Sự chuyển pha điện yếu trong mô hình zee babu và mô hình SU(3)c x SU(3)L x u(1)x x (1)n tt tiếng anh

MINISTRY OF EDUCATION
VIETNAM ACADEMY
AND TRAINING
OF SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY
———————————

NGUYEN CHI THAO

THE ELECTROWEAK PHASE
TRANSITION IN THE ZEE-BABU AND
SU (3)C ⊗ SU (3)L ⊗ U (1)X ⊗ U (1)N GAUGE MODELS
Major: Theoretical and Mathematical Physics
Code: 9 44 01 03

SUMMARY OF PHYSICS DOCTORAL THESIS

HA NOI- 2019


The thesis is completed at: GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY VIETNAM ACADEMY OF SCIENCE
AND TECHNOLOGY.

Supervisors:
1. Professor Dr. Hoang Ngoc Long
2. Assoc. Prof. Dr. Phung Van Dong

Reviewer 1: ........................................
Reviewer 2: .......................................
Reviewer 3: ........................................

The thesis will be defended in front of the institute doctoral thesis Assessment Council, held at the Graduate University of Science
and Technology - Vietnam Academy of Science and Technology at ...
o’clock, on day ... month ... year 2019.

The thesis can be found at:
- The library of Graduate University of Science and Technology.
- National Library of Vietnam.


INTRODUCTION
1. The imperative of the thesis
In physics baryon asymmetry is also known as material asymmetry. This issue is currently an interesting problem. Today, to explain baryon asymmetry, scientists use two mechanisms, Leptongenesis and Baryogenesis. A model having Baryogenesis must be satisfied
the three conditions of A.Sakharov [2].
The standard model (SM) has been successful in explaining
experimental results [4]. Source of CP violation in SM is smaller
Baryon asymmetry of Universe (BAU) and there is no strong phaseone transition with the Higgs mass mH = 125 GeV. In other words,
SM is not enough mH = 125 GeV the first order phase transition
[6-8].
2. Research targets of the thesis
We want to analyze the baryon asymmetry problem and determine the contribution role of newl particles in some extended standard models. We have done the thesis ”The electroweak phase
transition in the Zee-Babu and SU (3)C⊗SU (3)L⊗U (1)⊗U (1)N
gauge models”
3. The main research contents of the thesis
In addition to the introduction and conclusion of the thesis, there are
three chapters:
Chapter 1, investigation of weak phase transition in SM.
Chapter 2, investigating the weak electric phase transition
in the Landau gauge and ξ gauge in the Zee-Babu model.
Chapter 3, multiperiod structure of electroweak phase transition in the 3-3-1-1 model.

Chapter 1

OVERVIEW
1.1

The effective potential with the contribution of the
scalar field

The effective potential includes both thermal and quantum
contributions
Vef f = V +

¯
¯
m2φ (χ)
m4φ (χ)
T4
mφ
ln
+ 2 F− (
),
2
2
64π
µ
4π
T

(1.1)

in which
F− (

mφ
)=
T

−32m3 πT + 16m2 π 2 T 2 + 9m4 + 6m4 ln

ab T 2
m2

(1.2)

96T 4

where m ≡ mφ ; ln[ab ] = 2 ln[4π] − 2C ≈ 3.91.

1.2

The effective potential with the contribution of the
complex scalar and the gauge boson field
We have a general effective formula
Vef f = V (χ)
¯ +n

m4φ (χ)
¯

m2φ (χ)
¯
T4
mφ
ln
+
F− (
) ,
64π 2
µ2
4π 2
T

in which n is the degrees of freedom of these two fields.

(1.3)


1.3 The effective potential with the contribution of the fermion field

1.3

3

The effective potential with the contribution of the
fermion field

We have the effective potential with the contribution of the
fermion field
Vef f = V (χ)

¯ + 12

m4φ (χ)
¯
m2φ (χ)
¯
T4
mφ
+ 2 F+ (
) ,
ln
2
2
64π
µ
4π
T

(1.4)

in which ln[af ] = 2 ln[π] − 2C ≈ 1.14.

1.4

Effective potential in the SM

The effective potential in the standard model is:
m2W
m2z
1

4
4
3m
ln
+
6m
ln
z
W
64π 2
µ2
µ2
m2
m2t
4
(1.5)
−
12m
ln
+m4H ln H
t
µ2
µ2
T4
mz
mW
mH
mt
+ 2 3F− (
) + 6F− (

) + F− (
) + 12F+ ( ) .
4π
T
T
T
T
In Eq (1.5) we only consider the contribution of the t quark and Higgs
boson.
Vef f = V( χ)
¯ +

1.5

The electroweak phase transition in the SM

Electroweak phase transition is the transition from a symmetrical phase to an assymmetrical phase, the result of this process is the
particle mass generation. The essence of this phase transition is the
change
of VEV of the Higgs field from zero VEV to non-zero VEV
.
The effective potential in SM is determined as the following
Vc (φc , T ) = Λ + D T 2 − T02 φ2c − ET φ3c +

λ(T ) 4
φc
4

(1.6)

We define the phase transition strength S
S=

φmin(c)
2E
=
.
Tc
λ(Tc )

(1.7)

We obtain a phase transition graph of S in SM as shown in
the figure 1.1.
The EWPT is the first order transition when mH ≤ 47.3 GeV,
this contradicts the experimental Higgs mass of mH
125.5 GeV.
Therefore, explaining the baryon asymmetry we need to examine the
baryon asymmetry in the beyond SM.


1.6 Conclusion

4
S

1.8

1.6

1.4

1.2

1.0
m_H
25

30

35

40

45

50

Figure 1.1: The dashed line of S = 2E/λTc = 1, the solid line : 2E/λTc = 1.5.

1.6

Conclusion
The SM cannot explain the baryon asymmetry.


Chapter 2

ELECTROWEAK
PHASE TRANSITION

IN THE ZEE-BABU
MODEL
2.1

The mass of particles in the Zee-Babu model
The masses of h± and k ±± are given by
m2h± = p2 v02 + u21 , m2k±± = q 2 v02 + u22 .

(2.1)

Diagonalizing matrices in the kinetic components of the Higgs
potential, we obtain
2
m2H (v0 ) = −µ2 + 3λv02 , mZ (v0 ) = 41 (g 2 + g 2 )v02 = a2 v02 ,
m2G (v0 ) = −µ2 + λv02 , m2W (v0 ) = 41 g 2 v02 = b2 v02 .

2.2

(2.2)

Effective potential in the Zee-Babu model

2.2.1

Effective potential with Landau gauge

Vef f (v) = V0 (v) +

3
64π 2

+

1
64π 2

+

3T 4
4π 2

F− (

+

T4
4π 2

2F− (

m4Z (v)ln

2m4h± (v)ln

m2Z (v)
m2 (v)
m2 (v)
+ 2m4W (v)ln W 2 − 4m4t (v)ln t 2
2
Q

Q
Q

m2h± (v)
m2k±± (v)
m2 (v)
4
+
2m
(v)ln
+ m4H (v)ln H 2
±±
k
2
2
Q
Q
Q

mZ (v)
mW (v)
mt (v)
) + F− (
) + 4F+ (
)
T
T
T
mh± (v)
m ±± (v)

mH (v)
) + 2F− ( k
) + F− (
)
T
T
T


2.2 Effective potential in the Zee-Babu model

6

where vρ is a variable changing with temperature, and at T = 0,
vρ ≡ v0 = 246 GeV. Here
mφ
T

F±

mφ
T αJ 1 (α, 0)dα,
∓

=
0

∞
1
J∓

(α, 0)

= 2
α

2.2.2

1

(x2 − α2 ) 2
dx.
ex ∓ 1

Effective potential with ξ gauge

We are known that in high levels, the contribution of Goldstone boson cannot be ignored. Therefore, we must consider an effective potential in arbitrary ξ gauge,
V1T =0 (v) =

m2 ±
1
m2
1
(m2H )2 ln( H
) +
(m2h± )2 ln( h2 )
2
2
2
4(4π)
Q

4(4π)
Q

+

m2k±±
1
2×1
m2G + ξm2W
2
2
2
2 2
ln(
(m
)
+
(m
+
ξm
)
ln(
)
±± )
G
W
k
4(4π)2
Q2
4(4π)2

Q2

+

1
m2 + ξm2
2×3
m2
(m2G + ξm2Z )2 ln( G 2 Z ) +
(m2W )2 ln( W
)
2
2
4(4π)
Q
4(4π)
Q2

+

3
2×1
ξm2W
m2
2 2
2 2
Z
(m
)
ln(

)
−
(ξm
)
ln(
)
2
Z
W
Q
4(4π)2
4(4π)2
Q2

−

ξm2Z
1
2 2
(ξm
)
ln(
) ,
Z
4(4π)2
Q2

(2.3)

and

V1T =0 (v, T ) =
+

T4
m2H
J
B
2π 2
T2

+ JB

T4
m2G + ξm2W
2×JB
2
2π
T2

3T 4
m2W
+
2×J
B
2π 2
T2
−

m2h±
T2

+ JB
m2Z
T4

+ JB

T4
ξm2W
2×JB
2
2π
T2

+ JB

m2k±±
T2

+ 2 ×JB

ξm2Z
T2

m2G + ξm2Z
T2
+ JB

m2γ
T2

+ JB

ξm2γ
T2

in which
JB±

m2φ
T2

m2φ
T 2 αJ 1 (α, 0)dα.
∓

=
0

+ 4 ×JB
,

m2t
T2

(2.4)


2.3 Electroweak phase transition in Landau gauge

2.3

7

Electroweak phase transition in Landau gauge
The quartic expression in v
Vef f (v) = D(T 2 − T02 )v 2 − ET |v|3 +

λT 4
v ,
4

(2.5)

Tc critical temperature and phase transition strength are
Tc =

2E
T0
vc
=
.
,S=
2
T
λ
c
Tc
1 − E /DλTc

(2.6)

0 (v) are
The minimum conditions for Vef
f

0
Vef
f (v0 )
0 (v)
∂ 2 Vef
f

∂v 2

v=v0

= 0,
=

0 (v)
∂Vef
f

∂v

m2H (v)

v=v0

= 0,
(2.7)
2

v=v0

2

= 125 GeV .

To have a first-order phase transition, we require that the
strength is larger or equal to the unit (S ≥ 1). In Fig. 2.1, we have
plotted the transition strength S as a function of the new charged
scalars: mh± and mk±± .
As shown in Fig. 2.1, for mh± and mk±± being in the 0 −
350 GeV range, respectively, the transition strength is in the range
1 ≤ S < 2.4.
We see that the contribution of h± and k ±± are the same.
The larger mass of h± and k ±± , the larger cubic term (E) in the
effective potential but the strength of phase transition cannot be
strong. Because the value of λ also increases, so there is a tension
between E and λ to make the first order phase transition. In addition
when the masses of charged Higgs bosons are too large, T0 , λ will be
unknown or S −→ ∞.
500

mh GeV

400

300

200

100

0
0

100

200

300
mk

400

500

GeV

Figure 2.1: When the solid contour of S = 2E/λTc = 1, the dashed contour:
2E/λTc = 1.5, the dotted contour: 2E/λTc = 2, the dotted-dashed contour:
2E/λTc = 2.4, even and nosmooth contours: S −→ ∞.


2.4 Electroweak phase transition in ξ gauge

2.4

8

Electroweak phase transition in ξ gauge

The high-temperature expansions of the potential in Eq.(2.3)
and in Eq.(2.4) can be rewritten in a like-quartic expression in v
v
V = (D1 + D2 + D3 + D4 + B2 ) v 2
+ B1 v 3 + Λv 4 + f (T, u1 , u2 , µ, ξ),

(2.8)

in which
f (T, u1 , u2 , µ, ξ, v) = C1 + C2 ,
m2 +ξm2

(2.9)
m2 +ξm2

Expanding functions JB G T 2 W and JB G T 2 Z in Eq. (2.4),
we will obtain the term of mixing between ξ and v in B1 and B2 .
m2 +ξm2
m2 +ξm2
Therefore JB G T 2 W and JB G T 2 Z or B1 and B2 contain a
part of daisy diagram contributions mentioned in Ref. [22]. The
other part of ring-loop distribution comes to damping effect. On the
other hand, we see that the ring loop distribution still is very small,
it was approximated g 2 T 2 /m2 (g is the coupling constant of SU (2),
m is mass of boson), m ∼ 100 GeV, g ∼ 10−1 so g 2 /m2 ∼ 10−5 .

If we add this distribution to the effective potential, the D1 term
will give a small change only. Therefore, this distribution does not
change the strength of EWPT or, in other words, it is not the origin
of EWPT. The potential in Eq.(2.8) is not a quartic expression because B2 , D3 , D4 and f (T, u1 , u2 , µ, ξ, v) depend on v, ξ and T . It has
seven variables such as u1 , u2 , p, q, µ, λ and ξ. Therefore, the shape of
potential is distorted by u1 , u2 , p, q, ξ but not so much. If Goldstone
bosons are neglected and the gauge parameter is vanished (ξ = 0),
it will be reduced to Eq.(2.5) in the Landau gauge. The minimum
conditions for Eq.(2.8) are still like Eq.(2.7) but for this case, it holds:
m2H0 = −µ2 + 3λv02 = 1252 GeV. There are many variables in our
problem and some of them, for example, u1 , u2 , p, q and µ play the
same role. They are components in the mass of particles. It is emphasized that ξ and λ are two important variables and have different
roles. Therefore, in order to reduce number of variables, we have to
approximate values of variables, but must not lose the generality of
the problem

2.4.1

The case of small contribution of Goldstone boson

When the mass of Goldstone boson is small, it means that
µ2 ≈ λv02 and taking into account mH0 = 125 GeV, we obtain λ =
0.1297. We conduct a method yielding an effective potential as a
quartic expression in v through three steps.


2.4 Electroweak phase transition in ξ gauge

9

-The first approximate µ2 ≈ λv02 .
-In the second approximate step, we neglect u1 , u2 .
- In the third approximate step: replacing µ2 = λv02 and in the
square root term of B2 and C1 , we can approximate µ2 ∼ λv 2 .
After three approximate steps we rewrite the equation (2.8)
V

=

(D1 + D2 ) v 2 + Bv 3 + Λv 4 .

(2.10)

We obtain the strength of EWPT as shown in Fig.2.2. The
maximum of the strength is about 4.05.
80

60

Ξ

S
40
S=2

S=1.5

20
S=1

S=4.05
0
200

220

240

260
mk

280
h

300

320

340

GeV

Figure 2.2: The strength of EWPT with λ = 0.1297 and µ2 ∼ λv02

In fact, the mass of Goldstone boson is much smaller than
that of the W ± boson or the Z boson so the contribution of Goldstone boson must be very small in the effective potential. Hence, the
lines in Fig.2.2 are almost vertical or almost parallel to the axis ξ.
These results match those of Ref.[48].This shows that the strength
of EWPT is gauge independent. In addition, the new particles have
large masses, so they provide valuable contributions to the EWPT

in the Landau gauge or in an arbitrary gauge. The charge of these
particles increases their contributions.

2.4.2

Constraints on coupling constants in the Higgs
potential

In order to have the first order phase transition, the masses
of the new charged scalars mh± and mk±± must be smaller than 350
GeV. Therefore, we obtain the following limits: 0 < p < 1.22 and
0 < q < 1.22. However, to find these accurate values of mh± and
mk±± , other considerations are also needed.


2.5 Conclusion

2.5

10

Conclusion

In this chapter we have investigated the EWPT in the ZB
model by using the high-temperature effective potential. The EWPT
is strengthened by the new scalars, the phase transition strength is
from 1 to 4.15. The new charged scalars h± and k ±± are triggers for
the first-order EWPT.

Chapter 3

MULTI-PERIOD
STRUCTURE OF
ELECTROWEAK
PHASE TRANSITION IN
THE 3-3-1-1 MODEL
3.1
3.1.1

Brief review of the 3-3-1-1 model
The mass of the quarks
- The mass of top quarks and bottom quarks is as follows:
ht u
hb v
mt = √ , mb = √ ,
2
2

- The mass of the exotic quarks are determined as
ω
mU = √ hU ;
2

3.1.2

ω
mD1 = √ hD
11 ;
2

ω
mD2 = √ hD
22 .
2

The mass of the Higgs bosons
The mass terms of charged Higgs bosons are given by
m2H1 =

u2 + v 2
ω2 + v2
λ8 ; m2H2 =
λ7 .
2
2

The mass of neutral Higgs bosons is presented in Table 3.1

(3.1)


3.1 Brief review of the 3-3-1-1 model

12

Table 3.1: The neutral Higgs boson masses.

Neutral Higgs boson
Squared mass

3.1.3

S4
2λΛ2

Aη
λ9 ω 2
2

Aχ
λ9 u2
2

Sη
2λ3

u2

S

χ
2λ2 ω 2

Sρ
2λ1

v2

H3

λ9 (u2 +ω 2 )
2

Gauge boson sector

Because of the constraints u, v
ω, we have mW
mX
mY . The W boson is identified as the SM W boson. So we have:
u2 + v 2 = (246 GeV)2 .
Table 3.2: The mass of charged gauge bosons.
Gauge boson
Squared mass

W
+ v2)

g2
2
4 (u

Y
+ v2)

g2
2
4 (ω

X
+ u2 )

g2
2
4 (ω

From the above analysis, the phenomenological aspects of
the 3-3-1-1 model can be divided into two pictures corresponding to
different domain values of VEVs.
Picture (i): Λ ∼ ω
v∼u
We obtain the masses of neutral gauge bosons as follows
g 2 (u2 + v 2 )
,
4c2W

m2Z

(3.2)

g2
(3 + t2X )ω 2 + 4t2N (ω 2 + 9Λ2 )
18

m2Z1

+

((3 + t2X )ω 2 − 4t2N (ω 2 + 9Λ2 ))2 + 16(3 + t2X )t2N ω 4 , (3.3)

g2

(3 + t2X )ω 2 + 4t2N (ω 2 + 9Λ2 )
18

m2Z2

−

((3 + t2X )ω 2 − 4t2N (ω 2 + 9Λ2 ))2 + 16(3 + t2X )t2N ω 4 . (3.4)

From the experimental data ∆ρ < 0.0007 ones√get u/ω <
0.0544 or ω > 3.198 TeV [70] (provided that u = 246/ 2 GeV as
mentioned). Therefore, the value of ω results in the TeV scale as
expected.
Picture (ii): Λ
ω
v∼u
If we assume Λ
ω
u ∼ v, three gauge bosons are derived
as [5, 71, 72, 76]:
m2Z

g 2 (u2 + v 2 )
; m2Z1
4c2W

4g 2 t22 Λ2 ; m2Z2

g 2 c2W ω 2
(3 − 4s2W )

(3.5)

The W ± boson and Z boson are recognized as two famous
gauge bosons in the SM.


3.2 Effective potential in the model 3-3-1-1

3.2

13

Effective potential in the model 3-3-1-1
Hence the Higgs of the model, we obtain the tree-level po-

tential
V0 =

λφ4Λ
1
λ2 φ4ω
φ 2 µ2
1
λ3 φ4u
+ λ11 φ2Λ φ2ω +
+ Λ + µ22 φ2ω +
4
4
4

2
2
4
1
1 2 2
1
1
2 2
2 2
2 2
+ λ12 φΛ φu + λ6 φu φω + µ3 φu + λ5 φu φv
4
4
2
4
λ1 φ4v
1
1
1 2 2
2 2
2 2
+
+ λ10 φΛ φv + λ4 φv φω + µ1 φv .
4
4
4
2

(3.6)

Here V0 has quartic form as in the SM, but it depends on
four variables φΛ , φω , φu , φv , and has the mixing terms between them.
However, developing the Higgs potential in this model, we obtain four
minimum equations. Therefore, we can transform the mixing between
four variables to the form depending only on φΛ , φω , φu and φv .
Furthermore, importantly, there are the mixings of VEVs
because of the unwanted terms such as λ4 (ρ† ρ)(χ† χ), λ5 (ρ† ρ)(η † η),
λ6 (χ† χ)(η † η), λ7 (ρ† χ)(χ† ρ), λ8 (ρ† η)(η † ρ), λ9 (χ† η)(η † χ), λ10 (φ† φ)(ρ† ρ),
λ11 (φ† φ)(χ† χ) and λ12 (φ† φ)(η † η) in the Higgs potential. To satisfy
the generation of inflation with φ-inflaton [5,76], the values λ10,11,12
can be small, is about 10−10 − 10−6 . Thus, λ4,5,6,7,8,9 must be also
small to make the corrections of high order interactions of the Higgs
will not be divergent. In general, if we did not neglect these mixings,
V0 will have additional components Λv, Λω, ωv, uv. Considering
at the temperature T , for instance, a toy effective potential given
by
Vef f (v)

=

λv 4 − Ev 3 + Dv 2 + λk .ω 2 v 2 + λj .Λ2 v 2 + u2 .v 2
≈ λv 4 − Ev 3 + Dv 2 + λi .(ω 2 + Λ2 + u2 )v 2

(3.7)

The slices of the effective potential in (3.7) at ω 2 + Λ2 + u2 =
1 TeV as a function of v for some values of λi is plotted in 3.1.
From 3.1, we see that at arbitrary temperature T when λi , i =
4, .., 9 increases, the second minimum of the effective potential fades.
For a first order phase transition, the value of λi is not too large,

so that the potential still has two minima. We observe that if λi is
enough small to have a second minimum, at arbitrary temperature,
the shape of the effective potential remains the same in the absence
of λi . Therefore, we have one more reason to assume that λi must
be small and this mixing can be neglected. Hence, we can write
V0 (φΛ , φω , φu , φv ) = V0 (φΛ ) + V0 (φω ) + V0 (φu ) + V0 (φv ) and ignore
the mixing of different VEVs, otherwise our phase transitions will be
very complex or distorted.
2


3.3 Electroweak phase transition without neutral fermion

14

Veff TeV4

0.15

0.10

Λi
0.05

0.06
Λi

0.03
Λi

0.2

0.4

0.6

0.8

1.0

0

1.2

1.4

v TeV

Figure 3.1: The contours of the effective potential in (3.7) as a function of v for
some values of λi as λ = 0.3, D = 0.3, E = 0.6, Λ2 + ω 2 + v 2 = 1 TeV2

From the mass spectra, we can split the masses of particles
into four parts as follows
m2 (φΛ , φω , φu , φv ) = m2 (φΛ ) + m2 (φω ) + m2 (φu ) + m2 (φv ).

(3.8)

Taking into account Eqs. (3.6) and(3.8), we can also split
the effective potential into four parts
Vef f (φΛ , φω , φu , φv ) = Vef f (φΛ ) + Vef f (φω ) + Vef f (φu ) + Vef f (φv ).

We assume φΛ ≈ φω , φu ≈ φv over space-times. Then, the effective
potential becomes
Vef f (φΛ , φω , φu , φv ) = Vef f (φω ) + Vef f (φu ).

From the mass spectra, it follows that the squared masses
of gauge and Higgs bosons are split into three separated parts corresponding to three SSB stages.

3.3
3.3.1

Electroweak phase transition without neutral fermion
Two periods EWPT in picture (i)

1. Phase transition SU (3) → SU (2)
This phase transition involves exotic quarks, heavy bosons,
but excludes the SM particles. As a consequence, the effective potential of the EWPT SU (3) → SU (2) is Vef f (φω ).
2
Vef f (φω ) = Dω (T 2 − T0ω
)φ2ω − Eω T φ3ω +
2
T0ω
≡−

Fω
.
Dω

λω (T ) 4
φω ,

4

(3.9)
(3.10)


3.3 Electroweak phase transition without neutral fermion

15

3500
3000

S=1

mH3 GeV

2500
2000
1500

S=2
S=3

1000
S

500
0
0

1000

2000
mexotic

3000

4000

5000

quark Charged Higgs

6000

7000

GeV

Figure 3.2: The mass area corresponds to Sω > 1

The values of Vef f (φω ) at the two minima become equal at
the critical temperature and the phase transition strength are
Tcω

=

2Eω
T0ω

, Sω =
.
2
λ
Tcω
1 − Eω /Dω λTcω

There are nine variables: the masses of U, D1 , D2 , H2 , H3
and Aη , Sχ , S4 , Z1 . However, for simplicity, we assume mU = mD1 =
mD2 = mH2 ≡ O, mAη = mSχ = mH3 = mS4 ≡ P . Consequently,
the critical temperature and the phase transition strength are the
function of O and P ; therefore we can rewrite the phase transition
strength as follows
Sω =

2Eω
≡ Sω (O, P, Sω ).
λTcω

(3.11)

In Figs. 3.2 and 3.3, we have plotted the relation between
masses of the charged particles O and neutral particles P with some
values of the phase transition strength at ω = 6 TeV.
2500

S=1

mH3 GeV

2000

S=2

1500

S=3

1000
500
0
0

1000

2000

3000

4000

5000

6000

7000

mH1 GeV

Figure 3.3: The mass area corresponds to Sω > 1 with real Tc condition. The

gaps on the lines (S = 1, 2, 3) correspond to values making Tc to be complex.

The mass region of particles is the largest at Sω = 1, the


3.3 Electroweak phase transition without neutral fermion

16

mass region of charged particles and neutral particles are
0 ≤ mExoticQuark/ChargedHiggsboson ≤ 7000GeV ,
0 ≤ mH3 ≤ 2600 GeV .

From Eq. (3.11) the maximum of Sω is around 70.
2. Phase transition SU (2) → U (1)
√ In this period, the symmetry breaking scale equals to u =
246/ 2 and the masses of the SM particles and the masses of X, Y, H1 ,
H2 , H3 , Aχ , Sη are generated. There are six variables, the masses of
bosons H1 , H2 , Aχ , Aη , H3 , Sρ . For simplicity, we assume mH1 =
mH2 ≡ K, mAχ = mSη = mH3 ≡ L. The effective potential of
EWPT SU (2) → U (1) is given by
Vef f (φu ) =

λu (T ) 4
φu − Eu T φ3u + Du T 2 φ2u + Fu φ2u .
4

(3.12)

The minimum conditions are

Vef f (0) =

∂Vef f (φu )
∂φu

= 0;
u

∂ 2 Vef f (φu )
∂φ2u

= m2Aχ +m2H3 +m2Sη +m2Sρ ,
u

(3.13)
In Fig 3.5 we have plotted the relation between masses of the
charged particles K and neutral particles L with some values of the
phase transition strength.
600

mH3 GeV

S=1

S

500
400

S=1.2

300

S

200
S=2

100
S=3

0
200

400

600

800

1000

1200

m H1 GeV

Figure 3.4: The strength S =

2Eu
.

λTc

However, we can fit the mass of heavy particle one again when
considering the condition of Tc to be real, so that the maximum of
strength is reduced from 3 to 2.12.
With the mass region of neutral and charged particles given
in Table 3.3 the maximum phase transition strength is 2.12. This is
larger than 1 but the EWPT is not strong.


3.3 Electroweak phase transition without neutral fermion

17

200

S=1.2
150

mH3 GeV

S=1

S=1.3

100
50

S=2.1
0

200

250

300

350

400

450

mH1 GeV

Figure 3.5: The strength EWPT S =

2Eu
with Tc must be real.
λTc

Table 3.3: Mass limits of particles with Tc > 0.

Strength S
1.0-2.12

3.3.2

K[GeV ]
195 ≤ K ≤ 484.5

L[GeV ]
0 ≤ L ≤ 209.8

Three period EWPT in picture (ii)
- The first process is SU (3)L ⊗ U (1)X ⊗ U (1)N → SU (3)L ⊗

U (1)X .
- The second one is SU (3)L ⊗U (1)X → SU (2)L ⊗U (1)X .
The third process is SU (2)L → U (1)Q . The third process is
like SU (2) → U (1) in the picture (i).
The first process is a transition of the symmetry breaking of
U (1)N group. It generates mass for Z1 through Λ or Higgs boson S4 .
The third process is like the SU (2) → U (1) in picture (i) but it does
not involve Z1 and S4 .
The second process has the effective potential is like Eq. (3.9)
1000

S=1

mH3 GeV

800
600

S=2
S=3

400
200

S

0
0

1000

2000

mExotic Quark

3000

Charged Higgs

Figure 3.6: The strength EWPT Sω =

4000

GeV

2Eu
with ω = 6 TeV.
λTc


3.4 The role of neutral fermions in EWPT

18

When we import real Tc , the mass region of charged and
neutral particles are
0 ≤ mExoticquark/ChargedHiggsboson ≤ 4000 GeV ,
0 ≤ mH3 ≤ 1000 GeV .

The mass region of charged bosons is narrower than that in the
section 3.2. From Eq. (3.11), the maximum of S has been estimated
to be around 100.

3.4

The role of neutral fermions in EWPT

In the SU (3) → SU (2) if we add the contribution of neutral
fermions, then the maximum of S would decrease. However, the
neutral fermions do not lose the first-orde EWPT as shown in Table
3.4.
Table 3.4: Values of the maximum of EWPT strength with ω = 6 TeV.
Period

Picture

SU (3) → SU (2)
SU (3) → SU (2)

(i)
(ii)

mZ2 [TeV]

2.386
2.254

mN −R [TeV]

SM ax without NR

SM ax withNR

2.227
1.986

70
100

50
30

Looking at the Table 3.4, the following remarks are in order:
1. In case of the neutral fermion absence. In the picture (i), if
Z1 boson is involved in the SU (3) → SU (2) EWPT; the contribution
of Z1 makes increasing E and λ, but λ increases stronger than E.
2E
The strength S =
gets the value equals 70. For the picture (ii),
λTc
the mentioned value equals 100.
2. In case of the neutral fermion existence. When the neutral
fermions are involved in both pictures, Smax in picture (ii) decreases
faster than Smax in picture (i). The strength gets values equal to 50

and 30 for the picture (i) and (ii), respectively.
If the neutral fermions follow the Fermi-Dirac distribution
(i.e., they act as a real fermion but without lepton number), they
increase the value of the λ and D parameters. Thus, they reduce the
E
value of strength EWPT S, because S =
and E do not depend
2λTc
on the neutral fermions.
This suggests that DM candidates are neutral fermions (or
fermions in general) which reduce the maximum value of the EWPT
strength.
However, the EWPT process depends on bosons and fermions.
The boson gives a positive contribution (obey the Bose-Einstein distribution) but the fermion gives a negative contribution (obey the
Fermi-Dirac distribution). In order to have the first order transition,


3.5 Conclusion

19

the symmetry breaking process must generate mass for more bosons
than fermions.
In addition, in this model, the neutral fermion mass is generated from an effective operator. This operator which demonstrates
an interaction between neutral fermions and two Higgs fields. The
above neutral fermion is very different from usual fermions. The M
parameter has an energy dimension, and it may be an unknown dark
interaction. Thus, the neutral fermions only are effective fermions,
according to the Fermi-Dirac distribution, but their statistical nature
needs to be further analyzed with other data.

3.5

Conclusion

In the model under consideration, the EWPT consists of two
pictures. The first picture containing two periods of EWPT, has a
transition SU (3) → SU (2) at 6 TeV scale and another is SU (2) →
U (1) transition which is the like-standard model EWPT. The second
picture is an EWPT structure containing three periods, in which two
first periods are similar to those of the first picture and another one
is the symmetry breaking process of U (1)N subgroup. The EWPT is
the first order phase transition if new bosons with mass within range
of some TeVs. The maximum strength of the SU (2) → U (1) phase
transition is equal to 2.12 so the EWPT is not strong.
We have focused on the neutral fermions without lepton number being candidates for DM and obey the Fermi-Dirac distribution,
and have shown that the mentioned fermions can be a negative trigger
for EWPT. Furthermore, in order to be the strong first-order EWPT
at TeV scale, the symmetry breaking processes must produce more
bosons than fermions or the mass of bosons must be much larger than
that of fermions.
It is known that the mass of Goldstone boson is very small
[46] and the physical quantities are gauge indepen- dent so the critical temperature and the strength is gauge independent [44-46]. Consequently, the survey of effective potential in Landau gauge is also
sufficient, or other word speaking, it is just consider in determined
gauge. Thus, it is a reason why the Landau gauge is used in this
work. In this chapter, the structure of EWPT in the 3-3-1-1 model
with the effective potential at finite temperature has been drawn at
the 1-loop level; and this potential has two or three phases.
We have analyzed the processes which generate the masses
for all gauge bosons inside the covariant derivatives. After diagonalization, the masses of gauge bosons do not have mixing among

VEVs. Therefore, the EWPT stages are independent of each other
[62].
In conclusion, the model has many bosons which will be good
triggers for first-order EWPT. The situation is that as less heavy


3.5 Conclusion

20

fermion as the result will be better. However, strength of EWPT
can be reduced by many bosons (such as Z, Z1 , Z2 in the 3-3-1-1
model).
The new scalar particles playing a role in generation mass
for exotic particles, increase the value of EWPT strength. Because
these scalar fields follow the Bose-Einstein distribution, so that they
contribute positively to the effective potential. With the help of such
particles, the strength of phase transition will be strong. As mentioned above, their masses depend just on one VEV, so they only
participate in one phase transition. Moreover, among the neutral
fermions, they may be candidates for DM. From the point of view
of the early universe, the above particles can be an inflaton or some
product of the inflaton decay.


CONCLUSION
From the investigate content, we obtained the following results:
1. Investigation of weak phase transition in model Zee-Babu.
Considering the Landau gauge, this model has phase transition strength is in the range 1 ≤ S < 2.12, due to the contribution
of two mh± and mk±± particles. Their mass ranges in the range of
0 − 350 GeV.

- Considering the ξ gauge, the phase transition strength is in
the range 1 ≤ S < 4.15, more strong than the Landau gauge. Thus,
the phase transition strength will increase when the contribution of
gauge ξ. However, the ξ gauge is not the cause of the EWPT. This
leads to the fact that the calculation of EWPT in Landau gauge is
enough
cases. 2. We examined the EWPT in the 3-3-1-1 model with two
1. EWPT without neutral fermion.
We have two pictures in this case.
- The first picture has two phase transitions. Phase transition
SU (3) → SU (2) with value 5.856 TeV≤ ω ∼ Λ ≤ 6.654 TeV. Considering at ω = 6 TEV, we calculated the phase transition strength
in the range 1 GeV< Sω < 70 GeV. The mass region of particles is
the largest at Sω = 1, the mass region of exotic quarks and neutral
Higgs boson mH2 is between 0 and 7000 GeV.
246
Phase transition SU (2) → SU (1) at value u = √ TeV. The
2
maximum phase transition strength which must be 2.12. Mass limits
of particles: mH1 , mH2 in the range 195 GeV ≤ mH1 , mH2 ≤ 484.5
GeV and and the mass of particles: mAχ , mSη , mH3 in the range
0 ≤ mAχ , mSη , mH3 ≤ 209.8 GeV.
The second picture has three periods. Phase transitions occur
at Λ
ω = 6 TeV and ω
u ∼ v values.
The first process is SU (3)L ⊗ U (1)X ⊗ U (1)N → SU (3)L ⊗
U (1)X .
The second one is SU (3)L ⊗U (1)X → SU (2)L ⊗U (1)X .
The third process is SU (2)L → U (1)Q .

3.5 Conclusion

22

- The first process is a transition of the symmetry breaking
of U (1)N group. It generates mass for Z1 through Λ or Higgs boson
S4 . The third process is like the SU (2) → U (1) in picture (i) but it
does not involve Z1 and S4 .
- The second process is the phase strength S been estimated
to be around 100. The mass region of charged particles and neutral
particles are
0 ≤ mExoticquark/ChargedHiggsBoson ≤ 4000 GeV ,
0 ≤ mH3 ≤ 1000 GeV .

2. The role of neutral fermions in EWPT
Considering the phase transition SU (3) → SU (2), if we add
the contribution of neutral fermions, the maximum of S would decrease. However, the neutral fermions do not lose the first-orde
EWPT. In case of the neutral fermion absence, the contribution of Z1
makes increasing E and λ, but λ increases stronger than E. In case
of the neutral fermion existence, the phase trength decreases. If the
neutral fermions follow the Fermi-Dirac distribution, they increase
the value of the λ and D parameters. Thus, they reduce the value
of strength EWPT S. This suggests that DM candidates are neutral
fermions (or fermions in general) which reduce the maximum value
of the EWPT strength.
However, the EWPT process depends on bosons and fermions.
The boson gives a positive contribution but the fermion gives a negative contribution. In order to have the first order transition, the
symmetry breaking process must generate mass for more bosons than
fermions.

NEW CONTRIBUTIONS OF THE THESIS
1. We consider the baryogenesis picture in the Zee-Babu
model. Our analysis shows that electroweak phase transition (EWPT)
in the model is a first-order phase transition at the 100 GeV scale, its
strength ranges from1 to 4.15 and the masses of charged Higgs bosons
are smaller than 300 GeV. The EWPT is strengthened by only the
new bosons and this strength is enhanced by arbitrary ξ gauge. However, the ξ gauge does not break the first-order EWPT or, in other
words, the ξ gauge is not the cause of the EWPT. This leads to the
fact that the calculation of EWPT in Landau gauge is enough; and
the latter may provide baryon-number violation ( B-violation) necessary for baryogenesis in the relationship with nonequilibrium physics
in the early universe.
2.The EWPT is considered in the framework of 3-3-1-1 model.
The phase structure within three or two periods is approximated for
the theory with many vacuum expectation values (VEVs) at TeV
and electroweak scales. In the mentioned model, there are two pictures.
- The first picture containing two periods of EWPT, has a
transition SU (3) → SU (2) at 6 TeV scale and another is SU (2) →


3.5 Conclusion

23

U (1) transition which is the like-standard model EWPT.
- The second picture is an EWPT structure containing three
periods, in which two first periods are similar to those of the first
picture and another one is the symmetry breaking process of U (1)N
subgroup.
- Our study leads to the conclusion that EWPTs are the first
order phase transitions when new bosons are triggers and their masses

are within range of some TeVs. Especially, in two pictures, the maximum strength of the SU (2) → U (1) phase transition is equal to 2.12
so this EWPT is not strong.
- Moreover, neutral fermions, which are candidates for dark
matter and obey the Fermi-Dirac distribution, can be a negative trigger for EWPT. However, they do not make lose the first-order EWPT
at TeV scale.
- Furthermore, in order to be the strong first-order EWPT
at TeV scale, the symmetry breaking processes must produce more
bosons than fermions or the mass of bosons must be much larger than
that of fermions.