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.

v1: 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 (14)

• Holography from Conformal Field Theory
• Writing CFT correlation functions as AdS scattering amplitudes
• Spinning AdS Propagators
• Mellin amplitudes for $Ad{S}_5\times S^5$
• Conformal Bootstrap in Mellin Space
• Loop Corrections to Supergravity on $Ad{S}_5\times S^5$
• D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes
• All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation
• Genus-One String Amplitudes from Conformal Field Theory
• On the Polyakov-Mellin bootstrap
• High Energy Scattering in the AdS/CFT Correspondence
• Four-point functions of 1/2-BPS operators of any weights in the supergravity approximation

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