We review neutrino Yukawa textures with zeros within the framework of the type-I seesaw with three heavy right chiral neutrinos and in the basis where the latter and the charged leptons are mass diagonal. An assumed nonvanishing mass of every ultralight neutrino and the observed nondecoupling of any neutrino generation allow a maximum of four zeros in the Yukawa coupling matrix in family space. We show that the requirement of an exact symmetry, coupled with the observational constraints, reduces seventy-two allowed such textures to only four corresponding to just two different forms of the light neutrino mass matrix: one with an inverted and the other with a normal mass ordering. The masses and Majorana phases of ultralight neutrinos are predicted within definite ranges with laboratory and cosmological observational inputs. Within the same framework, we also study Yukawa textures with a fewer number of zeros, but with exact symmetry. We further formulate the detailed scheme of the explicit breaking of symmetry in terms of three small parameters for allowed four zero textures. The observed sizable mixing between the first and third generations of neutrinos is shown to follow for a suitable choice of these symmetry breaking parameters.

1. Introduction

The impressive experimental progress from neutrino oscillation studies [15] and the sharpening [6, 7] of the cosmological upper bound on the neutrino mass sum have underscored two fundamental but distinct puzzles. Why are the observed neutrinos so ultralight, that is, with masses in the sub-eV range? Why is the three neutrino mixing pattern of two large and one small (but measurable) angles so different from the sequentially small CKM mixing angles of quarks? There is a widespread feeling that the former is due to some kind of a seesaw mechanism [814] yielding ultralight Majorana neutrinos. It is our contention that the latter has to do with zeros in neutrino Yukawa textures plus a broken symmetry. Let us start with the simplest scheme of three weakly interacting flavored ultralight neutrinos discarding any possible light sterile ones mixing with them. We hold that there should be a fundamental principle behind a massless particle, as with gauge invariance and the photon. Since no such principle is identifiable with any single neutrino, we take each to have a nonzero mass. Though there are other types of proposed seesaw mechanisms, such as type-II [11, 12], type-III [13], and inverse seesaw [14, 15], in a minimalist approach we stick to the original type-I with three heavy right chiral electroweak singlet neutrinos denoted by the column vector .

We next turn to the issue of texture zeros. By a texture we mean a configuration of a Yukawa coupling matrix with some vanishing elements. Texture zeros have a long history in the quark sector where four zero Yukawa textures [1820] have had distinguished success in fitting the known quark masses and CKM parameters. The problem is simpler there since the Dirac quark mass matrix of a given charge, which is the corresponding Yukawa coupling matrix times the Higgs VEV, contains all information about physical quark masses. In the case of seesaw induced ultralight Majorana neutrinos, the elements of the Dirac mass matrix do not carry all information about physical neutrino masses. The latter are contained in the elements of the complex symmetric Majorana neutrino mass matrix which is related to through the standard seesaw formula. There have been initial as well as continuing efforts [2125] to assume the vanishing of certain elements in . But, we strongly feel that an occurrence of zeros must be linked to some fundamental symmetry [2628] or suppression mechanism [29] inherent in the Lagrangian itself. It seems more natural then to postulate the occurrence of such zeros in some elements of the neutrino Yukawa coupling matrix (equivalently ) which appears in the Lagrangian [16, 17, 3036]. There are ways [3739] to ensure the stability of such zeros under quantum corrections in type-I seesaw models.

An important point in the context of texture zeros is that of Weak Basis dependence. Both and change [40] under general (and different) unitary transformations of the left and right chiral fermion fields. In consequence, any Yukawa texture is basis dependent. It is further known that those fermions, which do not couple mutually in the Lagrangian, can be simultaneously put into a mass diagonal form by suitable basis transformations. Without loss of generality, we can therefore choose a Weak Basis in which the charged lepton fields and the very heavy right chiral neutrino fields are mass diagonal with real masses. The question arises as to how a flavor model, corresponding to a given set of texture zeros in such a basis, would be recognized in a different basis. It has been shown [40] that the vanishing of certain Weak Basis invariants would be a hallmark of those zeros. This is also related to the linkage of CP violation at low energies, probed in short or long baseline experiments, and at high energies, as relevant to leptogenesis. Though that linkage is a major motivation for postulating Yukawa texture zeros [3032], it is outside the scope of the present paper.

In this paper we focus on the role of texture zeros, occurring in , in understanding the observed pattern of neutrino masses and mixing angles. More generally, we show how they affect key aspects of low energy neutrino phenomenology. Four is shown to be the maximum number of such zeros allowed within our framework [30]. We classify all possible four zero textures, seventy two in total [30]. Then we introduce symmetry [16, 4166] as an invariance under the interchange of flavors () and () in the neutrino sector which is motivated by an automatic prediction of vanishing (maximal) mixing between the first (second) and third generations of neutrinos. This symmetry reduces the preceding seventy two textures to four which lead to only two distinct forms of whose phenomenological consequences are worked out [16, 17, 32]. Three zero textures with symmetry are also shown to have similar consequences, while textures with a lesser number of zeros have little predictivity [33]. We then discuss the general explicit breaking of symmetry in terms of three small parameters and show, within the lowest order of perturbation in those parameters, that the observed small mixing of first and third generations of neutrinos can be explained within our framework [33].

In Section 2 we set up our formalism. Section 3 contains the classification of all four zero textures and a discussion of symmetry. Section 4 addresses the consequent phenomenological implications. In Section 5 we discuss the realization of other symmetric texture zeros. Section 6 contains a general discussion of explicit symmetry breaking and how that fits observation. Finally, in Section 7 we summarize our conclusions.

2. Framework and Formalism

The relevant mass terms in our starting Lagrangian are where we have used the general definition of a conjugate fermion field ( being the charge conjugation matrix) and the identity Here , , and , respectively, denote the right chiral complex symmetric Majorana mass matrix, the neutrino Dirac mass, matrix and the charged lepton mass matrix in a three-dimensional family space. The superscripts “0” identifies the corresponding fields as flavor eigenstate ones. The complex symmetric neutrino mass matrix in the second line of (1) is denoted by , that is, The energy scale of is taken to be very high (>109 GeV), as compared with the electroweak scale  GeV.

The complete diagonalization of leads to where is a unitary matrix with blocks , , , and . In (4) and are three dimensional diagonal mass matrices, each with ultralight and heavy real positive entries, respectively:

Charged current interactions can then be written in terms of the semiweak coupling strength as well as the respective ultralight neutrino and heavy neutrino fields and : In an excellent approximation, the ultralight neutrino masses and mixing angles can now be obtained from Equation (8) is the well-known seesaw formula. We also choose to define the matrix and have

In (9), is the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) matrix admitting the standard parametrization: with , , and   being the yet unknown Dirac (Majorana) phase(s). We note for the sake of completeness that the unitary transformation between the column of mass eigenstate of left chiral neutrino fields and the corresponding flavor eigenstate is

The additional approximate relations to keep in mind are those between and the submatrices , of and of : Needless to add, we always neglect terms of order .

As mentioned earlier, without loss of generality, we can choose the Weak Basis in which and are with real positive entries. All CP-violating phases, stemming from , are contained in the Dirac mass matrix in this Weak Basis. As a consequence of (6) and (8), (9) can be written in the Casas-Ibarra form [67]: where is a complex orthogonal matrix: . An important comment on , following from (8), is that, our condition of no massless neutrino, that is , implies that . This means that textures of with one vanishing row or column or with a quartet of zeros (i.e., zeros in ,  ,  , and elements with and and  , or ) are inadmissible since they make vanish. Furthermore, in our Weak Basis, for any nonzero entry in with all other elements in its row or column being zero, from (8) develops a block diagonal form that is incompatible with the observed simultaneous mixing of three neutrinos. The same logic holds for any block diagonal texture of . Indeed, if any row in a texture of is orthogonal, element by element, to both the others, one neutrino family decouples and therefore makes such a texture inadmissible. These arguments have been shown [30] to be sufficient to rule out all textures in with more than four zeros. Four is then the maximum permitted number of zeros in a neutrino Yukawa texture.

3. Classification of Four Zero Textures and the Role of Symmetry

In this section we provide the classification of all possible four zero neutrino Yukawa textures and forms of the surviving textures, since these details were not given in [30, 68]. There are possible four zero neutrino Yukawa textures which can be classified into four classes [68]. In making this classification, we rule out the orthogonality between any two rows or columns by some artificial cancellation; orthogonality is to be ensured in terms of a vanishing product, element by element. We can now enumerate four cases.(i) and one family of neutrinos decouples: textures. For each texture of here, one row is orthogonal to the other rows. It follows that, in the neutrino mass matrix in our chosen basis with a diagonal , one neutrino family always decouples. So, though all neutrinos are massive here, these textures are to be discarded.(ii) and one family of neutrinos decouples: textures. Here each texture has a vanishing row and there are six such textures for every such row. Such a row generates a vanishing mass eigenvalue and the corresponding family decouples. Hence this class is also excluded.(iii) and no family decouples: textures [68]. Each of textures in this class has a vanishing column and each of the remaining has a quartet of zeros, leading to a vanishing . So, this class is rejected.(iv) and no family decouples: textures. These remaining textures are allowed by the criteria we have set up.

The retained textures are subdivided into two categories and . We wish to elaborate on this categorization [30]. Let us consistently use the complex parameters , , and for elements in belonging to the th column and the first, second, and third rows, respectively. The two categories then are as follows.

Category A. Here every texture has two mutually orthogonal rows (,   say, with ) and the corresponding derived has . Thus there are such textures divided into three sub-categories, each containing textures: () those with orthogonal rows and which generate ; () those with orthogonal rows and which generate ; () those with orthogonal rows and which generate . The explicit form of each of the textures in Category within the three sub-categories is shown in Table 1.

Category B. There are textures in this category. Each has two orthogonal columns, while no pair of rows is orthogonal. Invariably, then, it turns out that one row (, say) has two zeroes and the other two rows (say ) have one zero each. It is now a consequence of (8) that, in the derived neutrino mass-matrix , we have the relation Once again, one can make three subcategories with six entries each. has two zeros in the first row and one zero in each of the other two rows. has two zeros in the second row and one zero in each of the other two rows. has two zeros in the third row and one zero in each of the other two rows. All textures of Category are shown in Table 2 within the three subcategories.

We now raise the question of symmetry which we had explained in the Introduction. This symmetry is evidently invalid for the charged lepton mass terms. However, for elements in the Dirac mass matrix of neutrinos, it immediately implies the relations Moreover, for the masses of the very heavy right-chiral neutrinos, we have a result which is transparent as in our chosen basis. On account of (8) and (17) as well as (18), one is immediately led to the following relations among elements of the complex symmetric ultralight neutrino Majorana mass matrix : We take these as statements of a custodial symmetry in the ultralight neutrino sector. One can now invert (9) and explore the consequences of (19) in the parametrization of (11). An immediate consequence is the fixing of the two mixing angles pertaining to the third flavor: , . Since the measured former angle is compatible with within errors and the latter has been found to be small (), the occurrence of at least a broken symmetry in nature is a reasonable supposition that we adhere to. An interesting footnote to this discussion is the issue of tribimaximal mixing [69, 70] which subsumes symmetry but posits the additional relation leading to a fixation of the remaining mixing angle . However, we will not make use of (20).

An immediate consequence of the imposition of symmetry, via (17), is the drastic reduction of the seventy two allowed four zero textures of to only four [16]. This is seen just by inspection. The allowed symmetric textures are the following, each involving only three complex parameters.

Category A. One has

Category B. One has It may be noted that, in either category, any texture can be obtained from the other by the interchange of rows and or columns and . Because of symmetry, this means that the physical content of the two textures in each category is the same. Indeed, by use of (8), we obtain the same for either of them. Thus we have just two allowed ultralight neutrino Majorana mass matrices for Categories and , respectively.

4. Phenomenology with Symmetric Four Zero Yukawa Textures

Given symmetry, one automatically obtains that and . The current limits on these are and [1]. We shall later consider a small breaking of symmetry. But, for the moment, let us assume the latter to be the exact. The other mass and mixing parameters in the ultralight neutrino sector are kept free. Their experimentally allowed ranges to be used to constrain the nonzero elements of and in Table 3. We define , where () refer to the mass eigenstate neutrinos. It will now be convenient to reparametrize the elements of and in (23) and (24), respectively, in the way given in Table 3. Here , , , and   are real and positive quantities while , , , and   are phases. However, the phases and can be absorbed in the definition of the first family neutrino field for and respectively and therefore are not physical. Moreover, the overall phase in can also be absorbed by a further redefinition of all flavor eigenstate neutrino fields. So we can treat as real for further discussions. In addition, we have defined in Table 3 sets of derived real quantities which will be related to various observables.

With the reparametrization given in Table 3, and assume the simple forms These lead us, through the diagonalization of the matrix of (10), to the relations where for Categories , and

One can further make use of (9) to calculate [17] the ultralight masses in terms and also the Majorana phases , , compare (11), in terms of  ,  , and  . The former are given by and the latter by Here . The last quantity of physical interest that we calculate in this section is the effective mass appearing in the transition amplitude for the yet unobserved neutrinoless nuclear double beta decay. That is given by with as given in Table 3.

Feeding the experimental ranges from Table 4, we find that only the inverted mass ordering is allowed in Category while only the normal mass ordering is permitted for Category . Moreover, in the corresponding parameter plane [16], very constrained domains are allowed, as shown in Figure 1. The phases , are also severely restricted in magnitude, specifically and . These allow just a very limited region in the plane, leading to lower and upper bounds on the neutrino mass sum  , namely,    eV/  eV for an inverted/normal mass ordering [17]. It may be recalled that there is already a lower bound of  eV on the said sum from atmospheric neutrino data. Furthermore the general consensus [7] on the least model-dependent cosmological upper bound on it is  eV.

Turning to the individual neutrino masses /eV, /eV, and /eV, respectively, we obtain by use of (28) the respective intervals , , for Category and , , and for Category . However, there are correlated constraints among these masses. These are shown in the left-most panel of Figure 2. Given these allowed intervals and correlated constraints, it is not possible right now to distinguish between the hierarchical and quasi-degenerate possibilities. But a future reduction of these ranges and domains could pin this down. We next come to the Majorana phases , . One can ab initio restrict them to the interval to and utilize (29) as well as the expressions for , , , and in terms of the basic parameters (, , )/(, and ), depending on the category. The further application of the phenomenologically acceptable ranges of these parameters, as given above, leads to the allowed intervals , for Category and , for Category . Allowed values of and are shown in the middle panel of Figure 2.

Another quantity to be considered in this section is the double -decay effective mass , compare (30). The currently accepted upper bound [71] on it is  eV. In comparison, our allowed values versus the neutrino mass sum are shown in the rightmost panels of of Figure 2. More absolutely, we can say that for Category and for Category . The region near the upper bound in Category may be accessible in forthcoming experiments.

An interesting question pertains to the consequences of the effect of symmetry on couplings between the heavy right chiral and the ultralight left chiral neutrinos. The corresponding neutral gauge boson induced interactions are down by factor . On the other hand, the Higgs boson induced interactions affect leptogenesis modes and have been discussed in detail in [32]. Since leptogenesis is outside the scope of the present paper, we do not go into those discussions here.

5. Realization of Other Texture Zeros with Symmetry

Though four is the maximum number of allowed neutrino Yukawa texture zeros, we examine other textures with a lesser number of zeros for completeness [33]. Let us work in the same Weak Basis of real diagonal and . We wish to study only those textures that are compatible with symmetry which we believe to be approximately valid in the real world. The symmetric forms of and now are with , , , , and as complex numbers.

5.1. Three Zero Textures

We first identify possible three zero textures which are compatible with (31). Apart from , the other four complex parameters in come in pairs. So, for any texture with an odd number of zeros, must vanish. For three zero textures the remaining two zeros can be arranged in   ways. So, the four allowed three zero symmetric textures of are The last two textures have one vanishing row and one vanishing column, respectively. These can be discarded with our requirement of no massless neutrino, that is, , leaving only the first two textures as acceptable. In general, there can be   three zero textures. The conditions of symmetry, the non-zero value of , and the nondecoupling of any neutrino generation reduce this number to only two.

The first two textures of (32) have only three complex parameters each and we can just use , , and for both allowed textures of : Using the seesaw formula, we obtain an identical form of for both textures in (33), namely, Equation (34) can be written in the following form under a further reparametrization: with , , and . From , we can remove the phase and any phase in by rotating with the diagonal phase matrix  . Thus, has three real parameters, namely, , and , and only one phase . The interesting point to be noted is that the number of independent parameters in for symmetric three zero Yukawa textures is the same as that for symmetric four zero textures. We then have the same phenomenological expressions as in (26)–(30), but now with changed definitions of , namely,

5.2. Two Zero Textures

Again, looking at the symmetric form of in (31), we can conclude that, for any even zero textures, . Two zeros can be fitted to each of the remaining four pairs of parameters in four ways. The four possible two zero textures of are and are all allowed. So, the number of allowed symmetric two zero textures is the same as that of similar four zero textures. We need only the four parameters , , and to write down all four textures. The latter lead to three allowed forms of ; the first two such textures yield one form and the remaining two lead to two forms of . These are respectively given by Compared to four and three zero Yukawa textures, there are more independent parameters here. Apart from the overall mass scale, there will be three moduli and two irremovable phases. It is easier to fit the neutrino data with such a larger number of parameters and we do not discuss it any further.

5.3. One Zero

One zero textures represent the most trivial case among the symmetric neutrino Yukawa texture zeros. This is since, as an odd zero texture, it must have in (31). The single allowed texture of is and yields the following form of : Like the two zero textures, this allowed one zero texture to have six parameters: one overall real mass scale, three moduli, and two phases. These can easily fit the extant neutrino data.

6. The Breaking of Symmetry

As mentioned in previous sections, the results and are consequences of the custodial symmetry in . But present neutrino data from T2K, DOUBLE CHOOZ, RENO, and DAYA BAY experiments rule out by . So, the breaking of symmetry is an inevitable need in order to generate a nonzero value of . In addition, a departure from would arise from the same breaking. Another interesting consequence of a nonzero would be the observability of a CKM-type of CP violation in the lepton sector. Our previous expressions for , and will be modified if symmetry is broken.

This symmetry can be broken explicitly or spontaneously or dynamically as with the Renormalization Group evolution of Lagrangian parameters. Spontaneous breakdown generally requires the presence of extra scalars and needs a model with them. We do not go into a discussion of such models here. On the other hand, RG effects on neutrino masses can be incorporated with the methodology presented in [72, 73] in terms of the lepton mass arising through the running of the Yukawa coupling strength from the GUT scale to the weak scale . The effect is characterized by the parameter which has the 1-loop expression where is the ratio of the VEVs of the up-type and down-type neutral Higgs fields in the MSSM and is twice the sum of their squares. Even for a large , ()   is and cannot generate a of the order of .

We therefore turn to an explicit breaking of symmetry in . That can be realized in the following way.

Category A. One has Category B. One has Furthermore, where , are complex symmetry breaking parameters ( real) and is a real symmetry breaking parameter in . For these modified and , we have the following ’s.

Category A. One has Category B. One has The detailed diagonalization and expressions for mass differences and mixing angles are given in the appendix.

A nonzero arises after symmetry breaking. The value is possible for variations of , , , , compare the appendix. An appropriate choice of symmetry breaking parameters, that is, , and slightly shifted parameter spaces for in Category and in Category are needed. A nonzero violating effect can be effected through the Jarlskog invariant : where elements of and mass squared differences ,   are given in the appendix. Note that . A detailed treatment of explicitly broken symmetric four zero and three zero textures is given in [33].

7. Concluding Summary

We have reviewed neutrino Yukawa textures with zeros within the type-I seesaw with three heavy right chiral neutrinos and in the basis where the latter and the charged leptons are mass diagonal. The conditions of a nonvanishing mass of every ultralight neutrino and of the non-decoupling of any neutrino generation allow a maximum of four zeros in the neutrino Yukawa coupling matrix . There are seventy two such textures. We show that the requirement of an exact symmetry, coupled with observational constraints, reduces the seventy two allowed textures in such a to only four corresponding to just two different forms of the light neutrino mass matrix , resulting in an inverted/normal mass ordering. Apart from an overall mass scale, for every category has two real parameters and an irremovable phase. These parameters ,  , and for Category and ,  , and for Category get highly restricted, given the ranges of measured neutrino mass squared differences and mixing angles. Neutrino masses and Majorana phases are predicted within definite ranges with laboratory and cosmological inputs. The predicted respective masses /eV, /eV, and /eV are , , for Category and , , and for Category . The corresponding intervals of the Majorana phases are , for Category and , for Category . In addition, we predict the range of the mass scale associated with decay, most of which is well below the reach of planned experiments. We have also studied Yukawa textures with a fewer number of zeros, but with exact symmetry. Finally, we have formulated the detailed scheme of three-parameter explicit breaking of symmetry for allowed four zero textures. A value of can be arranged for a suitable choice of small values of these symmetry breaking parameters.


Expressions for Measurable Quantities

We can write the general form of a broken symmetric in the following way [33]: The explicit expressions of , , , , , and for four forms of neutrino mass matrices after symmetry breaking are given in Table 5.

We can now have Here Note that and have similar functional forms as . If we write in the following way then and will be

The diagonalization of yields and also expressions for five relevant measurable quantities. The latter are: , , , and . We will associate superscripts with all of these five quantities to distinguish them from their unperturbed expressions. The relevant functions for these physical quantities are given below where , , , being the unperturbed mixing angle in (26). There are also and ,  –6 which have similar functional forms as . If we write as then and will be

The final results with three symmetry breaking parameters are


The authors would like to thank Prof. Ambar Ghosal for his long collaboration in almost all works discussed in this review. P. Roy acknowledges partial support from a DAE Raja Ramanna Fellowship.