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\section{Limit on new physics}
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\label{sec:limit}
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%{\bf \color{red} The numbers in this Section need to be double checked.}
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\subsection{Limit on number of events}
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\label{sec:limnumevents}
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As discussed in Section~\ref{sec:results}, we see one event
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in the signal region, defined as SumJetPt$>$300 GeV and
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\met/$\sqrt{\rm SumJetPt}>8.5$ GeV$^{\frac{1}{2}}$.
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The background prediction from the SM Monte Carlo is 1.3 events.
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%, where the uncertainty comes from
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%the jet energy scale (30\%, see Section~\ref{sec:systematics}),
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%the luminosity (10\%), and the lepton/trigger
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%efficiency (10\%)\footnote{Other uncertainties associated with
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%the modeling of $t\bar{t}$ in MadGraph have not been evaluated.
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%The uncertainty on $pp \to \sigma(t\bar{t})$ is also not included.}.
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The data driven background predictions from the ABCD method
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and the $P_T(\ell\ell)$ method are $1.3 \pm 0.8({\rm stat}) \pm 0.3({\rm syst})$
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and $2.1 \pm 2.1({\rm stat}) \pm 0.6({\rm syst})$, respectively.
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These three predictions are in good agreement with each other
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and with the observation of one event in the signal region.
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We calculate a Bayesian 95\% CL upper limit\cite{ref:bayes.f}
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on the number of non SM events in the signal region to be 4.1.
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We have also calculated this limit using
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% a profile likelihood method
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% as implemented in
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the cl95cms software\cite{ref:cl95cms},
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and we also find 4.1. (This is not surprising, since cl95cms
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also gives baysean upper limits with a flat prior).
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These limits were calculated using a background prediction of $N_{BG} = 1.4 \pm 0.8$
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events, the error-weighted average of the ABCD and $P_T(\ell\ell)$ background
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predictions. The upper limit is not very sensitive to the choice of
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$N_{BG}$ and its uncertainty.
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To get a feeling for the sensitivity of this search to some
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popular SUSY models, we remind the reader of the number of expected
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LM0 and LM1 events from Table~\ref{tab:sigcont}: $8.6 \pm 1.6$
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events and $3.6 \pm 0.5$ events respectively, where the uncertainties
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are from energy scale (Section~\ref{sec:systematics}), luminosity,
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and lepton efficiency.
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\subsection{Outreach}
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\label{sec:outreach}
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Conveying additional useful information about the results of
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a generic ``signature-based'' search such as the one described
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in this note is a difficult issue.
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Here we attempt to present our result in the most general
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way.
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Models of new physics in the dilepton final state
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can be confronted in an approximate way by simple
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generator-level studies that
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compare the expected number of events in 34.0~pb$^{-1}$
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with our upper limit of 4.1 events. The key ingredients
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of such studies are the kinematical cuts described
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in this note, the lepton efficiencies, and the detector
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responses for SumJetPt and \met/$\sqrt{\rm SumJetPt}$.
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The muon identification efficiency is $\approx 95\%$;
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the electron identification efficiency varies from $\approx$ 63\% at
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$P_T = 10$ GeV to 91\% for $P_T > 30$ GeV. The isolation
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efficiency in top events varies from $\approx 83\%$ (muons)
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and $\approx 89\%$ (electrons) at $P_T=10$ GeV to
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$\approx 95\%$ for $P_T>60$ GeV.
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%{\bf \color{red} The following numbers were derived from Fall 10 samples. }
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The average detector
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responses for SumJetPt and $\met/\sqrt{\rm SumJetPt}$ are
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$1.02 \pm 0.05$ and $0.94 \pm 0.05$ respectively, where
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the uncertainties are from the jet energy scale uncertainty.
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The experimental resolutions on these quantities are 11\% and
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16\% respectively.
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To justify the statements in the previous paragraph
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about the detector responses, we plot
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in Figure~\ref{fig:response} the average response for
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SumJetPt and \met/$\sqrt{\rm SumJetPt}$ in MC, as well as the
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efficiency for the cuts on these quantities used in defining the
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signal region.
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% (SumJetPt $>$ 300 GeV and \met/$\sqrt{\rm SumJetPt} > 8.5$
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% Gev$^{\frac{1}{2}}$).
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%{\bf \color{red} The following numbers were derived from Fall10 samples }
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We find that the average SumJetPt response
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in the Monte Carlo is about 1.02, with an RMS of order 11\% while
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the response of \met/$\sqrt{\rm SumJetPt}$ is approximately 0.94 with an
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RMS of 16\%.
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%Using this information as well as the kinematical
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%cuts described in Section~\ref{sec:eventSel} and the lepton efficiencies
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%of Figures~\ref{fig:effttbar}, one should be able to confront
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%any existing or future model via a relatively simple generator
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%level study by comparing the expected number of events in 35 pb$^{-1}$
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%with our upper limit of 4.1 events.
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\begin{figure}[tbh]
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\begin{center}
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\includegraphics[width=\linewidth]{selectionEffDec10.png}
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\caption{\label{fig:response} Left plots: the efficiencies
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as a function of the true quantities for the SumJetPt (top) and
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tcMET/$\sqrt{\rm SumJetPt}$ (bottom) requirements for the signal
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region as a function of their true values. The value of the
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cuts is indicated by the vertical line.
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Right plots: The average response and its RMS for the SumJetPt
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(top) and tcMET/$\sqrt{\rm SumJetPt}$ (bottom) measurements.
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The response is defined as the ratio of the reconstructed quantity
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to the true quantity in MC. These plots are done using the LM0
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Monte Carlo, but they are not expected to depend strongly on
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the underlying physics.
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%{\bf \color{red} These plots were made with Fall10 samples. }
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}
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\end{center}
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\end{figure}
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%%% Nominal
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% -----------------------------------------
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% observed events 1
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% relative error on acceptance 0.000
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% expected background 1.400
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% absolute error on background 0.770
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% desired confidence level 0.95
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% integration upper limit 30.00
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% integration step size 0.0100
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% -----------------------------------------
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% Are the above correct? y
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% 1 16.685 0.29375E-06
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%
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% limit: less than 4.112 signal events
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%%% Add 20% acceptance uncertainty based on LM0
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% -----------------------------------------
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% observed events 1
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% relative error on acceptance 0.200
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% expected background 1.400
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% absolute error on background 0.770
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% desired confidence level 0.95
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% integration upper limit 30.00
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% integration step size 0.0100
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% -----------------------------------------
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% Are the above correct? y
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% 1 29.995 0.50457E-06
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%
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% limit: less than 4.689 signal events
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\subsection{mSUGRA scan}
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\label{sec:mSUGRA}
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We also perform a scan of the mSUGRA parameter space, as recomended
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by the SUSY group convenors\cite{ref:scan}.
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The goal of the scan is to determine an exclusion region in the
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$m_0$ vs. $m_{1/2}$ plane for
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$\tan\beta=3$,
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sign of $\mu = +$, and $A_{0}=0$~GeV. This scan is based on events
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generated with FastSim.
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The first order of business is to verify that results using
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Fastsim and Fullsim are compatible. To this end we compare the
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expected yield for the LM1 point in FullSim (3.56 $\pm$ 0.06) and
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FastSim (3.29 $\pm$ 0.27), where the uncertainties are statistical only.
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These two numbers are in agreement, which gives us confidence in
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using FastSim for this study.
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The FastSim events are generated with different values of $m_0$
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and $m_{1/2}$ in steps of 10 GeV. For each point in the
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$m_0$ vs. $m_{1/2}$ plane, we compute the expected number of
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events at NLO. We then also calculate an upper limit $N_{UL}$
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using cl95cms at each point using the following inputs:
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\begin{itemize}
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\item Number of BG events = 1.40 $\pm$ 0.77
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\item Luminosity uncertainty = 11\%
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\item The acceptance uncertainty is calculated at each point
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as the quadrature sum of
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\begin{itemize}
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\item The uncertainty due to JES for that point, as calculated
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using the method described in Section~\ref{sec:systematics}
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\item A 5\% uncertainty due to lepton efficiencies
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\item An uncertaity on the NLO cross-section obtained by varying the
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factorization and renormalization scale by a factor of two\cite{ref:sanjay}.
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\item A 13\% PDF uncertainty on the product of cross-section and acceptance.
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This uncertainty was calculated using the method of Reference~\cite{ref:pdf} for a
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number of points in the $m_0$ vs. $m_{1/2}$ plane, and was found to be
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approximately independent of mSUGRA parameters, see Table~\ref{tab:pdf}.
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\end{itemize}
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\item We use the ``log-normal'' model for the nuisance parameters
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in cl95cms
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\end{itemize}
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We actually calculate three different values of $N_{UL}$:
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\begin{enumerate}
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\item Observed $N_{UL}$ asssuming the NLO cross-section.
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\item Observed $N_{UL}$ asssuming the LO cross-section. In this case
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uncertainties due to PDFs and renormlization/factorization scales are not
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included.
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\item Expected $N_{UL}$ sssuming the NLO cross-section. This is
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calculated using the the CLA function also available in cl95cms.
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\end{enumerate}
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\begin{table}[hbt]
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\begin{center}
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\caption{\label{tab:pdf} PDF uncertainties on the product of
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cross-section and acceptance for a number of representative points
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in the mSUGRA plane.}
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\begin{tabular}{c|c|c|c|c|c}
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$\tan\beta$ & $m_0$ & $m_{1/2}$ & sign of $\mu$ & $A_0$ & uncertanity (\%) \\ \hline
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3 & 50 & 260 & + & 0 & $^{+13}_{-9}$ \\
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3 & 50 & 270 & + & 0 & $^{+13}_{-9}$ \\
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3 & 60 & 260 & + & 0 & $^{+14}_{-9}$ \\
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3 & 200 & 200 & + & 0 & $^{+12}_{-9}$ \\
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3 & 200 & 210 & + & 0 & $^{+13}_{-10}$ \\
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3 & 210 & 200 & + & 0 & $^{+11}_{-8}$ \\
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3 & 200 & 140 & + & 0 & $^{+16}_{-12}$ \\
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3 & 140 & 150 & + & 0 & $^{+08}_{-8}$ \\
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3 & 150 & 140 & + & 0 & $^{+14}_{-10}$ \\
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10 & 60 & 260 & + & 0 & $^{+16}_{-11}$ \\
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10 & 100 & 260 & + & 0 & $^{+14}_{-10}$ \\
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10 & 100 & 260 & + & 0 & $^{+12}_{-9}$ \\
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10 & 90 & 260 & + & 0 & $^{+15}_{-10}$ \\
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10 & 240 & 260 & + & 0 & $^{+10}_{-8}$ \\
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10 & 240 & 260 & + & 0 & $^{+13}_{-10}$ \\ \hline
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\end{tabular}
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\end{center}
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\end{table}
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An mSUGRA point is excluded if the resulting $N_{UL}$ is smaller
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than the expected number of events. Because of the quantization
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of the available MC points in the $m_0$ vs $m_{1/2}$ plane, the
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boundaries of the excluded region are also quantized. The excluded points
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are shown in Figure~\ref{fig:tanbeta3raw}; in this Figure we also show
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ad-hoc curves that represent the excluded regions.
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In Figure~\ref{fig:msugra} we show our results compared with
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results from previous experiments.
|
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|
| 239 |
|
| 240 |
\begin{figure}[tbh]
|
| 241 |
\begin{center}
|
| 242 |
\includegraphics[width=0.4\linewidth]{tanbeta3_NLO_observed.png}
|
| 243 |
\includegraphics[width=0.4\linewidth]{tanbeta3_NLO_expected.png}
|
| 244 |
\includegraphics[width=0.4\linewidth]{tanbeta3_LO_observed.png}
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| 245 |
\caption{\label{fig:tanbeta3raw}\protect Excluded points in the
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$m_0$ vs. $m_{1/2}$ plane for $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs.
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Top left: observed, using the NLO cross-section.
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Top right: expected using the NLO cross-section.
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Bottom left: observed, using the LO cross-section.
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The curves are meant to represent the excluded regions.}
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| 251 |
\end{center}
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| 252 |
\end{figure}
|
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|
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|
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\begin{figure}[tbh]
|
| 256 |
\begin{center}
|
| 257 |
\includegraphics[width=\linewidth]{exclusion.pdf}
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\caption{\label{fig:msugra}\protect Exclusion curves in the mSUGRA parameter space,
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assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs.}
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\end{center}
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\end{figure}
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\clearpage
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\subsubsection{Check of the nuisance parameter models}
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We repeat the procedure outlined above but changing the
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lognormal nuisance parameter model to a gaussian or
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gamma-function model. The results are shown in
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Figure~\ref{fig:nuisance}. (In this case,
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to avoid smoothing artifacts, we
|
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show the raw results, without smoothing).
|
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|
| 275 |
\begin{figure}[tbh]
|
| 276 |
\begin{center}
|
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\includegraphics[width=0.5\linewidth]{nuissance.png}
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| 278 |
\caption{\label{fig:nuisance}\protect Observed NLO exclusion curves in the
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| 279 |
mSUGRA parameter space,
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| 280 |
assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs
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using different models for the nuisance parameters. (Note: this
|
| 282 |
plot was made without the PDF uncertainties.}
|
| 283 |
\end{center}
|
| 284 |
\end{figure}
|
| 285 |
|
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We find that different assumptions on the PDFs for the nuisance
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parameters make very small differences to the set of excluded
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points.
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Following the recommendation of Reference~\cite{ref:cousins},
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we use the lognormal nuisance parameter model as the default.
|
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|
| 292 |
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% \clearpage
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|
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|
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\subsubsection{Effect of signal contamination}
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\label{sec:contlimit}
|
| 298 |
|
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Signal contamination could affect the limit by inflating the
|
| 300 |
background expectation. In our case we see no evidence of signal
|
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contamination, within statistics.
|
| 302 |
The yields in the control regions
|
| 303 |
$A$, $B$, and $C$ (Table~\ref{tab:datayield}) are just
|
| 304 |
as expected in the SM, and the check
|
| 305 |
of the $P_T(\ell \ell)$ method in the control region is
|
| 306 |
also consistent with expectations (Table~\ref{tab:victory}).
|
| 307 |
Since we have two data driven methods, with different
|
| 308 |
signal contamination issues, giving consistent
|
| 309 |
results that are in agreement with the SM, we
|
| 310 |
argue for not making any correction to our procedure
|
| 311 |
because of signal contamination. In some sense this would
|
| 312 |
be equivalent to using the SM background prediction, and using
|
| 313 |
the data driven methods as confirmations of that prediction.
|
| 314 |
|
| 315 |
Nevertheless, here we explore the possible effect of
|
| 316 |
signal contamination. The procedure suggested to us
|
| 317 |
for the ABCD method is to modify the
|
| 318 |
ABCD background prediction from $A_D \cdot C_D/B_D$ to
|
| 319 |
$(A_D-A_S) \cdot (C_D-C_S) / (B_D - B_S)$, where the
|
| 320 |
subscripts $D$ and $S$ refer to the number of observed data
|
| 321 |
events and expected SUSY events, respectively, in a given region.
|
| 322 |
We then recalculate $N_{UL}$ at each point using this modified
|
| 323 |
ABCD background estimation. For simplicity we ignore
|
| 324 |
information from the $P_T(\ell \ell)$
|
| 325 |
background estimation. This is conservative, since
|
| 326 |
the $P_T(\ell\ell)$ background estimation happens to
|
| 327 |
be numerically larger than the one from ABCD.
|
| 328 |
|
| 329 |
Note, however, that in some cases this procedure is
|
| 330 |
nonsensical. For example, take LM0 as a SUSY
|
| 331 |
point. In region $C$ we have a SM prediction of 5.1
|
| 332 |
events and $C_D = 4$ in agreement with the Standard Model,
|
| 333 |
see Table~\ref{tab:datayield}. From the LM0 Monte Carlo,
|
| 334 |
we find $C_S = 8.6$ events. Thus, including information
|
| 335 |
on $C_D$ and $C_S$ should {\bf strengthen} the limit, since there
|
| 336 |
is clearly a deficit of events in the $C$ region in the
|
| 337 |
LM0 hypothesis. Instead, we now get a negative ABCD
|
| 338 |
BG prediction (which is nonsense, so we set it to zero),
|
| 339 |
and therefore a weaker limit.
|
| 340 |
|
| 341 |
|
| 342 |
|
| 343 |
|
| 344 |
\begin{figure}[tbh]
|
| 345 |
\begin{center}
|
| 346 |
\includegraphics[width=0.5\linewidth]{sigcont.png}
|
| 347 |
\caption{\label{fig:sigcont}\protect Observed NLO exclusion curves in the
|
| 348 |
mSUGRA parameter space,
|
| 349 |
assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs
|
| 350 |
with and without the effects of signal contamination.
|
| 351 |
Note: PDF uncertainties are not included.}
|
| 352 |
\end{center}
|
| 353 |
\end{figure}
|
| 354 |
|
| 355 |
A comparison of the exclusion region with and without
|
| 356 |
signal contamination is shown in Figure~\ref{fig:sigcont}
|
| 357 |
(with no smoothing). The effect of signal contamination is
|
| 358 |
small, of the same order as the quantization of the scan.
|
| 359 |
|
| 360 |
|
| 361 |
\subsubsection{mSUGRA scans with different values of tan$\beta$}
|
| 362 |
\label{sec:tanbetascan}
|
| 363 |
|
| 364 |
For completeness, we also show the exclusion region calculated
|
| 365 |
using $\tan\beta = 10$ (Figure~\ref{fig:msugratb10}).
|
| 366 |
|
| 367 |
|
| 368 |
\begin{figure}[tbh]
|
| 369 |
\begin{center}
|
| 370 |
\includegraphics[width=0.4\linewidth]{tanbeta10_NLO_observed.png}
|
| 371 |
\caption{\label{fig:tanbeta10raw}\protect Excluded points in the
|
| 372 |
$m_0$ vs. $m_{1/2}$ plane for $\tan\beta=10$, sign of $\mu = +$ and $A_{0}=0$~GeVs.
|
| 373 |
This plot is made using the NLO cross-sections.
|
| 374 |
The curves is meant to represent the excluded region.}
|
| 375 |
\end{center}
|
| 376 |
\end{figure}
|
| 377 |
|
| 378 |
\begin{figure}[tbh]
|
| 379 |
\begin{center}
|
| 380 |
\includegraphics[width=\linewidth]{exclusion_tanbeta10.pdf}
|
| 381 |
\caption{\label{fig:msugratb10}\protect Exclusion curve in the mSUGRA parameter space,
|
| 382 |
assuming $\tan\beta=10$, sign of $\mu = +$ and $A_{0}=0$~GeVs.}
|
| 383 |
\end{center}
|
| 384 |
\end{figure}
|
| 385 |
|
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|
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|
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|
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|
