Axial Anomaly, Mismatched Fermi Surfaces and Vector Interaction in Dense Neutral Quark Matter

We report on effects of the q-$\bar{\rm q}$ vector interaction and/or the U(1)$_A$-anomaly-induced chiral-diquark coupling on the charge-neutral quark matter in $\beta$-equilibrium. We show that when the vector coupling is absent, there can appear a cross-over region sandwiched by two critical points in the intermediate temperature ($T$) region, while the phase transition in the low-$T$ region including zero temperature keeps being first order until the strength of the anomaly term is increased to have a critical value. On the other hand, when the vector coupling is also present, there appears a crossover region in the low-$T$ area including zero temperature with a new critical point, as was first demonstrated by Kitazawa et al and the present authors without and with the charge-neutral condition, respectively. We remark that the possible chromomagnetic instability is suppressed and can be even completely absent owing to the enhanced diquark coupling due to the anomaly term and/or by the vector interaction.