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\noindent {\bf Colour Coherence and a Comparison of the Fragmentation of
Gluon and Quark Jets}
~\\
Klaus Hamacher, Fachbereich C (Physik), Bergische Universität Wuppertal and DELPHI Collaboration\\
~\\
\noindent 
Coherence is an important property of gluon radiation in
the hadronic final state of inelastic interactions and a direct consequence
of the quantum nature of strong interactions. Theoretical calculations
observing coherence predict the evolution scales of jets inside of
multihadronic events. Typically these scales are transverse-momentum-like
and fulfill Lorentz-invariance in contrary to the often
used jet energies.

A clear demonstration of colour coherence is obtained from the measurement of the
 ratio
of the soft hadron multiplicity in narrow cones oriented perpendicular
to the plane of a three-jet event and the axix of a two jet event \cite{Abdallah:2004uu,drMS}.
This ratio is predicted as the product of the colour factor ratio $C_{A}/C_{F}$
and an event scale $r_{\perp}$. The $r_{\perp}$-depedence allows
to compare the amount of coherent soft radiation due to the gluon and the 
quark colour charge. The prediction without any free parameter describes
the data very well.  A destructive $1/N_{c}^{2}$
suppressed term included in $r_{\perp}$ is required to describe the
data.
The colour factor ratio when fitted to the data
is $C_{A}/C_{F}=2.211±0.014(stat.)±0.053(syst.)$. 
The reason for this good agreement of the leading order prediction
with the data lies in the soft, large angle hadrons used as testing
probes. These hadrons due to their large wavelength are insensitive
to higher order structures and also to finite energy and leading particle
effects.

A precision study of the overall multiplicity of three-jet events \cite{Abdallah:2005cy,drMS}
also relies on properly defined scales in order to describe the topology
dependence of the multiplicity. Two different choices are available
as the devision of the event multiplicity in a ${\rm q\bar{q}}$ and
a gluon part is ambigous. The most natural choice describes the data
whereas the choice minimising the amount of hadrons assigned to the
gluon jet (chosen by Opal) fails to agree. The colour factor ratio
was fit to the slope of the scale dependence of the multiplicity yielding
$C_{A}/C_{F}={2.261}\pm0.014_{\mathrm{stat.}}\pm0.036_{\mathrm{exp.}}\pm0.052_{\mathrm{theo.}}\pm0.041_{\mathrm{clus.}}$.
This result together with a measurement of the $\beta$-function of
the Thrust \cite{Abdallah:2003xz} which agrees to the one of the coupling in NLO restrict
the gauge group of strong interactions to SU(3). From the measurement
the multiplicity of gluon-gluon colour singlet events can be extracted.
This multiplicity rises about twice as fast with the energy scale
as in the ${\rm q\bar{q}}$-case clearly demonstrating the higher
colour charge of the gluon.

The precise agreement of the above discussed measurements to the colour
factor expectation leaves little room for further differences in the
particle production from gluons and quarks. The measurements mainly
admit differences for leading particles. Such are indeed observed
for baryons \cite{Abreu:2000nw}, where an excess of about a factor two (i.e. $\sim2\cdot C_{A}/C_{F}$)
is observed for leading particles. This excess is understood within
the string fragmentation model due to the higher number of possible
splitting processes into baryons. Cluster models here fail if no fundamental
splittings of gluons into diquark-anti-diquark pairs are foreseen.

It has been found in Monte Carlo studies that gluon fragmentation
functions obtained from three-jet events and hypothetical gluon-gluon
events disagree in specific parts of phase-space \cite{MAREK}. Therefore, the results
obtained from three-jet events were quantified as ``biased''.
This distinction seems unfounded as three-jet events form the really
accessible source of data. Moreover, if the observed differences were
essential they need to be brought into accord with the QCD factorisation
theorem. It turns out, however, that the discrepancies are consequence 
of hadronisation and the necessities of the measurement process \cite{Hamacher:2005rh}. 
The reconstruction of the parton kinematics due
to confinement must proceed by a direct identification of the jet
with the parton kinematics. However, the hadronisation transition
then leads to a smearing of the parton kinematics. Typically
this smearing vanishes $\propto1/E_{jet}$. The parton energy inferred
from the jets enters in the definition of the scaled momentum $x_{h}=p_{hadron}/p_{g/q}$
and in turn especially strongly varying fragmentation functions are
influenced by error propagation. In fact the effect is stronger for
the gluon compared to the quark fragmentation function due to the
stronger fall-off, as well as due to the somewhat increased hadronisation
smearing for gluon jets. A numerical calculation shows ratios of up
to a factor 2 between the smeared, measurable fragmentation function
and the underlying partonic one for gluons at high $x_{h}$ and small
$E_{jet}$. The difference detected for gluon fragmentation \cite{MAREK} can
be fully explained by the hadronisation/measurement process. It is
important to to note that similar problems unavoidably exist in any
measurement involving jets, e.g., when gluon fragmentation is measured
in ${\rm p\bar{p}\rightarrow 2~jets}$. The hadronisation smearing
cannot be properly included in an experimental unfolding as it will
depend on the fragmentation function to be measured. Still, it is
possible to use such measurements if the hadronisation smearing is
included in the phenomenological fitting procedure of the fragmention
functions. 
\begin{thebibliography}{1}
\bibitem{Abdallah:2004uu} J.~Abdallah \textit{et al.}
%``Coherent soft particle production in Z decays into three jets,''
 Phys.\ Lett.\ B \textbf{605} (2005) 37.
\bibitem{drMS} M. Siebel, PhD thesis, University of Wuppertal (WUB-DIS
2003-11), %http://elpub.bib.uni-wuppertal.de/edocs/dokumente/fbc/physik/diss2003/siebel
and references given therein.
\bibitem{Abdallah:2005cy} J.~Abdallah \textit{et al.} 
%``Charged particle multiplicity in three-jet events and two-gluon  systems,''
 Eur.\ Phys.\ J.\ C \textbf{44} (2005) 311;
K.~Hamacher, O.~Klapp, P.~Langefeld
and M.~Siebel [DELPHI], 
%``More about the multiplicity of symmetric three jet events,''
CERN-OPEN-2000-134,
P.~Abreu {\it et al.}
  Phys.\ Lett.\  B {\bf 449} (1999) 383.
\bibitem{Abdallah:2003xz}
  J.~Abdallah {\it et al.}
  %``A study of the energy evolution of event shape distributions and their
  %means with the DELPHI detector at LEP,''
  Eur.\ Phys.\ J.\  C {\bf 29}, 285 (2003).
\bibitem{Abreu:2000nw}
  P.~Abreu {\it et al.} 
  %``Identified Charged Particles in Quark and Gluon Jets,''
  Eur.\ Phys.\ J.\  C {\bf 17} (2000) 207.
\bibitem{MAREK} Compare talk by Marek Tasevsky at this meeting and references therein. 
\bibitem{Hamacher:2005rh} K.~Hamacher, %``Fragmentation at LEP,''
 Acta Phys.\ Polon.\ B \textbf{36} (2005) 433.
\end{thebibliography}

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