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%\newpage

\section{Dataset \label{sec:dataset}}

\subsection{\label{sub:datasample}\boldmath Data Sample}

\noindent For this analysis the framework used was vjets\_cafe v04-00-08 (Release p21.18.00)
and the data set consisted of 3JET skim produced by the commom samples group
\cite{3jet_data} and recorded between August 2002 and May 2010 (runs 151817 - 258547).


\begin{itemize}
\item   CSG$\_$CAF$\_3$JET$\_$PASS2$\_$p21.10.00
\item   CSG$\_$CAF$\_3$JET$\_$PASS4$\_$p21.10.00$\_$p20.12.00
\item   CSG$\_$CAF$\_3$JET$\_$PASS4$\_$p21.10.00$\_$p20.12.01
\item   CSG$\_$CAF$\_3$JET$\_$PASS4$\_$p21.10.00$\_$p20.12.02
\item   CSG$\_$CAF$\_3$JET$\_$PASS4$\_$p21.10.00$\_$p20.12.04
\item   CSG$\_$CAF$\_3$JET$\_$PASS4$\_$p21.12.00$\_$p20.12.05$\_$allfix
\item   CSG$\_$CAF$\_3$JET$\_$PASS4$\_$p21.10.00$\_$p20.16.07$\_$fix
\item   CSG$\_$CAF$\_3$JET$\_$PASS4$\_$p21.12.00$\_$p20.16.07$\_$summer2010
\end{itemize}



In this analysis we chose the three jets trigger JT2\_3JT15L\_IP\_VX.


This particular trigger was chosen based on our needs of looking for events with multiple jets and
the fact that it represents a gain of 20\% efficiency on signal selection if compared to previous p17 analysis.
Since the efficiencies for such trigger are not currently part of caf\_trigger package, 
in this analysis we benefit from the trigger modelling provided by the $hbb$ group \cite{bIDH_note}
for the $\phi b \rightarrow b\bar{b}b$ analysis. Trigger weight distributions for all
MC samples used in the analysis as a function of the number of b-tagged jets  are found in 
%Appendix \ref{app:trig_eff} and summarized in Table \ref{trig_weight}:
Appendix \ref{app:trig_eff} and Table II summarizes their mean values:



\begin{table}[h]
\begin{center}
\begin{tabular}{|c|r|r|r|r|} \hline
Process   & 0 tags  & 1 tag & 2 tags & 3 or more tags      \\ \hline

\hline


$t\overline{t}\rightarrow\tau+jets$ &\multicolumn{1}{c|}{0.7923}  &\multicolumn{1}{c|}{0.8620} &\multicolumn{1}{c|}{0.8953} &\multicolumn{1}{c|}{0.9039} \\

$t\overline{t}\rightarrow e+jets$ &\multicolumn{1}{c|}{0.7902}  &\multicolumn{1}{c|}{0.8599} &\multicolumn{1}{c|}{0.8933}&\multicolumn{1}{c|}{0.9020} \\

$t\overline{t}\rightarrow\mu+jets$ &\multicolumn{1}{c|}{0.7942}  &\multicolumn{1}{c|}{0.8639} &\multicolumn{1}{c|}{0.8973}&\multicolumn{1}{c|}{0.9058} \\

$t\overline{t}\rightarrow l+l$ &\multicolumn{1}{c|}{0.7274}  &\multicolumn{1}{c|}{0.7915} &\multicolumn{1}{c|}{0.8223} &\multicolumn{1}{c|}{0.8302} \\

$Wjj+jets\rightarrow$ $l\nu+jj+jets$ &\multicolumn{1}{c|}{0.5821}  &\multicolumn{1}{c|}{0.6337} &\multicolumn{1}{c|}{0.6586}&\multicolumn{1}{c|}{0.6652} \\

$Wbb+jets\rightarrow$ $l\nu+bb+jets$ &\multicolumn{1}{c|}{0.5948}  &\multicolumn{1}{c|}{0.6475}&\multicolumn{1}{c|}{0.6729}&\multicolumn{1}{c|}{0.6796} \\

$Wcc+jets\rightarrow$ $l\nu+cc+jets$ &\multicolumn{1}{c|}{0.5912}  &\multicolumn{1}{c|}{0.6435}&\multicolumn{1}{c|}{0.6687}&\multicolumn{1}{c|}{0.6754} \\

$Zjj+jets\rightarrow$ $ee+jj+jets$ &\multicolumn{1}{c|}{0.6769}  &\multicolumn{1}{c|}{0.7363}&\multicolumn{1}{c|}{0.7646}&\multicolumn{1}{c|}{0.7719} \\

$Zbb+jets\rightarrow$ $ee+bb+jets$ &\multicolumn{1}{c|}{0.4331}  &\multicolumn{1}{c|}{0.4712}&\multicolumn{1}{c|}{0.4895}&\multicolumn{1}{c|}{0.4943} \\

$Zcc+jets\rightarrow$ $ee+cc+jets$ &\multicolumn{1}{c|}{0.6167} &\multicolumn{1}{c|}{0.6746}&\multicolumn{1}{c|}{0.7035}&\multicolumn{1}{c|}{0.7127} \\

$Zjj+jets\rightarrow$ $\mu\mu+jj+jets$ &\multicolumn{1}{c|}{0.6641}  &\multicolumn{1}{c|}{0.7233}&\multicolumn{1}{c|}{0.7520}&\multicolumn{1}{c|}{0.7598} \\

$Zbb+jets\rightarrow$ $\mu\mu+bb+jets$ &\multicolumn{1}{c|}{0.6057}  &\multicolumn{1}{c|}{0.6598}&\multicolumn{1}{c|}{0.6860}&\multicolumn{1}{c|}{0.6931} \\

$Zcc+jets\rightarrow$ $\mu\mu+cc+jets$ &\multicolumn{1}{c|}{0.5817}  &\multicolumn{1}{c|}{0.6335}&\multicolumn{1}{c|}{0.6585}&\multicolumn{1}{c|}{0.6653} \\

$Zjj+jets\rightarrow$ $\tau\tau+jj+jets$ &\multicolumn{1}{c|}{0.5712} &\multicolumn{1}{c|}{0.6220}&\multicolumn{1}{c|}{0.6465}&\multicolumn{1}{c|}{0.6530} \\

$Zbb+jets\rightarrow$ $\tau\tau+bb+jets$ &\multicolumn{1}{c|}{0.6049}  &\multicolumn{1}{c|}{0.6586}&\multicolumn{1}{c|}{0.6845}&\multicolumn{1}{c|}{0.6914} \\

$Zcc+jets\rightarrow$ $\tau\tau+cc+jets$ &\multicolumn{1}{c|}{0.5889}  &\multicolumn{1}{c|}{0.6410}&\multicolumn{1}{c|}{0.6661}&\multicolumn{1}{c|}{0.6727} \\

$Zjj+jets\rightarrow$ $\nu\nu+jj+jets$ &\multicolumn{1}{c|}{0.5739}&\multicolumn{1}{c|}{0.6241}&\multicolumn{1}{c|}{0.6480}&\multicolumn{1}{c|}{0.6541} \\

$Zbb+jets\rightarrow$ $\nu\nu+bb+jets$ &\multicolumn{1}{c|}{0.6012} &\multicolumn{1}{c|}{0.6539}&\multicolumn{1}{c|}{0.6790}&\multicolumn{1}{c|}{0.6854} \\

$Zcc+jets\rightarrow$ $\nu\nu+cc+jets$ &\multicolumn{1}{c|}{0.6360}&\multicolumn{1}{c|}{0.6914}&\multicolumn{1}{c|}{0.7177}&\multicolumn{1}{c|}{0.7242} \\ \hline
\end{tabular}
\caption{Mean values of the trigger weight for all MC samples.}
\end{center}
\label{trig_weight} 
\end{table}


For this trigger we also measured the luminosity of our
data sample. Table \ref{lumi} shows the results for both v15 and v16 trigger versions.

%\newpage
%
\begin{table}[h]
%\begin{ruledtabular}
\begin{tabular}{|crrrr|}
%\begin{center}
\hline 
Trigger version&
Trigger name&
Delivered $\mathcal{L}$ ($pb^{-1})$& 
Recorded $\mathcal{L}$ ($pb^{-1})$& 
Reconstructed $\mathcal{L}$ ($pb^{-1})$
\tabularnewline
\hline
\hline 
V15.0 - V15.99& JT2\_3JT15L\_IP\_VX&            1682.08&  1544.71&  1385.99
\tabularnewline
V16.0 - V16.99& JT2\_3JT15L\_IP\_VX&            4059.92&  3887.95&  3565.86
\tabularnewline
\hline
T O T A L&         &              5742.00& 5432.66&  4951.85
\tabularnewline
\hline
%\end{center}
\end{tabular}
%\end{ruledtabular}
\caption{The results of luminosity calculation for the Run2b 3JET data skim for different D0 trigger list versions}
\label{lumi} 
\end{table}

\newpage

\subsection{\label{sub:background}\boldmath Backgrounds}

In this analysis the largest background sources are QCD ({}``fake
$\tau$''), which is estimated from data and  $W/Z$+jets, which are simulated Monte Carlo samples.
Other backgrounds that were not included in this analysis due to their small contribution are single top and diboson production.
A list of backgrounds sources is found in Section III of \cite{p17_note}.
In the following sections we describe both signal and background simulation.


\subsection{\label{sub:mcsample}\boldmath Monte Carlo Samples}

\noindent We use p20 certified MC samples as produced by CSG and caffed with p21.11.00 (version3) \cite{3jet_mc}. 
All $W/Z$ and $t\bar{t}$ were 
generated with ALPGEN v2.11 \cite{alpgen} interfaced with Pythia v6.409 \cite{pythia} 
for production of parton-level showers and hadronization.
EvtGen \cite{evtgen} is used to model b hadrons decays and TAUOLA \cite{tauola} used to model tau leptons decays.


\noindent ALPGEN is a leading order (LL) generator. In order to correct it to match with 
next-to-leading order (NLO) cross sections we apply {\it correction factors}  and then provide
a correct normalization. These correction factors were taken from {\tt vjets$\_$cafe} framework and are described 
in Ref.\cite{kfactor}. There are two kinds of correction factors: {\it k-factors}, which
are the result of the ratio between NLO and LL cross sections ($\sigma_{NLO}/\sigma_{LL}$) and
{\it heavy flavor factors}, which are in turn the ratio between k-factors for $HF+0lp(incl)$
and $2lp(incl)$ process from MCFM \cite{mcfm}. Here $HF$ denotes $Z + bb$, $Z + cc$, $W + bb$ or $W + cc$ and $lp$ stands
for {\it light parton}. Heavy flavor factors are applied on the top of k-factors in order to provide the correct
normalization for process where heavy quarks are present.
For $Z$ production, samples are split 
into $Z$ + light jets, $Z + bb$ and $Z + cc$. $Z$ + light parton
cross sections are multiplied by a k-factor of 1.3, while $Z + bb$ and $Z+ cc$ are multiplied by additional
heavy flavor factors of 1.52 and 1.67 respectively.
$W$ + jets samples are also split the same way: $W$ + light jets, $W + bb$ and $W + cc$. In $W$ + light jets
case a k-factor of 1.3 is applied while an additional heavy flavor factor of 1.47 is applied to 
both $W + bb$ and $W + cc$ samples.
%Table \ref{kxsec} summarizes factors applied.

Table IV summarizes the correction factors applied.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|r|} \hline
Process   & k-factor        \\ \hline

\hline


W + light partons     &  \multicolumn{1}{c|}{1.3}   \\ \hline


W + bb      &  \multicolumn{1}{c|}{1.3$\times$1.47} \\ \hline


W + cc     &  \multicolumn{1}{c|}{1.3$\times$1.47}   \\ \hline


Z + light partons     &  \multicolumn{1}{c|}{1.3}   \\ \hline


Z + bb      &  \multicolumn{1}{c|}{1.3$\times$1.52} \\ \hline


Z + cc     &  \multicolumn{1}{c|}{1.3$\times$1.67}   \\ \hline

\end{tabular}
\caption{k-factors for MC.}
\end{center}
\label{kxsec} 
\end{table}

All MC samples used in this analysis are shown in Table \ref{used_mc} with theirs respective cross sections 
and number of events. The cross sections shown are the averages of the cross-sections of
each set of MC process generated and are calculated from /caf$\_$mc$\_$util/mc$\_$sample$\_$info/MC.list

\clearpage

\begin{table}[h]
%\begin{center}
\begin{tabular}{|crr|}
\hline
Sample & $\sigma(pb)$ & \# of Events \\ \hline
$t+t+0lp-l\nu+2b+2lpc\_\mbox{excl}\_m172$  & 1.392196 &  $ 793267 $ \\ 
$t+t+1lp-l\nu+2b+3lpc\_\mbox{excl}\_m172$ & .576927 &  $ 456317 $  \\
$t+t+2lp-l\nu+2b+4lpc\_\mbox{incl}\_m172$ & .281831 &  $ 277912 $   \\
$\mbox{W}+0lp\rightarrow lnu+0lp\_\mbox{excl}$ & 4530.269741 &  $ 47070044 $ \\
$\mbox{W}+1lp\rightarrow lnu+1lp\_\mbox{excl}$  & 1283.094130 &  $ 20683540 $ \\ 
$\mbox{W}+2lp\rightarrow lnu+2lp\_\mbox{excl}$ & 306.073315 &  $ 19686862 $  \\ 
$\mbox{W}+3lp\rightarrow lnu+3lp\_\mbox{excl}$ & 73.494491 &  $ 4269023 $  \\ 
$\mbox{W}+4lp\rightarrow lnu+4lp\_\mbox{excl}$ & 16.958254 &  $ 3084707 $\\ 
$\mbox{W}+5lp\rightarrow lnu+5lp\_\mbox{incl}$ & 5.218917 &  $ 2565942 $ \\ 
$\mbox{W}+2b+0lp\rightarrow l\nu+2b+0lp\_\mbox{excl}$  & 9.315458 &  $ 1120570 $ \\
$\mbox{W}+2b+1lp\rightarrow l\nu+2b+1lp\_\mbox{excl}$ & 4.288365 &  $ 812095 $ \\
$\mbox{W}+2b+2lp\rightarrow l\nu+2b+2lp\_\mbox{excl}$ & 1.554786 &  $ 563315 $ \\ 
$\mbox{W}+2b+3lp\rightarrow l\nu+2b+3lp\_\mbox{incl}$ & 0.716175& $ 464475 $ \\
$\mbox{W}+2b+0lp\rightarrow l\nu+2c+0lp\_\mbox{excl}$ & 24.404153 &  $ 934253 $ \\
$\mbox{W}+2b+1lp\rightarrow l\nu+2c+1lp\_\mbox{excl}$ & 13.486806 &  $ 738709 $ \\
$\mbox{W}+2b+2lp\rightarrow l\nu+2c+2lp\_\mbox{excl}$ & 5.459005 &  $ 554236 $ \\
$\mbox{W}+2b+3lp\rightarrow l\nu+2c+3lp\_\mbox{incl}$ & 2.526973 &  $ 469900 $ \\
$\gamma \mbox{Z}+0lp\rightarrow ee+0lp\_\mbox{excl}\_75\_130$ & 132.086811 &  $ 1212214 $ \\
$\gamma \mbox{Z}+1lp\rightarrow ee+1lp\_\mbox{excl}\_75\_130$ & 40.060963 &  $ 599588 $ \\
$\gamma \mbox{Z}+2lp\rightarrow ee+2lp\_\mbox{excl}\_75\_130$ & 9.981935 &  $ 298494 $ \\
$\gamma \mbox{Z}+3lp\rightarrow ee+3lp\_\mbox{incl}\_75\_130$ & 3.297072 &  $ 150267 $ \\
$\gamma \mbox{Z}+2b+0lp\rightarrow ee+2b+0lp\_\mbox{excl}\_75\_130$ & 0.400826 &  $ 200121 $ \\
$\gamma \mbox{Z}+2b+1lp\rightarrow ee+2b+1lp\_\mbox{excl}\_75\_130$ & 0.173438 &  $ 97474 $ \\
$\gamma \mbox{Z}+2b+2lp\rightarrow ee+2b+2lp\_\mbox{incl}\_75\_130$ & 0.107248 &  $ 48269 $ \\
$\gamma \mbox{Z}+2c+0lp\rightarrow ee+2c+0lp\_\mbox{excl}\_75\_130$ & 0.900923 &  $ 182485 $ \\
$\gamma \mbox{Z}+2c+1lp\rightarrow ee+2c+1lp\_\mbox{excl}\_75\_130$ & 0.506337 &  $ 89293 $ \\
$\gamma \mbox{Z}+2b+2lp\rightarrow ee+2b+2lp\_\mbox{incl}\_75\_130$ & 0.285871 &  $ 47357 $ \\
$\gamma \mbox{Z}+0lp\rightarrow \mu \mu+0lp\_\mbox{excl}\_75\_130$ & 133.850906 &  $ 1553222 $ \\
$\gamma \mbox{Z}+1lp\rightarrow \mu \mu+1lp\_\mbox{excl}\_75\_130$ & 41.677185 &  $ 639392 $ \\
$\gamma \mbox{Z}+2lp\rightarrow \mu \mu+2lp\_\mbox{excl}\_75\_130$ & 9.822132 &  $ 446737 $ \\
$\gamma \mbox{Z}+3lp\rightarrow \mu \mu+3lp\_\mbox{incl}\_75\_130$ & 3.195801 &  $ 172628 $ \\
$\gamma \mbox{Z}+2b+0lp\rightarrow \mu \mu+2b+0lp\_\mbox{excl}\_75\_130$ & 0.424239 &  $ 210139 $ \\
$\gamma \mbox{Z}+2b+1lp\rightarrow \mu \mu+2b+1lp\_\mbox{excl}\_75\_130$ & 0.195271 &  $ 101055 $ \\
$\gamma \mbox{Z}+2b+2lp\rightarrow \mu \mu+2b+2lp\_\mbox{incl}\_75\_130$ & 0.099004 &  $ 49600 $ \\
$\gamma \mbox{Z}+2c+0lp\rightarrow \mu \mu+2c+0lp\_\mbox{excl}\_75\_130$ & 0.932203 &  $ 193928 $ \\
$\gamma \mbox{Z}+2c+1lp\rightarrow \mu \mu+2c+1lp\_\mbox{excl}\_75\_130$ & 0.548182 &  $ 92744 $ \\
$\gamma \mbox{Z}+2b+2lp\rightarrow \mu \mu+2b+2lp\_\mbox{incl}\_75\_130$ & 0.280795 &  $ 51277 $ \\
$\gamma \mbox{Z}+0lp\rightarrow \tau \tau+0lp\_\mbox{excl}\_75\_130$ & 131.564780 &  $ 1556389 $ \\
$\gamma \mbox{Z}+1lp\rightarrow \tau \tau+1lp\_\mbox{excl}\_75\_130$ & 40.300291 &  $ 595169 $ \\
$\gamma \mbox{Z}+2lp\rightarrow \tau \tau+2lp\_\mbox{excl}\_75\_130$ & 10.072067 &  $ 305312 $ \\
$\gamma \mbox{Z}+3lp\rightarrow \tau \tau+3lp\_\mbox{excl}\_75\_130$ & 3.089442 &  $ 205365 $ \\
$\gamma \mbox{Z}+2b+0lp\rightarrow \tau \tau+2b+0lp\_\mbox{excl}\_75\_130$ & 0.423679 &  $ 196943 $ \\
$\gamma \mbox{Z}+2b+1lp\rightarrow \tau \tau+2b+1lp\_\mbox{excl}\_75\_130$ & 0.196527 &  $ 103105 $ \\
$\gamma \mbox{Z}+2b+2lp\rightarrow \tau \tau+2b+2lp\_\mbox{incl}\_75\_130$ & 0.103561 &  $ 48476 $ \\
$\gamma \mbox{Z}+2c+0lp\rightarrow \tau \tau+2c+0lp\_\mbox{excl}\_75\_130$ & 0.898135 &  $ 260243 $ \\
$\gamma \mbox{Z}+2c+1lp\rightarrow \tau \tau+2c+1lp\_\mbox{excl}\_75\_130$ & 0.487548 &  $ 100802 $ \\
$\gamma \mbox{Z}+2b+2lp\rightarrow \tau \tau+2b+2lp\_\mbox{incl}\_75\_130$ & 0.297808 &  $ 50711 $ \\ 
$\gamma \mbox{Z}+0lp\rightarrow \nu \nu+0lp\_\mbox{excl}$ & 806.552968 &  $ 2368495 $ \\
$\gamma \mbox{Z}+1lp\rightarrow \nu \nu+1lp\_\mbox{excl}$ & 244.651772 &  $ 2591505 $ \\
$\gamma \mbox{Z}+2lp\rightarrow \nu \nu+2lp\_\mbox{excl}$ & 61.014112 &  $ 657110 $ \\
$\gamma \mbox{Z}+3lp\rightarrow \nu \nu+3lp\_\mbox{excl}$ & 14.091090 &  $ 194705 $ \\
$\gamma \mbox{Z}+4lp\rightarrow \nu \nu+4lp\_\mbox{excl}$ & 3.277295 &  $ 100158 $ \\
$\gamma \mbox{Z}+5lp\rightarrow \nu \nu+5lp\_\mbox{incl}$ & 0.936465 &  $ 49660 $ \\
$\gamma \mbox{Z}+2b+0lp\rightarrow \nu \nu+2b+0lp\_\mbox{excl}$ & 2.562976 &  $ 375572$ \\
$\gamma \mbox{Z}+2b+1lp\rightarrow \nu \nu+2b+1lp\_\mbox{excl}$ & 1.143703 &  $ 180558 $ \\
$\gamma \mbox{Z}+2b+2lp\rightarrow \nu \nu+2b+2lp\_\mbox{incl}$ & 0.617265 &  $ 91588 $ \\
$\gamma \mbox{Z}+2c+0lp\rightarrow \nu \nu+2c+0lp\_\mbox{excl}$ & 5.634504 &  $ 376456 $ \\
$\gamma \mbox{Z}+2c+1lp\rightarrow \nu \nu+2c+1lp\_\mbox{excl}$ & 3.002712 &  $ 199012 $ \\
$\gamma \mbox{Z}+2b+2lp\rightarrow \nu \nu+2b+2lp\_\mbox{incl}$ & 1.635746 &  $ 96147 $ \\\hline

\end{tabular}
\caption{MC Samples. Here $l$ stands for the three lepton flavor ($e$, $\mu$ and $\tau$). $\tau$ decays are not restricted.}  
%\end{center}
\label{used_mc}
\end{table}

\clearpage

\subsection{\label{sub:mcsample_xseccorr}\boldmath MC samples corrections}

Standard D\O\ corrections are applied to MC in order to obtain a better MC-data agreement \cite{top_sys}.


\noindent {\bf Trigger efficiency}: an additional scale factor (weight) is applied to MC to account for the trigger efficiency in data. 
Further details are given in Section \ref{sec:trig_param}.

\noindent {\bf Luminosity reweighting}: in order to reproduce luminosity effects from real data, simulated samples are overlaid to Zero Bias data
Due to a difference in intantaneous luminosity between the overlay and real data, the luminosity profile of all
MC samples is reweighted to match the luminosity profile in data \cite{lumireweight}.

\noindent {\bf Primary vertex reweighting}: $z$ vertex distributions are different between data and MC.This difference is corrected by reweighting
MC $z$ vertex distributions using the reweight processor from the {\tt caf\_mc\_util} package \cite{PVz_re}.

\noindent {\bf $W$ and $Z$ $p_{T}$ reweighting}: for both $W$ + jets and $Z$ + jets, the $p_{T}$ distribution from MC samples is reweighted to match 
the equivalent distribution in data, accordingly to the standard way \cite{WZPt_re}.

\noindent {\bf b fragmentation}: the systematics on the reweight of the b-fragmentation function from the default in Pythia
to the value tuned to reproduce collider data was assumed to be the symmetrized difference between
the AOD and SLD tunes \cite{bfrag}.

\noindent {\bf Jet Shifting Smearing and Removing (JSSR)}: due to differences in energy scale, resolution, reconstruction and identification
between data and MC, MC jets are shifted, smeared and possibly removed using standard JSSR 
processor \cite{jssr}. In this analysis shifting is turned off to signal $t\bar{t}$
and on to $W/Z$ + jets samples.


\noindent {\bf Tau Energy Scale (TES)}: due to the analysis sensitivity to any difference between data and MC
in the energy scale of taus decaying hadronically we apply a $E/p$ correction to this energy scale
as described in \cite{tes}.

%\subsubsection{\label{sub:hadtau_corr}\boldmath Hadronic $\tau$ corrections}

%\begin{itemize}
%\item E/p correction for hadronic tau relative tau energy scale from package {\tt caf\_util}.
%\item NN corrections from package {\tt tauid\_eff}.
%\item Track corrections from package {\tt muid\_eff}.
%\end{itemize}



%\clearpage