Fourth-order self-energy contribution to the Lamb shift

Abstract

Two-loop self-energy contributions to the fourth-order Lamb shift of ground-state hydrogenic ions are treated to all orders in $Z\alpha$ by using exact Dirac-Coulomb propagators. A rearrangement of the calculation into four ultraviolet finite parts, the $M,\hspace{0.5em}P,\hspace{0.5em}F$, and perturbed orbital (PO) terms, is made. Reference-state singularities present in the $M$ and $P$ terms are shown to cancel. The most computationally intensive part of the calculation, the $M$ term, is evaluated for hydrogenlike uranium and bismuth, the $F$ term is evaluated for a range of $Z$ values, but the $P$ term is left for a future calculation. For hydrogenlike uranium, previous calculations of the PO term give $-0.971$ eV: the contributions from the $M$ and $F$ terms calculated here sum to $-0.325$ eV.