Mellin amplitudes for 1d CFT

Abstract

We define a Mellin amplitude for CFT1 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 AdS2 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.

Multipoint correlators on the supersymmetric Wilson line defect CFT

“…Mellin space is known to offer an alternative, sometimes simpler, formulation for correlation functions. It would be interesting to extend [39] to multipoint correlators, and connect that formalism with the results presented here.…”

“…In addition, a closed-form expression for tree-level contact interactions of identical scalars at strong coupling was recently found in Mellin space through an inherently onedimensional Mellin formalism [39]. Mellin space is known to offer an alternative, sometimes simpler, formulation for correlation functions.…”

Selected Topics in Analytic Conformal Bootstrap: A Guided Journey

“…It would therefore be of great interest to find another Mellin representation that is intrinsic to the CFT 1 kinematics. Relevant works on the 1d Mellin representation include [109,110], but it is not yet clear how to establish such a formalism for general n-point functions.…”

Notes on $n$-point Witten diagrams in AdS${}_2$

“…This is already relevant for the four-point d = 1 correlator. In one dimension, several attempts were made to use the Mellin transform using the higher-dimensional formalism [10], or developing a one-dimensional formalism [35]. Using as a guide the principle that contact Witten diagrams correspond to constant Mellin amplitudes, the results in Section 3 may provide a starting point in generalising the onedimensional Mellin formalism developed in [35].…”

“…Natural extensions to this work include deriving position space results of contact diagrams in the case of higher external dimension and Polyakov blocks for higher exchanged dimension. A possible path for this is by extending the one-dimensional Mellin analysis developed in [35] to higher n-point functions using results from this study. In the former, Mellin amplitudes for higher ∆ were obtained, so that, in combination with these notes, results for all (n, ∆) may be achievable.…”