Two-loop Bethe logarithms for non-$S$ levels

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 ${\alpha }^8{m}_e{c}^2$, 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 ${(\alpha ∕\pi )}^2{\left(Z\alpha \right)}^6\ln \left[{\left(Z\alpha \right)}^{-2}\right]m_e{c}^2$ 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 ${(\alpha ∕\pi )}^2{\left(Z\alpha \right)}^6{m}_e{c}^2$ 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.

Figure 1

Location of the discrete pairs of frequencies $(k_1,k_2)$ which give rise to the squared decay rates for an $n=6$ state. The point $Z^2{R}_{\infty }={\left(Z\alpha \right)}^{2}m∕2$ denotes a “$Z$-dependent Rydberg constant” and marks the ionization energy of a nonrelativistic hydrogenlike atom system.

Figure 2

The two-loop diagrams which give rise to the $B_{60}$ coefficient naturally fall into four gauge-invariant subsets, labeled $\mathit{i}–\mathit{iv}$ (we follow the classification of diagrams according to Ref. 6). Our results for $B_{60}$ contain contributions from all of these diagrams; the separation into the four subsets is only relevant for the detailed formulas given in the Appendixes.