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

The 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 nontrivial 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 vanish at the critical point. The phenomenon has been found to have significance in the thermal properties of the mesonic spectra, especially concerning the $\eta -{\eta }^{\prime }$ system.

Figure 1

Mass spectrum for the zero anomaly case. Parametrization was carried out by fixing the $\pi$, $K$, and $\sigma$ masses. Pions are degenerated with the $\eta$.

Figure 2

Mass spectrum including the $U_A(1)$ anomaly. Parametrization was carried out by fixing the $\pi$, $K$, $\eta$, and ${\eta }^{\prime }$ masses.

Figure 3

Absolute value of the anomaly function $A_{k\to 0}$ as a function of $\sqrt{I_1}$ [note that $I_1\equiv (v_0^2+v_8^2)/2$]. The function increases everywhere with respect to the temperature, and deviates significantly from its value in the UV (dashed line).

Figure 4

Phase transition: Thermal evolution of the strange and nonstrange condensates. The dashed (solid) lines correspond to the zero (finite) anomaly case.

Figure 5

Absolute value of the anomaly at zero condensate and at the real ground state as a function of the temperature.

Figure 6

$\eta$ and ${\eta }^{\prime }$ masses as a function of the temperature. Solid (dashed) lines correspond to the temperature- and field-dependent (-independent) anomaly.