A Method to Determine the Lifetime of the Higgs Boson Based on Monte Carlo Simulations of the gg → H → ZZ → llll Decay Channels

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THE LIFETIME OF THE HIGGS BOSON

A Method to Determine the Lifetime of the Higgs Boson

Based on Monte Carlo Simulations of the

gg → H → ZZ → `

+

`

−

`` Decay Channels

Report Bachelor Project Physics and Astronomy, size 15 EC conducted between 21 - 04 - 2020 and 14 - 08 - 2020

National Institute for Subatomic Physics University of Amsterdam

Written By

Martijn van Hamersveld

Student nr. 11892374

Supervised By

Ass. Prof. dr. H.L. Snoek

Second Examiner

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Summary

Determining whether the lifetime of the Higgs boson agrees with the lifetime predicted by the Standard Model could give us more insight into the nature of Dark Matter and beyond the Standard Model physics. The goal of this project was to find a way to determine the Higgs lifetime by using a Monte Carlo generated dataset of the gg → H → ZZ → `+`−`+`− decay channel. To do this a new variable R was defined which is the ratio between the number of events in the off- and on-shell regions of the invariant mass distribution. This ratio R is linearly proportional to the Higgs width which itself is inversely proportional to the Higgs lifetime. This ratio R was then fitted to a dataset with known values of the Higgs width Γ_H giving the following relation R = 0.111(4)Γ_H + 1.061(13). This relation was then used to calculate ΓH of a different dataset giving ΓH = (2.90 ± 0.07)ΓSM_H . This was later compared

to the actual value of Γ_H = 2.8 ΓSM_H .

A similar analysis was done for the gg → H → ZZ → `+`−ν`ν` decay channel because of

its comparatively higher branching ratio. This higher branching ratio would result into more detected events and therefore more significant results when doing an analysis such as in this project experimentally. For this decay channel the ratio between the number of events in the off- and on-shell regions of the transverse mass distribution was used. The relation between this ratio and ΓH can then again be used to calculate the Higgs width of a dataset for this

decay channel.

Nederlandse Samenvatting

De Higgs boson is een belangrijk elementair deeltje dat een van de bouwstenen van ons universum is. Het bepalen van de levensduur van de Higgs boson is belangrijk voor het ontdekken van nieuwe natuurkunde zoals als de oorsprong van donkere materie of mogelijke nieuwe elementaire deeltjes. Zelfs de grootste deeltjesversneller van CERN in Genève is echter niet instaat om de levensduur van de Higgs boson direct te bepalen. Het doel van dit project was om een indirecte manier te vinden waarop deze levensduur toch bepaald zou kunnen worden. Dit werd gedaan door gesimuleerde data te gebruiken met verschillende Higgs levensduren die al wel bekend waren. Uit deze analyse volgde en relatie tussen de levensduur en een nieuwe variabele die gebruikt kan worden om de levensduur te bepalen voor andere datasets. Deze methode is getest op een dataset met een nog onbekende levensduur en lijkt redelijk goed te werken.

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1 Introduction

The discovery of the Higgs boson in 2012 by the ATLAS [1] and CMS [2] collaboration was very significant for our understanding of the universe. However after more than 8 years after the announcement of the discovery there is still much that remains to be learned about the properties of the Higgs boson. One of the questions that remains is what the lifetime of the Higgs boson is. Even though the Large Hadron Collider is the largest particle accelerator in the world its detectors do not have a high enough resolution to directly measure the lifetime of the Higgs boson. This is unfortunate because determining whether the lifetime agrees with the lifetime predicted by the Standard Model of 1.56 × 10−22s [3] could give us more insight into the nature of Dark Matter and beyond the Standard Model physics.

The goal of this project is to develop a different way of determining the lifetime of the Higgs boson based on Monte Carlo simulations of particle decay processes. Chapter 2 will give a general background of the LHC and the ATLAS & CMS experiments and how Higgs bosons are produced there. Chapter 3 will give more information about the importance of the Higgs boson and its lifetime. Here I will also analyse a Monte Carlo dataset and use it to develop a method to determine the lifetime from the gg → H → ZZ → `+`−`+`− decay channel. In chapter 4 I will do a similar analysis for the gg → H → ZZ → `+`−ν`ν` channel.

Chapter 5 will discuss the effectiveness of this method and compare my results with the actual value of the lifetime of the dataset. Finally chapter 6 will give a conclusion.

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2 The Large Hadron Collider

The Large Hadron Collider (LHC) is the most powerful particle accelerator in the world. It was built between 1998 and 2008 by CERN in collaboration with over 100 countries. It is a 27 km long accelerator with a beam energy of up to 6.5 TeV and thus a collision energy of 13 TeV [4]. To detect the results of these collisions there are four main detectors spread out along the LHC which are: ATLAS, CMS, ALICE and LHCb. The LHC is capable of accelerating a variety of charged particles like electrons and protons up to speeds close to the speed of light. It is most famous for detecting the Higgs boson in 2012 which was the last fundamental particle predicted by the Standard Model that remained to be detected experimentally.

2.1 ATLAS

ATLAS (A Toroidal LHC Apparatus) is the largest particle detector at the LHC. ATLAS is designed to be able to detect the high energy collisions and the massive particles that result from it. But it is also able to measure the energy, momentum and charge of a very broad range of signals. From this other properties like the mass, spin and lifetime of the detected particles can be reconstructed. The detector consists of many layers each designed to detect different kinds of particles. The four main parts are the inner detector, the calorimeters, the muon system and the magnet system [5].

The inner detector is closest to the beam and has the purpose of tracking the charged particles and measuring their momenta by tracking the curvature of the path as a result of the magnetic field in the detector. With the direction of the curve indicating the charge and the degree of curvature indicating the momentum of the particle.

The calorimeters are the next layer and their purpose is to measure the energy from the particles by absorbing them. The particle shower that results from this is an indication of the energy of the original particle that was absorbed.

The muon spectrometer is a relatively large part of the detector that is designed for detecting and tracking the muons. It has to be rather large because the muons are much harder to effect with the magnetic field and they easily pass through the two inner parts.

Lastly the magnet system uses superconducting magnets to bend the charged particles to measure their momenta. Particles with high momentum are curved less relative to particles with less momentum.

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Figure 1: Schematic of the ATLAS detector illustrating its main components [5].

Figure 2: Schematic cross-section of the CMS detector showing its main components and how the different particles interact with these components [6].

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2.2 CMS

The CMS (Compact Muon Solenoid) is the other of the two large detectors at the LHC. The main goals of the CMS detector are: to explore particle physics in the high energy scale, to detect and study the Higgs boson and to look for beyond the Standard Model physics [7]. It also has the purpose of complementing the ATLAS detector to give more certainty and weight to their findings. This is mainly because in the field of particle physics very many measurements (particle collisions) are being done so there must be a high level of certainty (5σ) to be able to for example claim the discovery of a new particle. Having two detectors with the same goal without them discussing their findings makes sure that they are able to do unbiased independent research that could then later be combined to confirm a discovery without them influencing each other unintentionally.

The CMS detector itself is made up of many layers each with a different purpose just like ATLAS, as seen in figures 1 & 2. The main parts are the inner tracker which measures the momentum of charged particles. The calorimeters which measure the energies of electrons, photons and hadrons. The magnet system which measures the momenta by curving the paths of the particles. And lastly the muon detector. In general a very similar design to the ATLAS detector which allows for this complementary research.

2.3 The luminosity of ATLAS

A way to characterize the performance of these particle detectors is to look at the luminosity (L) of the detector. The luminosity is defined as the ratio between the number of events that are detected (N ) in a certain timeframe (t) and the cross-section of the interaction (σ)

L = 1 σ

dN

dt . (1)

The luminosity relates the number of particle collisions to the interaction cross-section, which is a factor in determining the likelihood of a collision. Related to this is the integrated luminosity

Lint=

Z

Ldt (2)

which is a measurement of the amount of collected data and is an important value to char-acterize the performance of an accelerator and its detectors [8]. Multiplying this integrated luminosity by the cross-section of the events that are of interest gives the amount of events of interest that a detector is expected to detect [9]. Figure 3 shows this total integrated luminosity of ATLAS over time. It is split up into three different categories. First the total luminosity that is delivered by the LHC (green) which is the luminosity delivered that could

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Figure 3: Total integrated luminosity of ATLAS over time. Separated in what is delivered by the LHC, what is recorded by ATLAS and what part of this data can be used for physics research [10].

potentially be used. The part of that luminosity that is recorded by ATLAS is shown in yel-low. This is less than what is delivered by the LHC because ATLAS does not have a perfect efficiency in part due to data acquisition inefficiency and inefficiencies in the turning on of parts of the pixel system. Not all the data that is recorded can be used however. The part that is of good enough quality to be used for physics research is shown in blue.

2.4 Higgs boson production and decay

A Higgs boson can be produced in many different ways at the LHC, the most important of which are shown in figure 4. Of these the main processes are (a): ggF , (b): V BF , (c): qq and (f): tt/bbH [11]. Production channels with less interaction vertices are generally more likely to occur, depending also the type of interaction and their associated coupling constants.

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Figure 4: Feynman diagrams for the main Higgs boson production channels at the LHC [12].

A more precise description of the ratio of production channels is shown in figure 5. Here the cross-section of the Higgs production channels as predicted by the SM are plotted as a function of the Higgs mass.

In a similar way the cross-section of the decay channels of the Higgs boson can be plotted as a function of the the Higgs mass as seen in figure 6. For this project the ZZ → `+`−`+`− and ZZ → `+`−ν`ν` decay channels are most relevant with the latter being more prominent.

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Figure 5: Higgs boson production cross-sections at the LHC (√s = 14 TeV) for the most relevant production mechanisms as a function of the Higgs boson mass [13].

Figure 6: The SM Higgs-boson production cross-sections multiplied by decay branching ratios in pp collisions at √s = 8 TeV as a function of Higgs-boson mass [3]. For this project the ZZ → `+`−`+`− and ZZ → `+`−ν`ν` decay channels are most relevant with the later being

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3 Measuring the lifetime of the Higgs boson

Now that we know more about the LHC and how it produces Higgs bosons we can look deeper into how we can determine the Higgs lifetime from the data that we gather from the LHC. However I will first get a bit deeper into why the Higgs boson and its lifetime are so important to determine in the first place.

3.1 The importance of the Higgs boson and its lifetime

The Higgs boson was the last ‘piece’ of the Standard Model to be discovered experimentally. Discovering it gives a great confirmation of the validity of one of the best and most fundamen-tal theories of physics. The discovery of the Higgs boson and its properties was also hoped to be a good starting off point for finding new beyond the Standard Model physics, being the last predicted fundamental particle to be confirmed experimentally. The Higgs field of which the Higgs boson is the carrier also has the important role of explaining why particles have mass. Because the Higgs bosons helps explain how particles obtain mass it would also make sense that it plays an important role in explaining the nature of dark matter, the dominant form of matter in the universe. If the Higgs does couple to dark matter this would mean the Higgs boson has decay channels or interactions that are not anticipated by the SM and therefore its lifetime would be shorter than predicted by the SM [14][15]. This makes the lifetime of the Higgs boson very interesting to study further.

3.2 The resonance width

Because the lifetime of the Higgs boson is so short it is not possible to just measure it by somehow observing a Higgs boson and seeing how long it takes before it decays. The way to actually find it is to measure the width (Γ) of the resonance peak of the Higgs boson and calculate the lifetime (τ ) using the following equation

Γ = ~

τ. (3)

The derivation of this relation and the definition of the resonance is as follows. Resonance is a phenomenon associated with particles that decay very quickly, a lifetime in the order of about 10−23 seconds or less. The way we observe these resonances is by a peak in the scattering cross-section of a particle as a function of its invariant mass. One way to explain these peaks is to say that these peaks are the evidence for the actual short-lived particles that from in the intermediate steps of a particle collision [16]. The formation of these intermediate particles contributes to the cross-section of the particles in the collision, making it more likely

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at the energy level where these intermediate particles form. With these resonances we can determine the lifetime of a particle using the uncertainty principle for energy and time

∆E ∆t ≥ ~

2. (4)

By defining the width of the resonance as Γ we can rewrite this into an equation that relates this width Γ and the lifetime of the particle that we are ultimately looking for, resulting in equation 3. This also means most of the analysis will actually focus on the width and not on the lifetime of the Higgs boson as that is the quantity that we will actually see in the data.

3.3 The Breit-Wigner distribution

The Breit-Wigner distribution is a continuous probability distribution that is often used to model these resonances in high-energy physics. It is given by the following probability density function

f (E) = k (m2_4l− m2

H)2+ m2HΓ2

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where k is a constant of proportionality, equal to k = 2

√ 2mHΓφ π q m2 H+φ with φ = q m2 H(m2H + Γ2)

[17]. The width Γ is the full width at half the maximum value of the resonance peak. m_H is the mass of the resonance which in this case is the invariant mass of the Higgs boson. m4l is

the center of mass energy that produces the resonance which in this case will be the invariant mass of the four leptons that are the final state of the decay channel that will be focused on in this thesis.

3.4 The invariant mass

The invariant mass or rest mass is the amount of mass an object has in its own rest frame. As the name implies the invariant mass is an invariant quantity meaning that it is the same for all observers in all reference frames. This is in contrast to the relativistic mass which is dependent on the velocity of the observer and thus not invariant. Via the mass-energy equivalence the invariant mass can be defined as follows

m2₀c2 = E c

2

− ||p||2 ₍₆₎

with m₀ the invariant mass, E the energy of the particle and p the momentum of the particle. One useful property of the invariant mass is that it is conserved even after the decay of a particle. Therefore the total invariant mass of all the final state particles is equal to the invariant mass of the initial state particles. This is one way the properties of final state particles can be used to infer information about intermediate particles with short lifetimes that

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otherwise can’t be observed. When particles satisfy the mass-energy equivalence described by equation 6 they are said to be on-shell while particles that do not satisfy this relation are said to be off-shell. These off-shell particles are also referred to as virtual particles as it classically is not possible to violate the energy-momentum relation, the on-shell particles are therefore also referred to as ‘real’ particles. The virtual particles do also exist however but their existence is limited by the uncertainty principle so they only show up as intermediate particles and cannot be observed directly as final state particles.

3.5 The H → ZZ → ```` decay channel

To determine the lifetime of the Higgs boson we will need to look at a specific Higgs decay channel of which we can then analyse the properties of its final state particles like the invariant mass. The main decay channel that I will be looking at is gg → H → ZZ → ```` which I will refer to as the ‘Higgs’ decay channel as it actually involves an intermediate Higgs boson. The Feynman diagram of this decay is shown in figure 7. The `` here refers to one of the two lepton pairs which is either an electron and anti-electron pair, a muon and anti-muon pair or a pair of neutrinos. The decay into a pair of tau leptons will not be considered in this project as it brings some more complexity that is beyond the scope of this project. This is because the tau leptons themselves also decay rather quickly making it harder to reconstruct the decay channel that includes the Z boson to tau decay. This chapter will focus on the electron/muon channel while chapter 4 will discuss the decay channel where one of the lepton pairs is a neutrino-antineutrino pair.

The other channel I will be looking at will be referred to as the ‘continuum’ channel and is given by gg → ZZ → ````. The Feynman diagram of this decay is given in figure 8. This is a similar process as the Higgs decay, with the same initial and final state particles but without the intermediate Higgs. This channel provides a continuous background of leptons that were not produced via Higgs decay and that cannot be distinguished from the leptons produced by the Higgs mediated process. For the purposes of this analysis I will start off with simulated data that only shows one of these channels at a time to make the effect of the intermediate Higgs more obvious. The actual data that we would receive from a particle accelerator makes no distinction between these two sources however. The final analysis will therefore need to function using data that shows the combination of the two sources.

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Figure 7: Feynman diagram for gluon fusion into Higgs mediated production of two Z bosons that decay into two leptons each: gg → H → ZZ → ````. Referred to as the ‘Higgs’ channel for short. One of the Z bosons will be an off-shell Z boson which is denoted by Z∗ in the diagram.

Figure 8: Feynman diagram for direct Z boson production through gluon fusion with again decay into two lepton anti-lepton pairs: gg → ZZ → ````. This will be referred to as the continuum or ‘cont’ channel for short. This is because this provides a continuous background of leptons whose production was not mediated by an intermediate Higgs boson.

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3.6 The Monte Carlo dataset

The data that will be used to analyse these two decay channels is a set of Monte Carlo data generated using the gg2VV package [18]. To analyse this Monte Carlo data the python ROOT library [19] is used. The dataset includes many different plots of the cross-section or number of expected events as a function of a variety of different variables like the invariant mass or the transverse momentum. This dataset was generated for a Higgs rest mass of 126 GeV which the SM predicts to have a Higgs width of 4.21 MeV associated to it [3].

3.7 Higgs meditated Z boson production

The first part of the dataset I will analyse is the invariant mass distribution of two of the leptons that resulted from the decay of one of the two Z bosons in the full decay channel. This is plotted in figure 9. Analysing the two lepton invariant mass will help to better understand the full four lepton invariant mass plots later. The Z bosons have the same invariant mass as the lepton pair they decay into and they are referred to as Z₁ and Z₂. In figure 9 we can see two different peaks. One at around 91.5 GeV and a smaller one at around 30 GeV. The 91.5 GeV peak can be explained by looking at the invariant mass of the Z boson which is 91.2 GeV, approximately the center of the peak. Some difference is expected because a bin size of 0.5 GeV is used here. So this peak can be explained by an on-shell Z boson that decays into two leptons that have the same invariant mass as this on-shell Z boson that decayed and produced these two leptons. This means that for the other Z boson that is formed from the original 126 GeV Higgs boson, there is only 126 − 91.2 = 34.8 GeV left over. This explains that other peak that is seen at around 30 GeV when also taking the width of the other peak into account. These two leptons came from an off-shell Z boson, a virtual particle, that only had an invariant mass of about 30 GeV. The Z boson itself also has a width which is equal to about 2.5 GeV. There is also the possibility that both Z bosons are formed off-shell, but this is less likely. These two factors combined explain the events around and in between the two peak values.

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Figure 9: The cross-section for the Higgs mediated Z boson production as a function of the invariant mass of the two leptons that are the result of the decay of the Z boson. There is no clear distinction between the two Z bosons, both show two peaks at around 30 and 91.5 GeV. A bin size of 0.5 GeV is used.

This two lepton invariant mass helps to understand the decay of the Z boson into two leptons. However to find the width of the resonance of the Higgs boson the invariant mass of all four leptons needs to be used because together they are the final state of the Higgs decay channel. Therefore the plot of the m_4lvariable which describes the invariant mass of these four leptons, shown in figure 10, is used. Here it also becomes clear that the events from the continuum channel overshadow those from the Higgs channel. At this scale the Higgs resonance peak appears as two data points at the on-shell rest mass of the Higgs boson of 126 GeV.

In figure 11 only the Higgs channel is plotted, this time with a logarithmic vertical axis. This logarithmic axis more clearly shows the important features of the cross-section plot. We can again see the resonance peak at the Higgs rest mass. After that there is a significant bump starting around 180 GeV, which is twice the Z boson rest mass. This is similar to the peaks in the m2l cross-section plot in figure 9. The peak or resonance is evidence for the intermediate

particles that form during the decay, in this case a Higgs boson decaying in two Z bosons. Another similar peak can be identified starting at around 345 GeV. This is equal to twice the rest mass of the top quark of 173 GeV. Which again is caused by the two intermediate top quarks that fuse together into a Higgs boson as seen in the Feynman diagram of figure 7.

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Figure 10: Comparison between the cross-sections of the Higgs and continuum channels as function of the invariant mass of the four leptons. The cross-section of the continuum channel clearly dominates, the total cross-section of the continuum is over 8 times larger than that of the Higgs channel. Two data points near the resonance of the Higgs boson are also visible.

Figure 11: Same cross-section as function of the four lepton invariant mass but only for the Higgs channel and with a logarithmic vertical axis. Be-sides the Higgs resonance (126 GeV) also the reso-nances of the two intermediate Z bosons (180 GeV) and the two top quarks (345 GeV) can be identi-fied as a sharp peak and two wider bumps respec-tively.

Figure 12 shows this same m4l variable but zoomed in at the Higgs rest mass. Here the

reso-nance peak becomes clear. To determine the width of this peak a Breit-Wigner distribution is fit to the data. This BW fit gives a width of 4.433 ± 0.044 MeV. This is slightly larger than the 4.21 MeV width that is predicted by the SM but it is fairly close. Besides the parameters in the BW distribution, the integral over the histogram bins is also being used as a parameter in this fit.

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Figure 12: The cross-section of the Higgs mediated Z boson production channel plotted as function of the invariant mass of the 4 leptons (m_4l). To this a Breit-Wiger (BW) function is fitted as described by eq. 5. The results of the fit is a Higgs width of 4.433 ± 0.044 MeV.

3.8 The limited detector resolution

This is not the full story however as this width from the BW fit is determined from data that would come from an idealised detector with an almost perfect resolution. As was mentioned before the resolution of are actual detectors like ATLAS are limited. The effect of this limited resolution is shown in figure 13 where the m_4l variable is plotted zoomed in around the Higgs rest mass with the detector resolution taken into account. This dataset had an initial bin size of 2 MeV but was rebind to a bin size of 40 MeV to limit the effect of the noise and to more clearly demonstrate the shape of the distribution. This data can no longer be described by a BW distribution (blue) but instead is best fit by a Gaussian distribution (red). The width of this Gaussian distribution is 3.15 GeV. This means that we would only be able to measure the Higgs width directly using this method if it was equal or larger than 3.15 GeV. Since the SM value of the Higgs width is 4.21 MeV [3], the detector resolution would have to be almost 750 times greater to actually directly measure the Higgs width this way.

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Figure 13: cross-section as function of m_4l accounting for the resolution of the detector. The Breit-Wigner fit (blue) no longer is a good fit to the data when accounting for the resolution of the detector. Instead a Gaussian fit (red) is used that much better the effect that the limited detector resolution has on the data. This Gaussian fit gives a width (FWHM) of 3.15 GeV.

The reason why the distribution changes into a Gaussian one is because the ATLAS detector has many different layers and each layer adds its own bit of uncertainty when reconstructing the particle. When combining these many different individual sources of uncertainty together the resulting distribution naturally turns into a normal distribution.

3.9 Interference between the Higgs and continuum channels

This separate analysis of the Higgs mediated decay channel and continuous background chan-nel is useful for more clearly illustrating the different aspects of the Higgs decay. However as mentioned before when actual particle accelerator data is analysed we receive a combination of the Higgs mediated decay channel as well as continuous background data that can’t be distinguished from each other. The combination of these production and decay channels is however not just the naive addition of the Higgs and continuum channels, we also have to account for the interference between them. The effect of this interference is best illustrated by figure 14.

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Figure 14: The effect of the interference on the cross-section as a function of the invariant mass of the four leptons (m4l). Plotted in blue is the separate addition of the two squares of the two

channels. With ‘Higgs’ the Higgs mediated Z boson production: gg → H → ZZ → `+`−`+`− and ‘cont’ the continuum channel: gg → ZZ → `+`−`+_`−_{. Plotted in red is where the two}

channels are added up first and then squared, thus including the interference between them. The relative effect of the interference is most clear starting from about 320 GeV and above. The interference is destructive because the cross-section is lower for the channel that includes the interference.

Here we can see the difference between the addition of the Higgs and continuum channels separately (blue) and the channel where the effect of the interference term is included (red). The effect is mostly noticeable from about 320 GeV and above where we can see that the number of events from the channel that includes the interference is lower. This means that there is destructive interference between the two channels.

This interference can be explained more explicitly as follows. The Feynman diagrams de-scribing the two channels that are being considered here (figures 7 & 8) are a description of different quantum mechanical paths that can be taken form one particular initial state (gg) to reach the one particular final state (4`). Because both these paths have the same initial and final state we need to account for the interference between these different paths. This is very

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similar to a classical example of quantum inference, the double slit experiment. In the double slit experiment a photon displays its wave like properties by creating an interference pattern after moving through a double slit. Classically the photon would move through either one of the two slits and would create a simple pattern of just two lines behind the slits. What we actually see is an interference pattern that is created because the photon is better described not by a classical particle but by a quantum mechanical wave-function that can interfere with itself. Just like how the photon is described by a quantum mechanical wave function so too are the different paths of creating the final state of a particle interaction described quantum mechanically by Feynman diagrams. These Feynman diagrams are a graphical representation of complex probability amplitudes. To get a probability of a specific interaction occurring we take the square of this complex amplitude. If we write this out explicitly for the two different channels the quantitative effect of the interference on the cross-section becomes clear. The amplitudes of the two channels are here simply denoted by A1 and A2. For the cross-section

of the channel with interference

σ ∝ |A1+ A2|2 = |A1|2+ |A2|2+ Re(A1· A∗2+ A∗1· A2).

For the channel without interference it is simply the addition of the squares of the two amplitudes

σ ∝ |A1|2+ |A2|2.

Therefore the interference is given by the difference between these two equations: Re(A1· A∗2+ A

∗ 1· A2).

3.10 Defining a new variable correlated with the Higgs width

It has now become clear that because of the limited detector resolution the Higgs width cannot be measured directly. A different less direct way must therefore be developed which is the main goal of the project and this will be discussed in this chapter.

The production cross-section for a process like i → H → f can be described by σi→H→f ∼

g2_ig_f2 ΓH

(7) with gi,f the Higgs boson couplings to initial and final states i and f and ΓH the Higgs boson

width [20]. This equation is an approximation made using the narrow width approximation as reflected by the ∼ sign. This approximation can be made because the width of the Higgs boson (4.21 MeV) is much smaller than the rest mass of the Higgs boson (126 GeV). The limit of this approximation means this equation is only valid for on-shell Higgs production.

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The on-shell cross-section is invariant under a simultaneous re-scaling of the Higgs couplings and the Higgs width as follows: g → ξgSM and ΓH → ξ4ΓSM_H so that σon → σonSM [21].

A measurement of just the on-shell cross-section alone therefore allows for infinitely many solutions for the Higgs couplings and the Higgs width. We can however set the measurable σon fixed and independent of ΓH which instead makes σoff scale with ΓH. The on-shell

cross-section will then scale with just the coupling constants associated with the gg → H → ZZ decay channel

σon ∼ g2H→gggH→ZZ2 . (8)

To define a measurable variable that is dependent on ΓH we therefore also need to use the

off-shell cross-section σ_off given by the following integral

σoff∼ Z ds g2 ig2f (m2_4l− m2 H)2+ m2HΓ2H |_m2 4l m 2 H→ g_H→gg2 g2_H→ZZ m2_4l (9) where the off-shell condition of m_4l m2

H is used [21]. This shows that σoff∼ gH→gg2 g2H→ZZ,

where the initial and final states have also been changed to the channels that are being considered in this analysis. We can see that σoff does not directly depend on ΓH but it does

depend on the coupling constants. As we have seen before if we want to preserve the resonance cross-section while changing the Higgs coupling constants than we must also change ΓH which

means σ_off also changes proportionally to ΓH. More explicitly

σoff∼ gH→gg2 gH→ZZ2 → gH→gg2 gH→ZZ2 ξ4= gH→gg2 gH→ZZ2 ΓH ΓSM_H (10) using ξ4 ≡ ΓH ΓSM H

. Now to remove the dependence of the coupling constants I define a new variable R, that is the ratio between the off- and the on-shell cross-sections

R = σoff σon ∼ g 2 H→ggg2H→ZZΓH g2_H→ggg_H→ZZ2 ΓSM_H = ΓH ΓSM_H (11)

where I make the assumption that the coupling constants are the same for the on- and off-shell cross-sections which allows them to cancel out. It is not always possible to make this assumption if for example the HZZ vertex contains anomalous couplings [21] but none of these effects are assumed to be at play here. Basically what happens is that the on-shell cross-section which is normally related to the Higgs width now becomes fixed. The result of this is that our off-shell cross-section, which is normally independent on the width of the Higgs boson, now does become dependent on the width through this scale factor ξ that is itself directly related to the Higgs width.

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At this point it is also important to mention that unlike with the data that was discussed previously, the dataset form which I will determine the Higgs width will use the number of expected events N instead of the cross-section as before. However for the ratio between the on- and off-shell variables this actually ends up giving the same results. This is because the number of events is given as follows

N = σLint (12)

with Lint the integrated luminosity of the beam of the particle accelerator. Because this

luminosity will be the same for both the on- and off-shell events this factor will cancel out again when taking the ratio between them giving the same ratio as before.

R = Noff Non = σoffLint σonLint = σoff σon ∼ ΓH ΓSM H (13)

3.11 Definition of the on-shell and off-shell regions

Before using this new variable R that is dependent on the on- and off-shell events, first the on- and off-shell regions need to be defined so that the relevant events can be summed up. For defining the on-shell region I use figure 13 which shows that the Higgs resonance is most prominent from about 121 GeV to 131 GeV. Figure 15 also shows there is little to no dependence on Γ_H in this region as is expected of the on-shell region. The Monte Carlo data for the files with the known Higgs width has a bin size of 5 GeV for the m_4l variable so the on-shell region will be set to between 120 and 130 GeV.

For the off-shell region we know that we require that m2_4l m2

H but the precise bound

is not immediately clear. Figure 15 can again be used to see where the dependence on the Higgs width begins to become clear, this appears to be around 290 GeV. Based on reference [20] and [22] I decided to set the off-shell region as everything above 300 GeV. This is more inline with the convention and also greatly decreases the error margins compared to using an off-shell region that starts lower. The Monte Carlo dataset ends at 1000 GeV so the off-shell region will be set to 300 − 1000 GeV. When choosing the boundary for the off-shell region we would ideally like to put the lower boundary as low as possible because a large section of the data is wasted if the boundary is set too high and the number of events drops of quickly in the higher invariant mass range, this would increase the error margins in the final results. We also want to make sure that we are actually far enough away from on-shell region. The balance between these requirements is part of what determines this lower boundary of the off-shell region.

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Figure 15: Full plot of number of events as a function of the invariant mass of the four final state leptons m_4l for different values of Γ_H. Data from the combined Higgs and continuum channel. A logarithmic vertical axis is used to more clearly illustrate the effect of the different values of Γ_H. The Higgs resonance and on-shell region is seen as a sharp peak from 120 to 130 GeV. The off-shell region is seen as the region above 300 GeV where the dependence on ΓH becomes clear.

Now that I have defined a variable that is linearly dependent on Γ_H I will fit it to the Monte Carlo data with known values of ΓH. For this I looked at simulated data that used

both the continuum channel (gg → ZZ → `+`−`+`−) and the Higgs channel (gg → H → ZZ → `+_`−_`+_`−_{) aswell as the interference between them. The results of which are shown in}

figure 16. As well as the four given data points I have plotted a linear fit based on equation 11. The result of this fit is as follows

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Figure 16: Ratio between the number events in the on- and off-shell regions of m_4las function of the Higgs width. Fitted to this is a linear fit with two parameters and a reduced χ2 of 1.79.

Figure 17: Ratio between the number events in the on- and off-shell regions of m4las function

of the Higgs width as in figure 16 but with the data point for the file with the unknown Higgs width added in (blue). With a result of Γ_H = (2.90 ± 0.07)ΓSM_H .

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Now that I have derived a relation between this ratio of the number of off- and on-shell events and the Higgs width we can use this to determine the Higgs width of a dataset with a yet unknown Higgs width. Using the same definitions of the off- and on-shell regions I calculate the ratio between the number of events to be 1.385(8). Using eq. 11 this gives ΓH = (2.90 ± 0.07)ΓSMH for this file with the previously unknown Higgs boson width. These

results are also shown in figure 17. This method could be used for any similar data file with an unknown Higgs width that is in the same range as the known data that was used to develop the method. So the width should not be far outside the 1 – 4 ΓSM_H range for this method to work reliably. This will also only work for the gg → H → ZZ → `+`−`+`− decay channel. According to the PDG a Z boson only has a 3.36% chance to decay into a pair of electron and also only a 3.36% chance to decay into a pair of muons [23]. This means that the chance of both Z bosons decaying into either a pair of electrons or muons is 0.45%.

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4 Determining Γ

H

from the H → ZZ → `

+

`

−

ν

`

ν

`

decay channel

Another decay channel to consider that is similar to the 4 lepton channel is the channel where one of the two Z bosons decays into a neutrino pair: gg → H → ZZ → `+`−ν`ν`. We can’t

actually detect these neutrinos with ATLAS but we can still use this channel in an other way. The advantage of considering this decay channel is that the chance for a Z boson to decay into a neutrino pair is 20% compared to the 6.72% chance of decaying into an electron or muon pair. This means that this neutrino decay channel is almost 3 times a likely to occur compared to the 4 lepton decay channel as is also shown in figure 6. The neutrino decay channel would therefore be comparatively easier to detect and analyse, giving more significant results. A disadvantage is that we can’t determine the invariant mass of the final state anymore.

Studying neutrinos more generally using this method is also interesting because the uniquely low mass of the neutrinos may be a hint towards new beyond the Standard Model physics. This is because the SM initially predicted the neutrinos to be massless which we now know to be incorrect. This means there is either a yet to be discovered neutrino coupling to the Higgs boson or a different undiscovered Higgs boson type particle or even a completely different mechanism which provides mass [24].

4.1 The transverse mass

The way that was chosen to gather information about the Higgs width without being able to use m4l is to use the transverse mass mT. We don’t know the exact initial longitudinal

momentum of the particles colliding in our detectors but we do know that they collide head on with zero transverse momentum p_T. Using conservation of momentum we therefore know that the total transverse momentum of the final state particles must also be zero. So if we add up all of the p_T of the visible particles we know that the transverse momentum of the invisible neutrinos is the same but in the opposite direction:

X pT = X pvis_T +Xpmiss_T = 0 X p``_T = −Xpmiss_T ≡XEmiss_T

Here I have substituted the momentum of the two leptons for the visible energy and defined the missing energy as Emiss_T .

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This missing energy along with the missing momentum can be rewritten into the transverse mass mT ≡ s q (mP DG_Z )2_{+ |p}`` T|2+ q (mP DG_Z )2_{+ |p}miss T |2 2 − p`` T + pmissT 2 , (15)

where (mP DG_Z )2 is the PDG value of the Z boson rest mass, p``_T is the transverse momentum of the two leptons which we can detect and pmiss_T is the missing transverse of the neutrinos which we can’t detect [25]. This variable can be used instead of the m4l that was used in the

previous chapter.

4.2 Ratio analysis using the transverse mass

With this new m_T variable an analysis can be done in a very similar fashion to what I did with the ratio between the off- and on-shell events as a function of the four lepton invariant mass. Using a different set of Monte Carlo data for this gg → H → ZZ → `+`−ν`ν` decay

channel I first define the on- and off-shell regions again by looking at the plots, this time for the mT variable. The plot of the mT variable for the different Higgs widths can be seen in

figure 18. Here we can see that there are some differences between this and the m_4l variable that was used before. We can see mT has a minimum value equal to 2mZ. This is because

we now have two on-shell Z bosons that decay into massless neutrinos. Beyond this minimum value the number of events drops of quickly as it is limited by amount of initial energy that is turned into transverse momentum during the collision. We can also see that the shape of the curve is different to what we see with m_4l. It is not as simple as just moving the off-shell part of the m4l variable over, the distribution changes by using this different mass variable as

per its definition of equation 15. This also means that the definitions for the on- and off-shell regions need to be changed.

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Figure 18: Number of events as a function of the transverse mass for the gg → H → ZZ → `+`−ν`ν` decay channel for different values of the Higgs width. The on-shell region is defined

as the sharp peak which show little dependence on ΓH. The off-shell region is defined starting

from the point where the effect of the different values of Γ_H becomes clear.

4.3 Redefining the on- and off-shell regions for mT

Looking at the m_T plot there is a fairly clear distinction between the two regions. First we have a short high peak from about 180 to 205 GeV with little to no dependence on ΓH, this is

now the on-shell region. After that sharp drop off the distribution declines more slowly and the dependence on ΓH become clear. The off-shell region is defined as everything above 250

GeV here because the on-shell region has clearly ended beyond that point and this value also minimized the error for the ratio between m_T in the on- and off-shell regions. Just as before I have taken the ratio between the sum of the number of events in the on- and off-shell regions but now as function of the m_T variable. I have then fitted a linear fit again, as seen in figure 19, which gave the following relation

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Figure 19: The ratio between the number of events in the off- and on-shell regions of the transverse mass as function of the known values of the Higgs width. A linear fit is plotted to show this relation.

4.4 The alternative transverse mass

There is also an alternative definition of the transverse mass that is worth consideration. This alternative definition is also included in the dataset and is defined as follows

mT alt≡

q 2|p``

T||pTmiss|(1 - cosφ) (17)

with φ ≡ p``_T · pmiss

T . Where the fact that the two neutrinos are close to massless is used.

The full derivation is found in chapter 47 on kinematics of the ‘Review of particle physics’ by the Particle Data Group [25]. The potential benefits of this alternative definition of the transverse mass are that the difference between the on- and off-shell regions is a bit more clear and there is a larger on-shell region. Also this no longer requires that the two Z bosons have to both be on-shell.

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Figure 20: Number of events as a function of an alternative definition of the transverse mass for the gg → H → ZZ → `+`−ν`ν` decay channel for different values of the Higgs width.

The on-shell region is defined as the sharp peak which show little dependence on Γ_H. The off-shell region is defined starting from the point where the effect of the different values of ΓH

becomes clear.

I have repeated the analysis of taking the ratio between the off- and on-shell regions as before. For this we need to again define the on- and off-shell regions. For this is used figure 20 where the amount of events is plotted as a function of m_{T alt}. For the on-shell region I chose the region at the start with little to no dependence on ΓH which is from 0 to 90 GeV. The off-shell

region is defined as between 180 and 1000 GeV as this is far enough away from the on-shell region and the dependence on ΓH is strong here while the amount of events is still relatively

high. Then I again fitted a linear fit to the ratio between the number of events in these two regions as a function of the Higgs width which can be seen in figure 21. This gives the following fit result

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Figure 21: The ratio between the number of events in the off- and on-shell regions of the alternative transverse mass as function of the known values of the Higgs width. A linear fit is plotted to show this relation.

4.5 Comparing mT and mT alt

Comparing the relation between Higgs width and the ratios between the number of off- and on- shell events for the mT and mT alt variables (figures 19 and 21) we can see that figure 19

which uses the original definition of the transverse mass (eq.15) has a smaller relative error. When trying to determine ΓH from a dataset like the one used here this seems to be the

better definition to use.

Besides the fairly small differences between these two plots it is clear that using either definition of the transverse mass to determine Γ_H seems to work quite well. Using the trans-verse mass allows for other Higgs decay channels including those with invisible finale state particles like the gg → H → ZZ → `+`−ν`ν` decay channel to be investigated. Compared to

the use of the four lepton invariant mass m_4l used for the gg → H → ZZ → `+`−`+`− decay channel, the use of the transverse mass would appear to work better when working with a limited amount of actual experimental data. This is because the transverse mass is not only less restrictive in the types of decay channels that can be considered but in this particular case the branching ratio of the invisible channel was also significantly higher. This could be an important factor when the amount of available data is limited.

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5 Discussion

The method for determining the lifetime of the Higgs boson from the gg → H → ZZ → `+_`−_`+_`− _{decay channel gave a result of Γ}

H = (2.90 ± 0.07)ΓSMH . Because simulated data

was used the actual value used to generated this data could later be revealed after I had finished determining the width using the ratio method discussed in chapter 3.10. The actual value ended up being 2.8 ΓSM_H , so just a little outside of the margin of error. This means that there are a few factors to be discussed which may have caused this small discrepancy.

5.1 The on- and off-shell boundaries

First of all the determination of the lower bound of the off-shell regions is not very exact. The boundaries where set based on multiple factors including previous research, theoretical definitions of the off-shell regions and the data itself. This resulted in boundaries that where generally in line with convention and what was expected of the data but the boundaries where still not very exact. This uncertainty in the precise location of the boundaries was not factored into the error margin of the determined Higgs width. This could be one factor in explaining why the actual value was just outside of the error margin of the determined value. If the uncertainly of the boundaries was included this would result in a larger uncertainly for the Higgs width which could mean the actual value would fall within the margin of error. For the determination of the Higgs width from the gg → H → ZZ → `+`−ν`ν` decay channel

this same problem arises to a lesser degree. These results where however not compared to actual values used to generate the data.

5.2 Limitations of the developed method

Besides this potentially neglected source of uncertainly it is also important to place these results in a larger context. This method was developed to be used on data of just a single decay channel and background channel at a time. For research using actual experimental data there would be many more different sources of events that would have to be considered and differentiated from each other before being able to determine the Higgs width in this way. For this research I was also able to use a dataset that simulated different values of the Higgs width that I knew beforehand. Using the predictions of the Standard Model to simulate these datasets works well to develop a basic method like this. It is however not ideal to use predictions made by the Standard Model when looking for new beyond the Standard Model physics if for instance a different Higgs lifetime than the SM prediction is expected.

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Figure 22: The ratio between the number of off- and on-shell events as a function of the Higgs width. The fit used here includes and extra factor of√ΓH to account for the effects of the

interference between the Higgs and continuum channels.

5.3 Effect of the interference on the number of off-shell events

Something else which was not included in the determination of the Higgs width is the effect of the interference on the number of off-shell events. An extra factor ofq ΓH

ΓSM H

could be included in the fit of the relation between the ratio of on- and off-shell events and the Higgs width to include this effect [20]. In figure 22 a fit is used that includes this extra factor. However as there are now three parameters and only four data points it is hard to judge the validity of the extra term. The low value of the reduced χ2 of 0.12 shows this as well. To further test this, the analysis must be repeated with a larger dataset simulating more values of ΓH.

Further research could address any of the points mentioned above. In particular trying to expand the used data to for instance include the tau leptons in the gg → H → ZZ → `+`−`+`−decay channel instead of just the electrons and muons as in this project. This could be a step towards developing a method of investigating actual experimental data.

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6 Conclusion

The goal of this project was to find a way to determine the lifetime of the Higgs boson from a Monte Carlo generated dataset of the gg → H → ZZ → `+`−`+_`− _{decay channel}

with a previously unknown value of the Higgs lifetime. For this a dataset was used that simulated four different known values of the Higgs lifetime. This resulted in the following relation between the ratio of the number of off- and on-shell events R, and the width of the Higgs boson Γ_H: R = 0.111(4)Γ_H + 1.061(13). This width ΓH is inversely proportional to

the lifetime. This relation was then used to determine the Higgs width of a dataset with a previously unknown width. The result was a width of Γ_H = (2.90 ± 0.07)ΓSM_H compared to the actual value of 2.8 ΓSM_H that was revealed after I had determined my value. As the actual value is nearly within the error margin of the derived value, it can be concluded that this method works fairly well depending on the desired accuracy.

The gg → H → ZZ → `+`−ν`ν` decay channel was also looked into because of its higher

branching ratio compared to the four lepton channel. This higher branching ratio would result into more detected events and therefore more significant results when doing a similar analysis as in this project experimentally with actual data. For this the ratio of the number of events in the off- and on-shell regions as function of the transverse mass was used. This resulted in the following relation: RmT = 0.0100(5)ΓH+ 0.0531(14). This relation can than similarly be

used to calculate the Higgs width of a dataset for that decay channel. A different definition of the transverse mass, mT alt was also considered but showed no significant improvements over

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A Method to Determine the Lifetime of the Higgs Boson Based on Monte Carlo Simulations of the gg → H → ZZ → llll Decay Channels

THE LIFETIME OF THE HIGGS BOSON