1. Beam-beam effect with an external noise in LHC
K. Ohmi (KEK)
LHC LUMI 2006
Oct , 2006, Valencia
Thanks to W. Hofle and F. Zimmermann

2. Introduction Nonlinear system with noise
Beam-beam, beam-electron cloud interactions
Weak-strong and strong-strong effects (single particle issue and coherent motion)
Analyze the effects using a weak-strong and strong-strong simulations.
High statistics simulation to be sensitive for the emittance growth with the rate, De/e~1x10-9 (1day decay rate).
Crab cavity noise, RF cavity noise
Bunch by bunch feedback system

3. Two types of noise have been studied
Orbit fluctuation at collision point
Orbit diffusion and damping
d: random, but unique for every particles.

4. Fluctuation in collision due to the crab cavity and cavity noise
Noise of RF system. Deviation of RF phase, dj.
Phase error between two crab cavities.

5. Bunch by bunch feedback system of LHC (W. Hofle)
14 bit resolution, 214=16384.
Covered area is Dx=+-2 mm at b= m, resolution is dxmon=0.001s.
Effect of kick error is the same contribution, if an oscillation with Dx is damped by the damping rate of G with 14 bit system.
G: damping rate of the feedback system (feedback gain).
Beam fluctuation without beam-beam

6. Weak-strong effect Diffusion rate due to offset noise (T. Sen et al
Weak-strong effect Diffusion rate due to offset noise (T. Sen et al., PRL77, 1051 (1996), M.P.Zorzano et al., EPAC2000)

7. Strong-strong effect
Y.I. Alexahin, NIM391, 73 (1996)
Kick  Oscillation (s and p modes)  Decoherence  Emittance growth
dx: Kick error of the feedback system, ~G times monitor read error.

8. Simulation for the first type of noise
Orbit fluctuation at collision point
Use both of the weak-strong and strong-strong simulation.
My previous simulation was wrong. There was a mistake for the noise implementation.

9. Weak-strong simulation
This simulation is available for studying only the weak-strong effect.
The correlation time of the noise (tcor) is 1 turn.

10. Emittance growth rate and luminosity decrement in the weak-strong simulation
The correlation time of the noise (tcor) is 1 turn.
Hour-1=2.5x10-8 turn-1. Day-1=1x10-9 turn-1.
Tolerance is dx/sx=0.2% for one day decrement.

11. Strong-strong simulation, tcor= 1 turn
Dipole amplitude
Emittance growth
Luminosity decrement

12. Strong-strong simulation, tcor= 100 turn
Dipole amplitude
Emittance growth
Luminosity decrement

13. Emittance growth and luminosity decrement in the strong-strong simulation
The tolerance is more severe than that given by the weak-strong simulation.
The tolerance is slight less than 0.1% for tcor=1, but is 1% for tcor=100.
Build-up of the dipole oscillation is seen.
Bunch by bunch feedback may help the build-up of the dipole motion, therefore tolerance may be expected to be similar as that of weak-strong simulation.

14. Comparison with the simulation
DJ(a=1)=<DJ2>=2.3x10-27 m2/turn for dx=0.2 mm (0.012s) and t=100. De/e=4.5x10-9 (Tanaji’s formula).
The simulation gives De/e=2x10-9 at the same condition, dx=0.2 mm (0.012s) and t=100.
The agreement is good.

15. I have to apologize my mistake
Tolerance for Crab cavity noise is 10 times larger (easier).
Tolerance is now dx=0.2 mm(0.012s), df= 0.5 degree for t=100, and dx=0.02 mm (0.0012s), 0.05 degree for t=1, if luminosity life time ~ 1 day is required.

16. 2nd type of noise Orbit diffusion and damping
If the beam-beam effect is week,

17. Residual dipole amplitude and emittance growth
dx=0.012 sx.

18. dx=0.006 sx.

19. dx=0.003 sx.

20. dx= sx.

21. Residual dipole moment
<x2>~dx2/2G
Beam-beam interaction little affects the residual dipole motion.

22. Emittance growth rate and luminosity decrement in the strong-strong simulation
For G=0.1 and dxmon=0.1%s, dxkick=0.0002s, the luminosity life time is 1day.

23. Comparison with analytic theory
Agreement with the formula (Y. Alexahin) is very good.
Note: the decrement, 1e-9, is hard for simulation, because of the statistics.

24. Summary
Tolerance of crab cavity phase is dx=0.2 mm (0.012s) for tcor=100 turn, and dx=0.02 mm (0.0012s) for tcor=1 turn.
The effects of feedback noise is sensitive, but the resolution with14 bit system is sufficient.
Theory and simulation had good agreement.