Order Parameters and Color-Flavor Center Symmetry in QCD

Common lore suggests that $N$-color QCD with massive quarks has no useful order parameters that can be nontrivial at zero baryon density. However, such order parameters do exist when there are $n_f$ quark flavors with a common mass and $d\equiv gcd(n_f,N)>1$. These theories have a ${\mathbb{Z}}_d$ color-flavor center symmetry arising from intertwined color center transformations and cyclic flavor permutations. The symmetry realization depends on the temperature, baryon chemical potential, and value of $n_f/N$, with implications for conformal window studies and dense quark matter.

Figure 1

Possible phase structures of massless QCD as a function of $x=n_f/N$. The chiral and CFC symmetry realizations change at some $x=x_{\chi }$ and $x=x_{\mathrm{CFC}}$, respectively.

Figure 2

Sketch of a possible phase diagram of circle-compactified $SU(3{)}_V$ symmetric QCD at $m_q>0$, as a function of $T$ and $\mu$, in the large $L$ limit.

Figure 3

Contour plots of $V_f$ for $N=n_f=3$, with BCs (4), as a function of ${\theta }_1$, ${\theta }_2$ for two nearby values of $\mu L$ with $T/\mu ={10}^{-3}$, illustrating the quantum oscillations described in the text. Darker colors indicate lower values of $L^4{V}_{\mathrm{eff}}$. Regions outside the triangle shown are gauge equivalent to points within the triangle. The center-symmetric point $({\theta }_1,{\theta }_2,{\theta }_3)=(0,2\pi /3,4\pi /3)$ lies at the center of the triangle while the corners are the coinciding eigenvalue points (0,0,0) and $\pm (2\pi /3,2\pi /3,2\pi /3)$. Dots denote critical points of ${\widehat{V}}_f$. Results with BCs (4b) are similar.