# New universality class in three dimensions: The critical Blume-Capel model

## TL;DR

This paper investigates the critical behavior of the Blume-Capel model in fractional dimensions near three, using RG techniques to compute scaling exponents and explore its universality class, complementing conformal field theory results.

## Contribution

It introduces a three-loop RG analysis of the Blume-Capel universality class in fractional dimensions, providing new insights into its critical exponents and relation to other multicritical models.

## Key findings

- Computed scaling exponents at three-loop order.
- Estimated OPE coefficients to leading order.
- Identified a family of nonunitary multicritical models.

## Abstract

We study the Blume-Capel universality class in $d=\frac{10}{3}-\epsilon$ dimensions. The RG flow is extracted by looking at poles in fractional dimension of three loop diagrams using $\overline{\rm MS}$. The theory is the only nontrivial universality class which admits an expansion to three dimensions with $\epsilon=\frac{1}{3}<1$. We compute the relevant scaling exponents and estimate some of the OPE coefficients to the leading order. Our findings agree with and complement CFT results. Finally we discuss a family of nonunitary multicritical models which includes the Lee-Yang and Blume-Capel classes as special cases.