The other one is the Higgs field, and the Higgs mechanism works even at the tree-level. Taking the order parameters into account, we study the effective potential for $\vev{A_y}$ and $\vev{\Phi}$ parametrized by $$gL \vev{A_y}={\rm diag}(\theta_1, \theta_2, \cdots, \theta_N),~~
\vev{\Phi}={1\over\sqrt{2}}(v, 0, \cdots, 0)^T,
\label{vev}$$ where $\sum_{i=1}^{N}\theta_i=0$ and $v$ is a real constant. Here, we have arranged $\theta_i$ in such a way that $\abs{\hat{\theta}_1}\leq \abs{\hat{\theta}_2}\leq
\cdots\leq\abs{\hat{\theta}_{N}}$, where $\hat{\theta}_i = \theta_i$ mod $2\pi$ with $\hat{\theta}_i \le \pi$. This can be done without loss of generality. We showed in the appendix that the parametrization for the vacuum expectation value (VEV) of the Higgs field given in Eq.(\[vev\]) is enough to study the vacuum structure of the model. We assume $N_F$ is so large that the leading order correction comes from the fermion one-loop correction to $\vev{A_y}$ alone. Then, the effective potential is given by $$\begin{aligned}
V &= &-\half m^2 v^2+
{\lambda\over 8}v^4+
{{\hat{\theta}_1^2v^2}\over {2L^2}}
+{A\over {\pi^2 L^4}}\sum_{n=1}^{\infty}\sum_{i=1}^{N}{1\over n^4}
\cos(n\theta_i)\label{effpotu}\\
&=&{1\over L^4}
\left(-\half \mbar^2 \vbar^2+
{\lambda\over 8}\vbar^4+
\half\hat{\theta}_1^2 \vbar^2
+{A\over {\pi^2}}\sum_{n=1}^{\infty}\sum_{i=1}^{N}{1\over n^4}
\cos(n\theta_i)\right)\equiv \Vb L^{-4},
\label{effpotd}\end{aligned}$$ where $A\equiv 2^2 N_{F}$ and the number $2^2$ counts the physical degrees of freedom of a Dirac fermion. Here, we have introduced the dimensionless quantities, $\mbar\equiv mL, \vbar\equiv 