Order (Z\alpha)^4 m/M Ry Correction to Hydrogen Levels

Abstract

The first in m/M and fourth in Z\alpha (pure recoil) correction to a hydrogen energy level is found. This correction comprises two contributions, one coming from the atomic scale, the other from distances of the Compton wavelength order. Two different perturbation schemes are used to calculate the former. One of them exploits as unperturbed the solution to the Dirac-Coulomb problem, the nucleus' slow motion being the source of the perturbation. The alternative scheme treats both the electron and the nucleus as slow, while relativistic effects are considered perturbatively. The short-distance contribution is found in the Feynman and Coulomb gauges. Recent results for P levels are confirmed, in contrast with those for S levels. Numerically, the shift equals 2.77 kHz for the ground state and 0.51 kHz for 2S state.