Phenomenological improvement of the linear-$\sigma$ model in the large-N_c limit

TL;DR

This paper addresses a discrepancy in the linear-$\sigma$ model's critical temperature scaling in the large-$N_c$ limit by introducing a temperature-dependent parameter, aligning it with QCD expectations.

Contribution

The authors propose a phenomenological modification to the linear-$\sigma$ model to correct its large-$N_c$ scaling behavior of the critical temperature.

Findings

Corrected the $T_c$ scaling to be $N_c$-independent

Aligned the model's behavior with QCD expectations

Maintained agreement with other effective theories at finite $N_c$

Abstract

The linear-$\sigma$ model has been widely used to describe the chiral phase transition. Numerically, the critical temperature $T_{c}$ of the chiral phase transition is in agreement with other effective theories of QCD. However, in the large-$N_{c}$ limit $T_{c}$ scales as $\sqrt{N_{c}}$ which is not in line with the NJL model and with basic expectations of QCD, according to which $T_{c}$ is --just as the deconfinement phase transition- $N_{c}$-independent. This mismatch can be corrected by a phenomenologically motivated temperature dependent parameter.