On the textures of neutrino mass matrix for maximal atmospheric mixing angle and Dirac CP phase
Zhi-Cheng Liu Chong-Xing Yue Zhen-hua Zhao






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








Zhi-Cheng Liu, Chong-Xing Yue and Zhen-hua Zhao [E-mail: zhzhao@itp.ac.cn]   

Department of Physics, Liaoning Normal University, Dalian 116029, China







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 <cit.>. In the basis of charged lepton mass matrix M^_l being diagonal, the neutrino mixing matrix U <cit.> originates from diagonalization of the neutrino mass matrix M^_ν in a manner as

    U^† M^_ν U^* =  Diag( m^_1, m^_2, m^_3 )  ,

with m^_i (for i=1, 2, 3) being the neutrino masses.
In the standard parametrization, U reads

    U = P^_l O^_23 U^_13 O^_12 P^_ν ,

where P^_l =  Diag(e^ iϕ_e,  e^ iϕ_μ,  e^ iϕ_τ) and P^_ν =  Diag(e^ iρ,  e^ iσ, 1 ) are two diagonal phase matrices, and

    O^_23 = [       1       0       0;       0   c^_23   s^_23;       0 - s^_23   c^_23 ] , 
    U^_13 = [         c^_13             0  s^_13 e^- iδ;             0             1             0; - s^_13 e^ iδ             0         c^_13 ] , 
    O^_12 = [   c^_12   s^_12       0; - s^_12   c^_12       0;       0       0       1 ] ,

with c^_ij = cosθ^_ij and s^_ij = sinθ^_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 Δ 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 <cit.>

    Δ m^2_21 = (7.50^+0.19_-0.17)× 10^-5  eV^2 , 
    |Δ m^2_31| = (2.524^+0.039_-0.040)× 10^-3  eV^2 .

Note that the sign of Δ 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^2θ^_13 = 0.02166 ± 0.00075   , sin^2θ^_23 = 0.441 ± 0.024  , δ = 261^∘± 55^∘ ,

in the NH case, or

    sin^2θ^_13 = 0.02179 ± 0.00076   , sin^2θ^_23 = 0.587 ± 0.022  , δ = 277^∘± 43^∘ ,

in the IH case, while θ^_12 takes the value sin^2θ^_12 = 0.306 ± 0.012
in either case <cit.>. 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 <cit.> 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

    ν^_e ↔ν^c_e  , ν^_μ↔ν^c_τ ,
    ν^_τ↔ν^c_μ ,

and is characterized by

    M^_eμ = M^*_eτ ,  M^_μμ = M^*_ττ , 
    M^_ee  and M^_μτ  being  real ,

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 <cit.>

    ϕ^_e = π/2 , ϕ^_μ = - ϕ^_τ , θ^_23 = π/4 , δ = ±π/2 , ρ, σ = 0   or π/2 .

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 <cit.>. 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 μ-τ reflection symmetry. For these textures, some neutrino mass sum rules <cit.> 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 matrix M̅^_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 on M̅^_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 M̅^_ν by simply making the replacements M̅^_αβ = M^_αβ e^- i(ϕ^_α+ ϕ^_β) with M̅^_αβ being the αβ element of M̅^_ν. (2) Since M̅^_ν only has nine dfs, its twelve components R̅^_αβ =  Re(M̅^_αβ) and
I̅^_αβ =  Im(M̅^_αβ) are not all independent but subject to three constraint equations.

To proceed, we diagonalize M̅^_ν to give the expressions for the physical parameters in terms of R̅^_αβ =  Re(M̅^_αβ) and
I̅^_αβ =  Im(M̅^_αβ). From Eqs. (<ref>-<ref>), one gets

    O^T_12 U^†_13 O^T_23M̅^_ν O^_23 U^*_13 O^_12 =  Diag( m^_1 e^2 iρ, m^_2e^2 iσ, m^_3 )  .

In order to simplify the expressions, we define the following three matrices after the rotations O^_23, U^_13 and O^_12 are implemented in succession

    M^_1  = O^T_23M̅^_ν O^_23 , 
    M^_2 =  U^†_13 M^_1 U^*_13 , 
    M^_3  =  O^T_12 M^_2 O^_12 .

After taking θ^_23 = π/4 and δ = - π/2, the elements of these three matrices appear as

    M^11_1 = M̅^_ee ,  M^12_1 = M̅^_e μ - M̅^_e τ/√(2) ,   M^13_1 = M̅^_e μ + M̅^_e τ/√(2) ,  
           M^22_1 = M̅^_μμ + M̅^_ττ/2 - M̅^_μτ ,  M^23_1 = M̅^_μμ - M̅^_ττ/2 , 
    M^33_1 = M̅^_μμ + M̅^_ττ/2 + M̅^_μτ ,


    M^11_2 = c^2_13 M^11_1 -  isin 2 θ^_13 M^13_1 - s^2_13 M^33_1  , 
    M^12_2 = c^_13 M^12_1 -  i s^_13 M^23_1  , 
           M^13_2 = cos 2 θ^_13 M^13_1 -  i/2sin 2 θ^_13( M^11_1 + M^33_1 )  ,  M^22_2 = M^22_1  , 
           M^23_2 = c^_13 M^23_1 -  i s^_13 M^12_1  , 
    M^33_2 = c^2_13 M^33_1 -  isin 2 θ^_13 M^13_1 - s^2_13 M^11_1  ,

and

    M^11_3 = c^2_12 M^11_2 - sin 2 θ^_12 M^12_2 + s^2_12 M^22_2  , 
           M^12_3 = cos 2 θ^_12 M^12_2 + 1/2sin 2 θ^_12( M^11_2 - M^22_2)  ,  M^13_3 = c^_12 M^13_2 - s^_12 M^23_2  ,  
           M^22_3 = s^2_12 M^11_2 + sin 2 θ^_12 M^12_2 + c^2_12 M^22_2  ,    M^23_3 = s^_12 M^13_2 + c^_12 M^23_2  ,   M^33_3 = M^33_2  ,

where M^ij_k stands for the ij element of M^_k (for i,j,k=1,2,3). In order for the diagonalization process to work, one must have

    Re/Im( M^13_2 ) =  Re/Im( M^23_2 ) =  Re/Im( M^12_3 ) =  Im( M^33_3 ) = 0  ,

and

    m^_1 e^2 iρ = M^11_3 ,  m^_2 e^2 iσ = M^22_3 ,  m^_3 = M^33_3 .

The seven conditions in Eq. (<ref>) which will be referred to as A-G in order are explicitly written as

    A:   2 cos 2 θ^_13 R^13_1 = - sin 2 θ^_13( I^11_1 + I^33_1 )  , 
           B:   2 cos 2 θ^_13 I^13_1 =  sin 2 θ^_13( R^11_1 + R^33_1 )  , 
           C:   c^_13 R^23_1 = - s^_13 I^12_1  , 
           D:   c^_13 I^23_1 =   s^_13 R^12_1  , 
           E:   2 cos 2 θ^_12 R^12_2 = - sin 2 θ^_12( R^11_2 - R^22_2)  , 
           F:   2 cos 2 θ^_12 I^12_2 = - sin 2 θ^_12( I^11_2 - I^22_2)  , 
           G:  sin 2 θ^_13 R^13_1 = c^2_13 I^33_1 - s^2_13 I^11_1  ,

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 of M̅^_ν can be directly read from Eq. (<ref>). For example, equation E gives θ^_12 as

    tan 2 θ^_12 = -2 R^12_2/ R^11_2 - R^22_2 ,

while equation F gives θ^_12 as

    tan 2 θ^_12 = -2 I^12_2/ I^11_2 - I^22_2 .

By relating these two expressions for θ^_12, a constraint equation for the components of M̅^_ν arises as

    EF:    R^12_2 ( I^11_2 - I^22_2) = I^12_2 ( R^11_2 - R^22_2)  ,

where the symbol EF is used to indicate that this constraint equation results from equations E and F.
It can be expressed in terms of R̅^_αβ and I̅^_αβ by taking the expressions

    R^12_2     =    sgn(R̅^_e μ - R̅^_e τ) √(1/2(R̅^_e μ - R̅^_e τ)^2 + 1/4( I̅^_μμ - I̅^_ττ)^2 ) , 
    
    I^12_2
       =    sgn(I̅^_e μ - I̅^_e τ) √(1/2(I̅^_e μ - I̅^_e τ)^2 + 1/4( R̅^_μμ - R̅^_ττ)^2 ) , 
    
    I^11_2 - I^22_2     =    I̅^_ee - I̅^_μμ - I̅^_ττ , 
    
    R^11_2 - R^22_2     =   R̅^_ee + R̅^_μτ/2 - 3/4(R̅^_μμ + R̅^_ττ) +  sgn(I̅^_e μ + I̅^_e τ) 
          ×√(1/2( I̅^_e μ + I̅^_e τ)^2 + 1/4(R̅^_ee + R̅^_μτ + R̅^_μμ + R̅^_ττ/2)^2 ) .

In a similar way, one will arrive at the following constraint equations by relating the expressions for θ^_13 derived from equations A-D

    AB:    R^13_1 ( R^11_1 + R^33_1 ) = - I^13_1 ( I^11_1 + I^33_1 )  , 
           AC:    I^12_1 R^23_1 ( I^11_1 + I^33_1 ) = R^13_1 [ (I^12_1)^2 - ( R^23_1)^2 ]  , 
           AD:    I^23_1 R^12_1 ( I^11_1 + I^33_1 ) = - R^13_1 [ (R^12_1)^2 - ( I^23_1 )^2 ]  , 
           BC:    - I^12_1 R^23_1 ( R^11_1 + R^33_1 ) =  I^13_1 [ (I^12_1)^2 - ( R^23_1)^2 ]  , 
           BD:    I^23_1 R^12_1 ( R^11_1 + R^33_1 ) = I^13_1 [ (R^12_1)^2 - ( I^23_1 )^2 ]  , 
           CD:    R^23_1 R^12_1 = - I^23_1 I^12_1  ,

which in terms of R̅^_αβ and I̅^_αβ are respectively expressed as

    (R̅_e μ + R̅_e τ) (R̅_ee + R̅_μτ + R̅_μμ + R̅_ττ/2)
    = - (I̅_e μ + I̅_e τ) (I̅_ee + I̅_μτ + I̅_μμ + I̅_ττ/2)  , 
          (I̅_e μ - I̅_e τ) (R̅_μμ - R̅_ττ)
    (I̅_ee + I̅_μτ + I̅_μμ + I̅_ττ/2)
    = (R̅_e μ + R̅_e τ) [ (I̅_e μ - I̅_e τ)^2  
           - 1/2(R̅_μμ - R̅_ττ)^2 ]
     , 
          ( I̅_μμ - I̅_ττ) (R̅_e μ - R̅_e τ)
    (I̅_ee + I̅_μτ + I̅_μμ + I̅_ττ/2)
    = - (R̅_e μ + R̅_e τ) [ ( R̅_e μ - R̅_e τ)^2 
           - 1/2( I̅_μμ - I̅_ττ)^2 ]  , 
          (I̅_e μ - I̅_e τ) (R̅_μμ - R̅_ττ)
    (R̅_ee + R̅_μτ + R̅_μμ + R̅_ττ/2)
    = - (I̅_e μ + I̅_e τ) [ (I̅_e μ - I̅_e τ)^2 
           - 1/2(R̅_μμ - R̅_ττ)^2 ]
     , 
          ( I̅_μμ - I̅_ττ) (R̅_e μ - R̅_e τ)
    (R̅_ee + R̅_μτ + R̅_μμ + R̅_ττ/2)
    = (I̅_e μ + I̅_e τ) [ ( R̅_e μ - R̅_e τ)^2 
           - 1/2( I̅_μμ - I̅_ττ)^2 ]
     , 
          (R̅_μμ - R̅_ττ) (R̅_e μ - R̅_e τ)
    = - (I̅_μμ - I̅_ττ) (I̅_e μ - I̅_e τ)  .

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

    AG:    - I^11_1 + I^33_1 =  sgn(R^13_1) √(4 (R^13_1)^2 + (I^11_1 + I^33_1)^2) ,

by relating the expressions for θ^_13 derived from equations A and G.
Its expression in terms of R̅^_αβ and I̅^_αβ appears as

    - I̅^_ee + I̅^_μτ + I̅^_μμ + I̅^_ττ/2  = sgn(R̅^_e μ + R̅^_e τ) √(2 (R̅^_e μ + R̅^_e τ)^2 +  (I̅^_ee + I̅^_μτ + I̅^_μμ + I̅^_ττ/2)^2) .

To sum up, a total of five independent constraint equations for the components of M̅^_ν (i.e., Eqs. (<ref>, <ref>) and three independent ones from Eq. (<ref>)) will arise from the eliminations of θ^_12 and θ^_13 in Eq. (<ref>). 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 of M̅^_ν. At last, one can say that an M̅^_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. (<ref>) in Eq. (<ref>), the neutrino masses in combination with the Majorana CP phases are expressed as

    Re(m^_1 e^2 iρ)    =   R^11_2 + R^22_2/2
    -  sgn(R^12_2) √((R^12_2)^2 + 1/4(R^11_2 - R^22_2)^2 ) , 
     Im(m^_1 e^2 iρ)    =   I^11_2 + I^22_2/2
    -  sgn(I^12_2) √((I^12_2)^2 + 1/4(I^11_2 - I^22_2)^2 ) , 
     Re(m^_2 e^2 iσ)    =   R^11_2 + R^22_2/2
    +  sgn(R^12_2) √((R^12_2)^2 + 1/4(R^11_2 - R^22_2)^2 ) , 
     Im(m^_2 e^2 iσ)    =   I^11_2 + I^22_2/2
    +  sgn(I^12_2) √((I^12_2)^2 + 1/4(I^11_2 - I^22_2)^2 ) , 
    
    m^_3    =    - R̅^_ee - R̅^_μτ/2 + R̅^_μμ + R̅^_ττ/4 +  sgn(I̅^_e μ + I̅^_e τ) 
          ×√(1/2(I̅^_e μ + I̅^_e τ)^2 + 1/4(R̅^_ee + R̅^_μτ + R̅^_μμ + R̅^_ττ/2)^2) ,

where

    R^11_2 + R^22_2     =   R̅^_ee - 3R̅^_μτ/2 + R̅^_μμ + R̅^_ττ/4 +  sgn(I̅^_e μ + I̅^_e τ) 
          ×√(1/2( I̅^_e μ + I̅^_e τ)^2 + 1/4(R̅^_ee + R̅^_μτ + R̅^_μμ + R̅^_ττ/2)^2 ) , 
    
    I^11_2 + I^22_2
       =   I̅^_ee - 2 I̅^_μτ ,

while the expressions for R^12_2, I^12_2, R^11_2-  R^22_2 and R^11_2-  R^22_2 have been given in Eq. (<ref>).

We point out that the above results are derived in the general case where none of equations A-G has its two sides vanish. However, there is the possibility that M̅^_ν has such a special texture that one (or more) of equations A-G has its two sides vanish and thus always holds irrespective of the values of θ^_12 and θ^_13. Since the purpose of this paper is to derive the possible textures of neutrino mass matrix that can lead to θ^_23 =π/4 and δ = -π/2, such a possibility should of course be taken into account. 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. (<ref>) 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

    R^13_1 = I^11_1 = I^33_1 = 0  .

(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

    I^11_2 - I^22_2 = R^23_1 = I^12_1 =  0  .

(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., R^12_2 = R^11_2 - R^22_2 =0). The reverse is also true. So equations D and E always have their two sides vanish simultaneously. And in such a case one has

    R^11_2 - R^22_2= I^23_1 = R^12_1 =  0  .

(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.



§ VARIOUS TEXTURES


For later use, we give the expressions for the elements of M̅^_ν in terms of the physical parameters	

    M̅^_ee    =      m^_1 e^2  iρ c^2_12 c^2_13 + m^_2 e^2  iσ s^2_12 c^2_13 - m^_3 s^2_13 , 
    M̅^_eμ    =    1/√(2)[ m^_1 e^2  iρ c^_12(- s^_12 +  i c^_12 s^_13)
    + m^_2 e^2  iσ s^_12(c^_12+  i s^_12 s^_13) +  i m^_3 s^_13] c^_13 , 
    M̅^_eτ    =    1/√(2)[ m^_1 e^2  iρ c^_12(s^_12+  i c^_12 s^_13)
    + m^_2 e^2  iσ s^_12( - c^_12 +  i s^_12 s^_13) +  i m^_3 s^_13] c^_13 , 
    M̅^_μμ    =    1/2[ m^_1 e^2  iρ(s^_12- i c^_12 s^_13)^2
    + m^_2 e^2  iσ(c^_12+  i s^_12 s^_13)^2 + m^_3 c^2_13]  , 
    M̅^_μτ    =    1/2[ - m^_1 e^2  iρ(s^2_12 + c^2_12 s^2_13) - m^_2 e^2  iσ(c^2_12+s^2_12 s^2_13) + m^_3 c^2_13]  , 
    M̅^_ττ    =    1/2[ m^_1 e^2  iρ(s^_12+ i c^_12 s^_13)^2
    + m^_2 e^2  iσ(c^_12- i s^_12 s^_13)^2 + m^_3 c^2_13]  ,

which are obtained in the way as

    M̅^_ν = O^_23  U^_13 O^_12 Diag( m^_1 e^2 iρ, m^_2e^2 iσ, m^_3 ) O^T_12  U^T_13 O^T_23 .

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. (<ref>) and lead to the following relations for R̅^_αβ and I̅^_αβ

    R̅^_e μ = -R̅^_e τ , I̅^_ee = 0  ,  -2 I̅^_μτ = I̅^_μμ + I̅^_ττ .

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. (<ref>), one can easily get the desired neutrino mass sum rule as

    m^_1 c^2_12sin 2 ρ + m^_2 s^2_12sin 2 σ = 0  ,

from any one of the relations in Eq. (<ref>). This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>).
We discuss the implications of this sum rule in three cases:
(1) In the case of sin 2ρ = sin 2σ = 0, the neutrino masses become independent of the Majorana CP phases. As a result of Im(m^_1 e^2  iρ) =  Im(m^_2 e^2  iσ) = 0, an additional constraint equation

    I̅^_μτ = 0  ,

arises from Eq. (<ref>) under the condition of Eq. (<ref>). It is found that Eq. (<ref>) together with Eq. (<ref>) 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^2  iρ) =  Im(m^_1 e^2  iρ) =  Im(m^_2 e^2  iσ) = 0, two additional constraint equations
arise from Eq. (<ref>) under the condition of Eq. (<ref>). In this case, the effective neutrino mass

    | ⟨ m ⟩^_ee| = | m^_1 e^2 iρ c^2_12 c^2_13 + m^_2 e^2 iσ s^2_12 c^2_13 + m^_3 s^2_13 e^-2 iδ|  ,

which controls the rates of neutrinoless double beta decays takes a value of m^_2 s^2_12 c^2_13- m^_3 s^2_13≃ 0.002 eV for σ = 0 or m^_2 s^2_12 c^2_13 + m^_3 s^2_13≃ 0.004 eV for σ = π/2.
(3) In the case of sin 2ρ and sin 2σ≠ 0, we present sin2σ/sin2ρ 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, the value is very small for vanishingly small m^_1 but approaches -2.27 for 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 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

    I^13_1 = R^11_1 + R^33_1 = 0  ,

which lead to the following relations for R̅^_αβ and I̅^_αβ

    I̅^_e μ = - I̅^_e τ ,  -2 ( R̅^_ee + R̅^_μτ) =  R̅^_μμ + R̅^_ττ .

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. (<ref>), one can easily get the desired neutrino mass sum rule as

    m^_1 c^2_12cos 2 ρ + m^_2 s^2_12cos 2 σ + m^_3 = 0  ,

from any one of the relations in Eq. (<ref>). This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>).
With the help of the inequality

    m^_1 c^2_12cos 2 ρ + m^_2 s^2_12cos 2 σ≥ - m^_1 c^2_12  - m^_2 s^2_12 > - m^_2  ,

one can see that this sum rule can never be fulfilled in the NH case.
In the case of cos 2 ρ =0 (or cos 2 σ =0) where an additional constraint equation as

    - R̅_μτ (  or R̅_μτ ) =  sgn(R̅_e μ - R̅_e τ)
    √(1/2(R̅_e μ - R̅_e τ)^2 + 1/4( I̅_μμ - I̅_ττ)^2 + (R̅_ee + R̅_μτ)^2 ) ,

arises from Eq. (<ref>) under the condition of Eq. (<ref>),
σ (or ρ) and | ⟨ m ⟩^_ee| are presented as functions of m^_3 in Fig. 2. For vanishingly small m^_3, σ (or ρ) takes a value close to π/4 or 3π/4 while | ⟨ m ⟩^_ee| takes a value close to m^_1 c^2_12 c^2_13 + m^_2 s^2_12 c^2_13≃ 0.049 eV for ρ≃σ or m^_1 c^2_12 c^2_13 - m^_2 s^2_12 c^2_13≃ 0.019 eV for ρ≃σ + π/2. When m^_3 takes the upper value m^_2 s^2_12≃ 0.016 eV (or m^_1 c^2_12≃ 0.048 eV), σ (or ρ) becomes π/2 while | ⟨ m ⟩^_ee| becomes √((m^_1 c^2_12 c^2_13)^2 + (m^_2 s^2_12 c^2_13 + m^_3 s^2_13)^2)≃ 0.039 eV (or √((m^_2 s^2_12 c^2_13)^2 + (m^_1 c^2_12 c^2_13 + m^_3 s^2_13)^2)≃ 0.052 eV).






 §.§ 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. (<ref>) and lead to the following relations for R̅^_αβ and I̅^_αβ

    I̅^_ee = I̅^_μμ + I̅^_ττ , R̅^_μμ = R̅^_ττ , I̅^_e μ = I̅^_e τ .

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. (<ref>), one can easily get the desired neutrino mass sum rule as

    m^_1 sin 2 ρ - m^_2 sin 2 σ = 0  ,

from any one of the relations in Eq. (<ref>). This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>).
We discuss the implications of this sum rule in three cases:
(1) In the case of sin 2ρ = sin 2 σ = 0, the neutrino masses become independent of the Majorana CP phases. As a result of Im(m^_1 e^2  iρ) =  Im(m^_2 e^2  iσ) = 0, an additional constraint equation

    I̅^_μμ + I̅^_ττ  = 2 I̅^_μτ ,

arises from Eq. (<ref>) under the condition of Eq. (<ref>).
It is found that Eq. (<ref>) together with Eq. (<ref>) 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.
(2) In the case of m^_1 = 0, one immediately obtains σ = 0 or π/2.
As a result of Re(m^_1 e^2  iρ) =  Im(m^_1 e^2  iρ) =  Im(m^_2 e^2  iσ) = 0, two additional constraint equations
arise from Eq. (<ref>) under the condition of Eq. (<ref>). In this case, | ⟨ m ⟩^_ee|
takes a value of m^_2 s^2_12 c^2_13- m^_3 s^2_13≃ 0.002 eV for σ = 0 or m^_2 s^2_12 c^2_13 + m^_3 s^2_13≃ 0.004 eV for σ = π/2.
(3) In the case of sin2 ρ and sin 2σ≠ 0, we present sin2σ/sin2ρ as a function of the lightest neutrino mass in the NH and IH cases in Fig. 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. (<ref>) and lead to the following relations for R̅^_αβ and I̅^_αβ

    R̅^_ee + R̅^_μτ - 3/2(R̅^_μμ +R̅^_ττ)
    = - sgn(I̅^_e μ + I̅^_e τ)
    √(2 ( I̅^_e μ + I̅^_e τ)^2 + (R̅^_ee + R̅^_μτ + R̅^_μμ +R̅^_ττ/2)^2 ) , 
          I̅^_μμ = I̅^_ττ , R̅^_e μ =R̅^_e τ .

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. (<ref>), one can easily get the desired neutrino mass sum rule as

    m^_1 cos 2 ρ - m^_2 cos 2 σ = 0  ,

from any one of the relations in Eq. (<ref>). This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>).
We discuss the implications of this sum rule in three cases:
(1) In the case of cos 2ρ = cos 2σ = 0, the neutrino masses become independent of the Majorana CP phases. As a result of Re(m^_1 e^2  iρ) =  Re(m^_2 e^2  iσ) = 0, an additional constraint equation

    R̅^_μμ + R̅^_ττ = 2 R̅^_μτ ,

arises from Eq. (<ref>) under the condition of Eq. (<ref>). For the possible combinations [ρ, σ] = [π/4, π/4], [π/4, 3π/4], [3π/4, π/4] and [3π/4, 3π/4], we present | ⟨ m ⟩^_ee| as a function of the lightest neutrino mass in the NH and IH cases in Fig. 4: | ⟨ m ⟩^_ee| = √((m^_1 c^2_12 + m^_2 s^2_12)^2 c^4_13 + m^2_3 s^4_13) for ρ = σ or √((m^_1 c^2_12 - m^_2 s^2_12)^2 c^4_13 + m^2_3 s^4_13) for ρ≠σ. (2) In the case of m^_1 = 0, one immediately obtains σ = π/4 or 3π/4.
As a result of Re(m^_1 e^2  iρ) =  Im(m^_1 e^2  iρ) =  Re(m^_2 e^2  iσ) = 0, two additional constraint equations
arise from Eq. (<ref>) under the condition of Eq. (<ref>). In this case, | ⟨ m ⟩^_ee|
takes a value of √(m^2_2 s^4_12 c^4_13 + m^2_3 s^4_13)≃ 0.003 eV. (3) In the case of cos 2ρ and cos 2 σ≠ 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. (<ref>, <ref>) and lead to the following relations for R̅^_αβ and I̅^_αβ (see Eqs. (<ref>, <ref>))

    R̅^_e μ = -R̅^_e τ , I̅^_ee = 0  ,  -2 I̅^_μτ = I̅^_μμ + I̅^_ττ , 
          I̅^_e μ = - I̅^_e τ ,  -2 ( R̅^_ee + R̅^_μτ) =  R̅^_μμ + R̅^_ττ .

It is easy to see that these relations can be recombined into

    M̅^_e μ = -M̅^_e τ ,    -2 ( M̅^_ee + M̅^_μτ) =  M̅^_μμ + M̅^_ττ , M̅^_ee  being  real .

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. (<ref>, <ref>).
This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>).
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 π/2 and 0. As discussed in subsection 3.1, sin 2ρ = sin 2σ = 0 (which gives an additional constraint equation given by Eq. (<ref>)) 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. (<ref>, <ref>) can be reorganized into a single complex equation

    m^_1 c^2_12 e^2 iρ + m^_2 s^2_12 e^2 iσ + m^_3 = 0  ,

which immediately leads us to | ⟨ m ⟩^_ee| = m^_3.





 §.§ A&C&F&G


In the case of two sides of equations A, C, F and G being vanishing, equations AB, AC, AD, AG, BC, CD and EF become ineffective. We are left with only one constraint equation (i.e., equation BD). But there are six new constraint equations which are given by Eqs. (<ref>, <ref>) and lead to the following relations for R̅^_αβ and I̅^_αβ (see Eqs. (<ref>, <ref>))

    R̅^_e μ = -R̅^_e τ ,  I̅^_e μ = I̅^_e τ , R̅^_μμ = R̅^_ττ , I̅^_μμ = - I̅^_ττ , I̅^_ee = I̅^_μτ = 0  .

It is easy to see that these relations can be recombined into

    M̅^_eμ = - M̅^*_eτ , M̅^_μμ = M̅^*_ττ , M̅^_ee  and M̅^_μτ  being  real .

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 definition M̅^_αβ = M^_αβ e^- i(ϕ^_α+ ϕ^_β), the relations in Eq. (<ref>) will become those in Eq. (<ref>) by taking ϕ^_e =π/2 and ϕ^_μ = - ϕ^_τ. 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. (<ref>, <ref>), implying that the Majorana CP phases take trivial values (i.e., ρ, σ = 0 or π/2). This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>). For the possible combinations [ρ, σ] = [0, 0], [0, π/2], [π/2, 0] and [π/2, π/2], we present | ⟨ m ⟩^_ee| as a function of the lightest neutrino mass in the NH and IH cases in Fig. 6. In the NH case, three terms of | ⟨ m ⟩^_ee| add constructively to a maximal level for [ρ, σ] = [π/2, π/2] but will cancel out (i.e., | ⟨ m ⟩^_ee| = 0) at m^_1 = 0.002 eV (or 0.007 eV) for [ρ, σ] = [π/2, 0] (or [0, π/2]). In the IH case,
| ⟨ m ⟩^_ee| is dominated by the first two terms since the third one is highly suppressed. For ρ = σ (or ρ≠σ), | ⟨ m ⟩^_ee| approximates to m^_1 c^2_12 c^2_13 + m^_2 s^2_12 c^2_13 (or m^_1 c^2_12 c^2_13 -  m^_2 s^2_12 c^2_13) .





 §.§ A&D&E&G


In the case of two sides of equations A, D, E and G being vanishing, equations AB, AC, AD, AG, BD, CD and EF become ineffective. We are left with only one constraint equation (i.e., equation BC). But there are six new constraint equations which are given by Eqs. (<ref>, <ref>) and lead to the following relations for R̅^_αβ and I̅^_αβ (see Eqs. (<ref>, <ref>))

    R̅^_ee + R̅^_μτ - 3/2(R̅^_μμ +R̅^_ττ)
    = - sgn(I̅^_e μ + I̅^_e τ)
    √(2 ( I̅^_e μ + I̅^_e τ)^2 + (R̅^_ee + R̅^_μτ + R̅^_μμ +R̅^_ττ/2)^2 ) , 
          R̅^_e μ = R̅^_e τ = I̅^_ee = 0  , I̅^_μμ  = I̅^_ττ = - I̅^_μτ .

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. (<ref>, <ref>).
This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>). 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 subsection 3.4, cos 2ρ = cos 2σ = 0 gives an additional constraint equation given by Eq. (<ref>). At this lower value of m^_1, | ⟨ m ⟩^_ee| takes a value of m^_3 s^2_13. For m^_1 ≃ m^_2 ≃ 0.1 eV, Eqs. (<ref>, <ref>) give

    sin 2 σ/sin 2ρ≃ - c^2_12/s^2_12 , cos 2ρ≃cos 2σ .

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. (<ref>, <ref>) and lead to the following relations for R̅^_αβ and I̅^_αβ (see Eqs. (<ref>, <ref>))

    I̅^_ee = I̅^_μμ + I̅^_ττ , R̅^_μμ = R̅^_ττ = - ( R̅^_ee + R̅^_μτ)  , I̅^_e μ = I̅^_e τ = 0   .

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. (<ref>, <ref>).
This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>). 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 gets ρ≃σ or π/2 -σ from Eq. (<ref>). Eq. (<ref>) 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| approximates to m^_1 c^2_12 c^2_13 + m^_2 s^2_12 c^2_13 for these two mass ranges. For the particular value of m^_3 = 0.021 eV, one of the allowed solutions to Eqs. (<ref>, <ref>) is ρ = π/2 and σ = 0 in which case | ⟨ m ⟩^_ee| becomes m^_1 c^2_12 c^2_13 - m^_2 s^2_12  c^2_13 + m^_3 s^2_13≃ 0.020 eV. As discussed in subsection 3.3, sin 2ρ = sin 2σ = 0 (which gives an additional constraint equation given by Eq. (<ref>)) 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. (<ref>, <ref>) and lead to the following relations for R̅^_αβ and I̅^_αβ (see Eqs. (<ref>, <ref>))

    R̅^_e μ =R̅^_e τ , I̅^_e μ = - I̅^_e τ , R̅^_μμ = - R̅^_ττ , I̅^_μμ = I̅^_ττ , R̅^_ee = - R̅^_μτ .

It is easy to see that these relations can be recombined into

    M̅^_eμ = M̅^*_eτ , M̅^_μμ = - M̅^*_ττ , R̅^_ee = - R̅^_μτ .

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. (<ref>, <ref>).
This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>).
A combination of these sum rules yields

    m^_1 cos 2 ρ = m^_2 cos 2σ = -m^_3  .

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 c^2_13 + m^_2 s^2_12 c^2_13≃ 0.049 eV in the case of ρ≃σ (or m^_1 c^2_12 c^2_13 - m^_2 s^2_12 c^2_13≃ 0.019 eV in the case of ρ≃π/2+ σ) for vanishingly small m^_3 and m^_1 c^2_12 c^2_13 + m^_2 s^2_12 c^2_13+ m^_3 s^2_13 for m^_3 ≃ m^_1 ≃ m^_2 ≃ 0.1 eV.






 §.§ A&B&C&F&G


In the case of two sides of equations A, B, C, F and G being vanishing, equations AB, AC, AD, AG, BC, BD, CD and EF become ineffective. We are left with no constraint equation. But there are eight new constraint equations which are given by Eqs. (<ref>, <ref>, <ref>) and lead to the following relations for R̅^_αβ and I̅^_αβ (see Eqs. (<ref>, <ref>, <ref>))

    R̅^_e μ = -R̅^_e τ ,  I̅^_e μ = I̅^_e τ = I̅^_ee = I̅^_μτ = 0  , 
          I̅^_μμ = - I̅^_ττ , R̅^_μμ = R̅^_ττ = - ( R̅^_ee + R̅^_μτ)    .

It is easy to see that these relations can be recombined into

    M̅^_eμ = - M̅^*_eτ , M̅^_μμ = M̅^*_ττ ,
    R̅^_μμ = R̅^_ττ = - ( R̅^_ee + R̅^_μτ)  , 
          M̅^_ee , M̅^_eμ , M̅^_eτ  and M̅^_μτ  being  real ,

which suggest that this case can be viewed as a result of μ-τ reflection symmetry in combination with the condition of M̅^_eμ (and equivalently M̅^_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. (<ref>) and ρ, σ = 0 or π/2. This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>). 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. (<ref>) have a realistic solution m^_3 = 0.021 eV at which | ⟨ m ⟩^_ee| takes a value of 0.020 eV.



 §.§ A&B&D&E&G


In the case of two sides of equations A, B, D, E and G being vanishing, equations AB, AC, AD, AG, BC, BD, CD and EF become ineffective. We are left with no constraint equation. But there are eight new constraint equations given by Eqs. (<ref>, <ref>, <ref>) which lead to the following relations for R̅^_αβ and I̅^_αβ (see Eqs. (<ref>, <ref>, <ref>))

    R̅^_e μ =R̅^_e τ = I̅^_ee = 0  , I̅^_e μ = - I̅^_e τ , R̅^_ee = - R̅^_μτ , 
          R̅^_μμ = - R̅^_ττ , I̅^_μμ = I̅^_ττ = -I̅^_μτ .

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 Eqs. (<ref>, <ref>). This can be verified by taking the relations in Eq. (<ref>) in the expressions for the neutrino masses in combination with the Majorana CP phases in Eq. (<ref>). By taking the relations given by Eq. (<ref>) in Eq. (<ref>), one arrives at a neutrino mass sum rule as

    ( m^2_1 - m^2_3 ) c^4_12 =  ( m^2_2 - m^2_3 ) s^4_12 .

Unfortunately, this sum rule has no chance to be in agreement with the realistic results.


At this stage, the physical meaning of the various cases we have studied can be clarified in the language of μ-τ interchange symmetry <cit.>. This symmetry is defined as follows: In the basis of M^_l being diagonal, M^_ν should keep invariant under the μ-τ interchange operation
and is characterized by

    M^_eμ = M^_eτ ,  M^_μμ = M^_ττ .

Such a texture leads to the following predictions for the neutrino mixing parameters

    ϕ^_μ = ϕ^_τ + π , θ^_23 = π/4 , θ^_13 = 0   ,

implying that

    M̅^_eμ = - M̅^_eτ , M̅^_μμ = M̅^_ττ .

Before the measurement of θ^_13 which was widely believed to be negligibly small at that time, the μ-τ interchange symmetry was very popular for its predictions. But the observation of a relatively large θ^_13 <cit.> forces us to consider breaking of this symmetry. Nevertheless, we might have a partial μ-τ interchange symmetry where part of the following four relations given by Eq. (<ref>) still hold

    R̅^_eμ = - R̅^_eτ , I̅^_eμ = - I̅^_eτ , R̅^_μμ = R̅^_ττ , I̅^_μμ = I̅^_ττ .

From our results, it is found that: (1) In the case of two sides of equations A and G being vanishing, the relation R̅^_eμ = - R̅^_eτ still holds. (2) In the case of two sides of equation B being vanishing, the relation I̅^_eμ = - I̅^_eτ still holds. (3) In the case of two sides of equations C and F being vanishing, the relation R̅^_μμ = R̅^_ττ still holds. (4) In the case of two sides of equations D and E being vanishing, the relation I̅^_μμ = I̅^_ττ still holds. (5) In the case of two sides of equations A, B and G being vanishing, the relations R̅^_eμ = - R̅^_eτ and I̅^_eμ = - I̅^_eτ which can be recombined into M̅^_eμ = - M̅^_eτ still hold. (6) In the case of two sides of equations A, C, F and G being vanishing, the relations R̅^_eμ = - R̅^_eτ and R̅^_μμ = R̅^_ττ still hold. In this case, one can say that the real part of M̅^_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 I̅^_μμ = I̅^_ττ still holds, while R̅^_eμ and R̅^_eτ vanish. (8) In the case of two sides of equations B, C and F being vanishing, the relation R̅^_μμ = R̅^_ττ still holds, while I̅^_eμ and I̅^_eτ vanish. (9) In the case of two sides of equations B, D and E being vanishing, the relations I̅^_eμ = - I̅^_eτ and I̅^_μμ = I̅^_ττ still hold. In this case, one can say that the imaginary part of M̅^_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 relations R̅^_eμ = - R̅^_eτ and R̅^_μμ = R̅^_ττ still hold, while I̅^_eμ and I̅^_eτ vanish. In this case, one can say that the real part of M̅^_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 relations I̅^_eμ = - I̅^_eτ and I̅^_μμ = I̅^_ττ still hold, while R̅^_eμ and R̅^_eτ vanish. In this case, one can say that the imaginary part of M̅^_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 matrix M̅^_ν instead of M^_ν itself. Since the unphysical phases have cancelled out in M̅^_ν, its twelve components R̅^_αβ and
I̅^_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 for R̅^_αβ and I̅^_αβ. We derive these constraint equations (i.e., Eqs. (<ref>, <ref>) and three independent ones from Eq. (<ref>)) by eliminating θ^_12 and θ^_13 in Eq. (<ref>) 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 of M̅^_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. (<ref>) 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 of M̅^_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 <cit.>. 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.




Acknowledgments  This work is supported
in part by the National Natural Science Foundation of China under grant Nos. 11747318, 11875157 (Z.C.L. and C.X.Y.) and 11605081 (Z.H.Z.).

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