Exact Renormalization Group Equations. An Introductory Review

TL;DR

This review introduces the exact renormalization group equations (ERGE) for scalar theories, emphasizing different versions and the derivative expansion approximation, highlighting nonperturbative features of Wilson's renormalization group.

Contribution

It provides an introductory overview of ERGE, focusing on scalar theories and the derivative expansion method, clarifying nonperturbative aspects and variations of ERGE.

Findings

Different versions of ERGE exist and are useful.

The derivative expansion effectively captures nonperturbative features.

Leading order of the derivative expansion serves as a textbook example.

Abstract

We critically review the use of the exact renormalization group equations (ERGE) in the framework of the scalar theory. We lay emphasis on the existence of different versions of the ERGE and on an approximation method to solve it: the derivative expansion. The leading order of this expansion appears as an excellent textbook example to underline the nonperturbative features of the Wilson renormalization group theory. We limit ourselves to the consideration of the scalar field (this is why it is an introductory review) but the reader will find (at the end of the review) a set of references to existing studies on more complex systems.