Fourth post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach
J. Blümlein, A. Maier, P. Marquard, G. Schäfer
TL;DR
This work derives the motion of two-body gravitational systems to the fourth post-Newtonian order using an effective field theory framework with dimensional regularization in harmonic coordinates. By computing the classical two-body potential from Feynman amplitudes and performing a Legendre transform to a Hamiltonian, the authors show that intermediate $1/\varepsilon$ poles cancel in observables via canonical transformations, mapping results to pole-free ADM and EOB coordinates. They provide comprehensive 4PN Hamiltonians (including tail terms) and explicit generating functions for the required coordinate transformations, verifying consistency with existing ADM and EOB formulations and with circular-orbit observables such as $E(\Omega)$ and $J(\Omega)$, including the ISCO behavior. The work strengthens the bridge between EFT-based gravity calculations and widely used two-body formalisms, enabling high-precision predictions relevant for gravitational-wave modeling and data analysis.
Abstract
We calculate the motion of binary mass systems in gravity up to the fourth post--Newtonian order. We use momentum expansions within an effective field theory approach based on Feynman amplitudes in harmonic coordinates by applying dimensional regularization. We construct the canonical transformations to ADM coordinates and to effective one body theory (EOB) to compare with other approaches. We show that intermediate poles in the dimensional regularization parameter $\varepsilon$ vanish in the observables and the classical theory is not renormalized. The results are illustrated for a series of observables for which we agree with the literature.