\section{Results on the Data}
\label{sec:results}

We fit the events in the data using the procedure described in
Section~\ref{sec:fitting}. After detailing the results, we present
several additional results performed as crosschecks on the
primary measurement.

\subsection{Subsample Likelihood Curves}
\label{ssec:resultsSub}

The likelihood fit is first performed on each subsample separately.
\note{Fix this! Discuss \jes constraint?}
For a series of top quark masses and \jes values, the mass and \jes
parameters are fixed, while the likelihood is maximized with respect to
the remaining parameters ($n_s$ and $n_b$) using MINUIT. The resulting
likelihood contours in the \mtop-\jes plane are shown in \fig{f:lnlsubs}.

\begin{cfigure}
\includegraphics[width=.47\columnwidth]{plots/likelihoodcontour2d_4_prd2}
\includegraphics[width=.47\columnwidth]{plots/likelihoodcontour2d_3_prd2}
\includegraphics[width=.47\columnwidth]{plots/likelihoodcontour2d_2_prd2}
\includegraphics[width=.47\columnwidth]{plots/likelihoodcontour2d_1_prd2}
\caption[Subsample fits likelihood curves.]
{The contours of the likelihood in the \mtop-\jes plane for the
independent fit to each subsample in the data.
At each point in the plane, the likelihood is maximized with
respect to the other free parameters.
A crosshair shows the maximum likelihood point from the combined
fit, and contours are given at intervals of \sigunit{1},
where the \sigunit{k} curve is defined by $\Delta\ln L=0.5 k^{2}$.
Upper left: \twotag events; upper right: \onetagt events;
lower left: \onetagl events; lower right: \zerotag events.}
\label{f:lnlsubs}
\end{cfigure}

\subsection{Results of Combined Likelihood}
\label{ssec:resultsCombined}

Finally, the likelihood is maximized with respect to all parameters
using all four subsamples. The result, after scaling the
$\Delta(\ln L)=+1/2$ errors as described in section~\ref{ssec:methodCheck},
is a top quark mass of \gevcc{\measAStatJES{173.5}{3.7}{3.6}}.
The simultaneous measurement of the jet energy scale is
\sigunit{\measAErr{-0.10}{0.78}{0.80}}. The combined likelihood
as a function of the true top quark mass is shown in \fig{f:lnlcomb}.
For each value of the top quark mass and \jes,
the likelihood is maximized with respect to all other parameters.
This likelihood is not the simple product of the four likelihoods
shown in \fig{f:lnlsubs} because the \jes constraint term \ljes is
included in each of the subsample fits, but of course only once in
the combined fit.

\begin{cfigure}
\includegraphics{plots/likelihoodcontour2d_-1_prd2}
\caption[Combined fit likelihood contours.]
{The contours of the likelihood in the \mtop-\jes plane for the
combined fit to all four subsamples. At each point in the plane, the
likelihood is maximized with respect to the other free parameters.
The crosshair shows the best fit point, and contours are given at
intervals of \sigunit{1},
where the \sigunit{k} curve is defined by $\Delta\ln L=0.5 k^{2}$.
\note{Clarify about scaling of errors relative to this plot?}
}
\label{f:lnlcomb}
\end{cfigure}

The uncertainty on \mtop from the likelihood fit is a combination
of the statistical uncertainty in extracting a measurement of
\mtop and the systematic uncertainty due to allowed variations
of \jes. It is possible to get an idea of the size of each
contribution. Fixing \jes to its fitted value of \sigunit{-0.10} and
fitting for \mtop alone yields a top quark mass measurement of
\gevcc{\measAStatJES{173.5}{2.7}{2.6}}, corresponding to the
``pure statistical'' uncertainty. Subtracting this uncertainty
in quadrature from the full uncertainty gives an \mtop uncertainty
due to the jet energy scale of \gevcc{\pm2.5}.

The input constraints and fit results for the combined fit are
given in \tab{t:fitResults}. \Fig{f:fitConsistencyMreco} shows the
consistency of the reconstructed top quark mass distribution in each
subsample with the combined fit results, while \fig{f:fitConsistencyMjj}
shows the same for the reconstructed dijet mass.

\begin{table}
\caption[Combined fit results.]
{The input constraints and fitted values
are given for all free parameters in the
combined likelihood fit.}
\label{t:fitResults}
\begin{ruledtabular}
\begin{tabular}{llcccc}
\multicolumn{2}{l}{Category} & \twotag & \onetagt & \onetagl & \zerotag \\
\hline
\mtop & constr & \multicolumn{4}{c}{None} \\
      & fit & \multicolumn{4}{c}{\mathversion{bold}\gevcc{\measAStatJESBold{173.5}{3.7}{3.6}}}\\
      &     & \multicolumn{4}{c}{(\gevcc{\measAStatSepJES{173.5}{2.7}{2.6}{2.5}})} \\
\hline
\jes  & constr & \multicolumn{4}{c}{\sigunit{\measErr{0.0}{1.0}}} \\
      & fit & \multicolumn{4}{c}{\sigunit{\measAErr{-0.10}{0.78}{0.80}}} \\
$n_s$ & constr & \multicolumn{4}{c}{None} \\
      & fit & \measErr{23.5}{5.0} & \measErr{53.9}{7.9} &
              \measErr{14.3}{5.2} & \measErr{28.3}{8.3} \\
\hline
$n_b$ & constr & \measErr{1.89}{0.52} & \measErr{10.4}{1.72} &
                     \measErr{14.3}{2.45} & None \\
      & fit & \measErr{1.8}{0.5} & \measErr{10.1}{1.7} &
              \measErr{15.5}{2.2} & \measAErr{15.7}{8.0}{7.1} \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{cfigure}
\includegraphics{plots/datafit_mt_prd}
\caption[Subsample fits consistency checks: \mreco.]
{The reconstructed top quark mass distribution for each subsample is
shown overlaid with the expected distribution using the
top quark mass, jet energy scale, signal normalization,
and background normalization from the combined fit.}
\label{f:fitConsistencyMreco}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/datafit_mjj_prd}
\caption[Subsample fits consistency checks: \mjj.]
{The reconstructed dijet mass distribution for each subsample is
shown overlaid with the expected distribution using the
top quark mass, jet energy scale, signal normalization,
and background normalization from the combined fit.}
\label{f:fitConsistencyMjj}
\end{cfigure}

A set of pseudoexperiments is generated with a true top quark mass
of \gevcc{172.5} (close to the central value from the fit),
the nominal jet energy scale, and with the number
of events in each subsample equal to the number observed in our
data (\tab{t:eventTypes}). In \fig{f:expStatErr}, the positive and
negative uncertainties from the likelihood fits are plotted. Arrows
indicate the uncertainties from the fit to the data. Although smaller
than the median uncertainties from the pseudoexperiments, the
uncertainties on the data are reasonable---9.2\% of the pseudoexperiments
have smaller uncertainties than those returned by the fit to the data.
The distributions do not
change significantly if a true top quark mass value of \gevcc{165} or
\gevcc{180} is used. The better-than-expected uncertainties are
consistent with the sharpness of the reconstructed top quark mass
peaks in the \twotag and \onetagt subsamples, as shown in
\fig{f:fitConsistencyMreco}.

\begin{cfigure}
\includegraphics{plots/mtop_expect_uncert_prd}
\caption[Likelihood uncertainty distributions for data-like fit.]
{The distributions of positive and negative uncertainties from the
likelihood fit are shown, for pseudoexperiments generated with a
true top quark mass of \gevcc{172.5}, the nominal jet energy scale, and
the number of events in each subsample as observed in the data.
Arrows indicate the positive and negative uncertainties from the
likelihood fit to the data; 9.2\% of the pseudoexperiments have
smaller uncertainties.}
\label{f:expStatErr}
\end{cfigure}

\subsection{Alternate Fits}
\label{ssec:otherfits}

In addition to the primary result described above, a number of additional
fits are performed as crosschecks and to investigate the effect of
certain assumptions on our measurement. The differences between the
primary fit and the alternate fits are briefly described below,
along with the resulting top quark mass measurements. \Tab{t:alternateFits}
summarizes the results.

\subsubsection{Fit without \jes constraint}
\label{sssec:altFitNoJES}

In the primary fit, the \jes measurement is treated as an update
to the extrinsic calibration by including the \jes Gaussian constraint
\ljes in the likelihood. Here the \ljes
term is removed, so that \emph{all} the jet energy scale information
comes from the \emph{in situ} calibration to the resonance of the
hadronically decaying $W$ boson. The resulting top quark mass
measurement is \gevcc{\measStatJES{174.0}{4.5}}, and the simultaneous
fit for \jes gives \sigunit{\measStat{-0.25}{1.22}}. Although the
systematic uncertainties are not explicitly evaluated for this
approach, they are not expected to be significantly different from
those of the primary analysis.

\subsubsection{Traditional \mtop-only fit}
\label{sssec:altFit1D}

For this alternate result, the traditional fit for a single
variable, \mtop, is performed using a single reconstructed
quantity, \mreco.
This fit is virtually identical to the analysis performed in
\runi~\cite{Affolder:2000vy}. 
The event selection and \mreco reconstruction are exactly as
described earlier. With only one reconstructed quantity and one
measured quantity, the template parameterizations are simpler.
Signal and background p.d.f.'s for \mreco are fitted without any \jes
dependence, but otherwise identical in form to those described
above. The form of the likelihood used is also much simpler,
since the term \lshapemjj is absent and \lnev is greatly
simplified with only the sample of events after the \chisq cut
used. For each subsample, the likelihood is given by
\begin{eqnarray}
{\mathcal L} &=& \lshape \times \lbg; \nonumber \\
\lshape      &=& \frac{e^{-(n_{s}^{t}+n_{b}^{t})}(n_{s}^{t}+n_{b}^{t})^{r^{t}}}{r^{t}!} \times \nonumber \\
             & & \prod_{i=1}^{r^{t}} \frac{n_{s}^{t}P_{sig}(m_{i};\mtop) +
                 n_{b}^{t}P_{bg}(m_{i})}{n_{s}^{t}+n_{b}^{t}}; \nonumber \\
\lbg         &=& \exp\left(-\frac{(n_{b}^{t}-n_{b}^{t}(\text{const}))^2}{2\sigma_{n_{b}^{t}}^{2}}\right);
\end{eqnarray}
where $n_{s}^{t}$ and $n_{b}^{t}$ refer to the expected number of signal
and background events in the sample after the \chisq cut.
The combined likelihood is simply the product of the subsample likelihoods.

The fitted value of the top quark mass using this method is
\gevcc{\measAStat{173.2}{2.9}{2.8}}, with a central value very
close to the result from the primary measurement.
For this result, of course, since the jet energy scale systematic
uncertainty is not accounted for in the likelihood fit, its effect
on the top quark mass uncertainty must
be estimated separately and added in quadrature.
The negative log-likelihood curves for the \mtop-only fit in each
subsample are shown in \fig{f:lnlsubs1d}. As can be seen in the
lower-right pane, the \zerotag subsample contributes very little
to the overall measurement. This is because, absent a background
constraint, the fit prefers a small signal contribution, which
results in very little sensitivity to \mtop.

\begin{cfigure}
\includegraphics[width=.47\columnwidth]{plots/datafit3_con_lik}
\includegraphics[width=.47\columnwidth]{plots/datafit2_con_lik}
\includegraphics[width=.47\columnwidth]{plots/datafit1_con_lik}
\includegraphics[width=.47\columnwidth]{plots/datafit0_con_lik}
\caption[Subsample fits likelihood curves, \mtop-only fit.]
{The negative log-likelihood curves as a function of the true
top quark mass are shown for
the \mtop-only fit to each subsample in the data. Upper left:
\twotag events; upper right: \onetagt events; lower left: \onetagl events;
lower right: \zerotag events.}
\label{f:lnlsubs1d}
\end{cfigure}

\subsubsection{Traditional \mtop-only fit with additional tag category}
\label{sssec:altFitJetProb}

Events with two $b$ tags carry the most information about the top quark
mass because of their high purity and narrow reconstructed mass templates.
In this alternate analysis, we increase the number of events with two
$b$ tags by allowing one of the tags to come from the Jet Probability
tagger.
We establish a new category of events with exactly one secondary vertex
tag and an additional JPB tag; the former requirement ensures
that these events do not overlap with the \twotag subsample.
The events in this category are then removed from the \onetagt and
\onetagl samples so that all the subsamples remain disjoint.
Eighteen events are found in this category in the data sample:
4 out of 18 events were newly categorized from \onetagl
and 14 events from \onetagt in the default configuration.

The expected backgrounds in the new event category are estimated to be
$0.52 \pm 0.26$ events from $W\bbbar$, $Wc$, and $W\ccbar$ processes,
$0.15 \pm 0.08$ events from non-$W$ background,
$0.38 \pm 0.19$ events from mistagged \wjets events,
$0.08 \pm 0.04$ events from single top, and
$0.05 \pm 0.03$ events from the diboson processes $WW$ and $WZ$.
The total number of background events is thus estimated to be 
$1.2 \pm 0.6$ for the new subsample. The background estimates for the
exclusive one-tag subsamples change to account for the reduced acceptances.

The likelihood used to extract the top quark mass from this data is 
that described above in Section~\ref{sssec:altFit1D}, i.e.\ using
an \mtop-only fit. \Fig{f:jetProbResult} shows the reconstructed
top quark mass distribution for the events with one secondary vertex
tag and one JPB tag, along with the expected distribution using
parameters taken from the fit to only this set of events.
The inset shows the negative log-likelihood curve
for this subsample alone.
Using only the 18 events in this subsample, 
the measured top quark mass is \gevcc{\measAStat{173.3}{6.1}{6.5}}.

\begin{cfigure}
\includegraphics{plots/cfinal_5}
\caption[Secondary vertex tag + JPB tag sample consistency check.]
{The reconstructed top quark mass distribution for the 18 events
with one secondary vertex tag and one JPB tag, overlaid with
the expected distribution from the fit to this subsample.
The inset shows the shape of $- \Delta$ log ${\cal L}$ for the
fit to these events as a function of the true top quark mass.}
\label{f:jetProbResult}
\end{cfigure}

This result using this data sample can be combined with the other four categories
of events. A sensitivity study shows that the combined likelihood including the
new class of events improves the expected statistical uncertainty
by 2.6\%. In addition to the statistical improvement, increasing the number
of double-tagged events improves the jet energy systematic uncertainty.
The resulting combined top quark mass measurement on the
five subsamples is \gevcc{\measAStat{173.0}{2.9}{2.8}}.

\begin{table}
\caption[Alternate fit results.]
{The results of alternate fits are summarized. For the cases that
do not include the jet energy scale systematic effect in the likelihood
fit result, the independently determined systematic is given for
comparison (see Section~\ref{ssec:jetSystOnMtop} for more details).}
\label{t:alternateFits}
\begin{ruledtabular}
\begin{tabular}{lcc}
Method & \mtop fit result & \jes fit result \\
       & [\gevccnoarg]    & [$\sigma$] \\
\hline
Default & \measAStatJES{173.5}{3.7}{3.6} & \measAErr{-0.10}{0.78}{0.80} \\
No \jes constr & \measStatJES{174.0}{4.5} & \measErr{-0.25}{1.22} \\
\mtop-only & $\measAStat{173.2}{2.9}{2.8}\oplus3.1~(\jes)$ & N/A \\
\hspace{15pt}+ JPB & $\measAStat{173.0}{2.9}{2.8}\oplus3.0~(\jes)$ & N/A \\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Crosschecks on the results}
\label{ssec:xchecks}

The measurement of the top quark mass is checked by performing the
analysis in various subsamples and with different jet corrections and
background normalization constraints in order to ensure the robustness of
the result. In these crosschecks, we use the Traditional \mtop-only fit
described in Section~\ref{sssec:altFit1D} for simplicity.

\Fig{f:xcheck1} shows the resulting top quark mass measurement for various
modifications to the method. Any inconsistencies would most likely indicate
problems with the detector or the analysis method.
%, though physics beyond the
%Standard Model could appear here as well.
First the measurement using top-specific corrections (the default) and
using generic out-of-cone jet corrections is compared.
Next fits using the background constraints (the default) and
without the constraints are shown. For the remaining comparisons, the
dataset is divided into two subsamples. First results are shown from two
different run periods, then events with a primary electron vs those with a
primary muon, and finally positive-charge primary leptons vs negative-charge
primary leptons.
Except in the case of the generic jet corrections, the default reconstructed
mass templates are used. All the results are consistent with each other and
with the primary measurement.

\begin{cfigure}
\includegraphics{plots/prd_xcheck1}
\caption[Crosscheck measurements using different jet corrections, background constraints,
and lepton samples.]
{Top quark mass measurements using the \mtop-only fit are compared
for different assumptions (top-specific corrections (default) vs generic
jet out-of-cone corrections, constrained backgrounds vs unconstrained)
and different ways of subdividing the sample (two different run periods,
electrons vs muons, positive-charge leptons vs negative-charge leptons).
All results are consistent.}
\label{f:xcheck1}
\end{cfigure}

A series of top quark mass measurements using different subsamples of the
data is shown in \fig{f:xcheck2}. The primary effects of the increasingly tight
selection are, first, to increase the sample purity and thereby decrease
sensitivity to modeling of the background processes; and second, to select
events with low extra jet activity, decreasing sensitivity to modeling
of ISR and FSR. The list of samples used is as follows from top to bottom: 
all four subsamples (default);
\twotag, \onetagt, and \onetagl (removing \zerotag events); 
\twotag and \onetagt only;
\twotag and \onetagt only with any additional jets required to have $\et<\gev{15}$;
\twotag and \onetagt only with any additional jets required to have $\et<\gev{8}$.
For the last two cases, top mass templates are prepared with the additional
requirements. Again we find that all results are consistent, indicating that
background kinematics and extra jet activity are reasonably well modeled.

\begin{cfigure}
\includegraphics{plots/prd_xcheck2}
\caption[Crosscheck measurements using different subsamples and jet requirements.]
{Top quark mass measurements using the \mtop-only fit are compared
for various samples.
From top to bottom:
all four subsamples;
\twotag, \onetagt, and \onetagl subsamples;
\twotag and \onetagt subsamples only;
\twotag and \onetagt subsamples only, additional jets $\et<\gev(15)$;
\twotag and \onetagt subsamples only, additional jets $\et<\gev{8}$. 
All results are found to be consistent.}
\label{f:xcheck2}
\end{cfigure}

\subsection{Kinematic distributions}
\label{ssec:kinematic}

We compare various kinematic distributions for the \ttbar signal
candidate events with the Monte Carlo predictions for combined
signal and backgrounds. Comparisons of kinematic distributions
tell us how well the Monte Carlo models the data, which is very
important in this analysis.
This information could additionally be used to test 
whether the kinematic properties of the top quark we observe 
are consistent with Standard Model predictions.
For these distributions, we use only \twotag and \onetagt samples,
(73 events) in order to increase the signal purity.
All kinematic quantities are defined using the output of the
\chisq fitter, so that both jet-parton assignments and
the \pt of each object are taken at the minimum \chisq point.

\Fig{f:pt_top} and \fig{f:eta_top} show the \pt and rapidity
distributions of the reconstructed top quarks, respectively. 
The data distributions are in agreement with predictions using
HERWIG \ttbar signal events with top quark mass of \gevcc{172.5}
and simulated background events. 

\begin{cfigure}
\includegraphics{plots/Pt_top_ttopdh_comp-all_318}
\caption[Top \pt kinematic distribution.]
{The \pt distribution of the reconstructed top quarks
\kinblah}
\label{f:pt_top}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/Eta_top_ttopdh_comp-all_318}
\caption[Top $\eta$ kinematic distribution.]
{The rapidity distribution of the reconstructed top quarks
\kinblah}
\label{f:eta_top}
\end{cfigure}

We also find good agreement in the \pt distribution of the $b$ jets 
from top decays, shown in \fig{f:pt_bjet}. Good modeling of the
$b$-jet spectrum by the Monte Carlo simulation is one of the most
important things for a good determination of the top quark mass.
\Fig{f:pt_w} shows the \pt distribution of the reconstructed $W$ bosons.

\begin{cfigure}
\includegraphics{plots/Bjet_ttopdh_comp-all_318}
\caption[Kinematic distribution of $b$-jet \pt.]
{The \pt distribution of the reconstructed $b$ jets
\kinblah}
\label{f:pt_bjet}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/Pt_w_ttopdh_comp-all_318}
\caption[Kinematic distribution of $W$ \pt.]
{The \pt distribution of the reconstructed $W$ bosons 
\kinblah}
\label{f:pt_w}
\end{cfigure}

The \pt distribution of the \ttbar system is shown in
\fig{f:pt_ttbar}, which has good agreement between the data
and the prediction from simulated events.
This distribution is sensitive to the modeling of initial state radiation.
The distribution of the number of jets from data events
is also compared with the prediction from the Monte Carlos, as shown
in \fig{f:njets_ttbar}. To be counted in this plot, each jet is
required to have $\et>\gev{8}$ and $|\eta|<2.0$; note that this
distribution is sculpted by the selection requirements for the \twotag
and \onetagt subsamples. The data and prediction are in good agreement,
indicating that the number of extra jets (from hard initial and final
state radiation) is reasonably well modeled by HERWIG.

\begin{cfigure}
\includegraphics{plots/Pt_ttbar_ttopdh_comp-all_318}
\caption[Kinematic distribution of \ttbar \pt.]
{The \pt distribution of the reconstructed \ttbar system
\kinblah}
\label{f:pt_ttbar}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/Njets_ttopdh_comp-all_318}
\caption[Kinematic distribution of the number of jets.]
{The number of jets distribution 
\kinblah
Jets are required to have $\et>\gev{8}$ and $|\eta|<2.0$.
\note{Note sample selection requirements.}
}
\label{f:njets_ttbar}
\end{cfigure}
