1. Morimitsu Tanimoto Niigata University Morimitsu Tanimoto Niigata University Masses and Mixings of Quark-Lepton in the non-Abelian Discrete Symmetry VI th Rencontres du Vietnam August 9, 2006 VI th Rencontres du Vietnam August 9, 2006 This talk is based on collaborated work with E.Ma and H. Sawanaka This talk is based on collaborated work with E.Ma and H. Sawanaka

2. Plan of the talk 1 Introduction : Motivations 2 A 4 Symmetry 3 A 4 Model for Leptons 4 A 4 Model for Quarks 5 Summary 1 Introduction : Motivations 2 A 4 Symmetry 3 A 4 Model for Leptons 4 A 4 Model for Quarks 5 Summary

3. 1 Introduction : Motivations θ sol ~ 33°, θ atm ~ 45°, θ CHOOZ < 12° Neutrino Oscillation Experiments already taught us Δ m atm ~ 2×10 -3 eV 2, Δ m sol ~ 8×10 -5 eV 2, δ :unknown Two Large Mixing Angles and One Small MixingAngle (Δm sol / Δm atm ) 1/2 = 0.2 ≒ λ Ideas Two Large Mixing Angles and One Small MixingAngle (Δm sol / Δm atm ) 1/2 = 0.2 ≒ λ Ideas observed values structure of mass matrix flavor symmetry Θ ij, m i texture zeros, flavor democracy, μ-τ symmetry,... texture zeros, flavor democracy, μ-τ symmetry,... Discrete Symmetry S 3, D 4, Q 4, A 4... Discrete Symmetry S 3, D 4, Q 4, A 4... 22 22 ？

4. Quark/Lepton mixing Lepton : θ 12 = 30 〜 35°, θ 23 = 38 〜 52°, θ 13 < 12° by M.Frigerio 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 : θ 12 ~ 13°, θ 23 ~ 2.3°, θ 13 ~ 0.2° (90% C.L.)

5. Bi-Maximal Tri-Bi-Maximal Harrison, Perkins, Scott (2002) Barger,Pakvasa, Weiler,Whisnant (1998) θ 12 ≒ 35°

6. Bi - Maximal θ 12 = θ 23 =π/4, θ 13 =0 Tri - Bi-maximal θ 12 ≒ 35°, θ 23 =π/4, θ 13 =0

7. What is Origin of the maximal 2-3 mixing ? What is Origin of the maximal 2-3 mixing ? Discrete Symmetries are nice candidate. Flavor Symmetry S 3, D 4, Q 4, A 4... Tri-Bi-Maximal mixing is easily realized in A 4.

8. order 68101214... S N : permutation groups S3S3... D N : dihedral groupsD3D3 D4D4 D5D5 D6D6 D7D7... Q N : quaternion groupsQ4Q4 Q6Q6... T : tetrahedral groupsT(A 4 )... 2 A 4 Symmetry Non-Abelian discrete groups have non-singlet irreducible representations which can be assigned to interrelate families.

9. by E. Ma 1 1’ 1” 3

11. by E. Ma

12. 3 A 4 Model for Leptons L=( ν i, l i ) ～ 3 l i ～ 1, 1’, 1” （ Φ i, Φ i ） ～ 3 =v 1, v 2, v 3 L=( ν i, l i ) ～ 3 l i ～ 1, 1’, 1” （ Φ i, Φ i ） ～ 3 =v 1, v 2, v 3 0 － 0 E.Ma c M ν LL 3 ×3 L l c Φ 3 ×(1,1’,1”)× 3

13. Taking b=c, e=f=0, v 1 =v 2 =v 3 =v

15. Seesaw Realization L=( ν i, l i ) ～ 3 l i ～ 1, 1’, 1” （ Φ i, Φ i ） ～ 3 =v 1, v 2, v 3 L=( ν i, l i ) ～ 3 l i ～ 1, 1’, 1” （ Φ i, Φ i ） ～ 3 =v 1, v 2, v 3 0 － 0 Lν R Φ + ν Ri ν Rj χ k + M 0 ν Ri ν Rj （ Φ, Φ ）～ 1 ν Ri ～ 3 χ i ～ 3 0 0 He, Keum, Volkas hep-ph/0601001 c Another assignment: Altarelli, Feruglio, hep-ph/0512103 －

16. Quark Sector ? If the A 4 assignments are Q=(u i, d i ) ～ 3 d i, u i ～ 1, 1’, 1” （ Φ i, Φ i ）～ 3 = v 1, v 2, v 3 cc 0 with v 1 =v 2 =v 3 =v V CKM = U U† U D = I CKM mixings come from higher operators! 0 -

17. 4 A 4 Model for Quarks Ma, Sawanaka, Tanimoto, hep-ph/0606103 Quark-Lepton Unification in SU(5) 5* i (ν i, l i, d ic ) ～ 3 c 10 i ( l i, u ic, u ic, d ic ) ～ 1, 1’, 1” c c （ Φ i, Φ i ） D ～ 3 D = v 1D, v 2D, v 3D （ Φ i, Φ i ） E ～ 3 E = v 1E, v 2E, v 3E （ Φ 1,Φ 1 ） U ～ 1’ U = v 1U 0 0 0 With v 1E =v 2E =v 3E － － － 0 0 （ Φ 2,Φ 2 ） U ～ 1” U = v 2U 0 0 0 －

18. Parameters in Quarks: h i, v iD, μ 2, μ 3, m 2, m 3 v 1E =v 2E =v 3E in order to get Tri-Bi-maximal mixing v 1D << v 2D << v 3D in order to get quark mass hierarchy D 1 1’1”1 1’1” 1 1’1”1 1’1” 1’1’1’1’1’1’1”1”1”1”1”1”

20. O(λ) comes from A 4 phase ω ↑ Taking account in phase ω and Im(μ 3 ) CP violation is predicted. How to test the quark mass matrices : Since Vub depends on the phase of μ 3, We expect the correlation between Vub and sin2β.

22. 5 Summary A 4 Flavor Symmetry gives us Tri-Bi-maximal neutrino mixing and CKM Quark Mixings in the SU(5) unification of quarks / leptons. （ Φ i, Φ i ） E ～ 3 E = v 1E, v 2E, v 3E v 1E =v 2E =v 3E （ Φ i, Φ i ） D ～ 3 D = v 1D, v 2D, v 3D v 1D << v 2D << v 3D （ Φ 1,Φ 1 ） U ～ 1’ （ Φ 2,Φ 2 ） U ～ 1” ★ J CP comes from mainly A 4 phase ω. ★ Strong correlation between Vub and sin2β. A 4 Flavor Symmetry gives us Tri-Bi-maximal neutrino mixing and CKM Quark Mixings in the SU(5) unification of quarks / leptons. （ Φ i, Φ i ） E ～ 3 E = v 1E, v 2E, v 3E v 1E =v 2E =v 3E （ Φ i, Φ i ） D ～ 3 D = v 1D, v 2D, v 3D v 1D << v 2D << v 3D （ Φ 1,Φ 1 ） U ～ 1’ （ Φ 2,Φ 2 ） U ～ 1” ★ J CP comes from mainly A 4 phase ω. ★ Strong correlation between Vub and sin2β. -0 -00 0 00 --