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Renormalons and Analytic Properties of the βfunction

The presence or absense of renormalon singularities in the Borel plane is shown to be determined by the analytic properties of the Gell-Mann - Low function β(g) and some other functions. A constructive criterion for the absense of singularities consists in the proper behavior of the βfunction and its Borel image B(z) at infinity, β(g)\sim g^αand B(z)\sim z^αwith α\le 1. This criterion is probably fulfilled for the ϕ^4 theory, QED and QCD, but is violated in the O(n)-symmetric sigma model with n\to\infty.