1. Some Issues on Beam-Beam Interaction & DA at CEPC
Y. Zhang, J. Wu, N. Wang, D. Wang, Y.W. Wang
Nov. 2018, CEPC Workshop
IHEP, Beijing
Thanks: K. Ohmi, D. Shatilov, K. Oide, D. Zhou

2. Beam-Beam Parameter at CEPC & LEP2
CEPC-W
CEPC-H
CEPC-Z
R. Assmann
the damping rate per beam-beam interaction

3. If the machine parameter is reasonable
Limit of bunch population by beam-beam interaction
Beamstrahlung lifetime
If X-Z instability is suppressed
If asymmetric bunch current collision is stable
If there exist large enough stable working point space
If beam-beam parameter is safe enough
We do not focus on the optimization of the machine parameter

4. Bunch Current Limit @ Higgs
𝜉 𝑦 = 2 𝑟 𝑒 𝛽 𝑦 0 𝑁𝛾 𝐿 𝑓 0

5. Tune Scan Higgs X-Z instability @(0.535,0.61) 𝑄 𝑦 =0.61, 𝑄 𝑠 =0.035
K. Ohmi and etal., DOI: /PhysRevLett
X-Z
𝑄 𝑦 =0.61, 𝑄 𝑠 =0.035
The error bar shows the turn-by-turn luminosity difference.

6. W
Bunch Current W

7. Bunch Current Limit versus Horizontal TuneW
Bunch Current Limit versus Horizontal Tune
Collision is stable in the range of [0.552, 0.555]
Np=12× 10 10
@0.552 and 0.555, np > 15e10,
@0.560 and 0.565, np < 6e10

8. Tune W
Turn-by-Turn
Luminosity
Distortion
Luminosity
𝝈 𝒙
𝝈 𝒚

9. Tune W (Qy=0.590)

10. Bunch Current Limit versus Horizontal Tune
Z-3T
Bunch Current Limit versus Horizontal Tune
Criteria: 𝝈 𝒙 𝝈 𝟎 <1.1
Width of safe Qx ~ 0.006
Collision is stable in the range of [0.562, 0.568]
It is similar in case of Z-2T, with np=12× 10 10
Np=15× 10 10

11. 𝛽 𝑦 ∗ =1.5mm->1mm (Solenoid: 3T -> 2T)Z
𝛽 𝑦 ∗ =1.5mm->1mm (Solenoid: 3T -> 2T)
𝜉 𝑦 = 2 𝑟 𝑒 𝛽 𝑦 0 𝑁𝛾 𝐿 𝑓 0
With same beam current,
smaller 𝛽 𝑦 ∗ +weaker solenoid, luminosity increase by a factor of two.
bunch population increase from 8× to 12× , luminosity increase about 20%.

12. Crosstalk between Beam-Beam Interaction & Collective Effect
Bunch length caused by impedance => Gaussian Approximation
In the conventional case, it is fine since the longitudinal dynamics is not sensitive to the beam-beam interaction
In CEPC/FCC, the beamstrahlung effect will also lengthen the bunch
It is self-consistent to consider the longitudinal wake field and beamstrahlung

13. RMS size with longitudinal impedance

14. Beam-Beam Performance with longitudinal impedance

15. Check of 𝜈 𝑠 with impedance
A. Chao, “Physics of Collective Beam Instabilites in High Energy Accelerators”
Check of 𝜈 𝑠 with impedance

16. Dependence on Horizontal Tune
Ne=20e10

17. Moment Evolution w/ impedance
Ne=20e10
𝝂 𝒙 =𝟎.𝟓𝟓𝟓
Moment Evolution w/ impedance

18. Moment Evolution w/o impedance
Ne=20e10
𝝂 𝒙 =𝟎.𝟓𝟓𝟓
Moment Evolution w/o impedance

19. Moment Evolution w/ impedance
Ne=20e10
𝝂 𝒙 =𝟎.𝟓𝟓𝟎
Moment Evolution w/ impedance

21. Application of FMA to Beam-Beam Effects
J. Laskar, Icarus 88, 266 (1990).
D. Shatilov, E. Levichev, E. Simonov and M. Zobov, PRST-AB, 14, (2011)
Application of FMA to Beam-Beam Effects
Beam-beam resonances in the tune and amplitude
planes for DAFNE, crab~0.4.
The diffusion index is calculated as 𝑙𝑜𝑔10( 𝜎 𝜈 ), where 𝜎 𝜈 is the rms spread of tunes.

22. Could a So-Called Diffusion Map help us?
FMA does not work well with chromaticity
FMA will fail with strong synchrotron radiation and beamstrahlung effect
In a lattice, with strong non-linearity, beam-beam interaction, strong SR fluctuation, could we construct a Diffusion Map to do some analysis?

23. Diffusion Process caused by Beam-Beam Interaction
K. Ohmi, M. Tawada, K. Oide and S. Kamada,
“Study of the diffusion processes caused by the beam-beam interactions”, APAC 2004
Diffusion Process caused by Beam-Beam Interaction
Diffusion Equation with a stochastic kick:
𝜕 𝜕𝑠 Ψ 𝑥,𝑠 =𝐵 𝜕 2 𝜕 𝑥 2 Ψ 𝑥,𝑠
For initial condition, Ψ 𝑥,0 =𝛿 𝑥− 𝑥 0 , the solution is given as
Ψ= 1 4𝜋𝐵𝑠 exp − 𝑥− 𝑥 𝐵𝑠
which means Ψ is Gaussian and its rms value increase as 𝜎 2 =2𝐵𝑠
With damping 𝑑𝑥 𝑑𝑠 =−𝐷𝑥, the diffusion equation is replaced by Fokker-Plank equation
𝜕 𝜕𝑠 Ψ 𝑥,𝑠 =𝐷𝑥 𝜕 𝜕𝑥 Ψ 𝑥,𝑠 +𝐵 𝜕 2 𝜕 𝑥 2 Ψ 𝑥,𝑠
and the equilibrium solution is Ψ= exp − 𝑥 2 2𝐵/𝐷
In most of the circular accelerators, the betatron motion is much faster than the diffusion and damping, therefore the same discussion is applied by using the adiabatic invariant 𝐽 𝑥 instead of x.
If there are some diffusion mechanisms, which are independent each other, the total diffusion coefficient is summation of each diffusion coefficient, 𝐵= 𝐵 𝑖

24. MEASUREMENTS OF TRANSVERSE BEAM HALO DIFFUSION
FERMILAB-CONF APC
MEASUREMENTS OF TRANSVERSE BEAM HALO DIFFUSION
the particle loss rate at the collimator is equal
to the flux at that location:
L=−𝐷 𝜕 𝐽 𝑓 𝐽= 𝐽 𝐶
with phase-space density f(J, t) described by the diffusion equation, where J is the Hamiltonian action and D the diffusion coefficient in action space.

25. Action near Resonances
S. Y. Lee, Accelerator Physics

26. Synchrotron Radiation
K. Oide and etc., PRAB, 19, , 2016
K. Hirata and F. Ruggiero, “Treatment of Radiation for Multiparticle Tracking in Electron Storage Rings”, 1989

27. Diffusion with Different Model
200 particles are tacked,
With same initial coordinate: (3 𝝈 𝒙 ,11 𝝈 𝒚 )
A𝑖≝ 2 𝐽 𝑖 𝜀 𝑖
𝑖=𝑥,𝑦,𝑧
𝜎 𝑎 ≝ 𝜎 𝑎𝑥 2 + 𝜎 𝑎𝑦 2 + 𝜎 𝑎𝑧 2

28. Diffusion Map Analysis of Different Lattice
200 particles, 25 turns
𝐷≡ log 𝑡𝑢𝑟𝑛 𝜎 𝑎 2
The green lines show the boundary of D=1.6.
Slower diffusion indicates fewer halo particles.

29. Optimization based on the diffusion map
Constraint : 𝐷 1 𝜎 𝑥 ,10 𝜎 𝑦 <1.5
Case 0 :
Optimized:

30. Summary The beam-beam effect of CEPC CDR is briefly introduced
The longitudinal impedance shift the stable working point according to the initial result. Further check is needed.
We attempt to present a diffusion map analysis method, which help to judge if one lattice is good in a non-symplectic condition. The initial result shows agreement with halo particles distribution obtained with many particle, long turns tracking.