1. Physics 842, February 2006 Bogdan Popescu Presentation based on “Introduction to Elementary Particles” by David Griffiths WEAK INTERACTION (1)

2. Physics 842, February 2006 Bogdan Popescu Let’s start with...

7. Physics 842, February 2006 Bogdan Popescu - CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion - CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix WEAK INTERACTION (1)

8. Physics 842, February 2006 Bogdan Popescu CHARGED LEPTONIC WEAK INTERACTION The mediators of weak interactions are “intermediate vector bosons”, which are extremely heavy : The propagator for massive spin-1 particles is :, where M is M W or M Z In practice very often : The propagator for W or Z in this case :

9. Physics 842, February 2006 Bogdan Popescu CHARGED LEPTONIC WEAK INTERACTION The theory of “charged” interactions is simpler than that for “neutral” ones. We start by considering coupling of W’s to leptons. The fundamental leptonic vertex is : The Feynman rules are the same as for QED, except for the vertex factor : ( the weak vertex factor ) “Weak coupling constant” (analogous to g e in QED and g s in QCD) :

10. Physics 842, February 2006 Bogdan Popescu Example : Inverse Muon Decay ( lowest order diagram ) When the amplitude is : Applying Casimir’s trick we find :

11. Physics 842, February 2006 Bogdan Popescu Example : Inverse Muon Decay trace theorems using :

12. Physics 842, February 2006 Bogdan Popescu Example : Inverse Muon Decay In CM frame, and neglect the mass of the electron : where E is the incident electron (or neutrino) energy. The differential scattering cross section is : The total cross section :

13. Physics 842, February 2006 Bogdan Popescu - CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion - CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix WEAK INTERACTION (1)

14. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON As before : The amplitude :

15. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON In the muon rest frame : Let : Plug in :

16. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON The decay rate given by Golden Rule * : where : * a lot of work, since this is a three body decay

17. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON Perform integral : where : Next we will do the integral. Setting the polar axis along (which is fixed, for the purposes of the integration), we have :

18. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON Also : The integral is trivial. For the integration, let : and :

19. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON integration : where : The limits of E 2 and E 4 integrals :

20. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON Using :

21. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON

22. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON

23. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON (picture from Griffiths)

24. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON The total decay rate : Lifetime :

25. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON g W and M W do not appear separately, only in the ratio. Let’s introduce “Fermi coupling constant” : The muon lifetime :

26. Physics 842, February 2006 Bogdan Popescu DECAY OF THE MUON In Fermi’s original theory of beta decay there was no W; the interaction was a direct four-particle coupling. Using the observed muon lifetime and mass : and : “Weak fine structure constant” : Larger than electromagnetic fine structure constant

27. Physics 842, February 2006 Bogdan Popescu - CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion - CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix WEAK INTERACTION (1)

28. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON ( the same as in previous case )

29. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON In the rest frame of the neutron : We can’t ignore the mass of the electron. As before : where :

30. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON The integral yields : and : Setting the z-axis along (which is fixed, for the purposes of the integral), we have : and :

31. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON where : and :

32. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON The range of E 2 integral : E is the electron energy ( exact equation)

33. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON Approximations : Expanding to lowest order :

34. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON

35. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON (picture from Griffiths)

36. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON where : Putting in the numbers :

37. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON But the proton and neutron are not point particles. Replacement in the vertex factor : c V is the correction to the vector “weak charge” c A is the correction to the axial vector “weak charge”

38. Physics 842, February 2006 Bogdan Popescu DECAY OF THE NEUTRON Another correction, the quark vertex carries a factor of is the Cabibbo angle. Lifetime :

39. Physics 842, February 2006 Bogdan Popescu - CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion - CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix WEAK INTERACTION (1)

40. Physics 842, February 2006 Bogdan Popescu DECAY OF THE PION The decay of the pion is really a scattering event in which the incident quarks happen to be bound together. We do not know how the W couples to the pion. Use the “form factor”. “form factor”

41. Physics 842, February 2006 Bogdan Popescu DECAY OF THE PION

42. Physics 842, February 2006 Bogdan Popescu DECAY OF THE PION Experimental value : The decay rate : The following ratio could be computed without knowing the decay constant :

43. Physics 842, February 2006 Bogdan Popescu - CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion - CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix WEAK INTERACTION (1)

44. Physics 842, February 2006 Bogdan Popescu CHARGED WEAK INTERACTIONS OF QUARKS For leptons, the coupling to W + and W - takes place strictly within a particular generation : For example : There is no cross-generational coupling as : There are 3 generations of quarks : There exist cross-generational coupling as : Coupling within a generation :

45. Physics 842, February 2006 Bogdan Popescu CHARGED WEAK INTERACTIONS OF QUARKS (Cabibbo, 1963) (extra cos or sin in the vertex factor)

46. Physics 842, February 2006 Bogdan Popescu Example : Leptonic Decays l is an electron or muon. The quark vertex : Using a previous result : The branching ratio : Corresponding to a Cabibbo angle :

47. Physics 842, February 2006 Bogdan Popescu Example : Semileptonic Decays (semileptonic decay) (non-leptonic weak decay)

48. Physics 842, February 2006 Bogdan Popescu Example : Semileptonic Decays Neutron decay :Quark process : There are two d quarks in n, and either one could couple to the W. The net amplitude for the process is the sum. Using the quark wave functions : The overall coefficient is simply cos, as claimed before. In the decay :the quark process is the same But : we get an extra factor

49. Physics 842, February 2006 Bogdan Popescu Example : Semileptonic Decays The decay rate :

50. Physics 842, February 2006 Bogdan Popescu - CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion - CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix WEAK INTERACTION (1)

51. Physics 842, February 2006 Bogdan Popescu Cabibbo-GIM mechanism GIM = GLASHOW, ILIOPOULOS, MAIANI The decay is allowed by Cabibbo theory. Amplitude :, far greater. GIM introduced the fourth quark c (1970). The couplings with s and d :

52. Physics 842, February 2006 Bogdan Popescu Cabibbo-GIM mechanism Now the diagrams cancel.

53. Physics 842, February 2006 Bogdan Popescu Cabibbo-GIM mechanism The Cabibbo-GIM mechanism : Instead of the physical quarks d and s, the “correct” states to use in the weak interactions are d’ and s’ : In matrix form : The W’s couple to the “Cabibbo-rotated” states :

54. Physics 842, February 2006 Bogdan Popescu - CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion - CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix WEAK INTERACTION (1)

55. Physics 842, February 2006 Bogdan Popescu Kobayashi-Maskawa (KM) matrix The weak interaction quark generations KM is a generalization of Cabibbo-GIM for three generations of quarks. are related to the physical quarks states by Kobayashi-Maskawa (KM) matrix for example : Canonical form of KM matrix depend only on three generalized Cabibbo angles and one phase factor.

56. Physics 842, February 2006 Bogdan Popescu Kobayashi-Maskawa (KM) matrix The full matrix : Using the experimental values :

57. Physics 842, February 2006 Bogdan Popescu END OF PART ONE WEAK INTERACTION THANK YOU !