Classical equation of motion and Anomalous dimensions at leading order

TL;DR

This work develops a conformal-field-theory framework to compute leading anomalous dimensions of composite operators at Wilson-Fisher fixed points by marrying conformal constraints on two- and three-point functions with the classical equation of motion (a Schwinger-Dyson perspective without contact terms). The method is applied to $oxed{ ext{phi}}^6$ in $(3- abla)\epsilon$, $oxed{ ext{phi}}^4$ in $(4- abla)\epsilon$, and $oxed{ ext{phi}}^3$ in $(6- abla)\epsilon$ dimensions, as well as an $O(N) imes$scalar generalization with $oxed{ ext{phi}_i ext{phi}_i ext{chi}}$ interactions. In each case, the two- and three-point data fix the fixed-point coupling $g_*$ and yield leading-order anomalous dimensions $ g_n$ that agree with known perturbative results, all without explicit loop calculations. The approach clarifies how conformal constraints and equations of motion encode the leading critical behavior and suggests avenues for extending to higher orders and more complex theories.

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 $φ^3$-theory in $(6-ε)$ 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.