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. R

aimondi, M Zobov

Super bunch approachK. Takayama et al, PRL, (LHC)

F. Ruggiero, F. Zimmermann (LHC) P.Raimondi et al, (super B)

2

x x y y

NL

xx

N

yy

x x y

N

( 1)x x

z

2

z y y

NL

xx

z x

N

yy

z y

N

Short bunch Long bunch1x x

z

x is smaller due to cancellation of tune shift along bunch length

Overlap factor

y z x xy

Essentials of super bunch scheme

x xy

2

z y y

NL

xx

z x

N

yy

z y

N

Keep , and .y x xx

y x y

0y y

L • Bunch length is free.

• Small beta and small emittance are required.

Short bunch scheme

• Small coupling• Short bunch

• Another approach: Operating point closed to half integer

Keep , and .yx xy

2

x x y y

NL

xx

N

yy

x x y

N

y z

0y y

L

0.5x 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

• 1st orbit• 2nd tune, beta, crossing angle• 3rd chromaticity, transverse nonlinearity. z-

dependent chromaticity is now focused.

i i ij i j ijk i j kH a x b x x c x x x

( , , , , , ( / ))i x yx x p y p z E E x

1exp( : :) * [ , ] [ ,[ , ]] ...

2' ...i ij j ijk j k

M H H H H

a b x c x x

x x x x

Waist control-I traveling focus

• Linear part for y. z is constant during collision.

2I yH ap z

Iy

y

Hy y y azP

P

2y

Hap

z

2 2

1t

a z az

T Taz

1

0 1

azT

=0

: : : :0

I IH He eM Μ

Waist position for given z• Variation for s

• Minimum is shifted s=-az

22 2

( ) ( )1 1

t

a z az s az s az

M s M saz s az

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

• RFQ TM210•

21

2I yH p z

2

* 2

1

2

c pcV

e V~10 MV or more

Waist control-II crab waist (P. Raimondi et al.)

• Take linear part for y, since x is constant during collision.

2I yH axp

Iy

y

Hy y y axP

P

2

x x x y

Hp p p ap

x

1

0 1

axT

2 2

1t

a x ax

T Tax

: : : :0

I IH He eM Μ

Waist position for given x• Variation for s

• Minimum is shifted to s=-ax

22 2

( ) ( )1 1

t

a x ax s ax s ax

M s M sax s ax

Realistic example- II

• To complete the crab waist, a=1/, where is full crossing angle.

• Required H

• Sextupole strength

21I yH xp

*

2 *

1 '' 1 1

2 /x

y y x

B LK

p e

K2~30-50

Not very strong

Crabbing beam in sextupole• Crabbing beam in sextupole can give the nonlin

ear component at IP• Traveling waist is realized at IP.

2I yH ap z

**

( )( ) ( )

sz s x s

*

2 *

1 '' 1 1

2 /x

y y x

B LK

p e

K2~30-50

Super B (LNF-SLAC)Base PEP-III

C 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

/2 (mrad) 25 14

x 0.0025

y 0.1

( ) ( / 2)yy

x z

Fz s z dzds

y

Luminosity for the super B• Luminosity and vertical beam size as functions of K2• L>1e36 is achieved in this weak-strong simulation.

DAFNE upgradeDAFNE

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

/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+100.057,

0.097-0.01 5.19(3.3)

1.00E+110.057,

0.097-0.01 13.21(4.8)

() strong-strong , horizontal size blow-up

Super KEKBSuperKEKB Crab waist

x 9.00E-09 6.00E-09 6.00E-096.00E-

096.00E-09

y 4.50E-11 6.00E-11 6.00E-116.00E-

116.00E-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+105.50E+1

05.50E+1

03.50E+

103.50E+1

0

np 1.26E+111.27E+1

11.27E+1

18.00E+

108.00E+1

0

/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+356.70E+3

51.00E+3

63.95E+

354.80E+3

5

Lum (S.S.) 8.25E35 4.77E35 9E35 3.94E35 4.27E35

Horizontal blow-up is recovered by choice of tune. (M. Tawada)

• 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.!• Life time is improved.

x=24 nm y=0.18nm x=0.2m y=1mm z=3mm

Traveling waist

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

1 1 2 2 3 3( , , , , , ; ) '( , , , , , ; )x y zH x p y p z p s H J J J s

( ) ( ) 2s L s

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.

• Equilibrium distribution

( )2 ( )

d H Jd J

ds J

0 constdJ H

Jds

31 2

1 2 3

( ) exp : emittanceJJ J

J

y

py

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 freedomGaussian weak-strong beam-beam model

• Diffusion is seen even in sympletic system.

Diffusion due to crossing angle and frequency

spectra of <y2>

Only 2y signal was observed.

Linear coupling for KEKB• Linear coupling (r’s), dispersions, (, =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

5

10

15

20

nux

510

1520

nuy

00.00050.001

0.00150.002

5

10

15

20

nux0

0.00050.001

0.00150.002

2.5 5 7.5 10 12.5 15 17.5 20

2.5

5

7.5

10

12.5

15

17.5

20

(0.51,0.51)(0.51,0.51) (0.7,0.51)

(0.7,0.7)(0.51,0.7)

(0.7,0.51)

(0.51,0.7)

KEKB D=0.0005

3-D simulation including bunch length (z~y)Head-on collision

• Global structure of the diffusion rate. • Fine structure near x=0.5 Contour plot

5

10

15

20

nux

510

1520

nuy

00.000250.0005

0.000750.001

5

10

15

20

nux0

0.000250.0005

0.000750.001

2.5 5 7.5 10 12.5 15 17.5 20

2.5

5

7.5

10

12.5

15

17.5

20

(0.51,0.51)

(0.7,0.51)

•Good region shrunk drastically.

•Synchrobeta effect near y~0.5.

• x~0.5 region　 remains safe.

Crossing angle (z/x~1, z~y)• Good region is only (x, y)~(0.51,0.55).

5

10

15

20

nux

510

1520

nuy

00.00050.001

0.00150.002

5

10

15

20

nux0

0.00050.001

0.00150.002

2.5 5 7.5 10 12.5 15 17.5 20

2.5

5

7.5

10

12.5

15

17.5

20

(0.51,0.51)

(0.7,0.51)

(0.51,0.7)

Reduction of the degree of freedom.

• For x~0.5, x-motion is integrable. (work with E. Perevedentsev)

if zero-crossing angle and no error.

• Dynamic beta, and emittance

• Choice of optimum x

,0.5

lim 0x

C yD

2 2,0

0.5limx

xx

2

0.5limx

xp

1

(crossing angle)+(coupling)+(fast noise)x

L

Crab waist for KEKB• H=25 x py

2.

• Crab waist works even for short bunch.

Sextupole strength and diffusion rate 　　　　　　　　　　 K2=20

　　　　　　　　　 25 　　　　　　　　　　　　

　 30

5

10

15

20

nux

510

1520

nuy

00.00050.001

0.00150.002

5

10

15

20

nux0

0.00050.001

0.00150.002

5

10

15

20

nux

510

1520

nuy

00.00050.001

0.00150.002

5

10

15

20

nux0

0.00050.001

0.00150.002

5

10

15

20

nux

510

1520

nuy

00.00050.001

0.00150.002

5

10

15

20

nux0

0.00050.001

0.00150.002

Why the sextupole works?

• Nonlinear term induced by the crossing angle may be cancelled by the sextupole.

• Crab cavity

• Crab waist

Need study

22exp( : :) exp( : :) exp( : :) ...x yF p z K xp

exp(: :) exp( : :)BBF M F

exp( : :) exp( : :) exp(: :) 1x xF p z p z

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 bunc

h scheme.