Roberge-Weiss transitions at different center symmetry breaking patterns in a ${\mathbb{Z}}_3$-QCD model

Abstract

We study how the Roberge-Weiss (RW) transition depends on the pattern of center symmetry breaking using a ${\mathbb{Z}}_3$-QCD model. We adopt flavor-dependent quark imaginary chemical potentials, namely $({\mu }_u,{\mu }_d,{\mu }_s)/iT=(\theta -2\pi C/3,\theta ,\theta +2\pi C/3)$ with $C\in [0,1]$. The RW periodicity is guaranteed and the center symmetry of ${\mathbb{Z}}_3$-QCD is explicitly broken when $C\ne 1$ or/and quark masses are non-degenerate. For $N_f=3$ and $C\ne 1$, the RW transition occurs at $\theta ={\theta }_{RW}=(2k+1)\pi /3(k\in \mathbb{Z})$, which becomes stronger with decrease of $C$. When $C=1$, the ${\theta }_{RW}$ turns into $2k\pi /3$ for $N_f=2+1$, but keeps $(2k+1)\pi /3$ for $N_f=1+2$; in both cases, the RW transitions get stronger with the mass mismatch. For other $C\ne 0$ cases, the ${\theta }_{RW}$'s are not integral multiples of $\pi /3$. We find that the RW transition is more sensitive to the deviation of $C$ from one compared to the mass non-degeneracy and thus the strength of the traditional RW transition with $C=0$ is the strongest. The nature of RW endpoints and its implications to deconfinement transition are investigated.

Comments: 12 pages, 12 figures; rewritten version; some wrong conclusions and interpretations corrected; conferences added