Contents

Laboratoire de Physique Nucléaire et des Hautes Énergies

Strategies for precision measurements

of the charge asymmetry of the  boson mass

at the LHC within the ATLAS experiment

by

Florent Fayette

presented to the

Université de Paris VI – Pierre et Marie Curie

for the degree of Docteur de l’Université Paris VI

. This Ph.D. thesis was supervised by Dr. Mieczysław Witold Krasny at the LPNHE institute in Paris and it was eventually submitted to the Université de Paris VI - Pierre et Marie Curie to obtain the degree of Docteur de l’Université Paris VI on January 2009 in front of a committee consisting of Prof. Michael Joyce (President of the committee), Prof. Stefan Tapprogge (Referee), Prof. Fabian Zomer (Referee), Dr. Thomas LeCompte, Dr. Gavin Salam and Dr. Mieczysław Witold Krasny.

Abstract

This thesis dissertation presents a prospect for a measurement of the charge asymmetry of the  boson mass () at the LHC within the ATLAS experiment. This measurement is of primordial importance for the LHC experimental program, both as a direct test of the charge sign independent coupling of the bosons to the fermions and as a mandatory preliminary step towards the precision measurement of the charge averaged boson mass. This last pragmatic point can be understood since the LHC specific collisions will provide unprecedented kinematics for the positive and negative channels while the SPS and Tevatron collider produced and on the same footing. For that reason, the study of the asymmetries between and in Drell–Yan like processes (production of single decaying into leptons), studied to extract the properties of the boson, is described thoroughly in this document.

Then, the prospect for a measurement of at the LHC is addressed in a perspective intending to decrease as much as possible the systematic errors that will inevitably comes from the misunderstanding of both phenomenological and apparatus modeling. For that matter strategies have been devised specifically for the present measurement to display robustness with respect to the main uncertainties. These strategies consist of introducing new observables along with considering specific LHC running modes and configurations of the ATLAS tracker.

Eventually we show that the present (2009) precision can be improved at the LHC by a factor of and argue that such a precision is beyond the reach of the standard measurement and calibration methods imported to the LHC from the Tevatron program.

J’en ai plein le QCD.

Remerciements, podziekowania, acknowledgements, etc.

Mes premiers remerciements s’adressent à ceux qui ont œuvré pour que j’obtienne une bourse pour réaliser ma thèse au LPNHE, à savoir Anne-Marie Cazabat, Matteo Cacciari, Jean-Eudes Augustin, Philippe Schwemling et Mieczysław Witold Krasny mon directeur de thèse. En particulier je tiens à exprimer ma reconnaissance à Pascal Debu, Philippe Schwemling et Witold Krasny pour avoir dispensé avec parcimonie toutes injonctions pour rattraper ce léger retard que j’accusais ce qui m’aura permis de parachever ma thèse en toute quiétude. Enfin je tiens à remercier Stefan Tapprogge et Fabian Zomer pour avoir accepté d’en être les rapporteurs et pour avoir studieusement pris en considération mon travail pendant leurs vacances de Noël. Cette gratitude s’adresse par ailleurs aux autres membres du jury, Michael Joyce, Tom LeCompte, Gavin Salam et mon directeur.

Je remercie ce dernier, Mieczysław Witold Krasny, pour m’avoir offert l’opportunité de faire ma thèse avec lui et surtout pour avoir su composer, à la faveur de mes capacités et de mes aspirations, un sujet de thèse idoine à ma délicate personne. Par bien des aspects travailler sous sa direction sur le présent sujet fit largement honneur au travail de recherche tel que l’on peut se le figurer dans l’idéal. Ces remerciements ne sauraient rendre justice à ce que furent ces trois dernières années en sa compagnie si je ne devais mentionner sa sympathie, sa bienveillance et pour ne m’avoir jamais mis sous pression favorisant en cela le plaisir dans le travail à la production de résultats équivoques. Enfin l’originalité du personnage, à mi-chemin entre le diable de Tasmanie et le “road-runner”, son génie ainsi que son humour ont fait de cette collaboration une expérience unique.

Just like to fire there is water, or to vodka there is miód pitny to Witold Krasny there is Wiesław Płaczek. On many aspects I consider Wiesław Płaczek as my second supervisor for all of his helps which improved the quality of my work over these last years. I learned a lot from his rigor, his infallible skills in informatics and I am indebted for all the multiple times he helped me using the WINHAC Monte Carlo. Besides I always appreciated his zen attitude and I also do not forget that most of my stays in Kraków would not have been possibly that smooth if it had not been for his attentions and his hospitality. Za wszystko dziȩkujȩ Ci polski mistrzu Zen.

I am deeply grateful to Giorgos Stavropoulos whose help and support, while fighting with Athena, allowed me to reach my first concrete results in my thesis. His broad sympathy really made of this task an agreeable moment, which, in regard of what Athena is, stands as a huge compliment.

Jestem wdziȩczny Andrzejowi Siódmokowi. Indeed, Andrzej brought equilibrium to tackle the Krasny–Płaczek doublet which would have been hard to tame all by myself (especially the M.W.K. component), besides I am grateful to him for his sympathy and his hospitality in Kraków. I also do not forget Katarzyna Rejzner whose recent arrival in our team brought additional fun and more importantly allowed to decrease the amount of krasnic activity Andrzej and I used to deal with.

Je remercie les membres du groupe ATLAS du LPNHE pour leur accueil et plus précisément Frédéric Derue pour s’être montré toujours disponible pour m’aider lors de mes confrontations non-ASCII avec Athena ainsi qu’Emmanuel Busato dont les aides, même après son départ pour Clermont–Ferrand, auront grandemement contribuer à améliorer la qualité de mon code. Je tiens aussi à rendre hommage à Emmanuel “Mini Manu” Hornero, Jérôme “Jers” Glisse et Rui Pereira pour avoir toujours pris le temps de s’intéresser aux problèmes informatiques sur lesquels je butais. Je double mes louanges à l’endroit de Rui et Jers dont les aides en Matplotlib auront contribuer à améliorer la facture de ce document. Je triple mes louanges à Mini Manu pour tout le temps qu’il m’aura accordé pour résoudre les obstacles techniques qui se sont dressés sur mon chemin. Enfin remerciements particuliers à Pietro Cavalleri, Jérémie Lellouch et Jers (une fois de plus). Pietro, gentilshomme d’Italie dedans Paris, avec qui j’ai eu l’opportunité de passer de bons moments grace à son humour et un caractère unique. Jérémie, du groupe des zéros du LPNHE, pour nos échanges désinvoltes voire ouvertement oisifs qui, à l’aune de nos travaux, s’avérèrent salutaires. Jers enfin qui, outre toutes les aides en informatiques, m’aura prodigué à maintes reprises l’occasion de rire de bon cœur.

I thank the members of the ATLAS group of the Institute of Nuclear Physics in Kraków for their hospitality during my several stays amongst them, in particular Jolanta Olszowska and Janusz Chwastowski. I am grateful to the nice fellows I met in Kraków with whom I shared nice moments especially Zofia Czyczula, Ewa Stanecka, Adam Matyja, Sebastian Sapeta, Hayk and Meri. I have also a thought for all the poles from CERN especially Justyna and Janek. Enfin de manière plus générale je rends hommage au peuple Polonais pour son accueil et son savoir-vivre. Depuis les charmantes polonaises callipyges aux vieillards voutés par les ans en passant par les vendeurs ambulants de pierogi et de vodka c’est toujours avec grande courtoisie, force délicatesse et patience non feinte que l’on m’aura considéré. Ce faisant, tous mes séjours à Cracovie, sans exception, furent autant de pauses “civilisation” qui auront émaillées les vicissitudes inhérentes à la vie parisienne.

I would like to greet the people from the CERN Standard Model forum especially, Lucia DiCiacco, Nathalie Besson, Stefan Tapprogge, Thomas Nunneneman, Maarten Boonekamp and Troels Petersen. Special thanks to Tom LeCompte for sharing his experience with a guenuine vivid and communicative enthusiasm. I am also grateful to the following people with whom I had positive exchanges : David Rousseau, Andrea Dell’Acqua, Wojciech Wojcik, Paweł Brückman de Restrom, Mike Whalley, Borut Kersevan, Kristin Lohwasser, Fred Olness, Muge Karagoz Unel, Chris Hay, Ashutosh Kotwal, Mark Lancaster and Arkadiuz Weinstein.

Je suis reconnaissant à Benoit Loiseau pour son attention quant au bon déroulement de ma thèse et je l’associe aussi à ma gratitude envers Bertrand Desplanques tous deux m’ayant permis de trouver asile scientifique à Grenoble lorsque je m’étais mis en délicatesse avec une partie du milieu académique parisien. Outre le sauvetage d’un avenir potentiel dans les sciences, les explications de Bertrand Desplanques au cours de mon stage sous sa direction m’auront apportées beaucoup dans mon approche actuelle vis-à-vis de la physique des particules.

Je remercie aussi mes collègues du LPTHE : Matteo Cacciari, gran maestro di quark pesanti, pour le stage sous sa direction et pour m’avoir toujours fait profiter des ses explications limpides pendant le début de ma thèse quand mon esprit penait à se calibrer sur le mode de pensée polonais, Gavin Salam pour ses aides efficaces et pour sa relecture de mon manuscrit enfin Bruno Machet pour sa grande bonté, son enthousiasme et toutes ces agréables discussions qui, faisant écho aux discours de mes autres mentors, m’auront permis d’élargir mon ouverture d’esprit. Enfin une mention particulière à Rédamy Perez-Ramos pour ses aides et ses encouragements.

Je m’incline devant Souad Rey, Véronique Joisin, Jocelyne Meurguey et Annick Guillotau pour leur efficacité dans l’organisation de mes déplacements comme dans d’autres taches administratives.

Jestem wdzieczny mojej małej siostrze Ani “Pani Ruda” Kaczmarska i Mariusz Bucki whose sympathy brought me a lot during these last years especially when hustling my way through krasnic mazes or when I was feeling hyper-weak rather than electroweak. I cherish all these nice moments I shared with them in Kraków and thank them for the basics in polish they learned me, short, but yet enough to get myself into trouble.

Je rends un hommage particulier à Pascal Parneix dont la passion pour la physique et l’énergie mise à son service auront marqué une partie de mes études universitaires, pour ce stage bien sympathique passé sous sa direction ainsi que pour ses conseils et ses aides dans les moments difficiles.

Je remercie Alain “Docteur Folamour” Mazeyrat pour son soutien dans les passages à vide et salue sa dextérité qui sauva une carte mère dont la perte aurait grandement fait défaut à mon travail. Merci aussi à Jean Onésippe pour son aide, son écoute et pour avoir distillé un peu de sa grande sagesse antillaise dans mon esprit inquiet. Je suis reconnaissant à Olivier Destin pour m’avoir aidé bien des fois en C++ et en LaTeX avec une efficacité hors du commun. Enfin encore plus important je le remercie pour tous nos échanges passionants, divertissants ou hilarants qui m’auront permis de m’aérer l’esprit.

Pour conclure je remercie infiniment mes parents et ma sœur pour leur soutien au cours de mes longues années d’études sans lequel je n’aurais pas pu matérialiser mes aspirations. Chciałbym także podziȩkować mojej małej i subtelnej Paulince za jej czułość i ciepło oraz za wszystkie spȩdzone razem chwile, które tak bardzo pomogły mi przetrwać trudny czas przed obrona̧ pracy.

\@mainmatterfalse

## Chapter 1 Introduction

Là trois cent mille personnes trouvèrent place et bravèrent pendant plusieurs heures une température étouffante, en attendant l’arrivé du Français. De cette foule de spectateurs, un premier tiers pouvait voir et entendre ; un second tiers voyait mal et n’entendait pas ; quant au troisième, il ne voyait rien et n’entendait pas davantage. Ce ne fut cependant pas le moins empressé à prodiguer ses applaudissements.

De la Terre à la Lune
Jules Vernes

The actual description of the fundamental building blocks of matter and the interactions ruling them is called the Standard Model. It is believed not to be the most fundamental description but rather a phenomenological approximation for energies below the TeV scale. In this model, two types of particles are to be distinguished. First are the quarks and leptons building up the matter and, second, are the bosons that mediate the interactions among them. Of all the four known fundamental interactions –the gravitation, electromagnetic, weak and strong interactions– only the last three are now implemented within the mathematical formalism supporting the Standard Model. Amid these interactions the weak one, acting on both quarks and leptons, is mediated by the exchange of the massive neutral and two charged bosons. The and , which are antiparticle of each other, are the object of interest in this dissertation.

The boson has been observed in 1973 in the Gargamelle bubble chamber at CERN while the bosons were observed in single production in 1983 in the UA1 detector of the Super Proton Synchrotron (SPS) collider, again at CERN. This discovery confirmed the Glashow–Weinberg–Salam electroweak model. Since then, the have been studied for the last decades at the Large Electron Positron (LEP) collider ( pair production) and at the Tevatron collider (single production). These two experiments allowed to measure properties such as its mass or its width . These parameters are important since they provide, when combined to other Standard Model parameters, constraints on the Standard Model. For example the masses of the and of the top quark constrain the mass of the hypothetical Higgs boson. The specific study of this work aims at improving the experimental value of the charge asymmetry of the mass by studying single production at the Large Hadron Collider (LHC). This measurement has so far not received much attention and, as a consequence, displays an accuracy 10 times larger than the one on the absolute mass . With the new possibilities that the LHC collider should offer for the next years we considered the prospect for a drastic decrease of the experimental error on the value using the ATLAS multipurpose detector capabilities. The first motivation for such a measurement is to refine the confirmation of the invariance principle through the direct measurement of the masses, to complete the accurate test made by observing charged and life time decays. Other motivations will be given later. Besides, as it will be shown, the bosons, despite the fact they will be produced with the same process as at the SPS and the Tevatron colliders, will nonetheless display original kinematics due to the nature of the colliding beams. Indeed, while and are produced on the same footing in collision, this will not be the case anymore at the LHC. The first step of this work is to understand the and kinematics in collisions. After, this first stage providing all cards in our hands to understand the properties at the LHC, the rest of this work will focus on the improvement that could be provided to the measurement using the ATLAS detector at the LHC. Here, rather than reusing Tevatron tactics, we devised new strategies specific to this measurement and to the LHC/ATLAS context. The philosophy for these strategies –as it will be detailed– aims at being as independent as possible of both phenomenological and experimental uncertainties, that cannot be fully controlled. Eventually, we argue that the proposed strategies, could enhance the actual accuracy on by a factor of .

This work represents the second stage of a series of several publications aiming at providing precision measurement strategies of the electroweak parameters for the upcoming LHC era. In the same logic, next steps will provide dedicated strategies for the measurement of the absolute mass and width of the boson.

This work is the result of three years of collaboration in a team consisting of Mieczysław Witold Krasny, Wiesław Płaczek, Andrzej Siódmok and the author. Furthermore, the technical work that took place within the ATLAS software was made possible with substantial help from Giorgos Stavropoulos.

\@mainmattertrue

## Chapter 2 Phenomenological context and motivations

- Depuis lors, continua Aramis, je vis agréablement. J’ai commencé un poème en vers d’une syllabe ; c’est assez difficile, mais le mérite en toutes choses est dans la difficulté. La matière est galante, je vous lirai le premier chant, il a quatre cents vers et dure une minute.
- Ma foi, mon cher Aramis, dit d’Artagnan, qui détestait presque autant les vers que le latin, ajoutez au mérite de la difficulté celui de la brièveté, et vous êtes sûr au moins que votre poème aura deux mérites.

Les Trois Mousquetaires
Alexandre Dumas

This Chapter introduces the background of the present work. The first part reviews in a nutshell the Standard Model which is the present paradigm to describe the elementary particles and their interactions. Then, a parenthesis is made on the experimental setting to already provide to the reader the global vision necessary to understand the rest of the Chapter. For this purpose, the hadronic production of bosons and how their properties are extracted from leptonic decays are reviewed. Next, motivations for the measurement of the mass charge asymmetry are given.

The second part introduces the notations and conventions used throughout the document.

The third part describes the phenomenological formalism used to study  bosons produced in hadronic collisions and decaying into leptons, phenomenon also known as production of in Drell–Yan-like processes. This presentation, rather than being exhaustive, emphasises the relevant kinematics needed to understand the gist of physics in Drell–Yan-like processes and how, from such kinematics, the properties like its mass and its width are extracted.

Finally the Chapter closes on a presentation of the rest of the document.

### 2.1 The Standard Model in a nutshell

#### 2.1.1 Overview

Based on the experience that seemingly different phenomena can eventually be interpreted with the same laws, physicists came up with only four interactions to describe all known physics processes in our Universe. They are the gravitational, electromagnetic, weak and strong interactions. The electromagnetic, weak and strong interactions are described at the subatomic level by Quantum Field Theory (QFT), the theoretical framework that emerges when encompassing the features of both Special Relativity and Quantum Mechanics. In particular the weak and electromagnetic interactions are now unified in QFT into the Glashow–Weinberg–Salam electroweak theory. The description of the three interactions in QFT is called the Standard Model of elementary particles and their interactions, or simply Standard Model (SM). The “Standard” label means that it is the present day reference, which, although not believed to be the ultimate truth, is not yet contradicted by data. This amends for the term “phenomenology” used in some applications in the SM, where phenomena are described with non fundamental models and, hence, non fundamental parameters. Gravity, on the other hand, is not yet implemented in QFT, it is described by General Relativity. It mostly concerns Cosmology, that rules the behaviour of space–time geometry under the influence of massive bodies and in non-inertial frames. It explains the structure of the Universe and its components on large scales, and eventually leads to the Big Bang theory. In the Standard Model, the effects of gravity are negligible as long as the energies stay below the Planck scale ().

Before presenting the Standard Model in more details we present an overview of the particles properties and denominations. At this stage, we already adopt natural units where and . Table 2.1 displays the elementary (i.e. structureless) particles building the matter. They are fermions111Fermion is the generic term used to qualify all particles whose spin is a “half-value” of the Planck constant, i.e. in units of , being an integer. Boson on the other hand qualifies particles whose spin is an integer of the Planck constant. of spin and come in two types, the quarks and the leptons, both present in three generations of doublets : this leads to six different flavours of quarks or leptons. The only thing that differentiates each generation is the masses of the particles [1]. Quarks have fractional electrical charge –with respect to the charge of the electron– and are sensitive to all interactions. Charged leptons are sensitive to the weak and electromagnetic interactions, while neutral leptons (neutrinos) are only sensitive to the weak interaction. Interactions among all fermions are due to the exchange of elementary particles of boson type. The electromagnetic interaction is mediated by the exchange of neutral massless photons between all particles that have an electrical charge. The weak interaction is mediated by three massive vector bosons : the electrically charged and and the electrically neutral . All particles bearing a weak isospin charge are sensitive to the weak interaction. The strong interaction between particles having a color charge is mediated by massless colored gluons of eight different kinds.

The quarks and leptons were discovered on a time scale that span no less than a century. The electron was observed at the end of the century [2, 3, 4] while the muon and tau were observed respectively in the thirties [5] and the seventies [6]. The electron neutrino was found in the fifties [7], the ten years later [8] and the after another forty years later [9]. The up, down and strange quarks were observed in hadrons in the deep inelastic electron–nucleon and neutrino–nucleon scattering experiments [10, 11]. The discovery of the charmed quark occurred in the seventies [12, 13], the bottom was discovered a few years later [14], while the top, due to its very large mass, was discovered in the nineties at the Tevatron collider [15, 16]. Concerning the gauge bosons, the photon was discovered with the theoretical interpretation of the photo-electric effect [17] while the and bosons were isolated in collisions [18, 19, 20, 21]. The existence of gluons was deduced from the observation of hadronic jets generated in collisions at high energies [22, 23, 24].

Each particle mentioned so far has its own anti-particle with opposite quantum numbers. Anti-particles of electrically charged particles are noted with the opposite sign of the charge (e.g. is the anti-particle of ) while neutral particles are noted with a bar upon them (e.g. is the anti-particle of ). Some particles are their own anti-particles, e.g. the photon and the boson. The rest of this section presents the main features of the Standard Model. First, a glimpse at QFT is given, then we describe the electromagnetic, strong and weak interactions in QFT.

#### 2.1.2 The theoretical background of the Standard Model : Quantum Field Theory

The beginning of the century witnessed the emergence of two revolutions in physics : Quantum Mechanics and Special Relativity. Quantum Mechanics deals with phenomena below the atomic scale while Special Relativity, based on space–time homogeneity/isotropy, describes laws of transformations between inertial frames. Exploring deeper the subatomic world, the Heisenberg principle entails that, as the length scale decreases, the momentum (energy) increases. This increase of the energy amends for the High Energy Physics (HEP) terminology used to refer to the elementary particles and their interactions. In those conditions, taking into account Special Relativity is mandatory.

The Klein–Gordon and Dirac equations were the first relativistic generalisations of the Schrödinger equation. The problem with these equations is that they have negative energies solutions which are difficult to interpret. This problem can be overcome if one considers that a negative energy solution describes an anti-particle. An anti-particle can be interpreted as a positive energy particle travelling backward in time. Besides, in this relativistic and quantum context, a solution describes a quantum field whose elementary excitations are the particles. Then, the number of particles is no longer fixed and the relativistic mass–energy equivalence accounts for the annihilation/creation of pairs of particle–anti-particle. This new framework is the Quantum Field Theory.

QFT describes local interactions of supposedly point-like particles. It bears its own difficulties, with the arising of infinities, which need to be consistently taken care of in the process of renormalisation. Taking into account local Lorentz invariant QFT with the spin–statistic theorem leads to the symmetry, i.e. the conservation the product of the charge conjugation , parity and time inversion operators in any processes. In particular, symmetry shows that .

One method to quantify a field is to use the Feynman path integral. Path integral formulation of Quantum Mechanics relies on a generalisation of the least action principle of classical mechanics. It is based on the Lagrangian density from which the equations of the dynamics can be deduced. The Lagrangian density (called hereafter Lagrangian) of the field under study contains its kinetic and potential energies. Hence, the description of the dynamics of elementary particles under the influence of fundamental forces consists to formulate the right Lagrangian with : (1) the kinetic energy of the free fields of the spin- quarks/leptons and of the spin- bosons mediators and (2) the potential term displaying their mutual interactions and –if any– the bosonic field self-interaction terms. A term of interaction is proportional to a coupling constant, say , inherent to the interaction under study. Up to the fact that this constant is small enough, the calculus of the amplitude of probability can be developed into a perturbative expansion in powers of the coupling constant . All terms proportional to in the expansion involve processes with interaction points (vertices) between quarks/leptons and bosons. We can associate to them drawings, called Feynman diagrams. Each one entangles all processes sharing the same topological representation in momentum space.

Finally let us mention that QFT is also extensively used in condensed matter physics, in particular in many-body problems, and the interplay of ideas between this domain and HEP is very rich. With the basics features of QFT presented above we now describe the electromagnetic, strong and weak interactions.

#### 2.1.3 Quantum Electrodynamics

The theory that describes the electromagnetic interactions in QFT is called Quantum Electrodynamics (QED). The Lagrangian of QED includes the Dirac and Maxwell Lagrangians respectively to describe the kinetic energy of, say, the free electron field (its excitation are the electrons) and of the free gauge field (its excitation are the photons). It also contains an interaction term between the photon and electron fields which is proportional to , the charge of the electron. The charge is linked , the fine structure constant of QED, via . Since the expansion in power of is feasible. Still, when taking into account Feynman diagrams with loops, i.e. quantum fluctuations involving particles with arbitrary four-momenta, the calculus diverges. This reflects the conflict between the locality of the interactions inherited from Special Relativity and Quantum Mechanics that allows virtual processes to have arbitrary high energies. Divergences come from terms like , where is a temporary unphysical cut-off parameter and the mass of the electron.

The procedure to get rid of those divergences is called renormalisation. It consists in redefining the fundamental parameters of the theory by realising that the charge used so far is already the one resulting from all quantum loops. These loops affect the value of the bare charge that one would observe if there was no interaction. Expressing in a perturbative series as a function of and and, then, writing in the expression of the amplitude as a function of allows to get rid of the divergences when . The effect of these loops depends on the energy, for that reason a renormalisation scale energy is chosen close to a characteristic energy scale in the process. Since is arbitrary the charge must obeys an equation expressing the invariance of physics with respect to . This leads to the formulation of the renormalisation equations. Eventually the loops screen the bare charge. Then, the charge and mass of the electron, in our example, are no longer fundamental parameters but rather effective parameters (running coupling constants) which values depends of .

Gauge invariance is an essential tool in proving the renormalisability of a gauge theory. For QED, it appears that equations are invariant under a local transformation of the phase factor of the field . The invariance of the Lagrangian under this particular transformation directly governs the properties of the electromagnetic interaction. Group theory allows to describe these properties, in the case of QED this group is . Consequent to those developments, physicists tried to describe the remaining interactions with the help of gauge invariant theories, too.

#### 2.1.4 Quantum Chromodynamics

The description of the strong interaction by a gauge theory became sensible when the elementary structure of the hadrons was discovered in lepton–hadron deep inelastic scattering (DIS) experiments. Those experiments shed light on the partons, the hypothetical constituents of the hadrons, which become quasi free at high energies. The partons are of two types, the quarks and the gluons [25, 26] and, at first, the structure of the hadrons was found to satisfy some scaling properties.

The observation of hadrons made quarks of the same flavour and spin hinted at the existence of a new quantum number attached to quarks that could eventually help to anti-symmetrise the wave functions of such hadrons, and save the Fermi exclusion principle. It was then postulated that the group describing the strong interaction is , where c stands for color. Indeed, in analogy with the additive mixing of primary colors, quarks hold three primitive “colors” (“red”, “green” and “blue”) that, when combined, give “white” hadrons, which justifies the term of chromo in Quantum Chromodynamics (QCD), the QFT model of the strong interaction [27, 28, 29]. This group is non-abelian which pragmatically implies that the interactions carriers, the gluons, are colored and can interact among themselves, in addition to interacting with quarks. There are eight colored gluons to which we can associate the gauge field where .

This last property implies striking differences with respect to QED. In addition to gluon–quark vertices, equivalent to photon-lepton/quark vertices in QED, there are now three and four gluons vertices. During the renormalisation of the strong interaction coupling constant , while the quarks loops screen the color charge, the gluons loops magnifies it and eventually dominate. As a consequence , or equivalently , is a decreasing function of the renormalisation energy . This justifies a posteriori the asymptotic freedom hypothesis for the partons in DIS, and also, qualitatively, the confinement of quarks in hadrons. Hence, at some point, when the energy involved in a QCD process becomes small, gets large which prohibits the use of perturbative calculations. The energy scale marking this frontier is of the order of . Below this limit, other techniques have to be employed to make calculations (e.g. lattice QCD). Note, finally, that the change of with the energy implies that the formerly observed scaling property must be violated, which was observed and formalised in particular with DGLAP equations [30, 31, 32]. Note also that the renormalisation constraints impose to have as many quarks as there are leptons which twice allowed to predict new quarks : the charm to match, with the , and , the two first leptons families and eventually the top and bottom quark doublet when the tau lepton was discovered.

#### 2.1.5 Electroweak interactions

##### (a) Weak interactions and the path to the electroweak model

The weak interaction was discovered in nuclear –decay and recognised to drive as well the muon decay. The first model was a point-like interaction involving charged currents (to account for the change of flavor) and proportional to the Fermi constant . The peculiarity of this interaction is that it violates parity, leading to the vector minus axial-vector () nature of the weak currents. Hence, weak interactions couple only left-handed particles and right-handed anti-particles.

The problem of the Fermi model is that it collapses above by giving inconsistent predictions. Using unitarity constraints, one can shape the form of the more fundamental model ruling the weak interactions. First, based on the previous examples of QED and QCD, the weak interaction can be assumed to be mediated by the exchange of heavy charged vector bosons. In this new context the propagator of the boson damps the rise with energy. Assuming that the coupling strength of the to fermions is comparable to the one of QED, one finds that the mass of the must be around . Hence, due to the mass of the the range of the weak interaction is of the order of .

In addition to charged currents the theory needs neutral currents as well. Let us consider, for example, the process that occurs with an exchange of a neutrino. When the bosons are produced with longitudinal polarisation their wave functions grows linearly with the energy while the exchange of a neutrino predicts a growth quadratic with the energy. To overpass this new breakdown of unitarity a neutral boson, the , must take part in the process like with a tri-linear coupling to the and .

A consequence of this tri-linear coupling is that, now, scattering of vector bosons can be observed. In the case of the amplitude of longitudinally polarised bosons, built-up by the exchange of , grows as the fourth power of the energy in the center of mass of the collision. This leading divergence is canceled by introducing a quadri-linear coupling among the weak bosons. Still, unitarity is not yet restored for asymptotic energies since the amplitude still grows quadratically with the energy.

At this stage, two solutions can be envisaged. The scattering amplitude can be damped, either by introducing strong interactions between the bosons (technicolor model), or by introducing a new particle, the scalar Higgs boson , which interferes destructively with the exchange of weak bosons.

We have seen that the coupling strength of the weak and electromagnetic interactions are of the same order. Besides we have seen that the road to preserve unitarity is very much linked to the handling of longitudinally polarised bosons. We now give a brief presentation of the electroweak model that unifies electromagnetic and weak interactions.

##### (b) The electroweak model

Since weak interactions couple only left particles and right anti-particles, finding a gauge theory to cope with this implies that all elementary particles have to be considered, in a first approximation, to be massless. The path to the electroweak model can be presented in two steps. In the first step, the Glashow–Weinberg–Salam (GWS) model [33, 34, 35] unifies the weak and electromagnetic interactions up to the approximation that all quarks, leptons and vector bosons are massless, QCD can be added along to the GWS model, still with massless particles. The second step describes the mechanism that allows massive particles without explicitly breaking the gauge invariance constructed earlier. This is the Brout–Engler–Higgs–Kibble mechanism [36, 37, 38].

The fact that elementary fermions were found in doublets and the desire to unify the weak and electromagnetic interactions in a gauge invariant theory led to the gauge groups .

The takes into account the fact that left-handed fermions are found in weak isospin doublet, i.e. which, for the conventionally used third component, means , while right-handed fermions are found in singlets. Let us note also that the left-handed quarks eigenstates in weak interactions differ from the mass eigenstates ; the convention tends to write down the lower isospin states as linear combinations of the eigenstate masses , like , where and are the flavors of the quarks and the elements are the elements of the Cabibbo–Kobayashi–Maskawa (CKM) [39, 40] matrix. The non-abelian properties of the group ensures that some mediators of the interaction will have a charge and then will be assimilable to the bosons. At this stage, there are 3 fields, , and that couples to particles with a weak isospin with a coupling constant .

The group , although different from , is chosen so that, eventually, some of its components will give back QED. It governs the interaction of weakly hypercharged particles coupled to a gauge field via a coupling constant .

In this context one imposes the electric charge of a particle to be linked to the weak isospin and hypercharge via the Gell-Mann–Nishijima equation : . Things can then be recast in terms of electric charge to purposely make the bosons appear ; from the weak hypercharge and isospin terms we end up with three terms respectively displaying positively, negatively and neutrally charged currents. In this new basis the bosons fields are linear combinations of the and . We are left with a neutral term involving the gauge fields and . These fields, though, are not yet the photon and the . In fact, the latter appear to be admixtures of and . The angle that governs this mixture is called the Weinberg angle . Weak and electromagnetic interactions are now unified and described by three parameters : how they mix via and their individual coupling strength and .

Like mentioned previously QCD can be added such that three gauge theories finally account for the three interactions, i.e. . Nonetheless this model presents a few flaws. The problem of the unitarity above is partially fixed by introducing massive vector bosons but the unitarity still breaks down around in , where the are polarised longitudinally. Last but not least, some elementary particles are massive but if one enters mass terms explicitly in the equations the gauge invariance is broken. To keep the gauge invariance properties of the GWS model and take into account the masses a mechanism that spontaneously breaks the symmetry of the solutions had to be devised.

The solution was directly inspired by condensed matter physics, more precisely, from supra-conductivity where photons can become massive due to the non-symmetric fundamental state of the scalar field () describing electrons pairs. Here, the electroweak symmetry breaking (ESB) is realised by adding to the previous model a Higgs scalar field which, to respect , comes as a weak isospin doublet with an hypercharge. A part of the potential for the Higgs field is chosen so that the vacuum energy is degenerate. This complex Higgs doublet makes a total of four degrees of freedom. Three of these degrees mix with the and bosons and provide them with a third longitudinal spin state which makes them massive while the remaining one becomes the massive Higgs boson. The fermions acquire their masses via Yukawa couplings with the Higgs field. The Higgs boson is the last missing piece of the present day formulation of the Standard Model. If it exists and if it perturbatively interacts, its experimental observation will validate the Standard Model.

#### 2.1.6 Summary

We briefly presented the quantum and relativistic framework for the dynamics of the quarks and leptons sensitive to the electromagnetic, weak and strong interactions, as well as how massive particles gain their masses dynamically by interacting with the Higgs scalar field. The Standard Model has a highly predictive power in its actual form but its main problem still lies in the understanding of the origin and nature of the masses of the particles. Although the ESB mechanism accounts for them while respecting the gauge invariance, its addition is mainly ad hoc.

This summary of the Standard Model did not had the pretension to be exhaustive. In particular, the wide variety of experiments, such as hadron–hadron, lepton–nucleon or electron–positron colliders, that provided essential results, were not credited to keep this presentation as short as possible. As a consequence the reader is invited to refer to classical textbooks with more details and references to historical papers : for QFT/SM in order of accessibility Refs. [41, 42, 43, 44, 45], for details in QCD Ref. [46] and in the Electroweak Model Ref. [47] for example.

### 2.2 The W mass charge asymmetry in the Standard Model

#### 2.2.1 A first overview from the experimental point of view

The present work takes place in the context of collider physics, more specifically within the Large Hadron Collider (LHC) that should accelerate in a large ring counter-rotating hadrons –most of the time protons– and make them collide at several interaction points with an energy in the center of mass of . The observation of the particles emerging from these hadronic collisions is achieved by several detectors located in the vicinity of the interaction points. Among them is the ATLAS detector whose capabilities were used in this analysis. The LHC and the ATLAS detector will be described in more details in the next Chapter. Also, worth mentioning, is the Tevatron which, for the last decades before the LHC, has been the largest circular accelerator. The Tevatron collider accelerates counter rotating protons and anti-protons at energies in the center of mass of .

Amid all the difficulties entering such experimental analysis, two are to be noticed. The first, inherent to high energy physics, is that most of the exotic particles cannot be observed directly due to their short life time but are rather detected indirectly from the observations of their decays displaying specific kinematics. The second difficulty, specific to hadronic processes, is that from the theoretical point of view physicists speak in terms of quarks and gluons but from the experimental point of view only hadrons and their respective decays –if any– can be observed. When colliding hadrons this last problem is unavoidable due to the nature of the initial state.

In hadronic collisions the extraction of the and bosons properties are made studying their leptonic decays which display a distinguishable signal due to the high energy leptons in the final state. This, in particular, allows to get rid of the problems inherent to QCD in the final state. Let us remark the decay into the tau channel is not considered as the short life time of the latter makes it not “directly” observable in a detector.

In the case of the boson, where , things are simple as the direct observation of the two charged leptons gives access, via their invariant mass , to the invariant mass of the . In the case of the the presence of a neutrino in the final state complicates things a lot more. Indeed, because the neutrino does not interact with any part of the detector its kinematics can be only deduced from the overall missing energy for a given event, which will never be as precise as a direct measurement. Multipurpose detectors instruments are more precise in the transverse direction of the beam axis, since this is where interesting physics with high particles occur. The presence of the beam-pipe leaves the very forward region less hermetic in term of calorimetry, forbidding eventually to measure the longitudinal component of the neutrino. This leaves then only the transverse momenta of the two leptons to extract the  boson mass. As it will be shown in this Chapter, the shapes of these transverse momenta depends on properties such as its mass and its width . Nonetheless, because here we cannot access any Lorentz invariant quantities, the kinematics of the boson and of the leptons in the rest frame needs to be known with accuracy to model precisely enough the observed kinematics of the leptons in the laboratory frame. Hence the extraction of the properties proves to be a real challenge from both phenomenological and experimental point of views.

#### 2.2.2 Motivations for a measurement of the W mass charge asymmetry

As demonstrated by Gerhard Lüders and Wolfgang Pauli [48], any Lorentz-invariant quantum field theory obeying the principle of locality must be -invariant. For theories with spontaneous symmetry breaking, the requirement of the Lorentz-invariance concerns both the interactions of the fields and the vacuum properties. In -invariant quantum field theories, the masses of the particles and their antiparticles are equal.

The Standard Model is -invariant. In this model, the and bosons are constructed as each own antiparticle, which couple to leptons with precisely the same strength . The hypothesis of the exact equality of their masses is pivotal for the present understanding of the electroweak sector of the Standard Model. It is rarely put in doubt even by those who consider the invariant Standard Model as only a transient model of particle interactions. However, from a purely experimental perspective, even such a basic assumption must be checked experimentally to the highest achievable precision.

The most precise, indirect experimental constraint on equality of the and masses can be derived from the measurements of the life time asymmetries of positively and negatively charged muons [49, 50, 1]. These measurements, if interpreted within the Standard Model framework, verify the equality of the masses of the and bosons to the precision of . Such a precision cannot be reached with direct measurements of their mass difference. The experimental uncertainty of the directly measured mass difference from the first CDF run and reported by the Particle Data Group [1] is , i.e. about times higher. More recently the CDF collaboration [51] measured to be in the electron decay channel and in the muon decay channel. The Table 2.2 sums up the measured values at CDF for the last decades for both and at the time being. Note that in this thesis the experimental measurement will be focused on CDF results since up to this day it is the only collaboration that published experimental values for .

These measurements provide to this date the best model independent verification of the equality of the masses of the two charge states of the  boson. They are compatible with the charge symmetry hypothesis. It is worth stressing, that the present precision of the direct measurement of the charge averaged mass of the  boson derived from the combination of LEP and Tevatron results and under the assumption that is . It is better by a factor than the precision of the direct individual measurements of the masses of its charged states.

In top of the obligatory precision test of the -invariance of the spontaneously broken gauge theory with a priori unknown vacuum properties, we are interested to measure at the LHC for the following three reasons. Firstly, we wish to constrain the extensions of the Standard Model in which the effective coupling of the Higgs particle(s) to the  boson depends upon its charge. Secondly, contrary to the Tevatron case, the measurement of the charge averaged mass at the LHC cannot be dissociated from, and must be preceded by the measurement of the masses of the  boson charge states. Therefore, any effort to improve the precision of the direct measurement of the charge averaged mass of the  boson and, as a consequence, the indirect constraint on the mass of the Standard Model Higgs boson, must be, in our view, preceded by a precise understanding of the  boson charge asymmetries. Thirdly, we would like to measure the  boson polarisation asymmetries at the LHC. Within the Standard Model framework the charge asymmetries provide an important indirect access to the polarisation asymmetries. This is a direct consequence of both the conservation in the gauge boson sector and the purely ()-type of the conjugation () and parity () violating coupling of the  bosons to fermions. Any new phenomena contributing to the  boson polarisation asymmetries at the LHC must thus be reflected in the observed charge asymmetries.

The optimal strategies for measuring the charge averaged mass of the  boson and for measuring directly the masses of its charge eigenstates are bound to be different. Moreover, the optimal strategies are bound to be different at the LHC than at the Tevatron.

At the Tevatron, the nature of the colliding beam makes it so the production of is the same than up to a transformation. Then, producing equal numbers of the and bosons, the measurement strategy was optimised to achieve the best precision for the charge averaged mass of the  boson. For example, the CDF collaboration [51] traded off the requirement of the precise relative control of the detector response to positive and negative particles over the full detector fiducial volume, for a weaker requirement of a precise relative control of charge averaged biases of the detector response in the left and right sides of the detector. Such a strategy has provided the best up to date measurement of the charge averaged  boson mass, but large measurement errors of the charge dependent  boson masses as seen above in Table 2.2. More detailed explanations on this experimental development are given in Chapter 3 Appendix .5.

If not constrained by the beam transfer systems, the best dedicated, bias-free strategy of measuring of in proton–anti-proton colliders would be rather straightforward. It would boil down to collide, for a fraction of time, the direction interchanged beams of protons and anti-protons, associated with a simultaneous change of the sign of the solenoidal -field in the detector fiducial volume. Such a measurement strategy cannot be realised at the Tevatron leaving to the LHC collider the task of improving the measurement precision.

The statistical precision of the future measurements of the  boson properties at the LHC will be largely superior to the one reached at the Tevatron. Indeed, where this error was of at the Tevatron for an integrated luminosity of  [51] at the LHC, for the same measurement, in just one year of collisions at low luminosity () the statistical error should approximately reach . On the other hand, it will be difficult to reach comparable or smaller systematic errors. At the Tevatron they equalise to the statistical error, i.e. while at the LHC the primary goal is, to constrain the hypothesised Higgs mass, to reach . The measurements of the  boson mass and its charge asymmetry can no longer be factorised and optimised independently. The flavour structure of the LHC beam particles will have to be controlled with a significantly better precision at the LHC than at the Tevatron. While being of limited importance for the measurement at the Tevatron, the present knowledge of the momentum distribution asymmetries of : (1) the up and down valence quarks and (2) of the strange and charm quarks in the proton will limit significantly the measurement precision. The ‘standard candles’, indispensable for precise experimental control of the reconstructed lepton momentum scale – the  bosons and other “onia” resonances – will be less powerful for proton–proton collisions than for the net zero charge proton–anti-proton collisions. Last but not least, the extrapolation of the strong interaction effects measured in the  boson production processes to the processes of  boson production will be more ambiguous due to an increased contribution of the bottom and charmed quarks.

Earlier studies of the prospects of the charge averaged  boson mass measurement by the CMS [57] and by the ATLAS [58] collaborations ignored the above LHC collider specific effects and arrived at rather optimistic estimates of the achievable measurement precision at the LHC. In our view, in order to improve the precision of the Tevatron experiments, both for the average and for the charge dependent masses of the  boson, some novel, dedicated strategies, adapted to the LHC environment must be developed. Such strategies will have to employ full capacities of the collider and of the detectors in the aim to reduce the impact of the theoretical, phenomenological and measurement uncertainties on the precision of the  boson mass measurement at the LHC.

### 2.3 Notations and conventions

In this section notations and conventions used through out the rest of the document are introduced.

Let us remind to the reader that to simplify analytic expressions the natural unit convention and is adopted. In this notation energies, masses and momenta are all expressed in electron-Volt (eV). Nonetheless, although every MKSA units can be expressed in powers of eV, cross sections –noted – are expressed in powers of barns, where . Especially, unless stated otherwise, all differential cross sections for a scalar observable produced in this work are all normalised to , being the dimension of the observable .

Both Cartesian and cylindrical coordinate basis are considered in the inertia laboratory frame. They are defined already with respect to the experimental apparatus. The interaction point, located in the center of the ATLAS detector, corresponds to the origin of both coordinate systems. In the Cartesian basis, colliding hadrons move along the axis, points upward and to the center of the LHC accelerating ring. In the cylindrical basis, is the radius in the plane, the azimuthal angle with respect to the  direction, and the polar angle with respect to the  direction. Unit vectors along these different directions are noted where can stands for , , , or . The components of a vector observable along the axis is noted . All angles are expressed in radians unless stated otherwise.

To define the most relevant kinematics variables to collider physics we consider the example of a particle which four-momentum, energy, momentum and absolute momentum are noted respectively , , and . The four-momentum of a particle in its co-variant representation and in the Cartesian basis writes

 p=(E→p)=⎛⎜ ⎜ ⎜ ⎜⎝Epxpypz⎞⎟ ⎟ ⎟ ⎟⎠, (2.1)

where the energy and the momentum of the particle are related to its invariant mass by the relation . The form of Eq. (2.1) implies the same conventions for the time and space components of any other kind of four-vectors. The helicity of a particle is defined by the projection of its spin against the axis pointing in the same direction than the momentum of the particle , that is analytically

 λ≡→S⋅→p|→p|. (2.2)

The transverse component in the plane of the vector is defined by

 pT≡√p2x+p2y. (2.3)

In that notation transverses energies and missing transverse energies are written and . In ATLAS, and other multipurpose detectors, the central tracking sub-detector bathes in a solenoidal magnetic field . In this context, the transverse curvature , defined as the projection of a particle track on the plane, is related to the particle’s transverse momentum via

 ρT≡1/pT. (2.4)

A particle kinematics can be unequivocally described by its azimuth , its transverse momentum and its rapidity defined by

 y≡12ln(E+pzE−pz). (2.5)

The rapidity is additive under Lorentz transformations along the axis. For massless/ultra-relativistic particles the rapidity equals the pseudo-rapidity which is related to the particle polar coordinate by the following relation

 η≡−ln(tan(θ/2)). (2.6)

Finally to understand more deeply some important physics aspects it is better to consider them in the Rest Frame (RF) rather than in the laboratory frame (LAB) where fundamental dynamical patterns are blurred by the add-up of the effects of the boosts. Variables considered in the RF are labeled with a superscript while the one with no particular sign are to be considered in the laboratory frame. More details on relativistic kinematics can be found in Ref. [1].

Another useful variable that will be extensively used is the charge asymmetry, which, for a scalar , is noted and defined like

where the and refers to the electrical charge of the particle under consideration. Finally when the electrical charge is not made explicit it means we consider indifferently both positive and negative particles, i.e. hereafter and .

Different levels of understanding for the observables are considered, respectively the “true”, the “smeared” and the “reconstructed” levels. The true level, also called particle or generator level, refers to the phenomenological prediction of a model or, to be more precise, to the best emulation a given Monte Carlo simulation can produce. The smeared level refers to the true level convoluted with the finite resolution of a detector. For an observable , the link between the smeared distribution and the true one is

 Smear(a)≡∫dtTrue(t−a)Res(t;a), (2.8)

where is the function governing the response of the detector to an input of value for the observable . Here, the general resolution performances of a detector will be usually given using rough Gaussian parametrisation

 Res(t;a)=1√2πσae(−t2/2σ2a), (2.9)

where the variance characterises the resolution of the detector for the observable of the considered particle. Finally the collected data suffer from additional degradations coming from misalignment, miscalibration, limited accuracy of algorithms, etc. All those effects concur to give in the end a reconstructed observable deviating from the smeared value a perfect detector would provide. From the purely experimental point of view only the reconstructed level is relevant but the intermediate levels are used for both pedagogical and pragmatic purposes.

### 2.4 Generalities on the production of W boson in Drell–Yan like processes

This section starts with a short presentation of the decay of both unpolarised and polarised spins states of a real boson which leads to the computation of its width . These derivations will prove to be useful afterward and it allows as well to remind some basics related to the electroweak interaction. After, the whole process is presented. The goal is to give an intuitive comprehension of the kinematics relevant to the production. Among these kinematics is the charged lepton transverse momentum that is used in the present document for extracting the mass of the .

#### 2.4.1 W decay

##### (a) Unpolarised W bosons

The case of the leptonic decay of an unpolarised  boson is considered through the example of

 W−(P)→l−(p1)¯νl(p2), (2.10)

where , and are respectively the four-momenta of the boson, of the charged lepton and of the anti-neutrino . This process in the first perturbation order (Born level) is made of one Feynman diagram (Fig. 2.1). Here, and in all other Feynman diagrams, the time flow goes from the left to the right.

The leptonic decay can be of any type (electronic, muonic or tau) since all leptons can be considered to be massless. Indeed, in the Rest Frame (RF) each charged lepton have an energy of the order of , hence the charged leptons have a Lorentz factor of making out of them ultra-relativistic particles222In fact for the case of the , taking into account its mass eventually leads after computation to affect the width at the level of , but we consider this to be negligible in the context of our discussion.. The amplitude of probability of this process is

 Mλ=−ig√2ελμ(P)¯u(p1)γμ12(1−γ5)v(p2). (2.11)

In this last equation, the weak force is embedded in the strength that couples particles sensible to the weak interaction and by the nature of this coupling represented by the co-variant bi-linear term of Vector–Axial () form. The coupling constant is usually expressed as a function of the Fermi constant that was formerly used when modeling weak interactions as contact interactions when occurring at energies scales much lower than . For historical reasons they are linked through the relation

 GF√2≡g28M2W. (2.12)

The polarisation state of helicity is represented by the four-vector , is the spinor of and that of . We use here the convention which tends to consider that anti-fermions going forward in time can be treated on the algebraic level as fermions going backward in time. The spinors of backward traveling time fermions are noted for convenience. Let us note in the term the operator which, applied to a spinor , projects only its left-handed component

 12(1−γ5)u=uL, (2.13)

while the operator projects the right component of the latter

 12(1+γ5)u=uR. (2.14)

In other words, the presence of the term in the current couples only left-handed fermions and right-handed anti-fermions in electroweak interactions.

Usual “diracologic” calculus techniques allow to calculate , which corresponds to the Lorentz co-variant expression of the squared amplitude summed and averaged with respect to all possibles spins states for the and leptons. The differential decay rate writes in the RF

 dΓ=12MW¯¯¯¯¯¯¯¯¯¯¯¯¯¯|Mλ|2dLips, (2.15)

where , the Lorentz invariant phase space assuring energy-momentum conservation, can be reduced here to

 dLips=1128π2δ(E∗l−MW/2)dE∗ldΩ∗l, (2.16)

where the energy and solid angle are those of the charged lepton. After the integration over the whole accessible phase space to the particles, the partial width of this process reads

 Γ(W→lνl) = GF√2M3W6π, (2.17) ≡ Γ0W, (2.18)

which is a Lorentz scalar. The result is the same when considering the case of the decay. The reader willing to have further details on the previous development or how to undergo such perturbative calculation, can go to classic references in particle physics/quantum field theory such as Refs. [59, 45].

This first order expression for the partial width allows to calculate the total width of the or the . The can decay into leptons or into a pair of quark–anti-quark that in turn decay to observable hadrons with a probability of one

For the leptonic decay, in the ultra-relativistic approximation assumed so far, the width is simply

 Γ(W→leptons) = Γ(W→eνe)+Γ(W→μνμ)+Γ(W→τντ), (2.20) = 3Γ0W. (2.21)

For the hadronic decay, the calculus is similar to the one for the leptons except that here :  (1) CKM matrix element intervene and mix the quarks flavors and (2) in top of summing/averaging on the quarks spin states the sum and average on their color charge states need to be done as well. The can decay into all quarks flavors but the top which is forbidden by energy conservation since . The quarks possess masses such that they can be treated like the charged leptons as ultra-relativistic particles. Then studying gives for the partial width

where is the color number, the element of the CKM matrix governing the mixing angle between flavors and . Summing over all quarks flavors in Eq. (2.22) gives

 ∑q¯q′∣∣Vqq′∣∣2=∑q∑¯q′∣∣Vqq′∣∣2=1, (2.23)

where the sum on of the squared CKM elements translates the unitary of the CKM matrix and where is restricted to and flavors since the decay to the top is forbidden. Then in this context

Adding both leptonic and hadronic parts gives eventually in the Born level approximation

 ΓW=(3+2Nc)GF√2M3W6π, (2.25)

where has been replaced by its explicit form. Adding QCD corrections in quarks decays transforms the latter expression to

 ΓW=(3+2Nc[1+αs(MW)π])GF√2M3W6π, (2.26)

where is evaluated at the energy scale imposed by the mass. Following those developments it is assumed in the rest of this thesis that .

##### (b) Polarised W bosons

Considering the leptonic decay of a real  boson with a specific polarisation state proves to be very helpful to understand the angular decay behaviour of the quarks or leptons. The polarisation states of the two transverses () and the one longitudinal mode of a of momentum can be expressed in the laboratory frame like

 (2.27)
###### Transverse polarisation states.

The example of the leptonic decay of a is again considered, for a polarisation state of . Substituting the expression of in Eq. (2.11) gives the probability amplitude for . This amplitude squared and averaged on the leptons spins, gives the term from which the differential decay width, in the RF is

 dΓdcosθ∗W,l∝∣∣MW−(λ=+1)∣∣2. (2.28)

Developing that expression gives, in the end, an angular dependency of

 dΓdcosθ∗W,l∝(1−cosθ∗W,l)2, (2.29)

where the angle is defined like

 cosθ∗W,l≡→pW⋅→p∗l|→pW||→p∗l|. (2.30)

The angular dependency in Eq. (2.29) is an important key for understanding the leptons kinematics in the decay. It can actually be understood without going to refined calculations but using only the helicity conservation rules in the high energy limit imposed on the leptons by the mass of the . As it was seen previously the electroweak interactions couples only left-handed fermions and right-handed anti-fermions. In the high energy limit chirality and helicity becomes the same, which means that the helicity is a conserved quantum number and only negative (positive) helicity fermions (anti-fermions) are involved. This explain why sometime in the literature, as a shortcut, but only in the high energy limit, negative helicities states are referred to as “left” and positive states as “right”.

Now the previous example is treated from the helicity conservation point of view. The decay is depicted in the RF in the upper part of Fig. 2.2. The convention for the axis is the following. In the initial state the direction is parallel to the direction of the momentum of the in the laboratory frame while the direction points to the direction of the decaying charged lepton in the final state. In that context, the decay proceeds from an initial state with to a final leptonic state with and in both cases the system holds a total spin of , thus the amplitudes are proportional to the rotation matrices, it reads

 dJJf,z′,Ji,z(θ∗W,l)≡⟨J,Jf,z′∣∣e−iθ∗W,lJy∣∣J,Ji,z⟩, (2.31)

which expressions for commonly used spins are tabulated in many places, e.g. Ref. [1], then

 dΓdcosθ∗W,l ∝ ∣∣d1−1,1(θ∗W,l)∣∣2, (2.32) ∝ (1−cosθ∗W,l)2. (2.33)

Thus the most privileged configuration is the one where the initial and final spin projection and are aligned, while on the other hand the configuration where the would be emitted such that would be totally forbidden by helicity conservation.

The case of the decay of the can be deduced using the same kind of argument except this time the charged lepton is “right” and the neutrino is “left” as shown in the lower part of Fig. 2.2, the angular dependency is then

 dΓdcosθ∗W,l ∝ ∣∣d11,1(θ∗W,l)∣∣2, (2.34) ∝ (1+cosθ∗W,l)2. (2.35)
###### Longitudinal polarisations state.

For the decay of a possessing a longitudinal polarisation state a detailed calculus can be carried out using the expression of with and in the RF. Instead of doing so the angular dependency, that really matters, can be unraveled using again rotations matrices

 dΓdcosθ∗W,l ∝ ∣∣d1±1,0(θ∗W,l)∣∣2 (2.36) ∝ sin2θ∗W,l. (2.37)

In this equation in the case of a decay and for the one of a . Here there are no differences in the angular decay between the positive and negative channels.

###### Sum up for the decay of polarised W.

The previous derivations can be generalised to include in the formulae both the boson charge and helicity

 dΓW→lνldcosθ∗W,l ∝ (1+λQcosθ∗W,l)2, (2.38) dΓW→lνldcosθ∗W,l ∝ sin2θ∗W,l. (2.39)

The sign in front of can be deduced easily each time by deducing which direction is privileged for the charged lepton from helicity conservation arguments point of view.

#### 2.4.2 W in Drell–Yan-like processes at the LHC

The Drell–Yan [60] processes were originally defined, in hadron–hadron collisions within the parton model, before identifying partons with quarks and later on, with gluons. In these processes, a pair of partons, the hypothesised building blocks of nucleons, collide and annihilate giving in the final state a high invariant mass lepton pair . Within QCD, this partonic reaction proved to be achieved at first order, by the quark–anti-quark annihilation via . The main features of the formalism describing Drell–Yan survived to the rise of the QCD up to a few refinements. Still, for historical reasons even though the whole present discussion takes place within QCD the term parton is still used and refers to quarks or gluons.

By extension the production of high invariant mass pair through the production of an intermediate boson is referred to as “Drell–Yan like processes”, or –for convenience– Drell–Yan. Below reminders of the treatment of boson production in Drell–Yan is given. Let us remark that the formalism described is applicable to the original Drell–Yan and to a wide variety of over hard scattering processes including jet and heavy flavour production. Further general details on this topic can be found in Refs. [61, 46, 62].

##### (a) Overview

The production of a in Drell–Yan can be written

 HA(PA)HB(PB)→WX→l(p1)νl(p2)X. (2.40)

The associated Feynman like representation of this process is shown in Fig. 2.3, where and are hadrons of four-momenta and accelerated by the collider bringing a total energy in the center of mass. The collision produces a pair of leptons of four-momenta and . The other particles produced in this collision noted are not considered at all. The derivations that follow are applicable to the decay in the channel, nonetheless the analysis being restricted to the electronic and muonic decays, throughout the whole document unless stated otherwise.

The hadronic cross section for a given point of the quarks and leptons phase space, , and for a particular collision of partons and can be expressed using the factorisation “theorem” like

 d4σij(S)dx1dx2dp1dp2=fAi(x1,μf)fBj(x2,μf)d2^σij(x1x2S,p1,p2,μr), (2.41)

that is as the product of the partonic cross section and the parton distributions functions (PDFs) and .

The PDFs are the density probabilities and for two partons and , to carry before their hard scatter fractions and () of the four-momenta of the hadrons they respectively income from. Purely theoretical derivation of PDFs are not computable as their description falls in the non perturbative regime of QCD where is too large, therefore they are extracted from global fits to data from processes such deep inelastic scattering, Drell–Yan and jet production at the available energy range fixed by colliders. The implementation of QED and QCD radiative corrections from quarks in the initial state are universal, i.e. independent from the process. These corrections contain mass singularities that can be factorised and absorbed in a redefinition (renormalisation) of the PDFs. The singularities are removed in the observable cross section while the PDFs becomes dependent of a factorisation scale controlled by the DGLAP evolution equations [30, 31, 32]. This scale is to be identified –for example– to a typical scale of the process, like the transverse momentum or in the present case by the mass of the resonance .

The partonic cross section corresponds to the probability that the partons and , of four-momenta and , collide and create a resonance of mass which in turn decays into leptons and respectively of four-momenta and . The hard scattering occurs at such energies that partons can be seen as free, i.e. is small. This allows to calculate the cross section using perturbation theory. Up to a given energy scale higher order Feynman diagrams are not directly computed but accounted by factorising their effect in the value of the constant . This renormalisation scale is the one of the virtual resonance, which means here . The total partonic cross section is then developed in power of

 ^σij=^σ(0)ij+αs(MW)2π^σ(1)ij+(αs(MW)2π)2^σ(2)ij+O(α3s). (2.42)

In this equation the first term of this series is the leading-order (LO), usually called the Born level, the second the next-to-leading order (NLO) correction of order and so on the third term adds up next-to-next-to-leading order (NNLO) corrections of order . The last term contains all corrections above . In practice only the first corrections are brought to a calculus as the number of Feynman diagrams increase rapidly when going to higher order corrections. Let us remark that a correction term of order is to be implicitly apprehended as where

 δ(k)ij = 1if the ij collision contribute to ^σ(k)ij , δ(k)ij = 0if the ij collision do not contribute to ^σ(k)ij.

The total hadronic cross section is then deduced by embracing all available corrections to the partonic cross section and by successively integrating Eq. (2.41) over the accessible phase space to the leptons and partons. This means in the latter case integrating over all possibles and , but also by summing over all possible partonic collisions, which gives eventually

 σ(S)=∑i,j∫10dx1∫10dx2fAi(x1,μf)fBj(x2,μf)^σij(x1x2S,μr). (2.43)

From the experimentalist pragmatic point of view the matter of importance is to understand the kinematics of the leptons and how the properties can be extracted from them. Hence, in what follows, basics on the partonic cross section and on the PDFs are reminded with emphasis on the kinematics aspects rather than the dynamical issues. After that, an overview of the relevant kinematics in production is given.

##### (b) Partonic level

The production of a at the Born level is made through quark–anti-quark annihilation, which is illustrated here for the case of a