Investigation of the flat-beam model of the beam-beam interaction

At the interaction point of a storage ring collider each beam is subject to perturbations due to the electromagnetic field of the counterrotating beam. For flat beams, a well-known approximation models the beam by a current sheet which is uniform in the horizontal plane, restricting the particle motion to the vertical direction. In this classical model a water-bag beam distribution has been used to find working points and beam-beam tune shift parameters which lead to a stable beam distribution. We investigate the stability of a more realistic Gaussian equilibrium distribution. A linearized Vlasov equation written in action-angle variables is used to compute the radial and angular modes of a perturbation in two-dimensional phase space to first order in the displacement from the design trajectory. We find that the radial modes, which are often neglected, can have a stabilizing effect on the beam motion.