derive the following $\mathcal{O}(K^{3})$ equation for a two-point function.

![Feynman diagram for mesonic two-point functions for $\mathcal{O}(K^{3})$ self-consistent equation with the Hoelbling fermion. []{data-label="Two-H"}](HPEwHoelbling-Mass.eps){height="4cm"}

$$\begin{aligned}
\mathcal{S}(0,x) 
 = &\langle {{\bar{\chi}}}_0^a\chi_0^a {{\bar{\chi}}}_x^b \chi_x^b \rangle
{\nonumber}\\
 = &- \delta_{0x} N_c
{\nonumber}\\
 &-K^2 \langle {{\bar{\chi}}}_0^a \chi_0^a {{\bar{\chi}}}_0^c (\eta_{\mu,0})^2 
 \biggl[
   U_{\mu,0}^{cd} \chi_{\hat{\mu}}^d {{\bar{\chi}}}_{\hat{\mu}}^e 
   (U_{\mu,0}^{\dagger})^{ef}
 + (U_{\mu,-\hat{\mu}}^{\dagger})^{cd} 
   \chi_{-\hat{\mu}}^d 
 {{\bar{\chi}}}_{-\hat{\mu}}^e U_{\mu,-\hat{\mu}}^{ef}
 \biggr]
   \chi_0^f {{\bar{\chi}}}_x^b \chi_x^b \rangle
{\nonumber}\\
 & - \left( 2 K r i \displaystyle \frac{1}{2^3 \sqrt{3}} \right)^2
   \langle {{\bar{\chi}}}_0^a \chi_0^a {{\bar{\chi}}}_0^c \sum_{\alpha,\beta=\pm} \sum_{\mu \neq \nu} (\eta_{\mu\nu,0})^2 
   \biggl[
    (\mathcal{W}_{\alpha\mu\beta\nu,0}^{(2)})^{cd}
    \chi_{\alpha\hat{\mu}\beta\hat{\nu}}^d {{\bar{\chi}}}_{\alpha\hat{\mu}\beta\hat{\nu}}^e 
    (\mathcal{W}_{\alpha\mu\beta\nu,0}^{(2) \dagger})^{ef}
   \biggr]
   \chi_0^f {{\bar{\chi}}}_x^b \chi_x^b \rangle 
\ ,
\label{HPE-HoelTwo}   \end{aligned}$$

where $\mathcal{W}_{\alpha\mu\beta\nu,0}^{(2)}$ is defined in Table. \[Table:U2\]. Note that we consider only connected diagrams in Fig. \[Two-H\]. By integrating out the link variables in the strong-coupling limit, it is simplified as $$\begin{aligned}
\mathcal{S}(0,x)  \equiv
 \langle {{\bar{\chi}}}_0^a\chi_0^a {{\bar{\chi}}}_x^b \chi_x^b \rangle
 = - \delta_{0x} N_c &+ K^2 \sum_{\pm \mu} \langle 
 \chi_{\hat{\mu}}^a {{\bar{\chi}}}_{\hat{\mu}}^a 
  {{\bar{\chi}}}_x^b \chi_x^b \rangle
{\nonumber}\\
 & + \left( 2 K r i \displaystyle \frac{1}{2^3 \sqrt{3}} \right)^2
  \sum_{\substack{\pm \mu,\pm \nu \\ (\mu \neq \nu)}} 
  \langle \chi_{\hat{\mu}+\hat{\nu}}^a {{\bar{\chi}}}_{\hat{\mu}+\hat{\nu}}^a 
  {{\bar{\chi}}}_x^b \chi_x^b \rangle 
\ .
\label{Eq:HPE-Hoelbling2}\end{aligned}$$ 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} 