Study of the Six-Loop Beta Function of the $\lambda\phi^4_4$ Theory

TL;DR

This paper examines the six-loop beta function of the mbda 4 theory to determine if it shows signs of an ultraviolet zero, using Pade9 approximants, and finds no strong evidence for such a zero.

Contribution

It extends previous five-loop analyses to six loops and applies Pade9 approximants to assess the ultraviolet behavior of the theory.

Findings

No robust evidence for an ultraviolet zero at six loops

Perturbative calculations remain reliable within certain mbda ranges

Results support the absence of an ultraviolet fixed point in this regime

Abstract

We investigate whether the six-loop beta function of the $\lambda \phi^4_4$ theory exhibits evidence for an ultraviolet zero. As part of our analysis, we calculate and analyze Pad\'e approximants to this beta function. Extending our earlier results at the five-loop level, we find that in the range of $\lambda$ where the perturbative calculation of the six-loop beta function is reliable, the theory does not exhibit robust evidence for an ultraviolet zero.