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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 The PDF uncertainty on the product of cross-section and acceptance
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calculated using the method of Reference~\cite{ref: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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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. We smooth
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the boundaries using the method recommended by the SUSY
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group\cite{ref:smooth}. In addition, we show a limit
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curve based on the LO cross-section, as well as the
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``expected'' limit curve. The expected limit curve is
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calculated using the CLA function also available in cl95cms.
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Cross-section uncertainties due to variations of the factorization
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and renormalization scale are not included for the LO curve.
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The results are shown in Figure~\ref{fig:msugra}
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\begin{figure}[tbh]
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\begin{center}
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\includegraphics[width=\linewidth]{exclusion_noPDF.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. THIS IS STILL MISSING
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THE PDF UNCERTAINTIES. WE ALSO WANT TO IMPROVE THE SMOOTHING PROCEDURE.}
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\end{center}
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\end{figure}
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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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\begin{figure}[tbh]
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\begin{center}
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\includegraphics[width=0.5\linewidth]{nuissance.png}
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\caption{\label{fig:nuisance}\protect Exclusion curves in the
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mSUGRA parameter space,
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assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs
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using different models for the nuisance parameters.
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PDF UNCERTAINTOES ARE NOT INCLUDED.}
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\end{center}
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\end{figure}
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We find that the set of excluded points is identical for the
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lognormal and gamma models. There are small differences for the
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gaussian model. Following the recommendation of Reference~\cite{ref:cousins},
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we use the lognormal nuisance parameter model.
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\clearpage
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\subsubsection{Effect of signal contamination}
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\label{sec:contlimit}
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Signal contamination could affect the limit by inflating the
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background expectation. In our case we see no evidence of signal
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contamination, within statistics.
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The yields in the control regions
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$A$, $B$, and $C$ (Table~\ref{tab:datayield}) are just
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as expected in the SM, and the check
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of the $P_T(\ell \ell)$ method in the control region is
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also consistent with expectations (Table~\ref{tab:victory}).
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Since we have two data driven methods, with different
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signal contamination issues, giving consistent
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results that are in agreement with the SM, we
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argue for not making any correction to our procedure
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because of signal contamination. In some sense this would
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be equivalent to using the SM background prediction, and using
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the data driven methods as confirmations of that prediction.
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Nevertheless, here we explore the possible effect of
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signal contamination. The procedure suggested to us
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is the following:
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\begin{itemize}
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\item At each point in mSUGRA space we modify the
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ABCD background prediction from $A_D \cdot C_D/B_D$ to
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$(A_D-A_S) \cdot (C_D-C_S) / (B_D - B_S)$, where the
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subscripts $D$ and $S$ referes to the number of observed data
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events and expected SUSY events, respectively, in a given region.
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\item Similarly, at each point in mSugra space we modify the
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$P_T(\ell\ell)$ background prediction from
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$K \cdot K_C \cdot D'_D$ to $K \cdot K_C \cdot (D'_D - D'_S)$,
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where the subscript $D$ and $S$ are defined as above.
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\item At each point in mSUGRA space we recalculate $N_{UL}$
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using the weighted average of the modified $ABCD$ and
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$P_T(\ell\ell)$ method predictions.
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\end{itemize}
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\noindent This procedure results in a reduced background prediction,
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and therefore a less stringent $N_{UL}$.
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Note, however, that in some cases this procedure is
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nonsensical. For example, take LM0 as a SUSY
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point. In region $C$ we have a SM prediction of 5.1
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events and $C_D = 4$ in agreement with the Standard Model,
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see Table~\ref{tab:datayield}. From the LM0 Monte Carlo,
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we find $C_S = 8.6$ events. Thus, including information
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on $C_D$ and $C_S$ should {\bf strengthen} the limit, since there
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is clearly a deficit of events in the $C$ region in the
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LM0 hypothesis. Instead, we now get a negative ABCD
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BG prediction (which is nonsense, so we set it to zero),
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and therefore a weaker limit.
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\begin{figure}[tbh]
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\begin{center}
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\includegraphics[width=0.5\linewidth]{sigcont.png}
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\caption{\label{fig:sigcont}\protect Exclusion curves in the
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mSUGRA parameter space,
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assuming $\tan\beta=3$, sign of $\mu = +$ and $A_{0}=0$~GeVs
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with and without the effects of signal contamination.
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PDF UNCERTAINTIES ARE NOT INCLUDED.}
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\end{center}
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\end{figure}
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Despite these reservations, we follow the procedure suggested
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to us. A comparison of the exclusion region with and without
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the signal contamination is shown in Figure~\ref{fig:sigcont}
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(with no smoothing). The effect of signal contamination is
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small.
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\subsubsection{mSUGRA scans with different values of tan$\beta$}
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\label{sec:tanbetascan}
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For completeness, we also show the exclusion regions calculated
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using $\tan\beta = 10$ (Figure~\ref{fig:msugratb10}).
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\begin{figure}[tbh]
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\begin{center}
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\includegraphics[width=\linewidth]{exclusion_tanbeta10.pdf}
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\caption{\label{fig:msugratb10}\protect Exclusion curves in the mSUGRA parameter space,
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assuming $\tan\beta=10$, sign of $\mu = +$ and $A_{0}=0$~GeVs. THIS IS STILL MISSING
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THE PDF UNCERTAINTIES. WE ALSO WANT TO IMPROVE THE SMOOTHING PROCEDURE.}
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\end{center}
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\end{figure}
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