  The $O(N)$ model with scalar quartic interactions at its ultraviolet fixed
point, and the $O(N)$ model with scalar cubic interactions at its infra-red
fixed point are conjectured to be equivalent. This has been checked by
comparing various features of the two models at their respective fixed points.
Recently, the scaling dimensions of a family of operators of fixed charge $Q$
have been shown to match at the FPs up to $\cal{O}\left(\frac{1}{N^2}\right)$at
leading order (LO) and next-to-leading order (NLO) in $Q$ using a semiclassical
computation which is valid to all orders in the coupling. Here we perform a
complementary but overlapping comparison using a perturbative calculation in
six dimensions, up to three-loop order in the coupling, to compare these
critical scaling dimensions beyond NLO in $Q$, in fact to all relevant orders
in $Q$. We also obtain the corresponding results at
$\cal{O}\left(\frac{1}{N^3}\right)$ for the cubic theory.
