Two-loop self-energy for the ground state of medium-$Z$ hydrogenlike ions

The two-loop self-energy correction to the ground-state Lamb shift is calculated for hydrogenlike ions with the nuclear charge $Z=10–30$ without any expansion in the binding field of the nucleus. A calculational technique is reported for treatment of Feynman diagrams in the mixed coordinate-momentum representation, which yields significant improvement in numerical accuracy as compared to previous results. An extrapolation of the all-order numerical data yields a result for the higher-order remainder function for hydrogen. The previously reported disagreement between the all-order and the perturbative approaches is reduced to the marginal agreement.

Figure 1

The two-loop self-energy correction. The double line represents the electron propagator in the Coulomb binding field of the nucleus.

Figure 2

The $P$ term. The single line represents the free electron propagator. The dashed line with a cross indicates the interaction with the Coulomb field of the nucleus.

Figure 3

The higher-order remainder function of the two-loop self-energy correction. The cross on the $y$ axis indicates the analytical result.