On critical models with $N\leq 4$ scalars in $d=4-\epsilon$

TL;DR

This paper investigates the renormalization group flow of scalar field theories with quartic interactions in 4−ε dimensions for N=3 and N=4, identifying critical points and novel solutions with discrete symmetries.

Contribution

It provides a combined analytical and numerical analysis of critical points in scalar theories with N=3 and N=4, discovering new solutions with discrete symmetries and multiple anomalous dimensions.

Findings

N=3 admits three critical points: Wilson-Fisher, cubic, and biconical.

N=4 reveals new nontrivial solutions with discrete symmetries.

Identification of critical points with up to three distinct anomalous dimensions.

Abstract

We adopt a combination of analytical and numerical methods to study the renormalization group flow of the most general field theory with quartic interaction in $d=4-\epsilon$ with $N=3$ and $N=4$ scalars. For $N=3$, we find that it admits only three nondecomposable critical points: the Wilson-Fisher with $O(3)$ symmetry, the cubic with $H_3=(\mathbb{Z}_2)^3\rtimes S_3$ symmetry, and the biconical with $O(2)\times \mathbb{Z}_2$. For $N=4$, our analysis reveals the existence of new nontrivial solutions with discrete symmetries and with up to three distinct field anomalous dimensions.