Symmetry of Fermion Mixing
C.S. Lam
McGill and UBC, Canada
arXiv:0708.3665 (to appear in Phys. Lett)
Progress in particle physics relied heavily on
symmetry considerations
SU (3) F
SU (3)C  U (1)Q  SU (3)C  SU (2) I  U (1)Y  SO(10)?
(mb
mt )
1936. “ Who ordered it ?”
What about the generation problem?
(mu
mt , me
m )
(horizontal global) symmetry for mass matrices
mixing of fermions
Models of Fermion Mixing
•
Pick a (global) horizontal symmetry group G .
S3 (83), A4 (91), S4 (11)     
•
Assign IR to left-handed (L), right-handed (R), [ and heavy (H)]
fermions, and all the new Higgs fields 
•
Construct G -invariant mass terms. Coupling consts h
•
Assign vacuum expectation values
•
Compute the mixing matrix from the mass matrices
•
Tune the parameters
h, 

to break the symmetry
to get the desired mixing matrix
An Example (leptons)
Ma (hep-ph/0404199)
.
A4 [ IR : 3, 1, 1,1]
•
Pick a horizontal symmetry group G
•
Assign IR to left-handed (L), right-handed (R), [ and heavy (H)]
fermions, and all the new Higgs fields 
L  3; eR  1, 1,1; e  3 Iso-doublet
Iso-triplet   3, 1, 1,1
•
Construct G -invariant mass terms. Coupling consts
•
Assign vacuum expectation values

h
to break the symmetry
e  ve (1,1,1); 3  v (1,0,0)
h2
 h1
M e  ve  h1 h2

 h h 2
 1 2
h3 
h3 2 

h3 
a b  c
M  
a  b  c 2


d

•
Compute the mixing matrix from the mass matrices
•
Tune the parameters
h, 
2
1 
U
 1
6
 1



d

2
a  b  c 
to get the desired mixing matrix
2
2
2
0 

3 

 3 
(b  c)
A Systematic Study
•
Bottom-Up Approach:
given the mixing matrix U
• Trick: integrate out R and
H to study the effective Lmass matrices
†
u
†
d
†
e
M u M , M d M ; M e M , M
find G
• Top-Down Approach:
given G , find U
Bottom-Up Approach
M e M e†  diag (me2 , m2 , m2 )
F
U T M U  diag (m1 , m2 , m3 )
Gi
M u M u†  diag (mu2 , mc2 , mt2 )
U † M d M d†U  diag (md2 , ms2 , mb2 )
U  (v1 , v2 , v3 )
M vi  mi vi
residual symmetry
F  diag (1 , 2 , 3 )
Non-degenerate
finite group
Gi   I  2vi vi†
M d M d†vi  mi2vi
Gi vi  vi
[Gi , M ]  0
Gi2  1, G1  G2G3

n
G  SU (3)
Partial vs Full
Fn  I
[ F , M e M e† ]  0
Base independent !
Leptons
solar
atmospheric
reactor
 
 0.44 1 
s12  0.314 1 0.18
0.15
s1  0.333
s2 2
s1  0.50
0.41
0.22
2.3
s32  0.9 0.9
Fogli et al.
Harrison,Perkins,Scott
2
2
s3  0
2
1 
U
 1
6
 1

2
2
2
2
0 

3 

 3 
full, partial
Tri-bimaximal mixing
The Leptonic List for n=1,2,3
G  Z2 , G  Z2  Z2
i
1
0
1
G  {Z 2  Z 2 , D4 }, G
3
2
0,1,2
2
F degenerate
G  G  S 4 , H (12,3)
0
3
1
3
G  A4
2
3
G  S3 , H (6,3)
3
3
F non-degenerate

{Gi , F }  Gni
2
1 
U
 1
6
 1

2
2
2
0 

3 

 3 
H(6,3): 54 members
H(12,3): 216 members
finite or infinite list?
U discrete !!
Top-Down Approach
(Given a finite group G, find U )
•
G has to have an even order, with a 3-dim IR (exceptions)
•
 must be invariant under F  and Gi
• Number of parameters depends on the number of possible
May not be enough to give rise to realistic masses

Gi2  1
G has to have an even order, with a 3-dim IR (exceptions)
1
v  0
 
0
 
1


G 
1



1

No 3-dim IR
G33  S3 , H (6,3)
2
1 
U
 1
6
 1

2
2
2
0 

3 

 3 
1
v  0
 
0
 
1

G    


  


0
v  
 

 
 1

G 
 



  

 must be invariant under F  and Gi
L L C
L  L C
An Example (leptons)
Ma (hep-ph/0404199)
•
Pick a horizontal symmetry group G
.
A4 [ IR : 3, 1, 1,1]
G32
•
Assign IR to left-handed (L), right-handed (R), [ and heavy (H)]
fermions, and all the new Higgs fields 
L  3; eR  1, 1,1; e  3 Iso-doublet
Iso-triplet   3, 1, 1,1
•
Construct G -invariant mass terms. Coupling consts
•
Assign vacuum expectation values

h
to break the symmetry
e  ve (1,1,1); 3  v (1,0,0)
h2
 h1
M e  ve  h1 h2

 h h 2
 1 2
h3 
h3 2 

h3 
a b  c
M  
a  b  c 2


d

•
Compute the mixing matrix from the mass matrices
•
Tune the parameters
h, 
2
1 
U
 1
6
 1



d

2
a  b  c 
to get the desired mixing matrix
2
2
2
0 

3 

 3 
(b  c)
  0.2272
Quarks
0.0010
0.0010
  0.2262
0.0014
0.0014
A  0.818 0.007
0.017
A  0.815 0.013
0.013
  0.221
0.064
  0.235
0.031
  0.340
0.017
  0.349
0.020
0.028
0.045
 1  / 2


2
U 

1  / 2
 A 3 (1    i )  A 2

2
0.031
0.020
A (   i ) 

2
A


1

3
Remarks
•
Gisi only known numerically. It
is too hard to obtain a finite
horizontal group for three
generations.
G
{Gi , F }  G
i
n
n
• First consider two generations
(Cabibbo mixing)
 1 2
U 
 



1   2 

Cabibbo Mixing
 1 2
U 
 

 c2  s2 
G1  


s

c
2
 2
1

F 


1


non-degenerate !
 c2 s2 
G1F  


s
c
 2 2
G21
  c s


2 

s
c


1  

 G1 , F | G12 , F 2 ,(G1F ) m  Dm
 c2 m s2 m 
(G1F )  

  s2 m c2 m 
m
D7
C 

2m
  0.2225
PDG:
0.2262  0.0014
0.2272  0.0010
CKM Mixing
• More hopeful to use a top-down approach (in
progress)
• A discrete number of mixing matrices results
• Use one CKM parameter to decide which U to
use, then the other three CKM paramters are
determined (a purely symmetry calculation !).
Conclusion
• A systematic tool to study the horizontal mixing, both
bottom-up and top-down G U
• The tri-bimaximal neutrino mixing is well understood
in both approaches
• There is an exciting possibility to determine 3/4 CKM
parameters if quark mixing also has a finite horizontal
group (under way).
Yu, Luo River, circa 2100 BCE
2
9
4
5
3
8
1
2
1 
U
 1
6
 1

2
2
2
0 

3 

 3 
河圖洛書
7
6
The original 2-3 and magic symmetries