1. K. Ohmi (KEK-ACCL) eeFACT2016, Oct 24-27, 2016 Daresbury lab, UK
FCC-ee beam-beam strong-strong simulations for all working and mitigation
K. Ohmi (KEK-ACCL)
eeFACT2016, Oct 24-27, 2016
Daresbury lab, UK

2. Beam-beam limit 𝐿= 𝑁 2 𝑓 𝑟𝑒𝑝 4𝜋 𝜎 𝑥 𝜎 𝑦 𝑅 𝜎 𝑧 𝛽 𝑦 , 𝜃 𝑐 𝜎 𝑧 𝜎 𝑥
Luminosity
𝜎 𝑧 𝛽 𝑦 :hourglass, 𝜃 𝑐 𝜎 𝑧 𝜎 𝑥 : normalized crossing angle (Piwinski angle)
Tune shift
Increasing N, beam size especially vertical for flat beam increases. Tune shift is saturate at a certain value. Luminosity linearly increases for N, not N2. This situation is called Beam-beam limit.
How large tune shift is achieved in equilibrium? Do simulations predict the beam-beam limit?
𝐿= 𝑁 2 𝑓 𝑟𝑒𝑝 4𝜋 𝜎 𝑥 𝜎 𝑦 𝑅 𝜎 𝑧 𝛽 𝑦 , 𝜃 𝑐 𝜎 𝑧 𝜎 𝑥
N=N+=N-: bunch population
frep: collision freq.
qc: half crossing angle
𝜉 𝑦 =Δ 𝜈 𝑦 = 𝑁 𝑟 𝑒 2𝜋𝛾 𝛽 𝑦 𝜎 𝑦 ( 𝜎 𝑥 + 𝜎 𝑦 ) 𝑅 𝜎 𝑧 𝛽 𝑦 , 𝜃 𝑐 𝜎 𝑧 𝜎 𝑥
𝐿= 𝑁𝛾 𝑓 𝑟𝑒𝑝 2 𝑟 𝑒 𝛽 𝑦 𝜉 𝑦
𝜎 𝑥 ≫ 𝜎 𝑦

3. Beam-beam parameter, xL
Luminosity is calculate by the beam-beam simulation.
Beam-beam tune shift is estimated by the luminosity.
𝐿= 𝑁𝛾 𝑓 𝑟𝑒𝑝 2 𝑟 𝑒 𝛽 𝑦 𝜉 𝑦
𝜉 𝐿 = 2 𝑟 𝑒 𝛽 𝑦 𝑁𝛾 𝑓 𝑟𝑒𝑝 𝐿
Equilibrium beam-beam tune shift =Beam-beam parameter
Simulations are executed with scanning the bunch population;
initial beam-beam tune shift 𝜉 𝑦0 .
The beam-beam parameter is used for a index of the beam-beam limit.

4. Beam-beam limit in FCC-ee Higgs factory
E=120GeV
Ne=8x1010 , Nbunch=770
ex=0.61 nm, ey=1pm
bx=1 m, by=2mm
sz=2.4mm, sd=0.12% (include BS)
Study for small crossing angle, szqc/sx=0 or 0.5, qc=5mrad (half angle)
Study for large crossing angle, adjust crossing angle qc=20mrad so that szqc/sx=2.
Damping time ~150 turns

5. Large crossing angle and crab waist weak-strong simulation
Beam-beam parameter xL=0.6 is achieved for collision with crab waist in weak-strong simulation, (nx, ny=0.51,0.55).
Beam-beam parameter is saturated at xL=0.1 without crab waist.
Luminosity evolution for scanning bunch population

6. Equilibrium beam-beam parameter and beam size in weak-strong simulation
xmax~0.6 for (nx, ny)=(0.51,0.55)
xmax~0.2 for (nx, ny)=(0.54,0.61)
sy behavior correlates to Luminosity.

7. Strong-strong simulation for Large crossing anglezi zj
Two colliding bunches are divided into many slices, Nsl~10xszq/sx.
Sort slices with their positons zi+zj, collision order.

8. Several option of Strong-strong simulation
Gaussian approximation using turn-by-turn RMS values.
Gaussian approximation using turn-by-turn Gaussian fitting.
PIC for core part and Gaussian approximation for slice collision with large offset.
Complete PIC using shifted Green function
Example of shifted potential for collision with large offset.
Shifted Green function (J. Qiang)

9. Large crossing angle and crab waist Complete PIC based strong-strong simulation
𝜃 𝑐 𝜎 𝑧 𝜎 𝑥 =2
Large crossing angle, and crab waist.
Beam-beam parameter is saturated at xL=0.15 in strong-strong simulation.
Luminosity fluctuates x0>0.239, xL>0.12. Coherent <xz> oscillation is seen in both beams (in-phase mode).
tune (0.513,0.57)
Evolution of <xz>
Luminosity evolution for scanning bunch population
Different tune point.
(0.54,0.57)
Slower damping time, 300 turns.

10. Beam size Horizontal size increases twice.
Piwinski angle decrease. Perhaps hourglass effect is enhanced.

11. x limit for FCC-ee H
No clear difference for (0.54,0.61) from weak-strong xlim=0.2.
Big difference for (0.51,0.55), the limit in weak-strong is extremely high xlim=0.6 (ws), xlim=0.2 (ss).
limit is weakly dependent of tune in Strong-strong simulation.
It is possible to achieve xL=0.15 for tlep-H.
s.s
w.s

12. Large crossing angle and crab waist Gaussian (RMS) strong-strong simulation
𝜃 𝑐 𝜎 𝑧 𝜎 𝑥 =2
Large crossing angle, and crab waist.
Beam-beam parameter is saturated at xL=
Coherent <xz> motion is seen for x0>0.239, xL>0.15.
It is possible to achieve xL=0.15 for tlep-H.
xL is plotted every 10 turns, namely it actually strongly fluctuates turn-by-turn.

13. Beam-beam limit in FCC-ee Z factory
Systematic study was performed.
Only LPA/CW scheme is considered.
F.Z, 2016 Apr
𝜎 𝑧 𝜃 𝜎 𝑥 =2.6

14. Strong-strong simulation for Z factory
limit is around
Coherent instability is strong.
𝜉 target =0.17
𝐿 target =2.2× cm −2 s −1

15. Beam size behavior Horizontal size increases twice.
Piwinski angle decrease. Perhaps hourglass effect is enhanced.

16. x limit in Z factory Error bar is fluctuation due to coherent motion
The beam-beam parameter is limited 0.06, and coherent oscillation
Big difference from the weak-strong results xlim=0.2 at (0.51,0.55).
No big difference xlim=0.06(ss), 0.1(ws) in (0.54,0.61).
𝜉 𝑥 =0.13
s.s
𝜉 target =0.17
𝐿 target =2.2× cm −2 s −1
w.s

17. Animation of x-z motion
It is not simple m=1 x-z motion.
Collision frame

18. Tune scan Tune of both beams are changed 0.51-0.60.
Tune of electron beam is fixed 0.54, and that of positron beam is scanned.
Coherent motion is suppressed when tune go away form half integer, but luminosity is not good.
Effect of chromaticity is weak.

19. Study using simple model I
Collision of pan-cake and airbag beams.
The airbag is represented by N macro-particles.
Linear force due to beam-beam tune shift of projected positron bunch.
s=(ze,i-zp)/2
𝛽 𝑥 =1
𝑧 𝑒,𝑖 = 𝜎 𝑧 cos⁡(2𝜋( 𝜈 𝑠 𝑡+ 𝑖 𝑁 ))
∆ 𝑥 𝑒,𝑖 = 𝑧 𝑒,𝑖 𝜃
∆ 𝑥 𝑝 =− 𝑝 𝑝 𝑠
∆ 𝑥 𝑒,𝑖 = 𝑝 𝑒,𝑖 𝑠
∆ 𝑝 𝑝 =− 4π𝜉 𝑁 ( 𝑥 𝑝 − 𝑥 𝑒,𝑖 )
∆ 𝑝 𝑒,𝑖 =−4𝜋𝜉( 𝑥 𝑒,𝑖 − 𝑥 𝑝 )
∆ 𝑥 𝑝 =+ 𝑝 𝑝 𝑠
∆ 𝑥 𝑒 = −𝑝 𝑒 𝑠
∆ 𝑥 𝑒,𝑖 = −𝑧 𝑒,𝑖 𝜃

20. Result of the simple model I
Very strong head-tail motion.
High order mode、m~10
Tune scan showed tune near synchro-beta is worse.
x distribution after 1000 turn x x
One-Two particle model, which take into account of m=1 mode, is stable.
K. Ohmi, A. Chao, PRSTAB5, (2002).

21. Study using simple model II
Collision of two airbag beams
The airbag beam is represented by N macro-particles.
Beam-beam force between macro-particles is expressed by Bassetti- Erskine formula.
𝑧 𝑒,𝑖 = 𝜎 𝑧 cos⁡(2𝜋( 𝜈 𝑠 𝑡+ 𝑖 𝑁 ))
s=(ze,i-zp)/2
Sort zp,i+ze,j.
∆ 𝑥 𝑝,𝑗 = 𝑧 𝑝,𝑗 𝜃
∆ 𝑥 𝑒,𝑖 = 𝑧 𝑒,𝑖 𝜃
∆ 𝑥 𝑝,𝑗 = −𝑝 𝑝,𝑗 𝑠 𝑖𝑗
∆ 𝑥 𝑒,𝑖 = 𝑝 𝑒,𝑖 𝑠 𝑖𝑗
∆ 𝑝 𝑝,𝑗 =− 𝐹 𝐵𝐸 ( 𝑥 𝑝,𝑗 − 𝑥 𝑒,𝑖 )
∆ 𝑝 𝑒,𝑖 =− 𝐹 𝐵𝐸 ( 𝑥 𝑒,𝑖 − 𝑥 𝑝,𝑗 )
∆ 𝑥 𝑝,𝑗 = 𝑝 𝑝,𝑗 𝑠 𝑖𝑗
∆ 𝑥 𝑒,𝑖 = −𝑝 𝑒,𝑖 𝑠 𝑖𝑗
∆ 𝑥 𝑝,𝑗 = −𝑧 𝑝,𝑗 𝜃
∆ 𝑥 𝑒,𝑖 = −𝑧 𝑒,𝑖 𝜃

22. Simulated Beam distribution using simple model II
nx, ny=0.51,0.55
nx, ny=0.54,0.59
Lab frame (not collision frame)

23. Strong-strong simulation in SuperKEKB
𝜃 𝑐 𝜎 𝑧 𝜎 𝑥 =20
𝜉 𝑥/𝑦 =0.0028/0.088
𝜈 𝑠 =0.025
Strong-head-tail instability is seen only in limited tune. The stopband seems narrow.

24. Simulation with latest parameters
𝜎 𝑧 𝜃 𝜎 𝑥 =10 or 6 for Z

25. Beam-beam limit for Z factory
𝜃 𝑐 𝜎 𝑧 𝜎 𝑥 =6
𝜃 𝑐 𝜎 𝑧 𝜎 𝑥 =10

26. Comparison with SuperKEKB
Why big difference appear, SuperKEKB 𝜎 𝑧 𝜃 𝜎 𝑥 =20 and new parameter with 𝜎 𝑧 𝜃 𝜎 𝑥 =10 ?
The simple model I should not be sensitive for small detuning of parameters of two beams.
There is not big difference between 𝜎 𝑧 𝜃 𝜎 𝑥 =2.6 and 𝜎 𝑧 𝜃 𝜎 𝑥 =10 in tlep-Z.

27. Difference from SuperKEKB
The same sx=10mm, larger crossing angle for SuperKEKB.
SuperKEKB ex=3.2nm, bx=3.2cm FCCee-Z ex=0.2nm, bx =50cm
Change ex=2nm, bx =5cm.
Coherent instability disappears!

28. Summary
Beam-beam limit is evaluated for FCC-ee using strong-strong simulation
Coherent head-tail motion is induced by the beam-beam interaction in H and Z.
The instability seems manageable in H, but serious in Z.
Simplified model showed high order head-tail mode, m~10. Simulation results agree with excitation of the high order mode.
SuperKEKB, 𝜉 𝑥/𝑦 =0.0028/0.088, is stable. Latest FCCeeZ, 𝜉 𝑥/𝑦 = 0.025/0.16 or 0.05/0.13.
Changing 10ex and 0.1bx, the coherent instability disappears.
Detailed requirement to lattice design will be ready soon.
Studies and experiments in SuperKEKB are very important for FCC-ee.

29. Thank you for your attention

31. Weak-strong and strong-strong simulation
Weak-strong simulation
One (strong) beam is assumed to be fixed charge distribution, and the other (weak) beam is represented by macro-particles.
Beam-beam interaction is evaluated by tracking the macro-particles in the electro-magnetic field induced by the fixed charge distribution.
The strong beam is assumed to be Gaussian distribution in most cases.
Strong-strong simulation - Both beams are represented by macro- particles.
Beam distribution is represented on meshed space using Particle In Cell method. Arbitrary and self-consistent distribution of two beams are treated.
Statistical noise of macro-particles induces an fluctuation in potential calculated by PIC. The unphysical emittance growth by the noise is cared in the strong-strong simulation.
As an approximation, two beams are represented by Gaussian whose sizes are determined turn-by-turn. It is called Soft Gaussian approximation.
Strong-strong simulation based on PIC is more popular than the soft Gaussian approximation.
Quasi-strong-strong simulation
Repeat weak-strong simulation with keeping self-consistency.

32. Effect of chromaticity
Clear gain was seen in early stage, but it is small effect finally.

33. Crab waist Transfer matrix for y
ay depending on x appears. Waist depend on x. Particle with x collide with another beam core at its waist.
Waist shift is proportional to x.

34. Benchmark with parameters by D. Shatilov
Energy: 45.5 GeV, Emitt_x: 85 pm, Emitt_y: 1 pm
Bunch length: sz=5 mm
Energy spread: 6.9E-4
Crossing angle (full): 2qc=30 mrad
Working point: (0.54, 0.59)
Beta_y^*  1 mm
Beta_x^*  50, 100 cm
No beamstrahlung
ns=0.0075x2
szqc/sx=8

35. Evolution of Beam-beam parameter
Beam-beam parameter is saturated at 0.04
Coherent motion in <xz> appears.
Beam sizes sx and sy increase in equilibrium.
x0=0.048 at Ne=2.3x1010. independent of bx (.
bx=1m
bx=0.5m

36. Monochromatization (preliminary)
Initial beam distribution (opt’d).

37. Baseline
Design luminosity is achieved. No serious effect is seen.

38. Optimized
x-z coherent instability is seen in early stage and beam size blow up.
Residual x-z motion remains.
Luminosity is 60% of the design.

39. Beam distribution Instability affects betatron emittance growth
No beamstrahlung

40. Collision with No crossing angle or small crossing angle
szqc/sx=0 or 0.5

41. No crossing angle or small crossing angle weak-strong simulation
Beam-beam parameter xL=0.5 is achieved for collision with zero crossing angle in weak-strong simulation.
Beam-beam parameter is saturated at xL=0.27 for collision with small crossing angle
Luminosity evolution for scanning bunch population
Higgs factory,
damping time is 150 turns.
No crossing angle
Equilibrium Beam-beam parameter
Small crossing angle
bunch population
Initial tune shift,
𝜃 𝑐 𝜎 𝑧 𝜎 𝑥 =0.5

42. No crossing angle or small crossing angle strong-strong simulation
Collision with zero crossing angle.
Beam-beam parameter is saturated at xL=
Vertical emittance growths of e+ and e- are simultaneous and synchronized. Non Gaussian Collective emittance growth, K. Ohmi, PRL92, (1994).
No coherent motion, except vertical p mode seen only in x0=0.837.
x-y coupling is enhanced correctively.
Evolution of beam-beam parameter (luminosity)
Evolution of vertical beam size at IP
Evolution of horizontal beam size at IP

43. Collision with a Large Crossing angle and crab waist
szqc/sx=2

44. Summary
Beam-beam limit in lepton colliders are studied using Z-H factory parameters.
Two simulation methods, weak-strong and strong-strong, showed characteristic results in each.
Incoherent emittance growth can be studied by weak-strong. Coherent instability can be studied in strong-strong. The main difference seen in Z/H factories using LPA/CW scheme is due to coherent instability.
The instability may be caused by 2(nx+x-ns)=int. Hor. tune should be moved further far from half integer. (0.56,0.61) was not enough. nx>0.6
To make better precision for bam-beam limit prediction in FCC-ee, elaboration of strong-strong simulation is indispensable.
High LPA (szqc/sx>10) requires at least 10 times faster computer power.
The beam-beam limit for LPA/CW scheme will be examined in SuperKEKB.

45. Thank you for your attention

46. Beam hallo distribution given by w.s simulation
Usually Vertical hallo should be taken care. No horizontal tail.
Hallo is less serious for collision with zero crossing angle and with large crossing angle crab waist.
Small crossing angle
Zero crossing angle
Large crossing angle and no crab waist
Large crossing angle and crab waist

47. Lifetime given by weak-strong simulation
In equilibrium, particles escape a boundary is the same number as damping from the boundary. [M. Sands, SLAC-R-121 (1970)]
f(J): equilibrium beam distribution. For example f(J)=exp(-J/e) for Gaussian.
f(J)=N(J)/N0 in the last slide.
Zero crossing angle
Large crossing angle with crab waist
vertical aperture
vertical aperture

48. Beamstrahlung: essential for Higgs factory
Synchrotron radiation during beam-beam collision
Calculate trajectory interacting with colliding beam.
Particles emit synchrotron radiation due to the momentum kick dp/ds.
r=23.5/19.7m<< rbend= 6,094/11,000m (CEPC/TLEP)
uc= 164/194 MeV,, Ng=0.21/0.092

49. Weak-strong strong simulation
Bassetti-Erskine Formula (CERN-ISR-TH/ 80-06)- 2D Electric force induced by transverse Gaussian distribution
Round beam

50. Particle In Cell
Beam potential
Integrated Green function

51. Beam-beam force for flat beam
Integrated Green function is indispensable to reproduce correct beam-beam force for flat beam, sx/sy>100.
K. Ohmi, PRE62, 7287 (2000), PIC and Bassetti-Erskine formula
K. Yokoya had used Integrated Green function since 1980’.

52. Arc transformation Linear transfer matrix (6x6)
Take into account of dispersion and x-y coupling at IP
Crabbing and crossing angle, z
Important parameters
Flat beam bx, ay(waist) , by, nx, ny, hy, h’y, r1-r4, zx(crab angle),
Round beam + hx, h’x, ax

53. Chromaticity Effective Hamiltonian/generating function at IP.
Relations between a,b,c and chromaticity, n’,a’,b’.
Transformation for chromaticity

54. Crossing angle, crab cavity
Crossing angle is equivalent to collision of two beams with xz tilt.
xz tilt can be controlled by crab cavity.

55. Crab waist Transfer matrix for y
ay depending on x appears. Waist depend on x. Particle with x collide with another beam core at its waist.
Waist shift is proportional to x.

56. Synchrotron radiation
Simplest D=ti/T0 dij., t=()
Radiation matrix