our calculations.

The description of collective bound states can be facilitated if we introduce, as it was discussed above, the set of bosonic variables and pass to the semi-bosonized effective meson action which is responsible for the $\tau \to f_1\, \pi^{-} \nu_{\tau}$ decay, namely, the part which describes both the interactions of the ground pseudoscalar $\pi^{\pm}$, axial-vector $a_1^{\pm}(1260)$, $f_1(1285)$ mesons and their first radially excited states with light constituent quarks. In momentum representation the action takes the form $$\begin{aligned}
\label{Lagrangiane}
S &=&\!\int\!\frac{d^4p}{(2\pi )^4}\bar q(p) \left(\hat p-m \right)q(p)+ \Delta S_{mass}+ \Delta S_{int}, \nonumber \\
\Delta S_{int} &=&\! \int\!\frac{d^4p}{(2\pi )^4}\!\int\!\frac{d^4k}{(2\pi )^4}\, \bar{q}\left(k+\frac{p}{2}\right) \nonumber \\
&\times& \left[ \frac{1}{2} \gamma^{\mu}\gamma^{5}\left(A_{f_{1}}f_{1\mu}(p) + B_{f_{1}}f^{'}_{1\mu}(p)\right) \right.\nonumber \\
&+& \frac{1}{2} \gamma^{\mu}\gamma^{5} \vec\tau \left(A_{a_{1}}\vec a_{1\mu}(p) + B_{a_{1}}\vec a'_{1\mu}(p)\right) \nonumber \\
&+&  \left. \gamma^{5} \vec\tau \left(A_{\pi}\vec \pi (p) + B_{\pi}\vec \pi'\right)\right]q\left(k-\frac{p}{2}\right) .\end{aligned}$$ We shall not discuss here the mass part of the action $\Delta S_{mass}$ (see, for instance, [@Volkov17] or references therein). Our notations are as follows: $f_{1\mu}$, $\vec a_{1\mu}$ and $\vec \pi $ are the fields corresponding to the axial-vector and pseudoscalar mesons, the related excited states are marked with a prime. The constituent quark fields, $q$, have equal masses $m_{u} = m_{d} = 280$ MeV [@VolkovYu00; @Volkov17]. The 