parity-symmetric and parity-broken phases at $|K|=1/4$, which is consistent with the result from the one-point function in Eq. (\[cond\]). We note that the massless pion at the phase boundary is consistent with the scenario of second-order transition. 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. (\[MP\]) as $$\begin{aligned}
\cosh(m_\sigma a) &= 1 + \displaystyle \frac{1}{2K^2}
\ .
\label{HPE-Hoelsigma}\end{aligned}$$

Adams type {#subsec:HPE-A}
----------

We investigate the parity-phase structure for the Adams-type staggered-fermion by using $\mathcal{O}(K^{3})$ self-consistent equations in hopping parameter expansion. The approach is basically parallel to that of Hoelbling type. We just need to consider Feynman diagrams for this case. The action Eq. (\[AdamsS\]) is rewritten by redefining $\chi \rightarrow \sqrt{2K} \chi$ with $K=1/[2(M+r)]$ as, $$S =
\sum_{x} {{\bar{\chi}}}_x \chi_x + 2K \sum_{x, y} {{\bar{\chi}}}_{x}(\eta_{\mu}  
D_\mu )_{xy}\chi_{y}+ 2 K r \sum_{x,y} {{\bar{\chi}}}_{x} 
(M_{A})_{xy} \chi_{y}
\ ,
\label{HPE-Adams}$$ where $M_{A}$ is given in Eq. (\[AdamsM\]). In Fig. \[FR-A\], the Feynman rules in the HPE for this fermion are depicted.

![Feynman rules for the HPE with the Adams fermion. $a$ and $b$ stand for the color indices. We show the concrete forms of $\mathcal{W}_{\alpha\mu\beta\nu\gamma\rho\delta\sigma,x}^{(4)}$ in Table \[Table:U4\] of Appendix \[AdamsEff\].[]{data-label="FR-A"}](HPE-Feynman-rule-Adams.eps){height="6cm"}

