Improving Higgs decay width measurement using off-shell interference in gg(→H*)→ZZ→llνν

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MSc Physics and Astronomy

Gravitation, Astro-, and Particle Physics

Master Thesis

Improving Higgs decay width

measurement using off-shell

interference in gg → H

∗

_{→ ZZ → 2l2ν}

by

Bram Hoonhout

10201289

August 2020

60 ECTS

September 2017 – August 2020

Supervisor/Examiner

Dr. I.B. van Vulpen

Daily Supervisor:

Dr. H.L. Snoek

Second examiner:

Prof. Dr. M.P. Decowski

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Abstract

The Higgs decay width ΓH is related to the Higgs boson’s average lifetime,

and is equal to the width of its Breit-Wigner resonance peak located around its mass, mH = 125 GeV. The Standard Model prediction is ΓSMH = 4.07 MeV,

too small to be detected using current resolution (O(1 GeV)). There is how-ever a ΓH dependency present in the (sizable) interference of the gg (→ H∗) →

ZZ → 2l2νhannel: the ratio between the on-shell and (high mass) off-shell cross section. The qq → ZZ → 2l2ν background decreases the significance of the ΓH measurement and was proceeded to be eliminated through use of a

Multilayer Perceptron (MLP) neural network trained on ATLAS simulation data — making use of extensive preliminary toy model analysis regarding Data Selection Optimization and Machine Learning. Data Selection Optimization included exploring how to ensure that a discriminator variable (such as the one provided by the MLP) leaves the optimal amount of signal and background events in the reduced dataset, leading to the appointment of a well-working quantity for rating dataset reductions: Punzi score δp = _e/2+S√_B. Likewise, the

preparatory Machine Learning analysis provided usable information on tun-ing MLP hyperparameters, specifically to the case of an insufficient amount of training data being the limiting factor for event classification accuracy; this turned out to improve gg/q¯q separation, which in turn will allow for a more precise determination of ΓH in the data collected by the ATLAS

detec-tor. MLP employment (AUC: 0.893) on the ATLAS simulation data achieved a 72.67% reduction in q¯q-initiated background whilst preserving 80.75% of gg-initated (signal) events, associated with an approximate increase in ΓH

measurement error of ∼ 1.6 relative to the ΓH uncertainty in case of no

sim-ulations of 0-q¯q; down from a factor of ∼ 2.8 before the cut. Unfortunately, the currently available simulation data did not allow us to provide a quanti-tative expression for the gain in measurement accuracy following this thesis. However, an official ATLAS analysis using the same strategy as presented in this thesis will likely be able to provide a quantitative constraint on ΓH

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“Repugnant is a Creature who would squander the ability

to lift an eye to Heaven, conscious of his fleeting Time here”

— Maynard James Keenan

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List of Figures

2.1 Standard Model summary . . . 3

2.2 Example Feynman diagrams . . . 5

2.3 Possible interaction schemes for the Weak Nuclear Force. . . . 6

2.4 Possible interaction schemes for the Strong Nuclear Force. Gluons are drawn as curly lines. . . 7

2.5 Possible interaction schemes for the Higgs boson . . . . 8

2.6 “Mexican hat potential” for the Higgs field, which causes spontaneous symmetry breaking as soon as the energy density drops below the maxi-mum. This happens already early in the life of the universe; before that all particles were massless. . . 9

2.7 Graph showing the total Integrated Luminosity of the ATLAS detector for 2015-2019. . . 14

2.8 Higgs Branching Ratios / Decay Width, as function of mH . . . 15

2.9 Prominent Higgs production diagrams . . . 17

2.10 Higgs production cross sections, cross section × BR graphs . . . 18

2.11 Feynman diagrams for the main sig/bkg channels that allow for interference 19 2.12 Diagram of the accelerator complex in CERN . . . 24

2.13 Schematic cutaway diagram of the ATLAS detector . . . 24

2.15 Signatures within the detector of different types of particles. . . 27

3.1 Shapes and PDFs of toy models A and B . . . 34

3.2 Shapes and PDFs of toy models C and D . . . 36

3.3 Contour plots of SBR and Q for each model . . . 40

3.4 ROC curves for Q applied on Model B or D . . . 41

3.5 Qcut results for Model A . . . 45

3.6 Surviving S count for Model A after a cut on Q . . . 47

3.7 Survival rate of B for Model A after a cut on Q . . . 47

3.8 δ0 results for cut on Q versus 2 or 3 rectangular cuts on mod. A,B and D 49 3.9 Graphs of cuts on either S or B, and the corresponding δp values . . . . 52

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3.10 Graphical representation of Qcut as function of Nsim . . . 55

3.10 Graphical representation of Qcut as function of Nsim (cont.) . . . 56

3.11 δB/δA as function of N . . . 59

4.1 Schematic of a Boosted Decision Tree . . . 64

4.2 Schematic of the architecture of a MultiLayer Perceptron . . . 66

4.3 Trained classifier of a PDE-Foam implementation on Model A . . . 69

4.4 MLP architecture . . . 72

4.5 MLP response contour plots or different Nevents values . . . 73

4.6 MLP response histogram . . . 74

4.7 Flow chart describing which AUC stems from which dataset . . . 76

4.8 MLP: Histograms of AUCCand AUCT distributions for some (Nevents,ftrain )-combinations . . . 79

4.9 MLP: graphs of AUCC(Nevents; ftrain=0.5) . . . 80

4.10 MLP: all AUCC fits in one graph . . . 81

4.11 MLP: standard deviation of AUCCand AUCTas function of Nevents . . . 83

4.12 MLP: AUC differences histogram and 2D AUC histograms . . . 84

5.1 mZZ T line shape for µr = 1, showcasing interference effects . . . 89

5.2 mZZ T line shape for µr = 1, 2, 4, gg only . . . 90

5.3 Histograms of the (negated) interference contribution for various µr . . . 91

5.4 Histograms of Y distributions for various samples of µr . . . 93

5.5 Graphs showing Y as function of µr, including fits . . . 94

5.6 Contour plot of ∆Y relative to ∆Y . . . 96

5.7 MVA AUC for ATLAS gg(→ H∗_{) → ZZ → 2l2ν} simulation data . . . . 98

5.8 MVA response for ATLAS MC data w/Punzi significance calculation . . 99

5.9 mZZ T histograms before and after MVA value cut . . . 100

5.10 Combined plot linking the cut results to the relative increase in ∆Y due to q¯q . . . 100

A.1 ROC curves for small or large evaluation sets . . . 103

A.3 MLP: graphs of AUCC(Nevents; ftrain∈ {0.10, 0.25, 0.5, 0.75, 0.90}) . . . 103

A.5 MLP: More 2D AUC histograms, different parameter settings. . . 106

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List of Tables

3.1 Toy ModelP DF (x, y) designs for sig and bkg . . . 33 3.2 AUC results for Q . . . 42 4.1 MLP: Nevents and ftrain settings for analysis of AUC distributions . . . . 77

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Chapter 1 Introduction

In 2012, the ATLAS[1] and CMS[4] collaborations independently confirmed the existence of the Higgs boson with a mass of 125 GeV. Based on decay modes known to date, the Standard Model (SM) predicts that this Higgs boson has a specific average lifetime associated to its total decay width: τSM

H = ΓSM_H~ =

6.582 × 10−22_{MeV s}

4.07 MeV = 1.62 × 10 −22_s.

Performing a check on the Standard Model prediction value of ΓSM

H is very important,

as currently unknown decay modes (into beyond-SM particles) would contribute to the total decay width and therefore alter the average lifetime.

ΓSM_H also corresponds the width of the Breit-Wigner resonance peak of the Higgs boson’s invariant mass spectrum, but this peak is about 250 times smaller than the uncertainty with which its mass spectrum can be resolved, which makes it currently impossible to directly measure the decay width. This leads us to explore a different approach of measuring ΓSM

H , which is presented in this thesis in Ch. 2 along with other

relevant theory: ΓSM

H may be constrained by calculating ΓSMH -sensitive interference effects

in the high-mass region of the gg(→ H∗_{) → ZZ → 2l2ν} channel. A negative interference

between gg → H∗ _{→ ZZ → 2l2ν} and gg → ZZ → 2l2ν is theorized, which scales

linearly with the Higgs boson’s total decay width. Making use of ATLAS collision data to measuring the extent of this interference contribution as accurately as possible may lead to further constraining the limits onto ΓSM

H .

The channel of interest also features a q¯q-initiated counterpart which is considered background noise here. This channel will be reduced by making use of a Machine Learning algorithm trained to distinguish this channel from the gg-initiated channels. After the theory section, an introduction to how to properly reduce the dataset is given in Ch. 3, followed by a Machine Learning chapter in Ch. 4. We then apply all previously gathered knowledge onto ATLAS simulation data in Ch. 5, using a neural network to attempt to improve the accuracy with which ΓH is measured.

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Chapter 2 Measuring the Higgs boson’s lifetime

with the ATLAS detector

In this chapter we will give a brief introduction into the Standard Model of Particle Physics in §2.1, where we will focus on Higgs physics. In section §2.2 we will introduce a method for measuring the Higgs lifetime. We will introduce the Large Hadron Collider and ATLAS experiment in §2.3.

2.1 The Standard Model

The Standard Model (SM) is an elegant framework of physical theories that describes all known elementary particles and the interactions between them through fundamental forces. A particle is deemed ‘elementary’ when it cannot be subdivided into smaller par-ticles, unlike a composite particle such as a proton. This implies that no sub-structure can be discerned within an elementary particle even when you probe it using the great-est energies (and therefore smallgreat-est wavelengths) currently available, so that elementary particles can essentially be considered point-like.

The elementary particles can divided into fermions and bosons based on their “spin quantum number” or simply spin: fermions have a half-integer-spin (1⁄₂, 3⁄₂, ...) and

bosons have an integer-spin (0, 1, 2, ...). Based on this crucial division, the particles will be explained in the following subsections. For review of what’s about to follow, keep an eye on Fig.2.1 as it summarizes the different kinds of particles and forces.

After introducing fermions, bosons and the Higgs boson in particular, I will continue with information about the interpretation of Feynman diagrams, quantum transition rate and the calculation of particle production (∼ cross section).

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R/G/B2⁄₃ 1/2 2.2MeV up

u

R/G/B-1_⁄₃ 1/2 4.8MeV down

d

-1 1/2 511keV electron

e

–

1/2 <2eV eneutrino

ν

_e

R/G/B2⁄₃ 1/2 1.28GeV charm

c

R/G/B-1_⁄₃ 1/2 95MeV strange

s

-1 1/2 105.7MeV muon

µ

–

1/2 <190keV µneutrino

ν

_µ

R/G/B2⁄₃ 1/2 173GeV top

t

R/G/B-1_⁄₃ 1/2 4.7GeV bottom

b

-1 1/2 1.777GeV tau

τ

–

1/2 <18.2MeV τneutrino

ν

_τ

±1 1 80.4GeV

W

± 1 91.2GeV

Z

1 photon

γ

2× R/G/B 1 gluon

g

0 125.1GeV Higgs

H

graviton Strong N uclear F orce (Color ) Electromagne tic F orce (Char ge ) W eak N uclear F orce (Weak Isospin / Fla v or ) Gr a vit ational F orce (Mass ) charge color mass spin couples to 6 q uark s (+6 anti-q uark s) 6 lep tons (+6 anti-lep tons) 12 fermions (+12 anti-fermions)

Mass increases with generation →

5 bosons (+1 opposite charge W )

Standard Matter Exotic Matter Force Carriers

EWSB

provides Mass

Outside of the Standard Model

1

st

2

nd

3

_{rd · · ·} _generation

Figure 2.1: A diagram showing an overview the particle contents of the Standard

Model, grouped by the coupling to the four fundamental forces. Spin, charge, and color are indicated in the corners, as well as their masses.

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2.1.1 Fermions: matter

All matter is composed of fermions, which can be further subdivided into two elementary types: leptons and quarks. A quark comes with a color, that can either be Red, Blue, Green or an ‘anti’ version of R/G/B. Quarks never appear alone, but only in combinations (hadrons) that together are ‘neutral’ in color — meaning duos (mesons) of a color and the same anticolor or triplets (baryons) with one of each (anti)color. The lightest quarks are the up (u) and down (d) quarks, with respective electric charges of +2/3 or −1/3. A common example of stable combinations of up and down quarks are the proton (uud) or neutron (udd). Most other combinations are very short-lived.

The most familiar lepton is the electron (e−), with an electric charge of -1. The

electron is associated with another lepton: the electron neutrino (νe), a very light particle

with no charge. Whereas quarks have a color, leptons on the other hand, do not.

Three generations

So far, four particle flavours have been discussed: two quark (u, d) flavours and two lepton (e, νe) flavours. These particles are known as the first generation, while there are two

generations (or ‘families’) besides the first, so that there are a total of six quark flavours and six lepton flavours. Each next generation contains particles that are very similar but have increasingly greater masses; except for the neutrinos which barely have any mass in any generation. Whereas electrons have a mass of about 0.5 MeV, its generation-II sibling

muon (µ) has a mass of 105.7 MeV and their big generation-III brother tauon (τ) weighs

1777MeV – heavier than a proton (∼ 938 MeV). These heavier fermions are unstable, as only first-generation fermions seem to be present in the majority of visible nature. This is why the second/third-generation fermions are called ‘exotic’, since they only appear on earth through cosmic rays or in particle collider experiments.

The second and third generation quarks most similar to u (up-type quarks) are the

charm and the top (c, t) quarks, respectively; for down-type quarks there’s the strange (s)

and the bottom (b) quarks. Refer back to Fig. 2.1 to review the properties of all leptons and quarks.

Antiparticles

Besides coming in three generations, the SM fermions each have an antiparticle with the same mass and opposite charge. These particles can be considered normal particles travelling back in time, as per a symmetry of nature. An antiquark or antineutrino is denoted with with a bar, as follows: ¯u, ¯νµ — anti-up, anti-muon neutrino. Charged

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antileptons can be written with a flipped charge sign, for example the anti-electron also known as ‘positron’: e+.

2.1.2 Bosons: force-carriers

As mentioned before, all particles have various properties such as spin, mass, (electric) charge, and color. Each fundamental force is mediated through specific force-carrying bosons that couple to one of such properties. Currently, three out of four fundamental forces are represented in the SM, and will be discussed below along with their mathe-matical representation — described as symmetries of the following group:

SU (3)C ⊗ SU (2)L⊗ U (1)Y

The missing force is gravity, of which a fitting description has yet to be made, where the theorized graviton should connect particles by acting on a particle’s mass.

2.1.2.1 Electromagnetic force

The electromagnetic (EM) force acts on the electric charge of a particle, and is mediated through the electrically neutral photon (γ). Because photons act on the electric charge, neutral particles therefore cannot interact with photons. Photons couple to two of the same charged particles in a three-way crossing with a coupling constant of √αem, with

αem ' ₁₃₇1 . γ √ αEM Q Q

(a) Possible EM interactions. Here, Q has to have an

elec-tric charge. Fermions are

in-dicated with solid lines, bosons with nonzero spin (such as pho-tons) are drawn as wavy lines.

γ

e−

e+ _µ+

µ−

(b) Feynman diagram showing an electron and a positron (initial state) annihilating to create a

pho-ton which subsequently decays into a µ−_µ+ pair

(fi-nal state). Two instances of Fig. 2.2a have been stitched together in order to create this diagram. Note that electric charge is conserved. See §2.1.4 for more information.

Figure 2.2: On the left there is a diagram showing an elementary vertex of the possible

interactions for a photon with two of the same charged particles Q that have to have an electric charge. Such interactions are time-independent and thus can be rotated or flipped, and are used as ‘building blocks’ of physical processes of which an example is given on the right.

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Mathematically, photons arise from requiring that charged leptons are invariant after a position-dependent complex phase is added to the wave function. This can be summarized as a transformation of the U(1)EM group, which has only one generator: the photon field,

which arises naturally by adjusting the model to make it symmetrical under this U(1) transformation. The possible interactions the photon has with other particles (Fig. 2.2a) can then be read off the renewed Lagrangian. The photon has to be massless in order to preserve the symmetry.

2.1.2.2 Weak Nuclear force — Electroweak symmetry

The weak nuclear force is mediated by the W+, W−, and Z bosons. The (charged) W

bosons act on the flavor of a particle are so able to change the flavour of a fermion: be-tween up-type quark ↔ down-type quark, or charged lepton ↔ corresponding neutrino. When a quark undergoes a flavour-change through the weak interaction, the resulting new quark will likely belong to the same family as the initial quark, however due to ‘quark mixing’ sometimes quarks of adjacent generations can emerge, or in even more rare cases jumps across two generations (I ↔ III) occur.

Z √

αW

f f

(a) f can be any SM fermion

W+

D− 1/3

U3/2

(b) U can be any up-type quark (u, c, t), D can be any down-type quark (d, s, b)

W−

νl

l−

(c) l− can any charged

lep-ton (e, µ, τ), and νl the

cor-responding neutrino.

γ/Z

W+

W−

(d) Photons and Z boson couplings to W bosons

W+ _γ/Z

W−

γ/Z

(e) L being one of the SM leptons

Figure 2.3: Possible interaction schemes for the Weak Nuclear Force.

One can combine the EM and Weak nuclear force into one electroweak theory so that both are described together following the symmetries of SU(2)L⊗ U (1)Y, where Y is the

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looked up in literature such as the Particle Data Group’s “Review of Particle Physics”. Because the weak nuclear force is effectuated through short-lived massive particles, its reach is rather short, typically at the length scale of an atomic nucleus.

The possible interactions are shown in Fig. 2.3. The weak interaction is considerably weaker than the electromagnetic of strong nuclear interaction, with a coupling constant of αW = 10−6 ∼ 10−7.

Because neutrinos can only interact through the weak interaction, any interaction involving neutrinos is substantially less likely than interactions through other forces, and causes neutrinos to only rarely interact with matter so that they usually simply pass on through matter.

2.1.2.3 Strong Nuclear force

The strong nuclear force is what binds quarks together into hadrons, making use of gluons that act on the color charge contained by quarks or other gluons. There are eight different gluons, corresponding to the generators of SU(3)C where C stands for ‘colour charge’. The

associated coupling “constant” is relatively large with a value of αs' 0.12 at the energy

scale of the mass of a Z boson (91 GeV), and decreases further as the energy transferred by the quark increases (which is why calling it a constant is not truly justified). The possible elementary interactions are given in Fig. 2.4. Quarks bound by the strong nuclear force gain energy as the distance between them increases, until it reaches a point where the binding energy surpasses the rest mass energy of a quark-antiquark pair. Creating such quarks out of the vacuum at some point becomes energetically favourable, and leaves two bound quark pair of which each new quark is bound to one of the original quarks. This mechanism ensures that quarks cannot exist in an unbound state, as pulling them away from each other creates new ones in between.

g √

αs

q q

(a) q can be any quark

g g g (b) Gluon self-coupling g g g g

(c) Four-way gluon self in-teraction

Figure 2.4: Possible interaction schemes for the Strong Nuclear Force. Gluons are drawn

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2.1.3 The Higgs Boson — The Higgs mechanism

So far none of the forces gave an explanation for any of the elementary particles having a mass. This had been an unsolved problem for decades, while the existence of a particle that would give particles their mass has been theorized as early as in the ’60s by Peter Higgs[9], François Englert and Robert Brout[7]: the Higgs boson. Finally, around 2012, the Higgs boson was conclusively discovered through two independent efforts: the AT-LAS[1] and CMS[4] collaborations, with detectors located in the Large Hadron Collider. Its spin and electric charge are both 0, making it a scalar boson, and has as mass of 125 GeV. The allowed elementary Higgs interactions are given in Fig. 2.5.

H

l l

(a) Higgs coupling to

charged leptons

H

q q

(b) Higgs coupling to quarks

H W− W+ (c) Higgs coupling to W bosons H Z Z (d) Higgs coupling to Z bosons H H H

(e) Higgs self-coupling

W+

H W−

H

(f) Quartic (4-way) Higgs interaction with W bosons

H

Z Z

H

(g) Quartic Higgs interac-tion with Z bosons

H

H H

H

(h) Quartic Higgs self-interaction

Figure 2.5: Possible

interac-tion schemes for the Higgs

bo-son. The coupling strength in

three-way diagrams scales learly with the mass of the in-volved particles. The Higgs is a scalar boson (spin-0) and is therefore indicated with a dashed line.

The mathematical description of the Higgs boson arises through the so-called Higgs

mechanism by ‘spontaneously breaking’ the symmetry of the Higgs field that is

introduc-ing as a Mexican hat potential as visualized in Fig. 2.6. Any particle can only find itself in the center at high energies only present right after the Big Bang. So instead, particles will roll down the hill in any direction and will so be located in the ground state which

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Figure 2.6: “Mexican hat potential” for

the Higgs field, which causes spontaneous symmetry breaking as soon as the energy density drops below the maximum. This happens already early in the life of the universe; before that all particles were massless.

has a radial symmetry. This is called spontaneous symmetry breaking since the laws of nature are still symmetric, yet the system is forced into one of the many possible ground states. The spontaneous symmetry breaking grants mass to the W and Z bosons, but photons remain massless.

2.1.4 Physical interpretation of Feynman Diagrams — Matrix

Element

To better comprehend the processes of particle physics, the notation through Feynman diagrams is very useful. This paragraph describes the link between diagrams and how to interpret them physically. Furthermore, the transition rate is discussed, which introduces theory that leads to interference.

A particle physics process starts with a set of particles called the initial state and ends with another set of particles that shall be called the final state. The ‘final state’ does not imply that its particles are not subject to decay or that they may never interact again, but is only called as such to mark off a process so that we are able to do calculations about it up to and including the final state.

Such processes are often represented using Feynman diagrams, which are created us-ing elementary interactions given throughout §2.1.2 (such as Fig. 2.3) as ‘buildus-ing blocks’ by connecting multiple instances of these interactions to each other, following strict con-servation rules. Typically time flows from left to right, and antiparticles are understood to behave like particles going back in time in — so in Fig. 2.2b, the positron (e+) going

forward in time (to the right) is identical to an electron travelling backwards against the flow of time (to the left).

The initial state of a process is comprised of the incoming particles on the left side of the diagram, and the final state the outgoing particles on the right. What happens in between initial and final state determines how likely the process is compared to other processes, as each vertex (∼ junction) has an associated coupling strength. Taking into

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account all vertices in the process allows calcultation of the transition rate. A process of which the product of vertex couplings is weak (small number) results in the interaction being rare relative to processes with a stronger (greater) coupling.

The diagrams that use the least possible amount of vertices to get from an initial state to the final state are called “leading-order diagrams” (LO), so that “next-to-leading order” diagrams (NLO) would then be the diagrams that have the second-to-minimum number of vertices, and so on. Because every added vertex adds more coupling factors that are smaller than 1, diagrams will have smaller total coupling the higher the order of the diagram, despite that there are more NLO diagrams compared to lower orders.

Transition Rate – Quantum Interference

To derive the physical probability of a process to occur, one uses Fermi’s Golden Rule and calculates the transition rate of moving from an initial (i) to a final (f) quantum state: |Mi→f|2; which is the absolute square of the complex matrix element: Mi→f. The

matrix element is proportional to the sum all possible Feynman diagrams that connect initial state to final state — including higher-order diagrams. Next-order diagrams will contribute less and less than lower-order diagrams, due to the product of coupling fac-tors becoming smaller as more vertices are added. Because the sum of all diagrams is squared, quantum interference terms arise in the transition rate sum, which is an impor-tant quantum-mechanical implication that means, in a way, that a process can happen via two ways simultaneously.

It is exactly these interference terms that will play an important role in this research, which focuses on Higgs bosons decaying into two Z bosons which in turn decay into two leptons and two neutrinos: H → ZZ → 2l2ν. The initial state here are two partons, one from each proton, which somehow (§2.2.1) produce a Higgs boson that undergoes the specified process, until the final state of this process is reached: two leptons and two neutrinos (2l2ν or llνν).

2.1.5 Particle Lifetime, Decay Width, Branching Ratio

This section covers how particles decay, and in case of multiple decay options, how the options are weighed relative to each other, and how this relates a particle’s decay width to its lifetime. In the section that follows, the creation of the particles is discussed, since particles have to be created before they can decay.

After any unstable particle has been created (which will be covered in §2.1.6), it will at one point decay into other particles. A decaying particle only has a limited number of possibilities it may to decay into (decay modes), all of which are given by the very same three-way interactions described by the Standard Model. The coupling strengths of the

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possible decay modes dictate how likely a decay mode is relatively to other decay modes, but also how strongly a particle is urged to decay in general — ultimately determining how short-lived a particle is. Both these tendencies are described through the decay width Γ:

Γ(initial → f inal) ∝ |M(initial → f inal)|2

A relevant example for this thesis would be the case of a Higgs boson decaying decaying into two Z bosons:

Γ(H → ZZ) ∝ |M(H → ZZ)|2

In this case, Γ represents the partial decay width for a Higgs boson splitting into two Z bosons. For there are also other decay modes possible for the Higgs boson as given by Fig. 2.5a–e, and the sum of all partial decay widths yields the total decay width:

Γ(H) = Γ(H → ZZ) + Γ(H → e−e+) + Γ(H → µ−µ+) + Γ(H → tt) + ...

Finally, the ratio of a particular partial decay width (of a single decay mode) to the total decay width then describes the branching ratio (BR):

BR(H → ZZ) = Γ(H → ZZ)

Γ(H)

This value then tells you the odds of any created Higgs boson decaying into two Z bosons. A particle’s total decay width is linked to its mean lifetime τ as follows:

Γ ≡ ~/τ (2.1)

Here, “mean lifetime” implies the time it takes to reduce a number of particles to 1/e of the starting amount. Calculating the mean lifetime allows one to retrieve the total decay width, which is predicted by the Standard Model by summing all partial decay widths that follow from all currently known matrix elements (read: decay mode diagrams). Therefore, the calculation of the decay width acts as a check on the validity of the Standard Model, as any deviations could, for example, mean that some decay channels (into currently unknown particles) are missing from the total decay width sum. The coupling factors of the Higgs boson scale with the mass of the Higgs boson.

2.1.5.1 Invariant Mass, Resonance, Breit-Wigner

The invariant mass of a’single particle or system of particles is the inproduct of the sum of particle four-vectors (E, ~p) with itself. The result is invariant under Lorentz boosts and thus identical in every frame of reference; essentially it is the rest mass of the particle:

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Which in natural units (c = ~ = 1) looks like: m2₀ = E2− ||~p||2

To find out the invariant mass of a system of multiple particles, the energies and momenta of the decay products are summed:

M2 =XEi 2 − X ~ pi 2

The invariant mass is preserved during decays, allowing for the calculation the invariant mass of initial and intermediate particles by measuring the invariant mass of final state particles. Note that invariant mass for multiple particles is denoted as ‘M’ (capital M), distinct from lowercase m which stands for the (rest) mass of a particle.

When two particles fuse into an intermediate particle, say, a Higgs boson, this new Higgs boson lives only for a short time before decaying. In this case the Higgs boson is massive, so one can measure the invariant mass of its decay products. The measured invariant mass is (as the name suggests) invariant before, during, and after an interaction. This means that the invariant mass of the decay products will be identical to the invariant mass of the intermediate particle (here: Higgs boson) and identical to the invariant mass of its parental particles that initially fused together (here: two gluons or quarks, see Fig. 2.9).

This invariant mass should preferably be about equal to the intermediate particle’s mass (mass of the Higgs boson), but because of Heisenberg’s uncertainty principle and the short-livedness of the Higgs boson, it does not exactly have to be equal! Though, the more a particle’s invariant mass deviates from its actual mass, the more the production of the particle carrying that specific invariant mass is suppressed. The result would be that in a histogram of the invariant mass of a temporary massive particle, there is a peak around the particle’s mass. More particles are produced in that range because of the relative ease with which nature produces particles that have masses close to their actual mass. This phenomenon is called resonance, and the peak is shaped following the

Breit-Wigner distribution:

N ∝ f (E|M, Γ) = k

(E2_{− M}2₎2_{+ M}2_Γ2 (2.2)

with proportionality factor k = 2√2M Γγ

πpM2_+γ which makes use of γ = pM

2_(M2_{+ Γ}2₎. This

distribution has two shape parameters: the mass M of the particle which determines the position of the peak on the invariant mass spectrum; and the total decay width Γ which governs the width of the peak. Particles whose invariant mass lies reasonably close to their own mass are called on-shell, whereas particles with invariant masses differing

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greatly from the particle’s mass are deemed off-shell. Off-shell particles are noted with a ‘∗’ to their symbol, e.g. an off-shell Higgs boson would be H∗.

If an off-shell particle has a mass that is sufficiently far away from its true mass, |E − M | will be large which means the denominator of Eq. 2.2 will be dominated by its first term: (E2_{− M}2₎2. The contribution to the cross section of these particles can

therefore be reduced to:

Noff-shell ∝ 1

(E2_{− M}2₎2 (2.3)

And for the on-shell particles, the mass difference is small compared to the decay width: N_on-shell ∝ 1

M2_Γ2 (2.4)

Summarized: After creating a histogram of the invariant mass of a unstable particle,

one can see a resonance around the particle’s mass following the Breit-Wigner distribution of which the width is directly related to the average lifetime of the particle, which in turn is based on the particle’s total decay width, which can be predicted by summing all known decay mode diagrams.

2.1.6 Particle production in collision experiments

Accelerators such as the LHC (§2.3) speed up particles (in this case: protons) to extreme energy levels, and then let them smash into each other at one of the designated interaction points where opposing beams cross. The high energies allows the creation of new particles, which typically stay alive for a very short while before decaying into more stable particles. Eventually, particles will reach the parts of the detector that measure their energies by absorbing them, so that the physical processes in between production and decay can be analysed.

In §2.1.2 we covered which elementary interactions (Feynman diagram junctions) exist, which would then in §2.1.5 be applied to unstable particles to calculate the lifetime or total decay width of a particle. Such unstable particles need to be created first, and the physics of creating a particle are identical to those of the decay — the same ‘Feynman diagram building blocks’ are in use. The difference with particle production is that one does not start with a given unstable particle, but some other (usually stable) particles that are on a collision course — there is no guarantee that the particles will even collide. The calculation will shift more towards answering the question “How many X particles have been created in the collision experiment so far?” which could later be combined with the Branching Ratio of the decay channel of interest to end up with the number of expected particles. The answer to this question is calculated as follows:

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Figure 2.7: Graph showing the total In-tegrated Luminosity of the ATLAS detector for 2015-2019.

Here, L is the luminosity, a measure of how many high-energy particles (here: protons) the collider brings sufficiently close together each second, so that they may interact. This value depends on the accelerator and beam properties, and is mostly used in the form of (time-) integrated luminosity that builds up over time as the collider is in use: R L. Fig. 2.7 shows the integrated luminosity attained by the ATLAS detector (§2.3), which for the end of ‘Run II’ around the start of 2019 means that there’s a whopping 139 fb−1

worth of usable collision data at our disposal.

To get to the number of created particles of interest, this integrated luminosity is complemented by the cross section (σ). σ is a quantity expressed in units of “barns” that correspond to 1 × 10−28_m2, or more commonly picobarns (pb). Since N is dimensionless,

luminosity has units of pb−1_s−1. The value depends on the Feynman diagrams associated

with producing such a particle, as well as the collision energy (√s).

The dependence on collision energy is due in part to the fact that protons do not collide with each other as a whole, but rather only one parton (quark or gluon) of each proton undergoes the interaction. The partons that interact carry only a fraction of the energy of their parental proton, and how this energy fraction is distributed depends on the beam energy. Refer to Fig. 2.9e for a graph depicting the cross section of the Higgs boson at different center-of-mass energies. The total cross section is the topmost band and is comprised of many contributing processes below it, which will be discussed in §2.2.1.

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2.2 Measuring Higgs Lifetime

Applying the theory of the previous sections onto the Higgs boson in particular, we can calculate the SM prediction of its lifetime. For this we need all currently known decay diagrams of the Higgs contained the Standard Model, and the associated coupling strengths of each diagram. The relevant decay diagrams are already given by Fig.2.5a–e, some of which represent multiple analogous decay modes; for example, since there are three different charged leptons, Fig.2.5a can be expanded into a diagram for each of them, with a different coupling and partial decay width. So for a Higgs mass of 125 GeV, when the partial decay width of each possible decay diagram instance are summed up, we find:

ΓSM_H = 4.088+0.73%_−0.73%(theo)+0.99%_−0.98%(mq)+0.61%_−0.63%(αs) MeV [6]

Through Eq.2.1 (p.11) this value can be expressed as an average lifetime of τ_HSM= ~ ΓSM_H = 6.582 × 10−22MeV s 4.07 MeV = 1.62 × 10 −22 s

For other Higgs masses, this value changes as the coupling factors also change, ulti-mately influencing the longevity of the Higgs. The evolution of ΓH as function of mH is

shown in Fig. 2.8c. Following §2.1.5.1, finding a value of 4 MeV for ΓH means that the

[GeV]

H

M

80 100 120 140 160 180 200

Higgs BR + Total Uncert

-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 Zγ WW ZZ

(a) Graph of SM Higgs BR as

function of mH. These

val-ues, combined with the coupling strength of each decay mode, de-termine the (theoretical) value for

ΓH, which is given in Fig. 2.8c.

[GeV] H M 100 150 200 250 Branching Ratios -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 1 LHC HIGGS XS WG 2010 µ l = e, τ ν , µ ν , e ν = ν -e + e -e + e -µ + µ -e + e -l + l -l + l e ν -e e ν + e µ ν -µ e ν + e ν -l ν + l γ γ b b -τ + τ -µ + µ

(b) SM Higgs BR also including some second-level decays, as

func-tion of mH. Second level means

that the first decay products (such as ZZ or W W ) also decay, into the final states shown in this

graph. For mH = 125, BR(H → l−l+_{νν) = 1.055 × 10}−2_.[6] [GeV] H M 100 200 300 1000 [GeV]_H G -2 10 -1 10 1 10 2 10 3 10 LHC HIGGS XS WG 2010 500

(c) Plot of the total decay width of

the Higgs boson (ΓH) as function

of the mass of the Higgs boson. A

Higgs mass of mH = 125 GeV

cor-responds to a total decay width of

ΓH = 4.07 MeV.

Figure 2.8: Higgs Branching Ratios (BR, Fig. a+b) and decay width (ΓH, Fig. c) as

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width of the Higgs boson’s resonance peak in the invariant mass histogram will have a width of 4 MeV.

The Higgs boson lifetime is one of the fundamental properties of the particle. The prediction by the Standard Model is extremely accurate. However this prediction obvi-ously relies on the knowledge of all possible particle interactions the Higgs boson has. If other (beyond the SM) particles exist that interact with the Higgs boson, the lifetime of the Higgs boson may be substantially different from the SM predictions. Hence, the measurement of the Higgs boson lifetime is highly valuable.

The predicted Higgs boson decay width is rather small compared to the O(1) GeV resolution of the ATLAS detector — the uncertainties cause the resonance peak to appear ∼ 250× wider than the actual width of the resonance. Obviously this means that the width of the peak cannot be read off the invariant mass histogram directly until the resolution increases sufficiently. However, the resolution will never reach the required levels using the ATLAS detector LHC; constructing a new and even larger collider ring with a specialized detector would be necessary to achieve sufficient resolution, which would take few decades. For this reason, a different approach is taken in this research to allow for analysis now: examining the effects of quantum interference associated with the production of the Higgs boson in the high mass tail. To get started, first the decay channel of interest and the relevant diagrams for Higgs production at the LHC need to be covered.

2.2.1 The Channel of Interest: gg → H

∗

_{→ ZZ → 2l2ν}

The channel of interest in this thesis is a Higgs boson that decays into a Z boson pair, of which one splits into a lepton-antilepton pair consisting of either e or µ leptons, and the other decays into a neutrino-antineutrino pair. This is an interesting decay since it has a reasonable cross-section and it is relatively easy to reconstruct with the ATLAS detector. Since the charged leptons originate from the same Z boson, they should have the same flavour and have an opposite sign: e−_e+ or µ−_µ+; we require that these are not

τ leptons, as electrons and muons can be measured more accurately. For the flavour of the neutrinos the same flavour-conservation law applies. This means that the flavour is the same amongst the two neutrinos, yet does not have to be the same flavour as that of the leptons, although the flavour of the neutrinos cannot be determined in this decay channel. Some branching ratios relevant to this decay channel, for mH = 125 GeV: [6]

• BR(H → ZZ) = 2.619 × 10−2

• BR(H → ZZ → l−_l+_{νν | l ∈ {e, µ}) = 1.055 × 10}−2_{± 2.18%}

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t t t g g H

(a) ggF: “Gluon-gluon Fusion” gg → H, creating a Higgs boson from two gluons through a top-quark (or bottom-top-quark) loop.

W/Z

q q0/ q

H

q _q0_{/ q}

(b) VBF: “Vector Boson Fu-sion”, q¯q → q¯qH. The vector bosons V ∈ {W/Z} need to be of the same type, the incoming quarks do not. W/Z q q0/ q H W/Z (c) VH: q¯q → V H, describes two processes as V ∈ {W/Z}. t t g _t H g _t (d) ttH: gg → t¯tH. Analogously to this dia-gram, the top quarks can be replaced with bottom quarks. Higgs Production Process σ (fb) ggF 43.92 V BF 3.748 V H 2.250 ,→ W H 1.380 ,→ ZH 0.8696 qqH 1.0201 ,→ ttH 0.5085 ,→ bbH 0.5116

(e) Table of Higgs produc-tion cross secproduc-tions for vari-ous processes at a center of mass energy of 13 TeV and a

Higgs mass of 125 GeV. At

NNLO QCD (strong interac-tion) and NLO EW. The to-tal production cross section for Higgs is a little over 40 fb.

Figure 2.9: The most

prominent Feynman dia-grams for Higgs boson production at the LHC. The cross sections of these processes are tabulated in (2.9e). In Fig. b&c, q0 w.r.t to q indicates a transition from up-type to down-type or vice versa, which is only applicable if the vector bosons are as-sumed to be W (as Z does not change quark flavour.

• BR(H → ZZ → l− ala+l − b l + b | l ∈ {e, µ}) = 1.240 × 10 −4_{± 2.18%}

This last process was another candidate for this analysis because it has similar proper-ties and a cleaner final state, but is disfavoured because its branching fraction is about 200 times smaller than the H → ZZ → 2l2ν channel’s.

There are multiple ways of producing a Higgs boson in pp (proton-proton) collisions, of which the ones with the main contributing diagrams are shown in Fig. 2.9, includ-ing a table with the associated cross sections of these diagrams for √s = 13 TeV and mH = 125 GeV. Additionally, the cross section dependence on collision energy of several

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[TeV] s 6 7 8 9 10 11 12 13 14 15 H+X) [pb] ® (pp s 2 − 10 1 − 10 1 10 2 10 M(H)= 125 GeV LHC HIGGS XS WG 2016

H (N3LO QCD + NLO EW)

®

pp

qqH (NNLO QCD + NLO EW)

®

pp

WH (NNLO QCD + NLO EW)

®

pp

ZH (NNLO QCD + NLO EW)

®

pp

ttH (NLO QCD + NLO EW)

® pp bbH (NNLO QCD in 5FS, NLO QCD in 4FS) ® pp tH (NLO QCD, t-ch + s-ch) ® pp

(a) A graph of the total Higgs cross section (σ) and six contributing channels (excluding ggF ),

as function of collision energy √s. Tab. 2.9e

shows a tabulated version of the contributions

at √s = 13 TeV. The graph features 1-sigma

undertainty bands. [GeV] H M 100 150 200 250 BR [pb] × σ -4 10 -3 10 -2 10 -1 10 1 10 LHC HIGGS XS WG 2012 = 8TeV s µ l = e, τ ν , µ ν , e ν = ν q = udscb b b ν ± l → WH b b -l + l → ZH b ttb → ttH -τ + τ → VBF H -τ + τ γ γ q q ν ± l → WW ν -l ν + l → WW q q -l + l → ZZ ν ν -l + l → ZZ -l + l -l + l → ZZ

(b) This graph displays the Higgs cross sec-tion combined with the branching ratio (BR), which shows the effective cross section of pro-ducing a Higgs boson that goes down a specific decay path, two levels down (e.g. H → ZZ → Combined with the luminosity provided by the detector, these values here can be used to de-termine the number of Higgs bosons Important here is the teal-colored line showing the cross section of pp → H → ZZ → 2l2ν, which is the channel this research is focused on.

Figure 2.10: Higgs production Cross Section (σ) and σ× BR graphs. The latter also

takes into account the probability of creating a Higgs boson in first place, in this case it has been evaluated at a lower collision energy than the current LHC collision energy. H events would look like this:

Npp→H→ZZ→2l2ν =

Z

L dt · σ(pp → H) ·BR(H → ZZ → 2l2ν) · hefficiency factorsi (2.6) If one considers the main production channel, ggF , the combined Feynman diagram up to and including the creation of the Z bosons will be the one given in Fig. 2.11a. Since the masses of the two Z bosons exceed the nominal rest mass of a Higgs boson, one of these three bosons needs to be produced off-shell. As the Zs are taken to be on-shell, this means that the produced Higgs boson must therefore be off-shell, so that the full ggF signal process is denoted as:

gg → H∗ → ZZ → 2l2ν (2.7)

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H∗ t, b g _Z Z g (a) Signal: ggF → H∗_{→ ZZ} q q q q g _Z g _Z (b) Signal ‘gg’: gg → ZZ q q _Z q Z (c) Background ‘q¯q’: qq → ZZ

Figure 2.11: Feynman diagrams of the main signal and background channels for this

analysis (leading-order). The first two allow for interference effects to arise, since they have identical initial and final state.

Neutrinos: Missing energy

In the chosen channel, one of the two Z bosons decays into a neutrino and an antineutrino with the same flavour. Neutrinos cannot be detected by the ATLAS detector, and will therefore need to be reconstructed using the missing transverse energy, Emiss

T . The missing

transverse energy is based on the fact that energy is preserved in the transverse plane, as incoming protons have a zero net transverse energy before colliding. The momentum in the z-direction (along with the beam) is not zero by default, since the interacting partons each carry a fraction of their parental proton’s momentum and there is an infinitesimal chance of these momenta being identical.

Very similar to Emiss

T is the ‘missing transverse momentum’. This is based on the

same concept of conservation in the transverse plane, but with momentum adding up to zero instead of energy. The main difference lies in how the momenta and energies are calculated: The missing energy is based on summing all calorimeter measurements but lacks information about particles that do not fully deposit their energies, whilst the missing momentum arises from a vector sum of particles of which the trajectory can be reconstructed:

pmiss_T = −X

i

(~pi)T

Neutral particles that do deposit their energies in calorimeters (e.g. some hadrons) will not be able to be tracked this way. Like the associated ATLAS study, this research makes use of Emiss

T rather than the missing momentum — yet future studies will likely be using

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Background Channels

Two kinds of background are prominent in the ZZ → 2l2ν channel, respectively gg-initiated and q¯q-gg-initiated, which will be referred to as ‘gg’ and ‘q¯q’ respectively. Their diagrams are shown in Fig. 2.11b&c, and the relative ratio between these diagrams is roughly a factor 10 in favour of q¯q.[6]

2.2.2 Interference Contribution

As mentioned in §2.1.4, in the calculation of the transition from initial state to final state all relevant diagrams are summed and subsequently squared — diagrams for H as well as gg processes. The resulting sum will then contain pure H or gg contributions, but also cross-terms describing processes that are both H and gg simultaneously. The results of these interference terms are typically observed as a destructive contribution to the distribution of the Higgs boson’s invariant mass spectrum. These effects will be most prominent in the off-shell region at an invariant mass range considerably larger than the mass of the Higgs boson.

The interference occurs between processes of identical initial and final states, so when considering the largest contributing factors of H and gg, resp. Fig. 2.11a&b, the interfer-ence can be pictured in the following illuminating way:

Mgg(→H)→ZZ 2 ∝    H ₊    2 (2.8) =    H    2 | {z } Pure H contribution +       2 | {z } Pure gg contribution + 2    H          | {z } H×ggInterference term

As you can see, the transition amplitude contains pure H and gg contributions, as well as an interference term. The latter is crucial for this research. To describe the set of gg initiated processes that includes both H and gg as well as the interference, the following notations shall be used:

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The on-shell region is defined around mH = 125 GeV by a margin of 10 GeV to each

side. This margin is rather safe, as it corresponds to over 3000 times the width of the resonance peak (∼4 MeV). The off-shell region is thus marked off by (MH∗ =)M_ZZ >

140 GeV, but defining more regions can be of use, such as the High Mass (HM) off-shell region: > 200 GeV. The High Mass range will be compared to the on-shell region during the analysis.

The difference between cross sections of Higgs bosons produced on- or off-shell can be used as an entry point to constraining ΓSM

H , because the on-shell contributions are

depen-dent on ΓH as opposed to the off-shell contributions. If we rewrite respectively Eq. 2.2,

2.3 and 2.4 (p. 12) in terms of cross sections and the invariant mass of the ZZ bosons MZZ, we get: dσpp→H→ZZ dM2 ZZ ∼ g 2 Hgg· gHZZ2 (M2 ZZ − m2H) + m2HΓ2H (2.9) dσoff-shell_pp→H→ZZ dM2 ZZ ∼ g 2 Hgg· g 2 HZZ (M2 ZZ − m2H) (2.10) dσon-shell_pp→H→ZZ dM2 ZZ ∼ g 2 Hgg· gHZZ2 m2 HΓ2H (2.11) With the g-factors being the coupling constants for production/decay of SM Higgs bosons in this channel.1 _{We find that the off-shell cross-section has no dependence of Γ}

H.[3] At

this point it is unknown whether the values of g are different for on- or off-shell Higgs bosons, so these factors are be replaced with (potentially) shell-dependent κ, defined as a ratio relative to SM couplings. To this end we define:

µ_off-shell(ˆs) ≡ σ gg→H∗→ZZ off-shell (ˆs) σ_{off-shell, SM}gg→H∗→ZZ(ˆs) = κ 2 g,off-shell(ˆs) · κ2Z,off-shell(ˆs) (2.12)

where ˆs is the collision energy, and the κ’s describe coupling factors that belong to the production (gg → H∗) and decay (H∗ → ZZ) elements that make up the process. In

accordance with the diagrams of “Equation” 2.8, the interference terms should evidently be proportional to one factor of the coupling constants of each interfering component (opposed to the square of the factors one would get for a pure contribution), so that the

interference contribution in the off-shell region will be proportional to:

gHgg· gHZZ|_off-shell∝

√

µ_off-shell = κ_g,off-shell· κ_Z,off-shell (2.13)

1_{These g-factors may be increased by a factor ‘ξ’ pertaining to the uncertainty of the cross section of}

(QCD) processes, but [3] shows how these ξ factors can be absorbed by the (well-known) on-shell cross section through some mathematical trickery, thus eliminating them from the off-shell definition so that they may safely be ignored in this research.

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For the on-shell production, ΓH still plays a role, specifically: µon-shell(ˆs) ≡ σ gg→H∗→ZZ on-shell (ˆs) σgg→H_{on-shell, SM}∗→ZZ(ˆs) = κ2_g,on-shell(ˆs) · κ2_Z,on-shell(ˆs) ΓH/ΓSMH (2.14) This is the reason that in the analysis described in this thesis separate mass regions have been defined, as the differences in measured cross sections between these regions are used to constrain the Higgs decay width. The ratio between µoff-shell and µon-shell, here called

µr, thus relate to the Higgs decay width as follows:

µr(ˆs) = µoff-shell(ˆs) µ_on-shell(ˆs) = κ2_g,off-shell(ˆs) · κ2_Z,off-shell(ˆs) κ2 g,on-shell(ˆs) · κ2Z,on-shell(ˆs) · ΓH ΓSM_H (2.15)

Note that this works only if the couplings are assumed identical for both on- and off-shell production, or to the least that the on-shell couplings are not larger than the off-shell couplings since unknown decay modes can only increase the decay width, ΓH ≥ ΓSMH :

κ2_g,on-shell(ˆs) · κ2_Z,on-shell(ˆs) ≤ κ2_g,off-shell(ˆs) · κ2_Z,off-shell(ˆs) (2.16) This means that it is interesting to define an upper limit to ΓH. When the on- and

off-shell couplings are regarded as (identical) constants, Eq. 2.15 becomes µr =

ΓH

ΓSM_H or ΓH = µr· Γ

SM

H (2.17)

i.e. it is a dimensionless scale factor of the actual (or simulated) ΓH value with respect

to the Standard Model value for ΓH. This expression is useful when considering models

where ΓH has been magnified by a certain factor, as µr can directly assume the role of

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2.3 Large Hadron Collider

The Large Hadron Collider (LHC) in Geneva is an accelerator ring with a circumference of 26.7 km, and is the largest ring of the CERN’s accelerator complex which features many smaller linear or ring-shaped accelerators (see Fig. 2.12). Located in a tunnel previously occupied by the Large Electron Positron Collider (LEP) about 100 m below the surface, it is shielded from a large fraction of the cosmic radiation that would have been present at ground level.

The LHC features eight straight sections and eight curved sections, and contains two beam pipes — one for each direction. Protons (or sometimes ions) are brought up to speed by a sequence of the smaller accelerators, each one further accelerating the particles and passing them to larger rings, until the particles are finally injected into the LHC in both directions, at an energy of 0.45 TeV. The particles entering the ring are grouped in bunches consisting of up to 1011protons, spaced 25 ns apart. These bunches are bent in the curved

sections using superconducting magnets cooled with liquid helium, and are accelerated to the desired beam energy by superconducting radio frequency cavities employed in some points along the LHC. The LHC was designed to accelerate protons up to a center of mass energy of √s = 14TeV and reach a peak luminosity of L = 1034cm−2s−1 (more explanation on ‘luminosity’ follows). The current data-taking session (‘Run II’) runs at an energy which is slightly under the design limit: √s = 13TeV — which makes it the most powerful particle collider of the world to date. A collision energy of 13 TeV means that the protons carry an energy of 6.5 TeV in both directions.

Four of the straight sections of the LHC function as ‘interaction points’, where the beams cross each other and collisions are made possible. Around each of these interac-tion points detectors have been built into huge caverns in order to analyse the particles produced by collisions. The four primary detectors found at the interaction points are: ATLAS, CMS, LHCb and ALICE. The ATLAS detector will be discussed in detail in the next subsection.

2.3.1 The ATLAS experiment

The ATLAS[8] experiment (“A Toroidal LHC ApparatuS”) is a general purpose detector built around one of eight interaction points within the LHC, where opposing particle beams cross. It is cylindrical in shape, with a longitudinal axis measuring 44 m coinciding with the beam pipe and a diameter of 25 m. The detector has a forward-backward symmetry and is designed to have a close to full solid angle coverage around the interaction point, so that it may measure as many particles arising from collision events as possible. A cutaway diagram of the detector is visible in Fig. 2.13.

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Figure 2.12: Diagram of the accelerator complex in CERN, showing the LHC and the

accelerators leading up to it, as well as the location of several experiments along the LHC.

Figure 2.13: Schematic cutaway diagram of the ATLAS detector, displaying a variety

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2.3.1.1 Coordinate system

Measurements done by the ATLAS detector will make use of a suitable coordinate system to represent the data. The following list explains the coordinates and useful derived quantities, with the origin placed at the Interaction Point:

x: The positive x-axis points to the center of the LHC ring from the interaction point. y: The positive y-axis points upward, towards the Earth surface (since the

LHC/AT-LAS are underground).

z: The z-axis is defined as being along the beam pipe. The detector can be divided as having two endcaps called ‘A’ and ‘C’, and a barrel part called ‘B’. The A-side is used as having positive z values, which from a top-down view (such as Fig. 2.12) correspond to following the beam pipe in counterclockwise fashion.

θ: The angle away from the (positive) z-axis.

R: The distance away from the z-axis. Not commonly used to describe collisions, but may be used to specify detector components.

φ: The direction of a particle in the transverse plane, the x − y plane perpendicular to the beam, ranging from 0 to 2π. A value of 0 points coincides with the positive y-axis towards the center of the LHC ring, andπ⁄₂ points straight up.

pz: Momentum of a particle in the z-direction.

pT: Transverse momentum of a particle, i.e. momentum in the (φ-)plane perpendicular

to the beam. While the protons of the beam carry the same energy, the internal particles (partons) do not. This means that momentum along the z-axis virtually never sums up to a net value of exactly zero. However, the partons do have an initial transverse momentum of zero, and conservation of momentum dictates that the net transverse momentum of the final particles should also be zero. This concept can be exploited in order to reconstruct neutrinos which normally escape undetected. η: The pseudorapidity of a particle, η = − ln(tan(θ_/₂₎₎. Another definition, where θ

isn’t necessary directly is:

η = 1 2ln

E + pz

E − pz

This quantity is introduced because particles are often created in forward or back-ward direction, with angles close to the beam pipe. Using θ, this would lead the majority of particles having angles close to 0 or π, while η is Lorentz invariant in

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(a) Pseudorapidity η val-ues for some valval-ues of θ. A value of η = ±∞ coincides with the beam pipe. The sign of η matches that of z.

(b) Schematic cutaway diagram of the ATLAS Inner Detector (ID).

the z-direction so that particles particles are produced in the same rate for each equally sized slice of η. Examine Fig. 2.14a to get a feel for some values of η. ∆R: A quantity describing the angular separation of two particles, defined as ∆R ≡

p∆η2_{+ ∆φ}2. In some cases this quantity turns out to be a handy tool in telling

several processes apart.

2.3.1.2 Inner Detector

The inner part of the detector (“Inner Detector”, ID, see Ḟig. 2.14b) is 6.2 m long and has a diameter of 2.1 m. Its purpose is to track charged particles with extreme precision by using a silicon pixel tracker, silicon microstrip tracker and transition radiation tracker. The ATLAS ID has an very fine granularity, which is necessary because every bunch crossing yields about 1 000 newly created particles in the pseudorapidity range of |η| < 2.5, and those particles need to be distinguished from each other. The inner detector covers a range of |η| < 2.5, and is enveloped by a solenoid magnet providing a 2 T magnetic field. The magnetic field deflects eletrically charged particles into a helical, curved trajectory and enables the measurement of particle momenta and charge sign (+ or −).

2.3.1.3 Outer Detector

Moving more outwards, two layers of calorimeters follow which respectively measure the energy of electromagnetic and hadronic particles. For a full energy measurement, particles will need to deposit all of their energy into their corresponding calorimeter and so come to

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Figure 2.15: Signatures within the detector of different types of particles.

a complete standstill. The calorimeters therefore have different designs to accommodate for capturing different types of particles.

This concept can be more clear through Fig. 2.15, which shows the signature of a variety of particle types passing through the detector, especially how deeply a particle penetrates the detector outwards. Different particle types behave differently in each part of the detector, based on the particle’s characteristics. The Inner Detector only concerns with tracking the particles and is therefore constructed with the idea of minimal energy deposition in this part of the detector, so that the energy measurements of the outer detector can be made more precise.

The Electromagnetic Calorimeter (ECAL) captures and measures electromagnetic particles such as photons and electrons. Despite that protons or other charged hadrons do carry an electric charge, they are too massive to be stopped by this calorimeter and will therefore only leave a track. These particles will instead be measured by the Hadronic

Calorimeter (HCAL). Neutrons will pass on through the EM calorimeter without leaving

a track since they are neutral in terms of electric charge.

Jets are cones of hadronic particles that arises when quarks or gluons are created

during the collision. These will be measured by the hadronic calorimeter and all the jet’s comprising particles will grouped together so to be able to reconstruct the energy of the first quark, and sometimes even that quark’s flavour.

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subdetectors in the endcap that further provide a coverage of 3.1 < |η| < 4.9.

2.3.1.4 Muon system

The outermost layer consists of three layers of muon chambers, which track muons that usually pass through all previous layers — unlike their lighter electron relatives. Unique to this detector is that this section contains eight superconducting toroidal magnets, besides two smaller end-cap magnets. Together these provide a magnetic field in a different orientation than that of the inner detector, so that the momentum measurement of the muons (µ±) is made even more precise due to the muons undergoing a second magnetic

deflection. The toroidal magnets provide a magnetic field that covers a range of |η| < 1.4 and the end-cap magnets cover 1.6 < |η| < 2.7, while the gap in between will have a combination these two magnetic fields.

2.3.1.5 Neutrino reconstruction

Neutrinos will generally escape the detector without being directly registered by any part of the detector, and will therefore be reconstructed by calculating the “missing transverse energy” (Emiss

T ) or “missing transverse momentum” (pmissT ). As conservation

of energy(/momentum) in the direction perpendicular to the particle beam was zero beforehand, and should be zero after a collision as well. Summing the transverse momenta of all reconstructed particles could leave a gap, indicating that (one or more) neutrinos have escaped with a transverse momentum that complements Emiss

T . Refer to p.19 for a

bit more information about missing energy/momentum and the difference between these.

2.3.1.6 Trigger system

During Run II, the LHC has reached a peak luminosity of nearly 2 × 1034_cm−2_s−1, and

generates about 34 collisions per bunch crossing which occur at 25 ns intervals. Given that a single event is recorded using ∼ 1 MB of data, the huge flow of output data will need to be filtered as it cannot possibly all be stored. A large fraction of the data does not contain interesting physics and should be discarded right away. To accomplish this, two successive layers of triggers are used: ultra-fast Level-1 triggers and the slower Level-2 or “High Level Trigger” (HLT) system.

Level-1 (L1) triggers are specialized pieces of hardware that are designed to pass through only interesting collision events within a few milliseconds, based on hundreds of criteria calorimeter measurements need to fulfill. It needs to be as fast as possible to keep up with the production rate of the data. Effectively, the L1 triggers narrow the output rate of events down to about 100 kHZ by rejecting away events that (at first glance) do not contain interesting physics.

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After online preselection by the L1 triggers, data is sent to the HLTs based in a server cluster located further away from the detector and the accompanying radiation. These triggers do a more complex and rigorous analysis compared to the L1 system, and so attempt to reconstruct the travel path of all particles partaking in the collision event. The HLT system further reduces the output rate to about 1 kHz.

Improving Higgs decay width measurement using off-shell interference in gg(→H*)→ZZ→llνν

MSc Physics and Astronomy

Gravitation, Astro-, and Particle Physics

Master Thesis

Improving Higgs decay width

measurement using off-shell

interference in gg → H

∗

→ ZZ → 2l2ν

by

Bram Hoonhout

10201289

August 2020

60 ECTS

September 2017 – August 2020

Abstract

“Repugnant is a Creature who would squander the ability

to lift an eye to Heaven, conscious of his fleeting Time here”

Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Measuring the Higgs boson’s lifetime

with the ATLAS detector

2.1 The Standard Model

u

d

e

–

ν

e

c

s

µ

–

ν

µ

t

b

τ

–

ν

τ

W

Z

γ

g

H

1

2

3

2.1.1 Fermions: matter

2.1.2 Bosons: force-carriers

2.1.3 The Higgs Boson — The Higgs mechanism

2.1.4 Physical interpretation of Feynman Diagrams — Matrix

Element

2.1.5 Particle Lifetime, Decay Width, Branching Ratio

2.1.6 Particle production in collision experiments

2.2 Measuring Higgs Lifetime

2.2.1 The Channel of Interest: gg → H

→ ZZ → 2l2ν

2.2.2 Interference Contribution

2.3 Large Hadron Collider

2.3.1 The ATLAS experiment

_{→ ZZ → 2l2ν}

_e

_µ

_τ

_{→ ZZ → 2l2ν}