NEUTRINO MASSES AND MIXING IN THE STANDARD MODEL WITH a4 FLAVOR SYMMETRY

Proc. Natl. Conf. Theor. Phys. 35 (2010), pp. 255-259

NEUTRINO MASSES AND MIXING IN
THE STANDARD MODEL WITH A4 -FLAVOR SYMMETRY

NGO THI THU DINH
University of Natural Science, 334 Nguyen Trai, Thanh Xuan, Hanoi
PHUNG VAN DONG
Institute of Physics, VAST, 10 Dao Tan, Ba Dinh, Hanoi
Abstract. Although the Standard Model is very successful, it also leaves many questions unanswered, one of which is massless of neutrinos. In this talk we introduce A4 - flavour symmetry
into the Standard Model with appropriate extension of scalar representations. As a result, the
neutrinos gain naturely small masses in agreement with experiment. The neutrino mixing matrix
in terms of tribimaximal form is obtained.

I. INTRODUCTION
The neutrino experiments imply: masses of neutrinos are small, and tribimaximal
mixing neutrinos as proposed by Harrison-Perkins-Scott is given by:


UHPS = 

√2
6
− √16
− √16

√1
3
√1
3
√1

3

0
√1
2
− √12



.

(1)

The theories of neutrinos have recently been in trying to explain this form [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11].

II. THE MODEL
II.1. Flavour symmetry A4
The finite group of the even permutation of four objects,A4 , has 12 elements and
4 equivalence classes, with the number of elements 1, 4, 4, 3, respectively. This means that
there are 4 irreducible representations, with dimensions ni, such that Σi n2i = 12. There
is only one solution: n1 = n2 = n3 = 1 and n4 = 3, and the character table of the 4
representations is shown in Table below:


256

NGO THI THU DINH, PHUNG VAN DONG

Table 1. Character table of A4 .

Class χ1 χ1
C1
1
1
C2
1
ω
C3
1 ω2
C4
1
1

χ1
1
ω2
ω
1

χ3
3
0
0
−1

n
1
4
4

3

The complex number ω is the cube root of unity, i.e., e2πi/3 . Hence 1 + ω + ω 2 = 0.
Calling the 4 irreducible representations, and 3 respectively, we have the decomposition:
3 ⊗ 3 = 1(11 + 22 + 33) ⊕ 1 (11 + ω 2 22 + ω33)
⊕1 (11 + ω22 + ω 2 33) ⊕ 3(23, 31, 12) ⊕ 3(32, 13, 21)

(2)

The non-Abelian finite group A4 is also the symmetry group of the regular tetrahedron.
II.2. Lepton mass
Under A4, the fermions and scalars of this model transform as follows:
ψL =
l1R ∼ (1, −2, 1),

νL
eL

∼ (2, −1, 3),

l2R ∼ (1, −2, 1 ),

φ=

φ+
φ0

l3R ∼ (1, −2, 1 ),

∼ (2, 1, 3).

(3)
(4)

(5)

The Yukawa interactions are:
= h1 (ψL φ)1 l1R + h2 (ψL φ)1 l2R + h3 (ψL φ)1 l3R + H.c.
Llep
Y

(6)

The vacuum expectation value (VEV) of φ is (v1 , v2 , v3 ) under A4 . It is now assumed
that so that A4 is broken down to Z3 . The mass Lagrangian for the changed leptons is:
Llep
mass = h1 v(l1L + l2L + l3L )l1R
+h2 v(l1L + ωl2L + ω 2 l3L )l2R
+h3 v(l1L + ω 2 l2L + ωl3L )l3R
+H.c.
then the mass matrix is then diagonalized:
 √
 

3h1 v √ 0
0
me 0
0
 =  0 mµ 0  ,
UL−1 M lep UR = 

0
3h2 v √ 0
0
0 mτ
0
0
3h3 v

(7)

(8)

where



1 1
1
1
UR = 1, UL = √  1 ω ω 2  ≡ UlL .
3
1 ω2 ω

(9)


NEUTRINO MASSES AND MIXING IN THE STANDARD MODEL WITH...

257

II.3. Neutrino sector
To obtain arbitrary Majorana neutrino masses, four Higgs doublets are used:
σ=

+
0
σ11
σ12
+
++
σ12 σ22

∼ (3∗ , 2, 1),

(10)

s=

s011 s+
12
++
s+
12 s22

∼ (3∗ , 2, 3).

(11)

The Yukawa interactions are:
LνY

= x(ψ¯Lc ψL )1 σ + y(ψ¯Lc ψL )3 s + H.c.

(12)

The VEV of σ is:
σ =

u 0
0 0

.

(13)

The VEV of s is put as:
s = ( s1 , s2 , s3 )

(14)

with
s2 = s3 = 0, s1 =

t 0
0 0

,

(15)

so that it is broken down to Z2 in neutrino sector.
The mass Lagrangian for the neutrinos is:
LνY

c ν
c
c
c
= x(ν1L
1L + ν2L ν2L + ν3L ν3L )u + yν2L ν3L t + H.c.

The mass matrices are then obtained by




xu 0
0
xu + 21 yt 0
0
T
 UνL
0
xu
0
Mν =  0 xu 12 yt  = UνL 
,
1
1
0 2 yt xu

0
0 xu − 2 yt

(16)

(17)

where


ULν = 

0
√1
2
√1
2


1
0
0 √12 
.
0 − √12

(18)

The mismatch between Uν and UlL yields the tribimaximal mixing pattern as proposed by Harrison- Perkins- Scott:
√



2/3
1/√3
0√
√
†
UHP S = UlL
Uν =  −1/√6 1/√3 −1/√ 2  .
(19)
−1/ 6 1/ 3 1/ 2
This is a main result of the paper.


258

NGO THI THU DINH, PHUNG VAN DONG

II.4. Scalar Potential
We can separate the general scalar potential into
V = V φ + V σs ,

(20)

where
V φ = µ2φ (φ+ φ)1 + λφ1 [(φ+ φ)1 (φ+ φ)1 ] + λφ2 [(φ+ φ)1 (φ+ φ)1 ]
+λφ3 [(φ+ φ)3s (φ+ φ)3a ] + λφ4 [(φ+ φ)3s (φ+ φ)3s + H.c.],

(21)

V σs = V (σ) + V (s) + V (σ, s) + V (φ, σ) + V (φ, s) + V (φ, σ, s) + V ,

(22)

V (σ) = µ2σ T r(σ + σ) + λσ T r(σ + σ)2 + λ σ [T r(σ + σ)]2 ,

(23)

with

V (s) = T rV (φ → s) + λ1s T r(s+ s)1 T r(s+ s)1 + λ2s T r(s+ s)1 T r(s+ s)1
+λ3s T r(s+ s)3s T r(s+ s)3a + λ4s [T r(s+ s)3s T r(s+ s)3s + H.c.],

(24)

+
+
σs
+
+
V (σ, s) = λσs
1 T r[(σ s)3 (s σ)3 ] + λ1 T r(σ s)3 T r(s σ)3
+
+
σs
+
+
+λσs
2 T r[(σ σ)1 (s s)1 ] + λ2 T r(σ σ)1 T r(s s)1
+
+

σs
+
+
+[λσs
3 T r[(s s)3s (s σ)3 ] + λ3 T r(s s)3s T r(s σ)3
+
+
σs
+
+
+λσs
4 T r[(s s)3a (s σ)3 ] + λ4 T r(s s)3a T r(s σ)3 + H.c]
+
+
σs
+
+
+[λσs
5 T r[(s σ)3 (s σ)3 ] + λ5 T r(s σ)3 T r(s σ)3 + H.c],

(25)

φσ
+
+
+
+
V (φ, σ) = λφσ
1 T r[(σ φ)3 (φ σ)3 ] + λ1 T r(σ φ)3 T r(φ σ)3
φσ

+
+
+
+
+λφσ
2 T r[(σ σ)1 (φ φ)1 ] + λ2 T r(σ σ)1 T r(φ φ)1 ,

(26)

φs
+
+
+
+
V (φ, s) = λφs
11 T r[(φ s)1 (s φ)1 ] + λ11 T r(φ s)1 T r(s φ)1
φs
+
+
+
+
+λφs
12 T r[(φ s)1 (s φ)1 ] + λ12 T r(φ s)1 T r(s φ)1
φs
+
+
+
+
+λφs
13 T r[(φ s)3s (s φ)3a ] + λ13 T r(φ s)3s T r(s φ)3a

φs
+
+
+
+
+[λφs
14 T r[(φ s)3s (s φ)3s ] + λ14 T r(φ s)3s T r(s φ)3s + H.c]
φs
+
+
+
+
+λφs
21 T r[(φ φ)1 (s s)1 ] + λ21 T r(φ φ)1 T r(s s)1
φs
+
+
+
+
+λφs
22 T r[(φ φ)1 (s s)1 ] + λ22 T r(φ φ)1 T r(s s)1
φs
+
+
+
+
+λφs
23 T r[(φ φ)3s (s s)3a ] + λ23 T r(φ φ)3s T r(s s)3a
φs
+

+
+
+
+[λφs
24 T r[(φ φ)3s (s s)3s ] + λ21 T r(φ φ)3s T r(s s)3s + H.c.],

V (φ, σ, s) = µT r(φ+ σ + sφ) + H.c.,

(27)
(28)


NEUTRINO MASSES AND MIXING IN THE STANDARD MODEL WITH...

V = µ0 φT σφ + µ1 φT s1 φ.

259

(29)

III. CONCLUSION
We have shown the neutrinos gain naturely small masses in agreement with
experiment. The neutrino mixing matrix in terms of tribimaximal form is obtained. Based
on the flavour symmetry A4, we can understand neutrino experiments [1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11].
REFERENCES
[1]
[2]
[3]
[4]

[5]
[6]
[7]
[8]
[9]
[10]
[11]

P. V. Dong, L. T. Hue, H. N. Long, D. V. Soa, Phys. Rev. D 81 (2010) 053004.
E. Ma, arXiv:0908.3165 [hep-ph].
E. Ma, Phys. Lett. B 671 (2009) 366.
G. Altarelli, D. Meloni, J. Phys. G 36 (2009) 085005.
E. Ma, Mod. Phys. Lett. A 22 (2007) 101.
C. S. Lam, Phys. Lett. B 656 (2007) 193.
G. Altarelli, F. Feruglio, Nucl. Phys. B 741 (2006) 215.
E. Ma, Phys. Rev. D 73 (2006) 057304.
E. Ma, Mod. Phys. Lett. A 21 (2006) 2931.
P. F. Harrison, D. H. Perkins, W. G. Scott, Phys. Lett. B 530 (2002) 167.
E. Ma, G. Rajasekaran, Phys. Rev. D 64 (2001) 113012.

Received 30-09-2010.