luminosity and beam-beam effects in the large hadron ......the large hadron collider (lhc) tatiana...
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Luminosity and Beam-Beam Effects in the Large Hadron Collider (LHC)
Tatiana PieloniLaboratory of Particle Accelerator Physics Ecole Politechnique Federal de Lausanne
CAS@ESI: Basics of Accelerator Physics and Technology, 7-11 October 2019, Archamps, France
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Circular Accelerators: acceleration occurs at every turn!
E. Lawrence 1930
Target
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Circular Accelerators: acceleration occurs at every turn!
Target
Two Beams of
Beam 2 is a Target
7 TeV proton beam against fix target 115 GeV
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Colliders: higher energy
Anello di Accumulazione AdAB. Touschek 1960
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Colliders: higher energy
Anello di Accumulazione AdAB. Touschek 1960
Two Beams of
Beam 2 is a counter rotating beam
7 TeV proton beam colliding 14 TeV
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The Large Hadron Collider
27 Km lengthProtons or heavy ionsMaximum 14 TeV center of mass energy4 Interaction Regions for Experiments
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Circular colliders: Luminosity
Collider Luminosity
is the proportionality factor between
the cross section
and the number of events per second
Luminosity is a machine parameterIndependent of the physical reactionReliable procedure to compute and measure
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Luminosity calculation
The overlap integral of two bunches crossing each other head-on is proportional to the luminosity and it is given by:
Kinematic Factor
Time variable
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Luminosity formula
Uncorrelated densities in all planesFactorize the distribution density as:
For head-on collisions where “Kinematic Factor” K = 2To have the luminosity per second Needs to multiple by revolution frequency fIn the presence of many bunches nb
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Closed solution for Gaussian distributionsSimplest case assumptions:
• Gaussian distributions
• No dispersion at the collision point
• Head-on collision
Equal Transverse beams “Round” beams
Un-Equal Transverse beams “Flat” beams or optics
K = 2
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The LHC design parameters
LHC DesignN1 = N2 =1.15 1011 protons per bunchsx = sy =16.6 mmb* = 55 cm
L=1034 cm-2s-1
LHC RecordN1 = N2 =1.15 1011 protons per bunchsx = sy =9.5 mmb* = 30 cm
L=2 x 1034 cm-2s-1
High Luminosity Upgrade of LHCN1 = N2 =2.2 1011 protons per bunchsx = sy =7.0 mmb* = 6415 cm
L=(10-20) x 1034 cm-2s-1
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The LHC design parameters
LHC DesignN1 = N2 =1.15 1011 protons per bunchsx = sy =16.6 mmb* = 55 cm
L=1034 cm-2s-1
LHC RecordN1 = N2 =1.15 1011 protons per bunchsx = sy =9.5 mmb* = 30 cm
L=2 x 1034 cm-2s-1
High Luminosity Upgrade of LHCN1 = N2 =2.2 1011 protons per bunchsx = sy =7.0 mmb* = 6415 cm
L=(10-20) x 1034 cm-2s-1
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Different types of collisions
They occur when two beams get closer and collide
Two types
High energy collisions between two particles (wanted)Distortions of beam by electromagnetic forces (unwanted)
Unfortunately: usually both go together…0.001% (or less) of particles collide 99.999% (or more) of particles are distorted
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Proton Beams Electro Magnetic potential
Beam is a collection of charges Beam is an electromagnetic potential for other charges
Force on itself (space charge) and opposing beam (beam-beam effects)
Focusing quadrupole Opposite Beam
A beam acts on particles like an electromagnetic lens, but…
Single particle motion and whole bunch motion distorted
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Beam is a collection of charges Beam is an electromagnetic potential for other charges
Force on itself: space chargeeffects goes with 1/g2 factor for high energy colliders this contribution is negligible (i.e. force scales LHC 1/g2 = 1.8 10-8)
Focusing quadrupole Opposite Beam
A beam acts on particles like an electromagnetic lens, but…
Proton Beams Electro Magnetic potential
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Beam is a collection of charges Beam is an electromagnetic potential for other charges
Electromagnetic force from opposing beam (beam-beam effects)
Focusing quadrupole Opposite Beam
A beam acts on particles like an electromagnetic lens, but…
Single particle motion and whole bunch motion distorted
Proton Beams Electro Magnetic potential
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Beam-beam Force derivationGeneral approach in electromagnetic problems Reference[5] already applied to beam-beam interactions in Reference[1,3, 4]
Derive potential from Poisson equation for charges with distribution r
Then compute the fields
Solution of Poisson equation
From Lorentz force one calculates the force acting on test particle with charge q
Making some assumptions we can simplify the problem and derive analytical formula for the force…
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Beam-Beam Force for Round Gaussian distributions
Gaussian distribution for chargesRound beams: Very relativistic, Force has only radial component :
Beam-beam kick obtained integrating the force over the collision (i.e. time of passage)
Only radial component in relativistic case
Beam-beam Force
How does this force looks like?
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Beam-beam Force
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Beam-beam Force
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Why do we care?
Strongest non-linearity in a collider YOU CANNOT AVOID!
Pushing for luminosity means stronger beam-beam effects
Strong non-linear
electromagnetic
distortion impact on
beam quality (particle
losses and emittance
blow-up)
Physics fill lasts for many
hours 10h – 24h
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Crossing angle operation
Num. of maximum bunches
Head-On
3.7 m
A finite crossing angle has to be applied to avoid multiple collision points
Multi Bunch operations brings un-wanted interactions left and right of the 4 Experiments
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Crossing angle operation and beam-beam interactions
Head-On
Long range
Beam-beam force
Head-on
Long-range
Two type of interactions:Other beam passing in the center force HEAD-ON beam-beam interaction LHC has 4 corresponding to the 4
experiments ATLAS, CMS, Alice, LHCb
Other beam passing at an offset rLONG-RANGE beam-beam interactionLHC has up to 120 LR interactions
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Multiple bunch Complications
Num. of bunches :
Head-On
3.7 m
Due to the train structure of the beams different bunches will experience a different number of interactions!
PACMAN BUNCH
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Long-Range separations
Num. of bunches :
Multi Bunch operations brings un-wanted interactions left and right of the 4 Experiments
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Luminosity Geometric reduction factor
S is the geometric reduction factor
LHC design: f = 285 mrad, sx = 17 mm, ss = 7.5 cm, S=0.84LHC 2018: f = 320 mrad, sx = 9.3 mm, ss = 7.5 cm, S=0.61
Due to the crossing angle the overlap integral between the two colliding bunches is reduced!
Always valid for LHC and HL-LHC sx = 17-7 mm, ss = 7.5 cm
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Luminosity Geometric reduction factor
S is the geometric reduction factor
LHC design: f = 285 mrad, sx = 17 mm, ss = 7.5 cm, S=0.84LHC 2018: f = 320 mrad, sx = 9.3 mm, ss = 7.5 cm, S=0.61
Due to the crossing angle the overlap integral between the two colliding bunches is reduced!
Always valid for LHC and HL-LHC sx = 17-7 mm, ss = 7.5 cm
LHC
TD
R
LHC
To
day
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LHC operates at finite crossing angle
HL-LHC will have bunches of 2.2 1011 protons per bunchf = 590 mrad, sx = 9.3 mm, ss = 7.5 cm, S=0.26 73% of luminosity lost!
Crab Cavities used to tilt the bunches longitudinally and compensate for the crossing angle at the collision point!
Testing of crab cavities on-going in SPS!
Courtesy of R. Calaga
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Beam-Beam Force: single particle head-on collision
For small amplitudes: linear force
For large amplitude: very non-linear
The beam will act as a strong non-linear electromagnetic lens!
Lattice defocusing quadrupole Beam-beam force
Linear force
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Beam-Beam transverse kickGaussian distribution for chargesRound beams: Very relativistic, Force has only radial component :
Beam-beam kick obtained integrating the force over the collision (i.e. time of passage)
Only radial component in relativistic case
Radial deflection on single particle at r from the center of opposite beams
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Can we quantify the beam-beam strenght?
Beam-beam force
For small amplitudes: linear force
The slope of the force gives you the beam-beam parameter
Quantifies the strength of the force but does NOT reflect the nonlinear nature of the force
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Beam-Beam Parameter
For small amplitudes: linear force
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Beam-Beam parameter:
Parameters LHC TDR LHC 2012
Intensity Np,e/bunch 1.15 1011 1.8 1011
Energy GeV 7000 4000
Beam size H 16.6 mm 16.6 mm
Beam size V 16.6 mm 16.6 mm
bx,y* m 0.55-0.55 0.55-0.55
Crossing angle mrad 290 285
xbb 0.0037 0.007
For round beams: For non-round beams:
Examples:LHC 2012
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Parameters LEP (e+e-) LHC(pp)
Intensity Np,e/bunch 4 1011 1.15 1011
Energy GeV 100 7000
Beam size H 160-200 mm 16.6 mm
Beam size V 2-4 mm 16.6 mm
bx,y* m 1.25-0.05 0.55-0.55
Crossing angle mrad 0 285
xbb 0.07 0.0037
For round beams: For non-round beams:
Examples:
LHC 2012
1.7 1011
4000
18 mm
18 mm
0.6-0.6
290
0.009
Beam-Beam parameter:
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Linear Tune shift due to head-on collision
For small amplitude particles beam-beam can be approximated as
linear force as a quadrupole
Focal length is given by the beam-beam parameter:
Beam-beam matrix:
Beam-beam force
Equivalent to tune shift
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Perturbed one turn matrixFor small amplitudes beam-beam can be approximated as linear
force as a quadrupole
Focal length:
Beam-beam matrix:
Perturbed one turn matrix with perturbed tune DQ and beta function at the IP b*:
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Tune shift and dynamic beta
Solving the one turn matrix one can derive the tune shift DQ and the perturbed beta function at the IP b*:
Tune is changed
…how does the tune changes?
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Tune shift due to beam-beam interactionsTune shift as a function of tune
Larger x Strongest variation with Q
LHC design
HiLumi LHC 3 IPs
HL-LHC
Effects of multiple Interaction Points does not add linearly(phase advance between IP..)
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Linear head-on Tune shift
Tune shift in 2 dimensional case equally charged beamsand tunes far from integer and half
Zero amplitude particle will fill an extra defocusing term
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A beam is a collection of particles
Beam-beam force
Tune shift as a function of amplitude (detuning with amplitude or
tune spread)
Beam 2 passing in the center of force produce by Beam 1Particles of Beam 2 will experience different ranges of the beam-beam forces
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A beam will experience all the force range
Beam-beam force
Different particles will see different force
Beam-beam force
Second beam passing in the centerHEAD-ON beam-beam interaction
Second beam displaced offsetLONG-RANGE beam-beam interaction
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Detuning with Amplitude for head-on Instantaneous tune shift of test particle when it crosses the other beam is related to the derivative of the force with respect to the amplitude
For small amplitude test particle linear tune shift
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Beam with many particles this results in a tune spread
Mathematical derivation in Ref [3] using Hamiltonian formalism and in Ref [4] using Lie Algebra
Detuning with Amplitude for head-on
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Head-on detuning with amplitude1-D plot of detuning with amplitude for opposite and equally charged beams
And in the other plane? THE SAME DERIVATION
Maximum tune shift for small amplitude particlesZero tune shift for very large amplitude particles
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Head-on detuning with amplitude and footprints 1-D plot of detuning with amplitude
FOOTPRINT2-D mapping of the detuning with
amplitude of particles (1,0)
(2,0)
(3,0)(4,0)
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Long Range detuning with amplitude1-D plot of detuning with amplitude for opposite and equally charged beams
Maximum tune shift for large amplitude particlesSmaller tune shift detuning for zero amplitude particles and opposite sign
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2-D Long Range detuning with amplitude
Tune shift as a function of separation in horizontal planeIn the horizontal plane long range tune shiftIn the vertical plane opposite sign!
Long range tune shift scaling for distances
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Beam-beam tune shift and tune spread
Footprints depend on:
• number of interactions (124 per turn)• Type (Head-on and long-range)• Separation• Plane of interaction
Very complicated depending on collision scheme
Pushing luminosity increases this area while we need to keep it small to avoid resonances and preserve the stability of particles
Strongest non-linearity in a collider
(0,6)
(0,0)
(0,6)
Head-on and Long range interactions detuning with amplitude
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Beam-beam tune shift and spread
Higher Luminosity increases this area We need to keep it small to avoid resonances and preserve the long term stability of particles
Qx
Qy
(0,6)
(0,0)
(0,6)
The footprint from beam-beam sits in the tune diagram
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LHC Footprints and multiple experiments
LHC 2012
ATLAS+CMS+LHCb
ATLAS+CMS+LHCb+ALICE(LR)
ATLAS+CMS
LHCbLHCb+ALICE(LR)
…operationally it is even more complicated!…different intensities, emittances…
Luminosity Evolution
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Dynamical Aperture and Particle LossesDynamic Aperture: area in amplitude space with stable motionStable area of particles depends on beam intensity and crossing angle
Stable area depends on beam-beam interactions therefore the choice of running parameters (crossing angles, b*, intensity) is the result of
careful study of different effects!
Frequency Map Analysis
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Dynamical Aperture and Particle Losses
Stab
le A
rea
(s)
Beam-beam linear dependency with Intensity
Our goal: keep dynamical aperture above 6 s all particles up to 6 samplitude not lost over long tracking time (106 turns in simulation)
equivalent to 1 minute of collider
Example collider collision time : 24 hours
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53
Round optics15 cm, 590μrad: intensity scan
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54
Round optics15 cm, 590μrad: intensity scan
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Round optics15 cm, 590μrad: intensity scan
AT high intensity the beam-beam force gets too strong and makes particles unstable and eventually are lost
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56
Round 15cm, 2.2E11, 690μrad
Smaller beam-beam separation at parasitic long-range encounters stronger non linearities smaller dynamical aperture
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57
Round 15cm, 2.2E11, 650μrad
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58
Round 15cm, 2.2E11, 590μrad
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59
Round 15cm, 2.2E11, 540μrad
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60
Round 15cm, 2.2E11, 490μrad
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61
Round 15cm, 2.2E11, 440μrad
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Round 15cm, 2.2E11, 390μrad
Crossing angle changes the separation and the strength of BB-LR that strongly affect the tails. 0σ particle are almost not affected.
At small separation particles gets unstable and eventually lost
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dsep = 6 s
How does it look like in the LHC?
Particle losses follow number of Long range interactions
Small crossing angle = small separation
If separation of long range too small particles become unstable
and are lost proportionally to the number of long range
encounters
Beam-Beam separation at first LRRelative intensity decay 2012 experiment
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Do we see the particle losses?
Particle losses follow number of Long range interactionsMachine protection implication and beam lifetimes gets worse…
Best peformance of collider always a trade off between beam-beam and luminosity
Small crossing angle = small separation
Luminosity decays following the long range numbers… higher
number of long range interactions larger losses
Beam-Beam separation at first LRRegular Physics Fill of 2012 RUN LHC
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Long-range Beam-Beam effects: orbit
Long Range Beam-beam interactions lead to several effects…
In simple case (1 interaction) one can compute it analytically
Long range angular kick
For well separated beams
The force has several components at first order we have an amplitude independent contribution: ORBIT KICK
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Orbit effect as a function of separation
H in-plane scan
Closed Orbit effect:
Angular Deflections:
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Orbit effect as a function of separation
Closed Orbit effect:
Angular Deflections:
Orbit can be corrected but we should remember PACMAN effects
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LHC orbit effects
d =
0 –
0.2
un
its
of
be
am s
ize
Many long range interactions could become important effect!Holes in bunch structure leads to PACMAN effects this cannot be
corrected!
1-2% Luminosity loss due to beam-beam orbit effects
Self consistent evaluation
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…not covered here…
• Beam-Beam compensation schemes
• Landau damping and beam-beam
• Beam-Beam coherent effects
• Asymmetric beams effects
• Noise on colliding beams
• Van der Meer scans
• Leveling luminosity
• ….
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Thank you!
Questions?
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References:
[1] http://cern.ch/Werner.Herr/CAS2009/proceedings/bb_proc.pdf
[2] V. Shiltsev et al, “Beam beam effects in the Tevatron”, Phys. Rev. ST Accel. Beams 8, 101001 (2005)
[3] Lyn Evans “The beam-beam interaction”, CERN 84-15 (1984)
[4] Alex Chao “Lie Algebra Techniques for Nonlinear Dynamics” SLAC-PUB-9574 (2002)
[5] J. D. Jackson, “Classical Electrodynamics”, John Wiley & Sons, NY, 1962.
[6] H. Grote, F. Schmidt, L. H. A. Leunissen,”LHC Dynamic Aperture at Collision”, LHC-Project-Note 197, (1999).
[7] W. Herr,”Features and implications of different LHC crossing schemes”, LHC-Project-Note 628, (2003).
[8] A. Hofmann,”Beam-beam modes for two beams with unequal tunes”, CERN-SL-99-039 (AP) (1999) p. 56.
[9] Y. Alexahin, ”On the Landau damping and decoherence of transverse dipole oscillations in colliding beams ”, Part. Acc. 59, 43 (1996).
[10] R. Assmann et al., “Results of long-range beam-beam studies - scaling with beam separation and intensity ”
…much more on the LHC Beam-beam webpage:
http://lhc-beam-beam.web.cern.ch/lhc-beam-beam/
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From the potential of charge beam to the Beam-beam Force
We need to calculate the field E and B of opposing beam
In rest frame only electrostatic field:
In the lab frame the electric and magnetic fields can be obtained:
Lorentz force gives:
Ultra-relativistic case
Beam-Beam Effect is mainly a TRANSVERSE EFFECT
We can derive the electrostatic field
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Beam-beam potential and force
General approach in electromagnetic problems Reference[5]already applied to beam-beam interactions in Reference[1,3, 4]
Scalar Potential can be derived from Poisson equation which relates the potential to the
charge density r
Then compute the Electric Field from Gauss Law
We need to calculate the field E and B of opposing beam
In rest frame only electrostatic field:
Lorentz force gives:
Then back to the Lab frame we can compute the force
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Beam-beam potential
In the case of Gaussian Beam density distribution we can factorize the density distribution
N is the number of particles in bunch
The potential get’s the form:
This is difficult to solve but following [29] we can solve the diffusion equation.
Solution of Poisson equation
The poison equation can be formally solved using the Green’s function G(x,y,z,x0,y0,z0)method [25]
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Crossing angle effect
S is the geometric reduction factorFor small crossing angle
Examples: LHC (7 TeV): f = 285 mrad, sx = 17 mm, ss = 7.5 cm, S=0.84HL-LHC (7 TeV) f=590 mrad, sx = 7 mm, ss = 7.5 cm, S=0.3
70% loss of luminosity if not compensated
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S. Kheifets proposal
From the diffusion equation:
We obtain the potential U by going to the limit of A 0
Solving the diffusion equation instead of Poisson gives a Green’s function of the form:
We can then compute the potential
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S. Kheifets proposal
From Poisson Equation:
From Diffusion equation:
This allows to avoid the denominator in the integral and to collect the exponential which can be integrated
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The potential of charge beam: 2D case
Changing the independent variable t to q= 4t/A2 and using the three integrations:
Our potential assumes the form of:
Since we are interest in the transverse fields, in a two dimensional case
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2 dimensional problem
The two dimensional potential is then given by:
n is the line density of particles in the beame is the elementary chargee Is the permittivity of free space
From the potential we can derive the field
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Radial Force
Radial component
Azimuthal component
In cylindrical coordinates
From Lorentz Force Force has a radial component
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Beam-Beam Force: round beams
In cylindrical Coordinates
In Cartesian Coordinates:
For the case of q=-e opposite charges
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Beam-Beam Force
If we normalize the separations in units of the beam transverse rms size:
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Why do we care?
• Tune shift has opposite sign in plane of separation
• Break the symmetry between the planes, much more resonances are excited
• Mostly affect particles at large amplitude
• Cause effects on closed orbit, tune shift, chromaticity…
• PACMAN effects complicates the picture
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Dynamic beta effect and beating
• The beam-beam collision at the experiment changes also the optics of the machine
• This leads to changes in the phase Dm and to an “optical error” Db*
• Source of force at the position s, and the effect at position s0 in perturbation theory is given by:
If our case if optics changes → beam-beam force changes → optics changes → beam-beam force changes ...
Self-consistent calculation is required to evaluate the effect
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Dynamic Beta effect
In a simple case with one beam-beam interaction and seen as a perturbationAnd taking the effect at the source of the error (s=s0)
Dynamic beta
Dynamic beating
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Dynamic Beta effect single Interaction point
Sensitive to:• Beam-beam parameter: x• Tune : Q• Configuration (IPS) and optics (phase advance)
LHC case has 1-2 %HL-LHC 3-6 %…or more
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Dynamic beta-beating due to beam-beam effectsMaximum beta change as a function of unperturbed tune
Maximum beating as a function of tune
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Dynamic beta-beating due to beam-beam effects
From optics codes beating along the acceleratorHow will cleaning efficiency and machine protection deal with such beating?