Nonlinear Beam - Beam Resonances

In a colliding beam storage ring, a major source of nonlinear resonance excitation resides in the beam-beam collisions as the stored beams repeatedly cross each other. During such collisions, particles in each beam see the electromagnetic field generated by the counter rotating beam. The beam-beam collisions therefore perturb the particle motion, causing-1. the transverse beam size to blow up and loss of luminosity 2. the beam lifetime to be reduced 3. and rapid beam loss as beam intensity increases beyond a more or less distinct threshold. Considerable efforts, both experimental and theoretical, have brought some insight into the beam-beam instability problem and often led to improvements in luminosity. But the nature of the instability and its associated underlying mechanism(s) are not yet fully understood and remain an outstanding problem for the designers of colliding beam storage rings. The beam-beam problem will be reviewed by other authors of this school (Seeman on experiments , Schonfeld on models and Myers on simulations). Nonlinear resonance behavior is the underlying process in these reviews and it is the job of this lecture to identify the role of nonlin-ear resonances in the beam-beam problem. To reduce the scope of this lecture, we will consider only head-on collisions of bunched beams. For the sign convention of the beam-beam force, we assume the two colliding beams to have opposite charges. We also assume that the reader is at least somewhat familiar with Hamiltonian dynamics and some basic facts about the beam-beam interaction. For part of the discussions on coherent beam-beam effects, we assume the reader has some knowledge about the Vlasov equation.