Tau properties in $B\to D\tau\nu$ from visible final-state kinematics

In semi-leptonic $B$ decays with a tau lepton, features of the production process are imprinted on the tau helicity states. Since the tau momentum cannot be fully reconstructed experimentally, the available information on the tau properties is encoded in its visible decay products. Focusing on the process $B\to D\tau\nu$, we find explicit relations between the tau properties and the kinematics of the charged particles in the decays $\tau\to\pi\nu$, $\tau\to\rho\nu$, and $\tau\to\ell\nu\bar{\nu}$. In particular, we show that the perpendicular polarization, $P_\perp$, and the forward-backward asymmetry, $A_\tau$, of the tau lepton can simultaneously be extracted from an angular asymmetry of the charged particle against the $D$ meson. For the most sensitive decay channel, $\tau\to\pi\nu$, we expect a relative statistical precision of about $10\%$ for $P_\perp$ and $A_\tau$ in a measurement based on $50\,$ab$^{-1}$ of data at BELLE II.