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\begin{document}
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\begin{titlepage}
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\begin{tabular}{l r}
HEP'99 \# 1\_571 & \hspace{6cm} DELPHI 99-127 CONF 314 \\
Submitted to Pa 1 & 15 June 1999 \\
\hspace{2.4cm} Pl 1 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Precise Measurement of the Gluon Fragmentation Function
and a Comparison of the Scaling Violation in Gluon and Quark Jets }\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\end{center}
\vfill\vfill
\begin{center}
Paper submitted to the HEP'99 Conference \\
Tampere, Finland, July 15-21
\end{center}
\vspace{\fill}
\end{titlepage}
\newpage
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\pagebreak
\begin{titlepage}
\vspace*{0.6cm}
\end{titlepage}
\newpage
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%%% FROM HERE ON INSERT YOUR COMPLETE PAPER
% (including the title, the authors, and the abstract)
\begin{titlepage}
\pagenumbering{arabic}
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\begin{tabular*}{15.cm}{lc@{\extracolsep{\fill}}r}
{\bf DELPHI Collaboration} &
\hspace*{1.3cm} \epsfig{figure=dolphin_bw.eps,width=1.2cm,height=1.2cm}
&
%===================> DELPHI note number =====> To be filled <=====%
DELPHI 99-127 CONF 314
%========================================================================%
\\
& &
%===================> DELPHI note date =====> To be filled <=====%
%19 March, 1999
%\today, 1999 DRAFT 0.95
\today, 1999
%========================================================================%
\\
&&\\ \hline
\end{tabular*}
\vspace*{2.cm}
\begin{center}
\Large
{\bf \boldmath
%===================> DELPHI note title =====> To be filled <=====%
Precise Measurement of the Gluon Fragmentation Function
and a Comparison of the Scaling Violation in Gluon and Quark Jets
%========================================================================%
} \\
\vspace*{2.cm}
\normalsize {
%===================> DELPHI note author list =====> To be filled <=====%
{\bf K. Hamacher, O. Klapp, P. Langefeld, M. Siebel}\\
{\footnotesize Fachbereich Physik, University of Wuppertal, Postfach
100 127, DE-42097 Wuppertal, Germany}\\
%========================================================================%
}
%\vspace*{2.cm}
\end{center}
\vspace{\fill}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
The fragmentation functions of quarks and gluons
are measured in various three jet topologies in Z decays from the full
data set
collected with the {\sc Delphi} detector at the Z resonance
between 1992 and 1995.
The results at different values of transverse-momentum-like
scales are compared.
%\\
%Gluon jets are identified in three jet events containing primary heavy quarks
%by using impact parameter information.
%Comparable quark jet properties are obtained from light quark dominated
%three jet events.
\\
A precise measurement of the gluon fragmentation function has been performed
and a parameterization of the quark and gluon fragmentation
functions at a fixed reference scale is given.
The scale dependences of the quark and gluon fragmentation functions
show the predicted pattern of scaling violations.
The scaling violation for quark jets as a function of a
transverse-momentum-like scale is in a good agreement with that
observed in $\mbox{e}^+\mbox{e}^-$ annihilation.
For gluon jets it appears to be significantly stronger.
\\
The gluon and quark scale dependences are in good agreement with the prediction
of the DGLAP equations from which
the colour factor ratio $C_A/C_F$ is measured to be:
$$
\frac{C_A}{C_F} = 2.23 \pm 0.09_{stat.} \pm 0.06_{sys.},
$$
using the {\sc Durham} cluster algorithm.
\end{abstract}
\vspace{\fill}
\begin{center}
%==========> Proceedings.. presented at ..==> To be filled if needed<=====%
Paper submitted to the HEP'99 Conference \\
Tampere, Finland, July 15-21
%=========================================================================%
\end{center}
\vspace{\fill}
\end{titlepage}
%##################################################################### Text
%==================> DELPHI note text =====> To be filled <======%
%**************************************************************************
%**************************************************************************
\section{Introduction}
Collinear divergent terms appearing in perturbative calculations including
hadronic initial or final states can be absorbed in the definition of
structure- or fragmentation functions. So these experimentally measured
functions become integral part of inelastic cross-section calculations
including hadrons.
Compared to the large amount of measurements available for quark fragmentation
functions~\cite{quarkfragfun} the information on gluon fragmentation functions is sparse. The
reason is that gluons only appear as higher order contributions in the final
state of $e^+e^-$ annihilation or deep inelastic
scattering. Therefore also the
underlying hard scale of the measured gluon fragmentation function so far has
been left open. Precise measurements of the gluon fragmentation function at
defined underlying scales and a comparison of the evolution with scale of
gluon and quark fragmentation functions are the subject of this paper.
The extraction of the gluon (and quark) fragmentation function is performed
from three jet events observed in the hadronic Z decays. The assignment of
jets to individual gluons or quarks follows the evident analogy to tree level
graphs.
%This implies that higher order corrections (${\cal O}(\alpha_S/\pi)$)
%need to be neglected in this analysis.
As the centre-of-mass energy for all events entering this analysis is equal to
the Z-mass a scale dependence only becomes accessible from different three
jet topologies. The transverse momentum like scales applied in this analysis
are motivated by MLLA calculations of multihadron production
(for an overview see~\cite{khoze,greybook}), in particular
these scales follow from the coherence of gluon radiation.
The larger colour factor, $C_A$, relevant for bremsstrahlung from a gluon
compared to that for a quark, $C_F$, causes scaling violations of the gluon
fragmentation function to be stronger than that for quarks. This strong
dynamical dependence of the gluon fragmentation already demands to specify
the evolution scale. As a relatively large range of scales,
similar to that covered by the {\sc Petra} experiments, turned out to be
accessible in this analysis, a comparison of the scaling violation of gluon and
quark jets is feasible. This is used to demonstrate the correctness of the
scales employed and as an experimental cross-check of the colour factors. This
is an important check of QCD as the values of the colour factors are a direct
consequence of the $SU(3)$ group structure of QCD.
This paper is organized as follows. Section~\ref{sec_theo} introduces basic
definitions used throughout this paper and discusses the evolution scales and
other theoretical preliminaries. Section~\ref{sec_data} gives a brief
survey of the detector, the experimental data set, cuts, experimental
selections, procedures and corrections applied to extract the gluon and (light)
quark fragmentation function. Results on the gluon and quark fragmentation
functions are presented in section~\ref{sec_frag} including a short discussion
of experimental systematic errors. The corresponding $\xi$ distributions are
also discussed. Section~\ref{sec_scale} is devoted to the
comparison of the extracted quark fragmentation functions to those measured
in $e^+e^-$ annihilation at
lower energies and to gluon fragmentation functions obtained from symmetric and
non-symmetric three jet topologies. The chosen scales are validated from these
comparisons and from the behaviour of the jet broadening $\beta$.
In section~\ref{sec_fit} the gluon and quark fragmentation
functions are fit with a DGLAP evoluted ansatz for the fragmentation functions.
Parameterizations of the gluon fragmentation functions are given and scaling
violation for gluon and quark jets are compared in detail. A summary and
conclusion are given in section~\ref{sec_con}.
%******************************************************************************
%******************************************************************************
\section{Theory}
\label{sec_theo}
%*****************************************************************************
\subsection{Jet Scales}
\label{kappa_scale}
The assessment of scaling violations demands the specification of the
scale underlying the process under study.
This scale enters in the strong coupling and particularly specifies
the size of the available phasespace.
It is necessarily proportional to an outer scale like the
centre-of-mass energy.
As in the measurement of scaling violations the scale enters
logarithmically, i.e. only the relative change of scale matters (see
GDLAP Eqn.~\ref{gl_gdlap1} and~\ref{gl_gdlap2}), this outer
scale may be directly taken as the scale.
For this analysis the situation is different.
The fragmentation functions are studied for individual gluon and quark
jets in three jet events descending from Z decays
(See section~\ref{sec_ev-topo} for the definition of jets and event
topologies.).
As in these events the centre-of-mass energy is constant the relevant
scales for the individual jets need to be determined from the jet
energies and the event topology.
The event topology is especially important as due to the quantum
nature of QCD the soft radiation off the individual high energy
partons interferes.
In consequence the radiation attributed to a hard parton is limited to
opening angles determined by the angles between the hard
partons. This phenomenon is called angular ordering~\cite{angular_ord}.
It may be viewed as an effective reduction of the phasespace available
to soft radiation and thus can be absorbed in an appropriate
definition of the scale. So the scale relevant to this analysis
will be a product of jet energy (or momentum) and
angle of the hard partons, i.e. the scale will be transverse momentum like.
Studies of hadron production in processes with non-trivial topology
have shown that the characteristics of the parton cascade prove to
depend mainly on the scale $\kappa=2E_{jet}\sin \theta/2 \sim
E\theta$~\cite{ddt}.
$E_{jet}$ is the calculated jet energy and
$\theta$ the angle with respect to the closest jet.
For large angles ($\theta \rightarrow \pi$) this scale coincides with
the centre-of-mass energy $E_{CM}$.
$\kappa$ is proportional to $\sqrt{y}$ as defined with the Durham jet
finder.
Instead of using this scale definition we chose for convenience to
take the so called hardness scale
\begin{eqnarray*}
\kappa_H = E_{jet} \sin \frac{\theta}{2} \, ,
\end{eqnarray*}
as it corresponds more closely to a single jet scale as e.g.
the jet energy which is often taken
as the intuitive scale~\cite{paper,opal_pap}.
This definition leaves the relative change of the scale unaltered.
A calculation of the multiplicity of three jet events~\cite{khoze}
predicts the scale of a gluon jet, $\tilde{p}_1^T$, to be:
\begin{equation}
\tilde{p}_1^T = \sqrt{2 \frac{ (p_qp_g) (p_{\bar{q}}p_g) }
{ p_qp_{\bar{q}} } } \, .
\label{p1tdef}
\end{equation}
For symmetric three jet topologies the definition of $p_1^T = 1/2 \cdot
\tilde{p}_1^T$ and
$\kappa_H$ coincide.
The relative differences between these scales turn out to be below
10\% for the events accepted in this analysis.
As particle production from soft gluon radiation is the complementary
process to scaling violations besides $\kappa_H$ $p_1^T$ is
applied as the scale for gluon jets.
For further comparison also the transverse momentum with respect to
the leading jet
$
\kappa_T=E_{jet}\sin \vartheta
$, here $\vartheta$ is the smaller of the two angles w.r.t. to the axis of jet
one, and $E_{jet}$ have been tried.
\begin{figure}[p]
\breite 11.4cm
\breitemp 11.5cm
\begin{center}
\rotatebox{90}{
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{sc-plotscale}
\caption{Different jet scales for symmetric event topologies}
\label{fg_plotscale}
\end{center}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{kappa}
\caption{Separation of symmetric three jet event topologies
by the jet energy and hardness scales.}
\label{kappa_topo}
\end{center}
\end{minipage}
}
\end{center}
\end{figure}
Fig.~\ref{fg_plotscale} shows the different behaviour of the scales for
symmetric three jet event topologies and
Fig.~\ref{kappa_topo} compares the distribution of
the jet energies $E$ and hardnesses $\kappa_H$
for the jets of five symmetric event topologies.
The upper plot of Fig.~\ref{kappa_topo} shows the distribution of the
jet energies. As can be seen, there is a big overlap
among the energy distributions for fixed topologies.
The lower plot shows the distributions of $\kappa_H$.
The $\kappa_H$ distributions are clearly separated for the different
symmetric event topologies.
This can be understood easily: in the case of symmetric events
and massless jets, $\kappa_H$ falls steeply to $0$ with
increasing $\theta_2 (=\theta_3)$,
while the energy becomes nearly constant (see Fig.~\ref{kappa_topo}c).
This makes $\kappa_H$ less sensitive to small deviations
from the exact symmetric topology than the jet energy itself.
Finally it should be noted that the relative change of scales is bigger
in case of $\kappa_H$ compared to the jet energy.
%\clearpage
%****************************************************************************
\subsection{Scale Dependence of the Fragmentation Function of Quark and
Gluon Jets}
\begin{figure}[h]
\begin{center}
\mbox{\epsfig{file=il-feyn-abs.eps,width=13cm}}
\caption{Diagrams of the fundamental QCD couplings}
\label{qcdfeyn}
\end{center}
\end{figure}
The fundamental QCD couplings are illustrated in Fig.~\ref{qcdfeyn}.
The Casimir or colour factors $C_{F}$, $C_{A}$,
and $T_{F}$ determine the apparent coupling strengths of
gluon radiation from quarks (Fig.~\ref{qcdfeyn}a),
of the triple-gluon vertex (Fig.~\ref{qcdfeyn}b), and of gluon
splitting into a quark-antiquark pair (Fig.~\ref{qcdfeyn}c), respectively.
Within $SU(3)$, these factors are
$C_{F}=4/3$, $C_{A}=3$, and $T_{F}=1/2$, which has to be weighted
by the number of active quark flavors $n_{F}$ (here $n_{F}$ is taken to be 5).
The scale dependence (scaling violation) of gluon and quark
fragmentation functions $D^{H}_{g,q}(x_E,s)$
into a hadron $H$ as described by the DGLAP\footnote{{\bf D}okshitzer, {\bf
G}ribov, {\bf L}ipatov,
{\bf A}ltarelli, and {\bf P}arisi} equations~\cite{elli-pirelli}
(see Eqn.~\ref{gl_gdlap1} and~\ref{gl_gdlap2}).
is sensitive to the individual splittings depicted in Fig.~\ref{qcdfeyn}.
$s$ is relevant scale to be replaced by the scale $(2\kappa)^2$.
In the limit of large hadron energy fractions $x_E=E_{hadron}/E_{jet}$,
i.e. for $x_E \ge \frac{1}{2}$, the lower energy parton
in a splitting process cannot contribute.
In a $q \rightarrow qg$ splitting process
the lower energy parton is almost always the gluon. The $g\rightarrow q\bar{q}$
splitting is disfavoured w.r.t. $g\rightarrow gg$
(compare Eqn.~\ref{gl_split1}-\ref{gl_split4}).
The complete (leading order) evolution equations
for quarks and gluons are:
\begin{eqnarray}
\frac{dD^{H}_{g}(x_E,s)}{d\ln{s}} &=& \frac{\alpha_{s}(s)}{2 \pi} \cdot
\int_{x_E}^{1}\frac{dz}{z} \left[
P_{g \rightarrow gg}(z) \cdot D^{H}_{g}(\frac{x_E}{z},s) +
P_{g\rightarrow q\bar{q}}(z) \cdot D^{H}_{q}(\frac{x_E}{z},s)
\right]
\label{gl_gdlap1} \\
\frac{dD^{H}_{q}(x_E,s)}{d\ln{s}} &=& \frac{\alpha_{s}(s)}{2 \pi} \cdot
\int_{x_E}^{1}\frac{dz}{z} \left[
P_{q \rightarrow qg }(z) \cdot D^{H}_{q}(\frac{x_E}{z},s) +
P_{q \rightarrow gq }(z) \cdot D^{H}_{g}(\frac{x_E}{z},s)
\right] \, .
\label{gl_gdlap2}
\end{eqnarray}
These equations are the transpose of the DGLAP equations
for structure functions, however, this simple
relation does not persist at the next-to-leading
order. The relevant Altarelli Parisi splitting kernels
are:
\begin{eqnarray}
P_{q\rightarrow qg}(z) &=& C_F \cdot
\frac{1+z^2}{(1-z)_+} +
\frac{4}{3} \cdot \frac{3}{2}\delta(1-z) \\
\label{gl_split1}
P_{g\rightarrow gg}(z) &=& 2 C_A \cdot \left[
\frac{z}{(1-z)_+} + \frac{1-z}{z} \right] +
\left( \frac{11 N_C - 4 n_F T_F}{6}
\right) \delta(1-z) \\
\label{gl_split2}
P_{g\rightarrow q\bar{q}}(z) &=& 2 n_F T_F \cdot (z^2+(1-z)^2) \\
\label{gl_split3}
P_{q\rightarrow gq}(z) &=& C_F \cdot \frac{1+(1-z)^2}{z}
\label{gl_split4}
\end{eqnarray}
Here the 'plus' distribution is defined such that the integral with any
sufficiently smooth distribution $f$ is
\begin{eqnarray*}
\int_{0}^{1} dx \frac{f(x)}{(1-x)_+} &=&
\int_{0}^{1} dx \frac{f(x)-f(1)}{1-x}\, , \\
\m{and} \quad \frac{1}{(1-x)_+} &=& \frac{1}{1-x} \m{for}\, 0 \le x <1 \, .
\end{eqnarray*}
The 'plus' and the $\delta$ terms stem from virtual diagrams and
regularize the $1/(1-z)$ singularities.
The logarithmic slope $\frac{d\ln D^{H}_p(x_E,s)}{d\ln s}$
for each fragmentation function independently measures the product of the strong
coupling and the colour factors of the relevant splitting kernels.
Thus the ratio
\begin{eqnarray*}
r_{\cal S}(x_E) =
\frac{d\ln D^{H}_g(x_E,s)}{d\ln s}/\frac{d\ln D^{H}_q(x_E,s)}{d\ln s}
\end{eqnarray*}
is in the limit of large $x_E$ equal to the ratio of colour factors.
The slopes and the ratio can be predicted by solving the DGLAP equation
numerically. A detailed description is given in~\cite{evolve}.
Here a first order program is employed~\cite{prog_webber}.
The following ansatz has been used to parameterize the fragmentation
functions at a fixed reference scale $\kappa_0$ to start the evolution,
similar to other analyses~\cite{xfundelphi,xfunaleph}:
\begin{eqnarray}
D^{H}_p(x_E) = p_3 \cdot x_E^{p_1} \cdot (1-x_E)^{p_2}
\cdot \exp{(-p_4 \cdot \ln^2{x_E})}
\label{gl_dfrac}
\end{eqnarray}
The parameters $p_i^{q,g}$, $\Lambda_{QCD}$ and the colour factor $C_A$ shall
be fitted simultaneously.
The slope ratio $r_{\cal S}(x_E)$ can be used directly to
measure $\frac{C_A}{C_F}$.
In principle $r_{\cal S}(x_E)$ is also accessible by measuring the ratio
of the slopes of the gluon and quark fragmentation functions
$D_p^{F}(x_E,s)|_{x_E}$ independently. The method of the simultaneous fit
yields smaller errors, because the $x_E$ interval is extended.
%*****************************************************************************
%*****************************************************************************
\section{Data Analysis}
\label{sec_data}
This section describes the parts of
the {\sc Delphi} detector relevant to this analysis, the particle and event
selection, the jet reconstruction, the event topologies analysed, the impact
parameter tagging used for selecting gluon and quark jet samples,
and the subtraction method used
to extract the properties of pure light quark and gluon jet samples.
%*****************************************************************************
\subsection{The {\sc Delphi} Detector}
{\sc Delphi} is a hermetic detector with a solenoidal magnetic
field of 1.2\,T.
The tracking detectors, situated in front of the electromagnetic calorimeters
are a silicon micro-vertex detector VD,
a combined jet/proportional chamber inner detector ID,
a time projection chamber TPC as the major tracking device, and
the streamer tube detector OD in the barrel region. The forward region
is covered by the drift chamber detectors FCA and FCB.
The electromagnetic calorimeters are the high density projection chamber
HPC in the barrel, and the lead-glass calorimeter FEMC in the forward region.
Detailed information about the design and performance of
{\sc Delphi} can be found in~\cite{detector} and~\cite{delphiperform}.
%*****************************************************************************
\subsection{The Particle and Event Selection}
The full data collected by {\sc Delphi} during the years 1992 till 1995 are
considered in the present analysis.
In a first step of the selection procedure,
the quality cuts given in Tables~\ref{tb_spursch}
and~\ref{tb_neutsch}
are imposed on all charged particles and on all neutral particles detected
in the calorimeters respectively, in order to ensure a reliable
determination of their momenta and energies.
%($\epsilon$ is the impact parameter with
%respect to the primary vertex, $L_{track}$ the measured track length).
%-------------------------------------------------------------------------
\begin{table}[t,h,b]
\begin{minipage}[b]{15cm}
\begin{minipage}[b]{7cm}
\begin{center}
\small
\begin{tabular}{|c|c|c|}\hline
{\rul \bf Variable} & {\rul \bf Cut} & {\rul \bf \% Loss} \\
\hline
\hline
$p$ & $\ge 0.3\;GeV/c$
% $p$ & $\ge 0.3\;GeV/c$\footnote{It is additionally
% required that at least 5 charged tracks with $p\ge 0.4$ GeV
% exist.} & 19.8\\
& 19.8\\
\hline
$\theta_{polar}$ & $20^\circ - 160^\circ$ & 7.7 \\
\hline
$\epsilon_{xy}$ & $\le 5.0\;cm$ & 4.9\\
\hline
$\epsilon_{z}$ & $\le 10.0\;cm$ & 1.1\\
\hline
$L_{track}$ & $\ge 30\;cm$ & 2.4 \\
\hline
$\frac{\Delta p}{p}$ & $\le 100\%$ & 0.1\\
\hline
\end{tabular}
\caption{Track selection for charged particles}
\label{tb_spursch}
\end{center}
\end{minipage}
%\end{table}
%-------------------------------------------------------------------------
%-------------------------------------------------------------------------
\hfill
\begin{minipage}[b]{7.0cm}
%\begin{table}[t,h,b]
\begin{center}
\small
\begin{tabular}{|c|c|c|c|}\hline
{\rul \bf Detector} & {\rul \bf $E_{min}\,[GeV]$} &
{\rul \bf $E_{max}\,[GeV]$} & {\rul \bf \% Loss} \\
\hline
\hline
HPC & 0.5 & 50 & 3.5\\
\hline
EMF & 0.5 & 30 & 3.4 \\
\hline
HAC & 1.0 & 50 & 10.4 \\
\hline
\end{tabular}
\caption{Energy cuts for neutral particles}
\label{tb_neutsch}
\end{center}
\end{minipage}
\end{minipage}
\end{table}
%--------------------------------------------------------------------------
All charged particles are assumed to be pions and all neutral particles are
assumed massless.
A sample of hadronic events is then selected
using the cuts shown in Table~\ref{tb_ereigsch}.
\begin{table}
\begin{center}
\small
\begin{tabular}{|c|c|c|c|}\hline
{\rul \bf Variable} & {\rul \bf Cut} & {\rul \bf \% Loss} \\
\hline
\hline
$E_{ch.}^{hemi}$ & $\ge 3\%$ of $\sqrt{S}$ & 7.5\\
\hline
$E_{ch.}^{tot}$ & $\ge 15\%$ of $\sqrt{S}$ & 0.8\\
\hline
$N_{ch.}$ & $\ge 5$ & 0.2\\
\hline
$\theta_{Spericity}$ & $30^\circ - 150^\circ$ & 10.6\\
\hline
$p_{max}$ & 50\% of $\sqrt{S}$ & 0.6\\
\hline
\end{tabular}
\caption{Hadronic event selection}
\label{tb_ereigsch}
\end{center}
\end{table}
These demand a minimum charged multiplicity, $N_{ch}$, and a minimum visible
energy carried by charged particles, $E_{ch}^{tot}$, as well as
requiring the events to be well contained within the detector.
$E_{ch}^{hemi}$ denotes the sum of the energies
of charged particles in the forward or backward hemisphere of the {\sc Delphi}
detector. An event is discarded if the momentum of one of its charged particles
is greater than $p_{max}$. It is additionally
required that at least 5 charged tracks with $p\ge 0.4$ GeV
exist.The resulting hadronic event samples are
listed in Table~\ref{tb_topo}. The leptonic and two photon background
are negligible, especially after selecting three jet events for the further
analysis.
%*****************************************************************************
\subsection{The Jet Finding Algorithm}
After the hadronic event selection, three jet events are clustered using the
Durham algorithm~\cite{dur1} without requiring a fixed jet resolution parameter.
In this scheme, a jet resolution variable $y_{ij}$ is defined for every
pair of particles $i$ and $j$ in an event by:
\begin{eqnarray}
y_{ij} & = &
\frac{2\cdot{\mathrm{min}}\,(E_i^2,E_j^2)\cdot (1-\cos\Theta_{ij})}{E^2_{vis}}
\label{gl_durham}
\end{eqnarray}
where $E_{i}$ and $E_{j}$ are the energies of particles $i$ and $j$,
$\Theta_{ij}$ is the angle between them, and $E_{vis}$ is the sum of
all measured particle energies in the event.
The corresponding particle pair with
the lowest value of $y_{ij}$
is replaced by a pseudo-particle
with the sum of their four-momenta, ${\bf p}_{ij} = {\bf p}_i + {\bf p}_j$.
The procedure is then repeated (as in~\cite{multipap}),
re-evaluating the jet resolution
variables in each iteration, until only three four-momenta are left.
Each four-momentum vector remaining at the end of this
process is referred to as a ``jet''.
The properties of the jets depend both on the jet energy and the resolution
scale $k_T$. A fixed resolution scale would influence the jet properties,
futhermore the restriction implied by a fixed $k_T$ is rarely considered in
theoretical predictions~\cite{eden_gustavo}.
As a second algorithm the {\sc Cambridge} algorithm~\cite{cambridge} is used.
It uses the same jet resolution variable $y_{ij}$ but the particles and
sub-jets are merged in inverse angular order, the closest in angular are
combined first. A freezing of soft jets
is implemented to construct gluon jets uncontaminated by coherently emitted
particles. Once a soft jet is resolved, it is "frozen out", i.e. it gets no
extra multiplicity contribution.
Here also three jet events are clustered without using a specified $y_{cut}$.
This corresponds to the situation in~\cite{eden_gustavo}.
%*****************************************************************************
\subsection{Event Topologies}
\label{sec_ev-topo}
For a detailed comparison of quark and gluon jet
properties, it is necessary to obtain samples of quark and gluon jets with
nearly the same kinematics and the same scales to allow a direct
comparison of the
jet properties. To fulfill this condition,
different event topologies have been used,
as illustrated in Fig.~\ref{qcd_1_s}:
\begin{figure}[t,b]
\begin{center}
\mbox{\epsfig{file=il-ya-topo.eps,width=10cm}}
\caption{\label{qcd_1_s}Event topologies of symmetric Y events and
asymmetric events. $\theta_{j}$ are the angles between the jets after
projection into the event plane.}
\end{center}
\end{figure}
\begin{itemize}%\itemsep-.5ex
\item Basic three jet events with
$\theta_{2}, \theta_{3}
\in [135^\circ \pm 35^\circ]$.
\item Mirror symmetric events,
$\theta_{2}, \theta_{3}
\in (120^\circ,130^\circ,140^\circ,150^\circ,160^\circ) \pm5^\circ$,\\
subsequently called {\bf Y events}.
These Y events are a sub-sample of the basic three jet sample in which
the two low-energy jets should be more directly comparable.
\end{itemize}
The jet axes are projected into the event plane, which is defined
as the plane perpendicular to the smallest sphericity eigenvector as obtained
from the quadratic momentum
tensor ($M_{\alpha\beta}=\sum_{i=1}^n p_{i\alpha}p_{i\beta}$).
The jets are numbered in decreasing order of jet energy, where the energy
of each jet is calculated from the angles between the jets assuming massless
kinematics:
\begin{eqnarray}
\label{recalibration}
E_{j}\sp{calc} = { {\rm sin} \theta_{j} \over
{\rm sin} \theta_{1} +
{\rm sin} \theta_{2} +
{\rm sin} \theta_{3} } \sqrt{s},
\ \ \ \ j=1,2,3\, ,
\end{eqnarray}
where $\theta_{j}$ is the interjet angle as defined in Fig.~\ref{qcd_1_s}.
In order to enhance the contribution from events with three well-defined jets
attributed to $q\bar{q}g$ production, further cuts are applied to the
three jet event samples as summarized in Table~\ref{planar_cut}.
These cuts select planar events with each of the reconstructed jets well
contained within the sensitive part of the detector.
\begin{table}[htbp]
\begin{center}
%\footnotesize
\begin{tabular}{|l||c|c|} \hline
Measurement & Cuts & \% Loss \\\hline
\hline
Sum of angles between jets & $\ge 355^\circ$
& $\sim 0.06\%$ \\\hline
Polar angle of each jet axis & $30^\circ - 150^\circ$
& $\sim 12\%$ \\\hline
Visible jet energy per jet & $\ge 5$ GeV
& $\sim 0.11\%$ \\\hline
Number of particles in each jet & $\ge 2$ (charged or neutral)
& $\sim 0.38\%$ \\\hline
\end{tabular}
\end{center}
\caption{Planarity and acceptance selections for reconstructed jets and the \%
loss for the asymmetric events.}
\label{planar_cut}
%\renewcommand\arraystretch{1.}
\end{table}
From the initial $\sim$3,695,000 hadronic events collected by {\sc Delphi} (and
$\sim$10,507,000 Monte Carlo events), about 756,000
(2,500,000) asymmetric
three jet events remain (s. Table~\ref{tb_topo}).
%of which $\sim$xx,000 are symmetric events (s. Table~\ref{tb_topo}).
%-------------------------------------------------------------------------
\begin{table}[t,h,b]
\begin{center}
%\small
\begin{tabular}{|c|c|c||c|c|}\hline
& \mc{2}{c||}{\rul \bf Hadronic Events} &
\mc{2}{c|}{\rul \bf Asymmetric Events}\\
\hline
{\rul Year} & {\rul Data} & {\rul Simulation} &
{\rul Data} & {\rul Simulation} \\
\hline
\hline
1992 & 604490 & 1723829 & 164450 & 499380 \\
\hline
1993 & 608025 & 1605915 & 164609 & 464975 \\
\hline
1994 & 1198034 & 4304445 & 271343 & 1246130 \\
\hline
1995 & 577196 & 1029486 & 155750 & 297202 \\
\hline
\hline
total & 2987745 & 8663675 & 756152 & 2507687 \\
\hline
\end{tabular}
\caption{Samples of selected asymmetric events and the
initial hadronic event samples.}
\label{tb_topo}
\end{center}
\end{table}
%*****************************************************************************
\subsection{Quark and Gluon Jet Identification}
\label{sec_technics}
A sample rich in gluon jets is obtained from three jet events which originate from
Z decays to a $b\bar b$ pair.
The events are identified using a well established
lifetime-tag~\cite{bor-mar} technique,
and the gluon jets are tagged indirectly
by identifying the other two jets as $b$-quark jets
using the lifetime-tag.
The light (udsc) quark jets used for comparison to these gluon jets are taken
from events failing the event level lifetime-tag.
These jet samples, including the corresponding gluon jets,
are called ``normal mixture''
In effect, subtracting the small residual heavy quark contributions from
the tagged sample yields the pure gluon sample.
The properties of light quark jets are obtained
by subtracting the gluon distributions
from the distributions measured in the normal mixture jet sample.
In this way neither the gluon nor the light quark distributions
are significantly biased by the $b$-quark identification procedure.
However the jets identified as $b$-quark jets are biased.
More importantly, about half of the particles in $b$ jets
come from the weak decays of $B$-hadrons.
Thus $b$ jets cannot be used for a direct comparison with gluon jets
within a purely QCD framework neglecting these decays.
In the following, the selection of the gluon and the normal mixture jet samples
in asymmetric events is described in detail as well as the corrections applied
to obtain information on pure quark and gluon jets. A far more detailed
description can be found in~\cite{disoli,dispatty}.
%*****************************************************************************
\subsubsection{Lifetime Tags at Event and Jet Level}
\label{sec_tagprd}
The combined lifetime impact parameters and their error distributions
are used to construct an algorithm for tagging $b$
jets~\cite{bor-mar,bor-combined}. Basically, in this method, the
probability, $P_N$, for the hypothesis that all tracks arise directly from the
$e^+e^-$ annihilation point is evaluated for a given selection of $N$ tracks.
By construction, light quark events or jets have a flat distribution in $P_N$
while, because of the long lifetimes of $B$-hadrons, events or jets containing
$b$-quarks tend to give low values.
Events with a $b$-quark signature are selected as input to the gluon
identification by demanding that $P_E$, the value of $P_N$ evaluated for the
whole event, does not exceed $\lambda_E = -\log_{10}(P_E)+2=1.03$.
Events failing the $b$ tag are considered to be in the normal mixture sample.
The tracks corresponding to
each of the reconstructed jets are then used to construct a probability
$P_J$ per jet. Jets are finally classified according to the observed
values of $P_J$ following this selection strategy:
The most energetic jet is assumed to be a quark jet. Cuts on
$\lambda_J = -\log_{10}(P_J)+2$ are applied
to each of the two lower energy jets in order to decide which is the quark jet
and which is the gluon jet. The main criterion applied is to demand that one
of the two lower energy jets satisfies the condition $\lambda_J < 1.28$. The
remaining jet is then taken as the gluon jet provided its probability value
$\lambda_J$ is above 1.28. This ensures that the decay products of the
$B$-hadrons do not, in general, filter through to the selected sample of gluon
jets. In total, 142,413 gluon jets in the asymmetric event sample are selected
using this single jet tag method.
In Fig.~\ref{bjetprob}, measured the distribution of the jets, $\lambda_J$,
in Y events ($\theta_{2}, \theta_{3}
\in [150^\circ \pm 15^\circ]$)
is compared separately for normal mixture jets,
charm jets, $b$-quark jets, and gluon jets as predicted by the Monte Carlo
simulations.
\begin{figure}[t,b,h]
\begin{center}
\mbox{\epsfig{file=bo-lambda.eps,width=10cm}} \caption{Jet probabilities
as measured from the data and as predicted by the Monte Carlo for
normal mixture (in $uds$ events), $c$-quark, $b$-quark, and gluon jets (in $c$
and $b$ events) for Y events with $\theta_{2}, \theta_{3}
\in [150^\circ \pm 15^\circ]$. } \label{bjetprob}
\end{center}
\end{figure}
%*****************************************************************************
\subsubsection{Gluon and Quark Jet Purities}
Efficiency and purity calculations have been made using events generated
by the {\sc Jetset} 7.3 Monte Carlo~\cite{lun1} tuned to {\sc Delphi}
data~\cite{hamacher}, passed through the full simulation program ({\sc
Delsim}~\cite{delsim,delphiperform}) of the {\sc Delphi} detector and
the standard {\sc Delphi} data reconstruction chain.
Even in the Monte Carlo, the assignment of parton flavours to the jets is not
unique, as in parton models like {\sc Jetset} the decay history is interrupted
by the building of strings. Thus
two independent ways of defining the gluon jet in the fully
reconstructed Monte Carlo have been investigated~\cite{oliver,martin,alephps}:
\begin{itemize}%\itemsep-0.5ex
\item {\bf angle assignment:} The gluon induced jet is assumed to be the jet
making the largest angle with the nearest $B/D$-hadron originating from the
primary $b/c$-quarks.
%\item {\bf history assignment:} The jet containing the fewest decay particles
% from the heavy hadrons is assigned to the gluon.
\item {\bf PS assignment:}
First the partons are clustered to three jets if the event is accepted as
containing three well measured jets at detector level. Quarks are given a
weight of +1, antiquarks a weight of $-1$, and gluons a weight of 0.
Parton jets are identified as quark and gluon jets if the sum of the
flavour weights of all partons in a certain parton jet is +1, -1, and
0, respectively. The small amount (2\%) of events not showing this
expected pattern are discarded.
A gluon jet is identified as the parton jet which sum of
the parton flavours yield 0.
These parton jets are mapped onto the hadron jets in such a way
that the sum of the angles between the three hadron jets and their
belonging parton jets is minimized.
\end{itemize}
\begin{table}[t,b,h]
\begin{center}
\[
\begin{array}[c]{|c|c|ccc|}
%\cline{3-5}
\hline
\multicolumn{2}{|l|}{\mbox{\bf Method}} &
\multicolumn{3}{|c|}{\mbox{\bf Angle assignment}} \\ \cline{2-5}
\multicolumn{1}{|c|}{\ } &
\mbox{\bf gluon in:} &
\mbox{\ \ \ Jet 1} & \mbox{\ \ \ Jet 2} & \mbox{\ \ \ Jet 3}\\
\cline{2-5}
\hline
&\mbox{Jet 1} & 4.2\% & 0.01\% & 0.01\% \\
\mbox{\bf PS}&\mbox{Jet 2}
& 0.0\% & 25.5\% & 0.37\% \\
\mbox{\bf assignment} &\mbox{Jet 3}
& 0.01\% & 0.15\% & 69.8\% \\
\hline
\end{array}
\]
\caption{Correlation of angle and PS assignments. The table has been
obtained for arbitrary three jet events with
$\theta_{2}, \theta_{3} \in [110^\circ,170^\circ]$. These events also contain
the symmetric events. A similar behaviour is observed concerning
an assignment based on the history information of the Monte Carlo
models~\cite{oliver,martin}.}
\label{tb_matrix}
\end{center}
\end{table}
Tab.~\ref{tb_matrix} shows that the angle and PS assignments
give similar results and that
therefore the purities can be estimated with small systematic
uncertainties.
As in Monte Carlo events the gluon jets can be identified as well in
b/c-events as
in light quark events with the PS assignments, this method is used rather
than the hadron assignments.
With the tagging procedure described in this section,
gluon jet purities from 40\% to 90\% are achieved, depending on
the jet scale (see Fig.~\ref{fg_glupur}).
Here the purity is defined as the ratio of the
number of real tagged gluons (i.e. jets originating from gluons) to the total
number of jets tagged as gluons.
\begin{figure}[bth]
\begin{center}
\mbox{\epsfig{file=sc-plot_purmat2.eps,width=10cm}}
\caption{Gluon and quark fractions in identified gluon jets and in
normal mixture jets ($f_{nq}, f_{ng}, f_{gq}, f_{gg}$).
The decreasing fraction of gluon jets within normal mixture jets
is due to the fact that the probability of gluon Bremsstrahlung gets
smaller with increasing $p_{\perp}$ of the gluons.
Consequently a gluon fraction of $\simeq 35\%$ in the identified gluon set
for high $\kappa_H$ is a significantly enrichment compared to
the corresponding gluon fraction of $\simeq 15\%$ in the normal
mixture jets.}
\label{fg_glupur}
\end{center}
\end{figure}
%******************************************************************************
\subsubsection{Corrections}
\label{sec_nmix}
In a first step the pure generated light and $c$-quark, $b$-quark, and gluon
Monte Carlo distributions are mixed to represent the purities of the
data.
Secondly, the effects of finite resolution and acceptance of the detector
are corrected using a full simulation of the {\sc Delphi} detector.
A linear correction function $C$,
whereas at a time a factor for each bin $i$ of any distribution of an
observable adjusts the distribution
$R^{MC+det}$ to $R^{gen}$,
\begin{eqnarray*}
C_i^{acc} = \frac{R^{gen}}{R^{MC+det}}\ ,
\end{eqnarray*}
is determined.
Where $R^{gen}$ includes the total generated state before detector simulation
and $R^{MC+det}$ the state after detector simulation, which fulfills also
the data selection criteria. All effects caused by the detector are included
within this correction. By multiplication of the measured data distribution
$D_i^{meas}$ with the correction function one yields the acceptance
corrected data distribution:
\begin{eqnarray*}
D_i^{corr} = C_i^{acc} \cdot D_i^{meas}\ .
\end{eqnarray*}
%In the further analysis, bins with large acceptance correction factors are
%disregarded.
Finally, in order to achieve pure quark ($udsc$) and gluon jet distributions
the following equation has been solved by matrix inversion:
\begin{equation}
\left( \begin{array}{c} n \\ b \\ g
\end{array} \right)_{\m{\small measured}} =
\left( \begin{array}{ccc} f_{nq} f_{nb} f_{ng} \\
f_{bq} f_{bb} f_{bg} \\
f_{gq} f_{gb} f_{gg}
\end{array} \right)
\left( \begin{array}{c} n \\ b \\ g
\end{array} \right)_{\m{\small pure}}
\end{equation}
Here the $f_{ij}$ denote the relative fraction of a parton $j$ within the
identified measured distribution $i$. Here $i$ stands for a normal mixture
$(n)$, $b$-quark $(b)$ or gluon $(g)$ jet.
%\clearpage
%******************************************************************************
%******************************************************************************
\section{Results}
\label{sec_res}
\subsection{The General Behaviour of Gluon and Quark Fragmentation Functions}
\label{sec_frag}
%\begin{figure}[t,b,h]
%\begin{center}
%\mbox{\epsfig{file=null.eps,width=15.5cm}}
%\mbox{\epsfig{file=fanx-q.eps,width=15.5cm}}
%\caption{Scaled energy distributions for different topologies resp. scales}
%\label{pl_xe}
%\end{center}
%\end{figure}
\begin{figure}[p]
\breite 11.4cm
\breitemp 11.5cm
\begin{center}
\rotatebox{90}{
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{sc-fanx-q.eps}
\caption{Scaled quark energy distributions for different scale values
of $\kappa_H$ ({\sc Durham} algorithm).
The 1st order DGLAP fit is superimposed.}
\label{pl_xe2}
\end{center}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{sc-fanx-g.eps}
\caption{Scaled gluon energy distributions for different scale values
of $\kappa_H$ ({\sc Durham} algorithm).
The 1st order DGLAP fit is superimposed.}
\label{pl_xe1}
\end{center}
\end{minipage}
}
\end{center}
\end{figure}
Sizeable differences have been observed between the scaled energy
($x_E$)-distributions of stable hadrons produced in quark and gluon
jets~\cite{alephps,splitting_paper,xeopal}. In Fig.~\ref{pl_xe2} and~\ref{pl_xe1} the
fragmentation functions for quark and gluon jets in the overall sample of three
jet events are shown for three different values of $\kappa_H$. An approximately
exponential decrease of the fragmentation function with increasing $x_E$ is
seen, which is more pronounced in the gluon case.
A softening of the fragmentation functions with increasing $\kappa_H$ is
observed. This effect is more pronounced for gluon jets than for quark jets.
Tables of the quark fragmentation functions for different
values of $\kappa_H$ and the gluon fragmentation functions for different values
of $\kappa_H$ and $p_1^T$ with their statistical error
are available in the Durham/RAL
HEP-database~\cite{raldata} both for the {\sc Durham}
and {\sc Cambridge} cluster algorithms.
In order to study systematic uncertainties of the fragmentation functions the
dependence on the following sources of error have been studied:
\begin{enumerate}
\item {\bf Minimal number of tracks per jet} \\
The minimal number of tracks in each jet has been changed from two up to
four. This has no effect for the quark jets and a $\simeq \pm 2\%$ effect
is visible for the gluon jets.
\item {\bf Minimal angle between the jets and the beam} \\
The polar angle of each jet axis to the beam direction has been
changed from $30^\circ$ to $40^\circ$. This again
has no effect for the quark
jets and the effect is $\leq \pm 1\%$ for the gluon jets.
\item {\bf c depleted event sample} \\
The $b$ event tag has been varied to account for a different
composition ($b$ and $c$ depleted) of quark flavours in
the normal mixture sample. No effect has been observed.
\item {\bf Variation of the parton jets assignment} \\
Different quality cuts have been performed to map the three parton jets
to the three hadron jets.
We see a $\leq \pm 2\%$ effect for quarks and a $\leq 3\%$ effect
for gluons which only appears at scales $<10$ GeV.
% \item {\bf Using a specific $y_{cut}$} \\
% Instead of clustering all events to three jet events a $y_{cut} = 0.15$
% has been applied. On average a 3\% effect is seen and this effect is
% $\leq 5\%$, but well contained inside the statistical error. \\
% {\em BUT I don't know if this is a real systematic error!!}
% \item {\bf {\sc Cambridge} cluster algorithm instead of {\sc Durham}}
\end{enumerate}
Fig.~\ref{fg_xech2} compares the gluon and the quark fragmentation function
for symmetric three jet events with
$\theta_{2}, \theta_{3} \in [150^\circ \pm 15^\circ]$ ({\sc Durham} algorithm).
Quark jets fragment much harder than gluon jets.
The extra suppression at high
$x_E$ (by almost one order of magnitude) of gluon jets relative to quark jets
is expected because, contrary to the quark case, the gluon cannot be present as
a valence parton inside the produced hadron. The valence quarks of the hadrons
first have to be produced in a $g \rightarrow q\bar{q}$ splitting process.
The softer behaviour of the gluon fragmentation function may also be due to the
intrinsically larger scaling violations in gluon jets which, also in
combination with recoil effects
(for a thorough discussion see~\cite{eden_gustavo}), leads to a softer
hadron spectrum.
\begin{figure}[t,b,h]
\begin{center}
\mbox{\epsfig{file=to-xech.eps,width=10cm}}
\caption{\label{fg_xech2}Gluon and the quark fragmentation function of Y
events, $\theta_{2}, \theta_{3} \in [150^\circ \pm 15^\circ]$ ),
compared to the prediction of various fragmentation models.}
\end{center}
\end{figure}
In order to demonstrate qualitatively the connection between the strength of the
scaling violation at high $x_E$ and the increase in particle multiplicity which
happens predominantly at very small energies we compare in
Fig.~\ref{fg_xi_q} and~\ref{fg_xi_g} the $\xi$-distributions
$(\xi = -\ln{x_E})$ which are measured for particles assigned to individual
gluon (Fig.~\ref{fg_xi_g}) and quark (Fig.~\ref{fg_xi_q}) jets for different
values of $p_1^T$ respectively $\kappa_H$.
Jets here were defined using the
{\sc Cambridge} algorithm as this has an improved behaviour
to reconstruct the
gluon jets. As the emission of very soft (i.e. large $\xi$)
particles is expected to happen coherently from the $q\bar{q}g$ ensemble the
assignment of these particles to individual jets to some extend is
arbitrary. Neglecting this complication, from the behaviour of the data in
Fig.~\ref{fg_xi_q} and~\ref{fg_xi_g} it is evident that scaling violation as
well as the increase in multiplicity are stronger for gluon jets compared to
quark jets. This is a consequence of the higher colour factor of the gluons and
thus their apparent higher "coupling" to soft gluon radiation.
\begin{figure}[p]
\breite 11.4cm
\breitemp 11.5cm
\begin{center}
\rotatebox{90}{
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{sc-xi_q.eps}
\caption{$\xi$ quark distributions for different hardness scales
({\sc Cambridge} algorithm). The lines are simple gaussian
fits to the data applied in the region of the maxima of the
$\xi$ distributions.}
\label{fg_xi_q}
\end{center}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{sc-xi_g.eps}
\caption{$\xi$ gluon distributions for different $p_1^T$ scales
({\sc Cambridge} algorithm). The lines are simple gaussian
fits to the data applied in the region of the maxima of the
$\xi$ distributions.}
\label{fg_xi_g}
\end{center}
\end{minipage}
}
\end{center}
\end{figure}
The lines shown in Fig.~\ref{fg_xi_q} and~\ref{fg_xi_g} are simple gaussian
fits to the data applied in the region of the maxima of the $\xi$
distributions. The values of the central values $\xi^*$ of these fits together
with their statistical errors are given in Tab.~\ref{tb_xi}
and~\ref{tb_xi_cam} (both for the {\sc Durham}
and {\sc Cambridge} cluster algorithms). An approximately
linear increase of $\xi^*$ as a function of the scale $\kappa_H$ for quarks
or $\kappa_H$ or $p_1^T$ for gluons is observed similar to the behaviour in
overall $e^+e^-$ events. For gluons this increase differs significantly between
$\kappa_H$ and $p_1^T$ and also between {\sc Durham} and {\sc Cambridge}.
Due to the subtlety of assigning low energy particles to individual jets this
increase should, however, not be quantitatively compared to the overall
$e^+e^-$ data.
\begin{table}[t,b,h]
\begin{center}
%\footnotesize
\begin{tabular}{|c|c|c|c||c|c|c|c||c|c|c|c|}\hline
\mc{4}{|c||}{\rul \bf Quark Jets} &
\mc{8}{c|}{\rul \bf Gluon Jets} \\ \hline
$\kappa_H$ & $\xi^*$ & $\pm$ & $\chi^2/n.d.f.$ & $p_1^T$ & $\xi^*$ & $\pm$ &
$\chi^2/n.d.f.$ & $\kappa_H$ & $\xi^*$ & $\pm$ & $\chi^2/n.d.f.$ \\ \hline
\hline
5.73 & 2.49 & 0.04 & 0.93 & 5.36 & 2.62 & 0.02 & 2.06 & 5.73 & 2.60 & 0.02 & 0.54 \\
\hline
6.90 & 2.67 & 0.03 & 0.38 & 6.27 & 2.73 & 0.02 & 1.98 & 6.89 & 2.73 & 0.01 & 0.49 \\
\hline
8.07 & 2.74 & 0.04 & 1.14 & 7.19 & 2.80 & 0.01 & 1.12 & 8.06 & 2.81 & 0.01 & 2.13 \\
\hline
9.24 & 2.81 & 0.03 & 0.69 & 8.10 & 2.87 & 0.01 & 2.51 & 9.24 & 2.89 & 0.01 & 2.37 \\
\hline
10.41 & 2.89 & 0.03 & 0.65 & 9.03 & 2.91 & 0.02 & 1.47 & 10.41 & 2.95 & 0.01 & 1.14 \\
\hline
11.57 & 2.88 & 0.03 & 1.37 & 9.95 & 2.98 & 0.02 & 2.19 & 11.56 & 3.05 & 0.02 & 3.19 \\
\hline
13.30 & 3.05 & 0.02 & 2.05 & 11.29 & 3.06 & 0.01 & 3.16 & 13.27 & 3.09 & 0.01 & 1.78 \\
\hline
15.64 & 3.08 & 0.02 & 1.13 & 13.13 & 3.12 & 0.01 & 0.86 & 15.61 & 3.19 & 0.02 & 0.80 \\
\hline
17.98 & 3.14 & 0.02 & 0.23 & 14.96 & 3.20 & 0.02 & 1.90 & 17.97 & 3.26 & 0.02 & 0.48 \\
\hline
20.31 & 3.19 & 0.02 & 1.92 & 16.81 & 3.23 & 0.02 & 0.50 & 20.29 & 3.33 & 0.03 & 1.80 \\
\hline
22.65 & 3.23 & 0.02 & 0.87 & 18.64 & 3.27 & 0.03 & 0.82 & 22.64 & 3.40 & 0.04 & 1.50 \\
\hline
24.99 & 3.29 & 0.02 & 0.71 & 20.48 & 3.35 & 0.04 & 0.95 & 24.95 & 3.45 & 0.05 & 0.52 \\
\hline
27.27 & 3.33 & 0.03 & 0.78 & 22.30 & 3.36 & 0.04 & 0.77 & 27.20 & 3.49 & 0.07 & 1.22 \\
\hline
29.04 & 3.40 & 0.05 & 0.84 & 24.15 & 3.39 & 0.05 & 0.67 & 29.00 & 3.42 & 0.14 & 0.18 \\
\hline
%oldpstag
% 5.73 & 2.49 & 0.05 & 0.82 & 5.36 & 2.66 & 0.02 & 2.19& 5.73 & 2.63 & 0.02 & 0.35 \\
% 6.90 & 2.68 & 0.04 & 0.71 & 6.28 & 2.76 & 0.02 & 2.09& 6.89 & 2.75 & 0.01 & 0.97 \\
% 8.07 & 2.75 & 0.04 & 1.25 & 7.19 & 2.80 & 0.02 & 2.06& 8.06 & 2.82 & 0.01 & 2.56 \\
% 9.24 & 2.81 & 0.03 & 0.44 & 8.10 & 2.88 & 0.01 & 3.71& 9.24 & 2.90 & 0.01 & 2.49 \\
% 10.41 & 2.91 & 0.03 & 0.75 & 9.03 & 2.92 & 0.02 & 2.31& 10.40 & 2.95 & 0.01 & 1.21 \\
% 11.57 & 2.88 & 0.03 & 1.87 & 9.95 & 2.98 & 0.02 & 1.91& 11.57 & 3.05 & 0.02 & 4.02 \\
% 13.29 & 3.06 & 0.03 & 2.22 & 11.29 & 3.06 & 0.01 & 3.47& 13.27 & 3.09 & 0.01 & 2.32 \\
% 15.64 & 3.08 & 0.02 & 1.08 & 13.13 & 3.12 & 0.01 & 1.27& 15.61 & 3.19 & 0.02 & 0.64 \\
% 17.98 & 3.13 & 0.02 & 0.29 & 14.96 & 3.21 & 0.02 & 1.73& 17.96 & 3.26 & 0.02 & 0.90 \\
% 20.31 & 3.17 & 0.02 & 1.47 & 16.81 & 3.22 & 0.02 & 0.77& 20.29 & 3.35 & 0.03 & 1.15 \\
% 22.65 & 3.21 & 0.02 & 1.01 & 18.64 & 3.28 & 0.03 & 1.00& 22.65 & 3.42 & 0.04 & 1.59 \\
% 24.99 & 3.28 & 0.02 & 0.90 & 20.48 & 3.37 & 0.04 & 0.96& 24.95 & 3.45 & 0.05 & 0.84 \\
% 27.26 & 3.33 & 0.03 & 0.65 & 22.30 & 3.39 & 0.05 & 0.64& 27.20 & 3.48 & 0.07 & 0.79 \\
% 29.04 & 3.39 & 0.05 & 1.03 & 24.15 & 3.43 & 0.06 & 1.06& 28.99 & 3.43 & 0.14 & 0.30 \\
%\hline
\end{tabular}
\caption{$\xi^*$ values ({\sc Durham} algorithm)}
\label{tb_xi}
\end{center}
\end{table}
\begin{table}[t,b,h]
\begin{center}
%\footnotesize
\begin{tabular}{|c|c|c|c||c|c|c|c||c|c|c|c|}\hline
\mc{4}{|c||}{\rul \bf Quark Jets} &
\mc{8}{c|}{\rul \bf Gluon Jets} \\ \hline
$\kappa_H$ & $\xi^*$ & $\pm$ & $\chi^2/n.d.f.$ & $p_1^T$ & $\xi^*$ & $\pm$ &
$\chi^2/n.d.f.$ & $\kappa_H$ & $\xi^*$ & $\pm$ & $\chi^2/n.d.f.$ \\ \hline
\hline
5.74 & 2.33 & 0.07 & 0.64 & 5.37 & 2.49 & 0.02 & 0.85 & 5.74 & 2.48 & 0.02 & 1.70 \\
\hline
6.90 & 2.69 & 0.04 & 1.41 & 6.28 & 2.59 & 0.02 & 0.97 & 6.90 & 2.60 & 0.01 & 0.79 \\
\hline
8.07 & 2.74 & 0.04 & 0.86 & 7.19 & 2.69 & 0.02 & 0.74 & 8.07 & 2.72 & 0.01 & 0.99 \\
\hline
9.24 & 2.83 & 0.04 & 1.43 & 8.11 & 2.79 & 0.02 & 1.15 & 9.23 & 2.83 & 0.01 & 2.86 \\
\hline
10.41 & 2.94 & 0.04 & 0.86 & 9.03 & 2.86 & 0.02 & 1.64 & 10.41 & 2.89 & 0.01 & 1.59 \\
\hline
11.57 & 2.97 & 0.03 & 0.69 & 9.94 & 2.93 & 0.02 & 0.82 & 11.57 & 2.98 & 0.02 & 1.15 \\
\hline
13.30 & 3.07 & 0.03 & 1.46 & 11.30 & 3.00 & 0.01 & 0.93 & 13.28 & 3.06 & 0.01 & 1.23 \\
\hline
15.64 & 3.10 & 0.02 & 1.29 & 13.13 & 3.11 & 0.01 & 0.71 & 15.62 & 3.16 & 0.01 & 1.25 \\
\hline
17.98 & 3.13 & 0.02 & 0.37 & 14.96 & 3.18 & 0.02 & 1.45 & 17.96 & 3.25 & 0.02 & 0.74 \\
\hline
20.32 & 3.19 & 0.02 & 2.28 & 16.81 & 3.21 & 0.02 & 0.43 & 20.30 & 3.31 & 0.03 & 2.11 \\
\hline
22.66 & 3.23 & 0.02 & 0.88 & 18.64 & 3.28 & 0.03 & 0.73 & 22.64 & 3.38 & 0.04 & 0.71 \\
\hline
24.99 & 3.27 & 0.02 & 0.74 & 20.46 & 3.30 & 0.03 & 1.68 & 24.95 & 3.47 & 0.05 & 0.99 \\
\hline
27.27 & 3.32 & 0.02 & 0.58 & 22.29 & 3.34 & 0.04 & 0.84 & 27.19 & 3.49 & 0.07 & 0.36 \\
\hline
29.04 & 3.43 & 0.06 & 0.78 & 24.15 & 3.40 & 0.05 & 0.67 & 29.01 & 3.37 & 0.12 & 0.70 \\
\hline
\end{tabular}
\caption{$\xi^*$ values ({\sc Cambridge} algorithm)}
\label{tb_xi_cam}
\end{center}
\end{table}
\clearpage
%******************************************************************************
\subsection{Examining Jet Scales}
\label{sec_scale}
\begin{figure}[t,b,h]
\begin{center}
\mbox{\epsfig{file=sc-broad_single.eps,width=14cm}}
\caption{Mean values of the jet broadness $\beta$ as a function of different
jet scales. The dashed and solid lines are p4-fits.}
\label{fg_jetbroad}
\end{center}
\end{figure}
In Fig.~\ref{fg_jetbroad} the mean values of the jet
broadening,
\begin{equation}
\beta = \frac{\sum |\vec{p}_i \times \vec{r}_{jet}|}{2\sum|\vec{p}_i|}\, ,
\label{eq_beta}
\end{equation}
are shown as a function of the jet scale.
The $\vec{p}_i$ are the momenta of the tracks belonging to one jet and
$\vec{r}_{jet}$ is the corresponding jet direction.
The $\beta$-variable defined in Eqn.~\ref{eq_beta} is constructed analogous to
the event shape observable $B$~\cite{bobs} to give a
quantitative measurement of the angles or transverse momenta
of the particles w.r.t the jet axis i.e. the
"broadness" of the jet. Gluon jets are observed to be wider than quark
jets (see Fig.~\ref{fg_jetbroad} as expected from the
different quark and gluon colour structure.
The ratio of the $<\beta >$ values is typically
$~1.5$ for all scale values if $\kappa_H$ (or $p_1^T$) is taken as the scale.
Choosing instead the jet energy $E_{jet}$ as an intuitive
scale or alternatively the jet transverse momentum $\kappa_T$ a strong decrease
respectively increase is observed with increasing scale values (lines in
Fig.~\ref{fg_jetbroad}).
The approximate constance of $<\beta >$ for $\kappa_H$ as scale implies
that here the longitudinal and transverse momentum scale as would be expected for a relevant
scale. Thsi scale takes phase space effects properly into account and makes
jets of different topologies directly comparable.
This is
similarly so for the scale $p_1^T$, although here a slight increase with scale
is seen. A similar scaling
behaviour is observed for the event shape observable $B$
as a function of the centre-of-mass energy $\sqrt{S}$~\cite{event_broad}. The
observed small energy dependence in this case can be traced back to the running
of $\alpha_S$ and to power corrections. The obviously different behaviour for
$E_{jet}$ or $\kappa_T$ strongly indicates that these variables are unsuitable
as scales.
%\begin{figure}[p]
% \begin{center}
% \mbox{\epsfig{file=fan-low.eps,width=15.5cm}}
% \caption{Scale dependence and scaling violation of the quark
% fragmentation functions.
% The dashed line is the result of a power law fit.}
% \label{xe_low}
% \end{center}
%\end{figure}
%
%\begin{figure}[p]
% \begin{center}
% \mbox{\epsfig{file=fan-g-symm.eps,width=15.5cm}}
% \caption{Scale dependence and scaling violation of the gluon
% fragmentation functions.
% The dashed line is the result of a power law fit.}
% \label{xe_symg}
% \end{center}
%\end{figure}
\begin{figure}[p]
\breite 11cm
\breitemp 11.4cm
\begin{center}
\rotatebox{90}{
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{sc-fan-low.eps}
\caption{Scale dependence of the quark
fragmentation function ({\sc Durham} algorithm).
The reference data stem from
low and higher energy experiments.
The dotted line is the result of power law fits.}
\label{xe_low}
\end{center}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{sc-fan-g-2.eps}
\caption{Scale dependence of the gluon
fragmentation function ({\sc Durham} algorithm).
The dotted lines result from power law fits.}
\label{xe_symg}
\end{center}
\end{minipage}
}
\end{center}
\end{figure}
Fig.~\ref{xe_low} shows the comparison of the quark fragmentation function for
fixed $x_E$ as a function of the scale $\kappa_H$ (in the following called `fan'
plots) with the quark fragmentation
functions measured at lower energy $e^+e^-$ experiments and at high
centre-of-mass energies with
{\sc Delphi}.
These data distributions are obtained from $e^+e^-$ events scaled by
$\frac{1}{2}$. To account for the different number of primary partons.
A good agreement
is seen, both in normalization and slopes.
This also is an a posteriori justification of this analysis and yields
a new approach to study dynamical dependencies of hadron distributions.
Fig.~\ref{xe_symg} shows the gluon fragmentation functions for fixed $x_E$ as a
function of the scale $\kappa_H$. The results obtained from the overall data
set and from the symmetric events agree well. This is e.g. not the case if
$E_{jet}$ is chosen as the scale~\cite{dpf99}.
The good agreement of the three jet quark
distributions from $e^+e^-$ data and of the symmetric gluon jets with gluon
jets of any topology indicates again that $\kappa_H = E\cdot\sin\Theta/2$ is a
relevant scale for dynamical studies of jet properties.
The expected power behaviour, shown by a linear behaviour in the log-log
plots, is fitted by:
\begin{equation}
D(x_E,\kappa) = a \cdot \kappa^b,
\label{gl_linfit}
\end{equation}
indicated by the dotted lines in the fan plots~\ref{xe_low},\ref{xe_symg}.
The typical behaviour of scaling violations is observed in both plots,
namely a
strong fall off at large $x_E$ which diminishes with falling $x_E$. The
slope vanishes
around $x_E \sim 0.1$, and finally for small $x_E$ turns into a rise. The
rise at small $x_E$
causes the increase of multiplicity with the scale~\cite{multipap} (compare
also Fig.~\ref{fg_xi_q} and~\ref{fg_xi_g}). The
scaling violation behaviour is much stronger for gluons than for quarks.
This is expected due to the higher colour charge of gluons.
\clearpage
%******************************************************************************
\subsection{Scaling Violations}
\label{sec_fit}
%\begin{figure}[p]
% \begin{center}
% \mbox{\epsfig{file=fan-q.eps,width=15.5cm}}
% \caption{Scale dependence and scaling violation of the quark
% fragmentation functions.
% The full line is the fit of the DGLAP evolution of the
% fragmentation function.}
% \label{xe_scaleq}
% \end{center}
%\end{figure}
%
%\begin{figure}[p]
% \begin{center}
% \mbox{\epsfig{file=fan-g.eps,width=15.5cm}}
% \caption{Scale dependence and scaling violation of the gluon
% fragmentation functions.
% The full line is the expectation of the DGLAP evolution of the
% fragmentation function.}
% \label{xe_scaleg}
% \end{center}
%\end{figure}
\begin{figure}[p]
\breite 11cm
\breitemp 11.4cm
\begin{center}
\rotatebox{90}{
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{sc-fan-q.eps}
\caption{Scale dependence of the quark
fragmentation functions ({\sc Durham} algorithm).
The full line is the expectation of the DGLAP evolution of the
fragmentation function.}
\label{xe_scaleq}
\end{center}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{sc-fan-g.eps}
\caption{Scale dependence of the gluon
fragmentation functions ({\sc Durham} algorithm).
The full line is the fit of the DGLAP evolution of the
fragmentation function.}
\label{xe_scaleg}
\end{center}
\end{minipage}
}
\end{center}
\end{figure}
Figs.~\ref{xe_scaleq} and~\ref{xe_scaleg} show the measured quark and gluon
fragmentation function as a function of the scale $\kappa_H$.
A fit of the fragmentation functions including a simultaneous DGLAP
evolution is also shown.
\begin{table}[t,h,b]
\begin{center}
\begin{tabular}{|c|r@{}lcr@{}l||r@{}lcr@{}l|}\hline
& \mc{5}{c||}{\rul \bf Quark Jets} &
\mc{5}{c|}{\rul \bf Gluon Jets} \\
\hline
$p_1$ & -4.&74 &$\pm$& 0.&22 & -6.&27 & $\pm$& 0.&32 \\
\hline
$p_2$ & 0.&58 &$\pm$& 0.&10 & 0.&28 & $\pm$& 0.&19 \\
\hline
$p_3$ & 0.&059 &$\pm$& 0.&011 & 0.&011& $\pm$& 0.&003 \\
\hline
$p_4$ & 1.&024 &$\pm$& 0.&070 & 1.&29 & $\pm$& 0.&10 \\
\hline
\hline
\mc{11}{|c|}{$C_A = 2.97 \pm 0.12$} \\
\hline
\mc{11}{|c|}{$\Lambda_{\m{QCD}} = (397 \pm 113) MeV $} \\
\hline
\end{tabular}
\caption{Parameters of the simultaneously fitted fragmentation functions at
$\kappa_{H,0} = 5.5$ GeV ($\chi^2/n.d.f \sim 1.4$). The errors are
given neglecting any correlations between the parameters.}
\label{tb_dfrac}
\end{center}
\end{table}
For the evolution the fragmentation functions have
been parameterized at $\kappa_{H,0} = 5.5$ GeV over the $x_E$ range $0.15 \le x_E
\le 0.9$ to the form of Eqn.~\ref{gl_dfrac}. The parameters of the fit are
given in Tab.~\ref{tb_dfrac}. For the fit the range $6.5 GeV \le
\kappa_H \le 28 GeV$ has been used.
The systematic error stems from the sources listed in
Tab.~\ref{tb_casys} which have also been discussed in Sec.~\ref{sec_frag}.
To obtain systematic errors interpretable like statistical errors,
half the difference in the value obtained for $C_A/C_F$ when a parameter
is modified from its central value is quoted
as the systematic uncertainty.
The single errors are added quadratically.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|}\hline
{\rul \bf Source of systematic error} & {\rul \bf rel. Error} \\
\hline
Minimal number of tracks per jet & 0.5\% \\
\hline
Minimal angle between the jets and the beam & 0.7\% \\
\hline
c depleted event sample & 0.5\% \\
\hline
Variation of the parton jets assignment & 2.4\% \\
%\hline
% Fixed $y_{cut} = 0.15$ & 6.6\% \\
\hline
\hline
% Sum & 7.1\% \\
Sum & 2.6\% \\
\hline
\end{tabular}
\caption{Source of systematic errors}
\label{tb_casys}
\end{center}
\end{table}
From the fit the colour factor ratio
$C_A/C_F$ has been determined to be:
\begin{eqnarray}
% \frac{C_A}{C_F} = 2.23 \pm 0.09_{stat.} \pm 0.16_{sys.}
\frac{C_A}{C_F} = 2.23 \pm 0.09_{stat.} \pm 0.06_{sys.}
\label{eq_fitkap}
\end{eqnarray}
in a good agreement with the expectation of $C_A/C_F=2.25$.
$\Lambda_{\m{QCD}}$ has a fitted value in leading order of
$\Lambda_{\m{QCD}} = (397 \pm 113) MeV$ ,
with a $\chi^2/n.d.f \sim 1.4$ for 134 degrees of freedom.
%From the measured $\Lambda_{\m{QCD}}$ in first order the corresponding value
%of $\alpha_S(M_Z^2)$ yields:
%\begin{eqnarray*}
% \alpha_S(M_Z^2) = 0.116_{- 0.018}^{+0.010} \, ,
%% \alpha_S(M_Z^2) = 0.136_{- 0.007}^{+0.006} \, , n_f = 5
%\end{eqnarray*}
%consistent with the present world average.
The behaviour of the data for not too small values of $x_E$ is very well
represented by the DGLAP evolution. The good agreement fortifies the scaling
violation interpretation. The scale $E_{jet}$ again is disfavoured
because the fit in this case yields a rather unphysical result for
$\Lambda_{QCD}$ ($\Lambda_{QCD} \ge 2 GeV$). The scale $\kappa_T$ is disfavoured
because of a bad $\chi^2/n.d.f.$ of about 2. Another possible choice of
scales, namely a combination of the hardness $\kappa_H$ for the quark jets and
the scale $p_1^T$ of the gluon jets, as predicted by Eqn.~\ref{p1tdef},
yields:
\begin{eqnarray*}
\frac{C_A}{C_F} = 2.14 \pm 0.09_{stat.}
\end{eqnarray*}
in agreement with the result above. Here for the $p_1^T$ scale a range
$6 GeV \le
p_1^T \le 27.5 GeV$ has been used. $\Lambda_{\m{QCD}}$ has a fitted value of
$\Lambda_{\m{QCD}} = (404 \pm 1147) MeV$,
with a $\chi^2/n.d.f \sim 1.4$.
The corresponding results using the {\sc Cambridge} cluster algorithm are:
\begin{eqnarray*}
\frac{C_A}{C_F} &=& 2.44 \pm 0.12_{stat.}, \qquad \Lambda_{\m{QCD}} =
(280 \pm 102) MeV \mbox{and} \\
\frac{C_A}{C_F} &=& 2.35 \pm 0.12_{stat.}, \qquad \Lambda_{\m{QCD}} =
(292 \pm 104) MeV \, ,
\end{eqnarray*}
for the scale $\kappa_H$ and the mixed scale respectively with a comparable
$\chi^2/n.d.f$ of $\sim 1.4$.
The systematic uncertainties lead to similar results in $C_A/C_F$ as for
Eq.~\ref{eq_fitkap}.
\begin{figure}[t,b,h]
\begin{center}
\mbox{\epsfig{file=sc-fanrat-upper.eps,width=15cm}}
\caption{Comparison of scaling violation of quark and gluon jets.
The full lines are the fit of the DGLAP evolution. The grey
areas are obtained by changing $C_A$ in the range of the fit errors.}
%\caption{a) Comparison of scaling violation of quark and gluon jets (upper
% plot.
% b) Ratio of the scaling violations (lower plot).
% The full lines are the expectation of the DGLAP evolution. The grey
% areas are obtained by changing $C_A$ in the range of 2 to 4.}
\label{xe_ratio}
\end{center}
\end{figure}
In Fig.~\ref{xe_ratio}, the slopes as obtained from the fits
(Eqn.~\ref{gl_linfit}) to quark and gluon jets for the scale $\kappa_H$
are plotted as a function of $x_E$.
The typical scaling violation pattern
is directly evident. The data is very
well represented by the DGLAP expectation for quarks and for
gluons (see lines in Fig.~\ref{xe_ratio}).
Furthermore the slope of the quark fragmentation functions are in a very good
agreement with low energy data from {\sc Tasso}.
The stronger scaling violation for gluons compared to quarks is due to the
higher colour charge of the gluons. For gluons also the variation of the fitted
$C_A$ within the errors is shown as a grey area, indicating that
this measurement has a high sensitivity to the colour factor $C_A$.
%In Fig.~\ref{xe_ratio}b the ratio of the scaling violations for quarks and
%gluons is shown. The expectation from the DGLAP evolution is shown as a solid
%line. Again, as expected (see Eqn.~\ref{gl_limit}), it rapidly approaches the ratio
%$C_A/C_F = 2.25$ at large $x_E$. The pole at $x_E \sim 0.1$ is due to the
%vanishing of the quark scaling violation in this $x_E$ range. The measured
%values are nicely consistent with the QCD expectation. Within the large errors
%of the measurement, which are predominantly caused by the strong nonlinearity
%in the error propagation of the ratio, the ratio of the scaling violations for
%gluons to quarks directly measures the colour factor ratio:
%$$
% \frac{C_A}{C_F} = \frac{S_g}{S_q}\mid_{x_E > 0.5} = 2.7 \pm 0.7(stat.)
%$$
%\clearpage
%******************************************************************************
%****************************************************************************
\section{Conclusions}
\label{sec_con}
Light quark jets and gluon jets of similar transverse momentum like scales
have been selected from
planar symmetric three jet events measured with {\sc Delphi}.
Using impact parameter techniques, gluon jets have been selected
in heavy quark events. Light quark jets are obtained from heavy quark
depleted events.
Properties of pure quark and gluon jets have been obtained by
subtraction techniques.
A precise measurement of the quark and gluon fragmentation function
into stable charged hadrons has been presented as a function of the
jet scale $\kappa_H$ and $p_1^T$. A
good agreement between quark jet fragmentation functions
and those from lower (and high) energy data is observed indicating the proper
choice of scales. This is also validated from the study of the jet broadening
$<\beta >$ as a function of this scales.
Scaling violations are clearly observed for quark jets as well as for gluon
jets. The last presents evidence for the triple gluon coupling, a basic
ingredient of QCD. Scaling violations are observed to be much stronger for
gluon compared to quark jets. The colour factor ratio:
\begin{eqnarray*}
\frac{C_A}{C_F} &=& 2.23 \pm 0.09_{stat.} \pm 0.06_{sys.} \\
\frac{C_A}{C_F} &=& 2.25 \pm 0.09_{stat.} \pm 0.06_{sys.} \pm 0.11_{clus.}
\end{eqnarray*}
is measured from the scaling
violations in gluon to quark jets, obtained by the {\sc Durham}
resp. by the mean value of the {\sc Durham} and {\sc Cambridge}
cluster algorithm, by simultaneously fitting the quark and gluon
fragmentation functions as a function of the scale $\kappa_H$ resp. the mixture
of $\kappa_H$
and $p_1^T$
with a first order QCD DGLAP
equation. A parameterization given for the quark and gluon fragmentation
functions is given at a reference scale of $\kappa_H = 5.5$~GeV.
\subsection*{Acknowledgements}
\hspace{14pt}
We are greatly indebted to our technical collaborators and to the
funding agencies for their support in building and operating the {\sc Delphi}
detector, and to the members of the {\sc CERN-SL} Division for the
excellent performance of the {\sc Lep}
collider. We also thank Yu.L. Dokshitzer, V. Khoze and P. Kroll
for useful and illuminating discussions.
\clearpage
%******************************************************************************
%******************************************************************************
%\include{lit}
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%\small %\itemsep-0.5ex
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\end{thebibliography}
%****************************************************************************
%\clearpage
%\begin{appendix}
%\section{Quark and Gluon Fragmentation Functions}
%\label{app_xe}
%
%To be filled!!
%
%
%\end{appendix}
\end{document}