R(s) and hadronic τ-Decays in Order α4 s

Three important physical observables, namely, the ratio R(s) = σ(e e→hadrons) σ(e+e−→μ+μ−) , the hadronic decay rate of the Z-boson and the semileptonic branching ratio of the τ -lepton are expressed through the vector and axial-vector current correlators (see, e. g. reviews [4,5]). Perturbative QCD provides reliable predictions for these correlators in the continuum, i.e. sufficiently above the respective quark threshold and the respective resonance region. The O(αs) result for the massless vector correlator has been known since many years [6,7]. Recently the calculation of the next, order αs contribution to the vector correlator has been performed [1,2,3]. The aim of the present work is to discuss some technical aspects of our calculations as well as to provide some new arguments in favour of their correctness. Due to lack of space no phenomenological im-