# Spinning Witten Diagrams

## TL;DR

This paper presents a systematic method for computing conformal partial wave expansions of tree-level four-point Witten diagrams with spinning fields in AdS, advancing holographic duality understanding.

## Contribution

It introduces a framework for CPWE of Witten diagrams with arbitrary spin and mass, including diagonalization of conformal structures and cubic couplings, applied to higher-spin theories.

## Key findings

- Derived CPWE for generic exchange Witten diagrams with spinning fields.
- Diagonalized the map between conformal structures and AdS cubic couplings.
- Computed all exchange diagrams in higher-spin gauge theory on AdS.

## Abstract

We develop a systematic framework to compute the conformal partial wave expansions (CPWEs) of tree-level four-point Witten diagrams with totally symmetric external fields of arbitrary mass and integer spin in AdS$_{d+1}$. Employing this framework, we determine the CPWE of a generic exchange Witten diagram with spinning exchanged field. As an intermediate step, we diagonalise the linear map between spinning three-point conformal structures and spinning cubic couplings in AdS. As a concrete application, we compute all exchange diagrams in the type A higher-spin gauge theory on AdS$_{d+1}$, which is conjectured to be dual to the free scalar $O\left(N\right)$ model. Given a CFT$_d$, our results provide the complete holographic reconstruction of all cubic couplings involving totally symmetric fields in the putative dual theory on AdS$_{d+1}$.