Longitudinal instability thresholds of counter-rotating pp-bar or ee-bar beams in a collider

The longitudinal instability threshold for the multipole oscillations of two colliding beams is estimated; the beams are supposed to be counter-rotating in the same vacuum chamber. The eigenfrequencies of the usual bunch-to-bunch single beam oscillation modes are shown to be split into two sets due to beam-beam coupling through the resonance elements. The beam-beam coupling coefficient is found to depend upon the resonance element locations around the orbit with respect to the beam interaction points. 4 refs