Coulomb-gauge self-energy calculation for high-$Z$ hydrogenic ions

We present results from numerical calculations in the Coulomb gauge of the first-order self-energy shift of bound hydrogenic states in highly stripped ions. We apply the expressions for the renormalized free-electron self-energy and vertex operators obtained by G. S. Adkins [Phys. Rev. D 27, 1814 (1983); Phys. Rev. D 34, 2489 (1986)] to the evaluation of the zero- and one-potential terms. It is found that in this gauge the contribution from the many-potential term, which limits the overall accuracy, is significantly smaller than in the covariant Feynman gauge. This enables us to improve the accuracy of the self-energy prediction considerably compared to that obtained in the corresponding Feynman-gauge calculations.

Figure 1

Expansion of the bound self-energy operator in terms of a free particle scattered in the nuclear potential. The heavy lines represent bound states. The dashed lines corresponds to the scalar Coulomb interaction.

Figure 2

Contribution to the total self-energy of the $1s$ state from the zero- and one-potential terms in the two gauges.