Study of the Question of an Ultraviolet Zero in the Six-Loop Beta Function of the O($N$) $\lambda \mathrm{\mid }\vec{\varphi }{\mathrm{\mid }}^4$ Theory

We study the possibility of an ultraviolet (UV) zero in the six-loop beta function of an O($N$) $\lambda \mathrm{\mid }\vec{\varphi }{\mathrm{\mid }}^4$ 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.