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An Introduction to the Physics and Technology of e+e- Linear Colliders Lecture 7: Beam-Beam Effects Nick Walker (DESY) DESY Summer Student Lecture 31 st July 2002 USPAS, Santa Barbara, 16 th -27 th June, 2003
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Introduction Beam-beam interaction in a linear collider is basically the same Coulomb interaction as in a storage ring collider. But: –Interaction occurs only once for each bunch (single pass); hence very large bunch deformations permissible. –Extremely high charge densities at IP lead to very intense fields; hence quantum behaviour becomes important Consequently, can divide LC beam-beam phenomena into two categories: –classical –quantum
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Introduction (continued) Beam-Beam Effects –Electric field from a “flat” charge bunch –Equation of motion of an electron in flat bunch –The Disruption Parameter (D y ) –Crossing Angle and Kink Instability –Beamstrahlung –Pair production (background) E beam xx yy xx yy zz NeNe GeVmm nm mm 10 10 NLC25080.12452.71100.75 TESLA250140.455053002.0 CLIC150080.15431300.42
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Storage Ring Collider Comparison Linear beam-beam tune shift Putting in some typical numbers (see previous table) gives: Storage ring colliders try to keep
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Electric Field from a Relativistic Flat Beam Highly relativistic beamE+v B 2E Flat beam x y (cf Beamstrahlung) Assume infinitely wide beam with constant density per unit length in x ( (x)) Gaussian charge distribution in y: for now, leave (z) unspecified
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Electric Field from a Relativistic Flat Beam Use Gauss’ theorem: Assuming Gaussian distribution for z, the peak field is given by
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Electric Field from a Relativistic Flat Beam Note: 2 E y plotted Assuming a Gaussian distribution for (z)
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Electric Field from a Relativistic Flat Beam Assuming a Gaussian distribution for (z) effect of x width
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Equation Of Motion F = ma: Changing variable to z: why z (2z)?
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Linear Approximation and the Disruption Parameter Taking only the linear part of the electric field: Take ‘weak’ approximation: y(z) does not change during interaction y(z) = y 0 Thin-lens focal length: Define Vertical Disruption Parameter exact:
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Number of Oscillations Equation of motion re-visited: Approximate (z) by rectangular distribution with same RMS as equivalent Gaussian distribution ( z ) half-length!
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Example of Numerical Solution green: rectangular approximation black: gaussian
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Pinch Enhancement Self-focusing (pinch) leads to higher luminosity for a head-on collision. ‘hour glass’ effect Empirical fit to beam-beam simulation results Only a function of disruption parameter D x,y
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The Luminosity Issue: Hour-Glass = “depth of focus” reasonable lower limit for is bunch length z
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Luminosity as a function of y
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Beating the hour glass effect Travelling focus (Balakin) Arrange for finite chromaticity at IP (how?) Create z-correlated energy spread along the bunch (how?)
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Beating the hour glass effect Travelling focus Arrange for finite chromaticity at IP (how?) Create z-correlated energy spread along the bunch (how?)
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Beating the hour glass effect Travelling focus Arrange for finite chromaticity at IP (how?) Create z-correlated energy spread along the bunch (how?) ct
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Beating the hour glass effect foci ‘travel’ from z = 0 to z = chromaticity: travelling focus: NB: z correlated!
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1 6 5 4 3 2 The arrow shows position of focus for the read beam during travelling focus collision
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Kink Instability Simple model: ‘sheet’ beams with: Linear equation of motion becomes Need to consider relative motion of both beams in t and z: Classic coupled EoM.
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Kink Instability and substituting into EoM leads to the dispersion relation: Assuming solutions of the form Motion becomes unstable when 2 0, which occurs when
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Kink Instability Exponential growth rate: Maximum growth rate when Remember! Thus ‘amplification’ factor of an initial offset is: For our previous example with D y ~28, factor ~ 3
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Kink Instability
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Pinch Enhancement H= enhancement factor results of simulations:
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High D y example: TESLA 500
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Disruption Angle Remembering definition of D y The angles after collision are characterised by Numbers from our previous example give OK for horizontal plane where D x <1 For vertical plane (strong focusing D y > 1), particles oscillate: previous linear approximation:
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Disruption Angle: simulation results horizontal angle ( rad) vertical angle ( rad) Important in designing IR (spent-beam extraction)
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Beam-Beam Animation Wonderland Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (written by D. Schulte, CERN).
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Examples of GUINEAPIG Simulations NLC parameters D y ~12 Nx2 D y ~24
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NLC parameters D y ~12 Luminosity enhancement H D ~ 1.4 Not much of an instability Examples of GUINEAPIG Simulations
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Nx2 D y ~24 Beam-beam instability is clearly pronounced Luminosity enhancement is compromised by higher sensitivity to initial offsets Examples of GUINEAPIG Simulations
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Beam-beam deflection Sub nm offsets at IP cause large well detectable offsets (micron scale) of the beam a few meters downstream
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Beam-beam deflection allow to control collisions
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Examples of GUINEAPIG Simulations
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Beam-Beam Kick Beam-beam offset gives rise to an equal and opposite mean kick to the bunches – important signal for feedback! For small disruption ( D 1) and offset (y/ y <<1) For large disruption or offset, we introduce the form factor F:
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Long Range Kink Instability & Crossing Angle To avoid parasitic bunch interactions in the IR, a horizontal crossing angle is introduced: angle c parasitic beam-beam kick:
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Long Range Kink Instability & Crossing Angle y y IP small vertical offset ( = y / y ) gives rise to beam-beam kick resulting vertical offset at parasitic crossing gives next incoming bunches additional vertical kick: IP offset increases instability e+e+ ee l
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Long Range Kink Instability & Crossing Angle offset at IP of k-th bunch: distance from IP to encounter with l-th bunch (l < k) contribution from encounter with l-th bunch: NB. independent of l lk total offset:
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Long Range Kink Instability & Crossing Angle Assuming all bunches have same initial offset: For small disruption and small offsets, since thus where m b = number of interacting bunches example: D x,y = 0.1, 10 x = 220 nm z = 100 m c = 20 mrad C = 0.012 m b < 80 for factor of 2 increase
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Crab Crossing factor 10 reduction in L! x x use transverse (crab) RF cavity to ‘tilt’ the bunch at IP RF kick
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Beamstrahlung Magnetic field of bunch B = E/c Peak field: From classical theory, power radiated is given by For E = 250 GeV and z = 300 m: E/E ~ 12% note:
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Beamstrahlung Most important parameter is critical photon frequency Compton wavelength local bending radius beam magnetic field Schwinger’s critical field (= 4.4 GTesla) em field tensor electron 4-momentum
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Beamstrahlung: photon spectrum NOT classical synchrotron radiation spectrum! Need to use Sokolov-Ternov formula: classical quantum theoretical
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Beamstrahlung Numbers average and maximum photons per electron: average energy loss: BS and n (and hence ) talk directly to luminosity spectrum and backgrounds
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Beamstrahlung energy loss In lecture 1, we used the following equation to derive our luminosity scaling law Now we have this: with under what regime is our original expression valid?
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Luminosity scaling revisited low beamstrahlung regime 1: high beamstrahlung regime 1: homework: derive high BS scaling law
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Beamstrahlung Numbers: example TESLA
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Why Beamstrahlung is bad Large number of high-energy photons interact with electron (positron) beam and generate e + e pairs –Low energies ( 0.6), pairs made by incoherent process (photons interact directly with individual beam particles) –High energies (0.6 100), coherent pairs are generated by interaction of photons with macroscopic field of bunch. –Very high energies ( 100), coherent direct trident production e e e + e TESLA 0.5TeV ~ 0.06 NLC 1TeV ~ 0.28 CLIC 3TeV ~ 9 Beamstrahlung degrades Luminosity Spectrum
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Pair Production Incoherent e + e pairs 0.6 –Breit-Wheeler: –Bethe-Heitler: –Landau-Lifshitz: Coherent e + e pairs 0.6 100 –threshold defined by for >1 for intermediate colliders (E cm <1TeV), incoherent pairs dominate
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Pair Production 0.11 0.01 0.1 1 P T (GeV/c) (rad) pairs curl-up (spiral) in solenoid field of detector rdrd ldld dd vertex detector e + e pairs are a potential major source of background Most important: angle with beam axis ( ) and transverse momentum P T.
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Summary Single pass collider allows us to use very strong beam-beam to increase luminosity beam-beam is characterised by following important parameters: –D y = z /fdefines pinch effect (HD), kink instability, dynamics – QM effects, backgrounds, – BS [=f( av )]energy loss, lumi spectrum strong-strong regime requires simulation (e.g. GUINEAPIG). Analytical treatments limited.
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