Logarithmic tail contributions to the energy function of circular compact binaries

TL;DR

This paper derives explicit expressions for logarithmic tail contributions to the conservative dynamics of circular compact binaries up to 7PN order using PN, EFT, and SF methods, and resums leading logarithms via RG techniques.

Contribution

It provides the first explicit formulas for tail-induced logarithmic terms up to 7PN and introduces a resummation of leading logarithms at arbitrary PN orders.

Findings

Explicit 6PN logarithmic tail terms derived

All 7PN logarithmic terms determined, including tail-of-tail-of-tail

Leading logarithms resummed into a closed-form expression

Abstract

We combine different techniques to extract information about the logarithmic contributions to the two-body conservative dynamics within the post-Newtonian (PN) approximation of General Relativity. The logarithms come from the conservative part of non linear gravitational-wave tails and their iterations. Explicit, original expressions are found for conservative dynamics logarithmic tail terms up to 6PN order by adopting both traditional PN calculations and effective field theory (EFT) methods. We also determine all logarithmic terms at 7PN order, fixing a sub-leading logarithm from a tail-of-tail-of-tail process by comparison with self-force (SF) results. Moreover, we use renormalization group techniques to obtain the leading logarithmic terms to generic power $n$, appearing at $(3n+1)$PN order, and we resum the infinite series in a closed form. Half-integer PN orders enter the conservative dynamics starting at 5.5PN, but they do not generate logarithmic contributions up to next-to-next-to-leading order included. We nevertheless present their contribution at leading order in the small mass ratio limit.