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Morimitsu TANIMOTO Niigata University, Japan Prediction of U e3 and cosθ ２３ from Discrete symmetry XXXXth RENCONTRES DE MORIOND March ６, 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/ )

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

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★ 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

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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/

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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

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which has Z 2 Symmetry

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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

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

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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

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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 :

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D 4 × Z 2 model origin of lepton mixings

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|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.

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

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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

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

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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.

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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 |

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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

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

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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

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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/ A4A4 E.Ma and G.Rajasekaran, PRD 64 (2001) 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/ Q 8 (Q 4 )M. Frigerio, S. Kaneko., E. Ma and M. T, hep-ph/

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

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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 )

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Non-Abelian discrete groups order 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 :

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

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