Four-point correlators with higher weight superconformal primaries in the AdS/CFT Correspondence

TL;DR

This paper computes four-point correlators involving higher weight superconformal primaries in AdS/CFT, using harmonic polynomial formalism to handle complex tensor sums, and finds the supergravity Lagrangian is of sigma-model type with a split amplitude.

Contribution

It introduces an alternative harmonic polynomial formalism for evaluating tensor sums for higher conformal weights in AdS/CFT calculations.

Findings

Supergravity Lagrangian is of sigma-model type with no four-derivative couplings.

Connected amplitude splits into free and interacting parts.

Formalism extends to primaries with larger conformal dimensions.

Abstract

The four-point correlation function of two 1/2 BPS primaries of conformal weight $\Delta=2$ and two 1/2-BPS primaries of conformal weight $\Delta=n$ is calculated in the large 't Hooft, large $N$ limit. These operators are dual to Kaluza-Klein supergravity fields $s_k$ with masses $m^2=-4$ and $m^2=n(n-4)$. Given that the existing formalism for evaluating sums of products of SO(6) tensors that determine the effective couplings is only suitable for primaries with small conformal dimensions, we make us of an alternative formalism based on harmonic polynomials introduced by Dolan and Osborn. We then show that the supergravity lagrangian relevant to the computation is of sigma-model type (i.e., the four-derivative couplings vanish) and that the final result for the connected amplitude splits into a free and an interacting part, as expected on general grounds.