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\begin{document}
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%\copyrightheading{} %{Vol. 0, No. 0 (1993) 000--000}
%\vspace*{0.88truein}
\fpage{1}
\centerline{\bf FAST RECONSTRUCTION OF THE VERTEX OF ANTIPROTON}
\vspace*{0.035truein}
\centerline{\bf ANNIHILATION AT INTERMEDIATE ENERGIES,}
\vspace*{0.035truein}
\centerline{\bf BASED ON A ROTOR NEURAL NETWORK}
\vspace*{0.37truein}
\centerline{\footnotesize G. GIUGLIARELLI, L. SANTI}
\vspace*{0.015truein}
\centerline{\footnotesize\it Dipartimento di Fisica,
Universit\'a di Udine, Via delle Scienze 209}
\baselineskip=10pt
\centerline{\footnotesize\it Udine, I-33100,Italy}
\vspace*{0.225truein}
%\publisher{(received date)}{(revised date)}
\vspace*{0.21truein}
\abstracts{We constructed an algorithm with an higher
sampling rate, in which the global information
related to the geometrical correlation of the spatial points, is transformed
into a local information,
the tangent and the curvature of the related track in each point,
by means of a rotor neural network,
and this information is used to evaluate directly
the position of the vertex of the event.}{}{}
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\section{Introduction}
\noindent
In some fixed target experiments at intermediate energies
a continuous control of the conditions in which the
data are collected is necessary.
This is the situation in the particular in the case of measurement of
antiproton annihilation on gaseous targets, in the OBELIX
experiment (PS201, CERN). In this experiment, because of the relatively small
branching ratios of the reactions studied,
variable experimental conditions make data analysis
particularly difficult and therefore have to be carefully
controlled.
In the case of measurements with low target density
the area of $\overline{p}$ annihilation
can change significantly during the mean life time of a bunch in
the accumulation ring LEAR (about 1 hour in the usual conditions).
It is therefore the most crucial parameter, and
has to be maintained as constant as possible, in order to keep the
experimental parameters like angular
acceptance of the apparatus, {\it dose} fraction in target
and contamination by events for which the interaction occurred outside target,
stable and in a convenient range.
At present
the global monitor of the data acquisition that performs this control
is a program of {\it classical} event
reconstruction (pattern recognition of the
annihilation products, reconstruction of tracks and evaluation of the common
vertex), which analyzes the data from the tracking device
JDC\cite{99} (
a cylindrical multiwire drift chamber in an axial magnetic field).
The sampling frequency of this program is
about 1~events/s, which results in
long times (from 10 to 20 minutes) to accumulate a
sufficient statistics to evaluate the vertex distribution,
in the cases where a small fraction of events
occurred in the target, and therefore does not
guarantee a fast response to changes in the data taking conditions.
The low processing rate of the reconstruction program ( a very optimized
one) is due essentially to
\vspace{-0.2cm}
\begin{itemlist}
\item the requirement of precise pattern recognition of the tracks
for data sets affected by spurious hits
and by tracks (typically electrons) that spirals inside the detector;
\item the type of information given by the tracking device:
to convert
the drift time data from JDC into of the spatial position
of the hit with the precision
required ($\sim 200 \mu m$), the knowledge of the
local direction of track is
required; moreover, the side of the drift cell crossed by the track
is undetermined ({\it L/R} ambiguity).
These problems can be solved only by iterating the
procedure of track fitting and modifying the evaluation of point
position and the
L/R assignment at each step, according to the fit results.
\end{itemlist}
\vspace{-0.2cm}
The sensitivity of the performance of the monitor program to
these features, has
suggested us to search a different approach for the evaluation of the
vertex, in which the correlations of the
spatial distribution of points are used to obtain the local direction and
curvature of the tracks, without a full reconstruction.
The natural candidate for this type of algorithm (based on the
correlations existing in the input data and relatively insensitive to noise)
is a neural network.
In the following sections we will describe the structure of the neural network
used in our study (sect.~3,4),
the procedure to determine the position of the spatial
points from the drift times (sect.2) and the algorithm to evaluate the
vertex position (sect.5). Results are summarized in section 6.
\section{The geometry of JDC and the drift cell model used}
The JDC consists of two semi-cylindrical drift chambers, extending over a radius
of 20. to 80. cm and 150. cm long.
Both chambers are split by 42 drift wire planes into 41
identical sectors, $4^o$ in azimuth, each containing 40 sense wires. The sense
wires are staggered by $\pm 400 \mu m$, to resolve the L/R ambiguity, and
are grouped into three crowns.
The resulting geometry of the drift cells implies that a track crossing the
whole detector is sampled into three separated
clusters of hits, with an uniform radial step ($\Delta r = 0.8 cm$)
within a cluster and a large gap ($\sim 20. cm$) between the clusters.
The drift time $t_d$ of the electrons produced by an ionizing particle
is approximatively related to the distance $d$ of the track from the sense wire
by the expression
$$
d=\frac {cos(\beta+\gamma)(t_d v_d -r_{cell}) +r_{cell}} {cos(\beta)}
\eqno{(1)}
$$
where $\gamma$, $v_d$ and $r_{cell}$ are parameters that depend on the cell
and $\beta$ is the angular direction of the track in the cell.
To transform the raw information of the drift time into a spatial information,
we proceed as follow.
First, we partially resolved the L/R ambiguity of the hit by comparing
the drift time of three nearby sense wires and requiring consistency
with the staggering of sense wires. This type of assignment can be
preformed only for triplets within the same sector and crown, and for tracks
not crossing the plane of the sense wires within the considered cells.
In the second step, we calculated a rough estimate of the point coordinates
${\bf x}_i$, supposing $\beta=0$. After that, the matrix
$$
<({\bf x} - {\bf x}_i)({\bf x} - {\bf x}_i)^+> \eqno{(2)}
$$
is
calculated for each point ${\bf x}_i$, over all the points ${\bf x}$ for which
we resolved the L/R ambiguity and within
$r < r_{cut}=3.cm$ from ${\bf x}_i$. Then, we evaluated
the direction of the eingevector associated with
the maximum eingevalue of the matrix: this direction
represents the main axis of
the distribution of hits around the point ${\bf x}_i$ and
it is natural to assume it as
the most probable candidate for the tangent of the track crossing the
point. With this information, a more accurate evaluation of the point position
was calculated, by means of eq.(1).
Finally, for each point not accepted in step one, we decided the L/R
assignment choosing the one more close to the direction of the main axis
calculated previously for the nearby points.
\section{The Neural Network.}
Our approach to the calculation of the track vertex is based on the use of
a rotor neural network (RNN).
Its structure takes into
account of the fact that the projection of the particles traces on the plane
perpendicular to the axis of the detector (the $xy$ {\it plane})
are essentially segments of circles whose radius
is related to the particle momenta. On the basis of this fact
we arranged for the RNN to
work only with the distribution of points in the $xy$ plane which means that
the network determines only the features of the $xy$--projections of the
trajectories.
This type of network, first introduced by Peterson\cite{1},
is characterized by an energy function of the form:
$$
E=-{1\over 2} \sum_{ij} {\bf s}_i {\hat {\bf w}}_{ij} {\bf s}_j
\eqno(3)
$$
where \{${\bf s}_i$\} are the {\em rotor }
vectors associated with the set of data points \{$P_i$\}
for which the network is set up. These vectors corresponds to the local
direction of the trace crossing the local point. They interact with each other
through the weight matrices ${\hat {\bf w}}_{ij}$, that also determine the
pairs of points which are connected.
The explicit form of the matrix ${\hat {\bf w}}_{ij}$ depends on the type of
correlation between ${\bf s}_i$, ${\bf s}_j$ required.
In our case, this leads to the expression\cite{2}
$$
w_{ij}=\left(\matrix{ &\cos 2\varphi_{ij} &\sin 2\varphi_{ij} \cr
&\sin 2\varphi_{ij} &-\cos 2\varphi_{ij} \cr
}\right)
\eqno(3)
$$
With this choice, the term
${\bf s}_i {\hat {\bf w}}_{ij} {\bf s}_j$
reaches the minimum value
when the rotors
${\bf s}_i$ and ${\bf s}_j$ are directed as the tangents to a certain circle
passing through $P_i$ and $P_j$.
In such a way the problem of determining the
track which best fits a set of points is equivalent to that of determining
the configuration
of rotors that minimize the energy function (3).
To perform this task, because the long range
character of our network, we have used the dynamical approach
based on the mean field theory (MFT), which implies application of the
transformation;
\vspace{-0.1cm}
$$
\left\{ \begin{array} {l}\
{\bf U}_i= - \frac{\partial E({\bf s})} {\partial {\bf s}_i } \\
{\bf s}_i = \frac{{\bf U}_i} {|{\bf U}_i|} F({|{\bf U}_i|\over T})
\end{array} \right.
\eqno(4)
$$
\vspace{-0.1cm}
with $ F(x) = \frac{B_1(x)} {B_0(x)}$) until the system reaches a steady state.
The configuration obtained in the limit $ T \rightarrow 0$, minimizes
the energy function E.
For computational convenience, we implemented an updating
rule in our network, in which $F(x) = tanh(x)$, instead of the
expensive combination of Bessel functions.
We have verified that this simplification
does not appreciably affect the results.
The evolution of the network was performed by applying
the updating rule (4) asynchronously ,
and by decreasing the temperature T every time convergence is reached,
from an initial value $T_{in}$ to a final
$T_{fin}$.
The range of the temperature variation is determined by evaluating the critical
temperature $T_c$ of the network (i.e., the maximum eigenvalue of
the matrix $w_{ij}$), at which
the steady state has a transition to the
{\it trivial} solution
${\bf s}_i =0$, and setting $T_{in} = 0.90 T_c$ and $T_{fin} = 0.05 T_c$.
\section{The selection criteria of the network connections}
In principle the RNN could be able to treat the whole set of experimental
points without any preliminary
treatment of the corresponding connections. However, since
the computational times of the network is approximatively proportional to the
number of connections between the points, it is desirable to reduce it
with the help of some reasonable criteria.
The first selection criterion applied is to reject the connections between
very distant points ($d>d_{cut}=40.cm$); the value $d_{cut}$ chosen ensures
the connection of points associated to the same
track but lying in adjacent crowns.
The second selection criterion is to reject the points that are badly
correlated to the nearby hits.
For this purpose, the correlation matrix (2) is
calculated again for each point ${\bf x}_i$:
the ratio $\lambda_{max}/\lambda_{min}$ between the eingevalues of this matrix
is representative of the degree of correlation existing in the local
distribution of the hits and the selection is done requiring
$\lambda_{max}/\lambda_{min}>2.$.
Moreover, we chose to reject all the connections from a given point ${\bf x}_i$
to points not lying within a
cone of limited angular opening ($\Delta \theta < 12^o)$ around the direction
of main axis of matrix (2). A refinement of this criterion is done by
considering also the alignment of the points
in the $rz$ projection ($r^2=x^2+y^2$).
The application of the three selection criteria eliminate almost all the
connections between uncorrelated points (associated to different tracks or
isolated) and reduce the weight of points widely scattered around an
ideal track. Anyway,
we should remark that these procedures realize only a rough
pattern recognition of the
tracks, because the disconnected sets in which the points are partitioned
by means of the surviving connections (the {\it track candidates}),
may contain more than one track, or
segment of track, and sometimes one track is split into two sets.
\section{The evaluation of the vertex position.}
The configuration of active rotors (${\bf s}_i\neq0$)
in the final steady state of the network,
constitute an evaluation of the corresponding tangents to the tracks in each
JDC cell.
For each couple of points (${\bf x}_i,{\bf x}_j$)
in a track candidate, we can calculate a radius of curvature $\rho_{ij}$
by means of these estimated directions and associate to a single point
${\bf x}_i$ a mean curvature radius of track $\rho_i$, by averaging
$\rho_{ij}$ over the couples (${\bf x}_i,{\bf x}_j$) formed with all
points ${\bf x}_j$ connected to ${\bf x}_i$. In this way,
we can associate to any point the equation of the osculating
circle to the track in that point.
The circles associated to all the points of a given track generally
do not coincide and the variance of their equations' parameters
can be interpreted as the error occurring in the determination of that track.
This implies that the intersections of circles associated
with points of different
tracks have a spatial distribution with a mean position and a covariance
that represent, respectively, an evaluation of the position of the vertex
of the tracks and its error.
The algorithm for the evaluation of the vertex position
constructs the cluster of intersections of all possible couples of circle
associated with two given track candidate and calculates an averaged position
$c$ and the covariance matrix $\sigma_c$ of the spatial distribution of the
cluster (to reduce the systematic errors due to the non-uniformities of the
magnetic field, the energy loss and multiple scattering of the particle
in the detector, only the points lying in the inner crown are used).
After that, the evaluation the vertex common to all the tracks is done by
averaging the mean position $c$ of the clusters, weighted with $\sigma_c^{-1}$;
a further weight factor in this average came from the mean curvature associated
to the candidate tracks, to take into account the systematic errors in
the track determination, arising from the energy loss and multiple scattering
of the particles in the materials surrounding the target.
The error on the vertex position is then calculated by propagating
the covariance matrices of the clusters.
The presence of secondary vertices,
typically coming from gamma conversions in the inner
border of the detector, can deteriorate the estimator of the primary vertex
position. To reduce this effect, the averaging of the mean position
of the clusters is repeated, with a further weight factor
related to the distance of
the cluster from the former estimated vertex.
Finally, for each track candidate
a linear fit is performed in the plane rz ,and the z coordinate of the vertex
is determined by averaging the intersections of the resulting lines
with the z axis.
\section{Results and performances of the RNN algorithm}
The RNN algorithm has been tested on a set of 10000 events
($\overline{p} ^4He$), with a mean track multiplicity per event of 3.5
and a mean number of hits per event of 150.
In this set of events, the greater part comes from the annihilation
of $\overline{p}$ in the cylindrical vessel of the target, defining a sharp
peak of the spatial distribution both in the z and the radial
projections. We evaluated the error in the
estimated position of the vertex done by the RNN on both
the projections, from the FWHM of these peaks.
A check of the errors evaluated by the RNN was made by calculating
the RMS of
the ratio between the difference $\Delta$
of the estimated position of vertex and
the real position of the vessel wall, and the estimated error $\sigma$.
Tab.1 gives the results obtained, compared to those
from the present online monitor program ("classical").
\begin{table}[htbp]
\tcaption{Radial and longitudinal errors in the evaluation of the position
of the vertex and systematic deviations of the estimate errors (see text).}
\centerline{\footnotesize\smalllineskip
\begin{tabular}{lcccc}\\
\hline
{}& RNN & RNN & Class. & Class. \\
{}& Radial & Longit. &Radial & Longit. \\
\hline
FWHM peak (cm) & 1.2& 3. & 0.9 & 2.\\
$ {\Delta}/ {\sigma}$& 1.5 & 1.7 &1.2& 1.3\\
\hline
\end{tabular}}
\end{table}
The sampling rate of RNN has been tested, measuring the elapsed
time per event
for the execution of the different procedures (conversion of raw data,
setting of the network, execution of network and evaluation of vertex)
on a VaxStation Digital 4000-60. Results are reported in tab.2 and are
compared to the total elapsed time per event of the "classical"
program.
\begin{table}[htbp]
\tcaption{Elapsed time per event (s) for the RNN algorithm and for the
"classical" one, on a VaxStation Digital 4000-60}
\centerline{\footnotesize\smalllineskip
\begin{tabular}{lccccc}\\
\hline
{} &Conversion & Network setup & Network evol.& Vertex & Total \\
\hline
RNN & 0.058 & 0.079 & 0.131 & 0.028 & 0.296 \\
Class. &{} & {} &{} &{} & 1.40\\
\hline\\
\end{tabular}}
\end{table}
In conclusion, the resolution obtained with the algorithm based on RNN
is comparable
with that of the present online monitor program;
on the other hand, the sampling rate obtained ($4\div 5$ times greater)
will allow the monitoring of the data taking in more efficiency conditions.
\vskip0.7cm\noindent {\bf References}
\begin{thebibliography}{99}
\bibitem{1} C. Peterson, "Neural Networks and High Energy Physics",
in {\bibit Proc. of the Int. Workshop on Software Engineering,
Artificial Intelligence and Expert Systems for High Energy and Nuclear Physics},
Editions du CNRS, (1990).
\bibitem{2} A.A.Glazov, et al.,
{\bibit JINR Communication}, {\bibbf E10--92--352},
(1992).
\bibitem{99} F. Balestra et al., \bibit {NIM}, {\bibbf A323}, 523-527, (1992).
\end{thebibliography}
\end{document}