Equation of state of a quark-meson mixture in the improved Polyakov–Nambu–Jona-Lasinio model at finite chemical potential

We study the equation of state of QCD using an improved version of the three-flavor Polyakov–Nambu–Jona-Lasinio model beyond the mean-field approximation. It incorporates the effects of unquenched quarks into the Polyakov-loop effective potential, as well as mesonic contributions to the grand-canonical potential. We study in full detail the calculation of the thermodynamical potential in this approach and compare the resulting pressure and entropy density with the most-recent lattice-QCD calculations at zero baryochemical potential. Finally, we present some exploratory results at finite chemical potential which include the phase diagram of the model, the quark and meson masses, and finally, the thermodynamical pressure.

Figure 1

(a) ${\mathcal{U}}_{\mathrm{YM}}/T^4$ of Eq. (5) for several temperatures around $T_0=270$ MeV, where a first-order phase transition occurs. (b) Expectation value of the Polyakov loop as a function of the temperature for the case $N_f=0$ (pure gauge).

Figure 2

Gap equation for the quark mass in Eq. (10). We have explicitly included a finite spatial range for the four-point and six-point contact interactions to detail the topology of the diagrams.

Figure 3

(a) Expectation value of the Polyakov loop as a function of the temperature for the PNJL model. We assume the polynomial parametrization of the Polyakov-loop effective potential in Table 1, with $T_0=190$ MeV as suggested in Ref. [27]. (b) Effective potential ${\mathcal{U}}_{\mathrm{YM}}/T^4$ as a function of the Polyakov loop for three different temperatures around $T_0=190$ MeV. For each potential we mark the position of the EVPL with an arrow.

Figure 4

Quark masses (a) and EVPL (b) in the PNJL model as a function of the temperature at vanishing chemical potential for ${\mathcal{U}}_{\mathrm{YM}}$ with $T_0=190$ MeV (dashed lines) and for ${\mathcal{U}}_{\mathrm{glue}}$ with $T_{\mathrm{glue}}^{\mathrm{cr}}=190$ MeV (solid lines).

Figure 5

Renormalized quark condensate (22) as a function of the temperature vs $T/T_c$ for the PNJL model using ${\mathcal{U}}_{\mathrm{glue}}$ effective potential. The lattice-QCD result is taken from Ref. [13].

Figure 6

Pressure of the PNJL model at mean-field level using ${\mathcal{U}}_{\mathrm{YM}}$ with $T_0=190$ MeV, and using ${\mathcal{U}}_{\mathrm{glue}}$. We compare with lattice-QCD results denoted as Set 1 (taken from Ref. [15]) and Set 2 (from Ref. [16]).

Figure 7

(From left to right) Noninteracting, Hartree, and 't Hooft diagrams contributing at $O\left(N_c\right)$ to the thermodynamic potential, or ${\mathrm{\Omega }}_q^{(-1)}$. We have explicitly introduced a finite range for the interaction (dashed line).

Figure 8

Fock and ring diagrams contributing to $\mathcal{O}({(1/N_c)}^0)$ of the thermodynamic potential, or ${\mathrm{\Omega }}_q^{\left(0\right)}$. We have explicitly included a finite range for the interaction (dashed lines) and an effective four-point vertex (squares).

Figure 9

Pressure at ${\mu }_i=0$ of the PNJL model with the inclusion of the $0^{+},0^{-}$ meson states whose mass in vacuum is below 1 GeV: $\pi ,K,\overline{K},\eta ,{\eta }^{\prime },f_0$ [to be identified to the scalar $f_0\left(980\right)]$. (a) Sequential addition of mesonic contributions to the pressure. (b) Total pressure and Mott temperatures of mesons, indicating where they become unstable against decay into a $\overline{q}q$ pair. The blue and red lines are the total pressure below and above $T_c$, respectively.

Figure 10

Pressure and entropy density at ${\mu }_q=0$ of the PNJL model with effect of $0^{+},0^{-}$ meson states, compared to lattice-QCD calculations. (a) States whose mass in vacuum is below 1 GeV: $\pi ,K,\overline{K},\eta ,{\eta }^{\prime },f_0$. (b) All scalar and pseudoscalar states $\left(\pi ,K,\overline{K},\eta ,{\eta }^{\prime },a_0,K_0^{*},{\overline{K}}_0^{*},f_0,f_0^{\prime }\right)$.

Figure 11

Diagrams contributing at $\mathcal{O}\left({\alpha }_s\right)$ to the QCD pressure [35] (numerical prefactors are not displayed).

Figure 12

(a) Quark mass as a function of ${\mu }_q$ at $T=1$ MeV. (b) Mean-field thermodynamic potential corresponding to the masses of the left panel.

Figure 13

(a) Phase boundary of the PNJL model at mean field with the parametrization given in this work. In the shaded region we show the area where the presence of a critical point is unlikely according to Ref. [46]. (b) Quark masses as a function of the temperature for several quark chemical potentials. From top to bottom (at fixed temperature) they read ${\mu }_q=0,100,200,244,300$ MeV.

Figure 14

Pressure in the mean-field approximation of the PNJL model, for several chemical potentials.

Figure 15

$\pi$ (a) and $K,\overline{K}$ (b) masses functions of temperature and chemical potential.

Figure 16

Mott (or melting) temperature of $\pi ,K$, and $\overline{K}$ mesons vs chemical potential. The Mott temperature is calculated within an uncertainty of $\pm 3$ MeV.

Figure 17

Partial pressure of $\pi$ (a) and $K=(K^{+},K^0)$ (b) as a function of temperature and chemical potential. The diamonds indicate the Mott temperature for each ${\mu }_q$.

Figure 18

Partial pressure of $\overline{K}=({\overline{K}}^0,K^{-})$ (a) and total pressure of the system (quarks + Polyakov loop + mesons) (b) as a function of temperature and chemical potential. The diamonds indicate the Mott temperature for each ${\mu }_q$.

Figure 19

Pressure of a gas of quarks + Polyakov loop + $\pi +K+\overline{K}$ as a function of temperature and chemical potential. Lattice-QCD data are taken from Ref. [46].

Figure 20

Quark-antiquark scattering phase shift corresponding to the pion at $T=1$ MeV (a) and $T=300$ MeV (b). The phase shift is plotted as a function of the momentum and $s={\omega }^2-p^2$.

Figure 21

Landau and unitary cuts along the real axis of the complex-energy plane for the quark-antiquark propagator function ${\mathrm{\Pi }}_{q_1,q_2}(\omega ,\mathbf{p}=0)$.

Figure 22

Pion phase shift as a function of the energy at $\mathbf{p}=0$ and $T=1$ MeV.

Figure 23

Quark-antiquark scattering phase shift corresponding to the pion (a) and kaon (b) channels. The phase shift is plotted at $p=0$ as a function of the energy and temperature.