Renormalization and Asymptotic Expansion of Dirac's Polarized Vacuum

We perform rigorously the charge renormalization of the so-called reduced Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac operator, describes atoms and molecules while taking into account vacuum polarization effects. We consider the total physical density ρ_ph including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no 'real' electron. We show that ρ_ph admits an asymptotic expansion to any order in powers of the physical coupling constant α_ph, provided that the ultraviolet cut-off behaves as. The renormalization parameter 0 < Z₃ < 1 is defined by Z₃ = α_ph/α, where α is the bare coupling constant. The coefficients of the expansion of ρ_ph are independent of Z₃, as expected. The first order term gives rise to the well-known Uehling potential, whereas the higher order terms satisfy an explicit recursion relation.