1. Morimitsu TANIMOTO Niigata University, Japan Prediction of U e3 and cosθ ２３ from Discrete symmetry XXXXth RENCONTRES DE MORIOND March ６, 2005 @ La Thuile, Aosta Valley, Italy Based on the work by W.Grimus, A.Joshipura, S.Kaneko, L.Lavoura, H.Sawanaka and M.Tanimoto, Nucl. Phys. B, 2005 (hep-ph/0408123)

2. Neutrino Mixings are near Bi- Maximal I. Introduction tan θ sol = 0.33 - 0.49 (90% C.L.) 2 sin θ chooz < 0.057 (3σ) 2 θ 23 = 45° ±8° θ 12 = 33° ± 4° θ 13 < 12° sin 2θ atm > 0.92 (90% C.L.) 2

3. ★ Why are θ 23 and θ 12 so large ? Questions: ★ Why does sinθ 12 deviate from maximal although sinθ 23 is almost maximal ? ★ Why is θ 13 small ? How small isθ 13 ? Expectation: ★ Flavor Symmetry prevents non-zero θ 13

4. Plan of the talk 1. Introduction 2. Vanishing U e3 and Discrete Symmetry 4. Summary and Discussion s 3. Symmetry Breaking and Neutrino Mixing Angles Grimus, Joshipura, Kaneko, Lavoura, Sawanaka and Tanimoto Nucl. Phys. B(2005) hep-ph/0408123

5. 2. Vanishing Ue3 and Discrete Symmetry Basic Idea : Naturalness of Theory Suppose a dimensionless small parameter a. If a 0, the Symmetry is enhanced. Neutrino Mass Matrix is constructed in terms of neutrino masses and mixings m 1, m 2, m 3

6. which has Z 2 Symmetry

7. 2 2 2 2 Normal : m 1 < m 2 < m 3 (Δm sol = m 2 ー m １, Δm atm = m ３ ー m １ ) Inverted : m 3 < m 1 < m 2 (Δm sol = m 2 ー m １, Δm atm = m １ ー m ３ ) Quasi-Degenerate : m 1 ~ m 2 ~ m 3 2 2 222 222

8. In the limit γ=2θ 23 =π/2, θ 13 =0 μ-τ( Z 2 ) Interchange Symmetry Remark: θ 12 is arbitrary !

9. Assumption : θ 23 = 45°, θ 13 = 0° Framework : SM + 3ν R (seesaw model) θ 12 = arbitrary, no Dirac phase, two Majorana phases W.Grimus and L.Lavoura(2003) Neutrino mass matrix

10. D 4 × Z 2 model SM + 3ν R + 3φ + 2χ φ : gauge doublet Higgs χ : gauge singlet Higgs W.Grimus and L.Lavoura (2003) Charge assignment of D 4 :

11. D 4 × Z 2 model origin of lepton mixings

12. |ε|,|ε’| are constrained by experiments. W.Grimus, A.S.Joshipura, S.Kaneko., L.Lavoura H.Sawanaka and M.T (’0 ５ N.P.B) 3. Symmetry breaking and neutrino mixing angles Small perturbations ε,ε’ give non-zero Ue3 and cos 2θ 23 Two Independent Symmetry Breakings Terms should be considered.

13. In the hierarchical case (m 3 >> m 2 > m 1 ) with Majorana phases ρ=σ=0

14. ρ ＝ 0, σ=0 ρ ＝ π/4, σ=0ρ ＝ π/2, σ=0 Normal hierarcy of ν mass |ε|,|ε’| < 0.3 |U e3 | < 0.2 |cos2θ 23 | < 0.28 CHOOZ atm. symmetric

15. Inverted hierarcy of ν mass ρ ＝ π/4, σ=0 ρ ＝ 0, σ=0 ρ ＝ π/2, σ=0 CHOOZ atm.

16. Quasi-degenerate of ν mass ρ ＝ 0, σ=0 ρ ＝ π/4, σ=0 ρ ＝ π/2, σ=0 |ε| < 0.3, |ε’| < 0.03, m = 0.3 eV ρ ＝ 0, σ=π/2 CHOOZ atm.

17. 1. Normal ν mass hierarchy : small deviation of |U e3 | large deviation of |cos2θ 23 | 2. Inverted ν mass hierarchy : small / large deviation of |U e3 | (depend on Majorana phases) large deviation of |cos2θ 23 | 3. Quasi-Degenerate ν mass hierarchy : small / large deviation of |U e3 | (depend on Majorana phases) small deviation of |cos2θ 23 |

18. Model of Symmetry Breaking: Radiatively generated Ue3 and cos2θ atm Assumption : ε,ε’ are generated due to radiative corrections. MSSMSMGUT M EW MXMX Ue3 = ０ cos2θ 23 = 0 Ue3 = non-zero cos2θ 23 =non- zero

19. m = 0.3 eV, ρ ＝ 0, σ=π/2 tanβ is constrained : tanβ< 23 ~ Effect of Radiative correction is significant in Qusi-Degenerate case

20. We easily find the Neutrino Models based on Discrete Symmetry which predicts θ 13 = 0°, θ 23 = 45° in the symmetric limit. Discrete Symmetry: S 3, D 4, Q 4, Q 6 Q 4(8) Model (Talk of M. Frigerio) |U e3 | and |cos2θ 23 | are deviated from zero by small symmetry breaking (ex. Ｒ adiative correction) These deviations depend on Majorana phases ρ, σ. 4. Summary and Discussions

21. S3 S3 S.Pakvasa and H.Sugawara, PLB 73(1978)61. J.Kubo, A.Modragon, M.Mondragon and E.Rodrigues-Jauregui, Prog.Theor.Phys.109(2003)795. D4D4 W.Grimus and L.Lavoura, PLB 572 (2003) 189. Grimus, Joshipura, Kaneko, Lavoura, Sawanaka and M.T hep-ph/0408123 A4A4 E.Ma and G.Rajasekaran, PRD 64 (2001) 113012. K.S.Babu, E.Ma and J.W.F.Valle, PLB 552 (2003) 207. Models based on non-Abelian discrete groups Q 12 (Q 6 )K.S.Babu and J.Kubo, hep-ph/0411226. Q 8 (Q 4 )M. Frigerio, S. Kaneko., E. Ma and M. T, hep-ph/0409187.

22. Unify the lepton and quark sectors Higgs potential S-quark and S-lepton sector in SUSY Future

23. 2× ２ Decompositions of D 4 Mass matrix in D 4 doublet basis Higgs : H 1, H 2 lepton : ( l L1, l L2 ), ( l R1, l R2 )

24. Non-Abelian discrete groups order68101214... SNSN S3S3 DNDN D 3 (=S 3 )D4D4 D5D5 D6D6 D7D7... QNQN Q 8 (Q 4 )Q 12 (Q 6 )... TT(A 4 )... order : number of elements D 3 (=S 3 ) : rotations and reflections of △ D 4 : rotations and reflections of □ A 4 : rotations and reflections of tetrahedron Geometrical object :

25. Group D 4 21 3 4 C3C3 n : # of elements h : order of any elements in that class(g h =1) 14 2 3 Σ(dim. of reps.)^2 = # of elements # of reps. = # of classes