Anomalous dimensions of leading twist conformal operators

TL;DR

This paper introduces a new perturbative method for calculating anomalous dimensions and mixing matrices of leading twist conformal primary operators in conformal field theories, demonstrating its efficiency and equivalence to existing approaches.

Contribution

The paper develops a novel perturbative technique based on irreducibility breaking patterns to compute anomalous dimensions and mixing matrices, validated at two-loop order.

Findings

The method accurately computes anomalous dimensions at two-loop order.

It establishes equivalence with conformal Ward identity-based approaches.

The approach simplifies calculations of operator mixing in conformal field theories.

Abstract

We extend and develop a method for perturbative calculations of anomalous dimensions and mixing matrices of leading twist conformal primary operators in conformal field theories. Such operators lie on the unitarity bound and hence are conserved (irreducible) in the free theory. The technique relies on the known pattern of breaking of the irreducibility conditions in the interacting theory. We relate the divergence of the conformal operators via the field equations to their descendants involving an extra field and accompanied by an extra power of the coupling constant. The ratio of the two-point functions of descendants and of their primaries determines the anomalous dimension, allowing us to gain an order of perturbation theory. We demonstrate the efficiency of the formalism on the lowest-order analysis of anomalous dimensions and mixing matrices which is required for two-loop calculations of the former. We compare these results to another method based on anomalous conformal Ward identities and constraints from the conformal algebra. It also permits to gain a perturbative order in computations of mixing matrices. We show the complete equivalence of both approaches.