$\tau \rightarrow f_{1}(1285) \pi^{-}\nu_{\tau}$ decay in the extended Nambu-Jona-Lasinio model

Abstract.

Within the framework of the extended Nambu-Jona-Lasinio model, we calculate the matrix element of the $\tau \rightarrow f_{1}(1285) \pi^{-} \nu_{\tau}$ decay, obtain the invariant mass distribution of the $f_{1}\pi$-system and estimate the branching ratio $\mathrm{Br}(\tau \rightarrow f_{1} \pi^{-}\nu_{\tau}) = 4.0\times 10^{-4}$. 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 $\mathrm{Br}(\tau \rightarrow f_{1} \pi^{-} \nu_{\tau})=(3.9\pm 0.5)\times 10^{-4}$. The distribution function obtained for the decay $\tau \rightarrow f_{1}(1285) \pi^{-} \nu_{\tau}$ shows a clear signal of $a_{1}(1640)$ resonance which should be compared with future experimental data including our estimate of the decay width $\Gamma (a_{1}(1640) \rightarrow f_{1} \pi)=14.1$ MeV.