Nonrelativistic QED expansion for the electron self-energy

The recently proposed relativistic multipole expansion (RME) of the self-energy effect suggests some observations on the nonrelativistic expansion of the effect. First, the nature of the series for the one-loop self-energy of an electron bound by the Coulomb field of the nucleus is clarified. It is shown that the expansion of the energy shift caused by the self-energy effect contains terms of the form $\alpha {\left(Z\alpha \right)}^{7}ln\left(Z\alpha \right), \alpha {\left(Z\alpha \right)}^8{ln}^3\left(Z\alpha \right), \alpha {\left(Z\alpha \right)}^9{ln}^2\left(Z\alpha \right), \alpha {\left(Z\alpha \right)}^{10}{ln}^4\left(Z\alpha \right)$, and so on. Here $Z$ is the charge of the nucleus. The origin of these terms is traced back to the logarithmic divergence of the Dirac $S$-wave function at the origin. These terms eventually lead to breakdown of the nonrelativistic quantum electrodynamics approach. Second, at leading order relativistic multipole expansion requires an evaluation of the “extended Bethe logarithm” (EBL). When expanded in series in $Z\alpha$ EBL reduces at leading order to the ordinary Bethe logarithm. However, it is argued that it is both more accurate and easier to calculate the EBL than the ordinary Bethe logarithm. Both variants of the Bethe logarithm can be calculated by means of the pseudostate method. An improvement of this method is suggested. Finally, the contribution of the combined self-energy vacuum polarization contribution to the Lamb shift in muonic hydrogen for the $1s-4s$ and $2p-4p$ states by means of the EBL is calculated. For cases that had already been calculated the results reported here are more accurate than the previous ones.