The Nambu Jona-Lasinio model with density dependent U_A(1) anomaly

Whether the phase transition from chiral broken phase to chiral restored state is of the first order is an interesting question. To study this, we use the Nambu Jona-Lasinio model which is believed to be a low energy effective
theory of quantum chromodynamics (QCD). In this work, we consider the density dependent U_A(1) anomaly and study the critical surface in the quark mass and
chemical potential space. We have found that the surface shrinks when a quark chemical potential increases, which is the opposite result from the study by
using usual NJL model. The difference comes from the density dependence of the U_A(1) anomaly and our result is consistent to the recent lattice QCD study. Our
result suggest the restoration of the U_A(1) anomaly at high density, which seems to be physically reasonable because a colour superconductivity state may dominate
at intermediate baryon density where the U_A(1) anomaly should be negligible.