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Logarithmic tail contributions to the energy function of circular compact binaries

Luc Blanchet, Stefano Foffa, François Larrouturou, Riccardo Sturani

TL;DR

This work combines EFT and traditional PN techniques to derive and organize logarithmic tail contributions to the conservative dynamics of circular compact binaries, obtaining explicit 6PN log terms and completing 7PN logs by incorporating tail-of-tail-of-tail effects guided by self-force data. Through EFT proofs and non-local action formalism, the authors derive the log-tail corrections to the equations of motion and energy, connect them to the GW flux, and demonstrate consistency with SF results. Moreover, renormalization-group methods yield leading $( ext{log})^n$ contributions to the circular energy at arbitrary PN order and allow a compact resummation of these terms, providing a unified, all-orders perspective on logarithmic tail effects in PN gravity. The results enhance the precision of conservative dynamics for binaries, with explicit formulas for energy and angular momentum and a clear pathway to incorporate higher-order tail phenomena in waveform modeling.

Abstract

We combine different techniques to extract information about the logarithmic contributions to the two-body conservative dynamics within the post-Newtonian (PN) approximation of General Relativity. The logarithms come from the conservative part of non linear gravitational-wave tails and their iterations. Explicit, original expressions are found for conservative dynamics logarithmic tail terms up to 6PN order by adopting both traditional PN calculations and effective field theory (EFT) methods. We also determine all logarithmic terms at 7PN order, fixing a sub-leading logarithm from a tail-of-tail-of-tail process by comparison with self-force (SF) results. Moreover, we use renormalization group techniques to obtain the leading logarithmic terms to generic power $n$, appearing at $(3n+1)$PN order, and we resum the infinite series in a closed form. Half-integer PN orders enter the conservative dynamics starting at 5.5PN, but they do not generate logarithmic contributions up to next-to-next-to-leading order included. We nevertheless present their contribution at leading order in the small mass ratio limit.

Logarithmic tail contributions to the energy function of circular compact binaries

TL;DR

This work combines EFT and traditional PN techniques to derive and organize logarithmic tail contributions to the conservative dynamics of circular compact binaries, obtaining explicit 6PN log terms and completing 7PN logs by incorporating tail-of-tail-of-tail effects guided by self-force data. Through EFT proofs and non-local action formalism, the authors derive the log-tail corrections to the equations of motion and energy, connect them to the GW flux, and demonstrate consistency with SF results. Moreover, renormalization-group methods yield leading contributions to the circular energy at arbitrary PN order and allow a compact resummation of these terms, providing a unified, all-orders perspective on logarithmic tail effects in PN gravity. The results enhance the precision of conservative dynamics for binaries, with explicit formulas for energy and angular momentum and a clear pathway to incorporate higher-order tail phenomena in waveform modeling.

Abstract

We combine different techniques to extract information about the logarithmic contributions to the two-body conservative dynamics within the post-Newtonian (PN) approximation of General Relativity. The logarithms come from the conservative part of non linear gravitational-wave tails and their iterations. Explicit, original expressions are found for conservative dynamics logarithmic tail terms up to 6PN order by adopting both traditional PN calculations and effective field theory (EFT) methods. We also determine all logarithmic terms at 7PN order, fixing a sub-leading logarithm from a tail-of-tail-of-tail process by comparison with self-force (SF) results. Moreover, we use renormalization group techniques to obtain the leading logarithmic terms to generic power , appearing at PN order, and we resum the infinite series in a closed form. Half-integer PN orders enter the conservative dynamics starting at 5.5PN, but they do not generate logarithmic contributions up to next-to-next-to-leading order included. We nevertheless present their contribution at leading order in the small mass ratio limit.

Paper Structure

This paper contains 17 sections, 74 equations, 1 figure.

Table of Contents

  1. Motivations and overview
  2. Complete action for simple tail terms
  3. Action in Fourier and time domains
  4. Proof of the action by EFT methods
  5. Derivation of the dynamics for the log-tail contributions
  6. Equations of motion and conserved Noetherian energy
  7. Case of quasi-circular orbits
  8. Logarithmic and half-integer contributions in the circular energy
  9. Simple tail terms
  10. Tail-of-tail terms
  11. 7PN logs and tail-of-tail-of-tail terms
  12. Leading logarithms from renormalization group theory
  13. Renormalization Group equations
  14. Leading $\log^n$ terms at any PN order for circular orbits
  15. Conclusions
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Schematic representation of the relation between the tail self-energy amplitude (which includes also a real finite, local term understood here) and the purely imaginary self-energy diagram. The logarithmic term fully determines the non-local tail action \ref{['eq:StailFourier']}; the imaginary part relates to the tail contribution to the energy flux, which is proportional via a factor of $GMk_0\pi$ to the non-tail radiation flux; while the UV pole cancels the IR pole coming from near-zone conservative contributions JaraS13DJS14JaraS15DJS16BBBFMaBBBFMcMBBF17PR17FPRS19.