Critical point and Bose-Einstein condensation in pion matter

The Bose-Einstein condensation and the liquid-gas first-order phase transition are studied in the interacting pion matter. Two phenomenological models are used: The mean-field model and the hybrid model. Free model parameters are fixed by fitting the lattice QCD data on the pion Bose condensate density at zero temperature. In spite of some minor differences, the two models demonstrate an identical qualitative and very close quantitative behavior for the thermodynamic functions and electric charge fluctuations. A peculiar property of the considered models is an intersection of the Bose-Einstein condensation line and the line of the first-order phase transition at the critical endpoint.

Figure 1

(a) A line of the BEC $T_{\mathrm{BC}}$ for the ideal pion gas as a function of electric charge density $n_Q$ is shown by a solid line. A dashed line shows the same function when the ${\pi }^{-}$ presence is neglected. (b) Ratios of the ${\pi }^0$ and ${\pi }^{-}$ number densities to that of ${\pi }^{+}$ in the IdPG on the BEC line as functions of $T_{\mathrm{BC}}$.

Figure 2

(a) A fit of the LQCD data [17, 23] of the $n_{\mathrm{BC}}^{+}$ at $T=0$ as a function of $\mu$ in the mean-field model. A solid line shows the results with $A$ and $B$ given by Eq. (21), the dashed line corresponding to $A=0$ when attractive interactions are neglected, and the dotted vertical line presenting the IdPG results, i.e., when both $A=0$ and $B=0$. The leading order of the chiral perturbation theory ${\chi }_{\mathrm{PT}}$ from Ref. [21] is shown by a dashed-dotted line. A green box corresponds to the ${\mu }_0$ and $n_0$ values of the pion matter ground state. (b) The ratio of $p$ to $p_{\mathrm{id}}$ as functions of $T$ for $\mu =0$. A solid line corresponds to the mean-field model for $p$ with the parameters (21), a dashed line corresponds to the hybrid model discussed below.

Figure 3

(a) The phase diagram of the pion matter in the mean-field model with parameters (21) on the $(n_Q,T)$ plane. The left and right binodals are shown by solid lines, the BEC line by a dashed line. A green box denotes the ground state with $n_Q=n_0$ and $T=0$, and a red star denotes the CP with $T_c\cong 0.21m$. (b) The same as in panel (a), but for the $(\mu ,T)$ plane.

Figure 4

(a) A comparison of the BEC line in mean-field model at $T>0$ with the lQCD data [17]. (b) The charge density $n_Q$ from the mean-field model and the lQCD data [49] at $T=124$ MeV. A dashed line includes additional contribution (24) to $n_Q$ from ${\rho }^{\pm }$ mesons. Vertical dotted lines indicate $\mu =m$.

Figure 5

The scaled variance ${\omega }_Q$ (26) on the $(\mu ,T)$ plane in the mean-field model. The lines, green box, and red star symbols are the same as in Fig. 3.

Figure 6

(a) The FOPT and BEC lines in the mean field and hybrid models on the $(T,\mu )$ plane. A dotted line presents the pure repulsive interaction ($A=0$) when the FOPT is absent. (b) The $m^{*}/m$ ratio on the $(\mu ,T)$ plane.

Figure 7

The skewness (a) and kurtosis (b) in the hybrid model (see text for details).