Background field method in the large $N_f$ expansion of scalar QED

TL;DR

This paper calculates the beta function of scalar QED in the large $N_f$ limit using the background field method, revealing an analytic form with finite convergence and discussing renormalons in the Borel transform.

Contribution

It provides a novel calculation of the scalar QED beta function at large $N_f$ using two different methods, highlighting its analytic structure and renormalon issues.

Findings

Beta function expressed as a simple analytic function

Finite radius of convergence of the beta function

Presence of renormalons in the Borel transform

Abstract

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.