Anomalous dimensions at large charge for U(N) x U(N) theory in three and four dimensions

Recently it was shown that the scaling dimension of the operator phi n in lambda(phi over bar phi)2 theory may be computed semiclassically at the Wilson-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 and subleading n. This result was subsequently generalized to operators of fixed charge Q in O(N) theory and verified up to four loops in perturbation theory at leading and subleading Q. More recently, similar semiclassical calculations have been performed for the classically scale-invariant U(N) x U(N) theory in four dimensions, and verified up to two loops, once again at leading and subleading Q. Here we extend this verification to four loops. We also consider the corresponding classically scale-invariant theory in three dimensions, similarly verifying the leading and subleading semiclassical results up to four loops in perturbation theory.