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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

450 455 460 465 470 475 480 485 490 495 500

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]

355350345340335

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Example of Impact of Beamstrahlung

NlcTesla

in. stat. rad.without rad.

ECM [GeV]

σ[p

b]

354352350348346344342340

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

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

0

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

60

-400 -200 0 200 400

δne/

δθx

[106 ]

θx [µradian]

0

50

100

150

200

250

300

350

-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

0

500

1000

1500

2000

2500

3000

3500

0.001 0.01 0.1 1 10 100

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

0

500

1000

1500

2000

2500

3000

3500

0.001 0.01 0.1 1 10 100

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

0.01

0.1

1

10

100

1000

10000

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

0

0.5

1

1.5

2

2.5

3

3.5

0 200 400 600 800 1000 1200 1400 1600

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

0.94

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

0.2

0.4

0.6

0.8

1

1.2

1.4

-300 -200 -100 0 100 200 300

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