Here the pion mass becomes zero at $\mid{K}\mid = 1/4$ and becomes tachyonic in the range $\mid{K}\mid > 1/4$. It suggests that there occurs a second-order phase transition between parity-symmetric and broken phases at $|K|=1/4$, which is consistent with Eq. (\[condA\]). We can also derive the sigma meson mass by substituting $p=(i m_\pi a, 0, 0, 0)$ for $1/\mathcal{S}(p)=0$ in Eq. (\[MP2\]) as $$\begin{aligned}
\cosh(m_\sigma a) &= 1 + \displaystyle \frac{1}{6K^2}
\ .
\label{HPE-Asigma}\end{aligned}$$

Effective Potential Analysis {#sec:EPA}
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In the previous section, we have investigated the parity-phase structure in hopping parameter expansion. We found a strong sign of parity-broken phase for $\mid{K}\mid >1/4$. In order to judge whether the parity-broken phase is realized as a vacuum, the analysis of the gap solution in the hopping parameter expansion is not enough, and we need to investigate the effective potential for meson fields.

In this section, we consider the effective potential of meson fields for $SU(N)$ lattice gauge theory with staggered-Wilson fermions. In the strong-coupling limit and the large $N$ limit, effective action can be exactly derived by integrating the link variables [@KaS; @AokiP]. Then, by solving a saddle-point equation, we can investigate a vacuum and find meson condensations. In this 