Classical equation of motion and Anomalous dimensions at leading order

TL;DR

This paper introduces a method to calculate leading-order anomalous dimensions of composite operators in conformal field theories using only classical equations of motion and symmetry constraints, avoiding Feynman diagrams.

Contribution

It presents a novel, more perturbative approach to determine anomalous dimensions at leading order solely from classical equations and conformal symmetry.

Findings

Method determines leading anomalous dimensions without Feynman diagrams.

Applicable to models like $$-theory in $(6-)$ dimensions.

Relies on classical equations of motion and conformal constraints.

Abstract

Motivated by a recent paper by Rychkov-Tan \cite{Rychkov:2015naa}, we calculate the anomalous dimensions of the composite operators at the leading order in various models including a $\phi^3$-theory in $(6-\epsilon)$ dimensions. The method presented here relies only on the classical equation of motion and the conformal symmetry. In case that only the leading expressions of the critical exponents are of interest, it is sufficient to reduce the multiplet recombination discussed in \cite{Rychkov:2015naa} to the classical equation of motion. We claim that in many cases the use of the classical equations of motion and the CFT constraint on two- and three-point functions completely determine the leading behavior of the anomalous dimensions at the Wilson-Fisher fixed point without any input of the Feynman diagrammatic calculation. The method developed here is closely related to the one presented in \cite{Rychkov:2015naa} but based on a more perturbative point of view.