1. Aug 29-31, 2005M. Jezabek1 Generation of Quark and Lepton Masses in the Standard Model International WE Heraeus Summer School on Flavour Physics and CP Violation Dresden, 29 Aug – 7 Sep 2005

2. Aug 29-31, 2005M. Jezabek2 Preliminaries Metric: g 00 = -g 11 = -g 22 = -g 33 = 1 Dirac field Ψ (s = ½ ) { γ μ, γ ν } = 2g μν, { γ μ, γ 5 } = 0 (μ,ν = 0, 1, 2, 3) γ 5 † = γ 5, Tr γ 5 = 0, γ 5 2 = I 4 Left- (right-) handed fields: γ 5 Ψ L = - Ψ L, γ 5 Ψ R = Ψ R Ψ L = ½ (1 - γ 5 ) Ψ, Ψ R = ½ (1 + γ 5 ) Ψ

3. Aug 29-31, 2005M. Jezabek3 Weak charged current (one generation) and weak interaction (four fermion) Hamiltonian are invariant under chiral transformations Mass terms flip chirality and break the chiral invariance of the weak interaction theory

4. Aug 29-31, 2005M. Jezabek4 Weyl spinors In the chiral (Weyl) basis with

5. Aug 29-31, 2005M. Jezabek5 Dirac equation (m = 0): with: for m = 0 chirality ↔ helicity

6. Aug 29-31, 2005M. Jezabek6 Lorentz transformations with The generators of rotations J i and of boosts Κ i satisfy the commutation relations: In a more convenient basis

7. Aug 29-31, 2005M. Jezabek7 For Weyl fields ( ) with  For right – handed fermions → ( ½, 0 ) left– handed fermions → ( 0, ½ ) of

8. Aug 29-31, 2005M. Jezabek8 Parity For the generators of rotations and boosts (pseudo - vectors) (vectors) and Under parity transformation:

9. Aug 29-31, 2005M. Jezabek9 Charge conjugation For the Pauli matrices and Under a Lorentz transformation

10. Aug 29-31, 2005M. Jezabek10 Charge conjugated bi-spinor: with

11. Aug 29-31, 2005M. Jezabek11 Charge – conjugation matrix C For a Dirac particle in em field Complex conjugation flips the relative sign between and with

12. Aug 29-31, 2005M. Jezabek12 If one obtains The charge conjugation matrix fulfills the relations:

13. Aug 29-31, 2005M. Jezabek13 Fermion masses 1.Dirac mass term For two Weyl spinors and is invariant under Lorentz and parity transformations: In bi-spinor notation

14. Aug 29-31, 2005M. Jezabek14 2.Majorana mass term For a left – handed Weyl spinor is a right –handed object and is Lorentz invariant In bi-spinor notation: is a right – handed field

15. Aug 29-31, 2005M. Jezabek15 The Majorana mass term reads and for a right – handed field

16. Aug 29-31, 2005M. Jezabek16 GWS Theory of Electroweak Interactions local gauge symmetry doublet SU 2 singlet SU 2 The most general unitary transformation includes the lepton – number phase transformation which is not a local gauge symmetry.

17. Aug 29-31, 2005M. Jezabek17 Generators: - Pauli matrices - em charge operator The gauge fields interact with matter fields and through the covariant derivative where

18. Aug 29-31, 2005M. Jezabek18 Spontaneous Symmetry Breaking Scalar field: with non – zero vacuum expectation value In the unitary gauge

19. Aug 29-31, 2005M. Jezabek19 Higgs mechanism: Mass eigenstates: with:

20. Aug 29-31, 2005M. Jezabek20 and U(1) Q gauge symmetry remains unbroken The electromagnetic field couples to

21. Aug 29-31, 2005M. Jezabek21 Quarks

22. Aug 29-31, 2005M. Jezabek22 Fermion masses Yukawa couplings: where: Note: in some extended models and may be different Higgs doublets

23. Aug 29-31, 2005M. Jezabek23 respectively is invariant. For example: forrespectively

24. Aug 29-31, 2005M. Jezabek24 Under weak isospin transformations where is a unitary matrix and detU = 1 transforms as a SU 2 doublet. For the mass terms:

25. Aug 29-31, 2005M. Jezabek25 The vacuum expectation value of breaks chiral symmetry. For example: Dirac masses for charged leptons and quarks If a right – handed neutrino exists: Problem: why ?

26. Aug 29-31, 2005M. Jezabek26 Generations Three generations of quarks and leptons :

27. Aug 29-31, 2005M. Jezabek27 Yukawa couplings + SSB with: The mass matrices are complex and can be diagonalised by bi – unitary transformations

28. Aug 29-31, 2005M. Jezabek28 with D diagonal, and U and V unitary. Any (n x n) complex matrix can be diagonalised by a bi – unitary transformation Proof and hermitian The eigenvalues of and are the same real and non – negative unitary U and V: with D diagonal and real. The columns of AV and U are proportional

29. Aug 29-31, 2005M. Jezabek29 Diagonalisation of the mass matrices with diagonal leads to relations between the mass eigenstates and the weak interactions eigenstates : Weak charged current for quarks Quark mixing matrix (Cabibbo; Kobayashi, Maskawa) Note: in GWS theory only is observable

30. Aug 29-31, 2005M. Jezabek30 For (n x n) unitary matrix : n real parameters -(2n - 1) phases of and  p = ( n – 1 ) 2 observable real parameters A common convention: where rotation angles complex phases For n = 3 (e.g. PDG): with Parameters: and

31. Aug 29-31, 2005M. Jezabek31 Masses of neutrinos A.Dirac neutrinos Three right – handed sterile neutrinos B.Dirac neutrinos in Majorana form with

32. Aug 29-31, 2005M. Jezabek32 ↑ anticommutation of fermion fields The matrices of Majorana masses are symmetric: ↑ and are antisymmetric, anticommutation of fermion fields

33. Aug 29-31, 2005M. Jezabek33 Neutrino masses – general case with is a symmetric complex matrix of dimension (3 + n R ) x (3 + n R ) Note: for quarks and charged leptons due to electric charge conservation and

34. Aug 29-31, 2005M. Jezabek34 Any symmetric complex n x n matrix can be diagonalised by a transformation: where U is unitary, with real and Let where and are real and symmetric, and is a 2n x 2n real and symmetric matrix

35. Aug 29-31, 2005M. Jezabek35 Let and i. e.  Eigenvalues of are real and equal to and at least n of them are non - negative which implies that

36. Aug 29-31, 2005M. Jezabek36 Let where V and W are real matrices which fulfill the following system of equations: Solution: For k-th columns in one obtains It follows that is fulfilled and M is diagonalised by Unitarity of U:

37. Aug 29-31, 2005M. Jezabek37 Massive neutrinos in the Standard Model Before 1998 (SuperK): A simple extention of SM: The right – handed neutrinos are sterile. For singlets of gauge group SU 3 x SU 2 x U 1 explicit Majorana masses are allowed  a new mass scale|M R | Two mass scales: The Majorana masses of the active neutrinos are forbidden by the electroweak SU 2 x U 1 gauge symmetry M L = 0

38. Aug 29-31, 2005M. Jezabek38 Seesaw Mechanism For the mass spectrum splits into low and high mass parts: with a unitary matrix and with:

39. Aug 29-31, 2005M. Jezabek39 with  M / is in a block – diagonal form if

40. Aug 29-31, 2005M. Jezabek40 with

41. Aug 29-31, 2005M. Jezabek41 Low mass sector For

42. Aug 29-31, 2005M. Jezabek42