1. 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

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 (D y ) –Crossing Angle and Kink Instability –Beamstrahlung –Pair production (background) E beam xx yy xx yy zz NeNe GeVmm nm mm  10 10 NLC25080.12452.71100.75 TESLA250140.455053002.0 CLIC150080.15431300.42

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 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

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  E y plotted Assuming a Gaussian distribution for  (z)

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

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

10. 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:

11. Number of Oscillations Equation of motion re-visited: Approximate  (z) by rectangular distribution with same RMS as equivalent Gaussian distribution (  z ) 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 results Only a function of disruption parameter D x,y

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

15. Luminosity as a function of  y

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 6 5 4 3 2 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 and substituting into EoM leads to the dispersion relation: Assuming solutions of the form Motion becomes unstable when  2  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 D y ~28, factor ~ 3

24. Kink Instability

25. Pinch Enhancement H= enhancement factor results of simulations:

26. High D y example: TESLA 500

27. 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:

28. Disruption Angle: simulation results horizontal angle (  rad) vertical angle (  rad) 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 D y ~12 Nx2 D y ~24

31. NLC parameters D y ~12 Luminosity enhancement H D ~ 1.4 Not much of an instability Examples of GUINEAPIG Simulations

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

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 deflection allow to control collisions

35. 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/  y <<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  c parasitic beam-beam kick:

39. 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+ ee l

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 l lk total offset:

41. 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

42. Crab Crossing factor 10 reduction in L! x x use transverse (crab) RF cavity to ‘tilt’ the bunch at IP 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  z = 300  m:  E/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: classical quantum theoretical

46. Beamstrahlung Numbers average and maximum  photons per electron: average energy loss:  BS and n  (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 (  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

51. 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

52. 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 dd vertex detector e + e  pairs are a potential major source of background Most important: angle with beam axis (  ) and transverse momentum P T.

53. 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.