Doctor of Philosophy

This contributes in the neutrino mass matrices to deviate it from its tribimaximal configuration, ensuring the generation of non-zero θ13. Out of all the discrete groups studied in the literature, A4 (discrete group of equal permutations of four objects) turned out to be a special group which can reproduce this TBM pattern in a very economical way [7].

The Standard Model of Particle Physics

The Higgs scalar potential V(Φ) is responsible for the spontaneous symmetry breaking (SSM) SU(2)L×U(1)Y −→U(1)em. Under the condition λ > 0 and µ2 < 0, the neutral component of the scalar field (φ0) gets the expected vacuum value, which is not zero (vev),v/√.

Table 1.1: Fermionic field content in Standard Model, Here the subscripts L, R stands for the left-handed and right-handed particles.

Neutrinos in the Standard Model

Since both components of the lepton doublet in the first term of Eq. 1.10) transforms in the same way, all these unit rotation matrices (U`, V`) are essentially not reflected in the annual interactions written in Eq. Therefore, in the case of massive neutrinos, following the same procedure as discussed in Section 1.1, the interaction of the charged flux in the lepton sector can be written as

Evidences for Neutrino Mass

Here |mee| is known as the effective mass parameter, U is the PMNS matrix, and mi is the neutrino mass. Observing 0νββ will show that (a) neutrinos are Majorana particles and lepton number conservation is not a symmetry of nature, (b) measuring the effective mass parameter will help us understand the neutrino mass hierarchy.

Figure 1.1: Feynman diagram for neutrinoless double beta decay [20].

Theory of Massive neutrinos

Type-I Seesaw Mechanism
Type-II Seesaw Mechanism
Type-III Seesaw Mechanism
Inverse Seesaw Mechanism

Therefore, in the presence of Dirac and Majorana fermions, the Lagrangian responsible for the neutrino mass can be written as When the neutral component of the scalar takes vev (h∆0i=u∆), the neutrino mass matrix obtained through the type II saw can be written as.

Figure 1.2: Feynman diagrams for Type-I, Type-II and Type-III seesaw.

Origin of Lepton Mixing and Neutrino Oscillation

Neutrino Oscillation Parameters

Further, it is possible to determine the neutrino mass hierarchy by including the matter effect in the neutrino oscillation, which also depends on the magnitude of θ13. In Table 1.2, we have summarized the neutrino oscillation parameters from the recent global analysis by Forero et al.

Figure 1.3: Schematic diagram form neutrino mass hierarchies. Left-panel represents normal hierarchy whereas right panel represents inverted hierarchy [68].

Patterns of Lepton Mixing

Such a distinctive and unique structure of the TBM mixing matrix may indicate a special symmetry in the Lagrangian that is broken at high energy. Therefore, the TBM structure of the neutrino mixing matrix received great attention from the model building point of view.

Role of Flavor Symmetry in Explaining Lepton Mixing

The A 4 Group

After breaking A4 symmetry in the direction hφTi = (vT,0,0), the Lagrangian can be written as. Finally after breaking the electroweak symmetry we obtain the charged lepton mass matrix as.

Neutrinos and its possible connection with matter antimatter asymmetry 32

Therefore, in a nutshell, with complex CP phases in the Yukawa coupling, the decay of RH neutrinos out of equilibrium can cause lepton asymmetry. Second, when the single scalar triplet decays and heavy RH neutrinos enter the loop (M∆ > MR).

Figure 1.4: Feynman diagrams for decay of RH neutrinos into lepton and Higgs (a) tree level, (b) one-loop vertex level and (c) one-loop self energy level respectively

Structure of The Model

Then, in Section 2.3 we discuss the eigenvalues and phases involved in the RH neutrino sector. Finally, it sets the charged lepton coupling matrix as the diagonal in the forward order.

Table 2.1: Fields content and transformation properties under the symmetries imposed on the model

Neutrino Masses and Mixing

RH Neutrinos

The new scalar singlet ξ0 contributes to the RH neutrino heavy sector through the term xNξ0(NcNc) and the Majorana neutrino mass matrix takes the form of MRd.

Light Neutrino Masses and Mixing Angles

Horizontal blue shaded region stands for 3σ allowed range for sin2θ13 and the red shaded region inside represents 1σ range for sin2θ13. We have studied the variation of sin2θ13 against the parameter λ1 in Fig.2.1, where the 1σ and 3σ allowed regions for sin2θ13 obtained from [5] are also indicated in the same by red and blue horizontal shaded regions respectively for both NH and IH. These findings are listed in Table 2.2 and they are well within the 3σ allowed regions of sin2θ12 and sin2θ23 [5].

Figure 2.1: sin 2 θ 13 vs λ 1 (i.e. | d/a | ) plot. Horizontal blue shaded region stands for 3σ allowed range for sin 2 θ 13 and the red shaded region inside represents 1σ range for sin 2 θ 13

Constraints on parameters from neutrino oscillation data

Note that to do this, one must take into account the correct sign of cosφba in eqn (5.23) while considering the NH and IH cases. 1Eq.(2.29) describes a quadratic equation of |cosφba| after other parameters are adjusted. as observed from the shaded area of Fig.2.3, right panel. Two Majorana phases α21 and α31 can be investigated in the configuration in a similar way as done in [104].

Figure 2.3: Light neutrino masses m i ’s and their sum, P m i , as a function of λ 2 for NH (λ 1 = 0.37) and IH (λ 1 = 0.38)

Leptogenesis

Leptogenesis with fixed λ 1 and varying λ 2

In Fig.2.7 (left panel) we have plotted total baryon asymmetry YB (red continuous line) together with individual YB1,2,3 (orange large dashed, respectively green dots and blue dot dashed lines) modλ2 for Re(xC)= Re( xD) = 0.2 in the case of NH. As shown in Fig.2.7 (left panel), contribution from YB3 is suppressed (and with opposite sign). To check the possible values of Re(xC) and/or Re(xD) we have drawn a contour plot in Fig.2.8 (left panel) between Re(xD) and λ2, while Re(xC)=Re(xD) is taken as an example.

Figure 2.7: Baryon asymmetry of the Universe as function of λ 2 for NH (with λ 1 = 0.37, left panel) and IH (with λ 1 = 0.38, right panel)

Leptogenesis with fixed λ 2 and varying λ 1

This is because the term in the right-hand side of the resonance condition turns out to be in order 5×10−2κy[Re(xC) cosθ+ Re(xD) sinθ].

Chapter Summary

Therefore, the type-II saw mechanism consists of the conventional type-I saw contribution (mIN) together with the triple contribution (mIIν ) to the neutrino mass matrix. We have also studied the production of lepton asymmetry through the decay of the involved heavy triplet. In this case, the RH neutrinos are heavier compared to the triple mass involved.

The Model

Type-I Seesaw and Tribimaximal Mixing

We expect that the minimization of the potential involving φS and φT can produce this by properly tuning the parameters involved in the potential. Now we introduce two parameters α =b/a and k =v2y2/a which are real and positive since they are only part of the type-I contribution. This will be useful when we consider the decay of the RH neutrinos for leptogenesis in Section 3.4.

Triplet Contribution and Type-II seesaw

This would be useful when we will consider the decay of the RH neutrinos for leptogenesis in Section 3.4. 70 and leptogenesis Yukawa matrix for the triplet ∆ as given by,. Before discussing weaves for the ∆ field, let us describe the complete scalar potential V, including the triplet ∆ obeying the imposed symmetries, given by, . 3.12). Using Eqs.(3.10) and (3.13), the triplet contribution to the light neutrino mass matrix follows from the Lagrangian LII as.

Constraining parameters from neutrino mixing

Results for Case A

We also note that the part of the sin2θ13 contour for α <1 prefers the region with a relatively small value of β (<1). In the right panel of figure (3.4), we have plotted sin2θ13 contours corresponding to the upper and lower values (detonated with red dotted lines) allowed by the 3σ sin2θ13 range. When constructing the graph, only a narrow range of α corresponding to the 3σ variation of sin2θ13 is considered, as obtained in Figure (3.4), right panel (ie from P1 to P2).

Figure 3.3: Contour plots for both sin 2 θ 13 = 0.0234 (shown in red-dashed and red- red-continuous lines) and r = 0.03 (shown in blue-dashed and blue-continuous lines) in the α − β plane for various choices of δ with α < 1

Results for Case B

82 and leptogenesis where the solutions exist for all allowed values of δ and find that JCP in our model is expected to be 0.03<|JCP|<0.04. The right panel is for the contour plot of r with the best fit value r= 0.03 (shown in blue solid line) and 3σ range of sin2θ13 (shown by two red dashed lines). lower to the higher value, within the 3σ limit. With α > 1 we can now check that in obtaining the physical neutrino masses the contribution of type I dominates, just as in the case of α < 1. 5.25) ) and therefore the contribution of type II is almost an order of magnitude smaller compared to the contribution of type I.

Figure 3.9: Left panel contains contour plots for best-fit values of r (indicated by blue- blue-continuous lines) and sin 2 θ 13 (indicated by red-dashed line) for δ = 40 ◦ in α-β plane with α > 1

Leptogenesis

Therefore, there are two remaining options for creating successful lepton asymmetry [115, 200] in the present context; (I) from triplet decay, where the single-loop diagram includes virtual RH neutrinos, and (II) from RH neutrino decay, where the single-loop contribution includes a virtual triplet flowing in a loop. Provided that the triplet mass is small compared to all RH neutrinos (i.e. M∆< MRi), we consider option (I). The total decay width of the triplet ∆ (for ∆ → two leptons and ∆ → two scalar doublets) is given by.

Figure 3.12: Contours corresponding to different values of ∆ in the M ∆ − Λ plane withα < 1

Chapter Summary

We also analyze the effects of inhomogeneities on the lepton mixing matrix and their consequences in terms of decays that perturb the lepton aroma, etc. The diagonalizing matrix of the above form mν is representative of the underlying TBM mixing, where the charged lepton mass matrix is the diagonal. In section 4.2 below, we describe the construction of the model based on the framework symmetries.

The Model

96 nonunity of that LFV decay is vanishingly small due to exact cancellation of involved elements resulting from the specific flavor structure we considered. The detailed phenomenology that constrains the parameters of the model from the available data from neutrino experiments takes place in Sections 4.3 and 4.4. Now we choose the vev of φT as hφTi = vT so that the charged lepton mass matrix turns out to be diagonal in leading order and can be written as Ml =vvΛTdiag (ye, yµ, yτ).

Neutrino masses and Mixings

The real and positive mass eigenvalues (mi) can then be extracted using the following expressions. 4.25) The three phases associated with these mass eigenvalues are γi =φi+φ0, where φ0 is the common phase for ay12/y22 and φi is given by.

Constraining parameters from neutrino data

Case A: [φ ba = φ da = 0]
Case B: [φ ba = 0]
Case C: [φ da = 0]
Case D: [φ ba = φ da = φ a ]
General Case

Then, if we plot the contours of r and sinθ13 in the α, β plane, where simultaneously meeting the best-fit values of sinθ13 and r provides solutions for α and β with this specific choice of δ. Here in Table 4.3 we have given sets of values for (α, β, k) for different δ satisfying sinθ13= 0.153 and r= 0.03 obtained from neutrino oscillation experiments. Following the same rule as in case B, contours can be drawn for the best fit of sinθ13 and r values in the α, β plane.

Figure 4.1: [Left panel] Plot for sin θ 13 vs β . Here 3σ range for sin θ 13 fixes β in the range 0.328-0.413

Non-unitary effect

Case A: [φ ba = φ da = 0]
Case B: [φ ba = 0]
Case C: [φ da = 0]
Case D: [φ ba = φ da = φ]
General Case

The part of each contour line k that does not satisfy equation (4.52) is shown by the dotted segment. Using the non-unitarity constraints through Eq. in case A and case B, here we also show the forbidden part of the vf-Λ correlation. As discussed previously, here we can also plot the dependence of vf −Λ using Eq. 4.53) and evaluate the allowed regions for vf and Λ using Eq.

Table 4.6: Cutoff scale Λ for different C 1 (with φ da = φ ba = 0) when λ = 0.01, 0.1 and 1.

Lepton flavor violation

To find Uh, we use μM−1 as obtained in Eq. and ψ1,2 appearing in V1,2 as previously discussed. Then with the help of Mathematica4, we are able to find the diagonalizing matrix W (and hence also the matrix K) and evaluate the LFV decompositions. 3A common phase φ0 as described in the above discussion Eq. 4.26) in Section 4.3, is irrelevant to neutrino phenomenology and we therefore set it to zero.

Neutrinoless double beta decay and contribution of heavy neutrinos 123

In this present work, we propose a minimal extension of the SM by incorporating the U (1) flavor symmetry to establish a correlation between the relic dark matter abundance and the non-zero value of sinθ13. The total relic density of dark matter (say ψ1) with mass M1 can be approximately written as [244]. In Section 5.5 we derive the correlation between non-zero sinθ13 and Higgs portal dark matter coupling and conclude in Section 5.6.

Figure 5.1: Nonzero values of sin θ 13 predict Higgs portal couplings of DM via a U (1) flavor symmetry: a schematic presentation.

Structure of the model

Neutrino Sector

So the contribution to the effective neutrino mass matrix coming through the higher dimensional operator with respect to the considered symmetries can be written as. Note that this is the leading order contribution (and is proportional to 1/Λ) in the charged lepton mass matrix. The U(1) flavor symmetry considered here does not allow expressions involving φ, η(such as: `H`H(φ+η)/Λ2) as discussed (where φ and η are charged under U(1) , but the SM particles are not). 5.4) is the only relevant term up to 1/Λ2 order that contributes to the neutrino mass matrix (mν)0 ensuring its TBM structure as in Eq.

Dark sector and its interaction with neutrino sector

This typical flavor structure of the additional contribution to the neutrino mass matrix derives from the inclusion of the field η, which transforms as 10 under A. Here the field φ plays the role of the messenger field similar to that considered in [267]. Note that the field vev φ is also instrumental in producing the term in the neutrino mass matrix along with the vev ofη.

Phenomenology of the neutrino sector

Case A : φ ab = φ db = 0
Case B : φ db = 0
Case C : φ ab = 0
Case D : φ ab = φ db = β

All these findings are mentioned in Table 5.3 including the sum of the absolute masses (Σmi) of all three light neutrinos and effective neutrino mass parameter involved in neutrinoless double beta decay (|mee|) for different considerations of leptonic CP phase δ. Note that here we have plotted the sum of the three absolute light neutrino masses in accordance with the recent observation made by PLANCK, i.e. If we impose this restriction on the sum of absolute masses of the three light neutrinos, then the allowed range of y/k becomes further restricted.

Figure 5.2: Plot of sin θ 13 against . 3σ range [5] of sin θ 13 (indicated by the horizontal lines) fixes in the range: 0.328-0.4125 (indicated by vertical lines).

Phenomenology of DM Sector

The relic dark matter density (ψ1) is mainly dictated by annihilations through (i) ψ1ψ1 → W+W−, ZZ through SU(2)L gauge coupling, and (ii) ψ1ψ1 → hh through the Yukawa coupling introduced in Eq. Therefore the larger ∆M the smaller the co-annihilation and the larger the relic density. In the lower right panel, we also show the allowed relic density points through the blue dots.

Figure 5.9: Dominant Annihilation processes to Higgs and Gauge boson final states.

Correlation between Dark and Neutrino Sectors

The scarlet and blue dots simultaneously satisfy the relic density and direct search limits of Lux 2016. Combining the relic density limit and the direct search limits, we find the allowable area marked by the blue dots in the lower right panel of the figure. Therefore, the points in magenta satisfy the sinθd upper bound and represent the allowed range given the relic density and direct search limits.

Figure 5.20: Left Panel: Y vs M 1 scatter plot for correct relic density (Eq. 5.1). Here sin θ d (10 −6 -0.2) and ∆M (1-1100 GeV) varies simultaneously

Conclusions

An estimate of the effective mass parameter appearing in the neutrinoless double beta decay is also given. We also make predictions for the sum of the absolute mass of the light neutrinos, effective mass parameter appearing in the neutrinoless double beta decay and the JCP in this setup. In this thesis, we have some interesting results about δ in the study of spontaneous CP violation in a general type II seesaw.