Anomalous dimensions of leading twist conformal operators

TL;DR

This work develops a conformal-field-theory framework to efficiently compute anomalous dimensions and mixing matrices of leading-twist conformal primaries by relating operator divergences to conformal descendants through equations of motion. The key idea, Anselmi's trick, uses the ratio of two-point functions of descendants and primaries to extract the anomalous dimension, gaining one perturbative order. The authors apply the method to a six-dimensional $\phi^3$ theory and to ${\cal N}=4$ SYM, deriving explicit leading-order dimensions and mixing matrices, and show agreement with conformal Ward identities. They further elucidate the origin of mixing via conformal anomalies and provide a scheme that diagonalizes the anomalous-dimension matrix, with potential extensions to full QCD and higher-loop analyses for exclusive processes.

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.

Figures (4)

• Figure 1: Feynman diagrams for the evaluation of anomalous dimensions ($a$ and $b$) and mixing matrix ($c$) in $\phi^3$ theory. The symbol $\otimes$ stands for the conformal operator.
• Figure 2: Feynman diagrams for the evaluation of anomalous dimensions in maximally supersymmetric gauge theory.
• Figure 3: Feynman diagrams needed for the evaluation of the mixing matrix in $\mathcal{N} = 4$ SYM.
• Figure 4: The symbol $\otimes$ stands for the conformal operator and $\circ$ stands for the operator insertions $\mathcal{O}_{A_i}$.