Phenomenological improvement of the linear-σ model in the large-N_c limit

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.