الفهرس 
Standard model of elementary particle physicsOnly 14 pages are availabe for public view
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Abstract
Neutrinos are elementary spin 1
particles. Although they are one of the most abundant parti-
cles in our world, they are incredibly difficult to detect. They are neutral and colorless particles, thus they do not feel electromagnetic and strong interactions. They interact only via gravita- tional and weak interactions, as they have mass and isospin charge. There exist three types of neutrinos, which are called flavors: electron neutrino νe, muon neutrino νµ and tau neutrino ντ . Neutrino flavors are created associated with the charged leptons e, µ, and τ in the weak interactions. The experiments show that only the left-handed component of the neutrino con- tributes to the weak interactions. Therefore, the weak interactions maximally violate the parity symmetry.
The neutrino oscillation observations give clear and strong evidence that the neutrinos are massive, and their flavor states oscillate. The interactions of the neutrinos with other elementary particles are described within the frame of the standard model of the elementary particles. The standard model is SU (3)c ×SU (2)L ×U (1)Y gauge theory accompanied by the Higgs mechanism. The group SU (3)c describes the strong interactions between the quarks. However, the group SU (2)L × U (1)Y describes the electromagnetic and weak interactions between the elementary particles. The partial unification between electromagnetic and weak forces is described by SU (2)L × U (1)Y . The masses of the fermions and gauge bosons break explicitly the gauge invariance. Therefore, we introduce the scalar field is called a Higgs field, which gives a rise to the Higgs mechanism. When the Higgs field takes a vacuum expectation value, the spontaneous symmetry breaking SU (2)L × U (1)Y → U (1)em occurs, and the fermions together with gauge bosons acquire masses. As the gauge group of the electromagnetism U (1)em is not broken, the photon field remains massless in the model. The standard model is successful in the prediction of the values of the gauge bosons masses and also the weak neutral current interactions. Because only left-handed fermions feel weak interaction, there are no right-handed neutrinos in the standard model. Therefore, the neutrinos are massless in that model. However, this assumption contradicts the neutrino oscillation observations. The masses of the neutrinos are O(eV ) i.e. the neutrino masses are much smaller than the charged leptons and quarks. To have massive
neutrinos, we extend the standard model with right-handed neutrinos and assume the neutrino is a Dirac type. As to the Dirac neutrino, the neutrino and anti-neutrino are distinct particles, which have different quantum lepton numbers. The fine-tuning problem arises to get tiny massive neutrinos. To explain naturally how the observed neutrinos have tiny masses without fine-tuning the Yukawa couplings, we assume that the neutrinos are Majorana type. As to the Majorana neutrinos, the neutrinos and antineutrinos are the same particles. The Majorana particles open the door for the lepton number violating process, which is called the neutrinoless double beta decay. Because the right-handed neutrinos or left-handed anti neutrinos do not feel the weak interaction, we add a gauge singlet mass term is called a Majorana mass term. The mass of the observable neutrino is suppressed by the mass of the right-handed neutrino. The current experiments are not able to determine the nature of neutrino particles.
We assume that the neutrinos are Majorana type, and also we are working in the flavor basis, where the charged lepton mass matrix is diagonal. Therefore, the mixing comes solely from the neutrino sector. The neutrino mass matrix is parameterized by nine free parameters: three mixing angles (θ12, θ23, θ13), three complex phases (δ, ρ, σ), and three masses (m1, m2, m3). There exist experimental constraints on the three mixing angles, one Dirac phase, and two neutrino mass differences. To reduce the number of free parameters, some phenomenological have been presented, such as zero textures, zero minors, zero subtraces, equality textures, etc. We study analytically and phenomenologically the neutrino mass matrix characterized by single vanishing 2 × 2 subtrace. The vanishing sum condition reduces the number of free parameters in the neutrino mass matrix to seven. We throw N points of order 107 − 1010 in the 7-dim parameter space representing the parameters (θ12, θ13, θ23, δ, ρ, σ, δm2). We check by using Eqs. 3.3 and 3.4 the type of mass hierarchy, and then to see whether the bonds of
∆m2 together with those of Eq. 2.79 are satisfied. We find that all six non-singular possible textures can accommodate the experimental data for both hierarchy types. However, the four singular textures of inverted type are viable. We list in the tables 3.1 and 3.2 the allowed experimental ranges of the neutrino physical parameters (θ12, θ23, θ13, δ, ρ, σ, m1, m2, m3, mee, me and J ) for singular and non-singular cases with either hierarchy types at different confidence levels. The all possible correlation plots with either hierarchy types for each viable texture at the 2-σ level are also introduced. Finally, we reconstruct the neutrino mass matrix at one representative point at the 3-σ level. The point is chosen to be as close as possible to the best fit values for the mixing angles and the Dirac phase δ. Finally, we introduce different symmetry realization methods to enforce a vanishing subtrace condition in the neutrino mass matrix. The realization methods used in our study are called direct and indirect. In the direct method, we use Z2 × Z6, Z2 × Z2, and Z2 × Z4 × U (1)3 symmetries within type-I, type-II, and
type-I+II respectively in order to realize four viable textures. In the indirect method, We use
Z8 × Z2 and Z5 symmetries within type-I and type-II seesaw scenarios. However, Z12 × Z2 is used to realize some viable singular textures. We could not, however, find suitable assignments within either direct or indirect realization schemes to be applicable for C12 and C23 textures