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

Zoia A. Mandrykina, Zewen Sun, Natalia S. Oreshkina

TL;DR

This work benchmarks two independent approaches to compute Wichmann-Kroll corrections to vacuum polarization in hydrogen-like ions and muonic atoms across $Z=\{36,54,70,82,92\}$. It contrasts a Gaussian finite-basis-set (FBS) method with dual basis construction and analytic large-distance corrections against a semi-analytical Dirac Green function (GF) approach using a Fermi nuclear model. The study finds overall agreement at the $0.1\%$ level for electronic systems and up to $\sim1\%$ for muonic systems, with discrepancies mainly arising from nuclear-model choices and rms radii; it also provides systematic convergence analyses via partial-wave truncation and basis-size studies. The resulting WK reference data span sub-meV to hundreds of eV and serve as benchmarks for precision spectroscopy of exotic atoms and for guiding future QED calculations in strong Coulomb fields.

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.

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

TL;DR

This work benchmarks two independent approaches to compute Wichmann-Kroll corrections to vacuum polarization in hydrogen-like ions and muonic atoms across . It contrasts a Gaussian finite-basis-set (FBS) method with dual basis construction and analytic large-distance corrections against a semi-analytical Dirac Green function (GF) approach using a Fermi nuclear model. The study finds overall agreement at the level for electronic systems and up to for muonic systems, with discrepancies mainly arising from nuclear-model choices and rms radii; it also provides systematic convergence analyses via partial-wave truncation and basis-size studies. The resulting WK reference data span sub-meV to hundreds of eV and serve as benchmarks for precision spectroscopy of exotic atoms and for guiding future QED calculations in strong Coulomb fields.

Abstract

Wichmann-Kroll corrections are calculated in both hydrogen-like electronic ions and muonic systems (--) using two independent methods. The Gaussian finite basis set approach, enhanced with dual basis construction, analytical large-distance corrections, and -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.
Paper Structure (12 sections, 34 equations, 3 figures, 2 tables)

This paper contains 12 sections, 34 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic formalism
  3. Finite-basis-set Method
  4. Gaussian basis
  5. WK-induced charge density
  6. WK correction and convergence
  7. Green function method
  8. Construction of the Dirac Green functions
  9. Numerical implementation and partial-wave convergence
  10. Results and discussion
  11. Conclusion
  12. Numerical methods

Figures (3)

  • Figure 1: Feynman diagram for vacuum polarization correction decomposition. The double lines indicate the bound particle and virtual electron-positron state in the field of the nucleus, the wavy lines represent virtual photon emission.
  • Figure 2: WK charge density $\rho_\mathrm{VP}^{(3+)}(r)$ of $_{92}$U with different $\abs{\kappa}$ contributions at small (upper) and large (lower) distance, in natural (electronic) unit.
  • Figure 3: WK charge density $\rho_\mathrm{VP}^{(3+)}(r)$ of $_{92}$U calculated using dual basis construction (DBC) method.