New perturbation expansions in quantum chromodynamics and the determination of $\alpha_s$

Abstract

We consider a new class of perturbation expansions, which incorporate in a systematic way the available information about the divergent character of the perturbation series in QCD. The new expansion functions, which replace the powers of the coupling, are defined by the analytic continuation in the Borel plane, achieved through an optimal conformal mapping. We consider several possibilities of implementing the known behaviour of the Borel transform near the leading singularities in the complex plane and using the corresponding expansions. These expansions have the same asymptotic convergence rate, but differ at low orders. We show that the new expansions allow a precise determination of the strong coupling $\alpha_s$ from the hadronic decays of the $\tau$ lepton.