A handbook of holographic 4-point functions

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.

98 pp, 7 figs. v2: published version, new material on the OPE limit and exchange diagrams with derivative vertices

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Cited by in corpus (11)

• Bootstrapping Multi-Field Inflation: non-Gaussianities from light scalars revisited
• On the IR Divergences in de Sitter Space: loops, resummation and the semi-classical wavefunction
• The graviton four-point function in de Sitter space
• Renormalisation of IR divergences and holography in de Sitter
• Effective Field Theories and Cosmological Scattering Equations
• Momentum space CFT correlators of non-conserved spinning operators
• All plus four point (A)dS graviton function using generalized on-shell recursion relation
• Shift operators from the simplex representation in momentum-space CFT
• Shift Symmetries and AdS/CFT
• Soft Limits of Gluon and Graviton Correlators in Anti-de Sitter Space
• Notes on weight-shifting operators and unifying relations for cosmological correlators