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\begin{document}
%%% put your own definitions here:
\def\asmz{$\alpha_s(M_Z)$}
\def\ass{\alpha_s(\sqrt{s})}
\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{\mbox{$\alpha_s$}}
%\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 $\rightarrow 4\mathit{jets}$ events}
\newcommand{\Wtj}{WW $\rightarrow 2\mathit{jets} \, \ell \bar{\nu}$ events}
\def\Abreu{DELPHI Coll., P. Abreu {\it et al.,}\ }
\def\etal{{\it et al.,}\ }
\def\muldim{to be published in Proc. XXVIII International Symposium on
Multiparticle Dynamics, Frascati, September 1997.}
\include{particles}
%\include{/afs/cern.ch/user/n/neueld/tex/include/particles}
\renewcommand{\gevc}{$\mathrm{GeV/c}$ }
%\newcommand{\gevc}{$\mathrm{GeV/c}$ }
\newcommand{\rg}{\mbox{\( \mathrm{ C}_{5} {\mathrm F}_{12} \)}}
\newcommand{\rl}{\mbox{\( \mathrm{ C}_{6} {\mathrm F}_{14} \)}}
\newcommand{\Bmin}{\mbox{$B_N$}}
\newcommand{\spr}{\mbox{$\sqrt{s^\prime}$}}
\newcommand{\Leff}{\mbox{${\mathrm \Lambda_{\mathrm{eff}}}$}}
\newcommand{\zn}{\mbox{$ {\mathrm Z^0}$}~}
\newcommand{\pion}{\mbox{$ {\mathrm \pi}^{\pm} $}}
\renewcommand{\Kn}{\mbox{$ {\mathrm K}^{0}$}~}
\renewcommand{\k}{\mbox{$ {\mathrm K}^{+}$}~}
\renewcommand{\p}{\mbox{$ {\mathrm p}$}~}
\newcommand{\xip}{\mbox{$ {\mathrm{\xi_{p}}}$}}
%\newcommand{\Kn}{\mbox{$ {\mathrm K}^{0}$}~}
%\newcommand{\k}{\mbox{$ {\mathrm K}^{+}$}~}
%\newcommand{\p}{\mbox{$ {\mathrm p}$}~}
\newcommand{\xipeak}{\mbox{$ {\mathrm{\xi^{*}}}$}}
\newcommand{\xipkp}{\mbox{$ {\mathrm{\xi^{*}_{prim}}}$}}
\newcommand{\xipkh}{\mbox{$ {\mathrm{\xi^{*}}}$}}
\newcommand{\xipkd}{\mbox{$ {\mathrm{\xi^{*}_{decays}}}$}}
\newcommand{\xis}{\mbox{$ {\mathrm{\xi^{*}}}$}}
\newcommand{\xp}{\mbox{$ {x_{p}} $}}
\newcommand{\pmom}{\mbox{$ P $}}
\newcommand{\dsdxka}{\mbox{$\frac{1}{\sigma_{had}}\frac{d\sigma_{K}}{d x_p} $}}
\newcommand{\dsdxpr}{\mbox{$\frac{1}{\sigma_{had}}\frac{d\sigma_{p}}{d x_p} $}}
\newcommand{\dsdpka}{\mbox{$ \frac{1}{\sigma_{had}}\frac{d\sigma_{K}}{d P} $}}
\newcommand{\dsdppr}{\mbox{$ \frac{1}{\sigma_{had}}\frac{d\sigma_{p}}{d P} $}}
\newcommand{\ee}{\mbox{$ {\mathrm e}^+ {\mathrm e}^-$}}
\newcommand{\dedx}{\mbox{$ \frac{\mathrm dE}{\mathrm dX}$}}
\newcommand{\ftmark}{\raisebox{-0.25em}{*}}
\newcommand{\idhadrons}{\pip, \Kp, \Kn, \p\ and \La}
\begin{titlepage}
\pagenumbering{arabic}
%%% PLEASE NOTE : DO NOT CHANGE ANYTHING CONTAINED ON THIS PAGE
% EXCEPT THE TITLE, IF IT IS DIFFERENT FROM
% THE ONE APPEARING ON THE NEXT PAGE.
\begin{tabular}{l r}
HEP'99 \# 1\_229 & \hspace{6cm} DELPHI 99-119 CONF 306 \\
Submitted to Pa 1, 6 & 15 June 1999 \\
\hspace{2.4cm} Pl 1, 6 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Charged and Identified Particles from the hadronic
decay of W bosons and in \boldmath $e^+e^- \rightarrow q\bar{q}$ from 130 to 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}
\thispagestyle{empty}
\newpage*
\begin{titlepage}
\newpage
\pagebreak
\vspace{\fill}
\end{titlepage}
\pagebreak
\begin{titlepage}
\pagenumbering{arabic}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Charged and Identified Particles \\
from the Hadronic Decay of \\
W Bosons and in \boldmath $e^+e^- \rightarrow q\bar{q}$ from 130 to 189 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 N.~Anjos} $^1$,
{\bf A.~De Angelis} $^2$,
{\bf D.~Liko} $^{3,4}$,
{\bf N.~Neufeld} $^{3,4}$
{\bf Z.~Metreveli} $^5$,
{\bf R.~M\"oller} $^6$,
{\bf N.~Pukhaeva} $^7$,
{\bf F.~Verbeure} $^8$,
{\bf L. Vitale} $^{3,9}$
%========================================================================%
}
\vspace*{1cm}
\end{center}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
Inclusive distributions of charged particles in hadronic W decays
are experimentally investigated using the statistics
collected by the DELPHI experiment at LEP during 1997 and 1998,
at the centre-of-mass energies 183 and 189 GeV.
The possible effects of interconnection between
the hadronic decays of two Ws are discussed.
\\
Measurements of the average multiplicity for charged and identified
particles in $q\bar{q}$ and WW events at centre-of-mass energies from 130 to 189 GeV
and in W decays are presented.
\\
The results on the average multiplicity of identified particles
and on the position $\xi^*$ of the maximum of the
$\xi_p = -\mathrm{log} (\frac{2p}{\sqrt{s}})$
distribution have been compared with predictions
of JETSET and MLLA calculations.
%=========================================================================%
\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 (Area Rizzi), I-33100 Udine, Italy}
\par {\footnotesize $^3$ CERN, CH-1211 Geneva 23, Switzerland}
\par {\footnotesize $^4$ HEPHY, \"{O}sterr. Akad. d. Wissensch., Nikolsdorfergasse
18, A-1050 Vienna, Austria}
\par {\footnotesize $^5$ Institute for High Energy Physics of Tbilisi State
University, Georgia}
\par {\footnotesize $^6$ Niels Bohr Institute, Copenhagen, Denmark}
\par {\footnotesize $^7$ Joint Institute for Nuclear Research, Dubna, Russia}
\par {\footnotesize $^8$ Universitaire Instelling Antwerpen, Antwerpen,
Belgium}
\par {\footnotesize $^9$ Dipartimento di Fisica, Universit\`a di Trieste and
INFN, Area di Ricerca, Padriciano 99, I-34102 Trieste, Italy}
\end{titlepage}
\pagebreak
%==================> DELPHI note text =====> To be filled <======%
\section{Introduction}
At LEP II the study remains interesting and important, of
production properties of inclusive charged hadrons and identified
hadrons (\idhadrons). The interest in these studies is at least
twofold.
First they continue the respective measurements done at lower
centre-of-mass energies and allow the study of the energy dependence of
average multiplicities and single particle distributions. These can be
compared to event generators, which have been tuned using the ample
statistics available at the \Z\ pole and to analytic calculations
using perturbative QCD.
Second in the case of $W$ production they are expected to provide
insight into possible correlations and/or final state interactions
between the decay products of the two $W$s.
\subsection*{Hadron production at LEPII}
The way quarks and gluons transform into hadrons is complex and can not yet be completely
described by Quantum Chromo Dynamics (QCD). The most accurate description is
given by Monte Carlo simulations.
In the picture implemented in these simulations, the
hadronisation of a \q\qb\ pair
is split into 3 phases. In a first phase,
gluon emission and parton branching of the original
\q\qb\ pair take place. It is
believed that this phase can be described by perturbative QCD
(most of the calculations
have been performed in leading logarithmic approximation).
In a second phase,
at a certain virtuality cut-off scale $Q_0$, where $\alpha_s(Q_0)$ is still small,
quarks and gluons produced in
the first phase are
clustered in colour singlets and transform into mesons and baryons. Only
phenomenological models, which need to be tuned to the data,
are available to describe this process of fragmentation;
the models most
frequently used in \ep\emi\ annihilations are based on
string and cluster fragmentation.
In the third phase, the unstable states decay into hadrons which can
be observed or identified in the detector.
These models account correctly for the gross features of the $q\bar{q}$
events such as, for instance,
the average multiplicity and inclusive momentum spectra
up to the \Z\ energy.
%; with LEP~2,
%the energy range spanned in $e^+e^-$
%interactions is doubled (up to 183 $\mathrm{GeV}$), and it is interesting
%to check their validity.
A different and purely analytical approach (see e.g.~\cite{ochs}
and references therein)
giving quantitative predictions of hadronic spectra
are QCD calculations using the so-called Modified Leading Logarithmic
Approximation (MLLA) under the assumption
of Local Parton Hadron Duality (LPHD)~\cite{dok:mlla}.
In this picture multi-hadron production is described
by a parton cascade, and the virtuality cut-off $Q_0$ is lowered
to values of the order of 100 MeV, comparable to the hadron masses;
it is assumed that the results obtained for partons are proportional to the corresponding
quantities for hadrons.
The momentum spectra of hadrons can be calculated as functions of the variable
$\xip = -\ln{x_p}$ where $x_p = 2p/\sqrt{s}$ ($p$ being the particle's momentum and $\sqrt{s}$
the centre of mass energy):
\begin{equation}
\frac{1}{\sigma}\frac{d\sigma}{d\xi_p} = K_\mathrm{LPHD}
\cdot f(\xi_p,X,\lambda)
\label{eq:evol}
\end{equation}
with
\begin{eqnarray}
X= \log\frac{\sqrt{s}}{Q_{0}} & ; & \lambda =
\log\frac{Q_{0}}{\Lambda_{\mathrm{eff}}} \, .
\end{eqnarray}
The MLLA/LPHD predictions thus involve three parameters: an effective
scale parameter $\Lambda_\mathrm{eff}$, a virtuality cut-off $Q_0$ in the
evolution of the parton cascade and an overall normalisation factor $K_\mathrm{LPHD}$.
The function (\ref{eq:evol}) has the form of a ``humped-backed plateau''. It can be approximated by a distorted
Gaussian~\cite{dkt-1,bw:mlla}
\begin{equation}
\label{eq:dgauss}
D(\xi;N,\overline{\xi},\sigma,s_k,k)=
\frac{N}{\sigma\sqrt{2\pi}}\exp{\left(\frac{1}{8}k+\frac{1}{2}s_k\delta
-\frac{1}{4}(2+k)\delta^{2}+\frac{1}{6}s_k\delta^{3}
+\frac{1}{24}k\delta^{4}\right)}\ ,
\end{equation}
where $\delta =(\xi-\overline{\xi})/\sigma$ and $s_k$ and $k$ are the skewness
and kurtosis of the distribution. For an ordinary
Gaussian these last two terms would vanish. $\overline{\xi}$ coincides with the
peak of the distribution, \xis\, only up to next-to-leading order.
To check the validity of the MLLA/LPHD approach, it is interesting to study the evolution
of the position of the maximum, \xis, as a function of $\sqrt{s}$.
In the context of MLLA/LPHD the dependence of \xis on the centre-of-mass energy can be expressed
as~\cite{ochs,dkt-1}:
\begin{equation}
\xi^* = Y
\left(
\frac{1}{2} +\sqrt{C/Y}-C/Y
\right) +F_h (\lambda),
\label{eq:evol2}
\end{equation}
where
\begin{displaymath}
Y = \mathrm{log}\frac{\sqrt{s}/2}{\Leff}~~,~~
C =
\left(
\frac{11N_c/3 + 2n_f/(3N_c^2)}{4N_c}
\right)^2
\cdot
\left(
\frac{N_c}{11N_c/3 -2n_f/3}
\right),
\end{displaymath}
with $N_c$ being the number of colours and $n_f$ the active number of quark
flavours in the fragmentation process.
$F_h (\lambda)$ depends on the hadron type through the ratio
$\lambda = \mathrm{log}\frac{Q_0}{\Leff}$~\cite{ochs}.
%\begin{equation}
%F_h (\lambda) = -1.46\lambda + 0.207\lambda^2 \pm 0.06.
%\end{equation}
\subsection*{Interference and final state interactions in $W$ decays}
The possible presence of interference due to colour reconnection
and Bose-Einstein correlations (see for
example~\cite{sjre,gh,ww,were,elre,sjbe,moller} and
~\cite{yellow,balle} for a review)
in hadronic decays of \W\W\ pairs
%has been discussed on a theoretical basis,
%in the framework of the measurement of the W mass:
may provide information on hadron formation at a time scale
smaller than 1 fm/$c$. At the same time it
can induce a systematic uncertainty on the W
mass measurement in the 4-jet mode~\cite{yellow}
comparable with the expected accuracy of the measurement.
Interconnection can happen due to the fact that the lifetime of the W
($\tau_W \simeq \hbar /\Gamma_{W} \simeq 0.1$ fm/$c$) is an order of
magnitude smaller than the typical hadronisation times.
The interconnection between the products of the hadronic decays of
different Ws in WW pair events
can occur at several stages:
\begin{enumerate}
\item from colour rearrangement between the quarks
coming from the primary branching,
\item due to gluon exchanges during the parton
cascade,
\item in the mixing of identical pions due to Bose-Einstein correlations.
\end{enumerate}
The first two are QCD effects.
They can mix the two colour singlets and produce hadrons which cannot be
uniquely assigned to either $W$.
The perturbative effects are colour suppressed
such that the possible shift is expected to be only about 5 MeV in the W mass~\cite{sjre}.
Non-perturbative effects
need model calculations.
Several models have been proposed
\cite{balle} and some of them have already been included in
the widely used event generators
PYTHIA~\cite{lund}, ARIADNE~\cite{ari} and HERWIG\cite{HERWIG}.
In these models the final state quarks after the parton shower
can be rearranged to form colour singlets with probabilities which
in some cases can be free parameters (choices of these parameters are
directly suggested by the authors by simple considerations).
The shift on the W mass in these models is smaller than 50 MeV, but
other observables are affected by colour rearrangement.
Generally these models suggest a small effect on the total charge
multiplicity, of the order of $\pm$1-3$\%$ \cite{balle}.
Identified heavy particles, such as \k and \p\footnote{Unless
otherwise stated antiparticles are included as well.}, are expected to
be even more sensitive to interconnections.
The study of Bose-Einstein Correlations (BEC) is complicated by the fact that
a complete description would need the symmetrization of the amplitude for the
multiparticle system, which is computationally
difficult. One must thus make approximations
and build models~\cite{fialko,newsj}. Care must be taken not to
distort the multiplicity distribution by increasing the probability of final
states with large multiplicity.
Two-particle correlation function are usually used in BEC studies.
BEC could slightly increase the multiplicity for $(4q)$ events in
some models~\cite{moller}.
The WW events allow a comparison of
the characteristics of the W hadronic decays when
both Ws decay in hadronic modes
(in the following this shall often be
referred to as the $(4q)$ mode)
with the case in which the only one W decays
hadronically
($(2q)$ mode for brevity). These characteristics
should be the same in the absence of interference between the
products of the hadronic decay of the W bosons.
Previous experimental results based on the statistics collected by
LEP experiments during 1997 (see \cite{deafra,watvan,molvan} for reviews)
did not indicate at that level of statistics
the presence of interconnection or correlation effects.
This paper presents measurements of:
\begin{itemize}
\item
the charged particle multiplicities for the $q\bar{q}$ events
in the data sample
collected by the DELPHI experiment at LEP during 1997 and 1998,
at the centre-of-mass energies 183 and 189~GeV respectively;
\item
the charged particle multiplicity and inclusive distributions
for WW events at 183 and 189~GeV (values at 189~GeV are expected to be
slightly higher that at 183~GeV for increased phase space, but not
at the level of the precision of the measurements);
%For these energies the main background to WW processes
%is given by $q\bar{q}$ events.
% with a cross section (for an effective centre-of-mass
%energy larger than 10\% of the maximum annihilation energy) of about 105 pb.
\item
The average multiplicities for identified charged and neutral
particles (\idhadrons),
and the position $\xi^*$ of the maximum of the
$\xi_p = -\mathrm{log} (\frac{2p}{\sqrt{s}})$ distribution
for $q\bar{q}$ from 130 \gev\ to 189 \gev\ and WW events at 189~GeV.
\end{itemize}
\section{Data Sample and Event Preselection}
Data corresponding to
total luminosities of 157.7~pb$^{-1}$ (54.1 pb$^{-1}$) collected by DELPHI
at centre-of-mass energies around 189 (183) GeV
were analysed.
A description of the DELPHI detector can be found in ref.~\cite{deldet}; its
performance is discussed in ref.~\cite{perfo}.
A preselection of hadronic events was made, requiring at least 6
charged particles with momentum $p$ above 100 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 4 cm along the beam axis, and a
total transverse energy of all the particles above 0.2
times the centre-of-mass energy $\sqrt{s}$.
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/$c$ (200 MeV/$c$ in the charged identified particles analysis),
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 5.722 and JETSET 7.409~\cite{lund},
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 with simulated digitizations in each
detector. 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 samples collected at the Z during 1998 and 1997 were used.
From these samples, by integrating the distribution of
$\xi_E = -\ln(2E/\sqrt{s})$, where $E$ is the energy of the particle,
corrected bin by bin using the simulation,
the average charged particle multiplicities at the Z were measured to be
$20.91 \pm 0.03 (stat)$ and $20.66 \pm 0.03 (stat)$ respectively,
in satisfactory agreement with the world average of
$21.00 \pm 0.13$ \cite{pdg}. A
relative scale systematic uncertainty of 1\% was assumed on the multiplicity.
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, $\sqrt{s'}$,
the procedure described in ref.~\cite{sprimen} 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 elsewhere, the
energy of the ISR photon is computed from the jet directions
assuming massless kinematics. The effective centre-of-mass energy of the
hadronic system, \spr, is calculated as the invariant
mass of the system recoiling against 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.
The selection of the hadronic events
at the various energies, after the common preselection, depends on the
centre-of-mass energy due to the different background from
\Wp\Wm\ pairs.
\section{Analysis of charged particles in \boldmath{\q\qb} decays}
Events with $\sqrt{s'}/\sqrt{s}$ above 0.9
were used to compute the multiplicities.
A total of 3493 (1345) hadronic events were selected from the data at
189 (183) GeV, by requiring
that the multiplicity for charged particles (with $p>100$~MeV/$c$)
was larger than 9,
that the total transverse energy of the charged particles exceeded
$0.2\sqrt{s}$,
and that the narrow jet broadening~\cite{lepII}
was smaller than 0.065.
From the simulation it was calculated that the expected
background coming from WW and ZZ decays was 429+51 (128+21) events.
The contamination from double radiative returns to the Z,
within 10~GeV of the nominal Z mass, was estimated by
simulation to be below 5\%.
Other contaminations (from Zee, We$\nu$,
$\gamma\gamma$ interactions and Bhabhas) are below 2\% in total.
The average multiplicity of charged particles with $p > 0.1$~GeV/$c$
measured in the selected events at 189 GeV (183 GeV), after subtraction of the
WW and ZZ backgrounds estimated by simulation, was
$25.76 \pm 0.16 (stat)$ ($25.44 \pm 0.24 (stat)$), to be compared
to $25.85 \pm 0.05$ ($25.60 \pm 0.22 (stat)$) in the
q$\bar{\mathrm q}$ PS simulation including detector
effects.
The dispersion of the multiplicity distribution in the data was
$7.90 \pm 0.12 (stat)$ ($7.53 \pm 0.17 (stat)$), to be compared to
the dispersion from the q$\bar{\mathrm q}$ PS simulation of
$7.62 \pm 0.04$ ($7.60 \pm 0.16 (stat)$).
After correcting for detector effects, the average charge multiplicity
was found to be
$ = 27.21 \pm 0.17 (stat)$ ($ = 26.75 \pm 0.26 (stat)$),
and the dispersion to be $D = 8.71 \pm 0.13 (stat)$
($D = 8.23 \pm 0.19 (stat)$).
These values again include the products of the decays of particles with
lifetime $\tau < 10^{-9}$ s.
The average multiplicity was also computed
by integrating the distributions:
\begin{itemize}
\item of the rapidity with respect to the thrust axis
$y_T = \frac{1}{2}\ln\frac{E+p_{||}}{E=p_{||}}$ ($p_{||}$
is the absolute value of the momentum component on the thrust axis);
\item of $\xi_E = -\mathrm{log} (\frac{2E}{\sqrt{s}})$.
%\item of $p_T$ ??
\end{itemize}
all corrected bin by bin using the simulation.
The $\xi_E$ distribution was
integrated up to a value of 6.3, and the extrapolation to the region above
this cut was based on the simulation
at the generator level.
The average charge multiplicity of the selected events, including the
above corrections, is
27.01 and 26.97 (26.79 and 26.70)
respectively from the $y_T$ and $\xi_E$ distribution,
consistent with the value from the average observed multiplicity.
As a central value for the measurement of the charge multiplicity the
result of the integration of the $\xi_E$ distribution was taken, since
the detection efficiency depends mostly on the momentum of the particle.
The values
\begin{eqnarray}
_{\mathrm{189\,GeV}} = 26.97 \pm 0.17 (stat) \pm 0.37 (syst) \label{mulv189}\\
D_{\mathrm{189\,GeV}} = 8.71 \pm 0.13 (stat) \pm 0.24 (syst)\\
_{\mathrm{183\,GeV}} = 26.71 \pm 0.26 (stat) \pm 0.57 (syst) \label{mulv183}\\
D_{\mathrm{183\,GeV}} = 8.23 \pm 0.19 (stat) \pm 0.15 (syst)
\end{eqnarray}
were obtained for the average charge multiplicity and for the dispersion.
The systematic errors were obtained by adding in quadrature:
\begin{enumerate}
\item The scale uncertainty of $\pm 0.27$ ($\pm 0.27$) for the
multiplicity and $\pm 0.09$ ($\pm 0.08$) for the dispersion.
\item The effect of the cuts for the reduction of the background.
The value of the cut on the narrow jet
broadening was varied from 0.045 to 0.085 in steps of 0.010, in order to estimate the systematic error associated with the procedure of removing the
contribution from WW events.
The new values for the average charged particle
multiplicity and the dispersion were stable within these variations, and
half of the difference between the extreme values,
%the highest shifts with respect to the previous results,
0.05 and 0.08 (0.09 and 0.07) respectively, were added in quadrature to the
systematic error. The effect of the uncertainty on the WW cross-section was
found to be negligible.
\item The uncertainty on the
modelling of the detector response in the forward region. The analysis
was repeated by using only the tracks with $40^{\circ}<\theta<140^{\circ}$. The
values obtained for the multiplicity
and the dispersion were 26.73$\pm0.24$ and 8.50$\pm0.12$
(26.16$\pm0.37$ and 8.34$\pm$0.18) respectively,
and uncertainties of 0.17 and 0.20
(0.48 and 0.09) were computed after subtracting in quadrature the difference
between the statistical errors.
\item The systematic errors due to the statistics of the Monte Carlo samples,
0.04 (0.06) for the multiplicity and 0.04 (0.06) for the dispersion.
\item The uncertainty on the calculation of the
correction factors. The maximum shift in the multiplicity
calculated from the multiplicity distribution and from the integration
of the $y_T$ and $\xi_E$ distributions was 0.12 (0.05).
Half of the difference between the maximum and the minimum value
was taken as systematic error.
\item Half of the multiplicity extrapolated in the high-$\xi_E$ region,
0.14 (0.12).
\end{enumerate}
Fig.~\ref{mulea} shows the value of the average charged particle
multiplicity in $e^+e^- \rightarrow q\bar{q}$ events at 189 and 183~GeV
compared with lower energy points
from JADE~\cite{jade}, PLUTO~\cite{pateta}, MARK II~\cite{markII},
TASSO~\cite{tasso}, HRS~\cite{HRS}, and AMY~\cite{amy},
with DELPHI results in q$\bar{\mathrm q}\gamma$ events at the Z~\cite{qgjet},
with the world average at the Z~\cite{pdg}, and with the
averages of the LEP results at 133, 161 and 172 GeV
calculated according to the prescriptions in \cite{abreu}.
A point corresponding to the multiplicity observed by DELPHI in W decays
is also included at $\sqrt{s} = M_W$ and is discussed later in this paper.
The value at the Z has been lowered by 0.20, to account
for the different proportion of $b\bar{b}$ and $c\bar{c}$ events at the
Z with respect to the $e^+e^-\rightarrow\gamma^{*}\rightarrow
{\mathrm q}\bar{\mathrm q}$ \cite{dea}.
Similarly, the values at 133, 161, 172, 183 and 189~GeV were lowered by
0.15, 0.12, 0.11, 0.10 and 0.10 respectively.
%To the statistical errors
%on the charged multiplicities in q$\bar{\mathrm q}\gamma$ events at the
%Z~\cite{fuster},
%a systematic error assumed to be $\pm 0.50$ has been added in quadrature.
The QCD prediction for charge multiplicity has
been computed as a function
of $\alpha_s$ including the resummation of leading
(LLA) and next-to-leading (NLLA) corrections~\cite{weber,sprime}:
\begin{equation}\label{truffle}
(\sqrt{s}) = a [\ass]^b e^{c/\sqrt{\ass}}
\left[1+O(\sqrt{\ass})\right] \ ,
\end{equation}
where $s$ is the squared centre-of-mass energy and $a$ is a parameter
(not calculable from perturbation theory) whose value has been fitted from the
data. The constants $b=0.49$ and $c=2.27$ are predicted by
theory~\cite{weber} and $\ass$ is the strong coupling constant.
The data between 14~GeV and 189~GeV are plotted in Fig.~\ref{mulea},
with the curve corresponding to
$a = 0.069$ and $\alpha_s(m_{\mathrm{Z}})=0.113$ superimposed.
%A fit to data between 14~GeV and 189~GeV, excluding the results which are
%corrected for secondary decays (JADE and PLUTO),
%with $\alpha_s(\sqrt{s})$
%(expressed at next-to-leading order) and $a$ and $d$ as free
%parameters, is shown in Fig.~\ref{mulea}. The results of the fit are:
%$a = 0.069 \pm 0.004$, $d = 0.00 \pm 0.16$, and
%$\alpha_s(m_{\mathrm{Z}}) = 0.113 \pm 0.002 (stat)$.
The ratios of the average multiplicity to the dispersion
measured at 189~GeV and 183 GeV, $/D = 3.10 \pm 0.10 (stat+syst)$,
and $/D = 3.25 \pm 0.11$ respectively, are consistent
with the average from the
measurements at lower centre-of-mass energies (3.13 $\pm$ 0.04), as
can be seen in Fig.~\ref{muleb}.
From KNO scaling~\cite{kno} this ratio is predicted to be
energy-independent.
The ratio measured is also consistent with the
predictions of QCD including 1-loop Higher Order terms (H.O.)
\cite{webberkno}.
\section{Classification of the WW Events and Multiplicity Measurement}
About 4/9 of the WW events are \Wfj. At threshold,
their topology is that of two pairs of back to back jets, with no
missing energy; the constrained invariant mass of two
jet-jet systems is close to the W mass. Even at 183 and 189
GeV these characteristics allow a clean selection.
About 4/9 of the WW evens are \Wtj. At threshold,
their topology is 2-jets back-to-back, with a lepton and
missing energy opposite to it;
the constrained invariant mass of the
jet-jet system and of the lepton-missing energy system
equals the W mass.
\subsection{Fully Hadronic Channel (WW\boldmath{$\rightarrow 4j$})}
Events with both Ws decaying into q$\bar{\mathrm q}$ are
characterised by high multiplicity, large visible energy, and
tendency of the particles to be grouped in 4 jets.
The background is dominated by q$\bar{\mathrm q}(\gamma)$ events.
The events were pre-selected by
requiring at least 12 charged particles (with $p>100$~MeV/$c$), with a total
energy (charged plus neutral) above 20\% of the centre-of-mass energy.
To remove the radiative hadronic events, the hadronic centre-of-mass energy
was required to be above 110~GeV. The hadronic centre-of-mass energy was
computed as described in the previous section.
The particles in the event were then fitted to 4 jets using the LUCLUS
algorithm~\cite{durham}, and the events were kept if all jets had
multiplicity (charged plus neutral) larger than 3.
It was also required that the separation between the jets ($D_{min}$ value)
be larger than 6~GeV. The combination of these two cuts removed most of
the remaining semi-leptonic WW decays and
the 2-jet and 3-jet events of the q$\bar{\mathrm q}$ background.
A four constraint fit was applied, imposing energy and momentum conservation
and the equality of the two di-jet masses.
Of the three fits obtained by permutation of the jets, the one with the smallest
$\chi^2$ was selected.
Events were accepted only if
$$D_\mathrm{sel} =
\frac{E_{min}\theta_{min}}{E_{max}(E_{max}-E_{min})} > 0.004 \,
\mbox{rad GeV}^{-1}$$
where $E_{min}$ and $E_{max}$ are
respectively the the smallest and the largest
fitted jet energy,
and $\theta_{min}$ is the smallest angle between the fitted jet directions.
The details of the selection variable $D_\mathrm{sel}$ can be found in~\cite{Wxsec}.
The purity and the efficiency of the selected data sample
from the 189 (183) GeV data were estimated using simulation to be
about 77\% and 78\% (75\% and 80\%) respectively. The data sample
consists of 1241 (420) events, where 1219 (422) were expected from
the simulation.
The expected background was subtracted bin by bin from the observed
distributions, which were then corrected bin by bin using
scaling factors computed from the simulation.
Finally $$ was estimated by integrating the $\xi_E$ distribution.
The following values were obtained:
\begin{eqnarray}
_{\mathrm{189\,GeV}} & = & 38.32 \pm 0.32 (stat)
\pm 0.62 (syst) \label{mulha}\\
_{\mathrm{183\,GeV}} & = & 37.79 \pm 0.56 (stat)
\pm 0.57 (syst) \label{mulha2} \, .
\end{eqnarray}
The systematic errors account for:
\begin{enumerate}
\item Scale uncertainty 0.38 (0.38).
\item Variation of the selection criteria 0.42 (0.26).
\item Modelling of the detector in the forward region.
To investigate a possible dependence of the measured multiplicity on
the modelling of the detector response in the forward region, the analysis
was repeated by using only the tracks with $40^{\circ}<\theta<140^{\circ}$.
The value obtained for the multiplicity and dispersion
were $38.34 \pm 0.41$ and $8.43 \pm 0.23$ ($37.38 \pm 0.73$
and $8.77 \pm 0.41$).
The difference between this multiplicity and the reference ones
is $0.02 \pm 0.26$ ($0.41 \pm 0.46$),
assuming full correlation between this measurement
(which contains the same events and subsample of tracks)
and the reference in \ref{mulha} (\ref{mulha2})
for the calculation of the statistical errors.
Since this source of uncertainty is compatible with its statistical error,
it is assumed negligible.
\item Limited statistics in the simulated sample 0.04 (0.05).
\item Change of the q$\bar{\mathrm q}$ cross-sections within 5\%:
negligible.
\item Calculation of the correction factors 0.06 (0.23).
The value of $$ was also estimated:
\begin{itemize}
\item from the observed multiplicity distribution as 38.35 (37.75);
\item from the integral of the rapidity distribution (with respect to the
thrust axis) as 38.44 (37.33).
\item from the integral of the $p_T$ distribution (with respect to the thrust
axis) as 38.39 (37.77).
\end{itemize}
Half of the difference between the maximum and the minimum value
was taken as systematic error.
\item Uncertainty on the modelling of the background 0.24 (0.35).
The uncertainty on the modelling of the background
is the sum in quadrature of three contributions:
\begin{itemize}
\item Uncertainty on the $q\bar{q}$ multiplicity.
A relative uncertainty as in
Eq.~(\ref{mulv189}) (Eq.~(\ref{mulv183})
was assumed; this gives a multiplicity error of 0.09 (0.21).
\item Uncertainty on the modelling of
the 4-jet rate. The agreement between data
and simulation was studied in a sample of 4-jet events at the Z, with
$y_{cut}$ ranging from 0.003 to 0.005.
The rate of 4-jet events in the simulated sample was found to reproduce the
data within 10\%. The correction due to background subtraction was
correspondingly varied by 10\%, which gives an uncertainty of 0.00 (0.01).
\item Uncertainty on the multiplicity in 4-jet events.
The average multiplicity of 4-jet events selected
at the Z for a value of $y_{cut} = 0.005$
is larger by $(2.8 \pm 1.3 (stat))\%$ than the corresponding value in the
simulation. A shift by 2.8\% in the multiplicity for 4-jet events
induces a shift of 0.22 (0.28) on the value in Eq.~(\ref{mulha})
(Eq. (\ref{mulha2})).
\end{itemize}
\end{enumerate}
The presence of interference between the jets coming from the different Ws
could create subtle effects, such as to make the application of the
fit imposing equal masses inadequate.
For this reason a
different four constraint fit was performed, leaving the di-jet masses free
and imposing energy-momentum conservation.
Of the three possible combinations of the four jets into WW
pairs, the one with minimum mass difference was selected.
No $\chi^2$ cut was imposed in this case.
The average multiplicity obtained was
again fully consistent (within the statistical error) with
the one measured in the standard analysis.
The distribution of the observed charge multiplicity in $(4q)$ is shown in
Fig. \ref{wmul}c(\ref{wmul}a).
The value of the low momentum multiplicity in the range 0.1 and 1. GeV/$c$
was found to be 14.12$\pm0.20$ (14.06$\pm0.35$).
%14.12$\pm0.12$ (14.06$\pm0.21$).
%The dispersion of the
%multiplicity distribution in the data was $6.69 \pm 0.31(stat)$,
%to be compared to
%the dispersion from the q$\bar{\mathrm q}$ PS simulation of
%$6.39 \pm 0.04(stat)$.
After correcting for detector effects, the
dispersion was found to be:
\begin{eqnarray}
D^{(4q)}_{\mathrm{189\,GeV}} & = & 8.46 \pm 0.22 (stat) \pm 0.46 (syst) \\
D^{(4q)}_{\mathrm{183\,GeV}} & = & 8.59 \pm 0.40 (stat) \pm 0.20 (syst) \, .
\end{eqnarray}
In the systematic error:
\begin{enumerate}
\item 0.08 (0.09) accounts for the scale uncertainty;
\item 0.45 (0.17) accounts for the variation of the cuts;
\item the effect of the modelling of the detector in the forward region
is negligible;
\item 0.04 (0.05) from the limited Monte Carlo statistics.
\end{enumerate}
\subsection{Mixed Hadronic and Leptonic Final States
(WW\boldmath{$\rightarrow 2jl\nu$})}
Events in which one W decays into lepton plus neutrino and the other one into
quarks are characterised by two hadronic jets, one
energetic isolated charged lepton, and missing momentum resulting from
the neutrino. The main backgrounds to these events
are radiative q$\bar{\mathrm q}$ production and four-fermion
final states containing
two quarks and two oppositely charged leptons of the same flavour.
Events were selected by requiring seven or more charged particles, with
a total energy (charged plus neutral) above $0.2\sqrt{s}$
and a missing momentum larger than $0.1\sqrt{s}$.
Events in the q$\bar{\mathrm q}\gamma$ final state with ISR
photons at small polar angles, which would be lost inside the beam pipe,
were suppressed by requiring the polar angle of the
missing momentum vector to satisfy $|\cos\theta_{miss}|<0.94$.
Including the missing momentum as an additional massless neutral particle
(the candidate neutrino),
the particles in the event were fitted to 4 jets using the Durham algorithm.
%The distribution of the value for the jet resolution parameter
%$y_{\mbox{cut}}$needed to differentiate 3 to 4 jets (di-jet separation)
%is shown in Fig.~\ref{yc34}.
The jet for which the fractional jet energy carried by the
highest momentum charged particle was greatest was considered as the
``lepton jet''. The most energetic charged
particle in the lepton jet was considered the lepton candidate, and the
event was rejected if its momentum was smaller than 10~GeV/$c$.
%or greater than 65~GeV/$c$.
The ``neutrino jet''
was considered the jet clustered around the missing momentum.
The event was discarded if the invariant mass of the event
(excluding the lepton candidates) was smaller than 20~GeV/$c^2$ or
larger than 110~GeV/$c^2$.
At this point three alternative topologies were considered:
\begin{itemize}
\item Muon sample:
when the lepton candidate was tagged as a muon and its isolation
angle, with respect to other charged particles above 1~GeV/$c$, was
above 10$^{\circ}$,
the event was accepted either if the lepton momentum was greater
than 20~GeV/$c$, or if it was greater than 10~GeV/$c$ and the
value of the $y^{\mbox{cut}}_{3\rightarrow 4}$ parameter required by
the Durham algorithm to force the event from a 3-jet to a 4-jet configuration
was greater than 0.003.
\item Electron sample:
when the lepton candidate had associated electromagnetic energy larger than 20~GeV, deposited in
the calorimeters and an isolation angle (defined as
above) greater than 10 degrees, the event was accepted if the
required value of $y^{\mbox{cut}}_{3\rightarrow 4}$ was greater than 0.003.
\item Inclusive sample:
the events were also accepted if the ``lepton'' momentum was larger than
20~GeV/$c$ and greater than half of the jet energy, the missing momentum was
larger than $0.1\sqrt{s}$, the
required value of $y^{\mbox{cut}}_{3\rightarrow 4}$ was greater
than 0.003, and no other charged particle above 1~GeV/$c$ existed in the
lepton jet.
\end{itemize}
The purity and the efficiency of the selected data sample
from the 189 (183) GeV data were estimated using simulation to be
about 88\% and 55\% (87\% and 56\%) respectively. The data sample
consists of 701 (267) events, where 710 (245) were expected
from the simulation.
The expected background was subtracted bin by bin from the observed
distributions, which were then corrected bin by bin using
scaling factors computed from the simulation.
Finally the following values were obtained for the charged multiplicity
for one W decaying hadronically in a WW event
with mixed hadronic and leptonic final states by
integrating the $\xi_E$ distribution:
\begin{eqnarray}
_{\mathrm{189\,GeV}} & = & 19.62 \pm 0.29 (stat)
\pm 0.52 (syst) \label{mulle}\\
_{\mathrm{183\,GeV}} & = & 20.07 \pm 0.45 (stat)
\pm 0.48 (syst) \label{mulle2} \, .
\end{eqnarray}
In the systematic error:
\begin{enumerate}
\item 0.20 (0.20) accounts for scale uncertainty,
\item 0.19 (0.17) accounts for variation of the selection criteria,
\item 0.43 (negligible) accounts for modelling of the detector
in the forward region.
To investigate a possible dependence of the measured multiplicity on
the modelling of the detector response in the forward region, the analysis
was repeated by using only the tracks with $40^{\circ}<\theta<140^{\circ}$.
The values obtained for the multiplicity
and the dispersion were $19.14\pm0.37$ and $6.45\pm0.20$
($20.02\pm0.62$ and $6.83\pm0.34$) respectively.
The difference between these values and the reference ones
are $0.47 \pm 0.20$ ($0.05 \pm 0.35$).
The uncertainties of 0.43 and 0.00 were
obtained by subtracting in square the statistical error.
\item 0.03 (0.05) accounts for limited statistics in the simulated samples.
\item 0.01 (0.01) accounts for variation of the
q$\bar{\mathrm q}(\gamma)$ cross-sections within 5\%,
\item 0.10 (0.40) accounts for the uncertainty on the correction factors.
The value of $$ was also estimated:
\begin{itemize}
\item from the observed multiplicity distribution as 19.73 (20.30);
\item from the integral of the rapidity distribution (with respect to the
thrust axis) as 19.53 (19.50).
\item from the integral of the $p_T$ distribution (with respect to the thrust
axis) as 19.60 (20.04).
\end{itemize}
\end{enumerate}
The distribution of the observed charge multiplicity in $(2q)$ is shown in
Fig. \ref{wmul}d(\ref{wmul}b).
The value of the low momentum multiplicity in the range 0.1 and 1. GeV/$c$
was found to be 7.31$\pm0.18$ (7.59$\pm0.28$).
%7.31$\pm0.11$ (7.59$\pm0.17$).
After correcting for detector effects,
the dispersions were found to be:
\begin{eqnarray}
D^{(2q)}_{\mathrm{189\,GeV}} & = & 6.39 \pm 0.20 (stat) \pm 0.44 (syst) \\
D^{(2q)}_{\mathrm{183\,GeV}} & = & 6.17 \pm 0.31 (stat) \pm 0.27 (syst) \, .
\end{eqnarray}
In the systematic error:
\begin{enumerate}
\item 0.06 (0.06) accounts for the scale uncertainty;
\item 0.25 (0.26) accounts for the variation of the cuts;
\item 0.36 (negligible) accounts
for the modelling of the detector in the forward region;
\item 0.03 (0.05) from the limited Monte Carlo statistics.
\end{enumerate}
\section{Analysis of Interconnection Effects from
Charged Particle Multiplicity and Inclusive Distributions}
Most models predict that, in case of colour reconnection, the
ratio between the multiplicity in $(4q)$ events and twice the
multiplicity in $(2q)$ events is smaller than 1; the difference
is expected to be at the percent level.
We measure:
\begin{eqnarray}
\left(\frac{}{2}\right)_{\mathrm{189\,GeV}} & = &
0.977 \pm 0.017 (stat) \pm 0.027 (syst) \\
\left(\frac{}{2}\right)_{\mathrm{183\,GeV}} & = &
0.941 \pm 0.025 (stat) \pm 0.023 (syst) \, .
\end{eqnarray}
In the calculation of the systematic error on the ratio, all the
systematic errors were taken as uncorrelated
except for the scale error, for which full correlation was assumed.
A compatible value of $\pm 0.023$ ($\pm 0.018$) is obtained for the
systematic error when
correlations are explicitly computed for
modelling of the detector in the forward region,
%; 0.025$\pm0.014$=0.020
%(0.007$\pm0.020$=negligible)
change in the $q\bar{q}$ cross section and
computation of multiplicity using the multiplicity, $y$ and $p_T$
distributions. %$0.007(0.013)$
In case of interconnection,
the deficit of multiplicity is expected to be concentrated in the
region of low momentum.
The corrected momentum distribution in the $(4q)$ and in the
$(2q)$ cases is shown in Fig.s \ref{xp1} and \ref{xp2}
(Fig. \ref{xe1} shows the distributions in terms of the
%(Figuress~\ref{xe1} and \ref{xe2} show the distributions in terms of the
$\xi_E$ variable).
Assuming the systematic error in the momentum region between 0.1 and
1 GeV$/c$
to be proportional to the total systematic error, we measure:
\begin{eqnarray}
%(-2)_{\mathrm{189\,GeV}}^{0.1 < p < 1 \, {\mathrm GeV}/c}
%& = & -0.50 \pm 0.26 \pm 0.39 (syst)\\
\left.\frac{}{2}\right|_{\mathrm{189\,GeV}}^{0.1 < p < 1 \, {\mathrm GeV}/c}
& = & 0.966 \pm 0.027 (stat) \pm 0.027 (syst) \\
%(-2)_{\mathrm{183\,GeV}}^{0.1 < p < 1 \, {\mathrm GeV}/c}
%& = & -1.12 \pm 0.41 (stat) \pm 0.34 (syst) \, .
\left.\frac{}{2}\right|
_{\mathrm{183\,GeV}}^{0.1 < p < 1 \, {\mathrm GeV}/c}
& = & 0.926 \pm 0.041 (stat) \pm 0.023 (syst) \, .
\end{eqnarray}
Other inclusive distributions
(rapidity and $P_T$ for example) taking into account the
orientation of particles could display a larger sensitivity
with respect to interconnection effects. Special care should be taken,
since the definition of the thrust axis could introduce a bias
between $(2q)$ and $(4q)$ events. To estimate the effect of this
bias, the following procedure was used. First the distributions of
rapidity and $p_T$ were computed as in the case of the momentum for
the $(2q)$ and the $(4q)$.
Then, a set $(4q)$-like events was
constructed by mixing pairs of $(2q)$ events.
For each real $(4q)$ event each W was replaced by a
W from a $(2q)$, flying in
the same direction and with the same momentum,
and with the two reconstructed jets lying in the same plane.
If there would be no bias from the definition of the event axis
(or if the Monte Carlo correction could correctly account for the bias),
there would be no difference between the
$y_T$ and $p_T$ distributions in the mixed sample and twice the $(2q)$
sample. This difference has thus been taken as an estimator of the
systematic error from the bias, and added in quadrature to
the $(4q)$ distribution, to twice the $(2q)$ distribution, and to
their difference.
The distributions of transverse momentum with respect to
the thrust axis is
shown in Fig. \ref{xp3} and \ref{xp4}.
One can notice that the possible deficit of events in the
$(4q)$ sample is concentrated in the low-$p_T$ region. The rapidity
distribution, instead, does not display any particular feature.
Finally,
the dispersion in $(4q)$ events is consistent
at both energies with $\sqrt{2}$
times the dispersion in $(2q)$ events:
\begin{eqnarray*}
\left(\frac{D^{(4q)}}{\sqrt{2}D^{(2q)}}\right)_{\mathrm{189\,GeV}} & = &
0.94 \pm 0.04 \mathrm{(stat)}\pm 0.08 \mathrm{(syst)} \\
\left(\frac{D^{(4q)}}{\sqrt{2}D^{(2q)}}\right)_{\mathrm{183\,GeV}} & = &
0.98 \pm 0.07 \mathrm{(stat)}\pm 0.05 \mathrm{(syst)} \, .\\
\end{eqnarray*}
In conclusion, we observe a possible depletion of the multiplicity in
fully hadronic WW events with respect to twice the semileptonic events;
this depletion is at the percent level,
%and it is consistent with a statistical fluctuation,
with a significance less than two standard deviations at 183 GeV
and less than one standard deviation at 189 GeV.
\section{Identified Particles from \boldmath$\ep\emi\ra\q\qb$ }
This section describes the results obtained for \idhadrons\ with data
recorded by DELPHI at LEPII. After the event selection for
identified charged particles at
energies up to 189 GeV has been described the additional criteria for hadron
identification are explained.
\subsection{\boldmath Event selection at 130 and 136 \gev}
After the hadronic preselection described above,
events with a reconstructed effective centre
of mass energy, \spr ,
of at least 85\% of the nominal centre-of-mass energy have been
used for further analysis. The distribution of \spr\
is shown for data and simulation at $130$~GeV and $136$~GeV in
fig.~\ref{fig:sprime133}.
Data recorded at these two energies have been combined and are
referred to as the 133 \gev\ sample.
The total of $\sim 12~\mathrm{pb^{-1}}$ recorded by DELPHI yields
$1387$ events while $1406$ are expected from simulation.
\subsection{\boldmath Event selection at 161 \gev}
\label{sec:evsel161}
Selected events must have a minimum of 8 and a maximum of 40 charged tracks, a reconstructed centre of
mass energy of at least 85\% of $\sqrt{s}$ and a visible energy of at
least 50\% of $\sqrt{s}$. A cut is imposed on the polar angle $\theta$ of the thrust axis to
select events well within the acceptance of the detector. Finally \Wp\Wm\ events have been partially removed.
It has been found that a selection based on the
\emph{minimal (or narrow) hemisphere broadening}, \Bmin, is effective and minimises the bias introduced on the remaining
event sample.
At threshold one expects $\sim 30$~ \Wp\Wm\ events. Selecting events with
$\Bmin\leq 0.12$, reduces the background by 50\%.
Using this selection 342 events are expected from
simulation, while 357 were selected from the data,
with an estimated remaining \Wp\Wm\ background of 15 events.
\subsection{\boldmath Event selection at 172 \protect\gev}
In addition to the criteria described so far an event has been selected if it
had at most 38 charged tracks and a $\Bmin \leq 0.1$. This lead to
267 selected events, with 264 expected from simulation out of which 36
are \W\W\ background.
\subsection{\boldmath Event selection at 183 and 189 \protect\gev}
The event selection for charged identified particles at 189 (183) GeV follows very closely the procedure already
described in section~3. Events with $\spr/\sqrt{s}$ above $0.9$ were
used. \W\W\ background was suppressed by demanding $\Bmin \leq 0.1
(0.08)$, more than $9 (8)$ charged particles with $p\geq
200$~MeV$/c$. A total of $3617 (1122)$ were selected with an expected
background from \Z\Z\ and \W\W\ of $789 (146)$ events.
\subsection{Selection of charged tracks for identification}
A further selection has been applied to the charged
track sample to obtain \emph{well
identifiable} tracks. Two different momentum regions are considered,
above and
below 700 MeV respectively, which correspond to the separation of samples
identified solely by the ionization loss in the Time Projection Chamber (TPC) and by RICHes and TPC together respectively.
Below 700 MeV tighter cuts are applied, namely to eliminate secondary
protons. There had to be least 30 wire hits in the TPC associated
with the track and the
measured track length had be to larger than 100 cm. In addition
at least two associated VD layer hits in $r\phi$ are required
and an impact
parameter in the $r\phi$ plane of less then $0.1$.
If there are less then two
associated layers in $z$ the corresponding impact parameter has to be
less than $1$ cm, else less then $0.1$ cm.
Tracks above $0.7$ GeV/c are required to have a measured track
length bigger than 30 cm and good RICH quality, e.g.~presence of
primary ionization in the veto regions. Only tracks which are
well contained in the barrel region of DELPHI~($\mid\cos(\theta)\mid\leq 0.7$)
are accepted.
\subsection{Analysis}
For an efficient identification of charged particles over the full
momentum region information from the ionization loss in the TPC
(``dE/dX'') and information from the DELPHI~RICH-detectors has been combined.
The dedicated RICFIX, RICALI and RPROCO software-packages have been used.
%which are described
%in~\cite{rproco,ricali}.
RICFIX fine-tunes the Monte Carlo
simulation
concerning detector related effects (such as slight fluctuations in pressures
and refractive indices, background arising from photon feedback, crosstalk
between the MWPC readout strips, $\delta$-rays, track ionization
photoelectrons, etc.). RPROCO derives likelihoods from the specific energy
loss, the number of reconstructed photons and the mean reconstructed
\v{C}erenkov angles respectively. The likelihoods are then multiplied
and rescaled to one. From these likelihoods a set of ``tags'' which
indicate the likeliness for a particular mass hypothesis (\pio , \K, and
\p) are derived. Throughout this analysis leptons have not been discerned
from pions. There contribution to the pion sample has been
subtracted using simulation.
A matrix inversion formalism~\cite{emile} has been used to calculate the true particle
rates in the detector from the tagged rates.
The $3\times 3$ efficiency matrix is defined by
\begin{equation}
\label{def:effmat}
\mathcal{E}_{xx}^{yy}=\frac{\textit{Number of xx tagged as
yy}}{\textit{Number of xx}}
\end{equation}
It establishes the connection between the true particles in the RICH/TPC
and the tagged ones:
\begin{equation}
\label{eq:effmat}
\left( \begin{array}{c}N_{\pi}^{meas} \\
N_{K}^{meas} \\
N_{p}^{meas} \end{array} \right) =
\mathbf{\mathcal{E}}
\left( \begin{array}{c}N_{\pi}^{true} \\
N_{K}^{true} \\
N_{p}^{true} \end{array} \right)
\end{equation}
The inverse of the efficiency matrix works on the three classes of tagged
particles in two ways. First a particle can have multiple tags, that
means the information from the tagging is ambiguous. This is not
unlikely because in this analysis one is forced by the low statistics of
the data samples to apply rather loose selection criteria. Secondly a
particle can escape identification. Both effects can be corrected by
this method.
\Kn\ and \La\ candidates have been detected by their decay in flight into
$\pi^+\pi^-$ and $p \pi^-$ respectively. Candidate secondary decays, $V^0$,
in the selected sample of hadronic events were found by considering
all pairs of oppositely charged particles. The vertex defined by each such
pair was determined such that the $\chi^2$ of the hypothesis of a common
vertex was minimised. The tracks were then refitted to the common vertex.
The selection criteria were the ``standard'' ones described in \cite{perfo}.
The average detection efficiency from this procedure is about 36\% for
\Kn $\rightarrow \pi^+\pi^-$ and about 28\% for
$\Lambda \rightarrow p \pi^-$
in multi-hadronic events. The background under the invariant mass peaks
was subtracted, separately for each bin of $V^0$ momentum,
by linearly interpolating two sidebands in invariant mass which correspond to:
\begin{itemize}
\item the regions between 0.40 and 0.45~$\mathrm{GeV}/c^2$ and between
0.55 and 0.60~$\mathrm{GeV}/c^2$ for the \Kn;
\item the regions between 1.08 and 1.10~$\mathrm{GeV}/c^2$ and between
1.14 and 1.18~$\mathrm{GeV}/c^2$ for the $\Lambda$.
\end{itemize}
\subsection{Calibration of the efficiency matrix using \Z\ data}
\label{sec:effmat}
The \Z\ calibration data recorded during each year have been used to tune
the above described matrix for applying it to high energy data.
This is made possible by
the careful studies of the spectra done at the \Z\ pole~\cite{emile}
The previous DELPHI measurement at the \Z\ pole has established that
the exclusive particle spectra are well reproduced by the DELPHI
tuning of the generators with very high accuracy. Therefore deviations
of the rates of tagged particles
between data and simulation in this sample can be
interpreted as pure detector effects.
Comparison of the tag rates allows a validation of the efficiency matrix.
This can be compared to the method used at the \Z -pole where
the ample statistics collected allowed to use
very pure samples of charged particles originating from
$\Lambda$'s and \Kn's to measure the
efficiency matrix properly.
Unfortunately this is not possible with the limited data samples
at LEP II. The assumption here is that the simulation
describes the calibration samples correctly. The matrix is corrected
so that it
reproduces the simulated rates, assuming that the
correction factors are linear. The
discrepancies have been found to be smaller than 4\%. This is taken
into account when calculating the systematic uncertainties.
%Fig.~\ref{fig:effmat183} shows the efficiency matrix from 183~GeV data.
%The variation of identification efficiency and the rate of misidentification
%as a function of particle momentum is clearly visible.
As the high energy events are recorded over a long time period,
stability of the identification devices becomes a major
concern. Variations in the refractive index or the drift velocity in
the RICH's, may
significantly change the performance of the identification.
%This variations
%can not be taken into account by the alignment and fixing procedure which
%are done using the calibration sample at the \Z\ pole which
% is recorded in 2 weeks at the very beginning of the data taking.
To estimate the effect of these variations on the measurement
the radiative \Z 's are very well suited.
%They have been used for the 183~GeV data.
These have been selected in the following way.
Events which passed the hadronic
preselection were required in addition to contain
at least 8 charged tracks, a reconstructed effective centre-of-mass
energy below 130 GeV, and a total energy transversal with respect to
the thrust axis of more than
30~GeV. Fig.~\ref{fig:fracradz0183} shows the good
agreement for differential cross sections for this event sample which
may be taken as an
indication of the stability of the detector during the year at the few
percent level.
\subsection{\xip\ distributions and average multiplicities}
After background subtraction the tagged particle fractions have been unfolded
using the calibrated matrix.
The full covariance matrix has been calculated for the tag rates using
multinomial statistics. It has then been propagated to the true rates of
identified particles using the unfolding matrix.
%As a first result
%normalised production rates can be derived, which correspond to
%differential production cross sections. An example is shown in Fig.~\ref{fig:fracradz0183}.
The \xip\ distribution was corrected bin by bin (for detector
acceptance and selection efficiency) using the full detector
simulation (DELSIM and JETSET). The corrected \xip\ distributions for
\idhadrons\ are shown in figs.~\ref{fig:pions}, \ref{fig:kaons}, \ref{fig:protons}, \ref{fig:xineut}
respectively. In the figures the predictions from the DELPHI tuned
versions of the following generators
JETSET 7.4, HERWIG 5.8 and ARIADNE~4.8 as well as a fit to
expression~\ref{eq:dgauss} are also shown. Within the statistics of
the data samples analysed, the shape of the \xip\ distributions is
well described by the generators.
The multiplicity of the identified final states per hadronic event was
obtained by integration of the corresponding \xip\ distributions. The
results are shown in table~\ref{tab:multi}. The numbers given for
charged identified hadrons include decay products from particles with
a lifetime $\tau < 10^{-9}$ s. These numbers are compared with the
predictions from JETSET 7.4, HERWIG 5.8 and ARIADNE 4.8.
The following sources of systematic uncertainties
are taken into account.
\begin{itemize}
\item Uncertainties due to particle identification.
They are mainly due to the uncertainties in the modelling of the detector response.
In addition there are time dependent effects such as variations of the
drift velocity in the RICH's.
The unfolding matrix has been adjusted using the
\Z -calibration data recorded in the beginning of data taking in 1996
and 1997, as well as from the peak period in 1995.
This is described in section~\ref{sec:effmat}. For 1997 data also
radiative return \Z -events have been used. These are better
suited as they have been recorded under the same conditions as the
signal events.
Spectra have been obtained using the original and the adjusted matrices.
The difference between the
results obtained has been taken as the corresponding
systematic uncertainty.
\item Size of the subtracted $WW$ background (where applicable).
Variation of the selection criteria results in a 10\% uncertainty on the
fraction of the $W$ ~contamination in the \q\qb\ sample.
This corresponds to a 5\% uncertainty in the $W$ cross
section and the size of the $W$ background has been varied accordingly.
The maximal variation observed in the distributions
has been taken as the corresponding systematic uncertainty.
\item Particles with momenta below 0.2 GeV/c or above 50 GeV/c have not been
identified. Their contribution is taken from the simulation.
For the invisible momentum region the predictions from each of the 3
generators for each particle species are
compared. The maximal difference is added in quadrature
to the uncertainty of each
particle type.
\item For the pions this analysis relies on a subtraction of the
lepton contamination using simulation.
An extra uncertainty of 10\% of the JETSET
prediction for the total number of leptons is added.
\end{itemize}
\begin{center}
\begin{table}
\begin{tabular}{| c | c | c c c c |}
\hline
\small$\sqrt{s}$~[\gev] & Particle & \multicolumn{4}{c |}{$< n >$} \\ \cline{3-6}
& & \small JETSET~7.4 & \small HERWIG~5.8 &
\small ARIADNE~4.8 & \normalsize Data \\
\hline
% $133$ & $$ & $23.83$ & $23.78$ & $23.76$ &
% $24.04\pm 0.25 \pm 0.09 $ \\
$133$ & \pipm & $19.90$ & $19.99$ & $19.64$ &
$19.84\pm 0.29 \pm 0.08$ \\
& \Kn & $ 2.40 $ & 2.64 & &
$2.51\pm 0.21 \pm 0.14 $ \\
& \Kpm & $2.37$ & $2.31$ & $2.30$ &
$2.60\pm 0.26 \pm 0.13 $ \\
& \ppba & $1.11$ & $1.09$ & $1.29$ &
$1.56 \pm 0.25 \pm 0.09$ \\
& \La & $ 0.34 $ & 0.49 & &
$ 0.50\pm 0.07\pm 0.05 $ \\
\hline
% $161 $ & $$ & $25.70$ & $25.44$ & $25.52$ &
% $ 25.46\pm 0.45 \pm 0.37 $ \\
$161$ & \pipm & $21.24$ & $12.76$ & $21.12$ &
$ 20.75\pm 0.58 \pm 0.24$ \\
& \Kn & $2.56 $ & 2.78 & &
$2.56\pm 0.33\pm 0.25 $ \\
& \Kpm & $2.63 $ & $2.44$ & $2.44$ &
$ 2.87\pm 0.55 \pm 0.16$ \\
& \ppba & $1.26$ & $0.96$ & $1.40$ &
$ 1.21 \pm 0.48 \pm 0.01$ \\
& \La & & & &
\\
\hline
% $172$ & $$ & $26.33$ & $26.06$ & $26.19$ &
% $26.52 \pm 0.53 \pm 0.54$ \\
$172$ & \pipm & $21.77$ & $22.31$ & $21.68$ &
$21.79 \pm 0.68 \pm 0.16 $ \\
& \Kn & & & &
\\
& \Kpm & $2.68$ & $2.48$ & $2.50$ &
$2.09 \pm 0.74 \pm 0.29$ \\
& \ppba & $1.30$ & $0.99$ & $1.45$ &
$1.78 \pm 0.73 \pm 0.25$ \\
& \La & & & &
\\
\hline
% $183$ & $$ & $26.93 $ & $26.64$ & $26.77$ &
% $26.58 \pm 0.24 \pm 0.54$ \\
$183$ & \pipm & $22.28$ & $22.82$ & $21.18$ &
$21.79 \pm 0.36 \pm 0.13$ \\
& \Kn & $2.46$ & & &
$ 1.81\pm 0.14\pm 0.18 $ \\
& \Kpm & $2.74$ & $2.53$ & $2.55$ &
$2.83 \pm 0.37 \pm 0.03 $ \\
& \ppba & $1.33$ & $1.00$ & $1.48$ &
$1.32 \pm 0.34 \pm 0.01 $ \\
& \La & $0.39 $ & & &
$ 0.33\pm 0.04\pm 0.04 $ \\
\hline
$189$ & \pipm & $22.56$ & $23.10$ & &
$22.19\pm 0.24\pm 0.12$ \\
& \Kn & $2.49$ & & &
$ 2.10\pm 0.12\pm 0.44 $ \\
& \Kpm & $2.77$ & $2.55$ & &
$3.15\pm 0.21\pm 0.04$ \\
& \ppba & $1.35$ & $1.02$ & &
$1.19\pm 0.17\pm 0.01$ \\
& \La & $0.39 $ & & &
$ 0.40\pm 0.03\pm 0.02 $ \\
\hline
\end{tabular}
\caption{Average multiplicities of charged particles and \idhadrons\
at 133, 161, 172 and 183 \gev. The first uncertainty is statistical
the second systematic.
%The values for the inclusive multiplicities have been taken from
% refs.~\cite{mult133,mult161,mult183}.
}\label{tab:multi}
\end{table}
\end{center}
In Fig.~\ref{fig:mulev} the results are compared to the predictions
from JETSET~7.4 and HERWIG~5.8. The
results shown for energies below 133~\gev (open squares)
were extracted from ref.~\cite{pdg94}.
\subsection{\xis\ and its evolution}
An interesting aspect of the \xip -distribution is
the evolution of its peak position \xis with increasing centre-of-mass
energy. It is determined by
fitting a parametrisation of the distribution to the peak region.
A possible such parametrisation is the distorted
Gaussian in eq.~(\ref{eq:dgauss}).
% The mean, standard deviation,
%kurtosis and skewness have been calculated as functions of
%$\Lambda_{\mathrm{QCD}}$ and constant $O(1)$ term by Fong and
%Webber~\cite{fongwebber}, in the limiting spectrum scenario. The
%higher order terms in $Y$ are neglected.
%\begin{eqnarray}
%\label{eq:dgpara}
%\overline{\xi}=\frac{1}{2}Y(1+\frac{\rho}{24}\sqrt{\frac{48}{\beta
% Y}})+O(1) \nonumber \\
%\sigma=\sqrt{\frac{1}{3}Y}(\frac{1}{48}\beta Y)^{\frac{1}{4}}
% (1 - \frac{\beta}{64}\sqrt{\frac{48}{\beta Y}}) +
% O(Y^{-\frac{1}{4}}) \nonumber \\
%s=-\frac{\rho}{16}\sqrt{\frac{3}{Y}}(\frac{48}{\beta Y})^{\frac{1}{4}}
% + O(Y^{-\frac{5}{4}}) \nonumber \\
%k=-\frac{27}{5Y}(\sqrt{\frac{\beta
% Y}{48}}-\frac{1}{24}\beta)+O(Y^{-\frac{3}{2}})
%\end{eqnarray}
%Here $\rho =11+2N_f/N_c^3$ and $\beta = 11-2N_f/N_c$.
Another parametrisation is a standard Gaussian distribution.
While being a more crude approximation, it facilitates the analysis in
the case of limited statistics. This simpler parametrisation was used as well
The fitted mean
has been identified with the peak position with the statistical uncertainty
taken from the fit.
Sincd eq.~(\ref{eq:dgauss}) is
expected to describe well only the peak region, the fit range has
to be carefully chosen around the peak. Unfortunately the fit sometimes fails
due to limited statistics. Table~\ref{tab:xistar} shows the results with
fit range and statistical as well as systematic uncertainties.
The systematic uncertainty has the following contributions
which are added in quadrature to the statistical uncertainty of either
fit.
%The procedure is the same for the Gaussian and distorted Gaussian fit.
\begin{itemize}
\item Uncertain size of the background subtracted (above the $W$ threshold).
By the same argument as for the estimation of the uncertainty for the
multiplicities the maximal difference obtained by a variation of the
$WW$ background cross-section is added to the systematic
uncertainty.
\item Uncertainty due to the particle identification.
The fit is repeated using the calibrated matrices as described for the
measurement of average multiplicities the maximal difference obtains
thus is added in quadrature
\item Stability of the fit within limited statistics and dependence on
fit range.
Finally there is an uncertainty which arises from the combination of
the limited statistics, the resulting need to choose a coarse binning,
and the choice of the fit range. To estimate this effect systematic
shifts have been imposed on the data by variation within the
statistical uncertainty.
One standard deviation has been added to the values left of the
peak and one standard deviation has been subtracted form the values to its
right and vice versa. The maximum variation is taken as the contribution to
the systematic uncertainty.
The maximal difference obtained in this way has again been added in
quadrature
\end{itemize}
Fig.~\ref{fig:xistar} shows the fitted $\xi^*$ values as a function of the
centre-of-mass energy. Using MLLA this has been
predicted~\cite{ochs,brummer} to be
\begin{equation}\label{eq:xistar}
\xi^* = Y ( \frac{1}{2}+a\sqrt{\frac{\as(Y)}{32N_c\pi}}-
a^2\frac{\as(Y)}{32N_c\pi}+F_h(\lambda)),
\end{equation}
where
\[ Y = \ln{E_{\mathit{beam}}/\Lambda}, E_{beam}=E_{cms}/2,
\as(Y)=\frac{1}{b_0 Y}, \\
a = \frac{11}{3}N_c+\frac{2n_f}{3N_c^2}, b_0 =
\frac{11N_c-2n_f}{12\pi}.
\]
Setting $N_c, n_f$ to 3 one left with \Leff\ and $F_h$
as only free
parameters. Due to uncertainties from higher order corrections \Leff\
cannot be identified with $\Lambda_{\overline{MS}}$. $F_h = F_h(\lambda)$, $\lambda =
\mathrm{log}\frac{Q_0}{\Leff}$, has been calculated~\cite{ochs}:
\begin{equation}
F_h (\lambda) = -1.46\lambda + 0.207\lambda^2 \pm 0.06.
\end{equation}
Although the results suffer from the small size of the data samples,
the results are qualitatively in good agreement with the expectations.
\begin{table}
\rotatebox{90}{
\begin{tabular}{| c | c | c c | c c | c c | c c | c c |}
\hline
{\scriptsize $\sqrt{s}$} & &
\multicolumn{2}{c|}{\small fit range} &
\multicolumn{2}{c|}{\small JETSET} & \multicolumn{2}{c|}{\small HERWIG} &
\multicolumn{2}{c|}{\small ARIADNE} & \multicolumn{2}{c|}{Data} \\
\hline
& & {\scriptsize Gauss.} & {\scriptsize dist.~G.}
& {\scriptsize Gauss.} & {\scriptsize dist.~G.}
& {\scriptsize Gauss.} & {\scriptsize dist.~G.}
& {\scriptsize Gauss.} & {\scriptsize dist.~G.}
& {\scriptsize Gauss.} & {\scriptsize dist.~G.} \\
\hline
133 & \pipm & $[3.3:4.9]$ & $[2.5:5.5]$
& 3.9 & 4.11
& 3.9 & 4.12
& 4.00 & 4.15
& $4.04\pm 0.03$ & $4.32 \pm 0.37$ \\
& \Kpm & $[2.1:3.3]$ & $[2.2:3.2]$
& 2.85 & 2.99
& 2.86 & 2.99
& 2.85 & 2.99
& $2.90 \pm 0.3$ & - \\
& \Kn & - & $[0.6:5.4]$
& & 2.87
& & 3.15
& &
& & $ 2.86 \pm 0.43 $ \\
& \ppba & $[2.1:3.3]$ & $ [2.5:5.5]$
& 3.04 & 3.09
& 3.04 & 3.10
& 3.04 & $ 3.04 $
& $2.74\pm 0.33$ & - \\
& \La & - & $[0.6:4.8]$
& & 2.79
& & 2.99
& &
& & $2.81\pm 0.66 $ \\
\hline
161 & \pipm & $[3.5:5.1]$ & $[2.6:4.5]$ & 4.09 & 4.17 & 4.03 &
4.08 & 4.12 & 4.18 & $ 4.11 \pm 0.05 $ & $ 4.14 \pm 0.48 $ \\
& \Kpm & $[2.1:3.7]$ & $[2.1:3.1]$ & 2.99 & 2.98 & 3.00 &
3.00 & 3.01 & 2.98 & $3.16\pm 0.5$ & $3.21\pm 2.25$ \\
& \Kn & & & &
& & & & & & \\
& \ppba &$[2.1:3.7]$ &$[2.2:3.1]$& 3.01 & 3.14 & 3.2 & -
&3.14 &3.06 & $ 3.08 \pm 0.84$ & - \\
& \La & & & & & &
& & & & \\
\hline
172 & \pipm & $[3.5:5.1]$ & $[2.6:4.5]$ & 4.13 & 4.2 & 4.06 &
4.12 & 4.16 & 4.21 & $4.11\pm 0.03$ & $4.45\pm 0.4$ \\
& \Kpm & $[2.1:3.7]$ & $[2.1:3.1]$ & 3.04 & 3.02 & 3.03 &
3.01 & 3.05 & 3.01 & $2.98\pm 0.84$ & - \\
& \Kn & & & & & &
& & & & \\
& \ppba & $[2.1:3.7]$ & $[2.2:3.1]$ & 3.16 & 3.14 &
3.23 & 3.41 & 3.18 & 3.12 & $ 3.55 \pm 1.1$ & - \\
& \La & &
& &
& &
& &
& & \\
\hline
183 & \pipm & $[2.6:4.5]$ & $[3.5:5.1]$
& 4.16 &4.43
& 4.10 &4.3
& 4.19 &4.44
& $4.23 \pm 0.03$ & $4.63\pm 0.2$ \\
& \Kpm & $[2.1:3.1]$ & $[2.1:3.7]$
& 3.08 & 3.18
& 3.06 & 3.15
& 3.1 & 3.23
& $3.08\pm 0.29$ & - \\
& \Kn & $[2.4:4.8]$ & $[1.8:5.4]$
& 3.09 & 3.19
& &
& &
& $3.33 \pm 0.16$ & $3.35\pm 0.12$ \\
& \ppba & $[2.3:3.1]$ & $[2.1:3.7]$ & 3.21 & 3.32 & 3.31
& - & 3.21 & 3.39 & $3.29\pm 0.18$ & $3.29\pm 0.52$ \\
& \La & $[2.4:4.8]$ & $[0.6:6.6]$
& 3.02 & 3.01
& &
& &
& $3.25 \pm 0.13$ & $3.41 \pm 0.73$ \\
\hline
189 & \pipm & $[2.6:4.5]$ &
& $4.22$ &
& &
& &
& $4.18\pm 0.1$& \\
& \Kn & $[2.4:4.8]$ & $[1.2:6.0]$
& 3.13 & 3.02
& &
& &
& $3.06 \pm 0.14$ & $3.31\pm 0.80$ \\
& \ppba & $[2.3:3.1]$ &
& $3.11$ &
& &
& &
&$3.18\pm 0.12$& \\
& \La & $[2.4:4.8]$ & $[1.2:4.8]$
& 3.02 & 2.98
& &
& &
& $3.18 \pm 0.12$ & $3.19 \pm 0.11$ \\
\hline
\end{tabular}
}
\caption{Values of \xis\ for \idhadrons\ at 133, 161, 172, 183 and 189
\gev . The uncertainties include all contributions.}\label{tab:xistar}
\end{table}
The comparison with eq.~(\ref{eq:xistar}) and previous measurements
taken from~\cite{brummer} is
shown in fig.~\ref{fig:xistar}.
The fit of the data points to expression~\ref{eq:dgauss}
was used to extract the peak position of
the $\xi_p$ distribution, $\xi^*$.
In Fig.~\ref{fig:xistar} the evolution
of $\xi^*$ with the centre-of-mass energy is presented.
The data up to centre-of-mass energies of
91~$\mathrm{GeV}$ were taken from previous measurements~\cite{brummer}.
The fit to expression~\ref{eq:xistar}
is superimposed
to the data points (solid line). Fig.~\ref{fig:xistar} shows that
(within the statistics of the data samples analysed) the fitted
functions follow the data points rather well. This suggests that
MLLA+LPHD gives a good description
of the observed particle spectra.
From table~\ref{tab:xistar} it is shown that there is fair agreement
between
the data and the predictions from the generator (JETSET~7.4).
\section{Identified hadrons in~\boldmath \W\W\ events}
A different event selection was used for the
analysis of identified hadron production at 189 \gev .
A feed forward neural net improves the separation of $W^+W^- \rightarrow q \overline{q} q \overline{q}$ from 2-fermion (mainly
$Z^{0}/\gamma \rightarrow q \overline{q} (g)$) and
4-fermion background (mainly $ZZ \rightarrow \mathrm{anything}$).
The network is based on the JETNET package, uses the back-propagation
algorithm and consists of three layers with 13 input, 7 hidden and one
output node. Input variables are different jet or event observables which are described in detail in~\cite{Wxsec}.
The samples for training and testing the feed forward net consist of
PYTHIA simulation and real data. The
training is performed with 3500 events from signal and
$Z^{0}$/$\gamma$ simulation. Afterwards the network output is calculated for
other independent samples of WW, $Z^{0}$/$\gamma$ and ZZ MC and all real
data as test samples.
The selection efficiency and the remaining background are determined from
the simulation.
The efficiency is found to be $ 89.0 \pm 0.16$(stat)\%.
\subsection{Average multiplicity}
\label{anamul}
After event selection the analysis proceeds exactly along the same
lines as described for the $\q\qb$ analysis. Fully corrected \xip
-distributions are obtained and afterwards integrated to calculate the
multiplicity. The results for the average multiplicity are
shown in table~\ref{tbmultip}
and they are compared to the predictions
from JETSET~7.4 and HERWIG~5.8.
\begin{center}
\begin{table}
\begin{tabular}{|l||c|c|c|c|} \hline
event & Particle & \multicolumn{3}{|c|}{$$} \\ \cline{3-5}
type & & (Data) & JETSET 7.4 & HERWIG 5.8 \\ \hline
$\mathrm{WW} \rightarrow q \bar q q \bar q$
& \pipm &$31.65\pm 0.48\pm 0.19$ & $31.74$ & $31.55$ \\
& \Kpm &$4.38\pm 0.42\pm 0.12$ & $4.05$ & $3.63$ \\
& \ppba &$1.82\pm 0.29\pm 0.16$ & $1.89$ & $1.66$ \\ \hline
$\mathrm{WW} \rightarrow q \bar ql\nu$
& \pipm &$15.51\pm 0.38\pm 0.16$& $15.90$ & $16.08$ \\
& \Kpm &$2.23\pm 0.32\pm 0.17$ & $2.02$ & $1.82$ \\
& \ppba &$0.94\pm 0.23\pm 0.06$ & $0.95$ & $0.83$ \\ \hline
\end{tabular}
\caption{Average multiplicity for
$\pi^\pm$,
K$^\pm$ and protons
%\pipm,
%\Kpm\ and protons
for WW events at 189 GeV.
In the data the first uncertainty is statistical, the second is systematic.}
\label{tbmultip}
\end{table}
\end{center}
\subsection{\xip\ distributions for identified particles at \boldmath$M_W$}
For comparisons it is more meaningful to use the \xip\ distributions from \W\
decays and their peak positions when the spectra have been boosted
back to the rest frame of the \W . The mass of the \W\ boson is the
relevant energy scale which enters in the \xip\ variable.
Fig.~\ref{fig:xiwwrest} shows the \xip\ distributions obtained from
semi-leptonic \W\ events. The boosting procedure relies on the simulation for
the part of the spectrum which was not accessible before
the boosting. The distributions should thus be taken with a grain of
salt, nevertheless this does not affect so much the peak region of the
distributions and allows to extract \xis .
\begin{table}
\begin{tabular}{|c | c | c | c |}
\hline
& Jetset & data & fit range \\
\hline
\pipm & $3.66$ & $3.65\pm 0.02\pm 0.24$ & $[2.7:4.5]$ \\
\Kpm & $2.70$ & $2.61\pm 0.09\pm 0.33$ & $[1.10:3.86]$ \\
\ppba & $2.73$ & $2.86\pm 0.11\pm 0.44$ & $[1.10:3.86]$ \\
\hline
\end{tabular}
\caption{Values for \xis\ for \pipm , \Kpm\ and protons from a
Gaussian fit in the indicated range. The fit was done on the
spectra which have been boosted back in the case of the
data. The first uncertainty is statistical the second systematic.}
\label{tab:xisww}
\end{table}
Table~\ref{tab:xisww} shows the values of \xis\ in the rest frame of
the \W\ boson. Data from semi-leptonic \W\ decays at 189 GeV were
used. They are compared to the JETSET prediction. The \xis\ value was
obtained by a Gaussian fit in the indicated region.
The systematical uncertainties were obtained by varying the the bins
in the fit range within two times their statistical uncertainty and by
adding the difference between the value as obtained by a distorted and
an undistorted Gaussian fit.
Within the limited statistics of this sample the data are in good
agreement with the generator prediction as well as with the prediction
from MLLA~\ref{eq:xistar}
\section{Summary}
The charged particle multiplicity
and dispersion in $q\bar{q}$ events were measured to be:
\begin{eqnarray}
_{\mathrm{189\,GeV}} = 26.97 \pm 0.17 (stat) \pm 0.37 (syst)\\
D_{\mathrm{189\,GeV}} = 8.71 \pm 0.13 (stat) \pm 0.24 (syst)\\
_{\mathrm{183\,GeV}} = 26.71 \pm 0.26 (stat) \pm 0.57 (syst)\\
D_{\mathrm{183\,GeV}} = 8.23 \pm 0.19 (stat) \pm 0.15 (syst) \, .
\end{eqnarray}
The multiplicity values
are consistent with the QCD prediction on the
multiplicity evolution and the ratios of the multiplicity
to the dispersion are consistent with
independence of centre-of-mass energy~\cite{kno}.
For WW events the measured multiplicities in fully hadronic
events are:
\begin{eqnarray}
_{\mathrm{189\,GeV}} & = & 38.32 \pm 0.32 (stat)
\pm 0.62 (syst)\\
_{\mathrm{183\,GeV}} & = & 37.79 \pm 0.56 (stat)
\pm 0.57 (syst) \, ,
\end{eqnarray}
while for semileptonic events one has:
\begin{eqnarray}
_{\mathrm{189\,GeV}} & = & 19.62 \pm 0.29 (stat)
\pm 0.52 (syst) \\
_{\mathrm{183\,GeV}} & = & 20.07 \pm 0.45 (stat)
\pm 0.48 (syst) \, .
\end{eqnarray}
The PYTHIA Monte Carlo program
%without colour reconnection effects,
with parameters tuned to the DELPHI data at LEP1, predicts
multiplicities of 38.2 and 19.1 for the fully hadronic and
semileptonic events respectively.
A possible depletion of the multiplicity in
fully hadronic WW events with respect to twice the semileptonic events
is observed:
\begin{eqnarray}
\left(\frac{}{2}\right)_{\mathrm{189\,GeV}} & = &
0.977 \pm 0.017 (stat) \pm 0.027 (syst) \\
\left(\frac{}{2}\right)_{\mathrm{183\,GeV}} & = &
0.941 \pm 0.025 (stat) \pm 0.023 (syst) \, .
\end{eqnarray}
The depletion is at the percent level and concentrated in the
low-$p$ (and low-$p_T$) region.
%(as predicted by most colour reconnection models).
No significant difference is observed between the dispersion
in fully hadronic events and $\sqrt{2}$ times the dispersion
in semileptonic events.
%If the
%differences in multiplicity are due to a statistical fluctuation, one
%can average the results to
%obtain the best determination of the charged
%particle multiplicity from hadronic decays of the W,
%and of the dispersion of the distribution. The multiplicity in the
%$(4q)$ case has to be divided by a factor of 2, while the dispersion
%has to be
%divided by a factor $\sqrt{2}$.
%Considering the
%systematic errors as independent (excluding the scale error),
%one finally has:
Assuming correlated systematic errors for the two energies we obtain:
\begin{eqnarray}
& = & 19.22 \pm 0.12 (stat) \pm 0.31 (syst) \\
D^{(W)}
& = & 6.12 \pm 0.10 (stat)\pm 0.24 (syst) \, .
\end{eqnarray}
The value of $$ is plotted in Fig.~\ref{mulea} at an energy value corresponding to the W mass.
An increase of 0.35 units is applied
as described in section 3.
%to account for the different proportion of events with a $b$ or a $c$ quark
%in W hadronic decays than in continuum $e^+e^-$ events \cite{dea}.
The measurement lies on the same curve as the neutral current data.
The value of
$/D^{(W)}$ $3.14 \pm 0.13(stat+syst)$,
plotted in Figure \ref{muleb}, is also consistent with the
$e^+e^-\rightarrow\gamma^{*}\rightarrow
{\mathrm q}\bar{\mathrm q}$ average.
The production of \idhadrons\
%\pipm ,\Kpm\ and protons
from $q \bar q$ and WW events at 189 GeV
has also been studied.
% using data taken with the DELPHI detector at LEP.
The results on the average multiplicity of identified particles
and on the position $\xi^*$ of the maximum of the
%and on the position \xis\ of the maximum of the
$\xi_p = -\mathrm{log} (\frac{2p}{\sqrt{s}})$
distribution have been compared with predictions
of JETSET and with calculations based on MLLA+LPHD
approximations.
Within their uncertainties the data are in good agreement with the
prediction from the generator as well as with the predictions based on
the analytical calculations in the MLLA framework.
\subsection*{Acknowledgements}
We are greatly indebted to our technical
collaborators, to the members of the CERN-SL Division for the excellent
performance of the LEP collider, and to the funding agencies for their
support in building and operating the DELPHI detector.\\
We acknowledge in particular the support of \\
Austrian Federal Ministry of Science and Traffics, GZ 616.364/2-III/2a/98, \\
FNRS--FWO, Belgium, \\
FINEP, CNPq, CAPES, FUJB and FAPERJ, Brazil, \\
Czech Ministry of Industry and Trade, GA CR 202/96/0450 and GA AVCR A1010521,\\
Danish Natural Research Council, \\
Commission of the European Communities (DG XII), \\
Direction des Sciences de la Mati$\grave{\mbox{\rm e}}$re, CEA, France, \\
Bundesministerium f$\ddot{\mbox{\rm u}}$r Bildung, Wissenschaft, Forschung
und Technologie, Germany,\\
General Secretariat for Research and Technology, Greece, \\
National Science Foundation (NWO) and Foundation for Research on Matter (FOM),
The Netherlands, \\
Norwegian Research Council, \\
State Committee for Scientific Research, Poland, 2P03B06015, 2P03B03311 and
SPUB/P03/178/98, \\
JNICT--Junta Nacional de Investiga\c{c}\~{a}o Cient\'{\i}fica
e Tecnol$\acute{\mbox{\rm o}}$gica, Portugal, \\
Vedecka grantova agentura MS SR, Slovakia, Nr. 95/5195/134, \\
Ministry of Science and Technology of the Republic of Slovenia, \\
CICYT, Spain, AEN96--1661 and AEN96-1681, \\
The Swedish Natural Science Research Council, \\
Particle Physics and Astronomy Research Council, UK, \\
Department of Energy, USA, DE--FG02--94ER40817. \\
\hfill
%=========================================================================%
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\newpage
\begin{figure}[th]
\centerline{DELPHI preliminary}
\resizebox{\textwidth}{14cm}{\includegraphics{evol189.eps}}
\caption[]{Measured average charged particle multiplicity
in $e^+e^- \rightarrow {\mathrm q}\bar{\mathrm q}$ events as a function
of centre-of-mass energy $\sqrt{s}$. DELPHI high energy results
are compared with other experimental results
and with a fit to a prediction from
QCD in Next to Leading Order.
The average charged multiplicity in W decays
is also shown at an energy corresponding to the W mass.
The measurements have been corrected for the different
proportions of $b\bar{b}$ and $c\bar{c}$ events at the various energies.}
\label{mulea}
\end{figure}
\begin{figure}[th]
\centerline{DELPHI preliminary}
\resizebox{\textwidth}{!}{\includegraphics{knofig.eps}}
\caption[]{Ratio of the average charge multiplicity to the dispersion
in $e^+e^-\rightarrow {\mathrm q}\bar{\mathrm q}$ events
at 183~GeV, compared with lower
energy measurements. The ratio in W decays
is also shown at an energy corresponding to the W mass.
The straight solid and dotted lines represent
the weighted average of the data points and its error.
The dashed line represents the prediction from QCD (see text).}
\label{muleb}
\end{figure}
\newpage
\begin{figure}[t]
\centerline{DELPHI preliminary}
\begin{center}
\resizebox{0.5\textwidth}{0.44\textheight}{\includegraphics{nch_4q_183.eps}}\resizebox{0.5\textwidth}{0.44\textheight}{\includegraphics{nch_2q_183.eps}}
\resizebox{0.5\textwidth}{0.44\textheight}{\includegraphics{nch_4q_189.eps}}\resizebox{0.5\textwidth}{0.44\textheight}{\includegraphics{nch_2q_189.eps}}
\end{center}
\caption[]{
Charged particle multiplicity distribution for (a) the $(4q)$ events
and (b) the $(2q)$ events at 183 GeV, for (c) the $(4q)$ events
and (d) the $(2q)$ events at 189 GeV.
The shaded areas represent the background
contribution; the histograms are the sum of the expected signal
and background.}
\label{wmul}
\end{figure}
%\newpage
%\begin{figure}[ph]
%\resizebox{!}{15 cm}{\includegraphics{effmat183.eps}}
%\caption{Efficiency matrix for data at 183 GeV. The picture indicates
% the momentum dependence of efficiency and purity of the tagged samples.}
%\label{fig:effmat183}
%\end{figure}
\newpage
\begin{figure}[ph]
\centerline{DELPHI preliminary}
\begin{center}
\resizebox{\textwidth}{0.88\textheight}{\includegraphics{ww_2w_p_e189_stat.eps}}
\end{center}
\caption[]{\label{xp1} (a)
Corrected momentum
distributions for $(4q)$ events (closed circles) and
$(2q)$ events (open circles), compared to simulation
without colour reconnection, at 189 GeV.
The difference between $(4q)$ and twice $(2q)$ is shown in (b).}
\end{figure}
\newpage
\begin{figure}[ph]
\centerline{DELPHI preliminary}
\begin{center}
\resizebox{\textwidth}{0.88\textheight}{\includegraphics{ww_2w_p_e183_stat.eps}}
\end{center}
\caption[]{\label{xp2} (a)
Corrected momentum
distributions for $(4q)$ events (closed circles) and
$(2q)$ events (open circles), compared to simulation
without colour reconnection, at 183 GeV.
The difference between $(4q)$ and twice $(2q)$ is shown in (b).}
\end{figure}
\newpage
\begin{figure}[ph]
\centerline{DELPHI preliminary}
\begin{center}
%\mbox{\epsfxsize7.0cm\epsfig{file=ww_2w_y_189.eps}}
%\mbox{\epsfxsize7.0cm\epsfig{file=ww_2w_y_183.eps}}
\resizebox{\textwidth}{0.45\textheight}{\includegraphics{ww_2w_xsi_e189_stat.eps}}
\resizebox{\textwidth}{0.45\textheight}{\includegraphics{ww_2w_xsi_e183_stat.eps}}
\end{center}
\caption[]{\label{xe1} (a)
Corrected
%rapidity and
$\xi_E$
distributions for $(4q)$ events (closed circles) and
the $(2q)$ events (open circles), compared to simulation
without colour reconnection.
The difference between $(4q)$ and twice $(2q)$ is shown in (b).}
\end{figure}
%\newpage
%
%\begin{figure}
%\begin{center}
%\mbox{\epsfxsize7.0cm\epsfig{file=ww_2w_xsiqcd_e189.eps}}
%\mbox{\epsfxsize7.0cm\epsfig{file=ww_2w_xsiqcd_e183.eps}}
%\end{center}
%\caption[]{\label{xe2} (a)
%Corrected
%$\xi_E$
%distributions,
% in the rest frame of the parent W scaled to the W boson mass
%for $(4q)$ events (closed circles) and
%the $(2q)$ events (open circles), compared to simulation
%without colour reconnection.
%The difference between $(4q)$ and twice $(2q)$ is shown in (b).}
%\end{figure}
\newpage
\begin{figure}[ph]
\centerline{DELPHI preliminary}
\begin{center}
\resizebox{\textwidth}{0.85\textheight}{\includegraphics{ww_2w_pt_e189.eps}}
\end{center}
\caption[]{\label{xp3} (a)
Corrected $p_T$
distributions for $(4q)$ events (closed circles) and
$(2q)$ events (open circles), compared to simulation
without colour reconnection, at 189 GeV.
The difference between $(4q)$ and twice $(2q)$ is shown in (b).}
\end{figure}
\newpage
\begin{figure}[ph]
\centerline{DELPHI preliminary}
\begin{center}
\resizebox{\textwidth}{0.85\textheight}{\includegraphics{ww_2w_pt_e183.eps}}
\end{center}
\caption[]{\label{xp4} (a)
Corrected $p_T$
distributions for $(4q)$ events (closed circles) and
$(2q)$ events (open circles), compared to simulation
without colour reconnection, at 183 GeV.
The difference between $(4q)$ and twice $(2q)$ is shown in (b).}
\end{figure}
\newpage
\begin{figure}[t]
\begin{center}
\resizebox{12cm}{9cm}{\includegraphics{sprime133.eps}}
\caption{Reconstructed effective centre-of-mass energy, \spr , for data from
130 and 136 GeV.}\label{fig:sprime133}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\centerline{\resizebox{12cm}{9cm}{\includegraphics{fracradz0183.eps}}}
\caption{Fractions of identified particles in radiative \protect\Z\
events from 183 GeV data. The data have been taken under the same
conditions as the signal data. The data are in good agreement with
the measurements from the \Z\ pole over the full momentum range.}\label{fig:fracradz0183}
\end{center}
\end{figure}
%
\newpage
%
\begin{figure}[ph]
\centerline{DELPHI preliminary}
\resizebox{0.5\textwidth}{!}{\includegraphics{xipi133.eps}}
\resizebox{0.5\textwidth}{!}{\includegraphics{xipi161.eps}}
\resizebox{0.5\textwidth}{!}{\includegraphics{xipi172.eps}}
\resizebox{0.5\textwidth}{!}{\includegraphics{xipi183.eps}}
\caption{\xip\ distributions (fully corrected) for \pio\ at 133 \gev,
161 \gev, 172 \gev\ and 183 \gev : data (points), simulation using
JETSET\ (dashed dotted line), HERWIG\ (dotted line), ARIADNE\
(dashed line). The full (dotted) curve shows a distorted Gaussian
(Gaussian) fit to the data. The shaded area indicates the acceptance corrected background subtracted from the data.}\label{fig:pions}
\end{figure}
\newpage
\begin{figure}[ph]
\centerline{DELPHI preliminary}
\resizebox{0.5\textwidth}{!}{\includegraphics{xika133.eps}}
\resizebox{0.5\textwidth}{!}{\includegraphics{xika161.eps}}
\resizebox{0.5\textwidth}{!}{\includegraphics{xika172.eps}}
\resizebox{0.5\textwidth}{!}{\includegraphics{xika183.eps}}
\caption{\xip\ distributions (fully corrected) for charged Kaons at 133 \gev,
161 \gev, 172 \gev\ and 183 \gev : data (points), simulation using
JETSET\ (dashed dotted line), HERWIG\ (dotted line), ARIADNE\
(dashed line). The dotted curves show a Gaussian
fit to the data. The shaded area indicates the acceptance corrected background subtracted from the data.}\label{fig:kaons}
\end{figure}
\newpage
%\begin{figure}[ph]
%\centerline{DELPHI preliminary}
%\begin{center}
%\resizebox{0.45\textwidth}{0.42\textheight}{\epsfig{file=k0@133.eps}}
%\resizebox{0.45\textwidth}{0.42\textheight}{\epsfig{file=k0@161.eps}}
%\resizebox{0.45\textwidth}{0.42\textheight}{\includegraphics{k0@183.eps}}
%\resizebox{0.45\textwidth}{0.42\textheight}{\includegraphics{k0@189.eps}}
%\end{center}
%\caption{$\xi_p$ distributions (efficiency corrected) for neutral kaons
% at 133~\gev , 161~\gev ,
%183~\gev (c)
%and 189~\gev (d):
% data (points), simulation using JETSET\ (dashed-dotted line) and HERWIG\
% (dotted line).
% The full curves show the fit of the data to the distorted Gaussian.}\label{fig:kn}
%\end{figure}
%\newpage
\begin{figure}[ph]
\centerline{DELPHI preliminary}
\resizebox{0.5\textwidth}{!}{\includegraphics{xipr133.eps}}
\resizebox{0.5\textwidth}{!}{\includegraphics{xipr161.eps}}
\resizebox{0.5\textwidth}{!}{\includegraphics{xipr172.eps}}
\resizebox{0.5\textwidth}{!}{\includegraphics{xipr183.eps}}
\caption{\xip\ distributions (fully corrected) for protons and anti-protons at 133 \gev,
161 \gev, 172 \gev\ and 183 \gev : data (points), simulation using
JETSET\ (dashed dotted line), HERWIG\ (dotted line), ARIADNE
(dashed line). The dotted curves show a Gaussian
fit to the data. The shaded area indicates the acceptance corrected background subtracted from the data.}\label{fig:protons}
\end{figure}
%\newpage
%\begin{figure}[ph]
%\centerline{DELPHI preliminary}
%\resizebox{0.5\textwidth}{0.45\textheight}{\includegraphics{l0@133.eps}}
%\resizebox{0.5\textwidth}{0.45\textheight}{\includegraphics{l0@161.eps}}
%\resizebox{0.5\textwidth}{0.45\textheight}{\includegraphics{l0@183.eps}}
%\resizebox{0.5\textwidth}{0.45\textheight}{\includegraphics{l0@189.eps}}
%\caption{$\xi_p$ distributions (efficiency corrected) for neutral Lambdas
% at 133~\gev , 161~\gev ,
%183~\gev
%and 189~\gev :
% data (points), simulation using JETSET (dashed-dotted line) and HERWIG
% (dotted line).
% The full curves show the fit of the data to the distorted Gaussian.}
% \label{fig:la}
%\end{figure}
%\begin{figure}
%\centerline{DELPHI preliminary}
% \centerline{\epsfig{file=xikaww189qqlin.eps,width=0.92\linewidth,height=10cm}}
% \centerline{\epsfig{file=xiprww189qqlin.eps,width=0.92\linewidth,height=10cm}}
% \caption{$\xi_p$ distributions (efficiency corrected) for charged kaons
% and protons in fully hadronic WW events 189~GeV:
% data (points), simulation using JETSET (dashed-dotted line).
% The full curves show the fit of the data to a Gaussian, the shaded area shows the subtracted background.}
% \label{xikawwqq}
%\end{figure}
%%
%\begin{figure}
%\centerline{DELPHI preliminary}
% \centerline{\epsfig{file=xikaww189ql.eps,width=0.92\linewidth,height=10cm}}
% \centerline{\epsfig{file=xiprww189ql.eps,width=0.92\linewidth,height=10cm}}
% \caption{$\xi_p$ distributions (efficiency corrected) for charged kaons
% and protons in semileptonic WW events at 189~GeV:
% data (points), simulation using JETSET (dashed-dotted line).
% The full curves show the fit of the data to a Gaussian, the shaded area shows the subtracted background.}
% \label{xikawwql}
%\end{figure}
%
\newpage
\begin{figure}[t]
\resizebox{\textwidth}{0.85\textheight}{\includegraphics{csi.eps}}
\caption{$\xi_p$ distributions (efficiency corrected) for neutral Lambdas
and neutral Kaons at 183~\gev and 189~\gev respectively, data (points), simulation using JETSET (dashed-dotted line.
The full curves show the fit of the data to a Gaussian and the dashed ones to a distorted Gaussian.}
\label{fig:xineut}
\end{figure}
\newpage
\begin{figure}[ph]
\centerline{DELPHI preliminary}
\resizebox{0.5\textwidth}{0.44\textheight}{\includegraphics{xich_wwql189rest_bw.eps}}
\resizebox{0.5\textwidth}{0.44\textheight}{\includegraphics{xipi_wwql189rest_bw.eps}}
\resizebox{0.5\textwidth}{0.44\textheight}{\includegraphics{xika_wwql189rest_bw.eps}}
\resizebox{0.5\textwidth}{0.44\textheight}{\includegraphics{xipr_wwql189rest_bw.eps}}
\caption{\xip\ distribtions (efficiency corrected) for charged
particles, pions, kaons and protons in semileptonic \W\W\ events at
189~GeV. Data (points) which have been boosted back to the \W\
rest-frame, compared to the prediction from JETSET (solid line). Only
the statistical uncertainties are shown. The dashed dotted line
shows a fit to eq.~(\protect\ref{eq:dgauss}).}
\label{fig:xiwwrest}
\end{figure}
\newpage
\begin{figure}[ph]
\resizebox{\textwidth}{0.85\textheight}{\includegraphics{mulidc.eps}}
\caption{Average multiplicity of \K (a), \Kn (b), p
(c) and \La (d)
as function of the centre-of-mass energy (black squares).
Simulation using JETSET/PYTHIA tuned to DELPHI data (solid line) and HERWIG~5.8
(dashed line) are superimposed.}\label{fig:mulev}
\end{figure}
\newpage
\begin{figure}[ph]
\resizebox{!}{0.9\textheight}{\rotatebox{90}{\includegraphics{xistar99_bw.eps}}}
\caption{Evolution of $\xi^*$ in $e^+e^-\rightarrow q\bar q$ with increasing center of mass
energy. This measurement is compared with previous measurements from
DELPHI from refs.~\protect\cite{schy} and lower energy
experiments~\protect\cite{brummer}. The solid line is a fit to
eq.~(\protect\ref{eq:xistar}). For better legibility the data points
have been shifted from the nominal energy.}\label{fig:xistar}
\end{figure}
\end{document}