  We apply a self-consistent relativistic mean-field variational ``Gaussian
functional'' (or Hartree) approximation to the linear $\sigma$ model with
spontaneously and explicitly broken chiral O(4) symmetry. We set up the
self-consistency, or ``gap'' and the Bethe-Salpeter equations. We check and
confirm the chiral Ward-Takahashi identities, among them the Nambu-Goldstone
theorem and the (partial) axial current conservation [CAC], both in and away
from the chiral limit. With explicit chiral symmetry breaking we confirm the
Dashen relation for the pion mass and partial CAC. We solve numerically the gap
and Bethe-Salpeter equations, discuss the solutions' properties and the
particle content of the theory.
