Simple 3 - 3 - 1 model and 3 - 2 - 2 - 1 model for dark matter and neutrino masses

ristic that is the identity of three quark/ lepton generations. This means the physics of the generations is identical, then we can consider the interactions of only one generation (example the first generation), the interactions of other generations can be found out similarly. However, the experimental data of B meson decay anomalies have shown: RD∗ = Γ(B¯ → D∗ τ ν˜) Γ(B¯ → D∗ l ν˜) = 0.310± 0.015± 0.008 , RD = Γ(B¯ → D τ ν˜) Γ(B¯ → D l ν˜) = 0.403± 0.040± 0.024, l = e, µ, (1) with 3.5σ in comparison to the SM prediction: RD∗ = 0.252± 0.004, RD = 0.305± 0.012. (2) The above results provide hints for violation of the lepton flavor universality (LFU). Thus, a series of expanded models of MHC was developed in 2016 to further explain the newly published experimental results of the B meson decay anomalies. As mentioned above, these models must also fully explain the data of neutrinos and DM. One of those models is the G221. This model appropriately explains all data of the meson B decay anomalies. The G221 model contains the electroweak groups SU(2)1 and SU(2)2 in turn breaking at the high energy scales and then 7at the electroweak scale of SM. The initial fermions in the model included light fermions with left- and right-handed components transform as the singlets under the SU(2)1, while these left- and right-handed components respectively transform as the doublets and singlets under the SU(2)2, which is similar to SM. The heavy fermions with the left- and right-handed components are added in this model which they transform as singlets under the SU(2)2, dou- blets under the SU(2)1, so called vector-like fermions. Beside one Higgs doublet of the SU(2)2 as in SM, the model also has one dou- blet of the SU(2)1 and one self-dual bidoublet of both SU(2)1 and SU(2)2. Yukawa interactions between fermions and Higgs bosons give the fairly complex mass matrices of leptons and quarks. The consequence is the mixing matrices of fermion states give differ- ent mixing angles for different fermion generations. Thus, these physical fermions will interact differently with the gauge bosons in the model, thereby explaining the experimental results of the B meson decay. However, the neutrino mixing matrix in this model always gives the zero mass eigenvalue of light neutrinos, which is in conflict with the experimental results of neutrino oscillations. The above model also considers the Z2 symmetry, but this symmetry is softly-broken to ensure generating the right masses for the charged scalars in the model. Therefore, the model also does not contain the DM candidate. However, the possibility of DM in some of its extensions will be discussed. In order to solve the problem of neutrino mass in G221, we have found that the model contains two Higgs doublets, which are well suitable to the mechanism of neutrino mass generation as in the Zee model. An important thing is that when we apply the mechanism of neutrino mass generation as in the Zee model we only need to add a pair of singly charged scalars so that there is no new symmetry breaking scale. Thus, in addition to the neutrinos have very small masses produced by radiative corrections, all the mass, the mixing matrix, and the physical eigenstates of remaining particles are unaffected. Because of the current and urgent of the issues presented above, I chose emph “ The 3-3-1 simple model and the 3-2-2-1 model for dark matter and neutrino masses ”. The thesis focuses on two major issues: searching DM and generating of neutrino masses in the simple 3-3-1 model and the 3-2-2-1 model. 8The purposes of research: • Searching of dark matter in the proposed model called the simple 3-3-1 model (S331M). • Solving the neutrino mass problem and determine the Higgs spec- trum in the G221 model with lepton-flavor non-universality. The objects of research: • The candidates for dark Matter in the simple 3-3-1 model. • The neutrino masses and Higgs spectrum in the G221 model with lepton-flavor non-universality. The contents of research: • Propose the simple 3-3-1 model. • Introduce the inert scalar multiplets to SM331M for searching of DM candidates. • Generate neutrino masses from one loop corrections and study details the characteristics of the Higgs spectrum in the G221 model with lepton-flavor non-universality (LNU). The methods of research: • Quantum field theory. • The Feynman rule to calculate the amplitudes and the decay widths. • The Wolfram Mathematica software to solve the numerical cal- culation and to process the complex analytic reduction in the de- termination of interactive coefficients. The structure of thesis: In this thesis, apart from the introduction, conclusions and appen- dices, the main content of the thesis is presented in three chapters (according to the list of publications, Chapter 1 and Chapter 2 have the content of the first publication, Chapter 3 has the content of the third publication): 9Chapter 1: We constructed the simple 3-3-1 model, in which we have arranged the particle structure in the model, defined the physical Higgs scalar fields, the gauge bosons, the fermion masses and the proton stability as well as calculated the FCNCs. Chapter 2: We introduced the inert scalar multiplets to S331M for finding the DM candidates. Considering alternately S331M with the rho inert triplet and the S inert sextet as well as the replications of η and χ helps us identify the candidates for DM. We also made an estimate of the DM observations at the end of this chapter. Chapter 3: We abstracted the model based on the standard symmetry group SU(2)1⊗ SU(2)2⊗U(1)Y , in which we solved the problem of generating active neutrino masses from loop corrections and studied the detailed features of the gauge boson and Higgs sections in this model. In order to see the detailed features of S331M, we go into the contents of Chapter 1. 10 Chapter 1 The simple 3-3-1 model Based on the reduced minimal 3-3-1 model and the minimal 3-3-1 model we will build a new model with the minimal lepton and scalar sector—called the simple 3-3-1 model. The model has showed the experimental fit. 1.1 The particle structure of the model The fermion content which is anomaly free is defined as: ψaL ≡  νaLeaL (eaR) c  ∼ (1, 3, 0), QαL ≡  dαL−uαL JαL  ∼ (3, 3∗,−1/3), Q3L ≡  u3Ld3L J3L  ∼ (3, 3, 2/3) , (1.1) uaR ∼ (3, 1, 2/3) , daR ∼ (3, 1,−1/3) , JαR ∼ (3, 1,−4/3) , J3R ∼ (3, 1, 5/3) , (1.2) where a = 1, 2, 3 and α = 1, 2 are family indices. The quantum numbers in parentheses are given upon 3-3-1 symmetries, respec- tively. The electric charge operator takes the form Q = T3− √ 3T8+X, where Ti (i = 1, 2, ..., 8) are SU(3)L charges, while X is that of U(1)X . The new quarks possess exotic electric charges as Q(Jα) = −4/3 and Q(J3) = 5/3. 11 The model can work with only two scalar triplets: η =  η01η−2 η+3  ∼ (1, 3, 0), χ =  χ−1χ−−2 χ03  ∼ (1, 3,−1), (1.3) with VEVs 〈η〉 = 1√ 2  u0 0  , 〈χ〉 = 1√ 2  00 w  . (1.4) 1.2 Scalar sector The scalar potential of the model is given by: Vsimple(χ, η) = µ 2 1η †η + µ22χ †χ+ λ1(η †η)2 + λ2(χ †χ)2 + λ3(η †η)(χ†χ) + λ4(η †χ)(χ†η), (1.5) where µ1,2 have mass-dimensions while λ1,2,3,4 are dimensionless. Expanding η, χ around the VEVs, we get η =  u√20 0 +  S1+iA1√2η−2 η+3  , χ =  00 w√ 2 +  χ−1χ−−2 S3+iA3√ 2  , (1.6) The physical eigenstates are identified as follows h ≡ cξS1−sξS3, H ≡ sξS1 + cξS3, H ± ≡ cθη±3 + sθχ±1 . The physical eigenvalues are cor- respondingly m2h = λ1u2 + λ2w2 − √ (λ1u 2 − λ2w2)2 + λ23u2w2 ' 4λ1λ2−λ23 2λ2 u2, m2H = λ1u 2 +λ2w 2 + √ (λ1u 2 − λ2w2)2 + λ23u2w2 ' 2λ2w 2, m2 H± = λ4 2 (u2 +w2) ' λ4 2 w2. We have denoted cξ = cos(ξ), sξ = sin(ξ), cθ = cos(θ), sθ = sin θ. ξ and θ are alter- nately the mixing angles of S1 − S3 and χ1 − η3. There are eight massless scalar fields GZ ≡ A1, GZ′ ≡ A3, G±W ≡ η±2 , G ±± Y ≡ χ±±2 , G±X ≡ cθχ±1 − sθη±3 . In the effective limit, u  w, we have η '  u+h+iGZ√2G−W H+  , χ '  G − X G−−Y w+H+iGZ′√ 2  . (1.7) 1.3 Gauge sector The gauge bosons receive their masses from the following term of the Lagrangian, ∑Φ=η,χ(Dµ〈Φ〉)†(Dµ〈Φ〉), where Dµ = ∂µ + igstiGiµ + igTiAiµ+igXXBµ, with gs, g and gX are the gauge coupling constants, 12 while ti, Ti and X are respectively SU(3)C , SU(3)L and U(1)X charges; Giµ, Aiµ and Bµ are the gauge bosons, as associated with the 3-3-1 groups, respectively. The gluons Gi are massless and physical fields by themselves. The physical charged gauge bosons with masses are respectively given by W± ≡ A1 ∓ iA2√ 2 , m2W = g2 4 u2, X∓ ≡ A4 ∓ iA5√ 2 , m2X = g2 4 (w2 + u2), Y ∓∓ ≡ A6 ∓ iA7√ 2 , m2Y = g2 4 w2. (1.8) Two physical neutral gauge bosons (beside the photon) with their masses identified by m2Z1 = 1 2 [m2Z +m 2 Z′ − √ (m2Z −m2Z′)2 + 4m4ZZ′ ] ' g2 4c2W u2, m2Z2 = 1 2 [m2Z +m 2 Z′ + √ (m2Z −m2Z′)2 + 4m4ZZ′ ] ' g2c2W 3(1− 4s2W ) w2.(1.9) The model has shown that there is a mixing between two states, Z and Z ′, with the mixing angle t2ϕ = √ 3(1− 4s2W )3/2u2 2c4Ww 2 − (1 + 2s2W )(1− 4s2W )u2 ' √ 3(1− 4s2W )3/2 2c4W u2 w2 , (1.10) Because of ϕ  1, we have Z1 ' Z and Z2 ' Z′. The Z1 is the standard model like Z boson, while Z2 is a new neutral gauge boson with the mass in w scale. The contribution to the experimental ρ-parameter can be calculated as ∆ρ ≡ m 2 W c2Wm 2 Z1 − 1 ' m 4 ZZ′ m2Zm 2 Z′ ' ( 1− 4s2W 2c2W )2 u2 w2 . (1.11) Substituting s2W = 0.231 and ∆ρ 460 GeV. Since the other constraints yield w in some TeV, we conclude that the ρ-parameter is very close to one and suitable to the experimen- tal data Note that if we choose a model with two scalars, chi and rho, then the rho parameter is too large to fit the experiment. 13 1.4 Fermion masses and proton stability To generate mass for fermions, we construct Yukawa interac- tions by two scalar triplets η and χ, LY = hJ33Q¯3LχJ3R + hJαβQ¯αLχ∗JβR +hu3aQ¯3LηuaR + huαa Λ Q¯αLηχuaR +hdαaQ¯αLη ∗daR + hd3a Λ Q¯3Lη ∗χ∗daR +heabψ¯ c aLψbLη + h′eab Λ2 (ψ¯caLηχ)(ψbLχ ∗) + sνab Λ (ψ¯caLη ∗)(ψbLη ∗) +H.c., (1.12) where the Λ is a new scale (with the mass dimension) under which the effective interactions take place. heab is antisymmetric while sνab is symmetric in the flavor indices. The mass Lagrangian of quarks and charged leptons takes the form, −f¯aLmfabfbR + H.c., wheref = J, u, d, e. The mass of J3 is mJ33 = −hJ33w/ √ 2. The mass matrix of J1,2 is mJαβ = −hJαβw/ √ 2. They all get the masses propor- tional to w scale. The mass matrices of u, d and the charged lep- tons are respectively: mu3a = −hu3a u√2 , muαa = −huαa uw2Λ ; mdαa = −hdαa u√2 , md3a = h d 3a uw 2Λ ; and meab = √ 2u ( heab + h ′e ba w2 4Λ2 ) . Because of Λ ∼ w, they all have the masses in the weak scale u = 246 GeV. For top quark, we havemt = −hu33× 174 GeV leading to mt = 173 GeV if hu33 ≈ −1. The mass Lagrangian of neutrinos is given by − 1 2 ν¯caLm ν abνbL +H.c., where mνab = −sνab u 2 Λ . In fact, using Λ = 5 TeV, u = 246 GeV and mνab ∼ eV, we have sνab = h ∼ 10−10. The Yukawa coupling of electron is chosen h = he ∼ 10−6, the lepton number violating parameter is obtained  ∼ 10−4. The strength of the violating interaction for approximate lepton number is reasonably small in comparison to the ordinary inter- actions, and this may be the reason that the neutrino masses are observed to be small. The neutrino mass matrix is symmetric and generalized.Hence it can be compared to the neutrino mixing angles and squared mass differences, as the independent analysis of the model. 14 1.5 FCNC The tree-level FCNCs are discribed by the Lagrangian, LFCNC = − g√ 1− 3t2W (V ∗qL)3i 1√ 3 (VqL)3j q¯ ′ iLγ µq′jLZ ′ µ (i 6= j), (1.13) where we have denoted q as u either d. substituting Z′ = −sϕZ1 + cϕZ2, the effective Lagrangian for hadronic FCNCs can be derived via the Z1,2 exchanges as LeffFCNC = g2[(V ∗qL)3i(VqL)3j ] 2 3(1− 3t2W ) ( s2ϕ m2Z1 + c2ϕ m2Z2 ) (q¯′iLγ µq′jL) 2. (1.14) The contribution of Z1 is negligible. Therefore, only Z2 governs the FCNCs and we have LeffFCNC ' [(V ∗qL)3i(VqL)3j ] 2 w2 (q¯′iLγ µq′jL) 2. (1.15) We can see that this interaction is independent of the Landau pole 1/(1 − 4s2W ) (this is also an evidence pointing out that when the theory is encountered with the Landau pole, the effective interac- tions take place). The strongest constraint comes from the K0− K¯0 system, given by [(V ∗ dL)31(VdL)32] 2 w2 < 1 (104 TeV)2 . Assume that ua is flavor-diagonal. The CKM matrix is just VdL (i.e., VCKM = VdL). Therefore, |(V ∗dL)31(VdL)32| ' 3.6 × 10−4 and we have w > 3.6 TeV. This limit is still in the perturbative region of the model and is suitable to the recent experimental bounds . By contrast, if the first or second generation of quarks is arranged differently from the two others under SU(3)L FCNCs will be large unreasonably. 15 Chapter 2 Inert scalar and dark matter In order to find candidates for DM, we consider alternately the rest of scalars (ρ, S), even the replications of η, χ as the inert sector (Z2 odd) responsible for dark matter. 2.1 Simple 3-3-1 model with inert ρ triplet We can introduce into the theory constructed above an extra scalar triplet which transforms as an odd field under a Z2 symme- try, as follows ρ = ( ρ+1 , ρ 0 2, ρ ++ 3 )T ∼ (1, 3, 1). The normal scalar sector (η, χ) which consists of the VEVs, the conditions for parameters and the physical scalars with their masses as obtained above re- mains unchanged . For the inert sector, ρ has vanishing VEVs due to the Z2 conservation. Moreover, the real and imaginary parts of electrically-neutral complex field ρ02 = 1√2 (Hρ + iAρ) by themselves are physical fields. Any one of them can be stabilized if it is the lightest inert particle (LIP) among the inert particles resided in ρ due to the Z2 symmetry. we can show that Hρ and Aρcannot be a dark matter. Indeed, Hρ and Aρ are not separated (degenerate) in mass which leads to a scattering cross-section of Hρ and Aρ off nuclei due to the t-channel exchange by Z boson. Such a large contribution has already been ruled out by the direct dark matter detection experiments. 2.2 Simple 3-3-1 model with η replication The second hypothesis is that the model is added to the η repli- cation defined by η′ = ( η′01 , η ′− 2 , η ′+ 3 )T ∼ (1, 3, 0). Here, the η′ and η 16 have the same gauge quantum numbers. The η′ is assigned as an odd field under the Z2, η′ → −η′, whereas the η and all other fields of the simple 3-3-1 model are even. The scalar potential that is invariant under the gauge symmetry and Z2 is given by V = Vsimple + µ 2 η′η ′†η′ + x1(η ′†η′)2 + x2(η †η)(η′†η′) + x3(χ †χ)(η′†η′) +x4(η †η′)(η′†η) + x5(χ †η′)(η′†χ) + 1 2 [x6(η ′†η)2 +H.c.], (2.1) where µη′ has the dimension of mass while xi (i = 1, 2, 3, ..., 6) are dimensionless. The physical inert particle masses are determined by m2H′1 = M 2 η′ + 1 2 (x4 + x6)u 2, m2A′1 = M 2 η′ + 1 2 (x4 − x6)u2, m2η′2 = M 2 η′ , m 2 η′3 = M2η′ + 1 2 x5w 2, (2.2) where M2η′ ≡ µ2η′ + 12x2u2 + 12x3w2 and η′01 ≡ 1√2 (H ′1 + iA′1). The lightest inert particle (LIP) responsible for DM is H ′1 if x6 < Min{0, −x4, (w/u)2x5−x4}. Or alternatively A′1 if x6 > Max{0, x4, x4− (w/u)2x5}. Let us consider the case H ′1 as the dark matter candidate (or a LIP). The H ′1 transforms as a doublet dark matter under the standard model symmetry. 2.3 Simple 3-3-1 model with χ replication We introduce the χ replication (Z2 odd), χ′ = ( χ′−1 , χ ′−− 2 , χ ′0 3 )T ∼ (1, 3,−1). The scalar potential that is invariant under the gauge symmetry and Z2 is given by: V = Vsimple + µ 2 χ′χ ′†χ′ + y1(χ ′†χ′)2 + y2(η †η)(χ′†χ′) + y3(χ †χ)(χ′†χ′) +y4(η †χ′)(χ′†η) + y5(χ †χ′)(χ′†χ) + 1 2 [y6(χ ′†χ)2 +H.c.], (2.3) where χ′03 ≡ 1√2 (H ′3 + iA′3).Depending on the parameter regime, H ′ 3 or A′3 may be the LIP responsible for dark matter. Let us consider H ′3 as the LIP. The H ′ 3 is a singlet dark matter under the standard model symmetry. 2.4 Simple 3-3-1 model with inert scalar sextet We consider two cases in which two inert scalar sextet with X = 0 and X = 1 are introduced respectively. 17 2.4.1 Inert scalar sextet X = 0 The inert sextet with X = 0 has the following form S =  S011 S−12√ 2 S+13√ 2 S−12√ 2 S−−22 S023√ 2 S+13√ 2 S023√ 2 S++33  ∼ (1, 6, 0). (2.4) This sextet is odd under the Z2 (S → −S), whereas all the other fields are even. The scalar potential that is invariant under the gauge symmetry and Z2 is given by V = Vsimple + µ 2 STrS †S + z1(TrS †S)2 + z2Tr(S †S)2 +(z3η †η + z4χ †χ)TrS†S + z5η †SS†η + z6χ †SS†χ + 1 2 (z7ηηSS +H.c.), (2.5) Depending on the parameter space, HS, AS, H ′S and A ′ S may be dark matter candidates. However, HS andAS are similar to Hρ and Aρ which has already been ruled out by the direct dark matter detection experiments. By contrast,H ′S and A ′ S transform as dou- blets under the standard model symmetry and are separated in the masses, but they cannot be the LIP because both are much heavier than the H1 field. Therefore, they will rapidly decay that cannot be dark matter. To conclude, the scalar sextet S with X = 0 does not provide realistic dark matter candidates. 2.4.2 Inert scalar sextet X = 1 Let us introduce another sextet with X = 1 (Z2 odd), σ =  σ+11 σ012√ 2 σ++13√ 2 σ012√ 2 σ−22 σ+23√ 2 σ++13√ 2 σ+23√ 2 σ+++33  ∼ (1, 6, 1). (2.6) The scalar potential is given by V = Vsimple + µ 2 σTrσ †σ + t1(Trσ †σ)2 + t2Tr(σ †σ)2 +(t3η †η + t4χ †χ)Trσ†σ + t5η †σσ†η + t6χ †σσ†χ + 1 2 (t7χχσσ +H.c.). (2.7) Here, we obtain either the Hσor the Aσ can be regarded as the LIP responsible for dark matter. Without lost of generality, in the following let us consider Hσ as the dark matter candidate. 18 2.5 An evaluation of dark matter observables To be concrete, in the following we present for the case of the sextet dark matter (Hσ). There are various channels that might contribute to the relic density such as HσHσ → hh, ttc, W+W−, ZZ, as well as the annihilations HσH ± 1 → ZW±, AW±, t±2/3b±1/3 and H± 1 H ∓ 1 → hh, ttc, W+W−, ZZ, ZA, AA. They are given by the diagrams in Fig. 2.1 and Fig. 2.2 with respect to the Higgs and gauge portals, respectively. The annihilation cross-section times relative velocity is defined as Hσ(H + 1 ) Hσ(H − 1 ) h h h Hσ(H + 1 ) Hσ(H − 1 ) h h h Hσ(H + 1 ) Hσ(H − 1 ) t tc h Hσ(H + 1 ) Hσ(H − 1 ) W −, Z W+, Z Hσ h Hσ h Hσ H+1 h H−1 h H1 Figure 2.1: Contributions to Hσ or H±1 annihilation via the Higgs portal when they are lighter than the new particles of the simple 3-3-1 model. There are additionally two u-channels that can be derived from the corresponding t-channels above. 〈σv〉 ' α 2 (150 GeV)2 [( 2.3 TeV mHσ )2 + ( λ× 0.782 TeV mHσ )2] , (2.8) where λ ≡ t3 + t5/2, and α = 1/128. Note also that the quantity α2/(150 GeV)2 ' 1 pb has been factorized for a further convenience. The relic density can fit the data by this case if Ωh2 ' 0.1pb〈σv〉 ' 0.11, (2.9) 19 Hσ H±1 W± A3 W Hσ H±1 W± A3 W Hσ H±1 t±2/3 b±1/3 H1 Hσ H±1 W± A3 Hσ(H + 1 ) Hσ(H − 1 ) W− W+ H + 1 (H − 1 ) H+1 (H − 1 ) W +(W−) W+(W−) H+1 H−1 A3 A3 A3 H+1 H−1 W− W+ A3 H+1 H−1 t tc H1 H+1 H−1 A3 A3 Hσ H+1 H−1 W+ W− H1 Hσ Hσ W+ W− Figure 2.2: Contributions to Hσ or H±1 annihilation via the gauge portal when they are lighter than the new particles of the simple 3-3-1 model. There remain the u-channel contributions for H+1 H − 1 → A3A3 and HσHσ →W+W−, respectively, which can be extracted from the corresponding t-channel diagrams above. (where h is the reduced Hubble constant). ⇒ mHσ ' √ 5.29 + 0.61λ2 TeV. (2.10) • If |λ| = t3 + t5/2 1 ⇒ The dark matter gets the right abundance if it has a mass mHσ ' 2.3 TeV. • Otherwise, if |λ| & 1 ⇒ Due to the limit by the Landau pole, mHσ < 5 TeV (⇔ |λ| < 5.68 for the right abundance.) 20 The direct searches for the candidate Hσ measure the recoil energy deposited by the Hσ when it scatters off the nuclei of a large detector. This proceeds through the interaction of Hσ with the partons confined in nucleons. Because the Hσ is very non- relativistic, the process can be obtained by an effective Lagrangian as, Leff = 2λqmHσHσHσ q¯q. (2.11) where the scalar candidate has only spin-independent and even interactions (the interactions with gluons are loops induced that should be small). The above effective interaction is achieved by the t-channel diagram as mediated by the Higgs boson as Fig. 2.3. The Hσ-nucleon cross-section takes the form, Hσ Hσ q q h Figure 2.3: Dominant contributions to Hσ-quark scattering. σHσ−N ' ( 2.494λ′ TeV mHσ )2 × 10−44 cm2, (2.12) where λ′ ≡ t3 + t5 − λ32λ2 (t4 − t7). The current experimental bound: σHσ−N ' 10−44 cm2 ⇔ mHσ ' 2.494λ′ TeV. (2.13) the Hσ can get the right abundance by this case if we impose: λ′ ' mHσ/(2.494 TeV) ' √ 0.85 + 0.098λ2 ' 0.922÷ 2. (2.14) 21 Chapter 3 The SU(2)1 ⊗ SU(2)2 ⊗U(1)Y model with lepton-flavor non-universality (LNU) 3.1 Summary of the model The model based on the electroweak group SU(2)1 ⊗ SU(2)2 ⊗ U(1)Y with the following gauge couplings, fields and generators as follows, SU(2)1 : g1 ,W 1 i , T 1 i , SU(2)2 : g2 ,W 2 i , T 2 i , (3.1) U(1)Y : g ′ , B , Y , where i = 1, 2, 3 is the index used to distinguish the generators and gauge fields of the SU(2) group. All the left- and right-handed components of the SM fermions transform initially as follows, qL ∼ ( 3,1,2, 1 6 ) , `L ∼ ( 1, 1, 2,− 1 2 ) , uR ∼ ( 3, 1, 1, 2 3 ) , eR ∼ (1, 1, 1,−1) , dR ∼ ( 3, 1, 1,− 1 3 ) , (3.2) where the numbers in the brackets are the representations of the SU(3)C , SU(2)1, SU(2)2, and U(1)Y groups. All the above fermions are the singlets of SU(2)1 group. The electric charge operator is determined in the form Q = (T 13 +T 23 )+Y. The model has nV L genera- tions of vector-like fermions. They have the left- and right-handed components which transform as the doublets under the SU(2)1, and the singlets under the SU(2)2. They are denoted, namely QL,R ≡ ( U D ) L,R ∼ ( 3, 2, 1, 1 6 ) ; LL,R ≡ ( N E ) L,R ∼ ( 1, 2, 1,− 1 2 ) . (3.3) The model requires nV L ≥ 2 in order to explain successfully the LNU. In this thesis, we chose nV L = 2, which is suitable for the previous numerical illustration. 22 The Higgs sector includes: φ = ( ϕ+ ϕ0 ) ∼ ( 1,1,2, 1 2 ) , φ′ = ( ϕ′+ ϕ′0 ) ∼ ( 1,2,1, 1 2 ) , Φ = 1√ 2 ( Φ0 Φ+ −Φ− Φ˜0 ) ∼ (1, 2, 2˜, 0) , (3.4) where Φ = σ2Φ∗σ2; σ2 is the Pauli matrix; note that Φ˜0 = (Φ0)∗. The VEVs: 〈φ〉 = 1√ 2 ( 0 vφ ) , 〈φ′〉 = 1√ 2 ( 0 vφ′ ) , 〈Φ〉 = 1 2 ( u 0 0 u ) . (3.5) The spontaneous symmetry breaking (SSB) of the model follows the pattern SU(2)1 ⊗ SU(2)2 ⊗U(1)Y u−→ SU(2)L ⊗U(1)Y vφ, vφ′−→ U(1)Q . (3.6) With the above breaking chain, the VEVs are assumed to satisfy the relation u vφ, vφ′ . (3.7) Yukawa Lagrangian, fermion mass matrices, and diagonalization steps to construct physical states and masses of fermions were pre- sented in detail in the previous work. Hence, we will summarize here only important results and focus on new features of generating active neutrino masses from loop corrections. 3.1.1 Charged fermion masses The chiral fermions couple to the SM Higgs-like φ doublet by Yukawa interaction −Lφ = q¯L ydφdR + q¯L yu φ˜ uR + ¯`L y` φ eR + H.c., (3.8) where φ˜ ≡ iσ2φ∗. The mass matrices yd, yu, y` is 3 × 3 matrices. We combine the chiral and vector-like fermions as UIL,R ≡ (u i L,R, U k L,R) T , DIL,R ≡ (d i L,R,D k L,R) T , EIL,R ≡ (e i L,R,E k L,R) T , (3.9) where i = 1, 2, 3, k = 1, · · · , nV L and I = 1, · · · , 3 + nV L. After the SSB, the fermion mass Lagrangian has the form −Lfmass = U¯LMUUR + D¯LMDDR + E¯LMEER + H.c. (3.10) All above mass matrices are (3 + nV L) × (3 + nV L) and have the form MU =  1√2 yuvφ 12λqu 1√ 2 y˜uvφ′ MQ  ,MD =  1√2 ydvφ 12λqu 1√ 2 y˜dvφ′ MQ  ,ME =  1√2 y`vφ 12λ`u 1√ 2 y˜`vφ′ ML  . (3.11) 23 In the limit  = v/u 1, these matrices are blocked-diagonalized perturbatively via two steps. We recognize that the number of new lepton generation is nV L = 2. For the charged leptons, the physical masses (mei ,mEk ) related to ME in (3.11) by VeVLMEW †e = diag(mei , mEk ), the mass bases of left-and right-handed leptons E(d)IL,R ≡ (e(d)iL,R, E(d)kL