Beam-Beam Interaction D. Schulte (CERN) Pinch Effect Beamstrahlung - - PowerPoint PPT Presentation

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 CLIC Ecm GeV 500 3000 L 1034 cm−2s−1 2.0 5.7 N 109 20 3.7 σ∗

x

nm 655 40 σ∗

y

nm 5.7 1 σz µm 300 45 nb 2820 312 fr Hz 5 50 ∆z ns 300 0.5 θc mradian (20) 20 nγ 1.3 2 ∆E/E % 2.4 30 In the following, beam sizes are always given at the interaction point The beams are flat this in order to achieve 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 = HD N 2frnb 4πσxσy 1

• 1 +
• σz

σx tan θc 2

2

Ignore crossing angle and HD, yields L = N 4πσxσy Nfrnb ∝ N σxσy Pbeam 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 choice but 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 +
• σz

σx tan θc 2

2

This can be avoided using the “crab crossing” scheme

• a rotation is introduced into the bunch which makes it straight at

collision From the beam-beam point of view crab crossing can be treated as no crossing 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 σxσy Pbeam = N √βxǫx

βyǫy

Pbeam The emittances ǫx,y are beam properties, smaller ǫ is more demanding for the 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/β2

x,y

⇒ Lower limit β ≥ σz ⇒ We will see that this limit is important for the vertical plane, not for the horizontal

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 luminosity enhancement factor

Disruption Parameter

We consider the motion of one particle in the field of the oncoming bunch and 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 the focal 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 a thin lens on the other beam A large disruption parameter D ≫ 1 indicates that the particle oscillates in the field of the oncoming beam ⇒ the notion of the parameter as the ratio of focal length to bunch length is 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 particles and in collective field of oncoming bunch Condition for application of synchrotron radiation formulae is that the collective 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 by 1/γ ⇒ Number of coherence lengths η = γθx = Dx σx σz γ = 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 Nr2

eγ

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 = ω

ωc E E−¯ 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 still peaked at the nominal centre-of-mass energy But the reduction is very signififcant The importance will de- pend on the phyiscs pro- cess you want to mea- sure

0.1 0.2 0.3 0.4 0.5 0.6 450 455 460 465 470 475 480 485 490 495 500 L/L0 per bin Ecm [GeV]

Spectrum Quality vs. Luminosity

By modifying the hori- zontal beam size one can trade luminosity vs spec- trum quality Variation is around nom- inal ILC parameter ⇒ Need a way to determine which ∆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 [1034cm-2s-1] σx/σx,0 L L0.01 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 L0.01/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 fe

e(x, Q2)

⇒ the probability that the particle collides with a fraction x of its energy at a scale Q2 fe

e(x, Q2) = β

2(1 − x)(β

2−1)

  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 colliding particles For central production processes Q2 = s = 4E2

cm can be used

Comparison of Radiation Processes

Initial state radiation and beamstrahlung lead to similar reduction of the luminosity close to the nominal energy Initial State radiation can be calculated Beamstrahlung depends

• n beam parameters, re-

quires careful measure- ment of relevant param- eters

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/L0 per bin Ecm [GeV] all ISR BS

Relative luminosity spectrum, considering beamstrahlung (BS), initial state radiation (ISR) and both (all)

Example of Impact of Beamstrahlung: Top Threshold Scan

αS = 0.13 αS = 0.11 αS = 0.12 Ecm [GeV] σ [pb] 355 350 345 340 335 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Example of Impact of Beamstrahlung

Nlc Tesla

• in. stat. rad.

without rad. ECM [GeV] σ [pb] 354 352 350 348 346 344 342 340 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Keeping the Beams in Collision

The vertical beam size is very small (few nm) Even ground motion ef- fects become important at this level nm-offsets lead to tens

• f µradian deflection an-

gles ⇒ can be measured with BPM and used for feedback ⇒ in ILC intra-pulse feedback is possible ⇒ in CLIC this will be tough

• 200
• 150
• 100
• 50

50 100 150 200

• 6
• 4
• 2

2 4 6 ∆x’ [µradian] ∆x/σx,0 σx=σx,0 σx=2σx,0

• 150
• 100
• 50

50 100 150

• 6
• 4
• 2

2 4 6 ∆y’ [µradian] ∆y/σy,0 σy=σy,0 σy=2σy,0

Luminosity as a Function of Offset

Neglecting beam waist

• ne

can estimate for rigid bunches from the

• verlap of Gaussian dis-

tributions L L0 = exp

    −∆y2

4σ2

y

    

The beam-beam forces modify this ⇒ the beams attract each other

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1 2 3 4 5 6 L/L0 ∆y/σy D=72 D=18 D=4.5 analytic

⇒ If the disruption parameter is very large, we are more sensitive to beam

• ffsets

Choice of Disruption Parameter

Evidently a large disruption parameter makes the beam more sensitive to

• ffsets

⇒ one should limit the disruption But, the vertical disruption parameter is a function of the luminosity Assuming σx ≫ σy one can calculate L = HD N 4πσxσy Pbeam = a N σxσy L = a 2Nreσz γσy(σx + σy) γ 2reσz σx + σy σx L ≈ bDy σz ⇒ A long bunch requires a high vertical disruption parameter to reach high luminosity

The Banana Effect

At large disruption, cor- related

• ffsets

in the beam are important ⇒ offsets of parts of the beam lead to instabil- ity The emittance growth in the 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 projected emittances c) beam-beam calculation with full correlation

Mitigation of the Effect

Example with TESLA parameters is shown The effect can be cured by using luminosity opti- misation in a pulse While the effect can be cured by performing luminosity optimisation, this leads to a more com- plex tuning scheme

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 20 22 24 26 28 30 L [1034cm-2s-1] εy [nm] L1 Loff Lang approx.

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 relatively small angles

200 400 600 800 1000 1200

• 400
• 200

200 400 δnγEγ/δθx,y [1012GeV/radian] θx,y [µradian] hor. vert. 10 20 30 40 50 60

• 400
• 200

200 400 δne/δθx [106] θx [µradian] 50 100 150 200 250 300 350

• 400
• 200

200 400 δne/δθy [106] θy [µradian]

Incoherent Pair Production

Three different processes are important

• Breit-Wheeler
• Bethe-Heitler
• Landau-Lifshitz

The real photons are beamstrahlung 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 = 2πr2

em2

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 = 21 + (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 Weiz¨ acker-Williams) approximation the vir- tual photon in a process is treated as real and an equivalent photon flux is used The photon spectrum is given by d2fγ

e (x, Q2)

dxdQ2 = α 2π 1 + (1 − x)2 x 1 Q2 Since we neglect the virtuality we can integrate over Q2 dnγ

e(x)

dx = α 2π 1 + (1 − x)2 x ln ˆ Q2 ˇ Q2 The lower boundary is given by ˇ Q2 = x2m2 1 − x The upper boundary depends on the process, we use ˆ Q2 = m2 + p2

⊥

Spectrum of the Pairs

The Breit-Wheeler pro- cess produces the small- est amount of particles

• but they often have

larger angles The Landau-Lifshitz pro- cess is produces more soft and hard particles than the Bethe-Heitler process In the Bethe-heitler pro- cess usually the beam- strahlung photon is the hard photon

500 1000 1500 2000 2500 3000 3500 0.001 0.01 0.1 1 10 100 particles per bin E [GeV] BW BH LL

Beam Size Effect

The virtual photons with low transverse momen- tum are not well lo- calised The beams are very small ⇒ need to correct the cross section Model is to use the typical impact parameter b ≈ ¯ h/q⊥ If b > σy the process is suppressed Typical reduction of the production rate is a fac- tor two

500 1000 1500 2000 2500 3000 3500 0.001 0.01 0.1 1 10 100 particles per bin E [GeV] BW BH BH,bs LL LL,bs

Deflection by the Beams

Most of the produced particles have small an- gles The forward or backward direction is random The pairs are affected by the beam ⇒ some are focused some are defocused Maximum deflection θm =

• 4ln

D

ǫ + 1

• Dσ2

x

√ 3ǫσ2

z

0.1 1 10 100 0.001 0.01 0.1 1 Pt [MeV/c] θ

Vertex Detector

The vertex detector is the detector component closest to the interaction point 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 time into lighter quarks

• the tracks of these lighter parti-

cles will orign from the point where the b-quark decayed, not from the beam position The closer the vertex detector to the IP the better the resolution

Impact of the Pairs on the Vertex Detector

Hits of the pairs in the vertex detector can con- fuse the reconstruction

• f tracks

Can avoid this problem by combination of two means

• use sufficient opening

angle of the vertex detector

• confine pairs to small

radii by use of longi- tudinal magnetic field this exists in the de- tector anyway

0.01 0.1 1 10 100 1000 10000 5 10 15 20 25 30 35 40 45 50 particle density per train [mm-2] r [mm] Bz=1T Bz=2T Bz=3T Bz=4T Bz=5T Bz=6T

Impact on the Detector Design

A significant number of the pair particles can be hit something in the de- tector ⇒ their secondaries can be a problem Example: the

• ld

TESLA design with mask without mask t [ns] hits per ns 100 80 60 40 20 120 100 80 60 40 20

θ i θm

2m 4cm

quadrupole vertex detector

• instr. tungsten

interaction point graphite tungsten

Bremsstrahlung

Interaction

• f

particle with individual particle

• f other beam

Also called “radiative Bhabha” Soft scatter between two particles with emission

• f inital state radiation

Can be calculated as Compton scattering of vitual photon spectrum with beam particle Yields a relatively flat spectrum

with bs without bs analytical E [GeV] d N/d E [GeV−1] 200 150 100 50 10000 1000

Hadronic Background

A photon can contribute to hadron production in two ways

• direct

production, the photon is a real photon

• resolved production,

the photon is a bag full of partons Hard and soft events ex- ist e.g. “minijets”

Coherent Pairs

Coherent pairs are gen- erated by a photon in a strong electro-magnetic field Cross section depends exponentially on the field ⇒ Rate of pairs is small for centre-of-mass ener- gies below 1 TeV ⇒ In CLIC, rate is substan- tial

0.5 1 1.5 2 2.5 3 3.5 200 400 600 800 1000 1200 1400 1600 dNcoh/dE [106GeV-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 to luminosity ⇒ the emitting particle is not scattered much ⇒ it cannot be seper- ated from the beam by its angle ⇒ one needs to seperate it by it’s energy pairs bremsstr. beamstr. E [GeV] d N/d E [GeV−1] 200 150 100 50 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 luminosity time some function of the beam parameters Example shown is a scan

• f

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

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 L/L0, E/E0 Wy/sigmaz L L, fit E E, fit

Use of Beamstrahlung

Beamstrahlung is not proportional to luminos- ity at all Can use beamstrahlung to optimise knobs which modify one parameter at a time Need to identify correct combination of beam- strahlung

• sum of radiation of

both beams

• difference
• f radia-

tion of both beams

0.4 0.6 0.8 1 1.2 1.4 1.6

• 300
• 200
• 100

100 200 300 V/V0 waist shift [µm] L/L0 E1 C1 E2 C2 0.2 0.4 0.6 0.8 1 1.2 1.4

• 300
• 200
• 100

100 200 300 V/V0 waist shift [µm] L/L0 E1 C1 E2 C2

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