C_F + A_{VA}\frac{N}{2}-A_{SP}\frac{1}{2N} \right].\end{aligned}$$ Using the tree level result in eq.(\[eq:lambda\_1\_0\]) the vertex function up to the one-loop level is expressed as $$\begin{aligned}
\Lambda_1 &=&
\left[ 1 + g^2 \frac{1}{N}\left\{ 
<T> (N^2-1)+<A_{VA}>N^2-<A_{SP}>\right\}\right]
\Lambda_1^{(0)}.\end{aligned}$$ This result shows that the operator ${\cal O}_1$ is multiplicatively renormalizable in DWQCD, which is in contrast with the Wilson case[@pt_w4].

In a similar way we write the vertex corrections for ${\cal O}_2$. $$\begin{aligned}
I_2^a &=&  \frac{1}{2} g^2 K (T+A_{VA})
          \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] 
          \left[ \frac{1}{2} 1{\widetilde{\otimes}}1
                 - \frac{1}{2N} 1{\widetilde{\odot}}1 \right], \\
I_2^b &=& - \frac{1}{2} g^2 K (T+A_{VA})
          \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] 
          \left[ \frac{1}{2} 1{\widetilde{\otimes}}1
                 - \frac{1}{2N} 1{\widetilde{\odot}}1 \right], \\
I_2^c &=& \frac{1}{2} g^2 K (T+A_{SP})
          \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] 
          \left[ C_F 1{\widetilde{\odot}}1 \right]. \end{aligned}$$ The total contribution including those from Fig. $1a^\prime$, $1b^\prime$ and $1c^\prime$ becomes $$\begin{aligned}
2(I_2^a+I_2^b+I_2^c) &=& 2\frac{1}{2}g^2 
K \left[\gamma_\nu^L\otimes\gamma_\nu^R\right]_{\alpha\beta;\gamma\delta} 
\left[1{\widetilde{\odot}} 1 \right]^{ij;kl} \nonumber \\
&\times &
C_F\left[ T + A_{SP}\right],\end{aligned}$$ which leads to $$\begin{aligned}
\Lambda_2 &=&
\left[ 1 + g^2 \frac{N^2-1}{N}\left\{ <T> +<A_{SP}>\right\}\right]
\Lambda_2^{(0)}.\end{aligned}$$ We again find that the vertex correction is multiplicative up to the one-loop level as opposed to the Wilson case[@pt_w4].

The contribution from the fermion self-energy has already been evaluated [@Aoki-Taniguchi; @AIKT98] and the total lattice renormalization factor is now obtained: $$Z_\Gamma^{lat} = (1-w_0^2)^2 Z_w^2 Z_2^2 V_\Gamma,$$ 