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%==============================================================================
%==============================================================================
\begin{titlepage}
\pagenumbering{arabic}
\begin{tabular}{l r}
ICHEP'98 \#147 & \hspace{6cm} DELPHI 98-86 CONF 154 \\
Submitted to Pa 3 & 22 June, 1998 \\
\hspace{2.4cm} Pl 4 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Scaling Violations of Quark and Gluon Fragmentation Functions}\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\normalsize {
%===================> DELPHI note author list =====> To be filled <=====%
{\bf K.~Hamacher} $^1$,
{\bf O.~Klapp} $^1$,
{\bf P.~Langefeld} $^{1}$,
{\bf M.~Siebel} $^1$
%========================================================================%
}
\vspace*{1cm}
\end{center}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
The fragmentation functions of quarks and gluons
are measured in various 3-jet topologies in Z decays from the full data set
collected with the {\sc Delphi} detector at Z energies between 1992 and 1995.
The results at different values of a transverse-momentum-like
scale are compared.
\\
Gluon jets are identified in 3-jet events containing primary heavy quarks
by using impact parameter information.
Comparable quark jet properties are obtained from light quark dominated
3-jet events.
\\
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 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:
$$
\frac{C_A}{C_F} = 2.44 \pm 0.21_{stat.}
$$
A parameterization of the quark and gluon fragmentation
functions at a fixed reference scale is also given.
%=========================================================================%
\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. of Physics, University of Wuppertal,
Gau{\ss}stra{\ss}e 20, D~42097 Wuppertal, Germany}
\end{titlepage}
\pagebreak
%==============================================================================
%==================> DELPHI note text =====> To be filled <======%
%\include{document}
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\section{Introduction}
The observation of the scaling violations of the deep inelastic
nucleon structure function $F_2(x_{Bj},Q^2)$~\cite{prl35_14_901,prl38_25_1450},
and later the complementary measurement of scaling violations of the quark
fragmentation function $D_Q(x,Q^2)$~\cite{q_fragfun}, are among the fundamental
tests of Quantum Chromodynamics, QCD.
Evidently it is interesting to extend these studies
also to scaling violations in gluon fragmentation.
The presence of scaling violations in gluon fragmentation directly proves
the presence of the triple gluon coupling, one of the basic ingredients of
the non-abelian gauge theory QCD.
Comparing scaling violations in quark and gluon fragmentation also allows
to directly determine the relative coupling strengths of gluon radiation from
quarks and of the triple gluon coupling~\cite{splitting_paper}.
In this paper we present a detailed comparison of scaling violations in
quark and gluon fragmentation.
Quark and gluon initiated jets are identified in symmetric and asymmetric
three-jet events originating from hadronic Z decays.
This increases the statistical precision compared to
previous data based on symmetric topologies only.
Gluon jets are identified using heavy quark tagging.
Quark data are obtained from a quark gluon mixture, subtracting the
corresponding gluon data.
The identification of quark and gluon jets relies on the analogy to tree
level graphs. Higher order corrections affect this definition by terms of
${\cal O}(\alpha_S)$. The relevant scales are determined from the jet energies
and the event topology, as in studies of the scale dependence of the
multiplicity in quark and gluon jets~\cite{aleph_topo,multinotevanc}.
The paper is organized as follows. Section~\ref{sec_data} describes
briefly some technical aspects of the analysis.
Section~\ref{sec_theo} contains some theoretical
preliminaries and introduces the variables and distributions used for this
study. The results are presented in section~\ref{sec_res} and compared to
QCD calculations.
The colour factor ratio $C_A/C_F$ and $\Lambda_{\m{QCD}}$ are determined by a
simultaneous fit to the quark and gluon fragmentation functions.
Conclusions appear in section~\ref{sec_con}.
%*****************************************************************************
%*****************************************************************************
\section{Data Analysis}
\label{sec_data}
A detailed description of the {\sc Delphi} apparatus has been presented
in~\cite{detector}.
The full data collected by {\sc Delphi} are
considered in the present analysis.
Cuts applied to charged and neutral particles and to events in order
to select hadronic Z decays are identical to those given
in~\cite{splitting_paper,oliver,martin}. Three-jet events are clustered using the
Durham algorithm~\cite{dur1} with a jet resolution parameter $y_{cut}=0.015$.
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{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 events in which
the two low-energy jets should be directly comparable.
\end{itemize}
\begin{figure}[t,b,h]
\begin{center}
\mbox{\epsfig{file=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}
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\footnote{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 in~\cite{splitting_paper}.
From the initial $\sim$3,690,000 hadronic events collected by {\sc Delphi}
(and $\sim$10,510,000 Monte Carlo events),
there remain $\sim$450,000 events of which $\sim$65,000 are symmetric events.
The identification of gluon jets by anti-tagging of heavy quark jets
is identical to that described in~\cite{splitting_paper, paper}.
The 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 (or clusters in the case of {\sc Herwig}). Thus
three independent ways of defining the gluon jet in the fully
reconstructed Monte Carlo are investigated~\cite{oliver,martin}:
\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}
\[
\begin{array}[c]{|c|c|ccc|}
%\cline{3-5}
\hline
\multicolumn{2}{|l|}{\mbox{\bf Method}} &
\multicolumn{3}{|c|}{\mbox{\bf Angle assignment}} \\ \cline{2-5}
\multicolumn{1}{|c|}{\ } &
\mbox{\bf gluon in:} &
\mbox{\ \ \ Jet 1} & \mbox{\ \ \ Jet 2} & \mbox{\ \ \ Jet 3}\\
\cline{2-5}
\hline
&\mbox{Jet 1} & 4.5\% & 0\% & 0.03\% \\
\mbox{\bf PS}&\mbox{Jet 2}
& 0\% & 23.3\% & 0.3\% \\
\mbox{\bf assignment} &\mbox{Jet 3}
& 0.02\% & 0.13\% & 71.8\% \\
\hline
\end{array}
\]
\caption{Correlation of angle and PS assignments. The table has been
obtained for arbitrary three-jet events with
$\theta_{2}, \theta_{3} \in [90^\circ,170^\circ]$. These events also contain
the symmetric events. A similar behaviour is observed concerning the history
assignment~{\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\footnote{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.} from 40\% to 90\% are achieved, depending on
the jet scale (see Fig.~\ref{fg_glupur}).
\begin{figure}[bth]
\begin{center}
\mbox{\epsfig{file=purmat.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 $\kappa$ is a significantly enrichment compared to
the correspondingly gluon fraction of $\simeq 15\%$ in the normal
mixture jets.}
\label{fg_glupur}
\end{center}
\end{figure}
In order to achieve pure quark ($udsc$) and gluon jet distributions
the following equation has to be 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 as
$i$ identified measured distribution.
Moreover, the effects of finite
resolution and acceptance of the detector are corrected using a full simulation
of the {\sc Delphi} detector.
In the further analysis, bins with high acceptance correction factors are
disregarded.
%******************************************************************************
%******************************************************************************
\section{Theory}
\label{sec_theo}
\subsection{Jet Scales}
\label{kappa_scale}
In order to determine scaling violations, the scale underlying the evolution
of the corresponding jet needs to be known.
Usually this is taken to be the centre of mass energy, $E_{cm}$.
As all jets entering this analysis stem from Z decays, thus from fixed \
$E_{cm}$,
the scale has to be determined from the jet energy and the event topology.
In the present study we restrict ourselves to use the so called
hardness scale, $\kappa$~\cite{ddt}, the maximum allowed transverse momentum in a jet:
\begin{equation}
\kappa^i = E_{calc}^i \cdot \sin{\frac{\theta_{min}}{2}} \, ,
\label{kappadef}
\end{equation}
where $E_{calc}^i$ is the energy of jet $i$ as determined by
Eqn.~\ref{recalibration},
and $\theta_{min}$ is the angle with respect to the closest jet.
$\kappa$
is similar to the transverse momentum of the jet and is also
related to $\sqrt{y_{cut}}$, as used by the jet algorithms.
$\kappa$ is also used as a scale in the study
of the scale dependence of the multiplicity~\cite{multinotevanc}.
Fig.~\ref{kappa_topo} compares the distribution of
the jet energies $E$ and hardnesses $\kappa$
for the jets of the five symmetric event topologies.
\begin{figure}[t,b,h]
\begin{center}
\mbox{\epsfig{file=kappa.eps,width=10cm}}
\caption{Separation of symmetric event topologies
by the jet energy and hardness scales.}
\label{kappa_topo}
\end{center}
\end{figure}
The upper plot of Fig.~\ref{kappa_topo} shows the distribution of the
jet energies. As can be seen, there is a great overlap
among the energy distributions for fixed topologies.
The lower plot shows the distributions of $\kappa$.
The $\kappa$ distributions are clearly separated for the different
symmetric event topologies.
This can be understood easily: in the case of symmetric events
and massless jets, $\kappa$ falls steeply to $0$ with
increasing $\theta_2 (=\theta_3)$,
while the energy becomes nearly constant (see Fig.~\ref{kappa_topo}c).
This makes $\kappa$ less sensitive to small deviations
from the exact symmetric topology than the energy itself.
Finally it should be noted that the available range of scales is bigger
in case of $\kappa$ compared to the jet energy.
%Tab.~\ref{kappa_value} shows the geometrical calculated and measured
%values of the different jet scale definitions for the symmetric topologies.
%
%\begin{table}[t,h,b]
%\begin{center}
%
%\footnotesize
%\begin{tabular}{|c|r@{}l|r@{}l||r@{}l|r@{}l|}\hline
% & \mc{4}{c||}{\rul \bf $E$} &
% \mc{4}{c|}{\rul \bf $\kappa$} \\
%\hline
% {\rul Topology} & \mc{2}{c|}{\rul calc.} & \mc{2}{c||}{\rul meas.} &
% \mc{2}{c|}{\rul calc.} & \mc{2}{c|}{\rul meas.} \\
%\hline
%$120^\circ$ & 30.&40& 29.&66& 26.&33& 25.&08 \\
%\hline
%$130^\circ$ & 27.&76& 27.&64& 21.&26& 21.&11 \\
%\hline
%$140^\circ$ & 25.&82& 25.&75& 16.&60& 16.&48 \\
%\hline
%$150^\circ$ & 24.&44& 24.&21& 12.&22& 11.&96 \\
%\hline
%$160^\circ$ & 23.&51& 23.&24& 8.&04& 7.&90 \\
%\hline
%\end{tabular}
%\caption{Scale values of the low energy jets in symmetric topologies.}
%\label{kappa_value}
%
%\end{center}
%\end{table}
%The agreement between the calculated and the measured mean values of $\kappa$
%is excellent, taking into account that since gluon radiation is
%a bremsstrahlung effect $\theta_2$ and $\theta_3$ are shifted towards
%the higher limit of $\theta$ in a symmetric bin, thus yielding lower average
%values of $\kappa$.
%****************************************************************************
\subsection{Scale Dependence of the Fragmentation Function of Quark and
Gluon Jets}
\begin{figure}[bth]
\begin{center}
\mbox{\epsfig{file=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}.
Their relative coupling strength is determined by the colour factors
which are determined by the structure of $SU(3)$, the group underlying QCD.
The Casimir factors $C_{F}$, $C_{A}$,
and $T_{F}$ determine the coupling strengths of
gluon radiation from quarks, of the triple-gluon vertex, and of gluon
splitting into a quark-antiquark pair, respectively.
Within $SU(3)$, these coupling constants 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).
Jet splittings may be studied with respect to the energy sharing in a
splitting process. This analysis is connected to the analysis of the scale
dependence (scaling violation) of the
fragmentation functions $D^{H}_{p}(x_E,s)$ of a parton $p$ 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}, where $s$ is the
relevant scale to be replaced by $\kappa$.
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. $g\rightarrow gg$.
The (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] \\
\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_gdlap}
\end{eqnarray}
Observe that these equations are the transpose of the usual DGLAP equations
for structure functions. This 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) \\
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) \\
P_{g\rightarrow q\bar{q}}(z) &=& 2 n_F T_F \cdot (z^2+(1-z)^2) \\
P_{q\rightarrow gq}(z) &=& C_F \cdot \frac{1+(1-z)^2}{z}
\end{eqnarray*}
Here the 'plus' distribution is defined such that its 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{\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 a ratio of colour factors.
The slopes and the ratio can be predicted by solving the DGLAP equation
numerically~\cite{evolve}.
The following ansatz has been used to parameterize the fragmentation
function at a fixed scale $\kappa_0$ to start the evolution:
\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{x_E}^2)}
\label{gl_dfrac}
\end{eqnarray}
The parameters $p_i^{q,g}$, $\Lambda_{QCD}$ and the colour factor $C_A$ are
fitted simultaneously.
The slopes can be used directly to measure $\frac{C_A}{C_F}$.
$r_{\cal S}(x_E)$ is accessible by measuring the ratio
of the slopes of the $D_p^{F}(x_E,s)|_{x_E}$ curves independently.
%******************************************************************************
%******************************************************************************
\section{Results}
\label{sec_res}
%\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]
\newlength{\breite}\breite 11.4cm
\newlength{\breitemp}\breitemp 11.5cm
\begin{center}
\rotatebox{90}{
\begin{minipage}[t]{\breitemp}
\begin{center}
\includegraphics*[angle=0,width=\breite]{fanx-q.eps}
\caption{Scaled quark energy distributions for different scales.
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]{fanx-g.eps}
\caption{Scaled gluon energy distributions for different scales.
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{paper,xeopal,xealeph}.
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 different values of
$\kappa$.
An approximately exponential decrease of the
fragmentation function with increasing $x_E$ is seen, which is more
pronounced in the gluon case.
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 softening of the fragmentation functions with increasing
$\kappa$ is observed. This effect is more
pronounced for gluon jets than for quark jets.
%\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]
\newlength{\breitea}\breitea 11cm
\newlength{\breitempa}\breitempa 11.4cm
\begin{center}
\rotatebox{90}{
\begin{minipage}[t]{\breitempa}
\begin{center}
\includegraphics*[angle=0,width=\breitea]{fan-low.eps}
\caption{Scale dependence of the quark
fragmentation function. The reference data stem from
low and higher energy experiments.
The dashed line is the result of power law fits.}
\label{xe_low}
\end{center}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{\breitempa}
\begin{center}
\includegraphics*[angle=0,width=\breitea]{fan-g-symm.eps}
\caption{Scale dependence of the gluon
fragmentation function.
The dashed lines result from power law fits.}
\label{xe_symg}
\end{center}
\end{minipage}
}
\end{center}
\end{figure}
%\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]
\newlength{\breiteb}\breiteb 11cm
\newlength{\breitempb}\breitempb 11.4cm
\begin{center}
\rotatebox{90}{
\begin{minipage}[t]{\breitempb}
\begin{center}
\includegraphics*[angle=0,width=\breiteb]{fan-q.eps}
\caption{Scale dependence of the quark
fragmentation functions.
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]{\breitempb}
\begin{center}
\includegraphics*[angle=0,width=\breiteb]{fan-g.eps}
\caption{Scale dependence of the gluon
fragmentation functions.
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}
Fig.~\ref{xe_low} shows the comparison of the quark fragmentation function for
fixed $x_E$ as a function of the scale $\kappa$ (in the following called `fan'
plots) with the quark fragmentation
functions measured at lower $e^+e^-$ energy experiments and at high $E_{CM}$ with
{\sc Delphi}.
These data distributions are obtained from $e^+e^-$ events scaled by
$\frac{1}{2}$, because most of the events are two jet events. A good agreement
is seen. This is 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 $\kappa$.
The results obtained from the overall data set and from the symmetric events
agree well. 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 indicate that $\kappa = E\cdot\sin\Theta/2$
is a relevant scale for dynamical studies of jet properties.
to be used as the relevant scales
of the jets.
The expected power behaviour, indicated 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
well. The typical behaviour of scaling violations is observed in both plots,
namely a
strong fall off at large $x_E$ which diminishes with falling $x_E$. This
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{multinotevanc}. The
scaling violation behaviour is much stronger for gluons than for quarks.
This is expected due to the higher colour charge of gluons.
Figs.~\ref{xe_scaleq} and~\ref{xe_scaleg} show the measured quark and gluon
fan plot.
A fit of the fragmentation functions including a simultaneous DGLAP evolution
is also shown. For the evolution the fragmentation functions
have been parameterized at
$\kappa = 6.5$ 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 a $\kappa$ range $8 GeV \le \kappa \le 29 GeV$
has been used. From the fit the colour factor ratio $C_A/C_F$ has been
determined to be:
\begin{eqnarray*}
\frac{C_A}{C_F} = 2.44 \pm 0.21
\end{eqnarray*}
in a good agreement with the expectation of $C_A/C_F=2.25$.
$\Lambda_{\m{QCD}}$ has a fitted value of
\begin{eqnarray*}
\Lambda_{\m{QCD}} = (79 \pm 57) MeV \, .
\end{eqnarray*}
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} \, ,
\end{eqnarray*}
consistent with the present world average.
\begin{table}[t,b,h]
\begin{center}
\footnotesize
\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
$p_1$ & -3.&50 &$\pm$& 0.&28 & -4.&63 & $\pm$& 0.&37 \\
\hline
$p_2$ & 0.&71 &$\pm$& 0.&14 & 0.&78 & $\pm$& 0.&24 \\
\hline
$p_3$ & 0.&067 &$\pm$& 0.&017 & 0.&020& $\pm$& 0.&0085 \\
\hline
$p_4$ & 0.&925 &$\pm$& 0.&088 & 1.&09 & $\pm$& 0.&11 \\
\hline
\end{tabular}
\caption{Parameters of the simultaneously fitted fragmentation functions at
$\kappa_0 = 6.5$ GeV ($\chi^2/n.d.f = 1.058$). The errors are
given neglecting any correlations between the parameters.}
\label{tb_dfrac}
\end{center}
\end{table}
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.
\begin{figure}[t,b,h]
\begin{center}
%\mbox{\epsfig{file=fanrat.eps,width=15cm}}
\mbox{\epsfig{file=fanrat-iii.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 are plotted as a function of $x_E$.
The typical scaling violation pattern, as usually obtained for deep inelastic
scattering data, is directly evident. The data is very
well represented by the DGLAP expectation for quarks and for gluons.
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 scale
$\kappa$ 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 by gluon subtracting.
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 $\kappa$. A good agreement between quark jet fragmentation functions
and those from lower (and high) energy data is observed.
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:
$$
\frac{C_A}{C_F} = 2.44 \pm 0.21(stat.)
$$
is measured from the scaling violations in gluon to quark jets by
simultaneously fitting the quark and gluon fragmentation functions as a
function of the scale with a first order QCD DGLAP equation..
A parameterization given for the quark and gluon fragmentation
functions is given at a reference scale of $\kappa = 6.5$~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 P. Kroll for useful and illuminating discussions.
\clearpage
%******************************************************************************
%******************************************************************************
%\include{lit}
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