PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

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0.001 0.001 0.0001 0.0001 U e3 1e-05 1e-06 |0.5-sin 2 Θ 23 | 1e-05 1e-06 1e-07 1e-07 1e-08 -0.5 0 0.5 1 1.5 2 2.5 3 ε (GeV) 1e-08 -0.5 0 0.5 1 1.5 2 2.5 3 ε (GeV) Figure 3.3: Non-zero values of U e3 and |0.5 − sin 2 θ 23 | predicted when µ-τ symmetry is broken. Shown are the oscillation parameters against the µ-τ symmetry breaking parameter ǫ = M 3 −M 2 . Only points which reproduce the current neutrino observations within their 3σ C.L. are shown. The plot is generated at a fixed set of Yukawa couplings and heavy neutrino masses. where without loosing generality we have chosen to work in a basis where M is real and diagonal. Here the condition M 3 = M 2 is imposed due to the µ−τ symmetry. The above choice of Yukawa and heavy fermion mass matrix lead to the following form of the light neutrino mass matrix ⎛ ( ) ( ) ⎞ a 2 4 M 1 + 2a2 11 a M 2 a 4 11 M 1 + a 6+a 8 a M 2 a 4 11 M 1 + a 6+a 8 M ( ) 2 m ν ≃ v′2 a ⎜a 4 11 2 ⎝ M 1 + a 6+a 8 a 2 11 M 2 M 1 + a2 6 +a2 8 a 2 11 M 2 M 1 + 2a 6a 8 M ( 2 ⎟ ) ⎠ , (3.41) a a 4 11 M 1 + a 6+a 8 a 2 11 M 2 M 1 + 2a 6a 8 a 2 11 M 2 M 1 + a2 6 +a2 8 M 2 where we have used the seesaw formula given by Eq. (3.21). It is evident from the above mass matrix that the scale of the neutrino masses emerges as ∼ v ′2 a 2 /(2M), where a is a typical value of the Yukawa coupling in Eq. (3.39) and M the scale of heavy fermion masses. In this work, we restrict the heavy fermion masses to be less than 1 TeV in order that they can be produced at the LHC. Therefore in principle, neutrino masses of ∼ 0.1 eV could have been obtained with just the standard model Higgs doublet by reducing the Yukawa couplings to values ∼ 10 −6 . However with the addition of an extra Higgs doublet, it is possible to generate neutrino mass even with relatively large Yukawa coupling. We introduced a different Higgs doublet Φ 2 in our model, which couples only to the exotic fermions. On the other hand, Yukawa coupling of the standard Higgs Φ 1 with the exotic fermions was forbidden in our model by the Z 2 symmetry. Hence, only the VEV of this new Higgs doublet appears in Eq. (3.41). Since this Higgs Φ 2 is not coupled to any standard model particle, it could have a VEV which could be different. Therefore, we 44
demand that v ′ ∼ 10 5 eV in order to generate neutrino masses of ∼ 0.1 eV keeping the Yukawa couplings ∼ 1. We next turn to predictions of this model for the mass squared differences and the mixing angles. Since the neutrino mass matrix we obtained in Eq. (3.41) has µ-τ symmetry it follows that θ 13 = 0 and θ 23 = π/4. (3.42) To find the mixing angle θ 12 and the mass squared differences ∆m 2 21 and ∆m2 31 1 , one needs to diagonalize the mass matrix m ν given in Eq. (3.41). In fact, the form of m ν in Eq. (3.41) is the standard form of the neutrino mass matrix with µ-τ symmetry, and hence the expression of mixing angle θ 12 as well ∆m 2 21 and ∆m2 31 can be found in the literature (see for e.g. [26]). We show in Fig. 3.1 the variation of sin 2 θ 12 (upper panels) and R = ∆m 2 21/|∆m 2 31| (lower panels) with the Yukawa couplings a 4 , a 11 and a 6 . We do not show the corresponding dependence on a 8 since it looks almost identical to the panel corresponding to a 6 . The figure is produced assuming inverted mass hierarchy for the neutrino, i.e., ∆m 2 31 < 0. The neutrino mass matrix given by Eq. (3.41) could very easily yield ∆m 2 31 > 0 and hence the normal mass hierarchy (see for e.g. [26]). However, for the sake of illustration, we will show results for only the inverted hierarchy. In every panel of Fig. 3.1, all Yukawa couplings apart from the one plotted on the x-axis, are allowed to vary freely. The points show the predicted values of sin 2 θ 12 (upper panels) and R (lower panels) as a function of the Yukawa couplings for which all oscillation parameters are within the 3σ values given in [27], For this figure we take M 1 = M 2 for simplicity and define m 0 = v ′2 /(2M 1 ). The blue points are for m = 0.95 eV while the green points are for m 0 = 0.006 eV. Fig. 3.2 is a scatter plot showing the values of the Yukawa couplings which give all oscillation parameters within their 3σ allowed ranges given in [27]. Again as in the previous plot we assume M 1 = M 2 , define m 0 = v ′2 /(2M 1 ) and show the allowed points for m 0 = 0.96 eV (blue), 0.006 eV (green) and 0.0021 eV (red). All Yukawa couplings apart from the ones shown in the x-axis and y-axis are allowed to vary freely, in each panel. Since the µ andτ charged lepton masses aredifferent, we phenomenologically choose to not impose the µ-τ symmetry on the charged lepton mass matrix 2 . Hence, without loosing generality, the charged lepton Yukawa matrix can be taken as ⎛ Y l = ⎝ Y ⎞ e 0 0 0 Y µ 0 ⎠, . (3.43) 0 0 Y τ The masses of the light charged leptons can then be obtained from Eqs. (3.32) and/or (3.34). Forourchoice ofY Σ andM, itturnsoutthatm e ≈ vY e , m µ ≈ vY µ , andm τ ≈ vY τ , 1 We define ∆m 2 ij = m2 i −m2 j . 2 Our choice of the lepton masses and mixing are dictated solely by observations. 45
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Neutrinos and Some Aspects of Physi
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Declaration This thesis is a presen
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Acknowledgments First and foremost,
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iii
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trino masses are bounded within 0.1
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softly broken Z 2 symmetry, the mas
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tribimaximal mixing in the neutrino
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Bibliography [1] Two Higgs doublet
Page 23 and 24: Contents 1 Introduction 1 1.1 Stand
Page 25: 6.2.2 Charged Lepton Masses and Mix
Page 29 and 30: Chapter 1 Introduction 1.1 Standard
Page 31 and 32: For positive µ 2 and λ, the Higgs
Page 33 and 34: type of Yukawa interaction between
Page 35 and 36: interaction with the Higgs as λ f
Page 37 and 38: where ˆΦ is the MSSM chiral super
Page 39: (2004); A. Ghosal, hep-ph/0304090;
Page 44 and 45: active flavor. In recent years, sev
Page 46 and 47: Figure 2.1: The possible neutrino s
Page 48 and 49: Majorana mass term breaks lepton nu
Page 50 and 51: The kinetic term of the Higgs tripl
Page 52 and 53: ight handed neutrino N R can be exa
Page 54 and 55: alternating group which is not a di
Page 56 and 57: The group contains two one-dimensio
Page 58 and 59: from neutral-current interactions i
Page 60: [31] K. S. Babu and R. N. Mohapatra
Page 64 and 65: of producing the heavy fermion trip
Page 66 and 67: where Σ ± Ri = Σ1 Ri ∓iΣ2 Ri
Page 68 and 69: while U ν diagonalizes m ν and ˜
Page 70 and 71: It is evident that the masses of th
Page 74 and 75: if we neglect terms proportional to
Page 76 and 77: calculations for lepton flavor viol
Page 78 and 79: fb. Therefore, it is obvious that t
Page 80 and 81: Σ − -> e m 1 m h 0 Σ − -> e m
Page 82 and 83: We can also see that for Σ − m 1
Page 84 and 85: the sinα term. However, decay to h
Page 86 and 87: Σ − m 1 ->ν m1 W Σ 0 m 1 ->ν
Page 88 and 89: Γ 2HDM (Σ 0 m → ν m H 0 ) ≃
Page 90 and 91: Decay modes Σ ± m 1 Σ ± m 2 Σ
Page 92 and 93: III seesaw model. Clearly, for our
Page 94 and 95: Sl no Channels Effective cross-sect
Page 96 and 97: • The other dominant decay channe
Page 102 and 103: 3.9 Conclusions The seesaw mechanis
Page 104 and 105: Appendix A: The Scalar Potential an
Page 106 and 107: B: The Interaction Lagrangian B.1:
Page 108 and 109: C H0 ,L 1√ ll 2 (T 11Y † l S 11
Page 110 and 111: c Z,L ll c Z,L lΣ − c Z,L Σ −
Page 112 and 113: Bibliography [1] S. Weinberg, Phys.
Page 114: Lett. B 615, 231 (2005); Phys. Rev.
Page 118 and 119: violating λ ′′ coupling are co
Page 120 and 121: neutralino-neutrinomassmatrixandins
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4.3 Symmetry Breaking In this secti
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+(M 2 +m 2 Σ )ũ2 + ∑ m 2˜ν i
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Here M 1,2 are the soft supersymmet
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is violated, we have neutrino-neutr
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(M 1,2 ,M,µ,Y Σi ,v 1,2 ,ũ,u i )
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In our model in addition to the sta
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4.8 Conclusion In this work we have
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Y Σi √2 ĤuˆΣ 0 0c Rˆν L i
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while the parameter α 1 is α 1 =
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Bibliography [1] S. Roy and B. Mukh
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[15] M. C. Gonzalez-Garcia and J. W
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ν i A Σi • ˜ν i h,H,A × χ
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118
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of the form ⎛ ⎞ A B B m ν =
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triplet representation of A 4 , l =
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m 0 =0.016 m 0 =0.024 m 0 =0.032 2
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Higgs Neutrino mass matrix Eigenval
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5.3. The deviation of the mixing an
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Sin 2 θ 12 Sin 2 θ 13 Sin 2 θ 23
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Figure 5.5: Scatter plot showing th
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Figure 5.6: Same as Fig. 5.5 but fo
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mass matrix is then given as ⎛ m
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Bibliography [1] M. C. Gonzalez-Gar
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140
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142
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a way that the residual µ−τ sym
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where Λ is the cut-off scale of th
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m 2 . The eigenvectors are given as
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For λ ≃ 2 × 10 −2 ≃ λ2 c 2
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2 2 2 1 1 1 y 2 0 y 3 0 y 4 0 -1 -1
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0.08 0.25 0.06 0.2 10 -3 m νee (eV
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We show the values of |U e3 | ≡ s
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while the branching ratio for this
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6.4 The Vacuum Expectation Values I
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where we have defined B = (b+f 1 +f
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VEV alignments. Another way the VEV
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(2006); M. Maltoni, T. Schwetz, M.
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168
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metric standard model in chapter 1.
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mally two. However the R-parity vio
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sector contains the exact/approxima
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PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

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