Study of the Beam-Beam Effect for Crab Crossing in KEKB and Super KEKB

Crab crossing has been proposed to boost up the luminosity performance at KEKB. The luminosity is boosted up by removing x-z coupling due to the crossing angle. For a high beam-beam parameter (ξ > 0.05), linear couplings between degree of freedom induces symplectic diffusion, with the result that emittance growth and luminosity degradation arises. We have to care not only x − z coupling (crossing angle) but also x − y and y − z couplings. Chromaticity may be next order contribution of the luminosity degradation, External diffusion due to fast noises, for example kicker noise and phase jitter of cavities in crab crossing, could affect the luminosity performance. We present the tolerance for linear couplings, chromaticity and the fast noise in the crab crossing at KEKB.