Thus, we obtain that $$\mbox {vacuum configuration}=\left\{\begin{array}{lll}
\mbox{Hosotani phase} &\mbox{for}& \mbar < \mbar_4,\\
\mbox{coexisting phase} &\mbox{for}& \mbar_4 < \mbar < \mbar_3,\\
\mbox{Higgs phase} & \mbox{for}& \mbar_3 < \mbar.
\end{array}\right.
\label{sold}$$ The explicit form of the critical scale $\bar{m}_4$ is given by $$\begin{aligned}
\bar{m}_4 = 
2\pi\sqrt{\frac{1}{2a}\left( b + \sqrt{b^2 - 4ac}\right)} ,
\label{m_4}        \end{aligned}$$ where $$\begin{aligned}
a &=& 48 N^4 (N^2 -3N+3)^2 \biggl( 3(N-1)^3 +
      2N(N^2-3N+3)\frac{t}{16\pi^2} \biggr) ,\label{m_4a}\\
b &=& 24 N^2 (N-1)^2 \biggl( 3(N-1)^3 (N^2-N-1)(3N^2-5N+1)\nonumber\\
   & &\ \    + 2N^3 (2N-3)^2 (N^2-3N+3) \frac{t}{16\pi^2}\biggr) ,
      \label{m_4b}\\
c &=& (N-1)^3 \biggl( -9(N-1)^4 (N^2-N-1)^2
      - 6N(N-1)(N^2-N-1)\nonumber\\
   & &\ \   \times(11N^4-36N^3+33N^2-9)\frac{t}{16\pi^2}
      + 4N^2(N^2-3)^3 \biggl( \frac{t}{16\pi^2}\biggr)^2
      \biggr) . \label{m_4c}\end{aligned}$$ The phase transition at $\mbar=\mbar_3$ is the second order, while that at $\mbar=\mbar_4$ is the first order because the order parameters are not connected continuously.

(iii)  $ t \leq 24\pi^2{{(N-1)^3}\over {N^2(2N-3)}}$

Let us first compare the potential energy of the Hosotani phase with that of the Higgs phase. The scale $\mbar_5$ is the critical scale, at which $\Vb_{Hosotani}=\Vb_{Higgs}$ holds. Then, as shown in the appendix, we obtain that $$\begin{aligned}
\Vb_{Hosotani}< \Vb_{Higgs}\qquad {\rm for}
\qquad \mbar < \mbar_5,\label{hhu}\\
\Vb_{Hosotani}> \Vb_{Higgs}
\qquad {\rm for}\qquad \mbar > \mbar_5,
\label{hhd}\end{aligned}$$ where $$\mbar_5\equiv \left({{(N-1)(N^2-N-1)}\over N^3}
{\pi^2\over{24}}t\right)^{1\over 4}.
\label{regiont}$$ The parameter region 