CERN-THESIS-2016-213 30/09/2016 nthe in lcrwa ag oo Scattering Boson Gauge Electroweak Z W tteLreHdo Collider Hadron Large the at e auta ahmtkudNaturwissenschaften und Fakult¨at Mathematik der ilmPyie ei ohr(e.Thomas) (geb. Socher Felix Diplom-Physiker u ragn e kdmshnGrades akademischen des Erlangung zur hne ihteALSDetector ATLAS the with Channel eoe m0.Fbur18 nRiesa in 1986 Februar 01. am geboren e ehice Universit¨at Technischen Dresden der otrrrmnaturalium rerum Doctor DISSERTATION D.rr nat.) rer. (Dr. td of Study vorgelegt von
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1. Gutachter: Prof. Dr. Michael Kobel 2. Gutachter: Prof. Dr. Chara Petridou
Tag der m¨undlichen Pr¨ufung: 30.09.2016 Tag der Einreichung: 15.07.2016
Abstract
The Standard Model of particle physics is a very well tested gauge theory describing the strong, weak and electromagnetic interactions between elementary particles through the exchange of force carriers called gauge bosons. Its high predictive power stems from its ability to derive the properties of the interactions it describes from fundamental symmetries of nature. Yet, it is not a final theory as there are several phenomena it cannot explain. Furthermore, not all of its predictions have been studied with suf- ficient precision, e.g. the properties of the newly discovered Higgs boson. Therefore, further probing of the Standard Model is necessary and may result in finding possible indications for new physics.
The non-abelian SU(2)L U(1)Y symmetry group determines the properties of the elec- × tromagnetic and weak interactions giving rise to self-couplings between the electroweak gauge bosons, i.e. the massive W and Z boson, and the massless photon, via triple and quartic gauge couplings. Studies carried out over the past 20 years at various particle accelerator experiments have shed light on the structure of the triple gauge couplings but few results on quartic gauge couplings are available. The electroweak self-couplings are intertwined with the electroweak symmetry breaking and thus the Higgs boson through the scattering of massive electroweak gauge bosons. Both the W and Z boson couple to the Higgs boson and may interact with each other by exchanging it. Theory predictions yield physical results at high energies only if either both the self-couplings and Higgs boson properties are as described by the Standard Model or if they deviate from its predictions and contributions from new physics are present to render the calculations finite. This makes electroweak gauge boson scattering a powerful tool to probe the Standard Model and search for possible effects of new physics. The small cross section of massive electroweak gauge boson scattering necessitates high centre-of-mass energies and luminosities to study these processes successfully. The Large Hadron Collider (LHC) at CERN is a circular proton-proton collider equipped to supply a suitable environment for such studies with the colliding protons being the sources for the scattering of massive electroweak gauge bosons. The dataset collected in 2012 by the ATLAS detector at the LHC with a total lumi- nosity of 20.3 fb−1 and a centre-of-mass energy of 8 TeV is analysed in this work. The elastic scattering process WZ WZ is studied due to its clean signal properties. It → provides a complementary measurement to W ±W ± W ±W ± which reported the first → significant evidence for massive electroweak gauge boson scattering. Given the current data, WZ WZ scattering is not observed with large significantly. → A cross section upper limit of 2.5 fb at 95 % confidence level is measured, compatible with the cross section of 0.54 fb predicted by the Standard Model. In addition, distributions for several observables sensitive to electroweak gauge boson scattering are unfolded, removing effects caused by the measuring process. Physics beyond the Standard Model is probed in the framework of the electroweak chiral Lagrangian which expresses the size of effects from new physics in terms of strength parameters. The two strength parameters influencing the quartic gauge couplings are constrained to 0.44 < α4 < 0.49 and 0.49 < α5 < 0.47 thus limiting the possible − − size of new physics contributions.
Kurzdarstellung
Das Standardmodell der Teilchenphysik beschreibt die starken, schwachen und elektro- magnetischen Wechselwirkungen zwischen Elementarteilchen uber¨ den Austausch von Kraftteilchen, sogenannten Eichbosonen. Es ist eine anerkannte theoretische Beschrei- bung der Natur, da es in der Lage ist, aus fundamentalen Symmetrien die Charakterisi- ken der einzelnen Wechselwirkungen abzuleiten. Die so getroffenen Vorhersagen wurden durch eine Vielzahl von Experimenten erfolgreich uberpr¨ uft.¨ Dennoch ist es keine abge- schlossene Theorie, da es nicht alle in der Natur beobachteten Ph¨anomene beschreiben kann. Uberdies¨ konnten die von ihm gemachten Vorhersagen wie z.B. die Eigenschaften des kurzlich¨ gefundenen Higgs Bosons, noch nicht mit ausreichender Pr¨azision uberpr¨ uft¨ werden. Deshalb sind weitere Tests des Standardmodells notwendig. Die Eigenschaften der elektromagnetischen und schwachen Wechselwirkungen werden durch die nicht-abelsche Symmetriegruppe SU(2)L U(1)Y bestimmt. Eine direkte × Konsequenz ist die Existenz von Selbstwechselwirkungen zwischen den elektroschwa- chen Eichbosonen, den massiven W und Z Bosonen und dem masselosen Photon, die durch Dreier- und Vierer-Kopplungen beschrieben werden. Die Struktur der Dreier- Kopplungen ist in den letzten 20 Jahren an Teilchenbeschleunigern eingehend studiert worden. Erst seit kurzem sind durch neue Beschleuniger pr¨azise Untersuchungen der Vierer-Kopplungen m¨oglich. Das Higgs Boson koppelt an W und Z Bosonen da diese eine Masse haben. Damit kann, durch die Untersuchung der Streuung massereicher elektroschwacher Eichbosonen so- wohl die elektroschwache Selbstwechselwirkung, als auch die elektroschwache Symme- triebrechung untersucht werden. Die Vorhersagen des Standardmodells sind bei hohen Energien nur dann gultig,¨ wenn die Eigenschaften des Higgs Bosons jenen entsprechen, die vom Standardmodell vorhergesagt werden. Falls diese Bedingung nicht erfullt¨ ist, werden Beitr¨age neuer physikalische Prozesse ben¨otigt um unphysikalische Vorhersa- gen zu vermeiden. Somit ist die Streuung massereicher elektroschwacher Eichbosonen geeignet, das Standardmodell zu testen und nach neuer Physik zu suchen. Die kleinen Wirkungsquerschnitte fur¨ die zu untersuchenden Prozesse bedingen eine hohe Schwerpunktsenergie und hohe Luminosit¨aten um eine ausreichend große Daten- menge zu erhalten. Der Large Hadron Collider am CERN ist ein Kreisbeschleuniger der diese Voraussetzungen erfullt,¨ indem er Protonen, die Quellen fur¨ die streuenden elektroschwachen Eichbosonen sind, mit einer Schwerpunktsenergie von 8 TeV zur Kol- lision bringt. Diese Arbeit basiert auf dem im Jahr 2012 vom ATLAS Detektor auf- gezeichnete Datensatz, der einer Luminosit¨at von 20.3 fb−1 entspricht. Der untersuchte Prozess ist die elastische Streuung WZ WZ, welche komplement¨ar zum Prozess → W ±W ± W ±W ± ist, in dem erstmalig signifikante Hinweise auf die Streuung mas- → sereicher elektroschwacher Eichbosonen gefunden wurden. Mit der derzeit verfugbaren¨ Datenmenge kann WZ WZ nicht mit ausreichender → Signifikanz beobachtet werden. Fur¨ den Wirkungsquerschnitt wird eine obere Schranke von 2.5 fb mit 95 % Konfidenz gemessen, welche kompatibel mit der Standardmodell- vorhersage von 0.54 fb ist. Beitr¨age neuer Physik jenseits des Standardmodells k¨onnen generisch im Rahmen effek- tiver Feldtheorien durch St¨arkeparameter beschrieben werden. Die beobachteten Daten erm¨oglichen eine Einschr¨ankung der St¨arkeparameter α4 und α5, welche die Vierer- Kopplungen beeinflussen, auf die Bereiche 0.44 < α4 < 0.49 und 0.49 < α5 < 0.47. − −
Contents
1. Introduction 1
2. Theoretical Foundations 5 2.1. Introduction ...... 5 2.2. The Standard Model ...... 5 2.2.1. Local Gauge Theory ...... 7 2.2.2. Quantum Chromodynamics ...... 9 2.2.3. Electroweak Theory ...... 10 2.2.4. Electroweak Symmetry Breaking ...... 12 2.2.5. The Lagrangian of the Standard Model ...... 14 2.3. Electroweak gauge boson scattering ...... 14 2.3.1. Definition ...... 14 2.3.2. Motivation ...... 15 2.3.3. VBS Topology ...... 16 2.3.4. Choice Of Observation Channel ...... 18 2.4. Effective Field Theories ...... 23 2.4.1. Introduction ...... 23 2.4.2. Effective Theory of the Muon Decay ...... 24 2.4.3. Anomalous Quartic Gauge Couplings ...... 24 2.4.4. Electroweak Chiral Lagrangian ...... 24 2.4.5. Linear Symmetry Breaking Approach ...... 25 2.4.6. K-Matrix Unitarisation ...... 26
3. Experiment 29 3.1. CERN ...... 29 3.2. Large Hadron Collider ...... 29 3.3. The ATLAS Detector ...... 31 3.3.1. ATLAS coordinate system ...... 34 3.3.2. Inner Detector ...... 34 3.3.3. Electromagnetic Calorimeter ...... 36 3.3.4. Hadronic Calorimeter ...... 37 3.3.5. Muon Spectrometer ...... 38 3.3.6. Trigger System ...... 40 3.3.7. Luminosity Monitoring ...... 41 3.4. Object Reconstruction ...... 42 3.4.1. Muons ...... 42 3.4.2. Electrons ...... 44 3.4.3. Jets ...... 45 3.4.4. Missing Transverse Momentum ...... 47
4. Datasets 49 4.1. Introduction ...... 49 4.2. Real Data ...... 49 4.3. Simulated Data ...... 50 4.3.1. Introduction ...... 50 4.3.2. Event Generation ...... 51 4.3.3. Event Record ...... 55 4.3.4. Detector Simulation ...... 55 4.3.5. Data Format for Analysis ...... 56 4.3.6. Description of Used Generators ...... 56 4.4. Simulated Processes ...... 57 4.4.1. W ±Zjj-EW ...... 58 4.4.2. W ±Zjj-QCD ...... 59 4.4.3. Background processes ...... 60 4.4.4. Scaling Factors ...... 62
5. Object and Event Selection 63 5.1. Object Selection on Detector Level ...... 63 5.1.1. Electron Definition ...... 64 5.1.2. Muon Definition ...... 66 5.1.3. Jet Definition ...... 68 5.1.4. Missing Transverse Momentum ...... 69 5.2. Event Selection ...... 69 5.2.1. Detector Level Event Selection for the Inclusive Phase Space . . 69 5.2.2. Event Selection for the VBS Phase Space ...... 72 5.2.3. Event Selection for the aQGC Phase Space ...... 72 5.3. Object Selection on Particle Level ...... 73 5.3.1. Lepton Definition ...... 73 5.3.2. Jet Definition ...... 74 5.3.3. Neutrino Definition ...... 74 5.4. Event Selection on Particle Level ...... 75 5.4.1. Event Selection for the Inclusive Phase Space ...... 75 5.4.2. Event Selection for the VBS Phase Space ...... 76 5.4.3. Event Selection for the aQGC Phase Space ...... 77
6. Background Estimation 79 6.1. The Matrix Method ...... 80 6.1.1. Fake Ratio Estimation ...... 82 6.1.2. Application of the Matrix Method ...... 85 6.1.3. Matrix Method Results ...... 86
7. Systematics 89 7.1. Experimental Uncertainties ...... 89 7.1.1. Muon Uncertainties ...... 89 7.1.2. Electron Uncertainties ...... 90 7.1.3. Jet Uncertainties ...... 90 7.1.4. Missing Transverse Momentum Uncertainties ...... 91 7.1.5. Other Uncertainties ...... 91 7.2. Theoretical Uncertainties ...... 91 7.2.1. Scale Uncertainties ...... 91 7.2.2. Parton Distribution Function Uncertainties ...... 92 7.2.3. Parton Shower Uncertainties ...... 93 7.2.4. Lower Parton Multiplicity Uncertainties ...... 93 7.2.5. Theory Uncertainties for Signal Summary ...... 94 7.2.6. Theoretical Uncertainties on Backgrounds ...... 94
8. Measuring the VBS Cross Section 97 8.1. Phase Space Optimisation ...... 97 8.2. Event Yields in the VBS Phase Space ...... 100 8.3. Statistical Evaluation ...... 103 8.3.1. Cross Section Formula ...... 103 8.3.2. Profile Likelihood Method ...... 105 8.3.3. Technical Implementation ...... 107 8.3.4. Results ...... 108
9. Differential Distributions 111 9.1. General Approach ...... 111 9.2. Introduction to Bayesian Iterative Unfolding ...... 113 9.3. Analysis Implementation ...... 114 9.3.1. Technical Setup ...... 114 9.3.2. Systematic Uncertainties ...... 115 9.3.3. Optimising the Number of Iterations ...... 116 9.4. Results ...... 116 9.4.1. Jet Multiplicity ...... 117 9.4.2. Invariant Mass of the Dijet System ...... 119 9.4.3. Absolute difference in Rapidity of the Dijet System ...... 121
10.Setting Limits on Anomalous Quartic Gauge Couplings 125 10.1. Simulation ...... 125 10.2. Phase Space Optimisation ...... 127 10.2.1. Finding Variables for Optimisation ...... 127 10.2.2. Optimisation Approach ...... 131 10.3. Event Yields and Systematic Uncertainties ...... 132 10.4. Results ...... 134 10.4.1. Fiducial Cross Section Results ...... 134 10.4.2. Limits on Anomalous Quartic Gauge Couplings ...... 137
11.Summary 141
A. Particle Level Distributions for the WZ Channel 147
B. Additional Plots in the VBS Phase Space 151
C. Event Display for Event in VBS Phase Space 155
D. Additional Information on Unfolding Results 157
E. Additional Results on Anomalous Quartic Gauge Couplings 169 E.1. Optimisation Studies ...... 169 E.2. Results in the alternative aQGC Phase Space ...... 173 F. Simulated Data Samples 177
G. Software Framework 181
List of Figures 183
List of Tables 185
Bibliography 187 1. Introduction
Over the centuries many attempts have been made to create a complete description of the interactions between the most fundamental building blocks of matter. This process culminated in the Standard Model of particle physics1 [2–5]. Since its inception the Standard Model has been tested in numerous experiments without the observation of large discrepancies between its predictions and measured data cementing its reputation of being currently the best theoretical framework describing the interactions of funda- mental particles. Yet, it is also obvious that it cannot be a final theory for multiple reasons [6], one of them being that it only describes ordinary matter which only makes up 5 % of the known universe with dark matter and dark energy expected to provide the remaining majority. Furthermore, not all of its predictions have been studied with sufficient precision, e.g. the properties of the Higgs boson-like particle discovered in 2012 [7, 8]. Therefore, further probing of the Standard Model is necessary and may result in finding possible indications for new physics. The Standard Model is a quantum field theory describing the electromagnetic, weak, and strong interactions between particles. Thus it describes the fundamental interac- tions needed for understanding many of the physical phenomena observed in nature, e.g. the binding of molecules (electromagnetic interaction), particle decays such as the β decay (weak interaction), and the binding of atomic nuclei (strong interaction). Inter- actions between matter particles are mediated through the exchange of force carriers called gauge bosons. Thus, two matter particles may communicate with each other through the emission of a gauge boson by the first particle and the absorption of the gauge boson by the second particle. For example, scattering processes between elec- trons are described through the exchange of photons between the two particles. In the Standard Model, the force carriers are the photon for the electromagnetic force, the W +, W −, and Z bosons for the weak force and the gluon for the strong force. The properties of each interaction are governed by underlying symmetries which are incorporated in the Standard Model via the theoretical framework of gauge theory. The electroweak theory, a part of the Standard Model describing the electromagnetic and weak interaction, is based on the non-abelian SU(2)L U(1)Y symmetry group. × A direct consequence of the symmetry group being non-abelian is the prediction of self-couplings between multiple electroweak gauge bosons, i.e. the massive W and Z boson, and the massless photon. These self-couplings are direct interactions between three (four) electroweak gauge bosons without the exchange of an intermediate particle and are called electroweak triple (quartic) gauge coupling. The triple gauge coupling W ±W ±Z and the quartic gauge coupling W ±ZW ±Z are of immediate relevance to this work but are not the only triple and quartic gauge vertices in existence. Experiments at previous accelerators have examined these self-couplings and had suc- cess providing valuable insights in the structure of triple gauge couplings [9–14]. Results on the quartic gauge couplings have become available only recently [15–21] and further
1An alternative approach may be found in Ref. [1].
1 1. Introduction studies are needed to gain a better understanding of their properties. Another central part of the Standard Model is the electroweak symmetry breaking [22– 25] responsible for the masses of the massive particles and existence of the Higgs boson. The Higgs boson couples to all massive particles, introducing couplings of the form W ±W ±H, ZZH, and HHZZ among others. In consequence, two massive particles may interact via the exchange of a Higgs boson. The combination of the self-coupling vertices and Higgs boson exchanges defines the electroweak gauge boson scattering. Therefore, electroweak gauge boson scattering is sensitive towards the properties of both the electroweak theory and the electroweak symmetry breaking. The Large Hadron Collider (LHC) at CERN, a circular proton-proton collider of 27 km circumference and a design centre-of-mass energy of √s = 14 TeV provides the envi- ronment to study electroweak gauge boson scattering. Here, the colliding protons act as sources for the scattering electroweak gauge bosons. The event rate for a given process is directly proportional to its cross section and luminosity with the cross sec- tion being potentially dependent on the centre-of-mass energy. The low cross sections for electroweak gauge boson scattering predicted by the Standard Model necessitate high luminosities and centre-of-mass energies, otherwise a meaningful statistical anal- ysis cannot be carried out due to insufficient event statistics. Thus, the unprecedented centre-of-mass energy and luminosity of the LHC enable the study of electroweak gauge boson scattering for the first time. A first important goal is to measure the cross section of the electroweak gauge boson scattering as it has not been measured before representing an untested prediction of the Standard Model. Furthermore, the properties of the Higgs boson may be probed. The Standard Model predicts unnaturally high event rates for the scattering of massive electroweak gauge bosons at high centre-of-mass energies if the Higgs boson properties is not the one described by the Standard Model [26]. This is due to a residual direct dependence of the cross section to the centre-of-mass energy which is not present in the Standard Model prediction. In this case physics beyond the Standard Model would be needed to prevent these unnatural high event rates. However, the contributions by new physics would not necessarily lead to the cross sections for massive electroweak gauge boson scattering predicted in the Standard Model case. Therefore, deviations from the Standard Model predictions for massive electroweak gauge boson scattering may indicate new physics. A generic way to search for possible effects of new physics are effective field theories [27, 28]. These theories do not represent concrete physical models introducing new particles but aim to encompass all possible effects of new physics. The core assumption of these theories is that the particles associated with new physics are beyond the kinematic reach of the LHC and cannot be produced directly. Therefore, only low energy effects caused by the new physics may be observable. Effects of new physics are modelled by introducing new operators, each with an associated free parameter governing its overall contribution. These operators translate to new, anomalous couplings, e.g. anomalous triple and quartic gauge couplings, which emulate the possible behaviour of new physics beyond the Standard Model. Constraining the possible parameter space gives valuable insight into the nature of new physics and may help to build more concrete models. Being embedded in the field of diboson physics the electroweak gauge boson scattering is also an interesting test of perturbative QCD, as higher order corrections may not be covered by the usually applied theory uncertainty prescriptions [29, 30]. Thus it
2 provides motivation for further improvement of theory predictions. This work sets out to measure the cross section for the concrete electroweak gauge boson scattering process WZ WZ and to search for effects caused by physics beyond the → Standard Model. To achieve this, the dataset recorded by the ATLAS detector in the year 2012 with a total luminosity of 20.3 fb−1 at a √s = 8 TeV will be analysed. The analysis of this process is complementary to the study of the electroweak W ±W ± W ±W ± scattering → in which the first significant evidence for electroweak gauge boson scattering has been found [18, 19]. The signal signature contains three charged particles (electrons, positrons, muons, or anti-muons) and a flavour-matching neutrino plus two bundles of strongly interacting particles which stem from the quarks that emitted the scattering W and Z bosons. This final state may be abbreviated as l+l−l±νjj. Observables able to separate between the electroweak WZ WZ scattering and all → other Standard Model processes mimicking the same final state are identified. A region enriched with events caused by purely electroweak interactions will be defined using these observables and used to measure the cross section of the WZ WZ scattering. → Differential cross sections for variables sensitive to the scattering of electroweak gauge bosons will be provided and new physics beyond the Standard Model will be probed in the framework of the electroweak chiral Lagrangian [27]. This thesis is organised as follows. Chapter 2 will summarise the theoretical foundations introducing the Standard Model with a particular focus on electroweak gauge boson scattering and effective field theories. The choice of the W ±Zjj final state as the channel of interest will be discussed here also. The experimental setup, realised by the LHC and ATLAS detector, will be presented in Chapter 3. Chapter 4 will touch on the data acquisition and generation of simulated data detailing the samples used in the analysis. The selection strategies for both physical objects and collision events defining the phase spaces used in the subsequent chapters will be presented in Chapter 5. The estimation of the contributions of background processes to the selected events will be discussed in Chapter 6 followed by a review of the sources of theoretical and experimental uncertainties in Chapter 7. The results of this work will be discussed starting with Chapter 8 detailing the optimisation of the cross section measurement of the electroweak gauge boson scattering in the W ±Zjj final state. The determination of differential distributions providing unfolded results on multiple variables sensitive to electroweak gauge boson scattering is detailed in Chapter 9. Chapter 10 will complete the discussion on physics results by constraining the possible effects of physics beyond the Standard Model. A summary of this work will be given in Chapter 11.
3
2. Theoretical Foundations
2.1. Introduction
The ultimate goal of particle physics is the complete description of nature at all energies in one consistent theory. Four forces are known: The electromagnetic force describing attractive and repulsive forces between elec- • trically charged particles, the weak interaction responsible for phenomena such as the beta decay, • the strong interaction binding together nuclei and hadrons, and • gravity describing the attractive force between masses. • Any description of the physics of these interactions has to have certain properties, e.g. conservation of causality, energy, and momentum. As stated by Noether’s theo- rem [31, 32] such conversation principles stem from symmetries of the physical system under consideration. A popular example for this relation is that energy is conserved if the Hamiltonian describing the physical system in question is invariant under trans- formation in time. Therefore, the demanded properties may be expressed as the sym- metries that the Hamiltonian or Lagrangian of the theory has to follow. Examples for essential symmetries that a theory describing the interactions of elementary particles has to respect are: invariance under Lorentz transformations, which include global rotations, trans- • lations, and boosts in Minkowski space, invariance under the CPT transformation being a simultaneous application of the • parity, charge conjugation and time reversal operation, internal symmetries such as baryon symmetry. • The theoretical tools of choice for building such a theory are quantum field theo- ries [33–36]. Quantum field theories combine quantum mechanics describing subatomic systems with special relativity needed for describing objects at velocities comparable to the speed of light. Here, particles are described as the quantised excitations of quantum fields. The physics of a given quantum field theory is governed by the properties of its Lagrangian e.g., its symmetries and field content. Many Lagrangians may be for- mulated, however only those returning results that do not contradict the experimental data may be considered a possible description of nature. The Standard Model repre- sents such a quantum field theory and will be introduced in this chapter together with possibilities to search for physics not described by it.
2.2. The Standard Model
There are many pedagogical reviews of the Standard Model giving an excellent intro- duction into the matter [37, 38]. This work is not intended to add to the body of said
5 2. Theoretical Foundations
Generation el. weakly colour 1st 2nd 3rd charge charged charged electron muon tauon -1 yes no me = 0.511 MeV mµ = 105.65 MeV mτ = 1776.86 MeV
electron neutrino muon neutrino tau neutrino 0 yes no ---
2 up quark charm quark top quark + 3 yes yes mu ≈ 0.002 GeV mc ≈ 1.3 GeV mt = 173.21 GeV
1 down quark strange quark bottom quark − 3 yes yes mu ≈ 0.005 GeV ms ≈ 0.095 GeV mb = 4.66 GeV
Table 2.1.: Properties of elementary fermions. All fermions have half-integer spin. Two groups exist: Leptons (upper half) and quarks (lower half). Masses are only given where reasonably well measured/defined. No uncertainties on the mass measurements are given. Electromagnetic charge is listed explicitly while only the possession of a weak or strong charge is indicated. Values are taken from Ref. [39].
reviews but will only give an overview of the needed theoretical foundation. Over the past decades experiments in the field of particle physics have succeeded in finding a multitude of particles. These particles may be classified as elementary particles which do not have a permanent substructure and composite particles which are built from elementary particles. Hadrons are one class of composite particles and are made of quarks and anti-quarks which are bound together by the strong force. Hadrons are further subdivided into baryons consisting of three quarks or three anti-quarks and mesons consisting of a quark and an anti-quark. There are literally dozens of different composite particles [39]. In contrast, only 30 elementary particles have been discovered yet. These are the six quarks, six anti-quarks, six leptons, six anti-leptons, the photon, the gluon, the W + boson, the W − boson, the Z boson, and the Higgs boson. Quarks and leptons1 make up all of the ordinary matter. They have spin 1/2 and follow Fermi-Dirac statistics and are thus called fermions. The interactions between fermions are described as exchanges of bosons, elementary particles with integer spin, between the fermions. The couplings of fermions to bosons is described via so-called gauge vertices which are contact interactions taking place at a given spacetime point. One such interaction is the coupling of electrons to photons with a possible notation for the vertex being eeγ. Photon, gluons, W ± bosons, and Z bosons are the mediators of the electromagnetic, strong, and weak force and are responsible for the interactions between the fermions. The Higgs boson is a by-product of the Higgs mechanism which introduces masses to the Standard Model in a gauge-invariant way. Their integer spin designates them as bosons following Bose-Einstein statistics. Fermion properties are tabulated in Table 2.1 and boson properties can be found in Table 2.2 [39].
1Unless explicitly specified the terms quarks and leptons also encompass the anti-quarks and anti- leptons.
6 2.2. The Standard Model
el. weakly strongly mass charge charged charged −19 Photon mγ < 1 10 eV 0 no no × W boson mW = 80.385 GeV 1 yes no ± Z boson mZ = 91.188 GeV 0 yes no Gluon mg = 0 0 no yes
Table 2.2.: Properties of elementary bosons. All bosons have integer spin. No uncertainties on the mass measurements are given. Electromagnetic charge is given explicitly while only the possession of a weak or strong charge is indicated. Values are taken from Ref. [39].
The particle content of the Standard Model is not derived from first principles but is an experimental input. The nature of the interactions between these particles are likewise not derived from first principles but are rather a description of what is observed. However, it is not the case that new experimental insights are simply integrated into the Standard Model in a purely ad hoc way. In the Standard Model the formalism of local gauge theories is used to relate the properties of a given interaction with its associated symmetry group. The choice of the symmetry group determines many features of a theory at once giving the theoretical description its predictive power. Which symmetry group is associated with which interaction is not predicted by theory but has to be determined in experiments.
2.2.1. Local Gauge Theory
Maxwell’s equations represent a first example of a gauge theory, although they were not conceived in the framework of local gauge theories at that time. The idea of local gauge invariance was first formulated by Hermann Weyl in the late 1920s attempting to unify gravity and electromagnetism and later popularised by Wolfgang Pauli [40]. The formalism of local gauge theory was then used to formulate quantum electrodynamics (QED) which was achieved independently by Feynman [41, 42], Schwinger [43] and Tomonaga [44]. An account of the historical development of local gauge theory can be found in Ref. [45]. In order to illustrate the approach the Lagrangian of QED will be derived starting from the Lagrangian of the free fermion field as done in Ref. [37]. The Lagrangian for a fermionic field ψ with half-integer spin, electric charge Qe, and mass m without interactions is
µ = ψ¯(x)(i∂/ m)ψ(x) with ∂/ ∂µγ (2.1) L − ≡
† 0 µ with ψ¯ = ψ γ being the adjoint field and ∂/ = γ ∂µ the derivative in Feynman slash notation. The matrices γi are the Dirac matrices. One may consider subjecting the Lagrangian to a global U(1) transformation, corresponding to a shift in phase of the field:
7 2. Theoretical Foundations
ψ e−iQθψ, → ψ¯ ψe¯ iQθ, (2.2) → −iQθ ∂µψ e ∂µψ → with Q being the generator of the symmetry group and θ the continuous transformation parameter. This global symmetry leads to the conservation of the current Jµ
µ Jµ = ψγ¯ µeQψ with ∂µJ = 0, (2.3) which is invariant under the application of the global gauge transformation. The current may be interpreted as the electromagnetic current.2 However, this Lagrangian does not contain interactions. This can be achieved by chang- ing the global transformation into a local one by introducing a dependence on the space- time coordinate x to the continuous parameter θ. Thus the transformation operations change to:
ψ e−iQθ(x), → ψ¯ ψe¯ iQθ(x), (2.4) → −iQθ(x) −iQθ(x) ∂µψ e ∂µψ iQ(∂µθ(x))e ψ. → − The free Lagrangian in Equation (2.1) is not invariant under this transformation ne- cessitating the introduction of a new field Aµ(x). This field is interpreted as a gauge boson interacting with the field ψ and transforming under U(1) via
1 Aµ Aµ ∂µθ(x). (2.5) → − e
The transformation prescription for Aµ compensates for the terms spoiling the invari- ance of the Lagrangian under the local gauge transformation. An often chosen way to include the new field is to replace the derivative ∂µ with the covariant derivative Dµ:
Dµψ (∂µ ieQAµ)ψ. (2.6) ≡ − The covariant derivative transforms the same way as ψ:
−iQθ(x) Dµψ e Dµψ. (2.7) → The only term left to complete the theory is a kinetic term for the introduced gauge boson to describe its propagation. The field strength tensor for the gauge field providing the necessary kinetic term has the form:
Fµν = ∂µAν ∂νAµ. (2.8) − Thus, the U(1) gauge and Lorentz invariant Lagrangian can be written down:
2The c-number e was added to support this interpretation.
8 2.2. The Standard Model
1 µν = ψ¯(x)(iD/ m)ψ(x) Fµν(x)F (x) (2.9) LQED − − 4 As indicated by the subscript, this is the Lagrangian of quantum electrodynamics de- scribing the electromagnetic interaction of particles via the exchange of photons. The formulation of all the individual theories making up the Standard Model use the concept of local gauge theory as will be seen in Sections 2.2.2 and 2.2.3.
2.2.2. Quantum Chromodynamics
Quantum chromodynamics (QCD) provides a description for the strong interaction re- sponsible for the formation of hadrons and the stability of nuclei. Between the discovery of the neutron in 1932 by Chadwick [46] and the early 1960s many attempts were made to describe the strong interaction between the then known particles. Many of these strongly interacting particles had been found by the early 1960s and the idea that an underlying structure has to exist relating these particles became stronger and stronger. Gell-Mann [47] and Ne’eman [48] found that the SU(3) symmetry group would relate the known particles in a scheme they called the “Eightfold Way”. This underlying scheme motivated the introduction of new elementary particles, called quarks, with the up-, down-, and strange-quark representing a triplet. A new quantum number, colour, which has three possible values: red, green and blue [49] had to be introduced to ex- plain hadrons with three quarks of the same type, e.g. the ∆++ particle consisting of three up-quarks.3 Simultaneously, the triplet structure between the three quarks made way for the triplet structure of the colour charged quarks, e.g., (ur, ug, ub). Gluons en- tered the picture as the mediators of the strong interaction between the colour charged quarks [50].4 The formulation of QCD follows the same scheme as quantum electrodynamics. How- ever, the underlying symmetry group of the colour transformation SU(3)C is non- abelian meaning that its elements do not commute. This is in contrast to the abelian U(1) group of QED and leads to self-coupling terms among the gauge bosons of QCD, the gluons. The Lagrangian of QCD reads as:
X 1 a µν QCD = q¯(x)(iD/ mq)q(x) Fµν(x)Fa (x) (2.10) L q − − 4 with the covariant derivative being
λa a Dµq ∂µ igs G q (2.11) ≡ − 2 µ and the field strength tensor being
a a a aβγ F (x) = ∂µG (x) ∂νG (x) + gsf GµβGνγ. (2.12) µν ν − µ Here, the fields q are vectors of length 3 and dimension 1 containing the three quark fields for the individual colour charges red, blue, and green. The strong coupling
3Without colour, all the three quarks would have the same quantum state which is prohibited by the Pauli exclusion principle. 4An extended account of the history of QCD can be found in Ref. [51].
9 2. Theoretical Foundations
constant is denoted with gs and the SU(3) generating matrices are the eight Gell- a Mann matrices λa/2. The eight gluon fields have been written as Gµ and the structure constants as f abc(a, b, c = 1, ..., 8). Several properties are noteworthy. The gluons themselves are colour charged leading to self-coupling terms, a direct consequence of the non-abelian nature of the SU(3) symmetry group. This is a general property of so-called Yang-Mills theories [52] leading to the self-coupling of gluons which also has been observed experimentally [53]. Two further properties are not visible from analysing the Lagrangian: confinement and asymptotic freedom. Confinement [54] characterises the behaviour of the strong interaction at long distances describing the fact that no individual gluons or quarks but only colourless bound states are observed. When trying to separate a quark-anti-quark pair the energy stored in the strong field between them will increase with increasing distance. At some point the stored energy is sufficiently high to produce a new quark- anti-quark pair with the newly produced particles forming colourless hadrons with the original particles. It has to be noted that it has not been proven yet that confinement is predicted by QCD.5 The other phenomenon, asymptotic freedom, reflects that the strong coupling decreases at high energy/small distances. This is a consequence of the negative value of the β function of QCD describing the dependence of the strong coupling constant on the en- ergy scale. Asymptotic freedom enables calculations of matrix elements in perturbative theory for processes involving the strong interaction and was found independently by Gross and Wilczek [56, 57] and Politzer [58, 59].
2.2.3. Electroweak Theory
One of the great successes of the Standard Model is the unified description of the elec- tromagnetic and weak interaction by the electroweak theory. The underlying symmetry group of electroweak theory is SU(2)L U(1)Y which was suggested by Glashow in × 1961 [2]. The subscript L refers to the fact that weak gauge bosons only couple to left-handed particles while Y denotes the hypercharge to avoid confusion with the U(1) transformation of QED. The Lagrangian of the electroweak theory reads as:
1 a µν 1 µν = ¯liDl/ +qi ¯ Dq/ W W BµνB (2.13) LEW − 4 µν a − 4 with the covariant derivative
a Y Dµf = ∂µ igwTaW igY Bµ f (2.14) − µ − 2 and the field strength tensors
i i i ijk j k W = ∂µW ∂νW + g W W , (2.15) µν ν − µ µ ν Bµν = ∂µBν ∂νBµ. − 5The prove is in fact one of the millenium problems formulated by the Clay Mathematics Institute [55].
10 2.2. The Standard Model
field T3 QY 1 2 1 uL 2 3 3 1 1 1 dL 2 3 3 − − 2 4 ur 0 3 3 1 2 dr 0 3 3 1 − − νL 2 0 1 1 − eL 1 1 − 2 − − er 0 -1 -2
Table 2.3.: Charges of the fermions with respect to the SU(2)L U(1)Y symme- × try. Left-handed fermions transform as doublets under SU(2) which is reflected by the symmetric charges with respect to the weak isospin T3. Right-handed fermions transform as singlets and have no weak isospin.
The fermionic fields l and q denote the leptons and quarks which receive different treatment depending on their chirality. Left-handed fermions transform as doublets under SU(2)L, whereas right handed fermions transform as singlets as is reflected in Table 2.3.
The electroweak Lagrangian introduces three gauge fields for the SU(2)L symmetry (W1...3) and one for the U(1)Y symmetry (B). The generators of the SU(2)L symmetry are the three Pauli matrices divided by two. The coupling constants for the weak fields W1...3 and the hypercharge field B are free parameters of the theory and are denoted as gw and gY , respectively. As in QCD, the field strength tensor exhibits terms that lead to self-couplings between the gauge bosons which is of paramount importance to the study of electroweak gauge boson scattering and will be discussed further in a dedicated section. Four charges have been introduced alongside the gauge fields, the weak charges Ti and the weak hypercharge Y . The neutral weak charge T3 and the weak hypercharge Y are connected with the electric charge Q via the Gell-Mann-Nishijima formula [60, 61]:
Y Q = T3 + . (2.16) 2 The physically observed gauge bosons are a linear combination of the gauge fields present in the Lagrangian:
± 1 1 2 Wµ = Wµ iWµ , (2.17) √2 ∓ 3 Zµ = cos(θw)W sin(θw)Bµ, µ − 3 Aµ = sin(θw)Wµ + cos(θw)Bµ with θw being the weak mixing angle which governs the rotation of the gauge fields into ± the physical fields. The physical fields Wµ ,Zµ, and Aµ are obtained by requiring that the W ± act as ladder operators between the components of the left-handed doublets and that the photon does not couple to neutrinos.
11 2. Theoretical Foundations
A phenomenon not immediately visible from the Lagrangian is the CP violation of the electroweak interaction. This violation is introduced via the CKM matrix which describes the rotation of the interaction eigenstates of the down-type quarks into the respective mass eigenstates. The idea was first developed by Cabibbo in 1963 [62] for two generations of quarks and later extended to the three generation case by Kobayashi and Maskawa [63]. Decays of hadrons composed of quarks from different families (e.g. Λ0 = uds) are possible due to off-diagonal terms of the CKM matrix. The CP vio- lation observed for some electroweak processes is a result of a non-vanishing complex angle present in the CKM matrix which is only made possible by the fact that three generations of fermions exist. Measurements of the free parameters of the CKM matrix pose an important test of the Standard Model and are curated by the CKM Fitter group [64]. As of now no masses have been introduced to the electroweak theory. This also holds for the weak gauge bosons whose masses are of the order of 100 GeV. However, a naive introduction of mass terms is not possible as they would break the SU(2)L U(1)Y × symmetry. In the late 1960s Weinberg and Salam published papers combining the electroweak theory with the later to be discussed Higgs mechanism providing a theory that included massive gauge bosons [3,4] and enabled a gauge-invariant way to introduce fermion masses.
2.2.4. Electroweak Symmetry Breaking
In the early 60s of the last century three groups published proposals addressing the problem of the introduction of masses in the Standard Model in a very short timeframe: Peter Higgs [25], Robert Brout and Fran¸coisEnglert [24], and Gerard Guralnik, C.R. Hagen, and Tom Kibble [23].6 It is based on the concept of spontaneous symmetry breaking [22] describing the situa- tion that the physics determining the behaviour of a system has a symmetry that the system’s ground state7 does not have. Applied to quantum field theory it states that the Lagrangian of a system is invariant under a certain symmetry transformation but the ground state of the theory is not. In the case of the spontaneous symmetry breaking of a global symmetry the Goldstone Theorem [65] applies predicting one massless boson (called Goldstone boson) for each generator that does not destroy the vacuum. However, the introduction of massless bosons would not remedy the stated problem. Luckily, the Goldstone Theorem does not apply the same way in the case of gauge theories. For these, the Higgs Mechanism states that the Goldstone bosons introduced by the spontaneous symmetry breaking combine with the already present massless gauge bosons and become the longitudinal components of said bosons. As will be seen later the terms introducing the longitudinal components may be interpreted as mass terms for the respective massive electroweak
6 There is some debate how to call the mechanism based on the people who have pub- lished papers on it. Often used variants are Brout-Englert-Higgs-mechanism or En- glert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism or other variations. In this work, the short version Higgs mechanism will be used which does not intend to diminish the work of the other groups but solely adopts the general usage. In the end, it is rather improbable that the memory of humankind will ever remember things as they really were and in the end the discovery is important even if the true heroes may remain unsung. 7The ground state is the state that minimises the expectation value h0|Φ|0i.
12 2.2. The Standard Model gauge boson.
In the case of the Standard Model the SU(2)L U(1)Y group is broken down in the × ground state to U(1)em. The electroweak Lagrangian is extend by the following terms which introduce the spontaneous symmetry breaking:
† µ = (DµΦ) (D Φ) V (Φ) with (2.18) LEWSB − V (Φ) = µ2Φ†Φ + λ(Φ†Φ)2; λ > 0 (2.19) − with Φ being a complex doublet:
φ+ 1 φ + iφ Φ = = 1 2 (2.20) φ0 √2 φ3 + iφ4 and Dµ the covariant derivative of the electroweak theory. It has to be noted that µ2 > 0 and λ > 0 have to hold otherwise the vacuum would be symmetric and no spontaneous symmetry breaking would take place. Due to the U(1)em symmetry of the ground state an infinite number of ground states exists and one is free to choose a specific one for further calculations. The ground state is chosen to be:
1 0 Φ0 = (2.21) √2 v with v being the so-called vacuum expectation value. By applying unitarity gauge one can write the general form of equation (2.20) as
1 0 Φ(x) = (2.22) √2 v + H(x) with H being the newly introduced scalar boson dubbed the Higgs boson. The mass terms of the massive electroweak gauge bosons are encoded in the kinetic term of the Higgs field
2 2 2 2 2 † µ gwv + µ− 1 (gY + gw)v µ (DµΦ) (D Φ) = W W ZµZ + ..., (2.23) − 4 µ − 2 4 whereas the Higgs mass term itself can be found in the potential:
1 V (Φ) = (2µ2)H2 + ... . (2.24) 2 This leads to the following mass predictions
q 2 2 gwv gw + gY v MW = ; MZ = ; MH = √2µ. (2.25) 2 2 Using these relations and measured values for the masses of the gauge bosons and the electroweak couplings the vacuum expectation value v can be calculated and is found to be 246 GeV. The masses of the remaining fermions are introduced via Yukawa ≈ couplings which are shown in the full Standard Model Lagrangian in the next section. The numerical values for the Yukawa couplings are free parameters of the Standard Model and have to be determined experimentally.
13 2. Theoretical Foundations
2.2.5. The Lagrangian of the Standard Model
All necessary parts of the Standard Model are now in place and the three interactions can be written as one large Lagrangian which is invariant under the combined SU(3)C × SU(2)L U(1)Y symmetry group. The individual parts are: ×
= leptons + quarks + gauge + Higgs + Yukawa (2.26) L Lkin Lkin Lkin L L with
X leptons = iL¯j DL/ j + i¯lj Dl/ j , Lkin L L R R X quarks = iQ¯j DQ/ j + iu¯j Du/ j + id¯j Dd/ j , Lkin L L R R R R gauge 1 µν 1 µν 1 µν = BµνB Wa,µνW Ga,µνG , Lkin −4 − 4 a − 4 a Higgs µ † 2 † † 2 = (D Φ) (DµΦ) + µ Φ Φ λ(Φ Φ) , L − Yukawa X j j jk j c k jk j k = y j L¯ Φl Y Q¯ Φ u Y Q¯ Φd + h.c. L − l L R − u L R − d L R with j 1, 2, 3 being the summation index of the fermion generation. The non-diagonal ∈ jk Yukawa matrices Yx give rise to the the flavour changing currents described by the CKM matrix. The covariant derivative is
Y a λa a Dµ = ∂µ igY Bµ igwTaW igs G . (2.27) − 2 − µ − 2 µ
All terms necessary for theoretical calculations are present: Propagation terms for the individual fields, all allowed interaction terms between fermions and bosons as well as among the bosons themselves, the Higgs potential giving rise to the electroweak sym- metry breaking, and the Yukawa mass terms that contain the masses of the individual fermion fields.
2.3. Electroweak gauge boson scattering
2.3.1. Definition
Electroweak gauge boson scattering is also referred to as vector boson scattering (VBS) and the terms will be used interchangeably. It is the process VV VV with V = → W ±, Z, γ realised via electroweak triple and quartic gauge vertices, and interactions of the massive electroweak gauge bosons with the Higgs boson. In Section (2.2.3) the non-abelian structure of the electroweak theory was stated. This structure gives rise to self-couplings among the electroweak bosons, namely the W ± bosons, the Z boson and the photon [66]:
14 2.3. Electroweak gauge boson scattering
µ ν ν µ † µ ν† ν µ† = ie cot(θw) (∂ W ∂ W ) W Zν ∂ W ∂ W WµZν (2.28) L3 − − µ − − † µ ν ν µ ie cot(θw)WµW (∂ Z ∂ Z ) − ν − µ ν ν µ † µ ν† ν µ† ie (∂ W ∂ W ) W Aν ∂ W ∂ W WµAν − − µ − − † µ ν ν µ ieWµW (∂ A ∂ A ), − µ − 2 2 e † µ † µ† ν 4 = 2 WµW WµW WνW (2.29) L − 2 sin (θw) − 2 2 † µ ν † µ ν e cot (θw) W W ZνZ W Z WνZ − µ − µ 2 † µ ν † µ ν † µ ν e cot(θw) 2W W ZνA W Z WνA W A WνZ − µ − µ − µ 2 † µ ν † µ ν e W W AνA W A WνA − µ − µ Figure 2.1 exhibits the Feynman diagrams that result from these terms as well as the interactions of the Higgs boson with the massive electroweak gauge bosons which are not stated in the shown excerpts from the SM Lagrangian. These are the building blocks for the electroweak gauge boson scattering diagrams shown in Figure 2.2 which define the purely electroweak gauge boson scattering processes that will be examined in this work.
2.3.2. Motivation
The study of VBS poses a test of the gauge sector of the electroweak theory. When only subsets of the diagrams in Figure 2.2 are considered, the theory predictions for VBS scattering show an unbounded rise in cross section with rising centre-of-mass energy. Finite results are obtained only when all diagrams are taken into account with the Higgs being the one described by the Standard Model (see Figure 2.3). A first test of the Standard Model is the observation of VBS itself as it has not been observed significantly before. First evidence of massive electroweak gauge boson scat-
W + W + W + W + , Z