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Lattice study on QCD-like theory with exact center symmetry

Takumi Iritani, Etsuko Itou, Tatsuhiro Misumi

TL;DR

We address how center symmetry and chiral dynamics relate in a QCD-like theory by implementing exact $Z_{3}$ center symmetry on the lattice through flavor-dependent twisted temporal boundary conditions for three fundamental quarks in $SU(3)$. Our lattice study, conducted along the line $m_{PS}/m_V=0.70$ on $16^3 imes 4$ and $20^3 imes 4$ lattices, reveals a first-order center transition with hysteresis in the Polyakov loop and a concurrent, hysteretic decrease of flavor-diagonal chiral condensates, with flavor-symmetry breaking manifesting in the high-temperature phase. These results align with predictions from PNJL-type effective models and the $2\,pi/3$ periodicity associated with imaginary chemical potential, providing a nonperturbative link between center and chiral properties and offering a framework for further exploration of twisted-boundary gauge theories. The findings have potential implications for understanding the connections between confinement and chiral dynamics in QCD-like theories and motivate future studies toward the chiral limit, larger volumes, and topological aspects like fractional instantons.

Abstract

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.

Lattice study on QCD-like theory with exact center symmetry

TL;DR

We address how center symmetry and chiral dynamics relate in a QCD-like theory by implementing exact center symmetry on the lattice through flavor-dependent twisted temporal boundary conditions for three fundamental quarks in . Our lattice study, conducted along the line on and lattices, reveals a first-order center transition with hysteresis in the Polyakov loop and a concurrent, hysteretic decrease of flavor-diagonal chiral condensates, with flavor-symmetry breaking manifesting in the high-temperature phase. These results align with predictions from PNJL-type effective models and the periodicity associated with imaginary chemical potential, providing a nonperturbative link between center and chiral properties and offering a framework for further exploration of twisted-boundary gauge theories. The findings have potential implications for understanding the connections between confinement and chiral dynamics in QCD-like theories and motivate future studies toward the chiral limit, larger volumes, and topological aspects like fractional instantons.

Abstract

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 gauge theory with three fundamental Wilson quarks by twisting quark boundary conditions in a compact direction (-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 . 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.

Paper Structure

This paper contains 13 sections, 27 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. $Z_3$-QCD model
  3. Formulation as a continuum theory
  4. Chiral structure and flavor symmetry in $Z_3$-QCD model
  5. Expected properties from chiral model and QCD with a finite chemical potential
  6. Simulation setup
  7. Simulation results
  8. Polyakov loops and center symmetry
  9. Chiral condensates
  10. Summary and Discussion
  11. The flavor-dependent b.c. as imaginary chemical potential
  12. Comment on the other possible $Z_3$ symmetric theory
  13. PCAC mass and PS meson mass

Figures (6)

  • Figure 1: Polyakov loop distribution plot in $Z_{3}$-QCD (left) and the standard three-flavor QCD (right). Based on $16^3 \times 4$ lattice for $\beta=1.70,2.00,2.20$ with the same values of $\kappa$ in both panels.
  • Figure 2: $\beta$ dependence of the magnitude of Polyakov loop ($\langle |L| \rangle$) for the $Z_{3}$-QCD and standard three-flavor QCD on $16^3 \times 4$ (left) and $20^3\times 4$ (right) lattices. For the $Z_3$-QCD model, the data of $\langle |L| \rangle$ started with the cold start (triangle (blue) symbols) have a clear jump from zero to non-zero values around the region $1.55 \le \beta \le 1.60$ in both panels, while the jump occurs in $1.80 \le \beta \le 1.90$ (left) and $1.90 \le \beta \le 1.95$ (right) for the data generated by the hot start (circle (red) symbols). In the regions between these two jumps, the hysteresis exists in $Z_3$-QCD model. On the other hand, the data of the standard three-flavor ($N_f=3$) QCD (square (black) symbols) do not show such a jump from zero to nonzero nor hysteresis.
  • Figure 3: Polyakov loop susceptibility $\chi_{\langle | L | \rangle}$ as a function of $\beta$ on $20^3 \times 4$ lattices. Circle (red) and triangle (blue) symbols denote the data generated with hot and cold starts in $Z_3$-QCD model, respectively. Square (black) symbol shows the data for standard three-flavor ($N_f=3$) QCD model.
  • Figure 4: $\beta$ dependence of the expectation values of subtracted chiral condensates $\langle \bar{u} u \rangle_{\hbox{subt.}}$ for $Z_{3}$-QCD and three-flavor QCD on $20^3 \times 4$ lattices. 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, respectively. The error bar denotes the statistical error coming from the pseudo-scalar correlator in Eq.(\ref{['eq:def-chiral']}).
  • Figure 5: $\beta$ and flavor dependences of the expectation values of subtracted chiral condensates $\langle\bar{q}q \rangle$ for each flavor in $Z_{3}$-QCD model. The lattice size is $20^3 \times 4$. Circle (red), square (orange) and triangle (violet) symbols denote $u$-, $d$- and $s$-flavor generated with hot start, respectively.
  • ...and 1 more figures