Understanding QCD at high density from a $Z_3$-symmetric QCD-like theory

We investigate QCD at large $\mu /T$ by using $Z_3$-symmetric $SU(3)$ gauge theory, where $\mu$ is the quark-number chemical potential and $T$ is temperature. We impose the flavor-dependent twist boundary condition on quarks in QCD. This QCD-like theory has the twist angle $\theta$ as a parameter, and agrees with QCD when $\theta =0$ and becomes symmetric when $\theta =2\pi /3$. For both QCD and the $Z_3$-symmetric $SU(3)$ gauge theory, the phase diagram is drawn in $\mu$–$T$ plane with the Polyakov-loop extended Nambu–Jona-Lasinio model. In the $Z_3$-symmetric $SU(3)$ gauge theory, the Polyakov loop $\phi$ is zero in the confined phase appearing at $T\lesssim 200\mathrm{MeV}$ and $\mu \lesssim 300\mathrm{MeV}$. The perfectly confined phase never coexists with the color superconducting (CSC) phase, since finite diquark condensate in the CSC phase breaks $Z_3$ symmetry and then makes $\phi$ finite. When $\mu \gtrsim 300\mathrm{MeV}$, the CSC phase is more stable than the perfectly confined phase at $T\lesssim 100\mathrm{MeV}$. Meanwhile, the chiral symmetry can be broken in the perfectly confined phase, since the chiral condensate is $Z_3$ invariant. Consequently, the perfectly confined phase is divided into the perfectly confined phase without chiral symmetry restoration in a region of $\mu \lesssim 300\mathrm{MeV}$ and $T\lesssim 200\mathrm{MeV}$ and the perfectly confined phase with chiral symmetry restoration in a region of $\mu \gtrsim 300\mathrm{MeV}$ and $100\lesssim T\lesssim 200\mathrm{MeV}$. At low temperature, the basic phase structure of $Z_3$-symmetric QCD-like theory remains in QCD. Properties of the sign problem in $Z_3$-symmetric theory are also discussed. We discuss a numerical framework to evaluate observables at $\theta =0$ from those at $\theta =2\pi /3$.

Figure 1

$T$ dependence of $|{\mathrm{\Delta }}_l|$ at $\mu =340\mathrm{MeV}$ in the PNJL model with $\theta =2\pi /3$. The diquark condensates ${\mathrm{\Delta }}_1$, ${\mathrm{\Delta }}_2$ and ${\mathrm{\Delta }}_3$ are denoted by short dashed, solid and long dashed lines, respectively. Below $T=10\mathrm{MeV}$, all three diquark condensates are finite. In a region of $T=10\sim 80\mathrm{MeV}$, ${\mathrm{\Delta }}_1=0$ and ${\mathrm{\Delta }}_2={\mathrm{\Delta }}_3\ne 0$. Only the ${\mathrm{\Delta }}_3$ is finite in the region of $T=223\sim 419\mathrm{MeV}$.

Figure 2

$T$ dependence of $M$, $\mathrm{\Delta }$, and $\mathrm{\Phi }$ at $\mu =340\mathrm{MeV}$ in the PNJL model with $\theta =2\pi /3$. Here, $M$, $\mathrm{\Delta }$ and $\mathrm{\Phi }$ are denoted by short dashed, solid and long dashed lines, respectively, and $M$ and $\mathrm{\Delta }$ are normalized by the constituent quark mass $M_0$ at the vacuum.

Figure 3

$T$ dependence of $|{\mathrm{\Delta }}_l|$ at $\mu =340\mathrm{MeV}$ in the PNJL model with $\theta =0$. See Fig. 1 for the definition of lines. Below $T=32\mathrm{MeV}$, all three condensates are finite, while only the ${\mathrm{\Delta }}_3$ is finite in the region of $T=32\sim 86\mathrm{MeV}$.

Figure 4

$T$ dependence of $M$, $\mathrm{\Delta }$ and $\mathrm{\Phi }$ at $\mu =340\mathrm{MeV}$ in the PNJL model with $\theta =0$. Here $M$ and $\mathrm{\Delta }$ are divided by $M_0$. See Fig. 2 for the definition of lines.

Figure 5

Phase diagram in the PNJL model with $\theta =2\pi /3$. The CFL phase exists below the short dashed line. The solid line stands for the first-order chiral phase transition; $M$ is large on the left side of the solid line, but small on the right side. The dashed double-dotted line denotes the first-order deconfinement phase transition; the system is in the deconfinement phase above the line. The perfectly confined phase is labeled by “$\mathrm{\Phi }=0$, $\mathrm{\Delta }=0$”, while the CFL, uSC and 2SC phases are by “CFL”,“uSC” and “2SC”, respectively.

Figure 6

Phase diagram in the PNJL model with $\theta =0$. The solid line (thin dotted line) stands for the first-order (crossover) chiral transition. The thin dashed double-dotted line denotes the line of $\mathrm{\Phi }=0.5$. The almost-confined phase is labeled by “$\mathrm{\Phi }<0.5$”, while the deconfined phase is by “$\mathrm{\Phi }>0.5$” and the CFL and 2SC phases are by “CFL” and “2SC,” respectively.

Figure 7

Schematic figures of the allowed region in the complex $\mathrm{\Phi }$ plane. The three bold solid circles represent the values of $Z_3$ element. (a) In the ordinary QCD, the fermion determinant is real on the bold line. (b) In the theory with $Z_3$ symmetry, the fermion determinant is real on the three bold lines.

Figure 8

$\theta$ dependence of $\mathrm{\Phi }$ and the quark number density $n_q$ at $(\mu ,T)=(260\mathrm{MeV},100\mathrm{MeV})$ in the PNJL model. The solid line stands for $\mathrm{\Phi }$, while the dashed line stands for $n_q$. The $n_q$ is divided by $3n_0$, where $n_0$ is the normal nuclear matter density.

Figure 9

$\theta$ dependence of $\mathrm{\Phi }$ and the quark number density $n_q$ at $(\mu ,T)=(300\mathrm{MeV},150\mathrm{MeV})$ in the PNJL model. See Fig. 8 for the definition of lines.