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Towards the higher point holographic momentum space amplitudes II: Gravitons

Soner Albayrak, Savan Kharel

TL;DR

This work develops a momentum-space framework for AdS$_4$ gravitons that reorganizes the graviton bulk-to-bulk propagator to isolate a scalar factor per Witten diagram. By introducing bulk-point integrated objects and a commuting differential-operator structure, the authors reduce higher-point graviton amplitudes to scalar integrals with tensor information carried by a set of projectors, enabling explicit 3-, 4-, and 5-point tree-level results. They provide closed-form scalar factors, analyze flat-space and collinear limits, and demonstrate how these amplitudes replicate familiar S-matrix structures in the appropriate limits, with practical simplifications that avoid multi-dimensional bulk integrations. The approach offers a path toward algorithmic graviton calculations in AdS and highlights connections to flat-space amplitudes, cosmological bootstrap, and potential double-copy-type relations in holographic settings.

Abstract

In this follow up paper, we calculate higher point tree level graviton Witten diagrams in AdS$_4$ via bulk perturbation theory. We show that by rearranging the bulk to bulk graviton propagators, the calculations effectively reduce to the computation of a scalar factor. Analogous to the amplitudes for vector boson interactions we computed in the previous paper, scalar factors for the graviton exchange diagrams also become relatively simple when written in momentum space. We explicitly calculate higher point correlators and discuss how this momentum space formalism makes flat space and collinear limits simpler.

Towards the higher point holographic momentum space amplitudes II: Gravitons

TL;DR

This work develops a momentum-space framework for AdS gravitons that reorganizes the graviton bulk-to-bulk propagator to isolate a scalar factor per Witten diagram. By introducing bulk-point integrated objects and a commuting differential-operator structure, the authors reduce higher-point graviton amplitudes to scalar integrals with tensor information carried by a set of projectors, enabling explicit 3-, 4-, and 5-point tree-level results. They provide closed-form scalar factors, analyze flat-space and collinear limits, and demonstrate how these amplitudes replicate familiar S-matrix structures in the appropriate limits, with practical simplifications that avoid multi-dimensional bulk integrations. The approach offers a path toward algorithmic graviton calculations in AdS and highlights connections to flat-space amplitudes, cosmological bootstrap, and potential double-copy-type relations in holographic settings.

Abstract

In this follow up paper, we calculate higher point tree level graviton Witten diagrams in AdS via bulk perturbation theory. We show that by rearranging the bulk to bulk graviton propagators, the calculations effectively reduce to the computation of a scalar factor. Analogous to the amplitudes for vector boson interactions we computed in the previous paper, scalar factors for the graviton exchange diagrams also become relatively simple when written in momentum space. We explicitly calculate higher point correlators and discuss how this momentum space formalism makes flat space and collinear limits simpler.

Paper Structure

This paper contains 14 sections, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review: AdS Momentum Space Perturbation Theory
  4. Stripping off the tensorial part of the propagator
  5. Bulk point integrated expressions
  6. Amplitudes
  7. A basic Witten diagram: three point function
  8. Four point function
  9. Five point function
  10. Flat space and collinear limits
  11. Flat space limit:
  12. Collinear limit:
  13. Conclusion
  14. Scalar factor for five point diagram

Figures (3)

  • Figure 1: The three point graviton amplitude
  • Figure 2: The four point graviton amplitude
  • Figure 3: Five point graviton amplitude