Decay widths and total cross sections in perturbative QCD

The current status of analytic higher-order perturbative computations of total cross sections and decay widths in Quantum Chromodynamics is reviewed. Important issues are the methodology of renormalization-group evaluations, the ambiguities of the renormalization scheme and its scale, and the technical challenge of calculating many-loop diagrams. As examples, the authors consider the quantities ${\sigma }_{\mathrm{tot}}(e^{+}{e}^{-}\to \mathrm{hadrons})$ and $\Gamma ({\tau }^{-}\to {\nu }_{\tau }+\mathrm{hadrons})$ up to $O\left({\alpha }_s^3\right)$ as well as $\Gamma (H\to \mathrm{hadrons})$ up to $O\left({\alpha }_s^2\right)$. The evaluation of the four-loop QED beta function is also described. The problem of theoretical uncertainty estimates in perturbative calculations is briefly discussed.