Resummation of Renormalon Chains for Cross Sections and Inclusive Decay Rates

Abstract

Recently, we have developed a formalism to evaluate QCD loop diagrams with a single virtual gluon using a running coupling constant at the vertices. This corresponds to an all-order resummation of certain terms (the so-called renormalon chains) in a perturbative series and provides a generalization of the scale-setting prescription of Brodsky, Lepage and Mackenzie. In its original form, the method is applicable to Green functions without external gluons and to euclidean correlation functions. Here we generalize the approach to the case of cross sections and inclusive decay rates, which receive both virtual and real gluon corrections. We encounter nonperturbative ambiguities in the resummation of the perturbative series, which may hinder the construction of the operator product expansion in the physical region. The origin of these ambiguities and their relation to renormalon singularities in the Borel plane is investigated by introducing an explicit infrared cutoff. The ratios $R_{e^+ e^-}$ and $R_τ$ are discussed in detail.