More on the cubic versus quartic interaction equivalence in the $O(N)$ model

TL;DR

This paper computes scaling dimensions of fixed-charge operators in the $O(N)$ cubic model near six dimensions and confirms their equivalence with the quartic model at large $N$, providing insights into finite $N$ physics.

Contribution

It demonstrates the equivalence between cubic and quartic $O(N)$ models at their fixed points by comparing scaling dimensions across models and orders.

Findings

Cubic and quartic models have matching scaling dimensions at large $N$.

Results provide new finite $N$ insights from cubic model computations.

Reinforces the conjectured equivalence between the models.

Abstract

We compute the scaling dimensions of a family of fixed-charge operators at the infrared fixed point of the $O(N)$ model featuring cubic interactions in $d=6-\epsilon$ for arbitrary $N$ to leading and subleading order in the charge but to all orders in the couplings. The results are used to analyze the conjectured equivalence with the $O(N)$ model displaying quartic interactions at its ultraviolet fixed point. This is performed by comparing the cubic model scaling dimensions against the known large $N$ results for the quartic model and demonstrating that they match. Our results reinforce the conjectured equivalence and further provide novel information on the finite $N$ physics stemming from the computations in the cubic model just below 6 dimensions.