Beam-beam effect with an external noise in LHCK. Ohmi (KEK)LHC LUMI 2006Oct. 16-20, 2006, ValenciaThanks to W. Hofle and F. Zimmermann

IntroductionNonlinear system with noise Beam-beam, beam-electron cloud interactionsWeak-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 noiseBunch by bunch feedback system

Two types of noise have been studiedOrbit fluctuation at collision point

Orbit diffusion and dampingd: random, but unique for every particles.

Fluctuation in collision due to the crab cavity and cavity noiseNoise of RF system. Deviation of RF phase, dj.

Phase error between two crab cavities.

Bunch by bunch feedback system of LHC (W. Hofle)14 bit resolution, 214=16384.Covered area is Dx=+-2 mm at b=100-150 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

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

Strong-strong effectKick Oscillation (s and p modes) Decoherence Emittance growth

dx: Kick error of the feedback system, ~G times monitor read error.

Y.I. Alexahin, NIM391, 73 (1996)

Simulation for the first type of noiseOrbit 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.

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

Emittance growth rate and luminosity decrement in the weak-strong simulationThe 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.

Strong-strong simulation, tcor= 1 turnDipole amplitudeEmittance growthLuminosity decrement

Strong-strong simulation, tcor= 100 turnDipole amplitudeEmittance growthLuminosity decrement

Emittance growth and luminosity decrement in the strong-strong simulationThe 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.

Comparison with the simulationDJ(a=1)==2.3x10-27 m2/turn for dx=0.2 mm (0.012s) and t=100. De/e=4.5x10-9 (Tanajis 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.

I have to apologize my mistakeTolerance 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.

2nd type of noiseOrbit diffusion and damping

If the beam-beam effect is week,

Residual dipole amplitude and emittance growth dx=0.012 sx.

dx=0.006 sx.

dx=0.003 sx.

dx=0.0012 sx.

Residual dipole moment~dx2/2GBeam-beam interaction little affects the residual dipole motion.

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

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

SummaryTolerance 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.