Scaling dimensions at large charge for cubic $\phi^3$ theory in six dimensions

TL;DR

This paper computes the scaling dimensions of large charge operators in six-dimensional cubic $\

Contribution

It provides a perturbative three-loop calculation of large charge scaling dimensions in six-dimensional cubic $\

Findings

Matching of scaling dimensions with previous semiclassical results at leading and next-to-leading order.

Extension of calculations to three-loop order and $rac{1}{N^3}$ corrections.

Confirmation of the conjectured equivalence between $O(N)$ models with quartic and cubic interactions.

Abstract

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.