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 xx
1 Long bunch
z
N2 N2
L L
x x y y z y y
xx
x
N ( 1) N x
x z x
z x
Overlap factor
y N y
y N y
x x y z y
y z y
xx
x is smaller due to cancellation of
tune shift along bunch length
Essentials of super bunch scheme
N2
L y x xx
z y y Keep , and .
y x y
N x
x yy 0
z x
L
N y
y
z y
y
xx • Bunch length is free.
• Small beta and small
emittance are required.
Short bunch scheme
y
Keep x , x and .
N2
y
L yy 0
x x y y
x
N
x
L
• Small coupling
y • Short bunch
y N
x x y
y z • 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. z-
dependent chromaticity is now focused.
Waist control-I traveling focus
M e :H I :Μ 0 e:H I :
H I apy z
2
H I H
y y y azPy ap y
2
Py z
• Linear part for y. z is constant during collision.
a2 z 2 az
t
T T
az 1 =0
1 az
T
0 1
Waist position for given z
• Variation for s
az s az s az
2
a2 z 2
t
M ( s) M ( s)
az 1 s az 1
• Minimum is shifted s=-az
Realistic example- I
• Collision point of a part of bunch with z,
=z/2.
• To minimize at s=z/2, a=-1/2
• Required H
1 2
H I py z
2
• RFQ TM210
• 1 c 2 pc
V V~10 MV or more
2 * 2 e
Waist control-II crab waist
(P. Raimondi et al.)
M e :H I :Μ 0 e:H I :
H I axpy
2
H I H
y y y axPy px px px ap y
2
Py x
• Take linear part for y, since x is constant during
collision.
a2 x2 ax
t
T T
ax 1
1 ax
T
0 1
Waist position for given x
• Variation for s
ax s ax s ax
2
a2 x2
t
M ( s) M ( s)
ax 1 s ax 1
• Minimum is shifted to s=-ax
Realistic example- II
• To complete the crab waist, a=1/, where is
full crossing angle.
• Required H
1
HI 2
xp y
• Sextupole strength
1 B '' L 1 1 x*
K2 K2~30-50
2 p / e yy
*
x
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 apy z
2
( s)
z* ( s ) x( s )
*
1 B '' L 1 1 x
*
K2
2 p / e yy
*
x K2~30-50
Super B (LNF-SLAC)
Base PEP-III
Fy
C 3016 2200 y ( z ) ( s z / 2)dzds
y x f z
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
f/2 (mrad) 25 14
x 0.0025
y 0.1
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
f/2 (mrad) 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
Lum (W.S.) 8E+35 6.70E+35 1.00E+36 3.95E+35 4.80E+35
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.
• No big gain in Lum.! x=24 nm
• Life time is improved. y=0.18nm x=0.2m
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 beam-
beam 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 , p z ; 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 J const
ds
d H ( J )
ds
J d 2 ( J )
• Equilibrium distribution
J1 J 2 J 3
( J ) exp : emittance
1 2 3
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
Only 2y 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<
• 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 12.5
0.0015
20
0.001 10
0.0005 15
0 7.5
20 10 nux
15 5
(0.51,0.7) 10 5
nuy 5 2.5
KEKB D=0.0005 (0.51,0.51) 2.5 5 7.5 10 12.5 15 17.5 20
(0.51,0.51) (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.
•Synchrobeta
• Fine structure near x=0.5 Contour plot effect near
y~0.5.
• x~0.5 region
(0.7,0.51) remains safe.
20
17.5
0.001 15
0.00075
0.0005 20
12.5
0.00025 15
0 10
20 10 nux
15 7.5
10 5
nuy 5 5
2.5
(0.51,0.51)
2.5 5 7.5 10 12.5 15 17.5 20
Crossing angle (fz/x~1, z~y)
• Good region is only (x, y)~(0.51,0.55).
(0.7,0.51)
20
17.5
15
0.002 12.5
0.0015
20
0.001 10
0.0005 15
0
7.5
20 10 nux
15
5 5
(0.51,0.7) 10
nuy 5
2.5
(0.51,0.51)
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 2
2
x ,0 lim p 2
x
x 0.5 x 0.5
• Choice of optimum x
Crab waist for KEKB
• H=25 x py2.
• Crab waist works even for short bunch.
Sextupole strength
and diffusion rate 0.002
K2=20 0.0015
0.001
0.0005
20
15
0
20 10 nux
15
10 5
nuy 5
25 30
0.002 0.002
0.0015 0.0015
20 20
0.001 0.001
0.0005 15 0.0005 15
0 0
20 10 nux 20 10 nux
15 15
10 5 10 5
nuy 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 : xpy :) ...
2
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.