Search for an Ultraviolet Zero in the Seven-Loop Beta Function of the $\lambda \phi^4_4$ Theory

TL;DR

This study examines the seven-loop beta function of the mbda _4 theory to determine if an ultraviolet zero exists, using advanced mathematical techniques and confirming previous findings of no such zero.

Contribution

The paper provides a detailed seven-loop analysis of the mbda _4 theory's beta function, extending earlier five- and six-loop studies and exploring scheme transformation effects.

Findings

No evidence of an ultraviolet zero in the seven-loop beta function within the reliable perturbative range.

Use of Pade9 approximants supports the absence of an ultraviolet fixed point.

Scheme transformations do not alter the conclusion of no ultraviolet zero.

Abstract

We investigate whether the seven-loop beta function of the $\lambda \phi^4_4$ theory exhibits evidence for an ultraviolet zero. In addition to a direct analysis of the beta function, we calculate and study Pad\'e approximants and discuss effects of scheme transformations on the results. Confirming and extending our earlier studies of the five-loop and six-loop beta functions, we find that in the range of $\lambda$ where the perturbative calculation of the seven-loop beta function is reliable, the theory does not exhibit evidence for an ultraviolet zero.