Self-similar extrapolation in quantum field theory

Calculations in field theory arc usually accomplished by employing some variants of perturbation theory, for instance, using loop expansions. These calculations result in asymptotic series in powers of small coupling parameters, which as a rule are divergent for finite values of the parameters. In this paper, we describe a method allowing for the extrapolation of such asymptotic series to finite values of the coupling parameters and even to their infinite limits. The method is based on self-similar approximation theory. This theory approximates well a large class of functions, rational, irrational, and transcendental. We present a method resulting in self-similar factor approximants allowing for the extrapolation of functions to arbitrary values of coupling parameters from only the knowledge of expansions in powers of small coupling parameters. The efficiency of the method is illustrated by several problems of quantum field theory.