Dimensional regularization of local singularities in the 4th post-Newtonian two-point-mass Hamiltonian

TL;DR

This paper derives an explicit 4th post-Newtonian Hamiltonian for binary point masses, controlling local singularities via dimensional regularization, and determines the last stable orbit as a function of mass ratio.

Contribution

It provides the most complete explicit 4PN Hamiltonian with 51 out of 57 coefficients given exactly, improving upon previous results by including an additional six coefficients.

Findings

Explicit 4PN Hamiltonian with 51 coefficients

Controlled local divergences using dimensional regularization

Determined last stable circular orbit as a function of mass ratio

Abstract

The article delivers the only still unknown coefficient in the 4th post-Newtonian energy expression for binary point masses on circular orbits as function of orbital angular frequency. Apart from a single coefficient, which is known solely numerically, all the coefficients are given as exact numbers. The shown Hamiltonian is presented in the center-of-mass frame and out of its 57 coefficients 51 are given fully explicitly. Those coefficients are six coefficients more than previously achieved [Jaranowski/Sch\"afer, Phys. Rev. D 86, 061503(R) (2012)]. The local divergences in the point-mass model are uniquely controlled by the method of dimensional regularization. As application, the last stable circular orbit is determined as function of the symmetric-mass-ratio parameter.