# Non-local Lagrangians from Renormalons and Analyzable Functions

## TL;DR

This paper introduces a generalized Borel procedure to incorporate renormalons and renormalization, revealing that effective Lagrangians affected by renormalons are non-local in space but local in time, and connects analyzable functions with non-perturbative insights.

## Contribution

It presents a novel approach to understanding renormalons through a generalized Borel framework, linking analyzable functions to non-perturbative renormalization.

## Key findings

- Effective Lagrangians are non-local in space, local in time.
- Connection established between analyzable functions and the Callan-Symanzik equation.
- Insights into non-perturbative renormalization from perturbative methods.

## Abstract

We embed in a generalized Borel procedure the notion of renormalization and renormalons. While there are several efforts in literature to have a semi-classical understanding of the renormalons, here we argue that this is not the fundamental issue and show how to deal with the problem. We find that the effective Lagrangians describing the effects of renormalons are non-local in space but local in time. The quark-antiquark potential in QCD with an infinite number of fermions is also analyzed. The connection between the analyzable functions, the Callan-Symanzyk equation and the renormalons, provides an insight of a non-perturbative renormalization from the standard perturbative renormalization approach.