Quá trình phân rã của higgs boson h→zy và h→ trong một số mô hình 3 3 1 tt tiếng anh

INTRODUCTION The loop-induced decay of the standard model (SM)-like Higgs boson h → Zγ is attracting great interest from both theoretical and experimental aspects Namely, the SM predicts that one-loop level is the leading contribution to the decay amplitude The respective branching ratio (Br) is Br(h → Zγ) 1.54 × 10−3 (±5.7%), which has been searching by experiments Although this decay channel has not been observed, the recent upper constraints of the signal strength are established from the experiments at Large Hadron Collider (LHC) In the future projects, the sensitivities of these experiments will be improved so that the decay channel h → Zγ may be detected More interesting, if there is any of deviations between experimental results and those predicted by the SM, it will be new physics, in the meaning that there exist additional contributions from new particles in models Beyond the Standard Model (BSM) At the one loop level, the amplitude of the decay h → Zγ predicted by some BSM normally contains contributions of new particles that not appear in the SM framework Calculating these contributions is rather difficult in the usual ’t Hooft-Feynman gauge, because of the appearance of many unphysical states, namely Goldstone bosons and ghosts which always exist along with the gauge bosons They brings a very large number of Feynman diagrams In addition, their couplings are indeed model dependent, so it is hard to construct general formulas determining vector loop contributions using the t’ Hooft-Feynman gauge The technical difficulties caused by unphysical states will vanish if calculations are carried out in the unitary gauge In such a case, the number of Feynman diagrams as well as the number of necessary couplings become minimum, namely, only physical states are needed Then the Lorentz structures of these couplings are well defined, and hence the general analytic formulas of one-loop contributions from gauge boson loops can be constructed But in the unitary gauge we face the complicated forms of the gauge boson propagators, which generate many divergent terms Fortunately, many of them are excluded by the condition of on-shell photon in the decay h → Zγ The remaining ones will vanish systematically when loop integrals are written in terms of the Passarino-Veltman (PV) functions Our results can also be translated into the general analytic form used to calculate the amplitudes of the charged Higgs decay H ± → W ± γ which is also an interesting channel predicted in many BSM models Besides, signals of lepton-flavor-violating decays of the standard-model-like Higgs boson (LFVHDs) were investigated at the LHC not very long after its discovery in 2012 So far, the most stringent limits on the Br of these decays are Br(h → µτ, eτ ) < O(10−3 ), from the CMS Collaboration using data collected at a center-of-mass energy of 13 TeV The sensitivities of the planned colliders for LFVHD searches are predicted to reach the order of 10−5 The 3-3-1 models contain rich LFV sources which may result in interesting cLFV phenomenology such as charged lepton decays ei → ej γ In particular, it was shown that Br(µ → eγ) is large in these models and hence it must be taken into account to constrain the parameter space In addition, such rich LFV resources may give large LFVHD rates as promising signals of new physics The 3-3-1 model with inverse seesaw neutrino masses (331ISS) can predict a neutrino mass spectrum consistent with current experimental neutrino data, using the wellknown inverse seesaw (ISS) mechanism The model allows large Br(h → µτ, eτ ) ∼ O(10−5 ) in the allowed regions satisfying Br(µ → eγ) < 4.2 × 10−13 The most interesting allowed regions will also allow large LFVHD rates, which we will try to look for in this thesis Because of the above reasons, the studies on channels decay h → Zγ and h → µτ in the BSM have been being hot subjects to looking for new physics recently Research objective • Investigate general formulas of one-loop contributions to the amplitude of the decay h → Zγ • The structure and LFV source of the 331ISS • The Br(h01 → µτ ) predicted by the 331ISS Research objects and scope of the study • The decay h → Zγ in general and h01 in the 331ISS LFV couplings, Feynman diagrams and amplitude • Passarino Veltman functions (PV) for the decay h → Zγ and h01 → µτ Research content • The particle spectra relate with the general decay amplitude h → Zγ and the SM-like Higgs decay h01 → µ± τ ∓ predicted by the 331ISS • One-loop contribution on Br(h → Zγ), Br(h01 → µ± τ ∓ ) in the 331ISS • Compare the obtained results with the previous ones and illustrate interesting contributions in BSM Survey numerically of the Br(h01 ∓ ) in the 331ISS for predicting the ability to detect this decay at the LHC in the future • Determining allowed regions of the parameter space that satisfy all the theoretical and experimental constraints and give large Br(h01 → µ± τ ∓ ) in the 331ISS Research methods • Quantum field theory • Mathematica software for numerical calculation Structure of thesis: Chapter 1: We give summary discussion on relevant interactions of the SM Higgs boson that contribute to the decay h → Zγ, LFV sources in some well-known BSM and some elementary knowledge of the decays of the SM-like Higgs boson realting with recent experiments Chapter 2: Constructing general analytical formulas of one-loop contributions to calculate Br(h → Zγ) in the unitary gauge Chapter 3: Comparing and contrasting between the formulas obtained in chapter with some published results, and discussing some interesting contributions in the BSM that were ignored previously Chapter 4: Investigating numerically the Br(h01 → µ± τ ∓ ) in the 331ISS: find all couplings and one-loop Feynman diagrams in the unitary gauge, calculate in detail particular one loop contribution and prove the divergent cancelation in the total decay amplitude, draw plots to discuss numerically the results General conclusions: Review the main obtained results and propose future research directions Appendix: We present some detailed intermediate steps related to the calculations in the main part of this thesis Chapter OVERVIEW The SM describes successfully strong, weak and electromagnetic interactions based on the gauge symmetry group SU (3)C ⊗ SU (2)L ⊗ U (1)Y However, besides the success, there are still some issues that we need to expand the SM: the SM cannot unify all types of interaction (the gravitational force is not include in the SM), the SM does not answer the questions why are there three fermion generations? why is the top quark mass not well consistent with the experiment results? the SM describes neutrinos as massless while the experiments show that neutrinos are massive, Hence, many models beyond the SM have been introduced in order to explain the above shortages One of the new research directions for new physics in the BSM is studying the rare decay and the LFV decay In the SM, the couplings of the SM-like Higgs boson with other particles and among particles in the model contain the parameters that are measured by experiments Thus, the characteristics of the SM-like Higgs boson also have been determined in the SM and have been independently verified by experimental (LHC) For new particles in the BSM, the couplings will contain new unverified parameters Therefore, our study will contribute to further clarifying the allowed-value regions of these parameters They will be easily verified when they are taken to the limit of the SM The main source leads to LFV is due to the mixture of different generations of neutrinos, the new leptons that are added in the BSM There are such BSMs as super symmetry models, seesaw models, 3-3-1 models, However, in the framework of this thesis, we only focus on 3-3-1 models, particularly the 331ISS model Firstly, the LFV source from the new neutrinos Secondly, the LFV source from new interactions between the SM-like Higgs boson and the new gauge bosons, the SM-like Higgs boson and new charge Higgs boson These new particles create more one-loop contributions diagrams for the LFV decay Besides, the decays of LFVHDs have been investigated by experiments, these are new physical signals that is not existed in the SM All the decay channels had not been searched yet by the low energy accelerators The LHC is the first accelerator with high energy that enable to search for these decays In 2015, the upper bound Br(h → µτ ) was set up by CMS and ATLAS The Higgs boson is searched through the main channels: h → b¯b, c¯ c, τ + τ − , γγ, Γ(h→XY ) ZZ, W W + , gg The branching ratio is defines as Br(h → XY ) = Γtotal , where Γtotal 4.1 × 10−3 is the total decay width of SM-like Higgs boson The branching ratio of hγγ, hZγ are very small, about 2.10−3 with mh around 120 − 130 GeV In these decay channels, the channel with larger branching ratio has higher probability to be occurred in experimental measurements The decay h → Zγ and h → µτ are two channels that have been intensively searched by the experiments Therefore, these are the promoting subjects of new physics which have been hunted by recent experiments In this thesis, we focus mainly on issues related to these two decay channels Chapter ONE-LOOP CONTRIBUTIONS TO THE DECAY h → Zγ 2.1 Feynman diagrams and Feynman rules The amplitude of the decay h → Zγ is defined as follows µ∗ ν∗ ν∗ M(h → Zγ) ≡ M (Zµ (p1 ), γν (p2 ), h(p3 )) εµ∗ (p1 )ε2 (p2 ) ≡ Mµν ε1 ε2 , (2.1) The on-shell conditions are p21 = m2Z , p22 = and p23 = m2h The decay amplitude is generally written in the following form Mµν F00 gà + Fij pià pj + F5 ì i α β µναβ p1 p2 , (2.2) i,j=1 where µναβ is the totally antisymmetric tensor with 0123 = −1 and 0123 = +1, εν∗ p2ν = 0, F12,22 not contribute to the total amplitude (2.1) In addition, the Mµν in eq (2.2) satisfies the Ward identity, pν2 Mµν = 0, resulting in F11 = and (m2Z − m2h ) F21 (2.3) F00 = −(p1 p2 )F21 = Hence the amplitude (2.1) can be calculated through the form (2.2) via the following relations ν∗ M(h → Zγ) = Mµν εµ∗ ε2 , Mµν = F21 [−(p2 p1 )gµν + p2µ p1ν ] + F5 × i α β µναβ p1 p2 (2.4) The partial decay width then can be presented in the form m3 Γ(h → Zγ) = h × 32π m2 − Z2 mh |F21 |2 + |F5 |2 (2.5) The Feynman rules used in our calculations are listed in table 2.1 A new notation is Γµνλ (p0 , p+ , p− ) ≡ (p0 −p+ )λ gµν +(p+ −p− )µ gνλ +(p− −p0 )ν gλµ , where all momenta are incoming and p0,± are respective momenta of h and charged gauge and Higgs bosons ±Q ±Q with electric charges ±Q, denoted as Vi,j and Si,j , respectively The general case of four-gauge-boson coupling is (2, −1, −1) → (a1 , a2 , a3 ) and gZγVij = e Q gZVij Figure 2.1: One-loop diagrams contributing to the decay h → Zγ, where fi,j , Si,j and Vi,j are fermions, Higgs and gauge bosons, respectively Table 2.1: Couplings involving the decay of CP even neutral Higgs h → Zγ, in the unitary gauge Vertex Factor hfi fj −i Yhfij L PL + Yhfij R PR hSiQ Sj−Q , hSi−Q SjQ −iλhSij , −iλ∗hSij h(p0 )Si−Q (p− )VjQµ , h(p0 )SiQ (p+ )Vj−Qµ 2.2 ∗ ighSi Vj (p0 − p− )µ , −ighS (p0 − p+ )µ i Vj hVi−Qµ VjQν , hZ µ Z ν Aµ fi fi , Aµ SiQ Si−Q Aµ (p0 )ViQν (p+ )Vi−Qλ (p− ) Z µ fi fj Q Z µ Si (p+ )Sj−Q (p− ) Z µ ViQν Sj−Q , Z µ Vi−Qν SjQ ighVij gµν , ighZZ gµν ie Qγµ , ie Q(p+ − p− )µ −ieQΓµνλ (p0 , p+ , p− ) i gZfij L γµ PL + gZfij R γµ PR igZSij (p+ − p− )µ ∗ igZVi Sj gµν , igZV gµν i Sj Z µ (p0 )ViQν (p+ )Vj−Qλ (p− ) Z µ Aν ViQα Vj−Qβ −igZVij Γµνλ (p0 , p+ , p− ) −ie Q gZVij (2gµν gαβ − gµα gνβ − gµβ gνα ) Analytic formulas The contribution to F21 from all diagrams in Fig 2.1 are F21,Vijj = 2e Q ghVij gZVij 16π 8+ (m21 + m22 + m2h )(m21 + m22 − m2Z ) m21 m22 2(m21 − m22 )(m21 + m22 − m2Z ) (C1 + C2 ) m21 m22 2(m21 + 3m22 − m2Z )C0 + , (2.6) m22 e Q Nc (1) + + = F21 = − KLL,RR + KLR,RL + c.c (C12 + C22 + C2 ) 16π × (C12 + C22 + C2 ) + F21,fijj + + + +2 KLL,RR − KLR,RL + c.c (C1 + C2 ) + 2(KLL,RR + c.c.)C0 , F5,fijj = − × F21,Sijj = e Q Nc − − KLL,RR − KLR,RL − c.c 16π − (C1 + C2 ) − 2(KLL,RR − c.c.)C0 (2) F21 (3) F21,V SS = F21 = (2.7) e Q λ∗hSij gZSij + c.c [4(C12 + C22 + C2 )] , 16π ∗ g + c.c.) e Q (ghV −m22 + m2h i Sj ZVi Sj + = 16π m21 ×(C12 + C22 + C2 ) + 4(C1 + C2 + C0 )] , (4) F21,SV V = F21 = ∗ e Q (ghVj Si gZV j Si 16π + c.c.) ×(C12 + C22 + C2 ) − 4(C1 + C2 )] 1+ (2.8) (2.9) −m21 + m2h m22 (2.10) We have established the general analytic formulas expressing one-loop contributions to the amplitude of the Higgs boson decay h → Zγ, including those were ignored in the previous studies The analytic results are also expressed in terms of the PassarinoVeltman functions, following notations in the LoopTools library The main results of this chapter are published in Chapter DECAY h → Zγ IN SOME SPECIFIC MODELS 3.1 Decay h → γγ, Zγ in the SM The contribution of W bosons corresponds to (ghVij , gZVij , Q) → (g mW , g cW , 1) with m1 = m2 = mW , where mW is the W boson mass, g is the gauge coupling of the SU (2)L group, sW ≡ sin θW with θW being the Weinberg angle Then formula (2.6) is reduced to the simpler form h→Zγ,SM F21,W 2e g mW cW = 16π (2m2W + m2h )(m2W − m2Z ) 8+ m4W × (C12 + C22 + C2 ) + × + → (4m2W − m2Z )C0 mW αem g + 3t2 + 3(2t2 − t22 )f (t2 ) , 4πmW (3.1) (3.2) where we have used αem = e2 /(4π), e = g sW , m2h /m2W = 4/t2 , m2Z /m2W = 4/t1 , m2Z /m2W = 1/c2W = 1+t2W , sW = sin θW and tW = sW /cW Formula (3.3) is consistent with well-known result for the SM case given in (Phys Rev D 96, Nucl Phys B 299) which even has been confirmed using various approaches The right hand side of (2.6) can be proved to be completely consistent with the W contribution to the amplitude of the decay h → γγ The analytic form of this contribution is known (Westview Press; Sov J Nucl Phys 30), namely (3.3) h→γγ,SM F21,W = e g mW 16π 2 8+ 2+ t2 (2 − 0) (C12 + C22 + C2 ) +4 (4 − 0) C0 } αem g → + 3t2 + 3(2t2 − t22 )f (t2 ) 4πmW (3.3) 10 3.2 Decay H → Zγ, W γ in the GHU and Georgy-Machacek model For comparison with results in some BSM, we selected some specific models, first consider GHU The one-loop contribution from new gauge bosons in the GHU model was given in (Phys Rev D 92), where the unitary gauge was mentioned without detailed explanations Translated into our notation, the most important relevant part in (Phys Rev D 92) is GHU = m41 + m42 + 10m21 m22 E+ (m1 , m2 ) F21,V + (m21 + m22 )(m2h − m2Z ) − m2h m2Z E− (m1 , m2 ) − 4m21 m22 (m2h − m2Z ) + 2m4Z (m21 + m22 ) C0 + C0 , (3.4) where function C0 is determined by changing the roles of m1 and m2 and E± (m1 , m2 ) = + m2Z m2h − m2Z (2) (1) B0 − B0 ± (m22 C0 + m21 C0 ) (3.5) In the special case where Vi ≡ Vj , corresponding to m1 = m2 = m, C0 = C0 = −I2 (t2 , t1 )/m2 , C12 + C22 + C2 = I1 (t2 , t1 )/(4m2 ) and use some special formulas m2Z m2h − m2Z (2) (1) B0 − B0 = −1 − m2h − m2Z I1 (t2 , t1 ) + 2I2 (t2 , t1 ), 2m2W (3.6) In fact we find the agreement between eq (3.18) of (Phys Rev D 92) and our result, namely (3.4) is rewritten as follows GHU F21,V = m4 + m4 + 10m2 m2 E+ (m, m) + (m2 + m2 )(m2h − m2Z ) − m2h m2Z × E− (m, m) − 4m2 m2 (m2h − m2Z ) + 2m4Z (m2 + m2 ) (C0 + C0 ) −m2h − m2Z = 12m4 + 2m2 (m2h − m2Z ) − m2h m2Z I1 (t2 , t1 ) 2m2 + 4m2 (m2h − m2Z ) − m2h m2Z + 2m4Z I2 (t2 , t1 ) (3.7) So compared to the result of eq (3.18) in (Phys Rev D 92), we find the agreement but different by a factor of first m4Z This difference may be due to the confusion of the authors in (Phys Rev D 92) They differ by GHU δF21 = F21,V − 16π (F21,Vijj + F21,Vjii ) × −m21 m22 (m2h − m2z ) 2e Q ghVij gZVij = m1 =m2 Formula (3.4) equivalent to our results, namely F21,Vijj + F21,Vjii But two general results are not the same, i.e they differ by δF21 = −2 m21 C0 + m22 C0 m4Z In (Phys Rev D 92) we are considering them in general form but not use and use special 11 cases, if using their general formula, the results not match the SM results, so their general formula has errors Except F21,Vijj in eq (2.6), our formulas are consistent with the results given in (Phys Rev D 96), which were obtained by calculating the decay amplitude of charged Higgs boson H ± → W ± γ in the ’t Hooft-Feynman gauge for the Georgi-Machacek model The amplitude of the decay H ± → W ± γ, derived from (2.6) with mh → mH ± , mZ → mW , gZVij → gW Vij , ghVij → gHVij now has the following form H ±W ±γ F21,V ijj e Q gHVij gW Vij = 16π (m21 + m22 + m2H ± )(m21 + m22 − m2W ) 8+ m21 m22 2(m21 − m22 )(m21 + m22 − m2W ) (C1 + C2 ) m21 m22 2(m21 + 3m22 − m2W )C0 m22 × (C12 + C22 + C2 ) + + (3.8) We only need to focus on the part generated by the loop structures used to compare with the specific result given in (Phys Rev D 96) This case corresponds to m1 = mZ , m2 = mW = mZ cW and mH ± = m5 for the decay H5± → W ± γ Formula (3.8) now has the following form H5± W ± γ F21,V ijj (m2Z + m2Z c2W + m25 )(m2Z + m2Z c2W − m2Z c2W ) ∼ 8+ m2Z m2Z c2W 2(m2Z − m2Z c2W )(m2Z + m2Z c2W − m2Z c2W ) × (C12 + C22 + C2 ) + m2Z m2Z c2W × (C1 + C2 ) + 2(m2Z + 3m2Z c2W − m2Z c2W ) C0 m2Z c2W = 10(C12 + C22 + C2 ) + 6C0 + + m25 (C12 + C22 + C2 ) m2W s2W (C12 + C22 + 2C1 + 3C2 + 2C0 ) c2W (3.9) Which is different from the result given in (Phys Rev D 96) by the coefficient 10 instead of 12 in front of the sum (C12 + C22 + C2 ) We see that the two parts in our result with coefficients m25 /m2W and s2W /c2W are consistent with SGGG and SXGG in (Phys Rev D 96) respectively The difference in the remaining part might arise due to a missed sign of the ghost contribution Sghost 12 Decay H1 → Zγ in 331β0 model 3.3 In the unitary gauge, the decay H1 → Zγ is determined by Br331 (H1 → Zγ) = Γ331 (H1 → Zγ) , Γ331 H1 Γ331 (H1 → Zγ) = m3H1 32π 1− m2Z m2H1 331 | , |F21 (3.10) 331 (H → Zγ) where Γ331 H1 is the total decay width of the SM-like Higgs boson H1 and Γ 331 SM is the partial decay width predict by the 331β0 model The form factor F21 and F21 are written as SM SM SM 331 331 331 331 331 331 + F21,V + F21,S + F21,V = F21,f F21 Sjj + F21,SVjj , F21 = F21,W + F21,f , (3.11) ijj ijj ijj where particular contributions are derived based on the general formulas 331 F21,f F 331 ±1/2 21,H1,2 =− f+ e Qf Nc KLL,RR = KH1 S [16 (C12 + C22 + C2 ) + 4C0 ] , 16π ±1/2 × [4(C12 + C22 + C2 )] , S = H ± , H1,2 , (2m2G + m2H1 )(2m2G − m2Z ) 2(4m2G − m2Z )C0 331 + + (C12 + C22 + C2 ) , F21,G = KG × m2G m4G √ √ e 2g mZ cX (2sX sα + 2cX cα ) 331 F21,V SS = × 16π 2 −mS + mh × 1+ (C12 + C22 + C2 ) + 4(C1 + C2 + C0 ) , m2V √ √ 2g m c (2s s + 2cX cα ) e α Z X X 331 F21,SV × V = 16π 2 −mS + mh 1+ (C12 + C22 + C2 ) − 4(C1 + C2 ) , (3.12) m2V gm2f cα Tf3 − 2s2W Qf for fermions 2mW m2Ua sα s2W /(3u) for new quark Ua Other f+ where G = W, V, V are gauge bosons, KLL,RR =− in SM, m2Ea sα s2W /u for new lepton Ea and factors are λH1 H 1/2 e g(−c2X s2W + s2X c2W ) 2e g(1 − 2s2W ) λH1 H ± 1,2 ± × , K = × , KH1 H 1/2 = H1 H 1,2 16π 2cW 16π 2cW √ 2eg cW mW cα eg cW (2usα − 2vcα ) KW = − , K V = KV = (3.13) 16π 128π The signal strength of the decay is µ331 Zγ ≡ σ 331 (pp → H1 ) Br331 (H1 → Zγ) × σ SM (pp → H1 ) BrSM (H1 → Zγ) (3.14) In the SM, BrSM (H1 → Zγ) 1.57 × 10−3 and ΓSM 4.07.10−3 GeV with mH1 = H1 125.1 GeV Numerical illustrations for µ331 Zγ in the 331β0 are shown in Fig 3.1 13 λ1 =1, λ13=-1 λ1 =1, λ13=-0.5 1.00 1.01 0.95 μZγ μZγ 1.00 0.99 mF=0.2, mV=0.5 TeV mF=0.2, mV=0.5 TeV 0.90 mF=0.2, mV=1 TeV 0.98 mF=0.2, mV=1 TeV mF=1, mV=0.5 TeV mF=1, mV=1 TeV 0.97 0.5 mH± [TeV] 10 mF=1, mV=0.5 TeV mF=1, mV=1 TeV 0.85 0.5 10 mH± [TeV] Figure 3.1: Signal strength of the decay H1 → Zγ in 331 β0 model as a function of mH ± , horizontal lines corresponding to values given by SM 1, 0.99, 1.01 In the 331β0 model, the signal strength of the decay H1 → Zγ was investigated in the range from 100 GeV to O(10) TeV of the charged Higgs mass mH ± The Br(H1 → Zγ) is the same as the SM prediction at large mH ± On the other hand, small mH ± predicts µZγ < 1, implying that the signal of this decay channel is difficult to observe in future experiments, where the recent upper bound is µZγ < 3.4 Heavy charged boson effects on Higgs decays h → Zγ in the LR and HTM model Because new heavy charged gauge V ± and Higgs bosons S ± appear in non-trivial gauge extensions of the SM, they may contribute to loop-induced SM-like Higgs decays h → γγ and h → Zγ While the couplings hV V and hSS consisting of virtual identical charged particles always contribute to both decay amplitudes, the couplings hW V and hW S of the SM-like Higgs boson only contribute to the later These couplings may cause significant effects to Br(h → Zγ) in the light of the very strict experimental constraints of Br(h → γγ) When m2X m2W with X = S, V , the loop structures of the form factors with at least one virtual W boson have an interesting property that FW X ≡ F21,W F21,W XX + F21,XW W ∼ FW ≡ ∼O eQghXW gZXW /(16π ) eghW W gZW W /(16π ) m2W , the same order with the W loop contribution In contrast, the loop structure of a heavy gauge boson F21,V V V is FV ≡ F21,V V V ∼ O(m−2 V ), ghV V gZV V /(16π ) which is different from the SM contribution of the W boson by a factor m2W /m2V Numerical illustrations are shown in figure 3.2 where fW,X ≡ FW X /FW , fV ≡ FV /FW and mS = mV 14 0.5 fV m2 V /mW fW,S fW,V 0.2 0.1 100 500 1000 104 5000 mV [GeV] Figure 3.2: fV m2V /m2W , fW,S and fW,V as functions of the SU (2)R scale mV Hence, the large coupling product ghW X gZW X may give significant effects on the total amplitude of the decay h → Zγ But the contributions arising from this part were omitted in the literature, even with well known-models such as the left-right models (LR) and the Higgs Triplet models (HTM) Interesting studies on new charged gauge bosons W in the LR indicated that the couplings W W h, W W Z, W H ± Z result in important decays of W ± , which are being hunted at LHC These coupling also contribute to the decay h → Zγ We have used the Table 3.1: Vertex factors involved charged gauge and Higgs bosons contributing to one loop amplitude of the SM-like Higgs decay h → Zγ in the LR model Vertex SM LR 2 ghW W gZW W g m W cW , gL mW cW sin(β − α) sθ ghW W gZW W − gL gR mW cos(β + α) cW+ ghW W gZW W ghW + H − gZW − H + − ghW − +H− gZW −H+ s2 W −gR mW sin(β − α) cW − gR 2 mW cW sin(β + α) cos(2β)sθ+ g − 2R mW cW sin(β + α) cos(2β) − condition α = β − π/2 to guarantee that the coupling hW W is the same as that in the SM We ignore all suppressed terms having factors with orders larger than O( ), where = mW /mW and mW is the new heavy gauge boson mass, which can be considered as the breaking scale of the SU (2)R group, g ≡ gL and sθ+ tan θ+ = ggRL × sin(2β) and = mW /mW The couplings of the SM-like Higgs boson we discuss here are consistent with those in (JHEP 1605; JHEP 1510; J Phys G41) The triple gauge couplings are also consistent with (Phys Rev D86; Annals Phys 280) The decay 15 h → Zγ has contributions associated with charged gauge bosons estimated as follows LR F21,W WW LR F21,W W W SM SM F21,W F21,W LR LR F21,W W W + F21,W W W SM F21,W LR LR F21,HW W + F21,W HH SM F21,W 1, ∼− s2 gR W gL2 c2W , sin2 (2β) gR , ∼ 2gL2 c2W cos2 (2β) gR , ∼ 2gL (3.15) where ≡ mW /mW and α β − π/2 We can see that all quantities listed in (3.15) have the same order, although some of them are affected by the tiny mixing parameter sθ+ = O( ) between two charged gauge bosons Hence all of them must be taken into LR account This argument is different from previous treatment where only F21,W W W was mentioned The recent lower bounds of the SU (2)R scale give ≤ O(10−3 ), implying that the heavy charged Higgs and gauge contributions discussed here are suppressed The effects of heavy charged Higgs boson mH ± from F21,W SS and F21,SW W appear in simple models like the HTM They even appear in the simple HTM models extended from the SM by adding only one Higgs triplet ∆ The correlation of the two decays h → γγ and h → Zγ were investigated previously, but the contributions F21,W SS and F21,SW W mentioned here were ignored in (JHEP.05) because of the small product ghSW gZW S On the other hand, heavy neutral bosons H predicted by many BSM may have large gHW SS gZSS , for example the HTM In this case, contributions of F21,SW W , F21,W SS can reach the significant values of F21,W W W × O(v∆ /v) = F21,W W W × O(10−1 ) in the decay Br(H → Zγ), but they were ignored in previous works Our results would be useful for further studies of loop-induced decays of neutral and charged Higgs bosons H → Zγ, W γ, which have not been yet mentioned in many well-known BSM Namely, our results were applied to discuss on the decay of Higgs boson in the SM and BSM, including the Gauge-Higgs Unification model, the Higgs Triplet model, the left-right symmetric model, the Georgy-Machacek model and the 331β0 model We have calculated some contributions which have not been mentioned in the previous studies We have found that these contributions may be large and should be taken into account for fitting with upcoming experimental data The main results of this chapter are also published in Chapter DECAY OF SM-LIKE HIGGS BOSON h01 → µτ IN 331ISS 4.1 The 331ISS model Now we consider a 331ISS model as an extension of the above 331RHN model, where three right-handed neutrinos which are gauge singlets, NaR ∼ (1, 0), a = 1, 2, are added   νaL La =  eaL  ∼ 1, 3, − , eaR ∼ (1, 1, −1), c (NaR ) (4.1) ρ+ 1    , ρ =√ ρ= ∼ 1, 3, v1  , ρ ρ+ (4.2) v2 η10  −   , η =√ η= ∼ 1, 3, − η , η20 (4.3) χ01 1 − χ =  χ  ∼ 1, 3, − , χ = √   χ2 ω (4.4)             Now tree-level neutrino masses and mixing angles arise from the ISS mechanism Requiring that L is only softly broken, the additional Yukawa part is −LXR = Yab ψaL χXbR + (µX )ab (XaR )c XbR + H.c., (4.5) where àX is a ì symmetric matrix and L(XaR ) = L(XaR ) = −1 The last term in Eq (4.5) is the only one that violates both L and L and hence it can be assumed to be small, which is exactly the case in the ISS models The first term generates mass for 16 17 heavy neutrinos, resulting in a large Yukawa coupling Yab with SU (3)L Higgs triplets In addition, the ISS mechanism allows for large entries in the Dirac mass matrix mD , which is the opposite of the well-known requirement in the 331RHN model The two global symmetries-namely normal and new lepton numbers denoted respectively, as L and L were introduced They are related to each other by: L = √43 T8 + L The detailed values of nonzero lepton numbers L and L are listed in Table 4.1 Table 4.1: Nonzero lepton number L (left) and L (right) of leptons and Higgs bosons in the 331RHN Fields NL νL eL eR ρ+ η20 χ01 χ− Fields χ η ρ ψaL eaR 2 L -1 1 -2 -2 2 L 3 3 In the basis   νL  νL = NL  and νL (XR )c (νL )c = (NL )c  XR   c Neutrino mass term −Lνmass = νL M ν νL c + H.c., (4.6) where   mD ν  M = mTD MR  , MRT µX (4.7) ω MR is a × matrix (MR )ab ≡ Yab √ with a, b = 1, 2, For simplicity in the numerical study, we will consider the diagonal matrix MR in the degenerate case MR = MR1 = MR2 = MR3 ≡ k × z The parameter k will be fixed at small values that result in large LFVHD effects The total neutrino mass matrix in Eq (4.7) depends on only the free parameter z The heavy neutrino masses and the matrix U ν can be solved numerically, which is not affected by z because |µX | z In contrast, neutrino masses and mixing parameters defining the matrix mν ∗ in mν mD M −1 mTD ≡ UPMNS m ˆ ν UPMNS ,, which are used to calculate the matrix mD , are considered as free parameters 4.2 Couplings related to the SM-like Higgs decay h01 → µτ In this section, we present Higgs boson decay h01 → µτ in 331ISS, where h01 is identified with the SM-like Higgs boson (h) All of the couplings involved in LFV processes are listed in Table 4.2 ∓ The effective Lagrangian of the LFVHDs of the SM-like Higgs boson h01 → e± a eb is LLFVH = h01 ∆(ab)L ea PL eb + ∆(ab)R ea PR eb + H.c., 18 Table 4.2: Couplings related to the SM-like Higgs decay h01 → ea eb in the 331ISS model All momenta in the Feynman rules corresponding to these vertices are incoming Vertex Coupling igma h1 ea ea 2mW cα igcα 0∗ h01 ni nj 2mW λij PL + λij PR R,1∗ R,1 L,1∗ igcθ igcθ −m λL,1 bi PL + λbi PR , − mW λai PR + λai PL W H1+ ni eb , H1− ea ni H2+ ni eb , H2− ea ni ig − √2m R,2 √ ig λL,2 bi PL + λbi PR , − 2m W W ig √ U ν γ µ PL , bi ig √ Uν γ µ PL , (b+3)i Wµ+ ni eb , Wµ− ea ni Yµ+ ni eb , Yµ− ea ni H1+ h01 Yµ− Yµ+ H1− h01 h01 Wµ+ Wν− h01 Yµ+ Yν− h01 H1+ H1− iλ± H1 h01 H2+ H2− R,2∗ λL,2∗ PR + λai PL ig √ U ν∗ γ µ PL ig √ U ν∗ γ µ PL (a+3)i µ √ ig √ cα cθ + 2sα sθ pH − − ph01 2 µ √ ig √ cα cθ + 2sα sθ pH − − ph01 2 −igmW cα g µν √ igm √Y 2sα cθ − cα sθ g µν √ = −iω sα c2θ λ12 + 2sα s2θ λ2 − 2cα c2θ λ1 + cα s2θ λ12 tθ √ +i 2f cα cθ sθ √ sα ωλ12 + sα f iλ± H2 = −iv1 −2 2cα λ1 + v1 where the scalar factors ∆(ab)L,R arise from the loop contributions In the unitary gauge, the one-loop Feynman diagrams contributing to this LFVHD amplitude are shown in Fig 4.1 p1 e✁ a V h01 ea✁ ± (p1 + p2) Y h01 ni k V± h01 ni e+ b ea✁ ± e+ b H 1± p2 e✁ a H 1± h01 ni ± H 1,2 Y± (2) a ni h01 ± H 1,2 a ni h01 e+ b ni V± e✁ ± H 1,2 e✁ a h01 e+ b e+ b nj (5) (4) V± e✁ a h01 ± H 1,2 ni e+ b (7) (6) (8) e+ b nj (3) e✁ ea✁ ni V± e+ b ± H 1,2 (1) h01 ni h01 ni e+ b e✁ a e+ b (9) (10) Figure 4.1: One-loop Feynman diagrams contributing to the decays h01 → ea eb in the unitary gaue The partial width of the decay is + + − Γ(h01 → ea eb ) ≡ Γ(h01 → e− a eb ) + Γ(h1 → ea eb ) = mh01 |∆(ab)L |2 + |∆(ab)R |2 , (4.8) 8π with the condition mh01 ma,b Where ma,b are the masses of muon and tau, respectively The on-shell conditions for external particles are p21,2 = m2a,b and p2h0 ≡ (p1 + p2 )2 = m2h0 The corresponding branching ratio is Br(h01 → ea eb ) = Γ(h01 → 19 ea eb )/Γtotal , where Γtotal h0 h0 1 4.1 × 10−3 GeV The ∆(ab)L,R can be written as 10 (i)W ∆(ab)L,R ∆(ab)L,R = (i)Y ∆(ab)L,R + (4.9) i=1 i=1,5,7,8 Many of the contributions listed in Eq (4.9) are suppressed and hence they can be ignored in our numerical computation From now on, we just focus on the decay h01 → µτ and hence the simplified notations ∆L,R ≡ ∆(23)L,R will be used The decay h01 → eτ has similar properties, so we not need to discuss it more explicitly We µ ∆L can see that | ∆ | O m mτ In addition, we prove in the Appendix that the following R (1+5)W combinations are finite: ∆L,R (1+2+3+5)Y and (∆L,R (2) (6+9+10)Y H1 + ∆L,R (7+8)W (7+8)W , ∆L,R (4)Y H2 , ∆L,R ) With mµ,τ (6+9+10)Y H2 , ∆L,R (4)Y H1 , ∆L,R (1) (7+8)Y , ∆L,R (2) (2) mW , we have B1 + B1 , B1 − (4)Y H ± (7+8)Y B0 and hence ∆L,R , ∆L,R The two contributions ∆L,R 1,2 are also suppressed with a large mH2± for about a few TeV A very good approximate formula for this decay rate in the limit mµ , me → is Br(µ → eγ) = 12π |DR |2 , G2F (4.10) √ where GF = g /(4 2m2W ) and DR is the one-loop contribution from charged gauge H± H± W + D Y + D + D The analytic forms are and Higgs boson mediations, DR = DR R R R W DR eg =− 32π m2W Y DR =− H± DR k eg 32π m2Y eg fk =− 16π m2W + ν∗ ν Uai Ubi F (tiW ), i=1 ν∗ ν U(a+3)i U(b+3)i F (tiY ), i=1 i=1 L,k∗ R,k mni λai λbi × m2H ± k L,k λL,k∗ − 6tik + 3t2ik + 2t3ik − 6t2ik ln(tik ) λbi × 12(tik − 1)4 m2H ± k −1 + t2ik − 2tik ln(tik ) , 2(tik − 1)3 (4.11) where b = 2, a = 1, tiW ≡ m2ni m2ni m2ni , t ≡ , t ≡ , iY ik m2W m2Y m2H ± k ν f1 ≡ , f2 ≡ c2θ , λbiR,1 ≡ U(b+3)i , λbiR,2 ≡ Ubiν , 10 − 43x + 78x2 − 49x3 + 4x4 + 18x3 ln(x) F (x) ≡ − 12(x − 1)4 (4.12) Because all charged Higgs bosons couple with heavy neutrinos through the Yukawa coupling matrix hνab , this matrix is strongly affected by the upper bound O(10−13 ) on 20 Br(µ → eγ) In fact, our numerical investigation shows that the allowed regions with light charged Higgs masses are very narrow 4.3 Numerical discussions Other parameters can be calculated in terms of the above free ones, namely, √ 2mW mW 2mY √ , ω= v1 = v2 = , sθ = , g g cθ mY m2H ± g cθ m2H ± 2 (4.13) , mH ± = (t2θ + 1) f= 4mY Apart from that, the mixing parameter α of the neutral CP-even Higgs is determined as 4λ1 t2θ sα = t2θ + λ12 − cα = ω2 , f − λ12 ω √ − m2h0 4λ1 t2θ − f ω tθ f − λ12 ω t2θ + m2h0 ω2 4λ1 t2θ − (4.14) m2h0 ω2 The Higgs self-coupling λ2 is determined as t2 λ2 = θ m2h0 v1 − m2H ± 2ω λ12 − m2H ± + 4λ1 − 2 2ω m2h0 (4.15) v12 We reproduce the regions mentioned: mH2± ≥ 480 GeV, mW = 80.385 GeV, me = × 10−4 GeV, mµ = 0.105 GeV, mτ = 1.776 GeV, mh01 = 125.1 GeV, g 0.651, √ λ1 = 1, λ12 = −1, k ≥ 5.5, z = 50, 200, 400, 500 (z < π × v1 617), mY = 4.5T eV The respective regions of parameter space always satisfy the experimental bound on Br(µ → eγ) with large enough mH2± These regions are shown in Fig 4.2 with fixed z = 1, 5, 10, 100 and 500 GeV All allowed regions, those that satisfy the upper bound Br(µ → eγ) < 4.2 × 10−13 ) give a small Br(h01 → µτ ) < O(10−9 ) In general, for larger k we checked numerically that the values of the branching ratio of LFVHDs will decrease significantly and hence we will not discuss this further 21 MR=500z MR=500z 10-7 10-9 Br(h→μτ) Br(μ→eγ) 10-12 10-17 10-22 10-27 z=1 GeV z=100 GeV z=5 GeV z=500 GeV z=10 GeV 4.2×10-13 10-14 10-19 z=1 GeV z=100 GeV z=5 GeV z=500 GeV z=10 GeV 10-7 -24 10 10-32 0.5 10 50 0.5 mH±2 [GeV] 10 50 mH±2 [TeV] Figure 4.2: Br(µ → eγ) (left) and Br(h01 → µτ ) (right) as functions of mH ± with k = 500 MR=5.5z MR=9z 10-4 10-7 Br(μ→eγ) Br(μ→eγ) 10-9 10-14 z=50 GeV z=400 GeV z=200 GeV z=600 GeV z=300 GeV 4.2×10-13 10-17 10-19 z=50 GeV z=500 GeV z=200 GeV z=600 GeV z=300 GeV 4.2×10-13 -22 -24 10 10 0.5 10-2 10 50 0.5 mH±2 [TeV] mH±2 [TeV] MR=5.5z MR=9z z=50 GeV z=500 GeV z=200 GeV z=600 GeV z=300 GeV 10-4 10 z=50 GeV z=500 GeV z=200 GeV z=600 GeV z=300 GeV 10-5 50 10-2 10-4 Br(h→μτ) Br(h→μτ) 10-12 10-6 10-8 10-4 10-6 10-8 0.5 mH±2 [TeV] 10 50 0.5 10 50 mH±2 [TeV] Figure 4.3: Br(µ → eγ) (upper) and Br(h01 → µτ ) (lower) as functions of mH ± with k = 5.5 (left) and k = (right) With small values of k = 5.5 and 9, the dependence of both Br(µ → eγ) and Br(h → µτ ) on mH2± with fixed z are shown in Fig 4.3 Most regions of the parameter space are ruled out by the bound on Br(µ → eγ), except for narrow parts where particular contributions from charged Higgs and gauge bosons are destructive Furthermore, it predicts allowed regions that give a large Br(h01 → µτ ) In particular, the largest values can reach O(10−4 ) when k = 5.5 and z = 600 GeV, which is very close to the perturbative limit In general, the illustrations in two Figs 4.2 and 4.3 suggest that this branching ratio is enhanced significantly for smaller k and larger z, but changes slowly with the change of large mH2± In contrast, small mH2± plays a very important role in creating allowed regions that predict a large LFVHD Br(µ → eγ) does not depend on mH2± when it is large enough Furthermore, the branching ratio decreases with increasing k and it will go below the experimental 22 bound if k is large enough The allowed regions in Fig 4.3 are shown more explicitly in Fig 4.4, corresponding to k = 5.5 and k = 5.50 11.2 5.45 5.40 5.35 11.1 11.0 5.30 10.9 0.825 1.68 0.820 1.67 0.815 0.810 1.66 0.805 1.65 0.800 1.64 0.795 1.63 Figure 4.4: Density plots of Br(h01 → µτ ) and contour plots of Br(µ → eγ) (black curves) as functions of mH ± and z, with k = 5.5 (upper) and k = (lower) Only regions that give a large Br(h01 → µτ ) are mentioned They are bounded between two black curves representing the constant value of Br(µ → e) ì 1013 = Clearly, Br(h01 ) is sensitive to z and k, while it changes slowly with changing values of mH2± In contrast, the suppressed Br(µ → eγ) allows narrow regions of the parameter space, where some particular relation between mH2± and k and z is realized To understand how Br(h01 → µτ ) depends on the SU (3)L breaking scale defined by mY in this work, four allowed regions corresponding to the four fixed values mY = 3, 4, and TeV are illustrated in Fig 4.5 It can be seen that Br of LFVHD depends weakly on mY , namely, it decreases slowly with increasing mY Hence, studies of LFV decays will give useful information about heavy neutrinos and charged Higgs bosons besides the phenomenology arising from heavy gauge bosons discussed in many earlier works More interestingly, this may happen at large SU (3)L scales which the LHC cannot detect at present This chapter addressed a more attractive property, namely, the LFVHDs of the 23 5.55 5.50 5.50 5.45 5.45 5.40 5.40 5.35 5.35 5.30 5.50 5.45 5.45 5.40 5.40 5.35 5.35 5.30 5.30 Figure 4.5: Density plots of Br(h01 → µτ ) and contour plots of Br(µ → eγ) (black curves) as functions of mH ± and z, with k = 5.5, z around 500 GeV and different mY SM-like Higgs boson which are being investigated at the LHC The analytical formulas at the one-loop level to calculate these decay rates in the 331ISS model have been introduced The divergent cancellation in the total decay amplitudes of h01 → ea eb was shown explicitly From the numerical investigation, we have indicated that the Br(h01 → µτ ) predicted by the 331ISS model can reach large values of O(10−5 ) They are even very close to 10−4 , for example, in the special case with k = 5.5 and z 600 GeV, which is close to the perturbative limit of the lepton Yukawa couplings The main results of this chapter are published in CONCLUSION Using the unitary gauge to calculate the one-loop contributions to the decays of neutral Higgs bosons h → Zγ in a general BSM and SM-like Higgs boson h01 → µτ in the 331ISS, we obtain the following new results: • We have established the general analytic formulas expressing one-loop contributions to the amplitude of the Higgs boson decay h → Zγ, including those were ignored in the previous studies The analytic results are also expressed in terms of the Passarino-Veltman functions, following notations in the LoopTools library We have found that these contributions may be large and should be taken into account for fitting with upcoming experimental data • Our results would be useful for further studies of loop-induced decays of neutral and charged Higgs bosons H → Zγ, W γ, which have not been mentioned in many well-known BSM In this thesis, our results were applied to discuss on the decay of standard model-like Higgs boson in the SM and BSM, including the 331β0 model, the Higgs Triplet model and the left-right symmetric model • Analytical formulas of one loop contributions needed to calculate the branching ratios of the decays of SM-like Higgs boson h01 → µτ and µ → eγ in the 331ISS The divergent cancellation in the total decay amplitudes h01 → ea eb was pointed out • From the numerical investigation, we have indicated that Br(h01 → µτ ) predicted by the 331ISS model can reach large values of O(10−5 ) They are even very close to 10−4 For example, in the special case with k = 5.5 and z 600 GeV, which is close to the perturbative limit of the lepton Yukawa couplings These results may reach the sensitivities of the upcoming experiments 24 LIST OF PUBLICATIONS Trinh Thi Hong, Lam Thi Thanh Phuong and Nguyen Thi Lan Anh, “One-loop contributions of heavy charged fermions to decays of Seesaw III-Model-like Higgs”, Can Tho University, Number 43a, 11 (2017) L T Thuy, V T N Hien, T Y Mi, N T Phong, T T Hong, “One loop corrections to decay H → ea eb in economical 3-3-1 model”, Hanoi Pedagogical University 2, Number 50, 08 (2017) T T Hong, L T T Phuong, N T L Anh, and L T Hue, “Passsarino - Veltman function for decay rate h → Zγ at one loop lever”, Hanoi Pedagogical University 2, Number 50, 08 (2017) T T Hong, H T Hung, D P Khoi, L T M Phuong, H H Phuong, L T Hue, “Decay of standard model-like Higgs boson H1 → Zγ in the simplest 3-3-1 model”, Conference on Theoretical Physics, Number 43rd, 08 (2018) Nguyen Thi Kim Ngan, Trinh Thi Hong, Le Tho Hue, “The contribution of gauge bosons to decay H → γγ (revisited)”, Hanoi Pedagogical University 2, Number 54, 04 (2018) T Phong Nguyen, T Thuy Le, T T Hong, and L T Hue, “Decay of standardmodel-like Higgs boson h → µτ in a 3-3-1 model with inverse seesaw neutrino masses”, PHYSICAL REVIEW D 97, 073003 (2018) L T Hue, A B Arbuzov, T T Hong, T Phong Nguyen, D T Si, H N Long, “General one-loop formulas for decay h → Zγ”, EUROPEAN PHYSICAL JOURNAL C 78, 885 (2018) Trinh Thi Hong, Truong Tin Thanh, Le Tho Hue, Nguyen Thuy Nga, “Decay of standard-model-like Higgs boson h → Zγ in a inter Zee model”, Tan Trao University, Number 11, (2019) Main results of this thesis are published in and  ... 12 Decay H1 → Zγ in 33 1 0 model 3. 3 In the unitary gauge, the decay H1 → Zγ is determined by Br 3 31 (H1 → Zγ) = 33 1 (H1 → Zγ) , 33 1 H1 33 1 (H1 → Zγ) = m3H1 32 π 1 m2Z m2H1 33 1 | , |F 21 (3. 10 )... form factor F 21 and F 21 are written as SM SM SM 33 1 33 1 33 1 33 1 33 1 33 1 + F 21, V + F 21, S + F 21, V = F 21, f F 21 Sjj + F 21, SVjj , F 21 = F 21, W + F 21, f , (3. 11 ) ijj ijj ijj where particular contributions... ΓSM 4.07 .10 3 GeV with mH1 = H1 12 5 .1 GeV Numerical illustrations for 33 1 Zγ in the 33 1 0 are shown in Fig 3. 1  13 1 =1, λ 13 = -1 1 =1, λ 13 = -0.5 1. 00 1. 01 0.95 μZγ μZγ 1. 00 0.99 mF=0.2, mV=0.5