  The SU(2)_L x SU(2)_R chiral quark model consisting of the (sigma,pi) meson
multiplet and the constituent quarks propagating on the homogeneous background
of a temporal gauge field is solved at finite temperature and quark baryon
chemical potential mu_q using an expansion in the number of flavors N_f, both
in the chiral limit and for the physical value of the pion mass. Keeping the
fermion propagator at its tree-level, several approximations to the pion
propagator are investigated. These approximations correspond to different
partial resummations of the perturbative series. Comparing their solution with
a diagrammatically formulated resummation relying on a strict large-N_f
expansion of the perturbative series one concludes that only when the local
part of the approximated pion propagator resums infinitely many orders in 1/N_f
of fermionic contributions a sufficiently rapid crossover transition at mu_q=0
is achieved allowing for the existence of a tricritical point or a critical end
point in the mu_q-T phase diagram. The renormalization and the possibility of
determining the counterterms in the resummation provided by a strict large-N_f
expansion are investigated.
