On three-point correlation functions in the gauge/gravity duality

Abstract

We study the effect of marginal and irrelevant deformations on the renormalization of operators near a CFT fixed point. New divergences in a given operator are determined by its OPE with the operator $$ \mathcal{D} $$ that generates the deformation. This provides a scheme to compute the couplings $$ {a_{\mathcal{D}AB}} $$ between the operator $$ \mathcal{D} $$ and two arbitrary operators $$ {\mathcal{O}A} $$ and $$ {\mathcal{O}B} $$. We exemplify for the case of $$ \mathcal{N} = 4 $$ SYM, considering the simplest case of the exact Lagrangian deformation. In this case the deformed anomalous dimension matrix is determined by the derivative of the anomalous dimension matrix with respect to the coupling. We use integrability techniques to compute the one-loop couplings $$ {a{\mathcal{L}AB}} $$ between the Lagrangian and two distinct large operators built with Magnons, in the SU(2) sector of the theory. Then we consider $$ {a{\mathcal{D}AA}} $$ at strong coupling, and show how to compute it using the gauge/gravity duality, when $$ \mathcal{D} $$ is a chiral operator dual to any supergravity field and $$ {\mathcal{O}_A} $$ is dual to a heavy string state. We exemplify for the Lagrangian and operators $$ {\mathcal{O}_A} $$ dual to heavy string states, showing agreement with the prediction derived from the renormalization group arguments.