Electron positron pair density in charge renormalization

[boger]

I am trying to find out the density of electron positron pairs around a bare electron charge. In most texts, I saw that the treatment relates to the observed charge vs. the bare charge.
I wanted to know if there is a formula that describes the density of electron positron pairs that surround the electron or an estimate on the number of pairs that exist around it.

 

[Bob S]

The standard reference for vacuum polarization is Uehling's paper in Physical Review: Phys. Rev. 48, 55–63 (1935)
Polarization Effects in the Positron Theory
Abstract
Some of the consequences of the positron theory for the special case of impressed electrostatic fields are investigated. By imposing a restriction only on the maximum value of the field intensity, which must always be assumed much smaller than a certain critical value, but with no restrictions on the variation of this intensity, a formula for the charge induced by a charge distribution is obtained. The existence of an induced charge corresponds to a polarization of the vacuum, and as a consequence, to deviations from Coulomb's law for the mutual potential energy of point charges. Consequences of these deviations which are investigated are the departures from the Coulombian scattering law for heavy particles and the displacement in the energy levels for atomic electrons moving in the field of the nucleus.

 

[boger]

Uehling paper on vacuum polarization refers to the change in the observed charge and the columb potential.
I am looking for a formula of the total amount of virtual pairs that surround a charge and not just their net effect. Anyone has an idea where to find such description?

 

[Bill_K]

This is a FAMQ (Frequently-Asked Meaningless Question). Modification of the Coulomb field due to vacuum polarization results from a quantum superposition of any number of particle pairs.