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\begin{document}
%%% put your own definitions here:
\newcommand{\ee}{$\mathrm e^+e^-$}
\newcommand{\ecm}{E_{\mathrm CM}}
\newcommand{\ww}{{\rm WW}}
\newcommand{\zg}{{\rm Z}/\gamma}
\newcommand{\eps}{\varepsilon}
\newcommand{\fig}{Fig.~\ref}
\newcommand{\tab}{Table~\ref}
\newcommand{\as}{$\alpha_s$\hspace{0.1cm}}
\newcommand{\oas}{$\cal O$($\alpha_s^2$)}
\newcommand{\gev}{\mbox{\,\,Ge\kern-0.2exV }}
\newcommand{\mev}{\mbox{\,\,Me\kern-0.2exV }}
\newcommand{\zz}{{\rm ZZ}}
\newcommand{\bmin}{$B_{\mathrm min}$}
\newcommand{\mhigh}{$M_{\mathrm high}$}
\newcommand{\qcd}
{$e^+e^- \rightarrow {\rm Z}/\gamma \rightarrow q\bar{q}$\hspace{0.1cm}}
\newcommand{\DW}{Dokshitzer and Webber}
\newcommand{ \asb}{$\bar{\alpha}_0$}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\def\jetset{{\sc Jetset }}
\def\herwig{{\sc Herwig }}
\def\ariadne{{\sc Ariadne }}
\begin{titlepage}
\pagenumbering{arabic}
\begin{tabular}{l r}
ICHEP'98 \#138 & \hspace{6cm} DELPHI 98-83 CONF 151 \\
Submitted to Pa 3 & 22 June, 1998 \\
\hspace{2.4cm} Pl 4 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Energy Dependence of Inclusive Spectra in
\boldmath ${\mathrm e^+e^-}$ Annihilation}\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\normalsize {
%===================> DELPHI note author list =====> To be filled <=====%
{\bf O.Passon} $^1$
{\bf J.Drees} $^1$,
{\bf K.Hamacher} $^1$,
%========================================================================%
}
\vspace*{1cm}
\end{center}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
Inclusive charged hadron distributions as obtaind from the DELPHI measurements
at 130, 136, 161, 172 and 183 \gev are presented as a function of the
variables rapidity,
$\xi_p$, p and transversal momenta. Beside comparisons with
event generators the data are compared with MLLA calculations, in
order to examine the hypothesis of local parton hadron duality.
The differential momentum spectra show a new indication for coherence
effects in the production of soft particles. The relation between the energy
dependence of the charged multiplicity and the
rapidity distribution is examined.
%=========================================================================%
\end{abstract}
\vspace{\fill}
\begin{center}
Paper submitted to the ICHEP'98 Conference \\
Vancouver, July 22-29
\end{center}
\vspace{\fill}
\par {\footnotesize $^1$ Fachbereich Physik, Bergische Universit{\"a}t-GH
Wuppertal Gau\ss{}stra\ss{}e 20, 42097 Wuppertal, Germany}
%\par {\footnotesize $^2$ 2nd Institute address...}
%\par {\footnotesize $^3$ 3rd Institute address...}
\end{titlepage}
\pagebreak
%==================> DELPHI note text =====> To be filled <======%
%\input {intro.tex}
\section{Introduction}
From 1995 to 1997 LEP operated at centre of mass energies from 130 to 183\gev.
The number of $\approx$ 2500 hadronic events collected at this
energies together is small compared to the statistics near the
Z resonance. Therefore one main focus of LEP2 QCD analyses will be laid on the
study of energy dependence.
In this note inclusive spectra are under investigation.
The DELPHI data from 130 to 183 \gev in combination with low energy
results allow the examination of the energy dependence over more than one
order of magnitude.
Inclusive stable hadron spectra are highly sensitive to properties of
the hadronization process and to resonance decays as well as to
details of the parton shower.
They depend on the amount of large angle and collinear gluon
radiation as well as on the coherence of gluon radiation.
These measurements therefore provide rigid constraints to the
models of the hadronization process.
Direct comparisons of inclusive distributions with QCD calculations
suffer from their dependence to infrared and collinear divergencies.
Studies of the energy evolution of inclusive spectra provide a
new quality because
the divergent terms can be factorized out and the energy dependence can
be more directly compared to perturbative QCD predictions. The hypothesis that
inclusive hadron distributions are proportional to calculated parton
distributions is frequently refered to as ``local parton hadron duality''
(LPHD)\cite{ZPhysC27_65}.
In section 2
the selection of hadronic events, the reconstruction of
the centre of mass energy, the correction procedures applied to the data and
especially the corrections for $\mathrm{W^+W^-}$
events are briefly discussed.
In Section 3
the inclusive spectra as measured by DELPHI are compared with the predictions
of different Monte Carlo models, while the following sections deal in more
detail
with the energy dependence of the rapidity distribution and the distributions
of scaled- and absolute momenta.
%\input {selection.tex}
\section{Selection and correction of hadronic data\label{sec_select}}
The analysis is based on data taken with the DELPHI detector at 130, 136, 161,
172 and 183\gev. The measurements at 130 and 136 \gev are averaged and labeled
as 133~\gev. The integrated luminosity corresponds to
about $10\,\mbox{pb}^{-1}$ for the 133, 161 and 172\gev data and
$54\,\mbox{pb}^{-1}$ for the 183 \gev data.
DELPHI is a hermetic detector with a solenoidal magnetic
field of 1.2\,T. The tracking detectors, which lie in front of the
electromagnetic calorimeters, are a silicon micro vertex detector VD,
a combined jet/proportional chamber inner detector ID,
a time projection chamber TPC as the major tracking device, and
the streamer tube detector OD in the barrel region. The forward region
is covered by the drift chamber detectors FCA and FCB.
The electromagnetic calorimeters are the high density projection chamber
HPC in the barrel, the lead glass calorimeter FEMC in the forward region
and the STIC next to the beam pipe.
Detailed information about the construction and performance of
DELPHI can be found in \cite{NuclInstrMethA303_187,NuclInstrMethA378_57}.
% cut Tabelle vorstellen:
%~~~~~~~~~~~~~~~~~~~~~~~~~
In order to select well measured charged particle tracks
the cuts given in the upper part of \tab{cuts} have been applied.
The cuts in the lower part of the table have been used to
select \qcd\ events and to
suppress background processes such as two photon interactions, beam gas and
beam wall interactions, leptonic final states, and, most important for the
LEP2 analysis, initial state radiation (ISR) and WW pair production.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[tb]
\begin{center}
\begin{tabular}{ c|c c c c }\hline
track selection &\multicolumn{4}{c}{all energies} \\ \hline
p[GeV] &\multicolumn{4}{c}{$0.2\le p \le 100$} \\
$\Delta$p/p &\multicolumn{4}{c}{ $\le$ 1} \\
$\theta$ &\multicolumn{4}{c}{ $20^o \le \theta \le 160^o$ } \\
track length &\multicolumn{4}{c}{ $\ge$ 30cm} \\
$\Delta_{r \phi}$ &\multicolumn{4}{c}{$\le$ 4cm} \\
$\Delta_z$ &\multicolumn{4}{c}{ $\le$ 10cm } \\ \hline
event selection & \\ \hline
$\theta_{Thrust}$ &\multicolumn{4}{c}{ $30^o\le\theta\le 150^o$ } \\
$E_{tot}$ &\multicolumn{4}{c}{$\ge$ 50\% $E_{CM}$ } \\
$\sqrt{s^{\prime}}$ &\multicolumn{4}{c}{ $\ge$90\% $E_{CM}$} \\ \cline{2-5}
& 133 GeV & 161 GeV & 172 GeV & 183 GeV \\ \cline{2-5}
$N_{ch}$ & 7 $\le$ & $8\le N_{ch} \le$38 & $8\le N_{ch}
\le$40& $8\le N_{ch}\le$42 \\
$B_{min}$ & - & $\le$.09 & $\le$.08 & $\le$.07 \\ \hline
\end{tabular}
\end{center}
\caption{\label{cuts} Selection of tracks and events.
$p$ is the momentum, $\theta$ is the polar angle with respect
to the beam (likewise $\theta_{\mathrm Thrust}$ for the thrust axis),
$\Delta_{r \phi}$ and $\Delta_z$ are the distances to the interaction
point in r$\phi$ (radial distance to beam axis and azimuthal angle) and z
(distance along the beam axis) respectively.
$N_{\mathrm ch}$ is the number of charged particles,
$E_{\mathrm tot}$ the total energy carried by all particles, $s^{\prime}$ the
square of the reconstructed centre of mass energy, as reduced by initial state
radiation. $\ecm$ is the nominal LEP energy, \bmin\ is the minimal
jetbroadening.}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
At energies above the Z peak the initial fermions may
radiate off one or more photons before they interact,
such that the effective centre of mass energy for the collision is
the mass of the Z.
These ``radiative return events'' are the dominant part of the cross section.
The initial state radiation (ISR) is typically aligned along the beam
direction and the photons are only rarely identified inside the detector.
In order to evaluate the effective centre of mass energy of an event,
considering ISR, an algorithm is used that is based on a constraint fit method
using four--momenta of jets and taking energy and momentum conservation into
account. Several assumptions about the event topology are tested. The decision
is taken according to the the obtained $\chi^2$.
\fig{plot_isr} shows the spectra of the calculated energies for
simulated and measured events passing all cuts.
The agreement between data and simulation is good.
The plot shows a peak at about 91\gev\ , the mass of the Z.
The purpose of the $\sqrt{s'}$ cut in the lower part of \tab{cuts} is to
discard radiative return events.
%%%%%%%%%%%%%%%%%%%%%%%%%%% ISR PLOT
\begin{figure}[hbt]
\unitlength1cm
\unitlength1cm
\begin{center}
\begin{minipage}[t]{7.9cm}
\hspace{2.5cm}
\mbox{\epsfig{file=sprime_frame14_183.eps,width=8.1cm}}
\end{minipage}
\begin{minipage}[t]{7.9cm}
\hspace{2.5cm}
\mbox{\epsfig{file=bmin_frame14_183.eps,width=8.1cm}}
\end{minipage}
\end{center}
\caption{\label{plot_isr}
Left: Reconstructed centre of mass energy.
\label{plot_ww} Right: Discriminant variables $B_{\mathrm min}$ for
$\zg$ (dashed) $\ww$ (dotted) and $\zz$ events (dashed dotted). }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF ISR PLOT
Two photon events are strongly suppressed by the cuts. Leptonic
background also can be neglected in this analysis.
%However, for LEP2, the production of pairs of W is an important
%source of background in some bins of shape distributions and may
%bias their averages significantly.
%
%The signature of four jet $Z/\gamma$ and WW events is very
%similar, and thus any WW rejection implies a cut on this region.
%A bias on the shape distributions is expected. However one minimizes
%the problem by choosing a veto on one hemisphere only.%
%
%The bias introduced by \bmin\ (as defined in \cite{ZPhysC73_11}) is small;
%furthermore this variable has a fairly good background to signal
%ratio \fig{plot_ww}. Therefore this shape variable is used to reduce
%the WW contamination.
Above the threshold of 161\gev the production of W pairs occurs.
Since the topological signatures of QCD four jet events and hadronic
WW events (and other four fermion background) are very similar, no highly
efficient separation of the two classes of events is possible.
As a suitable discriminant variable for performing the separation the
shape \bmin\ (as defined in \cite{ZPhysC73_11}) is chosen. The
rejection due to \bmin\ is demonstrated in \fig{plot_ww} for the 183 \gev data.
The actual cut applied here is \bmin$ \le$~0.07.
The event selection cuts (table \ref{cuts}) are chosen in order to maximize the
purity with respect to the $\sqrt{s^{\prime}}$ selection, the efficiency of
collecting high energy QCD events and the WW rejection. Furthermore the QCD
bias due to the \bmin\ cut is minimized.
Table \ref{wwcut} shows the efficiency of the WW rejection and the remaining
contamination. This WW contribution was evaluated by Monte Carlo simulation
and subtracted from the data.
For the 183 GeV data also the small effect of Z pair production was taken into
account
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[b]
\begin{center}
\begin{tabular}{ c|c l l l }\hline
& 133 GeV & 161 GeV & 172 GeV & 183 GeV \\ \hline %\hline
$\sigma_{\mathrm WW}$[pb] &- &3.3 $\pm$ 0.3 &12.1 $\pm$ 0.2 & 15.5 $\pm$ 0.3\\
% \hline
WW rejection efficiency 763 &0.77 &0.82&0.84\\
%\hline
WW background &- &7.6 $\pm$ 0.3&21.7 $\pm$ 0.5&130.7 $\pm$
2.3 \\
%\hline
$\sqrt{s^{\prime}}$ purity& 0.87 & 0.94 & 0.93 & 0.95 \\ \hline
% \hline
selected events &763&332&253&1126 \\ \hline
\end{tabular}
\end{center}
\caption{\label{wwcut} Crosssection of W pair production, efficiency of WW
rejection and number of remaining WW events.These are subtraced with Monte Carlo
methods. The last row shows the number of selected QCD events entering in this
anlysis.}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The influence of detector effects was studied by passing
generated events (PYTHIA\cite{CompPhysComm39_347}) tuned by
DELPHI \cite{ZPhysC73_11} through a full
detector simulation (DELSIM \cite{NuclInstrMethA303_187}). This Monte Carlo
events are processed with the same cuts as real data.
They will be marked as ``acc''. The tuned generator
prediction will be denoted as ``gen''.
In order to correct for cuts, detector and ISR effects, the following
bin by bin correction factor (acceptance correction) was applied to the data:
\beq
C = \frac{h(f)_{\mathrm gen, no ISR}}
{h(f)_{\mathrm acc}}
\label{acc}
\eeq
where $h(f)_i$ represents the accepted or generated distribution.
For $h(f)_{\mathrm acc}$ all cuts are applied, for $h(f)_{\mathrm gen, no ISR}$
ISR of less than 1 \gev was demanded.
%
%\input {inclusive.tex}
\section{Results}
%\section{Inclusive Spectra and Modell Comparison}
Comparisons of inclusive distributions with model expectations are presented
as a function of the logarithm of the scaled
hadron momentum $\xi_p=\ln(1/x_p)$ (with $x_p=p/p_{\mathrm beam}$),
the rapidity with respect to the thrust axis
$y_t=\frac{1}{2}\ln\frac{E+p_{\|}}{E-p_{\|}}$ as well as the
transverse momenta ($p_t^{\mathrm in}$, $p_t^{\mathrm out}$)
with respect to the thrust axes.
$p$ and $E$ are the particle momenta and energies respectively.
All energies have been computed assuming the charged particles to be
pions.
Figures \ref{incl1} and \ref{incl2} show a selection of the $\xi_p$, $y_t$,
$p_t^{\mathrm in}$ and $p_t^{\mathrm out}$ spectra as determined from the high
energy data. The data are compared to the {\sc JETSET}, {\sc HERWIG}
and {\sc ARIADNE} fragmentation models as tuned by DELPHI \cite{ZPhysC73_11}.
The shaded areas display the size of the WW (above 161\gev) and ZZ
(above 183\gev) background which was subtracted from the data.
The upper inset in these plots indicates the size of the correction factor
applied to the data. The lower inset presents the ratio of the high energy
data to the corresponding results at the Z. This ratio is again compared to
the model predictions.
The models describe well all inclusive spectra measured at the high energies
and also the energy evolution from the Z peak (see lower
inset of Fig.~\ref{incl1} and \ref{incl2}). The most prominent
change in the $\xi_p$ distribution (Fig.~\ref{incl2} a and c) is an
increase at large $\xi_p$ (i.e. low $x_p$). In the rapidity distribution
(Fig.~\ref{incl2} b and d) the expected increase of the width of the
distribution is clearly observed together with a slight increase of the
plateau. These changes and the strong
increase in the transverse momentum distributions at large $p_{\perp}$
(Fig.~\ref{incl1}b and d)
are due to stronger gluon radiation of the higher energies.
%%% inklusiver 4er plot: ptin/out 133/172 GeV ***************************
\begin{figure}[hp]
\begin{center}
\vspace{-0.1cm}
\unitlength1cm
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=ptin133.eps,width=8.cm}}
\end{minipage}
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=ptout133.eps,width=8.cm}}
\end{minipage}
\end{center}
\vspace*{-1.2cm}
\begin{center}
\unitlength1cm
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=ptin172.eps,width=8.cm}}
\end{minipage}
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=ptout172.eps,width=8.cm}}
\end{minipage}
\caption{\label{incl1}{ $p_t^{\mathrm in}$ and $p_t^{\mathrm out}$ distributions
for 133 and 172\gev, the shaded area for the 172 \gev data
displays the acceptance corrected WW background which is subtracted
from the data.}}
\end{center}
\end{figure}
%%% inklusiver 4er plot xi und rap 161/183 *****************************
\begin{figure}[hp]
\begin{center}
\vspace{-0.1cm}
\unitlength1cm
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=logxp161.eps,width=8.cm}}
\end{minipage}
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=rap161.eps,width=8.cm}}
\end{minipage}
\end{center}
\vspace*{-1.2cm}
\begin{center}
\unitlength1cm
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=logxp183.eps,width=8.cm}}
\end{minipage}
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=rap183.eps,width=8.cm}}
\end{minipage}
\caption{\label{incl2}{ $\xi_p$ and rapidity distribution
at 161 and 183 GeV, the shaded area
displays the acceptance corrected WW background (and ZZ for the 183 \gev data)
which is subtracted from the data.}}
\end{center}
\end{figure}
%%% inklusiver 4er plot 183 *ende*******************************
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Energy Evolution of the Rapidity Distribution}
In Fig.\ref{rap} (left) rapidity distributions obtained from 14 to 183 \gev are
compared.
The full line displays the prediction of the {\sc jetset} modell
(tuned at Z energy \cite{ZPhysC73_11}),
wich agrees well with the DELPHI data.
While the 35\gev TASSO \cite{tasso88} and
35\gev MARK II \cite{mark2} data are well described, the AMY \cite{amy90} and
low energy TASSO data \cite{tasso84} deviate significantly from the model
prediction.
Contrary to the simple parton model expectation not only the length, but also
the height of the rapidity plateau increases with the centre of mass energy.
In order to get a more quantitative expression, measures for the height and
widht were defined.
The height of the plateau is determined by averaging over
the plateau points.The uncertainty in the precise position of the plateau
edge was incorporated in the systematic error. The width of the
distribution is taken to be the rapidity value at wich the
distribution is fallen off to half of the plateau height.
Figure \ref{rap} (right) shows the energy evolution of these parameters from
14 to 183 \gev (two lower curves). The dashed and dotted lines are
parametrisations with a polynomal of second order in $\log{\ecm}$. The upper
curve is the
product of those two parametrisations. Since the product of width and
length of the rapidity distribution is expected to give roughly the charged
multiplicity, this line is compared with various measurements of
$$. The double logarithmic presentation allows to compare the
relative
slope in width and height respectivly. A straight line fit of the
form $a_i\cdot E_{\mathrm CM}^{b_i}$ gives $b= 0.17 \pm 0.01$ for the width
and $b = 0.25 \pm 0.02$ for the height. The ratio of $b_h/b_w \approx 1.5$
indicates, that the multiplicity increase is mainly due to the increase of
particles with small rapidity, and to a lesser extend due to the growth of the
length of the rapidity distribution.
%%%%%%%%%%%%%%%%%%%%%%%%%%% rapplots
\begin{figure}[hbt]
\unitlength1cm
\unitlength1cm
\begin{center}
\begin{minipage}[t]{7.9cm}
\hspace{2.5cm}
\mbox{\epsfig{file=raps.eps,width=8.1cm}}
\end{minipage}
\begin{minipage}[t]{7.9cm}
\hspace{2.5cm}
\mbox{\epsfig{file=rappuzzle.eps,width=8.1cm}}
\end{minipage}
\end{center}
\caption{\label{rap}
Left: rapidity distributions for various energies. Right: Energy dependence
of $$, length and width of the
rapidity distribution. The two lower curves are simply parametrisations of this
measures, while the upper curve is the product of this two parametrisations.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Energy Dependence of $\xi_p$ and $\xi^*$}
The shape of the partonic $\xi_p$ distribution, as calculated in the Modified
Leading Log Approximation (MLLA),
exhibits the characteristic ``hump backed'' plateau due to suppression of soft
gluons \cite{ZPhysC55_107}.
The calculation in the ``limited spectrum'' approximation ($\Lambda_{\mathrm
eff} = Q_0$) can be well expressed by a distorted Gaussian
(\ref{distortedg}):
\begin{eqnarray}
\label{distortedg}
\frac{1}{N}\frac{dn}{d\xi_p}=\frac{N}{\sqrt{2\pi}}\exp{\left(\frac{k}{8}-
\frac{s\delta}{2}-\frac{(2+k)\delta^2}{4}+\frac{s\delta^3}{6}+
\frac{k\delta^4}{24}\right)}
\end{eqnarray}
with $\delta = \frac{\xi-<\xi>}{\sigma}$ ($<\xi>$ the mean of the distribution)
, $\sigma$ the width, $s$ the skewness and $k$ the kurtosis of
the distribution. $N$ is an energy dependent normalization factor.
%%% xi_p
\begin{figure}[t]
\begin{center}
\epsfig{file=xis_fongwebber.eps,height=12.0cm}
\caption{\label{xip}{ $\xi_p$ distributions for charged particles.
The full lines are the result of a simutanious fit of the Fong Webber
parametrisation. The curves are drawn in their fitrange ($\approx$ 50\% niveau).
}}
\end{center}
\end{figure}
With $\beta=11-2N_F/3$, $\rho=11+2N_F/27$ and
$\omega=1+N_F/27$ one obtains for {\em quark jets} \cite{PhysLettB229_289}:
\begin{eqnarray*}
<\xi>&=& \frac{1}{2}Y\left( 1+ \frac{\rho}{24}\sqrt
\frac{48}{\beta Y} \right)
\cdot \left(1-\frac{\omega}{6Y}\right) + {\cal O}(1) \\
\sigma&=& \sqrt{\frac{Y}{3}}\cdot\left( \frac{\beta Y}{48}\right)
^{1/4}\cdot \left( 1-\frac{\beta}{64}\sqrt{\frac{48}{\beta Y}} \right)
\cdot \left(1+\frac{\omega}{8Y}\right) +{\cal O}(Y^{-1/4})\\
s&=&-\frac{\rho}{16}\sqrt{\frac{3}{Y}}\cdot \left( \frac{48}
{\beta Y}\right)^{1/4} \cdot \left(1+\frac{\omega}{4Y}\right)
+{\cal O}(Y^{-5/4})\\
k&=&-\frac{27}{5Y}\left( \sqrt{\frac{\beta Y}{48}}-\frac{\beta}{24}
\right) \cdot \left(1+\frac{5\omega}{12Y}\right) +{\cal O}(Y^{-3/2})\\
\end{eqnarray*}
Here $Y=\ln (E_{\mathrm beam}/ \Lambda_{\mathrm eff})$ and
$\Lambda_{\mathrm eff}$ is an effective scale parameter.
$\Lambda_{\mathrm eff}$, the overall normalization N and the additional
constant term in $<\xi>$ are the free parameters in this expression wich
is valid in the region arround the maximum, and contains high energy
approximations.
Using the Local Parton Hadron Duality (LPHD)
hypothesis this shape can directly be adapted to the measured hadron
spectrum \cite{ZPhysC27_65}. In Fig. \ref{xip} the
$\xi_p$ distributions as determined from the LEP2 data are compared to the
results from low energy experiments \cite{ZPhysC47_187,PhysLettB345_335}.
The full lines are the result of a simultanious fit of the Fong-Webber
parametrisation to all but the Z data yielding $\chi^2/ndf$=92/89. The Z data
are left out because of the higher rate of $b\bar{b}$ events included in this
sample.
$N_F=3$ was choosen since light quarks dominate quark pair production in
the cascade.
For the energy independent
parameters one obtains $\Lambda_{\mathrm eff} = 210 \pm 8 $ MeV and $-0.55 \pm
0.03$ for the ${\cal O}(1)$ correction to $<\xi>$. (Including the Z
data one obtains a slightly smaller $\Lambda_{\mathrm eff}$ and a value of
$\chi^2/ndf$ of 129/101).
% xi_star ********************************
\begin{figure}[ht]
\begin{center}
\epsfig{file=xistern_lin_183.eps,height=9.0cm}
\caption{\label{xistar}{ Energy evolution of the $\xi_p$ peak position.
$\Lambda_{\mathrm eff}$ is obtained from a fit to the MLLA/LPHD
prediction. The phase space prediction is described in the text.}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
MLLA also provides a definite prediction for the energy evolution of the
maximum, $\xi^*$, of the $\xi$ distribution.
As hadronization and resonance decays are expected to act similarly at
different centre of mass energies,
the energy evolution of $\xi^*$ is expected to be
insensitive to non perturbative effects.
A small correction is to be expected, however, due to varying
contribution of heavy quark events.
These chain decays are known to shift $\xi^*$ in a way different from
ordinary resonance decays.
This shift also differs for the individual stable particle species due to their
different masses.
In this paper the influence of heavy decays is neglected.
The $\xi^*$ values entering in this analysis were determined by fitting a
distorted Gaussian whith the parameters s, k, $<\xi>$ and $\sigma$ given by
the Fong-Webber calculation.
The fit range in $\xi$ is restricted to the part of the
distribution close to the maximum with
1/N~dn/d$\xi_p$~$\ge$~0.6~(1/N~dn/d$\xi_p$)$_{\mathrm max}$.
To avoid systematic differences due to different strategies for the
$\xi^*$ determination this fit has also been performed to the low energy data
\cite{EPS-HEP-629,CERN-PPE96-186}.
The $\xi^*$ values obtained are given in the table \ref{xisterndep}.
The full line in Fig.\ref{xistar} shows a fit of the MLLA
expectation
\begin{eqnarray}
\xi^*=0.5\cdot Y+\sqrt{C}\cdot \sqrt{Y}-C+{\cal O}(Y^{-\frac{3}{2}})
\label{eq_lphd}
\end{eqnarray}
to the data.
From this fit $\Lambda_{\mathrm eff} = 200\pm 3\mev$.
The quantity $C$ depends weakly on the number of active flavours
( $C(N_F=3)=0.2915$, $C(N_F=5)= 0.3513$). The results are presented for
$N_F=3$.
The dashed line in Fig.~\ref{xistar} represents the slope of the
phase space expectation $\xi^* = a + Y$ \cite{hadronization}.
Due to angular ordering of gluon bremsstrahlung
the rise in $\xi^*$ is slower in the MLLA prediction.
\begin{table}[h]
\begin{center}
\begin{tabular}{c|cccc|}\hline
$E_{\mathrm CM}$ [GeV] & $\xi^*$ & $<\xi>$ &
$\xi^*$ - $<\xi>$ \\ \hline
14 & 2.453 $\pm$ 0.053 & 2.29 $\pm$ 0.04 & 0.16 $\pm$ 0.07 \\
22 & 2.738 $\pm$ 0.057 & 2.16 $\pm$ 0.18 & 0.58 $\pm$ 0.10 \\
35 & 3.072 $\pm$ 0.023 & 2.70 $\pm$ 0.05 & 0.37 $\pm$ 0.05 \\
44 & 3.174 $\pm$ 0.039 & 2.91 $\pm$ 0.05 & 0.26 $\pm$ 0.06 \\
91 & 3.701 $\pm$ 0.023 & 3.45 $\pm$ 0.02 & 0.26 $\pm$ 0.03 \\
133 & 3.908 $\pm$ 0.051 & 3.74 $\pm$ 0.07 & 0.17 $\pm$ 0.08 \\
161 & 4.085 $\pm$ 0.051 & 3.67 $\pm$ 0.20 & 0.42 $\pm$ 0.21 \\
172 & 4.088 $\pm$ 0.066 & 3.52 $\pm$ 0.53 & 0.57 $\pm$ 0.54 \\
183 & 4.126 $\pm$ 0.047 & 3.75 $\pm$ 0.15 & 0.38 $\pm$ 0.07 \\ \hline
meanvalue:& & & 0.35 $\pm$ 0.14 \\ \hline
\end{tabular}
\end{center}
\caption{\label{xisterndep}The peak position, $\xi^*$, and the mean value
$<\xi>$ of the $\xi_p$ distribution. The MLLA prediction for the deviation
is 0.351.}
\end{table}
The MLLA provides also a prediction for the deviation of $\xi^*$ and the mean
$<\xi>$. Its value is simply 0.351, independent of the $\ecm$
\cite{BasicsOfPertQCD}. Since the $\xi_p$ shape is symmetric in the DLA, this
deviation is in principle sensitive to the effect of single logarithmic terms.
It turns out that the determination of $<\xi>$ is not straight forward:
integrating over the whole $\xi_p$ spectrum would disregard that the
gaussian like shape is only valid in the maximum region.
Therefore $<\xi>$ was obtained by fitting the distorted gaussian with
$<\xi>$ entering as a free parameter in the maximum region i.e.
1/N~dn/d$\xi_p$~$\ge$~0.5~(1/N~dn/d$\xi_p$)$_{\mathrm max}$.
Table \ref{xisterndep} shows the results. The mean value of
$\xi^*$ - $<\xi>$ for the various energies results in 0.36 $\pm$ 0.15, wich is
in agreement with the LPHD/MLLA prediction.
\subsection{Energy Dependence of the Absolute Momentum}
As the scaled momentum distribution may veil effects with an absolute scale,
the evolution of the differential crossection in absolute momentum p is also
of interest.
In Fig.\ref{impuls1} the momentum spectra obtained for the different centre of
mass energies and the correponding predictions of the {\sc Jetset }
model are compared. For the low energy and Z data the spectra are obtained by
rescaling the $x_p$ distribution
\cite{ZPhysC73_11,ZPhysC47_187,PhysLettB345_335,PhysRevLett61_1263}.
%%%%%%%%%%%%%%%%%%%%%%%%%%% dndp plots
\begin{figure}[hbt]
\unitlength1cm
\unitlength1cm
\begin{center}
\begin{minipage}[t]{7.9cm}
\hspace{2.5cm}
\mbox{\epsfig{file=dndp_jetset_183.eps,width=8.1cm}}
\end{minipage}
\begin{minipage}[t]{7.9cm}
\hspace{2.5cm}
\mbox{\epsfig{file=dndp_ana_183.eps,width=8.1cm}}
\end{minipage}
\end{center}
\caption{\label{impuls1}
Left: 1/N dn/dp distribution compared to {\sc JETSET} for $p \le 80$ \gev and
Right: to MLLA/LPHD prediction for $p \le 2.5$ \gev }
\end{figure}
The energy evolution is well described by the model.
The most obvious feature is the increase of the distributions at large
hadron momentum with energy.
This is simply due to the enlarged phasespace.
A more interesting feature is the approximate CMS independence of hadron
production at very small momentum $p<1$~GeV.
This behaviour has been explained in \cite{Phys_LettB394} to be due to
the coherent emission of low energetic (i.e. long wavelength) gluons by the
total colour current.
This colour current is independent of the internal jet structure and
conserved under parton splittings.
Therefore low energy gluon emission is expected to be almost independent of
the number
of hard gluons radiated e.g. of the centre of mass energy.
As a consequence the number of produced hadrons at small momentum is
approximately constant.
The prediction \cite{Phys_LettB394} is quantitatively compared to the data
in Fig.\ref{impuls1} in its range of validity ($p \le 1$GeV). It depends on
the cutoff parameter $Q_0$, $\lambda=\log(Q_0/\Lambda_{\mathrm eff})$ and
the normalisation $K_H$ wich relates parton and hadron distributions according
to the LPHD hypothesis. It should be
noted, that in contrast to the $\xi_p$ calculations this prediction is {\em not}
obtained in the ``limited spectrum'' ($\Lambda_{\mathrm
eff} = Q_0$) approximation.
The curves in Fig.\ref{impuls1} are calculated using the parameters
$Q_0$ = 238 MeV, $K_H$ = 0.42
and $\lambda = 0.010$ (corresponding to $\Lambda_{\mathrm eff}=236\mev$).
These parameters were obtained by simultaneously fitting all DELPHI data from
91 to 183 \gev. Fitting all data or only the low energy data yields different
values (especially in the logarithmic variable $\lambda$), but it should be
noted that this parameters are highly correlated. Table \ref{dndppar} shows the
different fit results. The large $\chi^2/dof$ for the low energy data
compared to the
satisfactory $\chi^2/dof$ for the DELPHI data may indicate systematic
discrepancies between the measurements of the different low energy
experiments.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
\begin{center}
\begin{tabular}{ c|c c c c}\hline
fitted energies [GeV]& $Q_0$ [MeV] & $\lambda=\log(Q_0/\Lambda_{\mathrm eff})$
& $K_H$ & $\chi^2/dof$ \\ \hline
all (14 - 183) & 210 $\pm$ 4&0.095 $\pm$ 0.009&0.86 $\pm$ 0.04& 179/57 \\
low (14 - 58) & 228 $\pm$ 4&0.055 $\pm$ 0.008&0.71 $\pm$ 0.04& 118/34 \\
DELPHI (91.2 - 183)& 238 $\pm$ 3&0.010 $\pm$ 0.004&0.42 $\pm$ 0.04& 13.4/20 \\
\hline
%133 - 183 & 199 $\pm$ 9&0.083 $\pm$ 0.046&0.79 $\pm$ 0.17& 10.4/12 \\
%91.2 & 210 $\pm$ 8&0.059 $\pm$ 0.020&0.65 $\pm$ 0.09& 0.3/11 \\ \hline
\end{tabular}
\end{center}
\caption{\label{dndppar} Parameters obtained from simultaneous fits of the
LPHD/MLLA prediction to the momentum spectra at different energies. }
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input {zusammenfassung.tex}
\section{Summary}
Inclusive charged hadron distributions as obtaind from the DELPHI measurements
at 130, 136, 161, 172 and 183 \gev are presented. Fragmentation models tuned
at the Z also describe well the data at higher energies.
MLLA calculations in the limited spectrum approximation ($\Lambda_{\mathrm
eff}= Q_0$) allow to parametrize the $\xi_p$ distribution and the energy
dependence of its maximum. A new calculation \cite{Phys_LettB394} for the
soft part of the partonic momentum spectrum is able to describe the hadronic
$p$ distribution in the range of small momenta supporting the assumption
of local parton hadron duality.
The energy evolution of the rapidity distribution shows, that the increase in
mutliplicity is to a large extent due to the growth in the plateau height.
%=========================================================================%
\subsection*{Acknowledgements}
\vskip 3 mm
We are greatly indebted to our technical
collaborators and to the funding agencies for their
support in building and operating the DELPHI detector, and to the members
of the CERN-SL Division for the excellent performance of the LEP collider.
%=========================================================================%
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%=========================================================================%
\end{document}