Recursion and growth estimates in renormalizable quantum field theory

TL;DR

This paper establishes a relationship between the convergence bounds of divergent and convergent Green functions in renormalizable quantum field theory, linking the radius of convergence to the instanton radius and the beta function coefficient.

Contribution

It proves that the Lipatov bound for convergent Green functions extends to divergent ones, providing a formula for the radius of convergence in terms of known quantities.

Findings

Radius of convergence is min{ρ, 1/b_1}

Lipatov bound applies to divergent Green functions

Connects convergence bounds with instanton radius and beta function

Abstract

In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. The radius of convergence turns out to be ${\rm min}\{\rho,1/b_1\}$, where $\rho$ is the bound on the convergent ones, the instanton radius, and $b_1$ the first coefficient of the $\beta$-function.