Heavy Quark Expansion for the Inclusive Decay $\bar{B}\to \tau \bar{\nu }X$

We calculate the differential decay rate for inclusive $\bar B\to\tau,\bar\nu,X$transitionstoorder$1/m_b^2$ in the heavy quark expansion, for both polarized and unpolarized tau leptons. We show that using a systematic $1\mathrm{/}{m}_b$ expansion significantly reduces the theoretical uncertainties in the calculation. We obtain for the total branching ratio ${\rm BR}(\bar B\to\tau,\bar\nu,X)=2.30\pm 0.25%$,andforthetaupolarization$A_{\rm pol}=-0.706\pm0.006$. {}From the experimental measurement of the branching ratio at LEP, we derive the upper bound $\text{lo}\le 0.8{\text{gev}}^2$ for one of the parameters of the heavy quark effective theory.