1 Limits of Beam-Beam Interactions Ji Qiang Lawrence Berkeley National Laboratory Joint EIC2006 & Hot QCD Workshop, BNL, July

2 Outline Introduction Experimental observations Physical mechanisms Computational models Validation of computer codes Beam-beam issues in linac-ring colliders Summary

3 Beam Blow-Up during the Collision

4 Beam-Beam Interactions –Limit the peak luminosity –Reduce the beam lifetime –Cause extra background –Large number of particles loss may quench superconducting machine

5 Luminosity and Beam-Beam Parameter

6 First beam-beam limit Saturation of beam-beam parameter Luminosity scales linearly with current Second beam-beam limit Ultimate limit of luminosity Loss of particles and reduce of beam lifetime Beam-Beam Limits

7  & Luminosity vs. Current for e + e - Rings (J. Seeman, 1983) SPEAR CESRPETRA PEP 1 st beam-beam limit (max.  ) 2 nd b-b limit due to tails!

8 Background Noise and Scraper Location vs. Current (J. Seeman, 1983)

9 Luminosity vs. Current Square at PEP-II (J. Seeman et al, 2001)

10 Transverse Size vs. Current Square at PEP-II (J. Seeman et al, 2001)

11 Observation of Flip-Flop at PEP-II: Transverse Beam vs. Bunch Number (R. Holtzapple, et al (2002)

12 Observation of Flip-Flop at PEP-II: Luminosity vs. Bunch Number (R. Holtzapple, et al, 2002)

13 Observation of Flip-Flop at PEP-II: Horizontal Width vs. Snap Picture Number at LER R. Holtzapple, et al (2002)

14 Lepton Beam-Beam Tune shift vs. Proton Current at HERA (F. Willeke 2002)

15 2Q x +2Q y 4Q x 3Q y e+ Beam-Beam Limit at HERA tune footprint appears to be limited by 3 rd & 4 th order resonances beams separated in South IP luminosity in the North increases H1 and Zeus spec. lumi vs time

16 RHIC Working Point and Background (W. Fischer, 2003) Deuteron-gold collisions,  / IP  0.001, 4 head-on collisions Lowest order resonances are of order 9 between 0.2 and 0.25 High background rates near 9 th order resonaces Low background rates near 13 th resonances

17 7 th 5 th 12 th 5 th Contour Plots of Background Halo Rates for Protons and Antiprotons at Tevatron (V. Shiltsev et al 2005)

18 Beam-Beam Parameters in Hadron Accelerators (W. Fischer, 2003)

19 beam energy [GeV] tune shift per IP total tune shift damping decrement per IP LEP x10 -2 KEKB8, x10 -4 PEP-II3.1/ DAFNE Beam-Beam Parameters in Lepton Accelerators (F. Zimmermann, 2003)

20 Physical Mechanisms Collective/Coherent Resonance (Keil 1981, Dikansk and Pestrikov 1982, Chao and Ruth 1985, Hirata 1987, Krishnagopal and Siemann, 1991, Shi and Yao, 2000) –Dipole mode instability –Quadrupole model instability Flip-flop Period n oscillation higher mode instability Blow-up –Higher order modes

21 tune Stability Region of Coherent Dipole Mode (Keil, Chao, Hirata)

22  Max Mode =2 Unstable Max Mode =4 Unstable    Stability Diagram for Coherent Resonance up to 2 and 4 (Chao and Ruth, 1985)

23 IBS Touschek background scattering beam-beam bremsstrahlung orbit noise tune fluctuation quantumn excitation Arnold diffusion resonance overlap resonance trapping collision random fluctuation nonlinear resonance diffusion particle loss Particle Loss due to Incoherent Diffusion

24 Resonance Traping: Particle Transport by Slow Phase Space Topology Change (A. Chao 1979)

25 Resonance overlap: Phase Space Evolution vs. Increasing Beam-Beam Tune Shift (J. Tennyson 1979)

26 Computational Models Weak-Strong –One beam (weak beam) is subject to the electromagnetic fields of the other beam (strong beam) while the effects of weak beam on strong beam are neglected Strong-Strong –The electromagnetic fields from both oppositely rotating beams are included

27 Weak-Strong Model Advantages –Only one electromagnetic field calculation is needed. This model is fast and many macroparticles can be used in tracking studies. –The model is useful for halo/lifetime calculations or some quick machine parameter scan Disadvantages –Sensitive only to incoherent effects –Not self-consistent

28 Strong-Strong Model Advantages –Sensitive to both incoherent and coherent effects –Self-consistently modeling of beam-beam interaction Disadvantages –Electromagnetic fields from each beam have to be calculated at each collision –Computational expensive –Need advanced algorithms and computers

29 Strong-Strong Model Soft-Gaussian model –The particle distribution is assumed as a Gaussian distribution with 1 st and 2 nd moments updated after each collision Self-consistent model –PIC: electromagnetic fields are calculated at each collision point based on the charge distribution on a grid from macroparticle deposition –Direct numerical Vlasov-Poisson solver

30 Particle-In-Cell (PIC) Simulation Advance momenta using radiation damping and quantum excitation map Advance momenta using H beam-beam forces  Field solution on grid  Charge deposition on grid  Field interpolation at particle positions Setup for solving Poisson equation Initialize particles (optional) diagnostics Advance positions & momenta using external transfer map

31 Finite Difference Solution of Poisson’s Equation (S. Krishnagopal, 1996, Y. Cai, et al., 2001) Five point stencil with Fourier analysis by cyclic reduction (FACR) Reduced grid: –Before solving the Poisson equation, the potential on the reduced grid boundary is determined by a Green’s function method –Poisson solver uses FFT and cyclic reduction (FACR) Computational complexity: –Scales as N 2 log(N) within the domain: N – grid number in each dimension –Needs 4N 3 to find the boundary condition

32 Hybrid Fast Multipole Solution of Poisson’s Equation (W. Herr, M. P. Zorzano, F. Jones, 2001) Divided the solution domain into a grid and a halo area Charge deposition with the grid Multipole expansions of the field are computed for each grid point as well as for every halo particle Computation complexity: –Scales as PN 2 or PNp

33 Green Function Solution of Poisson’s Equation (K. Yokoya, K. Oide, E. Kikutani, 1990, E. Anderson et. al. 1999, K. Ohmi, et. al. 2000, J. Shi, et. al, 2000, J. Qiang, et. al. 2002, A. Kabel, 2003) ; r = (x, y) Direct summation of the convolution scales as N 4 !!!! N – grid number in each dimension

34 Green Function Solution of Poisson’s Equation (cont’d) Hockney’s Algorithm:- scales as (2N) 2 log(2N) - Ref: Hockney and Easwood, Computer Simulation using Particles, McGraw-Hill Book Company, New York, Shifted Green function Algorithm: - Ref: J. Qiang, M. Furman, R. Ryne, PRST-AB, vol. 5, (2002).

35 Green Function Solution of Poisson’s Equation Integrated Green function Algorithm for large aspect ratio: - Ref: K. Ohmi, Phys. Rev. E, vol. 62, 7287 (2000). J. Qiang, M. Furman, R. Ryne, J. Comp. Phys., vol. 198, 278 (2004). x (sigma) EyEy

36 Needs for High Performance Computers Number of particles per bunch: –10 10 – Number of turns: – Number of bunches per beam: –

37 Scaling on seaborg using strong-strong model (100Mp, 512x512x32 grid, 4 slices) # of processors Execution time/turn (sec) During the development of BeamBeam3D, several parallelization strategies were tested. The large amount of particle movement between collisions gives the standard approach (domain decomposition, bottom curve) poor scalability for the strong-strong model. A hybrid decomposition approach (top curve) has the best scalability. We are now able to perform 100M particle strong-strong simulation on 1024 processors

38 Stern and Valishev et. al. SciDAC2006 poster Synchrobetaron Mode Tunes vs. Beam-Beam Parameter Measurement vs Simulation (BeamBeam3D)

39 Specific Luminosity vs. beta* at HERA (J. Shi et al, 2003) 

40 Luminosity of a Routine Operation of PEP-II: Measurement vs Simulation (Y. Cai et. al. 2001)

41 Linac-Ring Beam-Beam Interaction Electron beam is re-injected from linac after each turn. This avoids the beam-beam tune shift limit or e-cloud limit to electron intensity inherent in storage ring. Issues: –Beam-beam head-tail instability –Electron disruption

42 Schematic Plot of Synchrobetatron Modes E. Perevedentsev and A. Valishev, PRSTAB, 4, (2001)

43 Synchrobetatron Mode Increments vs. Beam-Beam Parameter (Zero Chromaticity)

44 Threshold Value of D_  + / s vs. Disruption Parameter (R. Li et al, 2001)

45 Synchrobetatron Mode Increments vs. Beam-Beam Parameter (Finite Chromaticity 0.409) ( E. Perevedentsev and A. Valishev )

46 Summary Beam-beam limit has been improved by fine tuning of machine, lepton ~ 0.1, hadron ~ 0.01 per IP. Theoretical models provide a lot of insights to understand beam-beam limits. Computer codes can reasonably reproduce coherent spectrum and luminosity. However, prediction of beam lifetime is still a challenge. Linac-ring collider looks promising but detailed study of beam-beam limits including chromaticity and full 3d nonlinearity is needed.

47 Acknowledgements A. Chao, Y. Cai, W. Fischer, M. Furman, W. Herr, K. Hirata, R. Holtzapple, V. Lebedev, R. Li, L. Merminga, K. Ohmi, E. Perevedentsev, R. Ryne, J. Seeman, J. Shi, C. Siegerist, V. Shiltsev, E. Stern, J. Tennyson, A. Valishev, F. Willeke, F. Zimmermann