Investigation of the hadronic tau substructure and its application to the study of the CP properties of the Higgs boson with the ATLAS experiment at CERN LHC

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UNIVERSITÀ DEGLI STUDI DI PISA

Facoltà di Scienze, Matematiche, Fisiche e Naturali

Corso di Laurea in Fisica

Investigation of the hadronic tau substructure

and its application to the study of the CP

properties of the Higgs boson with the ATLAS

experiment at CERN LHC

Advisor

Prof. Vincenzo CAVASINNI

Candidate

Francesco LUCARELLI

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Nomenclature

Abbreviations and acronyms

2HDM Two-Higgs-Doublet Model

ALICE A Large Ion Collider Experiment

ATLAS A Toroidal LHC Apparatus

BDT Boosted Decision Tree

BEH Brout–Englert–Higgs

BSM Beyond Standard Model

CERN Conseil Européen pour la Recherche Nucléaire (European Organization for Nuclear Research)

CKM Cabibbo-Kobayashi-Maskawa

CMS Compact Muon Solenoid

CR Control Region

CS Central Solenoid

CSCs Cathode Strip Chambers

DAQ Data Acquisition System

DM Decay Mode

EF Event Filter

EMCal Electromagnetic calorimeter

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viii Nomenclature

FCal Forward calorimeter

FCNCs Flavour-Changing Neutral Currents

ggF gluon-gluon Fusion

HadCal Hadronic calorimeter

HLT High-Level Trigger

IBL Insertable B-Layer

ID Inner Detector

IP Impact Parameter

L1 Level 1 (Trigger)

LAr Liquid Argon

LHC Large Hadron Collider

LS1 Long Shutdown 1

MC Monte Carlo

MS Muon Spectrometer

MTDs Monitored Drift Tubes

NLO Next to Leading Order

NNLO Next to Next to Leading Order

OS Anti-ID CR Opposite Sign Anti ID Control Region

OS Opposite Sign

PDF Parton Density Function

PDG Particle Data Group

PMT Photomultiplier

PS Proton Synchrotron

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Nomenclature ix

QED Quantum Electrodynamics

RF Radio Frequency

ROS Readout System

RPCs Resistive Plate Chambers

SCT Semiconductor Tracker

SM Standard Model

SPS Super Proton Synchrotron

SR Signal Region

SS Anti-ID CR Same Sign Anti ID Control Region

SS ID CR Same Sign ID Control Region

SS Same Sign

TDAQ Trigger and Data Acquisition System

TGCs Thin Gap Chambers

TileCal Tile Calorimeter

TRT Transition Radiation Tracker

t ¯tH top-quark pair associated production

VBF Vector Boson Fusion

VEV Vacuum Expectation Value

VH W /Z associated production

ZMF Zero Momentum Frame

Symbols

C Charge conjugation

η Pseudorapidity (ATLAS coordinate system)

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x Nomenclature

MT Transverse mass

P Parity

φ Azimuthal angle (ATLAS coordinate system)

ϕCP Angle between the tau decay planes in h → τ τ

ϕ∗_CP Observable of the Higgs CP analysis

φτ CP-mixing angle

pT Transverse momentum

√

s Centre of mass energy

τhad Hadronically decaying tau

τhad−vis Visible part of hadronic tau decays

τlep Leptonically decaying tau

τ_vis Visible part of tau decays

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Introduction

The Higgs boson, discovered by the ATLAS and CMS collaborations at the Large Hadron Collider in 2012 [1, 2], is a fundamental ingredient in the Standard Model of particle physics: its existence, predicted in 1964 by Peter Higgs, François Englert and Robert Brout [3, 4], is needed to explain the mass of the gauge bosons and to retain the principle of gauge invariance within the theoretical framework.

After its discovery, much effort was made in measuring its properties. In the following I will focus on the CP quantum numbers of this particle, that is the topic of this thesis. The Higgs boson is predicted by the Standard Model to be a CP-even scalar particle, i.e. to have quantum numbers JP C= 0++. Alternative hypotheses to the Standard Model concerning pure CP-eigenstate Higgs bosons have been tested in the bosonic sector (h → ZZ∗, h → W W∗) and excluded at more than 99.9% confidence level by the analysis of the ATLAS and CMS collaborations [5, 6].

Nevertheless, the possibility still persists that the discovered Higgs boson is an admixture of a scalar (CP-even) and a pseudoscalar (CP-odd) component through a CP-mixing angle φτ. Such a hypothesis is supported by several theories, like the Two-Higgs Doublet Models

[7], that involve a second Higgs doublet for a total of five real scalar Higgs fields. Among them, three fields are neutral and can be identified as pure CP eigenstates, that do not necessarily coincide with the mass eigenstate. If this is the case, i.e. the Higgs mass eigenstates are a linear combination of the CP eigenstates, one ends up with a CP-violating model [7, 8].

The possible discovery of the CP violation in the Higgs sector due to the scalar/pseudoscalar mixing would open incredible perspectives of new physics and extensions of the Standard Model should be seriously taken into account. Furthermore such a mixing would help in explaining some of the currently unanswered questions in physics: it is well known in fact that the CP-violating parameters of the Standard Model are not enough to explain the baryon asymmetry observed in the Universe and that a new source of CP violation is needed [9–11]. Finally, it is important to remark that the CP properties of the Higgs boson have been studied so far only in the bosonic couplings (h → ZZ∗, h → W W∗), that are not sensitive at tree

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2 Introduction

level to a possible CP-odd component [12, 13]. For this reason an analysis carried out in the fermionic decay channel h → τ τ , as the one presented in this thesis, is crucial to confirm the properties of the Higgs boson or to investigate alternative hypotheses to the Standard Model.

The analysis of the tau leptons is one of the most sensitive for this purpose, since the h → τ τ decay has the second largest branching ratio in the fermionic sector of the Higgs decays and since the adoption of a new algorithm at the beginning of Run 2 improved the reconstruction performances of the hadronically decaying taus [14]. A detailed description of this algorithm, called Tau Particle Flow, and its performances is one of the preliminary tasks of this thesis and will be widely discussed in chapter 3.

The purpose of my thesis is to investigate the possibility of measuring the CP mixing angle φτ in the h → τhadτhad channel (where the taus are required to decay hadronically)

with the data collected by ATLAS in 2016 and in view of the future performances of High-Luminosity LHC [15]. The thesis is structured as follows: in chapter 1 a theoretical introduction to the Standard Model of particle physics and to the Higgs boson is given. Some details about the Higgs boson discovery in 2012 and the status of the h → τ τ analysis are also provided. Particular emphasis is put on the description of the Two-Higgs Doublet Models, that predict a possible mixing of the neutral Higgs bosons with the consequent violation of the CP symmetry.

The description of the Large Hadron Collider, the ATLAS detector and the coordinate system is given in chapter 2. Every component of the ATLAS detector and its performances are discussed in details.

The last two chapters are dedicated to my personal contribution to the analysis: chapter 3 is focused on the Tau Particle Flow algorithm used in the reconstruction of the hadronically decaying taus. The algorithm is designed to improve the angular and energy resolutions and to classify five different tau decay modes through the identification of the individual hadrons of the tau decays. I studied the resolutions and the classification purity provided by the algorithm and I evaluated the separation power of the variables used in the tau identification. Furthermore, to test the performances of the Tau Particle Flow I used data and Monte Carlo simulations to analyse the resonances of the ρ and a1 mesons in some of the tau decay

channels, with particular interest to the former since it is a key feature in determining the CP properties of the Higgs boson.

Finally, chapter 4 is devoted to the analysis of the Higgs CP-mixing angle. A theo-retical explanation of the investigated phenomenon is provided with the description of the experimental procedure to determine the observable and the mixing angle. The experimental observable is reconstructed with two different methods and a combination of them, depending on the tau decay modes. I performed a comparison of these methods to show the sensitivity

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Introduction 3

of each of them to the parameter of interest. Then I built and tested a data-driven algorithm to estimate the QCD background, since Monte Carlo simulations are not reliable in this case, that I used to realize the observable distributions using the data collected in 2016. In the last part of chapter 4 I used the information from 2016 data and Monte Carlo to derive the expected sensitivity to the mixing angle up to 1000 fb−1, which will be the luminosity collected in a three-year run of High-Luminosity LHC.

The large amount of available data at the end of Run 2 and the improved performances of tau reconstruction provided by the Tau Particle Flow algorithm are expected to open up perspectives of significant results in the analysis of the Higgs quantum numbers. This thesis is intended to provide an overview of the tau analysis at ATLAS and the experimental setup used to determine the CP-mixing angle of the Higgs bosons, with a particular interest to the expectations of the future performances of High-Luminosity LHC.

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Chapter 1 The Standard Model of Particle Physics

The present understanding of the structure of matter and the subatomic laws that govern the Nature has been a long process begun in the 1930s and incorporated into a coherent and satisfying model in the 1970s: the Standard Model (SM) of particle physics and fundamental interactions.

The SM describes both matter and forces in terms of fields and their quanta of excitation, represented by elementary particles. Furthermore, the SM has been developed within a successful context of gauge invariance which predicts the carriers of the interactions themselves to be particles. Masses are provided to particles by the Brout-Englert-Higgs mechanism, based on the spontaneous symmetry breaking, that introduces a new field in the model and hence a new particle, the Higgs boson, whose existence was experimentally confirmed in summer 2012 by ATLAS [1] and CMS [2] experiments.

The SM contemplates three fundamental interactions: the strong, the weak and the electromagnetic (EM) force. However, the last two interactions have been unified in a single model, the electroweak model, in 1970s. The fourth and last fundamental interaction, gravity, has not been described by a quantum field theory and has not been encapsulated in the SM so far.

The SM has provided successful explanations of several phenomena and has been tested at a very precise level. At present it is the most successful mathematical model developed in the attempt to describe the Nature at subatomic scale.

1.1 The Principle of Gauge Invariance

The symmetry of a quantum field theory is strictly related to the invariance of that theory under certain transformations. These transformations act on the fields and are called global if they involve the fields in the same way at all space-time points, or local if they act differently

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6 The Standard Model of Particle Physics

on different space-time points. Since global transformations imply the correlation of points of the space-time that are not causally connected, theories are generally required to be invariant under local transformations.

As an example, the Dirac Lagrangian that describes the field ψ of a fermion of mass m is

L = ψiγµ∂µψ − mψψ (1.1)

where γµ are the Dirac matrices [16]. This Lagrangian is clearly invariant under the global U(1) symmetry, i.e. under the field transformation

ψ(x)−−→ ψU(1) ′(x) = eieθψ(x) (1.2) where e and θ are real numbers. When the symmetry is required to be local, that is when the parameter of the transformation θ is a function of the space-time θ(x), one finds that a new field must be introduce to preserve the symmetry. The new field describes a new type of particle that can be interpreted as the mediator of the interaction related to the symmetry [17]. This is the content of the gauge principle, and theories based on it are called gauge theories. All the quantum field theories of the SM are gauge theories: in particular the Quan-tum Electrodynamics (QED), the quanQuan-tum field theory of the electromagnetic interaction described in section 1.3, offers a simple example of the application of the gauge principle.

1.2 Particles and Fields

A particle is elementary if it has no substructure, i.e. if it is not composed of other particles. In the SM elementary particles are described as excitations of the respective fields and are divided into two main categories: fermions and bosons (figure 1.1). Every particle of the SM has an antiparticle, which has exactly the same characteristics but opposite quantum numbers (such as charge, isospin, weak isospin, parity, etc.). Quantum numbers are properties of the particles that uniquely describe their state.

1.2.1 Fermions

Fermions are elementary spin-½ particles comprising leptons and quarks. The difference between these two categories of fermions is based on the interactions they undergo and hence the type of matter they produce:

• A lepton interacts solely via the EM and weak forces and can be found free or loosely bound, such as the electron in an atomic nucleus.

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1.2 Particles and Fields 7

• A quark undergoes all the fundamental interactions, including the strong force, and cannot be found free but only within bound states called mesons (quark-antiquark pairs, q ¯q) or baryons (three-quark states, qqq). Mesons and baryons are grouped together in a category of particles called hadrons. Each quark is characterized by a different flavour, a quantum number that is conserved by the strong and EM interactions but is not by the weak interaction.

Figure 1.1 Particles of the SM [18], divided into fermions and bosons. Fermions are further split into three generations of quarks and leptons, one for each column of the figure.

Both leptons and quarks are organised in three generations: the first generation, composed of the electron, the electron neutrino, the u quark and the d quark, are the constituents of the ordinary matter. Couplings between quarks of different generations are allowed by a mixing mechanism, parametrised by the Cabibbo-Kobayashi-Maskawa (CKM) matrix:

   d′ s′ b′   =    V_ud Vus Vub V_cd Vcs Vcb V_td Vts Vtb       d s b    (1.3)

d′, s′and b′are eigenstates of the weak interaction and are a linear combination of d, s and b, that are mass eigenstates. Table 1.1 shows the three generations of leptons and quarks and summarises their main properties.

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Fermions Generation Charge Interactions 1st 2nd 3rd Leptons νe νµ ντ 0 Weak e− µ− τ− −1 EM and weak u c t +2/3 Strong Quarks EM d s b −1/3 Weak

Table 1.1 Fermions of the SM and their properties. Leptons interact solely via the weak and electromagnetic forces whereas quarks undergo all the interactions.

1.2.2 Bosons

The bosons of the SM are integer spin particles that arise naturally from the principle of gauge invariance, and are therefore commonly referred to as gauge bosons. The global symmetry underlying the SM is SU(3)×SU(2)×U(1):

• The SU(3) component is the Quantum Chromodynamics (see section 1.4), which describes the strong interaction and is mediated by particles called gluons.

• The SU(2)×U(1) component is the unified electroweak model, which provides a good description of the EM and the weak interactions (see section 1.6). The former is mediated by photons, the latter by massive particles denoted as Z and W .

Interaction Strength Boson EM charge Mass [GeV1]

Strong 1 Gluon g 0 0

EM 10−3 Photon γ 0 0

Weak 10−8 W boson W

± _±1 _80.4

Z boson Z 0 91.2

Table 1.2 The table summarises the bosons of the SM and their properties, providing infor-mation on the interaction they carry and its relative strength [16]. The mass of the Z and W bosons are taken from reference [19].

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1.3 Quantum Electrodynamics 9

The list of the gauge bosons and the interaction they mediate is shown in table 1.2. The Higgs particle completes the family of the bosons of the SM: its existence is crucial to provide mass to particles (especially the gauge bosons W and Z) while keeping the gauge invariance of the theory. More details about the Higgs boson are provided in sections 1.7 and 1.8.

1.3 Quantum Electrodynamics

The QED is the quantum field theory that describes the interactions of the fermions and the EM field. The Dirac equation for a free fermion of mass m is

(iγµ∂µ− m)ψ = 0 (1.4)

where ψ is the four-component Dirac spinor representing the femionic field of a spin-½ particle.

The corresponding Lagrangian is provided in equation 1.1; as stated before the gauge principle requires the Lagrangian to be invariant under local U(1) symmetry. This local invariance is guaranteed by the replacement of the common derivative ∂µ with the covariant

derivative Dµ:

∂µ−→ Dµ= ∂µ+ ieAµ (1.5)

where Aµ is a new quantized field that is introduced in the theory and transforms as

Aµ−→ A′µ= Aµ− ∂µθ(x) (1.6)

The replacement 1.5 allows the Lagrangian 1.1 to be locally gauge invariant but introduces an extra term that can be interpreted as the interaction of the fermionic field ψ with the gauge field Aµand that accounts for the EM coupling:

Lint= −eAµψγµψ (1.7)

The field Aµdescribes the mediator of the EM force (the photons), while the constant e is

related to the strength of the interaction and corresponds to the electric charge of the fermion. To complete the Lagrangian of the EM interaction it is necessary to add a term that takes into account the kinetic energy of the electromagnetic field. This term involves the gauge-invariant EM field strength tensor Fµν, defined as Fµν = ∂µAν− ∂νAµ, and is

L_photons= −1 4F

µν_F

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The complete Lagrangian of the QED is then

LQED= ψ(iγµ∂µ− m)ψ | {z } Free particle − eψγµψAµ | {z } Interaction term − 1 4F µν_F µν | {z } EM kinetic term (1.9)

1.4 Quantum Chromodynamics

The strong force is described in the SM by the Quantum Chromodynamics (QCD): the name of such interaction derives from the quantum number involved, the colour, that can assume three different values (commonly referred to as blue, red and green), unlike the EM charge that consists only of two possible values. Quarks are the only fermions that carry a unit of colour, while antiquarks carry a unit of anticolour. The gluons, mediator of the strong interactions, carry a unit of colour and a different type of anticolour, so they are allowed to interact among themselves (unlike photons, that do not have EM charge). Colour is conserved in the strong interactions.

As already mentioned in section 1.2.1, quarks are not free in nature but bound in states called hadrons, i.e. mesons or baryons, that are globally colourless according to the principle of colour confinement. The hadronic structure was investigated in deep inelastic scattering experiments [20] and the emerging picture is that of composite particles made up not only by valence quarks, i.e. bound quarks that carry the quantum numbers of the hadron, but also by quark-antiquark pairs (sea quarks) and gluons. The constituents of the hadronic particles are generally called partons. Since these partons frequently interact inside the hadronic structure they do not carry a fixed value of momentum but only a fraction of the hadron momentum. The probability of finding the i-th parton with four-momentum fraction between a value x and x + dx is expressed in terms of a Parton Density Function (PDF) [21].

Hadronic interactions at high-energy colliders are basically parton-parton interactions. Thus, as a consequence of the colour confinement, neither quarks nor gluons can emerge from this hard scattering as isolated particles but they rather undergo an hadronisation process resulting in a cascade of hadrons called jet.

Formally the QCD is based on the SU(3) symmetry group: like the EM interaction, the strong interaction is simply derived by the request of invariance of the theory under the local gauge transformation

ψf SU(3)

−−−→ ψ′_f = eigsTaθa(x)_ψ

f (1.10)

where ψf is the quark field (the subscript f stands for the quark flavour), gs is the strong

coupling constant, θa(x) are eight local parameters (a is an index running from 1 to 8) and Ta

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1.5 Weak Interaction 11

fabc are the structure constants and a possible representation of the Ta is given by the

Gell-Mann matrices λa: Ta= λ₂a [16].

Since there are 6 different flavours of quarks the free Lagrangian is similar to the one expressed in equation 1.1 but summed over the quark flavours:

Lquarks=

∑

f

ψ_f(iγµ∂µ− mf)ψf (1.11)

The gauge invariance requires the replacement of the derivative ∂µ with the covariant

derivative

Dµ= ∂µ+ igsTaGa,µ (1.12)

where Ga,µ are the eight gluon fields, one for every coloured combination of colour and

anti-colour. Taking into account the Lagrangian of the gluon field

Lgluons= − 1 4G µν a Ga,µν (1.13) where Gµνa = ∂µGνa− ∂νG µ

a− gsfabcGµ_bGνc and grouping all the terms, the resulting

La-grangian of the QCD is then

LQCD= ψf(iγµ∂µ− m)ψf | {z } Free particles − gsψfγµψfTaGa,µ | {z } Interaction term − 1 4G µν a Ga,µν | {z }

Gluon kinetic term

(1.14)

1.5 Weak Interaction

The weak interaction, unlike the other fundamental interactions, affects all leptons and quarks and it is the only interaction that can change the quark flavours.

It is mediated by two bosons, W± and Z, the former being responsible for charged weak current and the latter for neutral weak current. Since they arise from a gauge symmetry, such as photons and gluons, both W±and Z are expected to be massless: a non-vanishing term of mass in the Lagrangian of the weak interaction would theoretically break the gauge invariance. Nevertheless, they were found to be massive mediators [22, 23]. How these particles acquire a mass while leaving the general framework of gauge theories unchanged is explained by the Brout–Englert–Higgs (BEH) mechanism [3, 4] described in section 1.7.

A basic concept in describing the weak interaction is the chirality: Dirac spinors that are eigenstates of the 1 + γ5(or 1 − γ5) matrix are defined as left- and right-handed chiral states. Any Dirac spinor can be decomposed into left- and right-handed chiral components

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by means of the chiral projection operators PL= 1₂(1 − γ5) and PR= 1₂(1 + γ5). Chirality

is a fundamental property of particles and the weak interaction, unlike the strong and EM ones, is not invariant under the exchange of the particles with opposite chiral states: in fact the W±boson only couples to left-handed particles and right-handed antiparticles, whilst the Z bosons couples differently to particles or antiparticles of different chirality.

The mathematical reasons of the different behaviour of Z and W± to chiral fermions are explained in terms of the V -A structure of the weak interaction. In fact both QED and QCD are vector interactions, and the particle current that interacts with the gauge field has the form jµ= ψγµψ and transforms as a four-vector under Lorentz transformations; on the contrary the weak current is a linear combination of a vector and an axial vector term:

jµ∝ ψ(gVγµ− gAγµγ5)ψ = gVj_Vµ+ gAj_Aµ (1.15)

As an example, the charged weak current has the form j_weakµ = ψγµ(1 − γ5)ψ and transforms partly as a vector and partly as an axial vector under Lorentz transformations. Since the 1 − γ5term of the charged weak current (absent in the EM and strong interaction) is proportional to the chiral projector, it selects only a specific chiral state and is responsible for the different weak coupling of chiral particles. This particular structure of the weak interaction is also responsible for the violation of parity and charge conjugation (see section 1.9).

1.6 Electroweak Unification

The electroweak model describes the unification of the EM and weak interactions and is based on the symmetry group SU(2)L×U(1)Y [16]. The subscript L in SU(2)L highlights

the fact that this symmetry (and hence the related interaction) only acts on left-handed particles, while the subscript Y in U(1)Y refers to the quantum number associated to the U (1)

symmetry, the hypercharge Y , defined via the Gell-Mann–Nishijima formula Y = 2(Q − T3),

where Q is the electric charge and T3is the third component of the weak isospin, a quantum

number related to the SU(2)L group.

The SU(2)L symmetry acts on the fields in the following way

ψL SU(2)L −−−−→ ψ′_L= eig2τaαa(x)_ψ L ψR SU(2)L −−−−→ ψ_R′ = ψR (1.16)

where g is the interaction coupling constant, τaare the Pauli matrices and αa(x) are three

functions of the space-time. SU(2)Lhas a well-known algebra and any representation of the

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1.6 Electroweak Unification 13

in the same way as the ordinary spin. Left-handed particles are representations of T = 1/2 and are grouped in weak isospin doublets (T3= ±1/2):

χ_L = νe e− ! L , νµ µ− ! L , ντ τ− ! L , u d′ ! L , c s′ ! L , t b′ ! L

Right-handed particles, being the trivial representation of the weak isospin algebra, are singlets (T = T3= 0):

e−_R, µ−_R, τ_R−, uR, cR, tR, dR, sR, bR

Right-handed neutrinos have never been observed so far and are excluded.

SU(2)L introduces three new gauge fields into the theory, Wµa(a = 1, 2, 3), that couple

with three weak isospin currents Jaµ = g₂χLγµτaχL. The fields of the physical charged

bosons W± can be derived by re-arranging the isospin currents in terms of charged currents: J_±µ= √1

2(J µ 1 ± iJ

µ

2). The W±’s fields are then

W_µ± =√1 2(W

1

µ∓ iWµ2) (1.17)

U(1)Y acts on both left- and right-handed fields as it was shown in the EM case (cf.

equation 1.2)

ψ−−−−→ ψU (1)Y ′= eig′2Y β(x)ψ _(1.18)

U(1)Y introduces a gauge field Bµ that mixes with the Wµ3field by means of the Weinberg

angle θWto produce the physical fields of the photon (Aµ) and Z (Zµ):

Aµ Zµ ! = cos θW sin θW − sin θW cos θW ! Bµ W_µ3 ! (1.19)

The Weinberg angle θW is defined by

cos θW=

g p

g2_{+ g}′2 (1.20)

where g and g′are the coupling constants defined respectively in equations 1.16 and 1.18. The Lagrangian of the electroweak model can be derived by replacing the derivative ∂µ

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14 The Standard Model of Particle Physics DL_µ = ∂µ+ i g 2τ a W_µa | {z } from SU(2)L + ig ′ 2Y Bµ | {z } from U(1)Y D_µR= ∂µ+ i g′ 2Y Bµ | {z } from U(1)Y (1.21)

In addition, the kinetic terms of the fields can be written by defining the field tensors:

W_µνa = ∂µWνa− ∂νWµa− gϵabcWµbWνc

Bµν= ∂µBν− ∂νBµ

(1.22)

The Lagrangian of the electroweak model is finally

L_Electroweak = ψ(iγµ∂µ− m)ψ | {z } Free particle − g 2ψLγ µ_τa_ψ LWµa− g′ 2Y ψLγ µ_ψ LBµ | {z } Left-handed interaction − g ′ 2Y ψRγ µ_ψ RBµ | {z } Right-handed interaction − 1 4W a,µν_Wa µν− 1 4B µν_B µν | {z } Kinetic term (1.23)

1.7 The Brout–Englert–Higgs Mechanism

Although the electroweak model described in section 1.6 provides a satisfying explanation of the unification of the EM and weak interactions, it leaves an open question: how can the W± and Z gauge bosons acquire a mass? The gauge theories require the gauge bosons to be massless and any insertion of a mass term in the Lagrangian by hand would violate the gauge invariance. A new mechanism, first described by Brout, Englert and Higgs in 1964 [3, 4], accounts for the mass of the vector bosons in the general framework of gauge invariant theories: they observed that a mass term can arise if the vacuum state is degenerate with respect to the symmetry group of the theory and the symmetry is spontaneously broken.

The BEH mechanism introduces a new complex SU(2)Ldoublet:

φ = √1 2 φ+ φ0 ! (1.24)

The Lagrangian of this doublet takes the form

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1.7 The Brout–Englert–Higgs Mechanism 15

where Dµis the covariant derivative of formula 1.21 and V (φ†φ) is the potential that gives

rise to the symmetry breaking:

V (φ†φ) = µ2φ†φ + λ(φ†φ)2 (1.26)

µ2and λ are two real parameters that define the shape of the potential. To ensure the existence of a global minimum λ is required to be positive. No restrictions are imposed on µ2: positive values of µ2 produce a single minimum at (φ†φ)min= 0 which is not degenerate, while

negative values of µ2produce a potential whose minima correspond to

(φ†φ)min= − µ2 2λ = v2 2 where v 2_{:= −}µ2 λ (1.27)

v is the vacuum expectation value of the field φ. An example of the Higgs potential 1.26 with a degenerate vacuum state is shown in figure 1.2.

Figure 1.2 A schematic representation of the Higgs potential 1.26 for λ > 0 and µ2< 0. Infinite minima are shown, corresponding to the values φ†φ = v2/2 and connected by the SU(2)L×U(1)Y symmetry.

The ground state is then degenerate with the same SU(2)L×U(1)Y symmetry of the

electroweak model: the spontaneous symmetry breaking arises when only one of the infinite minima of the potential is chosen. With no loss of generality the vacuum configuration can be chosen as φ = √1 2 0 v ! (1.28)

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16 The Standard Model of Particle Physics φ = √1 2 0 v + h(x) ! (1.29)

where h(x) is the Higgs field. This particular expansion corresponds to a specific choice of gauge, the unitarity gauge, where the remaining degrees of freedom give rise to the so called Numbu-Goldon bosons, which are non-physical and are absorbed by the gauge bosons to provide them longitudinal polarization.

Substituting 1.29 into the Lagrangian 1.25 the masses of the gauge bosons can be read off from the coefficients of the quadratic terms in the respective fields. The masses of W±, Z and γ are respectively

MW = 1 2vg MZ = 1 2v p g2_{+ g}′2 _M γ = 0 (1.30)

where g and g′are the coupling constants defined in section 1.6.

MW and MZ are not independent parameters of this model: comparing equations 1.20

and 1.30 they are found to be related by the Weinberg angle according to the following equation:

MW

MZ

= cos θW (1.31)

The excitations of the Higgs field correspond to a real spin-0 particle, the Higgs boson. Its mass can be expressed in terms of the parameters of the model, MH =

√

2λv, but its value is not fixed by the theory since λ is a free parameter. The vacuum expectation value is instead predicted from measurements of MW and g to be v = 246 GeV [16].

1.7.1 The Fermion Masses

The BEH mechanism also provides a good explanation of the fermion masses. The mass term of the Dirac Lagrangian in equation 1.1 is

− mψψ = −m(ψ_RψL+ ψLψR) (1.32)

and it is not invariant under SU(2)L×U(1)Y transformations, since left-handed fermions

are SU(2) doublets whereas right-handed fermions are SU(2) singlets. However, it can be shown [16] that the combination ψ_Lφ is invariant under SU(2)L; when combined with

a right-handed singlet, ψ_LφψR (and its hermitian conjugate ψRφ†ψL) is invariant under

SU(2)L and U(1)Y. Hence a term in the Lagrangian of the form −gf(ψLφψR+ ψRφ†ψL)

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1.8 Higgs Boson at the LHC 17

As an example let’s consider the electron. The Lagrangian term becomes

Le= − ge √ 2v(eLeR+ eReL) − ge √ 2h(eLeR+ eReL) (1.33) The first term corresponds to the electron mass term of the Dirac Lagrangian, but derived accordingly to the gauge symmetries of the theory, while the second term describes the coupling between the electron and the Higgs boson itself. By defining the electron mass in terms of the constant ge (known as the Yukawa coupling):

me=

ge

√

2v (1.34)

equation 1.33 takes the form

Le= −meee −

me

v hee (1.35)

which shows that the Higgs boson’s coupling to fermions is proportional to the their mass.

1.8 Higgs Boson at the LHC

1.8.1 Higgs Boson Production

The Higgs boson can be produced via different mechanisms. At the CERN’s LHC, which is a proton-proton collider, the main production modes are gluon-gluon Fusion (ggF), Vector-Boson Fusion (VBF), W /Z associated production (VH) and top-quark pair associated pro-duction (t ¯tH). Feynman diagrams for these modes are shown in figure 1.3.

The production cross sections of a 125 GeV Higgs are shown in figure 1.5a as a function of the centre of mass energy. At the LHC the dominant production mode is the ggF although it proceeds through loops. VBF has a much smaller cross section than ggF but is of particular interest at the LHC: in this mode the incoming quarks radiate two vector bosons (W or Z) that fuse to produce a Higgs boson. The Higgs boson’s decay products tend to be central in the detector while the remnant quarks produce two jets forward and backward in the detector. VH production mode, also called Higgs-strahlung, has a very low cross section compared to ggF and VBF but, unlike these two modes, benefits from a clear signature due to the presence of additional leptons in the final state. t ¯tH is the rarest production mode and is characterised by a complex signature composed of several jets in the final state.

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Figure 1.3 Feynman diagrams for the Higgs main production modes at the LHC: gluon-gluon Fusion (a), vector boson fusion (b), W /Z associated production (c) and top-quark pair associated production (d).

1.8.2 Higgs Boson Decay

Many channels are open for the Higgs decays, involving fermions and bosons. Although the Higgs boson couples only to massive particles, its decays into massless particles, like photons or gluons, are possible provided that they proceed through loops as shown in figure 1.4. Since the Higgs coupling depends on the mass of the particles, the dominant diagrams involve loops of top quarks (H → gg) or loops of top quarks and W bosons (H → γγ).

Figure 1.4 Feynman diagrams for the H → γγ decay through a loop of top quarks (left) and W bosons (right).

Figure 1.5b shows the branching ratios of the Higgs decays as a function of the Higgs mass between 120 GeV and 130 GeV. The partial width of a Higgs decaying into a couple fermion-antifermion (H → f ¯f ) of mass mf at tree level is [16]:

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1.8 Higgs Boson at the LHC 19 Γ_{f ¯}_f = NcmH 8πv2 m 2 f 1 − 4m2_f m2_H !3₂ (1.36)

where Ncis the number of colours (Nc= 3 for quarks, Nc= 1 for leptons) and mH is the

Higgs mass. Formula 1.36 shows that the decay width is larger for more massive fermions and explains why the H → τ+τ−decay has the second largest branching ratio in the fermion sector after the b quarks. t quarks are not considered since their mass is larger than Higgs mass and the H → tt decay, involving both off-shell t quarks, is strongly suppressed.

(a) (b)

Figure 1.5 (a) Higgs production cross section for single production modes and for a 125 GeV Higgs as a function of the centre of mass energy [25]: ggF (blue), VBF (red), VH (green and grey), t ¯tH (dark purple). (b) Branching ratios of the Higgs decays as a function of the Higgs mass in the range 120 GeV to 130 GeV [25].

1.8.3 Higgs Boson Discovery

The Higgs boson, predicted in 1964, was discovered only 50 years later by ATLAS and CMS [1, 2], the two general purpose experiments at the LHC. In July 2012 they announced the observation of a new particle of mass nearly 125 GeV in the search for a Higgs boson in the golden channels H → γγ and H → ZZ∗→ 4l. The mass value measured by the combined experiments was 125.09 ± 0.21(stat.) ± 0.11(syst.) GeV [26].

The new particle corresponds very well to the SM Higgs: the agreement between the SM expectations and the measured particle is parametrised by the signal strength µ which is

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defined as the ratio of the measured Higgs boson rate to its SM prediction (µ = σmeas/σSM). ATLAS and CMS combined analysis provided a signal strength of µ = 1.09+0.11_−0.10 [26], that denotes a good compatibility of the discovered particle with the SM expectations.

H → τ+τ− decays have been observed by ATLAS and CMS with a significance of 4.5 σ [27] and 5.9 σ [28] respectively. Figure 1.6a shows the invariant mass of the tau pair reconstructed by the ATLAS experiment with the Missing Mass Calculator (MMC) algorithm [29]. To highlight the most significant events, they are weighted by ln(1 + S/B), where S is the amount of signal and B of background in the bin of the event. An excess of data with respect to the background distributions is found around 125 GeV, that is exactly the measured value of the Higgs boson mass in its discovery channels.

(a) (b)

Figure 1.6 (a) Distribution of the reconstructed invariant τ τ mass [27] with the Missing Mass Calculator algorithm [29]. Events are weighted by ln(1 + S/B), where S is the amount of signal and B of background in the bin of the event. (b) Signal strength µ in the individual τ decay channels (from the bottom, H → τhadτhad, H → τlepτhad and H → τlepτlep) and in the

individual categories of the analysis (vbf and Boosted, described in section 4.1.2) [27]. The overall measurement, inclusive of all the channels and categories, is also shown.

The signal strength µ of the H → τ τ decays is shown in figure 1.6b. The analysis is split into three main channels, according to the nature of the tau pair decays: both hadronic

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1.9 Discrete Symmetries 21

(H → τhadτhad), both leptonic (H → τlepτlep), or one hadronic and one leptonic (H → τlepτhad).

Each channel is further split into two categories, the boosted and vbf, defined to fully exploit the signature of the different Higgs production modes (see section 4.1.2 for more details). The overall measurement of the signal strength, inclusive of all the channels and categories, is µ(H → τ τ ) = 1.4+0.4_−0.4, denoting that the observed particle is compatible with the SM Higgs boson within 1 σ.

1.9 Discrete Symmetries

Symmetries play an important role in physics since they are strictly related to conservation laws, as expressed by Noether’s theorem, and because they allow inferences on dynamical systems even when a complete theory is not available. U(1), SU(2) and SU(3) are examples of continuous symmetries, i.e. symmetries that can be reduced to the identity with continuity when the parameter of the transformation tends to zero. Quantum physics also contemplates discretesymmetries, in which the parameter of the transformation can assume only a discrete number of values and cannot be reduced to the identity with continuity.

Parity P is an example of such a discrete symmetry: when applied to a system it produces the inversion of the three spatial coordinate axes, or in other words it connects an object to its mirror image. Scalar and axial vector quantities remain unchanged under parity transformations while pseudoscalar and vector quantities change sign. A single particle can be an eigenstate of P in its rest frame (in this case parity is commonly referred to as intrinsic parity): since applying parity twice brings the system to its original state, i.e P2= 1, the intrinsic parity of a particle is +1 or −1. Although the laws of physics were thought invariant under parity transformations, the experiment carried out by madame Wu [30] in 1956 showed that parity in not an exact symmetry of Nature: the EM and strong forces conserve the parity of a system but the weak force does not. To explain such a violation the V -A structure of the weak interaction, already discussed in section 1.5, was successfully incorporated into the theory.

Charge conjugation C is another important discrete symmetry in particle physics: it changes the sign of all the internal quantum numbers of a system, i.e. it changes any particle into its antiparticle and vice versa. Since all the internal quantum numbers are exchanged by the charge conjugation, a single particle can be an eigenstate of C only if all its internal quantum numbers are 0. In this case the particle coincides with its antiparticle: the photon and the π0are examples of such particles. Also a particle-antiparticle system is an eigenstate of C in the centre of mass frame. Like the parity, C brings the system to its original state if applied twice: its possible eigenvalues are then +1 or −1. The aforementioned madame

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Wu’s experiment also proved that C is violated by the weak interaction, although it is known to be conserved by the strong and EM interactions.

1.9.1 CP Violation

P and C are separately violated by the weak interaction, but one may ask if the physics’ laws are the same under the combined application of the two transformations, namely if CP is conserved in physics. This is not obvious a priori and tests of CP are very challenging, but finally in 1964 an experiment carried out by Cronin and Fitch [31] on the decays of K mesons proved the violation of CP in Nature.

CP violation is accommodated by the theory provided that the CKM matrix contains at least an imaginary term. This requirement is satisfied only if the generations of fermions are equal to or more than three: nowadays it is well known that three generations exist and CP violation has its place in the theoretical framework.

1.10 CP in the Higgs Sector

In the SM the Higgs boson arises from a single doublet and is a CP-even scalar particle, i.e. it has quantum numbers JP C = 0++. CP properties of the Higgs boson have been studied at CERN in the bosonic sector using the channels H → γγ, H → ZZ∗→ 4ℓ and H → W W∗→ ℓνℓν, and a remarkable agreement has been found so far with the predictions of the SM. In particular, the spin-1 hypothesis is excluded by Landau-Yang theorem [32], since the decay H → γγ has been observed. Alternative hypotheses to the SM of pure CP-eigenstate Higgs bosons have been tested in the aforementioned bosonic channels and excluded at more than 99.9% confidence level in favour of the SM hypothesis [5, 6].

Even though a pure CP-odd state (JP C= 0+−) is excluded, the possibility still remains that the observed Higgs boson is an admixture of a scalar (CP-even) and a pseudoscalar (CP-odd) component. Such a mixing is accommodated for instance by SM extensions that introduce a second doublet in the Higgs sector and enlarge the Higgs family with additional bosons. It is important to remark that the bosonic sector of the Higgs decays is not sensitive to the CP-odd component at tree level [12, 13] and the CP-mixture has impact only in the fermion sector. The general features of these extended models that predict a second Higgs doublet, referred to as Two-Higgs-Doublet Models (2HDMs), are described in the following of this section.

The 2HDMs are minimal extensions of the SM that add the fewest new arbitrary parame-ters. From a theoretical point of view a plethora of motivations may be found to consider

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1.10 CP in the Higgs Sector 23

such extended models: the main one is probably that they might allow for a new source of CP violation that affects the Higgs sector [7] (unlike the CP-conserving SM Higgs). The CP-violating phase of the CKM matrix is not sufficient to explain the baryon asymmetry of the Universe, but the new source of CP violation introduced by the 2HDMs may do it [9–11], answering in such a way one of the most important open questions in physics nowadays. In addition, given some conditions, the 2HDMs provide the Higgs structure required in the low-energy supersymmetric models [12].

1.10.1 General Phenomenology of the 2HDMs

In a general 2HDM eight real scalar fields are arranged in two complex Higgs doublets φ1,2:

φ1= ϕ+₁ v1+ϕ_√1+iχ1 2 ! φ2= ϕ+₂ v2+ϕ_√2+iχ2 2 ! (1.37)

where ϕ1,2 and χ1,2are real fields, ϕ+_1,2are complex fields and v1,2are in general complex

values.

The most general 2HDM potential is then:

V =µ2₁(φ†₁φ1) + µ22(φ†2φ2) − h µ2₃(φ†₁φ2) + h.c. i +1 2λ1(φ † 1φ1)2+ 1 2λ2(φ † 2φ2)2+ λ3(φ†1φ1)(φ†2φ2) + λ4(φ†1φ2)(φ†2φ1) +1 2 h λ5(φ†₁φ2)2+ λ6(φ†₁φ1)(φ†₁φ2) + λ7(φ†₂φ2)(φ†₁φ2) + h.c. i (1.38)

where µ2₁, µ2₂ and λ1, . . ., λ4 are real parameters and µ23, λ5, λ6 and λ7 are complex in

principle.

If both doublets are allowed to interact with all fermions, the 2HDMs exhibit tree-level Higgs-mediated Flavour-Changing Neutral Currents (FCNCs) [8], which are strongly sup-pressed experimentally. Such a bothering problem can be avoided by invoking an appropriate Z2discrete symmetry and extending it to the fermion sector [33]. This symmetry is realized

for some choice of basis by the transformations of the doublets φ1→ φ1and φ2→ −φ2, and

gives rise to four different types of Yukawa interactions (and hence four different types of 2HDMs) depending on the transformations of the fermions with respect to the symmetry [7]. To remove the FCNCs from the model it is sufficient that all fermions of given charge couple to no more than one Higgs doublet [33].

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The Z2symmetry implies µ2₃= λ6= λ7= 0 in the potential of equation 1.38: however

the FCNCs are retained within the experimental boundaries even in case the symmetry is allowed to be softly broken, i.e. a basis exists in which λ6= λ7= 0 but µ2₃̸= 0 [8]. In the

following the softly-broken Z2symmetry scenario will be explored, since it allows for the

introduction of CP violation in a general 2HDM, provided that Im(λ5) ̸= 0 [7] as will be

explained in section 1.10.2.

The minimum of the potential in equation 1.38 occurs when

∂V ∂φ1 φ1=⟨φ1⟩ φ2=⟨φ2⟩ = 0 ∂V ∂φ2 φ1=⟨φ1⟩ φ2=⟨φ2⟩ = 0 (1.39)

where the Vacuum Expectaction Values (VEVs) ⟨φ1,2⟩ have been introduced. By employing

appropriate transformations of the fields and avoiding U(1)EM symmetry-breaking charge

vacuum solutions, it is always possible to write

⟨φ1⟩ = 1 √ 2 0 v1 ! ⟨φ2⟩ = 1 √ 2 0 v2eiξ ! (1.40)

where v1,2 are real and positive and 0 < ξ < 2π is a phase. It is convenient to introduce the β

angle defined as

v1:= v cos β v2:= v sin β (1.41)

where v2= v2₁+ v₂2= (√2GF)−1/2= (246 GeV)2. One is always free to rephase φ2in order

to set ξ = 0 [8]. In the following the Higgs field’s VEVs will be assumed real and positive. The conditions in equation 1.39 provide a relation between Im µ2₃ and Im λ5 (λi5 for

simplicity) [7]: Im µ2₃=v 2 2 λ i 5sβcβ (1.42)

where cβ and sβstand respectively for cos β and sin β. Therefore Im µ23is not independent

from λi₅and the latter may be regarded as the only source of CP violation, as it will be clearer in section 1.10.2.

1.10.2 Physical Higgs Fields and CP-Mixing

Not all the eight real fields of the 2HDMs are physical: three of them describe the unphysical Nambu-Goldston bosons, that provide polarization to the W± and Z bosons, and only the remaining five are related to the physical Higgs fields. Among them, two are charged (H±)

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1.10 CP in the Higgs Sector 25

and three are neutral: these three may be identified for instance as two even eigenstates of CP (ϕ1, ϕ2) and an odd eigenstate (A = − sin βχ1+ cos βχ2).

Two possible scenarios are now open: if the neutral CP-eigenstate Higgs fields also correspond to the physical fields, there is no CP violation in the model. On the contrary, if the opposite CP-eigenstate Higgs fields mix to produce the physical Higgs bosons, then the model exhibits CP violation.

To derive the physical neutral Higgs bosons, one defines first the real symmetric squared-mass-matrix M2, that is non-diagonal in the ϕ1, ϕ2, A basis if the CP-violating model

hypothesis holds [34]: M2=    M2 11 M212 − v2 2λi5sβ M2 12 M222 −v 2 2λ i 5cβ −v₂2λi₅s_β −v₂2λi₅c_β M2 33    (1.43)

The terms M2₁₃ and M2₂₃ suggest that the mixing between the CP-even (ϕ1, ϕ2) and the

CP-odd (A) neutral Higgs fields occurs if λi₅̸= 0, that is the condition given at the end of section 1.10.1.

The orthogonal transformation R that diagonalizes M2provides also the physical neutral Higgs states h1,2,3with corresponding squared-masses Mi2:

   h₁ h₂ h₃   = R    ϕ₁ ϕ₂ A    with RM 2_RT =    M₁2 0 0 0 M₂2 0 0 0 M₃2    (1.44)

The transformation R can be written as a product of three rotation matrices Riof angles

αi: the first one can be used to diagonalize the upper left 2 × 2 block of the matrix M2, that

produces a mixing of the CP-even fields ϕ1and ϕ2. By applying the first rotation R1of an

angle α1= α + π/22in the ϕ1-ϕ2subspace, the rotated fields are [7]

H = cos αϕ1+ sin αϕ2

h = − sin αϕ1+ cos αϕ2

(1.45)

At this stage the CP-odd field A remains unmixed and the 2 × 2 CP-even submatrix is rendered diagonal:

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26 The Standard Model of Particle Physics R1M2RT1 = M21=    M_h2 0 M′2₁₃ 0 M_H2 M′2₂₃ M′2₁₃ M′2₂₃ M_A2    (1.46)

The off-diagonal terms M′2₁₃ and M′2₂₃ of matrix 1.46 are still proportional to λi₅[8]. In the general 2HDM the fields h, H and A, that have definite CP quantum numbers, are intermediaries and do not necessarily correspond to physical particles. Only if λi₅= 0 the matrix 1.46 is diagonal, no additional rotations are needed and the fields h, H and A are also mass eigenstates. This is the case of a CP-conserving 2HDM.

Nevertheless, if λi₅̸= 0 at least an additional rotation is needed to diagonalize the matrix 1.46: as a consequence the mass-eigenstates are admixtures of the CP-even and CP-odd states and CP symmetry is broken.

1.10.3 Yukawa Lagrangian in the Neutral Higgs Sector

In the framework of a CP-violating 2HDM, the SM-like 125 GeV Higgs boson discovered at the LHC in 2012 may be regarded as the lightest boson among the three mass-eigenstates that have been derived in section 1.10.2, say h1, that for simplicity will be written h. Since it

is a CP-mixture, its tree-level interaction to leptons consists of a scalar and a pseudoscalar term, and hence the most general Yukawa Lagrangian may be written as follows [35]:

L_Y = −

_∑

ℓ=e,µ,τ m_ℓ v (cℓℓℓ + i ˜cℓℓγ 5_ℓ)h (1.47)

where mℓ is the lepton mass, ℓ is the lepton field and cℓ and ˜cℓare constant values which

depend on the rotation matrices Riand the β angle. The specific Yukawa Lagrangian of the

h → τ τ channel, that will be studied in this thesis, can be isolated from 1.47 and is

Lhτ τ= −

mτ

v κτ(cos φττ τ + i sin φττ γ

5_{τ )h} _(1.48)

where κτ is known as the reduced Yukawa coupling strength and φτ is the CP-mixing angle

that parametrises the relative contributions of the CP-even and CP-odd components to the h → τ τ coupling [36].

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Chapter 2 The ATLAS Experiment at the Large

Hadron Collider

The European Organization for Nuclear Research (CERN), founded in 1954 and located near the city of Geneva (Switzerland), is one of the largest scientific institutes for the research on particle physics. It hosts the world’s largest and most powerful particle collider, the Large Hadron Collider (LHC), which accelerates two beams of protons or heavy ions in opposite directions. The collisions are provided in four different locations of the ring, where the main detector experiments take place: ATLAS (A Toroidal LHC Apparatus) and CMS (Compact Muon Solenoid), two general purpose experiments devoted to the study of the Higgs boson and the phenomena beyond the SM, ALICE (A Large Ion Collider Experiment), designed to focus on the strong interaction in the quark-gluon plasma, and LHCb, an experiment performing precision measurements of the decays of the B mesons.

2.1 The Large Hadron Collider

The LHC is a proton-proton ring accelerator with a circumference of 27 km and designed to reach a nominal centre of mass energy of√s = 14 TeV and a peak luminosity of L = 1034cm−2s−1 [37]. Luminosity is a fundamental parameter of a particle accelerator and is defined by geometrical factors [38]:

L = NaNb 4πσxσy

kf (2.1)

where Na and Nb are the number of particles in the colliding bunches, k is the number of

bunches in the ring, f is the revolution frequency, σxand σy are the widths of the bunches

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28 The ATLAS Experiment at the Large Hadron Collider

(such as crossing angles, collision offset, non-Gaussian beam profile, non-zero dispersion at collision point, etc.) may modify equations 2.1 [39].

Luminosity is related to the expected rate of events dN/dt by the formula:

dN

dt = Lσ (2.2)

where σ is the cross-section of the process, containing all the information on the dynamics of the event. The cross-section has the dimensions of an area and is measured in barn (1 b = 10−24 cm2). L is known as the instantaneous luminosity; one may also define an integratedluminosity as

Lint=

Z

Ldt (2.3)

that is measured in b−1. Using the integrated luminosity, the number of events is simply given by N = Lintσ.

The LHC ring is the latest addition to the CERN’s accelerator complex [40] (figure 2.1). The proton’s acceleration process is based on the following steps:

• Protons are produced by stripping electrons from hydrogen atoms;

• The protons pass through the Linac2 and are injected into the Booster (a synchrotron ring) at an energy of 50 MeV;

• The Booster provides an acceleration of 1.4 GeV, after which the protons are sent to the Proton Synchrotron (PS) and are accelerated to 25 GeV;

• The next step in the acceleration process involves the Super Proton Synchrotron (SPS), where the protons reaches the energy of 450 GeV;

• Finally the protons enter the LHC (both in clockwise and anticlockwise direction) and are accelerated to 6.5 TeV.

Inside the LHC, the beam particles are organized in 2808 bunches with a time separation of 25 ns and travel in two separate pipes kept at ultrahigh vacuum. The particles are maintained in orbit thanks to a magnetic field above 8 T produced by superconducting electromagnets cooled to a temperature of 1.9 K by liquid helium. The acceleration is provided by a 400 MHz superconducting Radio Frequency (RF) cavity system. The beam pipes intersect in four different points of the LHC ring, making particle collisions possible. These interaction regions are surrounded by the detectors of the main experiments (ALICE, ATLAS, CMS, LHCb).

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2.1 The Large Hadron Collider 29

Figure 2.1 The CERN’s accelerator complex [41]. The acceleration starts in the Linac2 and continues in the Booster, PS and SPS. Finally the beam reaches the LHC at an energy of 450 GeV.

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Figure 2.2 Integrated luminosity delivered by the LHC (green) and recorded by ATLAS (yel-low) during the Run 2 by September 2018 [42]. The LHC’s Run 1 started on November

2009 and lasted by the end of 2012. During this period LHC operated at a centre of mass energy up to 8 TeV and collected data for an integrated luminosity of about 29 fb−1. On February 2013 it was shut down for a 2-year upgrade, called Long Shutdown 1 (LS1), to improve the centre of mass energy to the value of 13 TeV. The LHC restarted oper-ations for the Run 2 on April 2015 and is expected to produce data until the end of 2018. By September 2018, the integrated lu-minosity delivered by the LHC for the entire duration of the Run 2 is slightly more than 140 fb−1(figure 2.2).

2.2 General Layout of the ATLAS Experiment

ATLAS [43, 44] is a general purpose experiment located at point 1 of the LHC ring; some of its main objectives are the search of the Higgs boson (discovered at the end of the Run 1), tests of validity of the SM and search for signatures beyond the SM, precision measurements of cross sections, W and top masses and CP violation.

The requirements of the ATLAS detectors strictly depend on the features of the LHC. As an example, the high luminosity provided is necessary to allow the study of the very rare processes mentioned above, but one of its side effects is the production of a huge rate of events (about 109inelastic proton-proton events/s at design luminosity). Another remarkable aspect to consider is hidden in the nature of the hadronic collisions themselves: the QCD jet production and low transverse momentum events in fact dominate over the rare processes of interest and impose challenges on the capability of the detectors and on the identification of the experimental signatures of the characteristic processes. In particular, two phenomena interfere with the event reconstruction: the pile-up, i.e. objects associated to the event that do not come from the primary vertex where the process occurred, and the underlying events, that are the remnants of the scattering process. Therefore the main requirements of the ATLAS detectors are:

• Fast response, resistance to radiations and high granularity

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2.2 General Layout of the ATLAS Experiment 31

addition, high granularity is needed to handle the particle fluxes and to reduce the effects of overlapping events.

• Full coverage

Large angular acceptance is needed to fully and correctly reconstruct the energy balance of the events.

• Very good resolutions and particle identification

Very precise measurements are essential because of the rarity of the events of interest.

• Efficient trigger system

The 40 MHz event rate must be reduced to about 1 kHz to allow the collection and storage of data. It is therefore fundamental to have a sufficient background rejection.

Figure 2.3 shows the general layout of the ATLAS detector: it has a cylindrical symmetry and it extends 25 m in height and 44 m in length.

Figure 2.3 General layout of the ATLAS detector [45]. From the beam pipe to the exterior are shown in order: the pixel detector, the Semiconductor Tracker and the Transition Radiation Tracker (that compose the Inner Detector); the liquid argon electromagnetic calorimeter, the tile calorimeter, the end-cap liquid argon hadronic calorimeter and the forward calorimeter (which compose the calorimeter system); the solenoid and toroid magnets and finally the muon chambers.

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2.2.1 Coordinate system

The coordinate system used to describe an ATLAS event originates from the nominal interaction point. The z-axis is identified by the beam direction and the x-y plane is transverse to it. In particular, the y-axis is conventionally defined as pointing upward and the x-axis to the centre of the LHC. The azimuthal angle φ is defined on the transverse plane and is measured around the z-axis, whereas the polar angle θ is measured from the z-axis. In the analysis it is common to define the pseudorapidity1as

η = − ln tanθ 2 (2.4)

The distance ∆R in the η-φ plane angle space is defined as ∆R =p∆η2+ ∆φ2.

Most interactions of interest involve only a parton from each colliding proton, but since the partons might carry every fraction of the proton energy, the total energy of the interacting partons is unknown. However, the proton-proton collisions are head on, hence it is natural to assume that the total transverse momentum of the partons is close to zero. For this reason, many variables are defined in x-y plane (i.e. the plane transverse to the beam direction), such as the transverse momentum ppp_T = (px, py), the transverse energy EEET = (Ex, Ey) and

the missing transverse energy EEEmiss_T = (E_xmiss, E_ymiss). The magnitudes of these vectors are respectively referred to as pT, ET and E_Tmiss.

According to the ATLAS nomenclature, the transverse impact parameter d0is defined

as the transverse distance of the track to the beam axis at the point of closest approach, and the longitudinal impact parameter z0as the z coordinate of the track at the point of closest

approach.

2.3 Inner Detector

The ATLAS Inner Detector (ID) [44, 46–48], shown in figure 2.4, is mainly designed to provide track measurements, excellent momentum resolution and both primary and secondary vertex measurements for charged particles with pT > 0.5 GeV and within |η| < 2.5. It also

provides electron identification within |η| < 2.0. The ID is immersed in a solenoidal magnetic field of 2 T (see section 2.5).

The ID is the closest detector to the beam pipe and its overall dimensions are 2.1 m in diameter and 6.2 m in length. The ID is composed of three sub-detectors: the pixel detector, the Semiconductor Tracker (SCT) and the Transition Radiation Tracker (TRT).

1_{When the ultrarelativistic limit is not applicable, the rapidity, defined as y =}1 2ln _E +pz E−pz , is used instead.

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2.3 Inner Detector 33

Figure 2.4 General layout of the ATLAS ID (left) [49] and a view of the central sector (right) [47]. From the beam line to the exterior are shown in order: the 4-layer pixel detector (the innermost layer, called insertable B-layer, was added during the LS1, see section 2.3.1), the Semiconductor Tracker and the Transition Radiation Tracker.

The advanced technology implemented in the pixel and silicon microstrips allows to satisfy the requirements on vertex resolution and track reconstruction, whereas the TRT, the outer sub-detector, provides a large number of tracking points.

The ID consists of three units: a barrel (the central part) where the detector layers are arranged in concentric cylinders around the beam axis, and two identical end-caps (covering the remaining parts) where the detectors are shaped as disks perpendicular to the beam axis.

2.3.1 Pixel Detector

The silicon pixel detector [50–52] is the innermost ATLAS sub-detector and is designed to perform high-granularity measurements close to beam line. Four barrel layers of detectors, spanning the radial region of 33 mm-150 mm from the beam axis, and three disks on each side, between radii of 88 mm and 150 mm, allow to perform position measurements with the required level of resolution and reconstruction of secondary vertices from the decays of short-lived particles such as B mesons and τ leptons.

The innermost layer, called the Insertable B-Layer (IBL), was added to the three pre-existing layers during the LS1: it brought a large number of benefits to the data collected during the Run 2, such as improvements on track reconstruction against failure of pixel modules, higher b-tagging efficiency and better vertexing performances due to the closer location to the interaction point.

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All the modules are identical in the barrel and the disks and are 62.4 mm long and 21.4 mm wide. The three pre-existing layers, built for the Run 1, cover the region |η| < 2.5 and the pixel sensors mounted on them have typical dimensions of 50 × 400 µm2. The IBL covers the region |η| < 3.0 and consists of smaller pixel sensors (size 50 × 250 µm2). The IBL improved the impact parameter resolution for the Run 2 by nearly 40% compared to the Run 1, both in the longitudinal (σ(z0) ≃ 75 µm) and in the transverse direction (σ(d0) ≃ 10 µm),

as shown in figure 2.5.

Figure 2.5 Transverse (left) and longitudinal (right) impact parameter resolution measured from data in 2015,√s = 13 TeV, with the IBL as a function of pT for values of 0.0 < η < 0.2,

compared to the data in 2012,√s = 8 TeV, without the IBL [53].

2.3.2 Semiconductor Tracker

The Semiconductor Tracker (SCT) system uses silicon microstrip detectors and is designed to provide eight precision measurements per track, contributing to the impact parameter resolution and pattern recognition thanks to its high granularity.

The barrel consists of four cylindrical layers between radii of 30 cm and 52 cm, and the end-caps consist of nine wheels on each side whose radial range is adpted to cover the region |η| < 2.5. The microstrips are arranged in double-sided modules, and the sensors of each module are glued back-to-back and rotated of an angle of 40 mrad to provide measurements on two directions (Rφ and z for the barrel, Rφ and R for the end-caps).

Tracks can be distinguished by the SCT if separated by more than ∼200 µm. The spatial resolution is 17 µm in the Rφ direction and 580 µm in the z or R direction.

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2.4 Calorimetry 35

2.3.3 Transition Radiation Tracker

The Transition Radiation Tracker (TRT) [54] is a straw-tube detector designed to provide a large number of hits per track (typically 36) in the region |η| < 2.0. The tubes are arranged parallel to the beam axis in the barrel region and radially in the end-caps and are 4 mm in diameter, with a maximum length of 144 cm. They are equipped with a 30 µm diameter gold-plated W-Re wire and filled with a gas mixture of 70% Xe, 27% CO2and 3% O2. The

straw tube resolution (only in the Rφ direction) is around 130 µm.

The barrel covers the radial range from 56 cm to 107 cm and contains about 50 000 straws divided in two at the centre and read out at both ends. Each end-cap consists of 18 wheels perpendicular to the beam axis for an overall amount of 320 000 radial straws with the readout at the outer radius. The innermost 14 wheels cover the radial range from 64 cm to 103 cm, while the last four extend to an inner radius of 48 cm. Wheels 7 to 14 have half as many straws per cm in z as the others, to avoid an unnecessary increase of crossed straws and material at medium rapidity.

The TRT also provides electron identification via transition radiation from polypropylene fibres (barrel) or foils (end-caps) interleaved between the straws. A particle that traverses the boundary between two materials of different refraction index emits energy in the form of X-rays. Since the intensity of the emitted energy is proportional to the relativistic factor γ, a low mass particle (such as an electron) produces more radiation then a massive one (such as a hadron): hence the TRT is able to provide additional discrimination between electrons and hadrons thanks to the detection of transition-radiation photons in the xenon-based gas mixture of the straw tubes.

2.4 Calorimetry

The ATLAS calorimeter system is shown in figure 2.6 and is specifically designed to measure the energy of the particles (except muons and neutrinos), together with their position from the energy deposits. Its fine granularity allows also good jet reconstruction and E_Tmiss measurements. The calorimeter system consists of an electromagnetic calorimeter (EMCal) and a hadronic calorimeter (HadCal), plus forward calorimeters (FCal) to provide both electromagnetic and hadronic energy measurements in the region 3.2 < |η| < 4.9.

2.4.1 Electromagnetic Calorimeter

The EMCal is mainly dedicated to the measurements of electrons, photons and the electro-magnetic component of the jets in the range |η| < 3.2. It is composed of two half-barrels

Investigation of the hadronic tau substructure and its application to the study of the CP properties of the Higgs boson with the ATLAS experiment at CERN LHC

UNIVERSITÀ DEGLI STUDI DI PISA

Facoltà di Scienze, Matematiche, Fisiche e Naturali

Corso di Laurea in Fisica

Investigation of the hadronic tau substructure

and its application to the study of the CP

properties of the Higgs boson with the ATLAS

experiment at CERN LHC

Prof. Vincenzo CAVASINNI

Francesco LUCARELLI

Contents

Nomenclature

Introduction

Chapter 1

The Standard Model of Particle Physics

1.1

The Principle of Gauge Invariance

1.2

Particles and Fields

1.2.1

Fermions

1.2.2

Bosons

1.3

Quantum Electrodynamics

1.4

Quantum Chromodynamics

∑

1.5

Weak Interaction

1.6

Electroweak Unification

1.7

The Brout–Englert–Higgs Mechanism

1.7.1

The Fermion Masses

1.8

Higgs Boson at the LHC

1.8.1

Higgs Boson Production

1.8.2

Higgs Boson Decay

1.8.3

Higgs Boson Discovery

1.9

Discrete Symmetries

1.9.1

CP Violation

1.10

CP in the Higgs Sector

1.10.1

General Phenomenology of the 2HDMs

1.10.2

Physical Higgs Fields and CP-Mixing

1.10.3

Yukawa Lagrangian in the Neutral Higgs Sector

∑

Chapter 2

The ATLAS Experiment at the Large

Hadron Collider

2.1

The Large Hadron Collider

2.2

General Layout of the ATLAS Experiment

2.2.1

Coordinate system

2.3

Inner Detector

2.3.1

Pixel Detector

2.3.2

Semiconductor Tracker

2.3.3

Transition Radiation Tracker

2.4

Calorimetry

2.4.1

Electromagnetic Calorimeter

_∑