The O(N)-model within the Phi-derivable expansion to order lambda^2: on the existence, UV and IR sensitivity of the solutions to self-consistent equations

TL;DR

This paper examines the O(N)-model using the Phi-derivable expansion to order lambda^2, highlighting issues with UV sensitivity, IR behavior, and solution existence near phase transitions, impacting the model's predictivity.

Contribution

It provides a detailed analysis of the O(N)-model at order lambda^2, revealing UV/IR sensitivities and solution loss issues near critical temperatures, with a focus on the N=4 case.

Findings

Large sigma mass lowers Landau pole scale

Solution loss near the transition temperature in the chiral limit

Existence of unphysical solution branches affecting physical predictions

Abstract

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.