A handbook of holographic 4-point functions

TL;DR

This paper provides a detailed analysis of tree-level holographic 4-point functions of scalar operators in momentum space, including explicit formulas, renormalization procedures, and the use of weight-shifting operators to generate new correlators.

Contribution

It introduces explicit closed-form expressions for certain Witten diagrams, discusses their conformal properties, and develops a momentum-space derivation of weight-shifting operators with new insights.

Findings

Explicit formulas for Witten diagrams when 5=/2 are half-integers

Identification of new conformal anomalies and beta functions from renormalization

Development of a momentum-space derivation of weight-shifting operators

Abstract

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 $\beta = \Delta - d/2$ are half-integral, with $\Delta$ 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 $\Delta=2,3$. 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.