Relativistic nuclear recoil effects in hyperfine splitting of hydrogenic systems

Finite nuclear mass $(Z\,\alpha)^2\,m/M\,E_F$ corrections to the hyperfine splitting in hydrogenic systems are calculated using a combined relativistic heavy particle and nonrelativistic quantum electrodynamics. The obtained results are in disagreement with previous calculations by Bodwin and Yennie [Phys. Rev. D {\bf 37}, 498 (1988)]. The comparison of improved theoretical predictions with the corresponding measurements in hydrogen reveals $2\,\sigma$ discrepancy, which may indicate problems with the proton structure corrections.