Enforced neutrality and color-flavor unlocking in the three-flavor Polyakov-loop Nambu–Jona-Lasinio model

We study how the charge neutrality affects the phase structure of the three-flavor Polyakov-loop Nambu–Jona-Lasinio (PNJL) model. We point out that, within the conventional PNJL model at finite density, the color neutrality is missing because the Wilson line serves as an external colored field coupled to dynamical quarks. In this paper we heuristically assume that the model may still be applicable. To get color neutrality, one has then to allow nonvanishing color chemical potentials. We study how the quark matter phase diagram in $(T,m_s^2/\mu )$-plane is affected by imposing neutrality and by including the Polyakov-loop dynamics. Although these two effects are correlated in a nonlinear way, the impact of the Polyakov loop turns out to be significant in the $T$ direction, while imposing neutrality brings a remarkable effect in the $m_s^2/\mu$ direction. In particular, we find a novel unlocking transition, when the temperature is increased, even in the chiral $SU(3)$ limit. We clarify how and why this is possible once the dynamics of the colored Polyakov loop is taken into account. Also we succeed in giving an analytic expression for $T_c$ for the transition from two-flavor pairing (2SC) to unpaired quark matter in the presence of the Polyakov loop.

Figure 1

(a) Each color density of a finite density PNJL model with $m_s={\Delta }_{\eta }={\mu }_{e,3,8}=0$ as a function of $T$. $T_8$-color density is induced by the antitriplet color charge $l^{*}$ of Wilson line coupled to quarks. (b) The occupation number profile of quark with red, green, and blue, as a function of momentum at $T=0.4T_0$ which is indicated by the vertical dotted line in (a).

Figure 2

(a) Phase diagram in the $(\frac{m_s^2}{2\mu },T)$-plane at ${\Delta }_0=60\mathrm{MeV}$, $\mu =500\mathrm{MeV}$ with the Polyakov loop under charge neutrality constraints. The energy scales are normalized by ${\Delta }_0$. The bold line corresponds to the 1st order phase transition while the thin line indicates the 2nd order phase transition. For other instructions, see the text. (b) The same as (a) but without the Polyakov loop. In both figures, the bold square put on the $T$ axis indicates the weak coupling approximation of the critical temperature at chiral limit without $\Phi$, i.e., $T_c^0/{\Delta }_0=0.714$.

Figure 3

(a) Gaps (solid lines) and the Polyakov loop (long-dashed line) as functions of $T$ without the neutrality constraints, i.e., ${\mu }_{e,3,8}=0$. For the dashed line and the bold square, see the text. (b) The same as (a) but with the neutrality constraints being respected. Three charge chemical potentials are also depicted. ${\Delta }_3(ud)$ in (a) is also shown by the dashed line, just for comparison.

Figure 4

(a) The gaps ${\Delta }_{\eta }$ as a function of $m_s^2/2\mu$ at $T=0$. The dashed line is ${\Delta }_3$ for the 2SC solution. The vertical dotted line corresponds to the 1st order gCFL-to-g2SC transition; on the left of it the CFL is realized, while on the right of it the 2SC is realized as the ground state. (b) The same as (a) at finite temperature, $T=0.3{\Delta }_0$. All the transitions are of second order.

Figure 5

(a) The gaps ${\Delta }_{\eta }$, the chemical potentials ${\mu }_{e,8}$, and the traced Polyakov loop $\Phi$ as a function of $T$ at $m_s^2/2\mu =3.25{\Delta }_0$. (b) Comparison of (a) with simplified versions of the model. The bold full line is the same as ${\Delta }_3(T)$ in (a). The long-dashed line is the 2SC gap ${\Delta }_3(T)$ calculated in the pure NJL model without the Polyakov-loop dynamics. The dashed line is ${\Delta }_3(T)$ calculated with the Polyakov loop but without respecting charge neutralities, i.e., ${\mu }_{e,3,8}=0$.