Dynamics and thermodynamics of a nonlocal Polyakov--Nambu--Jona-Lasinio model with running coupling

A nonlocal covariant extension of the two-flavor Nambu and Jona-Lasinio model is constructed, with built-in constraints from the running coupling of QCD at high-momentum and instanton physics at low-momentum scales. Chiral low-energy theorems and basic current algebra relations involving pion properties are shown to be reproduced. The momentum-dependent dynamical quark mass derived from this approach is in agreement with results from Dyson-Schwinger equations and lattice QCD. At finite temperature, inclusion of the Polyakov loop and its gauge invariant coupling to quarks reproduces the dynamical entanglement of the chiral and deconfinement crossover transitions as in the (local) Polyakov-loop-extended Nambu and Jona-Lasinio model, but now without the requirement of introducing an artificial momentum cutoff. Steps beyond the mean-field approximation are made including mesonic correlations through quark-antiquark ring summations. Various quantities of interest (pressure, energy density, speed of sound, etc.) are calculated and discussed in comparison with lattice QCD thermodynamics at zero chemical potential. The extension to finite quark chemical potential and the phase diagram in the $(T,\mu )$-plane are also discussed.

Figure 1

Comparison of the form factor $\mathcal{C}$ used in the nonlocal NJL model (solid line) and the analogous form factor obtained from the instanton model (dashed line). The dotted-dashed line shows the function $\mathcal{C}(p)$ in an NJL model with four-momentum cutoff.

Figure 2

Momentum dependence of the constituent quark mass $M$ compared with lattice data extrapolated to the chiral limit (from Ref. 30). The dashed line shows the mass $M$ calculated fully within the IM for the chiral limit [28].

Figure 3

Solid curves: calculated temperature dependence of the chiral condensate $⟨\overline{\psi }\psi ⟩$ and of the Polyakov loop $\Phi$ normalized to the critical temperature $T_c=207\mathrm{MeV}$ as obtained in the nonlocal PNJL model. Also shown for orientation are lattice results from Ref. 12. The dashed lines show the chiral condensate in the chiral limit (left for the pure fermionic case) and of the Polyakov loop for the pure gluonic case, respectively.

Figure 4

Pressure calculated in the nonlocal PNJL model in units of $T^4$ as a function of temperature normalized to the critical temperature. Solid curve: full calculation (i.e. mean-field result plus mesonic corrections). Dotted curve: mean-field result (no mesonic corrections). Dashed curve: mean-field plus pionic modes (no sigma).

Figure 5

Comparison of the pressure for physical pion mass, $m_{\pi }=140\mathrm{MeV}$ (solid line), and heavy pion mass, $m_{\pi }=500\mathrm{MeV}$ (dashed line). Note that the crossing of the curves is due to the different critical temperatures they were normalized to.

Figure 6

Trace anomaly $(\epsilon -3P)/T^4$ shown as a function of temperature for the mean-field case and with mesonic correlations added. For orientation, lattice data are shown from Ref. 12.

Figure 7

Comparison of the energy density $\epsilon$ as a function of temperature for the mean-field case and with mesonic correlations.

Figure 8

Fraction of pressure and energy density $P/\epsilon$ as a function of 4th root of the energy density ${\epsilon }^{1/4}$.

Figure 9

Squared sound velocity $v_{\mathrm{s}}^2$ as a function of 4th root of the energy density ${\epsilon }^{1/4}$.

Figure 10

(Normalized) chiral order parameter $\overline{\sigma }/{\overline{\sigma }}_0$ shown in the $(T,\mu )$ plane. For small chemical potentials a crossover transition is manifest. For large chemical potentials a first-order phase transition is apparent, which terminates in the critical point (CEP) at $(T_{\mathrm{CEP}},{\mu }_{\mathrm{CEP}})=(167\mathrm{MeV},175\mathrm{MeV})$.