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Recently it was shown that the scaling dimension of the operator $\phi^n$ in$\lambda(\phi^*\phi)^2$ theory may be computed semi-classically at theWilson-Fisher fixed point in $d=4-\epsilon$, for generic values of $\lambda n$and this was verified to two loop order in perturbation theory at leading andsub-leading $n$. In subsequent work, this result was generalised to operatorsof fixed charge $Q$ in $O(N)$ theory and verified up to three loops inperturbation theory at leading and sub-leading order. Here we extend thisverification to four loops in $O(N)$ theory, once again at leading andsub-leading order. We also investigate the strong-coupling regime.

Anomalous dimensions at large charge in d=4 O(N) theory