Contributions of the vector-channel at finite isospin chemical potential with the self-consistent mean field approximation

The self-consistent mean field approximation of the two-flavor Nambu-Jona-Lasinio (NJL) model, which introduces a free parameter a (a reflects the weight of different interaction channels), is employed to investigate the contributions of the vector channel at a finite isospin chemical potential mu(I), zero baryon chemical potential mu(B), and zero temperature T. The calculations show that the consideration of the vector-channel contributions leads to a lower value of pion condensate in a superfluid phase, compared with the standard Lagrangian of the NJL model (alpha = 0). In a superfluid phase, we also obtain a lower isospin number density, and the discrepancy is getting larger with the increase of the isospin potential. Compared with the recent results from lattice QCD, the isospin density and energy density we obtained with alpha = 0.5 agree with the data of the lattice well. In the phase diagram in the T - mu(I) plane for it mu(B) = 0, we can see that the difference of the critical temperatures of a phase transition between the results with alpha = 0 and alpha = 0.5 is up to 3%-5% for a fixed isospin potential. All of these indicate that the vector channels play an important role in the isospin medium.