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

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 ${\alpha }^2(Z\alpha {)}^7$ were evaluated, where $Z$ is the nuclear charge number and $Z\alpha$ 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.