Derivation of local-in-time fourth post-Newtonian ADM Hamiltonian for spinless compact binaries

TL;DR

This paper derives the detailed local-in-time fourth post-Newtonian Hamiltonian for spinless compact binaries using canonical formalism, regularization techniques, and addresses divergences and distributional issues.

Contribution

It provides the first detailed derivation of the 4PN Hamiltonian with regularization of UV and IR divergences and addresses distributional derivative issues.

Findings

Explicit 4PN Hamiltonian expressions presented.

Regularization methods successfully handle divergences.

Addresses distributional derivative laws in regularization.

Abstract

The paper gives full details of the computation within the canonical formalism of Arnowitt, Deser, and Misner of the local-in-time part of the fourth post-Newtonian, i.e. of power eight in one over speed of light, conservative Hamiltonian of spinless compact binary systems. The Hamiltonian depends only on the bodies' positions and momenta. Dirac delta distributions are taken as source functions. Their full control is furnished by dimensional continuation, by means of which the occurring ultraviolet (UV) divergences are uniquely regularized. The applied near-zone expansion of the time-symmetric Green function leads to infrared (IR) divergences. Their analytic regularization results in one single ambiguity parameter. Unique fixation of it was successfully performed in T.Damour, P.Jaranowski, and G.Sch\"afer, Phys. Rev. D 89, 064058 (2014) through far-zone matching. Technically as well as conceptually (backscatter binding energy), the level of the Lamb shift in quantum electrodynamics is reached. In a first run a computation of all terms is performed in three-dimensional space using analytic Riesz-Hadamard regularization techniques. Then divergences are treated locally (i.e., around particles' positions for UV and in the vicinity of spatial infinity for IR divergences) by means of combined dimensional and analytic regularization. Various evolved analytic expressions are presented for the first time. The breakdown of the Leibniz rule for distributional derivatives is addressed as well as the in general nondistributive law when regularizing value of products of functions evaluated at their singular point.