Four-Point Functions in Momentum Space: Conformal Ward Identities in the Scalar/Tensor case

Abstract

We derive and analyze the conformal Ward identities (CWI's) of a tensor 4-point function of a generic CFT in momentum space. The correlator involves the stress-energy tensor $T$ and three scalar operators $O$ ($TOOO$). We extend the reconstruction method for tensor correlators from 3- to 4-point functions, starting from the transverse traceless sector of the $TOOO$. We derive the structure of the corresponding CWI's in two different sets of variables, relevant for the analysis of the 1-to-3 (1 graviton $\to$ 3 scalars) and 2-to-2 (graviton + scalar $\to$ two scalars) scattering processes. The equations are all expressed in terms of a single form factor. In both cases, we discuss the structure of the equations and their possible behaviors in various asymptotic limits of the external invariants. A comparative analysis of the systems of equations for the $TOOO$ and those for the $OOOO$, both in the general (conformal) and dual-conformal/conformal (dcc) cases, is presented. We show that in all the cases the Lauricella functions are homogenous solutions of such systems of equations, also described as parametric 4K integrals of modified Bessel functions.

arXiv comment46 pages, 5 figures

References in corpus (33)

• Implications of Conformal Invariance in Field Theories for General Dimensions
• Conformal Four Point Functions and the Operator Product Expansion
• Writing CFT correlation functions as AdS scattering amplitudes
• Conserved Currents and the Energy Momentum Tensor in Conformally Invariant Theories for General Dimensions
• Implications of conformal invariance in momentum space
• A Natural Language for AdS/CFT Correlators
• A Mellin space approach to the conformal bootstrap
• Conformal Bootstrap in Mellin Space
• Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies
• Solving the Conformal Constraints for Scalar Operators in Momentum Space and the Evaluation of Feynman's Master Integrals
• The Trace Anomaly and Massless Scalar Degrees of Freedom in Gravity
• Constraints from Conformal Symmetry on the Three Point Scalar Correlator in Inflation
• Renormalised 3-point functions of stress tensors and conserved currents in CFT
• Lectures on Yangian Symmetry
• Momentum space approach to crossing symmetric CFT correlators
• Conformal Anomalies and the Gravitational Effective Action: The $TJJ$ Correlator for a Dirac Fermion
• Evaluation of conformal integrals
• Momentum-space conformal blocks on the light cone
• New relation for AdS amplitudes
• Momentum space conformal three-point functions of conserved currents and a general spinning operator
• Tower of subleading dual BMS charges
• Momentum space approach to crossing symmetric CFT correlators II: General spacetime dimension
• Exact Correlators from Conformal Ward Identities in Momentum Space and the Perturbative $TJJ$ Vertex
• On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space
• The General 3-Graviton Vertex ($TTT$) of Conformal Field Theories in Momentum Space in $d=4$
• Graviton Vertices and the Mapping of Anomalous Correlators to Momentum Space for a General Conformal Field Theory
• Identities between Appell's and hypergeometric functions
• Renormalization, Conformal Ward Identities and the Origin of a Conformal Anomaly Pole
• Graviton Scattering and a Sum Rule for the c Anomaly in 4D CFT
• On Conformal Block, Crossing Kernel and Multi-variable Hypergeometric Functions
• Correlation functions of sourced gravitational waves in inflationary scalar vector models. A symmetry based approach
• de Sitter symmetries and inflationary correlators in parity violating scalar-vector models
• Two Lectures on QCD at Short Distances: Jets and Factorization

Cited by in corpus (14)

• Linking the Singularities of Cosmological Correlators
• Conformal correlators as simplex integrals in momentum space
• Double copy structure of parity-violating CFT correlators
• The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization
• Higher spin 3-point functions in 3d CFT using spinor-helicity variables
• Momentum space parity-odd CFT 3-point functions
• Relation between parity-even and parity-odd CFT correlation functions in three dimensions
• Constraining momentum space correlators using slightly broken higher spin symmetry
• A handbook of holographic 4-point functions
• Momentum space CFT correlators of non-conserved spinning operators
• ANEC in $\lambda {\varphi }^4$ theory
• $n$-point functions in Conformal Quantum Mechanics: A Momentum Space Odyssey
• The Conformal Anomaly Action to Fourth Order (4T) in $d=4$ in Momentum Space
• Gravitational coupling of QED and QCD: 3- and 4- point functions in momentum space