A conformal dispersion relation: correlations from absorption

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.