Six-Point Conformal Blocks in the Snowflake Channel

Abstract

We compute $d$-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two ${}_3{F}_2$-hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kampé de Fériet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions.

arXiv comment1+22 pages (+25 pages of appendixes), 9 figures

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• OPE statistics from higher-point crossing
• Gaudin Models and Multipoint Conformal Blocks: General Theory
• From Gaudin Integrable Models to $d$-dimensional Multipoint Conformal Blocks
• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• Recursion relations for 5-point conformal blocks
• Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation
• Six-point functions and collisions in the black hole interior
• Towards Feynman rules for conformal blocks
• Shadow Celestial Amplitude
• Improving the five-point bootstrap
• Feynman Rules for Scalar Conformal Blocks
• Dimensional reduction of higher-point conformal blocks
• Spectral representation in Klein space: simplifying celestial leaf amplitudes
• Conformal Blocks in Two and Four Dimensions from Oscillator Representations
• Operator growth and black hole formation