Constraints from spectrum of scalar fields in the 3-3-1 model with CKM mechanism

Communications in Physics, Vol. 29, No. 3SI (2019), pp. 357-369
DOI:10.15625/0868-3166/29/3SI/14281

CONSTRAINTS FROM SPECTRUM OF SCALAR FIELDS IN THE 3-3-1
MODEL WITH CKM MECHANISM
NGOC LONG HOANG1,† , VAN HOP NGUYEN2 , T. NHUAN NGUYEN1
AND HANH PHUONG HOANG3
1 Institute

of Physics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau
Giay, Hanoi, Vietnam
2 Department of Physics, Can Tho University, Vietnam
3 Faculty of Physics, Hanoi Pedagogical University 2, Phuc Yen, Vinh Phuc, Vietnam
† E-mail:



Received 20 August 2019
Accepted for publication 12 October 2019
Published 22 October 2019

Abstract. We explore constraints from positivity of scalar mass spectra in the 3-3-1 model with
CKS mechanism. The conditions for positivity of the diagonal elements are most important since
other constraints are followed from the first ones.
Keywords: extensions of electroweak gauge sector, extensions of electroweak Higgs sector, electroweak radiative corrections.
Classification numbers: 12.60.Cn; 12.60.Fr; 12.15.Lk.
I. INTRODUCTION
It is well known that the Higgs mechanism plays a very important role for production of
particle masses. In general, the Higgs potential has to be bounded from below to ensure its stability
[1]. In the Standard Model (SM) it is enough to have a positive Higgs boson quartic coupling
λ > 0. In the extended models with more scalar fields, the potential should be bounded from

below in all directions in the field space as the field strength approaches infinity. It is interesting
to note that the square scalar mass matrix is associated with the Hessian matrix Hi j determined at
the vacuum
∂ 2V
(H0 )i j =
.
(1)
∂ φi ∂ φ j φ =min
c 2019 Vietnam Academy of Science and Technology


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CONSTRAINTS FROM SPECTRUM OF SCALAR FIELDS IN THE 3-3-1 MODEL WITH CKM MECHANISM

The condition for the potential to be bounded from below also leads to positivity of the above
matrix [2]. The mentioned condition practically is the positivity of the principal minors. In this
paper we focus our attention on positivity of scalar mass spectra and intend to get the constraints
from it.
Let us remind the useful definition. A symmetric matrix M 2 of quadric form xT M 2 x for all
vector x in Rn with the following properties
xT M 2 x ≥ 0,
xT M 2 x > 0,

M2
M2

is called positive semidefinite,
is called positive definite .

(2)

If M 2 is 2 × 2 matrix with elements being Mi2j , i, j = 1, 2 then Eq.(2) leads to the following
conditions
2
> 0,
M11
2
+
M12

2
M22
> 0,

(3)

2 M2 > 0 .
M11
22

(4)

For 3 × 3 matrix we have [1]
2
M11
> 0,

2
M22

> 0,

2
M33
> 0,

(5)

2
M12
+

2 M2 > 0 ,
M11
22

(6)

2
+
M13

2 M2 > 0 ,
M11
33

(7)

2
M23

+

2 M2 > 0 ,
M22
33

(8)

and
2 M2 M2 + M2
M11
12
22 33

2 + M2
M33
13

2 + M2
M22
23

2
M11
> 0,

(9)

2
2

2
4
2
4
2
4
2
2
2
2
detM 2 = M11
M22
M33
− (M12
M33
+ M13
M22
+ M23
M11
) + 2M12
M13
M23
> 0.

(10)

For the matrices of rank 4 or 5. the reader is referred to Refs. [3, 4].
One of the main purposes of the models based on the gauge group SU(3)C × SU(3)L ×
U(1)X (for short, 3-3-1 model) [5, 6] is concerned with the search of an explanation for the number of generations of fermions. Combined with the QCD asymptotic freedom, the 3-3-1 models
provide an explanation for the number of fermion generations. To provide an explanation for the

observed pattern of SM fermion masses and mixings, various 3-3-1 models with flavor symmetries [7–9]and radiative seesaw mechanisms [7,12] have been proposed in the literature. However,
some of them involve non-renormalizable interactions [10], others are renormalizable but do not
address the observed pattern of fermion masses and mixings due to the unexplained huge hierarchy among the Yukawa couplings [8] and others are only focused either in the quark mass
hierarchy [8, 11], or in the study of the neutrino sector [12, 13], or only include the description of
SM fermion mass hierarchy, without addressing the mixings in the fermion sector [14].
It is interesting to find an alternative explanation for the observed SM fermion mass and
mixing pattern. The first renormalizable extension of the 3-3-1 model with β = − √13 , which explains the SM fermion mass hierarchy by a sequential loop suppression has been done in Ref. [15].
This model is called by the 3-3-1 model with Carcamo-Kovalenko-Schmidt (CKS) mechanism.


NGOC LONG HOANG et al.

359

The aim of this paper is to apply the procedure in (2) for the recently proposed 3-3-1 model with
CKS mechanism.
The further content of this paper is as follows. In Sect. II, we briefly present particle content
of scalar sector and spontaneous symmetry breaking (SSB) of the model. The Higgs sector is
considered in Sect. III. The Higgs sector consists of two parts: the first part contains lepton
number conserving terms and the second one is lepton number violating. We study in details the
first part and show that the Higgs sector has all necessary ingredients. We make conclusions in
Sect. IV.
II. SCALAR FIELDS OF THE MODEL
In the model under consideration, the Higgs sector contains three scalar triplets: χ, η and
ρ and seven singlets ϕ10 , ϕ20 , ξ 0 , φ1+ , φ2+ , φ3+ and φ4+ . Hence, the scalar spectrum of the model is
composed of the following fields
χ + χ ∼ 1, 3, −

χ =
χ

1
,
3

(11)

vχ T
1
, χ = χ10 , χ2− , √ (Rχ 0 − iIχ 0 )
0,0, √
3
3
2
2
T
1
2
ρ1+ , √ (Rρ − iIρ ) , ρ3+
∼ 1, 3,
,
3
2
1
η + η ∼ 1, 3, −
,
3

=

ρ =
η =

T

,

T
T
vη
1
√ ,0,0
, η = √ (Rη 0 − iIη 0 ) , η2− , η30 ,
1
1
2
2
0
0
ϕ1 ∼ (1, 1, 0),
ϕ2 ∼ (1, 1, 0),
+
+
φ1 ∼ (1, 1, 1), φ2 ∼ (1, 1, 1), φ3+ ∼ (1, 1, 1), φ4+ ∼ (1, 1, 1),
vξ
1
ξ 0 = ξ 0 + ξ 0 , ξ 0 = √ , ξ 0 = √ (Rξ 0 − iIξ 0 ) ∼ (1, 1, 0) .
2
2
The Z4 × Z2 assignments of the scalar fields are shown in Table 1.

η

=

(12)

Table 1. Scalar assignments under Z4 × Z2

χ

η

ρ

ϕ10

ϕ20

φ1+

φ2+

φ3+

φ4+

ξ0

Z4

1

1

−1

−1

i

i

−1

−1

1

1

Z2

−1

−1

1

1

1

1

1

−1

−1

1

The fields with nonzero lepton number are presented in Table 2. Note that the three gauge
singlet neutral leptons NiR as well as the elements in the third component of the lepton triplets,
c have lepton number equal to −1.
namely νiL


360

CONSTRAINTS FROM SPECTRUM OF SCALAR FIELDS IN THE 3-3-1 MODEL WITH CKM MECHANISM

Table 2. Nonzero lepton number L of fields

L

TL,R

J1L,R

J2L,R

c
νiL

eiL,R

EiL,R

NiR

ΨR

χ10

χ2+

η30

ρ3+

φ2+

φ3+

φ4+

ξ0

−2

2

2

−1

1

1

−1

1

2

2

−2

−2

−2

−2

−2

−2

i = 1, 2, 3

III. THE SCALAR POTENTIAL
The renormalizable potential contains three parts [16]. The first part is given by
4

2

VLNC = µχ2 χ † χ + µρ2 ρ † ρ + µη2 η † η + ∑ µφ2+ φi+ φi− + ∑ µϕ2i ϕi0 ϕi0∗ + µξ2 ξ 0∗ ξ 0
†

†

i

i=1
†

†

i=1
†

†

+ χ χ(λ13 χ χ + λ18 ρ ρ + λ5 η η) + ρ ρ(λ14 ρ ρ + λ6 η † η) + λ17 (η † η)2
+ λ7 (χ † ρ)(ρ † χ) + λ8 (χ † η)(η † χ) + λ9 (ρ † η)(η † ρ)
4

+ χ†χ

2

∑ λi φi+ φi− + ∑ λi ϕi0 ϕi0∗ + λχξ ξ 0∗ ξ 0
χφ

χϕ

i=1

i=1

4

+ ρ †ρ

2

ρφ

∑ λi

i=1

φi+ φi− + ∑ λi ϕi0 ϕi0∗ + λρξ ξ 0∗ ξ 0
i=1

4

+ η †η

2

ηφ

∑ λi

i=1
4

+

+

φi+ φi− + ∑ λi ϕi0 ϕi0∗ + ληξ ξ 0∗ ξ 0
ηϕ

i=1

4

∑ φi+ φi−

2

∑ λi j φ j+ φ j− + ∑ λi j ϕ 0j ϕ 0∗j + λi ξ 0∗ ξ 0
φφ

i=1

j=1

2

2

φξ

φϕ

j=1

∑ ϕi0 ϕi0∗ ∑ λi j

ϕϕ

i=1

+

ρϕ

0∗ 0
+ λξ (ξ 0∗ ξ 0 )2
ϕ 0j ϕ 0∗
j + λi ξ ξ
ϕξ

j=1

2
λ10 φ2+

+ w1 ϕ20

2

φ3−

2

+ λ11 φ2+

2

φ4−

2

2

+ λ12 φ3+

ϕ10 + w2 χ † ρφ3− + w3 η † χξ 0 + w4 ϕ20

2

φ4−

2

ϕ10∗ + w5 φ3+ φ4− ϕ10 + w6 φ3+ φ4− ϕ10∗

+ χρη(λ1 ϕ10 + λ2 ϕ10∗ ) + χ † ρφ4− λ15 ϕ10 + λ16 ϕ10∗ + λ3 η † ρφ3− ξ 0 + λ4 φ1+ φ2− ϕ20 ξ 0
+
+
+

λ19 φ3− φ4+ + λ20 φ3+ φ4−

ϕ20

2

+ λ21 ϕ10

3

ϕ10∗

4

2

i=1

i=1

λ22 χ † χ + λ23 ρ † ρ + λ24 η † η + ∑ λ61i φi+ φi− + ∑ λ62i ϕi0 ϕi0∗
0∗ 0

λ25 ξ ξ

(ϕ10 )2 + h.c.

(13)

The second part is a lepton number violating one (the subgroup U(1)Lg is violated) and the third
breaking softly Z4 × Z2 are given in Ref. [16].


NGOC LONG HOANG et al.

361

Expanding the Higgs potential around VEVs, ones get the constraint conditions at the tree
levels as follows
w3 = 0 ,

(14)

1
1
−µχ2 = v2χ λ13 + v2η λ5 + λχξ v2ξ ,
2
2
1
1

−µη2 = v2η λ17 + v2χ λ5 + ληξ v2ξ ,
2
2
1
1
−µξ2 =
λ v2 + λ v2 .
2 χξ χ 2 ηξ η

(15)

Applying the constraint conditions in (14), the charged scalar sector contains two massless fields:
η2+ and χ2+ which are Goldstone bosons eaten by the W + and Y + gauge bosons, respectively. The
other massive fields are φ1+ , φ2+ and φ4+ with respective masses
m2φ +
1

m2φ +
2

m2φ +
4

1 2 χφ
ηφ
φξ
v λ + v2η λ1 + v2ξ λ1 ,
1
2 χ 1
1

χφ
ηφ
φξ
= µφ2+ + v2χ λ2 + v2η λ2 + v2ξ λ2 ,
2
2
1
χφ
ηφ
φξ
= µφ2+ + v2χ λ4 + v2η λ4 + v2ξ λ4 .
4
2

= µφ2+ +

(16)

In addition, in the basis (ρ1+ , ρ3+ , φ3+ ), there is the mass mixing matrix


1
0
A + 21 v2η (λ6 +λ9 )
2 vη vξ λ3

2
√1 vχ w2 
0
A + 21 v2χ λ7 + v2η λ6

Mcharged
= 
,
2
1
1
2
√
v w
µφ + + B3
2 vη vξ λ3
2 χ 2

(17)

3

where we have used the following notations
A ≡ µρ2 +
Bi ≡

1 2
v λ18 + λρξ v2ξ ,
2 χ

1 2 χφ
ηφ
φξ
v λ + v2η λi + v2ξ λi
, i = 1, 2, 3, 4 .

2 χ i

(18)

The conditions in Eqs. (4 - 7) yield
1 2
1
v λ7 + v2η λ6 > 0 , µφ2+ + B3 > 0 ,
A + v2η (λ6 +λ9 ) > 0 , A +
3
2
2 χ
1
A + v2η (λ6 +λ9 )
2

A+

1 2
v λ7 + v2η λ6
2 χ

1
vη vξ λ3 +
2

1
A + v2η (λ6 +λ9 )
2

1
√ vχ w2 +
2

A+

1 2
v λ7 + v2η λ6
2 χ

> 0,

µφ2+ + B3 > 0 ,
3

µφ2+ + B3 > 0 .
3

(19)

(20)


362

CONSTRAINTS FROM SPECTRUM OF SCALAR FIELDS IN THE 3-3-1 MODEL WITH CKM MECHANISM

Note that in this case the constraints in (19) are just enough or other word speaking, if the conditions of semi-definition for diagonal elements are fulfilled then other ones are automatically
satisfied.
Now we turn into CP-odd Higgs sector. There are three massless fields: Iχ , Iη and Iξ 0 . The

field Iϕ2 has the following squared mass
m2Iϕ = µϕ22 + B2 ,

(21)

2

where

1 2 χϕ
ηϕ
v λ + v2η λ2 + v2ξ λnϕξ , n = 1, 2 .
2 χ n
There are other two mass matrices as follows: Firstly, in the basis (Iχ 0 , Iη 0 ), the matrix is
Bn ≡

1

2
mCPodd1
=

v2η

λ8
2

−vχ vη
v2χ

−vχ vη

(22)

3

.

(23)

The matrix in (23) provides two physical states
G1 = cos θa Iχ 0 + sin θa Iη 0 ,
1

3

A1 = − sin θa Iχ 0 + cos θa Iη 0 ,
1

(24)

3

where

vη
.
vχ
The field G1 is massless while the field A1 has mass as follows
tan θa =

(25)

λ8 v2χ
.
2 cos2 θa
Secondly, in the basis (Iϕ1 , Iρ ), the matrix is
m2A1 =

2
mCPodd2
=

µϕ21 −C + B1
1
2 vχ vη (λ1 − λ2 )

(26)

1
2 vχ vη (λ1 − λ2 )
A + λ26 v2η

,

(27)

where we have denoted
C ≡ v2χ λ22 + v2η λ24 + v2ξ λ25

(28)

The conditions in (4) yield
λ6 2
v > 0,
2 η

µϕ21 −C + B1 > 0 ,

A+

1
vχ vη (λ1 − λ2 ) +
2

µϕ21 −C + B1

(29)
A+

λ6 2
v
2 η

The above conditions provide the following constraints:
i) If λ1 < λ2 , then
µϕ21 −C + B1

A+

λ6 2
v
2 η

>

v2χ v2η
(λ1 − λ2 )2 .
4

ii) If λ1 > λ2 , there are only conditions given in (30).

> 0.

(30)


NGOC LONG HOANG et al.

363

Generally, physical states of matrix (27) are
A2
A3

=

cos θρ
− sin θρ

sin θρ
cos θρ

Iϕ1
Iρ

,

(31)

where the mixing angle is given by
tan 2θρ =

vχ vη (λ1 − λ2 )

,

(32)

µϕ21 −C + B1 − A − λ26 v2η

and their squared masses as follows
m2A2

=

1
2

A + D1 −

(A − D1 )2 + v2η 2(A − D1 )λ6 + v2η λ62 + v2χ (λ13 − λ14 )2

,

m2A3

=

1
2

A + D1 +

(A − D1 )2 + v2η 2(A − D1 )λ6 + v2η λ62 + v2χ (λ13 − λ14 )2

, (33)

where

1
(34)
D1 = µϕ21 + B1 −C + v2η λ6 .
2
Next, the CP-even scalar sector is our task. Ones have one massive field, namely Rϕ2 with

mass
m2Rϕ = m2Iϕ
2

2

= µϕ22 + B2

1 2 χϕ
ηϕ
ϕξ
vχ λ2 + v2η λ2 + v2ξ λ2
.
(35)
2
As mentioned in Ref. [15], the lightest scalar ϕ20 is possible DM candidate. Therefore from (35),
the following condition is reasonable
1
χϕ
ϕξ
.
(36)
µϕ22 = − v2χ λ2 + v2ξ λ2
2
In this case, the model contains the complex scalar DM ϕ20 with mass
1
ηϕ
m2Rϕ = m2Iϕ = v2η λ2 .
(37)
2
2
2
Other three mass matrices are
iii) In the basis (Rχ 0 , Rη 0 ), the matrix is

= µϕ22 +

1

3

2
mCPeven1
=

λ8
2

v2η
vχ vη

v χ vη
v2χ

.

(38)

This matrix is completely similar to that in (23). Thus, two physical states are
RG1

= cos θa Rχ 0 + sin θa Rη 0 ,
1

3

H1 = − sin θa Rχ 0 + cos θa Rη 0 ,
1

3

(39)

where RG1 is massless while the field H2 has mass as follows
m2H1

=

m2A1

λ8 v2χ
=
.
2 cos2 θa

(40)


364

CONSTRAINTS FROM SPECTRUM OF SCALAR FIELDS IN THE 3-3-1 MODEL WITH CKM MECHANISM

iV) In the basis (Rρ , Rϕ1 ), the matrix is
2
mCPeven2

=

A + λ26 v2η
− 21 vχ vη (λ1 + λ2 )
µϕ21 +C + B1
− 12 vχ vη (λ1 + λ2 )

.

(41)

As before, ones get
A+

λ6 2
v > 0,
2 η

µϕ21 +C + B1 > 0 ,

1
− vχ vη (λ1 + λ2 ) +
2

A+

λ6 2
v
2 η

(42)
µϕ21 +C + B1 > 0 .

(43)

Thus, if λ1 + λ2 > 0, then
λ6
A + v2η
2

µϕ21

+C + B1

v2χ v2η
>
(λ6 + λ2 )2 .
4

(44)

If λ1 + λ2 ≤ 0, there are only conditions in (42).
The physical states of matrix (41) are
H2
H3

=

cos θr sin θr
− sin θr cos θr

Rρ
Rϕ1

,

(45)

where the mixing angle is given by
tan 2θr =

vχ vη (λ1 + λ2 )

,

(46)

µϕ21 +C + B1 − A − λ26 v2η

and their squared masses are identified by
m2H2

=

1
2

A + D2 −

(A − D2 )2 + v2η 2(A − D2 )λ6 + v2η λ62 + v2χ (λ13 + λ14 )2

,

m2H3

=

1
2

A + D2 +

(A − D2 )2 + v2η 2(A − D2 )λ6 + v2η λ62 + v2χ (λ13 + λ14 )2

,
(47)

where
1
D2 = µϕ21 + B1 +C + v2η λ6 .
2
v) In the basis (Rχ , Rη , Rξ 0 ), the matrix is


2v2χ λ13
vχ vη λ5 λχξ vχ vξ
2
2v2η λ17 ληξ vη vξ  .
mCPeven3
=  vχ vη λ5

λχξ vχ vξ ληξ vη vξ
2λξ v2ξ

(48)

(49)


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365

Again, in this case the constraints in Eqs (4 - 9) are given by
λ13 > 0 , λ17 > 0 , λξ > 0 ,
vχ vη λ5 +

(50)

2v2η λ17 > 0 ⇒ λ5 > −2 (λ13 λ17 ) ,

2v2χ λ13

(51)

λχξ vχ vξ +

2v2χ λ13

2λξ v2ξ > 0 ⇒ λχξ > −2

λ13 λξ ,

(52)

ληξ vη vξ +

2v2η λ17

2λξ v2ξ > 0 ⇒ ληξ > −2

λ17 λξ ,

(53)

2v2χ λ13
+ληξ vη vξ

2λξ v2ξ + vχ vη λ5

2v2η λ17

2v2η λ17

2v2χ λ13 > 0

⇒ 2 λ13 λ17 λξ + λ5
2v2χ λ13

2λξ v2ξ + λχξ vχ vξ

2v2η λ17

λξ + λχξ

λ17 + ληξ

λ13 > 0 ,

(54)

2λξ v2ξ − [(vχ vη λ5 )2 2λξ v2ξ + (λχξ vχ vξ )2 2v2η λ17

+ 2v2χ λ13 (ληξ vη vξ )2 ] + 2vχ vη λ5 .λχξ vχ vξ .ληξ vη vξ > 0
⇒ 4λ13 λ17 λξ − [(λ5 )2 λξ + (λχξ )2 λ17 + (ληξ )2 λ13 ] + λ5 .λχξ .ληξ > 0 .

(55)

III.1. Special cases
To find solutions in Higgs sector, we should make some simplifications.
III.1.1. The SM-like Higgs boson
We consider now the matrix (49): with the basis (Rχ , Rη , Rξ 0 )


2v2χ λ13
vχ vη λ5 λχξ vχ vξ
2
2v2η λ17 ληξ vη vξ  .
mCPeven3
=  vχ vη λ5
λχξ vχ vξ ληξ vη vξ

2λξ v2ξ

(56)

Let us assume a simplified worth to be considered scenario which is characterized by the
following relations:
λ5 = λ13 = λ17 = λξ = λχξ = ληξ = λ ,

vξ = v χ .

The system of Eqs.(48 - 53) leads to another constraint, namely
√
λ > 0.
In this scenario, the squared matrix (49)
(Rη , Rχ , Rξ 0 ) takes the simple form:
 2
2x
2

x
mCPeven3 = λ
x

(57)

(58)

for the electrically neutral CP even scalars in the basis

x x

2 1  v2χ ,
1 2

x=

vη
.
vχ

(59)


366

CONSTRAINTS FROM SPECTRUM OF SCALAR FIELDS IN THE 3-3-1 MODEL WITH CKM MECHANISM

2
In this scenario, we find the that the physical scalars included in the matrix mCPeven3
are:


2
x
x




−1 + x9
3

3
Rη
h


1 
 H4  
0
− 12
Rχ  ,
2 
 √
Rξ 0
H5
2
1
1
3 x
2
2

(60)

where h is the 126 GeV SM like Higgs boson. Thus, we find that the SM-like Higgs boson h has
couplings very close to SM expectation with small deviations of the order of

v2η
.
v2χ

In addition, the

2
squared masses of the physical scalars included in the matrix mCPeven3
take the form:

4 2
λv ,
m2H4 λ v2χ ,
m2H5 3λ v2χ .
(61)
3 η
Taking into account the fact that mass of the SM Higgs boson is equal to 126 GeV, from (61) we
obtain
λ ≈ 0.187 .
(62)
Combining with the limit from the rho parameter in Ref. [16]
m2h

3.57 TeV ≤ vχ ≤ 6.9 TeV
yields
1.5 TeV ≤ mH4 ≤ 2.61 TeV ,
2.6 TeV ≤ mH5 ≤ 4.5 TeV .

(63)

III.1.2. The charged Higgs bosons
The charged scalar sector contains two massless fields: GW + and GY + which are Goldstone
bosons eaten by the longitudinal components of the W + and Y + gauge bosons, respectively. The
other massive fields are φ1+ , φ2+ and φ4+ with respective masses given in (18).

In the basis (ρ1+ , ρ3+ , φ3+ ), the squared mass matrix is given in (17). Let us make effort to
simplify this matrix. Note that µχ2 , µη2 , and µξ2 can be derived using relations (14) and (57). In
addition, it is reasonable to assume
v2χ χφ
v2χ
φξ
(λ18 + λρξ ) ≈ µη2 , µφ2+ = − (λ2 + λ2 ) ,
3
2
2
we obtain the simple form of the squared mass matrix of the charged Higgs bosons,


λ3
0
v
v
A + 21 v2η (λ6 +λ9 )
η
χ
2


1
1
2 λ + λ v2
2
.
√
0

v
v
w
Mchargeds
=
7
χ
2
6
χ
η
2
2


ηφ
λ3
1 2
√1 vχ w2
v
v
v
λ
η
χ
η
2
2
2
2

µρ2 = −

(64)

(65)

+
The matrix (65) predicts that there may exist two light charged Higgs bosons H1,2
with masses at
+
+
the electroweak scale and the mass of H3 which is mainly composed of ρ3 is around 3.5 TeV. In
addition, the Higgs boson H1+ almost does not carry lepton number, whereas the others two do.
Generally, the Higgs potential always contains mass terms which mix VEVs. However,
these terms must be small enough to avoid high order divergences (for examples, see Refs. [17,18])
and provide baryon asymmetry of Universe by the strong electroweak phase transition (EWPT).


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367

Ignoring the mixing terms containing λ3 in (65) does not affect other physical aspects, since
the above mentioned terms just increase or decrease small amount of the charged Higgs bosons.
Therefore, without lose of generality, neglecting the terms with λ3 satisfies other aims such as
EWPT.
Hence, in the matrix of (65), the coefficient λ3 is reasonably assumed to be zero. Therefore
we get immediately one physical field ρ1+ with mass given by
1
m2ρ + = v2η (λ6 + λ9 ) .

1
2

(66)

The other fields mix by submatrix given at the bottom of (65). The limit ρ1+ = H1+ when λ3 = 0 is
very interesting for discussion of the Higgs contribution to the ρ parameter.
Analysis of electroweak phase transition shows that the term of VEV mixing at the top-right
corner should be negligible [17, 18] or
λ3

0.

(67)

Therefore, from (17), it follows that ρ1+ is physical field with mass
1
m2ρ + = A + v2η (λ6 + λ9 ) ,
1
2

(68)

and two massive bilepton scalars ρ3+ and φ3+ mix each other by matrix at the right-bottom corner.
Taking into account the conditions in (4) yields
A+

1 2
v λ7 + v2η λ6 > 0 , µφ2+ + B3 > 0 ,
3

2 χ

1
√ vχ w2 +
2

A+

1 2
v λ7 + v2η λ6
2 χ

(69)

µφ2+ + B3 > 0 .

(70)

3

From (70) it follows that if w2 < 0, then
A+

1 2
v λ7 + v2η λ6
2 χ

µφ2+ + B3 >
3

v2χ w22
,
2

but if w2 > 0, there are only conditions in (69).
It is worth mentioning that the masses of three charged scalars φi+ , i = 1, 2, 4 are still not
fixed.
Let us deal with the charged Higgs boson sector by assuming
χφ

ηφ

φξ

λ6 = λ7 = λ9 = λ18 = λ3 = λ3 = λ3 = λ .

(71)

With this assumption, we have
3
λ
µχ2 = − (3v2χ + v2η ) − λ v2χ ,
2
2
2
2
2
µη = −λ (vη + vχ ) −λ v2χ ,
λ
µξ2 = − (v2χ + v2η )

2

1
− λ v2χ .
2

(72)


368

CONSTRAINTS FROM SPECTRUM OF SCALAR FIELDS IN THE 3-3-1 MODEL WITH CKM MECHANISM

In the basis (ρ1+ , ρ3+ , φ3+ ), the matrix in (17) becomes

µρ2 + λ v2χ + λ v2η
0

λ
2
2
0
µρ + 2 3v2χ + v2η
Mcharged = 

λ
√1 vχ w2
2 vη vχ
2



λ
2 vη v χ

1
√ vχ w2

2

µφ2+ + λχ2 + 21 v2η )
3

.

(73)

Next, assuming
µρ2 = µφ2+ = µη2 = −λ v2χ ,

(74)

3

we obtain


2
Mnewcharged

λ2
 0
= 
λ
2x

0
λ
(1
+ x2 )
2
√1 w2
2



λ
2x
1
√ w 2  v2 .
 χ
2
λ 2
2x

(75)

From (75), we get two charged Higgs bosons with masses at electoweak scale and one
massive with mass around TeV (∝ vχ ), in addition the Higgs boson composed mainly from ρ1+
does not carry lepton number, while the two others do.

IV. CONCLUSION
In this paper, we have applied the positivity of scalar mass spectra in the 3-3-1 model with
CKS mechanism. We show that for the Higgs squared mass matrices, the conditions for positivity
of the diagonal elements are most important since other constraints are followed from the first ones.
In the model under consideration, the above conclusion is very helpful for the fixing parameters.
Since there are a lot of Higgs fields in this model, so the vacuum stability will be considered
in the future study.
ACKNOWLEDGMENTS
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2017.356.
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