Nonlocal Polyakov–Nambu–Jona-Lasinio model and imaginary chemical potential

With the aim of setting constraints for the modeling of the QCD phase diagram, the phase structure of the two-flavor Polyakov-loop-extended Nambu and Jona-Lasinio (PNJL) model is investigated in the range of imaginary chemical potentials (${\mu }_{\mathrm{I}}$) and compared with available $N_f=2$ lattice QCD results. The calculations are performed using the advanced nonlocal version of the PNJL model with the inclusion of vector-type quasiparticle interactions between quarks, and with wave-function-renormalization corrections. It is demonstrated that the nonlocal PNJL model reproduces important features of QCD at finite ${\mu }_{\mathrm{I}}$, such as the Roberge-Weiss (RW) periodicity and the RW transition. Chiral and deconfinement transition temperatures for $N_f=2$ turn out to coincide both at zero chemical potential and at finite ${\mu }_{\mathrm{I}}$. Detailed studies are performed concerning the RW endpoint and its neighborhood where a first-order transition occurs.

Figure 1

The $p$ dependence of the quark mass function $M(p^2)$ and of the quasiparticle renormalization factor $Z(p^2)$ resulting from the distributions (2.7, 2.8) (solid lines). Solid triangles and open circles are LQCD data generated with a large quark mass, while solid squares show the extrapolation of the mass-function data to the chiral limit (from Ref. 41). Open triangles are LQCD results for $Z(p^2)$ taken from Ref. 62.

Figure 2

The $T$ dependence of the scalar mean field $\sigma$ and of the real part $\mathrm{Re}\Psi$ of the modified Polyakov loop. The dotted, dashed, and solid lines denote PNJL results at $\theta =0$, $\pi /6$, and $\pi /3$, respectively, using input set I.

Figure 3

The $T$ dependence of $\mathrm{Im}\Psi$ in the vicinity of the RW endpoint. The dotted, dashed, and solid lines are nonlocal PNJL results (using set I) at $\theta =\pi /3$ and its close neighborhood.

Figure 4

The $\theta$ dependences of the scalar mean field $\sigma$ [subfigure (a)], and of the modified Polyakov loop, $\mathrm{Re}\Psi$ and $\mathrm{Im}\Psi$ [subfigures (b) and (c)]. The dashed, dotted, and solid lines are results of nonlocal PNJL calculations (set I) at temperatures $T=220$, 230, and 250 MeV, respectively.

Figure 5

Phase diagram of the nonlocal PNJL model (set I) in the $T\mathrm{\text{−}}\theta$ plane, in comparison with LQCD results. The dotted and solid lines represent crossover and first-order transitions, respectively. Open squares and closed circles are LQCD data taken from Refs. 22, 25. The temperature is given in units of the pseudocritical temperature $T_c$ at $\mu =0$, while the imaginary chemical potential is expressed as $\theta ={\mu }_{\mathrm{I}}/T$ in units of $\pi /3$.

Figure 6

The $T$ dependence of $\mathrm{Im}\Psi$ in the nonlocal $\mathrm{PNJL}+G_v$ model at $\theta =\pi /3$. The dotted, dashed, and solid lines are results obtained with $G_v/G=0$, 0.4, and 0.6, respectively.

Figure 7

The $T$ dependence of the order parameters $\sigma$ and $\mathrm{Re}\Psi$ at $\theta =0$ and $\theta =\pi /3$ in the nonlocal PNJL model (set III) with the inclusion of the quark quasiparticle wave-function-renormalization factor $Z(p^2)$ (solid curves) as compared to the case with $Z\equiv 1$ (dotted and dashed lines).

Figure 8

The $G_v$ dependence of $\mathrm{Im}\Psi$ in the nonlocal PNJL model (set IV). The dashed and solid lines denote the results at $\theta =\pi /3$ with $G_v=0$ and $G_v=0.4G$, respectively.

Figure 9

Phase diagram of the nonlocal PNJL model with the inclusion of the quark quasiparticle renormalization factor $Z(p^2)$. Upper curves: without additional vector interaction (model set III). Lower curves: including vector coupling with $G_v/G=0.4$ (model set IV). The dotted and solid lines represent crossover and first-order phase transitions, respectively. Also shown for comparison is the result with $G_v/G=0.4$ but with no $Z$-factor effect (set II). Open squares and closed circles are LQCD data taken from Refs. 22, 25. The temperature scale $T_c$ is the pseudocritical temperature at $\mu =0$. The open circle at the triple point is the result of the set IV calculation with quark mass $m_0=12\mathrm{MeV}$.

Figure 10

Thermodynamical potential $\Omega$ of the nonlocal PNJL model (set IV with $G_v=0.4G$) as a function of the imaginary vector mean field $\mathrm{Im}\omega$. The lines are results of calculations at neighboring temperatures $T=216.5$, $T=217.5$, and 218.5 MeV, as indicated. The reference ${\Omega }_0$ is defined at $\mathrm{Im}\omega =0$.