\documentstyle[12pt,epsf,epsfig]{article}
\topmargin=-1cm
\oddsidemargin=0cm
\evensidemargin=0cm
\textwidth=16cm
\textheight=24cm
\raggedbottom
\sloppy
\begin{document}
%%% put your own definitions here:
\newcommand{\gevc}{$\mathrm{GeV/c}$ }
\newcommand{\rg}{\mbox{\( \mathrm{ C}_{5} {\mathrm F}_{12} \)}}
\newcommand{\rl}{\mbox{\( \mathrm{ C}_{6} {\mathrm F}_{14} \)}}
\newcommand{\zn}{\mbox{$ {\mathrm Z^0}$}~}
\newcommand{\pion}{\mbox{$ {\mathrm \pi}^{\pm} $}}
\newcommand{\Kn}{\mbox{$ {\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}}}$}}
\newcommand{\xp}{\mbox{$ {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}$}}
\newcommand{\ftmark}{\raisebox{-0.25em}{*}}
\begin{titlepage}
\pagenumbering{arabic}
\begin{tabular}{l r}
ICHEP'98 \#140 & \hspace{6cm} DELPHI 98-125 CONF 186 \\
Submitted to Pa 3, 5 & 22 June, 1998 \\
\hspace{2.4cm} Pl 4 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Production of \boldmath K$^\pm$, K$^0$, protons and
$\Lambda$s in q$\overline{\mbox{q}}$ and WW events at LEP~2}\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\normalsize {
%===================> DELPHI note author list =====> To be filled <=====%
{\bf P.~Abreu} $^1$,
{\bf A.~De~Angelis} $^2$,
{\bf R.~P.~Henriques} $^1$,
{\bf D.~Liko} $^{2,3}$,
{\bf N.~Neufeld} $^{2,3}$
%========================================================================%
}
\vspace*{1cm}
\end{center}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
The production of
charged and neutral kaons, protons and $\Lambda$s
at centre-of-mass energies above the Z$^0$
peak has been studied using data taken with the DELPHI detector at LEP.
The results on the average multiplicity of such 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 HERWIG, and with MLLA calculations.
%=========================================================================%
\end{abstract}
\vspace{\fill}
\begin{center}
Paper submitted to the ICHEP'98 Conference \\
Vancouver, July 22-29
\end{center}
\vspace{\fill}
\par {\footnotesize $^1$ LIP-IST-FCUL, Av. Elias Garcia, 14-1e, P-1000 Lisboa,
Portugal}
\par {\footnotesize $^2$ CERN, CH-1211 Geneva 23, Switzerland}
\par {\footnotesize $^3$ HEPHY, Osterr. Akad. d. Wissensch., Nikolsdorfergasse
18, AT-1050 Vienna, Austria}
\end{titlepage}
\pagebreak
%==================> DELPHI note text =====> To be filled <======%
\section{Introduction}
The way quarks and gluons transform into hadrons is complex and not entirely
understood by present theories; the most satisfactory description is
given by Monte Carlo simulations.
In the picture implemented in Monte Carlo simulations, the
hadronization of a $ q\overline{q} $ pair
is split into 3 phases. In a first phase,
gluon emission and parton branching of the original
$ q\overline{q} $ 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 $Q_0$, 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 $ e^{+}e^{-} $ 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 \zn 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 the particle yield 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 apply to hadrons as well.
The momentum spectra of particles produced can be calculated
as a function of the variable $\xi_p$, where
$\xi_p = -\mathrm{log} (\frac{2p}{\sqrt{s}})$
($p$ is the momentum of the particle and
$\sqrt{s}$ is the centre of mass energy):
\begin{equation}
\frac{1}{\sigma}\frac{d\sigma}{d\xi_p} = K_{LPHD}
\cdot f(\xi_p,X,\lambda)
\label{evol}
\end{equation}
with
\begin{eqnarray}
X= \mathrm{log} \frac{\sqrt{s}}{Q_{0}} & ; & \lambda =
\mathrm{log} \frac{Q_{0}}{\Lambda_{eff}} \, .
\end{eqnarray}
These MLLA+LPHD predictions involve three parameters: an effective scale
parameter $ \Lambda_{eff} $, a momentum
cut-off $Q_0$ in the evolution of the parton cascade
and an overall normalization factor $ K_{LPHD} $.
The function~\ref{evol}
has the form of a ``humped-backed plateau'', which is
approximately Gaussian in $\xi_p$~\cite{am:mlla}.
It can be approximated by a
distorted Gaussian over the entire spectrum,
as proposed in~\cite{dkt-1,bw:mlla}:
\begin{equation}\label{disga}
D(N,<\xi_p>,\sigma,\delta,s,k)
= \frac{N}{\sigma \sqrt{2 \pi}} \: \exp ({\frac{1}{8} k -
\frac{1}{2} s \delta - \frac{1}{4}(2+k) \delta^{2}
+ \frac{1}{6} s \delta^{3}
+ \frac{1}{24} k \delta^{4}})
\end{equation}
where $N$ is the average multiplicity,
$\delta = \frac{\xi_p -<\xi_p>}{\sigma}$ with $<\xi_p>$
being the mean of the $\xi_p$ distribution ($<\xi_p>$ coincides with the
peak position up to the next-to-leading order
only~\cite{ochs}), $\sigma$
is the width, $s$ the skewness and $k$ the kurtosis of the
distribution. For a pure Gaussian $s$ and $k$ vanish.
To check the validity of the MLLA+LPHD approach,
it is interesting to study the
evolution with the centre of mass energy of the maximum,
$\xi^*$, of the $\xi_p$ distribution. In the
framework of the MLLA+LPHD the dependence of $\xi^*$ on the centre
of mass energy can be expressed as~\cite{ochs,dkt-1,brummer}:
\begin{equation}
\xi^* = Y
\left(
\frac{1}{2} +\sqrt{C/Y}-C/Y
\right) +F_h (\lambda),
\label{evol2}
\end{equation}
where
\begin{displaymath}
Y = \mathrm{log}\frac{E_{beam}}{\Lambda_{eff}}~~,~~
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
flavors in the fragmentation process.
$F_h (\lambda)$ depends on the hadron type through the ratio
$\lambda = \mathrm{log}\frac{Q_0}{\Lambda_{eff}}$~\cite{ochs}:
\begin{equation}
F_h (\lambda) = -1.46\lambda + 0.207\lambda^2 \pm 0.06.
\end{equation}
In the present analysis the \k, \Kn, \p and
$\Lambda$\footnote{Unless
otherwise stated antiparticles are included as well.}
spectra
in $e^+e^-$ interactions recorded by DELPHI in the years 1995 to 1997
at energies above the \zn (around 133, 161, 172 and 183 $\mathrm{GeV}$)
were measured. W pair events, produced at an energy of 183 GeV were
selected as an independent sample.
The measurements were compared to the MLLA+LPHD calculations and to the
JETSET~7.4 and HERWIG~5.8 model predictions which use string
and cluster fragmentation respectively
\cite{lund,web:her}.
\section{Data Sample and Event Selection}
\label{sec:sample}
The DELPHI detector and its performance are
described in \cite{deldet,perfo}.
We used in the present analysis:
\begin{enumerate}
\item[(1a)] 6~$\mathrm{pb^{-1}}$ of data at centre-of-mass energy
around 130 $\mathrm{GeV}$ (3~$\mathrm{pb^{-1}}$ collected
during 1995 and another 3~$\mathrm{pb^{-1}}$
during 1997);
\item[(1b)] 6~$\mathrm{pb^{-1}}$ of data at centre-of-mass energy
around 136 $\mathrm{GeV}$ (3~$\mathrm{pb^{-1}}$
collected during 1995 and another 3~$\mathrm{pb^{-1}}$
during 1997);
\item[(2)] 9.96~$\mathrm{pb^{-1}}$ of data around
161~$\mathrm{GeV}$ (1996);
\item[(3)] 10.14~$\mathrm{pb^{-1}}$ of
data around 172~$\mathrm{GeV}$ (1996);
\item[(4)] 50.15~$\mathrm{pb^{-1}}$
of data around 183 $\mathrm{GeV}$ (1997).
\end{enumerate}
The samples (1a) and (1b), at 130 and 136 $\mathrm{GeV}$
respectively, were merged into a unique sample (1),
and attributed to the centre of mass energy of 133 $\mathrm{GeV}$.
A preselection of hadronic events in both samples was made, requiring
at least 6
charged particles with momentum $p$ above 400 MeV/$c$,
angle $\theta$ with respect to the beam direction
between 20$^\circ$ and 160$^\circ$,
a track length of at least 30 cm,
a distance of closest approach to the interaction point
less than 4 cm in the plane
perpendicular to the beam axis and
less than 10 cm along the beam axis, and a
total energy of the charged particles above 0.15
times the centre-of-mass energy $\sqrt{s}$.
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/$c$,
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 JETSET~7.4~\cite{lund},
with parameters tuned to fit LEP1 data in DELPHI \cite{tuning}.
The particles were followed through
the detailed geometry of DELPHI, giving simulated digitizations in each
sub-detector. These data were processed with the same
reconstruction and analysis programs as the real data.
The cross-section for $e^+e^- \rightarrow {\mathrm q}\bar{\mathrm q}(\gamma)$
above
the \zn 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 are not detected.
In order to compute the hadronic
centre-of-mass energy, the method described in \cite{sprime} was used.
The procedure clusters the particles into two jets using
the Durham algorithm~\cite{durham}, excluding candidate
ISR photons.
Assuming an ISR photon along the beam pipe if
no candidate ISR photon has been detected 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, $\sqrt{s^\prime}$, is calculated as the invariant
mass of the system recoiling from the ISR photon.
The method used to obtain the hadronic centre-of-mass energy overestimates the
true energy in the case of double hard radiation in the initial
state. For instance, if the two ISR photons are emitted back to back,
the remaining two jets may also
be back to back, but with energy much smaller than the beam energy.
Cutting on the total
energy measured in the detector
reduces the contamination from such events.
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
$\mathrm{W^+W^-}$ pairs
(table~\ref{tb1}).
For a centre of mass energy of 183~GeV, were the largest amount of data has
been collected, $\mathrm{W^+W^-}$ pairs have been selected
independently.
\begin{table}
\begin{center}
\begin{tabular}{|l||c|c|} \hline
$\sqrt{s}$ ($\mathrm{GeV}$) & $\sigma^{(h)}_{qq}$ ($\mathrm{pb^{-1}}$) &
$\sigma_{\mathrm{W^+W^-}}$ ($\mathrm{pb^{-1}}$) \\ \hline
133 & 74 & -- \\
161 & 35 & 3 \\
172 & 29 & 12 \\
183 & 25 & 16 \\ \hline
\end{tabular}
\end{center}
\caption{Hadronic cross section $\sigma^{(h)}_{qq}$
for effective centre-of-mass energy $ > 0.85 E_{cm}$
(from ZFITTER) compared to the $\mathrm{W^+W^-}$ cross section
$\sigma_{\mathrm{W^+W^-}}$.}
\label{tb1}
\end{table}
\subsection{Hadronic Selection at 133 $\mathrm{GeV}$}
Events with reconstructed hadronic centre-of-mass energy
($\sqrt{s'}$) above 122~$\mathrm{GeV}$, with total energy seen in the
detector above 0.15$\sqrt{s}$,
and with at least 7 charged particles with momentum
above 100~$\mathrm{MeV}/c$, were used.
A total of 830 events were selected in the data (790 $q\overline{q}$
predicted by the simulation).
\subsection{Hadronic Selection at 161 $\mathrm{GeV}$}
Events with reconstructed hadronic centre-of-mass energy
($\sqrt{s'}$) above 150~$\mathrm{GeV}$, with total energy seen in the
detector above
0.2$\sqrt{s}$, and with at least 9 charged particles with momentum
above 100~$\mathrm{MeV}/c^2$, were used.
A total of 326 events were selected (311
predicted by the simulation;
the estimated background from $\mathrm{W^+W^-}$
was 14 events).
\subsection{Hadronic Selection at 172 $\mathrm{GeV}$}
Events with reconstructed hadronic centre-of-mass energy
($\sqrt{s'}$) above 155~$\mathrm{GeV}$, with total energy seen in the
detector above
0.2$\sqrt{s}$, with at least 10 charged particles with momentum
above 100~$\mathrm{MeV}/c^2$,
and with narrow jet broadening\footnote{The narrow jet
broadening ($B$)
is defined as follows. For each hemisphere $j$ = 1, 2 with respect to
a plane perpendicular to the thrust axis:
$B_j = \frac{1}{2\cdot p_{tot}}\sum_i p_i \cdot sin \theta_{i-T}$,
where $p_{tot}$ is the sum of the
moduli of the momenta
of all the particles in the $j$-th hemisphere, $p_i$ is the
momentum of the particle $i$ and $\theta_{i-T}$ is the angle
between the direction of the particle and the thrust line.
The narrow jet broadening is the minimum between $B_1$ and $B_2$.}
smaller than 0.1 were used.
A total of 212 events were selected
(202 predicted by the simulation;
the estimated background from $\mathrm{W^+W^-}$
was 5 events).
\subsection{Hadronic Selection at 183 $\mathrm{GeV}$}
Events with reconstructed hadronic centre-of-mass energy
($\sqrt{s'}$) above 160~$\mathrm{GeV}$, with total energy seen in the
detector above
0.2$\sqrt{s}$, with at least 10 charged particles with momentum
above 100~$\mathrm{MeV}/c^2$, and with narrow jet broadening
smaller than 0.1 were used.
A total of 976 events were selected (1022
predicted by the simulation;
the estimated background from $\mathrm{W^+W^-}$
was 51 events).
\subsection{WW Event Selection at 183 GeV}
Events containing a W pair were selected following the procedure described
in~\cite{Wxsec}. To select events $W^+W^- \rightarrow q \bar q Q \bar Q$, jets
were reconstructed using the LUCLUS jet clustering algorithm with $d_{join}$
value of 6.5 GeV. Events with a reconstructed hadronic centre-of-mass
energy ( $\sqrt{s'}$ ) above 120~GeV
and at least 4 jets with 4 particles in each jet were selected.
These events were then forced in a 4 jet configuration. QCD background was reduced
by selecting those events fulfilling the cut
$D = ( E_{min}/E_{max} ) \theta_{min} / ( E_{max} - E_{min} ) < 0.0045$,
where $\theta_{min}$ is the minimum inter-jet angle and $E_{max}$ and
$E_{min}$ are the maximum and minimum jet energy respectively.
Events $W^+W^- \rightarrow q \bar q l \nu$ were selected by requiring
the presence of a hadronic system with large invariant mass, large missing
momentum pointing away from the beam pipe and either one energetic isolated
charged track or a low multiplicity jet. To reduce further the QCD background
contributions in the electron and tau channel a cut on the acoplanarity was
applied. These criteria gave 368 $W^+W^- \rightarrow q \bar q Q \bar Q$
candidates and 211 $W^+W^- \rightarrow q \bar q l \nu$ candidates.
(365 and 210 predicted by simulation; the estimated purity of the samples was
$80\pm 2$~\% and $95\pm 0.3$~\% respectively.)
\section{Analysis}
For the measurement of \k and \p, a tagging procedure
based on the combination of
the Cherenkov angle measurement in the RICH detector and the
ionization energy loss ($dE/dx$) in the TPC was applied.
The $dE/dx$ information was used for momenta below 0.7 \gevc for \k
(where no RICH information is available) and
up to 1.2 \gevc for proton identification.
%Due to the better
%resolution and separation between the particular expectation curves, the
%tagging performance of the RICH is higher compared to the $dE/dx$. Thus in
In the remaining momentum range the tagging was performed with the RICH
measurement. The analysis is restricted to the barrel RICH region
($41^{\circ} \le \, \theta_{track} \, \le 139^{\circ}$).
The RICH hadron identification was based on three standard (DELPHI-RICH)
software-packages: RIBMEAN, RICFIX and
NEWTAG~\cite{newtag}. RIBMEAN calculates an average Cherenkov angle
for the liquid and the gas radiator by application of a
clustering algorithm and simultaneously links a quality flag to each track
passing through the RICH. RICFIX corrects the RICH data and Monte Carlo
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.). In this analysis the standard NEWTAG
(RICH) selections were
applied to select charged kaons and protons.
%In the momentum range below
%0.9 \gevc the clearly separated bands
%corresponding to
%electron, pion, kaon and proton in the $dE/dx$ versus momentum
%were used for identification (muons cannot be distinguished from
%pions). Detailed calibration has been pursued as described in~\cite{DaRe}.
The efficiency was estimated from full detector simulation, and it is
$\sim$~56\% ($\sim$~46\%)
with a purity of $\sim$~75\% ($\sim$~92\%) for \k (\p).
\Kn and $\Lambda$ candidates were 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 minimized. 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 multihadronic 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}
The procedure described above relies on the Monte Carlo simulation to obtain
purity and efficiency of the particle identification. As a crosscheck also
$Z^0$ events, taken at the beginning of the datataking period, were used to
calibrate the detector performance. A matrix algorithm was then applied to
unfold the observed rates. This method profits from the calibration of the
detector performance on real data, but is difficult to apply to a low
statistic sample. At energies of 130 and 183 GeV compatible results have
been obtained.
\section {Results }
The $\xi_p$ distribution after background subtraction was corrected bin by bin
(for the detector acceptance and selection efficiency)
using the
simulation (JETSET).
The corrected
$\xi_p$ distributions for charged kaons, neutral kaons, protons and $\Lambda$s
at the various energies are shown in figure~\ref{xi1}, figure~\ref{xi1a},
figure~\ref{xi2} and figure~\ref{xi2a} respectively.
In the figures
the predictions from the generators JETSET~7.4 and HERWIG~5.8,
and the fit to expression~\ref{disga}, are also shown. In the fit to the data
distributions,
$s$ and $k$ were fixed to the values
obtained by fitting equation~\ref{disga}
to the corresponding JETSET~7.4 Monte Carlo distributions.
Within the statistics of
the data samples analysed, the shape of the $\xi_p$ distribution is well
described by both generators, JETSET~7.4 and HERWIG~5.8.
\subsection{$\xi^*$ distribution}
The fit of the data points to expression~\ref{disga}
was used to extract the peak position of
the $\xi_p$ distribution, $\xi^*$.
In figure~\ref{xista} and table~\ref{tbxis} the results on the evolution
of $\xi^*$ with the centre of mass energy are 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{evol2}, where $F_h$ ($Q_0$) was
taken as a free parameter and $\Lambda_{eff}$ was fixed at 150~$\mathrm{MeV}$
(this value of $\Lambda_{eff}$ comes from the description of the pion
spectra with
$\Lambda_{eff} = Q_0$, the limiting spectrum~\cite{dok:mlla}),
is superimposed
to the data points (solid line). Figure~\ref{xista} 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{tbxis}
and figure~\ref{xista} it is shown that there is good agreement
between
the data and the predictions from the generators JETSET~7.4
and HERWIG~5.8.
The systematic uncertainties on $\xi^*$ were obtained as the sum in
quadrature of the following contributions:
\begin{itemize}
\item
From the original $dN/d\xi_p$ distribution
two additional ``distorted'' distributions were
obtained: one by adding 1 standard deviation to the
values below $\xi^*$ and subtracting 1 standard deviation
to the values above, and the
other by subtracting 1 standard deviation to the
values below $\xi^*$ and adding 1 standard deviation
to the values above. From these
distributions, two new values were obtained for $\xi^*$. These were
compared to the values obtained from the normal distribution, and the larger
difference was taken to be systematic uncertainty.
\item The difference between the maximum using a distorted Gaussian fit
and the maximum using a simple Gaussian fit.
\end{itemize}
The parameter $Q_0$
obtained from the fit (figure~\ref{xista})
is presented in table~\ref{mer4}. It is observed that the $Q_0$ values
obtained for the different particles are consistent within the uncertainty,
although there is a systematic rise
of $Q_0$ with the particle mass (apart from the proton results).
%The values obtained for $F_h$ agree within the errors
%with the results from~\cite{brummer}, in which data with center of mass
%energies varying from 14~$\mathrm{GeV}$ to 91~$\mathrm{GeV}$ was used.
%This $\Lambda_{eff}$ corresponds to a value of $\alpha_s \sim 0.1$ when
%plugged in the evolution equation at leading order~\cite{brummer}:
%
%\begin{equation}
%\frac{\alpha_s}{2\pi} = \frac{1}{b\cdot
%\mathrm{log}(E_{beam}/\Lambda_{eff})}
%\end{equation}
%where
%\begin{equation}
%b = 11N_c/3 - 2n_f/3,
%\end{equation}
%$N_c$ is the number of colours and $n_f$ the active number of quark flavors in
%the fragmentation process.
%This suggests that MLLA+LPHD could possibly give a quantitative description
%of the observed particle spectra.
%The MLLA+LPHD approach is presented here as an
%attempt to understand some properties of multihadron production in
%terms of perturbative QCD, namely to test the possibility within this framework
%to determine hadron properties based on the corresponding parton distributions.
%It was shown that this approach can give predictions for properties of the
%momentum spectra that can be compared with experimental data.
\begin{table}
\begin{center}
\begin{tabular}{|l||c|c|c|c|c|} \hline
$\sqrt{s}$ & Particle & Fitting & \multicolumn{3}{|c|}{$\xi^*$} \\ \cline{4-6}
($\mathrm{GeV}$)& & range ($\mathrm{GeV}$) & (Data)& JETSET 7.4 &
HERWIG 5.8 \\ \hline
133 & \k & 1.2-5.4 &$2.95\pm0.06\pm0.13$&$2.84$&$3.12$\\
& $K^{0}$ & 0.6-5.4 &$2.86\pm0.14\pm0.41$&$2.87$&$3.15$\\
& p & 0.0-4.8 &$2.92\pm0.08\pm0.14$&$2.98$&$3.10$\\
& $\Lambda$ & 0.6-4.8 &$2.81\pm0.24\pm0.62$&$2.79$&$2.99$\\ \hline
161 & \k & 1.2-4.8 &$3.12\pm0.22\pm0.25$&$2.96$&$3.25$\\
& $K^{0}$ & -& - & - &-\\
& \p & 1.2-4.8 &$3.02\pm0.17\pm0.26$&$3.08$&$3.23$\\
& $\Lambda$ & -& - & - & -\\ \hline
172 & \k & 1.2-5.4 &$3.10\pm0.09\pm0.37$&$3.02$&$3.27$\\
& $K^{0}$ & - & - & - &-\\
& \p & 0.6-4.8 &$3.17\pm0.11\pm0.82$&$3.12$&$3.26$\\
& $\Lambda$ & - & - & - &-\\ \hline
183 & \k & 1.2-5.4 &$2.95\pm0.11\pm0.27$&$3.01$&$3.30$\\
& $K^{0}$ & 0.6-5.4 &$3.08\pm0.15\pm0.30$&$3.07$&$3.34$\\
& \p & 0.6-4.8 &$3.16\pm0.08\pm0.16$&$3.15$&$3.30$\\
& $\Lambda$ & 0.6-4.2 &$2.84\pm0.14\pm0.38$&$2.99$&$3.20$\\ \hline
$WW \rightarrow q\bar qQ\bar Q$
& \k & 1.3-5.4 &$3.03\pm 0.12\pm0.27$&$3.38$& - \\
& \p & 1.3-5.4 &$3.33\pm 0.20\pm0.43$&$3.39$& - \\ \hline
$WW \rightarrow q\bar ql\nu$
& \k & 1.3-5.4 &$3.40\pm 0.25\pm0.60$&$3.42$& - \\
& \p & 1.3-5.4 &$3.50\pm 0.26\pm0.50$&$3.41$& - \\ \hline
\end{tabular}
\end{center}
\caption{$\xi^*$ K$^\pm$, \Kn, protons
and $\Lambda$ in hadronic events at 133~$\mathrm{GeV}$, 161~$\mathrm{GeV}$,
172~$\mathrm{GeV}$ and 183~$\mathrm{GeV}$ and WW events at 183 GeV. In the
data the first uncertainty is statistical, the second is systematic.}
\label{tbxis}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|l||c|c|} \hline
Hadron & $Q_0$ & $\chi^2$ \\ \hline
\k & $0.325\pm0.009$& 2.5 \\ \hline
\Kn & $0.330\pm0.012$& 0.9 \\ \hline
\p & $0.314\pm0.011$& 0.5 \\ \hline
$\Lambda$ & $0.343\pm0.012$& 0.6 \\ \hline
\end{tabular}
\end{center}
\caption{\label{mer4} Results of the fit of the evolution
of $\xi^*$ with the centre of mass energy.}
\end{table}
\subsection{Average multiplicity}
\label{anamul}
The multiplicity of the identified final states
per hadronic event was obtained from the integration of the
distributions shown in figures~\ref{xi1} to~\ref{xi2a}
inside a range varying according to the particle type and
energy; outside this range the fraction of particles
was extrapolated using the JETSET~7.4 prediction.
The results are
shown in figure~\ref{xi3} and table~\ref{tbmultip}
and they are compared with the predictions
from JETSET~7.4 and HERWIG~5.8. In figure~\ref{xi3} the
results shown for energies below 133~$\mathrm{GeV}$ (open squares)
were extracted from~\cite{pdg94}.
\begin{table}
\begin{center}
\begin{tabular}{|l||c|c|c|c|c|} \hline
$\sqrt{s}$ & Particle & Integration &
\multicolumn{3}{|c|}{$$} \\ \cline{4-6}
($\mathrm{GeV}$)& & range ($\mathrm{GeV}$) & (Data)& JETSET 7.4 &
HERWIG 5.8 \\ \hline
133 & \k & 1.2-5.4 &$2.58\pm0.10\pm0.13$ & $2.49$&$2.74$\\
& $K^{0}$ & 0.6-5.4 &$2.51\pm0.21\pm0.14$ & $2.40$&$2.64$\\
& \p & 0.0-4.8 &$1.25\pm0.08\pm0.03$ & $1.15$&$1.08$\\
& $\Lambda$ & 0.6-4.8 &$0.50\pm0.07\pm0.05$ & $0.34$& $0.49$\\ \hline
161 & \k & 1.2-4.8 &$2.34\pm0.18\pm0.20$ & $2.63$&$2.90$\\
& $K^{0}$ & 0.6-4.2 &$2.56\pm0.38\pm0.25$ & $2.56$&$2.78$\\
& \p & 1.2-4.8 &$1.13\pm0.12\pm0.05$ & $1.25$&$1.19$\\
& $\Lambda$ & -& - & - &-\\ \hline
172 & \k & 1.2-5.4 &$2.95\pm0.23\pm0.17$ & $2.68$&$2.97$\\
& $K^{0}$ & - & - & - &-\\
& \p & 0.6-4.8 &$1.41\pm0.17\pm0.04$ & $1.30$&$1.22$\\
& $\Lambda$ & - & - & - &-\\ \hline
183 & \k & 1.2-5.4 &$2.87\pm0.12\pm0.17$ & $2.74$&$3.02$\\
& $K^{0}$ & 0.6-5.4 &$2.09\pm0.18\pm0.16$ & $2.66$&$2.91$\\
& \p & 0.6-4.8 &$1.32\pm0.09\pm0.03$ & $1.33$&$1.24$\\
& $\Lambda$ & 0.6-4.2 &$0.42\pm0.06\pm0.09$ & $0.39$&$0.55$\\ \hline
$WW \rightarrow q \bar q Q \bar Q$
& \k & 1.3-5.4 &$4.32\pm 0.48\pm 0.21$ & $4.04$& - \\
& \p & 1.3-5.4 &$2.22\pm 0.51\pm 0.06$ & $1.89$& - \\ \hline
$WW \rightarrow q \bar ql\nu$
& \k & 1.3-5.4 &$1.79\pm 0.41\pm 0.11$ & $2.02$& - \\
& \p & 1.3-5.4 &$1.09\pm 0.33\pm 0.03$ & $0.94$& - \\ \hline
\end{tabular}
\end{center}
\caption{Average multiplicity for K$^\pm$, \Kn, protons
and $\Lambda$ for hadronic events at 133~$\mathrm{GeV}$, 161~$\mathrm{GeV}$,
172~$\mathrm{GeV}$ and 183~$\mathrm{GeV}$ and WW events at 183 GeV.
In the data the first uncertainty is statistical, the second is systematic.}
\label{tbmultip}
\end{table}
The systematic uncertainties on the multiplicity were obtained as the sum in
quadrature of the following contributions:
\begin{itemize}
\item
The difference between the JETSET and HERWIG predictions
in the unseen region.
\item The uncertainty coming from
the particle identification.
A relative systematic uncertainty of 2\% was
estimated for protons and charged kaons (from the comparison between
the standard and the tight NEWTAG selections), as applied in~\cite{schy}.
A relative uncertainty of 3\% was estimated for
\Kn~\cite{al2} and 5\% for
$\Lambda$~\cite{al3}.
\item For the ${W^+W^-}$ sample the contribution due to the uncertainty in
the QCD background was included.
\end{itemize}
The Monte Carlo programs JETSET~7.4 and HERWIG~5.8
display in general a fair agreement with the
data. However the \Kn multiplicity at 183~$\mathrm{GeV}$ is
3.4 standard deviations (2.4 standard deviations)
below the HERWIG (JETSET) predictions;
HERWIG underestimates the results for protons at 133~$\mathrm{GeV}$ and
overestimates the results for charged kaons at 161~$\mathrm{GeV}$ by
2 standard deviations.
\section {Conclusions }
The production of \k , \Kn, p and $\Lambda$
at centre-of-mass energies above the \zn
peak has been studied using data taken with the DELPHI detector at LEP.
The results on the average multiplicity of such 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 HERWIG, and with calculations based on MLLA+LPHD
approximations.
The Monte Carlo programs JETSET~7.4 and HERWIG~5.8
display in general a fair agreement with the
data. However the measured \Kn multiplicity at 183~$\mathrm{GeV}$ is
3.4 standard deviations (2.4 standard deviations)
below the HERWIG (JETSET) predictions;
HERWIG underestimates the results for protons at 133~$\mathrm{GeV}$ and
overestimates the results for charged kaons at 161~$\mathrm{GeV}$ by
2 standard deviations.
For the $\mathrm{W^+W^-}$ sample the available statistic is small. A fair
agreement with the prediction of the generator has been found in both
multiplicities and $\xi^*$.
Within the statistics of
the data samples analysed the shape of the $\xi_p$ distribution is well
described by both generators, JETSET and HERWIG.
The parameter $Q_0$
was estimated. It is observed that the $Q_0$ values
obtained for the different particles are consistent within the uncertainties,
although there is a systematic rise
of $Q_0$ with the particle mass (excluding the proton results).
%The parameter $\Lambda_{eff} \sim 0.07$ was obtained from the data by
%fitting the evolution of $\xi^*$ versus $\sqrt{s}$ to the MLLA+LPHD
%predictions. This $\Lambda_{eff}$ corresponds to a value of
%$\alpha_s \sim 0.1$ when
%plugged in the evolution equation at leading order.
%This is in line with the idea
%that MLLA+LPHD could possibly give a quantitative description
%of the observed particle spectra.
\subsection*{Acknowledgements}
We are greatly indebted to our technical collaborators and to the funding
agencies for their support in building and operating the DELPHI detector.
Very special thanks are due to the members of the CERN-SL Division for the
excellent performance of the LEP collider. We are grateful to V. Khoze for
useful discussions.
\newpage
\begin{thebibliography}{99}
\def\Abreu{DELPHI Coll., P. Abreu et al.,\ }
\bibitem{ochs} V.A. Khoze and W. Ochs,
Int. J. Mod. Phys. {\bf A12} (1997) 2949.
\bibitem{dok:mlla}
Y.I. Azimov et al.,
Z. Phys. {\bf C27} (1985) 65 and ibid. {\bf C31} (1986) 213.
\bibitem{am:mlla}
A.H. Mueller, in Proc. 1981 Intern. Symp. on Lepton and Photon
Interactions at High Energies ed. W.Pfeil (Bonn 1981) 689;\\
Yu.L. Dokshitzer, V.S.Fadin and V.A. Khoze, Phys. Lett. B 115 (1982) 242.
\bibitem{dkt-1} Yu.L. Dokshitzer, V.A. Khoze and S.I. Troyan,
Int. J. Mod. Phys. {\bf A7} (1992) 1875.
\bibitem{bw:mlla}
C.P. Fong and B.R. Webber, {Phys. Lett. {\bf B229} (1989) 289}.
\bibitem{brummer} N.C.Brummer, Z. Phys. {\bf C66} (1995) 367.
\bibitem{lund} T. Sj\"{o}strand, Comp. Phys. Comm. {\bf 82} (1994) 74.
\bibitem{web:her}
G. Marchesini and B.Webber, {Nucl. Phys. {\bf B310} (1988) 461};\\
G. Marchesini et al., {Comp. Phys. Comm. {\bf 67} (1992) 465}.
\bibitem{deldet} \Abreu Nucl. Instr. Methods {\bf A303} (1991) 233.
\bibitem{perfo} \Abreu Nucl. Instr. Methods {\bf A378} (1996) 57.
\bibitem{tuning} \Abreu Z. Phys. {\bf C77} (1996) 11.
\bibitem{sprime} \Abreu Phys. Lett. {\bf B372} (1996) 172.
\bibitem{durham} S. Bethke et al., Nucl. Phys. {\bf B370} (1992) 310.
\bibitem{Wxsec} P.~Buschman et al.,``Measurement of W-pair Production
cross-section at $\sqrt{s}$=183 GeV'', DELPHI note 98-20 CONF 120.
\bibitem{newtag}
E.Schyns, ``NEWTAG, $\pi^{\pm}$, $K^{\pm}$ and $p\overline{p}$ tagging for
DELPHI RICHes'', DELPHI note 96-103 RICH-89.
%\bibitem{DaRe} J. Dahm, M. Reale and M. Elsing, DELPHI 95-48 TRACK 81.
\bibitem{pdg94} Review of Particle Properties 1994, Particle Data Group,
Phys. Rev. {\bf D50} n.3 (1994) 1173.
\bibitem{schy} DELPHI Collab., E. Schyns, DELPHI note 97-110 CONF
92, contribution n. 541 to the Jerusalem Conference on HEP (1997).
\bibitem{al2} DELPHI Collab., P. Abreu {\it et al.}, Z. Phys.
{\bf C65} (1995) 587.
\bibitem{al3} DELPHI Collab., P. Abreu {\it et al.}, Phys. Lett.
{\bf B318} (1993) 249.
\end{thebibliography}
%
\newpage
\begin{figure}
\centerline{\epsfig{file=xi1.eps,width=0.9\linewidth}}
\caption{$\xi_p$ distributions (efficiency corrected) for charged kaons
at 133~$\mathrm{GeV/c}$ (a), 161~$\mathrm{GeV/c}$ (b),
172~$\mathrm{GeV/c}$ (c)
and 183~$\mathrm{GeV/c}$ (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{xi1}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{file=xi1a.eps,width=0.9\linewidth}}
\caption{$\xi_p$ distributions (efficiency corrected) for neutral kaons
at 133~$\mathrm{GeV/c}$ (a), 161~$\mathrm{GeV/c}$ (b),
172~$\mathrm{GeV/c}$ (c)
and 183~$\mathrm{GeV/c}$ (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{xi1a}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{file=xi2.eps,width=0.9\linewidth}}
\caption{$\xi_p$ distributions (efficiency corrected) for protons
at 133~$\mathrm{GeV/c}$ (a), 161~$\mathrm{GeV/c}$ (b),
172~$\mathrm{GeV/c}$ (c)
and 183~$\mathrm{GeV/c}$ (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{xi2}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{file=xi2a.eps,width=0.9\linewidth}}
\caption{$\xi_p$ distributions (efficiency corrected) for $\Lambda$
at 133~$\mathrm{GeV/c}$ (a), 161~$\mathrm{GeV/c}$ (b),
172~$\mathrm{GeV/c}$ (c)
and 183~$\mathrm{GeV/c}$ (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{xi2a}
\end{figure}
\begin{figure}
\centerline{\epsfig{file=xsi.eps,width=0.9\linewidth}}
\caption{The maximum $\xi^*$ of the $ \xi_p $-distribution
is shown for \k (a), \Kn (b), p
(c) and $\Lambda$ (d)
as function of the centre-of-mass energy (closed squares).
The fit (solid line) is superimposed to the data points (see text).}
\label{xista}
\end{figure}
\begin{figure}
\centerline{\epsfig{file=mulid.eps,width=0.9\linewidth}}
\caption{Average multiplicity of \k (a), \Kn (b), p
(c) and $\Lambda$ (d)
as function of the centre-of-mass energy (black squares).
Simulation using JETSET~7.4 (open circles) and HERWIG~5.8
(open crosses) are superimposed.}
\label{xi3}
\end{figure}
\end{document}