1. Luminosity Optimization for FCC-ee: recent results
D. Shatilov (BINP) and K. Oide (CERN)
59th FCC-ee Optics Design Meeting
CERN, 25 August 2017

2. Outline
Brief review of previous results: 3D flip-flop, coherent X-Z instability, bootstrapping.
Current limitations
Injection rate, lifetime and asymmetry in the bunch currents
Table of parameters (new baseline)
Discussion and next steps

3. X-Z Instability and 3D Flip-Flop at 45.6 GeV
URF: 250 MV => 120 MV
Increase in s is very useful if it does not reduce (or better increase) z, so that x drops. We already did this by increasing the momentum compaction factor.
Now, reducing URF we increase z and decrease s in the same proportion.
If we want to keep L unchanged, Np also should be increased. Finally, x and s decrease in the same proportion.
The benefit: we increased the orders of synchro-betatron resonances located in the area of interest. This makes the resonances nearest to x weaker.
Both X-Z instability and 3D flip-flop are mitigated.
Luminosity can be increased.
2x - 6s = 1
2x - 8s = 1
x (cm) x
3D Flip-Flop
x (cm) x

4. Collision of short bunches (z without BS) at Z
x /x0
y /y0
z /z0
Turns
Finally, z increases due to BS, the X-Z instability disappears, and the beams stabilize.
But the transverse blowup in the beginning is very strong, so it would be better to increase the beam current gradually during collision => bootstrapping.

5. Bootstrapping z1 /z0 z2 /z0 x1 /x0 x2 /x0 Np = 4.01010

6. What are the Limitations Now?
Coherent X-Z instability and 3D flip-flop are seen at all energies except ttbar, where Piwinski angle is not large (  1.3) and damping is very strong. Both instabilities can be eliminated by the following steps:
Decrease in x (but this is limited by the energy acceptance).
Increase in momentum compaction factor (useful only at Z).
Decrease in RF voltage.
Increase and proper choice of x (in the range of 0.56  0.58).
At low energies the energy acceptance is smaller because of weaker damping and lower x. The natural energy spread is also smaller, but it is increased several times by the beamstrahlung.
Luminosity is limited by the energy acceptance (beamstrahlung lifetime) at all energies.
If there is an asymmetry in the bunch currents, the “weak” bunch’s lifetime becomes smaller.
The achievable asymmetry (less is better) depends on the lifetime and on the repetition rate of top-up injection cycles.

7. Injection Rate and Lifetime
We assume that pre-booster (PB) is SPS, which has 6.9 km perimeter and 9200 RF buckets with 400 MHz. Then we need 10 PB cycles to fill the main booster (MB) more or less uniformly. If PB is another ring with smaller perimeter, the number of PB cycles will be larger.
We assume the PB ramp (up+down) time is 1 second and injection energy is 20 GeV. The PB filling time is 5 seconds at Z (2000 bunches, linac 200 Hz, 2 bunches/pulse), 1 sec. at W, 0.2 sec. at H and 0.1 sec. at ttbar.
We assume the MB ramp (up+down) time is 5(E-20)/(175-20) seconds, that is 5 sec. at 175 GeV.
Finally, the time between injections to a given bunch (two MB cycles), tC:
Z) 122 sec.
W) 44 sec.
H) 31 sec.
tt) 32 sec.
The minimum allowable lifetime can be estimated as C = tC /, where  is the allowable asymmetry in the bunch currents (3 % corresponds to 0.06). This is very rough estimate, since the lifetime itself depends on asymmetry and it is much smaller for the “weak” bunch.

8. parameter Z W
H (ZH)
ttbar
beam energy [GeV]
45.6 80
120
175
arc cell optics
60/60
90/90
momentum compaction [10-5]
1.48
0.73
horizontal emittance [nm]
0.27
0.28
0.63
1.34
vertical emittance [pm]
1.0
1.3
2.7
horizontal beta* [m]
0.15
0.2
0.3 1
vertical beta* [mm]
0.8 2
length of interaction area [mm]
0.42
0.5
0.9
1.95
tunes, half-ring (x, y, s)
(0.569, 0.61, )
(0.577, 0.61, )
(0.565, 0.60, )
(0.553, 0.59, )
longitudinal damping time [ms]
414 77 23
7.5
SR energy loss / turn [GeV]
0.036
0.34
1.72
7.8
total RF voltage [GV]
0.10
0.44
2.0
9.5
RF acceptance [%]
1.9
2.3
5.0
energy acceptance [%]
1.5
2.5
energy spread (SR / BS) [%]
0.038 / 0.132
0.066 / 0.153
0.099 / 0.151
0.147 / 0.192
bunch length (SR / BS) [mm]
3.5 / 12.1
3.3 / 7.65
3.15 / 4.9
2.45 / 3.25
Piwinski angle (SR / BS)
8.2 / 28.5
6.6 / 15.3
3.4 / 5.3
1.0 / 1.33
bunch intensity [1011]
1.7
no. of bunches / beam
16640
2000
393 48
beam current [mA]
1390
147 29
6.4
luminosity [1034 cm-2s-1]
230 32
1.8
beam-beam parameter (x / y)
0.004 / 0.133
/ 0.118
0.016 / 0.108
0.095 / 0.157
luminosity lifetime [min] 70 50 42 39
time between injections [sec]
122 44 31
allowable asymmetry [%] 5 3
required lifetime by BS [min] 16 11 12
actual lifetime by BS (“weak”) [min]
> 200 20 24

9. Discussion & Next Steps
Since the 3D flip-flop was overcome, the luminosity dependence on the asymmetry is not so strong. For example, 2 % instead of 3 % leads to 45 % increase in the luminosity.
The bunch population Np (and the number of bunches Nb) should be chosen in accordance with the realistic asymmetry we can maintain. Otherwise the “weak” bunch can be lost.
We need feedback from the people responsible for the injection. What are the realistic injection rates for each energy?
Luminosity at Z can be further increased (larger Np, less Nb), but we need feedback from the people responsible for RF, impedances, electron clouds, etc.
Can we ask K. Ohmi to confirm these results for luminosity at all energies?
The next step: beam-beam simulations with nonlinear lattice.