Revisiting Extremal Couplings in AdS/CFT

TL;DR

This work analyzes cubic extremal bulk couplings in AdS$_{d+1}$/CFT$_d$ with $\Delta_3 = {\Delta_1}+{\Delta_2}$, showing the corresponding vertex diverges and can be renormalized using counterterms to yield a finite, renormalized on-shell action. The renormalization introduces a logarithmic contribution to the boundary three-point function and lifts the degeneracy between single- and double-trace operators, producing a leading-order anomalous dimension described by $\Delta_{\pm} = \Delta_1+\Delta_2 \mp \tilde{\gamma}$ with $\tilde{\gamma}$ given by a precise combination of gamma functions and the coupling $\lambda_{\rm ext}$. The analysis also extends to super-extremal and shadow-extremal cases, revealing similar renormalization structures and highlighting subtleties in interpreting the shadow case within a unitary CFT framework. Overall, the paper provides a concrete bottom-up holographic renormalization of extremal bulk couplings, clarifying how multi-trace mixing and anomalous dimensions arise in AdS/CFT and informing EFT constructions in holography.

Abstract

We consider an effective theory of massive scalar fields on a fixed AdS$_{d+1}$ background with a cubic extremal interaction among them. A bulk coupling is called extremal whenever the corresponding conformal dimension of any of the dual CFT$_d$ operators matches the sum of all the others. For cubic bulk couplings, this is $Δ_i+Δ_j=Δ_k$. These bulk interactions are often disregarded in the literature since they do not appear in traditional models of AdS/CFT. Turning them on yields a divergent vertex in the dual CFT, and here we show that these divergences can be regulated. Once renormalized, we demonstrate that this coupling introduces non-trivial mixing between single- and double-trace operators, and we compute the anomalous dimensions of the corrected operators to leading order in perturbation theory.