Simulating the uncertainty on the Higgs boson mass

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Simulating the uncertainty on

the Higgs boson mass

Author Hendrik Marinus Maarten Post Studentnumber 10192603

University University of Amsterdam

Faculty FNWI

Department NIKHEF

Supervisor Ivo van Vulpen 2th supervisor Marco van Woerden Thesis for Bachelor Physics

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Abstract

In this thesis the uncertainty on the invariant mass (σmH ) of four muons in the H → ZZ

∗ _→

4µ channel is simulated. This uncertainty is calculated from the uncertainty on the transverse momentum of the muons (σpt ). To simulate σpt an existing parametrisation, that relates ptto

σpt to also include a found relation between η and σpt . With this improved parametrisation the

average σmH is calculated for Higgs bosons with different masses. Here is found that the average

uncertainty increases with the Higgs boson mass.

Author Maarten Post Address Rodekruislaan 1212F

1111 XB Diemen

Telephone +31 6 39679841

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1 Introduction

1.1 How to find a Higgs Boson

In 2012 the Higgs boson was discovered by the ATLAS and CMS experiment at the CERN LHC accel-erator [2][5]. The Higgs particle is a boson named after Peter Higgs who proposed a mechanism which predicted a new particle [7]. After the discovery of the Higgs boson Peter Higgs and Francois Englebert, who had also proposed the mechanism, were awarded the Nobel prize in physics. The Higgs mechanism, more closely described in subsection 3.1, was needed to explain the masses of the W and Z gauge bosons. Some theories predict also more massive Higgs particles next to the one found in 2012.

To find the Higgs particle CERN used two beams of protons in a circular accelerator, the large hadron collider (LHC). These beams collide at interactions points where the ATLAS, CMS and other detectors are build around. When the proton beams collide a lot of particle interactions accrue, and all kinds of particles are produced. These particles can be detected by the detector. The detector measures the path or trajectory of the particles and it’s energy. With a magnetic field the track of charged particles is deflected. From this deflection the momentum of the particle is calculated. The ATLAS detector is discussed in more detail in subsection 4.

The mass of the Higgs Boson is determined by calculating the invariant mass of its decay products. Several decay products of the Higgs boson are known and are sought for in particle detectors after colli-sions. The acquired data is compared with simulated data to distinguish particles that originated from a Higgs boson from background (other processes that produced the same particles as a Higgs particle). The measurements of the detector are predicted using Monte Carlo generators. These generators use the cross section of all the particle interactions as predicted by the Standard Model. The data generation also take the properties of the detector and its interaction with the particles into account.

Data generated from a Standard Model without a Higgs boson is compared with the measured data of the ATLAS and CMS. In the measured data there where significantly more events with an invariant mass of 125 GeV. This means that there is a particle with this mass, this was the Higgs boson.

1.2 The scope of this Thesis

In this thesis a study is done into the uncertainty on the mass of the Higgs boson. By looking only into generated data from the Higgs boson decay channel H → ZZ∗ _{→ µ}+_µ−_µ+_µ−_{in which the Higgs boson}

decays into two Z bosons. Each of the Z bosons subsequently decay in a muon and an anti muon. This is done to investigate how the uncertainty on the mass of the Higgs boson changes as a function of the Higgs boson mass. Some theories predict other more massive Higgs particles so to know the uncertainty on the possible measurement of these particles is of interest to test these theories.

The momentum and direction of each of the muons are simulated. From the invariant mass of the muons, the mass of the Higgs boson is calculated. This means that the uncertainty on the muons measurements determines the uncertainty on the Higgs boson mass. Understanding the origin of the uncertainty on muon measurement might be helpful for improving particle detectors.

First an introduction in the physics (section 2) involved in this subject is given. After that the ex-perimental setup of the ATLAS detector is given in section 4. The CMS detector is not further described since the ATLAS and CMS detector a quite alike and use the same principles for particle detection. The H → ZZ∗→ µ+_µ−_µ+_µ− _{channel is discussed in section 3}

In section 6 the uncertainties on the muon measurements are related to the uncertainty on the Higgs boson mass. In the two sections following a faster way to calculate the uncertainty on the Higgs boson mass is described. This new method is used in section 10 to calculate the average uncertainty on the Higgs boson mass for different Higgs boson masses. In section a the origin of the uncertainty on the Higgs boson is found an in section 10 is investigate if the origin changes for different Higgs boson masses.

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2 Theory

In this section the physics involved in this thesis in discussed. The standard model of particle physics is a complicated theory and full understanding in not needed to understand this thesis so it is only briefly discussed. Further reading of the standard model can be in [10] or [6].

2.1 Relativistic Kinematics

Particle physics is often called high energy physics because research is done by accelerating particles near the speed of light. When the speed of the particles is this high special relativity is needed to describe the kinematics correctly. In particle physics it is common to use natural units, meaning c = ¯h = 1 and to express mass, energy and momentum in units of GeV. It is also convenient to use fourvectors x = (t, x, y, z) = (t, ~x), p = (E, px, py, pz) = (E, ~p), which inproducts pµpµ and xµxµ are Lorentz invariant.

The invariant mass of a system of objects is given by:

m2_inv=XE_i2−X|~pi|2 Note that for one particle minv is the particles rest mass (1)

= (Xpi)µ(

X

pi)µ (2)

This is an important quantity because when a particle decays into other particles the invariant mass of the decay products can be measured and is the same as the rest mass of the original particle. The invariant mass is the same for all observers.

In ATLAS two protons with the same energy collide head on. The strength of an accelerator is de-fined by the invariant mass of the two colliding proton beams. This is the energy available for new particle creation. √s:

√

s =p(E1+ E2)2− (|~pi| + |~pi|)2 E1= E2 (3)

=p(2E1)2 The total impuls is zero because ~p1= − ~p2 (4)

= 2E1 (5)

= 14 TeV (6)

This number also gives an idea at which speed the muons are traveling. The speed of the protons can be calculated from their Lorentz factor γ = _mE

pc2 =

7TeV

0.938MeV = 7463. When v is solved out of γ = 1 q

1−v2 c2

you find v = 99.993%c, almost the speed of light. In 2012 when the Higgs boson was discovered,√s was not higher than 8 GeV. After a year and a half of upgrading the LHC started running again in spring 2015 and is slowly rising√s to 14 TeV.

2.2 Standard Model

The standard model of particle physics gives a quantum mechanic description of three of the four fun-damental forces in nature in terms of particle interactions. It unifies electromagnetism with the weak and the strong nuclear force. Each of the three fundamental forces have a mediator particles, a boson that mediates the force between the particles that make up matter. The mediator particle of the elec-tromagnetic, weak and strong force is respectively the photon, gluon and the W and Z boson. The other particles of the standard model the fermions are the particles that make up matter and there heavier versions. For example the muon is the heavier version of an electron.

The formalism of the standard model relies on quantum field theory and is therefor only briefly dis-cussed. The Lagrangian of the Standard Model in a very compact form is given by.

L = −1 4FµνF µν ₍₇₎ + i ¯ψ /Dψ (8) + ψiyijψjφ + h.c. (9) + (Dµφ)∗(Dµφ) − µ2φ2− λφ4 (10)

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If all the terms are expanded the Lagrangian is a page long. The first term FµνFµν describes the

electro-magnetic, weak and strong force. The second line describes the how those forces act on fermions (quarks and leptons). The fourth line describes the dynamics of the Higgs field φ. The third line describes a interaction between the Higgs field φ and the fermion field ψ and also how the Higgs field gives mass to the different fermions. So in this Lagrangian all the particles of the Standard Model are accounted for by fields. The field strength tensor Fµν accounts for the force carriers (photon, gluon, W-boson and

Z-boson), the spinor field ψ accounts for the Quarks and Leptons and the Higgs field φ accounts for the Higgs boson. [6][8][10]

The Standard Model is a very accurate theory but it is incomplete. It can for example not explain gravity or the matter - anti matter asymmetry in the universe. To answer those and other question physicist search for beyond Standard Model particles. In some of those model the known Higgs boson is not alone and has heavier ’brothers’. To find those heavier Higgs bosons one must know what to expect to find in the detector. This can be done by running simulations of Higgs decay for different Higgs masses. In this thesis a simulation to see how the uncertainty on the Higgs boson mass (mH) change as a function

of the Higgs boson mass in the H → ZZ∗ _{→ µ}+_µ−_µ+_µ− _{channel. In section 4 a introduction to the}

ATLAS detector and data generation is given. In section 3 the decay modes of the Higgs boson are discussed with the focus on the Higgs to four muons channel. The simulated uncertainty on Higgs boson mass, with mHat 125 GeV, is discussed in section 6. In section 7 and 8 the simulated uncertainty from

section 6 is been recreated without doing a full ATLAS simulation, this gives a less accurate result but is much faster. The method of recreating the uncertainty on mHat 125 GeV is than used to calculate

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3 The Higgs boson and its decay

In this section a introduction into the Higgs boson is given. The Higgs mechanism is complicated and is only briefly discussed. Further reading can be done in [10].

3.1 Higgs Mechanism

In the Standard model the Higgs boson is the only boson that is not a force carrier. The Higgs mecha-nism was introduced to explain the mass of the W and Z boson. The Higgs field described in the Higgs mechanism is often explained as thick syrup where the W and Z boson pass through. The Higgs field syrup prevents the W and Z bosons form going with light speed. This means effective that the W and Z boson have mass.

In the Higgs mechanism the potential V (φ) = µ2_φ2_{+ λφ}4 _{(equation 10) is added to the Lagrangian.}

This results in an explanation for the mass of the W and Z boson without losing gauge invariance. This extra potential also gives rise to a extra mass term this the Higgs field and it corresponding particle is the Higgs particle.

3.2 Higgs boson creation and decay

A Higgs boson can be created trough a proton proton collision in several ways. The two most important are firstly gluon-gluon fusion where two gluons of the colliding protons fuse into a Higgs boson through a top quark loop. Secondly vector-boson fusion while where two quarks of the protons emit W/Z bosons that fuse into a Higgs boson.

When a Higgs boson is created it decays after a mean lifetime of 1.56 · 10−22seconds. The possibly decay channels are listed in order of likelyhood in table 1.

Higgs boson decay channel Branching ratio for mH = 125M eV

b¯b Bottom quark and an anti bottom quark 58.5%

W W W+ boson and a W− boson 21.6%

gg Two gluons 8.7%

τ ¯τ Tau and anti tau 6.4%

c¯c Charm quark and an anti charm quark 2.5%

ZZ∗ Two Z bosons (One Off-shell) 2.6%

γγ Two photons 0.2%

Zγ Z boson and a photon 0.2%

Table 1: The possible decay modes of the Higgs boson in order of branching ratio. Note that the Higgs boson only couples to mass so when it decays to a photon the Feynman diagram contains a loop For this thesis only the H → ZZ∗→ µ+_µ−_µ+_µ− _{channel is of importance and will be further discussed.}

The mass of the Higgs boson mass is around 125 GeV and the (on shell) mass of the Z boson is around 91 GeV. So in the rest frame of the Higgs boson it becomes clear that the mass of two Z bosons is too high. Nature solves this problem by producing one on shell Z boson with a mass around 91 GeV and a off-shell Z boson with a mass around 33 GeV denoted by Z∗. This is clearly visible in Figure 1. Because the Higgs boson has spin 0 the two Z bosons have opposite spin and all decay directions have the same likelyhood. The Z bosons both have spin 1 so the decay direction of the muons are not evenly likely.

3.3 The Z boson decay

The Z boson decays mostly to mesons q ¯q, with the exception of the top and anti top quark because they are to heavy. The Z boson can also decay to ν ¯ν(dineutriono) and l¯l (dilepton). The branching ratios are given in table 2 [10]. Only 3.3% of the Z bosons decay to µ+µ− which is the channal of interest in this thesis. Off all Higgs boson created only 0.003% decays in the channel H → ZZ∗→ µ+_µ−_µ+_µ− _.

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Br(Z → q ¯q) = 69% Br(Z → ν ¯ν) = 21% Br(Z → l¯l) = 10%

Table 2: The branching ratios of the Z boson. The Branching ratio for a specific lepton of neutrino is 1₃ of that in the table. The branching ratio for an up like quark is 15% and 12% for an down like quark.

3.4 Z boson mass distribution

The Higgs boson decays into two Z bosons, one on and one off shell. In Figure 1 the mass distribution of the two Z particles is displayed. The on shell boson has a mass around 91 GeV, the mass of the Z boson. The off shell boson has a mass around 32 GeV. Together they make up the 125 GeV mass of the Higgs boson.

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4 The Atlas Detector

The following section is a complete but short introduction into the workings of the ATLAS detector. First a summery is given of the detector parts and how they work and secondly the process of data generation is described. The data generation of ATLAS could calculate the uncertainty on the Higgs boson mass (σmH ) for different Higgs boson masses but, as will become clear would take too much time.

A faster way to calculate this is described in section 7 and 8. The ATLAS detector is discussed in more detail in [4].

4.1 Coordinate system

ATLAS measurements of particles contain three quantities pt,η and φ. ptis the momentum in the plane

(xy-plane) perpendicular to the beam pipe (z-axis). η is the (pseudo)rapidity angle of a particle relative to the beam axis. The pseudorapidity is defined as − ln[tan(θ₂)] where θ is the normal angle between the momentum and the beam axis. η is ∞ when a particle travels parallel to the beam pipe, 0 when travelling in the xy-plain and −∞ when travelling anti parallel to the beam pipe. φ is the normal angle in the xy-plane.

The Cartesian components of the momentum can by found by.

px=ptcos(φ) (11)

py=ptsin(φ) (12)

pz=ptsinh(η) (13)

|p| =ptcosh(η) (14)

4.2 Detector parts

The ATLAS detector can roughly be divided into three main parts. Closest to the beam pipe there is the tracker and solenoid magnet. Around the tracker is the calorimeter and the spectrometer and a toroidal magnets. The tracker consists of layers of silicon strips with electronics underneath. When a particle travels through the silicon it liberates a electron from the silicon molecules. These electrons are picked up by the electronics and the position and track of the particle can be reconstructed.

Covering the tracker is a solenoid magnet. The magnetic field produced by this magnet give a Lorentz force on charged particles flying through the detector so there track is bend. The curvature of a charged particle depends on the momentum of the particle. So by calculating the radius of curvature of the particles track the momentum of the particles can be determent.

The calorimeter absorbs most particles and measures their energy. The calorimeter consists of many layers of metal and sensors in between. If a particle interacts with a layer of metal is produces a shower of particles. Than These showers are detected and the total energy is measured. The ATLAS detector has an electromagnetic calorimeter and a hadronic calorimeter. In the electromagnetic calorimeter the shower particles liberate electrons from liquid argon that sits between the metal layers. The liberated electrons make a current and from the current intensity the energy of the particle can be calculated. In the hadronic calorimeter particles pass trough a plastic that emits light when a particle passes trough. The total intensity of the light is proportional to the energy of the particle.

Muons do not interact with the calorimeter as often as other particles so they pass trough the calorimeter. Outside the calorimeter is the muon spectrometer capable of detecting the muons. In ATLAS the drift chambers are installed cylindrical around the beam pipe and forwards. The forward muon spectrometer gives ATLAS the characteristic ’big wheel’ look.

The muon are detected with so called proportional counters. The metal tubes are gas-filled. Inside the tube there is a metal wire which is the anode. The gas is maintained at a negative potential and the anode at a positive. If a muon particle liberates an electron of one of the gas molecules it will feel a

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force due to the electric field. The electron itself will free other electrons and this signal can be picked up by the anode and processed.

4.3 p

t

from the radius of curvature

Inside the ATLAS detector is a strong magnetic field perpendicular to the transverse plain which bends charged particles how much a particle is bend depends on its momentum and charged. If the track of a particle can be reconstructed the radius of curvature (r) of the track is related to the transverse momentum in the following way.

pt= 0.3Br (15)

which gives the transverse momentum in GeV. Equation 15 is derived in [10] from the classical Lorentz force on moving charge, it also holds for the relativistic case where p = γmv.

4.4 Data Simulation

In all science when an experiment is done one must compare the predictions of a theory with the results of the experiment. In particle physics the standard model can not predict what will happen in a particle collision. This is because it is based on quantum mechanics a fundamental statistical theory. The stan-dard model can only predict the chance or cross subsection of a certain particle interaction. To predict what will be measured in the detector a data set is made which predicts what data will be measured in the ATLAS detector. To make this ’fake’ data set the Monte Carlo method is used.

The Monte Carlo method of calculation uses random numbers to approximate a calculation what can not be done, is very hard or time consuming to do analytic. In particle physics the Monte Carlo method is used in two ways. Firstly to predict what particle interactions happen in the Atlas detector with a pp col-lision at a certain energy. This is called the Monte Carlo simulation, more closely described in subsection 4.4.1. Secondly to predict how the detector material influence the particles and thus the measurement, more closely described in 4.4.2. The final stage to make the ’fake’ data set is the reconstruction more closely described in subsection 4.4.3

4.4.1 Monte Carlo Simulation

To predict the data of the ATLAS detector one must start at the collisions of the two protons. The out-come of the collision depends on several variables and can not be calculated directly. So the Monte Carlo method is applied using the cross subsections and distributions known very precise from the standard model

For example: inside the proton the quarks/partons do not all have the same momentum, the momentum of the quarks is distributed via the parton distribution functions (PDF) [10]. The trick is to pick a random momentum from this distribution so that if a great number of momentum is picked it represents the PDF. After this the Monte Carlo method is used again but now for another process. For example the cross subsection of the Higgs boson depends on the momenta of the colliding partons. The momenta of the colliding partons is already chosen and now a random number is picked to determine if a Higgs boson is created or not. If a Higgs boson is created, and also if it is not, the Monte Carlo method is used again now for example to determine how the Higgs boson decays and the angle between the decay products.

This process is repeated until the energy and momentum of the end products are known. For ex-ample the Higgs boson decayed to two Z bosons and those decayed both to two muon particle with a high angle in the detector. The next step is to calculate what influence the ATLAS detector itself has on the muon particles, and other end products. This is called the simulation.

4.4.2 Detector Simulation

When a particle travels through the detector it encounters the material of the ATLAS detector ranging from the construction steel to the electrical wires. The change that this particle interacts with the

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material depends on the energy of the particle, the thickness of the material and also on the material itself. How the energy and momentum of the particle and also the particles direction changes by the detector material is calculated in the detector simulation. To do this simulation again the Monte Carlo method is used. For example when a particle travels through a centimetre of steel the energy loss is chosen randomly from a distribution which depends on the energy of the particle and the properties of steel. To do the simulation correctly every part and its corresponding coordinates of the ATLAS detector is known. The detector simulation is ten times more time consuming than the Monte Carlo Simulations. 4.4.3 Reconstruction of the physics process

In the simulation all information of the particles is known, like what kind of particles are created. In the real detector there is no direct way to see what kind of particles pass trough the detector, only indica-tions. The charge of a particle can be determined with the curvature of the particles, because positive charged particles curve the other way around as negatively charged particles and neutral particles don’t curve at all. Charge, momentum and energy together with others are strong indicators of the kind op particles but can still be wrong. So if a particle that makes it through to the muon spectrometer mostly labelled as a muon because other particles are not likely to pass through the calorimeter. But if for example an electron passes through the muon spectrometer it is wrongly labelled.

In the reconstruction phase of the data generating process the data from the simulation is transformed from data containing all information to data how ATLAS would actually detect the data. This includes sometimes wrongly labelling particles. When comparing the simulated data with the reconstructed data one can learn how to improve the detector. Reconstructing the data takes more time than generating it but less than simulating it.

It has become clear that the simulation is the most time consuming part of the data generating process. In the next sections a way is described to improve this computation time for the calculation of the uncertainty of the Higgs boson mass. This is done by leaving out the simulation of the ATLAS detector and calculate the uncertainty direct from the Monte Carlo generated data, so only the fist part of the data generation. After this method is used at Monte Carlo generated data with other mHin the same

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5 Simulated data from the H → ZZ

∗

→ µ

+

_µ

−

_µ

+

_µ

−

_decay

The momentum and direction of the four muons that decayed from a Higgs boson are discussed in this section. This data of ten thousand Higgs bosons that decay via the H → ZZ∗→ µ+_µ−_µ+_µ− _channel

is generated with the Monte Carlo method described in section 4.4.1 and the Higgs boson mass is set at 125 GeV.

In the following sections the relation between the momentum and direction of the muons and the un-certainty on the invariant mass is discussed. The characteristic momentum and direction of muons that result in a high uncertainty on the invariant mass is discussed in section 9.In section 10 the average ptand

η from Monte Carlo generated data from different Higgs mass are compared to see how they change for different mHand how that influence the uncertainty on the invariant mass.

5.1 p

t

, η and φ of the four muons

When a Higgs decays in the H → ZZ∗→ µ+_µ−_µ+_µ− _{channel all the information about the Higgs boson}

must be deducted from the measurements of the four muons since the Higgs and Z bosons decay before they reach the detector material. The detector measures the momentum of a muon and its direction. The Higgs boson mass is than calculated from this quantities as done in appendix A.1.1.

In the sections to come it will become clear how ptand η are related to the uncertainty on the invariant

mass. In the rest of this section the pt, η and φ distribution are given in order to give the reader some

feeling for the decay of the Higgs boson in the H → ZZ∗→ µ+_µ−_µ+_µ− _channel.

5.1.1 pt distribution of the muons

When looking at the pt distribution of the muons in Figure 2(a), the mean pt is around 30 GeV. This is

the expected momentum of the muons from calculation 30. The distribution has a asymmetric shape. This is due to the fact that one of the Z bosons is off shell. If from the four muons the muon with the highest ptis associated with the highest ptanti muon to have come from the on shell Z boson this

becomes more clear. In Figure 2(b) the ptdistribution of muons that are thought to came from the on

and off shell Z boson are displayed. When doing the same calculation off appendix A.2 but only for a Z boson that decays in two muons instead of a Higgs boson that decays into four muons it becomes clear that it makes sense to combine the muons this way. This is because the mean ptis around 91GeV2

for muons from the on shell Z boson. Also the mean ptis around 30GeV₂ for muons from the off shell Z

boson.

(a) (b)

Figure 2: The ptdistribution for all muons left. In the right Figure the muons from the two Z bosons

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5.1.2 The η and φ distribution of the muons

In the rest mass of the Higgs boson the Z bosons decay in opposite direction. All directions are evenly likely. This means that the pseudo rapidity and the φ direction of the muons are distributed evenly. Where the pseudo rapidity is related to the angle of the particle in relation with the beam pipe as dis-cussed in section 4 and φ is the angle in the transverse plain.

From Figure 3(a) it becomes clear that the muons mostly travel perpendicular to the beam pipe and less in the forward and backward direction of the detector (very high and very low η ). This is thus not as expected. The η direction of the muons is related to there pz

p ratio. The fact that most muons have a

η around 0 tells that the initial momentum of the Higgs boson determines the direction of the muons. Since the two proton beams have no net pz, most Higgs bosons often also do not momentum in the Z

direction. This results in muons with also no pz.

The lack of event a precisely η = 0 comes from the fact that ATLAS has a blind spot at this place [11]. Figure 3(b) shows that the direction of the decay products is symmetric in the φ direction, as was expected.

(a) (b)

Figure 3: The distribution of the η and φ angle of the muons.

In the next section a closer look is taken at how the uncertainties on pt, η and φ are generated in the

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6 Simulated uncertainties of the H → ZZ

∗

→ 4µ decay

In this section the simulated uncertainties on pt and mHare discussed. In section 6.1 is described how the

uncertainty on the transverse momentum of particles is simulated. How the uncertainty on the invariant mass can be calculated from the error on ptis described in 6.2.

6.1 The uncertainty on p

t

from data generation

When data is generated for each quantity also an uncertainty is calculated. So for each value ptand

η also a σpt and ση is given. For each Higgs boson decay also the uncertainty on the invariant mass

is calculated from the uncertainties on ptand η . The uncertainties are calculated in the simulation

(subsection 4.4.2) of the data generating.

For the Monte Carlo generation a particle, say a muon, has a ’true’ value of pt . In the detector

simulation it is simulated how a muon passes through different detector layers. The ATLAS detector does not have the same number of detector layers in each direction so the accuracy on ptis different for

each muon. One muon could pass through several detector layer of the inner detector and also several of the muon spectrometer, so the track of the muon is measured very accurate. Because ptis calculated

from the radius of curvature of the track ptis calculated very accurate with a small σpt . If a muon only

passes through a few detector layers or only through the inner detector or muon spectrometer the track and ptare measured less accurate, this means the uncertainty on ptis higher.

Another effect is that particles can interact with the materials in the ATLAS detector. If a muon passes through material it can loose energy or scatter. If a muon looses energy its radius of curvature decreases and it is harder to reconstruct the initial pt. This results in a higher uncertainty in pt . If a

muon scatters the interaction points of the muons are different than the ’true’ points. If a track is fitted trough those point they have also a higher uncertainty.

6.2 From σ

p_ti

to σ

mH

The simulated uncertainty on ptis related in the detector construction as discussed in the previous

section. The invariant mass and thus the Higgs boson mass is calculated from the pt, η and φ of the

four muons with equation 1 and 11. In this section is described how to calculate the uncertainty on the invariant mass given the uncertainty on ptof all muons.

6.2.1 The error formula

When a quantities Z depends on the measured quantities {x1, x2, x3, ..., xn}, each with a uncertainty

σxi , by Z = f (x1, x2, x3, ..., xn) the uncertainty on Z is given by the error function [3][11].

σ_Z2 = N X i (σxi∂xif ) 2 ₍₁₆₎

There are two problem which occur when using the error formula. The first is that for many applications the function f a complicated functions and deriving it is hard or impossible. Secondly The error formula assumes that the measurements on the variables of f are normally distributed with a standard deviation of σxi. The solution to these problems is to calculated the uncertainty numerically, this technique is

described in subsection 6.2.3

The invariant mass of the four muons depend on ptand η . So given the error formula, equation 16,

the uncertainty on the invariant mass depends on the uncertainty on ptand η . The uncertainty can be

calculated directly using the error formula or numerically using a technique called ’trowing toys’. 6.2.2 Method 1: analytic

The uncertainty on invariant mass, shown in equation 17, is calculated by applying the error formula, equation 16 to the calculation of the invariant mass, equation 1. The full calculations are in shown

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appendix A.1. The full expression depends on σpt ,ση and σφ . The uncertainty on η and φ are assumed

to be zero because there are very small compared to σpt . This makes equation21 considerably simpler.

σ2mH = 4 X i=1  cosh ηi X i6=j pj(1 − cos θij) MH   2 σp2_ti (17) 6.2.3 Method 2: numerical

To calculate the uncertainty of the invariant mass numerically the invariant mass is calculated a high number of times. Each time randomly altering the ptvalues of the muons according to its uncertainty.

This means that for a given pt a random new ptnew is drawn from the uncertainty distribution of pt.

This does not have to be a normal distribution but this is assumed in this thesis. With the four new ptnew

the invariant mass is calculated and this process is repeated a high number of times. How the result of these calculations are distributed (again, this doesn’t have be Gaussian) around the central result is an indicator for the uncertainty on the invariant mass. Because it is assumed that the uncertainty on ptis

a Gaussian error, the invariant mass is also Gaussian distributed and so the standard deviation of this distribution equals the uncertainty on the invariant mass.

6.3 The distribution of σ

mH

The uncertainty on ptgive rise to an uncertainty on the invariant mass. The uncertainty of mHcalculated

from σpt from the full ATLAS simulation is distributed as in Figure 4. This distribution can be used to

learn what properties events with high and low σmH have. This is done in section 9. These properties

can give information about how to improve the detector in order to have a smaller uncertainty. That is why the next section focus on recreating the distribution and not only the mean uncertainty of mH. So

that also for other Higgs boson masses one knows how to improve the ATLAS detector in order to really find heavier Higgs particles.

As said the simulating of the data is the most time consuming part of the data generation. To research how this distribution changes for different Higgs boson mass would take too much time. In section 7 a way to approximate the uncertainty on pt directly, with the goal to recreate the distribution of the

uncertainty on the invariant mass, is described.

Figure 4: The distribution of the uncertainty on the invariant mass. The uncertainty on the invariant mass is calculated from full ATLAS simulated σpt .

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7 Parametrisation of the uncertainty on p

t

As described in section 6 the uncertainty of the invariant mass can be calculated from the uncertainty of the transverse momentum σpt (and pt, η , φ ). σpt is given by the simulation. In this section the

data from a full ATLAS simulation of the H → ZZ∗ → µ+_µ−_µ+_µ− _{channel, with m}

H= 125 GeV, is

studied to find a parametrisation to calculate σpt directly from pt. This parametrisation is than used

to calculated σpt and σmH directly from the Monte Carlo generated data. The σmH distribution found

this way is than compared with the σmH distribution from the full ATLAS simulation.

7.1 The p

t

dependence of σ

pt

Is there is a relation between the transverse momentum of a muon and the uncertainty on its transverse momentum? When a muon has a really high momentum it’s radius of curvature in the detector becomes really big meaning the muons almost travels at a straight line and it becomes harder to measure its ptcorrectly, so the uncertainty increases.

When looking at the simulated distribution of pt vs. σpt of the muons in Figure 5 this correlation between

and ptand σpt becomes clear. Higher ptmuons have higher σpt , on average. Can this information be

used to calculate the uncertainty on ptdirectly from pt? In the next subsection a formula is discussed

that computes σmH from ptbased on ATLAS measurements.

Figure 5: The distribution of σpt with respect to pt. The line through the ptdistribution represents

the parametrisation from subsection 7.1.1. The parametrisation is only correct for the mean of the uncertainty at a certain pt .

7.1.1 The parametrisation of the uncertainty on pt

When ascribing an uncertainty to a certain value of pta parametrisation of the momentum resolution is

used. This parametrisation is made by the best fit for the measured uncertainties on pt[1]. A combination

is made of a function to best fit the uncertainties of the inner detector and another for the uncertainty of muon spectrometer. The relative uncertainty on ptfor different parts of the detector is given by.

σpt pt = c1+ c2pt Inner Detector (18) σpt pt = c0 pt + c1+ c2pt Muon Spectrometer (19) σpt pt = c1+ c0pt p1 + (c3pt)2 + c2pt Combined resolution (20)

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Where c0, c1, c2and c3are coefficients which are shown in table 3. Only equation 20 will be used in the

rest of this thesis and will be called the original parametrisation. These parametrisations are shown in Figure 6. It is clearly that real high pt muon have a high uncertainty. This can be explained by the

fact that these muon are barely bend by the magnetic field and it’s harder to reconstruct there radius of curvature and therefor harder to determine there momentum. For low ptmuons the muon spectrometer

has a lower resolution this can be explain that the muon lose energy in the detector before reaching the muon spectrometer on the outside of the detector.

In 5 the parametrisation is shown next to the ptvs. σpt distribution. It is clear that the parametrisation

is a accurate way to describe the correlation between σpt and pt.

Figure 6: The data points and parametrisation of the resolution of ATLAS of the inner detector, muon spectrometer and the combined resolution from [1]

Original Parametrisation 0.0 < |η| < 0.5 0.5< |η| < 1.0 1.0 < |η| < 1.5 1.5 < |η|

c0 0.06 0.747 2.919 0.765 -10

c1 0.016 0.014 0.017 0.020 0.0243

c2 0.00023 0.00013 0.00011 0.000194 0.00016

c3 100.0 1296 15560 -272.8 20368

Table 3: The coefficients of equation 20 for the original and improved parametrisation. In the improved parametrisation the coefficients differ for different η regions.

7.1.2 The σmH distribution using the parametrisation

When the uncertainties on ptcalculated with the original parametrisation is used to calculated the

un-certainty on the invariant mass the mean unun-certainty on the invariant mass is the same as simulated. The distribution of the invariant mass found this way is shown in Figure 7, in relation with the simu-lated distribution the distribution has to loo little high uncertainty events. This can be explained from the fact that the original parametrisation only uses the mean uncertainty at a certain pt . This is a

bad approximation for events where all four muons have an uncertainty higher than that of the original parametrisation. The uncertainty on the invariant mass would have been higher than is calculated. This results in a distribution with too much mean uncertainty events and too little high uncertainty events. As discussed in section 6.3 is it important to not only find the average uncertainty of mHbut also the

distribution. In the next section the parametrisation used above is expanded so that the σmH distribution

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Figure 7: The distribution of the uncertainty on the invariant mass. The distribution of σmH calculated

from σpt given by the original parametrisation (green) has the same mean as the distribution of

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8 The improved parametrisation

In this section the parametrisation used in section 7 is expanded. This is done because the Higgs boson mass uncertainty distribution found in section 7 differs to much from the distribution of the simulation to learn what the origin of high and low uncertainty is.

In section 7.1.2 it becomes clear that the parametrisation should be expanded because it can not recreated the σmH distribution given by the full ATLAS simulation. Thus the parametrisation must be expanded

to also cover the events with a high uncertainty on pt. In this section a relation between η and high

uncertainty muons is discussed. Since the only parametrisation only depends on ptthis relation is left

out of the equation. To improve the parametrisation a η dependence is added in this section.

8.1 The η dependence of σ

pt

High η muons may not pass through as many detector layers as small η muons which also results in a higher σpt . From Figure 8(a) relation between η and σpt don’t immediately become clear. In Figure

8(b) it is shown that muons with a high σpt have a |η | of roughly between 1 and 2. In the rest of this

section a way is described to use this information to approximate σpt of a muon with a parametrisation

that depends on pt and η .

(a) (b)

Figure 8: The η distribution looks uniform. Events with |η | between 1 and 2 have a higher σpt this is

more visible in Figure 8(b).

8.2 Finding the improved parametrisaton

When looking at events with |η| < 0.5 and |η| > 1.5 in the ptvs. σpt distribution in Figure 9 it shows

that the parametrisation returns a too high uncertainty for |η| < 0.5 and a too low uncertainty for |η| > 1.5. To determine how the original parametrisation should change for different η , the distribution of the uncertainty for different ptand η is determent.

This is done in bins of pt = 5 GeV. In each pt-bin the events are divided into |η| > 1.5, 1.5 > |η| >

1.0, 1.0 > |η| > 0.5, 0.5>|η| . For each of those bins the mean uncertainty is determined and plotted in Figure 10(a). It is clear that the mean uncertainties of the bins have roughtly the same shape of the original parametrisation

As expected the muons with higher η have also a higher σpt , when looking at a certain pt. Here is

assumed that the uncertainties of the muons are normally distributed around the mean with a standard deviation that is shown in Figure 10(b). The coefficients {c0, c1, c2, c3} of the original parametrisation

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(a) (b)

Figure 9: In left Figure only the muons high η (|η| > 1.5) are shown while in the right Figure only the muons with low η (|η| < 0.5) are shown. From this Figures it becomes clear that the parametrisation is to low for high η muons and to high for low η muons.

are changed so the parametrisation passes through the means for different η . The coefficients are shown in table 3

(a) (b)

Figure 10: For different pt and η the mean uncertainty is fitted. From the left figur it becomes clear

that event with |η| > 1.0 have a higher uncertainty than from the original parametrisation and events with |η| < 1.0 Have a lower uncertainty. The right Figure shows that the uncertainties are become wider spread around the mean as ptincreases

8.2.1 The σmH distribution using the improved parametrisation

When the uncertainty on the invariant mass is determined with the uncertainties on ptfrom the improved

parametrisation the distribution of the uncertainty on the invariant mass is closer to that of the simulated uncertainty. This is shown in Figure 11 where the improvement relative to Figure 7 is clearly visible. This means that the improved parametrisation is a valid way to calculate σpt . With the improved

parametrisation σpt and σmH can be calculated directly from Monte Carlo generated data and the time

consuming ATLAS simulation can be left out. In the next section a closer look is taken to see how events with a low uncertainty of mHdiffer from events with a high uncertainty of mH. In section 10 the

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Figure 11: The distribution of the uncertainty on the invariant mass. The distribution of σmH calculated

from σpt given by the improved parametrisation (green) has the same mean and roughly the same

distribution as the distribution of σmH calculated from σpt given by the full ATLAS simulation (blue)

uncertainty on mHis calculated with the above described method for different Higgs boson masses. The

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9 Origin of the uncertainty on the Higgs Boson mass

The origin of the uncertainty of the Higgs boson mass is studied in this section. From the previous section it has become clear that the simulated σmH distribution can recreated with using the improved

parametrisation. Now the σmH distribution is studied to find the main difference between events with a

high uncertainty on the Higgs boson mass and events with a low uncertainty. This is done by looking at the σpt , ptand η distribution of the four muons separately in order of their σpt .

9.1 Characteristics of events with a high uncertainty on the invariant mass

The uncertainty on the transverse momentum of a single muon has a correlation with ptand |η| . One

Higgs boson decays to four muons each with different, almost random, ptand η . Say one muon from

the Higgs boson decay has a high ptand |η| and therefor also a high σpt . How does this influence the

uncertainty on the invariant mass? The effect of one muon with a high σpt could be cancelled by a muon

with a low σpt .

To further investigate this the events with a high uncertainty on the invariant mass (σmH > 2.5) are

separated from the events with a low uncertainty on the invariant mass (σmH < 1.3) and the properties

off the muons are ranked from highest to lowest uncertainty. The distribution of the σpt , pt and η are

shown in Figures 12 , 13 and 14. The distribution of the highest σpt muon is displayed in the (a) Figures.

Figure 12 shows that the muon with the highest uncertainty differs the most when comparing events with high and low σmH . The second and third muon from high σmH events have a higher uncertainty

on average than muon from low σmH events but the distribution don’t differs as much as the first muon.

The fourth muon from high σmH events almost has the same uncertainty distribution as the the fourth

muon from low σmH events.

When looking at the ptdistribution of the muons from high and low σmH events it becomes clear that

ptis not a indicator for high σmH events. Only the muon with the highest uncertainty has slightly higher

pt. The distributions in Figure 13b , 13c , 13d are almost the same for events with high and low σmH .

High and low σmH events differ mainly in the |η| distribution of their muons. In Figure 14 there can be

seen that all the muons from events with high σmH have a higher |η| than the muons from events with

low σmH . Where the low σmH muons mostly have a |η| > 1 , the high σmH muon have a |η| concentrated

between 1 and 2.7. Above 2.7 there are no events because there are no detector parts in this direction. So for an event with high σmH it is typical to have four muons with an high |η| . In Figure 15 the average

η of an events plotted against the uncertainty on the invariant mass from this plot it becomes clear that high σmH events have four muons in the same high η region.

9.2 The uncertainty on the invariant mass and the boost of the Higgs boson

In Figure 16 a relation is visible between the average η of a event and the gamma factor of the Higgs boson. If the Higgs boson has a high boost the pseudorapidity of the all four of the muons is high. There also exist a relation between the uncertainty on the average η and the uncertainty on the invariant mass (Figure 14 and 15).

This means the uncertainty on the invariant mass is strongly correlated with the gamma factor of the Higgs boson because when the Higgs boson has a high gamma factor the fraction pz

|p| is around one [11].

This means that the Higgs boson and the muons it decays to has a high |η| . Muons with a high |η| intend to have a higher σpt and this results in a higher σmH .

At mHequals 125 GeV events with a high σmH come from Higgs bosons that have a high boost. This

results in high |η| muons with high σpt . In the next section a study is done into how the average

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(a) (b)

(c) (d)

Figure 12: The σpt distribution for the four muons of high and low σmH events. The muons are ordered

to their uncertainty so (a) is the distribution of the muon with the highest σpt . Figure (a) and (b) show

a that the two highest error muon differ between low and high σmH events. The distribution in (c) and

(d) are similar for both cases.

change for different mH. This because the average |η| and boost of the Higgs boson are indicators for

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(a) (b)

(c) (d)

Figure 13: The ptdistribution for the four muons of high and low σmH events ordered from high to low

σpt muon. Only the muon with the highest σpt of an event has a slightly higher average pt. This means

that the main reason for a event to have a high uncertainty on the invariant mass is not that the muons have a high average pt.

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(a) (b)

(c) (d)

Figure 14: The η distribution for the four muons of high and low σmH events ordered from high to low

σpt muon. All muons from high σmH event have a higher |η| than muons from low σmH event. This is

the main effect that causes events to have a high uncertainty on the invariant mass

Figure 15: (a) The average η of a event vs the uncertainty on the invariant mass of that event. Events with a high average η have a higher σmH . (b) The normalised η distribution for high and low σpt muons.

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(a) (b)

Figure 16: The distribution of pseudorapidity versus the gamma factor of the Higgs boson. Left the mean η of the four muons is displayed. Right the η of muons is plotted with respect to the gamma factor of the Higgs boson

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10 The uncertainty on the Higgs boson mass for different Higgs

boson mass

The improved parametrisation from section 8 is used to simulated the uncertainty on the invariant mass of different Higgs boson masses. In the previous section it has become clear that the average |η| of the four muons and the boost of the Higgs boson are indicators for high σmH . So in this section the change

in σmH for different mHis compared with the change of average |η| and boost of the Higgs boson. This

is also done for average pt. The results are discussed in the following section.

Some beyond standard model theories that there are more than one Higgs boson. These other Higgs boson are expected to have a higher mass. These other Higgs bosons can also decay to four muons via two Z bosons. What is the effect of a different Higgs boson mass on the uncertainty on the invariant mass in this channel?

To simulate the effects on the uncertainty on the invariant mass the run time of the detector simulation for all the different Higgs boson masses would take about a week. The simulation can be simplified by using the improved parametrisation from chapter 8 to calculate σpt directly from ptand |η| , given by

Monte Carlo generated data. This way the run time of the simulation is reduced to only 1.5 hour.

10.1 The uncertainty for different Higgs boson mass

The average uncertainty on the invariant mass for the different masses of the Higgs boson is displayed in Figure 17. The average uncertainty increases with the Higgs boson mass. The relative uncertainty doubles from around 1% at mH = 200 GeV to around 2% at mH = 1000 GeV. The origin of the

uncertainty on the invariant event discussed in subsection 9.1. There is found that events with a high uncertainty on the invariant mass are often events where the Higgs boson has a high γ factor. The high γ factor relates to high |η| muons which have a higher σpt . The higher σpt of the muons means a higher

σmH on the invariant mass.

Figure 17: the average uncertainty on the invariant mass is displayed relative to Higgs boson mass.

10.2 Origin of the change in uncertainty

The average γ factor of the different Higgs particles is shown in Figure 18(b). The average γ decreases for higher Higgs boson masses. This could be explained from the fact that to produces a heavier Higgs boson both the partons of the colliding protons need a high momentum. When the partons both have similar momentum the boost of the Higgs boson is lower. Less high γ Higgs particles should relate to less muon with a high |η| . In Figure 18(a) the average |η| of the muons is displayed. While the uncertainty

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on the invariant mass increases the average |η| decreases. This was expected from the fact that more massive Higgs particles have lower average boost.

(a) (b)

Figure 18: In (a) the average η of the muons In (b) the average boost of the Higgs boson is displayed relative to the Higgs boson mass

The increase of σmH for more massive Higgs particles is not caused by a increase in average γ. What

other effect could caused the increase. Note that σmH depends only on σpt which is calculated from

ptand η . So if the average σpt did not increase because of muons with an average higher |η| an increase

in average ptis expected to account for the rise in σpt . This suspicion is confirmed in Figure 19 where

a clear increase in the average ptof the muons is visible.

Thus the change of average σmH is not caused by an increase in Higgs boson boost as was expected but

by a rise in average pt. This means that in order to decrease the uncertainty of the invariant mass in

the H → ZZ∗→ µ+_µ−_µ+_µ− _{channel ATLAS should improve the muon p}

t measurement. To learn more

about the change in origin of high σmH events one could use the same method as in section 9.

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11 Conclusion

11.1 Findings of this thesis

In this thesis in was accomplished to do a simulation of the ATLAS detector of the uncertainty on the Higgs boson mass in the H → ZZ∗ → µ+_µ−_µ+_µ− _{channel for different Higgs boson masses. This is}

done in three steps as described below. 11.1.1 Improved parametrisation

The main result of this thesis is the improved parametrisation of the uncertainty on the transverse momentum of muons. The improved parametrisation is an extension of the parametrisation from [9]. This original parametrisation was a function that depended only on pt. The improved parametrisation

uses the same function with different parameters for different regions of the pseudo rapidity of the muons. The uncertainty on the invariant mass of the four muons can be calculated from the uncertainty on ptfor each muon. If uncertainty on ptis determined with the improved parametrisation the calculated

uncertainty on the invariant mass is close to the uncertainty of the simulation. This means that the improved parametrisation can be used instead of the detector simulation. This is an advantage since the detector simulation is a time consuming process and calculating the uncertainties directly is not. 11.1.2 Origin of the uncertainty on the Higgs Boson

In this thesis it is found that the the uncertainty on the Higgs boson at 125 MeV is related to the boost of the Higgs boson. The boost of the Higgs is related to a high average |η| of the muons which have a higher uncertainty on pt than low |η| muons. This is in accordance with the findings of [11]

11.1.3 Evolution of the uncertainty at different mH

The uncertainty on the Higgs mass is positively related to its mass. At 125 MeV the uncertainty is related to the boost of the Higgs boson. So it was expected to see an increase in average boost. This is not the case, the average boost decrease for higher Higgs mass. The average ptincreases and is expected

to be the origin of the increasing uncertainty.

11.2 Dankwoord

Allereerst wil ik Ivo van Vulpen hartelijk bedanken voor zijn fantastische begeleiding van dit project. Ivo bedankt dat je zoveel tijd kon vrij maken voor het bespreken van de voortgang, op deze manier kwam het onderwerp echt tot leven en heb ik veel geleerd, zowel over dit project als daarbuiten. Marco bedankt dat jij ook een oogje in het zeil hield vanuit Genve.

Bedankt Olmo, dat je steeds de Linux-computer aan de praat kreeg. Lisa bedankt voor je strenge feedback op m’n scriptie. Annet en Klaartje bedankt voor jullie steun.

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11.3 Summery

In this thesis the uncertainty on the invariant mass (σmH ) of four muons in the H → ZZ

∗ _{→ 4µ}

channel is approximated at different Higgs boson mass. Full ATLAS simulated data, with mH = 125

MeV, is used to find the origin of the uncertainty. Then this information is used to make a approximate simulation for σmH at other mH.

The full ATLAS simulation produces the transverse momentum (pt) , the pseudorapidity (η ), the angle

(φ) and the uncertainty on pt (σpt ) of the muons and the uncertainty on the invariant mass. The

in-variant mass is calculated from the transverse momentum, the pseudorapidity and the angle the muons have. Each off those quantities also have a corresponding uncertainty. This means that the calculated invariant mass also has a certain uncertainty. This uncertainty on the invariant mass can be calculated from the uncertainties on pt, η and φ.

The uncertainties on η and φ are neglected because they are very small compared to the uncertainty on pt(σpt ). So the uncertainty on the invariant mass can be calculated from σpt . To simulated σpt and

thus σmH in the full ATLAS simulation takes a long computation time, this means that the simulation

of σmH for different Higgs boson masses can not be done because it would take too much computation

time. In this thesis a parametrisation is made for σpt so that is can be calculated directly from ptand

η . So that the simulation time reduces significantly

To calculate σpt directly from ptand η a known parametrisation of σpt that depends only on pt is

ex-panded. It was found that this parametrisation was too high for low |η| muons and too low for high |η| muons. To expand the parametrisation to also cover this behaviour the data from the muons was divided into 20 bins for ptand 4 bins for |η| . For each bin the average uncertainty was determined.

For each |η| the known parametrisation with unknown parameters was fitted through the average uncer-tainty. This means that instead of one parametrisation there is now one for each bin of |η| .

This improved parametrisation is used to approximate σpt and σmH for data generated (not simulated)

at other Higgs boson mass. A correlation if found between mH and σmH is found. It is found that at

mH = 125 MeV events that have a high uncertainty are events that have a high boost of the Higgs boson

which results in high |η| muons with high σpt . At mH= 1000 MeV the average boost of the Higgs boson

and the average |η| of the muons is lower, the average ptis higher. The rise of average pt is expected to

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11.4 Populair wetenschappelijke samenvatting

In wetenschappelijk onderzoek toetsen onderzoekers theorien aan de werkelijkheid met experimenten. Vanuit een theorie word berekend of bedacht wat voor resultaten een bepaald experiment zullen komen. Als dit overeenkomt betekend dit dat de theorie klopt en als dit niet overeenkomt is een nieuwe betere theorie nodig die de resultaten wel kan verklaren.

In onderzoek naar subatomaire deeltjes geldt dit principe ook. De onderzochte theorie is het ’stan-daard model van elementaire deeltjes’ en het experiment is de deeltjes versneller en detector in Genve. Waar protonen met hele hoge snelheid op elkaar worden gebotst en de deeltjes die daaruit ontstaan wor-den gedetecteerd Om uit te rekenen wat er verwacht wordt te vinwor-den in dit experiment worwor-den hele zware computerprogramma gebruikt. Die elke botsing nabootsen en voor elk deeltje stap voor stap berekenen hoe snel ze gaan en hoe ze door de detector vliegen.

In zo ’n berekende botsing zullen sommige deeltjes gaan toevallig door veel detectie-lagen andere missen ze net. Hoe minder vaak de deeltjes gedetecteerd worden, hoe minder nauwkeurig ze gemeten worden. Om deze nauwkeurigheid voor elk deeltje te bepalen kost heel veel tijd. In deze scriptie is onderzocht of je de nauwkeurigheid kan bepalen zonder de zware computerprogramma te gebruiken.

Door dit te doen kan je snel voorspellen wat er gebeurt als er deeltjes zijn die nu nog niet zijn ont-dekt. Er zijn namelijk andere theorien dan ’het standaard model’ die zo ook met een experiment worden getest.

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References

[1] G. Aad et al. Studies of the performance of the ATLAS detector using cosmic-ray muons. Eur.Phys.J., C71:1593, 2011.

[2] Georges Aad et al. Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys.Lett., B716:1–29, 2012.

[3] O. Behnke, K. Kr¨oninger, G. Schott, and T. Sch¨orner-Sadenius. Data Analysis in High Energy Physics: A Practical Guide to Statistical Methods. Wiley, 2013.

[4] Amos Breskin and Rdiger Voss. The CERN Large Hadron Collider: Accelerator and Experiments. CERN, Geneva, 2009.

[5] Serguei Chatrchyan et al. Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys.Lett., B716:30–61, 2012.

[6] A. Das Thomas Ferbel. Introduction to Nuclear and Particle Physics. World Scientific, 2003. [7] Peter W. Higgs. Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett., 13:508–509,

Oct 1964.

[8] Maarten Post and Olmo Kramer. Workshop higgs particle. Jan 2015.

[9] Antonio Salvucci. Measurement of muon momentum resolution of the ATLAS detector. EPJ Web Conf., 28:12039, 2012.

[10] M. Thomson. Modern Particle Physics. Modern Particle Physics. Cambridge University Press, 2013. [11] Danne van Roon. A study of the origin of the error in the reconstructed higgs boson mass at the

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A

Calculations

A.1 Analytic error on the Higgs Boson mass

In this appendix the full calculations made for this thesis are shown. This is done do increase readability. First is shown how to calculate the invariant mass from measured quantities. Secondly the error formula is applied to this expression to find a expression of the uncertainty of the invariant mass in terms of measured quantities and corresponding uncertainties.

A.1.1 Invariant mass

MH= v u u t( 4 X i=1 Ei)2− | 4 X i=1 ~ pi|2 | 4 X i=1 ~ pi|2= ( X pix) 2_{+ (}X piy) 2_{+ (}X piz₎2 = (Xpitsin(φi)) 2_{+ (}X pitcos(φi)) 2_{+ (}X pitsinh(ηi)) 2 Ei= q |~pi|2+ m2i =qp2 ix+ p 2 iy + p 2 iz+ m 2 i = q p2 it(cos 2_(φ i) + sin2(φi) + sinh2(ηi)) + m2i = q p2 itcosh(ηi) 2_{+ m}2 i = pitcosh(ηi) ( 4 X i=1 Ei)2= ( X pitcosh(ηi)) 2 MH= q (Xpitcosh(ηi)) 2_{− (}X_p itsin(φi)) 2_{− (}X_p itcos(φi)) 2_{− (}X_p itsinh(ηi)) 2 =√ℵ

A.1.2 The error formula

σ2_M_H =X i ∂MH ∂pti 2 σ_p2 ti + ∂MH ∂ηi 2 σ_η2_i+ ∂MH ∂φi 2 σ2_φ

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A.1.3 Derivative of the invariant mass with respect to pt , calculation 1 ∂p_tiMH = 1 2√ℵ∂ptiℵ ∂p_tiℵ = ∂p_ti( X pitcosh(ηi)) 2_{− (}X pitsin(φi)) 2_{− (}X pitcos(φi)) 2_{− (}X pitsinh(ηi)) 2 = 2(Xpitcosh(ηi)) cosh(ηi) − 2( X pitsin(φi)) sin(φi) − 2( X pitcos(φi)) cos(φi) − 2(Xpitsinh(ηi)) sinh(ηi) = 2pticosh 2_(η i) + 2 X i6=j

pticosh(ηi) cosh(ηj) − 2ptisin

2_(φ i) − 2 X i6=j ptisin(φi) sin(φj) − 2pticos 2 (φi) − 2 X i6=j

pticos(φi) cos(φj) − 2ptisinh

2 (ηi) − 2 X i6=j ptisinh(ηi) sinh(ηj) = 2pti(cosh 2

(ηi) − sinh2(ηi) − sin2(φi) − cos2(φi)) + 2

X

i6=j

pti(cosh(ηi) cosh(ηj)

− sin(φi) sin(φj) − cos(φi) cos(φj) − sinh(ηi) sinh(ηj))

= 2pti(1 − 1) + 2

X

i6=j

pti(cosh(ηi) cosh(ηj) − sin(φi) sin(φj) − cos(φi) cos(φj) − sinh(ηi) sinh(ηj))

= 2X

i6=j

pti(cosh(ηi) cosh(ηj) − cosh(ηi) cosh(ηj) cos(θij))

= 2 cosh(ηi)

X

i6=j

pticosh(ηj)(1 − cos(θij))

A.1.4 Derivative of the invariant mass with respect to pt , calculation 2

MH= s X i X i6=j pipj− X i X i6=j pipjcos θij [11] = s X i X i6=j ptiptjcosh(ηi) cosh(ηj) − X i X i6=j

ptiptjcosh(ηi) cosh(ηj) cos θij

∂p_tiMH=

2 cosh(ηi)Pi6=jptjcosh(ηj) − 2 cosh(ηi)

P

i6=jptjcosh(ηj) cos θij

2qP

i

P

i6=jptiptjcosh(ηi) cosh(ηj) −

P

i

P

i6=jptiptjcosh(ηi) cosh(ηj) cos θij

=

cosh(ηi)

h P

i6=jptjcosh(ηi) cosh(ηj) −

P

i6=jptjcosh(ηj) cos θij

i MH = cosh(ηi) P i6=jpj(1 − cos θij)) MH

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A.1.5 Derivative of the invariant mass with respect to pt , calculation 3 ∂MH ∂pi = ∂MH ∂pticosh(ηi) = 1 cosh ηi ∂MH ∂pti ∂MH ∂pi =X i6=j pj(1 − cos(θij)) MH [11] ∂MH ∂pti = cosh ηi ∂MH ∂pi = cosh ηi X i6=j pj(1 − cos(θij)) MH

A.1.6 Derivative of the invariant mass with respect to η

∂ηkmh=

∂ηk

P

i

P

i

P

i6=jptiptjcosh(ηi) cosh(ηj) cos θij

2mh

= ∂ηk P

i

P

i

P

i6=jptiptj(sin(φi) sin(φj) + cos(φi) cos(φj) + sinh(ηi) sinh(ηj)

2mh

= ptksinh ηk P

i6=kpticosh ηi− ptkcosh ηk

P

i6=kptisinh ηi

mh

= ptk P

i6=kpti(sinh ηkcosh ηi− cosh ηksinh ηi))

mh

= ptk( P

i6=kpti(sinh(ηk− ηi))

mh

A.1.7 Derivative of the invariant mass with respect to φ

∂φkMH=

∂φk(P P pp(sin φ sin φ + cos φ cos φ + sinh η sinh η))

MH

= P pp cos φksin φj− pp sin φkcos φj MH

= ptk

P

j6=kptjsin(φj− φk)

MH

A.1.8 The error formula

σ2_M_H =X i ∂MH ∂pti 2 σ_p2 ti + ∂MH ∂ηi 2 σ_η2_i+ ∂MH ∂φi 2 σ2_φ σ2_M H = X i cosh ηi P j6=ipj(1 − cos(θij)) MH 2 σ_p2 ti+ pti P j6=iptj(sinh(ηi− ηj)) MH 2 σ2_η i+ pti P j6=iptjsin(φj− φi) MH 2 σ_φ2 i (21)

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A.2 Expected momentum of the muons

Ebef ore = m2H (22) Eaf ter = 4 q p2 µ+ m2µ (23) (24) Ebef ore = Eaf ter (25)

m2_H = 4qp2 µ+ m2µ (26) (27) pµ= r m_H 4 2 − m2 µ (28) =mH 4 (29) = 31.2 GeV (30)

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Simulating the uncertainty on the Higgs boson mass