Mellin amplitudes for 1$d$ CFT

TL;DR

This paper introduces a Mellin amplitude framework for 1D conformal field theories, establishing its properties, deriving sum rules for operator data, and applying it to AdS$_2$ interactions to compute corrections.

Contribution

It defines a nonperturbative Mellin amplitude for 1D CFTs, explores its analytic properties, and applies it to compute corrections in AdS$_2$ field theories, providing new tools for CFT analysis.

Findings

Derived nonperturbative sum rules for CFT data.

Obtained closed-form Mellin transforms for contact interactions.

Calculated first-order corrections to operator dimensions.

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.