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

TL;DR

The study tackles the challenging nonperturbative calculation of the two-loop self-energy correction to the bound-electron $g$-factor by focusing on the $P$-term, which contains UV/IR-sensitive subgraphs. It develops and applies a mixed momentum-coordinate framework to derive explicitly finite formulas, separating UV and IR structures and performing careful renormalization anchored in Dirac-Coulomb Green functions. The authors implement extensive numerical computations, verify infrared-divergence cancellations, and provide precise results for high-$Z$ hydrogen-like ions (notably $^{118}$Sn, $Z=50$, and $^{209}$Bi, $Z=83$), improving theoretical accuracy for the bound-electron $g$-factor and aligning with nonperturbative expectations. This work paves the way for extending nonperturbative SESE calculations to heavier elements and lower-$Z$ systems, with implications for precision tests of bound-state QED and related experimental programs.

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 $Zα$ (where $Z$ is the nuclear charge number and $α$ 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.

Figures (4)

• Figure 1: Feynman diagrams representing the two-loop electron self-energy (SESE) correction for the bound-electron $g$ factor: ( a)-( c) loop-after-loop (LAL), ( d)-( f) nested (N), ( g)-( i) overlapping (O) diagrams. The double line denotes the electron in the presence of the binding nuclear field; the wavy line indicates the exchange of a virtual photon; the wavy line terminated by a triangle denotes the interaction with the external magnetic field.
• Figure 2: Diagrammatic representation of the perturbed-orbital $P$-term contributions. A double line denotes the bound electron propagator, the single line denotes the free electron propagator, the wave line denotes the photon propagator, the dashed line terminated by a stylized cross denotes the interaction with the Coulomb nuclear field, the wave line terminated by the stylized cross denotes the interaction with the external magnetic field.
• Figure 3: Diagrammatic representation of the derivative $P$-term contributions.
• Figure 4: Diagrammatic representation of the vertex $P$-term contributions.