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 E cm GeV 500 3000 10 34 cm − 2 s − 1 L 2.0 5.7 10 9 N 20 3.7 The beams are flat this in order to σ ∗ nm 655 40 x achieve high luminosity (small σ x × σ y ) σ ∗ nm 5.7 1 y and low beamstrahlung (large σ x + σ y ) σ z µ m 300 45 n b 2820 312 The luminosity is given by f r Hz 5 50 L = N 2 f r n b ∆ z ns 300 0.5 4 πσ x σ y θ c mradian (20) 20 so what does limit it? n γ 1.3 2 ∆ E/E % 2.4 30 In the following, beam sizes are always given at the interaction point
Luminosity Luminosity is given by (assuming rigid beams, no hour glass effect) N 2 f r n b 1 L = H D � � 2 4 πσ x σ y � � σ z σ x tan θ c � 1 + � 2 Ignore crossing angle and H D , yields N N L = Nf r n b ∝ P beam 4 πσ x σ y σ x σ y 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 θ c can lead to a luminosity reduction L 1 = � � 2 L 0 � � σ x tan θ c σ z � 1 + � 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 N N √ β x ǫ x L ∝ P beam = P beam � β y ǫ y σ x σ y The emittances ǫ x,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 � � β x,y ǫ x,y + z 2 /β x,y ǫ x,y = σ ∗ 1 + z 2 /β 2 σ x,y ( z ) = x,y x,y ⇒ Lower limit β ≥ σ z ⇒ 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 H D , 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 σ x and σ y - 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 2 Nr e x dy 2 Nr e y � � � � = − = − � � � � � � dz γσ x ( σ x + σ y ) dz γσ y ( σ x + σ y ) � � � final � final ⇒ Beam acts as a focusing lens We introduce the disruption parameter D x,y = σ z /f x,y , where f x,y is the focal length 2 Nr e σ z 2 Nr e σ z D x = D y = γσ x ( σ x + σ y ) γσ y ( σ x + σ y )
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 D x ≪ 1 and D y ≫ 1 ILC: D x ≈ 0 . 15 , D y ≈ 18 , CLIC: D x ≈ 0 . 2 , D y ≈ 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 σ x 2 Nr e η = γθ x = D x γ = σ z σ x + σ y ⇒ Usually of the order of several tens or hundreds ⇒ OK
Beamstrahlung Description • Synchrotron radiation is characterised by the critical energy γ 3 c ω c = 3 2 ρ ρ is bending radius • Beamstrahlung is often characterised using the beamstrahlung parameter Υ Υ = 2 hω c ¯ 3 E 0 Υ is the ratio of critical energy to beam energy (times 2/3) The average value can be estimated as (for Gaussian beams) Nr 2 � Υ � = 5 e γ 6 α ( σ x + σ y ) σ z
Emission Spectrum Sokolov-Ternov spectrum d ˙ w α 3 ( x ′ )d x ′ + ¯ hω hω ¯ � ∞ √ d ω = x K 5 hω K 2 3 ( x ) 3 πγ 2 E E − ¯ x = ω E ω c E − ¯ hω K 5 / 3 and K 2 / 3 are the modified Bessel functions For small Υ N N ∆ E ∝ Υ 2 σ z ∝ ( σ x + σ y ) ( σ x + σ y ) σ z ⇒ Use flat beams Typically the number of photons per beam particle n γ is of order unity, δE/E is of the order of a few percent
Luminosity Spectrum 0.6 0.5 The luminosity is still peaked at the nominal 0.4 centre-of-mass energy L/L 0 per bin 0.3 But the reduction is very signififcant 0.2 The importance will de- 0.1 pend on the phyiscs pro- cess you want to mea- 0 450 455 460 465 470 475 480 485 490 495 500 sure E cm [GeV]
Spectrum Quality vs. Luminosity 3.5 L 3 L 0.01 L [10 34 cm -2 s -1 ] 2.5 2 1.5 1 By modifying the hori- 0.5 zontal beam size one can 0.4 0.6 0.8 1 1.2 1.4 1.6 trade luminosity vs spec- σ x / σ x,0 trum quality 0.85 0.8 Variation is around nom- 0.75 0.7 inal ILC parameter 0.65 L 0.01 /L 0.6 ⇒ Need a way to determine 0.55 0.5 which ∆ E is acceptable 0.45 0.4 0.35 0.4 0.6 0.8 1 1.2 1.4 1.6 σ x / σ x,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 e e ( x, Q 2 ) ⇒ the probability that the particle collides with a fraction x of its energy at a scale Q 2 e ( x, Q 2 ) = β 1 + 3 − β 2(1 − x ) ( β f e 2 − 1) 8 β 4(1 + x ) ln Q 2 β = 2 α m 2 − 1 π The scale Q 2 depends on the actual interaction process of the colliding particles For central production processes Q 2 = s = 4 E 2 cm can be used
Comparison of Radiation Processes 0.7 Initial state radiation all ISR and beamstrahlung lead 0.6 BS to similar reduction of 0.5 the luminosity close to L/L 0 per bin the nominal energy 0.4 Initial State radiation 0.3 can be calculated 0.2 Beamstrahlung depends 0.1 on beam parameters, re- 0 quires careful measure- 450 455 460 465 470 475 480 485 490 495 500 ment of relevant param- E cm [GeV] eters Relative luminosity spectrum, considering beamstrahlung (BS), initial state radiation (ISR) and both (all)
Example of Impact of Beamstrahlung: Top Threshold Scan 1 α S = 0 . 12 α S = 0 . 11 0.9 α S = 0 . 13 0.8 0.7 0.6 σ [pb] 0.5 0.4 0.3 0.2 0.1 0 335 340 345 350 355 E cm [GeV]
Example of Impact of Beamstrahlung 0.9 without rad. in. stat. rad. 0.8 Tesla Nlc 0.7 0.6 0.5 σ [pb] 0.4 0.3 0.2 0.1 0 340 342 344 346 348 350 352 354 E CM [GeV]
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