Higgs Decay, $Z$ Decay and the QCD Beta Function

Recent developments in perturbative QCD, leading to the beta function in five-loop approximation are presented. In a first step the two most important decay modes of the Higgs boson are discussed: decays into a pair of gluons and, alternatively, decays into a bottom-antibottom quark pair. Subsequently the quark mass anomalous dimension is presented which is important for predicting the value of the bottom quark mass at high scales and, consequently, the Higgs boson decay rate into pair of massive quarks, in particular into $b\bar b$. In the next section the $\alpha_s^4$ corrections to the vector- and axial-vector correlator are discussed. These are the essential ingredients for the evaluation of the QCD corrections to the cross section for electron-positron annihilation into hadrons at low and at high energies, to the hadronic decay rate of the $\tau$ lepton and for the $Z$-boson decay rate into hadrons. Finally we present the prediction for the QCD beta-function in five-loop approximation, discuss the analytic structure of the result and compare with experiment at low and at high energies.