Spinning AdS Loop Diagrams: Two Point Functions

TL;DR

The work addresses the difficulty of evaluating AdS loop Witten diagrams by introducing a spectral representation that expresses loops as integrals over spectral parameters and products of higher-point tree diagrams. It concentrates on $2pt$ one-loop diagrams with totally symmetric fields of arbitrary mass and spin, using Mellin-Barnes type spectral integrals alongside known conformal integrals (Symanzik) to perform the calculations. The authors demonstrate the framework on scalar loops (bubbles and tadpoles) and extend it to parity-even cubic couplings of generic totally symmetric fields, ultimately linking the results to contributions to the anomalous dimensions of higher-spin currents in AdS. They also discuss applications to gravity loops and the type-A higher-spin theory, highlighting implications for higher-spin holography and quantum consistency in AdS/CFT beyond leading order.

Abstract

We develop a systematic approach to evaluating AdS loop amplitudes based on the spectral (or "split") representation of bulk-to-bulk propagators, which re-expresses loop diagrams in terms of spectral integrals and higher-point tree diagrams. In this work we focus on 2pt one-loop Witten diagrams involving totally symmetric fields of arbitrary mass and integer spin. As an application of this framework, we study the contribution to the anomalous dimension of higher-spin currents generated by bubble diagrams in higher-spin gauge theories on AdS.