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

We compute the N$^3$LO gravitational quadratic-in-spin interactions at $G^4$ in the post-Newtonian (PN) expansion 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. We carry out this computation within the effective field theory (EFT) of gravitating spinning objects, which requires extending the EFT of a spinning particle beyond linear order in the curvature to include higher-order operators that may be relevant at this PN order. We make use of a diagrammatic expansion in the worldline picture, and rely on our recent upgrade of the 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.