Leading Order Anomalous Dimensions at the Wilson-Fisher Fixed Point from CFT
Konstantinos Roumpedakis
TL;DR
The paper extends a symmetry-based conformal approach to compute leading-order anomalous dimensions at the Wilson-Fisher fixed point for spinning operators in $4-\\epsilon$ dimensions. By relating interacting operators to free-theory primaries through the equation of motion and carefully analyzing OPE data, the authors express anomalous dimensions in terms of free-theory coefficients and derive explicit results for operators with two and three fields, including spins $2$ and $3$. They construct primary operators with twist equal to the number of fields, address degeneracies among multi-field operators, and generalize the method to arbitrary spin, obtaining consistent results with known literature. The work solidifies a program where conformal symmetry and free-theory data largely determine leading corrections without exhaustive Feynman diagrammatics, with potential implications for AdS/CFT and critical phenomena. Overall, it provides a concrete, symmetry-driven framework for predicting anomalous dimensions at the WF fixed point across a broad class of operators.
Abstract
In this paper we consider $φ^4$ theory in $4-ε$ 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.