Quadratic-in-spin Hamiltonian at $ \mathcal{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 $$ \mathcal{O} $$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.