OPE inversion in Mellin space

TL;DR

This work proposes a Mellin-space analogue of Caron-Huot's OPE inversion by recasting the conformal block expansion in terms of Mellin blocks and exploiting their orthogonality. By introducing the amputated Mellin amplitude and a double-discontinuity kernel, it derives an inversion formula in Mellin space that isolates OPE data from the collinear region, and validates the method on simple examples: vacuum contributions recover free-theory OPE coefficients, while large-spin double-twist data yield explicit twist corrections. It further showcases higher-order corrections from scalar exchange and analyzes their large-$\beta$ behavior, aligning with known Regge-limit and large-spin results. The paper concludes with outlook toward a fully Mellin-space inversion framework and the exploration of contour deformations, aiming to deepen analytic control over CFT data through Mellin amplitudes.

Abstract

The fundamental ingredients that build the observables in conformal field theory are the spectrum of operators and the OPE coefficients, or equivalently, the two- and three-point functions of the theory. Recently an inversion formula solving the OPE coefficients by a convolution over the light-cone double-discontinuities of the correlator has been found by Simon Caron-Huot. Taking into account that the same OPE data determine the Mellin amplitude representation of the correlator, motivate us to look for an analogous inversion formula in Mellin space, which we develops partially on this paper.