  Crab crossing scheme is an essential collision scheme to achieve high
luminosity for the future colliders with large crossing angles. However, when
bunch length of one or both colliding beams is comparable with the wavelength
of the crab cavity voltage, the nonlinear dependence of the crabbing kick may
present a challenge to the beam dynamics of the colliding beams and impact the
beam quality as well as the luminosity lifetime. In this paper, the results of
nonlinear dynamics in the crab crossing scheme are presented, using both
analytical and numerical studies. The result indicates that higher-order
synchro-betatron resonances may be excited in the crab crossing scheme with
large crossing angle, which causes the beam quality deterioration and
luminosity degradation. The studies also reveal possible countermeasures to
suppress the synchro-beta resonance, hence mitigate the degradation of beam
quality and luminosity.
