QED vacuum polarization in the Coulomb field of a nucleus: a method of high-order calculation

TL;DR

This work advances the computation of QED vacuum-polarization corrections to the Coulomb potential of a pointlike nucleus by delivering a high-order two-loop analysis up to $α^2 (Zα)^7$ within a perturbative framework in $Zα$. The author develops a reduction to free-QED Feynman graphs (unfolding) and implements a Zimmermann forest renormalization in Feynman-parametric space to yield finite integrals, avoiding dimensional regularization. The finite, multi-loop parametric integrals are evaluated with a nonadaptive Monte Carlo method on GPUs, using a specially crafted sampling density to handle up to 17 variables and a wide range of external momenta; results include detailed graph-by-graph contributions for the VP terms $V_{23}$, $V_{25}$, and $V_{27}$. These high-precision momentum-space VP functions enable improved theoretical predictions for energy levels in high-$Z$ ions and illustrate a scalable approach for challenging bound-state QED calculations, albeit with limited direct applicability to general bound-state problems beyond this VP context.

Abstract

A calculation of the QED vacuum polarization potential in the Coulomb field of a pointlike nucleus was presented in an earlier publication by the author and his collaborators. Corrections up to order $α^2 (Zα)^7$ were evaluated, where $Z$ is the nuclear charge number and $Zα$ is treated as an independent variable. These corrections correspond to two-loop Feynman graphs with proper propagators of fermions in the external field. The calculation employed a reduction to free QED, leading to free QED Feynman graphs with up to eight independent loops. The method of calculation is described here in detail.

Figures (7)

• Figure 1: All Feynman graphs contributing to $V_{23}$. Solid lines denote free-fermion propagators, wavy lines denote photon propagators, and circled crosses denote Coulomb interactions with the nucleus.
• Figure 2: All Feynman graphs contributing to $V_{25}$.
• Figure 3: All Feynman graphs contributing to $V_{27}$.
• Figure 4: The unfolded Feynman graph corresponding to graph (6) in Fig. \ref{['fig25']}.
• Figure 5: Unfolded Feynman graphs of $\widetilde{V}_{13}$ (left) and $\widetilde{V}_{15}$ (right)