A VARIATIONAL PRINCIPLE IN ACTION? SYMMETRIES OF NEUTRINO MIXING: P. F. Harrison, D. H. Perkins and...

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 . UKCERN-TH-SEMINAR 13/01/06

TRI-BIMAXIMAL(“HPS”)-MIXING

EXTREMISATION

Phys. Lett. B 628 (2005) 93. hep-ph/0508012

SYMMETRIES“DEMOCRACY”

“MUTAUTIVITY”

2

1

3

1

6

12

1

3

1

6

1

03

1

3

2

321

e

U

TRIBIMAXIMAL (“HPS”) MIXING

AT LEAST APPROXIMATELY !!!!

c.f. G. Altarelliand F. Ferugliohep-ph/9807353with 31/ sin

HPS PLB 458 (1999) 79. hep-ph/9904297; WGS hep-ph/0010335

TRI-BIMAXIMAL (“HPS”) MIXING


21

31

61

21

31

61

31

32 0

321

e

2|| lU

IS PHASE-CONVENTION INDEPENDENT:

2||U



21

31

61

21

31

61

31

32 0

321

e

2|| lU

ROWS/COLUMNSSUM TO UNITY



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

M. Ishituka hep-ph/0406076

Oscillation 37.8/40 Decay 49.2/40Decoherence 52.4/40

2/1)( P0.11) 0.50 ||( 2

3 U



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

REACT.

/ KAMLANDVERDE- / PALOCHOOZ : ESPREACTORS

95)( eeP09.061.0

KAMLANDex/0212021-hep

al.et EguchiK.03.0 ||

||21 2

3

23

e

e

UUP

T. Araki et al. hep-ex/0406035

064.0 658.0



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

REACT.



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

SOLAR

REACT.

0502021/055502. (2005) 72 Phys.Rev.C

..

exnucl

aletAharmimB

0204008/

...

exnucl

aletAhmadRQ

011301 (2002) 89 ett.Phys.Rev.L

OD2 Pure SaltOD2

46.043.0

44.043.0

09.009.0

06.005.0

09.5

76.1

NC

CC28.027.0

19.019.0

11.011.0

05.005.0

81.4

72.1

NC

CC

036.0034.0

032.0031.0346.0

/

NCCC028.0029.0

021.0021.0358.0

/

NCCC

spectrum)-Bd undistorte assuming given thoseare here quoted (Results 8

ResultsSNO

028.0354.0 / NCCC)

(

errorssystematicinncorrelatio

ignoresaveragenaivemy

Solar Datadist. spect. B and . systs corr. ignoring -Salt NoSalt of

average naivemy is 03.035.0/point SNO8

NCCC

ph/9601346-hep 111 (1996) 374 PLB also see ;ph/0202074-hep 167 (2002) 530 PLB HPS

ph/9601346-hep 111. (1996) 374 PLB HPS3 Fig.

MIX.TRIMAX.IN DICTEDPRE ! ! ! !

THE “5/9-1/3-5/9” BATHTUB

TRIMAXIMAL MIXING:

33

1

3

33

1

3

3

1

3

1

3

1

e

1 2 3

U

“ 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)

)3/2exp( )3/2exp(

ii

MAXIMAL CP-VIOLATION !!

)36/(1 CPJ

MASS MATRICES:

33

1

3

33

1

3

3

1

3

1

3

1

e

1 2 3UUU l

abb

bab

bba

M l

xy

z

yx

M

0

00

0

2

10

2

10102

10

2

1

23

1

6

123

1

6

1

03

1

3

2

i

i

e

1 2 3

3 x 3 circulant 2 x 2 circulant

} diag{

mmmUMU

e

lll

} diag{

321 mmmUMU

Diagonalise: eigen-vecseigen-vals

) MMM (ASSUMED HERMITIAN

31

31

31

31

31

31

31

31

31

2||U

321

e

TRIMAXIMAL MIXING

21

31

61

21

31

61

031

32

2||U

321

e

TRI-BIMAX (“HPS”) MIXING

*31*

*31*

*31*

2||U

321

e

“S3 GROUP MIXING”

“MAGIC-SQUARE MIXING”

(GENERALISES TRIMAX.AND “HPS” MIXING)

UP-TO-DATE FITS

A. Strumia and F. Vissani Nucl.Phys. B726 (2005) 294. hep-ph/0503246

03.0/ 223

212 mm

12 IS THE BEST MEASURED MIXING ANGLE !!!

0.50) tan( 0.05 0.45 tan HPS 12 2

12 2

SYMMETRIES OF “HPS” MIXING

2

1

3

1

6

111

2

1

3

1

6

111

03

1

3

200

000

102

21

Mmm

JM = 0SUBSET

OF

CLEBSCH-GORDANCOEFFS.

e.g.

1 1 21 jj

COULD PERHAPS BE

A USEFUL REMARK ?!!

See: J. D. Bjorken, P. F. Harrison and W.G. Scott. hep-ph/0511201

2

1

3

1

6

12

1

3

1

6

1

03

1

3

2321

e

2||UVIOLATION-CP ZERO 1)

N'`REFLECTIO 2)

VEC.- ETRIMAX. ONE 3)

)!!( HOPEFULLYEAPPROXIMAT

ph/0305309-hepoura Grimus/Lav

ph/0210197-hep

219 (2002) 647 PLBHS

ph/0302025-hep 76 (2003) 537 PLBHS

ph/0203209-hep

163 (2002) 535 PLBHS

} ).'(` CPMAX

SYMMETTRIES OF “HPS” MIXING:

CYCLIC C3 GROUP:

001

100

010

)123(P

010

001

100

)321(P

100

010

001

I

NAT. REP.

C3 GROUP MATRIX:

abb

bab

bba

NAT. REP.

)321()123(2 PbbPaIM

CIRCULANT

TRIMAX. MIXING

)43( AC

SYMMETRIC S3 GROUP:

001

100

010

)123(P

010

001

100

)321(P

100

010

001

I

NAT. REP.

100

001

010

)12(P

010

100

001

)23(P

001

010

100

)31(P

S3 GROUP MATRIX:

abb

bab

bba

NAT. REP.

)321()123(2 PbbPaIM

RETRO-CIRC. CIRC.

)12()31()23( zPyPxP

zxy

xyz

yzx

(FLAVOUR BASIS)

'`

33

SQUARESMAGIC

ORTHOGONAL

S3 GROUP MIXING

(i.e. charged-leptons diagonal)

623

1

26

623

1

26

3

2

3

2

3

1

3

2

3

2

cisscsiscccisscsiscc

cisscsiscccisscsiscc

csiscssicc

U321

e

zxyzxyyxbbam

zyxbam

zxyzxyyxbbam

22223

2

22221

) 3(Im Re

Re2

) 3(Im Re

)2/()(32 tan

)/()(Im62 tan 222

xyxyz

zxyzxyzyxb

S3 GROUP MIXING “Magic-Square Mixing”

111

111

111

De

e

22 MUMU

DD 0, 2 MDD

DieU

31

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 “CLASSOPERATOR”)

: TABLE . MULTIPLCLASS S3

3

2

1

c

c

c

2133

3212

321

22

22/

cccc

cccc

ccc

321 ccc

NORMALISED

3

)31()23()12(

2

)321()123(:: 321

ccIc

: OPERATORS CLASS S3

Slightly Differently....

S3 CLASS ALGEBRA:

200

02/11

010

2c

021

200

100

3c

100

010

001

1c

NAT. REP.

S3 CLASS MATRIX:

qprr

rqpq

rqp

22

22/

NAT. REP.

3212 rcqcpcM l

TRI-BIMAXIMAL (“HPS”) MIXING!!

v MASS BASIS)(

(i.e. neutrino mass matrix diagonal)

MIXINGCLASS S3

:IDENTIFY

222

22222

3

2

6

32362

e

e

mmm

p

mmr

mmmq

2

1

3

1

6

12

1

3

1

6

1

03

1

3

2

e

321

U

).3.( TABLECHARACTERGROUPSTHEcf

NAT. REP. S3

010

100

001

)()2( PC

111

111

111

)()()( ePPeP

100

010

001

)1( IC

OPERATOREXCHANGE

MATRIXMASSDEMOCRATIC

OPERATORIDENTITY)1,'`( MMUTATIVITY

)3,0,'`( DDEMOCRACY

321 SSS

CLASS OPERATORS:

)1,'`( IIDENTITY

CLAS-MATRIX:

usutu

utusu

uuuts

NAT. REP. S3

)3()2()1(2 CuCtCsM

TRI-BI-MAX. MIXING

( FLAVOUR BASIS )321 SSS

):'(` 2iiii uDtMsImFORMULAMASS

MIXING32 SS

:IDENTIFY

322

21

22

23

21

23

21 mm

umm

tmm

s

2

1

3

1

6

12

1

3

1

6

1

03

1

3

2

e

321

U

!!!. AGAINBIMAXTRI

321

e

U

2

1

3

1

6

12

1

3

1

6

1

03

132

)1,0()1,1()1,0(

)iM,(Di

i

i

M

D

MUTATIVITY

DEMOCRACY

!! NUMBERS QUANTUM CONSERVED iiMD

SO FINALLY

010

100

001

PC: M

e

e

: Y"MUTAUTIVIT"

GENERAL IN UNITARY-ANTI IS MUTATIVITY NB.

NCONJUGATIOCOMPLEXC

P

EINTERCHANGP

ph/0403278-hep 324. (2004) 594 PLB HS

Matrix" Mass Neutrino Simplest The"

011

101

110

3010

100

001

100

010

001d

ixa2M

:PREDICTS

13.032sin 2213 atmsol mm

SYMM. C3

IMAG. PURE

EXTREMISATION(i.e. MAXIMISATIONOR MINIMISATION)

OF JARLSKOG INVARIANTS

b

s

d

1

1

1

tcu

( “WEAK-BASIS” INVARIANTS )

b

s

d

1

1

1

tcu

JARLSKOG INVARIANCE:

U(3)

Diagonal Non-Diagonald

u

MM Non-Diagonal

Diagonal d

u

MM

OBSERVABLES JARLKOG INVARIANT

FUNDAMENTAL LAWS JARLSKOG COVARIANT !!

Universal Weak Interact.

e.g. for the quarks:

Universal Weak Interact.

33333

22222

1

Tr :

Tr :

Tr :

mmmLL

mmmLL

mmmLL

e

e

e

FLAVOUR-SYMMETRIC

Charged-Leptons: Mass Matrix: lML :

JARLSKOG INVARIANT MASS PARAMETERS

} {

} { 321

mmm

LLL

e

33

32

31

33

23

22

21

22

3211

Tr : Tr : Tr :

mmmNNmmmNNmmmNN

} {

} {

321

321

mmm

NNN

MN : Neutrinos: Mass Matrix:

6/)23( Det

)/2( Pr

Tr

32131

221

1

LLLLmmmL

LLmmmmmmL

LmmmL

e

ee

e

THE CHARACTERISTC EQUATION

e.g. For the Charged-Lepton Masses:

0 ) (Det ) Pr( ) (Tr 23 LLL where:

The Disciminant:

222

613

31

23

22

21

321241

32

2

) ()()( 6/3/432/7

62/32/

ee mmmmmmLLLLLL

LLLLLLL

ALL JARLSKOG INVARIANT!!

z yxA

EXTREMISATION: A TRIVIAL EXAMPLE

In the SM:

NOT BAD!!

z y

x

mmme

GeV 180 2

v

Add to SM Action, the determinant :

0 y 0 0

xAzxAzyA

z

y

x

0 0 0

zyx

mmmL e Det (taken here to be dimensionless) i. e.

zyx , ,Yukawa couplings

HS PLB 333 (1994) 471. hep-ph/9406351

e.g.

νl ΔJΔNLiCC /3, Tr /3Tr Det 33 )-m)(m-m)(m-m (m Δ) -m)(m-m)(m-m (m Δ νeττμμel 133221

ssssssssscscs cJ 132312213

2/1223

2/121213

21323231212 )1()1()1(

2

3

1

2

1

2

1132312

ssss

i.e. LEADS TO TRIMAXIMAL MIXING!!

)],[ : ( NLiC

THE ORIGINAL FLAVOUR-SYMMETRIC

JARLSKOG MIXING INVARIANT:

The Determinant of the Commutator:

Extremising the Jarlskog Invariant J leads to:

MATRIX CALCULUS THEOREM:

XX / : TX AAX Tr

A any constant matrix, X a variable matrix

TL CNiC ],[ 3 Tr 23

TL CNiC ],[ 2 Tr 2

WHEREBY e.g:

) !! 0 Tr

],[ : (

C

NLiC

EXTREMISING Tr

0 ],[ 3/Tr 0 ],[ 3/Tr

23

23

TN

TL

CLiCCNiC

3C

2210

23

2210

23

],[ 3/Tr ],[ 3/Tr

NNICLiCLLICNiC

NNNT

N

LLLT

L

With No Constraints:

Differentiate Mass Constraints:

0 )(Tr 0 )(Tr 0 )(Tr 3 ) (Tr 2 ) (Tr ) (Tr

33

22

1

22

32

21

NNNNNNLLLLLLILL

LLL

LLL

23

32

21

23

22

1

3 )(Tr 2 )(Tr )(Tr 0 ) (Tr 0 ) (Tr 0 ) (Tr NNNNNNINN

LLLLLL

NNN

NNN

With Mass Constraints Implemented:NiLi / = Lagrange

Multipliers

(FOR FIXED MASSES)

cidxidy

idxbidz

idyidza

2M

e

e

MATRIX-MASS NEUTRINO HERM. COMPLEX ARB.

:WRITTEN BE ALWAYS MAY BASIS) FLAVOUR (IN

PHASING-RE BY ,,e!!! TRUE BUT

INCREDIBLE"CONVENTION-PHASE EPSILON"

BASIS" EPSILON"

EXTREMISING Tr

0)( )()( ))((

0)( )()( ))((

0)( )()( ))((

yxbayxmmmmd

zzacxzmmmmd

zycbzymmmmd

e

e

ee

3C

2210

23

2210

23

],[ 3/Tr ],[ 3/Tr

NNICLiCLLICNiC

NNNT

N

LLLT

L

Eq. 1, off-diagonal elements, Re parts:

)()()()()()(

bayxacxzcbzy

(CONTINUED)

zcybxa

MAGIC-SQUARECONSTRAINT!!

Non-Trivial Solution:

i.e.

EXTREMISING Tr 3C

0 )())(( ))((

0 )())(( ))((

0 )())(( ))((

222

222

222

zyxxydbammmm

yzzzxdacmmmm

xzyyzdcbmmmm

e

e

ee

2210

23

2210

23

],[ 3/Tr ],[ 3/Tr

NNICLiCLLICNiC

NNNT

N

LLLT

L

Eq.1 off-diagonal elements, Im parts:

(CONTINUED 2)

and and and

xzaczycbyxba

Non-Trivial Solution:

CIRCULANT MASS-MATRIXi.e. TRIMAXIMAL MIXING!!!

))()(2(2

))()(2(2

))()(2(2

22L10

2222

22L10

2222

22L10

2222

mmzyxdmmmmmd

mmzyxdmmmmmd

mmzyxdmmmmmd

LLee

LLee

eLeLe

32

31213

2

32

31222

21

41

L1

32

321

2212

31

51

0

Tr 3

299

Tr 3

6372/3

Tr 3

22/732/

CL

LLLL

CL

LLLLLL

CL

LLLLLLL

L

L

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!!

JARLSKOGSCALARS!!

diag diag /2, Tr /2Tr 22

11

KNLiCQ

lTl

)-mm-mm-m (m ) -mm-mm-m (m νμeeττμl 211332 ,, ,,

etccscsscscssccs cKcscsscsscsccs cK

e

e

))()(( ))()((

223

223131212

213

212

2122323

21323231

223

223131212

213

212

2122323

21323231

) !! 0 Tr ],[ : (

CNLiC

K-matrix

2/1 || 13233 ccU

THE SUM OF THE 2 x 2 PRINCIPAL MINOIRS:

The K-matrix is the CP-symmetric analogue of Jarlskog J:

Plaquette Products iJKUUUU llllll :: *1 11 1

*1 11 1

Extremise (in a hierachical approximation) wrt PDG:

2 x 2 MAX-MIX. ???

SO NOW TRY EXTREMISING Tr

0 ]],[,[ )2/Tr (0 ]],[,[ )2/Tr (

2

2

NLLCNLNC

TN

TL

0))(())(2(

0))(())(2(

0))(())(2(

2

2

2

zbammxydmmm

yacmmzxdmmm

xcbmmyzdmmm

ee

ee

e

2C

0))(())(2(0))(())(2(

0))(())(2(

bammdyxmmmdacmmdxzmmmdcbmmdzymmmd

ee

ee

e

Eq. 1, off-diagonal elements, Re parts:

Eq.1 off-diagonal elements, Im parts:

Triv. Solns: ,0 .. cbzydge 2 x 2 MAX. MIX. !!

0)] ],[,[

],[F

/2Tr

/YMMaxell ..(2

FA

fc

EXTREMISING Tr

)2)(2(

))(( ))((

)2)(2(

))(( ))((

)2)(2(

))(( ))((

ee

e

ee

e

ee

ee

mmmmmm

mmmmTTcbacz

mmmmmm

mmmmMMbacby

mmmmmm

mmmmEEacbax

2C

Non-Trivial Solution: (it turns out, we need only consider 0 d )

cba , , 321 , , mmmwith adjusted to give “observed”

Absolute masses not yet measured, but with the “minimalist” assumption of a normal classic fermionic neutrino spectrum

, 321 mmm we have a unique prediction for the mixing:

(CONTINUED)

NON-TRIVIAL CP-CONSERVING MIXING

0003.50409.49587.

6663.16257.17079.

.3333333333.33333.321

e || 2

lU

02/12/1

3/26/16/1

3/13/13/1

e

SUGGESTIVE, BUT NOT CONSISTENT WITH DATA !!

03.0/ 223

212 mm1

ca

abSetting:

THE ASSOCIATED LAGRANGE MULTIPLIERS

Fixing the Lagrange ,ultipliers:

)(2)(2

)(2)(2

)(2)(2

22L10

22

22L10

22

22L10

22

mmymmxmm

mmxmmzmm

mmzmmymm

LLe

LLe

eLeLee

)299(2

2},{Tr 3

)299(2

)2},{Tr 3)(2},{Tr 3(

)299(2

)2},{Tr 3)(},{Tr },{Tr (

31213

112

31213

11122

L1

31213

11212

0

LLLL

NLNL

LLLL

NLNLNLNL

LLLL

NLNLLNLLNL

L

L

These Lagrange Mults. are specific to the non-trivial soln.

i.e. they fail for the 2 x 2 Max. solution!!!

Assume the Non-TrivialSolution

A COMPLETE SET OF MIXING VARIABLES

22222222

222222

2222

],[Tr ],][,[Tr ],[Tr

],][,[Tr ],][,[Tr ],][,[Tr

],[Tr ],][,[Tr ],[Tr

2

1

NLNLNLNL

NLNLNLNLNLNL

NLNLNLNL

Q

diag ) (diag ) (diag diag 11 nl

mll

Tlmn KQ

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.

],[ ],[Tr , ],[ ],[Tr

],[ ],[Tr ,],[Tr 22

222

21

212

211

NLNLQNLNLQ

NLNLQNLQ

The Q-matrix is a moment-transform of the K-matrix:

(flavour-symmetric mixing variables!)

EXTREMISE IMPROVED “EFFECTIVE” ACTION

))(1()(

))(1()(

))(1()(

2

3

2

3

2

3

mmqmm

kzzc

mmqmm

kyyb

mmqmm

kxxa

ee

ee

2111 qQQA

N]]/2 ,[,[ ) (2/]]},[,[,{ N]]/2 ,[,[ ) (

2T21

2T21

LLQNLNLLNQ

N

L

{,}=AntiCommutator

Gives trajectory of solutions depending on the parameter qTo 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.

NON-TRIVIAL CP-CONSERVING MIXING

547.0333.0120.0

448.0333.0219.0

0.005333.0662.0

321

e

|| 2lU

2/13/16/1

2/13/16/1

03/13/2

e

i.e. APPROX. “HPS” MIXING !!!

Focus on pole at )(

1

mmq

)(

)1(

mm

q

and deviations

0.005 ||03.0/

23

223

212

eUmmSetting 2)03.0(

0

COVARIANTSTATEMENT

OF REALISTICMIXING!!!

07.0 sin 13

KOIDE’S RELATION:

020765664512 41

212

2213 LLLLLL

2/11561862072/27

1890108278181

612

41

22

21

32

42

513

31231

223

21

232

2321

LLLLLLLL

LLLLLLLLLLLLL

22532

22

525

424

323

22221202

12 )2(2 qLqLLL

qqqqqN

PPPP

LLLLLLL

And finally, the associated Lagrange Multipliers:

When we have the “perfect action” all LMs will vanish!!

where

3

2

)( 2

mmm

mmm

e

e

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 . UKCERN-TH-SEMINAR 13/01/06

TRI-BIMAXIMAL(“HPS”)-MIXING

EXTREMISATION

Phys. Lett. B 628 (2005) 93. hep-ph/0508012

SYMMETRIES“DEMOCRACY”

“MUTAUTIVITY”

SPARE SLIDES

623

1

26

623

1

26

sin32

3

1cos

32

si

csi

c

si

csi

c

i

U

321

e

MIXING MAX. TRIyx

d

2 tan

VIOL. CP MAX. MUTATIVITY 2/ PDG

TBM 0 d

zy MUTATIVITY IMPOSE

yxy

xyy

yyx

zxy

xyz

yzx

:THAT SUCH

PARAMETERS-3

dyx ,,

22 , atmsol

0,2 MM

:SET " SYMMETRY

LECTIONREF "

623

1

26

623

1

26

sin32

3

1cos

32

si

csi

c

si

csi

c

i

U

321

e

MIXING MAX. TRIyx

d

2 tan

VIOL. CP MAX. MUTATIVITY 2/ PDG

TBM 0 d

31

31

31

31

31

31

31

31

31

2||U

321

e

TRIMAXIMAL MIXING)

12sin

104.2

:ValuesFit New

2

232

eVm

NOON2004 ItshitsukaSK -



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

SOLAR

0309004/

...

exnucl

aletAhmedNS

????. (2003) ?? ett.Phys.Rev.L

0204008/

...

exnucl

aletAhmadRQ

011301 (2002) 89 ett.Phys.Rev.L

OD2 Pure SaltOD2

46.043.0

44.043.0

09.009.0

06.005.0

09.5

76.1

NC

CC

29.027.0

24.024.0

09.010.0

07.007.0

90.4

70.1

NC

CC

036.0034.0

032.0031.0346.0

/

NCCC

028.0028.0

022.0022.0347.0

/

NCCC

spectrum)-Bd undistorte assuming given thoseare here quoted (Results 8

ResultsSNO

03.035.0/ NCCC )

(

errorssystematicinncorrelatio

ignoresaveragenaivemy

*31*

*31*

*31*

2||U

321

e

S3 GROUP MIXING

(TRI- MAX. MIXING)

GENERALISES TBM:

*31*

*31*

*31*

2||U

321

e

S3 GROUP MIXING (TRI- MAX. MIXING)

GENERALISES TBM:

TRI-MAXIMAL MIXING:

33

1

3

33

1

3

3

1

3

1

3

1

e

1 2 3

U

“ We are probably far from this…. . but not very far…”

N. Cabibbo:

Lepton-Photon 2001

HPS PLB 349 (1995) 357N. Cabibbo PL 72B (1978) 222.

(cf. C3 CHARACTER TABLE)

SUMMARY

DATATHEWITH CONSISTENTTBM )1SYMMETRIES THREE TBM HAS )2

MIXING'GROUP `S3 )3

MIXING'CLASS `S3 )4

MIXING'S3 `S2 )5

) BASIS FLAVOUR(

) BASIS FLAVOUR(

) BASISMASS- (

*31*

*31*

*31*

) SNO ESP.(

CP, ZERO ( ) TRIMAX. ,REFLECTION 2

2||U

TBM

TBM

TBM

321

e

ph/0403278-hep ph/0308282-hep 157 (2004) 583 PLB Ma E.

Matrix" Mass Neutrino TriPartite The"

. ..).( 1 jiijc

kjik

ijjiij llf

llfllh

L

higgs

Isotriplet

higges

Isodoublet

i5 Dimension

higgs SM

Invariant "" 23 ZZ "democratic"

Invariant 3S

1 1 3

BBTB

kB

kTB

BTB

UUU

fUfU

hhUU

4/14/38/3

4/34/18/3

8/38/32/1BU

νl ΔJΔNLiCC /3, Tr /3Tr Det 33 )-m)(m-m)(m-m (m Δ) -m)(m-m)(m-m (m Δ νeττμμel 133221

ssssssssscscs cJ 132312213

2/1223

2/121213

21323231212 )1()1()1(

FLAVOUR-SYMMETRIC MIXING INVARIANTS:

1) The Determinant of the Commutator:

2) The Sum of the 2x2 Principal Minors:

diag diag

/2, Tr /2Tr 2211

KNLiCQ

lTl

)-mm-mm-m (m ) -mm-mm-m (m νμeeττμl 211332 ,, ,,

etccscsscscssccs cKcscsscsscsccs cK

e

e

))()(( ))()((

223

223131212

213

212

2122323

21323231

223

223131212

213

212

2122323

21323231

) !! 0 Tr ],[ : (

CNLiC

K-matrix

PDG wrt Extremise

ie. TRIMAX. MIX!!

,2/ 3/1 2/1 132312 ssss

.. PDG wrt Extremise ge 2/1 || 13233 ccU TRI-BIMAX ???

A VARIATIONAL PRINCIPLE IN ACTION? SYMMETRIES OF NEUTRINO MIXING: P. F. Harrison, D. H. Perkins and...

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Transcript of A VARIATIONAL PRINCIPLE IN ACTION? SYMMETRIES OF NEUTRINO MIXING: P. F. Harrison, D. H. Perkins and...