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

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.

💡 Research Summary

The paper presents a comprehensive study of the Wichmann‑Kroll (WK) vacuum‑polarization correction in both highly charged hydrogen‑like electronic ions and muonic atoms for nuclear charges Z = 36–92. Two independent computational approaches are implemented and compared: a Gaussian finite‑basis‑set (FBS) method and a semi‑analytical Dirac‑Green‑function (GF) method. The authors aim to assess the trade‑off between computational efficiency and systematic accuracy, and to provide reliable reference data for precision spectroscopy of exotic atoms.

In the theoretical background, the vacuum‑polarization energy shift ΔE is expressed as an integral of the bound‑state Dirac spinor with the VP potential V_VP(r), which itself is generated by the induced charge density ρ_VP(r). By expanding the Dirac Green function in powers of the nuclear potential, the first‑order term yields the well‑known Uehling contribution, while all higher odd‑order terms constitute the WK correction. The WK term is proportional to (Zα)^3 and higher, and becomes sizable for high‑Z systems and especially for muonic atoms where the muon’s Bohr radius is ≈200 times smaller than that of the electron.

The Gaussian FBS approach constructs the radial Dirac wave functions as linear combinations of Gaussian primitives π_±i(r) = N r^{d_±} exp(−ζ_i r^2). The exponents ζ_i are distributed geometrically between ζ_1 = 1/(2 r_1^2) and ζ_N = 1/(2 r_N^2), with r_1 = r_rms/20 and r_N = r_B/7, ensuring coverage of both the nuclear interior and the asymptotic region. The coefficients are obtained via a Rayleigh‑Ritz variational principle. To overcome two known deficiencies of the plain Gaussian basis—(i) spurious oscillations in the WK charge density for large |κ| values, and (ii) incorrect exponential decay at large radii—the authors introduce (a) a dual‑basis‑construction (DBC) scheme, where the full set of ζ_i is split into even and odd subsets, the WK density is computed twice, and the results are averaged, effectively canceling the basis‑dependent oscillations; and (b) an analytical large‑distance correction that stitches the numerically obtained WK potential to the known point‑nucleus WK potential (Ref. 24). Moreover, the WK charge density ρ^{(3+)}_VP(r) is represented by B‑splines on a dense radial grid, allowing rapid and accurate evaluation of the integral V_WK(r) = −4π α ∫ dr′ ρ^{(3+)}_VP(r′) / |r−r′|. Arbitrary‑precision arithmetic (mpmath) is employed to avoid linear‑dependence problems when the basis size N reaches 130–160. Convergence tests show that the WK energy shift stabilizes within a few 10⁻⁴ a.u. as N increases.

The Green‑function method builds the full Dirac Green function directly from regular solutions at the origin and at infinity. The regular solutions are expressed in terms of spherical Bessel functions (origin) and spherical Hankel functions (infinity) for the free case, and are numerically integrated for the case with a finite‑size Fermi nuclear charge distribution. Matching at a radius R where the nuclear potential transitions to a pure Coulomb field yields a continuous Green function G(ω, r, r′). The WK potential is then obtained from the contour integral V_WK(r) = α ∫ d³x ρ^{(3+)}_VP(x) / |r−x|, where ρ^{(3+)}_VP is extracted from the difference between the full Green function and the free‑particle Green function. This construction is free of basis‑set artifacts and automatically respects charge‑conjugation symmetry. The price is a higher computational load: the radial Dirac equation must be solved for a dense set of complex energies ω along the Feynman contour, and the matching procedure must be performed for each κ and each ω. Nevertheless, the method delivers systematic errors below 10⁻⁵ a.u. and reproduces known benchmark values for the WK correction in hydrogen‑like uranium and lead.

Results are presented for the 1s, 2s, and 2p₁/₂ states of electronic ions and the corresponding muonic levels. For electronic ions, the WK shift grows from a few eV at Z≈36 to ≈250 eV at Z=92. In muonic atoms the same states experience WK shifts of 0.5–1.5 keV, reflecting the stronger overlap of the muon wave function with the nuclear charge distribution. The Gaussian FBS method yields WK energies that agree with the GF results within 0.1 % for all examined Z and states, confirming that the DBC and large‑distance corrections effectively eliminate the main sources of systematic error. The GF method, however, provides a more robust reference: its results are independent of the choice of basis, and the residual discrepancy with literature values is below 0.02 %.

The authors discuss practical guidelines for choosing between the two approaches. The FBS method, especially with the DBC enhancement, is well suited for large‑scale surveys, multi‑electron systems, or when many nuclear charges must be processed quickly, because it reduces the computational time by up to two orders of magnitude compared with the GF approach. The GF method is recommended when the highest possible accuracy is required, such as in the extraction of nuclear charge radii from muonic spectroscopy, in tests of bound‑state QED at the few‑ppm level, or when benchmarking other numerical techniques.

In conclusion, the paper demonstrates that both the Gaussian finite‑basis‑set technique (with the newly introduced dual‑basis and analytic tail corrections) and the semi‑analytical Dirac‑Green‑function method can produce reliable WK corrections across a broad Z range. The presented data constitute a valuable reference set for future high‑precision experiments on highly charged ions and muonic atoms, and the methodological advances outlined here pave the way for extending vacuum‑polarization calculations to even higher orders (e.g., two‑loop contributions) and to more complex exotic systems.