Beam-beam effect with an external noise in LHC

Proton beam do not have any damping mechanism for an incoherent betatron motion. A noise, which kicks beam particles in the transverse plane, gives a coherent betatron amplitude. If the system is linear, the coherent motion remains in an amplitude range. Nonlinear force, beam-beam and beam-electron cloud interactions, causes a decoher- ence for the betatron motion with keeping an amplitude of each beam particle, with the result that an emittance growth arises. We focus only a fast noise, the correlation time is 1-100 turns. Slower noise is less serious, because it is regarded as an adiabatic change like closed orbit change. As sources of the noise, we consider the bunch by bunch feedback system and phase jitter of cavities which turns to transverse noise via Crab cavity.