Beam-beam effect in super-B
K. Ohmi
MAC2007 for KEKB,19-21, Mar. 2007

Two approach toward high luminosity• High current, high beam-beam parameter.

• Low emittance, low beta, low current, socalled super bunch collisionxsSuper B parameters
SuperKEKBMiddle

SuperB (LNF/SLAC)

εx

9.00E-09

6.00E-09

0.8E-09

εy

4.50E-11

6.00E-112E-12βx (mm)20050

9βy (mm)30.50.133

σz (mm)

366νs

0.025

0.01

0.012/0.026

ne

5.50E+10

5.50E+10

1.9E+10np
θ/2 (mrad)

1.26E+111.27E+113.3E+1001525

ξx

0.397

0.022

ξy0.794->0.24

0.179

Lum (W.S.)
Lum (S.S.)

8E+35
1.00E+36
8.25E35 8-9E35

1.00E+36

Comparison of two approach
ε xβx> 1 or θ = 0
θσ zHigh curent

Low emittance

ε xβx
<1
θσ z

Overlap factorL∼N2

ε x β xε y β y

ξx ∼N

εxβy
ξy ∼ Nε x β xε yβy > σ zL∼ξx ∼ξy ∼N2
1 + θ 2σ z ε y β yN
1 + θ 2σ zN
1 + θ 2σ zβy >θ: half crossing angleβxεxβyεyε xβxθξx is smaller due to cancellation oftune shift along bunch length

High current approachL∼βy
Keep ε x , β x and.
εyε yβy → 0N2ε x β xε y β yξx ∼L→∞Nεxβy
ξy ∼ Nε x β xε yβy > σ zβ y > σ z limits luminosity• High current, Small coupling• Choice of operating pointν x → +0.5 ξ y → ∞θ=0L→∞

N →∞Low emittance approachL∼ξx ∼ξy ∼βy >N2

θσ z ε y β yN

θσ zN

θσ zε xβxθβxεxβyεyKeepβy,εyε xβxβxand.
εxβy

ε yβy → 0L→∞
• Bunch length is free.
• Small beta and smallemittance are required.

High current approach• Reinforce RF system to store high current.
• Operation cost due to the high current.
• Heating of hardware, chamber, mask ....
• High beam-beam parameter with head-on
collision: high nonlinear system.
• Sensitive for noise and error due to highbeam-beam parameter.
• Limit of bunch length, instability andcoherent synchrotron radiation.

Low emittance approach• Low emittance, magnet system should bereinforced.
• Low beta, the strongly focused IR, may be limit
dynamic aperture.
• Single turn or multi turn injection. Multi-turninjection requires wide dynamic aperture, while
single turn injection requires a high speed
kicker (~2ns). A low emittance damping ring andhigh precision injection are required.
• High sensitivity due to the very low emittance.
• The main ring is similar to ILC damping ringequip with the final focus system of ILC.

• Crab crossing

Crab cavity• Crab waist
SextupolemagnetParticles collide anotherbeam at their waist point

Super B with high current approach
SuperKEKB

MiddleSuperB (LNF/SLAC)

εx

9.00E-09

6.00E-09

0.8E-09

εy

4.50E-11

6.00E-112E-12βx (mm)20050

9βy (mm)

30.5

0.133

σz (mm)

366

νs

0.025

0.01

0.012/0.026

ne

5.50E+10

5.50E+10

1.9E+10np
θ/2 (mrad)

1.26E+11

1.27E+11

3.3E+1001525

ξx0.3970.022

ξy0.794->0.24

0.179

Lum (W.S.)
Lum (S.S.)

8E+35
1.00E+36
8.25E35 8-9E35

1.00E+36

Beam-beam Interaction and crossing angle• Beam-beam interaction as a transformation ofdynamical variable.
X = exp ( − : (U 3 + U 4 + ...) :) x0• x4 , y4 term decrease but another coupling termincreases for crossing angle. A kind of symmetry
breaking degrade luminosity performance.
0.18
0.16
0.14

Crab crossing

xiy

0.12
0.1
0.08

Crossing angle=30 mrad

0.06
0.04
0.02
002

4

6

NHER (x E10)810

4-th order Coefficients due to crossing angle• Coefficient of U4(x,y,z).
x49e+068e+06
7e+06
6e+06
5e+06
4e+06
3e+06
2e+06
1e+06
0

1.8e+08
1.6e+08

5.5e101.4e+08

4e10
3e10
2e10
Ne=1e101.2e+08

5.5e10

4e10

1e+08

3e10

8e+07

2e10
Ne=1e10

6e+07
4e+07

0

510
15
x angle (half)

100210

20

025

15

20

25x3z

2020003000103000005.5e10250000200000

3e10
2e10

coef

40000010x angle (half)

5.5e104e105000005xpy2z600000coef

y4004000

coef

coef

4000003000002e10150000100000200000Ne=1e10Ne=1e10

50000

100000000

51015x angle (half)

2025

051015x angle (half)2025

Crab waist• Add sextupole magnets at both side ofcollision point for finite crossing collsion.
• Effective potential is expressed as follows,
U(x,y,z)δ(s-s*)+Kxpy2δ(s-s*-ε)-Kxpy2δ(s-s*+ε)
Put sextupole magnets both side of the collision
point at the vertical betatron phase difference,π(2n+1)/2 and horizontal mπ.

4-th order Coefficients as a function of crab
sextupole strength,
1.5e+07

1.2e+06
1e+06

1e+07

coefficient

coefficient800000

2020005e+06

2002000-5e+06

2011006000001002104000001020102000000-1e+07-1.5e+07-10-200000
010 20 30 40 50
K (xangle=2x15 mrad)-400000-1060

1011100

10 20 30 40 50
K (xangle=2x15 mrad)

60

30000025000030001015000010000050000
0-50000-10

210010,120010,030010
0

10 20 30 40 50
K (xangle=2x15 mrad)1.8e+06
60 1.6e+06
1.4e+06
1.2e+06
1e+06
8000006000004000002000000-200000-10

400000coefficient

coefficient

200000310000,220000,130000,040000
010 20 30 40 50
K (xangle=2x15 mrad)

60• H=K x py2/2, theoreticaloptimum, K=1/xangle.
• Clear structure- 220,121• Flat for sextupole
strength- 400, 301, 040

• Crab waist sextupoles cancel some coupling terms
induced by crossing angle, while keep the synchrobetatron terms.
• H=25 x py2. Luminosity for crab waist is comparable withthat of crab crossing for a parameter range.
Luminosities for crab crossing andcrab waist is not always same.
0.2

Equilibrium

0.15

xi

0 mradνs=0.016

0.1
11 mrad & K2=250.05

11 mrad002000 4000 6000 8000 10000 12000
turn

Tune scan, νx-νs-2 -1

L (cm s )crossing
crab waist

2e+31
1.5e+31
1e+31
5e+30

0.026
44.505
0.024
44.51
0.022
44.5150.02
44.52
44.525
νs0.018
νx
44.53
0.016
44.5350.014
44.54

σx (µm)• Efficient region of the Crabwaist scheme is not wide.
• Low νs.
• High νx degrades theperformance independentof crab waist. Synchrobeta dominant.

crossing
crab waist

σy (µm)crossing
crab waist

160
140

6120

4100

244.50544.51
44.51544.52
44.525νx
44.53
44.5350.026
0.024
0.022
0.02
νs
0.018
0.016
0.014

44.50544.51
44.51544.52
44.525νx
44.53
44.535

0.026
0.024
0.022
0.02
νs0.018
0.016
0.014

Horizontal tune half integer νx=0.5• Particles interact with fixed beam at x and –x mutually.
The phase space structure in y-py at x is the same asthat at -x, because of symmetry of the fixed beam.
• System is one dimension, beam-beam tune shift is 0.5• KEKB tries to realize a high luminosity with thistechnique.
• A high precision tuning is required.
yx

For crab waist at low emittance• Synchro-beta transformation term as x3z,
which is induced by crossing angle and is
not cancelled by the crab sextupole,
disturbs luminosity performance near thehalf integer tune.
• In KEKB low emittance operation, a beambeam instability has been observed.
• Operation strategy is completely different
from the present one.

Super B with meddle parameter
SuperKEKB

MiddleSuperB (LNF/SLAC)

εx

9.00E-09

6.00E-09

0.8E-09

εy

4.50E-11

6.00E-112E-12βx (mm)20050

9βy (mm)30.50.133

σz (mm)366

νs0.0250.01

0.012/0.026

ne

5.50E+10

5.50E+10

1.9E+10np
θ/2 (mrad)

1.26E+11

1.27E+11

3.3E+1001525

ξx0.3970.022

ξy0.794->0.24

0.179

Lum (W.S.)
Lum (S.S.)

8E+35
1.00E+36
8.25E35 8-9E35

1.00E+36

Strong strong simulation results for the
middle case, high current & low β
L
8 × 1035
6 × 1035
4 × 1035
2 × 1035
0864 nuy2(0.5,0.53) 46nux

2810(0.6,0.53)• Crab waist option ofSuper KEKB,
longitudinal slice,
Nslice=20, 1000 turn.
• Scan area
• 0.503<νx<0.603• 0.53<νy<0.6• 0.01 step

σx

σy0.00006
0.00005
0.00004
0.000030.00002

1.5 × 10−6862(0.5,0.53)

4 nuy
46nux

2

810(0.6,0.53)

1 × 10−6
5 × 10−72(0.5,0.53)

8
6
4 nuy

4
6nux

2

810(0.6,0.53)

Near integer tune: high current & low β
L1.75 × 1035
1.5 × 1035
1.25 × 1035
1 × 1035

86
4 nuy

2(0.03,0.03)

4nux268(0.1,0.03)• 0.033<νx<0.103• 0.03<νy<0.1• 0.01 step
• Lum. is better athigher tunes in both,but still low.

σxσy0.00006
0.00005
0.000042(0.03,0.03)

8(0.1,0.1)
6
4 nuy
4
nux

2

68(0.1,0.03)

1.6 × 10−6
1.4 × 10−6
1.2 × 10−6862(0.03,0.03)

4 nuy
4
nux

2

68(0.1,0.03)

Weak strong simulation results for the
middle case, high current & low β•

L

••
•

30

1 × 1036
7.5 × 1035
5 × 10352.5 × 10355
030(0.5,0.8)

2530
32020
nux20nuy•15

101010(0.5,0.5)

551015σx20

2530

Stopband is narrower
in the weak-strong
simulation. Note thatthe horizontal size of
one beam is fixed.

σy0.00010.000075
0.00005
0.000025
030(0.5,0.8)

Crab waist option ofSuperKEKB
0.503<νx<0.803
0.503<νy<0.803
0.01 step

30
320
nux20
nuy

1010

4 × 10−6
3 × 10−6
2 × 10−6
1 × 10−6030(0.5,0.8)(0.8,0.5)30320nuxnuy(0.5,0.5)

102010(0.5,0.5)

Super B with low ε and low β
SuperKEKBMiddle

SuperB (LNF/SLAC)

εx9.00E-09

6.00E-09

0.8E-09

εy

4.50E-11

6.00E-112E-12

βx (mm)20050

9βy (mm)

30.5

0.133

σz (mm)

366

νs

0.025

0.01

0.012/0.026

ne

5.50E+10

5.50E+10

1.9E+10np
θ/2 (mrad)

1.26E+11

1.27E+11

3.3E+1001525

ξx0.3970.022

ξy0.794->0.24

0.179

Lum (W.S.)
Lum (S.S.)

8E+35
1.00E+36
8.25E35 8-9E351.00E+36

Weak-strong simulation for low ε and low β(LNF/SLAC)• Longitudinal slice, Nslice=200• 0.503<νx<0.803, 0.503<νy<0.803, 0.01 step

1 × 1036
5 × 10350(0.5,0.5) 10
0.6 nux

1030 0.00004
0.00002
20
0nuy (0.5,0.5)−7

10

0.6 nux

20
30

σyσx

L

νx, νy=(0.8,0.5)

10
2030

(0.8,0.5)

30 1 × 10
5 × 10−8200nuy (0.5,0.5)

30
20nuy
10

0.6 nux1020
30(0.8,0.5)

• 0. 03<νx<0.303, 0.03<νy<0.303, 0.01 step
• Wide region with horizontal blow-up near νx~0. Similar
as strong-strong.

1 × 1036
7.5 × 1035
5 × 1035
2.5 × 10355
030(0.3,0.)

σyσx

L

2 × 10−630 1.5 × 10−−66
1 × 10 −7
5 × 10200
nuy

0.00004
30 0.00002
3020
nux
20
nuy

10(0,0.)

10
nux

10

1030(0,0.) 10
nux

20

20
nuy
10
20

30

(0,0.)(0.3,0.)Strong-strong simulation for the low ε and low β
design is difficult, because a bunch should be sliced
too much pieces in the longitudinal.
Stopband width in the strong-strong simulation may
be the interesting issue for the collision scheme.

30(0.3,0.)Summary• The crab crossing is being studied in KEKB now.
• We understand how accurate tuning we can and have to
do in KEKB.
• The crab waist scheme will be studied in DAFNE andKEKB in the near future.
• Synchro-betatron stop-band in the crab waist scheme,
which is remarkable in the strong-strong simulation, isimportant issue. The low tune shift may help thestopband width.
• It is difficult to simulate the low ε and low β parameterwith the strong-strong method now.
• We have to have a solution of dynamic aperture andinjection scheme in the low emittance scheme.
• It becomes clear how and which way we choose
gradually (soon).