Longitudinal instability in a 50 GeV×50 GeV muon collider ring

Simulations of the longitudinal dynamics in 50 GeV×50 GeV muon collider ring have been performed. Operation of the ring close to the slippage factor η₁≃10^-6, such that synchrotron motion is frozen, minimizes the need for RF to maintain the bunch length. Without appropriate RF compensation, however, the bunch wake induces an intolerable head-to-tail energy spread. This paper demonstrates that the bunch wake may be compensated by two RF cavities with low RF voltages. With this RF setup the small energy spread of the beam (δE/E=3×10^-5) in the 50 GeV×50 GeV muon collider ring can be maintained during the 1000 turn lifetime of the muons. These studies were made at the nominal design point, and sensitivities to errors were explored. The simulation also demonstrates that the computation of the wake field using bins of variable width (each with a constant number of macroparticles) accurately reproduces the wake and yields reduced computing time compared to the evaluation of the wake as the direct sum over the wakes of all preceding macroparticles