1. Crab crossing and crab waist at super KEKB K. Ohmi (KEK) Super B workshop at SLAC 15-17, June 2006 Thanks, M. Biagini, Y. Funakoshi, Y. Ohnishi, K.Oide, E. Perevedentsev, P. Raimondi, M Zobov

2. Super bunch approach K. Takayama et al, PRL, (LHC) F. Ruggiero, F. Zimmermann (LHC) P.Raimondi et al, (super B) Short bunch Long bunch  x is smaller due to cancellation of tune shift along bunch length Overlap factor

3. Essentials of super bunch scheme Bunch length is free. Small beta and small emittance are required.

4. Short bunch scheme Small coupling Short bunch Another approach: Operating point closed to half integer

5. We need L=10 36 cm -2 s -1 Not infinity. Which approach is better? Application of lattice nonlinear force Traveling waist, crab waist

6. Nonlinear map at collision point 1 st orbit 2 nd tune, beta, crossing angle 3 rd chromaticity, transverse nonlinearity. z- dependent chromaticity is now focused.

7. Waist control-I traveling focus Linear part for y. z is constant during collision.  =0

8. Waist position for given z Variation for s Minimum  is shifted s=-az

9. Realistic example- I Collision point of a part of bunch with z, =z/2. To minimize  at s=z/2, a=-1/2 Required H RFQ TM210 V~10 MV or more

10. Waist control-II crab waist (P. Raimondi et al.) Take linear part for y, since x is constant during collision.

11. Waist position for given x Variation for s Minimum  is shifted to s=-ax

12. Realistic example- II To complete the crab waist, a=1/ , where  is full crossing angle. Required H Sextupole strength K 2 ~30-50 Not very strong

13. Crabbing beam in sextupole Crabbing beam in sextupole can give the nonlinear component at IP Traveling waist is realized at IP. K 2 ~30-50

14. Super B (LNF-SLAC) BasePEP-III C3016 2200 xx 4.00E-102.00E-08 yy 2.00E-122.00E-10  x (mm) 17.810  y (mm) 0.080.8  z (mm) 410 ne2.00E+103.00E+10 np4.40E+109.00E+10  /2 (mrad) 2514 xx 0.0025 yy 0.1

15. Luminosity for the super B Luminosity and vertical beam size as functions of K2 L>1e36 is achieved in this weak-strong simulation.

16. DAFNE upgrade DAFNE C97.7 xx 3.00E-07 yy 1.50E-09  x (mm) 133  y (mm) 6.5  z (mm) 15 ne1.00E+11 np1.00E+11  /2 (mrad) 25 xx 0.033 yy 0.2479

17. Luminosity for new DAFNE L (x10 33 ) given by the weak-strong simulation Small s was essential for high luminosity netune s L(K2=10)1520 6.00E+100.53, 0.580.0124.273.79 6.00E+100.53, 0.58-0.014.353.93 1.00E+110.53, 0.58-0.014.075.665.53 6.00E+100.53, 0.580.012  z=10mm 4.322.47 6.00E+100.53, 0.58-0.01  z=10mm 4.652.7 6.00E+100.057, 0.097-0.015.19(3.3) 1.00E+110.057, 0.097-0.0113.21(4.8) () strong-strong, horizontal size blow-up

18. Super KEKB Crab waist xx 9.00E-096.00E-09 yy 4.50E-116.00E-11  x (mm) 2001005010050  y (mm) 310.51  z (mm) 36644 s 0.0250.01 ne5.50E+10 3.50E+1 0 np1.26E+111.27E+11 8.00E+1 0  /2 (mrad) 015 xx 0.3970.04180.0220.05470.0298 yy 0.794- >0.24 0.19850.1790.1780.154 Lum (W.S.)8E+356.70E+351.00E+36 3.95E+3 5 4.80E+35 Lum (S.S.)8.25E354.77E359E353.94E354.27E35 Horizontal blow-up is recovered by choice of tune. (M. Tawada)

19. Particles with z collide with central part of another beam. Hour glass effect still exists for each particles with z. No big gain in Lum.! Life time is improved.  x =24 nm  y =0.18nm  x=0.2m  y =1mm  z=3mm Traveling waist

20. Small coupling

21. Traveling of positron beam

22. Increase longitudinal slice  x =18nm,  y =0.09nm,  x =0.2m  y =3mm  z =3mm Lower coupling becomes to give higher luminosity.

23. Why the crab crossing and crab waist improve luminosity? Beam-beam limit is caused by an emittance growth due to nonlinear beam- beam interaction. Why emittance grows? Studies for crab waist is just started.

24. Weak-strong model 3 degree of freedom Periodic system Time (s) dependent

25. Solvable system Exist three J’s, where H is only a function of J’s, not of  ’s. For example, linear system. Particles travel along J. J is kept, therefore no emittance growth, except mismatching. Equilibrium distribution

26. y pypy

27. One degree of freedom Existence of KAM curve Particles can not across the KAM curve. Emittance growth is limited. It is not essential for the beam-beam limit.

28. Schematic view of equilibrium distribution Limited emittance growth

29. More degree of freedom Gaussian weak-strong beam-beam model Diffusion is seen even in sympletic system.

30. Diffusion due to crossing angle and frequency spectra of Only 2 y signal was observed.

31. Linear coupling for KEKB Linear coupling (r’s), dispersions, ( ,  =crossing angle for beam-beam ) worsen the diffusion rate. M. Tawada et al, EPAC04

32. Two dimensional model Vertical diffusion for  =0.136 D C <<D  for wide region (painted by black). No emittance growth, if no interference. Actually, simulation including radiation shows no luminosity degradation nor emittance growth in the region. Note KEKB D  =5x10 -4 /turn DAFNE D  =1.8x10 -5 /turn (0.51,0.51) (0.7,0.51) (0.7,0.7) (0.51,0.7) (0.7,0.51) (0.51,0.7) KEKB D  =0.0005

33. 3-D simulation including bunch length (  z ~  y ) Head-on collision Global structure of the diffusion rate. Fine structure near x =0.5 Contour plot (0.51,0.51) (0.7,0.51) Good region shrunk drastically. Synchrobeta effect near y~0.5. x~0.5 region remains safe.

34. Crossing angle (  z /  x ~1,  z ~  y ) Good region is only ( x, y )~(0.51,0.55). (0.51,0.51) (0.7,0.51) (0.51,0.7)

35. Reduction of the degree of freedom. For x~0.5, x-motion is integrable. (work with E. Perevedentsev) if zero-crossing angle and no error. Dynamic beta, and emittance Choice of optimum x

36. Crab waist for KEKB H=25 x p y 2. Crab waist works even for short bunch.

37. Sextupole strength and diffusion rate K2=20 25 30

38. Why the sextupole works? Nonlinear term induced by the crossing angle may be cancelled by the sextupole. Crab cavity Crab waist Need study

39. Conclusion Crab-headon Lpeak=8x10 35. Bunch length 2.5mm is required for L=10 36. Small beam size (superbunch) without crab-waist. Hard parameters are required  x =0.4nm  x =1cm  y =0.1mm. Small beam size (superbunch) with crab-waist. If a possible sextupole configuration can be found, L=10 36 may be possible. Crab waist scheme is efficient even for shot bunch scheme.