Nambu–Jona-Lasinio-based study of the QCD critical line

We employ a 3 flavor Nambu–Jona-Lasinio (NJL) model to stress some general remarks about the QCD critical line. The dependence of the critical curve on ${\mu }_q=({\mu }_u+{\mu }_d)/2$ and ${\mu }_I=({\mu }_u-{\mu }_d)/2$ is discussed. The quark masses are varied to confirm that, in agreement with universality arguments, the order of transition depends on the number of active flavors $N_f$. The slope of the critical curve vs chemical potential is studied as a function of $N_f$. We compare our results with those recently obtained in lattice simulations to establish a comparison among different models.

Figure 1

Phase diagram in the plane $({\mu }_q,T)$, for the physical $2+1$ case; ${\mu }_I$ and ${\mu }_s$ are set to zero. Dashed/solid lines indicate crossover/first order transitions; consequently, the dot in the picture labels the critical ending point.

Figure 2

Phase diagram in the plane $({\mu }_I,T)$, for the physical $2+1$ case; ${\mu }_q$ and ${\mu }_s$ are set to zero. The dashed line indicates a crossover transition for the scalar condensates, whereas the dotted line stands for genuine second order transition for the pion condensate. In the regions of a nonvanishing pion condensate, the discontinuous behavior of the scalar condensate turns into a continuous one, therefore the critical ending point is not present in this case.

Figure 3

Behavior of the light quark scalar condensates as a function of temperature at zero chemical potentials and $m_u=m_d=0$ (upper picture) and $m_s=8\mathrm{MeV}$ (lower picture). In the upper picture, $\chi \equiv {\chi }_u={\chi }_d={\chi }_s$, in the lower $\chi \equiv {\chi }_u={\chi }_d$. When the strange quark mass exceeds the critical value $m_s^C=10\mathrm{MeV}$, the discontinuous behavior turns into a genuine phase transition.

Figure 4

Phase diagram in the $({\mu }_q,m_s)$ space, at $m_u=m_d=0$ and ${\mu }_I={\mu }_s=0$. For every region in the diagram, the label I/II means whether, by increasing temperature and starting from $T=0$, the transition is of first or second order. Actually, the line in the diagram separating the two different regions follows the critical point by varying $m_s$.

Figure 5

Plot of the three cases $2,2+1,3$ in dimensionless units $(\mu /T_0,T/T_0)$, $\mu /T_0\le 1$. The curves relative to 2 and $2+1$ (the dashed one) almost overlap. In disagreement with lattice simulations, the curve relative to 3 stands slight above the others.