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Neutrino Yukawa Textures within Type-I Seesaw
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.
The impressive experimental progress from neutrino oscillation studies [1–5] 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 [8–14] 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 , 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 [18–20] 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 [21–25] to assume the vanishing of certain elements in . But, we strongly feel that an occurrence of zeros must be linked to some fundamental symmetry [26–28] or suppression mechanism  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, 30–36]. There are ways [37–39] 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  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  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 [30–32], 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 . We classify all possible four zero textures, seventy two in total . Then we introduce symmetry [16, 41–66] 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 . 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 .
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 : 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  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 . 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 . 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 . 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 . 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 . 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.
One can further make use of (9) to calculate  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 , 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 . 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  on the least model-dependent cosmological upper bound on it is eV.