Measurement of $\tau$ polarisation at LEP

Using the data collected with the L3 detector at LEP between 1990 and 1995, corresponding to an integrated luminosity of 149 pb$^{-1}$, the $\tau$ longitudinal polarisation has been measured as a function of the production polar angle using the $\tau$ decays \thad\ ($\rm h = \pi, \rho, \aone$) and \tlep\ ($\rm \ell = e, \mu$). From this measurement the quantities æl~and \at, which depend on the couplings of the electron and the $\tau$ to the Z, are determined to be $\ael = 0.1678 \pm 0.0127 \pm 0.0030 $ and $\at = 0.1476 \pm 0.0088 \pm 0.0062$, consistent with the hypothesis of e--$\tau$ universality. Under this assumption a value of $\al = 0.1540 \pm 0.0074 \pm 0.0044 $ is obtained, yielding the value of the effective weak mixing angle $\swsqb = 0.2306 \pm 0.0011$.