intensively discussed in [@Rev; @PdF].

Hopping Parameter Expansion {#sec:HPE}
===========================

In this section we investigate parity-phase structure in lattice QCD with staggered-Wilson fermions in the framework of hopping parameter expansion (HPE) in the strong-coupling regime [@AokiP]. In the hopping parameter expansion, we treat a mass term as a leading action while we perturbatively treat hopping terms. We thus perform expansion by a hopping parameter which essentially corresponds to inverse of a mass parameter. By using self-consistent equations we derive one- and two-point functions, and calculate meson condensates and meson mass for two types of staggered-Wilson fermions. We for simplicity drop the flavor indices until we discuss the two-flavor case in details in Sec. \[sec:Tf\]. However it is easy to recover the flavor indices for the field $\chi_{f}$, the mass parameter $M_{f}$ and the condensate $\Sigma_{f}$ ($f=1,2,...$).

Hoelbling type {#subsec:HPE-H}
--------------

![Feynman rules in hopping parameter expansion (HPE) with the Hoelbling-type staggered-Wilson fermion. $a$ and $b$ stand for the color indices. $\mathcal{W}_{\alpha\mu\beta\nu,x}^{(2)}$ is given in Table \[Table:U2\].[]{data-label="FR-H"}](HPE-Feynman-rule-Hoelbling.eps){height="6cm"}

   $\alpha$   $\beta$                $\mathcal{W}_{\alpha\mu\beta\nu,x}^{(2)}$
  ---------- --------- ---------------------------------------------------------------------
     $+$        $+$                       $U_{\mu,x}U_{\nu,x+\hat{\mu}}$
     $-$        $-$     $U_{\mu,x-\hat{\mu}}^\dagger U_{\nu,x-\hat{\mu}-\hat{\nu}}^\dagger$
     $+$        $-$              $U_{\mu,x}U_{\nu,x+\hat{\mu}-\hat{\nu}}^\dagger$
     $-$        $+$              $U_{\mu,x-\hat{\mu}}^\dagger U_{\nu,x-\hat{\mu}}$

![Feynman diagram for mesonic one-point functions in the $\mathcal{O}(K^{3})$ self-consistent equation of HPE with 