Dynamics of Binary Systems to Fourth Post-Minkowskian Order from the Effective Field Theory Approach

TL;DR

This paper computes the dynamics of non-spinning binary systems to the fourth Post-Minkowskian order using effective field theory, involving complex three-loop integrals and deriving Hamiltonians, with results matching recent amplitude-based and Post-Newtonian findings.

Contribution

It provides the first complete 4PM order potential interactions calculation for non-spinning binaries via EFT, including elliptic integrals and tail effects, extending previous lower-order results.

Findings

Computed scattering angle to ${ m O}(G^4)$ order.

Derived the 4PM Hamiltonian and radial actions.

Matched results with recent amplitude and Post-Newtonian calculations.

Abstract

We present the contribution from potential interactions to the dynamics of non-spinning binaries to fourth Post-Minkowskian (4PM) order. This is achieved by computing the scattering angle to ${\cal O}(G^4)$ using the effective field theory approach and deriving the bound radial action through analytic continuation. We reconstruct the Hamiltonian and center-of-mass momentum in an isotropic gauge. The (three-loop) integrals involved in our calculation are computed via differential equations, including a sector yielding elliptic integrals. Using the universal link between potential and tail terms, we also report: 1) The instantaneous energy flux at ${\cal O}(G^3)$; 2) The contribution to the 4PM unbound/bound radial action(s) depending on logarithms of the binding energy; 3) The (scheme-independent) logarithmic contribution to the 4PM non-local tail Hamiltonian for circular orbits. Our results in the potential region are in agreement with the recent derivation from scattering amplitudes. We also find perfect agreement in the overlap with the state-of-the-art in Post-Newtonian theory.