The effect of scalar fields on the flavor changing neutral currents in the s331 model and 3 3 1 1 model

INTRODUCTION The urgency of the thesis The Standard Model (SM) is currently considered as the orthodox theory to describe elementary particle interactions Several SM predictions, including the existence and properties of c, t quarks, and gauge bosons W ± and Z, were experimentally confirmed with high precision.The discovery of the Higgs boson at the LHC in 2012 is considered the final piece of the Standard Model picture However, there are many more issues that SM cannot solve, such as not explaining why the number of fermion generations is equal to 3, the neutrino masses, dark matter, dark energy, CP violation in QCD, and matter-antimatter asymmetry This implies that SM cannot be the end of line Particle physicists have been inspired to propose many BSM in which new physics states present at TeV-scale.The signals of these BSM are searched for at the accelerator as new resonances or as deviations from the SM prediction in specific observables In recent years, the process of modifying the predicate has garnered the most attention, as improvements in both nonperturbative techniques and data analysis have begun to reveal differences between the SM prediction and the experimental one These − 4σ deviations are known as flavor anomalies, fox examples: FCNC quark transitions b → sl+ l− of the B meson decays; the anomalous magnetic moment of the muon aµ There are assumptions that these anomalies arise as a result of our incomplete understanding of the non-perturbation effect, but in general, they are strongly implied about the origin of the new physics due to the large deviation and being very difficult to explain by SM itself There are three methods to build BSM models: via extending the spacetime dimension, the particle spectrum, and the electroweak gauge symmetry group In this thesis, we investigate current anomalies in two BSM model electroweak symmetry group extension: the simple 3-3-1 model (S331) and the 3-3-1-1 model The 3-3-1 model is based on the gauge group SU (3)C ⊗SU (3)L ⊗U (1)X , which explains a number of SM issues, including the family number, charged quantization, neutrino masses, CP violation in QCD, and dark matter The 33-1 models can be separated into numerous variants based on the arrangement of particle spectrum and the number of Higgs multiplets, whereas S331 model receiving the most attention It carries a unique Higgs spectrum featuring interactions at the tree level of the Higgs triplet with both leptons and quarks via generic Yukawa matrices, which are the source of lepton (quark) flavor violation decays of Standard Model like the Higgs boson (SMLHB), h → li lj , h → qi qj (i ̸= j), FCNC of the top quark t → qh (q = u, c), anomalous magnetic moment of the muon aµ In addition, the meson mixing systems ∆mK , ∆mBs , ∆mBd receive contributions from the new Higgs in addition to the contributions from the known gauge bosons The model, which is based on the gauge group SU (3)C ⊗SU (3)L ⊗U (1)X ⊗ U (1)N (3-3-1-1 model), is an extension of the 3-3-1 model with a gauged B − L symmetry, which not only inherits the advantages of 3-3-1 model but also has a naturally stable mechanism for dark matter, explains the inflation problem, matter-antimatter asymmetry In this model, there have been several phenomenological studies, one of which is the study of the FCNC process in meson mixing systems ∆mK , ∆mBs , ∆mBd However, only the FCNC contribution associated with the new gauge boson Z2 , ZN is taken into account, and not the FCNC of the new scalars In addition, the FCNC contributions influence rare decay of B meson such as Bs0 → µ+ µ− , B → K ∗ µ+ µ− , and B + → K + µ+ µ− at the tree level The 3-3-1-1 model additionally predicts new charged Higgs and new charged gauge bosons, and this is a new contributor to lepton and quark flavor violation decays, such as b → sγ, µ → eγ For the aforementioned reasons, we chose the topic ”The effect of scalar fields on the flavor-changing neutral currents in the S331 model and 3-3-1-1 model.” The objectives of the thesis ˆ In the S331 model, based on the lepton and quark flavor violation interactions of Higgs triplets, there are some phenomenologies studied, such as the LFVHD and QFVHD h → li lj , h → qi qj (i ̸= j), cLFV decay τ → µγ, anomalous magnetic moment of the muon aµ , FCNC top quark decay t → qh Also presented is the new contribution of the scalar component to the meson mixing systems: ∆mK , ∆mBs , ∆mBd ˆ In the 3-3-1-1 model, the study of FCNC-associated anomalies receives new contributions from the scalar part of meson mixing systems ∆mK , ∆mBs ,∆mBd and several rare decays of B meson: Bs → µ+ µ− , B → K (∗) µ+ µ− The main contents of the thesis ˆ The overview of SM and some BSM models We present some of the most recent experimental constraints and flavor anomalies discovered in colliders ˆ The summary of the S331 model We consider the influences of lepton and quark flavor violating interactions of Higgs triplets in some processes, namely LFVHD and QFVHD, cLFV decay, anomalous muon magnetic moment, FCNC top quark decay, and meson mixing systems ˆ The summary of the 3-3-1-1 model The contributions from new scalars into phenomenologies associated with FCNC include meson mixing systems, rare decays of B meson, and flavor violating radiative decays contributed by newly charged Higgs and gauge bosons CHAPTER OVERVIEW 1.1 The Standard Model The SM of elementary particle physics is a renormalizable quantum field theory that describes three of the four known interactions of nature, with the exception of gravity The particle spectrum in SM is represented as follows: ! ναL Leptons : ψαL = ∼ (1, 2, −1) , eαR ∼ (1, 1, −2), eαL ! uαL Quarks : QαL = ∼ (3, 2, 1/3) , dαL uαR ∼ (3, 1, 4/3), dαR ∼ (3, 1, −2/3), (1.1) where α = 1, 2, are the generation indexes In order to generate particle masses, SM must be spontaneously symmetry broken (SSB) or demand the Higgs mechanism The Higgs mechanism works with the following doublet ! ! + φ √ (1.2) ϕ= ∼ (1, 2, 1), ϕ = φ0 v After SSB, we have gauge bosons with masses and eigenstates as ′ Wµ± ′ Wµ1 ∓ iWµ2 √ = , gv , p v g + g ′2 gv mZ = = 2cW mW ± = ′ Zµ = cW Wµ3 − sW Bµ′ , ′ Aµ = sW Wµ3 + cW Bµ′ , mA = (1.3) Lagrangian Yukawa Llepton for leptons is Y Llepton Y = − X e¯aL Mlab ebR a,b Mlab ebR H + h.c., + e¯aL v (1.4) hl with Mlab = √ab2 v This matrix can be diagonalized using two unitary matrices, VL , VR Lagrangian Yukawa for quarks is X Lmass = − u ¯aL Muab ubR + d¯aL Mdab dbR Yukawa a,b − X u ¯aL a,b Muab Md ubR H + d¯aL ab dbR H + h.c., v v (1.5) u,d v √ is mixing quark mass matrices and can be diagonalized with Mu,d ab = hab u d , VL,R using unitay matrices VL,R Next, we consider the interactions between leptons and gauge bosons The charged current interactions have the following form: ′ ′ = g(Jµ1 Wµ1 + Jµ2 Wµ2 ) = Jµ− W −µ + Jµ+ W +µ , Llepton CC (1.6) where the currents Jµ± are defined as g X g X Jµ+ = √ ν¯aL γµ eaL , Jµ− = √ e¯aL γµ νaL a=1,2,3 a=1,2,3 (1.7) The neutral and electromagnetic currents are Llepton N C+em = eJµem Aµ + g Z µ J Z cW µ (1.8) with JµZ = X ¯la γµ [gL PL + gR PR ]la , Jµem a=1,2,3 = X Q(l)¯la γµ la , (1.9) a=1,2,3 with gL,R = T3 (lL,R ) − s2W Q(l) We it similarly for quark The charged current interactions of quarks are Lquark CC = g ′ µ √ u ¯iL γ Vij d′jL Wµ+ + h.c., (1.10) where V = VLu† VLd is an unitay matrix × 3, the so called CKM matrix The electromagnetic and neutral currents of quarks X em µ em Lquark = eJ A , J = Q(q)¯ qa γµ qa , q = u, d, em µ µ a=1,2,3 Lquark NC g Z µ = J Z , cW µ JµZ = X a=1,2,3 q¯a γµ [gL PL + gR PR ]qa (1.11) 1.2 Current experimental constraints and flavor anomalies 1.2.1 LFVHD and QFVHD Since the SM lacks right-handed neutrinos, the mass of Dirac neutrinos is zero As a result, the lepton number is conserved, which prevents the appearance of the cLFV decay Experiments have confirmed, however, that neutrinos have mass and that they oscillate among generations In the extended version of the Standard Model with right-handed neutrinos, νR , cLFV decay may exist but are heavily suppressed by the GIM mechanism Br(µ → eγ) < 10−54 Other cLFV decays, such as µ → 3e, τ → (e, µ)γ, similarly have extremely small branching ratios, and none of the present experiments have enough sensitivity to measure this value Currently, it is not established which cLFV signal is observed experimentally; rather, the upper limit of the branching ratio is given, namely Br(µ → eγ) < 4.2 × 10−13 (MEG experiment), Br(τ → eγ) < 3.3 ì 108 , Br( à) (Babar experiment) with 90% confidence level New physics may also manifest as Higgs boson properties different than those anticipated by SM, such as the LFVHD h → li lj (i ̸= j) In SM, only lepton-conserving decays h → f f¯ are allowed, whereas LFVHDs h → li lj (i ̸= j) are not permitted Current experimental limits for these LFVHDs are Br(h eà) < 6.1 ì 105, Br(h ) < 2.5 × 10−3 , and Br(h → eτ ) < 4.7 × 10−3 This shows that this may be an indication of the new physics 1.2.2 The anomalous muon magnetic moment The SM prediction for the anomalous muon magnetic moment aSM is µ aSM µ = 116591810(43) × 10−11 , (1.12) The very recent experiment result for aµ by g − experiment at FNAL reads aExp µ = 116 592061(41) × 10−11 (1.13) and shows the deviations with the SM one about 4.2σ ∆aµ −11 ≡ aExp aSM à = 251(59) ì 10 (1.14) The impressive accuracy of the SM prediction and experimental measurement provide aµ a highly accurate physics observable and one of the most sensitive channels for searching for a new physics signal If new physics is required to explain this ∆aµ discrepancy, it would appear in one-loop diagram contributions (new scalars, new vectors, or new fermions) 1.2.3 FCNC top quark decay t → qh (q = u, c) New physics effects are possible in the quark sector, but they are considerably complicated by interactions that contradict the Higgs predicate for the top quark The decays t → qh with q = u, c are one of the top-quark FCNC processes In the SM, Br(t → ch) ≃ 10−15 , Br(t → uh) = |Vub /Vcb |2 Br(t → uh) = |Vub /Vcb |2 Br(t → uh) ≃ 10−17 are extremely small Currently, CMS and ATLAS have not found any significant signals against the background for FCNC decays of top quarks, leading to upper limits for the branching ratios Br(t → qh) < 0.47% with a confidence level of 95% 1.2.4 The anomalies in semi-leptonic decays of B meson A crucial prediction of the SM is that different generations of charged leptons exhibit the same interaction (lepton flavor universality-LFU) Nonetheless, a few recent experiments have revealed the violation of LFU (LFUV), suggesting that it may be an indication of new physics One of the LFUV signals occurs in the FCNC quark transitions b → sl+ l− (l = e, mu) of the B meson, which differs from the prediction of the Standard Model ∼ 3σ: e.g branching ratio Br(B + → K + µ+ µ− ), Br(B → K 0∗ µ+ µ− ), Br(Bs0 → ϕµ+ µ− ); the P5′ coefficient in the decay B → K 0∗ µ+ µ− Due to the GIM mechanism, these LFUV observables cannot occur at the tree-level in the SM and are only present when considering the quantum corrections, such as penguin or box diagrams CHAPTER INVESTIGATION OF THE ANOMALOUS FCNC INTERACTIONS OF HIGGS BOSON IN THE SIMPLE 3-3-1 MODEL 2.1 The summary of the S331 model The S331 model is a combination of the reduced minimal 3-3-1 model and the minimal 3-3-1 model This model contains the following fermion spectrum:   νaL   ψaL ≡  eaL  ∼ (1, 3, 0), eaR ∼ (1, 1, −1) (eaR )c     dαL u3L     QαL ≡  −uαL  ∼ (3, 3∗ , −1/3), Q3L ≡  d3L  ∼ (3, 3, 2/3) , JαL J3L uaR ∼ (3, 1, 2/3) , daR ∼ (3, 1, −1/3) , JαR ∼ (3, 1, −4/3) , J3R ∼ (3, 1, 5/3) , (2.1) with a = 1, 2, and α = 1, are the generation indexes The third generation of quarks is arranged differently than the first two generations in order to obtain acceptable FCNC when the energy scale of the S331 model is suppressed by the Landau pole The scalar spectrum is   −   χ1 η10  −−   −  χ =  χ2  ∼ (1, 3, −1), (2.2) η =  η2  ∼ (1, 3, 0), η3+ χ03 with VEVs read ⟨η10 ⟩ = √u2 , ⟨χ03 ⟩ = √w2 In order to reveal candidates for DM, an inert scalar multiplet ϕ = η ′ , χ′ or σ is added Lagrangian Yukawa reads LY = ¯ 3L χJ3R hJ33 Q + ¯ αL χ∗ JβR hJαβ Q ¯ αL η ∗ daR + +hdαa Q + ¯ 3L ηuaR hu3a Q huαa ¯ + QαL ηχuaR Λ hd3a ¯ c Q3L η ∗ χ∗ daR + heab ψ¯aL ψbL η Λ sνab ¯c ∗ h′e ∗ ab ¯c + (ψaL ηχ)(ψbL χ ) + (ψaL η )(ψbL η ∗ ) + h.c., Λ Λ (2.3) 2.2 Research results of investigation of the anomalous FCNC interactions of the Higgs boson in the S331 model 2.2.1 LFV interactions of Higgs h → µτ Lagrangian for LFV interactions of Higgs reads ee ′ ⊃ e¯′R ghee e′L h + e¯′R gH eL H  ′¯ eν ′ ¯′ )c g νe e′ + ν¯′ L g νe e′ + (e′¯ )c g eν (ν ′ )c H + + h.c., + (eL )c gL νL + (νL L L R R R L R LY with ghee = URe† eν = (ULe )T gL   ee h′e ULe , gH cζ u1 Me − sζ √uw 2Λ2
ν νe uw e′ cθ he + sθ 2Λ h UL , gL = (ULν )T cθ he ULe , √ ULeT cθ √u2Λ sν URνT , eν gR (2.4)   e† uw ′e √ = UR sζ u Me + cζ 2Λ2 h ULe ,  heab h′e ab w 4Λ νe = ULν† cθ √u2Λ sν URe , gR  = (Me )ab = 2u + is the mixing mass matrix of charged leptons The branching ratio for the LFVHD is Br(h → ei ej )
mh e e e e |ghi j |2 + |ghj i |2 , 8πΓh = 10-4 L=500 GeV 0.001 L=2000 GeV L=3000 GeV 10-7 L=1000 GeV 10-6 Br Hh ® ΜΤL Br Hh ® ΜΤL 10-5 L=500GeV 10-5 L=1000 GeV L=2000 GeV 10-7 L=3000 GeV 10-8 10-9 L=4000 GeV 0.1 0.2 (2.5) L=4000 GeV 10-10 0.5 1.0 2.0 Λ3 5.0 10.0 0.1 20.0 0.2 0.5 1.0 2.0 Λ3 5.0 10.0 20.0 Λ2 Λ2 Figure 2.1: The branching ratio Br(h → µτ ) is as the function of the factor λλ32 with the different energy√ scale Λ The left and right panel are plotted by fix    m m ing (URe )† h′e ULe µτ = uµ τ , and (URe )† h′e ULe µτ = × 10−4 , respectively In the small region of Λ and the factor λλ23 > 1, Br(h → µτ ) ≃ 10−3 However, in this regime, S331 model may encounter the strong precision constraints of Higgs If Λ ∼ TeV but still is below the Landau pole, λ1 ∼ λ2 , the mixing angle ξ will be small and Br(h → µτ ) ≃ 10−5 τ → µγ The branching ratio of the cLFV decay τ → µγ has the following form Br(τ → µγ) = 48π α γ γ 2
|D | + |D Br(τ → µ¯ νµ ντ ), L R| G2F (2.6) γ where the factors DL,R are contributions from the one-loop and two-loop diagrams (see in the thesis for details) We now numerically study the contribution of each type of diagram for the branching ratio of τ → µγ � ���� ���� ������� � � ���� ���� ������� � � ���� � ���� × 10-7 × 10-8 ����� × 10-6 × 10-7 ����� × 10-8 × 10-9 × 10-7 × 10-8 Br(τ→μγ) Br(τ→μγ) � ���� × 10-6 × 10-7 � ���� × 10-8 × 10-9 1000 2000 3000 4000 5000 1000 2000 Λ (GeV) 3000 4000 5000 Λ (GeV) Figure 2.2: The dependence of Br(τ → µγ) on Λ with different contributions The green line is the current experimental constraint Br( à)Exp < 4.4 ì     m m µ τ 10−8 We fix (URe )† h′e ULe µτ = u and (URe ) he ULe = ì 104 , correspondingly for the left and right panel The factor both panels λ3 λ2 = is applied for The results shown in the Fig 2.2 indicate that the two-loop diagrams can   be the primary contribution for τ → µγ Depending on choice, (URe )† h′e ULe µτ = √   m m uµ τ , or (URe ) he ULe = 5ì104 , the two-loop contributions to Br(τ → µγ) can be larger or smaller than the one-loop contributions However, the   scenario (URe ) he ULe = ì 104 gives > 2.4 TeV, in agreement with Landau pole limit We compare Fig 2.2 and Fig 2.3, we find that the above statement changes slightly when the factor λλ32 raises 10 � ���� � ���� × 10-6 × 10-7 � ���� ���� ������� � � ���� � ���� ���� ������� � � ���� ����� ����� × 10-6 × 10-7 Br(τ→μγ) Br(τ→μγ) × 10-7 × 10-8 × 10-7 × 10-8 × 10-8 × 10-9 × 10-8 × 10-9 1000 2000 3000 4000 5000 1000 2000 3000 Λ (GeV) 4000 5000 Λ (GeV) Figure 2.3: Br(τ → µγ) when fixing λ3 λ2 =5 (g − 2)µ The S331 model contains FCNC, hence, it also contributes to the anomalous muon magnetic moment ! X  τ µ 2 mµ mτ m ϕ (∆aµ )M 331 ≃ gϕ ln − (2.7) 2 8π mϕ mτ ϕ Λ3 =5Λ2 ´ 10-14 DaM331 Μ Λ3 = 15 Λ2 Λ3 = 10 Λ2 ´ 10-14 ´ 10-14 ´ 10-15 ´ 10-15 Λ3 = Λ2 0.1 0.2 0.5 Λ2 1.0 2.0 331 Figure 2.4: The contribution of LFV interactions of Higgs to ∆aM as the µ λ3 function of the Higgs coupling λ2 with different factors λ2 The Fig 2.4 with the input parameters can explain the experimental con331 straint of Br(h → τ µ) but cannot deal ∆aM Nevertheless, (g − 2)µ in the µ S331 model also receives contribution from the doubly charged gauge boson ∆aµ (Y ±± ) ≃ 28 m2µ u2 + w2 (2.8) The energy scale that breaks the SU (3)L symmetry is 1.7 TeV < w < 2.2 TeV, and in this range, the anomalous muon magnetic moment can be explained (∆aµ )EXP-SM = (26.1 ± 8) × 10−10 11 (2.9) The LHC constraint for mass of Z ′ in the S331 model reads w > 2.38 TeV, and is very close to the parameter space of w that brings appropriate explanation for (∆aµ )EXP −SM In other words, in the parameter space explaining LHC result, the value of the anomalous muon magnetic moment is predicted, (aà )331 < 13.8 ì 1010 This limit is very close to the constraint given in (2.9) 2.2.2 QFV interactions of Higgs Meson mixing systems FCNC is caused not only by the exchange of the new neutral gauge boson (Z ′ ), but also by the exchange of SM’s Higgs boson and the new Higgs boson u ′ d ⊃ u¯′ R Ghu u′L h + d¯′ R Ghd d′L h + u¯′ R GH uL H + d¯′ L GH d′R H + h.c., (2.10) o  u
n d u d hu u hd u u u † d † with Gh = − (VR ) cξ u M + sξ Λ VL , Gh = − VR cξ u M − sξ Λ VLd , o  u
n d u hu u hd u u † d † u d sξ u M + cξ Λ VLd GH = − (VR ) sξ u M − cξ Λ VL , and GH = − VR The study of FCNC is associated with Zµ′ leading to mZ ′ > 4.67 TeV This limit is close to the Landau pole, the point at which the theory loses renormalizability With the choice (VdL )3a = 0, we have only FCNC associated Higgs,s which will be constrained by measurements of meson mixing systems K and Bs,d , LY f Lef F CN C = h i2 q   (Gh )ij   m2h h + h i2 q ∗   (Gh )ji q (GH )ij i2    m2H h   i2    (2.11)     i h i h i i h h q q q ∗ q ∗  (Gh )ij (GH )ij   (Gh )ji (GH )ji  +2 + +  mh mH   mh mH  + m2h + q ∗ (GH )ji (¯ qiR qjL ) m2H (¯ qiL qjR ) × (¯ qiR qjL ) (¯ qiL qjR ) h i2 (Ghq )ij m2H m2 We consider the ratio κ ≡  q i2 ≃ mH2 tan2 ξ < for w >> u This h [GH )ij mh suggests that the new scalar Higgs H gives more contributions FCNC than 12 SMLHB h The strongest constraint for New Physics coming from the system ¯s , it leads the limit of (G q )32 as : Bs –B h     λ3 u  d † d d  1 q | (Gh )32 |2 = + | (VR ) h VL 23 | < 1.8 × 10−6(2.12) 1+ κ κ λ2 w When λ3 /λ2 > and VRd , hd are matched, the new physical scale can be chosen to be positioned away from the Landau pole h → qi qj The S331 model predicts the branching ratio Br(h → qi qj ) as shown in the Table 2.1 The weakest constraint comes from b–s, Br(h − b¯ s) < 3.5 × 10−3 , which is too small to search at LHC because of the large QCD background, but these signals are expected to be observed in the near future at ILC Observables Oscillation D0 Constraints Br(h → u¯ c) ≤ Br(h → d¯b) ≤ Oscillation Bd0 Oscillation K Br(h → d¯ s) ≤ Br(h → s¯b) ≤ Oscillation Bs0 2×10−4 1+ κ 8×10−5 1+ κ 2×10−6 1+ κ 7×10−3 1+ κ Table 2.1: The upper bound for the flavor violation decays of SMLHB to light quarks with a confidence level of 95% from the measurements of meson mixing systems t → qh (q = u, c) The quark flavor violating interactions of SMLHB in the Eq (2.10) also lead to the non standard decays of top quark t → hui , ui = u, c, Γ(t → ui h) , Br(t → ui h) ≃ Γ(t → bW + )
u u m2 − h2 |Gi3 | + |G3i | t h Γ(t → hui ) = (2.13) 16π mt The LHC searches for Br(t → hc) < 0.16% and Br(t → hu) < 0.19% h with a i † confidence level of 95% In Fig 2.5, we draw Br(t → hc) when fixing (VRu ) hu VLu = 32 h i √ m m † (VRu ) hu VLu = uc t Br(t → ch) can be reached at 10−3 if the new 23 physics scale is about a few hundred GeV, and the factor 13 λ3 λ2 > In this parameter space, the mixing angle ξ is large Br(t → ch) decreases quickly when the factor w u increases With small mixing angle ξ, Br(t → ch) changes from 10−5 to 10−8 14 Br(t®hc)=10-7 12 u L 10 Br(t®hc)=10-6 Br(t®hc)=10-5 Br(t®hc)=10-4 2 Br(t®hc)=10-3 Λ3 10 Λ2 Figure 2.5: The branching ratio of top quark decays to hc 2.3 Conclusions We study constraints from the phenomenologies related to the flavor violating Yukawa interactions in the S331 model Both Higgs triplets couple with leptons and quarks, causing flavor violating signals in both lepton and quark sectors We have pointed out that this model provides large enough branching ratio for the lepton flavor violation decay of SMLHB h → µτ , and also agrees with other experimental constraints, including τ → µγ and (g − 2)µ The FCNC interactions, Higgs–quark–quark interactions, and meson mixing systems are discussed Br(h → qq ′ ) can be enhanced via the measurement of the meson mixing systems The branching ratio of t → qh can reach 10−3 , but also as low as 10−8 14 CHAPTER PHYSICAL CONSTRAINTS DERIVED FROM FCNC INTERACTIONS IN THE 3-3-1-1 MODEL 3.1 The summary of the 3-3-1-1 model The gauge symmetry group is SU (3)C × SU (3)L × U (1)X × U (1)N , the electrical and B − L operators are Q = T3 + βT8 + X, B − L = β ′ T8 + N, (3.1) Leptons and quarks are arranged as follows: (νaL , eaL , (NaR )c )T ∼ (1, 3, −1/3, −2/3), ψaL = νaR ∼ (1, 1, 0, −1), eaR ∼ (1, 1, −1, −1), QαL = (dαL , −uαL , DαL )T ∼ (3, 3∗ , 0, 0), Q3L = (u3L , d3L , UL )T ∼ (3, 3, 1/3, 2/3), uaR ∼ (3, 1, 2/3, 1/3), daR ∼ (3, 1, −1/3, 1/3), UR ∼ (3, 1, 2/3, 4/3), DaR ∼ (3, 1, −1/3, −2/3), (3.2) with a = 1, 2, 3, α = 1, are the generation indexes The scalar spectrum is ηT = (η10 , η2− , η30 )T ∼ (1, 3, −1/3, 1/3), ρT = + T (ρ+ , ρ2 , ρ3 ) ∼ (1, 3, 2/3, 1/3), χT = T (χ01 , χ− , χ3 ) ∼ (1, 3, −1/3, −2/3), ϕ ∼ (1, 1, 0, 2) (3.3) Their corresponding VEVs are u < η10 >= √ , v < ρ02 >= √ , w < χ03 >= √ , Λ < ϕ >= √ (3.4) The VEVs, u, v, break the electroweak symmetry and generate masses for the SM’s particles with the following condition: u2 + v = 2462 GeV2 The remaining VEVs, w, Λ, break SU (3)L , U (1)N and generate the masses for the new particles To be consistent with SM, we propose w, Λ ≫ u, v 15 3.2 Research results of physical constraints derived from FCNC interactions in the 3-3-1-1 model 3.2.1 Rare processes mediated by new gauge bosons and new scalars at the tree level The meson mixing systems Due to the different arrangement between the generations of quarks, SM quarks couple both Higgs triplets, leading to the FCNC associated neutral Higgs at the tree level, along with new gauge bosons Z2,N We are now looking at how the FCNCs caused by new gauge bosons and new scalars affect the meson oscillation systems in the 3-3-1-1 model The difference masses of mesons can be split as the sum of SM and new physics contributions (see in the thesis for details) ∆mK,Bd ,Bs = (∆mK,Bd ,Bs )SM + (∆mK,Bd ,Bs )NP , (3.5) We have the following constraints between new physics and experiments −0.3 < (∆mBd )NP (∆mBs )NP (∆mK )NP < 0.3, −0.4 < < 0.17, −0.29 < < 0.2 (3.6) (∆mK )exp (∆mBd )SM (∆mBs )SM Figure 3.1: The constraints for both w and u obtained from the differences of meson masses ∆mK ,∆mBs and ∆mBd The allowable region for ∆mK is whole panel, whereas the orange and green regimes are for ∆mBs and ∆mBd The Fig 3.1 shows the mixing parameters that are less affected by FCNC induced by new scalars Next, we compare the contributions by FCNC associated with new gauge bosons and new scalars to meson mixing parameters, shown in Fig 3.2 As a 16 result, the primary contribution comes from FCNC associated with new gauge bosons In Fig 3.1, we obtain the lower bound for the new physics scale satisfying constraints (3.6), w > 12 TeV This limit is much tighter and remarkably larger than the previously obtained limit, since in the previous studies, the authors ignored the SM contributions and just compared the New Physics prediction with measurements ����� -f=10000 GeV - ������� -f=1000 GeV - ������� ������ -f=10000 GeV -f=10000 GeV -f=5000 Gev - ����� Δ��� ���� /Δ��� ���� Δ�� ���� /Δ�� ���� -f=5000 Gev -f=5000 Gev - ������ Δ��� ���� /Δ��� ���� ������� - ����� -f=1000 GeV - ����� -f=1000 GeV - ������ - ������ - ����� - ������� - ������ ���� �� ��� �-��� �� ��� �� ��� - ����� ���� �� ��� �-��� �� ��� ���� �� ��� �� ��� �-��� �� ��� �� ��� Figure 3.2: The figure illustrates the dependence of the ratio H1 ,A Z2 ,ZN ∆mK,Bs ,Bd /∆mK,Bs ,Bd on the new physics scale w The decays Bs → µ+ µ− , B → K ∗ µ+ µ− and B + → K + µ+ µ− The effective Hamiltonian for the processed Bs → µ+ µ− , B → K ∗ µ+ µ− and B + → K + µ+ µ− are Heff = X 4GF (Ci (µ)Oi (µ) + Ci′ (µ)Oi′ (µ)) , − √ Vtb Vts∗ i=9,10,S,P (3.7) Their corresponding Wilson coefficients contain both SM and new physics contributions at the tree level ! ZN Z2 2 g g (f ) (f ) m (4π) g g V N V W C9NP = −Θ23 + , ∗ 2 cW Vtb Vts e g mZ2 g m2ZN ! ZN Z2 2 mW (4π) g2 gA (f ) gN gA (f ) NP C10 = Θ23 + (3.8) ∗ cW Vtb Vts e g m2Z2 g m2ZN ′ SM ′SM It is worth noting that CS,P = CS,P = Hence, CS,P , CS,P obtained by the New Physics contributions as follows:
d ∗ l d l Γ Γαα 8π Γ Γ 8π 32 23 αα ′NP CSNP = , C = , S e2 Vtb Vts∗ m2H1 e2 Vtb Vts∗ m2H1
l d ∗ d l Γ ∆αα 8π Γ ∆ 8π 32 23 αα ′NP CPNP = − , C = ,(3.9) P e Vtb Vts∗ m2A e2 Vtb Vts∗ m2A 17 where Γlαα = ∆lαa = reads u v mlα + − The branching ratio for the decay Bs → lα lα s Br(Bs → + − lα lα )theory 4m2lα τBs 2 ∗ = α GF fBs |Vtb Vts | mBs − 64π mBs ( ! 4m2lα m2Bs ′ × 1− (C − C ) (3.10) S S mBs mb + ms ) m Bs ′ + 2mlα (C10 − C10 )+ (CP − CP′ ) , mb + ms ¯s is considered, the predicted value If the oscillation effect of the system Bs − B of theory and experiment will be related by + − Br(Bs → lα lα )exp ≃ + − Br(Bs → lα lα )theory , − ys (3.11) For the decay of Bs → e+ e− , SM prediction and current experimental limit are Br(Bs → e+ e− )SM = (8.54 ± 0.55) × 10−14 , Br(Bs → e+ e− )exp < 2.8 × 10−7(3.12) The SM prediction for the branching ratio of Bs → e+ e− is very suppressed by the upper experimental constraint This is a potentially useful channel for searching for new physics signals In contrast with Bs → e+ e− , the very recent measurement result for the branching ratio of Bs → µ+ µ− is Br(Bs → µ+ µ− )exp = (3.09+0.46 −0.43 +0.15 −0.11 ) × 10−9 (3.13) Whereas, the SM predicts Br Bs → µ+ µ−
SM = (3.66 ± 0.14) × 10−9 (3.14) The New Physics effects in the decay of Bs → µ+ µ− will lead to significant constraints on the New Physics scale We now numerically investigate the decay Bs → µ+ µ− 18 3.6 ´ 10-9 -C9NP 0.01 3.2 ´ 10-9 NP C9,10,S,P Br3311 HBs ® ΜΜL 3.4 ´ 10-9 ´ 10-9 2.8 ´ 10-9 0.001 10-4 10-5 2.6 ´ 10-9 5000 NP C10 0.1 10 000 15 000 20 000 w - GeV 25 000 30 000 CS =CP 10-6 12 000 14 000 16 000 w - GeV 18 000 Figure 3.3: The left panel plots Br(Bs → µ+ µ− ): the red curve line shows the prediction value of the 3-3-1-1 model, and the gray line shows the central value of the SM prediction The blue and green lines represent, respectively, the upper and lower bounds of measurement The right panel predicts the new physics contributions to the Wilson coefficients Both panels are plotted by fixing Λ = 1000w, f = −w, u = 200 GeV Other input parameters are chosen for the section 3.2.1 From Fig 3.3, we have w > TeV, and this limit is lower than the one obtained in the above section As a result, the most appropriate limit ¯s − Bs ) and Br(Bs → µ+ µ− ) is fulfilling both the meson mixing system (B NP w > 12 TeV In the right panel of Fig 3.3,C9,10 >> CS,P For w > 12 TeV, NP C10 > 0, Br(Bs → µ+ µ− ) reduces about 5%, bringing the prediction value and experimental value close to each other The anomalies in B → K ∗ µ+ µ− can be explained if C9NP ≃ −1.1 With the following constraint w > 12 TeV, C9NP ≃ −0.01 Hence, the model 3-3-1-1 cannot deal with the anomalies in decay of B → K ∗ µ+ µ− Both two Wilson coefficients C9 , C10 account for Br (B + → K + µ+ µ− ) As NP predicted by the 3-3-1-1 model, w > 12 TeV, C9NP and C10 are too tiny, hence the anomalies in the decay B + → K + µ+ µ− also cannot be interpreted in the 3-3-1-1 model 3.2.2 The radiative decays b → sγ Apart from the charged currents induced by the SM gauge bosons, Wµ± , the 3-3-1-1 model also provides new charged currents, which couple with the new gauge boson Yµ± , two new charged Higgs H4± , H5± , v`a and FCNC associated by Z2,N All of these currents contribute to the decay b → sγ 19 20 000 The effective Hamiltonian for the decay b → sγ reads 4GF b→sγ Heff = − √ Vtb Vts∗ [C7 (µb )O7 + C8 (µb )O8 + C7′ (µb )O7′ + C8′ (µb )O8′ ],(3.15) The Wilson coefficients C7,8 (µb ) can be split as the sum of M and the 3-3-1-1 SM NP model contributions : C7,8 (µb ) = C7,8 (µb ) + C7,8 (µb ) The New Physics NP contributes to C7,8 via one-loop diagrams NP(0) C7,8 = H (0) C7,85 Z Y (0) (mH5 ) + C7,8 (mY ) + C7,82,N (0) (mZ2,N ), (3.16) The index indicates that these Wilson coefficients have not undergone QCD corrections It is important to stress that QCD corrections for b → sγ are essential and have to be included At the scale µ ∼ mb , w = 10 TeV, we Z have C7 2,N (µb ) ≃ O(10−5 ), much smaller compared with SM one, C7SM (µb ) = Z −0.3523 Consequently, in our calculation, C7 2,N can be ignored The QCD corrections for C7Y and C7H5 at the leading order (LO) are C7Y (µb ) = κ7 C7Y (mY ) + κ8 C8Y (mY ), C7H5 (µb ) = κ7 C7H5 (mY ) + κ8 C8H5 (mY ) (3.17) The branching ratio Br(b → sγ) is given by 6α |Vts∗ Vtb |2 (|C7 (µb )|2 + N (Eγ ))Br(b → ce¯ νe ), (3.18) πC |Vcb | Br(b → sγ) = 4.0 × 10-4 4.0 × 10-4 4.0 × 10-4 mH5 > mU > mY mH5 > mY > mU mU = 0.2 w tβ = 20 tβ = tβ = 10 3.4 × 10-4 tβ = 20 3.6 × 10-4 tβ = 10 tβ = 3.4 × 10-4 3.2 × 10-4 10 000 w - GeV 15 000 20 000 3.6 × 10-4 tβ = 20 tβ = tβ = 10 3.4 × 10-4 3.2 × 10-4 5000 mU = 0.9 w 3.8 × 10-4 Br (b → sγ) 3.6 × 10-4 mU > mH5 > mY mU = 0.5 w 3.8 × 10-4 Br (b → sγ) Br (b → sγ) 3.8 × 10-4 3.2 × 10-4 5000 10 000 w - GeV 15 000 20 000 5000 10 000 15 000 20 000 w - GeV Figure 3.4: The dependence of Br(b → sγ) on the new physics scale w, in +v the limit u, v ≪ −f u uv ∼ w ∼ Λ The dashed black line shows the current experimental constraints Br(b → sγ) = (3.32 ± 0.15) × 10−4 In Fig 3.4, we realize that the branching ratio is strongly affected by the value of tβ , whereas the term containing tβ comes from the C7H5 The lower bound of the new physics scale depends on tβ , namely, w ≥ TeV for tβ = 1; w ≥ 4.1 TeV for tβ = 10; w ≥ 7.7 TeV for tβ = 20 These bounds are lower than the bounds obtained in the previous sections 20 Similarly, we plot Br(b → sγ) in the scenario u, v ≪ −f ∼ w ∼ Λ in Fig 3.5 We see that the dependence of the branching ratio on tβ is not as strong as in Fig 3.4 In the condition w > 12 TeV, the influence of tβ to Br(b → sγ) becomes insignificant, and the predicted value of the model is close to the central value of measurement mH5 > mY > mU 3.8 × 10-4 tβ = 20 tβ = 10 3.6 × 10-4 3.4 × 10-4 tβ = 10 3.4 × 10-4 tβ = 1000 1500 2000 2500 3000 3500 4000 tβ = 3.2 × 10-4 1000 w - GeV tβ =20 tβ = 10 3.6 × 10-4 3.4 × 10-4 tβ = 3.2 × 10-4 3.2 × 10-4 mU = 0.9 w 3.8 × 10-4 tβ = 20 3.6 × 10-4 mU > mH5 > mY 4.0 × 10-4 mU = 0.5 w Br (b → sγ) Br (b → sγ) 3.8 × 10-4 mH5 > mU > mY 4.0 × 10-4 mU = 0.2 w Br (b → sγ) 4.0 × 10-4 1500 2000 2500 3000 3500 4000 1000 1500 2000 w - GeV 2500 3000 3500 4000 w - GeV Figure 3.5: The dependence of Br(b → sγ) on the new physics scale w in the case u, v ≪ −f ∼ w ∼ Λ cLFV decay The 3-3-1-1 model contains some new charged currents coupled with new ± charged particles Y ± , H4,5 , which give contributions to cLFV The effective Lagrangian expression for the process µ → eγ is µ→eγ Leff = eGF −4 √ mµ (AR e¯σµν PR µ + AL e¯σµν PL µ) F µν + H.c., (3.19) where the factors AL , AR are obtained via one-loop diagrams (see the thesis for details) The branching ratio Br(µ → eγ) is Br(µ → eγ) = 12π (|AL |2 + |AR |2 )Br(µ → eν˜e νµ ), GF 21 (3.20) 10-7 10-8 10-10 10-9 Br(μ→eγ)total Br(μ→eγ) �=� ��� 10-30 �=�� ��� 10-10 �=�� ��� 10-11 �� ����� �� ��� 10-12 ����� � �����+����� 10-50 10-13 2000 4000 6000 8000 10 000 w (GeV) 2000 4000 6000 8000 10 000 w (GeV) Figure 3.6: The left panel: the dependence of Br(µ → eγ) on the new physics scale w in each type of contribution The right panel: the dependence of Br(µ → eγ)total on the new physics scale w when fixing u = GeV, u = 10 GeV and u = 20 GeV, respectively The black line indicated the experimental upper bound The left panel of Fig 3.6 estimates the magnitude of each contributions to Br(µ → eγ) The primary contribution comes from newly charged gauge boson Y ± In order to satisfy the experimental constraint, the new physic scale needs w > 7.3 TeV, and this bound is quite similar to the one obtained when studying the decay b → sγ The right panel of Fig 3.6 shows Br(µ → eγ)total ,H5 depends very slightly on u It is worth noting that the factors AH are L,R affected by the electroweak scales u and v Therefore, this result suggests that the charged current and associated new charged Higgs have insignificant impacts on the decay µ → eγ and can be ignored 3.3 Conclusions In the 3-3-1-1 model, we study some phenomenologies related to FCNC The sources of the tree level FCNC come from both gauge bosons and Higgs, as clarified The experiments for meson oscillations will constrain the tree level FCNC tightly The obtained lower bound of the new physics scale is stronger than before, Mnew > 12 TeV In this limit, the tree level FCNC gives negligible contributions to Br(Bs → µ+ µ− ), Br(B → K ∗ µ+ µ− ) and Br(B + → K + µ+ µ− ) The branching ratio of the radiative decay b → sγ is affected by the factor uv via the diagrams mediated by the newly charged Higgs In contrast, the new charged current induced by the new gauge boson Yµ± gives the main contribution to the process sµ → eγ 22 GENERAL CONCLUSION THE EFFECT OF SCALAR FIELDS ON THE FLAVOR-CHANGING NEUTRAL CURRENTS IN THE S331 MODEL AND 3-3-1-1 MODEL A The S331 model: We study the constraints from some phenomenology associated with Yukawa interactions flavor-violating in the S331 model ˆ Both Higgs triplets interact with leptons and quarks, causing flavorviolating signals in the leptons and quarks sector We have shown that this model can give large contribution to LFVHD h → µτ and can reach agreement with other experimental constraints, such as cLFV decay τ → µγ and anomalous muon magnetic moment (g − 2)µ ˆ Contributions of flavor-changing neutral currents, Higgs–quark–quark interactions, meson mixing systems are studied Br(h → qq ′ ) can be increased according to the measurement of the meson mixing system, the branching ratio of t → qh can reach 10−3 , but can also be as small as 10−8 B The 3-3-1-1 model: In the 3-3-1-1 model, we discuss some of the phenomenology associated with FCNC ˆ The source of FCNC at the tree level coming from both the gauge and Higgs bosons is clarified Experiments for meson oscillations will constrain the tree level FCNC The lower limit of the newly obtained physic scale is tighter than before, Mnew > 12 TeV ˆ In this limit, the tree level FCNC gives a negligible contribution to Br(Bs → µ+ µ− ), Br(B → K ) ∗µ+ µ− ) and Br(B + → K + µ+ µ− ) ˆ The branching ratio of the radiative decay b → sγ is influenced by the coefficient factor uv via the diagrams mediated by the newly charged Higgs boson In contrast, the charged current of the new gauge boson Yµ± is the major contributor to the decay process µ → eγ 23 NEW FINDINGS OF THE THESIS We study the constraints from some phenomenology associated with Yukawa interactions violating flavor in the S331 model We have shown that this model can give large contribution to LFVHD h → µτ and can reach agreement with other experimental constraints, such as τ → µγ and (g − 2)µ The contributions of the flavor-changing neutral currents interaction, the Higgs–quark–quark interaction, and the meson mixing system are studied Br(h → qq ′ ) can be increased according to the measurement of the meson mixing system The branching ratio of t → qh can reach 10−3 , but can also be as small as 10−8 In the 3-3-1-1 model, we discuss some of the phenomenology associated with FCNC The source of FCNC at the tree level coming from both the gauge and Higgs bosons is clarified Experiments for meson oscillations will constrain the tree level FCNC The lower limit of the newly obtained physic scale is tighter than before, Mnew > 12 TeV In this limit, the tree level FCNC gives a negligible contribution to Br(Bs → µ+ µ− ), Br(B → K ) ∗µ+ µ− ) and Br(B + → K + µ+ µ− ) The branching ratio of the radiative decay b → sγ is influenced by the coefficient factor uv via the diagrams mediated by the newly charged Higgs boson In contrast, the charged current of the new gauge boson Yµ± is the major contributor to the decay process µ → eγ 24 LIST OF WORKS HAS BEEN PUBLISHED D T Huong, P V Dong, N T Duy, N T Nhuan and L D Thien, Investigation of dark matter in the 3-2-3-1 model, Physical Review D 98, 055033, 2018 D N Dinh, D T Huong, N T Duy, N T Nhuan, L D Thien, and Phung Van Dong, Flavor changing in the flipped trinification, Physical Review D 99, 055005, 2019 D T Huong, N T Duy, Investigation of the Higgs boson anomalous FCNC interactions in the simple 3-3-1 model, European Physical Journal C 80, 439, 2020 Duy Nguyen Tuan, Takeo Inami, Huong Do Thi, Physical constraints derived from FCNC in the 3-3-1-1 model, European Physical Journal C 81, 813, 2021 The main results used in the thesis are the third and fourth publications 25