Control of longitudinal collective behavior in the muon collider rings

The longitudinal single bunch collective effects in a Muon Collider ring are theoretically examined. The situation involves an intense bunch, a short bunch, a small momentum compaction, a rather large impedance compared with the stability threshold criterion, and a luminosity life time limited by muon decay to a thousand turns. Qualitative descriptions of stability are given and a scaling law for the instability threshold is derived. Numerical simulation results for the impedance-related instabilities are given for two cases of current interest-a 250 GeV×250 GeV demonstration machine and a 2 TeV×2 TeV high energy machine. The results of these simulations are in good agreement with the predictions of the scaling law and show that the longitudinal collective effects are controllable with a proper choice of parameters (viz. RF voltage, RF frequency, linear and non-linear longitudinal chromaticity)