A new lattice for 3 TeV c.o.m. energy with?* = 5mm was developed which follows the basic concept of the earlier 1.5 TeV design but uses quad triplets for the final focus in order to keep the maximum magnet strength and aperture close to those in 1.5 TeV case. Another difference is employment of combined-function magnets with the goal to lower heat deposition in magnet cold mass and to eliminate bending field free regions which produce 'hot spots' of neutrino radiation that can be an issue at higher energy. The proposed lattice is shown to satisfy the requirements on luminosity, dynamic aperture and momentum acceptance.

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.

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.

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.

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]*

A recent progress report on the lattice design of the 50-50 GeV muon collider is presented. The ring circumference needs to be as small as possible due to the short lifetime of the 50 GeV muons. The background at the detector is affected by the continuous decay of muons into electrons which requires a dipole between the high focusing quadrupoles and the detector. To obtain a luminosity on the order of 1 x 1033 cm−2 s−1 it is required to have beam intensities on the order of 1 x 1012 particles per bunch. The rms momentum spread of the beam is equal to 0.12% and the beta functions at the interaction point are equal to 4 cm. The maxima of the betatron functions at these quadrupoles are 1,300 m, resulting in large chromaticities which must be corrected by local chromatic correction. Pairs of horizontal and vertical chromatic sextupoles are located at locations where the corresponding betatron functions are 100 m and the values of the horizontal dispersion functions are 3 and 2 m, respectively. They are carefully placed so that most of their nonlinear effects are canceled. The dynamic aperture is larger than 7 times the mean size of the beam for the momentum offsets larger than -6 and +10 sigmas.

A design of a muon collider ring with the center of mass energy of 6 TeV is presented. The ring circumference is about 6.3 km, and the $\beta$ functions at collision point are 1 cm in both planes. The ring linear optics, the non-linear chromaticity correction scheme in the Interaction Region (IR), and the additional non-linear field orthogonal knobs are described. The IR magnet specifications are based on the maximum pole tip field of 20 T in dipoles and 15 T in quadrupoles. The results of the beam dynamics optimization for maximum dynamic aperture are presented.