A handbook of holographic 4-point functions

• Published in: The journal of high energy physics 2022-12, Vol.2022 (12), p.39-100, Article 39
• Main Authors: Bzowski, Adam, McFadden, Paul, Skenderis, Kostas
• Format: Article
• Language: English
• Subjects: AdS-CFT Correspondence; Anomalies; Classical and Quantum Gravitation; Conformal and W Symmetry; Conventions; Correlation; Correlators; Cosmology; Elementary Particles; Handbooks; High energy physics; Holography; Momentum; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Renormalization and Regularization; Scale and Conformal Symmetries; Spacetime; String Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP12(2022)039

A bstract We present a comprehensive discussion of tree-level holographic 4-point functions of scalar operators in momentum space. We show that each individual Witten diagram satisfies the conformal Ward identities on its own and is thus a valid conformal correlator. When the β = ∆ − d/ 2 are half-integral, with ∆ the dimensions of the operators and d the spacetime dimension, the Witten diagrams can be evaluated in closed form and we present explicit formulae for the case d = 3 and ∆ = 2 , 3. These correlators require renormalization, which we carry out explicitly, and lead to new conformal anomalies and beta functions. Correlators of operators of different dimension may be linked via weight-shifting operators, which allow new correlators to be generated from given ‘seed’ correlators. We present a new derivation of weight-shifting operators in momentum space and uncover several subtleties associated with their use: such operators map exchange diagrams to a linear combination of exchange and contact diagrams, and special care must be taken when renormalization is required.