Luận án Một số tính chất của neutrino thuận thang điện yếu

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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Neufeld (1986), “Relationship between longitudinally polarized vector bosons and their unphys- ical scalar partners”, Phys. Rev. D 34 3257. [89] S. L. Glashow (1961), “Partial symmetries of weak interactions”, Nucl. Phys. 22 579. [90] W. L. Benjamin, C. Quigg and H. B. Thacker (1977), “Weak interactions at very high energies: The role of the Higgs-boson mass”, Phys. Rev. D 16 1519. [91] J. Krog and C. T. Hill (2015), “Is the Higgs boson composed of neutrinos?”, Phys. Rev. D 92 9 [arXiv:1506.02843 [hep-ph]]. [92] T. P. Cheng and L. F. Li (1982), Gauge theory of elementary particle physics, Oxford. 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. 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