4mg_\pi \left\{ \left[I_{3}^{A_{f_{1}}} - \frac{C_{a_1}6m^4}{g_{a_1}M_{a_1}^2} I_4^{A_{f_{1}}A_{a_{1}}} \right] \right. \nonumber\\
&&+ \frac{C_{a_1}}{g_{a_1}} \left[I_{3}^{A_{f_{1}}A_{a_{1}}} - \frac{C_{a_1}6m^4}{g_{a_1}M_{a_1}^2} I_{4}^{A_{f_{1}}A_{a_{1}}A_{a_{1}}}\right] \nonumber \\
&&\times \frac{s - 6m^2}{M_{a_{1}}^2 - s - i \sqrt{s}\Gamma_{a_{1}}} \nonumber\\
&&+  \frac{C_{a'_1}}{g_{a_{1}}} \left[I_{3}^{A_{f_{1}}B_{a_{1}}} - \frac{C_{a_1}6m^4}{g_{a_1}M_{a_1}^2} I_{4}^{A_{f_{1}}B_{a_{1}}A_{a_{1}}}\right]  \nonumber\\
&&\times\left.\frac{s - 6m^2}{M_{a'_{1}}^2 - s - i \sqrt{s}\Gamma_{a'_{1}}} \right\}  e_{\mu\nu\alpha\beta} \epsilon^{\nu}(p_{f_{1}}) p^\alpha_{f_{1} }p^\beta_{\pi}.\end{aligned}$$ Here, in the first square brackets, the contact contribution is presented. The second and third square brackets contain the contributions of $a_1(1260)$ and $a_1(1640)$ resonances correspondingly.

The numerical integration over the $ f_1\pi\nu_\tau$ three-body phase space with the amplitude (\[Mtot\]) gives the branching ratio Br$(\tau\to f_1\pi^-\nu_\tau )=3.98 \times 10^{-4}$. The result can be schematically presented in terms of the individual contributions as follows $$\begin{aligned}
&&10^{4}\times \mbox{Br}(\tau\to f_1\pi^-\nu_\tau)= 3.98 \nonumber\\
&&= |c|^2+|a_1|^2+|a'_1|^2+2\mbox{Re}(a_1 c^*+a'_1 c^*+a'_1a_1^*) \nonumber\\
&&=1.62+5.92+0.45-5.22+0.33 +0.88,\end{aligned}$$ where $c, a_1, a'_1$ represent the contributions from the contact term, and from $a_1$ and $a'_1$ exchanges correspondingly. One can see that if one takes into account the $\pi -a_1$ transitions the branching ratio is in a good agreement with the experimental value Br$(\tau \to f_1\, \pi^{-} \nu_{\tau})=(3.9\pm 0.5) \times 10^{-4}$.

The differential decay distribution, $d\Gamma /d \sqrt s$, is shown in Fig. \[distribution\]. It reaches its maximum value near the $a_1(1640)$ resonance mass shell. Therefore, it 