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Advances in High Energy Physics
Volume 2018, Article ID 5806743, 16 pages
https://doi.org/10.1155/2018/5806743
Research Article

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 SCOAP3.

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 [16], 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 <2σ) CL. For recent reviews on decay see [813].

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 [1421] or some cofactors of neutrino mass matrix to be either zero or equal [19, 2227]. Specifically, mass matrices with zero textures (or cofactors) have been extensively studied [1418, 2224] 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 [3133]. 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].

Table 1: Current neutrino oscillation parameters from global fits at 3 confidence level [5, 6]. NO (IO) refers to normal (inverted) neutrino mass ordering.
Table 2: Sixty phenomenologically possible hybrid texture structures of at 3 C.L where the triangles “△” denote equal and nonzero elements (or cofactors) and “0” denotes the vanishing element (or minor). “×” denotes the nonzero elements or cofactor.
3.1. 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 . For the analysis, we have incorporated the assumptions of [28, 29], wherein authors have considered the lower bound on to be large (i.e., ). The upper bound on is chosen to be more conservative; that is, at 3 CL [10]. The input parameters () are generated by the method of random number generation. The three neutrino mixing angles and Dirac-type CP-violating phase are varied between to and to , respectively. However, the mass-squared differences () are varied randomly within their 3 experimental range [5, 6]. For the numerical analysis, we follow the same procedure as discussed in [20]. The main results and discussion are summarized as follows.

In Figures 1, 2, 3, 4, 5, and 6, it is explicitly shown that the octant of is well restricted for of 1TEE and 1TEC texture structures, respectively. However, for the remaining cases, the value of is unconstrained like other oscillation parameters. Apart from restricting the octant of , the analysis also ensures the quasi-degenerate mass ordering for these cases similar to the observation of [2830]. From Figures 1(a), 1(b), 3(a), and 3(b), it is clear that, for increasing value of , atmospheric mixing angle approaches to maximal value for the structure of 1TEE and 1TEC for both normal ordering (NO) and inverted ordering (IO). In Figures 2 and 4, it is explicitly shown that for cases and the quadrant of is already decided without the experimental input of the mixing angles. For 1TEE, we have for NO and for IO, whereas for 1TEC, for NO, while for IO (Figures 2(a), 2(b), 4(a), and 4(b)). Clearly the correlation plots of case are indistinguishable for 1TEE and 1TEC, if neutrino mass ordering is not considered as also pointed out earlier. Similar conclusion can be drawn for structure since both are related through 2-3 exchange symmetry (Figures 2(c), 2(d), 4(c), and 4(d)). Apart from the prediction of near maximality of , cases and also predict for 1TEE and 1TEC, respectively, if experimental range of mixing angles is considered as in Table 2. Figures 2(a) and 2(c) for NO and Figures 2(b) and 2(d) for IO depict the 2-3 interchange symmetry between cases and for 1TEE. Similar phenomenological observation is shown for 1TEC in Figures 4(a), 4(c), 4(b), and 4(d), respectively.

Figure 1: Correlation plots for textures ((a) NO and (b) IO) and for ((c) NO and (d) IO) at 3 CL for 1TEE. The symbols have their usual meaning. The horizontal line indicates the upper limit on effective neutrino mass term (i.e., ) at 90% CL, given in KamLAND-Zen experiment [7].
Figure 2: Correlation plots for textures ((a) NO and (b) IO) and for ((c) NO and (d) IO) at 3 CL for 1TEE. The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle , as given in Table 1.
Figure 3: Correlation plots for textures ((a) NO and (b) IO) and for ((c) NO and (d) IO) at 3 CL for 1TEC. The symbols have their usual meaning. The horizontal line indicates the upper limit on effective neutrino mass term (i.e., ) at 90% (<2σ)CL, given in KamLAND-Zen experiment [7].
Figure 4: Correlation plots for textures ((a) NO and (b) IO) and for ((c) NO and (d) IO) at 3 CL for 1TEC. The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle , as given in Table 1.
Figure 5: Correlation plots for textures ((a), (c)) and ((b), (d)) with IO for 1TEE at 3 CL. The symbols have their usual meaning. In (a) and (b), colored horizontal line indicates the upper limit on effective neutrino mass term (i.e., ) at 90 (<2σ)% CL, given in KamLAND-Zen experiment [7]. In (c) and (d), we have shown the upper limit on reactor mixing angle .
Figure 6: Correlation plots for textures ((a), (c)) and ((b), (d)) with NO for 1TEC at 3 CL. The symbols have their usual meaning. In (a) and (b), colored horizontal line indicates the upper limit on effective neutrino mass term (i.e., ) at 90 (<2σ)% CL, given in KamLAND experiment [7]. In (c) and (d), we have shown the upper limit on reactor mixing angle .

Similarly, cases and of 1TEE also predict near maximal atmospheric mixing angle () for IO (Figures 5(a) and 5(b)). Interestingly the parameter space of reactor mixing angle is found to be constrained between and (Figures 5(c) and 5(d)). In Figures 5(c) and 5(d), it is clear that, for the allowed experimental range of (), inches closer to . Similar predictions have been noted for cases and of 1TEC, however, for normal mass ordering (NO) (Figures 6(a), 6(b), 6(c), and 6(d)).

3.2. 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 do not limit the number of viable cases in 1TEE and 1TEC textures respectively. The number of viable cases obtained is the same as predicted in [21, 26, 27] for 1TEE and 1TEC, respectively. For executing the analysis, we vary the allowed ranges of three neutrino mixing angles () and mass squared differences () within their 3 confidence level. To facilitate the comparison, we have encapsulated the the predictions regarding three CP violating phases () and neutrino masses for all the allowed texture structures of 1TEE and 1TEC, respectively (Tables 3, 4, 5, and 6).

Category A. In Category A, there are 10 possible cases out of which only four () are allowed for 1TEE at 3 CL, and in addition inverted mass ordering (IO) is ruled out for all these cases. On the other hand, only three () are allowed for 1TEC with current oscillation data, while normal mass ordering (NO) is ruled out for these cases. For 1TEE, remain unconstrained; however, for 1TEC, only remains unconstrained, while Majorana phases ( are restricted near pertaining to viable cases. From Table 3, it is clear that lower bound on lowest neutrino mass ( (NO) or (IO)) is nearly equal or less than 1 meV for 1TEE and 1TEC.

Table 3: The allowed ranges of Dirac CP-violating phase , the Majorana phases , , and three neutrino masses , , for the experimentally allowed cases of Category A. Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.

Category B (C). In Category B, all the ten possible cases are allowed for both 1TEE and 1TEC, respectively, at 3 CL; however, cases allow only NO for 1TEE, while the same allow only IO for 1TEC (Table 4). Cases allow both NO and IO for 1TEE and 1TEC, respectively. As mentioned in [26, 27], cases of Category B are related to the cases belonging to Category C through permutation symmetry; therefore, we can obtain the results for Category C from B. We find that cases allow only NO for 1TEE, while the same allow IO for 1TEC.

Table 4: The allowed ranges of Dirac CP-violating phase , the Majorana phases , , and three neutrino masses , , for the experimentally allowed cases of Category B (C). Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.

Textures (NO), (NO, IO), (NO, IO), (NO), (NO), (NO), (NO), (NO), (NO), (NO), (NO, IO), (NO, IO), (NO), and (NO) held nearly no constraint on Dirac CP violating phase () for 1TEE and 1TEC, respectively, but with opposite neutrino mass ordering (Table 4). Only cases (NO), (IO), (NO), and (IO) for 1TEE and (IO), (NO), (IO), and (NO) for 1TEC show significant reduction in the parameter space of . It is found that is restricted near and for 1TEE and 1TEC, respectively (Table 4). These predictions are significant considering the latest hint on near [5, 6]. 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 3 CL for 1TEE and 1TEC, respectively (Table 5). However cases favor both NO and IO for 1TEE and 1TEC, while , , and are acceptable only for IO in case of 1TEE; however, the same cases allowed NO in case of 1TEC. Similarly, the results for cases belonging to Category F can be derived from Category D.

Table 5: The allowed ranges of Dirac CP-violating phase , the Majorana phases , , and three neutrino masses , , for the experimentally allowed cases of Category D (F). Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.

Cases (IO), (IO), (IO), (IO), (NO, IO), (IO), (NO, IO), (IO), (NO, IO), (IO), (IO), (IO), (IO), (IO), (NO, IO), (IO), (NO, IO), (IO), (NO, IO), and (IO) predict literally no constraints on for 1TEE. These cases give identical predictions for 1TEC, but for opposite mass ordering. For cases (NO), (NO), (NO), (NO), (NO), (NO), (NO), and (NO), the parameter space of is found to be well constrained for 1TEE. These cases give similar predictions regarding the parameter space of for 1TEC, but for IO.

Category E. In Category E, all the ten possible cases are allowed for 1TEE at 3 CL, while only nine other than are acceptable in case of 1TEC (Table 6). Cases allow only inverted mass ordering (IO) for 1TEE, while the same textures allow only normal mass ordering (NO) for 1TEC. Cases and allow both NO and IO for 1TEE; however is ruled out for both NO and IO for 1TEC at 3 CL. Similar to cases belonging to Category D, (IO) cover full range of for 1TEE, whereas the same cases (except ) give identical predictions for 1TEC, but for NO. For (NO) and (NO), phases are somewhat restricted at 3 CL for 1TEE, while only for (IO), the parameter space of seems to be restricted for 1TEC (Table 6).

Table 6: The allowed ranges of Dirac CP-violating phase , the Majorana phases , , and three neutrino masses , , for the experimentally allowed cases of Category E. Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.

To summarize our discussion, we have investigated all the viable cases of 1TEE and 1TEC texture structures in the limit of large effective neutrino mass . It is found that only four cases are able to produce near maximal atmospheric mixing for 1TEE and 1TEC, respectively. However, the predictions remain true irrespective of the experimental data on solar and reactor mixing angle. The observation also hints towards the indistinguishable feature of 1TEE and 1TEC texture structures, but for opposite mass ordering. In order to depict the indistinguishability, we have carried out a comparative study of 1TEE and 1TEC texture structures using the current experimental data at 3 CL. From our discussion we find that most of the cases belonging to 1TEE and 1TEC are almost indistinguishable as far as the neutrino oscillation parameters are concerned, but with opposite neutrino mass ordering. The indistinguishable nature of 1TEE and 1TEC is more prominent for quasi-degenerate mass ordering. For the cases where lower bound on lowest neutrino mass is very small (<1 meV), there is noticeable deviation in the predictions for 1TEE and 1TEC (Tables 3, 4, 5, and 6). This point is also discussed by Liao et. al. in [31]. In addition, the parameter space of for most of the cases belonging to 1TEE and 1TEC remains unrestricted, while only eight cases show maximal restriction for . Since no presently feasible experiment has been able to determine the neutrino mass ordering, therefore, we cannot distinguish 1TEE and 1TEC structures using the present oscillation data. However, the currently running and forthcoming neutrino experiments aimed at distinguishing the mass ordering of neutrinos will test our phenomenological results. Also the ongoing and future neutrinoless double beta decay experiments are capable of measuring term, which would, in turn, either confirm or rule out our assumption of large .

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author would like to thank the Director, National Institute of Technology Kurukshetra, for providing necessary facilities to work.

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