A Symplectic Model of Coherent Beam-Beam Quadrupole Modes

We consider a simple model to study the effects of the beam-beam force on the dynamics of colliding beams. We focus on the quadrupole modes of the coherent oscillation. The ingredients are: (1) linearized beam-beam kicks, (2) damping and noise due to synchrotron radiation, and (3) linear transport between beam-beam kicks. The dynamical variables are the 2nd-order moments of the canonical variables q, p, which include the rms bunch size. Our model is self-consistent in the sense that no higher order moments are generated by the linearized beam-beam kicks, and that the only source of violation of symplecticity is the radiation. We discuss the cases of round and flat beams. Depending on the values of the tune and beam-beam kick strength, we observe states in which otherwise identical bunches have sizes that are equal, or unequal, or periodic, or behave chaotically from turn to turn. Possible implications of luminosity saturation with increasing beam intensity are discussed.