Understanding of QCD at high density from Z3-symmetric QCD-like theory

TL;DR

The paper develops a Z3-symmetric QCD-like framework using flavor-dependent twist boundary conditions to dissect QCD at high density. By embedding FTBC in a PNJL model, it maps the μ–T phase diagram, revealing a perfectly confined phase (φ=0) that cannot host CSC, while chiral restoration can occur within confinement; CSC phases emerge at higher μ and low T, and this structure persists as a remnant in real QCD. It also discusses the sign problem, arguing it may be milder in the perfectly confined phase and proposes a θ-expansion lattice approach to relate Z3-QCD results to θ=0 QCD observables, with limitations in regions of diquark condensation. The work provides a conceptual and computational bridge between confinement, CSC, and chiral dynamics at high density, with implications for lattice QCD strategies.

Abstract

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=2π/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 varphi is zero in the confined phase appearing at T \lsim 200 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 varphi finite. When mu \gsim 300 MeV, the CSC phase is more stable than the perfectly confined phase at T \lsim 100 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 \lsim 300 MeV and T \lsim 200 MeV and the perfectly confined phase with chiral symmetry restoration in a region of μ\gsim 300 MeV and 100 \lsim T \lsim 200 MeV. The basic phase structure of Z_3-symmetric QCD-like theory remains in QCD. We show that in the perfectly confined phase the sign problem becomes less serious because of \varphi=0, using the heavy quark theory. We discuss a lattice QCD framework to evaluate observables at θ=0 from those at θ=2π/3.

Figures (8)

• Figure 1: $T$ dependence of $|\Delta_l |$ at $\mu =340$ MeV in the PNJL model with $\theta =2\pi /3$. The diquark condensates $\Delta_1$, $\Delta_2$ and $\Delta_3$ are denoted by dashed, solid and long dashed lines, respectively. Below $T=10$ MeV, three diquark condensates exist. In a region of $T=10\sim 80$ MeV, $\Delta_1=0$ and $\Delta_2=\Delta_3$.
• Figure 2: $T$ dependence of $M$, $\Delta$, and $\Phi$ at $\mu =340$ MeV in the PNJL model with $\theta = 2\pi /3$. Here, $M$, $\Delta$ and $\Phi$ are denoted by dashed, solid and long dashed lines, respectively, and $M$ and $\Delta$ are normalized by the constituent quark mass $M_0$ at the vacuum.
• Figure 3: $T$ dependence of $|\Delta_l |$ at $\mu = 340$ MeV in the PNJL model with $\theta =0$. See Fig. \ref{['delta_s_f']} for the definition of lines.
• Figure 4: $T$ dependence of $M$, $\Delta$ and $\Phi$ at $\mu =340$ MeV in the PNJL model with $\theta=0$. Here $M$ and $\Delta$ are divided by $M_0$. See Fig. \ref{['order_s_f']} for the definition of lines.
• Figure 5: Phase diagram in the PNJL model with $\theta =2\pi /3$. The CFL phase exists below the dotted 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 confinement phase below the line and in the deconfinement phase above the line. The perfectly confined phase is labeled by "$\Phi=0,~\Delta=0$", while the CFL, uSC and 2SC phases are by "CFL","uSC" and "2SC", respectively.