Two-Loop Bethe-Logarithm Correction in Hydrogenlike Atoms

We calculate the two-loop Bethe logarithm correction to atomic energy levels in hydrogenlike systems. The two-loop Bethe logarithm is a low-energy quantum electrodynamic (QED) effect involving multiple summations over virtual excited atomic states. Although much smaller in absolute magnitude than the well-known one-loop Bethe logarithm, the two-loop analog is quite significant when compared to the current experimental accuracy of the $1S$–$2S$ transition: It contributes $-8.19$ and $-0.84\mathrm{k}\mathrm{H}\mathrm{z}$ for the $1S$ and the $2S$ state, respectively. The two-loop Bethe logarithm has been the largest unknown correction to the hydrogen Lamb shift to date. Together with the ongoing measurement of the proton charge radius at the Paul Scherrer Institute, its calculation will bring theoretical and experimental accuracy for the Lamb shift in atomic hydrogen to the level of ${10}^{-7}$.