  In this paper, we present the effects of linear transverse-longitudinal
coupling on beam size at Interaction Point (IP) of a collider with 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 non-zero momentum dispersion at crab cavities and the non-ideal 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.
