Radiative Level Shifts. III. Hyperfine Structure in Hydrogenic Atoms

Abstract

A comprehensive calculation of the second-order quantum-electrodynamic corrections to the hyperfine splitting of $S$ states is presented. The gauge-invariant reduction of the self-energy expression given by Yennie and Erickson is used to systematically verify previous calculations of orders $\ensuremath{\alpha}$, $\ensuremath{\alpha}(Z\ensuremath{\alpha})$, $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{2}{\mathrm{ln}}^{2}{(Z\ensuremath{\alpha})}^{\ensuremath{-}2}$, and $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{2}\mathrm{ln}{(Z\ensuremath{\alpha})}^{\ensuremath{-}2}$ relative to the lowest order Fermi splitting ${E}^{F}$ and to obtain a result for the dominant contribution to order $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{2}$ for the $1S$ and $2S$ levels. The new contribution for the $1S$ state is $(\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}){(Z\ensuremath{\alpha})}^{2}[18.4\ifmmode\pm\else\textpm\fi{}5]{E}^{F}=[2.3\ifmmode\pm\else\textpm\fi{}0.6]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}{E}^{F},$where $\ensuremath{\alpha}$ is the fine-structure constant, $Z\ensuremath{\alpha}$ is the strength of the Coulomb potential, and the error limits are estimates of uncalculated terms. Our results for $n=2$ provide a substantial check of Zwanziger's calculation of the hyperfine splittings in the $1S$ and $2S$ levels.