HIGH-ORDER SIMULATION OF MUON COLLIDER INTERACTION REGION

1. HIGH-ORDER SIMULATION OF MUON COLLIDER INTERACTION REGION Pavel Snopok (SPbSU/MSU/FNAL) Carol Johnstone (FNAL) Martin Berz (MSU) Dmitry Ovsyannikov (SPbSU) Alexander Ovsyannikov (SPbSU)

2. Introduction We study the IR dynamics of the 50-on-50 GeV Muon Collider in the presence of deliberately introduced high-order correctors (sextupoles, octupoles and duodecapoles) aimed at maximizing the dynamic aperture of the collider

3. Advantages and disadvantages of muon colliders • Smaller size compared to proton machines due to the smaller rest mass • Reduced synchrotron radiation compared to the electron machines => smaller size • Muons are not stable => fast acceleration required => smaller size to avoid luminosity degradation

4. Luminosity goals • Luminosity: 1033 cm-2 s-1 • Momentum spread: 0.12% • Beta functions at the interaction point: 4 cm

5. Muon Accelerating Complex • Proton Production • Pion Production • Decay Channel • Muon Cooling • Acceleration • Muon Collider

6. Dynamic Aperture • Coordinate of the farthest from the origin stable particle • Particles are tracked for 1000 turns • The beam is assumed circular in (x-y) • Concentric grid with one sigma step, for 90π mm mrad normalized emittance σ=82·10-6 m

7. Optimization details • DA before the optimization is 7σ • To maximize the DA sextupole, octupole and/or duodecapole correctors are added to the IR superconducting quadrupoles • Strength of the correctors is chosen in different ways targeting different objectives and nonlinear terms in the transfer map

8. Optimization approaches I • Minimize some high order coefficients in the transfer map of one revolution => not effective because of the complicated connections between the nonlinear coefficients of the transfer map

9. Optimization approaches II • Minimize nonlinear resonances and/or tune shifts with amplitude=> more effective than the previous approach (7σ to 12σ), but requires further studies as the number of correctors is not sufficient to remove all the nonlinear resonances

10. Optimization approaches III • Straightforward way: use the DA itself as a figure of merit=>more time-consuming (recalculate the DA at each step which means tracking the grid of particles for 1000 turns), but most effective as of now (7σ to 13 σ, almost twice the aperture)

11. Qualitative results of the optimization (x-x’): (y-y’): Before: After:

12. Further optimization steps • Next study can be to add higher order correctors (decapoles, duodecapoles) to improve the DA further, though the preliminary study shows this approach to be not so effective (for one scheme with octupoles and duodecapoles particles at 14σ survive, but at 13σ – are lost) • A program is currently being developed to implement variational methods of optimization and try to find a better solution using the second approach (nonlinear resonances)

13. Summary • A variety of optimization methods aimed at improving the DA were studied, and we determined the one which worked in the most effective and efficient way and produced the best results • The results of this optimization using chosen method developed for this study were presented and the DA was found to increase by almost twice compared to the situation before optimization • Only two octupole correctors were required in the proximity of IR (at the peaks of beta functions in x and y planes) • Though we aimed at maximizing the DA, some of the higher order terms in the nonlinear resonance matrix decreased naturally by an order or two in magnitude. Their reduction serves as a strong, quantitative measure of the efficiency of this approach to optimization