Tune Shift Due to Crossing Collision and CRAB Collision

The use of crab cavities in the LHC may not only raise the luminosity, but it could also complicate the beam dynamics, e.g. crab cavities might not only cancel synchrobetatron resonances excited by the crossing angle but they could also excite new ones. In this paper, we use weakstrong beam-beam model to study the incoherent linear tune shift of the weak beam, for the crossing collision case and crab collision case with a nite crossing angle. The tune shift is also compared among the head-on collision, crossing collision and crab collision cases, both analytically and numerically.