Expansion of bound-state energies in powers ofm/Mand(1−m/M)

Elaborating on a previous Letter [A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 87, 013001 (2001)], we use a new approach to compute energy levels of a nonrelativistic bound state of two constituents, with masses m and M, by systematic expansions—one in powers of $m/M$ and another in powers of $(1\mathrm{}-m/M).\mathrm{}$ Technical aspects of the calculations are described in detail. Theoretical predictions are given for $\mathcal{O}\left(\alpha \right(Z\alpha {)}^5)$ radiative recoil and $\mathcal{O}\left(\right(Z\alpha {)}^6)$ pure recoil corrections to the average energy shift and hyperfine splitting relevant for hydrogen, muonic hydrogen, and muonium.