Self-energy correction to the bound-electron g factor in H-like ions: approximate treatment of the two-potential contribution

Methods of bound-state QED that treat the self-energy contributions to the g
factor of highly charged ions within the partial-wave expansion usually face the problem
of slow convergence of the latter. This work proposes a method aimed at accelerating
this convergence. Namely, we consider the vertex diagram contributing to the self-energy
correction to the bound-electron g factor. This diagram has an ultraviolet divergence. An
expansion of the electron propagators in terms of the binding potential V is used for its
analysis. The individual terms of this expansion are treated in coordinate and momentum
spaces [1]. There is no simple closed-form expression for the two-potential contribution
corresponding to the second power of V. The main idea of our method is to approximate
this contribution based on expressions for zero- or one-potential terms, which can be
calculated with high accuracy. This idea was first proposed for the self-energy correction
to energy levels [2]. Similar methods have also been successfully applied to the calculations of the two-electron [3] and two-loop [4, 5] self-energy contributions. A general analytical
derivation of momentum and angular integrals is presented, which makes it possible to
evaluate the discussed contribution for an arbitrary state of H-like ions.
1. V. A. Yerokhin et al., Phys. Rev. A 69, 052503 (2004).
2. J. Sapirstein, K. T. Cheng, Phys. Rev. A 108, 042804 (2023).
3. A. V. Malyshev et al., Phys. Rev. A 109, 062802 (2024).
4. V. A. Yerokhin et al., Phys. Rev. A 133 251803 (2024).
5. V. A. Yerokhin et al., Phys. Rev. A 111, 042820 (2025).