&- \delta_{0x} N_c + K^2 
  \sum_{\pm \mu} \langle \chi_{\hat{\mu}}^a {{\bar{\chi}}}_{\hat{\mu}}^a 
  {{\bar{\chi}}}_x^b \chi_x^b \rangle
{\nonumber}\\
 & - \left( 2 K r \displaystyle \frac{1}{4! \cdot 2^4} \right)^2
  \sum_{\substack{\pm \mu, \pm \nu, \pm \rho, 
  \pm \sigma \\ (\mu \neq \nu \neq \rho \neq \sigma)}} 
  \langle \chi_{\hat{\mu}+\hat{\nu}+\hat{\rho}
  +\hat{\sigma}}^a {{\bar{\chi}}}_{\hat{\mu}+\hat{\nu}+\hat{\rho}
  +\hat{\sigma}}^a {{\bar{\chi}}}_x^b \chi_x^b \rangle 
\ .
\label{Eq:HPE-Adams2}\end{aligned}$$ Then the self-consistent equation for $\mathcal{S}$ is given in the momentum space as $$\begin{aligned}
\mathcal{S}(p) &=
- N_c + 
 \biggl[- K^2 \sum_\mu \left( e^{-ip_\mu} + e^{ip_\mu} \right)
{\nonumber}\\  
 &+ 
  \left( 2 K r \displaystyle \frac{1}{4! \cdot 2^4} \right)^2
    \sum_{\alpha,\beta,\gamma,\delta=\pm} \sum_{\mu \neq \nu \neq \rho \neq \sigma} 
   e^{+i(\alpha p_\mu + \beta p_\nu + \gamma p_\rho + \delta p_\sigma)} 
 \biggr] 
\ .
\label{HPE-AdamsSelf2}\end{aligned}$$ We finally obtain the meson propagator as $$S(p) = N_c
\biggl[ - 2 K^2 \sum_\mu \cos p_\mu 
+ 16 \left( 2 K r \displaystyle \frac{1}{4! \cdot 2^4} \right)^2
 \sum_{\mu \neq \nu \neq \rho \neq \sigma} 
 \cos p_\mu \cos p_\nu \cos p_\rho \cos p_\sigma- 1 \biggr]^{-1}
\ .
\label{MP2}$$ Here we set $p=(i m_\pi a + \pi, \pi,\pi,\pi)$ for $1/\mathcal{S}(p)=0$ in Eq. (\[MP2\]), which gives the pion mass $m_\pi$ as $$\begin{aligned}
\cosh(m_\pi a) &= 1 + \displaystyle \frac{1-16K^2}{10K^2}
\ ,
\label{HPE-Adamspi}\end{aligned}$$ where we again set $r=16\sqrt{3}$ for simplicity. 