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

TL;DR

This paper introduces a method to measure the $K_1$ axial vector mixing angle through a specific tau decay channel, deriving the decay rate formula and performing a sensitivity analysis for Belle II to distinguish between two possible mixing angle solutions.

Contribution

The paper derives the first differential decay rate formula for the $ au^- o K_1^- u_ au$ decay mode and assesses its potential to determine the $K_1$ mixing angle at Belle II.

Findings

The decay spectrum can distinguish between $ heta_{K_1} hicksim 30^ ext{o}$ and $ hicksim 60^ ext{o}$ solutions.

Sensitivity study shows Belle II can effectively measure the $K_1$ mixing angle.

First derivation of the differential decay rate for this tau decay mode.

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.