Properties of 2+1-flavor QCD in the imaginary chemical potential region

We study properties of 2+1-flavor QCD in the imaginary chemical potential region. We clarify the conditions imposed on $\mu_{u}=\mu_{d}=\mu_{l}=i\theta_{l}T$ and $\mu_{s}=i\theta_{s}T$ that are needed to realize the Roberge-Weiss (RW) periodicity. The QCD thermodynamic potential exhibits the RW periodicity when $\theta_{l}$ and $\theta_{s}$ are not fixed to constant values, whereas the periodicity is broken if any one of $\theta_{l}$ and $\theta_{s}$ is fixed to a constant value. As a model possessing the same properties as QCD, the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model is introduced in order to visualize the properties mentioned above. By using the PNJL model, we calculate thermodynamic quantities and draw the QCD phase diagram as a function of $\theta_{l}$ for two conditions of $\theta_{s}=\theta_{l}$ and $\theta_{s}=0$. We also consider the cases $(\mu_{u},\mu_{d},\mu_{s})=(i\theta_{u}T,iC_{1}T,0)$ or $(iC_{2}T,iC_{2}T,i\theta_{s}T)$, where $C_{1}$ and $C_{2}$ are constants. For some choices of $C_{1}$ ($C_{2}$), the number density of up (strange) quark becomes smooth as a function of $\theta_{u}$ ($\theta_{s}$). This property is important, since it makes the analytic continuation more feasible.