the help of the extended NJL model. The square of this amplitude has a simple form $$|\mathcal{M}|^2=4G_{F}^2V_{ud}^2\left[2(QH)(Q'H)-H^2 (QQ')\right].$$ Thus one can easily find the decay width of the process $$\Gamma = \frac{1}{32M_\tau^3(2\pi )^3}\!\! \int\limits_{(M_{f_1}+M_\pi )^2}^{M_\tau^2}\!\!\!\!\! ds \int\limits_{t_-}^{t_+} dt \,| {\cal M} |^2$$ by performing the numerical integration over kinematical variables $t=(Q-p_{f_1})^2$ and $s=(p_{f_1}+p_\pi )^2$. Here a boundary of the physical region at fixed value of $s$ belongs to the interval $t_-\leq t\leq t_+$, where $$\begin{aligned}
&&t_\pm = \frac{1}{2} \left( t_0\pm\sqrt{D}\right), \nonumber \\
&& t_0= M_\tau^2+M_{f_1}^2+M_\pi^2 -s -\frac{M_\tau^2}{s}\left(M_{f_1}^2-M_\pi^2\right), \nonumber \\
&&\sqrt{D}=\frac{1}{s}\left(M_\tau^2-s\right)\sqrt{\lambda (s,M_{f_1}^2,M_\pi^2 )}, \nonumber \\
&&\lambda (x, y, z )=[ x - (\surd y-\surd z )^2] [ x - (\surd y+ \surd z )^2 ]. \end{aligned}$$

Finally we arrive at the following result for the branching ratio Br$(\tau\to f_1\pi^-\nu_\tau )=6.04 \times 10^{-4}$. It can be schematically presented in terms of individual contributions as follows $$\begin{aligned}
&&10^{4}\times \mbox{Br}(\tau\to f_1\pi^-\nu_\tau)= 6.04 \nonumber\\
&&= |c|^2+|a_1|^2+|a'_1|^2+2\mbox{Re}(a_1 c^*+a'_1 c^*+a'_1a_1^*) \nonumber\\
&&= 2.25+8.43+0.78-7.34+0.53+1.39.\end{aligned}$$ where $c, a_1, a'_1$ represent the contributions from the contact (direct) term, and from $a_1$ and $a'_1$ exchanges correspondingly. One can see that if one neglects the $\pi -a_1$ transitions (as we did here) the result overestimates the experimental value Br$(\tau \to f_1\, \pi^{-} 