Exact Amplitude Reconstruction in the Real Common-Phase Sector of Small-x Diffractive Energy Flow

Abstract

We develop an operator-level framework for reconstructing the angular structure of small-$x$ diffractive amplitudes from energy-flow measurements. Focusing on coherent diffractive dijet production in the leading-eikonal approximation, we establish the relation between the angular multipoles of the dipole amplitude, the gluon Wigner distribution, and the diffractive scattering amplitude, while keeping their distinct radial transforms explicit. We show that, in the real common-phase sector and away from diffractive zeros, the normalized scattering amplitude can be reconstructed directly, up to a global sign convention, from the square root of the normalized energy-flow distribution. For generic complex amplitudes, the measured intensity instead determines autocorrelations of the amplitude spectrum and phase retrieval is not unique without additional information. We further discuss corrections arising from harmonic-dependent phases, practical conditions for an EIC extraction, and effects beyond the leading-eikonal approximation.