%
%  Modified for JES measurement   Pekka March 2005
%
\section{Systematic Uncertainties}
\label{sec:sys}

Systematic uncertainties arise from uncertainties in our understanding of 
the detector response, and in the assumptions employed to infer a top
quark mass from the observed data.
The magnitudes of such uncertainties
are estimated using auxiliary data collected for this purpose, and
large samples of Monte Carlo simulated events that allow us to
estimate the sensitivity of the measurements to reasonable variations in 
analysis assumptions.

For each source of systematic uncertainty, the relevant quantities
or parameters are varied by $\pm 1 \sigma$, and new \gevcc{178}~\ttbar
signal and background Monte Carlo templates are produced by performing
event selection and mass reconstruction on the modified samples.
Events for pseudoexperiments (see section~\ref{ssec:methodCheck})
are taken from these new templates, but
the signal and background p.d.f.'s used in the analysis remain unchanged. 
The shift
in the median fitted top quark mass for a large ensemble of pseudoexperiments
is taken as the systematic uncertainty associated with a given assumption
or effect.
When the uncertainty on a given systematic shift due to the statistics
of the Monte Carlo sample is larger than the shift itself, the
uncertainty is used as the systematic uncertainty.

\subsection{Systematic Uncertainties Arising from the JES Calibration}
\label{ssec:jesSyst}

The use of the observed $W$\ boson mass to constrain the jet energy scale 
calibration essentially measures the average energy response of light
quark jets arising from the decay of the colorless $W$ boson.  
However, the top quark mass also depends on the energy response to $b$\ quark 
jets.  This introduces three possible sources of uncertainty:  i) uncertainties
in energy response arising from uncertainties in the decay properties of
bottom quarks, ii) uncertainties arising from possible variations in the 
fragmentation properties of bottom quarks, and iii) uncertainties in energy response arising from the 
different color flow associated with bottom quark jets produced in top
quark decay. 

We varied the bottom quark semi-leptonic branching fractions in our 
Monte Carlo models
to understand the effect of this uncertainty in the overall energy scale of
the bottom quark jet.   We found that this introduced an additional uncertainty
in the bottom quark jet energy scale of 0.4\%, resulting in an uncertainty
in the extraction of the top quark mass of \gevcc{0.4}.  
We used the high-statistics measurements of bottom quark fragmentation 
observed in $Z\rightarrow\bbbar$ decays at the LEP and SLC colliders to 
constrain the possible fragmentation models in our Monte Carlo calculations.
We found that this variation introduced an additional top quark mass 
uncertainty of \gevcc{0.4}.  In order to test the effects of possible 
variations in energy response due to different models of ``color flow'' in 
the top quark production and decay, we varied the parameters of the 
algorithms used to generate this color flow in both HERWIG and Pythia and
conservatively estimated that this could result in an uncertainty in the 
bottom quark jet energy scale of 0.3\%.  This results in an additional 
uncertainty in the top quark mass of \gevcc{0.3}.

We add these three contributions in quadrature and include an 
additional \gevcc{0.6} systematic uncertainty in the top quark mass
arising from the modeling of the bottom quark jets.

In this analysis, the jet energy scale is assumed to have the same
value for all jets in all events. However, the jet energy systematics have
contributions from many sources, and those component uncertainties
in general have different dependence on, for example, jet $\pt$ and
$\eta$, or on the event environment. 
To estimate the uncertainty arising from the assumption of
a monolithic jet energy scale, we produce samples in which the
components of the jet energy systematics are shifted independently and
in various combinations. The typical shift in the top quark mass
measurement is \gevcc{0.5}, which is taken as the largest part of
a ``method'' systematic. This systematic also includes the offset of
\sigunit{\measErr{-0.07}{0.02}}, which translates to \gevcc{0.3},
observed in the pull distributions of \fig{f:pulls}.

Finally, the effect of the jet energy scale uncertainty on background
events must be treated separately, since it is not included in the
background template parameterization. We find a small uncertainty
on \mtop, \gevcc{0.04}. However, this uncertainty is comparable in
effect to the previous two since it is added linearly
to the part of the uncertainty on \mtop from the likelihood
that is due to the jet energy scale (see Section~\ref{ssec:resultsCombined}).

\subsection{ISR/FSR/PDF Systematic Errors}
\label{ssec:isrFsrPdfSyst}

The systematic uncertainties due to initial state radiation, final
state radiation, and parton density functions are summarized in
this section.
%\tab{t:isrfsrpdfsyst}, and detailed below.

 Extra jets originating from the incoming partons and outgoing partons
 affect the measurement of \mtop when they are misidentified as
 jets from the final state partons or change the kinematics of the
 final state partons. ISR and FSR are controlled
 by the same DGLAP evolution equation that tells us the probability
 for a parton to branch~\cite{Dokshitzer:1977sg, Gribov:1972rt,
 Gribov:1972ri, Lipatov:1974qm, Altarelli:1977zs}. ISR is studied
 using Drell-Yan events in dilepton channels. The advantage 
 of Drell-Yan events is that there is no FSR, and they are produced
 by the \qqbar annihilation process, as are most ($\sim 85\%$)
 \ttbar pairs. 

 The level of ISR is measured as a function of the Drell-Yan mass
 scale and shows a logarithmic dependence on the Drell-Yan
 mass squared, as shown in \fig{f:isrplot}.
 By extrapolation, the ISR effect is then estimated
 at top pair production energies. Based on this measurement,
 two ISR systematic Monte Carlo samples ($+1\sigma_{ISR}$ and  $-1\sigma_{ISR}$)
 are produced using Pythia, by varying the value of $\Lambda_{QCD}$ and the
 $K$ factor to the transverse momentum scale for ISR showering.
 The parameters used are
 $\Lambda_{QCD}\text{(5 flavors)}=\mev{292}$, $K=0.5$ for $+1\sigma_{ISR}$
 and $\Lambda_{QCD}\text{(5 flavors)}=\mev{73}$, $K=2.0$ for $-1\sigma_{ISR}$.
 The corresponding curves of Drell-Yan dilepton $\left<\pt\right>$ vs
 invariant mass squared are shown in \fig{f:isrplot}.
 Although ISR is also sensitive to the choice of parton distribution
 function (PDF),
 the PDF uncertainty is not included as a part of the ISR uncertainty.
 Because a PDF change affects not only ISR but also hard scattering
 kinematics, the PDF uncertainty is treated separately.
 The largest top quark mass shift between default Pythia and the
 two ISR samples, \gevcc{0.4}, is taken as the ISR uncertainty.

 Since ISR and FSR shower algorithms are the same, the same variations
 in $\Lambda_{QCD}$ and $K$ are used to generate FSR systematic samples
 by varying a set of parameters specific to FSR modeling.
 The largest top quark mass shift between default Pythia and the two FSR samples,
 \gevcc{0.6}, is used as the FSR uncertainty.
\note{Doug: Should ISR and FSR systs be added linearly?}

\begin{cfigure}
\includegraphics{plots/pt_evol_gen_corr_v4_prd}
\caption[Figure supporting ISR uncertainty procedure.]
{The average $\pt$ of the dilepton system, which corresponds
to the level of ISR activity, shows a
logarithmic dependence on the dilepton invariant mass
$M_{ll}^2$. The data are compared with the predictions
of Pythia 6.2 and of the $+1\sigma_{ISR}$ and $-1\sigma_{ISR}$
samples.}
\label{f:isrplot}
\end{cfigure}

 The calculation of the top quark invariant mass does not depend
 directly on the choice of input PDF. However, changing the PDF changes
 the top quark $\eta$ and $\pt$ distributions as well as the size of ISR.
 This results in a change in the jet $\pt$ distributions and in the
 probability of selecting the correct jets, both of which affect the
 reconstructed top quark mass.
 
 To examine the systematic effect due to PDF uncertainties, 20 pairs of
 uncertainty sets based on CTEQ6M are used~\cite{Pumplin:2002vw,Stump:2003yu}.
 These PDFs provide ``$\pm 1 \sigma$'' variations for 20 independent
 eigenvectors, but do not include variation in  $\Lambda_{QCD}$.
 In addition, the MRST group~\cite{Martin:1999ww} provides PDFs with
 different assumptions for the value of $\Lambda_{QCD}$.
 The difference between the measured top quark mass using the
 MRST72 ($\Lambda_{QCD}=$228 MeV) and
 MRST75 ($\Lambda_{QCD}=$300 MeV) PDFs
 is taken as an uncertainty,
 as is the difference between leading order PDFs CTEQ5L and MRST72.
 Instead of 43 different, fully simulated sets of events, a single
 simulated sample is used, and mass templates are generated for the
 different PDF sets by weighting events according to the probability
 of observing their incoming partons using each PDF set. This technique
 also removes most of the uncertainty due to limited Monte Carlo statistics.
 A symmetrized uncertainty for each of the 20 pairs of CTEQ6 PDFs
 (determined by varying one eigenvector at a time)
 is added in quadrature to get one part of the PDF uncerainty.
 An additional systematic error of \gevcc{0.22} comes from the variation
 of $\Lambda_{QCD}$. This is consistent with the much less precise
 estimate using fully simulated samples. The total PDF uncertainty is \gevcc{0.3}.
\note{CTEQ vs MRST is $.08\pm.03$. Eigenvectors is .20. Want to include this detail?}

%%UK
  In order to check the sensitivity of the top quark mass measurement
  to a very different top quark \pt distribution,
  we have used a signal Monte Carlo sample with resonant \ttbar production,
  where the resonance occurs at \gevcc{700} and then top quarks decay according
  to the Standard Model. The measured top quark mass is shifted
  by only \gevcc{1.5}, demonstrating that this kinematic top mass fitter
  is nearly insensitive to the \pt of the top quark.

%%UK
\note{Need NLO systematic!}

\subsection{Other Systematic Errors}
\label{ssec:otherSyst}

The remaining sources of systematic uncertainty are described in this section.
%\Tab{t:othersyst} gives the summary.

% generator
  The difference in the top quark mass between HERWIG and Pythia samples
  is $\gevcc{\measErr{0.2}{0.2}}$. To be conservative, this
  difference is taken as another systematic uncertainty,
  although the differences
  in ISR and FSR between the two generators are already taken into
  account in the ISR and FSR uncertainties, and fragmentation
  effects are accounted for in the jet energy uncertainties.

% background shape
 The largest uncertainty in the shape of the reconstructed mass templates
 for background events is due to the uncertainty in the $Q^2$ scale
 that is used for the calculation of the hard scattering
 and for the shower evolution. Different background shapes
 are obtained for four different $Q^2$ scales ($4M_W^2$, $M_W^2$, $M_W^2/4$,
 and $M_W^2 + P_{TW}^2$) using ALPGEN MC samples.
 An ALPGEN $W\bbbar$ + 2 parton Monte Carlo sample is used for the
 tagged events and a $W$ + 4 parton sample for the 0-tag events.
 Half of the largest difference in top quark mass from pseudoexperiments
 using these samples is used as the systematic uncertainty, \gevcc{0.5},
 which is dominated by the uncertainty in the 0-tag sample.
 \note{Get gen5 number!}
 Smaller contributions to the background shape uncertainty are estimated by
 performing sets of pseudoexperiments in which background events are
 drawn not from the combined background template but from templates for
 one of the individual background processes, or from the templates
 derived from QCD-enriched data. Half of the largest difference observed
 in these pseudoexperiments is \gevcc{0.4} for the different background
 processes, and \gevcc{0.4} for the different models of the QCD background.
 Both of these are taken as additional systematic uncertainties on the top
 quark mass due to background shape modeling.
 \note{Want a sentence about correlations in bg estimates?}
 % Although the likelihood neglects correlations between the background
 % normalization uncertainties in the different subsamples, the effect
 % on the reported mass and error has been shown to be negligible.
 
%b tagging
 Different $b$-tagging efficiency in data and simulation can
 introduce a bias in the top quark mass measurement. The $\etjet$
 dependence of the $b$ tagging in data and simulation is consistent
 with being the same. But if a slope is introduced in the relative
 tagging efficiency as a function of $\etjet$, consistent at \sigunit{1}
 with the measured efficiencies, the shift in the top quark mass is
 \gevcc{0.1}, which is taken as a systematic uncertainty.
  
%MC statistics
 The analysis can have a systematic bias due to the finite statistics
 of Monte Carlo samples that are used to obtain the signal and
 background shape parameterizations. For a rough estimate of this
 uncertainty, sets of pseudoexperiments are performed with a series of
 fluctuated signal and background templates; in each fluctuated
 template, each bin is varied randomly according to Poisson statistics.
 For each fluctuated template, the median top quark mass measured by
 pseudoexperiments is shifted. The typical shift due to these
 statistical fluctuations, taken as a systematic uncertainty due
 to Monte Carlo statistics, is \gevcc{0.3}.

\note{Need a systematic due to chisq cut efficiency uncertainty?}

\subsection{Jet Systematic Errors}
\label{ssec:jetSystOnMtop}

The systematics on jet energy measurements are described in detail in
Section~\ref{ssec:jetSyst}. The primary analysis fits for the jet
energy scale, and the error from the likelihood fit includes a
contribution due to these systematics. For the \mtop-only fits,
however, this systematic uncertainty must be estimated
independently.

To determine the systematic uncertainty on the top quark mass measurement
given the various sources of uncertainty on the jet energy measurements,
the mass shifts for $+1\sigma$ and $-1\sigma$ perturbations in the jet
energies are extracted, and a symmetric uncertainty for each source
is defined as half the difference between the two shifts.
\Tab{t:jetenergysyst} lists the uncertainties obtained for
the \mtop-only measurement. The total systematic
uncertainty in the top quark mass due to jet energy measurements is
\gevcc{3.1} for the combined measurement. The corresponding
systematic uncertainties for an independent measurement in each subsample
are listed for comparison.

\begin{table}
\caption[Systematic uncertainties due to jet energy systematics.]
{The uncertainties on the \mtop-only top quark mass measurement are shown
for each jet energy systematic error. Estimates are
obtained for the independent subsamples as well as for the combined
measurement.
\note{Make sure the labels stay consistent with Section~\ref{sec:jetCorrSyst}.}
}
\label{t:jetenergysyst}
\begin{ruledtabular}
\begin{tabular}{ddddd}
\multicolumn{5}{l}{\textbf{Jet energy systematic}} \\
\multicolumn{5}{c}{$\Delta\mtop$ (\gevccnoarg)} \\
%\multicolumn{4}{c}{Subsamples} & \multicolumn{1}{c}{Combined} \\
\multicolumn{1}{c}{\twotag} & \multicolumn{1}{c}{\onetagt}
& \multicolumn{1}{c}{\onetagl} & \multicolumn{1}{c}{\zerotag}
%& \multicolumn{1}{c}{Measurement} \\
& \multicolumn{1}{c}{Combined} \\
\hline
\multicolumn{5}{l}{\textbf{Response relative to central}} \\
0.6 & 0.6 & 0.6 & 1.0 & 0.6 \\
\hline
\multicolumn{5}{l}{\textbf{Modeling of hadron jets (absolute scale)}} \\
2.0 & 2.3 & 2.0 & 2.7 & 2.2 \\
\hline
\multicolumn{5}{l}{\textbf{Modeling of parton showers (out-of-cone)}} \\
2.2 & 2.2 & 1.9 & 1.9 & 2.1 \\
\hline\hline
\multicolumn{5}{l}{\textbf{Total systematic due to jet energies}} \\
3.0 & 3.2 & 2.8 & 3.4 & 3.1 \\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Total Systematic Uncertainty}
\label{ssec:totalSyst}

The systematic uncertainties for the combined fit are
listed in \tab{t:uncertainties}.
The total systematic uncertainty is estimated to be \gevcc{1.5},
exclusive of the uncertainty due to jet energy scale that is
included in the likelihood error.
Also shown in \tab{t:uncertainties} are the systematic uncertainties
on the \jes measurement (\sigunit{0.36} total),
and the systematic uncertainties on \mtop for the \mtop-only
measurement without (\gevcc{3.4} total)
and with (\gevcc{3.3} total) JPB tags.

\begin{table}
\caption[Systematic uncertainty summary.]
{This table summarizes all systematic uncertainties for the
combined analysis and two alternate fits.
\note{Make sure to change BG number to gen5, all analyses.}
}
\label{t:uncertainties}
\begin{ruledtabular}
\begin{tabular}{ldddd}
Method &
\multicolumn{2}{c}{Primary} &
\multicolumn{1}{c}{\mtop-only} &
\multicolumn{1}{c}{\mtop-only} \\
\multicolumn{4}{c}{} &
\multicolumn{1}{c}{+ JPB} \\
&
\multicolumn{1}{c}{$\Delta\mtop$} &
\multicolumn{1}{c}{$\Delta\jes$} &
\multicolumn{1}{c}{$\Delta\mtop$} &
\multicolumn{1}{c}{$\Delta\mtop$} \\
&
\multicolumn{1}{c}{(\gevccnoarg)} &
\multicolumn{1}{c}{($\sigma$)} &
\multicolumn{1}{c}{(\gevccnoarg)} &
\multicolumn{1}{c}{(\gevccnoarg)} \\
\hline
Jet Energy 		  & \multicolumn{1}{c}{N/A} & \multicolumn{1}{c}{N/A}  & 3.1 & 3.0 \\
$b$-jet Energy  & 0.6 & 0.25 & 0.6 & 0.6 \\
Method          & 0.5 & 0.02 & \multicolumn{1}{c}{N/A} & \multicolumn{1}{c}{N/A} \\
ISR 			      & 0.4 & 0.08 & 0.4 & 0.3 \\
FSR 			      & 0.6 & 0.06 & 0.4 & 0.6 \\
PDFs 			      & 0.3 & 0.04 & 0.4 & 0.4 \\
Generators 		  & 0.2 & 0.15 & 0.3 & 0.2 \\
Bkgd Shape      & 0.8 & 0.17 & 0.8 & 0.8 \\
$b$ tagging 		& 0.1 & 0.01 & 0.2 & 0.3 \\
MC stats        & 0.3 & 0.05 & 0.4 & 0.4 \\
\hline
Total 			    & 1.5 & 0.36 & 3.4 & 3.3 \\
\end{tabular}
\end{ruledtabular}
\end{table}
