Beam-beam studies for Super KEKB

K. Ohmi & M Tawada (KEK)

Super B factories workshop in Hawaii

20-23 Apr. 2005

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 KEKBK. 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.

Parameter table of super KEKBHER LER HER LER

I 4.4A 10A 1.27A 1.7A

Nb 5000 5000 1293 1293

Ne 5.5x1010 1.26x1011 6.1x1010 8.2x1010

x 24nm 24nm 24nm 18nm

y 0.18nm 0.18nm 0.18nm 0.18nm

z 3mm 3mm 6mm 6mm

x 20-30cm 20-30cm 60cm 60cm

y 3mm 3mm 6mm 6mm

yR 0.16 0.16 0.05 0.085

Lb 0.8x1032 0.12x1032

L 4x1035 R=0.7 1.5x1034

KEKB

Tune scan• Bunch luminosity v.s. tune• Total luminosity = 5000x bunch luminosity• Green line sketches progr

ess of KEKB.

Ltot = 4x1035 cm-2 s-1

By M. Tawada

Luminosity evolution• Equilibrium state is realized for equal damping times.• Design damping time is 4000 and 6000 turns for HER an

d LER, respectively.• Slow luminosity decrease is observed for unequal dampi

ng times.

x=30 cm 20cm

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

What disturb to achieve the fundamental beam-beam limit?

• Nonlinear diffusion1. Linear coupling of arc2. Nonlinearity of arc lattice

• External diffusion 1. Phase jitter of crab and accelerating cavities

2. Feedback noise • Parasitic interaction• Other issues1. Heating, bunch lengthening, electron cloud …

Optics error at the collision point• One turn map is multiplication of beam-beam int

eraction and map of arc section.• Total luminosity performance is determined by th

e 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.

Vertical dispersion• Diffusion behavior due

to dispersion in a system without synchrotron radiation.

• Luminosity and beam size are degraded.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

hy (mm)

L/L0

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1 1.2

hy (mm)

y/

y0e-e+

Gaussian approx.

PIC simulation

X-y coupling

• Diffusion due to x-y coupling.

• Luminosity and beam size degradation.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25

r4

L/L0

0

1

2

3

4

0 0.05 0.1 0.15 0.2 0.25

r4

y/

y0e-e+

Gaussian approx.

PIC simulation

Crossing angle• Crossing angle is equivalent to x-z coupling.• Diffusion and luminosity degradation due to cros

sing angle

Gaussian approx. PIC simulation

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.

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.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1Dy/ y

L/L0

Tcor=1

0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1

Dy/ y

y/

y0e-e+

2( ) ( ') ( ')y t y t y t tD D D

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.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1Dy/ y

L/L0

Offset

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

Dy/ y

y/

y0e-e+

2( ) ( ') exp( )y t y t y t D D D

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.

tanRF

RF

cx

cos( ( *, ))tan tan

2sinx c

c cRF x RF

s sc cx

D

5(m) 2.5 10 (deg)x

Effect on the beam-beam performance of the phase jitter of RF’s

0

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12

Dx/ x

x/

x 0e-e+e-e+

• 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.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12Dx/ x

L/L0

Tcor=10Tcor=1

Parasitic collision• Nonlinear force (~1/r) with very large amplitude• Separation at the parasitic collision

• Dynamic beta and emittance (simulation)

,min 9 mm : half crossing anglesp crs crsx L D

,min,min 2

( 2) (0) 1 0.069 2.2 0.15 mm x=604sp

x sp x xx

LL

D

(0) 0.067 mm 0.028 (0.017 : analytic) m ( / 2) 0.93 mmx x x L

0.503, 0.5505x y

( 2)10 /xx LD

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.

Toward Higher luminosityHigher bunch current

with keeping total current

• Luminosity saturates for increasing bunch current. Beam-beam limit.

• Only HOM loss increases.

Toward higher luminosity, 1036 cm-2 s-1

• Two beam collision is limited by strong nonlinear diffusion coupled to synchrotron radiation around 4x1035 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.

4 eigenmodes of four-beam collision

No tune shift

=0

Focusing

=0

+

Defocusing

=0-

x,yz

Unstable modes

Rich unstable mode and no Landau damping

Summary

• Design and tolerance for Ltot = 4x1035 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.