A new model for the collective beam–beam interaction

Abstract

The collective beam–beam interaction is studied in the framework of maps with a ‘kick-lattice’ model in the four-dimensional (4D) phase space of the transverse motion. A novel approach to the classical method of averaging is used to derive an approximate map which is equivalent to a flow within the averaging approximation. The flow equation is a continuous-time Vlasov equation which we call the averaged Vlasov equation (AVE), the new model of this paper. The power of this approach is evidenced by the fact that the AVE has exact equilibria and the associated linearized equations have uncoupled azimuthal Fourier modes. The equation for the Fourier modes leads to a Fredholm integral equation of the third kind and the setting is ready-made for the development of a weakly nonlinear theory to study the coupling of the π and σ modes. The π and σ eigenmodes are calculated from the third kind integral equation. These results are compared with the kick-lattice model using our weighted macroparticle tracking code and a newly developed, density tracking, parallel, Perron–Frobenius (PF) code.