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NNNLO gravitational quadratic-in-spin interactions at the quartic order in G

Michèle Levi, Andrew J. McLeod, Matthew von Hippel

TL;DR

This work computes the complete static $N^3$LO gravitational interactions quadratic in spin at order $G^4$ in the PN expansion for inspiraling binaries using the EFT of gravitating spinning objects. It extends the EFTofPNG framework to include quadratic-in-curvature couplings and spin-induced finite-size effects, and performs a worldline diagrammatic expansion with $163$ graphs, including $52$ three-loop topologies. A key result is the cancellation of divergences, logarithms, and $\\zeta(2)$ terms in the final sum, yielding a finite rational contribution at $5$PN for maximally spinning objects. The findings push the precision frontier of spinning PN gravity and provide a robust computational pipeline for completing the remaining sectors, with future work to incorporate higher-order spin-curvature operators and cross-check with amplitudes-based approaches.

Abstract

We compute the N$^3$LO gravitational quadratic-in-spin interactions at $G^4$ in the post-Newtonian (PN) expansion via the effective field theory (EFT) of gravitating spinning objects for the first time. This result contributes at the $5$PN order for maximally-spinning compact objects, adding the spinning case to the static sector at this PN accuracy. This sector requires extending the EFT of a spinning particle beyond linear order in the curvature to include higher-order operators quadratic in the curvature that are relevant at this PN order. We make use of a diagrammatic expansion in the worldline picture, and rely on our recent upgrade of the \texttt{EFTofPNG} code, which we further extend to handle this sector. Similar to the spin-orbit sector, we find that the contributing three-loop graphs give rise to divergences, logarithms, and transcendental numbers. However, in this sector all of these features conspire to cancel out from the final result, which contains only finite rational terms.

NNNLO gravitational quadratic-in-spin interactions at the quartic order in G

TL;DR

This work computes the complete static LO gravitational interactions quadratic in spin at order in the PN expansion for inspiraling binaries using the EFT of gravitating spinning objects. It extends the EFTofPNG framework to include quadratic-in-curvature couplings and spin-induced finite-size effects, and performs a worldline diagrammatic expansion with graphs, including three-loop topologies. A key result is the cancellation of divergences, logarithms, and terms in the final sum, yielding a finite rational contribution at PN for maximally spinning objects. The findings push the precision frontier of spinning PN gravity and provide a robust computational pipeline for completing the remaining sectors, with future work to incorporate higher-order spin-curvature operators and cross-check with amplitudes-based approaches.

Abstract

We compute the NLO gravitational quadratic-in-spin interactions at in the post-Newtonian (PN) expansion via the effective field theory (EFT) of gravitating spinning objects for the first time. This result contributes at the PN order for maximally-spinning compact objects, adding the spinning case to the static sector at this PN accuracy. This sector requires extending the EFT of a spinning particle beyond linear order in the curvature to include higher-order operators quadratic in the curvature that are relevant at this PN order. We make use of a diagrammatic expansion in the worldline picture, and rely on our recent upgrade of the \texttt{EFTofPNG} code, which we further extend to handle this sector. Similar to the spin-orbit sector, we find that the contributing three-loop graphs give rise to divergences, logarithms, and transcendental numbers. However, in this sector all of these features conspire to cancel out from the final result, which contains only finite rational terms.

Paper Structure

This paper contains 6 sections, 13 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. EFT of gravitating spinning objects
  3. Extending the EFT of a spinning particle
  4. Perturbative expansion of quadratic-in-spin interactions
  5. N3LO gravitational quadratic-in-spin action at G4
  6. Conclusions

Figures (6)

  • Figure 1: Graph topologies at order $G^4$ classified according to their internal bulk vertices and the corresponding loop order in the worldline picture Levi:2020kvb: (a) $0$-loop; (b) One-loop; (c), (e) Two-loop; (d), (f), (g) Three-loop. Topology (e8) is a rank-two topology, whereas topologies (f5), (g4), and (g5), are rank-three topologies Levi:2020kvb.
  • Figure 2: Feynman graphs below three-loop order in the worldline picture, which contribute to the N$^3$LO spin$_1$-spin$_2$ static interaction at order $G^4$. All of the graphs in this figure and the following ones should be accompanied by their 'mirror' graphs, in which worldline labels are exchanged, namely $1\leftrightarrow2$.
  • Figure 3: Feynman graphs at three-loop order in the worldline picture, which contribute to the N$^3$LO spin$_1$-spin$_2$ static interaction at order $G^4$.
  • Figure 4: Feynman graphs below three-loop order in the worldline picture, which contribute to the N$^3$LO spin$_1$-spin$_1$ static interaction at order $G^4$.
  • Figure 5: Feynman graphs at three-loop order in the worldline picture, which contribute to the N$^3$LO spin$_1$-spin$_1$ static interaction at order $G^4$.
  • ...and 1 more figures