Properties of the linearised functional renormalization group

TL;DR

This paper investigates the properties and limitations of the linearized functional renormalization group, revealing how certain interactions behave well, while others lead to singularities and spontaneous interaction emergence.

Contribution

It provides a detailed analysis of the behavior of interactions under the linearized functional renormalization group, highlighting conditions for well-behaved flows and identifying challenges with faster-growing interactions.

Findings

Interactions slower than exponential of scalar field are well-behaved.

Flows can end prematurely in singularities for faster-growing interactions.

New interactions can spontaneously appear at any scale.

Abstract

Interactions growing slower than a certain exponential of the square of a scalar field, are well behaved when evolved under the functional renormalization group linearised around the Gaussian fixed point. They satisfy properties usually taken for granted, and reproduce standard perturbative quantisation. However, ever more challenging effects appear the more interactions grow faster than this. We show explicitly that firstly the flow no longer splits uniquely into operators of definite scaling dimension; then (linearised) flows to the infrared can end prematurely in a singularity; and finally new interactions can spontaneously appear at any scale.