Coordinate-space approach to vacuum polarization

The vacuum-polarization correction for bound electrons or muons is examined. The objective is to formulate a framework for calculating the correction from bound-state quantum electrodynamics entirely in coordinate space, including the Uehling potential, which is usually isolated and treated separately. Pauli-Villars regularization is applied to the coordinate-space calculation and the most singular terms are shown to be eliminated, leaving the physical correction after charge renormalization. The conventional derivation of the Uehling potential in momentum space is reviewed and compared to the coordinate-space derivation.

Figure 1

Expansion of the vacuum-polarization correction in powers of the binding potential $V$. Diagram (a) represents the complete vacuum polarization of order $\alpha$. Diagrams (b) through (e) depict the zero-, one-, two-, and three-potential contributions. Only diagrams (c) and (e) and higher-order diagrams with an even number of verticies in the loop are nonzero according to Furry's theorem.