  We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane
({\it J. Phys. B.}, 22, 3791--3814, 1989) 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 math-ph/0403005, 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 $\Lambda$ and the bare fine
structure constant $\alpha$. In the present work, we improve this result by
showing the existence of the minimizer by a variational method, for any cut-off
$\Lambda$ and without any constraint on the external field.
  We also study the behaviour of the minimizer as $\Lambda$ goes to infinity
and show that the theory is "nullified" in that limit, as predicted first by
Landau: the vacuum totally kills 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 $\alpha$, on a
simplified model where the exchange term is neglected.
