# Neutrino Yukawa Textures within Type-I Seesaw

Authors: Biswajit Adhikary; Probir Roy
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

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 exactμt 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 μt symmetry. We further formulate the detailed scheme of the explicit breaking of μt 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.


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## Body

## 1. Introduction

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. (1) Why are the observed neutrinos so ultralight, that is,  with masses in the sub-eV range? (2) 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 μτ 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 NR.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 MD do not carry all information about physical neutrino masses. The latter are contained in the elements of the complex symmetric Majorana neutrino mass matrix Mν which is related to MD through the standard seesaw formula. There have been initial as well as continuing efforts [21–25] to assume the vanishing of certain elements in Mν. But, we strongly feel that an occurrence of zeros must be linked to some fundamental symmetry [26–28] 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 MD) 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. BothMD and Mν 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 l and the very heavy right chiral neutrino fields NR 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 [30–32], it is outside the scope of the present paper.In this paper we focus on the role of texture zeros, occurring inMD, 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 μτ symmetry [16, 41–66] as an invariance under the interchange of flavors μ (2) and τ (3) 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 Mν whose phenomenological consequences are worked out [16, 17, 32]. Three zero textures with μτ 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 μτ 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 Section2 we set up our formalism. Section 3 contains the classification of all four zero textures and a discussion of μτ symmetry. Section 4 addresses the consequent phenomenological implications. In Section 5 we discuss the realization of other μτ symmetric texture zeros. Section 6 contains a general discussion of explicit μτ 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(1)-ℒm=νL0¯MDNR0+12N0LC¯MRNR0+lL0¯MllR0+h.c.=12(νL0¯N0LC¯)(0MDMDTMR)(ν0RCNR0)+lL0¯MllR0+h.c.,
where we have used the general definition of a conjugate fermion field ψC=γ0Cψ* (C being the charge conjugation matrix) and the identity
(2)ψ-LmψR′=ψ′LC¯mTψRC.
Here MR, MD, and Ml, 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 6×6 neutrino mass matrix in the second line of (1) is denoted by ℳ, that is,
(3)ℳ=(0MDMDTMR).
The energy scale of MR is taken to be very high (>109 GeV), as compared with the electroweak scale v≃246 GeV.The complete diagonalization ofℳ leads to
(4)𝒱†ℳ𝒱*=diag(d,D),
where 𝒱 is a 6×6 unitary matrix
(5)𝒱=(KGST),
with 3×3 blocks K, G, S, and T. In (4) d and D are three dimensional diagonal mass matrices, each with ultralight and heavy real positive entries, respectively:
(6)d=diag(m1,m2,m3),m1,2,3<1eV,D=diag(M1,M2,M3),M1,2,3>109GeV.Charged current interactions can then be written in terms of the semiweak coupling strengthg as well as the respective ultralight neutrino and heavy neutrino fields νi and Ni:
(7)ℒcc=-g2(liL¯γμKijνjL+liL¯γμGijNjL)Wμ.
In an excellent approximation, the ultralight neutrino masses and mixing angles can now be obtained from
(8)Mν≃-MDMR-1MDT,(9)U†MνU*=d.
Equation (8) is the well-known seesaw formula. We also choose to define the matrix hν=MνMν† and have
(10)U†hνU=d2.In (9), U is the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) matrix admitting the standard parametrization:
(11)UPMNS=(c12c13s12c13s13e-iδD-s12c23-c12s23s13eiδDc12c23-s12s23s13eiδDs23c13s12s23-c12c23s13eiδD-c12s23-s12c23s13eiδDc23c13)×(eiαM1/2000eiαM2/20001),
with cij=cosθij, sij=sinθij, and δD(αM1,αM2)  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 νL and the corresponding flavor eigenstate νL0 is
(12)νL0=UνL.The additional approximate relations to keep in mind are those betweenU and the submatrices K, G of 𝒱 and MD of ℳ:
(13)Kij=Uij+O(v2MR2),MkGjk=(MD)jk+O(v3MR2),knotsummed.
Needless to add, we always neglect terms of order v2/MR2.As mentioned earlier, without loss of generality, we can choose the Weak Basis in whichMl and MR are
(14)Ml=diag(me,mμ,mτ),MR=diag(M1,M2,M3),
with real positive entries. All CP-violating phases, stemming from ℳ, are contained in the Dirac mass matrix MD in this Weak Basis. As a consequence of (6) and (8), (9)  MD can be written in the Casas-Ibarra form [67]:
(15)MD=iUdRMR,
where R is a complex orthogonal matrix: RTR=RRT=I. An important comment on MD, following from (8), is that, our condition of no massless neutrino, that is detMν≠0, implies that detMD≠0. This means that textures of MD with one vanishing row or column or with a quartet of zeros (i.e., zeros in ij,  lk,  ik, and lj elements with i≠l and k≠j and  l=1,2, or 3) are inadmissible since they make detMD vanish. Furthermore, in our Weak Basis, for any nonzero entry in MD with all other elements in its row or column being zero, Mν 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 MD. Indeed, if any row in a texture of MD 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 MD 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 ofμτ 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 9C4=126 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)
detMD≠0 and one family of neutrinos decouples: 9 textures. For each texture of MD here, one row is orthogonal to the other rows. It follows that, in the neutrino mass matrix Mν in our chosen basis with a diagonal MR, one neutrino family always decouples. So, though all neutrinos are massive here, these textures are to be discarded.(ii)
detMD=0 and one family of neutrinos decouples: 18 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)
detMD=0 and no family decouples: 27 textures [68]. Each of 18 textures in this class has a vanishing column and each of the remaining 9 has a quartet of zeros, leading to a vanishing detMν. So, this class is rejected.(iv)
detMD≠0 and no family decouples: 72 textures. These remaining textures are allowed by the criteria we have set up.The retained textures are subdivided into two categoriesA and B. We wish to elaborate on this categorization [30]. Let us consistently use the complex parameters ak, bk, and ck for elements in MD belonging to the kth 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 (i,  j say, with i≠j) and the corresponding derived Mν has (Mν)ij=0. Thus there are 54 such textures divided into three sub-categories, each containing 18 textures: (A1) those with orthogonal rows 1 and 2 which generate (Mν)12=(Mν)21=0; (A2) those with orthogonal rows 2 and 3 which generate (Mν)23=(Mν)32=0; (A3) those with orthogonal rows 1 and 3 which generate (Mν)13=(Mν)31=0. The explicit form of each of the 54 textures in Category A within the three sub-categories is shown in Table 1.Table 1
Four zero Yukawa textures ofMD in Category A with subcategories A1, A2, and A3.
CategoryA1 (orthogonality between rows 1 and 2)
(
0
a
2
0
b
1
0
b
3
0
c
2
c
3
)
(
0
a
2
0
b
1
0
b
3
c
1
c
2
0
)
(
0
0
a
3
b
1
b
2
0
0
c
2
c
3
)
(
0
0
a
3
b
1
b
2
0
c
1
0
c
3
)
(
a
1
0
0
0
b
2
b
3
c
1
0
c
3
)
(
a
1
0
0
0
b
2
b
3
c
1
c
2
0
)
(
0
a
2
0
0
0
b
3
c
1
c
2
c
3
)
(
0
a
2
0
b
1
0
0
c
1
c
2
c
3
)
(
0
0
a
3
b
1
0
0
c
1
c
2
c
3
)
(
0
0
a
3
0
b
2
0
c
1
c
2
c
3
)
(
a
1
0
0
0
0
b
3
c
1
c
2
c
3
)
(
a
1
0
0
0
b
2
0
c
1
c
2
c
3
)
(
a
1
0
a
3
0
b
2
0
0
c
2
c
3
)
(
a
1
0
a
3
0
b
2
0
c
1
c
2
0
)
(
a
1
a
2
0
0
0
b
3
0
c
2
c
3
)
(
a
1
a
2
0
0
0
b
3
c
1
0
c
3
)
(
0
a
2
a
3
b
1
0
0
c
1
0
c
3
)
(
0
a
2
a
3
b
1
0
0
c
1
c
2
0
)
CategoryA2 (orthogonality between rows 2 and 3)
(
0
a
2
a
3
0
b
2
0
c
1
0
c
3
)
(
a
1
a
2
0
0
b
2
0
c
1
0
c
3
)
(
0
a
2
a
3
0
0
b
2
c
1
c
2
0
)
(
a
1
0
a
3
0
0
b
3
c
1
c
2
0
)
(
a
1
0
a
3
b
1
0
0
0
c
2
c
3
)
(
a
1
a
2
0
b
1
0
0
0
c
2
c
3
)
(
a
1
a
2
a
3
0
b
2
0
0
0
c
3
)
(
a
1
a
2
a
1
0
b
2
0
c
1
0
0
)
(
a
1
a
2
a
3
0
0
b
2
0
c
2
0
)
(
a
1
a
2
a
3
0
0
b
3
c
1
0
0
)
(
a
1
a
2
a
3
b
1
0
0
0
0
c
3
)
(
a
1
a
2
a
3
b
1
0
0
0
c
2
0
)
(
0
a
2
a
3
b
1
0
b
3
0
c
2
0
)
(
a
1
a
2
0
b
1
0
b
3
0
c
2
0
)
(
0
a
2
a
3
b
1
b
2
0
0
0
c
3
)
(
a
1
0
a
3
b
1
b
2
0
0
0
c
3
)
(
a
1
0
a
3
0
b
2
b
3
c
1
0
0
)
(
a
1
a
2
0
0
b
2
b
3
c
1
0
0
)
CategoryA3 (orthogonality between rows 1 and 3)
(
0
a
2
0
0
b
2
b
3
c
1
0
c
3
)
(
0
a
2
0
b
1
b
2
0
c
1
0
c
3
)
(
0
0
a
3
0
b
2
b
3
c
1
c
2
0
)
(
0
0
a
3
b
1
0
b
3
c
1
c
2
0
)
(
a
1
0
0
b
1
b
2
0
0
c
2
c
3
)
(
a
1
0
0
b
1
0
b
3
0
c
2
c
3
)
(
0
a
2
0
b
1
b
2
b
3
0
0
c
3
)
(
0
a
2
0
b
1
b
2
b
3
c
1
0
0
)
(
0
0
a
3
b
1
b
2
b
3
0
c
2
0
)
(
0
0
a
3
b
1
b
2
b
3
c
1
0
0
)
(
a
1
0
0
b
1
b
2
b
3
0
c
2
0
)
(
a
1
0
0
b
1
b
2
b
3
0
0
c
3
)
(
a
1
0
a
3
0
b
2
b
3
0
c
2
0
)
(
a
1
0
a
3
b
1
b
2
0
0
c
2
0
)
(
a
1
a
2
0
0
b
2
b
3
0
0
c
3
)
(
a
1
a
2
0
b
1
0
b
3
0
0
c
3
)
(
0
a
2
a
3
b
1
b
2
0
c
1
0
0
)
(
0
a
2
a
3
b
1
0
b
3
c
1
0
0
)Category B. There are 18 textures in this category. Each has two orthogonal columns, while no pair of rows is orthogonal. Invariably, then, it turns out that one row (i, say) has two zeroes and the other two rows (say k,l≠i) have one zero each. It is now a consequence of (8) that, in the derived neutrino mass-matrix  Mν, we have the relation
(16)detcofactor[(Mν)kl]=0.
Once again, one can make three subcategories with six entries each. B1 has two zeros in the first row and one zero in each of the other two rows. B2 has two zeros in the second row and one zero in each of the other two rows. B3 has two zeros in the third row and one zero in each of the other two rows. All 18 textures of Category B are shown in Table 2 within the three subcategories.Table 2
Four zero Yukawa textures ofMD in Category B with subcategories B1, B2, and B3.
CategoryB1
First row with two zeros, a pair of orthogonal columns, non-orthogonal rows
(
0
a
2
0
0
b
1
b
3
c
1
c
2
0
)
(
0
a
2
0
b
1
b
2
0
0
c
2
c
3
)
(
0
0
a
3
0
b
2
b
3
c
1
0
c
3
)
(
0
0
a
3
b
1
0
b
3
0
c
2
c
3
)
(
a
1
0
0
b
1
0
b
3
c
1
c
2
0
)
(
a
1
0
0
b
1
b
2
0
c
1
0
c
3
)
CategoryB2
Second row with two zeros, a pair of orthogonal columns, non-orthogonal rows
(
0
a
2
a
3
0
b
2
0
c
1
c
2
0
)
(
a
1
a
2
0
0
b
2
0
0
c
2
c
3
)
(
0
a
2
a
3
0
0
b
2
c
1
0
c
3
)
(
a
1
0
a
3
0
0
b
3
0
c
2
c
3
)
(
a
1
0
a
3
b
1
0
0
c
1
c
2
0
)
(
a
1
a
2
0
b
1
0
0
c
1
0
c
3
)
CategoryB3
Third row with two zeros, a pair of orthogonal columns, non-orthogonal rows
(
a
1
a
2
0
0
b
2
b
3
0
c
2
0
)
(
0
a
2
a
3
b
1
b
2
0
0
c
2
0
)
(
a
1
0
a
3
0
b
2
b
3
0
0
c
3
)
(
0
a
2
a
3
b
1
0
b
3
0
0
c
3
)
(
a
1
0
a
3
b
1
b
2
0
c
1
0
0
)
(
a
1
a
2
0
b
1
0
b
3
c
1
0
0
)We now raise the question ofμτ 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 MD of neutrinos, it immediately implies the relations
(17)(MD)12=(MD)13,(MD)21=(MD)31,(MD)23=(MD)32,(MD)22=(MD)33.
Moreover, for the masses of the very heavy right-chiral neutrinos, we have
(18)(MR)22=(MR)33,
a result which is transparent as M2=M3 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 Mν:
(19)(Mν)12=(Mν)13,(Mν)22=(Mν)33.
We take these as statements of a custodial μτ 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: θ23=π/4, θ13=0. Since the measured former angle is compatible with 45∘ within errors and the latter has been found to be small (≃9∘), the occurrence of at least a broken μτ 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 μτ symmetry but posits the additional relation
(20)(Mν)11+(Mν)13=(Mν)22+(Mν)23,
leading to a fixation of the remaining mixing angle θ12=sin-1(1/3)≃35.26∘. However, we will not make use of (20).An immediate consequence of the imposition ofμτ symmetry, via (17), is the drastic reduction of the seventy two allowed four zero textures of MD to only four [16]. This is seen just by inspection. The allowed μτ symmetric textures are the following, each involving only three complex parameters.Category A. One has
(21)MDA1=(a1a2a200b10b10),MDA2=(a1a2a20b1000b1).Category B. One has
(22)MDB1=(a100b10b2b1b20),MDB2=(a100b1b20b10b2).
It may be noted that, in either category, any texture can be obtained from the other by the interchange of rows 2 and 3 or columns 2 and 3. Because of μτ 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 Mν for either of them. Thus we have just two allowed ultralight neutrino Majorana mass matrices
(23)MνA=-(a12M1+2a22M2a2b1M2a2b1M2a2b1M2b12M20a2b1M20b12M2),(24)MνB=-(a12M1a1b1M1a1b1M1a1b1M1b12M1+b22M2b12M1a1b1M1b12M1b12M1+b22M2),
for Categories A and B, respectively.


## 4. Phenomenology withμτ Symmetric Four Zero Yukawa Textures

Givenμτ symmetry, one automatically obtains that θ23=π/4 and θ13=0. The current 3σ limits on these are 36.86∘<θ23<55.55∘ and 7.49∘<θ13<10.46∘ [1]. We shall later consider a small breaking of μτ 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 3σ ranges to be used to constrain the nonzero elements of MνA and MνB in Table 3. We define Δij2=mi2-mj2, where i,j (=1,2,3) refer to the mass eigenstate neutrinos. It will now be convenient to reparametrize the elements of  MνA and MνB in (23) and (24), respectively, in the way given in Table 3. Here k1, k2, l1, and  l2 are real and positive quantities while α, α′, β, and  β′ are phases. However, the phases α′ and β′ can be absorbed in the definition of the first family neutrino field νe for MνA and MνB respectively and therefore are not physical. Moreover, the overall phase in mA,B can also be absorbed by a further redefinition of all flavor eigenstate neutrino fields. So we can treat mA,B as real for further discussions. In addition, we have defined in Table 3 sets of derived real quantities  X1-4A,B which will be related to various observables.Table 3
Reparametrized quantities and relevant functions in CategoryA and Category B.
CategoryA
CategoryB
M
ν
A
M
ν
B
m
A
=
-
b
1
2
M
2,
m
B
=
-
b
2
2
M
2,
k
1
e
i
(
α
+
α
′
)
=
a
1
b
1
M
2
M
1,
l
1
e
i
β
′
=
a
1
b
2
M
2
M
1
k
2
e
i
α
′
=
a
2
b
1,
l
2
e
i
β
=
b
1
b
2
M
2
M
1
α
=
arg
a
1
a
2
β
=
arg
b
1
b
2
X
1
A
=
2
2
k
2
[
(
1
+
2
k
2
2
)
2
+
k
1
4
+
2
k
1
2
(
1
+
2
k
2
2
)
cos
2
α
]
1
/
2
X
1
B
=
2
2
l
1
l
2
[
(
l
1
2
+
2
l
2
2
)
2
+
1
+
2
(
l
1
2
+
2
l
2
2
)
cos
2
β
]
1
/
2
X
2
A
=
1
-
k
1
4
-
4
k
2
4
-
4
k
1
2
k
2
2
cos
2
α
X
2
B
=
1
+
4
l
2
2
cos
2
β
+
4
l
2
4
-
l
1
4
X
3
A
=
1
-
4
k
2
4
-
k
1
4
-
4
k
1
2
k
2
2
cos
2
α
-
4
k
2
2
X
3
B
=
1
-
(
l
1
2
+
2
l
2
2
)
2
-
4
l
2
2
cos
2
β
X
4
A
=
k
1
4
+
4
k
2
4
+
4
k
1
2
k
2
2
cos
2
α
X
4
B
=
l
1
4With the reparametrization given in Table3, MνA and MνB assume the simple forms
(25)MνA=mA(k12e2iα+2k22k2k2k210k201),MνB=mB(l12l1l2eiβl1l2eiβl1l2eiβl22e2iβ+1l22e2iβl1l2eiβl22e2iβl22e2iβ+1).
These lead us, through the diagonalization of the matrix hν of (10), to the relations
(26)Δ212=m2X,Δ322=m22(X3-X),tan2θ12=X1X2,
where m=mA,B for Categories A, B and
(27)X=X12+X22.One can further make use of (9) to calculate [17] the ultralight masses m1,2,3 in terms Δ212 and also the Majorana phases αM1, αM2, compare (11), in terms of  m1,  m2,  and  m3. The former are given by
(28)m1,2=|Δ212(2-X3∓X2X)|1/2,m3=|Δ212X|1/2
and the latter by
(29)cos(αM1-arg·Z)=|Z|2m32+m12sin4θ12-m22cos4θ122m1m3sin2θ12|Z|,cos(αM2-arg·Z)=|Z|2m32+m22cos4θ12-m12sin4θ122m2m3cos2θ12|Z|.
Here Z=[(ℳν)22+(ℳν)23][(ℳν)22-(ℳν)23]-1. The last quantity of physical interest that we calculate in this section is the effective mass mββ=|(Mν)11| appearing in the transition amplitude for the yet unobserved neutrinoless nuclear double beta decay. That is given by
(30)mββ=|Δ212X4X-1|1/2,
with X4 as given in Table 3.Feeding the experimental3σ ranges from Table 4, we find that only the inverted mass ordering Δ322<0 is allowed in Category A while only the normal mass ordering Δ322>0 is permitted for Category B. Moreover, in the corresponding k1-k2/l1-l2 parameter plane [16], very constrained domains are allowed, as shown in Figure 1. The phases α, β are also severely restricted in magnitude, specifically 89.0∘≤|α|<90∘ and 87.0∘≤|β|<90∘. These allow just a very limited region in the  X3-X plane, leading to 3σ lower and upper bounds on the neutrino mass sum  m1+m2+m3, namely,  [0.156,0.5]  eV/[0.074,0.132]  eV for an inverted/normal mass ordering [17]. It may be recalled that there is already a lower bound of 0.05 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 0.5 eV.Table 4
Input experimental values [1].
Quantity
Experimental3σ range
Δ
21
2
7.12
×
1
0
-
5
e
V
2
<
Δ
21
2
<
8.20
×
1
0
-
5
e
V
2
Δ
32
2
<
0
-
2.76
×
1
0
-
3
e
V
2
<
Δ
32
2
<
-
2.22
×
1
0
-
3
e
V
2
Δ
32
2
>
0
2.18
×
1
0
-
3
e
V
2
<
Δ
32
2
<
2.70
×
1
0
-
3
e
V
2
θ
12
31.3
0
∘
<
θ
12
<
37.4
6
∘Variation ofk1 and k2 in Category A and of l1 and l2 in Category B with μτ symmetry over the 3σ allowed ranges of Δ212,  |Δ322|, and θ12 [16].
(a)
(b)Turning to the individual neutrino massesm1/eV, m2/eV, and m3/eV, respectively, we obtain by use of (28) the respective 3σ intervals [0.0452,0.1682], [0.0457,0.1684], [0.077,0.1632] for Category A and [0.0110,0.0335], [0.0144,0.0345], and  [0.0485,0.0638] for Category B. 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 αM1, αM2. One can ab initio restrict them to the interval -π to π and utilize (29) as well as the expressions for Z, m1, m2, m3 and tan2θ12 in terms of the basic parameters (k1,k2,α)/(l1,l2 and β), depending on the category. The further application of the phenomenologically acceptable ranges of these parameters, as given above, leads to the allowed 3σ intervals -98.0∘≤αM1≤20.0∘, 9.2∘≤αM2≤36.4∘ for Category A and -88.6∘≤αM1≤7.97∘, 90.7∘≤αM2≤128.8∘ for Category B. Allowed values of αM1 and αM2 are shown in the middle panel of Figure 2.Allowedm2,3 versus  m1 (left), αM2 versus  αM1 (middle), and mββ versus  Σmi (right) for Category A (top) and Category B (bottom) [17].
(a)
(b)
(c)
(d)
(e)
(f)Another quantity to be considered in this section is the doubleβ-decay effective mass mββ, compare (30). The currently accepted upper bound [71] on it is mββ<0.35 eV. In comparison, our allowed values mββ versus the neutrino mass sum ∑i‍mi are shown in the rightmost panels of of Figure 2. More absolutely, we can say that 0.038≤mββ/eV≤0.161 for Category A and 0.003≤mββ/eV≤0.0186 for Category B. The region near the upper bound in Category A may be accessible in forthcoming experiments.An interesting question pertains to the consequences of the effect ofμτ 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 O(v2/MR2). 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 withμτ 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 Ml and MR. We wish to study only those textures that are compatible with μτ symmetry which we believe to be approximately valid in the real world. The μτ symmetric forms of MD and MR now are
(31)MD=(abbcdeced),MR=diag(M1,M2,M2),
with a, b, c, d, and e as complex numbers.


### 5.1. Three Zero Textures

We first identify possible three zero textures which are compatible with (31). Apart from a, the other four complex parameters in MD come in pairs. So, for any texture with an odd number of zeros, a must vanish. For three zero textures the remaining two zeros can be arranged in  4C1=4 ways. So, the four allowed three zero μτ symmetric textures of MD are
(32)(0bbc0ece0),(0bbcd0c0d),(000cdeced),(0bb0de0ed).
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, detMν≠0, leaving only the first two textures as acceptable. In general, there can be  9C3=84 three zero textures. The conditions of (1)μτ symmetry,  (2) the non-zero value of detMν, and (3) 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 b, c, and d for both allowed textures of MD:
(33)(0bbc0dcd0),(0bbcd0c0d).
Using the seesaw formula, we obtain an identical form of Mν for both textures in (33), namely,
(34)Mν=-(2b2M2bdM2bdM2bdM2c2M1+d2M2c2M1bdM2c2M1c2M1+d2M2).
Equation (34) can be written in the following form under a further reparametrization:
(35)Mν=m0(2k12ei2α1k1eiα1k1eiα1k1eiα1k22eiα2+1k22eiα2k1eiα1k22eiα2k22eiα2+1),
with m0=-d2/M2, k1eiα1=b/d, and k2eiα2=c/dM2/M1. From Mν, we can remove the phase α1 and any phase θm in m0 by rotating Mν with the diagonal phase matrix  diag(e-iα1,1,1)e-iθm/2. Thus, Mν has three real parameters, namely, m0, k1 and k2, and only one phase α2. The interesting point to be noted is that the number of independent parameters in Mν for μτ symmetric three zero Yukawa textures is the same as that for μτ symmetric four zero textures. We then have the same phenomenological expressions as in (26)–(30), but now with changed definitions of X1,2,3,4, namely,
(36)X1=22k1(1+2k12)2+4k24+4k22(1+2k12)cos2α2,X2=4k24+1+4k22cos2α2-4k14,X3=1-4k14-4k12-4k24-4k22cos2α2,X4=4k12.


### 5.2. Two Zero Textures

Again, looking at theμτ symmetric form of MD in (31), we can conclude that, for any even zero textures, a≠0. Two zeros can be fitted to each of the remaining four pairs of parameters in four ways. The four possible two zero textures of MD are
(37)(abbc0dcd0),(abbcd0c0d),(a00cbdcdb),(abb0cd0dc),
and are all allowed. So, the number of allowed μτ symmetric two zero textures is the same as that of similar four zero textures. We need only the four parameters a, b, c and d to write down all four textures. The latter lead to three allowed forms of Mν; the first two such textures yield one form and the remaining two lead to two forms of Mν. These are respectively given by
(38)Mν=-(a2M1+2b2M2acM1+bdM2acM1+bdM2acM1+bdM2c2M1+d2M2c2M1acM1+bdM2c2M1c2M1+d2M2),(39)Mν=-(a2M1acM1acM1acM1c2M1+d2M2+b2M2c2M1+2bdM2acM1c2M1+2bdM2c2M1+d2M2+b2M2),(40)Mν=-(a2M1+2b2M2bcM2+bdM2bcM2+bdM2bcM2+bdM2c2M2+d2M22cdM2bcM2+bdM22cdM2c2M2+d2M2).
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 theμτ symmetric neutrino Yukawa texture zeros. This is since, as an odd zero texture, it must have a=0 in (31). The single allowed texture of MD is
(41)(0bbcdeced),
and yields the following form of Mν:
(42)Mν=-(2b2M2beM2+bdM2beM2+bdM2beM2+bdM2c2M1+d2M2+e2M2c2M1+2edM2beM2+bdM2c2M1+2edM2c2M1+d2M2+e2M2).
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.


## 5.1. Three Zero Textures

We first identify possible three zero textures which are compatible with (31). Apart from a, the other four complex parameters in MD come in pairs. So, for any texture with an odd number of zeros, a must vanish. For three zero textures the remaining two zeros can be arranged in  4C1=4 ways. So, the four allowed three zero μτ symmetric textures of MD are
(32)(0bbc0ece0),(0bbcd0c0d),(000cdeced),(0bb0de0ed).
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, detMν≠0, leaving only the first two textures as acceptable. In general, there can be  9C3=84 three zero textures. The conditions of (1)μτ symmetry,  (2) the non-zero value of detMν, and (3) 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 b, c, and d for both allowed textures of MD:
(33)(0bbc0dcd0),(0bbcd0c0d).
Using the seesaw formula, we obtain an identical form of Mν for both textures in (33), namely,
(34)Mν=-(2b2M2bdM2bdM2bdM2c2M1+d2M2c2M1bdM2c2M1c2M1+d2M2).
Equation (34) can be written in the following form under a further reparametrization:
(35)Mν=m0(2k12ei2α1k1eiα1k1eiα1k1eiα1k22eiα2+1k22eiα2k1eiα1k22eiα2k22eiα2+1),
with m0=-d2/M2, k1eiα1=b/d, and k2eiα2=c/dM2/M1. From Mν, we can remove the phase α1 and any phase θm in m0 by rotating Mν with the diagonal phase matrix  diag(e-iα1,1,1)e-iθm/2. Thus, Mν has three real parameters, namely, m0, k1 and k2, and only one phase α2. The interesting point to be noted is that the number of independent parameters in Mν for μτ symmetric three zero Yukawa textures is the same as that for μτ symmetric four zero textures. We then have the same phenomenological expressions as in (26)–(30), but now with changed definitions of X1,2,3,4, namely,
(36)X1=22k1(1+2k12)2+4k24+4k22(1+2k12)cos2α2,X2=4k24+1+4k22cos2α2-4k14,X3=1-4k14-4k12-4k24-4k22cos2α2,X4=4k12.


## 5.2. Two Zero Textures

Again, looking at theμτ symmetric form of MD in (31), we can conclude that, for any even zero textures, a≠0. Two zeros can be fitted to each of the remaining four pairs of parameters in four ways. The four possible two zero textures of MD are
(37)(abbc0dcd0),(abbcd0c0d),(a00cbdcdb),(abb0cd0dc),
and are all allowed. So, the number of allowed μτ symmetric two zero textures is the same as that of similar four zero textures. We need only the four parameters a, b, c and d to write down all four textures. The latter lead to three allowed forms of Mν; the first two such textures yield one form and the remaining two lead to two forms of Mν. These are respectively given by
(38)Mν=-(a2M1+2b2M2acM1+bdM2acM1+bdM2acM1+bdM2c2M1+d2M2c2M1acM1+bdM2c2M1c2M1+d2M2),(39)Mν=-(a2M1acM1acM1acM1c2M1+d2M2+b2M2c2M1+2bdM2acM1c2M1+2bdM2c2M1+d2M2+b2M2),(40)Mν=-(a2M1+2b2M2bcM2+bdM2bcM2+bdM2bcM2+bdM2c2M2+d2M22cdM2bcM2+bdM22cdM2c2M2+d2M2).
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 theμτ symmetric neutrino Yukawa texture zeros. This is since, as an odd zero texture, it must have a=0 in (31). The single allowed texture of MD is
(41)(0bbcdeced),
and yields the following form of Mν:
(42)Mν=-(2b2M2beM2+bdM2beM2+bdM2beM2+bdM2c2M1+d2M2+e2M2c2M1+2edM2beM2+bdM2c2M1+2edM2c2M1+d2M2+e2M2).
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 ofμτ Symmetry

As mentioned in previous sections, the resultsθ13=0 and θ23=π/4 are consequences of the custodial μτ symmetry in Mν. But present neutrino data from T2K, DOUBLE CHOOZ, RENO, and DAYA BAY experiments rule out θ13=0 by 5σ. So, the breaking of μτ symmetry is an inevitable need in order to generate a nonzero value of θ13. In addition, a departure from θ23=π/4 would arise from the same breaking. Another interesting consequence of a nonzero θ13 would be the observability of a CKM-type of CP violation in the lepton sector. Our previous expressions for Δ212, Δ322 and tan2θ12 will be modified if μτ 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 τ lepton mass arising through the running of the Yukawa coupling strength from the GUT scale  Λ to the weak scale λ. The effect is characterized by the parameter Δτ which has the 1-loop expression
(43)Δτ≃mτ28π2v2(tan2β+1)ln(Λλ),
where tanβ is the ratio of the VEVs of the up-type and down-type neutral Higgs fields in the MSSM and v2 is twice the sum of their squares. Even for a large tanβ, (tanβ≃60)  Δτ is O(10-3) and cannot generate a θ13 of the order of 9∘.We therefore turn to an explicit breaking ofμτ symmetry in MD. That can be realized in the following way.Category A. One has
(44)MDA1=(a1a2a2(1-ϵ1eiθ)00b1(1-ϵ2eiϕ)0b10),MDA2=(a1a2a2(1-ϵ1eiθ)0b1(1-ϵ2eiϕ)000b1).Category B. One has
(45)MDB1=(a100b1(1-ϵ1eiθ)0b2(1-ϵ2eiϕ)b1b20),MDB2=(a100b1(1-ϵ1eiθ)b2(1-ϵ2eiϕ)0b10b2).
Furthermore,
(46)MR=(M1000M200M2(1-ϵ3)),
where ϵ1eiθ, ϵ2eiϕ are complex μτ symmetry breaking parameters (ϵ1,2 real) and ϵ3 is a real μτ symmetry breaking parameter in MR. For these modified MD and MR, we have the following Mν’s.Category A. One has
(47)MνA1=mA[(k12e2iα+2k22k2k2k210k201)MνA1a-ϵ1eiθ(2k22k20k200000)-ϵ2eiϕ(0k20k220000)MνA1a-ϵ3(-k22-k20-k2-10000)],MνA2=mA[(k12e2iα+2k22k2k2k210k201)MνA2a-ϵ1eiθ(2k220k2000k200)-ϵ2eiϕ(0k20k220000)MνA2a-ϵ3(-k220-k2000-k20-1)].Category B. One has
(48)MνB1=mB[(l12l1l2eiβl1l2eiβl1l2eiβl22e2iβ+1l22e2iβl1l2eiβl22e2iβl22e2iβ+1)-ϵ1eiθ(0l1l2eiβ0l1l2eiβ2l22e2iβl22e2iβ0l22e2iβ0)-ϵ2eiϕ(000020000)-ϵ3(0000-10000)],MνB2=mB[(l12l1l2eiβl1l2eiβl1l2eiβl22e2iβ+1l22e2iβl1l2eiβl22e2iβl22e2iβ+1)-ϵ1eiθ(0l1l2eiβ0l1l2eiβ2l22e2iβl22e2iβ0l22e2iβ0)-ϵ2eiϕ(000020000)-ϵ3(00000000-1)].
The detailed diagonalization and expressions for mass differences and mixing angles are given in the appendix.A nonzeroθ13 arises after μτ symmetry breaking. The value θ13≃9∘ is possible for 3σ variations of (Δ212)ϵ1,2,3, (Δ322)ϵ1,2,3, θ12ϵ1,2,3, θ23ϵ1,2,3, compare the appendix. An appropriate choice of symmetry breaking parameters, that is, ϵ1,2,3≃0.1, and slightly shifted parameter spaces for k1,k2,α in Category A and l1,l2,β in Category B are needed. A nonzero CP violating effect can be effected through the Jarlskog invariant JCP:
(49)JCP=ImH12H23H31(Δ212)ϵ1,2,3(Δ322)ϵ1,2,3(Δ312)ϵ1,2,3,
where elements of H and mass squared differences (Δ212)ϵ1,2,3,  (Δ322)ϵ1,2,3 are given in the appendix. Note that (Δ312)ϵ1,2,3=(Δ212)ϵ1,2,3+(Δ322)ϵ1,2,3. A detailed treatment of explicitly broken μτ 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 matrixYν. There are seventy two such textures. We show that the requirement of an exact μτ symmetry, coupled with observational constraints, reduces the seventy two allowed textures in such a Yν to only four corresponding to just two different forms of the light neutrino mass matrix MνA/MνB, resulting in an inverted/normal mass ordering. Apart from an overall mass scale, Mν for every category has two real parameters and an irremovable phase. These parameters k1,  k2, and |α| for Category A and l1,  l2, and |β| for Category B get highly restricted, given the 3σ ranges of measured neutrino mass squared differences and mixing angles. Neutrino masses and Majorana phases are predicted within definite ranges with 3σ laboratory and cosmological inputs. The predicted respective masses m1/eV, m2/eV, and  m3/eV are [0.0452,0.1682], [0.0457,0.1684], [0.077,0.1632] for Category A and [0.0110,0.0335], [0.0144,0.0345],  and [0.0485,0.0638] for Category B. The corresponding intervals of the Majorana phases are -98.0∘≤αM1≤20.0∘, 9.2∘≤αM2≤36.4∘ for Category A and -88.6∘≤αM1≤7.97∘, 90.7∘≤αM2≤128.8∘ for Category B. In addition, we predict the range of the mass scale associated with 0νββ 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 μτ symmetry. Finally, we have formulated the detailed scheme of three-parameter explicit breaking of μτ symmetry for allowed four zero textures. A value of θ13≃9∘ can be arranged for a suitable choice of small values of these symmetry breaking parameters.



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*Source: 324756-2013-02-20.xml*