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