Anomalous Dimensions at Large Charge in \(d=4\ \mathrm {O}(N)\) Theory

Recently it was shown that the scaling dimension of the operator $\phi^n$ n in $\lambda(\bar{\phi}\phi)^2$ theory may be computed semiclassically at the Wilson–Fisher fixed point in $d = 4 -e$, for generic values of λn, and this was verified to two-loop order in perturbation theory at leading and subleading n. In subsequent work, this result was generalised to operators of fixed charge $\bar{Q}$ in $\mathrm {O}(N)$ theory and verified up to three loops in perturbation theory at leading and subleading $\bar{Q}$. Here, we extend this verification to four loops in $\mathrm {O}(N)$ theory, once again at leading and subleading $\bar{Q}$. We also investigate the strong-coupling regime.