Neutrino masses and mixing in an extended standard model acquiring an A4 flavour symmetry are considered. The corresponding three-neutrino mixing matrix obtained via a pertur- bative method allows us to determine the Dirac CP violation phase (\delta_{CP}) as a function of the mixing angles (\theta_{12}, \theta_{23}, \theta_{13}). Then, numerical values and distributions of \delta_{CP} are given. The latter values are quite close to the global fits of the experimental data for both the normal ordering and inverse ordering of the neutrino masses

Van, Nguyen Thi Hong

Van, Phi Quang

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 26, No. 1 (2016), pp. 1-9
DOI:10.15625/0868-3166/26/1/7860
ON THE CP VIOLATION PHASE IN A NEUTRINO MIXING MODEL WITH
AN A4 FLAVOR SYMMETRY
PHI QUANG VAN AND NGUYEN THI HONG VAN†
Institute of Physics, Vietnam Academy of Science and Technology
†E-mail: nhvan@iop.vast.vn
Received 09 March 2016
Accepted for publication 14 April 2016
Abstract. Neutrino masses and mixing in an extended standard model acquiring an A4 flavor
symmetry are considered. The corresponding three-neutrino mixing matrix obtained via a per-
turbative method allows us to determine the Dirac CP violation phase (δCP) as a function of the
mixing angles (θ12,θ23,θ13). Then, numerical values and distributions of δCP are given. The latter
values are quite close to the global fits of the experimental data for both the normal ordering and
inverse ordering of the neutrino masses.
Keywords: neutrino mass and mixing, flavor symmetries, models beyond the standard model.
Classification numbers: 14.60.Pq, 11.30.Fs, 12.60.-i.
I. INTRODUCTION
Standard model (SM) [1–4] has been confirmed as an excellent model of elementary parti-
cles and their interactions, especially after the discovery of the Brout-Englert-Higgs boson (Higg
boson) by ATLAS and CMS [5, 6], which are the two biggest LHC collaborations (see [7] for a
review on the discovering of the Higgs boson). The SM, however, cannot explain a number of
phenomena including that of the neutrino masses and mixing [8, 9]. They, therefore, call for an
extension of the SM. One of the extensions of the SM which has attracted great interest for last
ten years is that with a flavor symmetry. A widely investigated flavor symmetry is based on the
discrete group A4 (see, for example, [10,11]). This symmetry can make a neutrino mixing scheme
tribi-maximal (TBM). The recent experimental data [12,13], however, shows that the mixing angle
θ13 and the Dirac CP-violation phase (CPV) δCP are non-zero. That means that the TBM scheme is
no longer valid. Consequently, many attempts to explain these new phenomena have been made.
Here, in this paper, as reported in [14], we use a perturbation method to diagonalize a general
neutrino mass matrix and obtain a Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing
matrix from which a relation between δCP and the mixing angles θi j can be derived. This relation
c©2016 Vietnam Academy of Science and Technology
2 PHI QUANG VAN AND NGUYEN THI HONG VAN
allows us, using experimental data of the mixing angles, to calculate δCP numerically, for both the
normal ordering (NO) and inverse ordering (IO) of neutrino masses. This means that the present
paper is an extended work of [14] in which only δCP for an NO was considered.
The plan of this paper is the following. In the next section we will make a quick introduction
to the representations of A4 group and an A4-symmetric neutrino mass model. Section 3 deals
with a perturbation approach to the problem of neutrino masses and mixing for this model, from
where a mass spectrum and a relation between δCP and θi j can be obtained. Then, numerical
values and distributions of δCP in both NO and IO are presented.
II. NEUTRINO MASS MODELWITH AN A4 FLAVOR SYMMETRY
Quick introduction to representations of A4 group
Let us give a concise introduction to representations of A4 which are widely given in the
literature (see, for example, [15, 16]). This group is composed of 12 elements and generated by
two basic permutations S and T,
S2 = T 3 = (ST )3 = 1. (1)
It has a three-dimensional unitary representation generated by
S =
 1 0 00 −1 0
0 0 −1
 , T =
 0 1 00 0 1
1 0 0
 , (2)
and three one-dimensional unitary representations 1, 1
′
and 1
′′
generated by
1 : S = 1 T = 1, (3a)
1
′
: S = 1 T = ei2pi/3 ≡ ω, (3b)
1
′′
: S = 1 T = ei4pi/3 ≡ ω2. (3c)
As in the case of an arbitrary group, applications of A4 often require to know the multiplication
rule of its (irreducible) representations
1×1 = 1, 1′×1′′ = 1, 1′′×1′ = 1, 1′×1′ = 1′′ , 1′′×1′′ = 1′ , (4a)
3×3 = 1+1′+1′′+31+32. (4b)
Let us explain the rule (4b) in more details, while the first ones are trivial. For two triplets, say
3a ∼ (a1,a2,a3) and 3b ∼ (b1,b2,b3), their direct product can be decomposed into irreducible
representations in the following way:
1 = a1b1+a2b2+a3b3, (5a)
1
′
= a1b1+ω2a2b2+ωa3b3, (5b)
1
′′
= a1b1+ωa2b2+ω2a3b3, (5c)
31 ∼ (a2b3,a3b1,a1b2), (5d)
32 ∼ (a3b2,a1b3,a2b1). (5e)
The information given here is used in constructing an A4-invariant action (Lagrangian) of the
considered model, including the Yukawa terms (6).
ON THE CP VIOLATION PHASE IN A NEUTRINO MIXING MODEL WITH AN A4 FLAVOR SYMMETRY 3
The model generality
In principle, an A4 flavor symmetry can be applied to a neutrino mixing model with an
arbitrary number of Majorana neutrinos but here we will consider, as an example, a model with
three active Majorana neutrinos, for which the neutrino mass matrix is symmetric and parametrized
by 6 independent parameters. Similar models have been studied widely before (see, for example,
in [17–19]) but the main goal of this paper (and [14]) is to concentrate on considering the Dirac
CPV phase (see also [20]). Besides the SM leptons transforming now also under A4 as a triplet
(left-handed leptons lL, l = e,µ,τ) or singlets (right-handed leptons eR, µR and τR), the lepton
sector of this model includes an A4 triplet N which is an SU(2)L singlet (right-handed neutrino).
The scalar sector consists of the SM Higgs φh which is an A4 singlet, and five new SU(2)L-singlet
scalars including two A4-triplets (denoted as φE , φN) and three A4-singlets (denoted as ξ , ξ
′
, ξ ′′).
The A4- and SU(2)L transformation rules of the leptons and the scalars in the present model are
listed in the table given below. It is enough for us to construct an A4-invariant Lagrangian.
Table 1. A4 and SU(2)L symmetry of leptons and scalars.
lL eR µR τR N φE φN ξ ξ
′ ξ ′′ φh
A4 3 1 1
′
1
′′
3 3 3 1 1
′
1
′′
1
SU(2)L 2 1 1 1 1 1 1 1 1 1 2
The Lagrangian of the model includes the new Yukawa term
−L newY = λe(lLφh)eR
φE
Λ
+λµ
(
lLφh
)′′
µR
φE
Λ
+λτ
(
lLφh
)′
τR
φE
Λ
+λN lLφ˜hN
+ cN
(
NcN
)
φN + cξ
(
NcN
)
ξ + cξ ′
(
NcN
)′′
ξ
′
+ cξ ′′
(
NcN
)′
ξ
′′
+H.c. (6)
From here, by denoting the VEVs of the scalars as
〈ξ 〉= ua, 〈ξ ′〉= ub, 〈ξ ′′〉= uc, 〈φE〉= (u1,u2,u3) , 〈φN〉= (v1,v2,v3) , 〈φh〉= v, (7)
the mass matrix of the charged leptons has the form
Mlept = v
 κeu1 κeu2 κeu3κµu1 ωκµu2 ω2κµu3
κτu1 ω2κτu2 ωκτu3
 , (8)
where
κe =
λe
Λ
, κµ =
λµ
Λ
, κτ =
λτ
Λ
. (9)
To keep maximally the SM Higgs VEV structure, we can assume 〈φE〉= (u,u,u), thus, the charged
lepton mass matrix becomes
Mlept =UL
 yev 0 00 yµv 0
0 0 yτv
≡UL
 me 0 00 mµ 0
0 0 mτ
, (10)
4 PHI QUANG VAN AND NGUYEN THI HONG VAN
where
UL = 1√3
 1 1 11 ω ω2
1 ω2 ω
 , (11)
ye = uκe, yµ = uκµ , yτ = uκτ , (12)
and me, mµ , mτ are the charged lepton masses.
In the neutrino subsector, we obtain the following Majorana mass matrix
MN =
 ca+ cb+ cd ε3 ε2ε3 ca+ωcb+ω2cd ε1
ε2 ε1 ca+ω2cb+ωcd
≡
 a b cb e d
c d f
 , (13)
where ca = cξua, cb = cξ ′ub, cd = cξ ′′uc, ε1 = cNv1, ε2 = cNv2, ε3 = cNv3, and Dirac mass matrix
MD =
 λN1v 0 00 λN2v 0
0 0 λN3v
≡
 x 0 00 y 0
0 0 z
 . (14)
The seesaw mechanism gives us the neutrino mass matrix,
Mν =−MTDM−1N MD, (15)
which, with (13) and (14) taken into account, can be written as
Mν =
−1
det(M)
 (d2− e f )x2 (−cd+b f )xy (−bd+ ce)xz(−cd+b f )xy (c2−a f )y2 (−bc+ad)yz
(−bd+ ce)xz (−bc+ad)yz (b2−ae)z2
 , (16)
with det(M) = 2bcd− c2e−b2 f −a(d2− e f ) being the determinant of MN .
Diagonalizing a neutrino mass matrix, such as the one in (16), is often a difficult task. To
solve this problem different methods and tricks have been proposed. Here we will use a perturba-
tion approach to solving this problem.
III. NEUTRINO MIXING AND THE CP VIOLATION PHASE
Let us work in the basis where the lepton mass matrix is diagonal. In this basis Mν becomes
Mν =UTL MνUL ≡
 A B CB E D
C D F
 . (17)
The current experimental data have shown that the PMNS neutrino mixing matrix UPMNS is a small
deviation from the TBM form and we have (see [14] and also [21])
Mν = M0+λV, (18)
with
M0 =
 A B −BB E −(A−E +B)
−B −(A−E +B) E
, λV =
 0 0 e10 0 e3
e1 e3 e2
 , (19)
ON THE CP VIOLATION PHASE IN A NEUTRINO MIXING MODEL WITH AN A4 FLAVOR SYMMETRY 5
where λ is a perturbation, i.e., small, parameter and M0 is a non-perturbative (TBM level) mass
matrix which can be diagonalized by the TBM neutrino mixing matrix
UT BM =

√
2
3
√
1
3 0
−
√
1
6
√
1
3
√
1
2√
1
6 −
√
1
3
√
1
2
×P∼ (|10〉, |20〉, |30〉)×P, (20)
as
diag(m01,m
0
2,m
0
3) =U
†
T BMM0UT BM. (21)
Here P is a matrix of Majorana phases which are not given explicitly because they play no role
in the CP violation process. Using the perturbation expansion of the basis |n〉 around the non-
perturbative one, |n0〉,
|n〉= |n0〉+λ ∑
k 6=n
|k0〉 Vkn
m0n−m0k
+O(λ 2); n,k = 1,2,3, (22)
where
Vkn = 〈k|V |n〉, (23)
we can diagonalize the matrix Mν by the PMNS matrix
UPMNS =

√
2
3 +
√
1
3 X
√
1
3 −
√
2
3 X −
√
2
3Y −
√
1
3 Z
−
√
1
6 +
√
1
3 X +
√
1
2Y
√
1
3 +
√
1
6 X +
√
1
2 Z
√
1
2 +
√
1
6Y −
√
1
3 Z√
1
6 −
√
1
3 X +
√
1
2Y −
√
1
3 −
√
1
6 X +
√
1
2 Z
√
1
2 −
√
1
6Y +
√
1
3 Z
×P+∆U, (24)
where ∆U is a higher order perturbative correction to UPMNS and
X =
√
2
6
(
2e3− e1− e2
m01−m02
)
, Y =
√
3
6
(
2e1+ e2
m01−m03
)
, Z =
1√
6
(
e1− e2
m02−m03
)
. (25)
Here we discuss only the CP-violation Dirac phase δCP, while the mass spectrum derived by the
above-described diagonalization of the neutrino mass matrix Mν is a subject of a future work be-
cause more analysis on the VEV’s of the scalars and the Yukawa coupling coefficients as well as
different phenomena and experimental results is needed. Since the CP violation is expected to be
small the symmetric matrixMν is near to a Hermitian matrix, therefore, the matrix (24) is near to
an unitary one.
Denoting the matrix elements of (24) by Ui j, i, j = 1,2,3, we get the relation
U21+
√
2U22−U31−
√
2U32 = 2U11−
√
2U12+∆Ui j, (26)
where ∆Ui j is a higher order perturbative correction. Fitting the matrix (24) with the elements of
the matrix UPMNS in the canonical form
UPMNS =
 c12c13 s12c13 s13e−iδ−c23s12− s13s23c12eiδ c23c12− s13s23s12eiδ s23c13
s23s12− s13c23c12eiδ −s23c12− s13c23s12eiδ c23c13,
×P, (27)
6 PHI QUANG VAN AND NGUYEN THI HONG VAN
where si j = sinθi j, ci j = cosθi j; i, j = 1,2,3, we obtain from (26) the following relation between
δCP ≡ δ and the mixing angles θi j:
cosδ tanθ13 =
(
√
2− tanθ12)
(1+
√
2tanθ12)(1− tanθ23)
(√
2
c23
− 1+ tanθ23
c13
)
. (28)
Solving (28) for δ ∈ [0,2pi] we can get two solutions: if a value δ0 is a solution of (28), the value
2pi − δ0 is the other solution. Therefore, we can choose to discuss one of these solutions, for
example, the bigger one.
In general, both theoretically and experimentally, it is not very easy to find δCP but the
relation (28) allows us to determine δCP numerically via the experimental data on the mixing
angles θi j. The distributions of δCP are plotted in Fig. 1 and Fig. 2 for a normal- and an inverse-
ordering. Here, a value of δCP is calculated event by event using equation (28) with the value
of sinθi j taken randomly (within 3σ range) based on a Gaussian distribution having the mean
and the standard deviation (σ ) to be the best fit value and the sigmas, respectively, of sinθi j
determined experimentally and given in [12, 13]. These figures show that δCP distributes in the
region 3.1 < δCP < 5.8 (for an NO) and 3.1 < δCP < 6.3 (for an IO). This distributions have a
mean value at δCP = 4.28 (for an NO) and at δCP = 4.56 (for an IO) which is close, between
1σ , to the global fit values (GFV’s) to the experimental data and have a maximum density around
δCP = 4.45 (for an NO) and δCP = 4.55 (for an IO) between 1σ region from the GFV [12,13]. For
more information, δCP versus sin2 θ13 are depicted in Fig. 3 for an NO and in Fig. 4 for an IO.
Mean    1.999
RMS    0.2568
 CPδ
0 1 2 3 4 5 6 7
En
tri
es
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Mean    4.284
RMS    0.2549
 (for an NO)CPδDistribution of 
Fig. 1. Distribution of δCP for a normal ordering.
ON THE CP VIOLATION PHASE IN A NEUTRINO MIXING MODEL WITH AN A4 FLAVOR SYMMETRY 7
Mean  
  1.722
RMS    0.5166
 CPδ
0 1 2 3 4 5 6 7
En
tri
es
0
200
400
600
800
1000
1200
Mean    4.561
RMS    0.5157
 (for an IO)CPδDistribution of 
Fig. 2. Distribution of δCP for an inverse ordering.
 (for an NO)13θ2 versus sinCPδ
 13θ
2sin
0.01 0.015 0.02 0.025 0.03 0.035 0.04
CPδ
0
1
2
3
4
5
6
7
Mean x  0.02339
Mean y 
  3.142
RMS x  0.00198
RMS y 
  1.171
Fig. 3. δCP versus sin2 θ13 for a normal ordering.
8 PHI QUANG VAN AND NGUYEN THI HONG VAN
 (for an IO)13θ2 versus sinCPδ
 13θ
2sin
0.01 0.015 0.02 0.025 0.03 0.035 0.04
CPδ
0
1
2
3
4
5
6
7
Mean x  0.02395
Mean y 
  3.141
RMS x  0.002081
RMS y 
  1.511
Fig. 4. δCP versus sin2 θ13 for an inverse ordering.
IV. CONCLUSION
A model of neutrino mass and mixing with an A4 flavor symmetry is considered. The mass
matrix is diagonalized and the mixing matrix is found by using a perturbation method. Then, a
relation between the CP-violation Dirac phase δCP and the mixing angles θi j can be established. It
allows us to determine δCP via the experimental data on θi j. The result is quite good (near the 1σ
region of the global fit) for both the normal- and inverse- ordering of the neutrino masses [12,13],
therefore, the neutrino mass and mixing matrices are more explicitly fixed.
In this paper a first order perturbation is used but a higher order perturbation is also being
considered.
ACKNOWLEDGMENT
This work is supported by Vietnam’s National Foundation for Science and Technology
Development (NAFOSTED) under the grant No 103.03-2012.49. The authors would like to thank
Dinh Nguyen Dinh and Nguyen Anh Ky for useful discussions and collaboration, and Nguyen
Anh Ky for reading the manuscript and comments.
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On the CP Violation Phase in a Neutrino Mixing Model with an A4 Flavor Symmetry

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