PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

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5.3. The deviation of the mixing angles from their TBM values can be seen to be D 12 ≃ 0, D 23 ≃ − ǫ 4d , U e3 ≃ − ǫ 2 √ 2d , (5.15) where D 12 = sin 2 θ 12 − 1/3 and D 23 = sin 2 θ 23 − 1/2. We show in the left hand panels of Fig. 5.3 the mixing angles sin 2 θ 12 (upper panel), sin 2 θ 13 (middle panel) and sin 2 θ 23 (lower panel) as a function of ǫ. We vary ǫ from large negative to large positive values and solve the exact eigenvalue problem numerically allowing the other parameters, m 0 , b and d, to vary freely. For ǫ = 0 of course we get TBM mixing as expected. For very small values of ǫ, the deviation of the mixing angles from their TBM values is reproduced well by the approximate expressions given in Eq. (5.15). For large ǫ of course the approximate expressions fail and we have significant deviation from TBM. The values of sin 2 θ 12 and sin 2 θ 13 increase very fast with ǫ, while sin 2 θ 23 decreases with it. We have only showed the scatter plots up to the 3σ allowed ranges for the mixing angles given in [20]. The mass dependent observables can be calculated upto first order in ǫ as ∆m 2 21 ≃ m2 0 (b+d+ ǫ 3ǫ )(3d−b+ 2 2 ), ∆m2 31 ≃ (4bd+2bǫ)m2 0 , (5.16) 〈m ee 〉 = m 0 2b 3 , m t = ∑ i |m i |, m 2 β ≃ m 2 0( 2b2 3 +2d2 − 4bd 2bǫ +2dǫ− 3 3 ) . (5.17) Of course, epsilon can be larger and we show numerical results for those cases. Normal Hierarchy The right panels of Fig. 5.3 show ∆m 2 21 (upper panel), ∆m 2 31 (middle panel) and m t (lower panel) as a function of ǫ assuming m 1 < m 2 ≪ m 3 . We show ∆m 2 21 and ∆m2 31 only within their 3σ [20] allowed range. We notice that while ∆m 2 21 and ∆m 2 31 are hardly constrained by ǫ, there appears to some mild dependence of m t on it. Fig. 5.4 gives the scatter plots showing the allowed parameter regions for this case. The upper, middle and lower panels show the allowed points projected on the b−c, d−c and b−d plane, respectively. Inverted Hierarchy For this case it is possible to obtain even inverted hierarchy. We show in Fig. 5.4 the scatter plots showing the allowed parameter regions for inverted hierarchy. We have allowed m 0 to vary freely and show the allowed points projected on the c−b, c−d and d−b planes. Onecancheck thatonly forthepointsappearing inthisplot, m 3 < m 1 < m 2 . 128
Higgs Neutrino mass matrix Eigenvalues Eigenvectors ⎛ ⎞ ⎛ a+ 2b c− b − b − 1 √3 1 6 − 1 ⎞ √ ξ,ξ ′′ 3 3 3 √ 2 m 0 ⎝ c− b 2b a− b ⎠ ⎜ 2 ⎟ a=c; 3 3 3 ⎝ 1√ − b a− b c+ 2b 3 3 0 ⎠ 3 3 3 −√ 1 √3 1 √2 1 ⎛ ⎞ 6 m 0 (b−a), ⎝ 2m 0 a, m 0 (a+b) ⎠ ξ,ξ ′ ξ ′ ,ξ ′′ m 0 ⎛ ⎝ m 0 ⎛ ⎝ a+ 2b 3 − b 3 d− b 3 − b 3 d− b 3 a− b 3 d+ 2b a− b 3 3 2b 3 2b 3 c− b 3 d− b 3 c− b 3 d+ 2b 3 − b 3 d− b 3 − b 3 c+ 2b 3 ⎞ ⎠ ⎞ ⎠ ⎛ ⎝ ⎛ ⎝ ⎛ ⎝ a=d; m 0 (b−a), 2m 0 a, m 0 (a+b) c=d; m 0 (b−c), 2m 0 c, m 0 (c+b) c = d+ǫ; ⎞ ⎠ ⎞ ⎠ m 0 (b−d− ǫ), ⎞ 2 m 0 (2d+ǫ), ⎠ m 0 (d+b+ ǫ) 2 ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ √ 2 3 − 1 √ 6 1 √3 − 1 √ 2 − 1 √ 6 1 √3 1 √2 √ 2 3 −√ 1 6 − √ 3ǫ 4 √ 2d −√ 1 6 + √ 3ǫ 4 √ 2d 1√ 3 0 ⎞ ⎟ ⎠ √ ⎞ 2 1√ 3 3 0 −√ 1 √3 1 6 −√ 1 ⎟ ⎠ 2 −√ 1 √3 1 √2 1 6 1√ 3 − ǫ 2 √ 2d 1√ 3 − 1 √ 2 + ǫ 1 √ 3 4 √ 2d √2 1 + ǫ 4 √ 2d ⎞ ⎟ ⎠ Table 5.3: Here we take two one dimensional A 4 Higgs at a time, and analytically display eigenvalues and eigenvectors of the neutrino mass matrix. 129
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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
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Contents 1 Introduction 1 1.1 Stand
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6.2.2 Charged Lepton Masses and Mix
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Chapter 1 Introduction 1.1 Standard
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For positive µ 2 and λ, the Higgs
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type of Yukawa interaction between
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interaction with the Higgs as λ f
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where ˆΦ is the MSSM chiral super
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(2004); A. Ghosal, hep-ph/0304090;
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active flavor. In recent years, sev
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Figure 2.1: The possible neutrino s
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Majorana mass term breaks lepton nu
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The kinetic term of the Higgs tripl
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ight handed neutrino N R can be exa
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alternating group which is not a di
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The group contains two one-dimensio
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from neutral-current interactions i
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[31] K. S. Babu and R. N. Mohapatra
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of producing the heavy fermion trip
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where Σ ± Ri = Σ1 Ri ∓iΣ2 Ri
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while U ν diagonalizes m ν and ˜
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It is evident that the masses of th
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0.001 0.001 0.0001 0.0001 U e3 1e-0
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if we neglect terms proportional to
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calculations for lepton flavor viol
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fb. Therefore, it is obvious that t
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Σ − -> e m 1 m h 0 Σ − -> e m
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We can also see that for Σ − m 1
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the sinα term. However, decay to h
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Σ − m 1 ->ν m1 W Σ 0 m 1 ->ν
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Γ 2HDM (Σ 0 m → ν m H 0 ) ≃
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Decay modes Σ ± m 1 Σ ± m 2 Σ
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III seesaw model. Clearly, for our
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Sl no Channels Effective cross-sect
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• The other dominant decay channe
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3.9 Conclusions The seesaw mechanis
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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
Page 122 and 123: 4.3 Symmetry Breaking In this secti
Page 124 and 125: +(M 2 +m 2 Σ )ũ2 + ∑ m 2˜ν i
Page 126 and 127: Here M 1,2 are the soft supersymmet
Page 128 and 129: is violated, we have neutrino-neutr
Page 130 and 131: (M 1,2 ,M,µ,Y Σi ,v 1,2 ,ũ,u i )
Page 132 and 133: In our model in addition to the sta
Page 134 and 135: 4.8 Conclusion In this work we have
Page 136 and 137: Y Σi √2 ĤuˆΣ 0 0c Rˆν L i
Page 138 and 139: while the parameter α 1 is α 1 =
Page 140 and 141: Bibliography [1] S. Roy and B. Mukh
Page 142 and 143: [15] M. C. Gonzalez-Garcia and J. W
Page 144 and 145: ν i A Σi • ˜ν i h,H,A × χ
Page 146 and 147: 118
Page 148 and 149: of the form ⎛ ⎞ A B B m ν =
Page 150 and 151: triplet representation of A 4 , l =
Page 152 and 153: m 0 =0.016 m 0 =0.024 m 0 =0.032 2
Page 154 and 155: Higgs Neutrino mass matrix Eigenval
Page 158 and 159: Sin 2 θ 12 Sin 2 θ 13 Sin 2 θ 23
Page 160 and 161: Figure 5.5: Scatter plot showing th
Page 162 and 163: Figure 5.6: Same as Fig. 5.5 but fo
Page 164 and 165: mass matrix is then given as ⎛ m
Page 166 and 167: Bibliography [1] M. C. Gonzalez-Gar
Page 168 and 169: 140
Page 170 and 171: 142
Page 172 and 173: a way that the residual µ−τ sym
Page 174 and 175: where Λ is the cut-off scale of th
Page 176 and 177: m 2 . The eigenvectors are given as
Page 178 and 179: For λ ≃ 2 × 10 −2 ≃ λ2 c 2
Page 180 and 181: 2 2 2 1 1 1 y 2 0 y 3 0 y 4 0 -1 -1
Page 182 and 183: 0.08 0.25 0.06 0.2 10 -3 m νee (eV
Page 184 and 185: We show the values of |U e3 | ≡ s
Page 186 and 187: while the branching ratio for this
Page 188 and 189: 6.4 The Vacuum Expectation Values I
Page 190 and 191: where we have defined B = (b+f 1 +f
Page 192 and 193: VEV alignments. Another way the VEV
Page 194 and 195: (2006); M. Maltoni, T. Schwetz, M.
Page 196 and 197: 168
Page 198 and 199: metric standard model in chapter 1.
Page 200 and 201: mally two. However the R-parity vio
Page 202 and 203: sector contains the exact/approxima
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PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

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