Nontopological soliton in the Polyakov quark-meson model

Within a mean-field approximation, we study a nontopological soliton solution of the Polyakov quark-meson model in the presence of a fermionic vacuum term with two flavors at finite temperature and density. The profile of the effective potential exhibits a stable soliton solution below a critical temperature $T\le T_{\chi }^c$ for both the crossover and the first-order phase transitions, and these solutions are calculated here with appropriate boundary conditions. However, it is found that only if $T\le T_d^c$ is the energy of the soliton $M_N$ less than the energy of the three free constituent quarks $3M_q$. As $T>T_d^c$, there is an instant delocalization phase transition from hadron matter to quark matter. The phase diagram together with the location of a critical end point has been obtained in the $T$ and $\mu$ plane. We notice that two critical temperatures always satisfy $T_d^c\le T_{\chi }^c$. Finally, we present and compare the result of thermodynamic pressure at zero chemical potential with lattice data.

Figure 1

The normalized chiral order parameter ${\overline{\sigma }}_v$ and the Polyakov-loop expectation values $\overline{\mathrm{\Phi }}, {\overline{\mathrm{\Phi }}}^{*}$ as functions of temperature for $\mu =0\mathrm{MeV}$ and $\mu =320\mathrm{MeV}$. The solid curves are for $\mu =0\mathrm{MeV}$ and the dashed curves are for $\mu =320\mathrm{MeV}$.

Figure 2

(a) The grand-canonical potentials ${\mathrm{\Omega }}_{\mathrm{MF}}$ as a function of the chiral order parameter $\sigma$ for $\mu =0\mathrm{MeV}$ by fixing the Polyakov loop on their expectation values. (b) The grand-canonical potentials ${\mathrm{\Omega }}_{\mathrm{MF}}$ as a function of the chiral order parameter $\sigma$ for $\mu =380\mathrm{MeV}$ by fixing the Polyakov loop on their expectation values. ${\mathrm{\Omega }}_{\mathrm{MF}}$ is scaled by a factor of $f_{\pi }^4$.

Figure 3

(a) The grand-canonical potential ${\mathrm{\Omega }}_{\mathrm{MF}}$ as a function of the Polyakov loop $\mathrm{\Phi }$ (or ${\mathrm{\Phi }}^{*}$) for $\mu =0\mathrm{MeV}$ by fixing the chiral order parameters on their expectation values. (b) The grand-canonical potential ${\mathrm{\Omega }}_{\mathrm{MF}}$ as a function of the Polyakov loop $\mathrm{\Phi }$ for $\mu =380\mathrm{MeV}$ by fixing the chiral order parameter on their expectation values. ${\mathrm{\Omega }}_{\mathrm{MF}}$ is scaled by a factor of $T^4$.

Figure 4

The quark fields in relative units and the $\sigma , \pi$ fields scaled with $f_{\pi }$ as function of the radius $r$ in vacuum.

Figure 5

(a) The quark fields in relative units and the $\sigma , \pi$ fields scaled with $f_{\pi }$ as function of the radius $r$ for $T=177\mathrm{MeV}$ as $\mu =0\mathrm{MeV}$. (b) The quark fields in relative units and the $\sigma , \pi$ fields scaled with $f_{\pi }$ as function of the radius $r$ for $T=56.1\mathrm{MeV}$, while $\mu =380\mathrm{MeV}$.

Figure 6

The total energy of system $M_N$ and the energy of three free constituent quark $3M_q$ as functions of temperature $T$. Here one set is for $\mu =0\mathrm{MeV}$ and another set is for $\mu =380\mathrm{MeV}$.

Figure 7

Two-flavor phase diagram in the $T-\mu$ plane in the Polyakov quark-meson model based on the nontopological soliton picture. The dash-dotted curve is the critical line for $T_d^c$, which characterizes the confinement phase transition, and the dashed lines are the critical line for conventional chiral phase transition in the region of crossover. The solid lines indicate the first-order phase transitions, and the solid circle indicates the CEPs for chiral phase transitions of $u$ and $d$ quarks.

Figure 8

The proton charge “rms” radius of a stable chiral soliton as a function of temperature $T$ at $\mu =0$ MeV and $\mu =380$ MeV. The solid curve is for $\mu =0$ MeV, while the dash-dotted curve is for $\mu =380$ MeV. The dotted curve is for the unstable baryonic phase existing in the crossover phase transition when $T_{\chi }^c\ge T>T_d^c$.

Figure 9

The normalized pressure variations with respect to temperature for the PQM model and the NS model at $\mu =0$, with $T_{\chi }^c\sim 201\mathrm{MeV}$. The solid line corresponds to the pressure in the PQM model, while the dash-dotted line is for the pressure in the NS model. All calculations are compared to lattice data ($N_{\tau }=6$) from Ref. [55].