%
%  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
measure 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 $\gevcc{175}~\ttbar$
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 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{Jet Systematic Errors}
\label{ssec:jetSyst}

There are significant uncertainties on many aspects of the measurement
of jet energies. Some of these are in the form of uncertainties
on the energy measurements themselves; some are uncertainties on
the detector simulation, which is used to derive many corrections,
and ultimately to extract the top mass; still others are best
understood as uncertainties on jet production and fragmentation
models used in the generators.
The jet energy systematic errors, as well as the procedure to evaluate
the resulting systematic uncertainty on the top mass measurement,
are described in this section.

The uncertainties discussed in this subsection are relevant to the 
top mass measurement made using 
only the jet energy scale calibration determined independently from the
top quark mass sample.  They are not relevant to the {\it in situ}\ jet energy
scale determination, which are described in the following section.
 
\begin{description}

\item[Response relative to central]
%The smoothing of the detector response as a function of $\eta$ is
%tested by photon-jet balancing in data and simulation.  This
%uncertainty takes
%into account residual effects of spline interpolation, biases in the
%dijet balancing procedure, especially in the crack region, variation
%of plug calorimeter response with time, differences seen in the
%photon-jet balancing, and half the difference between the corrections
%for data and simulation.  \Tab{t:jetrelative} summarizes the
%uncertainty that depends on $\eta$.
%\note{Mel: expand this para a bit.}
The systematic uncertainties in the calorimeter response relative to
the central calorimeter are summarized in \tab{t:jetrelative}. The
uncertainties for data account for the residual $\eta$ dependence
after dijet balancing, biases in the dijet balancing procedure,
especially near the uninstrumented regions, and the variation of
the plug calorimeter response with time. Half the difference
between the relative response in data jets and simulated jets is
taken as a systematic uncertainty for the simulated jets.
Photon-jet balancing is used to check the $\eta$ dependence after
corrections in data and simulated events (see \fig{f:gammaJet}).

\begin{table}
\caption[$\eta$-dependent jet energy systematics.]
{$\eta$-dependent systematic uncertainties on jet energies are
shown for data and simulated events.}
\label{t:jetrelative}
\begin{ruledtabular}
\begin{tabular}{ccc}
$\eta$ range & \multicolumn{2}{c}{Jet energy uncertainty} \\
 & Data & Simulation \\
\hline
$|\eta| < 0.2$       & 3\% & 1\% \\
$0.2 < |\eta| < 0.6$ & 0.5\% & 1\% \\
$0.6 < |\eta| < 1.0$ & 2\% & 1\% \\
$1.0 < |\eta| < 1.4$ & 4\% & 7\% \\
$1.4 < |\eta| < 2.0$ & 2\% & 6\% \\
$|\eta| > 2$         & 7\% & 7\% \\
\end{tabular}
\end{ruledtabular}
\end{table}

\item[Central Calorimeter Response]
A systematic uncertainty is associated with the
single-particle response of the central calorimeter.
The difference between \runi and \runii energy scales is studied
using photon-jet balancing and is measured to be 5\%, at least
$2\%$ of which comes from a difference in the CEM scale definition
and additional material in the \runii tracking volume, which results
in a softer particle spectrum due to external brehmsstrahlung and
conversions. The remaining 3\% difference is taken as a systematic
uncertainty on the data jet energies.

A separate comparison of the photon-jet balancing in data and
simulation (see \fig{f:gammaJet})
finds a maximal difference of 5\% between the two.
Subtracting the data uncertainty of 3\% in quadrature yields an
independent uncertainty of 4\% for the simulation energy scale.

Finally, a 1\% uncertainty for the stability of the central
calorimeter response is assigned.

\begin{cfigure}
\ifx\pdfoutput\undefined
\includegraphics[bb=92 398 239 732,clip,angle=270,width=.9\columnwidth]{plots/gammaJet}
\else
\includegraphics[viewport=92 398 239 732,clip,angle=270,width=.9\columnwidth]{plots/gammaJet}
\fi
\caption[$\gamma$-jet balancing vs $\eta$ after all corrections.]
{The balance factor $(\ptjet-\ptgam)/\ptgam$ is shown
as a function of $\eta$ for $\gamma$-jet events in data (red
circles) and simulation (blue triangles). The jet momenta
are corrected for all known effects.}
\label{f:gammaJet}
\end{cfigure}

\item[Correction to Hadrons]
This systematic error is due to uncertainties in the modeling
of jet showering and hadronization, which are tested by
varying relevant parameters
in the generator. Since the calorimeter response is non-linear,
different fragmentation can affect the overall jet energy scale.
This uncertainty only accounts for variations that affect the
energy inside the jet cone.

\item[Correction to Partons]
The uncertainty on the fraction of energy contained in the jet cone
(also due to jet fragmentation modeling) is estimated in two parts,
one between $R=0.4$ and $R=1.0$ and the other between $R=1.0$ and
$R=\infty$.

\end{description}

To determine the systematic uncertainty on the top quark mass measurement given
the various sources defined above, the mass shifts for $+1\sigma$
and $-1\sigma$ perturbations in the jet energies are extracted
as described above, 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 top mass measurement.  The total systematic
uncertainty in the top mass due to jet energy measurements is
$\gevcc{6.8}$ 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 top mass measurement are shown
for each jet energy systematic error. Estimates are
obtained for the independent subsamples as well as for the combined
measurement.
\robustnote{We do not have all these numbers for the currently blessed
analysis. Only the bottom right number is real, others are
guesses for now.}}
\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}} \\
2.5 & 3.0 & 2.8 & 3.5 & 3.0 \\
\hline
\multicolumn{5}{l}{\textbf{Central calorimeter response}} \\
4.0 & 5.0 & 4.7 & 6.0 & 5.0 \\
\hline
\multicolumn{5}{l}{\textbf{Corrections to hadrons (absolute scale)}} \\
2.0 & 2.2 & 2.1 & 2.5 & 2.2 \\
\hline
\multicolumn{5}{l}{\textbf{Corrections to partons (out-of-cone)}} \\
2.0 & 2.2 & 2.1 & 2.5 & 2.2 \\
\hline\hline
\multicolumn{5}{l}{\textbf{Total systematic due to jet energies}} \\
5.8 & 6.3 & 6.5 & 8.3 & 6.8 \\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Systematic Uncertainties Arising from the JES Calibration}
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 colourless $W$\ boson.  
However, the top 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 colour 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 $Z\rightarrow b\bar{b}$\ decays at the LEP Collider 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 ``colour flow'' in 
the top quark production and decay, we varied the parameters of the 
algorithms used to generate this colour 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 modelling of the bottom quark jets.

\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
\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 \ttbar pairs (85\% by \qqbar). 

 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 $\Lambda_{QCD}=\mev{384}$, $K=0.5$ for $+1\sigma_{ISR}$
 and $\Lambda_{QCD}=\mev{100}$, $K=2.0$ 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.
 Since ISR and FSR shower algorithms are the same, the same variations
 in $\Lambda_{QCD}$ and $K$ are used to generate FSR systematic samples.
 The largest top mass shift between default Pythia and the
 two ISR samples, $\gevcc{1.2}$, is taken as the ISR uncertainty.
 Using the same method, $\gevcc{1.2}$ is determined as the FSR uncertainty as well.

\begin{cfigure}
\includegraphics{plots/ISRPlot}
\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 in principle
 depend on the choice of input PDF, as long as the jets from final state
 \ttbar partons are used. However, changing the PDF changes
 the top quark $\eta$ and $\pt$ distributions as well as the size of ISR.
 The result is a change in the probability of selecting the correct jets,
 which affects the reconstructed invariant mass of the top quark.
 
 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  $\Lambda_{QCD}$.
 The difference between MRST72 ($\Lambda_{QCD}=$228 MeV) and
 MRST75 ($\Lambda_{QCD}=$300 MeV) is included as an addtional uncertainty.
 Instead of 42 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 the uncertainty due to limited Monte Carlo statistics.
 Half of the difference in top mass for each of the 20 pairs of CTEQ6 PDFs
 (varying one eigenvector at a time)
 is added in quadrature for the PDF uncerainty, with an additional systematic
 error of $\gevcc{0.076}$ from the variation of $\Lambda_{QCD}$.
 The total PDF uncertainty is $\gevcc{0.4}$.

\begin{table}
\caption[Systematic uncertainties due to ISR/FSR/PDF.]
{The systematic uncertainties on the top mass measurement are shown
corresponding to initial and final state radiation, and the choice
of parton density functions. Estimates are obtained for the
independent subsamples as well as for the combined measurement.
\robustnote{We do not have all these numbers for the currently blessed
analysis. Only the numbers in the right column are real, others are
guesses for now.}}
\label{t:isrfsrpdfsyst}
\begin{ruledtabular}
\begin{tabular}{ddddd}
\multicolumn{5}{l}{\textbf{Systematic source}} \\
\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{Initial state radiation}} \\
1.2 & 1.2 & 1.2 & 1.2 & 1.2 \\
\hline
\multicolumn{5}{l}{\textbf{Final state radiation}} \\
1.2 & 1.2 & 1.2 & 1.2 & 1.2 \\
\hline
\multicolumn{5}{l}{\textbf{Parton density functions}} \\
0.4 & 0.4 & 0.4 & 0.4 & 0.4 \\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Systematic Uncertainties Arising from the $W$\ Boson Mass}
The use of the observed $W$\ boson mass to constrain the jet energy scale 
calibration assumes that 

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

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

% generator
  The difference in the top mass between Herwig and Pythia samples
  is $\gevcc{\measErr{0.1}{0.3}}$. To be conservative, this
  difference is taken as another systematic uncertainty
  ($\gevcc{0.3}$), 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 uncertainty in the shape of the background top mass template is 
 dominated by 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 $W\bbbar$ + 1 parton Monte Carlo sample is used
 for the tagged events
 and a $W$ + 3 parton sample for the 0-tag events.
 Half of the largest difference in top mass is used as the systematic
 uncertainty, $\gevcc{1.0}$, which is dominated by the uncertainty
 in the 0-tag sample.

%jet resolution
 Although uncertainties in the jet energy scale have been accounted
 for above, there may also be inconsistencies in the energy resolution
 for data jets and simulated jets. In fact, photon-jet balancing
 studies show that simulated jets are roughly 15\% narrower than
 jets in the data. When jets in the simulated \ttbar sample are
 smeared by an extra 15\%, the shift in the top mass is $\gevcc{0.5}$,
 which is taken as a systematic uncertainty due to jet resolution.

%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. The effect of $\pm1\sigma$ in the $\et$
 dependence was found to be only $\gevcc{0.1}$ in
 \runi~\cite{Affolder:2000vy}. The same uncertainty is used here.

%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 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 $\gevcc{0.3}$.

\begin{table}
\caption[Systematic uncertainties due to ``other.'']
{The remaining systematic uncertainties on the top mass measurement
are shown. Estimates are obtained for the
independent subsamples as well as for the combined measurement.
\robustnote{We do not have all these numbers for the currently blessed
analysis. Only the numbers in the right column are real, others are
guesses for now.}}
\label{t:othersyst}
\begin{ruledtabular}
\begin{tabular}{ddddd}
\multicolumn{5}{l}{\textbf{Systematic source}} \\
\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{Background template shape}} \\
0.1 & 0.3 & 0.2 & 1.2 & 1.0 \\
\hline
\multicolumn{5}{l}{\textbf{Tagging efficiency}} \\
0.2 & 0.1 & 0.1 & \multicolumn{1}{c}{---} & 0.1 \\
\hline
%\multicolumn{5}{l}{\textbf{Top quark $\pt$}} \\
%0.4 & 0.4 & 0.4 & 0.4 & 0.4 \\
%\hline
\multicolumn{5}{l}{\textbf{Generators}} \\
1.2 & 1.2 & 1.2 & 1.2 & 1.2 \\
\end{tabular}
\end{ruledtabular}
\end{table}

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

The systematic uncertainties for the combined measurement
are summarized in \tab{t:uncertainties}.  The total
systematic uncertainty is estimated to be $\gevcc{7.1}$.

\begin{table}
\caption[Systematic uncertainty summary.]
{This table summarizes all systematic uncertainties for the combined analysis.}
%The numbers for the combined analysis are given.}
\label{t:uncertainties}
\begin{ruledtabular}
\begin{tabular}{ld}
Source of Systematics & \multicolumn{1}{c}{$\Delta\mtop$ ($\gevccnoarg$)} \\
\hline
Jet Energy 		& 6.8 \\
ISR 			    & 1.2 \\
FSR 			    & 1.2 \\
PDFs 			    & 0.4 \\
Generators 		& 0.3 \\
Background Shape 	& 1.0 \\
Jet Resolution & 0.5 \\
%Top $\pt$  & 0.6 \\
$b$ tagging 		& 0.1 \\ 
MC statistics & 0.3 \\
\hline
Total 			& 7.1 \\
\end{tabular}
\end{ruledtabular}
\end{table}
