On spinning loop amplitudes in Anti-de Sitter space

TL;DR

This work develops a momentum-space framework for spinning loop amplitudes in $AdS_{d+1}$, addressing the challenge of AdS loop integrals by expressing any loop Witten diagram as $W = \widetilde{\mathcal{D}} \mathcal{M}$, where $\widetilde{\mathcal{D}}$ encodes tensor structure and $\mathcal{M}$ is a scalar factor containing all integrals. The authors provide explicit constructions for $\mathcal{M}$ for gluon and graviton loops and introduce auxiliary polarization vectors to decouple tensorial from scalar parts, enabling an algorithmic, computer-amenable evaluation. They demonstrate the method on bubble, triangle, and box diagrams, showing how radial AdS integrals can be written in terms of Appell's $F_4$ functions and how loop momentum integrals can be reduced to one-dimensional forms, with the bubble example yielding a finite, 48-term structure in certain dimensions. This framework paves the way for scalable AdS loop computations and hints at connections to flat-space structures, symbol calculus, and cutting rules, potentially enriching holographic loop dynamics and their geometric interpretation.

Abstract

In this work we present a systematic study of AdS$_{d+1}$ loop amplitudes for gluons and gravitons using momentum space techniques. Inspired by the recent progress in tree level computation, we construct a differential operator that can act on a scalar factor in order to generate gluon and graviton loop integrands: this systematizes the computation for any given loop level Witten diagram. We then give a general prescription in this formalism, and discuss it for bubble, triangle, and box diagrams.

Figures (1)

• Figure 1: Example of various gluon loop diagrams