2. Transport formalism Taylor map: Third order Linear matrix elementsSecond order matrix elements Truncated maps Violation of the symplectic condition !

3. Lie algebraic treatment Symplectic Maps Lie Transformations Dragt-Finn factorization : [A. Dragt et al., Ann. Rev. Nucl. Part. Sci. 38 (1988) 455] Linear matrix produces T ijk and higher order terms (sextupole effects) produces third order and higher order terms (octupoles effects) Numerical methods for nonlinear optimization : PARTICLE TRACKING, Dynamic aperture scans, particle spectra… generators

4. Tracking codes: Simulations to show the feasibility Optics lattice MAD Placet SAD … Entrance: IP: Multiparticle tracking Beam-beam interaction Guinea-Pig performance [T. Asaka and J. Resta Lopez, CLIC-Note-637] transport Lie Importance of the benchmarking of codes

5. Nanometer-Size Beams in CLIC Beam profile at the IP: Some problems: Residual horizontal dispersion at the IP Nominal: σ x =40.12 nm; σ y =0.55 nm Simulations: σ x ≈47.3 nm; σ y ≈0.65 nm

6. Nanometer-Size Beams in CLIC Phase space at the IP: Particles with lower energy than the nominal one (1500 GeV) contribute strongly to the tails of the transversal phase space

7. Chromatic effects in phase space Chromatic aberrations study by means of tracking from matched initial ellipses at 1σ for the transversal plane X Red line: center ellipse movement in phase space up to third order !

8. Chromatic effects in phase space Chromatic aberrations study by means of tracking from matched initial ellipses at 1σ (figure on the left) and 3σ (figure on the right) for the transversal plane Y The particles at high position amplitude of several sigmas contribute to the population of the long tails. For the case of the ellipses at 3σ in the vertical phase space, it is possible to observe a strong deformation of the shape caused by the sextupoles located in the FFS.

9. Limits of the Luminosity L/L 0Placet Without SRWith SR Tolerable bandwidth up to 1 % energy spread The synchrotron radiation is a very important limitation factor for the luminosity

10. Collimation issues in CLIC

11. Beam-beam effects Simulations with Guinea-Pig: it includes beam-beam effects Disruption parameters: D y = 3.5 (CLIC) D y =19.4 (ILC) Luminosity versus vertical offset Analytic calculation considering a rigid gaussian beam:

12. Input LINAC BDS Beam-Beam Output Placet MatmerlinPlacet Guinea-Pig Simulink FB ILC integrated simulations Updated simulations: G. White version (2005):

13. Ground motion and FB system Nominal: L=2x10 34 cm -2 s -1 85 % of the nominal luminosity