+ I_{a'_{1}f_{1}\pi2}^{\lambda\nu}\right) }{M_{a'_{1}}^2 - s - i \sqrt{s}\Gamma_{a'_{1}}} 
\right\}.\end{aligned}$$

The first two integrals $I_{W f_1\pi 1}^{\mu\nu}$ and $I_{Wf_{1}\pi 2}^{\mu\nu}$ describe the direct contribution from the transition $W^\mu\to f_1^\nu\pi$ generated by the triangle quark loop (see Fig.\[Contact\]). In accord with two different directions for the loop momenta we specify them by indices 1 (clockwise) and 2 (counter-clockwise). Their expressions are $$\begin{aligned}
&& I_{Wf_{1}\pi1}^{\mu\nu} = N_c \int  \frac{\mathrm{d}^4k}{(2\pi)^4} A_{f_1}(k_\perp^2)  \\
&&\times\frac{ \mbox{tr}[\gamma^\nu\gamma^5(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m)\gamma^{5}(\hat{k} + m)]}{[(k + p_{f_{1}})^2 - m^2][(k - p_{\pi})^2 - m^2](k^2 - m^2)}, \nonumber \end{aligned}$$ $$\begin{aligned}
&&I_{Wf_{1}\pi2}^{\mu\nu} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4} A_{f_{1}} (k_\perp^2) \\
&&\times\frac{\mbox{tr}[\gamma^{5}(\hat{k} + \hat{p}_{\pi} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{f_{1}} + m)\gamma^{\nu}\gamma^{5}(\hat{k} + m)]}{[(k - p_{f_{1}})^2 - m^2][(k + p_{\pi})^2 - m^2](k^2 - m^2)} . \nonumber\end{aligned}$$

The other two integrals $I_{Wa_1}^{\mu\lambda}$ and $I_{Wa'_1}^{\mu\lambda}$ describe the $W^\mu\to a_1^\lambda$ and $W^\mu\to {a'_1}^\lambda$ transitions correspondingly (see Fig.\[Intermediate\]). The first of them is equal to $$\begin{aligned}
\label{IWa}
&&I_{Wa_{1}}^{\mu\lambda} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4}A_{a_{1}} (k_\perp^2) \nonumber \\
&&\times\frac{ \mbox{tr} [\gamma^{\lambda}\gamma^{5}(\hat{k} + \frac{\hat{q}}{2} + m)\gamma^{\mu}\gamma^5(\hat{k} - \frac{\hat{q}}{2} + m)]}{[(k + \frac{q}{2})^2 - m^2][(k - \frac{q}{2})^2 - m^2]}, \end{aligned}$$ with $q=Q-Q'=p_{f_1}+p_\pi$, and $q^2=s$. The second one can be obtained from (\[IWa\]) by the replacement $A_{a_{1}} (k_\perp^2) \to B_{a_{1}} (k_\perp^2)$.

The 