%
%  Summarize DLM method
%
%  Revisions:
%
%   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    Reduce length even further.
%   30 Aug 05    PKS    Implement changes for Draft 2.
%

The DLM technique, described in detail in \cite{r_Run2DLM}, defines a
likelihood for each event based on the
differential cross section per unit phase space volume of the final state
partons, $d\sigma_{\ttbar}/d\Phi$, 
as a function of \mtop.
Detector resolution effects are accounted for using \ttbar\ events 
generated by the HERWIG Monte Carlo program \cite{r_herwig}\ and full 
detector simulation to derive transfer functions (TF). 
The TF relates the transverse energies of the quarks,
denoted by $\bm{x}$, and the observed jets.
The parton kinematics of the \ttbar\ final state are 
constructed in the following way:  We first
generate a random value for the virtual mass squared of the $W$ boson
in the leptonic channel, $s_{W}$, according to the Breit-Wigner form.
We identify the momentum of the electron or muon daughter with the measured
value, and the neutrino transverse momentum with the measured
missing transverse energy.  We then
generate random values for the momenta of final state quarks
according to the TF probabilities.
We determine the $z$ component of the neutrino momentum, with a two-fold
ambiguity, using $s_W$ as a constraint.
Thus, for a given set of $\bm{x}$ and $s_{W}$, we fully determine
the event kinematics, and the event likelihood as a function of \mtop is given by
\begin{equation}
\bm L(\mtop) = N \sum_{I_{j}} \sum_{I_{\nu}}
                 \frac{d\sigma_{\ttbar}}{d\Phi}
                 (\mtop; \bm{x}, s_W),
\end{equation}
where the normalization factor, $N$, is independent of \mtop for a given
event, and the indices $I_j$ and $I_{\nu}$ run over the parton-jet
assignments and the two neutrino solutions, respectively.
The event likelihood is obtained by numerically integrating
over $\bm{x}$ given by the TF and $s_W$ given by the Breit-Wigner
distribution.

\Fig{f:DLMMaxLikelihood} shows the distribution of the top quark
mass value at the point of maximum likelihood in each event
compared with the expectation from simulated events.
An inset shows the joint log-likelihood as a function of \mtop,
formed by multiplying the likelihoods of the individual events together.
We account for the presence of background events by evaluating
the small shift of $+\gevcc{1.4}$\ they make in the measured top quark mass. 
From the joint likelihood we infer 
$\mtop = \gevcc{\measAStat{173.2}{2.6}{2.4}}$,
where the uncertainty is only statistical.
The systematic uncertainty due to the jet energy scale is estimated
as the shift in \mtop arising from a \sigcunit{1} change in \jes,
and is \gevcc{3.0}.

\begin{cfigure}
\includegraphics{plots/prl_figure1_v2}
\caption[DLM max-likelihood.]
{The value of the top quark mass at the maximum of the DLM
likelihood is plotted for each event. Data events (points)
are compared
to an expected distribution (histogram) comprising simulated \ttbar
(\mtop=\gevcc{172.5}) and background events.  The last point is an
overflow bin.  The inset shows the 
joint log likelihood for the 63 events, before accounting for the presence
of background.}
\label{f:DLMMaxLikelihood}
\end{cfigure}
