# Revisiting the dilatation operator of the Wilson-Fisher fixed point

## TL;DR

This paper re-examines the order epsilon dilatation operator at the Wilson-Fisher fixed point using symmetry principles, showing that all scaling dimensions at this order can be determined by conformal invariance.

## Contribution

It provides an algebraic, symmetry-based derivation of the order epsilon dilatation operator, connecting perturbative and conformal field theory results.

## Key findings

- All order-epsilon scaling dimensions can be fixed by symmetry.
- The form of the dilatation operator is determined up to an infinite set of coefficients.
- Recent CFT results confirm the coefficients without perturbation theory.

## Abstract

We revisit the order $\varepsilon$ dilatation operator of the Wilson-Fisher fixed point obtained by Kehrein, Pismak, and Wegner in light of recent results in conformal field theory. Our approach is algebraic and based only on symmetry principles. The starting point of our analysis is that the first correction to the dilatation operator is a conformal invariant, which implies that its form is fixed up to an infinite set of coefficients associated with the scaling dimensions of higher-spin currents. These coefficients can be fixed using well-known perturbative results, however, they were recently re-obtained using CFT arguments without relying on perturbation theory. Our analysis then implies that all order-$\varepsilon$ scaling dimensions of the Wilson-Fisher fixed point can be fixed by symmetry.