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\section{Blind analysis}
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We selected events for signal MC, and $B$ cocktail , prompt $J/ \psi$
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and $pp\rightarrow \mu\mu X$ background
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in the $B_s$ invariant mass region between $5.25$ and $5.65$ $GeV/c^2$
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and in the $\phi$ invariant mass region
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between $0.98$ and $1.06$ $GeV/c^2$. In this region we draw a
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two-dimensional plot to study, in each category,
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the correlation between the two variables
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(Fig.~\ref{fig:blind1},~\ref{fig:blind2},~\ref{fig:blind3},~\ref{fig:blind4}).
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We also project on each variable to evaluate which values of the
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variables will define our signal box and our sidebands
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(Fig.~\ref{fig:blind1a},~\ref{fig:blind2a},~\ref{fig:blind3a},
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~\ref{fig:blind4a},~\ref{fig:blind1b}
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,~\ref{fig:blind2b},~\ref{fig:blind3b},~\ref{fig:blind4b}):
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\begin{itemize}
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\item Signal box: $M_{B_s}\in [5.30;5.45]$ and $M_{\phi}\in [1.005;1.040]$
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\item Sidebands: $M_{B_s}\in [5.25;5.30]\cup [5.47;5.65]$ and
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$M_{\phi}\in [0.98;1.00]\cup [1.05;1.06]$
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\end{itemize}
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The correlation between the variables $B_s$ invariant mass and
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$\phi$ invariant mass, for each component, are
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reported in Table~\ref{tab:corr},~\ref{tab:corr2},~\ref{tab:corr3},
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~\ref{tab:corr4}:
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\begin{table}[ht]
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\begin{minipage}[b]{0.5\linewidth}\centering
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\begin{tabular}{l|cc}
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& $M_{B_s}$ & $M_{\phi}$ \\ \hline
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$M_{B_s}$ & $1$ & \\
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$M_{\phi}$ & $0.094$ & $1$ \\
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\end{tabular}
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\label{tab:corr}
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\caption{Summary table for the correlation between $B_s$
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invariant mass and $\phi$ invariant mass in signal.}
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\end{minipage}
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\hspace{0.5cm}
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\begin{minipage}[b]{0.5\linewidth}\centering
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\begin{tabular}{l|cc}
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& $M_{B_s}$ & $M_{\phi}$ \\ \hline
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$M_{B_s}$ & $1$ & \\
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$M_{\phi}$ & $-0.013$ & $1$ \\
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\end{tabular}
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\label{tab:corr2}
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\caption{Summary table for the correlation between $B_s$
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invariant mass and $\phi$ invariant mass in B background.}
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\end{minipage}
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\end{table}
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\begin{table}[ht]
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\begin{minipage}[b]{0.5\linewidth}\centering
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\begin{tabular}{l|cc}
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& $M_{B_s}$ & $M_{\phi}$ \\ \hline
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$M_{B_s}$ & $1$ & \\
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$M_{\phi}$ & $-0.093$ & $1$ \\
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\end{tabular}
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\label{tab:corr3}
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\caption{Summary table for the correlation between $B_s$
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invariant mass and $\phi$ invariant mass in prompt $J/ \psi$ background.}
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\end{minipage}
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\hspace{0.5cm}
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\begin{minipage}[b]{0.5\linewidth}\centering
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\begin{tabular}{l|cc}
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& $M_{B_s}$ & $M_{\phi}$ \\ \hline
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$M_{B_s}$ & $1$ & \\
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$M_{\phi}$ & $-0.020$ & $1$ \\
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\end{tabular}
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\label{tab:corr4}
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\caption{Summary table for the correlation between $B_s$
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invariant mass and $\phi$ invariant mass in $pp\rightarrow
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\mu\mu X$ background.}
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\end{minipage}
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\end{table}
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For each category we expect the number of events as reported
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for $1\ pb^{-1}$ in Table~\ref{tab:tabb} :
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\begin{table}[!h]
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\centering
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\begin{tabular}{|l|c|c|c|c|}
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\hline
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& $B_s\rightarrow J/ \psi \phi$ & $B$ background & Prompt
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$J/ \psi$ & $pp\rightarrow \mu \mu X$\\ \hline
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Luminosity ($pb^{-1}$) & \multicolumn{4}{c|}{1} \\ \hline
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Events in the region & $45$ & $59$ & $9$ & $25$ \\
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Events in the signal box & $42$ & $15$ & $1$ & $2$ \\
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Events in the sidebands & $0$ & $8$ & $1$ & $1$ \\ \hline
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\end{tabular}
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\label{tab:tabb}
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\caption{Summary table with the number of events in the signal
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box and in the sidebands.}
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\end{table}
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\clearpage
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\begin{figure}[!h]
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\vspace{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/blind_sig.eps}
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\caption{Reconstructed $\phi$ versus $B_s$ mass for signal.}
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\label{fig:blind1}
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\end{minipage}
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\hspace{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/blind_Bbkg.eps}
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\caption{Reconstructed $\phi$ versus $B_s$ mass for $B$ background.}
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\label{fig:blind2}
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\end{minipage}
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\end{figure}
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\begin{figure}[!h]
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\vspace{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/blind_prompt.eps}
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\caption{Reconstructed $\phi$ versus $B_s$ mass for prompt
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$J/ \psi$ background.}
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\label{fig:blind3}
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\end{minipage}
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\hspace{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/blind_ppmumux.eps}
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\caption{Reconstructed $\phi$ versus $B_s$ mass for
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$pp\rightarrow \mu \mu X$ background.}
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\label{fig:blind4}
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\end{minipage}
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\end{figure}
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\clearpage
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\begin{figure}[!h]
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\vspace{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/blind_sig_Bs.eps}
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\caption{Projection on $B_s$ mass in signal.}
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\label{fig:blind1a}
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\end{minipage}
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\hspace{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/blind_Bbkg_Bs.eps}
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\caption{Projection on $B_s$ mass in B background.}
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\label{fig:blind2a}
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\end{minipage}
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\end{figure}
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\begin{figure}[!h]
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\vspace{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/blind_prompt_Bs.eps}
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\caption{Projection on $B_s$ mass in prompt $J/ \psi$ background.}
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\label{fig:blind3a}
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\end{minipage}
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\hspace{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/blind_ppmumux_Bs.eps}
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\caption{Projection on $B_s$ mass in $pp\rightarrow \mu \mu X$
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background.}
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\label{fig:blind4a}
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\end{minipage}
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\end{figure}
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\clearpage
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\begin{figure}[h]
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\vspace{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/blind_sig_Phi.eps}
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\caption{Projection on $\phi$ mass in signal.}
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\label{fig:blind1b}
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\end{minipage}
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\hspace{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/blind_Bbkg_Phi.eps}
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\caption{Projection on $\phi$ mass in B background.}
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\label{fig:blind2b}
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\end{minipage}
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\end{figure}
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\begin{figure}[h]
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\vspace{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/blind_prompt_Phi.eps}
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\caption{Projection on $\phi$ mass in prompt $J/ \psi$ background.}
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\label{fig:blind3b}
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\end{minipage}
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\hspace{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/blind_ppmumux_Phi.eps}
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\caption{Projection on $\phi$ mass in $pp\rightarrow \mu \mu X$ background.}
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\label{fig:blind4b}
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\end{minipage}
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\end{figure}
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\clearpage
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