${\eta }^{\prime }$ multiplicity and the Witten-Veneziano relation at finite temperature

We discuss and propose the minimal generalization of the Witten-Veneziano relation to finite temperatures, prompted by STAR and PHENIX experimental results on the multiplicity of ${\eta }^{\prime }$ mesons. After explaining why these results show that the zero-temperature Witten-Veneziano relation cannot be straightforwardly extended to temperatures $T$ too close to the chiral restoration temperature $T_{\mathrm{Ch}}$ and beyond, we find the quantity which should replace, at $T>0$, the Yang-Mills topological susceptibility appearing in the $T=0$ Witten-Veneziano relation, in order to avoid the conflict with experiment at $T>0$. This is illustrated through concrete $T$-dependences of pseudoscalar meson masses in a chirally well-behaved, Dyson-Schwinger approach, but our results and conclusions are of a more general nature and, essentially, model-independent.

Figure 1

The relative-temperature dependences, on $T/T_{\mathrm{Ch}}$, of ${\tilde{\chi }}^{1/4}$, $⟨\overline{q}q{⟩}_0^{1/3}$, $f_{\pi }$ and $f_{s\overline{s}}$, i.e., the $T/T_{\mathrm{Ch}}$-dependences of the quantities entering in the anomalous contributions to various masses in the ${\eta }^{\prime }\mathrm{\text{−}}\eta$ complex—see Eq. (10) and formulas below it. The solid curve depicts ${\tilde{\chi }}^{1/4}$ for $\delta =0$ in Eq. (9), and the short-dashed curve is ${\tilde{\chi }}^{1/4}$ for $\delta =1$. At $T=0$, the both $\tilde{\chi }$’s are equal to ${\chi }_{\mathrm{YM}}=(0.1757\mathrm{GeV}{)}^4$, the weighted average [24] of various lattice results for ${\chi }_{\mathrm{YM}}$. The dotted (red) curve depicts $-⟨\overline{q}q{⟩}_0^{1/3}$, the dash-dotted (blue) curve is $f_{\pi }$, and the long-dashed (blue) curve is $f_{s\overline{s}}$.

Figure 2

The relative-temperature dependence, on $T/T_{\mathrm{Ch}}$, of the pseudoscalar meson masses for two $\tilde{\chi }(T)$, namely, Eq. (8) with $\delta =0$ (upper panel) and with $\delta =1$ (lower panel). The meaning of all symbols is the same on the both panels: the masses of ${\eta }^{\prime }$ and $\eta$ are, respectively, the upper and lower solid curve, those of the pion and nonanomalous $s\overline{s}$ pseudoscalar are, respectively, the lower and upper dash-dotted curve, $M_{{\eta }_0}$ and $M_{{\eta }_8}$ are, respectively, the short-dashed (red) and long-dashed (red) curve, $M_{{\eta }_{NS}}$ is the medium-dashed (blue), and $M_{{\eta }_S}$ is the dotted (blue) curve. The straight line $2\pi T$ is twice the lowest Matsubara frequency.