$tau to f_1 (1285)pi^{-} u_{tau}$ 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 $tau to f_1(1285) pi^{-} u_{tau}$ decay, obtain the invariant mass distribution of the $f_1pi$ -system and estimate the branching ratio Br$(tau to f_1 pi^{-} u_{tau})=4.0times 10^{-4}$. 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$(tau to f_1 pi^{-} u_{tau})=(3.9pm 0.5)times 10^{-4}$. The distribution function obtained for the decay $tau to f_1(1285) pi^{-} u_{tau}$ shows a clear signal of $a_1(1640)$ resonance which should be compared with future experimental data including our estimate of the decay width $Gamma (a_1(1640) to f_1 pi )=14.1,mbox{MeV}$.