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.
Highlights
On the strong coupling side, we know that this defect theory corresponds to a particular gauge fixing of the classically integrable nonlinear sigma model describing the motion of a string in AdS5 × S5
The study of integrable field theories in curved backgrounds is an active and largely unexplored research subject, which has recently witnessed some interesting developments [26,27,28,29]. It has been pointed out in many places, see for instance [7, 40], that a crucial ingredient for our understanding of integrability in curved space would be the analogue of flat space S-matrix factorization and we believe Mellin space may provide the correct setting to look for such a feature
We move the first step in this direction by defining an inherently one-dimensional Mellin transform
Summary
We will keep the ordering of the operators fixed, so that we will only be interested in the function f(z), but we will consider its analytic continuation to complex values This may seem unphysical thinking of line correlators, but from the perspective of the diagonal limit of higher dimensional correlators it would correspond to Lorentzian regimes for which z = z∗, but z = z. For some applications it will be useful to introduce the crossing symmetric function g(z) = z−2∆φf(z) , g(z) = g(1 − z) There is another interesting limit we will consider in the following, i.e. limit (we could take this limit along any direction excluding the real line to avoid the branch cuts, but for definiteness we take it along the imaginary axis).
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