%
%  Modifications for the JES fits.  16 March 2005   Pekka
%
\section{Top Mass Fitting}
\label{sec:fitting}

The two-dimensional distribution of reconstructed top quark mass and
dijet invariant mass for a particular Monte Carlo sample is referred to as
a template. The distribution of the reconstructed mass in data events will
be compared to Monte Carlo signal and background templates to extract the
measured top quark mass and the jet energy scale.   First, probability
density functions (p.d.f.'s) of the reconstructed masses are determined
from the templates for signal and background events; the signal p.d.f.'s
depend on the true top quark mass and jet energy scale.   We perform an
unbinned likelihood fit  to extract the best value of the top
quark mass and jet energy scale to describe the data. At the end of this
section, we describe a number of checks of the method using simulated
events.

\subsection{Parameterization of Signal and Background Shapes}
\label{ssec:parameterization}

Since templates are available only at discrete values of
true top quark mass and jet energy scale, the signal reconstructed mass
distributions are parameterized by a flexible functional form as a
function of true top quark mass and jet energy scale in order to smooth
the distributions and interpolate between the mass points. For background
events, there is no dependence on top quark mass or jet energy scale; a
single p.d.f.\ is used to describe the background reconstructed mass
shape.

In principle, the jet energy scale may change the shape of the background
templates.  However, we have determined from studies of the background
that the shape of the background template is insensitive to the jet
energy scale.  Rather, the overall rate of background events does show
some sensitivity to the jet energy response, and this uncertainty is
incorporated, where appropriate, into the uncertainty in the
rate of background events in the sample.

\subsubsection{Signal shape parameterization}
\label{sssec:signalparam}

Signal mass templates are produced using sets of Monte Carlo samples with
the input top quark mass at 2.5--5 $\gevccnoarg$ intervals from
$\gevcc{130}$ to $\gevcc{230}$\ and the jet energy scale varying from
-3.0 to +3.0 in steps of 0.5. An example of the template shape from each
event category is given in
\fig{f:sampletemp}.
\Tab{t:sampletemps} shows the evolution of the template mean, most
probable value, and RMS as a function of top quark mass using selected
generated mass samples and jet energy scales.

\begin{table}
\caption[Signal template evolution.]
{The evolution of the template parameters is demonstrated using selected
signal Monte Carlo samples with generated top quark mass of $\gevcc{145}$,
$\gevcc{165}$, $\gevcc{185}$, and $\gevcc{205}$, with jet energy scales
of -1.0, 0.0 and +1.0. The mean, most probable value (MPV), and RMS of the
templates are given for each subsample in each generated mass.}
\label{t:sampletemps}
\begin{ruledtabular}
\begin{tabular}{ccdddd}
 & $\mtop^{\textrm{gen}}$ &
 \multicolumn{1}{c}{\twotag} &
 \multicolumn{1}{c}{\onetagt} &
 \multicolumn{1}{c}{\onetagl} &
 \multicolumn{1}{c}{\zerotag} \\
% & $[\gevccnoarg]$ \\
\hline
Template        & 145 & 150.5 & 154.7 & 147.3 & 163.1 \\
Mean            & 165 & 165.6 & 167.3 & 160.7 & 172.9 \\
$[\gevccnoarg]$ & 185 & 178.5 & 180.5 & 172.2 & 183.9 \\
                & 205 & 193.2 & 192.6 & 185.1 & 194.6 \\
\hline
Template        & 145 & 144.9 & 144.0 & 141.6 & 147.7 \\
MPV             & 165 & 162.0 & 160.0 & 156.6 & 159.6 \\
$[\gevccnoarg]$ & 185 & 178.9 & 178.6 & 176.3 & 179.0 \\
                & 205 & 199.8 & 198.0 & 191.0 & 198.4 \\
\hline
Template        & 145 & 22.8 & 30.4 & 25.9 & 34.5 \\
RMS             & 165 & 25.4 & 30.6 & 28.4 & 36.7 \\
$[\gevccnoarg]$ & 185 & 26.7 & 33.5 & 29.3 & 35.2 \\
                & 205 & 28.3 & 33.1 & 33.2 & 37.0 \\
\end{tabular}
\end{ruledtabular}
\end{table}

%In \fig{f_templatemeans} are shown the means,
%medians, and most probable values of all 29 mass templates as a
%function of the generated top quark mass, as well as their slopes from a
%linear fit.
%\begin{cfigure}
%\includegraphics{plots/04042616-TemplateSlope}
%\caption[Mean, median, and most probable values of signal mass templates
%as a function of the generated top quark mass.]
%{Mean, median, and most probable values of mass templates as a function
%of the generated top quark mass between $\gevcc{130}$ and $\gevcc{230}$.
%The lines are linear fits to the points; slopes are given in the legend.}
%\label{f_templatemeans}
%\end{cfigure}

We derive from these distributions parametrized templates that
are a smoothly varying function of true top quark mass and jet energy
scale.  An arbitrary parameterization is used for the probability to
observe a particular reconstructed top quark mass given the true top quark
mass and jet energy scale.  For a given true top quark mass, the
parameterization consists of two Gaussians---intended to account for the
well measured events---plus the integrand of the Gamma
function---intended to account for events where the incorrect combination
is chosen, or where the four leading jets do not match the four partons
from the \ttbar decay.
%The second Gaussian is not well motivated a priori, but
%was added to the parameterization in order to
%fit better the large statistics Monte Carlo samples.
The 9 parameters necessary to specify this combination of functions
are themselves assumed to depend linearly on the true top quark mass and
jet energy scale, so that the full set of templates is specified by 27
parameters. This assumed functional form is certainly arbitrary as well,
but works well in the limited range of top quark masses and jet
energy scales considered; assuming quadratic dependence does not improve
the fit. Thus the parameterization is as follows:
\begin{eqnarray}
\label{e:sigparam}
P_{sig}(m;M,\jes) &=& \alpha_{7} \cdot \frac{\alpha_{2}^{1 + \alpha_{1}}}
	{\Gamma(1 + \alpha_{1})}
	\cdot (m - \alpha_{0})^{\alpha_{1}} e^{-\alpha_{2} (m - \alpha_{0})}
	\nonumber \\
&& + \alpha_{8} \cdot
	\frac{1} {\alpha_{4}\sqrt{2\pi}} \cdot
	e^{\frac{-(m-\alpha_{3})^{2}} {2\alpha_{4}^{2}}} \nonumber \\
&& + (1 - \alpha_{7} - \alpha_{8}) \cdot
	\frac{1} {\alpha_{6}\sqrt{2\pi}} \cdot
	e^{\frac{-(m-\alpha_{5})^{2}} {2\alpha_{6}^{2}}};
\end{eqnarray}
where
\begin{equation*}
\alpha_{i} = p_{i} + p_{i+9} \cdot (M-175) + p_{i+18} \cdot (\jes),
\end{equation*}
The variable $m$\ in $\gevccnoarg$\ refers to the reconstructed top quark
mass, $M$ in
$\gevccnoarg$ refers to the true top quark mass,
 and $\jes$\ refers to
the true shift in the jet energy scale from that determined from our
calibrations. These template parametrizations are normalized so that, for
a given true top quark mass
$M$\ and jet energy scale $\jes$, the integral over all reconstructed top
quark masses
$m$ is unity.  Similar parametrizations are developed for templates as a
function of the dijet invariant mass $\mjj$.

A binned likelihood fit is used to extract the 27 parameter values.
The \chisq is calculated between the MC samples and the
prediction from the fit, after rebinning to ensure that each bin
has at least five predicted events.  The resulting \chisq
values are given in \tab{t:parfits}, along with the
number of degrees of freedom. Although the corresponding
probabilities are quite small, due to the large statistics of
the templates, the fit matches the template shapes quite well.
This is demonstrated in \fig{f:fitvis}, where four signal templates
at varying generated masses are shown overlaid with the fitted
parameterization evaluated at each true mass. In this figure
can also be seen the changing shape of the reconstructed mass
templates as a function of true mass.  We show in \fig{f:fitmjjvis}\ the
similar distributions for the $\mjj$\ templates with varying jet energy
scale.  One sees that the location of the $W$ boson peak is sensitive to
the jet energy scale.

\begin{cfigure}
\includegraphics{plots/templates_prd3_3}
\caption[Four templates at different masses, with fits overlaid.]
{Four signal templates are shown with top quark masses ranging from
\gevcc{145} to \gevcc{205}, with $\jes$ set to 0. Overlaid are the fitted
parameterizations at each generated mass, taken from the full
parameterization given in Eq.~\ref{e:sigparam}.}
\label{f:fitvis}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/templates_mjj_prd3_3}
\caption[Four templates for $\mjj$\ at different jet energy scales, with
fits overlaid.] {Four $\mjj$\ signal templates are shown with various
$\jes$\ values and the true top quark mass set to \gevcc{180}. Overlaid
are the fitted parameterizations at each generated mass, taken from the
full parameterization given in Eq.~\ref{e:sigparam}.}
\label{f:fitmjjvis}
\end{cfigure}


\subsubsection{Background shape parameterization}
\label{sssec:bkgdparam}

Monte Carlo simulations of the various processes listed in
section~\ref{ssec:backgrounds} are used to model the reconstructed
observed top quark mass shape and dijet mass shape for background
processes. When possible, due to limited computing resources, a single
large-statistics sample is used to represent several background processes.

For the tagged backgrounds, the $W$ + heavy flavor
processes ($W\bbbar$, $W\ccbar$, $Wc$) all have similar reconstructed
mass shapes, as shown in \fig{f:bkgdcomp}, and thus are all modeled
with a high-statistics $W\bbbar$ simulated sample.
$WW$ and $WZ$ events, a negligible contribution to
the total expected background, are also included in this category.
The shapes for the three subsamples with tagged events are found by
reconstructing the simulated events exactly as is done for the data
and signal Monte Carlo. Similarly, the simulated s- and t-channel
single top quark events are used to obtain corresponding mass templates.

\begin{cfigure}
\includegraphics{plots/04100112-BkgdComp}
\caption[Plot comparing Wbb, Wcc, Wc templates.]
{The templates for $W\ccbar$ and $Wc$ background processes
are compared to the high-statistics $W\bbbar$ template for
all tagged events. The agreement is good in both cases,
so the $W\bbbar$ template is used to represent all $W$ + h.f.\
processes.}
\label{f:bkgdcomp}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/04100112-BkgdComp}
\caption[Plot comparing Wbb, Wcc, Wc templates.]
{The templates for $W\ccbar$ and $Wc$ background processes
are compared to the high-statistics $W\bbbar$ template for
all tagged events. The agreement is good in both cases,
so the $W\bbbar$ template is used to represent all $W$ + h.f.\
processes.}
\label{f:bkgdcomp}
\end{cfigure}

The mass templates for the $W$ + jets backgrounds in the tagged
subsamples, i.e.\ ``mistags,'' are not obtained using
the Monte Carlo $b$ tagging, which is not expected to model well the rate
or kinematic dependences of fake tags. Instead, a mistag matrix, derived
from the data, is used to give the probability for
a jet to be falsely tagged as a function of its $\et$, $\phi$, $\eta$,
number of tracks, and the $\Sigma\et$ for all jets in the event.
Then for each simulated event ($W$ + 4 partons, generated by ALPGEN
and showered by Pythia), every possible tag configuration on
the leading four jets is considered. For every tag configuration,
the best assignment of jets to partons is determined respecting the assumed
tags, and the appropriate mass
template is filled with a weight corresponding to the probability of
observing that set of tags. The result is a weighted template for the
mistag backgrounds.

The backgrounds that are least amenable to Monte Carlo modelling arise
from QCD background events, i.e.\ events with no real $W$ to produce the
isolated lepton and
$\met$. These events are difficult to simulate, but can be studied by
selecting events in the data with non-isolated leptons, which are enriched
in this type of background, but kinematically similar to events chosen
in the default selection.
Within the limited statistics available, \fig{f_qcdbkgd}\ and
\fig{f_qcdmjjbgd}\  show that the reconstructed mass distributions of the
QCD-enriched events are consistent with that of $W$ + jets events.  On the
left in \fig{f_qcdbgd}, the
\zerotag events are compared, and on the right, the combined tagged
events are compared with the combined mistag template. The shapes
appear to be very similar. A KS test gives a probability of 99\%
for the \zerotag events, and 6.5\% for the tagged events.
Given these similarities, the $W$ + jets reconstructed mass
templates are used also for the expected contributions from QCD for both
the reconstructed top quark mass and the dijet invariant mass.

\begin{cfigure}
\includegraphics[width=.35\columnwidth]{plots/nonisoalp}
\includegraphics[width=.55\columnwidth]{plots/jahred-qcdvwj-svx}
\caption[Comparison between $W$ + jets templates and QCD templates
modeled by data events with non-isolated leptons.]
{$W$ + jets background templates (histograms) and data events
with non-isolated leptons (points) are compared. The latter
are enriched in non-$W$ (QCD) background. At left,
the points are \zerotag events. At right, the points are
the three tagged categories combined to improve statistics.}
\label{f_qcdbkgd}
\end{cfigure}

\begin{cfigure}
\includegraphics[width=.35\columnwidth]{plots/nonisoalp}
\includegraphics[width=.55\columnwidth]{plots/jahred-qcdvwj-svx}
\caption[Comparison between $W$ + jets templates for the dijet invariant
mass and QCD templates modeled by data events with non-isolated leptons.]
{$W$ + jets background templates (histograms) and data events
with non-isolated leptons (points) are compared. The latter
are enriched in non-$W$ (QCD) background. At left,
the points are \zerotag events. At right, the points are
the three tagged categories combined to improve statistics.}
\label{f_qcdmjjbkgd}
\end{cfigure}

The background for the \zerotag subsample is treated separately from
the others. The dominant process is $W$ + jets, with a smaller
($\sim20\%$) contribution from non-$W$ (QCD) events. As above,
the reconstructed mass shape for events in the data with
non-isolated leptons matches well the simulated $W$ + jets template.
Thus the entire \zerotag background shape comes from $W$ + 3 parton
Monte Carlo events, simulated by ALPGEN and showered by Pythia.

We do not allow each background contribution to vary independently in
the final likelihood fit. Instead, for each subsample, the background
processes are combined in their expected ratios according to
\tab{t:backgroundnumber}. A single function is fit to the combined background
for each subsample and is used to describe the background shape in the
final likelihood fit (section~\ref{ssec:likelihood}). The overall
background normalization for each subsample is then permitted to vary in
the fit.

We determine the p.d.f.'s for the background reconstructed mass templates
using a parameterization similar in spirit to that of the signal, but
 simpler in form. First, there is no dependence on top quark
mass. Second, no narrow Gaussian peak is expected, so the full shape
is modeled by the integrand of the Gamma function. Specifically,
\begin{equation}
\label{e_bgparam}
P_{bg}(m) = \frac{p_{2}^{1 + p_{1}}} {\Gamma(1 + p_{1})}
	\cdot (m - p_{0})^{p_{1}} e^{-p_{2} (m - p_{0})},
\end{equation}
where $m$ in $\gevccnoarg$ is the reconstructed top quark mass.
In the case of the \zerotag background events, a slightly
more sophisticated function is used to achieve a good fit:
\begin{eqnarray}
\label{e_bgparamzerotag}
P_{bg0}(m) &=& p_{6} \frac{p_{2}^{1 + p_{1}}} {\Gamma(1 + p_{1})}
	\cdot (m - p_{0})^{p_{1}} e^{-p_{2} (m - p_{0})} \\ \nonumber
&+& (1-p_{6}) \frac{p_{5}^{1 + p_{4}}} {\Gamma(1 + p_{4})}
	\cdot (m - p_{3})^{p_{4}} e^{-p_{5} (m - p_{3})}.
\end{eqnarray}
%For our
%combined reconstructed mass templates
%the fitted parameters are given in Table~\ref{t:bkgdparfit}.
After rebinning so that each bin has at least five predicted events,
the $\chisq$ for the fits show reasonable agreement (\tab{t:parfits}).
The final background templates for the reconstructed top quark mass and
dijet mass for the four subsamples are shown in \fig{f:bkgdtemps}\ and
\fig{f:bkgbmjjtemps}, respectively.

\begin{table}
\caption[Parameterization fit information.]
{The \chisq and number of degrees of freedom are given for the signal
parameterization fits in each of the four subsamples.}
\label{t:parfits}
\begin{ruledtabular}
\begin{tabular}{cccc}
\twotag & \onetagt & \onetagl & \zerotag \\
\hline
\multicolumn{4}{c}{\chisq/n.d.o.f.\ for reconstructed top quark mass
(Eq.~\ref{e:sigparam})} \\
$13977/13081$ & $18736/17052$ & $14209/13444$ & $18791/16752$ \\
\hline
\multicolumn{4}{c}{\chisq/n.d.o.f.\ for dijet mass
(Eq.~\ref{e:sigparam})} \\
$19541/17000$ & $29410/25732$ & $23506/19827$ & $38315/30510$ \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{cfigure}
\includegraphics{plots/bkgdtemp_prd}
\caption[Reconstructed top quark mass distributions for background.]
{Reconstructed mass distributions of the combined backgrounds
in each subsample.
The contributions from different background templates are shown
stacked; overlaid are the fitted curves (see Eq.~\ref{e_bgparam}).}
\label{f:bkgdtemps}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/bkgdmjjtemp_prd}
\caption[Reconstructed dijet mass distributions for background.]
{Reconstructed dijet mass distributions of the combined backgrounds
in each subsample.
The contributions from different background templates are shown
stacked; overlaid are the fitted curves (see Eq.~\ref{e_bgparam}).}
\label{f:bkgdmjjtemps}
\end{cfigure}

\subsection{Likelihood Fit for Top Quark Mass}
\label{ssec:likelihood}

The reconstructed mass distributions from data are simultaneously compared
to the templates from signal and background sources using an unbinned
extended likelihood fit.
The likelihood involves parameters for the expectation values of
the number of signal and background events in each subsample,
and for the true top quark pole mass and jet energy scale.
For each subsample, the likelihood is given by:
\begin{eqnarray}
{\mathcal L} &=& {\mathcal L}^{m_t}_{shape} 
   \times {\mathcal L}^{m_jj}_{shape}
			 \times {\mathcal L}_{sig} \times {\mathcal L}_{bg};\nonumber\\
{\mathcal L}^{m_t}_{shape} &=&
\frac{e^{-(n_{s}+n_{b})}(n_{s}+n_{b})^{N}}{N!}\times\nonumber\\
									 	 & & \prod_{i=1}^{N} \frac{n_{s}P_{sig}(m_{i};\mtop) +
			 									 n_{b}P_{bg}(m_{i})}{n_{s}+n_{b}};\nonumber\\
{\mathcal L}_{sig} &=& e^{-\frac{(n_{s}-n_{s}^{exp})^2}
			 {2\sigma_{n_{s}}^{2}}};\nonumber\\
{\mathcal L}_{bg} &=& e^{-\frac{(n_{b}-n_{b}^{exp})^2}
			 {2\sigma_{n_{b}}^{2}}}.
\end{eqnarray}

\note{The above equation has to be revised for the dijet mass analysis
still.}

The most information on the true top quark mass is provided by the
products in
${\mathcal L}^{m_t}_{shape}$, the $i$th term of which gives the
probability of observing the $i$th data event with reconstructed mass
$m_i$, given the background template,
$P_{bg}(m_i)$, and the signal template with a true top quark mass of
$\mtop$\ and energy scale shift \jes,
$P_{sig}(m_i;\mtop, \jes)$.
The third term represents the information arising from the number of
signal and background events in the top quark mass and dijet mass samples,
which are correlated.  We denote the number of expected signal and
background events in the
$W\rightarrow jj$\ sample, $n_s^W$\ and $n_b^W$, respectively. The
expected numbers of signal and background events in the $m_t$\ sample
are given by $\epsilon_s n_s^W$\ and $\epsilon_b n_b^W$, respectively,
where the two parameters $\epsilon_s$\ and $\epsilon_b$\ represent the
efficiency of the $\chi^2$\ cut for signal and background events. The
efficiencies for signal range from $0.65$ to $0.92$\, while the
efficiencies for background range from $0.44$ to $0.83$.  The third term
in the likelihood, ${\mathcal L}_{nev}$, expresses the likelihood
associated with observing $r_W$\ and $r_t$\ events in the two samples
given the expected number of events and the expected efficiencies.  The
first sum expresses the Poisson probability to observe $r_s^W$\ signal
and $r_b^W$\ background events given Poisson means of $n_s^W$\ and
$n_b^W$, respectively.  The sum in the third term is over the sum of
those signal and background events that equal the observed number of
events in the $m_jj$\ sample:  $r_s^W + r_b^W = r^W$.  For each pair in
this sum, we then include the binomial probability to observe $r_s^t$\
signal events and $r_b^t$\ background events in the $m_t$\ sample given
the numbers of observed events in the $mjj$\ sample and the
$\chi^2_{min}$\ cut efficiencies.  The second sum in the ${\mathcal
L}_{nev}$\ is over the pairs of signal and background events in the
$m_t$\ sample that equal the observed number of events: $r_s^t + r_b^t=
r^t$.  

When independent estimates of background are available, the 
background normalizations are constrained in the likelihood fit by
Gaussian  terms with the form of $\mathcal{L}_{bg}$. The
background normalizations are constrained for the \twotag, \onetagt, and
\onetagl samples, and the signal normalization is constrained for
the \zerotag sample.
Both $n_s$ and $n_b$ are required to be greater than zero.

Independent detector calibrations and studies of other processes allow us
to independently determine the jet energy scale $\jes$, and this
information is used in the reconstruction of the data events and in the
determination of the signal templates.  We include in the likelihood fit
the knowledge of this independent jet energy calibration through a 
likelihood defined by a Gaussian constraint
\begin{eqnarray}
{\mathcal L}_{\jes} &=& e^{-\frac{(\jes-\jes^{exp})^2}
			 {2\sigma_{\jes}^{2}}} \\
  &=& e^{-\frac{\jes^2}{2}},
\end{eqnarray}
where the simplication arises because by definition the measured shift in
energy scale, $\jes^{exp}=0$\ and the uncertainty $\sigma_{\jes}=1.0$.

The total likelihood is given by the product of the likelihoods for
the four subsamples and the jet energy scale constraint
\begin{eqnarray}
\mathcal{L} = \mathcal{L}_{\twotag} \times \mathcal{L}_{\onetagt} \times
								\mathcal{L}_{\onetagl} \times \mathcal{L}_{\zerotag}
   \times \mathcal{L}_{\jes}.
\end{eqnarray}
The top quark pole mass $\mtop$\ and jet energy scale $\jes$\ are shared
between the four likelihoods and are free parameters in the fit. 
The likelihood is maximized with respect to all ten parameters
($n_s$ and $n_b$ for four subsamples, $\jes$\ and $\mtop$) using
the MINUIT package. A likelihood curve as a function of $\mtop$ is
found by maximizing the likelihood with respect to all other parameters
for a series of fixed $\mtop$. The statistical uncertainty from the fit
procedure is taken from the points $\mtop^+$ and $\mtop^-$ where
the log-likelihood changes by $+1/2$ unit from its minimum. The
positive and negative uncertainties are then scaled to achieve 68.3\%
coverage of the true mass value as described below in
section~\ref{ssec:methodCheck}.

We can also obtain a refinement in our measurement of the jet energy
scale by assuming a fixed $\mtop$\ and calculating confidence intervals
for $\jes$.  This provides a direct test of the consistency of our
independent jet energy scale determination with the $\mjj$\ distributions
observed in the mass candidate sample.

%We note that since we fit for both $\mtop$ and $n_{s}$, the
%expectation value of the number of signal events, we could extract the
%top mass and the $\ttbar$ cross-section simultaneously (given the
%efficiency of our $\chisq$ cut on signal events, plus various other
%quantities already available).  It would be interesting to do this
%exercise and see the cross-section value we observe, as well as its
%correlation with the top mass.

\subsection{Method Check}
\label{ssec:methodCheck}

The method described above is checked for any possible systematic
biases by running large numbers of ``pseudoexperiments,'' where we
create, using Monte Carlo signal and background events, samples of
signal and background events with an assumed value of the true top
quark mass and jet energy scale and with the same statistical properties
as our observed sample.  We then perform likelihood fits to each
pseudoexperiment and characterize the accuracy of the technique in
determining the true values.

For each pseudoexperiment, first the number of signal and background events
is determined, then a reconstructed top quark mass and dijet mass are
generated for each event. The number of background events in each
subsample is Poisson fluctuated around the central value given in
\tab{t:backgroundnumber}. 
The number of signal events is Poisson fluctuated around the number
observed in the data, minus the central value for the background
expectation, for each subsample.
For each event, a reconstructed mass is selected at random from the
template histogram corresponding to signal or background processes.
The resulting list of reconstructed masses is fit using exactly the
same machinery as will be used on the data, described in
section~\ref{ssec:likelihood}.  We note the correct statistical
correlations between the $\mjj$\ and the top quark mass sample are
modelled in this procedure.

For each pseudoexperiment, the likelihood fit provides a measured
top quark mass \mtop\ and jet energy scale $\jes$, as well as positive and
negative errors (\mterrp and
\mterrm) from the $\Delta\ln L=+1/2$ procedure. The pulls of the
measurement are checked using the symmetrized errors, where the
uncertainty on the top quark mass is taken to be $.5(\mterrp+\mterrm)$.
A pull distribution is generated for each of 12 input values for
the top quark mass, keeping $\jes$\ fixed to zero, where 10,000
pseudoexperiments are generated for each input mass value, and each pull
distribution is fitted using a Gaussian function. The mean and sigma of
the fitted functions are shown in
\fig{f:pullmeans} and \fig{f:pullwidths}, respectively. The pull means
show no overall bias, but the widths are slightly larger than one. 
We determine a similar set of pull distributions for the $\jes$\
parameter, keeping the true top quark mass fixed to $\gevcc{180}$.  The
results are shown in \fig{f:pullmeansjes}\ and \fig{f:pullwidthsjes}, and
demonstrate that the technique has no overall bias.

\begin{cfigure}
\includegraphics{plots/pullmean_vs_mass}
\caption[Means of pull distributions vs top quark mass.]
{The Gaussian mean of pull distributions from 10,000 pseudoexperiments
as a function of true top quark mass. The error bars come mostly from the
limited statistics of the Monte Carlo sample from which the pseudodata
is taken.}
\label{f:pullmeans}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/pullwidth_vs_mass}
\caption[Widths of pull distributions vs top quark mass.]
{The Gaussian width of pull distributions from 10,000 pseudoexperiments
as a function of true top quark mass. The error bars come mostly from the
limited statistics of the Monte Carlo samples from which the pseudodata
is taken.}
\label{f:pullwidths}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/pullmean_vs_mass}
\caption[Means of pull distributions vs jet energy scale.]
{The Gaussian mean of pull distributions from 10,000 pseudoexperiments
as a function of true jet energy scale. The error bars come mostly from
the limited statistics of the Monte Carlo sample from which the pseudodata
is taken.}
\label{f:pullmeansjes}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/pullwidth_vs_mass}
\caption[Widths of pull distributions vs jet energy scale.]
{The Gaussian width of pull distributions from 10,000 pseudoexperiments
as a function of true jet energy scale. The error bars come mostly from
the limited statistics of the Monte Carlo samples from which the
pseudodata is taken.}
\label{f:pullwidthsjes}
\end{cfigure}

The inflation of the pull widths is due to the low statistics of the
event sample. For a small number of events, and for templates such
as the ones described in section~\ref{ssec:parameterization}, the resulting
likelihood curve is typically non-Gaussian, and in fact, typically
shallower than Gaussian. This likelihood shape, using any definition of
the measurement error, in turn provokes large non-Gaussian
tails in a pull distribution, which lead to a fitted pull width greater
than one. The pull distributions become more Gaussian (with
width one) as pseudoexperiments with more events are performed. With ten
times the statistics, the pull widths are consistent with one.

For the current low-statistics data sample, the quoted measurement with
uncertainties is designed to include the true mass value 68.3\% of the time
by scaling the errors taken from $\Delta\ln L=+1/2$. The scale factor
is that necessary so that 68.3\% of an ensemble of pseudoexperiments
include the simulated top quark mass. The factor is determined using
pseudoexperiments for an input mass near the measured value, and uses
$\mterrp$ and $\mterrm$ rather than a symmetrized error.

The statistical errors from the ensembles of pseudoexperiments at various
top quark masses are depicted in \fig{f:expstaterr}. The mean, median,
68th percentile, and 90th percentile of the symmetrized error are
plotted as a function of the input top quark mass. The expected
statistical error increases with the top quark mass, but the fractional
uncertainty decreases. No scaling of the errors is performed in this plot.

\begin{cfigure}
\includegraphics{plots/errors_vs_mass}
\caption[Expected statistical errors vs input top quark mass.]
{The mean, median, 68th percentile, and 90th percentile of the
symmetrized statistical errors from 10,000 pseudoexperiments are
shown as a function of the input top mass.}
\label{f:expstaterr}
\end{cfigure}

