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ICHEP'98 \#150 & \hspace{6cm} DELPHI 98-90 CONF 158 \\
Submitted to Pa 3 & 22 June, 1998 \\
\hspace{2.4cm} Pl 4 & \\
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{\Huge \bf Identifying gluons event-by-event in
\boldmath $Z^0\to b\bar{b}g$ Mercedes events} \\
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\centerline{\large Preliminary}
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\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\normalsize {
%===================> DELPHI note author list =====> To be filled <=====%
{\bf M. Battaglia} $^1$
{\bf A. Kiiskinen} $^1$
{\bf V. Nomokonov} $^1$
{\bf R. Orava} $^1$
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}
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\begin{abstract}
\noindent
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We apply a novel method of reconstructing parton final states and
multi-element antenna patterns event-by-event \cite{letter} to double-tagged
$Z^0\to b\bar{b}g$ Mercedes events observed in the
DELPHI experiment at LEP~1. The approach allows reconstruction of colour
connectedness between the three partons and identifies the gluon in each event
with a purity of 90\%. We compare the charged particle
multiplicities attached to the $bg/\bar{b}g/\bar{b}b$ systems
in $|y|<1$ in the c.m.s. system of the pair
and measure the ratio $_{bg(\bar{b}g)}/_{b\bar{b}}=1.95\pm 0.08$
in good agreement
with the QCD based theoretical prediction \cite{formula}.
%=========================================================================%
\end{abstract}
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Paper submitted to the ICHEP'98 Conference \\
Vancouver, July 22-29
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\par {\footnotesize $^1$ Helsinki Institute of Physics, Helsinki (Finland)}
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\section{Introduction.}
A new method for reconstructing the event profiles of multi-parton final
states has been recently introduced \cite{letter}, which enables one to
determine the colour structure of a hard scattering final state,
event-by-event. In this paper we apply
this novel approach of analysing multi-parton final states to the double
b-tagged $Z^0$ decays into Mercedes type $b\bar{b}g$ final states.
In double b-tagged events the gluon is uniquely defined, and the directions
of the primary b-quarks are precisely known from the secondary decays of the
B-hadrons. We investigate the efficiency and purity with which the method
identifies the gluon and the angular precision with which the parton
directions are reconstructed. In addition to the studies with LEP~1 data
obtained with the DELPHI experiment, we analyse the method by using a large
sample of non-b Jetset \cite{jetset} Monte Carlo events processed through the
DELPHI detector simulation package \cite{delsim}.
We then proceed to investigate how the charged particle spectra associated
with the $bg/\bar{b}g$ and $b\bar{b}$ systems differ from each others.
\section{Event selection.}
For illustration of the validity of a new parton reconstruction procedure it
is useful to compare it with the data for which all kinematical and dynamical
quantities are known, event-by-event. Samples of double b-tagged
$b\bar{b}g$ Mercedes events measured by the DELPHI experiment at LEP~1,
and samples of Monte Carlo events at the $Z^0$ centre-of-mass energy
are selected for separate analysis for comparison purposes.
In these events, the directions and energies
of the final state quarks and gluons were defined from the hadrons that had
been either measured by the DELPHI experiment or, in the case of Monte Carlo
events, processed by the complete DELPHI detector simulator \cite{direction}
to match the experimental output of the DELPHI experiment.
In the selected event samples it was required that: the fraction of energy
measured for the charged particles was more than 30\% of the total visible
energy, the minimum number of charged particles with energy in excess of
0.7 GeV was more than 7 and the measured energy in each hemisphere was
larger than 15 GeV. The events were then required to be within the full
acceptance of the detector by removing the events in which the leading
particles were closer than $20^o$ relative to the beam direction. In order to
remove badly measured tracks the track length was required to be more than
70 cm, track momentum $p>$ 0.4 GeV, the relative error in momentum
measurement $\sigma_p/p<100\%$, and the impact parameter of the track
extrapolated to the beam spot $|d|<5.0$ cm.
In our sample of double b-tagged events two jets with impact parameter
probability \cite{btag}
of $p_{ip} < 0.02$, one jet with b-tagging probability of $p_b > 0.2$
was required. The measured $b(\bar{b})$ secondary vertex was chosen as the
direction of the $b(\bar{b})$ quark, and the gluon direction was determined by
using the peak energy flow (see Chapter 3). We required that the
quark directions, defined by two different methods coincide within $30^o$,
and that the
inter-parton angles $\Theta_{q\bar{q}}$, $\Theta_{qg}$, $\Theta_{\bar{q}g}$,
were equal to $120^o \pm 20^o$.
\section{Description of the algorithm.}
The clustering algorithm is based on first fixing the hard parton
directions from information given by the particles above a predetermined
energy threshold of $E_1 = 4~GeV$. The unit vector
$\vec p_i /|\vec p_i|$ of particle i with three-momentum $\vec p_i$ and
$E_i > E_1 =4~GeV$ is chosen as a parton direction
if there is at least one extra particle j, with
$E_j > E_2 = 2~GeV$ within an
angular cone of $20^o$ with respect to particle i. Secondly, the energies
of particles within the $20^o$ cones around each parton direction are summed
up. If the summed energy exceeds 75\% of the over-all visible energy of the
event the first phase of the algorithm is considered complete. Step (1) is
then repeated with another energy threshold
$E_1\to E'_1 = 3~GeV$ and $E_2 \to E'_2 = 1.5~GeV$.
In the Mercedes-type $q\bar{q}g$ events, where all the partons are well,
separated in angle, the CAMJET clustering algorithm \cite{camjet}
performs well and
we have used it as an alternative method for finding the parton directions
from the particle tracks and energy deposits with $E_i > 2~GeV$.
After defining the parton directions the probability of a hadron to be
correlated with each parton direction is calculated for every hadron in
the final state. The probability of a particle i to be associated with
a given cluster j, as a function of the particle energy $E_i$ and the angle
$\Theta_{ij}$ between the directions of the particle and the direction
cluster, is defined to be
\begin{equation}
w_{ij} = \frac{C_i}{k_{ij}^2} \label{label1}
\end{equation}
The variable $k_{ij}$ represents the squared transverse momentum
$k_{ij}^2 = 2 E_i^2 (1-cos\Theta_{ij})$ with respect to the direction cluster,
and the sum of the coefficients $\Sigma_j w_{ij}$
is normalised to unity.
The leading particles, i.e., the hadrons which are most likely to
carry the original partons as their constituents, are now defined in a
natural way as the ones with the probability $w_{ij}$ close to one,
$w_{ij} > 0.95$.
A particle, with the probability less than 0.95, is assumed to be a product
of an interaction between a number of final state partons. In the leading
order approximation with only two partons participating in the production,
each particle with $w_{ij} < 0.95$ is assigned to the cluster pair kl for
which the sum $w_{ikl} = w_{ik} + w_{il}$ is maximal. The colour connection
coefficient for each pair of clusters is defined as
\begin{equation}
W_{kl} = C \Sigma_i g(E_i)(w_{ik} + w_{il}), \label{label2}
\end{equation}
where the index i runs through all the particles assigned to the pair of
clusters, $g(E_i)$ is an energy dependent weight function \cite{comment},
and the sum
$\Sigma_{kl} W_{kl}$ is normalised to unity.
The method can be tested further by boosting the particles assigned
to a given pair of partons into the rest frame of the pair \cite{pt}.
In this frame a
particle i should exhibit a minimal amount of transversal momentum, $p_t$,
relative to the colour axis between the pair. In addition to a consistency
check, this transverse momentum could be used as an additional weight,
$W_{i,kl}^{pt} = C/p_{it}^{kl}$, in assigning particles to belong to a pair
of partons.
The clustering algorithm was applied to the 59 events fulfilling
the selection criteria, and the weight coefficients and colour connection
coefficients were calculated for the particles which were not included in
the direction clusters and which did not come from the secondary decays of
heavy B-particles.
\section{Results.}
In order to compare the reconstructed parton directions to the
original ones, known from the Monte Carlo generator, we plot the
difference between the pairs of directions in Figure \ref{angles}a. for the
perfect Mercedes events \footnote{We have produced a sample of
$q\bar{q}g$ events with exactly $120^o$ degrees between each pair
of partons, and processed the sample through the DELPHI simulation
package \cite{delsim}.} and in Figure \ref{angles}b.
for the parton shower (PS)
Mercedes events with interparton angles of $120\pm 20^o$. The peak energy
flow is seen to give the same level of angular accuracy as a preferred
jet-cluster method \cite{camjet}.
The weight distributions $W_{jk}$, obtained by using either (a)
$1/k_T^2$ or (b) $1/p_T$ weights, for all three parton combinations
in the double b-tagged event sample can be directly used to identify
the gluon in about 90\% of the events (Figure \ref{weights}).
The particles created in the secondary decays of
heavy B-particles were excluded from analysis with an estimated
$66\%$ efficiency and $88\%$ purity. We cross checked these efficiency
and purity values by using the following non direct method:
We boosted all particles to the $b\bar{b}$ rest system and calculated the
rapidity distribution for particles identified as coming from B-hadron decays.
We then fit this distribution with the sum of the two distributions of
particles known to come from the B-hadron decays and fragmentation,
respectively.
This method agrees well with the results obtained in our direct analysis.
By selecting
different values for the minimum weight, $min(W_{jk})$ $(min(W_{kl}^{pt}))$,
we calculate the efficiency and purity of the method as a function
of the minimum weight (Figure \ref{effpur}). At 80\% efficiency the purity of
the method is 90\%.
As a test of our method against a QCD prediction, we plot the
rapidity distributions $(1/N_{ev})(dN/dy)$ in the rest frames of
$bg,\bar{b}g$ and $b\bar{b}$ systems and compare them with the Monte Carlo
generated distributions (Figure \ref{rapidity}).
We used the decay products of the B-hadrons to determine the
primary b-quark direction. The gluon directions
were defined according to the procedure described in the previous section.
In calculating the rapidities all particles were assumed to be pions.
Depletion of particles at rapidities close to
zero in the $b\bar{b}$ system is clearly demonstrated. The ratio
of charged particle multiplicities in the interval $\Delta y = \pm 1$
gives: \\
$/ |_{\Delta y =\pm 1} = 1.95 \pm 0.08$
to be compared with the Monte Carlo prediction of $2.21\pm 0.09$
calculated for the same multiplicity ratio in the same rapidity interval
$\Delta y = \pm 1$.
The Jetset Monte Carlo without the detector effects
agrees with the analytical prediction \cite{explanation}.
\section{Discussion and Conclusions.}
By investigating a sample of double b-tagged events recorded by the DELPHI
experiment at $Z^0$, we have demonstrated that the colour structure of a hard
scattering final state can be determined
on an event-by-event basis. After first defining the parton directions,
the parton skeleton of the event, we proceed to calculate their
connectedness as a sum of energy weighted probabilities of particles
attached to a give pair of parton directions. We then perform a consistency
check of our procedure by a Lorentz transformation to the rest frame of
each connected pair of partons. In this frame the sum of particle
transverse momenta relative to the connection axis should be zero.
With the new method, a gluon emitted in a Mercedes event configuration
can be identified with a purity in excess of 90\%
by using the weighting procedure described above.
For the symmetric Mercedes-type $q\bar{q}g$ events we find that the
two weights procedures are strongly correlated. In analysing
multiparton final states we intend to combine the two procedures.
By measuring the rapidity distribution of particles in the rest frame of
the parton-parton systems, we find the event characteristics expected for
a colour connection between $qg$ and $\bar{q}g$ pairs and a
lack of connection between the $q\bar{q}$ pairs. The gluons are also
seen to produce particle multiplicities significantly higher than those
produced by quarks or antiquarks.
\begin{thebibliography}{999}
\bibitem{letter} R. Orava, V.A. Khoze, V. Nomokonov, R. Vuopionper\"a,
K. V\"ais\"anen and K. Lassila,
From jets to color portraits of hard
scattering final states, Helsinki Institute of Physics preprint
HIP-1997-66/EXP(1997).
\bibitem{formula} Ya.I. Azimov, Yu.L. Dokshitzer, V.A. Khoze and
S.I. Troyan, Phys.Lett. B165 (1985) 147.
\bibitem{jetset} T. Sj\"ostrand, Comp. Phys. Comm. 82 (1994)74.
\bibitem{delsim} Complete simulation of the DELPHI detector, see P.
Abreu et al., (DELPHI Collaboration), Nucl. Instr. and Meth. A378 (1996) 57.
\bibitem{direction} A procedure which combines track and energy information in
finding the direction clusters is described in: P. Eerola and R. Orava, Nucl.
Instr. and Meth. A295(1990)323.
\bibitem{btag} G. Borisov, Lifetime tags of $Z^0\to b\bar{b}$
with the DELPHI detector, DELPHI 94-125 PROG 208.
\bibitem{camjet} Yu.L. Dokshitzer, G.D. Leder, S. Moretti and B.R. Webber,
Better Jet Clustering Algorithms, Cavendish Preprint HEPP-97/06 (1997).
\bibitem{comment} The energy dependent weight function used in this
analysis is defined as $g(E_i)=E_i$. The same algorithm was also used with
the weight factor $g(E_i)=1$, which did not significantly change our
conclusions.
\bibitem{pt} E. Norrbin and T. Sj\"ostrand, PR D55(1997) R5.
\bibitem{explanation} The Jetset Monte Carlo does not inlcude the
interference effect proportional to $1/N_c^2$ and estimated to
decrease Monte Carlo value by about $10\%$ \cite{formula}.
\end{thebibliography}
\newpage
\begin{figure}
\psfull
\epsfig
{file=angles.ps,width=9.0cm,height=9.0cm,bbllx=-220,bblly=120,bburx=352,bbury=692,clip=}
\caption{Angular difference between the reconstructed parton directions and
the original ones, known from the Monte Carlo generator. Left figure (a)
corresponds to ``perfect mercedes'' Monte Carlo and the right one (b)
corresponds to parton shower Monte Carlo. Solid lines correspond to
directions, reconstructed using only particles with $E>2~GeV$,
dashed lines correspond to directions, reconstructed by Camjet algorithm.}
\label{angles}
\end{figure}
%
\begin{figure}
\psfull
\epsfig
{file=weights2.ps,width=9.0cm,height=9.0cm,bbllx=-220,bblly=120,bburx=352,bbury=692,clip=}
\caption{The weight distributions $W_{jk}$ for $b\bar{b}$ (dashed line), and
$bg(\bar{b}g)$ (solid line) for double b-tagged experimental data sample. The
plot was made for events with $W_{min}<0.2$. Figure a) corresponds to the
weights, calculated as $W_{kl} = C \Sigma_i g(E_i)(w_{ik} + w_{il})$
(see the text), and figure b) corresponds to the weights
$W_{i,kl}^{pt} = C/p_{it}^{kl}$.}
\label{weights}
\end{figure}
%
\begin{figure}
\psfull
\epsfig
{file=effpur_4.ps,width=9.0cm,height=9.0cm,bbllx=-220,bblly=120,bburx=352,bbury=692,clip=}
\caption{Purities and efficiencies of the method (applied
for double b-tagged experimental data sample) as a function of the upper
limit for the minimum weight. Solid and dashed lines (purity and
efficiency respectively) correspond to
$W_{kl} = C \Sigma_i g(E_i)(w_{ik} + w_{il})$ and
dotted and dashed-dotted lines (purity and efficiency respectively)
correspond to $W_{i,kl}^{pt} = C/p_{it}^{kl}$.}
\label{effpur}
\end{figure}
%
\begin{figure}
\psfull
\epsfig
{file=rapidity_3.ps,width=9.0cm,height=9.0cm,bbllx=-220,bblly=120,bburx=352,bbury=692,clip=}
\caption{The distribution on rapidity in the rest frames of $b\bar{b}$
(a) and of $bg(\bar{b}g)$ (b) pairs. Note that in the plot (b) quarks
and gluons are mixed up and only the zone around zero is significant.
Real data is shown by dots with error bars and Monte Carlo by solid
lines.}
\label{rapidity}
\end{figure}
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\end{document}