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 counter-rotating 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 is used to find working points and beam-beam tune shift parameters which lead to a stable beam distribution. We try to find stability criteria for a more realistic Gaussian equilibrium distribution. In order to analyze the instabilities, a linearized Vlasov equation is solved computing radial and angular modes to first order in the displacement from the design trajectory.