EXTREMISATION OF JARLSKOG INVARIANTS
W. G. SCOTT
RAL/SOTON
“WEAK-BASIS INV.” MEET: 3/3/06
JARLSKOG INVARIANCE: e.g. for the quarks:
Universal Weak Interact.
Universal Weak Interact.
u c t   1
 d 
u c t   1
  d 

 

 
 1  s
 1   s 

b

  b 
1
1

 

   U(3)
P. F. Harrison and W. G. Scott
Phys. Lett. B 628 (2005) 93. hep-ph/0508012
M u Diagonal
M d Non-Diagonal
M u Non-Diagonal
M d Diagonal
OBSERVABLES JARLKOG INVARIANT
FUNDAMENTAL LAWS JARLSKOG COVARIANT !!
IN THE STANDARD MODEL:
“Up” Mass Matrix
u
c t  




Mu
  u 
 
  c 
  t  
 
(“WEAK-BASIS”)
“Down” Mass Matrix
d 

s b 




Md
  d  
 
  s 
  b 
 
Universal Weak-Interaction

u
c t   1
  d  

 
 1   s 

  b 
1

 
You can have any 2 but NOT all 3 matrices diagonal!!
THE ARCHITYPAL JARLSKOG INVARIANT:
( C :  i [ L, N ])
THE JARLSKOG DETERMINANT:
The Determinant of the Commutator of mass matrices:
Det C  Tr C /3  Tr i L, N  /3  Δl J Δν
3
3
Δl  (me -mμ )(m μ -mτ )(mτ -me )
Δν  (m1-m2 )(m2-m3 )(m3-m1 )
2 1/ 2
J  c12 s12c23s23c132 s13s  (1  s122 )1/ 2 (1  s23
) (1  s132 ) s12 s23s13s
Extremising the Jarlskog Invariant J leads to:
1
s12 
2
1
s23 
2
1
s13 
3
s 

2
i.e. LEADS TO TRIMAXIMAL MIXING!!
TRIMAXIMAL MIXING
1  2  3
e
|U |  
2

1
 3
1
 3
 1
 3
1
1
1
Originally proposed for the quarks!!
HS PLB 333 (1994) 471. hep-ph/9406351
3
3
3
1 
3
1 
3
1 
3
TRI-BIMAX (“HPS”) MIXING
1  2  3
e
|U |  
2

2
 3
1
 6
1
 6
1
1
1
3
3
3
0 
1 
2
1 
2
“S3 GROUP MIXING”
(GENERALISES TRIMAX.
AND TRI-BIMAX MIXING)
|U | 
2
“MAGIC-SQUARE MIXING”
1  2  3
e  *

  *
  *
1
1
1
3
3
3
* 
* 

* 

S3 GROUP MIXING
“Magic-Square Mixing”
m1  a  Re b  3(Im b) 2  x 2  y 2  2  xy  yz  zx
m2  a  2Re b  x  y  z
m3  a  Re b  3(Im b) 2  x 2  y 2  2  xy  yz  zx
tan 2   6 (Im b) /( x 2  y 2  z 2  xy  yz  zx )
tan 2  3 ( z  y ) /( x  y  2 x)
1
e



2
2

c c  i
s s
3
3

 c c  is  s c s  is  c


6
2

 c c  is  s c s  is  c


6
2

2
1
3
1
3
1
3
U 
3
2
2

c s  i
s  c
3
3
c c  is  s c s  is  c


2
6
c c  is  s c s  is  c

2
6









TRI-BIMAXIMAL (“HPS”) MIXING
1
e 

2
| U l |    

 
SOLAR
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
REACT.
0

1
ATMOS.

2
(MINOS
1  SOON!)
2
Solar Data
SNO point CC / NC  0.35  0.03 is my naive average
of Salt  No Salt - ignoring corr. systs . and 8 B spect. dist.
HPS PLB 530 (2002) 167 hep - ph/0202074 ; see also PLB 374 (1996) 111 hep - ph/9601346
SNO Results
Pure D2O
Q. R. Ahmad et al.
Phys.Rev.Lett. 89 (2002) 011301
nucl  ex / 0204008
0.09
CC  1.76  00..06

05 0.09
0.46
NC  5.09  00..44

43 0.43
D2O  Salt
B. Aharmim et al.
Phys.Rev.C 72 (2005) 055502.
nucl  ex / 0502021
0.11
CC  1.72  00..05

05 0.11
0.28
NC  4.81  00..19

19 0.27
(Results quoted here are those given assuming undistorte d 8B - spectrum)
CC / NC
0.036
 0.346 00..032

031 0.034
CC / NC
0.028
 0.358  00..021

021 0.029
CC / NC   0.354  0.028
(my naive average  ignores
correlatio n in systematic errors )
THE “5/9-1/3-5/9” BATHTUB
Fig. 3 HPS PLB 374 (1996) 111. hep - ph/9601346
PRE DICTED
IN TRIMAX. MIX.
!!!!
UP-TO-DATE FITS
A. Strumia and F. Vissani
Nucl.Phys. B726 (2005) 294. hep-ph/0503246
2
m122 / m23
 0.03
tan  12  0.45  0.05
2
12
( tan  12 HPS  0.50)
2
IS THE BEST MEASURED MIXING ANGLE !!!
FLAVOUR-SYMMETRIC
JARLSKOG INVARIANT MASS PARAMETERS
Charged-Leptons:
Mass Matrix:
L1 : Tr L  me  m  m
2
2
2
2
L2 : Tr L  me  m  m
L3 : Tr L3  me3  m3  m3
Neutrinos:
{L1 L2 L3}

{ me m m }
Mass Matrix:
N1 : Tr N  m1  m2  m3
2
2
2
2
N 2 : Tr N  m1  m2  m3
N 3 : Tr N 3  m13  m23  m33
L : M l
N : M
{N1 N 2 N 3}

{m1 m2 m3 }
THE CHARACTERISTIC EQUATION
e.g. For the Charged-Lepton Masses:
3  (Tr L )2  (Pr L)  (Det L )  0
where:
Tr L  me  m  m  L1
Pr L  me m  m m  m me  ( L12  L2 )/2
Det L  me m m  ( L13  3L1 L2  2 L3 ) / 6
The Disciminant:
L2  L32 / 2  3L14 L2 / 2  6 L1 L2 L3
 7 L12 L22 / 2  3L23  4 L13 L3 / 3  L16 / 6
 (me  m ) 2 (m  m ) 2 (m  me ) 2
ALL JARLSKOG INVARIANT!!
EXTREMISATION: A TRIVIAL EXAMPLE
In the SM:
me  x   
m  y   
m  z   
Yukawa couplings x, y, z
v
  
 180 GeV
2
Add to SM Action, the determinant : Det L  me m m
(taken here to be dimensionless) i. e.
A  x yz
x A  y z  0
y A  x z  0
z A  x y  0
e.g.
x0
y0
z 0
HS PLB 333 (1994) 471. hep-ph/9406351
NOT BAD!!
MATRIX CALCULUS THEOREM:
 X
 X Tr AX  A
T
A any constant matrix,
:  / X 
X a variable matrix
WHEREBY e.g:
 L Tr C   3i [ N , C ]
3
2 T
 L Tr C   2i [ N , C ]
2
T
( C :  i [ L, N ]
Tr C  0 !! )
EXTREMISING Tr
C3
(FOR FIXED MASSES)
With No Constraints:
 L Tr C / 3   i[ N , C ]  0
 N Tr C 3 / 3   i[ L, C 2 ]T  0
3
2 T
Differentiate Mass Constraints:
 L (Tr L  L1 )  I
 L (Tr N  N1 )  0
 L (Tr L2  L2 )  2 L
 L (Tr N 2  N 2 )  0
 L (Tr L3  L2 )  3L2
 L (Tr N 3  N 3 )  0
 N (Tr L  L1 )  0
 N (Tr N  N1 )  I
 N (Tr L2  L2 )  0
 N (Tr N 2  N 2 )  2 N
 N (Tr L3  L2 )  0
 N (Tr N 3  N 3 )  3N 2
With Mass Constraints Implemented:
Li / Ni = Lagrange
Multipliers
 L Tr C 3 / 3   i[ N , C 2 ]T  L 0 I  L1 L  L 2 L2
 N Tr C 3 / 3   i[ L, C 2 ]T  N 0 I  N 1 N  N 2 N 2
ARB. COMPLEX HERM. NEUTRINO MASS - MATRIX
(IN FLAVOUR BASIS) MAY ALWAYS BE WRITTEN :
" EPSILON BASIS"
e
e  a
M 
2


BY RE - PHASING

 z  id
 y  id

e , , 
" EPSILON PHASE - CONVENTION"

z  id
b
x  id

y  id 

x  id 

c 
INCREDIBLE
BUT TRUE !!!
EXTREMISING Tr
C3
(CONTINUED)
 L Tr C 3 / 3   i[ N , C 2 ]T  L 0 I  L1 L  L 2 L2
 N Tr C 3 / 3   i[ L, C 2 ]T  N 0 I  N 1 N  N 2 N 2
Eq. 1, off-diagonal elements, Re parts:
 d (me  m )( m  me ) ( y  z )  (b  c)  ( y  z )  0
 d (m  m )( me  m ) ( z  x)  (c  a)  ( z  z )  0
 d (m  me )( m  m ) ( x  y )  (a  b)  ( x  y )  0
Non-Trivial Solution:
( y  z )  (b  c)
( z  x)  (c  a) i.e.
( x  y )  ( a  b)
a   x
b   y
c   z
MAGIC-SQUARE
CONSTRAINT!!
EXTREMISING Tr
C3
(CONTINUED 2)
 L Tr C 3 / 3   i[ N , C 2 ]T  L 0 I  L1 L  L 2 L2
3
2 T
2
 N Tr C / 3   i[ L, C ]  N 0 I  N 1 N  N 2 N
Eq.1 off-diagonal elements, Im parts:

 m ) (c  a)( d
 m ) (a  b)( d

 z ) y 0
 y )z 0
 (me  m )( m  me ) (b  c)( d 2  yz )  ( y 2  z 2 ) x  0
 (m  m )( me
 (m  me )( m


2
 zx )  ( z 2
2
 xy)  ( x 2
2
2
Non-Trivial Solution:
ab
bc
ca
and x  y
and y  z
and z  x
CIRCULANT MASS-MATRIX
i.e. TRIMAXIMAL MIXING!!!
Increibly, all the remaining equations are either redundant
or serve only to fix the lagrange multipliers
2d (m  m  2me )( m  m )( d 2  x 2  y 2  z 2 )  L 0  L1me  L 2 me2
2d (m  me  2m )( m  me )( d 2  x 2  y 2  z 2 )  L 0  L1m  L 2 m2
2d (me  m  2m )( me  m )( d 2  x 2  y 2  z 2 )  L 0  L1m  L 2 m2
 L15 / 2  3L13 L2  7 L1 L22 / 2  2 L12 L3
3
L 0 
Tr
C
3L2
3L14 / 2  7 L12 L2  3L22  6 L1 L3
JARLSKOG
3
L1 
Tr
C
3L2
SCALARS!!
 9 L3  9 L1 L2  2 L13
3
L 2 
Tr
C
3L2
Above remains true in all the extremisations we performed!!
THE SUM OF THE 2 x 2 PRINCIPAL MINOIRS:
( C :  i [ L, N ]
Tr C  0 !! )
Q11  Tr C 2 /2  Tr i L, N  /2
 Tl diag  l K diag  
2
 l  (mμ -mτ , mτ -me , me -mμ )
 ν  (m2-m3 , m3-m1 , m1-m2 )
The K-matrix is the CP-symmetric analogue of Jarlskog J:
Plaquette Products  l : U l 1 1U l*1 1U l 1 1U l*1 1 : K l  iJ
2
2
K e1  c23s23c132 (c23s23 (c122  s122 s132 )  c12 s12 s13 (c23
 s23
)c )
K-matrix
2
2
K e1  c23s23c132 (c23s23 ( s122  c122 s132 )  c12 s12 s13 (c23
 s23
)c ) etc
Extremise (in a hierachical approximation) wrt PDG:
| U 3 | c23c13  1 / 2
2 x 2 MAX-MIX. ???
SO NOW TRY EXTREMISING Tr C 2
( L Tr C 2 / 2)T   [ N , [ L, N ]]  0
( N Tr C 2 / 2)T   [ L, [ L, N ]]  0
Eq. 1, off-diagonal elements, Re parts:
(c. f . Maxell /YM
A  Tr F 2 /2
F  [  ,  ]

[  , [  ,  ] ]  0)
(m  m  2me )( d 2  yz )  (m  m )(b  c) x  0
(m  me  2m )( d 2  zx )  (m  me )(c  a) y  0
(me  m  2m )( d 2  xy)  (me  m )( a  b) z  0
Eq.1 off-diagonal elements, Im parts:
d (m  m  2me )( y  z )  d (m  m )(b  c)  0
d (m  me  2m )( z  x)  d (m  me )(c  a)  0
d (me  m  2m )( x  y )  d (me  m )( a  b)  0
Triv. Solns:
e.g. d  y  z  0, b  c
2 x 2 MAX. MIX. !!
2 x 2 MAXIMAL MIXING
1  2  3
e
|U |  
2

1

0

0

0
1
2
1
2

0 
1 
2
1 
2
Not Bad!! - but trivial 2 x 2 Max. solution excluded by solar data!!
EXTREMISING Tr
C2
(CONTINUED)
Non-Trivial Solution:
(it turns out, we need only consider d  0 )
x   (a  b)(c  a ) E
(me  m )( m  me )
E
y   (b  c)( a  b) M M 
z   (c  a )(b  c)T
with a, b, c
T
(me  m  2m )( m  me  2m )
(m  m )( me  m )
(m  m  2me )( me  m  2m )
(m  me )( m  m )
(m  me  2m )( m  m  2me )
adjusted to give “observed” m1 , m2 , m3
Absolute masses not yet measured, but with the “minimalist”
assumption of a normal classic fermionic neutrino spectrum
m1  m2  m3 , we have a unique prediction for the mixing:
NON-TRIVIAL CP-CONSERVING MIXING
Setting:
ba
1
ac
1
| U |  
l
2
e  .33333

  .17079
  .49587

e 1 / 3

 1 / 6
 1 / 2

2
m122 / m23
 0.03
2
3
.33333
.16257
.50409
1/ 3
1/ 6
1/ 2
.33333 

.6663 
.0003 
1/ 3 

2 / 3
0 
SUGGESTIVE, BUT NOT CONSISTENT WITH DATA
!!
THE ASSOCIATED LAGRANGE MULTIPLIERS
Fixing the Lagrange multipliers:
2(m  me ) y 2  2(me  m ) z 2  L 0  L1me  L 2 me2
2(me  m ) z 2  2(m  m ) x 2  L 0  L1m  L 2 m2
2(m  m ) x 2  2(m  me ) y 2  L 0  L1m  L 2 m2
L 0
Assume the
Non-Trivial
Solution
(Tr {L2 , N }L1  Tr {L, N }L2 )(3 Tr {L, N }  2 L1 N1 )

2(9 L3  9 L1 L2  2 L13 )
 (3 Tr {L2 , N }  2 L2 N1 )(3 Tr {L, N }  2 L1 N1 )
L1 
2(9 L3  9 L1 L2  2 L13 )
L 2 
3 Tr {L, N }  2 L1 N1
2(9 L3  9 L1 L2  2 L13 )
These Lagrange Mults. are specific to the non-trivial soln.
i.e. they fail for the 2 x 2 Max. solution!!!
A COMPLETE SET OF MIXING VARIABLES
 Tr [ L, N ]2

Tr [ L, N ][ L, N 2 ]
Tr [ L, N 2 ]2


1
2
2
2
2
2
2
Q   Tr [ L, N ][ L , N ] Tr [ L, N ][ L , N ] Tr [ L, N ][ L , N ] 
2

2
2
2
2
2
2
2 2
Tr
[
L
,
N
]
Tr
[
L
,
N
][
L
,
N
]
Tr
[
L
,
N
]


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.
Q11  Tr [ L, N ]2 ,
Q12  Tr [ L, N ] [ L, N 2 ]
Q21  Tr [ L, N ] [ L2 , N ] ,
Q22  Tr [ L, N ] [ L2 , N 2 ]
(flavour-symmetric mixing variables!)
The Q-matrix is a moment-transform of the K-matrix:
Qmn  Tl diag  l (diag l ) m1 K (diag l ) n1 diag  
EXTREMISE IMPROVED “EFFECTIVE” ACTION
A  Q11  qQ21
( L Q21 ) T  [ N , [ L2 , N]]/2  {L, [ N , [ L, N ]]} / 2
( N Q21 ) T   [ L2 , [ L, N]]/2
{,}=AntiCommutator
Gives trajectory of solutions depending on the parameter q
To locate realistic soln. impose “magic-square constraint”
a  x 
b  y 
c  z 
k3
x
(m  m ) 2 (1  q(m  m ))
k3
y
(m  me ) 2 (1  q(m  me ))
k3
z
(me  m ) 2 (1  q(me  m ))
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
1
 (1   )
Focus on pole at q 
and deviations q 
(m  m )
(m  m )
Setting   (0.03) 2
1
| U |  
l
2

e  0.662

  0.219
  0.120
e 2/3

  0  1/ 6
 1/ 6

i.e.
2
m122 / m23
 0.03
| U e3 |2  0.005  sin 13  0.07
2
3
0.005 

0.448 
0.333 0.547 
1/ 3
0  COVARIANT
 STATEMENT
1/ 3 1/ 2
OF REALISTIC

1/ 3 1/ 2
MIXING!!!
0.333
0.333
APPROX. “HPS” MIXING
!!!
And finally, the associated Lagrange Multipliers:
2
3
4
5



q


q


q


q


q
2 L 20
L 21
L 22
L 23
L 24
L 25
L 2  N1
2( L2P 2  2 LP 2 LP 3q  LP 5 q 2 ) 2
Where eg.
L 21  81L23 L2  27 L23 L12  108L3 L22 L1  90 L3 L2 L13  18L3 L15
 27 L42 / 2  207 L32 L12  186 L22 L14  56 L2 L16  11L81 / 2
When we have the “perfect action” all LMs will vanish!!
KOIDE’S RELATION:
me  m  m
2

2
3
( me  m  m )
512 L3 L1  64 L22  656 L2 L12  207 L14  0
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
e.g.
M=0
SUBSET
OF
m1
m2
J
M
2
0
0
0
2
3
1
6
1
6
1

3
1
3
1
3
1
0
j1  1  j2  1
CLEBSCHGORDAN
COEFFS.
COULD
PERHAPS BE
A USEFUL
REMARK ?!!
0
0
1 1
1 1
0
1
2
1

2
See: J. D. Bjorken, P. F. Harrison and W.G. Scott. hep-ph/0511201
TRI-BIMAXIMAL (“HPS”) MIXING
1
e 

2
| U l |    

 
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
0

1
ATMOS.

2
1 
2
W. G. SCOTT @ RL . AC . UK
CERN-TH-SEMINAR 13/01/06
SYMMETRIES OF NEUTRINO MIXING:
A VARIATIONAL PRINCIPLE IN ACTION?
P. F. Harrison, D. H. Perkins and W. G. Scott
TRI-BIMAXIMAL
Phys. Lett. B 530 (2002) 167. hep-ph/0202074 (“HPS”)-MIXING
P. F. Harrison and W. G. Scott
Phys. Lett. B 535 (2002) 163. hep-ph/0203209
SYMMETRIES
Phys. Lett. B 547 (2002) 219. hep-ph/0219197 “DEMOCRACY”
Phys. Lett. B 557 (2003) 76. hep-ph/0302025 “MUTAUTIVITY”
Phys. Lett. B 594 (2004) 324. hep-ph/0403278
Phys. Lett. B 628 (2005) 93. hep-ph/0508012 EXTREMISATION
TRI-BIMAXIMAL (“HPS”) MIXING
|U |2 IS PHASECONVENTION
INDEPENDENT:
1
e 

2
| U l |    

 
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
0

1
2
1 
2
TRIBIMAXIMAL (“HPS”) MIXING
HPS PLB 458 (1999) 79. hep-ph/9904297; WGS hep-ph/0010335
c.f. G. Altarelli
and F. Feruglio
1
2
3
hep-ph/9807353
 2

1
with sin   1/ 3

0 

e

U 


 3
 1
 6


1

 6

AT LEAST APPROXIMATELY !!!!
3
1
3
1
3


1 

2 
1 
2 
TRI-BIMAXIMAL (“HPS”) MIXING
1
e 

2
| U l |    

 
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
0

1
ATMOS.

2
1 
2
TRI
 MAX.MIXING
1
 TBM
d 0
e
U  











2
cos 
3
c
s
i
6
2
c
s
i
6
2
d
tan 2  
x y
2
3
1

2
i
sin  

3
c
s 


i
2
6
c
s 

i
2
6 
3
1
3
1
3
MUTATIVITY  MAX. CP VIOL.
 PDG   / 2
Oscillation
Decay
Decoherence
37.8/40
49.2/40
52.4/40
M. Ishituka
hep-ph/0406076
P(    )  1 / 2
(| U  3 |2  0.50  0.11)
IMPOSE MUTATIVITY
SET :
 x

 z
 y

y  z
z
y
x
SUCH THAT :
"  
REF LECTION
SYMMETRY "
 x
y 


x    y


z 
 y
x
M ,M   0
2

3 - PARAMETERS
x, y, d
y 

x 

y 
y
y
 , 
2
sol

2
atm
TRI
 MAX.MIXING
1
 TBM
d 0
e
U  











2
cos 
3
c
s
i
6
2
c
s
i
6
2
d
tan 2  
x y
2
3
1

2
i
sin  

3
c
s 


i
2
6
c
s 

i
2
6 
3
1
3
1
3
MUTATIVITY  MAX. CP VIOL.
 PDG   / 2
TRI-BIMAXIMAL (“HPS”) MIXING
1
e 

2
| U l |    

 
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
REACT.
0

1
ATMOS.

2
1 
2
TRIMAXIMAL MIXING)
1  2  3
e
|U |  
2

1
 3
1
 3
 1
 3
1
1
1
3
3
3
1 
3
1 
3
1 
3
REACTORS ESP : CHOOZ / PALO - VERDE / KAMLAND
P  1  2 | U e 3 |2
| U e3 |2  0.03
K. Eguchi et al.
hep - ex/0212021
KAMLAND
0.61  0.09
P(e  e)  5 9
T. Araki et al.
hep-ex/0406035
0.658  0.064
TRI-BIMAXIMAL (“HPS”) MIXING
1
e 

2
| U l |    

 
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
REACT.
0

1
ATMOS.

2
1 
2
New Fit Values :
m 2  2.4 10 3 eV 2
sin 2  1
2
- SK Itshitsuka NOON2004
TRI-BIMAXIMAL (“HPS”) MIXING
1
e 

2
| U l |    

 
SOLAR
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
0

1
ATMOS.

2
1 
2
TRI-BIMAXIMAL (“HPS”) MIXING
1
e 

2
| U l |    

 
SOLAR
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
REACT.
0

1
ATMOS.

2
1 
2
TRI-BIMAXIMAL (“HPS”) MIXING
1
e 

2
| U l |    

 
SOLAR
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
0

1
ATMOS.

2
1 
2
TRIMAXIMAL MIXING:
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.
3
1
2
N. Cabibbo PL 72B (1978) 222.
 
  exp( i 2 / 3)
  exp( i 2 / 3)
J CP  1 /(6 3 )
U
MAXIMAL
CP-VIOLATION !!
e


 1

 3

 3
 

 3
1
3
1
3
1
3

1 

3
 
3
 

3
(cf. C3 CHARACTER TABLE)
N. Cabibbo:
“ We are probably far from this…. . but not very far…”
Lepton-Photon 2001
SNO Results
Pure D2O
Q. R. Ahmad et al.
Phys.Rev.Lett. 89 (2002) 011301
nucl  ex / 0204008
D2O  Salt
S . N . Ahmed et al.
Phys.Rev.Lett. ?? (2003) ????.
nucl  ex / 0309004
0.09
CC  1.76  00..06

05 0.09
0.09
CC  1.70  00..07

07 0.10
0.46
NC  5.09  00..44

43 0.43
0.29
NC  4.90  00..24

24 0.27
(Results quoted here are those given assuming undistorte d 8B - spectrum)
CC / NC
0.036
 0.346 00..032

031 0.034
CC / NC  0.35  0.03
CC / NC
0.028
 0.347 00..022

022 0.028
(my naive average  ignores
correlatio n in systematic errors )
3 x 3 circulant
2 x 2 circulant
 a b b 
 

Ml  b
a b
 b b a 


 x 0 y


M   0 z 0 
 y 0 x


Diagonalise:
U  U U
e  1 1
3
 3
    1
U l M lU l 
diag{ me m m }

l

3

 
 3
M  MM  )
(ASSUMED HERMITIAN
MASS MATRICES:
3
1
3
U M U 
diag{ m1 m2 m3}
1  2  3
1  2  3
1 

3
 
3
 

3







1
2
0
1
2
0
1
0
1 

2
0 
1 
2 
eigen-vecs
eigen-vals
e












2
3
1
6
1
6
1
3
1
3
1
3

0 

i 

2
i 
2 
S3 GROUP MATRIX:
(FLAVOUR BASIS)
(i.e. charged-leptons diagonal)
NAT. REP.
M2  aI  bP(123)  bP(321)
 xP(23)  yP(31)  zP(12)
CIRC.
a
 
 b
b

b
a
b
RETRO-CIRC.
b 

b
a 

x

z
y

S3 GROUP MIXING
z
y
x
y

x
z 
3 3
ORTHOGONAL
`MAGIC  SQUARES '

S3 GROUP MIXING
(TRI-
GENERALISES TBM:
1  2  3
|U | 
2
e  *
 *

  *
1
1
1
MAX. MIXING)
3
3
3
* 
* 

* 


S3 GROUP MIXING
(TRI-
GENERALISES TBM:
1  2  3
e  *
|U |  
2
 *

  *
MAX. MIXING)
1
1
1
3
3
3
* 
* 

* 

An S3 GROUP MATRX Commutes with
THE “DEMOCRACY” OPERATOR:
(and the converse)
e
e 
1
D
1 
 1
3 
 1

 
1
1
1
1 

1 

1 
c.f. “The
Democratic
Mass matrix”
(S3 “CLASS
OPERATOR”)
DENICRACY SYMMETRY/INVARIANCE
D, M   0
2

Conserved Quantum Nos. etc.
U D M U  M
iD
UD  e
2

D
2
SO FINALLY
1
DEMOCRACY Di
(D i , Mi )
2
3
(0,1) (1,1) (0,1)
MUTATIVITY Mi
e

U 










2
3
1
1
3
1
1
6
1
3
1
2
1
6
3
2
0
Di Mi CONSERVED QUANTUM NUMBERS !!









TRI-MAXIMAL MIXING:
HPS PLB 349 (1995) 357
N. Cabibbo PL 72B (1978) 222.
U 
N. Cabibbo:
e


1  2  3
 1

 3

 3
 

 3
1
3
1
3
1
3
1 

3
 
3
 

3
(cf. C3 CHARACTER TABLE)
“ We are probably far from this…. . but not very far…”
Lepton-Photon 2001
TRI-BIMAXIMAL (“HPS”) MIXING
ROWS/COLUMNS
SUM TO UNITY
1
e 

2
| U l |    

 
2
2
3
1
6
1
6
AT LEAST APPROXIMATELY !!!!
1
3
1
3
1
3
3
0

1
2
1 
2
SUMMARY
1) TBM CONSISTENT WITH THE DATA ( ESP. SNO )
2) TBM HAS THREE SYMMETRIES
( ZERO CP,

REFLECTION ,
3) `S3 GROUP MIXING'  TBM
 2  TRIMAX. )
( FLAVOUR BASIS )
1  2  3
|U | 
2
4) `S3 CLASS MIXING'
5) `S2  S3 MIXING'
e


 *

 *

 *

 TBM
 TBM
1
1
1
3
3
3
* 
* 

* 

(  - MASS BASIS )
( FLAVOUR BASIS )
" The TriPartite Neutrino Mass Matrix"
E. Ma PLB 583 (2004) 157 hep - ph/0308282 hep - ph/0403278
Isotriplet

higgs 
Isodoublet
higges i
SM higgs 
Dimension 5







f
ij
k
c
L  hij .(li l j )1  f ij li . j lk 
li . l j .

" Z3  Z2 " Invariant
U hU B  h
T
B
U f UB  f
T
B
k
k
U BT U B  1 U B3  1
UB 
S3 Invariant
" democratic "
  1/2  3 / 8  3 / 8 


 3/8
1/ 4
 3/4 


 3/8  3/4
1/ 4 


FLAVOUR-SYMMETRIC MIXING INVARIANTS:
1) The Determinant of the Commutator:
( C :  i [ L, N ]
Tr C  0 !! )
Det C  Tr C /3  Tr i L, N  /3  Δl J Δν
3
3
Δl  (me -mμ )(m μ -mτ )(mτ -me )
Δν  (m1-m2 )(m2-m3 )(m3-m1 )
2 1/ 2
J  c12 s12c23s23c132 s13s  (1  s122 )1/ 2 (1  s23
) (1  s132 ) s12 s23s13s
Extremise wrt PDG  s12  s23  1 / 2 s13  1 / 3 s   / 2,
ie. TRIMAX. MIX!!
2) The Sum of the 2x2 Principal Minors:
Q11  Tr C 2 /2  Tr i L, N  /2
 Tl diag  l K diag  
2
 l  (mμ -mτ , mτ -me , me -mμ )
 ν  (m2-m3 , m3-m1 , m1-m2 )
2
2
K e1  c23s23c132 (c23s23 (c122  s122 s132 )  c12 s12 s13 (c23
 s23
)c )
2
2
K e1  c23s23c132 (c23s23 ( s122  c122 s132 )  c12 s12 s13 (c23
 s23
)c ) etc
Extremise wrt PDG  e.g. | U 3 | c23c13  1 / 2
K-matrix
TRI-BIMAX ???
S 2  S 3 MIXING
IDENTIFY :
m12  m32
s
2

m12  m32
t
2
1  2
e
U




 2

 3
 1
 6

 1
 6

TRI  BIMAX .
1
3
1
3
1
3

m22  m12
u
3
3

0 

1 
2 
1 
2 
AGAIN !!!