1. SuperB Dynamic Aperture A. Bogomyagkov, E. Levichev, P
SuperB Dynamic Aperture A.Bogomyagkov, E.Levichev, P.Piminov BINP, Novosibirsk 16 June 2009, Perugia

2. Content: 1. Correction of the high order aberrations in the – I transformation broken by the finite sextupole length 2. Theory 3. New approach 4. Simulation example 5. Conclusion

3. Y chromatic correction section (SuperB LER)

4. Sextupoles with finite length reduce DA
(1) Vertical chromatic sextupoles in IR limit both vertical and horizontal dynamic apertures (vertical and coupling terms)
(2) The main reason is the –I transformation break due to the non-zero length of the realistic sextupoles
(3) For the kick sextupoles with the exactly tuned –I phase advance in between, the dynamic aperture is infinitely large

5. -I recover (mechanism)
Pavel Piminov empirically
It was found that any distortion of the –I transformation can be recovered by an additional correction sextupole pair placed in some specific location relative to the main chromatic pair
The strength of the correction sextupoles is <10% of the main ones
The nature of this mechanism is not clear and further investigation is required (23 April 2009)

6. -I recover (results for the vertical direction)
Correction sextupoles with the 5% strength and opposite sign restore the –I transformation in the vertical IR chromatic section and increase the dynamic aperture from DA  30 x 80 (x×y) to DA  90 x 550 (x×y)

7. Study of the –I transformation restore
At this point we found that in the LER SuperB lattice an adjustment of the –I transformation is not perfect and in order to exclude this effect from the finite length sextupole study, we perform work below for the Tau-Charm Factory IR, which arrangement is close to the SuperB LER IR but with better tuning of –I.
When the SuperB lattice will be completed (more or less) we shall do the DA optimization for it also.

8. DA vs. sextupole length
Quadratic nonlinearity
Cubic nonlinearity

9. Theoretic background - binomial coefficient
Anton Bogomyagkov
- binomial coefficient
This solution is similar to the Lie series expansion but uses power series
instead of the exponential ones. The solution is symplectic if not truncated.

10. Solution for the finite length sextupole pair
Third order aberration of the sextupole pair –I
Pure octupole terms
Third order aberration of the thick sextupole pair with –I transformation
in between is similar to the octupole term but not identical (signs and
coefficients are different). So it could not be perfectly compensated
by octupoles.

11. 2+2 sextupoles (theory) l l
If the sextupoles are located on the one side of quadrupole
one set of coefficients can be zeroing and the other is reduced

12. 2+2 sextupoles main corr 8 cubic aberration terms are produced
Piminov’s empiric config. –I
Theory: 2 terms can be zero exactly and other 6 are reduced –I
Bogomyagkov’s theory
Predics better results
Theory: 4 terms can be zero exactly and other 4 are reduced –I –I
Pair of the correction sextupoles increases the on-energy DA substantially. The strength of the corr. sexts is 3-10% of the main ones.
No quadratic aberration terms appear
No influence on the nonlinear dispersion

13. 2+1 sextupoles 8 cubic aberration terms are produced
Again 4 terms can be zero exactly and other 4 are reduced but small quadratic terms (~S2L) appear

14. Simulation examples for CTF
The vertical chromatic section only
Black: main sextupoles with –I but finite length
Red: +correction sextupoles across the quad to the main ones (iniital approach): the vertical aberrations are cancelled but the coupling are remained
Blue: +correction sextupoles on the side of the main ones: better cancellation of the aberration terms (vertical and coupling)
Magenta: additional small numerical optimization

15. IR sextupoles optimization
LEB DA with the IR sextupoles only (no arc sextupoles and other nonlinearities)
Black: before optimization
Red: after optimization

16. LER DA tune scan
The tune point optimization should be done together with the beam-beam simulation and the luminosity/lifetime optimization
Before the IR sextupoles optimization
After the IR sextupoles optimization

17. Summary
Sextupoles in the IR chromatic sections limit dynamic aperture due to the high aberrations (mainly the third order) of the –I transform
Low strength (~3-10% of the main one) correction sextupole(s) can recover the dynamic aperture
The theory predicting the strength and location of the correction sextupoles are developed
Off-energy dynamic aperture ???
Correction octupoles instead of the correction sextupoles ???