\label{ch:analysis}
\chapter{Dataset and Simulated Samples}
\section{Dataset}
This analysis uses $pp$ collision data collected from 2015 to 2018 at $\sqrt{s}=13$ TeV, corresponding to 139/fb of data as shown in Figure \ref{fig:int_lumi} and \ref{fig:mu_profile}. From this dataset, only those events in which the tracker, calorimeters, and muon spectrometer have good data quality are used.  For a given event, the solenoid and toroidal magnets must also be operating at their nominal field strengths. Events with detector subsystem failures (e.g. events with LAr noise bursts, ECAL saturation, TileCal errors, and event recovery issues due to ID failures) are not used. Events with information missing from subsystems (usually from busy detector conditions) are rejected.  Events must also contain a primary vertex (PV) with at least two associated tracks, where the PV is the vertex with the largest $\sum p_{T}^{2}$ over tracks associated with the vertex with $p_{T}>0.5$ GeV.

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/lumi.png}
  \caption{Integrated luminosity for data collected from ATLAS from 2011 - 2018}.
  \label{fig:int_lumi}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/mu_profile.png}
  \caption{Mean number of interactions per crossing for data collected from ATLAS from 2011 - 2018}.
  \label{fig:mu_profile}
\end{figure}
\FloatBarrier

\section{Simulated Samples}
\label{Simulated Samples}
Samples are simulated in order to model backgrounds, evaluate signal acceptance, optimize event selection and estimate systematic and statistical uncertainties. The dominant backgrounds for this analysis are $W/Z$ + jets, diboson ($WZ$/$WW$), $t\bar{t}$, single top and multijet production.

$W/Z$+jet events are simulated using Sherpa 2.2.1 at NLO \cite{sherpa} and merged with the Sherpa parton shower using the ME+PS@NLO prescription \cite{me_ps}. These events are then normalized to NNLO cross sections. The $t\bar{t}$ and single-top backgrounds are generated with Powheg-Box with NNPDF3.0NLO PDF sets in the matrix element calculation \cite{pdfsets}. For all processes, the parton shower, fragmentation, and underlying event are simulated using Pythia 8.320 with the A14 tune set \cite{pdfsets}. Diboson processes are generated using Sherpa 2.2.1.

Signal samples are simulated using MadGraph 5-2.2.2 \cite{nnlo} and Pythia 8.186 with NNPDF230LO. RS Graviton samples are generated with $k/M_{PL}$=1. HVT Model A (B) samples are simulated with $g_{V}=1(3)$. To model VBF production of HVT signals, $g_{H}=1$ and $g_{f}=0$. Signal resonances are generated for masses between 300 GeV and 4~TeV.

\chapter{Objects}
\section{Electrons}
Electrons are reconstructed from electromagnetic showers in the ECAL. During reconstruction cells of $\Delta \eta \times \Delta \phi = 0.025 \times 0.025 $ are grouped into 3$\times$5 clusters. These clusters are then scanned for local maxima that seed electron clusters. These clusters must then be matched to a ID track from the PV. This requirement minimizes non-prompt electron and fake electron backgrounds. Electrons must pass identification and isolation requirements. Electron identification (loose, medium, tight) is based on a multivariate discriminant derived from a likelihood method. For this analysis, events are required to have one tight electron and no additional loose electrons. Electrons are also required to be isolated. The electrons are considered isolated if the quotient of the sum of the transverse momentum (of calorimeter energy deposits) in a cone around the electron of size $\Delta R = 0.2$ and the transverse momentum of the electron to be less than $0.015*p_{T}$ or 3.5 GeV, whichever is smaller. This requirement rejects non-prompt photons and fake leptons. Electrons in this analysis are also required to have $p_{T} > 30$ GeV and $|\eta| < 2.47$.

Electrons are calibrated with data-driven scale factors using $J/ \Psi \rightarrow ee$, $Z \rightarrow ee$, $Z \rightarrow \ell \ell \gamma$ processes. These corrections account for the  non-uniform response of the detector which introduces modeling and reconstruction uncertainties.

\section{Muons}
As muons traverse the entire detector, they are reconstructed from ID and MS tracks. For this analysis the muon identification and isolation working points are chosen to minimize the contributions from non-prompt muons. Towards this end, each selected event must contain exactly one muon that passes the medium identification working point, and no additional muons (that pass the loose working point). For the medium working point, two types of reconstructed muons are used: combined and extrapolated muons (CB and ME, respectively). For CB muons, ID and MS tracks are reconstructed independently and a combined track fit is performed by adding or removing MS tracks to improve the fit quality. ME muons are reconstructed from only MS tracks with hits in at least two layers, which ensures the track originates from the PV. ME muons extend the acceptance for muon reconstruction outside the ID from $2.5 < |\eta| < 2.7$.
The medium identification working point uses CB and ME tracks. CB tracks must have at least 3 hits in two MDT layers. ME tracks are required to have at least three MDT/CSC hits. To further minimize contributions from fake muons, the selected muons are required to be isolated from other tracks, as muons from $W,Z$ decays are often isolated from other particles. To insure the selected muons are isolated, the scalar sum of the transverse momentum of tracks in a cone of $\Delta R = 0.3$ compared to the transverse momentum of the muon must be less then 0.06. Muons are also required to have $p_{T} > 30$ GeV.

Muons are calibrated using well-studied resonances $J/ \Psi \rightarrow \mu \mu$ (for $p_{T}^{\mu}< 10$ GeV) and  $Z \rightarrow \mu \mu$ (for $p_{T}^{\mu} > 10$ GeV). Figure \ref{fig:muon_syst} shows the combined muon $p_{T}$ uncertainty from this calibration. The total systematic uncertainty is less then 1\% for all $p_{T}$ ranges considered in this analysis.

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/muon_syst.png}
  \caption{{This figure shows the breakdown of the muon reconstruction efficiency scale factor measured in $Z \rightarrow \mu \mu$ as a function of the muon $p_{T}$ \cite{muon_calib}. }}.
  \label{fig:muon_syst}
\end{figure}
\FloatBarrier

\section{Jets}
Three types of jets are used in this analysis: variable radius, small-R and large-R jets. Variable radius jets are used to reconstruct $Z$ bosons decaying to two $b$-jets in the jet catchment area of large-R jet in the merged regime. Small-R jets are used to to reconstruct the hadronically decaying $W/Z$ candidates in the resolved analysis and the forward jets from resonances produced through vector boson fusion. Large-R jets are used to reconstruct the hadronically decaying boson in the merged regime.

For these jet collections, the jet energy is calibrated sequentially as shown in Figure \ref{fig:jetcalib}. After the jet direction is corrected to point to the PV, the energy of the jet is corrected. First, the jet energy is corrected to account for pileup contributions based on the $p_{T}$ and area of the jet (these corrections are extracted from a $pp \rightarrow jj$ sample). Following this, another pileup correction is applied that scales with $\mu$ and $N_{PV}$.

MC-based corrections are then applied that are meant to transform the jet energy and $\eta$ back to particle level as detailed in \cite{jet_energy_calib}. These corrections account for the non-compensating nature of the ATLAS calorimeters and detector inhomogeneities. Following this, the Global Sequential Calibration is applied that reduces the flavor dependence of jet calibrations and accounts for energy leakage of jets outside the calorimeters. Finally, in-situ corrections are applied that account for differences in jet response between data and simulation ($\gamma /Z+$jet and fake lepton samples are used). These differences can arise from mismodeling the hard scattering process, pile-up, and jet formation.

To further reject jets not arising from the hadronically decaying boson, jets must pass quality requirements based on the following variables \cite{jet_selection}:

\begin{itemize}
\item[-] $f_{Q}^{LAr}$: fraction of energy of jet's LAr cells with poor signal shape
\item[-] $f_{Q}^{HEC}$: fraction of energy of jet's HEC cells with poor signal shape
\item[-] $E_{neg}$: sum of cells with negative energy
\item[-] $f_{EM}$: fraction of jet's energy deposited in EM calorimeter
\item[-] $f_{HEC}$: fraction of jet's energy deposited in HEC calorimeter
\item[-] $f_{max}$: maximum energy fraction in any single calorimeter layer
\item[-] $f_{ch}$: ratio of the scalar sum of the $p_{T}$ of a jet's charged tracks to the jet's $p_{T}$
\end{itemize}

Jets selected for the resolved analysis must pass one of the following criteria, to maximize jet efficiency:

\begin{itemize}
\item[-] $f_{HEC} > 0.5$ and $|f_{Q}^{HEC}| > 0.5$ and $\left\langle Q \right\rangle $ > 0.8, which minimizes jets formed from sporadic noise bursts in the HCAL endcap.
\item[-] $|E_{neg}| > 60$ GeV, which minimizes jets formed from sporadic noise bursts in the HCAL endcap.
\item[-] $f_{EM} > 0.95$ and $f_{Q}^{LAr} > 0.8$ and $\left\langle Q \right\rangle $ > 0.8 and $|\eta| < 2.8$, which minimizes jets formed from coherent noise or isolated pathological ECAL cells.
\item[-] $f_{max} > 0.99$ and $|\eta| < 2$, which minimizes jets mistakenly formed due to hardware issues, beam-induced backgrounds, and cosmic muon showers.
\item[-] $f_{EM} < 0.05$ and $f_{ch} < 0.05$ and $|\eta| < 2$,which minimizes jets mistakenly formed due to hardware issues, beam-induced backgrounds, and cosmic muon showers.
\item[-] $f_{EM} < 0.05$ and  $|\eta| > 2$, which minimizes jets mistakenly formed due to hardware issues, beam-induced backgrounds, and cosmic muon showers.
\end{itemize}

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/jetcalib.pdf}
  \caption{This diagram shows the calibration stages for EM jets \cite{jetcalib} .}
  \label{fig:jetcalib}
\end{figure}
\FloatBarrier

\subsection{Small-R jets}
Small-R jets are used to reconstruct the hadronically decaying $W/Z$ candidate when the two resulting jets are well-separated in $\eta$-$\phi$. Small-R jets are also used to identify forward jets from resonances produced through vector boson fusion. Small-R jets are constructed from topologically connected clusters of calorimeter cells (topoclusters), seeded from calorimeter cells with energy deposits significantly above the noise threshold.  These cells are then used as inputs to the $anti-k_{t}$ algorithm \cite{antikt} with a radius parameter, R = 0.4, implemented in the FastJet package \cite{fastjet}.

Small-R jets used in this analysis must have $p_{T} > 30$ GeV and $|\eta| < 2.5$. To further reduce jets not from the hadronically decaying boson, the jet-vertex-tagger (JVT) is used \cite{jvt}. The JVT uses two track-based variables, corrJVF and $R_{p_{T}}$ to calculate the likelihood that the jet originated from the PV. The corrJVF variable compares the scalar sum of the $p_{T}$ of tracks associated with the jet and PV to the scalar sum of the $p_{T}$ of tracks associated with the jet. This variable also includes a correction that reduces the dependency of corrJVF on the number of reconstructed vertices in the event. The other discriminant, $R_{p_{T}}$, is the ratio of the scalar sum of the $p_{T}$ of tracks associated with the jet and PV and the jet $p_{T}$. Both of these variables peak around zero for pileup jets, as these jets are unlikely to have tracks associated with the PV. JVT cuts are applied to all jets with $p_{T} > 120$ GeV. Central jets ($|\eta| < 2.4$) are required to have a JVT > 0.59 and forward jets ($2.4<|\eta| < 2.5$) are required to have JVT > 0.11.

\subsection{Large-R jets}
Large-R ($R = 1.0$) jets are used to reconstruct the hadronically decaying $W/Z$ candidate when the resulting jets are not well-separated in $\eta$-$\phi$ space, and overlap forming one large-R jet. Track-Calo Clusters (TCCs) are used to reconstruct these jets \cite{tcc}. These jets are constructed using a pseudo particle flow method using ID tracks matched to calorimeter clusters \cite{particleflow}. To remove contamination in the jet from pileup and the underlying event, jets are trimmed using a re-clustering algorithm. This algorithm removes subjets with $p_{T}^{subjet} < 0.1p_{T}^{jet}$.

The angular resolution of the calorimeter degrades sharply with jet $p_{T}$, but the jet energy resolution improves. The tracker has excellent angular resolution which improves with $p_{T}$. Therefore, by matching tracks to jets, TCCs have more precise energy and angular resolution than jets constructed from only calorimeter clusters. These jets are required to have $p_{T}>200$ GeV, $|\eta| < 2.0$ and $m_{J} > 50$ GeV.

TCC jets are trimmed as detailed in \cite{jet_trimming}, which suppresses pileup and soft radiation in the jet, the jet mass is calculated as the four-vector sum of the jet's constituents (assuming massless constituents). The jet mass peaks around the $W/Z$ boson mass for $W/Z \rightarrow qq$ jets, and more broadly for single-quark and single-gluon induced jets.

These jets are tagged as $W$ jet if the jet passes optimized jet mass and substructure ($D_{2}$) cuts for $W$ bosons, and a $Z$ jet if it passes the cuts for the $Z$ boson. The jet substructure variable $D_{2}$ is a ratio of energy correlation functions. These functions are derived from the energies and pair-wise angles of jet constituents (\cite{jet_observables}, \cite{boosted_boson}):

\begin{equation}
D_{2}^{\beta=1} = E_{CF3}\left(\frac{E_{CF1}}{E_{CF2}}\right)^{3}
\end{equation}

where the energy correlation functions are defined as:
\begin{equation}
E_{CF1}=\sum_{i}p_{T,i}
\end{equation}
\begin{equation}
E_{CF2}=\sum_{ij}p_{T,i}p_{T,j}\Delta R_{ij}
\end{equation}
\begin{equation}
E_{CF3}=\sum_{ijk}p_{T,i}p_{T,j}p_{T,k}\Delta R_{ij}\Delta R_{jk}\Delta R_{ki}
\end{equation}

A two-dimensional optimization of the jet mass and $D_{2}$ thresholds was performed to provide maximum sensitivity for this analysis. This optimization was done by maximizing the signal sensitivity (using  HVT $W'$ and $G_{KK}$ samples) against the single quark and gluon jet backgrounds in jet $p_{T}$ bins.  Figure \ref{fig:wztag_eff} shows the optimized thresholds on $D_{2}$ and jet mass as a function of $p_{T}$. Figure \ref{fig:boson_tagger_optimization} shows the efficiency of the optimized $W/Z$ taggers as a function of jet $p_{T}$.

\begin{figure}[h]
%\subfloat[][\label{fig:D2_W}]{
    \includegraphics[width=0.48\hsize]{figures/Analysis/D2_W_Fit.eps}
%}
%\subfloat[][\label{fig:Mass_W}]{
    \includegraphics[width=0.48\hsize]{figures/Analysis/Mass_W_Fit.eps}
%}

%\subfloat[][\label{fig:D2_Z}]{
    \includegraphics[width=0.48\hsize]{figures/Analysis/D2_Z_Fit.eps}
%}
%\subfloat[][\label{fig:Mass_Z}]{
    \includegraphics[width=0.48\hsize]{figures/Analysis/Mass_Z_Fit.eps}
%}
\caption{The upper cut on $D_2$ (a) and jet mass window cut i.e. the upper and lower boundary of the mass (b) of the $W$-tagger as a function of jet $p_{T}$. Corresponding values for $Z$-tagger are shown in (c) and (d). The optimal cut values for maximum significance are shown as solid markers and the fitted function as solid lines. Working points from $VV\to JJ$ \cite{VH} is also shown as dashed lines as a reference.}
\label{fig:wztag_eff}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=0.48\hsize]{figures/Analysis/sigeffW.pdf}
  \includegraphics[width=0.48\hsize]{figures/Analysis/sigeffZ.pdf}
  \caption{The left (right) plot shows the efficiency for the $W(Z)$ tagger mass and $D_{2}$ cuts as a function of large-R jet $p_{T}$.}
  \label{fig:boson_tagger_optimization}
\end{figure}
\FloatBarrier

\subsection{Variable Radius jets}
To accept more boosted $Z$ bosons decaying to $b\bar{b}$ that would normally be rejected due to topological cuts discussed \ref{Signal Region Definitions}, variable radius (VR) track jets are used to identify $b$-jets (discussed in \ref{Jet Flavor Tagging}) within the catchment area of large-R jets \cite{vrjets}. VR jets are constructed from ID tracks using the anti-$k_{t}$ algorithm with a radius parameter that depends on the $p_{T}$ of the track, shown in Equation \ref{vreq}.
\begin{equation}
R_{eff}(p_{T, i}) = \frac{\rho}{p_{T,i}}
\label{vreq}
\end{equation}
For this search $\rho=30$ GeV and a lower and upper limit on cone size are set to 0.02 and 0.4, respectively, to prevent unphysical asymptotic behavior of $\rho$. Collinear VR jets are possible, so track jets that are not separated by the the smaller jet's cone size are not used. Additionally, VR jets are required to have $p_{T} > $ 10 GeV and $|\eta| < 2.5$.

\subsection{Jet Flavor Tagging}
\label{Jet Flavor Tagging}
To more precisely identify jets, the small-R and VR jets originating from a $b$-quarks are classified using a multivariate $b$-tagging algorithm (BDT), MV2c10 \cite{atlas_btag}. This algorithm uses the impact parameters of the jet's ID tracks, secondary vertices (if they exist), and reconstructed flight paths of $b$ and $c$ hadrons in the jet to determine if the jet was induced by a $b$-quark. For this analysis, the 85\% efficient working point of this algorithm is used giving $c$, $\tau$, and light-flavor jet rejection of 3, 8, and 34 respectively in simulated $t\bar{t}$ samples.

\section{MET/Neutrinos}
Neutrinos are uncharged and colorless particles, so they do not leave tracks or jets in the detector. For this reason, neutrinos are reconstructed as the missing energy in the event, $E_{T}^{miss}$. Mathematically, $E_{T}^{miss}$ is the negative vector $p_{T}$ sum of all the physics objects and an extra soft term. The soft term accounts for energy deposits not associated with any of the objects in the event. In this analysis, the soft term is the $p_{T}$ sum of all ID tracks not associated with objects in the event. The selected tracks must be matched to the PV, which decreases pile-up contamination \cite{met_preformance}.

\section{Overlap Removal}
Reconstructed jets and leptons in this analysis can arise from the same energy deposits. For instance, a cluster of energy from an electron can also be a valid calorimeter seed for a jet. To mitigate this confusion of multiple objects originating from a single jet or lepton overlapping objects are removed via a procedure referred to a overlap removal.

The overlap selections used in this analysis are described below.

\begin{itemize}
\item[-] when an electron shares a track with another electron: the lower $p_{T}$ electron is rejected, as it is more likely to be a fake electron
\item[-] when a muon and electron share a track: the muon is rejected if it is a calo-muon, otherwise the electron is rejected
\item[-] when $\Delta R < 0.2$ for an electron and jet: the jet is rejected to maximize signal acceptance
\item[-] when $\Delta R > 0.2$ for an electron and jet: the electron is rejected as it likely originated from decays within the jet
\item[-] when $\Delta R <$min$(0.4, 0.04+10$GeV/$p_{T}^{\mu})$ for a muon and jet: the muon is rejected, again maximizing signal acceptance, otherwise the jet is rejected
\item[-] when $\Delta R < 1.0$ for the a large-R jet and electron: the jet is rejected

\end{itemize}

\section{Reconstructed Resonance Mass ($m_{WV}$)}
\label{mwv}
The $WV$ system mass, $m_{WV}$,is reconstructed from the lepton, neutrino, and hadronically-decaying boson candidate. The momentum of the neutrino along the $z$-direction is obtained by constraining the $W$ boson mass of the lepton neutrino system to be  80.3 GeV$/c^{2}$. For complex solutions to this constraint, $p_{Z}$ is taken as the real component of the solution. For real solutions, the one with the smaller absolute value is used. For the resolved analysis, $m_{WV}$ is reconstructed by constraining the $W(Z)$ dijet system in the SRs and TCRs (not the WCRs):
\begin{equation}
p^{corr}_{T,jj} = p_{T,jj} \times \frac{m_{W/Z}}{m_{jj}}
\end{equation}
\begin{equation}
m^{corr}_{jj}=m_{W/Z}
\end{equation}
where $m_{jj}$ and $m_{W/Z}$ are the reconstructed invariant mass of the hadronically-decaying W/Z boson and the PDG values of the $W/Z$ boson masses, respectively. This correction improves the $m_{WV}$ mass distribution by $\sim10\%$.

The reconstructed resonance mass is the final discriminating variable in this analysis. The distribution of this variable in SRs is used in the final likelihood fit to search for evidence of an excess of events due to BSM resonances. Distributions of $m_{WV}$ are shown in Figures \ref{fig:hvtww_cr_postfit}-\ref{fig:hvtwzvbf_tag_sr_postfit}.

\chapter{Event Selection and Categorization}
To effectively search for these new resonances, the simulated background and signal samples are analyzed to determine a series of optimized cuts used create SRs containing leptonic and hadronic resonance decay products. In these regions, the resonance mass is calculated as the combined system mass of the leptonic and hadronic systems as described in \ref{mwv}. The expected resonance mass distribution from the backgrounds and signal samples are compared to data to search for the existence of these BSM signals (also known as a bump hunt). Control regions enriched in the dominant backgrounds, $t\bar{t}$ and $W$+jets (TCR and WCR, respectively) are constructed to be orthogonal to SRs and used to determine the normalization of the $t\bar{t}$ and $W$+jets backgrounds in SRs.

Events are classified as produced via non-VBF or VBF processes using a Recursive Neural Network (RNN) described in \ref{rnn}. VBF $W'$ and $Z'$ and ggF $W'$ and $Z'$ resonances studied have unique SR and CR selections to maximize analysis sensitivity. RS Graviton signals are probed using the same selections as the ggF $Z'$ signal.

The more massive the resonance, the more likely $W/Z$ bosons are to be boosted in this search. As the boost of the hadronically decaying boson increases the separation of its hadronic decay products decreases. When the hadronically decaying boson has sufficient boost, the two quarks will overlap and not be identified separately. For this reason, a set of resolved selections are used when the hadronic decay products are reconstructed separately, and merged selections when the decay products overlap and identified as a single merged jet in the event. A $W/Z$ tagger identifies merged jets as originating from a $W/Z$ bosons based on jet substructure and mass cuts. However, the more boosted the jet is the less likely it is to pass the jet substructure cut, due to track merging. Consequently, the merged analysis uses a high purity region (HP), which requires that the jet pass both cuts, and low purity (LP) region where the jet can fail the jet substructure cut. These selections are summarized in \ref{Signal Region Definitions}.

The aforementioned SR definitions veto events with $b$-jets to minimize $t\bar{t}$ contamination. However, $b$-jets are anticipated from $W'$ resonances from the hadronically decaying $Z$ boson. To increase the signal acceptance of these resonances, a $Z\rightarrow bb$ tagger is used to construct additional SR and CRs called the tagged regions (and untagged if the event fails the $Z\rightarrow bb$ tagger).

\section{Event Pre-Selection}
Before applying topological cuts, preselection cuts are applied which include trigger and event requirements to reduce background contamination and the dataset size. Events must contain exactly one tight lepton (no additional loose leptons),  the $p_{T}^{\ell \nu} > 75$ GeV, and there must be at least two small-R jets or one large-R jet.
\section{Trigger Requirements}
The data were collected using the lowest unprescaled single-lepton or $E_{T}^{miss}$ triggers, as summarized in Table \ref{tbl:triggers}. Since the muon term is not considered in the trigger $E_{T}^{miss}$ calculation, the $E_{T}^{miss}$ trigger is fully efficient to events with high-$p_{T}$ muons. For this reason, the $E_{T}^{miss}$  trigger is used for events where $p_{T}^{\mu} > 150$ GeV, to compensate for the poor efficiency of the single muon trigger above $p_{T}^{\mu}>150$GeV.

%\begin{landscape}
\begin{table}[!htp]
\centering
\resizebox{\textwidth}{!}
{\begin{tabular}{|l|c|c|c|}
\hline
Data-taking period & $e\nu qq$ channel & $\mu\nu qq$ ($p_{T}(\mu\nu)<150$ GeV) channel & $\mu\nu qq$ ($p_{T}(\mu\nu) > 150$ GeV) channel  \\
\hline
\hline
\multirow{3}{*}{\centering {2015}} & HLT\_e24\_lhmedium\_L1EM20 OR & HLT\_mu20\_iloose\_L1MU15 OR & \multirow{3}{*}{ HLT\_xe70 } \\
 & HLT\_e60\_lhmedium OR & HLT\_mu50 & \\
 & HLT\_e120\_lhloose & & \\
\hline
\multirow{2}{*}{\centering {2016a (run $< 302919$)}} & HLT\_e26\_lhtight\_nod0\_ivarloose OR & HLT\_mu26\_ivarmedium OR  & \multirow{3}{*}{ HLT\_xe90\_mht\_L1XE50 } \\
 & HLT\_e60\_lhmedium\_nod0 OR & HLT\_mu50 &  \\
\multirow{2}{*}{($L<1.0\times10^{34}\,{\textrm cm}^{-2}\,{\textrm s}^{-1}$)} & HLT\_e140\_lhloose\_nod0 & & \\
 & HLT\_e300\_etcut & & \\
\hline
{\centering {2016b (run $\geq 302919$)}} & \multirow{2}{*}{same as above} & \multirow{2}{*}{same as above}  &  \multirow{2}{*}{HLT\_xe110\_mht\_L1XE50} \\
($L<1.7\times10^{34}\,{\textrm cm}^{-2}\,{\textrm s}^{-1}$) & & &\\
\hline
{\centering {2017}} & same as above & same as above  &  HLT\_xe110\_pufit\_L1XE55 \\
\hline
{\centering {2018}} & same as above & same as above  &  HLT\_xe110\_pufit\_xe70\_L1XE50  \\
\hline
\end{tabular}}
\caption{The list of triggers used in this analysis.} \label{tbl:triggers}
\end{table}
%\end{landscape}
\section{VBF/non-VBF RNN}
\label{rnn}
To classify events as originating from non-VBF or VBF processes a RNN is used \cite{rnn}. This approach is more powerful than a cut-based classification as it improves analysis sensitivity by exploiting correlations between variables that the RNN learns. In particular, a RNN architecture is ideal as it can handle variable numbers of jets in the events.

The RNN uses the four-momentum of candidate VBF jets to classify events as VBF or non-VBF topologies. Sometimes jets are incorrectly reconstructed, so the number of jets in the event is expected to vary across the input samples. VBF candidate jets are identified by removing jets from the event that are likely from $W/Z \rightarrow qq$. For the resolved regime this means removing the two leading small-R jets from the VBF candidate jet list. For the merged regime this means removing small-R jets separated by less than $\Delta R=1.0$ from the large-R jet. VBF candidate jets are also required to be within $|\eta| < 4.5$. From the list of remaining VBF candidate jets, the two highest-$p_{T}$ jets are chosen.

The architecture of the RNN is shown in Figure \ref{fig:rnn_architecture}. The RNN is composed of Long Short Term Memory Cells (LSTM) that extract meaningful information and retain it. The logic embedded in the LSTM is shown in Figure \ref{fig:lstm_struct}. LSTMs are useful for VBF event classification for events with two jets, where using the kinematic properties of both jets (and their correlations) will lead to more efficient event classification.

In this RNN architecture, the VBF candidates are first passed to a masking layer which checks the number of jets in the event. If there is only one jet, only one vertical LSTM layer is used. The output of masking is then passed to a LSTM, with a tanh activation function. The output of the LSTM is then passed to a second horizontal LSTM layer (and vertical LSTM layer if there are two jets in the event). Finally the output of the last LSTM cell is passed to a dense layer and then to a sigmoid activation layer, leading to an overall RNN score.

The weights and other parameters of the network are learned by training the network with HVT VBF and non-VBF signals and all simulated backgrounds over 200 epochs with an Adam Optimizer \cite{adamopt}. To prevent overfitting during training, dropout is applied to RNN weights and training is truncated if the network parameters are unchanged after ten iterations \cite{rnndropout}. Figure \ref{fig:rnn_roc} shows the ROC curve for the RNN using k-fold cross validation \cite{kfold}.

 Figure \ref{fig:rnn_score} shows the RNN discriminant for backgrounds, non-VBF signals, and VBF signals. The RNN score is $\sim$ 0 for non-VBF signals and background processes and $\sim$1 for VBF processes. Figure \ref{fig:rnn_limits} shows the limits for various signal processes based on the RNN cut applied. Requiring the RNN score to be $>$ 0.8 was chosen as it provided the best analysis significance for this final state and the $\nu\nu qq$ and $\ell \ell qq$ channels, which this channel will be combined with for future publications.

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/rnn_architecture.png}
  \caption{This figure shows the architecture of the RNN used to classify events as non-VBF/VBF. The two VBF candidate jet's variable are passed to a through two layers of LSTMs. The vector output of the final LSTM is combined to give the scalar output of the RNN used to classify the event as non-VBF/VBF. }
  \label{fig:rnn_architecture}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/LSTM_structure.jpeg}
  \caption{This figure shows the embedded logic in LSTM cells. This image was taken from \cite{lstmstruct}, where a more in depth discussion about LSTMs may be found. }
  \label{fig:lstm_struct}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/rnn.pdf}
  \caption{RNN Score distribution for ggF and VBF signals and backgrounds.}
  \label{fig:rnn_score}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/kFold_ROC.png}
  \caption{ROC curve using k-fold validation for RNN.}
  \label{fig:rnn_roc}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/rnn_limits.pdf}
  \caption{Comparison of ggF Z' limits for different RNN score selections. The bottom panel shows the ratio of the upper limits set for different RNN cuts to the cut-based analysis. In this panel smaller numbers, indicate that the expected upper limit is smaller than the cut-based analysis, which is desired.}
  \label{fig:rnn_limits}
\end{figure}
\FloatBarrier

\section{Signal Region Definitions}
\label{Signal Region Definitions}
Signal regions are constructed to be dominated by signal and used in the final likelihood fit to look for a bump in the reconstructed resonance mass distribution. Once an event is classified by the RNN, it must pass topological cuts that maximize $S/\sqrt{B}$. To efficiently select events with a $W\rightarrow \ell \nu$ candidate exactly one tight lepton is required and $E_{T}^{miss} > 100(60)$ GeV and $p_{T,\ell\nu} > 200(75)$ GeV in the merged (resolved) analysis to suppress the fake lepton backgrounds.

The resonances this search probes are expected to be produced approximately at rest with the two resulting bosons produced back-to-back. For this reason, it is required that the minimum value of $(p_{T, \ell \nu}, p_{T,J})/m_{WV} > 0.35(0.25)$ for the non-VBF (VBF) category.

To reduce $t\bar{t}$ contamination in the merged HVT $Z'$ and $G_{KK}$ analyses, events with at least one $b$-jet with $\Delta R > 1.0$ from the large-R jet are excluded. High purity signal regions require the $D_{2}$ and $W/Z$ mass window cut to be passed, whereas the low purity region only requires the  $W/Z$ mass window cut to be passed. More boosted jets, are more likely to fail the $D_{2}$ cut due to track merging. Therefore, by using high and low purity regions, the signal acceptance is increased.

The HVT $W'$ resonance search uses tagged and untagged regions to minimize backgrounds and increase signal acceptance. For events to be classified as tagged the large-R jet must contain exactly two $b$-tagged VR jets. Untagged events must have no more than one $b$-tagged jet matched to the large-R jet. These selections are shown in Table \ref{tab:SRdefinitions_1lep_merged}.

Events failing the merged selection are then re-analyzed in the resolved category. To enhance resolved signals, the event should contain one leptonic and one hadronic boson candidate that are back-to-back in $\phi$ as shown by the selections in Table \ref{tab:SRdefinitions_1lep_resolved}. Again, to suppress the $t\bar{t}$ backgrounds, events are required to have no additional $b$-jets for the HVT $Z'$ and $G_{KK}$ analyses. A summary of the resolved selections is shown in Table \ref{tab:SRdefinitions_1lep_resolved}.

The analysis cutflow in Figure \ref{fig:cutflow} shows how the different categories are prioritized. Events classified as VBF events are classified as merged high purity, merged low purity or resolved signal region selections sequentially. If the event does not pass any of these selections but passes a VBF control region selection it is classified as a VBF CR event.

If the event fails all VBF categories, it is then checked if it passes the merged high purity, merged low purity or resolved signal region selections (NB: for the $WZ$ decay modes all the regions have tagged and untagged categories). If the event fails all of the non-VBF signal region selections, it is then kept for non-VBF control region selections, if it passes those selections. Control region selection are discussed more in \ref{crs}.

Overall, for the DY HVT $Z'$ and ggF $G_{KK}$ signals there are 3 signal regions. For the DY HVT $W'$ signal there are 6 signal regions. For VBF HVT $W'$ and $Z'$ signals there are 3 signal regions.

\begin{table}[t]
\centering
\resizebox{\textwidth}{!}
{\begin{tabular}{|l|l|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|l|}{\multirow{2}{*}{Selection}} & \multicolumn{2}{c|}{SR}  &  \multicolumn{2}{c|}{$W$ CR (WR)}  & \multicolumn{2}{c|}{$t\bar{t}$ CR (TR1)} \\
\cline{3-8}
\multicolumn{2}{|l|}{} & HP & LP &HP & LP & HP & LP \\
\hline
\multirow{4}{*}{$W\rightarrow \ell\nu$} & Num of Tight leptons & \multicolumn{6}{c|}{ 1 } \\
\cline{2-8}
&Num of Loose leptons & \multicolumn{6}{c|}{ 0 }  \\
\cline{2-8}
&\vphantom{\Large B} $E_{T}^{miss}$& \multicolumn{6}{c|}{ $>100$ GeV } \\
\cline{2-8}
&$p_{T}(\ell\nu)$ & \multicolumn{6}{c|}{ $>200$ GeV } \\
\hline
\multirow{4}{*}{$W/Z\rightarrow J$} & Num of large-$R$ jets & \multicolumn{6}{c|}{ $\geq 1$ } \\
\cline{2-8}
& \vphantom{\Large B} $D_2$ cut & pass & fail & pass & fail & pass & fail \\
\cline{2-8}
 & $W/Z$ mass window cut & pass & pass & fail & fail & pass & pass\\
\cline{2-8}
 & Numb. of associated VR track jets $b$-tagged & \multicolumn{6}{c|}{ For $Z\to J$: $\leq 1$ ($= 2$) for untagged (tagged) category } \\
\hline
Topology cut & $\min \left( p_{T,\ell\nu}, p_{T,J} \right)/ m_{WV}$ & \multicolumn{6}{c|}{ $>0.35 (0.25)$ for DY/ggF (VBF) category } \\
\hline
Top-quark veto & Num of $b$-tagged jets outside of large-R jet & \multicolumn{4}{c|}{0} & \multicolumn{2}{c|} {$\geq 1$} \\
\hline
\multicolumn{2}{|c|}{Pass VBF selection} & \multicolumn{6}{c|}{ no (yes) for DY/ggF (VBF) category } \\
\hline
\end{tabular}}

\caption{Summary of selection criteria used to define the signal region (SR), $W$+jets control region ($W$ CR) and $t\bar{t}$ control region ($t\bar{t}$ CR) for merged 1-lepton channel.}
\label{tab:SRdefinitions_1lep_merged}
\end{table}

\begin{table}[t]
  \caption{The list of selection cuts in the resolved analysis for the $WW$ and $WZ$ signal regions (SR), $W$+jets control region (WR) and $ t\bar{t}$ control region (TR).} \label{tab:SRdefinitions_1lep_resolved}
\begin{center}
\resizebox{\textwidth}{!}{
\begin{tabular}{|l|l|c|c|c|}
\hline
\multicolumn{2}{|l|}{cuts} & SR & $W$ CR (WR) & $t\bar{t}$ CR (TR1) \\
\hline
\multirow{4}{*}{$W\rightarrow \ell\nu$ } & Number of Tight leptons & \multicolumn{3}{c|}{ 1 } \\
\cline{2-5}
&Number of Loose leptons & \multicolumn{3}{c|}{ 0 }  \\
\cline{2-5}
&$E_{T}^{miss}$ & \multicolumn{3}{c|}{ $> 60$ GeV } \\
\cline{2-5}
&$\$p_{T}(\ell\nu)$ & \multicolumn{3}{c|}{ $> 75$ GeV } \\
\hline
\multirow{7}{*}{$W/Z\rightarrow jj$ } & Number of small-R jets & \multicolumn{3}{c|}{ $\geq2$ } \\ %$\geq 2$ & $\geq 2$ & $\geq 2$  \\
\cline{2-5}
& Leading jet $p_{T}$ & \multicolumn{3}{c|}{ $> 60$ GeV}\\
\cline{2-5}
& Subleading jet $p_{T}$ & \multicolumn{3}{c|}{ $> 45$ GeV}\\
\cline{2-5}
 &$Z \rightarrow q\bar{q}$     &   $78 < m_{jj} < 105$ GeV & $50 <m_{jj} < 68$ GeV or & \multirow{2}{*}{$50 < m_{jj} < 150$ GeV} \\
 &$W \rightarrow q\bar{q}$     &   $68 < m_{jj} < 98$ GeV & $105 < m_{jj} < 150$ GeV & \\
\cline{2-5}
 & Num. of $b$-tagged jets &  \multicolumn{3}{c|}{ For $Z\to jj$: $\leq 1$ ($=2$) for untagged (tagged) category } \\
\hline
\multirow{5}{*}{Topology cuts} & $\Delta\phi(j,\ell)$ & \multicolumn{3}{c|}{ $>1.0$}\\
\cline{2-5}
& $\Delta\phi(j,E_{T}^{miss})$ & \multicolumn{3}{c|}{ $>1.0$}\\
\cline{2-5}
& $\Delta\phi(j,j)$ & \multicolumn{3}{c|}{ $<1.5$}\\
\cline{2-5}
& $\Delta\phi(\ell,E_{T}^{miss})$ & \multicolumn{3}{c|}{ $<1.5$}\\
\cline{2-5}
& $\min \left( p_{T,\ell\nu},  p_{T,jj} \right) / m_{WV}$ &\multicolumn{3}{c|}{$>0.35 (0.25)$ for DY/ggF (VBF) category}\\
\hline
Top veto &  Number of additional $b$-tagged jets & \multicolumn{2}{c|}{0} & $\geq 1$ \\
\hline
\multicolumn{2}{|c|}{Pass VBF selection} & \multicolumn{3}{c|}{ no (yes) for DY/ggF (VBF) category } \\
\hline
\end{tabular}
}
\end{center}
\end{table}

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/cutflow.pdf}
  \caption{This diagram shows the prioritization scheme used to classify events into the various SRs and CRs. The VBF regions are prioritized over the non-VBF regions and the merged analysis is prioritized over the resolved analysis.}

  \label{fig:cutflow}
\end{figure}
\FloatBarrier

\section{Selection Acceptance and Efficiency}
The signal acceptance is the ratio of the number of signal events selected to the number of signal events generated at truth level, which does not account for detector effects. The signal efficiency is the ratio of the number of reconstructed events selected and the number of truth events selected, which accounts for detector effects. The expected number of signal events is given by the product of these two quantities:

\begin{equation}
A \cdot \epsilon = \frac{N^{\mathrm{truth}}_{\mathrm{events\, selected}}}{N^{\mathrm{truth}}_{\mathrm{events\, generated}}} \cdot \frac{N^{\mathrm{reco}}_{\mathrm{events\, selected}}}{N^{\mathrm{truth}}_{\mathrm{events\, selected}}}
\\
= \frac{N^{\mathrm{reco}}_{\mathrm{events\, selected}}}{N^{\mathrm{truth}}_{\mathrm{events\, generated}}}
\end{equation}
The distributions of $A\cdot \epsilon$ as a function of the resonance mass for the different spin models are shown in Figures \ref{fig:accept_hvtww} - \ref{fig:accept_rsg}.

\begin{figure}[h!]
  \centering
\includegraphics[width=0.48\hsize]{figures/Analysis/signal_acceptance/Acc_times_Eff1lepDYHVT.pdf}
    \includegraphics[width=0.48\hsize]{figures/Analysis/signal_acceptance/Acc_times_Eff1lepVBFHVT.pdf}

      \caption{Selection acceptance times efficiency for the $W'\to WZ\to \ell \nu qq$ events from MC simulations as a function of the $W'$ mass for Drell-Yan (left) and VBF production (right), combining the merged HP and LP signal regions of the $WV\to \ell\nu J$ selection and the resolved regions of the $WV\to \ell\nu jj$ selection. Note: the VBF selection acceptance for the DY $W'$ is approximately zero in the left plot.}
  \label{fig:accept_hvtww}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
\includegraphics[width=0.48\hsize]{figures/Analysis/signal_acceptance/Acc_times_Eff1lepggFRSG.pdf}
    \includegraphics[width=0.48\hsize]{figures/Analysis/signal_acceptance/Acc_times_Eff1lepVBFRSG.pdf}

      \caption{Selection acceptance times efficiency for the $G\to WW\to \ell \nu qq$ events from MC simulations as a function of the $G$ mass for (a) Drell-Yan and (b) VBF production, combining the merged HP and LP signal regions of the $WV\to \ell\nu J$ selection and the resolved regions of the $WV\to \ell\nu jj$ selection. Note: the VBF selection acceptance for the ggF $G_{KK}'$ is approximately zero in the left plot.}
  \label{fig:accept_rsg}
\end{figure}
\FloatBarrier
\chapter{Background Estimate}
Backgrounds from $VV$, $t\bar{t}$, single-top, $W$+jets, $Z$+jets are simulated as described in \ref{Simulated Samples}. The dominant backgrounds for this search are from $W$+jet and $t\bar{t}$ processes. To more accurately model the $m_{WV}$ distribution from these backgrounds in the SRs, CRs are constructed for each as described in \ref{crs}. The $t\bar{t}$ and $W$+jets control regions are called TCR and WCR, respectively. There are separate control regions for VBF and non-VBF regions as well as for each region (merged HP, merged LP, resolved). For the HVT $W'$ search there are also tagged and untagged control regions (where tagged refers to events with two $b$-jets inside the large-R jet).

The shape of the aforementioned backgrounds containing real leptons are well-modeled with simulated samples. Backgrounds with fake leptons (also referred to as the multijet background) are not well-modeled with simulation. For this reason, the multijet background is extracted from data as described in \ref{fakelep}.

\section{Control Regions}
\label{crs}
The CRs are constructed to accurately model $W$+jets and $t\bar{t}$, the two dominant backgrounds in this search. These control regions are dominated by these processes and constrain the normalization of these backgrounds in the final likelihood fit
.
The TCRs use the same selections as the SRs, but must also contain at least one $b$-jet in the event (that is not within the catchement area of the large-R jet for the merged analysis). The WCR shares the SR selections as well, but uses different jet mass requirements. For the merged analyses, the large-R jet must fail the $W/Z$ tagger jet mass cut. In the resolved analyses, $m_{jj}$ must be $50 < m_{jj} < 68$ GeV or $105 < m_{jj} < 150$ GeV.

The TCR distributions of variables used in merged analysis (e.g. $m_{WV}$, $p_{T}(\nu)$, $p_{T}(J)$), in the non-VBF and VBF: HP and LP regions, are shown in Figures~\ref{fig:merged_hp_ww_TCR_datamc}-\ref{fig:merged_lp_wz_TCR_datamc}. The TCR distributions for the variables used in the resolved analysis (e.g. $m_{WV}$, $p_{T}(\nu)$, $p_{T}(j_{1}), p_{T}(j_{2})$) are shown in Figures~\ref{fig:resolved_ww_TCR_datamc} and \ref{fig:resolved_wz_TCR_datamc}. In general, in these plots the simulated distributions match the data well, which is necessary to have confidence in the prediction yields in the signal regions.

\begin{figure}[htbp]
  \centering
  \includegraphics{figures/Analysis/datamc/merged_hp_ww_tcr.pdf}
    \caption{Data MC comparison for the merged $WW$ HP TCR. The bottom panel shows the ratio of the difference between data and simulation to simulation. The red bands include the all systematic and statistical uncertainties on the background. }
  \label{fig:merged_hp_ww_TCR_datamc}
\end{figure}
\FloatBarrier

\begin{figure}[htbp]
  \centering
  \includegraphics{figures/Analysis/datamc/merged_lp_ww_tcr.pdf}
      \caption{Data MC comparison for the merged $WW$ LP TCR. The bottom panel shows the ratio of the difference between data and simulation to simulation. The red bands include the all systematic and statistical uncertainties on the background. }
  \label{fig:merged_lp_ww_TCR_datamc}
\end{figure}
\FloatBarrier

\begin{figure}[htbp]
  \centering
  \includegraphics{figures/Analysis/datamc/merged_hp_wz_tcr.pdf}
    \caption{Data MC comparison for the merged $WZ$ HP TCR. The bottom panel shows the ratio of the difference between data and simulation to simulation. The red bands include the all systematic and statistical uncertainties on the background. }
  \label{fig:merged_hp_wz_TCR_datamc}
\end{figure}
\FloatBarrier

\begin{figure}[htbp]
  \centering
  \includegraphics{figures/Analysis/datamc/merged_lp_wz_tcr.pdf}
      \caption{Data MC comparison for the merged $WZ$ LP TCR. The bottom panel shows the ratio of the difference between data and simulation to simulation. The red bands include the all systematic and statistical uncertainties on the background. }
  \label{fig:merged_lp_wz_TCR_datamc}
\end{figure}
\FloatBarrier

\begin{figure}[htbp]
  \centering
  \includegraphics{figures/Analysis/datamc/resolved_ww_tcr.pdf}
      \caption{Data MC comparison for the resolved $WW$ TCR. The bottom panel shows the ratio of the difference between data and simulation to simulation. The red bands include the all systematic and statistical uncertainties on the background. }
  \label{fig:resolved_ww_TCR_datamc}
\end{figure}
\FloatBarrier

\begin{figure}[htbp]
  \centering
  \includegraphics{figures/Analysis/datamc/resolved_wz_tcr.pdf}
      \caption{Data MC comparison for the resolved $WZ$ TCR. The bottom panel shows the ratio of the difference between data and simulation to simulation. The red bands include the all systematic and statistical uncertainties on the background. }
  \label{fig:resolved_wz_TCR_datamc}
\end{figure}
\FloatBarrier
\section{Fake Lepton Backgrounds}
\label{fakelep}
The fake lepton backgrounds for this search are not well-modeled with simulation. For this reason, this background is extracted from data. Fake electrons arise from fake jets and converted photons. Non-prompt muons often arise from heavy flavor decay products. This predominately occurs for lower lepton momentum, and therefore is only considered in the resolved analysis.

Fake electrons generally fail the electron ID criteria and fake muons fail the muon isolation requirement. Therefore, separate multijet samples are derived for the fake electron and muon samples. For each sample the $m_{WV}$ template shape is derived for the SR and WCR selections using the same SR and WCR cuts but with inverted lepton requirements as seen in Table~\ref{tab:TempMJCR}. NB: By inverting the lepton isolation/identification criteria the SRs and CRs are orthogonal.

To derive the multijet template in a given SR, first the multijet template in the WCR is derived, called the MJCR template. This template is calculated using events that pass the WCR selection but with the inverted lepton criteria. The $E_{T}^{miss}$ distribution for the MJCR is given by the difference between data and the simulated samples in the MJCR. The $E_{T}^{miss}$ distribution of those events is then added to the simulated backgrounds in the WCR. The floating background and multijet normalizations of the MJCR in this region are then fit to the data. The fitted MJCR is then used as the multijet sample in the WCR.

The fitted normalizations from the MJCR template are then used to construct the multijet template in the SR (MJSR). The MJSR is constructed from events that pass the SR selections but with the inverted lepton criteria. Again, the difference between the data and simulated backgrounds in this region gives MJSR template shape in $m_{WV}$. This shape is then scaled by the fitted normalizations from the MJCR. These fitted electron and muon muon multijet templates are then used as the multijet samples in the SRs. The normalizations of the electron and muon multijet samples are parameters in the final likelihood fit.

This template method was validated using WCR and full Run 2 data. The results of the fit are shown in Table \ref{tab:template_validation_CR}. The multijet contribution in the muon channel for $p_{T}^{W} > 150$ GeV is consistent with zero, and therefore neglected in the final fit. Applying the extracted normalization factor to MJCR in WCRs for various kinematic variables such as $E_{T}^{miss}$, $W$ transverse mass, lepton $p_{T}$, and the invariant mass as show in Figures \ref{fig:multijet_met_elec_ww} -\ref{fig:multijet_met_muon_wz_vbf}. These figures show good agreement between the data and background estimate.

%The template shape of the MJ background is determined by using a multijet validation region (MJVR) that requires the inverted lepton isolation/identification requirement and the two signal jets to satisfy the $m_{jj}$ requirement used in the $W$+jets CRs.
%The $E_{T}^{miss}$ distribution in MJCR is shown in Figure \ref{fig:multijet_met} for 2017 data. The template is then extracted by subtracting the data in the MJVR from the electroweak background processes. The resulting template and electroweak backgrounds are then fit to data. In this fit, %the $E_{T}^{miss}$ distribution compared to data to extract electroweak background, multijet electron and muon background normalizations.  The fitted scale factors from this MJVR template are then applied in the MJCR template. The electron and muon background normalizations in the M%JCR template are parameters in the final simultaneous fit. Technically, there should be a separate template for every CR and SR, but some MJ regions have insufficient statistics to do this. Additionally, the shapes for the MJ templates for VBF and ggF regions are found to be compatible within %statistical uncertainty. Therefore, the sample MJ template used for VBF and ggF CR/SRs, but with different pre-MJ-fit scale factors.
\begin{center}
  \begin{table}[ht]
  \centering
    \caption{Definitions of ``inverted'' leptons used in multijet control region. For the inverted muon selection, $ptvarcone30$ is given by sum of the $p_{T}$ of tracks in a cone around the muon candidate divided by the muon $p_{T}$. The size of the cone, $\delta R$ used is 10GeV/$p_{T}^{\mu}$ or 0.3, whichever is smaller. So, as the $p_{T}$ of the muon increases, the cone size used decreases. This is useful as more boosted muons are more likely to be produced in dense environments and using a smaller cone size more accurately determines the quality of the muon.}
  \small
  \begin{tabular}{|l|l|l|l|}
         \hline
         & Criterion       & signal lepton                        & inverted lepton
         \\
         \hline
  Electron & ID              & TightLH                   & \begin{tabular}[c]{@{}l@{}}MediumLH\\ !TightLH\end{tabular} \\
         & Calo Isolation  & FixedCutHighPtCaloOnlyIso  & FixedCutHighPtCaloOnlyIso                                                                           \\ \hline
  Muon   & ID              & WHSignalMuon              & WHSignalMuon                                                                                \\
         & Track Isolation & FixedCutTightTrackOnlyIso  & \begin{tabular}[c]{@{}l@{}}!FixedCutTightTrackOnlyIso\\ $ptvarcone30/pt<0.07^{*}$\end{tabular}\\ \hline
         &\multicolumn{3}{c|}{\small{ *Only applied to events with $pTW < 150GeV$}} \\ \hline
  \end{tabular}

  \label{tab:TempMJCR}
  \end{table}
\end{center}

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/multijet_met.pdf}
      \caption{The $E_{T}^{miss}$ distribution in MJCR for 2017 data in the electron channel(left), muon channel with W-boson pT < 150 GeV (center) and > 150 GeV (right). Multi-jet templates are given by the difference between the data and simulated distributions.}
  \label{fig:multijet_met}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_elec_ww.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the $WW$ electron channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_elec_ww}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_muon_ww.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the $WW$ muon channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_muon_ww}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_elec_wz_untag.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the $WZ$ untag electron channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_elec_wz_untag}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_muon_wz_untag.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the $WZ$ untag muon channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_muon_wz_untag}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_elec_wz_tag.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the $WZ$ untag electron channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_elec_wz_untag}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_muon_wz_tag.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the $WZ$ untag muon channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_muon_wz_untag}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_elec_vbf_ww.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the VBF $WW$ electron channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_elec_ww_vbf}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_muon_vbf_ww.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the VBF $WW$ muon channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_muon_ww_vbf}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_elec_vbf_wz.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the VBF $WZ$ electron channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_elec_wz_vbf}
\end{figure}
\FloatBarrier

\begin{figure}[htpb]
  \centering
  \includegraphics{figures/Analysis/multijet/mj_muon_vbf_wz.pdf}
      \caption{Postfit Data/MC comparison of distributions of $E_{T}^{miss}$, $m_{T}^{W}$, lepton and neutrino $p_{T}$, $m_{\ell \nu jj}$, lepton-$\nu$ angular distance in the VBF $WZ$ muon channel. The MJ template is obtained from the pre-MJ-fit.}
  \label{fig:multijet_met_muon_wz_vbf}
\end{figure}
\FloatBarrier

\begin{table}[ht]
    \centering
     \begin{tabular}{|c|c|c|cc|c|}
      \hline
     Region &Sample    & Yield   & R.U.    & SF    \\ \hline
     non-VBF WW WCR & Top\&W    & $650000\pm 1900$  & 0.31\%  & 0.99  \\
     &Z\&VV     & $24000$ & \multicolumn{2}{c|}{fixed} \\
     &MJ\_el    & $24000\pm 1200$  & 5.1\%  &4.0   \\
     &MJ\_mu    & $35000\pm 920$  & 2.6\%  &9.0  \\
      \hline
     non-VBF WZ untag & Top\&W    & $640000\pm 1900$  & 0.31\%  & 0.99  \\
     &Z\&VV     & $24000$ & \multicolumn{2}{c|}{fixed} \\
     &MJ\_el    & $24000\pm 1200$  & 5.1\%  &3.9   \\
     &MJ\_mu    & $36000\pm 920$  & 2.6\%  &8.7   \\
      \hline
     non-VBF WZ tag&Top\&W    & $71000\pm 690$  & 0.97\%  & 1.0  \\
     &Z\&VV     & $520$ & \multicolumn{2}{c|}{fixed} \\
     &MJ\_el    & $600\pm 450$  & 75\%  &0.094   \\
     &MJ\_mu    & $1200\pm 220$  & 19\%  &0.29   \\
      \hline
     VBF WW WCR & Top\&W    & $19000\pm 360$  & 1.9\%  & 0.93  \\
     &Z\&VV     & $1100$ & \multicolumn{2}{c|}{fixed} \\
     &MJ\_el    & $1400\pm 210$  & 15\%  &0.24   \\
     &MJ\_mu    & $1300\pm 160$  & 11\%  &0.31   \\
      \hline
     VBF WZ WCR & Top\&W    & $21000\pm 390$  & 1.8\%  & 0.94  \\
     &Z\&VV     & $1100$ & \multicolumn{2}{c|}{fixed} \\
     &MJ\_el    & $1400\pm 230$  & 16\%  &0.23   \\
     &MJ\_mu    & $1300\pm 160$  & 12\%  &0.31   \\
     \hline
     \end{tabular}\hfill%
     \\
\caption{\label{tab:template_validation_CR} Fit validation result in WCRs for 2015+16 data.
The fit is done in various WCRs, in order to obtain the corresponding scale factors for MJ templates: non-VBF resolved WCR for the $WW\to lvqq$ selection, non-VBF resolved untagged WCR for the $WZ\to lvqq$ selection,  non-VBF resolved tagged WCR for the $WZ\to lvqq$ selection,  VBF resolved WCR for the $WW\to lvqq$ selection,and VBF resolved WCR for the $WZ\to lvqq$ selection. Post-fit event yields for electroweak processes and MJ contributions are shown. The SF column shows the corresponding normalization scale factors for electroweak processes from the fit.
R.U. stands for relative uncertainty.
}
\end{table}

\chapter{Systematic Uncertainties}
This section describes the sources of systematic uncertainties of the $m_{WV}$ distribution. These uncertainties are divided into experimental and modeling uncertainties. Each systematic uncertainty is treated as a nuisance parameter in the final likelihood fit. The dominant systematics in this analysis arise from jet reconstruction and the generator choice for the $V$+jets backgrounds.

\section{Experimental Systematics}
The uncertainty on the integrated luminosity of the dataset used is 1.7\% and a systematic in the final fit. This uncertainty was calculated using $x-y$ beam separation scans \cite{lumi_measurement}.

Another source of systematic uncertainty is assigned to the pileup modeling in MC samples. This ensures simulated detector response and particle reconstruction conditions are as similar as possible. The distribution of the average number of interactions per bunch crossing applied to simulation is called the $\mu$ profile. The pileup modeling uncertainty is accounted for by reweighting simulated events so the average number of interactions per bunch crossing varies within its uncertainty due to systematics from vertex reconstruction \cite{vertexing}. The associated reweighting factors are propagated through the entire analysis chain to construct a systematic uncertainty on $m_{VV}$.

The single-lepton and $E_{T}^{miss}$ triggers used are not fully efficient, so scale factors are applied to simulation to more accurately model the data. These scale factors are given by the ratio of the distribution of offline objects before trigger selection and after trigger selection. The associated uncertainty on these scale factors are used in the final fit.

Uncertainties on small-R jet energy scale and resolution are measured in-situ by calculating the response between data and simulation. This analysis uses a reduced set of jet energy scale and resolution (JES and JER, respectively) uncertainties (totaling 30 and 8 systematics, respectively). This reduced set of systematics is determined using a principal component analysis, yielding largely uncorrelated independent systematics. The JES and JER systematics account for the dependence on $p_{T}$, $\eta$, $\mu$, flavor response and global sequential corrections. Systematic uncertainties associated with $b$-tagging are also considered. These systematics are evaluated as scale factor uncertainties, accounting for the difference in $b$-tagging efficiencies in data and MC, and the flavor dependence (between $b$, $c$, and light jets).

The $p_{T}$ scale uncertainty of the large-R jets is determined by comparing the jet's $p_{T}^{calo}$ to $p_{T}^{track}$ in di-jet simulation and data. In addition to this, uncertainties from tracking, modeling (Pythia vs Herwig), and statistical constraints are also used. The large-R jet $p_{T}$ resolution is given by smearing the jet $p_{T}$ with a Gaussian with a 2\% width.

The $W/Z$ tagging efficiency scale factor is estimated by comparing the tagging efficiency in simulation with that in data for four regions of the $W/Z$ tagger ($D_{2}$ fail, $m_{J}$ fail; $D_{2}$ pass, $m_{J}$ fail; $D_{2}$ fail, $m_{J}$ pass; $D_{2}$ pass, $m_{J}$ pass). Additionally, separate scale factors are determined for events with large-R jets from $W$ bosons and top backgrounds. A simultaneous template fit is used to fit the signal jets (jets initiated by $W/Z$ bosons or top quarks) and background jets (all other jets from the simulated backgrounds) to the data in the four regions using the $m_{J}$ distributions. The scale factor for a given region is then given by:

\begin{equation}
SF = \frac{ \epsilon_{data} = \frac{ N^{region}_{fitted-signal }}{N^{all-regions}_{fitted-signal}}}    {  \epsilon_{MC} = \frac{N^{region}_{signal}}{N^{all-regions}_{signal}}}
\end{equation}

The effects of experimental and theoretical uncertainties on the efficiency scale factor are determined by taking the ratio of efficiencies in data and simulation. By taking this ratio, uncertainties not arising for jet mass and $D_{2}$ cancel.

Lepton identification, reconstruction, isolation systematic uncertainties are determined by reconstructing the $Z$ mass peak with a tag and probe method (\cite{elec_syst},\cite{muon_syst}). The lepton energy and momentum scales are also measured with the Z mass peak. The effect of these systematics on the $m_{WV}$ distribution are $<5\%$.

As $E_{T}^{miss}$ is calculated using all the physics objects in the event, all those objects associated errors result in an uncertainty on $E_{T}^{miss}$. Additionally, the unassociated tracks used to construct $E_{T}^{miss}$ contribute to the uncertainty on $E_{T}^{miss}$.

\section{Theoretical Systematics}
Theoretical uncertainties for signal and background processes arise from uncertainties in MC parameters used. In particular for the $t\bar{t}$, $W/Z$+jets, diboson backgrounds and signal samples, the QCD scale, PDF, generator and hadronization uncertainties are considered. To assess the QCD scale uncertainty, the renormalization and factorization scales are scaled up and down by a factor of two at the event generation stage of sample production. Uncertainties from the PDF choice are evaluated by reweighting samples from the nominal PDF to a set of error PDFs which account for the uncertainty of the fits used to produce the PDF set. In addition to this, samples are re-weighted to different PDF sets to account for the arbitrariness of the PDF choice. The difference between the $m_{WV}$ distributions using different event generators is assessed by comparing samples generated with different generators. Similarly, the uncertainty in hadronization models is accounted for by comparing samples created using different hadronization models (e.g. $t\bar{t}$ Powheg is compared to aMC@NLO, $W$+jets compares Sherpa and MadGraph+Pythia samples). Figures \ref{fig:w_systs} - \ref{fig:ttbar_fsr_res} show the impact of these uncertainties on the $t\bar{t}$ and $W/Z$ + jets backgrounds. Additionally, contributions to the diboson background for the VBF analysis were found to be small and are accounted for by including a 10\% systematic in the diboson normalization in the final fit.

The normalization of the $t\bar{t}$ and $W$+jets processes impact the fake lepton template shape. The impact of these normalizations was assessed by including a shape systematic on the multijet background from varying the $t\bar{t}$ and $W$+jets normalization factors. The overall normalization of the template is a systematic in the final likelihood fit (account for other systematic effects on the template).

%fig:w_gen} - \ref{ttbar_fsr}

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/modelingsysts/w_syst.pdf}
      \caption{The W/Z+jet systematics for the a) Merged ggF, b) Resolved ggF, c) Merged VBF, and d) Resolved VBF regions. The top subplot shows the nominal and variation distributions/bands, the middle shows the ratio of the two, and the final shows just the shape of the envelope (the final uncertainty).}
  \label{fig:w_systs}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/modelingsysts/w_gen.pdf}
            \caption{The two-point generator comparison between Sherpa and MadGraph for the W/Z+jet samples in the a) Merged ggF, b) Resolved ggF, c) Merged VBF, and d) Resolved VBF regions. The normalization of the Madgraph sample is set to the Sherpa value to consider only shape effects. The bottom inlet shows the ratio of the two.}
  \label{fig:w_gen}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/modelingsysts/ttbar_gen_had_merg.pdf}
            \caption{Ratio between the variations of generator (red) and hadronization (blue) variations for the Merged regime for $t\bar{t}$ sample.}
  \label{fig:ttbar_gen_merg}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/modelingsysts/ttbar_gen_had_res.pdf}
            \caption{Ratio between the variations of generator (red) and hadronization (blue) variations for the Resolved regime for $t\bar{t}$ sample.}
  \label{fig:ttbar_gen_res}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/modelingsysts/ttbar_isr_merg.pdf}
            \caption{Ratio between the variations of ISR up (red) and down (blue) variations for the Merged regime for $t\bar{t}$ sample.}
  \label{fig:ttbar_isr_merg}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/modelingsysts/ttbar_isr_res.pdf}
            \caption{Ratio between the variations of ISR up (red) and down (blue) variations for the Resolved regime for $t\bar{t}$ sample.}
  \label{fig:ttbar_isr_res}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/modelingsysts/ttbar_fsr_merg.pdf}
            \caption{Ratio between the variations of FSR up (red) and down (blue) variations for the Merged regime for $t\bar{t}$ sample.}
  \label{fig:ttbar_fsr_merg}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Analysis/modelingsysts/ttbar_fsr_res.pdf}
            \caption{Ratio between the variations of FSR up (red) and down (blue) variations for the Resolved regime for $t\bar{t}$ sample.}
  \label{fig:ttbar_fsr_res}
\end{figure}
\FloatBarrier

\chapter{Statistical Analysis}
A statistical procedure based on a likelihood function is used to determine the compatibility of the data collected with the proposed resonances. This test compares the distribution of $m_{WV}$ for the background-only (which only considers SM processes, no new physics processes) hypothesis with the background and signal hypothesis (see Figures \ref{fig:hvtww_sr_postfit} - \ref{fig:hvtwzvbf_tag_sr_postfit} for $m_{WV}$ SR distributions). A discovery test is used to measure the compatibility of the observed data with the background-only hypothesis. If the observed data are sufficiently incompatible with the background only hypothesis, this could indicate a discovery. In the absence of discovery, upper limits on the signal strength parameter, $\mu$, are assessed using the CLs method. These $\mu$ limits are then translated into upper limits on the allowed cross section of new physics processes.

For signal masses below 500~GeV only the resolved analysis is used, as the merged analysis is not applicable for such small resonance masses. Similarly, it is unlikely that the two jets from the hadronically decaying boson will be well separated for signal masses exceeding 1~TeV. Therefore, only the merged analysis is used above 1~TeV. For signal masses between 500 - 1000 GeV, the merged and resolved analyses are combined.

\section{Likelihood Function}
The likelihood function used is product of Poisson probabilities over all $m_{WV}$ bins and the associated systematics:

\begin{equation}
\mathcal{L}(\mu,\bm{\theta})= \prod_{c} \prod_{i} \frac{(\mu s_{ci}(\bm{\theta}) + b_{ci}(\bm{\theta}))^{n_{ci}}}{n_{ci}!} e^{-(\mu s_{ci}(\bm{\theta})+b_{ci}(\bm{\theta}))}\prod_{k}(\theta^{'}_{k}|\theta_{k})
\end{equation}

Here $c$ are the analysis channels (e.g. merged SRs and CRs and resolved SRs and CRs) considered and $i$ runs over all the $m_{WV}$ bins used in the fit. The signal strength parameter, $\mu$, multiplies the expected signal yield in each analysis bin, $s_{ci}$. The background content for channel $c$ and bin $i$ is given by $b_{ci}$ . The dependence of signal and background predictions on systematic uncertainties is described by the aforementioned set of nuisance parameters $\bm{\theta}$, which are parameterized by Gaussian or log-normal priors, denoted here as $\theta_{k}$. Statistical uncertainties of the simulated bin contents are also included as systematic uncertainties. Most systematics are correlated among all the analysis regions and considered to be independent from other systematics. The validity of this assumption is checked by evaluating the covariance of nuisance parameters.

\section{Fit Configuration}
The binning of $m_{WV}$ in signal regions for the likelihood fit is depends on the signal mass resolution and statistical uncertainties of $m_{WV}$ bins. For each signal mass point, the signal mass resolution is set to the fitted Gaussian width in $m_{WV}$. The fitted signal widths are then fit to a line to give a parameterized signal mass width, as shown in Figures \ref{fig:resolved_sigwidth} and \ref{fig:merged_sigwidth}. Bin widths are set first to this parameterized signal mass resolution. Then if the statistical uncertainty of the data or simulated background is more than 50\% in any bin, bins are merged until the statistical uncertainty in all bins is less than 50\%. All control regions contain only a single bin.

\begin{figure}[h!]
  \centering
  \includegraphics[width=0.48\hsize]{figures/Analysis/signal_mass_resolution/sigres_resolved_1lephvt.pdf}
    \includegraphics[width=0.48\hsize]{figures/Analysis/signal_mass_resolution/sigres_resolved_1lephvtvbf.pdf}
 \caption{The HVT signal mass resolution as a function of mass fit with a straight line in the Resolved ggF region (left) and VBF (right) region. }
  \label{fig:resolved_sigwidth}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=0.48\hsize]{figures/Analysis/signal_mass_resolution/sigres_merged_1lephvt.pdf}
    \includegraphics[width=0.48\hsize]{figures/Analysis/signal_mass_resolution/sigres_merged_1lephvtvbf.pdf}
 \caption{The HVT signal mass resolution as a function of mass fit with a straight line in the Merged ggF region (left) and VBF (right) region. }
  \label{fig:merged_sigwidth}
\end{figure}
\FloatBarrier

For each signal model is fit simultaneously in the merged and resolved channels. The $W$+jets and $t\bar{t}$ normalizations are given by the best fit values in the overall fit and these fitted normalizations are then applied to those backgrounds in the SRs, as mentioned previously.

The $m_{WV}$ distributions for a given systematic may contain unphysically large fluctuations due to $m_{WV}$ bins with few events. This can lead to artificial pulls and/or constraints in the fit. To remove such issues a multi-step smoothing procedure is applied to all systematic variation distributions. First, distributions are rebinned until the statistical error per bin is at least 5\%. Next all local extrema are identified. The bins around smallest extrema are iteratively merged until only four local extrema remain. Then distributions are rebinned so that statistical uncertainties in each bin are $< 5\%$.

For some systematics, up and down variations may be in the same direction with respect to the nominal distributions. This causes the variations to not cover the nominal choice, and the interpretation of the confidence interval is skewed as the nominal distribution should be bracketed by the up and down variations. This asymmetry may also lead to underconstrained systematics in the fit. To handle such asymmetric systematics, if the up and down variation for a given systematic are in the same direction for at least three $m_{VV}$ bins, the variation is averaged for those bins. The averaging procedure replaces bin-by-bin the up and down variation bins by $b_{\pm}^{new}=b_{nom}\pm\frac{|b_{+}-b_{-}|}{2}$, where $b_{nom}$ is the nominal bin content and $b_{\pm}$ are the original up and down variation bin content. The same procedure is also applied to any variations where the integral of the difference between the up/down variation and the nominal distribution is twice that of the other down/up variation, further ensuring variations are symmetric around the nominal distribution.

Finally, systematics that have a negligible effect on the $m_{WV}$ distribution are not considered in the fit. Shape systematics where no bin in the variational distribution deviates more than $1\%$ from the nominal distribution (after normalizing all histograms to the nominal) are not included in the fit. Also, statistical bin uncertainties $< 1\%$ are ignored.

\section{Best Fit $\mu$}
The best-fit signal strength parameter is denoted by $\hat{\mu}$ and calculated by maximizing the likelihood function over the entire $m_{WV}$ distribution with respect to all systematics and $\mu$. The corresponding set of systematics that maximize the likelihood are given by $\bm{\hat{\theta}}$. The first term in the likelihood is maximized when the expected number of signal and background events over all $m_{WV}$ bins is equal to the number of events in data (per bin this means: $n_{ci}=\mu s_{ci} + b_{ci}$) . Thus, by maximizing the likelihood, the fit determines value of $\mu$ and $\bm{\theta}$ that give the best agreement between expected and measured event yields. The second term in the likelihood is a penalty term which decreases the likelihood when systematics are shifted from their nominal values. This prevents the fit from profiling (unexpectedly constraining or shifting the fitted systematic value far from its nominal value) systematics in unphysical ways to maximize the likelihood. The uncertainty on $\mu$ is calculated by varying $\mu$ up and down until the natural log of the likelihood function shifts by one-half.

\section{Discovery Test}
To determine if the observed dataset is consistent with a given signal model a likelihood ratio is constructed:

\begin{equation}
\lambda(\mu)=\frac{\mathcal{L}(\mu, \hat{\hat{\bm{\theta}}}_{\mu})}{\mathcal{L}(\hat{\mu}, \hat{\bm{\theta}})}
\end{equation}

The denominator in this equation is the maximized value of $\mathcal{L}$ over all systematics and $\mu$. The numerator is the maximized likelihood over all systematics for a given $\mu$ value, where the maximized systematics are given by $\hat{\hat{\bm{\theta}}}_{\mu}$. To test for the existence of signal the observed dataset the null hypothesis ($H_{0}$) is defined as the background only hypothesis and the alternate hypothesis includes signal and background ($H_{1}$). This test quantifies the compatibility of observed data with $H_{0}$ by calculating a p-value representing the probability of observing data as discrepant or more than the observed data under the $H_{0}$. The test statistic used to calculate this p-value is given by ($r_{0}$):

\begin{equation}
r_{0}=\left\{ \begin{array}{ll}
-2\ln \lambda (0), \hat{\mu} > 0\\
+2\ln \lambda (0), \hat{\mu} < 0
\end{array}
\right.
\end{equation}

The expected distribution of the the test statistic under $H_{0}$ ($f(r_{0}|0)$) is used to calculate the p-value:

\begin{equation}
p_{0}=\int_{r_0,obs}^{\infty} f(r_{0}|0)dr_{0}
\end{equation}

Small p-values indicate the observed data is poorly described by $H_{0}$. This equivalent Z-score of a given p-value is usually used to further quantify the agreement between the observed data and $H_{0}$. The Z-score is given by the number of standard deviations away from the mean of a Gaussian distribution, the integral of the upper tail of the distribution would equal the p-value. Mathematically:

\begin{equation}
Z = \Phi^{-1}(1-p_{0})
\end{equation}
where $\Phi$ is the Gaussian cumulative distribution function. The statistical significance of these tests are expressed as the $Z$-score. In particle physics, $3\sigma$ is considered evidence for new phenomena and $5\sigma$ is the threshold for discovery.
\section{Exclusion Limits}
In the absence of discovery, upper limits on the signal strength, $\mu$ are set using the CLs method \cite{cls}. The test statistic for this test, $q_{\mu}$, is constructed as:
\begin{equation}
\tilde{\lambda_{\mu}}=\left\{ \begin{array}{ll}
\frac{\mathcal{L}(\mu,\hat{\hat{\theta}}_{\mu})}{\mathcal{L}(\hat{\mu},\hat{\theta})},  \hat{\mu} > 0\\
\frac{\mathcal{L}(\mu,\hat{\hat{\theta}}_{\mu})}{\mathcal{L}(0,\hat{\hat{\theta}}_{0})},  \hat{\mu} < 0
\end{array}
\right.
\end{equation}

 \begin{equation}
\tilde{q}_{\mu}=\left\{ \begin{array}{ll}
-2\ln \tilde{\lambda}(\mu),  \hat{\mu} < \mu \\
+2\ln \tilde{\lambda}(\mu),  \hat{\mu} > \mu
\end{array}
\right.
\end{equation}

As defined, larger values of $q_{\mu}$ correspond to increasing incompatibility between the observed data and the background + signal hypothesis. The observed value of the test statistic, $q_{\mu, obs}$, is then compared to its expected distribution, $f$, to calculate p-values to assess the likelihood of the background+signal hypothesis. Using these distributions, $CL_{s}$ values are computed as:

 \begin{equation}
 CL_{s+b}=\int_{q_{\mu, obs}}^{\infty}f(q_{\mu}|\mu)dq_{\mu}\\
 \end{equation}
  \begin{equation}
CL_{b}=\int_{q_{0}^{obs}}^{\infty}f(q_{\mu}|\mu=0)dq_{\mu}\\
 \end{equation}
  \begin{equation}
CL_{s}=\frac{CL_{s+b}}{CL_{b}}
 \end{equation}

$CL_{s+b}$ is the p-value for the signal + background hypothesis and $CL_{b}$ is the p-value for the background only hypothesis. The $CL_{s}$ value is interpreted as the probability to observe the background + signal hypothesis normalized to the probability of background-only hypothesis. Normalizing by $CL_{b}$ prevents setting artificially strong exclusion limits due to downward fluctuations in data.

For a given signal hypothesis, $\mu$ values are scanned simultaneously over all $m_{WV}$ bins to find the $\mu$ value that yields $CL_{s}$=0.05, meaning the likelihood of finding data more incompatible with the background + signal hypothesis (relative to the background only hypothesis) is 5\%. The 95\% upper limit on the signal cross section is then calculated as the product of $\hat{\mu}$, BR, and theory cross section.


\section{Mutual Match}
\label{sec:mutual}

\Friending{} employs a matchmaking algorithm that creates a mutual match between participants after submitting their questionnaires.  \Friending{} does this by computing the comparison score of $Q \in Questions$ and $A,B \in Answers-to-Q$:
\begin{equation}
    CS(Q, A_1, A_2) = COMP(A_1, A_2) * CAT(Q) * IMP(A_1) * IMP(A_2) * IMP(Q)
\end{equation}

Where IMP is the importance rating from 1 to 3, CAT is the category weighting and COMP is the comparison as defined in section 4.5.5.   The comparison score for a questionnaire is then calculated based on the summation of the comparison score for all questions and their associated answers.
The compatibility score is computed from the comparison score of the questionnaire divided by the balanced comparison score for the questionnaire.  The balanced comparison score is $85\%$ of the best possible comparison score.

A mutual match is defined as a match between two participants who have a compatibility score equal to or above the minimum match threshold.

%% Comparison Score
%%
\subsection{Comparison Score}
\label{sec:score_comp}

The comparison score is computed by the function $COMP(A_1, A_2)$, which compares two user answers for a question. The matrix of comparison values is defined by the designer of the question. You can define this comparison matrix in the Question Builder as defined in section ~\ref{sec:matchcriteria}.

For example, given the set of options: [ \textit{tofu}, \textit{salad}, \textit{pizza}, \textit{burgers} ], a matrix of comparison responses is defined below.

\begin{center}
    \begin{tabular}{ l c c c r }
        & \textbf{tofu} & \textbf{salad} & \textbf{pizza} & \textbf{burgers} \\
        \textbf{tofu} & 1 & 0.75 & 0.25 & 0 \\
        \textbf{salad} & 0.5 & 0.75 & 0.75 & 0.5 \\
        \textbf{pizza} & \underline{0.5} & 0.5 & 0.5 & 0.5 \\
        \textbf{burgers} & 0 & 0.25 & 0.5 & 0.5 \\
    \end{tabular}
\end{center}

The usage $COMP(tofu, pizza)$ maps to the column by $A_1$ (i.e. \textit{tofu}), and row by $A_2$ (i.e. \textit{pizza}). The evaluation of the function would then be as such: $COMP(tofu, pizza) = 0.5$.

%% Category
%%
\subsection{Category}
\label{sec:score_category}

The category weighting contributes to the equation as $CAT(Q)$ where $Q$ is the question definition. From this question definition the weighting of the category can be returned using $\in [0, 1]$.

%% Importance
%%
\subsection{Importance}
\label{sec:score_importance}

The importance from the user is contributed to the equation as $IMP(A)$ where $A_i$ is the question definition. From this question definition the weighting set by the user can be returned with possible values $\in [ 0.0, 0.25, 0.5, 0.75, 1.0 ]$.

%% Question Importance
%%
\subsection{Question Importance}
\label{sec:scoreq_qimp}

The importance of a question contributes to the equation as $IMP(Q)$ where $Q$ is the question definition. From this question definition the importance of the question can be returned using $\in [ 0.0, 0.25, 0.5, 0.75, 1.0 ]$.


\documentclass[a4paper]{report}

\usepackage[margin=3.0cm]{geometry}
\usepackage{amsmath}
\usepackage[pdftex]{graphicx}
%\usepackage{graphics}
\usepackage{subfig}

\title{BLASFEO reference guide}
\author{Gianluca Frison}

\begin{document}

\maketitle
\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

BLASFEO - BLAS For Embedded Optimization.

BLASFEO is a library providing basic linear algebra routines performance-optimized for rather small matrices (fitting in cache).
A detailed introduction to BLASFEO can be found in the ArXiv paper at the URL
\begin{verbatim}
https://arxiv.org/abs/1704.02457
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{BLASFEO implementations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

BLASFEO comes in three implementations:
\begin{itemize}
\item reference (RF)
\item high-performance (HP)
\item wrapper to BLAS and LAPACK (WR)
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Matrix and vector data types}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The fundamental data types in BLASFEO are the C structures {\tt blasfeo\_dmat} and {\tt blasfeo\_dvec}, defining respectively a double-precision matrix and vector (and similarly {\tt blasfeo\_smat} and {\tt blasfeo\_sec} for single-precision matrix and vector).

Some structure members are common to all BLASFEO implementations, some others are specific to some BLASFEO implementation.
Therefore, some structure members should be considered public (typically, the common members), some others should be considered as private (and typically directly used only from advanced users and targeting specific BLASFEO implementations).
Below only the public members are documented.

%%%%%%%%%%%%%%%%
\section{{\tt blasfeo\_dmat} structure}
%%%%%%%%%%%%%%%%

The structure {\tt blasfeo\_dmat} defines the (double-precision) matrix type in BLASFEO.
Structures and routines for single-precision are analogue.

\subsection{{\tt blasfeo\_dmat} definition}

\begin{verbatim}
struct blasfeo_dmat
    {
    int m;
    int n;
    double *pA;
    int memsize;
    };
\end{verbatim}
where the structure members are
\begin{description}
\item[m] number of rows in the matrix
\item[n] number of columns in the matrix
\item[pA] pointer to the first element of the matrix
\item[memsize] size (in bytes) of the memory referenced by the structure
\end{description}

\subsection{{\tt blasfeo\_dmat} management}

\begin{verbatim}
void blasfeo_allocate_dmat(int m, int n, struct blasfeo_dmat *sA);
\end{verbatim}
Populates the structure defining a $m\times n$ matrix, referenced by {\tt sA} and internally dynamically allocated the memory referenced by the structure.
The use of this routine is intended for prototype and off-line use, and it not advised in performance-critical code, where the routine {\tt blasfeo\_create\_dmat} should be employed instead.

\begin{verbatim}
void blasfeo_free_dmat(struct blasfeo_dmat *sA);
\end{verbatim}
Frees the memory allocated by the routine {\tt blasfeo\_allocate\_dmat}.

\begin{verbatim}
int blasfeo_memsize_dmat(int m, int n);
\end{verbatim}
Computes the size (in bytes) of the memory referenced by the structure defining a $m\times n$ matrix.
The memory has to be externally allocated to use in the {\tt blasfeo\_create\_dmat} routine.

\begin{verbatim}
void blasfeo_create_dmat(int m, int n, struct blasfeo_dmat *sA, void *memory);
\end{verbatim}
Populates the structure defining a $m\times n$ matrix, referenced by {\tt sA}.
The memory referenced by the structure should be allocated externally and provided to the routine using the {\tt memory} pointer.
It should typically be aligned to 64-byte boundaries (the typical cache line size).
The amount of memory referenced by the structure is computed using the {\tt blasfeo\_memsize\_dmat} routine.

\subsection{{\tt blasfeo\_dmat} packing}

\begin{verbatim}
void blasfeo_pack_dmat(int m, int n, double *A, int lda, struct blasfeo_dmat *sA,
     int ai, int aj);
\end{verbatim}

\begin{verbatim}
void blasfeo_pack_tran_dmat(int m, int n, double *A, int lda, struct blasfeo_dmat *sA,
     int ai, int aj);
\end{verbatim}

\begin{verbatim}
void blasfeo_unpack_dmat(int m, int n, struct blasfeo_dmat *sA, int ai, int aj,
     double *A, int lda);
\end{verbatim}

\begin{verbatim}
void blasfeo_unpack_tran_dmat(int m, int n, struct blasfeo_dmat *sA, int ai, int aj,
     double *A, int lda);
\end{verbatim}

\subsection{{\tt blasfeo\_dmat} printing}

\begin{verbatim}
void blasfeo_print_dmat(int m, int n, struct blasfeo_dmat *sA, int ai, int aj);
\end{verbatim}

\begin{verbatim}
void blasfeo_print_tran_dmat(int m, int n, struct blasfeo_dmat *sA, int ai, int aj);
\end{verbatim}

%%%%%%%%%%%%%%%%
\section{{\tt blasfeo\_dvec} structure}
%%%%%%%%%%%%%%%%

The structure {\tt blasfeo\_dvec} defines the (double-precision) vector type in BLASFEO.
Structures and routines for single-precision are analogue.

\subsection{{\tt blasfeo\_dvec} definition}

\begin{verbatim}
struct blasfeo_dvec
    {
    int m;
    double *pa;
    int memsize;
    };
\end{verbatim}
where the structure members are
\begin{description}
\item[m] number of elements in the vector
\item[pA] pointer to the first element of the vector
\item[memsize] size (in bytes) of the memory referenced by the structure
\end{description}

\subsection{{\tt blasfeo\_dvec} management}

\begin{verbatim}
void blasfeo_allocate_dvec(int m, struct blasfeo_dvec *sx);
\end{verbatim}
Populates the structure defining a $m\times 1$ vector, referenced by {\tt sx} and internally dynamically allocated the memory referenced by the structure.
The use of this routine is intended for prototype and off-line use, and it not advised in performance-critical code, where the routine {\tt blasfeo\_create\_dvec} should be employed instead.

\begin{verbatim}
void blasfeo_free_dvec(struct blasfeo_dvec *sx);
\end{verbatim}
Frees the memory allocated by the routine {\tt blasfeo\_allocate\_dvec}.

\begin{verbatim}
int blasfeo_memsize_dvec(int m);
\end{verbatim}
Computes the size (in bytes) of the memory referenced by the structure defining a $m\times 1$ vector.
The memory has to be externally allocated to use in the {\tt blasfeo\_create\_dvec} routine.

\begin{verbatim}
void blasfeo_create_dvec(int m, struct blasfeo_dvec *sx, void *memory);
\end{verbatim}
Populates the structure defining a $m\times 1$ vector, referenced by {\tt sx}.
The memory referenced by the structure should be allocated externally and provided to the routine using the {\tt memory} pointer.
There are no alignment requirements for memory to be used by the vector structure. %It should typically be aligned to 64-byte boundaries (the typical cache line size).
The amount of memory referenced by the structure is computed using the {\tt blasfeo\_memsize\_dvec} routine.

\subsection{{\tt blasfeo\_dvec} packing}

\begin{verbatim}
void blasfeo_pack_dvec(int m, double *x, int incx, struct blasfeo_dvec *sy, int yi);
\end{verbatim}

\begin{verbatim}
void blasfeo_unpack_dvec(int m, struct blasfeo_dvec *sx, int xi, double *y, int incy);
\end{verbatim}

\subsection{{\tt blasfeo\_dvec} printing}

\begin{verbatim}
void blasfeo_print_dvec(int m, struct blasfeo_dvec *sx, int xi);
\end{verbatim}

\begin{verbatim}
void blasfeo_print_tran_dvec(int m, struct blasfeo_dvec *sx, int xi);
\end{verbatim}

\end{document}


\RequirePackage[l2tabu, orthodox]{nag}
\documentclass{article}

\usepackage[letterpaper, margin=1.3cm]{geometry}
\usepackage{booktabs}
\usepackage{mathtools}
\usepackage[final]{pdfpages}

\title{ECE 487 Assignment 2}
\author{Michael Kwok}
\begin{document}

\maketitle
\subsection*{Detection/Correction}
\begin{itemize}
    \item 6
    \item 7
\end{itemize}

\subsection*{Hamming Distance}
\begin{itemize}
    \item detected: 9, corrected: 4
    \item detected: 6, corrected: 3
\end{itemize}

\subsection*{Decoding}
\begin{itemize}
    \item \(11110 \oplus 11110 = 00000\), hamming distance 0, \verb|11|
    \item \(00010 \oplus 00000 = 00010\), hamming distance 1, \verb|00|
    \item \(00011 \oplus 01011 = 01000\), hamming distance 1, \verb|01|
    \item \(01110 \oplus 11110 = 10000\), hamming distance 1, \verb|11|
\end{itemize}

\subsection*{Linear block code}

This block code is a linear block code. This is due to the fact that every pair of codewords form another valid codeword.

\begin{align*}
    01111 \oplus 10001 & = 11110 \\
    01111 \oplus 11110 & = 10001 \\
    10001 \oplus 11110 & = 01111 \\
\end{align*}

\subsection*{Receiver Action}
\begin{itemize}
    \item Calculate syndrome: \((0 + 1 + 1 + 1 + 1) \mod 2 = 0\), Receiver will return \verb|0111|.
    \item Calculate syndrome: \((0 + 1 + 0 + 1 + 1) \mod 2 = 1\), Receiver will not return data.
\end{itemize}

\subsection*{2D Parity}

\begin{verbatim}
Sent:            Received:
1 1 1 1 1 1      1 0 1 1 0 1
1 0 1 1 1 0      1 0 1 1 1 0
0 1 1 1 0 1  ->  0 1 1 1 0 1
0 1 0 1 0 0      0 1 0 1 0 0
0 1 1 0 0 0      0 1 0 0 1 0
\end{verbatim}
The receiver will calculate the syndrome bits row by row and then column by column. The calculations will look like this:
\begin{itemize}
    \item r1: \((1 + 0 + 1 + 1 + 0 + 1) \mod 2 = 0\)
    \item r2: \((1 + 0 + 1 + 1 + 1 + 0) \mod 2 = 0\)
    \item r3: \((0 + 1 + 1 + 1 + 0 + 1) \mod 2 = 0\)
    \item r4: \((0 + 1 + 0 + 1 + 0 + 0) \mod 2 = 0\)
    \item r5: \((0 + 1 + 0 + 0 + 1 + 1) \mod 2 = 0\)
    \item c1: \((1 + 1 + 0 + 0 + 0) \mod 2 = 0\)
    \item c2: \((0 + 0 + 1 + 1 + 1) \mod 2 = 1\)
    \item c3: \((1 + 1 + 1 + 0 + 0) \mod 2 = 1\)
    \item c4: \((1 + 1 + 1 + 1 + 0) \mod 2 = 0\)
    \item c5: \((0 + 1 + 0 + 0 + 1) \mod 2 = 0\)
    \item c6: \((1 + 0 + 1 + 0 + 0) \mod 2 = 0\)
\end{itemize}

The errors here are detected, but cannot be corrected.
\end{document}