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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.

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(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!!

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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:

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

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TRI-BIMAX (HPS) MIXING

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

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S3 GROUP MIXING Magic-Square Mixing

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

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

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

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

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

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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.

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

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

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

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

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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!!

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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. ???

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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. !!

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

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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)

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

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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

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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!)

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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.

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

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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.

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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

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SPARE SLIDES

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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

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

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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

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

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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

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

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

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

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TRIMAXIMAL MIXING)

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T. Araki et al. hep-ex/0406035

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

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

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

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

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

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

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

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

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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)

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SO FINALLY

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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)

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

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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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