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\section{Overview and Analysis Strategy}
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\label{sec:overview}
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We are searching for a $t\bar{t}\chi^0\chi^0$ or $W \ell b W \ell \bar{b} \chi^0 \chi^0$ final state
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(after top decay in the first mode, the final states are actually the same). So to first order
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this is ``$t\bar{t} +$ extra \met''.
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We work in the $\ell +$ jets final state, where the main background is $t\bar{t}$. We look for
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\met\ inconsistent with $W \to \ell \nu$. We do this by concentrating on the $\ell \nu$ transverse
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mass ($M_T$), since except for resolution effects, $M_T < M_W$ for $W \to \ell \nu$. Thus, the
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initial analysis is simply a counting experiment in the tail of the $M_T$ distribution.
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The event selection is one-and-only-one high \pt\ isolated lepton, four or more jets, and
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some moderate \met\ cut. At least one of the jets has to be btagged to reduce $W+$ jets.
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The event sample is then dominated by $t\bar{t}$, but there are also contributions from $W+$ jets,
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single top, dibosons, etc.
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In order to be sensitive to $\widetilde{t}\widetilde{t}$ production, the background in the $M_T$
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tail has to be controlled at the level of 10\% or better. So this is (almost) a precision measurement.
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The $t\bar{t}$ events in the $M_T$ tail can be broken up into two categories:
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(i) $t\bar{t} \to \ell $+ jets and (ii) $t\bar{t} \to \ell^+ \ell^-$ where one of the two
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leptons is not found by the second-lepton-veto (here the second lepton can be a hadronically
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decaying $\tau$).
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For a reasonable $M_T$ cut, say $M_T >$ 150 GeV, the dilepton background is of order 80\% of
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the total. This is because in dileptons there are two neutrinos from $W$ decay, thus $M_T$
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is not bounded by $M_W$. This is a very important point: while it is true that we are looking in
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the tail of $M_T$, the bulk of the background events end up there not because of some exotic
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\met\ reconstruction failure, but because of well understood physics processes. This means that
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the background estimate can be taken from Monte Carlo (MC), after carefully accounting for possible
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data/MC differences. Sophisticated fully ``data driven'' techniques are not really needed.
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Another important point is that in order to minimize systematic uncertainties, the MC background
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predictions are always normalized to the bulk of the $t\bar{t}$ data, ie, events passing all of the
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requirements but with $M_T \approx 80$ GeV.
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This removes uncertainties
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due to $\sigma(t\bar{t})$, lepton ID, trigger efficiency, luminosity, etc.
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The $\ell +$ jets background, which is dominated by
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$t\bar{t} \to \ell $+ jets, but also includes some $W +$ jets as well as single top,
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is estimated as follows:
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\begin{enumerate}
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\item We select a control sample of events passing all cuts, but anti-btagged, i.e. b-vetoed. This
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sample is now dominated by $W +$ jets. The sample is used to understand the
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$M_T$ tail in $\ell +$ jets processes.
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\item In MC we measure the ratio of the number of $\ell +$ jets events in the $M_T$ tail to
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the number of events with $M_T \approx$ 80 GeV. This ratio turns out to be pretty much the
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same for all sources of $\ell +$ jets.
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\item In data we measure the same ratio but after correcting for the $t\bar{t} \to$ dilepton
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contribution, as well as dibosons etc. The dilepton contribution is taken from MC after
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the correction described below.
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\item We compare the two ratios, as well as the shapes of the data and MC $M_T$ distributions.
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If they do not agree, we try to figure out why and fix it. If they agree well enough, we define a
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data-to-MC scale factor (SF) which is the ratio of the ratios defined in step 2 and 3, keeping track of the
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uncertainty.
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\item We next perform the full selection in $t\bar{t} \to \ell +$ jets MC, and measure this ratio
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again (which should be the same as that in step 2).
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\item
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We perform the full selection in data. We count the number of events with $M_T \approx 80$ GeV, after subtracting off the dilepton contribution,
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and multiply this count by the ratio from step 5 times the data/MC scale factor from step 4.
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%We count the events with $M_T \approx 80$ GeV, we
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%subtract off the dilepton contribution, we multiply the subtracted event count by the ratio from step 5 (or from
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%step 2), and also by the data/MC SF from step 4.
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The result is the prediction for the $\ell +$ jets BG in the $M_T$ tail.
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\end{enumerate}
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Steps 1-4 above are all measurements on the b-vetoed samples in data and/or MC. Steps 5 and 6 are performed on the b-tagged sample.
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To suppress dilepton backgrounds, we veto events with an isolated track of \pt $>$ 10 GeV.
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Being the common feature for electron, muon, and one-prong
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tau decays, this veto is highly efficient for rejecting
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$t\bar{t}$ to dilepton events. The remaining dilepton background can be classified into the following categories:
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%The dilepton background can be broken up into many components depending
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%on the characteristics of the 2nd (undetected) lepton
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%\begin{itemize}
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%\item 3-prong hadronic tau decay
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%\item 1-prong hadronic tau decay
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%\item $e$ or $\mu$ possibly from $\tau$ decay
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%\end{itemize}
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%We have currently no veto against 3-prong taus. For the other two categories, we explicitely
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%veto events %with additional electrons and muons above 10 GeV , and we veto events
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%with an isolated track of \pt\ $>$ 10 GeV. This rejects electrons and muons (either from $W\to e/\mu$ or
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%$W\to \tau\to e/\mu$) and 1-prong tau decays.
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%(it turns out that the explicit $e$ or $\mu$ veto is redundant with the isolated track veto).
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%Therefore the latter two categories can be broken into
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\begin{itemize}
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\item lepton is out of acceptance $(|\eta| > 2.50)$
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\item lepton has \pt\ $<$ 10 GeV, and is inside the acceptance
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\item lepton has \pt\ $>$ 10 GeV, is inside the acceptance, but survives the additional isolated track veto
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\end{itemize}
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%Monte Carlo studies indicate that there is no dominant contribution: it is ``a little bit of this,
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%and a little bit of that''.
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The last category includes 3-prong tau decays as well as electrons and muons from W decay that fail the isolation requirement.
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Monte Carlo studies indicate that these three components populate the $M_T$ tail in the proportions of roughly 6\%, 47\%, 47\%.
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We note that at present we do not attempt to veto 3-prong tau decays as they are only 16\% of the total dilepton background according to the MC.
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The high $M_T$ dilepton backgrounds come from MC, but their rate is normalized to the
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$M_T \approx 80$ GeV peak. In order to perform this normalization in data, the $W +$ jets
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events in the $M_T$ peak have to be subtracted off. This introduces a systematic uncertainty.
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There are two types of effects that can influence the MC dilepton prediction: physics effects
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and instrumental effects. We discuss these next, starting from physics.
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First of all, many of our $t\bar{t}$ MC samples (eg: MadGraph) have
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BR$(W \to \ell \nu)=\frac{1}{9} = 0.1111$.
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PDG says BR$(W \to \ell \nu) = 0.1080 \pm 0.0009$. This difference matters, so the $t\bar{t}$ MC
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must be corrected to account for this.
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Second, our selection is $\ell +$ 4 or more jets. A dilepton event passes the selection only if there are
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two additional jets from ISR, or one jet from ISR and one jet which is reconstructed from the
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unidentified lepton, {\it e.g.}, a three-prong tau. Therefore, all MC dilepton $t\bar{t}$ samples used
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in the analysis must have their jet multiplicity corrected (if necessary) to agree with what is
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seen in $t\bar{t}$ data. We use a data control sample of well identified dilepton events with
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\met\ and at least two jets as a template to ``adjust'' the $N_{jet}$ distribution of the $t\bar{t} \to$
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dileptons MC samples.
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The final physics effect has to do with the modeling of $t\bar{t}$ production and decay. Different
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MC models could in principle result in different BG predictions. Therefore we use several different
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$t\bar{t}$ MC samples using different generators and dfferent parameters, to test the stability
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of the dilepton BG prediction. All these predictions, {\bf after} corrections for branching ratio
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and $N_{jet}$ dependence, are compared to each other. The spread is a measure of the systematic
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uncertainty associated with the $t\bar{t}$ generator modeling.
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The main instrumental effect is associated with the efficiency of the isolated track veto.
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We use tag-and-probe to compare the isolated track veto performance in $Z + 4$ jet data and
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MC, and we extract corrections if necessary. Note that the performance of the isolated track veto
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is not exactly the same on $e/\mu$ and on one prong hadronic tau decays. This is because
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the pions from one-prong taus are often accompanied by $\pi^0$'s that can then result in extra
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tracks due to phton conversions. We let the simulation take care of that.
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Note that JES uncertainties are effectively ``calibrated away'' by the $N_{jet}$ rescaling described above.
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%Similarly, at the moment
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%we also let the simulation take care of the possibility of a hadronic tau ``disappearing'' in the
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%detector due to nuclear interaction of the pion.
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%The sample of events failing the last isolated track veto is an important control sample to
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%check that we are doing the right thing.
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Finally, there are possible improvements to this basic analysis strategy that can be added in the future:
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\begin{itemize}
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\item Move from counting experiment to shape analysis. But first, we need to get the counting
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experiment under control.
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\item Add an explicit three prong tau veto
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\item Do something to require that three of the jets in the event be consistent with $t \to Wb, W \to q\bar{q}$.
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%This could help rejecting some of the dilepton BG; however, it would also loose efficiency for
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%the $\widetilde{t} \to b \chi^+$ mode
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This could help reject some of the dilepton BG in the search for $\widetilde{t} \to t \chi^0$,
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but is not applicable to the $\widetilde{t} \to b \chi^+$ search.
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\item Consider the $M(\ell b)$ variable, which is not bounded by $M_{top}$ in $\widetilde{t} \to b \chi^+$
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\end{itemize}
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