internal quark propagators appearing in the diagrams are multiplied by the damping factor which comes from eqs.[(\[eqn:qpsi\])]{} and [(\[eqn:psiq\])]{}. The following formula are useful. $$\begin{aligned}
\langle q \bar \psi_s\rangle \left( w_0^{s-1}P_L + w_0^{N_s-s} P_R\right)
=\left( w_0^{s-1}P_R + w_0^{N_s-s} P_L\right) \langle \psi_s\bar q \rangle 
&=& \frac{i\gamma_\mu \sin p_\mu}{{\widetilde{F}}\cdot {\widetilde{F}}_0 }\equiv {\overline{G}}, 
\label{eq:G_b} \\
\langle q \bar \psi_s\rangle \left( w_0^{s-1}P_R + w_0^{N_s-s} P_L\right)
=\left( w_0^{s-1}P_L + w_0^{N_s-s} P_R\right) \langle \psi_s\bar q \rangle 
&=& -\frac{1}{{\widetilde{F}}_0 } \equiv {\widetilde{G}}, 
\label{eq:G_t}\end{aligned}$$ where ${\widetilde{F}}=e^{-\alpha}-W$ and ${\widetilde{F}}_0 = e^\alpha-w_0$. Here we set $m=p_i=0$ for the internal propagator.

The contribution from Fig. 1a takes the form $$\begin{aligned}
I_\Gamma^a &=& \frac{1}{2} J_a^{AB}
\left\{{\overline{V}}_\mu(k) {\overline{G}}(k) + {\widetilde{V}}_\mu (k){\widetilde{G}}(k)
\right\}
\Gamma_X
\left\{{\overline{G}}(k) {\overline{V}}_\nu(k) + {\widetilde{G}}(k) {\widetilde{V}}_\nu(k)\right\}
\otimes \Gamma_Y
\, G_{\mu\nu}^{AB}(k),\end{aligned}$$ where $\Gamma_X = \gamma_\mu^L$, $\Gamma_Y = \gamma_\mu^L$ or $\gamma_\mu^R$, and the interaction vertices are $${\overline{V}}_\mu = -i g \gamma_\mu \cos(k_\mu/2),\quad
{\widetilde{V}}_\mu = -rg \sin(k_\mu/2).
\label{eq:V_bt}$$ The color factors are represented by $J_a^{AB}$, which are listed in Table \[tab:color\]. In a similar way the contributions from Fig. 1b and Fig. 1c are given by $$\begin{aligned}
I_\Gamma^b &=& \frac{1}{2} J_b^{AB}
\left\{{\overline{V}}_\mu(k) {\overline{G}}(k) + {\widetilde{V}}_\mu(k) {\widetilde{G}}(k)\right\}
\Gamma_X\otimes
\left\{{\overline{V}}_\nu (-k) {\overline{G}}(-k) + {\widetilde{V}}_\nu (-k){\widetilde{G}}(-k)\right\}
\Gamma_Y
\, G_{\mu\nu}^{AB}(k), \\
I_\Gamma^c &=& \frac{1}{2} J_c^{AB}
\left\{{\overline{V}}_\mu(k) {\overline{G}}(k) + 