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Binary dynamics from spin1-spin2 coupling at fourth post-Newtonian order

We calculate via the effective field theory (EFT) approach the next-to-next-to-leading order (NNLO) spin1-spin2 conservative potential for a binary. Hereby, we first demonstrate the ability of the EFT approach to go at NNLO in post-Newtonian (PN) corrections from spin effects. The NNLO spin1-spin2 interaction is evaluated at fourth PN order for a binary of maximally rotating compact objects. This sector includes contributions from diagrams, which are not pure spin1-spin2 diagrams, as they contribute through the leading-order spin accelerations and precessions, that should be first taken into account here. The fact that the spin is derivative-coupled adds significantly to the complexity of computations. In particular, for the irreducible two-loop diagrams, which are the most complicated to evaluate in this sector, irreducible two-loop tensor integrals up to order 4 are required. The EFT calculation is carried out in terms of the nonrelativistic gravitational (NRG) fields. However, not all of the benefits of the NRG fields apply to spin interactions, as all possible diagram topologies are realized at each order of $G$ included. Still, the NRG fields remain advantageous, and thus there was no use of automated computations in this work. Our final result can be reduced, and a corresponding Hamiltonian may be derived.