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Hidden simplicity in AdS spinning Mellin amplitudes via scaffolding

Song He, Xiang Li, Yuyu Mo, Dongyu Yang

TL;DR

This work develops a scaffolded Mellin-amplitude framework for AdS spinning correlators, showing that spinning $n$-point AdS amplitudes can be constructed by projecting $2n$-scalar scaffolding. Three-point spinning amplitudes with spins $(1,1,1)$ and $(2,2,2)$ reproduce flat-space massive amplitudes, with the AdS$_5\times S^5$ case yielding $\lambda=1$. In the AdS$_5$ YM–scalar sector, the authors bootstrap tree-level $n$-gluon amplitudes up to $n=6$ and uncover an emergent set of Feynman rules: all 3- and 4-vertices, including descendant levels, share the same flat-space form up to coefficients $\alpha_{m_1\dots m_k}$ that satisfy a factorial recursion, ensuring manifest flat-space limits. The resulting framework provides a compact, diagrammatic description of higher-point AdS spinning amplitudes, with potential extensions to supergravity, color–kinematics, and AdS string amplitudes.

Abstract

We uncover surprising hidden simplicity in Mellin amplitudes for tree-level AdS holographic correlators for spinning operators, such as AdS "gluons" and "gravitons" (spin 1 and 2). We define Mellin amplitudes with $n$ spinning operators via the so-called "scaffolding" of $2n$-scalar ones with specific projection operators for each spin state, which are rational functions of Mellin variables of $2n$ scalars generalizing flat-space scaffolding amplitudes. We classify possible three-point structures with spin 1 and 2 which take the same form as massive three-point amplitudes in flat space, and match with special solutions such as those extracted from 6-scalar ones in $\mathrm{AdS}_5\times S^3$ or $\mathrm{AdS}_5\times S^5$. Focusing on $\mathrm{AdS}_5$ gluons, we directly bootstrap spinning amplitudes in scaffolding form up to $n=6$ gluons (which amounts to $2n=12$ scalars) using factorizations, multi-linearity and flat-space limit. The results take a remarkably simple form in analogy with flat-space amplitudes, which can be constructed from familiar 3- and 4-vertices as well as propagators of massive spin-1 particles. Surprisingly, we find that vertices with any descendant levels are proportional to the primary ones with nice combinatorial coefficients, which makes manifest the correct flat-space limit in the simplest possible way.

Hidden simplicity in AdS spinning Mellin amplitudes via scaffolding

TL;DR

This work develops a scaffolded Mellin-amplitude framework for AdS spinning correlators, showing that spinning -point AdS amplitudes can be constructed by projecting -scalar scaffolding. Three-point spinning amplitudes with spins and reproduce flat-space massive amplitudes, with the AdS case yielding . In the AdS YM–scalar sector, the authors bootstrap tree-level -gluon amplitudes up to and uncover an emergent set of Feynman rules: all 3- and 4-vertices, including descendant levels, share the same flat-space form up to coefficients that satisfy a factorial recursion, ensuring manifest flat-space limits. The resulting framework provides a compact, diagrammatic description of higher-point AdS spinning amplitudes, with potential extensions to supergravity, color–kinematics, and AdS string amplitudes.

Abstract

We uncover surprising hidden simplicity in Mellin amplitudes for tree-level AdS holographic correlators for spinning operators, such as AdS "gluons" and "gravitons" (spin 1 and 2). We define Mellin amplitudes with spinning operators via the so-called "scaffolding" of -scalar ones with specific projection operators for each spin state, which are rational functions of Mellin variables of scalars generalizing flat-space scaffolding amplitudes. We classify possible three-point structures with spin 1 and 2 which take the same form as massive three-point amplitudes in flat space, and match with special solutions such as those extracted from 6-scalar ones in or . Focusing on gluons, we directly bootstrap spinning amplitudes in scaffolding form up to gluons (which amounts to scalars) using factorizations, multi-linearity and flat-space limit. The results take a remarkably simple form in analogy with flat-space amplitudes, which can be constructed from familiar 3- and 4-vertices as well as propagators of massive spin-1 particles. Surprisingly, we find that vertices with any descendant levels are proportional to the primary ones with nice combinatorial coefficients, which makes manifest the correct flat-space limit in the simplest possible way.
Paper Structure (12 sections, 50 equations, 3 figures, 3 tables)

This paper contains 12 sections, 50 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Spinning correlators from scaffolded and projected Mellin amplitudes
  3. Three-point amplitudes
  4. Bootstrap n-gluon amplitudes and Feynman rules
  5. Bootstrap
  6. Emergent Feynman rules
  7. Outlook
  8. A: Review of super-Yang-Mills and supergravity in AdS5 and results of mixed spin amplitudes
  9. Mellin amplitudes for super-Yang-Mills on AdS5S3
  10. Mellin amplitudes for Supergravity on AdS5S5
  11. B: Details on projectors and factorizations
  12. C: Recursion relations for coefficients and the flat-space limit

Figures (3)

  • Figure 1: Scaffolding from $2n=12$ scalars to $n=6$ gluons.
  • Figure 2: Mixture of spin-0 (left) and spin-1 (right) contribution in scaffolding residue on $\mathbf{X}_{1,3}$.
  • Figure 3: An example of 6-pt Feynman diagram ($3*3*4$ channel) where the two propagators have descendant levels $m_I=1$ and $m_J=2$, respectively.