QED effects in quadratic Zeeman splitting in highly charged hydrogen-like ions

TL;DR

The study tackles the accurate evaluation of quantum electrodynamic corrections to the quadratic Zeeman effect in highly charged hydrogen-like ions, focusing on the first-order QED contributions to the quadratic term for $1s_{1/2}$, $2s_{1/2}$, and $2p_{1/2}$ states across $Z=14$–92. Using a fully relativistic two-time Green's-function framework in the Furry picture, the authors compute the self-energy and vacuum-polarization corrections, carefully handling ultraviolet and infrared divergences, and they implement convergence-acceleration techniques for the partial-wave expansion. The results show that self-energy dominates the QED correction, with vacuum polarization providing a smaller but non-negligible contribution; for the $2p_{1/2}$ state, the rigorous result can be written as $g^{(2|1)}_{2p_{1/2}}=(\alpha/\pi) g^{(2|0)} F(\alpha Z)$, where $F(\alpha Z)$ is slowly varying and tends to 1 as $Z$ decreases, while simple effective-operator estimates align well only for this state at low $Z$. These findings benchmark bound-state QED in strong magnetic fields and lay the groundwork for extending the approach to interelectronic-interaction effects in more complex ions, relevant for precision spectroscopy and fundamental constants measurements.

Abstract

We present ab initio calculations of one-electron quantum electrodynamical corrections to the second-order Zeeman splitting for the $1s_{1/2}$, $2s_{1/2}$, and $2p_{1/2}$ states in highly charged hydrogen-like ions. The self-energy correction is evaluated using the rigorous QED approach. The vacuum polarization correction is evaluated within the electric-loop approximation. Calculations are performed for the wide range of nuclear charge number: $Z = 14 - 92$.

Figures (2)

• Figure 1: Self-energy diagrams. The double lines correspond to the electron propagators in the nuclear potential $V_{\rm nuc}$. The wavy lines denote the photon propagators. The mass-counterterm diagrams are omitted.
• Figure 2: Vacuum-polarization diagrams. The notations are the same as in Fig. \ref{['fig:se']}.