1. Crab Waist Collision Studies for e+e- Factories
M. Zobov, P. Raimondi, LNF INFN, Italy
D. N. Shatilov, BINP, Novosibirsk
K. Ohmi, KEK, Japan
CARE-HHH-APD Mini-Workshop IR’07,
INFN, Frascati (Italy), 7-9 November 2007

2. OUTLINE Crab Waist Concept Crab Waist Scheme for DAFNE Upgrade
1036 cm-2s-1 in SuperB Factory

3. Numerical Codes Used Weak-Strong Codes Strong-Strong Codes
BBC (K. Hirata, Phys.Rev.Lett.74, 2228 (1995))
LIFETRAC (D. Shatilov, Part.Accel.52, 65 (1996))
BBWS (K. Ohmi)
Strong-Strong Codes
BBSS, (K. Ohmi, PRSTAB 7, , (2004))
GUINEA-PIG (D. Schulte, CERN-PS LP) modified by P. Raimondi for storage rings
The codes have been successfully used for e+e- factories:
KEKB, DAFNE, PEP-II, BEPCII and colliders: VEPP4M, VEPP2000.

4. Crab Waist in 3 Steps Large Piwinski’s angle F = tg(q)sz/sx
Vertical beta comparable with overlap area by sx/q
Crab waist transformation y = xy’/(2q)
1. P.Raimondi, 2° SuperB Workshop, March 2006
2. P.Raimondi, D.Shatilov, M.Zobov, physics/

5. Crab Waist Scheme Sextupole IP (Anti)sextupole Sextupole strength
Equivalent Hamiltonian

6. x bY e- e+
4sx/q q
sz*q z
2sz
2sx

7. x bY e- e+
4sx/q q
sz*q z
2sz
2sx

8. Crab Waist Advantages F = tg(q)sz/sx by sx/q y = xy’/(2q)
Geometric luminosity gain
Very low horizontal tune shift
Large Piwinski’s angle
F = tg(q)sz/sx
2. Vertical beta comparable
with overlap area
by sx/q
3. Crabbed waist transformation
y = xy’/(2q)
Geometric luminosity gain
Lower vertical tune shift
Vertical tune shift decreases with oscillation amplitude
Suppression of vertical synchro-betatron resonances
Geometric luminosity gain
Suppression of X-Y betatron and synchro-betatron resonances

9. ..and besides,
There is no need to increase excessively beam current and to decrease the bunch length:
Beam instabilities are less severe
Manageable HOM heating
No coherent synchrotron radiation of short bunches
No excessive power consumption
The problem of parasitic collisions is automatically solved due to higher crossing angle and smaller horizontal beam size

10. Large Piwinski’s Angle
O. Napoly, Particle Accelerators: Vol. 40, pp ,1993
P.Raimondi, M.Zobov, DAFNE Technical Note G-58, April 2003
If we can increase N proportionally to F*:
L grows proportionally to F;
xy remains constant;
xx decreases as 1/F;
*F is increased by:
increasing the crossing angle q and increasing the bunch length sz for LHC upgrade (F. Ruggiero and F. Zimmermann)
increasing the crossing angle q and decreasing the horizontal beam size sx in crabbed waist scheme

11. Low Vertical Beta Function
Note that keeping xy constant by increasing the number of particles N proportionally to (1/by)1/2 :
(If xx allows...)

12. Vertical Synchro-Betatron Resonances
D.Pestrikov, Nucl.Instrum.Meth.A336: ,1993
tune shift
Resonance suppression factor
Angle = 0.00
0.0025
0.0050
0.01
Synchrotron amplitude in sz

13. Geometric Factors
Minimum of by along the maximum density of the opposite beam;
Redistribution of by along the overlap area. The line of the minimum beta with the crab waist (red line) is longer than without it (green line).

14. Crab Waist Collisions at f1 = -q, f2 = q

15. Geometric Luminosity Gain due to Crab Sextupoles
(DAFNE Example)
Strong-strong
DL, %
Weak-strong
Normalised sextupole strength
“..crabbed waist” idea does not provide the significant luminosity enhancement. Explanation could be rather simple: the effective length of the collision area is just comparable with the vertical beta-function and any redistribution of waist position cannot improve very much the collision efficiency…” (I. A. Koop, D.B.Shwatz)
Normalised sextupole strength

16. Suppression of X-Y Resonances
Horizontal oscillations
sextupole
Performing horizontal oscillations:
Particles see the same density and the same (minimum) vertical beta function
The vertical phase advance between the sextupole and the collision point remains the same (p/2)

17. X-Y Resonance Suppression
Much higher luminosity!
Typical case (KEKB, DAFNE etc.):
1. low Piwinski angle F < 1
2. by comparable with sz
Crab Waist On:
1. large Piwinski angle F >> 1
2. by comparable with sx/q

18. … and in the ideal case Crab Waist: DQy
Eliminates all (!) X-Y resonances
However, some horizontal synchrobetatron resonances appear
DQx
Here strong beam’s modulation is excluded
(100 times larger by and smaller ey)

19. Tails in SuperB
Bunch Current
Crab Sextupoles Off
Crab Sextupoles On

20. DAFNE Upgrade Parameters
FINUDA
DAFNE Upgrade
qcross/2 (mrad)
12.5 25
ex (mmxmrad)
0.34
0.20
bx* (cm)
170 20
sx* (mm)
0.76
FPiwinski
0.36
2.5
by* (cm)
1.70
0.65
sy* (mm)
5.4 (low current)
2.6
Coupling, %
0.5
Ibunch (mA) 13
Nbunch
110
sz (mm) 22
L (cm-2s-1) x1032
1.6 10
Larger Piwinski angle
Lower vertical beta
Already achieved

21. Weak-Strong Beam-Beam Simulation
for DAFNE Upgrade
With the present DAFNE parameters (currents, bunch length etc.) a luminosity in excess of 1033 cm-2 s-1 is predicted
With 2A on 2A more than 2x1033 is possible
Beam-beam limit is well above the reacheable currents

22. Luminosity vs tunes scan
Crab On  0.6/q
Crab Off
Lmax = 2.97x1033 cm-2s-1
Lmin = 2.52x1032 cm-2s-1
Lmax = 1.74x1033 cm-2s-1
Lmin = 2.78x1031 cm-2s-1

23. (Lifetrack code by D. Shatilov)
Beam-Beam Tails at (0.057;0.097)
(Lifetrack code by D. Shatilov)
ac > 0
ac < 0
Ax = ( 0.0, 12 sx); Ay = (0.0, 160 sy)

24. Siddharta IR Luminosity Scan
above half-integers
Lmax = 3.05 x 1033 cm-2s-1
Lmin = 3.28 x 1031 cm-2s-1

25. Strong-Strong Simulations
for DAFNE Upgrade
Single Bunch Luminosity
Single Bunch Luminosity
Crab Waist On
Crab Waist On
Crab Waist Off
tdamping = turns
tdamping = turns
x110 bunches = 1033 cm-2 s-1
(K. Ohmi, BBSS Simulations)

26. SuperB initial set of parameters
(June 2006)
Emit_x nm
0.8
Emit_y nm
0.002
Beta_x* mm
9.0
Beta_y* mm
0.080
Sigm_x* mm
2.67
Sigm_y* nm
12.6
Sigm_z mm
6.0
Sigm_e
1.0e-3
Cross_angle mrad
2*25
Np e10
2.5 Nb
6000
C km
3.0
s msec 10
Collision freq MHz
600
Luminosity e36
1.0
Defined a parameters set based on ILC-like parameters:
Same DR bunch length
Same DR bunch charges
Same DR damping time
Same ILC-IP betas
Same DR emittances
Crossing Angle and Crab Waist to minimize BB blowup

27. Luminosity and blowups vs current

28. The relation by  sx/q must be satisfied in all optimizations!
To achieve beam-beam limit for the initial set of parameters, Np should be increased by a factor of 2-3, that gives the luminosity exceeding 1037! Actually it means we have rather big margins to relax some critical parameters, and still get the desired luminosity L=1036. The list of parameters to optimize/relax is:
Damping time
Crossing angle
Bunch length
Bunch current
Number of bunches
Emittances
Betatron coupling
Beta-functions
The relation by  sx/q must be satisfied in all optimizations!

29. Optimization Results Relaxed damping time: 10msec=>16msec
Relaxed y/x IP bs: 80mm/9mm => 300mm/20mm
Relaxed y/x IP ss: 12.6nm/2.67mm => 20nm/4mm
Relaxed crossing angle: 2*25mrad => 2*17mrad
Possible to increase bunch length: 6mm => 7mm
Possible increase in L by further b’s squeeze
Possible to operate with half of the bunches and twice the bunch charge (same current), with relaxed requirements on ey: 2pm => 8pm (1% coupling)
Possible to operate with half of the bunches and twice the bunch charge (same current), with twice the emittances

30. SuperB Luminosity Tune Scan
DQy
Lmax = 1.21x1036 cm-2s-1
Lmin = 2.25x1034 cm-2s-1
DQx

31. SuperB with 2 IP (suggested by A. Variola)
Lmax = 1.05 x 1036 cm-2 s-1
Lmax = 6.17 x 1033 cm-2 s-1
Lmax = 1.03 x 1036 cm-2 s-1
Lmax = 7.01 x 1033 cm-2 s-1

33. L=1036 cm-2 s-1 Beam-Beam Blowup (weak-strong simulations) HER LER
Crab=0.8Geom_Crab
Crab=0.9Geom_Crab
HER
LER
L=1036 cm-2 s-1

34. Conclusions
We hope that now we understand how “Crab Waist” works
The expected luminosity increase due to “Crab Waist” is
a) at least, a factor of 6 for the DAFNE upgrade
b) about 2 orders of magnitude for the SuperB project
(with respect to the existing B-Factories)
3. Let us wait for the first DAFNE experimental results!
Thank you!