Neutrino seesaw mechanism with texture zeros

In the context of the Type I seesaw mechanism, we carry out a systematic study of the constraints that result from zeros in both the Dirac and right-handed Majorana neutrino mass matrices. We find that most constraints can be expressed in the standard form with one or two element/cofactor zeros alone, while there are 9 classes of nonstandard constraints. We show that all the constraints are stable under one-loop renormalization group running from the lightest right-handed neutrino mass scale to the electroweak scale. We study the predictions of the nonstandard constraints for the lightest neutrino mass, Dirac CP phase and neutrinoless double beta decay.


Introduction
Many Standard Model (SM) extensions predict the existence of right-handed (RH) Majorana neutrinos, which are often used to explain neutrino masses via the Type I seesaw mechanism [1]. The effective mass matrix of the light Majorana neutrinos is given by where M D and M R are the mass matrices for the Dirac and RH Majorana neutrinos, respectively. In the basis where the charged lepton mass matrix is diagonal, the light neutrino mass matrix is described by 9 real parameters and can be written as [2] M = V * diag(m 1 , m 2 , m 3 where the m i are real and nonnegative, V = U diag(1, e iφ 2 /2 , e iφ 3 After decades of neutrino oscillation experiments, five parameters (three mixing angles and two mass-squared differences) in the neutrino sector have been determined to rather good precision. However, the hierarchy of the three light neutrino masses is still unknown, and there are two possibilities: (i) m 1 < m 2 < m 3 for the normal hierarchy (NH) and (ii) m 3 < m 1 < m 2 for the inverted hierarchy (IH). The remaining four unknown parameters may be taken as the lightest mass (m 1 for the NH, or m 3 for the IH), the Dirac CP phase δ and two Majorana phases (φ 1 and φ 2 ). The Dirac phase and the mass hierarchy will be measured in future neutrino oscillation experiments, and the lightest mass can in principle be determined from beta decay experiments and cosmological observations.
Of the many neutrino mixing models proposed in the literature, texture zero models [3] are attractive because of the simple relations on the matrix elements that are found in these models. Texture zero constraints can be imposed on the light neutrino mass matrix directly. However, in the context of the Type I seesaw mechanism, since the light mass matrix is a product of the Dirac mass matrix and the RH neutrino mass matrix, it is natural to consider texture zeros in the Dirac and RH neutrino mass matrices [4]. Zeros in the Dirac and RH neutrino mass matrix can be realized from discrete Z N symmetries with suitable scalar singlets [5].
In this paper, we carry out a systematic study of the constraints that result from various zero textures in both the Dirac and RH Majorana mass matrices in the context of the Type I seesaw mechanism. We find that most cases lead to the standard constraints which can be expressed in the form of just one or two element/cofactor zeros, which have been studied extensively in the literature [6][7][8][9][10][11][12]. However, there are also 9 classes of nonstandard constraints, i.e., constraints which are not described by simple texture or cofactor zeros in the light neutrino mass matrix. We first show that both the standard and nonstandard constraints are stable under one-loop renormalization group equation (RGE) running from the lightest RH neutrino mass scale M 1 to the electroweak scale M Z . Then we study the phenomenological implications of the nonstandard constraints using recent data measured by neutrino oscillation experiments. We find that some cases are excluded, and for the rest we obtain preferred values for the lightest mass and Dirac CP phase, which will be probed in the next generation of neutrino experiments. We also study neutrinoless double beta decay for the nonstandard constraints.
This paper is organized as follows. In Section 2, we explore the constraints of zero textures in the seesaw mechanism. In Section 3, we discuss RGE effects on the constraints. In Section 4, we study the phenomenological implications of the nonstandard constraints and summarize our results in Section 5.

Classes of constraints
We consider three generations of RH Majorana neutrinos and we assume M R is not singular. Since the Majorana mass matrix is symmetric, there are 6 independent matrix elements in M R , and 9 independent elements in M D . For a case with N R zeros in M R and N D zeros in M D , there are should be at least N − 6 complex constraints on the elements of M, where N = N R + N D is the total number of zeros element. This may be understood as follows.
We start with 15 = 6 + 9 nonzero complex parameters in M R and M D , respectively. There are a total of N zeros in M R and M D , so that leaves 15 − N independent complex parameters. We can always insert a diagonal matrix between M R and M D (and the same matrix between M R and M T D ), which effectively absorbs 3 complex parameters (or, putting it another way, there are 3 relative normalizations that are redundant in the prediction of the light neutrino mass matrix from the seesaw mechanism). This leaves us with 12 − N independent complex parameters. In M there are 6 independent elements, so the net number of constraints on the elements of M is 6 − (12 − N) = N − 6. In terms of real parameters, we have 24 − 2N , from which we can subtract 3 unphysical phases, leaving us with 21 − 2N physical real parameters in M. Since there are 9 real parameters in the neutrino sector and 5 of them are well determined, in general, if N ≥ 9, the model will be over constrained and if N ≤ 6, the model will be unconstrained.
This counting rule only gives the minimum number of constraints. There are accidental constraints that result from some special combinations of texture zeros in M D and M R , e.g., if M D has a column or a submatrix composed of all zeros, then the constraint det M = 0 always exists regardless of the counting rule. So in order to obtain all the constraints for a given zero structure of M D and M R , we have to make sure all of the independent constraints are obtained. Our analysis proceeds as follows.
We scan all possible texture zero structures for M D . Since M is invariant under a permutation of the columns and rows of M R with a simultaneous permutation of the corresponding columns of M D , we only consider one representative structure for each set of M R that are equivalent under permutation. For each combination of M D and M R , we check all the elements and cofactors, and the determinant of M using Eq. (1) and look for independent constraints. We write the constraints in terms of the elements, cofactors or determinant of M in the simplest form. The experimental data does not allow more than two zeros in the elements or cofactors of the light mass matrix, hence we only consider cases that give less than three such zeros. The cases that give a block diagonal mass matrix are also ignored because they lead to a zero mixing angle, which is excluded by current experimental data. For the other cases, we analyze the elements of M, and make sure we obtain all of the independent constraints. We find that most constraints can be expressed in the standard form that can be written in terms of just one or two element/cofactor zeros, but there are also nine nonstandard constraints, which are listed in Table 1. The structures that lead to the nonstandard constraints are provided in Appendix A. Classes 2, 3, 4 and 5 are discussed in Ref. [4], with constraint equations in a different (albeit equivalent) form. Classes 1 and 9 are subsets of well-known models with the additional constraint det M = 0. Classes 6, 7 and 8 are new and not previously discussed in the literature.

Renormalization group running
The neutrino mixing parameters are measured at low energies, while the effective mass matrix of Eq. (1) can arise at a much high energy scale. The evolution of the light neutrino masses from the lightest RH neutrino mass scale M 1 to the electroweak scale M Z is described by the one-loop RGE [13],  3 + 5 C αα = 0 and C ββ C αγ = 2C αβ C βγ , is the renormalization scale and Y l = diag(y e , y μ , y τ ) with y α being the eigenvalues of the charged lepton Yukawa coupling matrix. In the SM, and in the Minimal Supersymmetric Standard Model (MSSM), where g 1 , g 2 are the gauge couplings, y t is the top quark Yukawa coupling, and λ is the Higgs self-coupling. The solution to Eq. (4) can be written in the form [14], where and for l = e, μ, τ . From Eq. (7), we see that RGE running can be described by multiplying the elements of the mass matrix by three factors, i.e., Since C αβ = det M(M −1 ) αβ , it is easy to verify that both the standard and nonstandard constraints are stable under one-loop renormalization group running from M 1 to M Z . Note that above the lightest RH neutrino mass, one must consider seesaw threshold effects, which can lead to large corrections to θ 12 and the Dirac and Majorana phases [13]. Due to these effects, some textures that are excluded by data may be allowed above the seesaw threshold [15]. Estimating the effect of RGE running above M 1 on low-energy parameters is beyond the scope of this paper because additional assumptions about the Yukawa coupling matrix and the RH Majorana mass matrix need to be invoked [16].
It has been shown that the two-loop RGE can change the rank of the light neutrino mass matrix [17], which implies that the constraint det M = 0 is not stable at two loop. In the SM, two-loop effects can be described by adding an additional term to Eq. (4) [18], so that The solution to the two-loop RGE is [19] M(M Z ) = I 0 where for m, n = e, μ, τ . We see that the nonstandard constraints are not stable under two-loop running. However, as shown in Ref. [18], the two-loop correction to the lightest mass is of order 10 −13 eV in the SM and 10 −10 (tan β/10) 4 eV in the MSSM. Two-loop effects can also generate a tiny value for θ 13 at the level of 10 −12 − 10 −14 in the SM [19]. Considering current uncertainties in the oscillation parameters, two-loop corrections do not affect our numerical results.

Phenomenology of the nonstandard constraints
The phenomenology of the standard constraints have been studied extensively in the literature; for recent analyses, see Ref. [6] for one element zero, Ref. [7] for one off-diagonal element/cofactor zero, and Ref. [8] for one diagonal element/cofactor zero. The results for two element/cofactor zeros can be found in Ref. [9]; other recent analyses can be found in Ref. [10] for two element zeros, Ref. [11] for two cofactor zeros, and Ref. [12] for one element zero and one cofactor zero. Here we focus on the phenomenology of the nonstandard constraints.

Classes 1, 2, 3, 9: det M = 0
Structures with the constraint det M = 0 are easiest to analyze. Since det M = 0 is equivalent to m 1 = 0 (m 3 = 0) for the NH (IH), the oscillation parameters are not affected, and these models are the same as models that have already been studied, with the additional constraint that the lightest mass is zero. For Class 9, det M = 0 is the only constraint and therefore any value of δ is allowed (see Section 5 for a discussion of the possible constraints from neutrinoless double beta decay). For models with constraints besides det M = 0, we only need to check the parameter space allowed by the additional constraints and see if m 1 = 0 (m 3 = 0) is allowed for the NH (IH). Class 1 constraints have the same number of parameters as the most economical seesaw model [20]. From Ref. [9], we know that if two diagonal elements are both equal to zero, only M μμ = M τ τ = 0 for the IH is allowed, but in this case the 2σ allowed region does not include m 3 = 0. Hence, Class 1 is not allowed by the experimental data at 2σ .
For Class 2, from Ref. [8] we know that only M μμ = 0 or M τ τ = 0 are allowed for the IH at 2σ when the lightest mass is zero. Similarly, for Class 3, from Ref. [7] we know that only M eμ = 0 or M eτ = 0 are allowed for the IH at 2σ when the lightest mass is zero. In all other cases with exactly one texture zero in the light neutrino mass matrix, a massless neutrino is excluded at 2σ . Hence only four cases in Classes 2 and 3 are allowed for the IH at 2σ , and none for the NH.  51)) as new constraints. The structures that lead to these conditions have six and three cofactor zeros, respectively. We do not count them as nonstandard constraints because a light neutrino mass matrix with more than two cofactor zeros has already been shown to be excluded by experimental data.
Since M αα = 0, Plugging M ββ M αγ = 2M αβ M βγ into the above equation, we get Together with M αα = 0 and M ββ = 0 (which is implicitly included in our constraints), we see Class 4 constraints are consistent with the conditions (52) in Ref. [4]. The form of the constraint in Table 1 has the merit of being quadratic instead of cubic (as in Ref. [4]), which allows for easier numerical analysis.
The constraints M αα = 0 and M ββ M αγ = 2M αβ M βγ can be written as and where Taking the absolute values of σ and ρ, we find m 2 /m 1 and m 3 /m 1 . Then m 1 can be written as where the plus (minus) sign in Eq. (22) is for the NH (IH). Our numerical analysis proceeds as follows. Since there are 5 independent parameters in the light mass matrix in this case, we take them to be θ 12 , θ 23 , θ 13 we choose a set of the five oscillation parameters within the 2σ range from a recent global fit [21]. Then we solve Eq. (22) by scanning over δ from 0 to 360 • . We sample 10 5 sets of oscillation parameters allowed at 2σ ; by keeping the values of m 1 and δ that are allowed, we obtain the allowed regions in m 1 − δ (m 3 − δ) plane for the NH (IH).
The allowed cases in Class 4 are as follows: Note that these allowed regions are consistent with the results obtained in Ref. [4], in which 3σ ranges of each parameter are used as input.

Class 5
Similar to Class 4, Class 5 constraints are consistent with the conditions (M −1 ) αα = 0, M ββ (M −1 ) ββ = 1 and M γ γ = 0 in Eq. (53) of Ref. [4]. We find that it is easier to use the quadratic form of Class 5 constraints for the numerical analysis.
Similar to Eqs. (18) and (19), the two constraints C αα = 0 and C ββ C βγ = 2C αβ C αγ can be written as and where ρ, σ , X i , A i and A ij are defined after Eq. (19). The analysis of this case follows that of Class 4 with the replacements, σ → 1/σ * and ρ → 1/ρ * . The allowed cases in Class 5 are as follows: 1. C ee = 0 and C μμ C eτ = 2C eμ C μτ . This case is allowed for the IH only. The allowed regions are shown in the left panel of Fig. 3. 2. C ee = 0 and C τ τ C eμ = 2C eτ C μτ . This case is allowed for the IH only. The allowed regions are shown in the right panel of Fig. 3, which is similar to the left panel with δ → δ + 180 • because of the approximate μ − τ symmetry.  3. C μμ = 0 and C τ τ C eμ = 2C μτ C eτ . This case is allowed for both hierarchies. The allowed regions are shown in Fig. 4. 4. C τ τ = 0 and C μμ C eτ = 2C μτ C eμ . This case is allowed for both hierarchies. The allowed regions are similar to Fig. 4 with δ → δ + 180 • for both hierarchies. The similarity can be explained by the approximate μ − τ symmetry of the experimental data.
Note that these allowed regions are consistent with the results obtained in Ref. [4]. Also, the Class 4 constraints are dual to the Class 5 constraints because they have the same functional form with the roles of elements and cofactors reversed [22]. From Ref. [22], we know that the allowed regions of Classes 4 and 5 should be similar for opposite mass hierarchies when the lightest mass is larger than about 20 meV. This can be seen from the left panel of Fig. 1

Class 6
Since det M = 0, the constraint M αα C αα = det M is equivalent to M αα (M −1 ) αα = 1, which can be written as Defining U αj = |U αj |e iβj and φ j = φ j + 2β j − 2β 1 , we absorb the phases of the U 's into φ j 's and write Eq. (25) as We now expand the above equation into real and imaginary parts, and after some simplification we get for the real part and for the imaginary part. The imaginary part can be put into the form where Equation (29) has the solution (as long as For a given φ 3 we solve for φ 2 and plug that into Eq. (27). Then for a fixed set of δ, m 1 (m 3 ) for the NH (IH), and a given set of oscillation parameters, we scan φ 3 to see where (if) Eq. (27) is satisfied. We also ignore those values of φ 3 that have no real solution for φ 2 , i.e., A 2 + B 2 < D 2 . If a solution is found, that point in m 1 − δ space is allowed for the corresponding set of oscillation parameters. With the value of φ 3 that gives the solution, φ 2 and φ 3 are also obtained.
The allowed cases in Class 6 are as follows: 1. M ee C ee = det M. This case is allowed for both hierarchies. There is no constraint on δ in this case. For the best-fit oscillation parameters, the allowed range of the lightest mass is m 1 > 0.0024 eV for the NH and m 3 > 0.00050 eV for the IH. The allowed range at 2σ is m 1 > 0.0020 eV for the NH and m 3 > 0.00037 eV for the IH. 2. M μμ C μμ = det M. This case is allowed for both hierarchies, and the allowed regions are shown in Fig. 5. 3. M τ τ C τ τ = det M. This case is allowed for both hierarchies, and the allowed regions are shown in Fig. 6. The allowed regions are similar to Fig. 5 with δ → δ + 180 • for both hierarchies because of the approximate μ − τ symmetry of the data.

Class 7
With x = e −iφ 2 and y = e −iφ 3 , the constraint M 2 αα C αα = 4M αβ M αγ C βγ can be written as For a fixed set of δ, m 1 (m 3 ) for the NH (IH), and a given set of oscillation parameters, we solve the cubic equation (32) to find x as a function of y, and then scan over φ 3 from 0 to 2π to find where (if) |x| = 1. If |x| = 1 is found, there is a solution for φ 2 and φ 3 . By keeping the values of m 1 (m 3 ) and δ that have a solution, we obtain the allowed regions in the m 1 − δ (m 3 − δ) space for the NH (IH). All the cases in Class 7 are allowed for the both hierarchies, and the allowed regions are shown in Fig. 7. The 2σ allowed regions in the second row of Fig. 7 are similar to the third row of Fig. 7 with δ → δ + 180 • for both hierarchies because of the approximate μ − τ symmetry of the data.

Class 8
Since det M = 0, the constraint M ββ det M = −M 2 αβ C αα is equivalent to Similar to Class 7, defining x = e −iφ 2 and y = e −iφ 3 , we can write the constraint as where a = The solutions for φ 2 and φ 3 then proceed as in Class 7. We find all the cases in Class 8 are allowed for the both hierarchies and the allowed regions are shown in Figs. 8 and 9. Due to the approximate μ − τ symmetry in the experimental data, the 2σ allowed regions in each row of Fig. 8 are similar to those in the corresponding row of Fig. 9 with δ → δ + 180 • for both hierarchies.

Discussion and conclusion
We carried out a systematic study of zeros in the Dirac and RH Majorana neutrino mass matrices in the context of the Type I seesaw mechanism. We derived complex constraints on the light neutrino mass matrix for various textures in both the Dirac and RH Majorana neutrino mass matrices. There are 9 nonstandard constraints besides the standard ones that can be expressed in the form of one or two element/cofactor zeros alone. We showed that both the standard and nonstandard constraints are stable under one-loop RGE running from the lightest RH neutrino mass scale M 1 to the electroweak scale M Z . In addition, we studied the phenomenological implications for the nonstandard constraints, and found that some cases for the normal or inverted hierarchy are excluded, and for the rest we obtain the allowed regions for the lightest mass and Dirac CP phase, which will be probed in the next generation of neutrino experiments. We found 12 new models (Classes 6, 7 and 8) not previously discussed in the literature which are allowed at 2σ for both the normal and inverted hierarchies. Once the lightest neutrino mass and Dirac CP phase are determined, the rate for neutrinoless double beta decay (0νββ), which depends on the magnitude of the ν e − ν e element of the neutrino mass matrix, is fixed.
In Table 2, we list the minimum and maximum values of |M ee | at the 2σ level for Classes 2, 3, 4 and 5 constraints. In Table 3, we list the minimum values for Classes 6, 7, 8 and 9 constraints with the best-fit oscillation parameters and the 2σ lower bounds. The maximum values for Classes 6, 7, 8 and 9 constraints are all above 1000 meV. The most stringent experimental up-  per limit on the effective mass |M ee | is 120-250 meV at 90% C.L. [23]. Some cases in Classes 3, 4 and 5 could be ruled out once the sensitivities to |M ee | reach about 50 meV for the currently running 0νββ experiments [24]. Nevertheless, the minimum values of |M ee | for Classes 6, 7, 8 and 9 constraints are all below 50 meV and can be completely probed only if the sensitivity of future 0νββ experiments is significantly improved. Note that the minimum value of |M ee | is about 15 meV for the IH and 0 for the NH. The mass hierarchy will also be determined if |M ee | can be pushed to 15 meV in the future. Recent global fits [21,25,26] indicate a slight preference for a nonzero Dirac CP phase, with a preferred value δ ∼ 270 • . Also, combined T2K and reactor analyses show a weak hint for the normal hierarchy [27]. Most viable cases of the nonstandard constraints allow δ = 270 • , except for two cases in Class 5: C ee = 0, C μμ C eτ = 2C eμ C μτ and C ee = 0, C τ τ C eμ = 2C eτ C μτ . These two cases may be ruled out if future measurements find δ to be close to 270 • . If the hint for the   Table 3 The minimum values of |M ee | (in meV) for Classes 6, 7, 8 and 9 constraints for the best-fit oscillation parameters, and the 2σ lower bounds. zeros in M, which is excluded by experimental data. Following Ref. [9], we define Class X constraints on M or C ≡ det(M)M −1 as one zero on diagonal and one off-diagonal zero sharing column and row, Class Y constraints as one zero on diagonal and one off-diagonal zero not sharing column and row, and Class Z constraints as two zeros on diagonal.
For N R = 4, there is only one structure for M R , where the symbol × denotes a nonzero matrix element. All other structures of M R can be obtained by a permutation of the columns and rows of the above structure. If M D has more than 5 zeros, M or C would have at least three zeros, so here we only consider N D ≤ 5.
which lead to the constraints, M ee = M μμ = 0 and det M = 0; which lead to the constraints, M ee = M τ τ = 0 and det M = 0; which lead to the constraints, M ee = 0 and det M = 0; which lead to the constraints, M μμ = 0 and det M = 0; which lead to the constraints, M τ τ = 0 and det M = 0. The remaining 12 structures lead to Class 4 nonstandard constraints: which lead to the constraints, M ee = 0 and M μμ M eτ = 2M eμ M μτ ; which lead to the constraints, M ee = 0 and M τ τ M eμ = 2M eτ M μτ ; which lead to the constraints, M μμ = 0 and M ee M μτ = 2M eμ M eτ ; which lead to the constraints, M μμ = 0 and M τ τ M eμ = 2M μτ M eτ ; which lead to the constraints, M τ τ = 0 and M μμ M eτ = 2M eμ M μτ ; which lead to the constraint, M ee C ee = det M; which lead to the constraint, M μμ C μμ = det M; which lead to the constraint, M τ τ C τ τ = det M. Also, there are 6 structures lead to Class 7 nonstandard constraints: which lead to the constraint, M 2 ee C ee = 4M eμ M eτ C μτ ; which lead to the constraint, M 2 μμ C μμ = 4M eμ M μτ C eτ ; which lead to the constraint, M 2 τ τ C τ τ = 4M eτ M μτ C eμ . The remaining one structure leads to the Class 9 constraint, det M = 0: (63)

A.2. N R = 3
For N R = 3, there are four structures of M R that are not equivalent under permutation. They are All other structures of M R can be obtained by a permutation of the columns and rows of the above structure. If M D has more than 5 zeros, M or C would have at least three zeros for all three structures of M R , so here we only consider N D ≤ 5.
which lead to the constraints, M ee = 0 and det M = 0; which lead to the constraints, M μμ = 0 and det M = 0; which lead to the constraints, M τ τ = 0 and det M = 0. The remaining 6 structures lead to Class 3 nonstandard constraints: which lead to the constraints, M eμ = 0 and det M = 0; which lead to the constraints, M eτ = 0 and det M = 0; which leads to the constraints, M ee = 0 and det M = 0; which leads to the constraints, M μμ = 0 and det M = 0; which leads to the constraints, M τ τ = 0 and det M = 0. Also, there are 6 structures that lead to Class 6 nonstandard constraints: which lead to the constraint, M ee C ee = det M; which lead to the constraint, M μμ C μμ = det M; which lead to the constraint, M τ τ C τ τ = det M. There are 6 structures lead to Class 8 nonstandard constraints: which leads to the constraint, M μμ det M = −M 2 eμ C ee ; which leads to the constraint, M τ τ det M = −M 2 eτ C ee ; which leads to the constraint, M ee det M = −M 2 eμ C μμ ; which leads to the constraint, M τ τ det M = −M 2 μτ C μμ ; which leads to the constraint, M μμ det M = −M 2 μτ C τ τ ; which leads to the constraint, M ee det M = −M 2 eτ C τ τ . The remaining 9 structures lead to the Class 9 constraint, det M = 0: which lead to the constraints, M ee = 0 and det M = 0; which lead to the constraints, M μμ = 0 and det M = 0; which lead to the constraints, M τ τ = 0 and det M = 0. The remaining 6 structures lead to Class 5 nonstandard constraints: which leads to the constraints, C ee = 0 and C μμ C eτ = 2C eμ C μτ ; which leads to the constraints, C ee = 0 and C τ τ C eμ = 2C eτ C μτ ; which leads to the constraints, C μμ = 0 and C ee C μτ = 2C eμ C eτ ; which leads to the constraints, C μμ = 0 and C τ τ C eμ = 2C μτ C eτ ; which leads to the constraints, C τ τ = 0 and C μμ C eτ = 2C μτ C eμ ; which leads to the constraints, C τ τ = 0 and C ee C μτ = 2C eτ C eμ . 2. N D = 4. Of the 126 different structures of M D in this case, 39 lead to at least three zeros in M or C, 6 lead to Class Z constraints in C, 18 lead to Class X constraints on M, and 3 lead to one zero in M and one zero in C, 12 lead to one diagonal zero in M, and 33 lead to one diagonal zero in C. In addition, there are 3 structures that lead to Class 2 nonstandard constraints: which leads to the constraints, M ee = 0 and det M = 0; which leads to the constraints, M μμ = 0 and det M = 0; which leads to the constraints, M τ τ = 0 and det M = 0. Also, there are 9 structures leads to Class 6 nonstandard constraints: which lead to the constraint, M ee C ee = det M; which lead to the constraint, M μμ C μμ = det M; which lead to the constraint, M τ τ C τ τ = det M. The remaining 3 structures lead to the Class 9 constraint, det M = 0: which lead to the constraints, M eμ = 0 and det M = 0; which lead to the constraints, M eτ = 0 and det M = 0; Since M R is diagonal in this case, the constraints mainly come from four zeros in M D , and they are consistent with the results discussed in Refs. [7,28].
A.4. N R = 1 For N D = 6, of the 84 different structures of M D in this case, 78 lead to more than three zeros in M or C, and 6 lead to one diagonal or off-diagonal zero in C. 22