#### Abstract

We investigate the novel possibilities of hybrid textures comprising a vanishing minor (or element) and two equal elements (or cofactors) in light neutrino mass matrix . Such type of texture structures leads to sixty phenomenological cases each, out of which only fifty-six are viable with texture containing a vanishing minor and an equality between the elements in , while fifty are found to be viable with texture containing a vanishing element and an equality of cofactors in under the current experimental test at 3 confidence level. Detailed numerical analysis of all the possible cases has been presented.

#### 1. Introduction

During the last two decades, our knowledge regarding the neutrino sector has been enriched to a great extent, thanks to solar, atmospheric, reactor, and accelerator based experiments which convincingly reveal that neutrinos have nonzero and nondegenerate masses and can convert from one flavor to another. While the developments over the past two decades have brought out a coherent picture of neutrino mixing, there are still several intriguing issues without which our understanding of neutrino physics remains incomplete. For instance, the present available data does not throw any light on the neutrino mass spectrum, which may be normal/inverted and may even be degenerate. In addition, nature of neutrino mass whether Dirac or Majorana particle, determination of absolute neutrino mass, leptonic CP violation, and Dirac CP phase are still open issues. Also the information regarding the lightest neutrino mass has to be sharpened further to pinpoint the specific possibility of neutrino mass spectrum.

After the precise measurement of reactor mixing angle in T2K, MINOS, Double Chooz, Daya Bay, and RENO experiments [1–5], five parameters in the neutrino sector have been well measured by neutrino oscillation experiments. In general, there are nine parameters in the lightest neutrinos mass matrix. The remaining four unknown parameters may be taken as the lightest neutrino mass, the Dirac CP-violating phase, and two Majorana phases. The Dirac CP-violating phase is expected to be measured in future long baseline neutrino experiments, and the lightest mass can be determined from beta decay and cosmological experiments. If neutrinoless double-beta decay () is detected, a combination of the two Majorana phases can also be probed. Clearly, the currently available data on neutrino masses and mixing are insufficient for an unambiguous reconstruction of neutrino mass matrices.

In the lack of a convincing fermion flavor theory, several phenomenological ansatz have been proposed in the literature as some elements of neutrino mass matrix are considered to be zero or equal [6–26] or some cofactors of neutrino mass matrix are considered to be either zero or equal [6, 27–35]. The main motivation for invoking different mass matrix ansatz is to relate fermion masses and mixing angles in a testable manner which reduces the number of free parameters in the neutrino mass matrix. In particular, mass matrices with zero textures (or cofactors) have been extensively studied [10–26, 29–35] due to their connections to flavor symmetries. In addition, texture specific mass matrices with one zero element (or minor) and an equality between two independent elements (or cofactors) have also been studied in the literature [7–9, 27, 28]. Out of sixty possibilities, only fifty-four are found to be compatible with the neutrino oscillation data [9] for texture structures having one zero element and an equal matrix elements in the neutrino mass matrix (also known as hybrid texture), while for texture with one vanishing minor and an equal cofactors in the neutrino mass matrix (also known as inverse hybrid texture) only fifty-two cases are able to survive the data [27, 28].

In the present paper, we propose the novel possibilities of hybrid textures where we assume one texture zero and an equality between the cofactors (referred as type X) or one zero minor and an equality between the elements (referred as type Y) in the Majorana neutrino mass matrix . Such type of texture structures sets two conditions on the parameter space and hence reduces the number of free parameters to seven. Therefore the proposed texture structures are as predictive as texture two zeros and any other hybrid textures.

In [6], it is demonstrated that an equality between the elements of can be realized through type-II seesaw mechanism [36–40] while an equality between cofactors of can be generated from type-I seesaw mechanism [41–44]. The zeros element (or minor) in can be obtained using flavor symmetry [29–35, 45]. Therefore the viable cases of proposed hybrid texture can be realized within the framework of seesaw mechanism.

In the present work, we have systematically investigated all the of sixty possible cases belonging to type X and type Y structures, respectively. We have studied the implication of these textures for Dirac CP-violating phase () and two Majorana phases (). We, also, calculate the effective Majorana mass and lowest neutrino mass for all viable hybrid textures belonging to type X and type Y structures. In addition, we present the correlation plots between different parameters of the hybrid textures of neutrinos for allowed ranges of the known parameters.

The layout of the paper is planned as follows: 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.

#### 2. Methodology

Before proceeding further, we briefly underline the methodology relating the elements of the mass matrices to those of the mixing matrix. In the flavor basis, where the charged lepton mass matrix is diagonal, the Majorana neutrino mass matrix can be expressed aswhere 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 (1) can be rewritten aswhere , , and For the present analysis, we consider the following parameterization of [46]:where and . Here, is a 3 3 unitary matrix consisting of three flavor mixing angles () and one Dirac CP-violating phase .

For the illustration of type X and Y structures, we consider a case , satisfying following conditions:andfor type X, while in case of type Y, it containsandorwhere denotes cofactor corresponding to row and column. Then can be denoted in a matrix form aswhere “” stands for nonzero and equal elements (or cofactors), while “0” stands for vanishing element (or minor) in neutrino mass matrix. “” stands for arbitrary elements.

##### 2.1. One Vanishing Minor with Two Equal Elements of

Using (1), any element in the neutrino mass matrix can be expressed in terms of mixing matrix elements aswhere run over e, , and , and is phase factor.

The existence of a zero minor in the Majorana neutrino mass matrix impliesThe above condition yields a complex equation asIt is observed that for any cofactor there is an inherent property as . Thus we can extract this total phase factor from the bracket in (13).

Hence (13) can be rewritten aswherewith () as the cyclic permutation of (1, 2, 3).

On the other hand, the condition of two equal elements in yields following:Equation (16) yields a following complex equation:where and

Orwhere and , and run over , , and .

Equation (18) can be rewritten aswhere ), ), and .

Solving (14) and (19) simultaneously leads to the following complex mass ratio in terms of :andUsing (14), (20), and (21), we obtain the relations for complex mass ratio in terms of andwhere and and . The magnitudes of the two neutrino mass ratios in (20), (21), (22), and (23) are given by and , while the Majorana CP-violating phases and can be given as .

##### 2.2. One Vanishing Element with Two Equal Cofactors of

If one of the elements of is considered zero [e.g., ], we obtain the following constraint equation:orwhere , , and .

The condition for two equal cofactors [e.g., ] in neutrino mass matrix impliesorwhere and due to inherent property of any cofactor. Thus we can writeorwhere

Equation (29) can be rewritten aswherewith () a cyclic permutation of (1, 2, 3).

Solving (25) and (30) simultaneously we obtain the analytical expressions of and

Using (30), (32), and (33), we get the relations for complex mass ratio in terms of andwhere .

The magnitudes of the two neutrino mass ratios are given by and , while the Majorana CP-violating phases and can be given as .

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 [8]The sign of is still unknown: or implies normal mass spectrum (NS) or inverted mass spectrum (IS). The lowest neutrino mass () is for NS and for IS. The experimentally determined solar and atmospheric neutrino mass-squared differences can be related to and asand the three neutrino masses can be determined using following relations:

From the analysis, it is found that cases belonging to type X (or type Y) exhibit the identical phenomenological implications and are related through permutation symmetry [36–40, 46]. 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 by 2-3 permutation. The following pairs among sixty possibilities of type X (or type Y) are related via permutation symmetry:

Clearly we are left with only thirty-two independent cases. It is worthwhile mentioning that , 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 level (CL) are given in Table 1. The effective Majorana mass relevant for neutrinoless double-beta () decay is given byThis effective mass is just the absolute value of component of the neutrino mass matrix. The observation of would establish neutrinos to be Majorana particles. Data from KamLAND-Zen experiment has presented an improved search for neutrinoless double-beta () decay [49–51] and it is found that at 90% (or ) CL. For recent reviews on decay, see [49–51].

In the present analysis, we consider more conservative upper bound on , i.e., at 3 CL [52]. We span the parameter space of input neutrino oscillation parameters () lying in their ranges by randomly generating points of the order of . Since the Dirac CP-violating phase is experimentally unconstrained at level, therefore, we vary within its full possible range []. Using (38) and the experimental inputs on neutrino mixing angles and mass-squared differences, the parameter space of , , and , and can be subsequently constrained.

In Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 we demonstrate the correlations for , , , and cases. Since there are large numbers of viable cases, therefore it is not practically possible to show all the plots. We have simply taken arbitrary independent cases from each category for the purpose of illustration of our results. The predictions regarding three CP-violating phases (), effective neutrino mass , and lowest neutrino mass for all the allowed cases of type X and type Y textures have been encapsulated in Tables 3, 4, 5, and 6. Before proceeding further, it is worth pointing out that the phenomenological results for , , and have been obtained using the two possible solutions of and , respectively [ (20), (21), (22), and (23)]. All the sixty phenomenologically possible cases belonging to type X and type Y texture structures have been divided into six categories A, B, C, D, E, and F (Table 2). Among them a large number of cases are found to overlap in their predictions regarding , , , and and are related via permutation symmetry as pointed out earlier. The main results and the discussion are summarized as follows.

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*Category A*. In Category A, all the ten cases , , , , and are found to be viable with the data at 3 CL for type X structure, and normal mass spectrum (NS) remainS ruled out for all these cases (Table 3). On the other hand, only four , , , and seem to be viable with current oscillation data for type Y, while inverted mass spectrum (IS) is ruled out for these cases.

For both types X and Y, no noticeable constraint has been found on the parameter space of CP-violating phases (). For type X, all the viable cases predict the value of in the range of 0.01eV to 0.05eV. This prediction lies well within the sensitivity limit of neutrinoless double-beta decay experiments [49–51]. On the other hand, for type Y, is predicted to be zero implying that neutrinoless double-beta decay is forbidden. Also the lower bound on lowest neutrino mass () is found to be extremely small ( or less) for all the viable cases of type X and type Y structure (Table 3). For the purpose of illustration, we have presented the correlation plots for indicating the parameter space of , , and lowest neutrino mass ( (Figures 1 and 2).

*Category B (C)*. In Category B, all the ten possible cases are allowed for both type X and type Y structure, respectively, at 3 CL (Table 4). Cases allow both NS and IS for type X, while cases allow both NS and IS for type Y. As mentioned earlier, cases of Category B are related to cases belonging to Category C via permutation symmetry; therefore we can obtain the results for Category C from B by using (41).

Type X cases (IS), (IS), (NS, IS), (IS), (NS, IS), (NS), (IS), (NS, IS), (IS), (IS), (IS), (IS), (NS, IS), (NS, IS), (NS), (IS), (IS), (NS, IS), and (IS) cover literally the complete range of . However, for (NS), (NS), (NS), (NS), (NS), (NS), (NS), (NS), (NS), and (NS) the parameter space of is found to be reduced to an appreciable extent (Table 4).

On the other hand, type Y cases (NS), (NS), (NS, IS), (NS), (NS, IS), (IS), (NS), (NS, IS), (NS), (NS), (NS), (NS), (NS, IS), (NS, IS), (IS), (NS), (NS), (NS, IS), and (NS) cover approximately the complete range of . For (IS), (IS), (IS), (IS), (IS), (IS), (IS), (IS), (IS), (IS), (IS), and (IS), the parameter space of is found to be constricted (Table 4).

From the analysis, it is found that textures , and belonging to type X predict near maximal Dirac type CP violation (i.e., and ) for NS. In addition, the Majorana phases and are found to be very close to for these cases. On the other hand, in case of type Y, and show almost similar constraints on the parameter space for however for opposite mass spectrum (Table 4). In Figures 3, 4, 5, and 6, we have complied the correlation plots for case for both types X and Y comprising the unknown parameters , , and lowest neutrino mass (. As explicitly shown in Figures 3(a), 3(b), and 6(b), and , while . The correlation plots between and have been encapsulated in Figures 3(c), 4(c), 5(c), and 6(c). The plots indicate the strong linear relation correlation between these parameters and, in addition, the lower bound on both the parameters is somewhere in the range from 0.001 to 0.01 eV. The prediction for the allowed space of for all the cases of category B is given in Table 4.

*Category D (F)*. In Category D, only nine cases are acceptable with neutrino oscillation data at 3 CL for both type X and type Y structures, respectively, while case is excluded for both of them (Table 5). Cases , , , , , , and show both NS and IS for type X and type Y, respectively, while and are acceptable for IS (NS) and NS(IS), respectively, in case of type X (type Y) structure. Similarly, the results for cases belonging to Category F can be obtained from Category D since both are related via permutation symmetry. It is found that only nine cases are allowed with data in category F, while is excluded at 3 CL.

Cases (NS), (NS, IS), (IS), (NS), (NS, IS), (NS), (NS), (NS), (NS), (NS), (NS, IS), (IS), (NS), (NS, IS), (NS), (NS), (NS), and (NS) predict literally no constraints on for type X texture. These cases give identical predictions for type Y as well, however for opposite mass ordering. On the other hand, for cases (IS), (IS), (IS), (IS), (IS), (IS), (IS), and (IS), is notably constrained for type X and similar observations have been found for these cases in type Y, however for opposite mass ordering (Table 5).

It is found that textures (IS), (IS), (IS), and (IS) belonging to type X predict near maximal Dirac CP violation (i.e., and ). In addition, the Majorana phases and are found to be very close to for these cases. The similar predictions hold for these cases belonging to type Y structure however for opposite mass spectrum.

The prediction on the allowed range of for all the cases of category D is provided in Table 5. As an illustration, in Figures 7, 8, 9, and 10 we have complied the correlation plots for case for type X and type Y structures. Figures 7(a), 7(b), 10(a), and 10(b) indicate no constraint on for NS(IS) corresponding to type X (type Y) structure at 3 CL. On the other hand, and , while and approach to for IS in case of type X structure (Figure 8). However, similar predictions for have been observed for type Y, however for NS (Figure 9). In Figures 7(c), 8(c), 9(c), and 10(c), we have presented the correlation plots between and indicating the linear correlation.

*Category E*. In Category E, only eight out of ten cases are allowed with experimental data for both type X and type Y structures at 3 CL (Table 6). Cases and are ruled out for both type X and type Y structures. Only and favor both NS and IS, while rest of the cases favor either NS or IS for type X and type Y structure (Table 6). From Table 6, it is clear that , cover literally full range of for type X. Same cases show identical prediction for type Y, however for opposite mass spectrum. For NS, cases belonging to type X predict the lower bound on effective mass to be zero, while for IS, cases predict larger lower bound (greater than 0.01eV) on (Table 6). However for type Y, all these cases show larger lower bound on (eV) for both NS and IS.

For the purpose of illustration, we have presented the correlation plots for indicating the parameter space of , , and lowest neutrino mass ( (Figures 11 and 12). As shown in Figures 11 and 12, remain literally unconstrained for both type X and type Y structures. In addition, there is a linear correlation among , and at 3 CL for type X structure (Table 6). Figure 11(c) indicates the strong linear correlation between and and, in addition, the lower bound on is predicted to be zero.

#### 4. Summary and Conclusion

To summarize, we have discussed the novel possibilities of hybrid textures in the flavor basis wherein the assumption of either one zero minor and an equality between the elements or one zero element and an equality between the cofactors in the Majorana neutrino mass matrix is considered. Out of sixty phenomenologically possible cases, only 56 are found to be viable for type X, while only 50 are viable with the present data for type Y at 3 CL. Therefore, out of 120 only 106 cases are found to be viable with the existing data. However only 38 seems to restrict the parametric space of CP-violating phases , , and , while 16 out of these predict near maximal Dirac CP violation, i.e., . The allowed parameter space for effective mass term related to neutrinoless double-beta decay and lowest neutrino mass term for all viable cases have been carefully studied. The present viable cases may be derived from the discrete symmetry. However the symmetry realization for each case in a systematic and self-consistent way deserves fine-grained research. The viability of these cases suggests that there are still rich unexplored structures of the neutrino mass matrix from both the phenomenological and the theoretical points of view.

To conclude our discussion, we would like to add that the hybrid textures comprising either one zero element and an equality between the elements or one zero minor and an equality between the cofactors lead to 106 viable cases; therefore there are now total 212 viable cases pertaining to the hybrid textures of in the flavor basis. Since most of these cases overlap in their predictions regarding the experimentally undetermined parameters, therefore we expect that only the future long baseline experiments, neutrinoless double-beta decay experiments, and cosmological observations could help us select the appropriate structure of mass texture.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

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