Particle physicists want a 10 tera-electron-volt (TeV) muon collider with very high collision rates. To get there they must make muon beams much narrower in position and momentum. The last stage of that beam shrinking, called final cooling, usually relies on very strong solenoid magnets (around 40 tesla). Those magnets are hard to build and operate. This paper designs a different support system for a previously proposed “wedge” cooling method that can meet the same transverse beam size goals without relying on ultra-high-field solenoids. The main target is to reduce the momentum-dependent beam offset (called dispersion) in the horizontal direction to about 0.001 meters (1 mm to 0.1 mm scale).

The authors modeled the wedge-based cooling lattice with the G4Beamline simulation code and an optimizer script. In their example the beam emerging from a thick diamond wedge had a mean momentum of about 87 MeV/c, a momentum spread of 7.15 MeV/c, and transverse emittances of roughly 42 micrometers in x and 146 micrometers in y. They designed a short magnet section made of a pair of quadrupoles (a focusing and a defocusing magnet) followed by a sector dipole magnet. The quadrupoles focus and match the beam, while the dipole corrects the remaining dispersion. The authors optimized parameters such as quadrupole gradients (about 4.5 and 6.7 T/m), a dipole field of about 1.74 T and a small bending angle, and found a configuration that reduces horizontal dispersion to about 0.0009 m and vertical dispersion to about -0.00076 m. Simulated particle transmission through the quadrupoles alone was about 93.5% and about 84.6% after the dipole.

At a basic level, dispersion is a correlation between a particle’s momentum and its transverse position: particles with different energies sit at different places in the beam pipe. The wedge cooling method relies on a position-dependent absorber thickness to transfer beam spread from the longitudinal direction (energy spread) into the transverse direction. Left unchecked, dispersion after the wedge means the beam position still depends on momentum, which spoils later RF manipulations that try to reduce energy spread. The designed magnet section removes most of that momentum-dependent offset so the following RF cavities can do an efficient energy-phase rotation and reduce the momentum spread as intended.