Dr Madar Romain

David Calvet

Samuel Calvet

Clermont Université

Auvergne

Docteur D'universite

Par « TNOURJI Abdellah

Soutenue Publiquement

M Monteil Stéphane

Mme Genest

M Ramón

Spécialité : physique des particules

Time calibration of the High Granularity Timing Detector and measurement of top quarks spin correlation with the ATLAS detector

Abdellah Tnourji

To cite this version:

Abdellah Tnourji. Time calibration of the High Granularity Timing Detector and measurement of top quarks spin correlation with the ATLAS detector.

Title: Time calibration of the High Granularity Timing Detector and measurement of top quarks spin correlation with 41 the ATLAS detector.

Keywords: HGTD, Top, Precision measurement, EFT, Entanglement, LHC Abstract: This thesis consists of two main parts. The first part corresponds to the timing calibration of all channels of the High Granularity Timing Detector (HGTD), a crucial new detector for the HL-LHC phase-II upgrade. HGTD is designed to enhance the capabilities of all-Silicon ITK by adding timing measurement to charged-particle trajectories as well as spatial information. The timing calibration procedure is proposed to calibrate the distribution of time hit, t hits , measured by each HGTD sensor. This procedure is intended to reduce the intrinsic time differences between the sensors to maintain a good temporal resolution of 30 ps-50 ps. The second part is on the measurements of top quark spin correlation observables in t t events which represents a unique possibility to test the standard model (SM) predictions and probe the new physics effects. Potential deviations from the SM expectations are parametrised within the framework of the Effective Field Theory (EFT). Furthermore, we present the first proposal for detecting entanglement in a pair of quarks , as well as the violation of Bell inequalities (BIs) at LHC.

Résumé

Au cours des cinquante dernières années, le Modèle Standard (SM) de la physique des particules a été le produit d'un immense effort expérimental et théorique, décrivant dans une théorie de jauge unifiée les interactions faibles, électromagnétiques et fortes entre les particules de base du modèle standard, les leptons et les quarks.

La force faible provoque la désintégration radioactive, la force électromagnétique est responsable du magnétisme et de l'électricité, et la force forte maintient les quarks ensemble pour former des protons, des neutrons et de grosses particules plus exotiques. La gravité -la force manquante -a un rôle vital à jouer dans notre vie quotidienne. Cependant, au niveau atomique, il est si faible qu'il peut presque toujours être ignoré. Avec le développement de la technologie au début des années 1970, il est devenu possible de construire des expériences qui étudient les particules du modèle standard et leurs interactions. Les prédictions du modèle standard ont été exceptionnellement précises par rapport aux données expérimentales. La dernière pièce du puzzle du modèle standard, le boson de Higgs, a été découverte en 2012 au grand collisionneur de hadrons (LHC), après près de quatre ans de collisions de protons. Le champ de Higgs joue un rôle unique et important dans le modèle standard.

C'est le seul champ scalaire fondamental et il est responsable de la génération de la masse des particules. Cela a confirmé la validité de notre compréhension des fondamentaux de l'univers au niveau quantique. Néanmoins, malgré le succès frappant apparent de la théorie de SM, il y a beaucoup de raisons pour lesquelles ce n'est pas la Une procédure de correction temporelle est proposée pour calibrer la distribution des temps d'impact des particules, t hits , mesuré par chaque capteur HGTD. Cette procédure est destinée à réduire les différences de temps intrinsèques entre les capteurs pour maintenir une bonne résolution temporel de 30 ps-50 ps. La précision de l'étalonnage dépend en partie de la durée de l'effet et en partie de la nature du composant affecté (ASIC, lpGBT, groupe d'ASIC sur une carte PEB, module). Pour cette raison, l'étalonnage est effectué en mesurant la constante d'étalonnage, ht hits i, à différents intervalles de temps de chaque canal en utilisant différentes granularités spéciales et des données déclenchées à un MHz dans le processeur FELIX. Chaque granularité spéciale cible une région particulière du détecteur HGTD, ainsi que des effets globaux ou plus largement corrélés à calibrer (toujours avec le compromis systématique/statistique). L'erreur statistique associée aux constantes d'étalonnage ht hits i pour plusieurs granularités et intervalles de temps est présentée. Une meilleure correction de cette valeur peut être obtenue avec des fenêtres d'étalonnage plus longues, car elles peuvent collecter plus de statistiques et, par conséquent, déterminer les constantes d'étalonnage plus précisément, ce qui se traduit par une plus grande précision de correction temporelle. Le calcul des constantes d'étalonnage à l'aide d'une petite fenêtre temporelle montre une grande erreur statistique. Malgré cela, on s'attend généralement à ce que des fenêtres d'étalonnage plus petites calibrent plus efficacement la distribution de t hits lorsque des variations à court terme affectent la mesure des occurrences temporelles. Une étude préliminaire montre qu'en utilisant un étalonnage par module, la distribution de t hits est calibrée avec une bonne précision, affichant une valeur de (0 ± 323 ps). L'étalonnage par module absorbe au mieux la variation temporelle de 1 kHz à intervalle régulier de petit événement. De plus, l'étalonnage par module est capable de corriger les effets géométriques, en particulier les effets de temps de vol (ToF).

Mesures des observables de spin du quark top dans des événements t t

Les quarks top du LHC sont principalement produits en paires t t via l'interaction forte, qui conserve la parité. Les quarks et les gluons de l'état initial ne sont pas polarisés, ce qui signifie que leurs spins ne sont préférentiellement alignés sur aucune direction donnée. Les quarks top produits par paires sont donc non polarisés. Cependant, les spins des quarks top et anti-top sont corrélés avec une force dépendant de l'axe de quantification des spins et du processus de production. Comme le quark top se désintègre avant l'hadronisation, les corrélations angulaires de la désintégration du quark top fournissent des informations sur son spin. Si les spins top et anti-top de la paire sont corrélés, alors la structure de spin de top quark peut être mesurée à l'aide d'observables angulaires de leurs produits de désintégration.

Cette deuxième partie de la thèse porte sur la mesure des observables de corrélation de spin dans la production de paires de quarks top à l'aide des données collectées par le détecteur ATLAS entre 2015 et 2018, correspondant à une luminosité intégrale de 139 fb 1 . Les échantillons de signal et de fond sont estimés à l'aide de simulation de Monte-Carlo. Pour les événements de signal, le canal di-lepton a été considéré, qui est principalement basé sur le fait qu'il y a deux leptons de charges opposées provenant des deux quarks top qui ont une charge opposée.

Les bruits de fond principaux sont single-top et V + jets, avec V = W ou Z. Plusieurs pondérations basées sur des variables cinématiques sont appliquées aux événements t tb b pour obtenir une meilleure modélisation. Les bruits de fonds issus de leptons non-prompts sont estimés à partir des données (ScaleFactor). Les leptons nonprompts sont principalement issus des désintégrations de hadrons à saveur lourde, de conversions de photons, d'une mauvaise identification des jets ou de désintégrations de mésons légers. D'autres bruits de fond bien connus sont issus directement de la simulation. La comparaison des données et des simulations MC de divers observables de corrélation de spin montre un bon accord.

Les observables de angulaire de spin mesuré avec le détecteur ATLAS est affecté par les imperfections de détecteur, telles que la résolution limitée de détecteur, l'acceptation du détecteur et les éventuelles sélections d'événements qui sont appliquées aux données. Cette distorsion varie avec la pseudorapidité (n), qui correspond aux changements dans la région active du détecteur, transformant les distributions sous-jacentes au niveau des particules (niveau vrai) en un ensemble de distributions au niveau du détecteur (niveau reconstruit). La méthode de correction des données du niveau du détecteur au niveau des particules est connue sous le nom de unfolding.

Généralement, l'unfolding est le processus de correction des effets du détecteur à partir d'une quantité mesurée en transformant les mesures au niveau reconstruit en un espace de phase au niveau des particules. Les résultats finaux de la mesure après unfolding peuvent être directement comparés aux différentes prédictions théoriques.

Interprétation EFT

Les observables angulaires de spin mesurés sont interprétés pour des signes potentiels de nouvelle physique (BSM) en utilisant le formalisme de la théorie des champs effectifs (EFT). La théorie quantique des champs fournit une procédure systématique pour calculer de manière perturbative l'impact de la nouvelle physique sur les distributions des observables angulaire de spin. Dans cette thèse, la théorie des champs effectifs de modèle standard (SMEFT) est utilisée, ce qui fournit un moyen indépendant du modèle pour paramétrer de nouveaux effets physiques. Nous avons produit des échantillons MC pour les prédictions EFT pour la matrice de densité de spin t t à précision LO et NLO en QCD, pour des énergies de centre de masse de 13 TeV. Nous avons discuté des informations sur la façon d'isoler les différentes contributions (SM pur, EFT pur et interférence) ainsi que les similitudes et les différences entre plusieurs modèles de simulation de prédiction EFT (modèle UFO) fréquemment utilisés dans l'interprétation des données du LHC.

Compte tenu de la particularité de cette analyse, l'implémentation MC du modèle SMEFT@NLO UFO utilisé dans la thèse doivent être validé. En particulier, nous avons examiné l'effet de la désintégration des paires de quarks top à l'aide du package MadSpin lors de la génération d'échantillons EFT à NLO. Nous avons montré que cette procédure capture toujours les effets de corrélation de spin lors des calculs NLO. Nous avons étudié l'impact de la précision NLO QCD sur l'opérateur Chromomagnetic Dipole Moment (CMDM), O tG et l'opérateur de couplages à quatre quarks, O 8 tq , en utilisant uniquement des observables de corrélation de spin dans la production de paires t t. Une amélioration significative est constatée entre les prédictions LO et NLO. Généralement, le couplage EFT est plus important en LO, mais comme nous le promouvons en NLO, cette dépendance est moins importante. Il est également intéressant de noter que les observables de corrélation de spin inter-diagonales sont les plus affectées par les opérateurs EFT. De plus, nous avons vérifié que la procédure d'unfolding n'introduirait pas de différence entre le spectre obtenu avec un échantillon de pseudo-données unfolded et le niveau de particule correspondant. Une légère différence est observée entre les distributions unfoled et au niveau des particules de l'échantillon SM, ce qui n'a qu'un faible effet sur l'interprétation EFT qui est bien couverte par l'incertitude totale. Les incertitudes statistiques et systématiques sont incluses dans cette analyse.

Interprétation quantique

Cette thèse présente, pour la première fois, une mesure de l'intrication quantique en utilisant les données Run-2 asimov. Essentiellement, l'intrication quantique implique que deux particules sont inextricablement liées, quelle que soit leur distance l'une de l'autre. Chaque fois que l'état de la particule change (e.g. spin), son "jumeau" change instantanément, comme s'il s'agissait d'un objet unique. L'intrication quantique marque la différence entre les mondes classique et quantique. L'objectif est de détecter l'intrication quantique entre une paire de quarks t t au LHC, ce qui constituerait une étape importante car il s'agirait de la première observation de l'intrication des quarks de plus haute énergie jamais explorée. Cela fournirait une base pour l'étude de la théorie de l'information quantique dans un collisionneur à haute énergie tel que le LHC.

Un élément clé pour détecter l'intrication est l'analyse des événements t t proches du seuil. La production de t t est produite principalement via la fusion de gluons au LHC. Au seuil, la paire de t t se révèle être dans un état spin-singulet. Remarquablement, le caractère invariant de l'état spin-singulet et la dominance de la fusion de gluons dans la production de t t au LHC impliquent également que la séparation angulaire entre les produits de désintégration leptonique fournit une signature d'intrication expérimentale directe. Cela peut être mesuré expérimentalement en utilisant l'observable D à partir de la distribution de la séparation angulaire entre les leptons, cos(q ``) tel que donné par la section efficace différentielle:

1 s ds d cos j = 1 2 (1 D cos j) (1)

A historical introduction

Since the dawn of time, people have wondered: What we are made of? Why does the world exist as it does? What are the fundamental elements of the world around us? In what way do they interact? The hope of understanding the world in terms of a few basic ingredients dates back to at least the 5th century BC.

Several cultures-Babylonians, Greeks and others invented a remarkably cohesive set of five "elements" from which everything else was made. The ones we know best are earth, air, fire, and water, but there was also a fifth heavenly element of earth, or quintessence. Among many ideas, this one was developed into a system devised by Aristotle. He suggested that each element sought a particular natural state; for example, the earth falls and the air rises. By mixing the elements in different combinations, we could explain the different substances that we see around us.

Democritus, a Greek philosopher who predates Aristotle, proposed that everything we know is made up of minuscule, indivisible pieces called "atoms". It was an unfortunate accident of history that this terminology was entered by John Dalton, a chemist who worked in the early 1800s, to refer to the parts that define the chemical elements. What we now consider to be an atom is not at all indivisible: most of the mass of an atom is concentrated in the tiny nucleus surrounded by a cloud of electrons. The nucleus in turn is formed of protons and neutrons, but these are not elementary either. They are actually composed of even smaller particles, called quarks.

Physics is used to synthesize many phenomena into a few theories. For example, in the early days, there were phenomena of motion and phenomena of heat; there were phenomena of sound, light and gravity. But it was soon discovered after Sir Isaac Newton explained the laws of motion, that some of these seemingly different things were aspects of the same things. For example, the phenomena of sound could be completely understood as the motion of atoms in the air. So sound was no longer seen as something in addition to movement. It was also discovered that the heating phenomena are easily understood from the laws of motion.

In this way, large globs of the theory of physics were synthesized into a single theory.

In 1873, James Clerk Maxwell combined the ideas from Oersted, Coulomb, Gauss, and Faraday into an overarching theory of electromagnetism, adding his own insights. In a ground-breaking theoretical contribution, the Scottish physicist unified electricity, magnetism, and optics. In his formulation of electromagnetism, the light is the propagation of electric and magnetic fields. As a general principle, he proposed the existence of electromagnetic radiation: a combination of electrical and magnetic fields travelling together at a speed equal to the speed of light.

Chapter 1. A historical introduction

In the past century, we have slowly built our understanding of particle physics, now called the Standard Model (SM), which provides a unified picture where the forces between particles are themselves described by the exchange of particles. Remarkably the SM provides a successful description of all current experimental data and represents one of the triumphs of modern physics. The standard model became the foundation of particle physics in the late 1940s. Theoretical physicists sought to embed quantum mechanics consistently into their models; the laws describing (sub)atomic particles, within the framework of special relativity;

the laws of motion for objects with very large velocities. Their efforts culminated in the development of Quantum Electrodynamics (QED); the quantum theory of the electromagnetic field.

In the atomic nucleus, protons and neutrons are united by the strong nuclear force, which is a manifestation of the fundamental theory of strong interaction, called Quantum Chromodynamics (QCD). The fundamental interactions of particle physics are completed by the weak force which is responsible for the nuclear b -decay of certain radioactive isotopes and in the nuclear fusion processes that fuel the Sun. In both nuclear b -decay and nuclear fusion, another particle, the nearly massless electron neutrinos (n e ) is produced. Physical phenomena can almost all be explained by describing the electron, electron neutrinos, proton and neutron, interacting by electromagnetic, strong and weak forces. The picture is completed by gravity which, although extremely weak, is still attractive and therefore responsible for the large-scale structure of the Universe.

As early as 1935, Yukawa published an attempt at writing a theory for the strong nuclear force [START_REF] Yukawa | On the Interaction of Elementary Particles I[END_REF]. In his 1934 article, Yukawa speculated that the nuclear strong force is carried by a particle with approximately one-tenth the proton mass. Shortly after his prediction, a particle with nearly this mass was identified in cosmic ray phenomena. In the beginning, Yukawa's predictions appeared to be incredibly accurate, but the particle found in the cosmic ray records caused problems. Even though its mass was 207 times greater than that of an electron, it had half-integral spin, not integral spin as Yukawa predicted the carrier of the strong nuclear force would have. It turned out that the cosmic ray particle was not the particle Yukawa was talking about. Instead the cosmic ray particle was essentially a heavy electron, which is now called the muon (µ).

Later three particles with masses approximately 270 times that of an electron were found. These did have the characteristics promised by Yukawa. One had a positive charge, one had a negative charge, and one had a neutral charge. Originally, called pi-mesons, they are now known as pions.

In order for particle theories to be valid, they must be consistent with special relativity. The combination of quantum mechanics, electromagnetism and special relativity led Dirac to the equation now universally known as the Dirac equation and, on quantising the fields, to quantum field theory. Quantum field theory had as its first triumph QED, which describes the interaction of the electron with the electromagnetic field.

The QED lies at the heart of our understanding of matter at the atomic level. It is a field theory; each kind of particle is described by a field extending through space. Photons are related to the electric and magnetic fields, and there is also an electron field. In these fields, particles can be likened to little packets of waves, which are short trains of waves, although the analogy is not perfect.

QED was invented in the 1930s and gave good results, but also ran into a serious problem [START_REF] Tom | The Standard Model of Particle Physics[END_REF]. The standard method of calculation, called perturbation theory, involves successive orders. First-order perturbation theory usually gave results within a few percent of the correct answer. Higher-order, meant to give more precise answers, actually gave infinity! Three people independently came up with the solution to that problem, in 1947, Richard Feynman and Julian Schwinger in the USA and in fact earlier [START_REF] Feynman | Space-Time Approach to Non-Relativistic Quantum Mechanics[END_REF][START_REF] Dyson | The Radiation Theories of Tomonaga, Schwinger, and Feynman[END_REF], in 1943, Sin-Itiro Tomonaga [START_REF] Tomonaga | On Infinite Field Reactions in Quantum Field Theory[END_REF][START_REF] Tomonaga | On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields*[END_REF] working all alone in wartime Japan. It was called renormalization theory. In essence, the theory of renormalization is a prescription which consistently isolates and removes all these infinities from the physically measurable quantities. The QED is full of infinities which need to be taken care of before the theoretical predictions can be compared with experimental measurements. Note that the need for renomalization is quite general and is not unique to the QED. The results obtained from QED, after renormalization, fit the experiments astonishingly well and rapidly became the most accurately verified theory in the history of physics.

As a consequence of QED's success, theoretical particle physicists sought knowledge about the two other types of forces that are less well-known since they are short-range, and relatively insignificant on scales larger than the atomic nucleus. These are the strong nuclear interaction and the weak nuclear interaction.

The story of weak interactions started in 1898 when Henri Becquerel discovered that the so-called Becquerel ray actually consisted of two distinct types of radiation: one that is readily absorbed which he called beta radiation, and another of a more penetrating character which he called alpha radiation. Then, in 1900, the Curies measured the electric charge of the beta particle and found it to be negative. That, at the turn of the century, began the history of the weak nuclear interaction. From the very start the road of discovery was tortuous, and the competition intense. Ernest Rutherford and others then categorized the nucleus decays into alpha, beta and gamma decays. But the real understanding of b -decay in the sense we know it now came only after Enrico Fermi invented a physical model for the b -decay process in 1934 [START_REF] Fermi | An attempt of a theory of beta radiation. 1[END_REF].

In essence, Wolfgang Pauli's work was the basis for Fermi's theory [START_REF] Lopes | A model of the universal fermi interaction[END_REF]. To solve the puzzle of the continuous energy spectrum of the electrons emitted in the b -decay of the nuclei, Pauli had suggested that along with the electron, an almost massless neutral particle also was emitted. Fermi succeeded in incorporating Pauli's suggestion and thus was born the theory of weak interactions. Fermi also named the particle as neutrino.

Analogizing electromagnetic interaction to photon emission at the quantum level, Fermi pictured the weak interaction responsible for the b -decay of the neutron as the emission of an electron-neutrino pair, the neutron converting itself into a proton in the process. Fermi theory was not renormalizable even after it was found to be highly successful in predicting observables in b -decay, such as the electron energy spectrum: as in QED, the infinities that plagued early calculations appear again, but unlike in QED, they cannot be removed.

Therefore it was abandoned as a fundamental theory.

An improvement came in the form of the intermediate vector boson model [START_REF] Lopes | A model of the universal fermi interaction[END_REF][START_REF] Feynman | Theory of Fermi interaction[END_REF], where, rather than a contact interaction, the decay was described as mediated by the exchange of vector bosons, completely analogously to the pions that mediated the strong force in Yukawa's theory, and the photons of QED. In this context, Fermi's point-contact interaction hypothesis, pictured to explain the b -decay, is very different from the actual interaction.

Astronomers found a large number of particles in cosmic rays and in particle accelerators [START_REF] Lattes | PROCESSES INVOLVING CHARGED MESONS[END_REF], so the question is, are these particles truly elementary? To make sense of this zoo, physicists looked for patterns, rather as chemists had done a century earlier in developing the periodic table. This revealed a lot of similarities and approximate symmetries. Werner Heisenberg had suggested a symmetry between protons and neutrons [START_REF] Tom | The Standard Model of Particle Physics[END_REF].

In his view, they are quite similar, with the same strong interactions, the same spin ( 1 2 ) and almost the same mass, and suggested that they might be regarded as two different states of a single entity, called the nucleon.

Another proposal of his was a reciprocal symmetry in which the two could be rotated into each other without changing anything. This symmetry is now called isospin, not because it has anything to do with real spin, but because the quantum mechanical description is mathematically similar to that of angular momentum.

But unlike angular momentum it is a dimensionless quantity, and is not actually any type of spin. Protons and neutrons do differ, because the proton has an electric charge and the neutron does not; the symmetry is broken by electromagnetism. But it is nevertheless a good approximation, for example in classifying energy levels of light atomic nuclei, because at short distances the strong interactions are much stronger than the electromagnetic.

There was also evidence, in 1961, that many of these newly discovered particles could be arranged into patterns in two dimensions, suggesting a wider, but even more approximate symmetry. This was discovered independently by Murray Gell-Mann and by Yuval Ne'eman. Gell-Mann called the symmetry the "eight-fold way", because of its characteristic patterns of octets [START_REF] Zweig | An SU(3) model for strong interaction symmetry and its breaking[END_REF][START_REF] Gell-Mann | A schematic model of baryons and mesons[END_REF].

The theory of strong interactions had a different problem; The strong interactions can be renormalizable just like the original Yukawa theory [START_REF] Yukawa | On the Interaction of Elementary Particles I[END_REF]. However, the perturbation theory is useless because of their strength, so they cannot be used to perform practical calculations. A deeper problem with our understanding of both the weak and the strong interactions was that there was no rationale for any of these theories. According to Salam and others of like mind [START_REF] Goldstone | Broken Symmetries[END_REF], the answer lies in a gauge theory. QED is the simplest example of a gauge theory. The simplest manifestation of this symmetry is that it is the difference in electric voltages, rather than the absolute voltages, that matters.

A gauge theory incorporating Heisenberg's isospin symmetry was introduced in 1954 by Chen-Ning Yang and Robert Mills [START_REF] Yang | Conservation of Isotopic Spin and Isotopic Gauge Invariance[END_REF][START_REF] Lee | Question of Parity Conservation in Weak Interactions[END_REF]. According to previous work, electromagnetic processes operate by exchanging Chapter 1. A historical introduction photons, in the same way, weak interactions operate by exchanging particles called W + and W , with the superscripts indicate positive and negative values for electric charge. W + and W are similar to the photon, in particular they all have spin 1, but with one important distinction. Since weak interactions have an extremely short range, it is necessary to assume the W particles are extremely massive, about 100 times heavier than proton. Schwinger suggested they might also be the "gauge bosons" of a gauge theory [START_REF] Anderson | Plasmons, Gauge Invariance, and Mass[END_REF], and indeed that there might be a unified theory of weak and electromagnetic interactions with the W + and W and photon (conventionally denoted by g) appearing symmetrically. But this symmetry could not be exact;

because of the big difference in mass, it had to be severely broken. Other proposals followed. In 1961, Salam [START_REF] Salam | On a gauge theory of elementary interactions[END_REF][START_REF] Salam | Electromagnetic and weak interactions[END_REF] proposed a modified theory. To solve another problem with Schwinger's scheme, concerning mirror symmetry, he added a fourth gauge boson, Z, electrically neutral like the photon. Glashow [START_REF] Sheldon | Partial-symmetries of weak interactions[END_REF] proposed essentially the same theory in 1964, unaware of Salam's work.

In the late 1950s, Yoichiro Nambu recognised that spontaneous symmetry breaking [START_REF] Anderson | Plasmons, Gauge Invariance, and Mass[END_REF][START_REF] Goldstone | Field theories with « Superconductor » solutions[END_REF], a process where a symmetric system ends up in an asymmetric state, could occur under certain conditions. In 1962 physicist Philip Anderson, working in the field of condensed matter physics, observed that symmetry breaking played a role in superconductivity, and may have relevance to the problem of gauge invariance in particle physics.

In 1963, this was shown to be theoretically possible, at least for some limited (non-relativistic) cases.

Following the 1962 and 1963 papers, groups of researchers independently published the 1964 PRL symmetry breaking papers [START_REF] Higgs | Broken Symmetries and the Masses of Gauge Bosons[END_REF][START_REF] Brout | Spontaneous Symmetry Breaking in Gauge Theories: a Historical Survey[END_REF][START_REF] Guralnik | The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles[END_REF] with similar conclusions and for all cases, not just some limited cases. They showed that the conditions for electroweak symmetry would be "broken" if a special type of field existed throughout the universe, and indeed, some fundamental particles would acquire mass. The field required for this to happen (which was purely hypothetical at the time) became known as the Higgs field (after Peter Higgs, one of the researchers) and the mechanism by which it led to symmetry breaking, known as the Higgs mechanism. A key feature of the necessary field is that it would take less energy for the field to have a non-zero value than a zero value, unlike all other known fields, therefore, the Higgs field has a non-zero value (or vacuum expectation) everywhere. This non-zero value could in theory break electroweak symmetry.

It was the first proposal capable of showing how the weak force gauge bosons could have mass despite their governing symmetry, within a gauge invariant theory.

By 1972, these ideas had been further developed into a comprehensive theory and had demonstrated their ability to offer "senseable" results that accurately described particles known at the time, and with remarkable predictive power. During the 1970s these theories rapidly became the SM of elementary physics.

As all of this was occurring, there were also remarkable developments in our understanding of strong interactions, starting with an effort to understand the eightfold-way symmetry. Gell-Mann and George Zweig suggested in 1963 [START_REF] Yang | Conservation of Isotopic Spin and Isotopic Gauge Invariance[END_REF][START_REF] Lee | Question of Parity Conservation in Weak Interactions[END_REF], that this pattern could be explained if strongly interacting particles were made up of three basic particles. Gell-Mann called them "quarks", distinguished as "up", "down" and "strange" (u , d, and s).

The proton and neutron for example were each formed from three quarks, the proton from (uud) and the neutron from (udd). Strange quarks, which were significantly heavier, were needed to make the more exotic members of the particle zoo. Viewed thus, isospin was a symmetry between u and d quarks, and the eightfold way a more approximate symmetry between u, d and s. The curious thing about quarks is that, unlike all previously observed particles, they have electric charges that are fractional multiples of proton charge e, +2 3 e for u and 1 3 e for d and s.

Despite this neat scheme, there was a flaw, most clearly illustrated by the existence of some particles referred to as D ++ , formed of (uuu). Since it has spin + 3 2 , the spins of the three u quarks (each + 1 2 ) must be pointing in the same direction. This means they are occupying the same state, in contradiction to the Pauli principle.

In 1964, Moo-Young Han, Yoichiro Nambu and Oscar Wallace Greenberg [START_REF] Greenberg | Spin and Unitary-Spin Independence in a Paraquark Model of Baryons and Mesons[END_REF][START_REF] Han | Three-Triplet Model with Double SU (3) Symmetry[END_REF] suggested a way to solve that problem -quarks come in three different colours, conventionally red, green and blue. The word colour in this context is purely colloquial and has no connection with the colour that we see with our eyes in everyday life. It is believed that all known particles are colour neutral, consisting of one quark of each colour (or a quark and an antiparticle). Thus the three quarks in (uuu) are of three different colours and so don't violate the Pauli principle. There is a perfect symmetry between the different colours.

As it turns out, quarks actually come in six types [START_REF] Tom | The Standard Model of Particle Physics[END_REF], not just three. Those three extra types are called "charms" (c), "bottom" (b) and "top" (t); more glamorous names for the last two, "beauty" and "truth", have not caught on. The six quarks can be arranged as three pairs, (u,d), (c,s), (t,b), each with the same pair of charges ( +2 3 e, 1 3 e). Each of the three pairs looks essentially the same, except for the fact that they get increasingly heavy; the top quark is actually heavier than the W and Z gauge bosons. The electron too has two very similar but heavier companions, the muon (µ) and the tau (t), and each of these three is paired with a massless neutrino with charge zero. Both quarks and what we call leptons ( e.g electron and muon) have weird three-generation structures.

In order to grasp the importance of the interactions, colour was crucial. Han and Nambu suggested that strong interactions might be mediated by an octet of gauge bosons, called "gluons", coupled specifically to colours. This idea forms the basis of the now accepted gauge theory of strong interactions, QCD.

In light of these discoveries, we can now call the Standard Model of particle physics complete: six quarks and six leptons, in three generations of increasing mass, make up the matter. The force carrier are the gauge bosons: the photon of electromagnetism, the W and Z of the weak interaction, and the gluon mediating the strong interaction. In order to realize all of this, the Higgs boson must be introduced. The Higgs boson breaks the electroweak symmetry, give mass to the W and Z, among other things. Despite its minimal nature, the model can explain all the observable matter content of the Universe, including its interactions (other than gravity).

Despite the Standard Model continues to triumph after over a century, physicists are looking for evidence of a new physics to take us beyond the current paradigm. Beyond Standard Model (BSM) is interpreted in many ways [27]. In some cases, it can refer to physical phenomena that have been observed experimentally operators: any BSM model can be matched to these, so that an analysis in terms of these can be matched to any BSM model.

The LHC, with a design centre-of-mass energy of 14 TeV, is the best tool for answering if there are new physics beyond SM at the TeV energy scale. Numerous benchmark analyses seeking new physics have already been published, and it is possible that there are manifestations of new physics in many of them.

A good question to ask is "Which EFTs operators describe the data the best?". Hence, it is becoming increasingly important for the LHC experimental analyses to properly parametrize BSM effects.

Among the potential new physics effects, The top quark sector is one of the well-motivated place to search for hints of BSM physics in the SM Effective Field Theory (SMEFT). In spite of the discovery of the top quark in 1995, it remained important in particle physics for a long time. The study of its properties has been extensively carried out at the LHC. Nevertheless, a few important questions still remain unanswered. Why is it so heavy? What function does the top quark play in the mechanism of electroweak symmetry breaking (EWSB)? Does the top quark play a role in non-SM physics?

Chapter 1. A historical introduction

The aims of present thesis are:

• The timing calibration of various channels of the High-Granularity Timing Detector (HGTD) is presented in light of ATLAS Phase II upgrade for the HL-LHC period where high pile-up is expected.

• A measurement of observables sensitive to spin correlations in top pair production is presented, using the Run-2 data set corresponding to an integrated luminosity of 139 fb 1 of proton-proton collisions at p s = 13 TeV with the ATLAS detector at the Large Hadron Collider (LHC). Spin correlation is a ideal property to test SM predictions. Previous measurement of spin correlation at the LHC with

Run-1 ATLAS pp data were consistent with the SM expectation and no evidence for new physics was found [START_REF] Aaboud | Measurements of top-quark pair spin correlations in the eµ channel at p s = 13 TeV using pp collisions in the ATLAS detector[END_REF][START_REF] Aad | Observation of Spin Correlation in tt Events from pp Collisions at p s = 7 TeV Using the ATLAS Detector[END_REF][START_REF] Bernreuther | A set of top quark spin correlation and polarization observables for the LHC: Standard Model predictions and new physics contributions[END_REF].

• The spin correlation measurements are interpreted using an Effective Field Theory (EFT) approach in the Warsaw basis [START_REF] Grzadkowski | Dimension-six terms in the Standard Model Lagrangian[END_REF].

• For the first time, the expected experimental sensitivity of quantum entanglement at the LHC using the full data from ATLAS Run 2 is presented. This fundamental property of quantum behaviour can be probed by measuring the differential normalized cross-section as function of the cos(j ``), where j ``is the angle between the two leptons in their respective top-quark parent rest-frame. Further, we present for the first time the expected sensitivity of violations of Bell's inequality (BIs) at the LHC.

The current analysis is not finalized, so the analysis is blinded. Hence, the statistical results shown in the chapter are expected results only and evaluated using the Asimov dataset.

The thesis is organized as follows. In chapter 2, a brief theoretical overview of the Standard Model, drawing particular attention to t t production, spin correlation and the theoretical foundations of the effective field theory (EFT) framework. Chapter 3 describes the LHC complex and ATLAS experimental setup. Chapter 4 is dedicated to the performance of timing calibration for the High-Granularity Timing Detector (HGTD) in ATLAS. Chapter 5 details the measurement of the spin correlation observables in top pair production as well as their interpretation within the effective field theory framework. Additionally, it presents a detailed search for the entanglement and violation of Bell's inequality (BIs) at the LHC. The conclusion of this thesis is given in Chapter 6.

Theoretical background

T

The Standard Model

To date, particle physics has been concerned with identifying what appear to be elementary units of matter and disclosing the nature of the forces that act between them. An elementary particle is defined as a particle that can not be sub-divided into smaller particles. The SM of particle physics provides a unified picture of elementary particles and the forces between them. In many ways, the standard model provides a successful description of most of the current experimental data with astounding accuracy and is one of the greatest triumphs of modern physics.

An introduction to the elementary particles of the SM and their fundamental interactions is given in this chapter. This chapter's second Section focuses on the top quark's properties, production mechanisms, and decay modes. Study of the q q ! t t and g ḡ ! t t amplitudes with polarized top quark production is also discussed. Section 3 examines the spin correlations in top quark pair productions. The last Section of this chapter discusses the EFT approach to top quarks. The following discussion heavily draws upon Mark

Thomson's Modern Particle Physics on gauge theories in the Standard Model [START_REF] Thomson | Modern particle physics[END_REF].

Elementary particles

Atoms are the fundamental units of ordinary matter that forms the observable universe. Every atom is composed of a nucleus and one or more negatively charged electrons (e ) orbiting around a central nucleus.

A nucleus consists of positively charged protons (p) and electrically neutral neutrons (n). The electrons interact with the nucleus through electromagnetic force, which is the manifestation of the fundamental theory of the electromagnetism, called QED. The nucleus are bound together by a short-ranged attractive potential called the residual strong force, which is a manifestation of the fundamental theory of strong interactions, called QCD. The fundamental interactions of particle physics is completed by the weak force, which describes the b -decay of unstable atoms and acts at very short distances (typically around 0.001 the size of the proton (10 18 m)). In b -decay, another particle, the nearly massless electron neutrinos (n e ) is Chapter 2. Theoretical background produced. At atomic and nuclear scales, all accounted physical phenomena can be explained by describing the electron, electron neutrinos, proton and neutron interacting by electromagnetic, strong and weak forces. The picture is completed by gravity which, although extremely weak, is still attractive and therefore responsible for the large-scale structure of the Universe.

At higher energy scales, a more complicated structure is observed, as illustrated in In brief, the picture is as follows. The matter units are fermions, with spin-1 2 (in units of h).They are two types, leptons, and quarks. The charged leptons are generalizations of the electron. As far as we know, they are elementary particles at the smallest distances currently probed by the highest-energy accelerators.

Leptons and quarks are further divided into three generations or families, of increasing mass: the electron, the electron neutrino, the up-quark, and the down-quark are known collectively as first generation. In high-energy scales, more complexity is revealed. For each of the four first generation particles, there are exactly two identical copies, with the only difference being their mass. These additional eight particles known as the second and third generations. For instance, the electron, the lightest charged lepton, is followed by the muon (µ ) with mass m µ ⇡ 200 m e , while the third generation tau-lepton (t ) is the heaviest with m t ⇡ 3500 m e . Though electron, muon, and tau have different masses, their fundamental properties are identical. The same generation pattern then repeats for the quarks. A down-quark has two heavier versions called the strange and bottom quarks, and an up-quark has two heavier versions called the charm and top quarks. The twelve elementary particles are summarized in detail in [START_REF]Review of Particle Physics[END_REF] While neutrinos are thought to be massless in the Standard Model scenario, it is now known that there are three discrete neutrino masses that differ and they do not correspond to the three flavours in the model [START_REF] Fukuda | Evidence for Oscillation of Atmospheric Neutrinos[END_REF][START_REF] Ahmad | Measurement of the Rate of n e + d ! p + p + e Interactions Produced by 8 B Solar Neutrinos at the Sudbury Neutrino Observatory[END_REF]. The neutrinos are emitted and absorbed in weak processes in flavour eigenstates but travel as mass eigenstates, this is referred to as neutrino oscillation.

In the Standard Model, every fermion has an associated antiparticle, with the same mass and opposite quantum numbers. So, for example, an electron has a negative electric charge and its antiparticle, the positron (e + ), is positive. A neutrino has no electric charge, allowing the possibility that it is its own antiparticle (also known as Majorana neutrinos). As of yet, neutrinos' exact nature is unknown.

Particle interaction

Four fundamental forces operate in the universe: the strong force, the weak force, the electromagnetic force, and the gravitational force. The gravitational force between two individual particles is extraordinarily small compared to even the weak force, and so can be neglected in the discussions of particle interaction. Also conceptually, gravity is still somewhat different from the other forces, which, as shown in the following sections, are quite similar. The fundamental forces act upon the fundamental twelve fermions and they can be classified based on the type of interaction they experience, as shown in A check-mark indicates that the particles are charged under the corresponding interaction and therefore participate in the associated force.

Modern particle physics describes every force and particle in terms of Quantum Field Theory (QFT), which means that elementary fermions interact through fields. The particles associated to the interaction fields are called gauge boson. In case of electromagnetism this is the theory of QED where the quanta of the electromagnetic interaction field between electrically charged fermions are the massless photons (g). The quanta of the weak interaction fields between fermions are the charged W ± bosons and the neutral Z boson, discovered at CERN in 1983. Since these carry mass, the weak interaction is short ranged. The quanta of the strong interaction field, the gluons (g), have zero mass and, like photons, might be expected to have infinite range. In contrast to the electromagnetic field, the gluon field, as well as quarks, are contained in composite particles called hadrons, which effectively limits the range of strong interactions to 1 ⇥ 10 15 meters, roughly the size of a proton.

Besides the gauge bosons, the SM contains one scalar (e.g. spin zero) boson with a mass of 124.92 The elementary particles of the Standard Model, including: the twelve fermions, the four gauge bosons, and the scalar Higgs boson. Though not listed, each of the quarks and leptons has an antiparticle partner that is identical except for its charge. Source Wikipedia

Standard Model Lagrangian

A fundamental result for particle physics is provided by Noether's theorem 1 . If an action is invariant under some group of transformations (symmetry), then there exist one or more conserved quantities (constants of motion) which are associated to these transformations. In this regard, Noether's theorem establishes that symmetries imply conservation laws. In a QFT, in order to have the theory invariant under local transformations, more fields are introduced, such that the extended theory is covariant (i.e. physical laws remain unchanged) with respect to the new extended group of local transformations

The Lagrangian of the Standard Model is constructed by postulating the set of symmetries of the system. The SM Lagrangian density3 can be decomposed into:

L SM = L QCD + L EW + L Higgs + L yukawa (2.1) 2.1.3.1 Quantum Electrodynamics
The Quantum Electrodynamics (QED) is perhaps the best fundamental physical theory ever built to describe Nature, which has become a prototype of a successful quantum field theory. The QED theory describes the interaction between spin-1/2 electrons and positrons as mediated by massless spin-1 photons. The gauge structure of QED follows a principle of invariance under local gauge transformations of the U(1) symmetry group, the real Lie group of unitary U † U = 1 matrices of dimension one. This symmetry requires the existence of massless spin-1 photons and the conservation of electrical charge, Q. As generators of this symmetry group commute, QED is an Abelian gauge theory, which implies that photons don't carry an electric charge and can't directly self-interact.

In order to determine these ground-breaking results of QED one must determine the Lagrangian of QED, which can be derived by imposing a local U(1) gauge invariance on the Lagrangian density of the free Dirac field, i.e. one without interactions. The Lagrangian density for a free Dirac field, y, can be written as:

L = i ȳg µ ∂ µ y m ȳy (2.2)
where ȳ corresponds to y † g 0 , g µ (µ = 0, 1, 2, 3) corresponds to the Dirac matrices and m is the fermion mass.

The free Dirac Equation 2.2 is clearly invariant under a global U(1) phase transformation:

y ! e ia y (2.3)

Chapter 2. Theoretical background

Suppose there is a fundamental symmetry of the Universe which requires physics to be invariant under local phase transformation defined by: y ! e iqa(x) y(x) (2.4) For this local U(1) phase transformation, the free-particle Dirac equation becomes:

ig µ ∂ µ + iq∂ µ a y = my (2.5)
Which differs from 2.2 by the term qg µ (∂ µ a)y. Hence, local phase invariance is not possible for a free theory. By adding a term to the Lagrangian density, it is possible to preserve invariance under local gauge transformation, such that the new Lagrangian density is equal to:

L = ȳ ig µ ∂ µ m y q ȳg µ A µ y (2.6)
The extra term q ȳg µ A µ y represents the interaction between a fermion with electric charge q and the photon field A µ , which under local gauge transformation, transforms as:

A µ ! A µ ∂ µ a (2.7)
The photon field A µ is related to the electromagnetic field tensor by

F µn = ∂ µ A n ∂ n A µ (2.8)
The kinematic (L Kin ) and mass (L m g ) terms for the photon can be written as:

L kin = 1 4 F µv F µv , L m g = 1 2 m 2 g A µ A µ (2.9)
Nevertheless, the existence of the mass term breaks the local gauge invariance:

1 2 m 2 g A 0 µ A 0µ ! 1 2 m 2 g A µ ∂ µ a (A µ ∂ µ a) 6 = 1 2 m 2 g A µ A µ (2.10)
As a result, the photon field is massless and the QED Lagrangian can be written as follows:

L QED = ȳ ig µ ∂ µ m y 1 4 F µn F µn q ȳg µ A µ y = L 0 + L Int (2.11)
where L 0 = ȳ ig µ ∂ µ m y 1 4 F µn F µn is the free field Lagrangian and L Int = q ȳg µ A µ y is the interaction term between two electron fields and the vector field of a photon as illustrated in Figure 2.3.

The QED Lagrangian is thus obtained by taking the Lagrangian for a free Dirac field and imposing local gauge invariance on it. This describes all of electrodynamics, free Dirac particles and their interactions. The quantum field theoretic description of the strong force is known as Quantum ChromoDynamics (QCD).

The QCD is invariant under SU(3), corresponding to the group of special (det(U) = 1) unitary matrices of dimension three, local phase transformations:

y(x) ! y(x) 0 = e ig s a(x) Ty (x) (2.12)
where T = {T a } are the eight generators of the SU(3) symmetry group, which are related to Gell-Mann matrices, l a , by:

T a = 1 2 l a (2.13)
Due to the fact that SU(3) generators are represented by 3 ⇥ 3 matrices, the wave-function y(x) must now include three additional degrees of freedom which can be represented with a three component vector analogous to the representation of SU(3) symmetry for the u, d, and s quarks. This new degree of freedom is termed colours with red, blue and green labelling the states, where colour is simply a label for the orthogonal states of SU(3) colour space. For the local gauge transformation of 2.12, the Dirac equation becomes:

ig µ ∂ µ + ig s (∂ µ a) T y = my (2.14)
Analogous to QED, the required local gauge invariance can be asserted by introducing eight new fields

G a µ (x)
, where the index a corresponds to the colour gauge field (1, ..., 8). The Dirac equation with the interactions with the new gauge fields:

ig µ ∂ µ + ig s G a µ T a y my = 0 (2.15)
The gluon field strength tensor G a µn (x), analogous to the field strength tensor of electromagnetism, is defined in terms of the gluon fields G a v as: The covariant derivative D µ is introduced as:

G a µv = ∂ µ G a v ∂ v G a µ g s f abc G b µ G c v ( 2
D µ i j = ∂ µ d i j ig s (T a ) i j G a µ (2.17)
Accordingly, the gauge invariant QCD Lagrangian density is:

L QCD = ȳi ⇣ ig µ D µ i j md i j ⌘ y j 1 4 G a µn G µn a (2.18)
The QCD theory has been very successful in the description of the interactions binding quarks to hadrons.

This theory has two important characteristics: colour confinement and asymptotic freedom.

Figure 2.5 shows the dependence of a s on the energy of the interaction. Within the range of energies relevant to particle physics, the dimensionless coupling strength of the strong interaction, a s , varies as follows; at low energies and large distance, the coupling strength is very large. In this regime, all particles that exhibit strong interactions are colourless bound states. These bound states are known as hadrons -formed either by a combinations with the quantum numbers of a quark q and a antiquark q pairs (mesons) or combinations of three quarks (baryons) in a colour-neutral state. This feature of QCD is explained by the colour confinement, in which colour particles are confined to singlet states and never appear as free particles. As a consequence, coloured partons produced in high-energy interactions manifest themselves experimentally as collimated streams of hadrons referred to as jets. In contrast with this, at high energies and short distances, the coupling strength is very small and quarks and gluons interact weakly with each other. As a result, partons (quarks and gluons) can be treated as free particles. The value of a s is generally small enough, of order O(0.1), to allow perturbative calculations -this is known as asymptotic freedom.

As an interesting aside, if nature had used a U(3) local gauge symmetry rather than SU(3), they would correspond to an additional U(1) generator. The ninth gluon would be colourless, unconfined, and behave like an interacting photon, leading to an infinite-range strong force; the Universe would be a very different place with strong interactions between all quarks in long-range. Throughout this Section, we provide an introduction to both the weak interactions, which is mediated by the massive W ± and Z bosons, and the unified picture of the electromagnetic and weak interactions known as Electroweak (EW) unification.

s (M Z 2 ) = 0.1179 ± 0.0009 August 2021 s (Q 2 ) Q [GeV] decay (N 3 LO) low Q 2 cont. (N

Weak interactions

In 1934, Fermi proposed the first quantum field theory of weak interaction, which was derived from Pauli's neutrino hypothesis; the theory explains the process of b -decay, n ! p + e + ne . It was subsequently referred to as effective theory of local current-current (CC) four-fermion interaction. The basic process was assumed to be the transition neutron!proton, which took place at a point-like vectorial current (V), with the emission of an e ! n pair, as shown in Figure 2.6 on the left hand side.

The breakthrough came in 1956, when [START_REF] Lee | Question of Parity Conservation in Weak Interactions[END_REF] [START_REF] Lee | Strength of Weak Interactions at Very High Energies and the Higgs Boson Mass[END_REF] suggested that parity was not conserved in all weak decays. An experiment conducted by Wu and colleagues in 1957 established for the first time that parity was violated in weak interactions, specifically in nuclear b -decay. Using only vector currents, the Fermi theory needs to be modified to accommodate this result. [START_REF] Sudarshan | Chirality Invariance and the Universal Fermi Interaction[END_REF] [START_REF] Sudarshan | Chirality Invariance and the Universal Fermi Interaction[END_REF],

and then Feynman and Gell-Mann (1958) [START_REF] Feynman | Space-Time Approach to Non-Relativistic Quantum Mechanics[END_REF][START_REF] Dyson | The Radiation Theories of Tomonaga, Schwinger, and Feynman[END_REF] and Sakurai (1958) [START_REF] Lynch | Measurement of the Asymmetry Parameter in µ e Decay[END_REF], proposed a specific form for the current-current interaction, namely the V A (pronounced 'V minus A') structure.

In this way the violation of parity comes from the combination of the vectorial and axial currents with the form V A. For example, in place of the leptonic combination ūe g µ u n , these authors proposed the form ūe g µ (1 g 5 )u n being the difference of a V-type and an A-type current. Due to the V A interaction only, the charged current interaction only involves left-handed (LV) fermions and right-handed (RH) anti-fermions, participate in the weak charge-current.

Due to the short range of the weak force, Fermi's theory gave an excellent approximation at low energy. It was nonetheless expected that Fermi's contact interaction theory would fail at high energy since it was not renormalizable. Thus, building on Fermi's theory of beta decay, work has been done to construct a gauge field theory of weak interaction which is invariant under a SU(2) L local phase transformation, where the subscript "L" indicates that only left-handed particles participate to these interactions.

Weak interactions can

Y = 2(Q T 3 ) (2.19)
The cornerstone of EW theory is the fact that it is a chiral theory, i.e. it distinguishes between LH and RH fermions, and via projection operators, P L,R = 1 2 1 ⌥ g 5 , the corresponding LH fermions and RH anti-fermions fields are as follows:

y L = 1 2 (1 g 5 ) y and y R = 1 2 (1 + g 5 ) y (2.20)
The y L and y R fields can be classified based on the weak isospin numbers T , T 3 (z component of T ), and weak hypercharge Y . Due to T = 1/2, the left-handed fermionic fields form a doublet under SU(2) L , with one part having T 3 = +1/2 and the other having T 3 = 1/2. Meanwhile, the right-handed fermionic fields have T = 0 and form singlets. The following are the structures of the doublet and singlet fields:

y leptons = ✓ n e L e L ◆ , e R , ✓ n µ L µ L ◆ , µ R , ✓ n t L t L ◆ , t R , (2.21)
y quarks = ✓ u L d L ◆ , u R , d R , ✓ c L s L ◆ , c R , s R , ✓ t L b L ◆ ,t R , b R (2.22)
Table 2.3 summarizes fermions and their electric charges (Q), the third component of weak isospin (T 3 ), and weak hypercharge (Y). The weak hypercharges of the particles in the weak isospin doublet has to be the same in order to be invariance under the SU(2) L ⇥U(1) Y gauge transformation. Each type of fermion has its own weak hypercharge. For example, left-handed charged leptons have a hypercharge of Y = 1, while right-handed charged leptons have a hypercharge of Y = 2.

As with the Dirac Lagrangian, new vector fields are introduced through imposing local gauge invariances.

The Lagrangian of the electroweak interaction consists of the kinetic part of both SU(2) and U(1) and the Chapter 2. Theoretical background

Q T 3 Y Quarks Q L = ✓ u L d L ◆ +2/3 1/3 +1/2 1/2 1/3 1/3 u R +2/3 0 4 /3 d R 1/3 0 2/3 Leptons L L = ✓ n L e L ◆ 0 1 +1/2 1/2 1 1 e R 1 0 2
Table 2.3: A summary of the first-generation fermion fields of SM fermions and their electric charge (Q), the third component of weak isospin (T 3 ), and weak hypercharge (Y). Second-generation leptons and quarks operate similarly.

interaction term between matter and fields as follows:

L EW = ȳ ig µ D µ y m y 1 4 W µn • W µn 1 4 B µn B µn (2.23)
where y is the fermionic field, g µ are the Dirac gamma matrices. The final two terms correspond to the free terms for the weak vector fields, W µ , and the hypercharge vector field, B µ . The covariant derivative in this case is given by

D µ ⌘ ∂ µ + ig W ⌧ • W µ + ig Y Y 2 B µ (2.24)
where g W and g Y are the coupling constants for each respective field and t represents the generators of the SU(2) L group.

Mass terms of gauge bosons are not gauge invariant, therefore they cannot be added to the Lagrangian. In this case, gauge bosons have massless gauge invariance. Yet, experimental evidence for massive gauge bosons requires that the particle masses also be included in the theory. Described next will be the mechanism that generates mass for the fields.

Electroweak Symmetry Breaking and Higgs Mechanism

The ElectroWeak (EW) Lagrangian with the SU(2) L ⌦U(1) Y symmetry successfully explains some properties of EW interactions. In contrast, the EW Lagrangian contains no mass terms, forcing gauge bosons to be massless. While this does not present a problem for photons and gluons, experimental data shows that the W ± and Z bosons have mass. Fortunately, this discrepancy can be resolved by spontaneous symmetry breaking and the Higgs mechanism4 , which modified the theory to incorporate massive gauge fields. The following is an introduction to both concepts.

The new model proposed a spontaneous breaking of the SU(2) L ⌦U(1) Y symmetry by the introduction of a scalar field in the electroweak Lagrangian. This mechanism is known as Higgs mechanism. The Lagrangian keeps its symmetry in this case, while the field does not adhere to its symmetry in the vacuum state. The phenomenon is known as spontaneous symmetry breaking. Upon symmetry breaking, the Higgs field breaks three out of the four degrees of freedom of the SU(2) L ⌦U(1) Y group 5 . These three degrees of freedom, the massless Goldstone bosons, are absorbed by the three W ± and Z bosons, and become only observable as components of these weak bosons, which now become massive. The remaining degree of freedom amounts for a massive scalar particle, the Higgs boson. The symmetries of the generators of the electromagnetic interaction, however, remain unbroken, leaving the field quantum (the photon) massless.

In the Standard Model, the Higgs mechanism is achieved by introducing weak isospin doublets of complex scalar fields:

f = ✓ f + f 0 ◆ = 1 p 2 ✓ f 1 + if 2 f 3 + if 4 ◆ (2.25)
As usual, the upper and down components of the doublet differ by one unit of the charge. The corresponding Lagrangian for this field is:

L Higgs = D µ f † (D µ f ) V f † f (2.26)
where V f † f is the Higgs potential:

V f † f = µ 2 f † f + l f † f 2 , l > 0 (2.27)
The Higgs potential depends on the parameter µ 2 :

µ 2 0 and l > 0 : ∂V ∂ f = 0 ! |f min | = 0 µ 2 < 0 and l > 0 : ∂V ∂ f = 0 ! |f min | = q µ 2 2l ⌘ v p 2
(2.28)

In the case of µ 2 0 the potential has a single point of minimum, or ground state. The Vacuum Expectation Value (VEV) vanishes and there is no spontaneous symmetry breaking. In the case of µ 2 < 0, the Higgs potential minimum is degenerate along a circumference, as shown in

chosen so that f 1 , f 2 , f 4 = 0 and f 3 = v = q µ 2
l , where v is called the VEV of the Higgs field. The ground state f 0 takes the form:

h0|f |0i = 1 p 2 ✓ 0 v ◆ (2.29)
where |0i represents the vacuum state. The Higgs field can be expanded around the minimal point using perturbative calculations. Thus, the f field is:

f (x) = 1 p 2 ✓ 0 v + h(x) ◆ (2.30)
where h(x) represents the massive Higgs boson in physical terms. This particular gauge choice is the basis of the Higgs mechanism. As a prelude to using this formulation of the scalar doublet to re-express the Lagrangian density of equation 2.26, it is useful to re-express the vector boson states as follows:

W ± µ = 1 p 2 W 1 µ ⌥ iW 2 µ ✓ A µ Z µ ◆ = ✓ cos (q W ) sin (q W ) sin (q W ) cos (q W ) ◆✓ B µ W 3 µ ◆ (2.31)
where the electroweak mixing angle, q W , is given by,

tan (q W ) = g W g Y (2.32)
Using these expressions, the Higgs Lagrangian of Equation 2.26 can be re-expressed as:

L Higgs = 1 2 ∂ µ h ∂ µ h + g 2 W 4 (v + h) 2 h W + µ 2 + W µ 2 i + g 2 W 8 cos 2 (q W ) (v + h) 2 Z µ 2 µ 2 2 (v + h) 2 l 4 (v + h) 4
(2.33)

Gauge boson mass

Both W ± and Z vector bosons have gained mass, whereas the photon is massless. W ± boson, Z boson, and photon physically observable states together with their masses, and weak mixing angle, q W , are as follows:

Z µ = cos q w W 3 µ sin q w B µ with m Z = 1 2 v q g 2 W + g 2 Y (2.34) W ± µ = 1 p 2 W 1 µ ± iW 2 µ with m W ± = 1 2 vg W (2.

35)

A µ = sin q w W 3 µ + cos q w B µ with m A = 0 (2.36)

q W = arctan ✓ g 0 g ◆ , sin q w = g Y q g 2 W + g 2 Y and cos q w = g W q g 2 W + g 2 Y (2.37)
where the masses of the gauge bosons can be extracted from the terms in L Higgs that are quadratic in the associated fields. By using the relation 2.36 and the measured values for m W and g W , the vacuum expectation value of the Higgs field is found to be 6 :

v = 246 GeV (2.

38)

Higgs mass

There is also one remaining scalar particle, h, in Equation 2.33, namely the Higgs boson, which is given by:

m h = p 2µ 2 = p 2l v (2.39)
The measurement of Higgs boson mass can be used to determine l , which is a free parameter of the theory.

As well as the mass term, the L Higgs contains the trilinear and quartic couplings of the Higgs scalar field, also referred to as self-interactions or self-couplings:

l hhh = 3m 2 h v and l hhhh = 3m 2 h v 2 (2.40)

Fermion mass

The SM Lagrangian needs one more component to be complete. The Higgs mechanism gives masses to the gauge bosons, but the fermions are still massless. Interestingly, the Higgs mechanism can also generate the mass of fermions. The masses of the fermions arise by adding mass terms via Yukawa couplings to the Dirac Lagrangian:

m ȳy = m ( ȳL y R + ȳR y L ) (2.41)
As the LH and RH fermions have different isospin quantum numbers, they cannot acquire a mass in a gauge-invariant way under isospin symmetry thus, it cannot be included in the Standard Model. Given a doublet of scalar field and after SSB, i.e. substituting the f field with the one in Equation 2.30, the Yukawa Lagrangian for Lepton and Quark can be expressed by:

L leptons Yukawa = y p2 v ¯L`R + ¯L`Rh + h.c. (2.42) L quarks Yukawa = 1 p 2 v y d dL d R + ỹu ūL u R 1 p 2 y d dL d R + ỹu ūL u R h + h.c. (2.43)
where h.c is the hermitian conjugate of fermion field. The fermion masses are given by the following relation 6 The VEV, v, was estimated before the discovery of the W boson, and can also be calculated as a function of Fermi Constant G F :

v = ⇣ p 2G F ⌘ 1 2 ⇠ 246GeV 32
Chapter 2. Theoretical background with the Yukawa coupling:

m f = y f v p 2 (2.44)
where y f is the Yukawa couplings. From the Equation 2.44, it can be seen that the Higgs coupling with fermions is proportional to the fermion mass. In the SM, Yukawa couplings are not predicted, therefore, the fermion masses must be measured experimentally. While neutrinos remain massless, their masses can be computed with the SM with a minimal expansion.

W ± bosons couple to leptons in the same generation, while in the quark sector, there are cross-generational couplings of the W ± boson. In 1963, Cabibbo proposed that the weak interactions do not couple to "physical" quarks, but to weak eigenstates that correspond to a linear combination of mass eigenstates with some mixing determined by the Cabibbo-Kobayashi-Maskawa (CKM) matrix as follows:

0 @ d 0 s 0 b 0 1 A = 0 @ V ud V us V ub V cd V cs V cb V td V ts V tb 1 A 0 @ d s b 1 A (2.45)
Determining the V i j is a goal of the LHCb collaboration at CERN and Belle II at KEK in Japan. The magnitudes of the CKM matrix entries have been determined experimentally. As of 2020, the best estimate of the magnitudes of the CKM matrix elements was [START_REF]Review of Particle Physics[END_REF]:

2 4 |V ud | |V us | |V ub | |V cd | |V cs | |V cb | |V td | |V ts | |V tb | 3 5 = 2 4
0.97370 ± 0.00014 0.2245 ± 0.0008 0.00382 ± 0.00024 0.221 ± 0.004 0.987 ± 0.011 0.0410 ± 0.0014 0.0080 ± 0.0003 0.0388 ± 0.0011 1.013 ± 0.030 3 5

(2.46)

There is no physical preference between up-and down-type quarks in the definition, and down-type quarks are simply included as a convention. Other conventions may also be used: the mass eigenstates u, c, and t of the up-type quarks can equivalently define the matrix in terms of their weak interaction partners u', c', and t'.

Tests of the Standard Model

The Standard Model predictions have been confirmed over the past few decades by different particle physics experiments, where the Standard Models can match experiments to an outstanding degree of accuracy. Figure 2.8 shows a summary of measured cross section values of several SM processes by the ATLAS experiment.

The measured cross sections of various Standard Model production processes have been observed to agree well with theoretical predictions.

The SM has many free parameters, which is one of the drawbacks. The values of these free parameters are not predicted by the SM but are calculated from various assumptions and measurements. The parameters are:

• Three coupling constants of the gauge interactions g e , q W , g s

• Two parameters describing the Higgs potential, the VEV (v) and the Higgs mass (m H ) or alternatively, the Higgs potential coupling constant l .

• A total of three mixing angles and the CP-violating7 phase of the CKM matrice. • Yukawa constants determines the masses of the nine charged fermions (six quarks, three leptons).

H ! 4H ! H ! ⌧⌧ (⇥0.25) H ! H!ZZ ⇤ H!WW ⇤ H ! H!bb (⇥0.5) Z W ttW ± ttZ H ! Zjj Wjj Z W ± total ZZ W ± W ± WZ 10 3
So far, these are the famous 18 parameters of the Standard Model. Since neutrinos have mass and oscillate (turn into one another), their mass eigenstates do not coincide with their eigenstates with respect to the weak interaction. Therefore, there must be another mixing matrix involved, which is called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. In the end, we have three additional mass inputs and four additional parameters for describing the cross-generational mixing of neutrinos oscillations. This brings the total number of parameters to 25, most of which are related to the flavour physics.

The discovery of the Higgs boson by the ATLAS [START_REF] Aad | Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC[END_REF] and CMS [START_REF] Chatrchyan | Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC[END_REF] collaborations strengthened the mechanism of electroweak symmetry breaking. However, the existence of the 125 GeV Higgs boson introduces a conceptual problem regarding the quantum correction of its mass. In the Standard Model, the predominant contribution comes from the top quark loop to the Higgs propagator to correct the Higgs boson mass.

Specifically, if the virtual particles' momentum is cut off at a scale L, then the correction to the observed Higgs mass is proportional to L 2 compared to the bare boson mass with no loop correction. The first order corrections to the Higgs mass are as follows:

d m H = l 2 f 16p 2  2L 2 + O ✓ m 2 f ln L m f ◆ (2.47)
where L is the scale of energy at which the model ceases to be valid. In this case, assuming the SM is valid up to the Plank scale, L P ⇡ 10 19 GeV, is problematic (e.g. if we assume it is part of a broader theory valid up to some very high energy level, like the Planck scale), as the mass correction will become very large. This problem is known as electroweak hierarchy problem or fine-tuning problem.

Chapter 2. Theoretical background

There are several other problems discussed in details in [START_REF] Morrissey | Physics searches at the LHC[END_REF][START_REF] Gershtein | New Particles Working Group Report of the Snowmass 2013 Community Summer Study[END_REF]. For instance, gravity is completely omitted from the theory, and efforts to reconcile quantum mechanics with Einstein's theory of gravity, referred to as general relativity, have failed repeatedly. Furthermore, many astronomical observations suggest that an invisible form of matter exists which is five times more abundant than normal matter. The Standard Model does not provide a description of this mysterious dark matter. Theories Beyond the SM (BSM) were developed in response to all of these problems. Chapter 2. Theoretical background

Top quark physics at hadron colliders

Top quark properties in the SM

Due to its heavy mass, the top quark is the only quark that can decay into an on-shell W boson. A large width of the top quark, 1.41 ± 0.17 GeV [START_REF]Review of Particle Physics[END_REF], leads to a very short lifetime of just over 5 ⇥ 10 25 s. Thus, the top quarks decay before forming hadronic bound stats since the characteristic hadronization time of QCD is longer than the lifetime of the top quark t had ' 1/L QCD ⇡ 3 ⇥ 10 24 s . In other words, the top quark is the only quark in the Standard Model that can be studied almost like a free quark. Accordingly, the decay products of the top quark retain the information of the top quark's polarization, which permits experiments to directly detect spin information transferred to its decay products independently from any non-perturbative effects (see Section 2.3 for details).

In the Standard Model, the t ! W q, where q is a down-type quark: d, s, or b, decay modes are proportional to the square amplitude of the corresponding terms in the CKM matrix. With |V tb | so close to unity, tops decay predominantly via t ! W b, where |V tb | denotes the relative probability that the top quark decays into a bottom quark via the emission of a W boson 10 .

The mass of the top quark is close to the electroweak symmetry breaking scale v = 246 GeV of the Higgs field, which gives a Yukawa coupling y t = p 2m t /v of almost unity. Several BSM theories state the top quark as a possible decay product of introduced particles with masses at TeV scale. Besides that, in some cases, these particles might be too heavy to be produced at the LHC, meaning that the observable effects may be the indirect ones through the modification of the SM particle properties 11 .

In the following, we present a brief review of the LHC's top quark production and decay, as well as an overview of the top quark production amplitudes for q q ! t t and gg ! t t at leading order (LO). Particular emphasis will be placed on the production of polarized top quarks, which is pertinent for the following Sections of the thesis. The top quark phenomenology is explained in detail in [START_REF]Review of Particle Physics[END_REF].

Top pair production

In the LHC and Tevatron, the top quarks are predominantly produced in pairs via quark-antiquark annihilation (q q ! t t) and gluon fusion (g ḡ ! t t) at the leading order (LO). Each process has a relative importance depending on the centre-of-mass collision energy ( p s) and the nature of the beam (pp or p p collisions).

In the Tevatron p p collider, with p s = 1.96 TeV, the antiquarks exist as valence quarks in an initial state, so q q annihilation was the main mode of production, producing almost 85% of the t t events. The remainder is produced by gluon fusion. In contrast, antiquarks only appear as sea quarks at the LHC pp collider, so nearly 90% of the t t pairs are produced via gluon fusion. Figure 2.10 illustrates the four LO Feynman diagrams for top-quark pair production via q q annihilation and gluon fusion.

Diagrams on the top and bottom indicate that top pairs are produced either via quark-antiquark annihilation or by gluon fusion. The top diagram can also be realized with an intermediary Z boson or photon, due to the fact that the t t threshold is far from the Z pole, intermediate Z bosons are negligible, and contributions from QED are reduced as a parametric effect of (a/a s ) 2 ⇡ 5.32 10 5 . 12The partonic cross section for quark-antiquark annihilation, averaging (summing) over initial (final) state 10 Due to the fact that in all production channels a top-charged current is involved, the respective cross-sections behave as

s t µ |V tb | 2 g 2 (2.49)
Figure 2.10: LO Feynman diagrams of t t production: the quark-antiquark annihilation (q q ! t t) and the gluon fusion (g ḡ ! t t).

spins and colours [START_REF] Carli | Studies of Quantum Chromodynamics at the LHC[END_REF][START_REF] Russell | Top quark physics in the Large Hadron Collider era[END_REF]:

s q q!t t = g 4 ŝ 108p ŝ b 3 b 2 (2.50)
Here ŝ denotes the centre-of-mass energy square 13 , b = p 1 4m 2 t / ŝ is the velocity of the top quark in the partonic centre-of-mass system (generically referred to as the 'threshold variable'). For gluon fusion, the results are as follows :

s gg!t t = g 4 ŝ 768p ŝ ✓ 31b 3 59b + 33 59b + 33 18b 2 + b 4 log 1 + b 1 b ◆ (2.51)
The hadronic cross section is obtained from the partonic cross section through a convolution with the parton distribution functions (PDFs) f i/H (x, µ F ) which, roughly speaking, describe the probability to find a parton i inside a hadron H with a momentum fraction between x and x + dx of the mother hadron. µ F is a factorization scale that separates hard and soft QCD regimes. The final formula for the hadronic cross section is given by

ds H 1 H 2 !t t+X = Â i, j Z dx 1 dx 2 f i/H 1 (x 1 , µ F ) f j/H 2 (x 2 , µ F ) ⇥ d ŝij!tt+X (x 1 P 1 , x 2 P 2 , k t , k t , a s (µ R ) , µ F ) (2.52)
where P 1 , P 2 are the momenta of the incoming hadrons, while k t , k t are the momenta of the outgoing top quarks. The sum is over all possible partons. The coupling constant a s is evaluated at the renormalisation scale µ R . More details are in the Chapter 5 of [START_REF] Carli | Studies of Quantum Chromodynamics at the LHC[END_REF], here only numerical results are discussed. 13 The centre-of-mass energy of the partonic interacion p ŝ is smaller than the proton-proton centre-of-mass energy s:

p ŝ = p x 1 x 2 s
where x 1 and x 2 are the momentum fractions of the interacting partons from the two protons.

Chapter 2. Theoretical background

Top quark decay

As the magnitude of the CKM matrix element V tb is close to unity, the top quark decays almost exclusively via electroweak charged current interaction into a b quark and W boson. The b quarks hadronize and form b-jets.

The W decays into either a pair of quarks ("hadronically") or into a lepton and a neutrino ("leptonically"). At Leading Order (LO), the Standard Model prediction for the total decay width of the top quark, G 0 t , is given by [START_REF]Review of Particle Physics[END_REF]:

G LO t = G F m 3 t 8p p 2 |V tb | 2 (2.53)
where G F denotes the Fermi coupling constant 14 and m t refers to the top-quark pole mass. Neglecting terms of order m 2 b /m 2 t , a 2 s , and (a s /p) M 2 W /m 2 t , the width predicted in the SM at next-to-leading order (NLO) is [START_REF]Review of Particle Physics[END_REF]:

G NLO t = G LO t ✓ 1 M 2 W m 2 t ◆ 2 ✓ 1 + 2 M 2 W m 2 t ◆ 2  1 2a s 3p ✓ 2p 2 3 5 2 ◆ (2.54) 14 G F = p 2 8 g 2 M 2 W
The NLO corrections reduce the top-quark decay width by approximately 10% compared to the LO prediction [START_REF] Gao | Top-Quark Decay at Next-to-Nextto-Leading Order in QCD[END_REF]. The t t processes are typically classified according to the final-state signatures, determined by the decay channels of the two W bosons, one W boson from the top quark decay and one from the anti-top quark decay.

The three different t t final-state signatures are:

• Full hadronic final state (alljets): Both W bosons from the t t pair further decay into quarks. The corresponding branching fraction is ⇡ 46%. A total of six jets is expected in the final state of the decay at LO, including the b quarks from the top and anti-top initial states.

• Semi-leptonic final state (lepton+jets): One of the W boson from the t t pair decays into quarks, while the second one decays leptonically. In all, four jets are created, including b quarks from the initial top and anti-top decays, a charged lepton, and a neutrino missing transverse energy. Hadronicallydecaying final state tau leptons, on the other hand, are classified separately from electron or muon final states. Only t ! en e n t and t ! µn µ n t are considered as `= e, µ. The total branching fraction of the semi-leptonic channel is around 45%.

• Dileptonic final state (dilepton): Both W bosons from the t t decay into a charged lepton and the corresponding (anti-)neutrino, respectively. At LO, two jets (b-jets), two oppositely charged leptons, and missing transverse energy are expected. This channel has 9% branching ratio.

In Figure 2.12, all t t decay branching fractions are shown. Despite the high number of neutrinos produced by the dileptonic mode, which makes top reconstruction difficult, and the relatively small branching ratio, the dileptonic mode delivers the cleanest experimental signal due to its two leptons in decay. Moreover, the LHC is a top-quark factory, which reduces the effect of the lower branching rate of the dilepton channel.

In semi-leptonic channels, branching ratios are high, however, the channel suffers from more background effects than dileptonic channels.

Polarized t t production

The purpose of this Section is to study the polarized t t production for different modes of production at Leading Order (LO), gluon fusion (g ḡ ! t t) and quark-antiquark annihilation (q q ! t t). The terminology used throughout this section should be defined before going into detail, since this section is devoted to a variety of topics.

Chapter 2. Theoretical background

Spin projection

The Quantum-mechanical spin is an intrinsic form of angular momentum carried by elementary particles. In the same way that angular momentum possesses both magnitude and direction, quantum-mechanical spin also has a definite magnitude and a definite direction. All elementary particles of a given kind have the same magnitude of spin angular momentum, though its direction may change (either up or down on the axis of rotation of the particle). These are indicated by assigning the particle a spin quantum number. As stated in quantum mechanics, a spin-s particle's angular momentum has only certain values along any direction [START_REF]Spin (physics) From Wikipedia, the free encyclopedia[END_REF]:

S i = hs i , s i 2 { s, (s 1),. .., s 1, s} (2.55)
where h = h 2p is the reduced Planck constant, S i is the spin component along the i-th axis (either x, y, or z), s i is the spin projection quantum number along the i-th axis, and s is the principal spin quantum number with s = n/2, where n can be any non-negative integer so, s 2 {0, 1/2, 1, 3/2, 2,...}. For the z axis:

S z = hs z , s z 2 { s, (s 1),. .., s 1, s} (2.56)
where S z is the spin component along the z axis, s z is the spin projection quantum number along the z axis. s z has a possible value of 2s + 1. The number 2s + 1 is the multiplicity of a spin system. For example, there are only two possible values for fermions, spin-1/2 particles: s z = +1/2 and s z = 1/2. These correspond to quantum states in which the spin component is pointing in the +z or z directions respectively, and are often referred to as spin up, " and spin down, #.

Top and anti-top quarks are spin-1/2 particles, so they can be in up/down spin states. When top and anti-top have the same spin projection quantum number along given axis, they are referred to as spin-like (or same spin), and otherwise they are referred to as unlike-spin (or opposite spin).

Spin polarization

As a general concept, spin polarization is any alignment of the components of a vectorial field with the nonrelativistic three-dimensional or relativistic four-dimensional regions over which it is defined. Particularly, spin polarization refers to the degree to which the spin is aligned with a given direction (or axis). The focus of our study is the spin polarization of individual particles along an axis, as well as the correlation between two particles' spin polarizations.

Helicity and Chirality

The helicity of a particle is defined as the projection of a spin vector on the direction of its momentum vector,

h ⌘ S •p | S||p| (2.57)
A particle's helicity states are illustrated in Figure 2.13. If a particle's spin vector points in the same direction as the momentum vector, the helicity is positive (right-handed), and if they point in opposite directions, the helicity is negative (left-handed). When two particles are produced with the same helicity, they are referred to as like-helicity (RR, LL) pair, and otherwise they are referred to as unlike-helicity (RL, LR) pair.

Is helicity an intrinsic property of the particle? Figure 2.13: Helicity states of a particle [59].

By using the concept of the mass of a particle, one can answer this question. A massless particle will have the same helicity in all reference frames since it will always travel at the speed of light. Therefore, its helicity cannot be flipped by changing the reference frame. On the other hand, if a particle has any mass, then helicity is not an intrinsic property since different observers (in valid reference frames) can measure different values for the helicity (left-or right-helicity). Although most particles have helicity, it is not one of their intrinsic properties.

Can there be any intrinsic property of a particle relating to its helicity?

The property is known as chirality. Like helicity, chirality can also be left-or right-handed for a particle.

For massless particles the chirality and helicity are the same. A massless chiral right(left)-handed particle also has right(left)-handed helicity. In contrast, massive particles have a specific chirality. Depending on the reference frame relative to the particle, a massive left-chiral particle may have either left-or right-helicity. In all reference frames the particle will still be left-chiral, no matter what helicity it is15 [60].

A major aspect of introducing the helicity concept is that any initial state of a process can be subdivided by its helicity structure. For instance, at Leading Order (LO), gluon fusion (g ḡ ! t t) and quark-antiquark annihilation (q q ! t t) can be further divided based on their helicity structure. This results in four different possibilities for t t production:

• q L qR and q R qL : Opposite-helicity quark anti-quark annihilation.

• q L qL and q R qR : Like-helicity quark anti-quark annihilation.

• g L g R and g R g L : Opposite-helicity gluon fusion.

• g L g L and g R g R : Like-helicity gluon fusion.

Basis parametrization

The purpose of this Section is to exploit the fact that, despite the very small net polarization of top and anti-top quarks in t t production, there is a very large asymmetry in the rate of producing the like-spin versus unlike-spin top quark pairs at hadron colliders if the appropriate spin axes are chosen. There are three distinct bases for computing the relevant amplitudes for the t t production based on the choice of axes along which the t and t spins are decomposed: the helicity basis, the beam-line basis and the off-diagonal basis. The choice of bases may not be ideal for all colliders. Interestingly, one is suitable for studies at the Tevatron, while the other is good for both the LHC and Tevatron.

In what follows, we demonstrate that the number of like-spin and unlike-spin top quark pairs can be made significantly different by choosing a spin basis [61,62,63,[START_REF] Parke | Spin Correlation Effects in Top Quark Pair Production[END_REF].

Generic basis

The study of the polarized top quark requires a definite spin basis. The description of a generic spin basis by Parke and Shadmi [62] is with top spin states defined in the top-quark rest frame, with the spin-quantization direction ŝt , which makes an angle x with the anti-top momentum in the clockwise direction. Similarly, the anti-top quark spin states are defined in the anti-top rest-frame, along the direction ŝt , which makes the same angle x with the top momentum also in the clockwise direction. See Based on the above convention for x , the following bases can be constructed:

Helicity basis

The helicity basis is built upon the helicities of the t and t. As top quarks are massive particles, decomposing them into like-helicity (LL, RR) and unlike-helicity (LR, RL) pairs will differ depending on which frame is utilized. The spin axis is chosen to correspond to the direction of the parent top and anti-top quarks in the t t zero momentum frame (ZMF) of the initial parton pair, e.g. cos x = 1, where we may describe the top pair production cross section in terms of the scattering angle q ⇤ between the top quark and the left moving beam, and the speed b of the top quark, as shown in Figure 2.15.

The helicity amplitudes given below are summed over the spins of the initial partons, as well as the colours of the initial and final states, for both quark-antiquark (q q) and gluon-gluon (gg) initial states. Spin-and colour-averaging factors have not been included [61,62].

For the q q initial state:

 LL,RR |M (q q ! t t)| 2 = 8g 4 1 b 2 sin 2 q ⇤ (2.58)  LR,RL |M (q q ! t t)| 2 = 8g 4 1 + cos 2 q ⇤ (2.59)
where b = p 1 4m 2 t / ŝ is the top quark speed. According to 2.58, in the high energy limit (b ! 1), like-helicity t t pairs is suppressed.

For the gg initial state 16 :

 LL,RR |M (gg ! t t)| 2 = 16 3 g 4 Y (b , cos q ⇤ ) 1 b 2 1 + b 2 + b 2 sin 4 q ⇤ (2.60)  LR,RL |M (gg ! t t)| 2 = 16 3 g 4 Y (b , cos q ⇤ ) b 2 sin 2 q ⇤ 1 + cos 2 q ⇤ (2.61)
Once again, the suppression of like-helicity t t pairs in the high energy limit. However, for low energies, unlike-helicity pair production is suppressed relative to the production of like-helicity pairs by a factor of b 2 .

Beam-line basis

The beam-line basis is composed of two spin axes p L for the top quark and p R for the anti-top quark, where p L and p R are light-like vectors parallel to the left and right moving beams, respectively. In this case:

cos x = cos q ⇤ + b 1 + b cos q ⇤ (2.62)
where q ⇤ is the top scattering angle with respect to the beam. It may be checked that in the ultra relativistic limit (b = 1), this basis reduces to the helicity basis. The basis also permits a frame-independent decomposition into like-spin ("", ##) and unlike-spin ("#, #") pairs.

For the q q initial state:

 "",## |M (q q ! t t)| 2 = 8g 4 b 2 1 b 2 sin 2 q ⇤ (1 b cos q ⇤ ) 2 (2.63)  "#,#" |M (q q ! t t)| 2 = 8g 4 " 1 + 1 b cos q ⇤ b 2 sin 2 q ⇤ 2 (1 b cos q ⇤ ) 2 # (2.64)
The factor b 2 1 b 2 in the like-spin pair amplitude suppresses this component for both small and large values of b . While, the amplitude of the pair of different spins depends on the first term 8g 4 .

 "",## |M (gg ! t t)| 2 = 16 3 g 4 Y (b , cos q ⇤ ) ⇥ 1 b 2 " 1 + b 2 cos 2 q ⇤ + 2b 3 sin 2 q ⇤ (b cos q ⇤ ) (1 b cos q ⇤ ) 2 # (2.65)  "#,#" |M (gg ! t t)| 2 = 16 3 g 4 Y (b , cos q ⇤ ) ⇥ b 2 sin 2 q ⇤ " 1 + 1 b 2 2 + 1 b cos q ⇤ b 2 sin 2 q ⇤ 2 (1 b cos q ⇤ ) 2 # (2.66)
The like-spin pairs resulting from the gluon-gluon fusion will be suppressed in the presence of large b , whereas unlike spin pairs are disfavoured at low b . 16 For the gg initial state we define the common spin-independent angular factor:

Y (b , cos q ⇤ ) ⌘ 7 + 9b 2 cos 2 q ⇤ (1 b 2 cos 2 q ⇤ ) 44
Chapter 2. Theoretical background

Off-diagonal basis

On the off-diagonal basis, the spin axis of the top quark makes an angle, W, relative to the top quark direction in the ZMF. This angle is defined as follows [62,[START_REF] Parke | Spin Correlation Effects in Top Quark Pair Production[END_REF]:

tan W = 1 b 2 tan q (2.67)
At threshold, the top quark spin is aligned along the beam line , W = q , and at ultra-relativistic energies, the top quark spin is aligned along the direction of the top quark motion W = 0. As a result, the off-diagonal basis smoothly transitions between these two extremes. The axis for the anti-top quark is always taken anti-parallel to the spin axis for the top quark, as shown in Figure 2.15. In addition, in this basis, top quark pairs are always in an up-down (UD) or down-up (DU) configuration because in the off-diagonal basis the amplitudes with like spins, UU and DD, vanish identically [START_REF] Parke | Spin Correlation Effects in Top Quark Pair Production[END_REF]. It should also be noted that as b ! 1, both the beam-line and off-diagonal bases converge to the helicity basis.

Therefore, for colliders with ultra-relativistic energies, all three bases are equivalent.

Brief synopsis

Tevatron's t t cross-section is dominated by quark-antiquark annihilation, accounting for approximately 85% of the total. This may be explained by amplitudes discussed above and the relative parton luminosities.

According to amplitudes 2.63, it is disfavoured to produce like-spin t t pairs in the beam-line basis from a q q initial state. Because of this, most of the t t pairs produced at the Tevatron have unlike spins in beam-line basis. Accordingly, the beam-line basis is more suitable for the Tevatron studies. With the off-diagonal basis at Tevatron, significant results are obtained [63], since the vanishing of amplitude for the production of like spin top pairs from quark-antiquark annihilation leads to a total cross section consisting of 92% unlike spin pairs. This is significantly better than the 70% unlike helicity pairs obtained in the helicity basis.

At .16: Fraction of t t MC particle level events passing a cut on t t invariant mass for like and unlike helicity gg and q q initial states.

The t t invariant mass regimes at the LHC

The production rate of like and unlike helicity gg and q q initial states is mostly dependent upon the resultant t t pair. Figure 2.17 shows particle level events of the t t invariant mass distribution, m t t , for the LHC with centre of mass energy 13 TeV, decomposed into like (SH) and unlike (OH) helicities in the zero momentum frame of the t t pair for both q q and gg initial states. Approximately 65% of gluons are like-helicity, and 35% are unlike-helicity gluons [START_REF] Mahlon | Spin correlation effects in top quark pair production at the LHC[END_REF]. The q q production process is dominated by the opposite-helicity case while the contribution of like-helicity q q annihilation is suppressed. The SM models used to generate the MC sample treat light quarks as ultra-relativistic (massless), and because the contributions to the cross section of like-helicity q q pairs are proportional to the mass, this mode of production has a net contribution of zero to the cross section. The Appendix A provides more details. 17 In the CM frame, the invariant mass is defined for q q/gg ! t t as: 2.17: Distribution of the t t invariant mass, m t t , for the LHC with centre of mass energy 13 TeV, decomposed into like (SH) and unlike (OH) helicities in the zero momentum frame of the t t pair for both q q and gg components. An important fact to note is that the threshold production for a t t pair is M t t = 2m t ⇡ 346 GeV, where m t = 173 GeV 17 In the 450-500 GeV region, the like-helicity and unlike-helicity contributions are equal, so it can be justified to select events with m t t less than some maximum value which will increase the purity of the dominant helicity configuration. Figure 2.18 shows how the cut affects the fraction of like-helicity and unlike-helicity pairs for gg and q q initial states. : Fraction of t t MC particle level events passing a cut on t t invariant mass for like and unlike helicity gg and q q initial states. Dots represent the fraction of evens that pass the selection M t t >cut.

M 2 t
When requiring the cut M t t < 450 GeV, the like-helicity purity of the sample will increase to 65% while the data sample is retained at 30%. Alternatively, requiring a cut M t t > 450 GeV will result in 47% unlike-helicity purity for gg and q q initial states, producing top quark pairs in left-right or right-left configurations. In this way, the phase space is split into two regions: low threshold, M t t < 450 GeV, and high threshold, M t t > 450 GeV. Although the cut of 450 GeV appears to lead to slightly more statistics in the low invariant mass region, nonetheless it is retained for ease of comparison with analyses of top-quark properties that also use this cut. The top quark mass has only a weak effect on the above fractions, varying by a few percent between the range 150 GeV < m top < 200 GeV [61].

Spin correlation

An interesting avenue for research consists in studying the angular correlations between the decay products of top quarks and anti-top quarks. As discussed earlier in this chapter, the top quark decays before hadronization.

Due to this, the angular correlations in top quark decay provide information about the spin of the top quark.

If the top and anti-top spins of the t t pair are correlated, then they will exhibit angular correlations in their decay products. At the Tevatron and LHC, several analyses have been conducted to confirm this picture by measuring the correlations in the spins of top-anti-top pairs. Spin correlations have been found at 3s level by D0 using matrix element techniques [START_REF] Abazov | Evidence for Spin Correlation inttProduction[END_REF], and at the 5s level by ATLAS by measuring the lab-frame Df (l + , l ) in the dileptonic channel [START_REF] Aad | Observation of Spin Correlation in tt Events from pp Collisions at p s = 7 TeV Using the ATLAS Detector[END_REF].

There are several well-known approaches to measuring spin correlations by studying decay products of the two top quarks. For instance, the matrix element method outlined above makes full use of the detailed information found in the six-body final state (dileptonic channel) [START_REF] Melnikov | Top quark spin correlations at the Tevatron and the LHC[END_REF][START_REF] Baumgart | A new twist on top quark spin correlations[END_REF]. Our analysis relies on a standard method for picking one particle from each decay and looking for correlations in the polar angles represented in their parent tops' rest frames [START_REF] Baumgart | A new twist on top quark spin correlations[END_REF]. Specifying with which axis to measure the polar angles is the first step in this method, and there are several well-motivated ways to do so, which we shall elaborate on below. Most spin correlation studies use this method, which can be optimized to detect Standard Model correlations at the LHC or Tevatron.

Spin-analysing power

In the SM, the decay products from a polarized top quark have their moment vectors correlated with the top quark spin quantization axis as follows:

1 G dG d cos q â = 1 + B âa cos q â 2 (2.68)
where q â is the angle between the top quark spin along a quantisation axis â and the decay product. B â is the polarisation of the top quark. a is the spin-analysing power that measures how sensitive the decay product of top quarks is to the spin information of their parent top quark. Standard Model values of a for different particles are in Table 2.4. The b-quark does not carry the full spin information of the top quark. The reason is because the b-quark is a massive particle, and it is always possible to define a reference frame in which the direction of the spins of these particles are flipped, diluting their spin-analysing power. The charged leptons and down-type quarks are the most effective spin analysers with a = 1 at LO. This is because charged leptons and down-type quarks can be approximated as massless particles, allowing for full spin information to be recovered. Although up quarks can be thought of as massless particles, they present an additional complication. This is because they can not be broken into separate angular and momentum-dependent parts and the spin information is diluted by this effect. Down-type quarks and charged leptons, on the other hand, can have their calculation split into only the angular dependent parts and the full spin information is available at LO [START_REF] Brandenburg | QCD-corrected spin analysing power of jets in decays of polarized top quarks[END_REF].

Considering the t t decay mode, the dileptonic and semi-leptonic channels: particles from both the top and the anti-top with a = 1 are contained in these decay modes, and they can thus access the full spin information.

Nevertheless, jets associated with down-type quarks cannot easily be distinguished from jets associated with up-type quarks experimentally. As a consequence, it is not possible to extract the full spin correlation information from hadronically decaying W-bosons. Accordingly, the dileptonic decay mode has the highest overall sensitivity to spin correlation among all of the t t decay modes. Due to this, the present analysis measures spin correlation in the dileptonic channel, when the decay of the t t pair leads to two oppositely charged leptons, The following will summarize the basic formalism for the top quark spin correlations, followed by a look at the impact of the invariant mass of the t t pair (m t t ) on spin correlations.

t t ! W + W b b ! `+v` vb

The angular observables

In t t events, spin correlation information are encoded in the squared matrix elements, with appropriate colour and spin summation implied, it can written as follows [START_REF] Bernreuther | A set of top quark spin correlation and polarization observables for the LHC: Standard Model predictions and new physics contributions[END_REF][START_REF] Baumgart | A new twist on top quark spin correlations[END_REF]71]:

M qq/gg ! tt ! `+v b` vb 2 µ rR r (2.69)
In this equation, `denotes an electron or muon. R represents the density matrix that describes how top quark pairs are produced in a specific spin configuration, while r, r are the decay spin density matrix of top and anti-top quarks into specific final states. Using a Pauli matrix basis, the production spin density matrix R in

C 2 ⌦ C 2 ,
can be decomposed into t and t spin spaces:

R = 1 4 (1 ⌦ 1 + B + i s i ⌦ 1 + B j 1 ⌦ s j +C i j s i ⌦ s j ) (2.70)
The first (second) factor in the tensor products of the 2 ⇥ 2 unit matrix 1 and of the Pauli matrices s i refers to the t (t) spin space. The expectation values for the t (t) spin projection s + (s ) are given by:

⌦ s + i ↵ = Tr [Rs i ⌦ 1] = B + i , (2.71)
D s j E = Tr [R1 ⌦ s j ] = B j , (2.72)
D s + i s j E = Tr [Rs i ⌦ s j ] = C i j (2.73)
Equations 2.71-2.73 give physical meaning to the parameters in R. The B + i and B i are three-dimensional vectors of functions that describe the longitudinal and transverse polarisation of the t quark and the t quark. The subscript +( ) refers to the t (t) quark, respectively. The C i j is a 3 ⇥ 3 matrix of functions that characterize the correlation between the top quark and antiquark spins (spin correlation coefficients).

In the dileptonic channel, the differential angular distribution for the two leptons follows from 2.69 and 2.70 is given by 1 s

d 4 s dW + dW = 1 (4p) 2 1 + B + • ˆ+ + B • ˆ ˆ+ •C • ˆ (2.74)
where, dW = d cos q df is the solid angles of the leptons in their parent top quark and antiquark rest frames.

The unit vector ˆ+ ( ˆ ) is the `+ (`+) direction of flight in the t (t) rest frame. The vectors B +, and the matrix C contain the information from the t t production and the t and t decay density matrices, which may be computed in the SM or in a New Physic (NP) model. Specifically, ˆ+ and ˆ are parametrized as spherical coordinates, but, at the same time, allow different reference axes, that is, â and b:

cos q + = ˆ+ • â, cos q = ˆ • b (2.75)
For the experimental analysis, integration of the azimuthal angles can be made to obtain a single-differential Chapter 2. Theoretical background cross section that depends only on the polarisations and spin correlation coefficients. For a choice of reference axis â and b, is as follows:

1 s The angle q â (q b) is defined as the angle between the positively (negatively) charged lepton and axis â ( b) in the rest frame of its parent top (anti-top) quark.

d 2 s d cos q â + d cos q b = 1 4 ⇣ 1 + B â + cos q â + + B b cos q b C( â, b) cos q â + cos q b ⌘ (2.
By changing variables and integrating out the Equation 2.76, the single-differential cross section with respect to cos q â + , cos q â and cos q â + cos q b are as follows:

1 s

ds d cos q â + = 1 2 1 + B â + cos q â + (2.77) 1 s ds d cos q â = 1 2 1 + B â cos q â (2.78) 1 s ds d cos q a + cos q b = 1 2 ⇣ 1 C( â, b)cosq â + cos q b ⌘ ln 0 @ 1 cos q â + cos q b 1 A (2.79)
where, in the Equation 2.77, for example, the differential cross section has a linear dependence on the polarisation B â + . The slope of distribution of 2.77-2.79 can provide a value of B â + , B b and C( â, b). One may also compute/measure these polarizations and correlations observables using the expectation value relations:

B â + = 3 < cos q â + > (2.80) B b = 3 < cos q b > (2.81) C( â, b) = 9 < cos q â + cos q b > (2.82)

Choice of reference axes â and b

For the reasons given in Section 2.2.4, and throughout this section and the rest of the thesis, unless otherwise specified, we define the reference axes â and b in the helicity basis, an obvious choice for the LHC. In this context, the spin analysing vector for the top quark, â, has opposite sign to the spin analysing vector for the anti-top quark, b, such as:

â = b = k (2.83)
where k denotes the direction of flight of the top quark in the centre-of-mass frame of the t t system. Defining p = (0, 0, 1) as recommanded in [START_REF] Bernreuther | A set of top quark spin correlation and polarization observables for the LHC: Standard Model predictions and new physics contributions[END_REF], which indicates the direction of the proton beam in the positive z direction in the laboratory frame, the three orthogonal spin quantisation axes are defined as follows:

2.3 Spin correlation 51 y = p • k, r = p 1 y 2 (2.84) r = sign(y) r ( p y k), n = sign(y) r ( p ⇥ k) (2.85)
The {r, k, n} is a right-handed orthonormal basis and is illustrated in Figure 2.19. Due to the Bose-symmetric nature of the initial t t state at the LHC (gluon-gluon fusion), the cos q calculations for the transverse axis are multiplied by the sign of the scattering angle. In addition, the calculations of cos q with respect to the negatively charged lepton, the axes are multiplied by -1.

n k r r ± k ± n ± ± beam anti-top top Figure 2.19:
The right-handed orthonormal basis {r, k, n} used for the spin correlation study. The k axis is defined by the top quark direction, measured in the t t CM frame, while r and n are defined in Equation 2.85.

The angle q â + (q b ) is defined as the angle between the positively (negatively, ` ) charged lepton, `+, and axis â ( b) in the rest frame of its parent top (anti-quark) quark.

The polarisation vector B ± and the 3 ⇥ 3 spin correlation matrix C can be expressed in terms of the orthonormal basis {r, k, n} as follows:

B ± = b ± r r + b ± k k + b ± n n (2.86) C = c rr rr > + c kk k k> + c nn n n> + c rk ⇣ r k> + kr > ⌘ + c kn ⇣ k n> + n k> ⌘ (2.87)
Chapter 2. Theoretical background BSM phenomena [START_REF] Bernreuther | A set of top quark spin correlation and polarization observables for the LHC: Standard Model predictions and new physics contributions[END_REF][START_REF] Sirunyan | Measurement of the top quark polarization and spin correlations using dilepton final states in proton-proton collisions[END_REF].

+c rn ⇣ r n> + nr > ⌘ + c r ⇣ k n> n k> ⌘ + c k ⇣ nr > r n> ⌘ + c n ⇣ r k> kr > ⌘ The coefficient functions b ± i , c i j ,

The angular observables

In summary, based on Equations 2.80-2.82 and 2.86, the spin density matrix decomposes into 15 polarisation and spin correlation observables that determine the t t cross section as follow:

• The three cos q â + and three cos q â to measure B â + and B â .

• The three cos q â + cos q â to measure C â â

• The six sum and difference terms cos q â + cos q b ± cos q b + cos q â to measure C âb ±C ba . The cross spin correlation coefficients for each pair of axes â 6 = b.

Where â, b 2 { k, r, n}. By measuring these set of 15 polarisation and spin correlation observables, the coefficient functions of the squared matrix element can be probed. ATLAS [START_REF] Aaboud | Measurements of top quark spin observables with the ATLAS detector[END_REF] and CMS [START_REF] Sirunyan | Measurement of the top quark polarization and spin correlations using dilepton final states in proton-proton collisions[END_REF] recently measured these 15 observables, and all of the results were in accordance with the SM theory.

The spin correlation matrix also exhibits sensitivity to additional observables:

cos j = ˆ+ • ˆ (2.88)
The single-differential cross section with respect to cos j: [START_REF] Afik | Entanglement and quantum tomography with top quarks at the LHC[END_REF]. The slope of distribution of 2.89 measures the value of D.

1 s ds d cos j = 1 2 (1 D cos j) (2.89) and D = 1 3 Tr[C] = (C kk +C rr +C nn )/3
One may also compute/measure the value of D using the relation D = 3hcos ji.

The particle-level distributions of all the angular observables are shown in Figures 2.20 and 2.21. In Table 2.5, all the observables are listed along with their corresponding measured coefficients, production spin density matrix coefficient function, as well as their NLO values. The uncertainty in the NLO calculations comes from varying µ R and µ F simultaneously up and down by a factor of 2 compared to their nominal value of m top . Observables Measured Coefficient Coefficient function NLO calculation

cos q k + B k + b k + 4.0 +1.7 1.2 ⇥ 10 3 cos q k B k b k 4.0 +1.7 1.2 ⇥ 10 3 cos q r + B r + b r + 1.6 +1.2 0.9 ⇥ 10 3 cos q r B r b r 1.6 +1.2 0.9 ⇥ 10 3 cos q n + B n + b n + 5.7 +0.5 0.4 ⇥ 10 3 cos q n B n b n 5.7 +0.5 0.4 ⇥ 10 3 cos q k + cos q k C kk c kk 0.331 +0.002
0.002 cos q r + cos q r C rr c rr 0.071 +0.008 0.006 cos q n + cos q n C nn c nn 0.326 +0.002 0.002 cos q n + cos q k + cos q k + cos q n C nk +C kn c kn 2.15 +0.04 0.07 ⇥ 10 3 cos q n + cos q k cos q k + cos q n C nk C kn c r 0 cos q n + cos q r + cos q r + cos q n C nr +C rn c nr 1.06 +0.01 0.01 ⇥ 10 3 cos q n + cos q r cos q r + cos q n C nr C rn c k 0 cos q r + cos q k + cos q k + cos q r C rk +C kr c rk -0.206 +0.002

0.002 cos q r + cos q k cos q k + cos q r C rk C kr c n 0 cos j D (c kk + c rr + c nn )/3 -0.243 +0.003 0.003
Table 2.5: List of all the observables along with their corresponding measured coefficients, production spin density matrix coefficient function. Calculated NLO values are listed with the associated scale uncertainties [START_REF] Bernreuther | A set of top quark spin correlation and polarization observables for the LHC: Standard Model predictions and new physics contributions[END_REF] at p s = 13 Tev. The first six observables show polarizations of the top and anti-top quarks along various axes, nine others show spin correlations and the last one displays the opening angle. In the NLO calculations, uncertainties are brought about by varying µ R and µ F simultaneously up and down by a factor of 2 from their nominal value of m top .

Chapter 2. Theoretical background

Effective Field Theory

Nature presents itself to us on a number of different scales, from macro-scale phenomena such as galaxies and planets to a microscopic ones such as atoms and nuclei. As it turns out, it is not necessary to understand the underlying theory of everything in order to comprehend how nature works at a particular scale18 [START_REF] Burgess | An Introduction to Effective Field Theory[END_REF].

Recent years have witnessed the emergence of practical theories that share this feature of nature. These theories are known as effective fields theories (EFTs). EFTs describe phenomena on a specific energy scale, therefore the theory has a built-in validity limit. The following is a summary of the basic ingredients of an EFT theory as a set of general principles.

The basis of Effective Field Theory 2.4.1.1 Power counting

Effective field theories efficiently analyse the physics at low-energy, without having to solve the details of dynamics at higher energy scales, where low is defined with respect to some energy scale L. In systems of energy E defined by two different energy scales, m ⌧ M, the EFT explicitly accounts for the relevant degrees of freedom at low energy scale m ⌧ L, but higher energy degrees of freedom are integrated out into an expansion of powers of E/L, known as the power counting parameter. Every EFT theory comes with a set of power counting rules that allow for a specific expansion of the calculations and a sizing of the most significant contributions.

Decoupling theorem

Why is it possible to understand what is happening on a scale without understanding everything all at once? This is explained by a fundamental property of nature described in the Decoupling Theorem [START_REF] Appelquist | Infrared singularities and massive fields[END_REF]. The decoupling Theorem states that effects of heavy physics (UV-physics) with mass M, decouple at low-energy (IR). The low-energy (IR) effects of heavy particles are either suppressed by inverse powers of the heavy masses, or they get absorbed into renormalization of the couplings and fields of the EFT obtained by removing the heavy particles .

If the low-energy Lagrangian is renormalizable [START_REF] Krause | Higgs Effective Field Theories -Systematics and Applications[END_REF], the influence of the heavy particles to low-energy observables decouples in the limit L ! •. A good example of this decoupling is Fermi's theory of the weak interaction (see Section 2.1.3.3).

Top-down vs. bottom-up approach

Effective Field Theories (EFTs) are used for two main reasons [START_REF] Georgi | Effective Field Theory[END_REF]:

• The high-energy (UV) theory is known, whereas we are interested in low-energy (IR) effects only.

The resulting EFT simplifies computations substantially, sometimes making them even feasible, this is know as Top-down approach. An example for a top-down EFT is the weak effective theory.

• The high-energy (UV) theory is unknown and EFT is used to universally parametrize the UV-physics effects, this is know as Bottom-up approach. An example of a bottom-up EFT is the SM-EFT described in Section 2.4.2.

The top-down and bottom-up approaches are illustrated in Figure 2.22. This thesis uses EFT models that utilize a bottom-up approach, therefore in the following, details about the bottom-up approach are presented [START_REF] Krause | Higgs Effective Field Theories -Systematics and Applications[END_REF].

Bottom-up EFT parameters are encoded in a series of operators composed of low-energy fields, which are arranged in a systematic expansion defined by power counting. The coefficients of the operators, called Wilson coefficients or low-energy constants (LECs), can be specified for a given model. Wilson coefficients encode information about the heavier degrees of freedom of the low-energy Lagrangian. In a model-independent analysis, they are free parameters that can be determined by experiment. Once they are determined, the leading and next-to-leading order Lagrangian can be used to make predictions of further experiments.

In general, the EFT Lagrangian can be pictured as follows:

L full f light , f heavy ! L full f light + Â i c i O d i i L d i 4
(2.90)

where O i are light field operators, and the couplings c i are Wilson coefficients. The operators O i are usually organized according to their dimension, d i , which fixes the dimension of their coefficients. The dimension of the coupling constant, c i , is given below using natural units, h = c = 1:

[O i ] = d i ! c i ⇠ 1 L d i 4
(2.91)

A scale L is introduced so that the coefficients c i are dimensionless. L is the high energy scale at which new physics occurs. At energies below L, the behaviour of the different operators is determined by their dimension.

Three types of operators can be distinguished:

• Relevant (d i < 4): Operators with dimensions less than four are more relevant at lower energies.

• Marginal (d i = 4): At all energy scales, dimension four operators have the same importance and are called marginal operators. They lie between relevancy and irrelevancy because quantum effects could modify their scaling behaviour on either side.

• Irrelevant (d i > 4): As their effects are suppressed by powers of E/L at low energies, they are called irrelevant operators. Irrelevant operators are weak at low energies.

When there is a large mass difference between the scale of the energy being analysed and the scale of any heavier states (i.e. m, E ⌧ M), the effects induced by irrelevant operators are always suppressed by powers of (E/M), (m/M), and can usually be neglected. The resulting EFT is called renormalizable, and it includes only relevant and marginal operators, such that its predictions are valid up to E/M corrections.

Toy model

The following illustrates an explicit example of the effective field theory. Let's consider the Yukawa theory of massless fermions, y, and a heavy scalar, F:

L Full = i ȳ 6 ∂ y + 1 2 D µ F 2 1 2 M 2 F 2 l F ȳy (2.92) = L (y) + L (y, F) (2.93)
Where L (y) contains only light field, while L (y, F) encodes the dependences on the heavy field F and its interactions with the lighter particles. The goal is to find the EFT theory that describes the physics below the M scale.

Integrating out the scalar F leads to the 4-fermion interaction, as illustrated in Figure 2.23. The following EFT Lagrangian provides the low-energy description of the full theory:

L EFT = i ȳ 6 ∂ y + c S M 2 1 2 ( ȳy)( ȳy) (2.94) = L (y) + d L (y) (2.95) Where, d L (y) = c S
Full: M full = ū (p 3 )( il )u (p 1 ) ū (p 4 )( il )u (p 2 ) " i (p 1 p 3 ) 2 M 2 # (3 $ 4) = U S il 2 M 2 ⇥ 1 + O q 2 /M 2 + •• • ⇤ where: U S = ū (p 3 ) u (p 1 ) ū (p 4 ) u (p 2 ) (3 $ 4) (2.96) 2.4 Effective Field Theory 59 EFT: c S M 2 1 2 ( ȳy)( ȳy) ! M eft = U S ic S M 2 (2.97)
Matching conditions apply at low M:

M full = M eft ! c s = l 2 (2.98)
Consequently, the EFT Lagrangian below scale M in Equation 2.94 becomes:

L EFT = i ȳ 6 ∂ y + l 2 2M 2 ( ȳy)( ȳy) (2.99)
The EFT Lagrangian in 2.99, can be systematically improved by higher dimension operators and higher order calculations. Matching at higher order calculations are beyond the scope of this section (this is covered in [START_REF] Mimasu | SMEFT and searches for new physics. 8th Edition of the Large Hadron Collider Physics Conference[END_REF]).

The Standard Model Effective Field framework

Without any collider signal of new physics, new particles are likely to be much heavier than the weak scale. As a result, the BSM particles can be integrated out of the theory, and their effects deduced using model-independent Effective Field Theory. Among the most popular EFT frameworks is the Standard Model Effective Field Theory (SMEFT). It is considered a bridge between the theories of new physics and experimental observation. SMEFT is a bottom-up EFT theory in which the new physics is decoupled from the SM. In this way, interactions with arbitrary large mass dimensions are permitted in the SMEFT, which results in the appearance of higher-dimensional operators in the SM Lagrangian that are suppressed by powers of L as follows:

L SMEFT = L (4) SM + Â i c (5) i O (5) i L + Â i c (6) i O (6) i L 2 + O ✓ 1 L 3 ◆ = L (4) SM + L (5) + L (6) + O ✓ 1 L 3 ◆ (2.100)
In the above, L

SM is the usual renormalizable part of the SM Lagrangian. In L (n) , the sum is over the full operator set at each mass dimension n, L is the cut-off scale of SMEFT, O dimensionless Wilson coefficients and multiplied by a factor of 1/L n 4 so that their overall combined dimension equals 4. Generally, the number of operators increases exponentially with each mass dimension [START_REF] Grzadkowski | Dimension-Six Terms in the Standard Model Lagrangian[END_REF]. The full set of operators that satisfy the symmetry constraint, such as Lorentz invariance and the full

SU(3) c ⇥ SU(2) L ⇥U(1)
Y SM gauge symmetry, can be derived for each mass dimension n.

In the following Sections, operators of dimension-five and dimension-six of the first orders of expansion in Equation 2.100 are derived. Prior to that, however, it is important to highlight the dimensions of the building blocks that built up these operators: Field strength tensors of dimension two X µn 2 G A µn ,W I µn , B µn , Higgs doublets of dimension one (denoted j), fermion fields Y of dimension 3/2 and various covariant derivatives D of dimension one.

Chapter 2. Theoretical background

Dimension-5 operators: fermion number violation

Let's start by discussing the very intriguing dimension-five operators. Several types of operators can be constructed with dimension five, but if the gauge symmetry constraints of the SM are imposed on it e.g.

L (5) , most of these operators are forbidden. For instance, operators based only on the Fermion are not allowed, as 5 is not a multiple of 3/2. Scalar-only operators are not permitted, since Higgs only appears in doublets, thus the number of scalars must be even. Dimensionally, two fermions and two scalars can compose an operator of dimension five. It is then be necessary that the two fermions combine to have a hypercharge zero 19 , so they need to be Hermitian conjugates of the same multiplet: y † y, which vanishes in the case of chiral fermions. Scalars multiply to give an SU(2) triplet, so the two fermions must also form a triplet in order to dot this into a scalar. Fermions must then be SU(2) doublet. This leads to the dimension-5 operators being written as [START_REF] Russell | Top quark physics in the Large Hadron Collider era[END_REF][START_REF] Grzadkowski | Dimension-Six Terms in the Standard Model Lagrangian[END_REF]:

Q nn = e jk e mn j j j m ⇣ l k p ⌘ T Cl n r ⌘ e j † l p T C j † l r (2.101)
where C is the charge conjugation matrix 20 . Interestingly, this is the only dimension-five operator allowed in the Standard Model effective theory [START_REF] Russell | Top quark physics in the Large Hadron Collider era[END_REF]. This fact was noted by S. Weinberg in [START_REF] Weinberg | Baryon-and Lepton-Nonconserving Processes[END_REF], so the dimension-five operator is sometimes referred to as the Weinberg operator. The Q nn operator violates the lepton number, whereas no terms in the SM Lagrangian violate this property. Furthermore, after the electroweak symmetry breaking, it generates neutrino masses. Neither L

SM nor L

SM terms can achieve this.

Adding heavy, right-handed neutrinos to the SM generates a model that generates this operator directly. Their quantum numbers allow a Yukawa interaction as well as a Majorana mass term. If the latter is very large, the right-handed neutrinos can be integrated out. The resulting low-energy EFT is given to first order by Q nn . In this mechanism, light left-handed neutrinos are produced by the introduction of heavy right-handed neutrinos, and it is known as seesaw mechanism [START_REF] Kayser | The Physics of massive neutrinos[END_REF].

Dimension-6 operators: parametrizing new physics

Dimension-six allows many more operators. Many new-physics models emerge to first approximation at the level of dimension-six operators, making it a very popular approach for searching for BSM physics at the LHC. As it is difficult to obtain the full set of dimensional-six operators, this section discusses only their main features and does not reproduce them completely.

There are several approaches to selecting a complete set of non-redundant dimension-six operators [START_REF] Buchmüller | Effective lagrangian analysis of new interactions and flavour conservation[END_REF][START_REF] Hagiwara | Low energy effects of new interactions in the electroweak boson sector[END_REF][START_REF] Francesco | The strongly-interacting light Higgs[END_REF]]. Yet it was only in 2010 that Grzadkowski et al. [START_REF] Grzadkowski | Dimension-Six Terms in the Standard Model Lagrangian[END_REF] presented a complete and non-redundant set of dimension-six operators. This basis is called Warsaw basis. There are three classes of operators with dimension-six [START_REF] Russell | Top quark physics in the Large Hadron Collider era[END_REF][START_REF] Grzadkowski | Dimension-Six Terms in the Standard Model Lagrangian[END_REF]:

Bosonic operators

A dimension-six bosonic operator field must contain an even number of Higgs fields, and an even number of covariant derivatives D, since all Lorentz indices must be contracted. These are the only possible contents for dimension-six's bosonic operator field:

X 3 , X 2 j 2 , X 2 D 2 , Xj 4 , XD 4 , Xj 2 D 2 , j 6 , j 4 D 2 and j 2 D 4 (2.102)
A few of the permitted bosonic operators may be excluded. The class Xj 4 is excluded because it is not Lorentz contracted. The class XD 4 can be moved to the X 2 D 2 class by the use of the identity

⇥ D µ , D n ⇤ ⇠ X µn .
Following is a review of the possible operators in classes j 2 D 4 , j 2 XD 2 and X 2 D 2 :

j 2 D 4 :
In this class, the use of the SM equations of motion (EOM) 21 ,

D µ D µ j j = m 2 j j l j † j j j ēG † e l j + e jk qk G u u dG † d q j D r G r µ A = g s qg µ T A q + ūg µ T A u + dg µ T A d D r W r µ I = g 2 ✓ j † i $ D I µ j + lg µ t I l + qg µ t I q ◆ ∂ r B r µ = g 0 Y j j † i $ D µ j + g 0 Â y2{l,e,q,u,d} Y y ȳg µ y (2.103)
moved us to bosonic classes j 2 XD 2 , j 4 D 2 and Y 2 FD 2 which belongs to the single-fermionic current operator class (This is defined later in the section). j 2 XD 2 : the class is moved to j 2 X 2 using again the identity X 2 D 2 : The EOM reduces the operator to bosonic classes X 3 , j 2 XD 2 or Y 2 XD.

⇥ D µ , D n ⇤ ⇠ X µn . Also,
As a result, the only remaining bosonic operator classes are X 3 , X 2 j 2 , j 6 and j 4 D 2 . These non-redundant operators are listed in columns 1, 2, and 4 of Figure 2.24.

Single-fermionic-current operator

The complete set of operators that belong to this class is as follows:

Y 3 D 3 , Y 2 jD 2 , Y 2 XD, Y 2 j 3 , Y 2 Xj and Y 2 j 2 D (2.104)
EOMs for quarks and leptons generally take the following form: 21 which follow from demanding a stationary action

i 6 Dl = G e ej, i 6 De = G † e j † l, i 6 Dq = G u u e j + G d dj, i 6 Du = G † u j † q, i 6 Dd = G † d j † q (2.
S SM = Z L SM d 4 x
with respect to a variation due to each SM field. and ( LR)( LR). The Fierz identity,

ȳL g µ y L ( cL g µ c L ) = ȳL g µ c L ( cL g µ y L ) (2.106)
or its right-handed counterpart, and

T A ab T A kl = 1 2 d al d kb 1 6 d ab d kl (2.

Higher dimension operators

Generally, operators above dimensions-six have sub-leading contribution compared to operators up to six dimensions. In the case of D = 7, the operator's set violates the lepton number, and many searches for lepton flavour change (LFV) appear to probe New Physics (NP) [START_REF] Ardu | What is Leading Order for LFV in SMEFT ?[END_REF]. For one (three) generations of fermions, the authors of [START_REF] Henning | Higher dimension operators in the SM EFT[END_REF] explicitly pointed out there are 30 (1542) independent operators at dimension seven and 993 (44877) independent operators at dimension eight. Further, as the observables are proportional to the square of the amplitude, the effects from dimension-six operators squared are of the same order as the interference effects of dimension-eight operators with the SM, making the dimension-eight terms important for future precision analyses. The dimension-six truncation is typically referred to as the Standard Model Effective Field Theory (SMEFT).

Effective Field Theory for top quark pair production

Given the scope of this study, the focus of our analysis is on top-philic new physics, i.e., new physics that manifests itself in the top sector. A specific focus of this thesis is the set of operators with dimension-six that modify the couplings to other SM fields to affect the production and decay of top quarks production at the LHC. To account for the fact that top-quark observables at the LHC are largely blind to the flavour of light quarks with identical quantum numbers, an U(2) q ⇥U(2) u ⇥U(2) d flavour symmetry is imposed among quarks of the first and second generation [START_REF] Brivio | O new physics, where art thou? A global search in the top sector[END_REF]. Using

q i = u i L , d i L u i = u i R , d i = d i R i = 1, 2 Q = (t L , b L ) t = t R , b = b R (2.108)
to denote left-and right-handed quarks of the first two generations and the third generation, respectively.

The gauge-invariant operators of dimension-6 are as follows:

Chapter 2. Theoretical background Operators with a top and an anti-top and one gluon, namely 22

O tG = Qs µn T A t e f G A µn (2.109)
8 four-quark operators with LL and RR chiral structure:

O 1,8 Qq = Qg µ T A Q qi g µ T A q i O 1,1 Qq = Qg µ Q ( qi g µ q i ) O 3,8 Qq = Qg µ T A t I Q qi g µ T A t I q i O 3,1 Qq = Qg µ t I Q qi g µ t I q i O 8 tu = tg µ T A t ūi g µ T A u i O 1 tu = tg µ t ( ūi g µ u i ) O 8 td = tg µ T A t di g µ T A d i O 1 td = (tg µ t) di g µ d i (2.110)
6 four-quark operators with LR and RL chiral structure To date, no experimental evidence currently exists that would allow one to definitively identify any of the above possible dimension-six operators as a signal of BSM physics. Rather, only more or less strong upper limits exist and these are shown in Table 2.6. 22 This thesis does not take into account interactions that only impact the standard gluon vertices, such as the operator's interactions

O 8 Qu = Qg µ T A Q ūi g µ T A u i O 1 Qu = Qg µ Q ūi g µ u i O 8 Qd = Qg µ T A Q di g µ T A d i O 1 Qd = Qg µ Q di g µ d i O 8 tq = qi g µ T A q i tg µ T A t O 1 tq = ( qi g µ q i ) tg µ t ;
O G = f abc G an µ G br n G cµ r .
See [START_REF] Cho | Looking for new gluon physics at the Tevatron[END_REF] for a study of its effects on top pair production. Table 2.6: The 68% and 95% confidence levels for dimension size operators contributing to top pair production [START_REF] Brivio | O new physics, where art thou? A global search in the top sector[END_REF].

The ATLAS experiment

This chapter provides an overview of the LHC [START_REF] Evans | LHC Machine[END_REF] and the ATLAS experiment [START_REF]The ATLAS Experiment at the CERN Large Hadron Collider[END_REF], with a focus on the detector systems that are most relevant to the analysis. The chapter ends with a description of the steps and principles for generating Monte Carlo (MC) samples for the purpose of data analysis and modeling.

The Large Hadron Collider

The European Organization for Nuclear Research (CERN) is home to the largest and most powerful particle accelerator in the world, the Large Hadron Collider (LHC). The LHC is a circular proton-proton (pp) and heavy-ion collider, which is located 100 meters beneath the French and Swiss borders near Geneva, Switzerland, in a tunnel with a circumference of 27 kilometres.

Following their passage through a series of pre-accelerators, proton beams are injected into the LHC at 450 GeV, accelerated in opposite directions within the beam pipes. Proton beams are intermittently transferred into the LHC in the form of proton bunches having a nominal number of 10 11 protons each, with a nominal spacing of 25 ns, which translates into a nominal crossing frequency of 40 MHz. A magnet system of 1232 dipoles and 392 quadruples holds the proton beams on their trajectory and focuses them. Magnets consisting of superconducting coils made of Niobium-Titanium alloy (NbTi) were used to produce the required high magnetic field and cooled by superfluid Helium operating at a temperature of 1.9K. Collisions take place at four points in the ring where the two beams are intercepted -these points are called Interaction Points (IP). In the same position, the caverns of the LHC's experiments are placed in order to detect the products of the particle's collisions. There are four experiments: ATLAS (A Toroidal LHC ApparatuS),

CMS (Compact Muon Solenoid), ALICE (A Large Ion Collider Experiment) and LHCb (Large Hadron

Collider Beauty). The ATLAS and CMS experiments are both general-purpose detectors. Their primary objective is the investigation of a wide range of physics, ranging from looking for the Higgs boson to looking for extra dimensions and particles that are responsible for dark matter. In ALICE, heavy-ions are used to study quark-gluon plasmas resulting from special lead-ion collisions. In LHCb, scientists aim to perform precise measurements of B-meson physics and to search for sources of CP violation in the SM.

Chapter 3. The ATLAS experiment

Luminosity and pile-up

The number of collisions per unit time is of paramount importance for the eventual physics analysis. This feature is quantitatively expressed as luminosity. The uncertainty in the luminosity is often considered a major systematic uncertainty in the measurements of cross-section. For a process with cross section s , the expected interaction rate per bunch crossing is given by:

N events = s L = s Z L dt (3.1)
where the quantity L is known as the instantaneous luminosity. The integrated luminosity, L, is measured in units of m 2 and is typically expressed in inverse femto-barn fb 1 , where 1b = 10 28 m 2 . The instantaneous luminosity and integrated luminosity are useful measures of a particle accelerator performance. Particularly, all collider experiments are aiming to maximize their integrated luminosities, since the higher the integrated luminosity, the more data is available for analysis. The formula for instantaneous luminosity is as follows:

L = N 2 b k b f rev g 4ps x s y F (3.2)
where N b is the number of colliding bunches, f rev is the revolution frequency of the accelerator and g is the relativistic factor of the accelerated particles. s x and s y are the horizontal and vertical widths of the beam spot profiles. They are typically around 2.5 µm at the LHC. F is a geometrical correction factor from the crossing-angle of the two beams at the interaction point. The instantaneous luminosity is measured in units of cm 2 s 1 . The LHC has a design maximum instantaneous luminosity of 10 34 cm 2 s 1 . The peak instantaneous luminosity of the LHC has increased gradually, exceeding the design limit, reaching 2.1 10 34 cm 2 s 1 in 2018.

With the high density of beam bunches and the high frequency of collisions in bunch crossings, many proton-proton interactions can occur simultaneously. This results in several interaction points which lead to a large number of particles. This phenomenon is referred to as pileup (PU), and is quantified by the average number of interactions per bunch crossing, hµi:

hµi = L bunch s inelastic f rev (3.3)
where L bunch is the instantaneous luminosity per bunch and s inelastic is the inelastic cross section for pp interaction. Pileup can be classified into two kinds: In-time and out-of-time. In-time pileup occurs when there are multiple inelastic pp collisions within the same bunch crossing. Meanwhile, out-of-time pileup occur when multiple bunch crossings occur during the time the detector takes to process a single event.

Pileup complicates the analysis of physics data as it results in additional energy deposits in the detectors and consequently complicates characterizing the various physics objects.

Run-2 performance of the LHC

The data used for these analyses were collected during the LHC data taken at p s = 13 TeV, also known as Run-2 data. In the period between 2015-2018, the LHC delivered proton-proton collisions equivalent to an integrated luminosity of 156 fb 1 among which 139 fb 1 of data were recorded with stable beams and detector conditions providing data that can be used for physics analysis in ATLAS. The integrated luminosity collected in Run-2 as a function of time is shown in Month in Year Chapter 3. The ATLAS experiment

J a n '1 5 J u l '1 5 J a n '1 6 J u l '1 6 J a n '1 7 J u l '1 7 J a n '1 8 J u l '1 8

The ATLAS detector

ATLAS is composed of three concentric cylinder sub-detector systems arranged around the IP. In each layer, different groups of particles interact with different sub-detector materials. These materials intercept almost all the particles created by collisions. In close proximity to the beam pipe, there is the Inner Detector (ID) built from silicon and transition-radiation gas detectors. Within a 2T magnet, the tracking detector reconstructs charged particles and measures their momentum based on their curved trajectory. It also identifies the vertices of interactions. There is next the electromagnetic and hadronic calorimeter (ECal and HCal), which measure the energy of incident particles through their electromagnetic and hadronic showers of secondary particles.

The muon spectrometer is the outermost layer, whose purpose is to identify and monitor muons with high precision. The cylindrical shape of the detector is divided into two parts: the central part, called barrel and the two edges of the cylinder, called endcaps, resulting in a geometric acceptance close to 4p steradians in solid angle. The following provide a brief description of each sub-detector.

ATLAS coordinate system

ATLAS detector coordinates are based on a right-handed Cartesian system (x, y, z), with the origin defined by the IP. In the LHC ring plane, the x-axis is orientated from the detector centre towards the centre and the y-axis upwards. The z-axis is taken to be parallel to beam pipe. The x-y plane defines the transverse plane to the beam axis. Considering the cylinderical nature of the ATLAS detector, the use of polar coordinates is more practical for describing particle direction: Th azimuthal angle f is measured around the beam axis, and polar angle q is measured from the beam axis. With this coordinate system, useful kinematic variables can be defined: pseudorapidity, h = ln tan q 2 , and the rapidity, y = 1 2 ln

⇣ E+p z E p z
⌘ where E is the particle energy, and p z its longitudinal momentum component. For relativistic particles with m ⌧ E ⇡ p, where p is the particle's three-momentum, the pseudorapidity (h) converges to rapidity (y) [START_REF]Pseudorapidity -Wikipedia, The Free Encyclopedia[END_REF]. It is worth noting that h = 0 defines the beam axis transverse direction, and h ! • the beam axis parallel direction; terms "central"

and "forward" (or "backward" where appropriate) refer to values of h near either of these two limits to the beam. The angular distance between two particles produced in the ATLAS detector is given by DR which is also invariant under a Lorentz boost along the z axis:

DR = q (Dh) 2 + (Df ) 2 (3.4)

The magnetic systems

The ATLAS magnet system [START_REF]ATLAS magnet system: Technical Design Report, 1[END_REF] consists of two magnetic components: First, there is the central solenoid magnet responsible for forcing the charged particles to curve their trajectory inside the Inner Detector. The solenoid produces an axial magnetic field of 2T in which charged particles follow a curved trajectory in the azimuthal plane. Second, there are three toroidal magnets: one in the barrel region and two in the endcap region, which provide a magnetic field to the muon spectrometer. The toroidal magnet provides an average magnetic field of 0.5T in the central barrel region and 1T for the muon systems in the endcap regions.

With the generated magnetic fields orthogonal to one another, combined with the inner tracker, they allow independent measurements of the muon momentum in the innermost and outermost parts of the ATLAS detector. Figure 3.3 illustrates the ATLAS magnet system.

s p T p T = 0.05% GeV p T 1% (3.5)
where the symbol represents quadrature sum.

The Pixel Detector

The silicon pixel detector [START_REF] Wermes | ATLAS pixel detector: Technical Design Report[END_REF] is the sub-detector closest to the beam pipes. It consists of layers of silicon pixels arranged to reach a very high level of granularity, which is needed for the resolution of primary and secondary interaction vertices. The barrel section contains four layers of detectors, while each endcap section has three layers of disks. In the area closest to the beamline is the Insertable B-Layer (IBL) [START_REF] Capeans | ATLAS Insertable B-Layer Technical Design Report[END_REF],

which is critical to identifying secondary vertices associated with B-hadrons. The detector consists of 1744 pixel sensors. Each of the sensors corresponds to a semiconductor sensor made from silicon with a size of 50 ⇥ 400 µm 2 and each sensor consists of 47232 pixels. The complete pixel detector contains approximately 82.3 million read-out channels, each one corresponding to a pixel. The resolution of the charged particle positions is of 10 µm in (R f ) and 115 µm in the z direction.

The Semiconductor Tracker (SCT)

The Semiconductor Tracker (SCT) Detector [START_REF]Technical Design Report for the ATLAS Inner Tracker Strip Detector[END_REF] is the middle section of the ID and is made of silicon microstrip sensors distributed over 4088 modules, 6.3 million channels, and with a larger size than those of the pixel detector. These modules are arranged in four layers in the barrel (|h| < 1.4) and nine discs on each endcaps (1.4 < |h| < 2.5). An average particle will have three measurements provided by the pixel detector, and eight by the SCT. The design resolution of SCT is 17 µm in the transverse plane, 580 µm along the z direction in the barrel and 580 µm in the R direction in the endcaps.

The Transition Radiation Tracker (TRT)

The Transition Radiation Tracker (TRT) is the outermost sub-detector system of the ID [START_REF] Mitsou | The ATLAS Transition Radiation Tracker. The ATLAS Transition Radiation Tracker[END_REF]. This detector is composed by straw tubes of 4 mm diameter, filled with a gas mixture of 70% Xe, 27%CO 2 , and 3% O 2 . A barrel region in the TRT consists of 73 layers of straws each measuring 144 cm long. In the endcap, there are 160 layers of 37 cm straws instead. An endcap straw is equipped with a gold-plated Tungsten wire with a radius of 0.015 mm. When a relativistic particle crosses many TRT straws, it emits photons with an energy of 5 30 KeV and is strictly proportional to the relativistic factor g = E/m of the incident particle. Thus for a given energy, particles with large g (like electrons) will produce many photons, while particle with small g (like pions) will produce few photons. The difference between electrons and pions in response is crucial to the discrimination between them. The photons (from the photoelectric effect) ionise the gas in the straw and cause currents in the Tungsten wire. The TRT can perform measurements in the R f coordinates with a precision of 170 µm per straw-tube.

Electromagnetic calorimeter

Essentially, the ECal consists of layers of lead, stainless steel and liquid Argon (LAr) surrounding the point of interaction. It measures the energy of non-hadronic, electromagnetically interacting particles (primarily electrons and photons). The calorimeter has two parts: a barrel that covers region |h| < 1.475, and an endcap that covers region 1.375 < |h| < 3.2. An electromagnetic shower is characterized by the longitudinal evolution of its radiation length X 0 . It refers to the distance after which an electron's energy is reduced by a factor of 1/e due to bremsstrahlung 1 and 7/9 of the mean free path for pair production from photons. To reliably capture most of the electron and photon energy, at least 22 radiation lengths were designed into the ECal everywhere. The target energy resolution of the ECal is:

s E E = 10%GeV 1/2 p E 0.7% (3.6)
where E is the particle energy, s E is the energy resolution. The first term is known as the sampling term, which is due to the stochastic nature of the shower cascades, in particular fluctuations in the showers and fluctuations arising from the calorimeter's sampling process. As for the second contribution, the constant term is primarily caused by non-uniformities in the detector.

The Hadronic Calorimeter 1 Here, e represents Euler's number, not the elementary charge.

Chapter 3. The ATLAS experiment

The purpose of the HCal is to provide a mean to measure the energy of jets originating from quarks and gluons.

The HCal consists of the following three parts: the Hadronic Endcap Calorimeter (HEC) [START_REF] Barillari | The ATLAS liquid argon hadronic end-cap calorimeter: construction and selected beam test results[END_REF], covering the region 1.5 < |h| < 3.2, the Forward Calorimeter (FCal) [START_REF] Artamonov | The ATLAS Forward Calorimeter[END_REF] that covers the region 3.1 < |h| < 4.9 and the Tile calorimeter (TileCal) [START_REF]ATLAS tile calorimeter: Technical Design Report[END_REF] in the barrel section.

• The HEC is a sampling calorimeter that uses a LAr technology with copper absorber plates. There are two separate wheels in each HEC, the first of which is composed of a flat copper layer that is 25 mm thick with 8.5 mm wide LAr gaps. In the second wheel, the major change is the thickness of the copper layers, which is 50 mm. The HEC is roughly 10 interaction lengths thick.

• The FCal is a sampling calorimeter based on LAr as an active medium. The detector is segmented into three layers in depth. The first layer uses copper absorber layers and targets the measurement of forward electromagnetic particles. The two other layers that are further downstream use tungsten absorbers. FCal absorbers are optimized for high absorption, and are about 10 interaction lengths deep.

• The TileCal hadronic calorimeter is a sampling calorimeter using steel as absorber material and scintillating plastic tiles as an active material. The energy measured in this calorimeter is the ultraviolet scintillation light that is produced when a charged particle crosses the active medium. For each tile, this scintillation light is collected by a wavelength-shifting optical fibre and is converted into visible light in this fibre. The output of the fibre is connected to a photo-multiplier where the signal is measured. The TileCal is placed in one central barrel and two extended barrels. The target energy resolution for the TileCal hadronic calorimeter is

s E E = 50%GeV 1/2 p E 3% (3.7)

Muon Spectrometer

The Muon Spectrometer (MS), shown in Figure 3.6, is comprised of toroidal superconducting magnets and a precision chamber designed to detect and measure muon momentum. A muon's properties are identified and measured in the outermost detector layer since muons minimally interact with the other subdetectors and have long lifetimes. It is, by far, the largest tracking system of ATLAS since it extends around the calorimeters by a radius of 4.25 m, out of the full radius of the detector, which measures 11 m. This detector system covers the region |h| < 2.7. It is also designed to trigger on muons in the region of |h| < 2.4. The MS is built from four kinds of components, all of which are gas detectors [START_REF]The ATLAS Experiment at the CERN Large Hadron Collider[END_REF]: Monitored Drift Tubes (MDTs), Cathode Strip Chambers (CSCs), Resistive Plate Chambers (RPCs) and the Thin Gap Chambers (TGCs). The magnetic field provided by toroidal magnets deflects muons in the h-direction but propagates along straight lines in the f -direction. Typical momentum resolutions of the MS is 10% for muons with 1 TeV of transverse momentum:

s p T p T = 10% p T GeV (3.8)

Trigger and Data Acquisition

There is 10 11 protons per bunch with hµi ⇡ 34 in the LHC beam bunches at a bunch-crossing rate of 40 MHz.

Each ATLAS event occupies a few MB of disk space, so it is impossible to store every single one. Further, the majority of the events correspond to well-established processes that are irrelevant to the ATLAS physics program. To maintain events with potential physics interest with the limited amount of storage, a trigger system is used. The ATLAS trigger and data acquisition system selects in real time events that are suitable for physics analysis to reduce the amount of data flowing. The trigger system operates in two stages: the level 1 hardware trigger (L1) and the high level software trigger (HLT). The L1 and HLT systems reduce the event rate to 100 kHz and 1 kHz, respectively, under nominal conditions.

Reconstruction of physics objects

Particles produced by proton-proton collisions interact with detector material, leaving different signals in different detector sections. A dedicated software algorithm converts these signals into tracks, energy clusters, and momenta. There are several stages in the object reconstruction process, and the final output is a collection of particles (also called physics objects) such as electrons, muons, jets, and missing transverse energy. This Section discusses how different physical objects can be defined at reconstruction levels based on certain signal patterns at detector level. This thesis focuses on electrons, muons, jets, specifically those originating from a b-quark, and missing transverse momentum.

Tracks and Vertices

The passage of charged particles, as described in Section 3.2.1, leaves several hits in the ID's sub-detectors, that are used to reconstruct the tracks. First, each pixel and SCT detector layer's hits are grouped into clusters. Three-dimensional space-points are created from these clusters, representing the point where the charged particle passed through. Three space-points define a track seed. This seed, combined with additional space-points from the remaining layers of the pixel and SCT detectors and compatible with the preliminary trajectory, can be used to build a track candidate. Tracks originating from pile-up effects and system noise are processed by a dedicated algorithm, which may reduce this by, for instance, requiring a hit in the IBL.

Similarly, vertices also play a crucial role in ATLAS particle reconstruction. The point at which hard scattering took place is called the Primary Vertex (PV). In the case that the particles produced within the PV last long enough to decay further away, a secondary vertex is created. Reconstruction of interaction vertices begins after track reconstruction. The vertex reconstruction process begins by calculating the crossing point of several different tracks [START_REF]Vertex Reconstruction Performance of the ATLAS Detector at s 13TeV[END_REF] and iteratively adding or discarding all tracks that are respectively compatible or incompatible with originating from this crossing point. In this process, the vertex position is updated with each change in track association. Usually, primary vertices are within the interaction region, but secondary vertices are not. Heavy flavoured hadrons usually have long lifetimes, so they can decay within millimetres of the collision point, so the secondary vertices are crucial information for heavy flavour identification.

Two important parameters summarize the final reconstructed tracks, the impact parameter d 0 and the longitudinal impact parameter z 0 . d 0 and z 0 , are defined as the point to the closest approach of the trajectory of a track to the PV in the transverse plane and z-direction, respectively

Electrons

The ATLAS experiment detects an electron produced in pp collision by a track in the ID as well as an energy deposit in the ECal. Electrons passing through the EM material interact with its material, producing electromagnetic showers in the cells. As the cells form a cluster, the electron energy is computed by adding up the energies of all the cluster cells. In the central region of the ID |h| < 2.5, tracks with a minimum of three hits in the ID layers are extrapolated to the second layer of the EM calorimeter cluster, which contains the most energy deposited by the electron. The reconstruction consists of three steps: seed-cluster reconstruction, track reconstruction, and electron candidate reconstruction [START_REF]Electron and photon energy calibration with the ATLAS detector using data collected in 2015 at p s = 13 TeV[END_REF][START_REF]Electron efficiency measurements with the ATLAS detector using the 2015 LHC proton-proton collision data[END_REF][START_REF]Electron reconstruction and identification in the ATLAS experiment using the 2015 and 2016 LHC proton-proton collision data at[END_REF]. A total of 200 ⇥ 256 towers are formed within the ECal in the h ⇥ f space, and the seed is built by searching for a local energy deposit within the towers. For track reconstruction, a hit in one of the ID tracking layers serves as a key component. Clusters formed from these hits are the basis for charged-particle reconstruction in the ID. The clusters are required to be within the range |h| < 2.47, excluding the transition region between the barrel and endcap calorimeters at 1.37 < |h| < 1.52. Beyond this range in |h|, candidate electrons are designated as "forward electrons," and they are applied to different identification algorithms; following that, only central electrons are considered. Electron candidates must also satisfy a transverse energy requirement of E T > 15GeV. Through these clusters, three-dimensional measurements (space-points) are constructed.

Track seeds are then formed from sets of space-points in the silicon-detector layers. Then the track candidate is obtained from a c 2 fit of the hits. The electron-reconstruction process ends with the matching of the track-candidate with the candidate calorimeter seed cluster and the calculation of their final size.

It is necessary to apply a number of additional requirements to distinguish electrons from other particles, like pions and converted photons, which are crucial for several analyses. These criteria are classified into three categories based on either independent selections of the discriminating variables (such as the shape of the energy shower (tower), the number of track hits, and the quality of comparison between the EM cluster and the ID track), referred to as cut-based identification. This identification level is referred to as Loose, Medium, and Tight, and has progressively higher background rejection power at a cost to electron efficiency. For all identification levels, Figure 3.8 shows electron identification efficiency measured with Z ! e + e events using data from 2016. Tight operating points have an identification efficiency of 55% at E T = 4.5GeV and 90% at E T = 100GeV. With the Loose operating point, identification efficiency varies from 85% at

E T = 20GeV to 99% at E T = 100GeV.
Requirements on the transverse and longitudinal impact parameters, such as d 0 /s (d 0 ) < 5.0 mm and z 0 sin(q ) < 0.5 mm, are established to make sure that electrons are reconstructed from the primary vertex, thus mitigating contributions from charge-misidentified electrons, non-prompt leptons, and pile-up.

Furthermore, electrons need to be isolated in order to distinguish prompt electrons (from hard-scattering events) from photon conversions and hadrons mis-reconstructed as electrons. For this reason, a tool referred to as the Prompt Lepton Isolation Tool is used to reduce the number of non-prompt electrons being reconstructed. The tool is specifically tuned to reject non-prompt leptons from hadronic jets generated by a bor c-quark. There are two working points defined for the tool: PLVTight and PLVLoose. The PLVTight working point has a higher rejection of fake electrons at the cost of a lower acceptance of real electrons.

For each year between 2015 and 2018, Figure 3.9 shows the efficiencies of the lowest unprescaled single electron triggers employed over Run-2. As the efficiency curves turn on around 25 GeV, p T > 27 GeV for the leading trigger-matched electron is a requirement of the analyses presented in this thesis.

Muons

Muon candidates are derived from track segments found in the muon spectrometers (MS) and the ID. The detector data from the individual sub-detectors is subsequently combined to form muon tracks. Muon tracks are reconstructed in the ID by using the same reconstruction algorithms as any charged particle [START_REF] T G Cornelissen | The global chi2 track fitter in ATLAS[END_REF].

Several quality requirements are imposed on muon candidates to reject misidentified muons, which originate primarily from pion and kaon decays. They are required to surpass a p T > 10 GeV and have a pseudorapidity of |h| < 2.4. The impact longitudinal and transverse parameters of muons are regulated similarly to electrons to ensure that their origin is at the primary vertex. These requirements are d 0 /s (d 0 ) < 3.0 mm and z 0 sin(q ) < 0.5 mm. Work points are divided into four categories: Loose, Medium, Tight and High-p T . The efficiency is computed with respect to offline isolated muon candidates which are reconstructed using standard ATLAS software and are required to pass Medium quality requirement. The MU20trigger requires that a candidate passed the 20 GeV threshold requirement of the L1 muon trigger system. The mu26_ivarmedium trigger is seeded by the MU20 trigger and requires that a candidate satisfied a 26 GeV HLT threshold and passed a medium isolation selection computed using inner detector tracks reconstructed online by the HLT within a cone with a variable size which depends on the p T of the muon. The MU50 trigger is seeded by MU20 at L1 and requires that a candidate satisfied a p T threshold of 50 GeV. The efficiency is measured using a tag-and-probe method with Z ! µµ candidates, with no background subtraction applied, in 13 TeV data from 2018. Only statistical data uncertainties are shown [START_REF] Morgenstern | Muon trigger efficiency in early 2018 data taking[END_REF].

Chapter 3. The ATLAS experiment

Jets

As described in Section 2.1, quarks and gluons cannot exist as freely propagating particles, but are instead collimated hadrons that form jets. It is these charged and neutral hadrons that interact with the detector components. Since four-momentum is conserved, the sum of the four-momentum of all the resulting particles must equal the four-momentum of the original parton. In this way, if the energy and momentum of a spray of hadrons composing a jet could be analysed, conservation of four-momentum could be used to fully describe the kinematics of the originating parton. The energy of these jets is measured in ECaL and in the HCal.

In variety of algorithms can be used to reconstruct a jet based on the fractional energies deposited by the calorimeters. In the ATLAS experiment, the anti-k t algorithm [START_REF] Cacciari | The anti-kt jet clustering algorithm[END_REF] is used to reconstruct jets on the basis of a set of inputs: groups of energy clusters from the calorimeters, a distances d i j between a jet candidate i and j, and a distance between a jet candidate i with respect to the beam d iB . These distances, d i j and d iB , are defined as:

d i j = min ⇣ p 2 T,i , p 2 T, j ⌘ (DR i j ) 2 R 2 , d iB = p 2 T,i (3.9)
where p T,i and p T, j are the transverse momentums of the two particles in the (h f ) plane. DR i j = q (Dh i j ) 2 + (Df i j ) 2 is the angular distance between these two particles. This document uses the radius parameter R = 0.4. The algorithm proceeds by finding the smallest distance between jet candidates d min i j

and between the jet with respect to the beam d iB . In the case of d min i j < d min iB , the two jets are combined to form a pseudo-jet by summing the four-momenta of jet i and j. Then, the new jet candidate is added back to the jet candidates list. Finally, the previous steps are repeated until d min i j becomes larger than d min iB , then the jet candidate i is considered as a final jet and removed from the input list. Additionally, a minimum p T of 30GeVand |h| < 4.5 are required for each jet. The reconstructed jets are corrected based on their energy scale (JES) [START_REF]Muon reconstruction performance of the ATLAS detector in protonproton collision data at $$\sqrt{s}$$ = 13 TeV[END_REF], which corrects the mismeasurement of calorimeter energy compared to the MC simulations as a function of p T and h, and resolution (JER) which is derived as the standard deviation from a Gaussian fit of the jet response function. The comparison of the average jet response in data and simulation is presented in Figure 3.12. The ratio of jet energies at the generator-and reconstruction-level is used as a correction. To reduce the number of reconstructed jets due to additional pileup interactions, the Jet Vertex Tagger (JVT) [START_REF]Tagging and suppression of pileup jets with the ATLAS detector[END_REF] is used. The JVT estimates the probability of a jet originating from a pileup or hard scattering based on the number of tracks associated with the PV. Further, jet cleaning techniques are employed to remove cosmic muon contributions, calorimeter noise and interaction with residual gas in the beampipe and non-detector material. The jet reconstruction is performed using Particle Flow algorithm [START_REF] Aaboud | Jet reconstruction and performance using particle flow with the ATLAS Detector[END_REF] which uses the combination of tracking and calorimeter information to reconstruct a jet. This method mimics the individual particle reconstruction procedure, such as leptons and vertices.

[GeV]

s = 0.4 (EM+JES) R t k Anti- +jet γ + jet ee → Z + jet µ µ → Z Multijet
Total uncertainty Statistical component Figure 3.12: Data-to-simulation ratio of the average jet p T response as a function of jet p T . The combined result is based on three in situ techniques: the Z+jet balance method (dielectron channel, upward-pointing triangles; and dimuon channel, downward-pointing triangles), g+jet balance method (open squares) and the multijet balance (open circles). The errors represent the statistical (inner error bars and small inner band) and the total uncertainty (statistical and systematic uncertainties added in quadrature, outer error bars and outer band). These results apply to anti-k t jets with R = 0.4, reconstructed from electromagnetic-scale topo-clusters. The calibration curve and its uncertainty are smoothed using a sliding Gaussian kernel [START_REF]Muon reconstruction performance of the ATLAS detector in protonproton collision data at $$\sqrt{s}$$ = 13 TeV[END_REF]. These in situ measurements were performed following methodology similar to that of [START_REF] Aaboud | Jet energy scale measurements and their systematic uncertainties in proton-proton collisions at p s = 13 TeV with the ATLAS detector[END_REF], and are taken as a residual correction.

Missing Transverse Energy

Particles that escape along the direction of the beamline or that do not interact with the materials of the sub-detectors are not detected by the ATLAS experiment, such as neutrinos. As neutrinos, or other invisible particles, are not directly detectable, they appear only as an imbalance in momentum, known as missing transverse momentum E miss T (or MET). With the proton beams aligned with the longitudinal axis, the transverse momentum of the partons inside is negligible. In this case, the transverse momentum of the initial state can be approximated with zero. This is not so for the longitudinal momentum since it is unknown what fraction of the momentum (x1 and x2) is carried by the two interacting partons. It is therefore possible to use momentum conservation only in the transverse plane for all produced particles. In this sense, missing transverse momentum can be defined as follows [START_REF] Aaboud | Performance of missing transverse momentum reconstruction with the ATLAS detector using proton-proton collisions at $$\sqrt{s} = 13˜\hbox{TeV }$$[END_REF] Ẽmiss

T = Â calibrated objects pT (3.10) E miss
T is computed as the negative vectorial sum of the momenta of all calibrated objects (individual leptons, photons and jets). Furthermore, they are also combined with the so-called MET soft-term, which is the soft hadronic activity measured by reconstructed charged-particle tracks matched to the PV that isn't associated with the previously counted reconstructed objects. The possibility of double counting of potential contributions from the reconstructed tracks of charged particles from the ID, the energy deposits in the calorimeter, and the reconstructed muons from the MS is avoided by using a signal ambiguity resolution procedure, which eliminates signals used previously when combining the different E miss T contributions.

Monte Carlo Event Generation

MC simulation method

Monte Carlo event generators (EVGEN) simulate particle collisions via the Monte Carlo method and provide fully detailed final states at the particle level. Monte Carlo simulations are used to compare data with simulation predictions and to estimate SM backgrounds when data-driven tools are not available. A key part of searches for new physics involves simulating the signal process, optimizing analysis sensitivity to the signal process, and interpreting results in a manner that is dependent on the model. Additionally, in an analysis, the use of generator level particle information, referred to as truth information, is important for determining the composition and behavior of signal and background. In order to arrive at a description at the level of observable particles, the event generation is split up into different stages that are sketched in Figure 3.13. In Monte Carlo event generators, the following steps are considered when predicting physics processes at the LHC: Figure 3.13: Pictorial representation of a t th event as produced by an event generator. The hard interaction (big red blob) is followed by the decay of both top quarks and the Higgs boson (small red blobs). Additional hard QCD radiation is produced (red) and a secondary interaction takes place (purple blob) before the final-state partons hadronise (light green blobs) and hadrons decay (dark green blobs). Photon radiation occurs at any stage (yellow) [START_REF] Gleisberg | Event generation with SHERPA 1.1[END_REF].

Hard process

The MC event generation begins with the hard process (dark red blobs), which is the process of highest momentum transfer in an event. Simulations and Matrix Element (ME) calculations for hard processes are performed with Leading Order (LO) accuracy using tree-level Feynman diagrams, which are usually sufficient for modeling distribution shapes. In the case of Next-to-Leading Order (NLO), or even Nextto-Next-to-Leading Order (NNLO), accuracy can be achieved by including all diagrams in the ME. The particles entering the ME in hard proton-proton collisions, like at the LHC, are not directly protons, but their constituents, partons (quarks and gluons). In such collisions, a proton is described by a set of Parton Distribution Functions (PDF). Various PDFs are available that describe parton momenta from protons at the energies expected at the LHC and can be used as input to the EVGEN process. This leads to slightly different fractional momenta carried by each constituent of the interacting protons. The analysis presented in this document will take into account uncertainties related to the choice of the PDF set.

Parton shower

Parton Shower (PS) (red lines) step describes the evolution of the QCD process of incoming and outgoing partons (quarks, gluons) involved in the hard scatter process. From the hard process to low momentum scales, the parton shower models the QCD Bremsstrahlung at perturbative scales to the point at which perturbation theory is invalid.

Chapter 3. The ATLAS experiment

Hadronization

As discussed in section 2.1, QCD becomes non-perturbative a s at long distances (low energy). In this regime, quarks and gluons are confined into colour neutral hadrons due to the increased strength of the strong force.

The process by which partons are bound into these states is referred to as hadronization. This is symbolized by the light green ellipses in Figure 3.13. Many phenomenological models are developed to describe the hadronization process. These models can be grouped into two categories: string models and cluster models.

In string models [START_REF] Andersson | Parton fragmentation and string dynamics[END_REF], color-connected parton pairs are pictured as string segments. Quarks are positioned at the end points of the string and gluons are pictured as kinks in the string. In cluster models [START_REF] Amati | Preconfinement as a property of perturbative QCD[END_REF], following the PS gluons are forced to split into quark-anti-quark pairs. After splitting, a set of color-singlets are constructed from quark-anti-quark pairs.

Underlying event

The Underlying Event (UE) (purple blob) represents the particle production process which is not associated with the leading hard process. These interactions are mostly due to soft interactions between proton remnants.

The steps for these interactions are the same as for hard interactions (e.g., parton shower shown in purple), and they produce particles on top of those produced in hard interactions. The UE are beyond QCD factorization theorem and are modelled by phenomenological models. To correct for the minimal bias contributions in the MC simulation, the number of vertices was taken as a measure for the number of additional proton-proton interactions, with the MC samples being re-weighted in order to fit the simulation to data. Dedicated event generators with parameters tuned to data are used for the underlying event and pile-up contributions.

MC Generators

Event generators come in a variety of types. In some cases, the event generator will only perform part of the process. Typically, these generators target either the hard scattering and matrix element stages or the parton shower and hadronization stages, while other generators handle the complete event generation process. ATLAS event generation framework integrates a wide range of generators, including general purpose generators such as PYTHIA [START_REF] Sjöstrand | PYTHIA 6.4 physics and manual[END_REF], HERWIG [START_REF] Michael | Matrix-element corrections to parton shower algorithms[END_REF], and SHERPA [START_REF] Gleisberg | SHERPA 1., a proof-of-concept version[END_REF], and specialized generators, as e.g. ALPGEN [START_REF] Michelangelo | ALPGEN, a generator for hard multiparton processes in hadronic collisions[END_REF], MC@NLO [START_REF] Frixione | Matching NLO QCD and parton showers in heavy flavour production[END_REF], and POWHEG [START_REF] Paul | The Monte Carlo event generator AcerMC versions 2.0 to 3.8 with interfaces to PYTHIA 6.4, HERWIG 6.5 and ARIADNE 4.1[END_REF]. Their usage depends on the physics process.

Detailed information about the generator used for the samples used in this study is provided in Section 5.1.

General Purpose Generators

The general-purpose generator produces complete events starting from the proton-proton initial state and can be used stand-alone or in combination with specialized generators (described below) to describe certain final states more accurately.

• PYTHIA: Extensively tested at both e + e and hadron colliders and providing a large number of built-in processes, it starts with a hard scattering process, calculated to the lowest order in QCD, where additional QCD and QED radiation can be added in form of shower approximations.

• HERWIG: Features an extensive number of built-in processes and was tested and tuned at the LHC.

Even though there are other generators used for specific final states, HERWIG and PYTHIA serve as benchmarks.

• SHERPA: The program implements the duplicated removal prescription described in [134] to match fixed-order matrix elements to QCD showers. As a result, it is expected to describe events with a large number of isolated jets better than HERWIG or PYTHIA. It is easy to incorporate new processes, and it can be integrated with a variety of external tools. In such cases, SHERPA is often used for simulating BSM physics scenarios.

Specialized Purpose Generators

In conjunction with general event generators, specialized generators are used to improve the accuracy of • ALPGEN: Generates events with a large number of well-separated hadronic final states. It is expected that the fixed order QCD matrix element will provide a better approximation than HERWIG and PYTHIA's shower approximation. Although, it interfaces with HERWIG and PYTHIA, which are used to generate parton showers and then hadronization.

• MC@NLO: Processes hard scattering at LO and NLO. The events it generates are used as input by HERWIG/PYTHIA. With top quark pair production, the transverse momentum distribution of top quarks is expected to be better modelled than with general purpose generators. As MC@NLO uses one-loop corrections, events with positive and negative weight are generated.

• POWHEG: Also includes the option of using either angular-ordered shower matching with cluster fragmentation, via the HERWIG generator, or momentum-ordered shower matching with string fragmentation, via the PYTHIA generator. Unlike MC@NLO, POWHEG generates only positive event weights. POWHEG's central formula is derived by manipulating and modifying the NLO differential cross section, by including higher order terms, in such a way as to have the same form as that given by PS.

A particle with a mean lifetime of more than 30 ps is usually considered stable in Monte Carlo event generators. The behaviour of the longer lived particles, their energy loss and further reactions taking place when entering and traversing the detector is modelled by a dedicated subsequent detector simulation, based on the GEANT4 software library [START_REF] Agostinelli | Geant4-a simulation toolkit[END_REF], which will not be described here. The fully detailed simulation of the ATLAS detector is referred to as FULL-SIM. The accuracy of this fully detailed simulation comes at a high cost in CPU, and the sensitivity of many physics analyses is already limited by the available Monte Carlo statistics and will become even more limited in the future. Therefore, sophisticated fast simulation tools have been developed refereed to as FASTSIM, which is less detailed and thus faster [START_REF] Gadatsch | Fast simulation methods in ATLAS: from classical to generative models[END_REF][START_REF] Jansky | The ATLAS Fast Monte Carlo Production Chain Project[END_REF].

4. High-Granularity Timing Detector: t 0 calibration

High Luminosity Large Hadron Collider (HL-LHC)

The High Luminosity Large Hadron Collider (HL-LHC) program aims to crank up the performance of the LHC in order to increase the potential for discoveries after 2027. The HL-LHC will deliver an integrated luminosity of up to 4 000 fb 1 . The ultimate luminosity will reach up to L ' 7.5 ⇥ 10 34 cm 2 s 1 compared to the current value of 2 ⇥ 10 34 cm 2 s 1 . Therefore an upgrade of the ATLAS detector [START_REF]ATLAS Phase-II Upgrade Scoping Document[END_REF] will be required prior to the start of this new phase (Phase-II upgrade).

The large increase in simultaneous interactions, pileup, will become a key experimental challenge of the HL-LHC, with an average of 200 simultaneous pp interactions per bunch crossing. The result is an average pileup density of 1.8 vertices/mm, where pileup density is defined as the number of collisions per length unit along the beam axis during one bunch crossing. This value must be compared to 0.24 vertices/mm for Run-2.

The ability to associate tracks to primary vertices depends on the longitudinal impact parameter resolution of the Inner Tracker (ITk).

High-Granularity Timing Detector (HGTD)

In this context, the High-Granularity Timing Detector (HGTD) is proposed. This new detector is designed to enhance the capabilities of the all-Silicon ITk by adding timing measurement to charged-particle trajectories as well as spatial information. This device will cover a pseudorapidity range between 2.4 and 4.0, an important region for improving physical object reconstruction performance and completing ITk in the forward region. The object-level reconstruction improvements increase the physical potential of ATLAS at the HL-LHC.

The HGTD will be located in the gap region between the barrel and the end-cap calorimeter, at a distance of approximately ±3.5m from the nominal interaction point. This volume is currently occupied by the Minimum-Bias Trigger Scintillators [START_REF] Sidoti | Minimum Bias Trigger Scintillators in ATLAS Run II[END_REF] which will be removed. The position of the two vessels for the HGTD within the ATLAS detector is shown in The two silicon sensor double-sides are foreseen to ensure the precise timing information for minimum ionizing particles with time resolution better than 35 ps per hits at the start of its lifetime and 70 ps at the end of HL-LHC operation. To achieve this timing resolution, two LGAD1 [START_REF] Pellegrini | Technology developments and first measurements of Low Gain Avalanche Detectors (LGAD) for high energy physics applications[END_REF] sensors connected to two ASICs will be used. The LGAD sensors will be divided into 15 ⇥ 15 pads. Each sensor is bump bonded to one ASIC to form an hybrid, or a bare module. When glued to the mother flex, two hybrids make one module.

There are in total 8032 modules in HGTD. This pads size will ensure a maximum occupancy of 10% at the highest expected level of pileup with small dead areas between pads. A global view of the various HGTD components is shown in Figure 4.4.

The readout rows are sets of modules whose flex cables (flexible PCB cables) are guided together towards larger radii to the peripheral on-detector electronics. They extend as a line of modules from lower to higher radii. The maximum length of the readout rows is limited by the manufacturing capabilities for the flexible circuits used for the data transmission. The maximum number of modules in a row is 19. The position of rows are chosen to have a maximum overlap between modules of both layers depending on the radius. The The various components are shown: hermetic vessel (front and rear covers, inner and outer rings), two instrumented double-sided layers (mounted in two cooling disks with sensors on the front and back of each cooling disk), two moderator pieces placed inside and outside the hermetic vessel. [START_REF]Technical Design Report: A High-Granularity Timing Detector for the ATLASPhase-ii Upgrade[END_REF] simulation study shows that the optimal overlap to achieve the required timing resolution: for r > 470 mm, an overlap of 20%, for 230 mm < r < 470 mm an overlap of 54% and for r < 230 mm, an overlap of 74%.

HGTD simulation

The HGTD full simulation is performed using dedicated software, the GEANT4 toolkit [START_REF] Agostinelli | A Simulation toolkit[END_REF], for the HL-LHC ATLAS upgrade programme. For MC samples, the same methodology is employed as for regular ATLAS offline analysis: event generation, detector simulation, digitization of simulated energy depositions into detector read-out data, and event reconstruction.

In previous calibration studies, the simulation step GEANT4 hits were stored as LArHits with a special granularity of 0.5 ⇥ 0.5 mm. In studies described in this note, GEANT4 hits are stored as SiHits with a special granularity of 1.3 ⇥ 1.3 mm. Further, the current simulation does not fit the HGTD model as described in the Section 4.2:

• In the HGTD simulation, a two-ring (r < 320 mm, 80% overlap, r > 320 mm, 20% overlap) layout Figure 4.8 shows the HGTD hits in simulated events for the two-ring layout, as a function of the radial distance from the beam axis. Compared with the inner layer, there is a slight increase in the outer layer, primarily caused by the increased probability of initiating showers due to hadronic interactions as more material is traversed.

• The number of modules in the HGTD simulation (7 894) is fewer than the detector description (8 032) 3 .

• In the HGTD simulation, modules consist of one sensor and two readout ASICs instead of two LGAD sensors connected to two ASICs, as specified in the detector description summarized in Section 4.2. The difference is due to the layout optimization undertaken in parallel to these simulation studies. The three-ring layout will be implemented in GEANT4 detector description in the future and will allow a study more consistent with the nominal design. In addition, the minor difference in the description of the detector geometry will have only a minor impact on the studies and analyses of physical objects due to the almost identical geometric coverage and the very similar number of HGTD hits simulated per track in different regions of pseudorapidity [START_REF]Technical Design Report: A High-Granularity Timing Detector for the ATLASPhase-ii Upgrade[END_REF].

Each HGTD end-caps is a combination of one hermetic vessel and two instrumented double-sided layers (mounted on two cooling/support disks), which gives 8 layers for both HGTD end-caps. The HGTD modules with one sensor, two readout ASICs and the flex PCB cable connecting the ASICs in peripheral electronics beyond r > 640 mm are also simulated. The module's position in each row, given the overlap between layers, is also simulated.

Timing simulation

The HGTD physics is highly dependent on the time of the different channels within an event. Although the detector will be regularly calibrated and adjusted, the measured time hits (t hits ) of charged particles in the HGTD, where t hits is defined as the elapsed time between the bunch crossing clock of an event and the hit in the HGTD, can vary across the pads (3.6M channels, see Section 4.2). This time can have unwanted variations which degrade the resolution unless they are understood and controlled. The origin of the t hits variations are as follows:

• The z position of charged particles within an event originating from the hard-scatter vertex and pileup vertices. This contribution to the resolution is of little importance.

• The variation of the time of flight (ToF) 4 , due to the position of the pads (radius dependence), the position z of the particle, and other factors such as the electromagnetic field generated by electromagnetic calorimeters. The variation of ToF as a function of the position of the pad is illustrated in Figure 4.10.

• The contribution from electronic read-out (jitters, time-walk). The irreducible and non deterministic clock contribution to the resolution is expected to be around 15 ps, coming mainly from the lpGBT clock jitter and the additional contribution from the flex cable and ASIC. For more details, see Section To summarize, the main contributing factors to the t hits variations are:

t hits = t BC + ToF(z, R, Kinematics) + electronics jitters (4.2)
Where t BC is the time of bunch crossing. The expected bunch crossing time offset for the ATLAS detector is included as a bunch-dependent variation. The collision time is expected to shift by a few ps per bunch.

t 0 time calibration

The preliminary plan for the timing correction procedure is to tune the t hits measured by HGTD pixels. Thus, all particles coming in from a particular vertex z would result in the same t hits , no matter what the channel position was. As a result, the t hits shift between pixels is only the result of the bunch crossing effect.

After time calibration procedure, t 0 represents the centre of the time distribution of vertices at z = 0. This t 0 differs from zero through the cumulative effects of time-of-flight, clock propagation delays, and dynamic shifts of the clock phase during data-taking. In addition, a perfect calibration will not eliminate the irreducible and nondeterministic contribution of the clock to the resolution which is expected to be around 15 ps, it is what should remain after a perfect calibration. The calibration procedure is described in more detail in Section 4.5.1.

Time simulation scheme

In the current simulation described in the last Section 4.2, the resolution contributed by the electronics is modelled by smearing the signal time with two Gaussian functions, one reflecting the clock jitter and time-walk contribution from the readout system (s = 25 ps with no irradiation) and the other for the clock distribution (s = 15 ps). The simulation of timing resolution is set up according to the specification detailed in Section 4.2, and the total resolution for the case of no irradiation contributed from the above sources is about 35 ps per hit. Nevertheless, there are other contributions to the resolution from the electronics that are not accounted for by the current simulation, such as time-dependent effects and static effects. In this section, more details are provided for understanding and injecting these effects into the simulation.

The time simulation scheme is shown in Figure 4.12. It consists of injecting the electronic jitters on top of t hits where the dynamic and static contributions to the time resolution have to be taken into account as described in Section 4.4.3. Afterward, an internal Phase Shifter (PS) is used to align events within the 2.5 ns approval window (see Section 4.4.4). ASIC per ASIC, the PS is tuned so that the maximum number of hits in a 2.5 ns window is achieved. The timing calibration procedure is then used to calibrate the output t hits .

Further details are available in Section 4.5.

Source of electronic jitters

The contribution of electronic jitters affects largely the t 0 knowledge of each individual channels (about 3.6 million channels). The jitter depends on the detector pads location, where the pads are assembled in readout rows 5 . These effects must be moderated and calibrated accordingly in order to minimise their effect on the time resolution. Electronic jitters can be static or time dependent:

Static effects:

The simulation propagates particles step-by-step inside the volumes in the detector as outlined in Section 4.4, which should affect the trajectory of the charged particles. These contribute to the resolution of time as geometric or static effects, i.e. those fixated by geometry or varying over time, as follows:

• Radially depend time-of-flight (ToF) and detector alignment, see Figure 4.10,

• Signal/clock distribution in the ASIC and in the flex cable of different length,

• Non-uniform clock propagation path within the ASIC to each TDC.

Time dependent effects:

The time dependent effects or dynamics effects contribution to the timing resolution are:

• The LHC radio frequency systems which compensate the beam loading and maintain bucket stability result in a periodic collision point time shift in the ATLAS Detector as shown in Figure 4.11. The collision time is expected to shift by a few ps per bunch,

• High-frequency jitter, noise in the flex cables,

• Low frequency day/night temperature changes.

In studying the time-dependent effects that contribute to the timing resolution, it is important to consider the length of the interval of events. For instance, the variation of the clock with time, can occur through a variety of mechanisms across a wide range of frequencies and depending on the period of these effects, and on the affected component and area (ASIC, module, group of ASICs of same lpGBT, PEB board), they may be calibrated with good accuracy.

Source of clock jitters:

A proposed HGTD architecture is shown in Figure 4.13 and can be broken down into two parts: on-detector electronics located in the experimental hall and off-detector electronics located in the USA15 counting room. The on-detector electronics consist of ALTIROC modules connected via flex cable to the Peripheral Electronics Board. The off-detector electronics is based on the general-purpose FELIX system, which is the main interface between the off-detector back-end and the on-detector electronics. The interface between on-detector and off-detector electronics is performed via optical links using lpGBT chip set and VTRx+ 6optoelectronics, which provides different data paths for Timing, Trigger and Control (TTC), DAQ and DCS 7 .

The main data stream is used for the propagation of clock, fast commands and configuration to the modules, as well as the data information for the ATLAS event processor. The contribution from irreducible and non • Front-end electronics: A random Gaussian-distributed 5 ps is injected to account for jitter in the ALTIROC.

• FLEX cable: A random Gaussian-distributed 5 ps is injected to account for jitter in FLEX (environment noise, inherent time jitters performance ...).

• lpGBT: A large non-Gaussian time jitter might be expected for lpGBT. However, any front-end chip with a phase-locked loop can filter these effect to a small 2.2 ps jitter [START_REF] Mastoridis | Cavity voltage phase modulation to reduce the high-luminosity Large Hadron Collider rf power requirements[END_REF]. Both of these scenarios are parametrized and included in t 0 calibration study and also shown in Figure 4.14.

• FELIX: A conservative 5.2 jitter is added to represent the worst jitter expected for the FELIX.

• A sinusoidal varying 100 ps time offset with variable period are injected to account for large term variation (heat cycle or other effects.) To stabilize the timing, a phase shifter is utilized to reduce the jitter of the clock and compensate for the The average t hits as function of hits distance to beam for different HGTD layers using simulated 1 434 t t events at a pileup of hµi = 200 in z > 0. The effects of ToF are mainly absorbed after applying the phase shifter, as can be seen by comparing with Figure 4.10 where the phase shifter is not applied. The sawtooth-shape for the points is cause by the phase shifter with step of 100 ps.

Timing Models

The time-of-arrival distribution of each measurement channel consists of a Gaussian core derived from the time dispersion of the LHC collisions convolved with the combined hit time resolution of the sensor and The study used the following method to add clock jitter sources one by one to the t hits distribution: Time model = t hits + jitters contribution (4.3)

Multiple models or scenarios of timing are created as a result of the different contributions of jitters to the t hits distribution. This is illustrated in Figure 4.17 and expressed as follows:

• TimeASIC: t hits + phase-shifter.

• TimeFull: TimeASIC + FLEX cable + FELIX + lpGBT + LHC clock jitters.

• TimeFull + LHC corrected: TimeFull but corrected from LHC clock using the estimation values shown in blue in Figure 4.11.

• TimeFull + 1 Hz/1 kHz: TimeFull + Sinusoidally varying 100 ps offset with period of 1 Hz (1 kHz).

Temperature variations (heat cycles or other effects) have a long-term effect on the measurement, and they could be estimated as 100 ps with variable frequency. This variable frequency is assumed to be 1 kHz for 1 000 events when the trigger rate is 1 MHz. The timing correction procedure is meant to reduce intrinsic time differences between the pads to maintain a good resolution of 30 ps-50 ps per pad. The hit time will be calibrated at regular intervals using the arithmetic mean of the t hits distribution. The timing correction process can be represented as follows:

t[calib, channel i] = t[raw, channel i] ht hits i (4.4)
Where ht hits i is the calibration constant. t[raw, channel i] is time hits in a given channel i before calibration.

For a given interval of events, the calibration constant is calculated based on the arithmetic mean of the t hits distribution in a particular detector region (see Section 4.5.2). The interval of events can be either long or short. A short time interval can correct for faster variations at the cost of reduced precision. A long time interval allows for better calibration constant precision, but it has a low performance of timing correction.

More details can be found in Sections 4.6.1 and 4.6.2.

Chapter 4. High-Granularity Timing Detector: t 0 calibration

Special granularity

As a way of evaluating the t 0 calibration, calibration constants are calculated using different methods for rearranging pixels groups. Each group of pixels is targeting a particular region of the detector, as well as targeting global or more broadly correlated effects to calibrate (always with the trade-off systematics/statistics):

• Calibration per module: calibration constants are computed at module 8 level (7 984 9 calibration constants).

• Calibration per row: calibration constants are per layer and per row, as defined by the readout electronics, see Figure 4.5.

• Calibration per ring: calibration constants are computed per ring, e.g. inner and outer ring as shown in Figure 4.8 (16 calibration constants).

• Calibration per layer: calibration constants are computed a layer level (8 calibration constants, 4 for each of the HGTD layers in the two end caps.).

• Calibration per circle: calibration constants are computed per layer and per circle defined by Dr = 40 mm.

• Inclusive calibration: calibration constants are computed at the level of 2 disks (1 calibration constant).

The following sections address each of these methods with the designation special granularity. Equation 4.4 calibrates the t hits distribution using calibration constants, ht hits i, that are computed using different special granularity approaches. Using these special granularities approaches allows us to estimate the performance of the time calibration procedure.

Depending on the length of the time offset between channels for a given special granularity, timing correction is thought to assume correlation or uncorrelation. For instance, the calibration per module is based on the assumption that time variations are not correlated between modules, on the other hand, The inclusive calibration computes calibration constant at the level of 2 disks, thus, time variations between channels are assumed to be correlated.

Figure 4.18 shows the variation of the number of hits applicable for the determination of the calibration constants given special granularity. The inclusive calibration collects hits from two disks to compute one calibration constant per event, for which a large number of hits is observed. On the other hand, the calibration per module calculates the calibration constant at the module level which gives rise to multiple calibration constants with a low number of hits. The other special granularities follow the same arguments.

Figure of merit of the calibration

The figure-of-merit of the calibration is to compute the mean of the distribution, ht calib i, with the smallest value of RMS using different special granularities approaches. The time dispersion of the hits encodes information between various clock jitters. The smaller the RMS of the distribution, the faster the absorption of these effects. Additionally, a small statistical error in the calibration constant ensures a fast correction of the timing variation, thus improving the accuracy t 0 .

Later sections discussed such a study. The figure-of-merits are:

• The t hits distribution before and after calibration for different windows of events,

• RMS (t[smeared+calibrated] -t[not smeared]),

• Calibrated t hits Vs Module Radius. 8 LGAD module contains 30 ⇥ 15 pads, where each module contains to two ASICs (with 225 channels each.) 9 Total number of HGTD modules in the simulation see Section 4.2 for more details. The calibration constants are computed as the arithmetic mean of t hits distribution over different time intervals and different special granularities approach 10 . For a small interval of events (10 events), calibration constants are calculated perfectly, and the injected time variation, which affects the time measurement, is visible as shown in Figure 4.19. Alternatively, when the interval of events becomes larger than the time variation injected (1 kHz), the shape becomes less apparent, as shown in Figure 4.20.

A shorter interval of events (or smaller calibration window sizes) are generally expected to calibrate the t hits distribution more effectively when shorter-term variations affect the measurement of time hits. Additionally, a longer calibration window can collect more statistics and ultimately determine the calibration constant more precisely, which will result in a more accurate hit time correction and better precision for t 0 .

Statistical precision on calibration constants

The statistical precision of calibration constants is calculated based on the formula s / p N hits where s represents the RMS of t hits distribution (TimeFull 1 kHz) and N hits is the number of hits used to compute The statistical error varies with different hit multiplicities in each layer of the HGTD (see Figure 4.8). For r > 320 mm, an overlap of 20% and for r < 320 mm, an overlap of 80% between modules where sensors are very close to the beam axis, thus are more likely to be exposed to minimum bias events, which allow for more hits. Therefore, this region has higher statistical precision. A direct impact of the N hits can be seen in the overall performance of the various special granularities. The calibration per module approach has the highest statistical error value compared to all other approaches.

The calibration constants are calculated at regular interval. In practice, the precision of t 0 calibration strongly depends on this interval. So knowing the variation of statistical error with respect to the event interval is helpful to determine what the best event interval is for computing and calibrating the t hits . Such correlation is shown in Figure 4.22, where the x-axis represents the variation period assuming a trigger rate of 1 MHz.

Better precision can be achieved through a longer averaging time, e.g. variation period greater than 10 µs, since it can collect more hits. Smaller calibration window sizes have a high error, especially for calibration per module. The use of shorter intervals, however, enables a fast correction of small variations in time hits, thereby reducing t 0 jitter and improving precision.

Timing correction

The t hits distribution before and after applying timing correcting procedure is shown in The statistical precision of calibration constants as a function of variation period, and different choices of special granularities using t t sample of 1 434 events with a pileup of hµi = 200. delays, and dynamic shifts of the clock phase during data-taking. Different special granularities 12 yield different t 0 values. Calibration per module performs better than other calibration methods, displaying a value of t 0 of (0 ± 323 ps). Figure 4.24 shows the average hits time variation as function of event index. The t hits distribution is calibrated using the calibration per module. In the y-axis, the (Time Full 1 kHz -Time ASIC) distribution represents all sources of time variation discussed above. For a variation period of fewer than 100 µs, the calibration procedure will always ameliorate the t 0 precision by absorbing the contribution of injected jitter and allow for a more precise time resolution.

In comparison to other calibration methods, the calibration per module corrects short-term variations faster, which are deemed to be uncorrelated per module across channels, but this is only relevant when using a smaller calibration window as illustrated in Figure 4.24. Figures showing the effects of the calibration window on the time calibration procedure, for inner and outer rings, can be found in Appendix B.1.

The RMS of calibrated time

The average RMS variation, t[TimeFull 1 kHz calibrated] -t[TimeASIC], as function of variation period for various special granularities is shown in Figure 4.25. Time ASIC is t hits taken from simulation after applying the phase shifter. Time Full 1 kHz is t hits with the contribution from the sensor and electronics jitters 13 . The RMS distribution represents the time dispersion of the hits in the entire HGTD and encodes information between various clock jitters. The smaller the RMS of the distribution, the faster the absorption of these effects. For that reason, using the module calibration approach and a small calibration window size reduces t 0 jitter.

Correction of time-of-flight effects

The time-of-flight (ToF) represents the path length of the particle's track between the origin of the track and the position of a given hit. Together with other sources of jitter, such as electronic jitters discussed above, 12 See Section 4.5.2 for more details. 13 Intrinsic limitation: To properly derive precision and bias of the calibration, one would need to study t[smeared+calibrated]t[not smeared] or in other words, t[obtained] -t[wanted]. The wanted time is not accessible in the used n-tuples when considering all smearing effects together, but it is when considering only the smearing effects at 1 kHz added on top of all other effects. The average time hits before and after timing correction procedure using short sample size w.r.t modules radius is shown in Figure 4.26. Different special granularities are used and the timing scenario chosen is TimeFull 1 kHz. The t hits are calibrated every 10 events following the Eq. 4.4. Before timing calibration procedure, the effect of ToF is visible due to its sensitivity to the path length of the particle's track.

As outlined in Section 4.4.4, the phase shifter removes some residual effects due to ToF, which could be viewed as a pre-calibration method. This distribution is shown in black (before calibration The average hits time variation as function of event index. The calibration per module is used. Time ASIC is t hits taken from simulation after applying the phase shifter. Time Full 1 kHz is t hits with the contribution from the sensor and electronics jitters and the collision time spread with additional sources of time jitter from the ASIC, FELIX, flex cable, lpGBT, and ATLAS collision time drift also, with an additional sinusoidally varying 100 ps offset of variable period. The Figure shows the variation of all injected jitters before (in black) and after the calibration per module using different event interval as function of event index. Calibrating t hits directly without splitting the event interval is called independent calibration. For a variation period of less than 100 µs, the calibration procedure will always improve the t 0 precision. For instance, when t hits is calibrated using an interval of a single event (in green), the hit time variation drops to approximately 0, with a small value when t hits is calibrated using an interval of 10 events (in blue). Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200. The average time hits before (in black) and after timing correction procedure using short sample size w.r.t modules radius. The calibration per module absorb all ToF contribution to the timing resolution. Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200.

Future of the t 0 calibration

Preliminary study of t 0 calibration shows that by using calibration per module, the t hits distribution is calibrated with good accuracy. The calibration per module best absorb the time variation of 1 kHz at regular interval of small event. Additionally, calibration per module is capable of correcting for geometrical effects, particularly time-of-flight effects. A lot of efforts have been done on the t 0 calibration but a more detailed review of the performance of the calibration procedure is needed. For instance:

• For the estimation of accuracy, some truth MC studies are needed, namely how to identify which particle produced the hit, and what is its real ToF. This is needed in order to know the targeted time that the calibration is suppose to get back when looking at the distribution of t[obtained] t[truth].

Currently truth informations are not available.

• The phase shifter (presented in Section 4.4.4) has been emulated and optimized by hand in these studies, and this should be revisited once a realistic procedure takes place.

• Design of a better figure of merit to assess quality of the calibration (with both precision and bias).

• Assess the accuracy of t 0 calibration procedure in the presence of systematic uncertainties.

• Provide an overview of typical online calibration workflows.

The first results of t 0 calibration allow for a good time resolution of 30 ps per pad, making the ATLAS detector six times more powerful to mitigate the effects of pileup contamination in the forward region at HL-LHC. As a results, the HGTD can be utilised to potentially enhance the precision of Standard Model measurements and the sensitivity to discover new physics. As an example, adding pileup jets can increase the number of background events passing a selection which reduces the effectiveness of kinematic variables to separate signals from backgrounds.

In light of this, the HL-LHC is particularly interesting for precise measurements of the top quark. As the heaviest particle in the SM, the top quark plays an important role for the electroweak symmetry breaking and becomes a sensitive probe for physics beyond the SM. Furthermore, the top quark is the only quark that decays before hadronisation, making top quark physics a unique playground to study its spin properties.

II 5 Spin Correlation with the full Run 2 dataset This chapter is organised as follows: an overview of the full analysis is given in 5.1. The data sample and Monte Carlo (MC) simulations used in this analysis are discussed in Section 5.2. Section 5.3 describes the reconstruction of the physics objects and event selection as well as the kinematic reconstruction of the t t system. Data-MC comparison is presented in Section 5.4. The unfolding and results are summarized in Section 5.5. The EFT interpretation of the results is discussed in Section 5.6. Section 5.7 discusses, for the first time, the prospect of studying the entanglement and BIs in top pair production with the ATLAS detector at the LHC.

Analysis strategy

The spin state of the top and anti-top quark, encoded in the spin density matrix, are transferred to the decay products and can be measured directly via their angular distributions. Several leptonic angular correlation observables are presented in this thesis, which probes all the independent coefficient functions of the top-spin dependent parts of the t t production spin density matrices. As discussed in Section Spin correlation, multiple methods are available for measuring spin angular correlation in t t events. In the helicity basis, the analysis relies on picking one particle from each decay and looking for correlations in the polar angles represented in their parent tops' rest frames. As charged leptons retain more information about the spin state of the top quarks e.g. higher spin analysing power (a), and as they can be precisely reconstructed, the measurement in this thesis is performed in the dileptonic final state of t t events. The charged leptons considered in this analysis are electrons or muons, either originating directly from W and Z decays, or indirectly through intermediate t when decaying leptonically1 .

Every experimental apparatus has limited acceptance and resolution, which affect the distributions of measured angular observables, leading to differences between detector(reconstructed)-level and particle(or parton)-level spectra. Using an unfolding technique, these effects can be removed from the distributions and the particle-level spectra can be recovered. In this way, the results can be directly compared with other experiment results and with particle-level theoretical predictions, without the need for simulation of the detector.

The LHC data analysed so far have not revealed any direct signals of BSM particles, meaning the phenomena might only be directly observed at an energy scale larger than that probed at the LHC. However, BSM physics could still indirectly manifest itself in new vertices and modified couplings. To this end, we analyse and compute the effects of new interactions on these observables in terms of a gauge-invariant effective Lagrangian relevant for t t production up to mass dimension six, one of the major contributions of the author.

Moreover, we present, for the first time, the expected sensitivity of quantum entanglement and violation of BIs at LHC, both of which are part of the author's main contributions.

Data and MC samples 5.2.1 Data

The analysis is performed using the full Run 2 proton-proton collision data sample at p s = 13 TeV recorded by the ATLAS detector. The data sample corresponds to an integrated luminosity of 139 ± 2.4 fb

Background modelling

The modelling of the background processes is summarized in this section. The analysis uses MC simulations, in particular to estimate the sample composition and to transform the measurement to particle-level. In order to correct for the mis-match between individual data taking periods and the simulated pile-up contribution in the MC, an event-based reweighting was performed, taking into account the expected and observed distribution of the average number of interactions per bunch crossing. This pile-up reweighting is performed for all MC samples using the officially recommended PileupReweightingTool. Each MC sample is normalised to the SM prediction for the sample process, and then is scaled to match the integrated luminosity of the data.

All MC samples are also corrected with scale factors for leptons, b-jets and pileup, which are all measured in data. A summary of all MC samples used in this analysis is given in Appendix A.

Signal samples

The

Single top samples

The single-top process Wt is considered as a background, however at NLO Wt single-top production overlaps with t t production. Two possibilities exist to account for this: the diagram-removal scheme and the diagramsubtraction scheme. The diagram-removal scheme removes the double-resonant diagrams from the signal definition, whereas the diagram-subtraction scheme uses a subtraction term that cancels the t t contribution.

The diagram removal scheme was employed to handle the interference with t t production in this thesis.

Single-top Wt associated production is modelled using the POWHEG BOX (version 2) generator at NLO in QCD in the five flavour scheme with the NNPDF3.0NLO PDF set. Single-top t-channel production can contribute to the estimation of fake background events and is modelled using the POWHEG BOX (version

2) generator at NLO in QCD in the four flavour scheme with the NNPDF3.0NLONF4 PDF set. Single-top s-channel production is modelled using the POWHEG BOXZ (version 2) generator at NLO in QCD in the five flavour scheme with the NNPDF3.0NLO PDF set. For all single-top processes, the events are interfaced with PYTHIA (version 8.230) using the A14 tune and the NNPDF23LO PDF set.

Diboson samples and V+jets

Samples of diboson final states (VV) are simulated with the SHERPA (version 2.2.1) or SHERPA (version 2.2.2) generator depending on the process, Fully leptonic final states and semileptonic final states, where one boson decays leptonically and the other hadronically, are generated using matrix elements at NLO accuracy in QCD for up to one additional parton and at LO accuracy for up to three additional parton emissions.

Samples for the loop-induced processes gg ! VV are generated using LO-accurate matrix elements for up to one additional parton emission for both cases of fully leptonic and semileptonic final states. The matrix element calculations are matched and merged with the Sherpa parton shower based on Catani-Seymour dipole using the MEPS@NLO [START_REF] Höche | QCD matrix elements + parton showers. The NLO case[END_REF] prescription. The NNPDF3.0NNLO set of PDFs is used, along with the dedicated set of tuned parton-shower parameters developed by the Sherpa authors.

Massive-vector-boson production plus QCD jets (V +jets, with V = W or Z) is simulated with the SHERPA (version 2.2.1) generator using NLO-accurate matrix elements for up to two jets, and LO-accurate matrix elements for up to four jets. They are matched with the Sherpa parton shower using the MEPS@NLO prescription using the set of tuned parameters developed by the Sherpa authors. The NNPDF3.0NNLO set of PDFs is used and the samples are normalised to a NNLO prediction.

t tX samples

Chapter 5. Spin Correlation with the full Run 2 dataset

The production of all most t tX events is modelled using the MADGRAPH5_AMC@NLO (version 2.3.3) generator at NLO with the NNPDF3.0NLO PDF set. The events are interfaced with PYTHIA (version 8.210) using the A14 tune and the NNPDF2.3LO PDF set. However, the production of t tW events is modelled using the SHERPA (version 2.2.1) generator with the NNPDF3.0NLO PDF set. The ME was calculated for up to one additional parton at NLO and up to two partons at LO and merged with the Sherpa parton shower using the MEPS@NLO prescription and a merging scale of 30 GeV. to be from single-top production in eµ channel and V + jets production in ee and µµ channels.

ATLAS

Event selection and reconstruction

In this analysis, the event selection is optimised to increase the fraction of t t events with a dileptonic final state. Events required to pass a set of selection requirements based on the physics objects, discussed in Section 3.2.2, and trigger selections. Selection requirements are designed to match the kinematic features of the signal process, efficiently selecting t t events and rejecting events originating from background sources.

In the following, the object and event selection criteria are summarized.

Event and object selection

Event Selection

Event-level selection cuts are used to suppress events not caused by dileptonic t t decay. In the following, the list of event selection that are used in this analysis are described along with the background processes that they are designed to suppress.

• The dilepton analysis requires events selected online by a high p T single lepton trigger (e or µ). At least one of the leptons must be associated with the high-level trigger object, to ensure that the event was triggered by the lepton. The trigger that is used depends upon the data period as described in Table 5.1. For the e + e and µ + µ channel the corresponding lepton trigger must have fired. For the e ± µ ± channel either an electron or a muon trigger must have fired.

Event selection and reconstruction 117

• In all three channels exactly two isolated oppositely-charged leptons (e + e , µ + µ and e ± µ ± ) are required. This suppresses backgrounds from any SM process that does not include two prompt leptons, for example single top production in the s or t channel. In the e ± µ ± channel, the requirement of exactly one muon and one electron has the additional effect of removing Z ! e + e and Z ! µ + µ events. At least one lepton in the event must be matched to the trigger lepton that is used to select the event.

• At least one b-jet is required with 85% efficiency working point. This cut removes most non-t t SM physics signals from the selection.

• The invariant mass of same lepton flavour (m ``) is required to be larger than 15 GeV. A reason for this cut is that low energy Z/g ⇤ ! ``events and meson production such as J/Y, can enter the selection by decaying into Leptons of opposite signs. Invariant mass for these processes is typically low, so they are rejected by this cut.

• Veto on events with m ``close to the Z-mass (e + e , µ + µ ) e.g. 81 101 GeV. This cut reduces a large amount of the Drell-Yan ! e + e or µ + µ background.

• Only in e + e and µ + µ channel, a cut on the E miss T is applied, requiring it to be larger than 30 GeV. This reduces the Drell-Yan contribution, since no neutrinos from hard scatter would result in messing energy. This cut also suppresses diboson events in which there are few or no neutrinos in their final state decay channels.

Object selection

Electrons

Electrons are reconstructed using a likelihood-based definition [START_REF] Aad | Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF]. Signal electrons are required to have minimum of p T of 15 GeV and pass the TightLH identification and to pass the PLVTight isolation requirements. The trigger-matched electrons are required to have a p T at least one GeV higher than the threshold of the triggers described in Table 5.1. Electrons that fall into the LAr crack region (1.37 < |n| < 1.52) are vetoed to reduce the contribution from fakes.

Muon

Muons are required to have 0.1 < |n| < 2.7 and p T > 15 GeV. The threshold of 15 GeV is chosen as this is the lower limit for muon reconstruction that can be derived from Z tag and probe events. Muons are selected using the Medium working point since it has smaller scale factor uncertainties than the Tight working point. Muons which are trigger matched are required to have a p T at least one GeV above the trigger threshold listed in Table 5.1.

Jets

Jets are reconstructed using the Anti-kt Pflow algorithm with a distance parameter R = 0.4. Baseline jets are required to have |n| < 2.5 and p T > 25 GeV. A JetVertexTagger (JVT) is applied to jets with p T < 120 GeV and |n| < 2.4 to suppress jets originating from pileup interactions. This jet collection and selection is the most-precise collection available in ATLAS, with per-jet uncertainties associated with JVT lower than 5% and, in central high-p T regions, below 1%. Jets that are positively identified by the DL1r algorithm, using a 85% b-tagging efficiency are considered b-jets.

Overlap removal

To prevent signals in the detector from being used to reconstruct more than one object and to ensure that signals from different objects do not interfere with each other, the following "overlap removal" procedure is used:

• Electrons that share an ID track with a muon are removed.

• A single jet is removed if it is close to an electron (within a DR < 0.2). If there is more than one jet within DR < 0.2), only the closest in DR) space is removed.

• Electrons within DR < 0.4 of a jet are removed.

• Muons within DR < 0.4 of any jet are removed. Where (p miss T ) Electrons, Muons, Jets refer to the vectorial sum of the calibrated electrons/muons/jets and pmiss T Soft is the vectorial sum built from all tracks not associated to any reconstructed object. The overlap between different objects is removed before calculating the pmiss T in order to reduce double counting.

Data-driven background estimation

The non-top dileptonic events with additional heavy flavour (HF) jets (principally from DY events) and non-prompt leptons (fake leptons) are not accurately model with simulation. A data-driven scale factor is used to correct the overall normalisation of various sources of heavy flavour and fake-leptons backgrounds.

The SF is derived by comparing MC to data in dedicated source control regions. The following describes the heavy flavour and fake-leptons modelling backgrounds.

Heavy Flavour background modelling

The background contribution from Z ! e + e or Z ! µ + µ in the e + e and µ + µ channels is modelled by MC generator. With dilepton selection requiring one (or more) b-tagged jets, two additional jets must be present, making the the theoretical uncertainties related to Z ! e + e and Z ! µ + µ large and since the modelling of E miss T in the Z +jets sample can be challenging, the resulting normalization after selections is not well described by MC. In this thesis, the SF is derived by comparing the data and MC in DY dominant regions. For this Control Region (CR), events are selected with the same dilepton selection criteria described in Section 5.3, but inverting the Z-mass window cut. The scale factor is defined as follows:

SF = Data MC except DY MC DY (5.2)
The scale factors are obtained by fitting the distribution corresponding to Equation in 5.

Fake lepton modelling

The background consisting of events with fake leptons can arise from many SM processes and would demand a large amount of MC statistics and suffer from a limited ability of proper MC modelling. These fake leptons can arise from heavy-flavour hadron decays, photon conversions, jet misidentification or light-meson decays. The main SM processes that contributed to this background are W production in association with jets, single top production in the s or t channel, and single lepton t t decays. Data with such events are poorly reproduced by the MC samples, which is why a scale factor is applied to correct their normalization. As with the standard e + e , µ + µ , and e ± µ ± event selections, the requirement for opposite sign electric charge leptons is inverted, so that both leptons must have the same sign in order to create a region dominated by fake leptons. This selection includes a small number of signal events, typically caused by charge-flipped electrons or conversion electrons, and the difference between the normalised MC and data is used as a scale factor to correct the MC in the signal region. As compared to the DY control region, the fake control region has relatively low statistical power, and so the scale factor is simply an inclusive one and is not parametrised as a function of any observable. The scale factors obtained by comparing data and MC in a fakes dominated same-sign region in a one b-tagged jet with 85% working point of the DL1r algorithm is presented in Table 5.3

e + e channel µ + µ channel e ± µ ± channel 1.16 ± 0.04 1.14 ± 0.02 1.12 ± 0.01 Table 5.3: The scale factors obtained by comparing data and MC in a fakes dominated same-sign region in a one b-tagged jet with 85% working point of the DL1r algorithm. The quoted uncertainties are statistical only.

Kinematic reconstruction of the t t system

The dileptonic t t final state consists of two charged leptons, two neutrinos and at least two jets originating from the top quark decay. As the neutrinos cannot be directly observed in the detector, the kinematics of the t t system, which is necessary to construct the observables, can not be simply reconstructed from the measured information. The two neutrinos can only be measured via one measurement of E miss T leading to an under-constrained system 2 . One therefore has to solve a set of under-constrained equations to retrieve the component of the momentum of each individual neutrino. Further complications arise from the lack of jet charge information, without which jets can't be differentiated between top and anti-top.

There are six unknowns (assuming neutrinos are massless): p n x , p n y , p n z , p nx , p ny , p nz . The kinematics of a dilepton decay can be expressed by six equations involving fixed values of W + (W ) and t(t) masses:

(p l + + p v ) 2 = m 2 W + , (p l + p v) 2 = m 2 W (5.3) (p l + + p v + p b ) 2 = m 2 t , (p l + p v + p b) 2 = m 2 t
(5.4)

E miss x = p n x + p nx , E miss y = p n y + p ny (5.5)
Equations 5.3 can be solved analytically for n and n momenta. There are three approaches to solving these equations, of which two are used in this thesis. Two of the methods are analytical: the Sonnenschein Method (SN) [START_REF] Sonnenschein | Analytical solution of tt dilepton equations[END_REF], named for its author and Ellipse Method (EM) [START_REF] Betchart | Analytic solutions for neutrino momenta in decay of top quarks[END_REF]. The basic idea of both methods is to reconstruct n and n momenta components followed by t and t momenta reconstruction. The third method is rather numerical and is called Neutrino Weighting (NW) [START_REF] Abbott | Measurement of the Top Quark Mass Using Dilepton Events[END_REF] which solves the system for different range of possible values of h n and h n . Each solution is then given a weight based on how well the reconstructed neutrinos agree with the measured E miss T . Depending on how the neutrino pseudo-rapidities are sampled, NW can be one of the most precise methods. However, this often comes at a cost of significant CPU time compared to methods such as EM and SN. The NW method was previously used for t t kinematics reconstruction in the t t Spin Density Matrix [START_REF] Aaboud | Measurements of top quark spin observables in top pair events using dilepton final states in 8 TeV pp collisions with the ATLAS detector[END_REF] and spin correlations [START_REF] Aaboud | Measurements of top-quark pair spin correlations in the eµ channel at p s = 13 TeV using pp collisions in the ATLAS detector[END_REF] measurements. A comparison of reconstruction efficiencies of the different top reconstruction methods is illustrated in Figure 5.3. In this figure, reconstruction efficiency is calculated as the fraction of events for which there is at least one solution at reco-level (including events that are either missing or have mis-identified a physics object). The efficiency of matching between the reconstructed top quark and the truth quark within DR < 0.4 is also shown.

In this analysis, the EM was found to be the best choice given its efficiency for reconstruction and the computation time, followed by the NW method, and the SN method. It is possible that sometimes the EM fails to find a solution, in which case the NW method is used to reconstruct. If this method also fails, a rudimentary pairing of each lepton with it's closest b-tagged jet (or closest b-tagged jet and highest p T light jet in the case of only 1 b-tagged jet in the event) is used. Approximately 85% of events are successfully reconstructed by the EM, 5% by NW, and the remaining 10% using the rudimentary pairing. Due to their similarities, when the EM fails the SN also fails in 99% of cases. Therefore, the SN method is not considered at all in the reconstruction procedure. Due to this, the EM was selected as the primary reconstruction method, with NW tried on events that fail 3 . The reconstruction of the t t system efficiency is an important point to consider in MC simulation and is often a limiting factor in the precision of the simulation.

2 system of polynomial equations is considered under-constrained if there are fewer equations than unknowns 3 A more detailed discussion of each method along with a comparison can be found in Appendix B of ref [START_REF] Madar | Evidence of quantum entanglement in events with top-quark pairs in ATLAS sq s 13 TeV data[END_REF]. combining ee, eµ and µµ are presented in Table 5.4. The combined statistical uncertainties is shown. The expected background yields are split into different background processes as discussed in Section ??. Overall, the number of predicted events is lower than the number of observed events, although both agreed within the statistical uncertainties in all channels. The ee channel is statistically limited compared to the µµ and eµ channels, and the eµ channel dominates in statistical sensitivity. In the ee and µµ channels, cuts are made to suppress Drell-Yan background that are not necessary for the em channel, hence the lower statistics. The ee channel has much lower statistics than µµ because electron isolation cuts are tighter than muon isolation cuts to suppress fake electrons. Data and expected events for each process show good agreement.

Angular observables distributions

In Figures 5.8, 5.9, 5.10, 5.11, 5.12, 5.13 distributions of all the reconstructed angular observables (defined in Section 2.3) are shown. Given the statistical uncertainties, the data and the sum of the expected signal and background contributions are in reasonable agreement. The e + e channel is statistically limited compared to the µ + µ and e ± µ ± channels, and the e ± µ ± channel dominates in statistical sensitivity in all variables. + distributions between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The first and last bin of the distribution corresponds to the underflow and overflow respectively. + distributions between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The first and last bin of the distribution corresponds to the underflow and overflow respectively. + distributions between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The first and last bin of the distribution corresponds to the underflow and overflow respectively. Data / SM (i) Figure 5.12: Comparison of the t t spin correlation observables between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The first and last bin of the distribution corresponds to the underflow and overflow respectively.

Unfolding

Any observable measured with the ATLAS detector is blurred by the imperfections of the measurement apparatus, such as limited resolution of the detector response, detector acceptance, and possible event and object selections that are applied to the data. This distortion vary with the pseudorapidity, which corresponds to changes in the active detector region, transforming the underlying particle-level (true) distributions into a set of detector-level (reconstructed) distributions. The method for correcting data from the detector-level (reconstructed level) to the particle level (true) is known as unfolding. Generally, unfolding is the process of correcting detector effects from a measured quantity by transforming the measurements from a detector-level phase space back into a particle-level phase space. By unfolding the data, one can directly compare the data from different experiments and predictions from theory, rather than having to, for instance, run detector simulation to process the prediction from the theory.

In mathematic terms, the measured quantity at detector level is equal to the true quantity with a function that represents the overall detector and selection acceptance and the detector response, it is commonly expressed as following:

n  i R i j • T i + B j = D j (5.6)
where D is a vector representing the expected number of events at the detector level (measured). T represents the expected number of events at truth level, and B represents the expected number of background events. Each vector consist of n components which correspond to the bins in the distribution. The transition or response matrix R contains the individual transition probabilities and hence the bin-to-bin migrations between the observed elements of the distribution and the corresponding true values. The response matrix has the simple interpretation as a conditional probability:

R i j = P( reconstructed in bin j | truth in bin i) (5.7)

Unfolding input

In this thesis, the unfolding techniques used are based on determining first a series of unfolding inputs that represent the expected response of the detector to the signal processes of interest. The inputs below were derived using MC samples as described in Section 5.2.

• Response matrix: The response matrix is a two-dimensional matrix, R i j , whose entries correspond to the probability that an event in particle-level bin i will be reconstructed in the detector-level bin j.

• Reconstruction efficiency: The reconstruction efficiency 4 , e eff , is a measurement of the fraction of events in MC which pass both detector-and particle-level selections (N reco^par ) to all those passing only the detector-level selection (N reco ):

e eff = N reco^par N reco (5.8)
The efficiency factor corrects for events which pass the detector-level selection but are not present at the particle-level.

• The acceptance correction: The fiducial fraction, f acc , is a measurement of the fraction of events in MC which pass both detector-and particle-level selections (N reco^par ) over the number of events that 4 sometimes called the "non-fiducial acceptance correction" pass only the particle-level selection (N part )

f acc = N reco^par N part (5.9)
The factor f acc corrects for events that are at particle-level but are not reconstructed in the detector.

When real data is involved, we don't have any information on distributions at the truth level, so in this case the idea is to calculate the inverse of the response matrix using the MC simulation in order to estimate the true physical distributions (T i ). From the equation 5.6, the unfolding procedure is then expressed as follows:

N i unfold = 1 f i acc  j R 1 i j e j eff ⇣ N j data N j bkg ⌘ (5.10)
where N i unfold is the number of unfolded events in bin i, N j data and N j bkg are the numbers of data and background reconstructed events in bin j. As defined above, f acc , e eff and R i j are the unfolding corrections and the response matrix. The obtained inverse matrix R 1 i j is applied to the distribution measured from N j data , and the respective unfolded distribution, N i unfold , is obtained as estimator for the true distribution based on the given measurement. The unfolding principle is illustrated in Figure 5.14 by showing an example distribution for a variable x at various stages of the measurement process. For unfolding, the most straightforward approach is to invert the detector response matrix, R 1 i j . Several procedures have been developed to do this. In the current analysis, the Profile Likelihood Unfolding (PLU) is used and implemented based on the TREXFITTER package [START_REF] Pinamonti | TRExFitter package[END_REF], which utilizes the HISTFACTORY tool [START_REF] Cranmer | HistFactory: A tool for creating statistical models for use with RooFit and RooStats[END_REF] within the ROOSTATS framework [START_REF] Moneta | The RooStats Project[END_REF]. The description of the statistical model is briefly explained and evaluated in the following Section. Detailed information about the various aspects of statistical analysis techniques that are relevant to the measurements discussed in this thesis is provided in the Appendix D.

Profile likelihood unfolding

The statistical analysis of the data is based on a binned likelihood function, which is constructed from a product of Poisson probability terms for all bins N of the distribution. As such, the likelihood used in the Chapter 5. Spin Correlation with the full Run 2 dataset unfolding takes the following schematic form:

L = ' r Poisson ✓ N data r | Â t µ t • N truth t • R tr + b r ◆ (5.11)
where the indices r and t denotes the bins of the observed and unfolded distributions, respectively. N data r and N truth t represent the number of event in the observed bin r and the number of event in the unfolded bin t, respectively. b r is the expected number of background events in bin r. R tr is the response matrix and gives the probability of an event generated in bin t to be reconstructed in bin r. The detector and selection effects are encompassed in R tr . By taking the response matrix connecting the three different analysis channels, ee, µµ and eµ, to the same truth spectrum, the channels are combined within the unfolding method. Instead of measuring directly the content of the unfolded bins, the signal strength µ t is chosen as parameter of interest (PoI) and represents a free parameter (an unconstrained parameter). The best estimate for µ t is obtained by maximizing the likelihood or equivalently minimizing the negative logarithm of the likelihood (negative log-likelihood).

Systematic uncertainties

The limitations of the simulation and of the detector response lead to uncertainties on the response matrix and the background expectations. These are accounted for in the likelihood by means of Nuisance Parameters (NP). For instance, a systematic effect, causing relative variations d tr s in the bin (t, r) of the response matrix and d r b in the bin r of the background expectation, is controlled by a NP q and implemented in the likelihood as:

L = ' r Poisson ✓ N data r | Â t µ t • N truth t • M tr • 1 + d tr s q + b r • (1 + d r b q ) ◆ • Gauss (0 | q , 1)
(5.12)

The variation d tr s is obtained by comparing the nominal reconstructed-level distribution, obtained by folding the nominal truth-level distribution with the nominal response matrix, and the systematically varied reconstructed-level distribution, obtained by folding the nominal truth-level distribution with the systematically varied response matrix. This is true for all systematic uncertainties, including modelling uncertainties arising from alternative MC simulations (see Section 5.5.2).

Re-parametrisation of the PoIs

In order to measure directly the spin parameters and their CIs from the fit, the PoIs of the likelihood are re-parametrised. Every spin parameter is obtained by multiplying the mean of the unfolded distribution of the corresponding observable by an integer factor. The mean of the unfolded distribution is measured in the fit and subsequently multiplied by the appropriate factor. It is defined:

< observable >=  N bin t=1 •bin_center t • N truth t • µ t  N bin t=1 N truth t • µ t (5.13)
One of the PoIs can be re-parametrised to become a function of the mean and of the other PoIs. In practice, the PoI controlling the first unfolded bin is chosen. It is modified as:

µ 1 = Â N bin t=2 ( bin_center t < observable >) • N truth t • µ t (< observable > bin_center 1 ) • N truth 1 (5.14)

Binning optimisation

The binning of spin correlation observables, in which a likelihood fit is performed, is of special importance.

The binning optimisation is performed to minimise the effect of statistical fluctuations in the data as well as to account for the experimental resolution. For instance, a high number of bins can increase the overall signal sensitivity, while it also decreases the MC statistics per bin.

There is no limit to how many bins can be used. Nevertheless, for a relatively stable unfolding, it is recommended that at least 50% of events at reco-level populate the correct truth bin i.e. the diagonal entries in the normalised migration matrix are at least 0.5. The binning optimisation leads to a four-bin configuration for the different correlation observables. The response matrix normalised per true bin are shown for spin correlation cosq k + cosq k in Figure 5.15 and one can observe that the diagonal entries are close to 0.5.

Figure 5.15: Response matrix of the cosq k + cosq k observable.

Systematic uncertainties

The measurement of the spin observables is affected in various ways by systematic uncertainties. Systematic uncertainties can adversely affect normalization and distributions of kinematic variables. Generally, a variable subject to systematic uncertainty is increased or decreased by one standard deviation to account for those effects. This is the up and down systematic uncertainty. Some of these variable variate in one direction and thus are referred to as one-sided systematic uncertainties. In order estimate their impact, the distribution of observables is calculated for the variation in one direction and then symmetrized around the nominal distribution [START_REF] Emin | Search for flavor-changing neutral-current interactions between the top quark and the Higgs boson in dileptonic same-charge final states with the ATLAS detector[END_REF]. The following describes three different types of systematic uncertainties: signal modelling systematics, detector uncertainties affecting the modelling of the physical objects and background modelling Chapter 5. Spin Correlation with the full Run 2 dataset systematics. The total systematic uncertainty is taken to be the sum of all of the individual sources added in quadrature. These systematic uncertainties are estimated using MC samples described in Section 5.1.

Signal modelling systematics

The systematic uncertainty resulting from t t modelling is the dominant source of systematic uncertainty for all correlation observables. Signal modelling systematics are considered in two forms; alternative generators and weights. For the alternative generator uncertainties, the difference between the measured observables values is taken as the systematic uncertainty. For the systematic uncertainties that use weights, the difference between the measured observables values for nominal and the sample shifted by the weights is taken as the uncertainty. These systematics are enumerated as follows:

• Matrix Element matching uncertainty: Samples generated with MADGRAPH5_AMC@NLO and POWHEG BOX V2, using NNPDF3.0NLO [START_REF] Ball | Parton distributions for the LHC run II[END_REF] parton distribution function (PDF) and interfaced with PYTHIA8 (version 8.230), were compared to assess the uncertainty in the matching of NLO matrix elements to the parton shower. Variation found in this comparison is taken as uncertainty.

• Parton shower uncertainty: a comparison between MADGRAPH5_AMC@NLO + HERWIG (version 7.1.3) and MADGRAPH5_AMC@NLO + HERWIG (version 7.13) is used to estimate the effect of the choice of parton shower algorithm on the measurement. Variation found in this comparison is taken as uncertainty.

• Scale µ r and Scale µ f : An estimation of the effect of renormalisation and factorisation scales increasing and decreasing by a factor of 2 in the nominal Powheg sample is determined using generator weights. Each of these scales is evaluated separately. The relevant generator weights are named (mu_F = 0.5, mu_R = 1.0), (mu_F = 2.0, mu_R = 1.0), (mu_F = 1.0, mu_R = 0.5) and (mu_F = 1.0, mu_R = 2.0.)

• ISR/FSR: The uncertainty due to initial-state radiation / final-state radiation (ISR/FSR) is evaluated using nominal Powheg sample within which the parameters controlling ISR and FSR are varied within a range consistent with data.

• PDF: The uncertainty due to the choice of PDF set is evaluated using the PDF4LHC [START_REF] Botje | The PDF4LHC Working Group Interim Recommendations[END_REF] recommendations, utilising 30 eigenvector shifts derived from fits to multiple PDF sets. The nominal Powheg sample contains these eigenvector variations as generator weights.

Object systematics

The experimental uncertainties are specific to each detector and are caused by detector responses, particle identifications, etc., which may affect the measurement of the spin correlation observables.

Luminosity

For processes estimated exclusively from simulations, the luminosity uncertainty is estimated by scaling the number of signal and background events simultaneously by the luminosity uncertainty [START_REF]Luminosity determination in pp collisions at p s = 13 TeV using the ATLAS detector at the LHC[END_REF]. The luminosity is not a dominant source of systematic uncertainty for any correlation observables.

Pileup

To account for possible deviations from data due to the estimation of the pile-up distribution, the reweighing to match simulation to data is varied within its uncertainty.

Electrons

The systematic uncertainties considered for electrons mainly arised due to the uncertainty on their reconstruction, identification and isolation and are measured using tag-and-probe techniques on Z and J/Y decays [161, With this form of the uncertainties, the constraints disappear and therefore this version is kept for the analysis.

Muon

The systematic uncertainties considered for muons arise due to the uncertainty on their identification and isolation and are measured using tag-and-probe techniques on Z and J/Y decays [START_REF] Aaboud | Electron reconstruction and identification in the ATLAS experiment using the 2015 and 2016 LHC proton proton collision data[END_REF][START_REF] Aad | Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF]. Each systematic has a separate statistical and systematic component.

Jet

There are three types of systematic uncertainty associated with jets: those related to the Jet Energy Scale (JES), for which the energy scale of b-tagged jets is considered separately (bJES), the Jet Energy Resolution (JER), and the Jet Vertex Tagging (JVT). In this analysis, jet energy scales (JES) and their uncertainty are derived from test-beam data and LHC collision data [START_REF] Aad | Jet energy measurement and its systematic uncertainty in proton ATLAS detector[END_REF], measuring to a precision of between 1 and 4% for jets with the uncertainty decreasing at higher p T . The systematic uncertainties considered for JES and JER are defined using the Category Reduction definition which reduces the number of uncertainties from a full breakdown to a reduced set of eignevector variations 2 . The uncertainty associated with the JVT is evaluated using the JetVertexTagger-Tool by varying the JVT cut up and down.

B-tagging

The DL1r b-tagging algorithm is used [START_REF] Aaboud | Jet energy scale measurements with the ATLAS detector[END_REF] which provides a large number of systematic variations. A total of 17 independent systematic variations; 9 related to B hadrons, 4 related to C hadrons, 4 related to light-jet misidentification, and two extrapolation uncertainties.

Trigger

This analysis uses a global trigger strategy, with associated uncertainties related to the input electron trigger and the input muon trigger (which is split into a systematic and statistical component).

MET

While all object-based uncertainties are fully-correlated with the MET object construction, there are other source of uncertainty specific to the construction of the MET object based on the soft tracks that not associated to leptons or jets. These uncertainties are propagated into the unfolded distribution uncertainty.

Background modelling systematics

As a result of the event selection and top-quark reconstruction, background events are an extremely small source of uncertainty in this measurement. The uncertainties due to the normalization of Wt channel single top, diboson and V+jets (V = W or Z) events are propagated to the uncertainty of the unfolded distribution.

Results

The unfolded distribution of cos q k + cos q k , using the response matrix in 5.15, is depicted in Figure 5.16. The corresponding spin parameter is measured as: The breakdown of uncertainties is given in Table 5.5. It is only mc16a that is shown here; Plots with the full Run-2 data will be added once they are available.

[ ] C(k, k) = 0.332

EFT interpretation

In top physics, the t t cross-section is one of the most accurately measured observable. To examine the sensitivity of the analysis to potential new physics in t t production, the measurements of spin correlation observables are interpreted in term of dimension-six operators within the EFT framework introduced in Section 2.4. In total, 43 Wilson coefficients can be systematically examined in order to identify those that can be constrained by differential cross-section measurements [START_REF] Aguilar Saavedra | Interpreting top-quark LHC measurements in the standardmodel effective field theory[END_REF]. The first step in EFT interpretation is to choose the relevant effective operators. In earlier studies [START_REF] Aad | Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF][START_REF]Top EFT summary plots[END_REF][START_REF] Englert | Giving top quark effective operators a boost[END_REF], the Chromomagnetic Dipole Moment (CMDM) operator, O tG , and the four-quark operator, O 8 tq , have been shown to have strong impact on spin correlation observations. Following is an overview of their impact:

• The modified top quark-gluon couplings described by parameters related to the CMDM operator, O tG , defined in the Equation 2.109, are tightly constrained by t t measurements [START_REF]Top EFT summary plots[END_REF]. Additionally, in the SM, the intrinsic spin of the top quark and its colour charge give it a small CMDM [START_REF] Bernreuther | A set of top quark spin correlation and polarization observables for the LHC: Standard Model predictions and new physics contributions[END_REF][START_REF] Bernreuther | Top quark spin correlations and polarization at the LHC: Standard model predictions and effects of anomalous top chromo moments[END_REF] which would result in a change in the overall rate of t t production and spin structure. The effect of ananomalous CMDM has been investigated in many studies [START_REF] Stange | Yukawa correction to top-quark production at the Fermilab Tevatron[END_REF][START_REF]Search for Anomalous Top Chromomagnetic Dipole Moments from angular distributions in ttbar Dileptonic events at 7 TeV with the CMS detector[END_REF].

• In general, the effect of four-quark coupling on the total cross-section of the top pair production are smaller than cross-section of O tG due to the difference in production rate between gluon and quark initial states. Nevertheless, at high transverse momenta, the PDF of the proton favours some operators among the four-fermion operators described in Section 2.4. In this analysis, the O 8 tq operator defined 5.6 EFT interpretation 139 in the Equation 2.110 is selected which is predicted to have the largest impact on the t t cross-sections at LO and result in additional t t events at high energy.

In the SMEFT framework, the contribution due to O tG and O 8 tq is parametrized by a dimensionless Wilson coefficients, C tG and C 8 tq respectively, divided by the square of the BSM scale, L, assumed to be large compared to the scales typically probed at the LHC. The conventional choice of L = 1 TeV is adopted throughout this Section. Schematic representation of the relevant operators for producing t t at hadron colliders using Feynman diagrams at LO is shown in Figure 5.17. In the following, guidelines for EFT prediction in presented. Section 5.6.2 discuss the search of new physics at NLO in QCD accuracy. Finally, Section 5.6.3 describes the impact of EFT on the unfolding results.

Guidelines for EFT prediction

In this Section, the technical aspects of the EFT prediction is presented. First a description of public SMEFT implementations coupled with their references is presented is Section 5.6.1.1. In Section 5.6.1.2, a few key concepts and current limitations of MC generation in an EFT context are outlined. Finally, a description of the reweighting framework and simulations of dedicated BSM processes in discussed in Section 5.6.1.3

Model implementation

For the presented results in this Section, the DIM6TOP UFO model and SMEFT@NLO UFO model are used to implement SMEFT in MC generators, using the Warsaw basis for the operators. The following are descriptions of both models: SMEFT@NLO [START_REF] Degrande | Automated one-loop computations in the standard model effective field theory[END_REF]: SMEFT@NLO is UFO model implementation of one-loop computations in the standard-model effective field theory at dimension six. Hence, this model allows for both LO and NLO QCD precision. SMEFT@NLO model implements a U(2) q ⇥U(3) d ⇥U(2) u symmetry in the quark sector and [U(1) l ⇥U(1) e ] 3 in the lepton sector. The SMEFT@NLO forbids all fermion masses and Yukawa couplings except that only of the top quark. The model therefore implements the fiveflavour scheme for PDFs. In addition, SMEFT@NLO UFO model does not implement CP violating operators. All EFT operators for all sectors (Top, Higgs, EW ...) are included in SMEFT@NLO,

Parametrization of the EFTs

For the purposes of this analysis, only one dimension-six operator is considered at a time, thereby reducing the EFT parametrization to a single effective coupling constant. This eliminates cross-terms in the matrix element expansion which depend on the product of two different EFT couplings. Given physics process, the matrix element would be given by

M full = M SM + C (6) L 2 M D6 (5.16)
which includes the SM and BSM contributions at LO in the perturbative expansion. The expression for the squared matrix element thus reads:

|M full | 2 = |M SM | 2 + 2 C (6) L 2 ¬(M ⇤ SM M D6 ) + C 2 (6)
L 4 |M D6 | 2 (5.17 the Lagrangian, which will modify the matrix element and, consequently, the observables for a given process.

This allows to schematically parametrise the dependence of a generic observable O (e.g. an inclusive or differential cross section) by a set of Wilson coefficients C as follows 8 :

O = O SM + C (6) L 2 a (6) + C 2 (6)
L 4 b (6) (5.18)
A simple quadratic fit to any observable of interest therefore yields the linear term a (6) , and the quadratic term b [START_REF] Tomonaga | On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields*[END_REF] . Following pure power counting, if dimension-8 operators are included in 5.16 e.g.

M full = M SM + C (6) L 2 M D6 + C 2 (8)
L 4 M D8 , the Equation 5.18 reads:

O = O SM + C (6) L 2 a (6) + 1 L 4 (C 2 (6) b (6) +C 2 (8) a (8) ) (5.19)
where terms arising from interference of dimension-8 operators with the SM are in the same order in 1/L as pure dimension-6 EFT terms. Therefore, when purely considering suppression order, those would also need to be included, which is not done in this analysis. Although the raison is purely technical, it is understood and outlined below.

EFT decompositions in MADGRAPH

Based on an SMEFT@NLO UFO model [START_REF] Degrande | UFO -The Universal FeynRules Output[END_REF] in MADGRAPH, one can generate the t t events at LO using the following command, with p represents the proton and t(t˜) represents top (anti-top) quark:

generate p p~> t t~QED=0 QCD=2 NP=2 (5.20)

where QED and NP9 specify the maximum number of electroweak and EFT couplings, respectively. The choice of NP=x, where x= {1, 2, 3...} is interpreted as NP  x and allows for the EFT insertion at amplitude level. In addition, NP 2 ==x, where x={1, 2, 3...} is defined at amplitude-squared (e.g. cross section) level.

This is used to fine-tune the exact contributions in perturbative (or EFT) expansion. Feynman diagrams for the EFT contributions of 5.20 is shown in 5.18. As such, the NP=2, understood as NP<=1, represents the number of EFT insertions at the amplitude-level: NP=0 for SM and NP=2 for pure EFT 10 . This produces a matrix element containing linear SM, SM-EFT interference, and EFT quadratic terms. To isolate these different terms, we include constraints on the number of squared-amplitude EFT insertions (NP 2 ), as shown in table 5.6.

full = SM + C (6) 2 D6 = SM + C (6) 2 (6) + C 2 (6) 4 (6)
Full |A SM + A EFT | 2 NP 2  4 EFT only |A EFT | 2 NP 2 == 4 SM + interference |A SM | 2 + 2¬[A SM A EFT ] NP 2  4 Interference only 2¬[A SM A EFT ] NP 2 == 2
Table 5.6: The EFT contributions at generation level for a given amplitude level.

In order to preserve t t spin correlations, top quark decays are performed with the MADSPIN package [START_REF] Artoisenet | Automatic spin-entangled decays of heavy resonances in Monte Carlo simulations[END_REF][START_REF] Frixione | Matching NLO QCD computations with parton shower simulations: the POWHEG method[END_REF]. Two major issues arise with MADSPIN:

• MADGRAPH does not support squared coupling orders (NP 2 ) together with MADSPIN (or decay chains). Due to this, it is currently not possible to separate the SM-EFT interference and quadratic EFT terms without sacrificing full spin correlations.

• For the decay of events generated at NLO QCD accuracy, MADSPIN retains spin correlation informations at tree-level (LO) accuracy only. This is problematic when considering global EFT fit since it may include operators that affect both generation and decay levels. It is hence necessary to bear such considerations in mind when preforming global EFT fit. On the other hand, fitting at NLO with one operator at a time is highly desirable and it provides more stringent constraints.

142

Chapter 5. Spin Correlation with the full Run 2 dataset

Reweighting technique

In this analysis, it is required to generate MC sample for a large number of points of EFT parameter space at LO and at NLO QCD accuracy. In order to produce independently samples for many models and benchmark points with full detector simulation, a large amount of computational and storage resources are required as well as it is time consuming. Fortunately, as the stages of a MC simulation are independent 11 , the BSM physics can be implemented at the parton level by changing the weight of each event in a sample to account for alternative theory or EFT benchmark point. This is then propagated through additional weight to all of the simulation chain, and saves one from performing a full simulation. This method is known as the reweighting technique [START_REF] Mattelaer | On the maximal use of Monte Carlo samples: re-weighting events at NLO accuracy[END_REF].

At LO QCD accuracy the new weight (W new ) is obtained from the original one (W orig ) by multiplying it by the ratio of the matrix-elements estimated on that event for both models, noted respectively M orig 2 and

|M new | 2 : W new = |M new | 2 M orig 2 •W orig (5.21)
The reweighting technique is also available at NLO.

Validation of the reweighting technique

The reweighting technique is validated in comparing distributions of simulated events at dedicated BSM points in the parameter space, refereed to as standalone, with those resulting from the reweighting procedure.

For this, two EFT samples are generated with AMC@NLO. One EFT sample is reweighted, and the other is generated at the specific BSM points. The EFT effects are added via the DIM6TOP UFO model with one insertion of Wilson coefficient C 8 tq = 0.2 (all other Wilson coefficients operators are set to zero.). Figure 5.19

shows the particle-level distribution of cosq k + cosq k using standalone method (in red) and using reweighting technique (in green). Both methods show good agreement within their statistical uncertainties. tq are set to zero when generating these events. In the bottom panel, the red error bars represent the Monte Carlo statistical uncertainties on these two distributions. Both standalone and reweighting procedure show excellent agreement within their statistical uncertainties.

Probing new physics at NLO

Large scale uncertainties are major problem with LO prediction. As discussed in Section 2.2.3, the NLO corrections reduce the top-quark decay width uncertainty by approximately 10% compared to the LO prediction and therefore an increase in cross-section precision. This Section studies the impact of NLO QCD accuracy on both the CMDM operator, O tG and the four-quark couplings O 8 tq operator in the t t pairs production. Reference [START_REF] Diogo | Probing top-quark chromomagnetic dipole moment at next-to-leading order in QCD[END_REF] contains the first theoretical study of how the NLO QCD correction affects chromomagnetic dipole moment operator O tG in top-quarks pair production, which is the basis for the analysis presented in this Section. This Section is structured as follows. The cross validation of MC implementation of SMEFT@NLO UFO model used to generate NLO EFT samples is investigated in Section 5.6.2.1. Probing the EFT impact on Spin correlation observables at NLO is discussed in Section 5.6.2.2.

SMEFT@NLO validation

SMEFT@NLO UFO model is currently the only model allowing for NLO prediction. As it is newly proposed [START_REF] Degrande | Automated one-loop computations in the standard model effective field theory[END_REF], we are considering cross-validating its MC implementations against DIM6TOP UFO model, which is commonly used in Top physics for EFT interpretation at LO. Hence, the comparison is primarily based on tree-level calculations. This study is important in order to assess the validity and interpretation using SMEFT@NLO UFO model.

SMEFT@NLO vs DIM6TOP

In general, cross-validation of MC implementations of new physics models is a very time-consuming process, especially for EFTs with a large number of new interactions. and parameters. In addition, a variety of default settings, electroweak input parameters 12 , particle decay width and tools 13 are used by each EFT model. It is 12 DIM6TOP model is more flexible than SMEEFT@NLO model. In DIM6TOP model, either (aEWM1,G f ,m Z ) or (G f ,m W ,m Z ) can be used, although, SMEEFT@NLO model allow only for (G f ,m W ,m Z ) to be used. Consequently, we are adopting (G f ,m W ,m Z ) as electroweak inputs for each model. 13 Such as Rosetta [START_REF] Falkowski | Rosetta: an operator basis translator for standard model effective field theory[END_REF] which allows for the translation between different bases of effective field theory operators. Each EFT model uses his own tools. therefore necessary to make internal changes in the implementation of both EFTs models in order for them to be able to predict the same physical distributions. The details of making both EFTs models compatible are given in Appendix F; only the most relevant differences are discussed here:

• SMEFT@NLO normalizes the top chromo-dipole operator (with Wilson coefficient C tG ) with a factor of the strong coupling constant a S which means that C DIM6TOP tg = a S ⇥C SMEFT@NLO tg14 .

• DIM6TOP can be configured to normalize all CMDM operator O tG with a factor of the strong coupling a S . Turning on this option would yield a cross section 20% bigger than the normal case (where this option is disable) 15 . Unfortunately, one need to specify the squared coupling order (NP 2 ) for this to work and since AMC@NLO does not currently support squared coupling orders together with MADSPIN (or decay chains), we can not use C tG for this study, thus we use C 8 tq .

• The SMEFT@NLO can only decay top quarks using the MADSPIN package to preserve spin correlation information, while DIM6TOP can generate EFT samples with spin correlation directly from the command line 16 .

Within this new setup of parameter matching, we can safely compare both models. The results are shown in Figure 5.20. At the particle level, the plot shows the value spin correlation observables for different values in C 8 tq generated using reweighting approach. Both methods are found to be compatible and predict the same physical behaviour of spin correlation observables within their 95% Confidence levels (CL) (roughly two standard deviations), indicating the validation of SMEFT@NLO MC implementation.

MADSPIN's Limitation

As the SMEFT@NLO MC implementation is successfully validated, one more cross-check is necessary before studying the NLO correction's effects. SMEFT@NLO UFO model can only decay top quarks by using MADSPIN package, in which case, the spin correlation is only computed at LO accuracy. Here, we examine the impact of generating event at NLO accuracy but the decay top quark are at LO accuracy only due to use of MADSPIN 17 . A study such as this is important, especially when setting limits on Wilson

p p~> t t˜QED=0 QCD=2 FCNC=0 DIM6=1 DIM6 2 ==1
Where the DIM6 2 (NP 2 ==2 when using SMEFT@NLO model) specifies the coupling orders with one insertion of EFT operator. The two samples should predict a cross section with 20% difference. 16 For instance: generate p p > t t DIM6=1 FCNC=0 ctG=1 QCD=2 QED=0, (t > b W+ DIM6=0, W+ > l+ vl DIM6=0), (t˜> b˜W-DIM6=0, W-> l-vl˜DIM6=0) 17 Reference [START_REF] Diogo | Probing top-quark chromomagnetic dipole moment at next-to-leading order in QCD[END_REF] mentions this issue but not examined in detail.

coefficients at NLO accuracy and still want to preserve spin correlation information in top quarks pair decay.

For this study, EFT samples for t t at LO and NLO in QCD accuracy using SMEFT@NLO UFO model are produced. One operator at a time is varied with the others set to zero. The reweighting framework described in Section 5.6.1.3 id used to weight events at the particle level to several non-zero value of Wilson coefficients C tG and C 8 tq :

[ 1.5, 1.2, 1, 0.5, 0.2, 0.09, 0, 0.09, 0.2, 0.5, 1, 1.2, 1.5](L/TeV) 2 (5.22)

Figure 5.21 shows the distribution of cos(q ) k + cos(q ) k as defined in Equation 2.79 in the helicity basis for the C tG /L 2 = 1/TeV 2 (left) and C 8 tq /L 2 = 1/TeV 2 (right). The differential bin-by-bin K-factor, the ratio of the NLO to LO cross section s NLO /s LO , is displayed in the lower panel. For C tG and C 8 tq , at the centre of the distribution, the differential K-factor is stable, while at the edges, it fluctuates. The K-factor distribution is similar to the SM distribution, using the SM K-factor to rescale the LO contribution of C tG can be a good The distribution of cos(q ) k + cos(q ) k of the t t system, with the differential bin-by-bin K-factor s NLO /s LO is displayed in the lower panel where error bars represent the Monte Carlo statistical uncertainties on these two distributions (small enough to be invisible).

approximation
The K-factor value can be used to encode the information from a NLO calculation. The variation of K-factor for different EFT coupling values is shown in Figure 5.22 where the cross section follows a quadratic form such as in Equation 5.18 as function of C tG /L 2 or C 8 tq /L 2 , thus reads:

s = s SM + C tG L 2 a C tG + ✓ C tG L 2 ◆ 2 b C tG s = s SM + C 8 tq L 2 a C 8 tq + C 8 tq L 2 ! 2 b C 8 tq (5.23)
where a i and b i , i = C tG /C 8 tq are the linear and quadratic terms respectively. To extract a (6) and b (6) terms we fit the cross sections of Equation 5.23. The result are quoted in Figure 5.22. Several patterns can be observed in the K-factor plot. For C tG , NLO corrections appear to be larger for negative value and tend to decrease for positive one. The K-factor of C 8 tq is generally close to unity, so impact of NLO corrections is minimal for EFT scenarios. In general, K-factor provides a good measure of the impact of NLO corrections, but it has some limitations.

According to the region of phase space studied, K-factor varies quite a bit. For instance, there may be a substantial difference in K-factors when stringent analysis cuts are applied to the total cross section of a given process. A fiducial cross section can be obtained by applying basic cuts. In these cases, the K-factor is dependent on the cuts' values [177]. In addition, the NLO corrections often result in a shape change. For this reason, the c 2 is used which quantify the shape difference between NLO and LO. The c 2 is defined as follows [START_REF] Gagunashvili | Comparison of weighted and unweighted histograms[END_REF]:

c 2 = Â i (h 1,i h 2,i ) 2 s 2 i (5.24)
where h 1 and h 2 correspond to NLO and LO histograms, respectively. s 2 i is the standard deviation of each bin content i. Figure 5.23 illustrates the impact of NLO on the spin correlation observables for different value of C tG and C 8 tq . The bigger the c 2 value, the larger the difference between the NLO and LO distribution or the spin correlation observable is more sensitive to NLO correction. For negative values of C tG , the diagonal spin correlation observables (top left plot) exhibit the highest c 2 value in particular for cos(q ) k + cos(q ) k which tends to be more sensitive to NLO correction with an increase in cross section of 11% while 7% increase for cos(q ) r + cos(q ) r and cos(q ) n + cos(q ) n . On the other hand, cross diagonal observables (top right plot) are strongly effected by NLO correction with an increase of 18% for cos(q ) n + cos(q ) rcos(q ) r + cos(q ) n and < 10% for the other observables. The positive value of C tG display low value of c 2 and thus less impacted by the NLO correction. For C 8 tq , the c 2 value is smaller for spin correlation observables thus are less sensitive to NLO correction with an improvement in cross section less than 5%.

Despite the fact that only tree-level matrix elements are used for decay of top quark pairs, this procedure does capture essentially spin correlation effects in NLO calculations. Even so, the NLO correction does not significantly alter the LO prediction, but instead improves precision, particularly it influences on the contribution of C tG on the cos(q ) k + cos(q ) k and cross-diagonal spin correlation distributions. Accordingly, in the following Section, NLO EFT samples are generated using the SMEFT@NLO UFO model with MADSPIN package which ensures the propagation of spin information to the decay products at LO accuracy. = 13 Tev, 139 fb s aMC@NLO + Pythia TeV 2 ⇤ (up plots) and C 8 tq /L 2 ⇥ TeV 2 ⇤ (bottom plots) respectively. The c 2 value is normalized to c 2 value of the SM.

2 - 1.5 - 1 - 0.5 - 0 0.5 1 1.5 2 tg C 0.9 0.95 1 1
t t → pp 2 - 1.5 - 1 - 0.5 - 0 0.5 1 1.5 2 tg C 0.6 0.8 1 1.2 1.4 1

EFT impact on Spin correlation at NLO

The NLO predictions are expected to significantly reduce the uncertainties from LO prediction and thus provide a more reliable estimation of the possible range of the Wilson coefficients C tG and C 8 tq [START_REF] Diogo | Probing top-quark chromomagnetic dipole moment at next-to-leading order in QCD[END_REF][START_REF] Englert | Effective field theory and scalar extensions of the top quark sector[END_REF].

In this context, spin correlation observables can serve as discriminators in case any deviation from the SM is observed. Thus, the angular observables can be used to set novel constraints on dimension-six effective field theory operators relevant to the t t interaction. To study the EFT impact on Spin correlation at NLO accuracy, we first look at the signal sensitivity, s EFT /s SM , for different EFT point of C tG and C 8 tq at LO and NLO QCD accuracy. The predicted behaviour is shown in Figure 5.24. On one hand, signal sensitivity at LO is significant when C tG is positive, with a small decrease after NLO corrections. On the other hand, with a value close to unity, C 8 tq shows no indication of deviation from SM.

C(k, k) SMEFT = C(k, k) SM + C i L 2 a i + ✓ C i L 2 ◆ 2 b i (5.25)
where a i and b i , i = C tG /C 8 tq are the linear and quadratic terms respectively. Using this Equation, one can easily extract the values of a (6) and b [START_REF] Tomonaga | On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields*[END_REF] . As expected, significant improvement can be noticed between LO and NLO predictions. Generally, EFT coupling is more important at LO, but as we promote to NLO, this dependency is less significant. To further quantify the effect to EFT coupling on spin correlation coefficients, we the use c 2 method defined in Equation 5.24 where h at NLO (LO) for cos(q ) k + cos(q ) k observable while that calculated for C 8 tq reaches ⇠ 6.2 (4) at NLO (LO)

i.e. a difference of 69 %. It is also interesting to note that the cross-diagonal terms are the most affected by EFT operators with c 2 reaching 37 for cos(q ) n + cos(q ) rcos(q ) r + cos(q ) n observable. The SM value of spin correlation observables and the value of a (6) and b [START_REF] Tomonaga | On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields*[END_REF] for O tG and O 8 tq operators are displayed in Table 5.24.

EFT unfolding

Any unfolding technique may bias the measured cross section. Section 5.5 details the the unfolding procedure used in this analysis. The study presented in this Section presents the first step towards EFT interpretation using spin correlation observables with full Run 2 data. The interpretation of the 95% CL uncertainties for C tG and C 8 tq Wilson coefficients requires some care and therefore is left for future analysis. The prime objective of this Section is to verify if the unfolding handles samples with EFT operators correctly. In the following section, an overview of ATLAS EFT sample generation is presented. In Section 5.6.3.2, the EFT impact of unfolding is discussed. samples will be left to future studies when it is possible to use SMEFT@NLO-NLO UFO model.

SMEFT signal

The production of t t events in the context of EFT is modelled using MADGRAPH and PYTHIA8 parton

shower. The EFT model is added to these calculations via the DIM6TOP at LO . It relies on a reweighting approach, described in Section 5.6. These value are chosen based on the available global bounds of Wilson coefficients [START_REF] Aguilar Saavedra | Interpreting top-quark LHC measurements in the standardmodel effective field theory[END_REF].

EFT impact on background processes

The background processes can have non-negligible EFT dependence which may impact the EFT interpretation.

It is important to account for such effects. The importance of potential EFT contributions strongly depends on the size of the individual backgrounds in the analysis. For large backgrounds, it is necessary to generate EFT samples with detector simulation for all relevant Wilson coefficients while if the backgrounds contribution is small, it may be feasible to make rough estimates of why the EFT contributions to the background are not likely to be significant, hence avoiding the need to generate additional background samples with non-zero Wilson coefficients.

The impact of EFT on background processes can be considered negligible, thus this study is irrelevant for the purpose of this analysis. The main backgrounds are expected to be from single-top and massive-vector-boson production plus QCD jets (V +jets, with V = W or Z). These backgrounds are discussed in detail in Section 18 One can for instance write in MADGRAPH import model SMEFTatNLO-NLO, which loads the default NLO restriction card. Be aware that it carries random values for all Wilson coefficients! ??. The EFT interpretation presented in this analysis allows only C tG and C 8 tq to be non-zero and hence we consider if these operators can effect the background. The 4-quark operator O 8 tq does not affect single-top at tree-level at all. The O tG operator can enter via the gt t vertex. The expected contributions by single-top events on spin correlation observables, do not exceed 5% throughout the distribution and in all channels.

For V+jets production, the measured cross-section agrees with the SM [START_REF]Athena, The ATLAS Collaboration[END_REF][START_REF]Measurement of the cross-section for producing a W boson in association with a single top quark in pp collisions at $$ \sqrt{s} = 13$$TeV withAT LAS[END_REF], thus it is implausible that EFT contributions could be of this level. Neither of the C tG and C 8 tq operators impact the V+jets process at tree-level. Due to this, we disregard any contribution of EFT to the background processes.

5.6.3.2 EFT impact on unfolding PLU unfolding is used in this analysis, and the full procedure and unfolded data distributions are described in Section 5.5. As unfolded spectra are utilized to extract limits on two EFT couplings e.g. C tG and C 8 tq , it is important to verify that the unfolding technique will not add any bias to the extracted values of the couplings or limits. The first step to validate this is to check that the unfolding procedure won't introduce a difference between the spectrum obtained unfolding a pseudo-data sample and the corresponding particle (or truth) level. This is done with the linearity test (or closure test), where two scenarios are analysed:

• FixRM: The response matrix which is built from the migration matrix and efficiency and acceptance corrections is fixed to the nominal SM prediction and is used to unfold the pseudo-data samples corresponding to the sum of expected background yields plus t t signal contribution that contain the non-zero EFT operators. There are two possible outcomes: either the unfolding works as intended and recover the truth level EFT prediction (unfolded values should coincide with the truth ones); or it biases us entirely towards the nominal SM prediction.

• FixData: SM-based efficiency and acceptance corrections may suffer from large corrections in an EFT scenario which could affect the unfolded results and therefore be biased towards the SM and degrade the findings of the EFT interpretation. To test such bias, the response matrix is derived separately for each EFT operator. Pseudo-data samples are derived from the sum of expected background yields plus t t signal fixed to the nominal SM prediction. Then if the unfolding is performing as expected and doesn't impart any shape bias, each unfolding should recover the SM truth value. + cosq k distribution where different EFT samples are unfolded following either FixData (top) or FixRM (bottom) approach. The red lines represent the SM sample where the solid and dashed correspond to the unfolded and particle level distributions, respectively. The black uncertainty band includes the limited MC statistics together with the systematic uncertainty related to t t modelling. The triangle coloured markers correspond to the particle level distribution, while those with circle marks represent the unfolded one. The lower panel shows the ratio between the particle level and unfolded particle level distributions and the uncertainty is computed from the statistical and systematic error on the unfolded histograms.

Quantum interpretation

When a pair of particles such as t t pair is created and then separated, they may remain connected and retain a non-separable state. This genuine feature of QM is know as entanglement. It follows that, for a system of two entangled particles, each of which can have one of two spin states (up and down). If we measure the direction of a first particle's spin, which is undetermined prior to experiment, let us assume it to be up, then the direction of second particle's spin must be downward. This comes about regardless of how far apart the two experiments happen. A wide variety of systems have been explored for detection of entanglement, such as photon [START_REF] Aspect | Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities[END_REF], atoms [START_REF] Hagley | Generation of Einstein-Podolsky-Rosen Pairs of Atoms[END_REF], mesons [START_REF] Go | Measurement of Einstein-Podolsky-Rosen-Type Flavor Entanglement in °(4S) ! B 0 B 0 Decays[END_REF], neutrinos [START_REF] Formaggio | Violation of the Leggett-Garg Inequality in Neutrino Oscillations[END_REF], macroscopic diamond [186], analogue Hawking radiation [START_REF] Steinhauer | Observation of quantum Hawking radiation and its entanglement in an analogue black hole[END_REF]. [START_REF] Stuart | Experimental Test of Local Hidden-Variable Theories[END_REF][START_REF] Aspect | Experimental Test of Bell's Inequalities Using Time-Varying Analyzers[END_REF] atoms [START_REF] Clauser | Bell's theorem. Experimental tests and implications[END_REF], superconducting systems [START_REF] Ansmann | Violation of Bell's inequality in Josephson phase qubits[END_REF]. All of the results have agreed with predictions based on QM.

The prospect of studying the entanglement and BIs using hadron colliders recently proposed by several theory papers such as Afik et al [START_REF] Afik | Entanglement and quantum tomography with top quarks at the LHC[END_REF], Fabbrichesi et al [START_REF] Fabbrichesi | Testing Bell Inequalities at the LHC with Top-Quark Pairs[END_REF], F. Maltoni et all [START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF], Alan J. Barr [195] and Takubo et al [START_REF] Takubo | Feasibility of Bell inequality violation at the ATLAS experiment with flavor entanglement of B 0 B 0 pairs from pp collisions[END_REF]. In this thesis, the measurement of quantum entanglement from differential cross-sections in dileptonic t t pair as well as the violation of BIs using Run 2 data at p s = 13 TeV with the ATLAS detector at the LHC. This Section is organised as follows. We briefly present the general theoretical background of entanglement, EFP paradox and BIs in top quarks pair production in Section 5.7.

|Yi = 1 p 2 (| "i| #i | #i| "i) (5.27)
is entangled. When we compare experimental results on the first and second particles, we find a perfect (negative) correlation. The quantum state like in Equation 5.27 are know as entangled pair 19 .

Density matrix and concurrence

For a system in which there are multiple states |y k i, with no knowledge as to which states describe our system, a probability p k is assigned to each state, such that the expectation value of any operator A in the system is thus calculated as follows [197]:

hAi = Â k p k hy k |A|y k i (5.28)
This state is described by the density matrix (r) operator:

r = Â k p k |y k ihy k | (5.29)
The state in Equation 5.29 is know as mixed state. If the system is describe by a specific state |yi, the density matrix is given by r = |yihy| (5.30) This state is known as pure state. Density matrices encode all our information about a state, pure or mixed, and allow to compute the expected outcome of any measurement. For a bipartite Hilbert space H formed by the direct product of two sub-systems A and B, H A N H B ; A and B are in separable state if the density matrix r can be written as:

r = Â k p k r A k ⌦ r B k , Â k p k = 1, p k 0 (5.31)
where r A,B k are density matrices for A and B, and the probability p k are positive and add to one. For the state to be entangled it has to be non-separable.

A necessary and sufficient condition for entanglement in the bipartite systems is given the Peres-Horodecki 160

Chapter 5. Spin Correlation with the full Run 2 dataset criterion20 [START_REF] Peres | Separability Criterion for Density Matrices[END_REF][START_REF] Horodecki | Separability criterion and inseparable mixed states with positive partial transposition[END_REF]. For partial transpose 21 .

r T B = (1 ⌦ T )r (5.32)
If r is entangled then all the eigenvalues of r T B are negative. Measurement of entanglement can also be done using the concurrence C[r] [START_REF] Afik | Entanglement and quantum tomography with top quarks at the LHC[END_REF] which is defined as

C[r] ⌘ max(0, l 1 l 2 l 3 l 4 ) (5.33)
where l i , i = 1, 2, 3, 4 are the eigenvalues, in decreasing order, of the matrix C[r] = p p r rp r with r = (s 2 ⌦ s 2 )r ⇤ (s 2 ⌦ s 2 ). r ⇤ is the complex conjugation of the density matrix. For a separable state, C[r] = 0; for a maximally entangled state, C[r] = 1.

A notable example of bipartite systems is provided by a pair of t t quarks. As detailed in Section 2.3, the density matrix r in H of dimension 2 ⇥ 2 can be written as:

r = I 4 + Â 3 i=1 B + i s i ⌦ I 2 + B i I 2 ⌦ s i + Â 3 i, j=1 C i j s i ⌦ s j 4 (5.34)
where I n is n ⇥ n identity matrix and s i represent Pauli matrices. B ± i and C i j are the the polarizations of each particles and spin correlations between the two particles 22 . The polarizations and spin correlations characterising the production spin density matrix are computed in helicity basis { k, r, n}

The partial transpose of spin density matrix r defined in Equation 5.34 is as follows [START_REF] Afik | Entanglement and quantum tomography with top quarks at the LHC[END_REF][START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF]:

r T B = (1 ⌦ T )r = 1 4 0 B B B @ 1 + B + n + B n +C n n B k +C nk + i B r +C nr B + k +C k n i B + r +C r n C kk +C r r + i C k r C rk B k +C nk i B r +C nr 1 + B + n B n C n n C kk C r r i C k r +C rk B + k C k n i B + r C r n B + k +C k n + i B + r +C r n C kk C r r + i C rk +C k r 1 B + n + B n C n n B k C nk + i B r C nr C kk +C r r + i C rk C k r B + k C k n + i B + r C r n B k C nk i B r C nr 1 B + n B n +C n n 1 C C C A (5.35)
According to the Peres-Horodecki criterion, if the eigenvalues of the matrix in Equation 5.35 are negative, then the corresponding spin density state is entangled. Specificity, if the eigenvalues of the matrix arising from r C = v T r T B v, where the 4-vectors v is defined as

v = (1, 0, 0, ±1) T , v = (0, 1, ±1, 0) T r C = 1 4 ✓ 1 + B n + Bn +C n n C kk +C r r + i C k r C rk C kk +C r r + i C rk C k r 1 B n Bn +C n n ◆ (5.36)
are negative, then the eigenvalues of matrix 5.35 are as well. This implies that det(r C ) < 1 is a sufficient condition for the presence of entanglement. Since 1  C n n  1, sufficient condition for entanglement is as follows:

|C kk +C r r| > 1 +C n n (5.37)

The above criteria is sufficient condition for entanglement, valid for arbitrary quantum states. The concurrence C[r] is given by:

C[r] = max(L, 0) 2 (5.38)
where L = C n n + |C kk +C r r| 1 > 0. There are only two regions of phase space in which the resulting t t pair is entangled [START_REF] Afik | Entanglement and quantum tomography with top quarks at the LHC[END_REF][START_REF] Aguilar-Saavedra | Improved tests of entanglement and Bell inequalities with LHC tops[END_REF], at threshold e.g. M t t ⇡ m top and at high transverse momentum top quarks p T e.g. M t t >> m top and q ⇡ p 2 where q is the production angle with respect to the beam line. The following examines the entanglement witness in each region.

Entanglement in t t threshold

The dominant production mechanism for t t at the LHC is gg fusion. The gluon is a massless spin-1 particle with two polarizations: +1/ 1. In this case, the initial state must have either a total spin of 0 or 2. The t t pair is consists of two fermions, thus at threshold t t is produced in an entangled spin-singlet state 0 [START_REF] Afik | Entanglement and quantum tomography with top quarks at the LHC[END_REF]:

|Y gg i = 1 p 2 (| " ni| # ni | # ni| " ni) (5.39)
At t t threshold, C kk + C r r < 0 (see Figure 5.32), thus from the inequality in Equation 5.38, the direct entanglement signature is provided by

C n n C kk C r r > 1 = Tr[C] < 1 (5.40)
where Tr[C] = C kk +C rr +C nn represent the trace of spin correlation matrix C. Interestingly, this trace can be directly measured using the observable D from the angular separation between the leptons, cos j as given by the single-differential cross section with respect to cos j:

1 s ds d cos j = 1 2 (1 D cos j) (5.41)
Were D = Tr[C]/3, thus the criterion 5.44 is can simply be written as:

D < 1 3 (5.42) if D < 1 3
, the two quarks are in a entangled (non-separable) quantum state, while D 5.41 as function of the m t t upper-bound defining the considered phase-space, at LO and NLO order [START_REF] Afik | Entanglement and quantum tomography with top quarks at the LHC[END_REF].

Entanglement in high m t t and q

In this region of phase space, the t t pair is produced in an entangled spin-triplet state as a consequence of conservation of angular momentum regardless of production channel (gg or q q):

|Yi = 1 p 2 (| " ni| # ni + | # ni| " ni) (5.43)
At high m t t and q , C kk +C r r > 0 [START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF], thus from the inequality in Equation 5.38, the direct entanglement signature is provided by

C n n +C kk +C r r > 1 (5.44)
Despite entanglement being present across a wide range of phase space, detecting entanglement is not trivial.

In this thesis, detection of entanglement is carried out in t t threshold region since more events are expected from the t t production cross section. In addition, the D observables provides a simple entanglement criterion and can be probed at current energy of LHC. cast in the form [START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF]:

C A,B  constant (5.45)
where C measures the correlation between non-interacting subsystems A and B. In 1969, Clauser, Horne, Shimony and Holt [START_REF] Clauser | Proposed Experiment to Test Local Hidden-Variable Theories[END_REF] proposed an inequality, now known as the CHSH inequality, which is an optimal version for Bell's inequality. Consider a measurements a, a 0 and b, b 0 on subsystems A and B23 , which are assigned to two possible eigenvalues ±1, the CHSH inequality reads:

|habi hab 0 i + ha 0 bi + ha 0 b 0 i|  2 (5.46)
For top and anti-top quarks which are spin-1/2 particles, CHSH inequality reads [START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF]:

|hm a mb i hm a m0 b i + hm 0 a mb i + hm 0 a m0 b i|  2 (5.47)
where the measurements in the CHSH inequality are the spin projections along four axes, m a , m 0 a for top quark, and mb , m0 b for anti-top quark. For a system described by the spin density (5.34), the maximum of the Chapter 5. Spin Correlation with the full Run 2 dataset CHSH inequality (5.47) is [START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF][START_REF] Takubo | Feasibility of Bell inequality violation at the ATLAS experiment with flavor entanglement of B 0 B 0 pairs from pp collisions[END_REF]:

max aa 0 bb 0 |hm a mb i hm a m0 b i + hm 0 a mb i + hm 0 a m0 b i| = 2 p l + l 0 (5.48)
where l and l 0 are the largest eigenvalues of C T C. thus the violation of any CHSH inequality requires l + l 0 > 1 (5.49)

Yet, using the large eigenvalues of C T C to test for CHSH violation may be biased due to the uncertainties involved in the measurement. As discussed in detail in [START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF], the eigenvalues of Following [START_REF] Afik | Entanglement and quantum tomography with top quarks at the LHC[END_REF][START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF][START_REF] Aguilar-Saavedra | Improved tests of entanglement and Bell inequalities with LHC tops[END_REF] and at high m t t and q , an alternative approach is to select a, a 0 , b, b 0 in a manner that leaves l.h.s. of CHSH inequality larger than 2. The optimal choice of axis is given by:

a k = d ki , a 0 k = d k j b i = b 0 i = ± 1 p 2 , b j = b 0 j = ± 1 p 2 , b k6 =i, j = 0 (5. 50
)
where i 6 = j. Using this basis, the CHSH inequality is reduced to the following form:

| C rr +C nn |  p 2 (5.51)
which is simply expressed in terms of diagonal spin correlation observables C rr and C nn . Using the Equation 5.49, the violation of the CHSH inequality (thus violation of BIs) can be evaluated for each point in phase space, in terms of m t t and q . The direct testing of BIs at threshold is challenging and represents a loophole in the analysis. This loophole is the so-called locality loophole. The measured t t spin correlation may not be the result of quantum effects, but could be caused by the t t pair exchanging classical information through hidden particles. This can be resolved by using events that are space-like separated, these measurements cannot influence each other and the two events have independent statistics [START_REF] Fabbrichesi | Testing Bell Inequalities at the LHC with Top-Quark Pairs[END_REF][START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF]. Figure 5.34 shows the Monte Carlo evaluation of the probability of space-like separated decays as a function of the t t invariant mass m t t . At threshold, half of the events are space-like separated. At m t t , 90-98% of the events are space-like separated. As a result, only high m t t is consider to test the violation of BIs.

Figure 5.34: Fraction of top and anti-top decays that are space-like separated as a function of m t t [START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF].

Expected experimental sensitivity of quantum entanglement at LHC

Detecting top quark entanglement at the LHC would be a milestone as it would be the first observation of highest energy quark entanglement ever explored. This would provide a basis for the study of quantum information theory at high-energy collider such as LHC. This Section aims at measuring the D observable in t t system and establish if it is significantly low than the entanglement limit of 1/3, this establish the presence of quantum entanglement.

The analysis uses the full ATLAS Run-2 data set and a profile likelihood unfolding (PLU) unfolding method is used. The parameter D is measured using two methods: the slope or the mean of the cosj distribution (5.41). The current analysis is not finalized, so the analysis is blinded. Hence, the statistical results shown in the chapter are expected results only and evaluated using the Asimov dataset. A detailed comparison of the two methods and the most appropriate method for determining observable D represents one of the author's most significant contribution. Details of the data generation in provided is Section 5.2.1. Signal and background simulations are described in Section 5.2.2. The event selection criteria are detailed in Section 5.3. Figures 5.35 depict the distribution of cosj ``in ee, µµ and eµ channel where j is the angle between the two leptons in their respective top-quark parent rest-frame. The cosj ``distribution is well modelled and agrees with the data in all channels. This Section is structured as follows: the unfolding procedure and the systematic uncertainties are described in Section 5.7.2.1, the results are described in Section 5.7.2.4.

Differential unfolding

In order to extract the D parameter and account for detector smearing effects, the data are unfolded to parton level. In this present analysis, the PLU is used, and the unfolding procedure is described in Section 5.5. As described in Section 5.7.1.1, in order to measure entanglement, this analysis targets the measurement of differential cross-section as function of the observable cosj ``in a low m t t region defined by an upper-bound of m t t threshold. A value of 380 GeV was chosen by running the full analysis chain for different values of m t t and optimising for the smallest systematic uncertainties and the largest significance. The value 380 GeV is close to t t threshold with twice the mass of the top-quark (⇡ 346 GeV). In order to account for event migrations near this m t t threshold, due to experimental resolution, the response matrices considered in this unfolding have the structure shown in As is to be expected, the f j acc values tend to be large for each bin because our selections efficiencies only accept only a small percentage of the total t t events. The low values for e eff are somewhat unique to this analysis, relative to previous similar dilepton analyses such as spin correlation, where they are typically above 0.9. The reason for this is that the m t t threshold cut creates and extremely narrow reco-level phase space (with a range of only 40 GeV) and suffers from significant smearing effects. Therefore, a large percentage of events populating the reco-level selection are not actually from this region at truth level which results in low values for e eff . The smearing observed in R i j is relatively symmetric and limited to the neighbouring bins.

Reparamaterisation of likelihood

As outline in Section 2.3, D can be obtained using two methods as follows:

• Mean method: Compute D using the mean of the normalized cross-section of cosj ``, such as

D = 3hcosj ``i.
• Compute D using from the slope of the normalized cross-section of cosj ``of a one-dimensional polynomial fit to the differential cross-section.

The likelihood can be directly expressed as a function of D so as to measure the D parameter directly, along with its confidence interval. This takes into account all uncertainties and correlations. This can be achieved by simply re-parametrising the PoIs in the likelihood. The following discusses reparameterization of likelihood using the mean and slope method.

Mean method: D observable can be obtained from the averaged cross-section over cosj ``:

D 3 = hcos j ``i =  i x centre i • N i • µ i  i N i • µ i (5.52)
where, µ i , i = 1, 2, 3, 4 is the fitted signal strength in data and the main parameter of interest (POI). N i is the number of events in bin i. x centre i is the value of the centre of bin i. One of the PoIs can be re-parametrised to become a function of the average and of the other PoIs. In practice, the PoI controlling the first unfolded bin (µ 1 ) is chosen to be modified as follows, from Equation 5.52:

µ 1 = Â i6 =1 x centre i D 3 • N i • µ i D 3 x centre i • N 1 (5.53)
Slope method: Following the same procedure, the normalization of each bin can be re-parametrised to become a function D. The single-differential cross section with respect to cos j is given by 1 s

ds d cos j ``= 1 2 • (1 D • cos j ``) (5.54)
The content of bin i of observable cos j, ds d cos j ``,i = µ i N i where N i is the truth prediction and µ i is the fitted signal strength in data. When integrating R ds d cos j ``dj ``= s , which in discrete form is s = Â i µ i N i w i , where w i is the width of bin i. The Equation 5.54 is now reads

1 s ds d cos j ``,i = N i • µ i N i • µ i • w i + Â j6 =i N j • w j (5.55) = 1 2 • 1 D • x centre i (5.56)
where x centre i is the bin centre of bin i. In practice, we take all of the normalization factors to be dependent on D in the following way:

µ i = f (x centre i ) • Â j6 =i N j • w j N i • (1 f (x centre i ) • w i ) (5.57) where f (x centre i ) = 1 2 (1 D • x centre i ), i = 1, 2, 3, 4 5.7.2.

Unfolded asimov-data

In order to characterize the sensitivity of the quantum entanglement search, the simultaneous likelihood fits is performed using an artificial dataset known as Asimov dataset. The asimov dataset [START_REF] Gross | Wald's theorem and the "Asimov" data set[END_REF] is generated by summing up the MC predictions for all processes. It can then be used to test the outcome of fits, asimov fit, based on the given prediction. This is very useful to test fit strategy and calculate expected results of the analysis. The signal region is constructed using all lepton combinations (ee, µµ, eµ). The result of the asimov fit is shown in Figure 5.37 where the nominal distribution of cosj ``(red line) with asimov data Figure 5.37: Post-fit distribution of the unfolded asimov-data (a) using slope method and (b) using mean method. All of the systematic uncertainties described in Section 5.5 are included in the fit.

The binning of distribution of cosj ``, in which a likelihood fit is performed, is of special importance. A high number of bins can increase the overall signal sensitivity, while it also decreases the MC statistics per bin. In order to determined the impact of number of bins on slope and mean methods, the unfolded procedure was repeated with 2, 4, 6, 8 and 10 bins. The upper value of the t t invariant mass, m t t , is selected to be 400 GeV (since the analysis is blinded and, though the final choice of the m t t cut is 380, the region between 380 and 400 is still quite sensitive to entanglement). The binning optimisation considers only t t modelling systematic uncertainties, the MCNLO and the POWH7 ones, as they are the dominant uncertainties and this should not change the overall results. Figure 5.38a shows the unfolded D value for different number of bins using the slope (violet dots) and mean (green dots) method. In the lower panel, the ratio between mean and slope method is displayed. A simple quadratic fit is performed to the unfolded D values. Mean method appears to be biased by the number of bins, while the slope method appears to be more stable. The mean method results in a bias because the mean of the distribution, for small numbers of bins, is not a good estimator of the mean of a population. Table 5.9: Gained in estimation of the statistical and systematic uncertainties between the slope and mean methods.

In Figure 5.38a, the value of the unfolded D is different between the mean and the slope method. This disagreement can be explain by the discrete nature of the bins. The mean method is based on the assumption of a continuous distribution, yet in the profile likelihood fit, the discrete bins are used which modifies these To estimate the significance 25 , likelihood had to be evaluated for the observed and non-entangled values 3241 of D. This later case is the null hypothesis with D = 1/3, which is the entanglement limit. Thus, the 3242 significance can be calculated as p q 0 , where:

3243 q 0 = 2 ln L (D = 1/3) L ( D) = 2(ln(L (D = 1/3)) ln(L ( D)) (5.59)
The measured D value using asimov data at an invariant mass cut of M t t < 380 GeV is given by: 25 The Appendix D provides more details about the test statics and the calculation of significance using profile likelihood ratio.

is a stronger and deeper criterion than the measurement of entanglement where it is important to stress that in general:

Correlation ; Entanglement ; Violation of BIs (5.61) correlations do not all result from entanglement and not all entangled states violate Bell inequality. The violation of BIS in t t events at the LHC has recently been shown in numerous studies [START_REF] Fabbrichesi | Testing Bell Inequalities at the LHC with Top-Quark Pairs[END_REF][START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF]. There were, however, discrepancies between the two groups' results. In [START_REF] Fabbrichesi | Testing Bell Inequalities at the LHC with Top-Quark Pairs[END_REF], the violation of BIs is possible with current ATLAS data at the 98% CL. Instead, in [START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF], they analysis concluded that violation of BIs is much more challenging: sufficiently strong correlations are found only for top quarks at very high-p T .

The purpose of this Section is to examine CHSH violation in t t system in order to validate the above results.

The present analysis uses the ATLAS MC perdition described in Section 5.1. This is the first feasibility study of violating Bell Inequalities using spin correlations in t t pairs carried out by ATLAS and one of the author's main contributions. This Section is organised as follows. Analysis procedure is presented in Section 5.7.3.1.

The results are shown in Section 5.7.3.2.

Analysis procedure

As shown in Section 5.7.1.2, CHSH inequality for t t system described by the spin density matrix (5.34) is given by max

aa 0 bb 0 |hm a mb i hm a m0 b i + hm 0 a mb i + hm 0 a m0 b i| = 2 p l + l 0 (5.62)
where l and l 0 are the largest eigenvalues of C T C. In order to test the CHSH inequality in Equation 5.62, one needs to determine the spin correlation matrix C in every phase space region of interest. The C matrix is evaluated for t t events generated using MC simulation described in Section 5.1. The mean method is used to compute the entries of C matrix: As a preliminary check, it must be ensured that there is no correlation between the spin correlation observables C kk , C r r, and C n n. In this manner, we can unfold each observable independently and perform the BIs test 5. [START_REF] Lynch | Measurement of the Asymmetry Parameter in µ e Decay[END_REF].

9hx â, bi = C â, b (5.63
at a detector level if there is a weak correlation. Figure 5.42 shows the correlations between diagonal spin observables. The error bars represent the Monte Carlo statistical uncertainties on these distributions. The correlation between C â â and C bb , â 6 = b, is low, hence this analysis assume no correlations.

Results

The spin correlations at threshold are sufficient to demonstrate entanglement but not sufficient to demonstrate a violation of the BIs. In contrast, BIs violations are expected at high invariant mass of the top quarks pair m t t and high angle w.r.t the beam q . In this region of phase space, the CHSH inequality can be tested using two methods as follows (see Section 5. • Compute l + l 0 (5.62). The violation of the CHSH inequality requires l + l 0 > 1. The method may introduce bias 27 , but we carried out the test for comparison with other studies ??.

The results are shown in Figure 5.43. In both tests, the bins in the high m t t and q region violates the CHSH inequality. Our next step is to assess the extent to which the values of l + l 0 > 1 and C rr +C nn > p 2 or < p 2 can be used to prove violations of the BIs. One need to consider how detector resolution, acceptance, efficiency and migration effects influence their determination of statistical uncertainties in particular the impact t t modelling systematic. For this, two differential unfolding in m t t and q phase space is required. The study would be the next step of the analysis. 27 The details are in Section 5. The value of l + l 0 in t t phase space, a value > 1 is violation. My study mainly focusses on the measurements of the top quark spin polarisation and correlation in top pair production. These measurements are interpreted in terms of fixed order predictions of the SM, potential signs of new physics using EFT formalism as well as fundamental quantum mechanical phenomena, quantum entanglement and Bell inequalities. The measurements take the form of differential cross-sections which are unfolded to both the full phase space at parton level or to a restricted phase space at the particle level.

Conclusion

Additionally, a detector performance study was conducted during the analysis preparation process. HL-LHC offers new physics opportunities through direct searches and precision measurements. To make full use of the HL-LHC collision data, ATLAS detector will need to be upgraded to cope with the intense radiation and high pileup conditions. On of the major ATLAS upgrade, the HGTD, a crucial detector to enhance the capabilities of all-Silicon ITK by adding timing measurement to charged-particle trajectories as well as spatial information. The detailed study of the time calibration procedure of each channels relying on a dedicated simulation is presented. The simulation is based on timing information provided by HGTD which is used to estimate the irreducible and non deterministic clock contribution to timing resolution coming mainly from the lpGBT clock jitter, FLEX cable, FILEX and ASIC. In addition, time-dependent effects and static effects are also simulated in this study. The timing correction procedure aims to calibrate t hits distribution. The accuracy of the calibration depend partly on the duration of the effect and partially on the nature of the affected component (ASIC, lpGBT, group of ASICs on a PEB board, module). For that reason, the calibration is performed by measuring the calibration constant, ht hits i, at different time intervals of each channel using different special granularities and data triggered at one MHz in the FELIX processor.

Each special granularity is targeting a particular region of the detector, as well as targeting global or more broadly correlated effects to calibrate (always with the trade-off systematics/statistics). Preliminary study shows that by using calibration per module, the t hits distribution is calibrated with good accuracy, displaying a value of t 0 of (0 ± 323 ps). The calibration per module best absorb the time variation of 1 kHz at regular interval of small event. Small interval of event can reduce the t 0 jitter when shorter-term variation affect the time hits. Additionally, calibration per module is capable of correcting for geometrical effects, particularly time-of-flight effects.

The second part of the work of this dissertation, a measurement of 15 top quark spin observables using full Run 2 ATLAS pp collision data. Two charged leptons with opposite charges and two high p T jets were required to select the signal region. For Drell-Yan contributions, additional cuts on the invariant mass and transverse missing energy of both leptons were required when both leptons are electrons or muons as detailed in thesis. Background events due to single-top, diboson and t tX were estimated using MC simulation. Data-driven methods were used to estimate background due to Drell-Yan leptons and fake leptons. A good agreement between data and prediction was observed. Each of the spin correlation distributions are moved back to the particle level. The systematic uncertainty resulting from t t modelling is the dominant source of systematic uncertainty for all correlation observables in the dilepton channel.

These measurements are first interpreted for potential signs of new physics using EFT formalism. In this case, the SM effective field theory (SMEFT) is used which provides a model independent way to parametrize new physics effects.We have produced a complete set of EFT predictions for the t t spin density matrix at LO and NLO precision in QCD, for centre of mass energies of 13 TeV. We have discussed information on how to isolate the different contributions (pure SM, pure EFT, and interference) as well as the similarities and differences between several UFO models. Given the particularity of this analysis, the MC implementation of SMEFT@NLO UFO model used in the thesis is cross-validated. In particular, we have examined the effect of decaying top quark pairs using the MADSPIN package while generating EFT samples at NLO. We showed that this procedure still captures the spin correlation effects at NLO calculations. We have studied the impact of NLO QCD accuracy on both the CMDM operator, O tG and the four-quark couplings O 8 tq operator using only spin correlation observables in the t t pairs production. Significant improvement is noticed between LO and NLO predictions. Generally, EFT coupling is more important at LO, but as we promote to NLO, this dependency is less significant. It is also interesting to note that the cross-diagonal observables are the most affected by EFT operators. In addition, we have checked that the unfolding procedure won't introduce a difference between the spectrum obtained unfolding a pseudo-data sample and the corresponding particle level. Slight difference is seen between the unfolded and the particle level distributions of the SM sample which has only a small effect on the EFT interpretation that is well covered by the total uncertainty. Both statistical and systematic uncertainties are included in this analysis. This thesis presents, for the first time, the expected sensitivity to the quantum entanglement using full Run 2 asimov data at an invariant mass cut of M t t < 380 GeV. Quantum theory links the value of both the mean and slope of the cos(q ``) distribution to D observable which is used to test the nature of the t t quantum state. if D < 1 3 , the two quarks are in a entangled (non-separable) quantum state, while D 1 3 means there is no entanglement. A detailed discussion of the unfolding process and the challenges involved in extracting the value of D observables using mean and slope methods was presented. The result shows the first expected experimental sensitivity of quantum entanglement at LHC with extracted D value of D = 0.4591 + 0.028 0.037. This corresponds to an expected significance of 4.9s . The dominant uncertainties are those arising from the modelling of the signal process. On the other hand, a different approach to addressing the quantum entanglement is to look at weather measurements on a given system can violate BIs in its CHSH version. Bell's inequality is a stronger and deeper criterion than the measurement of entanglement. Our study is performed at the parton level, without detector simulation or unfolding from detector-level information, since we limit ourselves to considering statistical uncertainties. We have showed that the BIs violations are expected at high invariant mass of the top quarks pair M t t and high angle w.r.t the beam q . This result is in accordance with the most recent theoretical result published to date [START_REF] Fabbrichesi | Testing Bell Inequalities at the LHC with Top-Quark Pairs[END_REF][START_REF] Severi | Quantum tops at the LHC: from entanglement to Bell inequalities[END_REF].

Future work

As a first step on the long road of analysis, this thesis showed that spin correlation observables are sensitive to a few dimension-six SMEFT operators relevant to t t production. Next up on this road is measuring the spin correlation observables with the full ATLAS 2015-2018 dataset and interpret the unfolded results using SMEFT@NLO at NLO in QCD precision, both at generation and decay levels. This will allow to shorten the space between the experimental and the theoretical expectations of the SM model and models beyond the SM. In the best scenario, this will allow finding a novel sensitivity to the operators associated with the Wilson coefficient c tg which modifies the gt t vertex.

A preliminary, blinded search for entanglement was set up for the first time with the ATLAS detector. A dedicated analysis strategy showed an expected experimental significance of quantum entanglement at LHC of 4.9s . These results suffer from the efficiency corrections which are low relative to previous similar spin correlation analyses, where they are typically above 0.9. The reason for this is that the m t t threshold cut creates an extremely narrow detector level phase space and suffers from significant smearing effects. Therefore, a large percentage of events populating at the detector level selection are not actually from this region at truth level which results in low values for the efficiency corrections. With the current setup, unfolding the distribution might not be a good idea (though it could be done, but significant bias corrections would be required). By improving the resolution of the reconstruction this can be corrected, but little improvement has been made since the Tevatron. The calibration curve method is proposed as an alternative method for correcting the data for detector effects. A calibration curve corrects a single value measured at reconstructed level R to a corresponding value at truth parton level T . Ideally, R is the same quantity as T but smeared by detector resolution and efficiency effects or is instead a parameter that is strongly correlated with T . To construct a calibration curve, many different hypotheses for T 0 with a corresponding R 0 are needed. Each pair of T 0 and R 0 values are plotted and then a polynomial is used to interpolate between the points. By construction, there must be at least n pairs of R 0 and T 0 to fit a polynomial of order n, and ideally many more. Once the curve has been constructed, any value for R 0 measured in data can be calibrated to its corresponding T 0 value using this curve. The results of this analysis are expected to be published the by the end of this year.

In this thesis we present the feasibility study for detecting Bell Inequalities violations using spin correlation observables in top quark pairs, marking the first step in a long road of the analysis. The next step is to consider how detector resolution, acceptance, efficiency and migration effects influence the violation of BIs measurements in particular the impact t t modelling systematic. For this, two differential unfolding in m t t and q phase space is required.

A. Like-Helicity quark-antiquark annihilation

Typically, SH and OH production asymmetry is primarily explained by relativistic parton luminosities in LHC and the appropriate choice of spin axes (which is mainly relevant when computing correlation observables), more detail are in Section 2.2.4.

At LHC, quark anti-quark annihilation, q q, is only 10% of the total cross section, dominated almost entirely by the opposite-helicity case. SH q q contribution is proportional to the mass. In the default SM mode we use to generate the MC sample in Section 2.2.4, all quarks are set to zero (4-flavour model), so SH q q has a net contribution of zero to the cross section. The workaround for this problem is to generate SM samples using a model where all quarks are different from zero and only generate the q q production mode in order to get more statistics for SH q q. For example, when using MADGRAPH5_aMC@NLO during the generation phase: import model sm-full generate b b~> t t~, (t > b W+, W+ > l+ vl), (t~> b~W-, W-> l-vl~) output ttbar_bb_SM where sm-full is the name of the SM model with all quark are different then zero and bb˜> tt˜is used to get more statistics for SH q q production mode. Figure ?? illustrate the contribution of SH and OH in the q q process for two scenarios pp > tt˜and bb˜> tt˜. For 100k event, the cross-section for opposite helicity is 2 ⇥ 1.634 while the one for SH b b is 2 ⇥ 0.0003708 (which correspond to only 22 events), adding the gluon sample in top of that, the sample size required will be even larger obviously. The figures display the t hits distribution before (black) and after the time correction procedure. Calibration constants are computed using multiple timing intervals (every 1, 100, 500 events) and using time hits from full, inner (r < 320 mm) and outer (r > 320 mm) layer. The timing model chosen for this study is TimeFull 1 kHz. Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200. The figures display the t hits distribution before (black) and after the time correction procedure. Calibration constants are computed using multiple timing intervals (every 1, 100, 500 events) and using time hits from full, inner (r < 320 mm) and outer (r > 320 mm) layer. The timing model chosen for this study is TimeFull. Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200.

A.1 2 - 1.5 - 1 - 0.5 - 0 0.5 1

D. Statistical inference D.1 Maximal likelihood estimators

Let suppose we have N different measurements, x = {x 1 , x 2 ,...,x N }, where the measurements are independent and identically distributed (i.i.d.). In this case, x i follows the probability distribution f (x; q ), where q = {q 1 , q 2 ,...q m } are the m parameters of the statistical model. For a specified value of q , the function, f (x; q ) can be thought of as the probability of all possible samples determined by the specified q . The case doesn't teach us anything about about the model. On the other hand, consider f (x; q ) as a function of q for an observed x i , in this case, f (x; q ) does not represent the probability density function of q because it does not normalize to unity. f (x; q ) expressed this way is called likelihood and can be written as

L (q ) = N ' i=1 f (x i ; q ) (D.1)
The Maximum-Likelihood Estimator (MLE) of the model parameter q = {q 1 , q 2 ,...q m } are refereed to as q and represents the parameter value that best fits the observed data. MLE for q are obtained by finding the global maximum of the likelihood function L (q ):

∂ lnL(q ) ∂ q i = 0, i = {1, 2,...m} (D.2)
For large samples, the likelihood can have many local maxima. Thus, finding the global maximum can be a major computational challenge. However, MLE offer unbiased and efficient results (i.e., with a minimum variance). In practice, the negative log-likelihood is used to avoid extremely small or large likelihood values.

Since the logarithm is a monotonic function, the maximum of L occurs at the minimum of ln(L).

Searches are commonly conducted with significance Z values instead of p-values:

Z 0 = F 1 (1 p 0 ) (D.11)
where F 1 is the inverse of the cumulative distribution of the standard Gaussian. As a general rule, a significance greater than 3s is considered evidence, and a significance greater than 5s indicates an observation of a signal above the background.

E. Normalisation to the integrated luminosity

In most physics analyses, we have to compare what we observed in real data with predictions based on simulated data. This requires normalizing the MC simulated samples based on the data sample's integrated luminosity. To accomplish this, each MC event is given the following weight:

w = s EFT • L • w EFT Â i w EFT i (E.1)
where, L is the integrated luminosity and for Run 2, L = 139 fb 1 . s EFT is the EFT cross-section of the simulated t t event at a particular EFT point. w EFT is the event weights at a particular EFT point. Further development of Equation E.1 can be accomplished by evaluating s EFT :

s EFT = s SM •  i w EFT i  i w SM i (E.2)
where s SM is the SM cross-section,  i w SM i is the sum of weights in the SM and  i w EFT i is the sum of weights at a particular EFT point. Therefore the final expression for EFT event weight should be:

w = s SM • L • w EFT Â i w SM i (E.3)
The predicted value of s SM for top quark pair cross section at NNLO+NNL precision at 13 TeV with top

F. Dim6Top and SMEFT@NLO F.1 Introduction

In this chapter, we propose a procedure to cross-validate Monte Carlo implementations of the standard model effective field theory based on a study of two models, SMEFT@NLO model and DIM6TOP model. We consider Leading Order (LO) calculation 1 . SMEFT@NLO and DIM6TOP models implementations are based on Lagrangian written in codes such as FeynRules [START_REF] Alloul | FeynRules 2.0-A complete toolbox for tree-level phenomenology[END_REF] and then exported through standard format to matrix-element-based MC generator, MadGraph5_aMC@NLO. The goal is to change SMEFT@NLO and DIM6TOP default setting so both models have the same physical predictions, this can be uniquely defined when the input parameters, the electroweak couplings input scheme employed and the order of the computations are set correctly in both models.

In the following section, a brief overview of SMEFT@NLO and DIM6TOP implementations with links and references will be given, then, a detailed description of the analysis procedure along with results will be given in section.F.3.

F.2 Model implementation

In this section, a brief description of public Standard Model Effective Field Theory (SMEFT) implementations along with their references:

• DIM6TOP is UFO model implementation of EFT interpretation of top-quark measurements at the LHC which can be used for Monte Carlo simulation [START_REF] Aguilar Saavedra | Interpreting top-quark LHC measurements in the standardmodel effective field theory[END_REF].

• SMEFT@NLO is UFO model implementation of one-loop computations in the standard-model effective field theory at dimension six. SMEFT@NLO can also be used for Monte Carlo simulation [START_REF] Degrande | Automated one-loop computations in the standard model effective field theory[END_REF].

F.3 Parameters matching

The MC implementations of new physics models is in general a very time-consuming exercise, in particular, for EFTs models which include a very large number of interactions and parameters. Furthermore, each EFT model rely on different default setting, operator bases, electroweak input parameters, particle decay width and tools 2 . The goal of our approach is make both EFTs models, DIM6TOP and SMEFT@NLO, compatible so they can predict the same physical distributions. In our analysis, we use cross-section and top pair spin observables to cross validate our implementations.

We propose the following procedure in order to make both EFTs models compatible :

• Each EFTs model uses different electroweak input scheme for instance, by default, DIM6TOP model uses (aEWM1, G f , m Z ) as electroweak inputs where aEWM1 is 1/a, the fine structure constant. Either this or m W (together with G f and m Z ) can be used as inputs to set the rest of electroweak parameters (weak mixing angle, g, g', etc.) which is the case for SMEEFT@NLO model. As inputs are different, different numerical results will be obtained in the two cases. Therefore, one needs to adopt that same electroweak inputs in both EFTs models.

DIM6TOP model is more flexible than SMEEFT@NLO model. In DIM6TOP model, either (aEWM1,G f ,m Z )

or (G f ,m W ,m Z ) can be used, although, SMEEFT@NLO model allow only for (G f ,m W ,m Z ) to be used.

Consequently, we are adopting (G f ,m W ,m Z ) as electroweak inputs for each model.

• The numerical values as input parameters : G f =1.166370.10 05 GeV 2 , m Z =91.18760 GeV, m W =79.82440

GeV, m H =125GeV, m T =172 GeV, a S (m Z )=0.1184, y m T =172 GeV, y t =0.0.

• The bottom quark is massive by default. Switching to the five-flavour factorization scheme by setting m B =0.0 and y m B =0.0.

• Set all dimension-six operator coefficients explicitly to zero especially for SMEFT@NLO model.

For our propose, we will set one operator to a non zero value (more details in the following section).

• MADSPIN package [START_REF] Artoisenet | Automatic spin-entangled decays of heavy resonances in Monte Carlo simulations[END_REF][START_REF] Frixione | Matching NLO QCD computations with parton shower simulations: the POWHEG method[END_REF] is used to perform tree level decays, accounting for leading order spin correlations. The same information about the branching fractions of the decayed particles should be included in both model.

• Same seed is used both in run card and MADSPIN card, ensuring that we get statistically dependent events.

• Same PDF set is used which is NNPDF30_nlo_as_0118 (with the LHAPDF ID = 260000).

SMEFT@NLO normalizes the top chromo-dipole operator (with Wilson coefficient c tg ) with a factor of the strong coupling constant a S which means that c DIM6TOP tg = a S ⇥ c SMEFT@NLO tg . Note that a S runs and that its value at the scale m Z depends on the parton distribution function employed.

DIM6TOP can be configured to normalize all chromo-dipole operator coefficients with a factor of the strong coupling a S . For instance, turning on this option would yield a cross section 20% bigger than the normal case (where this option is disable) 3 .

While using this option, one need to generate MC sample specifying squared coupling orders ("DIM6 2 ==1" and "NP 2 ==2") in the "generate" lines otherwise, the effect of normalising chromo-dipole operator (e.g c tg ) with a factor of the strong coupling a S is disabled. Unfortunately, Madgraph@NLO versions does not 2 Such as Rosetta which allows for the translation between different bases of effective field theory operators. Each EFT model uses his own tools 3 This can achieved by generating two samples (with and w/o normalize chromo-dipole operator) with the same following syntax (lets consider DIM6TOP model with c tg is the only non zero operator with a value equal to 1): p p > t t ⇠ QED=0 QCD=2 FCNC=0 DIM6=1 DIM6 2 ==1. Where the DIM6 2 (NP 2 when using SMEFT@NLO model) specifies the coupling orders with one insertion of EFT operator. The two samples should predict a cross section with 20% difference.

F.4 Sample generation 201

support squared coupling orders together with MADSPIN (or decay chains). As a result, we consider working with C tq8 coefficient instead of c tg coefficient to cross validate the new implementation 4 .

The above changes are only proven to be working using Madgraph@NLO version v2_9_1_2.

F.4 Sample generation

The new setup introduced in the previous section F. [START_REF] Feynman | Space-Time Approach to Non-Relativistic Quantum Mechanics[END_REF] should then be included in the restriction card of each model. In order to test the effect of our configuration in both EFTs models, we compute for each top pair spin observables at the particle level (e.g. C(k, k)), the linear a and quadratic b contributions of dimension-six operators, in addition to the SM contributions (e.g. C(k, k) SM ). Schematically, for C(k, k) observable, the expansion in powers of dimension-six operator coefficients can the be written as :

C(k, k) EFT ⇡ C(k, k) SM + a Ctq8 C tq8 L 2 + b Ctq8 C 2 tq8 L 4 (F.1)
Where C tq8 is four-quark Wilson coefficient chosen for this study. In general, the computation of a

F.5 Results

In this section, we present the effect of our approach to make DIM6TOP model and SMEFT@NLO model compatible, so they can predict the same value of cross section and the same value of a Ctq8 and b Ctq8 .

Cross section

A first possibility is to compare total cross-section for producing given final states. The total cross-section of different benchmarks point is shown in Figure F.1. The result shows a good agreement between DIM6TOP model and SMEFT@NLO model, in addition the total cross-section followed the correct quadratic behaviour as expected. The value of C(k, k), the value of a Ctq8 and b Ctq8 predicted by both models are compatible within statistical uncertainties. Despite having the same number of event, the same kinematic and the same seed, the generated data are statistically fully independent between benchmarking point of two models 5 which explains the difference between the value of C(k, k) predicted by DIM6TOP model and SMEFT@NLO model. This discrepancies could be solved by generating MC sample using reweighing method as it will be explained in the following section. 4 In our case, using C tq8 is a prefect choice for our comparison because, both models have the same definition of C tq8 operator as opposed c tg , where it is normalize with a factor of strong coupling in SMEFT@NLO model and it is not in DIM6TOP model. 5 On the contrary, samples that are generated within the same EFT model are 100% correlated between benchmarking point. ) is fixed while fitting in ?? and it is not fixed in ?? .

Spin observables

F.5.1 Reweighting

Reweighing consists in using a sample of events generated under given values of C tq8 and in associating with those events an additional weight that corresponds to a new value of C tq8 . In general, this has a cost in term of statistical precision since weighted events have larger statistical uncertainty that an unweighted sample (shown in section F.4, e.g standalone) with the same number of events.

Spin observables. Generating MC sample using Reweighing method and following our approach in matching both models, we can now see in perfect agreement the distribution of C(k, k) and the predicted value of a Ctq8 and b Ctq8 (e.g Figure ??). This confirms the difference seen in section F. The distribution of cos(q ) k + cos(q ) k of the t t system, with the differential bin-by-bin K-factor s NLO /s LO displayed in the lower panel. The distribution of cos(q ) k + cos(q ) k of the t t system, with the differential bin-by-bin K-factor s NLO /s LO displayed in the lower panel.

Cross-diagonal spin correlation observables

  théorie ultime. Il y a d'abord les observations expérimentales bien établies d'oscillations de neutrinos qui sont impossibles dans le SM. Deuxièmement, certaines valeurs des paramètres SM ne sont pas calculables dans la théorie, notamment la hiérarchie de masse des fermions. Par conséquent, la théorie a beaucoup trop d'arbitraire pour être l'histoire finale. Ces lacunes et d'autres du SM ont motivé l'effort de construire des théories au-delà du modèle standard (BSM). Pour collecter les particules produites par pp collisions, le LHC dispose de quatre détecteurs de particules: A Toroidal LHC ApparatuS (ATLAS) et Compact Muon Solenoid (CMS) sont des expériences généralistes tandis que LHCb et ALICE sont destinés respectivement à la physique des beaux hadrons et à l'étude des quarks et des gluons. ATLAS est le plus grand détecteur de particules au monde. L'un de ses objectifs les plus importants a été la recherche du boson de Higgs et de la recherche nouvelle physique. Le détecteur ATLAS a les dimensions d'un cylindre, 46 m de long, 25 m de diamètre, et se trouve dans une caverne à 100 m sous terre. Le détecteur ATLAS pèse 7 000 tonnes, soit le poids de la tour Eiffel. Le détecteur lui-même est un instrument à plusieurs couches conçu pour détecter certaines des particules les plus minuscules mais les plus énergétiques jamais créées sur terre. Près du centre, nous commençons par les détecteurs de piste internes (Inner Tracker, ITK), qui mesurent les trajectoires des particules chargées près du point d'interaction. Calorimètres électromagnétiques (EM) et hadroniques, qui mesurent l'énergie déposée par les électrons, les photons et les jets hadroniques. Les calorimètres sont entourés du spectromètre à muons, la couche la plus externe, qui est conçue pour mesurer la trajectoire des muons. Dans un effort pour étendre le potentiel de découverte et répondre aux questions les plus fondamentales du modèle standard et des scénarios qui en sont dérivés (tels que BSM), le LHC prévoit une mise à niveau pour augmenter le nombre total de collisions d'un facteur de 10 par rapport à sa valeur nominale initiale et collecter davantage de données. Il s'agit du projet LHC à haute luminosité (HL-LHC) qui devrait être opérationnel fin 2027. Avec le HL-LHC, les physiciens peuvent étudier plus en détail des mécanismes connus, tels que le boson de Higgs, observer d'éventuels nouveaux phénomènes rares et explorer des processus physiques qui vont au-delà du modèle standard (BSM), par exemple en recherchant des particules supersymétriques légères. La forte augmentation des interactions simultanées, connue sous le nom de PileUp (PU), est l'un des principaux défis expérimentaux clé du programme de physique du HL-LHC, avec une moyenne de 200 interactions pp simultanées par croisement de paquets. Le résultat est une densité pileup moyenne de 1.8 vertices/mm, où la densité pileup est définie comme le nombre de collisions par unité de longueur le long de l'axe du faisceau pendant le croisement d'un paquet. Cette valeur doit être comparée à 0.24 vertices/mm pour Run-2. Par conséquent Le détecteur ATLAS doivent être améliorés pour gérer cet afflux de signaux et l'augmentation de radiation. L'une des mises à niveau les plus importantes est le détecteur de temps hautement granulaire (High-Granularity Timing Detector, HGTD). HGTD réduira l'interaction de pileup en ajoutant une mesure temporelle de haute précision aux trajectoires des particules chargées ainsi que des informations spatiales qui amélioreront la reconstruction des objets devant, en particulier les jets et les leptons. Le HGTD sera situé dans la région de l'espace entre le barrel et le end-cap calorimeter, à une distance d'environ ±3.5m du point d'interaction nominal, ce volume est actuellement occupé par Minimum-Bias Trigger Scintillators qui sera enlevé. Détecteur de temps hautement granulaire: calibration du temps La première partie de la thèse est la calibration du temps d'impact des particules mesuré par chaque capteur HGTD sur la base d'une simulation dédiée. Le capteur utilise la technologie Low Gain Avalanche Diode (LGAD) qui fournit un gain modéré permettant une bonne résolution temporelle. Le capteur est lu par un ASIC dédié, connecté par un procédé de bump bonding, constituant un module nu. Ce module nu est ensuite collé sur un module FLEX sur lequel les signaux électriques sont reliés par câblage filaire. En utilisant les mesures du temps fournies par HGTD, la simulation a estimé la contribution irréductible et non déterministe de l'horloge à la résolution temporelle provenant principalement de la gigue d'horloge lpGBT, du câble FLEX, du FILEX et de l'ASIC. De plus, les effets dépendant du temps et les effets statiques sont également simulés dans cette étude. Tous les canaux HGTD sont censés être idéalement synchronisés par rapport à l'horloge de croisement de faisceaux.

  but are not described by the Standard Model; technically also anything that has to do with gravity, since gravity is not part of the Standard Model. BSM also refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the inability to explain the fundamental parameters of the standard model, the strong CP problem, neutrino oscillations, matter-antimatter asymmetry, and the nature of dark matter and dark energy. As a general definition, BSM can be interpreted as any possible extension of the Standard Model, regardless of whether it resolves any of the puzzles left unsolved by the Standard Model. Supersymmetry, string theory, M-theory, and extra dimensions are examples of theories that lie beyond of the Standard Model.To determine which theory is the correct or, at the very least, the 'best step' towards creating a Theory of Everything, experiments are required, and this is one of the most active areas of research in theoretical and experimental physics. For a large class of models beyond the SM, physics at energies below the mass scale L of the new particles can be parameterized by an Effective Field Theory (EFT) where the SM Lagrangian is supplemented by new operators with canonical dimensions D larger than 4. EFTs are characterized by their model-independence, so instead of focusing on every single microscopic BSM model, one can focus on the most relevant EFT

Figure 2 . 1 .

 21 The proton and neutron are the bound states of more fundamental particles named quarks, with the proton consisting of two up-quarks and a down-quark, p(uud), and the neutron consisting of two down-quarks and an up-quark, n(ddu). Unlike the constituents of atoms and nuclei, quarks have not been observed as stable isolated particles but are always confined to bound states called hadrons such as proton and neutron.

Figure 2 . 1 :

 21 Figure 2.1: From atomic physics to modern particle physics at TeV scale.

Figure 2 . 2

 22 Figure 2.2The following Sections of this chapter remind readers of the mathematical formalism and fundamental aspects of theories that construct the SM.

  The three forces described by the Standard Model each correspond to an underlying gauge symmetry, in which the Lagrangian remains invariant under a set of local phase transformations belonging to a Lie group, called gauge invariance. When the classical gauge field is quantized, the resulting quantum is referred to as a gauge boson, and the resulting Lagrangian obeys the symmetry group associated with the transformation.In the development to come, the strong interaction is invariant under the SU(3) c transformation group and the combination of the electromagnetic and weak forces (i.e. the electroweak force) is invariant under the nonabelian 2 special unitary group gauge transformation the SU(2) L ⌦U(1) Y term. As a result, the Lagrangian of the Standard Model is a non-abelian gauge theory with the symmetry group SU(3) c ⌦ SU(2) L ⌦U(1) Y .

Figure 2 . 3 :

 23 Figure 2.3: The predicted QED interaction vertex arising from the requirement of U(1) local gauge invariance.

Figure 2 . 4 :

 24 Figure 2.4: The predicted QCD interaction vertex arising from the requirement of SU(3) local gauge invariance. (a) quark gluon vertex; (b) 3 gluons vertex; (c) 4 gluons vertex. Source: Wikipedia

  Figure 2.6: b decay through the weak interaction

Figure 2 . 7 :

 27 Figure 2.7: An illustration of the Higgs potential for µ 2 0 and µ 2 < 0.

Figure 2 . 7 ,

 27 Figure 2.7, and there are an infinite number of ground states with a non-zero vacuum expectation value. The electroweak symmetry is spontaneously broken by selecting one of these vacuum states, which is usually

Figure 2 . 8 :

 28 Figure 2.8: Summary of several Standard Model total and fiducial production cross section measured by the ATLAS experiment, compared to the corresponding theoretical expectations [41].

Figure 2 . 9 :

 29 Figure 2.9: Summary of the ATLAS and CMS measurements from top quark decay. The results are compared with the LHC and Tevatron+LHC m top combinations [52].

Figure 2 .

 2 Figure 2.11 summarizes the cross sections measured at the LHC and Tevatron as a function of centre-of-mass energy. Based on these measurements, the measured cross section values agree well with the theoretical predictions. Furthermore, this figure shows that above the centre-of-mass of 10 TeV, the proton-proton and proton-antiproton processes have almost the same production cross sections.

Figure 2 . 11 :

 211 Figure 2.11: Summary of LHC and Tevatron measurements of the top-pair production cross section as a function of the centre-of-mass energy compared to the NNLO QCD calculation complemented with NNLL resummation (top++2.0). The theory band represents uncertainties due to renormalisation and factorisation scales, parton density functions and the strong coupling. The measurements and the theory calculation are quoted at m top = 172.5 GeV. Measurements made at the same centre-of-mass energy are slightly offset for clarity [55].

Figure 2 .

 2 Figure2.12: A categorization of the branching fractions for t t decays by the decay modes of W bosons[START_REF] Heinson | Feynman Diagrams for Top Physics Talks and Notes[END_REF]

Figure 2 Figure 2 . 14 :

 2214 Figure 2.14: The scattering process in the centre-of-mass frame (a), in the top quark rest-frame (b) and in the top anti-quark rest-frame (c). s t (s t ) is the top (anti-top) spin axis [62].

Figure 2 .

 2 Figure 2.15: (a) The non-zero spin configurations for top quark pair production via quark-antiquark annihilation or unlike helicity gluon fusion in the off-diagonal basis. (b) The non-zero spin configurations for top quark pair production via like helicity gluon fusion in the helicity basis [64].

Figure 2

 2 Figure2.16: Fraction of t t MC particle level events passing a cut on t t invariant mass for like and unlike helicity gg and q q initial states.

Figure 2 .

 2 Figure2.18: Fraction of t t MC particle level events passing a cut on t t invariant mass for like and unlike helicity gg and q q initial states. Dots represent the fraction of evens that pass the selection M t t >cut.

18

 18 Figure2.18: Fraction of t t MC particle level events passing a cut on t t invariant mass for like and unlike helicity gg and q q initial states. Dots represent the fraction of evens that pass the selection M t t >cut.

  [START_REF] Appelquist | Infrared singularities and massive fields[END_REF] where B â , B b and C( â, b) are the polarisations and spin correlation observables along the reference axis â, b.

  and c i are functions of the partonic CM energy squared p s and y = p • k which is equal to the top-quark scattering angle cos q . Due to the CP-invariance in the SM, the top quark and anti-quark have the same polarization coefficient functions b + i = b i . The coefficient functions b ± r and b ± k are induced by P-violating interactions and exhibit longitudinal t and t polarizations, while the coefficient functions b ± n are generated by P-and CP-even absorptive parts in the scattering amplitudes and leads to a transverse polarisation. At tree-level in QCD, b ± n disappears due to T-invariance (b ± n (y) ! b ± n (y)) [30]. CP-invariance of the SM depends on the C matrix being symmetric, which implies that the CP-odd coefficient functions vanish: c k = c r = c n = 0. The coefficient functions c kn and c rn are generated in SM by the absorptive parts of the 1-loop mixed QCD-weak corrections, and they are very small. Moreover, the diagonal correlations c kk , c rr , c nn , and c rk are generated by P-and CP-conserving interactions, in particular by QCD at the tree level, and therefore they are not zero at LO. Deviations from these expectations would indicate

Figure 2 . 20 :Figure 2 . 21 :

 220221 Figure 2.20: Distribution of the spin correlation and polarisations observables for particle level aMC@NLO events at p s = 13 TeV.

Figure 2 . 22 :

 222 Figure 2.22: Effective field theories: top-down vs. bottom-up [77]. v represents the low-energy scale.

M 2 1 2 (Figure 2 . 23 :

 22223 Figure 2.23: The tree-level elastic scattering of two light scalar fields yy ! yy, which proceeds through the exchange of a heavy scalar F (left). After integrating out scalar leads to 4-fermion interaction (right). The Wilson coefficients, c S of the new interactions can be determined by matching the high-energy and low-energy theories at the scale M. At Tree level, Matching can be done by comparing the amplitude of yy ! yy of the full theory and EFT:

i are a set of dimensionn

  operators that respect the SU(3) c ⇥ SU(2) L ⇥ U(1) Y gauge invariance of the L SM , and c (n) i are their

  taking advantage either of the gauge field EOMs or of the Bianchi identity, e.g. D [r X µn] = 0 moved the remaining operators of this class to the bosonic class j 4 D 2 or the fermionic class Y 2 FD 2 .

105 )Y 2 D 3 :Y 2 jD 2 : 4 Y 2 Y 2 j 3 , Y 2

 10523242232 By ordering of the derivatives, operator get reduced by the EOMs to the class Y 2 jD 2 . As previous cases, using ⇥ D µ , D n ⇤ ⇠ X µn and EOMs, this class is reduced to operator in the single fermionic current classes Y 2 j 3 , Y 2 Xj and Y XD: Taking advantage either of the gauge field EOM and of the Bianchi identity, all operators of this class are reduced to the bosonic class Y 2 Xj, Y 2 j 2 D and Y 4 . The remaining non-redundant operators of the single-fermionic current operator classes are thus all in the Xj and Y 2 j 2 D classes. They appear in columns 3, 5 and 6 of Figure 2.24.

Figure 2 .

 2 Figure 2.24: Dimension-six operators other than the four-fermion ones. The fermion generation indices are not explicitly displayed for readability purposes, but are denoted where relevant by the extra superscript, such as O 23 ej = j † j μ, nµ tj [80].

107 )

 107 are used to eliminate several cases in the ( LL)( LL) and ( RR)( RR) classes. Figure2.25 presents the nonredundant complete set of four-fermion operators. The tables in Figures 2.[START_REF] Greenberg | Spin and Unitary-Spin Independence in a Paraquark Model of Baryons and Mesons[END_REF] and 2.25 provide 59 baryonnumber conserving operators along with 5 baryon-number violating operators, organized into 12 subsets.By identifying different flavours of quarks and leptons, 2499 baryon-number-conserving operators can be specified. Seemingly, this presents a rather large basis comprised only of operators in dimension-six.

Figure 2 . 25 :

 225 Figure 2.25: Four-fermion operators. The fermion generation indices are not explicitly displayed for readability purposes, but are denoted where relevant by the extra superscript, such as O 1231 ll = ēg µ µ ( tg µ e). Colour indices are displayed only for operators that violate the baryon number B [80].

(2. 111 )

 111 Schematic representation of the relevant operators for producing t t at hadron colliders using Feynman diagrams is shown in Figure2.26.

Figure 2 . 26 :

 226 Figure 2.26: Example Feynman diagrams for the EFT effects on the production of two top quarks at LO. The dots indicate possible insertions of a dimension-6 operator. O 4q denotes the insertion of any of the four-quark operators.

  30,0.74] [-0.03,0.82] C 18 Qq [-0.79,0.15] [-1.11,0.49] C 38 Qq [-0.49,0.73] [-0.84,1.16] 41,0.35] [-0.58,0.63]

Figure 3 . 1 .

 31 During Run-2, the LHC delivered outstanding 3.2 The ATLAS experiment at the LHC 69 results, exceeding the design instantaneous luminosity and producing more integrated luminosity than what was expected. The increase of the instantaneous luminosity also resulted in a higher number of interactions per bunch crossing across the years with an average number of hµi = 33.7 for all of Run-2, compared to hµi ⇡ 20 in Run-1. The hµi distribution for each year in Run-2 is shown in Figure 3.1.

Figure 3 . 1 :

 31 Figure 3.1: Integrated luminosity versus time delivered to ATLAS (green) and recorded by ATLAS (yellow) during stable beams for pp collisions at 13 TeV centre-of-mass energy in the LHC Run-2 (left plot). Mean number of interactions per bunch crossing hµi per year in Run-2 (right plot) [92].

3. 2

 2 The ATLAS experiment at the LHC ATLAS (A Toroidal LHC Apparatus)[START_REF] Evans | LHC Machine[END_REF] is the biggest experiment on the LHC ring and a general-purpose detector. The structure is cylindrical, measuring 46 meters in length, 25 meters in diameter, and weighing about 7000 tonnes. The ATLAS detector is depicted in Figure3.2.

Figure 3 .

 3 Figure 3.2: A general scheme of the ATLAS detector showing its various sub-detectors [93].

Figure 3 . 3 :

 33 Figure 3.3: Schematic view of Barrel and Endcap toroids [96].

3. 2 . 1 . 3

 213 The Inner Detector (ID) The Inner tracking detector, or ID, is a very compact and highly sensitive part of ATLAS and it is the closest part to the interaction point. The ID provides a precise measurement of the parameters of charged particles trajectories with excellent vertex and momentum resolution. There are four distinct parts, each of which is based on a different technology depicted in Figure 3.4 and described in what follows. Based on the information gathered by the four sub-detectors, a transverse momentum resolution measure is obtained as follows:

Figure 3 . 4 :

 34 Figure 3.4: The structure of the ATLAS Inner Tracking Detector made of highly granular silicon pixels, silicon strips and straw tubes (left). A schematic representation of a charged particle crossing sub-detector planes (right). Red stars indicate every detector measurement (hit). The fitted track is shown with a black arrow. Green dots indicate the intersection of the track with each surface. The blue lines represent the track-to-hit residual of each layer [97].

Figure 3 .

 3 Figure 3.5 shows the ATLAS calorimetry system [102] located outside the ID. It is used to measure the energy of the particles by absorbing them and measuring their properties, making particle identification easier. Similarly to the rest of the detector, the calorimeter consists of barrel and endcap sections. There are two types of calorimeters in the ATLAS detector: The Electromagnetic Calorimeter (ECal) and the Hadronic Calorimeter (HCal). ATLAS calorimeters are sampling calorimeters, meaning they have active and passive layers alternated. Due to their density, the passive absorber layers create cascades of secondary particles, while the active layers measure the deposited energy. The energy of electrons and photons is measured in the ECal, whereas hadron showers usually begin to form in the ECal, but then proceed to the HCal.

Figure 3 . 5 :

 35 Figure 3.5: Computer generated image of the ATLAS calorimeter, showing its various subsystems [103].

Figure 3 . 6 :

 36 Figure 3.6: Schematic of the ATLAS Muons subsystem [107].

Figure 3 .

 3 7 depicts how different particles might interact with each layer of the detector.

Figure 3 . 7 :

 37 Figure 3.7: Representation of the interaction of various particles and the subsystems of the ATLAS detector [108].

Figure 3 . 8 :

 38 Figure 3.8: Measured electron-identification efficiencies in Z ! e + e events for the Loose (blue circle), Medium (red square), and Tight (black triangle) operating points as a function of E T . The vertical uncertainty bars (barely visible because they are small) represent the statistical (inner bars) and total (outer bars) uncertainties. The data efficiencies are obtained by applying data-to-simulation efficiency ratios that are measured in Jy ! e + e and Z ! e + e events to the Z ! e + e simulation. The bottom panel shows the data-to-simulation ratios [112].

Figure 3 . 9 :

 39 Figure 3.9: Efficiency of the lowest unprescaled single electron trigger combination in 2015 and 2016-2018 data, compared to Z ! ee Powheg Pythia Monte Carlo. A total of 47.8 fb 1 of proton-proton collision data at a center-of-mass energy of p s = 13TeV is used for 2018 data, whereas for 2015, 2016 and 2017 3.2 fb 1 , 32.9 fb 1 and 43.9 fb 1 are used, respectively. The efficiency is given as a function of the offline electron transverse energy, E T , where the offline electron fulfills a tight offline identification requirement as well as a tight offline isolation requirement. The trigger efficiency in 2015 is highest because no online isolation requirement and a looser online identification requirement was applied. The inefficiency observed in 2016 is due to too stringent online identification optimisation with respect to the previous offline selection criteria. The efficiency improvements in 2017-2018 are due to usage of more effective online algorithms (introduction of neural network based ringer selection) which allowed better alignment of the online selection (looser track-calorimeter matching, removal of likelihood-only calorimeter selection) with the final offline selection for Run 2[START_REF] Hryn | Single electron trigger 2018 performance plots[END_REF].

Figure 3 .

 3 Figure 3.10 illustrates the reconstruction efficiency for the Medium quality working point, used throughout this document, as a function of h for a calibration sample of Z ! µ + µ events. The identification efficiency varies from 60% at |h| < 0.1 to around 99% in 0.1 < |h| < 2.5 for the Medium operating point, while the efficiency for the Loose operating point is 96% at |h| < 0.1. As with electrons, the Prompt Lepton Isolation Tool is used to reject a large number of non-prompt muons.

Figure 3 . 10 :

 310 Figure 3.10: Muon reconstruction efficiency as a function of h measured in Z ! µ + µ events for muons with p t > 10GeV shown for Medium muon selections. In addition, the plot also shows the efficiency of the Loose selection (squares) in the region |h| < 0.1 where the Loose and Medium selections differ significantly. The error bars on the efficiencies indicate the statistical uncertainty. Panels at the bottom show the ratio of the measured to predicted efficiencies, with statistical and systematic uncertainties [115].

Figure 3 . 11 :

 311 Figure 3.11: Absolute efficiency of Level 1 (L1) MU20 trigger and absolute and relative efficiencies of the OR of mu26_ivarmedium with MU50 High Level Triggers (HLT) plotted as a function of p T of offline muon candidates in the barrel detector region.The efficiency is computed with respect to offline isolated muon candidates which are reconstructed using standard ATLAS software and are required to pass Medium quality requirement. The MU20trigger requires that a candidate passed the 20 GeV threshold requirement of the L1 muon trigger system. The mu26_ivarmedium trigger is seeded by the MU20 trigger and requires that a candidate satisfied a 26 GeV HLT threshold and passed a medium isolation selection computed using inner detector tracks reconstructed online by the HLT within a cone with a variable size which depends on the p T of the muon. The MU50 trigger is seeded by MU20 at L1 and requires that a candidate satisfied a p T threshold of 50 GeV. The efficiency is measured using a tag-and-probe method with Z ! µµ candidates, with no background subtraction applied, in 13 TeV data from 2018. Only statistical data uncertainties are shown[START_REF] Morgenstern | Muon trigger efficiency in early 2018 data taking[END_REF].

Chapter 3 .

 3 The ATLAS experimentb-tagged jetsThe jets that derive from a b-quark (so-called b-jets) have certain properties that set them apart from jets originating from lighter quarks or gluons. In particular, hadrons containing b-quarks have longer lifetimes, around 1.5 ⇥ 10 12 s, which allows them to travel several millimetres in the transverse direction before decaying. Efforts are made to identify b-jets at reconstruction levels. This is known as b-tagging.This analysis uses theDL1r algorithm for b-tagging [121]. The b-tagging algorithm is based on a deep feed-forward neural network (NN). The inputs to the DL1r tagger are the outputs of IP2D/IP3D 2 [121], SV1 3 [122], JetFitter 4 [123], and JetFitter c-tagger. The algorithm produces three output values, representing the probabilities for a given jet to have originated from a c-, b-or a light quark. These probabilities are then combined into a single score. This score is used as a continuous discriminant between b-tagged jets and nonb-tagged jets. Different operating points can be defined, each with a different b-tagging efficiency. Usually these working point are defined to provide 60%, 70%, 77% and 85% b-tagging efficiency. When a jet passes the selected b-tagging algorithm working point, it's called a b-tagged jet. The 85% efficiency working point is used in this thesis.

3. 3

 3 Monte Carlo Event Generation 87 specific decays or final states.

Figure 4 .

 4 [START_REF] Yukawa | On the Interaction of Elementary Particles I[END_REF] shows the resolution, s z 0 , of the longitudinal track impact parameter, z 0 , measured by the ITk as a function of |h|, for muons with p T = 1 GeV and p T = 10 GeV. For a good spatial separation between Hard Scattering (HS) and pileup, s z 0 , it should be significantly better than the inverse of the average pileup density, 600 µm.

Figure 4 .

 4 Figure 4.1 shows that, in the central region, s z 0 is well below this limit. In the forward region, however, the resolution exceeds the limit by a large factor, reaching 3 mm for particles with low transverse momentum at |h| ⇡ 4, due to the combination of geometric projection and increased material. The precise timing enables the separation of vertices when the z 0 resolution becomes higher than the distance between two vertices, thereby reducing the number of vertices that are considered for a certain track. The time dispersion as a function of the truth interaction of z is illustrated in Figure4.2.

Figure 4 . 1 :

 41 Figure 4.1: Resolution of the longitudinal track impact parameter, z 0 , as a function of |h| for muons of p T = 1 GeV and p T = 10 GeV using ITk alone. [139]

Figure 4 . 2 :

 42 Figure 4.2: Visualization of the truth interactions in a single bunch crossing in the z-t plane, showing the simulated Hard Scatter (HS) t t event interaction (in red) with pileup interactions superimposed (in black) for hµi = 200.

4. 2

 2 High-Granularity Timing Detector (HGTD) 91

Figure 4 . 3 .

 43 The active area of the detector is 120 to 640 mm from the axis of the beam, and its thickness should not exceed 125 mm including 50 mm of the moderator to protect the ITk and the HGTD from backscattered neutrons. Each HGTD end-cap is the integration of two instrumented double sided layers (mounted on two cooling/supporting disks), where the layers are rotated in opposite directions with respect to one another by 15 to 20 in order to maximize the uniformity in the number of hits per track.

Figure 4 . 3 :

 43 Figure 4.3: Position of the HGTD within the ATLAS Detector. The HGTD acceptance is defined as the surface covered by the HGTD between a radius of 120 to 640 mm at a position of z = ±3.5m along the beam-line, on both sides of the detector [139].

Figure 4 . 4 :

 44 Figure 4.4: Global view of the HGTD to be installed on each of two end-cap calorimeters (±3.5m).The various components are shown: hermetic vessel (front and rear covers, inner and outer rings), two instrumented double-sided layers (mounted in two cooling disks with sensors on the front and back of each cooling disk), two moderator pieces placed inside and outside the hermetic vessel.[START_REF]Technical Design Report: A High-Granularity Timing Detector for the ATLASPhase-ii Upgrade[END_REF]

Figure 4 .

 4 Figure 4.5 shows the position of the module on the front side (left plot) and back side (right plot).

Figure 4 . 5 :

 45 Figure 4.5: Position of modules and readout rows numbering for the front side of one disk (left plot) and for the back side of the same disk (right plot). The smallest radius at 120 mm and maximal radius at 660 mm are shown. The 660 mm radius is the minimal target for the external instrumented area. 230 to 470 mm radius are shown as typical limits of the different parts of the support units. [139]

Figure

4. 6 2 total = s 2 L + s 2 elec + s 2 clock 93 Figure 4 . 6 :

 622229346 Figure 4.6: Schematic drawing of two adjacent modules on the top side and one on the bottom side of the cooling plate [139].

2 94 Chapter 4 .

 944 lpGBT is a radiation tolerant ASIC for Data, Timing, Trigger and Control Applications in HL-LHC High-Granularity Timing Detector: t 0 calibration is used rather than a three-ring (r < 230 mm, 74% overlap, 230 mm < r < 470 mm, 54% overlap, r > 470 mm, 20%) layout. An overview of the module positions in both cases is shown in Figure4.7.

Figure 4 . 7 :

 47 Figure 4.7: Figures showing the placement of the modules in the two-ring (left) and the three-ring (right) layouts [139].

Figure 4 . 8 :

 48 Figure 4.8: Hits multiplicity distribution per module for a simulated 1 434 t t events at a pileup of hµi = 200. Distributions correspond to hits multiplicity for different HGTD layers where r < 320 mm has 80% overlap and r > 320 mm has 20% overlap.

Figure 4 . 9 :

 49 Figure 4.9: The trajectory of HGTD hits. The path length of the particle's track is assumed as a straight line between the origin of the track and the position of a given hit.

Figure 4 .

 4 Figure 4.9 shows the trajectory of HGTD hits. As the particles propagate through the HGTD geometry, the various interactions between the particles and the detector material are taken into account. The simulation propagates particles step-wise through the material of the volumes in the detector model and produces energy depositions at specific points in space and time.

Chapter 4 .

 4 High-Granularity Timing Detector: t 0 calibration

Figure 4 . 10 :

 410 Figure 4.10: Comparison of the average t hits as a function of hits distance to beam for different HGTD layers located at a position z > 0 (z = ±3.5m). The simulated 1 434 t t events at a pileup of hµi = 200 is used. The effects of ToF is clearly visible.

Figure 4 .

 4 Figure 4.11 displays the collision point time shift for the ATLAS detector. The measurement for each bucket corresponds to the average value over 100 turns, leading to a standard deviation of 6 ps. A variation of this sort will affect the t hits values measured by various HGTD pixels.

Figure 4 . 11 :

 411 Figure 4.11: Figure shows the timing variation of LHC. The measurement for each bucket corresponds to the average value over 100 turns, leading to a standard deviation of 6 ps [143].

Figure 4 . 12 :

 412 Figure 4.12: The Time simulation procedure scheme.

Figure 4 . 13 :

 413 Figure 4.13: The off-detector and on-detector architecture.

Figure 4 . 14 :

 414 Figure 4.14: Figure shows timing jitter assumed for lpGBT in the corrected (blue) and uncorrected (red) scenarios [143].

Chapter 4 .

 4 High-Granularity Timing Detector: t 0 calibration 100 ps stepped adjustment. In this way, the time and luminosity windows for each individual ASIC is centred on the bunch crossing clock. This means that the step of tuning the readout time window of the pixels is 100 ps. Figure4.15 depicts the impact of the PS on the t hits distribution.(a) t hits before applying PS.(b) t hits after applying PS.

Figure 4 . 15 :

 415 Figure 4.15: Figures showing the t hits before (a) and after (b) PS is applied. The effect of ToF is partially absorbed by the phase shifter, as shown in Figure 4.16. This Figure is to be compared with Figure 4.10 in which the phase shifter is not applied. Because t 0 calibration mainly involves subtracting ToF variation as well as electronics imperfections, this can be viewed as time pre-calibration.The phase shifter moves at a 100 ps step, thus creating a sawtooth shape on the points.

Figure 4 .

 4 Figure 4.16:The average t hits as function of hits distance to beam for different HGTD layers using simulated 1 434 t t events at a pileup of hµi = 200 in z > 0. The effects of ToF are mainly absorbed after applying the phase shifter, as can be seen by comparing with Figure4.10 where the phase shifter is not applied. The sawtooth-shape for the points is cause by the phase shifter with step of 100 ps.

4. 5

 5 t 0 calibration methodology 101 electronics. As introduced in Section 4.4, the source of clock jitters is parametrized within the MC simulation.

Figure 4 . 17 :

 417 Figure 4.17: Figures showing two timing scenarios as a function of event index using a t t sample of 1 434 events with a pileup of hµi = 200 in z > 0: (a) Time Full and (b) Time Full + 1 kHz.

Figure 4 . 18 :

 418 Figure 4.18: Comparison of the number of hits applicable for the determination of the calibration constants for various special granularities approaches in the t t sample of 1 434 events with a pileup of hµi = 200 in z > 0.

Figure 4 .

 4 Figure 4.19 shows the time-related calibration constant via the Event Index. The calculation of the calibration constants is carried out at a short time interval (every 10 events) with two different special granularities: the calibration per row, and the calibration per module.Figure 4.20 shows the same graphs with a time interval

Figure 4 .

 4 [START_REF] Goldstone | Field theories with « Superconductor » solutions[END_REF] shows the same graphs with a time interval of 50 events. The timing model chosen for this study is Time Full 1 kHz, described in detail in Section 4.4.5.Detailed studies are performed using a simulated sample of 1 434 t t events with an average of 200 pileup interactions overlaid.The Figure4.19 and 4.20 demonstrate the dependence of the calibration constant with the time variations.

Figure 4 . 19 :

 419 Figure 4.19: Figures showing the calibration constant computed every 10 events using (a) calibration per row and (b) calibration per module. The t t sample of 1 434 events with a pileup of hµi = 200 is used. The z-axis shows the number of entries.

Figure 4 . 20 :

 420 Figure 4.20: Figures showing the calibration constant computed every 50 events using (a) calibration per row and (b) calibration per module. The t t sample of 1 434 events with a pileup of hµi = 200 is used. The z-axis shows the number of entries.

4. 6

 6 Results 105 calibration constants per event.Figure 4.21 represents the error of the mean (s / p N hits ) as a function of the module radius.

Figure 4 . 21 :

 421 Figure 4.21: The statistical precision as a function of the module radius using t t sample of 1 434 events with a pileup of hµi = 200.

Figure 4 . 23 . 106 Chapter 4 .

 4231064 The t hits distribution before (in black) the timing correction procedure consists of a Gaussian core derived from the time dispersion of the LHC collisions convolved with the combined hit time resolution of the sensor, all sources of conservative jitters, timing variation of 100 ps bias, and time-of-flight, as explained in sections 4.4.3 and 4.4.5. Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200. The calibration framework for timing calibration procedure is developed11 .The mean value of the calibrated time distribution is centred at 0 for particles created at vertex z = 0, t 0 . Non-Gaussian tails arise from late particles, backscatter, cumulative effects of time-of-flight, clock propagation High-Granularity Timing Detector: t 0 calibration

Figure 4 .

 4 Figure 4.22:The statistical precision of calibration constants as a function of variation period, and different choices of special granularities using t t sample of 1 434 events with a pileup of hµi = 200.

Figure 4 . 23 :

 423 Figure 4.23:The t hits distribution before and after the calibration procedure. The Figure shows t hits distribution before (black) and after the timing correction procedure. The calibration per module shows a better value of t 0 of (0 ± 323 ps). Non-Gaussian tails arise from late particles, backscatter, and other effects. The t hits distribution (shown in black) is obtained from the HGTD simulation described in Section 4.2. Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200.

Figure 4 .

 4 Figure 4.24: The average hits time variation as function of event index. The calibration per module is used. Time ASIC is t hits taken from simulation after applying the phase shifter. Time Full 1 kHz is t hits with the contribution from the sensor and electronics jitters and the collision time spread with additional sources of time jitter from the ASIC, FELIX, flex cable, lpGBT, and ATLAS collision time drift also, with an additional sinusoidally varying 100 ps offset of variable period. The Figure shows the variation of all injected jitters before (in black) and after the calibration per module using different event interval as function of event index. Calibrating t hits directly without splitting the event interval is called independent calibration. For a variation period of less than 100 µs, the calibration procedure will always improve the t 0 precision. For instance, when t hits is calibrated using an interval of a single event (in green), the hit time variation drops to approximately 0, with a small value when t hits is calibrated using an interval of 10 events (in blue). Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200.

Figure 4 . 25 :

 425 Figure 4.25:The average RMS variation of calibrated time hits as function of variation period in t t sample of 1 434 events with a pileup of hµi = 200. Several choices of special granularities are used. The RMS is the time dispersion of hits in the entire HGTD.

Figure 4 .

 4 Figure 4.26: The average time hits before (in black) and after timing correction procedure using short sample size w.r.t modules radius. The calibration per module absorb all ToF contribution to the timing resolution. Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200.

  This chapter summarizes the measurements of the top quark spin polarisation and correlation in top pair production with a data set corresponding to an integrated luminosity of 139 fb 1 of proton-proton collisions at p s = 13 TeV with the ATLAS detector at the LHC. As discussed in Section 2.3, the 15 observables are each sensitive to a different coefficient of the top quark pair's spin density matrix, probing different symmetries in the production mechanism. An earlier study was carried out in ATLAS with LHC Run-1 proton-proton collision data. Data was consistent with SM prediction and no evidence for new physics was found. In this analysis, the author's contribution is Data-MC comparison, the extraction of the polarisation and spin correlation observables by unfolding the data to parton level to account for detector smearing effects, and the interpretation of the results in the context of effective field theory models.

Figure 5 . 1 :

 51 Figure 5.1: Background composition in ee, µµ and eµ channels.

Figure 5 .

 5 Figure 5.1 illustrate background composition in ee, µµ and eµ channels. The main backgrounds are expected

  based on the sum of the transverse momenta of electrons, muons and jets, as well as calorimeter energy deposits not included in reconstructed objects ("soft terms"). In this sense, missing transverse momentum can be defined as follows:

Figure 5 . 2 : 2 :

 522 Figure 5.2: Results of the fit to the p T observable in the (a) e + e and (b) µ + µ .

Figure 5 . 3 :

 53 Figure 5.3: A comparison of reconstruction efficiencies of the different top reconstruction methods [154].

Figure 5 . 4 :Figure 5 . 5 :

 5455 Figure 5.4: Comparison of the leading leptons and dilepton p T distributions between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The last bin of the distribution corresponds to the overflow.

Figure 5 . 6 :Figure 5 . 7 :

 5657 Figure 5.6: Comparison of the anti-top quark p T and h distributions between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The last bin of the distribution corresponds to the overflow.

Figure 5 . 8 :

 58 Figure 5.8: Comparison of the cosq k and cosq k+ distributions between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The first and last bin of the distribution corresponds to the underflow and overflow respectively.

Figure 5 . 9 :

 59 Figure 5.9: Comparison of the cosq r and cosq r+ distributions between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The first and last bin of the distribution corresponds to the underflow and overflow respectively.

Figure 5 . 10 :

 510 Figure 5.10: Comparison of the cosq n and cosq n+ distributions between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The first and last bin of the distribution corresponds to the underflow and overflow respectively.

Figure 5 . 11 :

 511 Figure 5.11: Comparison of the t t spin correlation observables between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The first and last bin of the distribution corresponds to the underflow and overflow respectively.

Figure 5 . 13 :

 513 Figure 5.13: Comparison of the t t spin correlation observables between data and MC prediction after the event selection and the kinematic reconstruction in the ee, eµ and µµ channels. The ratio between the data and prediction is also shown. The orange area shows the statistical uncertainty on the signal and background. The first and last bin of the distribution corresponds to the underflow and overflow respectively.

Figure 5 . 14 :

 514 Figure 5.14: Schematic description of the unfolding procedure The true distribution of an arbitrary variable x (left) is affected by acceptance effects (centre) and resolution effects (right) in the measurement process. The unfolding procedure aims to get the true distribution corresponding to the given measured distribution by reversing these effects.

5. 5

 5 Unfolding 137 162]. Earlier Studies showed that the default configurations for these uncertainties were too conservative, leading to large constraints in the fit. A simplified 1 version of these systematics are used where these uncertainties are parametrised as a function of the electron E T with binning [25 30 50 80 150 13000] GeV.

Figure 5 . 16 :

 516 Figure 5.16: The unfolded data distribution of cosq k + cosq k , including the statistical and systematic uncertainties summed in quadrature.

Figure 5 .

 5 Figure 5.17: Example LO Feynman diagrams for EFT contributions from C tG (left) and C 8 tq (right). The dots indicate possible insertions of a dimension-six operator.

•

  DIM6TOP [164]: DIM6TOP is a unique UFO model implementation dedicated to EFT studies in the top sector. Within the DIM6TOP model, there are numerous Wilson coefficients describing non-SM top-quark interactions. All operators of the Warsaw basis involving a top quark and satisfying U(2) q ⇥U(2) u ⇥U(2) d flavour conservation between the first two quark generations 7 are included. Baryon and lepton number violating operators are not included. DIM6TOP UFO model prediction are only at LO in QCD accuracy.

140 Chapter 5 .

 1405 Spin Correlation with the full Run 2 dataset thus this model should be considered where combinations with Higgs or Standard Model analyses are planned.

  ) Here |M SM | 2 is the dimension-four Standard Model squared matrix element. The term 2¬(M ⇤ SM M D6 ) is proportional to 1/L 2 and represents the interference between the Standard Model and the dimension-six operators in the EFT expansion. The quadratic term |M D6 | 2 is generated solely by new physics contributions, and is proportional to 1/L 4 . Additional EFT terms introduce additional vertices and Feynman Diagrams to

Figure 5 . 18 :

 518 Figure 5.18: NP coupling orders. The NP=0 (pure SM) and NP=2 (pure EFT) diagrams, give rise to a squared matrix element containing SM, SM-EFT interference, and EFT quadratic terms.

Figure 5 . 19 :

 519 Figure 5.19: Distribution of cosq k + cosq k using reweighting technique (green) is compared to dedicated simulation (red) for C 8 tq = 0.2. The event are generated using the DIM6TOP UFO model. All Wilson coefficients other than C 8tq are set to zero when generating these events. In the bottom panel, the red error bars represent the Monte Carlo statistical uncertainties on these two distributions. Both standalone and reweighting procedure show excellent agreement within their statistical uncertainties.

Figure 5 . 20 :

 520 Figure 5.20: The distribution of spin correlation observable for various value of C tq8 Wilson coefficient.

2 = 1 /TeV 2 Figure 5 .

 2125 Figure 5.21: The distribution of cos(q ) k + cos(q ) k of the t t system, with the differential bin-by-bin K-factor s NLO /s LO is displayed in the lower panel where error bars represent the Monte Carlo statistical uncertainties on these two distributions (small enough to be invisible).

  Figure 5.22: The K-factors s NLO /s LO in pp ! t t evaluated for different EFT scenarios. The error bars represent the Monte Carlo statistical uncertainties.

Figure 5 . 23 :

 523 Figure 5.23: The c 2 (NLO, LO) value for spin correlation observables as function of C tG /L 2 ⇥ TeV 2 ⇤ (up plots) and C 8 tq /L 2 ⇥ TeV 2 ⇤ (bottom plots) respectively. The c 2 value is normalized to c 2 value of the SM.

(a) C tG /L 2 (b) C 8 tq /L 2 Figure 5 . 24 :

 282524 Figure 5.24: Signal sensitivity s EFT /s SM in pp ! t t evaluated for different EFT point of C tG (left) and C 8 tq

1 and h 2 Figure 5 .

 25 Figure 5.28 illustrates the impact of C tG and C 8 tq on the spin correlation observables. Two different behaviours are observed. The observables are modified more for the negative values of C tG , whereas for the C 8 tq it is the positive values which have the most impact. Moreover, this c 2 test tells us that locally, i.e. for certain values of the Wilson coefficient, C tG has a greater absolute effect, with a value of c 2 reaching ⇠ 20 (22)

5. 6 EFT interpretation 149 5. 7

 61497 and 5.8 respectively. The value of a[START_REF] Tomonaga | On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields*[END_REF] and b (6) play import role in constraining operator coefficients.For instance, from the Equation5.25 and within some statistical framework, we can use the measured C(a, b) SMEFT , the estimated C(a, b) SM with their statistical and systematic uncertainties, and the value of a (6) and b (6) at NLO to derive global constraint on the operator coefficients[START_REF] Aguilar Saavedra | Interpreting top-quark LHC measurements in the standardmodel effective field theory[END_REF].

Figure 5 . 25 :

 525 Figure 5.25: The effects of C tG and C 8 tq on spin correlation coefficients at NLO and LO in QCD accuracy.

Figure 5 . 26 :Figure 5 . 28 :

 526528 Figure 5.26: The effects of C tG and C 8 tq on spin correlation coefficients at NLO and LO in QCD accuracy.

1 . 3 ,

 13 to modify the nominal event weights (corresponding to the SM) in order to yield a prediction according to a particular choice of EFT parameters. Specifically, for operators O tG and O 8 tq , four values of the Wilson coefficients are probed separately. The effect of these Wilson coefficients on spin correlation observables yields a full SM+EFT prediction of the form shown in Equation 5.25. The value of Wilson coefficients are C tG = [ 0.8, 0.4, 0, 0.4, 0.8](L/TeV) 2 C 8 tq = [ 3.2, 1.6, 0, 3.4, 3.8](L/TeV) 2

Figure 5 .Figure 5 . 29 :

 5529 Figure 5.[START_REF] Aad | Observation of Spin Correlation in tt Events from pp Collisions at p s = 7 TeV Using the ATLAS Detector[END_REF] shows the linearity test (or closure test) for the three diagonal spin correlation observables where different EFT samples are unfolded following either FixData (top) or FixRM (bottom) approach. The circle points (unfolded) should be compared with the triangle points (truth level) of the same colour. In the lower panel, the ratio between triangle and circle points is displayed, where it can be observed that it is covered by the uncertainty. The red lines represent the SM sample where the solid and dashed correspond to the unfolded and particle level distributions, respectively. The black uncertainty band includes the MC statistics together with the systematic uncertainty related to t t modelling described in Section 5.5. The FixData and FixRM linearity tests is shown in the top and bottom plots respectively. The linearity test for polarisation and cross diagonal spin correlation observables are shown in Appendix H. The slight difference seen between the unfolded and the particle level distributions of the SM sample (black), has only a small effect on the EFT interpretation that is well covered by the total uncertainty. the amount of non-linearity (non-closure) is negligible.The second step to validate the linearity test is presented in the same format as in Figure5.25. For diagonal spin correlation observables, the unfolded value and the expected particle level spin value is presented.The spin values are fitted with polynomial function of the form in Equation5.25. The confidence band in shown in the same colour as the dot points. The lower panel presents the ratio between the unfolded and the expected truth spin value for FixRM and FixData scenarios. The figures show that the bias induced by the unfolding for all the pseudo-data is much smaller than the MC statistical and systematic uncertainty related to t t modelling and for this reason no additional uncertainties are included to account for the potential bias.Additional plots for polarisation and cross diagonal spin correlation observables are presented in Appendix H.

FitFitFigure 5 . 30 :

 530 Figure 5.30: The unfolded spin correlation cosq k + cosq k distribution and their expected truth spin value given FixData and FixRM tests as a function of C tG and C 8 tq Wilson coefficients

1 3

 1 means there is no entanglement. Figure5.31 illustrates the evolution of D as function m t t . The entanglement is expected to happen when top quarks are produced at rest, near threshold.

Figure 5 . 31 :

 531 Figure 5.31: Evolution of the D coefficient from Equation5.41 as function of the m t t upper-bound defining the considered phase-space, at LO and NLO order[START_REF] Afik | Entanglement and quantum tomography with top quarks at the LHC[END_REF].

Figure 5 . 32 :

 532 Figure 5.32: The spin correlation observables C kk , C r r, and C n n in in t t phase space. C n n  0 and C kk , C r r have typically the same sign.

  C T C are likely to fluctuate positive than negative, thus the two largest eigenvalues of C T C are more likely to pick the ones that fluctuated up than the ones that fluctuated down. This can be illustrated by generating 3 ⇥ 3 matrices of the form C = 1 + d such as d is a random Gaussian variable with mean o and stander deviation s . The results are shown in Figure 5.33 where the bias towards the right is seen. It is possible that this bias effect suggests a violation of the CHSH inequality even when it is not present in the data or can artificially increase the statistical significance of a violation found. two largest eigenvalues C

Figure 5 .

 5 Figure 5.33: C T C's two largest eigenvalues for randomly generated spin correlation C matrices. The true value l + l 0 = 2, found for s = 0 is indicated by the dotted line.

Figure 5 . 35 :

 535 Figure 5.35: cosj ``distribution after selection in (a) ee , (b) µµ , and (c) eµ channel with statistical only uncertainties.

Figure 5 . 36 .

 536 The reconstruction efficiency e eff and the acceptance correction f j acc are shown in Figure5.36

Figure 5 .

 5 Figure 5.36: (a) Normalized response matrices for the combined final state (ee, em, mm). (b) The reconstruction efficiency and the acceptance correction.

(

  black dot) overlaid on top of it. Plot on the left uses slope method to compute parameter D, while plot on the right uses mean method to compute parameter D. All of the systematic uncertainties described in Section 5.5 are included in the fit. Since slope method re-parametrised all PoIs as a function of D (5.57), thus only the D parameter is entering the likelihood fits (two d.o.f), the systematic uncertainties is very small. Mean method shows more relaxed systematic uncertainties since it re-parametrizes PoIs controlling the first unfolded bin only (5.52) resulting in more free parameters in the likelihood fits (four d.o.f).

Figure 5 .

 5 Figure 5.38: (a) The unfolded D value for different number of bins. (b) The ratio of the relative uncertainty for the slope and mean methods, (d D slope /D slope )/(d D mean /D mean ), as function of number of the bins.

Figure 5 .

 5 Figure 5.38b represents the ratio of the relative uncertainty for the slope and mean methods as function of number of the bins. The red and orange dots represent the impact of MCNLO and POWH7 systematics respectively, while the red dots reflect the combined impact of MCNLO and POWH7 systematics. The slope method shows an improvement in the estimation of the statistical and systematic uncertainties. For instance, as shown in the Table5.9, using 4 bins we observed 39% improvement for the stat-only and 14%

Figure 5 . 39 :

 539 Figure 5.39: List of systematics under the assumption of 1% pruning. Only systematics with at least one green or yellow bin contribute to the overall uncertainty, and any uncertainty with all red are dropped entirely.

  )where x â, b = cosq âcosq b. The reference axes â and b are defined in helicity basis but the spin analysing vector for the top quark, â, has same sign to the spin analysing vector for the anti-top quark, b as shown in Table5.40. This differs from the convention used in the analysis so far where, { k, r, n} ! { k, sign(cos q )r, sign(cos q ) n}(5.64) this is because the CHSH inequalities (5.62) are developed under assumption â = b, hence the choice of the basis. The spin correlation matrix C in shown in Figure5.41. Spin correlation becomes negative as a result of this change in convention of the helicity basis26 .

Figure 5 . 40 :

 540 Figure 5.40: Definition of reference axes â and b following the definition of [74].

Figure 5 . 41 :

 541 Figure 5.41: Spin correlation C in the helicity basis { k, r, n} as defined in Table 5.40.

Figure 5 . 42 :

 542 Figure 5.42: The correlations between diagonal spin observables C kk , C r r, and C n n. The error bars represent the Monte Carlo statistical uncertainties on these distributions.

Figure 5 .

 5 Figure 5.43: (a) The value of CHSH inequality (5.49) in t t phase space. A value larger than p 2 or less than p 2 is a violation. (b) The value of l + l 0 in t t phase space, a value > 1 is violation.

A

  successful operation of the LHC with 13 TeV energy during Run 2 paves the way for ATLAS to collect 139 fb 1 of pp data between 2015 and 2018, providing greater precision when it comes to measuring Standard Model predictions as well as searching for phenomena BSM. The work presented in this thesis focused on top quark physics.

(((

  Figure B.1:The figures display the t hits distribution before (black) and after the time correction procedure. Calibration constants are computed using multiple timing intervals (every 1, 100, 500 events) and using time hits from full, inner (r < 320 mm) and outer (r > 320 mm) layer. The timing model chosen for this study is TimeFull 1 kHz. Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200.

(((

  Figure B.2:The figures display the t hits distribution before (black) and after the time correction procedure. Calibration constants are computed using multiple timing intervals (every 1, 100, 500 events) and using time hits from full, inner (r < 320 mm) and outer (r > 320 mm) layer. The timing model chosen for this study is TimeFull. Simulated hits are shown from t t sample of 1 434 events with a pileup of hµi = 200.

Ctq8 andb

  Ctq8 is performed by generating several MC samples for a given value of C tq8 , for which we compute spin observable C(k, k) and the value of a Ctq8 and b Ctq8 are extracted by fitting the parameter space with a second degree polynomial. In our study, we generated 13 MC samples, each sample corresponds to a different value of C tq8 coefficient, such as C tq8 = [-1.5, -1.2, -1, -0.5, -0.2, -0.09, 0, 0.09, 0.2, 0.5, 1, 1.2, 1.5]. These values of C tq8 are motivated by[START_REF] Brivio | O new physics, where art thou? A global search in the top sector[END_REF].

A

  second possibility is to compare the value of spin observables (e.g C(k, k)) and the value of a Ctq8 and b Ctq8 accordingly. The procedure of computing the value of a Ctq8 and b Ctq8 was explain in Section F.4. The Figure ?? shows a comparison between DIM6TOP model and SMEFT@NLO model, according to the guidelines mentioned in section F.3.

F. 2 202

 2 Figure F.1: The distribution of the total cross section for different benchmarks mode normalized to the total cross section of Stander Model (C tq8 =0.0)

Figure F. 2 :

 2 Figure F.2:The distribution of spin correlation observable C(k, k) for various value of C tq8 coefficient. The SM value of C(k, k) is fixed while fitting in ?? and it is not fixed in ?? .

  4 is only coming from statistical fluctuations.

3 F. 5 Figure F. 4 :Figure F. 6 :F. 5 Figure F. 8 :

 354658 Figure F.3: The distribution of spin correlation observable C(k, k) for various value of C tq8 coefficient. Standalone Vs Reweighing

  Figure G.2:The distribution of cos(q ) k + cos(q ) k of the t t system, with the differential bin-by-bin K-factor s NLO /s LO displayed in the lower panel.

FitFitFitFitFitFitFigure H. 4 : 4 C 4 C 4 C 4 CFigure H. 5 :Figure H. 6 :

 4444456 Figure H.4: The unfolded three diagonal spin correlation observables and their expected truth spin value given FixData and FixRM tests as a function of C tG and C 8 tq Wilson coefficients

Time calibration of the High Granularity Timing Detector and measurement of top quarks spin correlation with the ATLAS detector.

  Étalonnage temporel de détecteur de temps hautement granulaire et mesure de la corrélation de spin des quarks top avec le détecteur ATLAS.Mots clés: HGTD, Top, Mesure de précision, EFT, Intrication, LHC Résumé: Cette thèse se compose de deux parties principales. La première partie de la thèse est la calibration du temps d'impact des particules mesuré par chaque capteur de détecteur de temps hautement granulaire (HGTD) sur la base d'une simulation dédiée. HGTD est un nouveau détecteur crucial pour la mise à niveau de la Phase-II du HL-LHC et est conçu pour améliorer les capacités des sous-détecteurs ITK en a joutant une mesure du temps aux trajectoires des particules chargées ainsi que des informations spatiales. Une procédure de correction temporelle est proposée pour calibrer la distribution des temps d'impact des particules, t hits , mesuré par chaque capteur HGTD.Cette procédure est destinée à réduire les différences de temps intrinsèques entre les capteurs pour maintenir une bonne résolution temporel de 30 ps-50 ps. La deuxième partie porte sur les mesures des observables de corrélation de spin du quark top dans les événements t t, ce qui représente une possibilité unique de tester les prédictions du modèle standard (SM) et de sonder les nouveaux effets physiques. Les écarts potentiels par rapport aux attentes SM sont paramétrés dans le cadre de la théorie des champs effectifs (EFT). De plus, nous présentons la première proposition pour détecter l'intrication quantique dans une paire de quarks, ainsi que la violation des inégalités de Bell (BIs) au LHC.

	PH.D THESIS PH.D THESIS

Titre:

Table 2 .

 2 1.

			Leptons			Quarks
		Particle Charge mass/MeV	Particle Charge mass/MeV
	First generation electron (e )	-1	0.511	down	-1/3	4.67
		neutrino (n e )	0 < 1.1 10 6	up	+2/3	2.16
	Second generation	muon (µ )	-1	105.66	strange	-1/3	93
		neutrino (n µ )	0	< 0.19	charm	+2/3	1.27
	Third generation	tau (t )	-1	1776.86	bottom	-1/3	4.18
		neutrino (n t )	0	< 18.2	top	+2/3	173

Table 2 .

 2 1: Quarks and leptons in the Standard Model. Masses are taken from the 2020 summary tables provided by the Particle Data Group

Table 2

 2

		strong electromagnetic weak
	Quarks Leptons: charged neutrinos	p	p p	p p p

.2. The electromagnetic interaction of QED is experienced by all particles with electric charge, meaning all quarks and charged leptons. Only quarks carry the QCD equivalent of electric charge, called colour charge. Therefore, only the quarks are sensitive to the strong force. Lastly, the weak force affects all fermions.

Table 2 .

 2

2: Summary of the fundamental forces experienced by the twelve fermions of the Standard Model.

WG top LHC LHC comb. (Sep 2013)

	Kobayashi and Maskawa first proposed the existence of the top quark in 1973 to explain CP violations in
	decays of K mesons [46]. In 2008, Kobayashi and Maskawa won the Nobel Prize in Physics for the prediction
	of the top and bottom quarks. The Top quark 8 was discovered in 1995 at the Tevatron, the only collider with sufficiently high centre-of-mass energy to produce top quarks, by the CDF [47] and D0 / [48] experiments. ATLAS+CMS Preliminary = 7-13 TeV s summary, WG LHC top top m September 2021
	Similarly, the top quark was rediscovered in 2010 at CERN by the ATLAS and CMS collaborations [49, 50]. World comb. (Mar 2014) [2] stat total stat The Top quark is the heaviest particle in the Standard Model, which is as heavy as a gold atom, with a pole mass 9 [51]: total uncertainty syst) ± total (stat ± top m Ref. s
			m top = 173.34 ± 0.76 GeV	(2.48)
	This mass corresponds to the current LHC measurements illustrated in Figure 2.9. This measurement is
	a combination of top mass measurements from ATLAS and CMS experiments at different centre-of-mass
	energies [52].
	* Preliminary
	165	170	175	180	185
			top m	[GeV]

0.88) ± 0.95 (0.35 ± 173.29 World comb. (Mar 2014)

								7 TeV [1]
		173.34	±	0.76 (0.36	±	0.67)	1.96-7 TeV [2]
	ATLAS, l+jets	172.33	±	1.27 (0.75	±	1.02)	7 TeV [3]
	ATLAS, dilepton	173.79	±	1.41 (0.54	±	1.30)	7 TeV [3]
	ATLAS, all jets	175.1	±	1.8 (1.4	±	1.2)	7 TeV [4]
	ATLAS, single top	172.2	±	2.1 (0.7	±	2.0)	8 TeV [5]
	ATLAS, dilepton	172.99	±	0.85 (0.41	±	0.74)	8 TeV [6]
	ATLAS, all jets	173.72	±	1.15 (0.55	±	1.01)	8 TeV [7]
	ATLAS, l+jets	172.08	±	0.91 (0.39	±	0.82)	8 TeV [8]

ATLAS comb. (Oct 2018)

		172.69	±	0.48 (0.25	±	0.41)	7+8 TeV [8]
	ATLAS, leptonic invariant mass (*)	174.48	±	0.78 (0.40	±	0.67)	13 TeV [9]
	CMS, l+jets	173.49	±	1.06 (0.43	±	0.97)	7 TeV [10]
	CMS, dilepton	172.50	±	1.52 (0.43	±	1.46)	7 TeV [11]
	CMS, all jets	173.49	±	1.41 (0.69	±	1.23)	7 TeV [12]
	CMS, l+jets	172.35	±	0.51 (0.16	±	0.48)	8 TeV [13]
	CMS, dilepton	172.82	±	1.23 (0.19	±	1.22)	8 TeV [13]
	CMS, all jets	172.32	±	0.64 (0.25	±	0.59)	8 TeV [13]
	CMS, single top	172.95	±	1.22 (0.77	±	0.95)	8 TeV [14]

CMS comb. (Sep 2015) 7+8 TeV [13] 0.47) ± 0.48 (0.13 ± 172.44

	CMS, l+jets	172.25	±	0.63 (0.08	±	0.62)	13 TeV [15]
	CMS, dilepton	172.33	±	0.70 (0.14	±	0.69)	13 TeV [16]
	CMS, all jets	172.34	±	0.73 (0.20	±	0.70)	13 TeV [17]
	CMS, single top	172.13	±	0.77 (0.32	±	0.70)	13 TeV [18]

  LHC, nearly 90% of the t t pairs are produced through gluon fusion, and since LHC operates at ultrarelativistic energies, the helicity basis is suitable for LHC studies. With the off-diagonal basis, no significant improvements over the helicity basis are possible at the LHC primarily because the LHC is nearly dominated by like helicity gluon fusion.

	Fraction Events [%]	60 70 80 90 100	ATLAS Simulation Internal = 13 Tev s gg SH initial state	t M	t	> Cut	t M	t	< Cut				Fraction Events [%]	60 70 80 90 100	ATLAS Simulation Internal = 13 Tev s gg OH initial state	t M	t	> Cut	t M	t	< Cut
		50														50
		40														40
		30														30
		20														20
		10														10
		0	400	500	600	700		800			900		1000				0	400	500	600	700	800	900	1000
											t M	t	[Gev]						t M	t	[Gev]
					Fraction Events [%]	60 70 80 90 100		ATLAS Simulation Internal = 13 Tev s qq OH initial state		t M	t	> Cut	t M	t	< Cut
						50
						40
						30
						20
						10
						0		400			500	600	700			800	900	1000
																			t M	t	[Gev]

  t ⌘ s t t = (k t + kt ) 2 where, s t t is the Mandelstam variable, k t and kt are the top and antitop momenta. The threshold production for t t pair requires

					Chapter 2. Theoretical background
		Normalised Events	1 -10 1 10	ATLAS Simulation Internal aMC@NLO Particle Level t t → pp = 13 Tev s	Total gg OH	gg SH OH q q
			2 -10
			3 -10
			4 -10
			5 -10
				400 600 800 1000 1200 1400 1600 1800 2000
						t M	t	[Gev]
	Figure
		q
	b =	1 4m 2 t /M 2 t t = 0, where we can deduce, for m t = 173 GeV:
				M t t = 2m t ⇡ 346 GeV

  b.

	Particle a(LO) a(NLO)
	e + , µ + , t + d, s	1.00 1.00	0.998 0.930
	u, c -0.310	-0.310
	b -0.410	-0.390
	Table 2.4: Spin-analysing power at leading order and next-to-leading order for the decay products of the top
	quark, t ! bW + [70].

  ). Different special granularities show bias up to 40 ps, although the timing resolution should not exceed 30 ps. In the other hand, the only special granularity that can compensate for approximately all ToF effects and have targeting timing resolution is the calibration per module.

	Time Full 1kHz -Time ASIC [ns]	0.1 -0.05 -0 0.05 0.1 0.15 0.2 0.25	ATLAS Simulation Internal HGTD > = 200, nevnt : 1434 µ < t t z > 0	Before Calibration Independent Calibration Calib. every 500 evt Calib. every 1 evt Calib. every 10 evt Calib. every 100 evt
		0.15 -
		0 0.2 -	200 400 600 800 1000 1200 1400
				Event Index

Table 5 .

 5 1: A summary table of all of the trigger p T -thresholds used for each data taking period and lepton flavour. Individual p T selection with commas indicate a logical OR of multiple cuts.

		Year	Trigger
		2015	p T > 24, 60, 120 GeV
	Electron triggers	2016	p T > 26, 60, 140 GeV
		2017	p T > 26, 60, 140 GeV
		2018	p T > 26, 60, 140 GeV
		2015	p T > 20, 50 GeV
	Muon triggers	2016	p T > 26, 50 GeV
		2017	p T > 26, 50 GeV
		2018	p T > 26, 50 GeV

1

after requiring stable LHC beams and a fully operational detector. The event quality was checked to remove events with noise bursts or coherent noise in the calorimeters. Different trigger strategies are used in each channel, see table 5.1. Events are requested to fire single electron or single muon triggers and this is configured by the global trigger selection tool. A detailed list of all Data samples used in this analysis can be found in Appendix C.

  nominal t t signal MC sample is generated by POWHEG BOX (version 2) at NLO with the top quark mass set to 172.5 GeV and top-quark decays are simulated at LO using MADSPIN to preserve all spin correlations. The parameter h damp , that controls the matrix element/parton shower matching, and effectively regulates the high-pT radiation, is set to 1.5 m top[144]. The PDF set used is NNPDF3.0NLO[START_REF] Ball | Parton distributions for the LHC run II[END_REF]. The signal events are then showered with PYTHIA (version 8.230) using the A14 tune and the NNPDF23LO PDF set. Additional t t samples are used to evaluate signal modelling systematics. To assess the uncertainty due to the generator choice, the nominal generator setup is compared to an alternative

	setup that uses MADGRAPH5_AMC@NLO (version 2.6.0) with NNPDF3.0NLO for the calculation of
	the hard-scattering, interfaced with PYTHIA (version 8.230) using the A14 set of tuned parameters and

NNPDF23LO. The uncertainty coming from the choice of the h damp value is estimated by comparing the nominal samples with an alternative sample produced with an h damp value set to 3.0 m top . All alternative t t samples are generated using AF2

[START_REF] Coll | The simulation principle and performance of the ATLAS fast calorimeter simulation FastCaloSim[END_REF]

detector simulation and are always compared to an AF2 version of the nominal sample. All t t samples are normalised to the NNLO prediction

[147]

.

Table 5 .

 5

	Channel	ee	µµ	eµ
	t t	200000 ± 90 400000 ± 120 800000 ± 180
	Singletop	10000 ± 40	20000 ± 50	50000 ± 80
	Diboson	2000 ± 9	2600 ± 11	4700 ± 15
	Z +jets	43000 ± 280 100000 ± 600	4000 ± 50
	Fakes	3000 ± 21	1000 ± 28	9000 ± 40
	t tX	780 ± 3.2	3600 ± 4	3000 ± 6
	Total Expected 280000 ± 300 500000 ± 600 870000 ± 190
	Data	279286	541590	880794

4: Event yields of t t signal, background processes and data after the full event selection and the kinematic reconstruction. The given uncertainties correspond to the statistical uncertainties of the individual processes.

Table 5 .

 5 5: Breakdown of uncertainties (asimov, mc16a, eµ).

	Uncertainty source	D C(k, k)
	t t modelling	+0.043 / -0.058
	Weight systematics	+0.002 / -0.003
	Background MC stat. (gammas) +0.007 / -0.006
	Total systematic uncert.	+0.044 / -0.059
	Statistical uncert. Total	±0.032 +0.054 / -0.067

Table 5 .

 5 7: SM value of spin correlation observables, a (6) and b (6) at LO and NLO precision for CMDM operator O tG .

	)

Table 5 .

 5 8: SM value of spin correlation observables, a (6) and b (6) at LO and NLO precision for O 8 tq operator.

  For the unfolding study, EFT samples are generated using official ATLAS MADGRAPH distribution in which all events pass through the full simulation of the ATLAS detector. Three EFT MC16 campaigns are produced, each is matched to the condition of data taking in specific time-frame. MC16A is designed to correspond with 2015-2016 period, MC16d is designed to correspond with 2017 period, and MC16e is designed to correspond with 2018 period. All three MC16 campaigns together cover the full Run 2, 2015-2018, data taking period. SMEFT@NLO-NLO18 UFO model is not yet implemented in ATLAS MADGRAPH distribution for the EFT calculation at NLO QCD accuracy. This current limitation prevent the use of NLO corrections on the contribution of the O tG and O 8 tq operators. Due to this, the EFT model is added to the calculations via the DIM6TOP UFO model at LO QCD accuracy. The interpretation of EFT at NLO in QCD using ATLAS EFT

	5.6.3.1 ATLAS EFT sample

  The non-locality (A non-local physical theory is one that permits instantaneous effects at speeds much greater than the speed of light, v > c) feature of entanglement is one of the most non-intuitive features of QM. Due to this, Einstein, Podolsky and Rosen (EPR) argued in 1935 that quantum mechanics theory was incomplete in what is known as the EFP paradox[START_REF] Einstein | Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?[END_REF]. According to EPR, there is nothing miraculous or non-local in all of this. The entangled pair agree on outcomes ahead of time (Realism). This would indicate that the quantum state of two entangled pair is incomplete as there would be additional degree of freedom, refereed to as hidden variables, which indicate what the measurement will yield. Additionally, the hidden variables would be local, so no instantaneous action would be necessary.In 1964, John Stewart Bell, a theorist at CERN, propose inequalities know as Bell inequalities (BIs), that allowed local realism predictions to be tested experimentally against those of standard quantum physics.BIs are derived with the assumption that QM has hidden variables. The violation of BIs indicates that this assumption is simply not possible. Various experimental test have been performed to test the violation of BIs in many quantum systems such as photons

  Entanglement marked the difference between the classical and quantum worlds. This Section discuss the building blocks for entanglement quantum theory in t t pair. For the definition of entanglement, we consider a two subsystems A and B with quantum states are associated to Hilbert spaces H A and H B . The state of combined system A + B, whose Hilbert space is H A N H B is entangled if it can not be written in this form Equation5.26 is separeble such as states |y A i and |y B i are two independent parts. For a pair of spin-1/2 particles each can have one of two states, usually refereed to as qubit system, with associated Hilbert space is chosen to be the spin in z-direction with eigenstates spin up |"i or spin down |#i, the state

	1. Sections 5.7.2 and 5.7.3 present, for the first time, the expected sensitivity of quantum entanglement and violation of BIs at LHC respectively. 5.7.1 Entanglement and BIs 5.7.1.1 Entanglement theory [197]: |Yi = |y A i ⌦ |y B i (5.26) 159 where the state in 5.7 Quantum interpretation [197]:

  EPR thought experiment presents hidden variables as a solution to the mysterious spooky a distance to explain the strangeness of entanglement, yet our ignorance of these hidden variables is where the probability of quantum mechanics comes from. In 1964 John Bell suggested a mechanism to test the existence of hidden variables. That is, pre-existing (hidden) properties must be non-local in order to explain probabilities in quantum measurements. An inequality developed by Bell served as the basis for such a test. When two classical systems exchange information, they can establish a maximum correlation degree, which usually

										Ckk																Crr
	[TeV]	1.2	-0.612547	-0.131779		0.250861	0.492379		0.58202	0.667814		0.763706	0.689158	0.720047	0.766234	1		[TeV]	1.2	-0.0426215 -0.00215345 0.0158352	0.143348	0.192254	0.377705	0.463556	0.663577	0.766285	0.783743	1
	t																				t
	t Invariant Mass m	0.9 1 1.1	-0.721137 -0.720658 -0.708493	-0.483706 -0.418728 -0.332882	-0.154529 -0.040553 0.0583444	0.107682 0.183834 0.297248	0.261584 0.337605 0.46814	0.322845 0.440371 0.578928	0.391034 0.461806 0.532188	0.411069 0.49286 0.572692	0.399527 0.494162 0.576906	0.392912 0.528566 0.612356	0.5		t Invariant Mass m	0.9 1 1.1	-0.10247 -0.0655596 -0.0409495 -0.0245056 -0.0566125 -0.0207883 -0.0582431 -0.0356895 0.0123444	0.0594817 0.0854222 0.117808	0.138814 0.153617 0.123724	0.266915 0.284462 0.281436	0.401397 0.433634 0.48256	0.543465 0.595211 0.619982	0.640797 0.703722 0.763498	0.7323 0.762815 0.832625	0.5
		0.8																	0			0.8					0
			-0.764989	-0.529715	-0.239923 -0.00958769	0.124961	0.227401	0.267547	0.246878	0.261075	0.2813						-0.113356	-0.100027	-0.0490765	0.0139081	0.117881	0.247737	0.358241	0.503043	0.597463	0.668686
		0.7																				0.7
																							-0.144784	-0.140752	-0.088091	-0.0107884	0.0844976	0.190426	0.318767	0.452849	0.548084	0.609465
		0.6																	0.5 -			0.6	-0.20667	-0.18076		-0.142149	-0.0609673	0.0338523	0.138512	0.26241	0.377434	0.475035	0.531849	0.5 -
		0.5																				0.5	-0.271053	-0.248761	-0.195558	-0.133857	-0.0315928	0.0710799	0.177932	0.299123	0.378069	0.424719
																							-0.343397	-0.336246	-0.287199	-0.213669	-0.123123	-0.0349885	0.0886527	0.186861	0.266336	0.308296
		0.4																	1 -			0.4	-0.575035 -0.4526	-0.555121 -0.441298	-0.528248 -0.402646	-0.482653 -0.331479	-0.431145 -0.255658	-0.357747 -0.170359	-0.285917 -0.0626521	-0.221755 0.0264004	-0.173371 0.0891397	-0.150383 0.135311	1 -
																							-0.392013	-0.398235	-0.457782	-0.359237	-0.329195	-0.255171	-0.281344	-0.205555	-0.185226	-0.14608
		0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9 2 θ	/ CM	π	1			0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9 2 θ	/ CM	π	1
											[TeV]	1.2
													1.1
													1
													0.9
													0.8
													0.7
													0.6
													0.5
													0.4
													0	0.1	0.2	0.3		0.4	0.5	0.6	0.7	0.8	0.9 2 θ	/ CM	π	1
	5.7.1.2 Bell inequalities theory

Table 5 .

 5 [START_REF] Feynman | Theory of Fermi interaction[END_REF], using 4 bins we observed 39% improvement for the stat-only and 14% improvement for the systematic uncertainty. The reason for this is that a likelihood fit with systematic uncertainties has more free parameters than a stat-only analysis, and as a result, uncertainty gains are smaller.

	Additionally, when the number of bins is increased, the uncertainty gain between the two methods increases
	by 8%.
	NumOfBins Stat Only Sys. McNLO Sys. PowH7 Sys. McNLO,PowH7
	2	36%	31%	10%	12%
	4	39%	37%	18%	14%
	6	40%	41%	18%	17%
	8	39%	40%	21%	20%
	10	42%	40%	22%	20%

A German mathematician (1882-1935) who made fundamental contributions to abstract algebra and theoretical physics, Emmy Noether is regarded as one of the most significant mathematicians of all time.

A non-abelian is a group (G, ⇥) in which there exists at least one pair of elements a and b of G, such that a ⇥ b 6 = b ⇥ a, simply means that the generators of the group do not commute.

A Lagrangian density, denoted with a curly L , is used for this task because it has a relatively simple form which is classically understood, clearly demonstrates relativistic covariance and is well suited for testing various forms of symmetry. When integrated over all of space, it returns the Lagrangian, denoted by L.

also called Brout-Englert-Higgs (BEH) mechanism.

Since the massive gauge vector bosons have 3 degrees of polarization and massless ones have only 2, the mechanism which yields the mass terms is therefore required to have at least 3 degrees of freedom.

The CP symmetry is a combination of the charge conjugation symmetry and the parity symmetry which states that the laws of physics should be the same if a particle or a system of particles are interchanged with respective anti-particles (C symmetry) and when its spatial coordinates are inverted (P symmetry).

The top quark is also referred to as the truth quark

The pole mass is closer to the intuitive physical mass one measures in experiments. The pole mass is determined by the location of the poles in the propagator, and is independent of the energy scale of the experiment.

There are also direct searches that can be performed, such as searching for vector-like quarks.

At a distance of 1 femtometer (1 fm = 10 15 meters) or less, a s = 1, a = 1

In Section 2.1, for example, mentions weak interaction only coupling left-handed fermions, in this case, chirality, not helicity.

In engineering, for example, bridges can be designed based on parameters measured at macroscopic scales without having to know microscopic properties, such as weak interactions or the mass of the Higgs boson.

for this term to respect the U(1) Y part of the gauge symmetry.

In the Dirac representation C = ig 2 g 0 of gamma-matrices.

The IP2D and IP3D are impact parameter (IP) based algorithms which are complementary to each other.

A high performance algorithm for the identification of jets produced via fragmentation of bottom quarks.

The JetFitter algorithm is designed to reconstruct the full B-hadron decay chain inside a jet cone.

LGADs are n-on-p silicon detectors with an additional p-type doped layer containing charge multiplication to achieve an internal gain. The LGAD sensors aims to optimize an excellent timing resolution between 35 and 70 ps per hit over their lifetime.

That is the result of having a two-ring layout instead of a three-ring layout.

The readout rows are sets of modules whose flex cables (flexible PCB cables) are guided together towards larger radii to the peripheral on-detector electronics. They extend as a line of modules from lower to higher radii[START_REF]Technical Design Report: A High-Granularity Timing Detector for the ATLASPhase-ii Upgrade[END_REF].

VTRX+ is the Versatile Link+ optical module.

Another approach is to use particles coming from a vertex at z = 0 only, however, this is not feasible online, hits are not associated with any tracks (so, no vertex).

Events with leptons originating from intermediate tau leptons modified the spin density matrix, since their subsequent decays do not carry the full spin information of their parent top quark, thus reducing spin correlation information.

no restriction is imposed on the third-generation

In the DIM6TOP model, it is called DIM6 instead

Different from SMEFTSIM UFO model where NP=1.

Note that a S runs and that its value at the scale m Z depends on the parton distribution function employed.

This can achieved by generating two samples (with and w/o normalize chromo-dipole operator O tG ) with the same following syntax (lets consider DIM6TOP model with C tG is the only non zero operator with a value equal to 1):

Figure 5.27: The effects of C tG and C 8 tq on spin correlation coefficients at NLO and LO in QCD accuracy.

It is also refereed to as the PPT criterion

Partially implies that only part of the subsystems, A and B, is transposed.

More details are given in Section 2.3 where the spin density matrix is refereed to as R.

Such as A and B must be space-like separated[A j , B j ] = 0 i, j = 1, 2

This demonstrated by JUAN RAMON MUÑOZ DE NOVA jrmnova@fis.ucm.es, Departamento de Fısica de Materiales, Universidad Complutense de Madrid.

We consider LO calculation for two raison: first, DIM6TOP is LO model secondly, MADSPIN package performs only LO decay particle.

, for guiding me in my choices from the start and for your unconditional support throughout my studies, I would like to thank my father for his love and generosity and my mother for her support and encouragement.

dilep all_dilep_Phi_mcdata bTagSF_DL1r_85_eigenvars_C_0 bTagSF_DL1r_85_eigenvars_C_1 bTagSF_DL1r_85_eigenvars_C_2 bTagSF_DL1r_85_eigenvars_C_3 bTagSF_DL1r_85_eigenvars_Light_0 bTagSF_DL1r_85_eigenvars_Light_1 bTagSF_DL1r_85_eigenvars_Light_2 bTagSF_DL1r_85_eigenvars_Light_3 bTagSF_DL1r_85_extrapolation_from_charm bTagSF_DL1r_85_extrapolation JVT leptonSF_EL_SF_SIMPLIFIED_ID_0 leptonSF_EL_SF_SIMPLIFIED_ID_1 leptonSF_EL_SF_SIMPLIFIED_ID_2 leptonSF_EL_SF_SIMPLIFIED_ID_3 leptonSF_EL_SF_SIMPLIFIED_ID_4 leptonSF_EL_SF_SIMPLIFIED_ID_5 leptonSF_EL_SF_SIMPLIFIED_ID_6 leptonSF_EL_SF_SIMPLIFIED_ID_7 leptonSF_EL_SF_SIMPLIFIED_ID_8 leptonSF_EL_SF_SIMPLIFIED_ID_9 leptonSF_EL_SF_SIMPLIFIED_ID_10 leptonSF_EL_SF_SIMPLIFIED_ID_11 leptonSF_EL_SF_SIMPLIFIED_ID_12 leptonSF_EL_SF_SIMPLIFIED_ID_13 leptonSF_EL_SF_SIMPLIFIED_ID_14 leptonSF_EL_SF_SIMPLIFIED_ID_15 leptonSF_EL_SF_SIMPLIFIED_ID_16 leptonSF_EL_SF_SIMPLIFIED_ID_17 leptonSF_EL_SF_SIMPLIFIED_ID_18 leptonSF_EL_SF_SIMPLIFIED_ID_19 leptonSF_EL_SF_SIMPLIFIED_ID_20 leptonSF_EL_SF_SIMPLIFIED_ID_21 leptonSF_EL_SF_SIMPLIFIED_ID_22 leptonSF_EL_SF_SIMPLIFIED_ID_23 leptonSF_EL_SF_SIMPLIFIED_ID_24 leptonSF_EL_SF_SIMPLIFIED_ID_25 leptonSF_EL_SF_SIMPLIFIED_ID_26 leptonSF_EL_SF_SIMPLIFIED_ID_27 leptonSF_EL_SF_SIMPLIFIED_Iso_0 leptonSF_EL_SF_SIMPLIFIED_Iso_1 leptonSF_EL_SF_SIMPLIFIED_Iso_2 leptonSF_EL_SF_SIMPLIFIED_Iso_3 leptonSF_EL_SF_SIMPLIFIED_Iso_4 leptonSF_EL_SF_SIMPLIFIED_Iso_5 leptonSF_EL_SF_SIMPLIFIED_Iso_6 leptonSF_EL_SF_SIMPLIFIED_Iso_7 leptonSF_EL_SF_SIMPLIFIED_Iso_8 leptonSF_EL_SF_SIMPLIFIED_Iso_9 leptonSF_EL_SF_SIMPLIFIED_Iso_10 leptonSF_EL_SF_SIMPLIFIED_Iso_11 leptonSF_EL_SF_SIMPLIFIED_Iso_12 leptonSF_EL_SF_SIMPLIFIED_Reco_0 leptonSF_EL_SF_SIMPLIFIED_Reco_1 leptonSF_EL_SF_SIMPLIFIED_Reco_2 leptonSF_EL_SF_SIMPLIFIED_Reco_3 leptonSF_EL_SF_SIMPLIFIED_Reco_4 leptonSF_EL_SF_SIMPLIFIED_Reco_5 leptonSF_EL_SF_SIMPLIFIED_Reco_6 leptonSF_EL_SF_SIMPLIFIED_Reco_7 leptonSF_EL_SF_SIMPLIFIED_Reco_8 leptonSF_EL_SF_SIMPLIFIED_Reco_9 leptonSF_EL_SF_SIMPLIFIED_Reco_10 leptonSF_EL_SF_SIMPLIFIED_Reco_11 leptonSF_EL_SF_SIMPLIFIED_Reco_12 leptonSF_EL_SF_SIMPLIFIED_Reco_13 leptonSF_EL_SF_SIMPLIFIED_Reco_14 leptonSF_EL_SF_SIMPLIFIED_Reco_15 leptonSF_EL_SF_SIMPLIFIED_Reco_16 leptonSF_EL_SF_SIMPLIFIED_Reco_17 leptonSF_EL_SF_SIMPLIFIED_Reco_18 leptonSF_EL_SF_SIMPLIFIED_Reco_19 leptonSF_MU_SF_ID_STAT leptonSF_MU_SF_ID_STAT_LOWPT leptonSF_MU_SF_ID_SYST leptonSF_MU_SF_ID_SYST_LOWPT leptonSF_MU_SF_Isol_STAT leptonSF_MU_SF_Isol_SYST leptonSF_MU_SF_TTVA_STAT leptonSF_MU_SF_TTVA_SYST JET_EtaIntercalibration_TotalStat_1up JET_EtaIntercalibration_NonClosure_posEta JET_EtaIntercalibration_NonClosure_negEta JET_EtaIntercalibration_NonClosure_highE JET_EtaIntercalibration_NonClosure_2018data JET_EtaIntercalibration_Modelling JET_PunchThrough_MC16 JET_BJES_Response JET_Flavor_Response JET_Flavor_Composition JET_Pileup_PtTerm JET_Pileup_RhoTopology JET_Pileup_OffsetMu JET_Pileup_OffsetNPV JET_EffectiveNP_Mixed1 JET_EffectiveNP_Mixed2 JET_EffectiveNP_Mixed3 JET_EffectiveNP_Statistical1 JET_EffectiveNP_Statistical2 JET_EffectiveNP_Statistical3 JET_EffectiveNP_Statistical4 JET_EffectiveNP_Statistical5 JET_EffectiveNP_Statistical6 MET_SoftTrk_Scale MUON_SCALE MUON_MS MUON_ID EG_SCALE_ALL EG_SCALE_AF2 EG_RESOLUTION_ALL JET_EffectiveNP_Detector1 JET_EffectiveNP_Detector2 JET_EffectiveNP_Modelling1 JET_EffectiveNP_Modelling2 JET_EffectiveNP_Modelling3 JET_EffectiveNP_Modelling4 JET_SingleParticle_HighPt ttbar_MCNLOH7 ttbar_PowH704 PDFse_90901 PDFse_90902 PDFse_90903 PDFse_90904 PDFse_90905 PDFse_90906 PDFse_90907 PDFse_90908 PDFse_90909 PDFse_90910 PDFse_90911 PDFse_90912 PDFse_90913 PDFse_90914 PDFse_90915 PDFse_90916 PDFse_90917 PDFse_90918 PDFse_90919 PDFse_90920 PDFse_90921 PDFse_90922 PDFse_90923 PDFse_90924 PDFse_90925 PDFse_90926 PDFse_90927 PDFse_90928 PDFse_90929 PDFse_90930 MET_SoftTrk_ResoPara MET_SoftTrk_ResoPerp JET_JER_EffectiveNP_1_1up JET_JER_EffectiveNP_2_1up JET_JER_EffectiveNP_3_1up JET_JER_EffectiveNP_4_1up JET_JER_EffectiveNP_5_1up JET_JER_EffectiveNP_6_1up JET_JER_EffectiveNP_7_1up JET_JER_EffectiveNP_8_1up JET_JER_EffectiveNP_9_1up JET_JER_EffectiveNP_10_1up JET_JER_EffectiveNP_11_1up JET_JER_EffectiveNP_12restTerm JET_JER_DataVsMC_MC16 JET_JER_DataVsMC_MC16_PseudoData JET_JER_EffectiveNP_1_PseudoData JET_JER_EffectiveNP_2_PseudoData JET_JER_EffectiveNP_3_PseudoData JET_JER_EffectiveNP_4_PseudoData JET_JER_EffectiveNP_5_PseudoData JET_JER_EffectiveNP_6_PseudoData JET_JER_EffectiveNP_7_PseudoData JET_JER_EffectiveNP_8_PseudoData JET_JER_EffectiveNP_9_PseudoData JET_JER_EffectiveNP_10_PseudoData JET_JER_EffectiveNP_11_PseudoData JET_JER_EffectiveNP_12restTerm_PseudoData muRfac_1p0_fsr_muRfac_0p5 MUON_SAGITTA_RESBIAS ttbar_Var3c ttbar ttbar ttbar ttbar pileup bTagSF_DL1r_85_eigenvars_B_0 bTagSF_DL1r_85_eigenvars_B_1 bTagSF_DL1r_85_eigenvars_B_2 bTagSF_DL1r_85_eigenvars_B_3 bTagSF_DL1r_85_eigenvars_B_4 bTagSF_DL1r_85_eigenvars_B_5 bTagSF_DL1r_85_eigenvars_B_6 bTagSF_DL1r_85_eigenvars_B_7 bTagSF_DL1r_85_eigenvars_B_8 bTagSF_DL1r_85_eigenvars_C_0 bTagSF_DL1r_85_eigenvars_C_1 bTagSF_DL1r_85_eigenvars_C_2 bTagSF_DL1r_85_eigenvars_C_3 bTagSF_DL1r_85_eigenvars_Light_0 bTagSF_DL1r_85_eigenvars_Light_1 bTagSF_DL1r_85_eigenvars_Light_2 bTagSF_DL1r_85_eigenvars_Light_3 bTagSF_DL1r_85_extrapolation_from_charm bTagSF_DL1r_85_extrapolation JVT leptonSF_EL_SF_SIMPLIFIED_ID_0 leptonSF_EL_SF_SIMPLIFIED_ID_1 leptonSF_EL_SF_SIMPLIFIED_ID_2 leptonSF_EL_SF_SIMPLIFIED_ID_3 leptonSF_EL_SF_SIMPLIFIED_ID_4 leptonSF_EL_SF_SIMPLIFIED_ID_5 leptonSF_EL_SF_SIMPLIFIED_ID_6 leptonSF_EL_SF_SIMPLIFIED_ID_7 leptonSF_EL_SF_SIMPLIFIED_ID_8 leptonSF_EL_SF_SIMPLIFIED_ID_9 leptonSF_EL_SF_SIMPLIFIED_ID_10 leptonSF_EL_SF_SIMPLIFIED_ID_11 leptonSF_EL_SF_SIMPLIFIED_ID_12 leptonSF_EL_SF_SIMPLIFIED_ID_13 leptonSF_EL_SF_SIMPLIFIED_ID_14 leptonSF_EL_SF_SIMPLIFIED_ID_15 leptonSF_EL_SF_SIMPLIFIED_ID_16 leptonSF_EL_SF_SIMPLIFIED_ID_17 leptonSF_EL_SF_SIMPLIFIED_ID_18 leptonSF_EL_SF_SIMPLIFIED_ID_19 leptonSF_EL_SF_SIMPLIFIED_ID_20 leptonSF_EL_SF_SIMPLIFIED_ID_21 leptonSF_EL_SF_SIMPLIFIED_ID_22 leptonSF_EL_SF_SIMPLIFIED_ID_23 leptonSF_EL_SF_SIMPLIFIED_ID_24 leptonSF_EL_SF_SIMPLIFIED_ID_25 leptonSF_EL_SF_SIMPLIFIED_ID_26 leptonSF_EL_SF_SIMPLIFIED_ID_27 leptonSF_EL_SF_SIMPLIFIED_Iso_0 leptonSF_EL_SF_SIMPLIFIED_Iso_1 leptonSF_EL_SF_SIMPLIFIED_Iso_2 leptonSF_EL_SF_SIMPLIFIED_Iso_3 leptonSF_EL_SF_SIMPLIFIED_Iso_4 leptonSF_EL_SF_SIMPLIFIED_Iso_5 leptonSF_EL_SF_SIMPLIFIED_Iso_6 leptonSF_EL_SF_SIMPLIFIED_Iso_7 leptonSF_EL_SF_SIMPLIFIED_Iso_8 leptonSF_EL_SF_SIMPLIFIED_Iso_9 leptonSF_EL_SF_SIMPLIFIED_Iso_10 leptonSF_EL_SF_SIMPLIFIED_Iso_11 leptonSF_EL_SF_SIMPLIFIED_Iso_12 leptonSF_EL_SF_SIMPLIFIED_Reco_0 leptonSF_EL_SF_SIMPLIFIED_Reco_1 leptonSF_EL_SF_SIMPLIFIED_Reco_2 leptonSF_EL_SF_SIMPLIFIED_Reco_3 leptonSF_EL_SF_SIMPLIFIED_Reco_4 leptonSF_EL_SF_SIMPLIFIED_Reco_5 leptonSF_EL_SF_SIMPLIFIED_Reco_6 leptonSF_EL_SF_SIMPLIFIED_Reco_7 leptonSF_EL_SF_SIMPLIFIED_Reco_8 leptonSF_EL_SF_SIMPLIFIED_Reco_9 leptonSF_EL_SF_SIMPLIFIED_Reco_10 leptonSF_EL_SF_SIMPLIFIED_Reco_11 leptonSF_EL_SF_SIMPLIFIED_Reco_12 leptonSF_EL_SF_SIMPLIFIED_Reco_13 leptonSF_EL_SF_SIMPLIFIED_Reco_14 leptonSF_EL_SF_SIMPLIFIED_Reco_15 leptonSF_EL_SF_SIMPLIFIED_Reco_16 leptonSF_EL_SF_SIMPLIFIED_Reco_17 leptonSF_EL_SF_SIMPLIFIED_Reco_18 leptonSF_EL_SF_SIMPLIFIED_Reco_19 leptonSF_MU_SF_ID_STAT leptonSF_MU_SF_ID_STAT_LOWPT leptonSF_MU_SF_ID_SYST leptonSF_MU_SF_ID_SYST_LOWPT leptonSF_MU_SF_Isol_STAT leptonSF_MU_SF_Isol_SYST leptonSF_MU_SF_TTVA_STAT leptonSF_MU_SF_TTVA_SYST JET_EtaIntercalibration_TotalStat_1up JET_EtaIntercalibration_NonClosure_posEta JET_EtaIntercalibration_NonClosure_negEta JET_EtaIntercalibration_NonClosure_highE JET_EtaIntercalibration_NonClosure_2018data JET_EtaIntercalibration_Modelling JET_PunchThrough_MC16 JET_BJES_Response JET_Flavor_Response JET_Flavor_Composition JET_Pileup_PtTerm JET_Pileup_RhoTopology JET_Pileup_OffsetMu JET_Pileup_OffsetNPV JET_EffectiveNP_Mixed1 JET_EffectiveNP_Mixed2 JET_EffectiveNP_Mixed3 JET_EffectiveNP_Statistical1 JET_EffectiveNP_Statistical2 JET_EffectiveNP_Statistical3 JET_EffectiveNP_Statistical4 JET_EffectiveNP_Statistical5 JET_EffectiveNP_Statistical6 MET_SoftTrk_Scale MUON_SCALE MUON_MS MUON_ID EG_SCALE_ALL EG_SCALE_AF2 EG_RESOLUTION_ALL JET_EffectiveNP_Detector1 JET_EffectiveNP_Detector2 JET_EffectiveNP_Modelling1 JET_EffectiveNP_Modelling2 JET_EffectiveNP_Modelling3 JET_EffectiveNP_Modelling4 JET_SingleParticle_HighPt ttbar_MCNLOH7 ttbar_PowH704 PDFse_90901 PDFse_90902 PDFse_90903 PDFse_90904 PDFse_90905 PDFse_90906

present Kept Shape dropped Norm. dropped Dropped ttbar ttbar ttbar ttbar pileup bTagSF_DL1r_85_eigenvars_B_0 bTagSF_DL1r_85_eigenvars_B_1 bTagSF_DL1r_85_eigenvars_B_2 bTagSF_DL1r_85_eigenvars_B_3 bTagSF_DL1r_85_eigenvars_B_4 bTagSF_DL1r_85_eigenvars_B_5 bTagSF_DL1r_85_eigenvars_B_6 bTagSF_DL1r_85_eigenvars_B_7 bTagSF_DL1r_85_eigenvars_B_8 bTagSF_DL1r_85_eigenvars_C_0 bTagSF_DL1r_85_eigenvars_C_1 bTagSF_DL1r_85_eigenvars_C_2 bTagSF_DL1r_85_eigenvars_C_3 bTagSF_DL1r_85_eigenvars_Light_0 bTagSF_DL1r_85_eigenvars_Light_1 bTagSF_DL1r_85_eigenvars_Light_2 bTagSF_DL1r_85_eigenvars_Light_3 bTagSF_DL1r_85_extrapolation_from_charm bTagSF_DL1r_85_extrapolation JVT leptonSF_EL_SF_SIMPLIFIED_ID_0 leptonSF_EL_SF_SIMPLIFIED_ID_1 leptonSF_EL_SF_SIMPLIFIED_ID_2 leptonSF_EL_SF_SIMPLIFIED_ID_3 leptonSF_EL_SF_SIMPLIFIED_ID_4 leptonSF_EL_SF_SIMPLIFIED_ID_5 leptonSF_EL_SF_SIMPLIFIED_ID_6 leptonSF_EL_SF_SIMPLIFIED_ID_7 leptonSF_EL_SF_SIMPLIFIED_ID_8 leptonSF_EL_SF_SIMPLIFIED_ID_9 leptonSF_EL_SF_SIMPLIFIED_ID_10 leptonSF_EL_SF_SIMPLIFIED_ID_11 leptonSF_EL_SF_SIMPLIFIED_ID_12 leptonSF_EL_SF_SIMPLIFIED_ID_13 leptonSF_EL_SF_SIMPLIFIED_ID_14 leptonSF_EL_SF_SIMPLIFIED_ID_15 leptonSF_EL_SF_SIMPLIFIED_ID_16 leptonSF_EL_SF_SIMPLIFIED_ID_17 leptonSF_EL_SF_SIMPLIFIED_ID_18 leptonSF_EL_SF_SIMPLIFIED_ID_19 leptonSF_EL_SF_SIMPLIFIED_ID_20 leptonSF_EL_SF_SIMPLIFIED_ID_21 leptonSF_EL_SF_SIMPLIFIED_ID_22 leptonSF_EL_SF_SIMPLIFIED_ID_23 leptonSF_EL_SF_SIMPLIFIED_ID_24 leptonSF_EL_SF_SIMPLIFIED_ID_25 leptonSF_EL_SF_SIMPLIFIED_ID_26 leptonSF_EL_SF_SIMPLIFIED_ID_27 leptonSF_EL_SF_SIMPLIFIED_Iso_0 leptonSF_EL_SF_SIMPLIFIED_Iso_1 leptonSF_EL_SF_SIMPLIFIED_Iso_2 leptonSF_EL_SF_SIMPLIFIED_Iso_3 leptonSF_EL_SF_SIMPLIFIED_Iso_4 leptonSF_EL_SF_SIMPLIFIED_Iso_5 leptonSF_EL_SF_SIMPLIFIED_Iso_6 leptonSF_EL_SF_SIMPLIFIED_Iso_7 leptonSF_EL_SF_SIMPLIFIED_Iso_8 leptonSF_EL_SF_SIMPLIFIED_Iso_9 leptonSF_EL_SF_SIMPLIFIED_Iso_10 leptonSF_EL_SF_SIMPLIFIED_Iso_11 leptonSF_EL_SF_SIMPLIFIED_Iso_12 leptonSF_EL_SF_SIMPLIFIED_Reco_0 leptonSF_EL_SF_SIMPLIFIED_Reco_1 leptonSF_EL_SF_SIMPLIFIED_Reco_2 leptonSF_EL_SF_SIMPLIFIED_Reco_3 leptonSF_EL_SF_SIMPLIFIED_Reco_4 leptonSF_EL_SF_SIMPLIFIED_Reco_5 leptonSF_EL_SF_SIMPLIFIED_Reco_6 leptonSF_EL_SF_SIMPLIFIED_Reco_7 leptonSF_EL_SF_SIMPLIFIED_Reco_8 leptonSF_EL_SF_SIMPLIFIED_Reco_9 leptonSF_EL_SF_SIMPLIFIED_Reco_10 leptonSF_EL_SF_SIMPLIFIED_Reco_11 leptonSF_EL_SF_SIMPLIFIED_Reco_12 leptonSF_EL_SF_SIMPLIFIED_Reco_13 leptonSF_EL_SF_SIMPLIFIED_Reco_14 leptonSF_EL_SF_SIMPLIFIED_Reco_15 leptonSF_EL_SF_SIMPLIFIED_Reco_16 leptonSF_EL_SF_SIMPLIFIED_Reco_17 leptonSF_EL_SF_SIMPLIFIED_Reco_18 Chapter C. Data and MC samples (missing MC16e sample) 411289.PhPy8EG.DAOD_TOPQ1.e7948_e5984_a875_r9364_r9315_p4346 411289.PhPy8EG.DAOD_TOPQ1.e7948_e5984_a875_r10201_r10210_p4346 (missing MC16e sample) ME sample 410465.aMcAtNloPy8EvtGen.DAOD_TOPQ1.e6762_a875_r9364_p4346 410465.aMcAtNloPy8EvtGen.DAOD_TOPQ1.e6762_a875_r10201_p4346 410465.aMcAtNloPy8EvtGen.DAOD_TOPQ1.e6762_a875_r10724_p4346 Sherpa sample 700124.Sh.DAOD_TOPQ1.e8253_e7400_s3126_r9364_r9315_p4434 700124.Sh.DAOD_TOPQ1.e8253_e7400_s3126_r10201_r10210_p4434 700124.Sh.DAOD_TOPQ1.e8253_e7400_s3126_r10724_r10726_p4434 Single top Wt DR dilepton top 410648.PowhegPythia8EvtGen.DAOD_TOPQ1.e6615_s3126_r9364_p4346 410648.PowhegPythia8EvtGen.DAOD_TOPQ1.e6615_s3126_r10201_p4346 410648.PowhegPythia8EvtGen.DAOD_TOPQ1.e6615_s3126_r10724_p4346 Wt DR dilepton tbar 410649.PowhegPythia8EvtGen.DAOD_TOPQ1.e6615_s3126_r9364_p4346 410649.PowhegPythia8EvtGen.DAOD_TOPQ1.e6615_s3126_r10201_p4346 410649.PowhegPythia8EvtGen.DAOD_TOPQ1.e6615_s3126_r10724_p4346 Wt DR incl. top 410646.PowhegPythia8EvtGen.DAOD_TOPQ1.e6552_s3126_r9364_p4346 410646.PowhegPythia8EvtGen.DAOD_TOPQ1.e6552_s3126_r10201_p4346 410646.PowhegPythia8EvtGen.DAOD_TOPQ1.e6552_s3126_r10724_p4346 Wt DR incl. tbar 410647.PowhegPythia8EvtGen.DAOD_TOPQ1.e6552_s3126_r9364_p4346 410647.PowhegPythia8EvtGen.DAOD_TOPQ1.e6552_s3126_r10201_p4346 410647.PowhegPythia8EvtGen.DAOD_TOPQ1.e6552_s3126_r10724_p4346 t channel top 410658.PhPy8EG.DAOD_TOPQ1.e6671_s3126_r9364_p4346 410658.PhPy8EG.DAOD_TOPQ1.e6671_s3126_r10201_p4346 410658.PhPy8EG.DAOD_TOPQ1.e6671_s3126_r10724_p4346 t channel tbar 410659.PhPy8EG.DAOD_TOPQ1.e6671_s3126_r9364_p4346 410659.PhPy8EG.DAOD_TOPQ1.e6671_s3126_r10201_p4346 410659.PhPy8EG.DAOD_TOPQ1.e6671_s3126_r10724_p4346 s channel top 410644.PowhegPythia8EvtGen.DAOD_TOPQ1.e6527_s3126_r9364_p4346 410644.PowhegPythia8EvtGen.DAOD_TOPQ1.e6527_s3126_r10201_p4346 410644.PowhegPythia8EvtGen.DAOD_TOPQ1.e6527_s3126_r10724_p4346 s channel tbar 410645.PowhegPythia8EvtGen.DAOD_TOPQ1.e6527_s3126_r9364_p4346 410645.PowhegPythia8EvtGen.DAOD_TOPQ1.e6527_s3126_r10201_p4346 410645.PowhegPythia8EvtGen.DAOD_TOPQ1.e6527_s3126_r10724_p4346 Diboson WW `n`n PowhegPy8EG.DAOD_TOPQ1.e4616_s3126_r9364_p4344 PowhegPy8EG.DAOD_TOPQ1.e4616_s3126_r10201_p4344 PowhegPy8EG.DAOD_TOPQ1.e4616_s3126_r10724_p4344 W Z`n``PowhegPy8EG.DAOD_TOPQ1.e4475_s3126_r9364_p4344 PowhegPy8EG.DAOD_TOPQ1.e4475_s3126_r10201_p4344 PowhegPy8EG.DAOD_TOPQ1.e4475_s3126_r10724_p4344 W Zqq``PowhegPy8EG.DAOD_TOPQ1.e4711_s3126_r9364_p4344 PowhegPy8EG.DAOD_TOPQ1.e4711_s3126_r10201_p4344 PowhegPy8EG.DAOD_TOPQ1.e4711_s3126_r10724_p4344 ZZ````PowhegPy8EG.DAOD_TOPQ1.e4475_s3126_r9364_p4344 PowhegPy8EG.DAOD_TOPQ1.e4475_s3126_r10201_p4344 PowhegPy8EG.DAOD_TOPQ1.e4475_s3126_r10724_p4344 ZZnn``PowhegPy8EG.DAOD_TOPQ1.e4475_s3126_r9364_p4344 PowhegPy8EG.DAOD_TOPQ1.e4475_s3126_r10201_p4344 PowhegPy8EG.DAOD_TOPQ1.e4475_s3126_r10724_p4344 ZZqq``PowhegPy8EG.DAOD_TOPQ1.e4711_s3126_r9364_p4344 PowhegPy8EG.DAOD_TOPQ1.e4711_s3126_r10201_p4344 PowhegPy8EG.DAOD_TOPQ1.e4711_s3126_r10724_p4344 C.2 A Full list of MC samples and derivaitons 189 Drell-Yan 700127.Sh.DAOD_TOPQ1.e8118_s3126_r9364_r9315_p4344 700127.Sh.DAOD_TOPQ1.e8118_s3126_r10201_r10210_p4344 700127.Sh.DAOD_TOPQ1.e8118_s3126_r10724_r10726_p4344 700126.Sh.DAOD_TOPQ1.e8118_s3126_r9364_r9315_p4344 700126.Sh.DAOD_TOPQ1.e8118_s3126_r10201_r10210_p4344 700126.Sh.DAOD_TOPQ1.e8118_s3126_r10724_r10726_p4344 700125.Sh.DAOD_TOPQ1.e8118_s3126_r9364_r9315_p4344 700125.Sh.DAOD_TOPQ1.e8118_s3126_r10201_r10210_p4344 700125.Sh.DAOD_TOPQ1.e8118_s3126_r10724_r10726_p4344 700130.Sh.DAOD_TOPQ1.e8118_s3126_r9364_r9315_p4344 700130.Sh.DAOD_TOPQ1.e8118_s3126_r10201_r10210_p4344 700130.Sh.DAOD_TOPQ1.e8118_s3126_r10724_r10726_p4344 700129.Sh.DAOD_TOPQ1.e8118_s3126_r9364_r9315_p4344 700129.Sh.DAOD_TOPQ1.e8118_s3126_r10201_r10210_p4344 700129.Sh.DAOD_TOPQ1.e8118_s3126_r10724_r10726_p4344 700128.Sh.DAOD_TOPQ1.e8118_s3126_r9364_r9315_p4344 700128.Sh.DAOD_TOPQ1.e8118_s3126_r10201_r10210_p4344 700128.Sh.DAOD_TOPQ1.e8118_s3126_r10724_r10726_p4344 700133.Sh.DAOD_TOPQ1.e8118_s3126_r10201_r10210_p4344 700133.Sh.DAOD_TOPQ1.e8118_s3126_r10724_r10726_p4344 700132.Sh.DAOD_TOPQ1.e8118_s3126_r9364_r9315_p4344 700132.Sh.DAOD_TOPQ1.e8118_s3126_r10201_r10210_p4344 700132.Sh.DAOD_TOPQ1.e8118_s3126_r10724_r10726_p4344 700131.Sh.DAOD_TOPQ1.e8118_s3126_r9364_r9315_p4344 700131.Sh.DAOD_TOPQ1.e8118_s3126_r10201_r10210_p4344 700131.Sh.DAOD_TOPQ1.e8118_s3126_r10724_r10726_p4344 364204.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364204.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364204.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364205.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364205.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364205.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364206.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364206.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364206.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364207.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364207.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364207.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364208.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364208.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364208.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364209.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364209.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364209.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364198.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364198.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364198.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364199.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364199.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364199.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364200.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364200.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364200.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364201.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 Chapter C. Data and MC samples 364201.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364201.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364202.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364202.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364202.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364203.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364203.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364203.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364210.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364210.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364210.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364211.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364211.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364211.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364212.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364212.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364212.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364213.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364213.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364213.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364214.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364214.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364214.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 364215.Sherpa.DAOD_TOPQ1.e5421_s3126_r9364_p4344 364215.Sherpa.DAOD_TOPQ1.e5421_s3126_r10201_p4344 364215.Sherpa.DAOD_TOPQ1.e5421_s3126_r10724_p4344 W +jets 700143.Sh.DAOD_TOPQ1.e8118_s3126_r9364_p4344 700143.Sh.DAOD_TOPQ1.e8118_s3126_r10201_p4344 700143.Sh.DAOD_TOPQ1.e8118_s3126_r10724_p4344 700144.Sh.DAOD_TOPQ1.e8118_s3126_r9364_p4344 700144.Sh.DAOD_TOPQ1.e8118_s3126_r10201_p4344 700144.Sh.DAOD_TOPQ1.e8118_s3126_r10724_p4344 700145.Sh.DAOD_TOPQ1.e8118_s3126_r9364_p4344 700145.Sh.DAOD_TOPQ1.e8118_s3126_r10201_p4344 700145.Sh.DAOD_TOPQ1.e8118_s3126_r10724_p4344 700146.Sh.DAOD_TOPQ1.e8118_s3126_r9364_p4344 700146.Sh.DAOD_TOPQ1.e8118_s3126_r10201_p4344 700146.Sh.DAOD_TOPQ1.e8118_s3126_r10724_p4344 700147.Sh.DAOD_TOPQ1.e8118_s3126_r9364_p4344 700147.Sh.DAOD_TOPQ1.e8118_s3126_r10201_p4344 700147.Sh.DAOD_TOPQ1.e8118_s3126_r10724_p4344 700148.Sh.DAOD_TOPQ1.e8118_s3126_r9364_p4344 700148.Sh.DAOD_TOPQ1.e8118_s3126_r10201_p4344 700148.Sh.DAOD_TOPQ1.e8118_s3126_r10724_p4344 700149.Sh.DAOD_TOPQ1.e8118_s3126_r9364_p4344 700149.Sh.DAOD_TOPQ1.e8118_s3126_r10201_p4344 700149.Sh.DAOD_TOPQ1.e8118_s3126_r10724_p4344 700150.Sh.DAOD_TOPQ1.e8118_s3126_r9364_p4344 700150.Sh.DAOD_TOPQ1.e8118_s3126_r10201_p4344 700150.Sh.DAOD_TOPQ1.e8118_s3126_r10724_p4344 700151.Sh.DAOD_TOPQ1.e8118_s3126_r9364_p4344 700151.Sh.DAOD_TOPQ1.e8118_s3126_r10201_p4344 700151.Sh.DAOD_TOPQ1.e8118_s3126_r10724_p4344 C.2 A Full list of MC samples and derivaitons 191 t tX 410155.aMcAtNloPythia8EvtGen.DAOD_TOPQ1.e5070_s3126_r9364_r9315_p4346 410155.aMcAtNloPythia8EvtGen.DAOD_TOPQ1.e5070_e5984_s3126_r10201_r10210_p4346 410155.aMcAtNloPythia8EvtGen.DAOD_TOPQ1.e5070_e5984_s3126_r10724_r10726_p4346 410156.aMcAtNloPythia8EvtGen.DAOD_TOPQ1.e5070_s3126_r9364_r9315_p4346 410156.aMcAtNloPythia8EvtGen.DAOD_TOPQ1.e5070_e5984_s3126_r10201_r10210_p4346 410156.aMcAtNloPythia8EvtGen.DAOD_TOPQ1.e5070_e5984_s3126_r10724_r10726_p4346 410157.aMcAtNloPythia8EvtGen.DAOD_TOPQ1.e5070_s3126_r9364_r9315_p4346 410157.aMcAtNloPythia8EvtGen.DAOD_TOPQ1.e5070_e5984_s3126_r10201_r10210_p4346 410157.aMcAtNloPythia8EvtGen.DAOD_TOPQ1.e5070_e5984_s3126_r10724_r10726_p4346 346343.PhPy8EG.DAOD_TOPQ1.e7148_s3126_r9364_p4346 346343.PhPy8EG.DAOD_TOPQ1.e7148_s3126_r10201_p4346 346343.PhPy8EG.DAOD_TOPQ1.e7148_s3126_r10724_p4346 346344.PhPy8EG.DAOD_TOPQ1.e7148_s3126_r9364_p4346 346344.PhPy8EG.DAOD_TOPQ1.e7148_s3126_r10201_p4346 346344.PhPy8EG.DAOD_TOPQ1.e7148_s3126_r10724_p4346 346345.PhPy8EG.DAOD_TOPQ1.e7148_s3126_r9364_p4346 346345.PhPy8EG.DAOD_TOPQ1.e7148_s3126_r10201_p4346 346345.PhPy8EG.DAOD_TOPQ1.e7148_s3126_r10724_p4346 410081.MadGraphPythia8EvtGen.DAOD_TOPQ1.e4111_s3126_r9364_r9315_p4346 410081.MadGraphPythia8EvtGen.DAOD_TOPQ1.e4111_e5984_s3126_r10201_r10210_p4346 410081.MadGraphPythia8EvtGen.DAOD_TOPQ1.e4111_e5984_s3126_r10724_r10726_p4346 410080.MadGraphPythia8EvtGen.DAOD_TOPQ1.e4111_s3126_r9364_p4346 410080.MadGraphPythia8EvtGen.DAOD_TOPQ1.e4111_s3126_r10201_p4346 410080.MadGraphPythia8EvtGen.DAOD_TOPQ1.e4111_s3126_r10724_p4346 304014.MadGraphPythia8EvtGen.DAOD_TOPQ1.e4324_s3126_r9364_p4346 304014.MadGraphPythia8EvtGen.DAOD_TOPQ1.e4324_s3126_r10201_p4346 304014.MadGraphPythia8EvtGen.DAOD_TOPQ1.e4324_s3126_r10724_p4346 EFT 504488.MGPy8.DAOD_TOPQ1.e8276_a875_r9364_p4346 504488.MGPy8.DAOD_TOPQ1.e8276_a875_r10201_p4346 504488.MGPy8.DAOD_TOPQ1.e8276_a875_r10724_p4346

Chapter 5. Spin Correlation with the full Run 2 dataset results and introduced a bias such as 24 :

where N is the number of bins. Slope method, however, shows no such bias. In the following, the parameter D is measured by considering the slope of the cosj ``distribution.

Systematic uncertainties

Among the various systematic uncertainties discussed in Section 5.5 some might not have any influence on the fit. The fit can also be affected by limited statistics in a systematic. In order to ensure a stable likelihood fit, the number of NPs (and hence, systematics) are pruned such that only those with significant effects are considered. Systematics can be pruned to only include shape effects, normalisation effects, or completely removed. For a shape uncertainty to be pruned, it must differ by less than 1% from the nominal sample in every bin. Normalization uncertainties are pruned if the total number of events differs by less than 1% from the nominal number.

In this analysis, the vast majority of systematics are pruned and only a handful remain significant, regardless of the strength of the pruning that is chosen. The results of asimov fits are the same with the 1% pruning and with no pruning. This is to be expected as the observable is an leptonic angular one and the precision of such observables is generally very good and is not often sculpted by systematic uncertainties. An illustration of the pruning of the systematics in this analysis is shown in Figure 5.39. Only systematics with at least one green or yellow bin contribute to the overall uncertainty, and any uncertainty with all red are dropped entirely.

In general, the majority of systematic uncertainties that are considered are inconsequential to the measurement.

All electron and muon scale factor uncertainties are dropped by the pruning procedure (regardless of the pruning strength) as are many of the b-tagging uncertainties. Some jet-related uncertainties survive for weak pruning but have almost no effect on the final result when included. The dominant systematic uncertainties arise due to signal modelling.

Chapter A. Like-Helicity quark-antiquark annihilation

.1: Distribution of t t invariant mass, m t t , at particle level for same helicity and opposite helicity q q initial states in two scenarios: (a) pp > tt˜and (b) bb˜> tt˜.

B. HGTD's additional contributions B.1 Timing correction plots

This section provides additional plots showing the results of the t hits distribution before and after the application of the time correction procedure. The t hits distribution before (in black) the timing correction procedure consist of a Gaussian core derived from the time dispersion of the LHC collisions convolved with the combined hit time resolution of the sensor. All sources of conservative jitters, timing variation of 100 ps bias, and time-of-flight are injected as explained in sections 4.4.3 and 4.4.5.

Chapter B. HGTD's additional contributions B.2 Large tail investigation

The truth information of t hits distribution measured in HGTD using different pads are shown in Figure ??. Due to the dense environment expected for the HL-LHC, most of charged particles are originating from pileup and low secondary electrons refereed to as "Delta" whereas, Hard Scatter (HS) particles are responsible for a few hits. The non-Gaussian tail arias from late particles Delta as shown in Figure ??. Splitting the t hits distribution of Delta into inner and outer ring shows a weird effect where some hits have time large than > 1 000 [ns] at inner ring. This behaviours is been reported to the HGTD simulation group and it is under investigation.

where, the signal strength, µ = s obs /s th is defined as the ratio of a measured cross-section and the theory cross-section. Keeping with the particle physics context, The data is expected to follow a Poisson probability distribution around its expected number of events. As such, the Poisson probability can be used to determine the probability of finding n i data events in a bin with expected n(µ) events as follows:

In the statistical analysis related to this thesis, µ is the main parameter of interest (POI) (see Section 5.5).

D.2.2 Systematic uncertainties

In reality, the SM theory does not perfectly describe nature, so our best fit estimation of µ is biased. Nuisance parameters (NPs) are introduced to the model to account our ignorance. Despite not being of immediate interest, NPs must be taken into account when analysing the more important parameters of interest. To model the effect of the NPs, the expected number of background and signal counts are parametrized with additional parameters, q = {q 1 , q 2 ,...q m } as follows:

In this new parametrization, the likelihood is now dependent on the additional parameter q , which affects the estimate of uncertainty of the POI. Therefore, The q parameters themselves then need to be constrained.

For this raison, auxiliary measurements, y, are needed to reduce parameter space and constrain nuisance parameters. Combining auxiliary measurements with the combined likelihood will allow one to build a joint model.

The NPs are constrained by the probability density functions ((pdf)) p(y | ✓). The choice of the (pdf) for NP depends on the type of the uncertainty associated with it. The Gaussian (pdf) is a frequent choice for NPs, as

3 Test statistic 195 it described the uncertainties on parameters that can be both positive and negative. The nuisance parameter is expected to be distributed as a Gaussian with a width equal to the uncertainty on the parameter, as estimated from auxiliary measurements. However, the Gaussian (pdf) is not suited for positively defined observables such as luminosity, cross sections, efficiencies etc. The log-normal (pdf) is used for nuisance parameters associated to these observables, which affect only the normalization. The effect of limited statistics on simulation or background estimation methods is almost always included in the likelihood with a set of additional parameters for each bin. Those parameter variations are modelled with Gamma functions and are generally denoted as g-stat. The best estimate for µ and q is obtained by maximizing the likelihood or equivalently minimizing the negative logarithm of the likelihood (negative log-likelihood).

D.3 Test statistic

In the likelihood fit results, either the signal is detected, so its significance can be determined, or there is no

signal. An upper limit on the signal strength can be calculated. This can be accomplished by defining the following test statistic:

where the qµ is called the conditional maximum-likelihood (ML) estimator of ✓ and represents MLE of ✓ that maximizes L for a specified value of µ. The μ and ✓ are the unconditional ML estimators of µ and ✓, respectively. The profile likelihood ratio l µ 2 [0, 1]. For l µ = 1, µ is equal to the MLE μ, thus good agreement between data and the hypothesized value of µ. Deviation form the MLE μ results in a decrease of profile likelihood ratio l µ .

For convenience, The test statistic in Equation D.11 is re-defined as the log of the profile likelihood ratio multiplied by negative 2 and it reads:

The discovery of a new signal the statistic t µ with µ 0 is used to reject the background-only hypothesis (µ = 0). The test statistics, q 0 , is defined as follows:

Using the p-value, we can determined the agreement between data and null hypothesis and can be defined as follows:

where f (q 0 |0) is the pdf of the statistic q 0 under the assumption of background-only (µ = 0) hypothesis.

Chapter E. Normalisation to the integrated luminosity quark mass value of 172.5 GeV is adopted in this Section and is given by [START_REF]NNLO prediction[END_REF]:

where the fist error is theoretical and the second is for the PDF and a s . The black uncertainty band includes the limited MC statistics together with the systematic uncertainty related to t t modelling. The triangle coloured markers correspond to the particle level distribution, while those with circle marks represent the unfolded one. The lower panel shows the ratio between the particle level and unfolded particle level distributions and the uncertainty is computed from the statistical and systematic error on the unfolded histograms. The black uncertainty band includes the limited MC statistics together with the systematic uncertainty related to t t modelling. The triangle coloured markers correspond to the particle level distribution, while those with circle marks represent the unfolded one. The lower panel shows the ratio between the particle level and unfolded distributions and the uncertainty is computed from the statistical and systematic error on the unfolded histograms. The red lines represent the SM sample where the solid and dashed correspond to the unfolded and particle level distributions, respectively. The black uncertainty band includes the limited MC statistics together with the systematic uncertainty related to t t modelling. The triangle coloured markers correspond to the particle level distribution, while those with circle marks represent the unfolded one. The lower panel shows the ratio between the particle level and unfolded distributions and the uncertainty is computed from the statistical and systematic error on the unfolded histograms.