&g^2\frac{N+1}{2N}\varepsilon^{ijk} 
K \left[2T (\Gamma_X \otimes\Gamma_Y) +
\cos^2 (k_\mu/2) \sin^2 k_\alpha (\gamma_\mu\gamma_\alpha\Gamma_X+
\Gamma_X\gamma_\alpha\gamma_\mu)
\otimes (\Gamma_Y\gamma_\alpha\gamma_\mu) \right] \nonumber \\
& = &
g^2\frac{N+1}{2N}\varepsilon^{ijk} 
K \left[2T (\Gamma_X \otimes\Gamma_Y) +
\cos^2 (k_\mu/2) \sin^2 k_\alpha (\{\gamma_\mu\gamma_\alpha
+\gamma_\alpha\gamma_\mu \}\Gamma_X)
\otimes (\Gamma_Y\gamma_\alpha\gamma_\mu) \right ]\end{aligned}$$ for $\Gamma_X = P_R$ or $P_L$, therefore no Fierz transformation is necessary to simplify the spinor structure of the total contribution. Finally we obtain $$\begin{aligned}
I_{PD}^a+I_{PD}^b+I_{PD}^c &=& g^2\frac{N+1}{2N}\varepsilon^{ijk}
K \left[\Gamma_X\otimes\Gamma_Y\right] 
\left[ 3T + A_{SP} + 2A_{VA} \right].\end{aligned}$$ Compared with the tree level result of eq.(\[eq:PD\_tree\]) we find that the vertex correction is multiplicative up to the one-loop level: $$\Lambda_{PD} =
\left[ 1 + g^2 \frac{N+1}{2N}\left\{ 
3<T> + <A_{SP}> + 2<A_{VA}> \right\}\right]
\Lambda_{PD}^{(0)},$$ where $<X>$ $(X = T, A_{VA}, A_{SP})$ are defined in eq.(\[eq:integ\_X\]). We remark that in the Wilson case ${\cal O}_{PD}$ mixes with other operators which have different chiral structures under renormalization[@pt_w3].

Taking account of the contribution of the wave function the lattice renormalization factor for ${\cal O}_{PD}$ is expressed as $$Z_{PD}^{lat} = (1-w_0^2)^{3/2}Z_w^{3/2} Z_2^{3/2}V_{PD},$$ where $$\begin{aligned}
V_{PD} &=& 1+ \frac{g^2}{16\pi^2}\left[-\delta_{PD}\log (\lambda a)^2 +
v_{PD}\right],\end{aligned}$$ $$\begin{aligned}
v_{PD} &=& \frac{16\pi^2 (N+1)}{2N}\left[ 3 <T> + <<A_{SP}>> +
2 <<A_{VA}>> \right]+\delta_{PD}\log\pi^2\end{aligned}$$ with $\delta_{PD}=\displaystyle\frac{6(N+1)}{2N}$. Numerical values for $v_{PD}$, evaluated as before, are given in Table \[tab:4fermi\] 