Who Ordered the Muon ?(symmetry of three generations)

C.S. Lam

McGill and UBC, Canada

arXiv:0708.3665, PL B656 (2007) 193arXiv:0711.3795 arXiv: 0804.2622

discoveries from 1928 to 1936

Carl David Anderson

(1905—1991)

1928 Dirac equation

1930 Pauli: neutrino

1932 Anderson: positron

1932 Chadwick: neutron

1935 Yukawa: pion

1936 Anderson: muon

a famous ancient question from my

great-grand teacher

Carl David Anderson Isidor Issac Rabi

Who ordered that ?

1986 Columbia U.

(1898—1988)

the plot thickens

• now there are three generations of quarks and leptons. Why three?

deep meaning, or …….

the plot thickens

• now there are three generations of quarks and leptons. Why three?

• they have very different masses

• quark and neutrino mixings are vastly different

• Do they contain a key to solve the generation problem? Or at least a hint ?

symmetry may be the key

(1) (3)Q CU SU

(2) (1) (3)I Y CSU U SU

SUSY? GUTS? STRING?

motivated by symmetry, but no expt’l evidence

horizontal symmetry ?

generation problem

lots of expt’l data !!!

Quark/lepton masses & mixings

(Gell-Mann’s eightfold way)

what horizontal symmetry ?

“ with enough parameters, you can even wiggle the elephant’s tail”

3 4 4 5, , , , , , , ', (27), (3), (3), ......m m n nZ Z Z D S S A A T SU SO

true symmetry should reveal itself without tuning

need to find a clue, or the ‘modern Balmer series formula’

new (group theoretical) technique is needed

the modern Balmer seriesQuark

Lepton

Mass

Mixing

• masses: ,t u em m m m ( )t bm mSuggests a spontaneously broken symmetry like the SM

• quark mixing: small and somewhat irregular, could be a dynamical perturbation of no mixing

• neutrino mixing: large and regular (tri-bimaximal mixing), and that may be the clue .

a clue to Rabi’s question

(1) (3)Q CU SU

(2) (1) (3)I Y CSU U SU

SUSY? GUTS? STRING?

generation problem4S

any group containing

4S

S4 is the symmetry of the octahedron and the cube

presently known fermions

consequence

Other than the SM Higgs, there are a few more neutral Higgs particles to be found

(needed for symmetry breaking)

The coupling of SM Higgs to fermions is not proportional to their masses

Explains the miracle of tri-bimaximal mixing

Stepping stone for constructing GGUT (Great Grand Unified Symmetry)

Construction and phenomenology of -invariant dynamical models4S

masses and mixings

L

e

e

e

L

Integrate over all right-handed fermions

† † Te eH e M M e M

† †e e e eU M M U diag TU M U diag

†eU U U

eM diagonal

eU U

columns of columns of

non-alignment of eigenvectors

symmetry and mixing

G e eF†† †

e e e eF FM M M M TG M MG

† † Te eH e M M e M

†[ , ] 0e eM MF

,F †e eM M

same eigenvectors

2 1G

1F UG ,G Msame

eigenvectors

eU U

symmetry must be spontaneously brokencolumns of columns of

eigenvalues 1, 1, 1

charged leptons diagonal

Tri-bimaximal mixing matrix

2 2 01

1 2 36

1 2 3

U U

1 2 3

e

1 2 21

2 2 13

2 1 21G

1 2 212 1 2

32 2 1

2G

1 0 0

0 0 1

0 1 03G

also conversely, if F is non-degenerate

,eM F diagonal

Block diagonal and 3D IR

rules out 3, , ,n n m nZ Z Z D S

the horizontal group

F3G2G{ , , }n G 1nF

All eigenvalues different

n=3 : two distinct possibilities2 2

1

, 1F

43 4, 3.S SG

what about n=4,5,6,7,8,………?

1,2n

3 1

4 3( , )A F G

uniqueness All finite subgroups of SO(3) and SU(3) with 3-dim IR

(2)SU nZ nD 4A 4S 5A

T O I

(3)SU2 2,

(60), , (168),(360) (36) (72), (216

(3 (

)

6 ),

,

)n n

directno 3-dim IRn=3 or wrong F

breaking down to 4S 2 3, ,F G G

( , , , , , )A AL R L RH H e e N

AA AF

2,3AA AG

1,1',2,3,3'A

true for all couplings

( , , , , , )A AL R L RH H e e N

4S

2 3, ,F G G

allowed expectation values

32

1 2 212 1 2

32 2 1

G

33

1 0 0

0 0 1

0 1 0

G

2

3

1 0 0

0 0

0 0

F

AA AF

2,3AA AG

22 0

0F

23

0 1

1 0G

22

1 0

0 1G

3'2

1 2 212 1 2

32 2 1

G

3'3

1 0 0

0 0 1

0 1 0

G

2

3'

1 0 0

0 0

0 0

F

1 1 12 3 1F G G 1 1 1 1

2 0

31

0

0

2 1

1

3 0

3'1

1

1

3'1

0

0

1' 1' 1'2 3 1F G G 1' 1 1' 0

conclusion

• Present estimates for Higgs production and fermion-pair decay may be wrong

• The horizontal symmetry group is

• Dynamical details are model dependent

4S

1 2 21

2 2 13

2 1 21G

1 2 212 1 2

32 2 1

2G

1 0 0

0 0 1

0 1 03G