Decays $\tau \to f_0(\pi ,K){\nu }_{\tau }$ and $\tau \to 3\pi {\nu }_{\tau }$ accounting for the contribution of $f_0(500)$

In the $U(3)\times U(3)$ quark NJL model, $\tau$ lepton decays with the production of scalar mesons $f_0(\pi ,K])$ and neutrinos are studied, where $f_0=f_0(500),f_0(980)$. It is shown that these decays mainly occur via contact channels and channels with axial-vector mesons $a_1$, $K_1(1270)$ and $K_1(1400)$. All mesons are considered as quark-antiquark states in this case. The obtained results can be considered as predictions for future experiments. The obtained estimates for the branching fractions of the $\tau \to {\pi }^{-}2{\pi }^0{\nu }_{\tau }$ decay taking into account the contributions of the $\rho \pi$ and $f_0(500)\pi$ states are in satisfactory agreement with experimental data.