# Order parameters and color-flavor center symmetry in QCD

## TL;DR

This paper reveals the existence of non-trivial order parameters in N-color QCD with multiple flavors, linked to a color-flavor center symmetry, affecting the understanding of phase structure at various conditions.

## Contribution

It introduces the concept of a $ ext{Z}_d$ color-flavor center symmetry in QCD with multiple flavors and explores its implications for phase transitions and symmetry realization.

## Key findings

- Existence of $ ext{Z}_d$ symmetry in certain QCD theories.
- Dependence of symmetry realization on temperature and chemical potential.
- Implications for conformal window and dense quark matter studies.

## 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.