NNNLO gravitational spin-orbit coupling at the quartic order in G.

In this paper we derive for the first time the NNNLO gravitational spin-orbit coupling at the quartic order in G in the post-Newtonian (PN) approximation. 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, which is carried out within the effective field theory (EFT) of gravitating spinning objects. Our computation makes use of the publicly-available 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, and transcendental numbers, that the final result is also observed to have. We also find that this piece of the sector vanishes in the test particle limit. 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.