+ e^{ip_\mu} \right)
{\nonumber}\\  
  &+ \left( 2 K r \displaystyle \frac{1}{2^3 \sqrt{3}} \right)^2
  \sum_{\mu \neq \nu} \left( e^{-i(p_\mu+p_\nu)} + e^{i(p_\mu+p_\nu)}
   + e^{-i(p_\mu-p_\nu)} + e^{i(p_\mu-p_\nu)} \right)\biggr] \mathcal{S}(p)
\ .
\label{HPE-HoelSelf2}\end{aligned}$$ We finally obtain the meson propagator as $$\mathcal{S}(p) = N_c \biggl[ - 2 K^2 \sum_\mu \cos p_\mu 
+ 4 \left( 2 K r \displaystyle \frac{1}{2^3 \sqrt{3}} \right)^2
 \sum_{\mu \neq \nu} \cos p_\mu \cos p_\nu - 1 \biggr]^{-1}
\ .
\label{MP}$$ Here the pole of $\mathcal{S}(p)$ should give meson mass. Since $\chi$ is an one-component fermion, it may seem to be difficult to find the pion excitation from Eq. (\[MP\]). However, as we discussed, $\gamma_{5}$ in the staggered fermion is given by $\epsilon_{x}=(-1)^{x_{1}+...+x_{4}}$ and the pion operator is given by $\pi_{x}=\bar{\chi}_{x}i\epsilon_{x}\chi_{x}$. We therefore identify momentum of pion by measuring it from a shifted origin $p=(\pi,\pi,\pi,\pi)$. Here we set $p=(i m_\pi a + \pi, \pi,\pi,\pi)$ for $1/\mathcal{S}(p)=0$ in Eq. (\[MP\]). Then we derive the pion mass $m_\pi$ as $$\begin{aligned}
\cosh(m_\pi a) &= 1 + \displaystyle \frac{1-16K^2}{6K^2}
\ ,
\label{HPE-Hoelpi}\end{aligned}$$ where we again set $r=2\sqrt{2}$ for simplicity. In this result, the pion mass becomes zero at $|K|=1/4$, and tachyonic in the range $\mid{K}\mid > 1/4$. It implies that there occurs a phase transition between 