a non-trivial solution as $$\sigma = \displaystyle \frac{1}{16K^2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\pi = \pm \sqrt{ \displaystyle \frac{1}{16K^2} \left( 1- \displaystyle 
\frac{1}{16K^2} \right) }
\ .
\label{condA}$$ It indicates that parity-broken phase appears in the range of the hopping parameter as $\mid{K}\mid > 1/4$ or equivalently $-16\sqrt{3}-2<M<-16\sqrt{3}+2$.

Next, we discuss the meson mass from a two-point function of the meson operator $\mathcal{S}(0,x)\equiv\langle\mathcal{M}_{0}\mathcal{M}_{x}
\rangle$. From Fig. \[Two-A\] we derive the following equation for two-point functions,

![Feynman diagram for mesonic two-point functions for $ \mathcal{O}(K^{3})$ self-consistent equations of HPE with the Adams fermion.[]{data-label="Two-A"}](HPEwAdams-Mass.eps){height="4cm"}

$$\begin{aligned}
\mathcal{S}(0,x) 
 = &\langle {{\bar{\chi}}}_0^a\chi_0^a {{\bar{\chi}}}_x^b \chi_x^b \rangle
{\nonumber}\\
=&- \delta_{0x} N_c
{\nonumber}\\
 &-K^2 \langle {{\bar{\chi}}}_0^a \chi_0^a {{\bar{\chi}}}_0^c (\eta_{\mu,0})^2 
 \biggl[
   U_{\mu,0}^{cd} \chi_{\hat{\mu}}^d {{\bar{\chi}}}_{\hat{\mu}}^e 
   (U_{\mu,0}^{\dagger})^{ef}+ (U_{\mu,-\hat{\mu}}^{\dagger})^{cd} 
   \chi_{-\hat{\mu}}^d {{\bar{\chi}}}_{-\hat{\mu}}^e U_{\mu,-\hat{\mu}}^{ef}\biggr]
   \chi_0^f {{\bar{\chi}}}_x^b \chi_x^b \rangle
{\nonumber}\\
  & + \left( 2 K r \epsilon \eta_5 \displaystyle \frac{1}{4! \cdot 2^4} \right)^2
   \langle {{\bar{\chi}}}_0^a \chi_0^a {{\bar{\chi}}}_0^c 
   \sum_{\alpha,\beta,\gamma,\delta=\pm} \sum_{\mu \neq \nu \neq \rho \neq \sigma} 
   \biggl[
   \left( \mathcal{W}_{\alpha\mu\beta\nu\gamma\rho\delta\sigma,0}^{(4)} \right)^{cd}
   \chi_{\alpha\hat{\mu}\beta\hat{\nu}\gamma\hat{\rho}\delta\hat{\sigma}}^d 
    {{\bar{\chi}}}_{\alpha\hat{\mu}\beta\hat{\nu}\gamma\hat{\rho}\delta\hat{\sigma}}^e 
{\nonumber}\\
 & \times
\left( \mathcal{W}_{\alpha\mu\beta\nu\gamma\rho\delta\sigma,0}^{(4) \dagger} \right)^{ef}
 \biggr]
 \chi_0^f {{\bar{\chi}}}_x^b \chi_x^b \rangle
\ ,
\label{HPE-AdamsTwo}   \end{aligned}$$

where $\mathcal{W}_{\alpha\mu\beta\nu\gamma\rho\delta\sigma,x}^{(4)}$ is defined in Table. \[Table:U4\] of Appendix \[AdamsEff\]. By integrating out the link variables in the strong-coupling limit, it is simplified as $$\begin{aligned}
\mathcal{S}(0,x)  \equiv
 \langle {{\bar{\chi}}}_0^a\chi_0^a {{\bar{\chi}}}_x^b \chi_x^b \rangle
 = 