A global determination of at LEP

The value of the strong coupling constant,$$\alpha _s (M_{Z^0 } )$$, is determined from a study of 15 different observables in hadronicZ0 and τ decays. The study includes global event shape variables, jet production rates, energy correlations, theZ0 line shape and decay asymmetries and the hadronic branching fraction of τ-leptons. Differences between the αs values from the different observables can be consistently attributed to unknown higher order contributions to the calculations. These uncertaities may be parametrized by variations of the renormalization scale and of the parton virtuality to which the data are corrected, separately for each observable, resulting in a consistent description of the event shapes, jet rates and energy correlations with the value$$\alpha _s (M_{Z^0 } ) = 0.122_{ - 0.005}^{ + 0.006} $$ in. The error is dominated by the theoretical uncertainties. Application of recent calculations which include the resummation of leading and next-to-leading logarithms to all orders for some observables confirm this result with a reduced sensitivity to renormalization scale variations. TheZ0 line shapes and τ-lepton branching ratios yield$$\alpha _s (M_{Z^0 } ) = 0.148 \pm 0.021$$ and$$\alpha _s (M_{Z^0 } ) = 0.123_{ - 0.007}^{ + 0.006} $$, respectively, in. These measurements and their uncertainties are entirely independent of each other and from event shape and jet observables; the good agreement of the resulting αs values thus constitutes an important consistency check of the reliability of perturbative QCD.