%
%  Updated to 318 invpb and added JES event selection -- 11 Mar 05 Pekka
%
\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
$\sim100\%$ 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, a lepton candidate will refer
exclusively to an electron or a muon candidate.
Thus, interesting events 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).

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 assuming 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 thing 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 layers of
silicon at larger radii: one at \genunit{22}{cm} for $|\eta|<1$,
and two at \genunit{20}{cm} and \genunit{28}{cm} for $1<|\eta|<2$.
Besides providing extra measurements between the drift chamber and SVXII
to ease track reconstruction in the central region, the ISL also
allows tracks to be reconstructed at high $\eta$ with larger level arm, and
therefore better momentum resolution, since drift chamber
coverage drops off in the region $1<|\eta|<2$.

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{3} stereo angle),
each of which contains 12 wire layers, for a total of 96 layers.
There are 30,240 wires in total. The COT occupies a volume
\genunit{310}{cm} in length and between radii of \genunit{43}{cm} and
\genunit{130}{cm}.
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 $\frac{\sigma(\et)}{\et}=\frac{13.5\%}{\sqrt{\et}}\oplus1.5\%$.
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 ($|\eta|>1$) is new for \runii.
In principle, the plug calorimeters~\cite{r_NIMPCal}
are similar to the central ones,
but they use more modern technology. 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
$\frac{\sigma(E)}{E}=\frac{14.4\%}{\sqrt{E}}\oplus0.7\%$.
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
$2\%/\pt[\gevcnoarg]$.
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 heavily 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 $\pt>\gevc{50}$), 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 a cut that requires
the total 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, for an energy scale calibration,
and for the effects of multiple 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,
energy scale calibration, and multiple interactions.
See section~\ref{sssec:jetinputs} for further description
of jet corrections.

Jets can be identified as $b$ jets using a
displaced vertex tagger, which proceeds as follows.
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 $L_{xy}/\sigma(L_{xy})>3$, the jet is considered tagged.
This algorithm has an efficiency of about
55\% for tagging at least one $b$ jet in a \ttbar event.
More information concerning $b$ tagging is available
in~\cite{r_run2topxsec}.

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 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-parton 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 much 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
$\et>\gev{21}$.

A final selection cut is based on the quality of the event
mass reconstruction, detailed in section~\ref{sec:massRecon}.
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, 9.0, 9.0,
and 9.0 for \twotag, \onetagt, \onetagl, and \zerotag events,
respectively. This cut
helps to select events with reliable mass values, discriminating
against both background events and poorly recontructed signal
events. The jet and \chisq selection requirements for each of the
four event types are summarized in \tab{t:eventTypes}.

\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 are the expected signal to background
ratios for each category.}
\label{t:eventTypes}
%\begin{center}
\begin{ruledtabular}
%\begin{tabular}{|c|r||c|c|c|c|}\hline
\begin{tabular}{lccccc}
%\hline
\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}{$b$ tagging} & 2 tags & 1 tag & 1 tag & 0 tags \\
\hline
\multicolumn{2}{c}{Expected S:B} & 44:1 & 8.0:1 & 2.1:1 & 0.6:1 \\
\hline
\multicolumn{2}{c}{\chisq cut} & 9.0 & 9.0 & 9.0 & 9.0 \\
\end{tabular}
\end{ruledtabular}
%\end{center}
\end{table}

We find 135 \ttbar candidates in \invpb{318} of
data selected for good quality in all relevant subdetectors.
Events with $b$ tags come from the \invpb{162} of data with
good-quality silicon detector information. In the remaining
\invpb{31}, $b$ tagging is not available, so all events passing
other selection cuts are treated as \zerotag events.
\Tab{t:eventcounts} gives the number of events in each
subsample.

\begin{table}
\caption[Number of events in each subsample.]
{The number of events found and the luminosity used is given
for each subsample.}
\label{t:eventcounts}
%\begin{center}
\begin{ruledtabular}
%\begin{tabular}{|c||c|c|c|c|}\hline
\begin{tabular}{ccccc}
%\hline
Category & \twotag & \onetagt & \onetagl & \zerotag \\
Luminosity & \invpb{162} & \invpb{162} & \invpb{162} & \invpb{193} \\
\hline
Number of events & 2 & 18 & 8 & 40 \\
%\hline
\end{tabular}
\end{ruledtabular}
%\end{center}
\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 negligible.
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.

The procedures for each background type are described below in more
detail, and the resulting estimates for the background composition
are shown in \tab{t:backgroundnumber}.
\note{Correlations?}

\subsubsection{$W$ + multijet backgrouds}

Simulated samples of $W$ + multijet backgrounds are
obtained using the ALPGEN~\cite{r_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 is good for high jet multiplicity events,
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 tagged
$W$ + multijet contribution comes from the $W$ + multijet
events before tagging. But for \zerotag events, there is no
known independent normalization; for this reason, no background
constraint is used for the \zerotag sample.
\note{Could we use some theoretical W+jets x-sec with huge
uncertainties and at least get a loose constraint?}

%heavy flavor
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. 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 after non-$W$ background is subtracted.
Uncertainties on the estimate due to $Q^2$ scale and
NLO contributions are reduced by using this technique.

%light flavor
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{Non-$W$ (QCD) background}

For the non-$W$ background (QCD multijet events), a data-driven
technique estimates both the pre-tagged and tagged contributions
to the signal sample, because the pre-tagged sample
in the $W$ + multijet background estimation
needs to be corrected for the non-$W$ contribution.
The sideband regions 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 $\met$.
%This background is estimated separately for the electron and muon.

\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 contributions.
The NLO cross section values~\cite{Campbell:1999ah,Harris:2002md} 
are used for normalization.

\begin{table}
\caption[Background composition by process.]
{The sources and numbers of background events in the three
subsamples for which background estimates exist.}
\label{t:backgroundnumber}
%\begin{center}
\begin{ruledtabular}
%\begin{tabular}{|c||c|c|c|}
\begin{tabular}{cccc}
%\hline
Source & \multicolumn{3}{c}{Expected Background} \\
\cline{2-4}
 & \twotag & \onetagt & \onetagl \\
\hline%\hline
$W$ + 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}\\
\hline%\hline
Total               & \measErr{0.18}{0.04}& \measErr{3.9}{0.7}  & \measErr{2.7}{0.6}  \\
%\hline
\end{tabular}
\end{ruledtabular}
%\end{center}
\end{table}

