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 including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no `real’ electron. We show that it admits an asymptotic expansion to any order in powers of the physical coupling constant $\text{alphaph}$, provided that the ultraviolet cut-off behaves as $\mathrm{\Lambda }\sim e^{3\pi (1-Z_3)\mathrm{/}2\text{alphaph}}\gg 1$. The renormalization parameter $0