A handbook of holographic 4-point functions

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 = increment - d/2 are half-integral, with increment 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 increment = 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.

Automated summary

In this paper, we present a comprehensive discussion on the tree-level holographic 4-point functions of scalar operators in momentum space. We demonstrate that each Witten diagram satisfies the conformal Ward identities independently, making it a valid conformal correlator. Furthermore, we provide explicit formulas for evaluating Witten diagrams when certain conditions are met. Renormalization is necessary for these correlators and it leads to new conformal anomalies and beta functions. We also explore the idea of weight-shifting operators to link correlators of operators with different dimensions.