Landau anti-damping of the coherent beam–beam instability

Within the framework of a simplifying model and within the first approximation of the perturbation theory we discuss effects of the Landau damping on the stability of the coherent oscillations of short identical colliding bunches. Near the resonances n/(2m)n/(2m) where n and m are integers these oscillations are unstable. Comparing the stopbands calculated for monochromatic bunches with those that are calculated taking into account the beam–beam tunespreads of the bunches, we have found that the latter increases the widths of the stopbands of coherent modes, thus resulting in the Landau anti-damping of the coherent beam–beam oscillations.