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EXTREMISATION OF JARLSKOG INVARIANTS JARLSKOG INVARIANCE: U(3) Diagonal Non-Diagonal Diagonal OBSERVABLES JARLKOG INVARIANT FUNDAMENTAL LAWS JARLSKOG COVARIANT.

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Presentation on theme: "EXTREMISATION OF JARLSKOG INVARIANTS JARLSKOG INVARIANCE: U(3) Diagonal Non-Diagonal Diagonal OBSERVABLES JARLKOG INVARIANT FUNDAMENTAL LAWS JARLSKOG COVARIANT."— Presentation transcript:

1 EXTREMISATION OF JARLSKOG 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. Phys. Lett. B 628 (2005) 93. hep-ph/0508012 P. F. Harrison and W. G. Scott W. G. SCOTT RAL/SOTON MEET: 3/3/06 WEAK-BASIS INV.

2 (WEAK-BASIS)IN THE STANDARD MODEL: Universal Weak-Interaction Up Mass MatrixDown Mass Matrix You can have any 2 but NOT all 3 matrices diagonal!!

3 i.e. LEADS TO TRIMAXIMAL MIXING!! THE ARCHITYPAL JARLSKOG INVARIANT: THE JARLSKOG DETERMINANT: The Determinant of the Commutator of mass matrices: Extremising the Jarlskog Invariant J leads to:

4 TRIMAXIMAL MIXING HS PLB 333 (1994) 471. hep-ph/9406351 Originally proposed for the quarks!!

5 TRI-BIMAX (HPS) MIXING

6 S3 GROUP MIXING MAGIC-SQUARE MIXING (GENERALISES TRIMAX. AND TRI-BIMAX MIXING)

7 S3 GROUP MIXING Magic-Square Mixing

8 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. SOLAR REACT. (MINOS SOON!)

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11 THE 5/9-1/3-5/9 BATHTUB

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

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

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

15 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.

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

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

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

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

21 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!!

22 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. ???

23 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. !!

24 2 x 2 MAXIMAL MIXING Not Bad!! - but trivial 2 x 2 Max. solution excluded by solar data!!

25 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)

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

27 THE ASSOCIATED LAGRANGE MULTIPLIERS Fixing the Lagrange multipliers: 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

28 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!)

29 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.

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

31 KOIDES RELATION: And finally, the associated Lagrange Multipliers: When we have the perfect action all LMs will vanish!! Where eg. Y. Koide, Lett. Nuov. Cim. 34 (1982) 201.

32 CONCLUSIONS 1) Extremise Tr C^3 -> tri-max 2) Extremise Tr C^2 -> 2 x 2-max + non-trivial solution not in agreement with experiment

33 SPARE SLIDES

34 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

35 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! ATMOS.

36 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

37 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! IS PHASE- CONVENTION INDEPENDENT:

38 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

39 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! ATMOS.

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

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44 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. REACT.

45 TRIMAXIMAL MIXING)

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

47 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. REACT.

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50 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. SOLAR

51 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. SOLAR REACT.

52 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! ATMOS. SOLAR

53 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 !!

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

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

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

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

59 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)

60 SO FINALLY

61 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)

62 TRI-BIMAXIMAL (HPS) MIXING AT LEAST APPROXIMATELY !!!! ROWS/COLUMNS SUM TO UNITY

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65 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 ???

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