Dynamic Focusing to Suppress Emittance Transfer in Crab-Crossing Flat Beam Collisions

Flat hadron beam collisions, though expected to enhance peak luminosity by about an order of magnitude, have not yet been demonstrated. We identify a new mechanism in which the hourglass-induced synchrobetatron resonance, when coupled to unavoidable machine fluctuations, drives emittance transfer and degrades the flat-beam condition. This effect becomes a critical limitation for maintaining flat hadron beams on operational timescales. Using Electron-Ion Collider (EIC) parameters, we show that it imposes stringent tolerances on machine noise. We further demonstrate that a dynamic focusing scheme, implemented by combining sextupoles with crab cavities, suppresses the resonance and relaxes these constraints, providing a practical path toward robust flat-beam collisions in future lepton-hadron colliders.