Se p 20 00 The π 2 terms in the s – channel QCD observables

We analyze the effect of π 2 terms in the QCD perturbative expansions for the s–channel effective coupling and observables and discuss their influence on the numerical values of strong coupling ¯ α s as extracted from existing experiments. Our first result is that the common two-loop (NLO, NLLA) approximation widely used in the five-quark (10 GeV √ s 170 GeV) region for a shape analysis contains a systematic theoretical negative error of the 1–2 per cent order of magnitude for the extracted ¯ α (2) s values. The second comment is that usual three-loop (NNLO, NNLLA) theoretical expression for observables used in the four-quark and five-quark (2GeV √ s 40GeV) region for the time–like event analysis yields negative error of a few per cent level for the ¯ α (3) s values. Our summary conclusion is that strong invariant coupling value at M Z being extracted from HE f = 5 events is close to < ¯ α s (M 2 Z) > f =5 ≃ 0.124. Usually, physical quantities in the time-like channel, like the cross-section ratio of the inclusive e + e − → hadron annihilation or the τ –decay process, are presented in the form of two-or three-term perturbation expansion (our normalization of coefficients c k = C k π −k is different from the usually adopted, like in Refs.[1, 2, 3], one)