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We present the first measurement of the Michel parameter $\xi^\prime$ in the$\tau^-\to\mu^-\bar{\nu}_\mu\nu_\tau$ decay using the full data sample of$988\,\text{fb}^{-1}$ collected by the Belle detector operating at the KEKBasymmetric energy $e^+ e^-$ collider. The method is based on the reconstructionof the $\mu^- \to e^- \bar{\nu}_e\nu_\mu$ decay-in-flight in the Belle centraldrift chamber and relies on the correlation between muon spin and its daughterelectron momentum. We study the main sources of the background that can imitatethe signal decay, such as kaon and pion decays-in-flight and charged particlescattering on the detector material. Highly efficient methods of theirsuppression are developed and applied to select 165 signal-candidate events. Weobtain $\xi^\prime=0.22\pm0.94\pm0.42$ where the first uncertainty isstatistical, and the second one is systematic. The result is in agreement withthe Standard Model prediction of $\xi^\prime=1$.

Study of the muon decay-in-flight in the $\tau^-\to\mu^-\bar{\nu}_\mu\nu_\tau$ decay to measure the Michel parameter $\xi^\prime$