Two-loop vacuum polarization in a Coulomb field

TL;DR

This work advances bound-state QED by computing Coulomb corrections to the leading two-loop vacuum-polarization potential, deriving the momentum-space potentials $V_{23}$, $V_{25}$, and $V_{27}$ for three, five, and seven Coulomb insertions. A renormalization scheme based on Zimmermann's forest formula with a gauge-preserving projector $U$ yields finite, numerically tractable Feynman-parameter integrals, which are evaluated with large-scale Monte Carlo on GPUs and stored on grids. Energy shifts for H-like and Li-like ions are extracted via momentum-space expectation values, producing the function $G(Z\alpha)$ that encapsulates the $Z$-dependence and reducing the dominant part of the two-loop VP uncertainty. The results enable more precise tests of bound-state QED in heavy ions and improve determinations of nuclear charge radii, with excellent agreement for Bi and a small tension for U attributable to radii uncertainties. Future work could translate these potentials to coordinate space by Fourier transform to facilitate broader applications.

Abstract

The leading-order two-loop vacuum-polarization potential, linear in the Coulomb field of a nucleus, was first derived in the seminal 1955 work of Källén-Sabry. The higher-order two-loop vacuum-polarization corrections, however, have remained unknown until now. In this work, we compute Coulomb corrections to the Källén-Sabry potential, specifically those involving three, five, and seven Coulomb interactions inside the vacuum-polarization loop. The potentials are evaluated in momentum space and subsequently used to calculate one-electron energy shifts. Our results drastically reduce the theoretical uncertainty of the two-loop vacuum-polarization contribution to transition energies, which is required for next-generation tests of bound-state QED in heavy one and few-electron ions as well as for the determination of nuclear charge radii.

Figures (7)

• Figure 1: Feynman diagrams contributing to ${V}_{23}$. Solid lines denote free-electron propagators, wave lines denote the photon propagators, the circled crosses denote the Coulomb interactions with the nucleus.
• Figure 2: Feynman diagrams contributing to ${V}_{25}$.
• Figure 3: Feynman diagrams contributing to ${V}_{27}$.
• Figure 4: The unfolded Feynman diagrams corresponding to the graph (6) in Fig. \ref{['fig25']}.
• Figure 5: The values of $\widetilde{V}_{13}(p)$ (1-loop) and $\widetilde{V}_{23}(p)$ (2-loop). The numerical data have been smoothed to enhance visual clarity.