Nonlinear Dynamics Problems in the XXI Century High Energy Accelerator Physics Y. Alexahin

1. FERMI NATIONAL ACCELERATOR LABORATORY US DEPARTMENT OF ENERGY f Nonlinear Dynamics Problems in the XXI Century High Energy Accelerator Physics Y. Alexahin MAP collaboration meeting, IU Bloomington March 12, 2007

2. General trends • Will there be HEP? • Provided no global disaster happens (the deluge, asteroid hit, victorious Islamic jihad) the quest for fundamental knowledge will continue, the push to energies Ecom>100TeV can be expected. • The rumors of HEP demise spread some 15 years ago were exaggerated ! • Circular or Linear? • So far linear colliders were not competitive at the energy frontier, SLC was eclipsed by LEP • VLEPP, JLC, NLC were dropped, ILC will likely follow the “analytic continuation” • Laser acceleration, two-beam acceleration and other “new” schemes still not mature enough • Circular colliders will dominate HEP in the foreseeable future, the Muon Collider being an exciting perspective: • muons are point-like particles, the energy is not divided between the constituents • smaller spread in Ecom will enhance production rate of narrow resonances and allow to resolve close peaks • SR losses relatively small up to Ecom~100TeV (with diameter ~100km at B=20T) • offers a lot of interesting problems which graduate and PhD students can address! • (not limited in value to MC itself, but relevant to NF and hadron colliders as well) Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

3. Muon Collider Scheme Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

4. Muon Collider Parameters • Low emittance option (advanced): owing to ideas by Yaroslav Derbenev (HCC, PIC) much lower 6D emittances seem to be feasible than previously thought of. • High emittance option (baseline):conceptuallyfollows1999 PRSTAB Muon Collider Collaboration report Low Emitt.High Emitt. Energy (TeV) 0.75+0.75 (=7098.4) Average Luminosity (1e34/cm^2/s) 2.7 2 Average bending field (T) 10 6 Mean radius (m) 361.4 500 Number of IPs 4 (175m each) 2 P-driver rep.rate (Hz) 65 60 Beam-beam parameter/IP,  0.052 0.1  (cm) 0.5 1 Bunch length (cm), z 0.5 1 Number of bunches/beam, nb 10 1 Number of muons/bunch (1e11), N 1 11.3 Norm.transverse emittance (m), N 2.1 12.3 Energy spread (%) 1 0.2 Norm.longitudinal emittance (m), ||N 0.35 0.14 Total RF voltage (GV) at 800MHz 406.6 103c 5.6103c RF bucket height (%) 23.9 2.4 Synchrotron tune 0.723 103c 0.1103c + - in collision / proton 0.15 /2 0.15 8GeV proton beam power (MW) 1.1 0.6 Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

5. Road to High Luminosity f z /  “Hourglass factor” • Low transverse emittance to achieve the beam-beam limit with realistic N • Low  ( 1cm)  requires small bunch length z   therefore: • Low longitudinal emittance to keep E reasonable • Small momentum compaction factor (c ~ 10-4)  strongly focusing (hence nonlinear) lattice Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

6. Two Scenarios for Ionization Cooling Strawman LEMC REMEX-2 HCC REMEX-1 PIC Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

7. Ionization Cooling Principle Owing to ionization losses muon loses momentum (both P and P||) then it regainsP|| in RF, in the result • Problems • longitudinal oscillations are naturally anti-damped at p<300MeV/c (muons with higher energy lose less) • discrete (stochastic) nature of losses  longitudinal heating (straggling) • multiple scattering  transverse heating Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

8. Ionization Cooling Principle tilted solenoids straight solenoids  RF cavities • Solution • Generate dispersion to provide longitudinal damping (via path lengthening or wedge absorbers) • Strong focusing as possible to minimize effect of scattering (henceforth 50T solenoid) The simplest 6D cooling channel: FOFO with resonant dispersion generation by tilted solenoids • Bread for theorists: • Strongly nonlinear non-Hamiltonian system! • There are programs for tracking simulations (ICOOL, G4BL) but no optics code for strongly dissipative systems (untreatable as perturbed Hamiltonian system) • The necessary theory outlined in Leo Michelotti’s book and successfully used (but in perturbation approach) for LEP (Y.Alexahin, PAC99) Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

9. MC Ring Design Challenges • Low  ( 1cm) • Small momentum compaction factor (c ~ 10-4) • Focal length of LB quads can not be made small: shielding from decay electrons takes 3-5 cm, the first quad had to be far from IP (>6m at E=750GeV) • Huge chromatic aberrations very strong sextupoles higher order effects Chromatic functions with (x, x’) set (not with MAD variables): Typical values for IR parameters from the wishlist: MAD chromatic functions in IR w/o correction this means 400% - function modulation at p=0.001 (!) Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

10. MC Lattice Design Philosophy The only way to obtain a non-vanishing momentum acceptance is to cancel ax,y before they convert into bx,y : • sextupole correctors must be in the IR, not in a separate CC section! • dispersion must be present in IR, it may be generated 1) by dipoles in the IR so that Dx= Dx‘= 0 at the IP, 2) outside the IR so that Dx= 0 but Dx‘ 0 at the IP. We explore the first possibility (symmetric design) which has an apparent drawback – huge value of the dispersion invariant generated in the IR But we make a good use of it: • we can earn a negative contribution to c while suppressing this huge Jx • rest of the lattice can be simple FODO cells which helps to reduce the circumference (FODO has the largest dipole packing factor) • to maximize the dipole packing factor (and minimize the circumference) we choose a large phase advance per cell, 108 for now (plan to try 135) • Dipoles in IR provide detector protection from decay electrons (whether SR from this electrons presents a danger is the matter of ongoing study) Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

11. “Dipole First” MC Lattice Design Option x y DDx/100 Dx Wx Wy IR, negative dispersion and matching sections Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

12. MC Lattice Properties  CSIy [m] Qx p Qy c p  CSIx [m] The 1024 turns DA is only marginally sufficient for the high-emittance option: ~3 for N=12.5 m (O.K. for the low) Momentum acceptance of ± 0.7% is O.K. for the high-emittance option (not for the low) Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

13. Problems to Address Obviously the tracking simulations give the final verdict, but they are time consuming and provide little insight, semi-analytical methods (one has to resort to computer anyway) would be helpful. Strong nonlinearities - nonperturbative analysis? There were numerous attempts in the SSC era, mainly by SLAC people (R.Ruth & R.Warnock, S.Kheifets), but no viable tool developed that I know of, would be interesting to revisit the issue, direct integration of the Lie-transform + Dewar equations being a promising approach. With hadron colliders running in the forseeble future (LHC, RHIC, VLHC?) and possible -p option for the muon collider, the long-term effect of high order nonlinear resonances (driven by beam-beam interaction as well as nonlinear magnets) is important. Again, tools for express analysis would be highly helpful Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

14. Diffusion Due to Multiple Crossing of a Resonance (v) v At the Tevatron we observed strong diffusion on the 12th order resonances which were to weak for synchrotron satellite overlap. In search for the diffusion mechanism we rediscovered that of the multiple crossing first described by P.Sturrock in 50-s. Calculation of the action variance over one synchrotron period (two successive crossings) gives for the diffusion rate Dimensionless crossing velocity Fast crossing (Sturrock) Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

15. Diffusion Enhacement by Isolated Resonance Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

16. Total Picture of One Resonance trapping, D = Dm0 with constant crossing velocity, D = Dmult Contribution to the diffusion coefficients Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

17. Diffusion Equation Due to the following properties of the diffusion coefficients operator is self-adjoint on functions = 0 at the absorbing boundary and has therefore an orthogonal system of eigenfunctions Solution of the initial value problem Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

18. Evolution in Action Variables Space t = 0 t = 10min t = 1h Effect of sum resonances, with difference resonances the numerical algorithm is unstable Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007

19. References •  Leo Michelotti, “Intermediate Classical Dynamics with Applications to Beam Physics”. J.Wiley, 1995. • Methods and Complex of Programs for Radiating Particles 3DoF Nonlinear Dynamics Analysis. PAC’99, New York, 1999 • P.A.Sturrock, Ann. Physics, 3 (1958) 113 • Diffusion due to beam-beam resonances in hadron colliders. FERMILAB-CONF-05-177-AD (May 2005) Nonlinear Dynamics Problems in XXI Century - Y. Alexahin, FNAL March 12, 2007