A complete survey of texture zeros in the lepton mass matrices

We perform a systematic and complete analysis of texture zeros in the lepton mass matrices and identify all viable and maximally restrictive cases of pairs (M_\ell, M_D) and (M_\ell, M_L), where M_\ell, M_D and M_L are the charged-lepton, Dirac neutrino and Majorana neutrino mass matrices, respectively. To this end, we perform a thorough analysis of textures which are equivalent through weak-basis permutations. Furthermore, we introduce numerical measures for the predictivity of textures and apply them to the viable and maximally restrictive texture zero models. It turns out that for Dirac neutrinos these models can at most predict the smallest neutrino mass and the CKM-type phase of the mixing matrix. For Majorana neutrinos most models can, in addition, predict the effective neutrino mass for neutrinoless double beta decay. Apart from one model, which has marginal predictive power with respect to sin^2(theta_23), no other model can predict any of the already measured observables.

1 Introduction complex number. For example, the matrix  has four texture zeros and the five entries × denote arbitrary and independent complex numbers. Majorana mass matrices are understood to be symmetric, so if  represents a Majorana mass matrix, the 11, 12 and 33-entries are arbitrary complex numbers, while the 21-element must be equal to the 12-element. Since this matrix contains only three independent zero entries, it is counted as an instance of three texture zeros in a Majorana mass matrix. The paper is organized as follows. In section 2 we discuss basic constraints on the mass matrices. Section 3 is devoted to the relationship between texture zeros and weak-basis transformations. In section 4 we classify the possible inequivalent texture-zero models for Dirac and Majorana neutrinos. Section 5 is the main part of the paper: it contains the explanation of the numerical analyses, the definition of our predictivity measures and the results of our analyses. Finally, in section 6 the conclusions are presented. Some technical details of the numerics are deferred to an appendix.

Basic constraints on the lepton mass matrices
In this paper, one of our basic assumptions is that the lepton masses and the mixing matrix are obtained, with sufficient accuracy, by the tree-level mass matrices. Thus we require the charged-lepton mass matrix to have rank three, while the neutrino mass matrix can have rank three or two, since one neutrino mass being zero is compatible with all experimental data. For example, the mass matrix  will in general have rank two and thus does not represent an acceptable charged-lepton mass matrix. However, it represents a viable Dirac neutrino mass matrix.
In the case of Majorana neutrinos there is another possible constraint. Namely, the fact that the Majorana mass matrix is symmetric, in combination with texture zeros, can lead to matrices with two equal singular values, which we exclude. An example for such a texture in a Majorana neutrino mass matrix is  Fermions mass term masses rank(M) n 0,max Charged leptons −ℓ L M ℓ ℓ R + H.c. 0 < m e < m µ < m τ 3 6 Dirac neutrinos −ν R M D ν L + H.c. 0 ≤ m 1 < m 2 < m 3 or 0 ≤ m 3 < m 1 < m 2 ≥ 2 7 Majorana neutrinos 1 2 ν T L C −1 M L ν L + H.c.
0 ≤ m 1 < m 2 < m 3 or 0 ≤ m 3 < m 1 < m 2 ≥ 2 4 Table 1: The mass terms discussed in this paper. Since we consider only the Standard Model field content (plus three right-handed neutrinos in the case of Dirac neutrinos), all mass matrices are arbitrary complex 3 × 3-matrices. In addition, M L must be symmetric. By rank(M) we denote the required rank of the mass matrix M and n 0,max is the maximal number of (independent) texture zeros which can be imposed in M.
According to this line of reasoning, the above phenomenological requirements on the fermion masses directly exclude some types of texture zeros in the fermion mass matrices. In particular, there is a maximal number of texture zeros in the mass matrices. The fermion mass terms occurring in this paper and the maximal number of texture zeros in the mass matrices are listed in table 1. Note that in our convention the chiralities in the Dirac neutrino mass term are interchanged with respect to the charged-lepton mass term.
3 Texture zeros and weak-basis transformations 3

.1 Physical implications of texture zeros
In general, we always have the freedom to perform weak-basis transformations, i.e. field redefinitions which leave the form of the gauge and kinetic terms of the Lagrangian invariant [45,48,49]. In this paper we have in mind models with the same fermionic gauge multiplet structure as the Standard Model. If we consider an extension of the Standard Model by three right-handed gauge-singlet neutrino fields ν R and assume lepton number conservation, then neutrinos are Dirac particles and the lepton mass terms are given by (see table 1). Now we perform a field redefinition In order to leave the kinetic terms invariant, V In this new weak basis, the mass terms of equation (7) are given by with the new mass matrices A crucial observation in the case of Dirac neutrinos is that V R are independent transformation matrices, which is a consequence of the assumption that the right-handed fermion fields are gauge singlets. Obviously, we do not have this freedom in the case of Majorana neutrinos, where the weak-basis transformation on the mass matrices reads The fermion masses are the singular values of the mass matrices and are thus invariant under the biunitary transformation (10). Clearly, also the lepton mixing matrix U PMNS is invariant under this transformation. Consequently, whenever two types of texture zeros are related by a transformation of the form (10), they lead to the same physical constraints and are thus physically equivalent. This trivial statement has far-reaching consequences: on the one hand it allows to divide texture zero models into equivalence classes with the same physical consequences, on the other hand it relates to the existence of types of texture zeros, which do not impose any physical constraints [45,49,50], a fact which follows from the following well-known theorem of linear algebra.
• M W R is a lower triangular matrix.
These four statements are equivalent to the so-called QR, QL, RQ and LQ-decomposition of complex square matrices, respectively.
Indeed, theorem 1 assures us that beginning with any mass matrices M ℓ and M D (but not M L ) we can always perform a weak-basis transformation (10) such that, in the new basis, the mass matrices are upper or lower triangular matrices, i.e. M ℓ and M D are of the form  Consequently, any type of texture zeros equivalent to one of equation (11) is equivalent to the trivial case of no imposed texture zeros at all and does not impose any physical constraints. The textures  are even less restrictive than the ones of equation (11) and thus also do not impose any constraints on physical observables.

Equivalence of texture zeros
The number of possible cases of texture zeros which can be imposed in fermion mass matrices is huge, 1 however, as discussed above, not all of them lead to different physical predictions. Therefore, it is useful to divide the different patterns of texture zeros into equivalence classes with respect to weak-basis transformations. Regarding the arrangement of textures into equivalence classes, we are only interested in weak-basis transformations which leave the number of texture zeros in each individual mass matrix invariant.
In general the only weak-basis transformations fulfilling this requirement will be the ones where V L , V  and D is a diagonal phase matrix [27]. Weak-basis transformations where the three unitary matrices V L , V R are diagonal, leave all texture zeros invariant and can be used to eliminate unphysical phases in the elements of the mass matrices. In the numerical studies carried out for this paper, this rephasing freedom is used reduce the number of free parameters.
In sections 4.1 and 4.2 we will employ the weak-basis transformations based on the permutation matrices of equation (13) to divide the possible patterns of texture zeros in the lepton mass matrices into equivalence classes.
4 Classification of texture zeros in the lepton mass matrices

Dirac neutrinos
In this section we will use weak-basis transformation (10a) with V L , V (ℓ) being permutation matrices. For simplicity, such weak-basis transformations will be called weakbasis permutations.
Our strategy for constructing the inequivalent classes of texture zeros in M ℓ and M D is as follows. A weak-basis permutation can be expressed as a composition of the transformations and The crucial point in our approach to arrange the different patterns of texture zeros into classes is that the two above equations allow us to separately discuss texture zeros in M ℓ and M D . In the first step, making use of the weak-basis transformation (14) with V L and V Note that following the above prescription, we do in general not exploit the full freedom of weak-basis permutations. The reason is that for those M ℓ and permutation matrices leaves the positions of the zeros in M ℓ invariant, we have the freedom of multiplying M D with V L from the right: Thus, in addition to the permutation of rows in equation (15), there is this possibility to permute the columns of M D . This means that a further step is required to eliminate the remaining equivalent classes. We do this by brute force: Carrying out steps 1-4, we find the following: Step 1: using the requirement that M ℓ must be of rank three, one finds 247 different patterns of texture zeros in M ℓ . Representatives of each equivalence class are found by going through the 247 patterns, removing textures equivalent to an already found one. Moreover, one has to remove all cases which are equivalent to one of the textures of equations (11) and (12). Finally, one ends up with only nine classes of texture zeros in the charged-lepton mass matrix. Table 2 lists one representative of each class.
Step 2: as discussed before, two types of texture zeros in M D are equivalent if they are related by permutations of the rows of M D . Going through the 478 patterns of texture zeros of rank at least two, keeping only a representative of each class and discarding all textures which are equivalent to one of the textures of equations (11) and (12), we end up with 94 classes of texture zeros in M D . We list a representative of each class in tables 3 and 4.   in table 5. Finally, we want to emphasize an important issue concerning our treatment of texture zeros. This issue becomes most striking in comparison with the treatment of texture zeros in the literature when M ℓ is diagonal and assumed to be This charged-lepton mass matrix corresponds to our texture However, with our numerical procedure we cannot fix the order in which the chargedlepton masses m e , m µ , m τ appear on the diagonal of M ℓ and U (ℓ) L will in general be a permutation matrix. Therefore, the classification of texture zeros in M D of Hagedorn and Rodejohann in [27], which assumes the validity of equation (17) and which has, therefore, always U (ℓ) L = ½, has no unique correspondence to our notation. However, one may view the textures studied in [27] as special representatives of the classes of texture zeros shown in table 6. Table 3: Representatives for the 94 classes of texture zeros in the Dirac neutrino mass matrix M D . Part 1: four and less texture zeros. Table 4: Representatives for the 94 classes of texture zeros in the Dirac neutrino mass matrix M D . Part 2: five and more texture zeros.  Table 6: The types of texture zeros in the Dirac neutrino mass matrix, assuming a diagonal charged-lepton mass matrix studied by Hagedorn and Rodejohann in [27]. Left: notation of [27], right: corresponding class of textures in our notation. G 1 and G 9 of [27] have no correspondence in our paper, because these textures do not imply any physical constraints-see the discussion at the end of section 3.1.

Majorana neutrinos
In the case of Majorana neutrinos, the discussion of the possible patterns of texture zeros in M ℓ is the same as in section 4.1. However, under weak-basis permutations the Majorana neutrino mass matrix transforms as i.e. there is no unitary transformation V (ν) R . Thus, in contrast to the case of M D , after having already "used up" the freedom of choosing V L when dividing the textures of M ℓ into classes, there is no freedom left to divide the textures of M L . Consequently, we have to investigate all possible texture zeros in M L , taking into account that M L is symmetric. Note that, since now there is no freedom of performing weak-basis transformations, also theorem 1 does not apply to M L . However, the trivial case of no texture zeros at all still has to be excluded. Thus, M L must have at least one and at most four texture zerossee table 1. Furthermore, some textures with four zeros lead to two degenerate neutrino masses, which is phenomenologically excluded. Going through all possible patterns of texture zeros in M L , keeping only those which are of rank at least two and fulfill the above requirements, we find 50 possible textures, which are listed in table 7. Thus in total we find 9 × 50 = 450 types of texture zeros in M ℓ and M L . As in the case of Dirac neutrinos, by separately treating M ℓ and M L , we have ignored up to now the possibility that a weak-basis permutation with V L and V is allowed. Therefore, we have to go through all possible weak-basis permutations to eliminate the redundant classes. Doing so, we find 152 classes to be redundant, leaving a total of 298 classes of texture zeros in (M ℓ , M L ). The redundant classes are presented in table 8. Analogous to the case of Dirac neutrinos-see discussion at the end of section 4.1the textures with M ℓ ∼ 6 (ℓ) 1 have no one-to-one correspondence with the textures with M ℓ = diag(m e , m µ , m τ ) studied in [10]. Indeed the seven types of two texture zeros of [10] are special cases of classes of texture zeros discussed in the present paper: A 1 , A 2 , B 3 and B 4 belong to the same class 6 (ℓ) , the textures B 1 and B 2 are both contained in 6 (ℓ) , and C is a special case of 6 (ℓ)

The family tree of texture zeros
If a set of texture zeros in (M ℓ , M D ) or (M ℓ , M L ) is compatible with the experimental data, then any pattern of texture zeros with one or more zeros being replaced by free parameters will also be compatible with the data. Thus it is sufficient to discuss only those textures compatible with the experimental data which are maximally restrictive. By maximally restrictive we mean that one cannot place a further texture zero into one of the two mass matrices while keeping the model compatible with the data. For illustrational purposes we can arrange the textures in a "family tree," where less restrictive textures are the   "children" of the more restrictive ones. The family tree of texture zeros in M ℓ is shown in figure 1. By using the family trees for texture zeros in the pairs (M ℓ , M D ) and (M ℓ , M L ), we can easily find the maximally restrictive patterns. 2 The list of all allowed patterns of texture zeros can then be obtained by removing zeros from the maximally restrictive textures. In our results, tables 10 to 13, we present only the maximally restrictive pairs of charged-lepton and neutrino mass matrices. 5 Numerical analysis

χ 2 -analysis
We perform a χ 2 -analysis of the different patterns of texture zeros. Our χ 2 -function has the usual form We do not fit the Dirac phase δ. The central values and errors of the five neutrino oscillation parameters are taken from the global fit of oscillation data by Fogli et al. [4].
As central values of the charged-lepton masses we take the values from the Review of Particle Physics [51]. Since the experimental errors on the charged-lepton masses are so small that they can cause problems in the numerical χ 2 -analysis, we set them to one percent, i.e. σ(m ℓ ) = 0.01m ℓ for ℓ = e, µ, τ . For details of the numerical implementation of the χ 2 -minimization, we refer the reader to appendix A.
Results of the χ 2 -analysis: the only information we can obtain from the standard χ 2 -analysis is whether a given set of texture zeros in the mass matrices is excluded by the experimental data or not. We use the following criterion for a texture being compatible with the data: We call a set of textures in the lepton mass matrices compatible with the data if at the minimum of χ 2 the contribution of each observable to χ 2 is at most 25, i.e. the deviation of the observable from its experimental value is at most 5σ. This implies that χ 2 min ≤ 200.
We find that according to this criterion about three quarters of the classes of texture zeros comprise viable textures. In the case of Dirac neutrinos all textures fulfilling the criterion have χ 2 min < 10 −4 . In the case of Majorana neutrinos about 90 percent of the textures have χ 2 min < 10 −4 , the remaining 10 percent have χ 2 min < 30 (normal spectrum) and χ 2 min < 1 (inverted spectrum). Using the "family tree" of texture zeros discussed in section 4.3, we can identify the maximally restrictive textures among the compatible ones. This reduces the number of models to be investigated further to about 30 each for Dirac and Majorana neutrinos and for both neutrino mass spectra-see table 9. The compatible and maximally restrictive models are presented in tables 10 to 13. In only about a tenth of the compatible and maximally restrictive textures the charged-lepton mass matrix is diagonal, i.e. M ℓ ∼ 6 (ℓ) 1 .  Table 9: Results of the χ 2 -analysis.
To summarize, by means of a χ 2 -analysis and the "family tree" of texture zeros we have arrived at the list of maximally restrictive texture zero models compatible with the experimental or observational data. In the following we will further investigate each model with regard to its predictivity.

Predictivity analysis
The χ 2 -analysis of the previous section only tells us which textures are compatible with the observations, but does not yield any statement on their predictive power. Therefore, we have developed a numerical method to estimate the predictive power of texture zeros in the lepton mass matrices. The main idea behind this method is to find an answer to the question: Given matrices with a viable set of texture zeros and fixing the observables O j (j = i) to their experimentally observed values, how much can the remaining observable O i at most deviate from its experimental or best-fit value?
In the following, we will outline our attempt to answer the above question for the observables we are interested in. Technical details of the numerical implementation of the method are presented in appendix A.
Consider a model with parameters x making predictions P i (x) for the observables O i with mean values O i and errors σ i . For each observable O i , χ 2 (x) has the contribution 3 The contributions of all other observables to χ 2 (x) are then given by We define a measure for the maximal deviation of the observable O i from its experimentally observed value as where B i is defined as 25) and χ 2 min is the minimum of χ 2 (x) found in the χ 2 -analysis of section 5.1. The condition x ∈ B i fixes the other observables O j (j = i) to be close to their observed values O j . 4 The term δχ 2 is added to χ 2 min in order to improve convergence of the numerical maximization of χ 2 i (x) in equation (24). In this paper, depending on the observable, we use either δχ 2 = 0 or δχ 2 = 1-see appendix A. The quantity ∆ defined in equation (24) allows us to estimate the power of the studied set of texture zeros to predict O i . We will use this measure for a "predictivity analysis" of the five neutrino oscillation parameters and define: A set of texture zeros can correctly predict the observable O i , where O i is one of the observables ∆m 2 21 , ∆m 2 31 , sin 2 θ 12 , sin 2 θ 23 , In other words, we stipulate that a set of texture zeros is capable to predict an observable O i if its value can deviate from its central value O i by at most 5σ, while the other observables O j (j = i) are kept close to their experimental or best-fit values. 5 For the charged-lepton masses we use a different predictivity measure. Namely, for each charged-lepton mass m ℓ (ℓ = e, µ, τ ) we compute its minimal and maximal values and define: A set of texture zeros can correctly predict the charged-lepton mass m ℓ (ℓ = e, µ, τ ) if Here m exp ℓ denotes the mean experimental value of the charged-lepton mass m ℓ taken from [51]. In words, we call a model predictive if it predicts that m ℓ lies between m exp ℓ /2 and 2m exp ℓ . Finally, we also want to define a predictivity measure for those observables O which have not been measured up to now. In this case we compute the minimal and maximal value of O as where the parameter set B is defined as 4 This is done through the requirement χ 2 i (x) ≤ χ 2 min + δχ 2 . In addition, by the second requirement we demand that no observable O j is allowed to deviate from its central value by more than 5σ. 5 Relaxing the condition of equation (26)  Thus B is similar to B i with χ 2 i (x) replaced by the full χ 2 -function χ 2 (x). In the numerical analysis performed for this paper we have computed O min and O max for the observables m 0 , m ββ , δ, ρ and σ.
Here m 0 denotes the mass of the lightest neutrino, i.e. m 0 = m 1 in the case of a normal and m 0 = m 3 in the case of an inverted neutrino mass spectrum. The effective neutrino mass for neutrinoless double beta decay is given by The Dirac CP-phase δ and the two Majorana phases ρ and σ are defined via the decomposition where U stands for the standard parameterization of the mixing matrix [51]. The phases α, β and γ are not accessible by experimental scrutiny. By definition, the range of the phases δ, ρ and σ is [0, 2π). Since in all our numerical investigations we impose the constraint [51] 3 k=1 m k < 1 eV on the absolute neutrino mass scale, m 0 and m ββ can assume values between zero and about 1/3 eV. Given these bounds, we may define: A set of textures zeros can predict one of the observables m 0 , m ββ , δ, ρ, σ if Here range(m 0 ) = range(m ββ ) = 1/3 eV and range(δ) = range(ρ) = range(σ) = 2π.
Results of the predictivity analysis: We have performed the analysis explained above for all viable and maximally restrictive texture-zero models. The results of this paper, i.e. the viable and maximally restrictive textures and their predictions, are presented in four tables: i. In these tables, n denotes the number of real parameters of the model after removing as many phases as possible from the elements of the mass matrices by means of weak-basis transformations.
The results of the predictivity analysis may be summarized as follows.
• According to the criterion of equation (28), none of the investigated textures can predict any of the charged-lepton masses. Also relaxing the condition (28) for m e to m min e > 0.1 MeV, m max e < 5 MeV does not change this result.
• No set of texture zeros discussed in this paper fulfills the requirement of equation (26). Consequently, none of these textures can predict any of the five neutrino oscillation parameters.
• Most of the investigated textures can predict the smallest neutrino mass m 0 . In the case of Dirac neutrinos, all but two of the maximally restrictive viable textures have rank(M D ) = 2 and thus m 0 = 0.
• For all the maximally predictive and compatible textures for Dirac neutrinos we find δ min = 0 and δ max = π within the numerical accuracy. In fact, through a weak-basis being diagonal phase matrices, for all textures of tables 10 and 11 M ℓ and M D can be made real simultaneously, which implies δ ∈ {0, π}. Therefore, the maximally restrictive classes of viable texture zeros in the Dirac neutrino case do not admit CP violation in neutrino oscillations.
• Many of the textures for the Majorana neutrino case are predictive with respect to the Dirac phase δ. In contrast to the case of Dirac neutrinos, here also δ = 0, π is possible.
• None of the textures for Majorana neutrinos can predict any of the Majorana phases ρ and σ according to the condition of equation (35).
• Almost all maximally restrictive and compatible sets of textures in (M ℓ , M L ) predict m ββ .
• The effective mass m ββ can be big (larger than 0.1 eV) or small for normal ordering of the neutrino mass spectrum, however, for the inverted spectrum we always find m ββ < 0.1 eV.
• There are a few instances where M D or M L are diagonal and lepton mixing comes purely from M ℓ . Since we deal with maximally restrictive textures, in all of these instances we trivially have m 0 = 0.
Thus, the most interesting results of the predictivity analysis are the minima and maxima of m 0 , δ and m ββ , which we also show in tables 10 to 13. As for cos δ and Majorana textures with diagonal M ℓ -see the last three lines in tables 12 and 13, we have checked that our results agree with those of [18].
A note of caution on the interpretation of the results: The presented method for estimating the predictivity of a texture-zero model is based on maximal deviations from the observed value, see the definition of ∆ in equation (24), and maximal and minimal values of observables, see equations (27) and (29). Therefore, some of the models may still possess predictive power which can, however, not be measured by these quantities. For example, the maximally restrictive viable textures for Dirac neutrinos allow δ to assume only the values zero and π, i.e. we would call these models "predictive" with respect to δ. But in this case our analysis does not detect predictive power because equation (35) then reads

Conclusions
In this paper we have performed a thorough analysis of texture zeros in lepton mass matrices for both Dirac and Majorana neutrinos. It is instructive to summarize our basic assumptions. Firstly, we assume that there are three families of leptons, which means that the charged-lepton mass matrix M ℓ , the Dirac neutrino mass matrix M D and the Majorana neutrino mass matrix M L are all 3 × 3 matrices. Secondly, the tree-level mass matrices which contain the texture zeros should be compatible with our knowledge about the lepton masses and the lepton mixing matrix, which leads us to the requirements rank(M ℓ ) = 3 and rank(M D ) or rank(M L ) ≥ 2 and excludes some textures in M L with degenerate neutrino masses. Thirdly, the gauge-multiplet structure of the lepton fields is the same as in the Standard Model, i.e. left-handed doublets and right-handed singlets. This last assumption fixes the allowed forms of weak-basis transformation; for instance, in the case of Dirac neutrinos we are allowed to perform independent weak-basis transformations on the right-handed charged-lepton fields and the right-handed neutrino fields. One of the main points of this paper is that we admit all possible combinations of texture zeros in the pairs of mass matrices (M ℓ , M D ) and (M ℓ , M L ). In particular, we allow for non-diagonal M ℓ . Since there is a huge number of such combinations even after taking into account the second assumption above, we had to address the problem of equivalent texture zeros in the pairs of mass matrices, i.e. textures which are related through weakbasis transformations with permutation matrices (weak-basis permutations). We have found 570 inequivalent classes of texture zeros in the Dirac case and 298 classes in the Majorana case; for both cases about 75% of the classes are compatible with the data. However, if we consider only classes which are maximally restrictive-cf. section 4.3, we are down to about 30 classes of texture zeros for each of the four categories defined by Dirac/Majorana nature and normal/inverted ordering of the neutrino mass spectrum-see table 9.
The high value of 10 6 ensures that the minimum of ∆ i (x) respects the constraint x ∈ B i where B i is defined in equation (25). Adding χ 2 i (x) to 10 6 in the first line of the above equation drives the simplex towards a region respecting the desired constraints. The optimizations in equations (27) and (29) were done using the same technique.  Table 14: Details on the numerical analysis carried out for this paper. In line 2 the time limit is per texture and observable, in the other lines the limit is per texture and minimization and the same for maximization.