Advances in High Energy Physics

Volume 2018, Article ID 5806743, 16 pages

https://doi.org/10.1155/2018/5806743

## Investigating the Hybrid Textures of Neutrino Mass Matrix for Near Maximal Atmospheric Neutrino Mixing

Department of Physics, National Institute of Technology, Kurukshetra, Haryana 136119, India

Correspondence should be addressed to Madan Singh; moc.liamg@971nadamhgnis

Received 22 December 2017; Accepted 5 April 2018; Published 14 May 2018

Academic Editor: Jose W. F. Valle

Copyright © 2018 Madan Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We have studied that the implication of a large value of the effective Majorana neutrino mass in case of neutrino mass matrices has 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 CL.

#### 1. Introduction

In leptonic sector, the reactor mixing angle () has been established to a reasonably good degree of precision [1–6], and its nonzero 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. Another important issue is the determination of octant of atmospheric mixing angle , which may be greater than or less than or equal to . The determination of the nature of neutrinos whether Dirac or Majorana also remains an open question. The observation of neutrinoless double beta () decay would eventually establish the Majorana nature of neutrinos.

The effective Majorana mass term related to decay can be expressed asData from KamLAND-Zen experiment has presented an improved search for neutrinoless double-beta () decay [7] and it is found that at 90% (or <2*σ*) CL. For recent reviews on decay see [8–13].

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 that are considered to be zero or equal [14–21] or some cofactors of neutrino mass matrix to be either zero or equal [19, 22–27]. Specifically, mass matrices with zero textures (or cofactors) have been extensively studied [14–18, 22–24] 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 [20, 21, 26, 27]. 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 [21] for texture structures having one zero element and equal matrix elements in the neutrino mass matrix (1TEE), while for texture with one vanishing minor and equal cofactors in the neutrino mass matrix (1TEC) only fifty-two cases are able to survive the data [26, 27].

The purpose of present paper is to investigate the implication of large effective neutrino mass on 1TEE and 1TEC structures of neutrino mass matrix, while taking into account the assumptions of [28, 29]. The consideration of large is motivated by the extensive search for this parameter in the ongoing experiments. The implication of large has earlier been studied for the viable cases of texture two-zero and two-vanishing minor, respectively [28, 29]. Grimus et al. [30] 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 is 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 for 1TEE and 1TEC, respectively. In addition, the analysis also hints towards the indistinguishable feature of 1TEE and 1TEC. To present the indistinguishable 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 3 CL. The similarity between texture zero structures with one mass ordering and corresponding cofactor zero structures with the opposite mass ordering has earlier been noted in [31–33]. In [19], 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 the following manner. In Section 2, we shall discuss the methodology to obtain the constraint equations. Section 3 is devoted to numerical analysis. In the end we will summarize our result.

#### 2. Methodology

The effective Majorana neutrino mass matrix contains nine parameters which include three neutrino masses (, , ), three mixing angles (, , ), and three CP violating phases (, , ). In the flavor basis, the Majorana neutrino mass matrix can be expressed as follows:where is the diagonal matrix of neutrino masses and is the flavor mixing matrix, andwhere is diagonal phase matrix containing Majorana neutrinos . is unobservable phase matrix and depends on phase convention. Equation (2) can be rewritten aswhere For the present analysis, we consider the following parameterization of [20]:where , . Here, is a 3 × 3 unitary matrix consisting of three flavor mixing angles (, , ) and one Dirac CP-violating phase .

For hybrid texture structure (1TEE) of , we can express the ratios of neutrino mass eigenvalues in terms of the mixing matrix elements as [21]where is a phase factor. Similarly, in case of inverse hybrid texture structure (1TEC) of , we can express the ratios of mass eigenvalues as [26, 27] follows:wherewith () a cyclic permutation of (1, 2, 3) and is phase factor.

Using the above expressions, we can obtain the magnitude of neutrino mass ratios, and in each texture structure, and the Majorana phases () can be given as and .

The solar and atmospheric mass squared differences (), where corresponds to solar mass-squared difference and corresponds to atmospheric mass-squared difference, can be defined as [20]The experimentally determined solar and atmospheric neutrino mass-squared differences can be related to neutrino mass ratios asand the three neutrino masses can be determined in terms of , as

Among the sixty logically possible cases of 1TEE or 1TEC texture structures, there are certain pair, which exhibit similar phenomenological implications and are related via permutation symmetry [21, 26, 27]. This corresponds to permutation of the 2-3 rows and 2-3 columns of . The corresponding permutation matrix can be given byWith the help of permutation symmetry, one obtains the following relations among the neutrino oscillation parameters:where and denote the cases related to 2-3 permutation. The following pair among sixty cases are related via permutation symmetry: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;

Clearly we are left with only thirty-two independent cases. It is worthwhile to mention that cases , , , and are invariant under the permutations of 2 and 3 rows and columns.

#### 3. Numerical Analysis

The experimental constraints on neutrino parameters at 3 confidence levels (CL) are given in Table 1. The classification of sixty phenomenologically possible cases of 1TEE and 1TEC is done in the nomenclature, given by Wang et al. in [26, 27]. All the sixty cases are divided into six categories , , , , and (Table 2). In [26, 27], 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 ( and mass squared differences () at 3 CL [5, 6].