1. The LHC dynamic aperture saga: overview, ideas and recent developments Massimo Giovannozzi CERN – Beams Department Definition and physics/computational issues DA studies for LHC DA vs. time: models Benchmarking with real data Extensions Luminosity evolution model DA measurements Outlook Acknowledgements: A. Bazzani, S. Fartoukh, E. Laface, F. Lang, E. MacLean, W. Scandale, F. Schmidt, E. Todesco, R. Tomás, G. Turchetti, C. Yu

2. Fermilab APT seminar 5/11/2013 Definition and issues - I Massimo Giovannozzi - CERN2 The dynamic aperture is the region in phase in which bounded motion occurs. So far only tracking allows computing the DA of a given system. From a numerical point of view: A volume should be evaluated. This entails a scan over the phase space variables. An appropriate choice of the steps in the variables is required. NB: I will deal with protons -> symplectic dynamics! The tracking has to be long-term

3. Fermilab APT seminar 5/11/2013 Definition and issues - II Massimo Giovannozzi - CERN3 DA computations are CPU intensive. Fast tracking tools are required: Optimised codes (e.g., kick codes) Parallel approach (this is only possible over the initial conditions). As an alternative (maybe a dream…): find a dynamical quantity with a good correlation with DA, but less expensive in terms of CPU. A trade-off between number of turns and number of initial conditions might be possible (e.g., use a dense set of initial conditions iterated a small number of turns). Do not forget stable chaos and intermittency!

4. Fermilab APT seminar 5/11/2013 Definition and issues - III Massimo Giovannozzi - CERN4 Here are some examples of indicators: Lyapunov exponent Tune difference (this indicator triggered several studies on accurate computation of tunes in numerical simulations).

5. Fermilab APT seminar 5/11/2013 DA studies for LHC - I Massimo Giovannozzi - CERN5 The dynamic aperture studies for the LHC absorbed a lot of resources for about two decades in terms of: Theoretical studies in non-linear beam dynamics Software and analysis tools Specification magnetic field quality tables (with iterations with the magnet builders) The large amount of information gathered during the measurement stage of the magnets (warm and cold conditions) Special tools developed (e.g., WISE) This allows: Estimating the DA of the LHC as-built Estimating the impact of sorting on DA of the LHC as-built Generate realistic realisations of the magnetic field errors, based on Actual slot allocationActual slot allocation Actual field qualityActual field quality

6. Fermilab APT seminar 5/11/2013 DA studies for LHC - II Possibility to evaluate the DA for the machine as-built Possibility to evaluate impact of sorting on DA Massimo Giovannozzi - CERN6 Summary of DA at injection energy. The error bars represent the effect of 60 seeds Summary of DA at injection energy. The error bars represent the effect of 60 seeds

7. Fermilab APT seminar 5/11/2013 DA studies for LHC - III Possibility to evaluate the DA for the machine as-built Possibility to evaluate impact of sorting on DA Massimo Giovannozzi - CERN7 Summary of minimum DA for several running configurations.

8. Fermilab APT seminar 5/11/2013 DA studies for LHC - IV Possibility to evaluate the DA for the machine as-built Possibility to evaluate impact of sorting on DA Massimo Giovannozzi - CERN8 Impact of sorting. Selected generic seedsSelected generic seeds Each sequence of errors is re-ordered.Each sequence of errors is re-ordered. The various dynamical quantities are computed.The various dynamical quantities are computed. Yellow: all seeds (initial and re-ordered) Blue: selected seeds. Impact of sorting. Selected generic seedsSelected generic seeds Each sequence of errors is re-ordered.Each sequence of errors is re-ordered. The various dynamical quantities are computed.The various dynamical quantities are computed. Yellow: all seeds (initial and re-ordered) Blue: selected seeds.

9. Fermilab APT seminar 5/11/2013 DA studies for LHC - V Possibility to evaluate the DA for the machine as-built Possibility to evaluate impact of sorting on DA Massimo Giovannozzi - CERN9 Impact of sorting. Selected generic seedsSelected generic seeds Each sequence of errors is re-ordered.Each sequence of errors is re-ordered. The various dynamical quantities are computed.The various dynamical quantities are computed. Yellow: all seeds (initial and re-ordered) Blue: selected seeds. Red: average DA for as- built machine. Impact of sorting. Selected generic seedsSelected generic seeds Each sequence of errors is re-ordered.Each sequence of errors is re-ordered. The various dynamical quantities are computed.The various dynamical quantities are computed. Yellow: all seeds (initial and re-ordered) Blue: selected seeds. Red: average DA for as- built machine.

10. Fermilab APT seminar 5/11/2013 DA vs. time: models - I Massimo Giovannozzi - CERN10 Another strategy could be: is there a model to describe DA vs. time? In mathematical sense DA does not depend on time. Numerical simulations are performed with a specific maximum number of turns (N max ): the computed DA does depend on N max How does DA depend on N max in numerical simulations? ). Studies have been performed recently to review the functional dependence on  of fit model

11. Fermilab APT seminar 5/11/2013 Definition and issues - VII Massimo Giovannozzi - CERN11 Dynamic aperture of a model of the LHC ring (left) in physical space: The red points represent the initial conditions stable up to 10 5 turns The blue points represent unstable conditions and their size is proportional to the number of turns by which their motion is still bounded. The time-evolution of the DA is shown on the right. The markers represent the numerical results The continuous line shows the fitted inverse logarithmic law. The dotted line represents D 

12. Fermilab APT seminar 5/11/2013 DA vs. time: models - III Massimo Giovannozzi - CERN12 Is this a purely phenomenological fit? In fact not quite. The physical picture is: For r < D  The motion is governed by KAM theorem. Fully stable region (only Arnold diffusion for a set of initial conditions of small measure -> irrelevant from the physical point of view). For r > D  The motion follows Nekhoroshev theorem, i.e., the stability time N(r) of a particle at radius r is given by This provides a pseudo-diffusion.

13. Fermilab APT seminar 5/11/2013 DA vs. time: models - IV Massimo Giovannozzi - CERN13 Two regimes found in 4D simulations: D , b,  are always positive. This implies a stable region for arbitrary times. In 4D simulations with tune ripple or 6D simulations: There could be situations in which no stable region for arbitrary times exists. This corresponds to

14. Fermilab APT seminar 5/11/2013 DA vs. time: models - V Massimo Giovannozzi - CERN14 Fit of DA vs. time can lead to a number of extensions: Losses in hadron machines due to non-linear effects (single particle).

15. Fermilab APT seminar 5/11/2013 Benchmarking with real data – I Tevatron data: proton bunch at injection Estimates from purely diffusive model included. Estimates from purely diffusive model included. Massimo Giovannozzi - CERN 15 Nice agreement for all models! Experimental data from: T. Sen et al. “Beam Losses at Injection Energy and During Acceleration in the Tevatron”, IPAC03, p. 1754.

16. Fermilab APT seminar 5/11/2013 Benchmarking with real data – II SPS data: proton bunch at 55 GeV in coast Estimates from purely diffusive model included. Estimates from purely diffusive model included. Massimo Giovannozzi - CERN 16 Negative second order derivative cannot be reproduced by diffusive models!

17. Fermilab APT seminar 5/11/2013 Extensions - I Massimo Giovannozzi - CERN17 Evolution of DA in presence of beam-beam effects. N b =0.10×10 11 N b =1.15×10 11 N b =1.70×10 11 The proposed model holds for: -Non-linear single particle dynamics -Weak-strong beam-beam It is then tested on luminosity data from LHC.

18. Fermilab APT seminar 5/11/2013 Extensions – II Intensity and luminosity evolution during physics fills Massimo Giovannozzi - CERN18 LHC Tevatron

19. Fermilab APT seminar 5/11/2013 Luminosity evolution models - I The inverse logarithm model seems to fit well al data considered so far (LHC and other circular accelerators/colliders). For luminosity evolution: No consideration of, e.g., burn off effect: an “effective” fit has been considered and proved to work well, but the fit parameters might have little physical content. A correct approach would require disentangling pseudo- diffusive effects (inverse logarithm) from the rest. The boundary conditions: try to find a relatively simple model to allow analytical considerations. LHC Run I provided lots of data to probe new models… Massimo Giovannozzi - CERN19

20. Fermilab APT seminar 5/11/2013 Luminosity evolution models - II What do we know from the LHC Run I Massimo Giovannozzi - CERN20 2011 LHC data. The two curves refer to the change of  * occurred during the year.

21. Fermilab APT seminar 5/11/2013 Luminosity evolution models - III Is it possible to factor out the contribution of the pseudo-diffusive effects? Is it possible to normalise the data to find a “sort” of universal (at least for LHC) curve? The answer is positive: For the time being only the plain proton burn off has been included. Emittance evolution (e.g., radiation or rest gas interaction) can be included. Massimo Giovannozzi - CERN21

22. Fermilab APT seminar 5/11/2013 Luminosity evolution models - IV Massimo Giovannozzi - CERN22 2011 and 2012 LHC data. Dashed line: burn off only Blue squares: proposed model.

23. Fermilab APT seminar 5/11/2013 DA measurements - I What is the DA of the real machine? No lifetime problems or slow losses at injection. During aperture measurements (with beams probing high amplitudes) no sign of slow losses was found. This observation indicates that DA should be of the same order of mechanical aperture, i.e., about 10 . Measurement campaign launched: Two MD sessions organised (2011, 2012). Objective: benchmark numerical simulations against measurements (e.g., for HERA a factor of two was found). Massimo Giovannozzi - CERN23

24. Fermilab APT seminar 5/11/2013 DA measurements - II Two strategies applied: Beam 1: Blow up the beam until slow losses are observed. Record evolution of beam intensity. Fit beam intensity with proposed models. Compare with fit parameters from numerical simulations of DA. Beam 2: Kick the beam in order to push it towards high amplitudes until large losses are obtained (standard method). Requires rather strong kick (aperture kicker). Compare amplitude of beam losses with numerical simulations of DA. Massimo Giovannozzi - CERN

25. Fermilab APT seminar 5/11/2013 DA measurements: Beam 1 - III Two MD sessions: 2011 Almost half of the time lost… Q-kicker used to blow up the beam emittance: not easy to induce a symmetric blow up. Performed scans over the strength of MCOs using the same polarity in all sectors. 2012 Much more efficient MD… Transverse damper used to blow up the beam emittance: easy and reproducible blow-up. Performed scans over MCOs and MCDs. Used different alternating signs schemes (suggested by Stephane). Massimo Giovannozzi - CERN Strong chromatic effects Almost cancelled chromatic effects (but symmetry broken

26. Fermilab APT seminar 5/11/2013 DA measurements: Beam 1 – IV 2011 MD Massimo Giovannozzi - CERN 26

27. Fermilab APT seminar 5/11/2013 DA measurements: Beam 1 – V MD 2011 Massimo Giovannozzi - CERN 27 Effect of Q- kicker: large tails instead of a Gaussian beam Beam size evolution during the MD

28. Fermilab APT seminar 5/11/2013 DA measurements: Beam 1 - VI MD 2011 Massimo Giovannozzi - CERN I MCO =-40 A Another confirmation of the scaling law of intensity vs. time

29. Fermilab APT seminar 5/11/2013 Digression: simulations for Beam 1 DA measurements in 2011 Massimo Giovannozzi - CERN 29

30. Fermilab APT seminar 5/11/2013 DA measurement: Beam 1 – VII MD 2012 Massimo Giovannozzi - CERN 30 Same configuration as 2011 Scan MCDs Scan MCOs (alternating signs)

31. Fermilab APT seminar 5/11/2013 Digression: simulations for Beam 1 DA measurements in 2012 Massimo Giovannozzi - CERN 31 Negative Dinf found: all phase space unstable! MCO circuit in sector 1-2 was not available: simulations to be repeated and broken symmetry.

32. Fermilab APT seminar 5/11/2013 DA measurements: Beam 2 - VIII MD 2011-12 Massimo Giovannozzi - CERN Courtesy R. Tomás et al.

33. Fermilab APT seminar 5/11/2013 Conclusions - I Massimo Giovannozzi - CERN33 The LHC project has triggered several studies in non- linear beam dynamics. Several tools and techniques have been devised to achieve the goal of estimating the dynamic aperture. Latest proposals based on DA vs. time models Intensity vs. time Luminosity vs. time Luminosity models Next step: try to find a diffusive model providing an equivalent behaviour of the DA vs. time All confirmed by the analysis of available measured data.

34. Fermilab APT seminar 5/11/2013 Conclusions - II Massimo Giovannozzi - CERN34 The work done seems to have been fruitful: the LHC does not suffer from any single-particle non-linear effects. DA measurements have been performed and data analysis is in progress. HL-LHC will require even more sophisticated tools in view of new physics challenges!

35. Fermilab APT seminar 5/11/2013 Thank you for your attention Massimo Giovannozzi - CERN35