On the local Borel transform of Perturbation Theory

TL;DR

This paper proves the existence of the local Borel transform for the perturbative series in massive ^4-theory, providing sharper bounds and a simpler proof approach compared to previous work.

Contribution

It introduces a new, simpler induction method using flow equations to establish sharper bounds on the Borel transform in quantum field theory.

Findings

Sharper bounds on the Borel transform dependence on external momenta

Explicit bounds related to the number of external legs

A simplified proof technique avoiding large constants

Abstract

We prove existence of the local Borel transform for the perturbative series of massive $\vp_4^4$-theory. As compared to previous proofs in the literature, the present bounds are much sharper as regards the dependence on external momenta, they are explicit in the number of external legs, and they are obtained quite simply through a judiciously chosen induction hypothesis applied to the Wegner-Wilson-Polchinski flow equations. We pay attention not to generate an astronomically large numerical constant for the inverse radius of convergence of the Borel transform.