1. Simulation of the FCC-ee lattice with quadrupole and sextupole misalignments
Sergey Sinyatkin

2. Task (Quadrupole misalignment )
Quadrupoles are shifted randomly (Gauss) transversely with Sigma_x,y = 100 um, truncated at 2Sigma.
FF quads and those in V and H chromatic sections (extremely large beta) meanwhile are out of scope.
Q shift distorts CO, vertical dispersion and generates betatron coupling (COD at sextupoles)  vertical emittance degrades.
BPMs (4-5 per betatron wave) and steerers (3-4 per wave) are installed (the same scheme of correction for sextupoles misalignments).
Skew-quads are installed in the FF quads, IR vertical chromatic section and in every hundredth arc quads.
Steerers and skew-quads correct COD, coupling and vertical dispersion. What vertical emittance can be achieved after correction?

4. Correction MADX corrects COD, coupling and Dy.
The process is time consuming, so only 10 samples are available.
All sextupoles and bends are perfectly aligned.
The results are preliminary

5. Q shift and CO
For the 100 um Q shift the closed orbit w/o correction is hardly can be found (but can)
For the 10 um Q shift w/o correction the probability for CO is ~ 50% (presumably MADX problem).
For the 100 um Q shift after correction the CO exists for ~ 30-40% samples.
CO can not be found when the vertical tune moves to the half integer or integer resonance.

6. CO search by iterations
Correction of beam trajectory and betatron phase advance can be done step by step (BPM by BPM) like in beam line (not closed optics) starting from injection point. (very slowly with MADX)
Alternative is closed orbit correction iteratively with misalignments scaled from 0 um to 100 um.

7. CO search vertical tune limits the closed optics
For some seeds MADX failed with CO
E_y/Ex = 0.5-5x10-3
After skew quad correction. Q shift of 100 um (RMS).

8. COD after correction
COD around the ring (10 seeds) 100 um quadrupole shift produces average residue COD  70 um (x) and  50 um (y) after orbit correction

9. COD in the IR (after correction)
COD in the IR. Max horizontal COD  800 um

10. Optics distortion (9 samples) after correction
(βx_err/βx)max = 1.1 (βy_err/βy)max = 1.5
σDx= 1.5 cm σDy= 3.4 mm

11. Eigen mode oscillation (1 sample) after orbit correction
before Skew quads correction
εy/ εx = 1.6 %
after Skew quads correction
εy/ εx = 0.15 %

12. Vertical emittance excitation
CO correction
CO correction +
SQ in the IR +
SQ in the arcs
CO correction +
SQ in the IR

13. Steerers strength
σkickx= 6.6 urad σkicky= 6.5 urad

14. Skew quad strength
σK1sL= 79e-6 m-1

15. Q tilt (coupling)
Quadrupoles rotate randomly (Gauss) with Sigma_tilt = 100 urad, truncated at 2Sigma.
FF quads and those in H/V chromatic sections are not rotated.
Tilt excites vertical dispersion and generates betatron coupling  vertical emittance degrades.
Skew-Quads are installed in the FF quads, IR vertical chromatic section and in every hundredth arc quad.

16. Optics distortion Before SQ correction σDy= 2.6 mm After SQ correction

17. Vertical emittance excitation
Before correction
After correction by SQ
(E_y/E_x ~0.5x10-3)

18. Skew quad strength
σK1sL= 76e-6 m-1

19. Conclusion (Quads)
For the 100 um Q shift w/o correction CO exists for ~few% samples.
After COD correction ~30-40% samples are closed.
To have 100 % CO one has to correct CO and keep the vertical tune far from resonance.
For the 100 um Q shift CO correction by dipole steerers and coupling correction by skew quads provide the emittance ratio less then 0.1%.
100 urad Q tilt gives low betatron coupling ~0.2%.
Being supressed by skew quadrupoles it give the emittance ratio below 0.05%.

20. Task (Sextupoles misalignment )
All sextupoles are shifted randomly (Gauss) transversely with Sigma_x,y = 100 um, truncated at 2Sigma
The shift distorts CO, generates betatron coupling and vertical dispersion  vertical emittance degrades
BPMs (4-5 per betatron wave) and steerers (3-4 per wave) are installed
Skew-quads are installed in the FF quads, IR vertical chromatic section and in every fiftieth arc quads
Steerers and skew-quads correct COD, coupling and vertical dispersion. What vertical emittance can be achieved after correction?

21. COD COD around the ring (200 seeds)
100 um sextupole shift produces average COD  10 um (x) and  20 um (y)

22. COD in the IR
COD in the IR. Max vertical COD  400 um is in FF quad

23. Optics distortion (βx_err/βx)max = 1.7 (βy_err/βy)max = 2.3
σDx= 2.2 cm σDy= 5.2 cm

24. Vertical emittance excitation
Too bad samples. Shall not be counted?
Average: y  10% x

25. Tune spread
dQy ~ 0.22
dQx ~ 0.15

26. Correction COD, coupling and Dy correction was performed by MADX.
The process is time consuming, so not too many samples (10) are available by now.
The results are preliminary

27. COD after correction COD around the ring (10 seeds)
100 um sextupole shift produces average COD  0.1 um (x) and  0.2 um (y) m

28. Optics correction
A random sample before and after COD and coupling correction
(βx_err/βx)max = 1.7 (βy_err/βy)max = 2.3
σDx= 2.2 cm σDy= 5.2 cm
200 samples
(βx_err/βx)max = 1.1 (βy_err/βy)max = 1.9
σDx= 1.2 cm σDy= 0.8 cm
10 samples

29. Vertical emittance after correction
y/x = 310-3

30. Tune spread after correction
dQymax ~ 0.2
dQxmax ~ 0.1
Not big change
Lack of statystics???

31. Steering magnets

32. Example of skew quad correction
εy / εx = 6% -> 0.5 %
N ~ 10 (strong skew quads)

33. Conclusion
100 um misalignment of sextupoles provides unacceptable vertical emittance y/x  10%.
This is recovered to y/x = 310-3 by COD and coupling correction.
Vertical dispersion is reduced by skew quadrupoles (σDy= 5 cm  σDy= 0.8 cm), the some part of emittance blow up is formed by betatron coupling.
Another assumption is that number of samples is not sufficient.