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%  Updated to 318 invpb and added JES event selection -- 11 Mar 05 Pekka
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\section{Detector, Backgrounds, and Event Selection}
\label{sec:detBkgdSel}

This section begins with an explanation of the \ttbar event signature along
with a summary of the background processes that can mimic it.
The relevant parts of the CDF detector are briefly described. The
event selection and the separation into disjoint subsamples are
defined next. Finally, the expected number of background events is
discussed.

\subsection{Event Signature}
\label{ssec:signature}

The Standard Model top quark decays into a $W$ boson and a $b$ quark,
immediately ($\tau\approx4\times10^{-25}~\mathrm{s}$) and with
$\sim\!\!100\%$ branching ratio. The \ttbar event signature is
therefore determined by the decay products of the two $W$s, each
of which can produce two quarks or a charged lepton and a neutrino.
This analysis considers events in the lepton + jets channel, where
one $W$ decays to quarks and the other $W$ decays to $e\nu_{e}$ or
$\mu\nu_{\mu}$. In the following, ``lepton'' will refer
exclusively to a candidate electron or muon.
Thus, events of interest to this measurement have a high-$\pt$ isolated $e$ or $\mu$,
a neutrino, and four jets, two of which are $b$ jets.
More jets may be present due to hard gluon radiation from an
incoming parton (initial state radiation or ISR) or from a
final-state quark (final state radiation or FSR).
Events where a $W$ boson decays to $\tau\nu_{\tau}$ can also enter
the event sample when a secondary electron or muon from the tau
decay passes the lepton cuts---about 5\% of identified \ttbar
events have this decay chain. \note{Check that number!!!}

There are several non-\ttbar processes that have similar signatures
and enter into the event sample for this analysis. Events where a
leptonically decaying $W$ boson is found in association with QCD
production of at least four additional jets, sometimes including
a \bbbar pair, have the same signature and are an irreducible
background.
Singly produced top quarks, e.g. $q\bar{q}\rightarrow t\bar{b}$,
with a leptonic $W$ decay and additional jets produced via QCD
radiation, also have the same signature.
Additional background events enter the sample when
the \ttbar signature is faked. For example, a jet can fake an
isolated lepton with very low efficiency, a neutrino can be
mistakenly inferred when the missing energy in the event is
mismeasured, and a leptonically decaying $Z$ boson can look like
a $W$ if one lepton goes undetected.

\subsection{Detector}
\label{ssec:detector}

The Collider Detector at Fermilab is a general-purpose detector
observing \ppbar collisions at Fermilab's Tevatron.
The detector geometry is cylindrical, with the $z$ axis pointing
along a tangent to the Tevatron ring, in the direction of proton
flight in the accelerator.
Transverse quantities such as $\et$ and $\pt$
are projections into the plane
perpendicular to the $z$ axis. The coordinates $x$, $y$,
$r$, and $\phi$ are defined in this transverse plane,
with the $x$ axis pointing outward from the
accelerator ring, and the $y$ axis pointing straight up.
The angle
$\theta$ is the polar angle measured from the proton direction,
and $\eta=-\frac{1}{2}\ln(\tan\theta)$ is the pseudorapidity.
When $\eta$ is calculated using a reconstructed interaction
point ($z\neq 0$), it is referred to as $\etaevt$.
\Fig{f_CDFpic} shows an elevation view of the CDF detector.
The relevant subdetectors are described briefly below.
A more complete description of the CDF \runii detector is
provided elsewhere~\cite{r_cdfDetector}.

\begin{cfigure}
\includegraphics{plots/cdfelev}
\caption[Elevation view of the CDF detector.]
{An elevation view of the CDF \runii detector. From inside to
outside, CDF consists of a silicon strip detector, a tracking drift
chamber, an electromagnetic calorimeter, a hadronic calorimeter, and
muon chambers.}
\label{f_CDFpic}
\end{cfigure}

The CDF tracking system is the first detector element seen by a particle leaving
the interaction point.
The silicon detectors~\cite{r_NIMSVX} provide position measurements
with very high resolution for charged particles close to the interaction
region, allowing extrapolation of tracks back to the collision point and
reconstruction of secondary, displaced vertices.
There are three separate silicon microstrip subdetectors, in which
all but the innermost layer use
double-sided wafers of silicon to get measurements in both axial
and stereo directions for three-dimensional tracking. There are
a total of 722,432 channels, with a typical strip pitch
of 55--$65~\mu\mathrm{m}$ for axial strips, 60--$75~\mu\mathrm{m}$
for \degrees{1.2} small-angle stereo strips, and 125--$145~\mu\mathrm{m}$
for \degrees{90} stereo strips.
The innermost detector, layer 00 (L00),
is a single-sided layer of silicon mounted directly on the beampipe,
at a radius of about \genunit{1.6}{cm}, providing an axial
measurement as close to the collision point as possible.
The SVXII detector is \genunit{90}{cm} long and contains 12
wedges in $\phi$, each with 5 layers of silicon at radii from
\genunit{2.5}{cm} to \genunit{10.6}{cm}. One side of each layer
contains strips oriented in the axial direction,
and the other side contains \degrees{90} stereo strips in three cases,
and \degrees{1.2} small-angle stereo strips in two cases.
The Intermediate Silicon Layers (ISL) comprise three additional
layers of double-sided silicon at larger radii:
one at \genunit{22}{cm} for $|\eta|<1$,
and two at \genunit{20}{cm} and \genunit{28}{cm} for $1<|\eta|<2$.
Each layer of the ISL provides axial and small-angle stereo measurements.

The Central Outer Tracker (COT)~\cite{r_NIMCOT} measures particle
locations over a large radial distance, providing precise measurements
of track curvature. It is a large open-cell drift chamber
with 8 ``superlayers'' (4 axial and 4 with a \degrees{2} stereo angle),
each of which contains 12 wire layers, for a total of 96 layers.
There are 30,240 wires in total. The COT active volume is
\genunit{310}{cm} in length and covers \genunit{43}{cm} to
\genunit{132}{cm} in radius.
A magnetic field of $1.4~\mathrm{T}$ is provided by a superconducting
solenoid surrounding the silicon detectors and central drift chamber.

Particle energies are measured using sampling calorimeters. The
calorimeters are segmented into towers with projective geometry.
The segmentation of the CDF calorimeters is rather large, so that
often several particles contribute to the energy measured in one tower.

In the central region, i.e.\ $|\eta|<1.1$, the calorimeter
is divided into wedges subtending \degrees{15} in $\phi$. Each wedge has ten
towers, of roughly equal size in $\eta$, on each side of $\eta=0$.
The central electromagnetic calorimeter (CEM)~\cite{r_NIMCEM}
contains alternating
layers of lead and scintillator, making 18 radiation lengths of
material. The transverse energy resolution for high-energy electrons
and photons is $\frac{\sigma(\et)}{\et}=\frac{13.5\%}{\sqrt{\et[\gevnoarg]}}\oplus2\%$.
Embedded in the CEM is a shower maximum detector, the CES, which
provides good position measurements of electromagnetic showers
and is used in electron identification.
The CES consists of wire proportional chambers with wires and
cathode strips providing stereo position information.
The central hadronic calorimeter (CHA) and the end wall hadronic
calorimeter (WHA)~\cite{r_NIMCHAWHA} are of similar construction,
with alternating layers of steel and scintillator (4.7 interaction
lengths). The WHA fills a gap in the projective geometry between
the CHA and the plug calorimeter.

The calorimetry in the end plugs ($1<|\eta|<3.6$) is new for
\runii~\cite{r_NIMPCal}. The tower geometry is more
complicated in the plug, but the \degrees{15} wedge pattern is respected.
The plug electromagnetic
calorimeter (PEM) has lead absorber and scintillating tile read
out with wavelength shifting fibers. An electron traversing
the PEM sees 23.2 radiation lengths of material.
The energy resolution for high-energy electrons and photons is
$\frac{\sigma(E)}{E}=\frac{14.4\%}{\sqrt{E[\gevnoarg]}}\oplus0.7\%$.
There is
a shower maximum detector (PES), whose scintillating strips
measure the position of electron and photon showers. The plug hadronic
calorimeter (PHA) has alternating layers of iron and scintillating
tile, for a total of 6.8 interaction lengths.

Muon identification is performed by banks of single-wire drift
cells four layers deep. The central muon detector
(CMU)~\cite{r_NIMCMU} is located directly behind the hadronic
calorimeter in a limited portion of the central region ($|\eta|<0.6$).
The central muon upgrade (CMP) adds additional coverage in the central region and
reduces background with an additional \genunit{60}{cm} of steel
shielding, corresponding to 2.4 interaction lengths at \degrees{90}.
The central muon extension (CMX) covers the region $0.6<|\eta|<1.0$, and contains eight
layers of drift tubes, with the average muon passing through six.

A three-level trigger system is used to select interesting events
to be recorded to tape at $\sim\genunit{60}{Hz}$ from the bunch
crossing rate of $\genunit{1.7}{MHz}$.
This analysis uses data from triggers based on high-$\pt$ leptons,
which come from the leptonically decaying $W$ in the event.
The first two trigger levels perform
limited reconstruction using dedicated hardware, including
the eXtremely Fast Tracker (XFT), which reconstructs $r$-$\phi$
tracks from the COT with a momentum resolution of better than
$2\%/\pt[\gevcnoarg]$~\cite{Thomson:2002xp}.
The electron trigger requires a coincidence of an XFT track with
an electromagnetic cluster in the central calorimeter, while the
muon trigger requires that an XFT track point toward a set of hits
in the muon chambers.
The third level is a software trigger that performs full
event reconstruction. Electron and muon triggers at the third level
require fully reconstructed objects as described below, but with
looser cuts.

This analysis relies on the use of Monte Carlo (MC)
event generation and detector simulation.
Event generation is performed by HERWIG~\cite{r_herwig} for
\ttbar signal samples, and HERWIG, Pythia~\cite{r_pythia},
and ALPGEN~\cite{r_alpgen} for background and control samples.
A detailed description of the CDF detector is used in a simulation
that tracks the interactions of particles in each subdetector and
fills data banks whose format is the same as the raw
data~\cite{Gerchtein:2003ba}. The GEANT package~\cite{r_geant3}
provides a good description of most
interactions, and detailed models are developed and tuned to
describe other aspects (for example, the COT ionization and
drift properties) so that
high-level quantities like tracking efficiency and momentum
resolution from the data can be reproduced. The calorimeter
simulation is performed using a parameterized shower
simulation (GFLASH~\cite{Grindhammer:1989zg})
tuned to single particle energy response
and shower shapes from the data.

\subsection{Event Selection}
\label{ssec:selection}

The selection of \ttbar events in the lepton + jets channel begins by
requiring exactly one well identified lepton candidate in events recorded
by the high-$\pt$ lepton triggers. The lepton candidate
can be a central electron (CEM), or a muon observed in the CMU and
CMP detectors (CMUP) or a muon observed in the CMX detector (CMX).
The trigger efficiencies are high, $\sim96\%$ for electrons
and $\sim90\%$ for muons, and show negligible $\pt$ dependence
for leptons in the final sample.

Electrons are identified by a high-momentum track in the tracking
detectors matched with an energy cluster in the electromagnetic
calorimeter with $\et>\gev{20}$. Photon and hadronic fakes are
reduced by requiring the ratio of calorimeter energy to track
momentum to be no greater than 2 (unless $\pt>\gevc{50}$, in which
case this requirement is not imposed), and
by requiring the ratio of hadronic to electromagnetic energy in the
calorimeter towers to be less than $0.055+0.00045\cdot E_{EM}$.
Isolated electrons from $W$ decay are preferred over electrons from
$b$ or $c$ quark semi-leptonic decay by requiring
the additional calorimeter energy in a cone of
$\Delta R=\sqrt{\Delta\phi^2+\Delta\etaevt^2}=0.4$ around the cluster
to be less than 10\% of the cluster energy.
Electrons are rejected if they come from photon conversions to
$e^+e^-$ pairs that have been explicitly reconstructed.

Muons are identified by a high-momentum track in the tracking
detectors ($\pt>\gevc{20}$) matched with a set of hits in the
muon chambers. The calorimeter towers to which the track points
must contain energy consistent with a minimum ionizing particle.
An isolation cut is imposed, requiring the total calorimeter
energy in a cone of $\Delta R=0.4$ around the muon track (excluding
the towers through which the muon passed) to be less than
10\% of the track momentum. Cosmic ray muons are explicitly
identified and rejected. A complete description of electron and
muon selection, including all additional cuts used, can be found
elsewhere~\cite{r_run2wzxsec}.

A neutrino from the leptonic $W$ boson decay is inferred when the
observed momentum in the transverse plane does not balance.
The missing transverse energy, $\met$, is formed by projecting the
energy vectors for each tower in the central, wall, and
plug calorimeters into the plane transverse to the beams and
summing: $\met = -\left\|\sum_{i}\et^{i}\mathbf{n_{i}}\right\|$,
where $\mathbf{n_{i}}$ is the unit vector in the transverse plane
that points from the beamline to the $i$th calorimeter tower.
The $\met$ is corrected using the muon track momentum when
a muon is identified in the event. For clusters of towers that
have been identified as jets, the sum uses the jet energies
corrected for detector response relative to the fiducial central
region and for the effects of multiple \ppbar interactions.
We require the $\met$ to be at least $\gev{20}$.

Jets are identified by looking for
clusters of energy in the calorimeter using a cone algorithm,
{\tt JETCLU}, where the cone radius is $\Delta R=0.4$.
Towers with $\et>\gev{1}$ are used as seeds, then nearby towers
are added to the clusters, out to the maximum radius of 0.4.
A final step of splitting and merging is performed such that
in the end a tower does not contribute to more than one jet.
More details about the jet clustering are available
elsewhere~\cite{r_jetclu}.
Jet energies are corrected for relative detector response
and for multiple interactions, as described in
section~\ref{ssec:jetCorr}.

Jets can be identified as $b$ jets using a displaced
vertex tagging algorithm, which proceeds as follows.
The primary event vertex is identified using a fit to all
prompt tracks in the event and a beamline constraint.
Jets with $\et>\gev{15}$ are scanned for good-quality
tracks with both COT and silicon information. When a secondary
vertex can be reconstructed from at least two of those tracks,
the distance between the primary and secondary vertices in the
plane transverse to the beams ($L_{xy}$)
is calculated, along with its uncertainty ($\sigma(L_{xy})$).
If $L_{xy}/\sigma(L_{xy})>7.5$, the jet is considered tagged.
The per-jet efficiency for $b$ jets in the central region
is shown as a function of jet $\et$ in \fig{f:btageff};
the algorithm has an efficiency of about
60\% for tagging at least one $b$ jet in a \ttbar event.
More information concerning $b$ tagging is available
elsewhere~\cite{r_run2topxsec}.

\begin{cfigure}
\includegraphics{plots/btageff_prd}
\caption[Efficiency of $b$-tagging algorithm for central jets.]
{The efficiency of the secondary vertex $b$-tagging algorithm
is shown as a function of jet $\et$ for $b$ jets in the central
region of the detector ($|\eta|<1$), where the tracking
efficiency is high.
The shaded band gives the $\pm\sigunit{1}$ allowed region.
The efficiency is measured using a combination of data and
Monte Carlo simulated samples.}
\label{f:btageff}
\end{cfigure}

An additional $b$ tagging algorithm is used only in a cross-check
of this analysis, described in Section~\ref{ssec:otherfits}.
The Jet Probability (JPB) tagger~\cite{Abe:1994st,Buskulic:1993ka}
calculates the probability of observing the impact parameters of the
tracks in the jet with respect to the primary interaction
vertex, under the hypothesis that the jet does \emph{not} arise from
a heavy-flavor quark. In the check described later, a jet is
identified as a $b$ jet if it has a JPB value less than 5\%.
Since it uses much of the same information, the JPB tag
efficiency is correlated with the displaced vertex tag efficiency. 

We require at least four jets in the event with $|\eta|<2.0$
in order to reconstruct the \ttbar system.
The events are separated into four subsamples based on the
jet activity. These four categories of events are found to
have different background content and different shapes
in the reconstruction of the top quark mass for signal events.
By treating the subsamples separately, the statistical power
of the method is improved. Double-tagged (\twotag)
events have two $b$-tagged jets in the event.
These events have low background contamination, as well as
excellent mass resolution, since the number of allowed
jet-quark assignments is small. In this category,
we require three jets with $\et>\gev{15}$ and the fourth jet
with $\et>\gev{8}$. Tight single-tagged (\onetagt) events have exactly one $b$-tagged jet
in the event, and all four jets with $\et>\gev{15}$. Loose single-tagged (\onetagl)
events also have exactly one $b$ tag, but the fourth jet has
$\gev{8}<\et<\gev{15}$. These two categories have good mass
resolution, but \onetagl events have a higher background
content than \onetagt events. Finally, \zerotag events have no
$b$ tags. Here the background contamination is quite high, so
a tighter $\et$ cut is required: all four jets must have
$\et>\gev{21}$.

\note{Check the number of events. Against x-section for instance.}
We find 165 \ttbar candidates in \invpb{318} of data
selected for good quality in all relevant subdetectors.
The jet selection requirements for each of the four
event types are summarized in \tab{t:eventTypes}, which
also lists the expected signal to background ratio and
the number of each event type found in the data.
The expected S:B assumes a Standard Model top quark with
a mass of \gevcc{178} (the \runi world average)
and a \ttbar cross section of
\genunit{6.1}{pb}. Since in the \zerotag category we do
not have an independent background estimate, no estimate
of S:B is given; about 22 \ttbar events are expected.

\begin{table}
\caption[Selection requirements for the four event types.]
{The selection requirements for the four types of events are
given. The subsamples have different background content and
reconstructed mass shapes. The jet $\et$ requirements apply
to the leading four jets in the event, but additional jets
are permitted. Also shown for each category are the expected
signal to background ratio and the number of events observed
in \invpb{318} of data.}
\label{t:eventTypes}
\begin{ruledtabular}
\begin{tabular}{lrcccc}
\multicolumn{2}{c}{Category} & \twotag & \onetagt & \onetagl & \zerotag \\
\hline
Jet $\et$ & j1--j3 & $\et>15$ & $\et>15$ &
	$\et>15$ & $\et>21$ \\
cuts (\gevnoarg) & j4 & $\et>8$ & $\et>15$ &
 	$15>\et>8$ & $\et>21$ \\
\hline
\multicolumn{2}{c}{$b$-tagged jets} & 2 & 1 & 1 & 0 \\
\hline
\multicolumn{2}{c}{Expected S:B} & 10.6:1 & 3.7:1 & 1.1:1 & N/A \\
\hline
\multicolumn{2}{c}{Number of events} & 25 & 63 & 33 & 44 \\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Background Estimation}
\label{ssec:backgrounds}

Wherever possible, we obtain an independent estimate of the
background contamination in each subsample; adding this
information as a constraint in the likelihood fit improves
the result.

The amount and composition of the background contamination depends
strongly on the number of jets with $b$ tags. In the double
$b$-tagged sample, the background contribution is very small.
In the single $b$-tagged sample, the dominant backgrounds are $W$ +
multijet events and non-$W$ QCD events where the primary lepton
is not from a $W$ decay. In $W$ + multijet events, one of the jets
is either a heavy flavor jet or a light flavor jet mistagged
as a heavy flavor jet. In the events with no $b$ tag, $W$ + multijet
production dominates, and the jets are primarily light flavor since
there are no $b$ tags.

%\note{Can we eliminate this discussion and avoid talking about
%\chisq cut in this section? (Remove the line in the background
%table with post-\chisq-cut numbers.)}
%In the top quark mass reconstruction, detailed in
%section~\ref{ssec:chisq}, events are rejected if they
%are not compatible with the \ttbar hypothesis.
%\note{Awkward sentence:}
%The value, after minimization, of the \chisq expression giving
%consistency of the event with \ttbar decay (described in
%equation~\ref{eq_chi2}) is required to be less than 9.0.
%This cut helps to select events with reliable top quark mass
%values, discriminating against both background events and
%poorly recontructed signal events.
\Tab{t:backgroundnumber} gives estimates for the
background composition in each tagged subsample.
%, both before and after the cut on $\chisq<9.0$.
Note that some of the estimates in \tab{t:backgroundnumber} for
the various background processes are correlated, so the
uncertainty on the total background is not simply the sum in
quadrature of the component uncertainties.
The procedures for estimating each background type are
described in the following sections, and are detailed
elsewhere~\cite{r_run2topxsec}.

\begin{table}
\caption[Background composition by process.]
{The sources and expected numbers of background events in the three
subsamples with $b$ tags.
%The last line gives the number of
%events expected after the cut on $\chisq<9.0$, applied in the
%top quark mass reconstruction.
}
\label{t:backgroundnumber}
\begin{ruledtabular}
\begin{tabular}{lccc}
Source & \multicolumn{3}{c}{Expected Background} \\
\cline{2-4}
 & \twotag & \onetagt & \onetagl \\
\hline
Non-$W$ (QCD)       & \measErr{0.31}{0.08}  & \measErr{2.32}{0.50} & \measErr{2.04}{0.54}\\
$W\bbbar$+$W\ccbar$+$Wc$ 
										& \measErr{1.12}{0.43}  & \measErr{3.91}{1.23} & \measErr{6.81}{1.85}\\
$W$ + light jets    & \measErr{0.40}{0.08}  & \measErr{3.22}{0.41} & \measErr{4.14}{0.53}\\
$WW/WZ$             & \measErr{0.05}{0.01}  & \measErr{0.45}{0.10} & \measErr{0.71}{0.13}\\
Single top          & \measErr{0.008}{0.002}& \measErr{0.49}{0.09} & \measErr{0.60}{0.11}\\
\hline
Total               & \measErr{1.89}{0.52}  & \measErr{10.4}{1.72} & \measErr{14.3}{2.45}\\
%\hline
%\hline
%Source & \multicolumn{3}{c}{Expected Background, $\chisq<9.0$} \\
%\cline{2-4}
% & \twotag & \onetagt & \onetagl \\
%\hline
%$W$ + light jets    & \measErr{0.18}{0.04}  & \measErr{2.48}{0.32} & \measErr{3.19}{0.41}\\
%$W\bbbar$+$W\ccbar$+$Wc$           
%										& \measErr{0.37}{0.14}  & \measErr{2.70}{0.85} & \measErr{4.56}{1.24}\\
%$WW/WZ$             & \measErr{0.02}{0.004} & \measErr{0.31}{0.07} & \measErr{0.47}{0.09}\\
%non-$W$ (QCD)       & \measErr{0.14}{0.04}  & \measErr{1.79}{0.39} & \measErr{1.57}{0.42}\\
%Single top          & \measErr{0.003}{0.007}& \measErr{0.36}{0.07} & \measErr{0.40}{0.07}\\
%$\left<\epsilon_{\chisq<9.0}\right>$ & \measErr{0.38}{0.00} & \measErr{0.73}{0.00} & \measErr{0.71}{0.00}\\
%\hline
%\hline
%\multicolumn{4}{l}{In the top quark mass reconstruction, a cut of $\chisq<9.0$ is applied.}\\
%Total, $\chisq<9.0$ & \measErr{0.71}{0.18}  & \measErr{7.64}{1.24} & \measErr{10.2}{1.71}\\
\end{tabular}
\end{ruledtabular}
\end{table}

% Old division of background sample numbers
%$W$ + light jets    & \measErr{0.05}{0.02}& \measErr{1.1}{0.3}  & \measErr{0.8}{0.2}  \\
%$W\bbbar$           & \measErr{0.09}{0.03}& \measErr{0.6}{0.3}  & \measErr{0.4}{0.2}  \\
%$W\ccbar$           & \measErr{0.0}{0.0}& \measErr{0.3}{0.1}  & \measErr{0.1}{0.1}  \\
%$Wc$                & \measErr{0.0}{0.0}& \measErr{0.3}{0.1}  & \measErr{0.1}{0.1}  \\
%$WW/WZ$             & \measErr{0.0}{0.0}& \measErr{0.11}{0.02}& \measErr{0.08}{0.01}\\
%non-$W$ (QCD)       & \measErr{0.0}{0.0}& \measErr{1.2}{0.4}  & \measErr{1.2}{0.4}  \\
%Single top (S chan.)& \measErr{0.0}{0.0}& \measErr{0.11}{0.02}& \measErr{0.09}{0.01}\\
%Single top (T chan.)& \measErr{0.04}{0.02}& \measErr{0.08}{0.02}& \measErr{0.01}{0.01}\\

\subsubsection{Non-$W$ (QCD) background}

For the non-$W$ background (QCD multijet events), a data-driven
technique estimates the contribution to the signal sample.
The sideband regions (after subtracting the expected \ttbar
contribution) for the lepton isolation variable and the
$\met$ in the high-$\pt$ lepton sample are used to predict the
number of QCD multijet events in the signal region, assuming
no correlation between the isolation and $\met$.

\subsubsection{$W$ + multijet backgrouds}

Simulated samples of $W$ + multijet backgrounds are
obtained using the ALPGEN generator, which
produces multiple partons associated with a $W$ boson using
an exact leading order martrix element calculation. It is
interfaced with HERWIG for parton showering and hadronization.
This generator models well the kinematics of events with
high jet multiplicity, but suffers from a large theoretical
uncertainty in the normalization.
Thus, the normalization for these backgrounds is taken from
the data. The normalization for the $W$ + multijet background
in the subsamples requiring $b$ tags comes from the
$W$ + multijet events before tagging, after subtracting the
expected contributions for \ttbar and non-$W$ processes.
Since the \ttbar expectation scales with the top cross-section
\sigmatt, the estimate of the tagged background depends
on \sigmatt at second order.
For the same reason, in this scheme any estimate of the
$W$ + multijet background in \zerotag events would be
dependent on \sigmatt at first order; in order to minimize
the importance of the \sigmatt assumption, no background
constraint is used for the \zerotag sample.

The major contributions for the $W$ + heavy flavor backgrounds,
i.e.\ events with a $b$ tag on a real $b$ or $c$ jet,
come from the $W\bbbar$, $W\ccbar$, and $Wc$ processes.
The heavy flavor fraction for each process relative to the
inclusive $W$ + multijet process is estimated using the ALPGEN
Monte Carlo samples.
\note{Taka: Use ``ALPGEN showered by HERWIG''?
c.f.\ Sec.~\ref{sssec:bkgdparam}.}
The estimated number of each background
is then obtained by multiplying the heavy flavor fraction,
the tagging efficiency, and the pretagged number of $W$ + multijet
events in the data. Uncertainties on the estimate due to
$Q^2$ scale and next-to-leading order (NLO) contributions
are reduced by using this technique.

Another $W$ + multijet contribution comes
from events where a light flavor jet is misidentified
as a heavy flavor jet. Using generic jet data events,
a per-jet mistag rate is parameterized 
as a function of the number of tracks, $\et$, $\eta$, and
$\phi$ of the jet, and the scalar sum of $\et$ for all
jets with $\et>\gev{10}$ and $|\eta|<2.4$. 
The mistag rate is then applied to pretag data events
in the signal region to obtain the
$W$ + light flavor contribution.

\subsubsection{Other backgrounds}

There are other minor backgrounds: diboson production ($WW$, $WZ$, 
and $ZZ$) associated with jets, and single top production.
We use ALPGEN Monte Carlo samples to estimate their acceptance.
The NLO cross section values~\cite{Campbell:1999ah,Harris:2002md} 
are used for normalization.
