A PNJL model is built, in which the Polyakov-loop potential is explicitly $Z_{N_c}$-symmetric in order to mimic a Yang-Mills theory with gauge group $\mathrm{SU}(N_c)$. The physically expected large-$N_c$ and large-$T$ behaviors of the thermodynamic observables computed from the Polyakov-loop potential are used to constrain its free parameters. The effective potential is eventually U(1)-symmetric when $N_c$ is infinite. Light quark flavours are added by using a Nambu-Jona-Lasinio (NJL) model coupled to the Polyakov loop (the PNJL model), and the different phases of the resulting PNJL model are discussed in ’t Hooft’s large-$N_c$ limit. Three phases are found, in agreement with previous studies resorting to effective approaches of QCD. When the temperature $T$ is larger than some deconfinement temperature $T_d$, the system is in a deconfined, chirally symmetric, phase for any quark chemical potential $\mu$. When $T<T_d$ however, the system is in a confined phase in which chiral symmetry is either broken or not. The critical line $T_{\chi }(\mu )$, signalling the restoration of chiral symmetry, has the same qualitative features than what can be obtained within a standard $N_c=3$ PNJL model.

• Received 23 July 2011

DOI:https://doi.org/10.1103/PhysRevD.85.016009