Leading Order Anomalous Dimensions at the Wilson-Fisher Fixed Point from CFT

TL;DR

This paper extends a symmetry-based method to calculate leading order anomalous dimensions of operators at the Wilson-Fisher fixed point in conformal field theory, rederiving known results and providing new expressions for operators with various spins.

Contribution

It introduces an improved method for computing anomalous dimensions using symmetry arguments, applicable to operators with different spins and field numbers at the Wilson-Fisher fixed point.

Findings

Rederived known anomalous dimensions for certain operators.

Calculated new leading order anomalous dimensions for spin 2 and 3 operators.

Derived expressions for primary operators with arbitrary spin and fields.

Abstract

In this paper we consider $\phi^4$ theory in $4-\epsilon$ dimensions at the Wilson-Fisher fixed point where the theory becomes conformal. We extend the method in arXiv:1505.00963 for calculating the leading order term in the anomalous dimensions of some operators with spin. This method involves mostly symmetry arguments and reduces the process for calculating anomalous dimensions to some Wick contractions in the corresponding free theory. We apply this method in the case of operators with two and three fields whose twist is equal to the number of fields they contain, and we rederive known results for their anomalous dimensions. We also calculate the leading term in the anomalous dimensions of operators with spin two and three. In addition, we find expressions for the primary operators of the free theory, for arbitrary spin and number of fields, whose twist remains equal to the number of fields.