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\section{Simulated Event Samples}
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Table~\ref{tab:data} lists the sets of fully simulated events
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used in this study. Based on those samples we optimize
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the event selection and validate the analysis technique.
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All samples use PYTHIA~\cite{pythia} for simulation of the $pp$
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collisions. The signal and the $b \bar{b}$
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background are studied in the inclusively generated
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$b\rightarrow J/ \psi X$ sample.
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The EvtGen~\cite{evtgen} simulation takes care of all $b$ hadron decays.
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One of the two $b$ hadrons in the event is forced to decay to a final state
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with a $J/ \psi$ meson, directly or via intermediate excited charmonium states.
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%in the four inclusive datasets $B^0\rightarrow J/ \psi X$, $B_s\rightarrow J/ \psi X$, $B^+\rightarrow J/ \psi X$ and $\Lambda_b \rightarrow J/ \psi X$
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%which include EvtGen for the decay of all $b$ hadrons.
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Events with a $J/ \psi$ decaying into two muons are filtered at the
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generator level by requiring that the two muons have opposite charge
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and transverse momentum $p_T > 2.5$~GeV/c and a pseudo-rapidity $|\eta|<2.5$.
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For detailed studies we also use exclusively generated $B_S\to J/\psi \phi$
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and $B_d\to J/\psi K^*$ samples.
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For the prompt $J/ \psi$ background we use a sample of
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$pp\rightarrow J/ \psi X$-type events produced with the
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same generator-level muon filter as the $b$ hadrons samples.
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For the QCD background we study the inclusive
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$pp\rightarrow \mu \mu X$ sample with the same muon pre-
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selection requirements on $p_T$ and $\eta$ as for the other sets above.
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In the next iteration we will require that $b$-hadrons are decayed by
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EventGen~\cite{evtgen}.
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\begin{table}[htbp]
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\vspace{0.8cm}
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\centering
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\begin{tabular}{ccccc}
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\hline\hline
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Process & $N_{ev}$ & $\sigma$ (pb) & $\epsilon_{filter}$ $(\%)$ &
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\cal{L} $(pb^{-1})$ \\
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\hline
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%$B^0\rightarrow J/ \psi X$ & & & & \\
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%$B_s\rightarrow J/ \psi X$ & & & & \\
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%$B^+\rightarrow J/ \psi X$ & & & & \\
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%$\Lambda_b\rightarrow J/ \psi X$ & & & & \\
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$b\rightarrow J/ \psi X$ & $5\times 10^6$& $66.6\times 10^{6}$ &
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$0.024$ & $31$ \\
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$pp\rightarrow J/ \psi X$ & $10^7$ & $12.6\times 10^{6}$& $0.074$ & $11$\\
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$pp\rightarrow \mu \mu X$ & $10^6$ & $48.4$ & $1.12\times 10^{-5}$& $1.87$\\
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\hline\hline
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\end{tabular}
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\label{tab:data}
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\caption{For each event sample
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we list the total number of generated events $N_{ev}$,
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the production cross-section $\sigma$,
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the filter efficiency $\epsilon_{filter}$, and
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the integrated luminosity $L$.
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For the $b\rightarrow J/ \psi X$ sample we
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derive the integrated luminosity from the
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$B_s\rightarrow J/ \psi \phi$ sub-set.}
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\end{table}
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\clearpage
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