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Research Article | Open Access
New Possibilities of Hybrid Texture of Neutrino Mass Matrix
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.
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  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 , 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.
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 :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).
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 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 . 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.