Study of Beam-Beam Limit in Hadron Colliders

Yunhai Cai and Robert WarnockBeam Physics Department

SLAC

LARP collaboration meeting at SLACOctober 18, 2007

parameter symbolEarly Separation Large Piwinski Angle

transverse emittance [m] 3.75 3.75

protons per bunch Nb [1011] 1.7 4.9

bunch spacing t [ns] 25 50

beam current I [A] 0.86 1.22

longitudinal profile Gauss Flat

rms bunch length z [cm] 7.55 11.8

beta* at IP1&5 [m] 0.08 0.25

full crossing angle c [rad] 0 381

Piwinski parameter cz/(2*x*) 0 2.0

hourglass reduction 0.86 0.99

peak luminosity L [1034 cm-2s-1] 15.5 10.7

peak events per crossing 294 403

initial lumi lifetime L [h] 2.2 4.5

effective luminosity (Tturnaround=10 h)

Leff [1034 cm-2s-1] 2.4 2.5

Trun,opt [h] 6.6 9.5

effective luminosity (Tturnaround=5 h)

Leff [1034 cm-2s-1] 3.6 3.5

Trun,opt [h] 4.6 6.7

e-c heat SEY=1.4(1.3) P [W/m] 1.04 (0.59) 0.36 (0.1)

SR heat load 4.6-20 K PSR [W/m] 0.25 0.36

image current heat PIC [W/m] 0.33 0.78

gas-s. 100 h (10 h) b Pgas [W/m] 0.06 (0.56) 0.09 (0.9)

extent luminous region l [cm] 3.7 5.3

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Beam-Beam Effects in Hadron Machine

• Beam-Beam lifetime (T. Sen, A. Kabel )– Parasitic collision– Wire compensation (W. Fischer)– Electron lens compensation (V. Shiltsev)

• Emittance growth due to beam-beam collisions – Strong-weak, gives too small values– Strong-strong

• Gives unreliable values due the numerical noise• Predicts a beam-beam limit that is a factor of ten larger than

in the electron machine

Solve Poisson Equation with Reduced Region

• Assign potential on the reduced boundary:

• Solve Poisson’s equation with inhomogeneous boundary condition

At the collision point:

)','()','(''),( yxyyxxGdydxyx c

High resolution is achieved tocompute the beam distributionat the core of the beam

The solution is exact because of the uniqueness of the solution

Crossing Experiments at PEP-II• Simulation was carried out prior to

the experiments to make sure there was enough sensitivity.

• ‘By-4’ bunch pattern to avoid parasitic collision (30 x

- separation).• The orbit bump used to change the

angle. The knob was carefully calibrated against a pair of BPMs next to the IP.

• Luminosity feedbacks were on to align beams transversely after each change.

• Tune changes were necessary to compensate the optical errors introduced from the nonlinearity of the fringe field and magnets inside the bumps.

0.75

0.80

0.85

0.90

0.95

1.00

1.05

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

Half crossing-angle c (mrad)

Lsp

/ L

sp (

c=0)

Sim: 0.16/0.09 mA/b

Sim: 1.35/0.85 mA/b

by-4, 1.43/0.89 mA/b

by-4, fit

geometric

min16

5125.0

x

min1

5081.0

x

min

5152.0

x

min100

5142.0

x

Improving Strong-Strong Codes

• Numerical noise in PIC codes– Macro particle representation of density– Discrete Poisson solver

• Vlasov-Poisson approach to eliminate the noise due to macro particles– 1D code for microwave instabilities was proven very effective– 2D beam-beam code was developed by Andrey Sobol. Ported to computers at SLAC

• A new idea to solve Vlasov, with some advantages of PIC method (lower cost , while keeping low noise)– Forward tracking (similar to macro particles)– Smoothing by interpolation of data at quasi-random sites– Probability conserving algorithm at data sites

Goal and Plan

• Compare noise between PIC and Vlasov codes and quantify any improvements

• To understand emittance growth due to beam-beam in hadron machines at least in relative terms within two years

• Benchmark the codes against experiments in Tevatron and RHIC, understand the emittance growth at 10% level within five years and compare to the result from LHC

• Resource: – 2 FTE for two years possible extension to five years– At least one new FTE (post-doctoral)– Maybe a dedicated cluster later