Two-Loop Self-Energy Correction in High-$Z$ Hydrogenlike Ions

A complete evaluation of the two-loop self-energy diagrams to all orders in $Z\alpha$ is presented for the ground state of H-like ions with $Z\ge 40$.

Figure 1

Two-loop self-energy diagrams. Double line indicates an electron propagating in the Coulomb nuclear field. It is understood that the corresponding mass counterterms are subtracted from the diagrams.

Figure 2

Diagrammatic representation of the $M$ term. The dashed line denotes the interaction with the Coulomb nuclear field. $\Delta E_{\mathrm{S}\mathrm{E}}$ is the first-order self-energy correction.

Figure 3

Diagrammatic representation of the $P$ term. The last part should be counted twice, accounting for two equivalent terms.

Figure 4

Diagrammatic representation of the $F$ term. The last diagrams on the right in the first two rows should be counted twice, accounting for two equivalent diagrams.

Figure 5

The results of our numerical evaluation to all orders to $Z\alpha$ (dots) together with the first two terms of the $Z\alpha$ expansion (solid line) and all known terms of the $Z\alpha$ expansion (dashed line).