– Jona-Lasinio (NJL) model [@Eguchi76; @Volkov82; @Ebert83; @Volkov84; @Volkov86; @Ebert86; @Vogl91; @Klevansky92; @Volkov93; @Bijnens93; @Hatsuda94; @Volkov94; @Osipov96]. The first attempt to apply this model to describe the hadron part of the $\tau \to f_1 (1285) \pi^{-} \nu_{\tau}$ decay is presented in [@Vishneva14]. Though an analysis made there allows to reproduce the experimental value of branching ratio, the picture presented cannot be considered as a fully satisfactory description. Here we improve it on the following aspects.

First, as opposed to [@Vishneva14] we demonstrate that the pseudoscalar channel does not give contribution to the decay $\tau \to f_1 \pi^{-} \nu_{\tau}$. The latter is a direct consequence of the anomaly structure of the corresponding quark triangle diagram. Indeed, for the pseudoscalar-exchange channel the hadron axial-vector current, $J_\mu^A$, is proportional to a gradient of the pion field: $J_\mu^A\propto\partial_\mu\pi$. If the pion turns into the $f_1\pi$ couple, the corresponding $f_1\pi\pi$ vertex vanishes because it has an anomaly structure $e_{\mu\nu\alpha\beta}\epsilon^\mu_{f_1} p_1^\nu p_2^\alpha p_3^\beta$, where the antisymmetric Levi-Civita symbol is contracted with a polarization vector of the axial-vector field, $\epsilon^\mu_{f_1}$, and with three momenta $p_1, p_2, p_3$ of particles involved. However, due to the conservation of the total momenta, there are only two independent vectors. The linear 