arXiv:1106.3451v1 [hep-ph] 17 Jun 2011
Zero Textures of the Neutrino Mass Matrix from
Cyclic Family Symmetry
S. Dev∗, Shivani Gupta†and Radha Raman Gautam‡
Department of Physics, Himachal Pradesh University, Shimla 171005,
We present the symmetry realization of the phenomenologically viable Frampton-
Glashow-Marfatia (FGM) two zero texture neutrino mass matrices in the fla-
vor basis within the framework of the type (I+II) seesaw mechanism natural to
SO(10) grand unification. A small Abelian cyclic symmetry group Z3is used to
realize these textures except for class C for which the symmetry is enlarged to
Z4. The scalar sector is restricted to the Standard Model (SM) Higgs doublet
to suppress the flavor changing neutral currents. Other scalar fields used for
symmetry realization are at the most two scalar triplets and, in some cases,
a complex scalar singlet. Symmetry realization of one zero textures has, also,
Understanding the pattern of neutrino masses and mixings constitutes a major chal-
lenge for elementary particle physics. This pattern seems to be entirely different
from the observed hierarchical pattern of quark masses and mixings. In the absence
of flavor symmetries, fermion masses and mixings are, in general, undetermined in
gauge theories. All information about fermion masses and mixings is encoded in the
relevant fermion mass matrices which are important tools for the investigation of
the underlying symmetries and the resulting dynamics. The important first step in
this direction is the reconstruction of the neutrino mass matrix in the flavor basis.
However, this reconstruction results in a large variety of possible structures of the
neutrino mass matrix depending strongly on the neutrino mass scale, mass hierar-
chy and the CP violating phases. In fact, no presently conceivable set of feasible
experiments can determine the neutrino mass matrix completely. Therefore, in the
absence of sufficient data on neutrino masses and mixings, all possible mass matrix
structures need to be carefully scrutinized to find viable structures compatible with
the presently available data. Several interesting proposals have been made in the lit-
erature to restrict the form of the neutrino mass matrix and, thus, reduce the number
of free parameters. These include the presence of zero textures [1, 2, 3], vanishing
minors  and hybrid textures  to name just a few. Zero textures in the neutrino
mass matrix, in particular, have been extensively studied in the literature. The main
reason for such interest in zero textures is their implications for the possible existence
of family symmetries which require certain entries of the mass matrix to vanish. It
has been noted earlier  that the zeros of the Dirac neutrino mass matrix (MD) and
the heavy right-handed Majorana neutrino mass matrix (MR) may propagate as zeros
in the effective low energy neutrino mass matrix (Mν) through the seesaw mechanism
. Zero textures in the neutrino mass matrix, in general, may be obtained by im-
posing certain Abelian family symmetries  at the expense of an extended scalar
sector. Abelian family symmetries have been investigated systematically in  for
extremal mixings. General guidelines for the symmetry realization of zero textures
in both the quark and the lepton mass matrices have been propounded in  which
outline general methods for enforcing zero entries in the fermion mass matrices by
imposing Abelian family symmetries. However, specific guidelines for a simple and
minimal realization of each texture are still lacking.
Two zero texture Ans¨ atz is especially important since it can successfully describe
both the quark and lepton sectors including the charged lepton and the neutrino
masses. The two zero textures are, also, compatible with specific GUT models .
Furthermore, these mass matrices can accommodate the present values of sin2β .
In view of the phenomenological success of the two zero texture Ans¨ atz, it would be
interesting to examine the symmetry realization of two zero texture neutrino mass
matrices in the larger context of SO(10) GUT which includes both the type I and
type II contributions in the seesaw mechanism. Out of the seven phenomenologically
viable classes of two zero texture neutrino mass matrices (FGM), two classes have
been obtained in  from A4or its Z3subgroup in the context of type (I+II) seesaw
. Symmetry realization of all the phenomenologically viable classes of two zero
texture neutrino mass matrices was presented in  in the context of type II see-
saw mechanism. In the present work, we present the symmetry realization of all the
phenomenologically viable classes of two zero texture neutrino mass matrices in the
context of type (I+II) seesaw  without assuming the dominance of either type of
contributions. We keep just the Standard Model (SM) scalar doublet which trans-
forms trivially under the new family symmetry (Z3/Z4) thus, in effect, suppressing
the undesired flavor changing neutral currents. An additional advantage of a single
scalar doublet is the stability of zero textures in the neutrino mass matrix under
renormalization group evolution. The neutrino mass matrices at any two scales µ1
and µ2are related by [14, 16] Mν(µ1) = IMν(µ2)I, where I is diagonal and positive
leading to zeros in Mν at any other renormalizable scale. These zero textures are
enforced by extending the SM with at the most two scalar SU(2) triplets. However,
in some cases we are forced to introduce a complex scalar singlet which transforms
nontrivially under the family symmetry. The effective neutrino mass matrix contains
both type (I+II) seesaw contributions which is natural in SO(10) GUTs as
ν = ML− MDM−1
where ML is the left-handed Majorana mass matrix. Symmetry realization of one
zero texture neutrino mass matrices has, also, been discussed briefly.
2 Symmetry Realization of Two Zero Textures
The phenomenologically allowed two zero texture neutrino mass matrices are given
in Table 1. The cyclic symmetry Z3can be used to realize class A1when the leptonic
fields transform as
DLe→ ω2DLe, eR→ ω2eR, νR1→ ωνR1,
Table 1: Viable two zero texture neutrino mass matrices. X denote the non zero
where ω = exp(2iπ/3) is the generator of Z3. These assigned transformations gener-
ate the diagonal charged lepton mass matrix Ml. The bilinears DLjνRkand νRjνRk
relevant for MDand MRtransform as
Since, the SM Higgs doublet transforms trivially under Z3, the form of MDbecomes
Assuming a complex scalar singlet χ that transforms under Z3as χ → ω2χ, the form
After the type I seesaw we get
The bilinear DT
LjC−1DLkrelevant for MLtransforms under this symmetry as
We introduce a scalar SU(2) triplet ∆ written in 2 × 2 matrix notation as
which remains invariant under Z3. The vacuum expectation value (VEV) of Higgs
high mass of the scalar triplet [14, 17]. Thus, the type II seesaw contribution is given
√2. This induced VEV in the scalar potential is suppressed by the
The effective neutrino mass matrix Mνafter type (I+II) seesaw mechanism becomes
which is the class A1of FGM  two zero textures.
For class A2, the leptonic fields under Z3transform as
DLe→ ω2DLe, eR→ ω2eR, νR1→ ωνR1,
DLµ→ ωDLµ, µR→ ωµR, νR2→ νR2,
τR→ τR,νR3→ νR3.
In this case, the complex scalar singlet χ is assumed to transform as χ → ω2χ under
Z3resulting in the same structure of the right-handed Majorana mass matrix as of
A1and we obtain the same contribution from type I seesaw as in Eqn. (3). Assuming
the scalar triplet to remain invariant under Z3, the above given transformations of
the left-handed fields yield the type II seesaw contribution given by
Thus, type (I+II) seesaw results in class A2of FGM two zero textures. This Fritzsch-
type neutrino mass matrix with a nearly diagonal charged lepton mass matrix has
been obtained recently in  using S3× Z5× Z2symmetry and an extended Higgs
The requisite transformations of the leptonic fields for class B3are given by
DLµ→ ω2DLµ, µR→ ω2µR, νR2→ νR2,
With the complex scalar singlet χ transforming as χ → ω2χ under Z3, the type I
seesaw contribution is given by
eR→ eR,νR1→ ωνR1,
τR→ ωτR,νR3→ νR3.
and Z3invariant scalar triplet leads to the type II contribution given by
thus, yielding class B3of the FGM two zero texture neutrino mass matrices. The
required leptonic field transformations under Z3for class B4are given by
DLµ→ ωDLµ, µR→ ωµR, νR2→ νR2,
DLτ→ ω2DLτ, τR→ ω2τR, νR3→ νR3.
eR→ eR,νR1→ ωνR1,
The complex scalar singlet χ is required to transform under Z3as χ → ω2χ and the
resulting type I contribution for this case is given by
The Z3invariant scalar triplet leads to the type II contribution of the form
leading to class B4of the FGM neutrino mass matrices.
For classes B1and B2, the leptonic fields are required to transform under Z3as
DLµ→ ωDLµ, µR→ ωµR, νR2→ ωνR2,
DLτ→ ω2DLτ, τR→ ω2τR, νR3→ ω2νR3,
eR→ eR,νR1→ νR1,
resulting in a diagonal Dirac neutrino mass matrix MD. We do not require any scalar
singlet to realize MR. The resulting type I contribution becomes
However, in these cases the scalar triplet ∆ transforms under Z3as
∆ → ω2∆(forclassB1)
∆ → ω∆(forclassB2)
resulting in type II contribution of the form
for class B1and
for class B2. Type (I+II) seesaw yields classes B1 and B2 of the FGM two zero
neutrino mass matrices. The scalar potential for these two cases is given by
V (φ,∆) = m2φ†φ + M2tr(∆†∆) + (µφ†∆˜φ + h.c.) + ...,(22)
where˜φ = iτ2φ∗and the dots indicate quartic terms which respect Z3. The dimension
three term φ†∆˜φ is not allowed in the scalar potential under Z3. However, to obtain
a small non-zero VEV of the scalar triplet  we softly break the Z3symmetry by
including this term in the scalar potential.
Class C of FGM two zero textures cannot be realized by Z3symmetry, and the next
higher order group Z4is used to realize this texture. The leptonic fields are required
to transform under Z4as
DLτ→ −iDLτ, τR→ −iτR, νR3→ −iνR3,
leading to a diagonal charged lepton mass matrix, a diagonal Dirac neutrino mass
νR2→ iνR2, (23)
matrix MDand non diagonal right-handed Majorana mass matrix MR. The type I
contribution in this case is given by
However, for symmetry realization of class C we are constrained to introduce two
scalar triplets which transform as
under Z4leading to the type II contribution of the form
resulting in class C of FGM neutrino mass matrices.Again, we must break Z4
softly in the scalar potential to obtain small non-zero VEV of the scalar triplets. A
detailed phenomenological analysis of the FGM two zero textures has been presented
in . Class C of FGM textures was obtained from a horizontal symmetry Z4with
a scalar sector comprising of SM Higgs doublet and three scalar triplets . Class
C has, earlier , been obtained in a model based on non-Abelian group Q8with a
much more cumbersome scalar sector with two doublets and four triplets. Both these
scenarios have been discussed in the context of type-II seesaw.
3Symmetry Realization of One Zero Textures
Zero TextureDLe(er), DLµ(µr), DLτ(τr)
ω2, ω , 1
Mν12= 0ω2, 1, ωω2
Mν13= 0ω2, ω , 1ω2
Mν22= 01, ω2,ωω
Mν23= 0ω , 1, ω2
Mν33= 0 1, ω, ω2
Table 2: Leptonic field transformations under Z3for all six possible one zero textures.
One zero texture neutrino mass matrices can be realized from the cyclic Z3symmetry
by extending the Higgs sector. These textures are realized by extending SM with
two scalar triplets, three right-handed neutrino singlets and a complex scalar singlet.
Under Z3the complex scalar singlet transforms as χ → ω2χ, whereas one of the scalar
triplets (∆1) remains invariant. The right-handed neutrino fields transform as
The transformations of other fields for all classes of one zero textures in Mνare given
in Table (2). The scalar potential for the one zero textures is given by
V (φ,∆,χ) = m2φ†φ +
j∆k) + λχ∗χ + (µφ†∆1˜φ + h.c.) + ...,(29)
To obtain a small non-zero VEV of the scalar triplet ∆2 we softly break the Z3
symmetry by including dimension two terms in the scalar potential where j ?= k. A
detailed phenomenological analysis of one zero textures can be found in .
In the flavor basis, the zero textures of the neutrino mass matrix are realized using
the small Abelian cyclic symmetry group Z3/Z4. All the zero textures are realized
within the context of type (I+II) seesaw mechanism which is natural to SO(10) GUTs.
Since, only one Higgs doublet is used in the realization of these zero textures, the
flavor changing neutral current interactions are naturally suppressed. We introduced
three right-handed neutrino singlets and at the most two scalar triplets which acquire
small nonzero VEVs. However, in some cases we, introduced an additional complex
scalar singlet which transforms nontrivially under Z3symmetry. For one zero textures
and some of the two zero textures we softly break the relevant symmetry to obtain a
small non-zero VEV of scalar triplet. The zero textures in the neutrino mass matrix
remain stable under RG evolution as the symmetry realization involves only the SM
The research work of S. D. is supported by the University Grants Commission, Gov-
ernment of India vide Grant No. 34-32/2008 (SR). R. R. G. acknowledge the financial
support provided by the Council for Scientific and Industrial Research (CSIR), Gov-
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