Using the background field method, we, in the large N f approximation, calculate the beta function of scalar quantum electrodynamics at the first nontrivial order in 1/N f by two different ways. In the first way, we get the result by summing all the graphs contributing directly. In the second way, we begin with the Borel transform of the related two point Green's function. The main results are that the beta function is fully determined by a simple function and can be expressed as an analytic expression with a finite radius of convergence, and the scheme-dependent renormalized Borel transform of the two point Green's function suffers from renormalons.

The analytical method of the QCD running coupling constant is extended to a model with an all-order beta function which is inspired by the famous Novikov-ShifmanVainshtein-Zakharov beta function of N = 1 supersymmetric gauge theories. In the approach presented here, the running coupling is determined by a transcendental equation with non-elementary integral of the running scale μ. In our approach α an (0), which reads 0.30642, does not rely on any dimensional parameters. This is in accordance with results in the literature on the analytical method of the QCD running coupling constant. The new "analytically improved" running coupling constant is also compatible with the property of asymptotic freedom.