N$^3$LO gravitational spin-orbit coupling at order $G^4$

TL;DR

This work advances high-precision PN gravity by deriving the N$^3$LO gravitational spin-orbit coupling at order $G^4$ in the EFT of gravitating spinning objects, tackling the first spinning sector with three-loop worldline integrals. It extends the EFTofPNG framework with amplitude-inspired loop techniques to classify topologies, enumerate 388 graphs, and perform IBP-reduced, tensor-decomposed evaluations, revealing novel features such as poles, logarithms, and $\zeta(2)$ terms that survive in the final result. The main outcome is a concrete spin-orbit Lagrangian at this order, showing cancellations that limit the nonzero contributions to the $m_1^2m_2^2$ sector and placing the result at 4.5PN for maximally rotating binaries. Together with prior work, this paves the way toward completing PN accuracy at 4.5PN within the EFT framework and motivates cross-method validations. The computational framework and findings set the stage for releasing code updates and for further exploration of higher-order spin sectors.

Abstract

In this paper we derive for the first time the N$^3$LO gravitational spin-orbit coupling at order $G^4$ in the post-Newtonian (PN) approximation within the effective field theory (EFT) of gravitating spinning objects. This represents the first computation in a spinning sector involving three-loop integration. We provide a comprehensive account of the topologies in the worldline picture for the computation at order $G^4$. Our computation makes use of the publicly-available \texttt{EFTofPNG} code, which is extended using loop-integration techniques from particle amplitudes. We provide the results for each of the Feynman diagrams in this sector. The three-loop graphs in the worldline picture give rise to new features in the spinning sector, including divergent terms and logarithms from dimensional regularization, as well as transcendental numbers, all of which survive in the final result of the topologies at this order. This result enters at the 4.5PN order for maximally-rotating compact objects, and together with previous work in this line, paves the way for the completion of this PN accuracy.

Figures (11)

• Figure 1: The single graph topology at order $G$: One-graviton exchange with no self-interaction.
• Figure 2: Graph topologies at order $G^2$: (a) No self-interaction vertices. (b) One cubic vertex. This is a one-loop topology in the worldline picture.
• Figure 3: Graph topologies at order $G^3$: (a) No self-interaction vertices. (b) One cubic vertex. (c) One quartic vertex. (d) Two cubic vertices. Topologies (c) and (d) constitute two-loop topologies in the worldline picture. The topology (d3) is a rank-two topology as specified below.
• Figure 4: Graph topologies at order $G^4$: (a) No self-interaction vertices. (b) One cubic vertex. (c) One quartic vertex. (d) One quintic vertex. (e) Two cubic vertices. (f) One cubic vertex and one quartic vertex. (g) Three cubic vertices.
• Figure 5: Feynman graphs which contribute to the N$^3$LO spin-orbit coupling at $G^4$. This figure contains all graphs with topologies (a) and (b1)-(b4) of figure \ref{['G4topo']}. All of the graphs presented in this figure and the following ones should be accompanied by their 'mirror' graphs, in which the worldline labels are exchanged, i.e. $1\leftrightarrow2$.

Theorems & Definitions (2)

• Definition 1
• Definition 2