is a lepton current, where $Q$ and $Q'$ are momenta of the tau-lepton and neutrino; $\epsilon_{\nu}(p_{f_{1}})$ is a polarization vector of the $f_1$ meson with the momentum $p_{f_1}$; $s = (p_{f_1} + p_{\pi})^2$ is a square of the invariant mass of the $f_1\pi$-pair, the masses and widths of resonances are $M_{a_{1}} = 1230$ MeV, $M_{a'_{1}} = 1640$ MeV, $\Gamma_{a_{1}} \approx 400$ MeV, $\Gamma_{a'_{1}} = 254$ MeV [@Patrignani16]. The factors $C_{a_1}$ and $C_{a'_1}$ are the remaining of functions $A_{a_1}$ and $B_{a_1}$ (see eq.(\[coefficients\])) after integration over an internal momentum in the $W-a_1$ and $W-a_1'$ quark-loops (see Fig.\[Intermediate\]) correspondingly. $$\begin{aligned}
C_{a_1} & = & A^0_{a_1} \left( \sin\theta_{a_1}^+ +R_V\sin\theta_{a_1}^- \right), \nonumber \\
C_{a'_1} & = & -A^0_{a_1} \left(\cos\theta_{a_1}^+ +R_V\cos\theta_{a_1}^-\right), \nonumber \\
R_{V} & = & \frac{I_{2,1}}{\sqrt{I_{2,0}I_{2,2}}}\, .\end{aligned}$$ The integrals $I_3^{A_{f_1}\dots}$ come out from the evaluation of the one-loop-quark triangle diagrams of Fig.\[Contact\] and Fig.\[Intermediate\]. They correspond to the case $n=3$ of the general expression (in the next section the case $n=4$ will be also required) $$\label{In}
I_{n}^{A_{M}B_{M}\dots} =
\frac{-iN_c}{(2\pi)^4}\!\int\! \frac{A_M (k_\perp^2) B_M (k_\perp^2)\dots}{(m^2 - k^2)^n}\theta(\Lambda_3^2 - k_\perp^2)
\mathrm{d}^{4}k,$$ where $A_{M}, B_{M}$ are given in (\[coefficients\]).

The amplitude (\[M-without\]) has a conventional form $\mathcal{M}=G_{F}V_{ud}l^\mu H_\mu$, where we have found the hadron current $H_\mu$ with 