Polyakov loops, Gross-Witten like point and Hagedorn states

The phase transition for a finite volume system that incorporates the Polyakov loops and maintains the colorless state is explored using the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model. The order parameter for Polyakov loops is demonstrated to signal the appearance of a transition for $SU(3)_{c}$ analogous to Gross-Witten (GW-) phase transition instead of the deconfinement phase transition to quark-gluon plasma. The asymptotic restoration of Polyakov loops is conjectured to be a threshold production for meta-stable Hagedorn (or semi-QGP) states and this does not imply a direct deconfinement phase transition. In this context, the GW-like point is the point where the colorless states switches from the low-lying hadronic states to the meta-stable high-lying Hagedorn states. The chiral phase transition takes place within an extended GW-like point depending on the fireball's size. The deconfinement phase transition is determined by Hagedorn's temperature above GW-like temperature.