finding the higgs boson via vector boson fusion in …...de higgs-boson gevonden met een...
TRANSCRIPT
Finding the Higgs boson via Vector Boson Fusion in the
WW*→ `ν`ν decay channel with the ATLAS detector at√
s =
13 TeV with a misidentified jet
Maarten Hammer
10525319
July 5, 2017
Report Bachelor Project Physics and Astronomy, size 15 EC
Conducted between 10-4-2017 and 10-7-2017
Supervisors: Lydia Brenner Msc & Carsten Burgard Ph.D.
Second Assessor : dr. Ivo van Vulpen
Research institute: Nikhef
Wordcount : 3949
Abstract
Analysis of the VBF Higgs production in the decay mode H → WW ∗ → `µ`µ. In the
case where only one jet has been identified. The data is collected with ATLAS detector in
the LHC at a luminosity of 36fb−1 recorded at√s = 13 TeV. Higgs boson mass is assumed
to be 125 GeV. The data is analysed in three distinct cases. The first case assured as little
simulated signal loss as possible and resulted in a 2.2(0.7)σ significance. The second case had
aimed for a high signal over background ratio disregarding signal conservation and resulted
in a signal over background rate of 0.20 with a significance of 1.7(0.5)σ. In the last case a
balance was obtained between between VBF signal conservation and signal over background
ratio. The last scenario resulted in a significance of 1.9(0.5)σ and signal over background
ration of 0.09.
Populair wetenschappelijke samenvatting
Het Higgs-boson is in 2012 door de wetenschappers van de ATLAS en CMS samenwerk-
ingsverbanden bevestigd. Ze hebben dit deeltje ontdekt door in de Large Hadron Collider
in Geneve protonen met zeer hoge snelheden te laten botsen. Bij deze botsing vervallen de
deeltjes in de protonen op verschillende manieren naar andere deeltjes. Door deze verval-
processen te modelleren kunnen voorspellingen worden gedaan over deze vervalkettingen.
Deze voorspellingen zijn vergeleken met de data die de ATLAS en CMS detectors meten en
hieruit is geconcludeerd dat het Higgs-boson een massa heeft van 125 GeV. Dit onderzoek
hebben ze gedaan door naar veel verschillende vervalprocessen te kijken en de resultaten
samen te voegen in een groot onderzoek.
In mijn onderzoek heb ik gekeken of ik in een specifieke vervalketting de Higgs kon vin-
den. Ik heb gekeken naar het geval dat de protonen W- of Z-deeltjes uitstralen die fuseren
in het Higgs deeltje. Dit noemt men Vector Boson Fusion omdat W- en Z-deeltjes vector
bosonen zijn en deze deeltjes fuseren in een Higgs-boson. Het Higgs-boson heeft een korte
levensduur, en vervalt voor het gedetecteerd kan worden. Het Higgs-boson kan alleen wor-
den gezien door zijn vervalproducten te bekijken. Ik heb gekeken naar het vervalprocess
waarbij het Higgs-boson naar twee W-deeltjes vervalt. Die dan weer vervallen in elektronen,
muonen en neutrinos.
Bij deze vervalketting zijn er vier deeltjes meetbaar door de ATLAS detector, twee jets
(deeltjes van de protonen) en twee geladen leptonen (een muon en een elektron). De neu-
trinos zijn niet meetbaar door de detector. Neutrinos zijn wel te reconstrueren door te
kijken waar de detector energie en impuls mist. De detector meet niet altijd alle deeltjes
die meetbaar zijn, en gevallen waar een deel van de data mist wordt meestal weggegooid. Ik
kijk wel naar deze data. Mijn onderzoek kijkt naar de data waar een jet mist of verkeerd is
geidentificeerd.
Bij de botsingen in de LHC wordt veel meer gemaakt dan alleen het signaal waar ik
naar kijk. Andere vervalketens kunnen dezelfde eindproducten hebben als het signaal waar
ik naar kijk (een jet, twee geladen leptonen en missende energie). Deze andere vervalketens
zijn achtergrond data voor mijn onderzoek naar het Higgs-boson. Door de reacties te mod-
elleren kan er een voorspelling worden gemaakt hoe deze andere vervalprocessen zich gedra-
gen. Door in deze voorspellingen te kijken waar deze achtergrond groter is dan het signaal
(VBF Higgs-bosonen) en deze achtergrond weg te knippen blijft er een kleine dataset over
waar veel signaal in zit relatief tot de achtergrond. Deze kleine dataset wordt de signaal
regio genoemd.
Deze selectie criteria heb ik daarna ook op de gemeten data toegepast van de ATLAS
detector. Door de gemeten data en de gesimuleerde data te vergelijken met elkaar heb ik
de Higgs-boson gevonden met een statistische significantie van 2.3σ, dit komt neer op een
zekerheid van ongeveer 95%.
2
Contents
1 Introduction 4
2 Theory 5
2.1 Vector Boson Fusion Higgs boson production to W-Boson pair decay . . . . . 5
2.2 Data simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Data gathering in the ATLAS detector . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Significance determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 VBF Analysis 8
3.1 Preselection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Background rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Event selection example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Results 12
5 Conclusion 14
6 Discussion 14
7 Acknowledgements 15
3
1 Introduction
For the discovery of the Standard Model (SM) Higgs boson in 2012 by ATLAS [3] and CMS
[6], various production modes and decay channels have been used to reach a 5σ significance
for the discovery of the Higgs boson with a mass of 125 GeV. The analysis of the particle
has continued with higher center-of-mass energy to rediscover the Higgs boson in each in-
dividual production mode. In Vector Boson Fusion (VBF) this 5σ significance has not yet
been achieved.
In this report the VBF produced Higgs boson is analysed in the WW → `ν`ν decay
channel. In the case of a clean signal, the ATLAS detector can detect two quark jets, two
charged leptons and missing energy. The analysis has already been done in the clean signal
scenario and has resulted in a 1.9σ signal significance [5]. An analysis on data sets with
missing information has not been done yet.
The scenario where one jet has been misidentified is discussed in this report. In this
scenario the most effective event selection variables are unusable as they rely on data from
both jets. To reduce the background in this dataset a different approach has to be applied.
A set of 14 variables has been created to make an event selection on. Three of these variables
have been defined for this analysis specifically.
The analysis has been performed at a center-of-mass energy of√s =13 TeV with a 36
fb−1 luminosity using the ATLAS detector at the LHC. Due to large Drell-Yan background
of Z→ ee/µµ only states with opposite flavour and opposite sign have been considered. All
other Higgs boson production modes have been considered as background in this analysis
and a Higgs boson mass has been assumed of mH = 125 GeV.
Section 2 explains the production mode and decay channels of the SM VBF Higgs boson
as well as the data production. Section 3 explains three distinct selection processes used to
find the Higgs boson in this mode. In Section 4 the results are displayed and in 5 they are
presented. Finally, these results and future improvements to this analysis are discussed in
Section 6.
4
2 Theory
2.1 Vector Boson Fusion Higgs boson production to W-Boson pair
decay
Vector Boson Fusion (VBF) is one of four dominant production modes of the Higgs boson
in the LHC [4]. The other three dominant production modes are gluon-gluon fusion (ggF);
associated production with a vector boson (VH) and with a top-pair (ttH). The distinct
difference between VBF and other Higgs boson production modes is that the quark jet is
not consumed in the reaction. In VBF two quarks produce vector bosons (W bosons or Z
bosons) which fuse into other particles. One of the possible results is a Higgs boson.
The lifetime of the Higgs boson is around 10−22seconds [7], which has the effect that the
ATLAS detector can only measure the decay products. The Higgs boson has five dominant
decay channels in the ATLAS detector [3]. The Higgs boson decay into a W boson pair is
discussed here. The full VBF Higgs boson → WW → `µ`µ production chain is displayed in
Fig. 1. Other decay channels of the Higgs boson are the H→bb, H →ZZ and H→ ττ .
Figure 1: The Feynman diagram describing the VBF Higgs boson production if it decays in two
W bosons who then decay into a two lepton-neutrino pairs [8].
2.2 Data simulation
The dataset used in this analysis consists of two parts, a simulation with truth information
and observed data by the ATLAS detector. The simulated dataset is generated by Monte
Carlo (MC) generators at√s = 13 TeV with an intensity of 36 fb−1. The data is simulated
to contain ggF and VBF Higgs bosons at 125 GeV. ttH and bbH Higgs boson production
modes have been neglected because of their small cross section. The VH associated produc-
tion modes have been neglected due to the small cross section in the Nlep = 2 region. The
ggF Higgs boson production mode is considered as a background for this analysis.
The simulation is accurate to next-to-leading order in all signal channels as well as ac-
curate in next-to-next-to-leading order in ggF Higgs boson production. The background
signals analysed are events with two W bosons (WW events); non-Higgs boson VBF events;
5
single top-quark production events; W+jet production events; Z+jet production events as
well as other events with two bosons (other VV events) this includes Wγ, Wγ∗, WZ and
ZZ events. Further explanation on the background simulation, including the MC generators
used, can be found in Ref. [5].
2.3 Data gathering in the ATLAS detector
The observed data is measured by the ATLAS detector. A Toroidal LHC ApparatuS (AT-
LAS) is one of seven detectors in the Large Hadron Collider (LHC). The ATLAS detector is
a forward-backward symmetric cylindrical particle detector [2] and covers close to 4π solid
angle. It detects according to the coordinate system explained in Fig. 2.
The ATLAS detector consist of an inner tracking detector to determine the charge and
momentum of charged particles by tracking their movement through a strong magnetic field.
Around the inner detector hadronic and electromagnetic calorimeters determine the energy
of particles. In the outer shell large muon spectrometers determine the momenta and charge
of muons, which move through all other parts of the detector unhindered. Neutrinos are not
detectable by the ATLAS detector, but using the momentum and energy imbalance across
the 4π solid angle EmissT and pmissT can be determined which contains information about the
neutrinos [10].
Figure 2: The ATLAS detector uses a right-handed coordinate system with its origin at the
collision point in the beam pipe. The z-axis follows the beam pipe, the x-axis is directed to the
center of the LHC, the y-axis is points upward. The φ angle is measured with respect to the
x-axis. The θ angle is measured with respect to the z-axis. The pseudorapidity η is defined as
η = - ln tan(θ/2) [12].
6
2.4 Significance determination
With the final dataset the significance σobserved of the signal can be determined by the
normal Poisson significance given in Equations 1 and 2 [1]. This is under the assumption
NS << NB [1].
S1 =NS√NB
(1)
S2 =NS√
NS +NB(2)
If NS and NB are of the same order of magnitude Eq. 3 should be used instead [1].
2 ∗ S12 = 2 ∗ (√Ns +NB −
√NB) (3)
In these equations the NS is defined as NData −NB = NS for the observed significance
σobs. NB is defined to be the simulated background (Section 2.2) and NData is defined to be
the observed data by the ATLAS detector (Section 2.3). To calculated expected significance
σexp, NS is set as the amount of simulated VBF events.
VBF signal strength is defined as µ [5]:
µV BF =σobsσexp
(4)
7
3 VBF Analysis
Event selection processes have been applied to MC simulated data obtained in Section 2.2
and the observed data obtained in Section 2.3. The analysis is split in two parts. The
first part is preselection to get a dataset conforming with the channel selection VBF Higgs
boson→WW→ `ν`ν. The second part are arbitrary event selections made to reject as much
background as possible.
The background rejection selection criteria have been done in three distinct scenarios. In
the first scenario the criteria were applied such that as much signal is preserved as possible.
In the second scenario the criteria were applied very restrictively, aiming for a high signal to
background ratio. In the third scenario the criteria were balanced such that the final signal
region never drops below 3 simulated VBF events in the MC dataset. The final signal regions
(SR) have been named relaxed signal region, restrained signal region and semi-restrained
signal region. In Table 1 the cuts are shown in order of application to the final dataset with
respect to each signal region.
3.1 Preselection
To get a VBF Higgs boson→WW→ `ν`ν compatible event selection, a basic preselection
has been applied. Final state leptons are required to have a different flavour and different
charge. The most energetic lepton is classified as the leading lepton and is required to have
a pT > 22 GeV the other lepton is considered the sub-leading lepton and is required to have
a pT > 15 GeV. To reduce low-mass resonances, the invariant dilepton mass is limited at
m`` > 10GeV . To get events with one missing or misidentified jet Njet = 1 is set. Finally
a Nb−jet = 0 cut and a Z → ττ veto is applied to exclude top and Z-background from the
data. The Z → ττ veto eliminates most Z → ττ background by requiring the invariant ττ
mass (mττ ) to be away from the Z boson resonance [11].
3.2 Background rejection
After the preselection, the background rejection event selection is applied in the form of
data cuts. An overview of these cuts is displayed in Table 1. The restrained and semi-
restrained signal regions miss cuts because of the dependence of cuts. A harsh cut on
∆R`` =√
(∆φ``)2 + (∆η``)2, makes a cut on ∆φ`` or ∆η`` less effective or obsolete. In the
next paragraph each cut will be clarified in chronological order of application in the last
run. The order of application of the cuts has been changed repeatedly to check dependence
of cuts on each other.
The first applied cut is on the missing transverse energy EmissT caused by the immea-
surable neutrinos. This cut reduces mostly W+jet background which also produces neu-
trinos. The next cut is on the vectorial sum of the transverse mass of the two leptons
8
√(m`0
T )2 + (m`1T )2, this cut reduces the heavier decay products from the background. After
that a cut is made on the vectorial sum of the transverse momentum of the two leptons p``T .
This is a cut that reduces the relative strength of the WW and tt background. After this a
cut is applied on the transverse mass of the leading lepton m`0T . This cut has a similar effect
as the cut on the vectorial sum of the two transverse masses of the lepton described above,
which is why it is left out.
Next up are the cuts on the angles between the two leptons. The first cut in this cate-
gory is the ∆R`` =√
(∆φ``)2 + (∆η``)2 cut. This cut combines the azimuthal separation
∆φ`` and the rapidity separation ∆η``. These angles between the leptons are a result of the
H→WW split in the decay mode. For signal events these angles tend to be small. After the
cut on ∆R`` fine tuning cuts can be applied on ∆φ`` and ∆η``, but in the more restricted
scenarios this does not improve the data.
After the cuts on the separation of the two leptons a new set of cuts is defined. These
cuts are inspired by the outside lepton veto. The outside lepton veto requires the final state
leptons to exist in the rapidity gap spanned by the two jets in the final state [9]. As this
analysis only considers single jet events, this veto is not possible. To simulate the correla-
tion between the two jet system and the two lepton system three new variables have been
declared and cut on. ∆η``,jet compares the rapidity separation between the combined two
lepton system and the jet. ∆φ``,jet compares the azimuthal separation between the com-
bined two lepton system and the jet. ∆R``,jet =√
(∆φ``,jet)2 + (∆η``,jet)2 is a dependent
variable on the combination of the two. Te only link between the two lepton system and the
jet system is the VBF Higgs boson, so any correlation between the two systems is caused
by the Higgs boson.
Three final cuts have been applied. The cut on invariant mass of the dilepton system
m`` reduces WW and tt background. In VBF Higgs boson production the jet prefers to
move along the beam pipe and thus the rapidity ηjet is often higher than in scenarios where
non VBF events occur. A final cut is applied in the strict region to the momentum of the
leading lepton p`0T which reduces the W boson and ggF Higgs boson events at cost of large
signal strength.
9
Condition Relaxed SR Restrained SR Semi-Restrained SR
Preselection Two isolated leptons with different flavour and opposite sign
p`0T > 22 GeV, p`1T > 15 GeV
M llT > 10 GeV
Njets = 1
Nb−jets = 0
Z → ττ − veto
Background EmissT < 100 GeV < 150 GeV -
rejection√
(m`0T )2 + (m`1
T )2 < 140 GeV < 120 GeV < 100 GeV
p``T > 60 GeV > 60 GeV > 60 GeV
m`0T < 110 GeV < 110 GeV -
∆R`` < 2.1 < 1.0 < 1.0
∆φ`` < 2.0 - -
∆η`` < 1.3 < 0.4 -
∆R``,jet > 3.1 > 3.2 > 2.9
∆η``,jet > 1.7 > 2.1 > 2.1
∆φ``,jet > 1.5 > 2.55 > 2.55
m`` < 60 GeV - -
|ηjet| > 1.5 > 1.1 > 1.7
p`0T - > 60 GeV -
Table 1: Event selection criteria used in each of the three signal regions of the VBF analysis,
definitions of the variables are given in the text (Section 3.1 and 3.2).
10
3.3 Event selection example
In this subsection two examples have been given on event selection. All criteria have been
applied manually, which leaves the exact positions open for discussion. To determine where
to cut, the simulated data has been displayed as a minimalistic plot. This plot only shows
the background and the signal without specifying the breakdown of the contributions of
the background. In this minimalistic plots the signal (shown as a thick blue line) has been
re-scaled to be of the same order of magnitude as the background (shown as a thin line,
with hashed statistical deviation displayed).
The Fig. 3 and 4 display two scenarios in which the cuts have been made. Fig. 3 is an
example of a cut that allows for arbitrary tightness. If the goal is to preserve as much signal
as possible the cut is placed at 2.1, while a more restraining scenario could place it at 1.0.
In the second example (Fig. 4) the final cut has been placed around 3.0 in all three signal
regions as it is either very effective, or not effective at all.
(a) R`` distribution after the m`0T cut in
the relaxed SR.
(b) R`` distribution after the R`` cut in
the relaxed SR.
Figure 3: Example of a cut on ∆R``, VBF (MC) events have been scaled by 200, corresponds
with the ∆R`` event selection criteria in the relaxed signal region.
(a) R``,jet distribution after the ∆η``,jet
cut in the relaxed SR.
(b) R``,jet distribution after the ∆R``,jet
cut in the relaxed SR.
Figure 4: Example of a cut on ∆R``,jet, VBF (MC) events have been scaled by 90, corresponds
with the ∆R``,jet event selection criteria in the relaxed signal region.
11
4 Results
The event selection criteria defined in Table 1 and Sections 3.1 & 3.2 have been applied
to the MC simulated data and the observed data from the ATLAS detector. These event
selections resulted in three signal regions. The breakdown of final state contributions of the
background and signal is shown in Table 2 In the relaxed signal region MC data expects
8 VBF Higgs boson events with a signal over background of 0.05. In the restrained signal
region a signal over background ratio of 0.20 is expected with 1.5 VBF Higgs boson events.
In the semi-restrained signal region a balance of the two is found with 3 VBF Higgs boson
events with a signal over background ratio of 0.09.
Category Relaxed SR Restrained SR Semi-restrained SR
VBF Higgs bosons 8.17 ± 0.24 1.46 ± 0.10 3.07 ± 0.16
Other Higgs bosons 33.8 ± 33.83 3.30 ± 0.42 8.84 ± 0.62
WW 46.17 ± 1.72 1.46 ± 0.29 5.13 ± 0.59
Other VV 12.12 ± 2.96 0 ± 0 4.32 ± 1.93
Top-quark 44.52± 3.61 1.80 ± 0.61 6.72 ± 1.32
W+Jets 22.30 ± 14.3 0.93 ± 1.08 6.80 ± 4.77
Z+Jets 6.36 ± 2.93 0 ± 0 1.75 ± 1.6
Total background 165.3 ± 15.0 7.4 ± 1.2 33.6 ± 5.3
Observed Data 197 14 46
Table 2: MC and Data yields for each signal region defined in Section 3. The total background
can differ slightly from the sum of the contributions due to rounding. Other Higgs boson have
been considered background in this analysis.
An overview of the mT =√
(E``T + EmissT )2 − |p``T EmissT |2 with E``T =√|p``T |2 +m2
`` dis-
tributions of the final signal regions is given in Figures 5a, 5b and 5c.
Using Eq. 2 the observed and expected significance have been determined of the signal
regions and using Eq. 4 the signal strength is determined. The signal strength and signifi-
cance of each signal region is summarised in Table 3.
The 120-140 mT bin in Fig. 5a has been analysed individually on significance and signal
strength. This subset of the relaxed signal region shows a higher observed significance than
the total signal region. As result the signal strength is also stronger in this bin.
12
(a) Relaxed SR (b) Restrained SR
(c) Semi-restrained SR
Figure 5: mT distribution in the final state of the signal regions. MC data is displayed as
bars with contribution breakdown as displayed in the legend. The hatched band shows the
statistical uncertainty of the MC data. Data is shown as black crosses. The thick blue line is
the expected MC VBF Higgs boson signal. The difference between data and MC background
is used as observed VBF Higgs boson signal.
Signal Region Significance(expected) Signal Strength
Relaxed 2.3(0.6)σ 3.9µ
Restrained 1.7(0.5)σ 3.4µ
Semi-restrained 1.9(0.5)σ 3.8µ
Best single bin 2.6(0.5)σ 5.0µ
Table 3: Significance in all signal regions calculated using Eq. 2. The best single bin significance
is calculated using the 120-140 mT bin of the relaxed cut shown in Fig. 5a.
13
5 Conclusion
Event selection processes have been applied to data collected by the ATLAS detector to
search for the VBF Higgs boson → WW → `ν`ν signal among the background. Three sets
of event selection criteria have been applied to the data to form three signal regions. The
observed (expected) signal significance of the mH = 125 GeV VBF Higgs boson is 2.3(0.6)σ
for the relaxed signal region; 1.7(0.5)σ for the restrained signal region; and 1.9(0.5)σ for
the semi-restrained signal region. A single bin analysis has also been performed to find
a 2.6(0.5)σ significance in one bin of the final state mT distribution in the relaxed signal
region. The signal strength of the data is found to be 3.9µ for the relaxed signal region;
3.4µ for the restrained signal region and 3.8µ for the semi-restrained signal region.
6 Discussion
The Higgs boson was discovered with 5.0σ significance in 2012 by ATLAS using multiple
production modes and decay channels. The VBF Higgs boson has already been analysed
in the H → WW → `ν`ν decay channel, but it has never been done with a missing jet.
Previous SM VBF Higgs boson analysis has resulted in a 1.9σ significance [5]. The data and
event selection provided in this report could help the search for the Higgs boson in the VBF
production mode to reach the 5σ. Finding a signal in the mH =125 GeV standard model
Higgs boson is expected for this analysis and complies with a SM Higgs boson prediction.
Many improvements can be made to the analysis done in this report. One of which is
to search for a selection process that can be applied to reduce ggF produced Higgs bosons
from the data. The ggF Higgs boson is between 20 and 40 % of the background in the
final signal regions in the VBF Higgs boson analysis. Even though ggF Higgs bosons are
not background for Higgs boson analysis, a veto on this production mode could be useful in
VBF Higgs boson production analysis specifically.
Because the event selection cuts have been applied by hand, they are probably not the
most effective or efficient. To resolve this problem an automated multivariate analysis could
be applied around the most significant selection cuts found in this analysis. This analysis
compares the data simultaneously over all variables and searches for ideal event selection.
The final signal region in a multivariate analysis will have higher signal to background ratio
than a cut-based analysis. The variables to train this multivariate analysis on would be a
subset of φ``, η``, R``, φ``,jet, η``,jet, R``,jet, EmissT , m`0
T , m`1T , p``t , m`` and ηjet.
Another method to increase the significance of this research is to collect more data. In
this analysis the signal region with relaxed cuts has resulted in the highest significance. This
significance of a signal depends on amount of signal and the signal over background ratio.
In the more restrained signal regions the benefit of higher signal over background ratio does
not surpass the disadvantage a lower signal count brings which results in lower significance.
For future analyses done on higher luminosity the more restraining signal regions might
14
prove to be more effective.
In this analysis observed signal has been defined as NData − NB = NS . As result the
signal strength µ is high. This could indicate that some background is missing in the analysis
and the real significance should be lower. To test if the background is as stated, control
regions should be added to the analysis. A control region is, like a signal region, a subset
of your data. But in a control region the MC data is compared to the observed data to
compare the accuracy of your MC data. These subsets are taken in regions with good
predictions of the background. In Section 3.1 a cut is made using a Z→ ττ veto. This
veto is so effective because the Z→ ττ is a predictable background. By inverting the veto
you create a control region with a very strong Z→ ττ signal. If the MC data and observed
data vary greatly in this control region the background prediction is incorrect. To prove the
background prediction is correct, multiple control regions have to be checked.
7 Acknowledgements
I would like to thank my supervisors Lydia Brenner and Carsten Burgard for answering all
questions that came up. You have been the most supportive and excitable supervisors I
have ever been able to work with. I would also like to thank Ivo van Vulpen for taking the
time to read my thesis as second assessor.
15
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