Radiative Level Shifts. III. Hyperfine Structure in Hydrogenic Atoms

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 $\alpha$, $\alpha \left(Z\alpha \right)$, $\alpha {\left(Z\alpha \right)}^2{\ln }^2{\left(Z\alpha \right)}^{-2\mathrm{}}$, and $\alpha {\left(Z\alpha \right)}^2\ln {\left(Z\alpha \right)}^{-2\mathrm{}}$ relative to the lowest order Fermi splitting $E^F$ and to obtain a result for the dominant contribution to order $\alpha {\left(Z\alpha \right)}^2$ for the $1S$ and $2S$ levels. The new contribution for the $1S$ state is $\left(\frac{\alpha }{\pi }\right){\left(Z\alpha \right)}^2[18.4\mathrm{}\pm 5]E^F=[2.3\mathrm{}\pm 0.6]\times {10}^{-6\mathrm{}}{E}^F,$