Bound-electron self-energy calculations in Feynman and Coulomb gauges: detailed analysis

TL;DR

This work analyzes bound-electron self-energy (SE) corrections to the Lamb shift in hydrogen-like ions, focusing on the convergence of partial-wave expansions in the Feynman and Coulomb gauges. By working in the nonperturbative Furry picture and employing Green's-function and finite-basis-set methods, the authors renormalize UV divergences and dissect the SE into zero-, one-, and many-potential terms, with the many-potential part computed in coordinate space via PW expansions. A central contribution is the systematic evaluation and comparison of convergence-acceleration schemes—the two-potential and Sapirstein-Cheng (SC) methods—showing that the SC scheme in the Coulomb gauge yields the best overall accuracy for a broad range of $Z$, including light nuclei, and that Coulomb gauge PW terms converge notably faster than in Feynman gauge. The paper provides detailed computational strategies, extrapolation procedures, and nonpoint nucleus formulas, delivering high-precision SE data for several states and nuclear charges and offering practical guidance for applying these techniques to more complex QED corrections.

Abstract

The energy correction associated with the self-energy diagram is the leading (in magnitude) contribution to the Lamb shift in hydrogen-like ions. All conventional approaches to this correction rely on partial-wave expansions, which are a stumbling block limiting accuracy. To elucidate an issue, we perform a comparative analysis of partial-wave-expansion convergence in two gauges: Feynman and Coulomb. Some tricks for improving convergence are also discussed.

Figures (9)

• Figure 1: Self-energy diagram with the related mass counterterm. The double line indicates the electron propagator in the external field of the nucleus, the wavy line denotes the photon propagator, and the cross stands for the counterterm.
• Figure 2: Potential expansion of the self-energy diagram. The dashed line ended by a rhombus denotes the interaction with the nuclear field. The single line denotes the free-electron propagator. The mass counterterm is not shown.
• Figure 3: Diagrammatic representation of the quasi-two-potential term separation. The last term on the right side is calculated in momentum space without using any PW expansion.
• Figure 4: Deformed contour (blue line) in the complex plane $\omega$. Singularities and branching cuts of the electron Green's function (black lines and black dots) and photon propagator (black dashed lines) are shown.
• Figure 5: The extrapolation of the many-potential contribution to the self-energy correction for the $1S_{1/2}$ state of hydrogen-like xenon ($Z=54$) obtained within the framework of the DKB approach in Feynman (a) and Coulomb (b) gauges.