Summary of physics doctoral thesis: Some new physical effects in the 3−2−3−1 and 3−3−3−1 models

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Resolve the neutrino mass problem. Parameterize the parameters in the model 3−2−3−1 to seek for dark matter for each version of the model with q=0 and q=−1. Research for Z1 and Z01 at LEPII and LHC. Survey in detail the mass of gauge bosons, Higgs bosons, flavor-changing neutral current in the model 3−3−3−1 and calculate the branch ratio of the decay process mu - ey, u - 3e in the model.

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Nội dung Text: Summary of physics doctoral thesis: Some new physical effects in the 3−2−3−1 and 3−3−3−1 models

MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY - - - - - - - - *** - - - - - - - - NGUYEN THI NHUAN SOME NEW PHYSICAL EFFECTS IN THE 3 − 2 − 3 − 1 AND 3 − 3 − 3 − 1 MODELS Major: Theoretical and Mathematical Physics Code: 9 44 01 03 SUMMARY OF PHYSICS DOCTORAL THESIS Hanoi - 2019
INTRODUCTION 1. The urgency of the thesis The standard model (SM )of particle physics is based on two principal the- ories including electroweak theory with the SU (2)L × U (1)Y gauge symmetry and QCD theory with SU (3)c gauge symmetry. The SM describes elementary particles which create matter and their interactions which make the entire universe. In the SM, three interactions of particles are described successfully: strong interactions, electromagnetic interactions and weak interactions. Many predictions of the SM such as: the existence of W ± , Z boson, quark c, t, neutral currents ... have been verified with high fidelity by experiments. W, Z found in 1981 with the masses measured as the model proposed. There are many ways to expand SM such as introducing the spectral particles, extend gauge group,etc. Therefore, we would suggest two expansion models: the 3 − 2 − 3 − 1 and the 3−3−3−1 models. The 3−2−3−1 model is based on the gauge group SU (3)C ⊗ SU (2)L ⊗ SU (3)R ⊗ U (1)X . The 3 − 2 − 3 − 1 model solves the neu- trino masses problem, provides naturally candidate for dark matter, indicates the existence of FCNC at the approximate tree caused by the gauge bosons and Higgs, explains why are there three fermion generations. The 3 − 3 − 3 − 1 model is based on the gauge group SU (3)C ⊗ SU (3)L ⊗ SU (3)R ⊗ U (1)X . It unifies both the left-right and 3-3-1 symmetries, so it inherits all the good fea- tures of the two models. Therefore, the 3 − 3 − 3 − 1 model solves problems of fermion generation numbers, neutrino masses problems, dark matter problems, parity symmetry in electroweak theory. In particular, the model predicts lep- ton flavor violation of the charged lepton. Especially, on 4/7/2012, a particel was discovered at the Large Hadron Collider (LHC) located at the European Nuclear Research Center using two independent detectors, A Toroidal LHC ApparatuS (ATLAS) and Compact Muon Solenoid (CMS), with a mass mea- sured about 125 − 126 GeV. This particel has characteristics identical to the 1
boson Higgs predicted by SM that has not been found previously. It was the last piece for the picture called "Standard Model" to be completed. It can be stated that the SM of particle physics is very successful describing interactions in the Universe. However, the SM model can not explain some observed figures in the Universe and recent experimental results. Specifically: Why do neutri- nos have mass? SM does not identify the candidates for dark matter particles. SM does not explain some abnormal decay channels of mesons, higgs ... SM also does not answer the questions: Why are there three fermion generations? Why is asymmetrical between matter and antimatter? Why is mass graded in the fermion spectrum? ... So, an expansion is necessary. For the mentioned reasons, we choose the subject "Some new physics effects in the 3 − 2 − 3 − 1 and 3 − 4 − 1 models". 2. The objectives of the thesis • Resolve the neutrino mass problem. Parameterize the parameters in the model 3 − 2 − 3 − 1 to seek for dark matter for each version of the model with q = 0 and q = −1. Research for Z1 and Z10 at LEPII and LHC. • Survey in detail the mass of gauge bosons, Higgs bosons, flavor-changing neutral current in the model 3 − 3 − 3 − 1 and calculate the branch ratio of the decay process mu → eγ, µ → 3e in the model. 3. The main contents of the thesis • Overview of SM, flavor-changing neutral current, neutrino masses 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: ! νaL ψaL = ∼ (1, 2, −1), eaL eaR ∼ (1, 1, −2), a = 1, 2, 3. (1.1) Quarks: ! uaL 1 QaL = ∼ 3, 2, , daL 3 4 2 uaR ∼ 3, 1, , daR ∼ 3, 1, − , (1.2) 3 3 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 φ= = v+h+iG ∼ (1, 2, 1). (1.3) ϕ0 √ 2 Z After SSB, the received gauge bosons are: 1 Aµ = sW A3µ + cW Bµ , Zµ = cW A3µ − sW Bµ , Wµ± = √ (A1µ ∓ iA2µ ), 2 gv gv mA = 0, mZ = , mW ± = . (1.4) 2cW 2 3
The Yukawa interaction: − LY = heij ψ¯L i φejR + hdij Q ¯ iL φdj + huij Q R ¯ iL (iσ2 φ∗ )uj + H.c., R (1.5) for fermion mass matrices: Meij = heij √v2 , Mdij = hdij √v2 , và Muij = huij √v2 . 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: ! ! u uL Q1L = = , (1.6) dθc cosθc dL + sinθc sL L 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, ! ! uL uL Q1L = , Q2L = , cosθc dL + sinθc sL cosθc sL − sinθc dL uR , cR , dθc R , sθc R . (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: −iδ13   c12 c13 s12 c13 s13 e V =  −s12 c23 − c12 s23 s13 eiδ13 c12 c23 − s12 s23 s13 eiδ13 s23 c13  , (1.9) s12 s23 − c12 c23 s13 eiδ13 −c12 s23 − s12 c23 s13 eiδ13 c23 c13 where, cij = cosθij , sij = sinθij , i, j = (1, 2, 3). ¯ 0 mixing in SM 1.2.3. K 0 − K Since neutral kaons are the bound states of s and d quarks and their antin- ¯ 5 s), this mixing occur because there is a moving ¯ 0 ∼ dγ quarks, (K 0 ∼ s¯γ5 d, K 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: |∆S|=2 αGF X Lef f = √ (Vis∗ Vid )(Vjs ∗ Vjd )E(xi , yj )(¯ sγ µ PL d, (1.11) sγµ PL d)(¯ 2 4 2πsin θW i,j=c,t 5
where, PL = 1−γ 2 , Vis are CKM matrix elements and confficient function 5 E(xi , yj ) express the contributtions of two internal quarks with masses mi , mj m2 and xi ≡ M i2 . he confficient function E(xi , xj ): w 3 xi 3 1 9 1 3 1 E(xi ) ≡ E(xi , xi ) = − ( ) lnxi − xi [ − − ].(1.12) 2 xi − 1 4 4 xi − 1 2 (xi − 1)2 To get M1 2, we need to evaluate the matrix element of respect to kaons states: hK 0 |(¯ ¯ 0 i = 2 f 2 m2 B, sγ µ Ld)|K (1.13) 3 K K 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 K 0 and K¯0 . Hamiltonian is the mass-squared matrix reads as: ! 2 δm2 M 2M H= δm2 2 , (1.14) 2M M 2 this means M12 w δm2M . In the case of restricted two generatiob model, noting E(xc ) w −xc với xc
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:   ! νaR νaL 1 q−1 ψaL = ∼ 1, 2, 1, − , ψaR =  eaR  ∼ 1, 1, 3, , (2.1)   eaL 2 q 3 EaR   ! u3R u3L 1  d q+1 Q3L = ∼ 3, 2, 1, , Q3R =  3R  ∼ 3, 1, 3, , (2.2)  d3L 6 q+ 2 3 J3R 3   ! dαR uαL 1  −u ∗ q QαL = ∼ 3, 2, 1, , QαR =  αR  ∼ 3, 1, 3 , − , (2.3)  dαL 6 −q− 13 3 JαR q q+ 2 2 −q− 1 1 EaL ∼ (1, 1, 1, q), J3L 3 ∼ 3, 1, 1, q + , JαL 3 ∼ 3, 1, 1, −q − , (2.4) 3 3 2.2. Symmetry breaking schemes To break the gauge symmetry and generate the particle masses appropri- ately, the scalar content is introduced as ! 0 + −q S11 S12 S13 ∗ 2q + 1 S = − 0 −q−1 ∼ 1, 2, 3 , − , (2.5) S21 S22 S23 6 φ−q   1  −q−1  2q + 1 φ =  φ2  ∼ 1, 1, 3, − , (2.6) 3 φ03 7
Ξ− Ξq13   Ξ011 √12 2 √ 2  Ξ− −− Ξq−1  ∼ 2(q − 1) Ξ =  √12 Ξ22 23 √ 1, 1, 6, , (2.7)  2 2  3 Ξq13 Ξq−1 2q √ 2 23 √ 2 Ξ33 with vacuum expectation values (VEVs),    ! 0 Λ 0 0 1 u 0 0 1  1  hSi = √ , hφi = √  0  , hΞi = √  0 0 0(2.8) .   2 0 v 0 2 2 w 0 0 0 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: w SU (3)C ⊗ SU (2)L ⊗ SU (3)R ⊗ U (1)X −→ SU (3)C ⊗ SU (2)L ⊗ SU (2)R ⊗ U (1)B−L Λ u,v −→ SU (3)C ⊗ SU (2)L ⊗ U (1)Y ⊗ WP −→ 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)R0 ⊗ U (1)X 0 ⊗ WP0 w u,v −→ SU (3)C ⊗ SU (2)L ⊗ U (1)Y ⊗ WP −→ SU (3)C ⊗ U (1)Q ⊗ WP . The last case is w ∼ Λ, and the gauge symmetry is broken as: w,Λ SU (3)C ⊗ SU (2)L ⊗ SU (3)R ⊗ U (1)X −→ 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 parity WP 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: ¯ aL SΨbR + hE L ⊃ hlab Ψ ¯ † R ¯c † ab EaL φ ΨbR + hab ΨaR Ξ ΨbR + H.c. (2.9) 8
The neutral leptons get Dirac masses via u and right-handed Majorana masses c via Λ, given in the basis (νL , νR ) as follows ! 1 0 hl u Mν = − √ . (2.10) 2 (hl )T u 2hR Λ Because of u Λ, the type I seesaw mechanism applies and the active neutri- nos (∼ νL ) obtain small Majorana masses as u2 l R −1 l T mν ' √ h (h ) (h ) . (2.11) 2 2Λ √ Using hl = − 2ml /v and mν ∼ 0.1 eV, we evaluate R 1 u 2 ml 2 1010 GeV h ∼√ . (2.12) 2 v GeV Λ 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: Γ(µ− → e+ e− e− ) 1 Br(µ− → e+ e− e− ) ' − − = 2 4 |hR 2 R 2 eµ | |hee | , (2.13) Γ(µ → e νµ ν¯e ) GF mΞ22 In order to Br(µ− → e+ e− e− ) < 10−12 , we choose: hR ee,eµ = 10 −3 ÷ 1 so →mΞ22 = 1 ÷ 100T eV . 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: α 25 |(hR† hR )12 |2 Br(µ → eγ) ' , (2.14) 48π 16 MΞ422 G2F 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 hR eτ,µτ ' 0.03–3.16. 9
2.3.2. Search for Z1 and Z10 at colliders LEPII The LEPII at CERN searched for new neutral gauge boson signals that mediate the processes such as e+ e− → (Z1 , Z10 ) → f f¯, where f is ordinary fermion in the final state. From the neutral currents, we obtain effective inter- actions describing the processes, 2 gL µ I I ¯γ (a (f )PL + aI (f )PR )f µ I Leff = e ¯ γ (a L (e)P L + a R (e)P R )e f L R cos2W m2I Z10 Z10 ! 2 Z1 Z1 gL aL (e)aL (f ) aL (e)aL (f ) = 2 2 + 2 eγ µ PL e)(f¯γµ PL f ) (¯ cW mZ1 mZ 0 1 +(LR) + (RL) + (RR), (2.15) The cross-section for dilepton final states f = µ: ! 2 2 2 gL 1 (s sW + c cW βtX ) (c sW − cW s βtX ) 1 + < , (2.16) 4c2W t2R + β 2 t2X 2 mZ 2 m2Z 0 (6 TeV)2 1 1 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 → f f¯., where f is ordinary fermion in the final state. The cross-section for dilepton final states f f¯:   1 X dLqq¯ σ(pp → Z1 → f f¯) =  2 ˆ (q q¯ → Z1 ) × Br(Z1 → f f¯). (2.17) σ 3 dmZ1 q=u,d 10
*10 + *+ 2% Width + * 4 % Width 1 +* +* 8% Width + 16% Width *+ *+ 32%Width + * 0.1 +* + Model: Β=-1/ 3 +* + ΣHpp®Z1 ®llL *+ * Model : Β = 1 3 *+ + * +* 0.01 +* + *+ *+ + * + *+ *+ *+ *+ 0.001 + * + *+ *+ *+ *+ + * + *+ *+ *+ 10-4 *+ + * + *+ *+ *+ *+ + * + *+ *+ 10-5 * 1000 2000 3000 4000 5000 mZ1 Hình 2.1: The cross-section σ(pp → Z1 → l¯l) [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 0.1pb relic density agreement with the experiment Ωh2 ' ' 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 : E1 E1c → νν c , l− l+ , να ναc , lα − + lα , qq c , ZH1 . (2.18) where the first two productions have both t-channel by respective XR , YR and s-channel by Z1 , Z10 , 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 mZ10 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. 10 mZ1 = 4.13 TeV mZ2 = 81 TeV 1 Z1 Resonance W h2 ¯ 0.1 w = 9 TeV 0.01 ï= 100 TeV 500 1000 1500 2000 2500 3000 mE1 HGeVL Hình 2.2: The relic density of the fermion candidate as a function of its mass, mE 0 , in the limit Λ w, ở đây Z1 ≡ Z1 và Z2 ≡ Z10 . 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 10-44 Events HdaykgL 10-45 0.001 Σ E1 -Xe Hcm2 L 10-46 10-4 10-47 10-5 10-48 10-6 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000 mE1 HGeVL mE1 HGeVL 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, H1 H1 and f¯f since its mass is beyond the weak scale. The 12
annihilation cross-section is given by: " # α 2 600 GeV 2 x × 1.354 TeV 2 hσvi ' + , (2.19) 150 GeV mH6 mH6 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.However, when 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 Since H7 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 h H7 h H7 h H7 H2 h h H6 H7 H7 h H7 h H7 h H7 h H7 H3 h H7 H4 h H7 h H7 h H7 h H7 h Hình 2.4: Diagrams that describe the annihilation H7∗ H7 → H1 H1 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: 2 −2 m H 2 λ 5 λ 6 mH Ωh2 ' 0.1 7 λ¯− + λ0 3 . (2.20) 1.354 TeV 2(λ1Ξ + λ2Ξ ) 4mH7 − m2H3 2 • mH7 mH3 thì mật độ tàn dư: 2 m H7 Ωh2 ' 0.1 . (2.21) λeff × 1.354 TeV 13
Để Ωh2 ' 0.11 thì: mH7 ≤ |λeff | × 1.354TeV ∼ 1.354 TeV, We draw the graph: 0.20 BLE 0.15 STA P-UN Wh2 0.10 WIM 0.05 0.00 0.5 1.0 1.5 2.0 2.5 3.0 3.5 m H HTeVL 7 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 0 The mass of XR , YR : 2 2 gR gR m2XR 2 2 2 2 v2 + ω2 . = u + ω + Λ , mYR = (2.22) 4 4 0 In (2.22) show that the mass of the vector gauge boson XR is radically larger ± than that of the vector gauge boson YR . So, the vector gauge boson X 0 cannot be a dark matter candidate since it is unstable, entirely decaying into the YR± and standard model gauge bosons (W ∓ ). H8 scalar dark matter The scalar field, H80 , 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 H60 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 YR0 YR0∗ → W + W − , ZZ. The dark matter thermal relic 14
abundance is approximated as 2 −3 m2W ΩYR h ' 10 . (2.23) m2YR m2W Because the fraction m2Y is very small, their relic abundance is ΩYR h2 10−3 , R 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:     νaL νaR q−1 q−1 ψaL =  eaL  ∼ 1, 3, 1, , ψaR =  eaR  ∼ 1, 1, 3, ,     q 3 q 3 NaL NaR (3.2)     dαL dαR  −u ∗ q  −u ∗ q QαL = αL  ∼ 3, 3 , 1, − , QαR =  αR  ∼ 3, 1, 3 , − ,   1 −q− 3 3 −q− 31 3 JαL JαR (3.3)     u3L u3R  d q+1  d q+1 Q3L =  ∼ 3, 3, 1, , Q3R =  3R  ∼ 3, 1, 3, ,   3L 2 q+ 3 3 2 q+ 3 3 J3L J3R (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: 2 2 Lef f F CN C = −Υ ij ¯0 L q iL γ µ q 0 jL − Υ ij ¯0 R q iR γ µ q 0 jR , (3.5) where:  2 2  2 2 1 ∗ 3i g c − g s g s + g c ij 3j g 2 ξ3 3 ξ3 2 ξ3 3 ξ3 ΥL = VqL (VqL )  1 + +  , (3.6) 3 m2Z 0 m2Z m2Z 0 L R R   1 3i 2 g 2 s2 g 2 c2 ΥR ij = ∗ VqR (VqR ) 3j  4 ξ3 + 4 ξ3  . (3.7) 3 m2Z m2Z 0 R R Mass diference:: 2 012 < ΥL + Υ012 2 ∆mK = R mK fK , (3.8) 3 2 = < Υ013 013 mBd fB2 d , ∆mBd L + Υ R (3.9) 3 2 = < Υ023 023 mBs fB2 s . ∆mBs L + Υ R (3.10) 3 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 = − √13 , whereas w > 99 TeV, ΛR > 66 TeV for the model with β = √13 . 17
Hình 3.1: Contours of ∆mK , ∆mBs , and ∆mBd as a function of (w, ΛR ) ac- cording to β = − √13 (left panel) and β = √13 (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) = 384π 2 (4παem ) |AR |2 + |AL |2 , (3.15) where, αem = 1/128 and Form factors:   X 1 L L ∗ mk R L ∗ AR = − √  YH µk YH ek × F (Q) + YH µk YH ek × 3 × F (r, s k , Q)  192 2π 2 GF M 2 mµ H Q ,k H   1 Mw 2 ∗ ∗ m X  UL L Q R L k Q + Aµ U Aµ Gγ (λ k ) − U Aµ U Aµ R γ (λ k )  , (3.16) Q 32π 2 M 2 µk ek µk ek mµ Aµ ,k A µ   X 1 R R ∗ mk L R ∗ AL = − √  YH YH × F (Q) + YH YH × 3 × F (r, sk , Q) Q 192 2π 2 GF M 2 µk ek mµ µk ek H ,k H   1 2 2 Mw g R R ∗ ∗ m X R Q L R k Q + UA UA G (x) − UA UA Rγ (λk ) (,3.17) µ ek γ  32π 2 M 2 g2 µ µk µ µk µ ek m Q A L µ Aµ ,k µ 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 Wiµ , Hi± , Hi±± contribute mainly. We draw the graphs of the branch ratio: 18
± Hình 3.2: The branching ratio Br(µ → eγ) governed by intermediate W1,2 ± ±± gauge bosons (left panel) and Higgs bosons H1,2 and H1,2 (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 W1,2 gauge bosons contribute mainly, ΛR increases to a certain value, the branch ratio is almost unchanged. But ± ±± the right panel, with Higgs bosons H1,2 and H1,2 , the branch ratio decreases monotonically by ΛR . Comparing both graphs in the figure 3.2 shows, the contribution of the ± ± ±± gauge gauge W1,2 and the Higgs boson H1,2 and H1,2 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- ± ±(q+1) ±(q+1) proximations, the diagrams with Wiµ , Yiµ , Hi± , Hi±± , Hi 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- ±(q+1) cay µ → eγ by virtual charged Higgs H1,2 exchanges are extremely small ±(q+1) comparing to those by Y1,2 gauge bosons. 19