PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

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Higgs Neutrino mass matrix Eigenvalues Mixing Matrix ⎛ ⎞ ⎛ ⎞ ⎛ √ a+ 2b − b − b 2 m 3 3 3 0 (a+b), √ 1 ξ m 0 ⎝ − b 2b a− b 3 ⎠ ⎝ m 3 3 3 0 a, ⎠ ⎜ 3 0 ⎝ −√ 1 √3 1 6 −√ 1 2 m 0 (b−a) − b 3 a− b 3 2b 3 − 1 √ 6 1 √3 ⎞ ⎟ ⎠ √2 1 ξ ′′ m 0 ⎛ ⎝ 2b 3 c− b 3 − b 3 c− b 3 2b 3 − b 3 − b 3 − b 3 c+ 2b 3 ⎞ ⎠ ⎛ ⎝ m 0 (c+b), m 0 c, m 0 (b−c) ⎞ ⎠ ⎛ ⎜ ⎝ − 1 √ 6 1 √3 − 1 √ 2 −√ 1 √3 1 √2 1 √ 6 2 √ 1 3 3 0 ⎞ ⎟ ⎠ ξ ′ m 0 ⎛ ⎝ 2b 3 − b 3 d− b 3 − b 3 d− b 3 − b 3 d+ 2b − b 3 3 2b 3 ⎞ ⎠ ⎛ ⎝ m 0 (d+b), m 0 d, m 0 (b−d) ⎞ ⎠ ⎛ ⎜ ⎝ − √ 1 √3 1 6 −√ 1 ⎞ √ 2 2 √ 1 ⎟ 3 3 0 ⎠ −√ 1 √3 1 √2 1 6 Table 5.2: The mass matrix taking one Higgs at a time, its mass eigenvalues, and its mixing matrix. case are ∆m 2 21 = (−b 2 −2ab)m 2 0, ∆m 2 31 = −4abm 2 0 , (5.13) where m 0 = vu/Λ. 2 Since it is now known at more than 6σ C.L. that ∆m 2 21 > 0 [19,20], we have the condition that −2ab > b 2 . We have considered the parameters a and b as real. Since b 2 is a positive definite quantity the above relation implies that −2ab > 0, which can happen if and only if sgn(a) ≠ sgn(b). Inserting this condition into the expression for ∆m 2 31 gives us ∆m2 31 > 0 necessarily in this model. Therefore inverted neutrino mass hierarchy is impossible to get in the effective A 4 model originally proposed by Altarelli and Feruglio [14]. The conclusion is also valid for complex parameter a and b with a relative phase φ. However, going to the seesaw-realization given in [14], normal as well as inverted hierarchy is possible to realize. The sum of the absolute neutrino masses, effective mass in neutrinoless double beta decay and prediction for tritium beta decay are given respectively as m t = |m 1 |+|m 2 |+|m 3 |, 〈m ee 〉 = m 0 (a+2b/3), m 2 β = m 2 0 ( a 2 + 4ab ) 3 + 2b2 .(5.14) 3 In Fig. 5.1 we show the contours for the observables ∆m 2 21, ∆m 2 31 and m t in the a − b plane, for three different fixed values of m 0 . The details of the figure and description of the different lines can be found in the caption of the figure. 126
In Fig. 5.2 we present a scatter plot showing the points in the a−b parameter space which are compatible with the 3σ [20] allowed range of the mass squared differences. We have allowed m 0 to vary freely and taken a projection of all allowed points in the a−b plane. Whileaisrelatedto theVEVofthesinglet ξ, b isgiven interms oftheVEVs ofthe triplet φ S . The TBM form for the mixing matrix comes solely from the vacuum alignment ofφ S andξ isnotneeded for that. The singlet ξ isnecessary onlyforproducing thecorrect values of ∆m 2 21 and ∆m 2 31. However, it is evident from Fig. 5.2 that for a given value of a needed to obtain the right mass squared differences, the value of b is almost fixed. The relation between a and b can be obtained by looking at the ratio ∆m2 21 ∆m 2 31 = −b2 −2ab −4ab ≃ 0.03, where 0.03 on the right-hand-side (RHS) is the current experimental value. This gives us the relation b ≃ −1.88a. Therefore, in addition to the alignment 〈φ S 〉 = (v S ,v S ,v S ) to get TBM, one also needs a particular relation between the product of Yukawa couplings and VEVs of φ S and ξ in order to reproduce the correct phenomenology. Even if one includes the 3σ uncertainties on ∆m 2 21 and ∆m2 31 , |b| is fine tuned to |a| within a factor of about 10 −2 . 5.3.3 Two One-Dimensional A 4 Higgs If we take two Higgs fields at a time, belonging to two different one dimensional A 4 representations and allow for nonzero VEVs for them, then the m ν obtained for the three possible cases are shown in the first three rows of Table 5.3. One can again see that of the three possible combinations, only the ξ ′ , ξ ′′ combination gives a viable TBM matrix. The other two mass matrices exhibit e−τ (ξ, ξ ′′ )and e−µ (ξ, ξ ′ ) symmetry respectively and are ruled out. Note that we have chosen a = c for the ξ, ξ ′′ combination, a = d for the ξ, ξ ′ combination and c = d for the ξ ′ , ξ ′′ combination for the results given in Table 5.3. This is a reasonable assumption to make since the phenomenology of the three cases does not change drastically unless the VEVs of the singlet Higgs vary by a huge amount. In particular, by changing the relative magnitude of the VEVs, we do not expect the structure of the mixing matrix for the first two rows of Table 5.3 to change so much so that they could be allowed by the current data. In the limit that c = d, it is not hard to appreciate that the resultant matrix with ξ ′ and ξ ′′ would exhibit µ−τ symmetry, though the ξ ′ and ξ ′′ terms alone have e−τ and e−µ symmetry respectively. Since the ξ ′ , ξ ′′ combination is the only one which gives exact TBM mixing in the approximation that c = d, we perform a detailed analysis only for this case. Putting c = d is again contrived and would also lead to a certain fixed relation between them and b, as in the only ξ case. This would mean additional fine tuning of the parameters, unless explained by symmetry arguments. Hence, we allow the two VEVs to differ from each other so that c = d + ǫ. If ǫ is small we can solve the eigenvalue problem keeping only the first order terms in ǫ. The results for this case are shown in the final row of the Table 127
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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
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
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 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 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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