixth post-Newtonian accuracy

TL;DR

This work provides a complete analytic determination of the nonlocal-in-time contributions to the classical gravitational scattering angle at $O(G^7)$ and 6PN. By importing multi-loop techniques—Mellin transforms and harmonic polylogarithms—from quantum field theory, the authors analytically evaluate the previously intractable scattering integrals $A_{m n k}$, including the challenging $m=2$ and $m=3$ sectors. They obtain exact expressions for the $A_{2 n k}$ and $A_{3 n k}$, and compute the nonlocal coefficient $D=-12607/108$, along with the minimal f-h contributions, thereby fully fixing the 6PN nonlocal dynamics. The results demonstrate a fruitful synergy between classical GR and QFT methods and provide precise inputs for high-accuracy gravitational-wave modeling of binary encounters.

Abstract

A recently introduced approach to the classical gravitational dynamics of binary systems involves intricate integrals (linked to a combination of nonlocal-in-time interactions with iterated $\frac1r$-potential scattering) which have so far resisted attempts at their analytical evaluation. By using computing techniques developed for the evaluation of multi-loop Feynman integrals (notably Harmonic Polylogarithms and Mellin transform) we show how to analytically compute all the integrals entering the nonlocal-in-time contribution to the classical scattering angle at the sixth post-Newtonian accuracy, and at the seventh order in Newton's constant, $G$ (corresponding to six-loop graphs in the diagrammatic representation of the classical scattering angle).