Morimitsu Tanimoto Morimitsu Tanimoto Niigata UniversityNiigata University

Morimitsu Tanimoto Morimitsu Tanimoto Niigata UniversityNiigata University

Masses and Mixings of Quark-Leptonin

the non-Abelian Discrete Symmetry

Masses and Mixings of Quark-Leptonin

the non-Abelian Discrete Symmetry

VIVIthth Rencontres du VietnamRencontres du Vietnam

August 9 , 2006August 9 , 2006 VIVIthth Rencontres du VietnamRencontres du Vietnam

August 9 , 2006August 9 , 2006

This talk is based on collaborated work withThis talk is based on collaborated work withE.MaE.Ma and H. Sawanakaand H. Sawanaka

This talk is based on collaborated work withThis talk is based on collaborated work withE.MaE.Ma and H. Sawanakaand H. Sawanaka

Plan of the talkPlan of the talk

1 Introduction : Motivations

2 A4 Symmetry

3 A4 Model for Leptons

4 A4 Model for Quarks 5 Summary 　

1 Introduction : Motivations

2 A4 Symmetry

3 A4 Model for Leptons

4 A4 Model for Quarks 5 Summary 　

1 Introduction : Motivations1 Introduction : Motivations

θsol ~ 33°, θatm ~ 45°, θCHOOZ < 12°θsol ~ 33°, θatm ~ 45°, θCHOOZ < 12°

Neutrino Oscillation Experiments already taught us Neutrino Oscillation Experiments already taught us

Δmatm ~ 2×10-3 eV2, Δmsol ~ 8×10-5 eV2, δ :unknown Δmatm ~ 2×10-3 eV2, Δmsol ~ 8×10-5 eV2, δ :unknown

Two Large Mixing Angles and One Small MixingAngle 　

(Δmsol / Δmatm )1/2 = 0.2 ≒ λ

　　　　　　　　　　　　　　　 Ideas

Two Large Mixing Angles and One Small MixingAngle 　

(Δmsol / Δmatm )1/2 = 0.2 ≒ λ

　　　　　　　　　　　　　　　 Ideasobserved

valuesobserved

valuesstructure of mass matrixstructure of mass matrix

flavor symmetry

flavor symmetry

Θij , miΘij , mi

texture zeros,flavor democracy,μ-τ symmetry, ...

texture zeros,flavor democracy,μ-τ symmetry, ...

Discrete SymmetryS3, D4, Q4, A4...Discrete SymmetryS3, D4, Q4, A4...

2 2

2 2

？

Quark/Lepton mixingQuark/Lepton mixing

Lepton : θ12 = 30 〜 35°, θ23 = 38 〜 52°, θ13 < 12°Lepton : θ12 = 30 〜 35°, θ23 = 38 〜 52°, θ13 < 12°

by M.Frigerioby M.Frigerio

Quark ⇔ Lepton :Quark ⇔ Lepton : ● Comparable in 1-2 and 1-3 mixing.● Large hierarchy in 2-3 mixing. (Maximal 2-3 mixing in Lepton sector ?) Tri-Bi maximal mixing ?Tri-Bi maximal mixing ?

● Comparable in 1-2 and 1-3 mixing.● Large hierarchy in 2-3 mixing. (Maximal 2-3 mixing in Lepton sector ?) Tri-Bi maximal mixing ?Tri-Bi maximal mixing ?

Quark : θ12 ~ 13°, θ23 ~ 2.3°, θ13 ~ 0.2° (90% C.L.)Quark : θ12 ~ 13°, θ23 ~ 2.3°, θ13 ~ 0.2° (90% C.L.)

Bi-Maximal

Tri-Bi-MaximalHarrison, Perkins, Scott (2002)

Barger,Pakvasa,Weiler,Whisnant(1998)

θ12 ≒35°

Bi - Maximal θ12 = θ23 =π/4 , θ13 =0

Bi - Maximal θ12 = θ23 =π/4 , θ13 =0

Tri - Bi-maximalθ12 ≒35°, θ23 =π/4 , θ13 =0

What is Origin of the maximal 2-3 mixing ?

What is Origin of the maximal 2-3 mixing ? Discrete Symmetries are nice candidate. Discrete Symmetries are nice candidate.

Flavor SymmetryS3, D4, Q4, A4 ...

Tri-Bi-Maximal mixing is easilyrealized in A4 .

order 6 8 10 12 14 ...

SN : permutation groups

S3 ...

DN : dihedral groups D3 D4 D5 D6 D7 ...

QN : quaternion groups Q4 Q6 ...

T : tetrahedral groupsT(A4

)...

2 A4 Symmetry2 A4 Symmetry

Non-Abelian discrete groups have non-singlet irreducible representations which can be assigned to interrelate families. Non-Abelian discrete groups have non-singlet irreducible representations which can be assigned to interrelate families.

by E. Ma1 1’ 1” 3

by E. Ma

3 A4 Model for Leptons 3 A4 Model for Leptons

L=(νi , li ) ～ 3 li ～ 1, 1’, 1” （ Φi, Φi ）～ 3 < Φi, >=v1, v2, v3

L=(νi , li ) ～ 3 li ～ 1, 1’, 1” （ Φi, Φi ）～ 3 < Φi, >=v1, v2, v3

0 － 0

E.Ma

c

MνLL 3 ×3 L lcΦ 3 ×(1,1’,1”)× 3

Taking b=c , e=f=0 , v1=v2=v3=v

Seesaw Realization Seesaw Realization

L=(νi , li ) ～ 3 li ～ 1, 1’, 1” （ Φi, Φi ）～ 3 < Φi, >=v1, v2, v3

L=(νi , li ) ～ 3 li ～ 1, 1’, 1” （ Φi, Φi ）～ 3 < Φi, >=v1, v2, v3

0 － 0

LνR Φ + νRiνRj χk + M0νRiνRj

（ Φ, Φ ）～ 1νRi ～ 3 χi ～ 3 0

0

He, Keum, Volkas hep-ph/0601001

c

Another assignment: Altarelli, Feruglio, hep-ph/0512103

－

Quark Sector ?If the A4 assignments are

Q=(ui , di ) ～ 3 di , ui ～ 1, 1’, 1” （ Φi, Φi ）～ 3 <Φi> = v1, v2, v3

cc

0 with

v1=v2=v3=v

VCKM = UU† UD

= ICKM mixings come from higher operators!

0 -

4 A4 Model for QuarksMa, Sawanaka, Tanimoto, hep-ph/0606103

Quark-Lepton Unification in SU(5)

5*i (νi , li , dic ) ～ 3 c

10i ( li , uic , uic, dic) ～ 1, 1’, 1”cc

（ Φi, Φi ） D ～ 3 <Φi>D = v1D, v2D, v3D

（ Φi, Φi ） E ～ 3 <Φi>E = v1E, v2E, v3E

（ Φ1,Φ1 ） U ～ 1’ <Φ1>U = v1U

0

0

0 With v1E=v2E=v3E

－

－

－

0

0

（ Φ2,Φ2 ） U ～ 1” <Φ2>U = v2U

0 0

0

－

Parameters in Quarks: hi , viD , μ2 , μ3 , m2 , m3

v1E=v2E=v3E in order to get Tri-Bi-maximal mixing

v1D << v2D << v3D in order to get quark mass hierarchy

D

1 1’1” 1 1’1”

1’1’1’ 1”1”1”

O(λ)comes fromA4 phase ω

↑

Taking account in phase ω and Im(μ3 )CP violation is predicted.

How to test the quark mass matrices :Since Vub depends on the phase of μ3 ,We expect the correlation between Vub and sin2β.

5 Summary 　5 Summary 　 A4 Flavor Symmetry gives us

Tri-Bi-maximal neutrino mixing and CKM Quark Mixings in the SU(5) unification of quarks/leptons. （ Φi, Φi ） E ～ 3 <Φi>E = v1E, v2E, v3E

v1E=v2E=v3E

（ Φi, Φi ） D ～ 3 <Φi>D = v1D, v2D, v3D v1D<<

v2D<< v3D

（ Φ1,Φ1 ） U ～ 1’ （ Φ2,Φ2 ） U ～ 1”★JCP comes from mainly A4 phase ω.

★Strong correlation between Vub and sin2β.

A4 Flavor Symmetry gives us

Tri-Bi-maximal neutrino mixing and CKM Quark Mixings in the SU(5) unification of quarks/leptons. （ Φi, Φi ） E ～ 3 <Φi>E = v1E, v2E, v3E

v1E=v2E=v3E

（ Φi, Φi ） D ～ 3 <Φi>D = v1D, v2D, v3D v1D<<

v2D<< v3D

（ Φ1,Φ1 ） U ～ 1’ （ Φ2,Φ2 ） U ～ 1”★JCP comes from mainly A4 phase ω.

★Strong correlation between Vub and sin2β.

-0

- 00

0

0 0- -