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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}
A search for an excited quark state, called q*, is presented using data
recorded by CMS during the years 2016, 2017 and 2018. By analysing its
decay channels to qW and qZ, a minimum mass of 6.1 TeV resp. 5.5 TeV is
established. This limit is about 1 TeV higher than the limits found by a
previous research of data collected by CMS in 2016
\autocite{PREV_RESEARCH}, excluding the q* particle up to a mass of 5.0
TeV resp. 4.7 TeV. Also a comparison of the new DeepAK8
\autocite{DEEP_BOOSTED} and the older N-subjettiness \autocite{TAU21}
tagger is conducted, showing that the newer DeepAK8 tagger is currently
approximately at the same level as the N-subjettiness tagger, but has
the potential to further improve in performance.

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

{
\setcounter{tocdepth}{3}
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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.
Quarks are currently considered elementary particles by the Standard
Model. The composite quark models on the other hand predict that quarks
consist of particles unknown to us so far or can bind to other particles
using unknown forces. This could explain the 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. This thesis will search for their decay to a quark and a W/Z
boson. The W/Z boson then decays in the hadronic channel, to two more
quarks. The endstate of this decay has only quarks, making Quantum
Chromodynamics effects the main background.

In a previous research \autocite{PREV_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 \(\SI{137.19}{\per\femto\barn}\) of data usable
for research. 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. By using more data and new tagging techniques, it aims to
either confirm the existence of the q* particle or improve the
previously set lower limit of 5 TeV respectively 4.7 TeV for the decay
to qW respectively qZ on its mass to even higher values. It will also
directly compare the performance of this new tagging technique to an
older tagger based on jet substructure studies used in the previous
research.

In chapter 2, a theoretical background will be presented explaining in
short the Standard Model, its shortcomings and the theory of excited
quarks. Then, in chapter 3, the Large Hadron Collider and the Compact
Muon Solenoid, the detector that collected the data for this analysis,
will be described. After that, in chapters 4-7, 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, in chapter 8, the
results are presented and compared to previous research.

\newpage

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

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{sec:sm}{%
\subsection{Standard model}\label{sec:sm}}

The Standard Model of physics proved to be 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. Fermions are further classified into quarks
and leptons. Quarks and leptons can also be categorized into three
generations, each of which contains two particles, also called flavours.
For leptons, the three generations each consist of a lepton and its
corresponding neutrino, namely first the electron, then the muon and
third, the tau. The three quark generations consist of first, the up and
down, second, the charm and strange, and third, the top and bottom
quark. So overall, their exists a total of six quark and six lepton
flavours. 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.

The matter around us, is built from so called hadrons, that are bound
states of quarks, for example protons and neutrons. Long lived hadrons
consist of up and down quarks, as the heavier ones over time decay to
those.

\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.

Due to their high masses of 80.39 GeV resp. 91.19 GeV, the \(W^\pm\) and
\(Z^0\) bosons themselves decay very quickly. Either in the leptonic or
hadronic decay channel. In the leptonic channel, the \(W^\pm\) decays to
a lepton and the corresponding anti-lepton neutrino, in the hadronic
channel it decays to a quark and an anti-quark of a different flavour.
Due to the \(Z^0\) boson having no charge, it always decays to a fermion
and its anti-particle, in the leptonic channel this might be for example
a electron - positron pair, in the hadronic channel an up and anti-up
quark pair. This thesis examines the hadronic decay channel, where both
vector bosons essentially decay to to quarks.

The quantum chromodynamics (QCD) describes the strong interaction of
particles. It applies to all particles carrying colour (e.g.~quarks).
The force is mediated by gluons. These bosons carry colour as well,
although they don't carry only one colour but rather a combination of a
colour and an anticolour, and can therefore interact with themselves and
exist in eight different variants. As a result of this, processes, where
a gluon decays into two gluons are possible. Furthermore the strength of
the strong force, binding to colour carrying particles, increases with
their distance 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. Those
jets are cone like structures made of hadrons and other particles. The
effect is called Hadronisation.

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

While being very successful in describing the effects observed in
particle colliders or the particles reaching earth from cosmological
sources, 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 asymmetry}: 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 explain the symmetries between
particles 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 any boson. Quarks are smaller than
\(10^{-18}\) m. This corresponds to an energy scale of approximately 1
TeV. Therefore the excited quark states are expected to be in that
region. 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}. As explained in
sec.~\ref{sec:sm}, the vector boson can then decay either in the
hadronic or leptonic decay channel. This research investigates only the
hadronic channel with two quarks in the endstate. 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}.

The choice of only examining the decay of the q* particle to the vector
bosons is motivated by the branching ratios calculated for the decay
\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 decay to the vector bosons have the second highest branching ratio.
The decay to a gluon and a quark is the dominant decay, but virtually
impossible to distinguish from the QCD background described in the next
section. This makes the decay to the vector bosons the obvious choice.

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
has to be calculated. This can be achieved 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.

This theory has already been investigated in \autocite{PREV_RESEARCH}
analysing data recorded by CMS in 2016, excluding the q* particle up to
a mass of 5 TeV resp. 4.7 TeV for the decay to qW resp. qZ analysing the
hadronic decay of the vector boson. This thesis aims to either exclude
the particle to higher masses or find a resonance showing its existence
using the higher center of mass energy of the LHC as well as more data
that is available now.

\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 the QCD multijet backgroubd is the dominant
background of the signal described in sec.~\ref{sec:qs}.

\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}

\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 accelerate two beams of protons to an energy of 6.5 TeV resulting in
a collision with 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 recorded by the Compact Muon Solenoid
(CMS). It is one of the four main 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, as can be seen
in fig.~\ref{fig:cms_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 an
integrated luminosity of \(\SI{37.80}{\per\femto\barn}\). In 2017 it
collected \(\SI{44.98}{\per\femto\barn}\) and in 2018
\(\SI{63.67}{\per\femto\barn}\). Of that data, in 2016
\(\SI{35.92}{\per\femto\barn}\), in 2017 \(\SI{41.53}{\per\femto\barn}\)
and in 2018 \(\SI{59.74}{\per\femto\barn}\) were usable for research.
Therefore the combined integrated luminosity of data usable for research
is \(\SI{137.19}{\per\femto\barn}\).

\begin{figure}
\hypertarget{fig:cms_setup}{%
\centering
\includegraphics{./figures/cms_setup.png}
\caption{The setup of the Compact Muon Solenoid showing its onion like
structure, the different detector parts and where different particles
are detected \autocite{CMS_PLOT}.}\label{fig:cms_setup}
}
\end{figure}

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

Per convention, the z axis points along the beam axis in the direction
of the magnetic fields of the solenoid, the y axis upwards and the x
axis horizontal towards the LHC centre. 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 also introduced. The
coordinates are visualised in fig.~\ref{fig:cmscoords}. Furthermore, to
describe a particle's momentum, often the transverse momentum, \(p_t\)
is used. It is the component of the momentum transversal to the beam
axis. Before the collision, the transverse momentum obviously has to be
zero, therefore, due to conservation of energy, the sum of all
transverse momenta after the collision has to be zero, too. If this is
not the case for the detected events, it 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, closest to the collision is a
pixel detector and around that silicon strip sensors. They are used to
reconstruct the tracks of charged particles, measuring their charge
sign, direction and momentum. They are 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 and photodetectors. When
passed by particles, the crystal produces light in proportion to the
particle's energy. This light is measured by the 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. Those have are of higher energy and therefore only
leave some energy in the ECAL but are not stopped by it.

\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. It covers a total of \(0 < |\eta| < 2.4\).
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
\(R = \sqrt{\eta^2 - \phi^2}\) in the \(\eta\) - \(\phi\) plane forming
cone like jets. If two jets overlap, the jets shape is changed according
to its hardness in regards to the transverse momentum. 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:antiktcomparison}. 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:antiktcomparison}{%
\centering
\includegraphics{./figures/antikt-comparision.png}
\caption{Comparison 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
\autocite{ANTIKT}}\label{fig:antiktcomparison}
}
\end{figure}

Fig.~\ref{fig:antiktcomparison} shows, that the jets reconstructed using
the anti-\(k_t\) algorithm have the clearest cone like shape and is
therefore chosen for this thesis.

\newpage

\hypertarget{sec:moa}{%
\section{Method of analysis}\label{sec:moa}}

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}, the decay of the q* particle to a
quark and a vector boson with the vector boson then decaying
hadronically will be investigated. This is the second most probable
decay of the q* particle and easier to analyse than the dominant decay
to a quark and a gluon. Therefore it is a good choice for this research.
The decay q* \(\rightarrow\) qV + q \(\rightarrow q\bar{q}\) + q results
in two jets, because the decay products of the heavy vector boson are
highly boosted, causing them to be very close together and therefore be
reconstructed as one jet. The dijet invariant mass of those two jets,
which is identical to the mass of the q* particle, is reconstructed. The
only background considered is the QCD background described in
sec.~\ref{sec:qcdbg}. A selection using different kinematic variables as
well as a tagger to identify jets from the decay of a vector boson is
introduced to reduce the background and increase the sensitivity for the
signal. After that, it will be looked for a peak in the dijet invariant
mass distribution at the resonance mass of the q* particle.

The data studied were collected by the CMS experiment in the years 2016,
2017 and 2018. They are 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.

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 V-tagging mechanisms will be used to compare their
performance. One based on the N-subjettiness variable used in the
previous research \autocite{PREV_RESEARCH}, the other being a novel
approach using a deep neural network, that will be explained in the
following.

\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 function have also been studied but found to be
not able to reproduce the signal shape as they couldn't model the tails
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. It can be seen that the fit works very well and therefore confirms
the functions chosen to model signal and background. This is supported
by a \(\chi^2 /\) ndof of 0.5 and a found mean for the signal at 2999
\(\pm\) 23 \(\si{\giga\eV}\) which is extremely close to the expected
3000 GeV mean. Those numbers clearly show that the method in use is able
to successfully describe the data.

\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 reduce the background and increase the signal sensitivity, a
selection of events by different variables is introduced. It is divided
into two stages. The first one (the preselection) adds some general
physics motivated selection using kinematic variables and is also used
to make sure a good trigger efficiency is achieved. 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 by comparing the data
with the simulation in the signal as well as a sideband region, where no
signal events are expected.

\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 track reconstruction.
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. The
dijet invariant mass of those two jets will be used to reconstruct the
mass of the q* particle. 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. If this is the case, the two jets with
the highest \(p_t\) are used for the reconstruction of the q* mass. The
distributions of the number of jets before and after the selection 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}

The next selection is done using \(\Delta\eta = |\eta_1 - \eta_2|\),
with \(\eta_1\) and \(\eta_2\) being the \(\eta\) of the first two jets
in regards to their transverse momentum. 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 selection as in previous research of
\(\Delta\eta \le 1.3\) is used. A comparison of the \(\Delta\eta\)
distribution before and after the selection 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 selection 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 \(m_{jj}\) 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.

\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 of the
variables that were used for the preselection can be seen for year 2016
and the combined data of years 2016 to 2018. For analysing the real data
from the CMS, jet energy corrections have to be applied. Those are to
calibrate the ECAL and HCAL parts of the CMS, so the energy of the
detected particles can be measured correctly. The corrections used were
published by the CMS group. {[}source needed, but not sure where to find
it{]}

\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 no bias in the data and Monte
Carlo simulation is introduced. It is a region in which no signal event
is expected. 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. 105 GeV is well above the mass of 91 GeV of the Z boson, the
heavier vector boson. Therefore it is very unlikely that a particle
heavier than t 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 are in
good agreement 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 their performance. 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.

Both taggers provide a discriminator value to choose whether an event
originates in the decay of a vector boson or from QCD effects. This
value will be optimized afterwards to make sure the maximum efficiency
possible is achieved.

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

The N-subjettiness \autocite{TAU21} \(\tau_N\) 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 lower the \(\tau_{21}\) is, the more likely a jet is caused by the
decay of a vector boson. Therefore a selection will be introduced, so
that \(\tau_{21}\) of one candidate jet is smaller then some value that
will be determined by an optimization process described in the next
chapter. As candidate jet the one of the two highest \(p_t\) jets
passing the softdropmass window is used. If both of them pass, the one
with higher \(p_t\) is chosen.

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

The DeepAK8 tagger \autocite{DEEP_BOOSTED} uses a deep neural network
(DNN) to identify decays originating in a vector boson. It claims to
reduce the background rate by up to a factor of \textasciitilde10 with
the same signal efficiency compared to non-machine-learning approaches
like the N-Subjettiness method. This is supported by
fig.~\ref{fig:ak8_eff}, showing a comparision of background and signal
efficiency of the DeepAK8 tagger, with, between others, the
\(\tau_{21}\) tagger also used in this analysis.

\begin{figure}
\hypertarget{fig:ak8_eff}{%
\centering
\includegraphics[width=0.8\textwidth,height=\textheight]{./figures/deep_ak8.pdf}
\caption{Comparison of tagger efficiencies, showing, between others, the
DeepAK8 and \(\tau_{21}\) tagger used in this analysis. Taken from
\autocite{DEEP_BOOSTED}}\label{fig:ak8_eff}
}
\end{figure}

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.

The higher the discriminator value of the deep boosted tagger, the more
likely is the jet to be caused by decay of a vector boson. Therefore,
using the same way to choose a candidate jet as for the N-subjettiness
tagger, a selection is applied so that this candidate jet has a
WvsQCD/ZvsQCD value greater than some value determined by the
optimization presented next.

\hypertarget{sec:opt}{%
\subsection{Optimization}\label{sec:opt}}

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. It follows from the central limit theorem
that states, that for identical distributed random variables, their sum
converges to a gaussian distribution. The significance 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 resonance nominal mass is chosen. The significance
is then calculated for different selections on the discriminant of the
two taggers and then plotted in dependence on the minimum resp. maximum
allowed value of the discriminant to pass the selection for the deep
boosted resp. the N-subjettiness tagger.

The optimization process is done using only the data from year 2018,
assuming the taggers have similar performances on the data of the
different years.

\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, and for the deep boosted
tagger 26.

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{sec:extr}{%
\section{Signal extraction}\label{sec:extr}}

After the optimization, now the optimal selection for the N-subjettiness
as well as the deep boosted tagger is found and applied to the simulated
samples as well as the data collected by the CMS. The fit described in
sec.~\ref{sec:moa} is performed for all masspoints of the decay to qW
and qZ and for both datasets used, the one from 2016 und the combined
one of 2016, 2017 and 2018.

To extract the signal from the background, its cross section limit is
calculated using a frequentist asymptotic limit calculator. It performs
a shape analysis of the dijet invariant mass spectrum to determine an
expected and 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}}

For calculating the cross section of the signal, four sources of
uncertainties are considered.

First, the uncertainty of the Jet Energy Corrections. When measuring a
particle's energy with the ECAL or HCAL part of the CMS, the electronic
signals send by the photodetectors in the calorimeters have to be
converted to actual energy values. Therefore an error in this
calibration causes the energy measured to be shifted to higher or lower
values causing also the position of the signal peak in the \(m_{jj}\)
distribution to vary. The uncertainty is approximated to be 2 \%.

Second, the tagger is not perfect and therefore some events, that don't
originate from a V boson are wrongly chosen and on the other hand
sometimes events that do originate from one are not. It influences the
events chose for analysis and is therefore also considered as an
uncertainty, which is approximated to be 6 \%.

Third, the uncertainty of the parameters of the background fit is also
considered, as it might change the background shape a little and
therefore influence how many signal and background events are
reconstructed from the data.

Fourth, the uncertainty on the Luminosity of the LHC of 2.5 \% is also
taken into account for the final results.

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

This chapter will start by presenting the results for the data of year
2016 using both taggers and comparing it to the previous research
\autocite{PREV_RESEARCH}. It will then go on showing the results for the
combined dataset, again using both taggers comparing their performances.

\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.

As described in sec.~\ref{sec:extr}, the calculated cross section limits
are used to then calculate a mass limit, meaning the lowest possible
mass of the q* particle, by finding the crossing of the theory line with
the observed cross section limit. In fig.~\ref{fig:res2016} it can be
seen, that the observed limit in the region where theory and observed
limit cross is very high compared to when using the N-subjettiness
tagger. Therefore the two lines cross earlier, which results in lower
exclusion limits on the mass of the q* particle causing the deep boosted
tagger to perform worse than the N-subjettiness tagger in regards of
establishing those limits as can be seen in tbl.~\ref{tbl:res2016}. The
table also shows the upper and lower limits on the mass found by
calculating the crossing of the theory plus resp. minus its uncertainty.
Due to the theory and the observed limits line being very flat in the
high TeV region, even a small uncertainty of the theory can cause a high
difference of the mass limit.

\hypertarget{tbl:res2016}{}
\begin{longtable}[]{@{}lllll@{}}
\caption{\label{tbl:res2016}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 established by using the N-subjettiness tagger on the 2016
data 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 (as
can be seen in fig.~\ref{fig:res2016}). This could be caused by small
differences of the setup used or slightly differently processed data.
Comparing the expected limits, there is a difference between 3 \% and 30
\%, between the values calculated by this thesis compared to the
previous research. It is not, however, that one of the two results was
constantly lower or higher but rather fluctuating. Therefore it can be
said, that the results are in good agreement. The cross section limits
of the previous research can be 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{combined-dataset}{%
\subsection{Combined dataset}\label{combined-dataset}}

Using the combined data, the cross section limits seen in
fig.~\ref{fig:resCombined} were obtained. The cross section limits are,
compared to only using the 2016 dataset, almost cut in half. This shows
the big improvement achieved by using more than three times the amount
of data.

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}

The combination of the three years not just improved the cross section
limits, but also the limit for the mass of the q* particle. The final
result is 1 TeV higher for the decay to qW and almost 0.8 TeV higher for
the decay to qZ than what was concluded by the previous research
\autocite{PREV_RESEARCH}.

\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}

\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 already found to provide a higher significance in the
optimisation done in sec.~\ref{sec:opt}. The higher significance should
also result in lower cross section limits. Apparently, doing the
optimization only on data of the year 2018, was not the best choice. To
make sure, there is no mistake in the setup, also the expected cross
section limits using only the high purity category of the two taggers
with 2018 data are compared in fig.~\ref{fig:comp_2018}. There, the
cross section limits calculated using the deep boosted tagger are a bit
lower than with the N-subjettiness tagger, showing, that the method used
for optimisation was working but should have been applied to the
combined dataset.

Recently, some issues with the training of the deep boosted tagger used
in this analysis were also found, which might explain, why it didn't
perform much better in general.

\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}

\begin{figure}
\hypertarget{fig:comp_2018}{%
\centering
\includegraphics[width=0.6\textwidth,height=\textheight]{./figures/limit_comp_2018.pdf}
\caption{Comparision of deep boosted and N-subjettiness tagger in the
high purity category using the data from year
2018.}\label{fig:comp_2018}
}
\end{figure}

\clearpage
\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 the data analysis, the following selection was applied:

\begin{itemize}
\tightlist
\item
  \#jets \textgreater= 2
\item
  \(\Delta\eta < 1.4\)
\item
  \(m_{jj} >= \SI{1050}{\giga\eV}\)
\item
  \(\SI{35}{\giga\eV} < m_{SDM} < \SI{105}{\giga\eV}\)
\end{itemize}

For the deep boosted tagger, a high purity category of \(VvsQCD > 0.95\)
and a low purity category of \(VvsQCD <= 0.95\) was used. For the
N-subjettiness tagger the high purity category was \(\tau_{21} < 0.35\)
and the low purity category \(0.35 < \tau_{21} < 0.75\). These values
were found by optimizing for the highest possible significance of the
signal.

After the selection, the cross section limits were extracted from the
data and new exclusion limits for the mass of the q* particles set.
These are 6.1 TeV by analyzing the decay to qW, respectively 5.5 TeV for
the decay to qZ. Those limits are about 1 TeV higher than the ones found
in previous research, that found them to be 5 TeV resp. 4.7 TeV.

Two different taggers were used to compare the result. The newer deep
boosted tagger was found to not improve the result over the older
N-subjettiness tagger. This was rather unexpected but might be caused by
some training issues, that were identified lately.

This research can also be used 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}.

The optimization process used to find the optimal values for the
discriminant provided by the taggers, was found to not be optimal. It
was only done using 2018 data, with which the deep boosted tagger showed
a higher significance than the N-subjettiness tagger. Apparently, the
assumption, that the same optimization would apply to the data of the
other years as well, did not hold. Using the combined dataset, the deep
boosted tagger showed no better cross section limits than the
N-subjettiness tagger, which are directly related to the significance
used for the optimization. Therefore, with a better optimization and the
fixed training issues of the deep boosted tagger, it is very likely,
that the result presented could be further improved.

\newpage

\nocite{*}

\printbibliography

\newpage
\hypertarget{appendix}{%
\section*{Appendix}\label{appendix}}

\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}

\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}

\end{document}
