Beam-beam mode coupling in collision with a crossing angle

In recent e thorn e- colliders, a collision scheme with a large crossing angle has been designed and adopted. In two colliding beams, mode coupling between beam-beam oscillation modes is discussed in this paper. The typical sigma and pi modes are extended to include longitudinal motion. The tune shifts of the l 1/4 0 and l 1/4 1 modes are presented explicitly. The behavior of beam-beam modes is analyzed for different Piwinski angles. The instability due to the vacuum pipe impedance is discussed in terms of mode coupling between these beam-beam modes. Beam-beam simulations based on the strong-strong model have shown that a vertical beam-beam instability is induced by crosstalk between the beam-beam collision and the impedance. It is shown that the mode coupling theory for colliding beams reproduces the results of the beam-beam simulation. The impact of hourglass effects on the instability is also studied by simulation.