Crab dispersion and its impact on the CERN Large Hadron Collider collimation

Crab cavities are proposed to be used for a luminosity upgrade of the Large Hadron Collider (LHC). Crab cavities are rf cavities operated in a transverse dipole mode, which imparts on the beam particles a transverse kick that varies with the longitudinal position along the bunch. The crab cavity introduces another kind of dispersion to the particles which is $z$ dependent, and thus could complicate the beam dynamics and have an impact on the LHC collimation system. As for LHC, the off-momentum beta-beat and dispersion-beat already compromise the performance of the collimation system; the crab dispersion introduced by global crab cavities might do the same, and should be carefully evaluated. In this paper, we present a definition for the crab dispersion, and study its impact on the LHC collimation system.