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GuillermoFidalgo committed Mar 16, 2024
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56 changes: 28 additions & 28 deletions chapters/Chapter03.tex
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Expand Up @@ -12,31 +12,31 @@ \section{Background information on EJs}


\begin{figure}
\centering
\begin{subfigure}{.45\linewidth}
\includegraphics*[width=\textwidth]{pdfs/FlavoredSchematicOfEvent.pdf}
\caption{Flavor aligned}
\end{subfigure}
\begin{subfigure}{.45\linewidth}
\includegraphics*[width=\textwidth]{pdfs/UnflavoredSchematicOfEvent.pdf}
\caption{Unflavored}
\end{subfigure}
\caption{The Emerging Jets event models}
\label{fig:emj_production}
\centering
\begin{subfigure}{.45\linewidth}
\includegraphics*[width=\textwidth]{pdfs/FlavoredSchematicOfEvent.pdf}
\caption{Flavor aligned}
\end{subfigure}
\begin{subfigure}{.45\linewidth}
\includegraphics*[width=\textwidth]{pdfs/UnflavoredSchematicOfEvent.pdf}
\caption{Unflavored}
\end{subfigure}
\caption{The Emerging Jets event models}
\label{fig:emj_production}
\end{figure}

\begin{figure}
\begin{center}
\begin{subfigure}{.45\linewidth}
\includegraphics*[width=\linewidth]{pdfs/BSSWPairProduction_ggFusion.pdf}
\caption{gluon-gluon fusion}
\end{subfigure}
\begin{subfigure}{.45\linewidth}
\includegraphics*[width=\linewidth]{pdfs/BSSWPairProduction_qqAnnihilation.pdf}
\caption{quark anti-quark annihilation}
\end{subfigure}
\end{center}
\caption[Emergin jets production modes]{Feynman diagrams for pair production of dark mediator particles, with mediators decay to an SM quark and a dark quark}
\begin{center}
\begin{subfigure}{.45\linewidth}
\includegraphics*[width=\linewidth]{pdfs/BSSWPairProduction_ggFusion.pdf}
\caption{gluon-gluon fusion}
\end{subfigure}
\begin{subfigure}{.45\linewidth}
\includegraphics*[width=\linewidth]{pdfs/BSSWPairProduction_qqAnnihilation.pdf}
\caption{quark anti-quark annihilation}
\end{subfigure}
\end{center}
\caption[Emergin jets production modes]{Feynman diagrams for pair production of dark mediator particles, with mediators decay to an SM quark and a dark quark}
\end{figure}

In figure \ref{fig:emj_production} we see the production process of the emerging jets signature.
Expand All @@ -55,23 +55,23 @@ \section{Trigger Efficiency and Scale Factor studies}
The $H_T$ triggers chosen are the triggers with the lowest online $H_T$ threshold that are not pre-scaled. The configurations used for this analysis are:

\begin{itemize}
\item \verb|HLT_PFHT900_v* OR HLT_PFJet450_v*| for 2016. The addition of a jet trigger path in an OR configuration is the recommended path to mitigate an observed inefficiency at high values of $H_T$ caused by the Level-1 trigger firmware issues for 2016.
\item \verb|HLT_PFHT1050_v*| for 2017 and 2018.
\item \verb|HLT_PFHT900_v* OR HLT_PFJet450_v*| for 2016. The addition of a jet trigger path in an OR configuration is the recommended path to mitigate an observed inefficiency at high values of $H_T$ caused by the Level-1 trigger firmware issues for 2016.
\item \verb|HLT_PFHT1050_v*| for 2017 and 2018.
\end{itemize}

The calculation of the trigger efficiency is carried out by using the orthogonal \verb|HLT_Mu50_v*| trigger as the reference trigger. To determine the offline HT threshold at which the trigger can be considered to be fully efficient was estimated by fitting the trigger efficiency as a function of $H_T$ to an error function (erf) and an algebraic function ($f$):

\begin{align}
\text{erf}(H_T ;\ A,B,C) &= \frac A2 \left[1+ \text{erf}\left(\dfrac{H_T - |B|}{C}\right) \right]\label{eq:erf}\\
f(H_T ;\ A,B,C,D) &= A \dfrac{\frac{H_T - B}{C}}{1+ \left(\frac{H_T - B}{C}\right)^2} + D \label{eq:alg}
\text{erf}(H_T ;\ A,B,C) & = \frac A2 \left[1+ \text{erf}\left(\dfrac{H_T - |B|}{C}\right) \right]\label{eq:erf} \\
f(H_T ;\ A,B,C,D) & = A \dfrac{\frac{H_T - B}{C}}{1+ \left(\frac{H_T - B}{C}\right)^2} + D \label{eq:alg}
\end{align}

Where \eqref{eq:erf} and \eqref{eq:alg} are modeled after the sigmoid-like functions
\[
\text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \dd{t}
\text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \dd{t}
\]
\[
f(x)=\frac{x}{1+x^{2}}
f(x)=\frac{x}{1+x^{2}}
\]


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