observed recently by the COMPASS collaboration [@Adolph15]. Our study shows that even if there is a contribution due to $a_1(1420)$ exchange to the process discussed here, this contribution is negligible. It indicates that most probably $a_1(1420)$ is not the $q\bar q$ state. Some reasonings in favour of multi-quark structure of this state are recently given in [@Ivanov17].

Acknowledgments {#acknowledgments .unnumbered}
===============

We would like to thank A. B. Arbuzov for his interest to this work and useful discussions.

Appendix {#appendix .unnumbered}
========

Here we present the general expressions for the amplitude of $\tau\to f_1(1285)\pi^-\nu_\tau$ decay shown in Figs.\[Contact\]-\[pi-a1-intermediate\]. In the text we make derivative expansions of these expressions to obtain effective meson vertices in the long wavelength approximation. We consider the process $$\tau (Q)\to \nu_\tau (Q') f_1(p_{f_1}) \pi (p_\pi )$$ with the quantities in the parentheses denoting the four moments of the particles.

The amplitude, corresponding to diagrams of Fig.\[Contact\]-\[Intermediate\], has the following form $$\mathcal{M} =  G_{F} V_{ud} l_{\mu} \mathcal{H}^\mu ,$$ where a hadron current $\mathcal{H}^\mu$ is $$\begin{aligned}
\mathcal{H}^\mu & = & \frac{g_{\pi}}{2} \epsilon_{\nu}(p_{f_{1}}) 
\left\{ I_{Wf_{1}\pi 1}^{\mu\nu} + I_{Wf_{1}\pi 2}^{\mu\nu} \right. \nonumber\\
& + & \frac{I_{Wa_{1}}^{\mu\lambda}\left(I_{a_{1}f_{1}\pi1}^{\lambda\nu} + I_{a_{1}f_{1}\pi2}^{\lambda\nu}\right)}{M_{a_{1}}^2 - s - i \sqrt{s}\Gamma_{a_{1}}}  \nonumber \\
& + & \left.
 \frac{I_{Wa'_{1}}^{\mu\lambda}\left(I_{a'_{1}f_{1}\pi1}^{\lambda\nu} 