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MSc Physics
Master Thesis
Astroparticle physics at LHC
Dark matter search in ATLAS
by
Michaël Muusse
6171885
September 2015
60 EC
September 2013 – September 2015
Supervisor/Examiner:
Dr. D. Berge
Examiner:
Dr. M. Vreeswijk
... .
Astroparticle physics at the LHC
Dark matter search in ATLAS
Astrophysics at the LHC – Dark matter seach in ATLAS
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Astrophysics at the LHC – Dark matter seach in ATLAS
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Astroparticle physics at the LHC
Dark matter search in ATLAS
Master thesis in Physics
Michael Muusse
July 2015
Track: Grappa, University of Amsterdam
Supervisor: Dr. D. Berge
2nd supervisor: Dr. D. Salek
Second Reviewer: Dr. M. Vreeswijk
... .
Astrophysics at the LHC – Dark matter seach in ATLAS
Front cover image credit: [14] [178] edited.
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Abstract
Astroparticle physics at the LHC – Dark matter search in ATLAS
In astronomy and cosmology the hypothesized existence of dark matter and its properties
are inferred indirectly from observed gravitational effects. An explanation for the missing
mass problem in the study of motion of galaxies and clusters, the shape of galactic rotation
curves, observations on weak gravitational lens effects near clusters and the
correspondence between the cosmic microwave background anisotropies and the largescale structure of the Universe, could be given by a form of non electromagnetic interacting,
invisible matter, hence called dark matter.
An undetected heavy elementary relic particle that interacts only trough the gravitational and
weak forces is a leading candidate for dark matter. Furthermore elementary particles with
Weakly Interacting Massive Particle (WIMP) properties arise often in theories beyond the
standard model. Since WIMPs are ought to be produced in particle accelerator experiments,
important evidence could be found in the analysis of high energy collisions in the LHC.
In this thesis simulations of the pair production of WIMPs in the upcoming 2015 LHC
collisions with energy of 14 TeV are analysed and the sensitivity of a signature of large
missing energy accompanied by a jets is studied. A ROOT program was written which
optimises the selection criteria for signature events for all different scenarios of WIMP and
mediator particle properties. Different models with a contact interaction represented by the
D5 operator or an interaction trough a generic Z’ intermediate state with mediator masses in
the range between Mz’ = 100 GeV and Mz’ = 15 TeV and with a dark matter mass of Mx = 50
GeV and Mx = 400 GeV were studied and show harder selection cuts don’t always improve
signal to background ratios but a tuned asymmetric selection is preferred. Considering all 48
studied models in 62.9% of the cases significance was enhanced by an average of 3.0% with
respect to the application of any of the standard used selection methods, while there was no
situation in which significance was decreased. This confirms the viewpoint of multiple jets
being created opposed to the invisible WIMPs and shows the importance of using tuned
selection criteria to enhance significance.
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Astrophysics at the LHC – Dark matter seach in ATLAS
Table of contents
Front cover 1 ...................................................................................................................... i
Front cover 2 ...................................................................................................................... iii
Abstract .............................................................................................................................. v
Table of contents............................................................................................................... vii
Preface ............................................................................................................................... xi
Introduction ....................................................................................................... 1
Chapter 1. Dark matter in astrophysics ........................................................... 5
1.1
1.2.1
1.2.2
1.2.3
1.2.4
1.3
1.4
Introduction............................................................................................................ 5
Evidence for dark matter - Motion of galaxies and clusters ................................ 6
Evidence for dark matter - Gravitational lensing .................................................. 8
Evidence for dark matter - The CMB.................................................................... 9
Evidence for dark matter - Bullet Cluster ............................................................. 12
Dark matter candidates ......................................................................................... 13
Alternatives for dark matter .................................................................................. 16
Chapter 2. Dark matter in Big Bang cosmology .............................................. 17
2.1
2.2
2.3
2.4
2.5
2.6
2.7.1
2.7.2
2.7.3
2.8.1
2.8.2
2.8.3
2.8.4
2.8.5
2.8.6
2.8.7
2.8.8
2.8.9
2.8.10
2.8.11
Introduction............................................................................................................ 17
Spacetime & special relativity ............................................................................... 17
Spacetime and gravity, general relativity .............................................................. 19
The cosmological principle.................................................................................... 22
The expanding Universe ....................................................................................... 23
The Big Bang model.............................................................................................. 24
Problems with Big Bang theory: The horizon problem ......................................... 29
Problems with Big Bang theory: The flatness problem ........................................ 29
Inflation theory ....................................................................................................... 30
The thermal history of the Universe ...................................................................... 31
The Planck era ...................................................................................................... 31
Grand Unification .................................................................................................. 31
Inflation era ............................................................................................................ 32
Baryogenesis......................................................................................................... 32
Electroweak symmetry breaking ........................................................................... 33
The hadron epoch ................................................................................................. 33
The lepton epoch .................................................................................................. 34
Nucleosynthesis .................................................................................................... 34
Matter radiation equality ........................................................................................ 34
Recombination ...................................................................................................... 35
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Astrophysics at the LHC – Dark matter seach in ATLAS
2.8.12
2.8.14
2.8.15
2.8.16
2.8.17
2.9
Structure formation................................................................................................ 35
Reionization........................................................................................................... 35
Formation of galaxies ............................................................................................ 36
Formation of the Solar System ............................................................................. 37
Today..................................................................................................................... 37
ΛCDM .................................................................................................................... 37
Chapter 3. Dark matter in particle physics ...................................................... 39
3.1
3.2.1
3.2.2
3.2.3
3.3.1
3.3.2
3.3.3
3.4.1
3.4.2
3.5.1
3.5.2
3.5.3
3.5.4
3.6.1
3.6.2
Introduction............................................................................................................ 39
Symmetries and conservation laws ...................................................................... 39
External symmetries of spacetime ........................................................................ 39
Internal symmetries of spacetime ......................................................................... 40
The standard model of particle physics ................................................................ 42
Additional global symmetries in the Standard Model ........................................... 44
The Higgs mechanism .......................................................................................... 44
Problems with the Standard Model ....................................................................... 45
Extensions of the Standard Model ........................................................................ 50
WIMP detection ..................................................................................................... 53
Direct detection of WIMPs .................................................................................... 53
Indirect detection of WIMPs ................................................................................. 54
Accelerator searches ............................................................................................ 55
Dark matter search in lepton colliders .................................................................. 56
Dark matter search in proton colliders .................................................................. 57
Chapter 4. Dark matter at the LHC ................................................................... 59
4.1
4.2.1
4.2.2
4.2.3
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
4.3.6
4.3.7
4.4.1
4.4.2
4.5.1
4.5.2
4.5.3
Introduction............................................................................................................ 59
LHC ....................................................................................................................... 59
Event rate and luminosity in LHC ......................................................................... 61
LHC detectors ....................................................................................................... 64
The ATLAS Detector ............................................................................................. 65
ATLAS coordinate system .................................................................................... 66
The Inner detector ................................................................................................. 67
Calorimetry ............................................................................................................ 68
The muon spectrometer ........................................................................................ 68
The magnet system............................................................................................... 69
Computing grid ...................................................................................................... 70
Event reconstruction and identification ................................................................. 71
Particle identification ............................................................................................. 74
Transverse momentum and missing energy ........................................................ 76
Smallest angle ....................................................................................................... 77
Pseudorapidity ...................................................................................................... 77
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Chapter 5. Methods ........................................................................................... 79
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
General objectives and assumptions .................................................................... 79
Interactions in Effective Field Theory ................................................................... 81
Interactions trough a Z’ mediator .......................................................................... 84
Background events .............................................................................................. 87
Simulation methods............................................................................................... 91
Simulations of background.................................................................................... 93
WIMP production simulations ............................................................................... 94
Data samples ........................................................................................................ 95
General selection criteria ...................................................................................... 97
Data computation .................................................................................................. 99
Data analysis and display ..................................................................................... 101
Schematic overview of the analysis ...................................................................... 103
Chapter 6. Results ............................................................................................. 107
6.1
6.2
6.3
6.4
Results analysis A ................................................................................................. 107
Results analysis B ................................................................................................. 109
Results analysis C ................................................................................................. 116
Results analysis D ................................................................................................. 131
Chapter 7. Conclusion ...................................................................................... 151
7.1
7.2
7.3
7.4
7.5
7.6
7.6
Conclusion analysis A ........................................................................................... 151
Conclusion analysis B ........................................................................................... 152
Conclusion analysis C ........................................................................................... 155
Conclusion analysis D ........................................................................................... 159
General conclusion of the analysis ....................................................................... 161
General conclusion ............................................................................................... 162
Outlook .................................................................................................................. 163
References ......................................................................................................... 165
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Preface
In this thesis I performed an efficiency study of the dark matter detection in the ATLAS
experiment combining the fields of particle physics and cosmology. The thesis is written to
obtain a master’s degree in Physics at the University of Amsterdam and supervised by Dr.
David Berge, researcher at the Grappa institute and David Salek, researcher at ATLAS and
Nikhef.
I approached the topic by studying the vast literature about dark matter in particle physics
and cosmology and by taking part of courses in (particle) cosmology and quantum field
theory. Furthermore with this thesis I learned to become familiar with the ROOT Analysis
Framework, and develop and apply analytical methods to perform efficiency study and I
compared and inspected the results found in simulations to write the conclusions. By writing
this thesis I also hope to present an accessible overview of the dark matter problem in
physics.
Finalizing the project and thesis took almost two years and had a serious impact on my
personal life. Therefore I would like to thank first and especially my girlfriend Mariska Dijkstra
for supporting me so much the past years. Furthermore I would like to thank dr. D. Berge and
dr. D. Salek for giving the opportunity to investigate and complete this project, and for their
supervision of this thesis. Many thanks to my employer and colleagues of the Utrechts
Stedelijk Gymnasium and to my friends that supported me during the last years. Finally,
thanks to my family for their constant support and encouragement.
Michaël Muusse
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Introduction
Dark matter is a presumed form of invisible massive matter which makes up 83.9% of the
physical matter density in the Universe. The leading candidate for dark matter is an
undetected heavy elementary relic particle which interact only trough gravitation and the
weak force. Such a Weakly Interacting Massive Particle (WIMP) is often predicted in theories
beyond the standard model and is ought to be produced in particle collider experiments with
sufficient available energy. Important evidence could be found in the analysis of high energy
collisions in the Large Hadron Collider (LHC), the World’s biggest and most powerful particle
accelerator.
This research project aims to analyse simulations of WIMP pair production in the
upcoming 14 TeV LHC collisions, and develop a ROOT program which optimises the
selection criteria for signature events in ATLAS studying scenarios with different
WIMP and mediator particle properties.
The topic is approached by giving a short historical and introductional overview of the
physics behind dark matter in particle physics and cosmology. Present conceptions and
theories are mostly not attained in a straightforward manner, but can have different concepts
as foundation since science is not static but influenced by local traditions, cultural and social
tendencies and their context.
In chapter 1 the observational evidence for dark matter in astrophysics is described. Already
in the 19th century it was noticed from the study of the motion of celestial bodies that a form
of non-luminous mass exerting gravitation should be present. Evidence for the existence of
so called dark matter was prolonged in the 20th century by observations inferred from its
gravitational effects on the movement of galaxies and clusters, gravitational lenses and could
explain structure formation in the Universe and the observed anisotropies in the cosmic
microwave background (CMB). The leading candidate for dark matter are cold dark matter
particles that were produced by falling out of thermal equilibrium with the hot early Universe,
such as the WIMPs studied in this thesis. Observations show that the energy density of all
matter in the Universe consists of 83.9% dark matter and only for 16.1% of baryonic
(ordinary) matter.
In chapter 2 the foundations and concepts of the present standard model of cosmology in
which general relativity is a fundamental theory are explained. In Big Bang cosmology the
evolution of the Universe is described by its energy density content and curvature, starting
from a hot dense state that inflated rapidly to the accelerated expansion today. Remarkable
in the present cosmological model of the Universe (ΛCDM) 71.4% of the critical energy
density of the Universe consists of dark energy whereas 28.6% of the energy density is
determined by dark matter and ordinary baryonic matter.
Dark energy is the unknown form of energy responsible for the accelerated expansion of the
Universe. Extrapolating the expansion of the Universe back in time will give the description of
the thermal history of the Universe trough its different era’s and provide more insights in the
role of dark matter in the evolution of the Universe.
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The discovery of new elementary particles was always the realm of particle physics and in
this thesis we study the possibility to produce dark matter WIMPs in the high energy particle
collider LHC. Any produced dark matter particles would escape the detector unseen leaving
an energy imbalance as signature. This idea and its advantages and drawbacks will be
discussed in chapter 3, also in comparison with most dark matter research that aims to
detect dark matter directly interacting with normal matter (direct detection) or indirect via self
annihilation (indirect detection). Furthermore we describe how particles with WIMP like
properties often arise in theories beyond the Standard Model. The Standard Model combines
the quantum field theories describing the interaction of electromagnetism and the weak and
strong nuclear force and classifies all fundamental particles known.
In this thesis I will focus on the possibility WIMPs might be produced and detected by the
ATLAS detector when the LHC high energy particle accelerator has restarted in June 2015
running at a higher energy of 14 TeV. Large missing energy accompanied with jet production
is the signature of an event in which WIMPS could be produced. In chapter 4 we describe the
signature of such event considering the characteristics and limitations of the ATLAS
experiment.
For this thesis simulations of WIMP pair production and background processes are studied,
and an optimalisation of the selection criteria placing constraints on physical variables is
performed by a code written in ROOT using different scenarios of WIMP and mediator
particle properties. In chapter 5 the assumptions and constraints for this research are
presented with the method used to study the simulations. In chapter 6 the results for
optimising signal to background efficiencies are shown and in chapter 7 the conclusion is
discussed.
Dark matter interpreted as a heavy black opaque plush. [1]
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Chapter 1. Dark matter in astrophysics
1.1 Introduction
Cosmology, literally meaning in Greek “study of“ (λογία - logia) the “world“ (κόσμος kosmos), describes the physical laws of the origin, evolution, structure, dynamics and
ultimate fate of the Universe. Big Bang cosmology describes 13.8 billion years of cosmic
evolution and expansion, since the birth of the Universe from a singularity untill the present
and future. Astronomical observations and new discoveries about the laws of physics in the
early Universe could give us answers to fundamental questions unavoidable to mankind as
“How was matter created?“, ”What is the age of the Universe?“ and “How do the conditions
of our era compare to the history and future of the Universe?”.
As expansion cools the Universe down, we try to understand processes at earlier times by
extrapolating our theories to higher energies, describing the Universe when it was hot and
dense. At early times all matter interacted energetic and frequently and was bound by a
unified force. As this primordial plasma cooled while the Universe expanded, elements were
formed in a process called nucleosynthesis, and photons were able to stream freely trough
the Universe. These initial free photons can be seen today as the cosmic microwave
background radiation (CMB) and this “baby photo” of the Universe contains small
temperature fluctuations indicating tiny variations in the primordial density of matter, probably
originating from quantum fluctuations stretched to cosmic sizes during the early inflationary
epoch of rapid expansion. These primordial fluctuations are density variations in the early
Universe which are considered the beginning of all structure in the Universe, which under the
influence of gravitation let to the formation of stars, galaxies and clusters.
The evidence of the Big Bang comes from the observations of the expansion of the Universe,
the CMB and light element abundances which are in accordance with Big Bang
nucleosynthesis theory. From this evidence and other observational data it is calculated that
the majority of the Universe today consists of different forms of invisible energy and matter
called dark energy and dark matter. What the nature of dark energy and dark matter exactly
is remains still a mystery today.
Dark energy is an unknown form of energy present everywhere in space and is required to
exist in order to describe the observed accelerated expansion of the Universe. It is currently
believed 71.4 % of the critical energy density of the Universe consists of dark energy
whereas 28.6 % consists of matter (dark matter and normal baryonic matter). [3] [4]
Dark matter is an invisible form of matter which presence has been inferred from its
gravitational effects. Dark matter was first imposed to explain the motion of stars, but also
observations on the movement of our galaxy, galaxy rotation curves (movement of stars in
galaxies) and the velocity dispersion of galaxies in clusters, require a form of invisible matter
to be present. Dark matter could also give an explanation for cosmic structure formation and
its effects on the anisotropies observed in the cosmic microwave background. Observations
show that of all matter 83.9% is dark matter and 16.1% baryonic (ordinary) matter. From the
total critical energy density of the Universe 24.0% is composed of non baryonic dark matter
and only 4.6% of baryonic matter, see fig 1.1.
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Ordinary matter is in cosmology named baryonic matter since in atoms around 99% of the
mass is in the nucleus in the form of protons or neutrons which consist of 3 quarks (called a
baryon) while the mass contribution from electrons is small. From the 4.6% baryonic matter
an estimated 0.4% baryonic matter can be found in luminous matter like stars or hot gas
while 4.2% is in non-luminous baryonic matter, see fig. 1.1. [2] [4]
Fig. 1.1. Estimated distribution of dark energy, dark matter and baryonic matter in the
Universe. [5]
1.2.1 Evidence for dark matter - Motion of galaxies and clusters
Different astronomical bodies emit or absorb light in different ways with different efficiencies
and this is used to study the content of the Universe in astrophysics. The effectiveness of
emissivity is reflected in the mass to light (M/L) ratio of an object in a fixed photometric
system and the mass can be determined by the amount of light emission. A different way to
determine the mass of an astronomical object is by its gravitational effect on the motion of
other bodies nearby the studied object. For most objects the calculated mass from
gravitational effects exceeds the calculated mass by its M/L ratio, and the apparent nonluminous mass exerting a gravitational force is what is commonly referred to as dark matter.
A large amount of observational evidence of dark matter comes from the study of motion of
galaxies and clusters. The presence of dark matter was first noticed by Bessel in 1844 when
he showed that the positional measurements of Sirius and Procyon implied each star was
moving in orbit with another invisible object [6]. Jeans and Oort analysed in 1922 the vertical
motions of stars near the plane of our Galaxy and showed that the density of all stars near
the galactic plane was insufficient to justify the vertical motions. Therefore they assumed the
presence of two invisible stars next to each bright star. [7] [8]
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In 1933 Zwicky measured the redshift of galaxies in the Coma cluster and estimated the
clusters total mass by observing the motion of galaxies near its edge. He found the
calculated mass for the cluster was about 400 times larger than expected based on the
number of galaxies in the cluster and its total brightness. Therefore Zwicky concluded an
invisible form of matter or “missing mass“ exists, massive enough to keep the galaxies in the
cluster gravitationally bounded [9]. Babcock showed in 1939 that the galaxy rotation curves
of the Andromeda galaxy, which show the star velocity of rotation versus the distance from
the galactic center, could not be explained by the amount of visible matter. [10]
More precise estimates in line with modern observations were done in 1959 by Kahn &
Woltjer who noticed that while the light from most galaxies is redshifted due to the expansion
of the Universe, the Andromeda galaxy has a blueshift and thus moves towards our Galaxy
[11]. They concluded the Andromeda galaxy and our Galaxy form a two body system orbiting
around eachother and are currently approaching towards eachother. Calculation of the total
mass of the double system showed its mass to be about ten times higher then presumed.
Using the same method with modern data Bell & Einasto estimated in 1982 the total mass of
the Local Group (the galaxy group that includes our Galaxy and the Andromeda Galaxy) to
be in agreement with present measurements of the Andromeda Galaxy and our Galaxy
including their dark matter halos [12]. Rubin & Ford calculated in 1980 the rotation curve of
spiral galaxies with greater precision and showed that about 6 times as much mass from
dark matter is present than there is from visible stars. All these measurements indicate the
presence of a form of invisible matter with very large M/L ratio and large radius and mass in
galaxies and clusters [13]. Dark matter is assumed to be centrally concentrated and have a
roughly spherical symmetric distribution called a dark matter halo, see fig. 1.2.
Fig. 1.2. Artist impression of a large dark matter halo surrounding a galaxy. [14]
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1.2.2 Evidence for dark matter - Gravitational lensing
A gravitational lens is formed when clusters, galaxies or stars are so massive that their
excerted gravity bends and focuses light from distant objects that lie behind. Gravitational
lenses were hypothised by Einstein and Zwicky but not discovered until 1979 [15].
Gravitational lenses can be categorised into 3 classes. [16]
A strong lens shows an image with visible distortions such as Einstein rings, arcs and
multiple images, see fig 1.3. The mass of the cluster creating this distortion can be calculated
by the size of the distortions as the path light is bended near of the cluster. This mass
corresponds to dynamical estimates of the mass of clusters based on the movement of its
galaxies.
A weak lens distorts the image of background objects only slightly. Statistical analyses on
numerous objects can search for shear deformation and shape distortions in the image of the
adjacent background galaxies and determine the mass of the object causing the distortion.
[17]
The correspondence of the masses of objects measured by using the gravitational lens
techniques with other independent dark matter measurements as in the discovery of the
Bullet Cluster (Chapter 1.2.4) has convinced most physicists that dark matter exists as a
major component of the Universe. [18]
Microlenses have the property that no distorted image is visible, but the flux received from a
background object changes with time. Microlenses can lead to the discovery of dark objects
in our Galaxy and determine whether dark matter in our Galaxy is made of baryonic normal
matter.
Fig. 1.3. The strong lensing effect caused by the gravity of the luminous red galaxy (LRG 3757) distorted the image from a more distant blue galaxy such that the light bending resulted
in an Einstein Ring. Image from Hubble Space Telescope’s Wide Field Camera 3. [19]
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1.2.3 Evidence for dark matter – The CMB
The existence of the cosmic microwave background radiation (CMB) was predicted by Dicke
in 1965 in the same year that Penzias & Wilson discovered a signal by accident what turned
out to be the CMB radiation. The small fluctuations (order 10-5) in the CMB were first
detected by the COBE satellite (1992) but the experiment was not as accurate as the
observations from the Wilkinson Microwave Anisotropy Probe (WMAP) spacecraft (2009) or
the Planck satellite (2013). [20] For comparison the maps of the CMB made by the three
satellite experiments is shown in fig. 1.4.
Fig. 1.4. Maps of the CMB made by COBE, WMAP and Planck (from left to right). [21]
Using the data from WMAP the fluctuations in the CMB are reconstructed as is shown in fig.
1.5. The cosmic microwave background (CMB) has an average temperature of 2.725 K with
small fluctuations within 0.01% and appears to be smooth and isotropic at first sight, as can
be seen in the upper left part of fig. 1.5. If the average temperature is subtracted and the
remaining fluctuations are increased by a factor 400, the dipole pattern and the emission
from our Galaxy which dominate the red color in the picture are clearly visible in the upper
left part of fig. 1.5. The maximal temperature fluctuations have a 0.00335 K variation with one
hot pole and one cold pole which form the dipole pattern. After the dipole pattern is
subtracted and the contrast is enhanced by an extra factor 50, the intrinsic fluctuations in the
CMB are visible in the lower left part of fig. 1.5. When finally the emission from our Galaxy is
subtracted, the picture of the CMB as can be seen in the lower right part of fig 1.5 is obtained
in which the anisotropies are clearly visible. [20]
Fig. 1.5. Contrast enhancement in the WMAP data of the CMB showing in the lower right
figure a temperature fluctuation range of 0.0005 K from the coldest (blue) to the hottest (red)
parts of the CMB. [22]
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According to the present cosmological model the Universe was initially ionised and very hot,
and photons, electrons and quarks were tightly coupled. Perturbations of baryons oscillated
as sound waves in the electron-baryon plasma and these baryonic acoustic oscillations
(BAO) became embedded in the distribution of matter when the plasma condensed as the
Universe expanded. The largest possible amplitude of the BAOs is at a wavelength equal to
the size of the sound horizon at the moment of decoupling. This wavelength can be seen in
the first maximum of the angular power spectrum of the CMB while the following maxima
correspond to the overtones. When the Universe was 340.000 years old the photons were
able to decouple and stream freely trough the Universe. This “first light” can be detected
today as the cosmic microwave background radiation currently cooled due to the expansion
of the Universe to an average of 2.725 K. The fluctuations in the plasma are reflected by
temperature fluctuations in the CMB, and contain information about the large scale
distribution of the Universe at the time of decoupling. This distribution is highly dependent on
the amount of dark matter present in the early Universe. [23]
In March 2013 the Planck mission released the map of the cosmic microwave background
which showed the fluctuations with larger precision. From this data it was calculated that the
Universe is slightly older than thought (13.8 Gy) and show the fluctuations created as early
as 10−30 seconds after the Big Bang which are the seeds for the present structure of the
cosmic web, clusters and dark matter. [24]
The temperature fluctuations are described by a function written in multipole moments by
performing spherical harmonic decomposition by angle, thus depending on the direction on
the sky. If the fluctuations have Gaussian distributions they have no preferred direction and a
power spectrum (the intensity of each spectral component) can be made. Typically the
fluctuations in the CMB are showed in an angular power spectrum where the power per
natural logarithmic interval l is plotted against the temperature fluctuation in mK. The
reciprocal of l is the angular scale or angular wavelength of the fluctuation. For example l =
10 corresponds to about 10 degrees on the sky, and l = 100 to about 1 degree on the sky.
On the largest scales (l < 10) it’s difficult to make statistical statements about the scale of the
entire Universe if only a part of the Universe can be observed, and for low multipole
moments l this effect (cosmic variance) will lead to large errors. For smaller angular scales (l
> 100) the power spectrum shows acoustic peaks. On the smallest angular scales (l > 103)
the mean free path length of photons suppressed the growth of structures on small scales
and the power spectrum is damped by a photon diffusion damping scale called “Silk
damping”. This diffusion process limited the fluctuations to grow to the size of galaxies
observed today.
The position of the first peak of the power spectrum depends strongly on the total
matter/energy density and is in agreement with the predicted value of 1 in units of the critical
cosmological density, indicating the Universe is very close to spatially flat. From the acoustic
peaks also the baryon density can be estimated as baryons add mass to the photon-baryon
plasma. Acoustic peaks that are even numbered are related with the plasma “rebound”, the
odd numbered peaks are related to “compression” of the plasma. A large baryon density will
enhance the odd peaks over the even peaks. [23] [25]
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Furthermore increasing baryon density decreases the frequency of the BAOs and shifts the
position of the peaks to higher multipole moments l. A higher baryon density also results in
the power spectrum decreasing at higher multipole moments l. The power spectrum depends
on the baryon density in many independent ways and can therefore be calculated precisely.
With a resolution of 7 degrees on the sky COBE could only see the largest angle fluctuations
but combining the experiments of the 2015 data of Planck with measurements from baryon
acoustic oscillations, and the Hubble constant the best fit estimate for the baryon energy
density is 4.6% and for the cold dark matter energy density 24.0%. [24]
Fig. 1.6. The angular power spectrum of the CMB measured by the Planck experiment based
on data from 2015. On the left the error from cosmic variance is clearly shown. The red line
shows the current best fitting cosmological model. [24].
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1.2.4 Evidence for dark matter - Bullet Cluster
The Bullet Cluster is a system of two clusters which are colliding. Most of the baryonic mass
in the clusters is contained in the hot gas between the galaxies rather then in the mass of all
the stars of the galaxies. The baryonic mass is bound to the cluster by a dark matter halo
with an even greater mass. In most objects dark matter and baryonic matter are
concentrated around the same center as they experience mutual gravitational attraction.
However in the Bullet Cluster the collision has caused dark matter to separate from baryonic
matter. X-ray observations by Chandra show that the baryonic matter in the form of hot gas
is slowed down by electromagnetic interaction. Studying the weak gravitational lensing effect
of the clusters on background galaxies by measurements of the Hubble Space Telescope,
the European Southern Observatory's Very Large Telescope and the Magellan optical
telescopes, the location of the center of mass of the clusters was found to be inconsistent
with the position of the hot gas, as can be seen in fig 1.7. The interpretation is that dark
matter is non-collisional and that the dark matter halos of both clusters have collided and
passed eachother without slowing down, whereas baryonic matter behaved collisional and
has slowed down causing it to separate from the dark matter halos. The Bullet Cluster is
seen as direct evidence of the existence of dark matter independent of the exact laws of
gravity. From the temperature and density of the X-ray gas, pressure and gravity balance can
be estimated determinating the baryonic mass of the clusters at 12-15% of the total mass in
agreement with observations on other clusters. [18] [26]
Fig. 1.7. The Bullet Cluster: a colliding pair of galaxy clusters. Most of the baryonic mass is in
the X-ray emitting hot gas shown in red and yellow (left). In green the contours from the weak
gravitational lensing effect are shown indicating the gravitational effect from the system
dominated by dark matter which is centered on the observed distribution of galaxies (right). A
separation is visible between baryonic matter and dark matter. [26]
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1.3 Dark matter candidates
Besides the Bullet Cluster and the CMB power spectrum there are more observations that
show dark matter mostly consists of non baryonic matter. The theory of Big Bang
nucleosynthesis suggest baryonic matter accounts for about 4.6% of the critical energy
density of the Universe, and accurately predicts the observed abundance of elements as the
hydrogen/deuterium ratio which is discussed in more detail in Chapter 2. Non baryonic dark
matter accounts therefore for 24.0% of the energy density in order to combine to the 28.6%
of the matter density in the Universe. The amount of ordinary luminous matter in the objects
observed on the sky like stars and luminous gas and dust clouds is estimated to be 0.4% of
the critical density. This means 4.2% of the baryonic mass is in a non-luminous form of
baryonic matter called baryonic dark matter. [27][28]
Baryonic dark matter
Baryonic dark matter or missing baryonic matter was first proposed by Zwicky in 1937 and is
a form of normal baryonic matter which presence can be inferred from gravitational effects
only, as it does not emit light. Baryonic dark matter could consist of non-luminous interstellar
gas, cold gas, brown dwarfs (Jupiter’s) or Massive Astrophysical Compact Halo Objects
(MACHOs) such as neutron stars, white dwarfs, planet-like objects, black holes, etc.
MACHOs can be detected by gravitational microlensing if such a massive object passes near
the line of sight between Earth and distant stars. The gravitational field around the MACHO
will focus the light of background stars which appear to brighten for a period of time.
-8
3
Microlenses are sensitive to objects with mass m in the range 10 M☉< m <10 M☉ which is
in the range of the expected mass of MACHOs. The characteristic time profile of this
brightening is independent of the wavelength of the light and therefore distinguishable in
observations from other situations with brightning variations like variable stars. Signs of
microlensing were found in observations of the Magellanic Cloud and galactic bulge however
it’s unclear if the objects are a substantial part of the non luminous dark matter present in the
Universe. [2] [29] [30]
Non-baryonic dark matter
There exist a large number of non baryonic dark matter candidates but they can be classified
in the categories WIMPs, axions and neutrinos. All of these candidates are stable long lived
and electric and color neutral particles with halftime t > tUniverse. An important difference
between these candidates is the effect of dark matter on the formation of structures in the
Universe. We differentiate between cold dark matter (CDM) particle candidates that became
non-relativistic and hot dark matter (HDM) particle candidates that were relativistic at early
times in the Universe. The effect on structure formation is that hot dark matter has a large
free-streaming length and would have diffused fluctuations in the early Universe therefore
preventing the formation of structures in the Universe as all structure gets “washed out”. Hot
dark matter models would imply a top-down formation history of structure in the Universe
with the big structures (clusters, groups) formed first and smaller structures (galaxies, stars)
formed later in history. Instead cold dark matter would act as a compactor of structure pulling
matter together by its gravitational attraction enhancing the formation of filamentary
superclusters and voids in the Universe. The evolution of clusters and large scale filaments
in the presence of cold dark matter has been studied intensively with computer simulations
as showed in fig. 1.8 and fig. 1.9. [16] [31]
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Fig. 1.8. Computer simulation of an area of space of 140 million light years across presenting
the evolution of clusters and large-scale filaments in the Cold Dark Matter model with dark
energy from 100 million years after the Big Bang (left) untill the present (right). [32]
Observations suggest that structure formation in the Universe proceeded bottom up, with the
smallest structures being created first followed by galaxies and clusters of galaxies. The fact
that early formed high redshift galaxies are found in the Hubble Ultra Deep Field (HUDF) and
the fact that our galaxy appears to be older than the Local Group, support the bottom up
formation history of the Universe. The bottom up cold dark matter model of structure
formation in the Universe is favoured by studied which compare simulations of the structure
formation with observational data of galaxy surveys as the Sloan Digital Sky Survey (SDSS)
and 2dF Galaxy Redshift Survey (2dF). [32]
Fig. 1.9. A comparison of the time evolution
structure formation (showed from top to bottom)
using cold, warm and hot dark matter models
(showed from left to right). In the hot dark matter
model the smallest structures dissapear. [33]
Neutrinos as a form of dark matter are at first sight an interesting candidate since their cross
section is so small they are practically invisible, while in the last decade it is shown neutrinos
are not massless particles as predicted in the Standard model of particle physics, but have a
small mass of order of ~ eV. However it can be shown that neutrinos are not abundant
enough such that their relic density is the dominant component of dark matter. Furthermore
neutrinos are relativistic particles with a large free streaming length, and therefore would act
as hot dark matter. From the analysis of CMB anisotropies combined with large scale
structure data it can be shown the free streaming length of neutrinos is too large in
comparison with the galaxy scale perturbations for neutrinos to be a significant component of
dark matter. [34][16]
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The suggestion of the presence of cold dark matter in the Universe explains the observed
structure formation as it evolved from an initial hot dense state as illustrated in fig. 1.10.
However the nature of cold dark matter particles is still unknown. WIMPs are interesting
candidates as cold dark matter particles that would dominate dark matter in the Universe.
The leading hypothetical particle physics candidate for dark matter are WIMPs relic particles
that were produced by falling out of thermal equilibrium (frozen out) with the hot dense
plasma of the early Universe and interacted weakly with standard model particles via a force
similar in strength to the weak nuclear force. The foundations and concepts of Big Bang
cosmology which describes a Universe that was inflated from a hot dense state is described
in chapter 2. An overview is given of the thermal history of the Universe and the process of
the thermal creation of WIMPs is discussed in more detail.
Fig. 1.10. The history of 13.82 billion years of evolution of our Universe showing the main
events that occurred between the initial hot dense phase with tiny fluctuations, to the cosmic
large scale structure observed today. [35]
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1.4 Alternatives for dark matter
The large amount of unknown forms of matter has initiated attempts to construct theories
with modified laws of gravity at large distances, in comparison to Newton’s laws. Milgrom &
Bekenstein suggested in 1987 the theory of Modified Newtonian Dynamics (MOND) but other
theories exist as well. The aim of such theories is to explain the observational data without
presuming a form of invisible matter. The discovery of the Bullet Cluster is an argument
against MOND and other theories without dark matter. In such theories the lensing effect
would be centered on the baryonic matter which is dominated by the hot X-ray gas (see
chapter 1.2.4) and not outside these regions. This is interpreted as most of the mass in the
pair of clusters is in a form of non-luminous collisionless matter indicating the presence of
non-baryonic dark matter. The discovery of the Bullet Cluster has made theories with
modified gravity laws less popular. [36] [37]
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Chapter 2. Dark matter in Big Bang cosmology
2.1 Introduction
In this chapter the foundations and concepts of Big Bang cosmology are reviewed. In this
theory general relativity, the cosmological principle and the idea of a Universe that was
inflated from a hot dense state are used to describe the observable Universe as a whole
based on its cosmic inventory. Extrapolating the expansion back in time tells us the thermal
history of the Universe as forces arose, particles condensed and the structures we observe
today were formed.
2.2 Spacetime & special relativity
In physics the interaction of particles by forces moving in space and time is studied. Space
and time can be described by a single space-time continuum and Minkowski space is the
manifold combining three spatial dimensions and one time dimension. Incas described space
and time already by a single perception called “pacha” but its mathematical analysis was
unexplored untill the 19th century by the theory of electromagnetism (Maxwell) and in
Lagrangian mechanics (Lagrange and Hamilton).
Untill the end of the 19th century empty space was considered to be filled with a thin,
transparent medium called “aether” through which light and electromagnetic waves were
thought to propagate. However Michelson and Morley tried to measure with the famous
aether drift experiment in 1887 the relative difference in the speed of light using
measurements moving towards and away from the Sun as the Earth rotated. While it was
expected that as the Earth moved through the ether towards the Sun, the speed of light
increased and vice versa, no change in the velocity of light was observed [38]. Based on the
conclusions of this experiment Albert Einstein wrote in 1905 a paper that led to the
development of the theory of special relativity, describing the relation between space and
time based on two postulates; [39]
1. The laws of physics are invariant in all inertial systems
2. The speed of light in vacuum is equal for all observers
The consequences of this theory are relativity of simultaneity, time dilation and length
contraction, Lorentz transformations, and mass-energy equivalence described by formula
[2.1];
E = mc2
[2.1]
In special relativity it is described how with the increase of speed of an object one needs
more and more energy to make it move faster such that the maximum speed for any real
particle is approaching the speed of light. While inertial mass measures how hard it is to
accelerate an object, it is the true rest mass of the objects that remains constant and is an
intrinsic property of a particle and independent of the frame. It is the dynamical law that
relates momentum and energy that depends on velocity in special relativity and approaches
Newton’s laws in the low velocity limit.
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The finite speed of light means that the farther away an object is from the Earth, the longer it
takes for its light to reach us. Thus observing a more distant object means the observer is
actually looking back farther in time. Observing bodies at various distances allows us to
study their evolution at different epochs in time. The largest distance we can look back to is
the moment the CMB was created, and the first photons were able to stream freely trough
the Universe, see fig. 2.1.
Fig. 2.1. This illustration places the "look-back time" of the Hubble Ultra Deep Field (HUDF)
in the context of the history and age of the Universe. [40]
Everywhere in the Universe special relativity holds, and Minkowski visualized a light cone
can be drawn, defining the boundary between past and future accessible locations, since no
information is allowed to travel faster than the speed of light, see fig. 2.2.
Fig. 2.2 (left). Past and present lightcones. [41]
Fig. 2.3 (right). Lightcones in flat Minkowskian spacetime. [42]
The geometric and causal structure of spacetime is described by using a coordinate grid
called a metric that is laid down over all spacetime, which determines the distances between
specified points. Everywhere on the metric special relativity holds and light cones can be
drawn at every point on the metric, see fig. 2.3.
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In general relativity spacetime is not necessary flat and the shape of the Universe is
described by its local geometry, which mostly concerns the curvature of spacetime at any
arbitrary point of the observable Universe (averaged on a sufficiently large scale) and its
global geometry, which defines the topology of the Universe as a whole. Observations from
Planck have shown that the cosmological curvature parameter ΩK is 0.000±0.005, consistent
with a flat universe. [24]
The geometry and curvature of space-time can be mathematically described by the
Pythagorean theorem, see fig. 2.4. Zero curvature means spacetime is flat and the
Pythagorean theorem holds. If spacetime has positive curvature then the ythagorean
theorem is correct for triangles with their sum of angles larger then
. When spacetime
has negative curvature the Pythagorean theorem holds for triangles with their sum of angles
smaller then
.
Fig. 2.4. Three types of curvature of two-dimensional surfaces as an analogy to the 3dimensional structure of the Universe. [43]
2.3 Spacetime and gravity, general relativity
While special relativity only holds for non-accelerating inertial reference frames, Einstein
developed between 1907 and 1915 the theory of general relativity, applicable to any frame
and able to handle gravitational effects and general coordinate-transformations. The first
postulate in this theory is that local physics is still governed by the theory of special relativity
as is showed in fig. 2.5. The second postulate is the equivalence principle that states an
observer is not able to distinguish between gravity and acceleration. General relativity
generalizes Newton’s law of universal gravity and special relativity describing gravity as a
geometric property of spacetime directly relating the curvature of space-time to the local
energy and momentum density present within that spacetime, see fig 2.6. [44]
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Fig. 2.5 (left). Local lightcones near strongly curved spacetime near a black hole formed by
collapsing matter. [45]
Fig. 2.6 (right). When a small mass is close to a large mass, it will curve towards it since
spacetime itself is curved near the large mass. [46]
In 1915 Einstein did a Gedankenexperiment and imagined two elevators, one at Earth and
one accelerating in space, see fig. 2.7. A blinded observer inside an elevator would not be
able to distinguish between the gravitational force experienced while standing on Earth and
the force experienced by an observer in a non-inertial accelerated frame of reference. [83]
Fig. 2.7. Scientist on Earth (Newton, left) and in an elevator (Einstein, right) that can’t
distinguish between observing gravity or acceleration. [47]
Since Newton's equation of motion by the act of gravity is given by
Inertial mass x Acceleration = Gravitational mass x Intensity of the gravitational field
Einstein assumed in 1907 that “complete physical equivalence exist in the Universe between
a gravitational field and the corresponding acceleration of the reference system” [48]. From
this idea the weak principle is stated: “Gravitational mass and inertial mass are equal, and
different test particles which have no mutual gravitational interactions at equal spacetime
points in a gravitational field, will experience an equal acceleration, independent of their
properties including their rest mass”. [48] Therefore the gravitational motion of a small test
body depends only on its initial position in spacetime and velocity, and not on its
composition.
The strong equivalence principle is an even stronger statement: “Everywhere in the Universe
a local experiment (with or without mutual gravitational interactions) in a freely falling
laboratory is independent of the laboratory’s velocity and its position in spacetime”. [49] The
free falling object must be small such that local tidal forces can be neglected. The strong
equivalence principle applies to objects with significant internal gravitation interactions such
as stars, planets, black holes etc., and requires the gravitational constant to be equal
everywhere in the Universe.
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An immediate consequence of the equivalence principles is that gravity can bend the
trajectory of massless particles such as photons (light), see fig. 2.8. To visualize why this is
true one imagines a photon crossing the elevator that is accelerating into space. As the
photon crosses the elevator, the floor is accelerated upwards and the photon appears to fall
downwards. By the equivalence principle the same act must occur when the photon is
moving in a gravitational field. [52]
Fig. 2.8. The sun curves spacetime, here
represented by the grid. The preceived position of
the star A differs from its actual position B, an
observation that shows gravity can bend light. [50]
Thus in general relativity gravity can be described as motion caused by the geometric
properties of curved spacetime, and the relation is specified by the Einstein field equations,
described by Wheeler as “Matter tells spacetime how to curve, and spacetime tells matter
and light how to move” [51]. Furthermore space contracts near mass and dilates away from
it, time dilates near mass and contracts away from it, both effects that have been observed
with great precision. The strong equivalence principle suggests that the nature of gravity is
entirely geometrical and the metric solely determines the effect of gravity. Einstein proposed
three experiments that could be explained by general relativity, accurate measurements on
the perihelion precession of Mercury’s orbit, deflection of light by the Sun and gravitational
redshift of light, and was able to write down the field equations in 1915. [52] [53] [54]. The
first experimental confirmation of general relativity was made in 1919 when Eddington was
photographing a total solar eclipse and measurements of stellar positions near the solar limb
darkening proved Einstein was right. [55] Einstein furthermore showed that general relativity
agrees closely with the observed amount of perihelion shift of Mercury’s orbit and with the
gravitational redshift of light measured by the Pound-Rebka experiment in 1959. [56] Another
strong prove from modern observations came from the strong and weak lensing effects first
measured in respectively 1980 and 1986. Furthermore general relativity predicts a time
dilation in a gravitational field and this was confirmed by the Hafele Keating experiment with
atomic clocks flying airplanes in 1972. [57]
In 1916 Einstein predicted the existence of gravitational waves, ripples in the curvature of
spacetime that can propagate outwards from the source as waves and transport energy as
gravitational radiation. These gravitational waves have been indirectly detected in 1974 from
a binary pulsar by Hulse [58]. Accurate timing of the pulses showed that the binary pulsars
gradually spiral towards eachother demonstrating an energy loss in close agreement with the
predicted energy radiated by gravitational waves. [59]
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2.4 The cosmological principle
The Aristotelian model placing the Earth at a unique place in the center of the solar system
assuming everything is orbiting around it, was the commonly accepted viewpoint untill the
Middle Ages. At the end of the 16th century it was the idea of Copernicus that the Earth has
no special place in the Universe, and that the Sun was the center of our solar system with
the Earth in orbit around the Sun, while Galilei in the beginning of the 17th century showed
by observations on the moons of Jupiter and the motion of Venus, the Copernican
heliocentric viewpoint to be correct. [60] [61]. This new perspective had its implications on
religion, philosophy metaphysics and science, and encountered much resistance from the
Catholic Church.
On large scales, larger then galaxies and clusters, the Universe appears to look the same in
all directions by any observer, and the average density of matter is about equal at all places
and fairly smooth distributed. The Copernican viewpoint can therefore be extended by stating
that in the Universe no special directions (isotropy) and no special places (homogeneity)
exist. The assumption that on large scales the Universe is homogeneous and isotropic is
called the cosmological principle. Of course the Universe on smaller scales looks clumpy and
is far from isotropic and homogenous. [62]
Initially the cosmological principle was a postulate, but recent observations confirm this idea.
In 1997 the 2dF Galaxy Redshift survey started to collect data followed by the larger Sloan
Digital Sky Survey (SDSS) which began in 2000 to determine the large scale structure of the
local Universe, see fig. 2.9. The data of the observations done in New Mexico (USA) of
around 500 million objects covering 35% of the sky, were released in 2011, and the Universe
has been measured to be statistically homogeneous on scales larger than 70 Mpc at the
10% level. [63] In chapter 2.5 we will see that the cosmological principle singles out a unique
form of the spacetime geometry.
Fig. 2.9. A slice through the SDSS map of
the distribution of galaxies. The Earth is at
the center of the image and each point
represents a galaxy typically containing
about 100 billion stars. The colors of the
galaxies indicate the age of their stars with
the redder more strongly clustered points
showing galaxies that are made of older
stars. The outer circle is at a distance of two
billion light years from the centre. The region
between the wedges was not mapped by
the SDSS experiment because in these
directions dust in our own Galaxy obscures
the view of the more distant Universe [64]
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2.5 The expanding Universe
Einstein applied the equations of general relativity to the Universe as a whole, being the first
to describe a testable theory of the Universe and initiating the field of relativistic cosmology.
The evolution of the Universe is then described by its energy density content and curvature.
A model Universe is typically filled with different types of “matter” which are classified by their
energy density. Einstein found that a Universe filled with matter and radiation would exhibit
an infinite lasting expansion that will be slowed down by gravitation. This can be understood
considering expansion leads to a decrease of the energy density, further slowing down
expansion, etc. A slightly positive curved Universe would either effect in a recollapse, or a
slightly negative curved Universe would expand infinitely. The idea of an expanding Universe
originating from a smaller hotter state, was early in the 20th century an unusual idea. But the
idea of a Universe that would either expand forever or recollapse was so abnormal that it
made Einstein to develop a way to describe a static non-expanding Universe. Realizing
empty space is not “nothing” and that it is possible for more space to come into existence in
an expanding Universe, he made the prediction that empty space has its own energy. As
more space comes into existence this vacuum energy has the property that its energy
density remains constant. Adding such a cosmological constant or vacuum energy to the
content of the Universe, is uniquely allowed theoretically, and has an exponentially
expanding effect on the Universe. Einstein hoped to describe a static non expanding
Universe by assuming the cosmological constant would have the exact right value to balance
the slowing down expansion of the matter radiation content. However such a Universe could
be static non-expanding for a short period of time, but is fundamentally unstable since small
deviations would increase rapidly, leading again to fast expansion or collapse. Since the
model is fundamentally unstable and a perfectly balanced value for the cosmological
constant seems rather unlikely, it’s not applicable to our Universe.
Lemaitre (1927) and Hubble (1929) observed in 1929 that the velocities of faraway galaxies
were strongly correlated with their distance, see fig. 2.10. [65]. Nowadays the so called
Hubble constant is measured to be 67.74 km·s-1·Mpc-1 giving the speed in km/s of a galaxy
1Mpc (3.09·1019 km) away. Assuming that the Earth is not the center of the Universe the
interpretation is that all observable regions of the Universe moving away from all others with
a velocity proportional to their distance, proving the Universe is overall expanding. With
respect to the past viewpoint of the Universe Einstein therefore concluded the cosmological
constant to be his greatest blunder, and dropped it from his equations. Since we know that
the distance between galaxies increases today, reversing the argument means that galaxies
were closer located in the past. Thus the expansion of the Universe leads to the conclusion
the Universe was really denser (and hotter) in the past.
Fig. 2.10. Original plot of Hubble expansion
as published by Hubble in 1929. [a65]
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2.6 The Big Bang model
The cosmological principle implies that the Universe and therefore the metric are
homogeneous and isotropic on large scales as observed, and physical laws are universal
and independent of time leading to constant physical constants. Observations show that the
largest possible deviation of the fine structure constant over the age of the Universe is of
order 10−5. [81] The cosmological principle singles out the Friedmann–Lemaître–Robertson–
Walker metric (FLRW metric) as a solution from Einstein‘s equations. The FLRW metric
together with the Friedmann equation describes a homogeneous, isotropic expanding or
contracting Universe, and was developed independently by the authors in the 1920s and
1930s. It describes the observable Universe as a whole based on the mathematics of fluid
dynamics taking into account the inventory of the Universe as a perfect fluid. [25]
The greatest consequence of the cosmological principle is that it implies that all parts of
space are causally connected at some time in the past (although they may no longer be
connected today) and leads to the conclusion that the whole Universe appeared at a single
moment of time, a Big Bang. We note that due to isotropy there is not a point where the Big
Bang occurred since there is no definition of a center point in the Universe.
The metric now contains a scale factor a(t) which describes how the size of the Universe
expands or contracts as a function of time. This allows the choice of a coordinate system to
be made called comoving coordinates, which can be seen as a grid that expands along with
the Universe, see fig. 2.11 and fig. 2.12. Using co-moving coordinates objects which move
with the expansion of the Universe conveniently are fixed points on the metric. The
coordinate distance called comoving distance remains constant but the physical distance
between two comoving objects expands with the scale factor of the Universe. The FLRW
metric assumes a uniform distribution of mass and energy and therefore it applies to our
Universe only on large scales. Therefore local concentrations of matter such as galaxies are
gravitationally bound and do not expand in proportion to the expansion of the Universe,
however it is the space between galaxies that is expanding accelerated. The ratio between
the physical distance and the comoving distance is the scale factor a(t). The scale factor
increases for an expanding Universe, is dependent on time and determines the Hubble
constant by formula [2.2] which is the time dependent constant of proportionality between the
proper distance and its velocity.
H (t ) 
a '(t )
a(t )
[2.2]
The cosmic inventory is described by the categories (dark)matter, radiation and dark energy.
All matter consists of non-relativistic particles, and when a given volume of matter expands
by a factor a(t) as V ~ a3 the energy density dilutes as ρmatter ~ a-3. For relativistic particles or
radiation the energy density dilution contains an extra factor a-1 to account for the Doppler
effect and therefore ρradiation ~ a-4. The present radiation density comes from photons and
neutrinos and is neglectible compared to matter or dark energy densities. [25] [66]
The Universe today seems to expand accelerating and therefore is presumed to contain a
large component that can be described by a cosmological constant called dark energy. The
energy density of dark energy Λ is constant and a property of space itself, and doesn‘t dilute
for expanding space. The nature of dark energy is not known. [3]
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Fig. 2.11 (left). Expansion of the Universe. The comoving distance between points on the
grid remains constant as the Universe expands. The physical distance is proportional to the
comoving distance times the scale factor a(t) and hence gets larger as time evolves. [74]
Fig. 2.12 (right). Expansion visualized in three dimensions in which the space between the
objects expands. [68]
With the Friedmann equation [2.3] the behaviour of the scale factor and thus the Hubble
constant is described by matter and radiation energy density ρ, the curvature k of the
Universe and the dark energy density Λ.
8 G   k
 a'
H (t )    
  2
3
3 a
a
2
2
[2.3]
The curvature is indicative for the rate of expansion and whether or not the expansion rate is
increasing or decreasing and therefore implies the future fate of the Universe. When the
curvature is zero, the energy density of the Universe is equal to the so called critical density,
and the Universe will expand forever but with an ever decreasing rate. In a slightly positive
curved Universe gravitational attraction will eventually stop the expansion and recollapse the
Universe in a “Big Crunch”, a negative curved Universe will expand forever since no
sufficient energy density is present to stop the expansion. Combined observations from the
WMAP and Planck missions in 2008 and 2013 show the Universe is flat within 0.4% margin
of error and 13.798 billion years old, and using the measured Hubble constant today 67.8
km·s-1·Mpc-1 it is possible to calculate the critical density today, that is equal to the sum of all
contributions to the energy density, ρcrit = 1.9 10-29 g·cm-3 or about 5 hydrogen atoms per m 3.
[66] [69] [70]
When the dependence of the density of radiation and matter on the scale factor is used in
formula [2.3] and the critical density rescales the Friedmann equation with dimensionless
density parameters Ω
ρ
ρ
Ω
ρ
ρ
Ω
ρ
ρ
and Ωk the curvature density
parameter, the ratio of the Hubble constant at time t, H(t) and at present time H0 can be
expressed conveniently by formula [2.4]. [25]
H2
  R a 4   M a 3   K a 2   
H 02
[2.4]
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At present time this ratio is 1 and the scale factor is a(t) = 1. Considering the Universe is
measured to be flat at present and at earlier times, curvature is completely neglectable since
Ωk < . and the present the sum of density parameters ΩM , ΩR and ΩΛ must add up to 1.
Combined observations from the combined WMAP and Planck mission and baryon acoustic
oscillations (BAO’s) show ΩR = 9.4.10-5,, ΩM = 0.308 ,ΩΛ = 0.692, see fig. 2.13. [3] [70]
Fig. 2.13. Dark energy and dark matter content of the Universe. The intersection of the
supernova (SNe), cosmic microwave background (CMB) and baryonic acoustic oscillations
(BAOs) measurements indicate a topologically flat Universe with an energy density
composed 69.2% of dark energy (y-axis) and 30.8% of dark matter plus normal matter (xaxis). [71]
In the early Universe the scale factor was shorter meaning that the wavelength from CMB
photons was blue shifted, and energy and temperature were higher. The fact that the scale
factor is inversely proportional to temperature implies the initial state of the Universe was a
state of extreme high temperature and density. From formula [2.4] it can be seen that an
early hot and dense Universe is radiation dominated as the contribution to the energy density
from radiation increases most for small scale factors. With expansion the radiation density
drops faster than the matter density and the moment in time where radiation and matter
energy density were equal is called matter radiation equality and is estimated at 47.000
years after the Big Bang. The Universe remained optically thick to radiation untill the
Universe was about 378.000 years old and photons were last scattered composing the CMB.
Then a period of matter domination followed in which dark matter determined the growth of
the large scale of the Universe. Expansion was first slowed down by respectively radiation
and matter, but recently dark energy dominates the energy density of the Universe
accelerating its expansion, see fig. 2.14, fig. 2.15 and fig. 2.16. [3] [25] [66]
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Fig. 2.14. Examples of Universes with different matter density parameters from which ΩM =
0.30 ,ΩΛ = 0.70 is in close agreement with latest observations. It is showed how the Universe
recently started its accelerating expansion, whereas for models with no dark energy Ω Λ = 0
the Universe would expand with a constant speed ΩM = 0 or end with a Big Crunch ΩM > 0.
[66].
Fig. 2.15. Different epochs in the thermal history of the Universe. The radiation dominated
era is followed by a matter dominated era and the present dark energy dominated era. [72]
Astrophysics at the LHC – Dark matter search in ATLAS
Fig. 2.16. Cosmic evolution showed over the history of 13.7 billion years. [73]
28
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2.7.1 Problems with Big Bang theory: The horizon problem
An important problem of the Big Bang theory are the so called horizons. Since the Universe
has a finite age and light travels at a finite speed, there may exist events which occurred in
the past from which the radiated light has not sufficient time to reach us. The most distant
object observed therefore set the past horizon. On the other hand, because space is
expanding, light emitted by us today may never reach very distant objects. This defines a
future horizon, which for us limits the events in the future that we will be able to influence.
Remarkably the CMB reaching us from all directions in space is almost perfectly isotropic
with anisotropies in the CMB temperature within 0.01% (see chapter 1.2.3). This suggest
there must have been thermal contact in the past, however regions in space with opposite
directions from us have past light cones that don't overlap since the light only just had
enough time to travel to us. Nowadays for two most distant regions in space in opposite
direction from us, it would take light twice the age of the Universe to reach the other region.
However at earlier times when the photons were emitted both regions would have been 100
times the age of the Universe causally disconnected. These parts of the Universe are outside
eachothers horizon and never have been in causal contact. This applies for any two points in
the CMB that are separated by more than one degree. The horizon problem asks how those
regions could have been in thermal contact with eachother while their distance is too large to
be causally connected, see fig. 2.17. [74]
Fig. 2.17. Picture with the Astronomer’s view of the Universe showing opposed regions of the
CMB could not have been in thermal contact. [74]
2.7.2 Problems with Big Bang theory: The flatness problem
In 1929 Hubble measured the expansion speed of the Universe, from which the critical
density of the Universe can be calculated. The Universe described by the FLRW metric has a
closed, flat or open geometry if the mean energy density is respectively less, equal or larger
than the critical density. Departures from the critical density since the Planck time (10 −43
seconds after the Big Bang) will increase curvature with evolving time, untill the present age
of the Universe. However since the Universe has not expanded or recollapsed to infinite
sizes in its 13.8 billion years of existence, the departures from the critical density must have
been extremely small at early ages to grows to the almost flat Universe today. Even at the
relatively late epoch of nucleosynthesis (minutes after the Big Bang) departures are maximal
allowed to be of order 10-14 or the Universe would not exist today. [27]
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2.7.3 Inflation theory
Cosmic inflation as a theory to explain the horizon and flatness problem in cosmology was
pioneerd and developed since 1980 by Guth, Starobinsky and Linde. Cosmic inflation
assumes a period of incredibly rapid inflation existed very early in the Big Bang process
which increased the size of the Universe by a factor of 1030 while temperatures dropped in a
supercooled expansion by a factor 10 5. A homogeneous and isotropic scalar non-zero
inflaton field would have driven the inflation of the Universe as the inflaton rolls down a
potential energy hill with a speed slowly compared to the expansion of the Universe. After
this slow roll large amounts of potential energy are released which would have led to the
formation of matter in a process called reheating. During inflation, the Universe undergoes an
exponential expansion, and the particle horizon expands much faster than the speed of light.
This allows to understand the observed isotropy of the CMB as regions which are presently
on opposite sides of the observable Universe are inside eachothers horizon by this short era
of inflation.
Extrapolating the expansion of the Universe back in time will lead to a so called singularity, a
moment at a finite time in the past with infinite density and temperature. At this singularity all
present theories will breakdown and will be unable to describe the physics. However it’s
important to realize that such a singularity is not a point in space but a moment in time, with
infinite density across the whole Universe and collapsed space. The Big Bang is a
description of the expansion of space itself starting from this singularity described by the
doubling frequency of inflation, and is not an explosion of matter into space. The speed
objects in space are separating with is then proportional to their distance and objects can
separate faster than the speed of light when space itself expands. Therefore objects which
have been in thermal contact in the early Universe could at present be separated outside
eachothers horizon. Furthermore during inflation spacetime expanded such that its curvature
has been smoothed, and the present nearly flat state and critical density can be explained
with respect to the age of the Universe without assuming unnatural small departures from the
critical density at early ages [74] [25]
During the inflationary phase quantum thermal fluctuations would have been magnified to
cosmic scales and these primordial density fluctuations could be the seeds of all current
evolved structure in the Universe. These primordial fluctuations are observed to be Gaussian
and nearly scale invariant by measurements on the CMB as prognosticated by inflation
theory. In March 2014 the detection of inflationary gravitational waves in the CMB was
announced by the BICEP2 collaboration which would provide experimental evidence for
cosmic inflation, however in June 2014 the precision was lowered and the discovery
unconfirmed while the findings where attributed to cosmic dust. [75]
‘t Hooft showed in 1974 that Grand unification theories predict topological defects in space
that would manifest as magnetic monopoles which would have been produced numerously in
the early Universe. No monopoles have been found so far and cosmic inflation solves this
problem as all defects are removed from space in a similar way as the curvature is smoothed
by inflation. [76]
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2.8.1 The thermal history of the Universe
Extrapolating the expansion of the Universe back in time will lead to the description of the
thermal history of the Universe trough its different era’s.
2.8.2 The Planck era
t < 10-43 s, l < 10-35 m, T > 1032 K, T > 1019 GeV
Extrapolating the expansion of the Universe back in time will lead to the description of the
thermal history of the Universe trough its different era’s. During the lanck era the Universe
was extremely dense, hot and small. Before an age of 10-43 s we could speak of the Universe
as a singularity since its size and temperature were infinite large and not described by any
physics we know today. At these energies and temperatures the forces of nature become
symmetric and equal in strength. Before this unification all four forces are indistiguisable and
could be described by a single force described by a theory of everything (TOE). A
suggestion for the description of spacetime at this point was done by John Wheeler in 1955
who realized that at such high temperatures very energetic virtual particles are continuously
created and annihilated according to the Heisenberg uncertainty principle. These virtual
particles would have sufficient energy to curve spacetime giving it a foamy structure called a
space time quantum foam. At the Planck time (10 -43 s) at Planck energies (1019 GeV) when
the Universe had the size of the Planck length (10-35 m) and Planck temperature (1032 K) the
force of gravity becomes distinguishable from the other 3 fundamental forces which remain
unified in a theory called Grand Unification Theory (GUT). [23] [77]
2.8.3 Grand Unification
t ~10–43 - 10–35 s, l ~10-35 - 10-28 m, T ~ 1019- 1016 GeV
After the Planck time the Universe cooled down further untill 10–35 s The symmetry of the
grand unified group G breaks down to the Standard model gauge group
SU(3)CxSU(2)LxU(1)Y and the quark and lepton sector become separated. At the unification
point energies are sufficient low energy (10 16 GeV) for the force of the strong interaction to
become distinguishable from the force of electromagnetism and weak interaction. However
the energy of particles is still too high for the strong force to bind quarks into hadrons and the
Universe is still a soup of particles, a quark gluon plasma. The GUT symmetry breaking gives
rise to a less symmetric vacuum state and the large positive vacuum energy density would
have contributed to the total cosmological constant providing a mechanism to start driving
inflation. [23] [77]
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2.8.4 Inflation era
t ~10–40 - 10–35 s, l ~ 10-28 - 10-2 m, T ~ 1016 GeV
From 10–40 to 10–35 seconds after the Big Bang the Universe experienced an extremely rapid
exponential expansion called cosmic inflation triggered by the GUT phase transition. The
size of the early Universe increased during this fraction of a second by a factor of at least
1026 to around 10 centimeters. The quark-gluon plasma of particles is distributed across the
Universe and the geometry of the early Universe is smoothened out. After inflation in a
process called reheating the potential energy is converted into the creation of matter heating
it to pre-inflation temperatures. The total energy of the Universe remains zero as the created
matter/energy content is balanced against the created space that is being filled by the
gravitational field with negative potential energy. [74]
2.8.5 Baryogenesis
t ~ 10–35 - 10–12 s, T ~ 1016 - 103 GeV
When the era of inflation ends the Universe is permeated totally by a quark gluon plasma.
The strong force is already separated from the elektroweak forces and is the dominant force.
The strong force binds the quarks and antiquarks to baryons and antibaryons but these
hadrons are unstable since energies are still too high. There is currently no explanation why
the Universe contains far more baryons than antibaryons. In the early Universe there should
have been a small asymmetry or order 10 -10 in the baryon antibaryon density such that after
annihilation at the end of the era of baryogenesis a fraction of the baryons would remain
explaining the current amount of baryons today. If there was a complete symmetry there
would have no baryons today because of baryon antibaryon annihilation. Sakharov showed
that the Universe could be created with equal amounts of matter and antimatter, but can
develop into an asymmetric state when 3 conditions are satisfied (Sakharov conditions). The
conditions are 1) Violation of baryon number. 2) Violation of CP symmetry such that matter is
produced more commonly than anti-matter (which occurs in the Standard Model due to an
imaginary parameter in the weak interactions). 3) A Universe out of thermal equilibrium such
that the creation of matter from antimatter and vice versa happens at unequal rates (the
Universe is cooling off towards thermal equilibrium in Big Bang cosmology). The problem is
that CP violation occurs in the Standard Model quantified by an invariant quantity called the
Jarleskog invariant depending on the imaginary parameter in the weak interaction, which is
however too small to explain the current amount of matter today. Another explanation
suggests regions of the Universe exist in which antimatter dominates but such zones are
never observed and considered unlikely to exist. [23] [78]
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2.8.6 Electroweak symmetry breaking
t ~ 10–12 - 10–6 s, T ~ 1 TeV – 150 MeV
As the Universe cools down to the electroweak scale, an energy level of ~ 246 GeV, the
Higgs field spontaneously acquires a vacuum expectation value. This vacuum spontaneously
breaks the electroweak gauge symmetry denoted as SU(3)CxSU(2)LxU(1)Y into SU(3)CxU(1)Q
(see Chapter 3.3.3). As an effect of the symmetry breaking the weak force mediators
become massive which suppresses the range of the weak interaction, the Higgs particle
becomes massive and all the fermions become massive. The strong interaction and
electromagnetic force bosons (gluons and photon) remain massless. At the end of the
electroweak epoch all 4 forces are separated to their current form, while particles have
acquired their mass trough the Higgs mechanism. At temperatures larger than 100 GeV
thermal equilibrium existed keeping the number density of particle species in equilibrium. As
the Universe cooled below ~ 100 GeV creation of WIMPs became a rare process while
annihilation still occurred therefore decreasing the number density of WIMPs exponentially.
At some time and temperature the WIMP annihilation rate drops below the expansion rate of
the Universe and the WIMP dark matter particles are said to have experienced a freeze out
and were decoupled. If the WIMPs are stable particles they can now move freely trough the
Universe only experiencing the weak force and gravitation. The present dark matter
abundance ΩDM is inversely proportional to the annihilation cross section <σv> ann by formula
[2.5], since a larger cross section corresponds to lower abundances today. The present
scaled dark matter energy density of today ΩDM = 0.3 remarkably corresponds to a (weakly
interacting) cross section of σ ~ .3 pb for WIMPs with Tfreezeout ~ mWIMP ~ 100 GeV. This
coincidence is called the WIMP miracle and WIMP candidates with cross sections of this
order exist in numerous extensions of the Standard Model. [25]
 DM
3 1026 cm3 s 1
 0.110
  v ann
[2.5]
2.8.7 The hadron epoch
t ~ 10-6 – 1 s, T ~ 150 MeV – 1 MeV
At times before 1 microsecond after the Big Bang the temperature of the Universe is still too
high to bind the quarks from the quark-gluon plasma that is composing the Universe. During
the hadron epoch the temperature drops below the QCD phase transition scale of ~ 160 GeV
such that the strong interaction is strong enough to bind quarks into stable baryons such as
protons and neutrons. During the QCD phase transition chiral symmetry is the symmetry
which is spontaneously broken. [66]
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2.8.8 The lepton epoch
t ~ 1 s – 10 s, T ~ 1 MeV – 0.1 MeV
Around 1 second after the Big Bang neutrinos decoupled and began travelling freely through
space analogous to the WIMP thermal freeze out. About 10 seconds after the Big Bang the
Universe cooled down to an energy (511 keV) at which no new lepton pairs are created and
most lepton/antileptons are annihilated into photons. The temperature of the photons was
therefore larger then the temperature of the leptons (by a factor (4/11) 3). This temperature
difference and the difference in statistics (Bose-Einstein for photons and Fermi-Dirac for
leptons) results in an energy density of a single species of massless neutrinos of 0.23 times
that of photons. [25] [66]
2.8.9 Nucleosynthesis
t ~ 10 s – 20 min, T ~ 0.1 MeV – 1 keV
In the period after 10 seconds most leptons and anti-leptons are annihilated and the energy
of the Universe is dominated by photons. For a period of 380.000 years (also called the
photon epoch) these photons were interacting frequently with charged protons, electrons and
nuclei, and continued to do so for the next 380.000 years. At energies of ~ MeV no bound
nuclei or neutral atoms existed since radiation would have been energetic enough to
immediately destroy any nucleus. As the Universe cooled further, at the point most photons
have energies below ~ 0.1 MeV after t ~ 10 s it became possible for nuclei to be stable
particles. In a process called nucleosynthesis first deuterium is formed by fusing a proton and
neutron, after which more protons and neutrons would be bound to form He-3, He-4, Li and
Be. This process of nuclear fusion only occurs when the particle densities are sufficient high.
As the Universe expands further, the density will decrease and nuclear fusion will slow down.
Neutrons are unstable particles with a lifetime of t ~ 15 min unless bound inside a nucleus.
Therefore after ~ 20 min no individual neutrons are left and nuclear fusion will stop. The ratio
in which the light elements up to Li are formed depends on the expansion rate of the
Universe at that time (related to the total matter and radiation density) and the density of
protons and neutrons (baryonic matter density). The elements remained ionised untill the
temperature of the Universe cooled below the ionisation energy. [27]
2.8.10 Matter radiation equality
t ~ 46.000 years, T ~ 104 K, T ~ 1 eV
Since the energy density of radiation scales as a-4 while that of matter scales as a-3 the very
early Universe was radiation dominated. However at 46.000 years after the Big Bang the
energy density of non-relativistic matter (nuclei) and relativistic radiation (photons) were
equal. From this moment on gravitation will have started to play an important role since
gravitational attraction will balance pressure effects. Perturbations previously being wiped out
by free-streaming radiation will start to grow in amplitude. Cold dark matter is the dominating
form of matter, and its gravitational pull will lead to gravitational collapse and amplification of
the small imhomogeneties left by cosmic inflations. Dense regions became denser and voids
emptier, and dark matter is required to explain the rate of formation of large scale structures.
[25] [66]
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2.8.11 Recombination
t ~ 377.000 years, T ~ 3000 K, T ~ 0.25 eV
Before recombination, photons and electrons were coupled trough Compton scattering and
electrons and protons were coupled by Coulomb scattering. The photons, electrons and
protons formed a photon-baryon fluid in which the particles and light were coupled. When the
temperature droped below 0.25 eV electrons and protons will have combined to form neutral
hydrogen or helium. Hydrogen and helium are at first ionised with no electrons bound to the
nuclei, and therefore positively charged. Electrons get bound by the ions forming neutral
atoms in a fast process called recombination. At the end of the recombination era most of the
protons and electrons in the Universe were bound up in neutral atoms. Therefore the
photons mean free path became effectively infinite and the photons could travel freely trough
the Universe without interacting. The photons are said to have decoupled from matter and
the Universe became transparent. Today the most distant and oldest light observable are
photons from the “first light”, the cosmic microwave background created 377.000 years after
the Big Bang. At the time of recombination the baryon acoustic oscillations in the electronbaryon plasma were embedded in the distribution of matter and reflected in the CMB. [66]
2.8.12 Structure formation
t ~ 377.000 – 108 years, T ~ 3000 - 300 K
The formation of structures in the Universe is assumed to proceed hierarchically with smaller
structures being formed first, and larger structures formed later. First by the force of gravity
small initial fluctuations in the matter density already present at early times attracted matter
to regions with higher density and matter moved away from regions with lower density.
However this was a slow process and in order to explain the present observed structures the
density fluctuations must have had an amplitude that was too large to be consistent.
Therefore it is assumed a significant fraction of the matter in the Universe consists of nonbaryonic matter, and density fluctuations have started to grow before recombination. This
would occur since the interaction of non-baryonic matter and radiation is considerable
weaker than baryonic matter, such that radiation could not have slowed down early
fluctuations. Under this gravitational clustering of interstellar matter accelerated by dark
matter, temperatures were raised untill nuclear fusion ignites the center of the star and a
dynamical equilibrium formed between the thermal pressure and gravity. In this way quasars,
early stars and early active galaxies were formed as early as 150 million years after the Big
Bang ending the period called the Dark Ages and starting the period of structure formation. In
the first stars that are assumed to be formed light elements are fused into elements heavier
than lithium. [77]
2.8.13 Reionization
t ~ 108 - 109 years
After the first stars were formed their intense radiation released ionised matter in the
Universe such as hydrogen. This reionization ionised hydrogen again to an plasma at around
150 million years after the Big Bang, but remained transparent as matter had diffused
sufficient by the expansion of the Universe.
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2.8.14 Formation of galaxies
t > 5.108 years
The formation of galaxies began about 500 million years after the Big Bang and the top-down
and bottom-up model were discussed in Chapter 1. Structure formation proceeding bottom
up, with the smallest structures being created first followed by galaxies and clusters of
galaxies is consistent with the structures observed in the Universe by galaxy surveys, such
as the 2dF Galaxy Redshift Survey and the Sloan Digital Sky Survey.
In the early Universe dark matter clumps the size of superclusters are attracted by gravity
and form a network of filaments. At dense regions baryonic mass concentrates and galaxies
and clusters are formed. The dark matter distribution was modelled for example in the
Millennium Run which used more than 10 billion particles to model the evolution. A slice of
the dark matter density distribution modelled by the Millenium Run is shown in fig. 2.18. [79]
[80]
Fig. 2.18 (left). Slices of the simulated dark matter density distribution for a cubic region of
the Universe of 2 billion light-years (left) and subsequent panels to the right zoom in by a
factor of four with respect to the previous ones. From bottom to top the situation of
respectively 0.21, 1.0 and 4.7 Gy after the Big Bang is shown. [79]
Table 2.1 (right) Parameter 68% confidence limits for the ΛCDM model from 2015 Planck
CMB power spectra, in combination with lensing reconstruction (“lensing”) and external data
(“ext BAO+JLA+H0”) for TT, TE and EE polarization spectra. The 6 independent parameters
are physical baryon density Ωbh2, physical cold dark matter density Ωch2, age of the Universe
t0, scalar spectral index ns, curvature fluctuation amplitude Δr2 and reionization optical depth
τ. [24]
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2.8.15 Formation of the Solar System
t > 9·109 years
The Solar System is assumed to be formed about 4.6 billion years ago, 9 billion years after
the Big Bang, after the Galactic disk of the Milky Way was formed 8.8 billions years ago. A
cloud of mainly hydrogen with elements created by previous generations gravitationally
collapsed and ignited the Sun in its center and its surrounding accretion disk collapsed into
multiple smaller planets, asteroids and comets.
2.8.17 Today
t = 13.8·109 years
The current age of the Universe is estimated to be 13.8 billion years. Several billion years
ago the expansion of the Universe started to accelerate and the energy density of the
Universe has become dominated by dark energy instead of matter, see fig. 2.19.
2.9. ΛCDM
The ΛCDM (Lambda Cold Dark Matter) model describes a Big Bang cosmological model with
the presence of a cosmological constant or dark energy (Λ) and cold dark matter (CDM) in
order to explain the abundances of light elements formed in nucleosynthesis, the large scale
structure and accelerated expansion of the Universe and the anisotropies found in the CMB.
In this model the following 6 independent parameters are used to fit observations with ΛCDM
cosmology; physical baryon density Ωbh2, physical cold dark matter density Ωch2, age of the
Universe t0, scalar spectral index ns, curvature fluctuation amplitude Δr2 and reionization
optical depth τ. These parameters are estimated using large computer searches that include
multiple experiments like Planck, WMAP and observations of BAO and Type-Ia supernova.
In table 2.1 the values of the ΛCDM parameters and calculated values are given based on
the Planck 2015 data together with fixed parameters total density Ωtot = 1, equation of state
dark energy ω = 1 and a neglectible sum of neutrino masses. [27] [70]
Fig. 2.19. The different eras in the thermal history of the Universe are showed with according
timescale and temperature scale. It can be seen on the y-axis how the fraction of the energy
density consisting of dark energy, dark matter and baryonic matter changes in time. [84]
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3. Dark matter in particle physics
3.1 Introduction
The Standard Model is the theory that describes all known elementary particles and its
interaction trough the electromagnetic, weak and strong nuclear force. The theory was
formulated in the 1970s and explains experimental results with large precision and great
detail. The theory is based on quantum field theory in which the symmetries of the
Lagrangian and spacetime define the fundamental interactions and conservation laws.
However some phenomena are still unexplained by the Standard Model and interestingly
some beyond the Standard Model theories predict WIMP dark matter candidate particles.
Collider experiments operating at higher energies than ever reached before in history, such
as in the LHC phase 0 run, are sensitive to the presence of WIMPs and could give us more
insight in the nature of dark matter and the accompanying particle physics theory.
3.2.1 Symmetries and conservation laws
An important theorem in physics was proven by Emmy Noether in 1915 stating that any
symmetry of the action of a physical system is reflected by a conservation law and an
unobservable quantity. In mathematics a group is a set of elements together with an
operation that combines two elements to form a third element which is still contained in the
group. Every symmetry of a physical system is reflected by an operation and elements which
form a group. Since the action of a physical system is described by a Lagrangian density
function of a quantum field integrated over spacetime, symmetries of spacetime itself
(external symmetries) and symmetries of the quantum fields existing within spacetime
(internal symmetries) will give rise to conserved quantities. Furthermore using the
cosmological principle it is assumed that all conservation laws in physics are universal and
applicable to the whole Universe at any time in the past and future [87].
3.2.2 External symmetries of spacetime
Since we assume physics to be invariant when an event is translated in time and/or space,
we can state physics is invariant under the operation of the so called Galilei translations.
These external symmetries of spacetime are reflected by symmetries of the Galilei group and
give rise to the laws of non-relativistic energy and momentum conservation, while the
unobservable quantities are absolute position and time (as predicted by special relativity). A
subgroup of the Galilei group is rotational symmetry, since the Universe is isotropic and
nature is independent of the orientation of an experiment, physics is invariant under
orientation or direction. This leads to conservation of angular momentum while the
unobservable is the absolute angle. The center-of-mass theorem in 3 dimensions is a
consequence of momentum conservation since in non-relativistic physics momentum is the
product of mass and velocity.
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Assuming physics is invariant for any reference frame, Lorentz transformations are the
operations reflecting the according symmetry. The associated conservation law is
conservation of relativistic four-momentum while the unobservable quantity is the existence
of an absolute reference frame. A subgroup of the Lorentz group is the rotational group and
the rotational Noether charges are the four dimensional generalizations of angular
momentum while its conserved components yield the relativistic generalization of the center
of mass theorem. Physical quantities are said to posses Lorentz covariance if they transform
under a given representation of the Lorentz group. According to the representation theory of
the Lorentz group such quantities consist of scalars, four-vectors, four-tensors, and spinors.
Both the symmetries of the Galileo and Lorentz group can be described by the Poincare
group, and objects which are invariant under rotations in the Poincare group have complete
relativistic invariance. [86] [88] [89]
3.2.3 Internal symmetries of spacetime
In particle physics all elementary particles and their interactions under the three forces
electromagnetism, weak and strong nuclear force are described by the existence of
corresponding relativistic quantum fields in spacetime. The local gauge symmetries of these
fields are described by the corresponding Lagrangians are the internal symmetries of
spacetime and result in charge, weak isospin and color charge conservation laws such that
the interaction is well described. Particles can now be specified by representations of the
Poincare group (four momentum or mass squared) and their intrinsic quantum numbers. The
interaction is described trough the exchange of short lived virtual particles (respectively the
photon, W and Z bosons, and gluons) that exist according to the uncertainty principle
described by the short distance theory of quantum mechanics.
The reason all existing particles are described by quantum fields lies in the locality of nature.
Coulomb and Newton’s law describe forces by “action at distance” but this situation is
philosophically unsatisfactory and experimentally wrong. The theory of the Standard Model is
however correctly described using a quantum field with the requirement of locality and with
the interactions mediated in a local fashion by the field while for gravity no quantum field yet
exists. A second argument to use a quantum field theory is the existence of wave-particle
duality in nature. Regarding the particle’s field as fundamental wave like properties can be
described, while viewing a particle as an excitation of the field describes particle like
properties. In all three forces of the Standard Model interactions are mediated locally trough
the quantum field and particles are described by excitations of this field.
Particles and interaction should both satisfy quantum mechanics and Lorentz invariance, and
a first attempt was made constructing a valid theory by assuming the time evolution of a state
described by the Schrodinger equation and using relativistic motion, such that one obtains
the Klein Gordon equation. However the Klein Gordon equation contains a second order time
derivative and this can lead to negative energy densities. Dirac solved the problem for spin ½
fermions by introducing a discrete label to account for the spin of a particle and using the
empirical fact that fermions obey the Pauli exclusion principle. However negative energy
states are still present and interpreted as anti particles, and Dirac assumed all negative
energy states were occupied which make this Dirac sea unobservable.
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There exist a ground state for positive states and if a negative energy particle is excited into
a positive energy state it would leave behind a hole in the sea of negative states, appearing
as a positive charge. In 1928 Dirac predicted therefore the existence of the positron (the
antiparticle of the electron) before it was discovered in 1932. Particles can be annihilated or
created and Dirac described first the need of anti-particles, however the Dirac equation fails
to describe particles that don’t obey the auli exclusion principle such as bosons.
In nature particle number is not conserved with particle production in LHC as an extreme
example. However quantum mechanics describes only single particles while probing a
particle at high energies with short distances will reveal more particle antiparticles pair near
the original particle. Dirac was the first to understand that a relativistic quantum equation
would imply the existence of particles and antiparticles, and could describe states with an
unspecified number of particles. Furthermore at scales smaller then the Compton wavelength
the concept of a single pointlike particle breaks down completely and any theory based on
the single particle Schrodinger equation is not correct. Quantum field theory has turned out to
be the correct theory to describe an unspecified number of particles. In a quantum field
theory the fields are fundamental, unique and existing everywhere in space, while particles
are a derived concept. In such theory it is automatically assumed every particle of the same
type to be identical and indistinguishable anywhere in the Universe since particles will arise
as excitations of the same quantum field that exist throughout the Universe. [89] [90]
Schwinger first proofed the CPT theorem in 1951 stating that all physical phenomena obey
CPT symmetry, that is the simultaneous combination of the operations charge conjugation
(C), parity transformation (P) and time reversal (T). Any Lorentz invariant quantum field
theory with a hermitian Hamiltonian is therefore invariant under CPT transformations. If CP
were an exact symmetry the laws of Nature would be equal for matter and antimatter.
Gravity, electromagnetism and the strong interaction are C and P symmetric, but weak
interactions violate separate C and P symmetry maximal since it couples only to left handed
particles. However the combined CP symmetry is still preserved in most weak interactions.
[78] [89]
A consequence of the Poincare symmetry of spacetime is that changing 2 identical particles
will leave a state unchanged except from the possible sign which determines the statistics of
the particle. Using that fermions change sign under two C T reflections while bosons don’t,
the theorem that relates particle spin to the correct statistics called the spin statistics theorem
can be proven. The spin statistics theorem states that integer spin particles (bosons) obey
Bose statistics and the quantum mechanical wavefunction is symmetric under exchange of 2
such particles, while half-integer spin particles (fermions) obey Fermi statistics and the
wavefunction is anti-symmetric under the exchange of 2 such particles, and thus relates
particle spin to the correct statistics. [88] [89]
Since time and space are treated differently in the Schrodinger equation it is impossible to
find a theory that is quantum mechanically and relativistic correct. Therefore time and space
have to be treated at equal footing, by promoting time to an operator. Similar to the
quantization of a classical field where the classical degrees of freedom are promoted to
operators acting on a Hilbert space, the degrees of freedom in quantum field theory are
operator valued functions of space and time with an infinite number of degrees of freedom for
every point in spacetime, and space and time are treated at equal footing making the theory
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Lorentz invariant. A gauge theory is a quantum field theory in which the Lagrangian is
invariant under a continuous group of transformations between possible gauges, called
gauge transformations. These gauge transformations form a Lie group as the symmetry
group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of
group generators, and for each group generator a corresponding gauge field exist. Gauge
fields leave the Lagrangian invariant under group transformations which is called gauge
invariance.
Invariance of the Lagrangian under a transformation identically performed at every point in
the space has a global symmetry while the requirement of a transformation at a unique point
is a tighter constraint and has local symmetry. Global symmetries have no gauge bosons or
gauge forces associated with them, for example B-L symmetry has no gauge bosons. Local
symmetries however have gauge bosons as excitations of the gauge fields that will represent
gauge forces. If the symmetry group is non-commutative, the gauge theory is referred to as
non-abelian and the gauge bosons will have self-interactions while for commutative
symmetry groups no self coupling of bosons is allowed in the theory. [91]
3.3.1 The standard model of particle physics
The Standard Model of particle physics is a theory that describes effectively the kinematics
and the interactions of the fundamental constituents of matter. It is a combination of quantum
chromodynamics (QCD), the theory of quarks and the strong interaction, with the GlashowWeinberg-Salam (GWS) theory of the electroweak interaction.
The local gauge symmetry of quantum electrodynamics (QED) is described by an abelian
U(1)Q symmetry group such that all fermions of the standard model with a charge Q can
interact trough the non-self coupling photon wich is the only force carrying boson particle of
the theory.
The weak interaction has local gauge symmetries which are described by an non-abelian
U(2)L symmetry group such that all fermions of the standard model which come in left
handed (L) dupets (up- and down-type particle couples such as electron and its electron
neutrino or up and down quark) can change flavour by the exchange of one of the 3 selfcoupling bosons of the theory. The weak interaction only acts on left handed particles and
violates C and P symmetry maximal while CP is preserved in most but not all processes.
Electroweak theory or GWS combines the symmetries of QED and the weak interaction to
the local gauge theory of the electroweak interaction with symmetry group SU(2)LxU(1)Y. The
associated 4 boson fields and charges are rotated by the weak mixing angle such that the 3
self interacting gauge bosons describing the weak interaction are the W +, W- and Z0 bosons
while the photon describes the electromagnetic interaction. The coupling of quarks to W/Z
bosons in their mass eigenstates is governed by the Cabibbo–Kobayashi–Maskawa (CKM)
matrix which has a single CP violating parameter that is the origin of rare CP-violating
processes in the Standard Model that occur differently for matter then for antimatter. [78] [91]
Strong interactions are described by QCD, a local gauge theory with a non-abelian symmetry
group SU(3)C such that quarks which come in triplets with associated colorcharge C (each
quark in the Standard Model has 1 of the 3 colorcharges red, green or blue) can have
interactions trough 1 of the 8 self-coupling bosons of the theory called gluons.
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The Standard Model is a gauge theory with a structure described by the gauge symmetry
group SU(3)CxSU(2)LxU(1)Y and the Poincare group. The standard model is formulated in
terms of a Lagrangian density, which is written as a function of quantum fields and their
derivatives, and a Lagrangian with this structure gives us the complete particle spectrum and
all possible interactions for the electrodynamics, weak and strong interaction while
preserving the conservation laws of spacetime. Furthermore this Lagrangian gives rise to
Feynman rules that are used to compute probability amplitudes for different processes and
its predictions have passed all experimental tests to the present day with high accuracy. In
the Standard Model there exist a total of 12 massless gauge bosons that mediate
interactions between other bosons or mainly fermionic matter. The photon (γ), the W +, Wand Z0 bosons and the eight gluons (g) are all denoted as force carriers in fig. 3.1. [94]
Fig. 3.1. The particle spectrum in the standard model. [a15]
Fermionic matter comes in 3 families of flavours that differ by their mass and coupling. We
differentiate between elementary particles called quarks or leptons. Quark type particles are
the only particles subject to the strong force besides electromagnetism and the weak force,
and are either up-type (charge + 2/3) or down-type (charge -1/3). Leptons come in electron type
particles (charge -1) and massless neutrino type particles (charge 0). Electron type leptons
are subject to electromagnetism and the weak force while detecting neutrinos is very rare
since they are only sensitive to the weak force. For every particle in fig. 3.1 there exists also
a corresponding antiparticle with the same mass but opposite charge and quantum numbers.
Hadrons are particles composed of quarks and exist in 3 different types, (anti)baryons and
mesons (altough a pentaquark state was discovered recently). The heavy (anti)baryons that
consists of three (anti)quarks like the (anti)neutron that are not stable and will eventually
decay into (anti)protons, and mesons that consist of a quark, antiquark pair like pions and
kaons. QCD describes the strong interaction between the quarks and has the remarkable
property that the coupling strength decreases for increasing distances (chapter 4.4.1). [86]
This leads to the situation a quark can’t be separated and observed individually as pulling it
away from other quarks stores energy in the binding gluon field such that from the work
delivered only new quark antiquark pair are being created. No free quarks are therefore
assumed to exist and quarks are bound in hardons which is called confinement. On the other
hand on short distances inside the hadron or reached in high energy collisions above the
QCD scale of ~0.22 GeV the quarks and gluons are interacting weakly and form a quarkgluon plasma in a process called asymptotic freedom. It is worth noting that the observable
valance quarks in a hadron are bound by massless virtual gluons which have a large QCDbinding energy that gives rise to the existence of virtual sea quarks. The effective mass of a
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quark or a hadron is for 99% originating from the sea quarks surrounding the 3 valance
quarks. Since leptons are much lighter than quarks, most of the matter in the Universe is
actually originating from binding energy inside hadrons. One could state little is understood
about the origin of mass in normal matter around us let alone the origin of invisible mass in
the Universe, and that is might be to simple to assume dark matter is originating from the
mass of a particle species. [86] [93]
3.3.2 Additional global symmetries in the Standard Model
The standard model Lagrangian has additional global symmetries that are not imposed but
yet exist. Baryonnumber is a global U(1)B symmetry that leads to the conservation of baryon
number. Quark fields are given baryon number B = 1/3 while leptons have B = 0 which
guarantees that baryons like the proton cannot decay into lighter leptons which is verified
experimentally with great precision. A second accidental symmetry is the existence of 3
global lepton symmetries. Each lepton symmetry gives rise to conservation of the number of
charged and neutral leptons for each family. These symmetries forbid charged leptons to
decay into a lighter generations which is supported by experimental data, but are in
contradiction with experimental evidences of neutrino oscillations.
3.3.3 The Higgs mechanism
Local gauge symmetry requires all particles, fermions and gauge bosons, described in the
Standard Model to be massless and thus moving with the speed of light. Also some
interactions like massless WW scattering violate unitarity at high energies and make the
theory non-renormalizable. The Higgs mechanism is needed in the theory in order to
introduce mass terms for bosons and fermions in the Lagrangian while preserving the
symmetry of the theory. An additional scalar Higgs SU(2) field is added to the Lagrangian
that is present in all empty space without sources, while the full Lagrangian is kept invariant
under the original gauge symmetry group. All the standard model particles are coupled to the
Higgs field trough their Yukawa couplings. The Higgs field defines the properties of space
itself and is constant. Therefore no relative movement to this field can be observed and the
Higgs field is in rest for all observers, which allows for particle masses to be identical
throughout the Universe. With the existence of a Higgs field the potential term in the
Lagrangian has a degenerate ground state that became apparent at sufficient low energies
shortly after the Big Bang. Once a ground state is chosen in a process called spontaneous
symmetry breaking the Higgs field acquires a vacuum expectation value, and the original
gauge symmetry group of the Lagrangian SU(3)CxSU(2)LxU(1)Y is broken into SU(3)CxU(1)Q.
Spontaneous symmetry breaking describes a situation in which the full Lagrangian of the
theory remains invariant under the original gauge group, but will make the vacuum not
invariant under this symmetry, such that the vacuum state breaks the symmetry in question.
The expansion of the Higgs field around the vacuum describes interactions with the Higgs
particle, introducing interaction and mass terms for the gauge bosons associated with the
broken SU(2)L symmetry (massive W +, W- and Z0), a mass term for the Higgs particle itself
through the covariant derivative of the Higgs field, and mass terms for the fermions trough
their Yukawa couplings to the Higgs field. The gauge bosons associated with the unbroken
gauge symmetries, the photon and gluons, remain massless. An important feature of the
standard model with the higgs field is that all couplings of the interactions are dimensionless
in natural units and the whole theory is renormalizable and unitary such that the theory can
be extrapolated to high energies. [95] [96]
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3.4.1 Problems with the Standard Model
Although interactions described by the Standard Model are accurately verified
experimentally, the theory has some shortcomings and evidence exist that must be explained
by a new theory beyond the standard model.
QED zero point energy density and the cosmological constant problem
As more distant objects are moving away faster, expansion is assumed to be accelerating
instead of slowing down due to gravity as one might think. The energy driving this expansion
is of unknown form and called dark energy, which is accounted for to compensate the
difference between the observed critical matter/energy density of 28.6% and unity. This
assumption was already made by Einstein who added the cosmological constant to his
equations, a vacuum energy density with constant energy density per volume with the
property to drive cosmic expansion.
Dark energy as a vacuum energy can have different origins. In quantum field theories the
concept of empty space is replaced with a ground state or vacuum state defined to be the
state with the lowest possible energy density. Quantum fields obey the uncertainty principle
and can therefore have zero-point fluctuations, even in empty space. Virtual particles from
higher order diagrams without external lines thus contribute to the vacuum energy density
whereas they don‘t add to the scattering amplitudes in the quantum field theory. In a
quantum field theory only differences in vacuum energy are observable, since any vacuum
energy can be evaded by redefining the energy scale. However since vacuum energy is
positive and constant as a function of volume it contributes as a positive cosmological
constant which leads to negative pressure and therefore has an accelerating effect on the
scale factor in a FRLW model, driving cosmological expansion. [3] The problem arises as the
estimated contributions to the vacuum energy density from the quantum fields we assume to
exist in the Standard model exceed the small observational value of the cosmological
constant by large orders of magnitude (>1056). Clearly such discrepancy between the
prediction of quantum field theory and general relativity describing the cosmology of the
Universe show a fundamental problem in present physics which is called the cosmological
hierarchy problem.
Typically zero point fluctuations are considered in the energy range from zero up to the UV
cut-off scale set by the electroweak scale of 246 GeV (see Ch 2.8.5), leads to an estimate for
the QED zero point energy of [3.1];
 EW ~  246 GeV  ~ 109 GeV 4 ~ 1047 erg / cm3
4
[3.1]
This contribution alone exceeds the observational value of dark energy [3.2] [171], which can
be interpreted as the total vacuum energy density by an order of magnitude of a 10 56.
 ~ 1029 g / cm3  1047 GeV 4 ~ 109 erg / cm3
[3.2]
Speculating quantum field theory is a valid framework above energies of the electroweak
scale, assuming an UV cut-off scale for zero point fluctuations set by the GUT scale (10 16
GeV) lead to the vacuum energy of;
GUT ~ (1016 GeV )4 ~ 1064 GeV 4 ~ 10102 erg / cm3
[3.3]
1]
Clearly the experimental value is even more exceeded by 10111 orders of magnitude.
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Higgs vacuum and the cosmological constant
All the massive particles in the Standard Model are coupled to the Higgs field and their
masses are proportional to the non-zero vacuum expectation value of the Higgs field in the
broken phase. Multiple (complex) Higgs fields may exist in for example supersymmetry but in
the simplest case in the Standard Model the Higgs field is described by Mexican hat potential
V(ϕ) of the form [3.4] which is illustrated in Fig 3.2 where λ is a Higgs self coupling constant
and μ is the energy scale related to the vacuum expectation value of the Higgs field by [3.5].
1
1
V ( )  V0   2 †   ( † )2
2
4
 2  v 2
| | 
[3.4]
[3.5]
2
v

[3.6]
Fig. 3.2. The Higgs potential V(ϕ) [172]
This potential has a minimum at [3.6] and using [3.4] and the assumption that V0 = 0 and gh =
α (fine structure constant) we obtain for the minimum of the potential [3.7] which is negative
and 1053 orders of magnitude larger than the experimental value of the vacuum energy
density.

higgs ~ V0  v 4 ~ 106 GeV 4 ~ 1044 erg / cm3
4
[3.7]
1]
Leaving the assumption V0 = 0 but assuming a positive value, the negative Higgs vacuum
energy could be cancelled to obtain the experimental value of the vacuum energy density,
but this would require unnatural precise fine tuning. Furthermore the Higgs field potential has
temperature dependent correction terms, such that even for a close to zero contribution of
the Higgs vacuum expectation value to the vacuum energy density today, deviations would
have been large at early ages. [95] [96]
QCD vacuum structure and the cosmological constant
The vacuum structure of QCD is non-perturbative in the low energy regime and the ground
state is therefore not well described. Models for the QCD vacuum use that at low energies
non-zero vacuum expectation values of the gluons and quark exist which give rise to
condensates. These condensates allow to estimate the vacuum energy density which is
model dependent but of order ~λQCD4, where λQCD ~ 0.22 GeV is the characteristic QCD scale
where the strong coupling constant is of order 1, such that the vacuum energy density of the
QCD vacuum can be estimated by [3.8] which is 1044 orders of magnitude larger than the
observational value of the total vacuum energy density.
QCD ~ (0.2 GeV )4 ~ 103 GeV 4 ~ 1035 erg / cm3
[3.8]
1]
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The λQCD scale is also assumed to set the temperature scale for QCD phase transitions, in
which quark and gluon condensates in vacuum appear for decreasing temperatures and
disappear at high temperatures as quarks are no longer confined (deconfinement) within
hadrons but form a quark-gluon plasma. During such a QCD phase transition the
spontaneously broken symmetry is chiral symmetry.
In chapter 2 it was explained how within the Big Bang model the Universe experienced
epochs of expansion and cooling. As a consequence the Universe has passed trough
different phase transitions as the critical temperatures characteristic to the energy scales of
the phase transitions were reached. At every phase transition symmetry breaking occurred,
such that the Universe is emerged from a symmetric vacuum state towards a less symmetric
vacuum state at present. GUT symmetry breaking is assumed to have broken the grand
unified symmetry group to the standard model gauge group at the characteristic energy scale
of 1016 GeV. At 102 GeV electroweak symmetry is broken and chiral symmetry is broken at
0.2 GeV. These energy scales of the different symmetry breakings allow the contributions to
the vacuum energy density estimated to be respectively of order 1064 GeV4, 109 GeV4 and
10-3 GeV4 whereas the vacuum expectation value of the Higgs fields leads to a negative
vacuum energy density of -106 GeV. Assuming the gravitational effect of these vacuum
energy densities is described by the contributions to the effective cosmological constant,
cosmology is described by a hierarchy of contributing cosmological constants. Although the
present observational value of the cosmological constant is well constrained, little is known
about its value at earlier times. However it is speculated that the GUT phase transition could
have triggered inflation in the early Universe, whereas it is unknown which mechanism drives
the current expansion of the Universe.
Radiative corrections on Higgs propagator (hierarchy problem)
In the Standard Model the Higgs boson propagator receives infinitely many quantum
16
corrections from virtual fermion loops. The corrections are not small but of order Λ =
GeV for a theory valid up to the GUT scale, or Λ = 19 GeV for the Planck scale, while the
Higgs mass is not predicted by the Standard Model but by theory expected to be of the order
of the electroweak scale (246 GeV) while it is measured to have a mass of 125 GeV [118].
The fact that the sizes of the corrections to the Higgs mass are at least 1014 larger then the
value of the Higgs mass itself is a precision and fine-tuning which is highly unnatural and a
hierarchy problem in the theory. This philosophical issue arises generally in theories with
massive (m) spin 0 fields such as the Higgs field, while in “technically natural” theories with
higher-spin fields the action has more symmetry which typically prevents divergences. [95]
Charge quantization
Currently there is no explanation why the electron charge is exactly equal but opposite in
sign to the proton charge. In the standard model the lepton and quark sector are completely
separate parts and no coupling exists between quarks and leptons since the strong force
works only on colored particles and leptons are colorless. Grand Unified Theories (GUT’s)
assume nature has higher symmetries like SU(5), allowing interactions between quarks and
leptons trough X and Y bosons. However converting quarks to leptons and vice versa
violates the conservation of baryon and lepton number, but baryon number minus lepton
number (B-L) would still be conserved when a quark is changed to an anti-lepton and the
GUT is said to have a global U(1)B-L symmetry. However in these models proton decay is
possible while experimentally proton decay has never been observed, the lower limit on the
proton half life is 5.9·1033 year by observations searching for decay to K-mesons. [97] [98]
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The strong CP problem
In quantumchromodynamics CP symmetry breaking terms are allowed in the Lagrangian,
however no violation of the CP-symmetry in strong interactions has ever been observed. In
Peccei–Quinn theory the QCD Lagrangian is extended with a C violating term with a θ
parameter, and the massive goldstone boson that results from the spontaneously broken
Peccei–Quinn symmetry is a new particle called an axion. The existence of axions is purely
hypothetical but their properties of abundant creating during the Big Bang, small cross
section for strong and weak forces and a low mass preventing decay modes make it a
possible dark matter candidate experiments like XENON100, CDMS and others are
searching for.
Number of particle generations
The fermionic part of the particle spectrum comes in three generations of particles. Whereas
the number of bosonic generations is restricted to 1 by the gauge symmetries of the
Standard Model, little is understood about the origin of the discrete number of 3 generations
for the number of fermionic generations. [98]
Fermion masses and mixing angles
The masses of fermions are spread out in a large region from 0.511 MeV for the electron (1 st
generation lepton) to 175 GeV for a top quark (3 rd generation quark). The fermion masses
are fixed by the Yukawa couplings of the fermions to the Higgs field which are unpredicted
parameters in the Standard Model which have to be put in by hand to match experimental
results. Interestingly the CKM matrix describing the flavour mixing in the weak interaction has
a diagonal structure resembling the hierarchy found in the particle mass spectrum with
respect to the different generations. The correspondence between fermion masses and
mixing angles is however speculative and no accepted theory gives a satisfactory
explanation. [78] [92] [95]
Gauge couplings
The strong, weak and electromagnetic gauge couplings are not predicted by the standard
model and have large differences. Only in supersymmetric theories gauge coupling
unification exists.
Massive neutrinos
The experimentally verified neutrino oscillations require neutrinos to have a small but
nonzero mass. Massive neutrinos are not described in the standard model. The existence of
right handed massive neutrino representations can hypothetically be added to the Standard
model, but these particles with an implied fine-tuned small mass would exhibit the feature to
be completely sterile to all interactions except gravity. Sterile neutrinos with masses of ~ keV
are ruled out since such warm dark matter would rule out the observed dark matter structure.
Still sterile neutrinos are an interesting baryonic dark matter candidate and several
experiments aim to discover these neutral heavy leptons for example in FermiLab and LEPI3. [31]
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Besides the unsolved problems in the standard model some other unexplained phenomena
in nature as well in theoretical research raise big questions and motivates new theories.
Dark matter
Cosmological observations have shown that more than 85% of the matter in the Universe is
made of non-baryonic dark matter with leading candidate a particle with WIMP like
properties. Such particles are not predicted within the Standard model, but many WIMP
candidates occur in supersymmetric theories.
Matter-antimatter asymmetry in the Universe
Matter and antimatter should have been produced at equal amounts during the Big Bang,
however the observable Universe is almost entirely made of matter. There is currently no
explanation why the Universe contains far more baryons than antibaryons. In the early
Universe there should have been a small asymmetry or order 10 -10 in the baryon antibaryon
density such that after annihilation at the end of the era of baryogenesis a fraction of the
baryons would remain explaining the current amount of baryons today. However the origin of
the baryon asymmetry is unknown but most explanations involve violation of CP symmetry
such that matter is produced more commonly then antimatter (see Chapter 2.8.4). [78]
Gravity
Weak, strong and electromagnetic forces are described by quantum field theories and the
interaction of particles via force fields. It is noticeable that gravity is the only force not
described by a quantum field and gravity is a force that is much weaker than all others
interactions. The reason why the gravity is so much weaker then the other forces is not
known and is also called a hierarchy problem. The question remains how to incorporate
gravity with the other forces into a theory of everything (TOE). We could assume that in the
early Universe a unified force existed that included all known 4 forces thus also gravity, since
at higher energies gravity becomes stronger. However it is not known how to treat a theory of
gravity as a quantum theory at energies above this Planck scale. While observations proof
Newton’s gravity and general relativity to be correct, there is still no explanation for the true
origin of gravity. The Einstein equations of general relativity can be derived trough the
principle of least action from the Einstein-Hilbert action, which is based on regarding the
metric tensor as a dynamic variable, the principle of general covariance and equating the low
energy limit to Newton’s gravity. However what the origin of the Einstein-Hilbert action is and
why matter is able to tell spacetime to curve is unknown. Neither general relativity nor a
possible quantum gravity field theory provides an explanation about the origin of gravity and
the existence of space and time.
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3.4.2 Extensions of the Standard Model
Supersymmetry
A popular extension of the Standard Model is supersymmetry. Supersymmetry is an
assumed spacetime symmetry between bosons and fermions in which each particle has a
supersymmetric partner. Theoretically supersymmetry arises naturally since the Haag–
Lopuszanski–Sohnius theorem shows that the possible symmetries of a consistent 4dimensional quantum field theory do not only consist of internal symmetries and Poincaré
symmetry, but can also include supersymmetry. This theorem is a generalizing the
Coleman–Mandula theorem by a nontrivial extension of the Poincaré algebra, the
supersymmetry algebra. It is possible to have more than one type of supersymmetry
transformation N from N = 1 up to N = 8 at most in 4 dimensions, and such theories are
known as extended supersymmetric theories. Minimal supersymmetry with N = 1 doubles the
number of particles that would exist in nature and many new possible interactions are
assumed. Baryon and lepton number are therefore no longer automatically conserved, but
since they are tested observationally it is assumed the symmetry R-parity exist that forbids
couplings in the theory that allow baryon and lepton numbers to be violated. All standard
model particles have R-parity of +1 and all supersymmetric particles have R-parity of -1 such
that when R-parity is conserved the direct consequence is that the lightest supersymmetric
particle (LS ) is a stable particle that can’t decay, making it an interesting candidate for the
missing non-interacting stable mass in the Universe. R-parity as a symmetry naturally occurs
in SO(10) GUTs when the U(1)B-L continuous gauge symmetry is broken at a high energy
scale leaving a subgroup with R-parity conserving properties. [98] [99]
However it is notable that no superpartner particles or any new interactions predicted by
supersymmetry have ever been observed so far. Therefore superpartners and new bosons
are in an unnatural way assumed to be very heavy and yet undiscovered particles, existing
as a shadow part of the present particle spectrum, see fig 3.3. Since no superpartners have
yet been observed, if supersymmetry would exist it must be a symmetry that is broken while
preserving its features through a process called soft supersymmetry breaking. The simplest
models of this breaking mechanism require that the energy of the superpartners is not
extremely high and within reach of new collider experiments. Therefore supersymmetry is
expected to be observed in new experiments at the LHC. [100]
Furthermore in supersymmetric theories the fermion and boson contributions to the zero
point fluctuations are equally large and opposite in sign such that they would cancel and lead
to an exactly vanishing vacuum energy density. This would explain the small observational
value for the vacuum energy density, however since no superparticles are yet observed in
nature supersymmetry is clearly a broken symmetry. Therefore the fermion and boson
contributions are not equally large and without cancellation the vacuum energy density is
again orders of magnitude larger then its experimental value.
Supersymmetry also solves the hierarchy problem for the contributions to the Higgs mass, as
the radiative corrections from virtual fermion loops are cancelled by the radiative corrections
from the scalar supersymmetric partners of fermions the squarks (and sleptons).
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Fig. 3.3. Supersymmetric shadow particles as heavy yet undiscovered superpartner particles
from the normal standard model particles. [33]
We will quickly review some dark matter candidates that arise in supersymmetry. In N=1
supersymmetry 4 Higgs particle exist together with 4 supersymmetric Higgs particles,
Higgsino’s. While the Higgs has been detected at the LHC, no other Higgs or Higgsinos have
been discovered. However supersymmetry provides an interested dark matter candidate
since the neutral elektroweak gauge boson partners called the gauginos can mix with the
Higgsinos to form neutralinos. Neutralinos are Majorana fermions (particles that are its own
antiparticle) and in R-parity conserving theories the lightest neutralino is a stable particle
(LSP) that is an interesting dark matter candidate. Such a neutralino can be produced in the
decay chain of other sparticles and is an interesting dark matter candidate since it could have
been produced in the early Universe and then thermally frozen out as a relic particle when
the Universe cooled down. [101]
The superpartners of the Standard Model neutrinos, the sneutrino, have also long been
considered as dark matter candidates, but their scattering cross section is much larger than
the limits found by direct dark matter detection experiments.
The axinos are the superpartners of the axion and could be a dark matter candidate.
Supersymmetry is an interesting theory that predicts gauge coupling unification supporting
the idea that all elementary interactions had a unified origin when the Universe was created.
Supersymmetry has interesting dark matter candidates, however none of the particles or
interactions predicted by supersymmetry has ever been observed. New collider experiments
at energies in which supersymmetric particles are expected to be created will show the next
decades if supersymmetry is a true symmetry of nature. [101] [119]
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String theory
In string theory point-like particles are modelled as one-dimensional objects called strings
and the vibrational state of the string determines the characteristics of a particle. Superstring
theories describe both bosons and fermions by incorporating supersymmetry and include the
description of quantum gravity describing gravity by a vibrational state that is represented by
the graviton. Generalizing perturbation theory one-dimensional Feynman diagrams
representing the path of a point particle are replaced by two-dimensional surfaces
representing the motion of a string. The characteristic length scale of strings is assumed to
be of the order of the Planck length (10-35 meters), a scale at which the effects of quantum
gravity are believed to become significant. On larger length scales strings would be
indistinguishable from zero-dimensional point particles. String theories are described with
extra spacetime dimensions (superstring theory is 10 or 11 dimensional) and in order to
construct real physical phenomena these extra dimensions are compactified or expressed
using the brane world scenario. Compactification lowers the effective number of dimensions
as some dimensions are curled up with very small radii and shaped as a Calabi-Yau manifold
such that a physics model with the right properties is can be interpreted. The brane world
picture natural explains the weakness of gravity as standard model bosons are expressed by
strings in four dimensional space while gravity is propagating trough strings in surrounding
bulk space. String theory integrates all fundamental forces and could be a theory of
everything, however the current goal is to find a solution of the theory that describes the
observed particle spectrum and might explain additional unsolved problems like the small
cosmological constant or the mechanism behind cosmic inflation. [102] [103]
Entropic gravity
In the theory of entropic gravity the force of gravitation is probabilistically described as an
entropic force while the other 3 forces remain interpreted by quantum field theory. Already in
1970s the connection between gravity and thermodynamics was studied by Hawkings and in
2011 prof. Erik Verlinde published the paper “On the Origin of Gravity and the Laws of
Newton” using the holographic principle to describe the statistical behaviour of the
microscopic system by the projection of its information on a holographic screen. From the
viewpoint spacetime, particles and forces are emergent concepts a natural explanation is
given for presence of dark energy and dark matter in the Universe, although the theory was
received with scepticism. [104]
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3.5.1 WIMP detection
We have seen that there exist many dark matter candidates with WIMP properties such as
neutralinos, photinos, higgsinos, sneutrinos, massive neutrinos, axions and axinos. WIMPs
are searched for by direct and indirect detection experiments set to detect WIMPs from our
Galactic halo as they move past and through the Earth. Accelerator searches are not
influenced by astrophysical uncertainties and provide an alternative and more sensitive
discovering channel for detecting light WIMPs.
3.5.2 Direct detection of WIMPs
WIMP direct detection aims to detect WIMP particles in the laboratory as dark matter and
nuclei scatter and interact, see fig. 3.4a. From the dark matter abundance, the assumed
WIMP velocity and WIMP mass it can be estimated around 10 5 WIMPs pass trough every
square centimeter of the Earth per second, but their cross section is so small (σ ~ .3 pb)
that most WIMPs can travel trough the Earth undisturbed. In the rare event a WIMP is
elastically scattered off a nucleus the deposited energy is small (E ~ keV) and can only be
observed with sensitive detectors. Experiments aiming to observe WIMPs by direct detection
are mostly cryogenic detectors experiments such as CDMS which are operating at
temperatures below 0.1K to minimize background from thermal excitations and use
germanium or silicon crystals to detect vibrations in the crystal lattice and ionisation as the
WIMP scatters off a nucleus. Noble liquid detectors experiments such as XENON or ArDM
use drift chambers to detect the recoil of an ionised atom in a gas or by observing the
scintillation light produced by the particle collision in liquid xenon or argon. The complication
for direct detection experiments is the rarity of WIMP interactions and the many background
interactions which lead to similar signals. Therefore most direct detection experiments are
located in deep underground laboratories such as the Sudan mine and Gran Sasso National
Laboratory and shielded such that the background from cosmic rays is small. Under the best
conditions the background is reduced to order 1 event·kg-1·day-1 whereas the predicted
signal is dependent on the used dark matter particle candidate and model and mostly in the
range of 10-5 - 10 events·kg-1·day-1. Directional direct detection of dark matter such as the
DAMA experiments use a strategy where detector signals from the background using the fact
that in June the Earth’s orbit is aligned with the Sun’s motion in the Galaxy or anti-aligned in
December, which is expected to cause a periodic behaviour of the signal event rent while
leaving the background event rate undisturbed. Although it is possible some dark matter
candidates might be detected in the future, other candidates might be too rare for direct
detection experiments to be ever observed. [16] [105]
Astrophysics at the LHC – Dark matter search in ATLAS
Time
54
Time
Fig. 3.4a (left). The diagram for direct detection of a dark matter particle (χ) scattering of a
standard model particle (SM). [107]
Fig. 3.4b (right). The diagram for indirect detection of an annihilating dark matter (χ) pair
creating a standard model (SM) particle pair. [107]
3.5.3 Indirect detection of WIMPs
Indirect detection experiments search for the products of WIMP dark matter annihilation or
decay, which would create Standard Model particle antiparticle pairs (see fig. 3.4b) that
might annihilate to energetic gamma rays. Experiments like the Fermi Gamma Ray
Telescope and EGRET use gamma ray telescopes searching for an excess of gamma rays
originating from regions of high dark matter density. Some WIMPs in the galactic halo
would over the years be captured by the Sun or Earth and the annihilation among the
WIMPs could hypothetically lead to the production of neutrinos or other particles. Since
produced neutrinos are capable of escaping the Sun and reaching Earth, or travel trough
the Earth, high energy neutrino telescopes such as IceCube and Antares search for
energetic neutrinos originating from the Sun and center of the Earth. Detection of these
neutrinos is easier then solar neutrinos originating from that Sun since they have a much
higher energy then the solar neutrinos which compensates for their smaller number.
Neutrinos from the Sun or Earth are expected to produce muons however muons are also
created in great numbers by cosmic rays entering the Earth atmosphere leading to a
substantial background. Therefore these experiments are performed at night in location
deep underground and focuses on muons travelling upwards aiming to detect neutrinos
that have interacted in the Earth below the detector that would have produced muons while
minimizing contributions from the background process. The detectors used for indirect
detection are designed to have very large surface areas which are needed to reach the
sensitivity of direct detection experiments. [16] [106] [109]
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3.5.4 Accelerator searches
In history new fundamental particles were often discovered by high energy particle
accelerators as any new particle can be pair produced if the energy of the collision and
coupling strength of the colliding particles is sufficient. The assumption that dark matter is a
relic WIMP particle has let to the conclusion a (possible new) weak interaction can take place
between a WIMP and Standard Model particles, and therefore WIMPs could be pair
produced from collisions of Standard Model particles. To detect light WIMPs (mx < 1 TeV)
accelerator searches are more sensitive than direct or indirect experiments, however for
heavy dark matter WIMPs (mx > 1 TeV) direct detection is more receptive (if the interaction is
spin-independent). [109] [110] We note that accelerator experiments are not influenced by
astrophysical uncertainties such as the dark matte relic density, dark matter abundance near
the Earth or Sun or the dark matter velocity distribution. The signature event that could
indicate the presence of the creation of a non-interacting WIMP pair leaving the collider
detector unseen, can be obtained from a detailed study of the kinematics of the strong and
electromagnetic interacting particles produced in the collisions, as it would lead to a
momentum imbalance. [111] The background process is a Standard Model interaction in
which weakly interacting neutrino’s are produced, which leave similar momentum imbalance
signature. However these Standard Model processes have been measured with great
accuracy and are well understood, such that the WIMPs signal might be efficiently
discovered above the standard model backgrounds. For low mass WIMPs the energy
transfer in direct detection is proportional to the WIMP mass and confirmation of the potential
signal might be easier in accelerator experiments where a large production rate is predicted,
see Chapter 3.3. [31] [112]
3.6.1 Dark matter search in lepton colliders
Lepton colliders are sensitive to the coupling of the initial-state leptons to WIMP pairs, see
fig. 3.6. The interaction of leptons to WIMPs could be mediated trough different operators
with different suppression scales in comparison with the WIMP-quark interaction. Therefore
lepton colliders could complement information about to the WIMP particle.
The final product is a high momentum photon (mono-photon) from the initial leptons which is
used to indicate the missing momentum due to the creating of an invisible WIMP pair leaving
the detector unseen. The dominant background is the Standard Model production of neutrino
pairs trough a Z boson, with a photon originating from initial state radiation. Lepton collider
experiments offer two important advantages compared to similar studies at proton colliders.
First the polarization of the initial state can be controlled such that the WIMP signal and the
backgrounds can de distinguished. Three coupling scenarios between WIMPs and leptons
can be considered.
1) equal coupling where the couplings are independent of the helicity of the initial state.
2) helicity conserving coupling where the couplings conserve helicity and parity.
3) anti Standard Model coupling where WIMPs couple only to right-handed electrons (lefthanded positrons).
The last scenario allows the easiest distinction between the Standard Model backgrounds
and the WIMP signal. The second advantage of a lepton collider is its sensitivity to the WIMP
mass by the observed total energy of the photon. Furthermore the shape of the energy
Astrophysics at the LHC – Dark matter search in ATLAS
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spectrum of the photons and the ratio of cross sections measured with different beam
polarisations could show the helicity structure of the WIMP-lepton interaction giving insight in
the operator of the interaction. At LEP dark matter studies were performed but the small
integrated luminosity of the data and lack of control over beam polarizations has decreased
its sensitivity. [108] [113]
Time
Fig. 3.6. Dark matter (x) pair production from 2 leptons the associated initial state jet. [107]
3.6.2 Dark matter search in proton colliders
At the LHC the main interactions happen between the constituents of protons, the quarks
and gluons called partons. In hard scattering events the production of new particles with
large momentum takes place (see Chapter 4.4.1). Dark matter WIMPs could be produced
besides numerous other particles. In the dark matter searches at proton colliders WIMP pairs
are expected to be pair created and leave the detector unseen with their momentum
balanced against observable initial state jets hadronized from the outgoing gluon (g) as can
be seen in fig. 3.7a. The dominant Standard Model (SM) contribution to a final state of jets
with a momentum imbalance comes from Z boson plus jet production with the Z boson
decaying into a pair of undetected neutrinos [114] [156].
Time
Time
Fig. 3.7a (left). WIMP (x) pair production from colliding quarks (q) with the initial state jet from
the outgoing gluon (g). [107]
Fig. 3.7b (right). WIMP (x) pair production from colliding quarks (q) with the initial state
photon (γ). [107]
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When comparing mono-photon and mono-jet search we note that single photons can be
produced in LHC through diagrams similar to fig. 3.7a, but with the outgoing gluon replaced
by a photon (γ), see fig. 3.7b. Therefore the cross section for mono-photon production is
suppressed compared to mono-jet production by the ratio of the strong and electromagnetic
finestructure constants as well as a color factor. On the other hand cross section for
background events is equally smaller. Systematic uncertainties on the background prediction
are similar for mono-jet or mono-photon but the acceptance factor for mono-photons is lower
then for mono-jet because the photon is restricted to be measured in the barrel of the
electromagnetic calorimeter. Therefore the bounds set by mono-photon search are softer
than set by mono-jet searches. [116] [117] [176]
In chapter 4 the ATLAS experiment is illustrated and in chapter 5 the method used to study
the sensitivity of ATLAS for WIMP production is explained. The sensitivity of LHC for
discovering WIMPs using the mono-jet search is discussed further in chapter 4.2.2.
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Astrophysics at the LHC – Dark matter search in ATLAS
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Chapter 4. Dark matter at the LHC
4.1 Introduction
In particle accelerators with sufficient energy it is expected that dark matter WIMPs could be
pair produced. With the LHC restarted its operation in June 2015 operating at a record
breaking center of mass energy of
14 TeV, important evidence could be found in the
analysis of the data from the ATLAS detector. In this thesis a simulation of WIMPs and
background event creation in ATLAS is studied in order to investigate the dependence of
detection sensitivity on selection criteria. Since the WIMPs are invisible to the detectors, the
events can only be seen if there is associated initial-state radiation of a standard model
particle recoiled leaving a momentum energy imbalance as signature. In this chapter we
describe the ATLAS detector characteristics, the observable quantities and the expected
associated signature.
4.2.1 LHC
The Large Hadron Collider (LHC) is situated near Geneva, Switzerland and World‘s largest
particle collider was designed by the European Organization for Nuclear Physics (CERN).
The LHC is a synchrotron type accelerator, is constructed in an underground circular tunnel
with a diameter of 26.7 km and is capable of accelerating protons to a center-of-mass energy
of
7 TeV, see fig. 4.1. The construction of the LHC began in 1998 and after an accident
with an electronic connection LHC started operating in 2008, the first collisions took place in
November 2009. From 2010 till 2012 the LHC operated with an energy of about
4.0
GeV per beam and the discovery of a particle matching the Higgs boson was confirmed in
July 2012. The particles created by head-on collisions are expected to reveal information on
the behaviour of physics at conditions never reached before in a laboratory, and are detected
at the four particle detectors installed on the collision points. [121]
The LHC uses eight radio frequency cavities (RF) with a frequency of 400 MHz to create an
electric field to accelerate charged particles like protons and to correct small deviations of
spacing between the bunches of particles. Magnetic fields perpendicular to the trajectory
bend the beam and keep the particles in orbit, and the maximum energy to which the
particles can be accelerated is determined by the strength of the magnetic field created. To
collide particles two beams are travelling in separated pipelines with opposite magnetic fields
around them sharing a common magnet and cooling system, and meet in a collision point.
The LHC is not a perfect circle but consists of eight arcs and eight intersections (IS). In the
arcs the magnets are arranged such that the beam is repeatedly bended and focused. The
bunches are focussed to a diameter of 6 μm at the interaction points to increase the
probability of hard scattering. Curving the beams path is achieved by the Lorentz force
exerted by the magnetic field of dipole magnets, operating with a magnetic field strength of B
= 8.33 T using superconducting coils at a temperature of T = 1.9 K with a mean current of I =
11.5 kA. Quadrupole and multipole magnets focus the beam such that it stays well
constrained inside the vacuum beam pipeline.
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The IS have long straight sections and four are actual interaction points, two are used for
beam injection, one for RF accelerating cavities and one for beam dump, see fig. 4.2.
Fig. 4.1 (left) Overall view of the LHC experiments at Cern. [82]
Fig 4.2. (right) LHC consists of eight arcs and eight intersections (IS) with four interaction
points where the detectors are situated. [124]
In LHC heavy ions as well as protons are collided. For heavy ion physics an electrical field is
used to remove the electrons from the nucleus yielding only protons and neutrons. For
proton collisions the proton source is hydrogen gas, and with an electric field the electrons
are stripped from the hydrogen atoms to yield protons. Physicists prefer to quote heavy-ion
beam energies “per nucleon”, where protons and neutrons are both nucleons, as it allows for
an easy comparison with the energies of beams of protons and other types of ions. The
charged particles are accelerated by a succession of machines to increasingly higher
energies, as can been seen in fig. 4.3. The first accelerator in the chain Linac2 accelerates to
the energy of 50 MeV, the beam is then injected into the Proton Synchrotron Booster (PSB)
which accelerates to 1.4 GeV followed by the Proton Synchrotron (PS) which accelerates the
beam to 25 GeV. Particles are then sent to the Super Proton Synchrotron (SPS) where they
are accelerated to 450 GeV and finally transferred to the two beam pipes of the LHC where
the particles are moving at an ultra-relativistic speed of 0.999999991, the speed of light or
energy of 7 TeV. [174] After entering the LHC, it takes about 20 minutes to accelerate the
proton beams to reach the final energy of 7 TeV. Heavy ions are first accelerated at LINAC3
and the LowEnergy Ion Ring (LEIR). Then the beam is sent to the PS and follows the same
path as a proton beam. Lead ions will reach a final energy of 2.76 TeV per nucleon. [122]
[142] [175]
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Fig. 4.3. The Cern accelerator complex consists of a succession of machines to accelerate
particles to increasingly higher energies. [125]
4.2.2 Event rate and luminosity in LHC
Besides center of mass energy the instantaneous luminosity (L) is one of the most important
parameters of an accelerator and is a measurement of the number of collisions that can be
produced in a detector per m2 per second. The instantaneous luminosity contains the factors
that are fixed by the characteristics of the detector and quantities that can be controlled
experimentally, and is calculated by [4.1].
L
Nb 2 nb f 
F
4 n 
[4.1]
Here Nb is the number of particles per bunch, nb the number of bunches per beam, f the
revolution frequency, γ the relativistic gamma factor, F the geometric luminosity reduction
factor which depends on the crossing angle at the IP, the transverse beam size and the
bunch length, ɛn the normalized transverse beam emittance and β the beta function at the
collision point. The revolution frequency is fixed by the LHC radius and the speed of light the
particles are travelling with. The beam energy is mainly limited by the strength of the dipole
magnets. The emittance and the beta function arise as a bunch of particles, as it traverses
the magnets, experiences both longitudinal (synchrotron oscillation) and transversal
oscillations (betatron oscillations).
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The emittance describes the average spread of the bunch of particles in the transverse
plane, is constant along the trajectory and is proportional to the beam size. The beta function
describes the wavelength of the betatron oscillations and is related to the focus which can be
adjusted within a factor or 2. Quantities that can be more easily adjusted are the number of
bunches per beam and the number of protons per bunch.
The integral of the instantaneous luminosity L over time gives the integrated luminosity [4.2]
that characterizes the collected data size and the performance of the accelerator.
  L dt
[4.2]
The integrated luminosity has units of m -2 and is usually expressed in inverse barn units (b -1),
using the definition 1 barn = 10-28 m2. At full intensity each proton beam in the 27 km long
LHC consists of 2808 bunches containing 1.15 1011 protons, and the distance between
bunches is on average 9.6 m. Since the protons are moving with practically the speed of
light, head-on meetings between bunches at every collision point happen with a frequency of
31.2 MHz.
For a given interaction or process the cross section σ is defined as the interaction rate r
divided by the instantaneous luminosity L, by formula [4.3]. The cross-section contains the
intrinsic probability of the interaction to happen and is determined by the particles involved
and the nature of their interaction and has units of unit of m2.

r
L
[4.3]
The total proton-proton cross section at 7 TeV is approximately 100 mb and has
contributions from inelastic scattering (60 mb) and elastic scattering (40 mb). The cross
section from elastic scattering of the protons and diffractive events will not be seen by the
detectors as it is only the inelastic scatterings that give rise to particles at sufficient high
angles with respect to the beam axis. The event rate for inelastic scattering can then be
calculated multiplying LHC's nominal luminosity of 1034 m-2 s-1 and the cross section for
inelastic scattering 60 10-3 10-24 m2 yielding 600 million events per second. Since the
frequency of bunch crossings is 31.6 MHz there are typical μ = 19 inelastic events per bunch
crossing.
The amount of information that can be read out per event is about 25 MB of data but by
using hardware only the interesting events are triggered and selected reducing the data flow
to 1.3 MB per event, producing 1PB of data per second. This enormous amount of data
produced by the ATLAS detectors is reduced by a process of selecting interesting events
using a multi-level trigger system.
The LHC has finished its Phase 0 upgrade and after a two-year Long Shutdown (LS1) for
repairs and extensive upgrades it had restarted operating 5 April 2015. Actual collisions have
started in June 2015 and the proton proton collisions with a combined energy of
13 TeV
have already discovered the existence of a pentaquark in LHCb in July 2015. [126] By the
end of 2015 its design energy of
14 TeV is expected to be reached and an
34
-2 -1
instantaneous luminosity of L = 10 cm s or 25 fb-1 integrated luminosity per year with an
average of μ = 6 interactions per bunch crossing.
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In 2018 the Phase 1 upgrade is scheduled and during the Long Shutdown 2 (LS2) LHC is
upgraded to reach its ultimate design luminosity of L = 2·1034 cm-2 s-1 with μ = 60 and aims to
collect 300 fb-1 data by the end of 2022. [127] [128]
A phase 2 upgrade during the Long Shutdown 3 (LS3) called High-Luminosity LHC or HLLHC around 2022 could increase the instantaneous luminosity to L = 5 1034 cm-2 s-1 with μ =
140 aiming aiming to collect 3000 fb-1 in the years afterwards. [127] [129] [130
A sensitivity study for dark matter production at LHC using the D5 interaction (see chapter
5.2) shows that the upcoming LHC run with 14 TeV has a large potential for discovery WIMP
dark matter with mx = 50 GeV. Considering 5% systematic uncertainty on background and 25
fb-1 luminosity LHC could see a signal with mz’ up to 1.5 TeV at 5σ after one year. VL-HC and
HL-LHC could extent the reach of the produced WIMP mass to the TeV range and assuming
the final 1% precision of the systematic background the reach for a 5σ discovery extends to
mz’ = 2.2 TeV and mz’ = 2.6 TeV with 300 fb-1 and 3000 fb-1 of data respectively, see fig. 4.4.
[131]
Fig 4.4. The discovery potential for a dark matter signal using the D5 operator and mx = 50
GeV with a total luminosity of 25 fb-1 (top left), 300 fb-1 (top right) and 3000 fb-1 (bottom) [131
edited].
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4.2.3 LHC detectors
At the LHC four underground detection experiments are constructed around the four
intersections points of the LHC (see fig. 4.1 and fig. 4.3) namely;
ATLAS (A Toroidal LHC Apparatus)
CMS (Compact Muon Solenoid)
ALICE (A Large Ion Collider Experiment)
LHCb (Large Hadron Collider beauty)
The ATLAS and CMS detectors have a general purpose designed to be sensitive for the
broad spectrum of phenomena that might occur during the collisions of protons by good
electromagnetic and hadronic calorimeters and high precision muon momentum
measurements. CMS and ATLAS have a similar function but were designed by independent
teams making slightly different compromises. ATLAS has a slightly better performance for
calorimetric and momentum resolution for hadrons, and CMS is more sensitive for muons.
LHCb is designed to study the physics of b-hadrons produced in the proton proton (pp)
collisions investigating decays that are sensitive to matter antimatter symmetries. ALICE is
optimised to study heavy ion collisions that create a quark gluon plasma, a “fluid” described
by quantumchromodynamics that also existed shortly after the Big Bang.
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4.3.1 The ATLAS Detector
The ATLAS project involves roughly 3000 scientists and engineers from 175 institutions in 38
countries. The ATLAS detector is located in a cavern at 90 m under the surface around one
of the IP points and its construction was finished in 2008. The detector has a cylindrical
shape and is 45 m long and 25 m in diameter with a weight of about 7000 tonnes. ATLAS is
a 4π detector nearly covering the full solid angle around the collision point and is
symmetrically constructed in forward and backward direction with respect to the interaction
point. Respectively inwards out ATLAS consists of four segments, the inner tracking detector
surrounded by a thin superconducting solenoid, the electromagnetic and hadronic
calorimeters, and the external muon spectrometer with three large superconducting toroid
magnets. The inner detector tracks particles and measures the momentum of each charged
particle. The calorimeters measure the energy of easily stopped particles. The muon
spectrometer identifies and measures the momentum of highly penetrating muons. The
magnet systems bend charged particles in the inner detector and the muon spectrometer
such that their momentum can be calculated. To quickly select only interesting events in all
data a trigger system is used afer which the data is processed by offline analysis. [132]
The ATLAS experiment studies pp collisions and is sensitive to different phenomena such as
Improving the accuracy of Standard Model parameters measurements like W/Z boson and
top-quark masses.
Measure test and validate predictions of the Standard Model like cross-sections, decay rates.
Measurements of the Higgs boson and its characteristics (mass, couplings, spin, parity, etc.).
Measurements in the flavour sector of the SM that is sensitive to CP violation.
Searching for hints of physics beyond the Standard Model (supersymmetry, extra
dimensions).
Search for dark matter WIMPs.
These goals defined the design and decisions made constructing the ATLAS detector. The
construction goals included good electromagnetic and hadronic calorimeter systems and
high precision muon momentum measurements. The ATLAS detector is a complex system
and its four major components can be seen in the schematic view of the ATLAS detector in
fig. 4.5. [175]
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Fig 5.5. The components of the ATLAS detector and its size compared to the physicist in
front. [133]
Produced stable particles that are detected directly are neutrinos. Their presence is
measured if calculations show a momentum imbalance among the detected particles is
present. The same method can be used to find other invisible particles leaving the detector
without interacting, like the hypothesized dark matter WIMPs studied.
4.3.2 ATLAS coordinate system
Important for defining the observables measured by ATLAS is the definition of the coordinate
system. The coordinate system of ATLAS is a right-handed coordinate system with the x-axis
pointing towards the center and the z-axis along the tunnel, see fig 4.6. The y-axis is slightly
tilted with respect to vertical due the general tilt of the LHC tunnel. The x-y plane transverse
to the beam direction is called the transverse plane and all the quantities measured in this
plane are labelled as transverse quantities noted with a T index. Due to its cylindrical
geometry the coordinates in the transverse plane can also be represented by polar
coordinates (r,φ). The direction of a particle in the (y,z) plane is described by pseudorapidity
η (chapter 5.9) or by the angle θ of a particle relative to the beam axis further explained in
Chapter 4.5.3.
Fig. 4.6. The ATLAS coordinate
system with z in the beam
direction. [134]
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4.3.3 The Inner detector
The inner detectors (ID) of ATLAS are the detectors nearest to the beam axis. The ID
measures the tracks left by charged particles, revealing information about the momentum,
charge and type of particle. The magnetic field from the central solenoid surrounding the
entire inner detector causes charged particles to curve and the degree of curvature reveals
its momentum and the direction defines its electric charge. The ID is designed to have a very
good spatial resolution and is composed of three parts: the pixel detector (PD), the semiconductor tracker (SCT) and the transition radiation tracker (TRT).
The PD and the SCT use modules with p-n junctions to detect the passage of charged
particles by the formation of electron-hole pairs. The PD was constructed with 3 layers of
modules such that each particle should have three hits within the PD. During the upgrade
phase 0 an additional layer of pixel detectors is assembled between the beam pipe and the
first layer. The modules allow close placing such that the PD contains 1744 modules with
47232 pixels of 5 x 4 μm each. In total the D has over
million readout channels and
allows precise tracking of the high number of tracks going out of the IP of the collision. The
accuracy of the pixels is
μm in the (r,φ) direction and
5 μm in the z-direction. Placing
such large amount of modules closely together is a design and engineering challenge.
Furthermore due to its proximity to interaction point each PD component was radiation
hardened to withstand the radiation being exposed to. P-typed material becomes effectively
n-typed after exposure to a certain radiation dose, and since the effective doping is
dependent on the temperature the D are operating at a temperature of around - C.
The STC is the middle component of the inner detector and has 1512 long, narrow shaped
modules for tracking in the plane perpendicular to the beam and covering a larger area as
the PD of about 61 m 2. The STC has 6.3 millions channels and a coarser resolution then the
PD with an intrinsic accuracy of 7 μm in the (r,φ) direction and 5 μm in the z-direction.
The outermost component of the inner detector is the TRT which consists of around 300.000
drift tubes each with a length up to 144 cm and a diameter of 4 mm positioned such that the
transversal direction of the particle’s trajectory is observed and not the beam direction. The
position resolution is 3 μm in the (r,φ) direction and is less precise as the PD or SCT, but
covers a larger volume and therefore provides more measured points over a longer track
path. Each drift tube is filled with a gas mixture that consists of 70% Xe, 27% CO2 and 3% O2
and is operated at 5 - 10 mbar overpressure. The conductive tubes are held at ground
potential and function as a cathode while inside each drift tube a stretched wire functions as
anode, with an applied voltage of 1530 V between cathode and anode. When a charged
particle passes through the gas becomes ionised and negative ions created in the gas are
conducted into the wire, producing a current pulse that can be read out. A charged particle
passing trough will create a pattern of pulses that can be read out from the 35100 read-out
channels to allow precise reconstruction of the path of the particle.
The ID is surrounded by the inner solenoid magnet that produces a uniform 2 T magnetic
field surrounding it. The solenoid is designed to be as thin as possible to reduce the amount
of material in front of the calorimeters. It operates using superconducting coils with a nominal
current of 7.73 kA that are cooled down to a temperature of 2.5 K. [137] [151]
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4.3.4 Calorimetery
The calorimeters are placed outside the inner detector and measure a particle’s energy by
absorbing it, forcing the particle to deposit all its energy within the detector. A calorimeter is
designed to stop most of the particles from the collision and consist of layers of high density
absorbing material (lead) interleaved with layers with a sampling material such as liquid
argon. The inner electromagnetic calorimeter (ECAL) and the outer hadronic calorimeter
(HCAL) are sampling calorimeters. The ECAL absorbs energy from particles that interact via
the electromagnetic force such as charged particles (electrons, positrons) and photons.
Electrons and positrons ionise the absorbing material or lose energy by Brehmsstrahulung.
Photons interact with the absorbing material leading to pair production for photons with
energies above 5 MeV, or interact trough the photoelectric effect or via Compton scattering
for low energy photons. The created shower develops as more photons and electron-positron
pairs are being produced leading to a shower maximum. The calorimeter depth guarantees
that the shower is entirely contained in it. The created electron shower ionises the sampling
material producing electric signals that are collected by copper electrodes. Since the shower
depth is dependent on the energy of the incoming particle, the energy of the original particle
can be calculated in a wide energy range. The ECAL has a barrel (EMB), and end-cap
(EMEC). The electrodes of the EMB are accordion shaped to have a full uniform azimuthal
coverage. The angle of the particle's trajectory with respect to the z-direction and the (r,φ)
plane is measured with an accuracy of 0.025 radian. [138]
HCAL measures the energy of particles that interact via the strong interaction (mesons and
baryons) by the production of hadronic showers (jets). In the barrel steel is used as energy
absorbing material and in the end-caps copper and tungsten is used. For sampling HCAL
uses scintillators in the barrel and liquid argon in the end-cap regions. The interaction of the
produced charged particles will excite atoms and produce light in the plastic scintillators in
the barrel, or on electric signal in the endcap. The light in the scintillators is directed via
optical fibers and with photomultipliers converted to an electric signal. The HCAL is less
precise then the ECAL and the angle of a particle’s trajectory is measured to an accuracy of
0.1 radians. [139]
4.3.5 The muon spectrometer
The muon spectrometer is a large system that will measure the momentum of muon which
first go trough all more inwards parts of the ATLAS detector before reaching the
spectrometer. Considering the total energy of created particles in an event can not be
calculated if muons were ignored, it is necessary to measure the momentum of muons
accurately. Therefore it was necessary to build a large tracking system that consists of three
cylindrical shells surrounding the previous components with a radius of 5 m, 7.5 m and 10 m.
Monitored drift tubes (MDT) and cathode strip chambers (CSC), both ionisation gaseous
detectors, are used in combination with a superconducting outer toroidal magnet system to
curve the muons trajectory. The MDT acts as precision tracking chambers covering most of
the detector and the CDC are used in a small part of the forward region. Both are arranged
with a geometry such that most of the particles traverse the three layers providing precise
coordinate measurements of the hits left by muons. [135] [141]
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An important task of the muon detector is the triggering of muons, and both the Resistive
Plate Chambers (RPC) and Thin Gap Chambers (TGC) are fast response gaseous ionisation
detectors designed for triggering and complementing the measurements by the MDT and
CDC. The trigger is designed to accept muons in the pseudorapidity range |η| < 2.4 and full
azimuthal range. The muon spectrometer has about one million readout channels, its
detectors span a total area of 12.000 m2 and the momentum can be measured with an
accuracy of 3% for 100 GeV muons and 10% for 1 TeV muons. [140]
4.3.6 The magnet system
The ATLAS toroid magnet system consists of an air coil barrel toroid with eight barrel coils
mounted in separate cryostats and two end cap toroids with eight coils each, see fig. 4.7.
The toroid magnet system is situated outside the calorimeters and within the muon system
and operates with a magnetic field strength of 0.5T in the barrel and 1T in the endcap. [143]
Fig. 4.7. Three dimensional view of the bare windings of the ATLAS magnet system, the
central solenoid, 8 coils of the barrel toroid and the 2 x 8 coils of the end-cap toroids. [154]
4.3.7 Trigger system
When LHC is running at full luminosity, the event rate is around 600 MHz. The majority of the
produced events is not useful since new physics processes have small cross sections. A
trigger system of selecting interesting events is used such that data storage and bandwidth is
not overloaded. The Data AcQuisition system DAQ manages the data processing and has 3
levels.
The 1st level trigger (L1) triggers objects such as electrons, photons, muons or jets with high
transverse energy as well as events with large missing transverse energy. It is done by
custom made electronics that perform event selection on the electronic signals collected in
the calorimeters and muon spectrometer, based on the recognition of basic physics
signatures in the energy deposits. The time needed for a decision is shorter than 2.5 μs and
the FADC converts the analog signal into a digital signal.
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The selected events are called regions of interest (ROI) and are read out from the front end
electronics, into the readout drivers (ROD) and then into the readout buffers (ROB), which
are the starting point for the next trigger level. [144]
The 2nd (L2) and 3rd (L3) trigger are both software based, processing the events on computer
farms and using calibration and alignment parameters to reconstruct and identify particles
produced in the collision. The L2 has an event rejection factor of about 30, reducing the
event rate to 3.5 kHz. The L3 reduces the event rate to 200 Hz and the average size of a
stored event is about 1.3 MB. [145] [175]
4.3.7 Computing grid
To interpret data simulations of events and background events in the detector are calculated.
The simulations with the data of 1 billion events recorded per year are analysed and
distributed from Cern to 10 main computer centres in 3 continents where they are stored,
processed and further distributed to about 50 collaborating institutes for analysis via the
worldwide LHC computing grid. [145]
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4.4.1 Event reconstruction and identification
At the LHC the main interactions happen between the constituents of protons. Protons are
composite particles containing three valence quarks and sea quarks all tightly bound by the
strong force (gluons). All these constituents are called partons and each parton carries a
fraction of the protons momentum. The distribution of the protons momentum among its
constituents is described by parton distribution functions ( DF’s). Groups specialized in
quantumchromodynamics have fitted PDF models to existing data from deep inelastic
scattering experiments [86]. Since the momentum of a proton is distributed along its partons,
the longitudinal velocity of the center of momentum of each collision of partons is a priori not
known. Conservation of momentum can therefore only be applied in the transverse plane
since the partons had no or very small transverse momentum before the collision. [86] [146]
The collision between two protons can occur in two different ways, elastic or inelastic. In the
25% elastic events the two protons remain intact and a small transfer of momentum takes
place changing its direction and no energy is lost. Since no new particles are created these
events are not of interest. In the 75% of the inelastic event a transfer of momentum takes
place and there is a change in direction and energy of one or both protons, dissociating into
a system of particles. The protons can either interact on a large distance (soft scattering) with
small momentum transfer or at small distance (hard scattering) with large momentum
transfer. [86].
The typical behaviour of the strong force is that the strong coupling constant decreases in the
high energy limit and perturbation theory is applicable describing it in leading order by [4.5]
with Nc the number of color charges (Nc = 3), Nf the number of participating quark flavours
and Λ the QCD energy scale. [86]
s 
12
Q2
(11N c  2 N f ) ln

[4.5]
The physical “collision” can be viewed as a 2-step process. First a parton from one hadron
interacts with a parton from the other hadron in a hard scattering process leading to two
partons emerging with high speed from the collision. These interesting hard scattering events
happen when with sufficient proton energy a short distance interaction takes place and the
transferred momentum between the two protons is large enough to resolve the constituents
of the protons. The beam can then be described as partons with a different energy according
to their Parton Distribution Functions (PDF). The interacting partons will produce final state
particles characterized by high momentum transfer and the production of particles with high
transverse momentum, large mass and/or large scattering angle. For short distances
between the participating partons the coupling strength gets weaker and higher momentum
transfers Q2 occur. The limit Q2 →
is called asymptotic freedom since the quarks can
behave like free particles, and the perturbative approach of high energy QCD gives accurate
predictions describing interactions at short distances. In the second step the partons
produced in the hard scattering will experience an increasing effective interaction strength as
they are separated. Now the large distance non-perturbative domain of QCD is entered for
which no quantitative theory exist but parton fragmentation occurs. As a partons is emerging
from the collision more binding energy is stored in the gluon field and the probability
increases that the parton will radiate gluons, which will in turn radiate quark-antiquark pairs
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and so on, with each new parton shallow angled with respect to the originating parton. As
parton showering produces partons of successively lower energy, at some point the partons
condensate in colorless observable hadrons leaving the region of high energy perturbative
QCD. What is observed in a detector is a result of parton showering and combining of the
produced partons into stable hadrons (pions, protons, neutrons). The terms fragmentation
and hadronization are often used interchangeably in the literature to describe QCD radiation,
formation of hadrons, or both processes together. The narrow cone of stable hadrons and
the decay products of unstable particles (electrons, photons, etc) produced by hadronization
is called a jet. [147] [148]
In soft scattering collisions interactions occur when protons interact on large distances acting
as a whole. Soft scattering interactions with large distances between participating partons
correspond to low momentum transfers Q 2 and thus a stronger coupling. They are
characterized by a small momentum transfer and final state particles with a large longitudinal
momentum, small transverse momentum and small scattering angle, so most energy
escapes in the beam pipe. For energies below scale Q2 << Λ perturbation theory is not
applicable and soft scattering calculations rely on the infra-red low energy description of
QCD which is not easily calculable since perturbative theory breaks down. [147]
The hard interaction is accompanied by soft interactions between colliding partons that do
not participate in the hard scatter, and produce an underlying event with small scattering
angles.
The inelastic events can be divided in single, double and no diffractive processes. In
respectively single and double diffractive processes one or both of the protons remains intact
or disassociates into a mixture of particles with low energy compared to the original energy of
the proton, see fig 4.8. These events can be described by colorless momentum exchange.
Single diffractive (SD) and double diffractive (DD) processes occur in respectively 12% and
8%. Central diffractive (CD) processes (2%) will also leave two protons intact and nondiffractive (ND) processes (55%) both protons disassociate [149].
Time
Fig. 4.8. Diagrams for the different types of inelastic events and an elastic event. [149
edited].
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Inelastic events are triggered with a minimum bias trigger. Zero bias events have zero
requirements but a minimum bias event is an event with the requirement only some activity in
the detector happened with a minimal transverse momentum (p T) threshold of pT ~ 100 MeV.
A minimum bias event occurs in 99.999% of collisions. It is either a soft non-diffractive event,
soft single-diffractive event or soft double diffractive event, and is characterised by having no
objects with high transverse momentum (jets; leptons; photons), low transverse momentum
tracks at all angles, uniform energy deposits in calorimeter as function of rapidity and being
isotropic.
The soft underlying event (SUE) is everything in the event not originating from the hard
scattering, see fig. 4.9. Sometimes the minimum bias events are added to the SUE such that
it is a generic term defining all events that have no large transverse momentum. The
difference between soft and hard gluons is not exactly determined but scattering is generally
considered hard if its transverse momentum is large enough to hadronize into a jet.
Soft interactions can occur from the underling event, from Initial State Radiation (ISR) in
which incoming partons radiate gluons before the hard scattering, or from Final-State
Radiation (FSR) where outgoing partons radiate gluons after the hard scattering. Moreover
multiple scattering, interactions with the beam remnants, can lead to soft particle production.
Pile-up occurs mainly at low energies and happens when protons from the same bunch but
not participating in the hard interaction, interact via soft scattering with eachother.
Since the hard processes are usually accompanied by these soft interactions these
appearing processes must be included in order to produce accurate predictions. The
chances of producing more than one full-on proton-proton collision (hard scattering event)
per bunch crossing are low, but as the instantaneous luminosity per bunch crossing,
effectively the density of protons in the interaction region where the beams overlap goes up,
the likelihood of soft scattering between the constituent quarks and gluons of additional
proton-proton pairs increases. The challenge for ATLAS, especially at high luminosities, is
working out which tracks and energy deposits attribute to which interaction. [148] [150] [153]
Fig 4.9. A hard scattering proton-proton collision is shown with two jets in the final state, and
soft interactions like ISR, FSR and underlying events. [149]
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4.4.2 Particle identification
In a process called event reconstruction all electrical signals stored during the event are
processed with algorithms that associate hits together and produce observables that allow
identification of the produced particles.
In ATLAS tracks of charged particles in the ID, signals from clusters in the ECAL and HCAL,
and muon tracks from the muon spectrometer are observed. Identification is applied to
combine these signals with algorithms in order to understand which kind of particle was
created. Particles that are stable within the detector volume are considered a candidate if
they match their signature, see fig. 4.10.
An electron loses a small amount of its energy in the inner detector and deposits its
remaining energy in the electromagnetic calorimeter. An electron candidate signature is the
observation of a signal in the ECAL with a matching track in the ID.
Most photons give a signal in the calorimeter, but not in the inner detector. Some photons
convert into an electron-positron pair in the ID. A photon candidate is a signal in the ECAL
without a matching track, or a track that does not cover the whole ID. Since photons are not
absorbed by the scintillators neither ionising the gas in the TRT since the photon crosssection for photoelectric effect or Compton scattering is negligible, only pair production is
observed.
Muons leave only a small fraction of their energy in the calorimeters since the energy loss by
Brehmsstrahulung is inversely proportional to the mass of the moving particle. Muons travel
trough the calorimeters and give a signal in the ID and muon spectrometer and a small signal
in the calorimeter. Muon candidates have a signal in the ID matched to a track in the muon
spectrometer.
Other leptons and B and D mesons have lifetimes that allow them to travel some measurable
distance within the ID and they can be identified using secondary vertex information.
Hadrons shower deeper into the calorimeters than electrons and photons and a signal in
both the electromagnetic and hadronic calorimeter will be observed. A charged hadrons
signature is a signal in the inner detector and both calorimeters.
Jets are a combination of hadronic and leptonic particles and will leave a signal in the inner
detector, both calorimeters and in the muon spectrometer. Jets are identified by matching
ECAL and HCAL clusters with a corresponding track in the muon spectrometer and signal in
the ID.
Mini-jets are generally low pT jets associated with soft scattering from a double parton
interaction. By requiring a minimum transverse momentum for every jet only the interesting
hard scattering events are selected. The data contains the jets decreasingly ordered by pT
untill a lower limit and they are labelled increasingly with a index j, such that the first jet ( j =
0) is the jet with the highest pT and vice versa. Furthermore the total number of jets with
observance of the lower limit is stored in a variable called multiplicity labelled njet.
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Unstable particles are identified by their decay products. For example a Z boson is identified
when the decay into electron and positron is observed.
For each identified particle its track can be reconstructed and observables like momentum,
angle, rapidity, etc. are calculated and recorded as physics data.
Neutrinos have extremely low cross sections and neutrino candidates leave no trace in the
ATLAS detector. Also dark matter WIMPs will not interact with the detector. These invisible
particles are identified looking for a transverse momentum imbalance called missing energy.
Fig. 4.10. Signatures of different particles as they pass the inner detector (inner tracker),
calorimeter and muon spectrometer (outer tracker). [179]
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4.5.1 Transverse momentum and missing energy
In order to observe non-interacting particles which leave the detector unseen, at least one
hard object (jet, photon) is needed. Before a collision the transverse momentum of the initial
proton-proton system is zero or negligible compared to the longitudinal momentum, so
conservation of momentum implies that after the collision, the sum of all transverse
momentum of the produced particles must be zero. The transverse momentum of every jet is
denoted with pT[j] where j is the ordered transverse momentum index. However when
particles are created that are invisible to the detector and escape the detector unseen, these
particles can indirectly be detected since the vector sum computed over the momentum of all
the reconstructed particles and jets in the event (i), the net transverse momentum, will
generally not add up to zero. The vector quantity confusingly known as “missing energy” can
be calculated by taking the opposite vector to the net transverse momentum by formula [4.6]
and is noted with Emiss. Missing energy therefore represents the sum of transverse
momentum carried away by invisible particles, see fig. 4.11.
E Miss   PT ,i
[4.6]
i
Fig. 4.11. The momentum of 2 jets pT[0] and pT[1] is showed with 2 invisible WIM s (χ) in the
transverse plane of the detector. Missing energy Emiss is the vector quantity calculated by
taking the opposite vector to the net transverse momentum
and represents the sum
of transverse momentum carried away by invisible particles. Also the variable δφ is indicated
(chapter 4.5.2).
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4.5.2 Smallest angle
The variable deltaphi or δφ is defined as the smallest angle between missing energy and any
initial state jet, see fig. 4.11. A jet that is created opposed to missing energy has δφ = 180°.
4.5.3 Pseudorapidity
Pseudorapidity η is used to describe the angle of a particle or jet relative to the beam axis. It
is defined by formula [4.7];
   
   ln  tan   
2

[4.7]
 
where θ is the angle between the particle three-momentum p and the positive direction of
the beam axis. As a function of the three-momentum pand pL the longitudinal component of
the momentum along the beam axis the pseudorapidity is given by [4.8].


 
p
1  p  pL 
  ln
 arc tanh  L 
 p 
2  p  pL 


 
[4.8]
Pseudorapidity depends only on the polar angle of the particle's trajectory and not on its
energy. Differences in pseudorapidity Δη are additively, Lorentz invariant and are
independent on a reference frame. Particle production is constant as a function of
pseudorapidity. The “forward” direction in the LHC refers to regions of the detector with a
small angle θ and large η close to the beam axis in positive z-direction.
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79
Chapter 5. Methods
5.1 General objectives and assumptions
The general objective is to optimise the detection of WIMP pair production amongst
background events by studying the effect of using selection criteria on Monte Carlo simulated
events in ATLAS at
14 TeV.
By studying simulations of
14 TeV events in ATLAS the prospect is to make a profound
estimation about the physics that will be observed in ATLAS when LHC will restart after the
phase-0 upgrade in 2015. It is aimed that the study of 14 TeV simulations gives an accurate
prediction of real physics at
14 TeV. A study on final states containing a jet with high
transverse momentum and missing energy was performed using 10 fb−1 of pp collision data
with
8 TeV collected by the ATLAS detector showed a good agreement with the
simulated Standard Model predictions. [156]
14 TeV Monte Carlo simulations for the
standard model background samples were used which were produced by [155] with the
same methods as the existing reliable background samples for
8 TeV for consistency.
Simulations for dark matter WIMPs and their interactions at
14 TeV were produced by
[155] using an effective field theory approach and an interaction trough a simplified model
with a massive Z’ mediator. The signature of the in the collision created and detected
standard model particles and jets was generated with the same method as the background.
In this study we will first check if results from analysing the background simulations are in
agreement with previous studies. [155] [156] Then it is verified that for the used dark matter
simulations increasing the dark matter masses m x results in a smaller yield. Then we the
study different models with contact interaction represented by the D5 operator or interaction
trough a generic Z’ intermediate state (see chapter 5.2) with dark matter mass of respectively
mx = 50 GeV and m x = 400 GeV and with a range of mediator particle masses between m z‘ =
100 GeV and m z‘ = 15 TeV by optimizing significance by placing tuned selection criteria on
kinematic variables. The applied selection criteria on the data aim to filter out events with
large missing energy due to jets with a nonzero sum over transverse momentum. Making
cuts that concentrate on large missing energy are shown to be more efficient then enhancing
the overall luminosity. Exactly what the optimal cuts are in dark matter search is unclear from
previous studies, which have advocated a dedicated study to be carried out with tuned cuts
on transverse momentum and missing energy. While in most literature 2 of 3 selection
criteria are compared, sometimes labelled as low cut, medium cut, high cut, a more generic
method is aimed to determine the optimum selection criteria by investigating the space of
possible selection combinations. [116]
With the hypothetical observation of missing energy the question is raised whether this
originates from stable invisible dark matter particles being produced since numerous other
theories predict massive invisible particles such as Large Extra Dimensions and Split
supersymmetry. Furthermore the discovery in the jets plus missing energy channel would not
necessary prove it to be the astrophysical indirectly observed dark matter, neither could it
give information about the characteristics of dark matter such as its composition of single or
multi particle species, abundance or density profile. [157] [159]
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It might be to simple to assume 83.9% of the non baryonic dark matter content of the
Universe is explained by a single massive particle species, while for the 16.1% normal
baryonic matter in the Universe 99% of the mass is actually contained by a form of QCDbinding energy, with the remark that 91% of this baryonic matter is still unobserved (baryonic
dark matter) while only 9% is explained by the ordinary luminous matter in the objects
observed on the sky like stars and luminous gas and dust clouds. However since collider
experiments are assumed to have a great potential in producing and studying particles with
WIMP like properties in the near future, WIMP dark matter search at ATLAS is recognized as
one of the most important searches for new physics at the LHC. [127]
In the analysis used in this thesis WIMPs are assumed to be viable dark matter candidates
and an universal approach is used with respect to the nature of WIMP dark matter and it‘s
interaction with normal matter such that the results will be generic applicable. Numerous
WIMP and force mediating candidates exist in theoretical physics, but none has been
observed so far. Therefore we describe the interaction of the colliding partons, which are
standard model particles, with dark matter WIMPs, by a generic model.
The pre-assumption made is that WIMP particles are considered to be normal massive but
yet undetected Dirac fermionic spin ½ particles. The assumption that WIMPs are Dirac
fermionic matter consequently has the shortcoming that if dark matter is a different type of
matter, the optimisations are different. Dark matter WIMPs can also be presumed to be a
Majorana fermion or a more exotic particle (see Chapter 3), however a study analysing such
candidates is outside the scope this thesis.
The WIMPs are furthermore assumed to be stable massive non standard model particles
which have no charge, isospin or colorcharge, which interact trough a force comparable in
strength with the weak force with standard model particles in order to explain their
astrophysical existence as a thermal relic particle. Furthermore the WIMPs are amassive so
it is indirectly assumed a Yukawa coupling to a Higgs(like) fields exist that has the right
quantum numbers to form a singlet under SU(3)CxU(1)Q.
We furthermore assume that the WIMP coupling to the first generation of quarks is not
smaller than the coupling to other SM fermions. Since couplings to leptons cannot be directly
observed in a hadron collider environment like LHC such interactions are not taken into
consideration. [116] [177]
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5.2 Interactions in Effective Field Theory
If interactions between WIMPs and standard model particles involve a mediator particle Z’
that is too heavy to be resolved, the interaction can be modelled by an effective field theory
(EFT) as a non renormalizable four-point contact interaction. Herein it is assumed the
intermediate state is not accessible at the collider energies that are typically in the TeV scale.
In an effective field theory an explicit ultraviolet energy cutoff scale Λ is introduced for a
quantum field theory and its action. This cutoff energy scale is much larger than the
interacting particle mass, or any other energy scale of practical interest. Next the cutoff
energy is lowered by integrating out higher momentum degrees of freedom and as a result
the coefficients in the effective action will change. An ultraviolet cutoff Λ on the loop
momentum is induced such that all of the loop integrals in diagrams become finite. If in a
weakly coupled field theory (g << 1 and m2 << Λ2 ) the cutoff Λ scale is lowered via the
Wilson scheme the coupling constant will change according to its beta functions. The result
is that at an energy scale E well below the initial cutoff scale Λ0, the effective theory
described by the Wilsonian effective action is the same as predicted via renormalized
perturbation theory, up to small corrections by powers of E / Λ0. The Wilson scheme gives a
physical interpretation to non-renormalizable theories by regarding it as an effective action.
When describing an effective action with a cutoff scale Λ there exist a maximum possible
value for the cutoff scale Λmax such that it’s not possible to take the limit Λ → . If this limit is
applied the only possible value of g is g = and it is said the theory is “trivial” in the limit of an
infinite cutoff since there are no interactions. This is also true for quantum electrodynamics
and Λmax is known as the location of the so called Landau pole. However the cutoff can be
removed if the beta function for large g, grows no faster than g itself. Then the effective
coupling at increasing energies would remain fixed at g which is called an ultraviolet fixed
point of the renormalization group.
If the beta function is negative the theory is said to be asymptotically free and g decreases as
the cutoff energy is increased and the limit Λ → exist. In four spacetime dimensions the
only so called asymptotically free theories can be proven to be nonabelian gauge theories,
as is SU(3)C.
Using the EFT approach for standard model and Dirac WIMP interaction 16 different effective
operators with vector-, axial-vector, or tensor-type exchanges can be used, which are found
by considering all possible Lorentz structures that are consistent with SU(3)CxU(1)Q gauge
invariance.
The general interaction term in the Lagrangian for a vector mediator Z’ is given by [5.1]
where f is a generic Standard Model fermion field, with the sum implied over the quark and
lepton flavours and the kinetic and gauge terms in the Lagrangian being ignored and x is the
Dirac dark matter WIMP field.
  Z ' [ f   ( g Vf  g Af  5 ) f ]  Z ' [ x  ( g Vf  g Af  5 ) x]
f
[5.1]
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In the low energy limit this Lagrangian leads to the effective vector (V) and axial (A)
operators OV and OA describing the interaction between fermionic dark matter WIMPs fields x
and Standard Model quark fields q by respectively [5.2] and [5.3]. [112] [160]
OV
 x  x  q  q 

OA
 x   x  q 



2
 5

2

5 q


D5  x   x q   q




D8  x    5 x q    5 q
[5.2]

[5.3]
Here Λ is the EFT suppression scale up to which energy EFT is a valid approach. The pure
spin independent vector s-channel described by [4.2] is called D5 and the spin dependent
axial vector s-channel operator described by [5.3] is called D8.
Polarized beams are sensitive to the chirality of the underlying interaction, but in comparison
to lepton colliders LHC is not sensitive to the chirality of the main interaction. Therefore we
will consider only the spin independent D5 operator which is expected to be one of the
leading production channels of WIMPs in LHC. [161]
S-channel and t-channel refers to the most straightforward description in the UV and the
exchange of a new heavy colorless gauge boson Z’ features an s-channel mediator. Schannel mediators are not protected by WIMP stabilization symmetry since they can couple
to SM particles directly, and its mass can be smaller or larger then the WIMP mass itself. It is
assumed the mediator has equal coupling to all quarks of all flavours, no coupling to the
standard model bosons exist and is colorless. T-channel mediators and dark matter WIMPs
that are not Dirac fermions are protected by WIMP stabilization symmetry since they must
couple to at least one WIMP as well as other SM particles. The mediator mass should be
larger than the WIMP mass otherwise the WIMP would be able to decay. In the t-channel the
mediator carries color charge and can decay into a quark-DM pair. [116]
For Majorana dark matter WIMPs 10 leading effective operators exist which are consistent
with SU(3)CxU(1)Q gauge invariance and couple to quarks. The vector type interaction D5
would vanish in all cases, but interactions described trough the axial vector spin dependent
operator D8 will be non trivial and are studied in [7] [162].
Furthermore models exist in which dark matter WIMPs couple to the standard model
particles by exchange of a scalar mediator. In literature often 6 complex scalars and 4 real
scalar type effective operators are used and interactions can precede trough the s-channel or
t-channel [d8]. Dark matter of spin 1 and spin 3/2 is studied in [163] [164].
The EFT is a good approximation at low energies where the interacting heavy mediator Z’
with mass MZ’ can’t be probed. The interaction trough the D5 operator is parameterised by
the couplings gSM and gDM where gDM is the coupling to dark matter WIMPs x and gSM is the
coupling to Standard Model quarks q, see fig 5.1.
We focus on models where dark matter is a Dirac fermionic particle annihilating to Standard
Model fermions trough the s-channel via a Z’-type mediator.
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Time
Fig 5.1. D5 operator interaction of WIMPs and quarks.
These parameters are related to the EFT suppression scale Λ by the matching condition of
the ultra-violet (UV) theory with a mediator and its low-energy effective field theory by [5.4].

MZ'
g SM g DM
[5.4]
Perturbation theory implies a validity requirement of 0 < gSM < 4π and 0 < gDM <= 4π. At an
energy at which EFT is a reliable framework, a minimal constraint is the requirement that the
momentum transfer in the process Q is smaller than the mediator mass M Z’ such that the
validity requirement combined with [5.4] gives [5.5].
Q  g SM g DM 


Q  4 
[5.5]
It is natural to expect a higher momentum transfer Q for events with larger missing energy.
Therefore the validity of EFT can be tested by studying the fraction of events passing this
requirement as a function of the missing energy threshold used to define the signal region
investigated while the leading jet transverse momentum threshold is left fixed.
The following constraint to consider is the effect of the dark matter mass mx on the EFT
validity since mx and Q are not independent. If one assumes momentum transfer to occur in
the s-channel, kinematics require Q > 2 mx, such that [5.6] must be true in order to produce
the final WIMP particles.
2mx  Q  4



mx
2
[5.6]
This demonstrates the trend that an increasing dark matter mass mx results in a smaller
validity regime of EFT. [158]
Astrophysics at the LHC – Dark matter search in ATLAS
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5.3 Interactions trough a Z’ mediator
In the EFT framework it is assumed the particles that mediate the interaction between dark
matter and standard model particles are much heavier than the typical momentum
exchanged in mono-jet events, and the interaction is well approximated by a contact
operator. However as the LHC will be operating at record energy scales, hard scattering
events address the question of validity at high momentum transfer. We will investigate how
the predictions of effective theory are modified once a propagating particle is introduced and
describe a simplified model where the mediator is no longer integrated out.
For the mediating particle we consider an interaction trough a spin Z‘ vector boson as
intermediate state. Additional colorless Z0 like vector gauge bosons occur in extensions of
the Standard Model partly since additional abelian U(1) fields are harder to break by Higgs
mechanism than non-abelian fields. Z’ bosons can be assumed to exist at any energy scale
and couplings could be weak or strong, but we concentrate on ~TeV scale masses with
couplings similar to the electroweak force. Such a Z’ boson might therefore be observable at
the LHC in the future. [165] [166]
The discovery of a Z’ boson could lead to many new implications which can lead to further
searches at colliders. For example there should exist an associated Higgs(like) field to give
the Z’ mass. Discovering the corresponding Higgs particle would be much harder than
discovering the Z’, similar to the discovery of W and Z bosons was observed before the
Higgs boson. The understanding of the nature of Z’ couplings might give insight about its
embedding in more fundamental theories. Discovering a Z’ with the current Standard Model
fermions would be peculiar and could hint to possible new fermions that may be produced by
colliders. A Z’ boson can also play an important role in the dynamics of electroweak
symmetry breaking and the observation of a Z’ boson decay into SM gauge bosons could
give more insight in this process. For supersymmetry the discovery of a Z’ boson decaying
into superpartners can be an important discovery channel and Z’ could play an important role
in a solution of the μ problem in supersymmetry. [120] [127]
However we should also state that while the description of interactions by 3 of the 4
fundamental forces existing in nature are well described by the exchange of respectively a
foton, W/Z vector boson or gluon, no Z’ or 5th force has ever been observed. One can argue
it’s remarkable to imply a 5th force to explain matter that has been observed only indirectly
trough gravity. It is also worth to note that since the nature of gravitational interaction is not
known and most energy and matter in de Universe is of an unknown form, the assumption of
dark matter simply being a massive particle interacting trough a boson might turn out to be a
viewpoint that is a wrong extrapolation of our current best understanding of nature by the
theory of the standard model. While no accepted theories are yet able to answer what dark
matter actually is, LHC would within the next years be able to probe interactions of the
hypothetical WIMP dark matter and either confirm its existence or put limits on its
characteristics.
The Z’ mediator particle is assumed to be massive and hence we note that it is indirectly
assumed a non-minimal coupling between the Z’ field and a Higgs field exists such that the
original symmetry is spontaneously broken, and a mass term for the gauge boson Z’
connected to the broken symmetry appears together with a massive Higgs.
Astrophysics at the LHC – Dark matter search in ATLAS
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In the studied simplified model the mediator particle is Z’ that can couple to two quarks q in
the initial state (whether virtual or on-shell) with coupling constant g q. The Z’ can then decay
trough the s-channel into two WIMP particles x in the final state with coupling constant gx,
see Fig 5.2.
Time
Fig. 5.2. Interaction in a simplified model with interaction trough a Z’ boson and an outgoing
gluon (g).
The sensitivity of colliders will change when a simplified model with Z’ is used instead of a
EFT model, either suppressing or enhancing the signal. In the case of s-channel operators
resonance effects can enhance the production cross section once the mass of the s-channel
mediator is within the kinematic range and can be produced on-shell. The cross section for
the process
scales as [5.7].
 ( pp  xx  jets) ~
gq 2 g x 2
(q  M z ' )   / 4
2
2 2
2
E2
[5.7]
Here mz’ is the mass of the s-channel mediator Z’, Γ is the mediator decay width and s = q2 =
p12 + p22 is the partonic center-of-mass energy of parton 1 and 2 or the momentum transfer
trough the mediator.
For direct detection experiments (see chapter 3) the cross section of the process
with the reduced mass μxN of the dark matter x and target nucleus N scales as [5.8]
 ( xN  xN ) ~
At LHC operating at
gq 2 g x 2
M z'
4
 2 xN
14 TeV
[5.8]
has a broad distribution peaked areound a few
hunderd GeV such that if
~ mz’ and the condition 2 mx < mz’ resonant enhancement
favours collider production of dark matter in comparison with direct detection. However for
light mediators when mz’2 << q2 the cross section for collider WIMP production becomes
independent of m z’ while the cross section for direct detection increases for smaller m z’.
Therefore colliders have a relative disadvantage compared to direct detection experiments in
the case of a light mediator. [109] [116]
Astrophysics at the LHC – Dark matter search in ATLAS
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Using the simplified interaction model we can summarize different production types. For the
s-channel operator with a sufficient low mass mz’ kinematically in reach of the accelerator,
enhanced on-shell production leads to resonant enhancement of the cross section. For
heavier meadiators the production goed off-shell and is out of energy reach of the collider
and the interaction is suppressed. For the simulations it is expected that the interaction
trough the most massive mediators will have the most similar properties as an interaction
trough a contact interaction. [158]
Raising the collision energy will increase the average momentum transfer, allowing ineraction
trough larger mediator masses. Therefore increasing the collision energy will expand
sensetivety of WIMP production in colliders towards for larger mediator masses.
However colliders have the limitation of the available energy and to detect heavy dark matter
WIMPs (mx > 1 TeV) direct detection is more sensitive then accelerator searches (if the
interaction is spin-independent). [176]
In the simulations two different scenarios for dark matter masses were used; mx= 50 GeV
and mx = 400 GeV. For the simulation of the mediator Z‘ particle masses from mz’ = 100 GeV
up to mz’ =15 TeV is used and a 4 point contact interaction model described by the D5
operator.
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5.4 Background events
If a produced dark matter pair leaves the detector unseen its signature is missing energy.
First a lepton veto is introduced such that any event in which a charged lepton is created
automatically is disregarded as a signature event. Lepton veto is thus introduced in order to
remove Standard Model backgrounds. However the main background Z→νν does not have
leptons in the final state and the effect of the lepton veto on the signal is small. Therefore in
the simulations of [155] rather then using a lepton parameterisation for the
14 TeV
samples, the lepton veto efficiency from the
8 TeV analysis [156] is used to emulate the
14 TeV simulations (see chapter 5.7). This is justified by the numbers from the upgrade
performance studies showing that the lepton reconstruction efficiencies will be maintained.
[155] In our analysis we then check if the obtained background simulations have ratios that
are in agreement with [155].
Processes which produce background signals are standard model interactions where the
final state standard model particles are undetected leaving energy imbalance, while no
charged leptons were produced or reconstructed. The background consists of the following
processes noted in table 5.1;
Z  
W   l
main background
W   l
Z  ll
Single / pair t production
Di  boson production
with l or l not reconstructed
with ll not reconstructed
Tabel 5.1. Background events with dark matter production signature.
In the first three processes a W or Z boson is created that will decay into leptons that are
either neutrino type ν (ν e, ν μ, ντ) or electron-type l (e, μ, τ), see fig. 5.5a till 5.5d. The W- and
Z- bosons have very short lifetimes with a half life of 3×10−25 s. The main background consist
of Z → vv with both neutrinos not interacting and leaving an energy imbalance in the detector
while the lepton veto was passed due to the detector efficiency. In the second and third
interaction charged leptons were created but not detected due to the detection efficiency of
ATLAS, leaving an energy imbalance and passing the lepton veto. These two processes add
to the background but with a smaller fraction since the probability that 1 or 2 leptons are not
reconstructed in ATLAS is small. The process in which 2 leptons are created is ignored due
to its small contribution to the background signal [pub]. An overview of the contributions of
the main background processes W → μν, W → τν, W → eν and Z → νν can be seen in
chapter 5.4 and 5.6. [156]
Astrophysics at the LHC – Dark matter search in ATLAS
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Time
Fig. 5.5a. Z    background event.
Time
Fig. 5.5b. W  l background event.

Time
Fig. 5.5c. W   l background event.
Time
Fig. 5.5d. Z  l l background event.
Astrophysics at the LHC – Dark matter search in ATLAS
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The created tau particles can decay leptonically or hadronically. Tau leptons (τ-) have a
lifetime of 2.9×10−13 s and a mass of 1776.82 MeV/c2 and the tau lepton is the only lepton
that has sufficient mass to decay into hadrons. Because of its short lifetime it decays within
the detector volume trough the weak interaction hadronically into hadrons or leptonically into
leptons.
Tau leptons (τ-) that decay to v leptonically will produce a lepton pair with an antineutrino
(ve , v  ) and a charged lepton (e-, μ-) that triggers the lepton veto, see fig. 5.6. Leptonic
decay will give a similar background as the W decay and in the simulations where W bosons
are created from the pp collision this background is included.
Time
Fig. 5.6. Tau leptonic decay.
In tau (τ-) hadronic decay an invisible neutrino (vτ) and a pion (du, su ) are created, see fig.
5.7, the pion will just add to the numerous hadrons created in hadronization and the neutrino
will leave an energy imbalance. Hadronic decay should also be included in the background.
In the simulations where W bosons are created from the pp collision decay, the possible W
decay to a tau that itself can decay hadronically, is included.
Time
Fig. 5.7. Tau hadronic decay.
Furthermore events with single top quark production will contribute to the background. Top
quarks decay in about 5·10−25 s to a d, s or b quark and a W boson within the detector
volume, see fig. 5.8. A lepton pair is created from the W boson from which the neutrino will
leave an energy imbalance. When the charged lepton is not reconstructed this process can
add for about 0.5% to the background, but will be omitted due to its small contribution. [156]
Fig 5.8. Single top quark production
Time
Astrophysics at the LHC – Dark matter search in ATLAS
90
Also events in which a top antitop pairs ( t , t) are produced may occur and will add to the
background, see fig 5.9. However the contribution of these events is negligible compared to
the rest of the background events, and will be omitted. [156]
Time
Fig. 5.9. Top antitop production. [h1]
Di-boson production is a process in which 2 W or 2 Z bosons decay into a lepton pair l,v and
quark anti-quark pair qq , see fig. 5.10. The neutrino will give rise to measured energy
imbalance and the process will add to the background if the charged lepton is remained
undetected. At 14 TeV the diboson background is approximately 2.2 % of the total
background at Emiss > 300 GeV and 4.7% at Emiss > 800 GeV and will be omitted for
simplicity. [156]
Time
Fig. 5.10. Diboson production.[173]
Astrophysics at the LHC – Dark matter search in ATLAS
91
5.5 Simulation methods
To anticipate the detectors performance detailed simulations of potential processes are
studied. Monte Carlo Event Generators such as PYTHIA can simulate interactions and
detection. A Monte Carlo (MC) simulation of proton proton collisions at ATLAS consists of a
physics simulation of the colliding partons and generating the possible interactions that can
happen, in which besides the known and verified standard model interactions also the
possible dark matter interaction models can be specified. To simulate the physics of the
interaction a theoretical model is specified for the participating particles and possible
interactions and matrix element calculations predict final state hadrons and their kinematics.
A PDF must be implied to simulate the kinematics of the partons. After simulation of the hard
process and parton showering, hadronization is simulated atter which the underlying event
and unstable particle decays are produced. Finally the simulation can include detector
efficiencies and limitations.
The matrix element calculation can be performed at different orders. At leading order (LO)
calculations include no extra final state particles or virtual gluons in loops. At next to leading
order (NLO) also emission of real extra final state gluons and virtual gluon loops corrections
are considered, see fig. 5.3. At the next order (NNLO) also double virtual loops, double real
emission and real-virtual corrections are included, see fig. 5.4. Higher order corrections give
generally different normalisation, kinematic distributions and more accurate prediction of the
number of jets created. However higher order corrections quickly become very complex and
harder to calculate or even not yet performed (such as N3LO).
Virtual corrections (“loops”)
Real emission
Fig. 5.3. Different NLO diagrams. [167]
Double virtual
Real virtual
Fig. 5.4. Different NNLO diagrams [167
]
Double real
Astrophysics at the LHC – Dark matter search in ATLAS
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The Monte Carlo data consist of two sets of data, the truth level data and the reconstructed
level data.
In the reconstructed level data it is simulated how a hadron would have been detected by
ATLAS. The performance of the ATLAS detector (tracking, calorimetery and trigger) is
simulated and detector efficiency is taken into account. Detector output which only contains
indirect information about the parent particles is simulated using smearing functions that
relate true values to measured values on a statistical basis.
The truth-level describes an idealised representation of the interaction using MC simulations
of the jets following the decay chain back to the original partons. The energy at truth level is
calculating using MC studies and corresponds to the energy of the original parton from which
the jet originated. Truth simulated data provides any without detector limitations, and
comparison of this ideal truth data with reconstructed data is a test of the analysis made.
When analysis of the truth information has anomalies or inaccurate results, the analysis of
reconstructed data will result in improper conclusions.
CERN uses an analysis framework based on the C++ programming language called ROOT
developed by Rene Brun and Fons Rademakers which allows proper display and
calculations for Object Oriented analysis. [168] Both truth level data and reconstructed level
data can be analysed by ROOT in the same way as the data from a measured event in the
detector would be analysed. The data is recorded in the same format as actual data would
be stored, however to the simulated data there is the addition that all initial conditions of the
event are known and stored.
Since this thesis focuses on the analysis of different dark matter and mediator models and
not on practical detector properties or detection efficiency, simulations of data at truth level
are studied in all data samples.
When performing the missing energy calculation some factors are not provided such as low
energy jets or energy deposits which are not matched to any selected particle. Therefore the
variable sumEt which is based on the energy deposits in the calorimeter is used to estimate
missing energy. However if in the standard model background process a muon is created,
the muon is measured with the muon spectrometer and is not included in sumEt while it is
included in missing energy. Only in the case that the lepton is a muon therefore sumEt is
smaller then missing energy. In order to compare signal to background with the correct
missing energy, for the background data missing energy at truth level (metTruth) is
calculated with omitting the momentum of the muon (using the variable metTruth_noMuon)
while for the signal data missing energy at truth level is calculated including the momentum
of the muon (using the variable metTruth).
When in ATLAS an event passed the selections in the TDAQ system, the tracking and
calorimeter firmware records the information of interest and the output is recorded. Analysis
of data in high energy physics relies on correct interpreting information from the detector.
Astrophysics at the LHC – Dark matter search in ATLAS
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5.6 Simulations of background
The Standard Model contributions in the used simulations were generated by [155] using
Sherpa with the CT10 parton distribution function set. For the electroweak processes only
W→μν and W→τν MC simulations are available. Simulations for the dominant Z→νν
background as well as for the W→eν are not available. Emulation of the missing MC
background samples is done by using the two available backgrounds by scaling to the ratio
of processes at
8 TeV. The lepton veto selection efficiencies are emulated by applying
the veto efficiencies from the
8 TeV analysis. This is justified by the upgrade
performance study showing that the lepton reconstruction efficiencies will be maintained
[155]. The simulations are produced with a W/Z boson with transverse momentum of p T >140
GeV.
For the dominant background in the mono-jet analysis is Z→vv for which there is no Monte
Carlo available and to emulate the Z→νν background simulation ZννMC the MC simulation of
W→μν called WμνMC is used. The ZννMC is calculated by multiplying WμνMC with the ratioB of
the processes Z→νν / W→μν at
8 TeV for Emiss in the range [300 GeV, 1200 GeV]
before the lepton veto, a ratio of 0.831, and with the lepton veto selection efficiency of 0.9929
in the range for Emiss of [300 GeV, 1200 GeV].
The W→τν background is obtained by using WτνMC multiplied with the Wτν lepton veto
selection efficiency at
8 TeV of 0.5 for Emiss in the range [300 GeV, 600 GeV].
The W→μν background is obtained by using WμνMC multiplied with the Wτν lepton veto
selection efficiency at
8 TeV of 0.0929 for Emiss in the range [300 GeV, 600 GeV].
The W→eν background is obtained by using the previous calculated W→μν background
multiplied with the ratioA of the processes W→eν / W→μν at
8 TeV for Emiss in the range
[300 GeV, 1200 GeV] after the lepton veto, a ratio of 0.676.
An overview of how the background contributions are calculated can be seen in table 5.2.
[155]
Z
ratioB (0.8289)  Z lveto (0.9929)
W 
W lveto (0.5)
Z  W  MC 
W  W MC
W   W  MC W  lveto (0.0929)
We  W  
We
ratioA (0.0676)
W 
Table 5.2 Calculation of the background contributions.
Astrophysics at the LHC – Dark matter search in ATLAS
94
5.7 WIMP production simulations
The Monte Carlo simulations of
14 TeV that were used in this study were made by [155]
by using MadGraph [20] with the PDF set CTEQ6L1 and Pythia for parton showering using
MLM matching. Models with WIMP masses mx = 50 GeV and m x = 400 GeV are studied with
contact interactions with a EFT suppression scale Λ =
TeV while for the interaction trough
a massive mediator Z’ samples for various mediator masses were used.
For every simulation with a massive mediator only the decay width Γ
is used since for
the study of the effect of different selection criteria on significance of the same process the
absolute value of the cross section is not of interest, and in the study of the comparison of
significance with respect to the different mediator masses and dark matter masses used in
the models, all other characteristics of dark matter and the mediator have to be constant.
Therefore besides the decay width also the coupling constant gx between two WIMP particles
x and Z’ is assumed equal in all studied models.
Mediator masses with a mediator mass of m Z’ = 100 GeV, 300 GeV, 500 GeV, 1000 GeV,
3000 GeV, 5000 GeV, 10000 GeV and 15000 GeV were stored with the label “simulation
label” and is referred to throughout this thesis with the label “label” according to table 5.3.
Mx = 50 GeV
Mx = 400 GeV
’
’
mZ’
Simulation label
Label
Label
100 GeV
300 GeV
500 GeV
1000 GeV
3000 GeV
5000 GeV
10000 GeV
15000 GeV
DM50_MM100_W3
DM50_MM300_W3
DM50_MM500_W3
DM50_MM1000_W3
DM50_MM3000_W3
DM50_MM5000_W3
DM50_MM10000_W3
DM50_MM15000_W3
D5_ DM50_W3
DM50MM100
DM50MM300
DM50MM500
DM50MM1000
DM50MM3000
DM50MM3000
DM50MM10000
DM50MM15000
D5DM50
100 GeV
300 GeV
500 GeV
1000 GeV
3000 GeV
5000 GeV
10000 GeV
15000 GeV
mZ’
Label
DM400_MM500_W3
DM400_MM1000_W3
DM400_MM3000_W3
DM400_MM5000_W3
DM400_MM10000_W3
DM400_MM15000_W3
D5_ DM400_W3
DM400MM500
DM400MM1000
DM400MM3000
DM400MM3000
DM400MM10000
DM400MM15000
D5DM400
Table 5.3. Studied simulations of mediator models and their according labels.
Astrophysics at the LHC – Dark matter search in ATLAS
95
5.8 Data samples
Background samples
For the background samples of the 4 background processes mentioned in 5.4 and 5.6 which
dominantly contribute to the background signal, the following method was used to construct
the data files. The data of W→ μν and W → τν MC simulated events obtained from [155] were
contained in 9 files, see table 5.4. In order to have high statistics in all phase-space regions
the data had been divided in 3 regions depending on the transverse momentum of the W/Z
boson namely [140 ; 280] GeV, [280 ; 500] GeV and above 500 GeV. Each of those regions
had been divided as well into the three BFilter, CJetFilterBVeto and CJetVetoBVeto samples,
where the C/B filter(veto) means with or without any C/B-jets in the simulated event. This
was done to accurately predict b- and c-jet events in these samples.
To use the data in this study the 9 files per sample are added using lumiweight to construct 1
data set per background type. The lepton veto efficiencies and process ratios mentioned in
table 5.3 were applied and Z→νν as well as W→eν data files are constructed. Then the
different backgrounds are added using lumiweight (see chapter 4.10) to construct the total
Standard Model background.
SAMPLE
σ(nb)
ε
N
WμνMC
MONOJET.TRUTHCYCLE.MC.167773.WmunuMassiveCBPt140_280_BFilter
MONOJET.TRUTHCYCLE.MC.167774.WmunuMassiveCBPt140_280_CJetFilterBVeto
MONOJET.TRUTHCYCLE.MC.167775.WmunuMassiveCBPt140_280_CJetVetoBVeto
MONOJET.TRUTHCYCLE.MC.167782.WmunuMassiveCBPt280_500_BFilter
MONOJET.TRUTHCYCLE.MC.167783.WmunuMassiveCBPt280_500_CJetFilterBVeto
MONOJET.TRUTHCYCLE.MC.167784.WmunuMassiveCBPt280_500_CJetVetoBVeto
MONOJET.TRUTHCYCLE.MC.167791.WmunuMassiveCBPt500_BFilter
MONOJET.TRUTHCYCLE.MC.167792.WmunuMassiveCBPt500_CJetFilterBVeto
MONOJET.TRUTHCYCLE.MC.167793.WmunuMassiveCBPt500_CJetVetoBVeto
0.093958
0.093825
0.094027
0.0073782
0.0073799
0.0073774
0.00064106
0.00064225
0.00064117
0.076391
0.25341
0.67014
0.099807
0.27273
0.62733
0.12125
0.2845
0.59525
24999
25000
26000
25000
24999
25000
24949
24900
24950
WτνMC
MONOJET.TRUTHCYCLE.MC.167776.WtaunuMassiveCBPt140_280_BFilter
MONOJET.TRUTHCYCLE.MC.167777.WtaunuMassiveCBPt140_280_CJetFilterBVeto
MONOJET.TRUTHCYCLE.MC.167778.WtaunuMassiveCBPt140_280_CJetVetoBVeto
MONOJET.TRUTHCYCLE.MC.167785.WtaunuMassiveCBPt280_500_BFilter
MONOJET.TRUTHCYCLE.MC.167786.WtaunuMassiveCBPt280_500_CJetFilterBVeto
MONOJET.TRUTHCYCLE.MC.167787.WtaunuMassiveCBPt280_500_CJetVetoBVeto
MONOJET.TRUTHCYCLE.MC.167794.WtaunuMassiveCBPt500_BFilter
MONOJET.TRUTHCYCLE.MC.167795.WtaunuMassiveCBPt500_CJetFilterBVeto
MONOJET.TRUTHCYCLE.MC.167796.WtaunuMassiveCBPt500_CJetVetoBVeto
0.093944
0.093967
0.093978
0.0073745
0.0073768
0.007373
0.00064201
0.00064164
0.00063878
0.076351
0.25743
0.666
0.099829
0.27665
0.62348
0.12094
0.28774
0.59152
24900
24900
26000
24950
25000
24998
25000
24949
20000
Table 5.4 Background sample files. [169]
Dark matter samples
The data of the dark matter simulations from [15] had a different slicing of the samples. The
transverse momentum cut QCUT was chosen at values 200, 400 and 600 GeV to define
three regions. In order to make sure the samples are exclusive, a cut on the leading parton
pT was imposed;
QCUT = 200 GeV: leading parton pT within the range of [250 , 450] GeV,
QCUT = 400 GeV: leading parton pT within the range of [450 , 650] GeV,
QCUT = 600 GeV: leading parton pT above 650 GeV.
Astrophysics at the LHC – Dark matter search in ATLAS
96
The files for each dark matter sample are showed in table 5.5. The 3 files per dark matter
sample used in this study are added using lumiweight (see Chapter 4.10) to construct 1 data
set per dark matter type.
SAMPLE
σ(nb)
ε
N
D5, mx = 50GeV, Λ = 10 TeV
MONOJET.TRUTHCYCLE.MC.188408._D5_DM50_MS10000_QCUT200
MONOJET.TRUTHCYCLE.MC.188409._D5_DM50_MS10000_QCUT400
MONOJET.TRUTHCYCLE.MC.188410._D5_DM50_MS10000_QCUT600
4.3399e-08
1.053e-08
5.4569e-09
1.0
1.0
1.0
19999
19900
19950
D5, mx = 400GeV, Λ = 10 TeV
MONOJET.TRUTHCYCLE.MC.188411._D5_DM400_MS10000_QCUT200
MONOJET.TRUTHCYCLE.MC.188412._D5_DM400_MS10000_QCUT400
MONOJET.TRUTHCYCLE.MC.188413._D5_DM400_MS10000_QCUT600
2.9949e-08
8.1351e-09
4.5846e-09
1.0
1.0
1.0
19950
20000
20000
mx = 50GeV, mz’ [100,15000] GeV, Γ
MONOJET.TRUTHCYCLE.MC.188414._dmV_DM50_MM100_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188415._dmV_DM50_MM300_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188416._dmV_DM50_MM500_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188417._dmV_DM50_MM1000_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188418._dmV_DM50_MM3000_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188419._dmV_DM50_MM6000_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188420._dmV_DM50_MM10000_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188421._dmV_DM50_MM15000_W3_QCUT200
0.032403
0.016291
0.0073358
0.00089148
1.1477e-05
3.8143e-07
4.2505e-08
8.0758e-09
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
19948
20000
20000
19950
20000
20000
20000
19999
MONOJET.TRUTHCYCLE.MC.188422._dmV_DM50_MM100_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188423._dmV_DM50_MM300_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188424._dmV_DM50_MM500_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188425._dmV_DM50_MM1000_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188426._dmV_DM50_MM3000_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188427._dmV_DM50_MM6000_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188428._dmV_DM50_MM10000_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188429._dmV_DM50_MM15000_W3_QCUT400
0.0028214
0.0019689
0.00092809
0.0001948
3.0812e-06
9.6283e-08
1.0272e-08
1.9399e-09
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
20000
20000
19999
20000
20000
19948
20000
19950
MONOJET.TRUTHCYCLE.MC.188430._dmV_DM50_MM100_W3_QCUT60
MONOJET.TRUTHCYCLE.MC.188431._dmV_DM50_MM300_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188432._dmV_DM50_MM500_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188433._dmV_DM50_MM1000_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188434._dmV_DM50_MM3000_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188435._dmV_DM50_MM6000_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188436._dmV_DM50_MM10000_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188437._dmV_DM50_MM15000_W3_QCUT600
0 0.00070098
0.00052471
0.00029729
8.4294e-05
1.8051e-06
5.3073e-08
5.4028e-09
1.013e-09
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
19850
19950
19949
20000
19950
19950
19949
19999
mx = 400GeV, mz’ [500,15000] GeV, Γ
MONOJET.TRUTHCYCLE.MC.188462._dmV_DM400_MM500_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188463._dmV_DM400_MM1000_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188464._dmV_DM400_MM3000_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188465._dmV_DM400_MM6000_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188466._dmV_DM400_MM10000_W3_QCUT200
MONOJET.TRUTHCYCLE.MC.188467._dmV_DM400_MM15000_W3_QCUT200
0.00015878
0.00052236
9.7867e-06
2.8834e-07
3.0095e-08
5.6762e-09
1.0
1.0
1.0
1.0
1.0
1.0
19950
20000
19950
19950
19950
20000
MONOJET.TRUTHCYCLE.MC.188468._dmV_DM400_MM500_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188469._dmV_DM400_MM1000_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188470._dmV_DM400_MM3000_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188471._dmV_DM400_MM6000_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188472._dmV_DM400_MM10000_W3_QCUT400
MONOJET.TRUTHCYCLE.MC.188473._dmV_DM400_MM15000_W3_QCUT400
3.6603e-05
0.00012284
2.7837e-06
7.9632e-08
8.0976e-09
1.504e-09
1.0
1.0
1.0
1.0
1.0
1.0
20000
19950
20000
20000
19950
19999
MONOJET.TRUTHCYCLE.MC.188474._dmV_DM400_MM500_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188475._dmV_DM400_MM1000_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188476._dmV_DM400_MM3000_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188477._dmV_DM400_MM6000_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188478._dmV_DM400_MM10000_W3_QCUT600
MONOJET.TRUTHCYCLE.MC.188479._dmV_DM400_MM15000_W3_QCUT600
1.6637e-05
5.6528e-05
1.6927e-06
4.6744e-08
4.6093e-09
8.5597e-10
1.0
1.0
1.0
1.0
1.0
1.0
20000
19999
19950
20000
20000
20000
Table 5.5. Background files en samples. [169]
Astrophysics at the LHC – Dark matter search in ATLAS
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5.9 General selection criteria
To the simulated data some of dark matter and background first general selection criteria are
presumed and applied to all datasets of the standard model background and the dark matter
signal. The WIMP signature would manifest itself as an excess of jets plus an imbalance of
energy. Therefore we expect that if WIMPs were created, a selection made on the data
filtering out events will enhance the signal to background ratio. This allows a direct study of
the changes in background and signal yields.
Missing energy Emiss
ATLAS
8 TeV data analysis selects events with Emiss > 120 GeV. Due to higher pile-up
at
14 TeV, the thresholds for Emiss triggers should be increased. Since new physics is
expected to be seen at the higher end of the Emiss spectrum it is safe to assume that missing
energy should be at least 300 GeV to be well reconstructed, E miss > 300 GeV. Furthermore
events with missing energy of Emiss > 2 TeV are highly improbable. Since signal to
background ratio will decrease when scanning a region of E miss in which signal events are
improbable, we will set an upper limit of Emiss = 2 TeV. [155]
Transverse momentum pT [j]
It is assumed the jet are well defined objects only if their momentum is larger then pT = 30
GeV. This threshold also suppresses pile-up jets. Pile-up suppression is important to give
smaller contamination by pile-up jets. A pile-up study shows that for 14 TeV accuracy can be
gained by increasing the jet definition p T threshold from pT = 30 GeV to pT = 50 GeV in order
to suppress pile-up jets. However since this pT = 50 GeV threshold was not implied in the
simulations and to make the code workable the jet object definition with a threshold of pT =
30 GeV was assumed.
In order to have the missing energy to be well reconstructed and assuming events with no
hard scattering and little transverse momentum transfer will not enhance the significance,
events with a transverse momentum of the first jet p T [0] < 300 GeV are excluded. Also
events with pT [0] > 2 TeV are improbable and are excluded.
Pseudorapidity η[j]
An interaction with large transverse momentum transfer is characterized by an energetic jet
in the central region near η = 0. In the central region the accuracy of the ATLAS detector is
highest and data is better in agreement with simulations. In fig. 5.11 the pseudorapidity is
showed for different region in the ATLAS detector. For jets with transverse momentum of
approximately pT = 20 GeV the accuracy in pT between data and MC simulations is about 3%
for jets with |η|<2.1, 6% for 2.1 < |η| < 2.8 and 11% for 3.6 < |η| < 4.4. [170] Furthermore in
the data of the studied
14 TeV Monte Carlo simulations smearing factors are applied.
[155] The smearing factors for jets are available only up to |η| = 3.6 as it is difficult to predict
the performance of the jet reconstruction efficiency in the forward region. Therefore the
maximum limit for the pseudorapidity η of every jet is restricted to the acceptance region of
|η| = 3.6. [152]
Astrophysics at the LHC – Dark matter search in ATLAS
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Fig. 5.11. Pseudorapidity of ATLAS detector regions.[179]
Jet multiplicity, njet
The expectation is that when more jets are created, the energy imbalance can be less
precisely calculated compared to an event in which 1 energetic initial state jet is created
balanced against missing energy. Therefore it is expected that significance will decrease as
the number of jets increases. Studied are the scenarios in which n jet = 1, njet = 1 AND/OR njet
= 2, njet NOT 0, so respectively scenarios for only 1-jet, for 1 and/or 2 jets, and for any
number of jets, n-jets.
Multijets and deltaphi dφ
QCD multijets are a background process in which multiple jets are created with n jet
3. In
processes where multiple jets are created the topology of the event becomes complicated
and uncertainty in missing energy reconstruction increases. Additionally in the jets neutrinos
will be produced and emitted leading to more energy imbalance. Therefore multijet events
can lead to the false observation of missing energy while passing the lepton veto. A cut on
dφ between any jet and the leading jet of dφ > .5 is imposed to suppress background from
mis-reconstructed multi-jet events such that any further contributions can be omitted. [155]
Astrophysics at the LHC – Dark matter search in ATLAS
99
5.10 Data computation
Using a self written code in ROOT of 4 files containing each about 8000 lines, the simulation
files are automatically analysed, displayed and stored. The output contains tables with
calculated significance, and graphs of 6 variables for shape comparison for every simulated
dark matter model, for every possible combination of selections applied, with respect to the
background.
For every simulated sample a constant number of events n is generated by using the cross
section σ as intrinsic probability of the interaction to happen and a necessary generated
luminosity L. Since dark matter has not been detected directly there is no observational value
for the cross section of creating a WIMP pair and a theoretically predicted cross section is
used. In a simulation typically an event number of n = 25000 events are generated.
To study the effect of applying selection methods in order to increase the significance, we
have to compare simulations of background and signal interactions that will have different
generated luminosities. For each simulation we will therefore calculate the number of data
events ndata that is generated with equal and comparable amounts of data, L data. We therefore
weight the number of events nc by the generated luminosity of an event L gen, and normalise
this by the Ldata, see [5.10]. The ratio Ldata/Lgen is called lumiweight and applied as a factor to
the event number of signal (S) and background (B) by formula [5.9] and [5.10] respectively
automatically using the command lumiweight.
nSgen   S  LSgen
nS   S  Ldata 
nBgen   B  LBgen
nB   B  Ldata 
nSgen
LSgen
nBgen
LBgen
 Ldata
 Ldata
[5.9]
[5.10]
The signature of WIMP pair production is an access of events with large missing energy
compared to normal background events. Considering the amount of events and the fact that
the simulated missing energy varies in a range from E miss between 300 GeV and 2 TeV, we
have divided the total energy range of 1700 GeV in 34 regions of 50 GeV, and calculated the
number of data events per Emiss = 50 GeV bin region, and then plotted in a histogram for
background and signal events as [5.11];
N data
(events  0.02 GeV 1 ) vs. EMissbin (50 GeV )
EMissbin
[5.11]
A comparison is made how the number of events is distributed over missing energy, for
different samples and selections. For each simulation the number of events N data observed
within a range of missing energy [E miss_min ; Emiss_max] can be counted. This is done by
integration according to [5.12].
Emiss _ max
N data 

Emiss _ min
ndata dEmiss
[5.12]
Astrophysics at the LHC – Dark matter search in ATLAS
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We then study the effect of applying different selections on different variables, to increase the
signal Ndata_s to background Ndata_b ratio calculated over an interval of missing energy. We
quantify effect by calculating the significance
as a signal to background ratio with the
formula [5.13].

N data _ s
N data _ s  N data _ b
[5.13]
Furthermore a comparison is made how the number of events is distributed over the
transverse momentum of the first jet p T [0] for different samples and selections. The
transverse momentum can vary in a range from p T between 300 GeV and 2 TeV such that
the total momentum range of 1700 GeV is divided in 34 regions of 50 GeV, and we
calculated the number of data events per 50 GeV bin region. This quantity is then plotted in a
histogram in bins of pT = 50 GeV for background and signal events by [5.14].
N data
(events  0.02 GeV 1 ) vs. PTbin (50 GeV )
PT bin
[5.14]
For shape comparison also dφ is analysed and since dφ ranges from .5 rad till π rad the
histograms are showed using 50 bins as [5.15].
N data
bin
(events 
50
  0.5
rad 1 ) vs. bin (
rad )
  0.5
50
[5.15]
For the jet multiplicity in the range njet between 1 and 10 also histograms were made using 9
bins by [5.16].
N data
(events) vs. n jetbin (1)
n jet bin
[5.16]
Pseudorapidity ranges between -3.6 and 3.6 and by using 50 bins the histograms are
showed as [5.17].
N data
bin
(events  6.944) vs. bin (0.144)
[5.17]
Finally the ratio Emiss / pT[0] is calculated in the region between 0 and 3.0 using 50 bins and
plotted in histograms as [5.18].
Ndata (events 16.67) vs. Emiss / pT [0] (0.067)
[5.18]
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5.11 Data analysis and display
The simulations were analysed in 4 steps labelled analysis A till D. In all the analysis first the
general selection criteria from chapter 5.9 were applied after which the data was calculated
according chapter 5.10. A schematic overview of the involved steps is given in chapter 5.12.
A. In the first analysis A the rates of standard model background contributions are
compared to the ratios studied in other
14 TeV analysis with only the general
selection criteria applied as discussed in chapter 5.9. Since these previous 14 TeV
results are extrapolations of
8 TeV simulations that were observed to match
8 TeV observations, matching ratios would show this analysis to be correct. For
all variables the total background and its unscaled contributions are plotted for 1-jet, 1
and/or 2 and n-jet events.
B. In analysis B the simulations of interactions with different dark matter and mediator
models are compared for 1-jet, 1 and/or 2 jet, and n-jet events. The general selection
criteria were applied equal for all models, according to chapter 5.9. We expect to
conclude that models in which the mediator has a large mass have similar
interactions as contact interaction models. For all variables the mediator models are
plotted with respect to the background for 1-jet, 1 and/or 2 and n-jet events and for
DM50 and DM400 separately, and the maximums are scaled to the MM500 model
since the MM500 is the model available for DM50 and DM400 with the highest yield.
C In this analysis the effect of 6 different selections are studied on simulations of
interactions by all different mediator models assuming a dark matter mass of 50 GeV
and 400 GeV. First 6 types of selections were defined according to table 5.6. For
each type of selection all combinations of dark matter models and mediator models
were compared individually to the background and results. The procedure was done
separately for 1-jet, 1 and/or 2, and n-jet events. The D5 and MM500 model were
chosen to be plotted in order to do a shape comparison for an unresolved heavy
mediator model with respect to a light mediator model. MM500 is the model with
lightest mediator mass available for both DM50 and DM400. The maximums are
scaled with respect to the background of n-jet events.
1
Low cut
Emiss < 300 GeVPT [0] < 300 GeV
2
High cut
Emiss < 600 GeVPT [0] < 600 GeV
3
Assym Emiss
Emiss < 600 GeVPT [0] < 300 GeV
4
Assym PT
Emiss < 300 GeVPT [0] < 600 GeV
5
Large Δφ
Emiss < 300 GeVPT [0] < 300 GeV
π - 0.5 rad < Δφ < π rad
6
Small Δφ
Emiss < 300 GeVPT [0] < 300 GeV
π- 0.5 rad < Δφ < 1.0 rad
Table 5.6. The 6 selections used in analysis C.
Astrophysics at the LHC – Dark matter search in ATLAS
D
102
While in most literature 2 of 3 selection criteria are compared, sometimes labelled as
low cut, medium cut, high cut, a more generic method is aimed to determine the
optimum selection criteria by investigating the space of possible selection
combinations. A scan was performed for 1-jet, 1 and/or 2, and n-jet events, for all
combinations of dark matter models and mediator models, investigating the influence
of different selections on the significance. The selections involve all combinations of
cuts on the variables Emiss and PT [0] within the limits [5.19] and [5.20] with increments
of 50 GeV;
300 GeV < Emiss< 600 GeV
300 GeV < PT[0] < 600 GeV
[5.19]
[5.20]
3D Plots were made to show the significance as function of the lower selection limit
on Emiss and PT [0]. The maximum in these plots is the selection with the largest
significance, and is labelled as “Best”.
Since events with large dφ typically raise significance, the process was repeated with
the implication of the extra condition [5.21];
π - 0.5 rad < Δφ < π rad
[5.21]
and the selection with the largest significance was labelled was “Best + large Δφ”.
The results of analysis A till D are presented in Chapter 6.
Astrophysics at the LHC – Dark matter search in ATLAS
103
5.12 Schematic overview of the analysis
In fig. 5.12 a schematic overview is given of the performed analysis indicating the structure
used in the source code that was applied to produce the results automatically when applied
on the available simulations.
General assumptions
-Dark matter as a Dirac fermion
-Simulations
14 TeV
-D5 operator for EFT contact interaction model
-MM for Z‘ simplified interaction model with
-SM background
Studied are samples of MC simulations of the following models;
Dark matter interaction
D5 model
MM model
DM50
DM400
DM50
DM400
D5 DM50
D5 DM400
DM50 MM100
DM50 MM300
DM50 MM500
DM50 MM1000
DM50 MM3000
DM50 MM5000
DM50 MM10000
DM50 MM15000
Background process
SM
WμνMC
WτνMC
DM400 MM500
DM400 MM1000
DM400 MM3000
DM400 MM5000
DM400 MM10000
DM400 MM15000
Samples for each region are added using lumiweight to construct 1 data set for each model
Analysis A.
Calculate SM background ratio’s to match the
background composition of [pub].
Compare:
Zvv
Wμv
Wτv
Wev
Total BG
Plot variables:
Emiss
pT [0]
η
dφ
njet
Emiss/pT[0]
Emulation of the 4 main SM background
contributions
Z  W  MC 
Calculate
ratio’s.
Z
ratio B  Z lveto
W 
W  W  MC W  lveto
W  W  MC W  lveto
We  W  
We
ratio A
W 
Total background is constructed by addition.
General selection criteria discussed in Chapter 4.9 are implied on the data, selecting events
that meet the following conditions;
300 GeV > Emiss > 2 TeV
300 GeV > pT [0] > 2 TeV
|η| < 3.6
dφ > .5
Astrophysics at the LHC – Dark matter search in ATLAS
104
Number of jets. Seperately the following 3 conditions are implied for the number of jets in an
event producing 3 datasets for every model;
njet = 1
njet = 1 || njet = 2
njet ! 0
Analysis B. Comparison between the different dark matter interaction models;
1 jet
DM50
1 or 2 jet
DM400
n jet
Compare:
MM100*
MM300*
MM500
MM1000
MM3000
MM5000
MM10000
MM15000
Plot variables:
Emiss
pT [0]
η
dφ
njet
Emiss/pT[0]
Analysis C. Successively implying 6 additional selection criteria on the data;
1 Low cut
2 High cut
3 Assym Emiss
4 Assym PT
5 Large Δφ
6 Small Δφ
Emiss < 300 GeVPT [0] < 300 GeV
Emiss < 600 GeVPT [0] < 600 GeV
Emiss < 600 GeVPT [0] < 300 GeV
Emiss < 300 GeVPT [0] < 600 GeV
Emiss < 300 GeVPT [0] < 300 GeV
Emiss < 300 GeVPT [0] < 300 GeV
π - 0.5 rad < Δφ < π rad
π- 0.5 rad < Δφ < 1.0 rad
Comparison between dark matter (interaction) model and background.
1 jet
DM50
1 or 2 jet
DM400
n jet
MM100*
MM300*
MM500
MM1000
MM3000
MM5000
MM10000
MM15000
Selection
1
2
3
4
5
6
Compare to
total BG
Plot variables:
Emiss
pT [0]
η
dφ
njet
Emiss/pT[0]
Calculate
significance
Astrophysics at the LHC – Dark matter search in ATLAS
105
Analysis D. Successively implying additional selection criteria on variables Emiss and PT [0] within the
following limits with increments of 50 GeV;
300 GeV < Emiss< 600 GeV
300 GeV < PT [0] < 600 GeV
1.Comparison between dark matter (interaction) model and background.
1 jet
DM50
1 or 2 jet
DM400
n jet
MM100*
MM300*
MM500
MM1000
MM3000
MM5000
MM10000
MM15000
Selection
Compare to
total BG
Plot variables:
Emiss
pT [0]
η
dφ
njet
Emiss/pT[0]
Calculate
significance
Then the selections are compared to optimise for maximum significance and label as best.
Plot significance vs lower limit
on Emiss and PT [0]
Print optimised selection and
according significance.
2. Repeat the complete procedure of 1 with the additional requirement
π - 0.5 rad < Δφ < π rad
now using the label ‘best + large Δφ”.
Fig. 5.12. Schematic overview of the analysis performed in this thesis.
Astrophysics at the LHC – Dark matter search in ATLAS
106
Astrophysics at the LHC – Dark matter search in ATLAS
107
Chapter 6. Results
6.1 Results analysis A
In the first analysis the rates of standard model background contributions are compared to the ratios
studied in the ATLAS WIMP sensitivity study for
14 TeV [155]. Since those results are extrapolations
of
8 TeV simulations that were observed to match
8 TeV observations, matching ratios would
show this analysis to be correct. For all 6 variables discussed in chapter 5.10 the total background and its
unscaled contributions are plotted in fig. 6.2 for events with 1-jet, 1 and/or 2, and n-jet multiplicity.
The linecolors of the graphs of the different background contributions in fig 6.2 are shown by the
specification indicated in fig. 6.1. Here “BG Sum” stands for the summed data of the background
simulations whereas the individual contributions from processes W→μν, W→ τν, Z→νν and W→eν are
respectively indicated by “W→μν”, “W→ τν”, “Z→νν” and “W→eν”. The plots are shown with different
colors for 1-jet, 1 and/or 2, and n-jet events as indicated respectively by “1-jet”, “2-jet” and “n-jet”.
Number
of jets
Process
1-jet
1 and/or 2-jet
W→μν
Yield
(events)
7945
Fraction
W→τν
1.939·10
4
Z→νν
7.039·10
4
W→eν
5781
0.07683
Yield
(events)
4
1.608·10
0.1866
2.624·10
4
0.6807
9.463·10
4
0.05591
7773
n-jet
Fraction
0.07667
Yield
(events)
4
1.242·10
0.1883
3.073·10
4
0.6792
1.1·10
0.05570
9036
4
Fraction
0.07656
0.1894
0.6783
0.05571
Table 6.1 (left). Event yield of the different background contributions.
Fig 6.1 (right). Linecolors of the different background contributions and summed background (“BG Sum”).
The event yield of the different background contributions and their ratios with respect to the summed
background yield are shown in table 6.1. The results of table 6.1 are compared with the results from the
ATLAS Wimp dark matter sensitivity study [155] for the background composition using μ = 0, Emiss > 300
GeV for a sample of L = 20 fb-1 at
14 TeV and the results is shown in table 6.2 with and without
including other processes labelled as ‘Others’.
Process
Yield (events)
Ratio w Others
Ratio w/o Others
W→μν
1044.46
0.0750
0.0788
W→τν
2549.67
0.1867
0.1926
Z→νν
8934.86
0.6544
0.6748
W→eν
710.98
0.0520
0.0537
Others
412.67
0.0302
x
Table 6.2 Results from ATLAS WIMP dark matter sensitivity study [155].
Astrophysics at the LHC – Dark matter search in ATLAS
108
Below the total background and its unscaled contributions are plotted in fig. 6.2 for events with 1-jet, 1
and/or 2, and n-jets for all 6 variables discussed in chapter 5.10.
Fig. 6.2. DM50 model, 1-jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss , plot b (upper right) shows
the yield/PT[0] as a function of PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right)
shows the yield as a function of Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the
yield/dϕ as a function of ϕ.
Astrophysics at the LHC – Dark matter search in ATLAS
109
6.2 Results analysis B
In analysis B the simulations of different dark matter and mediator models are compared for events with 1jet, 1 and/or 2 jets, and n-jets using equal general selection criteria for all models as according to chapter
5.9. For all 6 variables discussed in chapter 5.10 the mediator models are plotted for 1-jet, 1 and/or 2 and
n-jet events and for DM50 and DM400 models separately in fig. 6.4 till fig. 6.9. In each model the
maximum is scaled to the MM500 model since MM500 is the only model available for both DM50 and
DM400 that has the highest yield. In all graphs of analysis B the linecolors of the different studied mediator
models are shown by the specification indicated in fig. 6.3.
D5
MM100
MM300
MM500
MM1000
MM3000
MM5000
MM10000
MM15000
Fig 6.3. The linecolors of the different mediator models used in the graphs of analysis B.
The unscaled yield of the DM50 and DM400 models for different mediators are shown in respectively table
6.3a and table 6.3b for events with 1-jet, 1 and/or 2 jets and n-jets.
DM50
Yield
n-jet
MM100
MM300
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
1.09E+08
9.44E+07
6.94E+07
1.25E+07
2.07E+05
6.52E+03
7.10E+02
1.35E+02
6.99E+02
1/2-jet
9.16E+07
7.89E+07
5.72E+07
1.06E+07
1.75E+05
5.52E+03
6.01E+02
1.14E+02
5.87E+02
1-jet
8.49E+07
5.46E+07
3.88E+07
7.35E+06
1.22E+05
3.85E+03
4.17E+02
7.84E+01
4.04E+02
Table 6.3a. The yield of 1-jet, 1 and/or 2 jet (“1/2-jet”) and n-jet events for DM50 simulations using only
general selection criteria.
DM400
Yield
n-jet
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
2.50E+06
7.97E+06
1.86E+05
5.37E+03
5.52E+02
1.03E+02
5.28E+02
1/2-jet
2.10E+06
6.71E+06
1.58E+05
4.55E+03
4.66E+02
8.67E+01
4.38E+02
1-jet
1.45E+06
4.63E+06
1.09E+05
3.17E+03
3.21E+02
6.00E+01
3.02E+02
Table 6.3b. The yield of 1-jet, 1 and/or 2 jet (“1/2-jet”) and n-jet events for DM400 simulations using only
general selection criteria.
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DM50 model, 1-jet events
Fig. 6.4. DM50 model, 1-jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss , plot b (upper right) shows
the yield/PT[0] as a function of PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right)
shows the yield as a function of Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the
yield/dϕ as a function of ϕ.
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DM400 model, 1-jet events
Fig. 6.5. DM400 model, 1-jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss , plot b (upper right)
shows the yield/PT[0] as a function of PT[0], plot c (middle left) shows the yield as a function of the Emiss/PT[0] ratio, plot d (middle
right) shows the yield as a function of Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the
yield/dϕ as a function of ϕ.
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DM50 model, 1 and/or 2 jet events
Fig. 6.6. DM50 model, 1 and/or 2 jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss , plot b (upper
right) shows the yield/PT[0] as a function of PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d
(middle right) shows the yield as a function of Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right)
shows the yield/dϕ as a function of ϕ.
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DM400 model, 1 and/or 2 jet events
Fig. 6.7. DM400 mode, 1 and/or 2 jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss , plot b (upper
right) shows the yield/PT[0] as a function of PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d
(middle right) shows the yield as a function of Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right)
shows the yield/dϕ as a function of ϕ.
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DM50 model, n-jet events
Fig. 6.8. DM50 model, n-jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss , plot b (upper right) shows
the yield/PT[0] as a function of PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right)
shows the yield as a function of Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the
yield/dϕ as a function of ϕ.
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DM400 model, n-jet events
Fig. 6.9. DM400 model, n-jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss , plot b (upper right)
shows the yield/PT[0] as a function of PT[0], plot c (middle left) shows the yield as a function of the Emiss/PT[0] ratio, plot d (middle
right) shows the yield as a function of Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the
yield/dϕ as a function of ϕ.
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6.3 Results analysis C
In analysis C the individual effect on the significance of the application of the 6 different selections
mentioned in table 5.6 is studied for all combinations of mediator and dark matter models, for 1-jet, 1
and/or 2, and n-jet events separately. Using the background data for all models the significance was
calculated, and the D5DM50 and D5DM400 model were chosen to be plotted in order to perform a shape
comparison between the 2 different dark matter models since the effect of the interaction trough different
mediators was already studied in analysis B.
The effect of application of the 6 selection criteria according to table 5.6 on the 6 variables discussed in
chapter 5.10 are shown for the D5DM50 simulation and D5DM400 in comparison to the background in fig.
5.11 till fig. 5.16 and fig. 5.17 till fig. 5.22 respectively.
The linecolors of the graphs of the different models plotted in analysis C are all colored by the
specification indicated in fig. 5.10. Here “BG” stands for the summed data of the background simulations
and “DM” stands for the data of the dark matter model studied in comparison. The plots are shown with
different colors for 1-jet, 1 and/or 2, and n-jet events as indicated respectively by “1-jet”, “2-jet” and “n-jet”.
Fig. 6.10. Linecolors of the graphs shown in analysis C
The calculated significance for events with n-jets is shown for DM50 and DM400 in respectively table 5.4a
and table 5.4b where the selection method that yields the largest significance is marked bold.
Significance
(cut)
Low cut
MM100
MM300
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
472.3
410.9
302.1
54.52
0.9018
0.02842
0.003097
0.0005866
0.003048
High cut
107.4
112
136.5
32.63
0.8313
0.02467
0.002522
0.0004777
0.002506
Assym Emiss
143.8
149.8
171.2
38.81
0.909
0.02714
0.002868
0.0005475
0.002855
Assym PT
120.8
120.9
144
29.06
0.6655
0.01997
0.002044
0.0003973
0.002033
Small Δφ
141.3
120.3
94.7
13.95
0.2166
0.006817
0.0007286
0.0001351
0.0007345
Large Δφ
350.3
303.9
214.4
42.83
0.7273
0.0231
0.002481
0.0004689
0.002408
Table 6.4a. Significance for DM50 models for n-jet events.
Significance
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM400
Low cut
10.88
34.73
0.8099
0.02342
0.002407
0.0004477
0.0023
High cut
7.51
23.1
0.7774
0.022
0.002168
0.0003966
0.002112
Assym Emiss
8.928
27.29
0.8581
0.02427
0.002427
0.0004528
0.002415
Assym PT
6.339
20.09
0.6241
0.01732
0.001732
0.0003188
0.001692
Small Δφ
2.722
9.222
0.1892
0.0018
0.0005718
0.0001063
0.0005636
Large Δφ
8.492
27.1
0.6545
0.01898
0.001912
0.0003593
0.00182
Table 6.4b. Significance for DM400 models, n-jet events.
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The significance for different mediator models using the dark matter DM50 and DM400 model for 1-jet or
2 jets events are shown in respectively table 6.5a and table 6.5b where the selection method that yields
the largest significance is marked bold.
Significance
MM100
MM300
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
Low cut
463.7
399.8
290.1
53.75
0.8878
0.02802
0.003049
0.0005782
0.00298
High cut
104.1
108.1
129.3
31.95
0.8376
0.02495
0.002531
0.0004776
0.002497
Assym Emiss
134.3
140.1
158.5
37.04
0.8964
0.02685
0.002831
0.000536
0.002781
Assym PT
115.4
116.4
136.9
29.4
0.6964
0.02091
0.002127
0.0004021
0.002098
Small Δφ
128.3
106
83.15
12.62
0.195
0.006043
0.0006318
0.0001241
0.000638
Large Δφ
350.1
303.8
214.3
42.77
0.7268
0.02306
0.002476
0.0004682
0.002403
Table 6.5a. Significance for DM50 models, 1 and/or 2 jet events.
Significance
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM400
Low cut
10.64
34.08
0.7999
0.0231
0.002364
0.0004401
0.002224
High cut
7.469
22.81
0.7818
0.02202
0.00216
0.0003982
0.002093
Assym Emiss
8.592
26.23
0.8418
0.02377
0.002397
0.0004428
0.002314
Assym PT
6.498
20.6
0.6509
0.01822
0.001798
0.000332
0.001742
Small Δφ
2.346
8.221
0.1713
0.004984
0.0005126
9.462e-05
0.0004762
Large Δφ
8.47
27.03
0.6533
0.01894
0.001909
0.0003585
0.001816
Table 6.5b. Significance for DM400 models, 1 and/or 2 jet events.
The significance for different mediator models using the dark matter DM50 and DM400 model for 1-jet
events are shown in respectively table 6.6a and table 6.6b.
Significance
MM100
MM300
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
Low cut
430
371.9
264.4
50.21
0.8311
0.02627
0.002847
0.0005354
0.002758
High cut
94.21
97.56
114.6
29.31
0.7997
0.02369
0.002403
0.000451
0.002347
Assym Emiss
113.1
119.6
135.1
43.84
0.8334
0.02473
0.002594
0.0004863
0.002505
Assym PT
103.3
104.5
121.5
28.29
0.712
0.02115
0.002157
0.0004051
0.002104
Small Δφ
96.68
81.67
62.62
9.699
0.1435
0.004473
0.0004756
9.05e-05
0.0004609
Large Δφ
349.2
302.1
211.7
42.14
0.7145
0.02267
0.002431
0.0004588
0.002361
Table 6.6a. Significance for DM50 models, 1-jet events.
Significance
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM400
Low cut
9.891
31.61
0.7477
0.02164
0.00219
0.0004098
0.00206
High cut
7.016
21.13
0.7409
0.02078
0.00205
0.0003778
0.001958
Assym Emiss
7.752
23.23
0.7721
0.02179
0.002184
0.000403
0.002088
Assym PT
6.46
20.04
0.6568
0.01848
0.001824
0.0003369
0.001741
Small Δφ
1.762
6.16
0.1304
0.003802
0.0003766
7.154e-05
0.0003553
Large Δφ
8.336
26.7
0.6404
0.01863
0.001873
0.0003525
0.001783
Table 6.6b. Significance for DM400 models, 1-jet events.
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The unscaled yield of the DM50 and DM400 models for different mediators is shown together with the
background yield in respectively table 6.7a and table 6.7b for events with 1-jet, 1 and/or 2 jets and n-jets
for the LowCut selection.
DM50
BG
MM100
MM300
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
n-jet
5.26E+10
1.09E+08
9.44E+07
6.94E+07
1.25E+07
2.07E+05
6.52E+03
7.10E+02
1.35E+02
6.99E+02
1/2-jet
3.88E+10
9.16E+07
7.89E+07
5.72E+07
1.06E+07
1.75E+05
5.52E+03
6.01E+02
1.14E+02
5.87E+02
1-jet
2.14E+10
8.49E+07
5.46E+07
3.88E+07
7.35E+06
1.22E+05
3.85E+03
4.17E+02
7.84E+01
4.04E+02
Table 6.7a. The yield of 1-jet, 1 and/or 2 jet (“1/2-jet”) and n-jet events for the background and DM50
simulations using only general selection criteria.
DM400
BG
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
n-jet
5.26E+10
2.50E+06
7.97E+06
1.86E+05
5.37E+03
5.52E+02
1.03E+02
5.28E+02
1/2-jet
3.88E+10
2.10E+06
6.71E+06
1.58E+05
4.55E+03
4.66E+02
8.67E+01
4.38E+02
1-jet
2.14E+10
1.45E+06
4.63E+06
1.09E+05
3.17E+03
3.21E+02
6.00E+01
3.02E+02
Table 6.7b. The yield of 1-jet, 1 and/or 2 jet (“1/2-jet”) and n-jet events for the background and DM50
simulations using only general selection criteria.
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In fig. 6.11 the effect of applying the Low cut selection criteria is shown for 1-jet, 1 and/or 2, and n-jet
events separately.
Fig. 6.11. Low cut selection effect on D5 DM50 vs. background model for 1-jet, 1 and/or 2, and n-jet
events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of PT[0],
plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of Njets,
plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.12 the 6 variables discussed in chapter 5.10 are shown for the D5 DM50 simulation in comparison
to the background, using the Asymmetric Emiss selection criteria according to table 5.6. For each simulation
the 1-jet, 1 and/or 2, and n-jet events are shown separately.
Fig. 6.12. Asymmetric Emiss selection effect on D5 DM50 vs. background model for 1-jet, 1 and/or 2, and njet events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of
PT[0], plot c (middle left) shows the yield as a function of the Emiss/PT[0] ratio, plot d (middle right) shows the yield as a function of
Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.13 the 6 variables discussed in chapter 5.10 are shown for the D5 DM50 simulation in comparison
to the background, using the Asymmetric PT selection criteria according to table 5.6. For each simulation
the 1-jet, 1 and/or 2, and n-jet events are shown separately.
Fig. 6.13. Asymmetric PT selection effect on D5 DM50 vs. background model for 1-jet, 1 and/or 2, and njet events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of
PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of
Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.14 the 6 variables discussed in chapter 5.10 are shown for the D5 DM50 simulation in comparison
to the background, using the High cut selection criteria according to table 5.6. For each simulation the 1jet, 1 and/or 2, and n-jet events are shown separately.
Fig. 6.14. High cut selection effect on D5 DM50 vs. background model for 1, 1 and/or 2, and n-jet events.
Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of PT[0], plot c
(middle left) shows the yield as a function of the Emiss/PT[0] ratio, plot d (middle right) shows the yield as a function of Njets, plot e
(lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.15 the 6 variables discussed in chapter 5.10 are shown for the D5 DM50 simulation in comparison
to the background, using the Small Δφ selection criteria according to table 5.6. For each simulation the 1jet, 1 and/or 2, and n-jet events are shown separately. The variable Δφ is shown in the region 0.5 < Δφ <
1.0 using 50 bins.
Fig. 6.15. Small Δφ selection effect on D5 DM50 vs. background model for 1-jet, 1 and/or 2, and n-jet
events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of PT[0],
plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of Njets,
plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.16 the 6 variables discussed in chapter 5.10 are shown for the D5 DM50 simulation in comparison
to the background, using the Large Δφ selection criteria according to table 5.6. For each simulation the 1jet, 1 and/or 2, and n-jet events are shown separately. The variable Δφ is shown in the region π-0.5 < Δφ
< π using 50 bins.
Fig. 6.16. Large Δφ selection effect on D5 DM50 vs. background model for 1-jet, 1 and/or 2, and n-jet
events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of PT[0],
plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of Njets,
plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.17 till fig. 5.20 the effect of the 6 selection criteria according to table 5.6 on the 6 variables
discussed in chapter 5.10 are shown for the D5 DM400 simulation in comparison to the background.
In fig. 6.17 the effect of applying the Low cut selection criteria is shown for 1-jet, 1 and/or 2, and n-jet
events separately.
Fig. 6.17. Low cut selection effect on D5 DM400 vs. background model for 1-jet, 1 and/or 2, and n-jet
events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of PT[0],
plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of Njets,
plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.18 the 6 variables discussed in chapter 5.10 are shown for the D5 DM400 simulation in
comparison to the background, using the Asymmetric Emiss selection criteria according to table 5.6. For
each simulation the 1-jet, 1 and/or 2, and n-jet events are shown separately.
Fig. 6.18. Asymmetric Emiss selection effect on D5 DM400 vs. background model for 1-jet, 1 and/or 2, and
n-jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of
PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of
Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.19 the 6 variables discussed in chapter 5.10 are shown for the D5 DM400 simulation in
comparison to the background, using the Asymmetric PT selection criteria according to table 5.6. For each
simulation the 1-jet, 1 and/or 2, and n-jet events are shown separately.
Fig. 6.19. Asymmetric PT selection effect on D5 DM400 vs. background model for 1-jet, 1 and/or 2, and njet events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of
PT[0], plot c (middle left) shows the yield as a function of the Emiss/PT[0] ratio, plot d (middle right) shows the yield as a function of
Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.20 the 6 variables discussed in chapter 5.10 are shown for the D5 DM400 simulation in
comparison to the background, using the High cut selection criteria according to table 5.6. For each
simulation the 1-jet, 1 and/or 2, and n-jet events are shown separately.
Fig. 6.20. High cut selection effect on D5 DM400 vs. background model for 1-jet, 1 and/or 2, and n-jet
events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of PT[0],
plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of Njets,
plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.21 the 6 variables are shown for the D5 DM400 simulation in comparison to the background, using
the Small Δφ selection criteria according to table 5.6. For each simulation the 1-jet, 1 and/or 2, and n-jet
events are shown separately. The variable Δφ is shown in the region 0.5 < Δφ < 1.0 using 50 bins.
Fig. 6.21. Small Δφ selection effect on D5 DM400 vs. background model for 1-jet, 1 and/or 2, and n-jet
events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of PT[0],
plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of Njets,
plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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In fig. 6.22 the 6 variables are shown for the D5 DM400 simulation in comparison to the background, using
the Large Δφ selection criteria according to table 5.6. For each simulation the 1-jet, 1 and/or 2, and n-jet
events are shown separately. The variable Δφ is shown in the region π-0.5 < Δφ < π using 50 bins.
Fig. 6.22. Large Δφ selection effect on D5 DM400 vs. background model for 1-jet, 1 and/or 2, and n-jet
events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of PT[0],
plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of Njets,
plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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6.4 Results analysis D
The influence of different selections on the significance is studied in a more generic way by investigating
combinations of applying lower limits on the variables Emiss and PT [0]. For 1-jet, 1 and/or 2, and n-jet
events, for all combinations of dark matter models and mediator models, significance was calculated for all
combinations of possible cuts on the variables Emiss and PT [0] within the limits of [5.19] and [5.20] using
increases with a step size of 50 GeV. The selection with the largest significance will be labelled as ‘Best’.
Together with the according lower limit labelled Best selection and denoted as (Emiss >, PT [0] >) where
Emiss> indicates the lower limit set on variable Emiss and PT[0]> indicates the lower limit set an variable PT
[0], the results are shown in table 6.8 till table 6.10.
The process was repeated with the extra condition [5.21] selecting only events with a large smallest angle
between missing energy and PT[0]. The according lower limit is labelled as ‘best + large Δφ selection” and
denoted similar as Best selection with the according significance is labelled as ‘best + large Δφ” in a
similar fashion as Best. The results can be found in table 6.8 till table 6.10 where the significance is
marked bold if the optimal selection has increased significance with respect to the significance found by
any method from analysis C, and significance was not marked bold if the optimal selection leads to the
same significance as any method used in analysis C.
The maximum significance and according selection criteria for different models in n - jet events are shown
in table 6.8a and 6.8b for DM50 and DM400 respectively.
Best
MM100
MM300
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
472.3
410.9
302.1
54.52
0.9638
0.02973
0.003186
0.0006096
0.003191
(400,300)
(400,300)
(400,300)
(400,300)
Best selection
(300,300)
(300,300)
(300,300)
(300,300)
(450,300)
Best+large Δφ
350.3
303.9
214.4
42.83
0.7454
0.02321
0.002481
0.0004689
0.002421
Best+large Δφ
selection
(300,300)
(300,300)
(300,300)
(300,300)
(350,350)
(300,300)
(300,300)
(400,300)
(350,300)
Table 6.8a. Optimised significance and according selection criteria for DM50 models and n-jet events.
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM400
Best
11.07
34.97
0.8879
0.02521
0.0026
0.0004804
0.002547
Best selection
(350,300)
(350,300)
(500,300)
(450,300)
(400,300)
(450,300)
(450,300)
Best+large Δφ
8.492
27.1
0.6809
0.01955
0.001958
0.0003631
0.001888
Best+large Δφ
selection
(300,300)
(300,300)
(500,300)
(400,400)
(400,350)
(400,300)
(400,300)
Table 6.8b. Optimised significance and according selection criteria for DM400 models and n-jet events.
The maximum significance and according selection criteria for different models in 1 and/or 2 - jet events
are shown in table 6.9a and 6.9b for DM50 and DM400 respectively.
MM100
MM300
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
Best
463.7
399.8
290.1
53.75
0.9442
0.02919
0.003117
0.000596
0.003107
Best selection
(300,300)
(300,300)
(300,300)
(300,300)
(450,300)
(400,300)
(400,300)
(400,300)
(400,300)
Best+large Δφ
350.1
303.8
214.3
42.77
0.7268
0.02306
0.002476
0.0004682
0.002403
Best+large Δφ
selection
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
Table 6.9a. Optimised significance and according selection criteria for DM50 models for 1 and/or 2 jet
events.
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MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM400
Best
10.79
34.17
0.8712
0.02472
0.002537
0.0004688
0.002441
Best selection
(350,300)
(350,300)
(500,300)
(400,300)
(500,300)
(450,300)
(450,300)
Best+large Δφ
8.47
27.03
0.6533
0.01894
0.001909
0.0003585
0.001816
Best+large Δφ
selection
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
Table 6.9b. Optimised significance and according selection criteria for DM400 models for 1 and/or 2 jet
events.
The maximum significance and according selection criteria for different models in 1 jet events are shown
in table 6.10a and 6.10b for DM50 and DM400 respectively.
MM100
MM300
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM50
Best
430
371.9
264.4
50.21
0.8753
0.0271
0.00288
0.0005457
0.002825
Best selection
(300,300)
(300,300)
(300,300)
(300,300)
(450,300)
(400,300)
(350,300)
(350,300)
(400,300)
Best+large Δφ
349.2
302.1
211.7
42.14
0.7145
0.02267
0.002431
0.0004588
0.002361
Best+large Δφ
selection
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
Table 6.10a. Optimised significance and according selection criteria for DM50 models for 1-jet events.
MM500
MM1000
MM3000
MM6000
MM10000
MM15000
D5 DM400
Best
9.934
31.62
0.8035
0.02284
0.002308
0.0004306
0.002211
Best selection
(350,300)
(300,300)
(500,300)
(400,300)
(400,300)
(450,300)
(450,300)
Best+large Δφ
8.336
26.7
0.6404
0.01863
0.001873
0.0003525
0.001783
Best+large Δφ
selection
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
(300,300)
Table 6.10b. Optimised significance and according selection criteria for DM400 models for 1-jet events.
The condition of large Δφ shows no further increase in significance (see chapter 7.4) and therefore no
further plots are shown. Next 3D plots were made to show the significance as a function of the lower
selection limit on the variables Emiss and PT [0], and to confirm the maximum is in agreement with the
optimised result labelled as Best. The 3D plots and the 6 variables discussed in chapter 5.10 are shown
for all the combinations of dark matter models DM50 and DM400 and massive mediator models D5,
MM3000 and MM500 with respect to background in fig. 6.24 till 6.46. The linecolors of the graphs of the
different models studied in analysis D are all shown by the specification indicated in fig. 6.23. Here “BG”
stands for the summed data of the background simulations and “DM” stands for the data of the dark
matter model studied in comparison. The plots are shown with different colors for 1-jet, 1 and/or 2, and njet events as indicated respectively by “1-jet”, “2-jet” and “n-jet”.
Fig. 6.23. Linecolors of the different graphs in analysis D.
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Significance as a function of the selection criteria for the D5 DM50 model for n-jet events is shown in fig.
6.24. The optimised selection maximizes the significance and is found to be the condition Emiss > 400 GeV
and PT [0] > 300 GeV.
Fig. 6.24. Significance for D5 DM50, n-jet events.
Significance as a function of the selection criteria for the D5 DM50 model for 1 and/or 2 jet events is
shown in fig. 6.25. The optimised selection maximizes the significance and is found to be the following
condition; Emiss > 400 GeV and PT [0] > 300 GeV.
Fig. 6.25. Significance for D5 DM50, 1 and/or 2 jet events.
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Significance as a function of the selection criteria for the D5 DM50 model for 1-jet events is shown in fig.
6.26. The optimised selection maximizes the significance and is found to be the condition; Emiss > 400 GeV
and PT [0] > 300 GeV.
Fig. 6.26. Significance for D5 DM50, 1-jet events.
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For the D5 DM50 model in events with the 1-jet, 1 and/or 2 and n-jets the optimised selection criteria are
for all multiplicities Emiss > 400 GeV and PT [0] > 300 GeV. In fig. 6.27 the effect of this optimal selection is
shown for the 6 variables discussed in chapter 5.10.
Fig. 6.27. Optimised selection applied to the D5 DM50 model and background for 1, 1 and/or 2, and n-jet
events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of PT[0],
plot c (middle left) shows the yield as a function of the Emiss/PT[0] ratio, plot d (middle right) shows the yield as a function of Njets,
plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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Next significance as a function of the selection criteria for the D5 DM400 model for n-jet events is shown
in fig. 6.28. The optimised selection maximizes the significance and is found to be the condition Emiss > 450
GeV and PT [0] > 300 GeV.
Fig. 6.28. Significance for D5 DM50, for n-jet events.
Significance as a function of the selection criteria for the D5 DM400 model for 1 and/or 2 jet events is
shown in fig. 6.29. The optimised selection maximizes the significance and is found to be the following
condition; Emiss > 450 GeV and PT [0] > 300 GeV.
Fig. 6.29. Significance for D5 DM400 for 1 and/or 2 jet events.
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Significance as a function of the selection criteria for the D5 DM400 model for 1-jet events is shown in fig.
6.30. The optimised selection maximizes the significance and is found to be the condition; Emiss > 450 GeV
and PT [0] > 300 GeV.
Fig. 6.30. Significance for D5 DM400 for 1-jet events.
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For the D5 DM400 model in events with the 1-jet, 1 and/or 2 and n-jets the optimised selection criteria are
equal, Emiss > 450 GeV and PT [0] > 300 GeV. In fig. 6.31 the effect of this selection which is equal to the
Assym Emiss selection is shown for the 6 variables discussed in chapter 5.10.
Fig. 6.31 Optimised selection applied to the D5 DM400 model and background for 1-jet, 1 and/or 2, and njet events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of
PT[0], plot c (middle left) shows the yield as a function of the Emiss/PT[0] ratio, plot d (middle right) shows the yield as a function of
Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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Then the significance as a function of the selection criteria for the DM50 MM3000 model for n-jet events is
shown in fig. 6.32. The optimised selection maximizes the significance and is found to be the condition
Emiss > 450 GeV and PT [0] > 300 GeV.
Fig. 6.32. Significance for DM50 MM3000 for n jet events.
The significance as a function of the selection criteria for the DM50 MM3000 model for 1 and/or 2 jet
events is shown in fig. 6.33. The optimised selection maximizes the significance and is found to be the
condition Emiss > 450 GeV and PT [0] > 300 GeV.
Fig. 6.33. Significance for DM50 MM3000 for 1 and/or 2 jet events.
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The significance as a function of the selection criteria for the DM50 MM3000 model for 1-jet events is
shown in fig. 6.34. The optimised selection maximizes the significance and is found to be the condition
Emiss > 450 GeV and PT [0] > 300 GeV.
Fig. 6.34. Significance for DM50 MM3000 for 1-jet events.
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For the DM50 MM3000 model in events with the 1-jet, 1 and/or 2 and n-jets the optimised selection is
equal, Emiss > 450 GeV and PT [0] > 300 GeV. In fig. 6.35 the effect of this selection is shown for the 6
variables discussed in chapter 5.10.
Fig. 6.35. Optimised selection applied to the DM50 MM3000 model and background for 1-jet, 1 and/or 2,
and n-jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a
function of PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a
function of Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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Now the significance as a function of the selection criteria for the DM400 MM3000 model for n-jet events
is shown in fig. 6.36. The optimised selection maximizes the significance and is found to be the condition
Emiss > 500 GeV and PT [0] > 300 GeV.
Fig. 6.36. Significance for DM40 MM3000 for n jet events.
The significance as a function of the selection criteria for the DM400 MM3000 model for 1 and/or 2 jet
events is shown in fig. 6.37. The optimised selection maximizes the significance and is found to be the
condition Emiss > 500 GeV and PT [0] > 300 GeV.
Fig. 6.37. Significance for DM400 MM3000 for 1 and/or 2 jet events.
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The significance as a function of the selection criteria for the DM400 MM3000 model for 1-jet events is
shown in fig. 6.38. The optimised selection maximizes the significance and is found to be the condition
Emiss > 500 GeV and PT [0] > 300 GeV.
Fig. 6.38. Significance for DM400 MM3000 for 1-jet events.
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For the DM400 MM3000 model in events with the 1-jet, 1 and/or 2 and n-jets the optimised selection is
equal, Emiss > 500 GeV and PT [0] > 300 GeV. In fig. 6.39 the effect of this selection is shown for the 6
variables discussed in chapter 5.10.
Fig. 6.39. Optimised selection applied to the DM400 MM3000 model and background for 1-jet, 1 and/or 2,
and n-jet events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a
function of PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a
function of Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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Now the significance as a function of the selection criteria for the DM50 MM500 model for n-jet events is
shown in fig. 6.40. The optimised selection maximizes the significance and is found to be the condition
Emiss > 450 GeV and PT [0] > 300 GeV.
Fig. 6.40. Significance for DM50 MM500 for n-jet events.
The significance as a function of the selection criteria for the DM50 MM500 model for 1 and/or 2 jet
events is shown in fig. 6.41. The optimised selection maximizes the significance and is found to be the
condition Emiss > 450 GeV and PT [0] > 300 GeV.
Fig. 6.41. Significance for DM50 MM500 for 1 and/or 2 jet events.
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The significance as a function of the selection criteria for the DM50 MM500 model for 1-jet events is
shown in fig. 6.42. The optimised selection maximizes the significance and is found to be the condition
Emiss > 450 GeV and PT [0] > 300 GeV.
Fig. 6.42. Significance for DM50 MM500 for 1-jet events.
For the DM50 MM500 model in events with the 1, 1 and/or 2 and n-jets the optimised selection is equal to
the general selection discussed in chapter 5.9, Emiss > 300 GeV and PT [0] > 300 GeV. In Fig. 6.4, 6.6 and
6.8 the effect of this selection (LowCut) is shown for the 6 variables discussed in chapter 5.10.
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Now the significance as a function of the selection criteria for the DM400 MM500 model for n-jet events is
shown in fig. 6.44. The optimised selection maximizes the significance and is found to be the condition
Emiss > 350 GeV and PT [0] > 300 GeV.
Fig. 6.44. Significance for DM400 MM500 for n-jet events.
The significance as a function of the selection criteria for the DM400 MM500 model for 1 and/or 2 jet
events is shown in fig. 6.45. The optimised selection maximizes the significance and is found to be the
condition Emiss > 350 GeV and PT [0] > 300 GeV.
Fig. 6.45. Significance for DM400 MM500 for 1 and/or 2 jet events.
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Last the significance as a function of the selection criteria for the DM400 MM500 model for 1-jet events is
shown in fig. 6.46. The optimised selection maximizes the significance and is found to be the condition
Emiss > 350 GeV and PT [0] > 300 GeV.
Fig. 6.46. Significance for DM400 MM500 for 1-jet events.
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For the DM400 MM500 model in events with the 1 and/or 2 and n-jets the optimised selection are Emiss >
350 GeV and PT [0] > 300 GeV. Fig. 6.47 is the last plot showing the effect of this for the 6 variables
discussed in chapter 5.10. (For 1-jet events the optimised selection is E miss > 300 GeV and PT [0] > 300
GeV. (The plot of the variables for this case can be found in fig. 6.5).
Fig. 6.47. Optimised selection applied to the DM400 MM500 model and background for 1 and/or 2, and njet events. Plot a (upper left) shows the yield/Emiss as a function Emiss, plot b (upper right) shows the yield/PT[0] as a function of
PT[0], plot c (middle left) shows the yield as a function of the E miss/PT[0] ratio, plot d (middle right) shows the yield as a function of
Njets, plot e (lower left) shows the yield/η as a function of η and plot f (lower right) shows the yield/dϕ as a function of ϕ.
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Chapter 7. Conclusion
7.1 Conclusion analysis A
From the results of the first analysis A we conclude by comparison of table 6.1 and table 6.2
that the ratios of the standard model contributions simulated in this study are in agreement
with the previous ATLAS WIMP sensitivity study, therefore validating the analysis performed
in this thesis. For 1-jet events the background contributions deviate at most 4.0% when
considering the 4 main background processes and omitting contributions from other small
background processes as discussed in chapter 5.5. Therefore the standard model
background is studied only by the processes W→μν, W→ τν, W→eν as well as Z→νν.
In the plot in fig. 6.2a showing the yield as a function of Emiss the background contributions
have a similar decreasing declining dependence of the yield as a function of Emiss and PT[0],
and their ratios differ only by a factor depending on the cross section. This is in agreement
with the expectation that for events with an increasing momentum transfer q2 and thus an
increasingly energetic intermediate state s2 the cross section of producing a final state with a
large missing energy signature decreases according to formula 5.7. Since missing energy is
balanced against the total momentum of the jets also the yield of events with increasing PT[0]
decreases. It is clearly shown in these unscaled plots how the ratio of the W→eν process
contributes about 5.6% of the sum of all background contribution, W→μν for 7.7% and W→τν
18.7%, while the dominant contribution comes from the process Z→νν with 68.0% (for 1-jet
events).
We expect due to momentum conservation that for single jet events the created jet and
missing energy have the same but opposite momentum in the center of mass frame, such
that they have a balanced but back-to-back topology in the transverse plane, see fig. 7.1.
Therefore the angle dφ between the only jet produced and missing energy will be peaked at
dφ ~ π (the creation of low momentum secondary jets that didn’t pass the treshold of P T = 30
GeV can result in value dφ < π). For events with multiple jets the minimum angle for any
produced jet and missing energy will be smaller such that π/2 < dφ < π, see fig. 7.2. This can
be seen from the data in the unscaled Δφ plot in fig. 6.2f as for small dφ the dominant
contribution to the event yield comes from the dataset with n-jet events followed by the 1-or-2
jet events, while the 1-jet event contribution is significantly lower. However for large dφ the 1
jet events yield the main contribution towards dφ ~ π.
Fig 7.1. For 1-jet events the jet pT[0] and
missing energy Emiss from the neutrino’s
(ν) are balanced back-to-back such that in
the transverse plane dφ = π.
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Fig 7.2 (left). For multi jet events the leading jet pT[0] and missing energy Emiss from the
neutrinos (ν) are balanced back-to-back but will in general have π/2 < dφ < π.
Fig 7.3 (right). A tau neutrino ντ carries away missing energy Emiss that is balanced against
the net transverse momentum
which is the momentum of the 2 detected initial state
jets pT[0], pT[2] and the momentum pT[1] of the jet created by the tau hadronic decay (jetτ).
Fig. 6.2c shows deviant dependence of the yield of the ratio E miss / PT[0] for the process
W→τν for any number of jets as the yield decreases in the region 0 < Emiss / PT[0] < 1 or
Emiss < PT[0]. This can be understood as tau hadronic decay will produce an invisible neutrino
leaving missing energy as signature, and a pion that will hadronize to a jet, both in opposed
direction of the initial state jets. Therefore missing energy Emiss will in general be smaller than
the momentum of the leading jet PT[0], see fig. 7.3.
The plot where the jet multiplicity is displayed (fig. 6.2d) shows the selection criteria set on
the jet multiplicity njet are in agreement with the obtained data. We see that the events that
have been selected to have 1 jet, dominate the njet = 1 bin, that the dataset with events
selected with the requirement of having 1 and/or 2 jet events lead the njet = 2 bin and the bins
with higher multiplicity yield only data from the selection with no requirement on njet.
The graph showing the pseudorapidity η (fig. 6.2e) shows that the production of jets in the
central region of the detector near |η| = 0 is most likely as it has the largest event yield. Since
differences in pseudorapidity Δη are Lorentz invariant a Lorentz boost can be applied to the
data by translating the graph along in the η-axis such that the directional distribution of the
jets can be studied from the center of mass frame of the colliding partons. The production of
jets is assumed to be constant as a function of pseudorapidity, however particles with large
pseudorapidity produced in the beam direction will not be detected even in the idealised
detector studied in the truth simulated data. Therefore the smooth and central directional
distribution around |η| = 0 of the jets is as expected.
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7.2 Conclusion analysis B
From fig. 6.4a till fig. 6.9a we conclude that for all dark matter models the specific mediator
model influences the slope of the distribution of the yield as a function of missing energy. The
distribution of the D5 model which describes the interaction trough an intermediate state that
can’t be resolved resembles that of the most massive mediator models mz’ ≥ 10 TeV as we
see the slopes of the most massive mediator models and D5 model overlap. The data of the
absolute yield for DM50 in table 6.3a and DM400 in table 6.3b shows that the yield is
smallest for the MM150000 model, followed by the MM10000 and D5 model which have
similar yields within 1.6% and 4.6% for DM50 and DM400 n-jet models respectively, and
increasing yield for models with smaller mediator mass with the only exception of the
DM400MM500. This is expected as the D5 simulations use an EFT suppression scale of Λ =
10 TeV and EFT can be seen as the high energy limit of the simplified interaction trough a
massive mediator. Furthermore for lighter mediators with mz’ < 10 TeV resonant
enhancement leads to the on shell production of the mediator, such that the interaction has a
renormalizable 2 body final state instead of 3 body state. Stronger resonant enhancement
appears for mediators with successfully lower mass as expected from formula [5.7]. The
only exception is the DM400MM500 model for which the yield has decreased with respect to
the DM400MM1000 model despite the smaller mediator mass. This can be interpreted since
for mz’ < 2 mx the mediator is not able to decay to a dark matter pair, so the only contribution
comes from a suppressed offshell produced mediator. We also conclude from table 6.3 that
the production of DM50 leads to slightly higher yields than DM400 for all mediator models
(with the exception of MM500). Table 6.3a and table 6.3b furthermore show that events with
lower multiplicity lead to a smaller yield. But as the yield of the background events will
decrease as well and only the calculation of the significance in analysis C and D can answer
the question how the detection sensitivity depends on multiplicity.
In the plot where jet multiplicity is shown it is clearly visible that the selection requirement on
the number of jets is in consensus with the observed multiplicity after placing the cut. The
graph showing pseudorapidity shows a smooth directional distribution with a maximum
around |η| = 0) as expected.
In fig. 6.3c we can see how Emiss / pT[0] peaks around Emiss / pT[0] ~ 1 while most events
have Emiss > pT[0]. This can be interpreted as events with a multiplicity njet > 1 will have a
signature in which the momentum of created jets is in same direction and opposed to the
missing energy from the momentum carried away by the produced dark matter pair (see fig.
7.4a), a conclusion supported by the plot showing dφ is maximal in the region π/2 < dφ < π.
Single jet events will have a balanced but exactly back-to-back topology for missing energy
and the created jet such that Emiss ~ pT[0]. The angle dφ between the only jet produced and
missing energy will be peaked around dφ ~ π which can be seen in fig. 7.4b.
However in some events jets are created opposed such that Emiss < pT[0] or Emiss / pT[0] < 1
and dφ < π/2, see fig. 7.4c. These events can be found in the region Emiss < pT[0].
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Fig 7.4a (upper). For 1-jet events the jet pT[0] and missing energy Emiss are balanced back-toback with Emiss ~ pT[0] such that in the transverse plane dφ ~ π.
Fig 7.4b (middle). For multi jet events the leading jet pT[0] and missing energy Emiss are
generally produced back-to-back and will have dφ < π such that Emiss > pT[0].
Fig 7.4c (lower). For some multi jet events the leading jet pT[0] and missing energy Emiss are
not produced back-to-back and will have dφ < π/2 such that Emiss < pT[0].
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7.3 Conclusion analysis C
Analysis C consistently with analysis B shows that the significance of the n-jet, 1 and/or 2jet
and 1-jet events respectively in table 6.4, table 6.5 and table 6.6, and the yield shown in table
6.7, is smallest for the MM150000 model followed by the MM10000 and D5 model. The yield
increases for models with smaller mediator mass with the only exception of the
DM400MM500, and since the yields of the different models are compared to the same
background the significance follows the same pattern. This in agreement with the
expectations as EFT is the high energy limit of the interaction trough a massive mediator and
resonant enhancement appears for mediators with successfully lower mass, while for models
with mz’ < 2 mx the mediator is not able to decay on-shell to a dark matter pair, so the only
contribution comes from an suppressed off-shell produced mediator. We also conclude that
the production of DM50 leads to higher yields than DM400 for all mediator models with the
exception of MM500, and the background yield is equal such that the significance follows the
same pattern.
Although the DM signal spectrum is harder than the background spectrum harder selection
cuts don’t always improve signal to background ratios. The selection methods High cut, Assym
PT, Small Δφ, Large Δφ don’t increase significance in 100% of the cases and clearly they are
constraints that should not be generally applied when searching for a possible signal in this
study and the signature events that are favoured in such selections are signatures that don’t
match the expected dark matter production events. From the data of the models with DM50
(table 6.4a, 6.5a and 6.6a) we conclude that for all mediator models except for MM3000
applying a tighter selection method than the method ‘Low Cut’ doesn’t enhance significance
in 100% of the cases. However significance is enhanced slightly for the DM50MM3000
model only when the Assym Emiss selection method is used by 4.1%, 0.97% and 0.27% in
respectively n-jet, 1 and/or 2 and 1 jet events. From the data of the models with DM400
(table 6.4b, 6.5b and 6.6b) we conclude that applying a tighter selection method than the
method Low Cut can enhance significance but only when the Assym Emiss selection method
is used. For models with mZ’ > 3000GeV in 100% of the n-jet and 1 and/or 2 jet events
significance is enhanced by maximal 6.0%. For 1 jet events significance is increased for the
MM3000, MM6000 and D5 mediator models by respectively 3.3%, 0.69% and 1.4%. Since
the Assym Emiss selection can increase significance is some cases it is a constraint that is
interesting to study in more detail (which is done in analysis D).
Selections using Assym Emiss will in some scenarios increase significance over LowCut and
thus it is interesting to check the signature of the events that are selected, which can be seen
by comparing fig 6.11a with fig. 6.12a or fig 6.17a with fig. 6.18a.
For pT[0] < 600 GeV the Assym Emiss cut will select events with Emiss > 600 GeV that have a
dominant jet with 300 GeV < pT[0] < 600 GeV pointing in the region opposed to missing
energy, as well as any smaller secondary jets which transverse momentum summed is
exactly the difference between PT[0] and missing energy and therefore also preferably in the
opposed direction of missing energy. The ratio Emiss / pT[0] will be consequently larger than 1,
and in fig. 6.12c or fig. 6.18c the Emiss / pT[0] plot it can be seen how the region Emiss / pT[0] >
1 has a larger yield than the region Emiss / pT[0] < 1. This type of event is illustrated in fig.
7.3b.
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For single jet events the secondary jets can account for the difference in momentum between
PT[0] and missing energy if, although with small probability, they are jets in the opposed
direction of missing energy that were not passing the 30 GeV threshold. In fig. 6.12b or 6.18b
this can be seen as for pT[0] < 600 GeV single jet events have the smallest yield. For multiple
jet events the condition that the sum of momentum of secondary jets is mainly in the direction
opposed to missing energy, leaves the possibility that small jets could be created opposed to
pT[0] but their momentum is compensated for by other secondary jets. This feature can nicely
be seen in the dφ plot (Fig. 6.12f) where it is shown how the single jet events peak around
dφ ~ π, and 1 and/or 2 jet events and n-jet events peak also peak around dφ ~ π but with an
larger width. When comparing the dφ plot of fig. 6.12f to fig. 6.11f, fig. 6.13f and fig. 6.14f it is
clearly showed how events with jets in the same direction as missing energy (0.5 < dφ < π/2)
are not selected by Assym Emiss. This can also be seen by comparing fig. 6.18f to fig. 6.17f,
fig. 6.19f and fig. 6.20f. Comparing the dφ plot of fig. 6.11f with fig. 6.14f we can see how
significance is decreased when the Large Δφ selection with the condition π - 0.5 < dφ < π is
used in comparison with Assym Emiss. This shows Assym Emiss is a more suitable method for
selecting events with the typical signature than only using a selection method based on the
minimum angle dφ as in Large Δφ.
For pT[0] > 600 GeV the Assym Emiss cut will select events that have Emiss > 600 GeV and a
leading jet with large transverse momentum pT[0] > 600 GeV pointing in the region opposed
to missing energy, as well as any smaller secondary jets which transverse momentum
summed is exactly the difference between p T[0] and missing energy and therefore preferably
in the direction of missing energy, see fig. 7.3c. This is not directly the typical signature we
would expect and therefore the yield in the range Emiss / pT[0] < 1 will be consequently smaller
than the yield in the region Emiss / pT[0] > 1. yield which can be seen in fig. 6.12c or 6.18c.
For single jet events these secondary jets should account for the difference in momentum
between pT[0] and missing energy. This could be jets with a total momentum pointing in the
direction of missing energy with an individual momentum that is not passing the 30 GeV
threshold. For multiple jet events the condition that the sum of momentum of secondary jets
is mainly in the direction towards missing energy, leaves the possibility open that secondary
jets are being be created from which at least one jet was in the direction of missing energy.
This feature can also be seen in the dφ plot of fig. 6.12f or 6.18f where for 0.5< dφ < π/2 the
yield is smaller than for dφ > π/2, but is dominated by multiple jet events and not single jet
events.
The fact that application of the Assym Emiss selection raises significance can also be seen in
the Emiss plot of fig. 6.11a or fig 6.17a. Closer examination of the region 300 GeV < Emiss <
400 GeV shows that in this region the signal yield is decreasing for decreasing Emiss,
opposed to the region Emiss > 400 GeV where signal yield is increasing for decreasing Emiss.
Consequently applying a selection with a tuned lower bound for Emiss could increase
significance further, if the factor with which the signal yield decreases is smaller than the
square root of the factor with which the background yield decreases. Whether this is the case
depends on the specific model used, which will be systematically studied in analysis D.
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Selections with a High cut favour the production of a energetic dark matter pair with large
Emiss balanced opposed to a dominant energetic jet with a large transverse momentum pT[0].
This is supported by the result that application of the selection method High cut decreases
significance least for events with single jet multiplicity. The selection method High cut
decreases significance in 100% of the cases and the signature event might consequently
have multiple jets besides a the dominant jet. This statement is supported by table 6.7
showing that in all cases the yield of events with a multiplicity njet = n is larger than for njet ≤ 2
which is subsequently larger than the yield of events with multiplicity njet = 1, which would not
necessary be true if the signature would be dominated by single jet production only.
Selections with Assym PT will remove events with a jet that have pT[0] < 600 GeV. Events
passing the selection can have multiple jets being created in the same direction with p T[0] >
600 GeV and therefore Emiss > 600 GeV. It is also possible multiple jets with p T[0] > 600 GeV
are created in opposed direction of each other such that Emiss < 600 GeV. Furthermore single
jet events with a large momentum pT[0] > 600 GeV and consequently Emiss > 600 GeV will
pass the selection.
First we note that studying the pT[0] graphs of fig. 6.11b (or 6.17b) in comparison with 6.13b
(or 6.18b) we confirm that applying a stronger selection on pT[0] will not enhance significance
as a result of the number of signal events is being reduced by a factor larger than the square
of the factor with which the background events are being reduced. From the Emiss graphs of
6.11a (or 6.17a) in comparison to 6.13a (or 6.19a) we note that for Emiss < 600 GeV the
selection has decreased the yield of signal events dramatically in comparison with the
background, since the condition Emiss < 600 GeV and pT[0] > 600 GeV would imply opposed
jets being created which is as expected not the typical signature event of a dark matter pair
produced opposed to hadronized jets.
For the region Emiss > 600 GeV the selection has decreased significance as the signal to
background ratio has decreased, although not as dramatically as in the region Emiss < 600
GeV. Therefore events with high momentum jets in the same direction as missing energy
could be possible but are not a significant part of the typical signature event and are not
effectively selected by the selection method Assym PT.
Last we note from the Emiss / pT[0] plot of fig 6.11c and 6.13c (or 6.17c with respect to 6.18c)
that the yield for background and signal single jet events with Emiss / pT[0] = 1 is 104 events
whereas after the selection the yield is respectively 800 and 10 3 events. Although the pure
signal to background ratio has increased and background yield has decreased, significance
has decreased from 71 to 24 by the reduced number of signal yield available. Therefore a
single jet event with Emiss / pT[0] =1 is not efficiently selected by the method Assym PT. We
conclude that the Assym PT selection method decreases significance in 100% of the cases
and the signature event will most likely not have multiple jets created in opposed direction to
each other with pT[0] > 600 GeV.
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The cut for small dφ Small Δφ uses the selection criteria 0.5 < dφ < 1.0 such that events
with a jet in the same direction as missing energy are selected. In 96% of all 48 combinations
of DM and the massive mediator models studied from table 6.4, 6.5 ad 6.6 it is shown that
the Small Δφ cut is the selection method with lowest significance in comparison to other
selection methods and in 4% it is the method with second lowest significance. Form these
results we conclude that the signature event has a preference for creating jets not in the
direction of missing energy.
Application of the selection Large Δφ implies the condition π - 0.5 < dφ < π such that events
with a no jets in the same direction as missing energy are selected. From the Emiss / pT[0] plot
in fig 6.16c in comparison with fig 6.11c (or 6.22c with respect to 6.17c) we can see how
signal events with Emiss / pT[0] < 1 are not selected, meaning events with any jets having a
component of its momentum opposed to pT[0] are not selected.
From table 6.4a till 6.6a and 6.4b till 6.6b it can be concluded that for respectively 92% of the
DM50 models and 50% of the DM400 models with mZ‘ < 3000 GeV the Large Δφ selection
results in a better significance in comparison with the methods High cut, Assym Emiss, Assym
PT and Small Δφ, however significance is not increased compared to Low cut. Therefore we
conclude that the large significance found for these light mediator models using Large Δφ
confirms the signature event is the creation of highly energetic jets in the opposed direction
of missing energy. However to obtain a significance that is larger than by using the to Low
cut more precisely tuned selection methods are required which will be studied in analysis D.
Table 6.7a and table 6.7b show for a given model selecting events with a lower multiplicity
leads to a smaller signal yields. But as the signal yield decreases when events with lower
multiplicities are selected, the background yield decreases even stronger, with the consistent
result that in all studied models the significance decreases when events with lower
multiplicities are selected.
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7.4 Conclusion analysis D
The results from analysis D show consistently with analysis C that selecting events with a
lower multiplicity leads to a smaller significance, the significance increases for models with
smaller mediator mass with the only exception of the DM400MM500 and that the production
of DM50 leads to higher significance than DM400 production independent of the mediator
models, with the exception of MM500.
Calculation of the significance for different combinations of cuts on the variables Emiss and PT
[0] resulted in the numbers labelled as Best selection presented in table 6.8, 6.9 and 6.10.
We notice that for models with mx = 400 GeV in 95.2 % of the studied models with different
mediator masses and multiplicity, the significance can be increased by an average 3.9% and
by as much as 6.9% using the optimised selection with respect to the best selection available
from analysis C. In 4.8% of the studied models the optimised selection is equal to the best
selection from analysis C. The criteria that led to this optimised significance are asymmetric
and require pT[0] > 300 GeV and a cut on Emiss varying from Emiss > 350 GeV to Emiss > 500
GeV. For models with mx = 50 GeV in 100% of the studied models with different multiplicity
and mz’ ≥ 3000 GeV significance can be increased by an average 3.7% and as much as
6.0% comparing to the best selection found in analysis C. The criteria that led to this
optimised significance are asymmetric and require pT[0] = 300 GeV and a cut on Emiss varying
from Emiss > 350 GeV to Emiss > 450 GeV. For models with mx = 50 GeV in 100% of the
studied cases with different multiplicities and mz’ < 3000 GeV significance is not increased
and equal to the best selection of analysis C, and the optimised selection is found to have
equal criteria as LowCut. In 100% of the studied models with mx = 50 GeV significance was
enhanced or kept at the same level when applying the optimised selection criteria found from
analysis D in comparison with using any of the selection methods used in analysis C and in
55.6% of the studied models significance was enhanced by an average 2.0% with respect to
analysis C.
The selection “Best + large Δφ selection” implies the extra condition [5.21] and thus select
only events with a large angle dφ while optimising the cuts on variables Emiss and PT[0]. This
method did not enhance significance over the Best selection method in 100% of the studied
cases. Furthermore as the criteria for Best selection already favoured an asymmetric cut that
is harder on Emiss than PT[0 the addition of the requirement of a large angle δφ reduced the
amount of data considerably and decreased the significance.
The 3D plots of fig. 6.24, 6.25 and 6.26 show the optimisation study for D5DM50 for
respectively n-jet, 1 and/or 2 jet and 1-jet events which is in agreement with the optimised
result labelled as Best. The optimised selection criteria for all multiplicities is Emiss > 400 GeV
and PT [0] > 300 GeV and in fig. 6.27 the effect of this optimal selection on the 6 variables
can be confirmed.
The 3D plots of fig. 6.28, 6.29 and 6.30 show the optimisation study for D5DM400 for
respectively n-jet, 1 and/or 2 jet and 1-jet events which is in agreement with the optimised
result labelled as Best. The optimised selection criteria for all multiplicities is Emiss > 450 GeV
and PT [0] > 300 GeV and in fig. 6.31 the effect of this optimal selection on the 6 variables
can be confirmed.
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The 3D plots of fig. 6.32, 6.33 and 6.34 show the optimisation study for DM50MM3000 for
respectively n-jet, 1 and/or 2 jet and 1-jet events which is in agreement with the optimised
result labelled as Best. The optimised selection criteria for all multiplicities is Emiss > 450 GeV
and PT [0] > 300 GeV and in fig. 6.35 the effect of this optimal selection on the 6 variables
can be confirmed.
The 3D plots of fig. 6.36, 6.37 and 6.38 show the optimisation study for DM400MM3000 for
respectively n-jet, 1 and/or 2 jet and 1-jet events which is in agreement with the optimised
result labelled as Best. The optimised selection criteria for all multiplicities uses the criteria
Emiss > 500 GeV and PT [0] > 300 GeV and is the most asymmetric cut found in analysis D. In
fig. 6.39 the effect of this optimal selection on the 6 variables can be confirmed.
The 3D plots of fig. 6.40, 6.41 and 6.42 show the optimisation study for DM50MM500 for
respectively n-jet, 1 and/or 2 jet and 1-jet events which is in agreement with the optimised
result labelled as Best. The optimised selection criteria for all multiplicities is Emiss > 450 GeV
and PT [0] > 300 GeV and in fig. 6.4, 6.6 and 6.8 the effect of this optimal selection on the 6
variables can be confirmed.
The 3D plots of fig. 6.44, 6.45 and 6.46 show the optimisation study for DM400MM500 for
respectively n-jet, 1 and/or 2 jet and 1-jet events which is in agreement with the optimised
result labelled as Best. The optimised selection criteria for events with 1 and/or 2 and n-jets
are Emiss > 350 GeV and PT [0] > 300 GeV. Fig. 6.47 shows this effect for the 6 variables
discussed in chapter 5.10. (For 1-jet events the optimised selection is Emiss > 300 GeV and PT
[0] > 300 GeV and the effect on the 6 variables for this case can be found in fig. 6.5.)
As in analysis C the signature event of the creation of highly energetic jets in the opposed
direction of missing energy is confirmed and the best tuned selection criteria have an
asymmetric preference towards cutting harder on E miss than on pT[0] for which in 100% of
these cases significance was enhanced or kept at the same level with respect to the LowCut.
Considering all 48 models in 62.9% significance is enhanced by an average 3.0% while there
was no situation in which significance is decreased.
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7.5 General conclusion of the analysis
First we showed that the rates of the standard model background contributions used in this
study are in agreement with the ratios obtained in the ATLAS WIMP sensitivity study for
14 TeV [155], validating the analysis performed on the simulations [156] used in this
thesis. Using models with a simplified interaction trough a massive mediator and EFT as the
high energy limit of this interaction [169], we have seen how stronger resonant enhancement
appears for mediators with successfully lower mass. Furthermore harder selection cuts don’t
always improve signal to background ratios and the selection methods High cut, Assym PT,
Small Δφ, Large Δφ don’t increase significance in 100% of the cases. Selections with Assym
Emiss will in some scenarios increase significance over LowCut. Study of the distributions of
Emiss, pT[0], Emiss / pT[0] and dφ show the dominant signature of a WIMP production event is
the creation of jets in the opposed direction to missing energy as expected. Using a more
generic method that optimised significance for different cuts on Emiss and pT[0] we have found
that the best tuned selection criteria have an asymmetric preference towards cutting harder
on Emiss than on pT[0] for which in 100% of these cases significance was enhanced or kept at
the same level with respect to the LowCut. For models with mx = 400 GeV in 95.2 % of the
studied models with different mediator masses and multiplicity, the significance can be
increased by an average 3.9% and by as much as 6.9% using the optimised selection with
respect to the best selection available from analysis C. In 4.8% of the studied models the
optimised selection is equal to the best selection from analysis C. For models with mx = 50
GeV in 100% of the studied models with different multiplicity and mz’ ≥ 3000 GeV significance
can be increased by an average 3.7% and as much as 6.0% comparing to the best selection
found in analysis C. For models with m x = 50 GeV in 100% of the studied cases with different
multiplicities and mz’ < 3000 GeV significance is not increased and equal to the best selection
found in analysis C. In general for models with m x = 50 GeV in 55.6% of the studied models
significance was enhanced by an average 2.0% with respect to analysis C. Considering all
48 models in 62.9% significance is enhanced by an average 3.0% while there was no
situation in which significance is decreased.
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7.6 General conclusion
In astronomy and cosmology invisible massive dark matter particles are hypothised to
explain the missing mass problem in astrophysical observations on the motion of galaxies
and clusters, the cosmic microwave background anisotropies and the large scale structure of
the Universe. Particles with WIMP like properties which would be thermally created in the
early Universe and have experienced a freeze out, are interesting dark matter candidates as
their relic density would be equal to the known dark matter density when their interaction
strength is comparable to the weak force. WIMP production might be in reach of accelerator
experiments which are a promising discovery channel as in June 2015 LHC has restarted its
operation with record breaking center of mass energy of
14 TeV. WIMP pair production
has the signature event of a momentum imbalance in the production of associated initialstate jets, called missing energy, which could be detected at ATLAS. Using selection criteria
that match the signature event it is possible to enhance the signal to background ratio. Using
ATLAS simulations of WIMP dark matter production and background processes at
14
TeV from [155] [169] an analysis was set-up to study the effect of placing tuned cuts on the
data to enhance significance. First we checked if the method is in agreement with previous
studies and verified that for the used WIMP simulations an increasing WIMP and mediator
particle Z’ mass resulted in a decreasing yield. Then we showed that for different models with
contact interaction represented by the D5 operator or for an interaction trough a generic Z’
intermediate state with mediator masses of respectively mz’ = 50 GeV and mz’ = 400 Gev and
with a range of WIMP masses between mx = 100 GeV and mx = 15 TeV harder selection cuts
don’t always improve signal to background ratios. The selection methods High cut, Assym PT,
Small Δφ, Large Δφ don’t increase significance in 100% of the cases. Using a more generic
method a developed program searched for the optimal selection criteria to maximize the
significance using different cuts on E miss and pT[0]. We have found that the best tuned
selection criteria have an asymmetric preference towards cutting harder on E miss than on pT[0]
for which in 100% of these cases significance was enhanced or kept at the same level with
respect to the common usage of general non-optimised cuts and there was no situation in
which significance was decreased. In 62.9% of all the studied 48 models significance was
enhanced by an average 3.0%. This confirms the viewpoint of multiple jets being created
opposed to the invisible WIMPs and shows the importance of using tuned selection criteria to
enhance significance.
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7.7 Outlook
In this outlook the limitations and assumptions for which the results of this study are valid,
and the possible extensions that could be performed will be described. The dark matter
search at ATLAS is independent of the astrophysical features of dark matter and is a
promising method to discover the hypothetical existence of invisible massive particles.
Although the discovery of a particle with WIMP like properties in LHC would be a
breakthrough in physics it should be noted that from this experiment no identification can be
made with the observed dark matter in astrophysics and cosmology, or its characteristics
such as its composition of single or multi particle species, abundance or density profile.
Nevertheless WIMP production at LHC is one of the most promising dark matter searches as
the event rate is expected to be larger than direct or indirect experiments and could give an
answer to the existence of light WIMP-like particles as early as 2016.
In order to simulate WIMP production at LHC some assumptions were made about its
characteristics. WIMPs were assumed to be a Dirac fermion, however other more exotic
WIMP candidates exist which could be interesting to study in future research but are beyond
the aim of this thesis. Also the type of interaction was simulated by the spin independent D5
operator or trough a simplified Z’ spin 1 massive mediator with m z’ in the range form 100 GeV
till 15.000 GeV and a succeeding study which optimises cuts for other types of interacting
mediators could give different results.
Using simulations of WIMP pair creation and background processes in ATLAS we have seen
how using tuned cuts optimises significance. The results of this study are however limited to
the type of simulations used and the range of the kinematic variables for which cuts are
applied. We have studied models in which mx = 50 GeV and mx = 400 GeV was assumed,
however when aiming for an independent discovering method no indication for the WIMP
mass can be set a priori. On the other hand including the assumption dark matter has WIMP
like properties from astrophysics, the estimate mx ~ 100 GeV from chapter 2.8.6 is a valid
argument to use the models studied.
The simulations used in this study were based on leading order calculations while higher
order correction generally would give a more accurate prediction of the kinematics of the
event and the number of jets created.
Sensitivity studies show that in LHC signals for WIMP interaction with mz’ < 1.5 TeV could be
discovered after one year of data taking which is a exciting prospect as LHC restarted
operating in June 2015. [131]
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