$\Gamma = \pm, 1,2$. Spinor indices are labeled by $\alpha,\beta,\gamma,\delta$ and color ones by $i,j,k,l$. Truncating the external quark propagators from $\langle {\cal O}_\Gamma \rangle$, where we multiply $\langle {\cal O}_\Gamma \rangle$ by ${i{p\kern-1ex /}_i + (1-w_0^2)m}$, we obtain the vertex functions, which is written in the following form up to the one-loop level $$(1-w_0^2)^4\left(\Lambda_\Gamma\right)_{\alpha\beta;\gamma\delta}^{ij;kl}
=(1-w_0^2)^4\left(\Lambda_\Gamma^{(0)}
+\Lambda_\Gamma^{(1)}\right)_{\alpha\beta;\gamma\delta}^{ij;kl},$$ where the superscript $(i)$ refers to the $i$-th loop level and the trivial factor $(1-w_0^2)^4$ is factored out for the convenience. We suppress the external momenta $p_i$ since the renormalization factor does not depend on them.

The tree level vertex functions $\Lambda_\Gamma^{(0)}$ are given by $$\begin{aligned}
&\Gamma = \pm, &\qquad
\frac{1}{2}\left[ \gamma_\mu^L 
\otimes \gamma_\mu^L\right]_{\alpha\beta;\gamma\delta}
\left[1{\widetilde{\otimes}} 1 \pm 1{\widetilde{\odot}} 1\right]^{ij;kl}, 
\label{eq:lambda_pm_0}\\
&\Gamma = 1, &\qquad
\frac{1}{2}\left[ \gamma_\mu^L 
\otimes \gamma_\mu^R \right]_{\alpha\beta;\gamma\delta}
\left[-N 1{\widetilde{\otimes}} 1+1{\widetilde{\odot}} 1\right]^{ij;kl},
\label{eq:lambda_1_0}\\
&\Gamma = 2, &\qquad
\frac{1}{2}\left[ \gamma_\mu^L 
\otimes \gamma_\mu^R \right]_{\alpha\beta;\gamma\delta}
\left[1{\widetilde{\odot}} 1 \right]^{ij;kl}, 
\label{eq:lambda_2_0}\end{aligned}$$ where $\otimes$ acts on the Dirac spinor space representing $[\gamma_X \otimes \gamma_Y ]_{\alpha\beta;\gamma\delta} \equiv
(\gamma_X)_{\alpha\beta}(\gamma_Y)_{\gamma\delta}$.

The one-loop vertex corrections are illustrated by six diagrams in Fig. 1, the sum of which yields the one-loop level vertex function $$\Lambda_\Gamma^{(1)}=
\int_{-\pi}^{\pi}\frac{d^4 k}{(2\pi)^4} 
\left(I_\Gamma^a+,\dots,+I_\Gamma^{c^\prime}\right).$$ In order to obtain the expressions for the integrands $I_\Gamma^a,\dots,I_\Gamma^{c^\prime}$ we should note that the 