case of ${\cal O}_{\pm}$. After simplifying the expressions of the color factors we obtain $$\begin{aligned}
I_\pm^a &=&  \frac{1}{2} g^2 
        K (T+A_{VA})\left[\gamma_\nu^L\otimes\gamma_\nu^L\right]
        \left[ (C_F\pm \frac{1}{2}) 1{\widetilde{\otimes}}1
               \mp \frac{1}{2N} 1{\widetilde{\odot}}1 \right], \\
I_\pm^b &=& -  \frac{1}{2} g^2 
        K (T+A_{SP})\left[\gamma_\nu^L\otimes\gamma_\nu^L\right]
        \left[ (-\frac{1}{2N}\pm \frac{1}{2}) 1{\widetilde{\otimes}}1
               +(\frac{1}{2}\mp \frac{1}{2N}) 1{\widetilde{\odot}}1 \right], \\
I_\pm^c &=&  \frac{1}{2} g^2 
        K (T+A_{VA})\left[\gamma_\nu^L\otimes\gamma_\nu^L\right]
        \left[ -\frac{1}{2N} 1{\widetilde{\otimes}}1
               +(\frac{1}{2}\pm C_F) 1{\widetilde{\odot}}1 \right]. \end{aligned}$$ The total contribution becomes $$\begin{aligned}
I_+^a+I_+^b+I_+^c &=&  \frac{1}{2} g^2 K 
\left[\gamma_\nu^L\otimes\gamma_\nu^L\right]_{\alpha\beta;\gamma\delta} 
\left[1{\widetilde{\otimes}}1+1{\widetilde{\odot}}1\right]^{ij;kl} \nonumber \\
&\times &
\{ T C_F + A_{VA}(C_F +\frac{1}{2} - \frac{1}{2N})
-A_{SP}(\frac{1}{2} - \frac{1}{2N}) \},
\label{eq:sumP}\end{aligned}$$ and $$\begin{aligned}
I_-^a+I_-^b+I_-^c &=&  \frac{1}{2} g^2 K 
\left[\gamma_\nu^L\otimes\gamma_\nu^L\right]_{\alpha\beta;\gamma\delta} 
\left[1{\widetilde{\otimes}}1-1{\widetilde{\odot}}1\right]^{ij;kl} \nonumber \\
&\times &
\{ T C_F + A_{VA}(C_F -\frac{1}{2} - \frac{1}{2N})
+A_{SP}(\frac{1}{2} + \frac{1}{2N}) \},
\label{eq:sumM}\end{aligned}$$ where $$A_{SP}=\cos^2(k/2)\cdot \sin^2 k.
\label{eq:A_SP}$$ We should note that the other three contributions $I_\pm^{a^\prime,b^\prime,c^\prime}$ from Fig. 1$a^\prime$, 1$b^\prime$ and 1$c^\prime$ are equal to $I_\Gamma^{a,b,c}$ respectively, therefore the factors $1/2$ in eqs. (\[eq:sumP\]) and (\[eq:sumM\]) disappear in the total contributions of all.

Comparing the one-loop results to the tree level ones we obtain $$\begin{aligned}
\Lambda_+ &=&
\left[ 1 + g^2 \frac{N-1}{N}\left\{ 
<T> (N+1)+<A_{VA}>(N+2)-<A_{SP}> \right\}\right]
\Lambda_+^{(0)}, 
\label{eq:1loop_+}\\
\Lambda_- &=&
\left[ 1 + g^2 \frac{N+1}{N}\left\{ 
<T> (N-1)+<A_{VA}>(N-2)+<A_{SP}>\right\}\right]
\Lambda_-^{(0)}
\label{eq:1loop_-} \end{aligned}$$ with $$< X > = \int_{-\pi}^{\pi}\frac{d^4 k}{(2\pi)^4} K(k) 