Analytical Evaluation of Dipole Performance Limits for a Muon Collider

Following the recommendation of the Updated European Strategy for Particle Physics, an International Muon Collider Collaboration has been formed and is currently studying the feasibility of a 10 TeV center-of-mass energy muon collider facility. Several technical challenges must be faced, mainly due to the limited muon lifetime at rest, 2.2
$\mu$
s. This extreme condition requires the use of ambitious magnets, RF systems, targets, shielding, and cooling. To avoid collimated neutrino beams from muon decay and remain below the natural radiation background that affects the area surrounding the facility, the straight lengths in the collider ring should be kept to an absolute minimum. To achieve this goal, the beam optics quadrupoles should be combined with the bending dipoles, featuring a high magnetic field (
$>$
10 T) and gradient (
$>$
100 T/m) in a large aperture (
$\sim$
150 mm). The need for a high field derives from the compactness requirement to achieve high luminosity via high crossing frequency. The large aperture is fundamental to allocate a radiation (W) beam screen, which will protect the superconductors from the muon decay products (a radiation heat load of 500 W/m due to electrons, positrons, and their synchrotron photons). All these constraints require cutting-edge technologies for the material choices, the mechanical layout, the quench protection, and the cooling. In this contribution, we show the performance limits of the possible candidate materials for such magnets (namely the LTS NbTi, Nb
$_{3}$
Sn, and the HTS ReBCO). The analysis is focused on dipoles, obtaining a relationship between maximum aperture and bore field determined by constraints including cost, critical current density, mechanical stress, and quench protection.