%
%  Add W->jj reconstruction and 318 pb-1	  16 March 2005    Pekka 
%
\section{Mass Reconstruction}
\label{sec:massRecon}

In this section, we describe the procedure for determining
the reconstructed top mass in each event and the procedure used to
reconstruct the $W$\ boson decaying into the dijet final state.  We then
discuss the results of these reconstruction techniques. It is
important to remember that the reconstructed mass is not an event-by-event
measurement of the top quark mass; rather it is a quantity whose
distribution in the data will be compared with simulated samples to
extract the top quark mass (see section~\ref{sec:fitting}).  The
distribution of the invariant dijet mass of the $W$\ boson candidate 
will be used to constrain the
calibration of the jet energy scale in the reconstructed events.

Throughout the mass reconstruction, each event is assumed to
be a \ttbar event decaying in the lepton + jets channel, and
the four leading jets are assumed to correspond to the four
quarks from the top and $W$ decays. First, the measured
four-vectors for the identified physics objects in the event
are corrected for known effects, and resolutions
are assigned where needed. Next, a \chisq fit is used to
extract the reconstructed mass.
After the procedure, each event has a particular value of
reconstructed mass and a corresponding \chisq value.
Some events ``fail'' the mass reconstruction and are discarded
when their minimized \chisq exceeds a cut value.

\subsection{Inputs to the mass reconstruction}
\label{ssec:inputs}

The \chisq fit takes as input the four-vectors of the various
physics objects identified in the event.  All known
corrections are applied to the jets and lepton, and Gaussian
uncertainties are established on the transverse momenta,
since the $\pt$ of each onne will be permitted to vary in the fit.
The $\met$ is a derived quantity, and does not have an uncertainty
independent of the other measured values. The \chisq considers
instead a related fundamental quantity, the unclustered energy,
which is described below.

\subsubsection{Jet inputs}
\label{sssec:jetinputs}

The jet four-vectors are the most important input to the \chisq fit.
Jets are the most numerous physics objects in the event, and they
are the most poorly measured. Identifying reconstructed jets with
partons from the \ttbar decay chain has both theoretical and
experimental pitfalls, and a series of corrections are made to
jet energies in order to best approximate the corresponding parton
energies. Measured jet energies have a large resolution, and are
permitted to vary in the fit. The angles of the reconstructed jets,
in contrast, are good approximations of the corresponding parton angles,
so they are used without corrections and are fixed in the fit.

Jets are first corrected with a set of ``generic'' jet corrections,
so called because they are intended to be independent of the
particular process under consideration. For these corrections,
when an assumption
must be made about the underlying parton $\pt$ spectrum, it is
taken to be flat. Since some of the corrections are a function of
jet $\pt$, and since the jet resolution is non-negligible, this
assumption has a considerable effect on the derived correction.

These generic jet corrections scale the measured jet four-vector to account
for various effects. First, a dijet balancing procedure is used to
determine and correct for variations in the calorimeter response to
jets as a function of $\eta$. These variations are due to different
detector technology, to differing amounts of material in the tracking
volume and the calorimeters, and to uninstrumented regions.
In dijet balancing, events are selected with two and only two jets, one
in the well understood central region ($0.2<|\eta|<0.6$). A correction
is determined such that the transverse momentum of the other jet, as
a function of its $\eta$, is equal on average to that of the central
jet. This relative correction ranges from about --10\% to +15\%.

Another sample of events, in which a photon and a jet are the two
reconstructed objects, allows an absolute measurement of the jet
response, since the electromagnetic energy of the photon is well
calibrated through $\zee$ decays. This $\gamma$-jet balancing is
used to set the overall jet energy scale to that of \runi, a
6.5\% correction, so that
a set of sophisticated corrections for calorimeter non-linearity,
derived in \runi for the same central calorimeter, can be applied.

After a small correction for the extra energy
deposited by multiple collisions in the same accelerator bunch
crossing, and the corrections for calorimeter non-linearity
(+10\% to +30\%, depending on jet $\pt$),
the jet energies correspond, on average, to the true in-cone
hadronic energy for jets with a flat underlying $\pt$ spectrum.
The various sources of uncertainty in the CDF jet energy scale and
simulation are described in section~\ref{ssec:jetSyst}.

Next, jet corrections are applied that have been
derived specifically for the \ttbar process.
There are separate corrections for light quark jets from $W$-boson
decay ($W$ jets) and $b$ jets, because of their
different fragmentation and decay properties.
In addition, these corrections account for the shift in the mean jet
energy due to the non-flat $\pt$ spectrum
of partons from \ttbar decay.  The \ttbar-specific corrections also
account for the energy falling outside the jet clustering cone. $W$ jets
and $b$ jets, which clearly have different response, are corrected using
different functions, but no separate correction is attempted for $b$
jets with identified semi-leptonic decays.
The \ttbar-specific corrections also provide uncertainties on the
jet energies (i.e.\ the measurement resolution, not systematic
uncertainties). Note
that, since these corrections depend on the flavor of the jet, they
must be applied after a hypothesis has been selected for the
assignment of the measured jets to partons from the \ttbar decay
chain.

The \ttbar-specific corrections are extracted from a large sample
of HERWIG \ttbar events in which the four leading jets in $\et$ are
matched within $\Delta R=0.4$ to the four generator-level partons
from \ttbar decay. The correction functions are consistent with
those extracted from a large Pythia sample.
The correction is defined as (the inverse of) the most probable value
(MPV) of the jet response curve, as a function of $\ptjet$ and $\etajet$.
The MPV is chosen, rather than the mean of the asymmetric distribution,
in order to accurately correct as many jets as possible in the core
of the distribution. This increases the number of events for which
the correct jet-quark assignment is chosen by the fitter (see below),
resulting in a narrower core for the reconstructed mass distribution
(\fig{f:sampletemp}).
A corresponding resolution is found by taking the symmetric window
about the MPV of the jet response that includes 68\% of the total area.
Figure~\ref{f_tscorrs} shows the corrections and resolutions
as a function of jet $\pt$ and $\eta$.

\begin{cfigure}
\includegraphics{plots/04082612_JFCorrections}
\includegraphics{plots/04082612_JFSigmas}
\caption{The \ttbar-specific corrections are shown for $W$ jets (left)
and $b$ jets (right) as a function of jet $\pt$ and $\eta$.
On the top is the correction factor, and
on the bottom is the fractional resolution passed to the fitter.}
\label{f_tscorrs}
\end{cfigure}

As a final step, the jet momentum is held fixed while the jet energy
is adjusted so that the jet has a mass according to its flavor
hypothesis.  A mass of $\gev{0.5}$ is used
for $W$ jets, and a mass of $\gev{5.0}$ is used for $b$ jets.
This is done for consistency with the Monte Carlo truth information
used to derive the \ttbar-specific corrections.

\subsubsection{Lepton Inputs}
\label{sssec:lepinputs}

The electron four-vector has energy determined by its electromagnetic
calorimeter cluster, and angles defined by the associated track.
The electron energy is corrected for differences in the calorimeter
response depending on where in the tower face the electron enters.
The electron mass is set to zero, and the angles
are taken as perfectly measured quantities.  The transverse momentum
$(\pte = p \sin \theta)$ of the electron has an uncertainty of
\begin{equation}
\frac{\sigma_{\pte}}{\pte} = \sqrt{\left( \frac{0.135}
	{\sqrt{\pte[\gevcnoarg]}}\right)^{2} + (0.02)^{2}}.
\end{equation}

The muon four-vector uses the three-vector of the associated track,
also with a mass of zero.  Muon curvature corrections due to chamber
misalignment are applied.
The angles and mass are given no uncertainty; the transverse
momentum has an uncertainty of
\begin{equation}
\frac{\sigma_{\ptmu}}{\ptmu} = 0.0011\cdot\ptmu[\gevcnoarg],
\end{equation}
The electron and muon transverse momentum uncertainties are taken
from studies of leptonic $Z^0$ decays.

\subsubsection{Neutrino Inputs: Unclustered Energy}
\label{sssec::neutrinoinputs}

The neutrino in a \ttbar event is not observed; its presence
is inferred by an imbalance in the observed transverse momentum.
Therefore, rather than treating the neutrino 4-vector
as an independent input to the \chisq fit, the measured quantities,
as varied in the fit, are used to dynamically calculate the
neutrino transverse momentum.

All of the transverse energy in the calorimeter (towers with
$|\eta| < 3.6$) that is not associated with the primary lepton
or one of the leading four jets is considered ``unclustered
energy.'' For towers clustered into a jet that has
$\et>\gev{8}$ and $|\eta|<2.0$, but that is not one of the leading
four jets, the tower momenta are replaced with the jet momentum
after the generic jet corrections described in
section~\ref{sssec:jetinputs}.
The rest of the tower momenta are multiplied by a scale factor of
1.4, which is roughly the generic jet correction factor for
$\gev{8}$ jets.
Finally, the unclustered energy includes the energy thought to
enter into the leading four jets from the underlying event,
and excludes the energy thought to fall outside the jet cones
of the leading four jets. This avoids double-counting of energy
that is included in the leading four jet energies after all
corrections.  Each transverse component
of the unclustered energy ($p_x^{UE}$, $p_y^{UE}$) is assigned an
uncertainty of $0.4 \sqrt{\sum\et^{unclus}}$, where
$\sum\et^{unclus}$ is the scalar sum of the transverse energy
excluding the primary lepton and leading four jets.
The uncertainty comes from studies of minimum bias events.

The unclustered energy is the observed quantity and the input to
the \chisq fit, but it is related to the missing energy through
the other measured physics objects in the event,
since the $\ppbar$ system has $\pt=0$.
The neutrino transverse momentum $\ptnu$ is
calculated at each step of the fit, using the fitted values of
lepton, jet, and unclustered transverse energies:
\begin{equation}
\vec{\ptnu} = -\left(\vec{\pt^{\ell}} + \sum\vec{\ptjet} +
				\vec{p_T^{UE}}\right)
\end{equation}
Note that this quantity, used in the mass fitting procedure,
is different from the missing energy described in
section~\ref{ssec:selection} and used in event selection,
where simpler calorimeter energy corrections are used.

Although this is a complicated input to the fit procedure,
and other treatments of the unclustered energy and missing energy
can be motivated, it has been shown that the $\met$ calculation
does not in fact have a large effect on the results of the \chisq fit.
Various other approaches to correcting the unclustered energy and
assigning resolution were tried, and no changes had any significant
effect on the top mass resolution.

As for the other components of the neutrino four-vector, the mass
is fixed at zero, and the longitudinal momentum, $\pz^{\nu}$, is a free
(unconstrained) parameter in the fit. The initial value of $\pz^{\nu}$
is calculated using the initial value of the lepton
four-vector and the initial $\ptnu$, and assuming their mother
$W$ has its pole mass.  Since these conditions yield a quadratic
equation, there are in general two solutions for the $\pz^{\nu}$;
a separate \chisq fit is done with each solution used as the initial
value of $\pz^{\nu}$.  When the two fits land in different minima, we
choose the one that provides the best fit using the \chisq values
as described below.

\subsection{Event \chisq fit}
\label{ssec:chisq}

Given the inputs described above, the event-by-event fit for the
reconstructed top mass proceeds as follows.  MINUIT is used to
minimize a \chisq where the top mass is a free parameter.
For each event, the \chisq is minimized once for each possible
way of assigning the leading four jets to the four quarks from
the \ttbar decay. Since the two $W$ daughter jets are indistinguishable
in the \chisq expresssion, the number of permutations is
$\frac{4!}{2}=12$. In addition, there are 2 solutions for the initial
value of the neutrino longitudinal momentum, so the minimization
is performed a total of 24 times for each event.
When $b$ tags are present, permutations that assign a tagged jet
to a light quark at parton level are rejected. In the case of
single-tagged events, the number of allowed permutations is six,
and for double-tagged events, it is two.

The \chisq expression has terms for the uncertain measurements of
jet, lepton, and unclustered energies, as well as terms for the
kinematic constraints applied to the system:
\begin{eqnarray}
\label{eq_chi2}
\chisq &=&
\sum_{i = \ell, 4 jets} \frac{(p_T^{i,fit} - p_T^{i,meas})^2}{\sigma_i^2} \nonumber \\
&&+ \sum_{j = x,y} \frac{(p_j^{UE,fit} - p_j^{UE,meas})^2}{\sigma_j^2} \nonumber \\
&&+ \frac{(M_{jj} - M_W)^2}{\Gamma_W^2}
+ \frac{(M_{\ell\nu} - M_W)^2}{\Gamma_W^2} \nonumber \\
&&+ \frac{(M_{bjj} - M_t)^2}{\Gamma_t^2}
+ \frac{(M_{b\ell\nu} - M_t)^2}{\Gamma_t^2}.
\end{eqnarray}

The first term constrains the $\pt$ of the lepton and four leading jets to their
measured values within their assigned uncertainties; the second term does
the same for both transverse components of the unclustered energy.
In the remaining four terms, the quantities $M_{jj}$, $M_{\ell\nu}$,
$M_{bjj}$, and $M_{b\ell\nu}$ refer to the invariant mass of the sum
of the four-vectors denoted in the subscripts. For example, $M_{jj}$
is the invariant mass of the vector sum of the $W$ daughter jets.
$M_{W}$ and $M_{t}$ are the pole masses of the $W$ and top quark.
$M_{W}$ is $\gevcc{80.41}$~\cite{r_PDBook}, and $M_{t}$ is the
free parameter that will be declared the reconstructed mass after the
\chisq is minimized. The fit is initialized with $M_{t} = \gevcc{175}$.
$\Gamma_W$ and $\Gamma_t$ are the total width of the $W$ boson and the
top quark. In order to use the \chisq formalism, the $W$ and
top Breit-Wigner lineshapes are modeled with Gaussian distributions, using the
Breit-Wigner full width at half maximum as the Gaussian sigma.
$\Gamma_{W}$ is $\gev{2.06}$~\cite{r_PDBook}, and $\Gamma_{t}$ is
$\gev{2.5}$. \note{Let's change this!! Be consistent with the generators.
Fit Gaussian to generator level mass distribution\dots}
\note{Reference?} Thus these terms provide constraints such that the $W$
masses come out correctly, and the $t$ and $\bar{t}$ masses come out
the same (modulo the Breit-Wigner distribution, here modeled by a
Gaussian, in both cases).

The jet-parton assignment (and $\pz^{\nu}$ solution) with the lowest
\chisq after minimization is selected for each event.  The \chisq
of this combination is denoted \chisqmin (or just \chisq when the
context is unambiguous), and the requirement $\chisqmin<\chisqcut$
is imposed.
The \chisq cut values given in \tab{t:eventTypes} were found to give
the best expected statistical uncertainty on the top mass, although
the dependence on the values of the cut is not strong.
So at this stage, after mass reconstruction, each event
either has been rejected or can be described by a single value of this
reconstructed mass variable. Indeed, the rest of the analysis can be
performed with nothing more than a list of the reconstructed masses
for events in various samples that have passed all selection cuts.

\subsection{Dijet Mass and Jet Energy Scale}
\label{ssec:mdijetRecon}

One of the largest systematic uncertainties in the top quark mass
measurement is the jet energy scale, which relates the observed energy
in the calorimeter associated with a reconstructed jet to the parent
parton that created the jet.  We use the $W$\ boson decay to two jets
as a direct means of constraining the accuracy of the jet energy scale. 
We do this by comparing the observed dijet invariant mass for the $W$\
boson daughter jets with the distribution expected for a $W$\ boson
decay assuming a shift in the jet energy scale.  

The shift in jet energy
scale is given by JES, which is the relative shift in the jet energy
scale in units of the nominal uncertainty in the jet energy scale
derived from the extrinsic calibration procedures.  
Although the jet energy scale uncertainty varies with energy, a one unit
shift in the JES parameter is equivalent to an approximately 3\%\
shift in the jet energy scale.

We reconstruct the dijet mass in the same data sample used to reconstruct
the observed top quark mass, with the exception that we relax the \chisq\
requirement on the jet-parton assignments under consideration, and we
elect to use all of the possible jet-parton assignments in each event. 
The imposition of the \chisq\ requirement imposes a bias in the dijet
masses being considered and therefore reduces the sensitivity of the
dijet mass distribution to JES.  Similarly, Monte Carlo studies have
shown that the sensitivity of the dijet mass distribution to the JES
parameter is maximized by considering all dijet mass combinations
possible in each event.  We note that the number of possible combinations
range from 1 (for double b-tagged events) to 6 (for 0 tagged events).

We simultaneously fit the reconstructed top quark mass distributions and
the dijet mass distributions to the Monte Carlo predictions, which are
parametrized as a function of the true top quark mass and JES.
 

\subsection{Mass reconstruction results}
\label{ssec:mrecoResults}

Typical reconstructed mass distributions for signal Monte Carlo
($\mtop=\gevcc{175}$) are shown for the four event categories
as the green histograms in \fig{f:sampletemp}. Each event in
the sample that passes both event selection and the $\chisq$
cut contributes exactly one entry to these histograms.
As expected, the distributions peak near the generated mass of
$\gevcc{175}$, but an exact correspondence between the generated
mass and, say, the mean of the reconstructed mass is neither sought
nor expected. Differences can arise when ISR/FSR jets are
selected instead
of the \ttbar decay products; even with the correct jets, the fit
may choose the wrong jet-parton assignment. In particular, the
broader shape, beneath the relatively sharp peak at $\gevcc{175}$,
comprises events where an incorrect permutation has been chosen in
the fit.  The red histograms in the same figure show the reconstructed
mass distributions for events where the four leading jets correspond to
the four partons from \ttbar decay, and where
the correct jet-parton assigment is chosen by the fit.
These histograms have much smaller tails than the overall distributions.

\begin{cfigure}
\includegraphics{plots/signaltemps_prd}
\caption[\gevcc{175} reconstructed mass templates.]
{In green, the reconstructed mass distribution for the $\gevcc{175}$ HERWIG
\ttbar sample.  Overlaid in red is the reconstructed mass distribution
for the correct parton-jet assignment, in the subset of events for which
the leading four jets are matched (within $\Delta R=0.4$)
to the four partons from the \ttbar decay.
Distributions are shown for \twotag (upper left), \onetagt (upper right),
\onetagl (lower left), and \zerotag (lower right) events.
\note{Remove stats boxes. Change terminology.}}
\label{f:sampletemp}
\end{cfigure}

Some results of the mass reconstruction on Monte Carlo \ttbar
signal ($\mtop=\gevcc{175}$) and background samples are given
in \tab{t:massRecoMC}. The four subsamples have significantly
different shapes for \ttbar signal, as evidenced by their
reconstructed mass mean and RMS values.
The \chisq cut efficiency is lowest for \twotag events,
especially in the background, because there are fewer allowed
jet-parton assignments and thus fewer chances to pass the \chisq
cut. The means of the background reconstructed
mass distributions are primarily driven by the jet cuts (see
\tab{t:eventTypes}).

\begin{table}
\caption[Result of mass reconstruction on Monte Carlo samples.]
{Monte Carlo samples of \ttbar signal and of background processes
with the expected relative weights are run through the \chisq
mass fitter. For signal and background in each of the four event
categories, the table shows the efficiency of the \chisq cut and
the mean and RMS of the resulting reconstructed mass distribution.}
\label{t:massRecoMC}
\begin{ruledtabular}
\begin{tabular}{lddd}
Sample
& \multicolumn{1}{c}{\chisq cut}
& \multicolumn{2}{c}{\mreco [\gevnoarg]} \\
Description
& \multicolumn{1}{c}{eff.}
& \multicolumn{1}{c}{Mean}
& \multicolumn{1}{c}{RMS} \\
\hline
\multicolumn{4}{l}{\textbf{Signal} ($\mtop=\gevcc{175}$)} \\
\twotag   & 0.67 & 172.1 & 25.5 \\
\onetagt  & 0.83 & 171.8 & 30.1 \\
\onetagl  & 0.86 & 166.2 & 30.3 \\
\zerotag  & 0.92 & 177.0 & 34.9 \\
\hline
\multicolumn{4}{l}{\textbf{Background}} \\
\twotag   & 0.37 & 164.9 & 36.1 \\
\onetagt  & 0.70 & 165.0 & 40.6 \\
\onetagl  & 0.70 & 153.6 & 36.1 \\
\zerotag  & 0.85 & 180.9 & 45.7 \\
\end{tabular}
\end{ruledtabular}
\end{table}

The reconstructed mass distribution for the 135 events found in
the data can be seen in \fig{f:datamasses}, separated into the
four subsamples. These events consist of both \ttbar 
signal and background events.
\Fig{f:chi2} shows the distribution of \chisq values in data and
simulated events, where the distribution from simulation contains
the expected mixture of signal and background events.
The distributions agree nicely, indicating that
kinematic quantities and resolutions are well simulated.

\begin{cfigure}
\includegraphics{plots/datamass_prd}
\caption[Reconstructed masses of data events.]
{The reconstructed mass for events in the data is shown,
with events from the four subsamples represented by
histograms in yellow
(\twotag, 2 events), blue (\onetagt, 18 events), green
(\onetagl, 8 events), and red (\zerotag, 40 events).
\note{Do we want this stacked as is? Or just one color,
all events combined? Or four plots, one for each subsample
(redundant with \fig{f:fitConsistency})?}}
\label{f:datamasses}
\end{cfigure}

\begin{cfigure}
\includegraphics{plots/04093013-Chi2Comp}
\caption[Comparison of \chisq distribution in data and MC.]
{The \chisq distribution is shown for data events and for
signal and background simulated events in the expected ratio.
In this plot only events with at least one $b$ tag are used.
\note{Should be all subsamples? Or one single subsample?}}
\label{f:chi2}
\end{cfigure}

The corresponding dijet mass distributions for the $W$\ boson
reconstruction are shown in \fig{f:datadijetmasses}\ for the four
subsamples.  One sees a clear $W$\ boson mass signal, with a peak near
the nominal $W$\ boson mass of $\gevcc{80.32\pm??}$.    The peak becomes
more evident with increasing numbers of b-tagged jets in the event, a
consequence of the decreasing number of combinations for $W$\ boson jet
daughters.

\begin{cfigure}
\includegraphics{plots/datadijetmass_prd}
\caption[Reconstructed dijet invariant masses for data events.]
{The reconstructed dijet mass for events in the data is shown,
with events from the four subsamples represented by
histograms in yellow
(\twotag, 2 events), blue (\onetagt, 18 events), green
(\onetagl, 8 events), and red (\zerotag, 40 events).}
\label{f:datadijetmasses}
\end{cfigure}

