Wichmann-Kroll Correction in Muonic Atoms and Hydrogen-Like Electronic Ions: a Comparative Study of Two Methods

TL;DR

This study compares two computational methods for calculating Wichmann-Kroll corrections in hydrogen-like ions and muonic atoms, highlighting their efficiencies and accuracies for precision spectroscopy.

Contribution

It introduces and compares a Gaussian finite basis set approach and a Green function method for calculating Wichmann-Kroll corrections, demonstrating their effectiveness and consistency.

Findings

Both methods yield results consistent with literature values.

The Gaussian basis set approach offers computational efficiency.

The Green function method provides higher systematic accuracy.

Abstract

Wichmann-Kroll corrections are calculated in both hydrogen-like electronic ions and muonic systems ($Z = \{36$--$92\}$) using two independent methods. The Gaussian finite basis set approach, enhanced with dual basis construction, analytical large-distance corrections, and $B$-spline representations, provides computational efficiency. The Green function method, based on semi-analytical construction from Dirac solutions with Fermi nuclear charge distributions, offers higher systematic accuracy and freedom from basis-dependent artifacts. Results are consistent with the literature values, providing reliable reference data for precision spectroscopy of exotic atoms.