X(k)
\label{eq:integ_X}$$ for $X = T, A_{VA}, A_{SP}$. We remark that $C_F< T+A_{VA} >$ and $C_F< T+A_{SP} >$ correspond to the one-loop vertex corrections to the (axial) vector current and the (pseudo) scalar density which are expressed as $(T_{VA}-1)$ and $(T_{SP}-1)$ in Ref. [@AIKT98]. The expressions of eqs.(\[eq:1loop\_+\]) and (\[eq:1loop\_-\]) show an important property of ${\cal O}_{\pm}$ in the DWQCD formalism: the one-loop vertex corrections are multiplicative. This is contrary to the Wilson case, in which the mixing operators with different chiralities appears at the one-loop level[@pt_w4].

We next turn to the case of ${\cal O}_{1,2}$. For ${\cal O}_1$ the vertex corrections of eqs.(\[eq:vtx\_a\]), (\[eq:vtx\_b\]) and (\[eq:vtx\_c\]) with $\Gamma_X = \gamma_\mu^L$ and $\Gamma_Y = \gamma_\mu^R$ are written as $$\begin{aligned}
I_1^a &=&  \frac{1}{2} g^2 K (T+A_{VA})
          \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] 
          \left[ (-N C_F+ \frac{1}{2}) 1{\widetilde{\otimes}}1
                 - \frac{1}{2N} 1{\widetilde{\odot}}1 \right], \\
I_1^b &=& - \frac{1}{2} g^2 K (T+A_{VA})
          \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] 
          \left[ (\frac{1}{2}+ \frac{1}{2}) 1{\widetilde{\otimes}}1
                 +(-\frac{N}{2}- \frac{1}{2N}) 1{\widetilde{\odot}}1 \right], \\
I_1^c &=&  \frac{1}{2} g^2 K (T+A_{SP})
          \left[\gamma_\nu^L\otimes\gamma_\nu^R\right]
          \left[ \frac{1}{2} 1{\widetilde{\otimes}}1
                 +(-\frac{N}{2}+ C_F) 1{\widetilde{\odot}}1 \right]. \end{aligned}$$ The total contribution including those from Fig. $1a^\prime$, $1b^\prime$ and $1c^\prime$ is given by $$\begin{aligned}
2(I_1^a+I_1^b+I_1^c) &=& 2\frac{1}{2} g^2 
K \left[\gamma_\nu^L\otimes\gamma_\nu^R\right]_{\alpha\beta;\gamma\delta} 
\left[- N 1{\widetilde{\otimes}} 1 +1{\widetilde{\odot}} 1\right]^{ij;kl} \nonumber
\\&\times &
\left[ T 