Lattice study on QCD-like theory with exact center symmetry

We investigate QCD-like theory with exact center symmetry, with emphasis on the finite-temperature phase transition concerning center and chiral symmetries. On the lattice, we formulate center symmetric $SU(3)$ gauge theory with three fundamental Wilson quarks by twisting quark boundary conditions in a compact direction ($Z_3$-QCD model). We calculate the expectation value of Polyakov loop and the chiral condensate as a function of temperature on 16^3 x 4 and 20^3 x 4 lattices along the line of constant physics realizing $m_{PS}/m_{V}=0.70$. We find out the first-order center phase transition, where the hysteresis of the magnitude of Polyakov loop exists depending on thermalization processes. We show that chiral condensate decreases around the critical temperature in a similar way to that of the standard three-flavor QCD, as it has the hysteresis in the same range as that of Polyakov loop. We also show that the flavor symmetry breaking due to the twisted boundary condition gets qualitatively manifest in the high-temperature phase. These results are consistent with the predictions based on the chiral effective model in the literature. Our approach could provide novel insights to the nonperturbative connection between the center and chiral properties.


I. INTRODUCTION
Strong dynamics based on Quantum ChromoDynamics (QCD) induces two main nonperturbative phenomena, "quark confinement" and "spontaneous chiral symmetry breaking", which are the main themes of research on strong-coupling dynamics. While both of the two phenomena are caused by infrared physics based on the asymptotic freedom, it is not fully understood how they are related with each other. Some of lattice QCD simulations show that these two phenomena are almost simultaneously turned off at a certain temperature [1][2][3], where the transition to the quark-gluon plasma (QGP) phase occurs. Regarding the connection between the two phenomena, we here have several questions: Do the transitions take place at the same temperature? Does one of the transitions induce the other?
In the presence of exact Z 3 center symmetry, one can study the confining/deconfining phase transition by use of the expectation value of the Polyakov loop. Since it is the order parameter of Z 3 center symmetry, it works as an indicator of confining/deconfining through the quark free energy. Quenched QCD (without dynamical quarks) [1] and adjoint QCD (with adjoint quarks instead of fundamental ones) [19] are examples of this case. However, their results on coincidence of the transition temperatures are inconsistent: For the quenched QCD it is reported that the transition temperatures coincides while they do not for the adjoint QCD. In the first place, these two theories are too far from the realistic QCD in a sense that one contains no dynamical quarks while the other contains an excessive amount of dynamical degrees of freedom of quarks.
For the physical N f = 2 + 1 QCD, where u-and d-quark masses are degenerated while s-quark has a heavier mass, things get more involved: the dynamical fundamental quarks explicitly break center symmetry, so what we can do the best is study the crossover tran-sition based on the expectation values of the Polyakov loop. In the literature [2,3], one finds inconsistent results on coincidence of the crossover temperatures of center and chiral transitions, depending on the scheme and setup of the lattice simulations: Some works argue that the two critical temperatures are very close [2] while the others show that they are not [3].
One ideal situation for studying this topic in details is that the exact center symmetry and the dynamical fundamental quarks are reconciled. As well-known, the existence of dynamical fundamental quarks breaks center symmetry explicitly, thus the above accommodation seems infeasible. However, appropriate boundary conditions for quarks enable us to realize such a situation. By imposing three different twisted boundary conditions on the three fundamental quarks (shifted by 2π/3) in the compact imaginary-time direction, we realize center symmetric SU(3) gauge theory with three dynamical quarks in fundamental representation on R 3 × S 1 (Z 3 -QCD model) [20][21][22][23][24]. By investigating this model, we could make progress in elucidating the connection between confining/deconfining and chiral transitions. The analytical study on this theory based on the Polyakov-loop extended chiral model [15,16,18] was initiated in Ref. [23], which shows that the center symmetry is spontaneously broken at certain temperature, associated by the manifestation of flavor symmetry breaking. Although the chiral model helps understand rough picture of the nonperturbative properties, we cannot fully eliminate the model artifacts.
In this paper, we numerically investigate finite-temperature Z 3 -QCD model on the lattice, with emphasis on center phase transition and its influence on the chiral properties.
We formulate center symmetric SU(3) lattice gauge theory with three fundamental Wilson quarks by twisting the boundary conditions in the imaginary time direction. We calculate the expectation value of Polyakov loop and the chiral condensate on 16 3 × 4 and 20 3 × 4 lattices with m P S /m V = 0.70 fixed. We find out the first-order center phase transition at certain temperature, where the hysteresis exists depending on the initial condition for the configuration generation. We show that the chiral condensate rapidly decreases around the center critical temperature and has the hysteresis too. We also verify manifestation of the flavor symmetry breaking in the flavor-diagonal meson sector in the high-temperature phase, which was predicted in the effective chiral model study.
The structure of the paper follows: In Sec. II, we briefly review the construction of Z 3 -QCD model in the continuum theory and introduce the expected properties based on the chiral model study. In Sec. III, we formulate Z 3 -QCD model on the lattice and note details of the simulation setup including the parameter fixing for finite-temperature simulations. In Sec. IV, we show our results for the phase transitions associated with the center and chiral symmetries. Sec. V is devoted to summary and discussion.

A. Formulation as a continuum theory
We give a brief review of Z 3 center-symmetric SU(3) gauge theory with three fundamental quarks on R 3 × S 1 proposed in Refs. [20]. Here, the compact dimension can be seen as imaginary time direction in the present work. We first consider the partition function Z in Euclidian spacetime with one compact dimension as where Ψ f is a quark field for degenerate three-flavor quarks with flavor index f = 1, 2, 3.
D µ ≡ ∂ µ − iA µ is a covariant derivative with the SU(3) gauge field A µ and the field strength is given by We here regard a compact dimension (τ ) as imaginary time direction, and impose an anti-periodic boundary condition on a quark field in the direction as By adopting this boundary condition, we now work on the finite-temperature system with It is known that the above action itself is invariant under the Z 3 center transformation, which is generated by the center elements of global color SU(3) transformation. However, this transformation eventually gives rise to 2π/3 shift of phase of the quark boundary condition as with k = 0, 1, 2. Thereby, the Z 3 center symmetry is explicitly broken via the fundamentalquark boundary conditions (3)(4) in the usual three-flavor SU(3) gauge theory, and of course in the realistic N f = 2 + 1 QCD.
We now consider a case that the three flavors have distinct boundary conditions as fol- lowing, for f = 1, 2, 3. It is nothing but SU(3) gauge theory in the presence of three fundamental quarks with flavor-dependent boundary conditions.
For this case, the Z 3 transformation in Eq. (4) shifts the quark boundary conditions as The twisted angles for the three flavors are changed into k + f − 1, but this can be straight- We here comment on flavor symmetry possessed by Z 3 -QCD model in the massless limit.
In Z 3 -QCD, due to the boundary conditions, the flavor-chiral SU(3) L × SU(3) R symmetry is explicitly broken to its Cartan subgroup U(1) 2 L × U(1) 2 R generated by λ 3 , λ 8 of the Gellmann matrix elements. However, this influence from the boundary condition disappears in the infinite compact-circumference limit, or in the zero-temperature limit. Thus, in this limit, Z 3 -QCD is reduced to the standard three-flavor QCD at zero temperature.
For nonzero temperature, the effects of the twisted boundary condition exists in principle, namely the flavor-chiral symmetry is broken to U(1) 2 L × U(1) 2 R at the action level. In this work, we regard this Cartan subgroup as the specific flavor-chiral symmetry of the Z 3 -QCD model. Thus, the pattern of the chiral symmetry breaking is basically given by U(1) 2 L × U(1) 2 R → U(1) 2 V in this model. While the chiral transition in the standard three-flavor QCD model is believed to be first-order in the chiral limit based on the universality class [25], the same discussion is probably not valid for the Z 3 -QCD model.
Here, we make a supplemental comment on the flavor-chiral symmetry U(1) 2 L × U(1) 2 R in the Z 3 -QCD model discussed above. It is notable that, indeed, we do not know at how high temperature the physical quantities start to be affected by the twisted boundary condition. It may be as soon as the temperature is turned on or may be at a certain nonzero temperature. Actually, as we will see in the next subsection, the study based on the chiral effective model indicates that the chiral condensate is insensitive to the boundary condition below the center phase transition temperature, where the flavor symmetry breaking due to the boundary condition is not manifest in the effective thermodynamic potential.
The significance of this model is that we can study the phase transition with respect to the Z 3 center symmetry by calculating expectation values of Polyakov loop, even in the presence of the dynamical fundamental quarks. We here have physical and theoretical questions: If the phase transition occurs, what is the order of the phase transition? How is the chiral condensate affected by the center phase transition? We study these topics in the next section numerically, although our simulations are not carried out in the chiral limit.
The study on such an ideal model with the exact center symmetry may help elucidate the relation of center and chiral properties from novel viewpoints.

C. Expected properties from chiral model and QCD with a finite chemical potential
In the study of Z 3 -QCD based on the Polyakov-loop-extended NJL (PNJL) model [15,16,18], the following three properties are predicted [23] (we call this model just as the effective chiral model in the present paper.): (i) Z 3 center symmetry is spontaneously broken in the high-temperature phase, where the order of the phase transition is first.
(ii) Although the SU(3) flavor symmetry the standard three-flavor QCD has is spoiled by the twisted boundary condition at the action level, the effective thermodynamical potential of the PNJL model in the low-temperature phase is not affected by the boundary condition.
It is expected that the Z 3 -QCD model becomes SU(3) flavor-symmetric in the phase. On the other hand, the flavor symmetry breaking becomes manifest in the high-temperature phase.
(iii) In a chiral limit, even above the critical temperature of the center symmetric phase transition, the chiral condensate has a nonzero value and the chiral symmetry is still broken.
It might be an artifact coming from the model cutoff in the chiral effective model. Instead, the value of chiral condensate has a specific jump at the center critical temperature.
The present model is also related to the works on QCD with finite imaginary chemical potential. As discussed in Refs. [26,27], the partition function has 2π/3 periodicity in the imaginary chemical potential. Thus we speculate that the critical temperature for the chiral phase transition in the present model, which also has 2π/3 periodicity of the temporal direction, could be the same as that of the standard three-flavor QCD.

III. SIMULATION SETUP
We utilize the Iwasaki gauge action with naive Wilson fermions in our lattice numerical simulation. The definition of the action is given by where β = 6/g 2 , in which g is a lattice bare gauge coupling constant, c 1 = −0.331, c 0 = 1−8c 1 and W 1×1 and W 1×2 denote the plaquette and rectangular, respectively. In Eq. (8), Here κ is the hopping parameter. We note that the value of κ in Z 3 -QCD model is universal for all flavors.
To realize the twisted boundary condition in Eq. (5) on the lattice, we introduce the following boundary conditions for the link variable only in the fermion action Eq. (8): These conditions from top to bottom are the same with those for the standard finitetemperature QCD with imaginary chemical potential; µ I = 0, 2π/3 and 4π/3, respectively.
We have to use the Rational Hybrid Monte Carlo (RHMC) algorithm to calculate the fermion determinant for each flavor, since three fermions have a different boundary condition with each other.
Firstly, we perform the zero-temperature simulation using 16 4 lattices to obtain the line of constant-physics. We carry out the simulations for several values of hopping parameter κ with each β value and measure the flavor-singlet pseudo-scalar mass (m P S ) and vector meson mass (m V ) for each flavor. The number of trajectories we generate is 2, 000 -3, 000, and we measure the correlator of these hadronic states every 10 Monte Carlo trajectories.
The estimated autocorrelation length is around 100 trajectories in this simulation. We summarize results on mass measurement in Table II and III in Appendix C.  We fix the ratio between m P S and m V constant, namely m P S /m V = 0.70, and tune the value of hopping parameter for each β shown in Table I. Using these parameter sets in Table I,

A. Polyakov loops and center symmetry
We first show the existence of Z 3 center symmetry in the present model based on the distribution plot of Polyakov loop (L), Here V denotes the spacial volume in a lattice unit. As shown in the left panel of  Table I. We generate Firstly, for the Z 3 -QCD model, we find hysteresis in the range of 1.55 < β < 1.90 depending on the initial conditions (cold or hot). On the other hand, we find that there are no hysteresis in the data for the standard three-flavor QCD. We note that the hysteresis is a signal of the first order phase transition.
Secondly, in the low-temperature phase, the magnitude of Polyakov loop is exactly zero for The meaning of colors of symbols is the same as the one in Fig. 2. The data for the Z 3 -QCD model has a relatively clear signal of peaks around β = 1.60 and β = 1.90 for cold and hot starts, respectively. The peak appears because of the co-existing states between the two phases associated with the phase transition [28]. On the other hand, the standard three-flavor QCD does not show a clear transition point. Actually, such a situation makes it difficult to determine the critical temperature in the physical QCD [29][30][31].
We also investigate the volume dependence of the critical β and find small finite volume effect, which we will not discuss further in the present paper. The precise determination of the critical temperature in the large volume and continuum limits remains as a future work.
Our results on center symmetric phase transition indicates the following points: (1) Z 3 -QCD model on the lattice possesses exact Z 3 center symmetry.
(2) Z 3 center symmetry is spontaneously broken in the high-temperature phase.
(3) The order of the center phase transition is first.

B. Chiral condensates
Here, we investigate the chiral property in the finite-temperature Z 3 -QCD mode, which is characterized by the chiral condensate. In the chiral limit, the non-vanishing chiral condensate is a signal of the spontaneous chiral symmetry breaking. As we discussed in Sec. II B, because of the twisted boundary condition, the symmetry breaking pattern is expected to On the other hand, as we have discussed in Sec. II C, the result of the chiral model implies that the flavor symmetry breaking due to the flavor-dependent twisted boundary condition gets manifest only in the high-temperature phase [23]. To investigate this conjecture, we will investigate the chiral condensate for each flavor independently.
Since our simulation is performed for heavy mass region m PS /m V = 0.70 and utilize the Wilson fermion, no chiral phase transition takes place. Nevertheless, we expect that, in the system with massive fermions, decrease of chiral condensate indicates effective and approximate restoration of chiral symmetry. Actually, several lattice numerical simulations in these years reveal the chiral property for the realistic N f = 2+1 QCD at finite temperature based on chiral condensate and its susceptibility as the effective order parameter. We note that the recent lattice results using the staggered fermions and domain-wall fermions give the consistent chiral critical temperature [29][30][31] .
We consider the following flavor-diagonal expectation value of the subtracted chiral condensate [32][33][34] for each flavor [35][36][37] (we do not consider flavor-mixing condensate in the present paper.), Here, m P CAC is partially conserved axial current (PCAC) mass, and P denotes the pseudo scalar state defined by P ≡ψ f γ 5 ψ f , for the corresponding flavor (f ) in LHS. The PCAC mass is defined via axial Ward identity; where A µ corresponds to the axial vector current defined by A µ =ψ f γ 5 γ µ ψ f . Here again, the label of flavor (f ) is fixed. The values of m PCAC are summarized in Appendix C.
Before showing our results of numerical simulation for chiral condensates for each flavor, we comment on the notation of flavor in this study. To fix the name of quarks, we firstly observe the complex phase (φ) of Polyakov loop given as L = |L|e iφ for each configuration.
As discussed in the previous section, in the center-symmetric phase the definition of the complex phase is meaningless since |L| = 0, while in the center broken phase we can define the value of φ [46]. Next, we measure the correlators in Eq. (13) using the Dirac operator given in Eq. (9), where the link valuable has the boundary condition given by where θ takes value 0 or ±2π/3. We define the flavor u as in φ + θ = 0 (mod 2π) for u-flavor, while the flavor d and s are defined with the following total phases; φ + θ = 4π/3 (mod 2π) for s-flavor. Figure 4 shows the β dependence of the chiral condensate for u-flavor of the Z 3 -and threeflavor QCD. Circle (red), triangle (blue) and square (black) symbols stand for those data associated with the hot start and cold start in Z 3 -QCD model and the standard three-flavor QCD at finite temperature, respectively. All results have common qualitative properties, where the chiral condensate gradually decreases as β increases. It is also notable that β dependence of chiral condensates in Z 3 -QCD model has hysteresis between the cold and hot starts as with that of the Polyakov loop. These results imply that effective restoration of the chiral symmetry is progressing from β = 1.55 to β = 1.95 for the Z 3 -QCD model. These values of β where the hysteresis exists are the same as those of the center phase transition.
Furthermore, the decreasing rate of chiral condensate in Z 3 -QCD model is relatively larger than the one in the standard three-flavor QCD.
According to the arguments in Refs. [26,27], the partition function for QCD has 2π/3 periodicity in the imaginary chemical potential, and it is expected that the chiral phase transition temperature in our model is the same as that of the standard three-flavor QCD.
On the other hand, since the flavor-chiral symmetry in Z 3 -QCD model in the chiral limit is broken to U(1) 2 L × U(1) 2 R due to the twisted boundary condition, we may have the smaller number of Nambu-Goldstone modes than the usual three-flavor QCD, which lifts the phase transition temperature in general [38]. (As we have discussed, the chiral effective model indicates that the full flavor symmetry is effectively preserved in the low-temperature phase, and in such a case we should have the common number of Nambu-Goldstone modes.) Our results in Fig. 4 indicate qualitatively the same chiral crossover temperature in Z 3 -QCD and three-flavor QCD, while, to be quantitative, the temperature in Z 3 -QCD seems slightly higher than that in the three-flavor QCD. We do not yet have sufficient ingredients to conclude on this question. Higher statistics and investigation of its susceptibility are necessary to determine the critical temperature in the massless limit. It is of well-known difficulty to determine the critical temperature of the chiral phase transition [39].
We also note that the absolute values of the chiral condensates in Z 3 -and three-flavor QCD are different in the low-temperature phase, which may indicate the qualitative difference of the chiral property between the two theories.
Next, we focus on the flavor symmetry breaking in the high-temperature phase. Figure 5 shows the expectation values of chiral condensates for each flavor. Here, circle (red), square (orange) and triangle (violet) symbols denote u-, d-and s-flavor generated with hot start, respectively. Three components of chiral condensate are degenerate in the low-temperature phase. On the other hand, in the high-temperature phase, there appears clear flavor symmetry breaking. Two of them, whose total complex phase (φ+θ) are nontrivial, are degenerated because of the momentum shift of the twisted boundary condition. It indicates that at least the Z 3 center of SU(3) flavor symmetry, which commutes with the Cartan subgroup, is effectively preserved in the low-temperature phase, while the breaking of this symmetry gets manifest in the high-temperature phase. Although our simulation cannot fully verify the conjecture of the chiral model shown in (ii) of Sec. II C, which states that the SU(3) flavor symmetry is not affected by the twisted boundary condition in the low-temperature phase, the above result on the Z 3 flavor symmetry is consistent with this conjecture. On the other hand, Fig. 4 shows the different values of chiral condensates between the Z 3 and three-flavor QCD, thus the two theories seem to have difference in chiral and flavor properties. Further study is required to understand the structure of flavor and chiral symmetry in the Z 3 -QCD model.

V. SUMMARY AND DISCUSSION
In this work we numerically simulate the finite-temperature Z 3 -QCD model on the lattice by introducing flavor-dependent twisted boundary conditions, with emphasis on center phase transition and its influence on the chiral properties. We calculate the finite-temperature expectation value of Polyakov loop and the flavor-diagonal chiral condensates on 16 3 × 4 and 20 3 × 4 lattices on the line of constant physics realizing m P S /m V = 0.70. We find out the first-order center phase transition at a certain temperature with the hysteresis depending on the initial conditions for configuration generation. The chiral crossover transition takes place around the critical temperature of center transition, where it also has hysteresis in the same range as that of the center phase transition. By comparing this approximate chiral restoration to that of the standard three-flavor QCD, we find that the decrease of chiral condensate in Z 3 -QCD is more rapid while temperatures of the two crossover transitions are almost the same. We also obtain the result which supports the manifestation of flavor symmetry breaking due to the boundary condition in the high-temperature phase, which was predicted in the study of the effective chiral model.
One of motivations for this work is study how the chiral condensate reacts to the center phase transition. Our preliminary result indicates not only that the chiral condensate has rapid decrease around the center critical temperature, but also that it has the hysteresis in the same range as that of the center phase transition. Based on this result, we argue that the chiral and center properties at finite temperature have a strong correlation at least in the present model. One possible reason for the correlation is that the first-order center phase transition might work as a trigger to the rapid decrease of chiral condensate. On the other hand, the argument in Ref. [27] implies that the chiral critical temperature in the present model is the same as that of the standard three-flavor QCD, thanks to the 2π/3 periodicity of imaginary chemical potential. Our result on comparison of the two chiral (crossover) transition temperatures is consistent with this conjecture. The precise study on the critical temperature in the Z 3 -QCD model may help determine chiral crossover temperature T c in the standard three-flavor QCD.
By comparing our results to those of the effective chiral model, we have supported the several conjectures based on the chiral model: first-order center phase transition, latency of flavor symmetry breaking in the low-temperature phase, and manifestation of flavor symmetry breaking in the high-temperature phase. In particular, we have observed that Z 3 center part of the SU(3) flavor symmetry seems to be intact in the low-temperature phase, while its breaking becomes manifest in the high temperature phase, accompanying the spontaneous breaking of Z 3 center of the SU(3) color symmetry. This is consistent with the conjecture of the effective chiral model, but is not sufficient to draw a conclusion. On the other hand, we could not find the specific jump of the chiral condensate right at the center critical temperature, which was seen in the chiral model as discussed in (iii) of Sec. II C. Although our simulation is not performed in the chiral limit, we may be able to interpret the jump seen in the chiral model as a model artifact and a remnant of the rapid decreasing of the chiral condensate seen in our simulation since the cut-off effect may be visible at the hightemperature in the effective model. Further study is required to clarify the flavor and chiral properties in the Z 3 -QCD model.
For future works, we address following points.
Towards a chiral limit: By approaching a chiral limit, the Z 3 -QCD model has both exact center and exact chiral symmetries in presence of fundamental quarks. We may be able to determine the both center and chiral critical temperatures using the exact order parameters, Polyakov loop and chiral condensate. Therein, we can compare the two critical temperatures and discuss relation of confining and chiral properties in more details.
Towards smaller lattice and larger volumes: To obtain the critical temperatures, we have to take a continuum limit to remove a lattice artifact and take large volume limit to estimate finite volume effects. Several works for the three-flavor real QCD near the physical points have been progressing, so that in principle it is doable at present. Comparison with the critical temperatures between the Z 3 -QCD and the standard three-flavor QCD in these limits must be interesting for understanding the center and chiral structure.
Towards topological objects (fractional instantons): In the presence of center-symmetric Polyakov loop holonomy, or equivalently the center-symmetric twisted boundary conditions, fundamental topological objects become fractional instantons with fractional topological charge as 1/N with N being the rank of gauge group [40,41]. In our simulations, we expect that 1/3 fractional instantons are present, and have influence on physical observables such as topological susceptibility. It could be valuable to investigate the topological properties in relation to the recent interest in the novel topological objects [42][43][44].
Towards further application of gauge theory with twisted boundary conditions: Our result is of significance in a sense that we first observed the center first-order phase transition in the lattice QCD simulation with dynamical fundamental quarks by introducing the twisted boundary condition (See also Appendix B). It is notable that the only difference between the standard three-flavor QCD and our Z 3 -QCD model is the twisted boundary condition, and this difference disappears in the zero-temperature limit. It means that, by choosing appropriate boundary conditions for quarks, we can realize an ideal situation that the confining/deconfining properties are well described by center symmetry and Polyakov loop.
We consider that such a technique using the twisted boundary conditions helps understand broader topics on nonperturbative aspects of strong-coupling physics.
Then, the action (2) is translated into This is the theory with the flavor-dependent imaginary chemical potentials and the usual anti-periodic boundary conditions Eq. (3).
Appendix B: Comment on the other possible Z 3 symmetric theory In the canonical ensemble, it is known that the Polyakov loop has exact zero expectation value [45], which is similar to that of Z 3 -QCD. Here, we briefly discuss the difference between the canonical ensemble and our Z 3 -QCD model.
The grand canonical partition function can be written by where µ denotes a quark (real) chemical potential. Using Z GC (T, µ), the canonical partition function can be expressed as with the quark number Q.

Appendix C: PCAC mass and PS meson mass
Here, we summarize pseudo-scalar (m PS ), vector (m V ), their ratio (m PS /m V ), and PCAC (m PCAC ) masses at the zero-temperature simulation using 16 4 lattice in Table II and III. Figure 6 shows the hopping parameter dependence of m PCAC and m 2 PS with the fit results using (1/κ − 1/κ c ). For smaller masses, both are proportional to (1/κ − 1/κ c ), and they become zero at almost the same κ up to O(am PCAC ), which are expected behaviors of Wilson type fermion [37]. For the tuning of the mass parameter at m PS /m V = 0.70, we interpolate m PS and m V as a function of (1/κ − 1/κ c ) and determine the line of constant physics.