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

We investigate QCD at large μ/T by using Z3-symmetric SU(3) gauge theory, where μ 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 θ as a parameter, and agrees with QCD when θ=0 and becomes symmetric when θ=2π/3. For both QCD and the Z3-symmetric SU(3) gauge theory, the phase diagram is drawn in μ-T plane with the Polyakov-loop extended Nambu-Jona-Lasinio model. In the Z3-symmetric SU(3) gauge theory, the Polyakov loop φ is zero in the confined phase appearing at T200 MeV and μ 300 MeV. The perfectly confined phase never coexists with the color superconducting (CSC) phase, since finite diquark condensate in the CSC phase breaks Z3 symmetry and then makes φ finite. When μ 300 MeV, the CSC phase is more stable than the perfectly confined phase at T100 MeV. Meanwhile, the chiral symmetry can be broken in the perfectly confined phase, since the chiral condensate is Z3 invariant. Consequently, the perfectly confined phase is divided into the perfectly confined phase without chiral symmetry restoration in a region of μ 300 MeV and T200 MeV and the perfectly confined phase with chiral symmetry restoration in a region of μ 300 MeV and 100T200 MeV. At low temperature, the basic phase structure of Z3-symmetric QCD-like theory remains in QCD. Properties of the sign problem in Z3-symmetric theory are also discussed. We discuss a numerical framework to evaluate observables at θ=0 from those at θ=2π/3.