Simplifying large spin bootstrap in Mellin space

Abstract

We set up the conventional conformal bootstrap equations in Mellin space and analyse the anomalous dimensions and OPE coefficients of large spin double trace operators. By decomposing the equations in terms of continuous Hahn polynomials, we derive explicit expressions as an asymptotic expansion in inverse conformal spin to any order, reproducing the contribution of any primary operator and its descendants in the crossed channel. The expressions are in terms of known mathematical functions and involve generalized Bernoulli (Norlund) polynomials and the Mack polynomials and enable us to derive certain universal properties. Comparing with the recently introduced reformulated equations in terms of crossing symmetric tree level exchange Witten diagrams, we show that to leading order in anomalous dimension but to all orders in inverse conformal spin, the equations are the same as in the conventional formulation. At the next order, the polynomial ambiguity in the Witten diagram basis is needed for the equivalence and we derive the necessary constraints for the same.

48 pages, 6 figures, typos corrected, to appear in JHEP

References in corpus (14)

• Holography from Conformal Field Theory
• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Writing CFT correlation functions as AdS scattering amplitudes
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Spinning AdS Propagators
• Conformal Bootstrap in Mellin Space
• A tauberian theorem for the conformal bootstrap
• Generalized Wilson-Fisher critical points from the conformal OPE
• A CFT Perspective on Gravitational Dressing and Bulk Locality
• The nuts and bolts of the BMS Bootstrap
• Geodesic bulk diagrams on the Bruhat-Tits tree
• Inverse Bootstrapping Conformal Field Theories
• On the Exchange Interactions in Holographic p-adic CFT
• Double Trace Interfaces

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