MINISTRY OF EDUCATION AND TRAINING VIET NAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF PHYSICS LE THO HUE LEPTON FLAVOR VIOLATING DECAYS IN SUPERSYMMETRIC 3-3-1 MODELS Major subject: Theoretical and mathematical Physics Code: 62 44 01 01 THE SUMMARY OF THE DOCTORAL DISSERTATION HA NOI - 2013 The research was finished at: INSTITUTE OF PHYSICS, VIET NAM ACADEMY OF SCIENCE AND TECHNOLOGY Supervisor : Prof Dr Hoang Ngoc Long Institute of Physics, Viet Nam Academy of Science and Technology Referee 1: Prof Dr Tran Huu Phat Referee 2: Prof Dr Nguyen Xuan Han Referee 3: Assoc Prof Dr Phan Hong Lien This dissertation will be defended in front of the evaluating assembly at academy level Place of defending: meeting room, floor , Institute of Physics, Viet Nam Academy of Science and Technology No 10, Dao Tan, Ba Dinh, Ha Noi Time: at…… … day… month…… year…… This thesis can be studied at the Vietnam National Library or at Library of the National Academy of Public Administration Introduction Urgency of the topic Particle physics are now strictly related with high energy colliders All models are waiting for experimental significance of New Physics to compare with all predictions as well as limiting parameter space of each model Specially, in the last 2012 and early 2013 the Large Hadron Collider (LHC), placed in CERNSwitzerland with two independent detectors CMS and ATLAS, has discovered a new scalar particle inheriting many properties of Higgs (Higgs-like) predicted by the Standard Model (SM) with mass around 125-126 GeV This is the last type of particle contained in SM which has already found by recent experiments In the coming time, when energy of LHC increases up to 14 TeV, physicists hope the appearance of New Physics significance beyond SM In particular, there may be Lepton Flavor Violating (LFV) decays of normal leptons Until now, SM is still the most successful model predicting all current experimental results except the experiments of neutrino oscillations They prove that neutrinos indeed have non-zero but very small masses and there appear the conversions among them This confirms that SM is still an effective theory of some more general theory The conversions among neutrinos gives an clear evidence for the LFV in the world of fundamental particles Therefore, one hopes that these effects may happen in the charged lepton sector While the Lepton Flavor (LF) numbers are absolutely conserved in the SM framework, then cLFV decays, if happen, will be a clear evidence of New Physics It was proved that the simplest model beyond SM, the Standard Model with right-handed neutrinos, has very suppressed cLFV effects that cannot be detected by recent experiments On the other hand, the minimal supersymmetric Standard Model (MSSM) which is the most attractive model studied by both theory and experiment, contains many interesting cLFV effects The model is also the simplest supersymmetric model with the least new particles as well as parameters Many publications for MSSM confirm that cLFV significance can be happen in colliders such as LHC In addition, some cLFV decays, such as cLFV decays of tauon , have limited the parameter space of this model, excluded many regions containing light superpartners This leads to the consequence that these superparticles are out of the detection of recent experiments For the supersymmeetric 3-3-1 (SUSY331) models, it is also necessary to predict and investigate the parameter space in order to compare with other models and experimental bounds This is the main reason why we concentrate on the study of the cLFV decays in the SUSY331 models and publish obtained results in this dissertation We will consider two models supersymmetric economical 3-3-1 (SUSYE331) and supersymmetric reduced minimal 3-3-1 (SUSYRM331) models Research purpose • We construct the SUSYRM331 • We study the lepton flavor violation in the SUSYE331 model through decays of Higgs, tau and Z boson Research object • LFV verteices in the SUSYE331 and SUSYRM331 • cLFV decays of H→ μτ , τ → μγ, τ → 3μ and Z → μτ in the SUSYE331 Research content • The supersymmetric reduced minimal 3-3-1 model • Properties of LFV vertices in SUSY331 models • The possibility of detection decay H→ μτ in recent colliders • Discussing on regions of parameter space of SUSYE331 satisfying the experimental bounds Research method • Quantum field theoretical method • Numerical investigation by Mathematica 7.0 Dissertation Structure The dissertation contains the introduction, four chapters presenting main content and the conclusion listing all of new results of our works In addition, the dissertation contains three more appendices which show in detail all necessary analytic formulas and computations The first chapter summaries properties of the 3-3-1 models, discusses on the LFV property of these models and the basis of the supersymmetric theory The second chapter concentrates on two models SYSYE331 and SUSYRM331 For SUSYE331, we just discuss on the LFV vertices which are the LFV sources creating all LFV decays considered in the next chapters The remain part of this chapter involves with constructing in detail the SUSYRM331 and the LFV vertices in the soft term of the model Two chapters three and four directly investigate some particular LFV decays in the frame work of the SUSYE331 Chương Review of 3-3-1 models The original 3-3-1 models are the minimal 3-3-1 (M331) and the 3-3-1 model with right-handed neutrinos These models are constructed by extending the SU(2)L gauge group in the SM to SU(3)L group A common property of these models are the appearance of LFV effects in the original structures them selves 1.1 The 3-3-1 model with right-handed neutrinos In this model, the SU(2)L group was extended SU(3)L by adding a right-handed neutrino to the bottom component of each lepton triplet In the quark sector, new quarks appearing as third components of quark (anti) triplets are called exotic quarks, with the lepton number L = One needs three Higgs (anti)triplets to generate masses of particles Because neutrinos and anti-neutrinos are in the same triplets then the lepton numbers are not conserved This is the common property of the 3-3-1 models But these models relate with a new conserved number called extended lepton number, denoted L It can be computed from the original L number by a formula (1.1) L = √ λ8 + LI More detail, the table 1.1 lists particular values of L and baryon number B = BI of multiplets contained in the considered model In addition, table 1.2 presents numerical values of components of all multiplets Bảng 1.1: Values of B and L for 3-3-1 models with right-handed neutrinos Multiplet B L χ η −2 ρ −2 Q3L −2 QαL 3 uaR daR TR −2 DαR faL laR Bảng 1.2: Non-rezo values of L for fields in the 3-3-1 models with right-handed neutrinos Field L NL −1 lL lR ρ+ −2 η3 −2 χ0 χ− 2 DαL DβL TL −2 TR −2 From table 1.2, we can see that neutral Higgses have zero values of VEVs unless they have L = The number of neutral Higgses can be reduced if neutral Higgses with non-zero values of L can get non-zero but enough small values of VEVs Because LFV is the natural property of the 3-3-1 models so this assumption is allowed, provided VEVs are enough small to satisfy experimental bounds This is the idea of authors P.V Dong, H.N Long, D.T Nhung and D.V Soa to construct the economical 3-3-1 (E331) model This model contains many proper properties comparing with other 3-3-1 models: The number of parameters are reduced, VEV values are limited from experiment bounds, the LFV in the neutral lepton sector can be explained, The simple property of the Higgs sector in the E331 model is used to construct the supersymmetric version which is widely investigated recently 1.2 The minimal 3-3-1 model The minimal 3-3-1 model (M331) model is constructed similar to the cases of above 3-3-1 models The difference is that all lepton triplets are well-known leptons in the SM and there need not any new leptons But the Higgs spectrum in this model is rather complicated because of the appearance of a Higgs sextet Table 1.3 lists values of B L for multiplets in the model Note that Bảng 1.3: Values of B and L in the M331 Đa tuyến Tích B Tích L χ ρ −2 η −2 S Q3L QαL 3 −2 uaR daR TR DαR 0 −2 3 3 faL formula (1.1) is still true for this model Table 1.3 shows that all Higgses with non-zero VEVs all have L = Although the M331 does not need extra leptons such as righthanded neutrios, the Higgs spectrum is complicated and it is hardly to find the exact mass eigenvalues of Higgses Hence, supersymmetric versions constructed from this model always has problems of Higgs diagonalization Recently, a new model with two Higgs triplets, called the reduced minimal 3-3-1 (RM331) model, was introduced The Higgs sector in this model is as simple as that of the E331, even the VEVs are fewer The supersymmetric version, called the supersymmetric reduced minimal (SUSYRM331) model, was also presented in 2013 by us and it is reviewed in the chapter of this dissertation Chương Suppersymmetric 3-3-1 models In this chapter, we concentrate on two supersymmetric models constructed from two 3-3-1 models with the simplest Higgs spectrums, namely the E331 and RM331 models The general basis of the supersymmetric theory is not summarized here 2.1 Supersymmetric economical 3-3-1 model The SUSYE331 was introduced in 2007 as the supersymmetric version of the E331 Similar to the case of MSSM, this model contains the double number of Higgs multiplets as those in the nonsupersymmetric model In previous works studying the SUSYE331, the LFV vertices in the soft terms are not considered In this dissertation we assume that the LFV sources only appear in the soft term of the lagrangian, namely −Lμτ = (˜∗ , τL ) μL ˜∗ ˜˜ + (˜c∗ , τL ) μL ˜c∗ m2 L ˜μ m∗2 ˜ Lμτ m2 R ˜μ m∗2μτ ˜R m2 μτ ˜L m2L ˜τ m2 μτ ˜R m2R ˜τ μL ˜ τ˜ L μc ˜L τL ˜c (2.1) Sleptons (˜L , τL ) and (˜c , τL ) are flavor eigenstates We denote μ ˜ μL ˜c and ˜R , ˜R The respective l l l mass eigenstates as ˜L , ˜L l (m2 , ˜L m2 ) ˜L (m2 , ˜R m2 ) ˜R masses are and The mass eigenstates are mixing of flavor eigenstates μ and τ The mixing is quan˜ ˜ titatively determined through new parameters sL and sR which satisfy, sL c L = m2 ˜ Lμτ m2 − m2 ˜L ˜L , sR c R = m2 ˜ Rμτ m2 − m2 ˜R ˜R (2.2) Eingenstates between two bases relate to each other by μL = ˜ ˜L − sL ˜L , τL = sL ˜L + cL ˜L , where cL = cos θL , sL = sin θL ; cL l l ˜ l l c μc = cR lR2 − sR lR3 , τL = sR lR2 + cR lR3 The lepton numbers are L conserved when sL = sR = Similar to the case of sneutrino sector, the mixing between flavor eigenstates is parameterized by sνL and sνR Four parameters sL , sR , sνL and sνR are four independent LFV sources, while the MSSM has only two independent LFV sources These sources will generate one-loop Feynman diagrams for cLFV decays considered in chapters and 2.2 Supersymmetric reduced minimal 3-31 model The SUSYRM331 was constructed in 2013 by D.T huong, L.T Hue, M.C Rodriguez and H.N Long (Nuclear Physics B 870 (2013) 293) Comparing with previous SUSY331s, the lepton and Higgs spectrums is simpler The lepton and Higgs (anti) triplets are as simple as those in the RM331 In addition, this supersymmetric version contains doubly charged bosons, instead of the non-hermitian neutral bosons in the SUSYE331 Both SUSYE331 and SUSYRM331 have some massless leptons and quarks at tree level Fortunately, recent works have indicated that the lepton and quark masses at one-loop are suitable with experiments Therefore, these models are realistic and they need to investigate more about phenomenology Chương H→ μτ decays in the SUSYE331 3.1 Effective operators and branching ratios The low energy effective operators in the general case are used from previous works The branching ratios of neutral Higgses are determined as BR(Φ0 → τ + μ− ) = BR(Φ0 → τ − μ+ ) = 2(1 + tan2 γ) | Δρ |2 + | Δρ |2 L R × BR(Φ0 → τ + τ − ), (3.1) where tan γ is the ratio of two Higgs VEVs tγ ≡ tan γ = v/v , Δρ and Δρ are one loop effective LFV coefficients, Φ0 denote L R the mass eigenstates of Higgses in the SUSYE331, Φ0 = ϕSa36 or φSa36 Feynmann digrams presenting contributions to Δρ and L Δρ are shown in Fig 3.1 The formulas are as follows, R Δρ L = Δρ + Δρ + Δρ + Δρ + Δρ + Δρ + Δρ , (3.2) La Lc Le Lb Ld Lf Lk where Δρ , Δρ , Δρ , Δρ , Δρ , Δρ Δρ receive one loop La Lc Le Lb Ld Lf Lk contributions from diagrams in Fig 3.1 They are computed in 10 (a) μ ˜Lα l λB ρ0 ρ ˜ ˜ (b) τc μ ρ0∗ (d) λ3 A λ8 A (e) νRα ˜ ρ0 ρ ˜ ˜ νLα ˜ ˜ ˜ ˜ ˜ τ c μ W +W − ρ+ ρ1− τ c ρ0∗ ρ0∗ ν Lα ˜ ˜ ˜ ˜ ˜ μ Y + Y − ρ+ ρ2− τ c μ (c) ˜Lα l ρ+ ˜2 ρ0∗ (f ) ν Rβ ˜ ρ2− ˜ ρ0∗ ν Rα ˜ ρ+ ˜1 τc μ ν Lβ ˜ ρ1− ˜ τc ρ0∗ (i) τ (k) ˜Rα l ρ ρ0 ˜ ˜ ˜Lα l λB μc μ ρ∗0 λB (l) ˜R l β ˜Lα l τc τ ρ∗0 λB ˜R l β μc ρ0∗ Hình 3.1: Feynmann diagrams contribute to Δρ [(a), (b), (c), (d), (e), (f ), (k)] and L Δρ [(i), (l)] R the dissertation They depend on only the function I3 (x, y, z) where the precise formula is, I3 (x, y, z) = 3.2 xy ln(x/y) + yz ln(y/z) + zx ln(z/x) (x − y)(y − z)(z − x) (3.3) Numerical investigation We just investigate the case of maximal mixing The LFV decays Br(H → μτ ) are large when tan γ is large enough so we choose tan γ = 50 Other parameters are assumed corresponding to the values shown in Figs refFDeltaRhoR1, 3.3, 3.4, 3.5 and 3.6 These Figs show that |Δρ |2 obtains the maximal value R ˜ ∼ 10−3 when |μρ |/mR is very large, the maximal values of |Δρ |2 L 11 are ∼ 5.10−3 when |μρ |/mL receive certain values |μρ |/mL ˜ ˜ ρ −3 Fig 3.4 draws then |ΔL | slowly approaches to the value of 10 the correlative ratio of left-handed and right-handed contributions to the branching ratio (3.1) This plot shows that |Δρ | is the L ˜ dominant contribution when |μρ |/mL is small |Δρ | gives large R contributions when |μρ |/mL are very large To compare the cor˜ 0.001 0.001 10 50 R 50 R 10 10 10 10 10 10 10 10 10 10 10 15 20 25 30 Μ Ρ mR 10 Μ Ρ mR |Δρ |2 depends on |μρ |/mR The respective parameters: 1) Blue–m = ˜ R Hình 3.2: mR = mL ; 2) green–3m = mR = mL ; 3) yellow- m = mR = 3mL ; 4) red–m = ˜ ˜ ˜ ˜ ˜ ˜ ˜ mR = mL /3 Two black lines correspond to two values 10−5 and 10−3 of |50Δρ |2 ˜ R The right (left) panel draws for ≤ μρ /mSUSY ≤ 10 (0 ≤ μρ /mSUSY ≤ 30) 0.01 0.01 0.001 10 10 10 50 L 50 L 10 15 20 25 30 Μ Ρ mL Hình 3.3: 10 10 10 10 10 0.001 10 10 Μ Ρ mL |Δρ |2 depends on |μρ |/mL The respective parameters: 1) blue–m = ˜ L mR = mL ; 2) green–3m = mR = mL ; 3) yellow– m = mL = 3mR ; 4) red– ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ m = mL = mR /3 The black lines correspond to values 10−3 of |50Δρ |2 L relation between ΔL and ΔR , we consider two Figs 3.4 and 3.5 In the left panel of the Fig 3.4, it can be see that if |μρ |/mL ≤ ˜ then |Δρ |2 R |Δρ |2 L is very small and the cLFV effects caused by only ΔL 12 0.1 50 L 10 0.001 10 10 10 10 0.001 0.1 50 R 50 R 50 L 1000 10 15 20 25 30 Μ Ρ mSUSY Μ Ρ mSUSY Plots of |Δρ |2 /Δρ |2 as function of |μρ |/mL Numerical values of pa˜ R L Hình 3.4: ˜ ˜ ˜ ˜ rameters: 1) blue –m = mR = mL ; 2) green–3m = mR = mL ; 3)yellow– m = mL = 3mR ; 4)red–m = mL = mR /3 The black line in the left panel ˜ ˜ ˜ ˜ ρ presents |ΔR |2 ρ |ΔL |2 and 0.1 for = Two black lines in the right panel present values of × 10−3 ρ |ΔR |2 ρ |ΔL |2 In the right panel, there is a region satisfying 10 ≤ |μρ |/mL ≤ 30 ˜ which ΔR are much larger than ΔR This is the region where diagrams cancel to each other and there appears minima which each minimum divides the plot into two regions Therefore, both ΔL and ΔR are very suppressed With |μρ |/mL ≥ 30, the dominant ˜ contributions to both ΔL and ΔR are pure gaugino diagrams (diagrams (l) and (k) in the Fig 3.1) and therefore the contributions of ΔL and ΔR to cLFV effects are the same In general the right contribution are large when μρ is very large In the Fig 3.5, the right panel shows that there not exist any regions of |Δρ |2 parameter space satisfying the condition |ΔR|2 ≥ 0.5 The left ρ |Δρ |2 R |Δρ |2 L L panel shows that the ratio is large when two following con˜ ˜ ditions are satisfied: 1) μρ is very large ;2) if mR > mL then the ratio mR /mL increases according to the increasing values of μρ , ˜ ˜ |Δρ |2 if mR < mL then |ΔR|2 increases when mR → mL ˜ ˜ ˜ ˜ ρ L Table 3.1 presents interaction vertices of neutral Higgses in the SUSYE331 comparing with those of the SM If neutral Higgses are lighter than exotic quarks, they can not decay to the exotic quarks Hence, light Higgses, which their masses depend on 13 0.01 0.001 0.5 0.1 3.0 2.0 mR mL mR mL 1.5 0.0005 0.01 0.1 1.0 0.50.001 00.001 0.1 0.0005 2.5 0.01 0.1 10 15 20 25 0.00 30 0.001 0.0 Μ Ρ mSUSY Hình 3.5: 0.00 10 0.0005 Μ Ρ mSUSY ρ Contour plots of |ΔR |2 , ρ |ΔL |2 mR /mL vs |μρ |/mSU SY with mR = mνR , ˜ ˜ ˜ ˜ ˜ ˜ m = mλ = mL = mνL = mSU SY The red region correspond to values of 0.5 ρ |ΔR |2 ρ |ΔL |2 ≥ Bảng 3.1: Higgs-fermion-fermion vertices in the SUSYE331 comparing with those in the SM Particle SM Higgs ϕSa36 φSa36 Up-fermion cα sα Down-fermion cα sα exotic up-quark sα /sγ cα exotic down-quark cα /sγ sα only VEVs v and v , can satisfy this condition Especially, the new Higgs with mass 125 GeV discoverd recently by LHC can decay to dominant decays of fermion-antifermion b¯ and τ τ For example, b ¯ ¯ the branching ratio of light Higgs ϕSa36 is Br(ϕSa36 → τ τ ) 8% This leads to Br(ϕSa36 → μτ ) × 10−3 % This is a significance which can be detected in recent colliders For heavy Higgses, the main decays are decays to other heavy boson such as W + W − , ZZ, so the LFV decays are very suppressed Numerical investigation in Fig 3.6 shows the region where the branching ratio BR(H → μτ )/BR(H → τ τ ) cực đại cỡ 10−3 when 0.1 ≤ |μρ |/MSU SY ≤ and 0.1 ≤ |mg |/MSU SY ≤ This ˜ branching ratio therefore can be detected by LHC in the coming time 14 3.0 0.001 2.5 0.003 2.0 mg MSUSY mg MSUSY 0.003 1.5 1.0 0.5 0.001 0.0 10 0.0 Μ Ρ MSUSY Hình 3.6: 0.5 1.0 1.5 2.0 2.5 3.0 Μ Ρ MSUSY Contour plots of BR(H → μτ )/BR(H → τ τ ) as function of mg and ˜ |μρ |/mSU SY Other parameters are fixed: m = mλ = mg and mR = mνR = mL = ˜ ˜ ˜ ˜ mνL = mSU SY In the right panel both green and yellow regions present the regions ˜ satisfying BR(H → μτ )/BR(H → τ τ ) ≥ O(10−3 ) 15 Chương cLFV decays of τ and Z boson in the SUSYE331 model Thank to the colliders such as LHC, BABAR, LEP, , many processes predicted by models beyond the SM are searching One of these is the class of cLFV decays which will be evidence of New Physics if detected We concentrate on only three cLFV decays with upper bounds established by recent experiments BR(τ − → μ− γ) < 4.4 × 10−8 , (4.1) BR(τ − → μ− μ+ μ− ) < 2.1 × 10−8 , (4.2) + − −5 BR(Z → μ τ ) < 1.2 × 10 (4.3) These three processes are studied at the same time because the experimental bounds are very clear and the analytic formulas calculating them closely relate to one another We will investigate these three decays in the framework of the SUSYE331 16 Γ b DL 10 GeV Γ b DL 300 2.5 2.5 5 GeV 200 10 250 mL3 GeV 250 mL3 GeV 10 0.5 0.25 200 0.25 150 150 0.5 0.5 15 100 100 5050 40 200 20 15 300 400 100 500 mΛ GeV Hình 4.1: m L2 = m ν L ˜ ˜ γ(b) 2 5 200 2.5 2.5 300 400 500 mΛ GeV Contour plots of DL = mνR ˜ 100 with tan γ = 3.0, mL3 = mνL = mνR ˜ ˜ ˜ 3 and = 300 GeV, θL = θνL = θνR = π/4 and μρ = 140 GeV ˜ ˜ (1TeV) for the left (right) panel The black and dashed curves correspond to values of mB = 300 GeV and mB = −300 GeV 4.1 Effective operators and branching ratios In the SUSYE331, the branching ratios of τ → μγ, Z → μτ , Z → τ μ and τ → 3μ in the limit of effective low energy theory have the same formulas as those of MSSM Contributions from Hμτ effective vertices to the branching ratio of τ → μμμ decays in our investigation is very suppressed Hence, they are ignored in this our calculation 4.2 Numerical calculation and discussion To guarantee the vacuum stability of the model constructed in previous works, we add the B/μ-term types in the soft term of the original model As the result, the charged Higgses naturelly satisfy the lower bounds of experiments Investigation of neutral Higgs sector shows that tγ can get small values These values can allow the existence of parameter space with light sleptons where limits of cLFV decays satisfy experimental bounds In this dissertation we only consider the small values of tan γ and light sleptons which can be detected by LHC 17 τ → μγ decay The branching ratio τ → μγ (4.1) have rather small upper γ bound, it corresponds to the condition |DL,R | ≤ 2.5×10−9 [GeV−2 ] γ(b) Two Figs 4.1 and 4.2 present numerical values of DL which is γ the dominate contribution to |DL,R | Fig 4.1 draws the case of light sleptons mL < 300 GeV To satisfy the experimental bounds ˜ of τ → μγ decay, masses of light sleptons approach to the degenerate values Fig 4.2 draws the regions of mass parameters containing heavy sleptons, order of TeV The results show that the satisfying regions are expanded in the regions of heavier sleptons masses Fig 4.3 presents left-right mixing Aτ as function of μρ in the Γ b DL 10 GeV Γ b DL 300 50 50 300 250 250 10 GeV 2.5 20 80 mL3 GeV mL3 GeV 20 200 10 10 200 15 150 15 150 30 100 100 100 100 80 200 40 300 40 400 30 Hình 4.2: 100 500 mΛ GeV 100 200 300 400 500 mΛ GeV γ(b) Contour plots of DL with tan γ = 3.0, mL2 = mνL = mνR ˜ ˜ ˜ 2 and mL2 = mνL = mνR = TeV, θL = θνL = θνR = π/4 and μρ = 140 GeV (1TeV) ˜ ˜ ˜ ˜ ˜ 2 for left (right) panel The black and dashed curves present for mB = 300 GeV and mB = −300 GeV, respectively presence of three maximal mixing angles With the very large γ values of DL , Aτ must be large to satisfy experiments as well as cancel tachyon sleptons In general, the existence of three LFV sources will exclude the mass parameter space of light sleptons If we consider the case of cLFV effects caused by only right√ γ handed charged slepton sector, sR = cR = 1/ 2, then only DR = Numerical investigation in Fig 4.4 indicates that the regions γ of parameter space containing light sleptons allow DR to easily 18 Γ DL 10 GeV 1000 2.5 Μ Ρ GeV 800 600 400 200 2.5 1000 500 500 1000 AΤ GeV Hình 4.3: γ Contour plots of DL with tan γ = 3., mL2 = TeV, θL = ˜ π/4, θR = θνL = θνR = 0, ALμ = Other parameters are chosen as ˜ ˜ τ (mB , mλ , mL3 , mR )[GeV]: (200, 300, 300, 200)–black, (100, 400, 100, 200)–dashed, ˜ ˜ γ (100, 500, 300, 100)–dotted In particular, the center curves correspond to DL = γ while two other curves limit |DL | ≤ 2.5 × 10−9 [GeV−2 ] satisfy the cLFV bounds of experiments This case is similar to that of MSSM, even the parameter space is wider Hence, sleptons still can be detected by LHC Correlation among effective operator coefficients The τ → 3μ decay includes all contributions of Feynman diagrams with photon, Higgs, Z, Z exchanges and also box diagrams Contributions from Higgs exchanges are very suppress because of small values of tan γ Therefore, there will appear constant ratios of branching ratio of τ → 3μ to one of those of the remain decays depending on which contributions are dominate Here we limit our work on two cases: i) LFV maximal mixing only in the left-handed charged slepton sector, namely cL = √ 1/ 2; ii) The same for the right-handed charged slepton sector To estimate relative contributions of effective vertices AZ and L γ DL into the branching ratio (τ − → μ− μ+ μ− ), we define two contribution coefficients fAZ and fDγ The precise formulas of these coefficients are shown in the dissertation We denote AZ are dominant contribution if 1.05 ≥ fAZ ≥ 0.95 The regions of parameter 19 Γ a DR 10 GeV Γ b DR 300 0.25 10 0.10 0.1 GeV 250 mR3 GeV mR3 GeV 250 200 150 2.5 200 2.5 150 2.5 0.25 100 300 200 100 100 100 200 300 300 4.5 200 200 m B GeV m B GeV Hình 4.4: 100 7.5 100 γ(a) Contour plots of DR γ(b) (left) and DR (right) as functions of two parameters mR3 and mB Other parameters are fixed as follows: tan γ = 3., mR2 = ˜ ˜ TeV, θL = θνL = θνR = 0, θR = π/4 and μρ = 150 GeV ˜ ˜ space satisfying this condditions are called the AZ -domination μL(R) regions ( in contribution to FL(R) ) If fDγ ≤ 0.05, the main conμ L(R) tributions to the Br(τ → 3μ) is FL(R) and the respective regions are called F −domination In contrast, if 1.05 ≥ fDγ ≥ 0.95 then the regions of parameter space are D-domination Maximal mixing in left-handed sector of (˜, τ ) μ ˜ √ sR A2Z = R This case corresponds to sL = cL = A2Z L = sνL = sνR = ˜ ˜ = In the Fig 4.5, and therefore leads to with small values of μρ the D-domination regions in both two panels are very narrow and they extend to the regions of large values of parameters The right panel shows that for large mλ the D-domination regions change rapidly according to the values of mL3 , and μρ > 900GeV The experiment excludes these regions ˜ the left panel indicates that for mλ is small, μρ has both upper and lower bounds because of the condition BR(τ → μγ) < 4.4 × 10−8 There not exist any AZ -domination regions Now we come to two decays Z → μτ and τ → μμμ with investigation shown in Fig 4.6 The mass parameter of gaugino mλ is chosen according to the limit bound of experiment Other parameters are chosen in order of O(100) GeV From Fig 4.6 the 20 300 f Az , fDΓ , BR Τ 800 ΜΓ 10 0.3 0.91 700 0.9 f Az , fDΓ , BR Τ 0.05 0.3 1000 500 mL3 GeV mL3 GeV ΜΓ 0.75 10 1.2 0.07 0.21.2 0.95 600 1.05 400 300 200 1200 1 4.4 800 0.7 1.05 600 400 0.2 0.07 200 0.05 0.05 100 100 150 200 250 300 350 400 Μ Ρ GeV Hình 4.5: 4.4 0.85 200 0.1 400 1.05 600 800 1000 Μ Ρ GeV μL(R) Correlations among AZ , FL L γ and DL with Aτ = The contour plots draw dashed, dotted and black corresponding to constant values of curves fAZ , fDγ and BR(τ → μγ) Two numerical values chosen for (mB , mλ , mL2 mL ) ˜ ˜ L R L are (100, 300, 1000, 100)[GeV] (left ) and (100, 500, 1000, 100) [GeV] (right) maximal value of BR(Z → μτ ) is about 5.10−10 and it is very small comparing with the recent experimental bounds While the values of BR(τ → 3μ) can approach the limit detection of colliders Fig 4.7 shows the numerical resulus of two branching ratios as functions of mB − μρ with Aτ = It is shown that the upper experimental bound of BR(τ → μγ) directly affects to the values of other cLFV decays Specifically, BR(τ → 3μ) < 0.5×10−9 and BR(Z → μτ ) < 10−10 On the other hand, if Aτ = and cancels values of Dγ , BRZ → μτ can be large but are still small than the upper bound of experiment In the case the maximal mixing happens only in the right√ handed charged sleptons, namely sR = 1/ 2, sL = sνL = sνR = ˜ ˜ and the mass parameter space still is in the O(100) GeV scale The obtained results are the same in both model SUSYE331 and MSSM For SUSYE331,the BR(τ → μγ) can approach the experiment bounds and it excludes the large values of BR(τ → 3μ), leading to BR(τ → 3μ) ≤ O(10−9 ) with Aτ = in addition, numerical investigation shows that fAZ as well as BR Z → μτ are still very small 21 10 10 1.5 10 10 1.0 10 10 10 200 300 400 500 600 700 800 mB Hình 4.6: 10 10 10 10 10 100 Μ Μ Μ 10 BR Τ 2.0 10 10 BR Z ΜΤ 3.0 10 10 5.0 10 11 10 11 100 200 300 400 500 600 700 mB Branching ratios Z → μτ (left panel) and τ → 3μ (right panel) as functions of mB The parameter space including (mλ , μρ , mL2 , mL3 , mR ) [GeV] is ˜ ˜ ˜ chosen for three cases : (300, 150, 1000, 100, 100)- black, (400, 200, 1000, 100, 100)green, (500, 150, 1000, 100, 100)- blue Results and discussion The main result of our work are summarized as follows We constructed the 3-3-1 supersymmetric model with the simplest particle content, called the SUSYRM331 We established and parameterize analytic formulas presenting cLFV vertices in the SUSYE331 This kind of vertices in the SUSYRM331 are also discussed In the frame work of the SUSYE331, we constructed analytic formulas of quantities needed for calculating branching ratios of the cLFV decays at one loop level, namely effective operators, Lagrangian and branching ratios of cLFV decays We numerically investigated four particular LFV decays: H → μτ , τ → μγ, Z → μτ and τ → 3μ The results are used to compare with bounds of experiments and find the allowed regions of the parameter space in the SUSYE331 We also predict that the branching ratios of H → μτ , τ → μγ and τ → 3μ are in the detectable limit of recent colliders In contrast the BRZ → μτ are very suppressed 22 800 Z ΜΤ 500 m B GeV 400 10 10 , Τ 4.4 Μ Μ Μ 10 , Τ ΜΓ 0.5 10 0.1 0.5 4.4 300 200 100 0.5 200 400 600 800 1000 Μ Ρ GeV Hình 4.7: Contour plots of branching ratios of τ − → μ− μ+ μ− (dotted), Z → μτ ( dashed) with τ → μγ (black) with Aτ = and (mλ , mL2 , mL3 , mR ) = ˜ ˜ ˜ (400, 150, 1000, 100, 200) with maximal value of 10−9 , out of the detection of experiments We discussed on the B/μ type terms which are not mentioned in the previous works on SUSYE331 and SUSYRM331 These terms are very necessary to cancel Higgs tachyons as well as guarantee the vacuum stability of the SUSY models These terms also affect directly to the Higgs spectrums This is a suggestion for next studies on the SUSY331 models Many formulas established for cLFV branching ratios in this dissertation can be used to calculate many other cLFV decays which we are studying and will publish in the near future 23 The author’s publication list Symmetry Factors of Feynman Diagrams for Scalar Fields, P.V Dong, L.T Hue, H.T Hung, H.N Long and N.H Thao (2010) Math.Phys, 1651500-1511 The 3-3-1 model with A4 flavor symmetr, P.V Dong, L.T Hue, H.N Long and D.V Soa, Phys.Rev D81 (2010) 053004 General formula for symmetry factors of Feynman Diagrams, L.T Hue, H.T Hung and H.N Long, Rept.Math Phys 69 (2012)331-351 Lepton-flavor violating decays of neutral Higgs to muon and tauon in supersymmetric economical 3-3-1 model, P.T Giang, L.T Hue, D.T Huong and H.N Long, Nuclear Physics B 864 (2012) 85 Supersymmetric reduced minimal 3-3-1 model, D.T Huong, L.T Hue, M.C Rodriguez and H.N Long, Nuclear Physics B 870 (2013) 293 Lepton flavor violating processes τ → μγ, τ → 3μ and Z → μτ in supersymmetric economical 3-3-1 model, L.T Hue, D.T Huong and H.N Long, Nucl Phys B 873 (2013) 207 The dissertation only uses publications [4], [5] and [6] 24 ... LFV decays in the frame work of the SUSYE331 Chương Review of 3-3-1 models The original 3-3-1 models are the minimal 3-3-1 (M331) and the 3-3-1 model with right-handed neutrinos These models are... 1.2 The minimal 3-3-1 model The minimal 3-3-1 model (M331) model is constructed similar to the cases of above 3-3-1 models The difference is that all lepton triplets are well-known leptons in the... multiplets Bảng 1.1: Values of B and L for 3-3-1 models with right-handed neutrinos Multiplet B L χ η −2 ρ −2 Q3L −2 QαL 3 uaR daR TR −2 DαR faL laR Bảng 1.2: Non-rezo values of L for fields in the 3-3-1