to a second derivative term as $\sim a\int dx^{4} \bar{\psi}D^{2}_{\mu}\psi$ up to $\mathcal{O}(a^2)$ errors. Thus we can regard them as cousins of Wilson fermion.

There are also non-trivial flavored-mass terms for staggered fermions, which split 4 tastes into branches and satisfy $\gamma_{5}$ hermiticity. Since $\gamma_{5}$ is expressed in spin-taste representation as $\gamma_{5}\otimes\gamma_{5}$ in this case, we only have two flavored-mass terms satisfying $\gamma_{5}$ hermiticity: ${{\bf 1}}\otimes\gamma_{5}$ and ${{\bf 1}}\otimes \sigma_{\mu\nu}$. (For larger discrete symmetry one needs to take a proper sum for $\mu,\nu$ in the latter case.) These spin-flavor representations translate into four- and two-hopping terms in the one-component staggered action up to $\mathcal{O}(a)$ errors. The first type is given by $$M_A= \epsilon\sum_{sym} \eta_{1}\eta_{2}\eta_{3}\eta_{4}
C_{1}C_{2}C_{3}C_{4}
= ({{\bf 1}}\otimes \gamma_{5}) + \mathcal{O}(a)
\ ,
\label{AdamsM}$$ with $$\begin{aligned}
(\epsilon)_{xy}&=(-1)^{x_{1}+...+x_{4}}\delta_{x,y}
\ ,
\\
(\eta_{\mu})_{xy}&=(-1)^{x_{1}+...+x_{\mu-1}}\delta_{x,y}
\ ,
\\
C_{\mu}&=(V_{\mu}+V_{\mu}^{\dag})/2
\ ,  
\\
(V_{\mu})_{xy}&=U_{\mu,x}\delta_{y,x+\mu}
\ .\end{aligned}$$ The second type is given by $$\begin{aligned}
M_H&={i\over{\sqrt{3}}}(\eta_{12}C_{12}+\eta_{34}C_{34}+\eta_{13}C_{13}+\eta_{42}C_{42}
+\eta_{14}C_{14}+\eta_{23}C_{23})
\ ,
\nonumber\\ 
&= [{{\bf 1}}\otimes (\sigma_{12}+\sigma_{34}+\sigma_{13}+\sigma_{42}+\sigma_{14}+\sigma_{23}) ] + \mathcal{O}(a)
\ ,
\label{HoelM}\end{aligned}$$ with $$\begin{aligned}
(\eta_{\mu\nu})_{xy}&=\epsilon_{\mu\nu}\eta_{\mu}\eta_{\nu}\delta_{x,y}
\ ,
\\
(\epsilon_{\mu\nu})_{xy}&=(-1)^{x_{\mu}+x_{\nu}}\delta_{x,y}
\ , 
\\
C_{\mu\nu}&=(C_{\mu}C_{\nu}+C_{\nu}C_{\mu})/2
\ .\end{aligned}$$ We refer to $M_A$ and $M_H$ as the Adams- [@Adams1] and Hoelbling-type [@Hoel], respectively. The former splits the 4 tastes into two branches with 