We study the regularization dependence on meson properties and the phase diagram of quark matter by using the two flavor Nambu-Jona-Lasinio model. We find that the meson properties and the phase structure do not show drastically difference depending
the regularization procedures. We also find that the location or the existence of the critical end point highly depends on the regularization methods and the model parameters. Then we think that regularization and parameters are carefully considered when one investigates the QCD critical end point in the effective model studies.
The Nambu--Jona-Lasinio model is investigated in the $1/N_c$ expansion with the dimensional regularization. At the four-dimensional limit the meson propagators have simple forms in the leading order of the $1/N_c$ expansion. Thus the next to leading
order calculation reduces to an ordinary one loop calculation. Here we obtain an explicit form of the $1/N_c$ correction and numerically evaluate the $N_c$ dependence for the gap equation.
We investigate the phase diagram on temperature-chemical potential plane in the Nambu-Jona-Lasinio model with the dimensional regularization. While the structure of the resulting diagram shows resemblance to the one in the frequently used cutoff regu
larization, some results of our study indicate striking difference between these regularizations. The diagram in the dimensional regularization exhibits strong tendency of the first order phase transition.
We study the phase structure of the unpolarized and polarized two-flavor quark matters at zero and finite temperatures within the Nambu--Jona-Lasinio (NJL) model. We focus on the region, which includes the coexisting phase of quark-antiquark and diqu
ark condensates. Generalizing the NJL model so as to describe the polarized quark matter, we compute the thermodynamic potential as a function of the quark chemical potential ($mu$), the temperature ($T$), and the polarization parameter. The result heavily depends on the ratio $G_D / G_S$, where $G_S$ is the quark-antiquark coupling constant and $G_D$ is the diquark coupling constant. We find that, for small $G_D / G_S$, the ferromagnetic phase is energetically favored over the paramagnetic phase. On the other hand, for large $G_D / G_S$, there appears the window in the ($mu, T$)-plane, in which the paramagnetic phase is favored.
