Vacuum polarization in a strong Coulomb field. III. Nuclear size effects

We compute the change in the vacuum polarization near a high-$Z$ nucleus arising from the finite extent of the nuclear charge density. A massless-electron technique is exploited to permit an analytical calculation of the effect to all orders in $Z\alpha$. Applications to muonic atoms are of particular interest as tests of quantum electrodynamics. The calculation of muonic energy level shifts is carried out to lowest order in the ratio of the nuclear radius to the radius of the muonic orbit. This approximation is appropriate to many muonic states with large orbits. Our analytical results are in excellent agreement with numerical studies by Gyulassy and give as a special case the order $\alpha {\left(Z\alpha \right)}^3$ calculation of Arafune. The effect on muonic x rays such as those produced in the $5g_{\frac{9}{2}}\to 4f_{\frac{7}{2}}$ transition in ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ is about 5 eV, increasing slightly the discrepancy between theory and experiment in such systems.