Branching Ratios and Spectral Functions of $\tau$ Decays: final ALEPH measurements and physics implications

The full LEP-1 data set collected with the ALEPH detector at the $Z$ pole during 1991-1995 is analysed in order to measure the $\tau$ decay branching fractions. The analysis follows the global method used in the published study based on 1991-1993 data, but several improvements are introduced, especially concerning the treatment of photons and $\pi^0$'s. Extensive systematic studies are performed, in order to match the large statistics of the data sample corresponding to over 300\,000 measured and identified $\tau$ decays. Branching fractions are obtained for the two leptonic channels and eleven hadronic channels defined by their respective numbers of charged particles and $\pi^0$'s. Using previously published ALEPH results on final states with charged and neutral kaons, corrections are applied to the hadronic channels to derive branching ratios for exclusive final states without kaons. Thus the analyses of the full LEP-1 ALEPH data are combined to yield a complete description of $\tau$ decays, encompassing 22 non-strange and 11 strange hadronic modes. Some physics implications of the results are given, in particular related to universality in the leptonic charged weak current, isospin invariance in $a_1$ decays, and the separation of vector and axial-vector components of the total hadronic rate. Finally, spectral functions are determined for the dominant hadronic modes and updates are given for several analyses. These include: tests of isospin invariance between the weak charged and electromagnetic hadronic currents, fits of the $\rho$ resonance lineshape, and a QCD analysis of the nonstrange hadronic decays using spectral moments, yielding the value $\alpha_s(m^2_\tau) = 0.340 \pm 0.005_{\rm exp} \pm 0.014_{\rm th}$. The evolution to the $Z$ mass scale yields $\alpha_s(M_Z^2) = 0.1209 \pm 0.0018$. This value agrees well with the direct determination from the $Z$ width and provides the most accurate test to date of asymptotic freedom in the QCD gauge theory.