A Conformal Dispersion Relation: Correlations from Absorption

TL;DR

This paper develops a conformal dispersion relation analogous to Kramers-Kronig relations, expressing four-point correlators in conformal field theories as integrals over their absorptive parts, with explicit kernels and broad applicability.

Contribution

It introduces a novel dispersion relation for conformal correlators, explicitly computes the kernel, and verifies it across various theories, including holographic models and the 3D Ising model.

Findings

Derived a simple kernel as an elliptic integral function

Validated the dispersion relation with perfect matches in multiple theories

Established a relation between inverted and ordinary conformal blocks

Abstract

We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its 'absorptive part', defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the 'inverted' conformal block with the ordinary conformal block.