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ICHEP'98 \#929 & \hspace{6cm} DELPHI 98-73 CONF 141 \\
Submitted to Pa 3 & 22 June, 1998 \\
\hspace{2.4cm} Pl 4 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf An Upper Limit for \boldmath ${\bf Br(Z^0\rightarrow ggg)}$
from Two and Three Jet Correlations in 3--jet ${\bf Z^0}$ Hadronic Decays}\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\normalsize {
%===================> DELPHI note author list =====> To be filled <=====%
{\bf L. Guerdioukov} $^1$
%========================================================================%
}
\vspace*{1cm}
\end{center}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
An upper limit for $Br(Z^0\rightarrow3g)$
is obtained from a correlation method, which distinguishes statistically
between quark and gluon jets by using
the difference in their charged particle multiplicity distributions.
From the sample of mirror symmetric
three jet events collected by the DELPHI experiment at LEP
during 1991-1995, the 95\% confidence level upper limit is deduced to be:
$Br(Z^0\rightarrow3g)\leq 2.4\times 10^{-2}$.
From the sample of threefold symmetric
three jet events
the 95\% confidence level upper limit is obtained to be:
$Br(Z^0\rightarrow3g)\leq 3.9\times 10^{-3}$.
%=========================================================================%
\end{abstract}
\vspace{\fill}
\begin{center}
Paper submitted to the ICHEP'98 Conference \\
Vancouver, July 22-29
\end{center}
\vspace{\fill}
\par {\footnotesize $^1$ Inst. for High Energy Physics, Serpukov
P.O. Box 35, Protvino, Russian Federation.}
\end{titlepage}
\pagebreak
%==================> DELPHI note text =====> To be filled <======%
% document.tex
%
\section{Introduction}
The measurement of branching ratio for the decay of the $Z^0$--boson
into three gluons is a good test for
the Standard Model which predicts a very small branching
ratio for the decay $Z^0\rightarrow 3g$
from quark loops \cite{Bij}:
%
\begin{eqnarray}
Br^{SM}(Z\rightarrow 3g)~\simeq 2.0\times 10^{-6}
\end{eqnarray}
%
and for the Model of Compositeness of the $Z$--boson which
would induce new couplings and decay modes
and a predicted branching ratio \cite{LEP}:
%
\begin{eqnarray}
Br(Z\rightarrow 3g)~\leq 2.0\times 10^{-3},
\end{eqnarray}
%
much larger than the standard model expectation.
In recent DELPHI paper \cite{UL} an upper limit for $Br(Z\rightarrow 3g)$
has been determined
from a sample of threefold symmetric 3 jet events
in which the angles between jets are in the range $120\pm 20^{\circ}$
(referenced below as M--events).
The analysis is based on the difference between the charged particle
multiplicity distributions of quark and gluon jets.
This difference is exploited by comparing the correlations present
between the jet multiplicities in symmetric 3 jet events, in
general consisting of two quark jets and one gluon jet, to those in uncorrelated
fake
events constructed by mixing jets from different real events.
The upper limit for $Br(Z^0\rightarrow3g)$ is
found to be equal to $1.6\times10^{-2}$.
This method, generally referred to as the correlation method,
has also been applied to the
study of the ratio of the mean charged particle
multiplicities in gluon and quark jets
in symmetric 3 jet events \cite{PPE}.
In present letter the correlation method is applied to the more abundant
sample of
mirror symmetric 3 jet events in which the two smaller energy jets have
approximately equal energy in the range $25\pm5$ GeV
(referenced below as Y--events). The method is modified
in such a way that only correlation between charged particle multiplicities
of smaller energy jets is considered.
The final sample of M--events is processed by using the
procedure described in previous work\cite{UL}.
The data
used were collected by the DELPHI experiment at LEP in the years 1991
to 1995 at centre-of-mass energies around 91.2 GeV.
They consist of about 3.5 million hadronic $Z^0$ decays.
\section{The correlation method}
The multiplicity correlation function is defined as :
%
\begin{eqnarray} \label{C}
C(n_{2},n_{3}) =
{{P(n_{2},n_{3})}\over {P_{uncor}(n_{2},n_{3})}},
\end{eqnarray}
%
where $P(n_{2},n_{3})$ is the probability of observing a
3 jet event
in which the charged particle multiplicities of the smaller energy
jets are equal to
$n_2$ and $n_3$. Jets will always be numbered such that
$n_2 \geq n_3$. The charged particle multiplicity of the biggest energy jet,
$n_1$, is ignored.
$P_{uncor}(n_{2},n_{3})$ is the
corresponding probability for uncorrelated jets
constructed using the mixed event technique: one mixed Y--event
was obtained from three different real 3 jet events by selecting
one jet at random from each event.
Assuming the multiplicities of the individual jets in a real event
to be uncorrelated,
the probability $P(n_{2},n_{3})$ can be expressed
through the multiplicity distributions for gluon jets, $G(n)$,
light ($uds$) quark jets, $Q(n)$,$c$--quark jets, $O(n)$,
and $b$--quark jets, $B(n)$, respectively:
%\setcounter{equation}{3}
\begin{eqnarray}\label{9}
&P(n_{2},n_{3})=
{{1-\beta-\zeta}\over 2}\{(1-R_c-R_b)[G(n_2)Q(n_3)+G(n_3)Q(n_2)]+ \\ \nonumber
& + R_c[G(n_2)O(n_3) + G(n_3)O(n_2)] +
R_b[G(n_2)B(n_3) + G(n_3)B(n_2)]\} + \\ \nonumber
& + \zeta\{(1-R_b-R_c)Q(n_2)Q(n_3)
+R_bB(n_2)B(n_3)+R_cO(n_2)O(n_3)\}
+ \beta G(n_2)G(n_3),
\end{eqnarray}
where
$\beta = N_{ggg}^{sym}/N_{3jet}^{sym}$
is the fraction of three--gluon events and $1-\beta$ the fraction
of $Z\rightarrow q\overline qg$ events in the symmetric 3 jet event sample,
$\zeta$ is the fraction of events with gluon jet carrying highest jet energy
in the event,
$R_c=\Gamma_{c\overline c}/\Gamma_{had}$ and
$R_b=\Gamma_{b\overline b}/\Gamma_{had}$ are
the $Z\rightarrow c\overline c$ and $Z\rightarrow b\overline b$
branching fractions.
%The particle multiplicity distribution of the gluon jet, $G(n)$,
%is assumed to be the same in 3--gluon events as in $q\overline qg$ events.
By construction, jets in the mixed event
sample are completely uncorrelated.
Therefore:
%
\begin{eqnarray}
P_{uncor}(n_{2},n_{3}) = J(n_{2})J(n_{3}),
\end{eqnarray}
%
where
\begin{eqnarray} \label{JN}
&J(n) = {{1-\beta-\zeta}\over 2}[G(n) + (1-R_c-R_b)Q(n)+R_cO(n)+R_bB(n)] + \\
& +\zeta[(1-R_c-R_b)Q(n)+R_cO(n)+R_bB(n)] + \beta G(n). \nonumber
\end{eqnarray}
The experimental correlation function $C(n_{2},n_{3})$
is determined by dividing the number of measured events with
given $n_2$ and $n_3$ by the normalized
number of such events from the mixed event sample.
The multiplicity correlation function for M--events
sample is defined as following:
%
\begin{eqnarray}\label{C3}
C(n_{1},n_{2},n_{3}) =
{{P(n_{1},n_{2},n_{3})}\over {P_{uncor}(n_{1},n_{2},n_{3})}},
\end{eqnarray}
%
where $P(n_{1},n_{2},n_{3})$ is the probability of observing a
M--event
in which the charged particle multiplicities of the jets are equal to
$n_1$, $n_2$ and $n_3$. Jets will always be numbered such that
$n_1 \geq n_2 \geq n_3$.
$P_{uncor}(n_{1},n_{2},n_{3})$ is the
corresponding probability for uncorrelated jets
constructed using the mixed event technique.
Under assumption that the multiplicities of the individual jets in a real event
are uncorrelated,
the probability $P(n_{1},n_{2},n_{3})$ can be expressed
through the multiplicity distributions for jets:
\begin{eqnarray}
&P(n_{1},n_{2},n_{3})=
\beta G(n_{1})G(n_{2})G(n_{3}) + {{1-\beta}\over 3}\times \\ \nonumber
&\times\{(1-R_c-R_b)[G(n_{1})Q(n_{2})Q(n_{3}) +
Q(n_{1})G(n_{2})Q(n_{3})
+ Q(n_{1})Q(n_{2})G(n_{3})] + \\ \nonumber
&+R_c[G(n_1)O(n_2)O(n_3)+O(n_1)G(n_2)O(n_3)+O(n_1)O(n_2)G(n_3)]\ + \\ \nonumber
&+R_b[G(n_1)B(n_2)B(n_3)+B(n_1)G(n_2)B(n_3)+B(n_1)B(n_2)G(n_3)]\}. \\ \nonumber
\end{eqnarray}
The probability of generating the mixed event with $n_1$, $n_2$ and $n_3$
is given by the following formula:
\begin{eqnarray}
P_{uncor}(n_{1},n_{2},n_{3}) = J(n_{1})J(n_{2})J(n_{3}),
\end{eqnarray}
where
\begin{eqnarray}\label{J3}
J(n) = {{1-\beta}\over 3}\{G(n) + 2[(1-R_c-R_b)Q(n)+R_cO(n)+R_bB(n)]\}
+ \beta G(n).
\end{eqnarray}
The experimental correlation function $C(n_{1},n_{2},n_{3})$
is determined by dividing the number of measured events with
given $n_1$, $n_2$ and $n_3$ by the normalized
number of such events from the mixed event sample.
In the analysis the particle multiplicity distributions
of gluon and quark jets, $G(n)$, $Q(n)$, $O(n)$ and $B(n)$, are
assumed to be described by Negative Binomial Distributions (NBD)\cite{NBD}.
To cross--check that the results are not unduly sensitive to this assumption,
a Poissonian parameterization
(PD) of the shapes of the multiplicity distributions was also tried.
The unknown parameters were determined from a fit
of the parametrized correlation function
$C(n_{2},n_{3})$ ($C(n_1,n_{2},n_{3})$)
as defined by equations \ref{C}--\ref{JN} (\ref{C3}--\ref{J3})
to the measured one.
The NBD parameters of $b$--quark jets,$\langle n\rangle_b$ and $k_b$,
were obtained from a separate fit of the
charged particle multiplicity distribution of the smaller energy
$b$--tagged\cite{BTAG} jets in Y-- and M--events samples.
The NBD parameters of gluon jets,$\langle n\rangle_g$ and $k_g$,
were obtained from a fit of the
charged particle multiplicity distribution of the smaller energy not tagged as
$b$ jets in $b$--tagged events with the second smaller energy jet tagged as $b$
jet.
The NBD parameters of light--quark jets,$\langle n\rangle_q$ and $k_q$,
were obtained from a fit of the
charged particle multiplicity distribution of the smaller energy jets in
$uds$--tagged events assuming that distribution is
a superposition of gluon and light--quark jets.
The parameter corresponding to
the difference in mean
multiplicity between $c$-quark and light quark jets was fixed according to
the published data\cite{B347,OPAL,OPALR,ALEPHR,SLD}.
The NBD width parameters of $c$--quark jets, $k_c$,
is assumed to be equal to that of light--quark jets, $k_q$.
Therefore the finally fitted parameter is
the fraction of 3--gluon events, $\beta$.
\section{Experiment and data selection}
A detailed description of the DELPHI detector can be found
elsewhere \cite{18}.
In this analysis only charged particles were used.
%Their energies were calculated from their momenta assuming the pion mass.
Their momenta were measured in the 1.2 T solenoidal magnetic field
by the following tracking detectors: the Micro Vertex Detector,
the Inner Detector, the Time Projection Chamber (TPC, the principal tracking
device of DELPHI), the Outer Detector
and the Forward Chambers A and B.
A charged particle was required to satisfy the following criteria :
\begin{itemize}
\item[--] momentum, $p$, greater than 0.2 GeV/$c$;
\item[--] error on $p < p$;
\item[--] polar angle, $\theta$, with respect to the beam
between $25^\circ$ and $155^\circ$;
\item[--] measured track length in the TPC greater than 50 cm;
\item[--] impact parameter with respect to the nominal beam crossing
point within 5 cm in the transverse $xy$ plane and 10 cm
along the beam direction ($z$-axis).
\end{itemize}
Hadronic events from $Z^0$ decays were then selected if
\begin{itemize}
\item[--] there were at least 5 charged particles;
\item[--] the total energy of charged particles (assuming a pion mass)
in each of the two hemispheres
defined with respect to the beam direction exceeded 3 GeV;
\item[--] the total energy of all charged particles
was greater than 15 GeV.
%\item[--] the angle between beam axis and each jet axis is between
% $40^\circ$ and $140^\circ$.
\end{itemize}
A total of $3.5\times10^6$ events satisfied these cuts. The contamination
from events due to beam-gas scattering and to $\gamma\gamma$ interactions
was estimated to be less than 0.1\% and the background from
$\tau^+\tau^-$ events to be less than 0.3\% of the accepted
events \cite{DELSIM}.
Samples of events with three jets were selected by applying
the DURHAM jet--finder (also known as
the $k_{\bot}$ algorithm),
with jet resolution parameter $y_{min}$=0.015 or 0.035.
The DURHAM jet--finder is well defined in
perturbation theory, allowing calculations to incorporate leading
terms to all orders \cite{OPAL}, and is
widely used in experimental work.
The value $y_{min}$=0.035 has an advantage with respect to smaller values of
$y_{min}$ because it gives the symmetric 3 jet event sample which is less
contaminated by the events without hard gluon emission artificially split
into 3 jet by the jet--finder.
Each reconstructed jet was
required to contain at least 2 charged particle, to have the jet axis
lying in the region $\vert \cos {\theta} \vert \leq$ 0.7,
to have a visible energy larger than 2 GeV and to have
the energy calculated from the
angular relation in the range 25$\pm$5 GeV or greater than 30 GeV.
To eliminate non-planar events, the sum of the angles between the three jets
was required to exceed $357^\circ$.
The total numbers obtained using the DURHAM algorithm are 82994 at
$y_{min}=0.015$ and 54371 at $y_{min}=0.035$.
Threefold symmetric 3--jet events of M--type where selected by projecting the
jets into the 3--jet event plane and requiring the angles between them to be in
the range $100^\circ$ to $140^\circ$.
The total numbers of events are 12030 at
$y_{min}=0.015$ and 13702 at $y_{min}=0.035$.
\section{Results}
The values of the NBD parameters for charged particle multiplicity
distributions in jets were obtained from the fit of respective distributions
for jets selected by using $b$--tagging technique.
The observed charged particle multiplicity distributions were fitted with the
convolution of NBD with the acceptance matrix:
\begin{eqnarray}
f(n)=\sum_{m=m_{min}}^{m_{max}}A_{nm}P^{NBD}_m,
\end{eqnarray}
where $A_{nm}$ is the probability to observe $n$ particles in the jet when the
multiplicity in the produced jet is equal to $m$.
It was calculated as a ratio of the number of jets reconstructed with
multiplicity $n$ after DELPHI detector simulation program DELSIM\cite{DELSIM}
to the number of jets generated by JETSET\cite{JETSET}
with multiplicity $m$ for each energy interval.
The NBD parameters of $b$--quark jets
were obtained from the NBD fit of the
charged particle multiplicity distribution of the smaller energy
jets with Negative Logarithm of Positive Probability (NLPP) for jet
greater than 4. The purity of the sample of $b$ jets is equal to 92\%.
The NBD parameters of gluon jets
were obtained from the NBD fit of the
charged particle multiplicity distribution of the smaller energy
jets with NLPP less than 1 when the second smaller energy jet is
tagged as $b$ jet.
To cross--check the results of the fit the sample of events with both $b$ jets
tagged was also used.
The NBD parameters of light--quark jets
were obtained from the fit of the
charged particle multiplicity distribution of the smaller energy jets in
$uds$--tagged events assuming that distribution is
a superposition of gluon and light--quark jets.
The sample of the $uds$ events was obtained requiring the Maximum NLPP (MNLPP)
in the event to be less than 1. The sample obtained with this cut consist
of 83\% $uds$, 14\% $c$ and 3\% $b$ events. The dependence of the purity of the
sample on MNLPP is shown in fig.\ref{fig1}.
The resulting values of the parameters of multiplicity distributions for $b$,
$uds$ and gluon jets are presented in table \ref{NBDJ}.
It is worthwhile to note that the variance of multiplicity distribution, $D$,
related with $\langle n\rangle$ and $k$ through the formula:
\begin{eqnarray}
\frac{D^2}{\langle n\rangle^2}=\frac{1}{\langle n\rangle}+\frac{1}{k},
\end{eqnarray}
for gluon jets is greater than for quark jets in the M--events sample,
as it is expected from the oscillations of cumulant moments of
parton multiplicity distributions inside a jet\cite{DREMIN}.
The average value of the difference between
the mean charged particle multiplicity in $c$--quark jets
and that in light quark jets,
$\delta_{cl}$, was calculated to be equal to 0.44$\pm$0.21,
the weighted average of the measurement
by OPAL\cite{OPAL} and SLD\cite{SLD}.
The $\zeta$ parameter was determined from HERWIG\cite{HERWIG} generated events
and found to be equal 0.05.
%----------------------------- tables -----------------------------
\begin{table}[h]
\caption{
Number of events, average energy of jet and NBD parameters of charged
particle multiplicity distribution in jets
for jet energy intervals $20\leq E_{jet}\leq 30GeV$ and
$25\leq E_{jet}\leq 35GeV$ and for different $y_{min}$ .}
\label{NBDJ}
\begin{center}
\begin{tabular}{||cccccc||}\hline
Jet & $N_{ev}$ & $E_{jet}$, $GeV$
& $ \langle n\rangle \pm\sigma_{stat}\pm\sigma_{syst}$
& $ k\pm\sigma_{stat}\pm\sigma_{syst}$
& $P(\chi^2)$ \\
\hline
&&\multicolumn{2}{l}{DURHAM, $y_{min}=0.015$ } && \\
$b$ & 9091 & 25.8 $\pm$ 2.8 & 8.80$\pm0.04\pm0.05$
& 80$\pm^{14}_{10}\pm^{10}_{5}$ &0.20\\
$g$ & 2172 & 24.0 $\pm$ 2.8 & 9.32$\pm0.08\pm0.05$
& 59$\pm^{16}_{10}\pm^{15}_{10}$ &0.51\\
$uds$ & 24851 & 25.0 $\pm$ 2.9 & 6.83$\pm0.05\pm0.05$
& 19.5$\pm^{2.1}_{1.7}\pm^{2.0}_{1.5}$ &0.07\\
$b$ & 7755 & 30.5 $\pm$ 2.8 & 9.25$\pm0.04\pm0.08$
& 74$\pm^{17}_{12}\pm^{11}_{8}$ &0.28\\
$g$ & 2470 & 29.0 $\pm$ 2.8 & 10.22$\pm0.08\pm0.05$
& 30$\pm^{4}_{3}\pm^{2}_{1.5}$ &0.69 \\
$uds$ & 13082 & 30.0 $\pm$ 2.9 & 6.25$\pm0.06\pm0.05$
& 72$\pm^{51}_{21}\pm^{30}_{10}$ &0.67\\
&&&&& \\
&&\multicolumn{2}{l}{DURHAM, $y_{min}=0.035$ } && \\
$b$ & 5938 & 25.9 $\pm$ 2.8 & 9.22$\pm0.05\pm0.05$
& 75$\pm^{25}_{15}\pm^{10}_{6}$ &0.42\\
$g$ & 1545 &24.4 $\pm$ 2.8 & 9.92$\pm0.07\pm0.05$
& 97$\pm^{83}_{31}\pm^{21}_{10}$ &0.15\\
$uds$ & 16216 & 25.1 $\pm$ 2.9 & 7.02$\pm0.07\pm0.05$
& 20.0$\pm^{4.5}_{3.2}\pm^{2.5}_{1.5}$ &0.07\\
$b$ & 5216 & 30.4 $\pm$ 2.8 & 9.63$\pm0.05\pm0.04$
& 85$\pm^{30}_{18}\pm^{10}_{5}$ &0.77\\
$g$ & 2039 & 29.1 $\pm$ 2.7 & 10.79$\pm0.09\pm0.06$
& 28$\pm^{4.5}_{3.4}\pm^{4}_{2}$ &0.93\\
$uds$ & 8538 & 29.9 $\pm$ 2.8 & 6.66$\pm0.07\pm0.05$
& 40$\pm^{16}_{9}\pm^{11}_{7}$ &0.17\\
\hline
\end{tabular}
\end{center}
\end{table}
In order to correct for the influence of imperfections of the DELPHI detector,
the correlation method was applied to the samples of
simulated events from the DELPHI detector simulation program
DELSIM \cite {DELSIM}.
In DELSIM, events were generated using the JETSET 7.3 PS program
\cite{JETSET} with DELPHI default parameters \cite{TUN}.
Particles were followed through the detector
and the resulting simulated digitizations were processed with the same
reconstruction programs as the experimental data.
Detector imperfections introduce
a systematic difference between
$C_J(n_{2},n_{3})$ for the events generated by JETSET and
$C_{D}(n_{2},n_{3})$ for the events reconstructed after DELSIM
(i.e. after the detector simulation).
In order to correct for this influence of the detector,
the correlation function
$C(n_{2},n_{3})$ observed for uncorrected data was
multiplied by the ratio
$K(n_{2},n_{3})=C_{J}(n_{2},n_{3})/C_{D}(n_{2},n_{3})$.
In order to take into account the imperfections of the jet finder algorithms,
a further correction factor was introduced. It was calculated as a ratio
$N(n_{2},n_{3})=C_{expected}(n_{2},n_{3})/
C_{observed}(n_{2},n_{3})$ for a normalisation sample
of events obtained by generating
symmetric $Z^0\rightarrow q\overline qg$ decays using JETSET.
This correction
is based on the fundamental property that the
correlation function should equal unity,
i.e. $C_{expected}(n_{2},n_{3})=1$, when the mixed events
are constructed from the same numbers of quarks and gluons as real events.
Indeed the probabilities $P(n_2,n_3)$ and $P_{uncor}(n_2,n_3)$ both
are described by the formula (\ref{9}) in this case.
The total correction factor $K\cdot N$ is typically between 0.9 and 1.1.
The corrected correlation function $C(n_{2},n_{3})$
is presented as a function of $n_3$
in Fig.\ref{fig2} for the DURHAM
jet--finder with $y_{min}=0.015$ for several $n_2$ values.
The curves in Fig.\ref{fig2} are the results of the fit for all values
of $2\leq n_2\leq25$. The numerical results of the fit
are as follows.
The value of $\beta$ is equal to -0.010$\pm$0.022 with probability
of the fit equal to 0.14 for 272 experimental points for $y_{min}$ equal to
0.015 and
$\beta$ is equal to -0.032$\pm$0.037 with probability
of the fit equal to 0.041 for 251 experimental points for $y_{min}$ equal to
0.035.
In order to estimate the
systematic errors due to the uncertainties in the values of the
fixed parameters, the fit was also performed for the central values of these
parameters plus or minus one standard deviation.
The corresponding systematic errors in $\beta$ are detailed in Tables
\ref{SYSY15} and \ref{SYSY35}.
\begin{table}[h]
\caption{
Contributions to the systematic error in $\beta$ from the uncertainties
in the parameters fixed in the fits for Y--events sample at $y_{min}=0.015$.}
\label{SYSY15}
\begin{center}
\begin{tabular}{||cccc||}\hline
Parameter value $\pm$ error
& \multicolumn {2}{c}{ $\sigma_{syst}$ } &\\
& NBD & PD & \\
\hline
$\langle n_g\rangle /\langle n_q\rangle =1.37\pm0.02$
& $^{+0.048}_{-0.054}$ & $^{+0.040}_{-0.044}$ &\\
$\delta_{bl}=1.97\mp0.09$
& $^{+0.013}_{-0.013}$ & $^{+0.010}_{-0.009}$ &\\
$\delta_{cl}=0.44\mp0.21$
& $^{+0.018}_{-0.018}$ & $^{+0.016}_{-0.016}$ &\\
$k_q=19.5\pm^{2.9}_{2.3}$ & $^{+0.011}_{-0.010}$ & &\\
$k_g=59\pm^{22}_{14}$ & $^{+0.013}_{-0.011}$ & &\\
$k_b=80\pm^{17}_{11}$ & $^{+0.001}_{-0.001}$ & &\\
\hline
Total
& $^{+0.056}_{-0.060}$ & $^{+0.044}_{-0.048}$ &\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\caption{
Contributions to the systematic error in $\beta$ from the uncertainties
in the parameters fixed in the fits for Y--events sample at $y_{min}=0.035$.}
\label{SYSY35}
\begin{center}
\begin{tabular}{||cccc||}\hline
Parameter value $\pm$ error
& \multicolumn {2}{c}{ $\sigma_{syst}$ } &\\
& NBD & PD & \\
\hline
$\langle n_g\rangle /\langle n_q\rangle =1.41\pm0.02$
& $^{+0.045}_{-0.052}$ & $^{+0.047}_{-0.041}$ &\\
$\delta_{bl}=2.20\mp0.11$
& $^{+0.013}_{-0.013}$ & $^{+0.011}_{-0.011}$ &\\
$\delta_{cl}=0.44\mp0.21$ & $^{+0.015}_{-0.016}$ & $^{+0.013}_{-0.013}$ &\\
$k_q=20.0\pm^{5.1}_{3.5}$ & $^{+0.015}_{-0.014}$ & &\\
$k_g=97\pm^{86}_{33}$ & $^{+0.018}_{-0.016}$ & &\\
$k_b=75\pm^{27}_{16}$ & $^{+0.001}_{-0.001}$ & &\\
\hline
Total
& $^{+0.054}_{-0.060}$ & $^{+0.050}_{-0.044}$ &\\
\hline
\end{tabular}
\end{center}
\end{table}
Further systematic errors were estimated taking into account the
variation of the results obtained with different
cuts on the jet multiplicity $n_2$
and the uncertainty in the values
of the total correction coefficients.
The resulting systematic bias in the values of $\beta$ does not exceed
0.004 and 0.002
for $y_{min}$ equal to 0.015 and 0.035 respectively.
Including the systematic errors in $\beta$ leads to the following
final results for $\beta$:
\begin{eqnarray*}
\beta &=& -0.010 \pm{0.022}(stat.) \pm^{0.056}_{0.060}(syst.)
\ \ \ \ ( y_{min}=0.015)\\
\beta &=& -0.032 \pm{0.037}(stat.) \pm^{0.054}_{0.060}(syst.)
\ \ \ \ ( y_{min}=0.035).
\end{eqnarray*}
%{\bf Calculation of an Upper limit for $Br(Z^0\rightarrow ggg)$ }
The branching fraction $Br(Z^0\rightarrow ggg)$ is calculated from
$\beta$ using the following formula:
\begin{eqnarray}
Br(Z^0\rightarrow3g)=\beta\cdot Br(Z^0\rightarrow hadr)\cdot
{{N^{sym}_{3jet}}\over{N_{hadr}}}\cdot
{{N_{\Upsilon}}\over{N^{sym}_{\Upsilon}}}, \label{BR}
\end{eqnarray}
where
%$\beta={{N^{sym}_{ggg}}/{N^{sym}_{3jet}}}$ is the fraction of 3
%gluon events in symmetric 3-jet events sample, \\
\mbox{${{N^{sym}_{3jet}}/{N_{hadr}}}$}
is the fraction of symmetric 3 jet events
in the hadronic event sample and
\mbox{${{N^{sym}_{\Upsilon}}/{N_{\Upsilon}}}$} is the fraction of symmetric
decays in an $\Upsilon$--like $1^{--}$ quarkonium state
to three gluons.
The latter ratio was calculated using JETSET 7.3.
The mass of the pseudo-onium was chosen to be equal to the $Z$ mass.
Due to the identical helicity structure of $Z^0\rightarrow ggg$ and
$\Upsilon \rightarrow ggg$ decays, the angular distributions for jets
from the two sources are expected to be identical.
Thus
${{N^{sym}_{ggg}}/{N_{ggg}}}$ should equal
${{N^{sym}_{\Upsilon}}/{N_{\Upsilon}}}$.
The numerical value of the factor relating $Br(Z\rightarrow3g)$ to $\beta$
in eq.(\ref{BR}) was found to be 0.263 at $y_{min}$=0.015
and 0.184 at $y_{min}$=0.035.
To calculate the 95\% confidence level upper limits on
the branching fraction $Br(Z^0 \rightarrow ggg)$,
%have been calculated by applying (\ref{BR}) to the value
%$\beta +\Delta^{95\%}\beta$.
the systematic errors were added in quadrature to the statistical errors
and unphysical negative values of $\beta$ were forced up to have $\beta=0$.
The calculation gave:
%
$$UL\{Br(Z\rightarrow3g)\}=0.032\ \ (y_{min}=0.015)$$
and
$$UL\{Br(Z\rightarrow3g)\}=0.024\ \ (y_{min}=0.035).$$
For the limit calculation, the systematic and statistical errors on $\beta$
were assumed to be uncorrelated.
The cross-check of using the Poissonian parametrisation
of the multiplicity distributions
gave similar estimates of the upper limit, namely
0.053 and 0.035 respectively with the probability of the fit at the
level of $3\times10^{-3}$ at $y_{min}=0.015$ and 0.07 at $y_{min}=0.035$.
The analysis of the increased statistics of the M--events
of the DELPHI experiment was performed.
In contrast with previous work\cite{UL} the difference in average charged
particle multiplicities in $c$--quark jet and light quark jet is taken into
account according to eqn.\ref{C3}-\ref{J3}.
The contributions to the systematic error in $\beta$ from the uncertainties
in the parameters fixed in the fits are collected
in Tables \ref{SYSM15} and \ref{SYSM35}.
\begin{table}[h]
\caption{
Contributions to the systematic error in $\beta$ from the uncertainties
in the parameters fixed in the fits for M--events sample at $y_{min}=0.015$.}
\label{SYSM15}
\begin{center}
\begin{tabular}{||cccc||}\hline
Parameter value $\pm$ error
& \multicolumn {2}{c}{ $\sigma_{syst}$ } &\\
& NBD & PD & \\
\hline
$\langle n_g\rangle /\langle n_q\rangle =1.64\pm0.02$
& $^{+0.010}_{-0.011}$ & $^{+0.003}_{-0.004}$ &\\
$\delta_{bl}=3.00\mp0.12$
& $^{+0.012}_{-0.012}$ & $^{+0.004}_{-0.004}$ &\\
$\delta_{cl}=0.44\mp0.21$ & $^{+0.002}_{-0.002}$ & $^{+0.002}_{-0.002}$ &\\
$k_q=72\pm^{60}_{23}$ & $^{+0.001}_{-0.001}$ & &\\
$k_g=30.0\pm^{4.5}_{3.4}$ & $^{+0.004}_{-0.003}$ & &\\
$k_b=74\pm^{20}_{14}$ & $^{+0.001}_{-0.001}$ & &\\
\hline
Total
& $^{+0.016}_{-0.017}$ & $^{+0.005}_{-0.005}$ &\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\caption{
Contributions to the systematic error in $\beta$ from the uncertainties
in the parameters fixed in the fits for M--events sample at $y_{min}=0.035$.}
\label{SYSM35}
\begin{center}
\begin{tabular}{||cccc||}\hline
Parameter value $\pm$ error
& \multicolumn {2}{c}{ $\sigma_{syst}$ } &\\
& NBD & PD & \\
\hline
$\langle n_g\rangle /\langle n_q\rangle =1.62\pm0.03$
& $^{+0.011}_{-0.014}$ & $^{+0.011}_{-0.013}$ &\\
$\delta_{bl}=2.97\mp0.11$
& $^{+0.009}_{-0.011}$ & $^{+0.008}_{-0.009}$ &\\
$\delta_{cl}=0.44\mp0.21$ & $^{+0.001}_{-0.003}$ & $^{+0.002}_{-0.003}$ &\\
$k_q=40\pm^{19}_{11}$ & $^{+0.001}_{-0.002}$ & &\\
$k_g=28\pm^{6}_{4}$ & $^{+0.003}_{-0.004}$ & &\\
$k_b=85\pm^{32}_{19}$ & $^{+0.001}_{-0.001}$ & &\\
\hline
Total
& $^{+0.015}_{-0.019}$ & $^{+0.014}_{-0.016}$ &\\
\hline
\end{tabular}
\end{center}
\end{table}
The fit and the systematic errors in $\beta$ give the following
final results for $\beta$:
\begin{eqnarray*}
\beta &=& -0.009 \pm{0.014}(stat.) \pm^{0.016}_{0.017}(syst.)
\ \ \ \ ( y_{min}=0.015)\\
\beta &=& -0.003 \pm{0.013}(stat.) \pm^{0.015}_{0.019}(syst.)
\ \ \ \ ( y_{min}=0.035).
\end{eqnarray*}
The probability
of the fit is equal to 0.97 for 235 experimental points for $y_{min}$ equal to
0.015 and 0.98 for 242 experimental points for $y_{min}$ equal to
0.035.
The numerical value of the factor relating $Br(Z\rightarrow3g)$ to $\beta$
in eq.(\ref{BR}) was found to be 0.144 at $y_{min}$=0.015
and 0.100 at $y_{min}$=0.035.
The 95\% confidence level upper
limit for the $Z^0 \rightarrow ggg$ branching ratio is obtained as follows:
%
$$UL\{Br(Z^0\rightarrow3g)\}=0.0060\ \ (y_{min}=0.015)$$
and
$$UL\{Br(Z^0\rightarrow3g)\}=0.0039\ \ (y_{min}=0.035).$$
%
The Poissonian parametrisation
of the multiplicity distributions
gave similar estimates of the upper limit, namely
0.0052 and 0.0040 respectively with
acceptable probability of the fit in both cases.
As a final result of the present study, the value of the upper limit from
the M--events sample at $y_{min}=0.035$ is accepted because in this sample the
jetfinder algorithm gives the best separation of the jets.
The upper limit is set four times lower the published paper\cite{UL} due to
increased statistics of the experiment and reduced number of free parameters of
the fit.
\section{Summary}
By using a correlation method
based on the difference between the
particle multiplicity distributions of
quark and gluon jets, an upper limit at 95\% confidence level
for the $Z^0 \rightarrow ggg$ branching ratio has been established
from the sample of Y--events:
%
$$Br(Z^0\rightarrow3g)\leq 3.2\times 10^{-2}$$
for the DURHAM jet--finder with $y_{min}$=0.015 and
$$Br(Z^0\rightarrow3g)\leq 2.4\times 10^{-2}$$
with $y_{min}$=0.035.
%
The correlation method modified for the case of two jet correlations
described in present paper can be applied in principle to esimate the possible
contribution in $2jet+\gamma$ event sample from the process
$Z^0\rightarrow\gamma gg$ which has the one order of magnitude larger
cross section than that for
$Z^0\rightarrow 3g$ in the compositeness model.
The correlations between multiplicities of 3 jets give more restrictive
values for the upper limit:
%
$$Br(Z\rightarrow3g)\leq 6.0\times 10^{-3}$$
for the DURHAM jet--finder with $y_{min}$=0.015 and
$$Br(Z\rightarrow3g)\leq 3.9\times 10^{-3}$$
with $y_{min}$=0.035.
%
At the present level of statistics, no signal
of the decay $Z^{0}\rightarrow ggg$ expected from the compositeness model
is observed.
The variance of multiplicity distribution
for gluon jets is greater than for quark jets in the M--events sample,
as it is expected from the oscillations of cumulant moments of
parton multiplicity distributions inside a jet.
\subsection*{Acknowledgements}
\vskip 3mm
We are greatly indebted to our technical staff and collaborators and our
funding agencies for their support in building and operating the DELPHI
detector, and to the members of the CERN-SL Division for the excellent
performance of the LEP collider.
%The author is greatly indebted to the Bergische
%Universit\"at -- GH Wuppertal for financial support and to the Wuppertal
%DELPHI group for the hospitality extended to him.
%========================================================================
%\newpage
\begin{thebibliography}{99}
\bibitem{Bij} E. W. N. Glover and J. J. van der Bij, CERN 89-04,
v.2, p.1, 1989.
\bibitem{LEP} F. Boudjema and F. M. Renard, CERN 89-04, v.2, p.182, 1989.
\bibitem{UL} DELPHI Collab., P. Abreu et al.,
Phys. Lett. {\bf B389} (1996) 405.
\bibitem{PPE}DELPHI Coll., P. Abreu et al., Z. Phys. {\bf C70} (1996) 179.
\bibitem{BTAG} G. V. Borisov, Preprint IHEP (Protvino) 94-98 (1994);
DELPHI Collab., P. Abreu et al., Z. Phys. {\bf C65} (1995) 555.
\bibitem{NBD} F. Bianchi, A. Giovannini, S. Lupia and R. Ugoccioni,
Z. Phys. {\bf C58} (1993) 71.
\bibitem{B347}DELPHI Coll., P. Abreu et. al., Phys. Lett. {\bf B347} (1995) 447.
\bibitem{OPAL}OPAL Coll., R. Akers et al., Phys. Lett. {\bf B352} (1995) 176.
\bibitem{OPALR} OPAL Collab., P. Acton et al., Z. Phys. {\bf C58} (1993) 387.
\bibitem{ALEPHR}ALEPH Coll., D. Buskulic et al.,
Phys. Lett. {\bf B384} (1996) 353.
\bibitem{SLD} SLD Coll., K. Abe et al., SLAC-PUB-7172(June 1996),
submitted to Phys. Lett. B.
\bibitem{18} DELPHI Collab., P. Aarnio et al., Nucl. Instr. and Meth.
{\bf A303} (1991) 233.
\bibitem{DREMIN}I.M. Dremin, Phys. Lett. {\bf B 313} (1993) 209.
\bibitem{DELSIM} DELPHI Collab., P. Abreu et al.,
Nucl. Instr. and Meth. {\bf A378} (1996) 57.
\bibitem{JETSET} T. Sj\"ostrand, Comp. Phys. Comm. {\bf 27} (1982) 243;
{\bf 28} (1983) 229; {\bf 39} (1986) 347;
T. Sj\"ostrand and M. Bengtsson,
Comp. Phys. Comm. {\bf 43} (1987) 367.
\bibitem{TUN} DELPHI Coll., P. Abreu et al., Z. Phys. {\bf C73} (1996) 11.
\bibitem{HERWIG}G. Marchesini and B. Webber, Nucl. Phys. {\bf B}310 (1988) 461;
I. G. Knowles, Nucl. Phys. {\bf B} 310 (1988) 571;
G. Marchesini et al., Comp. Phys. Comm. 67 (1992) 465.
\end{thebibliography}
%\bibitem{16} P. V. Chliapnikov, V. A. Uvarov and F. Bianchi, Phys. Lett.
% {\bf B300} (1993) 183.
%\bibitem{9} DELPHI Collab., P. Abreu et al., Z. Phys. {\bf C56} (1992) 63.
%\bibitem{25} DELPHI Collab., P. Abreu et al., Z. Phys.
% {\bf C52} (1991) 52.
%\bibitem{29} DELPHI Collab., P. Abreu et al., Preprint CERN-PPE/94-02 (1994)
% (submitted to Z. Phys.C).
%\bibitem{DN} F. Bianchi, P. Chliapnikov, L. Gerdyukov and A. Giovannini,
% DELPHI 95-40 PHYS 483, 1995.
%DELSIM User Guide, DELPHI 87-96 PROG-99 (Geneva, July 1989);
%DELSIM Reference Manual, DELPHI 87-98 PROG-100 (Geneva, July 1989).
%===============================================================================
\newpage
\begin{figure}
%\epsfig{file=usr:[gerdyukov.paper]fig1.ps,width=20cm,
\epsfig{file=fig1.ps,width=15cm,
bbllx=0cm,bblly=4.5cm,bburx=21cm,bbury=25cm}
\caption{
The purity of the sample of events
as a function of the maximum of negative logarithm of positive probability
for jets in the event.}
\label{fig1}
\end{figure}
\begin{figure}
%\epsfig{file=usr:[gerdyukov.paper]fig2.ps,width=20cm,
\epsfig{file=fig2.ps,width=15cm,
bbllx=0cm,bblly=4.5cm,bburx=21cm,bbury=25cm}
\caption{ The corrected correlation function
$C(n_{2},n_{3})$ as a function of
the smallest jet multiplicity $n_3$
for different values of the jet multiplicity $n_2$.
Symmetric 3--jet events are selected
from the sample of DELPHI data by using
the DURHAM jet--finder with $y_{min}$ equal to 0.015.
The curves are the result of the fit.}
\label{fig2}
\end{figure}
\end{document}