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%  Summarize Template method
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%  Revisions:
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%   13 Jul 05    PKS    Original version
%   13 Jul 05    EMB    Complete and add plots
%   16 Jul 05    PKS    Tidy up and incorporate 1st draft comments
%   20 Jul 05    PKS    Incorporate 2nd draft comments.
%   21 Jul 05    PKS    Further tighten length.
%   30 Aug 05    PKS    Implement chnage for Draft 2.
%

The template method is described in detail in~\cite{r_Run2TMT}.
We perform a \chisq minimization to fit the parton momenta from the \ttbar
daughters and determine \mreco for each event,
assuming that the final state arises from the decay of a 
\ttbar pair into $W$ bosons and $b$ quarks.
We use only the four leading jets in the mass reconstruction.
In the \chisq fit, both sets of $W$ decay daughters are constrained 
to have the invariant mass of the $W$ boson, and both $Wb$ states are
constrained to have the same mass.
The ambiguity arising from the different ways of assigning the 
jets to the four quarks
is resolved by selecting the assignment with the lowest \chisq,
taking into account the $b$-tagging information.
We construct a histogram of \mreco for each subsample, discarding
events with \chisq$>9$, corresponding to poorly reconstructed
or background events.

The parameter \jes\ is determined within this event sample by
removing the $W$ boson mass constraints, and identifying for each
event all pairs of jets that would be consistent
with the $W$ boson final state.
We form histograms of the invariant masses of these jet pairs for
each of the four event subsamples and compare these with what we expect 
given the precisely known $W$ boson mass~\cite{r_PDBook}.

We use these eight histograms to measure simultaneously \mtop\ and \jes. 
An unbinned likelihood fit is performed to
parameterized signal templates taken from simulated \ttbar events generated using
different values of \mtop and \jes, and background templates derived
from studies of the relevant background processes. 
We include in the fit a Gaussian constraint ($\jes=\sigcunit{\measErr{0}{1}}$)
from the extrinsic jet energy calibrations,
and we constrain the background rates in the 2-tag, 1-tag Tight,
and 1-tag Loose samples to the estimated background rates within
their uncertainties.
The background level in the 0-tag sample is determined in the fit using
the differences in predicted signal and background mass distributions to be
$\measAStat{15.7}{8.0}{7.1}$\ events.

The four reconstructed top quark mass distributions and the results of
the fit are shown in \fig{f:fitConsistencyMreco},
where we also show the background contributions. 
In all cases, we see agreement between the observed data
distributions and the fitted curves. We obtain 
$\mtop = \gevcc{\measAStat{173.5}{3.7}{3.6}}$,
where the uncertainty is statistical and incorporates the
uncertainty due to \jes, which we estimate contributes
$\sim\gevcc{2.5}$. \Fig{f:lnlcomb} shows the
likelihood in the \mtop-\jes plane for the combined measurement.
If we do not constrain \jes to the nominal value of zero, we obtain 
$\jes = \sigcunit{\measErr{-0.25}{1.22}}$, which indicates 
our nominal jet energy calibrations are in good agreement with 
information provided by the $W$ boson mass peak in the \ttbar decay.
This also demonstrates that the \jes\ constraint from the $W$\ boson decay
has comparable precision to the jet energy calibration.

\begin{cfigure}
\includegraphics{plots/datafit_mt_prd}
\caption[Subsample fits consistency checks: \mreco.]
{The \mreco distribution of the template method is shown
for each subsample overlaid with the expected distribution using the
top quark mass, jet energy scale, and signal and background
normalizations 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}
%
%\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.
%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}

%\begin{cfigure}[floatfix]
\begin{cfigure}
\includegraphics{plots/likelihoodcontour2d_1_prd_v2}
\caption[Combined fit likelihood contours.]
{Contours of the template method likelihood are shown in the
\mtop-\jes plane for the combined fit to all four subsamples.
The crosshair shows the best fit point. Contours are
given at intervals of $\Delta\ln L$, the change in
log-likelihood from its maximum.
\note{Clarify about scaling of errors relative to this plot?}
}
\label{f:lnlcomb}
\end{cfigure}
