Self-Energy Correction to the Hyperfine Structure Splitting of Hydrogenlike Atoms

A first testing ground for QED in the combined presence of a strong Coulomb field and a strong magnetic field is provided by the precise measurement of the hyperfine structure splitting of hydrogenlike ${}^{209}\mathrm{Bi}$. We present a complete calculation of the one-loop self-energy correction to the first-order hyperfine interaction for various nuclear charges. In the low- $Z$ regime we almost perfectly agree with the $Z\alpha$ expansion, but for medium and high $Z$ there is a substantial deviation.