Beam-Beam Interaction
Transcript of Beam-Beam Interaction
Beam-Beam Interaction
D. Schulte (CERN)
• Pinch Effect
• Beamstrahlung
• Imperfect Collisions
• Banana Effect
• Secondary Production
• Luminosity Monitoring
Supported by ELAN, EU contract number RII3-CT-2003-506395
Beam Parameters at Collision
Parameter Unit ILC nominal CLICEcm GeV 500 3000L 1034 cm−2s−1 2.0 5.7N 109 20 3.7σ∗
x nm 655 40σ∗
y nm 5.7 1σz µm 300 45nb 2820 312fr Hz 5 50∆z ns 300 0.5θc mradian (20) 20nγ 1.3 2
∆E/E % 2.4 30
In the following, beam sizes are alwaysgiven at the interaction point
The beams are flat this in order toachieve high luminosity (small σx×σy)and low beamstrahlung (large σx +σy)
The luminosity is given by
L =N 2frnb
4πσxσy
so what does limit it?
Luminosity
Luminosity is given by (assuming rigid beams, no hour glass effect)
L = HDN 2frnb
4πσxσy
1√
√
√
√1 +(
σzσx
tan θc2
)2
Ignore crossing angle and HD, yields
L =N
4πσxσyNfrnb ∝
N
σxσyPbeam
Can we ignore the crossing angle?
⇒ Need to minimise beam cross section, limits due to
hour glass effect
beamstrahlung
stability
Crossing Angle
A crossing angle between the beams can be required
- to minimise effects of parasitic crossings of bunches
- to be able to cleanly get rid of the spent beam
In a normal conducting machine, the short bunch spacing leaves no choicebut to have a crossing angle
In a superconducting machine one can in principle avoid a crossing angle
Crab Crossing
The crossing angle θc can lead to a luminosity reduction
L
L0=
1√
√
√
√1 +(
σzσx
tan θc2
)2
This can be avoided using the “crab crossing” scheme
• a rotation is introduced into the bunch which makes it straight atcollision
From the beam-beam point of view crab crossing can be treated as nocrossing angle
need to transform secondaries into laboratory frame
Beam Size Limitation 1: Hour Glass Effect
We can rewrite the beam size at the IP as
L ∝ N
σxσyPbeam =
N√βxǫx
√
βyǫyPbeam
The emittances ǫx,y are beam properties, smaller ǫ is more demanding forthe other systems
The beta-functions β are properties of the focusing system
Stronger focusing (lower β) can increase the luminosity
Too low β reduces luminosity due to hour glass effect
σx,y(z) =√
βx,yǫx,y + z2/βx,yǫx,y = σ∗x,y
√
1 + z2/β2x,y
⇒ Lower limit β ≥ σz
⇒ We will see that this limit is important for the vertical plane, not for thehorizontal
Beam Size Limitation 2: Beam-Beam Interaction
The beam is ultra-relativistic
⇒ the fields are almost completely transverse
Due to the high density the electro-magnetic beam fields are high
⇒ focus the incoming beam (electric and magnetic force add)
⇒ reduction of beam crossection leads to more luminosity
⇒ bending of the trajectories leads to emission of beamstrahlung
The increase in luminosity will be expressed by a factor HD, the luminosityenhancement factor
Disruption Parameter
We consider the motion of one particle in the field of the oncoming bunchand make the following assumptions
- the bunch transverse distribution is Gaussian, with widths σx and σy
- the particle is close to the beam axis
- the initial particle transverse momentum is zero
- the particle does not move transversely
We obtain for the final particle angle
dx
dz
∣
∣
∣
∣
∣
∣
∣final= − 2Nrex
γσx(σx + σy)
dy
dz
∣
∣
∣
∣
∣
∣
∣final= − 2Nrey
γσy(σx + σy)
⇒ Beam acts as a focusing lens
We introduce the disruption parameter Dx,y = σz/fx,y, where fx,y is thefocal length
Dx =2Nreσz
γσx(σx + σy)Dy =
2Nreσz
γσy(σx + σy)
Relevance of the Disruption Parameter
A small disruption parameter D ≪ 1 indicates that the beam acts as athin lens on the other beam
A large disruption parameter D ≫ 1 indicates that the particle oscillatesin the field of the oncoming beam
⇒ the notion of the parameter as the ratio of focal length to bunch lengthis no longer valid, the parameter is still useful
⇒ Since the particles in both beams will start to oscillate, the analytic esti-mation of the effects becomes tedious
⇒ resort to simulations
In linear colliders one usually finds Dx ≪ 1 and Dy ≫ 1
ILC: Dx ≈ 0.15, Dy ≈ 18, CLIC: Dx ≈ 0.2, Dy ≈ 7.6
Simulation Procedure
Two widely spread codes to simulate the beam-beam interaction are CAIN(K. Yokoya et al.) and GUINEA-PIG (D. Schulte et al.)
• The beam is represented by macro particles
• It is cut longitudinally into slices
• Each slice interacts with one slice of the other beam at a given time
• The slices are cut into cells
• The simulation is performed in a number of time steps in each of them
- The macro-particle charges are distributed over the cells
- The forces at the cell locations are calculated
- The forces are applied to the macro particles
- The particles are advanced
Beamstrahlung
Particles travel on curved trajectories
⇒ emitt radiation similar to synchrotron radiation
⇒ called beamstrahlung in this context
Beamstrahlung reduces the beam particle energy
⇒ particles collide at energies different from the nominal one
⇒ physics cross section are affected
⇒ threshold scans are affected
Beamstrahlung is not the only relevant process
Synchrotron Radiation vs. Beamstrahlung
Quantum mechanics: particle can scatter in field of individual particlesand in collective field of oncoming bunch
Condition for application of synchrotron radiation formulae is that thecollective field of the oncoming beam particles is important
- integrate over field of many particles during coherence length
- travel many coherence lengths during bunch passage
Beamstrahlung opening cone is roughly given by 1/γ
⇒ coherence length is the distance traveled while particle is deflected by1/γ
⇒ Number of coherence lengths
η = γθx = Dxσx
σzγ =
2Nre
σx + σy
⇒ Usually of the order of several tens or hundreds ⇒ OK
Beamstrahlung Description
• Synchrotron radiation is characterised by the critical energy
ωc =3
2
γ3c
ρ
ρ is bending radius
• Beamstrahlung is often characterised using the beamstrahlung parameterΥ
Υ =2
3
hωc
E0
Υ is the ratio of critical energy to beam energy (times 2/3)
The average value can be estimated as (for Gaussian beams)
〈Υ〉 =5
6
Nr2eγ
α(σx + σy)σz
Emission Spectrum
Sokolov-Ternov spectrum
d w
d ω=
α√3πγ2
∫ ∞x K5
3(x′)dx′ +
hω
E
hω
E − hωK2
3(x)
x = ωωc
EE−hω
K5/3 and K2/3 are the modified Bessel functions
For small Υ
∆E ∝ Υ2σz ∝N
(σx + σy)
N
(σx + σy)σz
⇒ Use flat beams
Typically the number of photons per beam particle nγ is of order unity,δE/E is of the order of a few percent
Luminosity Spectrum
The luminosity is stillpeaked at the nominalcentre-of-mass energy
But the reduction is verysignififcant
The importance will de-pend on the phyiscs pro-cess you want to mea-sure
0
0.1
0.2
0.3
0.4
0.5
0.6
450 455 460 465 470 475 480 485 490 495 500
L/L 0
per
bin
Ecm [GeV]
Spectrum Quality vs. Luminosity
By modifying the hori-zontal beam size one cantrade luminosity vs spec-trum quality
Variation is around nom-inal ILC parameter
⇒ Need a way to determinewhich ∆E is acceptable
0.5
1
1.5
2
2.5
3
3.5
0.4 0.6 0.8 1 1.2 1.4 1.6
L [1
034cm
-2s-1
]
σx/σx,0
LL0.01
0.35 0.4
0.45 0.5
0.55 0.6
0.65 0.7
0.75 0.8
0.85
0.4 0.6 0.8 1 1.2 1.4 1.6
L 0.0
1/L
σx/σx,0
Initial State Radiation
Colliding particles can emit photons during the collision
⇒ the collision energies are modified
⇒ e.g. important at LEP
The beam particles can be represented by a spectrum f ee (x, Q2)
⇒ the probability that the particle collides with a fraction x of its energyat a scale Q2
f ee (x,Q2) =
β
2(1 − x)(
β2−1)
1 +3
8β
− β
4(1 + x)
β =2α
π
ln
Q2
m2− 1
The scale Q2 depends on the actual interaction process of the collidingparticles
For central production processes Q2 = s = 4E2cm can be used
Comparison of Radiation Processes
Initial state radiationand beamstrahlung leadto similar reduction ofthe luminosity close tothe nominal energy
Initial State radiationcan be calculated
Beamstrahlung dependson beam parameters, re-quires careful measure-ment of relevant param-eters
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L/L 0
per
bin
Ecm [GeV]
allISRBS
Relative luminosity spectrum, considering beamstrahlung (BS), initial stateradiation (ISR) and both (all)
Example of Impact of Beamstrahlung: Top Threshold Scan
αS = 0.13αS = 0.11αS = 0.12
Ecm [GeV]
σ[p
b]
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0.9
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0.5
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0.3
0.2
0.1
0
Example of Impact of Beamstrahlung
NlcTesla
in. stat. rad.without rad.
ECM [GeV]
σ[p
b]
354352350348346344342340
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0.8
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Keeping the Beams in Collision
The vertical beam size isvery small (few nm)
Even ground motion ef-fects become importantat this level
nm-offsets lead to tensof µradian deflection an-gles
⇒ can be measured withBPM and used forfeedback
⇒ in ILC intra-pulsefeedback is possible
⇒ in CLIC this will betough
-200
-150
-100
-50
0
50
100
150
200
-6 -4 -2 0 2 4 6
∆x’ [
µrad
ian]
∆x/σx,0
σx=σx,0σx=2σx,0
-150
-100
-50
0
50
100
150
-6 -4 -2 0 2 4 6
∆y’ [
µrad
ian]
∆y/σy,0
σy=σy,0σy=2σy,0
Luminosity as a Function of Offset
Neglecting beam waistone can estimate forrigid bunches from theoverlap of Gaussian dis-tributions
LL0
= exp
−∆y2
4σ2y
The beam-beam forcesmodify this
⇒ the beams attracteach other
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 1.1
0 1 2 3 4 5 6
L/L 0
∆y/σy
D=72D=18D=4.5
analytic
⇒ If the disruption parameter is very large, we are more sensitive to beamoffsets
Choice of Disruption Parameter
Evidently a large disruption parameter makes the beam more sensitive tooffsets
⇒ one should limit the disruption
But, the vertical disruption parameter is a function of the luminosity
Assuming σx ≫ σy one can calculate
L = HDN
4πσxσyPbeam = a
N
σxσy
L = a2Nreσz
γσy(σx + σy)
γ
2reσz
σx + σy
σx
L ≈ bDy
σz
⇒ A long bunch requires a high vertical disruption parameter to reach highluminosity
The Banana Effect
At large disruption, cor-related offsets in thebeam are important
⇒ offsets of parts of thebeam lead to instabil-ity
The emittance growth inthe beam leads to corre-lation of the mean y po-sition to z
a)
b)
c)
a) shows development of beam in the main linac
b) simplified beam-beam calculation using projectedemittances
c) beam-beam calculation with full correlation
Mitigation of the Effect
Example with TESLAparameters is shown
The effect can be curedby using luminosity opti-misation in a pulse
While the effect canbe cured by performingluminosity optimisation,this leads to a more com-plex tuning scheme
22.22.42.62.8
33.23.43.63.8
4
20 22 24 26 28 30L
[1034
cm-2
s-1]
εy [nm]
L1Loff
Langapprox.
First angle and offset are corrected
Then luminosity is optimised
Approximate analytical scaling is L ∝ 1/√
ǫy
Secondary Production
We will focus on
- beamstrahlung (see above)
- incoherent pair production
- coherent pair production
- bremsstrahlung
- hadron production
Spent Beam and Beamstrahlung
Spent beam particles have relativelysmall angles
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200
400
600
800
1000
1200
-400 -200 0 200 400
δnγE
γ/δθ
x,y
[1012
GeV
/rad
ian]
θx,y [µradian]
hor.vert. 0
10
20
30
40
50
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-400 -200 0 200 400
δne/
δθx
[106 ]
θx [µradian]
0
50
100
150
200
250
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-400 -200 0 200 400
δne/
δθy
[106 ]
θy [µradian]
Incoherent Pair Production
Three different processesare important
- Breit-Wheeler
- Bethe-Heitler
- Landau-Lifshitz
The real photons arebeamstrahlung photons
The processes with vir-tual photons can be cal-culated using the equiv-alent photon approxi-mation and the Breit-Wheeler cross section
Breit-Wheeler Process
Collisions of two photons can produce electron positron pairs
dσ
dt=
2πr2em
2
s2
t − m2
u − m2+
u − m2
t − m2
− 4
m2
t − m2+
m2
u − m2
−4
m2
t − m2+
m2
u − m2
2
(1)
s = (k1 + k2)2, t = (k1 − p1)
2 and u = (k1 − p2)2 are Mandelstam
variables
In centre-of-mass system
dσ
d cos θ∝ 1 + β cos θ
1 − β cos θ+
1 − β cos θ
1 + β cos θ= 2
1 + (β cos θ)2
1 − (β cos θ)2
Cross section is peaked in forward and backward direction (cos θ ≈ 1)
⇒ pairs are usually produced with small transverse momentum
Equivalent Photon Approximation
In the equivalent photon (or Weizacker-Williams) approximation the vir-tual photon in a process is treated as real and an equivalent photon fluxis used
The photon spectrum is given by
d2fγe (x,Q2)
dxdQ2=
α
2π
1 + (1 − x)2
x
1
Q2
Since we neglect the virtuality we can integrate over Q2
dnγe(x)
dx=
α
2π
1 + (1 − x)2
xln
Q2
Q2
The lower boundary is given by
Q2 =x2m2
1 − x
The upper boundary depends on the process, we use Q2 = m2 + p2⊥
Spectrum of the Pairs
The Breit-Wheeler pro-cess produces the small-est amount of particles
- but they often havelarger angles
The Landau-Lifshitz pro-cess is produces moresoft and hard particlesthan the Bethe-Heitlerprocess
In the Bethe-heitler pro-cess usually the beam-strahlung photon is thehard photon
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part
icle
s pe
r bi
n
E [GeV]
BWBHLL
Beam Size Effect
The virtual photons withlow transverse momen-tum are not well lo-calised
The beams are verysmall
⇒ need to correct thecross section
Model is to use thetypical impact parameterb ≈ h/q⊥
If b > σy the process issuppressed
Typical reduction of theproduction rate is a fac-tor two
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part
icle
s pe
r bi
n
E [GeV]
BWBH
BH,bsLL
LL,bs
Deflection by the Beams
Most of the producedparticles have small an-gles
The forward or backwarddirection is random
The pairs are affected bythe beam
⇒ some are focusedsome are defocused
Maximum deflection
θm =
√
√
√
√
√
√
√
√
4ln
(
Dǫ + 1
)
Dσ2x√
3ǫσ2z
0.1
1
10
100
0.001 0.01 0.1 1
Pt [
MeV
/c]
θ
Vertex Detector
The vertex detector is the detectorcomponent closest to the interactionpoint
It should measure the production ver-tex
- e.g. if in an event a b-quark is pro-duced it will decay after a short timeinto lighter quarks
- the tracks of these lighter parti-cles will orign from the point wherethe b-quark decayed, not from thebeam position
The closer the vertex detector to theIP the better the resolution
Impact of the Pairs on the Vertex Detector
Hits of the pairs in thevertex detector can con-fuse the reconstructionof tracks
Can avoid this problemby combination of twomeans
- use sufficient openingangle of the vertexdetector
- confine pairs to smallradii by use of longi-tudinal magnetic fieldthis exists in the de-tector anyway
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1
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0 5 10 15 20 25 30 35 40 45 50
part
icle
den
sity
per
trai
n [m
m-2
]
r [mm]
Bz=1TBz=2TBz=3TBz=4TBz=5TBz=6T
Impact on the Detector Design
A significant number ofthe pair particles can behit something in the de-tector
⇒ their secondaries canbe a problem
Example: the oldTESLA design
with maskwithout mask
t [ns]
hits
per
ns
100806040200
120
100
80
60
40
20
0
θ i
θm
2m
4cm
quadrupole
vertex detector instr. tungsten
interaction point
graphite
tungsten
Bremsstrahlung
Interaction of particlewith individual particleof other beam
Also called “radiativeBhabha”
Soft scatter between twoparticles with emissionof inital state radiation
Can be calculated asCompton scattering ofvitual photon spectrumwith beam particle
Yields a relatively flat spectrum
with bswithout bsanalytical
E [GeV]
dN
/dE
[GeV
−1]
200150100500
10000
1000
Hadronic Background
A photon can contributeto hadron production intwo ways
- direct production,the photon is a realphoton
- resolved production,the photon is a bagfull of partons
Hard and soft events ex-ist
e.g. “minijets”
Coherent Pairs
Coherent pairs are gen-erated by a photon in astrong electro-magneticfield
Cross section dependsexponentially on the field
⇒ Rate of pairs is smallfor centre-of-mass ener-gies below 1 TeV
⇒ In CLIC, rate is substan-tial
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dNco
h/dE
[106 G
eV-1
]
E [GeV]
Need to foresee large enough exit hole (about 10mra-dian)
Luminosity Monitoring
Fast luminosity measurement is crucial for machine tuning
The detector will use Bhabha scattering
⇒ very good signal for accurate measurement
⇒ this signal is too slow for our luminosity optimisation
Need to use other signals, e.g.
- beamstrahlung/beam energy loss
- incoherent pairs
- bremsstrahlung
Two approaches
- try to reconstruct beam parameters from observables
- optimise one tuning knob after the other
Use of Bremsstrahlung
Bremsstrahlung is emit-ted into small angles
⇒ quite proportional toluminosity
⇒ the emitting particleis not scattered much
⇒ it cannot be seper-ated from the beamby its angle
⇒ one needs to seperateit by it’s energy
pairsbremsstr.beamstr.
E [GeV]
dN
/dE
[GeV
−1]
200150100500
10000
1000
Needs careful design of the spent beam line and it’s instrumentation
Use of Incoherent Pairs
The total energy of in-coherent pairs is pro-portional to luminositytime some function ofthe beam parameters
Example shown is a scanof the vertical waist,i.e. the longitudinal po-sition of the vertical fo-cal point 0.88
0.9
0.92
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0.96
0.98
1
1.02
1.04
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
L/L0
, E/E
0
Wy/sigmaz
LL, fit
EE, fit
Use of Beamstrahlung
Beamstrahlung is notproportional to luminos-ity at all
Can use beamstrahlungto optimise knobs whichmodify one parameter ata time
Need to identify correctcombination of beam-strahlung
- sum of radiation ofboth beams
- difference of radia-tion of both beams
0.4
0.6
0.8
1
1.2
1.4
1.6
-300 -200 -100 0 100 200 300
V/V
0
waist shift [µm]
L/L0E1C1E2C2
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1
1.2
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V/V
0
waist shift [µm]
L/L0E1C1E2C2
Conclusion
• High luminosity with limited beamstrahlung requires flat beams
• Beamstrahlung can significantly affect the experiments
• Beam-beam effects can generate background
- most important for the vertex detector
• Beam-beam background can be used for luminosity monitoring