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Limits of Beam-Beam Interactions

Ji Qiang Lawrence Berkeley National Laboratory

Joint EIC2006 & Hot QCD Workshop, BNL, July 17 - 22

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Outline

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

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Beam Blow-Up during the Collision

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Beam-Beam Interactions

– Limit the peak luminosity– Reduce the beam lifetime– Cause extra background– Large number of particles loss may quench

superconducting machine

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Luminosity and Beam-Beam Parameter

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21

22

21

21

2 yyxx

cfNNL

)(2 2221

1201

yxy

yy

Nr

yy

x

y IEL

)1(

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• 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

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& Luminosity vs. Current for e+e- Rings(J. Seeman, 1983)

SPEAR

CESR PETRA

PEP

1st beam-beam limit (max. ) 2ndb-b limit due to tails!

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

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

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

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

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

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Observation of Flip-Flop at PEP-II:Horizontal Width vs. Snap Picture Number at LER

R. Holtzapple, et al (2002)

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

e coh. bb tuneshift vs p Bunch current

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0 100 200 300 400 500 600

Ibp/mA

Df x

,y/k

Hz

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0.15 0.2 0.25 0.30.15

0.2

0.25

0.3

Explored Tune Region0.35

.15

q y i

R p q xi N( )

R m q xi N( )

R x q xi

R y q xi

.350.15 q xi

0.15 0.2 0.25 0.30.15

0.2

0.25

0.3

Explored Tune Region0.35

.15

q y i

R p q xi N( )

R m q xi N( )

R x q xi

R y q xi

.350.15 q xi

2Qx +2Q

y4Qx

3Qy

e+ Beam-Beam Limit at HERA tune footprint appears to be limited by 3rd & 4th order

resonances

beams separated in South IP

luminosity in the North increases

H1 and Zeus spec. lumi vs time

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RHIC Working Point and Background (W. Fischer, 2003)

Deuteron-gold collisions, / IP 0.001, 4 head-on collisions

Lowest order resonances are oforder 9 between0.2 and 0.25

High background ratesnear 9th order resonaces

Low background rates near 13th resonances

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7th 5th12th 12th 5th

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

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

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beamenergy[GeV]

tune shiftper IP

total tuneshift

dampingdecrementper IP

LEP 100 0.083 0.33 1.6x10-2

KEKB 8, 3.5 0.05-0.095

0.05-0.095

2x10-4

PEP-II 3.1/9 0.048-0.075

0.048-0.075

10-4

DAFNE 0.51 0.03 0.03 10-5

Beam-Beam Parameters in Lepton Accelerators(F. Zimmermann, 2003)

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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

21tune

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

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Max Mode =2

Unstable Max Mode =4

Unstable

0 1

0

0.10

0.20

Stability Diagram for Coherent Resonance up to 2 and 4(Chao and Ruth, 1985)

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IBSTouschek

background scattering

beam-beam bremsstrahlung

orbit noise

tune fluctuation

quantumnexcitation

Arnolddiffusion

resonance overlap

resonancetrapping

collision randomfluctuation

nonlinearresonance

diffusion

particle loss

Particle Loss due to Incoherent Diffusion

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

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

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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

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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

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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

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Strong-Strong Model• Soft-Gaussian model

– The particle distribution is assumed as a Gaussian distribution with 1st and 2nd 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

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Particle-In-Cell (PIC) Simulation

Advance momenta using radiation damping and quantum excitation map

Advance momenta using Hbeam-beam forces

Field solution on grid

Charge deposition on grid

Field interpolation at particle positions

Setup for solving Poisson equation

Initializeparticles

(optional)diagnostics

Advance positions & momenta using external transfer map

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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 N2log(N) within the domain: N – grid number in each dimension

– Needs 4N3 to find the boundary condition

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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 PN2 or PNp

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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) ')'()',()( drrrrGr

(ri) h G(rii '1

N

ri' )(ri' )

)log(21),( 22 yxyxG

Direct summation of the convolution scales as N4 !!!!N – grid number in each dimension

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Green Function Solution of Poisson’s Equation (cont’d)

F(r) Gs(r,r')(r')dr'Gs(r,r') G(r rs,r')

c(ri) h Gc(rii '1

2N

ri' )c(ri' )

(ri) c(ri) for i = 1, N

Hockney’s Algorithm:- scales as (2N)2log(2N)- Ref: Hockney and Easwood, Computer Simulation using Particles, McGraw-Hill Book Company, New York, 1985.

Shifted Green function Algorithm:- Ref: J. Qiang, M. Furman, R. Ryne, PRST-AB, vol. 5, 104402 (2002).

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Green Function Solution of Poisson’s Equation

c(ri) Gi(rii '1

2N

ri' )c(ri' )

Gi(r,r') Gs(r,r') dr'

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)

Ey

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Needs for High Performance Computers• Number of particles per bunch:

– 1010 – 1011

• Number of turns:– 109 - 1010

• Number of bunches per beam:– 1 - 1000

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Scaling on seaborg using strong-strong model (100Mp, 512x512x32 grid, 4 slices)

# of processors

Execution time/turn (sec)

16 18864 49256 14.771024 7.18

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

38Stern and Valishev et. al. SciDAC2006 poster

Synchrobetaron Mode Tunes vs. Beam-Beam Parameter Measurement vs Simulation (BeamBeam3D)

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

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

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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

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Schematic Plot of Synchrobetatron ModesE. Perevedentsev and A. Valishev, PRSTAB, 4, 024403 (2001)

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

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

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

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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.

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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