Some new physical effects in the 3 − 2 − 3 − 1 and 3 − 3 − 3 − 1 models

sses and dark matter problems in SM. • Investigate the 3 − 2 − 3 − 1 model with any charge of new leptons, neutrino masses, and identify dark matter candidates in the model and search for dark matter by the method direct search. • Investigate the model 3−3−3−1 with any charge of new leptons, gauge boson masses, Higgs mass, FCNCs, cLFV in decay process µ → eγ, µ→ 3e. 2 CHAPTER 1. OVERVIEW 1.1. The Standard Model SM describes strong, electromagnetic and weak interactions based on the gauge symmetry group SU(3)C⊗SU(2)L⊗U(1)Y (3−2−1). In particular, the gauge group SU(3)C describes strong interaction, gauge group SU(2)L⊗U(1)Y describes weak electrical interaction. The electric charge operator: Q = T3 + Y/2. The particles in SM are arranged under the gauge group as follows: Leptons: ψaL = ( νaL eaL ) ∼ (1, 2,−1), eaR ∼ (1, 1,−2), a = 1, 2, 3. (1.1) Quarks: QaL = ( uaL daL ) ∼ ( 3, 2, 1 3 ) , uaR ∼ ( 3, 1, 4 3 ) , daR ∼ ( 3, 1,−2 3 ) , (1.2) where a is the generation index. The SU(3)C ⊗ SU(2)L ⊗ U(1)Y gauge group is broken spontaneously via a single scalar field, φ = ( ϕ+ ϕ0 ) = ( ϕ+ v+h+iGZ√ 2 ) ∼ (1, 2, 1). (1.3) After SSB, the received gauge bosons are: Aµ = sWA 3 µ + cWBµ, Zµ = cWA 3 µ − sWBµ, W±µ = 1√ 2 (A1µ ∓ iA2µ), mA = 0, mZ = gv 2cW , mW± = gv 2 . (1.4) 3 The Yukawa interaction: − LY = heijψ¯iLφejR + hdijQ¯iLφdjR + huijQ¯iL(iσ2φ∗)ujR +H.c., (1.5) for fermion mass matrices: Meij = heij v√2 , Mdij = hdij v√2 , và Muij = huij v√2 . Diagonalization of these mass matrices will determine the physical fermion states and their masses. 1.2. GIM mechanism and CKM matrix 1.2.1. GIM mechanism If only three quarks exist: u, d, s with left-handed quarks arranged to dou- blet of SU(2)L group: Q1L = ( u dθc ) L = ( uL cosθc dL + sinθc sL ) , (1.6) and right-handed quarks arranged to singlet of SU(2)L group: uR, dθcR , sθcR (θ is flavor mixing angle, called Cabibbo angle), hence we have high flavor changing neutral current. This contradicts experiment. In 1970, Glashow, Iliopuolos and Maiani (GIM) proposed a new mechanism to solve this problem by introducing the two quark doublet which includes the four quark, which is now called the charm quark c, Q1L = ( uL cosθc dL + sinθc sL ) , Q2L = ( uL cosθc sL − sinθc dL ) , uR, cR, dθcR, sθcR. (1.7) and then we have no flavor changing neutral current at the tree level. Thus, the GIM mechanism came to the conclusion: to have a small FCNCs, there must be at least two quarks generations 1.2.2. CKM matrix In SM, if there were only two quark generations, scientists have no CP violation. To solve the CP symmetry violating problem, scientists supposed the existence of the third quark generation. The expansion of the model to three generations schemd, in order to accommodate the observed violation in KL decay, was first proposed by Kobayashi and Maskawa in 1973. The CP 4 violation via a phase in quark mixing matrix. The quark mixing matrix has three angles and one phase and is generalized from the Cabibbo mixing matrix into six quarks with three quark generations represented through the 3 × 3 matrix called the Cabibobo-Kobayashi-Maskawa matrix (CKM ). In 1977, the quark b was officially discovered, confirming the hypothesis of scientists has accurated. It also mark proposal of Kobayashi-Maskawa that is success befor finding the quark c of the second generation. By using three generations with a mixing angles: θ1, θ2, θ3 and CP violation phase, δ introduced by Kobayashi and Maskawa, the quark mixing matrix is as follows: V = R1(θ2)R3(θ1)C(0, 0, δ)R1(θ3), (1.8) Another parameterization of V is the so-called standard parameterization which is is characterized in terms of three angles θ12, θ23, θ13 and a phase δ13 as: V =  c12c13 s12c13 s13e−iδ13−s12c23 − c12s23s13eiδ13 c12c23 − s12s23s13eiδ13 s23c13 s12s23 − c12c23s13eiδ13 −c12s23 − s12c23s13eiδ13 c23c13  , (1.9) where, cij = cosθij , sij = sinθij , i, j = (1, 2, 3). 1.2.3. K0 − K¯0 mixing in SM Since neutral kaons are the bound states of s and d quarks and their antin- quarks, (K0 ∼ s¯γ5d, K¯0 ∼ d¯γ5s), this mixing occur because there is a moving process s¯d ↔ sd¯. In the FCNC processes of kaons, the strangeness changes |4S| = 2, while charge do not. Their mass diference: ∆mK ≡ mKL −mKS w 2M12, (1.10) According to Feynman rule, effective Lagrangian: L|∆S|=2eff = αGF 4 √ 2pisin2θW ∑ i,j=c,t (V ∗isVid)(V ∗ jsVjd)E(xi, yj)(s¯γµPLd)(s¯γ µPLd, (1.11) 5 where, PL = 1−γ52 , Vis are CKM matrix elements and confficient function E(xi, yj) express the contributtions of two internal quarks with masses mi,mj and xi ≡ m 2 i M2w . he confficient function E(xi, xj): E(xi) ≡ E(xi, xi) = −3 2 ( xi xi − 1) 3lnxi − xi[ 1 4 − 9 4 1 xi − 1 − 3 2 1 (xi − 1)2 ].(1.12) To getM12, we need to evaluate the matrix element of respect to kaons states: 〈K0|(s¯γµLd)|K¯0〉 = 2 3 f2Km 2 KB, (1.13) where, fk = 160 MeV is decay constant, mK is the mass of K-meson (mK w M) and B is the "bag-parameter", which parameterizes the ambiguity due to the non-perturbative QCD effects to form the bound states K0 and K¯0 . Hamiltonian is the mass-squared matrix reads as: H = ( M2 δm 2 2M δm2 2M M 2 ) , (1.14) this means M12 w δm 2 2M . In the case of restricted two generatiob model, noting E(xc) w −xc với xc << 1, we get: ∆mK w −GF√ 2 α 6pisin2θW B sin2θccos 2 θc m2c M2W . (1.15) The contribution of SM to K-meson mass different: bea ∆mK = 0.467.10 −2/ps. (1.16) According to recent calculations, B = 0.72±0.04, K-meson mass different: bea ∆mK = (3.483± 0.006)µeV = (5.292± 0.009).10−3/ps. (1.17) Thus, there is a difference in K-meson mass between SM theory and experiment. 6 CHAPTER 2. PHENOMENOLOGY OF THE 3− 2− 3− 1 MODEL 2.1. The anomaly cancellation and fermion content The electric charge operator: Q = T3L+T3R+βT8R+X. The right-handed fermions are arranged as: 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) 2.2. Symmetry breaking schemes To break the gauge symmetry and generate the particle masses appropri- ately, the scalar content is 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) 7 Ξ =  Ξ011 Ξ−12√ 2 Ξq13√ 2 Ξ−12√ 2 Ξ−−22 Ξq−123√ 2 Ξq13√ 2 Ξq−123√ 2 Ξ2q33  ∼ (1, 1, 6, 2(q − 1)3 ) , (2.7) with vacuum expectation values (VEVs), 〈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 spontaneous symmetry breaking is implemented through three possible ways. The first way assumes w  Λ  u, v, and the gauge symmetry is broken as: SU(3)C ⊗ SU(2)L ⊗ SU(3)R ⊗ U(1)X w−→ SU(3)C ⊗ SU(2)L ⊗ SU(2)R ⊗ U(1)B−L Λ−→ SU(3)C ⊗ SU(2)L ⊗ U(1)Y ⊗WP u,v−→ SU(3)C ⊗ U(1)Q ⊗WP . The second way assumes Λ  w  u, v, and the gauge symmetry is broken as: SU(3)C ⊗ SU(2)L ⊗ SU(3)R ⊗ U(1)X Λ→ SU(3)C ⊗ SU(2)L ⊗ SU(2)R′ ⊗ U(1)X′ ⊗W ′P w−→ SU(3)C ⊗ SU(2)L ⊗ U(1)Y ⊗WP u,v−→ SU(3)C ⊗ U(1)Q ⊗WP . The last case is w ∼ Λ, and the gauge symmetry is broken as: SU(3)C ⊗ SU(2)L ⊗ SU(3)R ⊗ U(1)X w,Λ−→ SU(3)C ⊗ SU(2)L ⊗ U(1)Y ⊗WP u,v−→ SU(3)C ⊗ U(1)Q ⊗WP . Conclusion: every symmetry breaking scheme leads to the matter parityWP as a residual gauge symmetry, which is not commuted with the beginning gauge symmetry. The normal particles have WP = 1. They are particles in SM. The wrong particles have WP = P+ or P−. They could be dark matter particles. 2.3. Research results of phenomenology of the 3− 2− 3− 1 model 2.3.1. Neutrino mass and lepton flavor violation Neutrino mass The Yukawa interaction: L ⊃ hlabΨ¯aLSΨbR + hEabE¯aLφ†ΨbR + hRabΨ¯caRΞ†ΨbR +H.c. (2.9) 8 The neutral leptons get Dirac masses via u and right-handed Majorana masses via Λ, given in the basis (νL, νcR) as follows Mν = − 1√ 2 ( 0 hlu (hl)Tu 2hRΛ ) . (2.10) Because of u Λ, the type I seesaw mechanism applies and the active neutri- nos (∼ νL) obtain small Majorana masses as mν ' u 2 2 √ 2Λ hl(hR)−1(hl)T . (2.11) Using hl = −√2ml/v and mν ∼ 0.1 eV, we evaluate hR ∼ 1√ 2 (u v )2 ( ml GeV )2 1010 GeV Λ . (2.12) The model predicts Λ ∼ 1010 GeV in the perturbative limit hR ∼ 1. Even relaxing the weak scale ratio as u/v = 1000–0.001, the B − L breaking scale is Λ = 1016–104 GeV. Lepton flavor violation The processes like µ → 3e happen at the tree level by the exchange of doubly-charged scalar (Ξ±±22 ). Branch ratio of the process µ→ 3e: Br(µ− → e+e−e−) ' Γ(µ − → e+e−e−) Γ(µ− → e−νµν¯e) = 1 G2Fm 4 Ξ22 |hReµ|2|hRee|2, (2.13) In order to Br(µ− → e+e−e−) < 10−12, we choose: hRee,eµ = 10−3 ÷ 1 so →mΞ22 = 1÷ 100TeV . The processes like µ → eγ does not exist at the approximate tree level. These processes are induced by one-loop corrections by exchange of doubly- charged scalar Higgs. Branch ratio of the process µ→ eγ: Br(µ→ eγ) ' α 48pi 25 16 |(hR†hR)12|2 M4Ξ22G 2 F , (2.14) where, α = 1/128. Taking the experimental bound Br(µ→ eγ) < 4.2× 10−13 leads to mΞ22 = 1–100 TeV for |(hR†hR)12| = 10−3–10, respectively. Compar- ing to the previous bound, this case translates to hReτ,µτ ' 0.03–3.16. 9 2.3.2. Search for Z1 and Z ′1 at colliders LEPII The LEPII at CERN searched for new neutral gauge boson signals that mediate the processes such as e+e− → (Z1,Z ′1) → ff¯ , where f is ordinary fermion in the final state. From the neutral currents, we obtain effective inter- actions describing the processes, Leff = g 2 L cos2Wm 2 I [ e¯γµ(aIL(e)PL + a I R(e)PR)e ] [ f¯γµ(aIL(f)PL + a I R(f)PR)f ] = g2L c2W ( aZ1L (e)a Z1 L (f) m2Z1 + a Z′1 L (e)a Z′1 L (f) m2Z′1 ) (e¯γµPLe)(f¯γµPLf) +(LR) + (RL) + (RR), (2.15) The cross-section for dilepton final states f = µ: g2L 4c2W 1 t2R + β 2t2X ( (ssW + ccWβtX) 2 m2Z21 + (csW − cW sβtX)2 m2Z′1 ) < 1 (6 TeV)2 , (2.16) we get: mZ1 > O(1) TeV. LHC The LEPII at CERN searched for new neutral gauge boson signals that mediate the processes such as pp→ Z1 → ff¯ ., where f is ordinary fermion in the final state. The cross-section for dilepton final states ff¯ : σ(pp→ Z1 → ff¯) = 1 3 ∑ q=u,d ( dLqq¯ dm2Z1 ) σˆ(qq¯ → Z1) × Br(Z1 → ff¯). (2.17) 10 ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 2% Width 4%Width 8% Width 16% Width 32%Width + Model: Β=-1/ 3 * Model : Β = 1 ‘ 3 1000 2000 3000 4000 5000 10-5 10-4 0.001 0.01 0.1 1 10 mZ1 Σ Hpp ® Z 1 ® llL Hình 2.1: The cross-section σ(pp→ Z1 → ll¯) [pb] as a function of mZ1 [GeV], where the points are the observed limits according to the different widths extracted at the resonance mass in the dilepton final state using 36.1 fb−1 of proton-proton collision data at √ s = 13 TeV with ATLAS detector. The star and plus lines are the theoretical predictions for β = ±1/√3, respectively. Experimental results show that a negative signal for new high-mass phe- nomena in the dilepton final state. It is converted into the lower limit on the Z1 mass, mZ1 > 4 TeV, for models with β = ±1/ √ 3. 2.3.3. Dark matter phenomenology A dark matter particle must satisfy the following conditions: Electrically neutral, colorless, the lightest mass of parity odd particles and the dark matter relic density agreement with the experiment Ωh2 ' 0.1pb ' 0.11. In this model, the dark matter candidates are: • q =0: E1, H6, H7, XR • q = -1: H8, YR E1 Fermion dark matter Dominated annihilator channels of E1: E1E c 1 → ννc, l−l+, νανcα, l−α l+α , qqc, ZH1. (2.18) where the first two productions have both t-channel by respective XR, YR and s-channel by Z1,Z ′1, while the remainders have only the s-channel. There may exist some contributions from the new scalar portals, but they are small and neglected. There is no standard model Higgs or Z portal. In Fig. 2.2 we display the dark matter relic density as a function of its mass. It is clear that the relic density is almost unchanged when mZ′1 changes. he stabilization of dark matter yields only a Z1 resonance regime. For instance, 11 w = 9 TeV, the dark matter mass region is 1.85 < mE1 < 2.15 TeV, given that it provides the correct abundance. mZ1 = 4.13 TeV mZ2 = 81 TeV Z1 Resonance ¯ w = 9 TeVï= 100 TeV 500 1000 1500 2000 2500 3000 0.01 0.1 1 10 mE1 HGeVL W h 2 Hình 2.2: The relic density of the fermion candidate as a function of its mass, mE0 , in the limit Λ w, ở đây Z1 ≡ Z1 và Z2 ≡ Z ′1. Currently, there are three ways to search for dark matter: search at the LHC, direct search and indirect search. The three methods have their own strengths. Using Micromegas software, we drawn a graph for the direct search process. The direct detection experiments measure the recoil energy deposited 500 1000 1500 2000 2500 3000 10-48 10-47 10-46 10-45 10-44 mE1 HGeVL Σ E 1 - X e Hcm 2 L 500 1000 1500 2000 2500 3000 10-6 10-5 10-4 0.001 mE1 HGeVL E v en ts Hd a y kg L Hình 2.3: The scattering cross-section (left-panel) and the total number of events/day/kg (right-panel) as functions of fermion dark matter mass. by the scattering of dark matter with the nuclei. This scattering is due to the interactions of dark matter with quarks confined in nucleons. Fig. 2.3 shows that the predicted results are consistent with the XENON1T experiment since the dark matter mass is in the TeV scale. H6 scalar dark matter The scalar H6 transforms as a SU(2)L doublet. The field H6 can annihilate into W+W−, ZZ,H1H1 and f¯f since its mass is beyond the weak scale. The 12 annihilation cross-section is given by: 〈σv〉 ' ( α 150 GeV )2 [(600 GeV mH6 )2 + ( x× 1.354 TeV mH6 )2] , (2.19) where x ∼ λSM ' 0.127. In order for H6’s density to reach the thermal abun- dance density or below the thermal abundance density, its mass must meet mH6 600 GeV is large, the scalar dark mat- ter can (co)annihilate into the new normal particles of the 3-2-3-1 model via the new gauge and Higgs portals similarly to the 3-3-1 model, and this can reduce the abundance of dark matter to the observed value, so H6 is not a good candidate for dark matter. H7 scalar dark matter SinceH7 is a singlet of the SU(2)L group, it has only the Higgs (H1,2,3,4,6,7), new gauge, and new fermion portals. The annihilation products can be the standard model Higgs, W,Z, top quark, and new particles. we chose the the parameter space to the primary annihilation channels is Higgs in SM through the new Higgs ports. H7 H6 h H7 h H7 H7 h H7 h H7 H7 h h h H7 H7 H2 h h H7 H7 H3 h h H7 H7 H4 h h H7 H7 h h Hình 2.4: Diagrams that describe the annihilation H∗7H7 → H1H1 via the Higgs portals, where and in the text we sometimes denote h ≡ H1 for brevity. We calculate the total amplitude of diagrams Feymman and build the ex- pression of the dark matter relic density as: Ωh2 ' 0.1 ( mH7 1.354 TeV )2( λ¯− λ5λ6 2(λ1Ξ + λ2Ξ) + λ′ m2H3 4m2H7 −m2H3 )−2 . (2.20) • mH7  mH3 thì mật độ tàn dư: Ωh2 ' 0.1 ( mH7 λeff × 1.354 TeV )2 . (2.21) 13 Để Ωh2 ' 0.11 thì: mH7 ≤ |λeff | × 1.354TeV ∼ 1.354 TeV, We draw the graph: W IM P -U N S T A B L E 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.00 0.05 0.10 0.15 0.20 mH7 HTeVL W h 2 Hình 2.5: The relic density depicted as a function of the scalar H7 mass. In figure 2.5: The straight line is experimental line correspond to Ωh2 ' 0.11, the resonance width mH7 ∼ 2.6 = mH3/2, Unstable bound is blocked by YR mass. XR gauge boson dark matter The mass of X0R, YR: m2XR = g2R 4 ( u2 + ω2 + Λ2 ) ,m2YR = g2R 4 ( v2 + ω2 ) . (2.22) In (2.22) show that the mass of the vector gauge boson X0R is radically larger than that of the vector gauge boson Y ±R . So, the vector gauge boson X 0 cannot be a dark matter candidate since it is unstable, entirely decaying into the Y ±R and standard model gauge bosons (W∓). H8 scalar dark matter The scalar field, H08 , is considered as a LWP. Because it transforms as the doublet of SU(2)L group, it directly couples to the standard model gauge boson and behaves like the H06 scalar field. So, H8 is not a good candidate for dark matter. YR gauge boson dark matter YR directly couples to the W±, Z gauge bosons, and the dominated anni- hilation channels are Y 0RY 0∗ R → W+W−, ZZ. The dark matter thermal relic 14 abundance is approximated as ΩYRh 2 ' 10−3 m 2 W m2YR . (2.23) Because the fraction m 2 W m2YR is very small, their relic abundance is ΩYRh 2  10−3, much lower than that measured by WMAP/PLANCK. 2.4. Conclusions The neutrino masses are naturally induced by a seesaw mechanism and the seesaw scale ranges from 104 GeV or 1016 GeV depending on the weak scale ratio u/v. At the low seesaw scale, the lepton flavor violation decays µ → 3e and µ→ eγ are dominantly induced by a doubly-charged Higgs exchange. The decay rates are consistent with the experimental bounds if the doubly-charged Higgs mass varies from few TeVs to hundred TeVs. The LEPII constrains the Z1 mass at O(1) TeV, while the LHC searches show that the Z1 mass is larger than 4 TeV for √ s = 13 TeV. The model q = 0 contains two types of dark matter, fermion and scalar fields. The model q = −1 there is no candidate for dark matter. 15 CHAPTER 3. PHENOMENOLOGY OF THE MINIMAL 3− 3− 3− 1 MODEL 3.1. The 3− 4− 1 model 3.1.1. Anomaly cancellation and fermion content The 3− 3− 3− 1 model, a framework for unifying the 3-3-1 and left-right symmetries is based on: SU(3)C ⊗ SU(3)L ⊗ SU(3)R ⊗ U(1)X , (3.1) gauge group. Fermion content: Các hạt được sắp xếp như sau: ψaL =  νaLeaL NqaL  ∼ (1, 3, 1, q − 1 3 ) , ψaR =  νaReaR NqaR  ∼ (1, 1, 3, q − 1 3 ) , (3.2) QαL =  dαL−uαL J −q− 1 3 αL  ∼ (3, 3∗, 1,− q 3 ) , QαR =  dαR−uαR J −q− 1 3 αR  ∼ (3, 1, 3∗,− q 3 ) , (3.3) Q3L =  u3Ld3L J q+ 2 3 3L  ∼ (3, 3, 1, q + 1 3 ) , Q3R =  u3Rd3R J q+ 2 3 3R  ∼ (3, 1, 3, q + 1 3 ) , (3.4) 3.2. Research results of phenomenology of the 3− 3− 3− 1 model 3.2.1. FCNCs As mentioned, the tree-level FCNCs arise due to the discrimination of quark generations, i.e. the third generations of left- and right-handed quarks Q3L,R 16 transform differently from the first two QαL,R under SU(3)L,R⊗U(1)X gauge symmetry, respectively. Hence, the neutral currents will change ordinary quark flavors that nonuniversally couple to T8L,R. The effective Lagrangian that these terms contribute to the meson mass mixing parameter as follows: LeffFCNC = −ΥijL ( q¯′iLγµq ′ jL )2 −ΥijR (q¯′iRγµq′jR)2 , (3.5) where: Υ ij L = 1 3 [( V ∗ qL )3i (VqL) 3j ]2  g21 m2 Z′ L + ( g2cξ3 − g3sξ3 )2 m2ZR + ( g2sξ3 + g3cξ3 )2 m2Z′ R  , (3.6) Υ ij R = 1 3 [( V ∗ qR )3i (VqR) 3j ]2  g24s2ξ3 m2ZR + g24c 2 ξ3 m2Z′ R  . (3.7) Mass diference:: ∆mK = 2 3 <{Υ′12L + Υ′12R }mKf2K , (3.8) ∆mBd = 2 3 <{Υ′13L + Υ′13R }mBdf2Bd , (3.9) ∆mBs = 2 3 <{Υ′23L + Υ′23R }mBsf2Bs . (3.10) The total mass differences can be decomposed as: (∆mM )tot = (∆mM )SM + ∆mM , (3.11) In the moedel: 0.37044× 10−2/ps < (∆mK)tot < 0.68796× 10−2/ps, (3.12) 0.480225/ps < (∆mBd)tot < 0.530775/ps, (3.13) 16.8692/ps < (∆mBs)tot < 18.6449/ps. (3.14) We make contours of the mass differences, ∆mK and ∆mBd,s in w-ΛR plane as Fig. 3.1. The viable regime (gray) for the kaon mass difference is almost entirely the frame. The red and olive regimes are viable for the mass differences ∆mBs and ∆mBd , respectively. Combined all the bounds, we obtain w > 85 TeV and ΛR > 54 TeV for the model with β = − 1√3 , whereas w > 99 TeV, ΛR > 66 TeV for the model with β = 1√3 . 17 Hình 3.1: Contours of ∆mK , ∆mBs , and ∆mBd as a function of (w,ΛR) ac- cording to β = − 1√ 3 (left panel) and β = 1√ 3 (right panel). 3.2.2. Charged LFV µ→ eγ process We are going to derive an expression for the branching decay ratio of µ→ eγ in the model 3 − 3 − 3 − 1. Similarly to the standard model, the decay µ→ eγ in the present model cannot occur at tree-level, but prevails happening through one-loop diagrams, which are contributed by new Higgs scalars, new gauge bosons, and new leptons. The branch ratio of the process µ→ e+ γ: Br(µ→ e+ γ) = 384pi2(4piαem) (|AR|2 + |AL|2) , (3.15) where, αem = 1/128 and Form factors: AR = − ∑ HQ,k 1 192 √ 2pi2GFM 2 H (Y LH )µk (Y LH )∗ek × F (Q) + mkmµ ( Y R H ) µk ( Y L H )∗ ek × 3 × F (r, sk,Q)  + ∑ A Q µ ,k 1 32pi2 M2w M2 Aµ (ULAµ ) µk ( U L Aµ )∗ ek G Q γ (λk) − ( U R Aµ ) µk ( U L Aµ )∗ ek mk mµ R Q γ (λk)  , (3.16) AL = − ∑ HQ,k 1 192 √ 2pi2GFM 2 H (YRH )µk (YRH )∗ek × F (Q) + mkmµ ( Y L H ) µk ( Y R H )∗ ek × 3 × F (r, sk,Q)  + ∑ A Q µ ,k 1 32pi2 M2w M2 Aµ g2R g2 L (URAµ ) µk ( U R Aµ )∗ ek G Q γ (x) − ( U L Aµ ) µk ( U R Aµ )∗ ek mk mµ R Q γ (λk)  ,(3.17) A. The µ→ eγ process when there is left-right asymmetry When there is left-right asymmetry, this mean wL = 0, at one-loop approx- imations, the diagrams with W±iµ, H ± i , H ±± i contribute mainly. We draw the graphs of the branch ratio: 18 Hình 3.2: The branching ratio Br(µ → eγ) governed by intermediate W±1,2 gauge bosons (left panel) and Higgs bosons H±1,2 and H ±± 1,2 (right panel), which is given as a function of ΛR for the selected values of their mixing angle ξw. The upper and lower blue lines correspond to the MEG current bound and near-future sensitivity limit. In figure 3.2 , the branch ratio depends strongly on the mixing angle and ΛR. When the mixing angle increases, the branch ratio increases and vice versa. The left panel shows that, with W±1,2 gauge bosons contribute mainly, ΛR increases to a certain value, the branch ratio is almost unchanged. But the right panel, with Higgs bosons H±1,2 and H ±± 1,2 , the branch ratio decreases monotonically by ΛR. Comparing both graphs in the figure 3.2 shows, the contribution of the gauge gauge W±1,2 and the Higgs boson H ± 1,2 and H ±± 1,2 is equivalent. B. The µ→ eγ process when there is left-right symmetry When there is left-right symmetry, this mean wL 6= 0, at one-loop ap- proximations, the diagrams with W±iµ, Y ±(q+1) iµ , H ± i , H ±± i , H ±(q+1) i contribute mainly. We draw the graphs of the branch ratio: If one uses the same values of the model’s parameters involved in the process, the contributions to the de- cay µ → eγ by virtual charged Higgs H±(q+1)1,2 exchanges are extremely small comparing to those by Y ±(q+1)1,2 gauge bosons. 19 Hình 3.3: Dependence of the branching ratio Br(µ → eγ), governed by the virtual Y ±(q+1)1,2 gauge boson exchanges (lef panel), and the virtual charged Higgs H±(q+1)1,2 exchanges (right panel) on wL for different values of the mixing angle ξY . The upper and lower lines correspond to the MEG current bound and the near future sensitivity limit. 3.2.3. µ→ 3e processes The effective Lagrangian as: Leff (µ→ 3e) = gLLLS (e¯cLµL) (e¯cLeL) + gRRRS (e¯cRµR) (e¯cReR) + gLRLS (e¯ c LµL) (e¯ c ReR) + g RL RS (e¯ c RµR) (e¯ c LeL) . (3.18) Here, we denote MHi (i = 1, 2) to be the masses of doubly charged Higgs bosons and gLLLS = − 2∑ i=1 2( MHi )2 (yLHi)eµ (yLHi)ee , gRRRS = − 2∑ i=1 2( MHi )2 (yRHi)eµ (yRHi)ee ,(3.19) gLRLS = − 2∑ i=1 1( MHi )2 (yLHi)eµ (yRHi)ee , gRLRS = − 2∑ i=1 1( MHi )2 (yRHi)eµ (yLHi)ee .(3.20) The branching ratio: Br(µ→ 3e) = 1 32G2F (|gLLLS |2 + |gRRRS |2 + |gLRLS |2 + |gRLRS |2) , (3.21) where GF = 1.166× 10−5GeV2 is the Fermi coupling constant. We draw graph of the branching ratio: 20 Hình 3.4: Branching ratio Br(µ → 3e) as a function of doubly charged Higgs masses. The three blue lines, Br(µ → 3e) = 10−12; 10−15; 10−16, correspond to the current experimental upper bound, the sensitivities of PSI and PSI upgraded experiments, respectively. The figure reveals a line of monotonically decreasing function as increasing of MH , which is consistent to the fact that the branching ratio is inversely proportional to M4