Thermal properties and evolution of the $U_A(1)$ factor for 2+1 flavors

Thermal evolution of the axial anomaly is investigated in the system of the linear sigma model for $2+1$ flavors. We explore the functional form of the effective potential and the coefficient of the `t Hooft determinant term. It is found that the latter develops a non-trivial structure as a function of the chiral condensate and grows everywhere with respect to the temperature. This shows that mesonic fluctuations strengthen the axial anomaly at finite temperature and it does not get vanished at the critical point. The phenomenon has been found to have significance in the thermal properties of the mesonic spectra, especially concerning the $\eta-\eta'$ system.