$\tau \to f_1(1285){\pi }^{-}{\nu }_{\tau }$ decay in the extended Nambu -- Jona-Lasinio model

Within the framework of the extended Nambu – Jona-Lasinio model, we calculate the matrix element of the $\tau \to f_1(1285){\pi }^{-}{\nu }_{\tau }$ decay, obtain the invariant mass distribution of the $f_1\pi$ -system and estimate the branching ratio Br$(\tau \to f_1 \pi^{-}\nu_{\tau})=4.0\times 10^{-4}$. The two types of contributions are considered: the contact interaction, and the axial-vector $I^G(J^{PC})=1^{-}(1^{++})$ resonance exchange. The latter includes the ground $a_1(1260)$ state, and its first radially excited state, $a_1(1640)$. The corrections caused by the $\pi -a_1$ transitions are taken into account. Our estimate is in a good agreement with the latest empirical result Br$(\tau \to f_1 \pi^{-} \nu_{\tau})=(3.9\pm 0.5)\times 10^{-4}$\mathrm{.}Thedistributionfunctionobtainedforthedecay$\tau \to f_1(1285) \pi^{-} \nu_{\tau}$showsaclearsignalof$a_1(1640)$ resonance which should be compared with future experimental data including our estimate of the decay width $\mathrm{\Gamma }(a_1(1640)\to f_1\pi )=14.1MeV$.