Self-consistent solution for the polarized vacuum in a no-photon QED model

We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane (1989J. Phys. B: At. Mol. Opt. Phys. 22 3791-814) which is a mean-field theory deduced from no-photon QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as the polarized vacuum and it solves a self-consistent equation. In a recent paper, we proved the convergence of the iterative fixed-point scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cut-off A and the bare fine structure constant a. In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cut-off A and without any constraint on the external field. We also study the behaviour of the minimizer as A goes to infinity and show that the theory is 'nullified' in that limit, as predicted first by Landau: the vacuum totally cancels the external potential. Therefore, the limit case of an infinite cut-off makes no sense both from a physical and mathematical point of view. Finally, we perform a charge and density renormalization scheme applying simultaneously to all orders of the fine structure constant a, on a simplified model where the exchange term is neglected.