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Multi-particle correlators with higher KK modes I: a bootstrap approach

Francesco Aprile, Stefano Giusto, Rodolfo Russo, Joao Vilas Boas

TL;DR

This work develops a double-particle bootstrap at tree level in AdS$_5 imes$S$^5$ to compute four-point holographic correlators with one external double-particle half-BPS operator and three single-particle half-BPS operators in ${ m N}=4$ SYM at large $N$ and large 't Hooft coupling. By exploiting the OPE data of double-particle operators and the partial non-renormalization theorem, the authors fix the connected $O(1/N^3)$ contribution as a sum of AdS contact diagrams (D-functions), despite degeneracies in the double-particle spectrum. They validate the results by matching to the double-particle limit of recently conjectured five-point tree-level single-particle correlators with arbitrary KK levels, providing strong evidence for the Fernandes:2025eqe proposal. The paper also develops a Mellin-space perspective, deriving explicit tree-level Mellin amplitudes for these correlators and analyzing their flat-space limits, while extending the analysis to broader families of double-particle operators and higher extremality. Overall, the results deepen our understanding of multi-particle operators in holographic CFTs and connect four- and five-point data in a concrete, testable framework.

Abstract

We bootstrap tree-level supergravity four-point correlators on AdS$_5\times$S$^5$ with one external half-BPS double-particle operator and three half-BPS single-particle operators. Our only input is the consistency of the operator product expansion of $SU(N)$ ${\cal N} = 4$ super Yang-Mills theory at large $N$ and large 't Hooft coupling. Even though the leading order OPE does not close on double-particle operators, but involves triple-particle operators, the CFT data of the double-particle operators, both long and protected, is sufficient to uniquely fix the correlators. We then verify that our results for the four-point correlators with one double-particle and three single-particle operators are reproduced by the appropriate double-particle limit of the five-point tree-level correlators of single-particle operators, with arbitrary Kaluza-Klein levels, recently conjectured in arXiv:2507.14124. Our study thus provides further evidence for the latter result.

Multi-particle correlators with higher KK modes I: a bootstrap approach

TL;DR

This work develops a double-particle bootstrap at tree level in AdSS to compute four-point holographic correlators with one external double-particle half-BPS operator and three single-particle half-BPS operators in SYM at large and large 't Hooft coupling. By exploiting the OPE data of double-particle operators and the partial non-renormalization theorem, the authors fix the connected contribution as a sum of AdS contact diagrams (D-functions), despite degeneracies in the double-particle spectrum. They validate the results by matching to the double-particle limit of recently conjectured five-point tree-level single-particle correlators with arbitrary KK levels, providing strong evidence for the Fernandes:2025eqe proposal. The paper also develops a Mellin-space perspective, deriving explicit tree-level Mellin amplitudes for these correlators and analyzing their flat-space limits, while extending the analysis to broader families of double-particle operators and higher extremality. Overall, the results deepen our understanding of multi-particle operators in holographic CFTs and connect four- and five-point data in a concrete, testable framework.

Abstract

We bootstrap tree-level supergravity four-point correlators on AdSS with one external half-BPS double-particle operator and three half-BPS single-particle operators. Our only input is the consistency of the operator product expansion of super Yang-Mills theory at large and large 't Hooft coupling. Even though the leading order OPE does not close on double-particle operators, but involves triple-particle operators, the CFT data of the double-particle operators, both long and protected, is sufficient to uniquely fix the correlators. We then verify that our results for the four-point correlators with one double-particle and three single-particle operators are reproduced by the appropriate double-particle limit of the five-point tree-level correlators of single-particle operators, with arbitrary Kaluza-Klein levels, recently conjectured in arXiv:2507.14124. Our study thus provides further evidence for the latter result.
Paper Structure (20 sections, 261 equations, 2 figures)

This paper contains 20 sections, 261 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Four-point correlators of half-BPS multiplets
  4. The space of double-particle operators
  5. The double-particle bootstrap for $\langle [{\cal O}_{r_1} {\cal O}_{r_2}] {\cal O}^{\space}_s {\cal O}_p {\cal O}_q\rangle$
  6. Overview of the method
  7. Algorithm at work in simple test cases
  8. More bootstrap results
  9. Higher extremality with ${\cal O}_2^2$
  10. Generic double-particle operators
  11. Mellin amplitudes with one double-particle operator
  12. All tree-level amplitudes $\langle {\cal O}_2^2\,{\cal O}_{p_2}{\cal O}_{p_3}{\cal O}_{p_4}\rangle$
  13. All tree-level amplitudes $\langle {\cal O}_r^2{\cal O}_{2}{\cal O}_{p}{\cal O}_{p}\rangle$
  14. The double-particle limit of the 5pt correlators
  15. Comments on the tree-level five-point Mellin amplitude of Fernandes:2025eqe
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Factorization of a five-point Mellin amplitude with $k=2$. The amplitude develops a simple pole at the twist $\Delta-J$ of a single-trace operator exchanged in the OPE and its conformal descendants. The residues are determined in terms of the lower-point amplitudes $M_L$ and $M_R$.
  • Figure 2: Tree-level Witten diagrams contributing to a five-point correlator. The red lines correspond to the exchange of single-trace operators in the OPE of the two external operators.