Coherent Beam - Beam kteraction with Four CoUiding Beams

The coherent beam-beam interaction in the absence of Ladau damping is studied with a computer simulation of four, space-charge compensate colliding beams. Results are presented for the modes, phase space structures, widths, and growth rates of coherent beam-beam resonances. These results are compared with solutions of the Vlasov equation, and with measurements made at the DCI storage ring which operated with space-charge compensated colliding beams. INTRODU~ION The luminosity of storage ring colliders is limited by the effects of the electromagnetic fields of one beam on the particles of the other beam. This beam-beam interaction is parametrized by the beam-beam strength parameter, (1) where re is the classical electron radius, N is the number of particles in the beam, ~~ is the vertical amplitude function at the interaction point, y is the beam energy in units of rest energy, md ax and Oy are the rms horizontal and vertical bem sizes at the interaction point. The beam-beam interaction is not linear in displacement, and, in the usual case of two colliding beams, those nonlinearities inmoduce single particle nonlinear resonances and a spread in transverse oscillation tunes. The vertical tune spread is equal to ~ which is sometimes denoted as Ey. The be~-beam luminosity limit could be due to the nonlinear resonances and the tune spread which are single particle, incoherent effats, or it codd be due to coherent instabihties.1 Coherent beam-beam instabilities are expected based on solutions of the Vlasov equation.2~3~4~5 They are characterized by rapid, turn-by-turn, correlated variations of the beam distributions. They have been seen in a two-beam simtiation that used particle-in-ce~ techniques to calculate electromagnetic fields.6 k this simulation there was qutitative agreement with Vlasov equation solutions for sixth and eighth order resonances, but higher order resonances were not seen, presumably because of Landau damping from the beam-beam tune spread. The instabilities that were observed occurred for ~ >0.05 and could be avoided by the appropriate choice of operating point. This led to the conclusion that the coherent beam-beam effect was not Wely to k important in operating or planned colfiders. h contras~ turn-by-turn variations of beam sizes have been observed at LW using a novel detector capable of imaging the beam on successive turns.7 More data are needed before drawing any conclusion about the relation between these variations and the beam-beam fimit. Incoherent beam-beam effects can be eliminated by colliding four beams in the field compensating cotilguratiofi …