Miami-2008, 17 Dec 2008Presented by: W. G. Scott, PPD/RAL.1 1) “Plaquette Invariants and the...

Miami-2008, 17 Dec 2008 Presented by: W. G. Scott, PPD/RAL. 1

1) “Plaquette Invariants and the Flavour-Symmetric…” P.F. Harrison, D. R. J. Roythorne, and W. G. Scott, Phys. Lett. B 657 (2007) 210. arXive:0709.1439 [hep-ph]2)“Real Invariant Matrices and Flavour-Symmetric…” P.F. Harrison, W. G. Scott and T. J. Weiler, Phys. Lett. B 641 (2006) 372. hep-ph/06073363)“Simplified Unitarity Triangles for the Lepton Sector…” J. D. Bjorken, P.F. Harrison, and W. G. Scott, Phys. Rev D 74 (2006) 073012. hep-ph05112014)“Covariant Extremisation of Flavour-Symmetric Jartlskog Invariants…” P.F. Harrison, and W. G. Scott Phys. Lett. B 628 (2005) 93. hep-ph/05080125) “The Simplest Neutririo Mass Matrix” P. F. Harrison and W. G. Scott Phys Lett. B B594 (2004) 324. hep-ph/0403278. ……..

A FLAVOUR-SYMMETRIC P. F. Harrison (U. of Warwick)W. G. Scott (STFC, PPD/RAL) Miami-2008 17 Dec 2008

“Tri-Bimaximal Lepton Mixing and the Neutrino Oscillation Data” P.F. Harrison, D. H. Perkins, W. G. Scott, Phys. Lett. B. 530 (2002) 167. hep-ph/0202074 (see also: HPS hep-ph/9904297 )

OUTLINE OF TODAYS TALK:

NOW OFFICIALLYA “FAMOUS” PAPER ( > 250 CITES).

“A TREMENDOUS ACHIEVEMENT!” T. D. LEE AT CERN - 30 AUG 2007 (CERN indico video min. 42!!)

PERSPECTIVE ON NEUTRINO MIXING

(emphasis on Flavour-Symmetry )

OF COURSE IT IS ACTUALLY THE EXPERIMENTS WHICH ARE TREMENDOUS!

“Review” ofpast few years2004-2007 of HS…

MILOS, 20 May 2008 Presented by: W. G. Scott, PPD/RAL. 2

T. D. LEE LECTURE AT CERN 30 AUG 2007

CERN video: http://indico.cern.ch/conferenceDisplay.py?confId=19674 (min. 42)

http://indico.cern.ch/conferenceDisplay.py?confId=19674


2/13/16/12/13/16/1

03/13/2 321

e

3/13/13/13/13/13/13/13/13/1

321

e

WE DID “ACHIEVE” SOMETHING HOWEVER:WE PREDICTED TWO (E)SM PARAMETERS!!:

Tri-Bi-Maximal (HPS 1999/2002)

Tri-Maximal Mixing (HS/HPS 1994/1995)

HS/BHS (2002-2006)

* 3/1* * 3/1* * 3/1*

321

e

via Tri-Phi-Maximal & Tri-Chi-Maximal (HS 2002)

“ -Trimaximal Mixing” “S3 Group Mixing”

“Magic-Square Mixing”

“Tri-χφ-Maximal”

3/122 UUe

CHOOZ EXPT. SAYS < 0.03 (not HS/HPS!!)

There was never a prediction from HPS/HS of exact Ue3≡0!

2ν

Please not just “tri-maximal”!!


21

31

61

21

31

61

031

32

|U| 2

}

Symmetries of TriBimaximal Mixing:

1) “CP symmetry” Zero CP violation J=0(hopefully approximate!)

2) “μτ-reflection symmetry” “Two rows equal” (=Max CPV!) |Uμi|=|Uτi| for all i=1-3.

3) “democracy symmetry”one trimaximal eigenvector |Uαi|=|1/3 for all α for some i.


YES – YOU’VE SEEN THESE NUMBERS BEFORE SOMEWHERE!

21

31

6111

21

31

6111

031

3200

000102

21

MmmJM = 0

SUBSETOF

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


HPS “Derivation” of TriBimaximal Mixing:

210

21

010

21-0

21

*

*

*

3ω

31

3

331

3ω

31

31

31

τ

μ

e

ν ν ν * * * 321

UUU l

abbbabbba

***

MM

* * *

ll

x0y0z0y0x

***

MM

* * *

νν

τωμe

τμe

τμe

lmωmm

ωmωmmmmm

***

M

2i

31

61

2i

31

61

03

132

τ

μ

e

ν ν ν 321

3 x 3 circulant(by definitionof the * basis)

2 x 2 circulant(determinesthe physics)

}m m{m UMMU

τμe

llll

diag2

} {diag

3212 mmm

UMMU

MM M

In the “circulant basis”: *

ω/3m/3ωm/3mb/3ωmω/3m/3mb

/3m/3m/3ma

2τ

2μ

2e

2τ

2μ

2e

2τ

2μ

2e

A popular choice:

Harrison, Perkins, Scott, Phys. Lett. B. 530 (2002) 167. hep-ph/0202074

† †


6cissc

2siscc

31

2cissc

6siscc

6cissc

2siscc

31

2cissc

6siscc

cs32isc

32

31ss

32icc

32

φχφχφχφχφχφχφχφχ

φχφχφχφχφχφχφχφχ

φχφχφχφχ

“ν2-Trimaximal Mixing”

6s

i2

c3

12

si

6c

6s

i2

c3

12

si

6c

s32i

31c

32

χχχχ

χχχχ

χχ

6s

2c

31

2s

6c

6s

2c

31

2s

6c

s32

31c

32

φφφφ

φφφφ

φφ

“Tri-χ-Maximal Mixing” “Tri-Φ-Maximal Mixing”

cosφcsinφscosχcsinχs

φ

φ

χ

χ

Exact μτ - Refl. Symm., J≠0 J=0, Break μτ-Symmetry

Χ→0Φ→0

“Tri-φχ-maximal mixing”, “S3 group mixing” “Magic-square mixing”, “BHS-mixing”… .

“Symmetries and Generalisations of Tri-Bimaxiaml Mixing” P.F. Harrison, and W. G. Scott Phys. Lett. B 535 (2002) 163. hep-ph/0203029

“Permutation Symmetry, Tri-Bimaximal Mixing and the S3 Group ...” P.F. Harrison, and W. G. Scott Phys. Lett. B 557 (2003) 76. hep-ph/0302025


Symmetric Group S3 (natural representation):

001100010

P(123)

010001100

P(321)

100010001

I

100001010

P(12)

010100001

P(23)

001010100

P(31)

zP(12)yP(31)xP(23) P(321)bbP(123)aI MM νν

odd"" even""

zxyxyzyzx

ννν

abbbabbba

ννν

ν ν ν ν ν ν

τ

μ

e

τ

μ

e

τμeτμe

Nature Plays Sudoku !!

Experiment tells us thatthe neutrino mass matrix² in the (charged-lepton) flavour basis can be writtenas a 3 x 3 Magic Square !!

All row/column sums equal !!

The most general such (hermitian) matrix may be constructed as an “S3 Group Matrix” in the natural representation of the S3 group ring

2x)yy)/(x(z32φ zx)yzxyzy(Imb)/(x62χ 222

tantan

Any “S3 Group Matrix” clearly has (at least) one trimaximal eigenvector:

111

31

ννMM

“circulant” “retro-circulant”

“ -Trimaximal Mixing”=“Magic-Square”/”S3 Group Mixing”=“Democracy Symmetry”

†

†

2

“Permutation Symmetry, Tri-Bimaximal Mixing and the S3 Group ...” P.F. Harrison, and W. G. Scott Phys. Lett. B 557 (2003) 76. hep-ph/0302025


Simplified Unitarity Traingles in the Lepton Sector

The Matrix* of UT angles:

“ν2.ν3”=“the ν1-triangle”

e3

e3

e3

321

U21C

21

31 C

61

U21C

21

31 C

61

U 3

1 C62

τ

μ

e

ν ν ν

τ3τ2τ1

μ3μ2μ1

e3e2e1321

φφφφφφφφφ

τμe

Φ

ν ν ν

“BHS” Mixing

Each angle Φαi appears inone row-based triangle and one column-based triangle

e

μ

τ

Uτ3

1σ

Uμ3Ue3

*Footnote [42] hep-ph/0511201 Note the natural “complementary” labelling of angles and triangles

(=“Tri-χφ-Mixing”)

J. D. Bjorken, P.F. Harrison, and W. G. Scott, Phys. Rev D 74 (2006) 073012. hep-ph0511201

CPe3

23e3 J

23U θ-

4π

2U

ImRe


“Simplified Unitarity Triangles for the Lepton Sector…”J. D. Bjorken, P.F. Harrison, and W. G. Scott, Phys. Rev D 74 (2006) 073012. hep-ph/0511201

1ν 1l*

1ν 1l

lν

*1ν 1l1ν 1l

321

UUΠ

UU

τμe

ν ν ν *

1ν 1l1ν 1l*

1ν 1l1ν 1llν UUUU: Π

J i K: Π lνlν

π π π π|π|π|

Π- ArgΠ- ArgΠ- ArgΠ- ArgΠ- ArgΠ- ArgΠ- ArgΠ- ArgΠ- Arg

τμe

φφφφφφφφφ

τμe

Φ

ν ν ν ν ν ν

τ3τ2τ1

μ3μ2μ1

e3e2e1

τ3τ2τ1

μ3μ2μ1

e3e2e1321321

UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU

τμe

Π

ν ν ν

*μ1

*e2μ2e1

*μ3

*e1μ1e3

*μ2

*e3μ3e2

*τ2

*e1τ1e2

*τ1

*e3τ3e1

*τ3

*e2τ2e3

*τ1

*μ2τ2μ1

*τ3

*μ1τ1μ3

*τ2

*μ3τ3μ2

321

We define the Matrix of UT Angles:*

From the Plaquette Products:

Form the Matrix of Plaquette Products:

*Footnote [42] hep-ph/0511201

3) (mod


ooo

o

o

o

4oo

ooo

ooo

tbtstd

cbcscd

ubussud

180 180 180

180|180|180|

)0(λ68112239067

157221

tcu

φγφφχβφαφχγφ

φβφχβφ

tcu

Φ

bs d b s d

UNITARITY TRIANGLES IN THE QUARK SECTOR

THE MATRIX OF UNITARITY TRIANGLES IN THE QUARK SECTPR

EQUIVALENT INFO. TO CKM MATRIX !!

χ

α

β+χγ -χ

s

d

bα

βγ

u t

c

“d.b”=“the s-triangle” “t.u”=“the c-triangle”

( in SM - see e.g. F. Muheim “Flavour in the Era of LHC” HEP Forum 21 June 2007)o1

!!!20 CDF/D0 o

Systematic “complemenatry” notation hereis a big improvement on existing notations!!


P.F. Harrison, D. R. J. Roythorn, and W. G. Scott, Phys. Lett. B 657 (2007) 210. arXive:0709.1439 [hep-ph]

1) Flavour Symmetry: A fundamental theory of flavour should be Flavour-Symmetric (ie. it should make no reference to explicit flavour indices).

The Principles which guide us:

Use Flavour-Symmetric Jarlskog Invariant variables!!The Architypal example:The Jarlskog CP-Invariant:

2) Jarlskog Invariance: A fundamental theory should be weak-basis independent(i.e. it should make no reference to any preferred weak-basis).

We define 6 New Flavour-Symmetric Jarlskog-Invariant mixing variables :

Independent, of plaquette choice l,ν hence “Plaquette Invariant”

νl S3S3 )11(

The Jarlskogian J is “odd-odd” under separate l and ν flavour permutations:

νl S3S3

33 CC l spanning theInvariant polynomial ring

(functions only of mixing angles)

“Plaquette Invariants and the Flavour-Symmetric …”

with odd/even symmetry under:

An `elemental” set - not all independent, e,g,

††


bsd

1

11

tcu

bsd

1

11

tcu

Jarlskog Invariance:

U(3)

(Also known as Weak-Basis Invaraince)

In any “weak” (“gauge”) basis the weak interaction is diagonal and universal (i.e proportional to the identity matrix)We often seem to choose to blame the mixing on the “down” quarks! weak basis

But we could equally choose to blame it on the “up”-type quarks! weak basis

Elsewhere in the Lagrangian: (i.e in the yukawa sector)

Mu is diagonal(Md is non-diagonal)

Md is diagonal(Mu is non-diagonal)

Mass²Matrices

CCweakint.

All observables are Jarlskog Invariant: e.g. masses, mixing angles: etc. J δ m m m

V m m m13 3 μe

2 ub t du

Note that the Jarlskogian J is (moreover) also Flavour-Symmetric !!


33333

22222

1

Tr : Tr : Tr :

mmmLLmmmLLmmmLL

e

e

e

FLAVOUR-SYMMETRIC

Charged-Leptons: Mass Matrix:

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

Neutrinos: Mass Matrix:

lll MMM : L †

ννν MMM : N †


6/)23( Det )/2( Pr

Tr

32131

221

1

LLLLmmmLLLmmmmmmL

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 are Flavour-Symmetric and Jarlskog Invariant!!


31

31

31

z31y

31

31

x31w

31

31

)U(P 2

xy)3(wzNL

T DetP Det)11( FΔΔ

(2)

)xy(wz)zyx(wz)yx(w1)/2P.P (Tr1) (1G 22222T(2)

Flavour-Symmetric Mixing Observables…P.F. Harrison, D. R. J. Roythorn, and W. G. Scott, Phys. Lett. B 657 (2007) 210. arXive:0709.1439 [hep-ph]

Six New FS Variables (“Plaquette Invariants”) A, B, C, D, F, G, analogous to Jarlskog J,order (n) with odd/even symmetry under - scalar or pseudoscalar.

z)]wz(wy)[xy(x29wxy)wxzwyz9(xyz1) (1C (3)

y)]xy(xz)[wz(w23y)]wz(xz)xy(wx)xz(z

y)yz(zy)wy(wx)3[wx(w)zyx2(w)11(A 3333(3)

x)]yyxwzz(w21wyz-xyz-wxzwxy

yz-zywxx[w331)1( B2222

2222(3)

y)]xxywzz(w21wxz-xyz-wyzwxy

xz-zxwyy[w33)1(1 D2222

2222(3)

/4FF/43GBDAC 2G2GFDCBA 32322222 Not all independent

)xS3(S3 νl

B, D are not l ↔νsymmetric

νl xS3S32 x 2 of


Plaquette Invariance (= Invariance)νl C3 x C3

xy-wzF/3

xy-wz yzy-xy-yw- zyzwy yw

z)yxy(w-y)z)(w(yF/322

“PLAQUETTE INVARIANT”!!!


Solving more generally for the P-matrix

Flavour-Symmetric Weak-Basis-Invariant Constraints on Mixing:

Democracy Symmetryie. one column=(1/3,1/3.1/3), iff: 0C 0F

“ μ–τ ” - Reflection Symmetry,ie. two rows (or columns) equal, iff:

0A 0F

Tri-Bi-Maximal Mixing, iff: 0JACF

1/6G in the limit F, A, C → 0 and 0 < G < 1/6, gives:


Ansatz F G C A Symm. 18J B D

Tri-Bi-Max. 0 1/6 0 0 Dem., μτ, CP 0 0 1/12√3

Tri-Max. Mix. 0 0 0 0 Dem., μτ 1/6 0 0Tri-χφ-Max. 0 - 0 - Dem.(ocracy) - 0 -2 Rows Eq. 0 - - 0 e.g. μ-τ - 0 -2 Cols. Eq. 0 - - 0 e.g. 1-2 - - 0Alt.-Feruglio 0 - (6G-1)/8 0 μτ, CP 0 0 -Tri-χ-Max. 0 - 0 0 Dem., μτ - 0 -Tri-φ-Max. 0 1/6 0 - Dem., CP 0 0 -Orig. Bi-Max. 0 1/8 -1/32 0 CP, μ-τ,1-2 0 0 0No Mixing 1 1 1 1 CP 0 0 0

Jarlskog J measures CP-violation (J=0 protects against violation of CP).

F measures the acoplanarity of the P-vectors in the flavour space(F=0 => Det <P(∞)> = 0, i.e. protects distant source against flavour analysis)

G = 3<<Pll(∞)>>-1 measures the flavour-averaged asymptotic survival prob….


22211

22112

22111

2122122

21121

211

2121112

211

311

(3)

CC CC CC CC CC CC CC CC C

CTrTrTrTrTrTrTrTrTr

C) NLC

NLT xy)3(wzP F

ΔΔ

(3)

ΔΔ3 Det (

DetDetDet

]N,i[LC nmmn Generalised Jarlskog Commutators:

The Matrix of Cubic Commutator Traces

The Jarlskog Commutator: N]i[L,C

3C C 3TrDet controls CP violation:

222120

121110

020100

A A A A A A A A A

21T

TrTrTrTrTrTrTrTrTr

The Matrix of Anti-Commutator Traces (traces of mass-matrix products):

}N,{LA nmmn And Anti-Commutators:

For example, F:

Directly in Terms of Mass Matrices: †

νννlll

MMMNMMML

†

In termsof Mass Matricesonly


nν

ml

nmmn m P. .mNL TrT

“Real Invariant Matrices and Flavour-Symmetric…”P.F. Harrison, W. G. Scott and T. J. Weiler, Phys. Lett. B 641 (2006) 372. hep-ph/0607336

νν1n

l1m

llTlmn Δ Δ diag )Σ (diag K )Σ (diag Δ diag Δ Q

The “P-matrix”: “T-matrix”

“Q-matrix”

The “K-matrix”Moment Transform:

Moment Transform:

*1ν 1l

*1ν 1l1ν 1l1ν 1llν

lνlνUUUUΠ

ΠRe K

2})PPP{P(PK 1ν 1l1ν 1l1ν 1l1ν 1llνlν ”permanent”

3) (mod

(invertible)

(invertible)


FFTT .T.N.LT - .P..P 6F Tr -Tr

N .T..LT .PP 12G GGTT Tr Tr

CCTT .Q.N.LT - .KP 612G32C Tr Tr

AATT .Q.N.LT .K..P 2F-2A Tr Tr

CATT .Q.N.LT .K.P 32B Tr Tr

ACTT .Q.N.LT .K.P 32D Tr Tr

0.P.P .P.P TT Tr Tr 011101110

Expressed as Traces

Two l ↔ν asymmetric cubic variables B,D:

No l ↔ν asymmetric quadratic variables:

Two quadratic variables G,F entirely in terms of Mass Matrices

Two l ↔ν symmetric cubic variables C,A:

etc. )L,L,(LLL 321GG


03L30LLL0

L1 L

1

2

12

ΔF

Expressed as Traces (cont.) The Mass-Polynomial Matrices Requd:

entirely in terms of Mass Matrices

etc. )L,L,(LLL 321GG

1

432

321

21

GLLLLLLLL3

L

221

2231

414 LL2L3L4L6L L

3 L0

Anti-symmetric Matrix

Symmetric Matrix

L L

AC

-1GΔ L DetL


*1ν 1l1ν 1l

1ν 1l*

1ν 1l

*1ν 1l1ν 1l

*1ν 1l1ν 1l

1ν 1l*

1ν 1l

*1ν 1l1ν 1l

UUUU

UU

UUUUUU

*1ν 1l1ν 1l

*1ν 1l1ν 1l

*1ν 1l1ν 1lI UUUUUU:Ω

1/9 G1/3 C2/9 ΩΩoddeven

*1ν 1l1ν 1l

*1ν 1l1ν 1llν UUUU: Π

J i K: Π lνlν

1ν 1l*

1ν 1l

lν

*1ν 1l1ν 1l

UUΠ

UU

J 9i 1)/2- (G Πlν

lν G)/2- (1 Klν

lν

Flavour-Summed Loop Amplitudes

Usual Plaquette Product:4-Plaquette

Hexaplaquette Product: 6-Plaquetteeven odd

purely real


More Flavour-Symmetric Constarints:

0AF)(CG27F8C 23

0DF)(BG27F8B 23

1/3|U| 2i

2i

2i |U||U|

Completely Symmetric CKM P-matrix:

DB 2

i2

i |V||V|

0|U| 2i 0J 0|K| 2

0V 0V 0V 0VJFCAJ)F,C,V(A,JFCA

2222

Extremise a “Potential”, e.g.:0JFCA

Tri-Bi-MaximalMixing !!!

oαi

2

90φ 0J 0|K|

!!!


“Covariant Extremisation of Flavour-Symmetric….”P.F. Harrison, and W. G. Scott Phys. Lett. B 628 (2005) 93. hep-ph/0508012

0 ]Ci[L, /3 C Tr 0]Ci[N, /3 C Tr

T23N

T23L

0 C]i[L, /2C Tr 0C]i[N, /2C Tr

T2N

T2L

/3C3 TrExtremising:

Extremising: - /2C2Tr

We extremise wrt Mass Matrices theselves:

N]i[L,C

νννlll

MM MNMM ML

†

†

The Jarlskog Commutator:

31

31

31

31

31

31

31

31

31

21

210

21

210

001

Extremising:

/2Cr/3C 23 Tr TrV(C)

0/3)CC /2)CC C)C 3223 TrTrTr (((Characteristic Equationn:

0C Tr

3 x 3 Max

2 x 2 Max

)V(C“The Simplest Neutrino Mass Matrix”P. F. Harrison and W. G. Scott PLB 594 (2004) 324. hep-ph/0403278

C) Det(

(=ΣPrincipal Minors C)

etperms.


X/ : X TX A AX Tr

) N][L, i : C (

0] ],[,[

],[F /2F Tr A

c.f.

νμμ

νμ

2

Mills-Yang / Maxell

Extremise wrt the Mass matrices themselves!

Exploit Matrix Calculus Theorem

0 ]Ci[L, /3 C Tr 0 ]Ci[N, /3 C Tr

T23N

T23L

Apply to Extremise Tr C³

Weak-BasisCovariant !!

Apply to Extremise Tr C²

0 C]i[L, /2C Tr 0 C]i[N, /2C Tr

T2N

T2L

Where A is any constant matrix and X is a variable matrix.


cidxidy

idxbidzidyidza

ννν

MMN

ν ν ν

τ

μ

e

νν

τμe

The “Epsilon” Phase Convention*The usual (charged-lepton) flavour basis has not been completely defined.

There remains the freedom to re-phase the fields such that he imaginary part of the neutrino mass matrix is proportional to the epsilon matrix

Incredible but true!!

Now the 7 parameters a, b, c, d, x, y, z encode directlythe 3 neutrino masses and the usual 4 mixing parameters.

*See Footnote 1 of: “The Simplest Neutrino Mass Matrix” P. F. Harrison & W. G. Scott Phys Lett. B B594 (2004) 324. hep-ph/0403278

†

01-1101-1-10

ε

ε N Im i.e. d

“the epsilom matrix”:


0b)z)(am(mxy))(d2mm(m0a)y)(cm(mzx))(d2mm(m

0c)x)(bm(myz))(d2mm(m

μe2

τμe

eτ2

μeτ

τμ2

eτμ

0b))(amd(my))(x2mmd(m0a))(cmd(mx))(z2mmd(m

0c))(bmd(mz))(y2mmd(m

μeτμe

eτμeτ

τμeτμ

cb , 0yx 0,dac , 0xz 0,dc b , 0z y0,d

Eq. 1 the of-diagonal Real Parts:

Eq. 1 the of-diagonal Imag Parts:

)2mm)(m2mm(m

)m)(mm(mT c)Ta)(b(cz

)2mm)(m2mm(m

)m)(mm(mM b)Mc)(a(by

)2mm)(m2mm(m

)m)(mm(mE a)Eb)(c(ax

eτμμeτ

τμeτ

τμeeτμ

μeτμ

μeττμe

eτμe

e.g. Extremise Tr C² 2 .Eq...........C]i[L, /2C Tr 1 Eq...........C]i[N, /2C Tr

T2N

T2L

2 x 2 Max. MixIn any sector!!

Easy Solutions:

Non-trivial Mass-Dependent Solution: d = 0

Fit a, b, c to “observed”

m ,m , m 321

ZeroCPV!


01/21/22/31/61/61/31/31/3

τμe

.0003.504.496.666.163.171.333.333.333

τμe

|U|

ν ν ν ν ν ν

2

321321 e

e

0.03/ΔΔΔmh 223

212 1

caab

Absolute neutrino masses not yet measured, but with the “minimalist” assumption of a normal classic fermionic neutrino spectrum we maymake a unique prediction for the MNS mixing:

The only operative parameter then becomes: (b-a)/(a-c) and setting:

In clear disagreement with experiment.

All the the right numbers in all the wrong places!!

Extremising Tr C² (non-trivial solution …continued)

X

313

1

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


/2)C( Tr r /3)(C Tr V(C) 23 N]i[L,C

Try a simple linear combination of the two:

)m)(mm(m)md(m

rdZ Z

ZXY z

)m)(mm(m)md(mrdY

YYZX y

)m)(mm(m

)md(mrdX

XXYZ x

τμeτ

μe2

μeτμ

eτ2

eτμe

τμ2

0.550.330.110.250.330.410.190.330.48

P

0.035h GeV0.163 r/d 2

With the “Magic-Square constraint” imposedthere are analytical solutions:

Take r to be a constant with dimensions of (mass)²

In general, for sufficiently extreme hierarcy h → 0, we are close to the pole at X →0, i.e. x→∞ and we have |x| >> y, z,whereby the “Simplest” assumption must hold.

In this sense this V(C) above points to the “Simplest Neutrino Mass Matrix”despite that in practice (in actuality!) the hierarchy h is too large!!

In practice:

X


”The Simplest Neutrino Mass Matrix”

P. F. Harrison and W. G. Scott Phys Lett. B594 (2004) 324. hep-ph/0403278.

0.030.13 m3m2

χ sin 2/3sinθ2atm

2sol

13

0Mν ,D“Democracy Symmetry”

111111111

D

“Mu-Tau Reflection Symmetry” (“mutautivity”)

ννT M) M( *EE

010100001

E

Finally, implementing the “Simplest” Condition:

In the charged-lepton flavour basis, ie. where lM Is diagonal, we impose:

the “democracyoperator”

Ie. commutes withνM

the “μτ-exchangeoperator”

Note definition includesa complex conjugation

dεxaIMν E


CONCLUSIONS

Again T. D. Lee’s lecture (a 2nd clip- from earlier in his talk)Inspirational for anyone working on fermion mixing and flavour etc. :“….these two 3 x 3 matrices (CKM and MNS) are the cornerstones of particle physics… ….but do we understand them???”

1) “Tri-BiMaximal Mixing” has useful partners “Tri-χ-Maximal Mixing”, and “Tri-φ-Maximal Mixing” and more generally “Tri- χφ-Maximal Mixing”(now “ν2-Trimaximal Mixing”) which are also consistent with the data.

2) We have introduced 6 New Flavour-Symmetric Mixing Observables, A,B,C,D,F,G which like the Jarlskogian J can be used to constrain the mixings in an entirely flavour-symmetric way.

3) A programme of Extremisiing Flavour Symmetric Jarlskog Invariants,Is under way with the aim of constraining both Mixings and Masses.Thus far the best that can be said is that our results point towards “The Simplest…” PLB 594 (2004) 324 (hep-ph/0403278) and Θ13 ~ 0.13.


T. D. Lee CERN colloquium Aug 2007


SPARE SLIDES AND SLIDES IN PROGRESS


Solar Data dist. 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


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


z yxAction

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

e.g.

P.F. Harrison and W. G. Scott Phys. Lett. B 333 (1994) 471. hep-ph/9406351

i.e. 2 zero mass 1 non-zero!

This notion appeared in:


ΔΔ33 N J L/3 NL, i Tr /3 C rT C Det

δ132312213

1/2223

1/2212δ13

21323231212 sss)ss(1)s(1)s(1sscscsc J

2πs

31s

21s

21s δ132312

ννl

Tl

22

Δ Δ diag K Δ diag Δ /2NL, i Tr /2C Tr C Pr

)m-m,m-m,m-(m Δ )m-m,m-m,m-(m Δ 211332νμeeττμl

etc. etc. ..... K ))cs(cssc)ss(cs(ccsc K

e2δ

223

223131212

213

212

2122323

2132323e1

21/cc |U| 1323τ3

ΔN ΔL NL, i- C ΔlΔ DiagDet DiagDet

Extremise Det C = Tr C³ /3 (wrt mixing angles, fixed masses)

Extremise the sum of The Principal Minors Tr C² /2 (wrt mixing angles)

3 x 3Max Mix. !!

2 x 2Max Mix. !!

(use hierachical approx.):


0y)(x b)(ay)(x )m)(mmd(m

0z)(z a)(cx)(z )m)(mmd(m 0z)(y c)(bz)(y )m)(mmd(m

τμeτ

μeτμ

eτμe

2 ..Eq.0......... ]Ci[L, /3 C Tr 1 ..Eq.0......... ]Ci[N, /3 C Tr

T23N

T23L

b)(ay)(xa)(cx)(zc)(bz)(y

zσcyσbxσa

Eq. 1 Off-Diagonal Real Parts:

Magic-SquareConstraint!!

0 )zy(xxy)b)(d(a )m)(mm(m

0 )yz(zzx)a)(d(c )m)(mm(m 0 )xz(yyz)c)(d(b )m)(mm(m

222τμeτ

222μeτμ

222eτμe

Eq. 1 Off-Diagonal Imag. Parts:

xz and acz yand

and

c byx ba Circulant mass matrix

i.e. 3 x 3 Maximal Mixing!!Maximal CPV (J=1/(6√3)

Extremise Tr C³


2τL2τL1L0

2222μeτμe

2μL2μL1L0

2222eτμeτ

2eL2eL1L0

2222τμeτμ

mλmλλ)zyx)(dm)(m2mm2d(mmλmλλ)zyx)(dm)(m2mm2d(mmλmλλ)zyx)(dm)(m2mm2d(m

32Δ

31213

L23

2Δ

31222

21

41

L1

32Δ

321

2212

31

51

L0

C Tr 3L

2LL9L9Lλ C Tr 3L

L6L3LL7L/23Lλ

C Tr 3L

L2L/2L7LL3L/2Lλ

0207LL656L64LL512L 41

212

2213 3

2)mmm(

mmm2

τμe

τμe

etc. etc. 0 )NN (Tr 0 )NN (Tr3L )L L (Tr 2L )L L (Tr I )L L (Tr

22

L1L

23

3L2

2L1L

2N2N1N0

T23N

2L2L1L0

T23L

NλNλIλ 0 ]Ci[L, /3 C Tr Lλ Lλ Iλ 0 ]Ci[N, /3 C Tr

Incredibly, all the remaining equations are either redundant or serve only to fix the Lagrange multipliers:

Differentiate the Mass Constraints

Our Equations get modified: (i.e. must add-in Lagrange multipliers λ)

but stillJarlskogCovariant

Solving explicitly:

If the action were the “right” one, the Lagrange multipliers would vanishfor the experimental mass values!

e.g.

Always True???

Koiderelation

JarlskogScalars!!

Miami-2008, 17 Dec 2008Presented by: W. G. Scott, PPD/RAL.1 1) “Plaquette Invariants and the...

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