Venice: 10 Mar 2009 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) Venice, Italy 10 Mar 2009
“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…
Venice: 10 Mar 2009 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)
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 3
2/13/16/1
2/13/16/1
03/13/2 321
e
3/13/13/1
3/13/13/1
3/13/13/1 321
e
WE DID “ACHIEVE” SOMETHING HOWEVER:WE PREDICTED TWO 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”!!
IMMEDIATELYGENERALISES
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 4
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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 5
ph/9601346-hep 111. (1996) 374 PLB HPS3 Fig.
MIX.TRIMAX.IN DICTEDPRE ! ! ! !
THE “5/9-1/3-5/9” BATHTUB
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 6
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.
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 7
HPS “Derivation” of TriBimaximal Mixing:
2
10
2
1
010
2
1-0
2
1
*
*
*
3
ω
3
1
3
33
1
3
ω3
1
3
1
3
1
τ
μ
e
ν ν ν * * * 321
UUU l
abb
bab
bba
*
*
*
MM
* * *
ll
x0y
0z0
y0x
*
*
*
MM
* * *
νν
τωμe
τμe
τμe
l
mωmm
ωmωmm
mmm
*
*
*
M
2
i
3
1
6
12
i
3
1
6
1
03
1
3
2
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
† †
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 8
YES – YOU’VE SEEN THESE NUMBERS BEFORE SOMEWHERE!
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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 9
6
cissc
2
siscc
3
1
2
cissc
6
siscc6
cissc
2
siscc
3
1
2
cissc
6
siscc
cs3
2isc
3
2
3
1ss
3
2icc
3
2
φχφχφχφχφχφχφχφχ
φχφχφχφχφχφχφχφχ
φχφχφχφχ
“ν2-Trimaximal Mixing”
6
si
2
c
3
1
2
si
6
c6
si
2
c
3
1
2
si
6
c
s3
2i
3
1c
3
2
χχχχ
χχχχ
χχ
6
s
2
c
3
1
2
s
6
c6
s
2
c
3
1
2
s
6
c
s3
2
3
1c
3
2
φφφφ
φφφφ
φφ
“Tri-χ-Maximal Mixing” “Tri-Φ-Maximal Mixing”
cosφc
sinφs
cosχc
sinχs
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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 10
Symmetric Group S3 (natural representation):
001
100
010
P(123)
010
001
100
P(321)
100
010
001
I
100
001
010
P(12)
010
100
001
P(23)
001
010
100
P(31)
zP(12)yP(31)xP(23) P(321)bbP(123)aI
MM νν
odd"" even""
zxy
xyz
yzx
ν
ν
ν
abb
bab
bba
ν
ν
ν
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
tan
tan
Any “S3 Group Matrix” clearly has (at least) one trimaximal eigenvector:
1
1
1
3
1
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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 11
Simplified Unitarity Traingles in the Lepton Sector
The Matrix* of UT angles:
“ν2.ν3”=“the ν1-triangle”
e3
e3
e3
321
U2
1C
2
1
3
1 C
6
1
U2
1C
2
1
3
1 C
6
1
U 3
1 C
6
2
τ
μ
e
ν ν ν
τ3τ2τ1
μ3μ2μ1
e3e2e1
321
φφφ
φφφ
φφφ
τ
μ
e
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
J. D. Bjorken, P.F. Harrison, and W. G. Scott, Phys. Rev D 74 (2006) 073012. hep-ph0511201
CPe3
23e3 J
23
U θ-
4π
2
U
ImRe
= “ν2-Trimaximal”
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 12
“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
UU
Π
UU
τ
μ
e
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- Arg
Π- ArgΠ- ArgΠ- Arg
Π- ArgΠ- ArgΠ- Arg
τ
μ
e
φφφ
φφφ
φφφ
τ
μ
e
Φ
ν ν ν ν ν ν
τ3τ2τ1
μ3μ2μ1
e3e2e1
τ3τ2τ1
μ3μ2μ1
e3e2e1
321321
UUUUUUUUUUUU
UUUUUUUUUUUU
UUUUUUUUUUUU
τ
μ
e
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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 13
ooo
o
o
o
4oo
ooo
ooo
tbtstd
cbcscd
ubussud
180 180 180
180|
180|
180|
)0(λ68112
239067
157221
t
c
u
φγφφ
χβφαφχγφ
φβφχβφ
t
c
u
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
b
α
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!!
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 14
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,
††
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 15
b
s
d
1
1
1
tcu
b
s
d
1
1
1
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 m
13 3 μe
2 ub t du
Note that the Jarlskogian J is (moreover) also Flavour-Symmetric !!
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 16
33333
22222
1
Tr :
Tr :
Tr :
mmmLL
mmmLL
mmmLL
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 †
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 17
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 are Flavour-Symmetric and Jarlskog Invariant!!
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 18
3
1
3
1
3
1
z3
1y
3
1
3
1
x3
1w
3
1
3
1
)U(P2
xy)3(wzNL
T DetP Det)11( F
ΔΔ
(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(x2
9wxy)wxzwyz9(xyz1) (1C (3)
y)]xy(xz)[wz(w2
3y)]wz(xz)xy(wx)xz(z
y)yz(zy)wy(wx)3[wx(w)zyx2(w)11(A 3333(3)
x)]yyxwzz(w2
1wyz-xyz-wxzwxy
yz-zywxx[w331)1( B2222
2222(3)
y)]xxywzz(w2
1wxz-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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 19
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”!!!
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 20
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:
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 21
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….
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 22
22211
22112
22111
2122122
21121
211
2121112
211
311
(3)
CC CC CC
CC CC CC
CC CC C
C
TrTrTr
TrTrTr
TrTrTr
C) NL
C
NL
T 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
2
1T
TrTrTr
TrTrTr
TrTrTr
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: †
ννν
lllMMMN
MMML
†
In termsof Mass Matricesonly
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 23
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
lνlν
UUUUΠ
ΠRe K
2})PPP{P(PK 1ν 1l1ν 1l1ν 1l1ν 1llνlν ”permanent”
3) (mod
(invertible)
(invertible)
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 24
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
011
101
110
Expressed as Traces
Two l ↔ν asymmetric cubic variables B,D:
No l ↔ν asymmetric quadratic variables:
Two quadratic variables G,Fentirely in terms of Mass Matrices
Two l ↔ν symmetric cubic variables C,A:
etc. )L,L,(LLL 321GG
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 25
03L
30L
LL0
L
1 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
G
LLL
LLL
LL3
L
221
2231
414 LL2L3L4L6L L
3 L0
Anti-symmetric Matrix
Symmetric Matrix
L L
A
C
-1GΔ L DetL
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 26
*1ν 1l1ν 1l
1ν 1l*
1ν 1l
*1ν 1l1ν 1l
*1ν 1l1ν 1l
1ν 1l*
1ν 1l
*1ν 1l1ν 1l
UU
UU
UU
UU
UU
UU
*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
UU
Π
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-Plaquette
even odd
purely real
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 27
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|
!!!
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 28
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:
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 29
“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 T23
N
T23L
0 C]i[L, /2C Tr
0C]i[N, /2C Tr T2
N
T2L
/3C3 TrExtremising:
Extremising: - /2C2Tr
We extremise wrt Mass Matrices theselves:
N]i[L,C
ννν
lllMM MN
MM 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.
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 30
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 T23
N
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 T2
N
T2L
Where A is any constant matrix and X is a variable matrix.
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 31
cidxidy
idxbidz
idyidza
ν
ν
ν
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
†
01-1
101-
1-10
ε
ε N Im i.e. d
“the epsilom matrix”:
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 32
/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(mrdZ
Z
ZXY z
)m)(mm(m
)md(mrdY
Y
YZX y
)m)(mm(m
)md(mrdX
X
XYZ x
τμeτ
μe2
μeτμ
eτ2
eτμe
τμ2
0.550.330.11
0.250.330.41
0.190.330.48
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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 33
”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”
111
111
111
D
“Mu-Tau Reflection Symmetry” (“mutautivity”)
ννT M) M( *EE
010
100
001
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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 34
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.
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 37
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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 38
01/21/2
2/31/61/6
1/31/31/3
τ
μ
e
.0003.504.496
.666.163.171
.333.333.333
τ
μ
e
|U|
ν ν ν ν ν ν
2
321321 e
e
0.03/ΔΔΔmh 223
212 1
ca
ab
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
3
1
3
1
P.F. Harrison, and W. G. Scott Phys. Lett. B 628 (2005) 93. hep-ph/0508012
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 39
0b)z)(am(mxy))(d2mm(m
0a)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 C²2 .Eq...........C]i[L, /2C Tr
1 Eq...........C]i[N, /2C Tr T2
N
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!
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 40
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
2
πs
3
1s
2
1s
2
1s 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
e2
δ223
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.):
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 41
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 T23
N
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
Venice: 10 Mar 2009 Presented by: W. G. Scott, PPD/RAL. 42
2τL2τL1L0
2222μeτμe
2μL2μL1L0
2222eτμeτ
2eL2eL1L0
2222τμeτμ
mλmλλ)zyx)(dm)(m2mm2d(m
mλmλλ)zyx)(dm)(m2mm2d(m
mλmλλ)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 (Tr
3L )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!!