${\eta }^{\prime }$ and $\eta$ mesons at high $T$ when the $U_A(1)$ and chiral symmetry breaking are tied

The approach to the ${\eta }^{\prime }\text{−}\eta$ complex employing chirally well-behaved quark-antiquark bound states and incorporating the non-Abelian axial anomaly of QCD through the generalization of the Witten-Veneziano relation is extended to finite temperatures. Employing the chiral condensate has led to a sharp chiral and $U_A(1)$ symmetry restoration, but with the condensates of quarks with realistic explicit chiral symmetry breaking—which exhibit a smooth, crossover chiral symmetry restoration in qualitative agreement with lattice QCD results—we get a crossover $U_A(1)$ transition with a smooth and gradual melting of anomalous mass contributions. In this way, we obtain a substantial decrease in the ${\eta }^{\prime }$ mass around the chiral transition temperature, but no decrease in the $\eta$ mass. This is consistent with current empirical evidence.

Figure 1

Axial-anomaly-induced, flavor-mixing transitions from hidden-flavor pseudoscalar states $P=q\overline{q}$ to $P^{\prime }=q^{\prime }{\overline{q}}^{\prime }$ including both possibilities $q=q^{\prime }$ and $q\ne q^{\prime }$. All lines and vertices are dressed. The gray blob symbolizes all possible intermediate states enabling this transition. The three bold dots symbolize an even [34] but otherwise unlimited number of additional gluons. As pointed out in Ref. [34], the diamond graph is just the simplest example of such a contribution.

Figure 2

The relative-temperature $T/T_{\text{Ch}}$ dependences of the pertinent order parameters calculated in our usual [9, 24] separable interaction model. The odd man out is the (third root of the absolute value of the) chiral condensate $⟨\overline{q}q{⟩}_0(T)$, which decreases steeply at $T=T_{\text{Ch}}$ and dictates similar behavior [9] to $\tilde{\chi }(T)$. All of the other displayed quantities exhibit smooth, crossover behaviors, which are smoother for heavier flavors: the dash-dotted and dashed curves are the (third roots of the absolute values of the) condensates $⟨\overline{s}s⟩(T)$ and $⟨\overline{u}u⟩(T)$, respectively, the thin solid curve is the resulting topological susceptibility $\chi (T{)}^{1/4}$, and the thick solid curve is the topological charge parameter $A(T{)}^{1/4}$. The decay constants $f_{\pi }(T)$ and $f_{s\overline{s}}(T)$ are, respectively, the lower dashed and dash-dotted curves.

Figure 3

$T$ dependence, relative to $T_{\text{Ch}}$, of various ${\eta }^{\prime }\text{−}\eta$ complex masses described in the text, the $\pi$ mass (thick, dash-dotted curve) for reference, the halved (to maintain clarity) total $U_A(1)$-anomaly-induced mass $\frac{1}{2}{M}_{U_A(1)}$ (short-dashed curve), and the topological charge parameter $A^{1/4}$ (solid curve). The straight line is 2 times the lowest fermion Matsubara frequency $2\pi T$.

Figure 4

$T/T_{\text{Ch}}$ dependence of pseudoscalar meson masses zoomed to the area important for the ${\eta }^{\prime }\text{−}\eta$ complex, for the simplest Ansatz ${\mathcal{C}}_m(T)=\text{constant}={\mathcal{C}}_m(0)$, which limits temperatures to $T\lesssim 1.6T_{\text{Ch}}$.

Figure 5

Relative $T$ dependence of the NS-S mixing angle $\varphi (T)$.