the renormalization factor does not depend on them.

At the tree level the vertex function takes the form $$\Lambda_\Gamma^{(0)}=\varepsilon^{ijk} \left[ \Gamma_X \otimes \Gamma_Y 
\right]_{\alpha\beta;\delta\gamma},
\label{eq:PD_tree}$$ where $[\Gamma_X \otimes \Gamma_Y]_{\alpha\beta;\delta\gamma} \equiv
(\Gamma_X)_{\alpha\beta}(\Gamma_Y)_{\delta\gamma}$.

The one-loop vertex corrections are shown in Figs. 2a, 2b and 2c, the sum of which gives the one-loop level vertex function $$\Lambda_{PD}^{(1)}=
\int_{-\pi}^{\pi}\frac{d^4 k}{(2\pi)^4} 
\left(I_{PD}^a+I_{PD}^b+I_{PD}^{c}\right).$$ Using the notations in eqs.(\[eq:G\_b\]), (\[eq:G\_t\]) and (\[eq:V\_bt\]) the integrands $I_{PD}^{a,b,c}$ are written as follows: $$\begin{aligned}
I_{PD}^a &=& \varepsilon^{abk}(-T^A)^T_{ia}T^B_{bj} {\nonumber}\\
&&\times \left\{{\overline{V}}_\mu(k) {\overline{G}}(k) 
+ {\widetilde{V}}_\mu (k){\widetilde{G}}(k)\right\}
\Gamma_X
\left\{{\overline{G}}(k) {\overline{V}}_\nu(k) 
+ {\widetilde{G}}(k) {\widetilde{V}}_\nu(k)\right\}
\otimes \Gamma_Y
G_{\mu\nu}^{AB}(k), \\
I_{PD}^b &=& \varepsilon^{ibc}T^A_{bj}T^B_{ck} {\nonumber}\\
&&\times \Gamma_X\left\{{\overline{G}}(k) {\overline{V}}_\nu(k) 
+ {\widetilde{G}}(k) {\widetilde{V}}_\nu(k)\right\}
\otimes\Gamma_Y\left\{{\overline{G}}(-k) {\overline{V}}_\nu(-k) 
+ {\widetilde{G}}(-k) {\widetilde{V}}_\nu(-k)\right\}
G_{\mu\nu}^{AB}(k), \\
I_{PD}^c &=& \varepsilon^{ajc}(-T^A)^T_{ia}T^B_{ck} {\nonumber}\\
&&\times \left\{{\overline{V}}_\mu(k) {\overline{G}}(k) + {\widetilde{V}}_\mu (k)
{\widetilde{G}}(k)\right\}
\Gamma_X\otimes
\Gamma_Y
\left\{{\overline{G}}(k){\overline{V}}_\nu(k)  
+ {\widetilde{G}}(k){\widetilde{V}}_\nu(k) \right\}
G_{\mu\nu}^{AB}(k).\end{aligned}$$ A little algebra yields $$\begin{aligned}
I_{PD}^a &=& g^2\frac{N+1}{2N}\varepsilon^{ijk} 
K \left[ T + A_{SP}\right]\left[\Gamma_X\otimes\Gamma_Y\right] , \\
I_{PD}^b &=& g^2\frac{N+1}{2N}\varepsilon^{ijk}  
K \left[T (\Gamma_X\otimes\Gamma_Y) +
\cos^2 (k_\mu/2) \sin^2 k_\alpha  (\Gamma_X\gamma_\alpha\gamma_\mu)
\otimes (\Gamma_Y\gamma_\alpha\gamma_\mu) \right], \\
I_{PD}^c &=& g^2\frac{N+1}{2N}\varepsilon^{ijk} 
K \left[T (\Gamma_X \otimes\Gamma_Y) +
\cos^2 (k_\mu/2) \sin^2 k_\alpha (\gamma_\mu\gamma_\alpha\Gamma_X)
\otimes (\Gamma_Y\gamma_\alpha\gamma_\mu) \right],\end{aligned}$$ where $K$, $T$ and $A_{SP}$ are given in eqs.(\[eq:K\]), (\[eq:T\]) and (\[eq:A\_SP\]). It is noted that a sum of $I_{PD}^b$ and $I_{PD}^c$ becomes $$\begin{aligned}
& 