The hadronic $\tau$ decay ${\tau }^{-}\to K_1^{-}{\nu }_{\tau }\to (K^{-}\omega ){\nu }_{\tau }\to (K^{-}{\pi }^{+}{\pi }^{-}{\pi }^0){\nu }_{\tau }$ 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.

14 pages, 2 figures

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