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\section{Trigger Efficiency Studies}
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Dependent on the instantaneous luminosity $l$ of LHC
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different dimuon triggers at Level-1 and at the
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High Level Trigger (HLT) farm will be implemented
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by CMS to adjust the data acquisition rate.
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The first scenario assumes $l_1\simeq 10^{30}$/cm$^2$ s,
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the second one a factor of 10 higher $l_2 = 10\cdot l_1$
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expected to occur after 10~pb$^{-1}$ of integrated luminosity
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has been recorded (i.e. likely after ICHEP 2010).
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For completeness we study preliminary HLT trigger scenarios~\cite{hlt}
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analoguously to the Level-1 trigger.
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We study three different triggers:
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\begin{itemize}
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\item \verb,L1DoubleMuOpen,, a
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double muon pass-through trigger, where no selection requirements
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beyond the Level-1 are applied,
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\item \verb,HLT_doubleMu0,, with at least two muons identified at the
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Level-3, without any $p_T$ requirement, and
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\item \verb,HLT_doubleMu3,, with at least two Level-3
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muons with $p_T>2.5$ GeV/c each.
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\end{itemize}
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Table~\ref{tab:trtab} summarizes the trigger details.
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\begin{table}[htbp]
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\vspace{0.5cm}
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\centering
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\begin{tabular}{cccc}
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\hline\hline
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Trigger & Pre-scale & Rate (Hz) & Signal $\epsilon$ $\%$\\
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\hline
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\verb,L1DoubleMuOpen, & $1$ $(l_1)$ & $5.9$ & $19.3$ \\
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& $50$ $(l_2)$ & $1.2$ & \\
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\hline
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\verb,HLT_doubleMu0, & $1$ $(l_1)$ & $0.4$ & $7.2$\\
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& $1$ $(l_2)$ & $3.8$ & \\
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\hline
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\verb,HLT_doubleMu3, & $1$ $(l_1)$ & $0.1$ & $4.2$\\
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& $1$ $(l_2)$ & $1.2$ & \\
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\hline\hline
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\end{tabular}
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\label{tab:trtab}
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\caption{Trigger scenarios. For each trigger we list
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the pre-scale factor and the rate for two the instantaneous luminosity
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scenarios ($l_1=10^{30}$/cm$^2$ s, $l_2 = 10\cdot l_1$) and the signal efficiency.}
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\end{table}
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For each trigger scenario we compare the transverse momentum
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distribution of the two exclusive channels $B_d\to J/\psi K^*$
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and $B_s\to J/\psi \phi$ with the original distribution
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provided by EvtGen~\cite{evtgen} (see Figs.~\ref{fig:f1},~\ref{fig:f2}).
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The distributions have been normalized to the event number in each
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selected sample.
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\begin{figure}[ht]
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\vspace{1.cm}
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\begin{minipage}[b]{0.5\linewidth}
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\centering
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\includegraphics[scale=0.4]{figure/BsPt.eps}
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\caption{$B_s$ transverse momentum distribution for the generated events
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and after application of the various triggers.}
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\label{fig:f1}
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\end{minipage}
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\hspace{0.5cm}
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\begin{minipage}[b]{0.5\linewidth}
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\centering
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\includegraphics[scale=0.4]{figure/BdPt.eps}
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\caption{$B_d$ transverse momentum distribution for the generated events
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and after application of the various triggers.}
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\label{fig:f2}
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\end{minipage}
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\end{figure}
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The $p_T$ dependent efficiency is defined as:
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\begin{equation}
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\epsilon(p_T^B) = \frac{N_{trg}(p_T^B)}{N_{gen}(p_T^B)}
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\end{equation}
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where $N_{trig}$ is the number of events passing the trigger in the given
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$p_T^B$ bin, and $N_{gen}$ is the number of events generated in the same bin.
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The trigger efficiencies for all three trigger scenarios as a function of
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$p_T^B$ are displayed in Fig.\ref{fig:trigeff}.
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\begin{figure}[ht]
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\vspace{1.cm}
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\centering
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\includegraphics[scale=0.4]{figure/trigeffplot.eps}
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\caption{Trigger efficiencies for $B_s$ events for the three trigger
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scenarios as a function of $p_T^B$.
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\label{fig:trigeff}
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}
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\end{figure}
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Since we use the proper decay length $c t$ as variable to discriminate
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signal and background, and to measure the proper decay time of the
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$B$ meson, we study the bias introduced by the different
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trigger scenarios. Fig.~\ref{fig:f5} displays the distribution in $c t$
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for the different trigger scenarios in a logarithmic event scale.
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Fig,~\ref{fig:f6} shows the ratio between the \verb,L1DoubleMuOpen,
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and the generated events with the statistical error per bin:
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we observe an insignificant distortion of the original distribution
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after application of the trigger requirements.
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\begin{figure}[ht]
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\vspace{0.4 cm}
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\centering
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\end{figure}
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\begin{figure}[ht]
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\begin{minipage}[b]{0.5\linewidth}
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\centering
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\includegraphics[scale=0.4]{figure/BsCtau.eps}
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\caption{$B_s$ proper decay length distribution for generated events
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and after application of various trigger scenarios.
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The colors are explained in the legend.}
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\label{fig:f5}
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\end{minipage}
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\hspace{0.5cm}
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\begin{minipage}[b]{0.5\linewidth}
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\centering
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\includegraphics[scale=0.4]{figure/trigRatio.eps}
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\caption{The ratio between signal events passing the
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L1DoubleMuOpen trigger requirements and generated events.
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This efficiency is $\epsilon = (19.7 \pm 0.3)\%$.}
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\label{fig:f6}
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\end{minipage}
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\end{figure}
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For completeness we also show the distribution of the total $B$
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momentum for the two major $B$ decay channels in Fig.~\ref{fig:f3},~\ref{fig:f4}.
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\begin{figure}[ht]
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%\vspace{-16.5cm}
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\begin{minipage}[b]{0.5\linewidth}
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\centering
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\includegraphics[scale=0.4]{figure/BsP.eps}
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\caption{$B_s$ total momentum for simulated $pp\to B_s \to J/\psi \phi$ events
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as generated and after trigger requirements were applied.
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The colors are explained in the legend.}
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\label{fig:f3}
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\end{minipage}
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\hspace{0.5cm}
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\begin{minipage}[b]{0.5\linewidth}
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\centering
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\includegraphics[scale=0.4]{figure/BdP.eps}
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\caption{$B_d$ total momentum distribution for simulated $pp\to B_d \to J/\psi K^*$ events
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as generated and after trigger requirements were applied.
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The colors are explained in the legend.}
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\label{fig:f4}
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\end{minipage}
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\end{figure}
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