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\begin{document}
%%% put your own definitions here:
\newcommand{\cacf} {\ifmmode{C_A/C_F} \else{$C_A/C_F$} \fi}
\newcommand{\lam} {\ifmmode{\Lambda_{QCD}} \else{$\Lambda_{QCD}$} \fi}
\newcommand{\epem} {\ifmmode{e^+e^-} \else{$e^+e^-$} \fi}
\begin{titlepage}
\pagenumbering{arabic}
\begin{tabular}{l r}
ICHEP'98 \#146 & \hspace{6cm} DELPHI 98-78 CONF 146 \\
Submitted to Pa 3 & 26 June, 1998 \\
\hspace{2.4cm} Pl 4 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf
The Scale Dependence of the\\
Hadron Multiplicity \\
in Quark and Gluon Jets and a\\
Precise Determination of
%\boldmath
{$\rm \bf C_A/C_F$}
}\\
\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 S.~Marti} $^2$,
{\bf M.~Siebel} $^1$
%========================================================================%
}
\vspace*{1cm}
\end{center}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
Data obtained at the Z resonance using the DELPHI detector at LEP are
used to determine
the hadron multiplicity in gluon and quark jets as a function
of a transverse momentum like scale.
The colour factor ratio $C_A/C_F$ is directly observed in the multiplicity
slopes.
The small multiplicity ratio in gluon to quark jets is understood by
differences in the hadronization of the leading quark or gluon.
From the dependence of the hadron multiplicity on the opening angle in
symmetric three jet events the colour factor ratio $C_A/C_F$ is
measured to be:
$$
\frac{C_A}{C_F} = 2.266 \pm 0.053~(stat.) \pm 0.055~(syst.) \pm 0.096~(theo.)
$$
%=========================================================================%
\end{abstract}
\vspace{\fill}
\begin{center}
Paper submitted to the ICHEP'98 Conference \\
Vancouver, July 22-29
\end{center}
\vspace{\fill}
\par {\footnotesize $^1$
Fachbereich Physik, University of Wuppertal, Gau\ss{}stra\ss{}e 20, D~42097
Wuppertal, Germany}
\par {\footnotesize $^2$ CERN, CH 1211 Geneva 23, Switzerland}
\end{titlepage}
\pagebreak
%==================> DELPHI note text =====> To be filled <======%
\sloppy
\flushbottom
%==============================================================================
% \newcommand{\ttbs}{\char'134}
%
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%------------------------------------------------------------------------------
%
\hyphenation{DELPHI}
%==============================================================================
\section{Introduction}
The gauge symmetry underlying the Lagrangian of an
interaction directly determines the
relative coupling of the vertices of the elementary fields
participating.
A comparison of the properties of quark and gluon jets, which are linked
to the quark and gluon couplings, therefore implies a direct and intuitive
test of Quantum Chromodynamics, QCD, the gauge theory of the strong interaction.
Hadron production is expected to proceed via a so-called parton shower,
a chain of successive bremsstrahl processes, followed by non-perturbative
hadron formation.
As bremsstrahlung is directly proportional to the coupling of the radiated
vector boson to the radiator the ratio of the
{\it radiated gluon multiplicity} from a
gluon and quark source is expected to be equal to the ratio of the
QCD colour factors $C_A/C_F=9/4$ \cite{brodskygunion}.
The higher multiplicity expected for gluon jets is directly complementary
to a stronger scaling violation of the gluon fragmentation function
\cite{splitting_paper,scaling_vancouver}.
It was however noted already in the first paper
on the gluon to quark multiplicity \cite{brodskygunion}
that this prediction not immediately applies to the {\it observed charged hadron
multiplicity} at finite energy
as this is also influenced by differences of the fragmentation
of the {\it primary} quark or gluon.
In fact these differences must be present due to the fact that quarks
are valence particles of the hadrons whereas gluons are not.
This is most clearly evident from the behaviour of the gluon
charged hadron fragmentation function at large scaled momentum.
Here it is suppressed by about one order of magnitude compared to the
quark fragmentation function \cite{scaling_vancouver}.
This also causes an
initially higher multiplicity to be expected from very low energetic quarks
compared to gluons.
Moreover, as low momentum, large wavelength gluons cannot resolve a hard
radiated gluon from the initial quark antiquark pair in the initial phase of an
event, soft radiation is further suppressed \cite{ochs,oliver}
compared to the naive expectation.
In a previous publication \cite{splitting_paper} it has
been shown that a reduction of the
primary splittings (splitting kernels) of gluons compared to the perturbative
expectation is indeed
responsible for the observed small gluon/quark multiplicity ratio.
If also heavy quark jets are included in the comparison a further
reduction of the multiplicity ratio is evident due to high number
of particles from the decays of the primary heavy particles.
Furthermore, the definition of quark and gluon jets in three-jet events in
$\mbox{e}^+\mbox{e}^-$ annihilation requires the usage of jet
algorithms which combine hadrons to jets.
Low energetic particles at large angles with respect to
the original parton direction are
likely to be assigned to a different jet.
As gluon jets are initially wider than quark jets this presumably
leads to a loss of multiplicity for gluon jets and a corresponding
gain for quark jets.
The effects discussed lead to an observed charged hadron multiplicity
ratio which is smaller than the radiated gluon multiplicity ratio.
So far these effects have mainly been neglected in experimental and
more elaborate theoretical investigations \cite{theorie_qg}.
However, as we will show in this paper,
at current energies these non-perturbative effects are still important
and need to be considered in a proper test of the prediction
\cite{brodskygunion}.
The stronger radiation in gluon jets is expected to become
directly evident from
a stronger increase of the gluon multiplicity with scale as compared to
quark jets.
In this way also the size of the non-perturbative terms can be directly
estimated from the quark and gluon multiplicity at very small scales.
A scale dependence of quark and gluon properties was first
demonstrated in \cite{gluon_paper_1} with the jet energy as intuitive
scale.
This result was later confirmed by other measurements
\cite{qg_edep_measurments}
and has recently been extended to a transverse momentum-like scale
\cite{aleph_qgmult}.
A study of the {\it total} multiplicity of symmetric three jet events as function
of the internal scales of the event avoids some of the complications mentioned
above.
In combination with a MLLA prediction of the 3-jet event
multiplicity, including coherence of soft radiation, a novel
precision measurement
of the colour factor ratio $C_A/C_F$ can be performed.
This note is based on a data analysis which is similar to that
presented in previous papers \cite{splitting_paper,gluon_paper_1}.
We therefore have chosen to restrict the experimental discussion in
chapter \ref{experim} to the relevant differences with respect to
these papers.
More detailed information can also be found in \cite{salva,martin}.
In chapter \ref{results} quark and gluon jet
multiplicities obtained from symmetric and non-symmetric event
topologies are compared.
The slopes of the multiplicities with scale are presented
yielding a direct measurement of $C_A/C_F$.
Then the precision measurement of the colour factor ratio from symmetric
three jet events is discussed.
Finally we summarize and conclude.
\section{Data Analysis \label{experim}}
The analysis presented in this letter includes the full hadronic
data set collected with the DELPHI detector at Z energies in the years 1992 to
1995.
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} for the $q\bar{q}g$ analysis and to
\cite{gluon_paper_1} for the $q\bar{q}\gamma$ analysis.
For the comparison of gluon and quark jets
three-jet events are clustered using the Durham-algorithm
\cite{durham_algo} with a jet resolution parameter $y_{cut}=0.015$.
In addition it was required that the angles $\theta_{2,3}$
of the low-energetic jets
with respect to the leading jet are in the range of $100^o$ to $170^o$
(see Fig. \ref{eventtopol}a).
The leading jet is not considered in gluon and quark jet analysis.
%From the overall set of events defined by these cuts which is dominated
%by non-symmetric topologies five symmetric
%reference samples of Y type events are extracted requiring
%$\theta_2 \approx \theta_3 = 120^\circ \pm 5^\circ , 130^\circ \pm 5^\circ ,
%140^\circ \pm 5^\circ , 150^\circ \pm 5^\circ ,160^\circ \pm 5^\circ $
%(see Fig.\ref{eventtopol}b).
\begin{figure}[htbp]
\begin{center}
\epsfig{file=p-topo.eps,width=14.cm}
\caption[eventtopology]
{
\label{eventtopol}
Definition of event topologies and angles used throughout this
analysis.
The length of the jet lines indicates the energies.
}
\end{center}
\end{figure}
The identification of gluon jets by anti-tagging of heavy quark jets
is identical to that described in \cite{splitting_paper,gluon_paper_1}.
Quark jets are taken from $q\bar{q}\gamma$ events
or $q\bar{q}g$ events which have been depleted from b
quark events using an impact parameter technique.
In order to achieve pure quark and gluon multiplicities, the data have
been corrected using purities from simulated events generated with
JETSET~7.3~\cite{pythia,tuningpaper}.
This is justified by the good agreement between data and simulation.
Furthermore the model independent techniques described in
\cite{splitting_paper} for symmetric events give results grossly compatible
to those obtained with the simulation correction \cite{martin}.
Moreover the effects of finite resolution and acceptance of the
detector are corrected using a full simulation of the DELPHI detector
and the analysis chain.
The correction for remaining b quark events
in the $q\bar{q}g$ sample
does not influence the
slope of the measured multiplicity with scale but only leads to a
shift of the absolute value.
In the simulation quark and gluon jets are identified on ``parton
level''.
Partons are clustered to three jets using the Durham algorithm.
Then for each jet the number of quarks and antiquarks
are summed over where quarks contribute with
weight +1 and antiquark with the weight -1.
These sums are expected to yield +1 for quarks jets, -1 for anti-quark
jets and 0 for gluon jets.
The small amount (2\%) of events not showing this expected pattern
were discarded.
Finally, the parton jets were mapped to the jets on hadron level by
requiring the sum of angles between the parton and hadron jets to be
minimal.
The gluon jet purities vary in the range of 95\% for low energetic
gluons and 46\% for the highest energy gluons.
Few bins with lower purities have been excluded from the analysis.
The quark purities are in the range of 43\% - 81\%.
For the analysis of the multiplicity of
symmetric three jet events all events were forced
to three jets using the Durham algorithm not requiring a minimal $y_{cut}$.
The angles between the jets were
then determined by rescaling the jet momenta to the centre of mass
energy as described in \cite{gluon_paper_1}. Symmetric events were
selected by demanding
$\bar{\Theta}_{bin} - \epsilon \le \Theta_{2,3} \le \bar{\Theta}_{bin}
+ \epsilon$.
Here $\epsilon$ is the angular bin width taken to be $1.25^{\circ}, 2.5^{\circ}$
or $5^{\circ}$ respectively.
The analysis has been performed for events of all flavours as well as for
b-depleted events. In both cases the measured multiplicity was corrected
for track losses due to detector effects and cuts applied. The correction
factor was calculated as ratio
of generated over accepted multiplicity using simulated events.
It varies smoothly, about polynomial,
between 1.15 at small angles and 1.3 at large $\Theta_{1}$.
\section{Results \label{results}}
In order to determine a scale dependence the scale underlying the
physics process needs to be specified.
In case that an outer scale like the centre of mass energy is varied
this is less important as the actual physical scale is necessarily
proportional to this outer scale.
As usually only the relative change in scale matters this outer scale
can directly replace the physical scale.
For this analysis the situation is different.
The jets entering the analysis stem from Z decays thus from a fixed
centre of mass energy.
So the relevant scales have to be determined from the properties of
the jets and the event topology.
From the above discussion the scale has to be proportional
to the jet energy as this quantity for similar events scales with the
energy in the centre of mass system.
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 hardness of the process producing the jet
\cite{stringeffect,pertQCD}:
\begin{equation}
\kappa = E_{jet} \sin \frac{\theta}{2}
\label{kappadef}
\end{equation}
which is similar to the transverse momentum of the jet and also
related to $\sqrt{y_{cut}}$ as used by the jet algorithms.
$E_{jet}$ is the energy of the jet and $\theta$ the angle with respect to
the closest jet.
It is also used as scale in the calculation of the energy dependence of the
hadron multiplicity in $e^+e^-$ annihilation
\cite{webber,colliderphysics}
to consider
the leading effect of coherence.
A similar scale, namely the geometric mean of the scales of the gluon
w.r.t. both quark jets, has recently been used in a study of
quark and gluon jet multiplicities \cite{aleph_qgmult}.
In Fig. \ref{qgcmp1}, the multiplicity for light quarks and gluons as obtained
from three jet events as function
of $\kappa$ is shown on a logarithmic x-scale.
For both multiplicities an approximately logarithmic increase with
scale is observed which is significantly stronger for gluon compared
to quark jets.
This was already noted in a previous paper
\cite{gluon_paper_1} where the jet energy was chosen as scale.
Meanwhile this observation has been confirmed also by other
measurements \cite{qg_edep_measurments} and has been extended
to other scales \cite{aleph_qgmult}.
Fragmentation models (not shown) predict an increase of the
multiplicities which is in good agreement with the data.
%It is in order to compare this measurement to previous data.
%In fact the measurement \cite{aleph_qgmult} is in reasonable agreement
%to this data for gluon jets, however shows a
%smaller slope and an overall higher multiplicity for quark jets.
%A study with a fragmentation model indicates that to a large part
%($~0.4$ units of multiplicity)
%the different normalization is due to the inclusion of heavy quark
%events in \cite{aleph_qgmult}.
As stated in the introduction we want to gain information on the
relative colour charges of quarks and gluons from the slopes of the
multiplicity with scale.
Assuming the validity of the perturbative QCD prediction, the ratio of the
gluon and quark multiplicities has to approach a constant value
(approximately the colour factor ratio) at large scale.
This trivially implies that also the ratio of the slopes of quark
and gluon multiplicities approaches the same limit.
This fact is a direct consequence of de l'H{\^o}spitals rule and is also
directly evident from the linearity of the derivative:
\begin{equation}
\mbox{at large scale:} ~~~ N_{gluon}(\kappa)=C \cdot N_{quark }(\kappa) ~~~\rightarrow~~~
\frac{dN_{gluon}/d\kappa}{dN_{quark}/d\kappa} = C~~~,
\label{trivialitaet}
\end{equation}
i.e. the QCD prediction for the ratio of multiplicities
applies equally well to the ratio of the slopes of the multiplicities.
In fact it is to be expected that the slope ratio is closer to the QCD
prediction as the multiplicity ratio as it should be less effected by
non-perturbative effects.
This contiguity has been cross checked using the HERWIG model~\cite{herwig}
which
allows to change the number of colours and thus by SU(n) group relations
the colour factor ratio $C_A/C_F$.
The predictions of HERWIG are found to directly follow
the expectation of the right hand side of Eqn. \ref{trivialitaet}.
This has also been confirmed in a recent theoretical calculation of this
quantity \cite{eden}.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=p-multi-fit.eps,width=14cm}
\caption[quark and gluon multiplicity fitted with the ansatz
\ref{offset_ansatz}]
{
\label{qgcmp1}
Upper plot:
Average charged particle multiplicity for quark and gluon jets
as function of $\kappa$ fitted with the ansatz \ref{offset_ansatz}.
Both QCD predictions for the multiplicity were used.
In the lower plot the ratio of the gluon to quark multiplicity is
plotted as well as the ratio of the slopes.
The fits coincide within the linewidth of the plot.
The corresponding parameters and their errors are given in
Tab. \ref{fit_mehrere}.
}
\end{center}
\end{figure}
It becomes immediately evident from Fig. \ref{qgcmp1}
that the slope for the gluon multiplicity is
about twice as big as for quarks thus already strikingly confirming the QCD
prediction.
In order to obtain quantitative information,
the following ansatz was fitted to the data:
\begin{eqnarray}
(\kappa) &=& N_0^q + N_{pert}(\kappa) \nonumber \\
(\kappa) &=& N_0^g + N_{pert}(\kappa) \cdot r(\kappa)
\label{offset_ansatz}
\end{eqnarray}
Here $N_0^{q,g}$ are non perturbative constants introduced to account for
the differences of the fragmentation of the leading quark or gluon as discussed
in detail in the introduction.
$N_{pert}$ is the perturbative prediction for the hadron multiplicity
as given in \cite{webber}:
\begin{eqnarray}
N_{pert}(\kappa) &=& K \cdot \alpha_s^b(\kappa)
\cdot \exp{ \frac{c} {\sqrt {\alpha_s {\kappa}}}}
\left[ 1 + O(\sqrt{\alpha_s}) \right ]\\
\label{mul_webber}
b&=&\frac{1}{4}+\frac{2}{3}\frac{n_f}{\beta_0}
\left(1-\frac{C_F}{C_A}\right) ~~;~~
c=\frac{\sqrt{32C_A\pi}}{\beta_0} ~~;~~
\beta_0=11-\frac{2}{3}n_f
\nonumber
\end{eqnarray}
or alternatively in \cite{mult_khoze}:
\begin{eqnarray}
N_{pert} &=& K \cdot \Gamma(B)\left (\frac{z}{2}\right)^{1-B} I_{1+B}(z)\\
B &=& \frac{a}{b} ~~~;~~~
a=\frac{11}{3}N_c + \frac{2n_f}{3N_c^2} ~~~;~~~
b=\frac{11}{3}N_c - \frac{2n_f}{3} \nonumber \\
z&=& 2 \log \frac{\kappa}{Q_0} \gamma_0 ~~~;~~~ \gamma_0 = \sqrt{\frac{2}{\pi}
C_A \alpha_s(\kappa)} \nonumber
\label{mul_khoze}
\end{eqnarray}
Here $\Gamma$ is the Gamma-function and $I_B$ the modified Bessel-function.
$K$ is a non-perturbative scale factor.
Finally:
\begin{eqnarray}
r(\kappa)&=& \frac{9}{4} ( 1 - r_1 \gamma_0 -r_2 \gamma_0^2 )
\label{mueller_r}
\end{eqnarray}
with:
\begin{eqnarray*}
r_1 &=& 1 + \frac{n_f}{C_A} - \frac{2 n_f C_F}{C_A^2} ~~;~~
r_2 = r_1
\left ( \frac{25}{8}
-\frac{3}{4}\frac{n_f}{C_A} - \frac{n_f C_F}{C_A^2}
\right ) ~~;~~
\gamma_0 =
\sqrt{\frac{\alpha (\kappa) C_A}{18 \pi}}
\end{eqnarray*}
is the perturbative prediction \cite{mueller2}
for the multiplicity ratio in back to back
quark or gluon jets, respectively.
The terms proportional to $r_1$ ($r_2$) correspond to the NLO and NNLO
prediction.
Numerically they correspond to corrections of about 8\% and 1\% , respectively.
The smallness of the higher order corrections indicates that the perturbative
series of the gluon to quark multiplicity ratio converges rapidly.
The fits represent the data well,
parameters of the fits are given in Tab. \ref{fit_mehrere}.
In the lower inset of Fig. \ref{qgcmp1} the ratio of the
gluon to quark multiplicity as calculated from the data and the fits
is shown as function of the scale as well as the ratio of the
slopes of the fits.
The ratio of the multiplicities of the data is between 1.1 at small
scale and 1.4 at the highest scales measured.
%{\bf nochmal altes resultat, cleo opal konsistent ?}
The ratio of the slopes for the different fit is about 2.1 corresponding to
a colour factor ratio of $C_A/C_F = 2.26 \pm 0.10$, e.g. nicely compatible with
the QCD expectation.
The fits further indicate that the quark multiplicity is bigger than
the gluon multiplicity for very small scale!
Consequently the constant terms attached to the
multiplicity due to the primary gluon or quark fragmentation
are larger for quarks (see Tab. \ref{fit_mehrere}).
The difference of these terms is about 2.6.
Choosing instead of $\kappa$ the transverse momentum w.r.t. the leading jet
as scale gives similar results. Taking the scale choice made in
\cite{aleph_qgmult} leads about to a 10\% increase of the measured colour factor
ratio and a corresponding increase of the difference of the non-perturbative
constants to 3.8.
In first sight a difference of the constant terms of the order of 2 to 3 units
in multiplicity looks unexpectedly large. It should however be noted,
that these constants also include the effects of the jet clustering.
Furthermore
stable hadron production to large extend proceeds via resonance decays, so that
the observed difference may only correspond to a difference of
about one primary particle.
Definitively the larger constant term for quarks compared to gluons
directly explains the different behaviour of the ratio of
multiplicities and the slope ratio in the lower part of Fig. \ref{qgcmp1}.
The observed behaviour would be expected from
non-perturbative effects in the fragmentation in the leading quark or gluon.
In the cluster fragmentation model, an additional gluon to quark antiquark
splitting is needed in the fragmentation of a gluon compared to a quark.
Also a suppression of the radiation of low-energetic gluons from a $q\bar{q}g$
due to coherence effects (angular ordering) would act similarly.
\begin{table}[t,h,b]
\begin{center}
\newcommand\rup[1]{\raisebox{1.5ex}[-1.5ex]{#1}}
\newcommand\rdw[1]{\raisebox{-1.5ex}[-1.5ex]{#1}}
\newcommand\mc[2]{\multicolumn{#1}{#2}}
\begin{tabular}{|l|c|c|}
\hline
\rdw{Parameter} & $N_{pert}$ from & $N_{pert}$ from \\
& Eqn. \ref{mul_webber}& Eqn. \ref{mul_khoze} \\
\hline
$\Lambda$ & 0.013 $\pm$ 0.003 & 0.047 $\pm$ 0.027 \\
K & 0.004 $\pm$ 0.000 & 0.067 $\pm$ 0.006 \\
$Q_0$ & -- & 0.003 $\pm$ 0.001 \\
$\cacf $ & 2.264 $\pm$ 0.089 & 2.264 $\pm$ 0.099 \\
$N_0^q$ & 2.774 $\pm$ 0.183 & 2.470 $\pm$ 0.175 \\
$N_0^g$ & 0.135 $\pm$ 0.169 & -0.363 $\pm$ 0.196 \\
$\chi^2$/n.d.f. & 2.083 & 2.214 \\
\hline
\end{tabular}
\caption{Results of the fits of the quark and gluon jet multiplicities
as a function of $\kappa$.}
\label{fit_mehrere}
\end{center}
\end{table}
The analysis presented so far, as most other comparisons of quark and gluon
jet multiplicities, has the disadvantage to rely on the identification
of (maybe low energetic) particles to jets.
Clearly this involves ambiguities and especially not considers coherent
soft gluon radiation from the initial $q\bar{q}g$ ensemble.
This can be avoided by studying the dependence of the total multiplicity
in three jet events as function of the quark and gluon scales.
In fact there is a definite MLLA prediction \cite{DKT} for this multiplicity
$N_{q\bar{q}g}$:
\begin{eqnarray}
N_{q\bar{q}g} = \left[2 N_q(Y^*_{q\bar{q}}) + N_g(Y^*_{g}) \right]
\cdot (1+{\cal O}(\frac{\alpha_s}{\pi}))
\label{3mul_0}
\end{eqnarray}
with:
\begin{equation}
Y^*_{q\bar{q}} = \ln \sqrt{ \frac{p_q \cdot p_{\bar{q}}} {2\Lambda^2} } =
\ln \frac{E^*_{q\bar{q}}}{\Lambda}
~~~;~~~
Y^*_g = \ln \sqrt{
\frac{ (p_qp_g) (p_{\bar{q}}p_g) }
{2\Lambda^2(p_qp_{\bar{q}} ) } } =
\ln \frac{ p_{1\perp} } {2\Lambda}
\end{equation}
$\Lambda$ is a scale parameter, the $p_i$ are the four-momenta of the quarks
and the gluon, respectively.
The three jet multiplicity depends on the
quark energy in the
centre of mass system of
the quark antiquark pair, $E^*$, and on the transverse momentum of the gluon,
$p_{1\perp}$.
For comparison with data, in \cite{khoze_ochs}, this is expressed in dependence
of the measured multiplicity in $e^+e^-$ events and a colour factor ratio
as given in Eqn. \ref{mueller_r}.
In addition to this identification we again choose to add
a constant term, $N_0$, to account
for differences in the fragmentation of quarks and gluons
as discussed above.
Thus omitting correction terms:
\begin{eqnarray}
N_{q\bar{q}g} = N_{e^+e^-}(2E^*) +
r(p_{1\perp}) ( \frac{1}{2} N_{e^+e^-}(p_{1\perp}) - N_0 )
~~~.
\label{3mul}
\end{eqnarray}
Although on a first sight this looks as just as the incoherent sum of the
multiplicity of the two quark jets
and the gluon jet, this formula includes coherence effects in the
exact definition of the scales of the $N_{e^+e^-}$ terms \cite{khoze_privat}.
Still in principle Eqn. \ref{3mul} requires the determination of the
quark antiquark and gluon scale independently.
Restricting however to symmetric, Y-type events (see Fig. \ref{eventtopol}b)
both scales can be expressed as functions of the opening angle $\theta_1$
only.
\mbox{$E^{*2}\propto E_q E_{\bar{q}} \sin^2 \theta_3/2$} for this
type of events is approximately constant (see upper full curve in Fig.
\ref{scale}) at fixed CMS energy.
$p_{1\perp}$, however, as it is proportional to the gluon transverse momentum
increases about linearly with the opening angle.
The $\theta$ dependence of the three jet multiplicity therefore mainly measures the
scale dependence of the multiplicity of the gluon jet.
\begin{figure}[htb]
\begin{center}
\epsfig{file=p-scale.eps,width=10cm}
\caption[scale]
{
\label{scale}
Variation of the scales $2E^*$ and $p_{1\perp}$ as function of the opening angle
$\theta$ in symmetric three jet events. Functions are the analytic expectation.
Points include a correction considering that sometimes the gluon forms the most
energetic jet.
}
\end{center}
\end{figure}
In a fraction of the events which strongly increases with opening angle
the gluon jet may be the most energetic jet.
This can be corrected for in different ways
when fitting Eqn. \ref{3mul} to the data
using Monte Carlo simulation.
Assuming an about logarithmic increase of the multiplicity with scale, which is
well supported by the data, the average scale at a given opening angle
can be expressed as the geometric mean of the resulting scales when the gluon
is low energetic or most energetic, respectively.
These corrected scales are for comparison also shown as points in Fig.
\ref{scale}.
The correction first increases with the opening angle but then decreases again
as the correction must vanish for fully symmetric events.
Alternatively, the fraction of events when the gluon is low/most energetic can
be considered separately in Eqn. \ref{3mul}.
To obtain information on the colour factor ratio $C_A/C_F$, the scale dependence
of the three jet multiplicity has to be compared to the multiplicity
in all $e^+e^-$ events.
This has been chosen to be taken from the DELPHI measurements with
hard photon radiation for energies below the Z mass and at 184 GeV \cite{delphi184}
and the LEP combined measurements at the intermediate energies \cite{lepcombi}.
Alternatively data from lower energy $e^+e^-$ experiments \cite{epemdata}
have also been included.
The DELPHI multiplicities in events with hard photon radiation
have been extracted as described in \cite{gluon_paper_1,salva}, but now contain
the full statistics available.
Small, energy dependent corrections ($<2\%$) to the $e^+e^-$ multiplicities
were applied to correct for the varying contribution of b quarks.
The so obtained multiplicities were fitted with the perturbative predictions
Eqs. \ref{mul_webber} or \ref{mul_khoze}, see Fig \ref{result_plot}a.
Both calculations describe the data equally well.
The parameters of the fits are given in Tab. \ref{finalfit}.
\begin{table}[t,h,b]
\begin{center}
\newcommand\rup[1]{\raisebox{1.5ex}[-1.5ex]{#1}}
\newcommand\rdw[1]{\raisebox{-1.5ex}[-1.5ex]{#1}}
\newcommand\mc[2]{\multicolumn{#1}{#2}}
\begin{tabular}{|l|c|c|c|}
\hline
\hline
\rdw{Parameter} & $N_{pert}$ from \cite{webber}
& $N_{pert}$ from \cite{mult_khoze}
&relevant\\
& (Eqn.\ref{mul_webber})
& (Eqn.\ref{mul_khoze}) &data\\
\hline
$\Lambda$ & 0.275 $\pm$ 0.070 & 0.146 $\pm$ 0.021 &data from\\
K & 0.026 $\pm$ 0.003 & 0.591 $\pm$ 0.034 &$e^+e^-$ and $q\bar{q}\gamma$\\
$Q_0$ & -- & 0.034 $\pm$ 0.002 &events\\
$\chi^2$/n.d.f. & 1.180 & 1.297 & \\
\hline
\hline
$C_A/C_F$ & 2.266 $\pm$ 0.053 & 2.292 $\pm$ 0.053 &data from\\
$N_0$ & 1.429 $\pm$ 0.092 & 1.512 $\pm$ 0.091 &symmetric\\
$\chi^2$/n.d.f. & 0.842 & 0.891 &3 jet events\\
\hline
\end{tabular}
\caption{Result of the fits of the \epem multiplicity
and the three jet event multiplicity.}
\label{finalfit}
\end{center}
\end{table}
\begin{figure}[htbp]
\begin{center}
\epsfig{file=p-khoze.eps,width=14cm}
\caption[result_plot]
{
\label{result_plot}
(a) Charged hadron multiplicity as a function of the centre of
mass energy fitted with the perturbative predictions Eqs.
\ref{mul_webber} or \ref{mul_khoze}.
(b) Charged hadron multiplicity in symmetric three jet events as a
function of the opening angle. The upper curve is a prediction using the ansatz
\ref{3mul} putting \cacf to ist default value and omitting
the constant offset, $N_0$.
The lower plot is a fit of the full ansatz \ref{3mul} to the data treating
\cacf and $N_0$ as free parameters.
}
\end{center}
\end{figure}
The measured, fully corrected multiplicity in all symmetric three jet events as
function of the opening angle is shown in Fig. \ref{result_plot}b.
A strong increase of the multiplicity from values around 14 for small opening
angle to about 29 at opening angles of 120$^\circ$
corresponding to fully symmetric events is observed.
Omitting the constant term, $N_0$ in Eqn. \ref{3mul} and putting
\cacf to its expected value predicts a similar increase
in multiplicity over this angular range (upper curve in \ref{result_plot}b).
The prediction is however higher by about three units of charged multiplicity.
This discrepancy is to be expected from the previously obtained result
due to differences in the fragmentation of the leading quark or gluon.
At small angles the difference between the primary QCD expectation and the
measurement increases.
Studies using Monte Carlo models have shown that this is due to an influence
of genuine two jet events which have accidentally clustered as
symmetric three jet events.
The models indicate that this contribution becomes negligible for angles above
30$^{\circ}$.
Fitting the full ansatz \ref{3mul} to the three jet multiplicity at
angles $\theta \geq 35^{\circ} $ using the 2 parameterizations in Eqs.
\ref{mul_webber} and \ref{mul_khoze} of the
multiplicity in $e^+e^-$ events keeping their parameters fixed as given
in Tab. \ref{finalfit} but varying $C_A/C_F$ and $N_0$ yields:
\begin{eqnarray}
\label{res_webber}
\frac{C_A}{C_F} &=& 2.266 \pm 0.053 ~~~~;~~~~ N_0 = 1.43 \pm 0.09\\
\frac{C_A}{C_F} &=& 2.292 \pm 0.054 ~~~~;~~~~ N_0 = 1.51 \pm 0.09 ~~~.
\end{eqnarray}
The offset term $N_0$ is bigger by about 0.35 units if only b-depleted events
are used.
The central result for \cacf, however, remains unchanged within errors.
This result with great precision confirms the QCD expectation
\cite{brodskygunion}
that the
ratio of the radiated multiplicity from gluon and quark jets is given by the
colour factor ratio \cacf.
In order to further validate the correctness of the ansatz \ref{3mul} and
to test the bias introduced by two jet events at small $\theta_1$, the lowest
angle used in the fit has been varied.
The resulting value for $C_A/C_F$ and the $\chi^2/Ndf$ respectively the
$\chi^2$ probability of the fit are show in Fig. \ref{stab_plot}.
It is observed that for $\theta_1 > 35 ^{\circ}$ satisfactory fits are
obtained.
For this angular range the fitted value of $C_A/C_F$ is stable within error.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=p-stable.eps,width=12cm}
\caption[stab_plot]
{
\label{stab_plot}
Upper plot (a): Stability of the result for \cacf against variation of the
smallest opening angle used in the fit.
Lower plot (b):
$\chi^2/Ndf$ and $\chi^2$ probability of these fits.
}
\end{center}
\end{figure}
The offset term $N_0$ in this analysis is slightly smaller compared to the
difference of the corresponding terms in the analysis presented in the
beginning of this letter.
This may be expected as the jet algorithms which were used only
in the first analysis
will tend to diminish the gluon multiplicity and correspondingly
increase the quark jet multiplicity.
The actual value of $N_0 \approx 1.8$ may correspond to less than one
primary particle.
This is indeed a reasonable value which had already been expected in
\cite{brodskygunion}.
Systematic uncertainties of the above result for the colour factor
ratio due to uncertainties in the three jet multiplicity data as well as
in the parameterization of the $e^+e^-$ charged multiplicity
and in the theoretical predictions are considered.
All relative systematic errors are collected in Tab. \ref{systematic}.
To obtain errors approximately comparable to statistical errors half the
difference of the result obtained with a modification w.r.t. the central result
is quoted as systematic error.
\begin{table}[t,h,b]
\begin{center}
%\small
\begin{tabular}{|rl|c|c|c|c|}
\hline
& Source & Sys. error & merged & merged & total \\
\hline
\hline
& \multicolumn{4}{l|}{Experimental uncertainties }&\multicolumn{1}{c|}{\ } \\
\cline{1-5}
1. & Min. particle momentum & $\pm~ 0.42 $~\% & & & \\
2. & Min. angle of jet w.r.t. beam & $\pm~ 0.38 $~\% & & & \\
3. & Min. number of tracks per jet & $\pm~ 0.02 $~\% & $ \pm~0.66$\% & & \\
4. & Corr. mode for gluon in jet 1 & $\pm~ 0.06 $~\% & & & \\
% \raisebox{1.5ex}[-1.5ex]{$\pm~ 0.66 $~\%} & & \\
5. & $b$ depletion & $\pm~ 0.34 $~\% & &
\raisebox{1.5ex}[-1.5ex]{$\pm~ 2.42$~\%} & \\
\cline{1-4}
6. & \epem data sets & $\pm~ 0.89 $~\% & & &
$\pm~ 4.88 $~\% \\
7. & Fit function & $\pm~ 1.31 $~\% & $\pm~ 2.33 $~\% & & \\
% \raisebox{1.5ex}[-1.5ex]{
8. & Starting point of fit & $\pm~ 1.70 $~\% & & & \\
\cline{1-5}
& \multicolumn{4}{l|}{Theoretical uncertainties }&\multicolumn{1}{c|}{\ } \\
\cline{1-5}
9. & Variation of $n_f$ & $\pm~ 1.30 $~\% & & & \\
10.& Calculation in 1st/2nd order & $\pm~ 4.00 $~\% & &
$\pm~ 4.24$\% & \\
11.& Setting $C_A$ fixed & $\pm~ 0.54 $~\% & & & \\
\hline
\end{tabular}
\caption{Systematic uncertainties on \cacf as derived from three jet event
multiplicities }
\label{systematic}
\end{center}
\end{table}
Results obtained from the individual data sets corresponding to the different
years of data-taking were found to be fully compatible within the statistical
error.
To estimate uncertainties in the three jet multiplicity the following
cuts which are sensitive to mis-representation of the data by the Monte Carlo
simulation have been varied:
\begin{enumerate}\itemsep-0.5ex
\item {
Cut on the minimal particle momentum
}\\
The cut on the minimal particle momentum has been lowered from 400 MeV
to 200 MeV and raised to 600 MeV.
\item {
Minimal angle of each jet with respect to the beam axis
}\\
This cut has been increased from 30$^\circ$ to 40$^\circ$ to test a possible
bias due to the limited angular acceptance.
\item {
Minimal number of particles per jet
}\\
The minimal number of particles per jet has been increased from 2 to 4 in
order to reject events which may not have a clear three jet structure.
\item {Correction mode for gluon in leading jet
}\\
Both types of corrections were made alternatively
to account for gluons in the most energetic jet.
\item {b contributions}
The analysis has also been performed for events depleted from b contributions.
%This changes $N_0$ but leaves $C_A/C_F$ almost unaltered.
\end{enumerate}
The following systematic uncertainties arise from uncertainties in the
experimental input other than from the three jet multiplicities
and from choices made for the fits of $N_{\epem}$.
These uncertainties are considered as experimental systematic uncertainties.
\begin{enumerate}\itemsep-0.5ex
\setcounter{enumi}{5}
\item {
Input of parameterization of $N_{e^+e^-}(\sqrt{s})$
}\\
To estimate the influence of an uncertainty in $N_{e^+e^-}$
different choices of input data were compared:
\begin{itemize}\itemsep-0.5ex
\item DELPHI multiplicities for 184 GeV and
from Z decays with hard photons combined with
LEP data for $90{\rm GeV}<\sqrt{s}<180{\rm GeV}$
\item DELPHI multiplicities from Z decays with hard photons.
\item \epem data taken at low centre of mass energies
\mbox{(TASSO, TPC, MARK-II, HRS, AMY)}
\item all available \epem Data between 10~GeV and 184~GeV
\mbox{(TASSO, TPC, MARK-II, HRS, AMY, LEP~combined, DELPHI)}
\end{itemize}
\item {
Choice of prediction used for fit
}\\
The fit functions \ref{mul_webber} and \ref{mul_khoze} were used alternatively.
\item {
Starting point of the fit
}\\
To consider a residual bias due to the starting point of the fit
the change of the result for \cacf between $30^\circ$ to $40^\circ$,
as well as changing the bin width $\epsilon$ from 1.25$^\circ$
to 5$^\circ$ enters into the systematic error.
\end{enumerate}
Finally systematic errors due to uncertainties in the theoretical prediction
were considered.
\begin{enumerate}\itemsep-0.5ex
\setcounter{enumi}{8}
\item {
Variation of $n_f$
}\\
The number of active quarks, $n_f$, relevant for the hadronic final state
is uncertain.
$n_f$ therefore has been varied from 3 to 5.
\item {
Order of calculation (LO - NNLO)
}\\
The prediction
$r(\kappa)$ (Eqn. \ref{mueller_r}) has been calculated for
back to back quarks or gluons.
As the jets are well separated it is to be
expected to also apply for this analysis.
Definitively when the gluon recoils with respect to the quarks the
prediction is exact.
Coherence effects (angular ordering) are in addition taken into
account in the definition of the scales $E^*$ and $p_{1\perp}$.
As the coupling for the triple gluon vertex is bigger than the coupling of
all other vertices it is however clear that the correction must be negative
as in the case of Eqn. \ref{mueller_r}.
The validity of the correction \cite{mueller2} is therefore assumed for
the whole range of angles considered. Conservatively half of the
difference obtained with the lowest order prediction $r=\cacf$ and the NNLO
prediction is considered as systematic uncertainty.
A leading order $\alpha_s$ was used together with the lowest order prediction
and a second order $\alpha_s$ in the other case.
Considering that in the three jet events mainly the gluon scale
$p_{1\perp}$ is varied
the resulting error estimate agrees with the one given in
\cite{khoze_ochs} (see Eqn. \ref{3mul_0}).
\item {Quantities influencing \cacf}\\
For the central result \cacf has been assumed variable in Eqn. \ref{mueller_r}
only.
The stability of the result was checked by also leaving $C_A$ variable
in some or all of the parameterizations of $\alpha_s$ and $N_{\epem}$.
%\begin{itemize}\itemsep-0.5ex
% \item Fix of $c_A$ in the parameterizing of the event multiplicities
% \item Fix of $c_A$ in the parameterizing of the $\beta$ functions
% \item Fix of $c_A$ in the parameterizing of $\alpha_s$ via the $\beta$ functions,
% but not in the $\beta$ functions themselves
% \item No fix of $c_A$ at all
%\end{itemize}
\end{enumerate}
As the parameterization Eqn. \ref{mul_webber} yields a more natural results for
$\alpha_s$ which also enter in Eqn. \ref{mueller_r} as final result we
consider Eqn. \ref{res_webber} in combination with the systematic errors
summarized in Tab. \ref{systematic}:
\begin{eqnarray}
\frac{C_A}{C_F} &=& 2.266 \pm 0.053 (stat.) \pm 0.055 (syst.) \pm 0.096
(theo.)
\end{eqnarray}
This result confirms the QCD expectation that gluon bremsstrahlung
is stronger from gluons compared to quarks by the colour factor ratio
\cacf
and is direct evidence for the triple gluon coupling.
Interpreted as a measurement of the colour factor ratio \cacf it yields
the most precise result obtained so far.
Even the most precise
measurements from four jet angular distributions
\cite{4jet}
suffer from
the relatively small number of four jet events available.
Furthermore, many of these measurements specify
no theoretical systematic error as they so far rely on leading order
calculations.
An offset term of about 1.8 charged hadrons between light quarks and gluons is
required
to consider differences in the quark and gluon fragmentation.
This may correspond to about one primary produced, unstable hadron.
This result also implies that Local Hadron Parton Duality (LPHD), that is
the proportionality of the number of gluons to hadrons, applies extremely
precisely if only the radiated gluons from a quark or gluon are considered.
\section{Summary}
In summary the dependence of the charged particle
multiplicity in quark and gluon jets
and in symmetric three jet events has been measured as a function of
the scale.
The ratio of the slopes
of gluon to quark multiplicity agrees with the expectation \cacf from QCD.
and directly
reflects the higher colour charge of gluons compared to quarks.
This can also be interpreted as direct evidence for the triple gluon
coupling, one of the basic ingredients of QCD.
It is of special importance that this evidence is due to very soft
radiated gluons and therefore complementary to the measurement of
the triple gluon
coupling in four-jet events at large momentum transfer.
The increase of the gluon to quark multiplicity ratio with increasing scale
is understood due to a difference in the fragmentation of the leading
quark or gluon.
The simultaneous description of the quark and gluon multiplicities
with scale also strongly supports the Local Parton Hadron Duality
hypothesis although
large non-perturbative terms for the leading quark or gluon are
responsible for the observed relatively small gluon to quark multiplicity
ratio.
Using the novel method of measuring the evolution with opening angle
of the multiplicity in symmetric three jet events
a precise result for the colour factor ratio is obtained:
$$
\frac{C_A}{C_F} = 2.266 \pm 0.053 (stat.) \pm 0.055 (syst.) \pm 0.096
(theo.)
$$
This measurement is superior in precision to the most precise
measurements from four jet events \cite{4jet}.
\section*{Acknowledgement}
We would like to thank V.A. Khoze for its interest in this analysis and
many helpful discussions and explanations.
We thank S. Lupia and W. Ochs for providing us with their program for
Eqn. \ref{mul_khoze}.
We are greatly indebted to our technical collaborators and to the funding
agencies for their support in building and operating the DELPHI detector
and to the CERN-SL-Division for the superb performance of the LEP collider.
\newpage
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\end{document}