Spin polarization and chiral symmetry breaking at finite density

We investigate the possibility of the spin polarization of quark matter at zero temperature and moderate baryon density.The Nambu-Jona-Lasinio (NJL) model including interactions in vector and axial-vector channel (coupling constant $G_2$) as well as scalar and pseudoscalar channel (coupling constant $G_1$) is used, and the self consistent equations for the spin polarization and chiral condensate are solved numerically in the framework of the Hartree approximation. Numerical calculations show that in the one flavor model the spin polarization is possible at finite density if the ratio $G_2/G_1$ is larger than O(1).We find that the interplay between the spin polarization and the chiral symmetry plays an important role.If the chemical potential of quarks reaches a certain finite value, the spin polarization begins to occur because the quark still has relatively large dynamical mass, which is generated by spontaneous chiral symmetry breaking.On the other hand, when the spin polarization occurs, it operates to lower the dynamical quark mass a little.If one increases the chemical potential furthermore, the spin polarization becomes weaker and finally disappears because the dynamical quark mass is reduced by restoring gradually the chiral symmetry.