\nu_{\tau})=(3.9\pm 0.5) \times 10^{-4}$.

The effect of $\pi -a_{1}$ transitions
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The replacement (\[pa-trans\]) in the quark-meson Lagrangian originates new vertices of the axial-vector type with the gradient of the pion field. This can be equivalently considered as a creation of the final pion via old $a_1$ meson field ($\pi -a_1$ transitions). To take them into account one has to append the diagrams shown in Fig. \[pi-a1-contact\] and Fig. \[pi-a1-intermediate\]. The corresponding additional contribution to the amplitude is given in the Appendix (see eq.(\[add-pi-a1\])). It can be simplified after the following observation.

Let us consider the trace of the quark triangles corresponding to these diagrams $$\mbox{tr}[\gamma^{\nu}\gamma^{5}(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m) \gamma^{\lambda}\gamma^{5}(\hat{k} + m)].$$ It can be written as a sum $$\begin{aligned}
\label{trace}
&& \mbox{tr}\,[\gamma^{\nu}(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m)\gamma^{\lambda}(\hat{k} + m) \nonumber\\
&& - 2m \gamma^{\nu}(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m)\gamma^{\lambda}].\end{aligned}$$ It is not difficult to see that the first term coincides with a trace from the triangle of the process $f_{1}(1285) \to \rho \gamma$ considered in our previous work [@Osipov17]. After integration over loop-momentum $k$ in the corresponding quark triangle expression (see, for instance, eq.(\[afa\])) one finds that it 