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UNIVERSITÀ DEGLI STUDI DI PISAFacoltà di Scienze, Matematiche, Fisiche e Naturali

Corso di Laurea in Fisica

Investigation of the hadronic tau substructureand its application to the study of the CP

properties of the Higgs boson with the ATLASexperiment at CERN LHC

AdvisorProf. Vincenzo CAVASINNI

CandidateFrancesco LUCARELLI

Academic year 2017/2018

Contents

Nomenclature vii

Introduction 1

1 The Standard Model of Particle Physics 51.1 The Principle of Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . 51.2 Particles and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Weak Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Electroweak Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 The Brout–Englert–Higgs Mechanism . . . . . . . . . . . . . . . . . . . . 14

1.7.1 The Fermion Masses . . . . . . . . . . . . . . . . . . . . . . . . . 161.8 Higgs Boson at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.8.1 Higgs Boson Production . . . . . . . . . . . . . . . . . . . . . . . 171.8.2 Higgs Boson Decay . . . . . . . . . . . . . . . . . . . . . . . . . . 181.8.3 Higgs Boson Discovery . . . . . . . . . . . . . . . . . . . . . . . 19

1.9 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.9.1 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.10 CP in the Higgs Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.10.1 General Phenomenology of the 2HDMs . . . . . . . . . . . . . . . 231.10.2 Physical Higgs Fields and CP-Mixing . . . . . . . . . . . . . . . . 241.10.3 Yukawa Lagrangian in the Neutral Higgs Sector . . . . . . . . . . . 26

2 The ATLAS Experiment at the Large Hadron Collider 272.1 The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . 27

iv Contents

2.2 General Layout of the ATLAS Experiment . . . . . . . . . . . . . . . . . . 302.2.1 Coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Inner Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.1 Pixel Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.2 Semiconductor Tracker . . . . . . . . . . . . . . . . . . . . . . . . 342.3.3 Transition Radiation Tracker . . . . . . . . . . . . . . . . . . . . . 35

2.4 Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.1 Electromagnetic Calorimeter . . . . . . . . . . . . . . . . . . . . . 352.4.2 Hadronic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Central Solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Muon Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.7 Trigger and Data Acquisition System . . . . . . . . . . . . . . . . . . . . . 432.8 Reconstruction of Taus with the ATLAS Detector . . . . . . . . . . . . . . 45

3 Hadronic Tau Substructure 473.1 Tau Leptons at ATLAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Event Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.1 Event Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . 493.2.2 Signal and Background Processes . . . . . . . . . . . . . . . . . . 503.2.3 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 OS-SS Background Estimation Method . . . . . . . . . . . . . . . . . . . 513.3.1 Signal and Control Regions . . . . . . . . . . . . . . . . . . . . . 533.3.2 Performance of the Method . . . . . . . . . . . . . . . . . . . . . 55

3.4 Tau Particle Flow Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 Pion Reconstruction and Identification . . . . . . . . . . . . . . . . 583.4.2 Decay Mode Classification . . . . . . . . . . . . . . . . . . . . . . 593.4.3 Tau Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Tau Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.6 ρ Resonance in 1p1n Decay Mode . . . . . . . . . . . . . . . . . . . . . . 663.7 a1 Resonance in 3p0n Decay Mode . . . . . . . . . . . . . . . . . . . . . . 69

4 CP Scenario in Higgs Decays to Tau Leptons 714.1 Event Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.1 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . 724.1.2 Event Selection and Categorisation . . . . . . . . . . . . . . . . . 73

4.2 Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2.1 Signal and Background Distributions . . . . . . . . . . . . . . . . 78

Contents v

4.3 Observable Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.1 Impact Parameter Method . . . . . . . . . . . . . . . . . . . . . . 794.3.2 ρ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3.3 Combined IP-ρ Method . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 Background Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.5.1 Expected sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 92

Conclusions 97

Appendix A Control Region Plots and Yields 99

Appendix B Performance of the Background Estimation Method 109

Appendix C Tau Identification Variables 117

Bibliography 121

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

EM Electromagnetic

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

QCD Quantum Chromodynamics

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

tt̄H 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)

EmissT Missing transverse energy

x Nomenclature

MT Transverse mass

P Parity

φ Azimuthal angle (ATLAS coordinate system)

ϕCP Angle between the tau decay planes in h→ ττ

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

τvis Visible part of tau decays

θ Polar angle (ATLAS coordinate system)

Introduction

The Higgs boson, discovered by the ATLAS and CMS collaborations at the Large HadronCollider 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, is neededto explain the mass of the gauge bosons and to retain the principle of gauge invariance withinthe theoretical framework.

After its discovery, much effort was made in measuring its properties. This thesis focuseson the CP quantum numbers of this particle. The Higgs boson is predicted by the StandardModel to be a CP-even scalar particle, i.e. to have quantum numbers JPC = 0++. Alternativehypotheses to the Standard Model concerning pure CP-eigenstate Higgs bosons have beentested in the bosonic sector (h → ZZ∗, h → WW ∗) and excluded at more than 99.9%confidence level by the analysis of the ATLAS and CMS collaborations [3, 4].

Nevertheless, the possibility still persists that the discovered Higgs boson is an admixtureof a scalar (CP-even) and a pseudoscalar (CP-odd) component through a CP-mixing angleφτ . Such a hypothesis is supported by several theories, like the Two-Higgs Doublet Models[5], that involve a second Higgs doublet for a total of five real scalar Higgs fields. Amongthem, three fields are neutral and can be identified as pure CP eigenstates, that do notnecessarily coincide with the mass eigenstate. If this is the case, i.e. the Higgs masseigenstates are a linear combination of the CP eigenstates, one ends up with a CP-violatingmodel [5, 6].

The possible discovery of the CP violation in the Higgs sector due to the scalar/pseudoscalarmixing would open incredible perspectives of new physics and extensions of the StandardModel should be seriously taken into account. Furthermore such a mixing would help inexplaining some of the currently unanswered questions in physics: it is well known in factthat the CP-violating parameters of the Standard Model are not enough to explain the baryonasymmetry observed in the Universe and that a new source of CP violation is needed [7–9].Finally, it is important to remark that the CP properties of the Higgs boson have been studiedso far only in the bosonic couplings (h→ ZZ∗, h→WW ∗), that are not sensitive at treelevel to a possible CP-odd component [10, 11]. For this reason an analysis carried out in the

2 Introduction

fermionic decay channel h→ ττ , as the one presented in this thesis, is crucial to confirm theproperties 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 theh→ ττ decay has the second largest branching ratio in the fermionic sector of the Higgsdecays and since the adoption of a new algorithm at the beginning of Run 2 improved thereconstruction performances of the hadronically decaying taus [12]. A detailed descriptionof this algorithm, called Tau Particle Flow, and its performances is one of the preliminarytasks 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 mixingangle 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 ofHigh-Luminosity LHC [13]. The thesis is structured as follows: in chapter 1 a theoreticalintroduction to the Standard Model of particle physics and to the Higgs boson is given. Somedetails about the Higgs boson discovery in 2012 and the status of the h→ ττ analysis are alsoprovided. 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 ofthe CP symmetry.

The description of the Large Hadron Collider, the ATLAS detector and the coordinatesystem is given in chapter 2. Every component of the ATLAS detector and its performancesare discussed in details.

The last two chapters are dedicated to my personal contribution to the analysis: chapter 3is focused on the Tau Particle Flow algorithm used in the reconstruction of the hadronicallydecaying taus. The algorithm is designed to improve the angular and energy resolutions andto classify five different tau decay modes through the identification of the individual hadronsof the tau decays. I studied the resolutions and the classification purity provided by thealgorithm 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 Carlosimulations to analyse the resonances of the ρ and a1 mesons in some of the tau decaychannels, with particular interest to the former since it is a key feature in determining the CPproperties 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 theexperimental procedure to determine the observable and the mixing angle. The experimentalobservable is reconstructed with two different methods and a combination of them, dependingon the tau decay modes. I performed a comparison of these methods to show the sensitivityof each of them to the parameter of interest. Then I built and tested a data-driven algorithm

Introduction 3

to estimate the QCD background, since Monte Carlo simulations are not reliable in thiscase, that I used to realize the observable distributions using the data collected in 2016. Inthe last part of chapter 4 I used the information from 2016 data and Monte Carlo to derivethe expected sensitivity to the mixing angle up to 1000 fb−1, which will be the luminositycollected in a three-year run of High-Luminosity LHC.

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

Chapter 1

The Standard Model of Particle Physics

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

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 withina successful context of gauge invariance which predicts the carriers of the interactionsthemselves to be particles. Masses are provided to particles by the Brout-Englert-Higgsmechanism, based on the spontaneous symmetry breaking, that introduces a new field inthe model and hence a new particle, the Higgs boson, whose existence was experimentallyconfirmed in summer 2012 by ATLAS [1] and CMS [2] experiments.

The SM contemplates three fundamental interactions: the strong, the weak and theelectromagnetic (EM) force. However, the last two interactions have been unified in a singlemodel, 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 SMso far.

The SM has provided successful explanations of several phenomena and has been testedat a very precise level. At present it is the most successful mathematical model developed inthe 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 theoryunder certain transformations. These transformations act on the fields and are called global ifthey involve the fields in the same way at all space-time points, or local if they act differently

6 The Standard Model of Particle Physics

on different space-time points. Since global transformations imply the correlation of points ofthe space-time that are not causally connected, theories are generally required to be invariantunder 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 [14]. This Lagrangian is clearly invariant under the globalU(1) symmetry, i.e. under the field transformation

ψ(x)U(1)−−→ ψ′(x) = eieθψ(x) (1.2)

where e and θ are real numbers. When the symmetry is required to be local, that is whenthe parameter of the transformation θ is a function of the space-time θ(x), one finds that anew field must be introduce to preserve the symmetry. The new field describes a new type ofparticle that can be interpreted as the mediator of the interaction related to the symmetry [15].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 quantum field theory of the electromagnetic interactiondescribed 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 aredivided into two main categories: fermions and bosons (figure 1.1). Every particle of the SMhas 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 theparticles that uniquely describe their state.

1.2.1 Fermions

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

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

1.2 Particles and Fields 7

• A quark undergoes all the fundamental interactions, including the strong force, andcannot be found free but only within bound states called mesons (quark-antiquark pairs,qq̄) or baryons (three-quark states, qqq). Mesons and baryons are grouped together in acategory 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 bythe weak interaction.

Figure 1.1 Particles of the SM [16], divided into fermions and bosons. Fermions are furthersplit 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, composedof the electron, the electron neutrino, the u quark and the d quark, are the constituents of theordinary matter. Couplings between quarks of different generations are allowed by a mixingmechanism, parametrised by the Cabibbo-Kobayashi-Maskawa (CKM) matrix:d′s′

b′

=

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

dsb

(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 andsummarises their main properties.


FermionsGeneration

Charge Interactions1st 2nd 3rd

Leptonsνe νµ ντ 0 Weak

e− µ− τ− −1 EM and weak

u c t +2/3 StrongQuarks EM

d s b −1/3 Weak

Table 1.1 Fermions of the SM and their properties. Leptons interact solely via the weak andelectromagnetic 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 ofgauge invariance, and are therefore commonly referred to as gauge bosons. The globalsymmetry underlying the SM is SU(3)×SU(2)×U(1):

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

• The SU(2)×U(1) component is the unified electroweak model, which provides a gooddescription of the EM and the weak interactions (see section 1.6). The former ismediated 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 80.4Z 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 [14]. The mass of the Z and Wbosons are taken from reference [17].

1Here and in the rest of this thesis masses and momenta are expressed in unit of c.

1.3 Quantum Electrodynamics 9

The list of the gauge bosons and the interaction they mediate is shown in table 1.2. TheHiggs particle completes the family of the bosons of the SM: its existence is crucial to providemass to particles (especially the gauge bosons W and Z) while keeping the gauge invarianceof 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 theEM 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 gaugeprinciple requires the Lagrangian to be invariant under local U(1) symmetry. This localinvariance is guaranteed by the replacement of the common derivative ∂µ with the covariantderivative 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 introducesan extra term that can be interpreted as the interaction of the fermionic field ψ with the gaugefield Aµ and that accounts for the EM coupling:

Lint =−eAµψγµψ (1.7)

The field Aµ describes the mediator of the EM force (the photons), while the constant e isrelated 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 thattakes into account the kinetic energy of the electromagnetic field. This term involves thegauge-invariant EM field strength tensor Fµν , defined as Fµν = ∂µAν −∂νAµ, and is

Lphotons =−14FµνFµν (1.8)


The complete Lagrangian of the QED is then

LQED = ψ(iγµ∂µ−m)ψ︸︷︷︸Free particle

− eψγµψAµ︸︷︷︸Interaction term

− 14FµνFµν︸︷︷︸

EM kinetic term

(1.9)

1.4 Quantum Chromodynamics

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

As already mentioned in section 1.2.1, quarks are not free in nature but bound in statescalled hadrons, i.e. mesons or baryons, that are globally colourless according to the principleof colour confinement. The hadronic structure was investigated in deep inelastic scatteringexperiments [18] and the emerging picture is that of composite particles made up not only byvalence quarks, i.e. bound quarks that carry the quantum numbers of the hadron, but also byquark-antiquark pairs (sea quarks) and gluons. The constituents of the hadronic particles aregenerally called partons. Since these partons frequently interact inside the hadronic structurethey 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 xand x+dx is expressed in terms of a Parton Density Function (PDF) [19].

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

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

ψfSU(3)−−−→ ψ′

f = eigsTaθa(x)ψf (1.10)

where ψf is the quark field (the subscript f stands for the quark flavour), gs is the strongcoupling constant, θa(x) are eight local parameters (a is an index running from 1 to 8) and Taare the eight generators of the group satisfying the commutation relations [Ta,Tb] = ifabcTc.

1.5 Weak Interaction 11

fabc are the structure constants and a possible representation of the Ta is given by theGell-Mann matrices λa: Ta = λa

2 [14].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

ψf (iγµ∂µ−mf )ψf (1.11)

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

Dµ = ∂µ+ igsTaGa,µ (1.12)

where Ga,µ are the eight gluon fields, one for every coloured combination of colour andanti-colour. Taking into account the Lagrangian of the gluon field

Lgluons =−14Gµν

a Ga,µν (1.13)

where Gµνa = ∂µGν

a − ∂νGµa − gsfabcG

µbG

νc and grouping all the terms, the resulting La-

grangian of the QCD is then

LQCD = ψf (iγµ∂µ−m)ψf︸︷︷︸

Free particles

−gsψfγµψfTaGa,µ︸︷︷︸

Interaction term

− 14Gµν

a Ga,µν︸︷︷︸Gluon kinetic term

(1.14)

1.5 Weak Interaction

The weak interaction, unlike the other fundamental interactions, affects all leptons and quarksand 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 weakcurrent 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-vanishingterm of mass in the Lagrangian of the weak interaction would theoretically break the gaugeinvariance. Nevertheless, they were found to be massive mediators [20, 21]. How theseparticles acquire a mass while leaving the general framework of gauge theories unchanged isexplained by the Brout–Englert–Higgs (BEH) mechanism [22, 23] described in section 1.7.

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


by means of the chiral projection operators PL = 12(1−γ

5) and PR = 12(1+γ

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

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

jµ ∝ ψ(gV γµ−gAγµγ5)ψ = gV j

µV +gAj

µA (1.15)

As an example, the charged weak current has the form jµweak = ψγµ(1 − γ5)ψ andtransforms partly as a vector and partly as an axial vector under Lorentz transformations.Since the 1−γ5 term 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 responsiblefor the different weak coupling of chiral particles. This particular structure of the weakinteraction is also responsible for the violation of parity and charge conjugation (see section1.9).

1.6 Electroweak Unification

The electroweak model describes the unification of the EM and weak interactions and isbased on the symmetry group SU(2)L×U(1)Y [14]. The subscript L in SU(2)L highlightsthe fact that this symmetry (and hence the related interaction) only acts on left-handedparticles, 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 T3 is the third component of the weak isospin, a quantumnumber related to the SU(2)L group.

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

ψLSU(2)L−−−−→ ψ′

L = eig2 τaα

a(x)ψL ψRSU(2)L−−−−→ ψ′

R = ψR (1.16)

where g is the interaction coupling constant, τa are the Pauli matrices and αa(x) are threefunctions of the space-time. SU(2)L has a well-known algebra and any representation of thegroup is described by a couple of numbers (the weak isospin T and its third component T3)

1.6 Electroweak Unification 13

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

χL =

(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, aresinglets (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, W a

µ (a= 1, 2, 3), that couplewith three weak isospin currents Jµ

a = g2χLγ

µτaχL. The fields of the physical chargedbosons W± can be derived by re-arranging the isospin currents in terms of charged currents:Jµ± = 1√

2(Jµ

1 ± iJµ2 ). The W±’s fields are then

W±µ =

1√2(W 1

µ ∓ iW 2µ) (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)

ψU(1)Y−−−−→ ψ′ = ei

g′2 Y β(x)ψ (1.18)

U(1)Y introduces a gauge field Bµ that mixes with the W 3µ field by means of the Weinberg

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

Zµ

)=

(cosθW sinθW

−sinθW cosθW

)(Bµ

W 3µ

)(1.19)

The Weinberg angle θW is defined by

cosθW =g√

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

in equation 1.1 with the covariant derivatives (for left- and right-handed particles) [24]


DLµ = ∂µ+ i

g

2τaW a

µ︸︷︷︸from SU(2)L

+ ig′

2Y Bµ︸︷︷︸

from U(1)Y

DRµ = ∂µ+ i

g′

2Y Bµ︸︷︷︸

from U(1)Y

(1.21)

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

W aµν = ∂µW

aν −∂νW a

µ −gϵabcW bµW

cν

Bµν = ∂µBν −∂νBµ

(1.22)

The Lagrangian of the electroweak model is finally

LElectroweak = ψ(iγµ∂µ−m)ψ︸︷︷︸Free particle

− g

2ψLγ

µτaψLWaµ − g′

2Y ψLγ

µψLBµ︸︷︷︸Left-handed interaction

− g′

2Y ψRγ

µψRBµ︸︷︷︸Right-handed interaction

− 14W a,µνW a

µν −14BµνBµν︸︷︷︸

Kinetic term

(1.23)

1.7 The Brout–Englert–Higgs Mechanism

Although the electroweak model described in section 1.6 provides a satisfying explanationof the unification of the EM and weak interactions, it leaves an open question: how canthe W± and Z gauge bosons acquire a mass? The gauge theories require the gauge bosonsto be massless and any insertion of a mass term in the Lagrangian by hand would violatethe gauge invariance. A new mechanism, first described by Brout, Englert and Higgs in1964 [22, 23], accounts for the mass of the vector bosons in the general framework of gaugeinvariant theories: they observed that a mass term can arise if the vacuum state is degeneratewith respect to the symmetry group of the theory and the symmetry is spontaneously broken.

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

φ=1√2

(φ+

φ0

)(1.24)

The Lagrangian of this doublet takes the form

LHiggs = (Dµφ)†(Dµφ)−V (φ†φ) (1.25)

1.7 The Brout–Englert–Higgs Mechanism 15

where Dµ is the covariant derivative of formula 1.21 and V (φ†φ) is the potential that givesrise to the symmetry breaking:

V (φ†φ) = µ2φ†φ+λ(φ†φ)2 (1.26)

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

(φ†φ)min =−µ2

2λ=v2

2where v2 :=−µ

2

λ(1.27)

v is the vacuum expectation value of the field φ. An example of the Higgs potential 1.26 witha 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 theSU(2)L×U(1)Y symmetry.

The ground state is then degenerate with the same SU(2)L×U(1)Y symmetry of theelectroweak model: the spontaneous symmetry breaking arises when only one of the infiniteminima of the potential is chosen. With no loss of generality the vacuum configuration canbe chosen as

φ=1√2

(0v

)(1.28)

and the field can be expanded around the minimum


φ=1√2

(0

v+h(x)

)(1.29)

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

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

MW =12vg MZ =

12v√g2 +g′2 Mγ = 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 followingequation:

MW

MZ= cosθW (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 isinstead predicted from measurements of MW and g to be v = 246 GeV [14].

1.7.1 The Fermion Masses

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

−mψψ =−m(ψRψL+ψLψR) (1.32)

and it is not invariant under SU(2)L×U(1)Y transformations, since left-handed fermionsare SU(2) doublets whereas right-handed fermions are SU(2) singlets. However, it canbe shown [14] that the combination ψLφ is invariant under SU(2)L; when combined witha right-handed singlet, ψLφψR (and its hermitian conjugate ψRφ

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

†ψL)

satisfies SU(2)L×U(1)Y gauge symmetry.

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 derivedaccordingly to the gauge symmetries of the theory, while the second term describes thecoupling between the electron and the Higgs boson itself. By defining the electron mass interms of the constant ge (known as the Yukawa coupling):

me =ge√

2v (1.34)

equation 1.33 takes the formLe =−meee−

me

vhee (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 isa 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 (tt̄H). 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 functionof the centre of mass energy. At the LHC the dominant production mode is the ggF althoughit proceeds through loops. VBF has a much smaller cross section than ggF but is of particularinterest 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 inthe 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 toggF and VBF but, unlike these two modes, benefits from a clear signature due to the presenceof additional leptons in the final state. tt̄H is the rarest production mode and is characterisedby a complex signature composed of several jets in the final state.


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 pairassociated production (d).

1.8.2 Higgs Boson Decay

Many channels are open for the Higgs decays, involving fermions and bosons. Althoughthe Higgs boson couples only to massive particles, its decays into massless particles, likephotons or gluons, are possible provided that they proceed through loops as shown in figure1.4. Since the Higgs coupling depends on the mass of the particles, the dominant diagramsinvolve 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) andW bosons (right).

Figure 1.5b shows the branching ratios of the Higgs decays as a function of the Higgsmass between 120 GeV and 130 GeV. The partial width of a Higgs decaying into a couplefermion-antifermion (H → ff̄) of mass mf at tree level is [14]:

1.8 Higgs Boson at the LHC 19

Γff̄ =NcmH

8πv2 m2f

(1−

4m2f

m2H

) 32

(1.36)

where Nc is the number of colours (Nc = 3 for quarks, Nc = 1 for leptons) and mH is theHiggs mass. Formula 1.36 shows that the decay width is larger for more massive fermionsand explains why the H → τ+τ− decay has the second largest branching ratio in the fermionsector after the b quarks. t quarks are not considered since their mass is larger than Higgsmass 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 GeVHiggs as a function of the centre of mass energy [25]: ggF (blue), VBF (red), VH (green andgrey), tt̄H (dark purple). (b) Branching ratios of the Higgs decays as a function of the Higgsmass 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 theobservation of a new particle of mass nearly 125 GeV in the search for a Higgs boson in thegolden channels H → γγ and H → ZZ∗ → 4l. The mass value measured by the combinedexperiments was 125.09±0.21(stat.)±0.11(syst.) GeV [26].

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


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], thatdenotes 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 of4.5 σ [27] and 5.9 σ [28] respectively. Figure 1.6a shows the invariant mass of the tau pairreconstructed 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), whereS is the amount of signal and B of background in the bin of the event. An excess of datawith respect to the background distributions is found around 125 GeV, that is exactly themeasured value of the Higgs boson mass in its discovery channels.

(a) (b)

Figure 1.6 (a) Distribution of the reconstructed invariant ττ mass [27] with the Missing MassCalculator algorithm [29]. Events are weighted by ln(1+S/B), where S is the amount ofsignal and B of background in the bin of the event. (b) Signal strength µ in the individual τdecay channels (from the bottom, H → τhadτhad, H → τlepτhad and H → τlepτlep) and in theindividual categories of the analysis (vbf and Boosted, described in section 4.1.2) [27]. Theoverall measurement, inclusive of all the channels and categories, is also shown.

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

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 exploitthe 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, isµ(H → ττ) = 1.4+0.4

−0.4, denoting that the observed particle is compatible with the SM Higgsboson within 1 σ.

1.9 Discrete Symmetries

Symmetries play an important role in physics since they are strictly related to conservationlaws, as expressed by Noether’s theorem, and because they allow inferences on dynamicalsystems even when a complete theory is not available. U(1), SU(2) and SU(3) are examplesof continuous symmetries, i.e. symmetries that can be reduced to the identity with continuitywhen the parameter of the transformation tends to zero. Quantum physics also contemplatesdiscrete symmetries, in which the parameter of the transformation can assume only a discretenumber 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 producesthe inversion of the three spatial coordinate axes, or in other words it connects an objectto its mirror image. Scalar and axial vector quantities remain unchanged under paritytransformations while pseudoscalar and vector quantities change sign. A single particle canbe an eigenstate of P in its rest frame (in this case parity is commonly referred to as intrinsicparity): since applying parity twice brings the system to its original state, i.e P2 = 1, theintrinsic parity of a particle is +1 or −1. Although the laws of physics were thought invariantunder parity transformations, the experiment carried out by madame Wu [30] in 1956 showedthat parity in not an exact symmetry of Nature: the EM and strong forces conserve the parityof a system but the weak force does not. To explain such a violation the V -A structure ofthe weak interaction, already discussed in section 1.5, was successfully incorporated into thetheory.

Charge conjugation C is another important discrete symmetry in particle physics: itchanges the sign of all the internal quantum numbers of a system, i.e. it changes any particleinto its antiparticle and vice versa. Since all the internal quantum numbers are exchangedby the charge conjugation, a single particle can be an eigenstate of C only if all its internalquantum numbers are 0. In this case the particle coincides with its antiparticle: the photonand the π0 are examples of such particles. Also a particle-antiparticle system is an eigenstateof C in the centre of mass frame. Like the parity, C brings the system to its original stateif applied twice: its possible eigenvalues are then +1 or −1. The aforementioned madame


Wu’s experiment also proved that C is violated by the weak interaction, although it is knownto 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 CPis 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 Kmesons proved the violation of CP in Nature.

CP violation is accommodated by the theory provided that the CKM matrix contains atleast an imaginary term. This requirement is satisfied only if the generations of fermions areequal to or more than three: nowadays it is well known that three generations exist and CPviolation 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 JPC = 0++. CP properties of the Higgs boson have beenstudied at CERN in the bosonic sector using the channels H → γγ, H → ZZ∗ → 4ℓ andH →WW ∗ → ℓνℓν, and a remarkable agreement has been found so far with the predictionsof 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 pureCP-eigenstate Higgs bosons have been tested in the aforementioned bosonic channels andexcluded at more than 99.9% confidence level in favour of the SM hypothesis [3, 4].

Even though a pure CP-odd state (JPC = 0+−) is excluded, the possibility still remainsthat 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 thatintroduce a second doublet in the Higgs sector and enlarge the Higgs family with additionalbosons. It is important to remark that the bosonic sector of the Higgs decays is not sensitiveto the CP-odd component at tree level [10, 11] and the CP-mixture has impact only in thefermion sector. The general features of these extended models that predict a second Higgsdoublet, referred to as Two-Higgs-Doublet Models (2HDMs), are described in the followingof 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

1.10 CP in the Higgs Sector 23

such extended models: the main one is probably that they might allow for a new source ofCP violation that affects the Higgs sector [5] (unlike the CP-conserving SM Higgs). TheCP-violating phase of the CKM matrix is not sufficient to explain the baryon asymmetry ofthe Universe, but the new source of CP violation introduced by the 2HDMs may do it [7–9],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 thelow-energy supersymmetric models [10].

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+

1v1+ϕ1+iχ1√

2

)φ2 =

(ϕ+

2v2+ϕ2+iχ2√

2

)(1.37)

where ϕ1,2 and χ1,2 are real fields, ϕ+1,2 are complex fields and v1,2 are in general complex

values.The most general 2HDM potential is then:

V =µ21(φ

†1φ1)+µ

22(φ

†2φ2)−

[µ2

3(φ†1φ2)+h.c.

]+

12λ1(φ

†1φ1)

2 +12λ2(φ

†2φ2)

2 +λ3(φ†1φ1)(φ

†2φ2)+λ4(φ

†1φ2)(φ

†2φ1)

+12

[λ5(φ

†1φ2)

2 +λ6(φ†1φ1)(φ

†1φ2)+λ7(φ

†2φ2)(φ

†1φ2)+h.c.

] (1.38)

where µ21, µ2

2 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) [6], which are strongly sup-pressed experimentally. Such a bothering problem can be avoided by invoking an appropriateZ2 discrete symmetry and extending it to the fermion sector [33]. This symmetry is realizedfor some choice of basis by the transformations of the doublets φ1 → φ1 and φ2 →−φ2, andgives rise to four different types of Yukawa interactions (and hence four different types of2HDMs) depending on the transformations of the fermions with respect to the symmetry [5].To remove the FCNCs from the model it is sufficient that all fermions of given charge coupleto no more than one Higgs doublet [33].


The Z2 symmetry 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 isallowed to be softly broken, i.e. a basis exists in which λ6 = λ7 = 0 but µ2

3 ̸= 0 [6]. In thefollowing the softly-broken Z2 symmetry scenario will be explored, since it allows for theintroduction of CP violation in a general 2HDM, provided that Im(λ5) ̸= 0 [5] as will beexplained 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 employingappropriate transformations of the fields and avoiding U(1)EM symmetry-breaking chargevacuum solutions, it is always possible to write

⟨φ1⟩=1√2

(0v1

)⟨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 βangle defined as

v1 := v cosβ v2 := v sinβ (1.41)

where v2 = v21 +v

22 = (

√2GF )

−1/2 = (246 GeV)2. One is always free to rephase φ2 in orderto set ξ = 0 [6]. In the following the Higgs field’s VEVs will be assumed real and positive.

The conditions in equation 1.39 provide a relation between Imµ23 and Imλ5 (λi5 for

simplicity) [5]:

Imµ23 =

v2

2λi5sβcβ (1.42)

where cβ and sβ stand respectively for cosβ and sinβ. Therefore Imµ23 is not independent

from λi5 and the latter may be regarded as the only source of CP violation, as it will be clearerin 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 unphysicalNambu-Goldston bosons, that provide polarization to the W± and Z bosons, and only theremaining five are related to the physical Higgs fields. Among them, two are charged (H±)

1.10 CP in the Higgs Sector 25

and three are neutral: these three may be identified for instance as two even eigenstates ofCP (ϕ1, ϕ2) and an odd eigenstate (A=−sinβχ1 + cosβχ2).

Two possible scenarios are now open: if the neutral CP-eigenstate Higgs fields alsocorrespond to the physical fields, there is no CP violation in the model. On the contrary, ifthe opposite CP-eigenstate Higgs fields mix to produce the physical Higgs bosons, then themodel 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 modelhypothesis holds [34]:

M2 =

M211 M2

12 −v2

2 λi5sβ

M212 M2

22 −v2

2 λi5cβ

−v2

2 λi5sβ −v2

2 λi5cβ M2

33

(1.43)

The terms M213 and M2

23 suggest that the mixing between the CP-even (ϕ1, ϕ2) and theCP-odd (A) neutral Higgs fields occurs if λi5 ̸= 0, that is the condition given at the end ofsection 1.10.1.

The orthogonal transformation R that diagonalizes M2 provides also the physical neutralHiggs states h1,2,3 with corresponding squared-masses M2

i :h1

h2

h3

=R

ϕ1

ϕ2

A

with RM2RT =

M21 0 0

0 M22 0

0 0 M23

(1.44)

The transformation R can be written as a product of three rotation matrices Ri of anglesαi: the first one can be used to diagonalize the upper left 2×2 block of the matrix M2, thatproduces a mixing of the CP-even fields ϕ1 and ϕ2. By applying the first rotation R1 of anangle α1 = α+π/22 in the ϕ1-ϕ2 subspace, the rotated fields are [5]

H = cosαϕ1 + sinαϕ2

h=−sinαϕ1 + cosαϕ2(1.45)

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

2The shift by π/2 is needed to match the standard convention used for the CP-conserving case [6].


R1M2RT1 =M2

1 =

M2h 0 M′2

13

0 M2H M′2

23

M′213 M′2

23 M2A

(1.46)

The off-diagonal terms M′213 and M′2

23 of matrix 1.46 are still proportional to λi5 [6].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 thematrix 1.46 is diagonal, no additional rotations are needed and the fields h, H and A are alsomass eigenstates. This is the case of a CP-conserving 2HDM.

Nevertheless, if λi5 ̸= 0 at least an additional rotation is needed to diagonalize the matrix1.46: as a consequence the mass-eigenstates are admixtures of the CP-even and CP-oddstates 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 discoveredat the LHC in 2012 may be regarded as the lightest boson among the three mass-eigenstatesthat have been derived in section 1.10.2, say h1, that for simplicity will be written h. Since itis a CP-mixture, its tree-level interaction to leptons consists of a scalar and a pseudoscalarterm, and hence the most general Yukawa Lagrangian may be written as follows [35]:

LY =− ∑ℓ=e,µ,τ

mℓ

v(cℓℓℓ+ ic̃ℓℓγ

5ℓ)h (1.47)

where mℓ is the lepton mass, ℓ is the lepton field and cℓ and c̃ℓ are constant values whichdepend on the rotation matrices Ri and the β angle. The specific Yukawa Lagrangian of theh→ ττ channel, that will be studied in this thesis, can be isolated from 1.47 and is

Lhττ =−mτ

vκτ (cosφτ ττ + isinφτ τγ5τ)h (1.48)

where κτ is known as the reduced Yukawa coupling strength and φτ is the CP-mixing anglethat parametrises the relative contributions of the CP-even and CP-odd components to theh→ ττ coupling [36].

Chapter 2

The ATLAS Experiment at the LargeHadron Collider

The European Organization for Nuclear Research (CERN), founded in 1954 and located nearthe city of Geneva (Switzerland), is one of the largest scientific institutes for the research onparticle physics. It hosts the world’s largest and most powerful particle collider, the LargeHadron Collider (LHC), which accelerates two beams of protons or heavy ions in oppositedirections. The collisions are provided in four different locations of the ring, where the maindetector experiments take place: ATLAS (A Toroidal LHC Apparatus) and CMS (CompactMuon Solenoid), two general purpose experiments devoted to the study of the Higgs bosonand the phenomena beyond the SM, ALICE (A Large Ion Collider Experiment), designedto focus on the strong interaction in the quark-gluon plasma, and LHCb, an experimentperforming 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 designedto reach a nominal centre of mass energy of

√s = 14 TeV and a peak luminosity of L =

1034 cm−2s−1 [37]. Luminosity is a fundamental parameter of a particle accelerator and isdefined by geometrical factors [38]:

L=NaNb

4πσxσykf (2.1)

where Na and Nb are the number of particles in the colliding bunches, k is the number ofbunches in the ring, f is the revolution frequency, σx and σy are the widths of the bunchesin the x and y directions in case of Gaussian profiles. Generally additional complications

28 The ATLAS Experiment at the Large Hadron Collider

(such as crossing angles, collision offset, non-Gaussian beam profile, non-zero dispersion atcollision 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 dynamicsof 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 anintegrated luminosity as

Lint =∫Ldt (2.3)

that is measured in b−1. Using the integrated luminosity, the number of events is simplygiven 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 synchrotronring) at an energy of 50 MeV;

• The Booster provides an acceleration of 1.4 GeV, after which the protons are sent tothe 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) andare accelerated to 6.5 TeV.

Inside the LHC, the beam particles are organized in 2808 bunches with a time separationof 25 ns and travel in two separate pipes kept at ultrahigh vacuum. The particles aremaintained in orbit thanks to a magnetic field above 8 T produced by superconductingelectromagnets cooled to a temperature of 1.9 K by liquid helium. The acceleration isprovided by a 400 MHz superconducting Radio Frequency (RF) cavity system. The beampipes 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).

2.1 The Large Hadron Collider 29

Figure 2.1 The CERN’s accelerator complex [41]. The acceleration starts in the Linac2 andcontinues in the Booster, PS and SPS. Finally the beam reaches the LHC at an energy of450 GeV.


Figure 2.2 Integrated luminosity delivered bythe LHC (green) and recorded by ATLAS (yel-low) during the Run 2 by September 2018 [42].

The LHC’s Run 1 started on November2009 and lasted by the end of 2012. Duringthis period LHC operated at a centre of massenergy up to 8 TeV and collected data for anintegrated luminosity of about 29 fb−1. OnFebruary 2013 it was shut down for a 2-yearupgrade, called Long Shutdown 1 (LS1), toimprove the centre of mass energy to thevalue of 13 TeV. The LHC restarted oper-ations for the Run 2 on April 2015 and isexpected to produce data until the end of2018. By September 2018, the integrated lu-minosity delivered by the LHC for the entireduration of the Run 2 is slightly more than140 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 ofits 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 measurementsof cross sections, W and top masses and CP violation.

The requirements of the ATLAS detectors strictly depend on the features of the LHC. Asan example, the high luminosity provided is necessary to allow the study of the very rareprocesses mentioned above, but one of its side effects is the production of a huge rate ofevents (about 109 inelastic proton-proton events/s at design luminosity). Another remarkableaspect to consider is hidden in the nature of the hadronic collisions themselves: the QCD jetproduction and low transverse momentum events in fact dominate over the rare processes ofinterest and impose challenges on the capability of the detectors and on the identification ofthe experimental signatures of the characteristic processes. In particular, two phenomenainterfere with the event reconstruction: the pile-up, i.e. objects associated to the event that donot come from the primary vertex where the process occurred, and the underlying events, thatare the remnants of the scattering process. Therefore the main requirements of the ATLASdetectors are:

• Fast response, resistance to radiations and high granularityFast and radiation hard electronics are required by the experimental conditions. In

2.2 General Layout of the ATLAS Experiment 31

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

• Full coverageLarge angular acceptance is needed to fully and correctly reconstruct the energy balanceof the events.

• Very good resolutions and particle identificationVery precise measurements are essential because of the rarity of the events of interest.

• Efficient trigger systemThe 40 MHz event rate must be reduced to about 1 kHz to allow the collection andstorage 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 symmetryand 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 areshown in order: the pixel detector, the Semiconductor Tracker and the Transition RadiationTracker (that compose the Inner Detector); the liquid argon electromagnetic calorimeter, thetile calorimeter, the end-cap liquid argon hadronic calorimeter and the forward calorimeter(which compose the calorimeter system); the solenoid and toroid magnets and finally themuon chambers.


2.2.1 Coordinate system

The coordinate system used to describe an ATLAS event originates from the nominalinteraction point. The z-axis is identified by the beam direction and the x-y plane is transverseto it. In particular, the y-axis is conventionally defined as pointing upward and the x-axisto the centre of the LHC. The azimuthal angle φ is defined on the transverse plane and ismeasured around the z-axis, whereas the polar angle θ is measured from the z-axis. In theanalysis it is common to define the pseudorapidity1 as

η =− ln(

tanθ

2

)(2.4)

The distance ∆R in the η-φ plane angle space is defined as ∆R=√

∆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 interactingpartons is unknown. However, the proton-proton collisions are head on, hence it is natural toassume 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 pppT = (px,py), the transverse energy EEET = (Ex,Ey) andthe missing transverse energy EEEmiss

T = (Emissx ,Emiss

y ). The magnitudes of these vectors arerespectively referred to as pT , ET and Emiss

T .According to the ATLAS nomenclature, the transverse impact parameter d0 is defined

as the transverse distance of the track to the beam axis at the point of closest approach, andthe longitudinal impact parameter z0 as the z coordinate of the track at the point of closestapproach.

2.3 Inner Detector

The ATLAS Inner Detector (ID) [44, 46–48], shown in figure 2.4, is mainly designed toprovide track measurements, excellent momentum resolution and both primary and secondaryvertex measurements for charged particles with pT > 0.5 GeV and within |η|< 2.5. It alsoprovides electron identification within |η|< 2.0. The ID is immersed in a solenoidal magneticfield of 2 T (see section 2.5).

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

1When the ultrarelativistic limit is not applicable, the rapidity, defined as y = 12 ln(

E+pzE−pz

), is used instead.

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 (theinnermost layer, called insertable B-layer, was added during the LS1, see section 2.3.1), theSemiconductor Tracker and the Transition Radiation Tracker.

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

The ID consists of three units: a barrel (the central part) where the detector layers arearranged in concentric cylinders around the beam axis, and two identical end-caps (coveringthe 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 toperform 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 eachside, between radii of 88 mm and 150 mm, allow to perform position measurements withthe required level of resolution and reconstruction of secondary vertices from the decays ofshort-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 collectedduring the Run 2, such as improvements on track reconstruction against failure of pixelmodules, higher b-tagging efficiency and better vertexing performances due to the closerlocation to the interaction point.


All the modules are identical in the barrel and the disks and are 62.4 mm long and21.4 mm wide. The three pre-existing layers, built for the Run 1, cover the region |η|< 2.5and the pixel sensors mounted on them have typical dimensions of 50×400 µm2. The IBLcovers the region |η|< 3.0 and consists of smaller pixel sensors (size 50×250 µm2). The IBLimproved the impact parameter resolution for the Run 2 by nearly 40% compared to the Run1, 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 measuredfrom 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 designedto provide eight precision measurements per track, contributing to the impact parameterresolution and pattern recognition thanks to its high granularity.

The barrel consists of four cylindrical layers between radii of 30 cm and 52 cm, and theend-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 eachmodule are glued back-to-back and rotated of an angle of 40 mrad to provide measurementson 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 spatialresolution is 17 µm in the Rφ direction and 580 µm in the z or R direction.

2.4 Calorimetry 35

2.3.3 Transition Radiation Tracker

The Transition Radiation Tracker (TRT) [54] is a straw-tube detector designed to provide alarge number of hits per track (typically 36) in the region |η|< 2.0. The tubes are arrangedparallel to the beam axis in the barrel region and radially in the end-caps and are 4 mm indiameter, with a maximum length of 144 cm. They are equipped with a 30 µm diametergold-plated W-Re wire and filled with a gas mixture of 70% Xe, 27% CO2 and 3% O2. Thestraw 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 000straws divided in two at the centre and read out at both ends. Each end-cap consists of 18wheels perpendicular to the beam axis for an overall amount of 320 000 radial straws withthe readout at the outer radius. The innermost 14 wheels cover the radial range from 64 cmto 103 cm, while the last four extend to an inner radius of 48 cm. Wheels 7 to 14 have halfas many straws per cm in z as the others, to avoid an unnecessary increase of crossed strawsand material at medium rapidity.

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

2.4 Calorimetry

The ATLAS calorimeter system is shown in figure 2.6 and is specifically designed to measurethe energy of the particles (except muons and neutrinos), together with their position fromthe energy deposits. Its fine granularity allows also good jet reconstruction and Emiss

T

measurements. The calorimeter system consists of an electromagnetic calorimeter (EMCal)and a hadronic calorimeter (HadCal), plus forward calorimeters (FCal) to provide bothelectromagnetic 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


Figure 2.6 The ATLAS calorimeter system [55], composed of a liquid argon electromagneticcalorimeter and a hadronic calorimeter.

separated by a small gap (6 mm) at η = 0, covering the region |η|< 1.5, and two end-caps,covering the region 1.4 < |η|< 3.2, each subdivided into two coaxial wheels. The EMCalis based on lead and liquid argon (LAr) as, respectively, absorber and active material: in-cident particles (mostly electrons and photons) interact with the lead layers producing anelectromagnetic shower of secondary particles, whose energy is then lost via ionization andcollected in the LAr layers.

The barrel is segmented into three regions, for a total depth of > 22 radiation lengths2:the innermost layer is finely segmented in η (granularity: ∆η×∆φ= 0.003×0.1) to optimizethe γ/π0 separation, the middle layer collects most of the energy deposited by electron andphoton showers (∆η×∆φ= 0.025×0.025), the third and last layer provides measurements ofthe tails of high-energy showers (∆η×∆φ= 0.050×0.025). A thin presampler layer of LAr(11 mm in depth) is placed in front of the first barrel layer in the range |η|< 1.8 and is usedto correct showers for upstream energy loss, with a granularity of ∆η×∆φ= 0.025×0.1. Aschematic view of a barrel module is depicted in figure 2.7.

In the precision region (1.5 < |η|< 2.5), the end-caps, as well as the barrel, are dividedinto three longitudinal layers, for a total depth of > 24 radiation lengths: the front layer,

2The radiation length is defined as the length traveled by a particle in an absorber material after which itsenergy is reduced by a 1/e factor due to bremsstrahlung.

2.4 Calorimetry 37

Figure 2.7 Sketch of a barrel module (η = 0) of the EMCal [56]. It is possible to see thepresampler and the three different layers. The granularity of each layer is also shown.

finely segmented with strips in the η direction (∆η×∆φ = 0.005×0.1), a middle layer tocollect most of the shower energy (∆η×∆φ= 0.025×0.025), and a back layer to providemeasurements of high-energy showers (∆η×∆φ= 0.050×0.025). The outermost region ofthe outer wheel (|η| < 1.5) and the inner wheel (2.5 < |η| < 3.2) are instead segmented inonly two longitudinal layers and have a coarser transverse granularity. A thin LAr presampleris implemented in front of the end-cap calorimeter to improve the energy measurements inthe range 1.5 < |η|< 1.8 (∆η×∆φ= 0.025×0.1).

The energy resolution of the EMCal is about

σEE

=10%√E

+0.2%

where E is the measured energy.The accordion geometry of the EMCal allows a perfect azimuthal coverage in φ; however,

the presence of transition regions in the η direction results in a degradation of the detectorperformances in specific ranges (crack regions):

• At η = 0, due to the 6 mm gap between the two half-barrels. The crack region extendsfor |∆η|< 0.01.


• At |η|∼1.5, corresponding to the transition between the barrel and the end-caps (1.37<|η|< 1.52).

• At |η|∼2.5, between the outer and inner wheel of the end-cap calorimeters, over aregion of size |∆η|< 0.01.

2.4.2 Hadronic Calorimeter

The HadCal surrounds the EMCal and is designed to provide measurements of energy andposition of hadrons: although charged hadrons deposit most of their energy in the EMCal,part of it is lost through strong nuclear inelastic processes and is detected by the HadCal.An important parameter of the hadronic calorimeters is their thickness: ATLAS HadCal hasa total thickness of 11 interaction lengths3 λ, which are enough to ensure good hadronicshower containment and to reduce the punch-through of hadrons into the muon system (seesection 2.6). The ATLAS HadCal is composed of three units based on different techniques:the central unit, the end-caps and the forward calorimeters.

Figure 2.8 Sketch of the ATLAS Tile Calorime-ter [44], consisting of iron absorbers and scin-tillating tiles. Wavelength shifting fibres shiftthe wavelength of the radiation and deliver itto the PMT’s, where the signal is amplified.

The central unit, commonly referred toas Tile Calorimeter (TileCal), consists ofa barrel (|η| < 1.0) and two extended bar-rels (0.8 < |η|< 1.7). It uses iron as the ab-sorber and scintillating tiles as the active ma-terial: both sides of the scintillating tiles areread out by wavelength shifting fibres, whichshift the radiation wavelength to larger val-ues and deliver it to two separate photomulti-pliers (PMT’s), that collect the photons fromthe detector and amplify the signal. TileCalextends from an inner radius of 2.28 m to anouter radius of 4.25 m and is segmented inthree layers in depth of thickness 1.5 λ, 4.1λ, 1.8 λ in the barrel and 1.5 λ, 2.6 λ, 3.3 λin the extended barrel. The resulting gran-ularity is ∆η×∆φ= 0.1×0.1 (0.2×0.1 inthe last layer). A 68 cm gap between the bar-rel and the extended barrels provides space

3The interaction length is defined as the mean free path for nuclear interactions: it is the length after whicha beam of hadrons is reduced by a factor 1/e in the matter.

2.5 Central Solenoid 39

for cables and services from the innermost detectors. A sketch of a TileCal section is depictedin figure 2.8.

Each end-cap consists of two independent wheels of outer radius 2.0 m and covers therange 1.5< |η|< 3.2. The end-caps are LAr-copper detectors: copper plates are interleavedwith a 8.5 mm LAr gap, which provides the active material. The granularity of the end-caphadronic calorimeters is ∆η×∆φ = 0.1×0.1 for 1.5 < |η| < 2.5 and ∆η×∆φ = 0.2×0.2for 2.5 < |η|< 3.2.

The forward calorimeter extends in the region 3.1< |η|< 4.9 (a very high-level radiationregion) and is designed to provide almost complete coverage: this is necessary to ensure goodmeasurements of Emiss

T , important for many physics signatures and Beyond Standard Model(BSM) searches. The FCal is segmented in depth in three layers and, like the end-caps, usesLAr as the active material and copper (tungsten) as the absorber material in the first layer(second and third layers). The granularity of the FCal is ∆η×∆φ≃ 0.2×0.2.

2.5 Central Solenoid

Figure 2.9 shows a scheme of the ATLAS magnet system, composed of the Central Solenoid(CS) described in this section and the toroid magnets used for the muon detection (see section2.6).

Figure 2.9 The ATLAS magnet system. The CS, the barrel toroids and the end-cap toroids(see section 2.6) are shown.

The barrel EMCal (see section 2.4.1) is contained in a barrel cryostat, which surroundsthe ID cavity and houses the CS. The CS extends over a length of 5.8 m and has an inner


(outer) diameter of 2.46 m (2.56 m). It is designed to provide a 2 T axial magnetic field alongthe z-axis. Since its position is right in front of the EMCal, minimum material requirementsare needed to achieve satisfying calorimeter performances (the CS covers a total length of∼0.66 radiation lengths at normal incidence). The coil is a single layer wound with a NbTisuperconductor specially developed to produce a high field while optimising thickness. Themagnets are cooled down to a temperature of 4.5 K by a helium cryogenic system.

Particles with momentum lower than 400 MeV are so tightly curved by the magneticfield that they do not move far enough away from the interaction point and are not detected.

2.6 Muon Spectrometer

Since ideally muons are the only charged particles able to traverse the detectors described inthe previous sections, the Muon Spectrometer (MS), depicted in figure 2.10, is the outermostdetector and is specifically designed for muons.

Figure 2.10 The ATLAS Muon Spectrometer [57]. The tracking (MDT and CSC) andtriggering (RPC and TGC) chambers, described in the following of this section, are shown.

It is based on the deflection of muon tracks in eight large superconducting air-core toroidmagnets, located symmetrically around the beam axis (figure 2.9): for |η|< 1 the magnetic

2.6 Muon Spectrometer 41

bending is provided by the large barrel toroid and for 1.4< |η|< 2.7 by two smaller end-capmagnets inserted into both ends of the barrel toroid. For 1.0 < |η| < 1.4 (the transitionregion) the magnetic deflection is provided by a combination of barrel and end-cap fields.The magnetic fields are designed to be mostly orthogonal to the muon trajectories. In thecentre of the detector (η = 0), a gap has been left open to allow for the passage of servicesand cables from the ID, the CS, and the calorimeters.

The MS has two main functions: triggering and high precision tracking. Over most of theη range (|η|< 2.0), track coordinates in the principal bending direction of the magnetic field(the so called precision coordinate) are measured by Monitored Drift Tubes (MDTs), whereasat larger pseudorapidities (2.0 < |η|< 2.7) and close to the interaction point Cathode StripChambers (CSCs) are used because of their capability to cope with higher background rates.The trigger system covers the pseudorapidity range |η| < 2.4: Resistive Plate Chambers(RPCs) are used in the barrel and Thin Gap Chambers (TGCs) in the end-cap regions. Thetrigger system is mainly dedicated to provide:

1. Bunch crossing identification;

2. Well defined pT thresholds;

3. Measurements of the muon coordinates in the direction orthogonal to that measured bythe precision chambers (transverse coordinate).

Monitored Drift Tubes

Precision measurements of the track coordinates in the region |η|< 2.7 are demanded to theMDT chambers [58], composed of two multilayers of tubes. Each tube, made out of Al, is30 mm in diameter and has a length from 70 cm to 630 cm. The central anode is a W-Rewire of 50 µm diameter. The chambers are filled with a mixture of gas (93% Ar, 7% CO2

and a small admixture of water vapor), which provides a maximum drift time of 700 ns and asingle wire space resolution of ∼80 µm.

Cathode Strip Chambers

In the range |η|> 2.0, where the rate of events per square centimeter exceeds the limit forsafe operation of the MDTs, CSCs [59] are used, which combine high spatial, time anddouble track resolution with high-rate capability and low neutron sensitivity. They consistof two disks of eight chambers each (eight small and eight large). The CSCs are multiwireproportional chambers with the wires oriented radially and both cathods segmented, one


with the strips perpendicular to the wires (providing the precision coordinate) and the otherparallel to the wires (providing the transverse coordinate). The chambers are filled with anon-flammable mixture of gas, 30% Ar, 50% CO2 and 20% CF4. The low sensitivity to theneutron background is explained by the absence of hydrogen in the gas mixture.

The position of the track is obtained by interpolation between the charges induced on theneighbouring cathode strips. The spatial resolution of this detector is ∼60 µm and the totalelectron drift time is 40 ns.

Resistive Plate Chambers

The RPCs are gaseous detectors adopted by the trigger system in the barrel region andconsisting of two parallel plates at a distance of 2 mm filled with a mixture of (94.7%C2H2F4, 5% Iso-C4H10 and 0.3% SF6). No wires are involved in these detectors. A view ofthe RPCs is presented in figure 2.11. They are located in three concentric cylindrical layers(called stations), each consisting of two independent detector layers measuring both η and φ.

Figure 2.11 A schematic view of the barrel sector of the MS showing the RPCs (coloured) andthe MDTs [44]. In the middle layer, RPC1 and RPC2 are below and above their respectiveMDT partner. In the outer layer, the RPC3 is above the MDT in the large and below theMDT in the small sectors.

The coincidence of the inner and outer RPCs permits to select high momentum tracksin the range 9–35 GeV (high-pT trigger), while the two inner chambers provide the low-pTtrigger in the range 6–9 GeV. The RPCs have a typical spatial-time resolution of 1 cm×1 ns.

2.7 Trigger and Data Acquisition System 43

Thin Gap Chambers

The TGCs provide the low-pT and high-pT trigger in the end-cap regions and measurethe transverse coordinate of the muons. The TGCs are designed similarly to multiwireproportional chambers, but with the anode wire pitch larger than the cathode–anode distance.The anode wires are arranged parallel to the MDT wires and provide the trigger information,while readout strips are arranged orthogonal to the wires and are also used to measure thetransverse coordinate. The chambers have a cathode-cathode distance (gas gap) of 2.8 mm,a wire pitch of 1.8 mm and a wire diameter of 50 µm. The quenching gas adopted for theTGCs is a mixture of 55% CO2 and 45% n-pentane (n-C5H12).

2.7 Trigger and Data Acquisition System

Due to the high luminosity of the LHC, the trigger system is required to select interestingevents while keeping a low rate of background. The Trigger and Data Acquisition (TDAQ)system [60], shown in figure 2.12, is implemented in two levels of online event selection, asdescribed below.

Level 1 Trigger

The hardware-based Level 1 (L1) trigger is the first step in the triggering process and isresponsible for reducing the huge event rate (∼109 Hz) to about 100 kHz (75 kHz for theRun 1) by searching for high transverse-momentum muons, electrons, photons, jets and largeEmissT . The L1 trigger exploits the coarse information from a subset of detectors, such as the

RPCs and TGCs for high-momentum muons, and reduced-granularity information from allthe calorimeters (EMCal and HadCal).

In each event, the L1 trigger also defines one or more Regions of Interest (RoI), i.e. theregions in η and φ within the detector where the selection has identified interesting features.The RoI are then elaborated by a second triggering level, the High-Level Trigger.

High-Level Trigger

The next triggering level, the High-Level Trigger (HLT), provides a further selection of theRoI, lowering the event rate to roughly 1.5 kHz (200 Hz for the Run 1). The HLT exploitsmore precise information than the L1 trigger.

The distinction between the Level 2 Trigger (L2) and the Event Filter (EF), two separatesteps that composed the HLT during the Run 1, has been removed for the Run 2 and the


processes have been conceptually merged [61]. This upgrade increases the capacity andflexibility of the system, allowing event building to occur even faster than the previousmaximum rate of 7 kHz.

Readout System

The Readout System (ROS) is the first stage of the Data Acquisition (DAQ) system andis designed to buffer the events that pass the L1 trigger selection and wait for the HLT toperform a more detailed processing. The events that also pass the HLT selection are sent tothe data logger for permanent storage.

Data Logger

The data logger is responsible for aggregating large volumes of event data accepted by theHLT for further processing into a standardised file format to be stored permanently at theCERN computer centre.

Figure 2.12 Scheme of the TDAQ system in ATLAS [62]. The L1 trigger operates a firstselection of the events and defines the RoI, whose information is buffered in the ROS. TheHLT performs a more detailed selection and instructs the ROS to remove the data or to sendthem to the data logger for the storage.

2.8 Reconstruction of Taus with the ATLAS Detector 45

2.8 Reconstruction of Taus with the ATLAS Detector

With a decay length of cτ = 87 µm [17], taus do not live enough to reach any detector, andsince they may decay both leptonically or hadronically, almost all detectors are involved inthe reconstruction of tau leptons. In particular, charged hadrons from hadronically decayingtaus are tracked in the ID and release their energy in the EMCal and in the HadCal; neutralhadrons, mostly consisting of π0’s, do not leave traces in the ID and are detected in theEMCal as pairs of photons produced from π0’s decays (cτ = 25 nm [17] for π0’s).

The detection of leptonically decaying taus depends on the leptons produced: electronsare tracked in the ID and release energy in the EMCal through an EM shower, muons aretracked in the ID, release a small amount of energy in the EMCal and are mainly detected inthe MS. All neutrinos involved in tau decays remain undetected, due to the low interactioncross section.

A complete description of the hadronically decaying tau reconstruction is provided inchapter 3, where the Tau Particle Flow algorithm implemented by the ATLAS collaborationfor this purpose is described.

Chapter 3

Hadronic Tau Substructure

Belonging to the third lepton generation, taus play an important role in search of new physics[63–65] or confirmations of the SM. Furthermore with a mass of 1.777 GeV [17] taus havethe highest Yukawa coupling to the Higgs boson in the leptonic sector and one of the highestat all.

At the beginning of the Run 2, the tau analysis at ATLAS benefited from the adoption ofa new reconstruction algorithm, the Tau Particle Flow, that combines calorimeter and trackerinformation to provide better resolution performances and to classify the tau’s hadronic decaymodes. The Tau Particle Flow has opened new perspectives in the tau sector: this thesiswill focus on one of them, the study of the CP properties of the Higgs boson in the h→ ττ

decays, a field of analysis that is expected to show significant improvements thanks to thenew algorithm.

This chapter is intended to provide a general overview of the hadronically decaying tauanalysis at ATLAS and refers to data collected by the experiment in 2017 with an integratedluminosity of 32.8 fb−1 at

√s = 13 TeV. The analysis is carried out on Z → ττ events

(referred to as signal for the purpose of this chapter), where a tau is required to decayhadronically and the other leptonically into a muon. After providing some information abouttau leptons and their properties in section 3.1, the event selection and the process sampleare discussed (section 3.2.1), followed by the description of the background estimationmethod for this analysis in section 3.3. The Tau Particle Flow algorithm (concerning thetau reconstruction and decay mode classification) and tau identification against QCD jetbackground are illustrated in details in section 3.4 and 3.5 respectively. Section 3.6 isdevoted to the study of the ρ meson resonance in the hadronic tau decays, whose importanceis crucial in the analysis of the CP properties of the Higgs boson in the tau sector (asexplained in chapter 4), and finally section 3.7 completes the study of the hadronic tau withthe reconstruction of the a1 resonance.

48 Hadronic Tau Substructure

3.1 Tau Leptons at ATLAS

Since it is heavier than other leptons and light hadrons such as pions and kaons, tau is allowedto decay into leptons (figure 3.1a) or hadrons (figure 3.1b): the former is referred to as τlep

and the latter as τhad. Moreover, the visible part of the tau decays, i.e. all particles butneutrinos, is denoted as τvis, and more specifically τhad−vis refers to the visible part of ahadronically decaying tau.

(a) (b)

Figure 3.1 Feynman diagrams of the leptonic (a) and an example of a hadronic (b) decay ofτ−.

The hadronic decays cover nearly 65% of all possible tau decays; among these 90%consists of pions and only 10% of other hadrons (mostly kaons). The main decay channelsof taus are listed in table 3.1 with the corresponding branching ratios; the last column shows,for each hadronic decay, the percentage relative to all the hadronic modes. In the followingany τhad will be supposed to decay into pions and contaminations of other hadrons in thefinal state will be neglected.

The tau analysis at ATLAS is based on five different types of τhad Decay Modes (DMs),classified according to the number of charged tracks (prongs) and neutral particles of thedecay. These modes are labelled as follows:

• 1p0n: τ → π± ντ

Decays where τhad−vis is only composed of a charged π±.

• 1p1n: τ → π±π0 ντ

Decays where τhad−vis is composed of a charged π± and only one π0.

• 1pXn: τ → π± ≥ 2π0 ντ

Decays where τhad−vis is composed of a charged π± and at least two π0’s.

• 3p0n: τ → 3π± ντDecays where τhad−vis is composed of 3 charged π±’s but no π0’s.

3.2 Event Sample 49

• 3pXn: τ → 3π± ≥ 1π0 ντ

Decays where τhad−vis is composed of 3 charged π±’s and at least one π0.

The classification in five DMs is one of the tasks of the Tau Particle Flow algorithm that,unlike the previous algorithm, combines tracker and calorimeter information to reconstructand identify the individual π0’s in the τhad decays.

A tau identification Boosted Decision Tree (BDT) is trained separately for 1-prong and3-prong τhad−vis to discriminate between τhad decays and QCD jet background: three workingpoints labelled loose, medium and tight are defined according to different tau identificationefficiencies [66], as described in more details in section 3.5. A fourth working point, labellednoID, is defined when a high identification efficiency is required (very poor QCD rejectionin this case).

Decay type Decay channel Branching ratioPercentage of

hadronic decays

Leptonicτ → e νe ντ 17.8% —τ → µ νµ ντ 17.4% —

Hadronic

τ → π± ντ 10.8% 16.7%τ → π±π0 ντ 25.5% 39.4%τ → π±2π0 ντ 9.3% 14.4%τ → π±3π0 ντ 1.0% 1.5%τ → 3π± ντ 9.0% 13.9%τ → 3π±π0 ντ 2.7% 4.2%τ → others 6.5% 10.0%

Table 3.1 Main tau decay channels with the corresponding branching ratios and, in case ofhadronic decays, the percentage relative to all hadronic channels [17].

3.2 Event Sample

3.2.1 Event Selection Criteria

The Z → τlepτhad event selection follows a tag-and-probe approach [12]: events are triggeredby the presence of a muon from a leptonic tau decay (tag) and are required to contain aτhad−vis candidate (probe) with pT > 20 GeV and a unit charge opposite to that of the muon.The τhad−vis candidates are built from jets reconstructed with the anti-kT algorithm [67] witha radius parameter ∆R < 0.4. The jet-cone is divided into a core region (∆R < 0.2 from


the initial jet-axis) and an isolation region (0.2 < ∆R < 0.4): the jet algorithm is seeded bythree-dimensional clusters of calorimeter cells, called TopoClusters, that satisfy pT > 10 GeVand |η| < 2.5 within the core region [68]. Tracks reconstructed in the inner detector arematched to the τhad−vis candidate if they are in the core region and satisfy the followingcriteria: pT > 1 GeV, at least two associated hits in the pixel detector and at least sevenhits in total in the pixel and SCT detectors. Additional requirements are imposed on thedistance of closest approach of the tracks to the tau primary vertex in the transverse plane(|d0|< 1.0 mm) and longitudinally (|z0 sinθ|< 1.5 mm) [68].

3.2.2 Signal and Background Processes

The events of interest in the analysis of the hadronic tau substructure are represented byZ → τhadτlep, that will be referred to as signal for the purposes of this chapter. The mainbackground sources are QCD processes, W+jets, Z→ ℓℓ (decays of Z into leptons ℓ differentfrom taus), single top quark and top pair tt (strongly suppressed by a very efficient b-veto).Figure 3.2 shows some of the Feynman diagrams of the W+jets and top backgroundsmentioned above.

(a) (b) (c)

Figure 3.2 Feynman diagrams for some of theW+jets (a), single top (b) and tt (c) background.

3.2.3 Monte Carlo Simulations

Comparisons between collected data and theoretical expectations, of crucial importance inthe physics analysis, are made possible by Monte Carlo (MC) simulations of signal andbackground events. The simulation of hadronic collisions at the LHC involves two maincomponents: the description of the elementary hard process and the parton shower andhadronisation. In both cases the fractions of the proton momentum carried by the partonsare described by Parton Density Functions (PDFs). The computation of the hard scatteringprocesses, i.e. the basic parton-parton interactions, involve the calculation of a MatrixElement (ME) based on the theoretical Lagrangian and on Feynman rules at a fixed order

3.3 OS-SS Background Estimation Method 51

in perturbation theory. This task is performed by a parton-level generator, for examplePOWHEG-BOX [69]. Although this provides satisfying descriptions of the hard scattering, thesame method cannot be exploited for the calculation of the parton shower which is composedof gluon radiation and quark hadronisation and is non-perturbative. The modelling of theparton shower is demanded to another generator, for example PYTHIA 8 [70], that uses a setof tuned parameters to simulate the final states of the hadronisation process.

For the generation of the single top samples [71], the POWHEG-BOX V1 [72] generatoris used for the hard scattering processes with the PDF set CT10f4 [73] for the Next toLeading Order (NLO) matrix element calculations. The parton shower, hadronisation andthe underlying events are simulated using PYTHIA 6.428 [74] with the PDF set CTEQ6L1[75] and the corresponding Perugia 2012 set of tuned parameters [76]. The tt samples[71] are generated by POWHEG-BOX V2 [72] interfaced to PYTHIA 8.186 [70] with thecorresponding A14 tune [77]. The NLO matrix element calculations use the CT10 [78] PDFset. In both cases EvtGen 1.2.0 [79] is used for the properties of b- and c-hadron decays. Thetop mass is set to 172.5 GeV.

The samples of Z→ ττ , Z→ ℓℓ and W+jets are generated using POWHEG-BOX V2 [72]interfaced to PYTHIA 8.186 [70] parton shower model. The CT10 [78] PDF set is used in thematrix element. The AZNLO set of tuned parameters [80] is used, with PDF set CTEQ6L1[75], for the modelling of non-perturbative effects. The EvtGen 1.2.0 program [79] is usedfor the decays of b- and c-hadrons. PHOTOS++ 3.52 [81] is used for the QED emissionsfrom electroweak vertices and charged leptons. The samples are normalized with the Next toNext to Leading Order (NNLO).

The QCD background is not simulated and its estimation exploits a data driven methoddiscussed in section 3.3. A full simulation of the ATLAS detector response and its perfor-mances is based on the toolkit GEANT4 [82].

3.3 OS-SS Background Estimation Method

The main source of background is composed of QCD jets misidentified as τhad candidates inmulti-jet events. This type of background is not simulated, due to the difficulties arising in itsmodelling, but it is estimated via a data-driven method. This method, called OS-SS method[83], consists of extrapolating the QCD background from the data themselves in a dedicatedregion that is orthogonal to the Signal Region (SR) where the signal analysis is performed.The definition of the SR is provided in table 3.2 of section 3.3.1.

Since in the SR the tau charges of the Z → ττ events have Opposite Sign (OS), the QCDbackground is extrapolated in a region defined similarly to the SR but with the charges of


the two reconstructed particles having the Same Sign (SS): the latter region is referred to asSame Sign Signal Region (SS SR) while the former as Opposite Sign Signal Region (OS SR).The QCD estimation is simply taken as the difference between data and MC in the SS SR.

Since the SS and OS QCD backgrounds may be not in a 1 to 1 ratio, two additionaldedicated Control Regions (CRs), the OS QCD CR and SS QCD CR, are defined to evaluatethe ratio rQCD of OS QCD events to SS QCD events. This ratio is defined as follows:

rQCD =Ndata

OS −∑XNMCX OS

NdataSS −∑XN

MCX SS

(3.1)

where X stands for all the simulated processes (X =W+jets, top, Z → ℓℓ, Z → ττ ). NdataOS

(NdataSS ) is the amount of collected data in the OS (SS) QCD CR and NMC

X OS (NMCX SS) is the

amount of simulated events of the process X with taus having opposite sign (same sign). Thebasic assumption is that the difference between data and simulated processes is solely dueto QCD events. More details about the definitions of the QCD CRs are provided in section3.3.1.

In addition, to correct for a possible mis-modelling of the W+jets MC normalization,two additional CRs (enriched of W+jets) are defined, separately for opposite sign and samesign events. The same is done for the top background. In each of these CRs a normalizationfactor (the k factor) is defined as the ratio of the data to the simulated events of that specificprocess:

kWOS =Ndata

OS −∑NMC otherOS

NMCW+jets OS

kWSS =Ndata

SS −∑NMC otherSS

NMCW+jets SS

ktopOS =Ndata

OS −∑NMC otherOS

NMCtop OS

ktopSS =Ndata

SS −∑NMC otherSS

NMCtop SS

(3.2)

where ∑NMC otherOS (∑NMC other

SS ) denotes the sum over all the simulated opposite sign(same sign) processes except the specific process of that CR. As an example, in the OS W CRthe kWOS is defined by taking all the opposite sign data, subtracting all the opposite sign MCexcept the opposite sign W+jets MC and then dividing by the amount of simulated oppositesign W+jets MC. If the amount of QCD contamination in these CRs is negligible and theprocesses are well simulated, the k factors are expected to be not too different from 1.


The expected amount of data in the OS SR is finally evaluated as follows:

NdataOS =rQCD (Ndata

SS −NMCZ→ττ SS −NMC

Z→ℓℓ SS −kWSSNMCW+jets SS −k

topSSN

MCtop SS)︸︷︷︸

SS QCD background

+NMCZ→ττ OS +N

MCZ→ℓℓ OS +k

WOSN

MCW+jets OS +k

topOSN

MCtop OS

(3.3)

Equation 3.3 reads as follows: the SS QCD background is estimated by subtracting the MCof all simulated SS processes (multiplied by the respective kSS factors for W+jets and top)from the data of the SS SR and is then multiplied by the rQCD factor to obtain the OS QCDbackground. Finally, the MC of the other OS processes are added, multiplied by the kOS

factors in case of W+jets and top events.For simplicity the terms are rearranged in the following manner:

NdataOS = rQCDN

dataSS +NMC

Z→ττ +NMCZ→ℓℓ+N

MCW + jets +N

MCtop (3.4)

whereNMC

Z→ττ =NMCZ→ττ OS − rQCDN

MCZ→ττ SS

NMCZ→ℓℓ =NMC

Z→ℓℓ OS − rQCDNMCZ→ℓℓ SS

NMCW+jets = kWOSN

MCW+jets OS − rQCDk

WSSN

MCW+jets SS

NMCtop = ktopOSN

MCtop OS − rQCDk

topSSN

MCtop SS

(3.5)

3.3.1 Signal and Control Regions

The signal region is a region enriched of Z → τlepτhad events with the τlep specificallydecaying into a muon. The SR is further split into an OS SR (where the study of the hadronictau substructure is performed) and a SS SR (where the QCD background is estimated) byrequiring the two taus to have respectively opposite sign or same sign. Table 3.2 lists the cutsadopted in the definition of the SR.

Most of the cuts in table 3.2 are self-explaining. The transverse mass MT between themuon and the Emiss

T is defined as

MT (µ,EmissT ) =

√2pT (µ)Emiss

T (1− cos∆φ(µ,EmissT )) (3.6)

while the visible mass Mvis(µ,τ) between the muon and the tau is defined as the invariantmass of the reconstructed visible muon four-momentum and the τhad−vis four-momentum.


Signal Region definition

no electrons, exactly one muonpT (µ)< 40 GeV

MT (µ,EmissT )< 50 GeV

at least one taupT (τ)> 20 GeV|η(τ)|< 2.47

b-vetoMuon isolation

45 <Mvis(µ,τ)< 120 GeV∆φ(µ,τ)> 2.4

cos∆φ(µ,EmissT )+ cos∆φ(τ,Emiss

T )>−0.15

Table 3.2 List of cuts of the SR. The SR is then split into OS SR and SS SR according to thesigns of the tau leptons.

Muon isolation is provided by two variables: one is pTvarcone, defined as the sum ofthe transverse momenta of the tracks inside a cone centred around the muon direction. Theradius ∆R of this cone gets smaller with the transverse momentum of the muon:

∆R= min(kTpT,R

)(3.7)

where kT = 10 GeV is a constant and R = 0.3 is the maximum radius of the cone used.pTvarcone is required to be less than 1% of the muon pT . The other variable is calledtopoETcone and is defined as the sum of the transverse energies of the topological clusterswithin a cone of radius ∆R = 0.2 around the muon cluster. TopoETcone is required to beless than 5% of the muon pT .

The QCD CR is defined by inverting the muon isolation requirement and is further splitinto an opposite sign QCD CR and a same sign QCD CR. The amount of QCD events ineach CR is computed by subtracting all the simulated processes to the data. The ratio of theOS QCD events to the SS QCD events defines the rQCD factor (see equation 3.1) that is usedto estimate the QCD background in the signal region as described in section 3.3. The rQCD

values are plotted in figure 3.3: for each decay mode, the rQCD factor seems to increase bytightening the working point, a trend that could be expected since the working point cuts offmore SS events than OS events.

The W CR is defined by the cuts listed in table 3.2 but removing the pT (µ) cut andrequiring:


Figure 3.3 rQCD values for each decay mode and each working point computed in the QCDCRs.

• MT (µ,EmissT )> 60 GeV

• cos∆φ(µ,EmissT )+ cos∆φ(τ,Emiss

T )< 0

• EmissT > 30 GeV

The W CR is split into an opposite sign W CR and a same sign W CR where two factors(kWOS and kWSS), one for each region, are computed as described in section 3.3 and plotted infigure 3.4.

Finally, the Top CR is defined by the same cuts listed in table 3.2 but removing the pT (µ)and Mvis(µ,τ) cuts, inverting the b-veto and requiring MT (µ,E

missT )> 40 GeV. Like the

other control regions, the Top CR is also split into opposite sign and same sign where thektopOS and ktopSS factors are evaluated respectively and plotted in figure 3.5.

To illustrate the composition of each control region, appendix A shows the muon-tauvisible mass Mvis(µ,τ) for some working points and some decay modes. Furthermore, theyield tables of each control region are shown for each decay mode and each working point.

3.3.2 Performance of the Method

The background estimation method described in this section is used in the following of thischapter. Plots of the OS SR, showing the tau pT and η distributions for each decay mode and


(a) (b)

Figure 3.4 Opposite sign (a) and same sign (b) k factors of W+jets for each decay mode andeach working point.

(a) (b)

Figure 3.5 Opposite sign (a) and same sign (b) k factors of Top for each decay mode andeach working point.

3.4 Tau Particle Flow Algorithm 57

each working point, are provided in appendix B to evaluate the performances of the method.Only the statistical uncertainty is reported.

The noID working point shows a reasonable agreement between data and MC for alldecay modes, with discrepancies no larger than 10%. The performances are also satisfyingfor all working points of 1p0n and 1p1n DM, with few disagreements up to 10%. Worseperformances are shown in case of 1pXn, 3p0n and 3pXn DMs with loose, medium and tightworking points, where discrepancies up to 20% can be found.

The pT distributions give an idea of the tau energy scale at ATLAS and show that tau pTis generally no greater than 70 GeV.

The η distributions are useful for some qualitative considerations: first, they show a gaparound |η|= 1.5, corresponding to the crack regions of the calorimeter between the barreland the end-caps (see section 2.4). In addition, 1p0n and, on a smaller scale, 1p1n decaymodes show a large contamination of Z → ℓℓ events around η = 0. The poorer performancesof discrimination between taus and muons for η = 0 are due to the presence of a gap thatdivides the two half barrels of the muon spectrometer (see section 2.6).

3.4 Tau Particle Flow Algorithm

The tau reconstruction algorithm used by ATLAS during the Run 1, called Baseline, exploitedonly calorimeter information. Such an approach has two important consequences: first, thereconstruction of the tau four-momenta is not optimized, due to the worse resolution of thecalorimeter compared to the tracker at tau energy scale (pT . 70 GeV, as can be seen fromthe plots in appendix B), and furthermore the τhad decays are classified only into one- orthree-prong events, and no information is provided about the number of neutral particlesinvolved in the decay.

During the LHC’s first years of operation, the Baseline algorithm was sensitive enoughto provide an evidence for the Yukawa coupling of the Higgs boson with tau leptons [84];however it turns out to be inadequate in case of more precise measurements of the Higgsboson properties. In particular, the CP analysis, that is the purpose of this thesis, relies on aprecise reconstruction of the tau four-momenta and on a classification of the tau decay modesas pure as possible, both tasks being unachievable with the Baseline approach.

A new algorithm was developed and adopted by ATLAS at the beginning of the Run 2,the Tau Particle Flow, designed to reconstruct τhad−vis’s with pT between 15 and 100 GeV.The basic concept of the new method, unlike the Baseline, is to combine all possibleinformation from the tracker and the calorimeter to reconstruct the (charged and neutral)individual particles from the tau decays and their four-momenta. The new algorithm has led


to significant improvements in the tau energy and directional resolutions and it has allowedthe classification of the decay modes mentioned in section 3.1. In the following of this sectiona general description of the Tau Particle Flow algorithm and its performances is provided.

3.4.1 Pion Reconstruction and Identification

The Tau Particle Flow algorithm aims to reconstruct the τhad−vis four-momenta from theindividual decay products’ four-momenta. All the decay products are assumed to be pionsand the pion mass hypothesis is applied to them. These pions typically have pT < 20 GeVand are extremely collimated, with an average separation of ∆R≈ 0.07 [12].

π±’s are reconstructed from their tracks in the tracker system, while the π0 reconstruction,requiring the disentanglement of the π0 energy deposits in the EM calorimeter from the π±

showers, is more challenging and consists of the following steps [12]:

1. Neutral pion candidates (π0cand’s) are created by clustering cells in the EM calorimeter

in the core region of the τhad−vis cone.

2. π0cand energy (deposited in the EM calorimeter) is corrected for contaminations fromπ±’s. The energy deposited by π±’s in the EM calorimeter EEM

π± is estimated as thedifference between the energy of the π±’s from the tracking system Etrk

π± and the energydeposited in the hadronic calorimeter EHAD

π± :

EEMπ± = Etrk

π± −EHADπ±

To calculate EHADπ± , all clustered energy deposits of the core region in the hadronic

calorimeter are assigned to the closest π±.

3. EEMπ± is subtracted from the energy of the closest π0

cand if it is within ∆R= 0.04 of theπ±. The subtraction is performed at cell level using average shower shapes derivedfrom simulations.

4. After the energy subtraction, the clustering procedure is repeated and different clustersmay be identified.

At this point many of the π0cand’s do not actually originate from π0’s, but rather from

π± remnants, pile-up or other sources. After the application of a minimum pT thresholdat ∼2.5 GeV, required to increase the purity of π0’s, the background is dominated by π±

remnants. The definitive π0 identification takes advantage of a Boosted Decision Tree (BDT)that exploits the properties of the π0 clusters. π0

cand’s identified as π0’s by the BDT will be


referred to as π0ID’s. More details about the pion identification BDT are provided in reference

[12].The two photons from a π0 decay are strongly collimated, due to the high boost of the π0,

and their energy deposits generally merge into a single cluster. Nevertheless, knowing theright number of photons of a τhad−vis candidate is of crucial importance to reconstruct thecorrect number of π0

cand’s and to classify the decay mode. For this reason, the Tau ParticleFlow also provides an algorithm to reconstruct the individual photons from a π0 decay. Thealgorithm exploits the finely segmented innermost layer of the EM calorimeter (see section2.4.1) and proceeds as follows: first, local energy maxima are searched for within the jetcore region (a local maximum is defined as a single cell with ET > 100 MeV whose nearestneighbours in η both have lower ET ); maxima found in adjacent φ cells are then combined,their energy is summed and the energy-weighted mean of their φ positions is used. In anycase, it is not rare that at high pT the two photons are boosted enough to produce a singlemaximum.

The information on photon clusters’ maxima is used to improve the decay mode classifi-cation: if a τhad−vis candidate contains a π0

ID with at least three associated maxima, then theτhad−vis candidate is classified as 1pXn decay mode.

3.4.2 Decay Mode Classification

The DM classification does not rely only on counting the number of π±’s and π0ID’s, but is

significantly improved by the kinematic analysis of the tau decay products performed viaadditional BDTs. Since the most challenging aspect of the classification is to determine thenumber of π0’s, three different BDTs are trained to distinguish between the following decaymodes: 1p0n VS 1p1n, 1p1n VS 1pXn, 3p0n VS 3pXn. Which of the three tests to apply to aτhad−vis candidate is chosen as follows:

• τhad−vis candidates with no reconstructed π0cand’s are always classified as 1p0n (if only

one track is associated to the τhad−vis candidate) or 3p0n (if three tracks are associated).

• τhad−vis candidates with one associated track and at least two π0cand’s, of which at least

one is π0ID, enter the 1p1n VS 1pXn test.

• τhad−vis candidates with one π0ID classified as 1pXn by counting the number of photons

as described at the end of section 3.4.1 retain their classification and are not consideredin the decay mode tests.

• The remaining τhad−vis candidates with one or three associated tracks enter the 1p0nVS 1p1n or 3p0n VS 3pXn tests respectively.


More information about the variables used for the training of each BDT can be found inreference [12]. The Tau Particle Flow algorithm is applied to 2016 data and the provideddecay mode classification is shown in figure 3.6 for each working point, together with thecontributions of the signal and backgrounds.

(a) (b)

(c) (d)

Figure 3.6 Reconstructed decay modes with noID (a), loose (b), medium (c) and tight (d)working points.

Figure 3.7 shows instead the purity matrices of the decay mode reconstruction, calculatedusing the MC sample of Z → τlepτhad. Each column of these matrices is normalized to theunity: for each reconstructed decay mode on the horizontal axis, the purity matrices provideinformation about the decay mode composition at truth level (vertical axis).

These matrices are almost block-diagonal, since the not infrequent misidentificationsof the number of π0’s produce contaminations among same-prong events, and since con-taminations between different-prong events are suppressed by the good performances of the


tracker system. To quantify the purity of the decay mode classification, the diagonal fraction,defined as the ratio of diagonal events to all events, is evaluated for the purity matrices ofeach working point and can be read in the same plots of figure 3.7. The diagonal fraction isestimated to be ∼75% and is not too dependent on the working point.

(a) (b)

(c) (d)

Figure 3.7 Decay mode purity matrices for noID (a), loose (b), medium (c) and tight (d)working points, realized with the MC sample of Z → ττ . Each column is normalized to theunity. The diagonal fraction is computed as the ratio of the diagonal events to all events andprovides an estimation of the classification purity.


3.4.3 Tau Reconstruction

The τhad−vis four-momentum is reconstructed by summing the four-momenta of the π±’s andof the first n π0

cand’s with the highest π0 identification BDT scores, where n is determinedfrom the decay mode classification. In any case no more than 2 π0

cand’s are considered inthe 1pXn decay mode and at most 1 π0 in the 3pXn decay mode. A pion mass hypothesis isapplied to the π0

cand’s with two exceptions: if the decay mode is classified as 1p1n but there aretwo π0

ID’s, the mass of each is set to zero and both are added to the τhad−vis four-momentum,as they are most likely photons from a π0 decay; or if the decay mode is classified as 1pXnbecause three or more photons are found in a single π0

cand, only this π0cand is added and its

mass is set to twice the π0 mass. Finally, a calibration is applied to the τhad−vis energy ineach decay mode as a function of ET , to correct for a possible π0

cand energy bias [83].Figure 3.8 shows the τhad−vis residual distributions ηreco − ηtruth, φreco −φtruth and the

relative residual distribution precoT /ptruth

T evaluated with the Tau Particle Flow on a MC sampleof Z → τlepτhad, where the superscript “reco” denotes the reconstructed variable and “truth”the MC truth level. The plots are inclusive of all the decay modes and are realized with thenoID working point.

(a) (b) (c)

Figure 3.8 τhad−vis η (a) and φ (b) residual distributions and relative pT (c) residual distribu-tion evaluated with the Tau Particle Flow algorithm on a MC sample of Z→ ττ . Distributionsare normalized so that the integral over the whole range is one.

The φ distribution of figure 3.8b shows a characteristic central narrow peak, that is due toevents with no π0’s (1p0n and 3p0n). Such events in fact, involving only charged pions, areeasier to reconstruct because they do not involve the neutral pion reconstruction proceduredescribed in section 3.4.1. Figure 3.9 shows for instance the φ residual distributions in the1p0n and 1p1n decay modes: the former is clearly narrower than the latter and the same


feature is shown in the 3p0n decay mode. A similar behaviour can be found, on a smallerscale, in case of η.

(a) (b)

Figure 3.9 τhad−vis φ residual distributions in 1p0n (a) and 1p1n (b) decay modes. Eventswith no π0’s show a narrower distribution and are responsible for the central peak of figure3.8b.

The core resolutions for η, φ and pT are defined as half of the central interval that covers68% of the integral of the respective distributions. Table 3.3 reports a comparison betweenTau Particle Flow and Baseline core resolutions: the new algorithm shows remarkableimprovements compared to the old one, having four times better angular performances andsignificantly reducing the pT resolution.

Core resolutions

σ(η) σ(φ) [rad] σ(pT ) [%]

Particle Flow 0.003 0.005 7%Baseline 0.012 0.020 15%

Table 3.3 Comparison between Tau Particle Flow and Baseline performances: the τhad−visη and φ core resolutions and the pT relative core resolution are shown. Tau Particle Flowresolutions are evaluated as described in this section, Baseline ones are taken from reference[12].


3.5 Tau Identification

The reconstruction of τhad−vis candidates provides very little rejection against the QCDjet background and a tau identification step is then necessary to discriminate between jetbackground and τhad decay products. Tau identification exploits a dedicated BDT trainedseparately for 1-prong and 3-prong events with the discriminating variables [66] listed below:

• Central energy fraction (fff cent): Fraction of the calorimeter transverse energy de-posited in the region ∆R < 0.1 with respect to all energy deposited in the region∆R < 0.2 around the τhad−vis candidate.

• Leading track momentum fraction (fff−1leadtrack): The transverse energy sum de-

posited in all cells belonging to TopoClusters in the core region of the τhad−vis candidate,divided by the transverse momentum of the highest-pT charged particle in the coreregion.

• Track radius (RRR0.2track): pT -weighted ∆R distance of the associated tracks to the

τhad−vis direction, using only tracks in the core region.

• Leading track IP significance (|||SSSleadtrack|||): Absolute value of transverse impactparameter (IP) of the highest pT track in the core region, calculated with respect to thetau vertex, divided by its estimated uncertainty.

• Fraction of tracks pppTTT in the isolation region (fff trackiso ): Scalar sum of the pT of tracks

associated with the τhad−vis candidate in the region 0.2< ∆R< 0.4 divided by the sumof the pT of all tracks associated with the τhad−vis candidate.

• Maximum ∆∆∆RRR (∆∆∆RRRMax): The maximum ∆R between a track associated with theτhad−vis candidate and the τhad−vis direction. Only tracks in the core region are consid-ered.

• Transverse flight path significance pppTTT in the isolation region (SSSflightT ): The decay

length of the secondary vertex in the transverse plane, calculated with respect to the tauvertex, divided by its estimated uncertainty. It is defined only for multi-track τhad−vis’s.

• Track mass (mmmtrack): Invariant mass calculated from the sum of the four-momenta ofall tracks in the core and isolation regions, assuming a pion mass for each track.

• Fraction of EM energy from charged pions (fff track-HADEM ): Fraction of the electro-

magnetic energy of tracks associated with the τhad−vis candidate in the core region.

3.5 Tau Identification 65

• Ratio of EM energy to track momentum (fffEMtrack): Ratio of the sum of cluster energy

deposited in the electromagnetic part of each TopoCluster associated with the τhad−vis

candidate to the sum of the momentum of tracks in the core region.

• Track-plus-EM-system mass (mmmEM+track): Invariant mass of the system composedof the tracks and up to two most energetic EM clusters in the core region, wherethe four-momentum of an EM cluster is calculated assuming zero mass and usingTopoCluster seed direction.

• Ratio of track-plus-EM-system to pppTTT (pppEM+trackT ///pppTTT ): Ratio of the τhad−vis pT ,

estimated using the vector sum of track momenta and up to two most energetic EMclusters in the core region to the calorimeter-only measurement of τhad−vis pT .

Signal and background distributions of these variables are shown in appendix C; thebackground distributions are data taken from the same sign signal region that mostly consistof QCD background. The 1-prong identification BDT uses all variables but ∆RMax, Sflight

T

and mtrack; the 3-prong identification BDT uses all variables but |Sleadtrack| and f trackiso .

variable S1p S3p

fcent 38.7% 26.5%f−1

leadtrack 33.8% 24.2%R0.2

track 37.0% 38.1%|Sleadtrack| 26.6% —f track

iso 41.9% —∆RMax — 39.8%S

flightT — 53.5%

mtrack — 53.0%f track−HAD

EM 36.1% 30.6%fEM

track 25.7% 30.5%mEM+track 19.5% 32.6%pEM+trackT /pT 50.4% 29.0%

Table 3.4 List of the variables used by the tauidentification BDTs with the correspondingseparation power as defined in equation 3.8 for1-prong (S1p) and 3-prong (S31p) events. If theseparation power is not shown, the correspond-ing variable is not used for that BDT.

The separation power of each discrimi-nating variable is defined according to thefollowing formula [85]:

S =

√√√√12 ∑bins

(Nsigi −Nbkg

i )2

Nsigi +N

bkgi

(3.8)

where N sigi and Nbkg

i are the entries in thei-th bin of the signal and background distri-butions respectively, plotted in appendix C.The separation power is zero for identicalsignal and background shapes and it is onefor shapes with no overlap. The separationpowers for 1-prong (S1p) and 3-prong (S3p)events are provided in table 3.4.

As it was already mentioned in section3.1, three working points, labelled loose,medium and tight are provided correspond-ing to different tau identification efficiencyvalues [66] as shown in table 3.5. The iden-tification efficiency is defined as the fraction


of 1-prong (3-prong) hadronic tau decays that are reconstructed as 1-track (3-track) τhad−vis

candidates which also pass the BDT selection criteria. A fourth working point called noID isdefined when a very high efficiency value is required (low jet rejection is provided in thiscase).

Loose Medium Tight

ID efficiency 1-prong 0.85 0.75 0.603-prong 0.75 0.60 0.45

Table 3.5 The three working points (loose, medium and tight) and the corresponding tauidentification efficiency values [66].

3.6 ρρρ Resonance in 1p1n Decay Mode

Figure 3.10 Feynman diagram of the ρ reso-nance in the 1p1n decay mode. A similar dia-gram can be drawn in case of τ+ by reversingall particle charges.

Among all the possible τhad decays, the de-cay mode 1p1n (τ± → ντπ

±π0) is one ofthe most important since it has the largestbranching ratio and it may proceed throughthe intermediate resonance of the ρ particle,as shown in the Feynman diagram of fig-ure 3.10, that decays into a pair of π’s inalmost 100% of cases. Such a resonanceis of crucial importance in the study of theCP properties of the Higgs boson, as itwill be clarified in chapter 4, and a dedi-cated section is devoted to the reconstruc-tion of its invariant mass. The ρ meson hasa mass of Mρ = 775 MeV and a full widthof Γρ = 150 MeV [17].

The reconstruction of the ρ invariant mass is made possible by the Tau Particle Flowalgorithm. To perform the analysis of the ρ meson two competitive aspects are considered:on the one hand the reconstructed 1p1n decay mode is made purer by tightening the cut onthe π0 identification BDT score; on the other hand the data sample size is reduced by thetighter cut.

3.6 ρ Resonance in 1p1n Decay Mode 67

Figure 3.11a shows the distributions of the invariant mass residuals M truthinv −M reco

inv

(difference between the true and the reconstructed invariant mass) for different cuts on theπ0 identification BDT score in the 1p1n decay mode. The distributions are produced with aMC sample of Z → ττ requiring noID working point. By tightening the BDT score cut, thedistributions get narrower and the tails are reduced. The black points in figure 3.11b showinstead the Root Mean Square (RMS) of the same distributions of figure 3.11a as a functionof the BDT score cut. For each value of the BDT score considered, the relative size of thedata sample is indicated by the red points, whose value can be read on the right axis.

(a) (b)

Figure 3.11 (a) Distributions of the difference M truthinv −M reco

inv between the true and thereconstructed invariant mass of the 1p1n decay mode for different cuts on the π0 identificationBDT score. The distributions, realized with a MC sample of Z → ττ with noID workingpoint, are normalized so that the integral over the whole range is one. (b) RMS (black points)of the same distributions as a function of the BDT score cut. The red points refer to the redright axis and show the relative size of the sample after the cut is applied.

Given the distributions in figure 3.11, a cut on the π0 identification BDT score graterthan 0.4 is chosen to be applied to study the rho resonance. Furthermore, τhad−vis candidatesare required to pass the medium working point. The invariant mass of the pions of the1p1n DM is then reported in figure 3.12: it shows a peak around the value of the expectedρ mass with a larger width than the natural one, suggesting that resolution effects are notnegligible. To obtain the central value of the peak Mfit

ρ and its full width Γfitρ , the background

MC distributions are subtracted from the data and the range around the peak of the resultingdistribution is fitted with a Gaussian function (figure 3.13). The fit parameters are Mfit

ρ =

798±3 MeV and Γfitρ = 307±7 MeV; even thought the fitted mass value is not statistically


consistent with the expected one, systematic effects and background mismodelling are notconsidered and they may account for the discrepancy. The fit shows a very good agreement,having a chi square of χ2 = 11.1 with 12 degrees of freedom.

Figure 3.12 Invariant mass distribution of the pion pair of the 1p1n decay mode with mediumworking point. The distribution shows a peak around the expected ρ mass value (775 MeV),but the width is enlarged by resolution effects.

Figure 3.13 Gaussian fit of the central peak of the 1p1n DM invariant mass after the back-ground is subtracted from the data. A good agreement is found (χ2/ndf = 11.1/12).

3.7 a1 Resonance in 3p0n Decay Mode 69

3.7 aaa111 Resonance in 3p0n Decay Mode

Figure 3.14 Feynman diagram of the a1 par-ticle resonance in the 3p0n decay mode. Anintermediate resonance of the ρ0 is also possi-ble in the decay of the a1. A similar diagramcan be drawn in case of τ+ by reversing allparticle charges.

The hadronic tau substructure shows anotherremarkable resonance in the 3p0n decaymode: the a1 meson (figure 3.14). This parti-cle was discovered in 1964 [86], but despitethe long time passed its basic parameters arenot well known so far [87]: the values ofthe a1 mass and width determined from dif-ferent processes or by different experimentsnot unusually contradict one another. Forthis reason there are no unique definitionsof the a1 parameters and the Particle DataGroup (PDG) provides a large uncertaintyon its mass (Ma1 = 1230±40 MeV) and fullwidth (Γa1 = 250 to 600 MeV) [17]. Eventhough the a1 meson is not used in the anal-ysis of the Higgs boson CP properties, it isan important and rich component of the τhad decays and this section is dedicated to the studyof its invariant mass.

The invariant mass of the τhad−vis’s classified as 3p0n and passing the medium workingpoint is reconstructed and plotted in figure 3.15: the agreement between data and MC is quitegood almost everywhere, with the exception of the two central bins where a 20% discrepancyis found. Such a discrepancy is not investigated further since the a1 meson, as mentioned, isnot relevant for the analysis of the Higgs CP properties and the Z → ττ MC, that is mostlyresponsible for the disagreement, is not used for the fit (see below). Despite the MC badmis-modelling of some events, the a1 resonant peak is clear from data.

Following the same procedure described in section 3.6, MC background distributions aresubtracted from data and the resulting distribution is plotted in figure 3.16. However in thiscase the Gaussian fit of the central peak is found unsatisfactory (χ2/ndf = 23/7), and the fitfunction is then chosen to be the convolution of a Gaussian (to reproduce resolution effects)and a Breit-Wigner (that is expected to be the natural distribution). Such a function is shownas a red line in figure 3.16 and the fit chi square turns out to be very good: χ2/ndf = 5.9/7.

The mass and natural full width (i.e. the width of the Breit-Wigner) of the a1 particleprovided by the fit procedure areMfit

a1= 1150±2 MeV and Γfit

a1= 232±18 MeV respectively,

in accordance with the values of reference [17] within twice the reported uncertainty in caseof the mass. The Gaussian width is found to be σfit = 210±5 MeV.


Figure 3.15 Invariant mass distribution of τhad−vis’s classified as 3p0n and passing the mediumworking point: despite the Z → ττ MC mis-modelling, the a1 resonant peak is clear.

Figure 3.16 Distribution of data after MC background subtraction: the central peak is fittedwith the convolution of a Breit-Wigner and a Gaussian (red line). The fit chi square, χ2 = 5.9with 7 degrees of freedom, shows a good agreement between data and fit function.

Chapter 4

CP Scenario in Higgs Decays to TauLeptons

In section 1.10 it was pointed out that the Higgs boson discovered at the LHC in July 2012showed a remarkable agreement with the SM expectations. In particular, its CP propertieshave been studied in the bosonic sector and alternative hypotheses to pure quantum numbersJPC = 0++ have been excluded at more than 99.9% confidence level [3, 4]. Even thoughthe SM seems to provide the correct description of Nature one more time, there is onepossibility left that the physical Higgs boson is actually an admixture of CP-even and CP-odd components, as predicted by some extensions of the SM such as the Two-Higgs-DoubletModels (see section 1.10).

It is important to remark that the CP properties of the Higgs boson have been studiedonly in its interactions with the massive gauge bosons W± and Z, whose couplings to thepossible CP-odd component of the Higgs boson are strongly suppressed [10, 11]. Thus ananalysis of the fermion sector of the Higgs decays to tau leptons may provide on the one handconfirmations of the SM in this sector, on the other hand unexpected results and proofs ofNew Physics. As an example, an additional source of CP violation is needed in the attemptto explain the baryon asymmetry of the Universe [7–9], that is not justified by the currentlyknown CP-violating parameters of the SM.

This chapter describes the analysis of the CP properties of the Higgs boson in the h→ ττ

decays and gives an overview of the expectations of High-Luminosity LHC, the futureCERN’s collider that aims to crank up the performance of the LHC after 2025, increasing theluminosity by a factor of 10 beyond the LHC’s design value [13]. The analysis is carried outon data collected by ATLAS in 2016, with a luminosity of 32.9 fb−1 and a centre of massenergy

√s = 13 TeV, and it focuses on events involving both hadronically decaying taus

72 CP Scenario in Higgs Decays to Tau Leptons

(h→ τhadτhad), that are found to be more sensitive than other tau decays (h→ τlepτhad orh→ τlepτlep) [36].

4.1 Event Sample

4.1.1 Monte Carlo Simulations

The signal of this analysis is provided by the h→ τhadτhad process. Z → τhadτhad eventscompose the irreducible and largest source of background: other backgrounds taken intoaccount are Z → ℓℓ, W+jets, diboson (illustrated for instance in figure 4.1), single top andtt. All these processes are simulated with MC techniques. Multi jet events, that composesthe second largest source of background after Z → τhadτhad, are not simulated, due tothe difficulties arising in their modelling, and are estimated through a data-driven methoddescribed in section 4.4.

(a) (b) (c)

Figure 4.1 Examples of Feynman diagrams of background diboson processes involving WW(a), WZ (b) and ZZ (c).

Signal samples in VBF production are simulated at Next to Leading Order (NLO)accuracy using POWHEG-BOX V2 [72] interfaced to PYTHIA 8.186 [70] for the decay andhadronisation processes. Production by ggF, that is the most abundant source of Higgsproduction at the LHC, is simulated at Next to Next to Leading Order (NNLO) accuracyusing the POWHEG NNLOPS program [88]. The PDF4LHC15 [89] parametrization of thePDF is used in the matrix elements of all production modes. the AZNLO [80] tune is used,with PDF set CTEQ6L1 [75], for the modelling of non-perturbative effects. PHOTOS++ 3.52[81] is used for QED emissions from electroweak vertices and charged leptons. The tau pairsin the signal samples are simulated with no spin correlation: such a correlation, that is thesignature of the Higgs CP (see section 4.2), is restored by the application of event weights

4.1 Event Sample 73

computed using TAUSPINNER [90]. In such a way it is possible to use the same simulatedsignal sample to build templates for any CP-mixing angle φτ hypothesis (cf. equation 1.48and see section 4.2).

The Z → ττ , Z → ℓℓ, W+jets and diboson backgrounds are simulated using SHERPA

2.2.1 [91]. Matrix elements are calculated at leading order using the Comix [92] andOpenLoops [93] generators and merged with the SHERPA parton shower [94] using theME+PS@NLO prescription [95]. The NNPDF30NNLO [96] PDF set is used in conjunctionwith dedicated parton shower tuning developed by the Sherpa authors

For tt background the POWHEG-BOX V2 [72] generator is used with the CT10 [78] PDFsets in the matrix element calculations. Single top events are generated with POWHEG-BOX

V1 [72] using the PDF set CT10f4 [73] in the matrix element calculations. The parton shower,hadronisation and the underlying events are simulated using PYTHIA 6.428 [74] with theCTEQ6L1 [75] PDF set and the corresponding Perugia 2012 tune [76].

For all samples a full simulation of the ATLAS detector response uses the GEANT4program [82].

4.1.2 Event Selection and Categorisation

Selected events consist of τhad−vis pairs matched to the same primary interaction vertex andwith opposite charges. The pT threshold is set to p1

T (τhad−vis)> 35 GeV and p2T (τhad−vis)>

25 GeV for the τhad−vis’s with the higher and lower pT respectively [97]. An additional jetwith pT > 25 GeV and |η|< 3.2 is also required at level 1 by the calorimeter trigger, due tothe rising instantaneous luminosity [97]. The leading jet must also fulfill the offline conditionpT > 70 GeV and |η|< 3.2 [97].

Further kinematic cuts are applied offline to select signal events and get rid of a goodpart of the background. The complete list of these cuts is reported in table 4.1. Most ofthe variables are self-explaining; mMMC

ττ is the invariant mass of the tau pair, includingneutrinos, reconstructed using the Missing Mass Calculator (MMC) algorithm [29]. Higgstransverse momentum pHT is reconstructed using the vector sum of the missing transversemomentum and the transverse momenta of τhad−vis’s [27]. The cuts listed in table 4.1 identifya preselection region, on which additional cuts are applied to divide the events into twocategories, the vbf and the boosted, designed to fully exploit the signatures of the Higgsproduction modes. The vbf category targets events with a Higgs boson produced mainly viavector boson fusion and turns out to be the poorest in terms of amount of events; the boostedcategory targets events with a Higgs boson produced mainly via gluon-gluon fusion and isthe richest one. Contaminations of these production modes in the “wrong” categories are


estimated by MC to be roughly 25% (ggF in vbf category) and 15% (vector boson fusion inboosted category).

Only events in which the τhad−vis’s decay modes are classified as 1p0n or 1p1n areretained for the analysis, since the experimental observable is reconstructed only for thesemodes (see section 4.3).

Preselection Region

Two τhad−vis candidates with opposite signBoth τhad−vis candidates associated to the same reconstructed primary vertex

Decay mode classified as 1p0n or 1p1npT (τ1)> 40 GeV, pT (τ2)> 30 GeV

Both τhad−vis candidates passing medium and at least one tight working pointNo electrons or muons

∆ηττ < 1.5 and 0.8 < ∆Rττ < 2.4EmissT > 20 GeV

70 <mMMCττ < 140 GeV

Vbf RegionPreselection region requirements


At least two jets with pj1T > 50 GeV and pj2

T > 30 GeVmjj > 400 GeV

∆ηjj > 3.0 and ηj1 ×ηj2 < 0Both τhad−vis candidates must lie between the two leading jets in pseudorapidity

Boosted RegionPreselection region requirements


Failed vbf selectionpHT > 100 GeV

Table 4.1 Definitions of the preselection region and the vbf and boosted categories used inthe analysis of this chapter. Vbf and boosted aim to target different Higgs production modes.

4.2 Observable

As stated at the beginning, this chapter investigates the possibility of measuring the CP-mixing of the neutral Higgs bosons in the 2HDMs described in section 1.10. The Higgs

4.2 Observable 75

CP nature can be derived from the τ -τ spin correlations of the h→ ττ decays as illustratedbelow.

The tau pair is a fermion-antifermion system in a CP state that depends on the total spinS, according to the formula [98]

CP = (−1)S+1 (4.1)

where the possible values of the spin are S = 0 or S = 1. The CP state of the tau pairproduced in the decay of a spin zero particle, such as the Higgs boson, affects the correlationbetween the transverse spin components [99] of the tau pair, i.e. the components of the tauspin perpendicular to the tau direction, and such a correlation is accessible from the angulardistributions of the tau decay products. In fact, due to their left-handed nature, neutrinosfrom τ− → ντπ

− decays are emitted preferably in the opposite direction of the tau transversespin polarization. For the same reason, due to their right-handed nature, antineutrinosfrom τ+ → ντπ

+ decays are emitted preferably in the same direction of the transverse spinpolarization. As a consequence, pions emerging from these decays are preferably antiparallelin case of S = 1 (figure 4.2a) and parallel in case of S = 0 (figure 4.2b). The information onthe pion relative directions, together with equation 4.1, is so used to infer the CP-parity ofthe decay.

(a) (b)

Figure 4.2 Configurations of tau decay products (1p0n decay mode) in case of total spinS = 1 (a) and S = 0 (b). When S = 1 (CP-even state) pions from tau decays are preferablyantiparallel, when S = 0 (CP-odd state) pions are preferably parallel.

What stated above is true, however, in the τ -τ Zero Momentum Frame (ZMF) and in caseof two-body τhad decays (1p0n decay modes). When the frame is different from the ZMFor more than two particles are involved in the decay, the effects of the tau spin correlationsare mitigated. Things are slightly more complicated if the hadron produced from the taudecay has non-zero spin, such as the resonant ρ meson (spin J = 1) in the 1p1n decay mode(see section 3.6). The hadron may in fact be longitudinally or transversely polarised, andconstraints related to (anti)neutrino handedness and conservation of angular momentum may


result in more complicated configurations with respect to the one described in the 1p0n case.As a result, the sensitivity to the tau spin correlation is weaker in the 1p1n than in the 1p0ndecay mode but some information can be retained, provided that the pion energies from the ρdecay are individually known.

The mathematical formalism that encodes the tau spin correlation in an experimentalobservable is now discussed for the 1p0n decay mode and then extended to the general case.The interaction Lagrangian of a CP-mixing Higgs boson h to tau leptons is recalled fromequation 1.48 in section 1 to be

Lhττ =−mτ

vκτ (cosφτ ττ + isinφτ τγ5τ)h (1.48)

φτ is the CP-mixing angle that parametrises the relative contributions of the scalar andpseudoscalar components to the h→ ττ coupling [36]. The pure CP-even hypothesis (that isthe SM one) is realized for φτ = 0, while the pure CP-odd hypothesis (excluded by previousexperiments [3]) corresponds to φτ = π/2. Other values of φτ accounts for a CP-violatingcoupling.

At the beginning of this section it was shown that the CP-mixing angle affects thecorrelations between the transverse spin components of the two taus. The differential decaywidth dΓhττ is so found to depend on the tau spin components and on the angle φτ [36]:

dΓhττ ∝ 1−s−z s+z + cos(2φτ )(s−⊥ · s+⊥)+ sin(2φτ )[(s−⊥× s+⊥) · k̂−] (4.2)

where s±z and s±⊥ are the longitudinal and transverse unit spin components of τ± in theirrespective tau rest frame and k̂− is the normalized τ− 3-momentum in the Higgs boson restframe. Due to the parity violation in weak decays, the tau spin directions are reflected in thedirections of the tau decay products, so that the angle ϕCP between the tau decay planes inthe Higgs boson rest frame relays information on φτ [100]:

1Γhττ

dΓhττ

dϕCP

∣∣∣∣1p0n

∝ 1− π2

16cos(ϕCP −2φτ ) (4.3)

ϕCP ∈ [0,2π] is the angle between the tau decay planes spanned by the τ± spatial momentumin the Higgs boson zero momentum frame (ZMF) and the hadron (π± in this case) spatialmomentum in the τ± rest frame, as illustrated in figure 4.3.

When other decay modes are considered, equation 4.3 can be generalised to the following[101–105]:

1Γhττ

dΓhττ

dϕCP∝

(1− b(E−)b(E+)

π2

16cos(ϕCP −2φτ )

)dE−dE+ (4.4)

4.2 Observable 77

Figure 4.3 Illustration of the ϕCP angle between the tau leptons decay planes spanned by theτ± spatial momentum in the Higgs zero momentum frame and the π± spatial momentum inthe τ± rest frame.

HereE± are the energies of the charged prongs (π±) from τ± decays in the respective tau restframe and the functions b(E±), referred to as spin-analysing functions, encode the strength ofthe spin correlation of τ±. This strength is maximal for direct decays to pions, i.e. in the 1p0ndecay mode where b(E±) = 1 [101]; for other decays the spin-analysing functions dependon the energy of the charged prongs and change sign at approximately E± = 0.55 GeV. Itcan be shown that contributions where E± > 0.55 GeV and E± < 0.55 GeV, i.e. whereb(E±) is respectively positive and negative, are equally important [101]. As a consequence,if equation 4.4 is integrated for all E± values without an appropriate prescription, thenthe cosine component is averaged to zero and the resulting distribution is flat, like the oneexpected for the background (see section 4.2.1). However the information on the tau spincorrelation can be retained if the energies of the individual pions are known, as described insection 4.3.2. Analytic expressions of the spin-analysing functions are provided in reference[101].

Equation 4.4 suggests that the effect of φτ is a shift of the distribution Γ−1hττ

dΓhττdϕCP

of anangle 2φτ : ϕCP is thus chosen as the observable of the analysis and information on theCP-mixing angle φτ is derived from the shape of the distribution.


4.2.1 Signal and Background Distributions

The signal distribution of ϕCP is cosine-shaped (cf. equation 4.4), with a maximum cor-responding to ϕCP = π if the pure CP-even SM hypothesis holds (φτ = 0). In case of apure CP-odd Higgs boson (φτ = π/2), the distribution of ϕCP is opposite to that of the SM,showing a minimum for ϕCP = π. Any other hypothesis, i.e. 0< φτ < π/2, results in a shiftof the distribution between the ones corresponding to a pure scalar and to a pure pseudoscalarHiggs.

The main background is composed of Z → ττ , which is an irreducible component, andmulti jet events faking τhad’s, that will be referred to as fake taus or simply fakes. In bothcases it can be shown [103] that the ϕCP distribution is flat for these background processes.

A schematic situation of the expected shapes for the distributions of signal and back-ground is depicted in figure 4.4. The signal is shown for three different values of the CP-mixing angle, corresponding to φτ = 0 (SM pure scalar Higgs), φτ = π/2 (pure pseudoscalarHiggs) and φτ = π/4.

Figure 4.4 Expected shapes of ϕCP in case of a pure scalar (blue line) or pseudoscalar (red)Higgs boson. The green line corresponds to a CP-mixing angle φτ = π/4. The Z → ττirreducible background is also shown as a flat black line. All shapes are reproduced on thesame scale and do not contain information on the absolute normalization of the processes.

4.3 Observable Reconstruction

The angle ϕCP between the tau decay planes described in section 4.2 is defined by means ofvariables in the Higgs boson rest frame and in the τ± rest frame. Nevertheless, such frames

4.3 Observable Reconstruction 79

are not experimentally accessible due to the presence of two undetected neutrinos and themeasurement must rely on slightly different observables, that are referred to with the symbolϕ∗CP to distinguish them from ϕCP . ϕ∗

CP is defined in the di-τhad−vis ZMF and the tau decayplanes are built from the spatial vectors of the charged and neutral pions of the tau decaysand their impact parameters. These proxies provide however the sensitivity to φτ required forthe measurement. According to the τ± decay modes, two different methods are used to buildthe observables and exploit the π± and ρ± produced respectively in the 1p0n and 1p1n taudecay modes. These methods, referred to as Impact Parameter and ρ methods, are describedin the followings. A combination of them is used in events where the decay modes are 1p0nfor a tau and 1p1n for the other.

4.3.1 Impact Parameter Method

Figure 4.5 Illustration of the ϕ∗CP an-

gle reconstructed with the Impact Pa-rameter method [36].

The Impact Parameter (IP) method [36, 100, 103] canbe applied to all decay modes but has the highestsensitivity to φτ in the decays that involve only onecharged pion. For this reason the IP method is usedonly when the decay modes of both τhad−vis’s areclassified as 1p0n.

In these cases the τ± decay planes are formedfrom the pion spatial momenta q± and the impactparameter vectors n± of the charged tracks with re-spect to the reconstructed primary interaction vertex.The impact parameter vectors, normalised to the unity(n̂±), are incorporated in four-vectors nµ± = (0, n̂±)

and boosted from the laboratory frame to the di-τhad−vis ZMF. The same boost is applied to the pionfour momenta: quantities in the di-τhad−vis ZMF aredenoted with a star, so for instance the boosted four-vectors are n∗µ± and q∗µ± . The spatial partsof n∗µ± are decomposed into their normalized components n̂∗±

∥ and n̂∗±⊥ , that are respectively

parallel and perpendicular to the normalised spatial momentum q̂∗±.

With these vectors one determines the unsigned angle ϕ∗ between the tau decay planes:

ϕ∗ = arccos(n̂∗+⊥ · n̂∗−

⊥)

0 < ϕ∗ < π (4.5)

To remove the ambiguity on the sign of ϕ∗, a CP-odd triple correlation O∗CP is defined:


O∗CP = q̂∗

− · (n̂∗+⊥ × n̂∗−

⊥ ) (4.6)

The sign of ϕ∗ is finally determined by the following prescription:

ϕ∗CP =

ϕ∗ if O∗CP ≥ 0

2π−ϕ∗ if O∗CP < 0

(4.7)

Figure 4.5 provides a schematic depiction of the angle ϕ∗CP reconstructed with the IP method.

MC distributions of ϕ∗CP are shown in figure 4.6 for pure scalar (φτ = 0) and pseudoscalar

(φτ = π/2) signal hypotheses and for the irreducible Z → ττ background, at truth (figure4.6a) and reconstruction (figure 4.6b) level. In both plots, the only applied cut regards thereconstructed decay mode that must be 1p0n for both taus.

(a) (b)

Figure 4.6 ϕ∗CP MC distributions at truth (a) and reconstruction (b) level for a SM Higgs

(φτ = 0), a pure pseudoscalar Higgs (φτ = π/2) and the irreducible background Z → ττusing the IP method. Error bars refer to the statistical uncertainty. All distributions arenormalized to the unity and the only applied cut is the reconstructed 1p0n decay mode forboth taus. Signal distributions are fitted with function 4.8 (fit results are shown in table 4.2),while the background distribution is fitted with a constant (χ2/ndf = 9.1/5 at truth level andχ2/ndf = 24.2/5 at reconstruction level). The bad χ2 value of the Z → ττ distribution atreconstruction level is mainly dependent on the external bins.

The ϕ∗CP signal distributions are fitted with a cosine function of the form

f(ϕ∗CP ) = w+ucos(ϕ∗

CP −2φ0) (4.8)


where u, w and φ0 are fit parameters. The normalization of f(ϕ∗CP ) is subjected to the

constraint [106]:

∫ 2π

0f(ϕ∗

CP )dϕ∗CP = 2πw = σaa′ (4.9)

where σaa′ is the Higgs production cross section including the branching fractions of the τ±

decays into the respective charged particles a and a′ (a, a′ = π in case of 1p0n decay mode,a, a′ = ρ in case of 1p1n decay mode). To quantify the size of the signal modulation and tocompare the reconstruction methods, the following asymmetry Aaa′ is defined:

Aaa′ =− 1σaa′

∫ 2π

0{dσaa′(ucos(ϕ∗

CP −2φ0)> 0)−dσaa′(ucos(ϕ∗CP −2φ0)< 0)}

(4.10)After integrating out the ϕ∗

CP dependence, the asymmetry turns out to be

Aaa′ =− 4u2πw

(4.11)

Equation 4.11 shows that Aaa′ does not depend on the value of φτ but rather on theproducts of the spin-analysing functions of a and a′. The larger Aaa′ , the larger the signalmodulation with respect to the flat background. As it was remarked in section 4.2, the1p0n decay mode has the highest sensitivity to the spin correlation and the correspondingasymmetry is expected to be the largest.

Impact Parameter Method

Level Signal process φ0 [deg] χ2/ndf Aππ

Truthφτ = 0 0±1 2.8/3 0.43±0.02φτ = π/2 91±2 0.3/3 0.33±0.02

Recoφτ = 0 3±2 21.9/3 0.19±0.02φτ = π/2 90±3 8.3/3 0.25±0.02

Table 4.2 Some of the fit results of figure 4.6 for the signal processes h→ ττ : the angularparameter φ0; the χ2 of the fit with the corresponding number of degrees of freedom (ndf); theasymmetry Aππ computed according to equation 4.11. The subscripted ππ in Aππ denotesthat both taus decay into a single pion (decay modes 1p0n for both).

Table 4.2 shows some of the fit results on the signal distributions in figure 4.6. Theangular parameters φ0 at reconstruction level are compatible with the corresponding angles


φτ of the signal processes within the statistical uncertainty. The asymmetry Aππ is partlydegraded after the reconstruction.

4.3.2 ρρρ Method


gle reconstructed with the ρ method[36].

When both decay modes are classified as 1p1n, theimpact parameter method can still be used but doesnot have the sensitivity required for the measurement.Thus in this case a different method is used, called ρmethod, that aims to build the tau decay planes fromthe spatial momenta of the charged and neutral pions[36, 107, 108]. This method, that requires to measurethe energies and momenta of the individual chargedand neutral pions in the ρ decays, fully exploits thefeatures of the Tau Particle Flow algorithm describedin chapter 3.

After being reconstructed, the charged and neu-tral pion four-momenta are boosted in the di-τhad−vis

ZMF: in this frame, the π−, π0 (π+, π0) spatial mo-menta are denoted as q∗−, q∗0− (q∗+, q∗0+). For eachneutral pion the normalized vectors q̂∗0±

⊥ are computed, that are perpendicular to the directionof the associated charged pion. An angle ϕ∗ and a triple-odd correlation O∗

CP are defined as

ϕ∗ = arccos(q̂∗0+⊥ · q̂∗0−

⊥)

O∗CP = q̂∗− · (q̂∗0+

⊥ × q̂∗0−⊥ ) (4.12)

where q̂∗− indicates the normalized q∗− vector defined above and ϕ∗ ∈ [0,π]. Again, theambiguity on the sign of ϕ∗ is removed by the prescription:

ϕ∗′ =



(4.13)

However ϕ∗′ is not yet the observable ϕ∗CP and things are slightly more complicated

in this case: since the spin-analysing functions b(E±) in equation 4.4 may be positive ornegative, ϕ∗′ is not sensitive to φτ unless the signs of the spin-analysing functions are takeninto account. By defining the variables

y− =Eπ− −Eπ0

Eπ− +Eπ0y+ =

Eπ+ −Eπ0

Eπ+ +Eπ0(4.14)


where Eπ±,0 are the pion energies in the laboratory frame, the CP-sensitive angle ϕ∗CP is

defined as

ϕ∗CP =

ϕ∗′ if y+y− ≥ 0

2π−ϕ∗′ if y+y− < 0(4.15)

A schematic depiction of the ϕ∗CP angle reconstructed with the ρ method is given in figure

4.7.Figure 4.8 shows the MC distributions of the pure scalar and pure pseudoscalar signal

and of the Z → ττ background at truth (figure 4.8a) and reconstruction (figure 4.8b) level.The only applied cut is on the reconstructed decay modes, that must be 1p1n for both taus.Signal distributions are fitted with the function in equation 4.8, background is fitted with aconstant.

(a) (b)


(φτ = 0), a pure pseudoscalar Higgs (φτ = π/2) and the irreducible backgroundZ→ ττ usingthe ρ method. Error bars refer to the statistical uncertainty. All distributions are normalized tothe unity and the only applied cut selects events where both tau decay modes are reconstructedas 1p1n. Signal distributions are fitted with function 4.8 (fit results are shown in table 4.3),while the background distribution is fitted with a constant (χ2/ndf = 1.6/5 at truth level andχ2/ndf = 9.8/5 at reconstruction level).

Some of the fit results and the asymmetry values Aρρ are reported in table 4.3. The fitparameters φ0 are compatible with the corresponding angles φτ within the uncertainty, atreconstruction level too; the asymmetry Aρρ is almost the same at truth and reconstruction


ρρρ Method

Level Signal process φ0 [deg] χ2/ndf Aρρ

Truthφτ = 0 −1±3 3.1/3 0.07±0.01φτ = π/2 94±5 1.8/3 0.06±0.01

Recoφτ = 0 0±2 0.8/3 0.10±0.01φτ = π/2 94±4 1.7/3 0.08±0.01

Table 4.3 Some of the fit results of figure 4.8 for the signal processes h→ ττ : the angularparameter φ0; the χ2 of the fit with the corresponding number of degrees of freedom (ndf);the asymmetry Aρρ computed according to equation 4.11. The subscripted ρρ in Aρρ denotesthat both taus decay into a ρ (decay modes 1p1n for both).

level. As expected, Aρρ <Aππ: the asymmetry is larger when the 1p0n decay modes areinvolved.

4.3.3 Combined IP-ρρρ Method


gle reconstructed with the combinedIP-ρ method [36].

In events in which one decay mode is reconstructedas 1p0n and the other as 1p1n, a combination of theIP and ρ methods is used [36]. To illustrate such amethod the τ+ will be supposed to decay into a ρmeson (1p1n decay mode) and the τ− into a singlepion (1p0n decay mode).

Consistently with the previous methods, ϕ∗CP is

built in the di-τhad−vis ZMF. In case of τ−, the nor-malized vectors q̂∗

− and n̂∗−⊥ are defined as described

in section 4.3.1; in case of τ+, the normalized vectorsq̂∗0+⊥ and q̂∗+

⊥ are defined as described in section 4.3.2.The angle ϕ∗ and the triple correlation O∗

CP are thendefined as follows:

ϕ∗ = arccos(q̂∗0+⊥ · n̂∗−

⊥)

O∗CP = q̂∗

− · (q̂∗+⊥ × n̂∗−

⊥ ) (4.16)

and combined in a single variable ϕ∗′:

ϕ∗′ =



(4.17)


Using the sign of y+, defined in equation 4.14 and computed in the laboratory frame, theangle ϕ∗

CP is finally built as:

ϕ∗CP =

ϕ∗′ if y+ ≥ 0

2π−ϕ∗′ if y+ < 0(4.18)

Figure 4.9 shows a schematic illustration of the ϕ∗CP angle reconstructed with the combined

IP-ρ method.The MC distributions of signal (pure scalar and pure pseudoscalar Higgs) and Z → ττ

background are plotted in figure 4.10 at truth (figure 4.10a) and reconstruction (figure 4.10b)level. The only applied cut selects events in which exactly one decay mode is reconstructedas 1p0n and the other as 1p1n. Signal distributions are fitted with the function in equation4.8, background is fitted with a constant.

(a) (b)


(φτ = 0), a pure pseudoscalar Higgs (φτ = π/2) and the irreducible backgroundZ→ ττ usingthe combined IP-ρ method. Error bars refer to the statistical uncertainty. All distributions arenormalized to the unity and the only applied cut selects events in which exactly one tau isreconstructed as 1p0n, the other as 1p1n. Signal distributions are fitted with function 4.8 (fitresults are shown in table 4.4), while the background distribution is fitted with a constant(χ2/ndf = 10.3/5 at truth level and χ2/ndf = 6.1/5 at reconstruction level).

Some of the fit results and the asymmetry values Aπρ are reported in table 4.4: once againthe fit parameters φ0 are compatible with the corresponding angles φτ within the uncertaintyand the asymmetries Aπρ, degraded after the reconstruction, lay between Aρρ and Aππ.


Combined IP-ρρρ Method

Level Signal process φ0 [deg] χ2/ndf Aπρ

Truthφτ = 0 2±1 1.2/3 0.18±0.01φτ = π/2 89±2 7.8/3 0.16±0.01

Recoφτ = 0 0±2 4.7/3 0.13±0.01φτ = π/2 89±3 11.6/3 0.12±0.01

Table 4.4 Some of the fit results of figure 4.10 for the signal processes h→ ττ : the angularparameter φ0; the χ2 of the fit with the corresponding number of degrees of freedom (ndf);the asymmetry Aπρ computed according to equation 4.11. The subscripted πρ in Aπρ denotesthat exactly one tau decays into a ρ (decay mode 1p1n), the other into a single pion (decaymode 1p0n).

4.4 Background Estimation

Figure 4.11 A schematic depictionof the ABCD background estimationmethod, where the CRs are built byreversing the requirements on thecharge signs (OS or SS) and/or onthe working points (ID or Anti ID)with respect to the SR.

The background estimation of all processes but fakesmakes use of MC simulations, as explained in section4.1.1. Fakes are estimated via a data driven method,since MC simulations are not reliable in this casedue to the difficulties arising in the modelling of theQCD processes. Such a method, that is commonlyknown as ABCD method, is based on the definitionof a Signal Region (SR), like the preselection, vbfand boosted shown in table 4.1, and three orthogonalControl Regions (CRs) defined by reversing some ofthe SR cuts.

To build the CRs two types of cuts are considered:the sign of the tau pair, that may be Opposite Sign(OS) or Same Sign (SS), and the required workingpoints, that are chosen to be medium for both tauswith at least one tight (ID) or, on the contrary, arechosen so that no tight working points are involved (Anti ID). The combination of thesecuts produces four possible regions, one corresponding to the SR, the others to three CRsas shown schematically in figure 4.11. CRs that are built by reversing only one of the twocuts just described, such as the Same Sign ID Control Region (SS ID CR) and the OppositeSign Anti ID Control Region (OS Anti-ID CR), respectively CR1 and CR2 of figure 4.11, are

4.4 Background Estimation 87

called close; the remaining CR, built by reversing both cuts, is called far and corresponds tothe Same Sign Anti ID Control Region (SS Anti-ID CR), CR3 of figure 4.11.

Once the CRs are defined, the procedure to estimate the SR fake distribution consists ofthe following steps:

• The distribution shape is derived in a close CR by subtracting all MC processes fromthe data: the basic assumption is that the MC simulations provide a good descriptionof the related processes and that the discrepancies between data and MC are due solelyto fakes (that are not simulated).

• The distribution derived in the close CR contains an absolute normalization whichmay be different from the one expected in the SR: to correct for such a difference,a relative factor, that analogously to chapter 3 is called rQCD, is computed in thetwo remaining CRs and is used to multiply the absolute normalization of the fakedistribution. Assuming for instance that the fake shape is derived in CR2 and thatrQCD is computed in CR1 and CR3, the factor is evaluated as follows:

rQCD =Ndata

CR1−∑XN

MCX CR1

NdataCR3

−∑XNMCX CR3

(4.19)

where NdataCR1

(NdataCR3

) is the amount of data in CR1 (CR3) and NMCX CR1

(NMCX CR3

) is theamount of simulated events of the process X in CR1 (CR3). If the fake shape is derivedin CR1 and rQCD is computed in CR2 and CR3, equation 4.19 still holds by replacingthe subscripted CR1 with CR2.

Since there are two close CRs, one or the other may be chosen to derive the fake shapeleading to two possible implementations of the ABCD method: the implementation thatderives the shape from the SS ID CR will be referred to as OS-SS implementation, whereasthe implementation that derives the fake shape from the OS Anti-ID CR will be referred to asID-AntiID implementation.

The performances of the two implementations are shown in figure 4.12, where the tau pairinvariant massmMMC

ττ is plotted at preselection. For both methods the agreement between dataand MC is evaluated via a chi square test: the ID-AntiID implementation (χ2/ndf = 16.5/20)is found to perform slightly better than the OS-SS implementation (χ2/ndf = 25.9/20) andis retained in the following of this chapter, whereas the latter is not used.

Table 4.5 illustrates the rQCD factors evaluated according to equation 4.19 in the ID-AntiID implementation. In the vbf, large errors on rQCD are due to the low amount of eventspresent in this category (see section 4.5).


(a) (b)

Figure 4.12 Plots of the tau pair invariant mass mMMCττ at preselection with fake background

derived by means of the ID-AntiID implementation (a) and OS-SS implementation (b).

rQCD

methodIP ρ Combined IP-ρ

preselection 1.87±0.20 0.82±0.04 1.27±0.06vbf 0.93±0.79 0.59±0.19 1.82±0.59

boosted 1.02±0.36 0.86±0.10 1.06±0.14

Table 4.5 rQCD factors evaluated according to equation 4.19 in the ID-AntiID implementationof the ABCD background estimation method. Large errors are shown in the vbf due to thepoor amount of events in this category.

4.5 Results

Plots of ϕ∗CP are shown in figure 4.13 at preselection and in the vbf and boosted categories

for each reconstruction method, assuming a SM Higgs (φτ = 0). The agreement betweendata and MC is satisfying in most cases, with statistical fluctuations of data generally fallingwithin the MC sample uncertainty. Plots show that vbf is the poorest category in terms of

4.5 Results 89

Figure 4.13 ϕ∗CP plots at preselection (first row) and in the vbf (second row) and boosted

(third row) categories. Plots are further divided according to the ϕ∗CP reconstruction method:

IP (first column), ρ (second column) and combined IP-ρ (third column).


events, but it has the highest signal percentage. MC processes of the IP method seem toshow some pathologies related to the first and last bins, where the data/MC disagreement isbeyond the uncertainty band. In particular, one of the bins of ϕ∗

CP (IP) distribution in vbfcontains negative entries, which is a non-physical situation. The point is that MC generatorsmay contemplate negative weights when higher order corrections in perturbation theory arenegative, and this may give rise to problems when a few events are involved.

Figure 4.14 is the sum of the ϕ∗CP distributions of all reconstruction methods and both

categories of figure 4.13. To validate the hypothesis of uniformly distributed background,the Z → ττ and fake distributions are separately taken from plot in figure 4.14 and a fit isperformed on them, using a constant (ffit = C) and a cosine (ffit = C−Acos

(ϕ∗CP −2φτ

))

function. A cosine fit is also performed on the signal distribution. Plots of the individual back-ground and signal distributions are shown in figure 4.15 with the corresponding fit functions.Fit results are shown in table 4.6: in both Z→ ττ and fake cases the oscillation amplitudes Aare compatible with zero and the χ2’s of the constant fits are perfectly reasonable, so that noevidences are provided to reject the hypothesis of uniform distributions for the background.

Figure 4.14 ϕ∗CP distribution inclusive of all reconstruction methods and both vbf and boosted

categories.

4.5 Results 91

(a) (b) (c)

Figure 4.15 ϕ∗CP distributions of Z → ττ (a), fakes (b) and h→ ττ (c) taken from figure

4.14 and fitted with a constant (only in case of background) and a cosine function. Thebackground distributions are expected to be flat while the signal distribution is expected toshow a cosine modulation. Fit functions are superimposed to data on the plots: the dashedlines denote the cosine fits, the continuous ones the constant fits. Fit values and results arereported in table 4.6

Signal and background fitFit function A C χ2

fit/ndf P (χ2 > χ2fit)

Z → ττConstant — 187±5 6.7/5 24%Cosine 8±7 188±5 5.5/3 14%

FakesConstant — 106±4 4.4/5 49%Cosine 8±6 106±4 2.7/3 44%

h→ ττ Cosine 2.2±0.2 10.2±0.1 2.5/3 47%

Table 4.6 Results of the fits in figure 4.15 for signal and background distributions, inclusiveof all methods and both categories. The table shows the oscillation amplitude A (only forcosine fit); the constant C of the fit functions; the fit chi square with the correspondingdegrees of freedom χ2

fit/ndf; the probability for the χ2 to be grater than χ2fit, P (χ

2 > χ2fit),

that gives an idea of the goodness of fit [109].

Finally, table 4.7 shows the per cent amount of each process of the inclusive plot in figure4.14, evaluated as the fraction of events of each MC process with respect to the total amountof MC events.


Fakes Others Z → ττ h→ ττ

(34.7±0.6)% (2.6±0.7)% (59.5±0.5)% (3.23±0.08)%

Table 4.7 Per cent composition of the processes in the inclusive plot of figure 4.14.

4.5.1 Expected sensitivity

Figure 4.14 shows that with a luminosity of 33 fb−1 the signal is hidden by the statisticalfluctuations of data, so that an appreciable cosine modulation does not clearly emerge fromthe distribution. Nevertheless, it may be interesting to estimate the expected sensitivity inview of the performances of High-Luminosity LHC that should provide a luminosity-per-yearof ∼330 fb−1, for a total of ∼1000 fb−1 in a three-year run. From the results at 33 fb−1

discussed in section 4.5, an extrapolation at 1000 fb−1 of the plot in figure 4.14 is performedbased on the following steps:

1. Given the amount of data at 33 fb−1 (Ndata33 ) from figure 4.14, the amount of data

expected at 1000 fb−1 (µdata1000) is evaluated via a simple proportion:

µdata1000 =

1000 fb−1

33 fb−1 Ndata33 (4.20)

2. To reproduce the equivalent plot of figure 4.14 at 1000 fb−1, a pseudo-data sample ofpseudo-random numbers is generated with the following probability density functionpdata

pdf :

pdatapdf =

1−fH

2π︸︷︷︸(1−fH)p

bkgpdf

+fH

2π

(1− A

Ccos(ϕ∗

CP −2φτ ))

︸︷︷︸fHp

sgnpdf

(4.21)

where fH is the fraction of signal events given in table 4.7, A and C are the sig-nal parameters given in table 4.6 and 2π is a normalization factor to ensure that∫pdata

pdf dϕ∗CP = 1. φτ is set to zero in this case (the pseudo-data sample is produced

under the SM hypothesis). As suggested in equation 4.21, pdatapdf can be seen as the sum

of the probability density function of the background (pbkgpdf ) and that of the signal (psgn

pdf),weighted by the respective percentage (1− fH for background, fH for signal). Thenormalization of the pseudo-data sampleNdata

1000 (i.e. the total amount of pseudo-randomnumbers to be generated) is a random integer from a Poisson distribution with meanµdata

1000.

4.5 Results 93

3. Samples of MC processes consist of pseudo-random numbers generated with theprobability density function pbkg

pdf in case of background and psgnpdf in case of signal. For

each process 106 pseudo-random numbers are generated and the resulting distributionis finally scaled to the expected normalization, that is computed as µdata

1000 times thepercentage of that process.

The procedure described above is thus used to obtain the expected distribution of ϕ∗CP at

1000 fb−1, that is shown in figure 4.16a in case of a SM Higgs.In order to establish the expected sensitivity to φτ (∆φτ ) at the given luminosity, plots

similar to the one in figure 4.16a are realized for different hypotheses of the CP-mixingangle, varying the φτ value in the generation of the signal MC from −90 deg to 90 deg insteps of 10 deg. A maximum binned likelihood method is then implemented as follows: foreach fixed value of φτ (i.e. for each plot realized with the aforementioned procedure), thebinned likelihood is defined as

L= ∏bins

µnii

ni!e−µi (4.22)

where µi is the amount of MC entries in the i-th bin and ni is the amount of pseudo-data inthat same bin. For ease of computation, the log-likelihood is used:

logL= ∑bins

ni logµi (4.23)

The terms ∑ log(ni!) and ∑µi are common to the binned likelihoods of the distributionsrealized for all φτ values and are omitted.

Finally, the delta negative log-likelihood ∆NLL is built as follows:

∆NLL(φτ ) = logL(φτ )− logLmin (4.24)

logL(φτ ) is the log-likelihood computed when the signal MC is generated with the valueφτ of the CP-mixing angle, logLmin is the minimal log-likelihood among them. ∆NLL isa function of φτ , whose minimum corresponds to the best estimator of the parameter φτ .The uncertainty ∆φτ on the CP-mixing angle φτ is evaluated as the semi-interval in which∆NLL< 0.5 [109]. The ∆NLL curve at 1000 fb−1 is depicted in figure 4.16b.

The aforementioned procedure is repeated for different values of luminosity to obtain acurve of sensitivity to φτ as a function of the luminosity. In order to reduce the fluctuationsof ∆φτ related to the choice of the seed used for the generation of the sequence of pseudo-random numbers, for each fixed value of luminosity ∆φτ is estimated several times ondifferent samples of pseudo-data and simulations, where the generation seed is varied. The


distributions of ∆φτ , one for each value of luminosity, are then built (as shown for instancein figure 4.17 at 1000 fb−1); since they have long tails at high values, ∆φτ ’s are chosen asthe most probable values from their respective distribution.

Finally, an uncertainty band, related to the choice of the parameters fH, A and C ofequation 4.21, is taken into account. For each value of luminosity, the aforementionedparameters are varied within their one-σ uncertainty to obtain the best signal condition(fH −→ fH +σfH , A −→ A+σA, C −→ C−σC) and the worst signal condition (fH −→fH −σfH , A−→A−σA, C −→ C+σC); in both conditions ∆φτ is estimated with the sameprocedure described above, providing respectively the upper and the lower bound of theuncertainty band. The resulting curve of sensitivity to φτ is shown in figure 4.18.

(a) (b)

Figure 4.16 (a) Expected distribution of ϕ∗CP at 1000 fb−1 realized with the procedure

described in this section. The MC signal distribution consists of a SM Higgs (φτ = 0). (b)∆NLL curve as a function of φτ . The best estimator for the CP-mixing angle is φτ = 0, withno surprise since the data sample is generated under the SM hypothesis. The red dashed lineidentifies the level ∆NLL= 0.5 and is used to estimate the uncertainty on the CP-mixingangle, ∆φτ .

4.5 Results 95

Figure 4.17 Distribution of ∆φτ at 1000 fb−1 obtained by varying the seed used in thegeneration of the pseudo-random numbers. The plot, containing 900 events, is normalized tobe a probability density function for ∆φτ , i.e. the integral over the whole range is one.

Figure 4.18 Sensitivity to φτ as a function of the luminosity. The shaded band correspondsto the one-σ uncertainty of the parameters in equation 4.21 and is built as discussed in thissection.

Conclusions

The tau analysis at ATLAS has seen significant improvements since the beginning of Run 2.The new Tau Particle Flow algorithm for the reconstruction of τhad is described in chapter3 of this thesis and its performances are studied. In particular the algorithm is found toprovide four times better angular resolutions and twice better energy resolution with respectto the old Baseline algorithm, used for the entire Run 1. The new algorithm, unlike theold one, is also designed to recognize five different types of τhad decay modes, achieving aclassification purity of around 75% with small contaminations among same-prong events(due to misidentifications of the number of π0’s) and negligible contaminations amongdifferent-prong events. To test the performances of the Tau Particle Flow algorithm, the ρ anda1 resonances, respectively in the 1p1n and 3p0n decay modes of τhad, are studied and theirinvariant masses are estimated by fit to be Mfit

ρ = 798±3 MeV and Mfita1

= 1150±2 MeV:the latter is found to be in accordance with the expectations, due to the large uncertaintyof the theoretical value, whereas the former is ∼20 MeV above the expected mass. Thediscrepancy cannot be explained in terms of statistical fluctuations but may depend onsystematic uncertainties that have not been taken into account.

In chapter 4 the CP nature of the Higgs boson is analysed in its h→ τhadτhad decays. Themethods used to infer the CP quantum numbers of the Higgs boson are described and thesize of the signal modulation in the ϕ∗

CP distribution is studied. The IP method is found toshow a larger signal modulation (asymmetry Aππ ≃ 0.20) than the ρ method (asymmetryAρρ ≃ 0.10). The combined IP-ρ method provides a size of the signal modulation betweenthat of the IP and that of the ρ methods (asymmetry Aπρ ≃ 0.13). The distribution of ϕ∗

CP

for the main background sources (Z → ττ and fakes) is uniform as expected. The amountof data collected by ATLAS in a one-year run is not enough to give an estimation of φτ ;however in view of LHC’s future developments, the expected sensitivity to φτ at 1000 fb−1

is estimated to be ∆φτ = 19.8+1.1−0.8 deg.

The analysis of the Higgs quantum numbers in its fermionic decays has a strategicimportance in the physics of elementary particles and in that of the Universe. From thispoint of view the tau channel of the Higgs decays is one of the most sensitive to the study

98 Conclusions

of its properties and could provide perspectives of significant results of New Physics orconfirmations of the SM. In both cases a complete analysis of h→ ττ involving the wholeRun 2 dataset is more than needed in the close future and could be strongly enhanced by theperformances of High-Luminosity LHC in some more years.

Appendix A

Control Region Plots and Yields

The plots shown in this appendix refer to the control regions described in section 3.3 andare provided to give an idea of their composition. The muon-tau visible mass Mvis(µ,τ)

is plotted as an example in each control region but only for some decay modes and someworking points. Figures A.1, A.2 and A.3 refer respectively to the QCD CR, W CR and TopCR. It is clear from figure A.1 that the QCD background, estimated as the difference betweendata and all MC simulations, is significantly reduced by tightening the working point, as itshould since the working point is related to the efficiency on rejecting the QCD background.

For each decay mode and each working point, yields of the control regions are shown intables A.1 (OS QCD CR), A.2 (SS QCD CR), A.3 (OS W CR), A.4 (SS W CR), A.5 (OSTop CR) and A.6 (SS Top CR).

100 Control Region Plots and Yields

Figure A.1 Plots of Mvis(µ,τ) for the 1p0n decay mode in the opposite sign (left column)and same sign (right column) QCD control region for noID (first row), loose (second row)and medium (third row) working points.

101O

ppos

iteSi

gnQ

CD

Con

trol

Reg

ion

1p0n

1p1n

1pX

n3p

0n3p

Xn

noID

Top

21±

592

±9

59±

769

±8

96±

10W

+jet

s21

00±

160

7990

±33

042

30±

250

6380

±32

051

80±

270

Z→ℓℓ

530±

2611

65±

3831

6±

2048

4±

2438

9±

21Z→ττ

1732

±69

6070

±13

019

46±

7124

30±

8211

45±

54D

ata

2004

0±

140

6844

0±

260

3607

0±

190

6583

0±

260

5744

0±

240

QC

D15

660±

230

5312

0±

440

2953

0±

320

5647

0±

420

5063

0±

365

loos

e

Top

11±

431

±5

13±

35±

25±

2W

+jet

s61

0±

9517

50±

150

426±

7443

1±

8318

4±

51Z→ℓℓ

340±

2237

0±

2249

±8

33±

718

±5

Z→ττ

1621

±67

5190

±12

014

07±

6119

94±

7671

9±

44D

ata

6398

±79

1725

0±

130

5819

±76

6056

±78

3531

±59

QC

D38

20±

140

9910

±23

039

20±

120

3590

±14

026

04±

90

med

ium

Top

11±

422

±4

10±

32±

13±

1W

+jet

s42

8±

8210

50±

120

201±

4819

8±

5954

±24

Z→ℓℓ

302±

2123

2±

1820

±5

9±

37±

3Z→ττ

1549

±67

4630

±12

011

70±

5716

29±

6945

2±

35D

ata

4829

±69

1125

0±

110

3259

±57

3463

±61

1580

±40

QC

D25

40±

130

5320

±20

018

59±

9516

30±

110

1064

±58

tight

Top

10±

413

±3

8±

31±

10.

6±

0.3

W+j

ets

318±

7452

4±

8537

±17

77±

4522

±11

Z→ℓℓ

251±

1811

0±

129±

42±

14±

2Z→ττ

1390

±64

3610

±11

080

0±

412

50±

6127

9±

27D

ata

3384

±58

6478

±80

1643

±41

2028

±45

757±

28

QC

D14

10±

120

2220

±16

078

9±

6569

8±

8845

2±

40

Table A.1 Opposite sign QCD control region’s yields for each decay mode and foreach working point (noID, loose, medium, tight). The QCD yields are computed as thedifference between data and all simulated processes.

102 Control Region Plots and YieldsSa

me

Sign

QC

DC

ontr

olR

egio

n

1p0n

1p1n

1pX

n3p

0n3p

Xn

noID

Top

3.6±

1.6

54±

739

±6

77±

965

±8

W+j

ets

1760

±17

051

90±

260

2760

±19

036

90±

220

3360

±22

0Z→ℓℓ

160±

1381

8±

3128

5±

1949

3±

2537

9±

22Z→ττ

16±

686

±19

56±

1323

±8

20±

6D

ata

1582

0±

130

5443

0±

230

3024

0±

170

5443

0±

230

4639

0±

220

QC

D13

880±

210

4828

0±

350

2709

0±

260

5014

0±

320

4257

0±

310

loos

e

Top

1±

19±

34±

25±

22±

1W

+jet

s32

4±

8193

0±

110

237±

5221

9±

5412

8±

50Z→ℓℓ

51±

821

4±

1649

±8

32±

627

±7

Z→ττ

14±

636

±9

26±

710

±6

6±

4D

ata

3687

±61

9895

±99

3701

±61

3023

±55

2025

±45

QC

D33

00±

100

8700

±15

033

85±

8127

57±

7718

22±

68

med

ium

Top

0.07

±0.

074±

20.

7±

0.4

4±

20.

2±

0.2

W+j

ets

167±

5057

0±

8813

9±

4211

0±

4071

±34

Z→ℓℓ

40±

711

5±

1226

±5

10±

31±

4Z→ττ

14±

727

±7

21±

73±

35±

4D

ata

2375

±49

5306

±73

1828

±43

1278

±37

794±

28

QC

D21

54±

7245

90±

120

1640

±62

1150

±55

706±

45

tight

Top

0.07

±0.

073±

10.

1±

0.1

3±

20.

2±

0.2

W+j

ets

71±

2826

7±

6247

±26

59±

3336

±23

Z→ℓℓ

26±

638

±6

6±

26±

25±

2Z→ττ

7±

415

±5

11±

50.

5±

0.5

3±

4D

ata

1402

±37

2366

±49

678±

2654

3±

2331

3±

18

QC

D12

97±

4720

43±

7961

5±

3747

4±

4026

8±

29

Table A.2 Same sign QCD control region’s yields for each decay mode and for eachworking point (noID, loose, medium, tight). The QCD yields are computed as thedifference between data and all simulated processes.

103

Figure A.2 Plots of Mvis(µ,τ) for the 1p1n decay mode in the opposite sign (left column)and same sign (right column) W control region for loose (first row), medium (second row)and tight (third row) working points.

104 Control Region Plots and YieldsO

ppos

iteSi

gnW

Con

trol

Reg

ion

1p0n

1p1n

1pX

n3p

0n3p

Xn

noID

Top

1555

±42

7708

±93

450±

7370

15±

8968

65±

86W

+jet

s13

8700

±14

0043

0100

±26

0021

2500

±18

0036

5800

±24

0032

0000

±22

00Z→ℓℓ

4897

±79

1436

0±

140

7060

±96

1084

0±

120

9330

±11

0Z→ττ

512±

3519

56±

7157

7±

3765

6±

4334

0±

30D

ata

1631

90±

400

4341

40±

660

2114

20±

460

3915

00±

630

3252

90±

570

QC

D17

500±

1500

−20

000±

2600

−13

300±

1900

7200

±25

00−

1120

0±

2300

loos

e

Top

779±

2925

94±

5495

1±

3280

9±

3143

4±

23W

+jet

s40

850±

800

1030

00±

1300

3019

0±

670

2779

0±

660

1783

0±

540

Z→ℓℓ

1487

±43

3336

±66

952±

3769

4±

2935

8±

21Z→ττ

483±

3416

21±

6540

6±

3149

2±

3818

1±

23D

ata

4614

0±

210

1098

90±

330

3537

0±

190

3345

0±

180

1969

0±

140

QC

D25

40±

830

−60

0±

1300

2870

±70

036

60±

680

890±

560

med

ium

Top

634±

2719

37±

4764

3±

2755

0±

2523

2±

17W

+jet

s27

730±

660

5943

0±

970

1545

0±

480

1193

0±

430

7150

±34

0Z→ℓℓ

1115

±37

1904

±50

433±

2529

4±

1913

4±

13Z→ττ

460±

3414

15±

6331

9±

2841

4±

3592

±15

Dat

a32

390±

180

6468

0±

250

1840

0±

140

1458

0±

120

7583

±87

QC

D24

50±

680

−6±

1000

1550

±50

013

90±

450

−26

±35

2

tight

Top

504±

2512

76±

3837

4±

2039

0±

2111

2±

12W

+jet

s16

840±

520

2681

0±

650

5700

±29

048

80±

280

2640

±20

0Z→ℓℓ

780±

3185

8±

3316

1±

1412

3±

1358

±9

Z→ττ

406±

3210

56±

5521

5±

2330

2±

3267

6±

14D

ata

2058

0±

140

3125

0±

180

7713

±88

6382

±80

3030

±55

QC

D20

60±

540

1250

±68

012

70±

300

680±

300

150±

210

Table A.3 Opposite sign W control region’s yields for each decay mode and for eachworking point (noID, loose, medium, tight). The QCD yields are computed as thedifference between data and all simulated processes.

105Sa

me

Sign

WC

ontr

olR

egio

n

1p0n

1p1n

1pX

n3p

0n3p

Xn

noID

Top

534±

2537

37±

6626

43±

5541

49±

6749

44±

75W

+jet

s77

500±

1100

2373

00±

1900

1321

00±

1400

2050

0±

1800

1819

00±

1600

Z→ℓℓ

4509

±77

1408

0±

140

7166

±96

1081

0±

120

9400

±11

0Z→ττ

7±

432

±9

13±

511

±5

9±

4D

ata

1032

60±

320

2621

70±

510

1310

50±

6023

0380

±48

019

4290

±44

0

QC

D20

700±

1100

7000

±19

00−

1080

0±

1500

1040

0±

1800

−20

00±

1700

loos

e

Top

112±

1155

0±

2525

1±

1618

0±

1415

7±

13W

+jet

s13

750±

460

4460

0±

810

1564

0±

490

1305

0±

440

7300

±34

0Z→ℓℓ

1173

±42

3174

±64

982±

3769

9±

3136

4±

23Z→ττ

7±

48±

48±

53±

35±

3D

ata

1821

0±

135

5226

0±

230

1776

0±

130

1520

0±

120

8931

±95

QC

D31

70±

480

3921

±84

088

0±

510

1260

±46

011

00±

350

med

ium

Top

83±

931

7±

1912

8±

1285

±10

63±

8W

+jet

s93

00±

380

2387

0±

590

7610

±34

051

60±

280

2760

±22

0Z→ℓℓ

801±

3517

59±

4846

4±

2428

5±

2014

2±

14Z→ττ

7±

47±

47±

53±

31±

1D

ata

1230

0±

110

2912

0±

170

8616

±95

6213

±29

031

80±

56

QC

D21

10±

390

3170

±62

040

0±

350

6213

±81

220±

220

tight

Top

48±

816

0±

1446

±8

43±

726

±5

W+j

ets

5110

±29

010

990±

410

3100

±23

019

00±

170

1070

±15

0Z→ℓℓ

495±

2878

2±

3216

0±

1311

0±

1248

±10

Z→ττ

7±

47±

44±

33±

31±

1D

ata

7480

±86

1309

0±

110

3399

±58

2434

±49

1134

±34

QC

D18

20±

300

1150

±43

090

±23

038

0±

180

−10

±15

0

Table A.4 Same sign W control region’s yields for each decay mode and for each workingpoint (noID, loose, medium, tight). The QCD yields are computed as the differencebetween data and all simulated processes.

106 Control Region Plots and Yields

Figure A.3 Plots of Mvis(µ,τ) for the 3p0n decay mode in the opposite sign (left column)and same sign (right column) Top control region for noID (first row), loose (second row) andmedium (third row) working points.

107O

ppos

iteSi

gnTo

pC

ontr

olR

egio

n

1p0n

1p1n

1pX

n3p

0n3p

Xn

noID

Top

6545

±89

2813

0±

190

1534

0±

140

2281

0±

170

2119

0±

160

W+j

ets

990±

150

3720

±28

018

00±

190

3380

±27

034

10±

290

Z→ℓℓ

105±

1429

7±

2312

6±

1718

9±

1922

7±

20Z→ττ

84±

1824

2±

3375

±19

92±

1825

±8

Dat

a82

06±

9132

526±

180

1758

0±

130

2875

0±

170

2668

0±

160

QC

D48

0±

200

140±

380

240±

270

2280

±60

1830

±37

0

loos

e

Top

3656

±67

1106

0±

120

3639

±65

3424

±65

1616

±45

W+j

ets

340±

110

950±

150

366±

8416

2±

5717

9±

51Z→ℓℓ

52±

976

±11

19±

511

±4

9±

4Z→ττ

74±

1721

4±

3061

±16

77±

1721

±7

Dat

a38

63±

6211

510±

110

3794

±62

3677

±61

1779

±42

QC

D−

260±

140

−79

0±

220

−29

0±

120

4±

110

−46

±81

med

ium

Top

3083

±63

8510

±11

025

79±

5525

10±

5792

4±

34W

+jet

s21

2±

7560

0±

120

133±

4752

±28

104±

39Z→ℓℓ

42±

827

±6

6±

35±

36±

3Z→ττ

74±

1718

0±

2756

±15

63±

1613

±5

Dat

a31

10±

5684

09±

9225

11±

5024

21±

4990

1±

30

QC

D−

300±

110

−92

0±

190

−26

3±

90−

208±

82−

146±

60

tight

Top

2495

±57

5899

±89

1586

±43

1866

±50

512±

25W

+jet

s13

3±

6127

4±

8187

±39

52±

2818

±18

Z→ℓℓ

29±

616

±5

3±

23±

23±

2Z→ττ

67±

1713

9±

2448

±15

53±

1512

±5

Dat

a23

91±

4956

89±

7514

72±

3817

13±

4146

8±

22

QC

D−

333±

98−

640±

140

−25

2±

71−

260±

73−

77±

38

Table A.5 Opposite sign Top control region’s yields for each decay mode and for eachworking point (noID, loose, medium, tight). The QCD yields are computed as thedifference between data and all simulated processes.

108 Control Region Plots and YieldsSa

me

Sign

Top

Con

trol

Reg

ion

1p0n

1p1n

1pX

n3p

0n3p

Xn

noID

Top

1744

±45

1051

0±

110

7630

±99

1154

0±

120

1263

0±

120

W+j

ets

520±

100

2070

±22

013

10±

160

1950

±21

017

10±

190

Z→ℓℓ

33±

724

9±

2015

2±

1726

1±

2223

2±

24Z→ττ

0.4±

0.4

0.8±

0.8

1.3±

1.3

2±

23±

3D

ata

3374

±58

1533

0±

120

9727

±99

1647

0±

130

1710

0±

130

QC

D10

750±

130

2500

±28

063

0±

210

2720

±27

025

30±

260

loos

e

Top

432±

2317

54±

4783

3±

3452

0±

2642

6±

22W

+jet

s67

±36

390±

110

60±

3613

1±

5185

±34

Z→ℓℓ

9±

348

±9

16±

57±

317

±6

Z→ττ

0.4±

0.4

0.4±

0.4

1±

11±

13±

3D

ata

737±

2725

56±

5111

36±

3481

7±

2959

3±

24

QC

D22

8±

5137

0±

130

225±

6015

9±

6461

±48

med

ium

Top

301±

1995

3±

3541

1±

2524

0±

1817

3±

15W

+jet

s23

±18

169±

7130

±31

55±

304±

4Z→ℓℓ

2±

218

±6

7±

34±

29±

5Z→ττ

0.4±

0.4

0.3±

0.3

1±

10.

8±

0.8

1±

1D

ata

488±

2214

16±

3855

1±

2336

5±

1923

7±

15

QC

D16

2±

3427

7±

8810

2±

4666

±40

51±

22

tight

Top

179±

1545

8±

2515

3±

1412

4±

1361

±9

W+j

ets

23±

1838

±22

21±

13±

44±

4Z→ℓℓ

2±

213

±4

5±

24±

22±

2Z→ττ

0.2±

0.2

0.1±

0.1

0.5±

0.5

0.4±

0.4

0.2±

0.2

Dat

a28

7±

1765

4±

2623

4±

1516

9±

1391

±10

QC

D84

±29

145±

4256

±24

37±

1924

±14

Table A.6 Same sign Top control region’s yields for each decay mode and for eachworking point (noID, loose, medium, tight). The QCD yields are computed as thedifference between data and all simulated processes.

Appendix B

Performance of the BackgroundEstimation Method

In this appendix the tau’s pT and η distributions are shown for the 1p0n (figure B.1), 1p1n(figure B.2), 1pXn (figure B.3), 3p0n (figure B.4) and 3pXn (figure B.5) decay modes in thesignal region. The plots are also split into the four working points (noID, loose, medium andtight) adopted in the analysis. The background is estimated with the method described insection 3.3: each distribution (Top, W+jets, Z → ℓℓ, Z → ττ , same sign) corresponds to oneof the terms of equation 3.4. Comments and observations about the plots are given in section3.3.2.

Table B.1 reports the yields os the opposite sign signal region for each decay mode andeach working point.

110 Performance of the Background Estimation Method

Figure B.1 Distributions of tau’s pT (left column) and η (right column) for the 1p0n decaymode in the signal region for noID (first row), loose (second row), medium (third row) andtight (fourth row) working point.

111



Figure B.3 Distributions of tau’s pT (left column) and η (right column) for the 1pXn decaymode in the signal region for noID (first row), loose (second row), medium (third row) andtight (fourth row) working point.

113



Figure B.5 Distributions of tau’s pT (left column) and η (right column) for the 3pXn decaymode in the signal region for noID (first row), loose (second row), medium (third row) andtight (fourth row) working point.

115O

ppos

iteSi

gnSi

gnal

Reg

ion

1p0n

1p1n

1pX

n3p

0n3p

Xn

noID

Top

(OS-

SS)

69±

1222

7±

3419

±28

96±

3152

±33

W+j

ets

(OS-

SS)

3514

±43

1042

8±

7048

60±

5010

092±

6682

33±

61Z→ℓℓ

(OS-

SS)

1774

±70

874±

119

−16

6±

68−

298±

85−

329±

77Sa

me

sign

1879

0±

150

5179

0±

242

2567

0±

170

4708

0±

230

3830

0±

210

Z→ττ

(OS-

SS)

9920

±18

032

490±

320

1009

0±

190

1301

0±

210

5840

±15

0

Dat

a32

630±

180

9255

0±

300

3858

0±

200

6720

0±

260

5102

0±

230

loos

e

Top

(OS-

SS)

34±

710

3±

166±

1024

±7

7±

6W

+jet

s(O

S-SS

)18

24±

2235

04±

3510

55±

2011

32±

1968

6±

15Z→ℓℓ

(OS-

SS)

1485

±53

374±

70−

35±

27−

75±

22−

36±

18Sa

me

sign

4404

±73

1244

0±

122

3857

±68

3532

±65

2146

±51

Z→ττ

(OS-

SS)

9520

±17

028

200±

290

7740

±15

411

840±

190

4070

±11

0

Dat

a16

270±

130

4221

0±

210

1129

0±

110

1492

0±

120

6062

±78

Med

ium

Top

(OS-

SS)

23±

669

±14

7±

717

±6

3±

4W

+jet

s(O

S-SS

)11

20±

1618

40±

2350

5±

1343

1±

1124

3±

8Z→ℓℓ

(OS-

SS)

1374

±49

347±

53−

17±

18−

43±

13−

31±

12Sa

me

sign

3042

±61

7076

±92

1980

±49

1469

±42

820±

31Z→ττ

(OS-

SS)

9280

±17

025

230±

280

6480

±14

099

10±

180

2847

±96

Dat

a13

920±

120

3237

0±

180

7835

±89

1072

0±

100

3372

±58

Tigh

t

Top

(OS-

SS)

18±

549

±12

4±

518

±5

4±

2W

+jet

s(O

S-SS

)78

2±

1410

09±

1723

7±

921

3±

811

3±

5Z→ℓℓ

(OS-

SS)

1205

±46

270±

37−

5±

9−

17±

9−

13±

8Sa

me

sign

1814

±46

3141

±61

806±

3158

6±

2628

0±

18Z→ττ

(OS-

SS)

8370

±16

019

370±

250

4540

±12

078

90±

160

1683

±75

Dat

a11

330±

110

2241

0±

150

4755

±69

7620

±87

1793

±42

Table B.1 Opposite sign signal region’s yields for each decay mode and for each workingpoint (noID, loose, medium, tight). Each process corresponds to one of the terms ofequation 3.4: Top, W+jets, Z → ℓℓ and Z → ττ are the difference between the OS andSS (OS-SS) processes, weighted by the respective k factor (for W+jets and Top) andrQCD (for SS), whereas Same sign is the amount of data in the SS signal region.

Appendix C

Tau Identification Variables

The distributions of the discriminating variables used for the tau identification BDTs andlisted in section 3.5 are shown in this appendix for 1-prong (figure C.1) and 3-prong (figureC.2) events, in case of signal (Z → ττ ) and same sign data (mostly consisting of QCDbackground). All the distributions, realized with noID working point, are normalized to theunity.

118 Tau Identification Variables

Figure C.1 Distributions of the tau identification variables for 1-prong τhad−vis. Each plotalso shows the separation power S as defined in equation 3.8.

119

Figure C.2 Distributions of the tau identification variables for 3-prong τhad−vis. Each plotalso shows the separation power S as defined in equation 3.8.

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