Massless picture, massive picture, and symmetry in the Gaussian renormalization group

TL;DR

This paper explores the structure of renormalization groups for Gaussian fields, revealing two equivalent formulations and a symmetry in the massive picture related to scale transformations and anomalous dimensions.

Contribution

It introduces the massless and massive pictures of Gaussian renormalization groups and demonstrates a symmetry in the massive picture linked to scale transformations.

Findings

Two equivalent formulations of the renormalization group: massless and massive pictures.

Existence of a symmetry in the massive picture involving scale transformations.

Relation between symmetry and anomalous dimensions is discussed.

Abstract

We consider renormalization groups of transformations composed of a Gaussian convolution and a field dilatation. As an example, we consider perturbations of a single component real Euclidean free field $\phi$ with covariance $(-\bigtriangleup)^{-1+\frac{\epsilon}{2}}$. We show that the renormalization group admits two equivalent formulations called massless picture and massive picture respectively. We then show in the massive picture that the renormalization group has a symmetry. The symmetry consists of global scale transformations composed with certain Gaussian convolutions. We translate the symmetry back to the massless picture. The relation between the symmetry and the notion of an anomalous dimension is briefly discussed.