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On the Apparent Ambiguities in the Post-Newtonian Expansion for Binary Systems

Rafael A. Porto, Ira Z. Rothstein

TL;DR

The paper tackles apparent ambiguities in the post-Newtonian expansion for binary systems by analyzing infrared and ultraviolet divergences in near and far zone calculations within an EFT framework (NRGR). It advocates zero-bin subtraction to remove double-counting between potential and radiation modes, showing that IR poles in the near zone convert into UV poles that cancel divergences in the tail/radiation sector. The authors argue that the only physical ambiguity parameters arise from unknown finite-size effects at 5PN for non-spinning bodies, and that at 4PN all ambiguities can be removed without external input. The work strengthens the PN EFT framework, delivering regulator-independent results and improving the reliability of high-accuracy gravitational-wave templates.

Abstract

We discuss the source of the apparent ambiguities arising in the calculation of the dynamics of binary black holes within the Post-Newtonian framework. Divergences appear in both the near and far zone calculations, and may be of either ultraviolet (UV) or infrared (IR) nature. The effective field theory (EFT) formalism elucidates the origin of the singularities which may introduce apparent ambiguities. In particular, the only (physical) 'ambiguity parameters' that necessitate a matching calculation correspond to unknown finite size effects, which first appear at fifth Post-Newtonian (5PN) order for non-spinning bodies. We demonstrate that the ambiguities linked to IR divergences in the near zone, that plague the recent derivations of the binding energy at 4PN order, both in the ADM and 'Fokker-action' approach, can be resolved by implementing the so-called 'zero-bin' subtraction in the EFT framework. The procedure yields ambiguity-free results without the need of additional information beyond the PN expansion.

On the Apparent Ambiguities in the Post-Newtonian Expansion for Binary Systems

TL;DR

The paper tackles apparent ambiguities in the post-Newtonian expansion for binary systems by analyzing infrared and ultraviolet divergences in near and far zone calculations within an EFT framework (NRGR). It advocates zero-bin subtraction to remove double-counting between potential and radiation modes, showing that IR poles in the near zone convert into UV poles that cancel divergences in the tail/radiation sector. The authors argue that the only physical ambiguity parameters arise from unknown finite-size effects at 5PN for non-spinning bodies, and that at 4PN all ambiguities can be removed without external input. The work strengthens the PN EFT framework, delivering regulator-independent results and improving the reliability of high-accuracy gravitational-wave templates.

Abstract

We discuss the source of the apparent ambiguities arising in the calculation of the dynamics of binary black holes within the Post-Newtonian framework. Divergences appear in both the near and far zone calculations, and may be of either ultraviolet (UV) or infrared (IR) nature. The effective field theory (EFT) formalism elucidates the origin of the singularities which may introduce apparent ambiguities. In particular, the only (physical) 'ambiguity parameters' that necessitate a matching calculation correspond to unknown finite size effects, which first appear at fifth Post-Newtonian (5PN) order for non-spinning bodies. We demonstrate that the ambiguities linked to IR divergences in the near zone, that plague the recent derivations of the binding energy at 4PN order, both in the ADM and 'Fokker-action' approach, can be resolved by implementing the so-called 'zero-bin' subtraction in the EFT framework. The procedure yields ambiguity-free results without the need of additional information beyond the PN expansion.

Paper Structure

This paper contains 7 sections, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The zero-bin subtraction
  3. The conservative dynamics at 4PN order
  4. The ambiguities
  5. Infrared divergences in the EFT framework
  6. Removing the double-counting
  7. Concluding remarks

Figures (1)

  • Figure 1: A contribution in NRGR to the gravitational potential from the near zone at 4PN order. We use the decomposition introduced in KK in terms of $(\phi,{\boldsymbol{A}}_i,\gamma_{ij})$ fields. As described in nrgr4pn2, the diagram may be cast as a 'three-loop' self-energy amplitude in gauge theory, which can be further decomposed in a series of master integrals.