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

Linda I. Uruchurtu

TL;DR

The paper addresses the four-point function of two $oldsymbol{ riangle=2}$ and two $oldsymbol{ riangle=n}$ 1/2-BPS primaries in $ ext{N}=4$ SYM at large $N$ and large $ ext{t Hooft}$ coupling via their AdS$_5 imes S^5$ duals, where the dual bulk fields are KK modes with masses $m^2=-4$ and $m^2=n(n-4)$. It introduces harmonic polynomials to manage generic $SO(6)$ tensor contractions and applies the insertion formula to constrain the dynamical piece to a single master function $oldsymbol{ rown}(u,v)$, enabling a clean comparison with CFT predictions. The supergravity calculation shows the on-shell action is of sigma-model type (no four-derivative quartic terms) and that the resulting four-point function splits into a free piece and an interacting piece, in line with CFT expectations. This work provides evidence for AdS/CFT in the KK sector and paves the way for computing higher-weight correlators and exploring stringy corrections such as $oldsymbol{R^4}$ terms.

Abstract

The four-point correlation function of two 1/2 BPS primaries of conformal weight $Δ=2$ and two 1/2-BPS primaries of conformal weight $Δ=n$ 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.

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

TL;DR

The paper addresses the four-point function of two $oldsymbol{ riangle=2}$ and two $oldsymbol{ riangle=n}$ 1/2-BPS primaries in $ ext{N}=4$ SYM at large $N$ and large $ ext{t Hooft}$ coupling via their AdS$_5 imes S^5$ duals, where the dual bulk fields are KK modes with masses $m^2=-4$ and $m^2=n(n-4)$. It introduces harmonic polynomials to manage generic $SO(6)$ tensor contractions and applies the insertion formula to constrain the dynamical piece to a single master function $oldsymbol{ rown}(u,v)$, enabling a clean comparison with CFT predictions. The supergravity calculation shows the on-shell action is of sigma-model type (no four-derivative quartic terms) and that the resulting four-point function splits into a free piece and an interacting piece, in line with CFT expectations. This work provides evidence for AdS/CFT in the KK sector and paves the way for computing higher-weight correlators and exploring stringy corrections such as $oldsymbol{R^4}$ terms.

Abstract

The four-point correlation function of two 1/2 BPS primaries of conformal weight and two 1/2-BPS primaries of conformal weight is calculated in the large 't Hooft, large limit. These operators are dual to Kaluza-Klein supergravity fields with masses and . 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.

Paper Structure

This paper contains 18 sections, 147 equations, 4 figures.

Table of Contents

  1. Introduction
  2. General Structure of the Four-Point Function
  3. The Insertion Formula
  4. Supergravity Calculation
  5. On-Shell Lagrangian
  6. Results for Exchange Integrals
  7. Contact Diagrams
  8. Results for the Four-Point Function
  9. Verifying the CFT Predictions
  10. Conclusions and Outlook
  11. Integrals over the Sphere
  12. Harmonic Polynomials
  13. Properties of $D$-Functions
  14. Exchange Diagrams
  15. Scalar Exchanges
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Propagator basis for the process $\langle \mathcal{O}_{2}(\vec{x}_1) \mathcal{O}_{2}(\vec{x}_2) \mathcal{O}_{n}(\vec{x}_3)\mathcal{O}_{n}(\vec{x}_4)\rangle$. The graphs are arranged in four equivalence classes. The symbol $n$ stands for the $n$ propagators coming out from the corresponding vertices.
  • Figure 2: Witten Diagrams for the $s$-channel process. (a) exchange by a scalar with $m^2=-4$(b) exchange by a massless vector (c) graviton exchange
  • Figure 3: Witten Diagrams for the $t$-channel process. (a) exchange by a scalar of mass $m^2=\Delta(\Delta-4)$(b) exchange by a vector of mass $m_{k}^2=k^2-1$(c) exchange by a tensor field of mass $f_k=k(k+4)$(d) Contact diagram.
  • Figure 4: Graphic representation of a $D$-function.