PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

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where Λ is the cut-off scale of the theory and the underline sign in the superscript represents the particular S 3 representation from the tensor product of the two S 3 doublets 1 . Since (D l D l ) andξ∆are2×2 productswhich couldgiveeither 1or2, andsince wecanobtain 1 either by 1×1 or 2×2, we have two terms coming from (D l D l ξ∆). The (D l D l )(ξ∆) as 1 ′ ×1 ′ term does not contribute to the neutrino mass matrix. In this model the presence of the Z 4 symmetry prevents the appearance of the usual 5 dimensional D l D l HH and l 1 l 1 HH Majorana mass term for the neutrinos. In fact, the neutrino mass matrix is completely independent of H due to the Z 4 symmetry. In addition, there are no Yukawa couplings involving the neutrinos and the flavon φ e due to Z 4 or/and Z 3 symmetry. The S 3 symmetry is broken spontaneously when the flavon ξ acquires a vacuum VEV: ( ) u1 〈ξ〉 = . (6.8) u 2 TheSU(2) L ×U Y (1)breaksattheelectroweak scalebytheVEVoftheSU(2)doublet Higgs H. The VEV of the Higgs triplet is ( ) ( ) 〈∆1 〉 0 0 〈∆〉 = , where 〈∆ 〈∆ 2 〉 i 〉 = . (6.9) v i 0 The neutrino get masses due to the VEV of the Higgs triplet field ∆ and as well as the standard model gauge singlet field ξ. The mass matrix of the neutrino is given as ⎛ ⎞ w 2y 4 2y Λ 3 v 2 2y 3 v 1 m ν = ⎝ u 2y 3 v 2 2y 2 v 2 w 1 2y Λ 2 ⎠ , (6.10) Λ w u 2y 3 v 1 2y 2 2y 1 v 1 Λ 1 Λ where w = u 1 v 2 +u 2 v 1 . For the VEV alignments the neutrino mass matrix reduces to the form ⎛ 2u 2y 1 v 1 4 2y Λ 3 v 1 2y 3 v 1 m ν = ⎝ u 2y 3 v 1 2y 1 v 1 2u 1 2y 1 v 1 Λ 2 Λ 2u 2y 3 v 1 2y 1 v 1 u 2 2y 1 v 1 Λ 1 Λ v 1 = v 2 , and u 1 = u 2 , (6.11) ⎞ ⎠ . (6.12) We discuss about the VEV alignments in section 6.4. Denoting u 1 Λ as u ′ 1 the mass matrix becomes ⎛ ⎞ 2y 4 u ′ 1 y 3 y 3 m ν = 2v 1 ⎝ y 3 y 1 u ′ 1 2y 2 u ′ ⎠ 1 , (6.13) y 3 2y 2 u ′ 1 y 1 u ′ 1 1 The term (l l D l ∆) denotes (l T l Ciτ 2D l ∆), where C is the charge conjugation operator. 146
where u ′ 1 = u 1 Λ and it is less than 1. Redefining 2y 4 u ′ 1 as y 4, y 1 u ′ 1 as y 1 and 2y 2 u ′ 1 as y 2, the final form of the mass matrix is m ν = 2v 1 ⎛ ⎝ ⎞ y 4 y 3 y 3 y 3 y 1 y 2 ⎠ . (6.14) y 3 y 2 y 1 In the above matrix, for the VEV u 1 = 0, we would obtain the matrix ⎛ ⎞ 0 1 1 m ν = 2v 1 y 3 ⎝ 1 0 0 ⎠ . (6.15) 1 0 0 Thisisaverywellknownformoftheneutrinomassmatrixwhichreturnsinvertedneutrino mass spectrum with eigenvalues {−2 √ 2v 1 y 3 ,2 √ 2v 1 y 3 ,0}, and bimaximal mixing with θ 23 = θ 12 = π/4 and θ 13 = 0. The family symmetry considered in the literature for obtaining the form of the mass matrix given by Eq. (6.15) is L e −L µ −L τ [8]. However, exact bimaximal mixing is ruled out by the solar neutrino and KamLAND data [9]. Besides, as one can see from the eigenvalues of this neutrino mass matrix, that ∆m 2 21 = 0. This is untenable in the light of the experimental data. In order to generate the correct ∆m 2 21 and deviation of θ 12 from maximal that is consistent with the data, one has to suitably perturb m ν (for instance, in [10] the authors have build a model in the framework of the Zee-Wolfenstein ansatz). In the S 3 model that we consider here, this is very easily obtained if we allow non-zero VEV for ξ. The strength of the additional terms is linearly proportional to u 1 /Λ and could be relatively small. In what follows, we will consider all values of u 1 /Λ from very small to ∼ 1. The eigenvalues of the most general matrix given by Eq. (6.14) are 2 ) m i = v 1 (y 1 +y 2 +y 4 − √y1 2 +y2 2 +y2 4 +8y2 3 +2y 1y 2 −2y 1 y 4 −2y 2 y 4 , (6.16) ) m j = v 1 (y 1 +y 2 +y 4 + √y1 2 +y2 2 +y4 2 +8y3 2 +2y 1 y 2 −2y 1 y 4 −2y 2 y 4 , (6.17) ( ) m 3 = 2v 1 y 1 −y 2 . (6.18) In the above, i,j = 1,2 and the only difference between m i and m j comes in the sign of the quantity within square root. We know that the solar neutrino data provides evidence for ∆m 2 21 > 0 at more than 6σ [9]. Therefore, the choice of m 1 and m 2 in Eqs. (6.16) and (6.17) is determined by the condition m 2 > m 1 viz., the larger eigenvalue corresponds to 2 For all analytical results given in this section we have assumed the model parameters to be real for simplicity. We check the phenomenological viability and testability of our model in the next section for complex Yukawa couplings. 147
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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
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B: The Interaction Lagrangian B.1:
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C H0 ,L 1√ ll 2 (T 11Y † l S 11
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c Z,L ll c Z,L lΣ − c Z,L Σ −
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Bibliography [1] S. Weinberg, Phys.
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Lett. B 615, 231 (2005); Phys. Rev.
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violating λ ′′ coupling are co
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neutralino-neutrinomassmatrixandins
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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
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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 156 and 157: 5.3. The deviation of the mixing an
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
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Page 172 and 173: a way that the residual µ−τ sym
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.
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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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