%Results and interpretation
\chapter{Results and Interpretation }
\label{ch:resint}
\section{Results}
\label{sec:results}
Our final prediction for the total background, including all sources utilizing the full data set and all control samples was compared to data corresponding to the integrated luminosity of 35.9 $\text{fb}^{-1}$ as shown in Fig.~\ref{fig:result}. Statistical uncertainties of data and systematic uncertainties of predictions are shown. The bottom plot shows the ratio of data over total background predictions. There is an overall agreement in all search bins between the SM prediction and data within the uncertainties.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{figure/Results/Results.pdf}
\caption{Observed data yields (black points) and prefit SM background predictions (filled solid areas) for the 84 search regions, where "prefit" means there is no constraint from the likelihood fit. The ratio of data to background prediction in each bin is shown in the bottom plot. The hatched bands correspond to the total uncertainty in the background prediction.}
\label{fig:result}
\end{center}
\end{figure}
%
%Fig.~\ref{fig:pull} shows the pull distribution versus search region bin number and all the 84 bin pull distribution. It is clear that the pull values are almost all within two sigmas and its distribution looks OK.
%\begin{figure}[!ht]
% \begin{center}
% \includegraphics[width=0.45\textwidth]{figure/Results/pull.pdf}
% \includegraphics[width=0.45\textwidth]{figure/Results/dist_pull.pdf}
% \caption{Pull calculated for each signal region(left) and pull distribution for all 84 bins(right).}
% \label{fig:pull}
% \end{center}
%\end{figure}
\section{Statistics}
\label{sec:stat}
Standard presentations of results in search experiments have always been a point of discussion among particle physicists. Bayesian credible intervals depend on some prior probability distribution and are often somewhat suggestive. In some cases where data do not overwhelm the prior probability, physicist want to summarize the observation independently. Frequentists on the other hand tend to draw conclusions based on compatibility of data with theory. In high statistics and signal dominated regions, both school of thoughts tend to converge. Unfortunately, experiments usually suffer from small signals buried in a large background. In such a case, misinterpretation of frequentist statements become a serious issue.\\\\
In this analysis, the null hypothesis is an observation that can be explained by SM contributions only, and the alternative hypothesis is the one where new physics is required to explain the observation. We aim to exclude the signal in it's absence and confirm it's existence in case of the actual signal. The sensitivity of the analysis to the different models discussed in Chapter~\ref{ch:analysis} is investigated by computing the expected limits as a function of the mass of the lightest supersymmetric particle (LSP) versus the mass of the top squark. For limit calculations, we use the Modified Frequentist ($CL_{s}$) statistical method~\cite{CLS1, CLS2}. \\\\
For $n$ different search channels, if $s_{i}$, $B_{i}$ and $d_{i}$ are signals predicted from MC, background and observed candidates in the $i^{th}$ channel, trespectively, hen the test-statistic $Q_{i}$ in that particular channel is given by,
\begin{equation}
Q_{i} = \frac{e^{-(s_{i} + b_{i})}(s_{i} + b_{i})^{d_{i}} }{d_{i}!} \Biggm/ \frac{e^{- b_{i}}( b_{i})^{d_{i}} }{d_{i}!},
\label{eq;teststat}
\end{equation}
and the test-statistic is the product of $Q_{i}$ all over $n$ channels. The test-statistic $Q$ is constructed for the entire experiment so that the confidence in the signal + background hypothesis is given by the probability that the test-statistic is less than or equal to the value observed in the entire experiment, $Q_{obs}$, i.e.,
\begin{equation}
CL_{s+b} = P_{s+b}(Q \leq Q_{obs}),
\end{equation}
where $P_{s+b}(Q \leq Q_{obs})$ is the sum of the Poisson probabilities. The confidence level for the background alone is,
\begin{equation}
CL_{b} = P_{b}(Q \leq Q_{obs}).
\end{equation}
Values of $CL_{b}$ very close to 1 indicate poor compatibility with the background hypothesis and favor the signal+background hypothesis. $CL_{s}$ is a ratio of confidence levels:
\begin{equation}
CL_{s} = \frac{CL_{s+b}}{CL_{b}}.\\
\label{eq:cls}
\end{equation}
The motivation behind the $CL_{s}$ method is to avoid excluding or discovering signals to which the analysis is not sensitive. For example, observing less than the mean expected background events could be accommodated best with a negative signal cross section. The exclusion may be as strong as to exclude the zero signal scenario at a given confidence level. This is a valid result in
terms of statistics, but it says more about fluctuations of the (known) background than about the searched signal. This effect is avoided by normalizing the confidence level of the signal
plus background hypothesis $CL_{s+b}$ to the confidence level in the background-only hypothesis
$CL_{b}$. For this analysis, the Higgs combination tool was used to calculate the limits, the full frequentist $CL_{s}$ limit calculation together with the LHC-like test statistics~\cite{CLs}.
\section{Systematic Uncertainties}
\label{sec:sysunc}
Sources of systematic uncertainties were studied on MC signal samples and their contributions were computed as follows:
\begin{itemize}
\item \textbf{MC Statistics:} statistical uncertainties from MC signal samples.
\item \textbf{Luminosity:} A 2.6$\%$ flat contribution is assigned as per suggestion from the CMS working group assigned to this task for the collaboration.
\item \textbf{Lepton Veto: } The number of signal events vetoed by each category are evaluated. Then the yields are varied by the corresponding veto category uncertainties and propogated to determine the relative changes to the signal yields in each search bin. The number of veto’ed events effectively reflects the various lepton selection efficiency.
\item \textbf{b-tag Efficiency: } The b-tagging and mistagging scale factors are functions of the jet $p_{\text{T}}$ and $\eta$. Scale factors were varied within their uncertainties and propagated to the signal bins to determine the signal uncertainties for each bin. We take the conservative assumption that the charm-mistagging scale factor uncertainty is correlated with the b-tagging scale factor. Both b-tagging and charm-mistagging scale factors were varied together, and the light flavor-mistagging scale factors were varied independently.
\item \textbf{b-tag FastSim Corrections: } The b-tagging and mistagging performance as derived from fast simulation was corrected to match the full simulation predictions. Separate correction factors are derived for b-jets, c-jets, and light-flavor-jets, as a function of the jet $p_{\text{T}}$ and $\eta$. As with the scale factors above, the correction factors for each type of jet are varied independently within their uncertainties and propagated to determine the uncertainty in the signal for each search bin.
\item \textbf{Trigger Efficiency: } The trigger efficiencies are measured using data as described in Sec.~\ref{sec:trigger}. The signal samples are corrected for the trigger inefficiencies. The effect of trigger efficiency uncertainties on the signal samples is at most 2.6$\%$ in the lowest $E_{\text{T}}^{\text{miss}}$ bins.
\item \textbf{Renormalization and Factorization Scales: } This uncertainty was calculated using the envelope of the weights obtained from varying the renormalization and factorization scales, $\mu_{R}$ and $\mu_{F}$, by a factor of two ~\cite{muRmuF, muRmuF2}. These effects on the shape of the signal are taken as the uncertainties. These uncertainties are considered as uncertainties on the signal cross section.
\item \textbf{Initial State Radiation: }An ISR correction is derived from $t\bar{t}$ events, with a selection requiring two leptons (electrons or muons) and two b-tagged jets, implying that any other jets in the event arise from ISR. The correction factors are 1.000, 0.882, 0.792, 0.702, 0.648, 0.601, 0.515 for $N_{jet}^{ISR}$ = 0, 1, 2, 3, 4, 5, 6+. The corrections are applied to the simulated signal jet samples with an additional normalization factor, typically $\sim$1.15 (depending on the signal model), to ensure the overall cross section of the sample remains constant. The systematic uncertainty in these corrections is chosen to be half of the deviation from unity for each correction factor.
\item \textbf{Jet Energy Corrections: }The jet energy corrections (JEC) are varied within the $p_{\text{T}}$ and $\eta$-dependent jet energy scale uncertainties available in the official CMS database. A different set of corrections and uncertainties are used in fast simulation samples. These variations are propagated into the jet-dependent search variables, such as: $N_{b-jets}$, $N_{tops}$, $E_{\text{T}}^{\text{miss}}$, $M_{\text{T2}}$, $H_{\text{T}}$, $\Delta\phi(E_{\text{T}}^{\text{miss}}, j_{i})$.
\item \textbf{Parton Distribution Functions:} The PDF4LHC prescription~\cite{pdf4lhc} for the uncertainty on the total cross section is included as $\pm$1 sigma bands in the results distributions. \\
\end{itemize}
Additional uncertainties such as data-MC difference scale factors and Fullsim/Fastsim scale factor for top quark reconstruction, uncertainties associated with $E_{\text{T}}^{\text{miss}}$ are considered. The signal systematic list and their typical range are shown in Tables~\ref{table:sysT2tt} and~\ref{table:sysT1tttt}.\\
\begin{table}[h]
\begin{center}
\caption{\small{In T2tt SMS, the signal systematic sources and their typical ranges as calculated. These are relative uncertainties.}}
\begin{tabular}{|c|c|}
\hline
FSource & Typical Value \\
\hline
MC Statistics & 1-100$\%$\\
\hline
Luminosity & 2.6$\%$ \\
\hline
Renormalization and factorization scales & 0-2.4$\%$ \\
\hline
“ISR” recoil & $\%$0-46\\
\hline
b-tagging efficiency, heavy flavor & 0-17$\%$ \\
\hline
b-tagging efficiency, light flavor & 0-17$\%$ \\
\hline
Lepton veto & 0-4.7$\%$ \\
\hline
Jet energy scale & 0-20$\%$ \\
\hline
MET uncerntainty & 0-24$\%$ \\
\hline
Trigger & 0-2.6$\%$ \\
\hline
Full/fastsim scale for top reco & 0-19$\%$ \\
\hline
top tagger efficiency data/MC difference & 0-14$\%$ \\
\hline
\end{tabular}
\label{table:sysT2tt}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\caption{\small{In T1tttt SMS, the signal systematic sources and their typical ranges as calculatd. These are relative uncertainties.}}
\begin{tabular}{|c|c|}
\hline
Source & Typical Value \\
\hline
MC Statistics & 1-100$\%$\\
\hline
Luminosity & 2.6$\%$ \\
\hline
Renormalization and factorization scales & 0-3.5$\%$ \\
\hline
“ISR” recoil & $\%$0-45\\
\hline
b-tagging efficiency, heavy flavor & 0-16$\%$ \\
\hline
b-tagging efficiency, light flavor & 0-21$\%$ \\
\hline
Lepton veto & 0-6.8$\%$ \\
\hline
Jet energy scale & 0-34 $\%$ \\
\hline
MET uncerntainty & 0-17$\%$ \\
\hline
Trigger & 0-2.6$\%$ \\
\hline
Full/fastsim scale for top reco & 0-24 $\%$ \\
\hline
top tagger efficiency data/MC difference & 0-11$\%$ \\
\hline
\end{tabular}
\label{table:sysT1tttt}
\end{center}
\end{table}
\section{Interpretation}
\label{sec:interpretation}
The exclusion limits on the model T2tt (a direct production of the top squark) with all of the different sources of the signal systematic uncertainties are shown in Fig.~\ref{fig:ResultT2tt}. For the 35.9 $\text{fb} ^{- 1}$ data we have substantial improvement of our exclusion to the top squark mass up to 1133 GeV and the LSP mass up to 480 GeV for expected limits with respect to the previous result~\cite{2.3fbpaper}. And for observed limits, we have exclusion up to 1022 GeV of the top squark mass and up to 430 GeV of the LSP mass.\\
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{figure/Results/T2tt.pdf}
\caption{The 95$\%$ CL upper limit on the production cross section of the T2tt simplified model as a function of the top squark and LSP masses. The solid black curves represent the observed exclusion contour with respect to NLO+NLL signal cross sections and the change in this contour due to variation of these cross sections within their theoretical uncertainties~\cite{CrossSecTeVScale}. The dashed red curves indicate the mean expected exclusion contour and the region containing 68$\%$ of the distribution of expected exclusion limits under the background-only hypothesis. No interpretation is provided for signal models for which $|m_{\tilde{t}}$ - $m_{\tilde{\chi}_{1}^{0}}$ - $m_{\text{T}} | \leq $ 25 GeV and $m_{\tilde{t}} \leq $ 275 GeV because signal events are essentially indistinguishable from SM $t\bar{t}$ events in this region, rendering the signal event acceptance difficult to model. }
\label{fig:ResultT2tt}
\end{center}
\end{figure}
The results were also interpreted for the gluino mediated top squark production (T1tttt SMS) model shown in Fig.~\ref{fig:ResultT1tttt}. All different sources of the systematic uncertainties were included in the limit calculations. For expected limits, the maximum exclusion of the gluino mass is 2028 GeV and the maximum LSP mass exclusion is 1154 GeV. For the observed limits, we have exclusion of the gluino mass up to 2038 GeV and 1154 GeV for the LSP mass. The results are a great improvement when compared to previous results based on data taken in the 2015 LHC run with an integrated luminosity of 2.3 $\text{fb}^{- 1}$~\cite{2.3fbpaper}.\\
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{figure/Results/T1tttt.pdf}
\caption{The 95$\%$ CL upper limit on the production cross section of the T1tttt simplified model as a function of the top squark and LSP masses. The meaning of the curves is explained in the ~\ref{fig:ResultT2tt} caption.}
\label{fig:ResultT1tttt}
\end{center}
\end{figure}
\section{Summary}
\label{sec:summary}
Results have been presented of a search for direct and gluino-mediated top squark production in proton-proton collisions at a center-of-mass energy of 13 TeV. The central feature of the analysis is our top quark identification algorithm that reconstructs hadronically decaying top quark across a wide spectrum of top quark transverse momentum $p_{\text{T}}$ with very good efficiency. The search uses all-hadronic events with at least four jets and a large imbalance in transverse momentum ($E_{\text{T}}^{\text{miss}}$), selected from data corresponding to an integrated luminosity of 35.9 fb$^{−1}$ collected with the CMS detector at the LHC in 2016. A set of 84 search regions where defined based on $E_{\text{T}}^{\text{miss}}$, $M_{\text{T2}}$, the number of top quark tagged objects, and the number of bottom quark jets. No statistically significant excess of events was observed when compared to the expectation from the standard
model predictins. \\\\
In simplified models of pair production of top squarks, that decay to a top quark and a neutralino, top squark masses of up to 1020 GeV and neutralino masses up to 430 GeV are excluded at the 95$\%$ confidence level. For models with gluino pair production, gluino masses of up to 2040 GeV, and neutralino masses up to 1150 GeV are excluded, for the T1tttt model. These results significantly extend those of our previous study~\cite{2.3fbpaper}. The use of top quark tagging was an exclusive approach taken in contrast to other analyses in CMS. A significant improvement of top quark tagging method over previous methods not only improved signal sensitivity, but also improved our background estimates. Top tagging provided a novel means to search for new phenomena at the LHC, yielding complementary sensitivity to other approaches.