1. Introduction
In the last two decades, neutrino experiments have illustrated that neutrinos oscillate and are massive. Nevertheless, according to the standard parametrization, the unitary lepton mixing matrix, which connects the neutrino mass eigenstates to flavor eigenstates is given by [1–3]
$\begin{eqnarray}\begin{array}{l}{U}_{\mathrm{PMNS}}=\,\left(\begin{array}{ccc}{c}_{12}{c}_{13} & {s}_{12}{c}_{13} & {s}_{13}{{\rm{e}}}^{-{\rm{i}}\delta }\\ -{s}_{12}{c}_{23}-{c}_{12}{s}_{23}{s}_{13}{{\rm{e}}}^{{\rm{i}}\delta } & {c}_{12}{c}_{23}-{s}_{12}{s}_{23}{s}_{13}{{\rm{e}}}^{{\rm{i}}\delta } & {s}_{23}{c}_{13}\\ {s}_{12}{s}_{23}-{c}_{12}{c}_{23}{s}_{13}{{\rm{e}}}^{{\rm{i}}\delta } & -{c}_{12}{s}_{23}-{s}_{12}{c}_{23}{s}_{13}{{\rm{e}}}^{{\rm{i}}\delta } & {c}_{23}{c}_{13}\end{array}\right)\\ \,\times \,\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & {{\rm{e}}}^{{\rm{i}}\rho } & 0\\ 0 & 0 & {{\rm{e}}}^{{\rm{i}}\sigma }\end{array}\right),\end{array}\end{eqnarray}$
where ${c}_{{ij}}\equiv \cos {\theta }_{{ij}}\mathrm{and}{s}_{{ij}}\equiv \sin {\theta }_{{ij}}$ (for i, j = (1, 2), (1, 3) and (2, 3)); δ is called the Dirac phase, analogous to the CKM phase and ρ, σ are called the Majorana phases, which are relevant for the Majorana neutrinos.Finally, consequences of the neutrino experiments, such as T2K [4, 5], RENO [6], DOUBLE-CHOOZ [7], and DAYA-BAY [8, 9] have indicated that there are a nonzero mixing angle θ13 which is small compared to the other two mixing angles and a possible nonzero Dirac CP-violation phase δCP. Therefore, the Tribimaximal (TBM) mixing matrix is rejected [10, 11]. The TBM mixing matrix is [12]
$\begin{eqnarray}{U}_{\mathrm{TBM}}=\left(\begin{array}{ccc}\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} & 0\\ \frac{-1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}}\\ \frac{-1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & \frac{-1}{\sqrt{2}}\end{array}\right),\end{eqnarray}$
where, regardless of the model, the mixing angles are: θ12 ≈ 35.26◦, θ13 ≈ 0, and θ23 ≈ 45° [13].Before this observation, models leading to the TBM mixing matrix, widely studied [14, 15]. Therefore, to produce θ13 ≠ 0 starting from an initial TBM mixing matrix, different approaches have been adopted [16]. One of the successful phenomenological neutrino mass models with flavor symmetry, which is an appropriate framework for understanding the family structure of charged-lepton and neutrino mass matrices [17, 18], is illustrated by group A4 [19–27]. The A4 is a symmetry group of the tetrahedron, was initially presented to illustrate the TBM mixing matrix [21]. Although the primary initial objective of the A4 models was illustrating the TBM mixing matrix [21], many efforts, e.g. [19, 20, 22–33], have been made to set up a model capable of describing the non-TBM mixing matrix phenomenology.
The present 3σ global fits for the existing and known neutrino oscillation parameters [34]
$\begin{eqnarray}\begin{array}{rcl}\delta {m}^{2}[{10}^{-5}{{\rm{eV}}}^{2}] & = & (6.94-8.14),\\ | {\rm{\Delta }}{m}^{2}| [{10}^{-3}{{\rm{eV}}}^{2}] & = & (2.47-2.63)-(2.37-2.53),\\ {\sin }^{2}{\theta }_{12} & = & (0.271-0.369),\\ {\sin }^{2}{\theta }_{23} & = & (0.434-0.610)-(0.433-0.608),\\ {\sin }^{2}{\theta }_{13} & = & (0.02000-0.02405)\\ & & -(0.02018-0.02424),\\ \delta & = & (128^\circ -359^\circ )-(200^\circ -353^\circ ),\end{array}\end{eqnarray}$
multiple sets of allowed ranges are stated, and the left and the right columns correspond to normal hierarchy and inverted hierarchy, respectively. $\delta {m}^{2}\equiv {m}_{2}^{2}-{m}_{1}^{2}$ and ${\rm{\Delta }}{m}^{2}\,\equiv {m}_{3}^{2}-{m}_{1}^{2}$.Despite the prevailing information about neutrino oscillation parameters equation (1.3 ), the mass and mixing problem in the lepton sector is still conceived as a fundamental problem.
In the current work, we mainly focused on the neutrinos based on the particular case of two-zero textures with A4 symmetry which we called it ${M}_{\nu }^{{S}_{7}}$ [35]. In [35], we have studied all seven possible two-zero textures with A4 symmetry, among which only two textures, the texture with (e, e) and (e, μ) vanishing element of mass matrix and its permutation symmetry, are consistent with the experimental data in the non-perturbation method.
In this paper, we intend to consider ${M}_{\nu }^{{S}_{7}}$ in perturbation method to generate (I) non-zero θ13, (II) CP violation phases δ along with two Majorana phases (ρ and σ) and (III) the deviation of θ23 from 45°. However, the discovery of the θ13, whose smallness (in comparison to other mixing angles) signifies modifying the neutrino mixing matrix using a small perturbation about the basic TBM mixing matrix. By employing different methods in a wide range of contexts, a lot of attempts have been made to generate some of the neutrino parameters in perturbation theory [36].
In the basis where the charged-lepton mass matrix is diagonal, a particular application of A4 is given by [18]
$\begin{eqnarray}{{ \mathcal M }}_{\nu }=\left(\begin{array}{ccc}a+\displaystyle \frac{2d}{3} & b-\displaystyle \frac{d}{3} & c-\displaystyle \frac{d}{3}\\ b-\displaystyle \frac{d}{3} & c+\displaystyle \frac{2d}{3} & a-\displaystyle \frac{d}{3}\\ c-\displaystyle \frac{d}{3} & a-\displaystyle \frac{d}{3} & b+\displaystyle \frac{2d}{3}\end{array}\right),\end{eqnarray}$
which also has magic symmetry.1(1 Magic symmetry is a symmetry in which the sum of elements in either any rows or any columns of the neutrino mass matrix are identical [37].)Various phenomenological textures, specifically texture zeros [38–47], have been investigated in both flavor and non-flavor bases. Such texture zeros not only cause a reduction in the number of free parameters of the neutrino mass matrix, but also contributes to establishing several simple and interesting relations between the parameters of mixing matrix. Therefore, in the current research, this allowed us to explore the effects of a specific case of two-zero texture on ${{ \mathcal M }}_{\nu }$ given by equation (1.4 ).
Moreover, assuming the Majorana nature of neutrinos, the present study strove to investigate the phenomenological implications of a specific case of two-zero texture of neutrino mass matrix along with A4 symmetry, based on a global fit of neutrino experimental data [34]. This special case of two-zero texture is ${M}_{\nu }^{{S}_{7}}$ with (μ, μ) = (τ, τ) = 0, which can expose the impressive phenomenological features of a defined Majorana neutrino mass matrix.
Our paper is hence organized as follows: in section 2 , the methodology is elaborated in two subsections. In subsection A, we reconstruct ${M}_{\nu }^{{S}_{7}}$ in the flavor basis as an unperturbed neutrino mass matrix, and also obtain the unperturbed neutrino mass matrix in the mass basis. In subsection 2.2 , we will propose the perturbed neutrino mass matrix as a complex symmetric non-Hermitian matrix in the mass basis. The first order of neutrino mass corrections and the third mass eigenstate are obtained in the mass basis to the first order corrections. Therefore, R&ngr; and the neutrino masses can be expressed. The first two flavor eigenstates are obtained by using the degenerate perturbation method and ∣&ngr;3⟩ in the flavor basis is rewritten. We find the mixing matrix corresponding to our model and thereby ${\sin }^{2}{\theta }_{13}$, CP violation phases (δ, ρ, and σ) and ${\tan }^{2}{\theta }_{23}$ are obtained. In section 3 , we compare the results of our work with those of the experimental data in two different cases. In each case, the complex elements of perturbation, α and β, are illustrated onto the allowed region of the parameter space and the allowed regions were found. In case I, where our parameter space and α are negative, the allowed regions of α and β are acceptable; therefore, the predicted regions of $\langle {m}_{{\nu }_{\beta \beta }}\rangle $ and θ23 are obtained which are consistent with the current experimental data and demonstrate the accuracy of our work. In case II, where our parameter space and α are positive, the obtained region of β is not acceptable; therefore the results of case II are ruled out. In section 4 , the conclusions are provided.
2. Methodology
2.1. The unperturbed neutrino mass matrix
Assuming the Majorana nature of neutrinos, the mass matrix ${{ \mathcal M }}_{\nu }$ is a complex symmetric matrix in equation (1.4 ). We examine in [35] the analysis of two-zero texture for the Majorana neutrino mass matrix based on A4 symmetry, ${{ \mathcal M }}_{\nu }$ in equation (1.4 ), that restricts the number of probably viable cases to seven. It is found that only the texture with (e, e) and (e, μ) vanishing elements of ${{ \mathcal M }}_{\nu }$ and its permutation symmetry, are consistent with the experimental data in the non-perturbation method. The texture with (e, τ) = (μ, μ) = 0, the texture with (e, μ) = (μ, μ) = 0 and their permutation symmetry, are not consistent with the experimental data at all in [35]. The seventh probably viable case of the two-zero texture of ${{ \mathcal M }}_{\nu }$ in with (μ, μ) = (τ, τ) = 0 is extremely interesting. We call it ${M}_{\nu }^{{S}_{7}}$ in [35] and is given by
$\begin{eqnarray}{M}_{\nu }^{{S}_{7}}=\left(\begin{array}{ccc}a+\displaystyle \frac{2}{3}d & -d & -d\\ -d & 0 & a-\displaystyle \frac{d}{3}\\ -d & a-\displaystyle \frac{d}{3} & 0\end{array}\right).\end{eqnarray}$
${M}_{\nu }^{{S}_{7}}$ in equation (2.1 ) is a magic matrix, which has μ − τ symmetry [48]. Consequently, it can lead to a TBM mixing matrix in equation (1.2 ) with θ13 = 0. Therefore, initially seems that ${M}_{\nu }^{{S}_{7}}$ in equation (2.1 ) is not allowed texture based on experimental data, but because θ13 is small; we believe that the perturbative treatment is a more accurate method to investigate the texture ${M}_{\nu }^{{S}_{7}}$.
A straightforward diagonalization procedure yields ${U}_{\mathrm{TBM}}^{T}{M}_{\nu }^{{S}_{7}}{U}_{\mathrm{TBM}}={M}_{\mathrm{diag}}^{{S}_{7}}$, where1.1 ), which ρ and σ are Majorana phases where neutrino oscillations are independent of them.
$\begin{eqnarray}\begin{array}{rcl}{m}_{1} & = & a+\displaystyle \frac{5}{3}d,\\ {m}_{2} & = & a-\displaystyle \frac{4}{3}d,\\ {m}_{3} & = & -a+\displaystyle \frac{1}{3}d,\end{array}\end{eqnarray}$
the mass eigenvalues can be complex; they can be presented positive and real by phase transformation, as (1, eiρ, eiσ) in equation (We reconstruct ${M}_{\nu }^{{S}_{7}}$ in equation (2.1 ) by using ${M}_{\nu }={U}_{\mathrm{TBM}}{M}_{\mathrm{diag}}{U}_{\mathrm{TBM}}^{T}$, where M&ngr; is a magic neutrino mass matrix with μ − τ symmetry. In this reconstruction, we define new parameters, as
$\begin{eqnarray}\begin{array}{rcl}m & \equiv & \displaystyle \frac{\sum {m}_{i}}{3}=\displaystyle \frac{({m}_{1}+{m}_{2}+{m}_{3})}{3},\\ {{\rm{\Delta }}}_{32} & \equiv & ({m}_{3}-{m}_{2}),\\ {{\rm{\Delta }}}_{31} & \equiv & ({m}_{3}-{m}_{1}).\end{array}\end{eqnarray}$
Also, because of the reported experimental results, which have shown δm2 is tiny and greater than zero [34], we approximate Δ31 ≃ Δ32 ≡ Δ. Therefore, the unperturbed mass matrix, the reconstructed ${M}_{\nu }^{{S}_{7}}$, in the flavor basis2(2 We work on a basis where the charged lepton mass matrix is diagonal, and thereby, the lepton mixing matrix is extracted from the neutrino mass matrix.) is2.4 ) has magic and μ − τ symmetries and can be diagonalized by UTBM3(3 Magic and μ − τ symmetries for the mass matrix become synonymous with the tribimaximal mixing matrix [37].) in equation (1.2 ). The mass spectrum of ${M}_{f\nu }^{(0)}$ is2.6 ) are as follows:1.2 ) are the unperturbed flavor eigenstates.
$\begin{eqnarray}{M}_{f\nu }^{0}\simeq \left(\begin{array}{ccc}m-\displaystyle \frac{{\rm{\Delta }}}{3} & 0 & 0\\ 0 & m+\displaystyle \frac{{\rm{\Delta }}}{6} & -\displaystyle \frac{{\rm{\Delta }}}{2}\\ 0 & -\displaystyle \frac{{\rm{\Delta }}}{2} & m+\displaystyle \frac{{\rm{\Delta }}}{6}\end{array}\right).\end{eqnarray}$
Still at this level, the unperturbed neutrino mass matrix ${M}_{f\nu }^{0}$ in equation ( $\begin{eqnarray}{m}_{1}^{(0)}={m}_{2}^{(0)}=m-\displaystyle \frac{{\rm{\Delta }}}{3},\qquad \mathrm{and}\qquad {m}_{3}^{(0)}=m+\displaystyle \frac{2{\rm{\Delta }}}{3}.\end{eqnarray}$
Here ${m}_{1}^{(0)}$, ${m}_{2}^{(0)}$ and ${m}_{3}^{(0)}$ are real and positive and ${m}_{1}^{(0)}$, and ${m}_{2}^{(0)}$ are the same. Hence, the unperturbed mass matrix in the mass basis is $\begin{eqnarray}{M}_{\mathrm{mass}}^{0}\simeq \left(\begin{array}{ccc}m-\frac{{\rm{\Delta }}}{3} & 0 & 0\\ 0 & m-\frac{{\rm{\Delta }}}{3} & 0\\ 0 & 0 & m+\frac{2{\rm{\Delta }}}{3}\end{array}\right).\end{eqnarray}$
In the mass basis the eigenstates of the unperturbed neutrino mass matrix M0mass in equation ( $\begin{eqnarray}| {\nu }_{1}^{(0)}\rangle =\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\,\,\,\,| {\nu }_{2}^{(0)}\rangle =\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),\,\,\,\,\,| {\nu }_{3}^{(0)}\rangle =\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right),\end{eqnarray}$
in which the first two mass eigenstates above are degenerate. It is noteworthy that the columns of UTBM in equation (Note that, up to now, the shortcomings are: (i) the absence of solar mass splitting, (ii) the neutrino masses ordering is unknown and (iii) the neutrino mixing matrix which is still UTBM. Thus, the main objective is generating the splitting of ${m}_{1}^{(0)}$, and ${m}_{2}^{(0)}$ by means of a mass perturbation, from which θ13 ≠ 0 and CP violation are also derived. Moreover, CP violation conditions are necessarily mandated in that μ − τ symmetry should be broken. An interesting question is: after breaking the μ − τ symmetry, will θ23 = 45° remain valid or not?
2.2. The perturbed neutrino mass matrix and perturbation
In this work, we propose the minimal symmetric perturbation neutrino mass matrix4(4 It could be the minimal form of perturbation mass matrix for magic unperturbed neutrino mass matrix with μ − τ symmetry where the first two mass eigenvalues are degenerate as we work the general form of it in [49].) in the mass basis as
$\begin{eqnarray}{M}_{\mathrm{mass}}^{\mbox{'}}\simeq {\rm{\Delta }}\left(\begin{array}{ccc}0 & 0 & \beta \\ 0 & \alpha & 0\\ \beta & 0 & 0\end{array}\right).\end{eqnarray}$
The dimensionless perturbation elements i.e. α, β can be real or complex and should be small compared to the elements of unperturbed neutrino mass matrix M0mass in equation (2.6 ) for a valid perturbation theory.
If ${M}_{\mathrm{mass}}^{\mbox{'}}$ is a complex symmetric non-Hermitian matrix given in equation (2.8 ), therefore we have to consider ${\left({M}_{\mathrm{mass}}^{0}+{M}_{\mathrm{mass}}^{^{\prime} }\right)}^{\dagger }({M}_{\mathrm{mass}}^{0}+{M}_{\mathrm{mass}}^{{\prime} })$ where we drop a term, which is ${ \mathcal O }({\alpha }^{2},\,{\beta }^{2})$. ${{M}_{\mathrm{mass}}^{0}}^{\dagger }{M}_{\mathrm{mass}}^{0}$ is the unperturbed Hermitian term and its eigenstates are the same as those of M0mass in equation (2.7 ) and its eigenvalues are ${\left({m}_{1}^{(0)}\right)}^{2}$, ${\left({m}_{2}^{(0)}\right)}^{2}$ and ${\left({m}_{3}^{(0)}\right)}^{2}$, therefore the perturbation term is ${M}_{\mathrm{mass}}^{p}\,={M}_{\mathrm{mass}}^{{0}^{\dagger }}{M}_{\mathrm{mass}}^{{\prime} }+{M}_{\mathrm{mass}}^{{{\prime} }^{\dagger }}{M}_{\mathrm{mass}}^{0}$ and the perturbation matrix is2.5 ) and first-order corrections to the neutrino masses, we have2.5 ), ${m}_{2}^{(1)}\,=2{\rm{\Delta }}{\rm{Re}}(\alpha )({m}_{2}^{(0)})$ must be positive.
$\begin{eqnarray}{M}_{\mathrm{mass}}^{p}\simeq {\rm{\Delta }}\,\left(\begin{array}{ccc}0 & 0 & {\rm{Re}}(\beta )(2m+\frac{{\rm{\Delta }}}{3})+{\rm{Im}}(\beta )(-{\rm{\Delta }})\\ 0 & {\rm{Re}}(\alpha )(2m-\frac{2{\rm{\Delta }}}{3}) & 0\\ {\rm{Re}}(\beta )(2m+\frac{{\rm{\Delta }}}{3})+{\rm{Im}}(\beta )({\rm{\Delta }}) & 0 & 0\end{array}\right).\end{eqnarray}$
The first-order corrections to the neutrino masses are obtained from ${m}_{i}^{(1)}{\delta }_{{ij}}=\langle {\nu }_{i}^{(0)}| {M}_{\mathrm{mass}}^{{0}^{\dagger }}{M}_{\mathrm{mass}}^{{\prime} }+{M}_{\mathrm{mass}}^{{{\prime} }^{\dagger }}{M}_{\mathrm{mass}}^{0}| {\nu }_{j}^{(0)}\rangle $. Therefore, by using equation ( $\begin{eqnarray}\begin{array}{rcl}{m}_{1}^{2} & = & {\left({m}_{1}^{(0)}\right)}^{2},\\ {m}_{2}^{2} & = & {\left({m}_{2}^{(0)}\right)}^{2}+{\rm{Re}}(\alpha )\left(2m-\displaystyle \frac{2}{3}{\rm{\Delta }}\right){\rm{\Delta }},\\ {m}_{3}^{2} & = & {\left({m}_{3}^{(0)}\right)}^{2}.\end{array}\end{eqnarray}$
The mass corrections arise from the first-order corrections with ${m}_{2}^{(1)}\ne 0$, and ${m}_{1}^{(1)}={m}_{3}^{(1)}=0$. Therefore, the splitting of m1, and m2 is equal to ${m}_{2}^{(1)}$. So far, Neutrino experimental data have definitely confirmed that $\delta {m}^{2}={m}_{2}^{2}-{m}_{1}^{2}\gt 0$. Therefore, due to the equation (From mass-squared in equation (2.10 ), we could obtain the ratio of two neutrino mass-squared differences ${R}_{\nu }=\tfrac{\delta {m}^{2}}{{\rm{\Delta }}{m}^{2}}$, as
$\begin{eqnarray}{R}_{\nu }={\rm{Re}}(\alpha )\displaystyle \frac{6m-2{\rm{\Delta }}}{6m+{\rm{\Delta }}},\end{eqnarray}$
where $\delta {m}^{2}\equiv {m}_{2}^{2}-{m}_{1}^{2}$ and ${\rm{\Delta }}{m}^{2}\equiv {m}_{3}^{2}-{m}_{1}^{2}$.Moreover, employing equation (2.5 ), equation (2.10 ), and equation (2.11 ) as well as the definitions associated δm2 and Δm2, the neutrino masses can be expressed with more convenient relations. More concretely, m1, m2, and m3 are related to $\tfrac{m}{{\rm{\Delta }}}$, and ${\rm{Re}}(\alpha )$, our parameter space, and the experimental parameters δm2 and R&ngr; as2.12 ), our prediction is normal neutrino mass hierarchy.
$\begin{eqnarray}\begin{array}{rcl}{m}_{1} & = & \sqrt{\delta {m}^{2}\,B},\\ {m}_{2} & = & \sqrt{\delta {m}^{2}(B+1)},\\ {m}_{3} & = & \sqrt{\delta {m}^{2}\left(\displaystyle \frac{1}{{R}_{\nu }}+B\right)},\end{array}\end{eqnarray}$
where $B\equiv \tfrac{\tfrac{m}{{\rm{\Delta }}}-\tfrac{1}{3}}{2{\rm{Re}}(\alpha )}$. Therefore, according to equation (It is worth noting that the lowest order of the first two flavor eigenstates in degenerate perturbation theory are
$\begin{eqnarray}| {\nu }_{1}{\rangle }_{\mathrm{flavor}}=\left(\begin{array}{c}\sqrt{\frac{2}{3}}\\ -\frac{1}{\sqrt{6}}\\ -\frac{1}{\sqrt{6}}\end{array}\right)+\left(\begin{array}{c}0\\ \frac{-\left(\frac{3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )\right)}{\sqrt{2}}\\ \frac{\left(\frac{3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )\right)}{\sqrt{2}}\end{array}\right)\end{eqnarray}$
and5(5 $({M}_{\mathrm{mass}}^{0}+{M}_{\mathrm{mass}}^{\mbox{'}})$ has magic symmetry in the flavor basis.) $\begin{eqnarray}| {\nu }_{2}{\rangle }_{\mathrm{flavor}}=\left(\begin{array}{c}\frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}\end{array}\right).\end{eqnarray}$
We reproduce the third mass eigenstate ∣&ngr;3 〉 , in the mass basis, to the first order corrections by
$\begin{eqnarray}| {\nu }_{3}{\rangle }_{\mathrm{mass}}=| {\nu }_{3}^{(0)}\rangle +\frac{\langle {\nu }_{j}^{(0)}| {M}_{\mathrm{mass}}^{p}| {\nu }_{3}^{(0)}\rangle }{{\left({m}_{3}^{(0)}\right)}^{2}-{\left({m}_{j}^{(0)}\right)}^{2}}| {\nu }_{j}^{(0)}\rangle ,\,\,(j\ne 3).\,\end{eqnarray}$
By inserting equation (2.5 ), equation (2.7 ), and equation (2.9 ) into equation (2.15 ), we obtain ∣&ngr;3⟩mass as
$\begin{eqnarray}| {\nu }_{3}{\rangle }_{\mathrm{mass}}=\left(\begin{array}{c}{\rm{Re}}(\beta )-\frac{3{\rm{i}}{\rm{\Delta }}}{6m+{\rm{\Delta }}}{\rm{Im}}(\beta )\\ 0\\ 1\end{array}\right).\end{eqnarray}$
Now, we rewrite ∣&ngr;3>mass in the flavor basis, as follows
$\begin{eqnarray}| {\nu }_{3}{\rangle }_{\mathrm{flavor}}=\left(\begin{array}{c}\sqrt{\frac{2}{3}}\left(\frac{-3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )\right)\\ \frac{1}{\sqrt{2}}-\frac{\frac{-3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )}{\sqrt{6}}\\ -\frac{1}{\sqrt{2}}-\frac{\frac{-3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )}{\sqrt{6}}\end{array}\right),\end{eqnarray}$
Using degenerate perturbation theory, according to equation (2.13 ), equation (2.14 ), and equation (2.17 ), we obtain the neutrino mixing matrix with δ ≠ 0, as
$\begin{eqnarray}U=\left(\begin{array}{ccc}\sqrt{\displaystyle \frac{2}{3}} & \displaystyle \frac{1}{\sqrt{3}} & \sqrt{\displaystyle \frac{2}{3}}\left(\displaystyle \frac{-3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )\right)\\ -\displaystyle \frac{1}{\sqrt{6}}-\displaystyle \frac{\left(\tfrac{3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )\right)}{\sqrt{2}} & \displaystyle \frac{1}{\sqrt{3}} & \displaystyle \frac{1}{\sqrt{2}}-\displaystyle \frac{\tfrac{-3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )}{\sqrt{6}}\\ -\displaystyle \frac{1}{\sqrt{6}}+\displaystyle \frac{\left(\tfrac{3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )\right)}{\sqrt{2}} & \displaystyle \frac{1}{\sqrt{3}} & -\displaystyle \frac{1}{\sqrt{2}}-\displaystyle \frac{\tfrac{-3{\rm{i}}{\rm{\Delta }}{\rm{Im}}(\beta )}{6m+{\rm{\Delta }}}+{\rm{Re}}(\beta )}{\sqrt{6}}\end{array}\right),\end{eqnarray}$
and $\begin{eqnarray}{U}_{\mathrm{tot}}=U\times \left(\begin{array}{ccc}1 & 0 & 0\\ 0 & {{\rm{e}}}^{{\rm{i}}\rho } & 0\\ 0 & 0 & {{\rm{e}}}^{{\rm{i}}\sigma }\end{array}\right).\end{eqnarray}$
We calculate the mixing angles from the mixing matrix U in equation (2.18 ), as follows
$\begin{eqnarray}\begin{array}{rcl}{\sin }^{2}{\theta }_{13} & = & | {U}_{13}{| }^{2}=\displaystyle \frac{1}{A}\left(\displaystyle \frac{2}{3}{\left({\rm{Re}}(\beta )\right)}^{2}+\displaystyle \frac{6{{\rm{\Delta }}}^{2}}{{\left(6m+{\rm{\Delta }}\right)}^{2}}{\left({\rm{Im}}(\beta )\right)}^{2}\right),\\ {\sin }^{2}{\theta }_{12} & = & \displaystyle \frac{| {U}_{12}{| }^{2}}{1-| {U}_{13}{| }^{2}}=\displaystyle \frac{A}{2+A},\\ {\tan }^{2}{\theta }_{23} & = & \displaystyle \frac{| {U}_{23}{| }^{2}}{| {U}_{33}{| }^{2}}=1-\displaystyle \frac{4\sqrt{3}{\rm{Re}}(\beta )}{2+A+2\sqrt{3}{\rm{Re}}(\beta )}.\end{array}\end{eqnarray}$
where $A=1+{\left({\rm{Re}}(\beta )\right)}^{2}+\tfrac{9{{\rm{\Delta }}}^{2}}{{\left(6m+{\rm{\Delta }}\right)}^{2}}{\left({\rm{Im}}(\beta )\right)}^{2}$.We add the perturbation matrix, ${M}_{\mathrm{mass}}^{\mbox{'}}$ in equation (2.8 ), in our texture with complex components in the mass basis which leads to the solar neutrino mass splitting, θ13 ≠ 0, the deviation of $\tan {\theta }_{23}$ from 1, nonzero the CP-violating phases (the Dirac phase δ, and two Majorana phases ρ and σ). In the following, a straightforward calculation yields these phases.
As mentioned in the appendix , we could obtain $\tan \delta $ by using the elements of the mixing matrix U in equation (2.18 ) and the mixing angles in equation (2.20 ), where we have dropped the terms of ${ \mathcal O }{\left({\rm{Re}}(\beta )\right)}^{2}$ and ${ \mathcal O }{\left({\rm{Im}}(\beta )\right)}^{2}$, as follows
$\begin{eqnarray}\tan \delta =\displaystyle \frac{(-3{\rm{\Delta }}){\rm{Im}}(\beta )}{(6m+{\rm{\Delta }}){\rm{Re}}(\beta )}.\end{eqnarray}$
In the basis where the charged lepton mass matrix is diagonal, by employing ${({M}_{\nu }^{0}+{M}_{\nu }^{\mbox{'}})}_{\mathrm{flavor}}={U}_{{tot}}{({M}_{\nu })}_{\mathrm{diag}}{{U}_{\mathrm{tot}}}^{T}$, we reorganize ${({M}_{\nu }^{0}+{M}_{\nu }^{\mbox{'}})}_{\mathrm{flavor}}$ as2.18 ), and neutrino masses mi, i = 1, 2, 3 given by equation (2.10 ).
$\begin{eqnarray}{({M}_{\nu }^{0}+{M}_{\nu }^{\mbox{'}})}_{\mathrm{flavor}}=U\left(\begin{array}{ccc}{m}_{1} & 0 & 0\\ 0 & {{\rm{e}}}^{2{\rm{i}}\rho }{m}_{2} & 0\\ 0 & 0 & {{\rm{e}}}^{2{\rm{i}}\sigma }{m}_{3}\end{array}\right){U}^{T},\end{eqnarray}$
where we adopted the mixing matrix U given by equation (We obtain six complex equations by using equation (2.22 ) regarding the elements of neutrino mass matrix, in the flavor basis, at (i, j) positions as follows
$\begin{eqnarray}{m}_{1}{U}_{i1}{U}_{j1}+{{\rm{e}}}^{2{\rm{i}}\rho }{m}_{2}{U}_{i2}{U}_{j2}+{{\rm{e}}}^{2{\rm{i}}\sigma }{m}_{3}{U}_{i3}{U}_{j3}={M}_{{ij}}.\end{eqnarray}$
By inserting the elements of mixing matrix U in equation (2.18 ) into equation (2.23 ) and solving dirty math equations, and neglecting terms of ${ \mathcal O }{\left({\rm{Re}}(\beta )\right)}^{2}$ and ${ \mathcal O }{\left({\rm{Im}}(\beta )\right)}^{2}$, we obtain ρ = −σ and the CP-violating Majorana phases ρ and σ are as follows
$\begin{eqnarray}\sin (2\rho )=-\sin (2\sigma )=\displaystyle \frac{12\,{\rm{\Delta }}\,{\rm{Re}}(\beta )\,{\rm{Im}}(\beta )\,{m}_{3}}{({\rm{\Delta }}+6m){m}_{2}}.\end{eqnarray}$
Up to this point, having used the perturbation method, we could obtain (I) the splitting of the two first neutrino masses, (II) the perturbed mass eigenstates in the CP violation case, therefore generating (III) θ13 ≠ 0, (IV) the Dirac phase δ, (V) the deviation rate of θ23 from 45◦, (VI) the mixing matrix corresponding to our texture, and (VII) the two Majorana phases ρ and σ. In the next section, by comparing the results of our work with those of the experimental data in equation (1.3 ), we will show that ${M}_{\nu }^{{S}_{7}}$ (2.1 ) could be allowed texture.
3. Comparison with experimental data
In this section, the results of the current study are compared with those of the experimental data in equation (1.3 ). As it was mentioned in the previous section, according to neutrino experimental data, we have $\delta {m}^{2}=2\,{\rm{\Delta }}\,{\rm{Re}}(\alpha ){m}_{2}^{(0)}\gt 0$. Therefore, since ${m}_{2}^{(0)}$ in equation (2.5 ) is real and positive, we have two cases; case (I) Δ and ${\rm{Re}}(\alpha )$ to be negative, case (II) Δ and ${\rm{Re}}(\alpha )$ to be positive.
Initially, we consider case I, when Δ < 0 and ${\rm{Re}}(\alpha )\lt 0$. The investigation of case I includes three steps; in the first step, we obtain the allowed range of ${\rm{Re}}(\alpha )$. We do this by mapping two of the constraints obtained from the experimental data onto our parameter space, $\tfrac{m}{{\rm{\Delta }}}$, as shown in figure 1. The two restricting sets of experimental data on neutrino mass come from the values of ${R}_{\nu }=\tfrac{\delta {m}^{2}}{{\rm{\Delta }}{m}^{2}}$ and ∑m&ngr;6(6 ∑m&ngr; < 0.12 eV is the significant experimental result reported by Planck's measurements of the cosmic microwave background (CMB) [50].). The overlap of the experimental data of R&ngr; and ∑m&ngr; and our model, equation (2.11 ) and equation (2.12 ), is restricted to a tiny region close to the bottom-right corner of the parameter space in figure 1, where we have magnified the overlap region in the zoomed box.
Figure 1. In this figure, the whole region of the ${\rm{Re}}(\alpha )-\tfrac{m}{{\rm{\Delta }}}$ plane which is allowed by our model is shown. The yellow (dark) area below the horizontal axis displays the allowed region of ${\rm{Re}}(\alpha )$ in case I, according to the experimental data of R&ngr;. In the zoomed box, we have magnified the blue (dark) tiny overlap region of the experimental values for R&ngr; and ∑m&ngr;, which is consistent with all of the experimental data. |
Therefore, as is shown in the zoomed box in figure 1, we could specify the allowed ranges of our parameter space, $\tfrac{m}{{\rm{\Delta }}}$, and ${\rm{Re}}(\alpha )$ as follows,
$\begin{eqnarray}\displaystyle \frac{m}{{\rm{\Delta }}}\approx \,(-0.015-0).\end{eqnarray}$
$\begin{eqnarray}{\rm{Re}}(\alpha )\approx \,-(0.01350-0.01320)\to \,-(0.01550-0.01405).\end{eqnarray}$
Having determined the unique overlap region in the above, we can predict the masses of the neutrinos in equation (2.12 ) as follows;
$\begin{eqnarray}\begin{array}{rcl}{m}_{1} & = & \{(0.02992-0.03241),(0.02991-0.03239)\}\,\mathrm{eV}\\ & \to & \,\{(0.02732-0.02959),(0.02869-0.03107)\}\,\mathrm{eV},\\ {m}_{2} & = & \{(0.03106-0.0336),(0.03104-0.03362)\}\,\mathrm{eV}\\ & \to & \{(0.02856-0.03093),(0.02987-0.03235)\}\,\mathrm{eV},\\ {m}_{3} & = & \{(0.05801-0.06066),(0.05800-0.05936)\}\,\mathrm{eV}\\ & \to & \{(0.05671-0.05920),(0.05738-0.05996)\}\,\mathrm{eV}.\end{array}\end{eqnarray}$
In the second step, we obtain the allowed range of $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}$, as is shown in the zoomed box in figure 2. We do this by inserting the experimental data of $\tan \delta $ into the equation of $\tan \delta $ in equation (2.21 ) and mapping $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}$ according to the allowed range of our parameter space in equation (3.1 ).
Figure 2. In this figure, the whole region of the $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}-\tfrac{m}{{\rm{\Delta }}}$ plane, which displays the region of $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}-\tfrac{m}{{\rm{\Delta }}}$ in case I, according to the experimental data of R&ngr; and $\tan \delta $. In the zoomed box, we have magnified the light blue (dark) tiny dark allowed region of the $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}$ according to the region of our parameter space, $\tfrac{m}{{\rm{\Delta }}}$ in equation ( |
Therefore, we obtain the allowed region of $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}$ onto our parameter space which is consistent with the experimental data as follows
$\begin{eqnarray}\displaystyle \frac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}\approx ((-1.21)-0.35)\to ((-1.31)-0.4),\end{eqnarray}$
accordingly, can write $\begin{eqnarray}{\rm{Im}}(\beta )\approx ((-1.21)-0.35){\rm{Re}}(\beta )\to ((-1.31)-0.4){\rm{Re}}(\beta ),\end{eqnarray}$
therefore in the second step, we obtain the ratio of $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}$.In the third step, we obtain the allowed range of ${\rm{Re}}(\beta )$, as is shown in the zoomed box in figure 3, and subsequently the allowed range of ${\rm{Im}}(\beta )$ due to equation (3.5 ). We do this by inserting the experimental data of ${\sin }^{2}{\theta }_{13}$ into the first equation of equation (2.20 ) and mapping ${\rm{Re}}(\beta )$ onto the allowed range of our parameter space in equation (3.1 ).
Figure 3. In this figure, the whole region of the ${\rm{Re}}(\beta )-\tfrac{m}{{\rm{\Delta }}}$ plane, which displays the region of ${\rm{Re}}(\beta )-\tfrac{m}{{\rm{\Delta }}}$ in case I, according to the experimental data of R&ngr; and ${\sin }^{2}{\theta }_{13}$. In the zoomed box, we have magnified the light blue (dark) tiny allowed region of the ${\rm{Re}}(\beta )$, below the line 0.099, according to the region of our parameter space, $\tfrac{m}{{\rm{\Delta }}}$ in equation ( |
According to perturbation theory, the perturbation elements i.e. α, and β in equation (2.8 ) should be small compared to the elements of unperturbed neutrino mass matrix m, and Δ in equation (2.6 ). Accordingly, we only accept those values of ${\rm{Re}}(\beta )$ that are less than 0.1.7(7 We choose ${\rm{Re}}(\beta )\lt 0.1$ because of the important experimental result for the sum of the three light neutrino masses that have been reported by the Planck measurements of the cosmic microwave background, which is ∑m&ngr; < 0.12 eV [50].) Therefore, the tiny region below the line 0.099 in the zoomed box in figure 3 displays the allowed region of ${\rm{Re}}(\beta )$ according to the allowed range of our parameter space, as
$\begin{eqnarray}{\rm{Re}}(\beta )\approx \,(0.05057-0.099)\to (0.046-0.099).\end{eqnarray}$
In the following, based on equation (3.5 ), and equation (3.6 ) the allowed region of ${\rm{Im}}(\beta )$ is obtained, as follows
$\begin{eqnarray}{\rm{Im}}(\beta )\approx \,((-0.099)-(0.0177))\to ((-0.099)-(0.0184)),\end{eqnarray}$
Having taken these three steps, we obtained the allowed region of our parameter space, $\tfrac{m}{{\rm{\Delta }}}$ in equation (3.1 ), and dimensionless perturbation elements ${\rm{Re}}(\alpha )$, ${\rm{Re}}(\beta )$, and ${\rm{Im}}(\beta )$, respectively in equation (3.2 ), equation (3.6 ), and equation (3.7 ).
Let us now proceed with our discussions by obtaining the range of predicted values of Majorana phases in case I for the texture ${M}_{\nu }^{{S}_{7}}$.
By taking $\tfrac{m}{{\rm{\Delta }}}$, ${\rm{Re}}(\beta )$, and ${\rm{Im}}(\beta )$ from equation (3.1 ), equation (3.6 ), and equation (3.7 ) respectively, our herein model yields the following values for the Majorana phases ρ and σ:
$\begin{eqnarray}\rho =-\sigma \approx \,((-3.50^\circ )-(1.24^\circ ))\to ((-3.11^\circ )-(1.19^\circ )),\end{eqnarray}$
which may be tested by future experiments.In the following, we outline further predictions of our model in case I, which can be a test of the accuracy of our predictions.
| • | The Majorana neutrinos can violate lepton number for example in the neutrinoless double beta decay procedure (ββ0&ngr;) [51]. Such a process has not been detected yet, but an upper bound has been set for the relevant quantity, i.e. $\langle {m}_{{\nu }_{\beta \beta }}\rangle $. Results from the first phase of the KamLAND-Zen experiment sets the following constraint $\langle {m}_{{\nu }_{\beta \beta }}\rangle \lt (0.061-0.165)\,\mathrm{eV}$ at 90 present CL [52]. The prediction of our model for $\langle {m}_{{\nu }_{\beta \beta }}\rangle $ is: $\begin{eqnarray}\langle {m}_{{\nu }_{\beta \beta }}\rangle \approx \,(0.02642-0.03264)\to (0.02332-0.03146)\,\mathrm{eV},\end{eqnarray}$ which consistent with the result of kamLAND-Zen experiment. |
| • | The value of ${\tan }^{2}{\theta }_{23}$ can be obtained by inserting equation ( $\begin{eqnarray}{\theta }_{23}\approx (41.80^\circ -43.47^\circ ),\end{eqnarray}$ which indicates the accuracy of predictions of the case I in our work. |
Figure 4. In this figure, the whole region of the ${\tan }^{2}{\theta }_{23}-\tfrac{m}{{\rm{\Delta }}}$ plane, which displays the region of ${\tan }^{2}{\theta }_{23}-\tfrac{m}{{\rm{\Delta }}}$ in case I, according to the experimental data of R&ngr; and ${\tan }^{2}{\theta }_{23}$. In the zoomed box, we have magnified the light blue (dark) tiny allowed region of the ${\tan }^{2}{\theta }_{23}$, according to the region of our parameter space $\tfrac{m}{{\rm{\Delta }}}$ in equation ( |
Now we consider our work according to case II when Δ and ${\rm{Re}}(\alpha )$ are both positive. In this case, similar to case I, we first obtain the allowed range of ${\rm{Re}}(\alpha )$ as is shown in the zoomed box in figure 5. We do this by mapping R&ngr; and ∑m&ngr;, two of the constraints obtained from the experimental data onto our parameter space8(8 At first glance, it seems that the allowed region of the parameter space in case II is $\tfrac{m}{{\rm{\Delta }}}\geqslant \tfrac{1}{3}$ which is obtained based on δ m2 > 0.) of case II, according to the experimental data in equation (1.3 ). The overlap of the experimental data of R&ngr; and ∑m&ngr; and our model, equation (2.11 ) and equation (2.12 ), is restricted to a tiny region inside the right triangle, which is shown in the zoomed box in figure 5.
Figure 5. In this figure, the whole region of the ${\rm{Re}}(\alpha )-\tfrac{m}{{\rm{\Delta }}}$ plane which is allowed by our model in the case II is shown. The area between the two blue curves displays the allowed region of ${\rm{Re}}(\alpha )$ in case II, according to the experimental data of R&ngr;. In the zoomed box, we have magnified a tiny overlap region of the experimental values for R&ngr; and ∑m&ngr; inside right triangle. |
Therefore in case II, as is shown in the zoomed box in figure 5, we could specify the allowed ranges of our parameter space, $\tfrac{m}{{\rm{\Delta }}}$, and ${\rm{Re}}(\alpha )$ as follows,
$\begin{eqnarray}\displaystyle \frac{m}{{\rm{\Delta }}}\approx \,(0.56-1.3).\end{eqnarray}$
$\begin{eqnarray}{\rm{Re}}(\alpha )\approx \,(0.042-0.09).\end{eqnarray}$
In the next step, we obtain the allowed range of $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}$ in case II as is shown in the zoomed box in figure 6. The same is case I, we do this by inserting the experimental data of $\tan \delta $ into the equation of $\tan \delta $ in equation (2.21 ) and mapping $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}$ according to the allowed range of our parameter space in equation (3.11 ).
Figure 6. In this figure, the whole region of the $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}-\tfrac{m}{{\rm{\Delta }}}$ plane, which displays the region of $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}-\tfrac{m}{{\rm{\Delta }}}$ in case II, according to the experimental data of R&ngr; and $\tan \delta $. In the zoomed box, we have magnified the light pink (dark) tiny allowed region of the $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}$ according to the region of our parameter space, $\tfrac{m}{{\rm{\Delta }}}$ in equation ( |
We obtain that the allowed region of $\tfrac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}$ onto our parameter space, in case II, as3.13 ) we could write2.20 ) and mapping ${\rm{Re}}(\beta )$ onto the allowed range of our parameter space of case II.
$\begin{eqnarray}\displaystyle \frac{{\rm{Im}}(\beta )}{{\rm{Re}}(\beta )}\approx \,(-(12)-3.4),\end{eqnarray}$
therefore, according to equation ( $\begin{eqnarray}{\rm{Im}}(\beta )\approx \,(-(12)-3.4){\rm{Re}}(\beta ).\end{eqnarray}$
we obtain the allowed range of ${\rm{Re}}(\beta )$, as is shown in the zoomed box in figure 7. As in case I, We do this by inserting the experimental data of ${\sin }^{2}{\theta }_{13}$ into the first equation of equation (
Figure 7. In this figure, the whole region of the ${\rm{Re}}(\beta )-\tfrac{m}{{\rm{\Delta }}}$ plane, which displays the region of ${\rm{Re}}(\beta )-\tfrac{m}{{\rm{\Delta }}}$ in case II, according to the experimental data of R&ngr; and ${\sin }^{2}{\theta }_{13}$. In the zoomed box, we have magnified a tiny allowed region of the ${\rm{Re}}(\beta )$, below the line 0.099, according to the region of our parameter space, $\tfrac{m}{{\rm{\Delta }}}$ in equation ( |
As mentioned in the previous case, the allowed range of ${\rm{Re}}(\beta )$ must be below 0.099; therefore, the area between two curves below the horizontal dashed line in the zoomed box in figure 7 displays the allowed range of ${\rm{Re}}(\beta )$ in case II as follows,
$\begin{eqnarray}{\rm{Re}}(\beta )\approx \,(0.07-0.099).\end{eqnarray}$
Afterwards, we map ${\rm{Im}}(\beta )$ based on equation (3.14 ) and equation (3.14 ), as is shown in the two figure 8 and, figure 9. We find that in case II the obtained value of ${\rm{Im}}(\beta )$ is much greater than ∣0.1∣, the red dashed line, which is not acceptable by the perturbation theory. Therefore, the obtained results in case II, with Δ > 0 and ${\rm{Re}}(\alpha )\gt 0$, are ruled out.
Figure 8. In this figure, the whole region of the ${\rm{Im}}(\beta )-\tfrac{m}{{\rm{\Delta }}}$ plane, which displays the region of ${\rm{Im}}(\beta )-\tfrac{m}{{\rm{\Delta }}}$ in case II, according to the value of ${\rm{Re}}(\beta )$ in equation ( |
Figure 9. In this figure, the whole region of the ${\rm{Im}}(\beta )-\tfrac{m}{{\rm{\Delta }}}$ plane, which displays the region of ${\rm{Im}}(\beta )-\tfrac{m}{{\rm{\Delta }}}$ in case II, according to the value of ${\rm{Re}}(\beta )$ in equation ( |
4. Conclusion
In assessing neutrino physics from a phenomenological point of view, the texture of the neutrino mass matrix is of particular relevance. The choice of symmetries for the mass matrix can lead to specific states in the mixing matrix, which may convey results consistent with the corresponding experimental data. Such consequences are significant because we can make additional predictions regarding neutrinos and their flavor symmetries.
One salient feature of investigating the neutrino mass matrix phenomena is that it could provide new indications for understanding the flavor symmetry and symmetry breaking simultaneously. Therefore, these symmetries and maybe the breaking of them could impact the neutrino mixing matrix, which has significant differences in the magnitude of (mixing) angles in contrast to the quark sector. Furthermore, studying the texture of the neutrino mass matrix could predict the answer to the following essential questions: (a) what are the masses of neutrinos? (b) what is the ordering of the three neutrino masses? (c) neutrinos are Dirac or Majorana Particles? (d) what are the values of three CP-violating phases of the PMNS mixing matrix? (e) how close to $\tfrac{\pi }{4}$ is θ23? (f) Do sterile neutrinos exist? certainly, these questions will be solved by future experiments.
In this work, by proposing a particular two-zero texture based on A4 symmetry along with magic and μ − τ symmetry, we have obtained predictions for all the previous questions that are consistent with the current experimental data.
We have studied the phenomenology of two-zero texture in the Majorana neutrino mass matrix along with A4 symmetry where the charged lepton mass matrix is diagonal. Therefore, there are seven possible two-zero textures. The seven viable textures are broadly categorized into two categories. To sum up, in general, textures which are consistent with the experimental data as ${M}_{\nu }^{{S}_{1}}$, ${M}_{\nu }^{{S}_{2}}$, ${M}_{\nu }^{{S}_{7}}$ and textures which are not consistent with the experimental data as ${M}_{\nu }^{{S}_{3}}$, ${M}_{\nu }^{{S}_{5}}$, and their permutation symmetry, as ${M}_{\nu }^{{S}_{4}}$, ${M}_{\nu }^{{S}_{6}}$ respectively.9(9 We have studied the texture ${M}_{\nu }^{{S}_{1}}$ and its permutation symmetry as texture ${M}_{\nu }^{{S}_{2}}$ in the non-perturbation method and find results that are exactly consistent with the experimental data. We also find that textures ${M}_{\nu }^{{S}_{3}}$, ${M}_{\nu }^{{S}_{5}}$ and their permutation symmetry are ruled out [35].)
In our work, We studied the texture ${M}_{\nu }^{{S}_{7}}$, with (μ, μ) and (τ, τ) vanishing elements, which has both magic and μ − τ symmetry. Therefore its corresponding mixing matrix is UTBM, with θ13 ≠ 0, δ = 0, and θ23 = 45°. Hence, we considered ${M}_{\nu }^{{S}_{7}}$ with the attendance of a small contribution as a perturbation matrix by employing the perturbation method10(10 As we know, the mixing angle θ13 is small compared to the other two angles, θ12 and, θ23, also the magnitude of the ratio of two neutrino mass-squared differences is ${R}_{\nu }=\tfrac{\delta {m}^{2}}{{\rm{\Delta }}{m}^{2}}\simeq {10}^{-2}$. Therefore, we proposed to modify the neutrino mixing matrix using a small perturbation about the basic tribimaximal structure that is one of the most appropriate ways to solve the neutrino problem.). By employing the texture ${M}_{\nu }^{{S}_{7}}$ we obtained the tribimaximal structure which led us to produce a mass matrix constrained by the elements of the tribimaximal mixing matrix. The mass matrix thus obtained (unperturbed mass matrix) loses the solar neutrino mass splitting whilst it has magic and μ − τ symmetry in the flavor basis. Our investigation proceeded in the diagonal mass matrix of the unperturbed mass matrix in the mass basis, which has degeneracy ${m}_{1}^{(0)}\,=\,{m}_{2}^{(0)}$. Therefore, the perturbation mass matrix is simultaneously responsible for the solar neutrino mass splitting and CP violation in the lepton sector, which generates a tiny δm2 and simultaneously small complex parameters in the neutrino mixing elements, such as U13, (θ13 and δ), and provides minor amendments to θ23. We proposed that the perturbation mass matrix be symmetric and complex in the mass basis, which is a minimal form of perturbation mass matrix for magic unperturbed neutrino mass matrix with μ − τ symmetry, and thereby is a non-Hermitian matrix. We obtained the neutrino mixing matrix corresponding to our texture by degenerate perturbation theory and therefore θ13, δ, ρ, σ, and δm2 that all could arise from a perturbation. The perturbation matrix also affects the atmospheric mixing angle θ23.
To get valuable predictions concerning neutrino masses, the Majorana phases, deviation of θ23 from 45°, and the effective neutrino mass for the neutrinoless double beta decay. we compared the results of our phenomenological model with the experimental data. In this regard, we have two different cases; in case I, Δ and ${\rm{Re}}(\alpha )$ are negative while in case II, both are positive.
Finally, we compare our predictions to the current experimental data in each case; in case I, We found that there is a good agreement. we obtained the allowed range of our parameter space, $\tfrac{m}{{\rm{\Delta }}}$ and the complex elements of perturbation mass matrix α, and β. In addition, the predictions for the texture ${M}_{\nu }^{{S}_{7}}$ agree with the observational data of the CMB and the neutrinoless double beta decay experiments. Furthermore, we found that our prediction of θ23 ≈ (41.80° − 43.47°) is quite satisfactory. We expect that our model results for neutrino masses, their hierarchy (normal), CP-violation parameters δ, ρ, and σ are in good agreement with future experiments.
In case II, we fail to obtain an acceptable region for β according to the rules of the perturbation theory. Therefore Δ and ${\rm{Re}}(\alpha )$ could not be positive, and case II is ruled out.
Acknowledgments
We would like to thank the research office of the Qazvin Branch, Islamic Azad University.
Appendix
As we know, in general, the value of δ can be extracted as follows
$\begin{eqnarray}\delta =\arg \left(\displaystyle \frac{{U}_{e2}{U}_{\mu 3}{{U}^{* }}_{e3}{{U}^{* }}_{\mu 2}-{{s}^{2}}_{12}{{c}^{2}}_{13}{{s}^{2}}_{13}{{s}^{2}}_{23}}{{c}_{12}{s}_{12}{{c}^{2}}_{13}{s}_{13}{c}_{23}{s}_{23}}\right).\end{eqnarray}$
We could obtain $\tan \delta $ by inserting the elements of the mixing matrix U in equation (2.18 ) and the equations in equation (2.20 ), into equation (5.1 ) as follows
$\begin{eqnarray}\tan \delta =\displaystyle \frac{3\left(\tfrac{{\rm{Im}}(\beta )\,{\rm{\Delta }}}{{\rm{\Delta }}+6m}+\tfrac{6{\rm{Im}}(\beta )\,{\rm{\Delta }}\,({\rm{\Delta }}+6m)}{(-3+9{\left({\rm{Im}}(\beta )\right)}^{2}+{\left({\rm{Re}}(\beta )\right)}^{2}){{\rm{\Delta }}}^{2}+12(-3+{\left({\rm{Re}}(\beta )\right)}^{2})\,m\,({\rm{\Delta }}+3m)}\right)}{{\rm{Re}}(\beta )}.\end{eqnarray}$
Because, ${\rm{Re}}(\beta )$ and ${\rm{Im}}(\beta )$ as perturbation elements must be too small compare to the elements of unperturbed neutrino mass matrix m and Δ, therefore we have dropped the terms ${ \mathcal O }{\left({\rm{Re}}(\beta )\right)}^{2}$ and ${ \mathcal O }{\left({\rm{Im}}(\beta )\right)}^{2}$ in equation (5.2 ), then after this approximation we obtain $\tan \delta $ as follow
$\begin{eqnarray}\tan \delta =\displaystyle \frac{-3{\rm{Im}}(\beta ){\rm{\Delta }}}{{\rm{Re}}(\beta )(6m+{\rm{\Delta }})}.\end{eqnarray}$