Noise instability in nonlinear beam-beam interaction

Diffusion instability of a beam induced by random fluctuations in the size of an opposite colliding beam has been studied. It is shown, that the considered beam-beam configuration is always unstable at any value of beam-beam tune shift, if two conditions are valid: (i) beam-beam kick is a nonlinear function of coordinate and (ii) beam-beam kick is subject to noise. Meanwhile, the value of beam emittance is conserved even in the noise regime if the beam-beam kick is a linear function of the coordinate. Simple formula has been derived, which demonstrates quadratic dependence of diffusion coefficient in emittance expansion on beam-beam parameter and noise amplitude of the opposing beam size in case of nonlinear beam-beam interaction.