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\def \mgev   {GeV/$c^{2}$}
\def \pgev   {GeV/$c$}
\def \meff    {\varepsilon}
\def \eff    {$\varepsilon$}
\def \stimes {\vec{\times}}
\def \et     {$E_{T}$}
\def \etx     {$E_{Tx}$}
\def \etm     {E_{T}}
\def \pt     {$P_{T}$}
\def \ete    {$E_{T}^{e}$}
\def \ptm    {$P_{T}^{\mu}$}
\def \miset  {$\,/\!\!\!\!E_{T}$}
\def \bsbbar  {$b/\bar{b}$}
\def \tstbar  {$t/\bar{t}$}
\def \ttbar  {$t\bar{t}$}
\def \ppbar  {$p\bar{p}$}
\def \emu    {$e\mu$}
\def \bbbar  {$b\bar{b}$}
\def \sp     {$\:$}
\newcommand{\figst}{\begin{figure}[p]}
\newcommand{\figed}{\end{figure}}

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\begin{document}

\LARGE

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\begin{small}
CDF/ANAL/TOP/CDFR/3189\\
\today\\
V2.0\\
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\begin{center}
\begin{large}
{\bf B-Jet Energy Correction for Top Mass Determination (II) \\
- as a Function of Top Mass -}\\
\vskip 0.2 in


        S. Aota, T. Kikuchi, S. Kim and K. Kondo \\
	
University of Tsukuba\\
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%\date{}

	We derived energy correction for b-jets in 
\ttbar\ events  
using HERWIG+QFL \ttbar\ production simulation data for top masses of
140, 160, 170, 180, 200,
220 and 240 
\mgev\ to determine
top quark mass with top candidates events accurately.

	The b-jet energy correction was obtained for 
dilepton +$\ge$2 jet events. We call this correction DLJSCO.

	The magnitude of the correction depends on top quark mass.
We parametrized
the correction coefficients as a quadratic function of top quark mass.
Using this parametrization, we obtained the top-mass-dependent b-jet energy
correction function.

\newpage
\section{ Reconstruction of simulated jets}
\hspace{\parindent}
	In order to obtain the deviation of b-jet
\pt\ from b-quark \pt, the following procedure was 
performed:
\begin{enumerate}
\item	Find \bsbbar\ quarks originating from \tstbar\ quark decay.
\item	We use only \bsbbar\ quark with \pt\ $>$ 5 \pgev.
\item   Find a b-jet from JETS banks with a jet cone size of 0.4  in 
$\eta-\phi$ plane as follows:
	    pick up all jets within a cone radius of 0.4 around a 
	    \bsbbar\ quark.
	    Apply the jet energy correction of 
QDJSCO\footnote{In this note, QDJSCO means JTC90S and
is always applied for a cone size of 0.4,
with underlying event energy correction and with out-of-cone correction.}
 to the jets.
	    Compare the ratio of b-quark \pt\ to jet \pt\ between the 
	    above jets.
	    We identify a jet with the ratio closest to 1 as a b-jet.
\item   Find soft lepton( e or $\mu$ ) with \pt\ $>$ 2 \pgev\ within a 
	    cone radius
	    of 0.4 around a \bsbbar\ quark.  
	    If the b-jet has 
	    at least one soft lepton, the b-jet is tagged with soft lepton.
\item   In the case of soft muon tagging, after we apply QDJSCO to the jet,
we deal with the jet in two ways:  
\begin{enumerate}
\item	    we do not apply any muon momentum correction.
\item	    we subtract muon energy deposit from the jet energy, and add
		    muon momentum to the jet energy,
	where muon energy deposit is 2.2 GeV which is the average energy
	    deposit in calorimeter by a minimum ionizing particle.
\end{enumerate}
\end{enumerate}

\newpage
\section{ Jet energy correction factors}
\hspace{\parindent}
	After the procedure of jet reconstruction described above, 
we obtained the 
ratio of b-quark \pt\ to b-jet \pt\ which we call R hereafter: 
$ R = P_{T}^{H}/P_{T}^{Q} $, where$P_{T}^{H}$ is b-quark \pt\ and 
$P_{T}^{Q}$ is the jet \pt\ corrected by QDJSCO.
We made the distributions of R for 4 \pgev\ bins of  $P_{T}^{Q}$
with $P_{T}^{Q}$ $\geq$ 12 \pgev.  
 The average value of R and its error
are plotted as a function of the b-jet \pt\ for b jets
\begin{enumerate}
\item	without lepton tagging ( generic b jets ), 
\item	with soft electron tagging, 
\item	with soft muon tagging and jet \pt\ not including muon momentum,
\item	with soft muon tagging and jet \pt\ including muon momentum .
\end{enumerate}

	For each plot, we fitted it with a function 
$Y = exp(p1 + p2\cdot$$P_{T}^{Q}) + p3$ in the same way as AAJSCO was 
obtained~\cite{AAJSCO}.

\section{ DLJSCO }
\hspace{\parindent}
	This Monte Carlo data was generated with
HERWIG, and simulated with QFL. We made 50,000  \ttbar\ $\to$ dilepton 
+ $\geq$ 2 jets
events for each of top masses of 140, 160, 170, 180, 200, 220 and
240 \mgev.
  Then we applied the following selection cuts of the top candidates:

\begin{itemize}
\item	Number of leptons $\geq$ 2 with \ete\ $>$ 20 GeV 
and \ptm\ $>$ 20 \pgev,
\item	\miset\ $>$ 25 GeV, 
\end{itemize}

	The distribution of b-quark \pt\ is shown for these simulation data in 
Fig.~\ref{exfig4}.
 We plotted the average value of R and its error
as a function of the b-jet \pt\ 
and fitted it with a function 
$Y = exp(p1 + p2\cdot$$P_{T}^{Q}) + p3$ as shown in
Figs.~\ref{fig5} and ~\ref{fig6}. 
The absolute magnitude of the correction is 35\% at b-jet \pt\ of 20 \pgev\
and less than 5\% at b-jet \pt\ above 40 \pgev\ for b-jet without lepton
tagging for a top mass of 170 \mgev. 
 We can obtain an overall
correction by making the R distribution accumulated over 
all b-jet \pt\ as shown in
Fig~\ref{exfig6}. The mean and RMS of R are 1.020$\pm$0.002 and 0.262, 
respectively, for generic b jets.

	We compared DLJSCO for a top mass of 170 \mgev\ with DLJSCO for 
top masses
of 140, 160, 180, 200, 220 and 240 \mgev\ 
in order to check the dependence of correction on
top mass.  
as shown in 
Fig.~\ref{fig8}.  The difference in DLJSCO errors are also shown in 
Fig.~\ref{fig9}.
We parametrized the correction coefficients $p1, p2$ and $p3$ as a quadratic
function of top mass, A + B$\cdot$$M_{top}$ + C$\cdot$$M_{top}^{2}$,
  to get the top-mass-dependent correction. 
The correction and its error
thus obtained are consistent with
 the original DLJSCO for top masses of 140, 160, 180, 200, 220 and 240 \mgev\ 
within an error of 2\% as shown in Figs.~\ref{fig12} and ~\ref{fig13}.

\newpage
\section{ Conclusions }
\hspace{\parindent}
	Energy correction DLJSCO for b-jets in \ttbar\ $\to$ dilepton events 
was derived
using HERWIG+QFL \ttbar\ production simulation data.
The absolute magnitude of the energy 
correction is 35\% at b-jet \pt\ of 20 \pgev\
and less than 5\% at b-jet \pt\ above 40 \pgev\ for b-jet without lepton
tagging for a top mass of 170 \mgev.	

	As we have a large dependence of DLJSCO 
on a top mass between 140 \mgev\ and 240 \mgev, 
we made the DLJSCO correction as a quadratic function of top mass.
The correction and its error
thus obtained were found to be consistent with
 the original DLJSCO for top masses of 140, 160, 170,
180, 200, 220 and 240 \mgev\ 
within an error of 2\%.

\newpage
\begin{thebibliography}{99}
\bibitem{QDJSCO} 
	J. Tonnison	CDF/ANAL/JET/CDFR/2189\\
	J. Tonnison	CDF/ANAL/JET/CDFR/2053\\
	L. Keeble and B. Flaugher CDF/ANAL/JET/CDFR/1513\\
	'Inclusive Jet Cross Section Measurement' F.Abe et al., Phys. Rev.
Lett. 68, 1104 (1992)\\
	T. Hessing, Ph.D. Thesis, DOE-ER40039-47, December 1990\\
\bibitem{AAJSCO} 
	A. Caner et al. CDF/ANAL/TOP/CDFR/2469
\bibitem{DLJSCO} 
	S. Kim	         CDF/ANAL/TOP/CDFR/2254\\
	S. Kim, S. Aota, K. Kondo and T. Uchida  CDF/ANAL/TOP/CDFR/2662
\end{thebibliography}

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