Some new physics effects in the 3 − 2 − 3 − 1 and 3 − 4 − 1 models

2φ∗)ujR +H.c., (1.4) then the fermions get masses as follows: Meij = heij v√2 , Mdij = hdij v√2 , and Muij = huij v√2 . Diagonalizing this mass matrix, we obtain the physical states and corresponding masses. Other results of the SM: • In SM, the lepton number conservation and it’s correct at every level of perturbation theory. Contemporaneous, the neutrinos are massless par- ticles in SM. But experimentally, the neutrinos have very small masses 3 (non-zero) and have transitions between different generations. This proves that there is a violation of the generation lepton number in the neutral lepton. • The SM contributions to the neutral meson mass differences at the one- loop level did not coincide with the experiment. • In SM, fermion generations perform the same (repeat) under gauge sym- metry. Therefore, SM does not explain why the fermion generation num- ber is 3. • There is no candidate for the dark matter in the SM. • The presented total width of the W boson is calculated at the tree level with electroweak and includes the QCD complementarity over the recent experimental data that is not identical. 1.2. The minimal left-right symmetry model (M3221) The M3221 based on the SU(3)C ⊗ SU(2)L ⊗ SU(2)R ⊗ U(1)B−L gauge group. The left-handed fermions are embedded as SU(2)L doublets and SU(2)R singlets, while the right-handed fermions are SU(2)L singlets and SU(2)R dou- blets. The M3221 usually works with a scalar as a SU(2)L,R bidoublet and two scalars triplet (one left and one right). The M3221 solves the neutrino mass problem well but does not explain is the existence of dark matter. The M3221 has been expanded. The propos- als that extended the gauge group can produce many interesting and reliable results as it is the most natural extension. 1.3. The 3− 4− 1 models These models are based on the SU(3)C ⊗ SU(4)L ⊗ U(1)X gauge group. Many of the problems have been explained by 3− 4− 1 models such as quan- tized charges, neutrino masses, .... However, the Higgs physics-currently the most important sector-has not received enough attention. Except for the su- persymmetric 3 − 4 − 1 model, the Higgs potential containing a decuplet is presented for the first time in this thesis. 4 CHAPTER 2. PHENOMENOLOGY OF THE 3− 2− 3− 1 MODEL 2.1. The model The electric charge operator: Q = T3L + T3R + βT8R + X. The fermion content: ψaL= ( νaL eaL ) ∼ ( 1, 2, 1,−1 2 ) , ψaR=  νaReaR EqaR ∼(1, 1, 3, q − 1 3 ) , (2.1) Q3L= ( u3L d3L ) ∼ ( 3, 2, 1, 1 6 ) , Q3R=  u3Rd3R J q+ 23 3R ∼(3, 1, 3, q + 1 3 ) , (2.2) QαL= ( uαL dαL ) ∼ ( 3, 2, 1, 1 6 ) , QαR=  dαR−uαR J −q− 13 αR ∼(3, 1, 3∗,−q 3 ) , (2.3) EqaL∼(1, 1, 1, q), J q+ 23 3L ∼ ( 3, 1, 1, q + 2 3 ) , J −q− 13 αL ∼ ( 3, 1, 1,−q − 1 3 ) , (2.4) The scalar multiplets are introduced as S = ( S011 S + 12 S −q 13 S−21 S 0 22 S −q−1 23 ) ∼ ( 1, 2, 3∗,−2q + 1 6 ) , (2.5) φ =  φ −q 1 φ−q−12 φ03  ∼ (1, 1, 3,−2q + 1 3 ) , (2.6) Ξ =  Ξ011 Ξ−12√ 2 Ξ q 13√ 2 Ξ−12√ 2 Ξ−−22 Ξ q−1 23√ 2 Ξ q 13√ 2 Ξ q−1 23√ 2 Ξ2q33  ∼ ( 1, 1, 6, 2(q − 1) 3 ) , (2.7) 5 〈S〉 = 1√ 2 ( u 0 0 0 v 0 ) , 〈φ〉 = 1√ 2  00 w , 〈Ξ〉 = 1√ 2  Λ 0 00 0 0 0 0 0 . (2.8) The total Lagrangian: L = Lkinetic + LYukawa − Vscalar, LYukawa = hlabψ¯aLSψbR + hRabψ¯caRΞ†ψbR + hqa3Q¯aLSQ3R + hqaβ ¯˜QaLS∗QβR + hEabE¯aLφ †ψbR + hJ33J¯3Lφ †Q3R + hJαβ J¯αLφ TQβR +H.c., (2.9) Vscalar = µ 2 STr(S †S) + λ1S [Tr(S†S)]2 + λ2STr(S†SS†S) + µ2ΞTr(Ξ †Ξ) + λ1Ξ[Tr(Ξ †Ξ)]2 + λ2ΞTr(Ξ†ΞΞ†Ξ) + µ2φφ †φ+ λφ(φ†φ)2 + λ1(φ †S†Sφ)+λ2Tr(S†SΞΞ†)+λ3(φ†ΞΞ†φ)+λ4(φ†φ)Tr(S†S) + λ5(φ †φ)Tr(Ξ†Ξ)+λ6Tr(Ξ†Ξ)Tr(S†S)+(fSφ∗S+H.c.). (2.10) In (2.10), the f , λ1,2,3 couplings have been imposed for generalization, which were skipped in the previous study. The SU(3)R anomaly cancellation and the QCD asymptotic freedom re- quires the fermion generation number is 3. The VEVs of S generate the Dirac masses for neutrinos, and the VEV of Ξ provides the Majorana masses for right-handed neutrinos. Subsequently, the small neutrino masses are induced via the type I seesaw mechanism. The model can provide the dark matter candidates. For the model with q = 0, the candidates are E0 or X0R or some combination of (φ 0 1, S 0 13, Ξ 0 13). For the model with q = −1, the candidates are Y 0R or some combination of (φ02, S 0 23). For the model with q = 1, the candidates are only Ξ 0 23. Particularly, the model contains a residual gauge symmetry W -parity as R-parity, which is stabilizing the dark matter. 2.2. Scalar sector The results obtained: H1 = uS1 + vS2√ u2 + v2 , H2 = −vS1 + uS2√ u2 + v2 , H3 = cϕS3 − sϕS4, H4 = sϕS3 + cϕS4, m2H1 = 2(λ1S + λ2S)u 2 − λ2Sv2, m2H2 = λ2(u 2 + v2)Λ2 2(v2 − u2) , m2H3 = λφw 2 + (λ1Ξ + λ2Ξ)Λ 2 − √ [(λ1Ξ + λ2Ξ)Λ2 − λφw2]2 + λ25w2Λ2, 6 m2H4 = λφw 2 + (λ1Ξ + λ2Ξ)Λ 2 + √ [(λ1Ξ + λ2Ξ)Λ2 − λφw2]2 + λ25w2Λ2. (2.11) A = vwA1 + uwA2 − uvA3√ (u2 + v2)w2 + u2v2 , GZ = −uA1 + vA2√ u2 + v2 , GZ1 = A4, GZ′1 = uv2A1 + u 2vA2 + w(u 2 + v2)A3√ (u2 + v2)(u2v2 + w2u2 + w2v2) , m2A = − [v2w2 + u2(v2 + w2)][2λ2S(u 2 − v2) + λ2Λ2] 2(u2 − v2)w2 . (2.12) m2 Ξ±±22 = λ2(v 2 − u2)− 2λ2ΞΛ2 2 , m2 Ξ±2q33 = λ3w 2 − λ2u2 − 2λ2ΞΛ2 2 , m2 Ξ ±(q−1) 23 = λ2(v 2 − 2u2) + λ3w2 − 4λ2ΞΛ2 4 . (2.13) H±5 = √ 2uΛS±12 + √ 2vΛS±21 + (v 2 − u2)Ξ±12√ 2(u2 + v2)Λ2 + (v2 − u2)2 , G ± W1 = −vS±12 + uS±21√ u2 + v2 , G±W2 = u(u2 − v2)S±12 + v(u2 − v2)S±21 + √ 2(u2 + v2)ΛΞ±12√ (u2 − v2)2(u2 + v2) + 2(u2 + v2)2Λ2 , m2 H±5 = λ2 4 [ v2 − u2 + 2(u 2 + v2)Λ2 v2 − u2 ] . (2.14) G±qX = uS±q13 − wφ±q1 + √ 2ΛΞ±q13√ u2 + w2 + 2Λ2 , H±q6 = cϕqH ′±q 6 − sϕqH ′±q7 , H±q7 = sϕqH ′±q6 + cϕqH ′±q7 , m2 H±q6 ' λ1(u 2 − v2)w2 − λ2u2Λ2 2(u2 − v2) , m 2 H±q7 ' λ3(w 2 + 2Λ2) 4 . (2.15) G ±(q+1) Y = −vS±(q+1)23 + wφ±(q+1)2√ v2 + w2 , H ±(q+1) 8 = wS ±(q+1) 23 + vφ ±(q+1) 2√ v2 + w2 , m2 H ±(q+1) 8 = − (v 2 + w2)[(u2 − v2)(2λ2Su2 − λ1w2) + λ2u2Λ2] 2(u2 − v2)w2 . (2.16) 2.3. Gauge sector The results obtained: X±qRµ = ( A4Rµ ± iA5Rµ ) / √ 2, Y ±(q+1) Rµ = ( A6Rµ ± iA7Rµ ) / √ 2, W±1µ = cξW ± Lµ − sξW±Rµ, W±2µ = sξW±Lµ + cξW±Rµ, m2XR = g2R 4 (u2 + w2 + 2Λ2), m2YR = g2R 4 (v2 + w2), m2W1 ' g2L 4 [ u2 + v2 − 4t 2 Ru 2v2 2t2RΛ 2 + (t2R − 1)(u2 + v2) ] , 7 m2W2 ' g2R 4 [ u2 + v2 + 2Λ2 + 4u2v2 2t2RΛ 2 + (t2R − 1)(u2 + v2) ] . (2.17) m2A = 0, m 2 Z ' g2L 4 { u2 + v2 c2W + 1(u2 − v2)κcW√ 3[t2R + t 2 X(1 + β 2)] − 2t 2 R(u 2 + v2)κcW [t2R + t 2 X(1 + β 2)] 3 2 } , m2Z1 ' g2L 6 { t2R(w 2 + 4Λ2) + t2X [β 2w2 + ( √ 3 + β)2Λ2] − √ [t2R(w 2 + 4Λ2) + t2X(β 2w2 + ( √ 3 + β)2Λ2)]2 − 12t2R[t2R + (1 + β2)t2R]w2Λ2 } , m2Z′1 ' g 2 L 6 { t2R(w 2 + 4Λ2) + t2X [β 2w2 + ( √ 3 + β)2Λ2] + √ [t2R(w 2 + 4Λ2) + t2X(β 2w2 + ( √ 3 + β)2Λ2)]2 − 12t2R[t2R + (1 + β2)t2R]w2Λ2 } ,  A3L A3R A8R B '  sW cW 0 0 sW tR − s 2 W tRcW √ t2 R +t2 X β2ssW tRtXcW − √ t2 R +t2 X β2csW tRtXcW βsW tR − βs 2 W tRcW t2RccW−βtXssW tRcW √ t2 R +t2 X β2 t2RscW+βtXcsW tRcW √ t2 R +t2 X β2 sW tX − s 2 W tXcW −βtXccW−ssW cW √ t2 R +t2 X β2 −βtXscW+csW cW √ t2 R +t2 X β2   A Z Z1 Z′1 . (2.18) We will use the approximation: Z = Z, ZR = ZR, and Z ′R = Z ′R. 2.4. Interactions 2.4.1. Fermion-gauge boson interactions We received four charged currents and the neutral current interactions, J−µ1W = − gLcξ√ 2 (ν¯aLγ µeaL + u¯aLγ µdaL) + gRsξ√ 2 (ν¯aRγ µeaR + u¯aRγ µdaR), J−µ2W = − gLsξ√ 2 (ν¯aLγ µeaL + u¯aLγ µdaL)− gRcξ√ 2 (ν¯aRγ µeaR + u¯aRγ µdaR), J−qµX = − gR√ 2 (E¯aRγ µνaR − d¯αRγµJαR + J¯3Rγµu3R), J −(q+1)µ Y = − gR√ 2 (E¯aRγ µeaR + u¯αRγ µJαR + J¯3Rγ µd3R), (2.19) LNC = −eQ(f)f¯γµfAµ − gL 2cW f¯γµ[gZV (f)− gZA(f)γ5]fZµ − gL 2cW f¯γµ[gZ1V (f)− gZ1A (f)γ5]fZ1µ − gL 2cW f¯γµ[g Z′1 V (f)− gZ ′ 1 A (f)γ5]fZ ′1µ, where f indicates every fermion. 2.4.2. Scalar–gauge boson interactions These vertex types are listed in Appendix A. 8 f gZV (f) g Z A(f) f g Z V (f) g Z A(f) νa 1 2 1 2 ea − 12 + 2s2W − 12 Ea −2s2W q 0 ua 12 − 43s2W 12 da − 12 + 23s2W − 12 Jα 2s2W (q + 13 ) 0 J3 −2s2W (q + 23 ) 0 Table 2.1: The couplings of Z with fermions. Ε2 = -0.001 Ε1 = -0.001 Ξ = -0. 001 Ε2= 0.001 DΡ = 0.0 00 16 DΡ = 0 .00 06 4 0 50 100 150 200 5000 10 000 15 000 20 000 u @GeVD L @GeV D Fig 2.1: The viable new physics regime for the case β = −1/√3. 2.5. New physics effects and constraints 2.5.1. ρ and mixing parameters The new-physics contribution to the ρ-parameter is evaluated as ∆ρ ' 1(v 2 − u2)c3Wκ√ 3(u2 + v2)[t2R + t 2 X(1 + β 2)] + 2t 2 Rc 3 Wκ [t2R + t 2 X(1 + β 2)]3/2 − 2u 2v2 (u2 + v2)Λ2 . (2.20) From the global fit, 0.00016 < ∆ρ < 0.00064. We make a contour for ∆ρ as in Fig 2.1 for the case β = −1/√3. For the mixing parameters, we make contours (solid line for 1, dashed line for 2, and short dashed line for ξ) for |ξ| = |1,2| = 10−3. The available parameter space lies above these third lines. The bounds for Λ is bounded by 6.6 TeV 210.4. The similar for the case β = 0 : 5.5 TeV 215. For the 9 case β = 1/ √ 3 : 4.6 TeV 222.3. 2.5.2. Flavor-changing neutral current Considering the interactions of quarks with scalars, we get tree-level FCNCs due to the contribution of H2, LH2FCNC = d¯′iLΓdijd′jRH2 + u¯′iLΓuiju′jRH2 +H.c., (2.21) Γdij = − √ u2 + v2 u2 (V †dLVuL)ik(M U )km(V ∗ uR)3m(VdR)3j , Γuij = √ u2 + v2 v2 (V †uLVdL)ik(M D)km(V ∗ dR)3m(VuR)3j . (2.22) Considering the interactions of quarks with gauge bosons, we get tree-level FCNCs due to the contribution of Z ′R, LZ′RFCNC = −ΘZ ′ R ij q¯ ′ iRγ µq′jRZ ′ Rµ (2.23) with i 6= j, where q′ is denoted as either u′ or d′, and ΘZ′Rij is defined as Θ Z′R ij = gL√ 3 √ t2R + β 2t2X(V ∗ qR)3i(VqR)3j . (2.24) The contribution of the new physics to the neutral meson Kaon mass diffier- ence, ∆mK = Re { 2 3 (Θ Z′R 12 ) 2 m2Z′R + 5 12 ( (Γd∗21) 2 m2H2 + (Γd12) 2 m2H2 )( mK ms +md )2 −Γ d∗ 21Γ d 12 m2H2 [ 1 6 + ( mK ms +md )2]} mKf 2 K . (2.25) Similarly, for neutral mesons Bd and Bs. The standard model contributions, (∆mK)SM = 0.467× 10−2/ps, (∆mBd)SM = 0.528/ps, (∆mBs)SM = 18.3/ps. (2.26) The total contributions, (∆mK,Bd,Bs)tot = (∆mK,Bd,Bs)SM + ∆mK,Bd,Bs . (2.27) The experimental values are (∆mK)Exp = 0.5292× 10−2/ps, (∆mBd)Exp = 0.5055/ps, 10 (∆mBs)Exp = 17.757/ps. (2.28) We require the theory to produce the data for the kaon mixing parameter within 30% and 5% for the B-meson mixing parameters, 0.37044× 10−2/ps < (∆mK)tot < 0.68796× 10−2/ps, (2.29) 0.480225/ps < (∆mBd)tot < 0.530775/ps, (2.30) 16.8692/ps < (∆mBs)tot < 18.6449/ps. (2.31) The Fig 2.4 for M = 5 TeV. The available region for ∆mK is the whole frame. The two separated regions are for ∆mBd . A lower half region is for ∆mBs . Hence, the available parameter space for ∆mK,Bd,Bs is only the (dark- est) region in the lower left corner of panel. We obtain constraints for the right- handed quark mixing matrix elements as |VuR| < 0.08 and |VdR| < 0.0015. The similar for M = 10 TeV: |VuR| < 0.2 and |VdR| < 0.003. In Fig 2.6, considering VuR = 0.05. The viable parameter space is the (darkest) region bounded in the upper left corner of panel. We obtainM > 2.8 TeV. The similar, M > 5.7 TeV for VuR = 0.1, and M > 8.2 TeV for VuR = 0.15. Note: (VdR)31 = (VdR)32 ≡ VdR, (VdR)233 = 1− 2V 2dR, and (VuR)33 ≡ VuR. 2.6. Summary • The model contain suitable spectra in gauge bosons and Higgs bosons, and the correct form of currents. All the SM particles and interactions are consistently recovered. • The model can explain the fermion generation number is 3, the small mass of the neutrino, and the existence of dark matter. • The available region for the new physics scale: M = 5–10 TeV. • For M = 5 TeV: |VuR| < 0.08 and |VdR| < 0.0015. For M = 10 TeV: |VuR| < 0.2 and |VdR| < 0.003. 11 Fig 2.4: The constraints for (VuR, VdR) coming from the meson mixing parameters ∆mK,Bd,Bs with respect to the new-physics scale, M = 5 TeV. Fig 2.6: The constraints for (M,VdR) coming from the meson mixing parameters ∆mK,Bd,Bs for VuR = 0.05. 12 CHAPTER 3. PHENOMENOLOGY OF THE MINIMAL 3− 4− 1 MODEL WITH RIGHT-HANDED NEUTRINO 3.1. The 3− 4− 1 model 3.1.1. Anomaly cancellation and fermion content For the class of the SU(3)C ⊗ SU(4)L ⊗ U(1)X (3− 4− 1) models the fol- lowing gauge anomalies must vanish: i) [SU(3)C ]2 ⊗ U(1)X , ii) [SU(4)L]3, iii) [SU(4)L] 2 ⊗ U(1)X ; iv) [Grav]2 ⊗ U(1)X ; and v) [U(1)X ]3. Exploit the rela- tion between charge operator and diagonal generators of the gauge symmetry SU(4)L, we proved that the five conditions reduce to two conditions only: [SU(4)L] 3 and [SU(4)L]2⊗U(1)X . This means that (i) the number of fermion quadruplets is equal to that of fermion antiquadruplets and (ii) the sum over electric charges of all left-handed fermions is zero. 3.1.2. Yukawa couplings and masses for fermions The fermions are arranged as faL = (νa , la , E q a , E ′q′ a ) T L ∼ ( 1, 4, q + q′ − 1 4 ) , a = 1, 2, 3, laR ∼ (1, 1,−1) , EqaR ∼ (1, 1, q) , E′q ′ aR ∼ (1, 1, q′). (3.1) Q3L = (u3 , d3 , T , T ′)TL ∼ ( 3, 4, 5 + 3(q + q′) 12 ) , u3R ∼ (3, 1, 2/3), d3R ∼ (3, 1,−1/3), TR ∼ ( 3, 1, 2 + 3q 3 ) , T ′R ∼ ( 3, 1, 2 + 3q′ 3 ) . (3.2) QαL = (dα ,−uα , Dα , D′α)TL ∼ ( 3, 4∗,−1 + 3(q + q ′) 12 ) , α = 1, 2, uαR ∼ (3, 1, 2/3), dαR ∼ (3, 1,−1/3), 13 DαR ∼ ( 3, 1,−1 + 3q 3 ) , D′αR ∼ ( 3, 1,−1 + 3q ′ 3 ) . (3.3) For SSB, we need four Higgs quadruplets, namely, Φ1 = ( Φ (−q′) 1 ,Φ (−q′−1) 1 ,Φ (q−q′) 1 ,Φ 0 1 )T ∼ ( 1, 4, q − 3q′ − 1 4 ) , Φ2 = ( Φ (−q) 2 ,Φ (−q−1) 2 ,Φ 0 2 ,Φ (q′−q) 2 )T ∼ ( 1, 4,−1 + 3q − q ′ 4 ) , Φ3 = ( Φ (+) 3 ,Φ 0 3 ,Φ (q+1) 3 ,Φ (q′+1) 3 )T ∼ ( 1, 4, 3 + q + q′ 4 ) , Φ4 = ( Φ04 ,Φ − 4 ,Φ (q) 4 ,Φ (q′) 4 )T ∼ ( 1, 4, q + q′ − 1 4 ) . (3.4) The Yukawa couplings, −LYukawa = hE ′ ab f¯aLΦ1E ′q′ bR + h E abf¯aLΦ2E q bR + h l abf¯aLΦ3lbR + h tQ¯3LΦ4u3R + hbQ¯3LΦ3d3R + h T Q¯3LΦ2TR + h T ′Q¯3LΦ1T ′ R + h d2 αβQ¯αLΦ † 4dβR + hu2αβQ¯αLΦ † 3uβR + h D2 αβ Q¯αLΦ † 2DβR + h D′2 αβ Q¯αLΦ † 1D ′ βR + H.c. (3.5) The fermions get masses as follows: (mE′)ab = h E′ ab V√ 2 , (mE)ab = h E ab ω√ 2 , (ml)ab = h l ab v√ 2 , mu3 = h t u√ 2 , md3 = h b v√ 2 , mT = h T ω√ 2 , mT ′ = h T ′ V√ 2 , (md2)αβ = h d2 αβ u√ 2 , (mu2)αβ = −hu2αβ v√ 2 , (mD2)αβ = h D2 αβ ω√ 2 , (mD′2)αβ = h D′2 αβ V√ 2 , (3.6) where V√ 2 , ω√ 2 , v√ 2 , and u√ 2 are VEVs of Φ01,Φ 0 2,Φ 0 3, and Φ 0 4, respectively. 3.1.3. Gauge boson masses The results obtained: m2W = g2(v2 + u2) 4 , m2W13 = g2(u2 + ω2) 4 , m2W23 = g2(v2 + ω2) 4 , m2W14 = g2(u2 + V 2) 4 , m2W24 = g2(v2 + V 2) 4 , m2W34 = g2(ω2 + V 2) 4 , m2A = 0, m 2 Z = O(m2W ), m2Z3 ' g2 4 [ 3s2αV 2 2s243 + [ 2 √ 2cαs43 + sαs32 (bs43c43t− 1) ]2 w2 6s243s 2 32 ] , 14 m2Z4 ' g2 4 [ 3c2αV 2 2s243 + [ 2 √ 2sαs43 − cαs32 (bs43c43t− 1) ]2 w2 6s243s 2 32 ] . (3.7) Wµ13 ≡ (Aµ1 − iAµ3 )/ √ 2 , ..., Aµ = sWA3µ + cW ( c32A8µ + c43s32A15µ + s43s32B ′′ µ ) , Zµ ' cWA3µ − sW ( c32A8µ + c43s32A15µ + s43s32B ′′ µ ) , Z3µ ' −s32cαA8µ + (c43c32cα − s43sα)A15µ + (s43c32cα + c43sα)B′′µ , Z4µ ' s32sαA8µ−(c43c32sα+s43cα)A15µ+(c43cα−s43c32sα)B′′µ . (3.8) 3.2. The minimal 3− 4− 1 model with right-handed neutrino 3.2.1. The model The fermions are arranged as faL = (νa , la , l c a , ν c a) T L ∼ (1, 4, 0) , a = e, µ, τ . (3.9) Q3L = (u3 , d3 , T , T ′)TL ∼ (3, 4, 2/3), u3R ∼ (3, 1, 2/3), d3R ∼ (3, 1,−1/3), TR ∼ (3, 1, 5/3), T ′R ∼ (3, 1, 2/3). (3.10) QαL = (dα ,−uα , Dα , D′α)TL ∼ (3, 4∗,−1/3), uαR ∼ (3, 1, 2/3), dαR ∼ (3, 1,−1/3), DαR ∼ (3, 1,−4/3), D′αR ∼ (3, 1,−1/3), α = 1, 2. (3.11) For SSB, we need four Higgs quadruplets, namely, χ = ( χ01 , χ − 2 , χ + 3 , χ 0 4 )T ∼ (1, 4, 0) , φ = (φ−1 , φ−−2 , φ03 , φ−4 )T ∼ (1, 4,−1), ρ = ( ρ+1 , ρ 0 2 , ρ ++ 3 , ρ + 4 )T ∼ (1, 4, 1) , η = (η01 , η−2 , η+3 , η04)T ∼ (1, 4, 0). (3.12) The Yukawa couplings for the quark sector are −LqYukawa = ht Q¯3L ηu3R+hb Q¯3Lρd3R+hT Q¯3L φTR+hT ′ Q¯3L χT ′ R+h d2 αβQ¯αLη †dβR + hu2αβQ¯αLρ †uβR + h D2 αβ Q¯αLφ †DβR + h D′2 αβ Q¯αLχ †D′βR + H.c. (3.13) The quarks get masses as follows: mu3 = h t u√ 2 , md3 = h b v√ 2 , mT = h T ω√ 2 , mT ′ = h T ′ V√ 2 , 15 (md2)αβ = h d2 αβ u√ 2 , (mu2)αβ = −hu2αβ v√ 2 , (mD2)αβ = h D2 αβ ω√ 2 , (mD′2)αβ = h D′2 αβ V√ 2 . (3.14) To produce masses for leptons, we introduce a symmetric decuplet, H = 1√ 2  √ 2H01 H − 1 H + 2 H 0 2 H−1 √ 2H−−1 H 0 3 H − 3 H+2 H 0 3 √ 2H++2 H + 4 H02 H − 3 H + 4 √ 2H04  ∼ (1,10, 0). (3.15) The Yukawa interaction for the lepton is given by −LlYukawa = h l ab√ 2 [ ν¯aL (√ 2νcbRH 0 1 + l c bRH − 1 + lbRH + 2 + νbRH 0 2 ) + l¯aL ( νcbRH − 1 + √ 2lcbRH −− 1 + lbRH 0 3 + νbRH − 3 ) + l¯caL ( νcbRH + 2 + l c bRH 0 3 + √ 2lbRH ++ 2 + νbRH + 4 ) + ν¯caL ( νcbRH 0 2 + l c bRH − 3 + lbRH + 4 + √ 2νbRH 0 4 )] + H.c. (3.16) Assuming that: H03 = v′+R H03 −iI H03√ 2 , H02 = +R H02 −iI H02√ 2 . The charged leptons get mass matrix given by (ml)ab = hlab√ 2 〈H03 〉 = h l ab v ′ 2 . The neutrinos obtain the Dirac mass given by (mν)ab = hlab√ 2 〈H02 〉 = h l ab  2 . The neutrino Majorana mass will follow from 〈H01 〉 and 〈H04 〉. 3.2.2. Gauge sector The results obtained: m2U±± = g2 4 (ω2 + v2 + 4v′2) , m2N0 = g2 4 (V 2 + u2 + 42), m2W± ' g2 4 (v2 + u2 + v′′2), m2K± ' g2 4 (V 2 + w2 + v′′2), m2X± ' g2 4 (V 2 + v2 + v′′2), m2Y± ' g2 4 (w2 + u2 + v′′2). m2A = 0, m 2 Z = g2(v2 + u2 + v′′2) 4c2W = m2W c2W , m2Z′3 = g2 24 [ 9s2αV 2 + ( sα − cα √ 8 + 3t2 )2 w2 ] + g2 24 [(√ 2cαs32 + sα )2 u2 + ( sα + (3t2 + 4)cαs32 2 √ 2 )2 v2 + 2 (√ 2sα − cαs32 )2 v′′2 ] , 16 m2Z′4 = g2 24 [ 9c2αV 2 + ( cα + sα √ 8 + 3t2 )2 w2 ] + g2 24 [( cα − √ 2sαs32 )2 u2 + ( cα − (3t 2 + 4)sαs32 2 √ 2 )2 v2 + 2 (√ 2cα + sαs32 )2 v′′2 ] . (3.17) N0 ≡W 014, U−− ≡W−−23 , Wµ = cos θW ′ µ − sin θK′µ , Kµ = sin θW ′µ + cos θK′µ, Yµ = cos θ ′ Y ′µ − sin θ′X ′µ , Xµ = sin θ′ Y ′µ + cos θ′X ′µ, Aµ = sWA3µ + cW c32A8µ + cW s32B ′′ µ , Zµ = cWA3µ − sW c32A8µ − sW s32B′′µ , Z′3µ = −s32cαA8µ − sαA15µ + c32cαB′′µ , Z′4µ = s32sαA8µ − cαA15µ − c32sαB′′µ . (3.18) U±± and Y ± are similar to the singly charged gauge bosons in M331, while N0 and X± play the similar role in ν331. The heaviest singly charged gauge bosonsK± are the completely new ones that couple with the exotic quarks and right-handed leptons only. In our assignment (and also in Voloshin’s paper), particles belonging to the minimal version are lighter than those in ν331. For the original 3− 4− 1 model, the above consequence is the opposite. 3.2.3. Currents From the Lagrangian Lfermion = i ∑ f f¯γ µDµf, we obtained: −LCC = g√ 2 ( Jµ−W W + µ +J µ− K K + µ +J µ− X X + µ +J µ− Y Y + µ +J µ0∗ N N 0 µ + J µ−− U U ++ µ + H.c. ) , where Jµ−W = cθ(ν¯aLγ µlaL + u¯3Lγ µd3L − u¯αLγµdαL) − sθ(−ν¯aRγµlaR + T¯LγµT ′L + D¯′αLγµDαL). (3.19) Jµ−K = cθ(−ν¯aRγµlaR + T¯LγµT ′L + D¯′αLγµDαL) + sθ(ν¯aLγ µlaL + u¯3Lγ µd3L − u¯αLγµdαL), Jµ−X = cθ′(ν¯ c aLγ µlaL + T¯ ′ Lγ µd3L − u¯αLγµD′αL) + sθ′(l¯ c aLγ µνaL + T¯Lγ µu3L + d¯αLγ µDαL), Jµ−Y = cθ′(l¯ c aLγ µνaL + T¯Lγ µu3L + d¯αLγ µDαL) − sθ′(ν¯caLγµlaL + T¯ ′Lγµd3L − u¯αLγµD′αL), Jµ−−U = l¯ c aLγ µlaL + T¯Lγ µd3L − u¯αLγµDαL, Jµ0∗N = ν¯aLγ µνcaL + u¯3Lγ µT ′L + D¯ ′ αLγ µdαL. (3.20) 17 The neutral current interactions, −LNC = eJµemAµ+ g4 2cW 3∑ i=1 Ziµ ∑ f {f¯γµ[g(V )(f)iV −g(A)(f)iAγ5]f}, (3.21) where e = g sin θW , t = g′ g = 2 √ 2 sin θW√ 1− 4 sin2 θW . (3.22) 3.2.4. Higgs potential In the limit of lepton number conservation, the potential can then be writ- ten as V (η, ρ, φ, χ,H) = V (η, ρ, φ, χ) + V (H). V (η, ρ, φ, χ) = µ21η †η + µ22ρ †ρ+ µ23φ †φ+ µ24χ †χ + λ1(η †η)2 + λ2(ρ†ρ)2 + λ3(φ†φ)2 + λ4(χ†χ)2 + (η†η)[λ5(ρ†ρ) + λ6(φ†φ) + λ7(χ†χ)] + (ρ†ρ)[λ8(φ†φ) + λ9(χ†χ)] + λ′9(φ †φ)(χ†χ) + λ10(ρ †η)(η†ρ) + λ11(ρ†φ)(φ†ρ) + λ12(ρ†χ)(χ†ρ) + λ13(φ †η)(η†φ) + λ14 (χ†η)(η†χ) + λ15 (χ†φ)(φ†χ) + (fijklηiρjφkχl + H.c.). (3.23) V (H) = µ25Tr(H †H) + λ16Tr[(H†H)2] + λ17[Tr(H†H)]2 + Tr(H†H)[λ18(η†η) + λ19(ρ†ρ) + λ20(φ†φ) + λ21(χ†χ)] + λ22(χ †H)(H†χ) + λ23(η†H)(H†η) + λ24(ρ†H)(H†ρ) + λ25(φ †H)(H†φ) + [f4χ†Hη∗ + H.c.]. (3.24) The results obtained: G±±U = √ 2v′H±±1 − √ 2v′H±±2 − vρ±±3 + wφ±±2√ w2 + v2 + 4v′2 , m2 h±±1 = λ24v 2 − λ25w2 4 = −m2 h±±2 , m2 h±±3 = w2 + v2 2 ( λ11 − fV u wv ) . (3.25) G±Y = −v′H±1 − wφ±1 + uη±3√ w2 + u2 + v′2 , G±X = −V χ±2 + v′H±4 + vρ±4√ V 2 + v2 + v′2 , G±K = wφ±4 − V χ±3 + v′H±3√ V 2 + w2 + v′2 , G±W = −uη±2 + vρ±1 + v′H±2√ u2 + v2 + v′2 , h±1 ≡ H±1 , h±2 ≡ H±2 , h±3 = vη ± 2 + uρ ± 1√ u2 + v2 , h±4 = uφ±1 + wη ± 3√ u2 + w2 , 18 h±5 = V φ±4 + wχ ± 3√ V 2 + w2 , h±6 = vχ±2 + V ρ ± 4√ V 2 + v2 , h±7 ≡ H±3 , h±8 ≡ H±4 . (3.26) m2 h±1 = 1 4 (λ23u 2 − λ25w2), m2h±2 = 1 4 ( λ23u 2 − λ24v2 ) , m2 h±3 = u2 + v2 2 ( λ10 − fwV uv ) , m2 h±4 = u2 + ω2 2 ( λ13 − fvV wu ) , m2 h±5 = V 2 + ω2 2 ( λ15 − fvu V w ) , m2 h±6 = V 2 + v2 2 ( λ12 − fwu V v ) , m2 h±7 = 1 4 ( λ22V 2 − λ25w2 ) , m2 h±8 = 1 4 ( λ22V 2 − λ24v2 ) . (3.27) The squared mass matrix of the CP-odd neutral Higgses is 10× 10 matrix. This matrix has a massless state of Im[H03 ] at the tree level. In the limit  = 0, we obtained: HA1 ≡ Im[H02 ], m2A1 = 1 4 ( λ22V 2 + λ23u 2 − 2λ16v′2 − λ24v2 − λ25w2 ) , HA2 ≡ Im[H01 ], m2A2 = 1 4 ( 2λ23u 2 − 2λ16v′2 − λ24v2 − λ25w2 ) , HA3 ≡ Im[H04 ], m2A3 = 1 4 ( 2λ22V 2 − 2λ16v′2 − λ24v2 − λ25w2 ) ,( GN0 HA4 ) =  V√V 2+u2 u√V 2+u2 − u√ V 2+u2 V√ V 2+u2 ( Im[χ01] Im[η04 ] ) ,  GZ1 GZ2 GZ3 HA5  =  − v√ v2+u2 0 0 v√ v2+u2 − vu2√ A(v2+u2) 0 √ (v2+u2V√ A − uv2√ A(v2+u2) −V 2u2v√ AB √ Aw√ B −V v2u2√ AB −V 2v2u√ AB V wu√ B V vu√ B wvu√ B V wv√ B   Im[ρ02] Im[φ03] Im[χ04] Im[η01 ] , m2A4 = V 2 + u2 2 ( λ14 − fwv V u ) , m2A5 = − f 2 [ V vu w + w ( V v u + u(V 2 + v2) V v )] , (3.28) where A = V 2v2 +u2(V 2 + v2), B = V 2v2(w2 +u2) +w2u2(V 2 + v2). We note that three Goldstone bosons absorbed by the three Hermitian gauge bosons, Zi(i = 1, 2, 3) are linear combinations of the above massless states, GZi . But the GZ1 mainly contributes to the Goldstone boson of the SM Z boson. In the neutral sector, the squared mass matrix separates into two submatri- ces. The firt mass matrix,M21H0 , has a massless state, which is the Goldstone boson N0∗. Besides, in the limit  = 0, three other mass values are m2h01 = 1 4 ( 2λ23u 2 − λ24v2 − 2λ16v′2 − λ25w2 ) , m2h02 = V 2 + u2 2 ( λ14 − fwv V u ) , 19 m2h03 = 1 4 ( 2λ22V 2 − λ24v2 − 2λ16v′2 − λ25w2 ) . (3.29) The mass eigenstates in this case are h01 ≡ Re[H01 ], h03 ≡ Re[H04 ], ( GN0∗ h02 ) =  − V√V 2+u2 u√V 2+u2 u√ V 2+u2 V√ V 2+u2 ( Re[χ01] Re[η04 ] ) . The second mass matrixM22H0 satisfy detM22H0 6= 0. But if v′ = 0,M22H0 has a massless value. In addition, if v′ = v = u = 0, the matrix has two massless values, implying that there may be two light CP-even neutral Higgs bosons. Hence one of them can be identified with the SM Higgs boson, meaning that the Higgs sector of the model under consideration is reliable. The main contributions to the four heavy Higgs bosons are m2h04 = −fwV, m2h05 = 1 4 ( λ22V 2 − λ25w2 ) , m2h06,7 = λ3w 2 + λ4V 2 ± √ (λ4V 2 − λ3w2)2 + λ′29 w2V 2. (3.30) To conclude, the model predicts many Higgs bosons with masses near the TeV range that today colliders can detect. Furthermore, the above investigation can be applied for the models where the 10S H is not included. 3.2.5. W boson and constraints 1. In the model under consideration, theW boson has the following normal main decay modes: W− → l ν˜l (l = e, µ, τ), ↘ ucd, ucs, ucb, (u→ c). (3.31) Due to the W −K mixing, we have other modes, namely, W− → lRν˜lR (l = e, µ, τ). (3.32) The predicted total width for the W d