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

EXTREMISATION OF JARLSKOG INVARIANTS

b

s

dtcu

1

1

1

b

s

dtcu

1

1

1

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.

Phys. Lett. B 628 (2005) 93. hep-ph/0508012P. F. Harrison and W. G. Scott W. G. SCOTT

RAL/SOTONMEET: 3/3/06 “WEAK-BASIS INV.”

b

s

dtcu

1

1

1

(“WEAK-BASIS”)

t

c

u

M

tcu

u

IN THE STANDARD MODEL:

Universal Weak-Interaction

b

s

d

M

bsd

d

“Up” Mass Matrix “Down” Mass Matrix

You can have any 2 but NOT all 3 matrices diagonal!!

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 ARCHITYPAL JARLSKOG INVARIANT:

THE JARLSKOG DETERMINANT:

The Determinant of the Commutator of mass matrices:

Extremising the Jarlskog Invariant J leads to:

31

31

31

31

31

31

31

31

31

2||U

321

e

TRIMAXIMAL MIXING

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

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 TRI-BIMAX MIXING)

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”

TRI-BIMAXIMAL (“HPS”) MIXING

AT LEAST APPROXIMATELY !!!!

21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

SOLAR

REACT.

(MINOSSOON!)

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

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

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

MIX.TRIMAX.IN DICTEDPRE ! ! ! !

THE “5/9-1/3-5/9” BATHTUB

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

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 CHARACTERISTIC 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.

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

21

210

21

210

001

2||U

321

e

2 x 2 MAXIMAL MIXING

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

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 multipliers:

)(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 eg.

3

2

)( 2

mmm

mmm

e

e

Y. Koide, Lett. Nuov. Cim. 34 (1982) 201.

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

SPARE SLIDES

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



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

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”



21

31

61

21

31

61

31

32 0

321

e

2|| lU

IS PHASE-CONVENTION INDEPENDENT:

2||U

2

1

3

1

6

12

1

3

1

6

1

03

1

3

2

321

e

U

TRIBIMAXIMAL (“HPS”) MIXING


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

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



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

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

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

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



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

REACT.

31

31

31

31

31

31

31

31

31

2||U

321

e

TRIMAXIMAL MIXING)

/ 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.

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



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

SOLAR

REACT.



21

31

61

21

31

61

31

32 0

321

e

2|| lU ATMOS.

SOLAR

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

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

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

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)

*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:

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”)

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

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)



21

31

61

21

31

61

31

32 0

321

e

2|| lU

ROWS/COLUMNSSUM TO UNITY

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

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Matrix" Mass Neutrino TriPartite The"

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higges

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1 1 3

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kB

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BTB

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4/14/38/3

4/34/18/3

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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

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

223

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213

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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 ???

MIXING32 SS

:IDENTIFY

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Documents

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