symmetry of fermion mixing c.s. lam mcgill and ubc, canada arxiv:0708.3665 (to appear in phys. lett)
TRANSCRIPT
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
(3) (1) (3) (2) (1) (10)?C Q C I YSU U SU SU U SO
What about the generation problem?
(horizontal global) symmetry for mass matricesmixing of fermions
(3)FSU
( )b tm m
( , )u t em m m m
1936. “ Who ordered it ?”
Models of Fermion Mixing
• Pick a (global) horizontal symmetry group .
• Assign IR to left-handed (L), right-handed (R), [ and heavy (H)] fermions, and all the new Higgs fields
• Construct -invariant mass terms. Coupling consts
• Assign vacuum expectation values to break the symmetry
• Compute the mixing matrix from the mass matrices
• Tune the parameters to get the desired mixing matrix
G
,h
G
3 4 4(83) (91) (11), ,S A S
h
An Example (leptons)Ma (hep-ph/0404199)
• Pick a horizontal symmetry group .
• Assign IR to left-handed (L), right-handed (R), [ and heavy (H)]
fermions, and all the new Higgs fields
• Construct -invariant mass terms. Coupling consts
• Assign vacuum expectation values to break the symmetry
G
G
4 [ : 3, 1, 1 ,1 ]A IR
h
3; 1, 1 ,1 ; 3R eL e 3, 1, 1 ,1
Iso-doublet
Iso-triplet
3(1,1,1); (1,0,0)e ev v
• Compute the mixing matrix from the mass matrices
• Tune the parameters to get the desired mixing matrix ,h
1 2 32
1 2 32
1 2 3
e e
h h h
M v h h h
h h h
2
2
a b c
M a b c d
d a b c
2 0
1
21
2
2
(
36
3 )
1
bU c
A Systematic Study
• Trick: integrate out R and H to study the effective L-mass matrices
† † †, ; ,u u d d e eM M M M M M M
• Bottom-Up Approach:
given the mixing matrix
find
• Top-Down Approach:
given , find G
G
U
U
Bottom-Up Approach† 2 2 2( , , )e e eM M diag m m m † 2 2 2( , , )u u u c tM M diag m m m
†2i i iG I v v
1 2 3( , , )TU M U diag m m m † † 2 2 2( , , )d d d s bU M M U diag m m m
1 2 3( , , )U v v vi i iM v m v
† 2d d i i iM M v m v
1 2 3( , , )F diag 2
1 2 31,iG G G G
i i iGv v[ , ] 0iG M
nF I†[ , ] 0e eF M M
Gfinite group
Partial vs Full
residual symmetry
(3)SUn
F
iG
Base independent !
Non-degenerate
Leptons2
1
21
23
0.333
0.50
0
s
s
s
0.18
0.15
0.41
0.22
2.3
0.9
21
22
23
0.314 1
0.44 1
0.9
s
s
s
solar
atmospheric
reactor
Harrison,Perkins,Scott
2 2 01
1 2 36
1 2 3
U
Fogli et al.
Tri-bimaximal mixing
full, partial
The Leptonic List for n=1,2,3
{ , } ii nG F G
01 2 1 2 2
3 0,1,22 2 2 4 2
,
{ , },
i Z Z Z
Z Z D
G G
G G2 2 0
11 2 3
61 2 3
U
0 1
3 3 4
23 4
33 3
, (12,3)
, (6,3)
S H
A
S H
G G
G
GH(6,3): 54 membersH(12,3): 216 members
finite or infinite list?
F degenerate
F non-degenerate U discrete !!
Top-Down ApproachG U
G
(Given a finite group , find )
• has to have an even order, with a 3-dim IR (exceptions)
• must be invariant under and
• Number of parameters depends on the number of possible
May not be enough to give rise to realistic masses
F iG
G has to have an even order, with a 3-dim IR (exceptions)
2 1iG
1
1
1
G
1
G
0
v
1
G
1
0
0
v
1
0
0
v
33 3 , (6,3)S HG
2 2 01
1 2 36
1 2 3
U
No 3-dim IR
must be invariant under and F iG
L L C
L L C
An Example (leptons)Ma (hep-ph/0404199)
• Pick a horizontal symmetry group .
• Assign IR to left-handed (L), right-handed (R), [ and heavy (H)]
fermions, and all the new Higgs fields
• Construct -invariant mass terms. Coupling consts
• Assign vacuum expectation values to break the symmetry
G
G
4 [ : 3, 1, 1 ,1 ]A IR
h
3; 1, 1 ,1 ; 3R eL e 3, 1, 1 ,1
Iso-doublet
Iso-triplet
3(1,1,1); (1,0,0)e ev v
23G
• Compute the mixing matrix from the mass matrices
• Tune the parameters to get the desired mixing matrix ,h
1 2 32
1 2 32
1 2 3
e e
h h h
M v h h h
h h h
2
2
a b c
M a b c d
d a b c
2 0
1
21
2
2
(
36
3 )
1
bU c
Quarks
2 3
2 2
3 2
1 / 2 ( )
1 / 2
(1 ) 1
A i
U A
A i A
0.0010
0.0010
0.007
0.017
0.064
0.028
0.017
0.045
0.2272
0.818
0.221
0.340
A
0.0014
0.0014
0.013
0.013
0.031
0.031
0.020
0.020
0.2262
0.815
0.235
0.349
A
Remarks
• is only known numerically. It is too hard to obtain a finite horizontal group for three generations.
• First consider two generations (Cabibbo mixing)
{ , } ii nG F G
nG
2
2
1
1U
iG
Cabibbo Mixing
2
2
1
1
c sU
s c
2 21
2 2
c sG
s c
1
1F
2 21
2 2
c sG F
s c
12G 2 2
1 1 1, | , ,( )m mG F G F G F D
2 21
2 2
( ) m mm
m m
c sG F
s c
2C m
7D 0.2225 0.2262 0.0014
0.2272 0.0010
PDG:
non-degenerate !
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
• 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).
UG
Yu, Luo River, circa 2100 BCE
河圖洛書4
5
2
8 1 6
3
9
7
The original 2-3 and magic symmetries
2 2 01
1 2 36
1 2 3
U