The quarkynic phase and the Z_{Nc} symmetry

We investigate the interplay between the Z_{Nc} symmetry and the emergence of the quarkyonic phase, adding the flavor-dependent complex chemical potentials \mu_f=\mu+iT\theta_f with (\theta_f)=(0, \theta, -\theta) to the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model. When \theta=0, the PNJL model with the \mu_f agrees with the standard PNJL model with the real chemical potential \mu. When \theta=2\pi/3, meanwhile, the PNJL model with the \mu_f has the Z_{Nc} symmetry exactly for any real \mu, so that the quarkyonic phase exists at small T and large \mu. Once \theta varies from 2\pi/3, the quarkyonic phase exists only on a line of T=0 and \mu larger than the dynamical quark mass, and the region at small T and large \mu is dominated by the quarkyonic-like phase in which the Polyakov loop is small but finite.

Understanding of the confinement mechanism is one of the most important subjects in hadron physics. Lattice QCD (LQCD) shows numerically that QCD is in the confinement and chiral symmetry breaking phase at low temperature (T ) and in the deconfinement and chiral symmetry restoration phase at high T . In the limit of infinite current quark mass, the Polyakov-loop is an exact order parameter for the deconfinement transition, since the Z Nc symmetry is exact there. The chiral condensate is, meanwhile, an exact order parameter for the chiral restoration in the limit of zero current quark mass. In the real world where u and d quarks have small current masses, the chiral condensate is considered to be a good order parameter for the chiral restoration, but there is no guarantee that the Polyakov-loop is a good order parameter for the deconfinement transition.
In the previous paper [1], we have proposed a QCD-like theory with the Z Nc symmetry. Let us start with the SU(N c ) gauge theory with N f degenerate flavors to construct the QCD-like theory.
The partition function Z of the SU(N c ) gauge theory is obtained in Euclidean space-time by with the action where q f is the quark field with flavor f and current quark mass m f , D ν = ∂ ν −iA ν is the covariant derivative with the gauge field A ν , g is the gauge coupling and The temporal boundary condition for quark is The fermion boundary condition is changed by the Z Nc transformation as [2,3] for integer k, while the action S 0 keeps the form of (2) in virtue of the fact that the Z Nc symmetry is the center symmetry of the gauge symmetry [2]. The Z Nc symmetry thus breaks down through the fermion boundary condition (3) in QCD.
Now we consider the SU(N) gauge theory with N degenerate flavors, i.e. N = N c = N f , and assume the twisted boundary condition (TBC) in the temporal direction [1]: with the twisted angles for flavors f labeled by integers from 1 to N, where θ 1 is an arbitrary real number in a range of 0 ≤ θ 1 < 2π. The action S 0 with the TBC is a QCD-like theory proposed in Ref. [1]. In fact, the QCD-like theory has the Z Nc symmetry, since f is changed into f +k by the Z N transformation but f + k can be relabeled by f . In the QCD-like theory, the Polyakov loop becomes an exact order parameter of the deconfinement transition. The QCD-like theory then becomes a quite useful theory to understand the confinement mechanism.
When the fermion field q f is transformed by with the twisted angle θ f and the Euclidean time τ , the action S 0 is changed into with the imaginary chemical potential µ f = iT θ f , while the TBC returns to the standard one (3).
The action S 0 with the TBC is thus identical with the action S(θ f ) with the standard one (3). In the limit of T = 0, the action S(θ f ) comes back to the QCD action S 0 with the standard boundary condition (3) kept. The QCD-like theory thus agrees with QCD at T = 0 where the Polyakov loop Φ is zero. One can then expect that in the QCD-like theory Φ is zero up to some temperature T c and becomes finite above T c , i.e, that the Z Nc symmetry is exactly preserved below T c but spontaneously broken above T c . Actually, this behavior is confirmed by imposing the TBC on the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The PNJL model with the TBC [1] is referred to as the TBC model in this paper. In the TBC model, the flavor symmetry is explicitly broken by the flavor-dependent TBC (5), but the flavor-symmetry breaking is recovered at T < T c . The TBC model is thus a model proper to understand the confinement mechanism.
A current topic related to the confinement is the quarkyonic phase [10,11,13,20,22]. It is a confined (color-singlet) phase with finite quark-number density n, that is, a phase with Φ = 0 and n = 0. The n-generation induces the chiral restoration; in fact, the two phenomena occur almost simultaneously in the PNJL model [21]. This fact indicates that the quarkyonic phase can be regarded as a chirally-symmetric and confined phase. It was suggested in Refs. [23,24] that the chirally-broken phase is enlarged toward lager µ by the chiral density wave. In this paper, for simplicity, we ignore inhomogeneous condensates such as the chiral density wave. Effects of the inhomogeneous condensate on the quarkynic phase and the interplay between the effects and the Z Nc symmetry are interesting as a future work. The concept of the quarkyonic phase was constructed in large N c QCD. In fact, the phase was first found at small T and large real quark-number chemical potential µ in large N c QCD. Recently, the PNJL model showed that a quarkyonic-like phase with Φ < 0.5 and n = 0 exists at small T and large µ for the case of N c = 3 [13,20]. This result may stem from the fact that the deconfinement transition is crossover for N c = 3. This suggests that the quarkyonic phase can survive even at N c = 3 in the QCD-like theory with the Z Nc symmetry. In this paper, we consider the PNJL model of N ≡ N c = N f = 3 with the flavor-independent real chemical potential µ and the flavor-dependent quark boundary condition (5) with instead of (6); see Fig. 1 for the boundary condition. The present system is the same as that with the standard boundary condition (3) and the flavor-dependent complex chemical potentials (9). The present model with the µ f is reduced to the standard PNJL model with the flavor-independent real chemical potential µ when θ = 0 and to the TBC model with the Z Nc symmetry when θ = 2π/3. Varying θ, one can see how the phase diagram is changed between the exact color-confinement in the TBC model and the approximate one in the standard PNJL model. The aim of this paper is to see this behavior. Our particular interest is the location of the quarkyonic and the quarkyonic-like phase in the µ-T plane.
In general, there is no guarantee that the QCD partition function with complex chemical potential is real. It is, however, possible to prove that the QCD partition function Z 0 (µ f ) with where the third equality is obtained by the relabeling of the f . The present system thus has the sign problem, but the partition function is real, since where the first equality is obtained by (10) and the second one by the charge conjugation. Also in the PNJL model with the µ f , the partition funciton is real, as shown later.
The three-flavor PNJL Lagrangian is defined in Euclidian space-time as where interaction [25,26], respectively. The KMT interaction breaks the U A (1) symmetry explicitly. The Polyakov-loop Φ and its conjugate Φ * are defined by with L = exp(iA 4 /T ) in the Polyakov gauge. We take the Polyakov potential of Ref. [8]: Parameters of U are fitted to LQCD data at finite T in the pure gauge limit. The parameters except T 0 are summarized in Table I. The Polyakov potential yields the first-order deconfinement phase transition at T = T 0 in the pure gauge theory [27,28]. The original value of T 0 is 270 MeV determined from the pure gauge LQCD data, but the PNJL model with this value yields a larger  value of the pseudocritical temperature T c at zero chemical potential than T c ≈ 160 MeV predicted by full LQCD [29][30][31]. We then rescale T 0 to 195 MeV so as to reproduce T c = 160 MeV [19].
Now we consider the flavor-dependent complex chemical potential µ f = µ + iθ f T . The thermodynamic potential (per volume) is obtained by the mean-field approximation as [16] with where Here the three-dimensional cutoff is taken for the momentum integration in the vacuum term [16]. Obviously, Ω is real. The dynamical quark masses M f and the mesonic potential U M are defined by where ǫ f gh is the antisymmetric symbol.
The PNJL model has six parameters, (G S , G D , m 1 , m 2 , m 3 , Λ). A typical set of the parameters is obtained in Ref. [32] for the 2+1 flavor system with m 1 = m 2 ≡ m l < m 3 . The parameter set is fitted to empirical values of η ′ -meson mass and π-meson mass and π-meson decay constant at vacuum. In the present paper, we set m 0 ≡ m l = m 3 in the parameter set of Ref. [32]. The parameter set thus determined is shown in Table II.
Taking the color summation in (16) leads to  where Note that F 2 (F2) is the complex conjugate to F 3 (F3), indicating that Ω is real.
In the case of θ = 2π/3, particularly, Ω is invariant under the Z 3 transformation, Namely, Ω possesses the Z 3 symmetry. When the exact color-confinement with Φ = 0 occurs, Ω is invariant for any interchange among E ± 1 , E ± 2 and E ± 3 . Namely, Ω has the flavor symmetry in the exact color-confinement phase. For θ = 2π/3, the deconfinement transition is no longer exact. As θ decreases from 2π/3 to zero, T dependence of Φ becomes slower, and near θ = π/2 the order of the deconfinement transition is changed from the first-order to crossover. T and µ M f becomes a quarkyonic-like phase with small but finite Φ and n = 0; the latter region is labeled by "Qy-like".
For small θ far from 2π/3, the deconfinement transition line declines as µ increases, but for θ = 2π/3 the line is almost horizontal at small µ and rises at intermediate µ, as seen in Fig. 3. The rising of the deconfinement transition line is a consequence of the Z 3 symmetry, as shown below.
The quark one-loop part of Ω, which is defined by Ω Q in (17), can be expanded into a Maclaurin series  and Ω = c 11 ΦΦ * + U in panel (b). Note that M f is fixed to 323 MeV and T dependence of Φ is determined from the Ω with the minimum condition. As µ increases, the transition temperature decreases for θ = 0, but increases for θ = 2π/3. The transition is first-order for the case of θ = 2π/3. These results are consistent with the qualitative discussion mentioned above.
In summary, we have investigated the interplay between the Z Nc symmetry and the emergence of the quarkyonic phase, adding the complex chemical potentials µ f = µ + iT θ f with (θ f ) = quarkyonic phase defined by Φ = 0 and n > 0 really exists at small T and large µ. Once θ varies from 2π/3 to zero, the Z Nc symmetry is broken. As a consequence of this property, the quarkyonic phase exists only on a line of T = 0 and µ M f , and the region at small T and large µ is dominated by the quarkyonic-like phase characterized by small but finite Φ and n > 0. The Z Nc symmetry thus plays an essential role on the emergence and the location of the quarkyonic phase in the µ-T plane, and the quarkyonic-like phase at θ = 0 is a remnant of the quarkyonic phase at θ = 2π/3. Since the Z Nc symmetry is explicitly broken at θ = 0, it is then natural to expand the concept of the quarkyonic phase and redefine it by a phase with small Φ and finite n. For this reason, the quarkyonic-like phase is often called the quarkyonic phase. The gross structure of the phase diagram thus has no qualitative difference between θ = 2π/3 and zero, if the concept of the quarkyonic phase is properly expanded. In this sense, the Z Nc symmetry is a good approximate concept for the case of θ = 0, even if the current quark mass is small.