On the role of power expansions in quantum field theory

Methods of summation of power series relevant to applications in quantum theory are reviewed, with particular attention to expansions in powers of the coupling constant and in inverse powers of an energy variable. Alternatives to the Borel summation method are considered and their relevance to different physical situations is discussed. Emphasis is placed on quantum chromodynamics. Applications of the renormalon language to perturbation expansions (resummation of bubble chains) in various QCD processes are reported and the importance of observing the full renormalization-group invariance in predicting observables is emphasized. News in applications of the Borel-plane formalism to phenomenology are conveyed. The properties of the operator-product expansion along different rays in the complex plane are examined and the problem is studied how the remainder after subtraction of the first $n$ terms depends on the distance from euclidean region. Estimates of the remainder are obtained and their strong dependence on the nature of the discontinuity along the cut is shown. Relevance of this subject to calculations of various QCD effects is discussed.