The hadronic $\tau $ decay $\tau ^-\rightarrow K_1^-\nu _\tau \rightarrow (K^-\omega ) \nu _\tau \rightarrow (K^- \pi ^+\pi ^-\pi ^0)\nu _\tau $ and the axial vector mixing angle
Résumé
We propose to measure the $\tau ^-\rightarrow K_1^-\nu _\tau \rightarrow (K^-\omega ) \nu _\tau \rightarrow (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 ^-\rightarrow K_1^-\nu _\tau \rightarrow (K^-\omega ) \nu _\tau \rightarrow (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.