1. Muon Coalescing 101 Chuck Ankenbrandt Chandra Bhat Milorad Popovic Fermilab NFMCC Meeting @ IIT March 14, 2006

2. Context for this talk Suppose that some day: A proton driver based on an 8-GeV linac exists; High-int. muon beams are available with low emittances in all three planes; The proton driver linac can be used to accelerate both protons and muons. Achieving low emittances requires parametric resonance ionization cooling. However, Slava Derbenev has found that PIC doesn’t work well for very intense bunches because of space-charge tune shifts. That led Rolland Johnson and Slava to develop scenarios that produce a large number of less intense bunches. In one specific scenario, each of ten equally spaced proton bunches produces a train of sixteen equally spaced muon bunches. That works as is for a neutrino factory; however, to achieve high luminosity in a collider, it is highly desirable to combine the bunches. That in turn led them to ask the question addressed in this talk: How can muon bunches be combined to enhance the luminosity of a muon collider?

3. General combining considerations Combining ought to be done after accelerating to high energy, where space charge is not a problem and adiabatic damping of beam sizes provides room to operate. At high energy, momentum-dependent path lengths work better than velocity differences for combining bunches. There are two bunch-combining techniques presently used operationally for protons at Fermilab: slip-stacking and coalescing. The specific implementations used for protons are much too slow for muons. The approach described here is a fast form of coalescing. Fast coalescing ignores slow niceties, so reducing the dilution of longitudinal emittance is a major consideration.

4. First-Order Ring Physics 1.Muon Decays in Rings Decay length So the number of turns to decay is given by where f is the fill factor

5. First-Order Ring Physics 2. Space Charge for Gaussian bunch Numbers: Compareat the same energy

6. First-Order Ring Physics 3. Slippage where and it’s easier to useHere, N c, number of turns to coalesce= Where Lo=half-length of bunch train Assuming momentum spread is constant

7. Schematic of the LINAC and Coalescing Ring Coalescing Ring 20 GeV Muon LINAC Bunch train with 1.3GHz structure Bunch LE~ 0.03 eVs dE~ 20 MeV vernier LINAC

8. Muon Coalescing Ring The following parameters are assumed for the Coalescing Ring: Injection beam : 1.3GHz bunch structure # of bunches/train = 17 Ring Radius = 52.33m; Revolution period= 1.09  s Energy of the muon = 20 GeV (gamma = 189.4) gamma_t of the ring = 4 If we assume Ring-Radius/rho (i.e., fill factor) = 2, then B-Field = 2.54T (This field seems to be reasonable) h for the coalescing cavity = 42, 84 Number of trains/injection = less than 37 (assuming ~100ns for injection/extraction) RF voltage for the coalescing cavity = 1.9 MV (h=42) = 0.38 MV (h=84) fsy ~ 5.75E3Hz Tsy/4 = 43.5us Number of turns in the ring ~40 Constraints: Muon mean-life = 2.2us (rest frame) Muon mean-life in lab = 418us for 20 GeV beam Time (90% survival) = 43.8us Radius=52.3m Injection extraction

9. Initial Simulation Results Three scenarios in a 20 GeV ring for up to 37 groups of 17 bunches of 1.3GHz Scenario1: rf cavities in the ring  takes 54  s Scenario2: vernier linac  takes about 46- 54  s Scenario3: vernier linac and rf cavities in the ring  takes about 38  s

10. 1 st Scenario Muon Bunch train from the LINAC

11. Muon Bunch train in the coalescing bucket T=0 sec dE~ 20 MeV

12. Muon Bunch train in the coalescing bucket T= 31.6  sec

13. Muon Bunch train in the coalescing bucket T= 54  sec dE~ 200 MeV Bunch Length~ 1.5ns

14. 2 nd Scenario A vernier-linac to give a tilt in the Longitudinal Phase-space Muon Bunches after pre-linac And next inject the beam into the Coalescing Ring Bunch train before the special purpose pre-linac

15. Muon Bunch train in the Coalescing Ring T=0 sec

16. Muon Bunch train in the Coalescing Ring T=46  sec dE~ 100 MeV Bunch Length~ 4ns

17. Muon Bunch train in the Coalescing Ring T=71  sec dE~ 60 MeV Bunch Length~ 3ns

18. Muon Bunch train in the Coalescing Ring T=0 sec 3 rd Scenario

19. Muon Bunch train in the Coalescing Ring T=38  sec dE~ 200 MeV Bunch Length~ 1.5ns

20. Summary and conclusions Fast coalescing requires: Short muon bunch trains (less than half the distance between proton bunches) A large momentum ‘ramp’ across each train Small transition gamma (weak focusing lattices?) Large radial acceptance in the ring The energy ramp can be generated with a vernier linac and/or with rf cavities in the ring. Coalescing leads to multiple constraints (on ring circumferences, bunch spacings, rf frequencies, etc.) Longitudinal emittance dilution is a concern. Of course, global optimization is required.

21. Mindset and motivation We are much more likely to get a proton driver if it can be designed and sited in such a way that it provides a versatile multistage upgrade path to transform existing facilities into sources of megawatt-class proton beams (as well as being an ILC testbed). We are much more likely to get a proton driver, a stopping muon program, a neutrino factory, and a muon collider if we can maintain synergy among all of them. In particular, the path to a neutrino factory should not diverge from the path to a muon collider. Even though a neutrino factory might be implemented with only modest muon cooling, early achievement of extreme muon cooling would have several important advantages: Muons could be accelerated in the proton driver linac; The rest of the neutrino factory (except cooling) would be easier to implement; The path from the neutrino factory to the muon collider would be much easier.