Scattering of polarised W bosons Measuring the polarised W bosons - Higgs coupling with ATLAS using charged lepton observables

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U

NIVERSITEIT VAN

A

MSTERDAM

MSc Physics

Track: Gravitation and astroparticle physics Amsterdam

Master Thesis

Scattering of polarised W bosons

Measuring the polarised W bosons - Higgs coupling

with ATLAS using charged lepton observables

By

Cornelis Ligtenberg

10447547

June 28, 2016 60 ECTS credits

The research was carried out between September 2015 and June 2016

Supervisors:

Prof. dr. ing. Bob van Eijk Magdalena Slawinska PhD

Examiners:

Prof. dr. ir. Paul J. de Jong Prof. dr. Robert Fleischer

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Abstract

In the standard model the masses of the gauge bosons are the result of electroweak symmetry breaking through the Higgs mechanism. Because of the W bosons gain mass, they also acquire a longitudinal polarisation. The Higgs boson contribution cancels a divergence of the cross section in the scattering of longitudinal W bosons. Therefore deviations from the standard model in these couplings will cause the diver-gence to re-emerge or to leave space for new physics. Electroweak symmetry break-ing is studied through the Higgs couplbreak-ings in the vector boson fusion H → W W∗ channel which contributes to W boson scattering. In this theses, we investigate the Higgs boson couplings to longitudinally and transversely polarised W bosons sepa-rately. Deviations may originate from various new physics scenarios such as models with a composite Higgs.

The possibilities for measurements on these couplings with the ATLAS detector in run 2 of the LHC at 13 TeV with an integrated luminosity of 300 fb−1are studied. We focus on lepton detections with two differently flavoured charged leptons to probe the longitudinal and transverse couplings in the HW W vertex. Events are simulated with hadronic and electromagnetic showering and hadronisation. Using a signal sig-nature with two hard jets and two charged leptons, an optimised set of cuts is found for the separation of the standard model signal from the background. The cuts are expected to select the vector boson fusion signal process from the background with a significance of 15.9 at parton level and 9.70 after showering and hadronisation.

These cuts are utilised to find deviations in lepton distributions resulting from anomalous couplings of the Higgs to the longitudinally and transversely polarised W bosons. The most promising observables are the leading lepton transverse momentum and the azimuthal angle between the two leptons. At parton level they are expected to set upper and lower limits on aLrespectively at 1.3 and 0.7. Although the expected

sensitivity is less than the for the jet observables, they can prove a valuable addition to an analysis on polarised W couplings.

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Introduction

The Standard Model of particle physics (SM) is a unified theory describing the fundamental particles and their interactions, with the exception of gravity. It describes nearly all matter and phenomenology down to the sub-atomic scale. Since the first steps in the 1960s and wider acceptance in the 1970s, the SM has come to constitute ”the first systematic and comprehensive theory of elementary particle dynamics” [1]. The discovery of the Higgs boson particle [2] is one of the latest discoveries that reaffirm the SM. For the agreement of its predictions with experimental results, it has been deemed ”one of the triumphs of modern physics” [3].

Nevertheless, some properties of the scalar sector of the SM have yet to be experimentally confirmed. The question remains if the SM Higgs boson solely responsible for the electroweak symmetry breaking and whether the scalar sector of the SM is truly the minimal one proposed in the SM.

Besides untested properties, there are experimental outcomes the contradict the SM and problems within the SM itself. On top of that, the theory of general relativity that covers gravity is not compatible with the SM. These are some of the reasons that many physicist argue that there must be a coherent theory of physics beyond the standard model. Precision test of the SM may elucidate some of the mysteries.

A few of the most pressing issues are also related to the recently discovered Higgs sector. The smallness of the Higgs mass compared to its radiative corrections is something that physicist regard as unnatural and is known as the hierarchy problem, see e.g. [4]. The problem is that the radiative corrections at high energies will have to be fine-tuned to acquire the light Higgs mass. One of the proposed solutions to the hierarchy problem is a composite Higgs particle. The composite Higgs are held together by a newly introduced force. The composite Higgs limits the fine-tuning to the energy scale at which the Higgs is disintegrates. Examples of the composite Higgs models are the little Higgs and holographic Higgs [5, 6].

One place where signs for these new models might be found is in the scattering of polarised W bosons. The longitudinal polarised W bosons are the result of the Higgs mechanism. In the scattering of longitudinal polarised W bosons, the Higgs boson cancels a divergence. A comparative measurement of the couplings between longitudinally polarised W bosons and the transversely polarised W bosons is a precision test probing new physics scenarios. The scattering with an s-channel Higgs boson, which is embodied in the VBF channel is specifically suitable, lending from its distinctive properties.

Previous efforts have already shown the possibilities for an determinations from a theoretical perspective for this channel [7]. Here this framework is extended to include an experimental prospect with a necessary description of the most important backgrounds. The ATLAS at the large hadron collider is one ongoing experi-ment that can perform such measureexperi-ments, wherefore a good selection of the signal is crucial. The possibilities for measurements in the lepton channel are specifically covered.

Given current experimental constraints, the energy required to study the new physics directly may not be within reach yet. The polarised W bosons at ATLAS provide a gateway to do precision tests the standard model and to probe possible physics at higher energy scales.

Organisation of this thesis

This thesis organised in 7 chapters. The first chapter gives an introduction to the standard model of particle physics. Particles and forces are outlined, and gauge theory and electroweak symmetry breaking is briefly described. The description there is not at all meant to be exhaustive, and only some relevant parts are treated.

In Chapter 2 there is a focus on the W bosons and their three polarisations. The scattering of polarised W bosons is discussed. A possible extension of the standard model in the form of a simple modification of couplings is looked into. Finally the VBF process is introduced and motivated.

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10 CONTENTS This process can be measured using the ATLAS detector at the large hadron collider. These are discussed in chapter 3. First the large hadron collider is shortly covered. Then the ATLAS detector and its sub-detectors are discussed in more detail, where attention is paid to matters relevant for the calculations in subsequent chapters. In chapter 4, the Monte Carlo generation of events is discussed. The Monte Carlo tools and how they work is discussed. In chapter 5 the definition of various objects and quantities is given. In this chapter the set of events is preselected for the analysis.

Chapter 6 is devoted to the actual selection of events. Many of the interesting properties of the signal are discussed here. The most important background processes are given. At two stages in the simulation it is explained how these backgrounds can be separated from the signal. This is done at parton level and after showering and hadronisation.

In chapter 7 the effect of the simple modification of couplings on the lepton observables is explored. Finally, the conclusions are presented.

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Chapter 1 The standard model of particle physics

In this chapter, the parts of the SM relevant for the analysis are discussed. First the particles of the standard model and their interaction are outlined. Then the gauge theory that introduces the electroweak gauge bosons is discussed. Thirdly, the Higgs mechanism that introduces the Higgs boson and the masses for these gauge bosons is treated. Lastly a short description is given on quantum chromodynamics, important for the forming of jets. Most of the content in this chapter is based on references [3, 8, 9].

1.1 The particles of the standard model

In the standard model there are two types of particles. The particles that make up most of the matter around us are the fermions that are characterised by half integer spin. In the SM the fundamental particles interact through the three fundamental forces, which are mediated by bosons, characterised by integer spin.

An overview of the fundamental particles and the forces they interact by, is given in figure 1.1. In addition to the shown particles, the SM contains an anti-particle for each particle with the same properties but opposite charge. Fermions are divided into leptons and quarks and both are divided into three generations with similar properties but different masses.

A generation of leptons is a pair consisting of a charged lepton and a neutrino. The charged leptons are charged particles with unit charge. The in total six leptons interact through the weak interaction mediated by the weak gauge bosons Z and W±. Like all charged particles, the charged leptons furthermore interact through the electromagnetic interaction mediated by the photon.

Quarks are fermions that carry a colour charge, and therefore interact through the strong force mediated by the gluon. The strong force causes quarks to always reside in colour neutral composite objects that can be made of a quark anti-quark pair called mesons, or three quarks or anti-quarks combinations called baryons. Quarks carry fractional electromagnetic charge, but the mesons and baryons always have integer charge.

The bosons that mediate the fundamental forces all have a spin of 1 in common, but their other properties differ. The weak gauge bosons mediating the weak force have masses, while the gluon and photon are massless. The W boson is the only gauge boson carrying an electromagnetic charge, while the gluon is the only boson carrying a colour charge. Finally, the Higgs boson allows masses of the fundamental particles to be introduced through electroweak symmetry breaking. Emphasising its special role in the SM, the Higgs is the only scalar (spin 0) particle.

Part of the theoretical framework of the standard model is provided by quantum field theory. Herein, particles are described as excitations of underlying fields and their interactions are described by interactions terms of these fields (see e.g. [11] ). The quarks and leptons of the SM are defined by fermion fields in quantum field theory, described by a Lagrangian. The gauge bosons and fermion interactions are the result of symmetries of the Lagrangian, which is known as gauge theory. The symmetries of the SM, the U (1) × SU (2)L× SU (3)

symmetries can be imposed on a fermion Lagrangian using Yang-Mills theory.

1.2 Electroweak gauge theory

The electroweak gauge bosons are introduced through the U (1) × SU (2)Lgauge symmetry. Imposing a local

U (1) × SU (2) invariance will lead to both the Bµ_{and the W}µ_{fields. This section imposes these symmetries}

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2 CHAPTER 1. THE STANDARD MODEL OF PARTICLE PHYSICS

Figure 1.1: All particles of the SM and some of their properties. The first three columns correspond to the first to third generation of particles. The shaded surfaces indicate the forces that affect the fermions and the connected bosons. Image taken from reference [10].

and it is laid out how the photon field Aµ, and electroweak W±and Z fields follow from it.

1.2.1 The U (1) and SU (2) transformations

The U (1) and SU (2) transformations are transformations of the special unitary group, the group of unitary n × n matrices with determinant 1. The transformation U on a wave function (doublet) is given by

ψ(x) → ψ0(x) = U (x)ψ(x), (1.1) where x is a four-vector. The U (1) symmetry is the transformation of a wave function by a local complex phase χ(x) and is given by

UU (1)(x) = eiχ(x). (1.2)

While the U (1) symmetry is represented by a single complex number, the SU (2) symmetry is represented by a 2 × 2 matrix, which can be written in terms of its generators, the pauli matrices σi. The general SU (2)

transformation matrix USU (2) for a doublet ψ is given by

USU (2)(x) = exp(

i

2α(x) · T), (1.3) where α(x) is a three-vector with constants, and T is a three-vector with the pauli matrices (σ1, σ2, σ3).

1.2.2 Invariance of the Lagrangian

The electroweak gauge bosons arise when the Lagrangian is required to be invariant under the U (1) and SU (2)L

transformation. The massless Dirac Lagrangian L of a bi-spinor doublet (e.g. a left handed lepton doublet ψL= (νee)L) is given by

L = ¯ψiγµ∂µψ, (1.4)

which is not U (1) nor SU (2) invariant, because the derivative transforms as

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1.2. ELECTROWEAK GAUGE THEORY 3 where we dropped any explicit x dependence and U is either the U (1) or SU (2) transformation. In order to acquire an invariant Lagrangian we must introduce a new field with specific properties. This is accomplished in two steps, the introduction of a new field and the definition of its properties, which will be calculated. Imposing the U (1) symmetry

Firstly the photon field Bµis introduced by replacing the derivative with a covariant derivative Dµ

∂µ→ Dµ= ∂µ+ iqBµ, (1.6)

where q is a constant. The U (1) invariant Lagrangian including interaction term is L = ¯ψiγµDµψ

= ¯ψiγµ∂µψ − q ¯ψγµBµψ.

(1.7) From an evaluation or using the strategy for the SU (2)Lsymmetry below, it is found that that the Lagrangian

is invariant if the Bµfield transforms as

Bµ→ Bµ0 = Bµ−

1

q∂µχ. (1.8) Imposing the SU (2)Lsymmetry

The procedure that makes the Lagrangian SU (2)L invariant is similar, but the SU (2)Lsymmetry is only

im-posed on left handed doublets. The first step introduces a new field Wµby replacing the derivative ∂µwith a

covariant derivative again

∂µ→ Dµ= ∂µ+ igWWµ· T , (1.9)

where gW is a constant, which results in the SU (2)Linvariant Lagrangian

L = ¯ψiγµDµψ

= ¯ψiγµ∂µψ − gWψiγ¯ µWµ· T ψ.

(1.10) The next step is to calculate how the Wµ fields transform such that this Lagrangian is invariant. Since ¯ψ

transforms as

¯

ψ → ¯ψ0= ¯ψU−1, (1.11) Dµshould transform as

Dµψ → Dµ0ψ0 = U (Dµψ) (1.12)

in order to keep the Lagrangian invariant. The covariant derivative from equation (1.9) can be substituted on the left hand side of equation (1.12)

D_µ0ψ0 = (∂µ+ igWWµ0 · T )ψ0

= U (∂µψ) + (∂µU )ψ + igWWµ0 · T U ψ,

(1.13) while the right hand side can be evaluated to

U (Dµψ) = U ∂µψ + igU Wµ· T ψ. (1.14)

Combining both equations and cancelling common terms leads to

igWWµ0 · T U ψ + (∂µU )ψ = igU Wµ· T ψ, (1.15)

which after noticing that it must hold for all ψ, results in the transformation properties of the Wµfield

W_µ0 · T = U W_µ· T U−1+ i gW

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4 CHAPTER 1. THE STANDARD MODEL OF PARTICLE PHYSICS

1.2.3 Kinetic terms

The Bµand Wµfields and their interactions enter the Lagrangian through the covariant derivative. In contrast,

the kinetic term has to be added by hand. From Quantum Electrodynamics(QED) the kinetic term of a spin 1 field is given by

L_kin= FµνFµν, where Fµν =

1

iq[Dµ, Dν]. (1.17) Evaluation for both covariant derivatives in equations (1.6) and (1.9) separately leads to

BµνBµν = BµνBµν, where Bµν = ∂µBν− ∂µBν (1.18)

FµνFµν = WµνWµν, where Wµν = ∂µWν− ∂µWν− gWWµ× Wν, (1.19)

and where BµνBµν and FµνFµν are kinetic terms. In comparison with Bµ, an additional cross product term

entered the tensor for the Wµ fields, because the generators σi do not commute, i.e. because the SU (2)L

symmetry is non-Abelian. This term is responsible for the W boson self interactions.

1.2.4 Mixing of the U (1) and SU (2)Lsymmetries

None of the Bµand Wµ fields correspond directly to the photon or Z field. The Bµ couples to hypercharge

instead of electromagnetic charge. In order to acquire the photon field Aµand the Z field, the Bµand Wµ3field

mix as

Aµ= Bµcos θW + Wµ3sin θW

Zµ= −Bµsin θW + Wµ3cos θW,

(1.20) where θW is the Weinberg angle (or weak mixing angle).

1.2.5 The physical gauge bosons

Similar to the the mixing of the Bµand Wµ3fields, the Lagrangian W1 and W2 terms have to be rewritten in

terms of the physical gauge boson terms W+and W−. This mixing is given by W±= √1

2(W

1_{∓ iW}2_), _(1.21)

which is motivated by noting that the interaction with the physical W±corresponds to a charge raising/lowering operator σ±on SU (2)Ldoublets. For example, an electron ψ ∝ (0e) can be transformed to a neutrino ψ ∝ (

νe 0 )

by the charge raising operator. The charge operators are given by σ+= 1 2(σ1+ iσ2) = 0 1 0 0 σ−= 1 2(σ1+ iσ2) = 0 0 1 0 , (1.22)

in which the operators were expressed using the σ1 and σ2 Pauli matrices that correspond to the W1 and W2

fields.

1.3 Higgs mechanism for electroweak gauge bosons

In the previous section the kinetic and interaction terms of the new fields Wµand Bµthat represent the gauge bosons, were introduced in the Lagrangian by imposing a U (1) × SU (2)Lsymmetry. Although it is

experimen-tally observed that gauge bosons have mass, the mass terms do not follow from these steps and in fact break the SU (2)Lgauge symmetry if they are added by hand like the kinetic terms. In accordance to observations,

masses follow from the Higgs mechanism, as well as a Higgs boson and Higgs-gauge boson-interactions. This section treats the Higgs mechanism for gauge bosons and is based on references [3, 9]1.

1

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1.3. HIGGS MECHANISM FOR ELECTROWEAK GAUGE BOSONS 5

Figure 1.2: The vacuum potential for µ2 < 0. Around the new vacuums represented as a circle, the field can be expanded using the h field. Figure adapted from reference [9].

1.3.1 The Higgs mechanism

Massive gauge bosons acquire their mass through the Higgs mechanism in four steps.

1. Add an isospin doublet to the Lagrangian. An isospin doublet automatically guarantees that the La-grangian retains its symmetries. A kinetic term for the new Higgs doublet φ enters the LaLa-grangian L as Lkin= (Dµφ)†(Dµφ), where φ = φ+ φ0 =φ1+ iφ2 φ3+ iφ4 , (1.23) φ+and φ0are complex , φiis real and Dµis the covariant derivative.

2. Add a potential that will break the symmetry. The simplest potential V (φ) that will break the symmetry is given by

Lpot = V (φ) = µ2(φ†φ) + λ(φ†φ)2, (1.24)

where λ and µ are complex parameters. If µ2 < 0 this potential shifts the minimum of the vacuum to the set φ†φ = −µ_2λ2 = v₂2. If the vacuum is electrically neutral φ+= 0, it is given by hφi0 = √1₂(0v).

3. Expand the vacuum around the new minimum using a new field h(x), which corresponds to the Higgs boson as shown in figure 1.2. The expansion is given by

φ = √1 2 φ1+ iφ2 v + h + iφ4 , (1.25)

where v is a constant and the remaining φi(x) terms can be interpreted as Goldstone bosons.

4. Choose a gauge, that absorbs the Goldstone terms. Using the new parameters ω = (ω1, ω2, ω3), the

expansion of the vacuum in equation (1.25) can be recast to φ(x) = √1 2 (ω2+ iω1)/2 v + h − iω3/2 = √1 2exp(i ω v · T 2) 0 v + h , (1.26)

where in the second line the ω terms were rewritten in the vicinity of the chosen vacuum. For v h, |ω| the second line reduces to the first. By performing the gauge transformation

U = exp(−iω · T /2v), (1.27) the Higgs doublet becomes

φ → φ0 = U φ = 0 v + h . (1.28)

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6 CHAPTER 1. THE STANDARD MODEL OF PARTICLE PHYSICS The Goldstone terms were eliminated from the Higgs field. They reappear as an additional degree of freedom for each of the gauge bosons after performing the same gauge transformation on the Wµfield:

Wµ· T → Wµ0 · T = U Wµ· T U−1+ i gW (∂µU )U−1 = U Wµ· T U−1− i gWv (∂µω · T ). (1.29)

Naturally the primed fields obey the same equations as the old Wµ and φ fields, because both were

constructed to be invariant under SU (2)Lgauge transformations.

1.3.2 Mass and interaction terms

The mass of the Higgs, the masses of the W gauge bosons, and the coupling between the them are now part of the Lagrangian, but in order to acquire explicit mass and interaction terms the scalar part of the Lagrangian

L_kin= (Dµφ)†(Dµφ), where φ = 0 v + h , (1.30) Dµ= ∂µ+ igWWµ· T (1.31) has to be evaluated.

1.4 Decay rates and cross sections in the standard model

Having introduced parts of the SM in this chapter, it only becomes applicable if chances for processes can be calculated. For a single particle initial state the decay rate Γf iis the probability per unit time to go from

initial state i to a final state f . For the transition from a multiparticle initial state to a final state, the likelihood likelihood is represented by the cross section σ defined as an effective area. Both have similar expressions

σ = 1 F Z |Mf i|2dΦ and Γf i= (2π)4 2Ei Z |Mf i|2dΦ, (1.32)

where |Mf i| is the Lorentz invariant matrix element which is a Lorentz invariant expression for the quantum

mechanical probability for the transition. The integral is over all the possible final states in the Lorentz invariant phase space Φ. Finally, the flux factor F accounts for the density of incoming states and the Eiis the energy of

the initial particle accounting for possible time dilation with respect to the observers frame. The matrix element |M_{f i}| which contains the SM physics can be calculated from the Lagrangian using quantum field theory or a shortcut can be taken using Feynmann rules, the rigorous derivation or exhibition whereof is left to alternative sources, e.g. reference [11].

1.4.1 Branching fractions of the Higgs

The Higgs couples to all fermions with a factor

− im_f gW 2mW

, (1.33)

where mf is the fermion mass. Because the coupling is proportional to mass, the Higgs primarily decays to the

heaviest fermion for which 2mf < mh, the bottom quark and anti-quark, constituting 57.8 % [13]. However, in

comparison to the fermion coupling, the coupling to two W bosons as will be given in equation (2.24) is large. Therefore, even though at least one of the W bosons is off-shell see figure 1.4, the Higgs decay to W bosons is the second largest branching fraction, constituting 21.6 %. An overview of the branching ratios is given in figure 1.3.

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1.5. QUANTUM CHROMODYNAMICS 7 [GeV] H M 120 121 122 123 124 125 126 127 128 129 130 H ig g s B R + T o ta l U n c e rt -4 10 -3 10 -2 10 -1 10 1 LHC HIGGS XS WG 2013 b b τ τ µ µ c c gg γ γ ZZ WW γ Z [GeV] W m 0 20 40 60 80 100 Events [A.U.] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 + W -W µ ν -µ e ν e+ → WW* → VBF H

Figure 1.3: Higgs boson branching ratios as a function of the Higgs mass MH. Figure taken

from [13].

Figure 1.4: W boson mass distribution for the VBF H → W W process from a Madgraph cal-culation using parameters given in table 4.2. Out of the two W bosons created in the Higgs decay, one is off shell.

1.4.2 Higgs production modes

Because the Higgs bosons coupling is proportional to mass for the fermions, the dominant Higgs boson pro-duction mode is gluon-gluon fusion through a heavy quark loop, having a cross section of 44 pb for a 125 GeV Higgs with 13 TeV proton-proton collisions [14]. The second production mode is vector boson fusion which has a cross section an order of magnitude smaller at 3.8 pb. Other production modes which include associated production with bosons and associated production with a top or bottom quark have a still smaller cross section.

1.5 Quantum chromodynamics

Besides the electromagnetic and weak force, the strong interaction is the third force described by the SM model. It acts on the quarks and is mediated by the massless gluon. The strong is important for the formation of jets.

1.5.1 SU (3) gauge symmetry

Similarly to the electromagnetic gauge bosons, the mediator of the strong force can be introduced through its own gauge symmetry, the SU (3) gauge symmetry. A discussion hereof (see e.g. [8]) is not reproduced here , but is limited to stating that the SU (3) symmetry is generated by the eight generators known as the Gell–Mann matrices. Similarly to the Pauli matrices of the SU (2)Lsymmetry, these do not commute giving the gluon self

interaction terms. The SU (3) symmetry introduces a charge similar to the electromagnetic charge, but with three components dubbed red, blue and green. In contrast to a photon which does not have an electromagnetic charge, there are eight gluons that carry a different colour charge.

1.5.2 Colour confinement and formation of jets

An isolated quark has never been observed, which is the result of colour confinement. This hypothesis states that colour charges are always combined into colour neutral objects. The energy stored in the gluon field between two quarks is given by a linear proportionality constant of approximately 1 GeV/fm. When the energy

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8 CHAPTER 1. THE STANDARD MODEL OF PARTICLE PHYSICS of the gluon field is large enough, a quark anti-quark pair q ¯q is created instead of increasing the energy of the field further. The newly created quarks can eventually combine the other quarks to form colour neutral objects. Partons with enough energy cause a shower of particles through this process. These streams of particles are called jets, which form when the quark anti-quark pairs q ¯q created in the vacuum are dragged along by the single quarks. Quarks created this way combine to hadrons, which are observed in jets.

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Chapter 2 The polarisation of W bosons

In the last chapter an overview of the SM was given. This chapter zooms in to an interest element thereof, the polarisation of the W bosons. First a description of the polarisation states of spin 1 particles is given. The third polarisation of the W bosons causes the cross section of W boson scattering to diverge at high energies. Then the connection to the Higgs boson and the scattering is made, and a method is described on how properties of the different polarisation can be modified separately. Finally a channel for investigation of these properties in a hadron collider is proposed.

2.1 Polarisation

The number of polarisation states of a spin 1 particle depends on the existence of its mass term [3]. A massless particle such as the photon has two polarisation states, and a massive particle such as the W±boson has three polarisation states, because of the extra degree of freedom added through the Higgs mechanism. The additional polarisation differs from the other two. Both categories are calculated in this section.

2.1.1 Polarisation states of massless spin 1 particles

For a massless spin 1 field Aµ, the free field equation is given by the Euler-Lagrange equation

∂ν∂νAµ= 0 (2.1)

and its solution is given by the plane wave

Aµ= µ(q) exp(−ixq), (2.2) where q is a momentum four vector, x a space four vector, and µ(q) is the polarisation four vector. Substituting solution (2.2) in equation (2.1), results in the constraint

∂ν∂ν(µ(q) exp(−ixq)) = −q2µ(q) exp(−iqx) = 0 ⇒ q2 = 0. (2.3)

On the massless field Aµthe Lorenz gauge condition ∂µAµ= 0 can be applied

0 = ∂µ(µexp(−iqx)) = µ(∂µexp(−iqx)) = −iµqµexp(−iqx) ⇒ qµµ= 0. (2.4)

The Lorenz gauge allows to make another gauge transformation of the form

Aµ→ A0µ= Aµ− ∂µΛ(x), (2.5)

for any ∂ν∂νΛ = 0. With the choice Λ(x) = −ia exp(−iqx), where a is a parameter, the transformation is

A0_µ= Aµ− ∂µ(−ia exp(−iqx)

= µexp(−ixq) + aqµexp(−iqx)

= (µ+ aqµ) exp(−iqx).

(2.6)

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10 CHAPTER 2. THE POLARISATION OF W BOSONS In the Coulomb gauge the parameter a is chosen such that the time component of the transformed polarisation 0_µis zero. Therefore, the constraint (2.4) becomes

· q = 0. (2.7) The polarisation has two degrees of freedom, and consequently only two polarisation states. For a particle travelling in the z-direction q = (0, 0, qz), the polarisation can be written as linearly polarised states

µ₁ = (0, 1, 0, 0) and µ₂ = (0, 0, 1, 0), (2.8) or circularly polarised states

µ₋ = √1 2(0, 1, −i, 0) and µ + = − 1 √ 2(0, 1, i, 0), (2.9) which correspond to the helicity eigenstates.

2.1.2 Polarisation states of massive spin 1 particles

The Lagrangian for a spin 1 particle with mass m given by L = −1₄FµνFµν+1₂mBµBµhas free field equation

(∂ν∂ν+ m2)Bµ− ∂µ(∂νBν) = 0, (2.10)

known as the Proca equation [15]. The Lorenz condition is automatically satisfied because acting ∂µ on this

equation

∂µ[(∂ν∂ν+ m2)Bµ− ∂µ(∂νBν)] = 0

∂ν∂ν∂µBµ+ m2∂µBµ− ∂µ∂µ∂νBν = 0

m2∂µBµ= 0.

(2.11)

With the Lorenz condition the Proca equation can be simplified to

(∂ν∂ν+ m2)Bµ= 0, (2.12)

which has the plain wave solutions

Bµ= µ(q) exp(−ixq). (2.13) Application of the Lorenz gauge condition ∂µBµ= 0 on this solution leads to the same constraint as it did for

massless particles

0 = ∂µ(µexp(−iqx)) = µ(∂µexp(−iqx)) = −iµqµexp(−iqx) ⇒ qµµ= 0. (2.14)

There is no freedom for a gauge transformation to reduce the degrees of freedom, because of the mass term. Unlike the free field equation (2.1), the Proca equation (2.12) is not invariant under the Lorenz gauge transfor-mation in equation (2.5). The massive spin 1 particles therefore have three polarisation states, corresponding to their three degrees of freedom. If two states are chosen to be the circular polarisation states

µ− = 1 √ 2(0, 1, −i, 0) and µ + = − 1 √ 2(0, 1, i, 0), (2.15) the third orthogonal state needs to take the form µ_L = (α, 0, 0, β). For a particle traveling in the z-direction qµ= (E, 0, 0, qz) and using the condition (2.14) the third polarisation state is

µ_L= 1 m(qz, 0, 0, E) Em −−−−→ µ_L= q µ m, (2.16) and in the high energy limit E m can be approximated by its momentum qµ.

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2.2. DIVERGENCES IN THE SCATTERING OF POLARISED W BOSONS 11

2.1.3 Projection operator for polarisation states

The transverse and longitudinal polarised parts of the vector bosons can be acquired using projection operators. The transverse part W_Tν of vector boson Wµcan be projected using operator Pµν as

W_Tν = PνµWµ, where Pµ0= P0ν = 0

and Pij = δij −p

i_pj

p2 .

(2.17)

The longitudal part W_Lν is orthogonal to the transverse part and therefore given by

W_Lν = (I − P)µνWν, (2.18)

where I is the identity matrix.

2.2 Divergences in the scattering of polarised W bosons

The scattering of W boson contains several divergences that cancel exactly in the SM. The dominant contribu-tions to the scattering are given by the interaccontribu-tions of longitudinal polarised W bosons exclusively, which are calculated in the first subsection. Separately the scattering of these longitudinal modes through γ, Z and 4 point interactions have a cross section that diverges with the energy to the power four, but when these contributions are summed the cross section is proportional to the energy squared. This divergence is cancelled by the Higgs scattering contribution which is also proportional to the energy. In the last part of this section, the s-channel Higgs contribution is looked into.

2.2.1 Divergences in W_L±W_L∓scattering

If the cross section of W±W∓scattering is calculated at high energies, a divergence is found. (An alternative approach is given in appendix A.) Because the longitudinal contribution is proportional to momentum q as shown in equation (2.16), it dominates the other polarisation states in the high energy limit. Using this ap-proximation, the matrix element may be calculated with the usual Feynman rules [16]. At tree level, the five dominant contributions to W_L±W_L∓scattering are given by

W_L+ W_L− W_L+ W_L− iM4= i g 2 4m4 W s2+ 4st + t2− 4m2 W(s + t) − 8m2 W s ut

Zγ W_L+ W_L− W_L+ W_L− iMZγs = −i g 2 4m4 W s(t − u) − 3m2_W(t − u)

Zγ W_L+ W_L− W_L+ W_L− iMZγ_t = −i_4mg24 W (s − u)t − 3m2_W(s − u) + 8m2W s u 2_,

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12 CHAPTER 2. THE POLARISATION OF W BOSONS where s, t, and u are the Mandelstam variables.

Effectively, the energy proportionality at leading order can be seen as M ∝ LLLL ∝ q4 ∝ s2 [17].

Luckily all s2 contributions of the 4W vertex are cancelled by the Z and γ contributions. What remains is a term asymptotically proportional to s,

iMHiggsless= −i

g2

4m2_Wu + O((E/mW)

0_), _(2.19)

which can still cause unitarity violation. The s and t-channel Higgs contributions are given by

H W_L+ W_L− W_L+ W_L−

H W_L+ W_L− W_L+ W_L− iMHiggs= −i g2 4m2_W (s − 2m2 W)2 s − m2_h + (t − 2m2_W)2 t − m2_h sm2_h −−−−→ i g 2 4m2_Wu, (2.20) where in the last step, the high energy limit of s m2_h, m2_W was taken. The Higgs contribution cancels the remaining divergence from the γ and Z boson exactly.

2.2.2 The W±W∓→ H → W±_W∓_process

In this section a closer look is taken at one of the scattering diagrams from the previous section. The scattering of W bosons with an s-channel Higgs is the dominant contribution in the region around the Higgs mass energy. The cross section of this process is calculated analytically in appendix B.4 to be

σ(W±W∓→ H → W±W∓) = 1 144πs g_w4 (s − m2_H)2_{+ Γ}2 Hm2H 3m2_W − s + s 2 4m4_W 2 . (2.21) This cross section is one of the elements graphed in figure 2.1, in which it can be seen that the cross section for this channel is largest near the Higgs mass of 125 GeV.

In equation (2.21) there is a constant term, a O(s) term and a O(s2) term between the parenthesis. The O(s2_{) term causes the cross section to diverge at high energies, but is cancelled by other W}±_W∓_{→ W}±_W∓

scattering terms, as was explained in section 2.2.1. If W boson fusion is calculated separately from other interfering process, the O(s2) term is not cancelled as is also shown in figure 2.1. Still despite the divergent term not being cancelled, the impact of the high energy contributions is minor compared to the Higgs mass resonance at 125 GeV, that is orders of magnitude larger than any currently attainable high mass divergence.

The third point of figure 2.1 is that the numerical calculation is found to be in excellent agreement with the analytical expression (2.21). In chapter 4 the numerical calculation will be discussed.

2.3 Higgs and polarised W bosons

In this section the polarised W boson-Higgs-interaction and the modification of the connected couplings is dis-cussed. First comments are made on the choice of reference frame, required for all calculations with polarised W bosons. By calculating the matrix element for the HWLWT coupling, it can shown to vanish for a specific

choice of frame. The fact that the mixed coupling HWLWT equals zero, is used to modify the couplings

di-rectly. This causes the cross section to depend on a gauge and a Lorentz frame. In the last part of this section, it is specified how the modification might fit in a complete theory that is Lorentz and gauge invariant.

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2.3. HIGGS AND POLARISED W BOSONS 13 [GeV] s 3 10 104 ] -2 [GeV σ 9 − 10 8 − 10 7 − 10 6 − 10 5 − 10 4 − 10 3 − 10 2 − 10 2 2 W 4m 2 s -s+ 2 W 3m 2 H Γ 2 H +m 2 ) 2 H (s-m 4 H g s π 144 1 = H σ WW simulation → H → Madgraph WW

.

2 -s 2 W 3m 2 H Γ 2 H +m 2 ) 2 H (s-m 4 H g s π 144 1 = H σ

Figure 2.1: The cross section for the process W±W∓ → H → W±_W∓ _{from the analytical expression and}

from a Madgraph calculation. The bottom line is the cross section without the O(s2) term, which is normally fully cancelled by other W boson scattering contributions. The cross section around the Higgs mass peaks outside the frame with a maximum of approximately 5 · 107 GeV−2. The statistical error from the Madgraph simulation is too small to be visible at this scale. The Higgs resonance around 125 GeV is unsuitable for numerical calculations because of the off shell W boson.

H

W+ W−

Figure 2.2: Feynmann diagram for the H →

W−W+_process

H

¯_b

b

Figure 2.3: Feynmann diagram for the H → b¯b process

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14 CHAPTER 2. THE POLARISATION OF W BOSONS

2.3.1 Choice of reference frame

Any calculation explicitly involving different polarisation states must use a specific reference frame, because the W boson polarisation states are not Lorentz invariant. The reference frame is chosen to be the Higgs rest frame, because in this frame the longitudinal and transverse polarisations do not mix.

The matrix element M for the process H → W−W+ shown in figure 2.2 can be written using the Feyn-mann rules as

M = −g_WmWgµνµ∗ν∗. (2.22)

The polarisation states of a single W boson such as given in equation (2.15) and (2.16) are orthogonal. For W bosons moving along the z-axis in the Higgs rest frame, the momentum of the W bosons is equal and opposite, i.e. qW +_z = −q_zW −. The difference between the polarisation states of the two W bosons is for the longitudinal polarisation in (2.16) an inversion of the momentum component, and for the circular polarisation states in (2.15) a change in direction, because helicity is defined as the projection of spin on the momentum. The polarisation states become µ∓(W±) = 1 √ 2(0, 1, −i, 0) µ_±(W±) = −1√ 2(0, 1, i, 0) µ_L(W±) = 1 mW √ 2(±qz, 0, 0, E) (2.23)

for the positively and negatively charged W boson. Since the polarisation states of one W boson are orthogo-nal, the longitudinal and transverse polarisation states of two W bosons moving in opposite direction are still orthogonal. As such, the matrix element in equation (2.22) is non-zero only in the case of two left handed, two right handed or two longitudinally polarised W bosons.

2.3.2 HW+_W− _{coupling strength}

The coupling strength in the centre of mass frame is acquired by explicitly calculating the matrix element in equation (2.22). Substituting the couplings from (2.23) in this equation results in

M−− =M++= −gWmW MLL = gW 2mW (q2_z+ E2) = gW mW (E2−m 2 W 2 ). (2.24)

2.3.3 A simple modification of the coupling strengths

In the following, the coupling strengths of longitudinally and transversely polarised W bosons are modified with respect to the SM through a scaling factor in the Lagrangian [7]. First the interaction term between the Higgs and W field is split into a transverse and longitudinal part by employing the projection operators defined in section 2.1.3. Secondly, reference frame is chosen to be the Higgs rest frame like in section 2.3.1 such that the mixed couplings HWLW+and HWLW−vanish. The couplings are then given by

L_{HW W} = gSMφW W = gSM(φWLWL+ φWTWT) , (2.25)

where W is one of the W boson fields and gSMis the standard model coupling strength. The actual modification

is performed through adding two real factors aLand aT in front of both terms, which changes the coupling of

the two polarisations to

LHW W = gSM(aLφWLWL+ aTφWTWT) . (2.26)

Despite this modification breaking both the SM gauge symmetry and Lorentz invariance, calculations are pos-sible and may give reasonable results. A more careful approach is given in the next section.

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2.4. THE VECTOR BOSON FUSION H → W W∗ → Eν_EµνµPROCESS 15

2.3.4 Modified couplings and effective field theory

Given that a direct modification of the couplings of polarised W bosons breaks the SM gauge symmetry and is not Lorentz invariant, it should not be viewed as an isolated extension of the SM. Rather, it should fit in a more extensive theory. A general approach to such new physics that reduces to the SM in the low energy limit is effective field theory.

The general Lagrangian of such a field theory can be written in terms of a SM contribution and higher order operators [18]. Suppression of the higher order terms at low energy is accomplished by an inverse proportionality to energy scale Λ at which new physics start affecting experimental outcomes. Accordingly, an extension of the 4 dimensional SM Lagrangian L(4)_SM to include 5 and 6 dimensional terms can be made

L = L(4)_SM+ L(5)+ L(6)+ ... = L(4)_SM+ 1 Λ X k C_k(5)Q(5)_k + 1 Λ2 X k C_k(6)Q(6)_k + O( 1 Λ3), (2.27)

where Q(n)_k and C_k(n)are n-dimensional operators and coupling constants respectively. Assuming both charge and parity conservation, the 5-dimensional operators involving gauge bosons are eliminated.

One of the 6-dimensional operators involving W bosons affects the transverse couplings of W boson more than the longitudinal couplings. This operator that changes the ratio between aLand aT, is given by

OW = (Dµφ)†Wˆµν(Dνφ), (2.28)

where Wµνis the field strength tensor

Wµν =

ig 2σ

a_(∂

µWνa− ∂νWµa+ gabcWµbWνc). (2.29)

Another one of the 6-dimensional operators involving Higgs fields can be used to scale all Higgs interactions and is given by

O_φ,2= 1 2∂µ(φ

†_φ)∂µ_(φ†_φ). _(2.30)

Therefore a combination of these two operators can set the longitudinal and transverse coupling parameters [7, 17].

2.4 The vector boson fusion H → W W

∗

→ eν

e

µν

µ

process

The vector boson fusion H → W W∗ channel is closest to a W±W∓ → H → W±W∓ channel that can be found at a hadron collider. The Vector Boson Fusion (VBF) process is the Higgs production though two vector bosons. The feynmann diagram for the VBF H → W W∗ → `ν``¯ν`process is shown in figure 2.4. The

VBF production of Higgs is the second production mode and its decay to two W bosons has the second largest branching fraction. Although W bosons only decay 33% leptonically [19], the channel were both W bosons decay to leptons gives the cleanest signal. Specifically the decay to two differently flavoured leptons (eµ/µe) stands out from other processes, because the equally flavoured (ee/µµ) channel suffers from a large Z/γ decay background. In this thesis the e+µ−combination is specifically chosen in order to save computational resources. The process is investigated at tree level and next to leading order terms are neglected. The analysis can easily be extended to the e−µ+state, which should result in a scaling of all result with a factor 2.

2.4.1 A test of the standard model

The cancelling of divergences in W boson scattering described in section 2.2.1 makes VBF especially interest-ing. As a matter of fact these divergences were one of the primary arguments for the existence of a Higgs boson and the basis behind the reasoning of the no-lose theorem for the LHC [20]. It was reasoned that either a Higgs boson should be found or some other mechanism should stop the W boson scattering from diverging. Since the discovery of the Higgs boson, it is known that the divergence is at least partially cancelled, but preciser measurements can give more secure confirmation. It is still possible that the discovered Higgs boson is only partly responsible for the electroweak symmetry breaking.

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16 CHAPTER 2. THE POLARISATION OF W BOSONS

V V H W+ W− q q j `+ ν` ¯ ν` `− j

Figure 2.4: Feynmann diagram of the vector boson fusion process.

2.4.2 A probe for new physics

Next to SM reasons for interest in this channel, the cross section near the strong resonance of VBF H → W W∗ can contain signs of new physics. Rf the Higgs boson is not exclusively responsible for cancelling the divergence [16], there is new physics connected to electroweak symmetry breaking. Most notably, additional Higgs particles can be responsible for the electroweak symmetry breaking such as predicted by 2 Higgs doublet models, or the the Higgs boson and gauge bosons can be composite particles.

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Chapter 3 The Large Hadron Collider and the ATLAS

detector

ATLAS located at the Large Hadron Collider (LHC) is one of the detectors that aims to measure the signal outlined in the previous section. First a short description of the LHC and some related notions is given, then the the ATLAS detector is described focusing on details relevant to give a prospect in later chapters.

3.1 The Large Hadron Collider

At the moment the LHC is the largest and most powerful particle collider [21]. The LHC is build near the CERN site, where existing infrastructure of Large Electron Proton Collider (LEP) was available. The LHC is designed to study physics at high energies with proton-proton collisions.

3.1.1 Design of the Large Hadron Collider

The LHC has eight straight sections and eight curved sections in order to fit in the former 27 km long LEP tunnel. The LHC accelerates the heavier protons in order to reach energies that were not attainable with elec-trons and posielec-trons at the former LEP. Strong magnets are required to bend high energy protons through a pipe. The LHC is foremost a proton-proton collider and therefore the two counter-rotating beams cannot share the same pipe. Motivated by space requirements and as a cost safer, the LHC has a twin-bore ring design, meaning the two pipes share a common magnet system. Superconducting magnets are used to direct the beam through the ring. 1232 helium cooled dipole magnets produce a magnetic field of 8.33 T bending the two beams. Additionally 392 quadrupole magnets stabilise and focus the beams.

Before protons enter the LHC, they are accelerated by a series of pre-accelerators. Hydrogen atoms from a tank are stripped of their electrons and first accelerated by the linear accelerator LINAC2. Protons supplied by LINAC2 are injected at a energy of 50 MeV into the Proton Synchrotron Booster (PSB), which accelerates the protons to 1.4 GeV. The protons are then further accelerated to 25 GeV and ordered into bunches by the Proton Synchrotron (PS). Before being injected into the LHC ring, the protons are accelerated to 450 GeV by the Super Proton Synchrotron (SPS). In the LHC rings the beams consisting of up to 2808 bunches of 1.15 · 1011protons, can eventually reach their goal energy. The LHC has of yet reached a centre of mass energy of√s = 13 TeV and plans to reach energies of√s = 14 TeV [22].

The LHC ring has 4 interaction points where the beams intersect, each studied by one of the 4 major experiments. ATLAS is one of the two high luminosity experiments, located in a specially dug cavern.

3.1.2 Luminosity and cross section

The cross section of a process σ is discussed in section 1.4 and depends on the energy of the two interacting partons ˆs. In an experiment, the most straightforward observable is the number of detected events n. It is proportional to the cross section which depends on a physics process and on the rate of events per effective area, the luminosity L that is a property of the accelerator and given by

n = Lσ. (3.1)

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18 CHAPTER 3. THE LARGE HADRON COLLIDER AND THE ATLAS DETECTOR In a proton-proton collider where Gaussian beams collide head on, the luminosity is given by

L = N

2 pNbf

4πσT

, (3.2)

where Np is the number of protons per bunch, Nb is the number of bunches per beam, f is the revolution

frequency of the protons, and σT is the transverse beam size [23]. The integrated luminosity is defined as

the total number of events per effective area during the operation time of an experiment, which is the relevant quantity for the total number of detections.

3.1.3 Pileup events

In the collision of two bunches multiple inelastic proton-proton interactions can be detected, the so called pileup events. There is out-of-time pileup and in-time pileup. Out-of-time pileup originates from detections that spill over from neighbouring bunches in the sub-detectors with a slower response time such as the transition radiation tracker. In-time pileup originates from multiple interactions per bunch crossing. The number of interactions per bunch collision follows a Poisson distribution with mean µ and is given by

µ = Lσinelastic nbf

, (3.3)

where σinelastic is the inelastic cross section. As this quantity is defined per bunch crossing, the number of

bunches nband the revolution frequency f are found in the denominator. Pileup events can give rise to unrelated

detections, hampering analysis of a single interaction. If pileup is high, methods to isolate the signal become important. In run 1 of ATLAS the mean interactions per bunch crossing was < µ >= 20.7 [24], while in the 2015 period it is only < µ >= 13.7 [25]. However for run 2 the interaction per beam crossing is expected to peak at values over 50 [26].

3.2 The ATLAS detector

The ATLAS detector capable of performing measurements on many kinds of particles, aims to do both mea-surements of SM physics and explore new physics. In attempting to detect ample particles, the detector has a toroidal design providing full coverage in the azimuthal direction and an good polar (pseudorapidity) coverage range. For the identification of particles, several detector systems are ordered in shells around the interaction point. The ATLAS detector can be divided in a few subdetectors, as is shown in figure 3.1. The interaction point of the two beams is surrounded by the inner detector, which provides track information of all charged particles. Around this the calorimeter system is build, which measures the energy of particles. The outermost subdetector is the muon spectrometer, which measures the tracks and energy of muons. Below an overview is given of the ATLAS subdetectors, which are detailed in reference [27].

3.2.1 The ATLAS coordinate system

Particles in the ATLAS detector are described in a coordinate system adapted for the cylindrical symmetry of the ATLAS detector. ATLAS uses cylindrical right handed coordinate system where the azimuthal angle φ is the angle around beam axis z. Motivated by the cylindrical symmetry of interactions, the transverse momentum pT is the component of the momentum transverse to the beam axis.

Pseudorapidity

In hadron colliders, the longitudinal momentum difference of outgoing particles is dependent on the unknown initial momenta of the colliding particles and is therefore unusable to study the interaction. The Lorentz invari-ant quinvari-antity that is used instead is the rapidity difference.

For a particle with momentum pzalong the z-axis and energy E, the rapidity y is defined as

y = 1 2ln E + pz E − pz . (3.4)

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3.2. THE ATLAS DETECTOR 19

Figure 3.1: Schematic overview of the ATLAS detector. Figure adapted from [28].

For (nearly) massless particles the momentum in the z-direction can be approximated by pz = E cos θ and thus

the rapidity y can be approximated by psuedorapidity η

y ≈ η = − ln(tan(θ/2)), (3.5) where θ is the polar-angle with the z-axis. The relation is shown in figure 3.2.

Distance in a detector

Since the separation in the longitudinal plane of two particles is better measured in psuedorapidity than in polar angle or longitudinal momentum, a new measure of the distance based on the psuedorapidity is defined. The general separation of particles ∆R can be defined as

∆R =p(∆φ)2_{+ (∆η)}2_, _(3.6) [Degree] θ 0 10 20 30 40 50 60 70 80 90 η 0 1 2 3 4 5

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20 CHAPTER 3. THE LARGE HADRON COLLIDER AND THE ATLAS DETECTOR

Figure 3.3: Schematic overview of the ATLAS inner detector. The insertable B layer is not shown on this figure. Figure taken from [29].

where φ is the azimuthal angle and η the psuedorapidity. A surface of constant ∆R with respect to a vector resembles an elliptical cone.

3.2.2 Inner detector

The inner detector reconstructs tracks of charged particles. A superconducting solenoid wraps around the inner detector, providing a 2 T magnetic field which bends the trajectories of the charged particles. The curvature of particle tracks aids identification and allows the momenta of charged particles to be deduced. The inner detector is divided in four systems, each of which has a cylindrical barrel part and a disc shaped end-cap part. The innermost is insertable B layer, surrounded by the pixel detector, the semiconductor tracker and the outermost part of the inner detector, the transition radiation tracker as shown in figure 3.3.

Insertable B layer

The insertable B layer is the innermost tracking detector at a radius of 33 mm and the newest addition to the ATLAS detector [30, 31]. It was installed especially for run 2 of the LHC primarily to improve tracking and b tagging efficiency, and to shield the pixel detector from high luminosities. Space for the insertable B layer was found by utilising a 8.5 mm gap and reducing the beam pipe radius by 4 mm. The subdetector consist of 14 staves around the beam pipe on which 20 silicon detectors are mounted in an angle with respect to the radial direction, overlapping to provide full φ coverage. The area |η| < 2.5 is covered with almost full coverage in the z direction using slim and active edges. Each stave has 12 centre modules containing 2 planar sensors and two sets of 4 edge modules containing a single 3D module. The newer 3D type sensor has higher acceptance due to active edges and can operate at higher temperatures with a lower bias voltage, but both 3D and planar type sensors have the same 50 µm ×250 µm sized pixels.

Pixel Detector

The pixel detector consist of 1744 identical silicon pixel sensors, covering the region |η| < 2.5. The sensors are arranged such that a typical track crosses tree pixel layers. The pixel sensors each containing 47232 pixels sized 50×400 µm2(∼90%) or 50×600 µm2(∼10%), provide a resolution of 10µm in the transverse direction, and 115 µm in the longitudinal direction.

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3.2. THE ATLAS DETECTOR 21

Figure 3.4: Schematic overview of the ATLAS insertable B layer. Figure taken from [30].

Semiconductor tracker

The semiconductor tracker is a third silicon based detector employing micro-strip sensors. The semiconductor tracker has 4088 modules mounted on 4 cylindrical shells and 9 end-cap disks. A module in the barrel region is sized 64×64 mm2 and has 768 pairs of active connected strips on each side that form a 126 mm long unit. The strips on one side of the module form an angle with the other side, hereby allowing to reconstruct the z-coordinate along the strip. The modules in the end-caps are similar, but have a trapezoid shape in order to assemble them in disks. The resolution is determined by the strip pitch, the distance between the centre of the strips, which is 80µm for the barrel modules and ranges between 57 and 90 µm for the end-cap modules. The resolution of the modules is 17 µm in their transverse direction (R in the barrel and φ in the end-caps) and 580 µm in their longitudinal direction (z in the barrel and R in the end-caps).

Transition radiation tracker

The outermost tracking detector is the transition radiation tracker, which is capable to discriminate between charged pions π± and electrons. The tracker is based on transition radiation that is emitted when a charged particle traverses a dielectric material with varying refractive index. In 4 mm wide drift tubes filled with a xenon-based gas, photons and charged particles ionise the gas and free electrons which are attracted to a 31µm cylindrical gold plated tungsten wire anode at the centre of the tube. Using timing on these electrons detections are made with a 130µm precision. Per track 36 detections are typically acquired. The transition radiation tracker is made of a cylindrical barrel with 73 layers of 144 cm long tubes and two disk shaped end-caps with 160 layers of 37 cm long tubes. Together they providing a coverage of |η| < 2.

3.2.3 Calorimeters

The energy of charged and neutral particles is measured in calorimeters, which is a detection system around the inner detector. The calorimeter system is divided in the electromagnetic calorimeter that measures the energy of electrons and photons, and the hadronic calorimeter that measures the energy of the more penetrative hadrons. Additionally, particles are detected before entering the calorimeter by a presampler. A calorimeter consist of alternating layers of absorbing material and active material. The energy is measured through both penetration depth and deposits in the active medium. It is important that most particles and showers lose all their energy in the calorimeters in order to accurately measure energies and to limit the number of particles leaking to the muon spectrometer.

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22 CHAPTER 3. THE LARGE HADRON COLLIDER AND THE ATLAS DETECTOR

Figure 3.5: Schematic overview of the ATLAS calorimeter. Figure taken from [32]. Electromagnetic calorimeter

The energy of electrons and photons is measured in the electromagnetic calorimeter, consisting of interleaved lead absorber plates and liquid Argon detection material arranged in an accordion geometry. The electrons lose their energy from deceleration by nuclei through brehmstrahlung, while high energy photons use their energy through e+e− pair production and through recoil taken by nuclei. The penetration is measured in radiation lengths X0, being the mean free path length of high energy electrons and 7/9th of the mean free path of high

energy photons. The liquid argon between the lead absorbers is ionised by charged particles. The free electrons are moved by an electric field towards the detecting copper electrodes placed at the centre of the liquid argon chambers.

The electromagnetic calorimeter consists of two half barrels each containing 16 modules covering the range |η| < 1.475 and two end-cap calorimeters divided into two discs containing 8 modules each covering 1.375 < |η| < 3.2. The design resolution is σE/E = 10%/pE[GeV] ⊕ 0.7% over the whole |η| < 3.2 range. The

modules all have three detection layers, one strip detection layer providing a precise η measurement with a depth of approximately 4 radiation lengths, a courser middle layer 16 X0 in thickness, and a back layer of

around 2X0 in depth with an even lower resolution. The electromagnetic calorimeter has a depth of at least

22 X0 in the whole range it covers. Hadrons deposit little energy in the electromagnetic calorimeter, having a

nuclear interaction depth of only approximately 1.5. Hadronic calorimeters

The energy of hadrons is measured in the hadronic calorimeter consisting of three subsystems. The tile calorimeter covers the barrel region up to |η| = 1.8, the hadronic end-cap calorimeter covers the range 1.5 < |η| < 3.2, and the forward calorimeter covers psuedorapidites 3.1 < |η| < 4.9. The design resolu-tion is σE/E = 50%/pE[GeV] ⊕ 3% in the region up to |η| < 3.2, and σE/E = 100%/pE[GeV] ⊕ 10% in

the forward region.

The tile calorimeter consist of a barrel and two extended barrels, each subdivided in 64 modules. The absorber material is iron and active components are scintillating tiles. Hadronic showers will cause light to be emitted in the scintillating tiles, which is captured by fibres at the edge of the tiles.

The hadronic end-cap calorimeter is comprised of two wheels per end-cap, having 32 modules each. The hadronic end-cap calorimeter uses copper as the absorbing material and liquid argon as the active material.

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electromag-3.2. THE ATLAS DETECTOR 23

Figure 3.6: Schematic overview of the ATLAS muon spectrometer. Figure taken from [33].

netic and hadronic showers. The forward calorimeter has three discs per end-cap. The innermost disc of each end-cap is optimised for electromagnetic showers and uses copper as the absorbent, while the two outermost discs use tungsten as the stopping material.

3.2.4 Muon spectrometer

Although similar to electrons, the heavier muons are not stopped by the tracking and calorimeter systems of the ATLAS detector, because they experience less acceleration and therefore do not emit as much bremsstrahlung. The calorimeters are found to sufficiently reduce punch-through to the muon spectrometer of particles other than muons. The specialised muon spectrometer system that detects the tracks and energy of muons with pT

between 3 GeV and 3 TeV, consist of fast trigger chambers and precision tracking chambers, see figure 3.6. The design resolution is 10% for 1 TeV muons. Tracking is provided by the monitored drift tubes in both the barrel and end-cap region and by the cathode strip chamber in the inner forward region. In the barrel region resistive plate chambers deliver tracking information, and in the end-caps this is provided by thin gap chambers. The muon spectrometer system features a airtight light and open design in order to minimise multiple scattering effects. The muon spectrometer is built as three cylindrical shells with radii between 5 m and 10 m, and four end-cap wheels located at distances between 7.4 m and 21.5 m from the interaction point. Two end-cap toroid and a barrel toroid produce a magnetic field required for the bending of the high energy muons.

Precision tracking chambers

Precision measurements by monitored drift tube chambers are performed on muons up to |η| < 2.7. Each chamber has three to eight layers of drift tubes, which are 1 to 6 m long, 30 mm wide aluminium pipes filled with an Argon/CO2 gas mixture. A charged particle traversing the tubes ionises the gas causing a charge

deposit on the centrally installed tungsten-rhenium anode wire. Through precise drift time measurements, the tubes achieve an average position resolution of 80 µm.

In the central forward region at psuedorapidity range 2 < |η| < 2.7, cathode strip chambers are used in place of monitored drift tube as the innermost tracking layer, because the counting rate exceeds the operation limit of the monitored drift tubes. The cathode strip chambers are multiwire proportionality chambers, which are chambers being similar to drift tubes but with all wires in the same volume. Readout is not performed on the

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24 CHAPTER 3. THE LARGE HADRON COLLIDER AND THE ATLAS DETECTOR Trigger Level 1 Typical offline selection

pT (GeV) pT (GeV) e 20 25 µ 15 21 e, e 10, 10 15, 15 e, µ 15, 10 19, 15 µ, µ 15 19, 10

Table 3.1: ATLAS level 1 triggers for run 2 [34, 35] on charged lepton transverse momentum pT. Both level

1 triggers are shown and the values used in a typical offline selection. The above value for a single µ assumes isolation, the transverse momentum threshold for non-isolated single µ is 20 GeV at level 1 and 42 GeV in the typical offline selection.

radially oriented wires, but on the walls of the chamber were strips are applied with an alternating transverse and radial orientation. The cathode strip chambers provide a resolution of 40 µm in the bending plane and approximately 5 mm in the transverse direction.

Trigger chambers

In order to be able to trigger on muon tracks, a fast detection system is chosen to complement the preci-sion tracking chambers. Fast track information is delivered by the trigger chambers, which are resistive plate chambers in the barrel region within the range |η| < 1.05, and thin gap chambers in the end-caps up to psue-dorapidities of |η| < 2.4. Resistive plate chambers are made of a parallel anode and a cathode plate separated by 2 mm with a C2H2F4based gas between them. A signal is detected by capacitively coupled metallic strips

mounted on the outside of the chambers. The thin gap chambers are also multiwire proportionality chambers, similar to the cathode strip chambers. These chambers add seven detection layers arranged in two doublet modules and one triplet module. In addition to trigger information, both systems provide a track coordinate in the non-bending plane.

3.2.5 The ATLAS Trigger system

In run 2 of the LHC, collisions in the ATLAS detector occur at a rate of 40 MHz. At approximately 2.4 MB per event, the ATLAS detector produces data at a staggering 96 TB/s. It is technically not feasible to register all events, therefore a trigger system is installed that selects the events of interest while data is being taken [34]. An overview of the rates is given in figure 3.7

The trigger system for run 2 consist of two parts. The level 1 trigger built on custom electronics, makes the first crude selection by only passing events that meet one or more simple threshold values, such as a high energy deposit or a high pT muon track. The relevant triggers for a VBF analysis are the charged lepton triggers, which

are summarised in table 3.1. Using the level 1 triggers, the event rate is reduced to 100 kHz.

A more thorough selection is made in the second step, the level 2 or high level trigger, in which a software analysis similar to offline algorithms can be performed. Selection is based on properties of events making use of calculated quantities such as ∆R separation between tracks. In the end, events are recorded at a rate of about 1 kHz.

3.2.6 Particle identification

Each subdetector has its own role in the identification of particles, see figure 3.8. Photons are detected in the electromagnetic calorimeter. Electrons are detected here as well and also leave a track in the inner detector. Muons are detected in the inner detector and the muon spectrometer. Hadrons are detected in the hadronic calorimeter and if they are charged they also leave a track in the inner detector. They are detected as showers which are usually clustered by the anti-kT clustering algorithm, which is described in section 5.1.1.

Scattering of polarised W bosons Measuring the polarised W bosons - Higgs coupling with ATLAS using charged lepton observables

U

NIVERSITEIT VAN

A

MSTERDAM

MSc Physics

Track: Gravitation and astroparticle physics Amsterdam

Master Thesis

Scattering of polarised W bosons

Measuring the polarised W bosons - Higgs coupling

with ATLAS using charged lepton observables

By

Cornelis Ligtenberg

10447547

Abstract

Contents

Introduction

Chapter 1

The standard model of particle physics

1.1

The particles of the standard model

1.2

Electroweak gauge theory

1.3

Higgs mechanism for electroweak gauge bosons

1.4

Decay rates and cross sections in the standard model

1.5

Quantum chromodynamics

Chapter 2

The polarisation of W bosons

2.1

Polarisation

2.2

Divergences in the scattering of polarised W bosons



2.3

Higgs and polarised W bosons

.





2.4

The vector boson fusion H → W W

→ eν

µν

process



Chapter 3

The Large Hadron Collider and the ATLAS

detector

3.1

The Large Hadron Collider

3.2

The ATLAS detector