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

Moreinteresting, if there is any of deviations between experimental results and those pre-dicted by the SM, it will be new physics, in the meaning that there exist additionalcontribution

at Large Hadron Collider (LHC) In the future projects, the sensitivities of these periments will be improved so that the decay channel h → Zγ may be detected Moreinteresting, if there is any of deviations between experimental results and those pre-dicted by the SM, it will be new physics, in the meaning that there exist additionalcontributions from new particles in models Beyond the Standard Model (BSM).

ex-At the one loop level, the amplitude of the decay h → Zγ predicted by some BSMnormally contains contributions of new particles that do not appear in the SM frame-work Calculating these contributions is rather difficult in the usual ’t Hooft-Feynmangauge, because of the appearance of many unphysical states, namely Goldstone bosonsand ghosts which always exist along with the gauge bosons They brings a very largenumber of Feynman diagrams In addition, their couplings are indeed model depen-dent, so it is hard to construct general formulas determining vector loop contributionsusing the t’ Hooft-Feynman gauge

The technical difficulties caused by unphysical states will vanish if calculations arecarried 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 physicalstates are needed Then the Lorentz structures of these couplings are well defined, andhence the general analytic formulas of one-loop contributions from gauge boson loopscan be constructed But in the unitary gauge we face the complicated forms of thegauge 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 Theremaining ones will vanish systematically when loop integrals are written in terms ofthe Passarino-Veltman (PV) functions Our results can also be translated into thegeneral 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

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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 cLFVphenomenology such as charged lepton decays ei → ejγ In particular, it was shownthat Br(µ → eγ) is large in these models and hence it must be taken into account toconstrain the parameter space In addition, such rich LFV resources may give largeLFVHD rates as promising signals of new physics

The 3-3-1 model with inverse seesaw neutrino masses (331ISS) can predict a neutrinomass spectrum consistent with current experimental neutrino data, using the well-known 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 mostinteresting allowed regions will also allow large LFVHD rates, which we will try tolook 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 h0

1→ µτ in the 331ISS

• LFV couplings, Feynman diagrams and amplitude

• Passarino Veltman functions (PV) for the decay h → Zγ and h0

1 → µτ Research content

• The particle spectra relate with the general decay amplitude h → Zγ and the SM-likeHiggs 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 tributions in BSM

con-• Survey numerically of the Br(h0

1 → µ±τ∓) in the 331ISS for predicting the ability todetect this decay at the LHC in the future

• Determining allowed regions of the parameter space that satisfy all the theoreticaland 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 Higgsboson that contribute to the decay h → Zγ, LFV sources in some well-known BSMand some elementary knowledge of the decays of the SM-like Higgs boson realting withrecent experiments

Chapter 2: Constructing general analytical formulas of one-loop contributions tocalculate Br(h → Zγ) in the unitary gauge

Chapter 3: Comparing and contrasting between the formulas obtained in chapter

2 with some published results, and discussing some interesting contributions in theBSM that were ignored previously

Chapter 4: Investigating numerically the Br(h01 → µ±τ∓) in the 331ISS: find allcouplings and one-loop Feynman diagrams in the unitary gauge, calculate in detailparticular one loop contribution and prove the divergent cancelation in the total decayamplitude, draw plots to discuss numerically the results

General conclusions: Review the main obtained results and propose future search directions

re-Appendix: We present some detailed intermediate steps related to the calculations

in the main part of this thesis

Chapter 1

OVERVIEW

The SM describes successfully strong, weak and electromagnetic interactions based onthe 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 alltypes of interaction (the gravitational force is not include in the SM), the SM does notanswer the questions why are there three fermion generations? why is the top quarkmass not well consistent with the experiment results? the SM describes neutrinos asmassless while the experiments show that neutrinos are massive, Hence, many modelsbeyond the SM have been introduced in order to explain the above shortages One ofthe new research directions for new physics in the BSM is studying the rare decay andthe LFV decay

In the SM, the couplings of the SM-like Higgs boson with other particles and amongparticles 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 SMand have been independently verified by experimental (LHC) For new particles in theBSM, the couplings will contain new unverified parameters Therefore, our study willcontribute to further clarifying the allowed-value regions of these parameters Theywill 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 ofneutrinos, the new leptons that are added in the BSM There are such BSMs as supersymmetry models, seesaw models, 3-3-1 models, However, in the framework of thisthesis, we only focus on 3-3-1 models, particularly the 331ISS model Firstly, theLFV source from the new neutrinos Secondly, the LFV source from new interactionsbetween the SM-like Higgs boson and the new gauge bosons, the SM-like Higgs bosonand new charge Higgs boson These new particles create more one-loop contributionsdiagrams for the LFV decay

Besides, the decays of LFVHDs have been investigated by experiments, these arenew physical signals that is not existed in the SM All the decay channels had not beensearched yet by the low energy accelerators The LHC is the first accelerator with highenergy that enable to search for these decays In 2015, the upper bound Br(h → µτ )

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was set up by CMS and ATLAS

The Higgs boson is searched through the main channels: h → b¯b, c¯c, τ+τ−, γγ,

ZZ, W W+, gg The branching ratio is defines as Br(h → XY ) = Γ(h→XY )Γ

total , where

Γtotal ' 4.1 × 10−3 is the total decay width of SM-like Higgs boson The branchingratio of hγγ, hZγ are very small, about 2.10−3 with mh around 120 − 130 GeV Inthese 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 whichhave been hunted by recent experiments In this thesis, we focus mainly on issuesrelated to these two decay channels

Chapter 2

ONE-LOOP CONTRIBUTIONS TO THE DECAY

h → Zγ

The amplitude of the decay h → Zγ is defined as follows

M(h → Zγ) ≡ M (Zµ(p1), γν(p2), h(p3)) εµ∗1 (p1)εν∗2 (p2) ≡ Mµνεµ∗1 εν∗2 , (2.1)The on-shell conditions are p21= m2Z, p22 = 0 and p23 = m2h The decay amplitude isgenerally written in the following form

where µναβ is the totally antisymmetric tensor with 0123 = −1 and 0123 = +1,

εν∗2 p2ν = 0, F12,22 do not contribute to the total amplitude (2.1) In addition, the Mµν

in eq (2.2) satisfies the Ward identity, pν2Mµν = 0, resulting in F11= 0 and

Γ(h → Zγ) = m

3 h32π ×



1 − m

2 Z

m2h

3

|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+)λgµν+ (p+−p−)µgνλ+ (p−−p0)νgλµ, where all momentaare incoming and p0,±are respective momenta of h and charged gauge and Higgs bosonswith electric charges ±Q, denoted as Vi,j±Q and Si,j±Q, respectively The general case offour-gauge-boson coupling is (2, −1, −1) → (a1, a2, a3) and gZγVij 6= e Q gZVij

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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 inthe previous studies The analytic results are also expressed in terms of the Passarino-Veltman functions, following notations in the LoopTools library The main results ofthis chapter are published in 7

2 + 3t2+ 3(2t2− t22)f (t2), (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 consistentwith 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 ofthe decay h → γγ The analytic form of this contribution is known (Westview Press;Sov J Nucl Phys 30), namely (3.3)

F21,Wh→γγ,SM= e g

2mW16π2



2 + 3t2+ 3(2t2− t22)f (t2) (3.3)

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in (Phys Rev D 92) is

F21,VGHU= m41+ m42+ 10m21m22
E+(m1, m2)+(m21+ m22)(m2h− m2Z) − m2hm2ZE−(m1, m2)

−

4m21m22(m2h− m2Z) + 2m4Z(m21+ m22) C0+ C00
, (3.4)where function C00 is determined by changing the roles of m1 and m2 and

E±(m1, m2) = 1 + m

2 Z

m2h− m2

Z



B0(2)− B0(1)± (m22C0+ m21C00) (3.5)

In the special case where Vi ≡ Vj, corresponding to m1 = m2 = m, C00 = C0 =

−I2(t2, t1)/m2, C12+ C22+ C2 = I1(t2, t1)/(4m2) and use some special formulas

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

F21,VGHU= m4+ m4+ 10m2m2
E+(m, m) +(m2+ m2)(m2h− m2Z) − m2hm2Z

× E−(m, m) −4m2m2(m2h− m2Z) + 2m4Z(m2+ m2)(C0+ C0)

= −m2

h− m2 Z2m2



12m4+ 2m2(m2h− m2Z) − m2hm2ZI1(t2, t1)+ 44m2(m2h− m2Z) − m2hm2Z + 2m4ZI2(t2, t1) (3.7)

So compared to the result of eq (3.18) in (Phys Rev D 92), we find the agreementbut different by a factor of 2 first m4Z This difference may be due to the confusion ofthe authors in (Phys Rev D 92) They differ by

δF21 = F21,VGHU−



16π22e Q ghVijgZVij(F21,Vijj + F21,Vjii)



×−m21m22(m2h− m2z)

m 1 =m 2

= 0

Formula (3.4) equivalent to our results, namely F21,Vijj + F21,Vjii But two generalresults are not the same, i.e they differ by δF21= −2 m21C0+ m22C00
m4Z In (Phys.Rev D 92) we are considering them in general form but do not use and use special

cases, if using their general formula, the results do not match the SM results, so theirgeneral formula has errors Except F21,Vijj in eq (2.6), our formulas are consistentwith the results given in (Phys Rev D 96), which were obtained by calculating thedecay amplitude of charged Higgs boson H± → W±γ in the ’t Hooft-Feynman gaugefor the Georgi-Machacek model The amplitude of the decay H± → W±γ, derivedfrom (2.6) with mh → mH±, mZ → mW, gZVij → gW Vij, ghVij → gHVij now has thefollowing form

m2W(C12+ C22+ C2)

+ s

2 W

c2W(C12+ C22+ 2C1+ 3C2+ 2C0). (3.9)Which is different from the result given in (Phys Rev D 96) by the coefficient 10instead of 12 in front of the sum (C12+ C22+ C2) We see that the two parts in ourresult 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

1 is the total decay width of the SM-like Higgs boson H1 and Γ331(H1→ Zγ)

is the partial decay width predict by the 331β0 model The form factor F21331 and F21SM

are written as

F21331 = F21,f331ijj+ F21,V331ijj+ F21,S331ijj + F21,V S331 jj + F21,SV331 jj, F21SM = F21,WSM + F21,fSM, (3.11)

where particular contributions are derived based on the general formulas

F21,f331 = −e QfNcK

f + LL,RR16π2 [16 (C12+ C22 + C2) + 4C0] ,

√2cXcα)4

√2cXcα)4

asαs2W/u for new lepton Ea and m2U

asαs2W/(3u) for new quark Ua Otherfactors are

K

H 1 H1,21/2 =

λ

H 1 H1,21/216π2 × e g(−c

2

Xs2W + s2Xc2W)2cW , KH1 H ± = λH1 H ±

16π2 ×2e g(1 − 2s

2

W)2cW ,

KW = −2eg

2cWmWcα16π2 , KV = KV0 = eg

mF=0.2, mV=0.5 TeV mF=0.2, mV=1 TeV mF=1, mV=0.5 TeV mF=1, mV=1 TeV

0.85 0.90 0.95 1.00

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± TheBr(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 < 6

the LR and HTM model

Because new heavy charged gauge V± and Higgs bosons S± appear in non-trivialgauge 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 identicalcharged 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 maycause significant effects to Br(h → Zγ) in the light of the very strict experimentalconstraints of Br(h → γγ) When m2X  m2

W with X = S, V , the loop structures ofthe form factors with at least one virtual W boson have an interesting property that

∼ FW0 ≡

F21,W

eghW WgZW W/(16π2)

the same order with the W loop contribution

In contrast, the loop structure of a heavy gauge boson F21,V V V is

FV0 ≡ F21,V V V

ghV VgZV V/(16π2) ∼ O(m−2V ),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 X0 /FW0 , fV ≡ FV0 /FW0and mS = mV

are written as

F 21< /sub>33 1< /sup> = F 21, f33 1< /sup>ijj+... F 21, V33 1< /sup>ijj+ F 21, S33 1< /sup>ijj + F 21, V S33 1< /sup> jj + F 21, SV33 1< /sup> jj, F 21< /sub>SM... class="text_page_counter">Trang 12

1< /small> is the total decay width of the SM-like Higgs boson H1< /sub> and Γ33 1< /sup>(H1< /sub>→ Zγ)

is