Quadratic-in-spin Hamiltonian at O (G2) from scattering amplitudes

We obtain the quadratic-in-spin terms of the conservative Hamiltonian describing the interactions of a binary of spinning bodies in General Relativity through O(G2) and to all orders in velocity. Our calculation extends a recently-introduced framework based on scattering amplitudes and effective field theory to consider non-minimal coupling of the spinning objects to gravity. At the order that we consider, we establish the validity of the formula proposed in 1 that relates the impulse and spin kick in a scattering event to the eikonal phase.