Analysis of ${\mathrm{U}}_A(1)$ symmetry breaking and restoration effects on the scalar-pseudoscalarmeson spectrum

We explore patterns of effective restoration of the chiral ${\mathrm{U}}_A(1)$ symmetry using an extended three-flavor Nambu-Jona-Lasinio model that incorporates explicitly the axial anomaly through the ’t Hooft interaction, and assuming that the coefficient of the anomaly term is temperature and density dependent. The special case of explicit breaking of chiral symmetry without ${\mathrm{U}}_A(1)$ anomaly is also considered, since this scenario can provide additional information allowing to understand the interplay between the ${\mathrm{U}}_A(1)$ anomaly and (spontaneous) chiral symmetry-breaking effects. The pseudoscalar and scalar sectors are analyzed in detail bearing in mind the identification of chiral partners and the study of its convergence. We also concentrate on the behavior of the mixing angles that give us relevant information on the issue under discussion. In the region of temperatures (densities) studied, we do not observe signs indicating a full restoration of $\mathrm{U}(3)\otimes \mathrm{U}(3)$ symmetry as, for instance, the degeneracy of both $a_0$ and $f_0$ mesons with the pion. As we work in a real world scenario ($m_u=m_d\ll m_s$), we only observe the return to symmetries of the classical QCD Lagrangian in the nonstrange sector.

Figure 1

Temperature dependence of $⟨g_D{⟩}_u$ (left panel) and $⟨g_D{⟩}_s$ (right panel) for the different cases.

Figure 2

Temperature dependence of meson masses and of the continuum thresholds (dotted lines) $2M_u$, $2M_s$ (left panels) and $M_u+M_s$.

Figure 3

Left panel: scalar and pseudoscalar mixing angles as a function of temperature for the three cases presented in Table  and for the ideal mixing. Right panel: Topological susceptibility as a function of temperature for the three cases. The lattice data results are plotted with error bars [28].

Figure 4

Density dependence of meson masses and of limits of the Dirac sea continua (dotted lines) defining $q\overline{q}$ thresholds for the mesons ${\eta }^{\prime },a_0,\kappa$. The $f_0$ meson is always a resonance state.

Figure 5

Left panel: scalar and pseudoscalar mixing angles as a function of density for the three cases presented in Table  and for the ideal mixing ($g_D=0$). Right panel: topological susceptibility as a function of density for the three cases.

Figure 6

Density dependence of $⟨g_D{⟩}_u$ (left panel) and $⟨g_D{⟩}_s$ (right panel) for the different cases.

Figure 7

Chiral asymmetry parameter as a function of density for the three cases presented in Table  and the case $g_D=0$.

Figure 8

Density dependence of meson masses and of limits of the Dirac sea continua (dotted lines) defining $q\overline{q}$ thresholds for the mesons. The low-lying solutions also are included. The anomaly coupling constant is kept constant (Case I).

Figure 9

Left panel: scalar and pseudoscalar mixing angles as a function of density for the three cases and for the ideal mixing. Right panel: topological susceptibility as a function of density for the three cases.

Figure 10

Density dependence of meson masses and of limits of the Dirac sea continua (dotted lines) defining $q\overline{q}$ thresholds for the mesons. The low-lying solutions also are included. The anomaly coupling is a Fermi function (Case II).

Figure 11

Density dependence of meson masses and of limits of the Dirac sea continua (dotted lines) defining $q\overline{q}$ thresholds for the mesons. The low-lying solutions also are included. The anomaly coupling is a decreasing exponential (Case III).

Figure 12

Density dependence of meson masses and of limits of the Dirac sea continua (dotted lines) defining $q\overline{q}$ thresholds for the mesons. The low-lying solutions also are included. The axial anomaly is absent ($g_D=0$).