\label{regionofcoexist1}\\
\mbar_1 \leq \mbar \le \mbar_3 && \quad{\rm for}\quad
t < 48\pi^2\frac{(N-1)^3}{N(N^2-3)} ,
\label{regionofcoexist2}\end{aligned}$$ where $\bar{m}_1$ is the critical scale above which the realty condition is furnished and is given by Eq.(\[m\_1\]) in the appendix.

Now, we are ready to determine the phase structure of the model. As shown in Fig.1, the lines $\bar{m}_i (i=1,2,3)$ divide the $\bar{m}$-$t$ plane into the several regions. Some region allows only one phase, which is nothing but the vacuum configuration. There are, however, overlapping regions in which two of the three phases remain as candidates of the vacuum configuration. In this case, one has to determine which phase gives the lowest energy among them. Fig.1 will help us understand the phase structure of the model.

Since $\bar{m}_1=\bar{m}_2$ at $t=t_1\equiv 
48\pi^2\frac{(N-1)^3}{N(N^2-3)}$ and $\bar{m}_2=\bar{m}_3$ at $t=t_2\equiv 24\pi^2\frac{(N-1)^3}{N^2(2N-3)}$ (no other intersections of the curves $\bar{m}_i$ for $t>0$), it is convenient to consider separately the three parameter regions of $t$: $$\begin{aligned}
48\pi^2{{(N-1)^3}\over{N(N^2-3)}} < & t, &
\hspace{3.9cm}(\mbar_2 < \mbar_3) ,\label{cou}\\
24\pi^2{{(N-1)^3}\over {N^2(2N-3)}}
< &t & \leq  48\pi^2{{(N-1)^3}\over{N(N^2-3)}},~~~
(\mbar_1 \leq \mbar_2 < \mbar_3),\label{cod}\\
& t & \leq 24\pi^2{{(N-1)^3}\over {N^2(2N-3)}},~~
(\mbar_1 < \mbar_3 \leq \mbar_2).\label{cot}\end{aligned}$$ Here, the relative magnitude of $\bar{m}_i$ for each parameter region of $t$ is shown in 