Two-loop Bethe logarithms for non-Slevels

Abstract

Two-loop Bethe logarithms are calculated for excited $P$ and $D$ states in hydrogenlike systems, and estimates are presented for all states with higher angular momenta. These results complete our knowledge of the $P$ and $D$ energy levels in hydrogen at the order of ${\ensuremath{\alpha}}^{8}{m}_{e}{c}^{2}$, where ${m}_{e}$ is the electron mass and $c$ is the speed of light, and scale as ${Z}^{6}$, where $Z$ is the nuclear charge number. Our analytic and numerical calculations are consistent with the complete absence of logarithmic terms of order ${(\ensuremath{\alpha}∕\ensuremath{\pi})}^{2}{(Z\ensuremath{\alpha})}^{6}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}[{(Z\ensuremath{\alpha})}^{\ensuremath{-}2}]{m}_{e}{c}^{2}$ for $D$ states and all states with higher angular momenta. For higher excited $P$ and $D$ states, a number of poles from lower-lying levels have to subtracted in the numerical evaluation. We find that, surprisingly, the corrections of the ``squared decay-rate type'' are the numerically dominant contributions in the order ${(\ensuremath{\alpha}∕\ensuremath{\pi})}^{2}{(Z\ensuremath{\alpha})}^{6}{m}_{e}{c}^{2}$ for states with large angular momenta, and provide an estimate of the entire ${B}_{60}$ coefficient for Rydberg states with high angular momentum quantum numbers. Our results reach the predictive limits of the quantum electrodynamic theory of the Lamb shift.