  We discuss various aspects of the O(N)-model in the vacuum and at finite
temperature within the Phi-derivable expansion scheme to order lambda^2. In
continuation to an earlier work, we look for a physical parametrization in the
N=4 case that allows to accommodate the lightest mesons. Using zero-momentum
curvature masses to approximate the physical masses, we find that, in the
parameter range where a relatively large sigma mass is obtained, the scale of
the Landau pole is lower compared to that obtained in the two-loop truncation.
This jeopardizes the insensitivity of the observables to the ultraviolet
regulator and could hinder the predictivity of the model. Both in the N=1 and
N=4 cases, we also find that, when approaching the chiral limit, the
(iterative) solution to the Phi-derivable equations is lost in an interval
around the would-be transition temperature. In particular, it is not possible
to conclude at this order of truncation on the order of the transition in the
chiral limit. Because the same issue could be present in other approaches, we
investigate it thoroughly by considering a localized version of the
Phi-derivable equations, whose solution displays the same qualitative features,
but allows for a more analytical understanding of the problem. In particular,
our analysis reveals the existence of unphysical branches of solutions which
can coalesce with the physical one at some temperatures, with the effect of
opening up a gap in the admissible values for the condensate. Depending on its
rate of growth with the temperature, this gap can eventually engulf the
physical solution.
