gauge group is changed. We find some models whose phase structures do not depend on the parameters of the models and also the size of the extra dimension. The final section is devoted to conclusions and discussions. Details of calculations will be given in Appendix.

$SU(N)$ Gauge-Higgs Model
=========================

We study the vacuum structure of an $SU(N)$ gauge-Higgs model with $N_F$ massless fundamental fermions. The Higgs field also belongs to the fundamental representation under $SU(N)$. We take our space-time to be $M^3\otimes S^1$ in order to perform analytic calculations, where $M^3$ and $S^1$ stand for three-dimensional Minkowski space-time and a circle with radius $R$, respectively. Our action is $$S=\int d^3x \int_0^Ldy\left(
-\half{\rm tr}F_{{\hat\mu}{\hat\nu}}F^{{\hat\mu}{\hat\nu}}+
\sum_{I=1}^{N_F}{\bar\psi}_I i\Gamma^{{\hat\mu}}D_{{\hat\mu}}\psi_I
+(D^{{\hat\mu}}\Phi)^{\dagger}D_{{\hat\mu}}\Phi - V(\Phi^{\dagger}, \Phi)\right),$$ where the Higgs potential is given by $$V(\Phi^{\dagger}, \Phi)=-m^2\Phi^{\dagger}\Phi
+{\lambda\over 2}(\Phi^{\dagger}\Phi)^2.
\label{potential}$$ We have used a notation such as $x^{{\hat\mu}}\equiv (x^{\mu}, y)$ and the length of the circumference of $S^1$ by $L=2\pi R$.

As stated in the introduction, there are two kinds of the order parameters for gauge symmetry breaking. One is the component gauge field $A_y$ for the $S^1$ direction, which is related with the Hosotani mechanism, and the mechanism is essentially caused by quantum corrections in the extra dimension. 