Dimensional regularization of the IR divergences in the Fokker action of point-particle binaries at the fourth post-Newtonian order

TL;DR

This work resolves infrared ambiguities in the 4PN Fokker action for point-particle binaries by implementing dimensional regularization for both IR and UV divergences. By computing the DR-HR difference for the instantaneous gravitational part, incorporating the non-local tail term in $d$ dimensions, and performing near-zone/far-zone matching, the authors show that the IR poles cancel against a tail-induced UV pole, fixing the two ambiguity parameters. The tail calculation yields a finite, gauge-consistent contribution with a coefficient matching EFT results, and the final DR Lagrangian reproduces the previously identified values $oldsymbol{eta}_1=-2179/315$ and $oldsymbol{eta}_2=192/35$, in agreement with GSF constraints. Overall, the paper confirms that the 4PN dynamics derived from the Fokker action in harmonic coordinates is complete and consistent with independent self-force and EFT analyses, highlighting dimensional regularization as a robust tool for classical GR calculations.

Abstract

The Fokker action of point-particle binaries at the fourth post-Newtonian (4PN) approximation of general relativity has been determined previously. However two ambiguity parameters associated with infra-red (IR) divergencies of spatial integrals had to be introduced. These two parameters were fixed by comparison with gravitational self-force (GSF) calculations of the conserved energy and periastron advance for circular orbits in the test-mass limit. In the present paper together with a companion paper, we determine both these ambiguities from first principle, by means of dimensional regularization. Our computation is thus entirely defined within the dimensional regularization scheme, for treating at once the IR and ultra-violet (UV) divergencies. In particular, we obtain crucial contributions coming from the Einstein-Hilbert part of the action and from the non-local tail term in arbitrary dimensions, which resolve the ambiguities.