Crab crossing and crab waist
at super KEKB
K. Ohmi (KEK)
Super B workshop at SLAC
15-17, June 2006
Thanks, M. Biagini, Y. Funakoshi, Y.
Ohnishi, K.Oide, E. Perevedentsev, P.
Raimondi, M Zobov
Super bunch approach
K. Takayama et al, PRL, (LHC)
F. Ruggiero, F. Zimmermann (LHC)
P.Raimondi et al, (super B)
Short bunch
L
Long bunch
N2
 x  x y  y
x
y
 xx
1
 z
N
x
y
N
 x  x y
y   z
L
 xx

( 1)
 z
x
Overlap factor
y
N2
 z  y  y
x
x
N
 z
N
 z
y 
y
y
 xx

x is smaller due to cancellation of
tune shift along bunch length
Essentials of super bunch scheme
L
x
y
y 
N2
 z  y  y
N
 z
N
 z
 xx

x
x
y
y
Keep
y
,
y
x
and
x
 xx
.
y
 yy  0
L 
• Bunch length is free.
• Small beta and small
emittance are required.
Short bunch scheme
L
N2
 x  x y  y
x
y
N
x
y
N
 x  x y
y   z
y
Keep  x ,  x and
.
y
 yy  0
L 
• Small coupling
• Short bunch
• Another approach: Operating
point closed to half integer
 x  0.5  y  
L
We need L=1036 cm-2s-1
• Not infinity.
• Which approach is better?
• Application of lattice nonlinear force
• Traveling waist, crab waist
Nonlinear map at collision point
H  ai xi  bij xi x j  cijk xi x j xk
x  xi  ( x, px , y, p y , z ,  (  E / E ))
1
M  exp( : H :)x*  x  [ H , x]  [ H ,[ H , x]]  ...
2
 ai  bij x j  c 'ijk x j xk  ...
• 1st orbit
• 2nd tune, beta, crossing angle
• 3rd chromaticity, transverse nonlinearity. zdependent chromaticity is now focused.
Waist control-I traveling focus
M  e:H I :Μ 0 e:H I :
H I  ap y2 z
y  y
H I
 y  azPy
Py
  
H
   ap y2
z
• Linear part for y. z is constant during collision.
 

 
 
 
  T 
 
 
 1 az 
T 

0
1



a2 z 2

  t 


T

 az
 



az 

 
1 

 
=0
Waist position for given z
• Variation for s

a2 z 2
 


M ( s)
 az



2

az 
s  az 

 

  t

M (s)  

1 
s  az



 


• Minimum  is shifted s=-az
s  az 

 
1 

 
Realistic example- I
• Collision point of a part of bunch with z,
<s>=z/2.
• To minimize  at s=z/2, a=-1/2
• Required H
1 2
H I   py z
2
• RFQ
•
TM210
c 2 pc
V
2 *   2 e
1
V~10 MV or more
Waist control-II crab waist
(P. Raimondi et al.)
M  e:H I :Μ 0 e:H I :
H I  axpy2
H I
y  y
 y  axPy
Py
px  px 
H
 px  ap y2
x
• Take linear part for y, since x is constant during
collision.
 

 
 
 

T


 
 
 1 ax 
T 

0
1



a2 x2

  t 


T

 ax
 



ax 

 
1 

 
Waist position for given x
• Variation for s

a2 x2
 

M ( s) 
 ax



2

ax 
s  ax 

 

  t

M ( s)  

1 
s  ax



 


• Minimum  is shifted to s=-ax
s  ax 

 
1 

 
Realistic example- II
• To complete the crab waist, a=1/, where  is
full crossing angle.
• Required H
HI 
1
xp y2

• Sextupole strength
1 B '' L 1 1
K2 

2 p / e   y*  y
 x*
x
K2~30-50
Not very strong
Crabbing beam in sextupole
• Crabbing beam in sextupole can give the
nonlinear component at IP
• Traveling waist is realized at IP.
H I  ap y2 z
z* 
 ( s)
 ( s ) x( s )
*

1 B '' L 1 1
K2 

2 p / e   y*  y
 x*
x
K2~30-50
Super B (LNF-SLAC)
Base
C
PEP-III
3016
2200
x
4.00E-10
2.00E-08
y
2.00E-12
2.00E-10
x (mm)
17.8
10
y (mm)
0.08
0.8
z (mm)
4
10
ne
2.00E+10
3.00E+10
np
4.40E+10
9.00E+10
25
14
f/2 (mrad)
x
0.0025
y
0.1

Fy

y
y  
 ( z ) ( s  z / 2)dzds
x f z
Luminosity for the super B
• Luminosity and vertical beam size as functions of K2
• L>1e36 is achieved in this weak-strong simulation.
DAFNE upgrade
DAFNE
C
97.7
x
3.00E-07
y
1.50E-09
x (mm)
133
y (mm)
6.5
z (mm)
15
ne
1.00E+11
np
1.00E+11
f/2 (mrad)
25
x
0.033
y
0.2479
Luminosity for new DAFNE
• L (x1033) given by the weak-strong simulation
• Small s was essential for high luminosity
ne
tune
s
L(K2=10)
15
20
6.00E+10 0.53, 0.58
0.012
4.27
3.79
6.00E+10 0.53, 0.58
-0.01
4.35
3.93
1.00E+11 0.53, 0.58
-0.01
4.07
5.66
5.53
6.00E+10 0.53, 0.58
0.012 z=10mm
4.32
2.47
6.00E+10 0.53, 0.58
-0.01 z=10mm
4.65
2.7
6.00E+10 0.057, 0.097
-0.01
5.19(3.3)
1.00E+11 0.057, 0.097
-0.01
13.21(4.8)
() strong-strong , horizontal size blow-up
Super KEKB
SuperKEKB
Crab waist
x
9.00E-09
6.00E-09
6.00E-09
6.00E-09
6.00E-09
y
4.50E-11
6.00E-11
6.00E-11
6.00E-11
6.00E-11
x (mm)
200
100
50
100
50
y (mm)
3
1
0.5
1
0.5
z (mm)
3
6
6
4
4
s
0.025
0.01
0.01
0.01
0.01
ne
5.50E+10
5.50E+10
5.50E+10
3.50E+10
3.50E+10
np
1.26E+11
1.27E+11
1.27E+11
8.00E+10
8.00E+10
0
15
15
15
15
x
0.397
0.0418
0.022
0.0547
0.0298
y
0.794->0.24
0.1985
0.179
0.178
0.154
8E+35
6.70E+35
1.00E+36
3.95E+35
4.80E+35
f/2 (mrad)
Lum (W.S.)
Lum (S.S.)
8.25E35 4.77E35
9E35
3.94E35
4.27E35
Horizontal blow-up is recovered by choice of tune. (M. Tawada)
Traveling waist
• Particles with z collide with central part of
another beam. Hour glass effect still exists for
each particles with z.
x=24 nm
• No big gain in Lum.!
y=0.18nm x=0.2m
• Life time is improved.
y=1mm z=3mm
Small coupling
Traveling of positron beam
Increase longitudinal slice
x=18nm, y=0.09nm, x=0.2m y=3mm z=3mm
Lower coupling becomes to give higher luminosity.
Why the crab crossing and crab
waist improve luminosity?
• Beam-beam limit is caused by an
emittance growth due to nonlinear beambeam interaction.
• Why emittance grows?
• Studies for crab waist is just started.
Weak-strong model
• 3 degree of freedom
• Periodic system
• Time (s) dependent
H ( x, px , y, p y , z , pz ; s )  H '( J1 , 1 , J 2 , 2 , J 3 , 3 ; s)
 ( s  L)   ( s)  2
Solvable system
• Exist three J’s, where H is only a function of J’s,
not of ’s.
• For example, linear system.
• Particles travel along J. J is kept, therefore no
emittance growth, except mismatching.
dJ
H

0
ds

d H ( J )

ds
J
J  const
 d  2 ( J )
• Equilibrium distribution
 J1 J 2 J 3 
 ( J )  exp     
 1  2  3 
 : emittance
py
y
One degree of freedom
• Existence of KAM curve
• Particles can not across the KAM curve.
• Emittance growth is limited. It is not essential for
the beam-beam limit.
• Schematic view of equilibrium distribution
• Limited emittance growth
More degree of freedom
Gaussian weak-strong beam-beam model
• Diffusion is seen even in sympletic system.
Diffusion due to crossing
angle and frequency
spectra of <y2>
Only 2y signal
was observed.
Linear coupling for KEKB
• Linear coupling (r’s), dispersions, (h, =crossing angle
for beam-beam) worsen the diffusion rate.
M. Tawada et al, EPAC04
Two dimensional model
• Vertical diffusion for =0.136
• DC<<D for wide region (painted by black).
• No emittance growth, if no interference. Actually, simulation including
radiation shows no luminosity degradation nor emittance growth in
the region.
• Note KEKB D=5x10-4 /turn DAFNE D=1.8x10-5 /turn
(0.51,0.7)
(0.7,0.7)
20
(0.7,0.51) 17.5
15
0.002
0.0015
0.001
0.0005
0
20
(0.51,0.7)
12.5
20
10
15
7.5
10 nux
15
5
10
nuy
KEKB D=0.0005
5
2.5
5
(0.51,0.51)
2.5
(0.51,0.51)
5
7.5
10
12.5
15
17.5
20
(0.7,0.51)
3-D simulation including bunch length (z~y)
Head-on collision
•Good region
shrunk
drastically.
• Global structure of the diffusion rate.
• Fine structure near x=0.5
Contour plot
(0.7,0.51)
•Synchrobeta
effect near
y~0.5.
• x~0.5 region
remains safe.
20
17.5
0.001
0.00075
0.0005
0.00025
0
20
15
20
12.5
15
10 nux
15
10
7.5
10
nuy
5
5
5
2.5
(0.51,0.51)
2.5
5
7.5
10
12.5
15
17.5
20
Crossing angle (fz/x~1, z~y)
• Good region is only (x, y)~(0.51,0.55).
(0.7,0.51)
20
17.5
15
0.002
0.0015
0.001
0.0005
0
20
12.5
20
10
15
10 nux
15
(0.51,0.7)
10
nuy
5
5
(0.51,0.51)
7.5
5
2.5
2.5
5
7.5
10
12.5
15
17.5
20
Reduction of the degree of freedom.
• For x~0.5, x-motion is integrable.
(work with E. Perevedentsev)
lim DC , y  0
 x 0.5
if zero-crossing angle and no error.
1
L
 x  (crossing angle)+(coupling)+(fast noise)
• Dynamic beta, and emittance
lim
 x 0.5
x
2

2
x ,0
• Choice of optimum x
lim
 x 0.5 
p
2
x

Crab waist for KEKB
• H=25 x py2.
• Crab waist works even for short bunch.
Sextupole strength
and diffusion rate
K2=20
0.002
0.0015
0.001
0.0005
0
20
20
15
10 nux
15
5
10
nuy
25
30
0.002
0.0015
0.001
0.0005
0
20
20
15
10 nux
15
10
nuy
5
5
5
0.002
0.0015
0.001
0.0005
0
20
20
15
10 nux
15
10
nuy
5
5
Why the sextupole works?
• Nonlinear term induced by the crossing angle
may be cancelled by the sextupole.
• Crab cavity
exp( : F :)  exp( :  px z :) exp(:  px z :)  1
• Crab waist
exp( : F :)  exp( :  px z :)exp(K2 : xpy2 :)  ...
Need study
exp(: F :)M BB exp( : F :)
Conclusion
• Crab-headon
Lpeak=8x1035. Bunch length 2.5mm is required
for L=1036.
• Small beam size (superbunch) without crab-waist.
Hard parameters are required x=0.4nm x=1cm
y=0.1mm.
• Small beam size (superbunch) with crab-waist.
If a possible sextupole configuration can be found,
L=1036 may be possible.
• Crab waist scheme is efficient even for shot
bunch scheme.