Hidden Simplicity in AdS Spinning Mellin Amplitudes via Scaffolding

We uncover surprising hidden simplicity in Mellin amplitudes for tree-level anti–de Sitter (AdS) holographic correlators for AdS “gluon” (spin-1) operators. We define Mellin amplitudes with n spinning operators via the so-called scaffolding of 2 n -scalar ones with specific projection operators for each spin state, which are rational functions of Mellin variables of 2 n scalars generalizing flat-space scaffolding amplitudes. We classify possible three-point structures with spin 1, which take the same form as massive three-point amplitudes in flat space, and match them with special solutions such as those extracted from 6-scalar ones in AdS 5 × S 3 . Focusing on AdS 5 gluons, we directly bootstrap spinning amplitudes in scaffolding form up to n = 6 gluons (which amounts to 2 n = 12 scalars) using factorizations, multilinearity, 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.