Vanishing Beta Function curves from the Functional Renormalisation Group

TL;DR

This paper develops a method using vanishing beta function curves within the Functional Renormalisation Group framework to analyze fixed points in O(N) theories across various dimensions, revealing known and new fixed points, including in higher dimensions.

Contribution

It introduces a systematic approach to derive vanishing beta function curves for exploring fixed points in O(N) theories across arbitrary dimensions and field components.

Findings

Restoration of Mermin-Wagner theorem in D≤2

Presence of Wilson-Fisher fixed point in 2<D<4

Identification of a new fixed point in 4<D<6 dimensions

Abstract

In this paper we will discuss the derivation of the so-called vanishing beta function curves which can be used to explore the fixed point structure of the theory under consideration. This can be applied to the O($N$) symmetric theories, essentially, for arbitrary dimensions ($D$) and field component ($N$). We will show the restoration of the Mermin-Wagner theorem for theories defined in $D\leq2$ and the presence of the Wilson-Fisher fixed point in $2<D<4$. Triviality is found in $D>4$. Interestingly, one needs to make an excursion to the complex plane to see the triviality of the four-dimensional O($N$) theories. The large-$N$ analysis shows a new fixed point candidate in $4<D<6$ dimensions which turns out to define an unbounded fixed point potential supporting the recent results by R. Percacci and G. P. Vacca in: "Are there scaling solutions in the O($N$) models for large-$N$ in $D>4$?" [Phys. Rev. D 90, 107702 (2014)].