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

Piotr Jaranowski, Gerhard Schäfer

TL;DR

This paper advances the 4th post-Newtonian (4PN) two-body Hamiltonian for binary point masses by delivering 51 of 57 coefficients exactly in the center-of-mass frame, using dimensional regularization to regulate local divergences and isolating one numerically known coefficient tied to near-zone infrared structure. The approach builds on the ADM formalism extended to $d$ spatial dimensions, employing a DR-Cauchy framework to obtain a pole-free local 4PN Hamiltonian. New DR-derived contributions, including $C_{42}$, $C_{22}$, and $c_{02}$, are presented, while remaining coefficients are identified for future determination. The paper then computes the 4PN binding energy for circular orbits, exposes the logarithmic term, and analyzes the LSCO location as a function of symmetric mass ratio, providing refined predictions for gravitational-wave modeling and EOB calibration.

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/Schäfer, 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.

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

TL;DR

This paper advances the 4th post-Newtonian (4PN) two-body Hamiltonian for binary point masses by delivering 51 of 57 coefficients exactly in the center-of-mass frame, using dimensional regularization to regulate local divergences and isolating one numerically known coefficient tied to near-zone infrared structure. The approach builds on the ADM formalism extended to spatial dimensions, employing a DR-Cauchy framework to obtain a pole-free local 4PN Hamiltonian. New DR-derived contributions, including , , and , are presented, while remaining coefficients are identified for future determination. The paper then computes the 4PN binding energy for circular orbits, exposes the logarithmic term, and analyzes the LSCO location as a function of symmetric mass ratio, providing refined predictions for gravitational-wave modeling and EOB calibration.

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/Schäfer, 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.

Paper Structure

This paper contains 5 sections, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The ADM canonical approach in $d$ space dimensions
  3. Dimensional regularization of local divergences
  4. Center-of-mass Hamiltonian
  5. Circular orbits

Figures (1)

  • Figure 1: The location $x_\text{LSCO}$ of the LSCO as a function of the symmetric mass ratio $\nu$ for successive PN approximations from 1PN up to 4PN. Solid curves correspond to 1PN, dashed curves to 2PN, dotted curves to 3PN, and dashed/dotted curves to 4PN. Left panel: $x_\text{LSCO}$ is computed as the minimum of the function $E(x;\nu)$. Right panel: $x_\text{LSCO}$ is obtained by means of the $j$-method.