# Conformal invariance and vector operators in the $O(N)$ model

## TL;DR

This paper investigates whether the $O(N)$ model exhibits conformal invariance by computing the scaling dimensions of vector operators using multiple approximation schemes, finding results that support conformal invariance in these models.

## Contribution

It provides a comprehensive analysis of vector operator scaling dimensions in the $O(N)$ model across three approximation methods, supporting the conjecture of conformal invariance.

## Key findings

- All considered vector operators have scaling dimensions much larger than -1.
- Results support the existence of conformal invariance in the $O(N)$ model.
- Generalized correlation function inequalities provide bounds for vector operator dimensions.

## Abstract

It is widely expected that, for a large class of models, scale invariance implies conformal invariance. A sufficient condition for this to happen is that there exists no integrated vector operator, invariant under all internal symmetries of the model, with scaling dimension $-1$. In this article, we compute the scaling dimensions of vector operators with lowest dimensions in the $O(N)$ model. We use three different approximation schemes: $\epsilon$ expansion, large $N$ limit and third order of the Derivative Expansion of Non-Perturbative Renormalization Group equations. We find that the scaling dimensions of all considered integrated vector operators are always much larger than $-1$. This strongly supports the existence of conformal invariance in this model. For the Ising model, an argument based on correlation functions inequalities was derived, which yields a lower bound for the scaling dimension of the vector perturbations. We generalize this proof to the case of the $O(N)$ model with $N\in \left\lbrace 2,3,4 \right\rbrace$.