A precise measurement of the $\tau$ polarisation at LEP-1

The \(\tau\) polarisation has been studied with the \({\rm e^+e^-}\to \tau^+\tau^-\) data collected by the DELPHI detector at LEP in 1993, 1994 and 1995 around the Z resonance firstly through the exclusive decay channels \({\rm e}\nu\bar{\nu}\), \(\mu\nu\bar{\nu}\), \(\pi\nu\), \(\rho\nu\) and \({\rm a}_1\nu\) and secondly with an inclusive hadronic analysis which benefits from a higher efficiency and a better systematic precision. The results have been combined with those previously published on 1990 to 1992 DELPHI data, to produce results which reflect the full LEP-1 statistics. The fit of the \(\tau\) polarisation dependence on the production angle yielded the polarisation parameters \({\cal A}_{_{\scriptstyle \!\tau}}=0.1359 \pm 0.0096\) and \({\cal A}_{_{\scriptstyle \!e}}=0.1382 \pm 0.0116\). From these results the ratio of the vector and axial-vector effective couplings \(\bar{v}_{\tau} / \bar{a}_{\tau} = 0.0683\pm0.0048\) and \(\bar{v}_{\rm e} / \bar{a}_{\rm e} = 0.0694\pm0.0058\) have been derived, compatible with e\(-\tau\) universality. With the assumption of lepton universality, the ratio of vector to axial-vector effective couplings for leptons \(\bar{v}_l / \bar{a}_l = 0.0687\pm0.0037\) was obtained, implying a value of the effective weak mixing angle \(\sin^{2}\theta^{\rm lept}_{\rm eff} = 0.23282\pm0.00092\).