Critical O(N) models above four dimensions: Small-N solutions and stability

TL;DR

This paper investigates O(N) models in dimensions between 4 and 6, revealing an ultraviolet fixed point for all N in 5D, and explores their stability and relation to known universality classes.

Contribution

It provides a functional RG analysis showing the existence of a fixed point down to N=1 in 5D and clarifies the embedding of O(N) classes within cubic models.

Findings

Fixed point exists for all N in 5D.

Cubic models share critical exponents with O(N) universality classes.

Fixed point potential is globally stable.

Abstract

We explore O(N) models in dimensions $4<d<6$. Specifically, we investigate models of an O(N) vector field coupled to an additional scalar field via a cubic interaction. Recent results in $d=6-\epsilon$ have uncovered an interacting ultraviolet fixed point of the renormalization group (RG) if the number N of components of the vector field is large enough, suggesting that these models are asymptotically safe. We set up a functional RG analysis of these systems to address three key issues: Firstly, we find that in $d=5$ the interacting fixed point exists all the way down to N=1. Secondly, we show that the standard O(N) universality classes are actually embedded in those of the cubic models, in that the latter exhibit the same values for (most of) the critical exponents, but feature an additional third RG relevant direction. Thirdly, we address the critical question of global stability of the fixed-point potential to test whether the fixed point can underly a viable quantum field theory.