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

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