Roberge-Weiss Transition in the Polyakov-loop Extended Quark-meson Model

${\mathbb{Z}}_3$-QCD is a QCD-like theory with strict center symmetry. We use the Polyakov-loop extended quark meson model (PQM) as a low-energy effective theory of ${\mathbb{Z}}_3$-QCD to study the RW transitions in different center symmetry breaking patterns. The flavor-dependent imaginary chemical potentials, namely $(\mu_{\rm{u}},\,\mu_{\rm{d}},\,\mu_{\rm{s}})= {\rm{i}}T(\theta-2C\pi/3,\theta,\theta+2C\pi/3)$ are adopted, which guarantees the RW periodicity. The traditional and quark improved Polyakov-loop potentials are used, respectively. For $N_{\rm{f}}\!=\!3$ with $C\!\ne\!1$, the RW transition occurs at $\theta\!=\!\pi/3$ (mod $2\pi/3$), which gets stronger when $C$ declines from one to zero. When $C\!=\!1$, the RW transition happens at $\theta\!=\!2\pi/3$ (mod $2\pi/3$) for $N_{\rm{f}}\!=\!2+1$, but $\theta\!=\!\pi/3$ (mod $2\pi/3$) for $N_{\rm{f}}\!=\!1+2$. We find that all RW transition endpoints are triple points when $C\!=\!1$. We confirm that the RW transition becomes weaker and the deconfinement temperature gets lower when taking into account the quark back-reaction effect. However, the modification of the gluon sector due to the quark effect does not change the main conclusions mentioned above.