On the textures of neutrino mass matrix for maximal atmospheric mixing angle and Dirac CP phase

In this paper, we derive in a novel approach the possible textures of neutrino mass matrix that can lead us to maximal atmospheric mixing angle and Dirac CP phase which are consistent with the current neutrino oscillation data. A total of eleven textures are thus found. Interestingly, the specific texture given by the $\mu$-$\tau$ reflection symmetry can be reproduced from one of the obtained textures. For these textures, some neutrino mass sum rules which relate the neutrino masses and Majorana CP phases will emerge.


Introduction
Thanks to the enormous neutrino oscillation data, a framework of three-flavor neutrino mixing has been established [1]. In the usually used basis of charged lepton mass matrix M l being diagonal which is also adopted here, the neutrino mixing matrix U [2] originates from diagonalization of the neutrino mass matrix M ν in a manner as with m i (for i = 1, 2, 3) being the neutrino masses. In the standard parameterization, U reads where P l = Diag e iφe , e iφµ , e iφτ and P ν = Diag e iρ , e iσ , 1 are two diagonal phase matrices, and Here we have used the standard notations c ij = cos θ ij and s ij = sin θ ij for the mixing angles θ ij (for ij = 12, 13, 23). As for the phases, δ is known as the Dirac CP phase and responsible for the CP violation effects in neutrino oscillations, while ρ and σ are known as the Majorana CP phases and control the rates of neutrinoless double beta decays which are used to testify the Majorana nature of neutrinos. φ e,µ,τ are called unphysical phases since they can be removed by the redefinitions of charged lepton fields. In addition, neutrino oscillations are also dependent on the neutrino mass squared differences ∆m 2 ij = m 2 i − m 2 j (for ij = 21, 31). The experimental results for the neutrino masses are given by [3] ∆m 2 21 = 7.50 +0.19 −0.17 × 10 −5 eV 2 , |∆m 2 31 | = (2.524 +0.039 −0.040 ) × 10 −3 eV 2 .
Note that the sign of ∆m 2 31 has not yet been determined, thereby allowing for two possible neutrino mass orderings: m 1 < m 2 < m 3 (the normal hierarchy and NH for short) and m 3 < m 1 < m 2 (the inverted hierarchy and IH for short). And the absolute neutrino mass scale or the lightest neutrino mass (m 1 in the NH case or m 3 in the IH case) remains unknown. On the other hand, the mixing parameters θ 13 , θ 23 and δ take the values sin 2 θ 13 = 0.02166 ± 0.00075 , sin 2 θ 23 = 0.441 ± 0.024 , δ = 261 • ± 55 • , in the NH case, or sin 2 θ 13 = 0.02179 ± 0.00076 , sin 2 θ 23 = 0.587 ± 0.022 , δ = 277 • ± 43 • , in the IH case, while θ 12 takes the value sin 2 θ 12 = 0.306 ± 0.012 in either case [3]. However, information about ρ and σ is still lacking. It is interesting to note that the current neutrino oscillation data is consistent with maximal atmospheric mixing angle (θ 23 = π/4) and leptonic CP violation (δ = −π/2). These remarkable relations may point towards some special texture of M ν . In this regard, the specific texture given by the neutrino µ-τ reflection symmetry [4]- [6] serves as a unique example. It is defined by M eµ = M * eτ , M µµ = M * τ τ , M ee and M µτ being real , where M αβ denotes the αβ element of M ν (for α, β = e, µ, τ ), and can be attributed to the invariance of M ν under a combination of the µ-τ interchange and CP conjugate operations Such a texture gives θ 23 = π/4 and δ = ±π/2 as well as φ e = π/2, φ µ = −φ τ and trivial Majorana CP phases (i.e., ρ, σ = 0 or π/2) [7]. The purpose of this paper is to derive in a novel approach the possible textures of neutrino mass matrix that can naturally generate maximal atmospheric mixing angle and leptonic CP violation [8]. Such a study may help us reveal the underlying flavor symmetries in the lepton sector. A total of eleven textures are thus found. Interestingly, one of the obtained textures can reproduce the specific texture given by the neutrino µ-τ reflection symmetry. For these textures, some neutrino mass sum rules which relate the neutrino masses and Majorana CP phases will emerge. The rest part of this paper is organized as follows. In section 2, we formulate our approach of deriving the desired textures. In section 3, the derived textures and the resulting neutrino mass sum rules are discussed one by one in some detail. Finally, a summary of our main results is given in section 4.

The approach
A 3×3 complex symmetric neutrino mass matrix generally contains twelve degrees of freedom (dfs). After the diagonalization process, three dfs will emerge as the unphysical phases φ e,µ,τ while nine dfs as the physical parameters θ ij , δ, ρ, σ and m i . Therefore, one would suffer some uncertainties due to the unphysical phases when retrodicting the textures of M ν based on the physical parameters. In comparison, the effective neutrino mass matrixM ν = P † l M ν P * l where the unphysical dfs cancel out only consists of nine physical dfs. For this reason, we choose to work onM ν instead of M ν itself so that the uncertainties created by the unphysical phases can be evaded. Two immediate comments are given as follows: (1), One can recover the results for M ν from those forM ν by simply making the replacementsM αβ = M αβ e −i(φ α +φ β ) withM αβ being the αβ element ofM ν . (2), SinceM ν only has nine dfs, its twelve componentsR αβ = Re(M αβ ) andĪ αβ = Im(M αβ ) are not all independent but subject to three constraint equations.
To proceed, we diagonalizeM ν to give the expressions for the physical parameters in terms of R αβ = Re(M αβ ) andĪ αβ = Im(M αβ ). From Eqs. (1)(2), one finds In order to simplify the related expressions, we define the following matrices after the rotations O 23 , U 13 and O 12 are implemented in succession where M ij k stands for the ij element of M k (for i, j, k = 1, 2, 3). By comparing two sides of Eq. (9), one obtains Re/Im M 13 2 = Re/Im M 23 2 = Re/Im M 12 and Explicitly, the seven equations in Eq. (14) with R ij k = Re M ij k and I ij k = Im M ij k . The expressions for θ 12 and θ 13 in terms of the components ofM ν can be directly read from Eq. (16). For example, equation E gives θ 12 as By relating these two expressions for θ 12 , a constraint equation for the components ofM ν arises as It can be expressed in terms ofR αβ andĪ αβ by taking the expressions In a similar way, one will arrive at the following constraint equations by relating the expressions for θ 13 derived from equations A-D where the symbol AB (and so on) is used to denote the constraint equation resulting from equations A and B (and so on). In terms ofR αβ andĪ αβ , they are expressed as But not all of these six constraint equations are independent. For example, equation BC can be derived from equations AB and AC. In fact, at most three of them can be independent. A set of three independent constraint equations (e.g., AB, AC and AD) can be chosen in such a way that each of equations A-D has been used at least once in deriving them. Finally, we obtain a constraint equation as by relating the expressions for θ 13 derived from equations A and G. Its expressions in terms ofR αβ andĪ αβ appears as To sum up, a total of five independent constraint equations for the components ofM ν (i.e., Eqs. (20, 24) and three independent ones from Eq. (22)) will arise from the eliminations of θ 12 and θ 13 in Eq.
(16). This can be understood from the fact that two more conditions (i.e., θ 23 = π/4 and δ = −π/2) have been imposed on the basis of three inherent constraint equations for the components ofM ν . At last, one can say that anM ν with its components satisfying these constraint equations will necessarily produce θ 23 = π/4 and δ = −π/2. By taking the expressions for θ 12 and θ 13 derived from Eq. (16) in Eq. (15), the neutrino masses in combination with the Majorana CP phases are obtained as Im m 1 e 2iρ = I 11 2 + I 22 Im m 2 e 2iσ = I 11 2 + I 22 where while the expressions for R 12 , I 12 , R 11 2 − R 22 2 and R 11 2 − R 22 2 have been given in Eq. (20). However, it should be pointed out that the above results are derived in the general case where none of equations A-G has its two sides vanish. When an equation has its two sides vanish, it fails to give an expression for θ 12 or θ 13 and thus the constraint equation(s) resulting from it will become ineffective. In such kind of case, the number of independent constraint equations will get increased compared to in the general case. (For example, in the case of two sides of equation E being vanishing, the expression for θ 12 in Eq. (17) and thus the constraint equation in Eq. (19) become ineffective. But there are two new constraint equations R 12 2 = R 11 2 − R 22 2 = 0. So, in effect, the number of independent constraint equations in this case gets increased by one compared to in the general case.) When this number gets increased by one (and so on), there will correspondingly be one (and so on) neutrino mass sum rules as we will see. In the next section, all the possible cases where one or more equations in Eq. (16) have their two sides vanish will be examined. Before doing that, we make several observations: (1), In the case of two sides of equation A being vanishing (i.e., R 13 1 = I 11 1 + I 33 1 = 0), two sides of equation G are necessarily also vanishing (i.e., R 13 1 = c 2 13 I 33 1 − s 2 13 I 11 1 = 0). The reverse is also true. It is easy to see that a combination of c 2 13 I 33 1 − s 2 13 I 11 1 = 0 and I 11 1 + I 33 1 = 0 leads us to I 11 1 = I 33 1 = 0. So equations A and G always have their two sides vanish simultaneously. And in such a case one has (2), In the case of two sides of equation C or D being vanishing (i.e., R 23 1 = I 12 1 = 0 or I 23 1 = R 12 1 = 0), as a result of the relation I 12 2 = c 13 I 12 1 − s 13 R 23 1 or R 12 2 = c 13 R 12 1 + s 13 I 23 1 , two sides of equation F or E are necessarily also vanishing (i.e., I 12 2 = I 11 2 − I 22 2 = 0 or R 12 2 = R 11 2 − R 22 2 = 0). The reverse is also true. So equations C and F or D and E always have their two sides vanish simultaneously. And in such a case one has or By taking these relations, the expressions for the surviving constraint equations in Eqs. (20,22) can be simplified to some extent. In total, the number of independent constraint equations in the case under study gets increased by one compared to in the general case. So one neutrino mass sum rule will arise. The neutrino masses in combination with the Majorana CP phases in Eq. (25) can also be simplified to some extent by taking the relations in Eq. (30). Thereinto, Im m 1 e 2iρ and Im m 2 e 2iσ can be written as from which it is easy to get m 1 c 2 12 sin 2ρ + m 2 s 2 12 sin 2σ = 0 .
In the particular case of sin 2ρ = sin 2σ = 0 (i.e., ρ, σ = 0 or π/2), the neutrino masses will become independent of the Majorana CP phases. For this case, an additional constraint equation results from Eq. (31). It is found that Eq. (33) together with Eq. (30) would lead us to a situation where two sides of equations C and F are also vanishing. This happens to be the case of two sides of equations A, C, F and G being vanishing which will be discussed in subsection 3.6. For the case of sin 2ρ, sin 2σ = 0, we present sin 2σ/ sin 2ρ as a function of the lightest neutrino mass in the NH and IH cases in Fig. 1. In the IH case, it takes a value close to −c 2 12 /s 2 12 −2.27 in the whole mass range as a result of m 1 m 2 . In the NH case, its value is very small in the range of m 1 being vanishingly small but approaches −2.27 in the range of m 1 m 2 0.1 eV.

B
In the case of two sides of equation B being vanishing, equations AB, BC and BD become ineffective. We are thus left with four independent constraint equations (i.e., Eqs. (19, 23) and two of equations AC, AD and CD). But there are two new constraint equations which lead to the following relations forR αβ andĪ αβ By taking these relations, the expressions for the surviving constraint equations in Eqs. (20,22,24) can be simplified to some extent. In total, the number of independent constraint equations in the case under study gets increased by one compared to in the general case. So one neutrino mass sum rule will arise.
The neutrino masses in combination with the Majorana CP phases in Eq. (25) can also be simplified to some extent by taking the relations in Eq. (35). Thereinto, Re m 1 e 2iρ , Re m 2 e 2iσ and m 3 can be written as Re m 1 e 2iρ = −R µτ + R ee +R µτ 1 cos 2θ 12 , from which it is easy to get m 1 c 2 12 cos 2ρ + m 2 s 2 12 cos 2σ + m 3 = 0 .
With the help of the inequality one can see that this sum rule can never be fulfilled in the NH case. Hence we discuss the implications of this sum rule in the IH case. In Fig. 2, we show the correlation between ρ and σ for some representative values of m 3 . For the particular value of m 3 = 0, an interesting solution to Eq. (37) is cos 2ρ = cos 2σ = 0 (i.e., ρ, σ = π/4 or 3π/4) which gives two additional constraint equations from Eq. (36). In this case, the effective neutrino mass σ or ρ as well as | m ee | can be presented as a function of m 3 as shown in Fig. 3. In the range of m 3 being vanishingly small, σ and ρ take a value close to π/4 or 3π/4 while | m ee | takes a value close to 0.049 eV or 0.019 eV. This is consistent with the results in the case of m 3 = cos 2ρ = cos 2σ = 0. When m

C&F
In the case of two sides of equations C and F being vanishing, equations AC, BC, CD and Eq. (19) become ineffective. We are thus left with three independent constraint equations (i.e., Eq. (23) and two of equations AB, AD and BD). But, as discussed at the end of section 2, there are three new constraint equations given by Eq. (28) which lead to the following relations forR αβ andĪ αβ By taking these relations, the expressions for the surviving constraint equations in Eqs. (22,24) can be simplified to some extent. In total, the number of independent constraint equations in the case under study gets increased by one compared to in the general case. So one neutrino mass sum rule will arise.
The neutrino masses in combination with the Majorana CP phases in Eq. (25) can also be simplified to some extent by taking the relations in Eq. (42). Thereinto, Im m 1 e 2iρ and Im m 2 e 2iσ become Im m 1 e 2iρ = 1 2 from which it is easy to get m 1 sin 2ρ − m 2 sin 2σ = 0 .
In the particular case of sin 2ρ = sin 2σ = 0 (i.e., ρ, σ = 0 or π/2), the neutrino masses will become independent of the Majorana CP phases. In this case, an additional constraint equation same as that in Eq. (33) results from Eq. (43). It is found that Eq. (33) together with Eq. (42) would lead us to a situation where two sides of equations A and G are also vanishing. As mentioned in subsection 3.1, this happens to be the case of two sides of equations A, C, F and G being vanishing which will be discussed in subsection 3.6. For the case of sin 2ρ, sin 2σ = 0, we present sin 2σ/ sin 2ρ as a function of the lightest neutrino mass in the NH and IH cases in Fig. 4. In the IH case, it takes a value close to 1 in the whole mass range as a result of m 1 m 2 . In the NH case, its value is very small in the range of m 1 being vanishingly small but approaches 1 in the range of m 1 m 2 0.1 eV.

D&E
In the case of two sides of equations D and E being vanishing, equations AD, BD, CD and Eq. (19) become ineffective. We are thus left with three independent constraint equations (i.e., Eq. (23) and two of equations AB, AC and BC). But, as discussed at the end of section 2, there are three new constraint equations given by Eq. (29) which lead to the following relations forR αβ andĪ αβ By taking these relations, the expressions for the surviving constraint equations in Eqs. (22,24) can be simplified to some extent. In total, the number of independent constraint equations in the case under study gets increased by one compared to in the general case. So one neutrino mass sum rule will arise. The neutrino masses in combination with the Majorana CP phases in Eq. (25) can also be simplified to some extent by taking the relations in Eq. (45). Thereinto, Re m 1 e 2iρ and Re m 2 e 2iσ become Re m 1 e 2iρ = 1 2 from which it is easy to get In the particular case of cos 2ρ = cos 2σ = 0 (i.e., ρ, σ = π/4 or 3π/4), the neutrino masses will become independent of the Majorana CP phases. In this case, an additional constraint equation results from Eq. (46). For the case of cos 2ρ, cos 2σ = 0, one can also present cos 2σ/ cos 2ρ as a function of the lightest neutrino mass. The result is the same as that for sin 2σ/ sin 2ρ in subsection 3.3.

A&B&G
In the case of two sides of equations A, B and G being vanishing, equations AB, AC, AD, BC, BD and Eq. (23) (20,22) can be simplified to some extent. In total, the number of independent constraint equations in the case under study gets increased by two compared to in the general case. So two neutrino mass sum rules will arise. By taking the relations in Eqs. (30,35), the neutrino masses in combination with the Majorana CP phases in Eq. (25) can be written as in Eqs. (31,36). So the desired neutrino mass sum rules are the same as those in Eqs. (32, 37). As discussed in subsection 3.2, these sum rules can only be fulfilled in the IH case. In Fig. 5, we present ρ, σ and | m ee | as functions of the lightest neutrino mass m 3 . There is a lower value 0.021 eV of m 3 at which ρ and σ respectively take the values 0 and π/2 (or π/2 and 0). As discussed in subsection 3.1, sin 2ρ = sin 2σ = 0 (which gives an additional constraint equation given by Eq. (33)) would lead us to a situation where two sides of equations C and F are also vanishing. This happens to be the case of two sides of equations A, B, C, F and G being vanishing which will be discussed in subsection 3.10. On the other hand, | m ee | is found to be equal to m 3 : The sum rules in Eqs. (32, 37) can be reorganized into a single complex equation which immediately tells us that | m ee | = m 3 .

A&D&E&G
By taking the relations in Eqs. (30,45), the neutrino masses in combination with the Majorana CP phases in Eq. (25) can be written as in Eqs. (31,46). So the desired neutrino mass sum rules are the same as those in Eqs. (32,47). In Figs. 7-8, we present ρ, σ and | m ee | as functions of the lightest neutrino mass in the NH and IH cases. In the NH case, there is a lower value 0.004 eV of m 1 at which ρ and σ respectively take the values π/4 and 3π/4 (or 3π/4 and π/4). As discussed in Only for ρ π − σ 0 or π/2, can these two relations be fulfilled simultaneously. Consequently, | m ee | approximates to m 1 c 2 12 + m 2 s 2 12 in this mass range. In the IH case, we have m 1 m 2 and thus ρ π − σ 0 or π/2 and | m ee | m 1 c 2 12 + m 2 s 2 12 in the whole mass range.

B&C&F
In  (35,42). By taking these relations, the expressions for the surviving constraint equations in Eqs. (22,24) can be simplified to some extent. In total, the number of independent constraint equations in the case under study gets increased by two compared to in the general case. So two neutrino mass sum rules will arise. By taking the relations in Eqs. (35,42), the neutrino masses in combination with the Majorana CP phases in Eq. (25) can be written as in Eqs. (36,43). So the desired neutrino mass sum rules are the same as those in Eqs. (37,44). As discussed in subsection 3.2, these sum rules can only be fulfilled in the IH case. In Fig. 9, we present ρ, σ and | m ee | as functions of the lightest neutrino mass m 3 . As a result of m 1 m 2 in the IH case, one finds ρ σ or π/2 − σ from Eq. (44). Eq. (37) further leads us to ρ σ π/4 or 3π/4 for vanishingly small m 3 and ρ σ π/2 for m 3 m 1 m 2 0.1 eV. Consequently, | m ee | takes a value close to m 1 c 2 12 + m 2 s 2 12 for these two mass ranges. Finally, for m 3 0.02 eV, it is found that one of the allowed solutions to Eqs. (37,  (22,24) can be simplified to some extent. In total, the number of independent constraint equations in the case under study gets increased by two compared to in the general case. So two neutrino mass sum rules will arise. By taking the relations in Eqs. (35,45), the neutrino masses in combination with the Majorana CP phases in Eq. (25) can be written as in Eqs. (36,46). So the desired neutrino mass sum rules are the same as those in Eqs. (37,47). A combination of these sum rules further yields Apparently, these relations can only be fulfilled in the IH case. In Fig. 10, we present ρ, σ and | m ee | as functions of the lightest neutrino mass m 3 . In consideration of m 1 m 2 in the IH case, the results for ρ and σ are presented by the same lines. One has ρ, σ π/4 or 3π/4 for vanishingly small m 3 and ρ σ π/2 for m 3 m 1 m 2 0.1 eV. Consequently, | m ee | takes a value close to m 1 c 2 12 + m 2 s 2 12 0.05 eV in the case of ρ σ (or m 1 c 2 12 − m 2 s 2 12 0.02 eV in the case of ρ π/2 + σ) for vanishingly small m 3 and m 1 c 2 12 + m 2 s 2 12 for m 3 m 1 m 2 0.1 eV.

A&B&C&F&G
In 42, 35). In total, the number of independent constraint equations in the case under study gets increased by three compared to in the general case. So three neutrino mass sum rules will arise.
The neutrino masses in combination with the Majorana CP phases in Eq. (25) can be simplified to some extent by taking the relations in Eqs. (30,42,35). Thereinto, Im m 1 e 2iρ and Im m 2 e 2iσ become vanishing while Re m 1 e 2iρ , Re m 2 e 2iσ and m 3 can be written as in Eq. (36). So the desired neutrino mass sum rules are given by Eq. (37) and ρ, σ = 0 or π/2. As discussed in subsection 3.2, these sum rules can only be fulfilled in the IH case. It turns out that only for the combination [ρ, σ] = [π/2, 0] can Eq. (37) have a realistic solution (m 3 0.02 eV) in which case | m ee | takes a value of 0.02 eV.

A&B&D&E&G
In the case of two sides of equations A, B, D, E and G being vanishing, Eqs. (19, 21, 23) become ineffective. We are thus left with no constraint equations. But there are eight new constraint equations given by Eqs. (27,29,34) which lead to the relations forR αβ andĪ αβ in Eqs. (30,45,35). In total, the number of independent constraint equations in the case under study gets increased by three compared to in the general case. So three neutrino mass sum rules will arise.
The neutrino masses in combination with the Majorana CP phases in Eq. (25) can be simplified to some extent by taking the relations in Eqs. (30,42,35). Thereinto, Re m 1 e 2iρ and Re m 2 e 2iσ becomeR ee while Im m 1 e 2iρ , Im m 2 e 2iσ and m 3 can be written as in Eqs. (31,36). So the desired neutrino mass sum rules are given by Eqs. (32, 52). By taking the relations given by Eq. (52) in Eq. (32), one arrives at a neutrino mass sum rule as Unfortunately, this sum rule has no chance to be in agreement with the experimental results.

Summary
Motivated by the fact that the current neutrino oscillation data is consistent with maximal atmospheric mixing angle and leptonic CP violation, we derive in a novel approach the possible textures of neutrino mass matrix that can naturally lead us to θ 23 = π/4 and δ = −π/2. In order to evade the uncertainties created by the unphysical phases, we work on the effective neutrino mass matrixM ν instead of M ν itself. Since the unphysical phases have cancelled out inM ν , its twelve componentsR αβ andĪ αβ are not all independent but subject to three constraint equations. After imposing the conditions θ 23 = π/4 and δ = −π/2, there will be five independent constraint equations forR αβ andĪ αβ . We derive these constraint equations (i.e., Eqs. (20, 24) and three independent ones from Eq. (22)) by eliminating θ 12 and θ 13 in Eq. (16) in the general case where none of equations A-G has its two sides vanish. On the basis of this, we further study the possible textures ofM ν by considering that one or more of equations A-G may have their two sides vanish.
When an equation has its two sides vanish, the constraint equation(s) resulting from it will become ineffective. But the fact that its two sides vanish itself brings about two new constraint equations. So the number of independent constraint equations gets increased compared to in the general case. When this number gets increased by one (and so on), there will correspondingly be one (and so on) neutrino mass sum rules relating the neutrino masses and Majorana CP phases.
Thanks to the observations that equations A and G (or C and F or D and E) always have their two sides vanish simultaneously and equations E and F (or A, B, C, D and G) are not allowed to have their two sides vanish simultaneously, one just needs to consider the cases where equations A&G, B, C&F, D&E, A&B&G, A&C&F&G, A&D&E&G, B&C&F, B&D&E, A&B&C&F&G or A&B&D&E&G have their two sides vanish. In the case of two sides of equations A&G, B, C&F or D&E being vanishing, there is one neutrino mass sum rule. In the case of two sides of equations A&B&G, A&C&F&G, A&D&E&G, B&C&F or B&D&E being vanishing, there are two neutrino mass sum rules. In the case of two sides of A&B&C&F&G or A&B&D&E&G being vanishing, there are three neutrino mass sum rules. The neutrino mass sum rule Eq. (37) arising from the vanishing of two sides of equation B can only be fulfilled in the IH case. By taking φ e = π/2 and φ µ = −φ τ , the texture ofM ν obtained in the case of two sides of equations A&C&F&G being vanishing can reproduce the specific texture given by the neutrino µ-τ reflection symmetry. In the case of two sides of equations A&B&C&F&G, the unknown neutrino parameters can be completely determined: the neutrino masses are of the inverted hierarchy with m 3 0.02 eV while the Majorana CP phases are [ρ, σ] = [π/2, 0]. But in the case of two sides of equations A&B&D&E&G, the resulting neutrino mass sum rule has no chance to be in agreement with the experimental results.
Finally, we note that the results obtained in this work can be further studied from two aspects: On the one hand, one can study the origins of these special textures from some underlying flavor symmetries in the lepton sector [10]. On the other hand, one can study the breaking effects of these special textures so as to accommodate the deviations of θ 23 and δ from π/4 and −π/2 [11].