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.
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Carmi, D., Caron-Huot, S. A conformal dispersion relation: correlations from absorption. J. High Energ. Phys. 2020, 9 (2020). https://doi.org/10.1007/JHEP09(2020)009
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DOI: https://doi.org/10.1007/JHEP09(2020)009