The derivative expansion of the renormalization group

TL;DR

This paper develops a derivative expansion method for the renormalization group flow equations of the effective action, enabling the study of non-perturbative continuum limits and phase transitions in scalar field theories across multiple dimensions.

Contribution

It introduces a derivative expansion approach to the flow equations of the effective action, providing a robust tool for analyzing non-perturbative fixed points and critical phenomena.

Findings

Effective equations derived for fixed points and anomalous dimensions.

Application to scalar field theories in 2, 3, and 4 dimensions.

Method successfully identifies and characterizes phase transitions.

Abstract

By writing the flow equations for the continuum Legendre effective action (a.k.a. Helmholtz free energy) with respect to a particular form of smooth cutoff, and performing a derivative expansion up to some maximum order, a set of differential equations are obtained which at FPs (Fixed Points) reduce to non-linear eigenvalue equations for the anomalous scaling dimension $\eta$. Illustrating this by expanding (single component) scalar field theory, in two, three and four dimensions, up to second order in derivatives, we show that the method is a powerful and robust means of discovering and quantifying non-perturbative continuum limits (continuous phase transitions).