Order parameters and color-flavor center symmetry in QCD

TL;DR

This paper presents a theory of N-color QCD with massive quarks that has a Z_{d} color-flavor center symmetry arising from intertwined color center transformations and cyclic flavor permutations, with implications for conformal window studies and dense quark matter.

Abstract

Common lore suggests that $N$-color QCD with massive quarks has no useful order parameters which can be non-trivial 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.

Figures (4)

• Figure 1: Possible phase structures of massless QCD as a function of $x = {n_{\rm f}}/N$. The chiral and CFC symmetry realizations change at some $x = x_{\chi}$ and $x = x_{\rm CFC}$, respectively.
• Figure 2: (Color online.) 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: (Color online.) Contour plots of $V_{\rm f}$ for $N = {n_{\rm f}} = 3$, with BCs \ref{['eq:BCa']}, 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 $L^4 V_{\rm eff}$. 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 $\widehat{V}_{\rm f}$. Results with BCs \ref{['eq:BCb']} are similar.
• Figure 4: (Color online.) Contour plots of $L^4 V_{\rm eff}(\Omega)$ as a function of $\theta_1$ and $\theta_2$, for $N = {n_{\rm f}} = 3$ and $TL = 0$, with BCs \ref{['eq:BCa']}. Shown are four nearby values of $\mu L$, illustrating the existence of quantum oscillations as a function of $\mu L$. Darker colors indicate lower values of $L^4 V_{\rm eff}$, and regions outside the triangle shown are gauge-equivalent to points within the triangle. Dots denote critical points of $V_{\rm f}$. Results with BCs \ref{['eq:BCb']} are similar.