The dominant quantum electrodynamics correction to the hyperfine splitting of energy levels in hydrogen-like ions comes from a set of self-energy (SE) diagrams. The nonperturbative in alpha*Z (alpha is the fine-structure constant and Z is the nuclear charge number) evaluation of the corresponding correction has a long history; see, e.g., Ref. [1] and references therein. To the best of our knowledge, all previous calculations were performed using the Feynman gauge only. Nevertheless, the advantages of applying the Coulomb gauge in the case of the Lamb-shift calculations are well known [2]. The present work has two primary goals. First, we want to numerically check the gauge invariance of the set of SE diagrams for the hyperfine splitting. Second, we aim to study the benefits of using the Coulomb gauge for hyperfine-splitting calculations.
The self-energy correction is conveniently divided into three parts [1]: irreducible, reducible, and vertex parts. The treatment of the irreducible part is reduced to an evaluation of a nondiagonal matrix element of the first-order self-energy operator. Therefore, its calculation in the Coulomb gauge is straightforward. The renormalization of the reducible and vertex parts is performed following the results presented in Refs. [3]. Currently, our work is focused on the initially ultraviolet finite many-potential contribution, which is considered in coordinate space within the partial-wave expansion approach. Once completed, conclusions will be drawn regarding improvements in accuracy due to the use of the Coulomb gauge.

[1] V. A. Yerokhin and U. D. Jentschura, PRA 81, 012502 (2010).
[2] D. Hedendahl, J. Holmberg, PRA 85, 012514 (2012); V. A. Yerokhin et al., PRA 111, 012802 (2025).
[3] G. S. Adkins, PRD 27, 1814 (1983); 34, 2489 (1986).