N$^3$LO Quadratic-in-Spin Interactions for Generic Compact Binaries

We derive the third subleading (N$^3$LO) corrections of the quadratic-in-spin sectors via the EFT of spinning objects in post-Newtonian (PN) gravity. These corrections consist of contributions from $4$ sectors for generic compact binaries, that enter at the fifth PN order. One of these contributions is due to a new tidal interaction, that is unique to the sectors with spin, and complements the first tidal interaction that also enters at this PN order in the simple point-mass sector. The evaluation of Feynman graphs is carried out in a generic dimension via advanced multi-loop methods, and gives rise to dimensional-regularization poles in conjunction with logarithms. At these higher-spin sectors the reduction of generalized Lagrangians entails redefinitions of the position beyond linear order. We provide here the most general Lagrangians and Hamiltonians. We then specify the latter to simplified configurations, and derive the consequent gauge-invariant relations among the binding energy, angular momentum, and frequency. We end with a derivation of all the scattering angles that correspond to an extension of our Hamiltonians to the scattering problem in the simplified aligned-spins configuration, as a guide to scattering-amplitudes studies.