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\begin{document}
%%% put your own definitions here:
%%% 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}
%%% 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\_225 & \hspace{6cm} DELPHI 99-123 CONF 310 \\
Submitted to Pa 1, 3 & 15 June 1999 \\
\hspace{2.4cm} Pl 1, 3 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Inclusive Spectra at 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}
\newpage
\vspace{\fill}
\pagebreak
%==================> Put the text =====> To be filled <======%
%%% FROM HERE ON INSERT YOUR COMPLETE PAPER
% (including the title, the authors, and the abstract)
\begin{titlepage}
\pagenumbering{arabic}
\begin{tabular}{l r}
HEP'99 \# 1\_225 & \hspace{6cm} DELPHI 99-123 CONF 310 \\
Submitted to Pa 1, 3 & 15 June 1999 \\
\hspace{2.4cm} Pl 1, 3 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Inclusive Spectra at 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 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 from the DELPHI
measurement at {189\gev} are presented as a function of the
variables rapidity, $\xi_p$, $p$ and transverse momenta.
Data are compared with event generators and with calculations in the framework
of the Modified Leading Logarithmic Approximation (MLLA), in
order to examine the hypothesis of local parton hadron duality (LPHD).}
%=========================================================================%
\end{abstract}
\vspace{\fill}
\begin{center}
Paper submitted to the HEP`99 Conference \\
Tampere, Finnland, July 15-21\\
\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 <======%
\section{Introduction}
This paper presents inclusive charged hadron spectra measured by DELPHI
at a centre-of-mass energy of 189 \gev. The results are compared to
previous Z and LEP2 data and to measurements from lower energies. It is an
update of the analysis \cite{olli_p}, which might be consulted for further
details.
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 and on the coherence of gluon radiation.
The measurements therefore provide rigid constraints on
models of the hadronization process.
Direct comparisons of inclusive distributions with QCD calculations
suffer from their dependence on infrared and collinear divergences:
finite predictions in the framework of perturbative QCD can only be obtained
by introducing a cut-off in momentum.
Moreover, to compare calculations of spectra to experimental data
one has to assume that parton
distributions are proportional to inclusive hadron distributions; this is
frequently referred 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 the background from $\mathrm{W^+W^-}$
events are briefly discussed.
In Section 3
the inclusive spectra as measured by DELPHI are compared with the predictions
from different Monte Carlo models. Results concerning the
energy dependence of the distributions
of scaled and absolute momenta are presented.
A brief summary is given in Section 4 .
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Selection and correction of hadronic data\label{sec_select}}
The analysis is based on data taken with the DELPHI detector at 189 \gev.
The integrated luminosity corresponds to about $157\,\mbox{pb}^{-1}$ .
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 outer 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}.
The standard
system of coordinates of DELPHI is also defined in
\cite{NuclInstrMethA378_57}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[tb]
\begin{center}
\begin{tabular}{ | c | c | }\hline
track selection & \\ \hline
$p$[GeV] &{$0.2\le p \le 100$} \\
$\Delta p/p$ &{ $\le$ 1} \\
$\theta$ &{ $20^{\circ} \le \theta \le 160^{\circ}$ } \\
track length &{ $\ge$ 30cm} \\
$\Delta_{r \phi}$ &{$\le$ 4cm} \\
$\Delta_z$ &{ $\le$ 10cm } \\ \hline
event selection & \\ \hline
$\theta_{\mathrm{Thrust}}$ &{ $25^{\circ}\le\theta\le
155^{\circ}$ } \\
$E_{\mathrm{tot}}$ &{$\ge$ 50\% $E_{CM}$ } \\
$\sqrt{s^{\prime}}$ &{ $\ge$90\% $E_{CM}$} \\
$N_{\mathrm{ch}}$ &{42$\ge$ N$_{ch}$ $\ge$ 7 } \\
$B_{\mathrm{min}}$ &{$\le$ 0.08 } \\ \hline
\end{tabular}
\end{center}
\caption{\label{cuts} Selection of charged particles and events.
$p$ is the momentum, $\theta$ is the polar angle with respect
to the beam (likewise $\theta_{\mathrm{Thrust}}$ for the thrust axis
$n_{\mathrm T}$),
$\Delta_{r \phi}$ 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 selected particles,
$s^{\prime}$ the square of the reconstructed centre-of-mass energy,
reduced by initial state
radiation. $\ecm$ is the nominal LEP energy, \bmin\ is the minimal
jet broadening (see text).}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 hadronic events \qcd\ 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, events with large initial state radiation (ISR) and W pair
production.
At energies above the Z peak, initial fermions may
radiate 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 ($\sqrt{s'}$) of an
event,
considering ISR, an algorithm is used that is based on a constrained fit method
using four--momenta of jets and taking energy and momentum conservation into
account \cite{abreusp}.
The production of W pairs occurs above the threshold of 161\gev~.
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, which is defined as \bmin\ = min\{$B_+ , B_-$\} with:
\begin{displaymath}
B_{\pm}=\frac{\sum_{\pm \vec{p}_i\cdot \hat{n}_{\mathrm T}>0} |\vec{p}_i
\times \hat{n}_{\mathrm{T}}|}{2 \sum |\vec{p}_i|} \, ,
\end{displaymath}
where the sum runs over all selected particles. The cut applied to the maximum
charged multiplicity also discards W events. The event selection cuts
(table \ref{cuts}) are optimized in order to maximize the
purity with respect to the $\sqrt{s^{\prime}}$ selection, the efficiency of
collecting high energy $q\bar{q}$ events and the WW rejection.
Furthermore the QCD bias introduced by 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.
The small effect of Z pair production was also taken into
account.
Two photon events are strongly suppressed by the cuts. Leptonic
background can also be neglected in this analysis.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[t]
\begin{center}
\begin{tabular}{ |c|c| }\hline
& 189 GeV \\ \hline %\hline
${\cal L}$ & 156.6 $pb^{-1}$ \\ \hline
$\sigma_{\mathrm {q\bar{q}}}$ & 99.8 $pb$ \\
$\sigma_{\mathrm {q\bar{q}}}$ ($\sqrt{s^{\prime}}>0.9 \cdot \sqrt{s}$) &
18.2 $pb$ \\
$\sqrt{s^{\prime}}$ purity & 0.94 \\ \hline
$\sigma_{\mathrm {WW}}$ & 16.65 $pb$\\
WW rejection efficiency & 0.81 \\
WW background events & 477 \\ \hline
$\sigma_{\mathrm {ZZ}}$ & 1.59 $pb$ \\
ZZ rejection efficiency & 0.84 \\
ZZ background events & 41 \\ \hline
selected events & 3382 \\ \hline
\end{tabular}
\end{center}
\caption{\label{wwcut} Integrated luminosity, cross-- sections of quark, Z and
W pair production, efficiency of rejecting the last two event classes, and
number of remaining background events. These are subtracted assuming the
background to be distributed as in the simulation.
Monte Carlo studies demonstrate that other background sources give negligible
contributions. The last row shows the number of {\em selected} hadronic events
(including background).}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The influence of detector effects was studied by passing
generated events (PYTHIA\cite{CompPhysComm39_347} tuned by
DELPHI \cite{ZPhysC73_11}) through the full
detector simulation (DELSIM \cite{NuclInstrMethA378_57}). Such simulated
events were processed with the same cuts as real data.
They are identified by the subscipt ``acc'' in the following. The tuned
generator prediction is denoted as ``gen''.
In order to correct for cuts, detector and ISR effects, a
bin by bin correction factor (acceptance correction) was applied to the data.
For bin $f$ of histogram $h$ it is defined as:
\beq
C = \frac{h(f)_{\mathrm {gen, no ISR}}}
{h(f)_{\mathrm{acc}}} \, .
\label{acc}
\eeq
For $h(f)_{\mathrm{acc}}$ all cuts were applied, for
$h(f)_{\mathrm{gen, no ISR}}$
a total ISR of less than 1 \gev~was demanded.
%
%\input {inclusive.tex}
\section{Results}
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
$y_t~=~0.5~\cdot~\ln((E+p_{\|})/(E-p_{\|}))$ as well as the
two momentum components ($p_t^{\mathrm {in}}$, $p_t^{\mathrm {out}}$)
transverse to the thrust axis which lie in and perpendicular to the plane of the
event.
$E$ and $p$ are the particle momenta and energies respectively.
All energies have been computed assuming the charged particles to be pions, and
the momenta of neutral particles assuming them to be massless.
Figures \ref{incl1} show the $\xi_p$, $y_t$,
$p_t^{\mathrm {in}}$ and $p_t^{\mathrm {out}}$ spectra as determined from the
189 {\gev} data.
The data are compared to the {\sc Jetset 7.4},
{\sc Herwig 5.8} and {\sc Ariadne 4.08} fragmentation models as
tuned by DELPHI \cite{ZPhysC73_11}.
The shaded areas display the size of the WW and ZZ
background which was subtracted from the data.
The upper inset in these plots shows 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 energy
and also the energy evolution from the Z peak (see the lower
inset of Fig.~\ref{incl1}).
The most prominent
change in the $\xi_p$ distribution (Fig.~\ref{incl1} a) is an
increase at large $\xi_p$ (i.e. low $x_p$). In the rapidity distribution
(Fig.~\ref{incl1} b) the expected increase of the width of the
distribution is clearly observed together with a slight increase of the
plateau height. These changes and the strong
increase in the transverse momentum distributions at large p$_{\perp}$
(Fig.~\ref{incl1} d)
are due to stronger gluon radiation at the higher energies.
%%% inklusiver 4er plot: logxp, rap ptin ptout @ 189 gev *******************
\begin{figure}[hp]
\begin{center}
\vspace{-0.1cm}
\unitlength1cm
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=logxp189.eps,width=8.cm}}
\end{minipage}
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=rap189.eps,width=8.cm}}
\end{minipage}
\end{center}
\vspace*{-1.2cm}
\begin{center}
\unitlength1cm
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=ptin189.eps,width=8.cm}}
\end{minipage}
\begin{minipage}[t]{7.4cm}
\mbox{\epsfig{file=ptout189.eps,width=8.cm}}
\end{minipage}
\caption{\label{incl1}{ inclusive spectra at 189 \gev.
The shaded areas
display the WW -- respectively ZZ background which has been subtracted
from the data. The upper insets in these plots show the correction factor
applied to the data. The lower insets present the ratio of the high energy
data to the corresponding results at the Z.}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Energy Dependence of \boldmath{$\xi_p$} and \boldmath{$\xi^*$}}
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
gluon radiation \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_{\mathrm {event}}}\frac{dn}{d\xi_p}=\frac{N(Y)}{\sigma \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 $\delta = (\xi-<\xi>)/(\sigma)$, $<\xi>$ the mean,
$\sigma$ the width, $s$ the skewness and $k$ the kurtosis of
the distribution. $N(Y)$ is an energy dependent normalization factor.
%%% xi_p
\begin{figure}[t]
\begin{center}
\epsfig{file=LOGXPTASSO.eps,height=12.0cm}
\caption[]{\label{xip}{ $\xi_p$ distributions for charged particles.
The full lines present the results of a simultaneous fit of the Fong-Webber
parametrisation (\ref{distortedg}). The curves are drawn over the range of the
fit ($\approx$ 50\% of the maximum height).}}
\end{center}
\end{figure}
With the number of participating quark flavours, $N_F$,
$\beta=11-2N_F/3$, $\rho=11+2N_F/27$ and
$\omega=1+N_F/27$ one obtains for {\em quark jets} \cite{PhysLettB229_289}:
\begin{eqnarray*}
<\xi>&=& \frac{1}{2}Y \cdot \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} \cdot \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 of ${\cal O}(1)$ in $<\xi>$ are the free parameters in this
expression which
is valid in the region around the maximum, and contains high energy
approximations.
% xi_star ********************************
\begin{figure}[bt]
\begin{center}
\epsfig{file=XISTERN_dla.eps,height=10.0cm}
\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 and DLA predictions are described in the text.}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 LEP2 data are compared to the DELPHI
results at the Z \cite{ZPhysC73_11} and to other experiments
\cite{ZPhysC47_187,PhysLettB345_335}.
The full lines are the results of a simultaneous fit of the Fong-Webber
parametrisation (\ref{distortedg}) to all but the Z data yielding
$\chi^2/\mathrm{dof}$=99.6/97. The Z data
are left out because of the higher rate of $b\bar{b}$ events included in this
sample, which leads to a shift of
the maximum of the $\xi$ distribution, $\xi^*$, to smaller values.
Values of $\Lambda_{\mathrm{eff}}~=~210~\pm~8$ MeV and $-0.55~\pm~0.03$ for the
${\cal O}(1)$ correction to $<\xi>$ are obtained for the energy independent
parameters.
Here $N_F=3$ was chosen since light quarks dominate quark pair production in
the cascade.
MLLA also provides a definite prediction for the energy evolution of $\xi^*$.
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
less sensitive to nonperturbative 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 with 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 at the other energies
\cite{EPS-HEP-629,CERN-PPE96-186}.
For the 189{\gev} data one obtains $\xi^* = 4.157 \pm 0.030$, where the
statistical and systematic error are added in quadrature.
The full line in Fig.~\ref{xistar} shows a fit of the MLLA prediction
\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 $\Lambda_{\mathrm {eff}} = 200\pm 2\mev$. This is in good
agreement with the $\Lambda_{\mathrm {eff}}$ value obtained from the fit to
the whole spectra.
The quantity $C=\frac{\rho^2}{48\beta}$ depends 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.
The dotted line represents the fit of only the first term of equation
\ref{eq_lphd}, which is the result in the double logarithmic approximation
(DLA).
\subsection{Energy Dependence of the Momentum Distribution}
As the scaled momentum distribution may veil effects connected to an absolute
scale, the evolution of the differential cross--section in absolute momentum
$p$ is also of interest.
In Fig.~\ref{impuls1} a) the momentum spectra obtained for the different
centre-of-mass energies and the corresponding predictions of the {\sc Jetset}
model are compared.
%%%%%%%%%%%%%%%%%%%%%%%%%%% dndp plots
\begin{figure}[hbt]
\unitlength1cm
\unitlength1cm
\begin{center}
\begin{minipage}[t]{7.9cm}
\hspace{2.5cm}
\mbox{\epsfig{file=P3JETSET.eps,width=8.1cm}}
\end{minipage}
\begin{minipage}[t]{7.9cm}
\hspace{2.5cm}
\mbox{\epsfig{file=P3MLLA.eps,width=8.1cm}}
\end{minipage}
\end{center}
\caption[]{\label{impuls1}
a) 1/N dn/d$p$ distribution compared to {\sc Jetset} for $p\le 55$ \gev~and
b) to MLLA/LPHD prediction for $p \le 2.5$ \gev. The parameter values of this
fit are given in the last row of table \ref{dndppar}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}.
The energy evolution is well described by the fragmentation model.
The most obvious feature is the increase of the distributions at large
hadron momentum with energy.
This is simply due to the enlarged phase space.
A more interesting feature is the approximate $E_{\mathrm {CM}}$ 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 energy (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 and hence of the centre-of-mass energy.
As a consequence, the number of produced hadrons at small momentum is
approximately constant.
In its range of validity ($p \le 1$GeV) this prediction \cite{Phys_LettB394}
is compared to the data in Fig.~\ref{impuls1}~b).
It depends on
the cutoff parameter $Q_0$, $\lambda=\log(Q_0/\Lambda_{\mathrm {eff}})$ and
the normalisation $K_H$ which 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} b) correspond to a simultaneous fit of all
DELPHI data from 91 to 189 \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 these parameters are highly correlated. Table \ref{dndppar} shows
the different fit results. The large $\chi^2/\mathrm{dof}$ for the fits of
the low energy data compared to the satisfactory $\chi^2/\mathrm{dof}$ for
the DELPHI data may indicate systematic discrepancies between the
measurements of the different experiments. Also the agreement with the
MLLA/LPHD prediction is only expected to be qualitive, since many
non-perturbative effects should be beyond its predictive power.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{center}
\begin{tabular}{ c|c c c c}\hline
fitted energies [GeV]& $Q_0$ [MeV] & $\lambda=\log(Q_0/\Lambda_{\mathrm {eff}})$
& $K_H$ & $\chi^2/\mathrm{dof}$ \\ \hline
low (14 - 58) & 228 $\pm$ 4&0.055 $\pm$ 0.008&0.71 $\pm$ 0.04& 118/34 \\
DELPHI (91.2 - 189)& 226 $\pm$ 3&0.020 $\pm$ 0.004&0.53 $\pm$ 0.04& 17/26 \\
\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}
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\section{Summary}
Inclusive charged hadron distributions as obtained from the DELPHI measurement
at 189~\gev~are presented. Fragmentation models tuned
at the Z describe the data well at higher energies.
MLLA calculations in the limited spectrum approximation ($\Lambda_{\mathrm
{eff}}= Q_0$) allow the $\xi_p$ distribution and the energy
dependence of its maximum to be parameterised.
The 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. All these measurements yield
values for $\Lambda_{\mathrm {eff}}$ from 200 to 221 \mev. This supports the
assumption of local parton hadron duality.
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\subsection*{Acknowledgements}
\vskip 3 mm
We thank W. Ochs for helpful discussions and S. Lupia for providing us
with the code for the $\frac{dn}{dp}$ calculation.\\
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.\\
\pagebreak
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