Polyakov loop, diquarks, and the two-flavor phase diagram

An updated version of the PNJL model is used to study the thermodynamics of $N_f=2$ quark flavors interacting through chiral four-point couplings and propagating in a homogeneous Polyakov loop background. Previous PNJL calculations are extended by introducing explicit diquark degrees of freedom and an improved effective potential for the Polyakov loop field. The mean field equations are treated under the aspect of accommodating group theoretical constraints and issues arising from the fermion sign problem. The input is fixed exclusively by selected pure-gauge lattice QCD results and by pion properties in vacuum. The resulting $(T,\mu )$ phase diagram is studied with special emphasis on the critical point, its dependence on the quark mass and on Polyakov loop dynamics. We present successful comparisons with lattice QCD thermodynamics expanded to finite chemical potential $\mu$.

Figure 1

Left: Fit to scaled pressure, entropy density and energy density as functions of the temperature in the pure gauge sector, compared to the corresponding lattice data taken from Ref. 5. Right: Resulting effective potential (6) that drives spontaneous Z(3) symmetry breakdown at $T=T_0$.

Figure 2

Using the fit of the Polyakov loop (dotted line) to lattice results taken from Ref. 6 in the pure gauge sector (empty symbols), the PNJL model predicts the Polyakov loop behavior as a function of temperature in the presence of dynamical quarks (solid line). This prediction is compared to lattice data in two flavors (full symbols) taken from Ref. 7.

Figure 3

The spontaneous chiral symmetry breaking mechanism of the PNJL model generates a temperature dependent chiral condensate $⟨\overline{\psi }\psi ⟩$ (solid line), which is compared here to lattice QCD results in two flavors shown in Ref. 18.

Figure 4

Second, fourth, sixth and eighth moments of the pressure difference with respect to the chemical potential, plotted as functions of the temperature. (Note that the temperature scales of the upper and lower graphs are different.) We compare to lattice data (diamonds with errorbars) taken from Ref. 4.

Figure 5

Upper left panel: comparison of the phase diagrams of NJL and PNJL model. The crossover of the chiral condensate is drawn solid, first-order lines are dashed and second order lines dotted. Upper right panel: comparison of the phase diagram at different current quark masses with inclusion of diquark degrees of freedom. (Note the scale on the temperature axis.) Lower panel: comparison of the PNJL model with and without inclusion of diquarks.

Figure 6

Dependence of the gap $\Delta$ on the presence of the Polyakov loop. The solid lines are the solutions to the self consistency equations of the NJL and the PNJL model at $\mu =0.4\mathrm{GeV}$. The dashed lines are obtained by enforcing fixed values for the Polyakov loop. Note that the PNJL model with the Polyakov loop fixed at $\Phi =1$ (deconfinement) coincides with the self-consistent solution of the NJL model.

Figure 7

Left panel: Squared speed of sound (dashed) and ratio $\frac{p}{\epsilon }$ (solid), and their dependence on the temperature at vanishing chemical potential. The two panels to the right: Studies of the sound velocity in the vicinity of the critical endpoint in the PNJL model without diquarks (at chemical potential $\mu ={\mu }_{\mathrm{CEP}}$). The PNJL model without diquarks is drawn with solid lines, the PNJL model with diquarks with dashed lines. Center panel: square of the velocity of sound. Right panel: pressure over energy density $\frac{p}{\epsilon }$.