# Proposed Economical and Stable Lepton Mass Matrices with Texture Zeros

Authors: Shu-jun Rong
Journal: Advances in High Energy Physics
 (2017)

---

## Abstract

There are many viable combinations of texture zeros in lepton mass matrices. We propose an economical and stable mass texture. Analytical and numerical results on mixing parameters and the effective mass of neutrinos are obtained. These results satisfy new constraints from neutrinos oscillation experiments and cosmological observations. Their stabilities are also examined through the perturbations to some structure parameters. Our proposition reveals that, in the complex forest of neutrinos mixing models, a simple and stable one is still possible.


---

## Body

## 1. Introduction

Neutrinos oscillation is one of most mysterious phenomena in particle physics, which reveals that neutrinos are massive and mixing structure of them is nontrivial. However, how to interpret the origin of masses and special mixing pattern of neutrinos is still an open question. As useful phenomenological scenarios, special textures of lepton mass matrices and corresponding flavor groups are introduced to predict mixing parameters of neutrinos. In particular, lepton mass matrices with texture zeros [1–8] or structures alike [9–12] are widely employed in mixing models of neutrinos, which can be realized in Abelian or non-Abelian flavor groups; see [13–15], for example. As an important progress, the recent work [16] has performed a complete survey of possible combinations of texture zeros in mass matrices of charged leptons and Majorana or Dirac neutrinos. There are so many types of possible mass matrices with texture zeros that we do not know what is the true texture chosen by nature. In order to evaluate importance of classes of texture zeros, we propose two criteria: First, the mass matrices with texture zeros should be as economical as possible; second, predictions of texture-zeros models should be stable under perturbations of mass matrices. We consider that an economical model of texture zeros can give clear dependence of observables on parameters of lepton mass matrices. And texture-zeros models with robust predictions are more sustainable, because they can react to the progress of neutrinos experiments by introduction of small modifications or perturbations.In this paper, we propose a special combination of texture zeros of leptons mass matrices with an economical and stable structure. The mass matrix of charged leptons is Hermitian with Fritzsch texture [17–19]; that is,(1)
M
l
=
0
A
l
0
A
l
∗
0
B
l
0
B
l
∗
C
l
,where C
l is real and A
l and B
l are complex parameters. And the mass matrix of Majorana neutrinos with one zero is as follows:(2)
M
ν
=
a
+
c
b
0
b
a
c
0
c
a
+
b
,where a is complex and b and c are real parameters. As for M
l, because of its special structure, resorting to a diagonal phase matrix, it can be transformed to a real symmetric matrix with three parameters determined by masses of charged leptons. So it can be diagonalized by a matrix with a simple analytical expression. As for M
ν, it is magic, which predicts a trimaximal mixing pattern (T
M
2) [20] if the mass matrix of charged leptons is diagonal. The structure of M
ν is characterised by the ratio b
/
c. So it can be diagonalized by a matrix with one real parameter. Furthermore, if a is a real parameter, M
ν would be more economical. However, in this case, ranges of mixing parameters would be smaller. So we introduce a as a complex parameter in this paper.In the following sections, the analytical expression of the lepton mixing matrix is given. Numerical predictions of the mixing parameters are obtained by scanning parameter space of mass matrices. Stabilities of these predictions are examined by introduction of perturbations that break the original structure of the neutrino mass matrix. Finally, a conclusion is presented.


## 2. Analytical Framework

In this section, we consider the analytical expression of mixing matrix and dependence of observables on parameters in lepton mass matrices.


### 2.1. Analytical Expression of Mixing Matrix

The lepton mixing matrix with the standard parametrization is expressed as(3)
U
=
diag
⁡
e
i
δ
e
,
e
i
δ
μ
,
e
i
δ
τ
·
V
·
diag
⁡
e
-
i
φ
1
/
2
,
e
-
i
φ
2
/
2
,
1
,where δ
e
,
μ
,
τ are unphysical phases, φ
1,2 are Majorana phases, and V is written as [22](4)
V
=
c
12
c
13
s
12
c
13
s
13
e
-
i
δ
-
s
12
c
23
-
c
12
s
13
s
23
e
i
δ
c
12
c
23
-
s
12
s
13
s
23
e
i
δ
c
13
s
23
s
12
s
23
-
c
12
s
13
c
23
e
i
δ
-
c
12
s
23
-
s
12
s
13
c
23
e
i
δ
c
13
c
23
,where s
i
j
≡
sin
⁡
θ
i
j, c
i
j
≡
cos
⁡
θ
i
j, and δ is the Dirac CP violating phase. This standard mixing matrix can be obtained through U
=
U
l
+
U
ν, where U
l in our model comes from diagonalization of the Hermitian mass matrix M
l; that is,(5)
U
l
+
M
l
U
l
=
diag
⁡
m
e
,
m
μ
,
m
τ
.
U
ν comes from diagonalization of mass matrix of Majorana neutrinos; that is,(6)
U
ν
T
M
ν
U
ν
=
diag
⁡
m
1
,
m
2
,
m
3
.In detail, U
l
=
P
l
O
l, where(7)
P
l
=
diag
⁡
e
i
α
,
e
i
β
,
1
,where α
=
arg
⁡
[
A
l
]
+
arg
⁡
[
B
l
], β
=
arg
⁡
[
B
l
], and elements of O
l are written as [23, 24](8)
O
l
11
=
x
l
-
z
l
1
+
x
l
1
-
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
12
=
-
i
1
+
z
l
x
l
3
1
+
x
l
x
l
+
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
13
=
1
-
x
l
z
l
3
1
-
z
l
x
l
+
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
21
=
x
l
-
z
l
1
+
x
l
1
-
z
l
1
/
2
,
O
l
22
=
i
x
l
1
+
z
l
1
+
x
l
x
l
+
z
l
1
/
2
,
O
l
23
=
z
l
1
-
x
l
1
-
z
l
x
l
+
z
l
1
/
2
,
O
l
31
=
-
x
l
z
l
1
-
x
l
1
+
z
l
1
+
x
l
1
-
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
32
=
-
i
z
l
1
-
x
l
x
l
-
z
l
1
+
x
l
x
l
+
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
33
=
x
l
1
+
z
l
x
l
-
z
l
1
-
z
l
x
l
+
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,where x
l
=
m
e
/
m
μ and z
l
=
m
e
/
m
τ. Because mass hierarchy of charged leptons is strong, approximation of O
l is(9)
O
l
≈
1
-
i
x
l
1
/
2
z
l
3
/
2
x
l
-
1
x
l
1
/
2
i
z
l
1
/
2
x
l
-
1
/
2
-
z
l
1
/
2
-
i
z
l
1
/
2
x
l
-
1
/
2
1
.Furthermore, mass parameters of M
l could also be expressed with masses of charged leptons, that is, [23](10)
A
l
=
m
e
m
μ
m
τ
m
e
-
m
μ
+
m
τ
1
/
2
,
B
l
=
m
e
-
m
μ
m
μ
-
m
τ
m
e
+
m
τ
m
e
-
m
μ
+
m
τ
1
/
2
,
C
l
=
m
e
-
m
μ
+
m
τ
.As for neutrinos,U
ν
=
O
ν
P
ν, where(11)
O
ν
=
O
1
ν
O
2
ν
O
3
ν
,
(12)
O
1
ν
=
N
1
ν
-
1
+
x
ν
-
1
-
x
ν
+
x
ν
2
x
ν
+
1
-
x
ν
+
x
ν
2
1
,
O
2
ν
=
1
3
1
1
1
,
O
3
ν
=
N
3
ν
-
1
+
x
ν
+
1
-
x
ν
+
x
ν
2
x
ν
-
1
-
x
ν
+
x
ν
2
1
,where x
ν
=
b
/
c and N
1
ν and N
3
ν are normalization factors expressed as(13)
N
1
ν
=
1
+
-
1
+
x
ν
-
1
-
x
ν
+
x
ν
2
2
+
x
ν
+
1
-
x
ν
+
x
ν
2
2
-
1
,
N
3
ν
=
1
+
-
1
+
x
ν
+
1
-
x
ν
+
x
ν
2
2
+
x
ν
-
1
-
x
ν
+
x
ν
2
2
-
1
.And the diagonal phase matrix P
ν is expressed as(14)
P
ν
=
diag
⁡
e
-
i
ϕ
1
/
2
,
e
-
i
ϕ
2
/
2
,
e
-
i
ϕ
3
/
2
,where ϕ
1
=
arg
⁡
[
a
-
b
2
-
b
c
+
c
2
], ϕ
2
=
arg
⁡
[
a
+
b
+
c
], and ϕ
3
=
arg
⁡
[
a
+
b
2
-
b
c
+
c
2
]. Employing expressions of O
l, P
l, O
ν, and P
ν listed above, elements of the lepton mixing matrix can be written as(15)
U
i
j
=
O
l
+
i
1
O
ν
1
j
e
-
i
ϕ
j
/
2
+
α
+
O
l
+
i
2
O
ν
2
j
e
-
i
ϕ
j
/
2
+
β
+
O
l
+
i
3
O
ν
3
j
e
-
i
ϕ
j
/
2
.Obviously, ϕ
j
s are just relevant to Majorana phases of the mixing matrix.


### 2.2. Dependence of Observables on Parameters in Lepton Mass Matrices

The mixing angles of leptons can be obtained from comparison of expression ofU
i
j in (15) with the form of U in the standard parametrization in (3) and (4). In detail, for θ
13,(16)
sin
⁡
θ
13
=
U
13
=
O
l
+
11
O
ν
13
e
-
i
α
+
O
l
+
12
O
ν
23
e
-
i
β
+
O
l
+
13
O
ν
33
.Because O
l is dependent on masses of charged leptons that are determined by experiments very well, θ
13 in fact is dependent on three real effective mass parameters: α, β, and x
ν. The same observation can also hold for θ
23, θ
13, and δ. Therefore, three mixing angles and Dirac CP violating phase are not independent of each other.As for masses of neutrinos, that is,(17)
m
1
=
a
-
b
2
-
b
c
+
c
2
,
m
2
=
a
+
b
+
c
,
m
3
=
a
+
b
2
-
b
c
+
c
2
,they are dependent on four real parameters of M
ν. And the effective mass of neutrinoless double-beta decay, that is,(18)
m
e
e
=
m
1
U
11
2
+
m
2
U
12
2
+
m
3
U
13
2
,is dependent on all 6 real parameters in M
l and M
ν.


## 2.1. Analytical Expression of Mixing Matrix

The lepton mixing matrix with the standard parametrization is expressed as(3)
U
=
diag
⁡
e
i
δ
e
,
e
i
δ
μ
,
e
i
δ
τ
·
V
·
diag
⁡
e
-
i
φ
1
/
2
,
e
-
i
φ
2
/
2
,
1
,where δ
e
,
μ
,
τ are unphysical phases, φ
1,2 are Majorana phases, and V is written as [22](4)
V
=
c
12
c
13
s
12
c
13
s
13
e
-
i
δ
-
s
12
c
23
-
c
12
s
13
s
23
e
i
δ
c
12
c
23
-
s
12
s
13
s
23
e
i
δ
c
13
s
23
s
12
s
23
-
c
12
s
13
c
23
e
i
δ
-
c
12
s
23
-
s
12
s
13
c
23
e
i
δ
c
13
c
23
,where s
i
j
≡
sin
⁡
θ
i
j, c
i
j
≡
cos
⁡
θ
i
j, and δ is the Dirac CP violating phase. This standard mixing matrix can be obtained through U
=
U
l
+
U
ν, where U
l in our model comes from diagonalization of the Hermitian mass matrix M
l; that is,(5)
U
l
+
M
l
U
l
=
diag
⁡
m
e
,
m
μ
,
m
τ
.
U
ν comes from diagonalization of mass matrix of Majorana neutrinos; that is,(6)
U
ν
T
M
ν
U
ν
=
diag
⁡
m
1
,
m
2
,
m
3
.In detail, U
l
=
P
l
O
l, where(7)
P
l
=
diag
⁡
e
i
α
,
e
i
β
,
1
,where α
=
arg
⁡
[
A
l
]
+
arg
⁡
[
B
l
], β
=
arg
⁡
[
B
l
], and elements of O
l are written as [23, 24](8)
O
l
11
=
x
l
-
z
l
1
+
x
l
1
-
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
12
=
-
i
1
+
z
l
x
l
3
1
+
x
l
x
l
+
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
13
=
1
-
x
l
z
l
3
1
-
z
l
x
l
+
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
21
=
x
l
-
z
l
1
+
x
l
1
-
z
l
1
/
2
,
O
l
22
=
i
x
l
1
+
z
l
1
+
x
l
x
l
+
z
l
1
/
2
,
O
l
23
=
z
l
1
-
x
l
1
-
z
l
x
l
+
z
l
1
/
2
,
O
l
31
=
-
x
l
z
l
1
-
x
l
1
+
z
l
1
+
x
l
1
-
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
32
=
-
i
z
l
1
-
x
l
x
l
-
z
l
1
+
x
l
x
l
+
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,
O
l
33
=
x
l
1
+
z
l
x
l
-
z
l
1
-
z
l
x
l
+
z
l
x
l
-
z
l
+
x
l
z
l
1
/
2
,where x
l
=
m
e
/
m
μ and z
l
=
m
e
/
m
τ. Because mass hierarchy of charged leptons is strong, approximation of O
l is(9)
O
l
≈
1
-
i
x
l
1
/
2
z
l
3
/
2
x
l
-
1
x
l
1
/
2
i
z
l
1
/
2
x
l
-
1
/
2
-
z
l
1
/
2
-
i
z
l
1
/
2
x
l
-
1
/
2
1
.Furthermore, mass parameters of M
l could also be expressed with masses of charged leptons, that is, [23](10)
A
l
=
m
e
m
μ
m
τ
m
e
-
m
μ
+
m
τ
1
/
2
,
B
l
=
m
e
-
m
μ
m
μ
-
m
τ
m
e
+
m
τ
m
e
-
m
μ
+
m
τ
1
/
2
,
C
l
=
m
e
-
m
μ
+
m
τ
.As for neutrinos,U
ν
=
O
ν
P
ν, where(11)
O
ν
=
O
1
ν
O
2
ν
O
3
ν
,
(12)
O
1
ν
=
N
1
ν
-
1
+
x
ν
-
1
-
x
ν
+
x
ν
2
x
ν
+
1
-
x
ν
+
x
ν
2
1
,
O
2
ν
=
1
3
1
1
1
,
O
3
ν
=
N
3
ν
-
1
+
x
ν
+
1
-
x
ν
+
x
ν
2
x
ν
-
1
-
x
ν
+
x
ν
2
1
,where x
ν
=
b
/
c and N
1
ν and N
3
ν are normalization factors expressed as(13)
N
1
ν
=
1
+
-
1
+
x
ν
-
1
-
x
ν
+
x
ν
2
2
+
x
ν
+
1
-
x
ν
+
x
ν
2
2
-
1
,
N
3
ν
=
1
+
-
1
+
x
ν
+
1
-
x
ν
+
x
ν
2
2
+
x
ν
-
1
-
x
ν
+
x
ν
2
2
-
1
.And the diagonal phase matrix P
ν is expressed as(14)
P
ν
=
diag
⁡
e
-
i
ϕ
1
/
2
,
e
-
i
ϕ
2
/
2
,
e
-
i
ϕ
3
/
2
,where ϕ
1
=
arg
⁡
[
a
-
b
2
-
b
c
+
c
2
], ϕ
2
=
arg
⁡
[
a
+
b
+
c
], and ϕ
3
=
arg
⁡
[
a
+
b
2
-
b
c
+
c
2
]. Employing expressions of O
l, P
l, O
ν, and P
ν listed above, elements of the lepton mixing matrix can be written as(15)
U
i
j
=
O
l
+
i
1
O
ν
1
j
e
-
i
ϕ
j
/
2
+
α
+
O
l
+
i
2
O
ν
2
j
e
-
i
ϕ
j
/
2
+
β
+
O
l
+
i
3
O
ν
3
j
e
-
i
ϕ
j
/
2
.Obviously, ϕ
j
s are just relevant to Majorana phases of the mixing matrix.


## 2.2. Dependence of Observables on Parameters in Lepton Mass Matrices

The mixing angles of leptons can be obtained from comparison of expression ofU
i
j in (15) with the form of U in the standard parametrization in (3) and (4). In detail, for θ
13,(16)
sin
⁡
θ
13
=
U
13
=
O
l
+
11
O
ν
13
e
-
i
α
+
O
l
+
12
O
ν
23
e
-
i
β
+
O
l
+
13
O
ν
33
.Because O
l is dependent on masses of charged leptons that are determined by experiments very well, θ
13 in fact is dependent on three real effective mass parameters: α, β, and x
ν. The same observation can also hold for θ
23, θ
13, and δ. Therefore, three mixing angles and Dirac CP violating phase are not independent of each other.As for masses of neutrinos, that is,(17)
m
1
=
a
-
b
2
-
b
c
+
c
2
,
m
2
=
a
+
b
+
c
,
m
3
=
a
+
b
2
-
b
c
+
c
2
,they are dependent on four real parameters of M
ν. And the effective mass of neutrinoless double-beta decay, that is,(18)
m
e
e
=
m
1
U
11
2
+
m
2
U
12
2
+
m
3
U
13
2
,is dependent on all 6 real parameters in M
l and M
ν.


## 3. Predictions of Lepton Mass Matrices and Their Stabilities

### 3.1. Predictions of Lepton Mass Matrices

We take masses of charged leptons from Particle Data Group [25]:(19)
m
e
=
0.51
M
e
V
,
m
μ
=
105.658
M
e
V
,
m
τ
=
1776.86
M
e
V
.Then using the Mixing-Parameters-Tools package [26, 27], we scan 6 real parameters in lepton mass matrices, that is, arg
⁡
[
A
l
], arg
⁡
[
B
l
], arg
⁡
[
a
], |
a
|, b, and c. And their ranges at 3
σ level [21] are shown in Table 1. The numerical results are shown in Figure 1, including predictions for mixing angles (θ
13, θ
12, and θ
23), Dirac CP violating phase δ, mass ratios (m
1
/
m
2, m
2
/
m
3), the mass of the lightest neutrino m
1, sum of masses of neutrinos ∑
m
i, and the effective mass of neutrinoless double-beta decay m
e
e. Some comments are given as follows:(i)
Different from the case where both of mass matrices of charged leptons and neutrinos are of the form of Fritzsch texture [28], the octant of θ
23 is uncertain in our case.
(ii)
Because of the strong mass hierarchy of charged leptons, the range of the Dirac CP violating phaseδ is narrow, that is, 0
∘
±
5
0
∘ or 18
0
∘
±
5
0
∘.
(iii)
The upper limit of the mass of the lightest neutrino is small, that is, about 23.5 meV and so is that of the effective mass of double-beta decay, that is, about 24 meV. Accordingly, the rate of neutrinoless double-beta decay may be too small to detect in the present stage [29].
(iv)
The mass ordering of neutrinos is normal. And the hierarchy of them is not strong; so the upper limit of the sum of masses of neutrinos is also small, that is, approximating 102 meV, which satisfies new stringent constrains from cosmological observations; for example,∑
m
i
<
146
m
e
V at the 2σ level in [30].Table 1
Parameter ranges of lepton mass matrices at3
σ level.
arg
⁡
[
A
l
]
arg
⁡
[
B
l
]
arg
⁡
[
a
]
a
/
1
0
-
2
e
V
b
/
1
0
-
2
e
V
c
/
1
0
-
2
e
V
[
0
,
2
π
]
[
0
,
2
π
]
[
0
,
2
π
]
[
2
,
3
]
[
-
1.2
,
1.8
]
[
-
1.6
,
-
3
]Figure 1
Predictions for mixing angles (θ
13, θ
12, and θ
23), Dirac CP violating phase δ, mass ratios (m
1
/
m
2, m
2
/
m
3), the mass of the lightest neutrino m
1, sum of masses of neutrinos ∑
m
i, and the effective mass of neutrinoless double-beta decay m
e
e. Here the unit of angles is degree.


### 3.2. Stabilities of Predictions under Perturbations

The above predictions are obtained on the basis of the special combinations of texture zeros and other special properties of leptons mass matrices. In order to examine stabilities of these predictions, we introduce perturbations that break the original structure of the mass matrix. Because the mass parameters of the charged leptons are well determined by the experimental data through (10), we just consider the perturbations to the neutrinos mass matrix here.In general, the neutrinos mass matrix under perturbations could be expressed as(20)
M
ν
=
M
ν
0
+
δ
M
ν
=
a
+
c
b
0
b
a
c
0
c
a
+
b
+
ϵ
1
ϵ
4
ϵ
0
ϵ
4
ϵ
2
ϵ
5
ϵ
0
ϵ
5
ϵ
3
,where M
ν
(
0
) is the original mass matrix and ϵ
i is a real perturbation. By redefinitions of parameters a
,
b
,
c, and ϵ
i, the matrix of perturbations could be reduced to the following expression:(21)
δ
M
ν
=
0
0
ϵ
0
0
ϵ
0
0
ϵ
0
0
0
+
ϵ
1
0
0
0
ϵ
2
0
0
0
ϵ
3
.We note that the first part of the perturbation with the parameter ϵ
0 breaks zeros while the second part with ϵ
i
(
i
=
1,2
,
3
) breaks the magic property of the original mass matrix. Together with original mass parameters at 3
σ level, these four parameters are scanned in turn. Their ranges at 3
σ level are shown in Table 2. And their impacts on neutrinos oscillation parameters are shown in Figures 2–5.Table 2
Ranges of the parameters of perturbations at3
σ level.
ϵ
0
/
a
ϵ
1
/
a
ϵ
2
/
a
ϵ
3
/
a
[
-
1
%
,
0.99
%
]
[
-
3.9
%
,
0.68
%
]
[
-
2.2
%
,
2.53
%
]
[
-
2.29
%
,
3.07
%
]Figure 2
Correlations of neutrinos oscillation parameters before and after the perturbation with the parameterϵ
0. Here the superscripts “0” in the plots denote original oscillation parameters (before perturbations). The original mixing parameters are at 3σ level from [21].Figure 3
Correlations of neutrinos oscillation parameters before and after the perturbation with the parameterϵ
1.Figure 4
Correlations of neutrinos oscillation parameters before and after the perturbation with the parameterϵ
2.Figure 5
Correlations of neutrinos oscillation parameters before and after the perturbation with the parameterϵ
3.Some comments are given as follows:(i)
The size of the range ofϵ
0 is about half of that of the parameter ϵ
i with i
=
1,2
,
3.
(ii)
The parameterϵ
0 at 3
σ level has notable impact on the squared mass difference δ
m
21
2. It could bring the modification to δ
m
21
2 as large as 5% of the original value. And its impacts on the mixing angles θ
12 and θ
23 and the Dirac CP phase δ are negligible. The impacts of ϵ
0 on δ
m
31
2 and θ
13 are moderate. The maximal modifications to them are about 2% of the original values.
(iii)
Different fromϵ
0, ϵ
1 has obvious impact on the mixing angles θ
12. The maximal modification to θ
12 could be 10% of the original value. Furthermore, its impact on δ
m
21
2 is more notable than that of ϵ
0. The maximal modification to δ
m
21
2 is about 10% of the original value. Its impacts on other oscillation parameters are similar to those in the case of ϵ
0.
(iv)
The impacts ofϵ
2 or ϵ
3 on the oscillation parameters are similar to those in the case of ϵ
1 except that the modifications to θ
12 in these two cases are relatively moderate.On the basis of above observations, we find that although some of the oscillations parameters after perturbations are not dependent on their original values linearly, the relative modification of them is approximately of the same order as that of the perturbation in both the cases ofϵ
0 and ϵ
i with i
=
1,2
,
3. So the special assumption on the structure of neutrinos mass matrix including zeros and magic property could be broken moderately while the predictions on oscillation parameters are still in their 3
σ ranges.


## 3.1. Predictions of Lepton Mass Matrices

We take masses of charged leptons from Particle Data Group [25]:(19)
m
e
=
0.51
M
e
V
,
m
μ
=
105.658
M
e
V
,
m
τ
=
1776.86
M
e
V
.Then using the Mixing-Parameters-Tools package [26, 27], we scan 6 real parameters in lepton mass matrices, that is, arg
⁡
[
A
l
], arg
⁡
[
B
l
], arg
⁡
[
a
], |
a
|, b, and c. And their ranges at 3
σ level [21] are shown in Table 1. The numerical results are shown in Figure 1, including predictions for mixing angles (θ
13, θ
12, and θ
23), Dirac CP violating phase δ, mass ratios (m
1
/
m
2, m
2
/
m
3), the mass of the lightest neutrino m
1, sum of masses of neutrinos ∑
m
i, and the effective mass of neutrinoless double-beta decay m
e
e. Some comments are given as follows:(i)
Different from the case where both of mass matrices of charged leptons and neutrinos are of the form of Fritzsch texture [28], the octant of θ
23 is uncertain in our case.
(ii)
Because of the strong mass hierarchy of charged leptons, the range of the Dirac CP violating phaseδ is narrow, that is, 0
∘
±
5
0
∘ or 18
0
∘
±
5
0
∘.
(iii)
The upper limit of the mass of the lightest neutrino is small, that is, about 23.5 meV and so is that of the effective mass of double-beta decay, that is, about 24 meV. Accordingly, the rate of neutrinoless double-beta decay may be too small to detect in the present stage [29].
(iv)
The mass ordering of neutrinos is normal. And the hierarchy of them is not strong; so the upper limit of the sum of masses of neutrinos is also small, that is, approximating 102 meV, which satisfies new stringent constrains from cosmological observations; for example,∑
m
i
<
146
m
e
V at the 2σ level in [30].Table 1
Parameter ranges of lepton mass matrices at3
σ level.
arg
⁡
[
A
l
]
arg
⁡
[
B
l
]
arg
⁡
[
a
]
a
/
1
0
-
2
e
V
b
/
1
0
-
2
e
V
c
/
1
0
-
2
e
V
[
0
,
2
π
]
[
0
,
2
π
]
[
0
,
2
π
]
[
2
,
3
]
[
-
1.2
,
1.8
]
[
-
1.6
,
-
3
]Figure 1
Predictions for mixing angles (θ
13, θ
12, and θ
23), Dirac CP violating phase δ, mass ratios (m
1
/
m
2, m
2
/
m
3), the mass of the lightest neutrino m
1, sum of masses of neutrinos ∑
m
i, and the effective mass of neutrinoless double-beta decay m
e
e. Here the unit of angles is degree.


## 3.2. Stabilities of Predictions under Perturbations

The above predictions are obtained on the basis of the special combinations of texture zeros and other special properties of leptons mass matrices. In order to examine stabilities of these predictions, we introduce perturbations that break the original structure of the mass matrix. Because the mass parameters of the charged leptons are well determined by the experimental data through (10), we just consider the perturbations to the neutrinos mass matrix here.In general, the neutrinos mass matrix under perturbations could be expressed as(20)
M
ν
=
M
ν
0
+
δ
M
ν
=
a
+
c
b
0
b
a
c
0
c
a
+
b
+
ϵ
1
ϵ
4
ϵ
0
ϵ
4
ϵ
2
ϵ
5
ϵ
0
ϵ
5
ϵ
3
,where M
ν
(
0
) is the original mass matrix and ϵ
i is a real perturbation. By redefinitions of parameters a
,
b
,
c, and ϵ
i, the matrix of perturbations could be reduced to the following expression:(21)
δ
M
ν
=
0
0
ϵ
0
0
ϵ
0
0
ϵ
0
0
0
+
ϵ
1
0
0
0
ϵ
2
0
0
0
ϵ
3
.We note that the first part of the perturbation with the parameter ϵ
0 breaks zeros while the second part with ϵ
i
(
i
=
1,2
,
3
) breaks the magic property of the original mass matrix. Together with original mass parameters at 3
σ level, these four parameters are scanned in turn. Their ranges at 3
σ level are shown in Table 2. And their impacts on neutrinos oscillation parameters are shown in Figures 2–5.Table 2
Ranges of the parameters of perturbations at3
σ level.
ϵ
0
/
a
ϵ
1
/
a
ϵ
2
/
a
ϵ
3
/
a
[
-
1
%
,
0.99
%
]
[
-
3.9
%
,
0.68
%
]
[
-
2.2
%
,
2.53
%
]
[
-
2.29
%
,
3.07
%
]Figure 2
Correlations of neutrinos oscillation parameters before and after the perturbation with the parameterϵ
0. Here the superscripts “0” in the plots denote original oscillation parameters (before perturbations). The original mixing parameters are at 3σ level from [21].Figure 3
Correlations of neutrinos oscillation parameters before and after the perturbation with the parameterϵ
1.Figure 4
Correlations of neutrinos oscillation parameters before and after the perturbation with the parameterϵ
2.Figure 5
Correlations of neutrinos oscillation parameters before and after the perturbation with the parameterϵ
3.Some comments are given as follows:(i)
The size of the range ofϵ
0 is about half of that of the parameter ϵ
i with i
=
1,2
,
3.
(ii)
The parameterϵ
0 at 3
σ level has notable impact on the squared mass difference δ
m
21
2. It could bring the modification to δ
m
21
2 as large as 5% of the original value. And its impacts on the mixing angles θ
12 and θ
23 and the Dirac CP phase δ are negligible. The impacts of ϵ
0 on δ
m
31
2 and θ
13 are moderate. The maximal modifications to them are about 2% of the original values.
(iii)
Different fromϵ
0, ϵ
1 has obvious impact on the mixing angles θ
12. The maximal modification to θ
12 could be 10% of the original value. Furthermore, its impact on δ
m
21
2 is more notable than that of ϵ
0. The maximal modification to δ
m
21
2 is about 10% of the original value. Its impacts on other oscillation parameters are similar to those in the case of ϵ
0.
(iv)
The impacts ofϵ
2 or ϵ
3 on the oscillation parameters are similar to those in the case of ϵ
1 except that the modifications to θ
12 in these two cases are relatively moderate.On the basis of above observations, we find that although some of the oscillations parameters after perturbations are not dependent on their original values linearly, the relative modification of them is approximately of the same order as that of the perturbation in both the cases ofϵ
0 and ϵ
i with i
=
1,2
,
3. So the special assumption on the structure of neutrinos mass matrix including zeros and magic property could be broken moderately while the predictions on oscillation parameters are still in their 3
σ ranges.


## 4. Conclusion

There are many viable combinations of texture zeros in lepton mass matrices. According to the criteria that the structure of leptons mass matrices should be economical and the prediction of them should be robust, a special combination of texture zeros of lepton mass matrices with other special properties is proposed. The analytical expression of lepton mixing matrix is obtained, which shows clear dependence of an observable on model parameters. By scanning of parameter space of mass matrices of leptons, numerical predictions for mixing angles, Dirac CP violating phase, and masses and effective masses of neutrinos are given. These results satisfy constraints from neutrinos oscillation experiments and those from new cosmological observations. The predictions on the oscillation parameters are still in their3
σ ranges when the special assumption on zeros or magic property of the mass matrix of neutrinos is moderately broken. Therefore, in the complex forest of neutrinos mixing models, a simple and stable one is still possible.



---

*Source: 8785076-2017-01-30.xml*