The hadronic $$\tau $$ τ decay $$\tau ^-\rightarrow K_1^-\nu _\tau \rightarrow (K^-\omega ) \nu _\tau \rightarrow (K^- \pi ^+\pi ^-\pi ^0)\nu _\tau $$ τ - → K 1 - ν τ → ( K - ω ) ν τ → ( K - π + π - π 0 ) ν τ and the axial vector mixing angle

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.