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

27 trang |

Chia sẻ: honganh20 | Ngày: 07/03/2022 | Lượt xem: 284 | Lượt tải: 0
Bạn đang xem trước 20 trang tài liệu **The electroweak phase transition in the zee - Babu and SU(3)c ⊗ SU(3)l ⊗ U(1)x ⊗ U(1)n gauge models**, để xem tài liệu hoàn chỉnh bạn click vào nút DOWNLOAD ở trên

c = 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± = p
2v20 + u
2
1, m
2
k±± = q
2v20 + u
2
2. (2.1)
Diagonalizing matrices in the kinetic components of the Higgs
potential, we obtain
m2H(v0) = −µ2 + 3λv20 ,
m2G(v0) = −µ2 + λv20 ,
m2Z(v0) =
1
4(g
2 + g′2)v20 = a2v20 ,
m2W (v0) =
1
4g
2v20 = b
2v20 .
(2.2)
2.2 Effective potential in the Zee-Babu model
2.2.1 Effective potential with Landau gauge
Veff (v) = V0(v) +
3
64pi2
(
m4Z(v)ln
m2Z(v)
Q2
+ 2m4W (v)ln
m2W (v)
Q2
− 4m4t (v)lnm
2
t (v)
Q2
)
+
1
64pi2
(
2m4h±(v)ln
m2h±(v)
Q2
+ 2m4k±±(v)ln
m2k±±(v)
Q2
+m4H(v)ln
m2H(v)
Q2
)
+
3T 4
4pi2
{
F−(
mZ(v)
T
) + F−(
mW (v)
T
) + 4F+(
mt(v)
T
)
}
+
T 4
4pi2
{
2F−(
mh±(v)
T
) + 2F−(
mk±±(v)
T
) + F−(
mH(v)
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
F±
(mφ
T
)
=
∫ mφ
T
0
αJ1∓(α, 0)dα,
J1∓(α, 0) = 2
∫ ∞
α
(x2 − α2) 12
ex ∓ 1 dx.
2.2.2 Effective potential with ξ gauge
We are known that in high levels, the contribution of Gold-
stone boson cannot be ignored. Therefore, we must consider an ef-
fective potential in arbitrary ξ gauge,
VT=01 (v) = 1
4(4pi)2
(m2H)
2[ ln(m2H
Q2
)
]
+
1
4(4pi)2
(m2h±)
2[ ln(m2h±
Q2
)
]
+
1
4(4pi)2
(m2k±±)
2[ ln(m2k±±
Q2
)
]
+
2× 1
4(4pi)2
(m2G + ξm
2
W )
2[ ln(m2G + ξm2W
Q2
)
]
+
1
4(4pi)2
(m2G + ξm
2
Z)
2[ ln(m2G + ξm2Z
Q2
)
]
+
2× 3
4(4pi)2
(m2W )
2[ ln(m2W
Q2
)
]
+
3
4(4pi)2
(m2Z)
2[ ln(m2Z
Q2
)
]− 2× 1
4(4pi)2
(ξm2W )
2[ ln(ξm2W
Q2
)
]
− 1
4(4pi)2
(ξm2Z)
2[ ln(ξm2Z
Q2
)
]
, (2.3)
and
VT 6=01 (v, T ) =
T 4
2pi2
[
JB
(m2H
T 2
)
+ JB
(m2h±
T 2
)
+ 2×JB
(m2k±±
T 2
)]
+
T 4
2pi2
[
2×JB
(m2G + ξm2W
T 2
)
+ JB
(m2G + ξm2Z
T 2
)]
+
3T 4
2pi2
[
2×JB
(m2W
T 2
)
+ JB
(m2Z
T 4
)
+ JB
(m2γ
T 2
)
+ 4×JB
(m2t
T 2
)]
− T
4
2pi2
[
2×JB
(ξm2W
T 2
)
+ JB
(ξm2Z
T 2
)
+ JB
(ξm2γ
T 2
)]
,
(2.4)
in which
JB±
(
m2φ
T 2
)
=
∫ m2φ
T 2
0
αJ1∓(α, 0)dα.
2.3 Electroweak phase transition in Landau gauge 7
2.3 Electroweak phase transition in Landau gauge
The quartic expression in v
Veff (v) = D(T
2 − T 20 )v2 − ET |v|3 + λT
4
v4, (2.5)
Tc critical temperature and phase transition strength are
Tc =
T0√
1− E2/DλTc
, S =
vc
Tc
=
2E
λTc
. (2.6)
The minimum conditions for V 0eff (v) are
V 0eff (v0) = 0,
∂V 0eff (v)
∂v
∣∣∣
v=v0
= 0,
∂2V 0eff (v)
∂v2
∣∣∣
v=v0
=
[
m2H(v)
] ∣∣∣
v=v0
= 1252 GeV2.
(2.7)
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 −→∞.
0 100 200 300 400 500
0
100
200
300
400
500
mk±± @GeVD
m
h±
@Ge
VD
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 8
2.4 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) v2
+ B1v3 + Λv4 + f(T, u1, u2, µ, ξ),
(2.8)
in which
f(T, u1, u2, µ, ξ, v) = C1 + C2, (2.9)
Expanding functions JB
(
m2G+ξm
2
W
T 2
)
and JB
(
m2G+ξm
2
Z
T 2
)
in Eq. (2.4),
we will obtain the term of mixing between ξ and v in B1 and B2.
Therefore JB
(
m2G+ξm
2
W
T 2
)
and JB
(
m2G+ξm
2
Z
T 2
)
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 g2T 2/m2 (g is the coupling constant of SU(2),
m is mass of boson), m ∼ 100 GeV, g ∼ 10−1 so g2/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 be-
cause 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 + 3λv20 = 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 em-
phasized 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 bo-
son
When the mass of Goldstone boson is small, it means that
µ2 ≈ λv20 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 ≈ λv20.
-In the second approximate step, we neglect u1, u2.
- In the third approximate step: replacing µ2 = λv20 and in the
square root term of B2 and C1, we can approximate µ2 ∼ λv2.
After three approximate steps we rewrite the equation (2.8)
V = (D1 +D2) v2 + Bv3 + Λv4. (2.10)
We obtain the strength of EWPT as shown in Fig.2.2. The
maximum of the strength is about 4.05.
S=1
S=1.5
S=2
S=4.05
S¥
200 220 240 260 280 300 320 340
0
20
40
60
80
mk±±h± @GeVD
Ξ
Figure 2.2: The strength of EWPT with λ = 0.1297 and µ2 ∼ λv20
In fact, the mass of Goldstone boson is much smaller than
that of the W± boson or the Z boson so the contribution of Gold-
stone 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 10
2.5 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 Brief review of the 3-3-1-1 model
3.1.1 The mass of the quarks
- The mass of top quarks and bottom quarks is as follows:
mt =
htu√
2
, mb =
hbv√
2
,
- The mass of the exotic quarks are determined as
mU =
ω√
2
hU ; mD1 =
ω√
2
hD11 ; mD2 =
ω√
2
hD22.
3.1.2 The mass of the Higgs bosons
The mass terms of charged Higgs bosons are given by
m2H1 =
u2 + v2
2
λ8; m
2
H2 =
ω2 + v2
2
λ7. (3.1)
The mass of neutral Higgs bosons is presented in Table 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 S4 A′η Aχ Sη S′χ Sρ H3
Squared mass 2λΛ2 λ9ω
2
2
λ9u
2
2
2λ3u2 2λ2ω2 2λ1v2
λ9(u
2+ω2)
2
3.1.3 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 + v2 = (246 GeV)2.
Table 3.2: The mass of charged gauge bosons.
Gauge boson W Y X
Squared mass
g2
4 (u
2 + v2) g
2
4 (ω
2 + v2) g
2
4 (ω
2 + u2)
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
m2Z ' g
2(u2 + v2)
4c2W
, (3.2)
m2Z1 '
g2
18
(
(3 + t2X)ω
2 + 4t2N (ω
2 + 9Λ2)
+
√
((3 + t2X)ω
2 − 4t2N (ω2 + 9Λ2))2 + 16(3 + t2X)t2Nω4
)
, (3.3)
m2Z2 '
g2
18
(
(3 + t2X)ω
2 + 4t2N (ω
2 + 9Λ2)
−
√
((3 + t2X)ω
2 − 4t2N (ω2 + 9Λ2))2 + 16(3 + t2X)t2Nω4
)
. (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 + v2)
4c2W
; m2Z1 ' 4g2t22Λ2; m2Z2 '
g2c2Wω
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 13
3.2 Effective potential in the model 3-3-1-1
Hence the Higgs of the model, we obtain the tree-level po-
tential
V0 =
λφ4Λ
4
+
1
4
λ11φ
2
Λφ
2
ω +
λ2φ
4
ω
4
+
φ2Λµ
2
2
+
1
2
µ22φ
2
ω +
λ3φ
4
u
4
+
1
4
λ12φ
2
Λφ
2
u +
1
4
λ6φ
2
uφ
2
ω +
1
2
µ23φ
2
u +
1
4
λ5φ
2
uφ
2
v
+
λ1φ
4
v
4
+
1
4
λ10φ
2
Λφ
2
v +
1
4
λ4φ
2
vφ
2
ω +
1
2
µ21φ
2
v.
(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
Veff (v) = λv
4 − Ev3 +Dv2 + λk.ω2v2 + λj .Λ2v2 + u2.v2
≈ λv4 − Ev3 +Dv2 + λi.(ω2 + Λ2 + u2)v2 (3.7)
The slices of the effective potential in (3.7) at ω2 +Λ2 +u2 =
1 TeV2 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.
3.3 Electroweak phase transition without neutral fermion 14
Λi = 0
Λi = 0.03
Λi = 0.06
0.2 0.4 0.6 0.8 1.0 1.2 1.4
vHTeVL
0.05
0.10
0.15
Veff ITeV
4M
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,Λ
2 + ω2 + v2 = 1 TeV2
From the mass spectra, we can split the masses of particles
into four parts as follows
m2(φΛ, φω, φu, φv) = m
2(φΛ) +m
2(φω) +m
2(φu) +m
2(φv). (3.8)
Taking into account Eqs. (3.6) and(3.8), we can also split
the effective potential into four parts
Veff (φΛ, φω, φu, φv) = Veff (φΛ) + Veff (φω) + Veff (φu) + Veff (φv).
We assume φΛ ≈ φω, φu ≈ φv over space-times. Then, the effective
potential becomes
Veff (φΛ, φω, φu, φv) = Veff (φω) + Veff (φu).
From the mass spectra, it follows that the squared masses
of gauge and Higgs bosons are split into three separated parts corre-
sponding to three SSB stages.
3.3 Electroweak phase transition without neutral fermion
3.3.1 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 po-
tential of the EWPT SU(3)→ SU(2) is Veff (φω).
Veff (φω) = Dω(T
2 − T 20ω)φ2ω − EωTφ3ω + λω(T )
4
φ4ω, (3.9)
T 20ω ≡ − Fω
Dω
. (3.10)
3.3 Electroweak phase transition without neutral fermion 15
S=1
S=2
S=3
S¥
0 1000 2000 3000 4000 5000 6000 7000
0
500
1000
1500
2000
2500
3000
3500
mexotic-quarkCharged Higgs@GeVD
m
H 3
@Ge
VD
Figure 3.2: The mass area corresponds to Sω > 1
The values of Veff (φω) at the two minima become equal at
the critical temperature and the phase transition strength are
Tcω =
T0ω√
1− E2ω/DωλTcω
, Sω =
2Eω
λTcω
.
There are nine variables: the masses of U,D1, D2, H2, H3
and A′η, S′χ, S4, Z1. However, for simplicity, we assume mU = mD1 =
mD2 = mH2 ≡ O, mA′η = mS′χ = 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
λTcω
≡ Sω(O,P, Sω). (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.
S=1
S=3
S=2
0 1000 2000 3000 4000 5000 6000 7000
0
500
1000
1500
2000
2500
mH1@GeVD
m
H 3
@Ge
VD
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 ofX,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
Veff (φu) =
λu(T )
4
φ4u − EuTφ3u +DuT 2φ2u + Fuφ2u. (3.12)
The minimum conditions are
Veff (0) =
∂Veff (φu)
∂φu
∣∣∣∣
u
= 0;
∂2Veff (φu)
∂φ2u
∣∣∣∣
u
= m2Aχ+m
2
H3 +m
2
Sη+m
2
Sρ ,
(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.
S=1
S=1.2
S=2
S=3
S¥
S¥
200 400 600 800 1000 1200
0
100
200
300
400
500
600
mH1@GeVD
m
H 3
@Ge
VD
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
S=1 S=1.3
S=1.2
S=2.1
200 250 300 350 400 450
0
50
100
150
200
mH1@GeVD
m
H 3
@Ge
VD
Figure 3.5: The strength EWPT S =
2Eu
λTc
with Tc must be real.
Table 3.3: Mass limits of particles with Tc > 0.
Strength S K[GeV ] L[GeV ]
1.0-2.12 195 ≤ K ≤ 484.5 0 ≤ L ≤ 209.8
3.3.2 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)
S=1
S=2
S=3
S¥
0 1000 2000 3000 4000
0
200
400
600
800
1000
mExotic QuarkCharged Higgs@GeVD
m
H 3
@Ge
VD
Figure 3.6: The strength EWPT Sω =
2Eu
λTc
with ω = 6 TeV.
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 mZ2 [TeV] mN−R[TeV] SMax without NR SMax withNR
SU(3)→ SU(2) (i) 2.386 2.227 70 50
SU(3)→ SU(2) (ii) 2.254 1.986 100 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.
The strength S =
2E
λTc
gets the value equals 70. For the picture (ii),
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
value of strength EWPT S, because S =
E
2λTc
and E do not depend
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 dis-
tribution) 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 gen-
erated 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 num-
ber 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 criti-
cal temperature and the strength is gauge independent [44-46]. Con-
sequently, 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 diago-
nalization, 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 men-
tioned 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 transi-
tion 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
2. We examined the EWPT in the 3-3-1-1 model with twocases.
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. Con-
sidering 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.
Phase transition SU(2)→ SU(1) at value u = 246√
2
TeV. The
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 ,

Các file đính kèm theo tài liệu này:

- the_electroweak_phase_transition_in_the_zee_babu_and_su3c_su.pdf