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The top quark is the heaviest known elementary particle, with a mass
that is approximately 35 times that of the other quarks and leptons.
Because of this anomalously large mass, measurements of the properties
of the top quark may provide insight into our understanding of mass in
general, and test various theories that explain the large mass
heirarchy among quarks and leptons, and the interactions that give
rise to such disparate properties.
Within the context of the Standard Model of particle physics, as the
heaviest fermion in the theory, its mass is correlated with the mass
of the $W$\ intermediate vector boson and the Higgs boson, the latter
object being the key to our understanding of the origin of mass.  A
precision measurement of the top quark and $W$\ boson masses provide a
indirect measurement of the Higgs boson mass, and tests the
consistency of the Standard Model, and in particular the Higgs
mechanism for spontaneous symmetry breaking in the theory.  

A precision measurement of the top quark mass is therefore of critical
importance to particle physics today, and has been one of the goals of
the experiments being performed at the Fermilab Tevatron Collider
using 2 TeV proton-antiproton collisions.

This letter reports two measurements of the top quark mass in the
lepton + jets decay channel using the Collider Detector at Fermilab (CDF) 
with $\invpb{318}$ of collision data collected in \runii, between
March 2002 and August 2004.
We focus on the lepton + jets decay channel, which results from the
pair production of a top quark and anti-top quark and the subsequent
fully hadronic decay of one heavy quark and the semi-leptonic decay of
the other, as it is the largest decay channel with good
signal-to-noise, and previous work has shown that the most accurate
mass measurements are possible in this decay mode.  
We select events consistent with this decay topology, and then analyze
them two complementary ways.  The first technique uses an
event-by-event likelihood analysis employing the full matrix element
for the production and decay to extract a likelihood distribution as a
function of the true top quark mass for each event.  The likelihood
distributions are combined to determine the best estimate of the top
quark mass.
This technique, known as the ``dynamical likelihood method'' or DLM,
was developed by the CDF collaboration \cite{CDF DLM}\ and 
is similar to the method used by the \DZero\ collaboration to make the 
single most
precise measurement of the top quark mass using data from the previous
Tevatron data-taking run \cite{Abazov:2004cs,r_run2dlm}.
A second technique, based on a method developed by the CDF
collaboration \runi~\cite{Affolder:2000vy}, determines a top quark
mass from each event, and uses the distribution of these reconstructed
top quark masses compared with template distributions derived from
model calculations with differing top quark masses to estimate the
top quark mass.  This technqiue, known as the ``template method,'' has
been improved by using the fact that the $W$ boson daughters
should form a dijet invariant mass that is consistent with the precisely
know $W$ boson.  This gives an independent constraint on the dominant
systematic uncertainty in this measurement, the jet energy scale, that
allows us to improve the overall accuracy of the measurement.

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%  Overview of Experiment, data selection and contamination
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The data used in this analysis were collected by CDF during the first
thirty months of the Run II phase of the Tevatron Collider.  A total
of $\invpb{318}$\ of events were analyzed and recorded. 
The detector is a general-purpose charged and neutral particle
detector located at one of the three interaction points along the
Tevatron Collider, and is described in detail elsewhere \cite{CDF Detector}.
It consists of a solenoidal charged particle spectrometer, consisting of a
seven-layer silicon microstrip detector array and a cylindrical drift
chamber immersed in a 1.4 T magnetic field.
A segmented sampling calorimeter with scintillator tile readout measures
energy flow to a pseudorapidity $|\eta|<3.6$\ and provides electron and
photon identification.  A set of charged particle detectors outside the
calorimeter are used to identify muon candidates.
Due to the high intrinsic data rate of collisions, the experiment
employs an on-line, three-level trigger system that only selects
approximately one collision in 100,000 to record for subsequent
analysis, using criteria such as the total observed transverse energy in the
detector, the presence of jet candidate with $\et>120$\ GeV, or an electron 
or muon candidate with $\pt > \gevc{20}$.  
Events for this analysis were identified by the presence of a 
charged lepton candidate with $\pt > \gevc{20}$, with a trigger 
efficiency in excess of 90\%.

The resulting event sample, consisting of ?? electron and muon
candidates with transerve energy $\et>20$\ GeV, were 
subsequently reduced by requiring in the event the
presence of three or more energetic jets with $\et > 15$, and the
a high-energy neutrino candidate by requiring that the missing
transverse energy in the event exceed 20 GeV.  To reduce backgrounds
further, we required either a fourth jet with $\et>15$\ GeV or
that one of the jets be identified as a bottom quark through the
presence of a displaced vertex in the middle of the jet.  In the
latter case, we also required that there be at least a fourth jet in
the event with $\et>8$\ GeV.  In all cases, the jets were required to
have pseudorapidity $|\eta| < 2.0$.
This selection resulted in ?? events that, based on our measurements,
was signicantly enriched in \ttbar\ events.

The two analysis methods employ subsets of these data.  The DLM technique
uses the subset consisting of four and only four jets with $\et > 15$\
with at least one of the four jets having a displaced vertex b tag.  This
results in 63 events.  The template technique subdivides these data into
four subsamples that have different reconstructed top quark mass
distributions and background levels.  Ordered by the resulting statistical
power, the four samples are 1) events with two b tagged jets (2-tag sample
with 16 events),
events, 2) events with at least four jets with $\et > 15$\ GeV and one b
tagged jet (1-tag Tight sample with 57 events), 3) events with a fourth jet 
with $8 < \et < 15$\ GeV and one
be tagged jet (1-tag Loose sample with 25 events), and 4) events with four 
jets with $\et > 15$\ and no b tagged jets (0-tag sample with 40 events).   

Need more details on sample composition and background.  Probably a
table here.  Note $13.1 \pm 2.2$\ events in DLM sample, dominated by
mistags.  There are $1.9\pm0.5$, $10.4\pm1.7$\ and $14.3\pm2.5$\ events
background in the three template samples with a b-tag, in this case the
backgrounds being evenly split between mistags and $W b\bar{b}$\ 
and $W c\bar{c}$.

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%  Summarize DLM method
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The DLM method, described in detail in \cite{Affolder: DLM PRD}, defines a
likelihood for each event based on the differential cross section for the
event as a function of the top quark mass, taking into account the
detector resolution and combinatorial factors by introducting ``transfer
functions'' that reflect these experimental effects.  The joint likelihood
for the 63 event sample is formed by multiplying the event likelihoods
together.  We take into account the presence of background events by modelling the
effect of the background as a shift on the top quark mass and making an
appropriate correction.  This introduces a systematic uncertainty that is
constrained by our understanding of the modelling of top quark production
and decay.  An additional mapping function is employed to convert 
the top quark mass determined through this procedure to the true top quark
mass, again using our model of top quark production and decay to constrain
the uncertainties on this transformation.

The resulting likelihood function plotted versus the true top quark mass
is shown in Fig. 1, from which we infer that the maximum likelihood
estimate for the top quark mass is 
$\mtop = \gevcc{\measAStat{171.8}{2.2}{2.0}}$,
where the uncertainties are only statistical.

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%  Summarize Template method
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The template method takes each event in the sample and using a chi-squared
fitting method makes a specific parton-to-jet assignment of the expected
four quark daughters from the \ttbar\ decay, taking into account any b-tag
information and constraining the $W$\ decay daughters to form an invariant
$W$\ boson mass.  The assignment with the lowest chi-square is selected, and
the a reconstructed top mass is determined for each event.  A histogram of
the reconstructed top masses for each subsample is then created.  At the
same time, the events are analyzed in a slightly different way, removing
the $W$\ boson mass constraint and independently determining for each
event all combinations of parton-jet assignments that are consistent with
the b-tag information and determining a set of dijet invariant masses
arising from the $W$ boson decay.  This provides a second histogram for
each sample, that we then use to independently determine a jet energy
scale correction factor, JES.  We use all eight histograms and fit them
simultaneously to the true top quark mass, to JES and the expected
background levels.  We include in the
fit a constraint on JES from the jet energy calibrations done using both
in situ data and calibration information and constrain the background
rates in the 2-tag, 1-tag Tight and 1-tag Loose samples to the estimated
background rates.  The background rate in the 0-tag sample is determined
in the fit using the difference in predicted signal and background
distributions.

The four reconstructed top quark mass distributions, and the results of
the fit are shown in Fig. 2, where we also note the contributions from
background sources.  We show in Fig. 3 the distributions of the dijet
invariant mass for the four subsamples, as well as the result of the fits.
In all cases, we see excellent agreement between the observed data
distributions and the predictions.
We obtain from the fit 
$\mtop = \gevcc{\measAStat{173.5}{3.7}{3.6}}$,
where the uncertainties are only statistical and incorporates the
uncertainties on the JES measurement.
If we do not constrain JES to the nominal value of zero, we obtain 
${\rm JES} = -0.76 \pm 1.00$\ which indicates our nominal jet energy
calibrations are in good agreement with the energy scale information
provided by the $W$\ boson mass peak in the \ttbar\ decay.

%  Discuss systematic uncertainties that are common
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There are number of sources of systematic uncertainty that affect both
analyses.  These include
\begin{enumerate}
\item PDF
\item Background shape
\item Generator
\end{enumerate}
Have one table that summarizes these?

%  Discuss DLM systematics, derive final measurement
%
In addition to these sources, the DLM technique has additional
uncertainties that arise from the jet energy scale (3.0 \gevcc), the use
of the transfer functions (0.2 \gevcc).  Together with the common sources
noted above, the systematic uncertainty on the DLM mass measurement is
\gevcc{3.3}, which gives a top quark mass measurement of
$\mtop = \gevcc{\measAStat{173.8}{2.7}{2.5}{3.3}}.$

%  Discuss Template systematics, derive final measurement
%%
If we combine all the systematic uncertainties for the template technique
together, we obtain a total systematic uncertainty of \gevcc{1.7}.  This
gives a top quark mass measurement, using the template technique, of
$\mtop = \gevcc{\measAStatSyst{173.5}{3.7}{3.6}{1.7}}$.

The template technique has an additional uncertainty if one attempts to
use this method to estimate JES, namely that it would depend relatively
weakly on the same sources of systematic uncertainties noted above and on
the  actual top quark mass.  The total systematic uncertainties sum in
quadrature to 0.36, while aAssuming a top quark mass
uncertainty of \gevcc{5}\ would result in an additional systematic
uncertainty on the JES measurement of 0.36.
%
%  Conclusion
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We have made two new measurements of the top quark mass,
one using the DLM technique and resulting in a mass of
$\mtop = \gevcc{\measAStat{173.8}{2.7}{2.5}{3.3}}$.
and the other using the template technique and resulting in a mass of
$\mtop = \gevcc{\measAStatSyst{173.5}{3.7}{3.6}{1.7}}$. 
There is a large statistical correlation between these measurements,
so that we only choose to quote as a result the single most accurate
measurement, the mass obtained using the template method, assuming we 
can combine systematic uncertainties in
quadrature with the statistical uncertainties.
This provides one of the most precise single constraints on this 
important physical parameter.

Combining this measurement with the most precise Run I top quark and
$W$\ boso mass measurements
from the \DZero\ and CDF experiments, we further constrain the Higgs
boson mass to lie within ?? and ??.    The relatively low Higgs boson
mass has implications for searches for it, both at the Tevatron
Collider and the Large Hadron Collider.

These measurements constitute just part of a rich top physics program
at the TeVatron.   
As the luminosity acquired increases from the current \invpb{318}
to an expected 4000--7000 $\invpbnoarg$ for \runii,
the overall uncertainties on this mass measurement will be reduced
till we approach overall uncertainties of approximately $\gevcc{2.0}$.

\begin{acknowledgments}
We thank the Fermilab staff and the technical staffs of the
participating institutions for their vital contributions. This
work was supported by the U.S. Department of Energy and National
Science Foundation; the Italian Istituto Nazionale di Fisica
Nucleare; the Ministry of Education, Culture, Sports, Science and
Technology of Japan; the Natural Sciences and Engineering Research
Council of Canada; the National Science Council of the Republic of
China; the Swiss National Science Foundation; the A.P. Sloan
Foundation; the Bundesministerium fuer Bildung und Forschung,
Germany; the Korean Science and Engineering Foundation and the
Korean Research Foundation; the Particle Physics and Astronomy
Research Council and the Royal Society, UK; the Russian Foundation
for Basic Research; the Comision Interministerial de Ciencia y
Tecnologia, Spain; and in part by the European Community's Human
Potential Programme under contract HPRN-CT-20002, Probe for New
Physics.
\end{acknowledgments}
