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\begin{document}
%%% put your own definitions here:
\def\asmz{$\alpha_s(M_Z)$}
\def\ass{\alpha_s(E)}
\def\bb{{b\bar{b}}}
\def\cc{{c\bar{c}}}
\def\ll{{l\bar{l}}}
\newcommand{\kos}{\ifmmode {{\mathrm K}^{0}_{S} } \else
${\mathrm K}^{0}_{S}$ \fi}
\newcommand{\kpm}{\ifmmode {{\mathrm K}^{\pm}} \else
${\mathrm K}^{\pm}$\fi}
\newcommand{\ko}{\ifmmode {{\mathrm K}^{0}} \else
${\mathrm K}^{0}$ \fi}
\def\as{$\alpha_s$}
\def\asb{$\alpha_s\sp{b}$}
\def\asc{$\alpha_s\sp{c}$}
\def\asuds{$\alpha_s\sp{udsc}$}
\def\Lam{$\Lambda$ }
\def\ZP{Z.\ Phys.\ {\bf C}}
\def\PL{Phys.\ Lett.\ {\bf B}}
\def\PR{Phys.\ Rev.\ {\bf D}}
\def\PRL{Phys.\ Rev.\ Lett.\ }
\def\NP{Nucl.\ Phys.\ {\bf B}}
\def\CPC{Comp.\ Phys.\ Comm.\ }
\def\NIM{Nucl.\ Instr.\ Meth.\ }
\def\Coll{Coll.,\ }
\def\Rmu{$R_3(\mu)/R_3(had)$\ }
\def\Re{$R_3(e)/R_3(had)$\ }
\def\Rmue{$R_3(\mu +e)/R_3(had)$\ }
\def\ee{$e\sp{+}e\sp{-}$}
\newcommand{\Wfj}{WW $\rightarrow 4jets$ events}
\newcommand{\Wtj}{WW $\rightarrow 2jets \, \ell \bar{\nu}$ events}
\def\Abreu{DELPHI Coll., P. Abreu {et al.,}\ }
\def\etal{{\it et al.,}\ }
\def\muldim{to be published in Proc. XXVIII International Symposium on
Multiparticle Dynamics, Frascati, September 1997.}
\begin{titlepage}
\pagenumbering{arabic}
\begin{tabular}{l r}
ICHEP'98 \#141 & \hspace{6cm} DELPHI 98-17 CONF 118 \\
Submitted to Pa 3, 7 & 22 June, 1998 \\
\hspace{2.4cm} Pl 4, 9 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Fragmentation Properties of \boldmath $b$
\unboldmath Quarks compared to Light Quarks \\in
\boldmath $q\bar{q}$ \unboldmath
Events at \boldmath $\sqrt{s} = $ \unboldmath 183 GeV}\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\normalsize {
%===================> DELPHI note author list =====> To be filled <=====%
{\bf P.~Abreu} $^1$,
{\bf A.~De~Angelis} $^2$,
{\bf R.~N\'obrega} $^1$,
{\bf M.~Pimenta} $^1$,
{\bf L.~Vitale} $^3$
%========================================================================%
}
\vspace*{1cm}
\end{center}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
The DELPHI detector at LEP has collected during 1997 54pb$^{-1}$ of data
at a centre-of-mass energy about 183 GeV. We have used these data to
measure the average charged particle multiplicity $$ in $e^+e^-
\rightarrow b\bar{b}$ events and the difference $\delta_{bl}$
between $$ and the
multiplicity in generic light quark (u,d,s) events:
\begin{displaymath}
\delta_{bl} = 4.23 \pm 1.09 (stat) \pm 1.40 (syst) \, .
\end{displaymath}
This result is remarkably in agreement with QCD predictions, while it is
inconsistent with calculations assuming that the multiplicity
accompanying the decay of a heavy quark is independent of the mass of
the quark itself.
%=========================================================================%
\end{abstract}
\vspace{\fill}
\begin{center}
Paper submitted to the ICHEP'98 Conference \\
Vancouver, July 22-29
\end{center}
\vspace{\fill}
\par {\footnotesize $^1$ LIP-IST-FCUL - Av. Elias Garcia, 14-1e,
P-1000 Lisboa, Portugal}
\par {\footnotesize $^2$ CERN, CH-1211 Geneva 23, Switzerland}
\par {\footnotesize $^3$ Dipartimento di Fisica, Universit\`a di Trieste and
INFN, Via A. Valerio 2, I-34127 Trieste, Italy}
\end{titlepage}
\pagebreak
%==================> DELPHI note text =====> To be filled <======%
\section{Introduction}
The study of the properties of
the fragmentation of heavy quarks compared to
light quarks offers new
insights in perturbative QCD. Particularly important is
the difference in charged particle multiplicity between
light quark and heavy quark initiated events in
$e^+e^-$ annihilations. QCD predicts
that this difference is energy independent; this is motivated by
mass effects on the gluon radiation
(see \cite{schumm,DOKBM,petrov,deus} and \cite{khoze} for a recent review).
The QCD prediction is somehow counter-intuitive \cite{kisselev}.
The present experimental tests, although preferring the QCD-motivated
scenario, were not conclusive
(see \cite{schumm} and references therein,
\cite{delphi,opal,sld,tristan}).
At LEP 2 energies, however, the difference
between the QCD prediction and the model ignoring mass effects
is large, and the experimental measurement can firmly distinguish between the
two hypotheses.
\section{Analysis and Results}
Data corresponding to
a total luminosity of 54.0 pb$^{-1}$ collected by DELPHI
at centre-of-mass energies around 183~GeV during 1997 were analysed.
A description of the DELPHI detector can be found in \cite{deldet}; its
performance is discussed in \cite{perfo}.
A preselection of hadronic events was made, requiring
at least 6
charged particles with momentum $p$ above 400 MeV/$c$,
angle $\theta$ with respect to the beam direction
between 20$^\circ$ and 160$^\circ$,
a track length of at least 30 cm,
a distance of closest approach to the interaction point
less than 4 cm in the plane
perpendicular to the beam axis and
less than 10 cm along the beam axis, and a
total energy of the charged particles above 0.15
times the centre-of-mass energy $E_{cm}$.
In the calculation of the energies $E$, all
charged particles were assumed to have the pion mass.
Charged particles were then used in the analysis if they had $p > 100$ MeV/$c$,
a relative error on the momentum measurement $\Delta p/p < 1$, polar angle
$20^\circ < \theta < 160^\circ$, a track length of at least 30 cm,
and a distance of closest approach to the primary vertex smaller than
3 cm in the plane perpendicular to the beam axis and 6 cm along the beam axis.
The influence of the detector on the analysis was studied with
the full DELPHI simulation program, DELSIM~\cite{perfo}.
Events were generated with
PYTHIA 5.7 and JETSET 7.4~\cite{lund},
with parameters tuned to fit LEP1 data from DELPHI \cite{tuning}. The
Parton Shower (PS) model was used.
The particles were followed through
the detailed geometry of DELPHI giving simulated digitisations in each
subdetector. These data were processed with the same
reconstruction and analysis programs as the real data.
%To check the ability of the simulation to model the efficiency for the
%reconstruction of charged particles,
%the sample
%collected at the Z during 1997 was used.
%From this sample, by integrating the distribution of
%$\xi_E = -\ln(2E/\sqrt{s})$ corrected bin by bin using the simulation,
%the average charged particle multiplicity at the Z was measured to be
%$20.61 \pm 0.04 (stat)$, to be compared with the world average of
%$21.00 \pm 0.13$ \cite{pdg}. A scale uncertainty of $1.8\%$
%was thus assumed on the measured multiplicities.
The cross-section for $e^+e^- \rightarrow {\mathrm q}\bar{\mathrm q}(\gamma)$
above
the Z peak is dominated by radiative q$\bar{\mathrm q}\gamma$ events;
the initial state radiated photons (ISR photons) are
generally aligned along the beam direction and not detected.
In order to compute the hadronic
centre-of-mass energy, the procedure described in \cite{sprime} was used.
The procedure clusters the particles into two jets using
the Durham algorithm~\cite{durham}, excluding candidate
ISR photons.
Assuming an ISR photon along the beam pipe if
no candidate ISR photon has been detected, the
energy of the ISR photon is computed from the jet directions
using massless kinematics. The effective centre-of-mass energy of the
hadronic system, $\sqrt{s^\prime}$, is calculated as the invariant
mass of the system recoiling from the ISR photon.
%The method used to obtain the hadronic centre-of-mass energy overestimates the
%true energy in the case of double hard radiation in the initial
%state. For instance, if the two ISR photons are emitted back to back,
%the remaining two jets may also
%be back to back, but with energy much smaller than the beam energy.
%Cutting on the total
%energy measured in the detector
%reduces the contamination from such events.
Events with reconstructed hadronic centre-of-mass energy
($\sqrt{s'}$) above 170~GeV
were used. A total of 1031 hadronic events were selected by requiring that
the multiplicity for charged particles (with $p > 100$ MeV/$c$)
was larger than 9, that the total energy of the charged
particles exceeded $0.2\sqrt{s}$, and that the narrow jet broadening was
smaller than 0.1.
A 90\% $b$ enriched sample has then been obtained by requiring that the
$b$-event tagging variable $y$ defined as in Ref. \cite{perfo} was lower
than $5 \times 10^{-5}$.
% efficiency is .59
From this sample the average $\bb$ multiplicity has been measured by
unfolding via simulation for detector effects, selection criteria
and initial state radiation.
The value obtained was $ = 29.54 \pm 0.79 (stat)$.
The relative systematic uncertainty on $$ was assumed as in
\cite{delphi}, so that:
\begin{equation}
= 29.54 \pm 0.79 (stat) \pm 0.63 (syst) \, .
\end{equation}
It was verified that the value above is stable within $\pm$ 0.13 units with
respect to variations of the cut on the $y$ variable (Figure \ref{stab};
notice that the errors in the plot
are correlated, since the samples at
larger $y$
include the samples at smaller $y$).
The difference $\delta_{bl}$ between the $\bb$ multiplicity
and the multiplicity in generic light quark $l = u,d,s$ events
can then be extracted by using the equation:
\begin{equation}
= R_b + R_c + R_l \, ,
\end{equation}
where the fractions $R_b=0.16$, $R_c=0.26$, $R_l=0.58$ are the ones
predicted by the Standard Model at $\sqrt{s} = 183$ GeV.
For the $\cc$ multiplicity, the PYTHIA prediction of 27.6 was
assumed, with a systematic error of $\pm 1.5$ units.
For $$, we used the result published by DELPHI in \cite{noi}:
\begin{equation}\label{mul183}
= 26.58 \pm 0.24 (stat) \pm 0.54 (syst) \, .
\end{equation}
Finally we obtain:
\begin{equation}\label{delbl}
\delta_{bl} = 4.23 \pm 1.09 (stat) \pm 1.40 (syst) \, .
\end{equation}
The value assumed for $$ is consistent within better than 1.5
standard deviations with both $$ and $$,
neglecting the errors on $$, $$: we thus
verified a-posteriori that the systematic error assumed for
$$ was conservative.
The dependence of $\delta_{bl}$ on $$ is
anyhow small, and the systematic error assumed
gives a contribution of $\pm 0.67$ units
to the systematic error on $\delta_{bl}$.
%The dominant contributions on systematic error in (\ref{delbl}) are the
%systematics from $$ (0.93) and from $$ (0.80).
\section{Comparison with Models and QCD Predictions}
{\bf{Flavour-Independent Fragmentation ---}}
In a model in which the hadronization is independent of the mass,
one can assume that the non-leading multiplicity
in an event,
i.e., the light quark multiplicity which accompanies
the decay products of the primary hadrons,
is governed by the effective energy available
to the fragmentation system following the
production of the primary hadrons \cite{kisselev}.
One can thus write:
\begin{eqnarray}
\delta_{bl}(E_{cm}) & = & 2 +
\int_0^1 dx_B f(x_B) \int_0^1 dx_{\bar{B}} f(x_{\bar{B}}) \, \,
n_{l\bar{l}}\left( \left(1-\frac{x_B+x_{\bar{B}}}{2} \right)
E_{cm}\right)\nonumber \\
& - & n_{l\bar{l}}(E_{cm}) \, ,
%\nonumber \\
% & \simeq & 2 +
% n_{l\bar{l}}\left( \left(1-\right)
% E_{cm}\right)\nonumber \\
% - n_{l\bar{l}}(E_{cm})
\end{eqnarray}
where
$$ is the average number of charged
particles coming from the
decay of a $B$ hadron,
$x_B$ ($x_{\bar{B}}$) is the fraction of the beam energy taken by
the $B$ ($\bar{B}$) hadron, and $f(x_B)$ is the $b$ fragmentation function.
We assumed $2 = 11.0 \pm 0.2$ \cite{schumm},
consistent with the average $ = 5.7 \pm 0.3$
measured at LEP \cite{vietri}.
For $f(x_B)$, we assumed a Peterson function of average $0.70 \pm 0.02$
\cite{pdg}.
The value of $n_{l\bar{l}}(E)$ was computed from the fit
to a perturbative
QCD formula \cite{webber} including the resummation of leading
(LLA) and next-to-leading (NLLA) corrections,
which reproduces well the
measured charged multiplicities \cite{noi}, with appropriate
corrections to remove the effect of heavy quarks \cite{dea}.
The prediction of the model in which the hadronization
is independent of the mass is plotted in Figure \ref{nice}
as a function of the centre-of-mass energy.
%the dominant uncertainty comes from
%$$.
There are several variations of the
model in the literature,
leading to slightly different predictions
(see \cite{vietri} and references therein).
\noindent{\bf{QCD Calculation ---}}
The large mass of the $b$ quark,
in comparison to the scale of the strong interaction,
$\Lambda \approx 0.2$ GeV, results in a natural
cut off for the emission of gluon bremsstrahlung.
Furthermore, where the centre-of-mass energy
greatly exceeds the scale of the $b$ quark mass,
the inclusive spectrum of heavy quark production is
expected to be well described by perturbative QCD
in the Modified Leading Logarithmic Approximation (MLLA, \cite{book}).
The value of $\delta_{bl}$
has been calculated in perturbative QCD\cite{schumm,petrov}:
\begin{equation}
\delta_{bl} =
2
- + O(\alpha_s(m_b)) \, .
\end{equation}
Although the last term in the equation limits the accuracy in
the calculation of $\delta_{bl}$, one can conclude that
$\delta_{bl}$ is fairly
independent of $E_{cm}$.
The calculation of the actual value of
$\delta_{bl}$ in \cite{schumm}
on the basis of the first two terms in (6) gives
a value of $5.5 \pm 0.8$; an approximation including the last term in the
above equation gives a value of 3.68 \cite{petrov}.
This demonstrates the importance of the contribution
proportional to $\alpha_s(m_b)$.
A condition less restrictive is the calculation of upper limits.
An upper limit $\delta_{bl} < 4.1$ is given in \cite{petrov}, while
from phenomenological arguments,
$\delta_{bl} < 4$ is predicted in Ref. \cite{deus}.
In Figure \ref{nice} the high energy prediction from QCD is taken from the
average of the experimental values of $\delta_{bl}$ up to $m_Z$
included, $<\delta_{bl}> = 2.96 \pm 0.20$.
Our measurement of $\delta_{bl}$ is fully consistent with the
prediction of energy independence based on perturbative QCD, and
about three standard deviations larger than
predicted by the naive model presented in the beginning of this
section.
\section{Conclusions}
We measured the average charged particle multiplicity
$$ in $e^+e^- \rightarrow b\bar{b}$ events
at centre-of-mass energy of 183 GeV to be:
\begin{equation}
= 29.54 \pm 0.79 (stat) \pm 0.63 (syst) \, .
\end{equation}
The difference $\delta_{bl}$ between $$
and the multiplicity in generic light quark $l = u,d,s$ events
has also been measured to be:
\begin{displaymath}
\delta_{bl} = 4.23 \pm 1.09 (stat) \pm 1.40 (syst)
\end{displaymath}
which is remarkably in agreement with QCD predictions, while it is
about three standard deviations larger than calculations assuming that
the multiplicity accompanying the decay of a heavy quark is independent of the
mass of the quark itself.
\subsection*{Acknowledgements}
We are greatly indebted to our technical collaborators and to the funding
agencies for their support in building and operating the DELPHI detector.
Very special thanks are due to the members of the CERN-SL Division for the
excellent performance of the LEP collider. We are grateful to Jorge Dias
de Deus, Valery Khoze and Torbj\"orn Sj\"ostrand for
useful discussions.
%=========================================================================%
\newpage
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\end{thebibliography}
\newpage
\begin{figure}
\mbox{\epsfxsize16cm\epsffile{stab.ps}}
\caption[]{
Stability of $$ with respect to variations of the cut on the
b-tagging
variable, $y$. Notice that the errors
in the plot are correlated (see text).}\label{stab}
\end{figure}
\newpage
\begin{figure}
\mbox{\epsfxsize16cm\epsffile{db.ps}}
\caption[]{
The present measurement of $\delta_{bl}$
compared to previous measurements as a function of the centre-of-mass energy,
to the QCD prediction, and to the prediction from
flavour-independent
fragmentation.}\label{nice}
\end{figure}
\end{document}