New possibilities of hybrid texture of neutrino mass matrix

In this paper, we investigate the noval possibilities of hybrid textures comprising a vanishing minor (or element) and two equal elements (or cofactors) in light neutrino mass matrix $M_{\nu}$. Such type of texture structures lead 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 $M_{\nu}$, while fifty are found to be viable with texture containing a vanishing element and an equality of cofactors in $M_{\nu}$ under the current experimental test at 3$\sigma$ confidence level. Detailed numerical analysis of all the possible cases have been presented.


Introduction
During the last two decades, our knowledge regarding the neutrino sector has enriched to a great extent. Thanks to solar, atmospheric, reactor and accelerator based experiments which convincingly reveal that neutrinos have non-zero and non degenerate 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 δ are still open issues. Also the information regarding the lightest neutrino mass has to be sharpened further to pin point the specific possibility of neutrino mass spectrum.
After the precise measurement of reactor mixing angle θ 13 in T2K, MINOS, Double Chooz, Daya Bay and RENO experiments [1][2][3][4][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 (0νββ) 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 ansatze have been proposed in the literature such as some elements of neutrino mass matrix are considered to be zero or equal [6][7][8][9][10] or some co-factors of neutrino mass matrix to be either zero or equal [8,11,12]. The main motivation for invoking different mass matrix ansatze 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 [6,11] 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 [7,9,10,12]. Out of sixty possiblities, only fifty four are found to be compatible with the neutrino oscillation data [10] 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 [12].
In the present paper, we propose the noval 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 M ν . 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. There are total sixty such possibilities in each case which have been summarized in Table 1.
In Ref. [8], it is demonstrated that an equality between the elements of M ν can be realized through type-II seesaw mechanism [13] while an equality between cofactors of M ν can be generated from type-I seesaw mechanism [14]. The zeros element(or minor) in M ν can be obtained using Z n flavor symmetry [11,15]. 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 (δ) and two Majorana phases (ρ, σ). 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 3σ 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.

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 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, and where P ν is diagonal phase matrix containing Majorana neutrinos ρ, σ. P l is unobservable phase matrix and depends on phase convention. Eq. 1 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 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 the illustration of type X and Y structures, we consider a case A 1 , satisfying following conditions and for type X, while in case of type Y , it contains and where C ij denotes cofactor corresponding to i th row and j th column. Then A 1 can be denoted in a matrix form as where "∆" stands for nonzero and equal elements (or cofactors), while "0" stands for vanishing element (or minor) in neutrino mass matrix. The "×" stands for arbitrary elements.

One Vanishing minor with Two Equal Elements of M ν
Using Eq. 1, any element M pq in the neutrino mass matrix can be expressed in terms of mixing matrix elements as where p, q run over e, µ and τ , and e i(φp+φq) is phase factor. The existence of a zero minor in the Majorana neutrino mass matrix implies The above condition yields a complex equation as It is observed that for any cofactor there is an inherent property as φ p +φ q +φ r +φ s = φ t + φ u + φ v + φ w . Thus we can extract this total phase factor from the bracket in Eq.13. Hence Eq.13 can be rewritten as where with (i, j, k) as the cyclic permutation of (1, 2, 3). On the other hand, the condition of two equal elements in M ν yields following equation Eq. 16 yields a following complex equation where P 1 = e i(φa+φ b ) and P 2 = e i(φc+φ d ) . or i=1,2,3 where P ≡ P 1 P 2 = e i(a+b−c−d) and a, b, c, d run over e, µ and τ . Eq. 18 can be rewritten as Solving Eqs. 14 and 19 simultaneously lead to the following complex mass ratio in terms of (λ 13 ) ± and Using Eqs. 14, 20 and 21, we obtain the relations for complex mass ratio in terms of (λ 23 ) ± and where . The magnitudes of the two neutrino mass ratios in Eqs. 20, 21, 22 and 23, are given by ξ ± = |(λ 13 ) ± |, ζ ± = |(λ 23 ) ± |, while the Majorana CP-violating phases ρ and σ can be given as ρ = 1 2 arg(λ 13 ) ± , σ = 1 2 arg(λ 23 ) ± .
Since the Dirac CP-violating phase δ is experimentally unconstrained at 3σ level, therefore, we vary δ within its full possible range [0 • , 360 • ]. Using Eq. 38 and the experimental inputs on neutrino mixing angles and mass-squared differences, the parameter space of δ, ρ, σ and |M| ee and m 0 can be subsequently constrained. In Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 we demonstrate the correlations for A 1 , B 2 , D 7 and E 1 cases. Since there are large number 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 (ρ, σ, δ), effective neutrino mass |M| ee and lowest neutrino mass m o for all the allowed cases of type X and type Y textures have been encapsulated in Table 3,4 ,5, 6. Before proceeding further, it is worth pointing out that the phenomenological results for ρ, σ, δ, |M| ee and m o have been obtained using the two possible solutions of λ 13 and λ 23 respectively[Eqs. are summarized as follows: Category A: In Category A, all the ten cases A 1 , A 2 , A 3 , A 4 , A 5 , A 6 , A 7 , A 8 , A 9 , A 10 are found to be viable with the data at 3σ CL for type X structure, and normal mass spectrum (NS) remain ruled out for all these cases [ Table 3]. On the other hand, only four A 1 , A 4 , A 5 , A 6 seem to be viable with current oscillation data for type Y, while inverted mass spectrum (IS) is ruled out for these cases.
For both type X and Y, no noticeable constraint has been found on the parameter space of CP violating phases (ρ, σ, δ). For type X , all the viable cases predict the value of |M| ee in the range of 0.01eV to 0.05eV. This prediction lies well within the sensitivity limit of neutrinoless double beta decay experiments as mentioned above. On the other hand, for type Y, |M| ee is predicted to be zero implying that neutrinoless double beta decay is forbidden. Also the lower bound on lowest neutrino mass (m o ) is found to be extremely small (∼ 10 −3 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 A 1 indicating the parameter space of ρ, σ, δ, Category B (C): In Category B, all the ten possible cases are allowed for both type X and type Y structure, respectively at 3σ CL [Table4]. Cases B 2,3,4,5, 8,9,10 allow both NS as well as IS for type X, while cases B 1,2,3,4,5,8,9,10 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 Eq.41.
and C 10 (NS) the parameter space of δ is found to be reduced to an appreciable extent [ Table 4].
On the other hand, type Y cases  Table 4].
From the analysis, it is found that textures B 2 , B 4 , C 5 and C 7 belonging to type X predict near maximal Dirac type CP violation (i.e. δ ≈ 90 0 and 270 0 ) for NS. In addition, the Majorana phases ρ and σ are found to be very close to 0 0 for these cases. On the other hand, in case of type Y, B 1 , B 2 , B 4 , B 6 , C 1 , C 4 , C 5 and C 7 show almost similar constraints on the parameter space for δ however for opposite mass spectrum [ Table 4]. In Figs. 3,4, 5, 6, we have complied the correlation plots for case B 2 for both type X and Y comprising the unknown parameters ρ, σ, δ, |M| ee and lowest neutrino mass (m o ). As explicitly shown in Figs. 3(a, b) and 6(a, b), δ ≈ 90 0 and 270 0 , while ρ, σ ≈ 0 0 . The correlation plots between |M| ee and m o have been encapsulated in Figs. 3(c), 4(c), 5(c), 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 |M| ee 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 3σ CL for both type X and type Y structures respectively, while case D 8 is excluded for both of them [ Table 5]. Cases D 1 , D 2 , D 4 , D 5 , D 6 , D 7 , D 9 show both NS and IS for type X and type Y respectively, while D 3 and D 10 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 F 7 is excluded at 3σ CL. F 10 (NS) predict literally no constraints on δ 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 D 6 (IS), D 4 (IS), D 7 (IS), D 9 (IS), F 3 (IS), F 6 (IS), F 8 (IS), F 9 (IS) δ is notably constrained for type X, and similar observation have been found for these cases in type Y, however for opposite mass ordering [ Table 5]. It is found that textures D 7 (IS), D 9 (IS), F 6 (IS) and F 8 (IS) belonging to type X predict near maximal Dirac CP violation (i.e. δ ≈ 90 0 and 270 0 ). In addition, the Majorana phases ρ and σ are found to be very close to 0 0 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 |M| ee for all the cases of category D is provided in Table 5. As an illustration, in Figs. 7, 8, 9, 10 we have complied the correlation plots for case D 7 for type X and type Y structures. Figs. 7(a, b)(10(a, b)) indicate no constraint on δ, ρ, σ for NS(IS) corresponding to type X (type Y) structure at 3σ CL. On the other hand, δ ≈ 90 0 and 270 0 , while ρ and σ approach to 0 0 for IS in case of type X structure [ Fig. 8]. However, similar predictions for δ, ρ, σ have been observed for type Y, however for NS [ Fig.9]. In Figs. 7 (c), 8(c), 9(c), 10(c), we have presented the correlation plots between |M| ee and m 0 indicating the linear correlation.
Category E: In Category E, only eight out of ten cases are allowed with exper- imental data for both type X and type Y structures at 3σ CL [ Table 6]. Cases E 7 and E 8 are ruled out for both type X and type Y structures. Only E 5 and E 10 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 E 1 , E 2 , E 3 , E 4 , E 5 , E 10 cover literally full range of δ for type X. Same cases show identical prediction for type Y, however for opposite mass spectrum. For NS, cases E 1 , E 2 , E 5 , E 9 , E 10 belonging to type X predict the lower bound on effective mass |M| ee to be zero , while for IS, cases E 3 , E 4 , E 5 , E 10 predict larger lower bound (greater than 0.01eV) on |M| ee [ Table 6]. However for type Y, all these cases show larger lower bound on |M| ee (≥ 0.01eV) for both NS and IS. For the purpose of illustration, we have presented the correlation plots for E 1 indicating the parameter space of ρ, σ, δ, |M| ee and lowest neutrino mass (m o ) [Figs. 11,12]. As shown in Figs.11, 12, ρ, σ, δ remain literally unconstrained for both type X and type Y structures. In addition, there is a linear correlation among ρ, σ and δ at 3σ CL for type X structure [ Table 6]. Fig. 11(c) indicates the strong linear correlation between |M| ee and m o and in addition, the lower bound on |M| ee is predicted to be zero.

Summary and Conclusion
To summarize, we have discussed the noval 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 3σ CL. Therefore, out of 120 only 106 cases are found to be viable with the existing data. However only 38 seems to restrict the parameteric space of CP violating phases δ, ρ, σ, while 16 out of these predict near maximal Dirac CP violation i.e. δ ≃ 90 0 , 270 0 . The allowed parameter space for effective mass term |M| ee related to neutrinoless double beta decay as well as 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 deserve 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 theoretical points of view.
To conclude our discussion, we would like 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 M ν 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 longbaseline experiments, neutrinoless double beta decay experiments and cosmological observations could help us to select the appropriate structure of mass texture.              Table 3: The allowed ranges of Dirac CP-violating phase δ, the Majorana phases ρ, σ, effective neutrino mass |M| ee , and lowest neutrino mass m 0 for the experimentally allowed cases of Category A at 3σ CL. The predictions corresponding to (λ 13 ) − and (λ 23 ) − neutrino mass ratios have been put into brackets.     Table 5: The allowed ranges of Dirac CP-violating phase δ, the Majorana phases ρ, σ, effective neutrino mass |M| ee , and lowest neutrino mass m 0 for the experimentally allowed cases of Category D(F) at 3σ CL. The predictions corresponding to (λ 13 ) − and (λ 23 ) − neutrino mass ratios have been put into brackets.  Table 6: The allowed ranges of Dirac CP-violating phase δ, the Majorana phases ρ, σ, effective neutrino mass |M| ee , and lowest neutrino mass m 0 for the experimentally allowed cases of Category E at 3σ CL. The predictions corresponding to (λ 13 ) − and (λ 23 ) − neutrino mass ratios have been put into brackets.