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HEP'99 \# 1\_510 & \hspace{6cm} DELPHI 99-126 CONF 313 \\
Submitted to Pa 1 & 15 June 1999 \\
\hspace{2.4cm} Pl 1 & \\
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\Huge {\bf Testing of the New Parton Final State Reconstruction Method
Using \boldmath $Z^0\to b\bar{b}g$ Mercedes Events }\\
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\centerline{\large Preliminary}
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\large {DELPHI Collaboration \\}
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Paper submitted to the HEP'99 Conference \\
Tampere, Finland, July 15-21
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{\Huge \bf Testing of the New Parton Final State Reconstruction Method
Using \boldmath $Z^0\to b\bar{b}g$ Mercedes Events} \\
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\large {DELPHI Collaboration \\}
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\normalsize {
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{\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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A novel method of reconstructing parton final states and
multi-element antenna patterns event-by-event was applied
to double b-tagged
$Z^0\to b\bar{b}g$ Mercedes events observed in the
DELPHI experiment at LEP~1. The approach allows the reconstruction of colour
connectedness between the three partons and identifies the gluon in each event
with a purity of 60\%. The charged particle
multiplicities of the parton pairs
$bg(\bar{b}g)/\bar{b}b$ are compared within the c.m.s. rapidities of
$|y|<1$ and the ratio $<~n~>_{bg(\bar{b}g)}/<~n~>_{b\bar{b}}$ is measured
as a demonstration of the depletion of particles in $b\bar{b}$ interjet
region.
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\par {\footnotesize $^1$ Helsinki Institute of Physics, Finland}
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\section{Introduction}
A method for reconstructing the event profiles of multi-parton final
states has been recently introduced \cite{letter}, which enables to
determine the colour structure of a hard scattering final state,
event-by-event. In this paper
this novel approach of analysing multi-parton final states was applied
to the double
b-tagged $Z^0$ decays into Mercedes type (i.e. a threefold symmetric
topology) $b\bar{b}g$ final states, where
the gluon is uniquely defined.
The efficiency and the purity with which the method
identifies the gluon were investigated by matching a parton found to
have two colour connections with the not b-tagged one. The method
exploits the well known QCD drag effect \cite{book}
which results in the suppression of gluon radiation in the region between
the quark jets giving a surplus of radiation in the $qg$ and $\bar{q}g$
valleys that subsequently leads to higher particle multiplicities
in the (anti)quark-gluon interjet regions as compared to the quark-quark
region.
The method was applied to
LEP~1 data recorded by the DELPHI experiment, and compared with a large
sample of events generated by the
Jetset 5.7 Parton Shower Monte Carlo program \cite{jetset} and
processed through the
full DELPHI detector simulation \cite{delsim}.
The difference of the charged particle multiplicities in the
region of rapidities close to zero, associated
with the $bg(\bar{b}g)$ and $b\bar{b}$ systems was investigated.
\section{Event selection}
To illustrate the validity of the new parton reconstruction procedure,
the method was applied for events for which all kinematical and dynamical
quantities are known, on the event-by-event basis. Samples of double b-tagged
$b\bar{b}g$ Mercedes events recorded by the DELPHI experiment at LEP~1
in 1994 and 1995,
and samples of Monte Carlo events at the $Z^0$ centre-of-mass energy
were selected. The directions and energies
of the final state quarks and gluons were obtained
by using the particles reconstructed in the DELPHI detector.
In order to
remove poorly measured tracks, the track length was required to be more than
20 cm, the track momentum above 0.4 GeV, the relative error in momentum
measurement less than $100\%$, the
angle relative to the beam direction more than $10^o$ and the
projection of the impact parameter relative to the interaction point
in the plane transverse to the beam direction less than $4.0$ cm.
Hadronic events were selected by requiring five or more charged particles
and a total energy in charged particles larger than $12\%$ of the collision
energy, assuming all charged particles to be pions.
The Durham \cite{Durham} clustering
algorithm with $y_{cut}=0.01$ was used and events with three jets
were selected. It was also required that
the jet energy carried by leptons must not exceed $60\%$.
To select a sample enriched in events containing b quarks,
a well established b-tagging method was used \cite{btag}.
The technique is based on the measurement of the impact parameter and
tests the compatibility of each charged particle track
in a jet to originate from the primary vertex of the event.
The b-jets are characterised by a low value of this probability due to
presence of secondary particles.
In the sample of double b-tagged events
two jets with impact parameter probability
of $P_{jb} < 0.02$ and one jet with $P_{jb} > 0.2$ was required.
The fraction of hadronic $Z^0$ decays not containing b~quarks
was estimated to be $9\%$.
The simulation study has shown that in $5\%$ of genuine $b\bar{b}g$
events the gluon jet was misidentified.
It was required that the
inter-jet angles $\theta_{b\bar{b}}$, $\theta_{bg}$, $\theta_{\bar{b}g}$
were equal to $120^o \pm 20^o$.
The number of events remaining after applying each of
the selection criteria and their reduction factors for real data
and for simulated events are listed in the Table
\ref{table1}.
\begin{table}
\caption{The number of events remaining after applying different
selection criteria and corresponding reduction factors for
real data and for simulated events.}
\begin{tabular}{lrll} \\ \hline
Selection criteria & Num. of events & Reduction &
Reduction \\
& & factor, data & factor, simulation \\ \hline
Initial number of events & 8623000 & & \\
Hadronic event selection & 1982000 & 4.4 & \\
$N_{jets} = 3$ & 513200 & 3.9 & 3.7 \\
Jet quality & 430500 & 1.2 & 1.2 \\
$100^o<\theta_{interjet}<140^o$ & 19110 & 22.5 & 21.0 \\
b-tag cuts & 757 & 25.2 & 25.7 \\
\hline
\end{tabular}
\label{table1}
\end{table}
%
\section{Description of the algorithm}
A clustering algorithm, independent on the results of the Durham
clustering applied before, has been developed.
The iterative algorithm starts by choosing the hard cluster
directions as the directions of those particles with the energy
above a predetermined threshold $E_1 = 4~GeV$. 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 cluster 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 $i$. Then, the energies
of the particles within the $20^o$ cones around each cluster direction
are summed.
If the summed energy exceeds 75\% of the visible energy of an event,
the cluster direction is retained. Otherwise, this procedure
is repeated with new energy thresholds
$E_1 = 3~GeV$ and $E_2 = 1.5~GeV$. The cluster directions are used to
approximate the parton directions.
In this specific case, where combining particles in jets is needed for
the b-tagging purposes the directions obtained by the
Durham clustering algorithm were used to avoid an
ambiguity of the jet definition.
In the Mercedes-type $q\bar{q}g$ events, where all the clusters are well
separated in angular space, the directions reconstructed by both methods
have been found to agree within $3^o$.
After having defined the cluster directions,
the probability of a hadron to be correlated with
each of these directions is calculated for every charged hadron in
the final state.
Since the B-hadron decay products are not created in
the colour field between the initial partons ($bg$, $\bar{b}g$), they
must be excluded from the analysis.
The exclusion was performed using the impact parameter method.
The efficiency and purity values of this exclusion procedure
was estimated. It was found that
$49\%$ of all charged B-hadron decay products were correctly tagged. The
percentage of B-hadron decay products among the tagged particles is found
to be $77\%$.
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 direction of the particle and that of the
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
of the cluster.
The sum of the coefficients $\Sigma_j w_{ij}$ over the jets $j$
was normalised to unity. $C_i$ is a normalisation coefficient.
The leading hadrons which define the energy flow directions and which
are less sensitive to the coherence effects, 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.
Assuming 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 $W_{kl}$ for each pair of clusters
is defined as
\begin{equation}
W_{kl} = C_{kl} \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, and the sum
$\Sigma_{kl} W_{kl}$ is normalised to unity via adjusting the normalisation
coefficients $C_{kl}$.
The energy dependent weight function used in this
analysis is defined as $g(E_i)=E_i$.
The method can be tested further by using the alternative definition of
the coefficient $W_{kl}$ following the procedure
described below \cite{pt}.
All the particles are boosted into the rest frame of each pair of clusters.
A particle $i$ should exhibit a minimal amount of transverse momentum, $p_T$,
relative to the colour axis between the pair, (in the rest frame of the
parent parton pair) as compared to the two other pairs.
Thus, to perform a consistency
check, this transverse momentum could be used in an alternative colour
connection coefficient construction.
Each particle with a predetermined weight value of $w_{ij} < 0.95$
is assigned to the cluster pair $kl$ for
which the transverse momentum, calculated in the rest frame of
the clusters $k$ and $l$, $p_{T_{i}}^{kl}$ is minimal.
The colour connection coefficient can then be defined as
\begin{equation}
W_{kl}^{p_{T}} = C_{kl}\Sigma_i g(E_i) /p_{T_{i}}^{kl}.
\label{label3}
\end{equation}
The algorithm was applied to the 757 $b\bar{b}g$ Mercedes
events fulfilling the selection criteria described above.
The four-momenta of clusters were defined using the reconstructed jet
directions, using
energy and momentum conservation constraints and assuming the partons
to be massless.
\section{Results}
The colour connection coefficient distributions $W_{kl}$,
obtained by using either
$1/k_T^2$ (Eqn. \ref{label2}) or $1/p_T$ (Eqn. \ref{label3})
colour connection coefficient (ccc), for all three parton combinations
in the double b-tagged event sample are shown on Fig. \ref{weights}.
The colour connection coefficients can be directly used for the gluon
identification since the $\bar{b}g$ and $bg$ pairs should exhibit higher
values of the ccc as compared to the
$\bar{b}b$ pair. It was found that the ccc corresponding to the quark-antiquark
pair $W_{\bar{b}b}$ has the smallest value among the three pair
coefficients in about 60\% of the events.
\begin{figure}[h!]
\begin{center}
\includegraphics[bb=20 150 580 700, scale=0.5]{bbg_tamp_wei.ps}
\end{center}
\caption{The colour connection coefficient
distributions $W_{jk}$ for $b\bar{b}$ (solid line), and
$bg(\bar{b}g)$ (dashed line) for double b-tagged experimental data sample. The
plots show events with $W_{min}<0.2$. Figure a) corresponds to
$W_{kl} = C_{kl}\Sigma_i g(E_i)(w_{ik} + w_{il})$,
and figure b) corresponds to
$W_{kl}^{p_{T}} = C_{kl}\Sigma_i g(E_i) /p_{T_{i}}^{kl}$.}
\label{weights}
\end{figure}
By selecting
different values for the minimal ccc,
$min(W_{kl})$ $(min(W_{kl}^{P_{T}}))$,
the efficiency and purity of the method can be varied.
Fig. \ref{effpur} shows the purity as a function of the efficiency
for coefficients, calculated as $W_{kl} =
C_{kl}\Sigma_i g(E_i)(w_{ik} + w_{il})$
(boxes), and for coefficients, calculated as
$W_{kl}^{p_{T}} = C_{kl}\Sigma_i g(E_i) /p_{T_{i}}^{kl}$ (triangles).
Both methods give very similar results.
The efficiency is defined
here as a fraction of events remaining from the initial sample of 757
events after requiring $W_{min}$ to be below some predetermined
value.
\begin{figure}[h!]
\begin{center}
\includegraphics[bb=20 150 580 700, scale=0.5]{bbg_tamp_eff.ps}
\end{center}
\caption{Purity {\it vs} efficiency of the gluon identification
obtained by varying the minimum colour connection coefficient
$W_{min}$. Boxes correspond to
$W_{kl} = C_{kl}\Sigma_i g(E_i)(w_{ik} + w_{il})$ and triangles
correspond to $W_{kl}^{p_{T}} = C_{kl}\Sigma_i g(E_i) /p_{T_{i}}^{kl}$.
The efficiency is defined in the text.
The $W_{min}$ is changing from $0.07$ to $0.33$ with equal steps.
The solid line shows the value of $33\%$, which is
the natural purity of the sample.
}
\label{effpur}
\end{figure}
As a demonstration of the depletion of particles in the $b\bar{b}$
inter-parton region with respect to the $bg(\bar{b}g)$ region
at rapidities close to zero, the ratio
of charged particle multiplicities in the parton's rest systems
in the interval $\Delta y = \pm 1$
was calculated:
\begin{equation}
R_{\Delta y =\pm 1}=\frac{1}{2}(+
)/
|_{\Delta y =\pm 1} = 1.81 \pm 0.09_{stat}
\end{equation}
In calculating the rapidities
cluster momenta were used as the axes. The partons were
identified using the b-tagging procedure for jets.
All particles were assumed to be pions and
particles were attached to the pairs of partons using the method
described above.
A depletion of particles at rapidities close to
zero in the $b\bar{b}$ system is clearly demonstrated.
(The rapidity distributions are shown on the Fig.\ref{y_plot}.)
\begin{figure}[h!]
\begin{center}
\includegraphics[bb=20 150 580 700, scale=0.5]{bbg_tamp_rap.ps}
\end{center}
\caption{
The distribution on rapidity in the rest frames of $b\bar{b}$
(solid line) and of $bg(\bar{b}g)$ (dashed line) pairs. Only charged particles
not tagged as B-hadron decay products were taken into account.
The depletion of particles in the $b\bar{b}$
inter-parton region with respect to the $bg(\bar{b}g)$ region
at rapidities close to zero is clearly visible.}
\label{y_plot}
\end{figure}
Although the Jetset Monte Carlo reproduces the basic features
of the data, a small discrepancy was observed between the
model prediction and the data.
The number of charged particles with $w_{ij}<0.95$
attached to the $b\bar{b}$
pair and not tagged as B-hadron decay products
is higher in experimental data than predicted by the Parton Shower
model, while the number of
particles attached to the $bg(\bar{b}g)$ pairs is correctly predicted. This
effect does not appear in the total multiplicity measurements because of
the small absolute number of particles attached to the quark-antiquark
pairs. An average number of particles of $2.06\pm 0.09$
was measured to be connected with the
quark-antiquark pairs, while an average of
$1.77\pm 0.05$ was predicted by the Jetset Monte Carlo model.
This results in a systematically higher purity of the method as predicted
by the model and also in a higher multiplicity ratio
$R_{\Delta y =\pm 1} = 2.33 \pm 0.07_{stat}$.
This measurement is based on 2315 selected Monte Carlo events.
The comparison of the Jetset and Herwig \cite{Herwig} Monte Carlo programs
without the detector effects also demonstrates a similar disagreement
between the models with the Herwig model
predicting more particles to be attached to the $b\bar{b}$ pair.
A number of cross checks was performed.
To investigate the influence of the B-hadron decay product exclusion
procedure, two event samples were selected, namely with
more (less) than five excluded particles in the first (second) sample.
The purities of the gluon identification were measured to be
$62\pm 2\%$ for the first sample and $57\pm 2\%$ for
the second one respectively, well above the natural sample purity of
$33\%$. The exclusion procedure, therefore, does not
generate the observed effect, though it naturally improves the results.
Using both charged and neutral particles, without any exclusion,
results in a gluon identification purity of $54\pm 2\%$.
The efficiency and the purity of the B-hadron decay products exclusion
were cross checked by using the following indirect method.
All particles were boosted into the $b\bar{b}$ rest system and
the particle rapidities were calculated for the
particles identified as coming from the B-hadron decays.
The measured rapidity distribution is then compared to the sum of the two
distributions for particles known to come from the B-hadron decays and b-quark
fragmentation, weighted with the corresponding coefficients.
This method yields purities well consistent with the results obtained
in the direct analysis.
Using $g(E_i)=1$ in Eqn.(\ref{label2}) and (\ref{label3})
did not significantly change the performance of the algorithm.
\section{Discussion and Conclusions}
By investigating a sample of double b-tagged events recorded by the DELPHI
experiment at the $Z^0$ energy,
it has been 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, their
connectedness is calculated
as a sum of energy weighted probabilities of particles
attached to a given pair of parton directions.
A consistency check of the procedure is then performed by implementing
an alternative procedure, where the particle colour connection coefficients
are defined as
inverse transverse momenta in the rest frame of a given pair of partons.
For the symmetric Mercedes-type $q\bar{q}g$ events it is found that the
two procedures are strongly correlated and
a gluon emitted in a Mercedes event configuration
can be identified with a purity in excess of 60\%
by using either one of the two weighting procedures.
In analysing multiparton final states it may therefore be useful to combine
the two procedures.
By measuring the rapidity distribution of particles in the rest frame of
the parton-parton systems, a
lack of connection between the $q\bar{q}$ pairs was found.
%\newpage
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\end{document}