Anomalous dimensions at large charge for $U(N)\times U(N)$ theory in three and four dimensions

TL;DR

This paper extends the semiclassical computation of anomalous dimensions for operators with large charge in $U(N)\times U(N)$ theories to four loops in both three and four dimensions, confirming previous leading and subleading results.

Contribution

It provides a four-loop perturbative verification of semiclassical predictions for large charge operators in $U(N)\times U(N)$ theories in three and four dimensions.

Findings

Four-loop verification of semiclassical results in four dimensions

Four-loop verification in three dimensions

Confirmation of leading and subleading large charge behavior

Abstract

Recently it was shown that the scaling dimension of the operator $\phi^n$ in $\lambda(\bar\phi\phi)^2$ 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 generalised 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 $U(N)\times U(N)$ 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.