Resurgence of the Renormalization Group Equation

TL;DR

This paper demonstrates how renormalons naturally arise from the renormalization group equation without Feynman diagrams by linking it to resurgent equations, revealing insights into non-perturbative effects on running couplings.

Contribution

It introduces a novel perspective by connecting renormalons to resurgent equations, providing a diagram-free derivation and implications for gauge theories.

Findings

Renormalons emerge from the RG equation as resurgent functions.

A one-to-one correspondence exists between resurgent equations and Borel resummation of renormalons.

Non-perturbative effects influence the behavior of running couplings.

Abstract

We show how the renormalons emerge from the renormalization group equation with a priori no reference to any Feynman diagrams. The proof is rather given by recasting the renormalization group equation as a resurgent equation studied in the mathematical literature, which describes a function with an infinite number of singularities in the positive axis of the Borel plane. Consistency requires a one-to-one correspondence between the existence of such kind of equation and the actual (generalized) Borel resummation of the renormalons through a one-parameter transseries. Our finding suggests how non-perturbative contributions can affect the running couplings. We also discuss these concepts within the context of gauge theories, making use of the large number of flavor expansion.