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Physics 842, February 2006

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

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Physics 842, February 2006

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

  2. WEAK INTERACTION (1) - 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 Physics 842, February 2006 Bogdan Popescu

  3. 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 MW or MZ In practice very often : The propagator for W or Z in this case : Physics 842, February 2006 Bogdan Popescu

  4. 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 ge in QED and gs in QCD) : Physics 842, February 2006 Bogdan Popescu

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

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

  7. 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 : Physics 842, February 2006 Bogdan Popescu

  8. WEAK INTERACTION (1) - 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 Physics 842, February 2006 Bogdan Popescu

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

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

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

  12. 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 : Physics 842, February 2006 Bogdan Popescu

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

  14. DECAY OF THE MUON integration : where : The limits of E2and E4 integrals : Physics 842, February 2006 Bogdan Popescu

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

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

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

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

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

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

  21. 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 Physics 842, February 2006 Bogdan Popescu

  22. WEAK INTERACTION (1) - 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 Physics 842, February 2006 Bogdan Popescu

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

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

  25. 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 : Physics 842, February 2006 Bogdan Popescu

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

  27. DECAY OF THE NEUTRON The range of E2integral : E is the electron energy ( exact equation) Physics 842, February 2006 Bogdan Popescu

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

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

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

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

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

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

  34. WEAK INTERACTION (1) - 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 Physics 842, February 2006 Bogdan Popescu

  35. 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” Physics 842, February 2006 Bogdan Popescu

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

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

  38. WEAK INTERACTION (1) - 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 Physics 842, February 2006 Bogdan Popescu

  39. 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 : Coupling within a generation : There exist cross-generational coupling as : Physics 842, February 2006 Bogdan Popescu

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

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

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

  43. 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 Physics 842, February 2006 Bogdan Popescu

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

  45. WEAK INTERACTION (1) - 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 Physics 842, February 2006 Bogdan Popescu

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

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

  48. 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 : Physics 842, February 2006 Bogdan Popescu

  49. WEAK INTERACTION (1) - 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 Physics 842, February 2006 Bogdan Popescu

  50. Kobayashi-Maskawa (KM) matrix KM is a generalization of Cabibbo-GIM for three generations of quarks. The weak interaction quark generations 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. Physics 842, February 2006 Bogdan Popescu

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