Net-baryon number fluctuations in the hybrid quark-meson-nucleon model at finite density

We study the mean-field thermodynamics and the characteristics of the net-baryon number fluctuations at the phase boundaries for the chiral and deconfinement transitions in the hybrid quark-meson-nucleon model. The chiral dynamics is described in the linear sigma model, whereas the quark confinement is manipulated by a medium-dependent modification of the particle distribution functions, where an additional scalar field is introduced. At low temperature and finite baryon density, the model predicts a first-, second-order chiral phase transition, or a crossover, depending on the expectation value of the scalar field, and a first-order deconfinement phase transition. We focus on the influence of the confinement over higher-order cumulants of the net-baryon number density. We find that the cumulants show a substantial enhancement around the chiral phase transition; they are not as sensitive to the deconfinement transition.

Figure 1

The mean fields in the QMN model for $\alpha b_0=300\mathrm{MeV}$ (left) and $\alpha b_0=390\mathrm{MeV}$ (right) at temperature $T=10\mathrm{MeV}$.

Figure 2

Net-baryon density fractions from Eq. (29), plotted as functions of baryon chemical potential at $T=10\mathrm{MeV}$. The top panel shows the case with $\alpha b_0=300\mathrm{MeV}$. The result in the bottom panel is obtained with $\alpha b_0=390\mathrm{MeV}$.

Figure 3

Thermodynamic pressure plotted against the baryon density ${\rho }_B$ in the units of saturation density ${\rho }_0$, at $T=10\mathrm{MeV}$ for two values of the $\alpha$ parameter. Also, the chiral and deconfinement transition points are indicated by the arrows.

Figure 4

Left: Phase diagram of the hybrid QMN model in the ($\alpha$, ${\mu }_B$) plane at several fixed $T$. At low temperatures, the liquid-gas transition is of first order and develops the critical point (CP; not shown here) at around $T=27\mathrm{MeV}$ [13], and eventually becomes a crossover. Right: Corresponding phase diagram at $T=10\mathrm{MeV}$ in the ($\alpha$, ${\rho }_B$) plane. In the left panel, the first-order phase transitions are indicated by the solid curves. The black areas in the right panel correspond to the density jump associated with the first-order phase transition. In both panels, the broken-dashed lines indicate the crossover transition. The circles indicate the critical points on the chiral transition lines. Also, in the left panel, the diamonds indicate the points above which the constraint introduced in Sec. 2c breaks down (see text).

Figure 5

The logarithm of the normalized net-baryon number fluctuations ${\chi }_2$ as a function of logarithm of $|{\mu }_B-{\mu }_B^c|$ near the $CP$ in the mean-field approximation at $T=10\mathrm{MeV}$. The solid line indicates the linear fit to the values obtained numerically.

Figure 6

Net-baryon number fluctuations normalized by ${\mu }_B^2$ as a function of baryon chemical potential in three models. The parameters for the parity-doublet and NJL models used are tabularized in Table 2. The right panel corresponds to set A, and the left panel to set B.

Figure 7

The diagonal contributions to the second-order cumulant from the $\sigma$ and $b$ mean fields, for the case of $\alpha b_0=390\mathrm{MeV}$. The inset figure shows the details in the vicinity of the deconfinement transition.

Figure 8

First four cumulants in the hybrid QMN model for different values of the parameter $\alpha$ plotted as a function of baryon chemical potential normalized by corresponding values of the chiral transition ${\mu }_B^c$ in each case.