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

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