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\begin{document}

\title{Separating $\Upsilon(4\mbox{S})$ decays from continuum events
using a neural network}

\author{Fabrizio Salvatore and Glen Cowan}

\address{Physics Dept., Royal Holloway, University of London,
Egham, Surrey, TW20 0EX}


\begin{abstract}
We report preliminary results on separating 
$q\overline{q}$ continuum background events where $q = u,d,s$ or $c$
from $b\bar{b}$ signal events produced in $\mbox{e}^+\mbox{e}^-$ 
interactions at the $\Upsilon(4\mbox{S})$ resonance.  The analysis uses
an Artificial Neural Network (NN) and is based on a Monte Carlo 
simulation without consideration of detector effects.  It was found 
that for a signal efficiency of 90\%, only $\sim16\%$ of the background 
events survive the NN cut.
\end{abstract}

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\section{Introduction}

Artificial Neural Networks (NN) have been applied to a variety of problems 
in high energy physics in order to discriminate between different 
classes of events.  Here the technique is used to separate 
$b\bar{b}$ events from $\Upsilon(4\mbox{S})$ decays
(`signal') from the continuum background for 
$\mbox{e}^+\mbox{e}^-$ interactions at a centre-of-mass energy
$E_{\rm cm} = 10.58$ GeV, i.e., at the $\Upsilon(4\mbox{S})$ resonance. 
The analysis relies on the different topologies for
signal and background at a B Factory: while $b\bar{b}$ events are 
more `spherical', $uds$ and $c\bar{c}$ events are more `jet-like'. 
This is due to the fact that in the $\Upsilon(4\mbox{S})$ rest frame, the 
momenta of the produced B mesons are small and they decay
isotropically.  On the other hand, events where light quarks 
($u\overline{u}$, $d\overline{d}$ or $s\overline{s}$) 
are produced are characterized by a preferred direction with hadron
jets following roughly the quark and antiquark. A similar 
jet-like structure is also present in $c\overline{c}$ events, 
although there it is less pronounced.

For this study, two samples of 600k events of the type
$\mbox{e}^+\mbox{e}^- \rightarrow q\overline{q}$ have been generated using  
PYTHIA \cite{pythia} all at a centre-of-mass energy equal to 
the mass of the $\Upsilon(4\mbox{S})$, where the quark flavour $q$ is 
a mixture of $u$, $d$ and $s$ in the first set
and $c$ in the second.  A third data sample of the same size consisted
of $\Upsilon(4\mbox{S})$ decays.  The analysis has been performed at the 
level of the generated particles and no attempt has been made at this
stage to simulate detector effects.

\section{The shape variables}
\label{sec:shape}

The variables used to discriminate jet-like from isotropic events are 
described below.  Further information on the variables can be
found in \cite{babar, aleph}.

\begin{itemize}

\item[(a)] The ratio {\bf $R_2 = H_2 / H_0$} of the 2nd to the 0th 
Fox-Wolfram moments.  Neglecting particle 
masses, 4-momentum conservation requires that $H_0 = 1$. For a two-jet 
event, $H_1=0$ and $H_l \sim 1$ for $l$ even and $H_l \sim 0$ for $l$ odd. 
The $R_2$ distribution is shown in Fig.~\ref{fig:fig1}, 
where the different behaviour for signal and background events can be 
observed.

\item[(b)] The thrust {\bf $T = \sum_i |\hat{T}\cdot{\bf p_i}| / 
\sum_i |{\bf p_i}|$}, where $\hat{T}$ is the thrust axis of the event 
\cite{babar}.  For an isotropic event one has {\bf $T=0.5$} while 
{\bf $T\sim 1$} for a highly directional one, as shown in Figs.~\ref{fig:fig2}
(a) and (c).

\end{itemize}

\setlength{\unitlength}{1.0 cm}
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\begin{figure}[htbp]
\begin{picture}(40.0,6.)
\put(3.5,-2.5)
{\special{psfile=r2.ps
hscale=38 vscale=38}}
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\caption{\footnotesize $R_2$ distribution for (a) $\Upsilon(4\mbox{S})$ 
signal and (b) continuum 
events.}
\label{fig:fig1}
\end{figure}
\renewcommand{\baselinestretch}{1}
\footnotesize\normalsize

The variables defined above are quite powerful for identifying events 
where the two-jet topology is well defined and clearly distinguishable 
from the isotropic case.  The signal to background separation 
decreases, however, when one or more gluons are emitted, which is more 
likely to happen when light quarks are produced. In this case there 
are three or more jets, so that the event shape resembles more that
of the signal.

To discriminate further between signal and background, the Durham 
clustering algorithm \cite{durham_alg} is applied to group particles 
into jets.  For every pair of particles $i$ and $j$, the `distance' 
$y_{ij}$ is computed as

\begin{equation}
y_{ij} = \frac{2~\mbox{min}(E_i^2,E_j^2)~(1- \cos \theta_{ij})}{E_{\rm vis}^2}
\,,
\label{eqn:eq1}
\end{equation}

\noindent where $E_i$ and $E_j$ are the particles' energies, 
$\theta_{ij}$ is their opening angle and $E_{\rm vis}$ is the total 
(visible) energy in the event.  The pair with the smallest $y_{ij}$
is replaced by a pseudo-particle with four-momentum 
$p^{\mu} = p^{\mu}_{i} + p^{\mu}_{j}$. The procedure is repeated 
until $N$ jets  have been found.  In our case $N=3$ or 4 and the 
smallest $y_{ij}$ values are called $y_{3}$ and $y_{4}$, respectively.
Once the clustering has been done, the following discriminating 
variables can be added to the list:

\begin{itemize}

\item[(c)] {\bf $y_{3}$}, {\bf $y_{4}$};

\item[(d)] {\bf $\theta_{\rm BZ}$}, the angle between the highest and 
second highest energy jet;

\item[(e)] {\bf $\theta_{\rm KS}$}, the angle between the lowest and 
second lowest energy jet;

\item[(f)] {\bf $\theta_{\rm NR}$}, the angle between the planes defined 
by the highest and lowest energy jets and by the other two jets;

\item[(g)] {\bf $\theta_{34}$}, the angle between the planes defined 
by the highest and second highest jets and by the other two jets;

\item[(h)] the QCD four-jet matrix element squared, {\bf $|M_{QCD}|^2$}, 
averaged over all possible assignments of the jets to the final-state partons
\cite{ERT}.

\end{itemize}

As an example, the distribution of $\ln|M_{QCD}|^2$ is shown in 
Fig.~\ref{fig:fig2}(b) and (d) for signal and background events.

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\begin{picture}(40.0,8.2)
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{\special{psfile=thrust_qcd.ps
hscale=50 vscale=50}}
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\caption{\footnotesize Distribution of (a) thrust and (b) 
$\ln |M_{QCD}|^2$ (right) for signal events; (c) and (d) show the same
for continuum events.}
\label{fig:fig2}
\end{figure}
\renewcommand{\baselinestretch}{1}
\footnotesize\normalsize

\section{The analysis results}
\label{sec:res}

All the variables defined in the previous section have been combined in a 
Neural Network using the package JETNET 3.51 \cite{jetnet}. The network is a
3 layered feed-forward one, with 9 input nodes, 12 hidden nodes and 
one output.  The network has been trained using 100k $b\bar{b}$, 
100k $uds$ and 100k $c\bar{c}$ events. Each input variable has been 
scaled linearly so that it lies in the range [-1,1];  in this way at 
the start of the training no single variable dominates the inputs 
to any neuron.  The training algorithm is `error back-propagation'.

After the training, the NN has been used with an independent sample of 
events, 500k for each type, and the output distributions are shown in 
Fig.~\ref{fig:fig3}. The signal and background curves are very well 
separated, with continuum background events accumulating around 0 and 
signal events around 1. 

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hscale=52 vscale=52}}
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\caption{\footnotesize Distributions of the NN output for 
(a) signal and (b) background events. The distributions 
are superimposed in (c) after being normalized to unity.}
\label{fig:fig3}
\end{figure}
\renewcommand{\baselinestretch}{1}
\footnotesize\normalsize

To evaluate the rejection power of the NN we can set a cut on the output
such that the efficiency for the signal is 90\%; this corresponds
to requiring an output greater than $0.35$. The fraction of background 
events surviving the cut is 14\% for $uds$ events and 18\% for 
$c\bar{c}$ events.

\section{Conclusions}

A feed-forward neural network has been applied to separate $q\overline{q}$
continuum events from $\Upsilon(4\mbox{S})$ decays produced in 
simulated $\mbox{e}^+\mbox{e}^-$ interactions at the $\Upsilon(4S)$ 
resonance.  Preliminary results are quite encouraging. For a signal 
efficiency of 90\%, only $\sim 16\%$ of background events survive the 
NN cut.  More work needs to be done to verify that the results are still 
valid after a full detector simulation; nevertheless, the NN technique seems 
a promising option for this kind of analysis.

%\section*{Acknowledgements}
%
%We would like to thank the organizers of the Durham workshop,
%especially Nicolo de Groot, Mike Whalley and Adrian
%Signer, for an enjoyable and fruitful meeting.  

\section*{References}

\begin{thebibliography}{[88]}

\bibitem{pythia} T.~Sj\"ostrand, {\it Computer Physics Commun.}
{\bf 82} (1994) 74

\bibitem{babar} P.F.~Harrison and H.R.~Quinn (eds.),
{\it The BaBar Physics Book: Physics at an Asymmetric B Factory}, 
SLAC-R-504 (1998) 183

\bibitem{aleph} R.~Barate {\it et al.} (ALEPH Collaboration), 
{\it Physics Reports} {\bf 294} (1998) 1

\bibitem{durham_alg} W.~J.~Stirling {\it et al.}, {\it J. Phys.} {\bf G}:
{\it Nucl. Part. Phys.} {\bf 17} (1991) 1567

\bibitem{ERT} R.~K. Ellis, D.~A. Ross and
A.~E.~Terrano, {\it Phys.\ Rev.\ Lett.} {\bf 45} (1980) 1225; {\it Nucl.\
Phys.} {\bf B178} (1981) 421

\bibitem{jetnet} C.~Peterson, T.~R\"ognvaldsson, L.~L\"onnblad 
{\it Computer Physics Commun.} {\bf 81} (1994) 185 

\end{thebibliography}

\end{document}

