1. 1 Crab Waist Studies for SuperB and KEKB Y. Ohnishi/KEK SuperB Workshop V Paris 10/May/2007

2. 2 Crab Crossing and Crab Waist Head-on collision is effectively realized by crab cavity, while crab waist controls nonlinear interaction induced by crossing angle. For high current scheme, either scheme will work, but not both at once. For low emittance scheme, only crab waist is applicable because beam-beam tune shift becomes too high. Crab waist can make luminosity comparable to that of crab crossing.

3. 3 Crab Waist Scheme Hamiltonian can be considered at IP as follows: Then, the vertical position changes as: Transformation of phase space can be written by: If drift space is also considered, total transformation: This Hamiltonian can be made by sextupoles.

4. 4 Crab Waist Scheme (cont'd) Twiss parameters at IP can be transformed by M: The waist moves along the x position: When an appropriate K is chosen as follows, particle collides with other beam at the waist point. The strength of sextupoles are determined by above formula. s x 2x2x

5. 5 Crab Waist Scheme (cont'd) In order to make H=xp y 2 at IP using sextupoles, an appropriate betatron phase advance is needed. The Hamiltonian of sextupoles is: X and Y are reference coordinate (x and y are physical coordinate) : Transformation of sextupoles(SX1 → IP and SX2 ← IP): m, n, k, l are integer and arbitrary. This term makes crab waist.

6. 6 Crab Waist Scheme (cont'd) When  x = ,  y =  /2 is chosen, effect of the crab waist becomes: In order to make crab waist, strength of sextupoles is: On the other hand, x 3 term becomes: 2  x =34 mrad,  x *=20 mm,  y *=0.3 mm,  x,sext =14 m,  y,sext =140 m → K=26.5 m -2 F 3,x 3 = ~80000 F 3,crab = 29 Very large effect !

7. 7 Crab Waist Studies for SuperB

8. 8 SuperB-HER new lattice by Pantaleo

9. 9 Lattice Parameter betatron tunes

10. 10 Final Focus - Old(left) & New(right) OCTX0(K 3 =-8) DECY0(K 3 =1)OCTX4(K 3 =-23)DECY4(K 3 =-6) -I' SFX5 K 2 =0 -I' crab waist sextupole

11. 11 Final Focus with Crab Waist  x = 6.0   y = 5.5   x crab /  y crab =14 m/140 m Sextupole(thin) for crab waist  x */  y * = 20 mm/200  m

12. 12 Interaction Region 30 cm OCT0 magnet (multipole) IP close-up of IR

13. 13 Chromaticity Correction

14. 14 SuperB-HER Chromatic Functions DDX WX WY WX,WY DDX s (m) IP

15. 15 Chromatic Functions (cont'd) DDX WX WY WX,WY DDX s (m) IP

16. 16 Dynamic Aperture for SuperB-HER  p/p 0 n x =x/  x J y /J x =0.25% (fixed) Synchrotron oscillation: ON Radiation damping: OFF Quantum excitation: OFF #turns: 1000 x / y =48.57/23.60  x */  y *=20 mm/200  m  x crab /  y crab =14 m/140 m (thin-lens sextupole) 7 GeV Crab waist K 2 =0 Crab waist K 2 =+20/-20 Fringe effect for all magnets Dashed line: OFF Solid line: ON Crab waist:K 2 =0, Fringe effect:OFF, Synch. oscillation:OFF Fringe field reduces dynamic aperture.

17. 17 Effects of Fringe Field(Maxwellian Fringe) All magnets fringe on FF region fringe on QD0 only fringe on All magnets fringe off  p/p 0 (2J x /  x ) 1/2 J y /J x =0.25% Blue: Crab waist off/Red: Crab waist on Fringe of QD0 strongly affects dynamic aperture.

18. 18 Local Compensation of Final Focus Fringe OCT0 K 3 Score(are of dynamic aperture) K 3 =16 K 3 =8 K 3 =24 K 3 =4 K 3 =0 K 3 =-4 K 3 =12  p/p 0 (2J x /  x ) 1/2 J y /J x =0.25% OCT0 is turned on All magnets include nonlinear fringe effects Octupole near QD0 increases dynamic aperture(on-momentum) by 30%. Crab waist ON

19. 19 Summary of Studies at SuperB SAD is consistent with MAD for the case of no fringe field. If the fringe field is ignored, the 50  x (70  x for K 2 =0) can be achieved in the dynamic aperture for on- momentum. However, fringe field is turned on, the dynamic aperture decreases to 18  x. Final focus doublet(QD0) affects the dynamic aperture significantly. Octupoles near QD0 increase the dynamic aperture(on- momentum) by 30%. Cure of x 3 term should be considered ?

20. 20 Crab Waist Studies for KEKB

21. 21 Crab Waist Optics at KEKB Beam-beam simulation by K. Ohmi has shown that the crab waist scheme will boost the luminosity of the present KEKB, as well as the crab cavity. Several lattice design was tried on the computer. As the result, drastic degradation of the dynamic aperture was found. No good solution has been obtained so far. sextupole waist

22. 22 LER Crab Waist Oho-Nikko Version  X 12.5*2   y 11.25*2   X 12.5*2   y 11.75*2  K2 9.3 K2 -9.3 NX 25 NY 23 I - transformation Nikko Oho Tsukuba LER H. Koiso

23. 23 Crab Waist Oho-Nikko Version The dynamic aperture is estimated by tracking in the 6D phase space. 1000 turns (1/4 of the longitudinal damping time) Crab waist sextupoles significantly decrease the dynamic aperture. x/  x  p/p 0

24. 24 Crab Waist Tsukuba Version 1  X 1.5*2   y 1.75*2   X 2.0*2   y 1.75*2  SX K 2 9.3 SX K 2 9.3 Crab waist sextupoles (SCWTR/L) are located in IR. -I’ transformation between the sextupoles  X /  Y =21/150 m at SCWTR/L Those sextupoles decrease the dynamic aperture that is the same as Oho-Nikko version. K2 ±9.3 K2=0 SL0 を off RF ON Two additional quadrupoles to match the condition for the crab waist.

25. 25 Crab Waist Tsukuba Version 2  X 1.5*2   y 1.75*2   X 2.0*2   y 1.75*2  SX K2 6.6 SX K2 6.6 Sextupoles both in IR and in other straight sections to cancel x 3 term at IP.  x/  y=10/300m @ SCWTR/L  x/  y=60/15m @ SCWN/O Removed solenoid and multipole components of the final quadrupoles (QCS, QC2) Thin sextupoles  X 11.5*2   y 10.5*2   X 11*2   y 10*2  SCWTR/L: K2=+6.6 SCWN: K2=-.449 SCWO: K2=+.449

26. 26 Crab Waist Tsukuba Version 2 Sextupoles(SCWTR/L) for the crab waist are located in the Tsukuba straight section(IR). The sextupoles in the Oho and Nikko section cancel out the x 3 term.  x /  y =10/300 @ SCWTR/L  x /  y =60/15 @ SCWN/O SCWTR/L: K 2 =+6.6 SCWN : K 2 =-.449 SCWO : K 2 =+.449 To make cancellation of x 3 term (K. Oide) 300 - 10/60×15 x3x3 xp y 2

27. 27 Crab Waist Tsukuba Version 2 Optics: 27JAN07C No solenoid, skew quads SCWTR/L:  x /  y = 10/300 m |K 2 |=6.6 SCWN/O :  x /  y = 60/15 m |K 2 |=0.449 J y =0 J y /J x =10 % K 2 =0 J y /J x =10 % RF ON

28. 28 Crab Waist Tsukuba Version 2 ・ Optics: 27JAN07C ・ RF OFF ・ J y /J x =10 % ・ SCWTR/L K 2 =0, 0.66, 1.1, 2.2, 3.3, 6.6 ・ SCWN/O K 2 =0 Thin SCWN/O NZ=0 NZ=+2 x/  x

29. 29 Crab Waist Simple IP Version K2 6.6 Quadruples only for the region between SCWTR and SCWTL SCWTR/L: thin lens.

30. 30 Crab Waist Simple IP Version Jy/Jx=10 % RF OFF SCWTR/L K 2 = 6.6, 0 Quadrupoles between two sextupoles for the crab waist are thin lens. Elements between two sextupoles are investigated. kinematical terms of drift spaces are removed. Nonlinear fringe fields of quadrupoles between crab waist sextuples are removed. OFF crab waist sextupoles ON x/  x

31. 31 HER Model Optics for Crab Waist Sextupoles are installed in the Nikko and Oho section. *2  yy xx yy xx A. Morita

32. 32 Dynamic Aperture of HER Octupoles(more than 30) are located at RF sections. These octupoles can not recover the dynamic aperture. J y /J x = 10 %

33. 33 Summary of Studies at KEKB Several kinds of the optics for the crab waist at KEKB are studied: Sextupoles in Oho and Nikko for the crab waist. Sextupoles in Tsukuba for the crab waist. Sextupoles for the crab waist in Tsukuba and sextupoles to cancel the x 3 term in Oho and Nikko section. Solenoid, QCS offset, QCS & QC2 mutlipole elements are removed from the realistic optics. Replace the realistic IR with the simple low-beta IR. Optimization of octupoles for the crab waist optics in HER Maxwellian fringe of quadrupoles, kinematics terms of drift space etc. reduce the dynamic aperture. There is no good solution found so far.