Loops in the Bulk

TL;DR

This paper develops a recursion-based method to construct Mellin amplitudes for AdS Witten diagrams at arbitrary loop order using the split representation. It introduces a Mellin pre-amplitude M and a factorized kernel K that generate loop-level recursions on Mandelstam-like variables for scalar phi^m theories. One-loop necklaces are analyzed with a practical kernel involving auxiliary xi variables, and higher loops are handled via an effective chain of non-trivial OPE channels. As a concrete test, the 4-point triangle is worked out, yielding a sixfold Mellin integral and a pole structure in the S-channel consistent with double-trace operators, with residues factorizing into products of 3-point pre-amplitudes; a companion paper will provide full derivations and contour analyses.

Abstract

We initiate a systematic investigation of Mellin amplitudes of Witten diagrams to all loop levels, by introducing integral recursion relations among them. Focusing on the scalar effective theories in AdS with the simplest type of interactions, the integral kernel that triggers the recursion obeys universal rules. As a first application, analytic properties of a 4-point triangle diagram is analyzed with this method.

Figures (4)

• Figure 1: Loop formation.
• Figure 2: Constructing a necklace diagram (the AdS boundary and all bulk-to-boundary propagagtors are omitted). Examples of channels are shown by red and orange dashed curves.
• Figure 3: Effective diagram from a chosen chain of non-trivial channels. Double lines denote the effective tree propagator for the channels.
• Figure 4: Constructing the 4-point triangle diagram.