Hadronic structure in the decay ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{\pi }^0{\pi }^0$ and the sign of the tau neutrino helicity

Based on a sample corresponding to $4.3\times {10}^6$ produced $\tau$-pair events, we have studied hadronic dynamics in the decay ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{\pi }^0{\pi }^0$ in data recorded by the CLEO II detector operating at the CESR $e^{+}{e}^{-}$ collider. The decay is dominated by the process ${\tau }^{-}\to {\nu }_{\tau }{a}_1^{-}\mathrm{}\left(1260\right),$ with the $a_1^{-}$ meson decaying to three pions predominantly via the lowest dimensional (mainly S-wave) ${\rho }^{-}{\pi }^0$ Born amplitude. From model-dependent fits to the Dalitz plot and angular observables in bins of $3\pi$ mass, we find significant additional contributions from amplitudes for $a_1$ decay to $\sigma \pi ,$ $f_0\mathrm{}\left(1370\right)\mathrm{}\pi ,$ and $f_2\mathrm{}\left(1270\right)\mathrm{}\pi ,$ as well as higher dimensional $a_1\to \rho \pi$ and ${\rho }^{\prime }\pi$ amplitudes. Notably, the squared $\sigma \pi$ amplitude accounts for approximately 15% of the total ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{\pi }^0{\pi }^0$ rate in the models considered. The data are well described using couplings to these amplitudes that are independent of the $3\pi$ mass. These amplitudes also provide a good description for the ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{\pi }^{+}{\pi }^{-}$ Dalitz plot distributions. We have searched for additional contributions from ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{\prime }-\mathrm{}\left(1300\right).$ We place $90\%$ confidence level upper limits on the branching fraction for this channel of between $1.0\times {10}^{-4}$ and $1.9\times {10}^{-4},$ depending on the ${\pi }^{\prime }$ decay mode considered. The ${\pi }^{-}{\pi }^0{\pi }^0$ mass spectrum is parametrized by a Breit-Wigner form with a mass-dependent width which is specified according to the results of the Dalitz plot fits plus an unknown coupling to an $a_1\to K^{*}K$ amplitude. From a ${\chi }^2$ fit using this parametrization, we extract the pole mass and width of the $a_1,$ as well as the magnitude of the $K^{*}K$ coupling. We have also investigated the impact of a possible contribution from the $a_1^{\prime }\mathrm{}\left(1700\right)\mathrm{}$ meson on this spectrum. Finally, exploiting the parity-violating angular asymmetry in $a_1\to 3\pi$ decay, we determine the signed value of the $\tau$ neutrino helicity to be $h_{{\nu }_{\tau }}=-1.02\pm 0.13\hspace{0.5em}\left(\mathrm{stat}\right)\pm 0.03\hspace{0.5em}(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}+\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}),$ confirming the left-handedness of the $\tau$ neutrino.