Elements of the Continuous Renormalization Group

TL;DR

This paper discusses recent advances in the exact non-perturbative renormalization group, focusing on derivation, continuum limits, fixed points, and new insights into renormalizability and renormalons, with pedagogical explanations.

Contribution

It introduces new demonstrations of non-perturbative renormalizability and discusses ultraviolet renormalons within the framework of the exact renormalization group.

Findings

Demonstration of non-perturbative renormalizability

Analysis of ultraviolet renormalons

Clarification of fixed points and eigenoperators

Abstract

These two lectures cover some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will describe how real non-perturbative calculations can be done, particularly within derivative expansion approximations. An effort has been made to keep the lectures pedagogical and self-contained. Topics covered are the derivation of the flow equations, their equivalence, continuum limits, perturbation theory, truncations, derivative expansions, identification of fixed points and eigenoperators, and the role of reparametrization invariance. Some new material is included, in particular a demonstration of non-perturbative renormalizability, and a discussion of ultraviolet renormalons.