Hadronic vacuum polarization correction to atomic energy levels

TL;DR

This paper evaluates the impact of hadronic vacuum polarization on atomic energy levels in hydrogenlike ions and muonic hydrogen using a semiempirical approach based on experimental data, deriving formulas for energy shifts.

Contribution

It introduces a semiempirical method to calculate hadronic vacuum polarization effects on atomic energy levels, including formulas for extended and point-like nuclei, and compares with non-relativistic results.

Findings

Calculated energy shifts for low-lying levels in hydrogenlike ions and muonic hydrogen.

Derived closed-form formulas for hadronic Uehling potential for extended nuclei.

Compared relativistic and non-relativistic results, highlighting differences.

Abstract

The shift of atomic energy levels due to hadronic vacuum polarization is evaluated in a semiempirical way for hydrogenlike ions and for muonic hydrogen. A parametric hadronic polarization function obtained from experimental cross sections of $e^-e^+$ annihilation into hadrons is applied to derive an effective relativistic Uehling potential. The energy corrections originating from hadronic vacuum polarization are calculated for low-lying levels using analytical Dirac-Coulomb wave functions, as well as bound wave functions accounting for the finite nuclear size. Closed formulas for the hadronic Uehling potential of an extended nucleus as well as for the relativistic energy shift in case of a point-like nucleus are derived. These results are compared to existing analytic formulas from non-relativistic theory.