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  pdftitle={Search for excited quark states decaying to qW/qZ},
  pdfauthor={David Leppla-Weber},
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\title{Search for excited quark states decaying to qW/qZ}
\author{David Leppla-Weber}
\date{}

\begin{document}
\maketitle
\begin{abstract}
Abstract.
\end{abstract}
\begin{abstract}
Abstract 2.
\end{abstract}

{
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}
\newpage
\pagenumbering{arabic}

\hypertarget{introduction}{%
\section{Introduction}\label{introduction}}

The Standard Model is a very successful theory in describing most of the
effects on a particle level. But it still has a lot of shortcomings that
show that it isn't yet a full \enquote{theory of everything}. To solve
these shortcomings, lots of theories beyond the standard model exist
that try to explain some of them.

One category of such theories is based on a composite quark model. They
predict that quarks consist of particles unknown to us so far or can
bind to other particles using unknown forces. This could explain some
symmetries between particles and reduce the number of constants needed
to explain the properties of the known particles. One common prediction
of those theories are excited quark states. Those are quark states of
higher energy that can decay to an unexcited quark under the emission of
a boson. These decays are the topic of this thesis.

In previous research, a lower limit for the mass of an excited quark has
already been set using data from the 2016 run of the Large Hadron
Collider with an integrated luminosity of
\(\SI{35.92}{\per\femto\barn}\). Since then, a lot more data has been
collected, totalling to \(\SI{137.19}{\per\femto\barn}\). This thesis
uses this new data as well as a new technique to identify decays of
highly boosted particles based on a deep neural network to further
improve this limit and therefore exclude the excited quark particle to
even higher masses. It will also compare this new tagging technique to
an older tagger based on jet substructure studies used in the previous
research.

First, a theoretical background will be presented explaining in short
the Standard Model, its shortcomings and the theory of excited quarks.
Then the Large Hadron Collider and the Compact Muon Solenoid, the
detector that collected the data for this analysis, will be described.
After that, the main analysis part follows, describing how the data was
used to extract limits on the mass of the excited quark particle. At the
very end, the results are presented and compared to previous research.

\newpage

\hypertarget{theoretical-background}{%
\section{Theoretical background}\label{theoretical-background}}

This chapter presents a short summary of the theoretical background
relevant to this thesis. It first gives an introduction to the standard
model itself and some of the issues it raises. It then goes on to
explain the background processes of quantum chromodynamics and the
theory of q*, which will be the main topic of this thesis.

\hypertarget{standard-model}{%
\subsection{Standard model}\label{standard-model}}

The Standard Model of physics proofed very successful in describing
three of the four fundamental interactions currently known: the
electromagnetic, weak and strong interaction. The fourth, gravity, could
not yet be successfully included in this theory.

The Standard Model divides all particles into spin-\(\frac{n}{2}\)
fermions and spin-n bosons, where n could be any integer but so far is
only known to be one for fermions and either one (gauge bosons) or zero
(scalar bosons) for bosons. The fermions are further divided into quarks
and leptons. Each of those exists in six so called flavours.
Furthermore, quarks and leptons can also be divided into three
generations, each of which contains two particles. In the lepton
category, each generation has one charged lepton and one neutrino, that
has no charge. Also, the mass of the neutrinos is not yet known, only an
upper bound has been established. A full list of particles known to the
standard model can be found in fig.~\ref{fig:sm}. Furthermore, all
fermions have an associated anti particle with reversed charge. Multiple
quarks can form bound states called hadrons (e.g.~proton and neutron).

\begin{figure}
\hypertarget{fig:sm}{%
\centering
\includegraphics[width=0.5\textwidth,height=\textheight]{./figures/sm_wikipedia.pdf}
\caption{Elementary particles of the Standard Model and their mass
charge and spin.}\label{fig:sm}
}
\end{figure}

The gauge bosons, namely the photon, \(W^\pm\) bosons, \(Z^0\) boson,
and gluon, are mediators of the different forces of the standard model.

The photon is responsible for the electromagnetic force and therefore
interacts with all electrically charged particles. It itself carries no
electromagnetic charge and has no mass. Possible interactions are either
scattering or absorption. Photons of different energies can also be
described as electromagnetic waves of different wavelengths.

The \(W^\pm\) and \(Z^0\) bosons mediate the weak force. All quarks and
leptons carry a flavour, which is a conserved value. Only the weak
interaction breaks this conservation, a quark or lepton can therefore,
by interacting with a \(W^\pm\) boson, change its flavour. The
probabilities of this happening are determined by the
Cabibbo-Kobayashi-Maskawa matrix:

\begin{equation}
  V_{CKM} =
    \begin{pmatrix}
      |V_{ud}| & |V_{us}| & |V_{ub}| \\
      |V_{cd}| & |V_{cs}| & |V_{cb}| \\
      |V_{td}| & |V_{ts}| & |V_{tb}|
    \end{pmatrix}
  =
    \begin{pmatrix}
      0.974 & 0.225 & 0.004 \\
      0.224 & 0.974 & 0.042 \\
      0.008 & 0.041 & 0.999
    \end{pmatrix}
\end{equation} The probability of a quark changing its flavour from
\(i\) to \(j\) is given by the square of the absolute value of the
matrix element \(V_{ij}\). It is easy to see, that the change of flavour
in the same generation is way more likely than any other flavour change.

The quantum chromodynamics (QCD) describe the strong interaction of
particles. It applies to all particles carrying colour (e.g.~quarks).
The force is mediated by the gluon. This boson carries colour as well,
although it doesn't carry only one colour but rather a combination of a
colour and an anticolour, and can therefore interact with itself and
exists in eight different variant. As a result of this, processes, where
a gluon decays into two gluons are possible. Furthermore the strong
force, binding to colour carrying particles, increases with their
distance r making it at a certain point more energetically efficient to
form a new quark - antiquark pair than separating the two particles even
further. This effect is known as colour confinement. Due to this effect,
colour carrying particles can't be observed directly, but rather form so
called jets that cause hadronic showers in the detector. An effect
called Hadronisation.

\hypertarget{sec:qcdbg}{%
\subsubsection{Quantum Chromodynamic background}\label{sec:qcdbg}}

In this thesis, a decay with two jets in the endstate will be analysed.
Therefore it will be hard to distinguish the signal processes from QCD
effects. Those can also produce two jets in the endstate, as can be seen
in fig.~\ref{fig:qcdfeynman}. They are also happening very often in a
proton proton collision, as it is happening in the Large Hadron
Collider. This is caused by the structure of the proton. It not only
consists of three quarks, called valence quarks, but also of a lot of
quark-antiquark pairs connected by gluons, called the sea quarks, that
exist due to the self interaction of the gluons binding the three
valence quarks. Therefore in a proton - proton collision, interactions
of gluons and quarks are the main processes causing a very strong QCD
background.

\begin{figure}
\centering
\feynmandiagram [horizontal=v1 to v2] {
    q1 [particle=\(q\)] -- [fermion] v1 -- [gluon] g1 [particle=\(g\)],
    v1 -- [gluon] v2,
    q2 [particle=\(q\)] -- [fermion] v2 -- [gluon] g2 [particle=\(g\)],
};
\feynmandiagram [horizontal=v1 to v2] {
    g1 [particle=\(g\)] -- [gluon] v1 -- [gluon] g2 [particle=\(g\)],
    v1 -- [gluon] v2,
    g3 [particle=\(g\)] -- [gluon] v2 -- [gluon] g4 [particle=\(g\)],
};
\caption{Two examples of QCD processes resulting in two jets.} \label{fig:qcdfeynman}
\end{figure}

\hypertarget{shortcomings-of-the-standard-model}{%
\subsubsection{Shortcomings of the Standard
Model}\label{shortcomings-of-the-standard-model}}

While being very successful in describing mostly all of the effects we
can observe in particle colliders so far, the Standard Model still has
several shortcomings.

\begin{itemize}
\tightlist
\item
  \textbf{Gravity}: as already noted, the standard model doesn't include
  gravity as a force.
\item
  \textbf{Dark Matter}: observations of the rotational velocity of
  galaxies can't be explained by the known matter. Dark matter currently
  is our best theory to explain those.
\item
  \textbf{Matter-antimatter assymetry}: The amount of matter vastly
  outweights the amount of antimatter in the observable universe. This
  can't be explained by the standard model, which predicts a similar
  amount of matter and antimatter.
\item
  \textbf{Symmetries between particles}: Why do exactly three
  generations of fermions exist? Why is the charge of a quark exactly
  one third of the charge of a lepton? How are the masses of the
  particles related? Those and more questions cannot be answered by the
  standard model.
\item
  \textbf{Hierarchy problem}: The weak force is approximately
  \(10^{24}\) times stronger than gravity and so far, there's no
  satisfactory explanation as to why that is.
\end{itemize}

\hypertarget{sec:qs}{%
\subsection{Excited quark states}\label{sec:qs}}

One category of theories that try to solve some of the shortcomings of
the standard model are the composite quark models. Those state, that
quarks consist of some particles unknown to us so far. This could
explain the symmetries between the different fermions. A common
prediction of those models are excited quark states (q*, q**,
q***\ldots). Similar to atoms, that can be excited by the absorption of
a photon and can then decay again under emission of a photon with an
energy corresponding to the excited state, those excited quark states
could decay under the emission of some boson. Quarks are smaller than
\(10^{-18}\) m, due to that, excited states have to be of very high
energy. That will cause the emitted boson to be highly boosted.

\begin{figure}
\centering
\feynmandiagram [large, horizontal=qs to v] {
  a -- qs -- b,
  qs -- [fermion, edge label=\(q*\)] v,
  q1 [particle=\(q\)] -- v -- w [particle=\(W\)],
  q2 [particle=\(q\)] -- w -- q3 [particle=\(q\)],
};
\caption{Feynman diagram showing a possible decay of a q* particle to a W boson and a quark with the W boson also
decaying to two quarks.} \label{fig:qsfeynman}
\end{figure}

This thesis will search data collected by the CMS in the years 2016,
2017 and 2018 for the single excited quark state q* which can decay to a
quark and any boson. An example of a q* decaying to a quark and a W
boson can be seen in fig.~\ref{fig:qsfeynman}. The boson quickly further
decays into for example two quarks. Because the boson is highly boosted,
those will be very close together and therefore appear to the detector
as only one jet. This means that the decay of a q* particle will have
two jets in the endstate (assuming the W/Z boson decays to two quarks)
and will therefore be hard to distinguish from the QCD background
described in sec.~\ref{sec:qcdbg}.

To reconstruct the mass of the q* particle from an event successfully
recognized to be the decay of such a particle, the dijet invariant mass,
the mass of the two jets in the final state, can be calculated by adding
their four momenta, vectors consisting of the energy and momentum of a
particle, together. From the four momentum it's easy to derive the mass
by solving \(E=\sqrt{p^2 + m^2}\) for m.

\newpage

\hypertarget{experimental-setup}{%
\section{Experimental Setup}\label{experimental-setup}}

Following on, the experimental setup used to gather the data analysed in
this thesis will be described.

\hypertarget{large-hadron-collider}{%
\subsection{Large Hadron Collider}\label{large-hadron-collider}}

The Large Hadron Collider is the world's largest and most powerful
particle accelerator \autocite{website}. It has a perimeter of 27 km and
can collide protons at a centre of mass energy of 13 TeV. It is home to
several experiments, the biggest of those are ATLAS and the Compact Muon
Solenoid (CMS). Both are general-purpose detectors to investigate the
particles that form during particle collisions.

Particle colliders are characterized by their luminosity L. It is a
quantity to be able to calculate the number of events per second
generated in a collision by \(N_{event} = L\sigma_{event}\) with
\(\sigma_{event}\) being the cross section of the event. The luminosity
of the LHC for a Gaussian beam distribution can be described as follows:

\begin{equation}
  L = \frac{N_b^2 n_b f_{rev} \gamma_r}{4 \pi \epsilon_n \beta^*}F
\end{equation} Where \(N_b\) is the number of particles per bunch,
\(n_b\) the number of bunches per beam, \(f_{rev}\) the revolution
frequency, \(\gamma_r\) the relativistic gamma factor, \(\epsilon_n\)
the normalised transverse beam emittance, \(\beta^*\) the beta function
at the collision point and F the geometric luminosity reduction factor
due to the crossing angle at the interaction point: \begin{equation}
  F = \left(1+\left( \frac{\theta_c\sigma_z}{2\sigma^*}\right)^2\right)^{-1/2}
\end{equation} At the maximum luminosity of
\(10^{34}\si{\per\square\centi\metre\per\s}\),
\(N_b = 1.15 \cdot 10^{11}\), \(n_b = 2808\),
\(f_{rev} = \SI{11.2}{\kilo\Hz}\), \(\beta^* = \SI{0.55}{\m}\),
\(\epsilon_n = \SI{3.75}{\micro\m}\) and \(F = 0.85\).

To quantify the amount of data collected by one of the experiments at
LHC, the integrated luminosity is introduced as \(L_{int} = \int L dt\).

\hypertarget{compact-muon-solenoid}{%
\subsection{Compact Muon Solenoid}\label{compact-muon-solenoid}}

The data used in this thesis was captured by the Compact Muon Solenoid
(CMS). It is one of the biggest experiments at the Large Hadron
Collider. It can detect all elementary particles of the standard model
except neutrinos. For that, it has an onion like setup. The particles
produced in a collision first go through a tracking system. They then
pass an electromegnetic as well as a hadronic calorimeter. This part is
surrounded by a superconducting solenoid that generates a magenetic
field of 3.8 T. Outside of the solenoid are big muon chambers. In 2016
the CMS captured data of a integrated luminosity of
\(\SI{35.92}{\per\femto\barn}\). In 2017 it collected
\(\SI{41.53}{\per\femto\barn}\) and in 2018
\(\SI{59.74}{\per\femto\barn}\). Therefore the combined dataset of all
three years has a total integrated luminosity of
\(\SI{137.19}{\per\femto\barn}\).

\hypertarget{coordinate-conventions}{%
\subsubsection{Coordinate conventions}\label{coordinate-conventions}}

Per convention, the z axis points along the beam axis, the y axis
upwards and the x axis horizontal towards the LHC centre. Furthermore,
the azimuthal angle \(\phi\), which describes the angle in the x - y
plane, the polar angle \(\theta\), which describes the angle in the y -
z plane and the pseudorapidity \(\eta\), which is defined as
\(\eta = -ln\left(tan\frac{\theta}{2}\right)\) are introduced. The
coordinates are visualised in fig.~\ref{fig:cmscoords}. Furthermore, to
describe a particles momentum, often the transverse momentum, \(p_t\) is
used. It is the component of the momentum transversal to the beam axis.
It is a useful quantity, because the sum of all transverse momenta has
to be zero. Missing transverse momentum implies particles that weren't
detected such as neutrinos.

\begin{figure}
\hypertarget{fig:cmscoords}{%
\centering
\includegraphics[width=0.6\textwidth,height=\textheight]{./figures/cms_coordinates.png}
\caption{Coordinate conventions of the CMS illustrating the use of
\(\eta\) and \(\phi\). The Z axis is in beam direction. Taken from
https://inspirehep.net/record/1236817/plots}\label{fig:cmscoords}
}
\end{figure}

\hypertarget{the-tracking-system}{%
\subsubsection{The tracking system}\label{the-tracking-system}}

The tracking system is built of two parts, first a pixel detector and
then silicon strip sensors. It is used to reconstruct the tracks of
charged particles, measuring their charge sign, direction and momentum.
It is as close to the collision as possible to be able to identify
secondary vertices.

\hypertarget{the-electromagnetic-calorimeter}{%
\subsubsection{The electromagnetic
calorimeter}\label{the-electromagnetic-calorimeter}}

The electromagnetic calorimeter measures the energy of photons and
electrons. It is made of tungstate crystal. When passed by particles, it
produces light in proportion to the particle's energy. This light is
measured by photodetectors that convert this scintillation light to an
electrical signal. To measure a particles energy, it has to leave its
whole energy in the ECAL, which is true for photons and electrons, but
not for other particles such as hadrons and muons. They too leave some
energy in the ECAL.

\hypertarget{the-hadronic-calorimeter}{%
\subsubsection{The hadronic
calorimeter}\label{the-hadronic-calorimeter}}

The hadronic calorimeter (HCAL) is used to detect high energy hadronic
particles. It surrounds the ECAL and is made of alternating layers of
active and absorber material. While the absorber material with its high
density causes the hadrons to shower, the active material then detects
those showers and measures their energy, similar to how the ECAL works.

\hypertarget{the-solenoid}{%
\subsubsection{The solenoid}\label{the-solenoid}}

The solenoid, giving the detector its name, is one of the most important
features. It creates a magnetic field of 3.8 T and therefore makes it
possible to measure momentum of charged particles by bending their
tracks.

\hypertarget{the-muon-system}{%
\subsubsection{The muon system}\label{the-muon-system}}

Outside of the solenoid there is only the muon system. It consists of
three types of gas detectors, the drift tubes, cathode strip chambers
and resistive plate chambers. The system is divided into a barrel part
and two endcaps. Together they cover \(0 < |\eta| < 2.4\). The muons are
the only detected particles, that can pass all the other systems without
a significant energy loss.

\hypertarget{the-trigger-system}{%
\subsubsection{The Trigger system}\label{the-trigger-system}}

The CMS features a two level trigger system. It is necessary because the
detector is unable to process all the events due to limited bandwidth.
The Level 1 trigger reduces the event rate from 40 MHz to 100 kHz, the
software based High Level trigger is then able to further reduce the
rate to 1 kHz. The Level 1 trigger uses the data from the
electromagnetic and hadronic calorimeters as well as the muon chambers
to decide whether to keep an event. The High Level trigger uses a
streamlined version of the CMS offline reconstruction software for its
decision making.

\hypertarget{the-particle-flow-algorithm}{%
\subsubsection{The Particle Flow
algorithm}\label{the-particle-flow-algorithm}}

The particle flow algorithm is used to identify and reconstruct all the
particles arising from the proton - proton collision by using all the
information available from the different sub-detectors of the CMS. It
does so by extrapolating the tracks through the different calorimeters
and associating clusters they cross with them. The set of the track and
its clusters is then no more used for the detection of other particles.
This is first done for muons and then for charged hadrons, so a muon
can't give rise to a wrongly identified charged hadron. Due to
Bremsstrahlung photon emission, electrons are harder to reconstruct. For
them a specific track reconstruction algorithm is used. After
identifying charged hadrons, muons and electrons, all remaining clusters
within the HCAL correspond to neutral hadrons and within ECAL to
photons. If the list of particles and their corresponding deposits is
established, it can be used to determine the particles four momenta.
From that, the missing transverse energy can be calculated and tau
particles can be reconstructed by their decay products.

\hypertarget{jet-clustering}{%
\subsection{Jet clustering}\label{jet-clustering}}

Because of the hadronisation it is not possible to uniquely identify the
originating particle of a jet. Nonetheless, several algorithms exist to
help with this problem. The algorithm used in this thesis is the
anti-\(k_t\) clustering algorithm. It arises from a generalization of
several other clustering algorithms, namely the \(k_t\),
Cambridge/Aachen and SISCone clustering algorithms.

The anti-\(k_t\) clustering algorithm associates hard particles with
their soft particles surrounding them within a radius R in the \(\eta\)
- \(\phi\) plane forming cone like jets. If two jets overlap, the jets
shape is changed according to its hardness. A softer particles jet will
change its shape more than a harder particles. A visual comparison of
four different clustering algorithms can be seen in
fig.~\ref{fig:antiktcomparision}. For this analysis, a radius of 0.8 is
used.

Furthermore, to approximate the mass of a heavy particle that caused a
jet, the softdropmass can be used. It is calculated by removing wide
angle soft particles from the jet to counter the effects of
contamination from initial state radiation, underlying event and
multiple hadron scattering. It therefore is more accurate in determining
the mass of a particle causing a jet than taking the mass of all
constituent particles of the jet combined.

\begin{figure}
\hypertarget{fig:antiktcomparision}{%
\centering
\includegraphics{./figures/antikt-comparision.png}
\caption{Comparision of the \(k_t\), Cambridge/Aachen, SISCone and
anti-\(k_t\) algorithms clustering a sample parton-level event with many
random soft \enquote{ghosts}. Taken from}\label{fig:antiktcomparision}
}
\end{figure}

\newpage

\hypertarget{method-of-analysis}{%
\section{Method of analysis}\label{method-of-analysis}}

This section gives an overview over how the data gathered by the LHC and
CMS is going to be analysed to be able to either exclude the q* particle
to even higher masses than already done or maybe confirm its existence.

As described in sec.~\ref{sec:qs}, an excited quark q* can decay to a
quark and any boson. The branching ratios are calculated to be as
follows \autocite{QSTAR_THEORY}:

\begin{longtable}[]{@{}llll@{}}
\caption{Branching ratios of the decaying q* particle.}\tabularnewline
\toprule
decay mode & br. ratio {[}\%{]} & decay mode & br. ratio
{[}\%{]}\tabularnewline
\midrule
\endfirsthead
\toprule
decay mode & br. ratio {[}\%{]} & decay mode & br. ratio
{[}\%{]}\tabularnewline
\midrule
\endhead
\(U^* \rightarrow ug\) & 83.4 & \(D^* \rightarrow dg\) &
83.4\tabularnewline
\(U^* \rightarrow dW\) & 10.9 & \(D^* \rightarrow uW\) &
10.9\tabularnewline
\(U^* \rightarrow u\gamma\) & 2.2 & \(D^* \rightarrow d\gamma\) &
0.5\tabularnewline
\(U^* \rightarrow uZ\) & 3.5 & \(D^* \rightarrow dZ\) &
5.1\tabularnewline
\bottomrule
\end{longtable}

The majority of excited quarks will decay to a quark and a gluon, but as
this is virtually impossible to distinguish from QCD effects (for
example from the qg \(\rightarrow\) qg processes), this analysis will
focus on the processes q* \(\rightarrow\) qW and q* \(\rightarrow\) qZ.
In this case, due to jet substructure studies, it is possible to
establish a discriminator between QCD background and jets originating in
a W/Z decay. They still make up roughly 20 \% of the signal events to
study and therefore seem like a good choice.

The data studied was collected by the CMS experiment in the years 2016,
2017 and 2018. It is analysed with the Particle Flow algorithm to
reconstruct jets and all the other particles forming during the
collision. The jets are then clustered using the anti-\(k_t\) algorithm
with the distance parameter R being 0.8. Furthermore, the calorimeters
of the CMS detector have to be calibrated. For that, jet energy
corrections published by the CMS working group are applied to the data.

To find signal events in the data, this thesis looks at the dijet
invariant mass distribution. The data is assumed to only consist of QCD
background and signal events, other backgrounds are neglected. Cuts on
several distributions are introduced to reduce the background and
improve the sensitivity for the signal. If the q* particle exists, the
dijet invariant mass distribution should show a resonance at its
invariant mass. This resonance will be looked for with statistical
methods explained later on.

The analysis will be conducted with two different sets of data. First,
only the data collected by CMS in 2016 will be used to compare the
results to the previous analysis \autocite{PREV_RESEARCH}. Then the
combined data from 2016, 2017 and 2018 will be used to improve the
previously set limits for the mass of the q* particle. Also, two
different tagging mechanisms will be used. One based on the
N-subjettiness variable used in the previous research, the other being a
novel approach using a deep neural network.

\hypertarget{signal-and-background-modelling}{%
\subsection{Signal and Background
modelling}\label{signal-and-background-modelling}}

To make sure the setup is working as intended, at first simulated
samples of background and signal are used. In those Monte Carlo
simulations, the different particle interactions that take place in a
proton - proton collision are simulated using the probabilities provided
by the Standard Model by calculating the cross sections of the different
feynman diagrams. Later on, also detector effects (like its limited
resolution) are applied to make sure, they look like real data coming
from the CMS detector. The q* signal samples are simulated by the
probabilities given by the q* theory \autocite{QSTAR_THEORY} and
assuming a cross section of \(\SI{1}{\per\pico\barn}\). The simulation
was done using MadGraph. Because of the expected high mass, the signal
width will be dominated by the resolution of the detector, not by the
natural resonance width.

The dijet invariant mass distribution of the QCD background is expected
to smoothly fall with higher masses. It is therefore fitted using the
following smooth falling function with three parameters p0, p1, p2:
\begin{equation}
\frac{dN}{dm_{jj}} = \frac{p_0 \cdot ( 1 - m_{jj} / \sqrt{s} )^{p_2}}{ (m_{jj} / \sqrt{s})^{p_1}}
\end{equation} Whereas \(m_{jj}\) is the invariant mass of the dijet and
\(p_0\) is a normalisation parameter. It is the same function as used in
the previous research studying 2016 data only.

The signal is fitted using a double sided crystal ball function. It has
six parameters:

\begin{itemize}
\tightlist
\item
  mean: the functions mean, in this case the resonance mass
\item
  sigma: the functions width, in this case the resolution of the
  detector
\item
  n1, n2, alpha1, alpha2: parameters influencing the shape of the left
  and right tail
\end{itemize}

A gaussian and a poisson have also been studied but found to not fit the
signal sample very well as they aren't able to fit the tail on both
sides of the peak.

An example of a fit of these functions to a toy dataset with gaussian
errors can be seen in fig.~\ref{fig:cb_fit}. In this figure, a binning
of 200 GeV is used. For the actual analysis a 1 GeV binning will be
used.

\begin{figure}
\hypertarget{fig:cb_fit}{%
\centering
\includegraphics{./figures/cb_fit.pdf}
\caption{Combined fit of signal and background on a toy dataset with
gaussian errors and a simulated resonance mass of 3
TeV.}\label{fig:cb_fit}
}
\end{figure}

\newpage

\hypertarget{preselection-and-data-quality}{%
\section{Preselection and data
quality}\label{preselection-and-data-quality}}

To separate the background from the signal, cuts on several
distributions have to be introduced. The selection of events is divided
into two parts. The first one (the preselection) adds some general
physics motivated cuts and is also used to make sure a good trigger
efficiency is achieved. It is not expected to already provide a good
separation of background and signal. In the second part, different
taggers will be used as a discriminator between QCD background and
signal events. After the preselection, it is made sure, that the
simulated samples represent the real data well.

\hypertarget{preselection}{%
\subsection{Preselection}\label{preselection}}

First, all events are cleaned of jets with a
\(p_t < \SI{200}{\giga\eV}\) and a pseudorapidity \(|\eta| > 2.4\). This
is to discard soft background and to make sure the particles are in the
barrel region of the detector for an optimal detector resolution.
Furthermore, all events with one of the two highest \(p_t\) jets having
an angular separation smaller than 0.8 from any electron or muon are
discarded to allow future use of the results in studies of the semi or
all-leptonic decay channels.

From a decaying q* particle, we expect two jets in the endstate.
Therefore a cut is added to have at least 2 jets. More jets are also
possible, for example caused by gluon radiation of a quark causing
another jet. The cut can be seen in fig.~\ref{fig:njets}.

\begin{figure}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/2016/v1_Cleaner_N_jets_stack.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/2016/v1_Njet_N_jets_stack.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/combined/v1_Cleaner_N_jets_stack.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/combined/v1_Njet_N_jets_stack.eps}
\end{minipage}
\caption{Number of jet distribution showing the cut at number of jets $\ge$ 2. Left: distribution before the cut. Right:
distribution after the cut. 1st row: data from 2016. 2nd row: combined data from 2016, 2017 and 2018. The signal curves 
are amplified by a factor of 10,000, to be visible.}
\label{fig:njets}
\end{figure}

Another cut is on \(\Delta\eta\). The q* particle is expected to be very
heavy in regards to the center of mass energy of the collision and will
therefore be almost stationary. Its decay products should therefore be
close to back to back, which means the \(\Delta\eta\) distribution is
expected to peak at 0. At the same time, particles originating from QCD
effects are expected to have a higher \(\Delta\eta\) as they mainly form
from less heavy resonances. To maintain comparability, the same cut as
in previous research of \(\Delta\eta \le 1.3\) is used as can be seen in
fig.~\ref{fig:deta}.

\begin{figure}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/2016/v1_Njet_deta_stack.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/2016/v1_Eta_deta_stack.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/combined/v1_Njet_deta_stack.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/combined/v1_Eta_deta_stack.eps}
\end{minipage}
\caption{$\Delta\eta$ distribution showing the cut at $\Delta\eta \le 1.3$. Left: distribution before the cut. Right:
distribution after the cut. 1st row: data from 2016. 2nd row: combined data from 2016, 2017 and 2018. The signal curves 
are amplified by a factor of 10,000, to be visible.}
\label{fig:deta}
\end{figure}

The last cut in the preselection is on the dijet invariant mass:
\(m_{jj} \ge \SI{1050}{\giga\eV}\). It is important for a high trigger
efficiency and can be seen in fig.~\ref{fig:invmass}. Also, it has a
huge impact on the background because it usually consists of way lighter
particles. The q* on the other hand is expected to have a very high
invariant mass of more than 1 TeV. The distribution should be a smoothly
falling function for the QCD background and peak at the simulated
resonance mass for the signal events.

\begin{figure}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/2016/v1_Eta_invMass_stack.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/2016/v1_invmass_invMass_stack.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/combined/v1_Eta_invMass_stack.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/combined/v1_invmass_invMass_stack.eps}
\end{minipage}
\caption{Invariant mass distribution showing the cut at $m_{jj} \ge \SI{1050}{\giga\eV}$. It shows the expected smooth 
falling functions of the background whereas the signal peaks at the simulated resonance mass.
Left: distribution before the
cut. Right: distribution after the cut. 1st row: data from 2016. 2nd row: combined data from 2016, 2017 and 2018.}
\label{fig:invmass}
\end{figure}

After the preselection, the signal efficiency for q* decaying to qW of
2016 ranges from 48 \% for 1.6 TeV to 49 \% for 7 TeV. Decaying to qZ,
the efficiencies are between 45 \% (1.6 TeV) and 50 \% (7 TeV). The
amount of background after the preselection is reduced to 5 \% of the
original events. For the combined data of the three years those values
look similar. Decaying to qW signal efficiencies between 49 \% (1.6 TeV)
and 56 \% (7 TeV) are reached, wheres the efficiencies when decaying to
qZ are in the range of 46 \% (1.6 TeV) to 50 \% (7 TeV). Here, the
background could be reduced to 8 \% of the original events. So while
keeping around 50 \% of the signal, the background was already reduced
to less than a tenth. Still, as can be seen in fig.~\ref{fig:njets} to
fig.~\ref{fig:invmass}, the amount of signal is very low and, without
logarithmic scale, even has to be amplified to be visible.

\hypertarget{data---monte-carlo-comparison}{%
\subsection{Data - Monte Carlo
Comparison}\label{data---monte-carlo-comparison}}

To ensure high data quality, the simulated QCD background sample is now
being compared to the actual data of the corresponding year collected by
the CMS detector. This is done for the year 2016 and for the combined
data of years 2016, 2017 and 2018. The distributions are rescaled so the
integral over the invariant mass distribution of data and simulation are
the same. In fig.~\ref{fig:data-mc}, the three distributions that cuts
were applied on can be seen for year 2016 and the combined data of years
2016 to 2018.

\begin{figure}
\begin{minipage}{0.33\textwidth}
\includegraphics{./figures/2016/DATA/v1_invmass_N_jets.eps}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\includegraphics{./figures/2016/DATA/v1_invmass_deta.eps}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\includegraphics{./figures/2016/DATA/v1_invmass_invMass.eps}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\includegraphics{./figures/combined/DATA/v1_invmass_N_jets.eps}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\includegraphics{./figures/combined/DATA/v1_invmass_deta.eps}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\includegraphics{./figures/combined/DATA/v1_invmass_invMass.eps}
\end{minipage}
\caption{Comparision of data with the Monte Carlo simulation.
1st row: data from 2016.
2nd row: combined data from 2016, 2017 and 2018.}
\label{fig:data-mc}
\end{figure}

The shape of the real data matches the simulation well. The
\(\Delta\eta\) distributions shows some offset between data and
simulation.

\hypertarget{sideband}{%
\subsubsection{Sideband}\label{sideband}}

The sideband is introduced to make sure there are no unwanted side
effects of the used cuts. It is a region in which no data is used for
the actual analysis. Again, data and the Monte Carlo simulation are
compared. For this analysis, the region where the softdropmass of both
of the two jets with the highest transverse momentum (\(p_t\)) is more
than 105 GeV was chosen. Because the decay of a q* to a vector boson is
being investigated, later on, a selection is applied that one of those
particles has to have a mass between 105 GeV and 35 GeV. Therefore
events with jets with a softdropmass higher than 105 GeV will not be
used for this analysis which makes them a good sideband to use.

In fig.~\ref{fig:sideband}, the comparison of data with simulation in
the sideband region can be seen for the softdropmass distribution as
well as the dijet invariant mass distribution. As in {[}fig:data-mc{]},
the histograms are rescaled, so that the dijet invariant mass
distributions of data and simulation have the same integral. It can be
seen, that in the sideband region data and simulation match very well.

\begin{figure}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/2016/sideband/v1_SDM_SoftDropMass_1.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/2016/sideband/v1_SDM_invMass.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/combined/sideband/v1_SDM_SoftDropMass_1.eps}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/combined/sideband/v1_SDM_invMass.eps}
\end{minipage}
\caption{Comparison of data with the Monte Carlo simulation in the sideband region. 1st row: data from 2016. 2nd row: 
combined data from 2016, 2017 and 2018.}
\label{fig:sideband}
\end{figure}

\newpage

\hypertarget{jet-substructure-selection}{%
\section{Jet substructure selection}\label{jet-substructure-selection}}

So far it was made sure, that the actual data and the simulation match
well after the preselection and no unwanted side effects are introduced
in the data by the used cuts. Now another selection has to be
introduced, to further reduce the background to be able to extract the
hypothetical signal events from the actual data.

This is done by distinguishing between QCD and signal events using a
tagger to identify jets coming from a vector boson. Two different
taggers will be used to later compare the results. The decay analysed
includes either a W or Z boson, which are, compared to the particles in
QCD effects, very heavy. This can be used by adding a cut on the
softdropmass of a jet. The softdropmass of at least one of the two
leading jets is expected to be within \(\SI{35}{\giga\eV}\) and
\(\SI{105}{\giga\eV}\). This cut already provides a good separation of
QCD and signal events, on which the two taggers presented next can
build.

\hypertarget{n-subjettiness}{%
\subsection{N-Subjettiness}\label{n-subjettiness}}

The N-subjettiness \(\tau_n\) is a jet shape parameter designed to
identify boosted hadronically-decaying objects. When a vector boson
decays hadronically, it produces two quarks each causing a jet. But
because of the high mass of the vector bosons, the particles are highly
boosted and appear, after applying a clustering algorithm, as just one.
This algorithm now tries to figure out, whether one jet might consist of
two subjets by using the kinematics and positions of the constituent
particles of this jet. The N-subjettiness is defined as

\begin{equation} \tau_N = \frac{1}{d_0} \sum_k p_{T,k} \cdot \text{min}\{ \Delta R_{1,k}, \Delta R_{2,k}, …, \Delta
R_{N,k} \} \end{equation}

with k going over the constituent particles in a given jet, \(p_{T,k}\)
being their transverse momenta and
\(\Delta R_{J,k} = \sqrt{(\Delta\eta)^2 + (\Delta\phi)^2}\) being the
distance of a candidate subjet J and a constituent particle k in the
\(\eta\) - \(\phi\) plane. It quantifies to what degree a jet can be
regarded as a jet composed of \(N\) subjets. Experiments showed, that
rather than using \(\tau_N\) directly, the ratio
\(\tau_{21} = \tau_2/\tau_1\) is a better discriminator between QCD
events and events originating from the decay of a boosted vector boson.

The \(\tau_{21}\) cut is applied to the one of the two highest \(p_t\)
jets passing the softdropmass window. If both of them pass, it is
applied to the one with higher \(p_t\).

\hypertarget{deepak8}{%
\subsection{DeepAK8}\label{deepak8}}

The DeepAK8 tagger uses a deep neural network (DNN) to identify decays
originating in a vector boson. It is supposed to give better
efficiencies than the older N-Subjettiness method.

The DNN has two input lists for each jet. The first is a list of up to
100 constituent particles of the jet, sorted by decreasing \(p_t\). A
total of 42 properties of the particles such es \(p_t\), energy deposit,
charge and the angular momentum between the particle and the jet or
subjet axes are included. The second input list is a list of up to seven
secondary vertices, each with 15 features, such as the kinematics,
displacement and quality criteria. To process those inputs, a customised
DNN architecture has been developed. It consists of two convolutional
neural networks that each process one of the input lists. The outputs of
the two CNNs are then combined and processed by a fully-connected
network to identify the jet. The network was trained with a sample of 40
million jets, another 10 million jets were used for development and
validation.

In this thesis, the mass decorrelated version of the DeepAK8 tagger is
used. It adds an additional mass predictor layer, that is trained to
quantify how strongly the output of the non-decorrelated tagger is
correlated to the mass of a particle. Its output is fed back to the
network as a penalty so it avoids using features of the particles
correlated to their mass. The result is a largely mass decorrelated
tagger of heavy resonances. As the mass variable is already in use for
the softdropmass selection, this version of the tagger is to be
preferred.

Just like the \(\tau_{21}\) cut, the cut on the discriminator introduced
by the DeepAK8 tagger is applied on the one of the two highest \(p_t\)
jets passing the softdropmass window.

\hypertarget{optimization}{%
\subsection{Optimization}\label{optimization}}

To figure out the best value to cut on the discriminators introduced by
the two taggers, a value to quantify how good a cut is has to be
introduced. For that, the significance calculated by
\(\frac{S}{\sqrt{B}}\) will be used. S stands for the amount of signal
events and B for the amount of background events in a given interval.
This value assumes a gaussian error on the background so it will be
calculated for the 2 TeV masspoint where enough background events exist
to justify this assumption. This follows from the central limit theorem
that states, that for identical distributed random variables, their sum
converges to a gaussian distribution. The value therefore represents how
good the signal can be distinguished from the background in units of the
standard deviation of the background. As interval, a 10 \% margin around
the masspoint is chosen.

\begin{figure}
  \begin{minipage}{0.5\textwidth}
    \includegraphics{./figures/sig-db.pdf}
  \end{minipage}
  \begin{minipage}{0.5\textwidth}
    \includegraphics{./figures/sig-tau.pdf}
  \end{minipage}
\caption{Significance plots for the deep boosted (left) and N-subjettiness (right) tagger at the 2 TeV masspoint.}
\label{fig:sig}
\end{figure}

As a result, the \(\tau_{21}\) cut is placed at \(\le 0.35\), confirming
the value previous research chose and the deep boosted cut is placed at
\(\ge 0.95\). For the deep boosted tagger, 0.97 would give a slightly
higher significance but as it is very close to the edge where the
significance drops very low and the higher the cut the less background
will be left to calculate the cross section limits, especially at higher
resonance masses, the slightly less strict cut is chosen. The
significance for the \(\tau_{21}\) cut is 14.0818, and for the deep
boosted tagger 25.6097. For both taggers also a low purity category is
introduced for high TeV regions. Using the cuts optimized for 2 TeV,
there are very few background events left for higher resonance masses,
but to reliably calculate cross section limits, those are needed. As low
purity category for the N-subjettiness tagger, a cut at
\(0.35 < \tau_{21} < 0.75\) is used. For the deep boosted tagger the
opposite cut from the high purity category is used: \(VvsQCD < 0.95\).

\hypertarget{signal-extraction}{%
\section{Signal extraction}\label{signal-extraction}}

To extract the signal from the background, its cross section limit is
calculated using a frequentist asymptotic limit calculator. It uses a
fit to the simulated samples to calculate expected limits for all the
available masspoints and then uses a fit to the actual data to determine
an observed limit. If there's no resonance of the q* particle in the
data, the observed limit should lie within the \(2\sigma\) environment
of the expected limit. After that, the crossing of the theory line,
representing the cross section limits expected, if the q* particle would
exist, and the observed data is calculated, to have a limit of mass up
to which the existence of the q* particle can be excluded. To find the
uncertainty of this result, the crossing of the theory line plus,
respectively minus, its uncertainty with the observed limit is also
calculated.

\hypertarget{uncertainties}{%
\subsection{Uncertainties}\label{uncertainties}}

The following uncertainties are considered:

\begin{itemize}
\tightlist
\item
  \emph{Luminosity}: the integrated luminosity of the LHC has an
  uncertainty of 2.5 \%.
\item
  \emph{Jet Energy Corrections}: for the Jet Energy Corrections, an
  uncertainty of 2 \% is assumed.
\item
  \emph{Tagger Efficiency(?)}: 6 \% (TODO!)
\item
  \emph{Parameter Uncertainty of the fit}: The CombinedLimit program
  used for determining the cross section varies the parameters used for
  the fit and therefore includes their uncertainties to calculate the
  final result.
\end{itemize}

\hypertarget{results}{%
\section{Results}\label{results}}

In this chapter the results and a comparison to previous research will
be shown as well as a comparisos n between the two different taggers
used.

\hypertarget{section}{%
\subsection{2016}\label{section}}

Using the data collected by the CMS experiment on 2016, the cross
section limits seen in fig.~\ref{fig:res2016} were obtained. The
extracted cross section limits are:

\begin{longtable}[]{@{}lllll@{}}
\caption{Cross Section limits using 2016 data and the N-subjettiness
tagger for the decay to qW}\tabularnewline
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endhead
1.6 & 0.10406 & 0.14720 & 0.07371 & 0.08165\tabularnewline
1.8 & 0.07656 & 0.10800 & 0.05441 & 0.04114\tabularnewline
2.0 & 0.05422 & 0.07605 & 0.03879 & 0.04043\tabularnewline
2.5 & 0.02430 & 0.03408 & 0.01747 & 0.04052\tabularnewline
3.0 & 0.01262 & 0.01775 & 0.00904 & 0.02109\tabularnewline
3.5 & 0.00703 & 0.00992 & 0.00502 & 0.00399\tabularnewline
4.0 & 0.00424 & 0.00603 & 0.00300 & 0.00172\tabularnewline
4.5 & 0.00355 & 0.00478 & 0.00273 & 0.00249\tabularnewline
5.0 & 0.00269 & 0.00357 & 0.00211 & 0.00240\tabularnewline
6.0 & 0.00103 & 0.00160 & 0.00068 & 0.00062\tabularnewline
7.0 & 0.00063 & 0.00105 & 0.00039 & 0.00086\tabularnewline
\bottomrule
\end{longtable}

\begin{longtable}[]{@{}lllll@{}}
\caption{Cross Section limits using 2016 data and the deep boosted
tagger for the decay to qW}\tabularnewline
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endhead
1.6 & 0.17750 & 0.25179 & 0.12572 & 0.38242\tabularnewline
1.8 & 0.11125 & 0.15870 & 0.07826 & 0.11692\tabularnewline
2.0 & 0.08188 & 0.11549 & 0.05799 & 0.09528\tabularnewline
2.5 & 0.03328 & 0.04668 & 0.02373 & 0.03653\tabularnewline
3.0 & 0.01648 & 0.02338 & 0.01181 & 0.01108\tabularnewline
3.5 & 0.00840 & 0.01195 & 0.00593 & 0.00683\tabularnewline
4.0 & 0.00459 & 0.00666 & 0.00322 & 0.00342\tabularnewline
4.5 & 0.00276 & 0.00412 & 0.00190 & 0.00366\tabularnewline
5.0 & 0.00177 & 0.00271 & 0.00118 & 0.00401\tabularnewline
6.0 & 0.00110 & 0.00175 & 0.00071 & 0.00155\tabularnewline
7.0 & 0.00065 & 0.00108 & 0.00041 & 0.00108\tabularnewline
\bottomrule
\end{longtable}

\begin{longtable}[]{@{}lllll@{}}
\caption{Cross Section limits using 2016 data and the N-subjettiness
tagger for the decay to qZ}\tabularnewline
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endhead
1.6 & 0.08687 & 0.12254 & 0.06174 & 0.06987\tabularnewline
1.8 & 0.06719 & 0.09477 & 0.04832 & 0.03424\tabularnewline
2.0 & 0.04734 & 0.06640 & 0.03405 & 0.03310\tabularnewline
2.5 & 0.01867 & 0.02619 & 0.01343 & 0.03214\tabularnewline
3.0 & 0.01043 & 0.01463 & 0.00744 & 0.01773\tabularnewline
3.5 & 0.00596 & 0.00840 & 0.00426 & 0.00347\tabularnewline
4.0 & 0.00353 & 0.00500 & 0.00250 & 0.00140\tabularnewline
4.5 & 0.00233 & 0.00335 & 0.00164 & 0.00181\tabularnewline
5.0 & 0.00157 & 0.00231 & 0.00110 & 0.00188\tabularnewline
6.0 & 0.00082 & 0.00126 & 0.00054 & 0.00049\tabularnewline
7.0 & 0.00050 & 0.00083 & 0.00031 & 0.00066\tabularnewline
\bottomrule
\end{longtable}

\begin{longtable}[]{@{}lllll@{}}
\caption{Cross Section limits using 2016 data and deep boosted tagger
for the decay to qZ}\tabularnewline
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endhead
1.6 & 0.16687 & 0.23805 & 0.11699 & 0.35999\tabularnewline
1.8 & 0.12750 & 0.17934 & 0.09138 & 0.12891\tabularnewline
2.0 & 0.09062 & 0.12783 & 0.06474 & 0.09977\tabularnewline
2.5 & 0.03391 & 0.04783 & 0.02422 & 0.03754\tabularnewline
3.0 & 0.01781 & 0.02513 & 0.01277 & 0.01159\tabularnewline
3.5 & 0.00949 & 0.01346 & 0.00678 & 0.00741\tabularnewline
4.0 & 0.00494 & 0.00711 & 0.00349 & 0.00362\tabularnewline
4.5 & 0.00293 & 0.00429 & 0.00203 & 0.00368\tabularnewline
5.0 & 0.00188 & 0.00284 & 0.00127 & 0.00426\tabularnewline
6.0 & 0.00102 & 0.00161 & 0.00066 & 0.00155\tabularnewline
7.0 & 0.00053 & 0.00085 & 0.00034 & 0.00085\tabularnewline
\bottomrule
\end{longtable}

Using the deep boosted tagger, the observed limit in the region where
theory and observed limit cross is very high compared to when using the
N-subjettiness tagger. This causes the tagger to perform worse than the
older tagger as the crossing of the two lines therefore happens earlier.

\begin{longtable}[]{@{}lllll@{}}
\caption{Mass limits found using the data collected in
2016}\tabularnewline
\toprule
Decay & Tagger & Limit {[}TeV{]} & Upper Limit {[}TeV{]} & Lower Limit
{[}TeV{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Decay & Tagger & Limit {[}TeV{]} & Upper Limit {[}TeV{]} & Lower Limit
{[}TeV{]}\tabularnewline
\midrule
\endhead
qW & \(\tau_{21}\) & 5.39 & 6.01 & 4.99\tabularnewline
qW & deep boosted & 4.96 & 5.19 & 4.84\tabularnewline
qZ & \(\tau_{21}\) & 4.86 & 4.96 & 4.70\tabularnewline
qZ & deep boosted & 4.49 & 4.61 & 4.40\tabularnewline
\bottomrule
\end{longtable}

\begin{figure}
  \begin{minipage}{0.5\textwidth}
  \includegraphics{./figures/results/brazilianFlag_QtoqW_2016tau_13TeV.pdf}
  \end{minipage}
  \begin{minipage}{0.5\textwidth}
  \includegraphics{./figures/results/brazilianFlag_QtoqW_2016db_13TeV.pdf}
  \end{minipage}
  \begin{minipage}{0.5\textwidth}
  \includegraphics{./figures/results/brazilianFlag_QtoqZ_2016tau_13TeV.pdf}
  \end{minipage}
  \begin{minipage}{0.5\textwidth}
  \includegraphics{./figures/results/brazilianFlag_QtoqZ_2016db_13TeV.pdf}
  \end{minipage}
\caption{Results of the cross section limits for 2016 using the $\tau_{21}$ tagger (left) and the deep boosted tagger 
(right).}
\label{fig:res2016}
\end{figure}

\hypertarget{previous-research}{%
\subsubsection{Previous research}\label{previous-research}}

The limit is already slightly higher than the one from previous
research, which was found to be 5 TeV for the decay to qW and 4.7 TeV
for the decay to qZ. This is mainly due to the fact, that in our data,
the observed limit at the intersection point happens to be in the lower
region of the expected limit interval and therefore causing a very late
crossing with the theory line when using the N-subjettiness tagger. This
could be caused by small differences of the setup used or slightly
differently processed data. In general, the results appear to be very
similar to the previous research, seen in fig.~\ref{fig:prev}.

\begin{figure}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/results/prev_qW.png}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/results/prev_qZ.png}
\end{minipage}
\caption{Previous results of the cross section limits for q\* decaying to qW (left) and q\* decaying to qZ (right). 
Taken from \cite{PREV_RESEARCH}.}
\label{fig:prev}
\end{figure}

\hypertarget{section-1}{%
\subsection{2016 + 2017 + 2018}\label{section-1}}

Using the combined data, the cross section limits seen in
fig.~\ref{fig:resCombined} were obtained. It is quite obvious, that the
limits are already significantly lower than when only using the data of
2016. The extracted cross section limits are the following:

\begin{longtable}[]{@{}lllll@{}}
\caption{Cross Section limits using the combined data and the
N-subjettiness tagger for the decay to qW}\tabularnewline
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endhead
1.6 & 0.05703 & 0.07999 & 0.04088 & 0.03366\tabularnewline
1.8 & 0.03953 & 0.05576 & 0.02833 & 0.04319\tabularnewline
2.0 & 0.02844 & 0.03989 & 0.02045 & 0.04755\tabularnewline
2.5 & 0.01270 & 0.01781 & 0.00913 & 0.01519\tabularnewline
3.0 & 0.00658 & 0.00923 & 0.00473 & 0.01218\tabularnewline
3.5 & 0.00376 & 0.00529 & 0.00269 & 0.00474\tabularnewline
4.0 & 0.00218 & 0.00309 & 0.00156 & 0.00114\tabularnewline
4.5 & 0.00132 & 0.00188 & 0.00094 & 0.00068\tabularnewline
5.0 & 0.00084 & 0.00122 & 0.00060 & 0.00059\tabularnewline
6.0 & 0.00044 & 0.00066 & 0.00030 & 0.00041\tabularnewline
7.0 & 0.00022 & 0.00036 & 0.00014 & 0.00043\tabularnewline
\bottomrule
\end{longtable}

\begin{longtable}[]{@{}lllll@{}}
\caption{Cross Section limits using the combined data and the deep
boosted tagger for the decay to qW}\tabularnewline
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endhead
1.6 & 0.06656 & 0.09495 & 0.04698 & 0.12374\tabularnewline
1.8 & 0.04281 & 0.06141 & 0.03001 & 0.05422\tabularnewline
2.0 & 0.03297 & 0.04650 & 0.02363 & 0.04658\tabularnewline
2.5 & 0.01328 & 0.01868 & 0.00950 & 0.01109\tabularnewline
3.0 & 0.00650 & 0.00917 & 0.00464 & 0.00502\tabularnewline
3.5 & 0.00338 & 0.00479 & 0.00241 & 0.00408\tabularnewline
4.0 & 0.00182 & 0.00261 & 0.00129 & 0.00127\tabularnewline
4.5 & 0.00107 & 0.00156 & 0.00074 & 0.00123\tabularnewline
5.0 & 0.00068 & 0.00102 & 0.00046 & 0.00149\tabularnewline
6.0 & 0.00038 & 0.00060 & 0.00024 & 0.00034\tabularnewline
7.0 & 0.00021 & 0.00035 & 0.00013 & 0.00046\tabularnewline
\bottomrule
\end{longtable}

\begin{longtable}[]{@{}lllll@{}}
\caption{Cross Section limits using the combined data and the
N-subjettiness tagger for the decay to qZ}\tabularnewline
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endhead
1.6 & 0.05125 & 0.07188 & 0.03667 & 0.02993\tabularnewline
1.8 & 0.03547 & 0.04989 & 0.02551 & 0.03614\tabularnewline
2.0 & 0.02523 & 0.03539 & 0.01815 & 0.04177\tabularnewline
2.5 & 0.01059 & 0.01485 & 0.00761 & 0.01230\tabularnewline
3.0 & 0.00576 & 0.00808 & 0.00412 & 0.01087\tabularnewline
3.5 & 0.00327 & 0.00460 & 0.00234 & 0.00425\tabularnewline
4.0 & 0.00190 & 0.00269 & 0.00136 & 0.00097\tabularnewline
4.5 & 0.00119 & 0.00168 & 0.00084 & 0.00059\tabularnewline
5.0 & 0.00077 & 0.00110 & 0.00054 & 0.00051\tabularnewline
6.0 & 0.00039 & 0.00057 & 0.00026 & 0.00036\tabularnewline
7.0 & 0.00019 & 0.00031 & 0.00013 & 0.00036\tabularnewline
\bottomrule
\end{longtable}

\begin{longtable}[]{@{}lllll@{}}
\caption{Cross Section limits using the combined data and deep boosted
tagger for the decay to qZ}\tabularnewline
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Mass {[}TeV{]} & Exp. limit {[}pb{]} & Upper limit {[}pb{]} & Lower
limit {[}pb{]} & Obs. limit {[}pb{]}\tabularnewline
\midrule
\endhead
1.6 & 0.07719 & 0.10949 & 0.05467 & 0.14090\tabularnewline
1.8 & 0.05297 & 0.07493 & 0.03752 & 0.06690\tabularnewline
2.0 & 0.03875 & 0.05466 & 0.02768 & 0.05855\tabularnewline
2.5 & 0.01512 & 0.02126 & 0.01080 & 0.01160\tabularnewline
3.0 & 0.00773 & 0.01088 & 0.00554 & 0.00548\tabularnewline
3.5 & 0.00400 & 0.00565 & 0.00285 & 0.00465\tabularnewline
4.0 & 0.00211 & 0.00301 & 0.00149 & 0.00152\tabularnewline
4.5 & 0.00118 & 0.00172 & 0.00082 & 0.00128\tabularnewline
5.0 & 0.00073 & 0.00108 & 0.00050 & 0.00161\tabularnewline
6.0 & 0.00039 & 0.00060 & 0.00025 & 0.00036\tabularnewline
7.0 & 0.00021 & 0.00034 & 0.00013 & 0.00045\tabularnewline
\bottomrule
\end{longtable}

The results for the mass limits of the combined years are as follows:

\begin{longtable}[]{@{}lllll@{}}
\caption{Mass limits found using the data collected in 2016 -
2018}\tabularnewline
\toprule
Decay & Tagger & Limit {[}TeV{]} & Upper Limit {[}TeV{]} & Lower Limit
{[}TeV{]}\tabularnewline
\midrule
\endfirsthead
\toprule
Decay & Tagger & Limit {[}TeV{]} & Upper Limit {[}TeV{]} & Lower Limit
{[}TeV{]}\tabularnewline
\midrule
\endhead
qW & \(\tau_{21}\) & 6.00 & 6.26 & 5.74\tabularnewline
qW & deep boosted & 6.11 & 6.31 & 5.39\tabularnewline
qZ & \(\tau_{21}\) & 5.49 & 5.76 & 5.29\tabularnewline
qZ & deep boosted & 4.92 & 5.02 & 4.80\tabularnewline
\bottomrule
\end{longtable}

\begin{figure}
  \begin{minipage}{0.5\textwidth}
  \includegraphics{./figures/results/brazilianFlag_QtoqW_Combinedtau_13TeV.pdf}
  \end{minipage}
  \begin{minipage}{0.5\textwidth}
  \includegraphics{./figures/results/brazilianFlag_QtoqW_Combineddb_13TeV.pdf}
  \end{minipage}
  \begin{minipage}{0.5\textwidth}
  \includegraphics{./figures/results/brazilianFlag_QtoqZ_Combinedtau_13TeV.pdf}
  \end{minipage}
  \begin{minipage}{0.5\textwidth}
  \includegraphics{./figures/results/brazilianFlag_QtoqZ_Combineddb_13TeV.pdf}
  \end{minipage}
\caption{Results of the cross section limits for the three combined years using the $\tau_{21}$ tagger (left) and the 
deep boosted tagger (right).}
\label{fig:resCombined}
\end{figure}

The combination of the three years has a big impact on the result. The
final limit is 1 TeV higher than what could previously be concluded.

\hypertarget{comparison-of-taggers}{%
\subsection{Comparison of taggers}\label{comparison-of-taggers}}

The previously shown results already show, that the deep boosted tagger
was not able to significantly improve the results compared to the
N-subjettiness tagger. For further comparison, in
fig.~\ref{fig:limit_comp} the expected limits of the different taggers
for the q* \(\rightarrow\) qW and the q* \(\rightarrow\) qZ decay are
shown. It can be seen, that the deep boosted is at best as good as the
N-subjettiness tagger. This was not the expected result, as the deep
neural network was supposed to provide better separation between signal
and background events than the older N-subjettiness tagger. Recently,
some issues with the training of the deep boosted tagger used in this
analysis were found, so those might explain the bad performance.

\begin{figure}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/limit_comp_w.pdf}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\includegraphics{./figures/limit_comp_z.pdf}
\end{minipage}
\caption{Comparison of expected limits of the different taggers using different datasets. Left: decay to qW. Right: 
decay to qZ}
\label{fig:limit_comp}
\end{figure}

\newpage

\hypertarget{summary}{%
\section{Summary}\label{summary}}

In this thesis, a limit on the mass of the q* particle has been
successfully established. By combining the data from the years 2016,
2017 and 2018, collected by the CMS experiment, the previously set limit
could be significantly improved. For that, a combined fit to the QCD
background and signal had to be performed and the cross section limits
extracted. Also, the new deep boosted tagger, using a deep neural
network, was compared to the older N-subjettiness tagger and found to
not significantly change the result, neither to the better nor to the
worse. Due to some training issues identified lately, there is still a
good chance, that, with that issue fixed, it will be able to further
improve the results. Also previously research of the 2016 data was
repeated and the results compared. The previous research arrived at a
exclusion limit up to 5 TeV resp. 4.7 TeV for the decay to qW resp. qZ,
this thesis at 5.4 TeV resp. 4.9 TeV. The difference can be explained by
small differences in the data used and the setup itself. After that,
using the combined data, the limit could be significantly improved to
exclude the q* particle up to a mass of 6.2 TeV resp. 5.5 TeV. With the
research presented in this thesis, it would also be possible to test
other theories of the q* particle that predict its existence at lower
masses, than the one used, by overlaying the different theory curves in
the plots shown in fig.~\ref{fig:res2016} and
fig.~\ref{fig:resCombined}.

\newpage

\nocite{*}

\printbibliography

\end{document}
