Properties and the role of right-handed neutrino in the EWνR
model were clearly analysed in every chapter of the thesis. Explicitly,
right-handed neutrino belongs to the doublet SU(2)W; is non-sterile
particle and interacts with weak boson W and Z; can be producted
and detected in colliders such as LHC and ILC with decay products
consisting of two like-sign leptons in the SM; condenses at the order
of O(TeV) when ανR = ανCR = π; condensate state of right-handed
neutrino is one of the agents of DEWSB, is directly related to the mass
generation for the fundamental Higgs χ0 and indirectly related to that
of the others: gauge boson W; Z and fermions in the EWνR model;
right-handed neutrino especially plays an important role in explaining
for smallness of neutrino masses since right-handed neutrino has a
direct and indirect connection with Majorana mass and Dirac mass,
respectively.

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HUE UNIVERSITY
COLLEGE OF EDUCATION
PHYSICS DEPARTMENT
NGUYEN NHU LE
PROPERTIES OF RIGHT-HANDED
NEUTRINOS
Speciality: Theoretical and Mathematical Physics
Code: 62 44 01 03
SUMMARY OF DOCTORAL THESIS
HUE - 2016
The doctoral thesis is accomplished at:
Physics Department, College of Education, Hue
University.
Supervisers:
1. Prof. Dr. Pham Quang Hung, University of
Virginia, United States of America
2. Dr. Vo Tinh, Physics Departent, College of
Education, Hue University
Referee 1: Assoc. Prof. Dr. Nguyen Quynh Lan,
Physics Department, Hanoi University of Education
Referee 2: Assoc. Prof. Dr. Nguyen Anh Ky,
Theoretical Center, Vietnam Institute of Physics
The thesis was defensed at Hue University of
Education
The thesis can be found at:
HUE – 2016
iTABLE OF CONTENTS
Table of contents . . . . . . . . . . . . . . . . . . . . . . . i
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 1. FUNDAMENTAL KNOWLEGDE . . . . . . . 4
1.1 Gauge theory . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The SM for the electroweak interaction . . . . . . . . . 5
Chapter 2. THE EWνR MODEL . . . . . . . . . . . . . . 6
2.1 The neutrino particle . . . . . . . . . . . . . . . . . . 6
2.2 The neutrino mass . . . . . . . . . . . . . . . . . . . . 6
2.3 The see-saw mechanism . . . . . . . . . . . . . . . . . 7
2.4 The Left-Right symmetric model . . . . . . . . . . . . 7
2.5 The EWνR model . . . . . . . . . . . . . . . . . . . . 7
Chapter.CONDENSATE STATES IN THE EWνR MODEL 10
3.1 Non-relativity theory for condensate states . . . . . . . 11
3.2 SD equations approach to condensate states of fermions
in the EWνR model . . . . . . . . . . . . . . . . . . . 11
3.3 One-loop β function for Yukawa couplings in the EWνR
model . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter4. DYNAMICAL ELECTROWEAK SYMMETRY
BREAKING IN THE EWνR MODEL . . . . . . . . . 15
4.1 Dynamical electroweak symmetry breaking . . . . . . . 16
4.2 Dynamical electroweak symmetry breaking in the EWνR
model . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3 The mass of the Higgs particle . . . . . . . . . . . . . 19
4.4 Neutrino mass . . . . . . . . . . . . . . . . . . . . . . 20
LIST OF PUPLISHED RESEARCH PAPERS INCLUDED
IN THE THESIS . . . . . . . . . . . . . . . . . . . . . 23
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . 25
1INTRODUCTION
1. Significance of the study
The neutrino oscillation discovered by the Super Kamiokande collab-
oration [8] is one of the experimental evidence supporting physics be-
yond the standard model (SM). The EWνR model [13] prosposed by
Pham Quang Hung is one of several models in which neutrinos can
obtain masses. The EWνR model keeps the same gauge group as the
SM but increases its fermion as well as its Higgs content to satisfy
the condition: the massive state of neutrino has a small mass which
scale is of the order of the electroweak scale ΛEW . The right-handed
neutrino can be then detected and its Majorana nature can be tested
in an experiment. The EWνR model is supposed to be validity in theo-
retical modern particle physics since this model passes the electroweak
precision data test very well and satisfies the experiment data of the
125 GeV Higgs boson [17]. This is a clear indication that, the full the-
oretical construction of the EWνR model plays a quintessential role
in explaining phenomena of high energy physics. The first version of
EW-scale νR model was introduced the see-saw mechanism in which
the focus of interest was an explanation of the tiny mass of neutrino.
Nevertheless, a theory of dynamical electroweak symmetry breaking
(DEWSB) by which the Higgs fields develop VEVs has not been ex-
plicitly discussed. Properties and the role of right-handed neutrino in
the generating mass mechanism have not been investigated yet. With
these hotly-debated topics, I chose the thesis title: “Some proper-
ties of right-handed neutrinos” .
2. Statement of the problem
2It is generally agreed that the SM is an incomplete theory since there
remains many questions of which necessitates a framework that goes
beyond the SM (BSM). The nature of electroweak symmetry breaking
is one of them. The SM is spontaneously broken by the Higgs potential
of the form V (φ) = −µ2φ+φ + λ (φ+φ)2, where φ is an elementary
scalar field. This leaves many often-asked questions such as: why µ2
is positive, or the hierarchy problem: why the electroweak scale v is
smaller than the Plank scale MP by many orders of magnitude. The
most popular way to deal with this problem is to use the cancellation
between the quadratically-divergent contributions of fermion and that
of boson proposed in some interesting models come along with the
Supersymmetry (SUSY), Little Higgs, Twin Higgs, etc... [23]. Another
idea can be found in the idea of Large Extra Dimensions, Higgsless
models [23] where the extra dimensions play an essential rule to avoid
the hierarchy problem. Another class of models that does not involve
elementary scalar fields is one in which symmetry breaking is realized
dynamically through condensates of bilinear fermion fields. There are
many models of this type such as: composite Higgs models, Technicolor
(TC), Extended TC, top-color, etc...[23] and the EW-scale νR model
follows this direction in a framework based on the presented in Ref. [32].
At a high energy scale, right-handed neutrinos and mirror fermions in
our model will condense through Yukawa interactions of Higgs triplet
and Higgs doublet, respectively.
3. Objectives of the study
• Find the conditions for which right-handed neutrinos and mirror
fermions in the EWνR model condense and the corresponding en-
3ergy scales.
• Construct the DEWSB mechanism for the EWνR model. Give
an "explanation" for the smallness of neutrino mass through the
DEWSB mechanism.
• Analyze properties and the role of right-handed neutrinos in every
each chapter.
4. Contents of the study
• Find the conditions for Yukawa couplings at which the correspond-
ing condensate states get formed.
• Find the one-loop beta functions for Yukawa couplings of right-
handed neutrinos and mirror fermions.
• Find the numerical solutions to the renornalization group equa-
tions and the energy scale at which right-handed neutrinos and
mirror quarks condense.
• Construct the DEWSB mechanism in the EWνR model; Describe
the mass generation for neutrino though the see-saw mechanism
in the EWνR model.
5. Limitation of the study
We restrict our discussion to the electroweak interactions in the EWνR
model.
6. Method of the study
Research approachs of the thesis consist of methods of quatum field
theory such as gauge principle, Green functions, renormalization group
equations, Feynman rules and numerical methods.
47. Scientific and practice value of the study
The results of the study undoubtedly contribute to an important part
of an attempt to investigate the nature of the Higgs mechanism, the
mass generation for matter. The DEWSB mechanism is proposed and
the smallness of neutrinos masses is dynamically explained. Moreover,
the study results give an orientation and valuable imformation for ex-
perimental physicists to detect fermions in the EWνR model.
8. Thesis organization
Beside the important parts such as: introduction, conclusion, list of fig-
ures, list of published research papers included in the thesis, references
and appendix, the thesis content is presented in 4 chapter. The gauge
theory and the SM in general are performed in chapter 1. Chapter 2
gives an overview on the models that generate masses for neutrinos
and the EWνR model. In chapter 3, we study the properties of the
condensates and the scales at which fermions in the EWνR model con-
dense through analytical formula and numerical results of the one-loop
β functions for the Yukawa couplings in the EWνR model. Finally, the
study of DEWSB in the EWνR model is included. In this way one
achieves the properties and the role of right-handed neutrinos.
Chapter 1
FUNDAMENTAL KNOWLEGDE
1.1 Gauge theory
In quantum field theory and particle physics, the Noether’s theorem
implies the symmetry [1]: if an action is invariant under some group
5of transformations, then there exist one or more conserved quantities
which are associated to these transformations. In this sense, Noether’s
theorem establishes that symmetries imply conservation laws or sym-
metry could imply dynamics.
1.2 The SM for the electroweak interaction
The SM for the electroweak interaction can be investigated in the SM
Lagrangian, in which properties of matter fields, gauge boson, Higgs
boson and their mutual interactions are fully treated.
Lgauge + Lvh = −1
4
FµνF
µν − 1
2
W+µνW
−µν + M 2WW
+
µ W
−µ
−1
4
ZµνZ
µν + M 2ZZ
+
µ Z
−µ +
1
2
∂µH∂
µH
−1
2
M 2HH
2 + W+W−A + W+W−Z
+W+W−AA + W+W−ZZ + W+W−AZ
+W+W−W+W− + HHH + HHHH
+W+W−H + W+W−HH + ZZH
+ZZHH , (1.130)
Llepton + Ley =
∑
e
e¯ (i 6 ∂ −me) e +
∑
νe
ν¯e(i 6 ∂)νe
+ e¯eA + ν¯eeW
+ + e¯νeW
−
+ e¯eZ + ν¯eνeZ + e¯eH , (1.131)
Lquark =
∑
q=u,...,t
q¯(i 6 ∂)q, (1.132)
6LqY = −
3∑
i,j=1
[
guiju¯Ri
(
Φ˜+qLj
)
+ gdiju¯Ri
(
Φ+qLj
)]
(1.133)
Lquark + LqY =
∑
q=u,...,t
q¯(i 6 ∂ −mq)q + q¯qA + u¯d′W+
+ d¯′uW− + q¯qZ + q¯qH . (1.134)
Chapter 2
THE EWνR MODEL
2.1 The neutrino particle
The existence of the neutrino particle was first postulated by W. Pauli
in a letter to the physics meeting in Tubinge, December 4, 1930. The
original purpose was to desperately save the energy conservation law in
the β-decay process. In 1998, the neutrino oscillations were discovered
by the Super-Kamiokande laboratory [8]. This result would be regarded
as the first experimental evidence supporting a non-zero mass for the
neutrino.
2.2 The neutrino mass
In the SM, neutrino masses are zero since right-handed neutrinos do
not exist. The right-handed neutrinos are then introduced into the SM.
Unlike charged leptons and quarks, neutrinos can have two types of
masses, Dirac mass and Majorana mass. In both two cases, the values
7of Yukawa couplings, however, are at the order of gνe ∼ O(10−11) or
v∆ has to be very tiny.
2.3 The see-saw mechanism
There has been a variety of theories going beyond the SM to explain
the neutrino mass, among which most elegently are the ideas of the
models of the see-saw mechanism [12]. These models can be categorized
into three types I, II, and III. In both three types, the massive states
of neutrinos or the additional Higgs fields, however, have huge masses
then they can not be detected in the current colliders.
2.4 The Left-Right symmetric model
In the left-right (LR) symmetric extension of the SM, parity is a fun-
damental symmetry. In order to break it spontaneously, ones have to
enlarge the gauge group. Nevertheless, according to the recent results
of the CMS laboratoty [54], the gauge boson WR in the LR model can
not be tested with an experiment since the mass of WR is very large,
MWR ≥ 3 TeV.
2.5 The EWνR model
Phạm Quang Hưng proposed the EWνR model which gauge group is
the same as that of the SM: SU(3)C × SU(2)W × U(1)Y [13]. The
right-handed neutrino mass has the order of the electroweak scale and
8Bảng 2.4: Fermion fields in the EW-scale νR model.
Fermion SU(2)W × U(1)Y Fermion SU(2)W × U(1)Y
trong SM gương
lL =
νL
eL
(2,−1
2
)
lMR =
νR
eMR
(2,−1
2
)
eR (1,−1) eML (1,−1)
qL =
uL
dL
(2, 1
6
)
qMR =
uMR
dMR
(2, 1
6
)
uR
(
1,
2
3
)
uML
(
1,
2
3
)
dR
(
1,−1
3
)
dML
(
1,−1
3
)
νR can be detected in the current colliders. The fermion contents are
listed in the table 2.4.
In the electroweak see-saw mechanism, the existence of the Majo-
rana mass term ofMRν
T
Rσ2νR breaks the electroweak gauge symmetry.
The bilinear lM,TR σ2l
M
R contains the term ν
T
Rσ2νR and transforms un-
der the SU(2)W × U(1)Y symmetry as
(
1 + 3,
Y
2
= −1
)
. The Higgs
which couples to this bilinear, therefore, cannot be a singlet of SU(2)W
which carries the quantum number of
(
1,
Y
2
= +1
)
since this singlet
charged scalar cannot develop a VEV. This leaves the triplet Higgs
as a suitable scalar which can couple to the aforementioned bilinear.
Explicitly, χ˜ is given as
χ˜ =
1√
2
~τ · ~χ =
(
1√
2
χ+ χ++
χ0 − 1√
2
χ+
)
. (2.1)
9However, if our model contains only one aforementioned triplet then
ρ =
1
2
. This immedietly leads to the fact that at tree level, the
ρ ≈ 1 constraint is no longer satisfied. To preserve custodial sym-
metry [55], the EWνR model introduced an additional Higgs triplet,
ξ =
(
3,
Y
2
= 0
)
. To generate a neutrino Dirac mass term, a sin-
glet Higgs field φS was introduced [13]. SM quark and charged lepton
masses are obtained by a coupling to a Higgs doublet Φ2 [13] and those
of mirror quarks and charged leptons come from a coupling to a second
Higgs doublet Φ2M [19]. The latter was needed [19] in order to accom-
modate the discovery of the 125-GeV scalar at the LHC. Higgs fields
transforming under SU(3)c × SU(2)W × U(1)Y are listed as follows.
χ˜ =
1√
2
~τ · ~χ =
(
1√
2
χ+ χ++
χ0 − 1√
2
χ+
)
=
(
1, 3,
Y
2
= 1
)
, (2.57)
ξ =
ξ+
ξ0
ξ−
= (1, 3, Y2 = 0
)
. (2.58)
These two Higgs triplets can be combined as following [19]
χ =
χ0 ξ+ χ++
χ− ξ0 χ+
χ−− ξ− χ0∗
, (2.59)
Φ1 =
(
φ+1
φ01
)
=
(
1, 2,
Y
2
=
1
2
)
, (2.60)
Φ2 =
(
φ+2
φ02
)
=
(
1, 2,
Y
2
=
1
2
)
, (2.61)
10
φS = (1, 1,
Y
2
= 0). (2.62)
The interaction between fermion and Higgs fields has the form
LY SM = −gijΨ¯LiΦ2ΨRj + h.c., (2.63)
LeM = −gMe l¯MR Φ2MeML + h.c., (2.64)
LdM = −gMd q¯MR Φ2MdL − gMd q¯MR Φ˜2MdML + h.c., (2.65)
LνR = gM lM,TR σ2τ2χ˜lMR , (2.66)
LSe = −gSel¯LlMR φS + h.c., (2.67)
LSq = −gSqq¯MR qLφS − g′Sqq¯ML qRφS + h.c.. (2.68)
The reason why the EWνR model is highly evaluated by particle
physicsists and validity in theoretical modern particle physics mostly
lies on the characteristics of right-handed neutrinos. Right-handed neu-
trinos in the EWνR model have the following main properties in mind:
• Right-handed neutrinos belong to the doublet SU(2)W and their
parteners are mirror charged leptons.
• Right-handed neutrinos in the EWνR model are non-sterile and
couple to the Z and W bosons.
• In the see-saw mechanism of the EWνR model, right-handed neu-
trino is a Majorana particle and its mass has the order of the
electroweak scale, ΛEW then it can be produced in the colliders
such as LHC and ILC.
Chapter 3
CONDENSATE STATES IN THE EWνR MODEL
11
3.1 Non-relativity theory for condensate states
Non-relativity theory states that when fermions have sufficient large
masses, the corresponding condensate states will get formed through
the Yukawa interaction with a scalar field. The fourth generations of
quarks and leptons satisfy conditions of forming condensates as the
Yukawa couplings are sufficient large. This undoubtedly occurs be-
cause the fourth generations of fermions are massive which order is
ΛEW . Since mirror fermions and right-handed neutrino masses have
the same order with that of fourth generations, conditions of conden-
sation in non-relativity limit for fermions in the EWνR model can be
obtained similar to the case of the fourth generations. However, par-
ticles considered in my thesis are relativity. Then the condensation of
fermions in the EWνR model will be studied by using SD equations
approach.
3.2 SD equations approach to condensate states of fermions
in the EWνR model
The formation of condensate states of fermions in the EWνR model
will be studied in the framework presented in Ref. [32]. The role of
the condensation of fermions in the EWνR model in DEWSB will be
discussed in detail in chapter 4.
Since top quark is too light to form condensates, then SD equa-
tions for self-energy of fermions in the SM will not be considered in
this section. Beside, according to the recent study of lepton number vi-
olating processes, µ→ eγ [21], gSe is constrained to be less than 10−3
12
and if we assume gSe ∼ gSq ∼ g′Sq, then equations (2.67) and (2.68)
will not be investigated. For this reason, in the following analysis we
have made the following assumptions:
• It will be assumed in our discussion of condensate formation that
the fundamental Higgs fields are massless. These fields then have
no VEV at tree level.
• To preserve custodial symmetry SU(2)D (will be presented in
detail in chapter 4), VEV of quark bilinears satisfy the condi-
tion:
〈
U¯ML U
M
R
〉
=
〈
D¯ML D
M
R
〉
. For this reason, we asume that
guM = gdM = gqM .
• Yukawa couplings of mirror leptons are not sufficiently large to
form condensates.
Hence, two types of condensates are considered here: that which is
generated by the exchange of the fundamental Higgs triplet χ˜ between
two right-handed neutrinos and the other which is generated by the
exchange of the fundamental Higgs douplet Φ2M between two mirror
quarks.
ΣνR(p) =
g2M
(2pi)4
∫
d4q
1
(p− q)2
ΣνR(q)
q2 + Σ2νR(q)
. (3.18)
SD equation for self-energy of mirror quark is given by
ΣqM (p) = 2×
gM2q
(2pi)4
∫
d4q
1
(p− q)2
ΣqM (q)
q2 + Σ2
qM
(q)
. (3.32)
When αgM , αqM are larger than the corresponding critical value α
c
νR
=
pi, αcqM =
pi
2
, the solutions to equations (3.18) and (3.32) have the form
of bound states.
13
The energy scale of condensate has a relationship with vχ and
vΦ2M
〈νTRσ2νR〉 ∼ O(−v3χ), (3.37)
〈q¯ML qMR 〉 ∼ O(−v3φ2M ), (3.38)
where vχ and vΦ2M (will be presented in chapter 4) are VEV of χ and
Φ2M . The two quantities given in equations (3.37) and (3.38) will be
the agents of DEWSB, give masses to the Higgs fields and fermions in
the EWνR model.
The energy cutoff of the fermion condensation in the EWνR
model does not appear the fine tuning picture, explicitly, Λ ∼ O(TeV).
DEWSB at TeV scale is a certain consequence of the fact that the
EWνR model is based on fermion condensate states and Yukawa cou-
plings of corresponding fermions satisfy the condensation conditions at
this scale. How to investigate the evolution of Yukawa couplings? This
issue will be performed in the following section.
3.3 One-loop β function for Yukawa couplings in the EWνR
model
The beta functions βgM , βgqM and βgeM are given as [83]
βgM =
dgM
dt
=
13g3M
32pi2
+
g2
eM
gM
32pi2
, (3.49)
βg
qM
=
dgqM
dt
=
3g3
qM
8pi2
, (3.56)
βg
eM
=
dgeM
dt
=
11g3
eM
32pi2
+
3g2MgeM
64pi2
. (3.63)
14
The differential equations (3.49), (3.56) and (3.63) can be solved
numerically [83] and the results depend on initial values of Yukawa cou-
plings. However, one may query the implications of the initial Yukawa
couplings chosen once solutions to RGEs was found. Why some observ-
able quantities such as masses were not used as initial values in this
situation. The answer would rely on our current paper’s hypothesis
that matter acquires no mass until DEWSB. Therefore, these initial
Yukawa couplings would infer some physical quantities from a differ-
ent point of view where EWSB occurs, say, the naive masses [32].
From now, the solution to RGEs can be investigated by using initial
values of naive mirror fermion and right-handed neutrino masses in
the EWνR model. Explicitly, Fig. 3.17 shows the situation where the
Hình 3.17: The evolution of Yukawa couplings where the initial masses of νR, e
M and qM are
200 GeV, 102 GeV and 202 GeV, respectively. The blue and green arrows indicate the energy
values where the Higgs triplet χ and Higgs doublet Φ2M correspondingly get VEVs [84].
initial masses of νR, e
M and qM are 200 GeV, 102 GeV and 202 GeV,
15
respectively. As shown in Fig. 3.17, the Yukawa couplings increase dra-
matically as energy increases and Ladau pole singularities appear at
t = 1.50 (E = 2.89 TeV). By using the critical values of Yukawa cou-
plings found above, we immediately find that the values of t at which
condensates of right-handed neutrino and mirror quark arise are esti-
mated to be 1.19 and 1.09, respectively, i.e. at the order of O(1 TeV).
Then the Higgs fields χ,Φ2M will get VEV through condensations of
fermions.
The properties and the role of right-handed neutrino in studying
condensate states of fermions in the EWνR are shown as followings
• The SD equations for self-energy of right-handed neutrino state
that when the critical Yukawa coupling αcνR = pi, the Yukawa
interaction system between χ˜ and right-handed neutrino becomes
condenstes.
• Since right-handed neutrino belongs to doublet SU(2)W , the beta
function βgM in equation (3.49) depends on Yukawa coupling of
mirror charged lepton geM and vise versa, βeM given in (3.63) is
also in terms of gM . The energy scales at which two fermions
form condensates are mutually independent and at the order of
O(1 TeV).
• Since right-handed neutrino interacts with the triplet Higgs field
χ˜ through Yukawa coupling, then the corresponding condensate
state is directly related to the VEV of χ0 and one of the agents of
DEWSB in the EWνR model.
16
Chapter 4
DYNAMICAL ELECTROWEAK SYMMETRY
BREAKING IN THE EWνR MODEL
4.1 Dynamical electroweak symmetry breaking
The concept of DEWSB is throughoutly discussed in theoretical mod-
ern particle physics. The underlying physics of this issue would come
from pairing interaction of fermion through a scalar and DEWSB in
the EWνR model follows this direction.
The scale of EWSB can be determined by studying the scarttering
WLWL, explicitly
Λ2SB ≤
8
√
2pi
3GF
≈ (1.0 TeV)2. (4.11)
Information about the symmetry breaking sector is to be found by
considering WLWL-scattering because the longitudinal modes of the
gauge bosons are exactly part of the new physics which breaks the
electroweak symmetry.
4.2 Dynamical electroweak symmetry breaking in the
EWνR model
To study DEWSB, the elementary scalar fields in the EW-scale νR
model are assumed to have no VEVs at tree level [32]. The scale-
invariant potential for fundamental Higgs can be written as [84]
Vf = Vf(Φ2,Φ2M , χ) = λ1
[
TrΦ+2 Φ2
]2
+ λ2
[
TrΦ+2MΦ2M
]2
17
+λ3
[
Trχ+χ
]2
+ λ4
[
TrΦ+2 Φ2 + TrΦ
+
2MΦ2M + Trχ
+χ
]2
+λ7
[(
TrΦ+2 Φ2
) (
TrΦ+2MΦ2M
)− (TrΦ+2 Φ2M) (TrΦ+2MΦ2)]
+λ6
[ (
TrΦ+2MΦ2M
) (
Trχ+χ
)− 2(TrΦ+2M τ a2 Φ2M τ b2
)
× (Trχ+T aχT b) ] + λ8 [3Trχ+χχ+χ− (Trχ+χ)2]
+λ5
[ (
TrΦ+2 Φ2
) (
Trχ+χ
)− 2(TrΦ+2 τ a2 Φ2τ b2
)
× (Trχ+T aχT b) ], (4.12)
from which the Higgs fields have no mass terms. However, under the
condensate scale, the Higgs potential is no longer scale-invariant and
the Higgs fields will acquire masses due to the appearance of νR and
mirror quark condensates. Explicitly, the negative effective mass squared
terms for the fields χ0 and φ02M in the Higgs effective potential can be
written as
1
2
g2M
ΣνR(0)
〈νTRσ2νR〉
∣∣χ0∣∣2 , (4.16)
g2
qM
ΣqM (0)
〈u¯ML uMR 〉
∣∣φ02M ∣∣2 . (4.17)
The Feynman diagrams which correspond with equations (4.16) and
Hình 4.1: Diagram giving masses to (a) χ0, (b) φ02M [84].
(4.17) are illustrated in Fig. 4.1.
18
In the case of the fundamental Higgs ξ which has no interaction
with fermions, a mass term for ξ0 can be obtained through quadratic
interactions with χ0 và φ02M . The corresponding Feynman diagrams
are shown in Fig. 4.19a. Contributions of induced µ2 of ξ0 to the Higgs
effective potential involving the neutral fundamental scalar ξ0 are as
follow
1
2
{
g2M
ΣνR(0)
〈νTRσ2νR〉I(1)χ +
2g2
qM
ΣqM (0)
〈u¯ML uMR 〉I(1)φ2M
}∣∣ξ0∣∣2 . (4.18)
Similar, the Feynman diagrams giving masses to φ02 are shown in
Hình 4.19: Diagrams give masses to (a) ξ0, (b) φ02 [84].
Fig. 4.19b and one has the same form for
∣∣φ02∣∣2 term as follow
1
2
{
g2M
ΣνR(0)
〈νTRσ2νR〉I(2)χ +
2g2
qM
ΣqM (0)
〈u¯ML uMR 〉I(2)φ2M
}∣∣φ02∣∣2 . (4.21)
19
One can see that the VEV’s vanish when the condensates vanish.
The presence of the terms in Eqs. (4.16), (4.17), (4.18) and (4.21) en-
ables the fundamental Higgs fields to get non-zero VEVs. The VEVs of
χ, ξ,Φ2 and Φ2M are assumed to be vχ, vξ, vΦ2 and vΦ2M , respectively.
When χ, Φ2 and Φ2M get VEVs, the symmetry is dynamically broken
from global SU(2)L⊗ SU(2)R down to the custodial SU(2)D. At tree
level, the gauge boson masses are obtained by kinetic part of the Higgs
Lagrangian.
4.3 The mass of the Higgs particle
After the spontaneous breaking of SU(2)W × U(1)Y → U(1)EM , the
Higgs fields of the EWνR model are as follows
• One five-plet: H±±5 , H±5 , H05 .
• Two triplets: H±3 , H03 và H±3M , H03M .
• Three singlets: H01 , H01M , H0
′
1 .
In light of the discovery of the 125-GeV SM-like scalar, it is imper-
ative that any model beyond the SM (BSM) shows a scalar spectrum
that contains at least one Higgs field with the desired properties as
required by experiment. As discussed in Ref. [19], both scenarios Dr.
Jekyll and Mr. Hyde satisfy the experiment data of 125-GeV Higgs
boson.
20
4.4 Neutrino mass
The Dirac mass term of neutrino is given in eq. (2.67) as
LSe = −gSel¯LlMR φS + h.c.
= −gSe
(
ν¯LνR + e¯Le
M
R
)
φS + h.c., (4.49)
where VEV of φS is 〈φS〉 = vS. The Dirac mass then has the form
mDν = gSevS. (4.50)
The Majorana mass of neutrino can be derived from Yukawa in-
teraction given in (2.66) when χ get VEV
MR = gMvχ. (4.57)
The tiny mass of neutrino coming from see-saw mechanism in the
EWνR model is given as
mν =
(
mDν
)2
MR
=
g2Sev
2
S
MR
. (4.63)
When condensate states get formed, the fundamental Higgs singlet φS
simultaneously develops VEV which gives a Dirac mass to the neutrino.
Since the light neutrino masses ∼ m2D/MR are constrained to be <
O(eV) and MR ∼ O(ΛEW ), it follows that mD = gSlvS < O(105 eV).
From [21], gSl is constrained to be less than 10
−3 using the present
upper bound on the rate of µ→ eγ. This implies that vS < 100 MeV <
ΛEW . The smallness of vS is a certain consequence of gSe ∼ gSq ∼
g′Sq ≤ 10−3. The tiny neutrino mass and the hierarchy of vS and ΛEW
can be dynamically explained through DEWSB in the EWνR model.
Right-handed neutrino νR plays a crucial rule in DEWSB and
see-saw mechanism of the EWνR model. Explicitly,
21
Hình 4.22: Diagram giving VEV to φS: (a) from right-handed neutrino self-energy, (b) from
mirror quark self-energy. [84].
• The condendate state of νR is one of the agents of DEWSB. The
fundamental Higgs χ0 develops VEV through µ2 term of 〈νTRσ2νR〉
in the Higgs effective potential.
• Condendate states of νR and mirror quark generate mass for fun-
damental Higgs field φ02 (interacts with fermions in SM) and ξ
0
(do not interact with fermions) through quadratic interactions.
• The condendate state of νR involves in developing VEV of singlet
Higgs field φS through DEWSB in the EWνR model. The smallness
of vS is "naturally" a result of small value of Yukawa coupling
gSl. This can be dynamically explained by considering Feynman
diagrams giving VEV to φS.
• Right-handed neutrino plays an important role in explaining the
smallness of neutrino since its mass is of the order of ΛEW and
directly related to neutrino Dirrac mass.
22
CONCLUSION
By using numerical method and methods of quantum field theory such
as gauge principle, Green functions, renormalization group equation
and Feynman rules, we definitely gain the aims of the thesis. Main
results obtained can be briefly presented as follows
1. Solved SD equations for self-energy of mirror quark and right-
handed neutrino. The critical Yukawa couplings where the correspond-
ing condensate states get formed have been found. Explicitly, when
the Yukawa couplings are sufficiently large and exceed critical values:
αcνR = pi and α
c
qM =
pi
2
, solutions of SD equations satisfy condensate
conditions.
2. Obtained analytical fomula of β functions for Yukawa cou-
plings of mirror fermions and right-handed neutrino. Found numer-
ical solutions to renormalization group equations. The results state
that fermions in the EWνR model, explicitly, right-handed neutrino
and mirror quark satisfy conditions of the condensation at the scale
O(TeV). With this scale, there does not appear a fine-tuned picture of
momentum cutoff in our model.
3. Constructed the DEWSB mechanism and presented the under-
lying physics of the Higgs mechanism, the mass generation for matter.
Explicitly, the Higgs fields χ,Φ2,Φ2M and φS will acquire masses when
right-handed neutrino and mirror quark condense at the scale O(TeV).
The symmetry SU(2)L×SU(2)R of the EWνR model is then dynami-
cally broken down to SU(2)D. And, as a result, the weak bosonsW,Z
and fermions in our model will acquire masses.
4. Explained the nature of origin of neutrino masses and why
23
they are so tiny through the see-saw mechanism in the EWνR model.
The small VEV of the singlet Higgs φS is a certain consequence of
the smallness of Yukawa couplings gSe, gSq và g
′
Sq. Hence, we avoid
difficulties of the privious models in the "explanation" for tiny neutrino
masses and the hierarchy between the VEV of φS and the electroweak
scale ΛEW in the EWνR model. This result plays an important role in
modern theoretical physics.
5. Properties and the role of right-handed neutrino in the EWνR
model were clearly analysed in every chapter of the thesis. Explicitly,
right-handed neutrino belongs to the doublet SU(2)W ; is non-sterile
particle and interacts with weak boson W and Z; can be producted
and detected in colliders such as LHC and ILC with decay products
consisting of two like-sign leptons in the SM; condenses at the order
of O(TeV) when ανR = αCνR = pi; condensate state of right-handed
neutrino is one of the agents of DEWSB, is directly related to the mass
generation for the fundamental Higgs χ0 and indirectly related to that
of the others: gauge boson W,Z and fermions in the EWνR model;
right-handed neutrino especially plays an important role in explaining
for smallness of neutrino masses since right-handed neutrino has a
direct and indirect connection with Majorana mass and Dirac mass,
respectively.
Beside contributions to the content, the validity of direction using
condensate states to construct DEWSB is also presented in my the-
sis. Explicitly, the Higgs fields used in DEWSB are composites. This
direction is more particularly attractive since there is also a recent in-
terest concerning the possibility that the Higgs boson is a composite
of neutrinos [91].
24
LIST OF PUPLISHED RESEARCH PAPERS
INCLUDED IN THE THESIS
1. Nguyen Nhu Le, Pham Quang Hung (2014), “One-Loop β Func-
tions for Yukawa Couplings in the Electroweak-Scale Right-Handed
Neutrino Model”, J. Phys.: Conf. Ser. 537 012016.
2. Nguyen Nhu Le, Pham Quang Hung (2016), “Schwinger-Dyson
equations for fermions self-energy in the electroweak-scale right-
handed neutrino model”, Hue University’s Journal of Natural
Science 116 02.
3. Pham Quang Hung, Nguyen Nhu Le (2016), “Dynamical Elec-
troweak Symmetry Breaking in the model of electroweak-scale
right-handed neutrinos”, Int. J. Mod. Phys. A 31 1650065.
25
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