Abstract
We derive the effective Hamiltonian of nonrelativistic quantum electrodynamics up to order $m{\ensuremath{\alpha}}^{8}$ by using a procedure of scattering matching. Our results are in agreement at order $m{\ensuremath{\alpha}}^{6}$ with those achieved by applying the Foldy-Wouthuysen transformation [K. Pachucki, Phys. Rev. A 71, 012503 (2005)]. 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 $m{\ensuremath{\alpha}}^{8}$. The energy shift due to the photon-exchange interactions is determined by studying the pole structure of the Green's function.
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