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

The tau polarisation has been studied with the e(+)e(-) --> 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 ev (v) over bar, mu v (v) over bar, pi v, rho v and a(1)v 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 A(tau) = 0.1359 +/- 0.0096 and A(epsilon) = 0.1382 +/- 0.0116. From these results the ratio of the vector and axial-vector effective couplings (v) over bar/(a) over bar(tau) = 0.0683 +/- 0.0048 and (v) over bar/(a) over bar(e) = 0.0694 +/- 0.0058 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 (v) over bar(l)/(a) over bar(l) = 0.0687 +/- 0.0037 was obtained, implying a value of the effective weak mixing angle sin(2) theta(eff)(lept) = 0.23282 +/- 0.00092