Beam instability issues of the 50GeV×50GeV muon collider ring

Single bunch instabilities for the 50GeV×50GeV muon collider are discussed. An impedance budget of the collider is estimated. A phase-slip factor of |η|=1×10^-6 is desired to avoid excessive rf systems. Potential-well distortion of a smooth bunch can be compensated by rf cavities. Accumulated growth in energy due to imperfections and noises in the muon bunch can be reduced by smoothing the bunch distribution before injection. The growth due longitudinal microwave instability is small because of the compensated rf cavities, the finite lifetime of the muons, and the choice of a small |η|. Beamloadings in the compensating rf cavities are large and suitable feed-forward cancellation is required. Transverse microwave instability can be damped by chromaticities and octupoles. Beam breakup can be cured by Balakin-Novokhatsky-Smirnov damping in principle, but is nontrivial in practice. When beam breakup is small, it can possibly be damped by a betatron tune spread in the beam.