Mellin amplitudes for 1$d$ CFT

Abstract

We define a Mellin amplitude for CFT${}_1$ four-point functions. Its analytical properties are inferred from physical requirements on the correlator. We discuss the analytic continuation that is necessary for a fully nonperturbative definition of the Mellin transform. The resulting bounded, meromorphic function of a single complex variable is used to derive an infinite set of nonperturbative sum rules for CFT data of exchanged operators, which we test on known examples. We then consider the perturbative setup produced by quartic interactions with an arbitrary number of derivatives in a bulk AdS${}_2$ field theory. With our formalism, we obtain a closed-form expression for the Mellin transform of tree-level contact interactions and for the first correction to the scaling dimension of "two-particle" operators exchanged in the generalized free field theory correlator.

arXiv comment48 pages, 2 figures, 1 Mathematica notebook attached; v2: enlarged text in Sections 1,3,5, matches published version in JHEP

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• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• Analytic bootstrap for the localized magnetic field
• Multipoint correlators on the supersymmetric Wilson line defect CFT
• Conformal dispersion relations for defects and boundaries
• Dispersive Sum Rules in AdS${}_2$
• Celestial Mellin Amplitude
• Notes on $n$-point Witten diagrams in AdS${}_2$
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• Bootstrability in Defect CFT: Integrated Correlators and Sharper Bounds
• 1d Conformal Field Theory and Dispersion Relations
• From conformal correlators to analytic S-matrices: CFT${}_1$/QFT${}_2$
• Moments and saddles of heavy CFT correlators