The muon collider would ex-tend limitations of the e[sup+] e- colliders and provide new physics potentials with a possible discovery of the heavy Higgs bosons. At the maximum energy of 2 TeV the projected luminosity is of the order of 10[sup 35] cm[sup[minus]2]s[sup[minus]1]. The colliding[mu][sup+][mu][sup[minus]] bunches have to be focused to a very small transverse size of few tenths of[mu]m which is accomplished by the betatron functions at the crossing point of[beta]*= 3mm. This requires the longitudinal space of the same length 3 mm. These very short bunches at 2 TeV could circulate only in a quasi-isochronous storage ring where the momentum compaction is very dose to zero. We report on a design of the muon collider isochronous lattice. The momentum compaction is brought to zero by having the average value of the dispersion function through dipoles equal to zero. This has been accomplished by a combination of the FODO cells together with a low beta insertion. The dispersion function oscillates between negative and positive values.

We discuss a preliminary design for a high luminosity 4 TeV center of mass[mu][sup+][mu][sup -] collider ring. A possible lattice for a muon collider has been described. The design satisfies most of the collider requirements, although it is not fully realistic. Error and tolerance analyses are yet to be performed as well as tracking to determine the dynamic aperture and achievable luminosity. In order to make the final-focus design realistic, drift spaces must be introduced between all magnet elements, and the lengths of the insertions will have to be increased in order to achieve the required dispersion values in the sextupoles with reasonable dipole fields.

Muon collider is a promising candidate for the next energy frontier machine. However, in order to obtain peak luminosity in the 1035/cm2/s range the collider lattice design must satisfy a number of stringent requirements, such as low beta at IP ([beta]*

Two modes are being considered for a 50 on 50-GeV muon collider: one being a high-luminosity ring with broad momentum acceptance (dp/p of (approximately) 0.12%, rms) and the other lower luminosity with narrow momentum acceptance (dp/p of (approximately) 0.003%, rms). To reach the design luminosities, the value of beta at collision in the two rings must be 4 cm and 14 cm, respectively. In addition, the bunch length must be held comparable to the value of the collision beta to avoid luminosity dilution due to the hour-glass effect. To assist the rf system in preventing the bunch from spreading in time, the constraint of isochronicity is also imposed on the lattice. Finally, the circumference must be kept as small as possible to minimize luminosity degradation due to muon decay. Two lattice designs will be presented which meet all of these conditions. Furthermore, the lattice designs have been successfully merged into one physical ring with mutual components; the only difference being a short chicane required to match dispersion and floor coordinates from one lattice into the other.