Double-logarithmic two-loop self-energy corrections to the Lamb shift

Abstract

Self-energy corrections involving logarithms of the parameter $Z\ensuremath{\alpha}$ can often be derived within a simplified approach, avoiding calculational difficulties typical of the problematic nonlogarithmic corrections (as customary in bound-state quantum electrodynamics, we denote by Z the nuclear charge number, and by $\ensuremath{\alpha}$ the fine-structure constant). For some logarithmic corrections, it is sufficient to consider internal properties of the electron characterized by form factors. We provide a detailed derivation of related self-energy ``potentials'' that give rise to the logarithmic corrections; these potentials are local in coordinate space. We focus on the double-logarithmic two-loop coefficient ${B}_{62}$ for P states and states with higher angular momenta in hydrogenlike systems. We complement the discussion by a systematic derivation of ${B}_{62}$ based on nonrelativistic quantum electrodynamics. In particular, we find that an additional double logarithm generated by the loop-after-loop diagram cancels when the entire gauge-invariant set of two-loop self-energy diagrams is considered. This double logarithm is not contained in the effective-potential approach.