Ambiguity-Free Completion of the Equations of Motion of Compact Binary Systems at the Fourth Post-Newtonian Order

TL;DR

This work provides an ambiguity-free derivation of the conservative equations of motion for a two-body system of non-spinning compact objects at $4$PN order using a Fokker action in harmonic coordinates. It achieves this by a detailed near-zone/$d$-dimensional far-zone matching of PN and multipolar fields, together with a nonlocal tail term computed via $oldsymbol{\varepsilon\eta}$ regularization to handle divergences. The key result is the fixation of the last ambiguity parameter, $\kappa=\frac{41}{60}$, consistent with gravitational self-force and EFT calculations, and the demonstration that UV/IR poles cancel in the final, scale-free Lagrangian. This confirms the consistency of the EFT and GSF pictures and provides a fully specified 4PN conservative dynamics that aligns with independent approaches. The techniques, including the explicit treatment of tails and nonlocality, bolster confidence in high-precision gravitational-wave templates for binary inspirals.

Abstract

We present the first complete (i.e., ambiguity-free) derivation of the equations of motion of two non-spinning compact objects up to the 4PN order, based on the Fokker action of point particles in harmonic coordinates. The last ambiguity parameter is determined from first principle, by resorting to a matching between the near zone and far zone fields, and a consistent computation of the 4PN tail effect in d dimensions. Dimensional regularization is used throughout for treating IR divergences appearing at 4PN order, as well as UV divergences due to the model of point particles describing compact objects.