\chapter{Interpretations: 3rd generation squarks}
    \label{chapter:Interpretations}

In this chapter, the model independent upper limits on the visible cross section are presented for the monojet analysis.
The monojet results are then interpreted in terms of searches for new physics in different models with stops or sbottoms in the final state.
When it is possible, these interpretations are then compared to the bounds set by other searches.

\section{Model independent upper limits}
    \label{sec:ModelIndependentLimits}

The agreement between the data and the Standard Model predictions for the different selections is interpreted in terms of model independent 95\% confidence level (CL) upper limits on the visible cross section.
The visible cross section, $\sigma_{\text{vis}}$, is defined as:

\begin{equation}
\sigma_{\text{vis}} = A\times\epsilon\times\sigma = N / L,
\label{eq:visibleXSec}
\end{equation}

\noindent where $\sigma$ is the production cross section, $A$ is the acceptance of the selection (without considering detector effects), $\epsilon$ is the experimental efficiency to select the signal events, $N$ are the expected events for the process and $L$ is the integrated luminosity.

Values of $\sigma \times A \times \epsilon$ in the range between 96~fb and 5.2~fb are excluded at 95\% CL.
The limits are derived with pseudo-experiments and with the asymptotic approximation (see Chapter~\ref{chapter:StatisticalModel}), leading to similar results with both approaches, as shown in Table \ref{tab:modelIndependent}.

\begin{table}[tb]
\begin{center}
\begin{tabular}{cccccc}
\hline
{\bf Signal channel} & & $\mathbf{\langle A\times\epsilon\times\sigma\rangle^{95}_\text{\textbf{obs}}\,\text{\textbf{[fb]}}}$ & $\mathbf{S^{95}_\text{\textbf{obs}}}$ & $\mathbf{S^{95}_\text{\textbf{obs}}}$ & $\mathbf{CL_b}$ \\
\hline
\multirow{2}{*}{M1} & asymp & $95.42$ & $1934.8$ & $1953.6^{847.2}_{291.7}$ & $0.49$ \\
                    & toy   & $96.23$ & $1951.2$ & $1962.0^{839.8}_{319.3}$ & $0.49$ \\
\multirow{2}{*}{M2} & asymp & $28.67$ & $581.4$  & $596.9^{200.4}_{122.7} $ & $0.48$ \\
                    & toy   & $28.36$ & $575.0$  & $590.6^{205.0}_{117.2} $ & $0.48$ \\
\multirow{2}{*}{M3} & asymp & $9.63$  & $195.3$  & $193.9^{68.8}_{54.1}$    & $0.49$ \\
                    & toy   & $9.63$  & $195.3$  & $193.7^{68.7}_{53.2}$    & $0.49$ \\
\multirow{2}{*}{M4} & asymp & $13.16$ & $266.8$  & $262.4^{89.5}_{68.5}$    & $0.52$ \\
                    & toy   & $13.22$ & $268.0$  & $264.4^{90.6}_{70.1}$    & $0.52$ \\
\multirow{2}{*}{M5} & asymp & $5.45$  & $110.5$  & $84.2^{33.0}_{23.3}$     & $0.79$ \\
                    & toy   & $5.45$  & $110.5$  & $84.2^{33.0}_{23.3}$     & $0.79$ \\
\multirow{2}{*}{M6} & asymp & $5.22$  & $105.8$  & $61.7^{21.7}_{16.3}$     & $0.96$ \\
                    & toy        & $5.22$  & $105.8$  & $61.9^{21.2}_{16.1}$     & $0.96$ \\
\hline
\end{tabular}
\end{center}
\caption[Model independent 95\% CL limits for the different signal regions.]{Observed and expected 95$\%$ CL limits on the visible cross section, defined as cross sections times acceptance time efficiency for the different signal selections.}
\label{tab:modelIndependent}
\end{table}

\section{Notes on the computation of the limits}
    \label{sec:ComputationOfLimits}

For the models that will be studied in this chapter, the signal regions M1-M3 are considered (see Section~\ref{sec:EventSelection}), and the one providing the best expected $CL_s$ (best exclusion) is used for the results that are reported.

The computation of the limits for these models is based on the profile likelihood method discussed in Section \ref{sec:StatisticalTreatment}.
A simultaneous fit of the MC expectations to the data in the signal and control regions is performed including statistical and systematic uncertainties.
Uncertainties on the signal acceptance times efficiency, the background predictions and the luminosity are considered, and correlations between systematic uncertainties on signal and background predictions are taken into account.
The fit accounts for any potential contamination of signal events in the control regions which a priory has been estimated to be very small.

A statistical test is performed and the $CL_s$ value is computed for each signal model.
Those signal models with a $CL_s<0.05$ are considered excluded.
Once the $CLs$ values are known for all the signal models generated, a linear interpolation between them in the parameter space is performed, so that a continuous exclusion plane can be generated.

\section{Signal samples simulation}
    \label{sec:SignalSimulation}

The stop pair production with $\stoptocharm$ is simulated using {\madgraph} with one additional jet from the matrix element and \pythia-6 for the showering.
CTEQ6L1 PDFs and AUET2B tune are used for the parton distribution functions and the simulation of the underlying event, respectively.
Cross sections are calculated to NLO in the strong coupling constant, adding the resummation of soft gluon emission at NLO+NLL accuracy.
The renormalization and factorization scales are set to the mass of the stop.
The samples are produced with stop masses between $\unit[100]{GeV}$ and $\unit[400]{GeV}$ and neutralino masses between $\unit[70]{GeV}$ and $\unit[390]{GeV}$.
The difference between the $\stopone$ and the $\ninoone$ masses, $\Delta m$, varies between $\unit[2]{GeV}$ and $\unit[82]{GeV}$, in a maximum step size of $\unit[30]{GeV}$. Cases in which $\Delta m < \unit[2]{GeV}$ have not been considered, since in this regime the stop can become long-lived, leading to different signatures, as for example, the one studied in Ref.~\cite{Johansen:2010ac}.

The stop pair production with $\stopfourbody$ is simulated with the same prescriptions as the $\stoptocharm$ MC simulation.
For this process, samples with stop mass ranges between $\unit[110]{GeV}$ and $\unit[300]{GeV}$ and $\Delta m$ that varies between $\unit[10]{GeV}$ and $\unit[80]{GeV}$ are produced.

Samples with asymmetric decay are generated with $\BR{\stoptocharm}=0.5$ and $\BR{\stopfourbody}=0.5$ for stops with masses $\unit[110]{GeV}$, $\unit[200]{GeV}$, $\unit[250]{GeV}$ and $\unit[300]{GeV}$, and with $\Delta m$ equal to $\unit[10]{GeV}$ and $\unit[80]{GeV}$.
These samples, combined with the $\stoptocharm$ and $\stopfourbody$ exclusive decays, allow to simulate different branching fraction scenarios.

Finally, the sbottom pair production with $\sbottomtob$ is simulated similarly, with sbottom masses between $\unit[100]{GeV}$ and $\unit[350]{GeV}$, and neutralino masses between $\unit[1]{GeV}$ and $\Delta m = \unit[10]{GeV}$.

\section{Systematic uncertainties on the signal}
    \label{sec:SysUncertaintiesSignal}

A complete study on systematic uncertainties, including both experimental and theoretical uncertainties, have been performed for each of the models studied.
Similar systematic effects have been found for all the models.
Here, the uncertainties for the stop pair production model with $\stoptocharm$ are presented.

The experimental uncertainties account for effects related to the estimation of the jet and $\met$ reconstruction, energy scale and resolution, pileup and jet vertex fraction mismodeling, and the luminosity.
These uncertainties introduce variations in the signal yield of the order of 3\% to 7\% depending on the third generation squark and neutralino mass configuration, and the signal selection under consideration.

The theoretical uncertainties account for effects related to the modeling of the processes.
These uncertainties can affect either the acceptance or the cross section of the model.
The theoretical uncertainties on the acceptance are parametrized with nuisance parameters in the fit, in a similar way as the experimental uncertainties are modeled.
Instead, a different procedure is followed to account for the theoretical uncertainties on the cross section.
In this case, the fit is performed three times, for the nominal and for the $\pm 1 \sigma$ variations on this uncertainty.
The theoretical uncertainties affecting the acceptance and the cross section are listed in Tables~\ref{tab:signalsyst_mono} and~\ref{tab:signalxsecuncert} respectively, and include:

\paragraph{Scale variations:} The uncertainties on the factorization and renormalization scales are the dominant theoretical uncertainties and affect mainly the cross section.
They are computed by varying both scales by factors two and one-half.
These uncertainties introduce variations between 13\% and 15\% on the cross section depending on the stop mass.
The effect of these uncertainties on the acceptance is found to be between 1\% and 6\% depending on the stop and neutralino mass configuration and the selection under consideration.

\paragraph{ISR/FSR:} The uncertainty on the modeling of the initial- and final-state radiation (ISR/FSR) is evaluated by varying the parameters that regulate the parton shower in a range that is consistent with the experimental data.
Altogether, these uncertainties introduce a variation between 3\% and 6\% on the cross sections.
The uncertainty on the ISR also introduces variations on the acceptance up to 8\% for configurations in which the masses of the stop and the neutralino are similar, whereas its effect is negligible for configurations in which this mass difference is large.
The FSR uncertainty introduces effects on the acceptance between 1\% and 9\% depending on the stop and neutralino mass configurations and the signal region under study.

\paragraph{PS to ME matching scale:}The impact in the signal yields from the variation of the parton shower to matrix element matching is also considered.
The parameters regulating this matching in the MC generator are varied by a factor of two up and down.
The effect of this uncertainty on the cross section can be up to 5\%.
This systematic uncertainty also introduces an effect up to 10\% on the signal acceptance, as the leading jet $\pt$ and the $\met$ requirements tighten.

\paragraph{PDF:} The uncertainty due to PDFs are evaluated using the Hessian method described in Ref.~\cite{Pumplin:2001ct} with the PDF error sets associated with CTEQ6L1.
This uncertainty affects the cross section of the model up to 8\%, while its effect is negligible in the acceptance.

\paragraph{}

%--- \ref{tab:signalsyst_mono}
\input{Interpretations/Tables/SignalAcceptanceUncertainties.tex}

%--- \ref{tab:signalxsecuncert}
\input{Interpretations/Tables/SignalXSecUncertainties.tex}

The $\met$ distributions for the nominal samples and the samples with renormalization/factorization scale variations in the signal region M1 are shown in Figure~\ref{fig:signalscalesysta6}.
Figures~\ref{fig:signalisrsysta6} (\ref{fig:signalfsrsysta6}) show the impact of ISR (FSR) variations in the missing transverse energy.
Finally, Figure~\ref{fig:signalmlmsysta6} shows the $\met$ distribution for the nominal sample and the samples with the matching scale variations.

\begin{figure}[tb]
\begin{center}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_100_70_factScale.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_100_95_factScale.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_200_125_factScale.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_200_195_factScale.eps}
\end{center}
\caption[Impact of the renormalization/factorization scale uncertainties on several signal models.]{Impact of the renormalization/factorization scale uncertainties on the missing transverse
  energy for a signal with a scalar
  stop mass of $m_{\tilde{t}}$ = 100\,GeV\ and LSP mass of
  $m_{\tilde{\chi}^0_1}$ = 70\,GeV\ (top left), $m_{\tilde{t}}$ = 100\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 95\,GeV\ (top right), $m_{\tilde{t}}$ = 200\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 125\,GeV\ (bottom left) and $m_{\tilde{t}}$ = 200\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 195\,GeV\ (bottom right).  All plots are
  shown for signal region M1.}
\label{fig:signalscalesysta6}
\end{figure}

\begin{figure}[tb]
\begin{center}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_100_70_Alpha_s.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_100_95_Alpha_s.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_200_125_Alpha_s.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_200_195_Alpha_s.eps}
\end{center}
\caption[Impact of the ISR uncertainty on several signal models.]{Impact of the ISR uncertainty on the missing transverse
  energy for a signal with a scalar
  stop mass of $m_{\tilde{t}}$ = 100\,GeV\ and LSP mass of
  $m_{\tilde{\chi}^0_1}$ = 70\,GeV\ (top left), $m_{\tilde{t}}$ = 100\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 95\,GeV\ (top right), $m_{\tilde{t}}$ = 200\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 125\,GeV\ (bottom left) and $m_{\tilde{t}}$ = 200\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 195\,GeV\ (bottom right). All plots are
  shown for signal region M1.}
\label{fig:signalisrsysta6}
\end{figure}

\begin{figure}[tb]
\begin{center}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_100_70_FSR.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_100_95_FSR.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_200_125_FSR.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_200_195_FSR.eps}
\end{center}
\caption[Impact of the FSR uncertainty on several signal models.]{Impact of the FSR uncertainty on the missing transverse
  energy for a signal with a scalar
  stop mass of $m_{\tilde{t}}$ = 100\,GeV\ and LSP mass of
  $m_{\tilde{\chi}^0_1}$ = 70\,GeV\ (top left), $m_{\tilde{t}}$ = 100\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 95\,GeV\ (top right), $m_{\tilde{t}}$ = 200\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 125\,GeV\ (bottom left) and $m_{\tilde{t}}$ = 200\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 195\,GeV\ (bottom right).  All plots are
  shown for signal region M1.}
\label{fig:signalfsrsysta6}
\end{figure}

\begin{figure}[tb]
\begin{center}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_100_70_Q.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_100_95_Q.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_200_125_Q.eps}
  \includegraphics[width=0.49\textwidth]{Interpretations/Figures/stop_200_195_Q.eps}
\end{center}
\caption[Impact of the MLM matching scale uncertainty on several signal models.]{Impact of the matrix element to parton shower matching scale uncertainty on the missing transverse energy for a signal with a scalar
  stop mass of $m_{\tilde{t}}$ = 100\,GeV\ and LSP mass of
  $m_{\tilde{\chi}^0_1}$ = 70\,GeV\ (top left), $m_{\tilde{t}}$ = 100\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 95\,GeV\ (top right), $m_{\tilde{t}}$ = 200\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 125\,GeV\ (bottom left) and $m_{\tilde{t}}$ = 200\,GeV\ and
  $m_{\tilde{\chi}^0_1}$ = 195\,GeV\ (bottom right).  All plots are
  shown for signal region M1.}
\label{fig:signalmlmsysta6}
\end{figure}

\clearpage
\section{Direct stop pair production}
    \label{sec:DirectStopProduction}

The results of the monojet analysis are translated into exclusion limits on the pair production of top squarks as a function of the stop mass for different neutralino masses.

\subsection{Stop decaying to a charm quark and a neutralino}

In this model, each top squark produced is assumed to decay in a charm-quark and a neutralino, $\stoptocharm$, with a branching fraction of $100\%$.
A Feynman diagram for this process is shown in Figure~\ref{fig:Diagrams3rdGen} (left).
This final state is characterized by the presence of two jets from the hadronization of the charm quarks, and missing transverse energy from the two undetected LSPs.
However, given the relatively small difference between the stop and the neutralino masses, $\Delta m$, both the transverse momenta of the two charm jets and the $\met$ are low, making it very difficult to extract the signal from the large multijet background.
Instead, the presence of initial-state radiation jets is required to boost the squark-pair, leading to larger $\met$.

The monojet analysis is expected to be sensitive in the very low $\Delta m$ region of the phase space, where the charm jets are not boosted enough to be detected.
Figure~\ref{fig:modelIndependent} shows the fiducial cross section, $\sigma\times A \times \epsilon$, as a function of the mass of the stop for different $\Delta m$ configurations in each selection.
For illustration, the model independent limits from Table~\ref{tab:modelIndependent} are included.
The stop and neutralino mass configurations excluded by the monojet analysis can already be approximately inferred from this figure.

\begin{figure}[!ht]
  \begin{center}
    \mbox{
      \includegraphics[width=0.495\textwidth]{MonojetAnalysis/Figures/ModelIndependent_Stop_M1.eps}
      \includegraphics[width=0.495\textwidth]{MonojetAnalysis/Figures/ModelIndependent_Stop_M2.eps}
    }
    \mbox{
      \includegraphics[width=0.495\textwidth]{MonojetAnalysis/Figures/ModelIndependent_Stop_M3.eps}
      \includegraphics[width=0.495\textwidth]{MonojetAnalysis/Figures/ModelIndependent_Stop_M4.eps}
    }
    \mbox{
      \includegraphics[width=0.495\textwidth]{MonojetAnalysis/Figures/ModelIndependent_Stop_M5.eps}
      \includegraphics[width=0.495\textwidth]{MonojetAnalysis/Figures/ModelIndependent_Stop_M6.eps}
    }
  \end{center}
  \caption[Model independent 95\% CL limits for the different signal regions.]{Observed and expected 95\% CL model independent limits on the visible cross section for the regions M1-M6 compared to the $\stoptocharm$ predictions as a function of the stop mass for different $\Delta m$.}
  \label{fig:modelIndependent}
\end{figure}

The 95\% CL limits on this model are computed with the $CL_s$ method described in Section~\ref{sec:ComputationOfLimits}, which properly accounts for the correlations on the systematic uncertainties among the different signal and background processes.
Observed and expected limits are computed separately in the different signal regions, and the one with best expected limit is adopted as the nominal result.
The signal region that gives the best expected limit for each stop and neutralino mass is shown in Figure~\ref{fig:ExclusionStoptocharm} (top).
The selection M1 drives the exclusion limits for low stop masses, while M2 and M3 enhance the sensitivity for very low $\Delta m$ as the stop mass increases.
Figure~\ref{fig:ExclusionStoptocharm} (bottom) shows the exclusion plane at 95\% CL for the stop pair production with $\stoptocharm$ as a function of the $m_{\stop}$ and $m_{\ninoone}$.
The 95\% CL observed limits corresponding to the $\pm 1 \sigma$ variations on the SUSY theoretical cross sections are also added.
In the region of phase space where the stop and the neutralino masses are almost degenerated, stop masses up to $\unit[260]{GeV}$ are excluded.
The sensitivity of the analysis reduces as the $\Delta m$ increases, as a consequence of the maximum jet multiplicity requirement in the monojet selection.
Large $\Delta m$ scenarios can be excluded if the mass of the stop is smaller than $\unit[170]{GeV}$.
These results significantly extend the previous exclusion limits from LEP~\cite{Aaltonen:2012tq} and CDF~\cite{Abazov:2008rc} in this channel, as shown in the figure.

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotStop_Stop_combined_M1_M2_M3_BestRegion.eps}
}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotStop_Stop_combined_M1_M2_M3_.eps}
}
\end{center}
\caption[Exclusion plane at 95\% CL for stop pair production with $\stoptocharm$ as a function of the $m_{\stop}$ and $m_{\ninoone}$, combining the selections M1 to M3.]{Exclusion plane at 95\% CL as a function of stop and neutralino masses. The observed (red line) and expected (blue line) upper limits from this analysis are compared to previous results from Tevatron experiments~\cite{Abazov:2008rc}, and from LEP experiments~\cite{Aaltonen:2012tq} at CERN with squark mixing angle $\theta=0^{\circ}$. The dotted lines around the observed limit indicate the range of observed limits corresponding to $\pm 1 \sigma$ variations on the NLO SUSY cross section predictions. The shaded area around the expected limit indicates the expected $\pm 1 \sigma$ ranges of limits in the absence of a signal. A band for $m_{\stopone} - m_{\ninoone}< \unit[2]{GeV}$ indicates the region in the phase space for which the stop can become long-lived \protect\cite{Aad:2014nra}.}
\label{fig:ExclusionStoptocharm}
\end{figure}

The monojet analysis results can be combined with the results of a dedicated analysis optimized for moderate $\Delta m \gt \unit[20]{GeV}$.
In this regime, the charm jets receive a large enough boost to be detected.
For this reason, in addition to the requirements on the presence of an initial-state radiation, the identification of jets containing the decay products of charm hadrons is used.
This analysis is referred as ``charm-tagged'', and is detailed in Ref.~\cite{Aad:2014nra}.
In this region of the phase space, the charm-tagged C1 and C2 selections (see Appendix~\ref{app:CharmTaggedAnalysis}) give the best expected limits, as Figure~\ref{fig:ExclusionStoptocharmCombinedAll} (top) indicates.
The combination of both analyses leads to the exclusion limits shown in Figure~\ref{fig:ExclusionStoptocharmCombinedAll} (bottom).
The charm-tagged analysis complements the monojet analysis and increases the exclusion region for moderate and large $\Delta m$.
After the combination, masses for the stop up to 240~GeV are excluded at 95\%~CL for arbitrary neutralino masses, within the kinematic boundaries.
For neutralino masses of about 200~GeV, stop masses below 270~GeV are excluded at 95\%~CL.

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotStop_Stop_combined_M1_M2_M3_C1_C2_BestRegion.eps}
}
\mbox{
\includegraphics[width=0.795\textwidth]{Appendix_CharmTagged/Figures/limitPlotStop_Stop_combined_M1_M2_M3_C1_C2_.eps}
}
\end{center}
\caption[Exclusion plane at 95\% CL for stop pair production with $\stoptocharm$ as a function of the $m_{\stop}$ and $m_{\ninoone}$, combining the monojet and the charm-tagged approaches]{Exclusion plane at 95\% CL as a function of stop and neutralino masses for the monojet and charm-tagged approaches (see Appendix~\ref{app:CharmTaggedAnalysis}), combined. The observed (red line) and expected (blue line) upper limits from this analysis are compared to previous results from Tevatron experiments~\cite{Abazov:2008rc}, and from LEP experiments~\cite{Aaltonen:2012tq} at CERN with squark mixing angle $\theta=0^{\circ}$. The dotted lines around the observed limit indicate the range of observed limits corresponding to $\pm 1 \sigma$ variations on the NLO SUSY cross section predictions. The shaded area around the expected limit indicates the expected $\pm 1 \sigma$ ranges of limits in the absence of a signal. A band for $m_{\stopone} - m_{\ninoone}< \unit[2]{GeV}$ indicates the region in the phase space for which the stop can become long-lived \protect\cite{Aad:2014nra}.}
\label{fig:ExclusionStoptocharmCombinedAll}
\end{figure}

\subsection{Stop decaying to a $b$-quark, two fermions and a neutralino}

The monojet results are also interpreted in terms of exclusion limits on the stop pair production, with each stop decaying into a bottom quark, two fermions (either leptons or quarks) and a neutralino, $\stopfourbody$, with 100\% branching fraction.
The Feynman diagram for this process are shown in Figure~\ref{fig:Diagrams3rdGen} (center).
The exclusion limits are computed with the same $CL_s$ approach used above, with the region giving the best expected $CL_s$ taken as the nominal.
The selections M1 to M3 are combined, as indicated in Figure~\ref{fig:ExclusionStopToFourbody} (top)

Figure \ref{fig:ExclusionStopToFourbody} (bottom) presents the 95\% CL limits as a function of the stop and neutralino masses.
Stop masses up to $\unit[255]{GeV}$ can be excluded.
This result is similar to the exclusion found for the $\stoptocharm$ decay, since in a mass-degenerated scenario the decay products of the squarks are too soft to be identified in the final state, and the signal selection only relies on the presence of an ISR jet.
For large $\Delta m$, the bottom jets and the fermions receive a larger boost, which allows them to be detected.
The increase in the fermion multiplicity decreases the sensitivity of the monojet analysis to this final state, due to the jet multiplicity requirement and the lepton veto in the selection.

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotFourBody_Stop_combined_M1_M2_M3_BestRegion.eps}
}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotFourBody_Stop_combined_M1_M2_M3_.eps}
}
\end{center}
\caption[Exclusion plane at 95\% CL for stop pair production with $\stopfourbody$ as a function of the $m_{\stop}$ and $m_{\ninoone}$]{Exclusion plane at 95\% CL as a function of stop and neutralino masses for the decay channel $\stopfourbody$ (BR=100\%). The dotted lines around the observed limit indicate the range of observed limits corresponding to $\pm 1 \sigma$ variations on the NLO SUSY cross section predictions. The shaded area around the expected limit indicates the expected $\pm 1 \sigma$ ranges of limits in the absence of a signal. A band for $m_{\stopone} - m_{\ninoone}< \unit[2]{GeV}$ indicates the region in the phase space for which the stop can become long-lived \protect\cite{Aad:2014nra}.}
\label{fig:ExclusionStopToFourbody}
\end{figure}

\subsection{Mixed scenarios}

The exclusion limits shown in Figures \ref{fig:ExclusionStoptocharm} and \ref{fig:ExclusionStopToFourbody} are produced assuming a 100\% branching ratio to $\stoptocharm$ and $\stopfourbody$, respectively.
In the following, the monojet analysis is interpreted in terms of stop pair production, considering the stops decays $\stoptocharm$ or $\stopfourbody$, with different branching ratios, and assuming that $\text{BR}(\stoptocharm)$ + $\text{BR}(\stopfourbody)$ = 1.
For this purpose, new samples are generated, following the prescriptions detailed in Section~\ref{sec:MCSamples}, and assuming each stop to decay in a different final state.
The expected number of events for a model with a $\text{BR}(\stoptocharm) = \alpha$ can be computed with the following expression:

\begin{equation}
\begin{split}
N_{\alpha} &= \alpha^2 N_{\stopone\stopone \;\rightarrow \;c\ninoone \; c\ninoone} + 2\alpha(1-\alpha)N_{\stopone\stopone \;\rightarrow \;c\ninoone \; bff'\ninoone} \\
&+ (1-\alpha)^2 N_{\stopone\stopone \;\rightarrow \;bff'\ninoone \;bff'\ninoone},
\end{split}
\label{eq:MixedSamplesCombination}
\end{equation}

\noindent where $N_{\stopone\stopone \;\rightarrow \;c\ninoone \; c\ninoone}$, $N_{\stopone\stopone \;\rightarrow \;c\ninoone \; bff'\ninoone}$, and $N_{\stopone\stopone \;\rightarrow \;bff'\ninoone \;bff'\ninoone}$ are the number of events from the stop pair production assuming BR$(\stoptocharm)=1$, BR$(\stopfourbody)=1$, and $\text{BR}(\stoptocharm) = \text{BR}(\stopfourbody) = 0.5$, respectively.

Figure \ref{fig:ExclusionMixed} shows the 95\% CL upper cross section limits as a function of the stop mass and the branching fractions (red lines) for two different $\Delta m$ configurations.
These upper limits can be compared to the nominal $\stopone$ pair production cross section (blue line).

\begin{figure}[!t]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/upperLimitXSectionBR_Stop_dM_10_M1_M2_M3.eps}
}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/upperLimitXSectionBR_Stop_dM_80_M1_M2_M3.eps}
}
\end{center}
\caption[Upper limits on the stop pair production cross section at 95\% CL as a function of $m_{\stopone}$ for different branching ratios.]{95\% CL upper cross section limits as a function of the stop mass (in red) compared to the $\stopone$ pair nominal cross section (in blue). $\Delta m = \unit[10]{GeV}$ and $\Delta m = \unit[80]{GeV}$ models are considered in the top and bottom figures respectively.}
\label{fig:ExclusionMixed}
\end{figure}

When $\Delta m = \unit[10]{GeV}$, the available phase space for the products of the stop decay is reduced, and thus the selection relies only on the production of an ISR jet.
The exclusion is independent on the branching ratio, when the stop and the neutralino are almost degenerated in mass.
Instead, when $\Delta m = \unit[80]{GeV}$ the Standard Model decay products of each stop are boosted enough to be reconstructed.
The $\stopfourbody$ decay of the stop is more affected by the jet multiplicity requirement and the lepton vetoes, than the $\stoptocharm$.
For this reason, less stringent limits on the mass of the stop can be set, as the branching ratio to $\stopfourbody$ increases.

\section{Direct sbottom pair production}
    \label{sec:DirectSbottomProduction}

In the case of bottom squark pair production, it is assumed a SUSY particle mass hierarchy such that the sbottom decays exclusively into a bottom quark and a neutralino, $\sbottomtob$.
Figure~\ref{fig:Diagrams3rdGen} (right) shows a Feynman diagram for this decay.
The expected signal for the direct sbottom pair is characterized by the presence of two energetic jets from the hadronization of the bottom quarks and large $\met$ from the two LSPs in the final state.
The monojet results are interpreted in terms of this search, $\sbottomtob$, in compressed scenarios.
The sbottom and the neutralino masses are almost degenerated, leading to two soft $b$-jets and an energetic ISR in the final state.

Signal regions M1 to M3 are used, and for each mass point the one with best expected $CL_s$ is chosen, as shown in Figure~\ref{fig:ExclusionSbottomtob} (top).
Figure \ref{fig:ExclusionSbottomtob} (bottom) shows the exclusion limits at 95\% CL for the $\sbottomtob$ model, as a function of the sbottom and neutralino masses.
The fact that the exclusion for very low or very high $\Delta m$ is better than for medium $\Delta m$ has to do with the acceptance, with the negotiation between mass and momentum investment for the neutralino, and the phase space available for extra radiation.
The cross section is independent of the $\Delta m$ for fixed sbottom masses, so the $CL_s$ values for the different mass configurations depend exclusively on acceptance and efficiency.
For all neutralino masses, the sbottoms are boosted by an initial-state radiation.
Scenarios with small $\Delta m$ (large neutralino mass), are characterized by soft $b$-jets (hardly ever identified) and little extra ISR/FSR.
As the mass of the neutralino decreases (medium $\Delta m$), the two $b$-jets are reconstructed and there is more phase space available for extra jet radiations.
Therefore, more events fail the jet veto selection and the sensitivity of the monojet analysis to this region of the phase space decreases.
For large $\Delta m$ configurations (low neutralino masses), the loss of sensitivity due to extra jet radiation is compensated by the increase in $\met$ due to the highly boosted neutralinos.

Sbottom masses below $\unit[180]{GeV}$ can be excluded for arbitrary neutralino masses.
In the case of sbottom and neutralino degenerated in mass, this analysis excludes sbottom masses up to $\unit[255]{GeV}$, thus expanding the exclusion limits set by other searches \cite{Aaltonen:2010dy,Abazov:2010wq,Aad:2013ija}.
For very low neutralino masses, the analysis also excludes sbottom masses up to $\unit[255]{GeV}$.

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotSbottom_Stop_combined_M1_M2_M3_BestRegion.eps}
}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotSbottom_Stop_combined_M1_M2_M3_.eps}
}
\end{center}
\caption[Exclusion plane at 95\% CL for sbottom pair production with $\sbottomtob$ as a function of the $m_{\sbottom}$ and $m_{\ninoone}$]{Exclusion plane at 95\% CL as a function of sbottom and neutralino masses for the decay channel $\sbottomtob$ (BR=100\%). The observed (red line) and expected (blue line) upper limits from this analysis are compared to previous results from CDF~\cite{Aaltonen:2010dy}, D0~\cite{Abazov:2010wq}, and ATLAS~\cite{Aad:2013ija}. For the latter, the area below the dashed-dotted line is excluded. The dotted lines around the observed limit indicate the range of observed limits corresponding to $\pm 1 \sigma$ variations on the NLO SUSY cross section predictions. The shaded area around the expected limit indicates the expected $\pm 1 \sigma$ ranges of limits in the absence of a signal. A band for $m_{\sbottomone} - m_{\ninoone}< \unit[2]{GeV}$  indicates the region in the phase space for which the sbottom can become long-lived. \protect\cite{Aad:2014nra}.}
\label{fig:ExclusionSbottomtob}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\cleardoublepage
\chapter{Interpretations: inclusive squarks or gluinos}
    \label{chapter:SquarkGluinoProduction}

This chapter presents the interpretation of the monojet analysis in terms of models involving the direct production of inclusive squarks, or gluinos.
Only the signal regions M1 to M3 are considered, and the one giving the best expected exclusion is used for the results.

\section{Inclusive squark pair production}
    \label{sec:InclusiveSquarkProduction}

This section presents the interpretation of the analysis in terms of pair production of degenerated light-flavour squarks, with each squark decaying into a light quark and a neutralino (see Figure \ref{fig:DiagramsInclusiveProduction} left).
The MC samples for this model have been simulated with \madgraph{} and \pythia{}, using the PDF set CTEQ6L1.
The renormalization and factorization scales are set to the mass of the mean mass of the participating particles, $Q=(m_{\squark}+m_{\gluino})/2$.
The AUET2B tune has been used for the simulation of the underlying event, while the MLM matching scheme is used with up to one additional jet in the \madgraph{} matrix element.
More detailed information on these samples can be found in Ref.~\cite{Aad:2014wea}.
Different mass points have been generated for this process, in a grid with squark masses ranging between $\unit[87]{GeV}$ and $\unit[1225]{GeV}$, and neutralino masses between $\unit[0]{GeV}$ and those corresponding to a $\Delta m = m_{\squark} - m_{\ninoone}$ equal to $\unit[10]{GeV}$.
The signal cross sections are calculated to NLO in the strong coupling constant, adding the resummation of soft gluon emission at next-to-leading-logarithmic pQCD accuracy (NLO+NLL).

Experimental and theoretical systematic uncertainties for the different mass configurations have been computed, as explained in Section~\ref{sec:SysUncertaintiesSignal}.
Figure~\ref{fig:signalAcceptanceSquark} shows the impact of the scale variations on the signal acceptance for different squark masses as a function of $\Delta m$.
The uncertainties on the factorization and renormalization scales can be modeled as shown in Ref.~\cite{Aad:2014wea}.
The validity of this parametrization, shown in the figure by the dashed blue line, has been carefully checked for the monojet analysis, and is finally adopted.
In the case of the matching scale uncertainty, a flat 10\% is considered, based on the studies from Fig.~\ref{fig:signalAcceptanceSquark} (right).
Altogether, the systematic uncertainty on the acceptance for the signal is parametrized as:

\begin{equation}
\left(\frac{\Delta N}{N}\right)_{\text{signal}} = 0.15 \times e^{-{\Delta m / 250}} \oplus 0.20 \times e^{-{\Delta m / 250}} \oplus 0.1.
\label{eq:systematicAcceptanceSquark}
\end{equation}

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.32\textwidth]{Interpretations/Figures/signalUncert_mu_M1.eps}
\includegraphics[width=0.32\textwidth]{Interpretations/Figures/signalUncert_Q_M1.eps}
\includegraphics[width=0.32\textwidth]{Interpretations/Figures/signalUncert_Matching_M1.eps}
\end{center}
\caption[Parametrization of the theoretical systematic uncertainties for the squark pair production when $\squarktoq$.]{The red points show the impact of (left) renormalization/factorization $\mu$, (center) $Q(\alpha _s)$ and (right) matching scales used in \madgraph{}+\pythia{} on the number of expected events $N$ for a simplified model with $\squark$ pair production (\squark$\rightarrow q+$\ninoone). The relative effect $\Delta N/N$ after proper normalization to the same total cross section is shown as a function of $\Delta m = m_{\tilde{q}} - m_{\tilde{\chi}_1^0}$. The dashed lines show the parameterization used to compute the uncertainties on the signal acceptance~\cite{Aad:2014wea}. The points in black and green show similar measurements in the monojet M1 signal region done with \madgraph{} samples for the low $\Delta m$ points used in the monojet-like analysis.}
\label{fig:signalAcceptanceSquark}
\end{figure}

\subsection{Exclusion Limits at 95\% CL}

The exclusion limits at 95\% CL for the first- and second-generation squark pair production are shown in Figure~\ref{fig:ExclusionSquarktoq}, as a function of the squark and neutralino masses.
The shape of the exclusion is related to the acceptance of the monojet analysis for each mass configuration, and follows the same arguments as for the sbottom pair production with $\sbottomtob$, shown in Section~\ref{sec:DirectSbottomProduction}.

Squark masses up to $\unit[320]{GeV}$ are excluded at 95\% CL for arbitrary neutralino masses.
For very compressed scenarios, the monojet analysis excludes squark masses up to $\unit[440]{GeV}$, thus extending the exclusion limits of the analysis in Ref.~\cite{Aad:2014wea} and shown in the figure.
Masses up to $\unit[660]{GeV}$ are also excluded for neutralino masses below 20~GeV.

\begin{figure}[!t]
  \begin{center}
    \mbox{
      \includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotSquark_Stop_combined_M1_M2_M3_.eps}
    }
  \end{center}
  \caption[Exclusion plane at 95\% CL for sbottom pair production with $\squarktoq$ as a function of the $m_{\squark}$ and $m_{\ninoone}$]{Exclusion plane at 95\% CL as a function of the squark and neutralino masses for the $\squarktoq$ process. The dotted lines around the observed limit indicate the range of observed limits corresponding to the $\pm 1 \sigma$ variations on the cross section predictions. The shaded area around the expected limit indicates the expected $\pm 1 \sigma$ ranges of limits in the absence of a signal.
  The upper limits from this analysis are also compared to the previous results from ATLAS~\cite{Aad:2014wea}, shown in the figure with a dot-dashed red line.}
  \label{fig:ExclusionSquarktoq}
\end{figure}

\section{Gluino pair production}
    \label{sec:GluinoProduction}

Similarly, the results have been interpreted in terms of final states involving the production of pairs of gluinos.
Two different decay modes of the gluino have been considered.
First, the gluino is assumed to decay with 100\% branching fraction into a bottom quark and a virtual sbottom, which then decays into another bottom quark plus a neutralino, $\gluinotobb$.
In the second decay mode under consideration, the gluino decays exclusively to a gluon and a neutralino, $\gluinotog$, via a loop in which the interchange of a quark is involved.
The Feynman diagrams for both processes are shown in the middle and right panes of Figure~\ref{fig:DiagramsInclusiveProduction}.
The calculation of experimental and theoretical uncertainties follows the procedure explained in Section~\ref{sec:SysUncertaintiesSignal}.

\subsection{Gluino decaying to two $b$-quarks and a neutralino}

Samples for gluino pair production with $\gluinotobb$ have been simulated using \madgraph{} with one additional jet from matrix element and \pythia{-6} for the showering, following the same prescriptions as for the $\squarktoq$ simulated samples.
A grid of points with the gluino mass between $\unit[200]{GeV}$ to $\unit[1600]{GeV}$ and neutralino masses between $\unit[1]{GeV}$ and values corresponding to a $\Delta m = \unit[25]{GeV}$ has been produced.

The 95\% CL exclusion limits as a function of the gluino and neutralino masses are shown in Figure~\ref{fig:ExclusionGluinoGbb}.
The monojet analysis allows to expand the excluded parameter space in Ref.~\cite{TheATLAScollaboration:2013tha} towards compressed gluino and neutralino mass configurations.
Gluino masses up to $\unit[580]{GeV}$ can be excluded for low $\Delta m = m_{\gluino} - m_{\ninoone}$ values.
As the difference between the gluino and neutralino masses increases, the bottom quarks from the gluino decays are more boosted and the $b$-jets can be reconstructed.
Therefore, the analysis looses sensitivity to these configurations due to the jet veto requirement in the selections.
Gluino masses up to $\unit[420]{GeV}$ are excluded for very low neutralino masses.

\begin{figure}[!ht]
  \begin{center}
    \mbox{
      \includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotGluinoGbb_Stop_combined_M1_M2_M3_.eps}
    }
  \end{center}
  \caption[Exclusion plane at 95\% CL for gluino pair production with $\gluinotobb$ as a function of the $m_{\gluino}$ and $m_{\ninoone}$]{Exclusion plane at 95\% CL as a function of the gluino and neutralino masses for the $\gluinotobb$ process. The dotted lines around the observed limit indicate the range of observed limits corresponding to the $\pm 1 \sigma$ variations on the cross section predictions. The shaded area around the expected limit indicates the expected $\pm 1 \sigma$ ranges of limits in the absence of a signal.
  The upper limits from this analysis are also compared to the previous results from ATLAS~\cite{TheATLAScollaboration:2013tha}, shown in the figure with a dot-dashed red line.}
  \label{fig:ExclusionGluinoGbb}
\end{figure}

\subsection{Gluino decaying to a gluon and a neutralino}

Finally, a second grid for gluino pair production with $\gluinotog$ has been produced with the same prescriptions as the $\gluinotobb$ samples.
The production consists of several samples with gluino masses between $\unit[150]{GeV}$ and $\unit[1500]{GeV}$, and neutralino masses between $\unit[0]{GeV}$ and values corresponding to $\Delta m = \unit[50]{GeV}$.
Figure \ref{fig:ExclusionGluinoGg} shows the 95\% CL exclusion plane as a function of the gluino and neutralino masses.
Gluinos with masses up to 600~GeV are excluded at 95\% CL for the very compressed scenario.
For massless neutralinos, gluino masses up to 850~GeV can be excluded.

\begin{figure}[!ht]
  \begin{center}
    \mbox{
      \includegraphics[width=0.795\textwidth]{Interpretations/Figures/limitPlotGluinoGg_Stop_combined_M1_M2_M3_.eps}
    }
  \end{center}
  \caption[Exclusion plane at 95\% CL for gluino pair production with $\gluinotog$ as a function of the $m_{\gluino}$ and $m_{\ninoone}$]{Exclusion plane at 95\% CL as a function of the gluino and neutralino masses for the $\gluinotog$ process. The dotted lines around the observed limit indicate the range of observed limits corresponding to the $\pm 1 \sigma$ variations on the cross section predictions. The shaded area around the expected limit indicates the expected $\pm 1 \sigma$ ranges of limits in the absence of a signal.}
  \label{fig:ExclusionGluinoGg}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\cleardoublepage
\chapter{Interpretations: Dark Matter related}

This chapter presents interpretations of the monojet analysis in terms of models involving the direct production of potential Dark Matter candidates.
This includes models based on effective theories, simplified models involving the pair production of Weakly Interacting Massive Particles, or the production of gravitinos in Gauge Mediated SUSY breaking scenarios.
Sensitivity studies of the monojet analysis to models involving the direct production of charginos or neutralinos are also collected in Appendix~\ref{sec:CharginoNeutralinoProduction}.

%------------------
%--- Section for the WIMPs, wrongly included first as an appendix!!!
%------------------
\input{Appendix_WIMPs/Appendix_WIMPs.tex}
%------------------
%------------------
%------------------

\section{Gravitino production in GMSB}
    \label{sec:GravitinoProduction}

In Gauge Mediated SUSY breaking scenarios, the gravitino mass gives direct access to the scale of the SUSY breaking, and can potentially contribute to the total amount of Dark Matter in the Universe.
In this section, the monojet results are interpreted in the context of gravitino production in association with a squark or a gluino in the final state.
Figure~\ref{fig:DiagramsGravitinoGMSB} shows some of the Feynman diagrams for this process.
A simplified SUSY model is used for which the squark or the gluino decays to a gravitino, and a quark or a gluon in the final state (see Figure~\ref{fig:DiagramsVertexGravitinoGMSB}), thus leading to a monojet signature.

Monte Carlo samples corresponding to gravitino production in association with a gluino or a squark in the final state, $\pp\rightarrow\squark\gravino+X$ and $\pp\rightarrow\gluino\gravino+X$ are generated at LO using \madgraph{}, interfaced with \pythia{} for the showering.
The ATLAS detector simulation is provided by the ATLAS fast simulation, while the PDF set used is CTEQ6L1.
The renormalization and factorization scales are set to the average of the mass of the final state particles involved in the hard interaction $(m_{\gravino} + m_{\squark , \gluino})/2 \simeq m_{\squark , \gluino}/2$.
A grid with different mass configurations has been generated with $m_{\squark, \gluino}$ from $\unit[50]{GeV}$ to $\unit[2.6]{TeV}$ and $m_{\squark}/m_{\gluino} = 0.25, 0.5, 1, 2, 4$, and a gravitino mass $m_{\gravino} = \unit[5 \times 10^{-4}]{eV}$.
Both experimental and theoretical systematic uncertainties for the different mass configurations are computed as for the previous models discussed in Section~\ref{sec:SysUncertaintiesSignal}.
Experimental uncertainties result into a 4.6\% to 2.9\% effect on the signal yield in M3, and a 16\% to 3\% effect in M6 for squark and gluino masses of 200~GeV and 2.4~TeV, respectively.
The theoretical uncertainties on the acceptance introduce a variation in the signal yield of about 15\%, while the theoretical uncertainties on the cross section contribute altogether to a 24\% to 55\% on the signal yield for different squark and gluino masses.

\subsection{Exclusion Limits at 95\% CL}

In this case, the 95\% CL limits on the visible cross section of the monojet analysis shown in Table~\ref{tab:modelIndependent} are used to extract the limits on the gravitino mass as a function of the masses of the squark or the gluinos.
The best sensitivity to the gravitino production is obtained for the selections M3, M5 and M6, and depends on the squark and gluino mass configuration.
Figure~\ref{fig:modelIndependentGravitino} shows, for the signal region M5, the fiducial cross section as a function of the squark and gluino mass, for different gravitino masses.
For comparison, the model independent limits from Table~\ref{tab:modelIndependent} are shown.
The intersection between the model independent limit and the signal fiducial cross section determines the exclusion in terms of the parameters of the model.
The following limits are calculated:

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.995\textwidth]{Interpretations/Figures/ModelIndependentGravitino_mGVariable_Stop_A9.eps}
}
\end{center}
\caption[Fiducial cross section for the $\gravino + \squark / \gluino$ production as a function of the squark/gluino mass for degenerate squark and gluinos in the signal region M5.]{Fiducial cross section, $\sigma \times A \times \epsilon$, for the $\gravino + \squark / \gluino$ production as a function of the squark/gluino mass for degenerate squark and gluinos in the signal region M5. Different values of the gravitino mass are considered and the predictions are compared to the model independent limits (see Table \ref{tab:modelIndependent}).}
\label{fig:modelIndependentGravitino}
\end{figure}

\begin{itemize}
\item{Observed:}
intersection between the observed model independent limit and the signal visible cross section.
\item{Observed $- 1\sigma_{\text{total}}^{\text{signal}}$:}
intersection between the observed model independent limit and the signal visible cross section $- 1\sigma$ of the total uncertainty on the signal.
The total uncertainty is computed by summing in quadrature the experimental uncertainties and both the theoretical uncertainties on the acceptance and on the cross section.
\item{Observed $- 1\sigma_{\text{exp}}^{\text{signal}}$:}
intersection between the observed limit and the signal visible cross section $-1\sigma$ of the experimental uncertainty on the signal together with the effects of the modeling uncertainty on the signal acceptance (no cross section uncertainty is considered in this case).
\item{Expected:}
intersection between the expected model independent limit and the signal visible cross section.
\item{Expected $\pm 1\sigma$ or $\pm 2\sigma$:}
intersection between the signal visible cross section and the expected limit with $\pm 1\sigma$ or $\pm 2\sigma$ experimental uncertainty on the Standard Model background.
\end{itemize}

This approach does not take into account the correlations between the signal and the background uncertainties.
The $CL_s$ computation for each of the mass configurations in the grid would require a huge computational power, thus making the analysis very time consuming.
Tests performed for several cases showed that the exclusions using the model independent limits or using the $CL_s$ method return compatible, almost identical, results.

Figure~\ref{fig:GravitinoMassExclusion_mqmg} shows the 95\% CL limits on the gravitino mass, $m_{\gravino}$, for equal squark and gluino masses.
Gravitino masses below $3.5\times10^{-4}$~eV, $3\times10^{-4}$~eV and $2\times10^{-4}$~eV are excluded at 95\% CL for squark/gluino masses of 500~GeV, 1~TeV and 1.5~TeV.
For very high squark/gluino masses the narrow-width approximation (NWA) employed is violated since the partial width for the gluino and squark to decay into a gravitino and a parton becomes more than 25\% of its mass.
In this case, other decay channels for the gluino and squarks should be considered, leading to a different final state.
Figures~\ref{fig:GravitinoMassExclusion_xmqmg} and~\ref{fig:GravitinoMassExclusion_mqxmg} show the limits on the gravitino mass, for $m_{\gluino} = 2\times m_{\squark}$ and $m_{\gluino} = 4\times m_{\squark}$; and $m_{\gluino} = m_{\squark}/2$ and $m_{\gluino} = m_{\squark}/4$, respectively.
In this case, lower bounds on gravitino mass in the range between $5\times10^{-4}$ and $5\times10^{-5}$ are set depending on the squark and gluino masses.

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_mGLimit_Stop_A4_A9_A10.eps}
}
\end{center}
\caption[95\% CL lower limits on the gravitino mass as a function of the squark mass for equal squark and neutralino masses.]{Observed (solid line) and expected (dashed line) 95\% CL lower limits on the gravitino mass as a function of the squark mass for equal squark and neutralino masses. The dotted line indicates the impact on the observed limit of the $\pm1\sigma$ LO theoretical uncertainty. The shaded bands around the expected line indicate the expected $\pm1\sigma$ and $\pm2\sigma$ ranges of limits.
  The region above the black dotted line defines the validity of the narrow-width approximation (NWA) for which the decay width is smaller than 25\% of the squark/gluino mass.}
\label{fig:GravitinoMassExclusion_mqmg}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_mGLimit_Stop_A4_A9_A102mqmg.eps}
}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_mGLimit_Stop_A4_A9_A104mqmg.eps}
}
\end{center}
\caption[95\% CL lower limits on the gravitino mass as a function of the squark mass for $m_{\gluino} = 2 \times m_{\squark}$ and $m_{\gluino} = 4 \times m_{\squark}$.]{Observed (solid line) and expected (dashed line) 95\% CL lower limits on the gravitino mass as a function of the squark mass for $m_{\gluino} = 2 \times m_{\squark}$ (top) and $m_{\gluino} = 4 \times m_{\squark}$ (bottom). The dotted line indicates the impact on the observed limit of the $\pm1\sigma$ LO theoretical uncertainty. The shaded bands around the expected line indicate the expected $\pm1\sigma$ and $\pm2\sigma$ ranges of limits.
  The region above the black dotted line defines the validity of the narrow-width approximation (NWA) for which the decay width is smaller than 25\% of the squark/gluino mass.}
\label{fig:GravitinoMassExclusion_xmqmg}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_mGLimit_Stop_A4_A9_A10mq2mg.eps}
}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_mGLimit_Stop_A4_A9_A10mq4mg.eps}
}
\end{center}
\caption[95\% CL lower limits on the gravitino mass as a function of the squark mass for $m_{\gluino} = 1/2 \times m_{\squark}$ and $m_{\gluino} = 1/4 \times m_{\squark}$.]{Observed (solid line) and expected (dashed line) 95\% CL lower limits on the gravitino mass as a function of the squark mass for $m_{\gluino} = 1/2 \times m_{\squark}$ (top) and $m_{\gluino} = 1/4 \times m_{\squark}$ (bottom). The dotted line indicates the impact on the observed limit of the $\pm1\sigma$ LO theoretical uncertainty. The shaded bands around the expected line indicate the expected $\pm1\sigma$ and $\pm2\sigma$ ranges of limits.
  The region above the black dotted line defines the validity of the narrow-width approximation (NWA) for which the decay width is smaller than 25\% of the squark/gluino mass.}
\label{fig:GravitinoMassExclusion_mqxmg}
\end{figure}

The limits on the gravitino mass shown in Figures~\ref{fig:GravitinoMassExclusion_mqmg} to~\ref{fig:GravitinoMassExclusion_mqxmg} can be translated into 95\% CL upper limits on the breaking scale of SUSY, $\sqrt{\langle F \rangle}$.
These limits are shown in Figures~\ref{fig:GravitinoSqrtFExclusion_mqmg} to~\ref{fig:GravitinoSqrtFExclusion_mqxmg}, for the different squark and gluino mass configurations.
Values of the $\sqrt{\langle F \rangle}$ below 1~TeV can be excluded for squark/gluino masses of 1~TeV.

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_sqrtFLimit_Stop_A4_A9_A10.eps}
}
\end{center}
\caption[95\% CL lower limits on the SUSY breaking scale $F$ as a function of the squark mass for equal squark and neutralino masses.]{Observed (solid line) and expected (dashed line) 95\% CL lower limits on the SUSY breaking scale $F$ as a function of the squark mass for equal squark and neutralino masses. The dotted line indicates the impact on the observed limit of the $\pm1\sigma$ LO theoretical uncertainty. The shaded bands around the expected line indicate the expected $\pm1\sigma$ and $\pm2\sigma$ ranges of limits.
  The region above the black dotted line defines the validity of the narrow-width approximation (NWA) for which the decay width is smaller than 25\% of the squark/gluino mass.}
\label{fig:GravitinoSqrtFExclusion_mqmg}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_sqrtFLimit_Stop_A4_A9_A102mqmg.eps}
}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_sqrtFLimit_Stop_A4_A9_A104mqmg.eps}
}
\end{center}
\caption[95\% CL lower limits on the SUSY breaking scale $F$ as a function of the squark mass for $m_{\gluino} = 2 \times m_{\squark}$ and $m_{\gluino} = 4 \times m_{\squark}$.]{Observed (solid line) and expected (dashed line) 95\% CL lower limits on the SUSY breaking scale $F$ as a function of the squark mass for $m_{\gluino} = 2 \times m_{\squark}$ (top) and $m_{\gluino} = 4 \times m_{\squark}$ (bottom). The dotted line indicates the impact on the observed limit of the $\pm1\sigma$ LO theoretical uncertainty. The shaded bands around the expected line indicate the expected $\pm1\sigma$ and $\pm2\sigma$ ranges of limits.
  The region above the black dotted line defines the validity of the narrow-width approximation (NWA) for which the decay width is smaller than 25\% of the squark/gluino mass.}
\label{fig:GravitinoSqrtFExclusion_xmqmg}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_sqrtFLimit_Stop_A4_A9_A10mq2mg.eps}
}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentGravitino_combined_sqrtFLimit_Stop_A4_A9_A10mq4mg.eps}
}
\end{center}
\caption[95\% CL lower limits on the SUSY breaking scale $F$ as a function of the squark mass for $m_{\gluino} = 1/2 \times m_{\squark}$ and $m_{\gluino} = 1/4 \times m_{\squark}$.]{Observed (solid line) and expected (dashed line) 95\% CL lower limits on the SUSY breaking scale $F$ as a function of the squark mass for $m_{\gluino} = 1/2 \times m_{\squark}$ (top) and $m_{\gluino} = 1/4 \times m_{\squark}$ (bottom). The dotted line indicates the impact on the observed limit of the $\pm1\sigma$ LO theoretical uncertainty.
  The shaded bands around the expected line indicate the expected $\pm1\sigma$ and $\pm2\sigma$ ranges of limits.
  The region above the black dotted line defines the validity of the narrow-width approximation (NWA) for which the decay width is smaller than 25\% of the squark/gluino mass.}
\label{fig:GravitinoSqrtFExclusion_mqxmg}
\end{figure}

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\cleardoublepage
\chapter{Interpretations: ADD Large Extra Dimensions}
    \label{chapter:ADDGravitonProduction}

This chapter presents the results of the monojet analysis interpreted in the context of the LED ADD scenario discussed in Section \ref{sec:ADD}.
This model postulates the presence of $n$ extra spacial dimensions of size $R$, with only the graviton field being able to propagate through them.
This results in a reduction of the gravitational strength, with $M_D$, the fundamental Planck scale in $4+n$ dimensions, close to the electroweak scale for large enough $R$, and thus solving the hierarchy problem.
The agreement between the data and the MC background simulation for the selections M1 to M6 is translated into $95\%$~CL limits on the parameters of this model.

\section{ADD LED signal samples and systematic uncertainties on the signal}

Monte Carlo samples for different $n$ and $M_D$ parameter configurations of the ADD LED model, are generated using \exograviton{}\footnote{\exograviton{} is a dedicated module of \pythia{8}} and the CTEQ6.6 PDFs set.
The renormalization and factorization scales are set to $\sqrt{m_G^2 / 2 + \pt^2}$, where $m_G$ is the graviton mass and $\pt$ denotes the transverse momentum of the recoiling parton \cite{ATLAS:2012zim}.

Different sources of systematic uncertainties on the ADD signals are considered, as detailed in Section~\ref{sec:SysUncertaintiesSignal} for the case of third generation SUSY searches.
Experimental uncertainties include: uncertainties on the jet and $\met$ energy scales and resolutions; uncertainties on the simulated lepton identification, energy scales and resolutions; and the uncertainty on the total integrated luminosity.
The uncertainty on the PDFs; the uncertainty on the factorization, renormalization and matching scales; and the uncertainty on the initial- and final-state gluon radiation constitute the theoretical uncertainties, that affect both the acceptance and the cross section of the model.
The theoretical uncertainties on the acceptance introduce a 10\% effect on the total signal yield, inspired by the previous studies found in Ref.~\cite{ATLAS:2012zim}.
This reference also provides a computation for the theoretical uncertainty on the cross section, which is also adopted for this analysis.
This uncertainty results into a $36\%$ to $62\%$ in all the signal regions for $n$ increasing from~2 to~6.

\section{Exclusion Limits on $M_D$ and $n$}

The interpretation of the LED ADD model follows the same strategy as the light gravitino production in GMSB scenarios explained in Section~\ref{sec:GravitinoProduction}.
The exclusion in terms of the number of extra dimensions, $n$, and the fundamental Planck scale, $M_D$ is computed from the intersection between the model independent limit on visible cross section (in Table \ref{tab:modelIndependent}) and the ADD LED signal fiducial cross section for the different parameter configurations.
The uncertaities on the backgrounds and the signal are considered as independent in this simplified approach, and therefore no correlation between them is taken into account.
As an illustration, Figure \ref{fig:ADDModelIndependentLOM5} shows the fiducial cross section as a function of $M_D$ for $n=2, 4, 6$ in the signal region M5.
The band around the signal represents the total uncertainty (experimental, modeling effect on acceptance and on cross section all together).

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentADD_LO_Stop_A9.eps}
}
\end{center}
\caption[Fiducial cross section, $\sigma \times A \times \epsilon$, for the ADD LED model as a function of $M_D$ parameter for $n=2$, $n=4$ and $n=6$ (LO signal cross sections) compared to the observed and expected model independent limits in M5.]{Fiducial cross section, $\sigma \times A \times \epsilon$, as a function of $M_D$ parameter for $n=2$, $n=4$ and $n=6$ (LO signal cross sections) compared to the observed and expected model independent limits in the signal region M5. The colored band on the signal curves represent the total uncertainty (experimental and modeling uncertainties on acceptance and cross section).}
\label{fig:ADDModelIndependentLOM5}
\end{figure}

The best sensitivity for this model is obtained for the signal regions M3, M5 and M6, depending on $n$ and $M_D$.
The limits on $M_D$ parameter versus $n$ of the ADD model at leading order (LO) are reported in Table~\ref{tab:ADD_Limits_LO} and shown in Figure \ref{fig:ADDExclusionLOCombined}.
The green and yellow bands represent the $\pm 1\sigma$ and $\pm 2\sigma$ experimental uncertainty on the SM background yield respectively.
The limits on $M_D$ have been significantly improved with respect to the previous analysis in ATLAS~\cite{ATLAS:2012zim}, performed with $\unit[10]{fb^{-1}}$.

%--- ref{tab:ADD_Limits_LO}
\input{Interpretations/Tables/ADD_LO.tex}

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/plotExclusionADD_LO_combined_Stop_A4_A9_A10.eps}
}
\end{center}
\caption{The 95\% CL lower limits on the $M_D$ parameter of the ADD model for a number of extra dimensions $n$, considering LO signal cross sections.}
\label{fig:ADDExclusionLOCombined}
\end{figure}

The next-to-leading order cross section (NLO) for ADD signal is obtained by applying the scale factors extracted from Ref.~\cite{CMS:rwa}.
These scale factors have values of $1.5$ for $n=2, 3$ and $1.4$ for $n=4, 5, 6$.
As discussed in Ref.~\cite{ATLAS:2012ky}, the analysis partially probes the phase space region with $\sqrt{\hat{s}}>M_D$, where $\sqrt{\hat{s}}$ is the center-of-mass energy of the interaction.
This challenges the validity of the lower bounds on $M_D$, since they depend on the unknown ultraviolet behavior of the effective theory.
For this reason, the 95\% CL limits are re-computed after suppressing all the events with $\sqrt{\hat{s}}>M_D$.
Figure~\ref{fig:ADDModelIndependentNLOtruncatedM5} shows the variation of the visible cross section for different ADD models as a function of $M_D$, after suppressing the events with $\hat{s} > M_D^2$ for signal region M5.
The limits on $M_D$ as a function of the number of extra dimensions are reported on Table~\ref{tab:ADD_Limits_NLO_truncated} and shown in Figure~\ref{fig:ADDExclusionNLOtruncatedCombined}.
This figure also shows that the limits are not affected by the truncation of the events with $\hat{s} > M_D^2$, and compares the results obtained from this analysis to the latest CMS results in Ref.~\cite{CMS:rwa}.

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/ModelIndependentADD_NLO_truncated_Stop_A9.eps}
}
\end{center}
\caption[Fiducial cross section, $\sigma \times A \times \epsilon$, for the ADD LED model as a function of $M_D$ parameter for $n=2$, $n=4$ and $n=6$ (NLO signal cross sections, removing events with $\hat{s} > M_D$) compared to the observed and expected model independent limits in the signal region M5.]{Fiducial cross section, $\sigma \times A \times \epsilon$, as a function of $M_D$ parameter for $n=2$, $n=4$ and $n=6$ (NLO signal cross sections, removing events with $\hat{s} > M_D^2$) compared to the observed and expected model independent limits in M5. The colored band on the signal curves represent the total uncertainty (experimental and modeling uncertainties on acceptance and cross section).}
\label{fig:ADDModelIndependentNLOtruncatedM5}
\end{figure}

%--- ref{tab:ADD_Limits_NLO_truncated}
\input{Interpretations/Tables/ADD_NLO_truncated.tex}

\begin{figure}[!ht]
\begin{center}
\mbox{
\includegraphics[width=0.795\textwidth]{Interpretations/Figures/plotExclusionADD_ALLDistr_NLO_combined_Stop_A4_A9_A10.eps}
}
\end{center}
\caption{The 95\% CL lower limits on the $M_D$ parameter of the ADD model for a number of extra dimensions $n$, considering NLO signal cross sections and removing events with $\hat{s} > M_D^2$.}
\label{fig:ADDExclusionNLOtruncatedCombined}
\end{figure}

\clearpage


\section{Model-based Analysis}

As described in the introduction, prefetching has the potential to enable visualization systems to respond with interactive latencies.  However, current approaches are designed for interfaces with limited options for user interaction~\cite{}; many have suggested that increasing the expressivity of the interface (e.g., the number of possible user actions) makes the prediction task significantly more challenging.  In this section, we seek to understand the minimum accuracy for model needs in order to support low user perceived latencies, as well as the conditions that affect this accuracy.

\subsection{The Model}

Our goal is to estimate the minimum prediction accuracy $\alpha$ needed in order to ensure that the user perceived latency $l_{user}$ is below a fixed threshold (e.g. 100ms~\cite{} or 500ms).
Let $T=0$ be the time.
We assume that the user will perform her request at $T=t$ where $t \ge 0$, and that the cost of answering a request consists of fetching and rendering the results.
$l_{user}$ is defined as the time between $t$ and the change reflected in the visualization.
Let $l_{net}$ be the latency to execute a request, transfer the results across the network, and render it on screen.   In a typical system, $l_{user} = l_{net}$, which can be undesirable when query processing or the network latency are high.
Prefetching allows the client to proactively answer a request at $T=0$ and store the results in a client cache.
If a future request accessed data in the cache, then it can take much less time $l_{cache} < l_{net}$ and result in a more responsive user perceived experience.

We now model the expected user perceived latency as $l_{cache} + max(0, l_{net} - t)$ if model accurately predicted the request, and $l_{net}$ if it mispredicted.  The $max()$ operation accounts for cases when the cost of a cache miss is larger than $t$:
$$l_{user} = (l_{cache} + max(0, l_{net}-t))\times \alpha + l_{net}\times(1-\alpha) $$
Rearranging the terms, we can derive the minimum prediction accuracy in order to maintain $l_{user}$.
%$$\alpha = \frac{l_{net} - l_{user}}{l_{net} - l_{cache}}$$
$$\alpha = \frac{l_{user} - l_{net}}{l_{cache} + max(0, l_{net}-t) - l_{net}}$$

Prior perception and interactive visualization literature have used both $100$ and $500$ millisecond response times as thresholds for what is considered interactive~\cite{liu2014effects}.   Similarly, we assume request latencies can vary from $0$ to $2$ seconds~\cite{}, which embodies latencies that stem from query processing, serialization, network latencies and client overheads.  Finally, it is typically challenging to make accurate predictions more than several hundred milliseconds into the future~\cite{pasqual2014mouse}; we vary the ratio $\frac{t}{l_{net}}$ between $0.5$ and $1$.   We use the above constants in our analysis below.

Figure~\ref{fig:model_base} plots the model accuracy for two commonly cited $l_{user}$ thresholds (100 and 500ms).  The x-axis varies $l_{net}$ and each line represents the percentage of the prefetching costs that the user will experience.  For instance, when $\frac{t}{l_{net}}=1$, the user initiates the request at $T=t=l_{net}$ and does not experience any of the prefetching costs, whereas when $\frac{t}{l_{net}}=0.5$, the prefetching request is only half complete by the time the user initiates her request at $T=0.5\times l_{net}$.  We use a vertical line to mark the network latency at $1$ second.

The first observation is that when $l_{net}$ is smaller than the threshold, the user perceived accuracy is independent of the model accuracy.  When the threshold is $100$, the minimum model accuracy quickly exceeds $80\%$ by even a request latency of $500$ms; although the minimum accuracy is lower under the $500$ms threshold, the accuracy converges to $70\%$.
It is challenging to guarantee such high user prediction accuracy for a single request---the accuracy in existing literature is slightly above $50\%$ for a constrained user interface with 9 interaction options~\cite{battle2016dynamic}.

\ewu{Argue that assuming $t < l_{net}$ is a reasonable assumption.}

\begin{figure}[ht]
	\centering
	\includegraphics[width=1\columnwidth]{figures/model_base}
 	\caption{Minimum $\alpha$ vs network latency (x-axis), $\frac{t}{l_{net}}$ ratios (lines), and two thresholds (facets).}
    \label{fig:model_base}
\end{figure}

\stitle{Vary Prefetch Concurrency}
Most modern data processing systems are able to execute multiple concurrent requests~\cite{ebenstein2016fluxquery,giannikis2012shareddb}, thus we now modify the model to vary the number of concurrent prefetch requests $N$.
The $(1-\alpha)^N$ term is the probability that none of the prefetch requests match the user's actual request at $T=t$:
%
$$l_{user} = (l_{cache} + max(0, l_{net} - t)\times (1-(1-\alpha)^N) + l_{net}\times(1-\alpha)^N $$
%
Rearranging the terms results in the following minimum prediction accuracy:
%
$$\alpha = 1 - \left(\frac{l_{cache}+max(0,l_{net}-t)-l_{user}}{max(0,l_{net}-t)-l_{net}}\right)^{1/N}$$

\begin{figure}[h]
	\centering
	\includegraphics[width=1\columnwidth]{figures/model_concurrency}
 	\caption{Minimum $\alpha$ vs network latency (x-axis), concurrency (lines), and two latency thresholds (facets).}
  \label{fig:model_concurrency}
\end{figure}

Figure~\ref{fig:model_concurrency} shows that increasing the number of concurrent requests has an immediate effect on $\alpha$ ($t=l_{net}$ in these plots).    With $N=20$, a prediction model need only be $12\%$ accurate to ensure an interactive latency of $l_{user}=100$, while ensuring  $l_{user}=500$ only requires an accuracy of $3.5\%$.  At the limit, sending concurrent requests for all possible queries (assuming the cost is $l_{net}$) means the user latency is independent of prediction accuracy.  These results suggest that increasing the concurrency is an effective counter balance to model accuracy.

\stitle{Network Latency Variance}
Figure~\ref{fig:model_std} simulates $l_{net}$ drawn from a gaussian distribution with standard deviation of $std \in \{0, 100, 500\}$ milliseconds.  We set $N=20$, $t=l_{net}$, and the y-axis is from 0 to $0.4$.  Although the request variance affects the required model accuracy when $l_{net}$ is low, the curves for each threshold converge to the same levels independent of $std$; this is simply because $l_{net}$, rather than $std$, is the dominant factor as it increases.

\begin{figure}[h]
	\centering
	\includegraphics[width=1\columnwidth]{figures/model_std}
 	\caption{Minimum $\alpha$ vs network latency (x-axis), concurrency (lines), and two latency thresholds (facets).}
  \label{fig:model_std}
\end{figure}

\begin{figure}[h]
	\centering
	\includegraphics[width=1\columnwidth]{figures/model_partial}
 	\caption{Required prediction accuracy as a function of network latency, under progressive conditions where partial responses are sufficient.}
    \label{fig:model_partial}
\end{figure}

\stitle{Progressive Results}
The visualization and database communities have argued for approximate visualizations that are slightly inaccurate but potentially much faster to compute.  Techniques such as Online Aggergation~\cite{control,wanderjoin} or streaming wavelet compression~\cite{} are considered {\it progressive} because they are able to immediately return (and render) approximate results that improve over time; thus providing a natural trade-off between latency and result quality.  Our final analysis studies progressive results that require a fraction of the full request latency to be ``sufficiently good''.
\ewu{Use sampling and immens arguments as basis for 25\% partial results.} We find that $5\%$ accuracy is enough to maintain user perceived latency of $100$ when $25\%$ of the results is sufficient; at threshold of $500$, the perceived latency is independent of the model accuracy when $l_{net}\le 2000$.

\subsection{Discussion}
We find that a careful combination of existing techniques (concurrency and progressive results) can dramatically reduce the required model accuracy to enable low user perceived latency---in some extreme cases when the threshold is $500$, the model accuracy can even be $0\%$.
\ewu{Argue that given  fixed network throughput between client and server, increasing concurrency beyond a certain point naturally requires partial results.  Thus the two go hand in hand.   We also see that the prediction model simply needs to be ``good enough'' with an accuracy of $5-10\%$.  When the number of possible interactions are limited to $10-20$ possible interactions, then even a random model is sufficient.  However in rich interfaces with potentially dozens or a hundred possible interactions, there may simply be too many choices to learn a good model.}


\input{permve-ntnu-latex-assignment.tex}

\usepackage{float}

\title{
\normalfont \normalsize
\textsc{Norwegian University of Science and Technology\\IT3105 -- Artificial Intelligence Programming}
\horrule{0.5pt} \\[0.4cm]
\huge Module 4:\\ Using Minimax with\\ Alpha-Beta Pruning to play 2048\\
\horrule{2pt} \\[0.5cm]
}

\author{Per Magnus Veierland\\permve@stud.ntnu.no}

\date{\normalsize\today}

\begin{document}

\fancyfoot[C]{}
\maketitle

\newpage
\fancyfoot[C]{\thepage~of~\pageref{LastPage}} % Page numbering for right footer
\setcounter{page}{1}

\section*{Branching factor}

The aim of module~4 is to achieve a winning score in the game \textsc{2048}. The game consists of a two-dimensional $4\times 4$ grid where each tile in the grid is either empty or has a number between $2^1$ and $2^{17}$ (inclusive). An upper bound of the game's branching factor can be found by multiplying the number of possible player moves~(4) by the maximum number of empty tiles~(16) by the number of possible tile spawns~(2):

\begin{displaymath}
b_{\textit{upper}} = 4 \cdot 16 \cdot 2 = 128
\end{displaymath}

\section*{State representation}

Given such a large branching factor for each player move it is evident that a large number of states must be evaluated to perform even a shallow search involving a low number of moves. To enable a large number of states to be considered an efficient state representation scheme is beneficial.

Even if the theoretical maximum tile is $2^{17}$, the largest tile required to win the game, 2048, is only $2^{11}$. Given that the number 1 is not present in the game, it is possible to represent the empty tile and tiles for powers of two between $2^1$ and $2^{15}$ by using 4~bits to represent each tile. With 4~bits reserved for each tile, all 16~tiles can be represented by a single 64-bit integer. Using a commonly available 64-bit processor architecture, each board state can be held in a single CPU register.

The resulting representation with example values is shown in Figure~\ref{figure:represent} showing tile values and the corresponding bit values.

\begin{figure}[H]
\centering
\includegraphics[scale=1.0]{images/2048_state}
\caption{64-bit integer state representation}
\label{figure:represent}
\end{figure}

\section*{Lookup tables}

Another benefit from the compact state representation is that it allows for efficient use of lookup tables. The logic necessary to compute the successor state when given a move requires loops and branches. The effect of performing a move is also local to each row or column, depending on the direction of the move. Each row or column can then be considered in isolation and depend on the exact same logic to produce their successor. The number of possible rows or columns is $2^{16}=65536$, so representing a lookup table for a single direction will take $2^{16}*8=524288$~bytes, or $0.52$~MB. By using four lookup tables, one for each direction, successor states can be evaluated by performing four lookups, one for each row or column.

Lookup tables can also be used to evaluate both board heuristics and board scores. The board heuristic may be a costly function to evaluate, making it very beneficial to pre-compute. However, unlike the state transitions which can be considered as functions considering each row and column individually, the heuristic function evaluates the entire board state. Since a lookup table containing entries for all states would be $2^{64}*8=18.45$~exabytes, this is currently infeasible. However by imposing the limitation that the heuristic function can only consider a single row or column at a time, only a single lookup table with $2^{16}$ entries is needed. This limits the information available to the heuristic function, but the implemented heuristic function is still able to provide sufficient guidance to the search. When evaluating the heuristic value for a state; 8~lookups are performed, one for each row and column, and the result is summed. The advantage of this approach is that the heuristic evaluation can use very costly operations in the pre-computation, since its effective use consists only of lookups.

Evaluating the score of a given state is also done using lookup tables. This is a simple convenience such that the scoring of a state can be performed using four row lookups.

\section*{Expectimax}

The algorithm used to solve \textsc{2048} is \textit{Expectimax} with pruning based on a probability threshold and a transposition table. \textit{Expectimax} was chosen over \textit{Minimax} since the ``adversary'' is stochastic and not actively minimizing the expected outcome. The code centers around the two functions \texttt{score\_player\_node} and \texttt{score\_chance\_node} which calls each other recursively with successive board evaluations, until either the search is pruned due to the node having a probability beneath the probability threshold, or because the depth exceeds the depth threshold, or if the current board state is found in the transposition table.

When traversing downwards into the search, the current probability is tracked such that nodes falling beneath a probability threshold is pruned and heuristically evaluated instead of diving further down their state space. This is done to avoid spending time evaluating unlikely cases, such as when a $4$-tile spawns three times in a row. Using pruning based on probability is very effective, and the pruning rate can typically be 60-80\% without harming the effectiveness of the algorithm.

In addition to pruning based on probability, a configurable depth limit is also imposed to control the algorithm.

When evaluating the game state space, there will be many overlaps where different moves and/or different chance scenarios will end up resulting in the same game state. To improve the algorithm, a transposition table is used where evaluated states in chance nodes are added with their score and depth to a transposition table. If later on, a node is evaluated with a board state which exists in the transposition table, and where the stored depth is greater or equal to the current depth of the new search node, meaning that the current node's state has been evaluated at least as deep as what what the current search would do; the existing score stored in the transposition table is used instead of evaluating the node's state space again. The transposition table typically has a hit rate of 60-70\%.

\section*{Heuristic}

The \textit{Expectimax} algorithm requires a heuristic to evaluate the quality of a given state as it is unfeasible to fully evaluate the full state space for a given node due to a high branching ratio and a typical ``win'' state after approximately 2000~plies.

The chosen heuristic depends on four metrics and six constants. Since a lookup approach is used with the heuristic, the heuristic function can only evaluate a single row or column in isolation. For a given row the number of direct merges, $M_1$, secondary merges, $M_2$, and ternary merges, $M_3$, are recorded. E.g. for a row ``2 2 4 8'' the number of direct merges possible is one (``2'' with ``2'' resulting in ``4''), and the number of secondary merges is one (the resulting ``4'' with the existing ``4'', resulting in ``8''). The number of ternary merges is one (the resulting ``8'' with the existing ``8''). The last metric tracked is the number of ``bad tiles''; $B$. This is simply all non-zero tiles in a row which are not part of a merge.

The first constant, $k_L$, is a base used to distinguish losing states from non-losing states. Since losing states, i.e. states where there are no possible moves, are evaluated to zero -- the heuristic should result in a positive number for any non-losing state.

The three next constants is used to value the different merges, while the constants $k_{B1}$ and $k_{B2}$ is used to adjust the value of ``bad tiles''. A base constant is raised to the power of the number of bad tiles; penalizing each additional ``bad tile'' increasingly.

\begin{displaymath}
H(\textit{row}) = k_L + k_1 \cdot M_1 + k_2 \cdot M_2 + k_3 \cdot M_3 - \frac{{k_{B1}}^{B}}{k_{B2}}
\end{displaymath}

More sophisticated heuristics can be constructed which takes into account the monotonicity of the board state, the values of the tiles involved, and the number of empty tiles. However, the implemented heuristic performs well for shallow depths and is able to achieve a ``2048'' tile on ~80\% of runs with a maximum depth of 4 in under 3 minutes. It also regularly accomplishes the ``4096'' tile; but has not been able to score ``8196'', despite attempts with a maximum depth of 10.

The following constants have been found by thought, trial and error. Achieving good constants for the heuristic will likely involve a meta-search using e.g. a gradient descent approach to tune the algorithm.

\begin{displaymath}
k_L = 100, k_1 = 10, k_2 = 5, k_3 = 2.5, k_{B1} = 5, k_{B2} = 0.08
\end{displaymath}

\end{document}


\section{Tests}
	Tests have been taken from the ones used in laboratories and from TSPLIB, a library of sample instances for the TSP from various sources and of various types. A total of 13 instances, ranging in size from 12 to 100, have been tested 10 times for each method. Higher size tests should have been included but ILP solver already starts to slow at 100 nodes and the time to test was not so much.

	\subsection{ILP}
		Results obtained testing the ILP solver gives expected results, as it always finds optimal solution, and execution times grow with problem size. Average solution times on 10 runs are displayed in Table 1.\\

		\begin{center}
			\begin{tabular}{ | l | c | r | }
				\hline
				\multicolumn{1}{|c|}{Test} & Size & \multicolumn{1}{|c|}{Avg Time (s)}\\
				\hline
				tsp12.dat		& 12		& 0,133\\
				tsp15.dat		& 15		& 0,062\\
				tsp17.dat		& 17		& 0,162\\
				tsp26.dat		& 26		& 0,298\\
				tsp29a.dat		& 29		& 0,633\\
				tsp29b.dat		& 29		& 0,668\\
				tsp42.dat		& 42		& 2,462\\
				tsp48a.dat		& 48		& 8,371\\
				tsp48b.dat		& 48		& 10,357\\
				tsp48c.dat		& 48		& 2,275\\
				tsp60.dat		& 60		& 27,066\\
				tsp100a.dat	& 100	& 101,527\\
				tsp100b.dat	& 100	& 128,637\\
				\hline
			\end{tabular}
		\end{center}
		\captionof{table}{ILP tests results}
		\vspace{1em}
		We can see that while solving small instances is very efficient, bigger ones require more time and going from size 60 to 100 (5/3 ratio) it gets way slower (more than 5 times), and the trend continue growing in size. This is due to exponential growth of the solution space the solver has to explore to find and prove optimality.

	\subsection{Genetic Algorithm}
		Genetic algorithms, due to their random nature, give different results at each execution, so an average of 10 runs is considered. Obtained solutions and computation time are also greatly related to set parameters, which are reported for each instance. Parameters don't differ so much, but more generations and bigger populations are used for bigger instances, and to keep computational times low populations evaluated are reduced. Results obtained with specified tests are reported in Table 2.

		\begin{center}
			\hspace*{-1.2cm}
			\begin{tabular}{ | l | c | r | r | c | c | c | c |}
				\hline
				\multicolumn{1}{|c}{\multirow{2}{*}{Test}} & \multicolumn{1}{|c|}{\multirow{2}{*}{Size}} & Average & Avg. opt. & Opt. & \multicolumn{3}{c|}{Parameters} \\
				\cline{6-8}
				& & \multicolumn{1}{c}{time} & \multicolumn{1}{|c|}{distance} & found & Populations & Pop. size & Generations\\
				\hline
				tsp12.dat		& 12		& 0,405	& 0,00\%	& 10/10	& 5	& 500	& 200\\
				tsp15.dat		& 15		& 0,401	& 0,00\%	& 10/10	& 5	& 500	& 200\\
				tsp17.dat		& 17		& 0,502	& 0,02\%	& 9/10	& 5	& 500	& 250\\
				tsp26.dat		& 26		& 1,037	& 0,36\%	& 9/10	& 10	& 500	& 250\\
				tsp29a.dat		& 29		& 1,038	& 0,68\%	& 2/10	& 10	& 500	& 250\\
				tsp29b.dat		& 29		& 1,049	& 0,40\%	& 3/10	& 10	& 500	& 250\\
				tsp42.dat		& 42		& 1,246	& 0,75\%	& 2/10	& 10	& 500	& 300\\
				tsp48a.dat		& 48		& 1,779	& 2,30\%	& 0/10	& 7	& 1000	& 300\\
				tsp48b.dat		& 48		& 1,777	& 1,65\%	& 0/10	& 7	& 1000	& 300\\
				tsp48c.dat		& 48		& 1,859	& 2,91\%	& 0/10	& 7	& 1000	& 300\\
				tsp60.dat		& 60		& 1,802	& 3,19\%	& 0/10	& 7	& 1000	& 300\\
				tsp100a.dat	& 100	& 3,053	& 7,46\%	& 0/10	& 7	& 1000	& 500\\
				tsp100b.dat	& 100	& 3,083	& 6,33\%	& 0/10	& 7	& 1000	& 500\\
				\hline
			\end{tabular}
		\end{center}
		\captionof{table}{Genetic algorithm tests results}
		\vspace{1em}
		Note that computation time is highly dependent on set parameters, so if time is less important parameters could be increased to get better solutions. A direct consequence is that with genetic algorithms it's possible to get the desired precision/efficiency ratio. However as GAs don't provide an optimality proof, if an optimal solution is found, all remaining computation is executed anyway.

		Another thing to consider is that, because of GAs random nature, solution effectiveness might vary a lot from run to run. This is however not the case as the solutions found, even in bigger instances, are not more than 2\% over the average. This means that implemented genetic algorithm is reliable even if random.

	\subsection{Tests details}
		Here have been shown the most important results, all recorded data can be found in this spreadsheet:

		\href{https://docs.google.com/spreadsheets/d/1wJ98av-soxw_okteJ7X8zMJUfesh_16ECjA9uRbP8Hg/edit?usp=sharing}{\color{blue}{https://docs.google.com/spreadsheets/d/1wJ98av-soxw\_okteJ7X8zMJUfesh\_16ECjA9uRbP8Hg/edit?usp=sharing}}