O (N) model within the Φ -derivable expansion to order λ2: On the existence and UV/IR sensitivity of the solutions to self-consistent equations

We discuss various aspects of the O(N)-model in the vacuum and at finite temperature within the Φ-derivable expansion scheme to order λ2. In continuation of an earlier work, we look for a physical parametrization in the N=4 case that allows us 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 Φ-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 Φ-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.