Properties of derivative expansion approximations to the renormalization group

TL;DR

This paper reviews how derivative expansion approximations, combined with reparametrization invariance, transform the renormalization group equations into nonlinear eigenvalue problems, aiding the discovery of non-perturbative continuum limits in quantum field theory.

Contribution

It provides a comprehensive review of derivative expansion methods and their convergence properties within the renormalization group framework, emphasizing reparametrization invariance and non-perturbative analysis.

Findings

Derivative expansions lead to nonlinear eigenvalue equations at fixed points.

Reparametrization invariance plays a crucial role in the approximation process.

The convergence of the derivative expansion is analyzed and discussed.

Abstract

Approximation only by derivative (or more generally momentum) expansions, combined with reparametrization invariance, turns the continuous renormalization group for quantum field theory into a set of partial differential equations which at fixed points become non-linear eigenvalue equations for the anomalous scaling dimension $\eta$. We review how these equations provide a powerful and robust means of discovering and approximating non-perturbative continuum limits. Gauge fields are briefly discussed. Particular emphasis is placed on the r\^ole of reparametrization invariance, and the convergence of the derivative expansion is addressed.