1. Beam-beam studies for Super KEKB K. Ohmi & M Tawada (KEK) Super B factories workshop in Hawaii 20-23 Apr. 2005

2. Luminosity limit Fundamental limit due to the beam-beam interaction. Super B factories target the fundamental limit. How high is the beam-beam limit? The beam-beam limit is discussed in two papers, K. Ohmi et al, PRL 92, 214801 (2004). Beam-beam limit in Super KEKB K. Ohmi et al, PRST 7, 104401 (2004). Crossing angle effect in KEKB The beam-beam limit can be understood by using strong-strong simulation (based on the Particle in cell method). Design of super KEKB is discussed from the view of the beam-beam limit using the simulation.

3. Parameter table of super KEKB HERLERHERLER I4.4A10A1.27A1.7A NbNb 5000 1293 NeNe 5.5x10 10 1.26x10 11 6.1x10 10 8.2x10 10 xx 24nm 18nm yy 0.18nm zz 3mm 6mm xx 20-30cm 60cm yy 3mm 6mm yRyR 0.16 0.050.085 LbLb 0.8x10 32 0.12x10 32 L4x10 35 R=0.7 1.5x10 34 KEKB

4. Tune scan Bunch luminosity v.s. tune Total luminosity = 5000x bunch luminosity Green line sketches progress of KEKB. L tot = 4x10 35 cm -2 s -1 By M. Tawada

5. Luminosity evolution Equilibrium state is realized for equal damping times. Design damping time is 4000 and 6000 turns for HER and LER, respectively. Slow luminosity decrease is observed for unequal damping times.  x =30 cm 20cm

6. Difference of the damping time Beam size asymmetry and luminosity decrease arise. Emittance equalization is required. Equal damping time (6000 turn) 4000 (HER) & 6000 (LER) turn

7. What disturb to achieve the fundamental beam-beam limit? Nonlinear diffusion 1.Linear coupling of arc 2.Nonlinearity of arc lattice External diffusion 1.Phase jitter of crab and accelerating cavities 2.Feedback noise Parasitic interaction Other issues 1.Heating, bunch lengthening, electron cloud …

8. Optics error at the collision point One turn map is multiplication of beam-beam interaction and map of arc section. Total luminosity performance is determined by the map of arc section, which is controllable by us. X-y coupling, dispersion (xy-  coupling) and crossing angle (x-z coupling) A symplectic diffusion is induced by the couplings. Mixing of degree of freedom seems to enhance Arnold diffusion.

9. Vertical dispersion Diffusion behavior due to dispersion in a system without synchrotron radiation. Luminosity and beam size are degraded. Gaussian approx. PIC simulation

10. X-y coupling Diffusion due to x-y coupling. Luminosity and beam size degradation. Gaussian approx. PIC simulation

11. Crossing angle Crossing angle is equivalent to x-z coupling. Diffusion and luminosity degradation due to crossing angle Gaussian approx. PIC simulation

12. Nonlinear terms Effect of chromaticity, d /d , d  /d , d  /d , has been studied. The effect was very weak for d /d  ~7. Life time degradation may be issue. Higher nonlinearity using a Taylor map will be included in the beam-beam simulation. Detailed studies will be done.

13. External diffusion: Vertical offset noise Since the beam-beam system is chaotic, such noise enhances the diffusion of the system. Luminosity degradation for the noise without correlation between turns.

14. Orbit offset (static) Static vertical offset. Tolerance is easier than the fast noise. For slower variation than radiation damping time, emittance can be an adiabatic invariant.

15. Phase jitters of RF systems: horizontal offset noise Noise of RF system. Deviation of RF phase, . Phase error between two crab cavities. The transverse offset affects the beam-beam performance.

16. Effect on the beam-beam performance of the phase jitter of RF’s Luminosity and beam size as functions of  x. Correlation time of the jitter, 1 or 10 turns, is important for the degradation. Since Q=200,000 and H=5120, the correlation time will be larger than 10 turns. Tolerance is 0.05 degree.

17. Parasitic collision Nonlinear force (~1/r) with very large amplitude Separation at the parasitic collision Dynamic beta and emittance (simulation)

18. Luminosity with or without parasitic interaction In fixed parasitic beam model, no effect is observed. When parasitic beam is treated as soft Gaussian, a stable solution was not obtained by beam loss at the early stage (unmatched). It may be critical situation, especially for the beam life time or injection.

19. Toward Higher luminosity Higher bunch current with keeping total current Luminosity saturates for increasing bunch current. Beam-beam limit. Only HOM loss increases.

20. Toward higher luminosity, 10 36 cm -2 s -1 Two beam collision is limited by strong nonlinear diffusion coupled to synchrotron radiation around 4x10 35 cm -2 s -1. Compensation scheme is one of limited choices. Our first study (Ohnishi & Ohmi) did not give better results than that of the two beam collision.

21. 4 eigenmodes of four-beam collision No tune shift = 0 Focusing = 0 + Defocusing = 0 - x,y z Unstable modes Rich unstable mode and no Landau damping

22. Summary Design and tolerance for L tot = 4x10 35 cm -2 s -1 were studied. Reduce optics error at the collision point. Maybe acceptable. Reduce external diffusions especially those with fast frequency component. Arc nonlinearity and life time issues will be studied soon by collaboration with BINP (D. Shatilov). Efforts on higher luminosity are continued.