New textures for the lepton mass matrices

We study predictive textures for the lepton mass matrices in which the charged-lepton mass matrix has either four or five zero matrix elements while the neutrino Majorana mass matrix has, respectively, either four or three zero matrix elements. We find that all the viable textures of these two kinds share many predictions: the neutrino mass spectrum is inverted, the sum of the light-neutrino masses is close to 0.1 eV, the Dirac phase $\delta$ in the lepton mixing matrix is close to either $0$ or $\pi$, and the mass term responsible for neutrinoless double-beta decay lies in between 12 and 22 meV.


Introduction
The origin of neutrino masses, the reasons behind their smallness, and the structure of lepton mixing are still unanswered questions. There has been a great deal of theoretical work in this area, trying to provide answers based on such diverse ideas as, for instance, seesaw mechanisms, radiative generation of the neutrino masses, Abelian and non-Abelian symmetries imposed on the leptonic sector, and 'textures' for the leptonic mass matrices. In the past few years, a wealth of experimental data concerning neutrino oscillationsin particular the recent confirmation [1,2,3] of a non-zero value for the mixing angle θ 13 -became available, allowing theorists to test their models and discard those that do not conform to the experimental discoveries. Here, we shall consider new textures for the leptonic mass matrices and investigate what the most recent and stringent phenomenological data say about them and their predictive power.
In this paper we work in the context of a model with three light neutrinos which are Majorana particles. The lepton mass terms are given by where C is the charge-conjugation matrix in Dirac space. The three lightneutrino fields in the column-vector ν are left-handed. The neutrino mass matrix M ν acts in flavour space and is symmetric. Let the two mass matrices be bi-diagonalized as where U L , U R , and U ν are 3 × 3 unitary matrices. Then, the lepton mixing matrix is Even though M ℓ is more fundamental, in practice we only need to consider since it is its diagonalization that fixes the matrix U L which appears in U: Let M ν denote the neutrino mass matrix in the basis where the chargedlepton mass matrix is diagonal. Then, There have been many attempts at using non-Abelian symmetries to constrain lepton mixing [4]- [21]. This is usually done with the goal of obtaining 'mass-independent schemes', wherein the constraints on U do not depend on the values of the lepton masses. However, those attempts appear to have reached their limits [22]. A simpler avenue, at least in group-theoretical terms, is provided by Abelian symmetries. In appropriate bases for the lepton and Higgs fields, they enforce 'texture zeros' in the lepton mass matrices, but they cannot enforce relationships among their nonzero matrix elements. In the pioneering work of ref. [23], M ℓ was assumed to be diagonal, hence to have six zero matrix elements, while M ν had two zero matrix elements. This was later generalized to the situation wherein M ℓ is diagonal and M −1 ν has two zero matrix elements [24]; mixed situations in which both M ν and M −1 ν have one zero matrix element, while M ℓ remains diagonal, were considered in ref. [25].
In this work we propose new textures for the lepton mass matrices which are in principle as predictive as the ones considered in refs. [23,24,25]. Let (m, n) denote a class of textures with m nonzero matrix elements in M ℓ and n nonzero matrix elements in M ν . 1 Then, the textures mentioned at the end of the previous paragraph are in the (3,4) class. In this paper we consider predictive, viable textures in the (4, 3) and (5, 2) classes. Those textures are in principle just as predictive as the ones in class (3,4); each of them has eight degrees of freedom-seven moduli and one rephasing-invariant phasein the matrices H and M ν . Those eight degrees of freedom are meant to fit ten observables-the three charged-lepton masses m e,µ,τ , the three neutrino masses m 1,2,3 , the three lepton mixing angles θ 12,13,23 , and the Dirac phase δ. (We do not care about the Majorana phases in U because they are not observable in neutrino oscillations. However, we shall specify the predictions of our textures for the mass term responsible for neutrinoless double-beta decay, m ββ ≡ |(M ν ) ee |.) So, in principle each texture yields two predictions, which may conveniently be taken to be one prediction for the overall scale of the neutrino masses and one prediction for cos δ.
It has long been known [26] that any mass-matrix texture, in particular any set of matrices M ℓ and M ν with some zero matrix elements, can be implemented in a suitable extension of the Standard Model of the electroweak interactions, furnished with both additional scalar multiplets and appropriate Abelian symmetries. We rely on this fact to assert that all the textures in this paper may be implemented in renormalizable models. However, we shall not attempt here to construct a specific model for any of the textures; we also do not attempt to search for the simplest model which might justify any given texture [27].
We emphasize that all the textures will be analyzed in this paper only at the 'classical' level, i.e. we shall neglect both quantum corrections to the mass matrices and renormalization-group effects.
The texture-zero approach for the mass matrices pursued in this paper is inherently limited in its scope and objectives. Even if it were found that the experimental data fully agree with the predictions of some texture, we would not be sure that the mass matrices indeed have that texture, because there are many sets of mass matrices leading to the same observables-in particular, any two sets of mass matrices connected among themselves through a weakbasis transformation lead to the same observables. Further studies would be necessary in order to identify specific models that lead to mass matrices with that texture and also to identify other observable predictions of those models, viz. extra particles and interactions that they may feature. So, the study of textures may be looked upon as just the first part of a longer search for models of 'new physics'. Still, that study has some relevance in itself, since it may suggest the most likely ranges for some observables-for instance, knowing whether the phase δ is more likely to be large or small-and which correlations among observables may be expected and are enforceable through well-defined renormalizable models. This paper is organized as follows. In section 2 we derive all the viable (5, 2) textures and briefly survey their predictions. We do the same for (4, 3) textures in section 3. A listing of all the viable textures that we have found, and a summary of their predictions, is provided in section 4.

(5, 2) textures
Since all three charged leptons are massive, the determinant of M ℓ cannot vanish. Therefore, through an appropriate permutation of the columns of M ℓ -this permutation changes U R but does not change U L , hence it leaves U invariant-one may always obtain the (1, 1), (2, 2), and (3, 3) matrix elements of M ℓ to be nonzero. Since in a (5, 2) texture M ℓ has five nonzero matrix elements, there are then (6 × 5) / 2 = 15 possibilities: Equations ( Both eqs. (14a) and (14c) lead to two degenerate neutrinos and are therefore incompatible with experiment. With eq. (14b) lepton mixing originates fully in M ℓ ; indeed, one then has U = U † L but for possible reorderings of the columns of U. For two physical neutrinos ν i and ν j with i = j, So, Similarly, Phenomenologically, there are no two columns i and j of U such that either | are allowed to be so much smaller than unity as indicated by eqs. (16) and (17). Therefore, with eq. (14b) either H ij = 0 or (H −1 ) ij = 0 are phenomenologically forbidden for i = j. If, together with eq. (14b), the form of M ℓ is as in one of eqs. (7), then lepton mixing would only be 2 × 2, which is also incompatible with experiment. Therefore, eq. (14b) must be excluded, just as eqs. (14a) and (14c). We shall therefore concentrate on eq. (14d). With that form for M ν , one neutrino is massless; this is one of the predictions of viable (5,2) textures.
If M ν is as in eq. (14d) while M ℓ is as in one of eqs. (7), then the matrix U has one vanishing matrix element. This contradicts the phenomenology. Therefore, we may exclude eqs. (7) and concentrate exclusively on the other possibilities for M ℓ . As we have seen, they can be subsumed in six different possibilities: Then, the permutation group of three objects is represented by Let Z be any of the six matrices in S 3 . Those matrices are orthogonal, hence Z −1 = Z T . Interchanging the rows and columns of M ν is equivalent which means a reordering of the rows and columns of H.
So, a reordering of the rows and columns of M ν is equivalent to an analogous reordering of the rows and columns of H. Therefore, instead of considering separately each of the three conditions H 12 = 0, H 13 = 0, and H 23 = 0, one may consider only the condition H 12 = 0 provided one allows for all the possible reorderings of the rows and columns of M ν . We thus conclude that there are twelve potentially viable (5, 2) textures: together with the six textures that result from substituting H 12 = 0 by (H −1 ) 12 = 0 in each of eqs. (20).
With either H 12 = 0 or (H −1 ) 12 = 0, the matrix H can always be made real through a rephasing, i.e. there is always a diagonal unitary matrix Y ℓ such that has real matrix elements. The real and symmetric matrix H real is diagonalized by an orthogonal matrix O ℓ : Since either H real or its inverse has one vanishing matrix element, it contains only five degrees of freedom; three of them correspond to the three chargedlepton masses and the remaining two are implicitly contained in O ℓ . Thus, O ℓ is not fully general-a general 3 × 3 orthogonal matrix has three degrees of freedom, not just two. Similarly, phases may be withdrawn from the matrix M ν in eq. (14d): where f and r are non-negative real and Y ν = diag e iξ , e i(ρ/2−φ) , e −iρ/2 , the phase ξ being arbitrary. Therefore, the lepton mixing matrix always ends up being where O b is the real, orthogonal matrix that diagonalizes the real version of M ν while X is a diagonal unitary matrix containing only one phase. This is because the arbitrariness of the phase ξ in Y ν allows one to absorb one phase in X.
Let us define With a massless neutrino there are two possibilities for the neutrino mass spectrum: either it is 'normal' (which we call "case n"), and then 3 or it is 'inverted' (which we call "case i"), and then Suppose the initial M ν was as in eq. (14d). Then, after withdrawing phases from it, we would have, in case n while, in case i 3 We use in this paper the quantity √ ∆ ≈ 0.5 eV as the unit for all neutrino masses.
The diagonalization of the real matrices M n and M i proceeds as We see that the mixing angle θ b appears in O b . If b is n, then If b is i, then Since ε ∼ 1/30 is small, θ n ∼ 20 • is smallish. On the other hand, θ i is very close to 45 degrees, viz. almost maximal. It turns out that, because the mixing angle θ n is so small, case n is not much different from the one, treated in eqs. (15)- (17), in which lepton mixing originates fully in M ℓ . Because of this, a normal neutrino mass spectrum does not work with (5,2) textures.
For case i, one may write down the six possible forms of the lepton mixing matrices. They are In the forms for U in eqs. (34), the matrix O ℓ is the real orthogonal matrix that diagonalizes H according to eq. (22). The matrix O ℓ contains two degrees of freedom because H satisfies either H 12 = 0 or (H −1 ) 12 = 0. The matrix U depends on three degrees of freedom: one of them is the phase ℵ and the other two are contained in O ℓ . So, there is one non-trivial constraint on U.
For the mass term responsible for neutrinoless double-beta decay one finds the formula where the values for the indices i and j are given in table 1.
We have found numerically that all six eqs. (34) are able to fit U provided H 12 = 0, but they are unable to achieve that fit when (H −1 ) 12 = 0. Furthermore, the predictions of eqs. (34a)-(34c) (with H 12 = 0) are all very similar (but not really identical) among themselves. In all those cases, one must have a rather large solar mixing angle, sin 2 θ 12 0.3. The prediction of eqs. (34a)-(34c) for the Dirac phase is cos δ −0.6, while eqs. (34d)-(34f) make the symmetric prediction cos δ 0.6. The prediction for neutrinoless double-beta decay is 0.24 m ββ √ ∆ 0.4. We can see these predictions displayed in fig. 1, in which we plot m ββ √ ∆ against cos δ. Each point in the plot corresponds to some definite values for the parameters of the model-neutrino oscillation observables, phase ℵ, and two entries of the matrix H). For definiteness, these predictions are based on the use of the phenomenological 3σ data in ref. [29]; other phenomenological fits to the data-see refs. [30] and [31]-can hardly yield much too different results.

(4, 3) textures
Since there are no massless charged leptons, the determinant of M ℓ must be nonzero. Therefore, after an adequate reordering of the rows and columns of M ℓ , Therefore, where γ ≡ arg (t 2 t * 3 ). From eq. (5), the columns of U L are the normalized eigenvectors of H. It is clear from eq. (37) that one of the eigenvalues of H is |t 1 | 2 and the corresponding normalized eigenvector is (1, 0, 0) T . Therefore, either where X is a diagonal unitary matrix containing the phases of the eigenvectors of H; those phases are meaningless. Equation (38a) holds if |t 1 | = m e , eq. (38b) holds if |t 1 | = m µ , and eq. (38c) holds if |t 1 | = m τ . The angle θ is fixed by We assume that only three out of the six independent matrix elements of M ν are nonzero. Therefore there are (6 × 5 × 4)/ 3! = 20 possible forms for M ν . They are Let Z be any of the six matrices in the group S 3 of eq. (19). Those matrices are orthogonal, hence Z −1 = Z T . Interchanging the rows and columns of M ν is equivalent to making M ν → ZM ν Z T . But U ν → ZU ν when this happens. Therefore U → U † L ZU ν . This is equivalent to letting U L → Z † U L , which corresponds to a reordering of the rows of U L . We conclude that, provided one allows for a reordering of the rows of the three possibilities for U L in eqs. (38), one is free to avoid considering separately two matrices M ν which differ only by an interchange of their rows and columns. In this way, out of the 20 forms for M ν in eqs. (40-59), one only needs to consider the following six: The first three of these forms for M ν are excluded when taken in conjunction with the U L matrices in eqs. (38). Indeed, eq. (60a) leads to U with four zero matrix elements; either eq. (60b) or eq. (60c) lead to U with one zero matrix element; and both those situations are phenomenologically excluded. The only viable forms of M ν are those that give rise to genuine 3 × 3 mixing in M ν , viz. eqs. (60d-60f). One may, without lack of generality, assume the three nonzero matrix elements of M ν to be real and positive, because, for each of the three matrices in eqs. (60d-60f), there is a diagonal unitary matrix Y ψ = diag e iψ 1 , e iψ 2 , e iψ 3 such that Y ψ M ν Y ψ is real and has positive nonzero matrix elements. We may thus write the matrices where a, b, and d are positive. One has M ν = Y * ψ M K Y * ψ , where K may be either A, B, or C.
The matrix M K is diagonalized by the orthogonal matrix O K : The real numbers µ k (k = 1, 2, 3) are the eigenvalues of M K ; |µ k | = m k are the neutrino masses. From eq. (2b), the matrix Y ′ is a diagonal unitary matrix which affects the transformation µ k → m k in the following way: where U L is either one of the matrices in eqs. (38) or one of them with the rows interchanged.
The matrix U † L Y ψ contains four phases-one phase γ in U † L and three phases ψ 1,2,3 in Y ψ . One may pull three of those phases to the left-hand side of U † L , leaving at its right-hand side only one phase-let χ denote it. Suppose for instance that eq. (38a) holds, then where χ ≡ ψ 3 − ψ 2 + γ. The matrix X ′ ≡ X * diag e iψ 1 , e iψ 2 , e i(ψ 2 −γ) contains unphysical phases. Thus, there are 18 possible forms for U in (4, 3) textures. Let Kp denote those 18 forms, where K may be either A, B, or C. If K = B, then the number p may be 1, 2, . . . , 9: The real orthogonal matrix O B diagonalizes the real symmetric matrix M B , see eqs. (61) and (62). When one interchanges the second and third rows and columns in the matrix M A one obtains the same matrix with b and d interchanged; this is just a meaningless renaming of parameters. Similarly, any permutation of the rows and columns of M C is equivalent to a renaming of the parameters a, b, and d. Therefore, there are nine more possible forms for U: In eqs. (65) and (66) the angle θ and the phase χ are free parameters, to be adjusted in order to obtain a good fit of U. The diagonal unitary matrix Y ′ is in practice irrelevant for phenomenology.

Forms A1-6
We first consider the matrix M A in the first eq. (61) and its diagonalizing matrix O A . It is convenient to define f ≡ √ b 2 + d 2 and the angle ϕ: The matrix M A has vanishing determinant. Therefore, one neutrino is massless and eqs. (26)(27)(28) apply. In case n, In case i, It is clear from eqs. (66a-66f) that one row of U must coincide, but for the phases contained in Y ′ , with a row of O A . But, no row of the matrix O A in eq. (69b) may possibly coincide with a row of U, therefore case n is excluded. This is because: 1. The first row of O A in eq. (69b) contains a zero matrix element, while no matrix element of U vanishes.
2. In the second and third rows of eq. (69b), the second entry is larger in modulus than the third entry by a factor ε −1/4 ≈ 2.4; this factor is much too small for what is observed in the first row of U and much too large for what is observed in the second and third rows of U.
Coming to case i, either the second row or the third row of O A in eq. (70b) may coincide with either the second row or the third row of U. This is because those rows of O A feature a first entry which is larger in modulus than the second entry by a factor (1 + ε) 1/4 ≈ 1; this is compatible with what occurs in either the second or third row of U. Therefore, models A5, 6 are viable (although with some deviation from the mean values of the mixing angles) in case i.
For form A6 of U the results are analogous to those of form A5, except that cos δ is negative instead of positive and θ 23 is preferred to be in the first octant instead of in the second one.
are negative. Thus, in case n . Model A5, inverted hierarchy Figure 3: sin 2 θ 12 vs sin 2 θ 23 for form A5. The numeric scan was made using the 3 σ data of ref. [29]. Therefore, According to eqs. (66g-66i), if the PMNS matrix is of form Ci (i = 1, 2, 3) then its i'th row coincides, in the moduli of its matrix elements, with the first row of O C . It follows from eq. (62) that Equation (74), together with the normalization of the first row of O C , yield Thus, when U has the form Ci, One may use the expressions of µ 1,2,3 in either eqs. (71) or eqs. (72)-for cases n and i, respectively-together with |U i3 | 2 to compute |U i1 /U i2 | 2 through eq. (76). One can in this way find out for which values of ε and of the parameters of U the form Ci agrees with experiment. We have found that form C1 is incompatible with the phenomenology, while both forms C2 and C3 are viable, but only for the case of an inverted hierarchy. Form C2 predicts cos δ 0.67 while form C3 predicts cos δ −0.67; both forms predict 0.24 ≤ m ββ √ ∆ ≤ 0.34; furthermore, these forms only work for sin 2 θ 12 0.325, cf. fig. 4).

Forms B1-9
The mass matrix M B in the second eq. (61) is of 'Fritzsch type' [32]. The exact diagonalization of a Fritzsch mass matrix has been known for a long time [33]. The use of Fritzsch-type mass matrices in the lepton sector has been proposed before [34]. With If the matrix U has form Bp, then one of its rows coincides, in the moduli of its matrix elements, with a row of O B . Considering the absolute values of the matrix elements in the third column of O B , one finds that none of them can be equal to either sin θ 23 cos θ 13 or cos θ 23 cos θ 13 -they are either too large or too small for that. On the other hand, either (O B ) 23 or (O B ) 33 may coincide with sin θ 13 . However, whenever this happens the other two matrix elements in the corresponding row of O B are practically equal in absolute value, which means that θ 12 would be close to maximal, contradicting phenomenology. We thus conclude that the forms B1-9 for U are not viable in case n.
one of the M ℓ textures in eqs. (8). As for (4, 3) textures, there are eight of them which agree with present-day phenomenology; the corresponding forms the lepton mixing matrix are given in eqs. (65b), (65c), (65h), (65i), (66e), (66f), (66h), and (66i); the corresponding textures of M ν are those in eqs. (60d)-(60f). 5 Even though there such a large variety of viable textures, the same cannot be said about the ensuing predictions, which are broadly similar for all of them: all the viable textures only tolerate • an inverted neutrino mass spectrum, • an overall scale of the neutrino masses given by (m 1 + m 2 + m 3 ) √ ∆ in the range [2.0, 2.1], • cos δ far away from 0, i.e. close to either +1 or −1, and  √ ∆ vs sin 2 θ 23 for form B8. The numeric scan was made by using the 3 σ data of ref. [29].
We thus conclude that texture-zero models of the (5, 2) and (4, 3) varieties are quite monotonous in their predictive power.