Two-Zero Textures Based on A₄ Symmetry and Unimodular Mixing Matrix

Abstract

We propose a phenomenological model of two-zeros Majorana neutrino mass matrix based on the $A_4$ symmetry, where the structure of mixing matrix is a unimodular second scheme of trimaximal $T{M}_2$, and the charged lepton mass matrix is diagonal. We show that, among seven possible two-zero textures with $A_4$ symmetry, only two textures, namely the texture with $M_{ee}=0$ and $M_{e\mu }=0$ and its permutation, are acceptable in the non-perturbation method, since the results associated with these two textures are consistent with the experimental data. We obtain a unique relation between our phases, namely $\rho +\sigma =\varphi \pm \pi$, and an effective equation ${sin}^2{\theta }_{13}=\frac{2}{3}{R}_{\nu }$ where $R_{\nu }=\frac{\delta m^2}{\Delta m^2}$. Then, only by using the experimental ranges of $R_{\nu }$, we obtain the allowable range of the unknown parameter $\varphi$ as the phase of $T{M}_2$ mixing matrix, which leads to obtaining not only the ranges of all neutrino oscillation parameters of the model (which agree well with experimental data) but also with the masses of neutrinos, the Dirac and Majorana phases and the Jarlskog parameter, and to predict the normal neutrino mass hierarchy. Finally, we show that all the predictions regarding our two specific textures agree with the corresponding data reported from neutrino oscillation, cosmic microwave background and neutrinoless double beta decay.

1. Introduction

One of the successful phenomenological neutrino mass models with flavor symmetry, which is an appropriate framework towards understanding the family structure of charged-lepton and of neutrino mass matrices, is based upon the group $A_4$ [1,2,3,4,5,6,7,8,9,10,11,12]. The $A_4$ is a symmetry group of the tetrahedron, whose introduction was primarily motivated so that a tribimaximal (TBM) [13] mixing matrix [6] could be considered to explore the implications of the mentioned charged-lepton and neutrino mass matrices. The TBM mixing matrix is

$$U_{TBM}=\left(\begin{array}{ccc}-\sqrt{\frac{2}{3}}& \frac{1}{\sqrt{3}}& 0\\ \frac{1}{\sqrt{6}}& \frac{1}{\sqrt{3}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{6}}& \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{2}}\end{array}\right),$$

(1)

where, regardless of the model, the mixing angles are ${\theta }_{12}\approx {35.26}^{\circ }$, ${\theta }_{13}\approx 0$, and ${\theta }_{23}\approx {45}^{\circ }$ [13]. In the last decade, significant consequences were extracted from neutrino experiments, such as T2K [14], RENO [15], DOUBLE-CHOOZ [16], and DAYA-BAY [17,18], which have indicated that there are a nonzero mixing angle ${\theta }_{13}$ (at a significance level higher than $8\sigma$) and a possible nonzero Dirac CP-violation phase ${\delta }_{CP}$. Therefore, the TBM mixing matrix as above had to be rejected [19,20]. This consequence is in our opinion of particular interest, being at the core motivation and purpose of our paper, which we elaborate as follows.

According to the standard parametrization, the unitary lepton mixing matrix, which connects the neutrino mass eigenstates to flavor eigenstates, is given by [21,22,23]

$$U_{PMNS}=\left(\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}& s_{13}{e}^{-i\delta }\\ -s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}{e}^{i\delta }& c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}{e}^{i\delta }& s_{23}{c}_{13}\\ s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}{e}^{i\delta }& -c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}{e}^{i\delta }& c_{23}{c}_{13}\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& e^{i\rho }& 0\\ 0& 0& e^{i\sigma }\end{array}\right),$$

(2)

where $c_{ij}\equiv cos{\theta }_{ij}\mathrm{and}{s}_{ij}\equiv sin{\theta }_{ij}$ (for $i,j=(1,2),(1,3)\mathrm{and}(2,3)$); $\delta$ denotes the Dirac phase similar to the CKM phase; $\rho$ and $\sigma$ stand for the Majorana phases that are applicable to the Majorana neutrinos. Furthermore, as reported from experiments, the number of the known available neutrino oscillation parameters approaches five. In Table 1, information concerning neutrino masses and mixing provided is summarized [24].

Table 1. The experimental data associated with the neutrinos oscillation parameters. When multiple sets of allowed ranges are stated, the upper row and the lower row correspond to normal hierarchy and inverted hierarchy, respectively ($\delta m^2\equiv m_2^2-m_1^2$ and $\Delta m^2\equiv m_3^2-m_1^2$).

Parameter	The Experimental Data
	$\mathbf{3}\mathbf{\sigma }$Range	bfp±$\mathbf{1}\mathbf{\sigma }$
$\delta m^2\left[{10}^{-5}e{V}^2\right]$	6.94–8.14	7.30–7.72
$|\Delta m^2|\left[{10}^{-3}e{V}^2\right]$	2.47–2.63	2.52–2.57
	2.37–2.53	2.42–2.47
${sin}^2{\theta }_{12}$	0.271–0.369	0.302–0.334
${sin}^2{\theta }_{23}$	0.434–0.610	0.560–0.588
	0.433–0.608	0.561–0.568
${sin}^2{\theta }_{13}$	0.02000–0.02405	0.02138–0.02269
	0.02018–0.02424	0.02155–0.02289
$\delta$	${128}^{\circ }$–${359}^{\circ }$	${172}^{\circ }$–${218}^{\circ }$
	${200}^{\circ }$–${353}^{\circ }$	${256}^{\circ }$–${310}^{\circ }$

In order to meet these experimental results, several models with a discrete flavor symmetry [25,26,27,28], including an $A_4$ flavor symmetry, have been proposed [1,2,4,5,6,7,8,9,10,11,12,25,29,30,31,32,33,34,35,36,37]. Although the original objective of the $A_4$ models was to substantiate a TBM mixing matrix [6], in view of the disagreeing observational data [14,15,16,17,18], considerable efforts have been made to set up a description conveying instead a non-TBM mixing matrix; see, e.g., [4,5,7,8,9,10,11,12,25,30,31,33,34,35,36,38].

$A_4$ is the smallest non-Abelian group; it is the group of even permutations of four objects with a three one-dimensional, irreducible representation. It has 12 elements and 4 irreducible representations: 1, $1{}^{\prime }$, $1{}^{″}$, and 3, with the multiplication rule

$$3\times 3=1+1{}^{\prime }+1{}^{″}+3+3,$$

(3)

which is the reason why it is one of the most popular groups in neutrino mass models.

The Yukawa interaction of the left-handed lepton doublets fields of the model $SU{\left(3\right)}_C\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y\times A\left(4\right)$ and a Higgs scalar triplet (Neutrinos are massless in the standard model because there is no Higgs scalar triplet and there are no right-handed singlet neutrinos; therefore, one of the easiest ways to expand the standard model is to add a Higgs scalar triplet [39]) is:

$${\mathcal{L}}_{Yukawa}={\mathcal{L}}_{Yukawa}^{SM}-\frac{1}{2}{y}_{ij}{L_i}^c\Delta L_j+h.c.$$

(4)

The second term in (4) would generate a non-zero Majorana neutrino mass matrix if the neutral component of the triplet Higgs ${\Delta }^0$ has a small non-zero vacuum expectation value. Here, the leptons are assumed to transform under $A_4$ as $L_i=({\nu }_i,l_i)\sim \underline{3}(i=1,2,3)$, $l_{1R}\sim \underline{1}$, $l_{2R}\sim \underline{1}{}^{\prime }$, $l_{3R}\sim \underline{1}{}^{″}$ and the Higgs doublets ${\varphi }_i=({{\varphi }_i}^0,{{\varphi }_i}^{-})\sim \underline{3}(i=1,2,3)$.

Now,

$$\begin{array}{lll}\hfill {\mathit{L}}_{Chargedleptons}& =& -[h_1\left(\overline{L_1}{\varphi }_1\right)l_{1R}+h_1\left(\overline{L_2}{\varphi }_2\right)l_{1R}+h_1\left(\overline{L_3}{\varphi }_3\right)l_{1R}+h_2\left(\overline{L_1}{\varphi }_1\right)l_{2R}\hfill \\ & +& \omega \left\{h_2\left(\overline{L_2}{\varphi }_2\right)l_{2R}\right\}+{\omega }^2\left\{h_2\left(\overline{L_3}{\varphi }_3\right)l_{2R}\right\}+h_3\left(\overline{L_1}{\varphi }_1\right)l_{3R}\hfill \\ & +& {\omega }^2\left\{h_3\left(\overline{L_2}{\varphi }_2\right)l_{3R}\right\}+\omega \left\{h_3\left(\overline{L_3}{\varphi }_3\right)l_{3R}\right\}]+h.c.\hfill \end{array}$$

(5)

Spontaneous symmetry breaking leads to ${\upsilon }_i=⟨{{\varphi }_i}^0⟩$. Therefore, the charged lepton mass matrix is given by

$$M_l=\left[\begin{array}{ccc}{h}_1{\upsilon }_1& h_2{\upsilon }_1& h_3{\upsilon }_1\\ h_1{\upsilon }_2& h_2\omega {\upsilon }_2& h_3{\omega }^2{\upsilon }_2\\ h_1{\upsilon }_3& h_2{\omega }^2{\upsilon }_3& h_3\omega {\upsilon }_3\end{array}\right],$$

(6)

where the minimum of the Higgs potential is given by ${\upsilon }_1={\upsilon }_2={\upsilon }_3=\upsilon$ [1]. In that case, the charged lepton mass matrix $M_l$ is diagonalized by the transformation

$$M_{ld}={U_L}^{†}{M}_l{U}_R,.$$

(7)

$M_{ld}$ is the diagonal form of $M_l$, $U_R=I$ and

$$U_L=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right],$$

(8)

where $\omega =exp\left(\frac{2\pi i}{3}\right)=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$.

Therefore, we can obtain $M_{ld}$ as

$$\begin{array}{lll}\hfill M_{ld}& =& \frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right]\left[\begin{array}{ccc}{h}_1\upsilon & h_2\upsilon & h_3\upsilon \\ h_1\upsilon & h_2\omega \upsilon & h_3{\omega }^2\upsilon \\ h_1\upsilon & h_2{\omega }^2\upsilon & h_3\omega \upsilon \end{array}\right]=\left[\begin{array}{ccc}\sqrt{3}{h}_1\upsilon & 0& 0\\ 0& \sqrt{3}{h}_2\upsilon & 0\\ 0& 0& \sqrt{3}{h}_3\upsilon \end{array}\right]\hfill \\ & =& \left[\begin{array}{ccc}{m}_e& 0& 0\\ 0& m_{\mu }& 0\\ 0& 0& m_{\tau }\end{array}\right].\hfill \end{array}$$

(9)

Let ${\Delta }_1\sim \underline{1}$, ${\Delta }_2\sim \underline{1}{}^{\prime }$, ${\Delta }_3\sim \underline{1}{}^{″}$, and ${\Delta }_i\sim \underline{3}(i=4,5,6)$, where ${\Delta }_i=({{\Delta }_i}^{++},{{\Delta }_i}^{+},{{\Delta }_i}^0)$. Now, we have

$$\begin{array}{lll}\hfill {\mathit{L}}_{NeutrinoMajorana}& =& -\frac{1}{2}[({\Delta }_1+{\Delta }_2+{\Delta }_3){\nu }_{1L}^T{C}^{-1}{\nu }_{1L}+({\Delta }_1+\omega {\Delta }_2+{\omega }^2{\Delta }_3){\nu }_{2L}^T{C}^{-1}{\nu }_{2L}\hfill \\ \hfill & +& ({\Delta }_1+{\omega }^2{\Delta }_2+\omega {\Delta }_3){\nu }_{3L}^T{C}^{-1}{\nu }_{3L}+{\Delta }_5{\nu }_{1L}^T{C}^{-1}{\nu }_{2L}\hfill \\ \hfill & +& {\Delta }_5{\nu }_{2L}^T{C}^{-1}{\nu }_{1L}+{\Delta }_4{\nu }_{2L}^T{C}^{-1}{\nu }_{3L}+{\Delta }_4{\nu }_{3L}^T{C}^{-1}{\nu }_{2L}\hfill \\ \hfill & +& {\Delta }_6{\nu }_{3L}^T{C}^{-1}{\nu }_{1L}+{\Delta }_6{\nu }_{1L}^T{C}^{-1}{\nu }_{3L}]\hfill \end{array}$$

(10)

Therefore, the Majourana mass matrix in the original basis is:

$${\mathcal{M}}_{\nu }{}^0=\left(\begin{array}{ccc}a+b+c& 0& 0\\ 0& a+\omega b+{\omega }^{2}c& d\\ 0& d& a+{\omega }^{2}b+\omega c\end{array}\right).$$

(11)

Let us assume we have a natural minimum of Higgs potential for a continuous range of parameter values. Therefore, we can set $⟨{{\Delta }_1}^0⟩=a$, $⟨{{\Delta }_2}^0⟩=b$, $⟨{{\Delta }_3}^0⟩=c$, $⟨{{\Delta }_4}^0⟩=d$, and $⟨{{\Delta }_5}^0⟩=⟨{{\Delta }_6}^0⟩=0$.

Let us continue by choosing a basis where the charged-lepton mass matrix is diagonal; a particular representation for $A_4$ is [3]:

$${\mathcal{M}}_{\nu }={U_L}^{†}{{\mathcal{M}}_{\nu }}^0{U}_L=\left(\begin{array}{ccc}a+\frac{2d}{3}& b-\frac{d}{3}& c-\frac{d}{3}\\ b-\frac{d}{3}& c+\frac{2d}{3}& a-\frac{d}{3}\\ c-\frac{d}{3}& a-\frac{d}{3}& b+\frac{2d}{3}\end{array}\right).$$

(12)

${\mathcal{M}}_{\nu }$ is invariant under the transformation $G_u$, i.e., $G_u^T{\mathcal{M}}_{\nu }{G}_u={\mathcal{M}}_{\nu }$, where $G_u=1-2u{u}^T$. The transformation $G_u$ corresponds to the magic symmetry [40] (Magic symmetry is a symmetry in which the sum of elements in either any row or any column of the neutrino mass matrix is equal [41]). Thus, ${\mathcal{M}}_{\nu }$ also has magic symmetry. Therefore, the mixing matrix corresponding to ${\mathcal{M}}_{\nu }$ (as given by (12)) could be the second scheme of trimaximal mixing (The mixing matrix corresponding to the magic symmetry is called the second scheme of trimaximal mixing) $\left(T{M}_2\right)$ [42], i.e.,

$$U_{T{M}_2}=\left(\begin{array}{ccc}\sqrt{\frac{2}{3}}cos\theta & \frac{1}{\sqrt{3}}& \sqrt{\frac{2}{3}}sin\theta \\ -\frac{cos\theta }{\sqrt{6}}+\frac{e^{-i\varphi }sin\theta }{\sqrt{2}}& \frac{1}{\sqrt{3}}& -\frac{sin\theta }{\sqrt{6}}-\frac{e^{-i\varphi }cos\theta }{\sqrt{2}}\\ -\frac{cos\theta }{\sqrt{6}}-\frac{e^{-i\varphi }sin\theta }{\sqrt{2}}& \frac{1}{\sqrt{3}}& -\frac{sin\theta }{\sqrt{6}}+\frac{e^{-i\varphi }cos\theta }{\sqrt{2}}\end{array}\right),$$

(13)

where $\theta$ and $\varphi$ are two free parameters. The first matrix on the right-hand side of (13) represents $U_{T{M}_2}$, which corresponds to the magic symmetry and, for the particular case where $\theta =0$ and $\varphi =0$, reduces to $U_{TBM}$ given by (1).

In (12), by assuming the Majorana type nature of neutrinos and an $A_4$ based symmetry for ${\mathcal{M}}_{\nu }$, at least nine free real parameters can be obtained: three flavor mixing angles $({\theta }_{13},{\theta }_{12},{\theta }_{23})$, three CP violating phases $(\delta ,\rho ,\sigma )$ and three neutrino masses $(m_1,m_2,m_3)$. Additional predictions are produced when we combine an $A_4$ symmetry with additional constraints applied to the elements of ${\mathcal{M}}_{\nu }$ as given by (12)presence of zeros in ${\mathcal{M}}_{\nu }$. Various phenomenological textures, specifically texture zeros [43,44,45,46,47,48,49,50,51], have been investigated in both flavor and non-flavor basis. Such texture zeros not only cause the number of free parameters of neutrino mass matrix to be reduced, but also assists in establishing important relations between mixing angles. Recently, by employing the zero texture introduced in [52] as well as the texture proposed in [53], several parameters have been extracted as well as computed within a novel phenomenological approach to neutrino physics.

Within the context conveyed through the preceding paragraphs, the purpose of our paper is to investigate effects arisen from using the two-zero textures on ${\mathcal{M}}_{\nu }$ given by (12). Specifically, assuming a Majorana (However, we should mention that establishing the nature of neutrinos is still a controversial subject, which could eventually be decided by experimental observation. In particular, by means of the nonzero magnetic dipole moment of neutrinos ruling out Majorana neutrinos or neutrinoless double beta decay [54] ruling out Dirac neutrinos) nature for neutrinos, where the charged-lepton mass matrix is diagonal, we aim to explore the phenomenological implications of seven two-zero textures of neutrino mass matrix together with $A_4$ symmetry, in a scenario where $|detU|=+1$. This is a valuable procedure that enables obtaining a unique relation between the phases present in the $U_{T{M}_2}$ mixing matrix, therefore allowing for extracting the parameters based on a global fit of the neutrino oscillation data [24]. This is the main contribution of our work. Moreover, let us also point out that it has been believed that a two-zero texture of $A_4$ symmetry can further assist with explaining a Majorana neutrino mass matrix. Therefore, in our paper, we also proceed systematically by $\left(i\right)$ employing two-zero textures of $A_4$ symmetry and $\left(ii\right)$ comparing them with experimental data, so that, consequently, we additionally show that only the predictions for two-zero textures $M_{\nu }^{S_1}$$(M_{ee}=M_{e\mu }=0)$, and $M_{\nu }^{S_2}$$(M_{ee}=M_{e\tau }=0)$ are consistent with the experimental data, whilst the results of others are not.

Our paper is hence organized as follows: In Section 2, we consider a methodology by which we reconstruct the Majorana neutrino mass matrix with $A_4$ symmetry when the charged-lepton mass matrix is diagonal and imposes two-zero textures. Specifically, we study all seven possible two-zero textures of $A_4$ symmetry. In Section 2.1, we will investigate texture $M_{\nu }^{S_1}$ along with a unimodular condition, by which we obtain constraints on Majorana phases. Moreover, we obtain some useful relations for neutrino masses, Majorana phases and mixing angles. Subsequently, we not only compare the consequences of the texture $M_{\nu }^{S_1}$ with the recent experimental data but also present our predictions based on the actual masses and CP-violation parameters. In Section 2.2, we will discuss and explore the texture $M_{\nu }^{S_2}$ as well as the permutation symmetry between it and the $M_{\nu }^{S_1}$. Furthermore, by applying a numerical analysis, we will discuss the predictions of the texture $M_{\nu }^{S_2}$ for neutrino parameters. In Section 2.3 and Section 2.4, the other two-zero textures will be studied. We will show that their corresponding consequences are not in agreement with the experimental data. In Section 3, we present our conclusions.

2. Methodology

By considering the Majorana nature of neutrinos, the mass matrix ${\mathcal{M}}_{\nu }$ in (12) is a complex symmetric matrix. In this respect, we have shown that applying the analysis of two-zero texture for the Majorana neutrino mass matrix based on $A_4$ symmetry, the number of distinct cases of ${\mathcal{M}}_{\nu }$ in (12) will be restricted to seven. In what follows, respecting the distinguishing properties of these seven two-zero textures, we would classify them into three categories. Here, we first introduce them, briefly. Then, in the following subsections, we will explain in detail how we can establish their corresponding models.

• Category I:
  In this category, by applying the two-zero texture of $A_4$ symmetry for ${\mathcal{M}}_{\nu }$ in (12), we will consider only $M_{\nu }^{S_1}$ and $M_{\nu }^{S_2}$ textures, which are obtained by imposing $M_{ee}=M_{e\mu }=0$ and $M_{ee}=M_{e\tau }=0$, respectively:

  $$M_{\nu }^{S_1}=\left(\begin{array}{ccc}0& 0& c-\frac{d}{3}\\ 0& c+\frac{2}{3}d& -d\\ c-\frac{d}{3}& -d& d\end{array}\right)\mathrm{and}{M}_{\nu }^{S_2}=\left(\begin{array}{ccc}0& c-\frac{d}{3}& 0\\ c-\frac{d}{3}& d& -d\\ 0& -d& c+\frac{2}{3}d\end{array}\right).$$

  (14)
  It has been shown that there is a permutation symmetry between $M_{\nu }^{S_1}$ and $M_{\nu }^{S_2}$, such that the phenomenological predictions of texture $M_{\nu }^{S_2}$ can be generated from those of the texture $M_{\nu }^{S_1}$ [51].
• Category II:
  In this category, we propose four two-zero textures based on $A_4$ symmetry for ${\mathcal{M}}_{\nu }$ in (12). Namely, the textures $M_{\nu }^{S_3}$, $M_{\nu }^{S_4}$, $M_{\nu }^{S_5}$ and $M_{\nu }^{S_6}$, which are constructed from imposing $M_{e\mu }=M_{\mu \mu }=0$, $M_{e\tau }=M_{\tau \tau }=0$, $M_{e\mu }=M_{\tau \tau }=0$ and $M_{e\tau }=M_{\mu \mu }=0$, respectively:

  $$M_{\nu }^{S_3}=\left(\begin{array}{ccc}a+\frac{2}{3}d& 0& -d\\ 0& 0& a-\frac{d}{3}\\ -d& a-\frac{d}{3}& d\end{array}\right),M_{\nu }^{S_4}=\left(\begin{array}{ccc}a+\frac{2}{3}d& -d& 0\\ -d& d& a-\frac{d}{3}\\ 0& a-\frac{d}{3}& 0\end{array}\right),$$

  (15)

  $$M_{\nu }^{S_5}=\left(\begin{array}{ccc}a+\frac{2}{3}d& 0& c-\frac{d}{3}\\ 0& c+\frac{2}{3}d& a-\frac{d}{3}\\ c-\frac{d}{3}& a-\frac{d}{3}& 0\end{array}\right),M_{\nu }^{S_6}=\left(\begin{array}{ccc}a+\frac{2}{3}d& c-\frac{d}{3}& 0\\ c-\frac{d}{3}& 0& a-\frac{d}{3}\\ 0& a-\frac{d}{3}& c+\frac{2}{3}d\end{array}\right).$$

  (16)
  We should note that the textures $M_{\nu }^{S_3}$ and $M_{\nu }^{S_5}$ are related through permutation symmetry to $M_{\nu }^{S_4}$ and $M_{\nu }^{S_6}$, respectively.
• Category III:
  Finally, another two-zero texture based on $A_4$ symmetry for ${\mathcal{M}}_{\nu }$ in (12), $M_{\nu }^{S_7}$, is obtained from assuming $M_{\mu \mu }=M_{\tau \tau }=0$:

  $$M_{\nu }^{S_7}=\left(\begin{array}{ccc}a+\frac{2}{3}d& -d& -d\\ -d& 0& a-\frac{d}{3}\\ -d& a-\frac{d}{3}& 0\end{array}\right),$$

  (17)

  which has $\mu -\tau$ symmetry.

2.1. Formalism of Texture $M_{\nu }^{S_1}$

In the basis where the charged lepton mass matrix is diagonal, by employing ŀinebreak $M_{\nu }^{S_1}=U_{T{M}_2}^{*}{\left(M_{\nu }^{S_1}\right)}_d{U}_{T{M}_2}^{†}$, we reorganize the neutrino mass matrix of the texture $M_{\nu }^{S_1}$ as

$$M_{\nu }^{S_1}=U_{T{M}_2}^{*}\left(\begin{array}{ccc}{\lambda }_1& 0& 0\\ 0& {\lambda }_2& 0\\ 0& 0& {\lambda }_3\end{array}\right)U_{T{M}_2}^{†},$$

(18)

where we adopted the mixing matrix $U_{T{M}_2}$ given by (13); and ${\lambda }_1=m_1$, ${\lambda }_2=e^{2i\rho }{m}_2$ and ${\lambda }_3=e^{2i\sigma }{m}_3$. Now, using assumptions ${\left(M_{\nu }\right)}_{ee}=0$ and ${\left(M_{\nu }\right)}_{e\mu }=0$, associated with the texture $M_{\nu }^{S_1}$, provides two complex equations. Using the former yields

$$m_1=\left(\frac{sin2(\rho -\sigma )}{2sin2\sigma {cos}^2\theta }\right)m_2,$$

(19)

and

$$m_3=-\left(\frac{sin2\rho }{2sin2\sigma {sin}^2\theta }\right)m_2.$$

(20)

From Equations (19) and (20), we can obtain the ratio of two neutrino mass-squared differences $R_{\nu }=\frac{\delta m^2}{\Delta m^2}$ (where $\delta m^2\equiv m_2^2-m_1^2$ and $\Delta m^2\equiv m_3^2-m_1^2$) as

$$R_{\nu }=\frac{-{sin}^{2}2(\rho -\sigma )+4{cos}^4\theta {sin}^{2}2\sigma }{{cot}^4\theta {sin}^{2}2\rho -{sin}^{2}2(\rho -\sigma )}.$$

(21)

We should note that $R_{\nu }$ is independent of $T{M}_2$ phase parameter, $\varphi$.

Moreover, reemploying Equations (19) and (20) gives

$$\frac{m_1}{m_3}=\frac{cot2\rho -cot2\sigma }{csc2\sigma {cot}^2\theta }.$$

(22)

Furthermore, complex equation $({\left(M_{\nu }\right)}_{ee}={\left(M_{\nu }\right)}_{e\mu })=0$ yields relations

$$\frac{m_1}{m_3}=\frac{\sqrt{3}tan\theta sin2\sigma +sin(2\sigma +\varphi )}{sin\varphi }$$

(23)

and

$$cot2\sigma =-\left(cos2\theta cot\varphi +\frac{sin2\theta }{\sqrt{3}sin\varphi }\right).$$

(24)

By inserting (23) and (24) into (22), we obtain

$$cot2\rho =cot\varphi +\frac{cot\theta }{\sqrt{3}sin\varphi }.$$

(25)

Substituting the expressions associated with two Majorana phases from (24) and (25) into (21), we obtain an interesting relation between $T{M}_2$ mixing angle parameter $\left(\theta \right)$ and $R_{\nu }$:

$$sin\theta =\sqrt{R_{\nu }},$$

(26)

which plays an essential role within our work, as we will now elaborate.

Employing (26), we can rewrite relations (24) and (25) in terms of $R_{\nu }$ and $\varphi$:

$$tan2\sigma =-\frac{\sqrt{3}sin\varphi }{2\sqrt{R_{\nu }(1-R_{\nu })}+\sqrt{3}(1-2R_{\nu })cos\varphi }$$

(27)

and

$$cot2\rho =cot\varphi +\frac{1}{\sqrt{3R_{\nu }}sin\varphi }.$$

(28)

Let us also impose $|det{U}_{T{M}_2}|=1$ (For the unitary neutrino mixing matrix, without loss of generality, we can impose the condition $|detU]|=1$. This is a unimodularity condition of mixing matrix [55,56,57,58,59,60]). Concretely, in our herein paper, the physics of neutrino will be governed by the mixing matrix $U_{T{M}_2}$ of (13), which is unitary, unimodular and rephasing invariant. Therefore, we obtain an important relation between the phases of $U_{T{M}_2}$, $\varphi ,\rho$ and $\sigma$, which is:

$$\rho +\sigma =\varphi \pm n\pi ,$$

(29)

where $-\pi \le \varphi \le \pi$ and $n=0,1,\cdots$. We should note that Equation (29), which is obtained only from imposing unimodularity condition for the mixing matrix $U_{T{M}_2}$, is independent of the neutrino mass zero texture.

Substituting Majorana phases (27) and (28) into (29), the most significant consequence of our model is obtained:

$$\begin{array}{ll}\frac{1}{2}{tan}^{-1}\left[-\frac{\sqrt{3}sin\varphi }{2\sqrt{R_{\nu }(1-R_{\nu })}+\sqrt{3}(1-2R_{\nu })cos\varphi }\right]\hfill & \\ \\ +\frac{1}{2}{cot}^{-1}\left[cot\varphi +\frac{1}{\sqrt{3R_{\nu }}sin\varphi }\right]=\varphi \pm n\pi ,\hfill \end{array}$$

(30)

which is rewritten as a functions of only $T{M}_2$ phase parameter $\varphi$. Our endeavors have shown that Equation (30) yields acceptable results for only $n=1$; see, for instance, Figure 1.

Figure 1. In this figure, we show that $\rho +\sigma$ coincides with the lines $\varphi +{180}^{\circ }$ and $\varphi -{180}^{\circ }$, in which the coincident points illustrate the allowed range of the $T{M}_2$ phase parameter $\left(\varphi \right)$. The black dotted line indicates the line $\varphi +{180}^{\circ }$, and the green dot-dashed line indicates the line $\varphi -{180}^{\circ }$. The blue solid curve and the red dashed curve display $\rho +\sigma$ for $R_{\nu }=2.64\times {10}^{-2}$ and $R_{\nu }=3.29\times {10}^{-2}$, respectively. All phases and angles are in degrees.

Moreover, employing (27), (28), (19) and (20) as well as the definitions associated with $\delta m^2$ and $\Delta m^2$, the neutrino masses can be expressed with more convenient relations. More concretely, $m_1$, $m_2$ and $m_3$ are related to the unknown $T{M}_2$ phase parameter $\varphi$ and the experimental parameters $\delta m^2$ and $R_{\nu }$ as

$$\begin{array}{lll}\hfill m_1& =& \sqrt{\delta m^2(A-1)},\hfill \\ \hfill m_2& =& \sqrt{\delta m^{2}A},\hfill \\ \hfill m_3& =& \sqrt{\delta m^2\left(\frac{1}{R_{\nu }}+A-1\right)},\hfill \end{array}$$

(31)

where $A\equiv \left(4-4R_{\nu }\right){\left(3-6R_{\nu }-\frac{2R_{\nu }cos\varphi }{\sqrt{\frac{R_{\nu }}{3-3R_{\nu }}}}\right)}^{-1}$. Therefore, according to (31), our prediction is normal neutrino mass hierarchy.

Furthermore, from comparing Equations (13) and (2) and using (26), we easily obtain all the mixing angles ${\theta }_{13}$, ${\theta }_{12}$ and ${\theta }_{23}$ in terms of $R_{\nu }$ and $\varphi$:

$$\begin{array}{lll}\hfill {sin}^2{\theta }_{13}& =& \frac{2}{3}{R}_{\nu },\hfill \\ \hfill {sin}^2{\theta }_{12}& =& \frac{1}{3(1-{sin}^2{\theta }_{13})}=\frac{1}{3-2R_{\nu }}.\hfill \end{array}$$

(32)

According to (32), the deviation of ${\theta }_{12}$ from ${35}^{\circ }$ depends on the value of ${\theta }_{13}$, where ${\theta }_{13}$ depends only on $R_{\nu }$. Moreover, we obtain

$${sin}^2{\theta }_{23}=\frac{1}{2}+\frac{\sqrt{3R_{\nu }(1-R_{\nu })}cos\varphi }{3-2R_{\nu }},$$

(33)

which implies that the deviation of ${\theta }_{23}$ from ${45}^{\circ }$ depends on the $T{M}_2$ phase parameter $\left(\varphi \right)$. Using (33), we can easily show $\frac{1}{2}-\frac{\sqrt{3R_{\nu }(1-R_{\nu })}}{3-2R_{\nu }}$$\le {sin}^2{\theta }_{23}\le$$\frac{1}{2}+\frac{\sqrt{3R_{\nu }(1-R_{\nu })}}{3-2R_{\nu }}$.

Moreover, $\delta \ne 0$ and ${\theta }_{13}\ne 0$ are the necessary conditions to obtain CP-violation within the standard parametrization given by (2). Four independent CP-even quadratic invariants have been known, which can conveniently be chosen as $U_{11}^{*}{U}_{11}$, $U_{13}^{*}{U}_{13}$, $U_{21}^{*}{U}_{21}$ and $U_{23}^{*}{U}_{23}$. Furthermore, there is an independent CP-odd quadratic invariant, which is called Jarlskog re-phasing invariant parameter J [61]. The Jarlskog parameter is relevant to the CP violation in lepton number conserving processes like neutrino oscillations:

$$J\equiv \mathcal{I}m\left(U_{11}{U}_{12}^{*}{U}_{21}^{*}{U}_{22}\right).$$

(34)

By parameterizing the mixing matrix $U_{PMNS}$ given by (2), the analytical expression for J can be rewritten as

$$J=sin\delta sin{\theta }_{12}sin{\theta }_{23}sin{\theta }_{13}cos{\theta }_{12}cos{\theta }_{23}{cos}^2{\theta }_{13}.$$

(35)

In addition, in the scheme of the $T{M}_2$ of mixing matrix given by (13), the analytical expression for J is:

$$\begin{array}{l}\hfill J=\frac{1}{6\sqrt{3}}cos\varphi sin2\theta =\frac{1}{3\sqrt{3}}cos\varphi \sqrt{R_{\nu }(1-R_{\nu })},\end{array}$$

(36)

where we have used (26).

Comparing relations (35) and (36), as well as reemploying (26), the expression for the CP violating Dirac phase $\delta$, in the scheme of the $T{M}_2$ of mixing matrix, can be written as

$$\delta ={tan}^{-1}\left[\left(\frac{3-2R_{\nu }}{3-4R_{\nu }}\right)tan\varphi \right].$$

(37)

In the present work, since we have considered massive neutrinos as the Majorana particles, we can therefore obtain nine physical parameters: three neutrino masses given by (31); three flavor mixing angles given by (32) and (33); one CP-violating Dirac phase given by (37); two CP-violating Majorana phases given by (27) and (28). Surprisingly, solving Equation (30) leads to the prediction of the range of all nine physical neutrino parameters, which were mentioned earlier in the texture $M_{\nu }^{S_1}$. Let us be more precise. The value of the $T{M}_2$ phase parameter $\left(\varphi \right)$ can be calculated by using two experimental data $\delta m^2$ and $\Delta m^2$, which yield $R_{\nu }$. By substituting the value of $R_{\nu }=(2.64-3.29)\times {10}^{-2}$ [24] into Equation (30), we obtain the allowed range for the $T{M}_2$ phase parameter $\left(\varphi \right)$ as

$$\varphi \approx \pm (128.7^{\circ }-129.8^{\circ }).$$

(38)

Moreover, in order to depict the allowed range of the $T{M}_2$ phase parameter $\left(\varphi \right)$, let us plot $\rho +\sigma$ and $\varphi \pm \pi$ against $\varphi$ according to (30); see Figure 1. Obviously, the allowed range of $T{M}_2$ phase parameter $\left(\varphi \right)$ seen in Figure 1 is exactly the same as the one specified in (38).

By substituting $R_{\nu }=(2.64-3.29)\times {10}^{-2}$ and $\varphi$ from (38) into relations (27), (28), (31), (33), (36) and (37), we can not only obtain the ranges of the five neutrino oscillation parameters (it is seen that these are consistent with the experimental range of neutrino oscillation parameters in Table 1), but we can also predict the masses of the neutrinos, the CP violation parameters, the Dirac phase $\delta$, the Majorana phases $\rho$ and $\sigma$ and the Jarlskog invariant parameter (which may be measured by the future neutrino experiments).

Let us now proceed our discussions by obtaining the range of predicted values of neutrino oscillation parameters for the texture $M_{\nu }^{S_1}$.

By taking $A\approx (1.2096-1.2213)$ and $\varphi$ form (38), our herein model yields the following values for five neutrino oscillation parameters:

$$\begin{array}{lll}\hfill {sin}^2{\theta }_{13}& \approx & (0.01760-0.02119),\hfill \\ \hfill {sin}^2{\theta }_{12}& \approx & (0.3393-0.3408),\hfill \\ \hfill {sin}^2{\theta }_{23}& \approx & (0.4326-0.4411),\hfill \\ \hfill \delta m^2& \approx & (6.94-8.14)\times {10}^{-5}e{V}^2,\hfill \\ \hfill \Delta m^2& \approx & (2.47-2.63)\times {10}^{-3}e{V}^2,\hfill \end{array}$$

(39)

which are in agreement with the available experimental data for neutrino parameters in Table 1.

Moreover, as mentioned, our model yields the following consequences, which may be tested by future experiments:

$$\begin{array}{lll}\hfill m_1& \approx & (0.003918-0.004130)eV,\hfill \\ \hfill m_2& \approx & (0.009206-0.009923)eV,\hfill \\ \hfill m_3& \approx & (0.049912-0.051421)eV,\hfill \\ \hfill \delta & \approx & \mp (50.{84}^{\circ }-51.{80}^{\circ }),\hfill \\ \hfill \rho & \approx & \pm (7.{46}^{\circ }-8.{40}^{\circ }),\hfill \\ \hfill \sigma & \approx & \mp (58.{52}^{\circ }-58.{77}^{\circ }),\hfill \\ \hfill \left|J\right|& \approx & (0.0193-0.0220).\hfill \end{array}$$

(40)

Consequently, according to the allowed ranges for the values of three neutrino masses in (40), our model successfully predicts that the neutrino mass hierarchy is normal. However, the corresponding relations obtained from (31) emphasizes enough this fact. Note that the results of the texture $M_{\nu }^{S_1}$ endorse our prediction for the neutrino mass hierarchy, which subsequently pinpoints the corresponding relevant neutrino parameters for that mass hierarchy.

It is worth mentioning that the texture $M_{\nu }^{S_1}$, together with using $R_{\nu }\equiv \frac{\delta m^2}{\Delta m^2}$, assisted us with predicting all the neutrino parameters (see relations (39) and (40)), which are in good agreement with the available experimental data. It is worth noting that such an ability is a distinguishing feature of the neutrino mass matrix models.

In what follows, let us outline further predictions of our herein model which can be a test on the accuracy and precision of our predictions in (40)

• Regarding the sum of the three light neutrino masses, it should be noted that the significant experimental results were reported by Planck’s measurements of the cosmic microwave background (CMB) [62]:

  $$\sum m_{\nu }<0.12eV(\mathrm{Plank}+\mathrm{WMAP}+\mathrm{CMB}+\mathrm{BAO}).$$

  (41)
  In our model, this quantity is predicted as $\sum m_{\nu }\approx (0.063965-0.064546)$ eV, which is in agreement with (41).
• As for the flavor eigenstates, we are able to obtain solely the expectation values of the masses:

  $$⟨m_{{\nu }_i}⟩=\sum _{j=1}^3|U_{ij}{|}^2|m_j|,$$

  (42)

  where $i=e,\mu ,\mathrm{and}\tau$. Regarding these expectation values, our predictions are:

  $$\begin{array}{lll}\hfill ⟨m_{{\nu }_e}⟩& \approx & (0.006517-0.007065)\mathrm{eV},\hfill \\ \hfill ⟨m_{{\nu }_{\mu }}⟩& \approx & (0.025432-0.026265)\mathrm{eV},\hfill \\ \hfill ⟨m_{{\nu }_{\tau }}⟩& \approx & (0.031468-0.0317636)\mathrm{eV}.\hfill \end{array}$$

  (43)
• The Majorana neutrinos can violate lepton number, for instance, the neutrinoless double beta decay $\left(\beta \beta 0\nu \right)$ was referred [54]. Such a process has not been observed yet, but an upper bound has been set for the relevant quantity, i.e., $⟨m_{{\nu }_{\beta \beta }}⟩$. For instance, the results associated with the first phase of the KamLAND-Zen experiment set a constraint as $⟨m_{{\nu }_{\beta \beta }}⟩<(0.061-0.165)$ eV at 90 present CL [63]. Concerning this quantity, our model predicts: $⟨m_{{\nu }_{\beta \beta }}⟩\approx (0.005086-0.005332)$ eV, which is consistent with the result of the kamLAND-Zen experiment.

Up to now, our herein predictions of the texture $M_{\nu }^{S_1}$ may suggest it as an appropriate neutrino mass model. Notwithstanding, it would be considered as a more successful model if its predictions will also be supported by the cosmological and the neutrinoless double beta decay forthcoming experiments.

It is worth noting that the results associated with the texture $M_{\nu }^{S_1}$ are only applicable for the Majorana neutrinos, whilst they are not valid for the Dirac neutrinos.

2.2. Formalism of Texture $M_{\nu }^{S_2}$

There is a 2–3 permutation symmetry between the textures (The 2–3 permutation symmetry explains that $M_{\nu }^{S_1}$ and $M_{\nu }^{S_2}$ are related by the exchange of 2–3 rows and 2–3 columns of the neutrino mass matrix.) $M_{\nu }^{S_1}$ and $M_{\nu }^{S_2}$. Concretely, the corresponding permutation matrix is

$$P_{23}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right).$$

(44)

The 2–3 permutation symmetry given by (44) indicates the following relations among their corresponding oscillation parameters [51]:

$${\left({\theta }_{13}\right)}_{s_2}={\left({\theta }_{13}\right)}_{s_1},{\left({\theta }_{12}\right)}_{s_2}={\left({\theta }_{12}\right)}_{s_1},{\left({\theta }_{23}\right)}_{s_2}={90}^{\circ }-{\left({\theta }_{23}\right)}_{s_1},{\left(\delta \right)}_{s_2}={\left(\delta \right)}_{s_1}-{180}^{\circ }.$$

(45)

Moreover, textures $M_{\nu }^{S_1}$ and $M_{\nu }^{S_2}$ have the same eigenvalues ${\lambda }_i$ (for $i=1,2,3$). Consequently, except for ${sin}^2{\theta }_{23}$ and $\delta$, the other predictions for neutrino oscillation parameters associated with the texture $M_{\nu }^{S_2}$ (calculated by our model) are the same as those predicted by the texture $M_{\nu }^{S_1}$ (cf Section 2.1). These exceptions in the texture $M_{\nu }^{S_2}$ are:

$${\left({sin}^2{\theta }_{23}\right)}_{s_2}\approx (0.5588-0.5673),-{\left(\delta \right)}_{s_2}\approx {180}^{\circ }\pm (50.{84}^{\circ }-51.{80}^{\circ }).$$

(46)

2.3. Formalism of Textures $M_{\nu }^{S_3}$, $M_{\nu }^{S_4}$, $M_{\nu }^{S_5}$, and $M_{\nu }^{S_6}$

The mass matrix of textures $M_{\nu }^{S_3}$ (see Equation (15)) has two conditions $(M_{e\mu }=0$ and $M_{\mu \mu }=0)$, which imply the following complex equations

$$m_1=(1+\frac{3sin\theta (1+e^{2i\varphi })}{\sqrt{3}{e}^{i\varphi }cos\theta -3sin\theta })e^{2i\rho }{m}_2,$$

(47)

and

$$m_3=-\frac{3e^{i\varphi }-\sqrt{3}tan\theta }{3+\sqrt{3}{e}^{i\varphi }tan\theta }{e}^{i(\varphi +2\rho -2\sigma )}{m}_2,$$

(48)

by which we can calculate $R_{\nu }$. In Figure 2, by depicting the experimental value of $R_{\nu }$ as a function of $\theta$ and $\varphi$, we have obtained the allowed range of $\theta$ around $\theta \approx \left({23}^{\circ }--{70}^{\circ }\right)$ and $\left({110}^{\circ }--{157}^{\circ }\right)$. We substitute the value of $\theta$ in the expression of ${sin}^2{\theta }_{13}$, which, in turn, is obtained from comparing the $U_{e3}$ in Equations (2) with (13) as ${sin}^2{\theta }_{13}=\frac{2}{3}{sin}^2\theta$. Finally, for the texture $M_{\nu }^{S_3}$, we obtain ${sin}^2{\theta }_{13}\approx \left(0.102--0.589\right)$, which is inconsistent with the experimental data. From a phenomenological point of view, the consequences associated with the textures $M_{\nu }^{S_4}$ and $M_{\nu }^{S_3}$ are equivalent. For the experimental values of $R_{\nu }$, we have shown that these textures predict a very large values of ${\theta }_{13}$, which is not allowed.

Figure 2. In this figure, we show the experimental value of $R_{\nu }$ as a function of $\theta$ and $\varphi$ for the texture $M_{\nu }^{S_3}$. $\theta$ and $\varphi$ are in degrees.

Concerning the textures $M_{\nu }^{S_5}$ and $M_{\nu }^{S_6}$ (see (16)), we find that they predict $m_1=m_3\ne 0$, which is not allowed.

Consequently, all the textures associated with the Category II are ruled out completely by the experimental data listed in Table 1 [24].

2.4. Formalism of Texture $M_{\nu }^{S_7}$

Concerning the texture $M_{\nu }^{S_7}$ in Equation (17), we see that the mass matrix also has $\mu -\tau$ symmetry. Therefore, it implies the TBM mixing matrix with $sin{\theta }_{13}=0$, which is inconsistent with the experimental data listed in Table 1 [24].

3. Discussion and Conclusions

Matrix models are of particular importance for the phenomenological evaluation of neutrino physics. The choice of symmetries for the mass matrix can lead to specific states in the mixture matrix, which can lead to results consistent with the corresponding experimental data. Such consequences are significant as we can make additional predictions about neutrinos and their flavor symmetries.

A salient feature of the study of the neutrino mass matrix phenomenon is that it could, in principle, provide new clues for understanding the flavor problem; in particular, its mixing matrix, which, in contrast to the quark sector, has large (mixing) angles. In addition, the discrepancy between the masses of charged neutrinos and leptons are more pronounced than the corresponding features in the quark sector. Indeed, the mass and mixing problem in the lepton sector is a fundamental problem. Furthermore, the following important questions should be answered by future experiments: What are the masses of the different neutrinos? What is the nature of neutrinos? How close to ${45}^{\circ }$ is ${\theta }_{23}$? What are the values of three CP-violating phases associated with the neutrino mixing matrix (i.e., the Dirac phase $\delta$ and the Majorana phases $\rho$ and $\sigma$)?

In our work, we applied two-zero textures within the neutrino mass matrix with $A_4$ symmetry, along with imposing $|detU|=+1$ on the neutrino mixing matrix, where the charged lepton mass matrix is diagonal and the nature of neutrinos are Majorana. Concretely, we have retrieved seven viable two-zero textures such that the mixing matrix could be the second scheme of trimaximal $T{M}_2$ mixing matrix. Then, assuming the unimodular property of the $T{M}_2$, we determined algebraic relations for the Majorana phases $\rho \mathrm{and}\sigma$, together with the $T{M}_2$ phase parameter $\left(\varphi \right)$; cf. relation (29).

Based on the common physical properties of these seven textures, we classified them into three categories. We investigated the phenomenological properties of all these textures and then compared them with the available experimental data. Among these textures, we have shown (in the non-perturbation method) that only $M_{\nu }^{S_1}$ and $M_{\nu }^{S_2}$ have properties that may agree with the experimental data. It is worth noting that applying a perturbation analysis to $M_{\nu }^{S_7}$ may result in agreement with the experimental data. However, such an investigation was not in the scope of our current work.

Let us be more precise. Regarding the texture $M_{\nu }^{S_1}$, we have shown that (i) $sin\theta =\sqrt{R_{\nu }}$ and (ii) $\rho +\sigma =\varphi \pm (+1)\pi$. This is an original result that leads to an innovative and simple way of calculating accurate predictions for neutrino parameters. Subsequently, employing the allowed ranges $R_{\nu }$ and $\delta m^2$, we have obtained the allowed ranges of $\varphi$. Then, we presented the predictions of our model for the values of neutrino parameters such as mixing angles, the neutrino masses, the expectation value of neutrino masses in the flavor bases i.e., $(⟨m_{{\nu }_e}⟩,⟨m_{{\nu }_{\mu }}⟩,⟨m_{{\nu }_{\tau }}⟩)$, the CP violation parameters $\delta$, $\rho$, $\sigma$, and J. We emphasize that the values of all these parameters are retrieved by merely using the allowed ranges of $R_{\nu }$ and $\delta m^2$ and nothing else. Finally, we compare our predictions to recently reported data. We found that there is a good agreement. In addition, the predictions for the texture $M_{\nu }^{S_1}$ agree with the observational data of the CMB and the neutrinoless double beta decay experiments; see relations (40). Furthermore, concerning the texture $M_{\nu }^{S_1}$, we found that our prediction for neutrino mass hierarchy is quite satisfactory.

We expect that our model results for neutrino masses, their hierarchy, CP–violation parameters $\delta$, $\rho$, $\sigma$ and J are in good agreement with future experiments. We have shown that there is a 2–3 permutation symmetry between the textures. Disregarding the values of ${\theta }_{23}$ and $\delta$, the mentioned symmetry yields a similarity for the rest of predictions associated with the textures $M_{\nu }^{S_1}$ and $M_{\nu }^{S_2}$.

In summary, by applying the $A_4$ symmetry, two-zero texture assumption, and specially including the unimodular feature of the $T{M}_2$ mixing matrix, we have provided the textures $M_{\nu }^{S_1}$ and $M_{\nu }^{S_2}$.

In our next investigation into neutrino physics, we will focus on perturbation theory to assess states ruled out by experimental data in other frames. More specifically, we will examine the texture $M_{\nu }^{S_7}$ in the perturbation method to assess whether or not the corresponding one agrees with the experimental data.