Two-loop electron self-energy in bound-electron $g$ factor: diagrams in momentum-coordinate representation

TL;DR

This paper advances the calculation of the two-loop electron self-energy correction to the bound-electron g factor by evaluating complex Feynman diagrams in a mixed momentum-coordinate framework, reducing theoretical uncertainties.

Contribution

It introduces a novel method for calculating difficult Feynman diagrams in the two-loop self-energy correction without expansion in Zα.

Findings

Successful calculation of complex Feynman diagrams in mixed representation

Reduction of uncertainties in bound-electron g factor predictions

Method applicable to other high-order QED corrections

Abstract

The two-loop electron self-energy correction is one of the most problematic QED effects and, for a long time, was the dominant source of uncertainty in the theoretical prediction of the bound-electron $g$ factor in hydrogen-like ions. A major breakthrough was recently achieved in [B. Sikora et al. Phys. Rev. Lett. 134, 123001 (2025)], where this effect was calculated without any expansion in the nuclear binding strength parameter $Z\alpha$ (where $Z$ is the nuclear charge number and $\alpha$ is the fine-structure constant). In this paper, we describe our calculations of one of the most difficult parts of the two-loop self-energy, represented by Feynman diagrams that are treated in the mixed momentum-coordinate representation.