All-Order Wichmann and Kroll Contribution in Heavy Electronic and Exotic Atoms

TL;DR

The paper tackles precision Lamb-shift predictions in high-$Z$ and exotic atoms, where nonperturbative all-order vacuum polarization is required. The authors develop an all-order in $\alpha Z$ bound-state QED framework based on the Dirac-Coulomb Green's function, isolating $V_{VP}^{(3+)}(r)$ by subtracting the leading $V_{VP}^{(1)}(r)$ and evaluating along the imaginary energy axis. Applying this to electronic, muonic, and antiprotonic systems across multiple nuclear models, they quantify the VP contributions $E_{VP}$ for low-lying and circular Rydberg states, finding strong enhancement in exotic atoms and near independence of circular Rydberg states from nuclear structure. The work informs upcoming high-precision spectroscopy (e.g., PAX) by clarifying the importance of all-order VP, the limit of nuclear-model uncertainties for excited states, and the small role of the antiproton's finite size.

Abstract

We present a theoretical study of the Wichmann and Kroll correction to the one-loop vacuum polarization (VP) to all-orders in $αZ$. We consider electronic, muonic, and antiprotonic atoms for a wide range of nuclear charge numbers and explicitly investigate the influence of finite nuclear size effects and different nuclear models. Moreover, we place special emphasis on circular Rydberg states in the exotic atoms as they have recently attracted interest as a tool to perform isolated tests of strong-field QED. We find that the higher-order vacuum polarization is strongly enhanced in exotic atoms and remains large enough in highly excited Rydberg states that an accurate treatment is crucial for the analysis of upcoming spectroscopy experiments like PAX. Moreover, our calculations show that the VP contribution to the Lamb shift in these exotic Rydberg states has almost no dependence on the structure of the nucleus.

Figures (2)

• Figure 1: Leading-order in $\alpha$ and all-order in $\alpha Z$ Feynman diagram for the vacuum polarization contribution to the Lamb shift (left side of the equation) and its expansion into powers of $\alpha Z$ (right side of the equation).
• Figure 2: Higher-order vacuum polarization charge distribution \ref{['Eq:VP3+']} (left panel) for [208][82]Pb and shell (black solid line), sphere (blue dotted line), Fermi (orange dash-dotted line) as well as point (red dashed line) nuclear models. The differences of the nuclear models to the shell model are shown on the right panel. The summation over $\kappa$ was carried out until $\kappa = \pm 12$ .