Perturbed Orbital Contribution to the Two-Loop Lamb Shift in Hydrogen

Abstract

A part of the two-loop Lamb shift called the perturbed orbital term is evaluated with exact Dirac-Coulomb propagators. It is shown to be the most nonperturbative function of $Z\ensuremath{\alpha}$ yet encountered in QED, so much so that even at $Z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ the leading term in its $Z\ensuremath{\alpha}$ expansion is of the opposite sign from the complete answer. The higher order contributions are $\ensuremath{-}71(1)\phantom{\rule{0ex}{0ex}}\mathrm{kHz}$ to the ground state Lamb shift in hydrogen and $\ensuremath{-}8.9(1)$ and $\ensuremath{-}346(1)\phantom{\rule{0ex}{0ex}}\mathrm{kHz}$ to the $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ Lamb shift in hydrogen and ${\mathrm{He}}^{+}$, respectively.