Vacuum polarization in a strong Coulomb field. II. Short-distance corrections

We continue our study of the behavior of the vacuum polarization near the nucleus, the region of importance for heavy muonic atoms. We show that the vacuum polarization potential induced by a point nuclear charge can be expanded in certain integral and nonintegral powers of the radius $r$, an expansion involving $r^{-1\mathrm{}}$, $r$, $r^{2\lambda }$, and higher terms. The coefficient of the $r^{-1\mathrm{}}$ term was computed in the previous paper, analytically, to all orders of $Z\alpha$. In this paper we compute the coefficients of the $r$ and $r^{2\lambda }$ terms analytically, to all orders in $Z\alpha$. Our results agree in third order with earlier calculations, which had been done only to this order. Parts of our calculation are considerably simplified by an expansion in powers of the electron mass using formal operator and determinantal techniques.