Investigation of the Flat-Beam Model of the Beam-Beam Interaction

Report: Cornell CBN-03-12 PAC’03, Portland/OR 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. BEAM EVOLUTION We model the flat beam as a current sheet which is uniform in the horizontal direction, x, and consider only motion in the vertical direction, y. Consider one-dimensional phase space distributions ψ1 and ψ2 of the two beams which are normalized to unity. Then the impulse from the second (first) on the first (second) beam is ∆y 1,2 = −Iψ2,1(y, s) (1) where we define Iψ(y, s) ≡ 4πNre γ ∞