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 μ-τ 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 basis of charged lepton mass matrix M l being diagonal, the neutrino mixing matrix U [2,3] 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 parametrization, 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 with 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 that can be used to testify the Majorana nature of neutrinos. And φ e,µ,τ are called unphysical phases since they can be removed by the redefinitions of charged lepton fields. Furthermore, 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 neutrino oscillation experiments by now give the following results for the neutrino mass squared differences [4]  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). Besides, the absolute neutrino mass scale or equivalently the lightest neutrino mass (m 1 in the NH case or m 3 in the IH case) also 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 • , (1.5) in the NH case, or sin 2 θ 13 = 0.02179 ± 0.00076 , sin 2 θ 23 = 0.587 ± 0.022 , δ = 277 • ± 43 • , (1.6) in the IH case, while θ 12 takes the value sin 2 θ 12 = 0.306 ± 0.012 in either case [4]. But 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 Dirac CP phase (δ = −π/2). These remarkable results may point towards some special texture of M ν . In this regard, the specific texture given by the µ-τ reflection symmetry  serves as a unique example. This symmetry is defined as follows: in the basis of M l being diagonal, M ν should keep invariant under a combination of the µ-τ interchange and CP conjugate operations and is characterized by where M αβ denotes the αβ element of M ν (for α, β = e, µ, τ ). Such a texture leads to the following predictions for the neutrino mixing parameters [35,36] φ e = π 2 , φ µ = −φ τ , θ 23 = π 4 , δ = ± π 2 , ρ, σ = 0 or π 2 . (1.9) The purpose of this paper is to derive in a novel approach the possible textures of neutrino mass matrix that can give θ 23 = π/4 and δ = −π/2 [37][38][39][40][41]. Such a study may help us reveal JHEP10(2018)106 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 µ-τ reflection symmetry. For these textures, some neutrino mass sum rules [42][43][44][45][46][47] 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 characteristics of 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 due to the unphysical phases can be evaded. Two immediate comments are given as follows: (1) One can recover the results for M ν from those for M 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 ofR αβ = Re(M αβ ) andĪ αβ = Im (M αβ

JHEP10(2018)106
where the symbol EF is used to indicate that this constraint equation results from equations E and F. 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 which in terms ofR αβ andĪ αβ are respectively expressed as (2.14) 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.

JHEP10(2018)106
Finally, we obtain a constraint equation as by relating the expressions for θ 13 derived from equations A and G. Its expression in terms ofR αβ andĪ αβ appears as (2.16) To sum up, a total of five independent constraint equations for the components ofM ν (i.e., eqs. (2.12), (2.16) and three independent ones from eq. (2.14)) will arise from the eliminations of θ 12 and θ 13 in eq. (2.8). 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 intrinsic 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

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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. But the fact that its two sides vanish itself will bring about two new constraint equations. It turns out that 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. (2.9) and thus equation EF 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 rule as we will see. In the next section, all the possible cases where one or more of equations A-G have their two sides vanish will be examined. Before doing that, we make a few 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 being vanishing (i.e., R 23 1 = I 12 1 = 0), as a result of the relation I 12 2 = c 13 I 12 1 − s 13 R 23 1 , two sides of equation F are also vanishing (i.e., I 12 2 = I 11 2 − I 22 2 = 0). The reverse is also true. So equations C and F always have their two sides vanish simultaneously. And in such a case one has (3) In the case of two sides of equation D being vanishing (i.e., I 23 1 = R 12 1 = 0), as a result of the relation R 12 2 = c 13 R 12 1 + s 13 I 23 1 , two sides of equation E are also vanishing (i.e., . The reverse is also true. So equations D and E always have their two sides vanish simultaneously. And in such a case one has (4) Equations E and F (or A, B, C, D and G) are not allowed to have their two sides vanish simultaneously. Otherwise, θ 12 (or θ 13 ) would be free of any constraint and have no reason to take the measured value. For these observations, we just need 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.

JHEP10(2018)106 3 Various textures
For later use, we give the expressions for the elements ofM ν in terms of the physical parameters which are obtained in the way as In the calculations, θ 23 = π/4 and δ = −π/2 have been input.

A&G
In the case of two sides of equations A and G being vanishing, equations AB, AC, AD and AG become ineffective. We are left with three independent constraint equations (i.e., equation EF and two independent ones of equations BC, BD and CD). But, as discussed at the end of section 2, there are three new constraint equations which are given by eq. (2.19) and lead to the following relations forR αβ andĪ αβ By taking these relations, the expressions for the surviving constraint equations can be simplified to some extent. In total, the number of independent constraint equations gets increased by one compared to in the general case. So one neutrino mass sum rule will arise. With the help of eq. (3.1), one can easily get the desired neutrino mass sum rule as  arises from eq. (2.17) under the condition of eq. (3.3). It is found that eq. (3.5) together with eq. (3.3) will 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. (2) In the case of m 1 = 0, one immediately obtains σ = 0 or π/2. As a result of Re(m 1 e 2iρ ) = Im(m 1 e 2iρ ) = Im(m 2 e 2iσ ) = 0, two additional constraint equations arise from eq.

B
In the case of two sides of equation B being vanishing, equations AB, BC and BD become ineffective. We are left with four independent constraint equations (i.e., equations AG, EF and two independent ones 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 can be simplified to some extent. In total, the number of independent constraint equations gets increased by one compared to in the general case. So one neutrino mass sum rule will arise.
With the help of eq. one can see that this sum rule can never be fulfilled in the NH case. In the case of cos 2ρ = 0 (or cos 2σ = 0) where an additional constraint equation as

C&F
In the case of two sides of equations C and F being vanishing, equations AC, BC, CD and EF become ineffective. We are left with three independent constraint equations (i.e., equation AG and two independent ones of equations AB, AD and BD). But, as discussed at the end of section 2, there are three new constraint equations which are given by eq. (2.20) and lead to the following relations forR αβ andĪ αβ By taking these relations, the expressions for the surviving constraint equations can be simplified to some extent. In total, the number of independent constraint equations gets increased by one compared to in the general case. So one neutrino mass sum rule will arise. With the help of eq. (3.1), one can easily get the desired neutrino mass sum rule as arises from eq. (2.17) under the condition of eq. (3.12). It is found that eq. (3.14) together with eq. (3.12) will 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. 0.004 eV for σ = π/2. (3) In the case of sin 2ρ and sin 2σ = 0, we present sin 2σ/ sin 2ρ as a function of the lightest neutrino mass in the NH and IH cases in figure 3. 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 for vanishingly small m 1 but approaches 1 for 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 EF become ineffective. We are left with three independent constraint equations (i.e., equation AG and two independent ones of equations AB, AC and BC). But, as discussed at the end of section 2, there are three new constraint equations which are given by eq.  and lead to the following relations forR αβ andĪ αβ By taking these relations, the expressions for the surviving constraint equations can be simplified to some extent. In total, the number of independent constraint equations gets increased by one compared to in the general case. So one neutrino mass sum rule will arise. With the help of eq. In the case of cos 2ρ and 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, AG, BC and BD become ineffective. We are left with two constraint equations (i.e., equations CD and EF). But there are five new constraint equations which are given by eqs. (2.19), (3.7) and lead to the following relations forR αβ andĪ αβ (see eqs.
It is easy to see that these relations can be recombined intō By taking these relations, the expressions for the surviving constraint equations can be simplified to some extent. In total, the number of independent constraint equations gets increased by two compared to in the general case. So two neutrino mass sum rules will arise. It turns out that the desired neutrino mass sum rules are the same as those in eqs. (3.4), (3.9). This can be verified by taking the relations in eq. (3.18) in the expressions for the neutrino masses in combination with the Majorana CP phases in eq. (2.17). As discussed in subsection 3.2, these sum rules can only be fulfilled in the IH case. In figure 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 π/2 and 0. As discussed in subsection 3.1, sin 2ρ = sin 2σ = 0 (which gives an additional constraint equation given by eq. (3.5)) will 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. which immediately leads us to | m ee | = m 3 .

A&C&F&G
In It is easy to see that these relations can be recombined intō By taking these relations, the expression for the surviving constraint equation can be simplified to some extent. It is interesting to find that the texture thus obtained can reproduce the specific texture of M ν given by the µ-τ reflection symmetry: in view of the definitionM αβ = M αβ e −i(φ α +φ β ) , the relations in eq. (3.22) will become those in eq. (1.8) by

A&D&E&G
In  In total, the number of independent constraint equations gets increased by two compared to in the general case. So two neutrino mass sum rules will arise. It turns out that the desired neutrino mass sum rules are the same as those in eqs. (3.4), (3.16). This can be verified by taking the relations in eq. (3.23) in the expressions for the neutrino masses in combination with the Majorana CP phases in eq. (2.17). In figures 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 subsection 3.4, cos 2ρ = cos 2σ = 0 gives an additional constraint equation given by eq. 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 c 2 13 + m 2 s 2 12 c 2 13 ± m 3 s 2 13 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 c 2 13 + m 2 s 2 12 c 2 13 ± m 3 s 2 13 in the whole mass range.

B&C&F
In the case of two sides of equations B, C and F being vanishing, equations AB, AC, BC, BD, CD and EF become ineffective. We are left with two constraint equations (i.e., equations AD and AG). But there are five new constraint equations which are given by eqs. (3.7), (2.20) and lead to the following relations forR αβ andĪ αβ (see eqs. (3.8), (3.12)) By taking these relations, the expressions for the surviving constraint equations can be simplified to some extent. In total, the number of independent constraint equations gets increased by two compared to in the general case. So two neutrino mass sum rules will arise.
It turns out that the desired neutrino mass sum rules are the same as those in eqs. (3.9), (3.13). This can be verified by taking the relations in eq. (3.25) in the expressions for the neutrino masses in combination with the Majorana CP phases in eq. (2.17). As discussed in subsection 3.2, these sum rules can only be fulfilled in the IH case. In figure 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 gets ρ σ or π/2 − σ from eq. 0.020 eV. As discussed in subsection 3.3, sin 2ρ = sin 2σ = 0 (which gives an additional constraint equation given by eq. (3.14)) will lead us to a situation where two sides of equations A and G 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.

B&D&E
In the case of two sides of equations B, D and E being vanishing, equations AB, AD, BC, BD, CD and EF become ineffective. We are left with two constraint equations (i.e., equations AC and AG). But there are five new constraint equations which are given by eqs. (3.7), (2.21) and lead to the following relations forR αβ andĪ αβ (see eqs. (3.8), (3.15)) It is easy to see that these relations can be recombined intō By taking these relations, the expressions for the surviving constraint equations can be simplified to some extent. In total, the number of independent constraint equations gets increased by two compared to in the general case. So two neutrino mass sum rules will arise. It turns out that the desired neutrino mass sum rules are the same as those in eqs. (3.9), (3.16). This can be verified by taking the relations in eq.
It is easy to see that these relations can be recombined intō which suggest that this case can be viewed as a result of µ-τ reflection symmetry in combination with the condition ofM eµ (and equivalentlyM eτ ) being real if we take φ e = π/2 and φ µ = −φ τ . In total, the number of independent constraint equations gets increased by three compared to in the general case. So three neutrino mass sum rules will arise. It turns out that the desired neutrino mass sum rules are given by eq. (3.9) and ρ, σ = 0 or π/2. This can be verified by taking the relations in eq. (3.29) in the expressions for the neutrino masses in combination with the Majorana CP phases in eq. (2.17). As discussed in subsection 3.2, these sum rules can only be fulfilled in the IH case. It is found that only for the combination [ρ, σ] = [π/2, 0] can eq. (3.9) have a realistic solution m 3 = 0.021 eV at which | m ee | takes a value of 0.020 eV. In this case, one can say that the real part ofM ν still respects the µ-τ interchange symmetry. (7) In the case of two sides of equations A, D, E and G being vanishing, the relation I µµ =Ī τ τ still holds, whileR eµ andR eτ vanish. (8) In the case of two sides of equations B, C and F being vanishing, the relationR µµ =R τ τ still holds, whileĪ eµ andĪ eτ vanish. (9) In the case of two sides of equations B, D and E being vanishing, the relationsĪ eµ = −Ī eτ and I µµ =Ī τ τ still hold. In this case, one can say that the imaginary part ofM ν still respects the µ-τ interchange symmetry. (10) In the case of two sides of equations A, B, C, F and G being vanishing, the relationsR eµ = −R eτ andR µµ =R τ τ still hold, whileĪ eµ andĪ eτ vanish. In this case, one can say that the real part ofM ν still respects the µ-τ interchange symmetry. (11) In the case of two sides of equations A, B, D, E and G being vanishing, the relationsĪ eµ = −Ī eτ andĪ µµ =Ī τ τ still hold, whileR eµ andR eτ vanish. In this case, one can say that the imaginary part ofM ν still respects the µ-τ interchange symmetry.

Summary
Motivated by the fact that the current neutrino oscillation data is consistent with maximal atmospheric mixing angle and Dirac CP phase, we derive in a novel approach the possible textures of neutrino mass matrix that can 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 are five independent constraint equations forR αβ andĪ αβ . We derive these constraint equations (i.e., eqs. (2.12), (2.16) and three independent ones from eq. (2.14)) by eliminating θ 12 and θ 13 in eq. (2.8) 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 some 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. (3.9) 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 JHEP10(2018)106 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 being vanishing, the unknown neutrino parameters can be completely determined: the neutrino masses are of the inverted hierarchy with m 3 = 0.021 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 being vanishing, the resulting neutrino mass sum rules have no chance to be in agreement with the experimental results. As discussed at the end of section 3, the various cases we have studied can find a motivation from the partial µ-τ interchange symmetry.
Finally, we point out 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 [54,55]. 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.