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

Authors: Madan Singh
Journal: Advances in High Energy Physics
 (2018)

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## 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 3σ CL.


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## Body

## 1. Introduction

In leptonic sector, the reactor mixing angle (θ13) 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 θ23, which may be greater than or less than or equal to 450. The determination of the nature of neutrinos whether Dirac or Majorana also remains an open question. The observation of neutrinoless double beta (0νββ) decay would eventually establish the Majorana nature of neutrinos.The effective Majorana mass term related to0νββ decay can be expressed as(1)Mee=m1c122c132e2iρ+m2s122c132e2iσ+m3s132.Data from KamLAND-Zen experiment has presented an improved search for neutrinoless double-beta (0νββ) decay [7] and it is found that Mee<(0.061–0.165)eV at 90% (or <2σ) CL. For recent reviews on 0νββ 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 massMee on 1TEE and 1TEC structures of neutrino mass matrix, while taking into account the assumptions of [28, 29]. The consideration of large Mee is motivated by the extensive search for this parameter in the ongoing 0νββ experiments. The implication of large Mee 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 θ23 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 3σ 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 Section2, 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(Mν) contains nine parameters which include three neutrino masses (m1, m2, m3), three mixing angles (θ12, θ23, θ13), and three CP violating phases (δ, ρ, σ). In the flavor basis, the Majorana neutrino mass matrix can be expressed as follows:(2)Mν=PlUPνMdiagPνTUTPlT,where Mdiag=diag⁡(m1,m2,m3) is the diagonal matrix of neutrino masses and U is the flavor mixing matrix, and(3)Pν=eiρ000eiσ0001,Pl=eiϕe000eiϕμ000eiϕτ,where Pν is diagonal phase matrix containing Majorana neutrinos ρ,σ. Pl is unobservable phase matrix and depends on phase convention. Equation (2) can be rewritten as(4)Mν=PlUλ1000λ2000λ3UTPlT,where λ1=m1e2iρ,λ2=m2e2iσ,λ3=m3. For the present analysis, we consider the following parameterization of U [20]:(5)U=c12c13s12c13s13-c12s23s13-s12c23e-iδ-s12s23s13+c12c23e-iδs23c13-c12c23s13+s12s23e-iδ-s12c23s13-c12s23e-iδc23c13,where cij=cos⁡θij, sij=sin⁡θij. Here, U is a 3 × 3 unitary matrix consisting of three flavor mixing angles (θ12, θ23, θ13) and one Dirac CP-violating phase δ.For hybrid texture structure (1TEE) ofMν, we can express the ratios of neutrino mass eigenvalues in terms of the mixing matrix elements as [21](6)λ1λ3=PUa3Ub3Uα2Uβ2-Ua2Ub2Uα3Uβ3+Ua2Ub2Uc3Ud3-Ua3Ub3Uc2Ud2PUa2Ub2Uα1Uβ1-Ua1Ub1Uα2Uβ2+Ua1Ub1Uc2Ud2-Ua2Ub2Uc1Ud1,λ2λ3=PUa1Ub1Uα3Uβ3-Ua3Ub3Uα1Uβ1+Ua3Ub3Uc1Ud1-Ua1Ub1Uc3Ud3PUa2Ub2Uα1Uβ1-Ua1Ub1Uα2Uβ2+Ua1Ub1Uc2Ud2-Ua2Ub2Uc1Ud1,where P=ei(ϕα+ϕβ-ϕc-ϕd) is a phase factor. Similarly, in case of inverse hybrid texture structure (1TEC) of Mν, we can express the ratios of mass eigenvalues as [26, 27] follows:(7)λ1λ3=A1B2-A2B1A2B3-A3B2,λ2λ3=A1B2-A2B1A3B1-A1B3,where(8)Ai=UpjUqjUrkUsk-UtjUujUvkUwk+j⟷k,Bi=-1m+nQUajUbjUckUdk-UejUfjUgkUhk--1m′+n′Ua′jUb′jUc′kUd′k-Ue′jUf′jUg′kUh′k+j⟷k,with (i,j,k) a cyclic permutation of (1, 2, 3) and Q=ei(ϕa+ϕb+ϕc+ϕd-ϕa′-ϕb′-ϕc′-ϕd′) is phase factor.Using the above expressions, we can obtain the magnitude of neutrino mass ratios,α≡λ1/λ3 and β≡λ2/λ3 in each texture structure, and the Majorana phases (ρ,σ) can be given as ρ=1/2arg⁡λ1/λ3 and σ=1/2arg⁡λ2/λ3.The solar and atmospheric mass squared differences (δm2,Δm2), where δm2 corresponds to solar mass-squared difference and Δm2 corresponds to atmospheric mass-squared difference, can be defined as [20](9)δm2=m22-m12,Δm2=m32-12m12+m22.The experimentally determined solar and atmospheric neutrino mass-squared differences can be related to neutrino mass ratios (α,β) as(10)Rν≡δm2Δm2=2β2-α22-β2+α2,and the three neutrino masses can be determined in terms of α, β as(11)m3=δm2β2-α2,m2=m3β,m1=m3α.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 Mν. The corresponding permutation matrix can be given by(12)P23=100001010.With the help of permutation symmetry, one obtains the following relations among the neutrino oscillation parameters:(13)θ12X=θ12Y,θ23X=90∘-θ23Y,θ13X=θ13Y,δX=δY-180∘,where X and Y denote the cases related to 2-3 permutation. The following pair among sixty cases are related via permutation symmetry: (A1,A1);  (A2,A8);  (A3,A7);  (A4,A6); (A5,A5); A9,A10; (B1,C1)(B2,C7);  (B3,C6);  (B4,C5);  (B5,C4); (B6,C3)(B7,C2); (B8,C10)(B9,C9);  (B10,C8);  (D1,F2);  (D2,F1);  (D3,F4); (D4,F3)(D5,F5);  (D6,F9);  (D7,F8);  (D8,F7);  (D9,F6); (D10,F10)(E1,E2);  (E3,E4);  (E5,E5);  (E6,E9);  (E7,E8); (E10,E10).Clearly we are left with only thirty-two independent cases. It is worthwhile to mention that casesA1, A5, E5, and E10 are invariant under the permutations of 2 and 3 rows and columns.


## 3. Numerical Analysis

The experimental constraints on neutrino parameters at 3σ 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 A, B, C, D, and E (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 (θ12,θ23,θ13,δm2) and mass squared differences (Δm2,δ) at 3σ CL [5, 6].Table 1
Current neutrino oscillation parameters from global fits at 3σ confidence level [5, 6]. NO (IO) refers to normal (inverted) neutrino mass ordering.
Parameter Best fit 3σ δm210-5eV2 7.50 7.03–8.09 Δ m 31 2   1 0 - 3  e V 2 (NO) 2.52 2.407–2.643 Δ m 31 2   1 0 - 3  e V 2 (IO) 2.52 2.39–2.63 θ12 33.5 6 ° 31.3°–35.99° θ 23 (NO) 41 . 6 ° 38 . 4 ° – 52 . 8 ° θ 23 (IO) 50 . 0 ° 38 . 8 ° – 53 . 1 ° θ 13 (NO) 8.4 6 ° 7.9 9 ° – 8.9 0 ° θ 13 (IO) 8.4 9 ° 8.0 3 ° – 8.9 3 ° δ (NO) 26 1 ° 0 ° – 36 0 ° δ (IO) 27 7 ° 14 5 ° – 39 1 °Table 2
Sixty phenomenologically possible hybrid texture structures ofMν at 3σ 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.
A B C D E F 1 0 △ △ △ × × △ × × △ 0 △ 0 × × △ × × △ △ 0 △ × × 0 × × △ △ × △ 0 × × × × △ △ × △ × 0 × 0 × △ △ × △ × × × × 0 2 0 △ × △ △ × × × × × 0 △ 0 △ × △ × × △ × 0 × △ × 0 × × △ × △ × 0 × △ × × △ × △ × × 0 △ 0 × △ × △ × × × △ × 0 3 0 × △ × × △ △ △ × × 0 △ 0 × △ △ △ × △ × 0 × × △ 0 △ × △ × 0 × 0 △ 0 △ × △ × × × △ 0 × 0 × △ × × × △ × × × 0 4 0 × × × △ △ × △ × × 0 △ 0 × × △ × △ △ × 0 × × × 0 × △ △ × × × 0 × × × △ △ × × × × 0 × 0 △ △ × × × × △ × △ 0 5 0 × × × △ × × × △ △ 0 × 0 △ × × × × × △ 0 △ △ × 0 × × × △ △ △ 0 × △ × × × △ △ △ × 0 △ 0 × × △ △ △ × × △ × 0 6 0 × × × × △ × △ △ △ 0 × 0 × △ × △ × × △ 0 △ × △ 0 △ × × △ × △ 0 △ × △ × × △ × △ △ 0 × 0 × × △ × △ △ × × × 0 7 0 △ × △ × △ × △ × △ 0 × 0 × × × × △ × △ 0 △ × × 0 × △ × △ × △ 0 × × × △ × △ × △ × 0 × 0 △ × △ × △ × △ × △ 0 8 0 × △ × × × △ × △ × 0 × 0 △ △ × △ × × × 0 × △ △ 0 △ × × × △ × 0 △ △ △ × × × △ × △ 0 △ 0 × × × △ × △ × △ × 0 9 0 × △ × △ × △ × × × 0 × 0 △ × × × △ × × 0 × △ × 0 × △ × × △ × 0 × △ × △ × × △ × × 0 △ 0 △ × × △ × × △ △ △ 0 10 0 △ × △ × × × × △ × 0 × 0 × △ × △ △ × × 0 × × △ 0 △ △ × × × × 0 △ × △ △ × × × × △ 0 × 0 △ × × × × △ △ × △ 0


### 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 largeMee. For the analysis, we have incorporated the assumptions of [28, 29], wherein authors have considered the lower bound on Mee to be large (i.e., Mee>0.08eV). The upper bound on Mee is chosen to be more conservative; that is, Mee<0.5eV at 3σ CL [10]. The input parameters (θ12,θ23,θ13,δm2,Δm2,δ) are generated by the method of random number generation. The three neutrino mixing angles and Dirac-type CP-violating phase δ are varied between 00 to 900 and 00 to 3600, respectively. However, the mass-squared differences (δm2,Δm2) are varied randomly within their 3σ 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 Figures1, 2, 3, 4, 5, and 6, it is explicitly shown that the octant of θ23 is well restricted for B2,C7,D3,F4 of 1TEE and 1TEC texture structures, respectively. However, for the remaining cases, the value of θ23 is unconstrained like other oscillation parameters. Apart from restricting the octant of θ23, the analysis also ensures the quasi-degenerate mass ordering for these cases similar to the observation of [28–30]. From Figures 1(a), 1(b), 3(a), and 3(b), it is clear that, for increasing value of Mee, atmospheric mixing angle θ23 approaches to maximal value for the structure B2 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 B2 and C7 the quadrant of θ23 is already decided without the experimental input of the mixing angles. For 1TEE, we have θ23<450 for NO and θ23>450 for IO, whereas for 1TEC, θ23>450 for NO, while θ23<450 for IO (Figures 2(a), 2(b), 4(a), and 4(b)). Clearly the correlation plots of case B2 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 C7 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 θ23, cases B2 and C7 also predict δ≃900,2700 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 B2 and C7 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 texturesB2 ((a) NO and (b) IO) and C7 for ((c) NO and (d) IO) at 3σ CL for 1TEE. The symbols have their usual meaning. The horizontal line indicates the upper limit on effective neutrino mass term Mee (i.e., Mee<0.165eV) at 90% CL, given in KamLAND-Zen experiment [7].
(a) (b) (c) (d)Figure 2
Correlation plots for texturesB2 ((a) NO and (b) IO) and C7 for ((c) NO and (d) IO) at 3σ CL for 1TEE. The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle θ13<8.90, as given in Table 1.
(a) (b) (c) (d)Figure 3
Correlation plots for texturesB2 ((a) NO and (b) IO) and C7 for ((c) NO and (d) IO) at 3σ CL for 1TEC. The symbols have their usual meaning. The horizontal line indicates the upper limit on effective neutrino mass term Mee (i.e., Mee<0.165eV) at 90% (<2σ)CL, given in KamLAND-Zen experiment [7].
(a) (b) (c) (d)Figure 4
Correlation plots for texturesB2 ((a) NO and (b) IO) and C7 for ((c) NO and (d) IO) at 3σ CL for 1TEC. The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle θ13<8.90, as given in Table 1.
(a) (b) (c) (d)Figure 5
Correlation plots for texturesD3 ((a), (c)) and F4 ((b), (d)) with IO for 1TEE at 3σ CL. The symbols have their usual meaning. In (a) and (b), colored horizontal line indicates the upper limit on effective neutrino mass term Mee (i.e., Mee<0.165eV) at 90 (<2σ)% CL, given in KamLAND-Zen experiment [7]. In (c) and (d), we have shown the upper limit on reactor mixing angle θ13.
(a) (b) (c) (d)Figure 6
Correlation plots for texturesD3 ((a), (c)) and F4 ((b), (d)) with NO for 1TEC at 3σ CL. The symbols have their usual meaning. In (a) and (b), colored horizontal line indicates the upper limit on effective neutrino mass term Mee (i.e., Mee<0.165eV) at 90 (<2σ)% CL, given in KamLAND experiment [7]. In (c) and (d), we have shown the upper limit on reactor mixing angle θ13.
(a) (b) (c) (d)Similarly, casesD3 and F4 of 1TEE also predict near maximal atmospheric mixing angle (θ23) for IO (Figures 5(a) and 5(b)). Interestingly the parameter space of reactor mixing angle θ13 is found to be constrained between 00 and 350 (Figures 5(c) and 5(d)). In Figures 5(c) and 5(d), it is clear that, for the allowed experimental range of θ13 (8.50-9.80), θ23 inches closer to 450. Similar predictions have been noted for cases D3 and F4 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 (θ12,θ23,θ13) and mass squared differences (δm2,Δm2) within their 3σ confidence level. To facilitate the comparison, we have encapsulated the the predictions regarding three CP violating phases (ρ,σ,δ) and neutrino masses m1,2,3 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 (A1,4,5,6) are allowed for 1TEE at 3σ CL, and in addition inverted mass ordering (IO) is ruled out for all these cases. On the other hand, only three (A1,4,6) are allowed for 1TEC with current oscillation data, while normal mass ordering (NO) is ruled out for these cases. For 1TEE, ρ,σ,δ remain unconstrained; however, for 1TEC, only δ remains unconstrained, while Majorana phases (ρ,σ) are restricted near 00 pertaining to viable cases. From Table 3, it is clear that lower bound on lowest neutrino mass (m1 (NO) or m3 (IO)) is nearly equal or less than 1 meV for 1TEE and 1TEC.Table 3
The allowed ranges of Dirac CP-violating phaseδ, the Majorana phases ρ, σ, and three neutrino masses m1, m2, m3 for the experimentally allowed cases of Category A. Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.
Cases 1TEE 1TEC NO IO NO IO A1 ρ = - 9 0 0 – 9 0 0 × × ρ=-0.02770–-0.02200⊕0.02140–0.02740 σ = - 9 0 0 – 9 0 0 × × σ = - 0.027 3 0 – 0.027 1 0 δ=00–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.00147 – 0.0106 × × m 1 = 0.0430 – 0.0534 m 2 = 0.00849 – 0.0139 × × m 2 = 0.0439 – 0.0541 m 3 = 0.0437 – 0.0551 × × m 3 = 0.000904 – 0.00504  A2(A8) × × × ×  A3(A7) × × × ×  A4(A6) ρ=-900–900 × × ρ=-0.02750–-0.02220⊕0.02150–0.02690 σ = - 9 0 0 – 9 0 0 × × σ = - 0.025 0 0 – 0.026 8 0 δ=00–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.00148 – 0.0106 × × m 1 = 0.0431 – 0.0534 m 2 = 0.00850 – 0.0139 × × m 2 = 0.0439 – 0.0540 m 3 = 0.0437 – 0.0551 × × m 3 = 0.000950 – 0.00504  A5(A5) ρ=-900–900 × × × σ = - 9 0 0 – 9 0 0 × × × δ=00–3600 × × × m 1 = 0.00163 – 0.0105 × × × m 2 = 0.00854 – 0.0137 × × × m 3 = 0.0438 – 0.0545 × × ×  A9(A10) × × × ×Category B (C). In Category B, all the ten possible cases are allowed for both 1TEE and 1TEC, respectively, at 3σ CL; however, cases B1,6,7 allow only NO for 1TEE, while the same allow only IO for 1TEC (Table 4). Cases B2,3,4,5,8,9,10 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 C1,2,3 allow only NO for 1TEE, while the same allow IO for 1TEC.Table 4
The allowed ranges of Dirac CP-violating phaseδ, the Majorana phases ρ, σ, and three neutrino masses m1, m2, m3 for the experimentally allowed cases of Category B (C). Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.
Cases 1TEE 1TEC NO IO NO IO B1(C1) ρ = - 9 0 0 – 9 0 0 × × ρ=-0.02770–0.02790 σ = - 9 0 0 – 9 0 0 × × σ = - 0.028 4 0 – 0.028 4 0 δ=00–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.00550 – 0.0298 × × m 1 = 0.0439 – 0.0597 m 2 = 0.0101 – 0.0310 × × m 2 = 0.0447 – 0.0604 m 3 = 0.0443 – 0.0605 × × m 3 = 0.000754 – 0.0297  B2(C7) ρ = - 2.7 6 0 – 2.8 1 0 ρ=-7.740–7.750 ρ = - 9 0 0 – 9 0 0 ρ=-4.460–4.400 σ = - 10.4 8 0 – 10.2 2 0 σ = - 8.8 4 0 – 9.2 2 0 σ = - 9 0 0 – 9 0 0 σ = - 5.4 1 0 – 5.3 9 0 δ=77.30–94.270⊕266.30–2840 δ = 84.0 1 0 – 98 . 7 0 ⊕ 262.1 9 0 – 276 . 6 0 δ = 86.2 2 0 – 273 . 3 0 δ = 81.5 6 0 – 96.4 8 0 ⊕ 263 . 7 0 – 278 . 4 0 m 1 = 0.0235 – 0.314 m 1 = 0.0453 – 0.277 m 1 = 0.00548 – 0.267 m 1 = 0.0507 – 0.419 m 2 = 0.0242 – 0.0310 m 2 = 0.0465 – 0.274 m 2 = 0.00806 – 0.316 m 2 = 0.0508 – 0.419 m 3 = 0.0485 – 0.0315 m 3 = 0.0117 – 0.271 m 3 = 0.0428 – 0.320 m 3 = 0.0249 – 0.414  B3(C6) ρ = - 9 0 0 – - 5 . 9 0 ⊕ 5 . 9 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – - 5 . 9 0 ⊕ 5 . 9 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=7.470–3560 δ = 8 . 3 0 – 17 3 0 ⊕ 18 7 0 – 35 1 0 δ = 5.1 6 0 – 17 8 0 ⊕ 183 . 7 0 – 353 . 6 0 δ = 0 0 – 36 0 0 m 1 = 0.0153 – 0.331 m 1 = 0.0459 – 0.335 m 1 = 0.00504 – 0.485 m 1 = 0.0427 – 0.505 m 2 = 0.0161 – 0.331 m 2 = 0.0461 – 0.334 m 2 = 0.00920 – 0.483 m 2 = 0.0445 – 0.503 m 3 = 0.0445 – 0.0335 m 3 = 0.0136 – 0.330 m 3 = 0.0416 – 0.480 m 3 = 0.00884 – 0.500  B4(C5) ρ = - 19.0 7 0 – 19.7 5 0 ρ=-4.800–4.660 ρ = - 43.1 3 0 – 40.1 2 0 ρ=-16.380–18.80 σ = - 9 0 0 – 9 0 0 σ = - 2.8 3 0 – 3.1 2 0 σ = - 4 3 0 – 40 . 1 0 σ = - 41.4 3 0 – 40.5 9 0 δ=00–1000⊕260.20–3600 δ = 84.6 8 0 – 94 . 7 0 ⊕ 265.0 8 0 – 275 . 5 0 δ = 88.0 4 0 – 124 . 4 0 ⊕ 230.0 9 0 – 27 2 0 δ = 5.5 6 0 – 13.8 2 0 ⊕ 255 . 7 0 – 307 . 3 0 m 1 = 0.00681 – 0.318 m 1 = 0.0488 – 0.314 m 1 = 0.0105 – 0.373 m 1 = 0.0444 – 0.420 m 2 = 0.00969 – 0.0317 m 2 = 0.0486 – 0.312 m 2 = 0.0129 – 0.369 m 2 = 0.0447 – 0.416 m 3 = 0.0421 – 0.0321 m 3 = 0.0202 – 0.310 m 3 = 0.0435 – 0.374 m 3 = 0.0103 – 0.418  B5(C4) ρ = - 20.8 3 0 – 20.7 9 0 ρ=-900–-11.30⊕11.30–900 ρ = - 9 0 0 – 9 0 0 ρ=-24.190–24.450 σ = - 36 . 3 0 – 3 6 0 $ σ = - 9 0 0 – - 14 . 3 0 ⊕ 14 . 8 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 27 . 7 0 – 27 . 4 0 δ=00–177.50⊕1820–3600 δ = 0 0 – 16 8 0 ⊕ 192 . 5 0 – 36 0 0 δ = 9.3 3 0 – 172 . 4 0 ⊕ 187.20 – 350 . 4 0 δ = 0 0 – 179 . 3 0 ⊕ 180 . 6 0 – 36 0 0 m 1 = 0.00965 – 0.330 m 1 = 0.0761 – 0.330 m 1 = 0.0603 – 0.498 m 1 = 0.0440 – 0.498 m 2 = 0.0113 – 0.0330 m 2 = 0.0762 – 0.330 m 2 = 0.0608 – 0.496 m 2 = 0.0447 – 0.499 m 3 = 0.0421 – 0.0332 m 3 = 0.0603 – 0.328 m 3 = 0.0629 – 0.500 m 3 = 0.0103 – 0.499  B6(C3) ρ = - 9 0 0 – 9 0 0 × × ρ=-900–900 σ = - 9 0 0 – 9 0 0 × × σ = - 9 0 0 – 9 0 0 δ=00–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.0147 – 0.324 × × m 1 = 0.0452 – 0.454 m 2 = 0.00161 – 0.324 × × m 2 = 0.0451 – 0.451 m 3 = 0.0445 – 0.324 × × m 3 = 0.00161 – 0.450  B7(C2) ρ = - 26.2 5 0 – 26.4 3 0 × × ρ=-30.920–30.420 σ = - 50.2 8 0 – 50 . 2 0 × × σ = - 41 . 9 0 – 43 . 8 0 δ=00–177.50⊕1820–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.00883 – 0.331 × × m 1 = 0.0447 – 0.498 m 2 = 0.00121 – 0.329 × × m 2 = 0.0451 – 0.498 m 3 = 0.0421 – 0.334 × × m 3 = 0.00804 – 0.498  B8(C10) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–166.50⊕196.50–3600 δ = 45.0 5 0 – 13 5 0 ⊕ 224 . 5 0 – 314 . 6 0 δ = 45.8 6 0 – 315.0 5 0 δ = 0 0 – 36 0 0 m 1 = 0.00127 – 0.279 m 1 = 0.0418 – 0.279 m 1 = 0.00498 – 0.381 m 1 = 0.0427 – 0.303 m 2 = 0.00691 – 0.277 m 2 = 0.0429 – 0.307 m 2 = 0.00736 – 0.382 m 2 = 0.0428 – 0.302 m 3 = 0.0409 – 0.280 m 3 = 0.00398 – 0.305 m 3 = 0.0407 – 0.384 m 3 = 0.00342 – 0.298  B9(C9) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=5.550–3540 δ = 1 2 0 – 171 . 7 0 ⊕ 189 . 7 0 – 352 . 7 0 δ=8.320–175.90⊕1860–350.470 δ = 0 0 – 36 0 0 m 1 = 0.00338 – 0.279 m 1 = 0.0435 – 0.319 m 1 = 0.00622 – 0.487 m 1 = 0.0447 – 0.486 m 2 = 0.00831 – 0.277 m 2 = 0.0445 – 0.319 m 2 = 0.00681 – 0.486 m 2 = 0.0451 – 0.487 m 3 = 0.0423 – 0.282 m 3 = 0.00398 – 0.317 m 3 = 0.00914 – 0.486 m 3 = 0.00458 – 0.480  B10(C8) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–3600 δ=44.640–136.50⊕224.60–314.20 δ=1.920–313.040 δ = 0 0 – 36 0 0 m 1 = 0.00163 – 0.294 m 1 = 0.0425 – 0.277 m 1 = 0.00516 – 0.348 m 1 = 0.0427 – 0.365 m 2 = 0.00691 – 0.293 m 2 = 0.0423 – 0.275 m 2 = 0.00806 – 0.346 m 2 = 0.0453 – 0.365 m 3 = 0.0409 – 0.299 m 3 = 0.00380 – 0.272 m 3 = 0.0416 – 0.348 m 3 = 0.00879 – 0.391TexturesB1 (NO), B3 (NO, IO), B5 (NO, IO), B6 (NO), B7 (NO), B8 (NO), B10 (NO), C1 (NO), C2 (NO), C3 (NO), C4 (NO, IO), C6 (NO, IO), C8 (NO), and C10 (NO) held nearly no constraint on Dirac CP violating phase (δ) for 1TEE and 1TEC, respectively, but with opposite neutrino mass ordering (Table 4). Only cases B2 (NO), B4 (IO), C7 (NO), and C5 (IO) for 1TEE and B2 (IO), B4 (NO), C7 (IO), and C5 (NO) for 1TEC show significant reduction in the parameter space of δ. It is found that δ is restricted near 900 and 2700 for 1TEE and 1TEC, respectively (Table 4). These predictions are significant considering the latest hint on δ near 2700 [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 3σ CL for 1TEE and 1TEC, respectively (Table 5). However cases D1,2,4,5,6,7,9 favor both NO and IO for 1TEE and 1TEC, while D3, D8, and D10 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 phaseδ, the Majorana phases ρ, σ, and three neutrino masses m1, m2, m3 for the experimentally allowed cases of Category D (F). Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.
Cases 1TEE 1TEC NO IO NO IO D1(F2) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=30.1850–322.60 δ = 0 0 – 36 0 0 δ=00–3600 δ=31.790–318.660 m 1 = 0.0541 – 0.224 m 1 = 0.00432 – 0.252 m 1 = 0.00331 – 0.603 m 1 = 0.0731 – 0.501 m 2 = 0.0538 – 0.224 m 2 = 0.00437 – 0.251 m 2 = 0.00800 – 0.606 m 2 = 0.0719 – 0.500 m 3 = 0.0679 – 0.230 m 3 = 0.00343 – 0.248 m 3 = 0.0371 – 0.604 m 3 = 0.0593 – 0.496  D2(F1) ρ = - 50.2 8 0 – 60.4 8 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-63.30–640 σ=-900–900 σ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 89.3 7 0 – 86.2 5 0 δ=87.190–297.40 δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 δ = 70.5 0 0 – 295.7 7 0 m 1 = 0.124 – 0.293 m 1 = 0.0422 – 0.263 m 1 = 0.00542 – 0.0277 m 1 = 0.128 – 0.581 m 2 = 0.125 – 0.294 m 2 = 0.0430 – 0.262 m 2 = 0.00993 – 0.0291 m 2 = 0.130 – 0.582 m 3 = 0.132 – 0.295 m 3 = 0.00552 – 0.257 m 3 = 0.0440 – 0.0600 m 3 = 0.120 – 0.579  D3(F4) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0439 – 0.308 m 1 = 0.0161 – 0.474 × × m 2 = 0.0445 – 0.308 m 2 = 0.0164 – 0.471 × × m 3 = 0.00888 – 0.304 m 3 = 0.0416 – 0.469 ×  D4(F3) ρ = - 58.7 1 0 – 59.3 5 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-61.40–61.50 σ = - 88.9 0 0 – 87.1 8 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=75.870–285.580 δ = 0 0 – 36 0 0 δ = 18.1 2 0 – 318.7 5 0 δ=77.860–286.150 m 1 = 0.0574 – 0.287 m 1 = 0.00429 – 0.254 m 1 = 0.0102 – 0.795 m 1 = 0.0711 – 0.424 m 2 = 0.0575 – 0.286 m 2 = 0.00430 – 0.254 m 2 = 0.0147 – 0.796 m 2 = 0.0740 – 0.424 m 3 = 0.0679 – 0.291 m 3 = 0.00969 – 0.202 m 3 = 0.0369 – 0.790 m 3 = 0.0564 – 0.418  D5(F5) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–3600 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 m 1 = 0.0337 – 0.334 m 1 = 0.0431 – 0.335 m 1 = 0.00255 – 0.509 m 1 = 0.0538 – 0.505 m 2 = 0.0338 – 0.334 m 2 = 0.0434 – 0.334 m 2 = 0.00370 – 0.505 m 2 = 0.0539 – 0.506 m 3 = 0.0541 – 0.334 m 3 = 0.00940 – 0.330 m 3 = 0.0351 – 0.510 m 3 = 0.0305 – 0.500  D6(F9) ρ = - 65.0 9 0 – 58.4 8 0 ρ = - 9 0 0 – 9 0 0 ρ = - 23.0 3 0 – - 7.7 4 0 ⊕ 7.6 6 0 – 22.1 4 0 ρ=-18.420–-6.560⊕6.770–18.270 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 68.5 3 0 – - 38.7 5 0 ⊕ 37.8 6 0 – 69 . 0 0 σ = - 71 . 1 0 – - 46 . 2 0 ⊕ 46 . 2 0 – 70.4 8 0 δ=64.20–288.10 δ = 0 0 – 36 0 0 δ = 109.6 8 0 – 150.0 5 0 ⊕ 210 . 4 0 – 250 . 2 0 δ = 118.1 9 0 – 154 . 2 0 ⊕ 206 . 7 0 – 242 . 3 0 m 1 = 0.0974 – 0.323 m1=0.0446–0.277 m1=0.0202–0.456 m1=0.108–0.398 m 2 = 0.0965 – 0.0321 m2=0.0448–0.277 m2=0.0204–0.456 m2=0.109–0.396 m 3 = 0.107 – 0.0326 m3=0.00888–0.324 m3=0.0451–0.456 m3=0.0962–0.396  D7(F8) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–3600 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 m 1 = 0.0478 – 0.273 m 1 = 0.0439 – 0.333 m 1 = 0.00663 – 0.436 m 1 = 0.0647 – 0.437 m 2 = 0.0479 – 0.273 m 2 = 0.0454 – 0.333 m 2 = 0.0103 – 0.434 m 2 = 0.0649 – 0.438 m 3 = 0.0644 – 0.300 m 3 = 0.00806 – 0.330 m 3 = 0.0416 – 0.438 m 3 = 0.0439 – 0.491  D8(F7) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0451 – 0.0938 m 1 = 0.00941 – 0.0803 × × m 2 = 0.0459 – 0.0941 m 2 = 0.0125 – 0.0804 × × m 3 = 0.00121 – 0.0790 m 3 = 0.0445 – 0.0953 ×  D9(F6) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–3600 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 m 1 = 0.0492 – 0.300 m 1 = 0.0455 – 0.310 m 1 = 0.00769 – 0.436 m 1 = 0.0647 – 0.411 m 2 = 0.0496 – 0.300 m 2 = 0.0456 – 0.310 m 2 = 0.00932 – 0.434 m 2 = 0.0638 – 0.412 m 3 = 0.0638 – 0.303 m 3 = 0.00969 – 0.306 m 3 = 0.0426 – 0.438 m 3 = 0.0458 – 0.407  D10(F10) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0432 – 0.102 m 1 = 0.00201 – 0.0793 × × m 2 = 0.0442 – 0.102 m 2 = 0.00870 – 0.0795 × × m 3 = 0.00301 – 0.0901 m 3 = 0.0435 – 0.0932 ×CasesD1 (IO), D2 (IO), D3 (IO), D4 (IO), D5 (NO, IO), D6 (IO), D7 (NO, IO), D8 (IO), D9 (NO, IO), D10 (IO), F1 (IO), F2 (IO), F3 (IO), F4 (IO), F5 (NO, IO), F6 (IO), F7 (NO, IO), F8 (IO), F9 (NO, IO), and F10 (IO) predict literally no constraints on δ for 1TEE. These cases give identical predictions for 1TEC, but for opposite mass ordering. For cases D1 (NO), D2 (NO), D4 (NO), D6 (NO), F1 (NO), F2 (NO), F4 (NO), and F6 (NO), the parameter space of δ is found to be well constrained for 1TEE. These cases give similar predictions regarding the parameter space of δ for 1TEC, but for IO.Category E. In Category E, all the ten possible cases are allowed for 1TEE at 3σ CL, while only nine other than E5 are acceptable in case of 1TEC (Table 6). Cases E1,2,3,4,6,7,8,9 allow only inverted mass ordering (IO) for 1TEE, while the same textures allow only normal mass ordering (NO) for 1TEC. Cases E5 and E10 allow both NO and IO for 1TEE; however E5 is ruled out for both NO and IO for 1TEC at 3σ CL. Similar to cases belonging to Category D, E1,2,3,4,5,6,7,8,9 (IO) cover full range of ρ,σ,δ for 1TEE, whereas the same cases (except E5) give identical predictions for 1TEC, but for NO. For E5 (NO) and E10 (NO), phases ρ,σ,δ are somewhat restricted at 3σ CL for 1TEE, while only for E10 (IO), the parameter space of ρ,σ,δ seems to be restricted for 1TEC (Table 6).Table 6
The allowed ranges of Dirac CP-violating phaseδ, the Majorana phases ρ, σ, and three neutrino masses m1, m2, m3 for the experimentally allowed cases of Category E. Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.
Cases 1TEE 1TEC NO IO NO IO E1(E2) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0511 – 0.0804 m 1 = 0.00283 – 0.0629 × × m 2 = 0.0516 – 0.0805 m 2 = 0.00878 – 0.0634 × × m 3 = 0.0264 – 0.0678 m 3 = 0.0440 – 0.0809 ×  E3(E4) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0439 – 0.333 m 1 = 0.00914 – 0.499 × × m 2 = 0.0457 – 0.333 m 2 = 0.00115 – 0.504 × × m 3 = 0.00888 – 0.333 m 3 = 0.0432 – 0.500 ×  E5 ρ = - 9 0 0 – - 156 ⊕ 29.3 – 9 0 0 ρ=-900–900 × × ρ=-900–-156⊕29.3–900 σ=-900–900 × × δ=29.30–153.50⊕206.50–3340 δ=00–3600 × × m 1 = 0.130 – 0.332 m 1 = 0.0439 – 0.269 × × m 2 = 0.132 – 0.330 m 2 = 0.0451 – 0.270 × × m 3 = 0.137 – 0.334 m 3 = 0.0103 – 0.262 × ×  E6(E9) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 36 0 0 × × m 1 = 0.0511 – 0.333 m 1 = 0.00914 – 0.499 × × m 2 = 0.0517 – 0.333 m 2 = 0.00115 – 0.504 × × m 3 = 0.0273 – 0.333 m 3 = 0.0432 – 0.500 ×  E7(E8) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 36 0 0 × × m 1 = 0.0511 – 0.0908 m 1 = 0.0225 – 0.0469 × × m 2 = 0.0517 – 0.0911 m 2 = 0.0238 – 0.0480 × × m 3 = 0.0273 – 0.0746 m 3 = 0.0494 – 0.0708 ×  E10 ρ = - 9 0 0 – - 28 . 6 0 ⊕ 26.0 1 0 – 9 0 0 ρ = - 18.1 5 0 – 18.0 8 0 ρ = - 9 0 0 – 9 0 0 ρ=-900–-190⊕19.50–900 σ=-900–-28.60⊕26.010–900 σ = - 9 0 0 – - 6.4 1 0 ⊕ 6.0 9 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – - 1 9 0 ⊕ 19 . 5 0 – 9 0 0 δ=25.350–151.920⊕206.970–332.30 δ = 0 0 – 17 7 0 ⊕ 184 . 3 0 – 357 . 5 0 δ = 0 0 – 36 0 0 δ = 23.7 5 0 – 161.9 7 0 ⊕ 198 . 9 0 – 33 9 0 m 1 = 0.136 – 0.331 m1=0.0431–0.334 m1=0.00156–0.481 m1=0.141–0.481 m 2 = 0.137 – 0.331 m2=0.0445–0.334 m2=0.00571–0.480 m2=0.142–0.492 m 3 = 0.143 – 0.334 m3=0.00969–0.330 m3=0.0404–0.484 m3=0.133–0.491To summarize our discussion, we have investigated all the viable cases of 1TEE and 1TEC texture structures in the limit of large effective neutrino massMee. 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 3σ 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 δ for most of the cases belonging to 1TEE and 1TEC remains unrestricted, while only eight cases show maximal restriction for δ. 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 Mee term, which would, in turn, either confirm or rule out our assumption of large Mee.


## 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 largeMee. For the analysis, we have incorporated the assumptions of [28, 29], wherein authors have considered the lower bound on Mee to be large (i.e., Mee>0.08eV). The upper bound on Mee is chosen to be more conservative; that is, Mee<0.5eV at 3σ CL [10]. The input parameters (θ12,θ23,θ13,δm2,Δm2,δ) are generated by the method of random number generation. The three neutrino mixing angles and Dirac-type CP-violating phase δ are varied between 00 to 900 and 00 to 3600, respectively. However, the mass-squared differences (δm2,Δm2) are varied randomly within their 3σ 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 Figures1, 2, 3, 4, 5, and 6, it is explicitly shown that the octant of θ23 is well restricted for B2,C7,D3,F4 of 1TEE and 1TEC texture structures, respectively. However, for the remaining cases, the value of θ23 is unconstrained like other oscillation parameters. Apart from restricting the octant of θ23, the analysis also ensures the quasi-degenerate mass ordering for these cases similar to the observation of [28–30]. From Figures 1(a), 1(b), 3(a), and 3(b), it is clear that, for increasing value of Mee, atmospheric mixing angle θ23 approaches to maximal value for the structure B2 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 B2 and C7 the quadrant of θ23 is already decided without the experimental input of the mixing angles. For 1TEE, we have θ23<450 for NO and θ23>450 for IO, whereas for 1TEC, θ23>450 for NO, while θ23<450 for IO (Figures 2(a), 2(b), 4(a), and 4(b)). Clearly the correlation plots of case B2 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 C7 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 θ23, cases B2 and C7 also predict δ≃900,2700 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 B2 and C7 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 texturesB2 ((a) NO and (b) IO) and C7 for ((c) NO and (d) IO) at 3σ CL for 1TEE. The symbols have their usual meaning. The horizontal line indicates the upper limit on effective neutrino mass term Mee (i.e., Mee<0.165eV) at 90% CL, given in KamLAND-Zen experiment [7].
(a) (b) (c) (d)Figure 2
Correlation plots for texturesB2 ((a) NO and (b) IO) and C7 for ((c) NO and (d) IO) at 3σ CL for 1TEE. The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle θ13<8.90, as given in Table 1.
(a) (b) (c) (d)Figure 3
Correlation plots for texturesB2 ((a) NO and (b) IO) and C7 for ((c) NO and (d) IO) at 3σ CL for 1TEC. The symbols have their usual meaning. The horizontal line indicates the upper limit on effective neutrino mass term Mee (i.e., Mee<0.165eV) at 90% (<2σ)CL, given in KamLAND-Zen experiment [7].
(a) (b) (c) (d)Figure 4
Correlation plots for texturesB2 ((a) NO and (b) IO) and C7 for ((c) NO and (d) IO) at 3σ CL for 1TEC. The symbols have their usual meaning. The horizontal line indicates the upper limit on reactor mixing angle θ13<8.90, as given in Table 1.
(a) (b) (c) (d)Figure 5
Correlation plots for texturesD3 ((a), (c)) and F4 ((b), (d)) with IO for 1TEE at 3σ CL. The symbols have their usual meaning. In (a) and (b), colored horizontal line indicates the upper limit on effective neutrino mass term Mee (i.e., Mee<0.165eV) at 90 (<2σ)% CL, given in KamLAND-Zen experiment [7]. In (c) and (d), we have shown the upper limit on reactor mixing angle θ13.
(a) (b) (c) (d)Figure 6
Correlation plots for texturesD3 ((a), (c)) and F4 ((b), (d)) with NO for 1TEC at 3σ CL. The symbols have their usual meaning. In (a) and (b), colored horizontal line indicates the upper limit on effective neutrino mass term Mee (i.e., Mee<0.165eV) at 90 (<2σ)% CL, given in KamLAND experiment [7]. In (c) and (d), we have shown the upper limit on reactor mixing angle θ13.
(a) (b) (c) (d)Similarly, casesD3 and F4 of 1TEE also predict near maximal atmospheric mixing angle (θ23) for IO (Figures 5(a) and 5(b)). Interestingly the parameter space of reactor mixing angle θ13 is found to be constrained between 00 and 350 (Figures 5(c) and 5(d)). In Figures 5(c) and 5(d), it is clear that, for the allowed experimental range of θ13 (8.50-9.80), θ23 inches closer to 450. Similar predictions have been noted for cases D3 and F4 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 (θ12,θ23,θ13) and mass squared differences (δm2,Δm2) within their 3σ confidence level. To facilitate the comparison, we have encapsulated the the predictions regarding three CP violating phases (ρ,σ,δ) and neutrino masses m1,2,3 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 (A1,4,5,6) are allowed for 1TEE at 3σ CL, and in addition inverted mass ordering (IO) is ruled out for all these cases. On the other hand, only three (A1,4,6) are allowed for 1TEC with current oscillation data, while normal mass ordering (NO) is ruled out for these cases. For 1TEE, ρ,σ,δ remain unconstrained; however, for 1TEC, only δ remains unconstrained, while Majorana phases (ρ,σ) are restricted near 00 pertaining to viable cases. From Table 3, it is clear that lower bound on lowest neutrino mass (m1 (NO) or m3 (IO)) is nearly equal or less than 1 meV for 1TEE and 1TEC.Table 3
The allowed ranges of Dirac CP-violating phaseδ, the Majorana phases ρ, σ, and three neutrino masses m1, m2, m3 for the experimentally allowed cases of Category A. Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.
Cases 1TEE 1TEC NO IO NO IO A1 ρ = - 9 0 0 – 9 0 0 × × ρ=-0.02770–-0.02200⊕0.02140–0.02740 σ = - 9 0 0 – 9 0 0 × × σ = - 0.027 3 0 – 0.027 1 0 δ=00–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.00147 – 0.0106 × × m 1 = 0.0430 – 0.0534 m 2 = 0.00849 – 0.0139 × × m 2 = 0.0439 – 0.0541 m 3 = 0.0437 – 0.0551 × × m 3 = 0.000904 – 0.00504  A2(A8) × × × ×  A3(A7) × × × ×  A4(A6) ρ=-900–900 × × ρ=-0.02750–-0.02220⊕0.02150–0.02690 σ = - 9 0 0 – 9 0 0 × × σ = - 0.025 0 0 – 0.026 8 0 δ=00–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.00148 – 0.0106 × × m 1 = 0.0431 – 0.0534 m 2 = 0.00850 – 0.0139 × × m 2 = 0.0439 – 0.0540 m 3 = 0.0437 – 0.0551 × × m 3 = 0.000950 – 0.00504  A5(A5) ρ=-900–900 × × × σ = - 9 0 0 – 9 0 0 × × × δ=00–3600 × × × m 1 = 0.00163 – 0.0105 × × × m 2 = 0.00854 – 0.0137 × × × m 3 = 0.0438 – 0.0545 × × ×  A9(A10) × × × ×Category B (C). In Category B, all the ten possible cases are allowed for both 1TEE and 1TEC, respectively, at 3σ CL; however, cases B1,6,7 allow only NO for 1TEE, while the same allow only IO for 1TEC (Table 4). Cases B2,3,4,5,8,9,10 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 C1,2,3 allow only NO for 1TEE, while the same allow IO for 1TEC.Table 4
The allowed ranges of Dirac CP-violating phaseδ, the Majorana phases ρ, σ, and three neutrino masses m1, m2, m3 for the experimentally allowed cases of Category B (C). Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.
Cases 1TEE 1TEC NO IO NO IO B1(C1) ρ = - 9 0 0 – 9 0 0 × × ρ=-0.02770–0.02790 σ = - 9 0 0 – 9 0 0 × × σ = - 0.028 4 0 – 0.028 4 0 δ=00–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.00550 – 0.0298 × × m 1 = 0.0439 – 0.0597 m 2 = 0.0101 – 0.0310 × × m 2 = 0.0447 – 0.0604 m 3 = 0.0443 – 0.0605 × × m 3 = 0.000754 – 0.0297  B2(C7) ρ = - 2.7 6 0 – 2.8 1 0 ρ=-7.740–7.750 ρ = - 9 0 0 – 9 0 0 ρ=-4.460–4.400 σ = - 10.4 8 0 – 10.2 2 0 σ = - 8.8 4 0 – 9.2 2 0 σ = - 9 0 0 – 9 0 0 σ = - 5.4 1 0 – 5.3 9 0 δ=77.30–94.270⊕266.30–2840 δ = 84.0 1 0 – 98 . 7 0 ⊕ 262.1 9 0 – 276 . 6 0 δ = 86.2 2 0 – 273 . 3 0 δ = 81.5 6 0 – 96.4 8 0 ⊕ 263 . 7 0 – 278 . 4 0 m 1 = 0.0235 – 0.314 m 1 = 0.0453 – 0.277 m 1 = 0.00548 – 0.267 m 1 = 0.0507 – 0.419 m 2 = 0.0242 – 0.0310 m 2 = 0.0465 – 0.274 m 2 = 0.00806 – 0.316 m 2 = 0.0508 – 0.419 m 3 = 0.0485 – 0.0315 m 3 = 0.0117 – 0.271 m 3 = 0.0428 – 0.320 m 3 = 0.0249 – 0.414  B3(C6) ρ = - 9 0 0 – - 5 . 9 0 ⊕ 5 . 9 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – - 5 . 9 0 ⊕ 5 . 9 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=7.470–3560 δ = 8 . 3 0 – 17 3 0 ⊕ 18 7 0 – 35 1 0 δ = 5.1 6 0 – 17 8 0 ⊕ 183 . 7 0 – 353 . 6 0 δ = 0 0 – 36 0 0 m 1 = 0.0153 – 0.331 m 1 = 0.0459 – 0.335 m 1 = 0.00504 – 0.485 m 1 = 0.0427 – 0.505 m 2 = 0.0161 – 0.331 m 2 = 0.0461 – 0.334 m 2 = 0.00920 – 0.483 m 2 = 0.0445 – 0.503 m 3 = 0.0445 – 0.0335 m 3 = 0.0136 – 0.330 m 3 = 0.0416 – 0.480 m 3 = 0.00884 – 0.500  B4(C5) ρ = - 19.0 7 0 – 19.7 5 0 ρ=-4.800–4.660 ρ = - 43.1 3 0 – 40.1 2 0 ρ=-16.380–18.80 σ = - 9 0 0 – 9 0 0 σ = - 2.8 3 0 – 3.1 2 0 σ = - 4 3 0 – 40 . 1 0 σ = - 41.4 3 0 – 40.5 9 0 δ=00–1000⊕260.20–3600 δ = 84.6 8 0 – 94 . 7 0 ⊕ 265.0 8 0 – 275 . 5 0 δ = 88.0 4 0 – 124 . 4 0 ⊕ 230.0 9 0 – 27 2 0 δ = 5.5 6 0 – 13.8 2 0 ⊕ 255 . 7 0 – 307 . 3 0 m 1 = 0.00681 – 0.318 m 1 = 0.0488 – 0.314 m 1 = 0.0105 – 0.373 m 1 = 0.0444 – 0.420 m 2 = 0.00969 – 0.0317 m 2 = 0.0486 – 0.312 m 2 = 0.0129 – 0.369 m 2 = 0.0447 – 0.416 m 3 = 0.0421 – 0.0321 m 3 = 0.0202 – 0.310 m 3 = 0.0435 – 0.374 m 3 = 0.0103 – 0.418  B5(C4) ρ = - 20.8 3 0 – 20.7 9 0 ρ=-900–-11.30⊕11.30–900 ρ = - 9 0 0 – 9 0 0 ρ=-24.190–24.450 σ = - 36 . 3 0 – 3 6 0 $ σ = - 9 0 0 – - 14 . 3 0 ⊕ 14 . 8 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 27 . 7 0 – 27 . 4 0 δ=00–177.50⊕1820–3600 δ = 0 0 – 16 8 0 ⊕ 192 . 5 0 – 36 0 0 δ = 9.3 3 0 – 172 . 4 0 ⊕ 187.20 – 350 . 4 0 δ = 0 0 – 179 . 3 0 ⊕ 180 . 6 0 – 36 0 0 m 1 = 0.00965 – 0.330 m 1 = 0.0761 – 0.330 m 1 = 0.0603 – 0.498 m 1 = 0.0440 – 0.498 m 2 = 0.0113 – 0.0330 m 2 = 0.0762 – 0.330 m 2 = 0.0608 – 0.496 m 2 = 0.0447 – 0.499 m 3 = 0.0421 – 0.0332 m 3 = 0.0603 – 0.328 m 3 = 0.0629 – 0.500 m 3 = 0.0103 – 0.499  B6(C3) ρ = - 9 0 0 – 9 0 0 × × ρ=-900–900 σ = - 9 0 0 – 9 0 0 × × σ = - 9 0 0 – 9 0 0 δ=00–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.0147 – 0.324 × × m 1 = 0.0452 – 0.454 m 2 = 0.00161 – 0.324 × × m 2 = 0.0451 – 0.451 m 3 = 0.0445 – 0.324 × × m 3 = 0.00161 – 0.450  B7(C2) ρ = - 26.2 5 0 – 26.4 3 0 × × ρ=-30.920–30.420 σ = - 50.2 8 0 – 50 . 2 0 × × σ = - 41 . 9 0 – 43 . 8 0 δ=00–177.50⊕1820–3600 × × δ = 0 0 – 36 0 0 m 1 = 0.00883 – 0.331 × × m 1 = 0.0447 – 0.498 m 2 = 0.00121 – 0.329 × × m 2 = 0.0451 – 0.498 m 3 = 0.0421 – 0.334 × × m 3 = 0.00804 – 0.498  B8(C10) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–166.50⊕196.50–3600 δ = 45.0 5 0 – 13 5 0 ⊕ 224 . 5 0 – 314 . 6 0 δ = 45.8 6 0 – 315.0 5 0 δ = 0 0 – 36 0 0 m 1 = 0.00127 – 0.279 m 1 = 0.0418 – 0.279 m 1 = 0.00498 – 0.381 m 1 = 0.0427 – 0.303 m 2 = 0.00691 – 0.277 m 2 = 0.0429 – 0.307 m 2 = 0.00736 – 0.382 m 2 = 0.0428 – 0.302 m 3 = 0.0409 – 0.280 m 3 = 0.00398 – 0.305 m 3 = 0.0407 – 0.384 m 3 = 0.00342 – 0.298  B9(C9) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=5.550–3540 δ = 1 2 0 – 171 . 7 0 ⊕ 189 . 7 0 – 352 . 7 0 δ=8.320–175.90⊕1860–350.470 δ = 0 0 – 36 0 0 m 1 = 0.00338 – 0.279 m 1 = 0.0435 – 0.319 m 1 = 0.00622 – 0.487 m 1 = 0.0447 – 0.486 m 2 = 0.00831 – 0.277 m 2 = 0.0445 – 0.319 m 2 = 0.00681 – 0.486 m 2 = 0.0451 – 0.487 m 3 = 0.0423 – 0.282 m 3 = 0.00398 – 0.317 m 3 = 0.00914 – 0.486 m 3 = 0.00458 – 0.480  B10(C8) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–3600 δ=44.640–136.50⊕224.60–314.20 δ=1.920–313.040 δ = 0 0 – 36 0 0 m 1 = 0.00163 – 0.294 m 1 = 0.0425 – 0.277 m 1 = 0.00516 – 0.348 m 1 = 0.0427 – 0.365 m 2 = 0.00691 – 0.293 m 2 = 0.0423 – 0.275 m 2 = 0.00806 – 0.346 m 2 = 0.0453 – 0.365 m 3 = 0.0409 – 0.299 m 3 = 0.00380 – 0.272 m 3 = 0.0416 – 0.348 m 3 = 0.00879 – 0.391TexturesB1 (NO), B3 (NO, IO), B5 (NO, IO), B6 (NO), B7 (NO), B8 (NO), B10 (NO), C1 (NO), C2 (NO), C3 (NO), C4 (NO, IO), C6 (NO, IO), C8 (NO), and C10 (NO) held nearly no constraint on Dirac CP violating phase (δ) for 1TEE and 1TEC, respectively, but with opposite neutrino mass ordering (Table 4). Only cases B2 (NO), B4 (IO), C7 (NO), and C5 (IO) for 1TEE and B2 (IO), B4 (NO), C7 (IO), and C5 (NO) for 1TEC show significant reduction in the parameter space of δ. It is found that δ is restricted near 900 and 2700 for 1TEE and 1TEC, respectively (Table 4). These predictions are significant considering the latest hint on δ near 2700 [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 3σ CL for 1TEE and 1TEC, respectively (Table 5). However cases D1,2,4,5,6,7,9 favor both NO and IO for 1TEE and 1TEC, while D3, D8, and D10 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 phaseδ, the Majorana phases ρ, σ, and three neutrino masses m1, m2, m3 for the experimentally allowed cases of Category D (F). Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.
Cases 1TEE 1TEC NO IO NO IO D1(F2) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=30.1850–322.60 δ = 0 0 – 36 0 0 δ=00–3600 δ=31.790–318.660 m 1 = 0.0541 – 0.224 m 1 = 0.00432 – 0.252 m 1 = 0.00331 – 0.603 m 1 = 0.0731 – 0.501 m 2 = 0.0538 – 0.224 m 2 = 0.00437 – 0.251 m 2 = 0.00800 – 0.606 m 2 = 0.0719 – 0.500 m 3 = 0.0679 – 0.230 m 3 = 0.00343 – 0.248 m 3 = 0.0371 – 0.604 m 3 = 0.0593 – 0.496  D2(F1) ρ = - 50.2 8 0 – 60.4 8 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-63.30–640 σ=-900–900 σ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 89.3 7 0 – 86.2 5 0 δ=87.190–297.40 δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 δ = 70.5 0 0 – 295.7 7 0 m 1 = 0.124 – 0.293 m 1 = 0.0422 – 0.263 m 1 = 0.00542 – 0.0277 m 1 = 0.128 – 0.581 m 2 = 0.125 – 0.294 m 2 = 0.0430 – 0.262 m 2 = 0.00993 – 0.0291 m 2 = 0.130 – 0.582 m 3 = 0.132 – 0.295 m 3 = 0.00552 – 0.257 m 3 = 0.0440 – 0.0600 m 3 = 0.120 – 0.579  D3(F4) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0439 – 0.308 m 1 = 0.0161 – 0.474 × × m 2 = 0.0445 – 0.308 m 2 = 0.0164 – 0.471 × × m 3 = 0.00888 – 0.304 m 3 = 0.0416 – 0.469 ×  D4(F3) ρ = - 58.7 1 0 – 59.3 5 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-61.40–61.50 σ = - 88.9 0 0 – 87.1 8 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=75.870–285.580 δ = 0 0 – 36 0 0 δ = 18.1 2 0 – 318.7 5 0 δ=77.860–286.150 m 1 = 0.0574 – 0.287 m 1 = 0.00429 – 0.254 m 1 = 0.0102 – 0.795 m 1 = 0.0711 – 0.424 m 2 = 0.0575 – 0.286 m 2 = 0.00430 – 0.254 m 2 = 0.0147 – 0.796 m 2 = 0.0740 – 0.424 m 3 = 0.0679 – 0.291 m 3 = 0.00969 – 0.202 m 3 = 0.0369 – 0.790 m 3 = 0.0564 – 0.418  D5(F5) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–3600 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 m 1 = 0.0337 – 0.334 m 1 = 0.0431 – 0.335 m 1 = 0.00255 – 0.509 m 1 = 0.0538 – 0.505 m 2 = 0.0338 – 0.334 m 2 = 0.0434 – 0.334 m 2 = 0.00370 – 0.505 m 2 = 0.0539 – 0.506 m 3 = 0.0541 – 0.334 m 3 = 0.00940 – 0.330 m 3 = 0.0351 – 0.510 m 3 = 0.0305 – 0.500  D6(F9) ρ = - 65.0 9 0 – 58.4 8 0 ρ = - 9 0 0 – 9 0 0 ρ = - 23.0 3 0 – - 7.7 4 0 ⊕ 7.6 6 0 – 22.1 4 0 ρ=-18.420–-6.560⊕6.770–18.270 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 68.5 3 0 – - 38.7 5 0 ⊕ 37.8 6 0 – 69 . 0 0 σ = - 71 . 1 0 – - 46 . 2 0 ⊕ 46 . 2 0 – 70.4 8 0 δ=64.20–288.10 δ = 0 0 – 36 0 0 δ = 109.6 8 0 – 150.0 5 0 ⊕ 210 . 4 0 – 250 . 2 0 δ = 118.1 9 0 – 154 . 2 0 ⊕ 206 . 7 0 – 242 . 3 0 m 1 = 0.0974 – 0.323 m1=0.0446–0.277 m1=0.0202–0.456 m1=0.108–0.398 m 2 = 0.0965 – 0.0321 m2=0.0448–0.277 m2=0.0204–0.456 m2=0.109–0.396 m 3 = 0.107 – 0.0326 m3=0.00888–0.324 m3=0.0451–0.456 m3=0.0962–0.396  D7(F8) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–3600 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 m 1 = 0.0478 – 0.273 m 1 = 0.0439 – 0.333 m 1 = 0.00663 – 0.436 m 1 = 0.0647 – 0.437 m 2 = 0.0479 – 0.273 m 2 = 0.0454 – 0.333 m 2 = 0.0103 – 0.434 m 2 = 0.0649 – 0.438 m 3 = 0.0644 – 0.300 m 3 = 0.00806 – 0.330 m 3 = 0.0416 – 0.438 m 3 = 0.0439 – 0.491  D8(F7) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0451 – 0.0938 m 1 = 0.00941 – 0.0803 × × m 2 = 0.0459 – 0.0941 m 2 = 0.0125 – 0.0804 × × m 3 = 0.00121 – 0.0790 m 3 = 0.0445 – 0.0953 ×  D9(F6) ρ = - 9 0 0 – 9 0 0 ρ=-900–900 ρ = - 9 0 0 – 9 0 0 ρ=-900–900 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 δ=00–3600 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 δ = 0 0 – 36 0 0 m 1 = 0.0492 – 0.300 m 1 = 0.0455 – 0.310 m 1 = 0.00769 – 0.436 m 1 = 0.0647 – 0.411 m 2 = 0.0496 – 0.300 m 2 = 0.0456 – 0.310 m 2 = 0.00932 – 0.434 m 2 = 0.0638 – 0.412 m 3 = 0.0638 – 0.303 m 3 = 0.00969 – 0.306 m 3 = 0.0426 – 0.438 m 3 = 0.0458 – 0.407  D10(F10) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0432 – 0.102 m 1 = 0.00201 – 0.0793 × × m 2 = 0.0442 – 0.102 m 2 = 0.00870 – 0.0795 × × m 3 = 0.00301 – 0.0901 m 3 = 0.0435 – 0.0932 ×CasesD1 (IO), D2 (IO), D3 (IO), D4 (IO), D5 (NO, IO), D6 (IO), D7 (NO, IO), D8 (IO), D9 (NO, IO), D10 (IO), F1 (IO), F2 (IO), F3 (IO), F4 (IO), F5 (NO, IO), F6 (IO), F7 (NO, IO), F8 (IO), F9 (NO, IO), and F10 (IO) predict literally no constraints on δ for 1TEE. These cases give identical predictions for 1TEC, but for opposite mass ordering. For cases D1 (NO), D2 (NO), D4 (NO), D6 (NO), F1 (NO), F2 (NO), F4 (NO), and F6 (NO), the parameter space of δ is found to be well constrained for 1TEE. These cases give similar predictions regarding the parameter space of δ for 1TEC, but for IO.Category E. In Category E, all the ten possible cases are allowed for 1TEE at 3σ CL, while only nine other than E5 are acceptable in case of 1TEC (Table 6). Cases E1,2,3,4,6,7,8,9 allow only inverted mass ordering (IO) for 1TEE, while the same textures allow only normal mass ordering (NO) for 1TEC. Cases E5 and E10 allow both NO and IO for 1TEE; however E5 is ruled out for both NO and IO for 1TEC at 3σ CL. Similar to cases belonging to Category D, E1,2,3,4,5,6,7,8,9 (IO) cover full range of ρ,σ,δ for 1TEE, whereas the same cases (except E5) give identical predictions for 1TEC, but for NO. For E5 (NO) and E10 (NO), phases ρ,σ,δ are somewhat restricted at 3σ CL for 1TEE, while only for E10 (IO), the parameter space of ρ,σ,δ seems to be restricted for 1TEC (Table 6).Table 6
The allowed ranges of Dirac CP-violating phaseδ, the Majorana phases ρ, σ, and three neutrino masses m1, m2, m3 for the experimentally allowed cases of Category E. Masses are in eV. “×” denotes the nonviability of case for a particular mass ordering.
Cases 1TEE 1TEC NO IO NO IO E1(E2) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0511 – 0.0804 m 1 = 0.00283 – 0.0629 × × m 2 = 0.0516 – 0.0805 m 2 = 0.00878 – 0.0634 × × m 3 = 0.0264 – 0.0678 m 3 = 0.0440 – 0.0809 ×  E3(E4) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 64.3 4 0 ⊕ 295 . 7 0 – 36 0 0 × × m 1 = 0.0439 – 0.333 m 1 = 0.00914 – 0.499 × × m 2 = 0.0457 – 0.333 m 2 = 0.00115 – 0.504 × × m 3 = 0.00888 – 0.333 m 3 = 0.0432 – 0.500 ×  E5 ρ = - 9 0 0 – - 156 ⊕ 29.3 – 9 0 0 ρ=-900–900 × × ρ=-900–-156⊕29.3–900 σ=-900–900 × × δ=29.30–153.50⊕206.50–3340 δ=00–3600 × × m 1 = 0.130 – 0.332 m 1 = 0.0439 – 0.269 × × m 2 = 0.132 – 0.330 m 2 = 0.0451 – 0.270 × × m 3 = 0.137 – 0.334 m 3 = 0.0103 – 0.262 × ×  E6(E9) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 36 0 0 × × m 1 = 0.0511 – 0.333 m 1 = 0.00914 – 0.499 × × m 2 = 0.0517 – 0.333 m 2 = 0.00115 – 0.504 × × m 3 = 0.0273 – 0.333 m 3 = 0.0432 – 0.500 ×  E7(E8) × ρ=-900–900 ρ = - 9 0 0 – 9 0 0 × × σ=-900–900 σ = - 9 0 0 – 9 0 0 × × δ=00–3600 δ = 0 0 – 36 0 0 × × m 1 = 0.0511 – 0.0908 m 1 = 0.0225 – 0.0469 × × m 2 = 0.0517 – 0.0911 m 2 = 0.0238 – 0.0480 × × m 3 = 0.0273 – 0.0746 m 3 = 0.0494 – 0.0708 ×  E10 ρ = - 9 0 0 – - 28 . 6 0 ⊕ 26.0 1 0 – 9 0 0 ρ = - 18.1 5 0 – 18.0 8 0 ρ = - 9 0 0 – 9 0 0 ρ=-900–-190⊕19.50–900 σ=-900–-28.60⊕26.010–900 σ = - 9 0 0 – - 6.4 1 0 ⊕ 6.0 9 0 – 9 0 0 σ = - 9 0 0 – 9 0 0 σ = - 9 0 0 – - 1 9 0 ⊕ 19 . 5 0 – 9 0 0 δ=25.350–151.920⊕206.970–332.30 δ = 0 0 – 17 7 0 ⊕ 184 . 3 0 – 357 . 5 0 δ = 0 0 – 36 0 0 δ = 23.7 5 0 – 161.9 7 0 ⊕ 198 . 9 0 – 33 9 0 m 1 = 0.136 – 0.331 m1=0.0431–0.334 m1=0.00156–0.481 m1=0.141–0.481 m 2 = 0.137 – 0.331 m2=0.0445–0.334 m2=0.00571–0.480 m2=0.142–0.492 m 3 = 0.143 – 0.334 m3=0.00969–0.330 m3=0.0404–0.484 m3=0.133–0.491To summarize our discussion, we have investigated all the viable cases of 1TEE and 1TEC texture structures in the limit of large effective neutrino massMee. 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 3σ 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 δ for most of the cases belonging to 1TEE and 1TEC remains unrestricted, while only eight cases show maximal restriction for δ. 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 Mee term, which would, in turn, either confirm or rule out our assumption of large Mee.



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