Optimisation of a multivariate analysis technique for the ttbar background rejection in the search for Higgs boson pair production in bbtautau decay channel with the CMS experiment at the LHC

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Universit`a degli Studi di Pisa

DIPARTIMENTO DI FISICA “E. FERMI”

Tesi di Laurea Magistrale in Scienze Fisiche

Curriculum “Fisica delle Interazioni Fondamentali”

Aprile 2018

Optimisation of a multivariate analysis technique

for the t¯t background rejection in the search

for Higgs boson pair production in b¯

b⌧

+

⌧

decay channel

with the CMS experiment at the LHC

Candidato

Angela Giraldi

Relatore

Dott. Giuseppe Bagliesi

Correlatore

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Abstract

This thesis reports the optimisation of a multivariate analysis technique for the search for Higgs boson pair (HH) production. HH production gives an access to the Higgs boson trilinear self-coupling and is sensitive to the presence of physics beyond the Standard Model. Both resonant and nonresonant production mechanisms are investigated exploring events with one Higgs boson decaying into two b

quarks and the other decaying into two t leptons (HH !b¯bt+_t _{). This process is studied through}

the examination of the three decay modes of the t+_t _{system, with one or two t decaying into}

hadrons in the final state. The search uses proton-proton collision data collected atps=13 TeV with

the CMS experiment at the CERN LHC, corresponding to an integrated luminosity of 35.9 fb 1.

The main effort has been devoted to design and develop a multivariate technique to separate the signal from the t¯t background. This technique has been applied for the first time to all the three final states and has proved to be an essential element to enhance the sensitivity. No evidence for the presence of a signal has been found and results are found to be consistent, within uncertainties, with the standard model background predictions. Upper limits are set at 95% confidence level on resonant and nonresonant HH

production. The expected and observed upper limits are about 15⇥s_HHSMand 10⇥s_HHSM, respectively,

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4.1.1 Classification Analysis . . . 84 4.2 Tree-Based Methods . . . 86 4.2.1 Classification Trees . . . 86 4.2.2 Growing a Tree . . . 87 4.2.3 Tree Parameters . . . 88 4.2.4 Splitting a Node . . . 89 4.2.5 Variable Selection . . . 90 4.2.6 Tree (In)Stability . . . 91 Overtraining . . . 91 Pruning a Tree . . . 91

Averaging Several Trees . . . 92

4.3 Committee Machines . . . 92

4.3.1 Bagging . . . 92

4.3.2 Boosting . . . 93

AdaBoost . . . 93

4.3.3 Random Forest . . . 94

4.4 Boosted Decision Trees . . . 94

4.4.1 Gradient Boosting . . . 94

5 MVA procedure for t ¯t background rejection 97 5.1 General Strategy . . . 97

5.2 Search for BDT Inputs . . . 100

5.2.1 Initial Kinematic Variables . . . 100

5.2.2 Study of the Compatibility among Signal Samples . . . 104

Identification of Mass Regions . . . 109

5.2.3 Algorithm to Find the Most Discriminating Variables . . . 109

Selected Variables . . . 113

5.2.4 Parametrized Learning . . . 113

5.3 BDT Optimization . . . 118

5.3.1 Hyperparameter Space . . . 118

5.3.2 BDT Overtaining Check: Cross-Validation . . . 119

Selected Hyperparameters . . . 121

5.3.3 Performance Evaluation . . . 122

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CONTENTS

6 Results and Conclusions 129

6.1 Background Estimation . . . 129

6.2 Systematic Uncertainties . . . 130

6.3 Statistical Interpretation . . . 131

6.3.1 Signal Extraction . . . 131

6.3.2 Validation of the Statistical Model . . . 133

6.4 Results . . . 133

6.5 Conclusions . . . 141

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Introduction

The theory that describes the current knowledge of elementary particles and their interaction is the Standard Model of particle physics (SM) [1]. The SM is currently the most precise theoretical explanation of the fundamental building blocks of matter and it has encountered a tremendous success because of its high predictive power. So far experimental observations and direct tests at collider experiments well corroborate the SM: a further important confirmation has been received with the recent discovery of the Higgs boson. However, the SM theory has some critical points and it leaves some phenomena unexplained: from not including gravitational force and not taking into account neutrino masses to being in contrast with experimental evidence without explaining dark matter and dark energy. It is reasonable to believe the SM is only a low-energy approximation of a more extended physics model and new possibilities are open for searches beyond the Standard Model (BSM). The nature of BSM physics is still an open question, as no clear signs of its presence have been found so far.

The Large Hadron Collider (LHC) was built to explore the properties of the electroweak symmetry breaking (EWSB) mechanism and explore the physics at the TeV energy scale. It is designed to collide protons at a centre-of-mass energy up to 14 TeV in four interactions points, where detectors are located. The discovery of the Higgs boson by A Toroidal LHC Apparatus (ATLAS) and Compact Muon Solenoid (CMS) collaborations [2, 3] marked a milestone in the history of physics, not only because it completes the SM but mainly because it opens up the study of a new sector of the theory, the scalar sector. A very precise determination of Higgs couplings and properties is now of utmost importance. In this context, the Higgs boson self-interactions represent one of the leading elements to completely characterise the scalar sector of SM. On the other hand, precise description of rare production and decay modes are of particular interests because the Higgs boson provides a preferential way to search for deviations from the SM.

The work presented in this thesis is situated in this context and aims to study the Higgs sector properties and explore the extensions of SM through the study of Higgs boson pair (HH) produc-tion. This process directly gives access to the trilinear self-coupling of the Higgs boson. Thus, its observation represents a crucial test of the SM validity. This measurement at LHC is particularly challenging due to the small cross-section. Nevertheless, HH production is very sensitive to the presence of BSM contributions. BSM effects could be revealed either directly by new states decaying to a HH pair (resonant production), or by contributions in the quantum loops that would modify its cross-section and kinematic properties (nonresonant production). The search of HH production explored in this thesis consists in the investigation of both resonant and nonresonant gluon fusion production mechanisms, exploiting events where one Higgs boson decays into two b quarks and the

other into two t lepton. The investigation of the HH !b¯bt+_t _{process explores 3 different final}

states: the fully-hadronic channel HH !bbt_h±t_h⌥ where t_hmeans a t which decays into hadrons

plus a n_t, and two semi-leptonic channels HH !bbt±

l th⌥ where one t decays to a lighter lepton

e or µ (tl) while the other decays hadronically (th). The fully-hadronic channel represents the 42% of

the di-tau decay modes, the semi-leptonic occurs in the 46% of the cases and they are complementary to remaining 12% of the decays where both taus decay into muons or electrons.

The b¯bt+_t _{final state is one of the most sensitive decay channels with its sufficiently large branching}

fraction of 7.3% and the relatively small contamination from SM backgrounds. The data used for this

search were collected using proton-proton collision atps=13 TeV by the CMS experiment in 2016,

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CONTENTS

and developed new multivariate techniques (MVA) applied for the first time to the three final states to optimally separate the signal from the dominant t¯t background. The final goal of this analysis is to measure an excess of events above the background or, otherwise, to set upper limits both on resonant and nonresonant HH production.

The first parts of the thesis are introductory from a theoretical and experimental point of view,

discussing the motivations and challenges of a search for HH!b¯bt+_t _{production and presenting}

the experimental apparatus used to perform the search. In addition, there is a description of how the information from different sub-detectors is used to reconstruct the physical objects of interest to the final state investigated in this thesis. The analysis strategy for event selection is then explained, focusing on the experimental signature of the signal processes and on the main background sources, and introducing the techniques developed to select the events.

Resonant production is explored using signal events produced via gluon-gluon fusion for narrow width resonances in the mass range from 250 GeV to 900 GeV, for both spin-0 and spin-2 hypothesis. Nonresonant production is considered in the context either of SM and BSM frameworks. Multiple

sources of background affect the b¯bt+_t _{search: the main irreducible one is t¯t}

! b¯bW±_W⌥ _! b¯bl±_n

lt⌥nt(l = e, µ, t), with the same particle content in the final state as the b¯bt+t signal.

Re-ducible backgrounds instead arise from inaccurate object identification. Although the misidentification probabilities are usually small, there is a sizeable contribution because the cross-section of the back-ground processes is many orders of magnitude larger than the one expected for the signal. The rejection of these two classes of background requires the implementation of different strategies: for the reducible backgrounds, tight object quality criteria are applied to reduce the misidentification rate; instead, the suppression of the irreducible background sources arise from the exploitation of their kinematic differences with respect to the signal processes. The estimation of the signal and background processes is performed with a combination of Monte Carlo simulation and data-driven methods.

The main steps, that help to improve the discrimination from the background, correspond to the trig-ger requirements, object pre-selections, event categorisation, and definition of the signal regions. The trigger requirements are necessary to store events offline for subsequent analysis and pre-selections of final objects allow having optimised constraints on the quality of reconstructed objects. Objects

satisfy-ing these quality criteria are combined into H!t+t and H!b¯b candidates, whose properties are

used to classify the events depending on their t+_t _{final state and bb topology. Finally, in each event}

categories, the separation from the background is increased applying dedicated selections on the

invariant mass of the t+_t _{and bb pairs and exploiting their kinematic properties with a multivariate}

analysis technique.

My personal contribution has been the development of a dedicated multivariate technique, in-cluding definition of discriminant in the form of a boosted decision tree (BDT) algorithm, to improve the suppression of the t¯t background and consequently to increase the sensitivity in the exclusion limits. Firstly, in Chapter 4, there is a general presentation of how to construct decision trees and how to enhance the performance of typically weak methods. Emphasis has been set on the method used during the analysis performed in this thesis: the gradient boost algorithm.Then, in Chapter 5, the design of the entire MVA strategy is presented. The starting point is the definition of an extensive set of kinematic observables, which can potentially provide good discrimination between signal and background. Then, the probability density functions (PDFs) of these kinematic variables are studied for each signal samples as well as for background. The further step is the identification of three mass regions, studying with proper statistical methods the compatibility among the variable PDFs at different mass points. The resonant samples are divided into the low mass (LM) region, for masses from 250 GeV to 320 GeV, the medium mass (MM) region, with masses from 340 GeV to 400 GeV, and the high mass (HM) region, which covers the mass range up to 900 GeV. In order to increase the statistic, inside these regions all the signal samples are joined together considering both spin hypotheses and the three final states. Afterwards, a specific algorithm based on statistical methods is implemented to list the variables according to their discrimination power and it is applied both for the nonresonant sample and for each region of resonant samples. The twenty most discriminant variables are considered as the inputs of the BDT. In addition, a study is performed to understand whether the separation between signal and background remain optimal after the samples are joined together inside the aforementioned regions. To keep trace of the origin of the events, a parametrised learning is introduced adding information from variables like the mass of the resonance and the final state

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CONTENTS channel. The toolkit used in this thesis to implement the multivariate classifier algorithm is TMVA [4], integrated into the ROOT analysis framework [5]. The performance and the stability of the BDT are explored depending on the values of the setting parameters. These hyperparameters are chosen with a cross-validation technique that avoids overtraining and ensures that the method performs properly also on unknown data. The trainings are finally evaluated for each signal point and each final state and they are compared using specific classifier methods, the ROC curves. The comparison is also

performed with respect to the results of the last published analysis [6], arising a 30% improvement.

The results obtained with this new setup of MVA techniques are promising and interesting for future developments since they outperform the existing ones.

The comparison between background predictions and data are presented in the last part of the thesis together with the results and their interpretation. The signal extraction is based on the output of the BDT both for resonant and nonresonant samples. The upper limits for the resonant samples, both for spin-0 and spin-2 hypothesis, gained an improvement up to 50%. For nonresonant sample, the expected and observed upper limits are respectively about 15 and 10 times the SM prediction, significantly improved over the previous limits and corresponding to the most stringent limits set so far at the LHC.

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Chapter

1

Theoretical Overview

T

his chapter provides the theoretical background for the search performed in this thesis. It begins with

a brief outline of the Standard Model (SM), a renormalizable quantum field theory which provides a unified description of the strong, weak and electromagnetic forces. The SM incorporates a minimal scalar sector that is at the origin of the spontaneous breaking of the electroweak symmetry and of the masses of the fermions. So far experimental observations and direct experimental tests at collider experiments well corroborate the SM, which received further confirmation with the recent discovery of the Higgs boson. However, some theoretical considerations and observations from the subnuclear to the astrophysical scales, suggest that the SM may be incomplete and there is the necessity to extend the theory Beyond the SM (BSM). The Higgs boson discovery allows to explore the interaction of BSM particles with the SM scalar sector. This chapter focuses on the importance of the study of HH production in the context of both SM and BSM physics, that can result in resonant or nonresonant Higgs boson pair (HH) signatures. For example, in the context of BSM models, it is possible to interpret the discovered scalar neutral boson H as a particle with properties similar to the SM Higgs boson, that can be produced in pair by the decay of the heavy neutral resonance X, which is potentially observable with the currently available data at LHC. These models are explained, with particular attention on

the scalar sector and the theoretical and experimental reasons for the investigation of HH!bbtt production.

Finally, a presentation of the phenomenology in collider experiments and results previously obtained at the LHC are reported.

1.1 The Standard Model

The Standard Model of particle physics [7, 8] is the current best understanding and most precise theoretical explanation of the fundamental building blocks of matter: it summarises the current knowl-edge of elementary particles and their interactions. The SM elegantly describes matter through funda-mental spin-1/2 particles, the so-called fermions grouped into two classes, quarks and leptons, according to the interactions they undergo. The fermions interact among themselves via spin-1 particles known

as bosons, and they are regulated by three of the fundamental interactions1: the electromagnetic force,

the weak force, and the strong force.

Mathematically, the SM is formulated based on the Quantum Field Theory (QFT) and the local

SU(3)_C⌦SU(2)_L⌦U(1)_Ygauge symmetry.

1.1.1 Fundamental Particles and Interactions

The fermions, containing six leptons and six quarks, are the elementary building blocks of matter and are classified into three generations according to their mass, starting from the lightest and most stable particles. Quarks participate in all the three types of the interactions mentioned above, while leptons do not experience the strong force. It is possible to identify doublets of leptons and quarks

1_{The gravitational force cannot be included in the SM description because it is 40 orders weaker than the other three forces.}

It is foreseen to become relevant at the so-called Planck scale, of the order of 1019GeV and in the context of the Grand Unification

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1 Theoretical Overview R/G /B 2/3 1/2 2.3 MeV up

u

R/G /B 1/3 1/2 4.8 MeV down

d

1 1/2 511 keV electron

e

1/2 < 2 eV e neutrino

⌫

e

R/G /B 2/3 1/2 1.28 GeV charm

c

R/G /B 1/3 1/2 95 MeV strange

s

1 1/2 105.7 MeV muon

µ

1/2 < 190 keV µ neutrino

⌫

µ

R/G /B 2/3 1/2 173.2 GeV top

t

R/G /B 1/3 1/2 4.7 GeV bottom

b

1 1/2 1.777 GeV tau

⌧

1/2 < 18.2 MeV ⌧ neutrino

⌫

⌧

±1 1 80.4 GeV

W

± 1 91.2 GeV

Z

1 photon color 1 gluon

g

0 125.1 GeV Higgs

H

graviton st ron g n u cle ar for ce (c ol or ) el ec tr om agn et ic for ce (c h ar ge ) w eak n u cl ear for ce (w eak is os p in ) gr av it at ion al for ce (m as s) charge colors mass spin 6 q u ar k s (+ 6 an ti-q u ar k s) 6 le p ton s (+ 6 an ti-le p ton s) 12 fermions (+12 anti-fermions) increasing mass! 5 bosons (+1 opposite charge W ) standard matter unstable matter force carriers

Goldstone bosons

outside standard model

1

st

2

nd

3

rd generation

Figure 1.1:The Standard Model table of elementary particles. Shown are the three generations of matter particles, the

fermions, and the force carrier particles, the gauge bosons, in addition to the Higgs boson. Electric charge, spin, colour and the mass of each particle are also noted. This table is an attempt to show in particular how particles relate to the four principal forces and how the recently discovered Higgs boson fits in [9].

in each generation for both families, and it is the same for the antiparticles. According to Dirac’s relativistic wave equation [10], which describes the behaviour of fermions, all quarks and leptons have a corresponding anti-particle. This prediction implies the presence of anti-matter and has been experimentally confirmed by observing the positron (the anti-particle of the electron) [11]. Anti-particles share the same properties (mass, spin and lifetime) with their corresponding partner particle but with opposite electric charge. It is worth remembering that a fermion field can be divided into its

left-handed part y_Land its right-handed part y_Ras y=y_L+y_R.

A charged massive lepton and a neutral particle called neutrino, assumed massless in the SM

theory2, compose the couplet of leptons. The same organisation represents the quarks doublets: they

can be of six flavours, are massive and have a fractional electric charge. Positively and negatively charged quarks are divided into up-type and down-type groups respectively. Moreover, quarks have the colour charge, which exists in three states “red” (R), “green” (G) and “blue” (B). Quarks do not exist as free particles in nature: they combine in “white” colour charge states, called hadrons, divided in mesons, a quark-antiquark couple, and in barions, made of three quarks. In particular, leptons and quarks can be represented as in the following notation:

ylepton_L = ✓ n l ◆ L yquark_L = ✓ u d ◆ L (1.1)

The classification of the elementary particles is illustrated in Figure 1.1, where it is also possible to understand how the fermions interact among themselves: it happens through the exchange of five gauge bosons corresponding to different types of interactions. A massless boson, the photon g, is the mediator of the electromagnetic interactions for all fermions except for the massless and elec-trically neutral neutrinos. When a strong interaction between quarks occurs, eight gluons mediate the interaction among them; gluons are massless, electrically neutral and have a colour charge.

2_{According to the observation of the neutrino flavour oscillation, different neutrino flavours have different masses, although}

these masses have been shown to be very tiny by the experiments [12, 13]. 6

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1.1 The Standard Model

Three gauge bosons can carry the weak force for all the fermions, the charged W±_{and the neutral Z}

0. An additional boson, the Higgs boson H, predicted by the SM was observed in July 2012 at the LHC [2, 3] completing the picture of the SM and will be discussed more widely in Section 1.1.3.

1.1.2 Standard Model as Gauge Theory

The SM is a quantum field theory, formulated relying on the construction of a local invariant

Lagrangian3under the non-abelian group SU(3)_C⌦SU(2)_L⌦U(1)_Yto explain the strong, weak, and

electromagnetic interactions [7] .

The strong force is the one responsible for holding ordinary matter together. It is described by

the non-abelian gauge theory of quantum chromodynamics (QCD) which is based on the SU(3)_C

symmetry group. The theory was introduced by Yang and Mills [14]. The strong interaction between quarks is carried by eight gluons, corresponding to eight physical gauge fields described by eight generators, the Gell-Mann matrices. An additional charge known as the colour charge is required by the QCD to preserve the Pauli exclusion principle. The colour charge has three eigenstates (red, green and blue) for the quarks and gluons. Since gluons carry a colour charge, they can interact with each other and this corresponds to the non-abelian property of the groups. The coupling strength of the

strong interaction a_s varies depending on the momentum transfer of the interaction. More details

about QCD can be founds in [1, 7].

The relativistic quantum field theory of electrodynamics (QED), an abelian gauge theory with

the symmetry group U(1)_L, describes the electromagnetic force. It represents the phenomenon of

photon exchange upon interaction of charged fermions. The abelian U(1)_Lgroup is generated by the

electric charge and the gauge boson Aµ, representing the photon, is associated to this. As a result of

the gauge invariance, the Aµboson is massless and because of the abelian property of the group it is

not self-interacting. Photons can only couple to charged particles, with the coupling strength given

by the constant a = _4pe¯hce2 ⇡ 1371 ⌧1. The QED Lagrangian for a 12spin field interacting with the

electromagnetic field is given in natural units by:

L = ¯y(igµDµ m)y

1

4FµnFµn (1.2)

where Dµ⌘∂µ+iQAµis the gauge covariant derivative, Q can be interpreted as the electric charge

of the spinor field and m its mass, Aµis the covariant four-potential of the electromagnetic field and

Fµnis the electromagnetic field tensor.

The radioactive decays are governed by the weak force described by the theory of quantum

flavour-dynamics (QFD), under the U(1)_Ysymmetry group. The weak force is mediated by two charged (W±)

and one neutral (Z₀) gauge bosons. Due to their large mass, m_W± =80.4 GeV and m_Z₀ =90.2 GeV,

and short lifetime under 10 24s, there are range restrictions of the weak interactions. The Z₀and W±

bosons were discovered by UA1 and UA2 experiments in 1983 at CERN [15–18]. Also, the fermions have a determined mass while the only particle massless is the photon and this property has been confirmed experimentally so far [19].

The SU(2)_L⌦U(1)_Ygroup describes the unified electromagnetic and weak interactions (EWK),

as theorised by Glashow, Weinberg and Salam [20–22]. Although both interactions are different at low

energies, they can be merged into one gauge theory at high energy regimes, beyond⇠100 GeV. The

three isospin weak operators4,~_T_{⌘ (}_T₁_{, T}₂_{, T}₃₎_{, and the weak hypercharge Y are the generators of the}

groups. The fermions interact with gauge bosons, which, as a consequence of the non-abelian property,

also interact with each other. Moreover the third component of the isospin (T3) and hypercharge Y

are related to the electric charge Q = T3+Y/2. It is possible to associate theW~µ ⌘ (Wµ1, Wµ2, Wµ3)

3_{A system is considered invariant if it remains unchanged, despite the changes of its properties, under a local transformation,}

which depends on space-time coordinates or under a global transformation, not depending on the space-time coordinates.

4_T

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1 Theoretical Overview

gauge bosons to the isospin operators and Bµgauge boson to the hypercharge operator thanks to the

gauge invariance. These four gauge bosons are massless. However, measurements have shown that the mediators of the weak force are massive.

The full electroweak Lagrangian can be expressed in a compact form:

L =iygµ_D µy 1 4W~µn· ~Wµn 1 4BµnBµn, (1.3) where now Dµ=∂µ+ig 0

2YBµ+ig~T· ~Wµ. Furthermore, g0and g are the coupling constants introduced

for U(1)_Yand SU(2)_Lgauge group, and their relative field BµandW~µ.

The resulting EWK Lagrangian describes all the interactions between the gauge fields and the fermions but crucially does not contain mass terms for the weak force bosons or any of the fermions. Attempts to add gauge boson mass terms or fermion mass terms break the gauge invariance of the Lagrangian.

To confirm the experimental results the spontaneous breaking of the electroweak symmetry5was

theorised and formulated in 1964 by P. W. Higgs [23, 24], and by F. B. Englert and R. Brout [25], and is now an essential ingredient of the SM. In the following there will be a summary of the key points of the mechanism, indispensable to understand the beyond Standard Model searches in the Higgs sector.

1.1.3 Spontaneous Symmetry Breaking in the Brout-Englert-Higgs Mechanism

The Brout-Englert-Higgs (BEH) mechanism [23–25] predicts the existence of a scalar boson, the Higgs boson, to give mass to the fermions and the weak bosons. The mechanism is founded on the spontaneous breaking of the electroweak symmetry [26], which allows the symmetry of the gauge theory to be broken without disrupting the gauge invariance: the Lagrangian remains invariant under the symmetry but the ground state of the theory no. This leads the ground state to be degenerate, and so it is necessary to choose one eigenstate, breaking the symmetry. The Higgs mechanism is based on Goldstone’s theorem [1]. The Higgs field breaks the three out of the four degrees of freedom of the

SU(2)_L⌦U(1)_Ygroup. These three degrees of freedom, the massless Goldstone bosons, are absorbed

by the three W±_{and Z}

0bosons, and become only observable as components of these weak bosons,

which now become massive. The remaining degree of freedom amounts for a massive scalar particle, the Higgs boson H.

Introducing a complex scalar field f with a non-zero vacuum expectation value can break the symmetry and, at the same time, preserves the gauge invariance once added to the EWK Lagrangian:

f= ✓ f+ f ◆ = p1 2 ✓ f₁+if2 f₃+if4 ◆ , (1.4)

it is an isospin doublet, a linear combination of four real scalar field, and with hypercharge Y=1. The

field must be scalar to satisfy space isotropy, otherwise the expectation value on the vacuum would be frame-dependent. Moreover, the expectation value on the vacuum must be constant to satisfy space

homogeneity. As long as the field has Y = 1 its covariant derivative is Dµ = ∂µ+ig

0

2Bµ+igW~µis2i, thus the BEH lagrangian can be written as:

LBEH = (Dµf)†(D µ

f) V(f†f). (1.5)

The scalar potential which describes the evolution of the f field can be expressed as:

V(f) =µ2f+f+l(f+f)2. (1.6)

It depends on the mass parameter µ2, considered<0 to lead to a non-vanishing vacuum state, and on

the dimensionless coupling parameter l, needed to be>0 to have the potential bounded from below.

The ground state is not uniquely identified but the hypersphere|f|2= _2lµ2 is where the minima lie

(see Figure 1.2). For the particular minimum chosen:

hfi = p1 2 ✓₀ v ◆ , (1.7)

5_{The mechanism is based on the concept of spontaneous symmetry breaking, a phenomenon that is often observed in Nature}

whenever individual ground states of a system do not satisfy the symmetries of the system itself. 8

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Figure 1.2:Shape of the Higgs potential for µ2<0 [27].

the vacuum expectation value(VEV) is v=

q

µ2/l. A positive vacuum expectation value

automati-cally fixes the masses of the gauge bosons. Three of the four gauge bosons acquire a positive mass,

while the photon remains massless as the U(1)_emsymmetry remains unbroken. The expansion of the

field f around the vacuum statehfiintroduces four scalar fields: the massless Goldstone bosons

Q_1,2,3and the scalar and neutral Higgs field h(x). As a consequence, the f field expansion around this

minimum is written as:

f(x) = p1 2 ✓ ₀ v+h(x) ◆ . (1.8)

Finally the potential V(f)can be rewritten as:

V(f) = lv2h2(x) lvh3(x) 1

4lh

4

(x), (1.9)

where the second and third terms represent the cubic and quartic self-coupling of the Higgs boson, respectively, from the first term instead the value of the Higgs mass is obtained:

m2h=2lv2= 2µ2 =) lHHH=lHHHH =l= m

2 h

2v2. (1.10)

An important remark is that both Higgs boson self-couplings are directly related to the parameters of the scalar potential and are entirely determined from the Higgs boson mass and the VEV. Their measurement thus represents a test of the validity and coherence of the SM. From a wider perspective, the Higgs boson self-couplings are not comparable to anything else in the SM: the Higgs boson self-interactions are purely related to the scalar sector of the theory in contrast to the weak boson self-interactions, that have a gauge nature. Moreover, the Higgs boson self-coupling are are responsible for the mass of the Higgs boson itself.Their experimental determination is thus crucial to reconstruct the Higgs boson potential and explore the nature of the electroweak breaking symmetry.

The massive gauge vector bosons can be obtained replacing the vacuum expectation value of the f field in the EWK Lagrangian with the addition invariant term due to scalar field part. The achieved masses can be expressed regarding the weak coupling constant g and the vacuum expectation value v as:

m_W = 1

2vg mZ=

mW

cos q_W mg=0, (1.11)

where q_W is known as the electroweak mixing angle, the Weinberg angle. The mixing angle and

the vacuum expectation value are both obtained from the precise experimental measurements of

mWand mZ. Another relation experimentally verified [28], can evaluate the relative strength of weak

nuclear interactions with neutral and charged current: r⌘ m2W

m2_Zcos 2q_W =1.

The coupling between the fermion field and the Higgs field can lead fermions to acquire their masses. This coupling is known as the Yukawa coupling, which after symmetry breaking results in

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the original mass terms proportional to the vacuum expectation value of the Higgs field: mf =

l_fv p

2. The masses of the fermions are parameters in the theory and are not predicted by the SM, and the input is given from experimental values. It is also possible to determine the value of the Higgs vacuum

expectation v by measuring the Fermi constant G_F [29], related to the W boson mass in this way

GF

p

2 =

g

8m2_W. The resulting value of v is [30]:

v2= p1

2GF ⇡ (

246GeV)2. (1.12)

From Eq. 1.9 and Eq. 1.10 it is possible to write the SM Higgs self-couplings:

g_h3 =3mh

2

v gh4 =3

m_h2

v2 . (1.13)

For the couplings to gauge bosons and fermions one should calculate their masses from the Lagrangian

mass termsLmV ⇠m 2 V ⇣ 1+ h_v⌘2,Lmf ⇠mf ⇣

1+h_v⌘and arrive to the Higgs boson couplings to gauge

bosons and the Yukawa coupling to fermions: gh f f = mf v , ghVV = 2 m2V v , hhhVV= 2 m2V v2 . (1.14)

The proportionality to mass in the couplings of gauge bosons and fermions is in agreement with the experimental measurements of Higgs boson properties [31]. In Figure 1.3 there are the Feynman diagrams of the SM Higgs interactions.

H ¯ f f (a) H H Vµ V⌫ (b) H Vµ V⌫ (c)

SM Higgs - Fermions interaction. SM Higgs - Vector bosons interactions.

H H H (d) H H H H (e) SM Higgs - Self interactions.

Figure 1.3:Feynman diagrams of the SM Higgs interactions.

1.1.4 The Higgs Boson: Phenomenology and Experimental Status

The Large Hadron Collider (LHC) at CERN (Section 2.1) has been built with the primary aim of testing the SM theory and discovering new physics beyond Standard Model (BSM). In particular, the discovery and study of the characteristics of the SM Higgs boson are the purposes of the LHC. In the next sections there are outlines of the production mechanisms, of the decay processes and the discovery of the SM Higgs boson by CMS and ATLAS experiments.

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Production Mechanisms and Decay Channels

Since the theory does not predict the SM Higgs boson mass, its production cross-section and its

branching fraction6are calculated as functions of its mass, as shown in Figure 1.4 (a) at the centre of

massps=13 TeV. They are calculated at leading order (LO) considering the QCD and electroweak

corrections at next-to-leading-order (NLO) [32, 33].

(a) [TeV] s 6 7 8 9 10 11 12 13 14 15 H+ X ) [pb] → (pp σ 2 − 10 1 − 10 1 10 2 10 M(H)= 125 GeV L HC HI G G S X S W G 2016 H (N3LO QCD + NLO EW) → pp qqH (NNLO QCD + NLO EW) → pp WH (NNLO QCD + NLO EW) → pp ZH (NNLO QCD + NLO EW) → pp ttH (NLO QCD + NLO EW ) → pp bbH (NNLO QCD in 5FS, NLO QCD in 4FS) → pp tH (NLO QCD, t-ch + s-ch) → pp (b) (c)

Figure 1.4:The SM Higgs boson production cross-sections (a) and the theoretical branching fraction (c) as a function of

m_Hin the full mass range. (b) Higgs boson production cross-section as a function ofps for different production

mechanisms. The coloured bands around the lines show the respective uncertainties, estimated considering both the theoretical and the parametric uncertainties [32, 34].

The most probable production mechanism is gg!H, the gluon fusion (ggF). Because the Higgs

boson does not couple directly to the massless gluons the mechanism is mediated by a loop of quark (Figure 1.5 (a)), dominated by the loop of the most massive quark, the top. The next mechanism is

the vector boson fusion (VBF) qq! V⇤_V⇤ _!_{qqH, with V} ₌_{W, Z and where the Higgs boson is}

produced in association with a jet pair (Figure 1.5 (b)). Despite the cross-section is one order less than the gluon fusion, it can be well discriminated against the background thanks to some characteristics: the two jets have the tendency to be produced along the beam line with a high pseudo-rapidity

|h| < 4, and they are highly energetic with high invariant mass (mjj 700 GeV). In the low mass region other two mechanisms, the associate production with vector bosons (VH) (Figure 1.6) and with heavy quarks (tH or t¯tH) (Figure 1.7), can be used for the SM Higgs discovery. Despite the lower cross-section the first process, also called Higgsstrahlung, has a quite clear signature thanks to the presence of one or two isolated leptons coming from the vector boson. On the contrary, the production in association with heavy quarks, beauty or top, is quite challenging due to the presence of a lot of hadronic jets in the event. Despite this, it is decisive for directly measure the Higgs Yukawa couplings to t/b quarks. Studying the rare VH and VBF production mechanisms allows for probing the Higgs

boson coupling to vector bosons. The cross sections of these production modes as a function ofps are

summarized in Figure 1.4 (b).

The SM theory predicts that the Higgs boson has a mean lifetime of⇡10 22s, too small to be

detected except through its decay products. The inverse of the lifetime is proportional to the theoretical

branching fractions G_H, from this and the partial decay width of H!y it is possible to compute the

branching fraction:

B(H!y) = G(H!y)

G_H (1.15)

As shown in Figure 1.4 (b) also the theoretical branching fractions of the Higgs depend on the mass of the Higgs boson. The Higgs boson tends to decay in the heaviest particles and the particle decay widths are primarily proportional to the masses of the decay particles and of course also to the one of the Higgs itself. In Figure 1.4 (c) it is clear how some decay channels are more favoured than others

in a particular mass region, for example in the low mass region mH 130 GeV the decay H!gg

has a clean signature, because it is produced through a W boson loop, but it is highly suppressed.

In a slightly higher level the decays H ! gg through a top quark loop, H ! ttand H ! cc are

at the same level while the dominant decay is H ! bb (B ⇡ 57%) which is quite challenging to

6_{The cross-section is proportional to the probability that a particular process occurs while the branching fraction is the}

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1 Theoretical Overview p p g t/b g H H (a) p p q W⇤_/Z⇤ W⇤/Z⇤ q0 q q0 H (b)

Figure 1.5:Leading Feynman diagrams for Higgs boson production via gluon-gluon fusion (a) and vector boson fusion (b).

p p q q W⇤_{, Z}⇤ W, Z H (c) p p g t/b g H Z (d) p p g t/b g Z H Z (e)

Figure 1.6:Leading diagrams for the VH production channel. Gluon initiated processes (d,e) are important when looking

for a high-pTHiggs decaying to hadrons.

p p q, g q, g g t, b t, b t, b H p p g g t/b t/b H

Figure 1.7:Leading diagrams for associated production with heavy flavour quark pairs.

identify due to a considerable presence of jets in a hadronic environment but at the same time very important to study the coupling of Higgs with fermions. In the high mass region the prevalent decays

are H!WW and H!ZZ, are amongst the rarest but experimentally advantageous because of the

high signal-to-background (S/B) ratio and the excellent invariant mass resolution. If mH 2mVthere

is the possibility of the decay into real vector boson, below this kinematic threshold it is possible that the Higgs boson decays into one or two virtual gauge bosons, decaying into fermions. Higgs boson

decays branching fractions are summarized in Table 1.1 for a Higgs boson of mass mH=125.09 GeV.

Experimental Searches and Discovery

The search for the Higgs boson started more than 40 years ago, and it has been a great effort for physicists to prove or deny its existence[36]. In the years after its theorisation, in the early 1970s, there were only a few restraints on the existence of the Higgs boson coming from the absence of observation related effects in nuclear physics, neutron stars, and neutron scattering experiments.

In the early planning studies for the Large Electron-Positron Collider (LEP) [37] at CERN, the Higgs boson played no role: it does not appear to be mentioned in any of the reports until 1979 [38]. The first detailed study examining the possibilities of discovering the Higgs boson at LEP appeared only in 1986 [39]. After that, the search for the Higgs boson became firmly established within the LEP

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1.1 The Standard Model Decay mode B[%] H!b¯b 58.09+0.72_0.73 H!W±W⌥⇤ 21.52+0.33_0.33 H!gg 8.18+0.42_0.42 H!t+t 6.27+0.10_0.10 H!c ¯c 2.88+0.16_0.06 H!ZZ⇤ _2.641+0.040 0.040 H!gg 0.2270+0.0047_0.0047 H!Zg 0.1541+0.0090_0.0090 H!µ+µ 0.02171+0.000036_0.00037

Table 1.1:Branching fractions of the main Higgs boson decay modes for a SM Higgs boson of mass

mH=125.09 GeV [35]. Theoretical uncertainties combine the uncertainties on the Higgs boson partial width,

on the value of a_s, and on the quark masses.

program and that is how the search started. The Large Electron-Positron collider operated at CERN

from 1989 to 2000 with a centre of massps between 90 and 209 GeV. The three most important ways

in which electron-positron collisions could lead to the production of a Higgs boson were: the vector boson fusion, the Higgsstrahlung and the exchange of a W or Z boson which emits a Higgs boson along the way. Direct and indirect searches were performed but without any evidence of the presence of the SM Higgs boson. The inclusive results of the search for the Higgs boson at LEP established a

final lower bound of m_H =114.4 GeV at the 95% of confidence level [19, 40].

After the shutdown of LEP, the lead in Higgs searches was taken by the CDF and D0 experiments,

at the Tevatron collider atps = 1.96 TeV, where the dominant production mechanism was

Hig-gsstrahlung in association with a W±_{or Z. They arrived to exclude a range of Higgs masses between}

156 and 177 GeV [41], as well as a range of lower masses in the range excluded by LEP. Since the most

important decay channels for the Tevatron experiments were H!bb and H!W+_{W , which have}

relatively poor mass resolution, the small excess of Higgs candidate event in the mass range between 120 and 135 GeV was not strong enough to be considered as a hint [42].

The discovery of a new particle, the Higgs boson, with a mass around 125 GeV, was announced the 4th July 2012 by ATLAS [43] and CMS [44] collaborations using the data collected at centre of mass

energy of 7 and 8 TeV. Thanks to the clean signature the discovery was performed in the H!gg

and H!ZZ decays, despite the low branching fractions: there was an excess of events of 5 standard

deviations with respect to the background [2, 3]. Three years later, the combination of the ATLAS and CMS results established the currently most precise measurement of the Higgs boson mass to be

125.09±0.21(stat)±0.11(syst.)GeV [45]. The existence of this scalar particle is now firmly established

and further confirmed with the data collected atps = 13 TeV (Run II), as shown in Figure 1.8.

However, the observation of this new particle only represents the first step in the exploration of the

scalar sector. After the end of the Run I, and with the first data atps=13 TeV, it has been possible to

study some properties of the Higgs boson. The measurements of ATLAS and CMS experiments for the mass gave the results shown in Figure 1.9.

Further studies conducted by ATLAS and CMS confirmed that the particle discovered is compatible with the SM Higgs boson, in particular the Higgs boson couplings to other boson and fermions

(H !gg, H!ZZ, H!WW, H!bb and H!tt) are consistent within±1s confidence interval

with the SM predictions. The agreement between the SM prediction and the relative measurement is

described by the signal strength parameter µ. For each production and decay channel i!H! f the

production and decay signal strengths are defined as:

µ_i = si s_iSM µf = Bf Bf ,SM, µ f i =µi⇥µf. (1.16)

The global signal strength measurement, performed assuming the same µ_iand µ_f for each process,

gives as a result of a best-fit value:

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1 Theoretical Overview (GeV) l 4 m 70 80 90 100 110 120 130 140 150 160 170 E vent s / 2 G eV 0 10 20 30 40 50 60 70 (13 TeV) -1 35.9 fb CMS Data H(125) * γ ZZ, Z → q q * γ ZZ, Z → gg Z+X (a) H!ZZ⇤ _!_l+_{l l}0+_l0 _{channel (CMS)} [GeV] γ γ m 110 120 130 140 150 160 weight s - fit ted bkg ∑ 20 − 10 − 0 10 20 30 40 50 weight s / G eV ∑ 100 200 300 400 500 600 700 Data Background Signal + Background Signal Preliminary ATLAS -1 = 13 TeV, 36.1 fb s = 125.11 GeV H m ln(1+ s/b) weighted sum (b) H!gg(ATLAS)

Figure 1.8:(a) Four-lepton invariant mass [46] and (b) photon pair invariant mass [31] distribution in events collected at_p

s=13 TeV.

[GeV]

H

m

123 124 125 126 127 128 129

Total Stat. Syst.

CMS and ATLAS Run 1 LHC _T_o_ta_{l St at .} _Syst. l +4 γ γ CMS + ATLAS 125.09 ±0.24 ( ±0.21 ±0.11) GeV l 4 CMS + ATLAS 125.15 ±0.40 ( ±0.37 ±0.15) GeV γ γ CMS + ATLAS 125.07 ±0.29 ( ±0.25 ±0.14) GeV l 4 → ZZ → H CMS 125.59 ±0.45 ( ±0.42 ±0.17) GeV l 4 → ZZ → H ATLAS 124.51 ±0.52 ( ±0.52 ±0.04) GeV γ γ → H CMS 124.70 ±0.34 ( ±0.31 ±0.15) GeV γ γ → H ATLAS 126.02 ±0.51 ( ±0.43 ±0.27) GeV (a) [GeV] H m 124 124.5 125 125.5 126 126.5

Total Stat. Syst.

Preliminary ATLAS -1 = 13 TeV, 36.1 fb s Tot al St a t. Sy s t . Combined 124.98 ±0.28 ( ±0.19 ±0.21) GeV γ γ → H 125.11 ±0.42 ( ±0.21 ±0.36) GeV l 4 → ZZ* → H 124.88 ±0.37 ( ±0.37 ±0.05) GeV LHC Run 1 125.09 ±0.24 ( ±0.21 ±0.11) GeV (b)

Figure 1.9:In (a) there is a summary of Higgs boson mass measurements from the individual analyses of ATLAS and CMS

and the combined analysis [45]. In (b) the summary of the Higgs boson mass measurements from the individual and combined analyses by ATLAS [31, 47], compared to the combined Run 1 measurement by ATLAS and CMS. The systematic (magenta-shaded bands), statistical (yellow-shaded bands), and total (black error bars) uncertainties are indicated. The (red) vertical line and corresponding (grey) shaded column indicate the central value and the total uncertainty of the combined measurement, respectively.

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1.1 The Standard Model Analogous measurements are performed treating each production signal strength independently, assuming SM branching ratios, and each branching ratio signal strength, assuming SM production cross-sections. The best fit results are reported in Figure 1.10.

(a) (b)

Figure 1.10:Best-fit results for the production signal strength parameters (a) and the decay signal strengths (b) for the

ATLAS and CMS combination. The results of each experiment are also shown [48].

The measurements of the Higgs charge-parity (CP) properties, its spin (JP = 0+), and the

up-per limit on the Higgs decay width G_h, are also consistent with the predictions of the SM [49–54].

The G_hpredicted is almost equal to 4 MeV, which is significantly lower than the experimental mass

resolution, so upper limits of G_h  22 MeV have been set at the 95% of confidence level by both

experiments [53, 54].

One of the most significant achievements of the LHC Run 1 is represented by the discovery of the Higgs boson and the measurement of its properties. The characterisation of the Higgs is however far from has been complete, and improvements are expected already by the end of the second Run of the LHC. A precision measurement of other parameters in the Higgs sector is a further check of the predictions of the SM. In addition to the Higgs mass and the couplings to the fermions and other bosons one would like to measure the Higgs self-coupling (Eq. 1.13), accessible in multi-Higgs production processes. With its mass now known with precision, the value of the Higgs self-coupling

can be computed from Eq. 1.10 to be l_HHH ⇡ 0.13, completely determined in the SM from m_H

and v. Experimentally the measure of l_HHHwould provide a test of the consistency of the SM. As

this coupling is responsible for the Higgs boson mass itself, it is related to the very fundamental

properties of the EWSB and of the BEH mechanism. The l_HHHcoupling can be directly probed in

Higgs boson pair (HH) production. Similarly, the measurement of the quadrilinear coupling lHHHH,

a further probe of the BEH potential, requires the study of triple Higgs final state. The production

of the latter is however extremely rare in the SM, with a cross-section of about 80 ab atps=14 TeV,

out of the experimental reach of the LHC. In contrast, HH production, although challenging, can be

experimentally probed at the LHC. The direct determination of l_HHHfrom HH production is thus an

essential step in the understanding of the BEH mechanism and, for this reasons, it represents one of the main goals of the LHC physics programme.

Observation of Higgs pair production is a significant long-term objective of the LHC physics program as it will shed light on the scalar potential of the Higgs field and the nature of EWSB. However, the data collected so far at LHC by ATLAS and CMS are insensitive to the self-coupling in the SM [55–57], because of the expected small signal rates and large backgrounds. The double Higgs

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production, though enhanced in the Run 2 because of the higher centre of mass energy, is expected to be accessible only at higher integrated luminosities (HL-LHC) [58].

On the other hand, the available measurements do not exclude the possibility that the discovered Higgs boson belongs to the Higgs sector predicted by models beyond the SM (BSM). Various models that consider resonant and nonresonant double Higgs production mechanisms are described in Section 1.2. Some parts of the parameter spaces of these models, which was not yet excluded from the previous measurements, are accessible with the data collected by LHC during 2016 Run.

1.1.5 Higgs boson pair production in the SM

The measurement of the Higgs boson pair production cross-section can be useful for the extraction

of the value of the trilinear coupling l_HHH7. However, among all the possible interactions the l_HHH

coupling represents only one contribution to HH production. At the leading order, a pair of on-shell Higgs bosons can be produced in the final state of a collision through any of the following diagrams:

H H H V V H H t H H V H H

From different combinations of these interactions, where the l_HHHcontribution is entangled to

other effects, it is possible to derive the main mechanisms for Higgs boson pairs production at hadron colliders. Here they are listed in decreasing order of their cross-section, with some representative Feynmann diagrams illustrating the Higg boson couplings involved:

• gluon fusion production gg!HH

It involves both the production of a Higgs boson pair through the trilinear Higgs boson self-coupling and the radiation of two on-shell Higgs bosons froma a heavy quark loop. The

cross-section thus depends on l_HHHand on the top quark Yukawa coupling8.

g t g H H H g t H H g

• vector boson fusion (VBF) production qq0 !jjHH

This process depends not only on the Higgs boson trilinear coupling, but also on the quadriliner coupling of a Higgs boson pair to a vector boson pair as well as on the single Higgs boson coupling to vector bosons. Its cross-section is one order of magnitude smaller than the gluon fusion one, but the two final state jets provide a clean signature useful to discriminate signal events from background.

q0 q V⇤ V⇤ H H q0 q q0 q H H q0 q q0 q H H q0 q

7_{The importance of the trilinear coupling was highlighted since the first computation of the cross section, before the Higgs}

boson discovery itself.

8_{The contribution from b quarks is smaller than 1% at the leading order and can be neglected given the current accuracy of}

the theoretical computations and the experimental sensitivity. 16

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• top quark pair associated production qq0/gg!t ¯tHH

It is a HH variant of the single Higgs boson pair production in association to a top quark pair (ttH), where the two Higgs boson are produced from the Higgs boson self-coupling or are both radiated from the top quarks. Its cross-section exceeds the one from VBF HH production at high transverse momenta of the HH pair and for high centre-of-mass energies.

g g H H t ¯ t ¯q q g H H ' q¯0 q g H H

• vector boson associated production qq0 !V HH, V =W±, Z

It involves the same Higgs boson coupling as VBF production, but now in the final state an on-shell vector boson is present.

¯ q0 q V⇤ H V H H _¯ q0 q V⇤ V H H ¯ q0 q V⇤ V H H

• single top quark associated production qq0!tjHH

It can proceed through either t- or s-channel as illustrated in the top and bottom row of the diagrams below, respectively. It is the only process contemporary sensitive to the HH couplings to vector boson and to top quarks and their relative phase. However, its cross-section is so small that it can hardly be investigated at the LHC, but could be studied in a future higher energy collider. q W b q0 t H H H q W b q0 t H H q W b q0 t H H H q W b q0 t H H q W b q0 t H H q q0 W b t H H H q q0 W b t H H q q0 W b t H H q q0 W b t H H q q0 W b t H H

The cross sections of these production mechanism at different centre-of-mass energies are graphically compared in Figure 1.11. As long as HH production is very rare at LHC, experimental searches focus on

the dominant gluon fusion production (s_HH(ggF) =33.49 fb forps=13 TeV and mH =125.09 GeV)9

[59]. Nevertheless there is an interest on VBF HH production to have access to the VVHH interaction, which is currently unexplored.

The two leading order mechanisms in the gluon fusion production have amplitudes of the same order of magnitude, but they interfere destructively. This is the reason why the cross-section is unfortunately small, but at the same time this makes HH production extremely sensitive to BSM physics. The destructive interference might be altered by BSM physics contributions, producing large modification that can be probed with the current LHC data. Thus the HH production can be seen both as a test of the SM and as a probe of BSM physics.

9_{Whenever not ambiguous,in the following the symbol s}

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1 Theoretical Overview 10-3 10-2 10-1 100 101 102 103 104 8 1314 25 33 50 75 100 σNLO [f b] √s[TeV]

HH production at pp colliders at NLO in QCD

MH=125 GeV, MSTW2008 NLO pdf (68%cl) pp→HH(EFT loop-improved) pp→HHjj (VBF ) pp→ttHH pp→WHH pp→tjHH pp→ZHH MadGraph5_aMC@NLO

Figure 1.11:Total cross-section for HH production in pp collisions for different production modes [60]. The cross sections

are computed at the next-to-leading order accuracy and the bands shown the linear combination of the theoretical errors on the scale and parton distribution uncertainties.

1.2 Beyond the Standard Model

The SM has a huge success in answering many questions and describing the strong and electroweak interactions as well as the Higgs mechanism and its properties. However, while the SM has been extensively tested at the energies accessible by the present experiments, its validity for the energy scales above TeV is an open question [61]. There are fundamental physical phenomena in nature, which the SM fails to explain and there are many theoretical problems, which have no solution in the SM. Some of these open topics are:

• It does not describe one of the four known forces in nature, the gravity. It is not possible to merely add a gravity force mediator, the so-called graviton, and be consistent with experimental observations without considerable modifications to the theory.

• It does not explain the baryogenesis [62] and it does not include mechanism that could describe the dark energy and dark matter phenomena, which combined contribute to around 95% of the mass of the universe [63, 64].

• SM assumes massless neutrinos, while neutrino oscillation experiments have shown the contrary [13, 65]. It is also unclear if the neutrino masses arise in the same way as for other fundamental particles.

• Observations have shown a big asymmetry between matter and antimatter [66], while in the SM a much smaller amount is predicted.

• It is based on the symmetry group SU(3)_C⌦SU(2)_L⌦U(1)_Y which does not provide the

unification of the gauge coupling constants below the Planck scale, the so-called unification problem.

• It cannot provide an explanation for the existence of three families of fermions, identical under all aspects but for their coupling with the Higgs boson.

• The mass of the Higgs boson is not predicted by any fundamental symmetry of the SM and it is subject to quadratically divergent radiative corrections, which cancellation require a fine tuning of other parameters.

It exist many Beyond Standard Model (BSM) theories that propose the solutions to some (or even to all) of the above-listed problems of the SM framework. It is natural in this context to think that the

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1.2 Beyond the Standard Model SM is only the manifestation of a more extended theory beyond it.

As already specified at the end of the previous Section (see Section 1.1.4) the current measurements do not exclude the possibility that the discovered Higgs boson belongs to some BSM scalar sector. This is the primary motivation for the experimental data analysis presented in this thesis: HH production is both a probe for BSM physics, and a tool to discriminate between possible alternatives. If the scale of BSM physics can be reached by the LHC, new states can be directly produced and subsequently decay to a HH pair. The BSM theories considered in this thesis can be divided into two categories: ones that predict existence of a heavy resonance that decay into a pair of Higgs bosons (resonant HH production, Section 1.2.2) and effective filed theories (EFT) that model enhancement of the nonresonant

HH production through anomalous couplings of Higgs boson (Section 1.2.3).

1.2.1 Supersymmetry

The Higgs mechanism may be realised with an arbitrary number of scalar fields, but as shown in Section 1.1.3 in the Standard Model the most straightforward case is considered by introducing only

one scalar field. A theory called supersymmetry (SUSY)10claims that in Nature there could be more

than one scalar fields. On the one hand, SUSY solves most of the issues that the Standard Model leaves open. On the other hand, SUSY is more appealing than the SM from the point of view of mathematical consistency, because, as was proved by Haag, Sohnius and Lopuszanski, the supersymmetry algebra is the only graded Lie algebra of symmetries of the S-matrix consistent with relativistic quantum field theory [69]. Until now there has been no experimental evidence for SUSY, but there is still a significant part of the SUSY parameter space that is not yet excluded by the experimental measurements. Supersimmetry relates and transforms a half-spin particle to an integer spin particle, and vice versa. Irreducible representations of the supersymmetry algebra are called supermultiplets. Each supermul-tiplet is composed of both fermion and boson states, which are commonly known as superpartners of each other, and contains an equal number of fermion and boson degrees of freedom. Fermions may only have scalar superpartners because their left-handed and right-handed parts transform differently under the gauge group, and therefore they are represented only within the chiral supermultiplets. They are indeed organised in superfields of two types: chiral, a combination of a complex scalar field

f_iwith a left-handed Weyl fermion y_i; and gauge, a combination of a spin-1 gauge boson and its

spin-1/2 superpartner (gauginos).

The possible form of a renormalizable Lagrangian has constraints and restrictions from invariance under the supersymmetry transformations, so the part that corresponds to the interactions of chiral supermultiplets should be written in the following general form:

DLint= 1₂(Wijyiyj+Wij⇤y†iy†j) WiWi⇤, (1.18)

where W is the superpotential of the scalar field f and:

Wi= dW

df_i W

ij₌ d2W

df_idf_j. (1.19)

The form of the superpotential W for a SUSY theory without gauge singlet chiral supermultiplets is:

W= 1₂Mijf_if_j+1₆yijkf_if_jf_k (1.20)

where Mijis the symmetric mass matrix for the fermion fields and yijk is the symmetric tensor of

Yukawa couplings of a scalar f_kand two fermions f_if_j . The presence of superpartners cures the

quadratic divergences of the Higgs mass squared and, if the supersymmetry is an exact symmetry, the masses of the particles and their superpartners would be the same. The only freedom present in this theory is the superpotential W which gives form to the scalar field potential and the Yukawa interactions between fermion and scalar fields [61]. Since no experimental evidence for a scalar particle with the same mass of the known fermions has been found, the SUSY cannot be an exact symmetry. The symmetry has to be broken in a way that the hierarchy problem is not reintroduced and that the gauge

10_{A detailed description of SUSY could be found in Supersymmetry and supergravity by Wells and Bagger [67] or in A}

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invariance and the renormalizability of the theory are preserved. There is no commonly accepted dynamical way to break SUSY; this can be done introducing terms manually in the Lagrangian that break the symmetry to construct an effective low-energy SUSY model. The Minimal Supersymmetric Standard Model (MSSM) is one such model, which extends the SM by introducing the minimal number of new particles needed to be compatible with supersymmetry [61, 68]. Without fine tuningof the model, the MSSM requires that the “superpartners” of the SM particle have masses below TeV scale.

1.2.2 Resonant BSM HH Production

There are BSM physics models predicting the existence of a new resonance X of mass mX 2mH

that couple significantly to the Higgs boson. To probe these models it is possible to use final state with a Higgs boson pair. In the search carried out for this thesis, a model-independent approach has been performed considering a narrow width resonance approximation. The model-indepentent representation of the results provide a general and efficient way to explore a broad class of BSM physics models.

In this section some of this BSM physics scenarios are presented, focusing the attention on the models which predict a resonant HH production and that can simultaneously be probed, despite their different

theoretical assumptions11. The profound relation between HH production and the properties of the

BEH mechanism suggest that resonant HH signatures can appear in models with an extended scalar sector, or in models with warped extra dimensions that alter the relation between the Higgs field and the matter fields.

The Minimal Supersymmetry Model

MSSM is one of the simplest extension to SM in the supersymmetry context and it is based

on the group SU(3)C⌦SU(2)L⌦U(1)Yand on the R-parity conservation. The supersymmetry is

broken by explicitly adding mass terms and trilinear couplings for scalar particles without introduc-ing other quadratic divergences (soft-SUSY-breakintroduc-ing) but introducintroduc-ing a huge number of unknown parameters (105). These parameters can be reduced making assumptions like no CP-violation and

no flavour-changing neutral currents (FCNC)12 at three level. To describe the phenomenological

properties only a constrained part of the parameters phase space is considered. This allows defining the so-called benchmark scenarios, to test the predictions of the model and to apply experimental constraints. The benchmark scenario considered in this thesis will be presented in the following. The lepton and baryon numbers are conserved by imposing the conservation of a discrete symmetry called R-parity:

R_p= ( 1)2s+3B+L (1.21)

where s, B and L are, respectively, the spin, the baryon and the leptonic quantum numbers.

The R-parity is equal to +1 for the SM particles and 1 for their supersymmetric partners. Its

conservation leads to the SUSY particles to be produced in pairs and that the lightest SUSY particle is stable.

In the MSSM the spin-1 gauge bosons and their spin-1/2 partners, the gauginos, are in vector supermultiplets. Only three generations of spin-1/2 quarks and leptons are present, which belong to chiral superfields together with their spin-0 superpartners, the squarks and sleptons. Finally two

chiral superfields ˆH₁Hˆ₂with hypercharges equal to 1 and+1 respectively are needed to delete the

quadratic divergences, and their scalar components H₁and H₂give masses to fermions while the two

doublets fields lead to five Higgs particles, as will be described later. A summary is in Table 1.2. The SUSY superpotential W (Eq. 1.20) depends only on the superfields, but not on their conjugates. The form of the SUSY Lagrangian (Eq. 1.18) implies that in SUSY, one scalar field cannot provide masses for both 1/2 and -1/2 isospin fermions without explicitly breaking the SUSY symmetries.

11_{Depending on the model, the mass of the resonance can range from the kinematic threshold of 250 GeV up to several TeV.}

Experimentally, this requires the development of complementary analysis methods and the analysis of several final states to ensure a high acceptance over the entire mass range

12_{The flavour-changing neutral currents (FCNCs) represent the transition of a fermion which changes its flavour and not the}

charge. These processes are suppressed at tree-level in the SM and as well at higher orders by the Glashow-Iliopoulos-Maiani (GIM) mechanism [70].

Optimisation of a multivariate analysis technique for the ttbar background rejection in the search for Higgs boson pair production in bbtautau decay channel with the CMS experiment at the LHC

Universit`a degli Studi di Pisa

DIPARTIMENTO DI FISICA “E. FERMI”

Tesi di Laurea Magistrale in Scienze Fisiche

Curriculum “Fisica delle Interazioni Fondamentali”

Aprile 2018

Optimisation of a multivariate analysis technique

for the t¯t background rejection in the search

for Higgs boson pair production in b¯

b⌧

+

⌧

decay channel

with the CMS experiment at the LHC

Candidato

Angela Giraldi

Relatore

Dott. Giuseppe Bagliesi

Correlatore

Abstract

Contents

Introduction

Chapter

1

Theoretical Overview

T

1.1 The Standard Model

1.1.1 Fundamental Particles and Interactions

u

d

e

⌫

e

c

s

µ

⌫

µ

t

b

⌧

⌫

⌧

W

Z

g

H

1

2

3

1.1.2 Standard Model as Gauge Theory

1.1.3 Spontaneous Symmetry Breaking in the Brout-Englert-Higgs Mechanism

1.1.4 The Higgs Boson: Phenomenology and Experimental Status

1.1.5 Higgs boson pair production in the SM

1.2 Beyond the Standard Model

1.2.1 Supersymmetry

1.2.2 Resonant BSM HH Production