Chiral and $U(1{)}_A$ restoration for the scalar and pseudoscalar meson nonets

We analyze the restoration pattern of the members of the scalar and pseudoscalar meson nonets under chiral $O(4)$ and $U(1{)}_A$ symmetries. For that purpose, we exploit QCD Ward identities (WI), which allow one to relate susceptibilities with quark condensates, as well as susceptibility differences with meson vertices. In addition, we consider the low-energy realization of QCD provided by $U(3)$ chiral perturbation theory (ChPT) at finite temperature to perform a full analysis of the different correlators involved. Our analysis suggests $U(1{)}_A$ partner restoration if chiral symmetry partners are also degenerated. This is also confirmed by the ChPT analysis when the light chiral limit is reached. Partner degeneration for the $I=1/2$ sector, the behavior of $I=0$ mixing and the temperature scaling of meson masses predicted by WI are also studied. Special attention is paid to the connection of our results with recent lattice analyses.

Figure 1

Different susceptibilities combinations from the lattice data in [4] for $32^3\times 8$ lattice size. (a) The four light susceptibilities. (b) Scalar and pseudoscalar pure strange susceptibilities.

Figure 2

Comparison between the scaling of ${\chi }_{5,\mathrm{disc}}$ and the subtracted condensate ${\mathrm{\Delta }}_{ls}(T;T_0)$ in (32), with respect to the reference temperature $T_0=139\mathrm{MeV}$. Data are taken from [4] for $32^3\times 8$ lattice size and $\widehat{m}/m_s=0.088$. We include also the comparison with $\sqrt{{\mathrm{\Delta }}_{l,s}}$ for the reasons explained in the main text.

Figure 3

Different susceptibility combinations from the lattice data in [4] for $32^3\times 8$ lattice size, related to the analysis of the $\eta -{\eta }^{\prime }$ mixing angle. (a): Susceptibility combination related to the vanishing of the $\eta -{\eta }^{\prime }$ mixing angle with $\widehat{m}/m_s=0.088$ [4], where we also plot $-{\chi }_P^{ls}$ according to (28). (b): Partner degeneration in the scenario of small $X_{ls}$ and $X_{08}$ with $X=P$, $S$ discussed in the text, according to (39).

Figure 4

Comparison of pseudoscalar screening mass ratios and subtracted condensates for the four channels $\pi$, $K$, $\overline{s}s$ and $\kappa$ with reference values $r_1^3{⟨\overline{q}q⟩}_l^{\mathrm{ref}}=0.750$, $r_1^3⟨\overline{s}s{⟩}^{\mathrm{ref}}=1.061$. The lattice data are taken from [40] (masses) and [60] (condensates) with the same lattice action and resolution, $T_0=145\mathrm{MeV}$, $T_c\simeq 196\mathrm{MeV}$ and $r_1\simeq 0.31\mathrm{fm}$ used in [60].

Figure 5

Susceptibilities calculated in $U(3)$ ChPT to NNLO for the physical pion mass. Left:The four susceptibilities of the $I=0$, 1 sector. Right: $I=1/2$ sector susceptibilities.

Figure 6

Different partner degeneration temperatures as the light chiral limit $m_{\pi }\to 0^{+}$ is approached.

Figure 7

Behavior of susceptibilities in the decomposition (43) close to the chiral limit in $U(3)$ ChPT.

Figure 8

Ratio of connected to total scalar susceptibility in the combined limits $M_0\to 0^{+}$, $m_{\pi }\to 0^{+}$ with $\alpha =M_0/m_{\pi }$.

Figure 9

Comparison of temperature scaling of ${\chi }_{5,\mathrm{disc}}$ and ${⟨\overline{q}q⟩}_l(T)$ for different values of the pion mass.

Figure 10

Temperature dependence of scalar and pseudoscalar mixing angles according to the definitions given in the main text.