Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at $O(G^4)$

TL;DR

This paper advances the understanding of two-body gravitational scattering at fourth post-Minkowskian order by computing the conservative dynamics including tail effects, using modern amplitude techniques and effective field theory.

Contribution

It provides the first complete calculation of $O(G^4)$ conservative binary dynamics including radiative tail effects using scattering amplitudes and effective field theory.

Findings

Derived the classical amplitude involving elliptic integrals and polylogarithms.

Obtained the radial action from the amplitude and matched it with post-Newtonian results.

Extended the scattering amplitude methods to include tail effects at high order.

Abstract

We complete the calculation of conservative two-body scattering dynamics at fourth post-Minkowskian order, i.e. $O(G^4)$ and all orders in velocity, including radiative contributions corresponding to the tail effect in general relativity. As in previous calculations, we harness powerful tools from the modern scattering amplitudes program including generalized unitarity, the double copy, and advanced multiloop integration methods, in combination with effective field theory. The classical amplitude involves complete elliptic integrals, and polylogarithms with up to transcendental weight two. Using the amplitude-action relation, we obtain the radial action directly from the amplitude, and match the known overlapping terms in the post-Newtonian expansion.