On the vacuum-polarization Uehling potential for a Fermi charge distribution

TL;DR

This paper derives exact analytical formulas for the vacuum-polarization Uehling potential considering a Fermi charge distribution of the nucleus, facilitating precise calculations in quantum electrodynamics.

Contribution

It provides new analytical expressions for the Uehling potential with a Fermi distribution, including higher-order K"all ten-Sabry contributions, using special functions and derivatives.

Findings

Exact formulas for Uehling potential with Fermi distribution

Expressions in terms of Bickley-Naylor and Bessel functions

Cusp and asymptotic behaviors derived

Abstract

We present analytical formulas for the vacuum-polarization Uehling potential in the case where the finite size of the nucleus is modeled by a Fermi charge distribution. Using a Sommerfeld-type development, the potential is expressed in terms of multiple derivatives of a particular integral. The latter and its derivatives can be evaluated exactly in terms of Bickley-Naylor functions, which connection to the Uehling potential was already pointed out in the pure Coulomb case, and of usual Bessel functions of the second kind. The cusp and asymptotic expressions for the Uehling potential with a Fermi charge distribution are also provided. Analytical results for the higher-order-contribution K\"all\`en-Sabry potential are given.