  The question of thermodynamic consistence and $\Phi$-derivability of the
asymptotic limit of high spatial dimensions for quantum itinerant models is
addressed. It is shown that although the irreducible $n$-particle Green
functions are local, reducible vertex functions retain different momentum
dependence. As a consequence, the vertex corrections to conductivity do not
generally vanish in the mean-field limit. The mean-field theory is a
$\Phi$-derivable approximation only if regular nonlocal or anomalous local
external sources are admitted.
