Importance of beam-beam tune spread to collective beam-beam instability in hadron colliders

In hadron colliders, electron-beam compensation of beam-beam tune spread has been explored for a reduction of beam-beam effects. In this paper, effects of the tune-spread compensation on beam-beam instabilities were studied with a self-consistent beam-beam simulation in model lattices of Tevatron and Large Hodron Collider. It was found that the reduction of the tune spread with the electron-beam compensation could induce a coherent beam-beam instability. The merit of the compensation with different degrees of tune-spread reduction was evaluated based on beam-size growth. When two beams have a same betatron tune, the compensation could do more harm than good to the beams when only beam-beam effects are considered. If a tune split between two beams is large enough, the compensation with a small reduction of the tune spread could benefit beams as Landau damping suppresses the coherent beam-beam instability. The result indicates that nonlinear (nonintegrable) beam-beam effects could dominate beam dynamics and a reduction of beam-beam tune spread by introducing additional beam-beam interactions and reducing Landau damping may not improve the stability of beams.