1. Beam-beam limit for e+e- factories
K. Ohmi (KEK)
e+e- Factories
13-16 Oct. 2003

2. Introduction Luminosity Beam-beam limit is discussed.
The beam-beam limit influences the design of factory machines.
I: RF, Vacuum, instability
by: IR Optics

3. Courtesy J. Rogers HALO03

4. DAFNE
VEP2K
BEPC
BEPCII
CESR-c
E (GeV)
0.51 1
1.4
1.89
1.5
C (m)
97.69
24.38
240.4
237.5
768
ex(nm)
600
136
580
144
180
Ne(1010)
4.7/3.1 10 23
4.9
6.4
sz(cm) 3 5
bx (cm)
170
120
100 70
by(cm)
2.7
1.1 xx
0.04
0.1
0.058
0.05
Lb (1030)
1.6
(100)
11.2
(11)
(6.7) xy
0.02
(0.1)
0.041
(0.04)
(0.056)

5. PEPII
SPEPII
KEKB
SKEKB
CESR
E (GeV)
3.1/9
3.5/8
5.3
C(m)
2199
3016
768
ex(nm)
33/49
19.5
18/24 24
206
Ne(1010)
9.3/5
10/4.5
7/5.2
13/5.5
12.7
sz(cm)
1.2
0.175 7 3
1.8
bx (cm) 40 15 60 20 96
by(cm)
1.1
0.15
0.7
0.3 xx
0.07
0.12
0.2
0.026
Lb (1030) 6
(145) 8
(50) 29 xy
0.048
(0.15)
0.05
(0.1)

6. Source of beam-beam limit
Current limit, due to hardware heating etc.
Collision offset. (1st order　in H)
Optics error (2nd order), nonlinearity (3rd or higher)
Fundamental limit
We study the beam-beam limit using computer simulations.

7. One turn map including beam-beam interaction
Hamiltonian
Lattice
Beam-beam
See next slide.
Equation of motion (one turn map)

8. Integration of the beam-beam interaction
Lattice one turn map
V0(s,0) : map of IP to collision point of the particle
Collision point : s+=(z+-z-)/2 s-=- s+
integration
IP (s=0)

9. Beam-beam potential
f: potential given by solution of 2D Possion equation at every s and every z.

10. Numerical integration
Discretize the interaction. In visual expression, slice the target bunch.
It is easy to evaluate exp(-:f(z):), because it includes only x,y,z not px…. Potential kick.
But …

11. How to estimate f
The potential f have to be solved for r at every s and every z.
Gaussian approximation : exact forms of r and f are given as functions of s: that is, synchro-beam mapping (Hirata).
Particle in cell : it is difficult to get r and f for arbitrary s(z), because the potential has to be solved for each particle (z): that is, target particles are transferred to s=(zi-z)/2 and the potential have to be solved.
To reduce the process, the potential is interpolated in the sliced region.

12. Other notes
r is affected by preceding interaction in the strong-strong model: disruption effect.
Collision should be evaluated along the right time (s) order for both beams.

13. Slice dependence (strong-strong)
Luminosity converge at Nsl=10.
Nsl=5 is 5% less, but is not bad.
Nsl=5 is used in this presentation.
KEKB parameter
Weak strong also shows similar behavior for slicing

14. Convergence for the slice number
By M. Tawada
If we do not use this algorithm
All particles in i-th slice are kicked by φcp
Interpolation

15. Two types of beam-beam limit
Particle motion and/or distribution function are determined by one turn lattice map (revolution matrix for the linear part) at the collision point, V0 and beam-beam map at IP region.
Lattice dependent beam-beam limit
If V0 does not include any coupling, one turn map is represented only by tune, radiation damping, excitation and the beam-beam parameters.
Fundamental beam-beam limit

16. Lattice dependent Beam-beam limit One turn Lattice map at IP, V0
Tune
b, a at IP.
x-y coupling r1-r4 at IP.
xy-z coupling hx, h’x, hy, h’y, zx, z’x, zy, z’y
Crossing angle is regarded as zx.
Chromaticity, amplitude dependent tune shift…

17. How do the parameters lead the beam-beam limit?
Resonance
Contradict of normal axis for lattice and beam-beam interaction.
…..
Analytical approaches are limited to qualitative discussions.
Numerical approaches are relied: weak-strong and strong-strong simulations.

18. Linear optics error DAFNE
weak-strong codes: BBC (Hirata) & LIFETRAC (Shatilov)
M. Zobov, Beam dynamics news letter 31, 2003

19. Optics tuning at KEKB Waist of by at IP. RF phase.
Vertical dispersion at IP
X-y coupling R1-R4 at IP.
These are tuned everyday.

20. Optics parameters are controlled by local bumps

21. X-y coupling LER R4

22. Parameter dependence given by simulation

23. Crossing angle Crossing angle is equivalent to zx at IP.
X-z coupling is induced in one turn map.
Resonance. Mixing of normal mode.

24. Specific luminosity for various current product (I+I-).
f = 0mrad and 11mrad
Weak strong strong-strong

25. Beam-beam parameter
** The beam-beam limit in finite crossing is determined by a weak-strong effect in this parameter.

26. Crossing angle and luminosity
Sharp peak near zero crossing angle

27. Effect of crossing angle
We discuss later.
The beam-beam parameters are >0.2 for weak-strong and 0.1 for strong-strong at the head-on collision.
The weak-strong and strong-strong show similar results x~0.06 for f =11 mrad.
Crossing angle degrades the luminosity.

28. Crossing and crab cavity
Crab cavity creates zx=-q at IP.
Completely agree with head-on.
The difference of sx is only apparent.

29. Lattice nonlinearity
DAFNE
M. Zobov, Beam dynamics news letter 31,2003

30. Fundamental beam-beam limit
Coherent motion
p mode or higher order instability
Studies coherent mode spectrum (Yokoya, Alexahin et al.).
Two stream type (Perevedentsev, Valishev)
Incoherent effects
Quantitative analysis is relied on numerical approaches.

31. Beam-beam limit due to coherent motion
In strong-strong simulations, coherent p mode limits luminosity for short bunch and 2D model.
Coherent motion is observed for sz<by/2.
No coherent motion
Coherent motion

32. Coherent motion seen in some codes
J. Qiang (LBL)
Y. Cai (SLAC) K.Ohmi (KEK)

33. Coherent beam-beam limit for short bunch
The beam-beam limit is about for rings with the damping time ~5000 turns (B factory level).
In actual machine, sz~by, the coherent motion may be smeared by tune spread along the bunch.
Even for short bunch, Landau damping caused by many unknown sources may smear the coherent motion.
Tune and/or intensity difference of two rings also suppress the coherent motion.

34. Other notes
Strong-strong simulation with Gauss approximation is insensitive for the p mode instability.
p mode instability
s mode always dominate.
???
PIC xlim~0.06-7
Gauss xlim>0.2

35. Different tune
No coherent motion. Luminosity is recovered to that by 3D-PIC level.
The luminosity is closed to that obtained by 3D PIC strong-strong.

36. Two stream type of instability (Perevedentsev, Valishev)
For high disruption parameter and low beta, two stream type of instability is caused by the beam-beam interaction.
Disruption parameter
w: beam-beam oscillation freq.
It is serious for ny closed to half integer.

37. Example of beam-beam two-stream instability
nx= ny= ns=0.02
sz <yz>

38. Rogers, beam-beam 03

39. Beam-beam limit due to incoherent effects
Tune scan using weak-strong.
How large beam-beam limit can we achieved if no coherent instability occurs.
Are weak-strong and Gaussian strong-strong simulations accurate to predict the beam-beam limit (x>0.2).
Do all type of simulations show the unique beam-beam limit?
Can we understand the beam-beam limit?

40. Tune scan for BEPC II
P.D.Gu et al., Beam-dynamics News Letter 31, 2003
BBC (Hirata)

41. Tune survey by strong-strong at KEKB (Tawada)

42. Golden operating point
In many machines, tune operating point around horizontal, nx= , and vertical, ny= , is the best region of the tune space.
Horizontal beam size is squeezed by the dynamic beta effect.
ny= is bad due to the two-stream effect.

43. How high beam-beam parameter can be achieved
Strong-strong simulation with Gauss approximation and Particle In Cell method
Remarkable difference. Gauss x>0.2 PIC x~0.1

44. Check using some PIC codes
LBL (J. Qiang)

45. SLAC (Cai) and KEK (Ohmi) codes50

46. Luminosity evolution by PIC
What is the sudden dip?
Such dips are observed randomly for high beam-beam parameter. Once or none in a simulation

47. An example with a sudden dip: particles are initialized with final distribution of Gaussian simulation
Only sy
Does this behavior have a hint of the beam-beam limit?

48. Change of particle distribution No coherent motion during the growth
200 final

49. Is any small coherent motion contribute?
Weak-strong simulation with PIC.
The strong-beam was given the final distribution of the strong-strong simulation.
Basically the beam-beam limit is caused by weak-strong effect due to distorted beam.
There is a luminosity difference 15% between them.

50. weak-strong with PIC strong-strong
Vertical beam size
weak-strong with PIC strong-strong
Weak-strong with PIC & Gauss distr.
Incoherent effect dominates for the beam-beam limit.
There are small other effects (DL=15%): coherent motion?

51. Equilibrium distribution of two beams
Vlasov equation
Fokker Plank equation
D:rad. damping
B:rad. excit.

52. Does an equilibrium distribution exist?
If the Hamiltonian system is solvable,
an equilibrium distribution exists.
Function property of the distribution for J is determined by Fokker Plank term.
The distribution depends only on emittance, B/D, basically.

53. Nonlinear diffusion of beam-beam system
If the Hamiltonian system is not solvable, equilibrium distribution does not exist.
Arnord’s diffusion is caused by the nonlinearity (Chirikov, Tennyson et al.).
The beam-beam interaction is not solvable, even in weak-strong model.
Many discussions have been done for halo formation.
In this strong-strong case, diffusion for coupled two beams seems to be important.

54. Quasi-strong-strong simulation
The equilibrium distribution can be given by more simple way.
Switch the PIC weak-strong simulation with a certain period.
Talk at Working group

55. Round beam (VEPP2K)
Beam-beam limit, ~0.2, is higher than that of the flat beam.
To gain the luminosity for xr=2xy, br<4by is required. It is not easy.

56. Summary Luminosity beam-beam limit is caused by various reasons.
Linear coupling including crossing angle and nonlinearity of lattice degrade the luminosity.
Coherent motion may limit the luminosity, but Landau damping suppress the motion. Whether the coherent motion limits, depends on the bunch length and the operating point.
Tune scan shows the best operating point, nx= , ny=

57. Summary
Two stream type of beam-beam effect is serious for tune closed to half integer in the plane with high disruption (vertical).
Incoherent beam-beam limit caused by equilibrium distribution of two beams appear at ~0.1 for B factories.
The limit depends on the damping time.
The beam-beam limit does not appear in simulations with Gauss approximation.　Gauss Approximation is not suitable for study of the beam-beam limit.

58. Thanks Y Cai Y. Funakoshi J. Qiang K. Oide J. Rogers M. Tawada
A. Valishev
M. Zobov