Combined effects of crab dispersion and momentum dispersion in colliders with local crab crossing scheme

In this paper, we present the effects of linear transverse-longitudinal coupling on beam size at the interaction point (IP) of a collider with a local crab crossing scheme, when time-dependent transverse deflection (crab kicks) and dispersive orbit intertwine near IP. The analytic propagation formula and the closed orbit form of the crab dispersion and momentum dispersion are derived. The nonzero momentum dispersion at crab cavities and the nonideal phase from crab cavities to IP are detailed with the derived propagation formula to predict the beam size distortion at IP with or without the beam-beam interaction. The linear results are compared with nonlinear simulation using the weak-strong beam-beam code.