$U_A(1)$ symmetry restoration at high baryon density

We study the relation between chiral and $U_A(1)$ symmetries in the quark-meson model. Although quarks and mesons are described in mean field approximation, the topological susceptibility characterizing the $U_A(1)$ breaking comprises two components: one controlled by the condensate and the other by the meson fluctuation. The $U_A(1)$ restoration is governed by the competition of these components. In a hot medium, the condensates melt. However, the fluctuation is enhanced. Therefore, the $U_A(1)$ symmetry cannot be solely restored via the temperature effect. Nevertheless, the baryon density reduces the condensates and fluctuation, and thereby, the $U_A(1)$ symmetry can only be restored in a dense or dense and hot medium. The strange condensate plays a weak role in the susceptibility, and the chiral and $U_A(1)$ symmetry restorations occur almost at the same critical point.