Recursion Relations in Witten Diagrams and Conformal Partial Waves

Abstract

We revisit the problem of performing conformal block decomposition of exchange Witten diagrams in the crossed channel. Using properties of conformal blocks and Witten diagrams, we discover infinitely many linear relations among the crossed channel decomposition coefficients. These relations allow us to formulate a recursive algorithm that solves the decomposition coefficients in terms of certain seed coefficients. In one dimensional CFTs, the seed coefficient is the decomposition coefficient of the double-trace operator with the lowest conformal dimension. In higher dimensions, the seed coefficients are the coefficients of the double-trace operators with the minimal conformal twist. We also discuss the conformal block decomposition of a generic contact Witten diagram with any number of derivatives. As a byproduct of our analysis, we obtain a similar recursive algorithm for decomposing conformal partial waves in the crossed channel.

arXiv commentv1: 48 pages, 2 figures; v2: updated references, added in Appendix B explicit expressions for the direct channel decomposition coefficients; v3: published version

References in corpus (54)

• Holography from Conformal Field Theory
• Conformal Partial Waves and the Operator Product Expansion
• Writing CFT correlation functions as AdS scattering amplitudes
• Analyticity in Spin in Conformal Theories
• Projectors, Shadows, and Conformal Blocks
• Conformal Regge theory
• A Natural Language for AdS/CFT Correlators
• Emergent Spacetime and Holographic CFTs
• Light-ray operators in conformal field theory
• Large Spin Perturbation Theory
• Spinning AdS Propagators
• Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks
• Mellin amplitudes for $Ad{S}_5\times S^5$
• Loops in AdS from Conformal Field Theory
• Towards Feynman rules for Mellin amplitudes in AdS/CFT
• A Mellin space approach to the conformal bootstrap
• AdS/CFT 4-point functions: How to succeed at z-integrals without really trying
• A spacetime derivation of the Lorentzian OPE inversion formula
• Conformal Partial Waves: Further Mathematical Results
• How to Succeed at Holographic Correlators Without Really Trying
• Conformal Bootstrap in Mellin Space
• Gravitational S-matrix from CFT dispersion relations
• Loop Corrections to Supergravity on $Ad{S}_5\times S^5$
• Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory
• D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes
• Analytic Bounds and Emergence of ${\text{AdS}}_2$ Physics from the Conformal Bootstrap
• All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry
• Weight Shifting Operators and Conformal Blocks
• Quantum Gravity from Conformal Field Theory
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices
• The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation
• Unmixing Supergravity
• General Scalar Exchange in AdS_d+1
• Genus-One String Amplitudes from Conformal Field Theory
• Crossing Symmetry in Alpha Space
• Loop corrections for Kaluza-Klein AdS amplitudes
• The double-trace spectrum of N=4 SYM at strong coupling
• Holographic Reconstruction of AdS Exchanges from Crossing Symmetry
• Conformal Bootstrap in the Regge Limit
• On the Polyakov-Mellin bootstrap
• The Mellin Formalism for Boundary CFT${}_d$
• Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications
• On Superconformal Four-Point Mellin Amplitudes in Dimension $d>2$
• Holographic Four-Point Functions in the (2, 0) Theory
• Simplifying large spin bootstrap in Mellin space
• Anomalous Dimensions from Crossing Kernels
• High Energy Scattering in the AdS/CFT Correspondence
• Anomalous dimensions at finite conformal spin from OPE inversion
• Four-point functions of 1/2-BPS operators of any weights in the supergravity approximation
• Double-Trace Deformations of Conformal Correlations
• On Mellin Amplitudes in SCFTs with Eight Supercharges
• Towards 4-point correlation functions of any 1/2-BPS operators from supergravity
• Four-point functions of all-different-weight chiral primary operators in the supergravity approximation

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