Spinning Black Hole Binary Dynamics, Scattering Amplitudes and Effective Field Theory

We describe a systematic framework for finding the conservative potential of compact binary systems with spin based on scattering amplitudes of particles of arbitrary spin and effective field theory. An arbitrary-spin formalism is generally required in the classical limit. By matching the tree and one-loop amplitudes of four spinning particles with those of a suitably-chosen effective field theory, we obtain the spin1-spin2 terms of a two-body effective Hamiltonian through O(G^2) and valid to all orders in velocity. Solving Hamilton's equations yields the impulse and spin changes of the individual bodies. We write them in a surprisingly compact form as appropriate derivatives of the eikonal phase obtained from the amplitude. It seems likely this structure persists to higher orders. We also point out various double-copy relations for general spin.