The hadronic τ decay τ-→K1-ντ→(K-ω)ντ→(K-π+π-π0)ντ and the axial vector mixing angle

Abstract

We propose to measure the ${\tau }^{-}\to K_1^{-}{\nu }_{\tau }\to (K^{-}\omega ){\nu }_{\tau }\to (K^{-}{\pi }^{+}{\pi }^{-}{\pi }^0){\nu }_{\tau }$decay in order to determine the $K_1$ axial vector mixing angle ${\theta }_{K_1}$. We derive, for the first time, the differential decay rate formula for this decay mode. Using the obtained result, we perform a sensitivity study for the Belle (II) experiment. We will show that the $K^{-}{\pi }^{+}{\pi }^{-}{\pi }^0$ spectrum of the ${\tau }^{-}\to K_1^{-}{\nu }_{\tau }\to (K^{-}\omega ){\nu }_{\tau }\to (K^{-}{\pi }^{+}{\pi }^{-}{\pi }^0){\nu }_{\tau }$decay can discriminate the two solutions ${\theta }_{K_1}=\sim {30}^{\circ }$ or $\sim {60}^{\circ }$ observed in the other measurements.