Investigating the hybrid textures of neutrino mass matrix for near maximal atmospheric neutrino mixing

In the present paper, we have studied that the implication of a large value of the effective Majorana neutrino mass in case of neutrino mass matrices having either two equal elements and one zero element (popularly known as hybrid texture) or two equal cofactors and one zero minor (popularly known as inverse hybrid texture) in the flavor basis. In each of these cases, four out of sixty phenomenologically possible patterns predict near maximal atmospheric neutrino mixing angle in the limit of large effective Majorana neutrino mass. This feature remains irrespective of the experimental data on solar and reactor mixing angles. In addition, we have also performed the comparative study of all the viable cases of hybrid and inverse hybrid textures at 3$\sigma$ CL.


Introduction
In leptonic sector, the reactor mixing angle (θ 13 ) has been established to a reasonably good degree of precision [1][2][3][4][5], and its non zero and relatively large value has not only provided an opportunity in exploring CP violation and the neutrino mass ordering in the future experiments, but has also highlighted the puzzle of neutrino mass and mixing pattern. In spite of the significant developments made over the years, there are still several intriguing questions in the neutrino sector which remain unsettled. For instance, the present available data is unable to throw any light on the neutrino mass spectrum, which may be normal/inverted and may even be degenerate.
In the lack of any convincing theory, several phenomenological ideas have been proposed in the literature so as to restrict the form of neutrino mass matrix, such as some elements of neutrino mass matrix are considered to be zero or equal [11][12][13][14] or some co-factors of neutrino mass matrix to be either zero or equal [12,[15][16][17].
Specifically, mass matrices with zero textures (or cofactors) have been extensively studied [11,15] due to their connections to flavor symmetries. In addition, texture structures with one zero element (or minor) and an equality between two independent elements (or cofactors) in neutrino mass matrix have also been studied in the literature [13,14,17]. Such form of texture structures sets to one constraint equation and thus reduces the number of real free parameters of neutrino mass matrix to seven. Hence they are considered as predictive as the well-known two-zero textures and can also be realised within the framework of seesaw mechanism. Out of sixty possibilities, only fifty four are found to be compatible with the neutrino oscillation data [14] for texture structures having one zero element and an equal matrix elements in the neutrino mass matrix (1TEE), while for texture with one vanishing minor and an equal cofactors in the neutrino mass matrix (1TEC) only fifty two cases are able to survive the data [17].
The purpose of present paper is to investigate the implication of large effective neutrino mass |M | ee on 1TEE and 1TEC structures of neutrino mass matrix, while taking into account the assumptions of Refs. [18,19].
The consideration of large |M | ee is motivated by the extensive search for this parameter in the ongoing 0νββ experiments. The implication of large |M | ee has earlier been studied for the viable cases of texture two zero and two vanishing minor, respectively [18,19]. Grimus et. al [20] also predicted the near maximal atmospheric mixing for two zero textures when supplemented with the assumption of quasi degenerate mass spectrum. However the observation made in all these analyses are independent of solar and reactor mixing angles. Motivated by these works, we find that only four out of sixty cases are able to predict near maximal θ 23 for 1TEE and 1TEC, respectively. In addition, the analysis also hints towards the indistinguisble feature of 1TEE and 1TEC.
To present the indistingusible nature of the 1TEE and 1TEC texture structures, we have then carried out a comparative study of all the viable cases of 1TEE and 1TEC at 3σ CL. The similarity between texture zero structures with one mass ordering and correponding cofactor zero structures with the opposite mass ordering has earlier been noted in Refs. [21,22]. In Ref. [12], the strong similarities have also been noted between the texture structures with two equalities of elements and structures with two equalities of cofactors in neutrino mass matrix, with opposite mass ordering.
The rest of the paper is planned in following manner: In Section 2, we shall discuss the methodology to obtain the constraint equations. Section 3 is devoted to numerical analysis. Section 4 will summarize our result.

Methodology
The effective Majorana neutrino mass matrix (M ν ) contains nine parameters which include three neutrino masses (m 1 , m 2 , m 3 ), three mixing angles (θ 12 , θ 23 , θ 13 ) and three CP violating phases (δ, ρ, σ). In the flavor basis, the Majorana neutrino mass matrix can be expressed as, where M diag = diag(m 1 , m 2 , m 3 ) is the diagonal matrix of neutrino masses and U is the flavor mixing matrix, where P ν is diagonal phase matrix containing Majorana neutrinos ρ, σ. P l is unobservable phase matrix and depends on phase convention. Eq. (2) can be re-written as where λ 1 = m 1 e 2iρ , λ 2 = m 2 e 2iσ , λ 3 = m 3 . For the present analysis, we consider the following parameterization of U [13]: where, c ij = cos θ ij , s ij = sin θ ij . Here, U is a 3 × 3 unitary matrix consisting of three flavor mixing angles (θ 12 , θ 23 , θ 13 ) and one Dirac CP-violating phase δ.
For hybrid texture structure (1TEE) of M ν , we can express the ratios of neutrino mass eigenvalues in terms of the mixing matrix elements as [14] where P = e i(φα+φ β −φc−φ d ) is a phase factor. Similarly, in case of inverse hybrid texture structure (1TEC) of M ν , we can express the ratios of mass eigenvalues as [17] follows where with (i, j, k) a cyclic permutation of (1, 2, 3) and is phase factor.
The solar and atmospheric mass squared differences (δm 2 , ∆m 2 ), where δm 2 corresponds to solar masssquared difference and ∆m 2 corresponds to atmospheric mass-squared difference, can be defined as [13] The experimentally determined solar and atmospheric neutrino mass-squared differences can be related to neutrino mass ratios (α, β) as and the three neutrino masses can be determined in terms of α, β as Among the sixty logically possible cases of 1TEE or 1TEC texture structures, there are certain pair, which exhibits similar phenomenological implications and are related via permutation symmetry [14,17]. This corresponds to permutation of the 2-3 rows and 2-3 columns of M ν . The corresponding permutation matrix can be given by With the help of permutation symmetry, one obtains the following relations among the neutrino oscillation where X and Y denote the cases related by 2-3 permutation. The following pair among sixty cases are related (E 1 , E 2 ); (E 3 , E 4 ); (E 5 , E 5 ); (E 6 , E 9 ); (E 7 , E 8 ); (E 10 , E 10 ).
Clearly we are left with only thirty two independent cases. It is worthwhile to mention that cases A 1 , A 5 , E 5 and E 10 are invariant under the permutations of 2-and 3-rows and columns.

Numerical analysis
The experimental constraints on neutrino parameters at 3σ confidence levels (CL) are given in Table 1. The classification of sixty phenomenologically possible cases of 1TEE and 1TEC is done in the the nomenclature, given by W. Wang in Ref. [17]. All the sixty cases are divided into six categories A, B, C, D and E [Table2].
In [17], it is found that the phenomenological results of cases belonging to 1TEC (or 1TEE) are almost similar to each other due to permutation symmetry. For the purpose of calculation, we have used the latest experimental data on neutrino mixing angles (θ 12 , θ 23 , θ 13 , δm 2 ) and mass squared differences (∆m 2 , δ) at 3σ CL [5].

Near maximal atmospheric mixing for 1TEE and 1TEC texture structures
As a first step of the analysis, all the sixty cases of 1TEE and 1TEC have been investigated in the limit of large |M | ee . For the analysis, we have incorporated the assumptions of Refs. [18,19], wherein authors have considered  The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle θ 13 < 8.9 0 , as given in Table 1.
the lower bound on |M | ee to be large (i.e. |M | ee > 0.08eV ). The upper bound on |M | ee is choosen to be more conservative i.e. |M | ee < 0.5eV at 3σ CL [9]. The input parameters (θ 12 , θ 23 , θ 13 , δm 2 , ∆m 2 , δ) are generated by the method of random number generation. The three neutrino mixing angles and Dirac-type CP-violating phase δ are varied between 0 0 to 90 0 and 0 0 to 360 0 , respectively. However, the mass-squared differences (δm 2 , ∆m 2 ) are varied randomly within their 3σ experimental range [5]. For the numerical analysis, we follow the same procedure as discussed in [13]. The main results and discussion are summarized as follows: In Fig

Comparing the results for 1TEE and 1TEC texture structures
In this subsection, we compare the results of all the viable structures of 1TEE and 1TEC in neutrino mass matrix. It is worthwhile to mention that the present refinements of the experimental data does not limit the number of viable cases in 1TEE and 1TEC textures repectively. The number of viable cases obtained are same as predicted in Refs. [14] and [17] for 1TEE and 1TEC, respectively. For executing the analysis, we vary the allowed ranges of three neutrino mixing angles (θ 12 , θ 23 , θ 13 ) and mass squared differences (δm 2 , ∆m 2 ) within their 3σ confidence level. To facilitate the comparison, we have encapsulated the the predictions regarding three  The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle θ 13 < 8.9 0 , as given in Table 1.
CP violating phases (ρ, σ, δ) and neutrino masses m 1,2,3 for all the allowed texture structures of 1TEE and 1TEC respectively [ Textures and 1TEC respectively, however with opposite neutrino mass ordering [     Table   4]. These predictions are significant considering the latest hint on δ near 270 0 [5]. Therefore all the above cases discussed are almost indistinguishable for 1TEE and 1TEC, if neutrino mass ordering is not considered.
Category D (F): All the ten possible cases belonging to Category D are acceptable with neutrino oscillation data at 3σ CL for 1TEE and 1TEC, respectively [   On the other hand, for D 1 (NO), D 2 (NO), D 4 (NO), D 6 (NO), F 1 (NO),F 2 (NO), F 4 (NO), F 6 (NO) δ is notably constrained for 1TEE, and same is true for 1TEC, however for opposite mass ordering.

Category E:
In Category E, all the ten possible cases are allowed for 1TEE at 3σ CL, while only nine other than E 5 are acceptable in case of 1TEC [    term, which would, in turn either confirm or rule out our assumption of large |M | ee .

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper. Table 2: Sixty phenomenologically possible hybrid texture structures of M ν at 3σ C.L where the triangles "△" denote equal and non-zero elements (or cofactors), and "0" denotes the vanishing element (or minor)." × " denote the non-zero elements or cofactor.