  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.
