Tune-shift with amplitude due to nonlinear kinematic effect

Tracking studies of the Muon Collider 50 on 50 GeV collider ring show that the on-momentum dynamic aperture is limited to around 10 sigma even with the chromaticity sextupoles turned off. Numerical results from the normal form algorithm show that the tune-shift with amplitude is surprisingly large. Both analytical and numerical results are presented to show that nonlinear kinematic effect originated from the large angles of particles in the interaction region is responsible for the large tune-shift which in turn limits the dynamic aperture. A comparative study of the LHC collider ring is also presented to demonstrate the difference between the two machines. (14 refs).