Study of the Question of an Ultraviolet Zero in the Six-Loop Beta Function of the O($N$) $\lambda |\vec \phi|^4$ Theory

TL;DR

This paper investigates whether a UV zero exists in the six-loop beta function of an O(N) λ|φ|^4 theory in four dimensions, finding evidence against its presence within the reliable perturbative regime.

Contribution

It provides a six-loop analysis of the beta function for O(N) λ|φ|^4 theory, extending previous lower-loop studies and assessing the existence of a UV zero.

Findings

No evidence of a UV zero in the six-loop beta function for the considered range

Supports the view that such a UV zero does not occur in this theory within perturbation theory

Extends understanding of the renormalization group flow in scalar field theories

Abstract

We study the possibility of an ultraviolet (UV) zero in the six-loop beta function of an O($N$) $\lambda |\vec \phi|^4$ field theory in $d=4$ spacetime dimensions. For general $N$, in the range of values of $\lambda$ where a perturbative calculation is reliable, we find evidence against such a UV zero in this six-loop beta function.