Spin(1)spin(2) effects in the motion of inspiralling compact binaries at third order in the post-Newtonian expansion

We use effective field theory techniques to compute the potentials due to spin-spin and spin-orbit effects, from which the O(S₁S₂) contribution to the motion of spinning compact binaries to third Post-Newtonian (PN) order follow. We use a formalism which allows us to impose the spin supplementary condition (SSC) in a canonical framework to all orders in the PN expansion. We explicitly show the equivalence with our previous results, obtained using the Newton-Wigner SSC at the level of the action for spin-spin and spin-orbit potentials reported in arXiv:gr-qc/0604099 and arXiv:0712.2032 , respectively.