Scattering Amplitudes and Conservative Binary Dynamics at ${\cal O}(G^4)$

TL;DR

This paper computes the conservative binary dynamics in general relativity at fourth post-Minkowskian order using scattering amplitudes, advanced multiloop techniques, and effective field theory, extending previous lower-order results.

Contribution

It introduces a novel calculation of the ${ m O}(G^4)$ potential in binary systems using modern scattering amplitude methods and derives the Hamiltonian directly from the amplitude.

Findings

Agreement with known post-Newtonian results up to sixth order

Derived the radial action and Hamiltonian in isotropic gauge

Determined energy loss from radiation at ${ m O}(G^3)$

Abstract

Using scattering amplitudes, we obtain the potential contributions to conservative binary dynamics in general relativity at fourth post-Minkowskian order, ${\cal O}(G^4)$. As in previous lower-order 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 polylogarithms with up to transcendental weight two and elliptic integrals. We derive the radial action directly from the amplitude, and determine the corresponding Hamiltonian in isotropic gauge. Our results are in agreement with known overlapping terms up to sixth post-Newtonian order, and with the probe limit. We also determine the post-Minkowskian energy loss from radiation emission at ${\cal O}(G^3)$ via its relation to the tail effect.