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 α⁸m_ec₂, where m_e is the electron mass and c is the speed of light, and scale as Z⁶, where Z is the nuclear charge number. Our analytic and numerical calculations are consistent with the complete absence of logarithmic terms of order (α/π)²(Zα)⁶ ln[(Zα)⁻²]m_ec² 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 (α/π)²(Zα)⁶m_ec² for states with large angular momenta, and provide an estimate of the entire B₆₀ 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