Simple But Efficient Tests of QCD from Tau Decays$^{*)}$

Abstract

We review the determinations of the QCD coupling $\alpha_s$ from the inclusive and exclusive modes of $\tau$-decays. The most recent $\tau$-data provide the average value $ \alpha_ s \left(M^2_\tau \right)\simeq 0.347\pm 0.030 $ corresponding to $ \alpha_ s \left(M^2_Z \right) \simeq 0.121\pm 0.003 \pm 0.001.$ The values of the QCD vacuum condensates extracted from the inclusive weighted-moment distributions are consistent with the ones from QCD spectral sum rules. Accurate estimates and further precision tests should be reached in the next $\tau$C/B-factory machines. The $M_\tau$-stability test from the $ e^+e^- \longrightarrow I=1 $ hadron data is also discussed.