aainstitutetext: Department of Physics, Tay Nguyen University, 567 Le Duan,
Buon Ma Thuot, Vietnambbinstitutetext: Institute of Physics, VAST, 10 Dao Tan,
Ba Dinh, Hanoi, Vietnam

# The $T_{7}$ flavor symmetry in 3-3-1 model with neutral leptons

V. V. Vien b H. N. Long<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We construct a 3-3-1 model based on non-Abelian discrete symmetry $T_{7}$
responsible for the fermion masses. Neutrinos get masses from only anti-
sextets which are in triplets $\underline{3}$ and $\underline{3}^{*}$ under
$T_{7}$. The flavor mixing patterns and mass splitting are obtained without
perturbation. The tribimaximal form obtained with the breaking
$T_{7}\rightarrow Z_{3}$ in charged lepton sector and both $T_{7}\rightarrow
Z_{3}$ and $Z_{3}\rightarrow\\{\mathrm{Identity}\\}$ must be taken place in
neutrino sector but only apart in breakings
$Z_{3}\rightarrow\\{\mathrm{Identity}\\}$ (without contribution of
$\sigma^{\prime}$), and the upper bound on neutrino mass $\sum_{i=1}^{3}m_{i}$
at the level is presented. The Dirac CP violation phase $\delta$ is predicted
to either $\frac{\pi}{2}$ or $\frac{3\pi}{2}$ which is maximal CP violation.
From the Dirac CP violation phase we obtain the relation between Euler’s
angles which is consistent with the experimental in PDG 2012\. On the other
hand, the realistic lepton mixing can be obtained if both the direction for
breakings $T_{7}\rightarrow Z_{3}$ and
$Z_{3}\rightarrow\\{\mathrm{Identity}\\}$ are taken place in neutrino sectors.
The CKM matrix is the identity matrix at the tree-level.

##  1 Introduction

Despite the great success of the Standard Model (SM) of the elementary
particle physics, the origin of flavor structure, masses and mixings between
generations of matter particles are unknown yet. The neutrino mass and mixing
is one of the most important evidence of beyond Standard Model physics. Many
experiments show that neutrinos have tiny masses and their mixing is sill
mysterious altar1 ; altar2 .

The tri-bimaximal form for explaining the lepton mixing scheme was first
proposed by Harrison-Perkins-Scott (HPS), which apart from the phase
redefinitions, is given by hps1 ; hps2 ; hps3 ; hps4

$\displaystyle
U_{\mathrm{HPS}}=\left(\begin{array}[]{ccc}\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\end{array}\right),$
(4)

can be considered as a good approximation for the recent neutrino experimental
data.

The most recent data are a clear sign of rather large value $\theta_{13}$
smirnov  . The data in PDG2012 PDG2012 ; PDG1 ; PDG2 ; PDG3 ; PDG4  imply:

$\displaystyle\sin^{2}(2\theta_{12})=0.857\pm
0.024,\,\,\,\sin^{2}(2\theta_{13})=0.098\pm
0.013,\,\,\sin^{2}(2\theta_{23})>0.95,$ $\displaystyle\Delta
m^{2}_{21}=(7.50\pm 0.20)\times 10^{-5}\mathrm{eV}^{2},\,\,\Delta
m^{2}_{32}=(2.32^{+0.12}_{-0.08})\times 10^{-3}\mathrm{eV}^{2}.$ (5)

These large neutrino mixing angles are completely different from the quark
mixing ones defined by the Cabibbo- Kobayashi-Maskawa (CKM) matrix CKM ; CKM1
. This has stimulated work on flavor symmetries and non-Abelian discrete
symmetries are considered to be the most attractive candidate to formulate
dynamical principles that can lead to the flavor mixing patterns for quarks
and lepton. There are many recent models based on the non-Abelian discrete
symmetries, such as $A_{4}$ A41 ; A42 ; A43 ; A44 ; A45 ; A46 ; A47 ; A48 ;
A49 ; A410 ; A411 ; A412 ; A413 ; A414 ; A415 ; A416 ; A417 ; A418 ; dlsh  ,
$A_{5}$ A51 ; A52 ; A53 ; A54 ; A55 ; A56 ; A57 ; A58 ; A59 ; A510 ; A511 ;
A512 ; A513  , $S_{3}$S31 ; S32 ; S33 ; S34 ; S35 ; S36 ; S37 ; S38 ; S39 ;
S310 ; S311 ; S312 ; S313 ; S314 ; S315 ; S316 ; S317 ; S318 ; S319 ; S320 ;
S321 ; S322 ; S323 ; S324 ; S325 ; S326 ; S327 ; S328 ; S329 ; S330 ; S331 ;
S332 ; S333 ; S334 ; S335 ; S336 ; S337 ; S338 ; S339 ; S340 ; S341 ; S342  ,
$S_{4}$ S41 ; S42 ; S43 ; S44 ; S45 ; S46 ; S47 ; S48 ; S49 ; S410 ; S411 ;
S412 ; S413 ; S414 ; S415 ; S416 ; S417 ; S418 ; S419 ; S420 ; S421 ; S422 ;
S423 ; S424 ; S425 ; S426 ; S427 ; S428 ; S429  , $D_{4}$ D41 ; D42 ; D43 ;
D44 ; D45 ; D46 ; D47 ; D48 ; D49 ; D410 ; D411 ; D412 , $D_{5}$ D51 ; D52  ,
$T^{\prime}$ Tp1 ; Tp2 ; Tp3 ; Tp4 ; Tp5 ; Tp6 ; Tp7 ; Tp8 ; Tp9 ; Tp10 ; Tp11
; Tp12 , $T_{7}$ T71 ; T72 ; T73 ; T74 ; T75 ; T76 ; T77  and so forth.

Among the possible extensions of SM, a curious choice are the 3-3-1 models
which encompass a class of models based on the gauge group
$\mbox{SU}(3)_{C}\otimes\mbox{SU}(3)_{L}\otimes\mbox{U}(1)_{X}$ 331m1 ; 331m2
; 331m3 ; 331m4 ; 331m5 ; 331r1 ; 331r2 ; 331r3 ; 331r4 ; 331r5 ; 331r6 ;
e3311 ; e3312 ; e331v1 ; e331v2 ; e3313 ; e3314 , that is at first
spontaneously broken to the SM group
$\mbox{SU}(3)_{C}\otimes\mbox{SU}(2)_{L}\otimes\mbox{U}(1)_{Y}$ and then
undergoes the spontaneously broken to
$\mbox{SU}(3)_{C}\otimes\mbox{U}(1)_{Q}$. The extension of the gauge group
with respect to SM leads to interesting consequences. The first one is that
the requirement of anomaly cancelation together with that of asymptotic
freedom of QCD implies that the number of generations must necessarily be
equal to the number of colors, hence giving an explanation for the existence
of three generations. Furthermore, quark generations should transform
differently under the action of $SU(3)_{L}$. In particular, two quark
generations should transform as triplets, one as an antitriplet.

A fundamental relation holds among some of the generators of the group T76 ;
T77 :

$\displaystyle Q=T_{3}+\beta T_{8}+X$ (6)

where $Q$ indicates the electric charge, $T_{3}$ and $T_{8}$ are two of the
$SU(3)$ generators and $X$ is the generator of $U(1)_{X}$. $\beta$ is a key
parameter that defines a specific variant of the model.

The model thus provides a partial explanation of the family number, as also
required by flavor symmetries such as $T_{7}$ for 3-dimensional
representations. In addition, due to the anomaly cancelation one family of
quarks has to transform under $\mathrm{SU}(3)_{L}$ differently from the two
others. $T_{7}$ can meet this requirement with three inequivalent
representations
$\underline{1},\underline{1}^{\prime},\underline{1}^{\prime\prime}$. Note that
$T_{7}$ has not been considered before in the kind of the 3-3-1 model.

There are two typical variants of the 3-3-1 models as far as lepton sectors
are concerned. In the minimal version, three $\mathrm{SU}(3)_{L}$ lepton
triplets are $(\nu_{L},l_{L},l^{c}_{R})$, where $l_{R}$ are ordinary right-
handed charged-leptons 331m1 ; 331m2 ; 331m3 ; 331m4 ; 331m5  . In the second
version, the third components of lepton triplets are the right-handed
neutrinos, $(\nu_{L},l_{L},\nu^{c}_{R})$ 331r1 ; 331r2 ; 331r3 ; 331r4 ; 331r5
; 331r6 . To have a model with the realistic neutrino mixing matrix, we should
consider another variant of the form $(\nu_{L},l_{L},N^{c}_{R})$ where $N_{R}$
are three new fermion singlets under standard model symmetry with vanishing
lepton-numbers dlshA4 ; dlsvS4 ; dlnvS3 ; vlD4  .

In our previous works dlshA4 ; dlsvS4 ; dlnvS3 ; vlD4  , the discrete
symmetries have been explored to the 3-3-1 models. In Ref. dlsvS4  we have
studied the 3-3-1 model with neutral fermions based on $S_{4}$ group, in which
most of the Higgs multiplets are in triplets under $S_{4}$ except $\chi$ lying
in a singlet111$\chi$ is the unique singlet under $S_{4}$ defined in
expression (25) in Ref.dlsvS4 , and the exact tribimaximal form hps1 ; hps2 ;
hps3 ; hps4  is obtained, where $\theta_{13}=0$. As we know, the recent
considerations have implied $\theta_{13}\neq 0$ A41 ; A42 ; A43 ; A44 ; A45 ;
A46 ; A47 ; A48 ; A49 ; A410 ; A411 ; A412 ; A413 ; A414 ; A415 ; A416 ; A417
; A418 ; S31 ; S32 ; S33 ; S34 ; S35 ; S36 ; S37 ; S38 ; S39 ; S310 ; S311 ;
S312 ; S313 ; S314 ; S315 ; S316 ; S317 ; S318 ; S319 ; S320 ; S321 ; S322 ;
S323 ; S324 ; S325 ; S326 ; S327 ; S328 ; S329 ; S330 ; S331 ; S332 ; S333 ;
S334 ; S335 ; S336 ; S337 ; S338 ; S339 ; S340 ; S341 ; S342 ; S41 ; S42 ; S43
; S44 ; S45 ; S46 ; S47 ; S48 ; S49 ; S410 ; S411 ; S412 ; S413 ; S414 ; S415
; S416 ; S417 ; S418 ; S419 ; S420 ; S421 ; S422 ; S423 ; S424 ; S425 ; S426 ;
S427 ; S428 ; S429 , but small as given in (5). This problem has been improved
in this type of the model in Ref. dlnvS3 ; vlD4  by adding new $SU(3)_{L}$
multiplets and one of them is regarded as a small perturbation. The model
therefore contains up to eight Higgs multiplets, and the scalar potential of
the model is quite complicated dlnvS3 ; vlD4 .

CP violation plays a crucial role in our understanding of the observed baryon
asymmetry of the Universe CPvio . In the SM, CP symmetry is violated due to a
complex phase in the CKM matrix CKM ; CKM1 . However, since the extent of CP
violation in the SM is not enough for achieving the observed BAU, we need new
source of CP violation for successful BAU. On the other hand, CP violations in
the lepton sector are imperative if the BAU could be realized through
leptogenesis. So, any hint or observation of the leptonic CP violation can
strengthen our belief in leptogenesis CPvio .

The violation of the CP symmetry is a crucial ingredient of any dynamical
mechanism which intends to explain both low energy CP violation and the baryon
asymmetry. Renormalizable gauge theories are based on the spontaneous symmetry
breaking mechanism, and it is natural to have the spontaneous CP violation as
an integral part of that mechanism. Determining all possible sources of CP
violation is a fundamental challenge for high energy physics. In theoretical
and economical viewpoints, the spontaneous CP breaking necessary to generate
the baryon asymmetry and leptonic CP violation at low energies brings us to a
common source which comes from the phase of the scalar field responsible for
the spontaneous CP breaking at a high energy scale CPvio .

In this paper, we investigate another choice with $T_{7}$, the smallest group
with two non-equivalent 3-dimensional irreducible representations, contains
two triplet irreducible representations and three singlets which play a
crucial role in consistently reproducing fermion masses and mixing. As we will
see, $T_{7}$ model has some new features since fewer Higgs multiplets are
needed in order to allow the fermions to gain masses and to break symmetries
and the physics we will see is different from the former. The CP violation is
the first time considered under $SU(3)_{L}\times U(1)_{X}$ model based on
$T_{7}$ flavor symmetry in which the $T_{7}$ symmetry avoids the mass
degeneracy of lepton masses. The light neutrino masses can be generated at
tree level, and the vacuum alignment problem which arises in the presence of
two $T_{7}$-triplet scalar fields $\underline{3},\underline{3}^{*}$ can
naturally explain the measured value of $\theta_{13}$ and thereby the
hierarchy of neutrino masses. The seesaw mechanism can explain the smallness
of the measured neutrino masses and the maximal Dirac CP violation.

The rest of this work is organized as follows. In Sec. 2 and 3 we present the
necessary elements of the 3-3-1 model with the $T_{7}$ symmetry as well as
introducing necessary Higgs fields responsible for the charged lepton masses.
In Sec. 4, we discuss on quark sector. Sec. 5 is devoted for the neutrino mass
and mixing. We summarize our results and make conclusions in the section 7.
Appendix A presents a brief of the $T_{7}$ theory. Appendix B provides the
lepton number ($L$) and lepton parity ($P_{l}$) of particles in the model.

##  2 Fermion content

The gauge symmetry is based on
$\mathrm{SU}(3)_{C}\otimes\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}$, where
the electroweak factor $\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}$ is
extended from those of the Standard Model (SM), and the strong interaction
sector is retained. Each lepton family includes a new neutral fermion
$(N_{R})$ with vanishing lepton number $L(N_{R})=0$ arranged under the
$\mathrm{SU}(3)_{L}$ symmetry as a triplet $(\nu_{L},l_{L},N^{c}_{R})$ and a
singlet $l_{R}$. The residual electric charge operator $Q$ is therefore
related to the generators of the gauge symmetry by

$\displaystyle Q=T_{3}-\frac{1}{\sqrt{3}}T_{8}+X,$ (7)

where $T_{a}$ $(a=1,2,...,8)$ are $\mathrm{SU}(3)_{L}$ charges with
$\mathrm{Tr}T_{a}T_{b}=\frac{1}{2}\delta_{ab}$ and $X$ is the
$\mathrm{U}(1)_{X}$ charge. This means that the model under consideration does
not contain exotic electric charges in the fundamental fermion, scalar and
adjoint gauge boson representations.

Since the particles in the lepton triplet have different lepton number (1 and
0), so the lepton number in the model does not commute with the gauge symmetry
unlike the SM. Therefore, it is better to work with a new conserved charge
$\mathcal{L}$ commuting with the gauge symmetry and related to the ordinary
lepton number by diagonal matrices dlshA4 ; dlsvS4 ; dlnvS3 ; clong ; vlD4

$\displaystyle L=\frac{2}{\sqrt{3}}T_{8}+\mathcal{L}.$ (8)

The lepton charge arranged in this way, i.e. $L(N_{R})=0$, is in order to
prevent unwanted interactions due to $\mathrm{U}(1)_{\mathcal{L}}$ symmetry
and breaking due to the lepton parity to obtain the consistent lepton and
quark spectra. By this embedding, exotic quarks $U,D$ as well as new non-
Hermitian gauge bosons $X^{0}$, $Y^{\pm}$ possess lepton charges as of the
ordinary leptons: $L(D)=-L(U)=L(X^{0})=L(Y^{-})=1$.

Under the
$[\mathrm{SU}(3)_{L},\mathrm{U}(1)_{X},\mathrm{U}(1)_{\mathcal{L}},\underline{T}_{7}]$
symmetries as proposed, the fermions of the model transform as follows

$\displaystyle\psi_{L}$ $\displaystyle\equiv$
$\displaystyle\psi_{1,2,3L}=\left(\nu_{L}\hskip 14.22636ptl_{L}\hskip
14.22636ptN^{c}_{R}\right)^{T}\sim[3,-1/3,2/3,\underline{3}],$ $\displaystyle
l_{1R}$ $\displaystyle\sim$ $\displaystyle[1,-1,1,\underline{1}],\hskip
14.22636ptl_{2R}\sim[1,-1,1,\underline{1}^{\prime}],\hskip
14.22636ptl_{3R}\sim[1,-1,1,\underline{1}^{\prime\prime}],$ $\displaystyle
Q_{1L}$ $\displaystyle\equiv$ $\displaystyle\left(d_{1L}\hskip 14.22636pt-
u_{1L}\hskip
14.22636ptD_{1L}\right)^{T}\sim[3^{*},0,1/3,\underline{1}^{\prime}],$
$\displaystyle Q_{2L}$ $\displaystyle\equiv$ $\displaystyle\left(d_{2L}\hskip
14.22636pt-u_{2L}\hskip
14.22636ptD_{2L}\right)^{T}\sim[3^{*},0,1/3,\underline{1}^{\prime\prime}],$
(9) $\displaystyle Q_{3L}$ $\displaystyle=$ $\displaystyle\left(u_{3L}\hskip
14.22636ptd_{3L}\hskip
14.22636ptU_{L}\right)^{T}\sim[3,1/3,-1/3,\underline{1}],$ $\displaystyle
u_{R}$ $\displaystyle\sim$ $\displaystyle
u_{1,2,3R}=[1,2/3,0,\underline{3}],\hskip
14.22636ptd_{R}\sim[1,-1/3,0,\underline{3}^{*}],$ $\displaystyle U_{R}$
$\displaystyle\sim$ $\displaystyle[1,2/3,-1,\underline{1}],\hskip
14.22636ptD_{1R}\sim[1,-1/3,1,\underline{1}^{\prime\prime}],\hskip
14.22636ptD_{2R}\sim[1,-1/3,1,\underline{1}^{\prime}].$

where the subscript numbers on field indicate to respective families which
also in order define components of their $T_{7}$ multiplets. $U$ and $D_{1,2}$
are exotic quarks carrying lepton numbers $L(U)=-1$ and $L(D_{1,2})=1$, known
as leptoquarks. In the following, we consider possibilities of generating the
masses for the fermions. The scalar multiplets needed for the purpose are also
introduced.

##  3 Charged lepton masses

The charged lepton masses arise from the couplings of
$\bar{\psi}_{L}l_{1R},\bar{\psi}_{L}l_{2R}$ and $\bar{\psi}_{L}l_{3R}$ to
scalars, where $\bar{\psi}_{L}l_{iL}\,(i=1,2,3)$ transforms as $3^{*}$ under
$\mathrm{SU}(3)_{L}$ and $\underline{3}^{*}$ under $T_{7}$. To generate masses
for charged leptons, we need a $SU(3)_{L}$ Higgs triplets that lying in
$\underline{3}$ under $T_{7}$ and transforms as $3$ under
$\mathrm{SU}(3)_{L}$,

$\displaystyle\phi_{i}=\left(\begin{array}[]{c}\phi^{+}_{i1}\\\
\phi^{0}_{i2}\\\ \phi^{+}_{i3}\\\
\end{array}\right)\sim[3,2/3,-1/3,\underline{3}]\hskip 14.22636pt(i=1,2,3).$
(13)

Following the potential minimization conditions, we have the followings
alignments:

  * (1)

The first alignment:
$\langle\phi_{1}\rangle=\langle\phi_{2}\rangle=\langle\phi_{3}\rangle$ then
$T_{7}$ is broken into $Z_{3}$ consisting of the elements {$e,b,b^{2}$}.

  * (2)

The second alignment:
$\langle\phi_{1}\rangle\neq\langle\phi_{2}\rangle\neq\langle\phi_{3}\rangle$
or $\langle\phi_{1}\rangle\neq\langle\phi_{2}\rangle=\langle\phi_{3}\rangle$
or
$\langle\phi_{2}\rangle\neq\langle\phi_{1}\rangle\neq\langle\phi_{3}\rangle$
or
$\langle\phi_{3}\rangle\neq\langle\phi_{1}\rangle\neq\langle\phi_{2}\rangle$
then $T_{7}$ is broken into $\\{\mathrm{Identity}\\}$.

  * (3)

The third alignment:
$0=\langle\phi_{1}\rangle\neq\langle\phi_{2}\rangle=\langle\phi_{3}\rangle\neq
0$ or
$0=\langle\phi_{2}\rangle\neq\langle\phi_{3}\rangle=\langle\phi_{1}\rangle\neq
0$ or
$0=\langle\phi_{3}\rangle\neq\langle\phi_{1}\rangle=\langle\phi_{2}\rangle\neq
0$ then $T_{7}$ is broken into $\\{\mathrm{Identity}\\}$.

  * (4)

The fourth alignment:
$0=\langle\phi_{1}\rangle\neq\langle\phi_{2}\rangle\neq\langle\phi_{3}\rangle\neq
0$ or
$0=\langle\phi_{2}\rangle\neq\langle\phi_{1}\rangle\neq\langle\phi_{3}\rangle\neq
0$ or
$0=\langle\phi_{3}\rangle\neq\langle\phi_{2}\rangle\neq\langle\phi_{1}\rangle\neq
0$ then $T_{7}$ is broken into $\\{\mathrm{Identity}\\}$.

  * (5)

The fifth alignment:
$0=\langle\phi_{1}\rangle=\langle\phi_{2}\rangle\neq\langle\phi_{3}\rangle\neq
0$ or
$0=\langle\phi_{1}\rangle=\langle\phi_{3}\rangle\neq\langle\phi_{2}\rangle\neq
0$ or
$0=\langle\phi_{2}\rangle=\langle\phi_{3}\rangle\neq\langle\phi_{1}\rangle\neq
0$ then $T_{7}$ is broken into $\\{\mathrm{Identity}\\}$.

In this work, we argue that only the first alignment of VEV in charged -
lepton sector is taken place, i.e, $T_{7}\rightarrow Z_{3}$, and this can be
achieved by the Higgs triplet $\phi$ with the VEV alignment
$\langle\phi\rangle=(\langle\phi_{1}\rangle,\langle\phi_{1}\rangle,\langle\phi_{1}\rangle)$
under $T_{7}$, where

$\langle\phi_{1}\rangle=\left(0\hskip 14.22636ptv\hskip
14.22636pt0\right)^{T}.$ (14)

The Yukawa interactions are

$\displaystyle-\mathcal{L}_{l}$ $\displaystyle=$ $\displaystyle
h_{1}(\bar{\psi}_{L}\phi)_{\underline{1}}l_{1R}+h_{2}(\bar{\psi}_{L}\phi)_{\underline{1}^{\prime\prime}}l_{2R}+h_{3}(\bar{\psi}_{iL}\phi)_{\underline{1}^{\prime}}l_{3R}+H.c$
(15) $\displaystyle=$ $\displaystyle
h_{1}(\bar{\psi}_{1L}\phi_{1}+\bar{\psi}_{2L}\phi_{2}+\bar{\psi}_{3L}\phi_{3})l_{1R}$
$\displaystyle+$ $\displaystyle
h_{2}(\bar{\psi}_{1L}\phi_{1}+\omega^{2}\bar{\psi}_{2L}\phi_{2}+\omega\bar{\psi}_{3L}\phi_{3})l_{2R}$
$\displaystyle+$ $\displaystyle
h_{3}(\bar{\psi}_{1L}\phi_{1}+\omega\bar{\psi}_{2L}\phi_{2}+\omega^{2}\bar{\psi}_{3L}\phi_{3})l_{3R}+H.c.$

The mass Lagrangian for the charged leptons is then given by

$\displaystyle-\mathcal{L}^{\mathrm{mass}}_{l}$ $\displaystyle=$
$\displaystyle
h_{1}v\bar{l}_{1L}l_{1R}+h_{2}v\bar{l}_{1L}l_{2R}+h_{3}v\bar{l}_{1L}l_{3R}$
(16) $\displaystyle+$ $\displaystyle
h_{1}v\bar{l}_{2L}l_{1R}+h_{2}v\omega^{2}\bar{l}_{2L}l_{2R}+h_{3}v\omega\bar{l}_{2L}l_{3R}$
$\displaystyle+$ $\displaystyle
h_{1}v\bar{l}_{3L}l_{1R}+h_{2}v\omega\bar{l}_{3L}l_{2R}+h_{3}v\omega^{2}\bar{l}_{3L}l_{3R}+H.c.$

The mass Lagrangian for the charged leptons reads

$\displaystyle-\mathcal{L}^{\mathrm{mass}}_{l}=(\bar{l}_{1L},\bar{l}_{2L},\bar{l}_{3L})M_{l}(l_{1R},l_{2R},l_{3R})^{T}+H.c,$
(17)

where

$M_{l}=\left(\begin{array}[]{ccc}h_{1}v&\,\,\,h_{2}v&\,\,\,h_{3}v\\\
h_{1}v&\,\,\,\,\,\,\,h_{2}v\omega^{2}&\,\,\,\,h_{3}v\omega\\\
h_{1}v&\,\,\,\,\,h_{2}v\omega&\,\,\,\,\,\,h_{3}v\omega^{2}\\\
\end{array}\right).$ (18)

This matrix can be diagonalized as,

$\displaystyle
U^{\dagger}_{L}M_{l}U_{R}=\left(\begin{array}[]{ccc}\sqrt{3}h_{1}v&0&0\\\
0&\sqrt{3}h_{2}v&0\\\ 0&0&\sqrt{3}h_{3}v\\\
\end{array}\right)\equiv\left(\begin{array}[]{ccc}m_{e}&0&0\\\ 0&m_{\mu}&0\\\
0&0&m_{\tau}\\\ \end{array}\right),$ (25)

where

$\displaystyle
U_{L}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}1&\,\,\,1&\,\,\,1\\\
1&\,\,\,\omega^{2}&\,\,\,\omega\\\ 1&\,\,\,\omega&\,\,\,\omega^{2}\\\
\end{array}\right),\hskip 14.22636ptU_{R}=1.$ (29)

As will see in section 5, in this case, the exact tribimaximal mixing form is
obtained, by choosing the right vev’s in the neutrino sector.

The experimental values for masses of the charged leptons at the weak scale
are given as PDG2012  :

$\displaystyle m_{e}=0.511\,\textrm{MeV},\hskip 14.22636pt\ m_{\mu}=105.658\
\textrm{MeV},\hskip 14.22636ptm_{\tau}=1776.82\,\textrm{MeV}$ (30)

from which it follows that $h_{1}\ll h_{2}\ll h_{3}$. On the other hand, if we
choose the VEV $v=100GeV$ then $h_{1}\sim 10^{-6},\,h_{2}\sim
10^{-4},\,h_{3}\sim 10^{-2}$.

##  4 Quark masses

Table 1: List of couplings which form a singlet from the invariance under the $T_{7}$ Couplings | Higgs multiplets
---|---
$\bar{Q}_{3L}U_{R}\sim\left(3^{*},\frac{1}{3},-\frac{2}{3},\underline{1}\right)$ | $\chi\sim\left(3,-\frac{1}{3},\frac{2}{3},\underline{1}\right)$
$\bar{Q}_{2L}D_{2R}\sim\left(3,-\frac{1}{3},\frac{2}{3},\underline{1}\right)$ | $\chi^{*}\sim\left(3^{*},\frac{1}{3},-\frac{2}{3},\underline{1}\right)$
$\bar{Q}_{1L}D_{1R}\sim\left(3,-\frac{1}{3},\frac{2}{3},\underline{1}\right)$ | $\chi^{*}\sim\left(3^{*},\frac{1}{3},-\frac{2}{3},\underline{1}\right)$
$\bar{Q}_{3L}d_{R}\sim\left(3^{*},-\frac{2}{3},\frac{1}{3},\underline{3}^{*}\right)$ | $\phi\sim\left(3,\frac{2}{3},-\frac{1}{3},\underline{3}\right)$
$\bar{Q}_{1L}u_{R}\sim\left(3,\frac{2}{3},-\frac{1}{3},\underline{3}\right)$ | $\phi^{*}\sim\left(3^{*},-\frac{2}{3},\frac{1}{3},\underline{3}^{*}\right)$
$\bar{Q}_{2L}u_{R}\sim\left(3,\frac{2}{3},-\frac{1}{3},\underline{3}\right)$ | $\phi^{*}\sim\left(3^{*},-\frac{2}{3},\frac{1}{3},\underline{3}^{*}\right)$
$\bar{Q}_{3L}u_{R}\sim\left(3^{*},\frac{1}{3},\frac{1}{3},\underline{3}\right)$ | $\eta\sim\left(3,-\frac{1}{3},-\frac{1}{3},\underline{3}^{*}\right)$
$\bar{Q}_{1L}d_{R}\sim\left(3,-\frac{1}{3},-\frac{1}{3},\underline{3}^{*}\right)$ | $\eta^{*}\sim\left(3^{*},\frac{1}{3},\frac{1}{3},\underline{3}\right)$
$\bar{Q}_{2L}d_{R}\sim\left(3,-\frac{1}{3},-\frac{1}{3},\underline{3}^{*}\right)$ | $\eta^{*}\sim\left(3^{*},\frac{1}{3},\frac{1}{3},\underline{3}\right)$

To generate masses for quarks with a minimal Higgs content, we additionally
introduce the following Higgs triplets:

$\displaystyle\eta_{i}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\eta^{0}_{i1}\\\ \eta^{-}_{i2}\\\
\eta^{0}_{i3}\\\
\end{array}\right)\sim\left[3,-1/3,-1/3,\underline{3}\right]\hskip
14.22636pt(i=1,2,3),$ (34) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\chi^{0}_{1}\\\ \chi^{-}_{2}\\\
\chi^{0}_{3}\\\ \end{array}\right)\sim\left[3,-1/3,2/3,\underline{1}\right].$
(38)

The Higgs content and Yukawa couplings in the quark sector are summarized in
Table 1.

The Yukawa interactions are

$\displaystyle-\mathcal{L}_{q}$ $\displaystyle=$ $\displaystyle
h^{d}_{3}\bar{Q}_{3L}(\phi
d_{R})_{1}+h^{u}_{1}\bar{Q}_{1L}(\phi^{*}u_{R})_{1^{\prime\prime}}+h^{u}_{2}\bar{Q}_{2L}(\phi^{*}u_{R})_{1^{\prime}}$
(39) $\displaystyle+$ $\displaystyle h^{u}_{3}\bar{Q}_{3L}(\eta
u_{R})_{1}+h^{d}_{1}\bar{Q}_{1L}(\eta^{*}d_{R})_{1^{\prime\prime}}+h^{d}_{2}\bar{Q}_{2L}(\eta^{*}d_{R})_{1^{\prime}}$
$\displaystyle+$ $\displaystyle f_{3}\bar{Q}_{3L}\chi
U_{R}+f_{1}\bar{Q}_{1L}\chi^{*}D_{1R}+f_{2}\bar{Q}_{2L}\chi^{*}D_{2R}+H.c$
$\displaystyle=$ $\displaystyle
h^{d}_{3}\bar{Q}_{3L}(\phi_{1}d_{1R}+\phi_{2}d_{2R}+\phi_{3}d_{3R})$
$\displaystyle+$ $\displaystyle
h^{u}_{1}\bar{Q}_{1L}(\phi^{*}_{1}u_{R}+\omega^{2}\phi^{*}_{2}u_{2R}+\omega\phi^{*}_{3}u_{3R})$
$\displaystyle+$ $\displaystyle
h^{u}_{2}\bar{Q}_{2L}(\phi^{*}_{1}u_{R}+\omega\phi^{*}_{2}u_{2R}+\omega^{2}\phi^{*}_{3}u_{3R})$
$\displaystyle+$ $\displaystyle
h^{u}_{3}\bar{Q}_{3L}(\eta_{1}u_{1R}+\eta_{2}u_{2R}+\eta_{3}u_{3R})$
$\displaystyle+$ $\displaystyle
h^{d}_{1}\bar{Q}_{1L}(\eta^{*}_{1}d_{1R}+\omega^{2}\eta^{*}_{2}d_{2R}+\omega\eta^{*}_{3}d_{3R})$
$\displaystyle+$ $\displaystyle
h^{d}_{2}\bar{Q}_{2L}(\eta^{*}_{1}d_{1R}+\omega\eta^{*}_{2}d_{2R}+\omega^{2}\eta^{*}_{3}d_{3R})$
$\displaystyle+$ $\displaystyle f_{3}\bar{Q}_{3L}\chi
U_{R}+f_{1}\bar{Q}_{1L}\chi^{*}D_{1R}+f_{2}\bar{Q}_{2L}\chi^{*}D_{2R}+H.c.$

We suppose that $T_{7}$ is broken into $Z_{3}$ like the case of the charged
lepton sector, i,e, the VEVs of $\eta$ and $\chi$ are given as
$\langle\eta\rangle=(\langle\eta_{1}\rangle,\langle\eta_{1}\rangle,\langle\eta_{1}\rangle)$
with

$\displaystyle\langle\eta_{1}\rangle$ $\displaystyle=$
$\displaystyle\left(u\hskip 14.22636pt0\hskip 14.22636pt0\right)^{T},$ (40)

and

$\displaystyle\langle\chi\rangle$ $\displaystyle=$ $\displaystyle\left(0\hskip
14.22636pt0\hskip 14.22636ptv_{\chi}\right)^{T}.$ (41)

The mass Lagrangian for quarks is given by

$\displaystyle-\mathcal{L}^{mass}_{q}$ $\displaystyle=$
$\displaystyle-h^{u}_{1}v_{1}\bar{u}_{1L}u_{1R}-h^{u}_{1}v_{2}\omega^{2}\bar{u}_{1L}u_{2R}-h^{u}_{1}v_{3}\omega\bar{u}_{1L}u_{3R}$
(42) $\displaystyle-$ $\displaystyle
h^{u}_{2}v_{1}\bar{u}_{2L}u_{1R}-h^{u}_{2}v_{2}\omega\bar{u}_{2L}u_{2R}-h^{u}_{2}v_{3}\omega^{2}\bar{u}_{2L}u_{3R}$
$\displaystyle+$ $\displaystyle
h^{u}_{3}u_{1}\bar{u}_{3L}u_{1R}+h^{u}_{3}u_{2}\bar{u}_{3L}u_{2R}+h^{u}_{3}u_{3}\bar{u}_{3L}u_{3R}$
$\displaystyle+$ $\displaystyle
h^{d}_{1}u_{1}\bar{d}_{1L}d_{1R}+\omega^{2}h^{d}_{1}u_{2}\bar{d}_{1L}d_{2R}+\omega
h^{d}_{1}u_{3}\bar{d}_{1L}d_{3R}$ $\displaystyle+$ $\displaystyle
h^{d}_{2}u_{1}\bar{d}_{2L}d_{1R}+\omega
h^{d}_{2}u_{2}\bar{d}_{2L}d_{2R}+\omega^{2}h^{d}_{2}u_{3}\bar{d}_{2L}d_{3R}$
$\displaystyle+$ $\displaystyle
h^{d}_{3}v_{1}\bar{d}_{3L}d_{1R}+h^{d}_{3}v_{2}\bar{d}_{3L}d_{2R}+h^{d}_{3}v_{3}\bar{d}_{3L}d_{3R}$
$\displaystyle+$ $\displaystyle
f_{3}v_{\chi}\bar{U}_{L}U_{R}+f_{1}v_{\chi}\bar{D}_{1L}D_{1R}+f_{2}v_{\chi}\bar{D}_{2L}D_{2R}+H.c.$

The exotic quarks get masses

$\displaystyle m_{U}=f_{3}v_{\chi},\hskip
14.22636ptm_{D_{1,2}}=f_{1,2}v_{\chi}.$ (43)

The mass matrices for ordinary up-quarks and down-quarks are, respectively,
obtained as follows:

$\displaystyle
M_{u}=\left(\begin{array}[]{ccc}-h^{u}_{1}v&\,\,\,-h^{u}_{1}v\omega^{2}&-h^{u}_{1}v\omega\\\
-h^{u}_{2}v&\,\,-h^{u}_{2}v\omega&\,-h^{u}_{2}v\omega^{2}\\\
\,\,\,\,h^{u}_{3}u&\,\,\,h^{u}_{3}u&h^{u}_{3}u\\\
\end{array}\right),\,\,\,M_{d}=\left(\begin{array}[]{ccc}h^{d}_{1}u&\,\,\,\,\,h^{d}_{1}u\omega^{2}&\,\,\,h^{d}_{1}u\omega\\\
h^{d}_{2}u&\,\,\,\,h^{d}_{2}u\omega&\,\,\,\,\,h^{d}_{2}u\omega^{2}\\\
h^{d}_{3}v&h^{d}_{3}v&h^{d}_{3}v\\\ \end{array}\right).$ (50)

The structure of the up- and down-quark mass matrices in (50) is similar to
those in A419 , i.e, in the model under consideration there is no CP violation
in the quark mixing matrix. The mass matrices $M_{u},M_{d}$ in (50) are
diagonalized as follows

$\displaystyle U^{u+}_{L}M_{u}U^{u}_{R}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}-\sqrt{3}h^{u}_{1}v&0&0\\\
0&-\sqrt{3}h^{u}_{2}v&0\\\ 0&0&\sqrt{3}h^{u}_{3}u\\\
\end{array}\right)=\left(\begin{array}[]{ccc}m_{u}&0&0\\\ 0&m_{c}&0\\\
0&0&m_{t}\\\ \end{array}\right),$ (57) $\displaystyle
U^{d+}_{L}M_{d}U^{d}_{R}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\sqrt{3}h^{d}_{1}u&0&0\\\
0&\sqrt{3}h^{d}_{2}u&0\\\ 0&0&\sqrt{3}h^{d}_{3}v\\\
\end{array}\right)=\left(\begin{array}[]{ccc}m_{d}&0&0\\\ 0&m_{s}&0\\\
0&0&m_{b}\\\ \end{array}\right).$ (64)

where

$\displaystyle
U^{u}_{R}=U^{d}_{R}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}1&1&1\\\
\omega&\omega^{2}&1\\\ \omega^{2}&\omega&1\\\ \end{array}\right),\hskip
14.22636ptU^{u}_{L}=U^{d}_{L}=1$ (68)

The unitary matrices, which couple the left-handed up- and down-quarks to
those in the mass bases are unit matrices. Therefore we get the Cabibbo-
Kobayashi-Maskawa (CKM) matrix

$\displaystyle U_{\mathrm{CKM}}=U^{d\dagger}_{L}U^{u}_{L}=1.$ (69)

Note that the property in (69) is common for some models based on the discrete
symmetry groups dlsvS4 ; dlshA4 .

The up and down quark masses are

$\displaystyle m_{u}$ $\displaystyle=$
$\displaystyle-\sqrt{3}h^{u}_{1}v,\hskip
14.22636ptm_{c}=-\sqrt{3}h^{u}_{2}v,\hskip
14.22636ptm_{t}=\sqrt{3}h^{u}_{3}u,$ $\displaystyle m_{d}$ $\displaystyle=$
$\displaystyle\sqrt{3}h^{d}_{1}u,\hskip
14.22636pt\,\,\,\,\,m_{s}=\sqrt{3}h^{d}_{2}u,\hskip
14.22636pt\,\,\,\,\,m_{d}=\sqrt{3}h^{d}_{3}v.$ (70)

The current mass values for the quarks are given by PDG2012 :

$\displaystyle m_{u}$ $\displaystyle=$ $\displaystyle 2.3^{+0.7}_{-0.5}\
\textrm{MeV},\,\,m_{c}=1.275\pm 0.025\ \textrm{GeV},\,\,m_{t}=173.5\pm 0.6\pm
0.8,\ \textrm{GeV}$ $\displaystyle m_{d}$ $\displaystyle=$ $\displaystyle
4.8^{+0.7}_{-0.3}\ \textrm{MeV},\,\,m_{s}=95\pm 5\ \textrm{MeV},\hskip
14.22636pt\hskip 14.22636pt\,\,\,m_{b}=4.18\pm 0.03\ \textrm{GeV}.$ (71)

It is obvious that if $|u|\sim|v|$, the Yukawa coupling hierarchies are
$|h^{u}_{1}|\ll h^{u}_{2}\ll h^{u}_{3}$, $h^{d}_{1}\ll h^{d}_{2}\ll h^{d}_{3}$
and the couplings between up-quarks $(h^{u}_{2},h^{u}_{3})$ and Higgs scalar
multiplets are slightly heavier than those of down-quarks $(h^{d}_{2}\ll
h^{d}_{3})$, respectively.

##  5 Neutrino mass and mixing

The neutrino masses arise from the couplings of $\bar{\psi}^{c}_{L}\psi_{L}$
to scalars, where $\bar{\psi}^{c}_{L}\psi_{L}$ transforms as $3^{*}\oplus 6$
under $\mathrm{SU}(3)_{L}$ and
$\underline{3}\oplus\underline{3}^{*}\oplus\underline{3}^{*}$ under $T_{7}$.
It is worth mentioning that, with the $T_{7}$ group,
$\underline{3}\times\underline{3}\times\underline{3}$ has two invariants and
$\underline{3}\times\underline{3}\times\underline{3}^{*}$ has one invariant.
For the known scalar triplets $(\phi,\chi,\eta)$, there is no available
interaction because of the $\mathcal{L}$-symmetry. We will therefore propose
new SU(3)L anti-sextets instead coupling to $\bar{\psi}^{c}_{L}\psi_{L}$
responsible for the neutrino masses. To obtain a realistic neutrino spectrum,
the antisextets transform as follows222 Note that in the model under
consideration, if the choice is a $SU(3)_{L}$ triplet [for example
$\rho\sim(3,2/3,-4/3,\underline{3}^{*})$] instead of $SU(3)_{L}$ anti-sextet
$\sigma$ as in (72), there will be a contribution from term of
$(\bar{\psi}^{c}_{L}\psi_{L})_{\underline{3}^{*}}\rho$ added to the elements
$(M_{D})_{11},(M_{D})_{22},(M_{D})_{33}$ of the matrix $M_{D}$ which is the
same order. The lepton mixing matrix therefore can only reach to $U_{HPS}$ but
not $\theta_{13}\neq 0$

$\sigma_{i}=\left(\begin{array}[]{ccc}\sigma^{0}_{11}&\sigma^{+}_{12}&\sigma^{0}_{13}\\\
\sigma^{+}_{12}&\sigma^{++}_{22}&\sigma^{+}_{23}\\\
\sigma^{0}_{13}&\sigma^{+}_{23}&\sigma^{0}_{33}\\\
\end{array}\right)_{i}\sim[6^{*},2/3,-4/3,\underline{3}^{*}]\hskip
14.22636pt(i=1,2,3),$ (72)

Following the potential minimization conditions, we have the followings
alignments:

  * (1)

The first alignment:
$\langle\sigma_{1}\rangle=\langle\sigma_{2}\rangle=\langle\sigma_{3}\rangle$
then $T_{7}$ is broken into $Z_{3}$ consisting of the elements {$e,b,b^{2}$}.

  * (2)

The second alignment:
$\langle\sigma_{1}\rangle\neq\langle\sigma_{2}\rangle\neq\langle\sigma_{3}\rangle$
or
$\langle\sigma_{1}\rangle\neq\langle\sigma_{2}\rangle=\langle\sigma_{3}\rangle$
or
$\langle\sigma_{2}\rangle\neq\langle\sigma_{1}\rangle\neq\langle\sigma_{3}\rangle$
or
$\langle\sigma_{3}\rangle\neq\langle\sigma_{1}\rangle\neq\langle\sigma_{2}\rangle$
then $T_{7}$ is broken into $\\{\mathrm{Identity}\\}$.

  * (3)

The third alignment:
$0=\langle\sigma_{1}\rangle\neq\langle\sigma_{2}\rangle=\langle\sigma_{3}\rangle\neq
0$ or
$0=\langle\sigma_{2}\rangle\neq\langle\sigma_{3}\rangle=\langle\sigma_{1}\rangle\neq
0$ or
$0=\langle\sigma_{3}\rangle\neq\langle\sigma_{1}\rangle=\langle\sigma_{2}\rangle\neq
0$ then $T_{7}$ is broken into $\\{\mathrm{Identity}\\}$.

  * (4)

The fourth alignment:
$0=\langle\sigma_{1}\rangle\neq\langle\sigma_{2}\rangle\neq\langle\sigma_{3}\rangle\neq
0$ or
$0=\langle\sigma_{2}\rangle\neq\langle\sigma_{1}\rangle\neq\langle\sigma_{3}\rangle\neq
0$ or
$0=\langle\sigma_{3}\rangle\neq\langle\sigma_{1}\rangle\neq\langle\sigma_{2}\rangle\neq
0$ then $T^{\prime}$ is broken into $\\{\mathrm{Identity}\\}$.

  * (5)

The fifth alignment:
$0=\langle\sigma_{1}\rangle=\langle\sigma_{2}\rangle\neq\langle\sigma_{3}\rangle\neq
0$ or
$0=\langle\sigma_{1}\rangle=\langle\sigma_{3}\rangle\neq\langle\sigma_{2}\rangle\neq
0$ or
$0=\langle\sigma_{2}\rangle=\langle\sigma_{3}\rangle\neq\langle\sigma_{1}\rangle\neq
0$ then $T_{7}$ is broken into $\\{\mathrm{Identity}\\}$.

To obtain a realistic neutrino spectrum, in this work we argue that both the
breakings $T_{7}\rightarrow Z_{3}$ and
$T_{7}\rightarrow\\{\mathrm{identity}\\}$ (Instead of
$Z_{3}\rightarrow\\{\mathrm{identity}\\}$) must be taken place in neutrino
sector. However, the VEVs of $\sigma$ does only one of these tasks. The
$T_{7}\rightarrow Z_{3}$ can be achieved by a $SU(3)_{L}$ anti-sextet $\sigma$
given in (72) with the VEVs is set as
$\langle\sigma\rangle=(\langle\sigma_{1}\rangle,\langle\sigma_{1}\rangle,\langle\sigma_{1}\rangle)$
under $T_{7}$, where

$\displaystyle\langle\sigma_{1}\rangle=\left(\begin{array}[]{ccc}\lambda_{\sigma}&0&v_{\sigma}\\\
0&0&0\\\ v_{\sigma}&0&\Lambda_{\sigma}\\\ \end{array}\right).$ (76)

To achieve the second direction of the breakings
$T_{7}\rightarrow\\{\mathrm{Identity}\\}$ (equivalently to
$Z_{3}\rightarrow\\{\mathrm{Identity}\\}$), we additionally introduce another
$SU(3)_{L}$ anti-sextet Higgs scalar which is either put in $\underline{3}$ or
$\underline{3}^{*}$ under $T_{7}$. This is equivalent to breaking the subgroup
$Z_{3}$ of the first direction into $\\{\mathrm{Identity}\\}$, and it can be
achieved within each case below.

  1. 1.

A new $SU(3)_{L}$ anti-sextet $s$ which is put in the $\underline{3}$ under
$T_{7}$,

$\displaystyle
s_{i}=\left(\begin{array}[]{ccc}s^{0}_{11}&s^{+}_{12}&s^{0}_{13}\\\
s^{+}_{12}&s^{++}_{22}&s^{+}_{23}\\\ s^{0}_{13}&s^{+}_{23}&s^{0}_{33}\\\
\end{array}\right)_{i}\sim[6^{*},2/3,-4/3,\underline{3}],$ (80)

with the VEVs given by $\langle s\rangle=(\langle s_{1}\rangle,0,0)^{T}$,
where

$\displaystyle\langle
s_{1}\rangle=\left(\begin{array}[]{ccc}\lambda_{s}&0&v_{s}\\\ 0&0&0\\\
v_{s}&0&\Lambda_{s}\\\ \end{array}\right).$ (84)

  2. 2.

Another $\mathrm{SU}(3)_{L}$ anti-sextet $\sigma^{\prime}$ is put in the
$\underline{3}^{*}$ under $T_{7}$, with the VEVs chosen by

$\displaystyle\sigma^{\prime}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\sigma^{\prime
0}_{11}&\sigma^{\prime+}_{12}&\sigma^{\prime 0}_{13}\\\
\sigma^{\prime+}_{12}&\sigma^{\prime++}_{22}&\sigma^{\prime+}_{23}\\\
\sigma^{\prime 0}_{13}&\sigma^{\prime+}_{23}&\sigma^{\prime 0}_{33}\\\
\end{array}\right)\sim[6^{*},2/3,-4/3,\underline{3}^{*}],$ (88)
$\displaystyle\langle\sigma^{\prime}_{1}\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\lambda^{\prime}_{\sigma}&0&v^{\prime}_{\sigma}\\\
0&0&0\\\ v^{\prime}_{\sigma}&0&\Lambda^{\prime}_{\sigma}\\\
\end{array}\right),\,\langle\sigma^{\prime}_{2}\rangle=\langle\sigma^{\prime}_{3}\rangle=0.$
(92)

Note that $\sigma^{\prime}$ differs from $\sigma$ only in the VEVs alignment.
Combining both cases, after calculation, we obtain the Yukawa interactions:

$\displaystyle-\mathcal{L}_{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}x(\bar{\psi}^{c}_{L}\sigma)_{{\underline{3}}^{*}}\psi_{L}+y(\bar{\psi}^{c}_{L}s)_{{\underline{3}}^{*}}\psi_{L}+\frac{z}{2}(\bar{\psi}^{c}_{L}\sigma^{\prime})_{{\underline{3}}^{*}}\psi_{L}+H.c.$
(93) $\displaystyle=$
$\displaystyle\frac{1}{2}x(\bar{\psi}^{c}_{1L}\sigma_{2}\psi_{1L}+\bar{\psi}^{c}_{2L}\sigma_{3}\psi_{2L}+\bar{\psi}^{c}_{3L}\sigma_{1}\psi_{3L})$
$\displaystyle+$ $\displaystyle
y(\bar{\psi}^{c}_{2L}s_{3}\psi_{1L}+\bar{\psi}^{c}_{3L}s_{1}\psi_{2L}+\bar{\psi}^{c}_{1L}s_{2}\psi_{3L})$
$\displaystyle+$
$\displaystyle\frac{z}{2}(\bar{\psi}^{c}_{1L}\sigma^{\prime}_{2}\psi_{1L}+\bar{\psi}^{c}_{2L}\sigma^{\prime}_{3}\psi_{2L}+\bar{\psi}^{c}_{3L}\sigma^{\prime}_{1}\psi_{3L})+H.c.$

The mass Lagrangian for the neutrinos is given by

$\displaystyle-\mathcal{L}^{\mathrm{mass}}_{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}x(\lambda_{\sigma}\bar{\nu}^{c}_{1L}\nu_{1L}+v_{\sigma}\bar{\nu}^{c}_{1L}N^{c}_{1R}+v_{\sigma}\bar{N}_{1R}\nu_{1L}+\Lambda_{\sigma}\bar{N}_{1R}N^{c}_{1R})$
(94) $\displaystyle+$
$\displaystyle\frac{1}{2}x(\lambda_{\sigma}\bar{\nu}^{c}_{2L}\nu_{2L}+v_{\sigma}\bar{\nu}^{c}_{2L}N^{c}_{2R}+v_{\sigma}\bar{N}_{2R}\nu_{2L}+\Lambda_{\sigma}\bar{N}_{2R}N^{c}_{2R})$
$\displaystyle+$
$\displaystyle\frac{1}{2}x(\lambda_{\sigma}\bar{\nu}^{c}_{3L}\nu_{3L}+v_{\sigma}\bar{\nu}^{c}_{3L}N^{c}_{3R}+v_{\sigma}\bar{N}_{3R}\nu_{3L}+\Lambda_{\sigma}\bar{N}_{3R}N^{c}_{3R})$
$\displaystyle+$ $\displaystyle
y(\lambda_{s}\bar{\nu}^{c}_{3L}\nu_{2L}+v_{s}\bar{\nu}^{c}_{3L}N^{c}_{2R}+v_{s}\bar{N}_{3R}\nu_{2L}+\Lambda_{s}\bar{N}_{3R}N^{c}_{2R})$
$\displaystyle+$
$\displaystyle\frac{1}{2}z(\lambda^{\prime}_{\sigma}\bar{\nu}^{c}_{3L}\nu_{3L}+v^{\prime}_{\sigma}\bar{\nu}^{c}_{3L}N^{c}_{3R}+v^{\prime}_{\sigma}\bar{N}_{3R}\nu_{3L}+\Lambda^{\prime}_{\sigma}\bar{N}_{3R}N^{c}_{3R})+H.c.$

The neutrino mass Lagrangian can be written in matrix form as follows

$-\mathcal{L}^{\mathrm{mass}}_{\nu}=\frac{1}{2}\bar{\chi}^{c}_{L}M_{\nu}\chi_{L}+h.c.,$
(95)

where

$\displaystyle\chi_{L}$ $\displaystyle\equiv$
$\displaystyle\left(\nu_{L}\hskip 14.22636ptN^{c}_{R}\right)^{T},\hskip
14.22636pt\,\,\,M_{\nu}\equiv\left(\begin{array}[]{cc}M_{L}&M^{T}_{D}\\\
M_{D}&M_{R}\\\ \end{array}\right),$ (98) $\displaystyle\nu_{L}$
$\displaystyle=$
$\displaystyle(\nu_{1L},\nu_{2L},\nu_{3L})^{T},\,\,N_{R}=(N_{1R},N_{2R},N_{3R})^{T},$
(99)

and the mass matrices are then obtained by

$M_{L,R,D}=\left(\begin{array}[]{ccc}a_{L,R,D}&\hskip 14.22636pt0&\hskip
14.22636pt0\\\ 0&\hskip 14.22636pta_{L,R,D}&\hskip 14.22636ptb_{L,R,D}\\\
0&\hskip 14.22636ptb_{L,R,D}&\hskip 14.22636pta_{L,R,D}+c_{L,R,D}\\\
\end{array}\right),$ (100)

with

$\displaystyle a_{L}$ $\displaystyle=$ $\displaystyle\lambda_{\sigma}x,\hskip
14.22636pta_{D}=v_{\sigma}x,\hskip 14.22636pta_{R}=\Lambda_{\sigma}x,$
$\displaystyle b_{L}$ $\displaystyle=$ $\displaystyle\lambda_{s}y,\hskip
14.22636ptb_{D}=v_{s}y,\hskip 14.22636ptb_{R}=\Lambda_{s}y,$ $\displaystyle
c_{L}$ $\displaystyle=$ $\displaystyle\lambda^{\prime}_{\sigma}z,\hskip
14.22636ptc_{D}=v^{\prime}_{\sigma}z,\hskip
14.22636ptc_{R}=\Lambda^{\prime}_{\sigma}z.$ (101)

Three observed neutrinos gain masses via a combination of type I and type II
seesaw mechanisms derived from (95) and (100) as

$M_{\mathrm{eff}}=M_{L}-M_{D}^{T}M_{R}^{-1}M_{D}=\left(\begin{array}[]{ccc}A&0&0\\\
0&B_{1}&C\\\ 0&C&B_{2}\\\ \end{array}\right),$ (102)

where

$\displaystyle A$ $\displaystyle=$ $\displaystyle
a_{L}-\frac{a^{2}_{D}}{a_{R}},$ $\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle
a_{L}-\frac{a_{R}b^{2}_{D}-2a_{D}b_{D}b_{R}+a^{2}_{D}(a_{R}+d_{R})}{a^{2}_{R}-b^{2}_{R}+a_{R}d_{R}}$
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle
B_{1}+d_{L}+\frac{2(b_{D}b_{R}-a_{D}a_{R})d_{D}+(a^{2}_{D}-b^{2}_{D})d_{R}-a_{R}d^{2}_{D}}{a^{2}_{R}-b^{2}_{R}+a_{R}d_{R}},$
$\displaystyle C$ $\displaystyle=$ $\displaystyle
b_{L}-\frac{(a^{2}_{D}+b^{2}_{D})b_{R}-(2a_{D}a_{R}+a_{D}d_{R})b_{D}+(a_{D}b_{R}-a_{R}b_{D})d_{D}}{a^{2}_{R}-b^{2}_{R}+a_{R}d_{R}}.$
(103)

We can diagonalize the mass matrix (102) as follows
$U^{T}_{\nu}M_{\mathrm{eff}}U_{\nu}=\mathrm{diag}(m_{1},m_{2},m_{3})$, with

$\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(B_{1}+B_{2}+\sqrt{(B_{1}+B_{2})^{2}+4C^{2}}\right),$
$\displaystyle m_{2}$ $\displaystyle=$ $\displaystyle A,$ (104) $\displaystyle
m_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(B_{1}+B_{2}-\sqrt{(B_{1}+B_{2})^{2}+4C^{2}}\right),$

and the corresponding neutrino mixing matrix:

$\displaystyle U_{\nu}=\left(\begin{array}[]{ccc}0&1&0\\\
\frac{1}{\sqrt{K^{2}+1}}&0&\frac{K}{\sqrt{K^{2}+1}}\\\
-\frac{K}{\sqrt{K^{2}+1}}&0&\frac{1}{\sqrt{K^{2}+1}}\\\
\end{array}\right).\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&i\\\
\end{array}\right),$ (111)

where

$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{B_{1}-B_{2}-\sqrt{4C^{2}+(B_{1}-B_{2})^{2}}}{2C}.$ (112)

Combining (29) and (111), we get the lepton mixing matrix:

$\displaystyle
U^{\dagger}_{L}U_{\nu}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}\frac{1-K}{\sqrt{K^{2}+1}}&1&\frac{1+K}{\sqrt{K^{2}+1}}\\\
\frac{\omega(1-K\omega)}{\sqrt{K^{2}+1}}&1&\frac{\omega(\omega+K)}{\sqrt{K^{2}+1}}\\\
\frac{\omega(\omega-K)}{\sqrt{K^{2}+1}}&1&\frac{\omega(K\omega+1)}{\sqrt{K^{2}+1}}\\\
\end{array}\right).\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&i\\\
\end{array}\right).$ (119)

It is worth noting that in our model, $K$ given in (112) is an arbitrary
number. Hence in general the lepton mixing matrix given in (119) is different
to $U_{HPS}$ in (4), but similar to the original version of trimaximal mixing
considered in TBM2  which is based on the $\Delta(27)$ group. In the case
where $T_{7}$ is broken in $\mathrm{Identity}$ (Instead of
$Z_{3}\rightarrow\mathrm{Identity}$) only by $s$, i.e, without contribution of
$\sigma^{\prime}$ (or
$\lambda^{\prime}_{\sigma}=v^{\prime}_{\sigma}=\Lambda^{\prime}_{\sigma}=0$),
the lepton mixing matrix (119) being equal to $U_{HPS}$ as given in (4). This
is a good features of $T_{7}$ with tensor product
$\underline{3}\otimes\underline{3}$ given in (160).

In the standard Particle Data Group (PDG) parametrization, the lepton mixing
matrix ($U_{PMNS}$) can be parametrized as

$U_{PMNS}=\begin{pmatrix}c_{12}c_{13}&-s_{12}c_{13}&-s_{13}e^{-i\delta}\\\
s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&-s_{23}c_{13}\\\
s_{12}s_{23}+c_{12}c_{23}s_{13}e^{i\delta}&c_{12}s_{23}+s_{12}c_{23}s_{13}e^{i\delta}&\,\,c_{23}c_{13}\\\
\end{pmatrix}\times P.$ (120)

where $P=\mathrm{diag}(1,e^{i\alpha},e^{i\beta})$, and
$c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$ with $\theta_{12}$,
$\theta_{23}$ and $\theta_{13}$ being the solar angle, atmospheric angle and
the reactor angle respectively. $\delta$ is the Dirac CP violating phase while
$\alpha$ and $\beta$ are the two Majorana CP violating phases.

From the (119) and (120) we rule out $\alpha=0,\beta=\frac{\pi}{2}$, and the
lepton mixing matrix in (119) can be parameterized in three Euler’s angles
$\theta_{ij}$ as follows:

$\displaystyle s_{13}e^{-i\delta}$ $\displaystyle=$
$\displaystyle\frac{-1-K}{\sqrt{3}\sqrt{K^{2}+1}},$ (121) $\displaystyle
t_{12}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{K^{2}+1}}{K-1},$ (122)
$\displaystyle t_{23}$ $\displaystyle=$
$\displaystyle-\frac{\omega+K}{1+K\omega}.$ (123)

Substituting $\omega=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$ into (123) yields:

$\displaystyle K$ $\displaystyle=$ $\displaystyle k_{1}+ik_{2},$
$\displaystyle k_{1}$ $\displaystyle=$
$\displaystyle\frac{t^{2}_{23}-4t_{23}+1}{2(t^{2}_{23}-t_{23}+1)},\hskip
14.22636ptk_{2}=\frac{\sqrt{3}(t^{2}_{23}-1)}{2(t^{2}_{23}-t_{23}+1)}.$ (124)

The expression (124) tells us that $k^{2}_{1}+k^{2}_{2}\equiv|K|^{2}=1$.
Combining (121) and (122) yields:

$\displaystyle e^{-i\delta}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}s_{13}t_{12}}\frac{1+K}{1-K}=\frac{1}{\sqrt{3}s_{13}t_{12}}\left[\frac{1-k_{1}^{2}-k_{2}^{2}}{[(1-k_{1})^{2}+k_{2}^{2}]}+\frac{2k_{2}}{[(1-k_{1})^{2}+k_{2}^{2}]}i\right]$
$\displaystyle=$
$\displaystyle-\frac{i(1-t_{23})}{s_{13}t_{12}(1+t_{23})}\equiv\cos\delta-i\sin\delta$

or

$\displaystyle\cos\delta$ $\displaystyle=$ $\displaystyle 0,\hskip
14.22636pt\sin\delta=\frac{1-t_{23}}{s_{13}t_{12}(t_{23}+1)}.$ (125)

Since $\cos\delta=0$ so that $\sin\delta$ must be equal to $\pm 1$, it is then
$\delta=\frac{\pi}{2}$ or $\delta=\frac{3\pi}{2}$. Thus, our model predicts
the maximal Dirac CP violating phase which is the same as in Refs. TBM2 ;
MaximalCP , and this is one of the most striking prediction of the model under
consideration.

Up to now the precise evaluation of $\theta_{23}$ is still an open problem
while $\theta_{12}$ and $\theta_{13}$ are now very constrained PDG2012 . From
(125), our model can provide constraints on $\theta_{23}$ from $\theta_{12}$
and $\theta_{13}$ which satisfy PDG2012  as follows.

  * (i)

In the case $\delta=\frac{\pi}{2}$, from (125) we have the relation among
three Euler’s angles as follows:

$t_{23}=\frac{1-s_{13}t_{12}}{1+s_{13}t_{12}}.$ (126)

In Fig. 1, we have plotted the values of $t_{23}$ as functions of $s_{13}$ and
$t_{12}$ with $s_{13}\in(0.1585,0.1590)$, $t_{12}\in(0.691,0.692)$ . If
$s_{13}=0.1585\,(\theta_{13}=9.11^{o})$ we have the relation between $t_{23}$
and $t_{12}$ as shown in Fig. 2.

Figure 1: $t_{23}$ as functions of $s_{13}$ and $t_{12}$. (a)
$s_{13}\in(0.1585,0.1590)$, $t_{12}\in(0.691,0.692)$, (b)
$s_{13}\in(0.1585,0.1590)$, $t_{12}\in(1.447,1.448)$. Figure 2: $t_{23}$ as a
function of $t_{12}$ with $s_{13}=0.1585$ and $t_{12}\in(0.691,0.692)$.

For the best fit values of $\theta_{12}$ and $\theta_{13}$ given in PDG2012 ,
$s_{13}=0.1585,\,\,t_{12}=0.691$ we obtain
$t_{23}=0.802576\,(\theta_{23}=38.75^{o})$, and

$\displaystyle K=-0.930528-0.366221i,\hskip 14.22636pt(|K|=1).$ (127)

The lepton mixing matrix in (119) then takes the form:

$U\simeq\left(\begin{array}[]{ccc}0.831597&\hskip 14.22636pt0.57735&\hskip
14.22636pt0.157754\\\ -0.552417&\hskip 14.22636pt0.57735&\hskip
14.22636pt-0.799061\\\ -0.27918&\hskip 14.22636pt0.57735&\hskip
14.22636pt0.641307\\\ \end{array}\right).$ (128)

These results also implies that in the model under consideration, the value of
the Jarlskog invariant $J_{CP}$ which determines the magnitude of CP violation
in neutrino oscillations is determined Jarlskog :

$J_{CP}=\frac{1}{8}\cos\theta_{13}\sin 2\theta_{12}\sin 2\theta_{23}\sin
2\theta_{13}\sin\delta=0.03527.$ (129)

  * (ii)

In the case $\delta=\frac{3\pi}{2}$, we have the relation among three Euler’s
angles as follows:

$t_{23}=\frac{1+s_{13}t_{12}}{1-s_{13}t_{12}}.$ (130)

In Fig. 3, we have plotted the values of $t_{23}$ as a function of $s_{13}$
and $t_{12}$ with $s_{13}\in(0.1585,0.1590)$, $t_{12}\in(0.691,0.692)$.

Figure 3: $t_{23}$ as a function of $s_{13}$ and $t_{12}$ with
$s_{13}\in(0.1585,0.1590)$ and $t_{12}\in(0.691,0.692)$.

For the best fit values of $\theta_{12}$ and $\theta_{13}$ given in PDG2012 ,
$s_{13}=0.1585,\,\,t_{12}=0.691$ we obtain
$t_{23}=1.24599\,(\theta_{23}=51.25^{o})$, and

$K=-0.930527+0.366223i,\hskip 14.22636pt(|K|=1).$ (131)

The lepton mixing matrix in (119) in this case takes the form:

$U\simeq\left(\begin{array}[]{ccc}0.831597&\hskip 14.22636pt0.57735&\hskip
14.22636pt-0.157755\\\ -0.279179&\hskip 14.22636pt0.57735&\hskip
14.22636pt-0.641306\\\ -0.552418&\hskip 14.22636pt0.57735&\hskip
14.22636pt0.799061\\\ \end{array}\right),$ (132)

and the value of the Jarlskog invariant $J_{CP}$ is determined Jarlskog :

$J_{CP}=\frac{1}{8}\cos\theta_{13}\sin 2\theta_{12}\sin 2\theta_{23}\sin
2\theta_{13}\sin\delta\simeq-0.03527.$ (133)

Until now values of neutrino masses (or the absolute neutrino masses) as well
as the mass ordering of neutrinos is unknown. The tritium experiment Wein ;
Lobashev  provides an upper bound on the absolute value of neutrino mass

$\displaystyle m_{i}\leq 2.2\,\mathrm{eV}$ (134)

A more stringent bound was found from the analysis of the latest cosmological
data Tegmark

$m_{i}\leq 0.6\,\mathrm{eV},$ (135)

while arguments from the growth of large-scale structure in the early Universe
yield the upper bound Weiler

$\sum^{3}_{i=1}m_{i}\leq 0.5\,\mathrm{eV}.$ (136)

The neutrino mass spectrum can be the normal mass hierarchy
($|m_{1}|\simeq|m_{2}|<|m_{3}|$), the inverted hierarchy
($|m_{3}|<|m_{1}|\simeq|m_{2}|$) or nearly degenerate
($|m_{1}|\simeq|m_{2}|\simeq|m_{3}|$). The mass ordering of neutrino depends
on the sign of $\Delta m^{2}_{23}$ which is currently unknown. In the case of
3-neutrino mixing, in the model under consideration, the two possible signs of
$\Delta m^{2}_{23}$ correspond to two types of neutrino mass spectrum can be
provided. Combining (104) and the two experimental constraints on squared mass
differences of neutrinos as shown in (5) and the values of $K$ in (127) or in
(131), we have the solutions as shown bellows.

###  5.1 Normal case ($\Delta m^{2}_{23}>0$)

####  5.1.1 The case $\delta=\frac{\pi}{2}$

In this case, combining (112) and the values of $K$ in (127), we obtain

$B_{1}=B_{2}-(1.67146\times 10^{-7}+0.732442i)C.$ (137)

Substituting $B_{1}$ from (137) into (104) and combining with the two
experimental constraints on squared mass differences of neutrinos as shown in
(5), we get the solutions (in [eV]) given in Appendix C.

From (137), (161) and (• ‣ C) we see that $A$ must be a real number in order
to make the light neutrino masses $m_{1,2,3}$ to be real. In general,
$B_{1,2},C$ are complex numbers, $\alpha$ is also a complex number but
$\mathrm{Im}(\alpha)\ll\mathrm{Re}(\alpha)$; $m_{1,2}$ are real numbers,
$m_{3}$ is a complex number with $\mathrm{Im}(m_{3})\ll\mathrm{Re}(m_{3})$ but
as will see below in the regions of the model parameters $m_{3}$ is real
number, too.

The solutions in equations from (161) to (164) have the same absolute values
of $m_{1,2,3}$, the unique difference is the sign of $m_{1,3}$. So, here we
only consider in detail the case in (161) 333The expressions from (161) to
(164) show that $m_{i}\hskip 14.22636pt(i=1,2,3)$ depends only on a parameter
$A=m_{2}$ so we consider $m_{1,3}$ as functions of $m_{2}$. However, to have
an explicit hierarchy on neutrino masses, in the following figures, $m_{2}$
should be included. Using the upper bound on the absolute value of neutrino
mass (135) we can restrict the values of $A$: $A\leq 0.6\,\mathrm{eV}$.
However, in the case in (161), $A\in(0.0087,0.01)\,\mathrm{eV}$ or
$A\in(-0.01,-0.0087)\,\mathrm{eV}$ are good regions of $A$ that can reach the
realistic neutrino mass hierarchy. In this region of $A$, $B_{1,2}$ and $C$
are complex numbers. The real parts and the imaginary part of $B_{1,2}$ and
$C$ as functions of $A$ (or $m_{2}$) are plotted in Figs. 4a and 4b,
respectively.

Figure 4:  (a) The real part of $B_{1,2}$ and $C$ as functions of $m_{2}$, (b)
The imaginary part of $B_{1,2}$ and $C$ as functions of $m_{2}$ in the case of
$\Delta m^{2}_{23}>0$.

In Figs. 5a and 5b, we have plotted the value $m_{1,3}$ as functions of
$m_{2}$ with $m_{2}\in(0.00867,0.05)\,\mathrm{eV}$ and
$m_{2}\in(-0.05,-0.00867)\,\mathrm{eV}$. These figures shown that there exist
allowed regions for values $m_{2}$ (or $A$) where either normal or quasi-
degenerate neutrino masses spectrum achieved. The quasi-degenerate mass
hierarchy obtained when $A$ lies in a region [$0.05\,\mathrm{eV},+\infty$]
($A$ increases but must be small enough because of the scale of $m_{1,2,3}$).
As shown in Figs. 6a and 6b, the normal mass hierarchy will be obtained if $A$
takes the values around $(0.0087,0.01)\,\mathrm{eV}$ or
$(-0.01,-0.0087)\,\mathrm{eV}$. The Figs. 7a and 7b give the sum
$\sum^{3}_{i=1}m_{i}$ and $\sum^{3}_{i=1}|m_{i}|$ with
$m_{2}\in(0.0087,0.05)\,\mathrm{eV}$, respectively.

Figure 5: The $m_{1,3}$ and $|m_{1,3}|$ as functions of $m_{2}$ in the case of
$\Delta m^{2}_{23}>0$. (a) The $m_{1,3}$ as a function of $m_{2}$ with
$m_{2}\in(0.00867,0.05)\,\mathrm{eV}$, (b) The $m_{1,3}$ as a function of
$m_{2}$ with $m_{2}\in(-0.05,-0.00867)\,\mathrm{eV}$; (c) The $|m_{1,3}|$ as a
function of $m_{2}$ with $m_{2}\in(0.00867,0.05)\,\mathrm{eV}$, (d) The
$|m_{1,3}|$ as a function of $m_{2}$ with
$m_{2}\in(-0.05,-0.00867)\,\mathrm{eV}$. Figure 6: The $m_{1,3}$ and
$|m_{1,3}|$ as functions of $m_{2}$ in the case of $\Delta m^{2}_{23}>0$. (a)
The $m_{1,3}$ as a function of $m_{2}$ with
$m_{2}\in(0.00867,0.01)\,\mathrm{eV}$, (b) The $m_{1,3}$ as a function of
$m_{2}$ with $m_{2}\in(-0.01,-0.00867)\,\mathrm{eV}$. Figure 7: (a) The sum
$\sum^{3}_{i=1}m_{i}$ as a function of $A$ with
$A\in(0.00867,0.05)\,\mathrm{eV}$; (b) The sum $\sum^{3}_{i=1}|m_{i}|$ as a
function of $A$ with $A\in(0.00867,0.05)\,\mathrm{eV}$ in the case of $\Delta
m^{2}_{23}>0$.

From the expressions (119) , (131) and (161), it is easily to obtain the
effective mass $\langle m_{ee}\rangle$ governing neutrinoless double beta
decay betdecay1 ; betdecay2 ; betdecay3 ; betdecay4 ; betdecay5 ; betdecay6 ,

$\displaystyle\langle m_{ee}\rangle$ $\displaystyle=$
$\displaystyle\mid\sum^{3}_{i=1}U_{ei}^{2}m_{i}\mid$ (138) $\displaystyle=$
$\displaystyle\frac{A}{3}-(0.333333-8.90616\times
10^{-18}i)\sqrt{4A^{2}-0.0003}$ $\displaystyle+$
$\displaystyle(0.0248863-1.26619\times 10^{-8}i)\sqrt{\Gamma},$
$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle 0.002245+(2-2.6469\times
10^{-23}i)A^{2}-(1.73176+1.22425\times 10^{-7}i)\sqrt{\gamma},$ (139)
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle(1.88579\times
10^{-7}+1.33377i)(2.44973\times 10^{-19}+0.00866025i-Ai)$ (140)
$\displaystyle\times$ $\displaystyle(4.79026\times
10^{-19}+0.0481664i+A)(1.84151\times 10^{-18}-0.0481664i+A)$
$\displaystyle\times$ $\displaystyle(0.00866025-6.69766\times 10^{-26}i+A).$
$m_{\beta}=\sqrt{\sum^{3}_{i=1}|U_{ei}|^{2}m_{i}^{2}}\,,$ (141)

where

$\displaystyle\sum^{3}_{i=1}|U_{ei}|^{2}m_{i}^{2}$ $\displaystyle=$
$\displaystyle 2.13672\times 10^{-6}+1.09955A^{2}$ $\displaystyle-$
$\displaystyle(0.0430971+3.0467\times
10^{-9}i)\sqrt{\gamma^{\prime}}+0.0248863\sqrt{4A^{2}-0.0003}\sqrt{\Gamma^{\prime}},$

with

$\displaystyle\gamma^{\prime}$ $\displaystyle=$ $\displaystyle-2.32077\times
10^{-7}+3.28127\times 10^{-14}i+(0.00299432-4.23359\times 10^{-10}i)A^{2}$
$\displaystyle+$ $\displaystyle(1.33377-1.88579\times 10^{-7}i)A^{4},$
$\displaystyle\Gamma^{\prime}$ $\displaystyle=$ $\displaystyle
0.002245+(2-2.64698\times 10^{-23}i)A^{2}-(1.73176+1.22425\times
10^{-7}i\sqrt{\gamma^{\prime}}).$ (143)

We also notice that in the normal spectrum, $|m_{1}|\approx|m_{2}|<|m_{3}|$,
so $m_{1}$ given in (161) is the lightest neutrino mass. Hence, it is denoted
as $m_{1}\equiv m_{light}$. In Figs. 8a and 8b, we have plotted the value
$|m_{ee}|$, $|m_{\beta}|$ and $|m_{light}|$ as functions of $m_{2}$ with
$m_{2}\in(0.0087,0.05)\,\mathrm{eV}$ and
$m_{2}\in(-0.05,-0.0087)\,\mathrm{eV}$, respectively.

Figure 8: The $|m_{ee}|$, $|m_{\beta}|$ and $|m_{light}|$ as functions of
$m_{2}$ from (161) in the case of $\Delta m^{2}_{23}>0$. (a)
$m_{2}\in(0.00867,0.05)\,\mathrm{eV}$, (b)
$m_{2}\in(-0.05,-0.00867)\,\mathrm{eV}$.

To get explicit values of the model parameters, we assume
$m_{2}=10^{-2}\,\mathrm{eV}$, which is safely small. Then the other neutrino
masses are explicitly given as $m_{1}\simeq-5.298\times
10^{-3}\,\mathrm{eV},\,m_{2}\simeq
10^{-2}\,\mathrm{eV},\,m_{3}\simeq-4.95\times 10^{-2}\,\mathrm{eV}$ and
$|m_{ee}|\simeq 1.09\times 10^{-3}\,\mathrm{eV},\,|m_{\beta}|\simeq
1.178\times 10^{-2}\,\mathrm{eV}$. This solution means a normal neutrino mass
spectrum as mentioned above and consistent with the recent experimental data
PDG2012 ; expbet1 ; expbet2 . It follows that

$\displaystyle C$ $\displaystyle\simeq$ $\displaystyle 0.0237465-8.39362\times
10^{-10}i\simeq 0.0237465\,\mathrm{eV},$ $\displaystyle B_{1}$
$\displaystyle=$ $\displaystyle-0.0270968-0.00869645i,\hskip
14.22636ptB_{2}=-0.0276928+0.0232392i.$ (144)

Furthermore, by assuming that

$\lambda_{s}=-\lambda_{\sigma}=-\lambda_{\sigma^{\prime}}=-1\,\mathrm{eV},\,v_{s}=v_{\sigma}=-v^{\prime}_{\sigma},\,\Lambda_{s}=-\Lambda_{\sigma}=-\Lambda^{\prime}_{\sigma}=-v^{2},$
(145)

we obtain

$\displaystyle A$ $\displaystyle=$ $\displaystyle 2x,\hskip
14.22636ptC=y\left(-2+\frac{4x^{2}}{x^{2}-y^{2}+xz}\right),$ $\displaystyle
B_{1}$ $\displaystyle=$ $\displaystyle
x\left(2+\frac{4y^{2}}{x^{2}-y^{2}+xz}\right),\hskip
14.22636ptB_{2}=2(z-x)+\frac{4x^{3}}{x^{2}-y^{2}+xz}.$ (146)

Combining (144) and (146) yields: $x\simeq 5\times 10^{-3}$,
$y\simeq(-4.52717-7.71265i)\times 10^{-3}$, $z\simeq(-10.4861+5.89481i)\times
10^{-3}$.

Quite similar, the value $m_{1,3}$ and $|m_{1,3}|$ as functions of $m_{2}$
with $m_{2}\in(0.00867,0.5)\,\mathrm{eV}$ and
$m_{2}\in(-0.5,-0.00867)\,\mathrm{eV}$ in the case in (162) was plotted in
Figs. 9a, 9b, 9c and 9d. In this case, if $m_{2}=5\times
10^{-2}\,\mathrm{eV}$, which is safely small, the other neutrino masses are
explicitly given as $m_{1}\simeq-4.925\times
10^{-2}\,\mathrm{eV},\,m_{2}\simeq
10^{-2}\,\mathrm{eV},\,m_{3}\simeq-1,679\times 10^{-1}\,\mathrm{eV}$. It
follows that $C\simeq 0.0637652-2.2539\times 10^{-9}i\simeq
0.0637652\,\mathrm{eV},\,B_{1}\simeq-0.10858-0.0233521i,\,B_{2}\simeq-0.10858+0.0233521i$.
Furthermore, with the assuming (145) we obtain $x\simeq 2.5\times 10^{-2}$,
$y\simeq(-1.244-4.172i)\times 10^{-2}$, $z\simeq(-5.305+1.013i)\times
10^{-2}$.

Figure 9: The $m_{1,3}$ and $|m_{1,3}|$ as functions of $m_{2}$ from (162) in
the case of $\Delta m^{2}_{23}>0$. (a) The $m_{1,3}$ as a function of $m_{2}$
with $m_{2}\in(0.00867,0.5)\,\mathrm{eV}$, (b) The $m_{1,3}$ as a function of
$m_{2}$ with $m_{2}\in(-0.5,-0.00867)\,\mathrm{eV}$; (c) The $|m_{1,3}|$ as a
function of $m_{2}$ with $m_{2}\in(0.00867,0.5)\,\mathrm{eV}$, (d) The
$|m_{1,3}|$ as a function of $m_{2}$ with
$m_{2}\in(-0.5,-0.00867)\,\mathrm{eV}$.

In Figs. 10a, 10b, 10c and 10d we have plotted the value $m_{1,3}$ and
$|m_{1,3}|$ as functions of $m_{2}$ with $m_{2}\in(0.00867,0.01)\,\mathrm{eV}$
and $m_{2}\in(-0.01,-0.00867)\,\mathrm{eV}$.

Figure 10: The $m_{1,3}$ and $|m_{1,3}|$ as functions of $m_{2}$ from (163) in
the case of $\Delta m^{2}_{23}>0$. (a) The $m_{1,3}$ as a function of $m_{2}$
with $m_{2}\in(0.00867,0.01)\,\mathrm{eV}$, (b) The $m_{1,3}$ as a function of
$m_{2}$ with $m_{2}\in(-0.5,-0.00867)\,\mathrm{eV}$; (c) The $|m_{1,3}|$ as a
function of $m_{2}$ with $m_{2}\in(0.00867,0.01)\,\mathrm{eV}$, (d) The
$|m_{1,3}|$ as a function of $m_{2}$ with
$m_{2}\in(-0.01,-0.00867)\,\mathrm{eV}$.

In this case, if we assume $m_{2}=10^{-2}\,\mathrm{eV}$, which is safely
small. Then the other neutrino masses are explicitly given as $m_{1}\simeq
5\times 10^{-5}\,\mathrm{eV},\hskip 14.22636ptm_{3}\simeq-3,92\times
10^{-2}\,\mathrm{eV}$. It follows that $C\simeq 0.0237465-8.39362\times
10^{-10}i\simeq
0.0237465\,\mathrm{eV},\,B_{1}\simeq-0.0170968-0.00869645i,\,B_{2}\simeq-0.0170967+0.00869645i$.
Furthermore, with the assuming (145) we obtain $x\simeq 5\times 10^{-3}$,
$y\simeq(-4.11978-5.98325i)\times 10^{-3}$, $z\simeq(-8.36324+2.45329i)\times
10^{-2}$.

In the case (164), the normal neutrino masses spectrum achieved with $A\in$
(0.00867, 0.05) or $A\in(-0.05,-0.00867)$ as shown in Figs. 11a, 11b, 11c and
11d.

Figure 11:  The $m_{1,3}$ and $|m_{1,3}|$ as functions of $m_{2}$ from (164)
in the case of $\Delta m^{2}_{23}>0$. (a) The $m_{1,3}$ as a function of
$m_{2}$ with $m_{2}\in(0.00867,0.05)\,\mathrm{eV}$, (b) The $m_{1,3}$ as a
function of $m_{2}$ with $m_{2}\in(-0.05,-0.00867)\,\mathrm{eV}$; (c) The
$|m_{1,3}|$ as a function of $m_{2}$ with
$m_{2}\in(0.00867,0.05)\,\mathrm{eV}$, (d) The $|m_{1,3}|$ as a function of
$m_{2}$ with $m_{2}\in(-0.05,-0.00867)\,\mathrm{eV}$.

In this case, if we assume $m_{2}=10^{-2}\,\mathrm{eV}$, which is safely
small. Then the other neutrino masses are explicitly given as $m_{1}\simeq
1.003\times 10^{-2}\,\mathrm{eV},\hskip 14.22636ptm_{3}\simeq 5.83\times
10^{-2}\,\mathrm{eV}$. It follows that $C\simeq 0.0291198-1.02929\times
10^{-9}i\simeq
0.0291198\,\mathrm{eV},\,B_{1}\simeq-0.0320968-0.0106643i,\,B_{2}\simeq-0.0286423+0.0296257i$.
Furthermore, with the assuming (145), we obtain $x\simeq 5\times 10^{-3}$,
$y\simeq(-5.48977-7.44641i)\times 10^{-3}$, $z\simeq(-9.65158+8.47899i)\times
10^{-2}$.

####  5.1.2 The case $\delta=\frac{3\pi}{2}$

In this case, combining (112) and the values of $K$ in (131), we obtain

$B_{1}=B_{2}+(2.01498\times 10^{-7}+0.732446i)C.$ (147)

Substituting $B_{1}$ from (147) into (104) and combining with the two
experimental constraints on squared mass differences of neutrinos as shown in
(5), we obtain four solutions (in [eV]) given in Appendix D.

Similar to the case $\delta=\frac{\pi}{2}$ in subsection 5.1.1, in this case
we also have four solutions in which $m_{1,3}$ have the same absolute values,
the unique difference is the sign of $m_{1,3}$. So, we only consider one
solution in the case (166). In this case, the value $m_{1,3}$ and $|m_{1,3}|$
as functions of $m_{2}$ Figs. 12a, 12b, 12c, 12d.

Figure 12: The $m_{1,3}$ and $|m_{1,3}|$ as functions of $m_{2}$ from (166) in
the case of $\Delta m^{2}_{23}>0$. (a) The $m_{1,3}$ as a function of $m_{2}$
with $m_{2}\in(0.00867,0.05)\,\mathrm{eV}$, (b) The $m_{1,3}$ as a function of
$m_{2}$ with $m_{2}\in(-0.05,-0.00867)\,\mathrm{eV}$; (c) The $|m_{1,3}|$ as a
function of $m_{2}$ with $m_{2}\in(0.00867,0.05)\,\mathrm{eV}$, (d) The
$|m_{1,3}|$ as a function of $m_{2}$ with
$m_{2}\in(-0.05,-0.00867)\,\mathrm{eV}$.

In this case, if we assume $m_{2}=10^{-2}\,\mathrm{eV}$, which is safely
small. Then the other neutrino masses are explicitly given as $m_{1}\simeq
8.997\times 10^{-3}\,\mathrm{eV},\hskip 14.22636ptm_{3}\simeq 5.00\times
10^{-3}\,\mathrm{eV}$. It follows that $C\simeq 2.37465-1.01188\times
10^{-7}i\,\mathrm{eV},\,B_{1}\simeq-2.70967+0.869651i,\,B_{2}\simeq-2.76928-2.32392i$.
Furthermore, with tha assuming (145), we obtain $x\simeq 5\times 10^{-3}$,
$y\simeq(-4.52717+7.71265i)\times 10^{-3}$, $z\simeq(-10.4861-5.8948i)\times
10^{-2}$.

###  5.2 Inverted case ($\Delta m^{2}_{23}<0$)

Similar to the normal case, in this case we also have four solutions in which
$m_{1,3}$ have the same absolute values, the unique difference is the sign of
$m_{1,3}$. So, in the case $K=-0.930528-0.366221i$, we only consider one
solution in the form:

$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle B_{2}-(1.67146\times
10^{-7}-0.732442i)C,$ $\displaystyle C$ $\displaystyle=$ $\displaystyle
0.5\sqrt{\alpha-2\sqrt{\beta}},$ $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-0.5\sqrt{4A^{2}-0.0003}+(8.3573\times 10^{-8}+0.366221i)C$
$\displaystyle-$ $\displaystyle 0.5\sqrt{(3.46353+2.4485\times
10^{-7})C^{2}},$ $\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle-0.5\sqrt{4A^{2}-0.0003},\hskip 14.22636ptm_{2}=A,$ (148)
$\displaystyle m_{3}$ $\displaystyle=$ $\displaystyle-0.5\sqrt{4A^{2}-0.0003}$
$\displaystyle-$ $\displaystyle\sqrt{-0.002395+2.58494\times
10^{-26}i+(2-2.64698\times 10^{-23}i)A^{2}-(1.73176+1.22425\times
10^{-7}i)\sqrt{\beta_{1}}}.$

where

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle-0.00276597+1.95536\times 10^{-10}i+(2.30978-1.63287\times
10^{-7}i)A^{2},$ $\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle(2.32077\times 10^{-7}-3.28127\times
10^{-14}i)-(0.00319439-4.51646\times 10^{-10}i)A^{2}$ (149) $\displaystyle+$
$\displaystyle(1.33377-1.88579\times 10^{-7}i)A^{4}.$

In Figs. 13a and 13b, we have plotted the real and the imaginary part of
$B_{1,2}$ and $C$ in (148) as functions of $m_{2}$ with
$m_{2}\in(0.0482,0.05)\,\mathrm{eV}$, respectively.

Figure 13: (a) The real part of $B_{1,2}$ and $C$ as functions of $m_{2}$, (b)
The imaginary part of $B_{1,2}$ and $C$ as functions of $m_{2}$ in the case of
$\Delta m^{2}_{23}<0$ and $K=-0.930528-0.366221i$.

The $m_{1,3}$ and the absolute value $|m_{1,3}|$ as functions of $m_{2}$ with
$m_{2}\in(0.0482,0.1)\,\mathrm{eV}$ are plotted in Figs. 14a and 14b,
respectively. These figures show that there exist allowed regions for value of
$m_{2}$ (or $A$) where either inverted or quasi-degenerate neutrino mass
hierarchy achieved. The quasi-degenerate mass hierarchy obtained when $A$ lies
in a region [$0.1\,\mathrm{eV},+\infty$] 0r [$-\infty,-0.1\,\mathrm{eV}$]
($|A|$ increases but must be small enough because of the scale of
$m_{1,2,3}$). The inverted mass hierarchy will be obtained if $A$ takes the
values around $(0.0482,0.05)\,\mathrm{eV}$ or $(-0.05,-0.0482)\,\mathrm{eV}$
as shown in Figs. 15a, 15b, 15c, 15d.

Figure 14: (a) The $m_{1,3}$ as functions of $m_{2}$ with
$m_{2}\in(0.0482,0.1)\,\mathrm{eV}$ in the case of $\Delta m^{2}_{23}<0$ and
$K=-0.930528-0.366221i$, (b) The $m_{1,3}$ as functions of $m_{2}$ with
$m_{2}\in(-0.1,-0.0482)\,\mathrm{eV}$ in the case of $\Delta m^{2}_{23}<0$ and
$K=-0.930528-0.366221i$; (c) The $|m_{1,3}|$ as functions of $m_{2}$ with
$m_{2}\in(0.0482,0.1)\,\mathrm{eV}$ in the case of $\Delta m^{2}_{23}<0$ and
$K=-0.930528-0.366221i$, (d) The $|m_{1,3}|$ as functions of $m_{2}$ with
$m_{2}\in(-0.1,-0.0482)\,\mathrm{eV}$ in the case of $\Delta m^{2}_{23}<0$ and
$K=-0.930528-0.366221i$. Figure 15: (a) The $m_{1,3}$ as functions of $m_{2}$
with $m_{2}\in(0.0482,0.05)\,\mathrm{eV}$ in the case of $\Delta m^{2}_{23}<0$
and $K=-0.930528-0.366221i$, (b) The $m_{1,3}$ as functions of $m_{2}$ with
$m_{2}\in(-0.05,-0.0482)\,\mathrm{eV}$ in the case of $\Delta m^{2}_{23}<0$
and $K=-0.930528-0.366221i$; (c) The $|m_{1,3}|$ as functions of $m_{2}$ with
$m_{2}\in(0.0482,0.05)\,\mathrm{eV}$ in the case of $\Delta m^{2}_{23}<0$ and
$K=-0.930528-0.366221i$, (d) The $|m_{1,3}|$ as functions of $m_{2}$ with
$m_{2}\in(-0.05,-0.0482)\,\mathrm{eV}$ in the case of $\Delta m^{2}_{23}<0$
and $K=-0.930528-0.366221i$.

The Figs. 16a and 16b give the sum $\sum^{3}_{i=1}m_{i}$ and
$\sum^{3}_{i=1}|m_{i}|$ with $m_{2}\in(0.0482,0.05)\,\mathrm{eV}$,
respectively.

Figure 16: (a) The sum $\sum^{3}_{i=1}m_{i}$ as a function of $m_{2}$ with
$m_{2}\in(0.0482,0.05)\,\mathrm{eV}$ in the case of $\Delta m^{2}_{23}<0$ and
$K=-0.930528-0.366221i$, (b) The sum $\sum^{3}_{i=1}|m_{i}|$ as a function of
$m_{2}$ with $m_{2}\in(0.0482,0.05)\,\mathrm{eV}$ in the case of $\Delta
m^{2}_{23}<0$ and $K=-0.930528-0.366221i$.

In similarity to the normal case, to get explicit values of the model
parameters, we assume $m_{2}=5\times 10^{-2}\,\mathrm{eV}$, which is safely
small. Then the other neutrino masses are explicitly given as
$m_{1}\simeq-4.925\times 10^{-2}\,\mathrm{eV}$ and $m_{3}\simeq 8.507\times
10^{-2}\,\mathrm{eV}$. It follows that $C\simeq 0.0192514-6.80475\times
10^{-10}i\,\mathrm{eV},\,B_{1}\simeq-0.0671582-0.00705026i\,\mathrm{eV},\,B_{2}\simeq-0.0671582+0.00705026i\,\mathrm{eV}.$
Furthermore, with the assuming (145), we obtain $x\simeq 2.50\times 10^{-2}$,
$y\simeq(-0.365272-3.79824i)\times 10^{-2}$,
$z\simeq(-3.41173-0.126634i)\times 10^{-2}$.

##  6 Remark on breaking, VEVs and rho parameter

To obtain a realistic neutrino spectrum, in this work we argue that both the
breakings $T_{7}\rightarrow Z_{3}$ and
$T_{7}\rightarrow\\{\mathrm{identity}\\}$ must be taken place in neutrino
sector while only the breaking $T_{7}\rightarrow Z_{3}$ is taken place in
charged lepton and quark sectors. The quark masses at the tree-level can be
fitted but then the CKM matrix is diagonal.

A breaking of the lepton parity due to the odd VEVs
$\langle\eta^{0}_{3}\rangle$, $\langle\chi^{0}_{1}\rangle$, or a violation of
$\mathcal{L}$ and/nor $S_{3}$ symmetry in terms of Yukawa interactions will
disturb the tree level matrix resulting in mixing between SM and exotic quarks
and/or possibly providing the desirable quark mixing pattern dlsvS4 ; dlnvS3 ;
vlD4 . To get a realistic pattern of SM quarks mixing, we should add radiative
correction or use the effective six dimensional operators (for details, see
Ref. S3quark ). However, detailed study on this problem is out of the scope of
this work.

Note that $\Lambda_{\sigma},\Lambda_{s},\Lambda_{\sigma^{\prime}}$ are needed
to the same order and not to be so large that can naturally be taken at TeV
scale as the VEV $v_{\chi}$ of $\chi$. This is because $v_{\sigma},v_{s}$ and
$v_{\sigma^{\prime}}$ carry lepton number, simultaneously breaking the lepton
parity which is naturally constrained to be much smaller than the electroweak
scale dlshA4 ; dlsvS4 ; dlnvS3 ; e331v1 ; e331v2 . This is also behind a
theoretical fact that $v_{\chi}$, $\Lambda_{\sigma}$ are scales for the gauge
symmetry breaking in the first stage from
$\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}\rightarrow\mathrm{SU}(2)_{L}\otimes\mathrm{U}(1)_{Y}$
in the original form of 3-3-1 models e331v1 ; e331v2 ; clong . They will
provide masses for the new gauge bosons: $Z^{\prime}$, $X$ and $Y$. Also, the
exotic quarks gain masses from $v_{\chi}$ while the neutral fermions masses
arise from $\Lambda_{\sigma},\Lambda_{s},\Lambda_{\sigma^{\prime}}$. The
second stage of the gauge symmetry breaking from
$\mathrm{SU}(2)_{L}\otimes\mathrm{U}(1)_{Y}\rightarrow\mathrm{U}(1)_{Q}$ is
achieved by the electroweak scale VEVs such as $u,v$ responsible for ordinary
particle masses. In combination with those of type II seesaw as determined, in
our model, the following limit is often taken into account dlshA4 ; dlsvS4 ;
dlnvS3 ; e331v1 ; e331v2 :

$(\mathrm{eV})^{2}\sim\lambda^{2}_{\sigma},\lambda^{2}_{s},\lambda^{2}_{\sigma^{\prime}}\ll
v^{2}_{\sigma},v^{2}_{s},v^{2}_{\sigma^{\prime}}\ll u^{2},v^{2}\ll
v_{\chi}^{2}\sim\Lambda^{2}_{\sigma}\sim\Lambda^{2}_{s}\sim\Lambda^{2}_{\sigma^{\prime}}\sim(\mathrm{TeV})^{2}.$
(150)

Our model contains a lot of $\mathrm{SU}(3)_{L}$ scalar triplets that may
modify the precision electroweak parameter such as $S,T,U$ stu and $\rho$
parameters. The most serious one can result from the tree-level contributions
to the $\rho$ parameter. To see this let us approximate $W,Z$ mass and $\rho$
parameter 444We have used the notation
$s_{W}=\sin\theta_{W},c_{W}=\cos\theta_{W},t_{W}=\tan\theta_{W}$, and the
continuation of the gauge coupling constant $g$ of the $\mathrm{SU}(3)_{L}$ at
the spontaneous symmetry breaking point DongHLT ; DongLongdh ; vlD4 ,
$t=\frac{3\sqrt{2}s_{W}}{\sqrt{3-4s^{2}_{W}}}$ was used.:

$\displaystyle M^{2}_{W}$ $\displaystyle\simeq$
$\displaystyle\frac{g^{2}}{2}v^{2}_{W},\hskip
14.22636ptM^{2}_{Z}\simeq\frac{g^{2}v^{2}_{W}}{2c^{2}_{W}},$ $\displaystyle
M^{2}_{Y}$ $\displaystyle\simeq$
$\displaystyle\frac{g^{2}}{2}\left(6\Lambda^{2}_{\sigma}+2\Lambda^{2}_{s}+2\Lambda^{2}_{\sigma^{\prime}}+v_{\chi}^{2}\right),$
(151)

and

$\displaystyle\hskip 14.22636pt\rho=\frac{M^{2}_{W}}{c^{2}_{W}M^{2}_{Z}}\simeq
1+\frac{\lambda^{2}_{s}}{v^{2}_{W}},$ (152)

where $v^{2}_{W}\simeq(3u^{2}+3v^{2})=(246\,\textrm{GeV})^{2}$ is naturally
given according to (150) with $u\sim v\sim 100\,\mathrm{GeV}$. Since
$\lambda_{s}=6v^{2}_{\sigma}+2v^{2}_{s}+2v^{2}_{\sigma^{\prime}}$ is in eV
scale responsible for the observed neutrino masses, the $\rho$ in (152) is
absolutely close to the unity and in agreement with the data PDG2012 .

The mixings between the charged gauge bosons $W-Y$ and the neutral ones
$Z^{\prime}-W_{4}$ are in the same order since they are proportional to
$\frac{v_{\sigma}}{\Lambda_{\sigma}}$, and in the limit
$\lambda_{s},\lambda_{\sigma},v_{s},v_{\sigma}\rightarrow 0$ these mixing
angles tend to zero. In addition, from (150) and (151), it follows that
$M^{2}_{W}$ is much smaller than $M^{2}_{Y}$.

##  7 Conclusions

In this paper, we have constructed the $T_{7}$ model based on
$\mathrm{SU}(3)_{C}\otimes\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}$ gauge
symmetry responsible for fermion masses and mixing. Neutrinos get masses from
only anti-sextets which are in triplets $\underline{3}$ and
$\underline{3}^{*}$ under $T_{7}$. The flavor mixing patterns and mass
splitting are obtained without perturbation. The number of Higgs multiplets
needed in order to allow the fermions to gain masses are less than those of
$S_{3},S_{4}$ and $D_{4}$ dlnvS3 ; dlsvS4 ; vlD4 . The tribimaximal form
obtained with the breaking $T_{7}\rightarrow Z_{3}$ in charged lepton sector
and both $T_{7}\rightarrow Z_{3}$ and
$Z_{3}\rightarrow\\{\mathrm{Identity}\\}$ must be taken place in neutrino
sector but only apart in breakings $Z_{3}\rightarrow\\{\mathrm{Identity}\\}$
(without contribution of $\sigma^{\prime}$), and the upper bound on neutrino
mass $\sum_{i=1}^{3}m_{i}$ at the level is presented. From the Dirac CP
violation phase we obtain the relation between Euler’s angles which is
consistent with the experimental in PDG 2012\. On the other hand, the
realistic lepton mixing can be obtained if both the direction for breakings
$T_{7}\rightarrow Z_{3}$ and $Z_{3}\rightarrow\\{\mathrm{Identity}\\}$ are
taken place in neutrino sectors. The CKM matrix is the identity matrix at the
tree-level. The Dirac CP violation phase $\delta$ is predicted to either
$\frac{\pi}{2}$ or $\frac{3\pi}{2}$ which is maximal CP violation.

## Acknowledgments

This research is funded by Vietnam National Foundation for Science and
Technology Development (NAFOSTED) under grant number 103.01-2011.63.

##  Appendix A  $T_{7}$ group and Clebsch-Gordan coefficients

The tetrahedral group $A_{4}$ has 12 elements and four equivalence classes
with three inequivalent one-dimensional representations and one three-
dimensional one, which is the smallest group with only a real 3
representation. The Frobenius group $T_{7}$ has 21 elements and five
equivalence classes with three inequivalent one-dimensional representations
and two three-dimensional once, which is the smallest group with a pair of
complex 3 and $\underline{3}^{*}$ representations. It is generated by

$\displaystyle a$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\rho&0&0\\\ 0&\rho^{2}&0\\\
0&0&\rho^{4}\end{array}\right),\hskip
14.22636ptb=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right),$ (159)

where $\rho=\exp(2\pi i/7)$, so that $a^{7}=1$, $b^{3}=1$, and $ab=ba^{4}$.
The character table of $T_{7}$ (with $\xi=-1/2+i\sqrt{7}/2$) is given in table
2.

Table 2: Character table of $T_{7}$ group

class $n$ $h$ $\chi_{1}$ $\chi_{1^{\prime}}$ $\chi_{1^{\prime\prime}}$
$\chi_{3}$ $\chi_{3^{*}}$ $C_{1}$ 1 1 1 1 1 3 3 $C_{2}$ 7 3 1 $\omega$
$\omega^{2}$ 0 0 $C_{3}$ 7 3 1 $\omega^{2}$ $\omega$ 0 0 $C_{4}$ 3 7 1 1 1
$\xi$ $\xi^{*}$ $C_{5}$ 3 7 1 1 1 $\xi^{*}$ $\xi$

Let us put $\underline{3}(1,2,3)$ which means some $\underline{3}$ multiplet
such as $x=(x_{1},x_{2},x_{3})\sim\underline{3}$ or
$y=(y_{1},y_{2},y_{3})\sim\underline{3}$ and so on, and similarly for the
other representations. Moreover, the numbered multiplets such as
$(...,ij,...)$ mean $(...,x_{i}y_{j},...)$ where $x_{i}$ and $y_{j}$ are the
multiplet components of different representations $x$ and $y$, respectively.
In the following the components of representations in l.h.s will be omitted
and should be understood, but they always exist in order in the components of
decompositions in r.h.s. All the group multiplication rules of $T_{7}$ as
given below.

$\displaystyle\underline{1}\otimes\underline{1}$ $\displaystyle=$
$\displaystyle\underline{1}(11),\hskip
14.22636pt\underline{1}\otimes\underline{1}^{\prime}=\underline{1}^{\prime}(11),\hskip
14.22636pt\,\,\,\underline{1}\otimes\underline{1}^{\prime\prime}=\underline{1}^{\prime\prime}(11),$
$\displaystyle\underline{1}^{\prime}\otimes\underline{1}^{\prime\prime}$
$\displaystyle=$ $\displaystyle\underline{1}(11),\hskip
14.22636pt\underline{1}^{\prime}\otimes\underline{1}^{\prime}=\underline{1}^{\prime\prime}(11),\hskip
14.22636pt\underline{1}^{\prime\prime}\otimes\underline{1}^{\prime\prime}=\underline{1}^{\prime}(11),$
$\displaystyle\underline{1}\otimes\underline{3}$ $\displaystyle=$
$\displaystyle\underline{3}(11,12,13),\,\,\underline{1}^{\prime}\otimes\underline{3}=\underline{3}(11,\omega
12,\omega^{2}13),\,\,\underline{1}^{\prime\prime}\otimes\underline{3}=\underline{3}(11,\omega^{2}12,\omega
13),$ $\displaystyle\underline{1}\otimes\underline{3}^{*}$ $\displaystyle=$
$\displaystyle\underline{3}^{*}(11,12,13),\,\,\underline{1}^{\prime}\otimes\underline{3}^{*}=\underline{3}^{*}(11,\omega
12,\omega^{2}13),\,\,\underline{1}^{\prime\prime}\otimes\underline{3}^{*}=\underline{3}^{*}(11,\omega^{2}12,\omega
13),$ $\displaystyle\underline{3}\otimes\underline{3}$ $\displaystyle=$
$\displaystyle\underline{3}(33,11,22)\oplus\underline{3}^{*}(23,31,12)\oplus\underline{3}^{*}(32,13,21),$
(160) $\displaystyle\underline{3}^{*}\otimes\underline{3}^{*}$
$\displaystyle=$
$\displaystyle\underline{3}^{*}(33,11,22)\oplus\underline{3}(23,31,12)\oplus\underline{3}(32,13,21),$
$\displaystyle\underline{3}\otimes\underline{3}^{*}$ $\displaystyle=$
$\displaystyle\underline{1}(11+22+33)\oplus\underline{1}^{\prime}(11+\omega
22+\omega^{2}33)$ $\displaystyle\oplus$
$\displaystyle\underline{1}^{\prime\prime}(11+\omega^{2}22+\omega
33)\oplus\underline{3}(21,32,13)\oplus\underline{3}^{*}(12,23,31).$

Note that $\underline{3}\times\underline{3}\times\underline{3}$ has two
invariants and $\underline{3}\times\underline{3}\times\underline{3}^{*}$ has
one invariant.

##  Appendix B The numbers

In the following we will explicitly point out the lepton number ($L$) and
lepton parity ($P_{l}$) of the model particles (notice that the family indices
are suppressed):

Particles | $L$ | $P_{l}$
---|---|---
$N_{R}$, $u$, $d$, $\phi^{+}_{1}$,$\phi^{\prime+}_{1}$, $\phi^{0}_{2}$,$\phi^{\prime 0}_{2}$, $\eta^{0}_{1}$,$\eta^{\prime 0}_{1}$, $\eta^{-}_{2}$,$\eta^{\prime-}_{2}$ $\chi^{0}_{3}$, $\sigma^{0}_{33}$, $s^{0}_{33}$ | 0 | 1
$\nu_{L}$, $l$, $U$, $D^{*}$, $\phi^{+}_{3}$,$\phi^{\prime+}_{3}$, $\eta^{0}_{3}$, $\eta^{\prime 0}_{3}$, $\chi^{0*}_{1}$, $\chi^{+}_{2}$, $\sigma^{0}_{13}$, $\sigma^{+}_{23}$, $s^{0}_{13}$, $s^{+}_{23}$ | $-1$ | $-1$
$\sigma^{0}_{11}$, $\sigma^{+}_{12}$, $\sigma^{++}_{22}$, $s^{0}_{11}$, $s^{+}_{12}$, $s^{++}_{22}$ | $-2$ | 1

##  Appendix C The solutions with $\delta=\frac{\pi}{2}$ in the normal case

  * •

The first case:

$\displaystyle C$ $\displaystyle=$ $\displaystyle
0.5\sqrt{\alpha-2\sqrt{\beta}},$ $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-0.5\sqrt{4A^{2}-0.0003}+(8.3573\times 10^{-8}+0.366221i)C$
$\displaystyle-$ $\displaystyle 0.5\sqrt{(3.46353+2.4485\times
10^{-7})C^{2}},$ $\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle-0.5\sqrt{4A^{2}-0.0003},\hskip 14.22636ptm_{2}=A,$ (161)
$\displaystyle m_{3}$ $\displaystyle=$ $\displaystyle-0.5\sqrt{4A^{2}-0.0003}$
$\displaystyle-$ $\displaystyle\sqrt{0.002245+(2-2.64698\times
10^{-23}i)A^{2}-(1.73176+1.22425\times 10^{-7}i)\sqrt{\beta}}.$

  * •

The second case:

$\displaystyle C$ $\displaystyle=$ $\displaystyle
0.5\sqrt{\alpha+2\sqrt{\beta}},$ $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-0.5\sqrt{4A^{2}-0.0003}+(8.3573\times 10^{-8}+0.366221i)C$
$\displaystyle-$ $\displaystyle 0.5\sqrt{(3.46353+2.4485\times
10^{-7})C^{2}},$ $\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle-0.5\sqrt{4A^{2}-0.0003},\hskip 14.22636ptm_{2}=A,$ (162)
$\displaystyle m_{3}$ $\displaystyle=$ $\displaystyle-0.5\sqrt{4A^{2}-0.0003}$
$\displaystyle-$ $\displaystyle\sqrt{0.002245+(2-2.64698\times
10^{-23}i)A^{2}+(1.73176+1.22425\times 10^{-7}i)\sqrt{\beta}},$

  * •

The third case:

$\displaystyle C$ $\displaystyle=$
$\displaystyle-0.5\sqrt{\alpha-2\sqrt{\beta}},$ $\displaystyle B_{2}$
$\displaystyle=$ $\displaystyle 0.5\sqrt{4A^{2}-0.0003}+(8.3573\times
10^{-8}+0.366221i)C$ $\displaystyle-$ $\displaystyle
0.5\sqrt{(3.46353+2.4485\times 10^{-7})C^{2}},$ $\displaystyle m_{1}$
$\displaystyle=$ $\displaystyle 0.5\sqrt{4A^{2}-0.0003},\hskip
14.22636ptm_{2}=A,$ (163) $\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle 0.5\sqrt{4A^{2}-0.0003}$ $\displaystyle-$
$\displaystyle\sqrt{0.002245+(2-2.64698\times
10^{-23}i)A^{2}+(1.73176+1.22425\times 10^{-7}i)\sqrt{\beta}},$

  * •

The fourth case:

$\displaystyle C$ $\displaystyle=$ $\displaystyle
0.5\sqrt{\alpha+2\sqrt{\beta}},$ $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle 0.5\sqrt{4A^{2}-0.0003}+(8.3573\times 10^{-8}+0.366221i)C$
$\displaystyle-$ $\displaystyle 0.5\sqrt{(3.46353+2.4485\times
10^{-7})C^{2}},$ $\displaystyle m_{1}$ $\displaystyle=$ $\displaystyle
0.5\sqrt{4A^{2}-0.0003},\hskip 14.22636ptm_{2}=A,$ (164) $\displaystyle m_{3}$
$\displaystyle=$ $\displaystyle 0.5\sqrt{4A^{2}-0.0003}$ $\displaystyle-$
$\displaystyle\sqrt{0.002245+(2-2.64698\times
10^{-23}i)A^{2}+(1.73176+1.22425\times 10^{-7}i)\sqrt{\beta}},$

where

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle(0.00259273-1.8329\times
10^{-10}i)+(2.30978-1.63287\times 10^{-7}i)A^{2},$ $\displaystyle\beta$
$\displaystyle=$ $\displaystyle-2.32077\times 10^{-7}+3.28127\times 10^{-14}i$
$\displaystyle+$ $\displaystyle(0.00299432-4.23359\times
10^{-10}i)A^{2}+(1.33377-1.88579\times 10^{-7}i)A^{4}.$

##  Appendix D The solutions with $\delta=\frac{3\pi}{2}$ in the normal case

  * •

The first case:

$\displaystyle C$ $\displaystyle=$ $\displaystyle
0.5\sqrt{\alpha^{\prime}-2\sqrt{\beta^{\prime}}},$ $\displaystyle B_{2}$
$\displaystyle=$ $\displaystyle-0.5\sqrt{4A^{2}-0.0003}+(1.00749\times
10^{-7}+0.366223i)C$ $\displaystyle-$ $\displaystyle
0.5\sqrt{(3.46352+2.95173\times 10^{-7})C^{2}},$ $\displaystyle m_{1}$
$\displaystyle=$ $\displaystyle-0.5\sqrt{4A^{2}-0.0003},\hskip
14.22636ptm_{2}=A,$ (166) $\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle-0.5\sqrt{4A^{2}-0.0003}$ $\displaystyle-$
$\displaystyle\sqrt{0.002245+2.58494\times
10^{-26}i+2A^{2}-(1.73176+1.47587\times 10^{-7}i)\sqrt{\beta^{\prime}}}.$

  * •

The second case:

$\displaystyle C$ $\displaystyle=$ $\displaystyle
0.5\sqrt{\alpha+2\sqrt{\beta}},$ $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-0.5\sqrt{4A^{2}-0.0003}+(1.00749\times 10^{-7}+0.366223i)C$
$\displaystyle-$ $\displaystyle 0.5\sqrt{(3.46352+2.95173\times
10^{-7})C^{2}},$ $\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle-0.5\sqrt{4A^{2}-0.0003},\hskip 14.22636ptm_{2}=A,$ (167)
$\displaystyle m_{3}$ $\displaystyle=$ $\displaystyle-0.5\sqrt{4A^{2}-0.0003}$
$\displaystyle-$ $\displaystyle\sqrt{0.002245+2.58494\times
10^{-26}i+2A^{2}+(1.73176+1.47587\times 10^{-7}i)\sqrt{\beta^{\prime}}}.$

  * •

The third case:

$\displaystyle C$ $\displaystyle=$ $\displaystyle
0.5\sqrt{\alpha^{\prime}-2\sqrt{\beta^{\prime}}},$ $\displaystyle B_{2}$
$\displaystyle=$ $\displaystyle 0.5\sqrt{4A^{2}-0.0003}-(1.00749\times
10^{-7}+0.366223i)C$ $\displaystyle-$ $\displaystyle
0.5\sqrt{(3.46352+2.95173\times 10^{-7})C^{2}},$ $\displaystyle m_{1}$
$\displaystyle=$ $\displaystyle 0.5\sqrt{4A^{2}-0.0003},\hskip
14.22636ptm_{2}=A,$ (168) $\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle 0.5\sqrt{4A^{2}-0.0003}$ $\displaystyle-$
$\displaystyle\sqrt{0.002245+2.58494\times
10^{-26}i+2A^{2}-(1.73176+1.47587\times 10^{-7}i)\sqrt{\beta^{\prime}}},$

  * •

The fourth case:

$\displaystyle C$ $\displaystyle=$ $\displaystyle
0.5\sqrt{\alpha^{\prime}+2\sqrt{\beta^{\prime}}},$ $\displaystyle B_{2}$
$\displaystyle=$ $\displaystyle 0.5\sqrt{4A^{2}-0.0003}-(1.00749\times
10^{-7}+0.366223i)C$ $\displaystyle-$ $\displaystyle
0.5\sqrt{(3.46352+2.95173\times 10^{-7})C^{2}},$ $\displaystyle m_{1}$
$\displaystyle=$ $\displaystyle 0.5\sqrt{4A^{2}-0.0003},\hskip
14.22636ptm_{2}=A,$ (169) $\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle 0.5\sqrt{4A^{2}-0.0003}$ $\displaystyle-$
$\displaystyle\sqrt{0.002245+(2-2.64698\times
10^{-23}i)A^{2}+(1.73176+1.47587\times 10^{-7}i)\sqrt{\beta^{\prime}}},$

where

$\displaystyle\alpha^{\prime}$ $\displaystyle=$
$\displaystyle(0.00259274-2.20962\times 10^{-10}i)+(2.30979-1.96848\times
10^{-7}i)A^{2},$ $\displaystyle\beta^{\prime}$ $\displaystyle=$
$\displaystyle-2.32078\times 10^{-7}+3.95569\times 10^{-14}i$ $\displaystyle+$
$\displaystyle(0.00299433-5.10375\times 10^{-10}i)A^{2}+(1.33378-2.27338\times
10^{-7}i)A^{4}.$

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