Aspects of the Functional Renormalisation Group

TL;DR

This paper explores the structural aspects of the functional renormalisation group, including symmetry relations, optimization, and applications to gauge theories and practical truncation schemes.

Contribution

It provides a comprehensive analysis of the functional renormalisation group, including symmetry relations, optimization criteria, and applications to gauge theories and composite operators.

Findings

Derived flow equations for correlation functions

Established symmetry relations in regularised theories

Proposed optimization and truncation schemes

Abstract

We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being NPI quantities. We discuss optimisation and derive a functional optimisation criterion. Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson-Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation.