On the Question of an Ultraviolet Zero of the Beta Function of the $\lambda (\vec{\phi}^2)^2_4$ Theory

We investigate the possibility of an ultraviolet (UV) zero in the $n$-loop beta function of a $\lambda (\vec{\phi}^2)^2$ field theory with an $N$-component scalar field, $\vec{\phi}$, in four spacetime dimensions, up to the level of $n=5$ loops. Although the two-loop beta function has a UV zero, we find that the $n$-loop beta function for $n=3, \ 4, \ 5$ either does not have a UV zero or does not have one at a value of $\lambda$ in approximate agreement with the two-loop calculation. Similar results are obtained after application of scheme transformations to the beta function and via calculation of Pad\'e approximants. We thus conclude that in the range of $\lambda$ where the perturbative calculation of the $n$-loop beta function is reliable, the theory does not exhibit robust evidence of a UV zero up to the level of $n=5$ loops.