Nonrelativistic quantum electrodynamic Hamiltonian and photon-exchange interaction up to m α8 order

We derive the effective Hamiltonian of nonrelativistic quantum electrodynamics up to order m α⁸ by using a procedure of scattering matching. Our results are in agreement at order m α⁶ with those achieved by applying the Foldy-Wouthuysen transformation [K. Pachucki, Phys. Rev. A 71, 012503 (2005), 10.1103/PhysRevA.71.012503]. Based on the obtained Hamiltonian, we further derive the photon-exchange interactions in the nonretardation approximation, as well as the retardation corrections up to order m α⁸ . The energy shift due to the photon-exchange interactions is determined by studying the pole structure of the Green's function.