[Paper Review] Loops in the Bulk

Ellis Ye Yuan|arXiv (Cornell University)|Oct 3, 2017

Black Holes and Theoretical Physics16 citations

TL;DR

This paper introduces universal integral recursion relations for Mellin amplitudes of Witten diagrams in AdS, enabling a systematic all-loop analysis of scalar effective field theories with simple interactions. The method reveals universal kernel rules and analytically determines the analytic structure of a 4-point triangle diagram at one loop.

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.

Motivation & Objective

• To develop a systematic framework for analyzing Mellin amplitudes of Witten diagrams at all loop levels.
• To identify universal recursion relations that govern Mellin amplitudes in scalar effective field theories in AdS.
• To understand the analytic properties of higher-point and higher-loop amplitudes through integral kernels.
• To apply the recursion to a concrete 4-point triangle diagram and extract its analytic structure.

Proposed method

• Introduce integral recursion relations that relate Mellin amplitudes across different loop orders.
• Identify a universal integral kernel that governs the recursion, independent of the specific diagram.
• Apply the recursion to a 4-point scalar triangle diagram in AdS with simple interactions.
• Use the kernel's universal rules to derive analytic properties of the Mellin amplitude without explicit diagram evaluation.
• Leverage Mellin space techniques to encode bulk Witten diagram amplitudes in a form amenable to recursive analysis.
• Utilize the structure of scalar effective field theories in AdS to simplify the recursion to a universal kernel form.

Experimental results

Research questions

• RQ1 What universal structure underlies Mellin amplitudes of Witten diagrams across all loop levels?
• RQ2 How can integral recursion relations be constructed to relate Mellin amplitudes at different loop orders?
• RQ3 What analytic properties emerge for a 4-point triangle diagram in AdS using the proposed recursion?
• RQ4 Can the recursion kernel be universally applied across different scalar effective theories in AdS?
• RQ5 How does the recursion reveal the singularity structure and analyticity of Mellin amplitudes?

Key findings

• The integral kernel driving the recursion is universal across scalar effective theories in AdS with simple interactions.
• The recursion enables a systematic all-loop analysis of Mellin amplitudes without explicit loop integration.
• The method successfully determines the analytic structure of a 4-point triangle diagram at one loop.
• The universal kernel structure simplifies the computation of Mellin amplitudes across loop orders.
• The approach reveals hidden analytic properties of the amplitude through recursive constraints.
• The framework provides a new tool for studying higher-point and higher-loop amplitudes in AdS/CFT.