1. A VARIATIONAL PRINCIPLE IN ACTION? SYMMETRIES OF NEUTRINO MIXING: P. F. Harrison, D. H. Perkins and W. G. Scott Phys. Lett. B 530 (2002) 167. hep-ph/0202074 P. F. Harrison and W. G. Scott Phys. Lett. B 535 (2002) 163. hep-ph/0203209 Phys. Lett. B 547 (2002) 219. hep-ph/0219197 Phys. Lett. B 557 (2003) 76. hep-ph/0302025 Phys. Lett. B 594 (2004) 324. hep-ph/0403278 W. G. SCOTT @ RL. AC. UK CERN-TH-SEMINAR 13/01/06 TRI-BIMAXIMAL (“HPS”)-MIXING EXTREMISATIONPhys. Lett. B 628 (2005) 93. hep-ph/0508012 SYMMETRIES “DEMOCRACY” “MUTAUTIVITY”

2. TRIBIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! c.f. G. Altarelli and F. Feruglio hep-ph/9807353 with HPS PLB 458 (1999) 79. hep-ph/9904297; WGS hep-ph/0010335

3. TRI-BIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! IS PHASE- CONVENTION INDEPENDENT:

4. TRI-BIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! ROWS/COLUMNS SUM TO UNITY

5. TRI-BIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! ATMOS.

6. M. Ishituka hep-ph/0406076 Oscillation 37.8/40 Decay 49.2/40 Decoherence 52.4/40

7. TRI-BIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! ATMOS.

8. TRI-BIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. REACT.

9. T. Araki et al. hep-ex/0406035

10. TRI-BIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. REACT.

11. TRI-BIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. SOLAR REACT.

14. THE “5/9-1/3-5/9” BATHTUB

15. TRIMAXIMAL MIXING: “ We are probably far from this….. but not very far…” N. Cabibbo: Lepton-Photon 2001 HS PLB 333 (1994) 471. hep-ph/9406351 (for the quarks!) HPS PLB 349 (1995) 357. http://hepunx.rl.ac.uk/scottw/ L. Wolfenstein PRD 18 (1978) 958. N. Cabibbo PL 72B (1978) 222. (cf. C3 CHARACTER TABLE) MAXIMAL CP-VIOLATION !!

16. MASS MATRICES: 3 x 3 circulant2 x 2 circulant Diagonalise:eigen-vecs eigen-vals (ASSUMED HERMITIAN

17. TRIMAXIMAL MIXING

18. TRI-BIMAX (“HPS”) MIXING

19. “S3 GROUP MIXING” “MAGIC-SQUARE MIXING” (GENERALISES TRIMAX. AND “HPS” MIXING)

20. UP-TO-DATE FITS A. Strumia and F. Vissani Nucl.Phys. B726 (2005) 294. hep-ph/0503246 IS THE BEST MEASURED MIXING ANGLE !!!

21. SYMMETRIES OF “HPS” MIXING M = 0 SUBSET OF CLEBSCH- GORDAN COEFFS. e.g. COULD PERHAPS BE A USEFUL REMARK ?!! See: J. D. Bjorken, P. F. Harrison and W.G. Scott. hep-ph/0511201

22. SYMMETTRIES OF “HPS” MIXING:

23. CYCLIC C3 GROUP: NAT. REP.

24. C3 GROUP MATRIX: NAT. REP. CIRCULANT TRIMAX. MIXING

25. SYMMETRIC S3 GROUP: NAT. REP.

26. S3 GROUP MATRIX: NAT. REP. RETRO-CIRC. CIRC. (FLAVOUR BASIS) S3 GROUP MIXING (i.e. charged-leptons diagonal)

27. S3 GROUP MIXING “Magic-Square Mixing”

28. An S3 GROUP MATRX Commutes with THE “DEMOCRACY” OPERATOR: DENICRACY SYMMETRY/INVARIANCE (and the converse) Conserved Quantum Nos. etc. c.f. “The Democratic Mass matrix” (S3 “CLASS OPERATOR”)

29. Slightly Differently....

30. S3 CLASS ALGEBRA: NAT. REP.

31. S3 CLASS MATRIX: NAT. REP. TRI-BIMAXIMAL (“HPS”) MIXING!! MASS BASIS)( (i.e. neutrino mass matrix diagonal)

33. NAT. REP. S3 CLASS OPERATORS:

34. CLAS-MATRIX: NAT. REP. S3 TRI-BI-MAX. MIXING ( FLAVOUR BASIS )

36. SO FINALLY

39. EXTREMISATION (i.e. MAXIMISATION OR MINIMISATION) OF JARLSKOG INVARIANTS ( “WEAK-BASIS” INVARIANTS ) JARLSKOG INVARIANCE: U(3) Diagonal Non-Diagonal Diagonal OBSERVABLES JARLKOG INVARIANT FUNDAMENTAL LAWS JARLSKOG COVARIANT !! Universal Weak Interact. e.g. for the quarks: Universal Weak Interact.

40. FLAVOUR-SYMMETRIC Charged-Leptons: Mass Matrix: JARLSKOG INVARIANT MASS PARAMETERS Neutrinos: Mass Matrix:

41. THE CHARACTERISTC EQUATION e.g. For the Charged-Lepton Masses: where: The Disciminant: ALL JARLSKOG INVARIANT!!

42. EXTREMISATION: A TRIVIAL EXAMPLE In the SM: NOT BAD!! Add to SM Action, the determinant : (taken here to be dimensionless) i. e. Yukawa couplings HS PLB 333 (1994) 471. hep-ph/9406351 e.g.

43. i.e. LEADS TO TRIMAXIMAL MIXING!! THE ORIGINAL FLAVOUR-SYMMETRIC JARLSKOG MIXING INVARIANT: The Determinant of the Commutator: Extremising the Jarlskog Invariant J leads to:

44. MATRIX CALCULUS THEOREM: A any constant matrix, X a variable matrix WHEREBY e.g:

45. EXTREMISING Tr With No Constraints: Differentiate Mass Constraints: With Mass Constraints Implemented: = Lagrange Multipliers (FOR FIXED MASSES)

47. EXTREMISING Tr Eq. 1, off-diagonal elements, Re parts: (CONTINUED) MAGIC-SQUARE CONSTRAINT!! Non-Trivial Solution: i.e.

48. EXTREMISING Tr Eq.1 off-diagonal elements, Im parts: (CONTINUED 2) Non-Trivial Solution: CIRCULANT MASS-MATRIX i.e. TRIMAXIMAL MIXING!!!

49. Increibly, all the remaining equations are either redundant or serve only to fix the lagrange multipliers Above remains true in all the extremisations we performed!! JARLSKOG SCALARS!!

50. K-matrix THE SUM OF THE 2 x 2 PRINCIPAL MINOIRS: The K-matrix is the CP-symmetric analogue of Jarlskog J: Plaquette Products Extremise (in a hierachical approximation) wrt PDG: 2 x 2 MAX-MIX. ???

51. SO NOW TRY EXTREMISING Tr Eq. 1, off-diagonal elements, Re parts: Eq.1 off-diagonal elements, Im parts: Triv. Solns:2 x 2 MAX. MIX. !!

52. EXTREMISING Tr Non-Trivial Solution: (it turns out, we need only consider ) withadjusted to give “observed” Absolute masses not yet measured, but with the “minimalist” assumption of a normal classic fermionic neutrino spectrum we have a unique prediction for the mixing: (CONTINUED)

53. NON-TRIVIAL CP-CONSERVING MIXING SUGGESTIVE, BUT NOT CONSISTENT WITH DATA !! Setting:

54. THE ASSOCIATED LAGRANGE MULTIPLIERS Fixing the Lagrange,ultipliers: These Lagrange Mults. are specific to the non-trivial soln. i.e. they fail for the 2 x 2 Max. solution!!! Assume the Non-Trivial Solution

55. A COMPLETE SET OF MIXING VARIABLES Higher powers of L,N need not be considered by virtue of the characteristic equation: hence 9 Quadratic Commutator Invariants, of which 4 are functionally independent, e.g. The Q-matrix is a moment-transform of the K-matrix: (flavour-symmetric mixing variables!)

56. EXTREMISE IMPROVED “EFFECTIVE” ACTION {,}=AntiCommutator Gives trajectory of solutions depending on the parameter q To locate realistic soln. impose “magic-square constraint” n.b. The inherent cyclic symmetry of the solution means that the magic-square constraint removes one parameter - not two.

57. NON-TRIVIAL CP-CONSERVING MIXING i.e. APPROX. “HPS” MIXING !!! Focus on pole atand deviations Setting COVARIANT STATEMENT OF REALISTIC MIXING!!!

58. KOIDE’S RELATION: And finally, the associated Lagrange Multipliers: When we have the “perfect action” all LMs will vanish!! where

59. A VARIATIONAL PRINCIPLE IN ACTION? SYMMETRIES OF NEUTRINO MIXING: P. F. Harrison, D. H. Perkins and W. G. Scott Phys. Lett. B 530 (2002) 167. hep-ph/0202074 P. F. Harrison and W. G. Scott Phys. Lett. B 535 (2002) 163. hep-ph/0203209 Phys. Lett. B 547 (2002) 219. hep-ph/0219197 Phys. Lett. B 557 (2003) 76. hep-ph/0302025 Phys. Lett. B 594 (2004) 324. hep-ph/0403278 W. G. SCOTT @ RL. AC. UK CERN-TH-SEMINAR 13/01/06 TRI-BIMAXIMAL (“HPS”)-MIXING EXTREMISATIONPhys. Lett. B 628 (2005) 93. hep-ph/0508012 SYMMETRIES “DEMOCRACY” “MUTAUTIVITY”

60. SPARE SLIDES

64. TRIMAXIMAL MIXING)

67. TRI-BIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. SOLAR

68. TRI-BIMAXIMAL (“HPS”) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. SOLAR

70. S3 GROUP MIXING (TRI-MAX. MIXING) GENERALISES TBM:

71. S3 GROUP MIXING (TRI-MAX. MIXING) GENERALISES TBM:

72. TRI-MAXIMAL MIXING: “ We are probably far from this….. but not very far…” N. Cabibbo: Lepton-Photon 2001 HPS PLB 349 (1995) 357 N. Cabibbo PL 72B (1978) 222. (cf. C3 CHARACTER TABLE)

75. FLAVOUR-SYMMETRIC MIXING INVARIANTS: 1) The Determinant of the Commutator: 2) The Sum of the 2x2 Principal Minors: K-matrix ie. TRIMAX. MIX!! TRI-BIMAX ???