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\begin{document}
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\begin{titlepage}
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\begin{tabular}{l r}
HEP'99 \# 1\_220 & \hspace{6cm} DELPHI 99-118 CONF 305 \\
Submitted to Pa 1, 3 & 15 June 1999 \\
\hspace{2.4cm} Pl 3 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Hadronization Properties of b Quarks compared to Light
Quarks\\in \boldmath $q\bar{q}$ Events\\at
$\sqrt{s}$ = 183 GeV and 189 GeV }\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\end{center}
\vfill\vfill
\begin{center}
Paper submitted to the HEP'99 Conference \\
Tampere, Finland, July 15-21
\end{center}
\vspace{\fill}
\end{titlepage}
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\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}}}
\def\lbl{{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.}
\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 and 189 GeV}\\
\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 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 54 pb$^{-1}$ of data
at a centre-of-mass energy of about 183 GeV during 1997,
and 158 pb$^{-1}$ of data
at a centre-of-mass energy of about 189 GeV during 1998.
We have used these data to
measure the average charged particle multiplicity in $e^+e^-
\rightarrow b\bar{b}$ events, $_\bb$, and the difference $\delta_{bl}$
between $_\bb$ and the
multiplicity, $_\ll$, in generic light quark (u,d,s) events:
\begin{eqnarray*}
\delta_{bl}(183 \, GeV) &=& 5.07 \pm 1.28 (stat) \pm 1.07 (syst) \\
\delta_{bl}(189 \, GeV) &=& 3.97 \pm 0.83 (stat) \pm 0.68 (syst) \, .
\end{eqnarray*}
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}
\par {\footnotesize $^1$ LIP-IST-FCUL - Av. Elias Garcia, 14-1e,
P-1000 Lisboa, Portugal}
\par {\footnotesize $^2$ Dipartimento di Fisica, Universit\`a di Udine and
INFN, Via delle Scienze 208, I-33100 Udine, Italy}
\par {\footnotesize $^3$ Dipartimento di Fisica, Universit\`a di Trieste and
INFN, Via A. Valerio 2, I-34127 Trieste, Italy}
\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 existing 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 luminosity of 54 pb$^{-1}$ collected by DELPHI
at centre-of-mass energies around 183~GeV during 1997,
and to a luminosity of 158 pb$^{-1}$ collected
at centre-of-mass energies around 189~GeV during 1998,
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 10
charged particles with momentum $p$ above 100 MeV/$c$ and
less than 1.5 times the beam energy,
%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 4 cm along the beam axis,
a relative error on the momentum measurement $\Delta p/p < 1$,
and a
total transverse energy of the charged particles above 0.2
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
and computes the effective centre-of-mass energy of the
hadronic system, $\sqrt{s^\prime}$, as the invariant
mass of the system recoiling against an ISR photon, possibly unseen.
Events with reconstructed hadronic centre-of-mass energy
($\sqrt{s'}$) above 90\%
of the centre-of-mass energy ($\sqrt{s}$) were used.
The selected 1997 (1998) data sample consisted of 1699 (4583) hadronic events.
For each years's data, two samples enriched in
(1) $b-$ events and in (2) $uds-$ events were selected by means of the
$b$ tagging variable $y$ defined as in Ref. \cite{perfo}.
To select the samples (2), it was required in addition that the narrow
jet broadening defined as in Ref. \cite{lep2yr} is smaller than 0.065,
to remove the background due to WW events. The purities estimated with
simulation after applying the event
criteria and removal of the WW background were approximately 91\%
($b-$ events) in sample (1), and 79\%
($uds-$ events) in sample (2). The average charge multiplicity was
computed for each sample.
A third sample (3) containing
the nominal quark flavour ratios
was taken into account by considering the measurement of multiplicity
described in \cite{noi}.
The measured mean multiplicities together with the event
probability cuts applied and the corresponding fractions of $q$-type quarks
$f_q$, as determined from the simulation, are shown in Table~\ref{puri}.
Factors $C\sp{(i)}_q$ multiplying the coefficients $f_q$
were introduced to account for the acceptance corrections and for
possible biases introduced by the
application of the $b$ probability and the jet broadening cuts;
these factors were computed by means of the simulation, and are also shown
in Table~\ref{puri}.
\begin{table}[htbp]
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\hline
\multicolumn{9}{|c|}{Data at 183 GeV}\\ \hline
{\rm Sample} & b-tag prob. & $f\sp{(i)}_b$ & $C\sp{(i)}_b$ & $f\sp{(i)}_{uds}$
& $C\sp{(i)}_{uds}$ & $f\sp{(i)}_c$ & $C\sp{(i)}_c$ & $< n >^{(i)}$ \\ \hline
${\rm (1) }$ & $P_E<0.00001$ & 0.914 & 0.955
& 0.017 & 1.29 & 0.069 & 0.934 & $ 27.90 \pm 0.83 $ \\ \hline
${\rm (2) }$ & $0.2^{(i)}$ \\ \hline
${\rm (1) }$ & $P_E<0.00001$ & 0.910 & 0.962
& 0.014 & 1.16 & 0.076 & 0.949 & $ 28.74 \pm 0.49 $ \\ \hline
${\rm (2) }$ & $0.2$,
in three event samples of different flavour content, $f_q$, and correction
factors $C_q$.}
%selected by applying cuts on the b-tag probability, ${ P_E}$. }
\end{table}
In each of the three samples, the average multiplicity $$ is a linear
combination of the unknowns $< n >_\bb$, $< n >_{\ll}$ and $< n >_\cc$.
One can thus formulate a set of three simultaneous equations to compute these
unknowns:
\begin{eqnarray}
< n >\sp{\rm (1)}
&= f\sp{(1)}_b C\sp{(1)}_b < n >_\bb
+ f\sp{(1)}_{uds} C\sp{(1)}_{uds} < n >_{\ll}
+ f\sp{(1)}_c C\sp{(1)}_c < n >_\cc\ \, ,\\
< n >\sp{\rm (2)}
&= f\sp{(2)}_b C\sp{(2)}_b < n >_{\bb}
+ f\sp{(2)}_{uds} C\sp{(2)}_{uds} < n >_{\ll}
+ f\sp{(2)}_c C\sp{(2)}_c < n >_\cc\ \, ,\\
< n >\sp{\rm (3)}
&= f\sp{(3)}_b < n >_\bb
+ f\sp{(3)}_{uds} < n >_{\ll}
+ f\sp{(3)}_c < n >_\cc\ .
\end{eqnarray}
Solving the above equations gave the following mean
charged particle multiplicities at 183 GeV:
\begin{eqnarray*}
\ & < n>_\bb(183\,GeV) & = 29.04 \pm 1.08\ , \\
\ & < n>_\cc(183\,GeV) & = 31.47 \pm 3.94\ , \\
\ & < n>_{\lbl}(183\,GeV) & = 23.07 \pm 1.32\ , \\
\ & \delta_{bl}(183\,GeV) & = 5.07 \pm 1.28\, ,
\end{eqnarray*}
and at 189 GeV:
\begin{eqnarray*}
\ & < n>_\bb (189\,GeV) & = 30.14 \pm 0.65\ , \\
\ & < n>_\cc(189\,GeV) & = 26.82 \pm 2.61\ , \\
\ & < n>_{\lbl}(189\,GeV) & = 26.17 \pm 0.91\ , \\
\ & \delta_{bl}(189\,GeV) & = 3.97 \pm 0.83\, .
\end{eqnarray*}
The relatively large uncertainty of the
measured mean multiplicities for charm
stems from the inability of the $P_E$ variable
to extract a c-enriched sample of events.
Systematic uncertainties affect the shape of the impact
parameter distribution of tracks used in the
construction of the $b$-tag probability, $P_E$,
and consequently affect the determination of the flavour content
as a function of $P_E$. Such uncertainties were investigated by means of
the simulation.
The main physics sources of these uncertainties
arise from the assumed lifetime of b-hadrons
($\tau_b = 1.564 \pm 0.014$ ps) \cite{pdg},
and the D$\sp{+},$ D$\sp{0}$ lifetimes and production rates
\cite{pdg}.
%Uncertainties in the determination of the
%acceptance and bias correction factors,
%$C\sp{i}_q$, were also investigated.
%In the latter case, the uncertainty was estimated by
%solving the simultaneous equations assuming
%no bias, i.e.\ with $C\sp{i}_q=1$.
%The larger systematic error on
%$\delta_{bl}$, in comparison to $< n >_b$,
%is a consequence of
%the large overlap in the $P_E$ distribution between
%c and uds events, making the
%extraction of the light quark multiplicity less precise than that of the b.
%In addition, uncertainties due to the detector acceptance and
%the track and event selections were investigated by varying
%the selection criteria.
%A further systematic error, due to
%photon conversions in the material of the detector, was estimated
%by varying their number in the simulation
%by $\pm 10\%$. These uncertainties affect mainly the absolute
%multiplicity values rather than the multiplicity difference,
%$\delta_{bl}$, and are estimated to be
%about $\pm 0.22$ for $< n >_b$.
The analysis was repeated
with different cuts applied to the $b$-tag probability, $P_E$,
and the results for the $\delta_{bl}$
were found to be quite stable (see
figure~\ref{stab189}).
%and the results obtained did not differ significantly.
A systematic error was evaluated as
half of the difference between the greatest and the smallest
multiplicity values obtained from
varying the cut on $P_E$ from $0.5\times10^{-5}$ to $1.5\times10^{-5}$.
The uncertainty due to the event selection in sample (2) was investigated by
repeating the analysis after variation of the narrow jet broadening cut,
from 0.05 to 0.08.
Half of the differences between the greatest and the smallest multiplicities
were added in quadrature to the systematic error previously calculated.
The propagated systematic error in the total multiplicity
in equation (3) was also added in quadrature to the systematic error.
The final mean values of
the event multiplicity in b events, $_{\bb}$,
and the multiplicity
difference between $\bb$ and light quark-antiquark events, are:
\begin{eqnarray}\label{e:mulc}
\ & _\bb(183\,GeV) & = 29.04 \pm 1.08 \pm 0.41\ , \\
\ & _\bb(189\,GeV) & = 30.14 \pm 0.65 \pm 0.33\, \\
\ & \delta_{bl}(183\,GeV) & = 5.07 \pm 1.28 \pm 1.07\ , \\
\ & \delta_{bl}(189\,GeV) & = 3.97 \pm 0.83 \pm 0.68\ .
\end{eqnarray}
These values include the products of K$^0_S$ and $\Lambda$ decays.
\section{Comparison with Models and QCD Predictions}
{\bf{Flavour-Independent Fragmentation ---}}
In a model in which the hadronization is independent of the
mass of the quarks,
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_{E_{cm}}(x_B) \int_0^1 dx_{\bar{B}} f_{E_{cm}}(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 $f_{E_{cm}}(x_B)$ 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 $f_{E_{cm}}(x_B)$, we assumed a Peterson function
%of average $0.70 \pm 0.02$
with hardness parameter $\epsilon_p=0.0047^{+0.0010}_{-0.0008}$
\cite{pdg}, evolving with energy as in~\cite{lund} to take into account the
effects of scaling violations.
The value of $n_{\ll}(E)$ 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} and leading
particles.
The prediction of the model in which the hadronization
is independent of the quark mass is plotted in Figure \ref{nice}.
%the dominant uncertainty comes from
%$$.
There are several variations of the
model in the literature,
leading to slightly different predictions
(see \cite{vietri} and references therein).
The result from substituting in Eq.~(5)
$n_{l\bar{l}}\left( \left(1-\frac{x_B+x_{\bar{B}}}{2} \right) E_{cm}\right)$
with
$n_{l\bar{l}}\left( E_{cm}\sqrt{(1-x_B)(1-x_{\bar{B}})} \right)$ as
in~\cite{opal}, or approximating the Peterson fragmentation function with a
Dirac delta function at $$, are within the errors. Also by using for
$n_{l\bar{l}}$ the expression in~\cite{opal} one stays within the band
in Figure~\ref{nice}.
\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 (9) gives
a value of $5.5 \pm 0.8$. This demonstrates the importance of the contribution
proportional to $\alpha_s(m_b)$.
A different calculation of $\delta_{bl}$
gives a value of 3.68 \cite{petrov}.
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
more than three standard deviations larger than
predicted by the naive model presented in the beginning of this
section, at 189 GeV (2.97 standard deviations at 183 GeV).
\section{Conclusions}
We measured the average charged particle multiplicity
$_\bb$ in $e^+e^- \rightarrow b\bar{b}$ events
at centre-of-mass energy of 183 and 189 GeV to be:
\begin{eqnarray*}
_\bb(183\,GeV) &=& 29.04 \pm 1.08 (stat) \pm 0.41 (syst)\\
_\bb(189\,GeV) &=& 30.14 \pm 0.65 (stat) \pm 0.33 (syst) \, .
\end{eqnarray*}
The difference $\delta_{bl}$ between $_\bb$
and the multiplicity in generic light quark $l = u,d,s$ events
has also been measured to be:
\begin{eqnarray*}
\delta_{bl} &=& 5.07 \pm 1.28 (stat) \pm 1.07 (syst)\\
\delta_{bl} &=& 3.97 \pm 0.83 (stat) \pm 0.68 (syst) \, .
\end{eqnarray*}
which 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.
\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, Vladimir Petrov, Valery Khoze and Torbj\"orn Sj\"ostrand for
%useful discussions.
%=========================================================================%
\newpage
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Editions Fronti\`{e}res, Gif-sur-Yvette, France, 1991.
\end{thebibliography}
\newpage
\begin{figure}
\mbox{\epsfxsize16cm\epsffile{stab189.eps}}
\caption[]{
Stability of $\delta_{bl}=_\bb-_{l\bar{l}}$ with respect to variations
of the
cut on the b-tagging variable, $y$. Notice that the errors
in the plot are correlated (see text). The arrow indicates the
value used in the analyses.}\label{stab189}
\end{figure}
\newpage
\begin{figure}
\mbox{\epsfxsize16cm\epsffile{dbsca189.eps}}
\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}