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.

arXiv comment98 pp, 7 figs. v2: published version, new material on the OPE limit and exchange diagrams with derivative vertices

References in corpus (51)

• Lecture Notes on Holographic Renormalization
• Correlation functions in the CFT(d)/AdS(d+1) correpondence
• On graviton non-Gaussianities during inflation
• Thermodynamics of Asymptotically Locally AdS Spacetimes
• Graviton exchange and complete 4-point functions in the AdS/CFT correspondence
• On Soft Limits of Inflationary Correlation Functions
• HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters
• Implications of conformal invariance in momentum space
• Cosmological Collider Physics
• Non-Gaussianity in the modulated reheating scenario
• Bootstrapping Inflationary Correlators in Mellin Space
• On Four-point Functions in the CFT/AdS Correspondence
• The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds
• A Mellin Space Approach to Cosmological Correlators
• Analyticity and Unitarity for Cosmological Correlators
• CMB from CFT
• Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies
• Late-time Structure of the Bunch-Davies De Sitter Wavefunction
• Weight Shifting Operators and Conformal Blocks
• Holography for inflation using conformal perturbation theory
• From AdS to dS Exchanges: Spectral Representation, Mellin Amplitudes and Crossing
• From dS to AdS and back
• Four-point Functions of Lowest Weight CPOs in N=4 SYM_4 in Supergravity Approximation
• Linking the Singularities of Cosmological Correlators
• Comments on 4-point functions in the CFT/AdS correspondence
• Conformal Partial Wave Expansions for N=4 Chiral Four Point Functions
• Renormalised 3-point functions of stress tensors and conserved currents in CFT
• The Inflationary Wavefunction from Analyticity and Factorization
• Conformal $n$-point functions in momentum space
• On duality of color and kinematics in (A)dS momentum space
• Renormalised CFT 3-point functions of scalars, currents and stress tensors
• Color/Kinematics Duality in AdS${}_4$
• Momentum space approach to crossing symmetric CFT correlators
• Evaluation of conformal integrals
• Conformal correlators as simplex integrals in momentum space
• Cosmological Scattering Equations
• Diffeomorphisms, Anomalies and the Fefferman-Graham Ambiguity
• An étude of momentum space scalar amplitudes in AdS
• New relation for AdS amplitudes
• Momentum space approach to crossing symmetric CFT correlators II: General spacetime dimension
• AdS Weight Shifting Operators
• On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space
• Higher spin 3-point functions in 3d CFT using spinor-helicity variables
• Momentum space parity-odd CFT 3-point functions
• On the power spectrum of inflationary cosmologies dual to a deformed CFT
• Universal Features of Holographic Anomalies
• Four-Point Functions in Momentum Space: Conformal Ward Identities in the Scalar/Tensor case
• Momentum space CFT correlators for Hamiltonian truncation
• TripleK: A Mathematica package for evaluating triple-K integrals and conformal correlation functions
• Conformal partial waves in momentum space
• Dimensional regularization for holographic RG flows

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