{\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}$$ After a little algebra the expressions of $I_\Gamma^{a,b,c}$ are reduced to $$\begin{aligned}
I_\Gamma^a &=&  \frac{1}{2} g^2 J_a^{AA} 
K \left[ T + A_{VA}\right]\left[\Gamma_X\otimes\Gamma_Y\right], 
\label{eq:vtx_a}\\
I_\Gamma^b &=& - \frac{1}{2} g^2 J_b^{AA} 
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_\mu\gamma_\alpha\Gamma_Y) \right], 
\label{eq:vtx_b}\\
I_\Gamma^c &=&  \frac{1}{2} g^2 J_c^{AA} 
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],
\label{eq:vtx_c}\end{aligned}$$ where $$\begin{aligned}
&&
K =\displaystyle \frac{1}{{\widetilde{F}}^2{\widetilde{F}}_0^2 (4\sin^2(k/2)+\lambda^2)},
\label{eq:K}\\&&
T=r^2 \sin^2(k/2) {\widetilde{F}}^2+r \sin^2 k {\widetilde{F}},
\label{eq:T}\\&&
A_{VA}= \sum_\mu \cos^2(k_\mu/2) \sin^2 k_\mu.
\label{eq:A_VA}\end{aligned}$$

In order to rewrite the second term of $I_\Gamma^{b,c}$ we apply the Fierz transformation: $$\begin{aligned}
(\gamma_\mu\gamma_\alpha\gamma_\nu^L)\otimes 
(\gamma_\mu\gamma_\alpha\gamma_\nu^L) &=&
-\gamma_\nu^L\odot\gamma_\nu^L
 = \gamma_\nu^L\otimes\gamma_\nu^L,
\\
(\gamma_\mu\gamma_\alpha\gamma_\nu^L)\otimes 
(\gamma_\nu^L\gamma_\alpha\gamma_\mu) &=&
-\gamma_\nu^L\odot\gamma_\nu^L (1-2\delta_{\alpha\nu})(1-2\delta_{\mu\nu}) 
\\&=&
-\gamma_\nu^L\otimes\gamma_\nu^L (1-2\delta_{\alpha\nu})
(1-2\delta_{\mu\nu})
+2\gamma_\nu^L\otimes\gamma_\nu^L \delta_{\alpha\mu},
\\
(\gamma_\mu\gamma_\alpha\gamma_\nu^L)\otimes 
(\gamma_\mu\gamma_\alpha\gamma_\nu^R) &=&
2P_R\odot P_L \delta_{\mu\alpha}
 = \gamma_\nu^L \otimes \gamma_\nu^R \delta_{\mu\alpha},
\label{eq:lr1_fierz}\\
(\gamma_\mu\gamma_\alpha\gamma_\nu^L)\otimes 
(\gamma_\nu^R\gamma_\alpha\gamma_\mu) &=&
2P_R\odot P_L
 = \gamma_\nu^L \otimes \gamma_\nu^R .
\label{eq:lr2_fierz}\end{aligned}$$ where $[\gamma_X \odot \gamma_Y ]_{\alpha\beta;\gamma\delta} \equiv
(\gamma_X)_{\alpha\delta}(\gamma_Y)_{\gamma\beta}$, and summation over $\nu$ is taken. The Fierz transformation is again used for the second equality. We omit the tensor term in eq.(\[eq:lr1\_fierz\]) since it vanishes in the integral.

Choosing $\Gamma_{X,Y} = \gamma_\mu^L$ in eqs.(\[eq:vtx\_a\]), (\[eq:vtx\_b\]) and (\[eq:vtx\_c\]) we first consider the 