Dynamical Generation of a Repulsive Vector Contribution to the Quark Pressure

Lattice QCD results for the coefficient $c_2$ appearing in the Taylor expansion of the pressure show that this quantity raises with the temperature towards the Stefan-Boltzmann limit. On the other hand, model approximations predict that when a vector repulsion, parametrized by $G_V$, is present this coefficient reaches a maximum just after $T_c$ and then deviates from the lattice predictions. Recently, this discrepancy has been used as a guide to constrain the (presently unknown) value of $G_V$ within the framework of effective models at large-$N_c$ (LN). In the present investigation we show that, due to finite $N_c$ effects, $c_2$ may also develop a maximum even when $G_V=0$ since a vector repulsive term can be dynamically generated by exchange type of radiative corrections. Here we apply the the Optimized Perturbation Theory (OPT) method to the two flavor Polyakov--Nambu--Jona-Lasinio model (at $G_V=0$) and compare the results with those furnished by lattice simulations an by the LN approximation at $G_V=0$ and also at $G_V \ne 0$. The OPT numerical results for $c_2$ are impressively accurate for $T \lesssim 1.2\, T_c$ but, as expected, predict that this quantity develops a maximum at high-$T$. After identifying the mathematical origin of this extremum we argue that such a discrepant behavior may naturally arise within these effective quark models (at $G_V=0$) whenever the first $1/N_c$ corrections are taken into account. We then interpret this hypothesis as an indication that beyond the large-$N_c$ limit the correct high temperature (perturbative) behavior of $c_2$ will be faithfully described by effective models only if they also mimic the asymptotic freedom phenomenon.