Classical Hamiltonian of Reissner-Nordstr\"om black holes at second post-Minkowskian order from scattering amplitudes
We employ scattering amplitudes in Einstein-Maxwell theory to compute the classical Hamiltonian of a binary system of two charged, non-spinning compact objects. The effective Hamiltonian is valid to all orders in velocity and up to second post-Minkowskian order (2PM), i.e. $\mathcal{O}(G^2)$. The classical interaction potential is extracted via matching the four-point tree and one-loop amplitudes to those of a suitably chosen classical effective field theory. We derive the 2PM scattering angle from the Hamiltonian and find complete agreement with nown results. Using also the third order result for the scattering angle computed by Wilson-Gerow, we expand the scattering angle at second post-Newtonian order (2PN). Additionally, we provide expressions for the binding energy and periastron advance at 2PN order via the boundary-to-bound dictionary introduced by K\"alin and Porto and find agreement with previous findings.