DESY Summer Student Lecture 31 st July 2002

1. An Introduction to thePhysics and Technologyof e+e- Linear CollidersLecture 7: Beam-Beam EffectsNick Walker (DESY) DESY Summer Student Lecture31st July 2002 USPAS, Santa Barbara, 16th-27th June, 2003

2. 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

3. Introduction (continued) • Beam-Beam Effects • Electric field from a “flat” charge bunch • Equation of motion of an electron in flat bunch • The Disruption Parameter (Dy) • Crossing Angle and Kink Instability • Beamstrahlung • Pair production (background)

4. Storage Ring Collider Comparison Linear beam-beam tune shift Putting in some typical numbers (see previous table) gives: Storage ring colliders try to keep

5. Electric Field from a Relativistic Flat Beam • Highly relativistic beam E+vB  2E • Flat beam sx  sy (cf Beamstrahlung) • Assume • infinitely wide beam with constant density per unit length in x ( r(x)) • Gaussian charge distribution in y: • for now, leave r(z) unspecified

6. Electric Field from a Relativistic Flat Beam Use Gauss’ theorem: Assuming Gaussian distribution for z, the peak field is given by

7. Electric Field from a Relativistic Flat Beam Note: 2Ey plotted Assuming a Gaussian distribution for r(z)

8. Electric Field from a Relativistic Flat Beam effect of x width Assuming a Gaussian distribution for r(z)

9. Equation Of Motion F = ma: Changing variable to z: whyrz(2z)?

10. exact: 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) = y0 Thin-lens focal length: Define Vertical Disruption Parameter

11. Number of Oscillations Equation of motion re-visited: Approximate r(z) by rectangular distribution with same RMS as equivalent Gaussian distribution (sz) half-length!

12. Example of Numerical Solution green: rectangular approximation black: gaussian

13. Pinch Enhancement • Self-focusing (pinch) leads to higher luminosity for a head-on collision. ‘hour glass’ effect Empirical fit to beam-beam simulation resultsOnly a function of disruption parameter Dx,y

14. The Luminosity Issue: Hour-Glass b = “depth of focus” reasonable lower limit for b is bunch length sz

15. Luminosity as a function of by

16. Beating the hour glass effect Travelling focus (Balakin) • Arrange for finite chromaticity at IP (how?) • Create z-correlated energy spread along the bunch (how?)

17. Beating the hour glass effect Travelling focus • Arrange for finite chromaticity at IP (how?) • Create z-correlated energy spread along the bunch (how?)

18. Beating the hour glass effect Travelling focus • Arrange for finite chromaticity at IP (how?) • Create z-correlated energy spread along the bunch (how?) ct

19. Beating the hour glass effect foci ‘travel’ from z = 0 to z = chromaticity: travelling focus: NB: z correlated!

20. 1 2 3 6 5 4 The arrow shows position of focus for the read beam during travelling focus collision

21. 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.

22. Kink Instability Assuming solutions of the form and substituting into EoM leads to the dispersion relation: Motion becomes unstable when w2 0, which occurs when

23. Kink Instability Exponential growth rate: Maximum growth rate when Remember! Thus ‘amplification’ factor of an initial offset is: For our previous example with Dy ~28, factor ~ 3

24. Kink Instability

25. Pinch Enhancement H= enhancement factor results ofsimulations:

26. High Dy example: TESLA 500

27. Numbers from our previous example give OK for horizontal plane where Dx<1 For vertical plane (strong focusing Dy > 1), particles oscillate: Disruption Angle Remembering definition of Dy The angles after collision are characterised by previous linear approximation:

28. Disruption Angle: simulation results horizontal angle (mrad) vertical angle (mrad) Important in designing IR (spent-beam extraction)

29. Beam-Beam Animation Wonderland Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (written by D. Schulte, CERN).

30. Examples of GUINEAPIG Simulations NLC parameters Dy~12 Nx2 Dy~24

31. Examples of GUINEAPIG Simulations NLC parameters Dy~12 Luminosity enhancement HD ~ 1.4 Not much of an instability

32. Examples of GUINEAPIG Simulations Nx2 Dy~24 Beam-beam instability is clearly pronounced Luminosity enhancement is compromised by higher sensitivity to initial offsets

33. Beam-beam deflection Sub nm offsets at IP cause large well detectable offsets (micron scale) of the beam a few meters downstream

34. Beam-beam deflectionallow to control collisions

35. Examples of GUINEAPIG Simulations

36. Examples of GUINEAPIG Simulations

37. 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/sy<<1) For large disruption or offset, we introduce the form factor F:

38. Long Range Kink Instability & Crossing Angle To avoid parasitic bunch interactions in the IR, a horizontal crossing angle is introduced: angle fc parasitic beam-beam kick:

39. Long Range Kink Instability & Crossing Angle y IP small vertical offset (D = dy/sy) gives rise to beam-beam kick e- e+ y l resulting vertical offset at parasitic crossing gives next incoming bunches additional vertical kick: IP offset increases instability

40. 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 llk total offset:

41. Long Range Kink Instability & Crossing Angle Assuming all bunches have same initial offset: For small disruption and small offsets, since where mb = number of interacting bunches thus example:Dx,y = 0.1, 10sx = 220 nmsz = 100 mmfc = 20 mrad C = 0.012  mb < 80 for factor of 2 increase

42. Crab Crossing x factor 10 reduction in L! use transverse (crab) RF cavity to ‘tilt’ the bunch at IP x RF kick

43. Beamstrahlung Magnetic field of bunch B = E/c Peak field: From classical theory, power radiated is given by For E = 250 GeV and sz = 300mm: DE/E ~ 12% note:

44. 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

45. Beamstrahlung: photon spectrum NOT classical synchrotron radiation spectrum!Need to use Sokolov-Ternov formula: quantum theoretical classical

46. Beamstrahlung Numbers average and maximum  photons per electron: average energy loss: dBS and ng (and hence ) talk directly to luminosity spectrum and backgrounds

47. 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?

48. Luminosity scaling revisited low beamstrahlung regime 1: high beamstrahlung regime 1: homework: derive high BS scaling law

49. Beamstrahlung Numbers: example TESLA

50. 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 (U>100), coherent direct trident production e  e e+ e-TESLA 0.5TeV  ~ 0.06NLC 1TeV  ~ 0.28CLIC 3TeV  ~ 9 • Beamstrahlung degrades Luminosity Spectrum