Scaling and superscaling solutions from the functional renormalization group

TL;DR

This paper investigates the fixed points and scaling solutions of scalar models with and without supersymmetry using functional renormalization group methods, revealing their connection to conformal field theories and universality classes across different dimensions.

Contribution

It introduces new analytic and numerical techniques to analyze the fixed points, spectrum, and critical exponents of scalar models, including supersymmetric cases, across various dimensions.

Findings

Identification of fixed points as minimal models in 2D

Connection of solutions to Wilson-Fisher universality in 3D

Determination of critical dimensions where universality classes change

Abstract

We study the renormalization group flow of $\mathbb{Z}_2$-invariant supersymmetric and non-supersymmetric scalar models in the local potential approximation using functional renormalization group methods. We focus our attention to the fixed points of the renormalization group flow of these models, which emerge as scaling solutions. In two dimensions these solutions are interpreted as the minimal (supersymmetric) models of conformal field theory, while in three dimension they are manifestations of the Wilson-Fisher universality class and its supersymmetric counterpart. We also study the analytically continued flow in fractal dimensions between 2 and 4 and determine the critical dimensions for which irrelevant operators become relevant and change the universality class of the scaling solution. We also include novel analytic and numerical investigations of the properties that determine the occurrence of the scaling solutions within the method. For each solution we offer new techniques to compute the spectrum of the deformations and obtain the corresponding critical exponents.