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\begin{document}
%%% put your own definitions here:
%===================> ADD here your LATEX definitions
\newcommand \ps {\ifmmode {\rm ps} \else ps\fi}
\newcommand \ipb {\ifmmode {\rm pb}$^{-1}$ \else pb$^{-1}$\fi}
\newcommand \gev {\ifmmode {\rm GeV} \else GeV \fi}
\newcommand \mev {\ifmmode {\rm MeV} \else MeV \fi}
\newcommand \ee {\ifmmode e^+e^- \else $e^+e^-$ \fi}
\newcommand {\lala}{\ifmmode \Lambda\Lambda \else $\Lambda\Lambda$ \fi}
\newcommand {\alal}{\ifmmode \bar{\Lambda}\bar{\Lambda} \else $\bar{\Lambda}\bar{\Lambda}$ \fi}
\newcommand {\lal} {\ifmmode \Lambda\bar{\Lambda} \else $\Lambda\bar{\Lambda}$ \fi}
\newcommand \al {\ifmmode \Lambda \else $\Lambda$ \fi}
\newcommand \bl {\ifmmode \bar{\Lambda} \else $\bar{\Lambda}$ \fi}
\newcommand \cl {\ifmmode \Lambda^0 \else $\Lambda^0$ \fi}
\newcommand \dl {\ifmmode \bar{\Lambda^0} \else $\bar{\Lambda^0}$ \fi}
\newcommand \zz {\ifmmode Z^0 \else $Z^0$ \fi}
\newcommand \ys {\ifmmode y^* \else $y^*$ \fi}
\newcommand \Q {\ifmmode Q}
%========================================================================%
\begin{titlepage}
\pagenumbering{arabic}
\begin{tabular}{l r}
ICHEP'98 \#154 & \hspace{6cm} DELPHI 98-114 CONF 176 \\
Submitted to Pa 9 & 22 June, 1998 \\
\hspace{2.4cm} Pl 6, 11 & \\
\end{tabular}
\vspace*{1.0cm}
\begin{center}
\Huge {\bf Determination of the spin composition of
\boldmath $\Lambda\bar{\Lambda}$ and $\Lambda\Lambda$
($\bar{\Lambda}\bar{\Lambda}$) pairs
in hadronic Z decays}\\
\vspace*{0.8cm}
\centerline{\large Preliminary}
\vspace*{1.0cm}
\large {DELPHI Collaboration \\}
\vspace*{0.6cm}
\normalsize {
%===================> DELPHI note author list =====> To be filled <=====%
{\bf T.~Lesiak} $^1$,
{\bf H.~Palka} $^1$
%========================================================================%
}
\vspace*{1cm}
\end{center}
\begin{abstract}
\noindent
%===================> DELPHI note abstract =====> To be filled <=====%
The spin composition of the $\Lambda\bar{\Lambda}$
and $\Lambda\Lambda$ ($\bar{\Lambda}\bar{\Lambda}$)
pairs produced inclusively in hadronic Z decays has been
measured as a function of Q, the difference of $\Lambda$'s
four-momenta. The fraction of spin 1 component
for the $\Lambda\bar{\Lambda}$ pairs was found to
be consistent with 0.75 over the full measured Q range.
The spin composition of the $\Lambda\Lambda$
($\bar{\Lambda}\bar{\Lambda}$) system is
different from a statistical spin mixture for $Q<1.8$~GeV,
being consistent with a dominance of spin 0 component.
Such an enhancement of the spin 0 component for low $Q$ values is
expected from the Pauli exclusion principle in the absence of other
dynamical correlations. The observed effect allowed us to estimate
the spatial dimension of the $\Lambda\Lambda$ emitter to be:
$R = 0.11^{+0.05}_{-0.03}(stat)\pm 0.01(syst)$~fm .
%=========================================================================%
\end{abstract}
\vspace{\fill}
\begin{center}
Paper submitted to the ICHEP'98 Conference \\
Vancouver, July 22-29
\end{center}
\vspace{\fill}
\par {\footnotesize $^1$ H.Niewodniczanski Institute of Nuclear Physics, Krakow, Poland}
\end{titlepage}
\pagebreak
%==================> DELPHI note text =====> To be filled <======%
\section{Introduction}
%#####################################################################
%\vskip 1cm
Despite of the successes in understanding of general features
of multi-particle production, many important aspects of the transition from
partons to observable hadrons are still scarcely known. Amongst them are
the space-time structure of the source of hadrons, spin phenomena and baryon
production processes.
Correlation studies of pairs of particles can shed some light on properties
of the emitter of hadrons. Such correlation studies for bosons which are close
to each other in phase space revealed the existence of an enhancement of
the number of identical bosons over that of non-identical bosons
(the Bose-Einstein Correlation - BEC ).
The BEC effect was used, besides studies of its quantum mechanical aspect,
to estimate the dimension of the emitting source of the identical bosons.
The radius of the spherical source of Gaussian density was measured to be
typically of the order of 1.0 fm.
Analogous phenomenon is expected for a pair of identical fermions in the triplet
spin state \cite{LIPKIN}. The number of fermion-fermion pairs with parallel spins
and close to each other in phase space, could be depleted due to the Pauli
exclusion principle. The observation of this effect would allow to extract the
dimension of the identical fermion emitter, without the need of constructing
uncorrelated reference samples which plagues the BEC studies for bosons.
The above tempted us to study spin-spin correlations of pairs of \al
hyperons produced inclusively in $Z^{0}$ decays, with the main aim to
investigate the underlying baryon production processes.
In \zz decay baryons are
produced at much lower rates than pions or kaons, and hyperons production is even
less abundant due to strangeness suppression. On the other hand however,
the relative fraction of primary \al's is much higher than mesons, thus
they are well suited to study
parton-hadron transitions.
The \al provides also the unique possibility of spin analysis, owing to
its weak decay self-analysing the spin state of the parent particle.
The spin-spin correlations for \al pairs have been studied previously
by OPAL~\cite{OPAL} and ALEPH~\cite{ALEPH}.
The determination of the fractions of the S~=~0 and S~=~1 spin states
of the \lala (\lal)
\footnote{unless explicitly stated, any symbol of particle refers
at the same time to its anti-particle; thus by the `\lala'
sample we understand both \lala and
\alal pairs}
pairs is based on formalism proposed by G. Alexander
and H.J. Lipkin~\cite{LIPKIN}. The information concerning the spin composition
is extracted from the distribution of the cosine of the angle (\ys) between the two
hyperons' decay protons momenta, each measured in its parent hyperon rest frame.
From the Wigner-Eckart theorem the \ys distribution for the S~=~0
and S~=~1 spin states for the \lal system reads~\cite{LIPKIN}
\begin{equation}
\left.\frac{dN}{d\ys}\right|_{\rm S=0} = 1 + \alpha_{\al}^2\cdot\ys
~~~~~~~~~~~~~~~{\rm and}~~~~~~~~~~~~~~~
\left.\frac{dN}{d\ys}\right|_{\rm S=1} = 1 - \frac{\alpha_{\al}^2}{3}\cdot\ys.
\label{LAL}
\end{equation}
The respective formulae for the \lala pairs are
\begin{equation}
\left.\frac{dN}{d\ys}\right|_{\rm S=0} = 1 - \alpha_{\al}^2\cdot\ys
~~~~~~~~~~~~~~~{\rm and}~~~~~~~~~~~~~~~
\left.\frac{dN}{d\ys}\right|_{\rm S=1} = 1 + \frac{\alpha_{\al}^2}{3}\cdot\ys.
\label{LL}
\end{equation}
$\alpha_\Lambda=0.642\pm 0.013$~\cite{PDG} is the decay asymmetry
parameter of the $\Lambda$ hyperon.
Denoting the fraction of
\lala (\lal)
states with S~=~1 by $\epsilon$, the expression
for the overall \ys distribution becomes:
\begin{equation}
\frac{dN}{d\ys} =
(1-\epsilon)\left.\frac{dN}{d\ys}\right|_{\rm S=0}
+\epsilon\left.\frac{dN}{d\ys}\right|_{\rm S=1}.
\label{IDEAL}
\end{equation}
For a statistical mixture of spins 0 and 1 each spin state probability
is weighted by the factor 2S+1. Then the parameter $\epsilon$ is equal
to 0.75 which yields a constant $dN/dy^*$ distribution.
The formulae of the Eq.~\ref{LAL}-\ref{IDEAL} are strictly valid
only close to the di-hyperon threshold
($Q = \sqrt{m^2_{\Lambda\Lambda}-4m_{\Lambda}^2}\approx 0$),
being only approximate for larger Q values.
It has been verified in the data that this approximation works well
up to Q $\approx$ 5.0 GeV, by checking that the results of the analysis
do not depend on
whether the angle between decay protons is determined from the momenta
calculated in the parent \al rest frames or the di-hyperon centre of mass system.
%#####################################################################
\section{Data selection}
%#####################################################################
The sample of hadronic \zz decays was selected using the standard
criteria~\cite{DELPHI} based on the charged track multiplicity and the visible
energy of the event. In total $3.48\times 10^6$ events were classified as
hadronic in the data collected in 1992-95. The full study described below was also performed
on the sample of $4.6\times 10^6$ $Z^{0}\rightarrow qq$ events generated
with the JETSET 7.3
parton shower model~\cite{JETSET} with parameters tuned as in~\cite{DELTUN}.
These events have been passed through the DELPHI detector simulation and
subjected to the
full event reconstruction the same as for the data.
The \al hyperons were selected by reconstructing their decay into $p\pi$
using the DELPHI standard V0 package described in~\cite{DELPHI},~\cite{RECV0}.
To remove (a very small) fraction of \al baryons containing
a primary $s$-quark from the decays $Z^0\to s\bar{s}$,~it was required that
the relative energy $x_E^{\al} = E_{\al}/E_{beam}$ do not exceed 0.3.
The purity
of the selected sample was further improved by demanding that the
higher
momentum track from the \al decay is not identified as a pion.
For this purpose the identification information from the RICH and the ionisation
energy loss in the TPC was used.
The effect of the proton identification is shown in Fig.~\ref{MASSL0}.
The background under the \al signal has been reduced by a factor of 5,
at the expense of 40~\% signal loss. This background suppression was necessary
to keep background to signal ratio for \lala low, down to very small Q values.
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(100,100)
\put(10,20){\bf $\downarrow$}
\put(25,20){\bf $\downarrow$}
\put(33,30){\bf $\downarrow$}
\put(42,30){\bf $\downarrow$}
\put(50,20){\bf $\downarrow$}
\put(65,20){\bf $\downarrow$}
\put(80,-2){\LARGE\bf $m(p\pi)~(GeV)$}
\put(-8,98){\LARGE\bf $\frac{dN}{m(p\pi)\cdot 0.001}$}
\epsfxsize=100mm
\epsffile{l0_fig1.eps}
\end{picture}
\end{center}
\caption{Invariant mass spectrum ($p\pi$) before (solid histogram)
and after (shaded histogram) demanding the pion's veto for the proton's track.
The shaded histogram corresponds to the final \al selection. The
vertical arrows mark the `signal' and `background' regions as
defined in the text.}
\label{MASSL0}
\end{figure}
%
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(100,100)
\put( 25,-2){\LARGE\bf $m_{\Lambda\Lambda}~(GeV)$}
\put( 85,-2){\LARGE\bf $Q~(GeV)$}
\put(-9,115){\LARGE\bf $\frac{dN}{m_{\Lambda\Lambda}\cdot 0.1}$}
\put(53,115){\LARGE\bf $\frac{dN}{Q\cdot 0.2}$}
\epsfxsize=120mm
\epsffile{qmasl0l0_fig1a.eps}
\end{picture}
\end{center}
\caption{The distributions of the invariant mass of the \al pairs
$m_{\Lambda\Lambda}$ (left plot) and of the difference of \al's four
momenta $Q$ (right plot). The data points (shaded histogram) correspond
to the \lal (\lala) pairs, respectively.}
\label{QMASL0L0}
\end{figure}
%
The pairs of \al hyperons being attributed to the same hemisphere
(with respect to the thrust axis)
have been used in the study of the~\ys distribution.
Fig~\ref{QMASL0L0} shows the distributions of the four-momentum difference
of the pair $Q$ and of the invariant mass of the \al pairs.
The purities $P$ of the \lala, \lal samples were determined from the data
itself using the method described in~\cite{PURITY}.
Denoting by $m_i$, i = 1,2 the masses of the first (second) \al
hyperon respectively, the absolute values of their deviations
from the nominal mass $m_{\Lambda}$ given by the Particle Data Group~\cite{PDG}
have been defined
\begin{equation}
\Delta m_i = \left| m_i - m_{\Lambda} \right|~,~~i~=~1,~2.
\end{equation}
Each pair is then represented as a point
in the plane ($\Delta m_1$,$\Delta m_2$) (Fig~\ref{PLANE}).
The population $N$ of the `signal region' defined by
$\Delta m_1, \Delta m_2 < a$ was decomposed into the sum of four contributions:
\begin{itemize}
\item $N_{tt}$, the number of pairs in the `signal region' in which both
hyperons are true
\item $N_{tf}$, the number of pairs in the `signal region' in which the first
$\Lambda$ is true and the second fake
\item $N_{ft}$, the number of pairs in the `signal region' in which the first
$\Lambda$ is fake and the second true
\item $N_{ff}$, the number of pairs in the `signal region' in which both
$\Lambda$s are fake
\end{itemize}
\begin{equation}
N = N_{tt} + N_{tf} + N_{ft} + N_{ff}.
\end{equation}
The three background contributions were extracted from the numbers of
pairs $N_1$, $N_2$ and $N_3$ in the respective `control regions' defined as:
$$\begin{tabular}{ll}
$N_1 = kN_{ft} + kN_{ff}$ & $\Delta m_2~<~a~,~b~<~\Delta m_1~<~b + ka$ \\
$N_2 = kN_{tf} + kN_{ff}$ & $\Delta m_1~<~a~,~b~<~\Delta m_2~<~b + ka$ \\
$N_3 = k^2N_{ff}$ & $b~<~\Delta m_1~<~b + ka$,
$b~<~\Delta m_2~<~b + ka$\\
\end{tabular}$$
The purity of the sample was determined from the formula:
\begin{equation}
P = \frac{N_{tt}}{N} = 1-\frac{1}{N}\frac{N_1+N_2}{k}+\frac{N_3}{k^2}.
\label{PURITY}
\end{equation}
The parameter $a$ controls the width of the `signal region', $b$ defines the
lower bound and $k$ the width of the `control regions', respectively.
Their values have been chosen to be $a~=~6$~MeV, $b~=~16$~MeV, $k=~3$
(cf. Fig~\ref{MASSL0}).
It has been checked that the results of the analysis were stable while varying the
above parameters within the reasonable range.
The full information on the statistics of \al pairs sample is collected
in the Table~\ref{PAIRS}.
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(120,120)
%
\put(30,5){\Large\bf a}
\put(49,5){\Large\bf b}
\put(101,5){\Large\bf b+ka}
%
\put(3,30){\Large\bf a}
\put(3,50){\Large\bf b}
\put(-6,105){\Large\bf b+ka}
%
\put(-12,85){\huge\bf $\Delta m_2$}
\put(75,-5){\huge\bf $\Delta m_1$}
%
\put(18,18){\huge\bf N}
\put(16,78){\huge\bf N$_1$}
\put(72,18){\huge\bf N$_2$}
\put(72,78){\huge\bf N$_3$}
\epsfxsize=120mm
\epsfysize=120mm
\epsffile{plane_fig2.eps}
\end{picture}
\end{center}
\caption{The plane ($\Delta m_1$, $\Delta m_2$) with the definitions of the
`signal' and `control' regions as used in the determination of the purity
of $\Lambda\Lambda$ pairs.}
\label{PLANE}
\end{figure}
\begin{table}[ht]
\caption[]
{The statistics of \al pairs selected in data and in
simulation.}
$$\begin{tabular}{|c|r|r|}
\hline % table header
\multicolumn{1}{|c}{Sample} &
\multicolumn{2}{|c|}{Number of pairs in the `signal region'} \\
\hline % table header
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{~~~~~~data~~~~~~} &
\multicolumn{1}{c|}{simulation} \\
\hline
& & \\
$\Lambda\bar{\Lambda}$ & 3030 & 5154 \\
& & \\
\hline
& & \\
$\Lambda\Lambda$ and $\bar{\Lambda}\bar{\Lambda}$ & 620 & 1029 \\
$\Lambda\Lambda$ & 323 & 562 \\
$\bar{\Lambda}\bar{\Lambda}$ & 297 & 467 \\
& & \\
\hline
\end{tabular}$$
\label{PAIRS}
\end{table}
%#####################################################################
\section{Measurement of the spin composition}
%#####################################################################
The Eq.~\ref{IDEAL} has been modified as follows to account for the background
contribution in the signal region:
\begin{equation}
\left.\frac{dN}{d\ys}\right|_{signal} =
P\cdot \{ (1-\epsilon)\left.\frac{dN}{d\ys}\right|_{\rm S=0}
+\epsilon\left.\frac{dN}{d\ys}\right|_{\rm S=1} \}
+(1-P)\cdot (1+\kappa\cdot\ys).
\label{SIGNAL}
\end{equation}
The $N_1+N_2+N_3$ pairs from all three `control regions'
have been added together
to account for the background described by the last term in Eq.~\ref{SIGNAL}.
%\begin{equation}
%\left.\frac{dN}{d\ys}\right|_{bckg.} = (1+\kappa\cdot\ys).
%\label{BCKG}
%\end{equation}
The parametrisation of Eq.~\ref{SIGNAL} allows
the background to have the linear dependence
on~\ys with the slope $\kappa$ and the fit was performed simultaneously
to both signal and background angular distributions.
The slope of the background distribution was found however, to be consistent
with zero in the whole analysed Q region.
Therefore it was neglected in the final fits to reduce the number of unknowns
to be determined.
The fit was performed on event-by-event basis using the
maximum likelihood method in four Q bins.
The widths of bins in $Q$
have been chosen such that the population of
\al pairs is approximately equal in each of them.
% Alternatively to control the systematic
%uncertainties resulting from the fitting procedure the binned fit using
%the $\chi^2$ method was also carried on.
The results of the fits are shown in Fig~\ref{YSTAR} as dashed lines superimposed on
the experimental~\ys distributions.
The fit to the \lal sample in all Q intervals yields the $\epsilon$ values close
to 0.75, being consistent with flat angular distributions.
For the \lala sample the fitted $\epsilon$ values for $Q<1.8 GeV$ are different from
0.75, being consistent with the falling down \ys distributions of the binned data.
The fits to the \lala in the low Q bins with the $\epsilon$ fixed to 0.75 resulted
in the increased value of the logarithm of the likelihood function by more than
two units.
The values of the $\epsilon$
parameter determined from the fits together with the numbers of
\al pairs and the purities in five Q intervals
are collected in the Table~\ref{EPSILON} for the real and simulation data.
The systematic uncertainties due to
the statistical error of the sample's purity and
the decay asymmetry parameter $\alpha_{\Lambda}$
were studied by varying $P$ and $\alpha_{\Lambda}$ by
one standard deviation. As a measure of the uncertainty resulting
from the slope of the background the fit was performed to the signal and
background spectra simultaneously with the slope $\kappa$ as a free parameter.
The full information about the systematic uncertainties of the S~=~1
contribution is collected in the Table~\ref{SYST}. The individual sources have
been added in quadrature to estimate the overall systematic error
(cf. the last row of the Table~\ref{SYST}).
%%Denoting the
%%first(second) hyperon as $\Lambda_1$($\Lambda_2$), respectively the absolute
%%values of their mass differences w.r.t. to the value given by the Particle
%%Data Group (PDG)~\cite{PDG} were defined:
%%\begin{equation}
%%\delta m_i = \left| m_i - m_{PDG} \right|,~~~~~~~~~~~~~~~{\rm i}~=~1,2
%%\end{equation}
\begin{table}[ht]
\caption[]
{The values of the fraction of spin 1 contribution (parameter
$\epsilon$)
in the systems $\Lambda\bar{\Lambda}$ and
$\Lambda\Lambda$ resulting from the fits to the~\ys distributions
together with
the numbers of hyperon' pairs and the purities of their samples.}
$$\begin{tabular}{|l|r|r|r|r|r|}
% \hline % table header
\multicolumn{6}{c}{$\Lambda\bar{\Lambda}$ data} \\
\hline % table header
$Q$ range (GeV) &0.0 -- 1.2 &1.2 -- 1.8& 1.8 -- 2.4& 2.4 -- 3.0& 3.0 -- 5.0 \\
\hline
Number of pairs & 1219 & 816 & 459 & 237 & 238 \\
\hline
Purity (\%) & 96$\pm$4 & 96$\pm$5 & 95$\pm$7 & 95$\pm$9 & 93$\pm$9 \\
\hline
$\epsilon$ & 0.76 & 0.80 & 0.67 & 0.88 & 0.89 \\
$\Delta\epsilon$ (stat) & 0.09 & 0.12 & 0.16 & 0.22 & 0.23 \\
$\Delta\epsilon$ (syst) & 0.02 & 0.02 & 0.03 & 0.02 & 0.03 \\
\hline
\end{tabular}$$
$$\begin{tabular}{|l|r|r|r|r|r|}
% \hline % table header
\multicolumn{6}{c}{$\Lambda\Lambda$ data} \\
\hline % table header
$Q$ range (GeV) &0.0 -- 1.2 &1.2 -- 1.8& 1.8 -- 2.4& 2.4 -- 3.0& 3.0 -- 5.0 \\
\hline
Number of pairs & 109 & 111 & 103 & 91 & 138 \\
\hline
Purity (\%)& 73$\pm$12& 81$\pm$13& 81$\pm$13& 84$\pm$14& 93$\pm$11 \\
\hline
$\epsilon$ & 0.29 & 0.19 & 0.69 & 0.81 & 0.78 \\
$\Delta\epsilon$ (stat) & 0.42 & 0.35 & 0.37 & 0.37 & 0.29 \\
$\Delta\epsilon$ (syst) & 0.12 & 0.11 & 0.02 & 0.07 & 0.03 \\
\hline
\end{tabular}$$
$$\begin{tabular}{|l|r|r|r|r|r|}
% \hline % table header
\multicolumn{6}{c}{$\Lambda\bar{\Lambda}$ simulation} \\
\hline % table header
$Q$ range (GeV) &0.0 -- 1.2 &1.2 -- 1.8& 1.8 -- 2.4& 2.4 -- 3.0& 3.0 -- 5.0 \\
\hline
Number of pairs & 2041 & 1545 & 777 & 336 & 341 \\
\hline
Purity (\%) & 97$\pm$3 & 97$\pm$4 & 97$\pm$5 & 96$\pm$8 & 94$\pm$8 \\
\hline
$\epsilon$ & 0.78 & 0.70 & 0.75 & 0.76 & 0.95 \\
$\Delta\epsilon$ (stat) & 0.07 & 0.08 & 0.12 & 0.18 & 0.18 \\
\hline
\end{tabular}$$
$$\begin{tabular}{|l|r|r|r|r|r|}
% \hline % table header
\multicolumn{6}{c}{$\Lambda\Lambda$ simulation} \\
\hline % table header
$Q$ range (GeV) &0.0 -- 1.2 &1.2 -- 1.8& 1.8 -- 2.4& 2.4 -- 3.0& 3.0 -- 5.0 \\
\hline
Number of pairs & 187 & 205 & 172 & 143 & 216 \\
\hline
Purity (\%)& 73$\pm$12& 81$\pm$13& 81$\pm$13& 84$\pm$14& 93$\pm$11 \\
\hline
$\epsilon$ & 0.95 & 0.97 & 0.63 & 1.17 & 0.75 \\
$\Delta\epsilon$ (stat) & 0.28 & 0.26 & 0.27 & 0.29 & 0.26 \\
\hline
\end{tabular}$$
\label{EPSILON}
\end{table}
\begin{table}[ht]
\caption[]
{The systematic uncertainties of the fraction of spin 1 contribution
$\epsilon$ in the systems $\Lambda\bar{\Lambda}$ and $\Lambda\Lambda$.}
$$\begin{tabular}{|l|r|r|r|r|r|}
% \hline % table header
\multicolumn{6}{c}{$\Lambda\bar{\Lambda}$ data} \\
\hline % table header
$Q$ range (GeV) &0.0 -- 1.2 &1.2 -- 1.8& 1.8 -- 2.4& 2.4 -- 3.0& 3.0 -- 5.0 \\
\hline
$\Delta P~$ & 0.00 & 0.01 & 0.01 & 0.01 & 0.02 \\
$\Delta\kappa$ & 0.01 & 0.01 & 0.02 & 0.01 & 0.02 \\
$\Delta\alpha_{\Lambda}$ & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 \\
\hline
$\Delta\epsilon$ & 0.02 & 0.02 & 0.03 & 0.02 & 0.03 \\
\hline
\end{tabular}$$
$$\begin{tabular}{|l|r|r|r|r|r|}
% \hline % table header
\multicolumn{6}{c}{$\Lambda\Lambda$ data} \\
\hline % table header
$Q$ range (GeV) &0.0 -- 1.2 &1.2 -- 1.8& 1.8 -- 2.4& 2.4 -- 3.0& 3.0 -- 5.0 \\
\hline
$\Delta P~$ & 0.09 & 0.10 & 0.01 & 0.01 & 0.01 \\
$\Delta\kappa$ & 0.08 & 0.03 & 0.01 & 0.07 & 0.03 \\
$\Delta\alpha_{\Lambda}$ & 0.02 & 0.03 & 0.01 & 0.01 & 0.01 \\
\hline
$\Delta\epsilon$ & 0.12 & 0.11 & 0.02 & 0.07 & 0.03 \\
\hline
\end{tabular}$$
\label{SYST}
\end{table}
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(140,200)
\put(35,200){\huge\bf $\Lambda\bar{\Lambda}$}
\put(105,200){\huge\bf $\Lambda\Lambda$}
\put(55,185){\Large\bf $ 0 < Q < 1.2$~GeV}
\put(55,140){\Large\bf $ 1.2 < Q < 1.8$~GeV}
\put(55, 95){\Large\bf $ 1.8 < Q < 2.4$~GeV}
\put(55, 50){\Large\bf $ 2.4 < Q < 3.0$~GeV}
\put(69,5){\huge\bf $y^*$}
\epsfxsize=150mm
\epsfysize=200mm
\epsffile{fiteps_fig3.eps}
\end{picture}
\end{center}
\caption{The $y^*$ distributions (points with error bars) for the
$\Lambda\bar{\Lambda}$ (left part) and $\Lambda\Lambda$ (right part) pairs
in four intervals of the four-momentum transfer $Q$. The dashed lines
represent the results of an unbinned maximum likelihood fit described in the
text.}
\label{YSTAR}
\end{figure}
%
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(140,140)
\put(5,130){\huge\bf $\epsilon$}
\put(85,113){\large\bf DELPHI}
\put(127,2){\Large\bf $Q$~(GeV)}
\epsfxsize=140mm
\epsfysize=140mm
\epsffile{epsplot_fig4.eps}
\end{picture}
\end{center}
\caption{The distribution of the measured fraction of the S~=~1 spin state
contribution ($\epsilon$) to the $\Lambda\Lambda$ system versus the
four-momentum difference $Q$. The error bars correspond to the statistical
and systematic uncertainties added in quadrature.
The solid line represents the result of an unbinned maximum likelihood fit
to the Goldhaber parametrisation. The dashed line
corresponds to the statistical spin mixture.}
\label{EPSPLOT}
\end{figure}
%
%#####################################################################
\section{Summary and Conclusions}
We have measured the spin composition of \lal, \lala and \alal pairs
in the Q region extending from 0. to 5.0 GeV. For the \lal pairs the
spin composition was found to be compatible with a statistical mixture
of the S~=~1 and S~=~0 states. This disfavours the colour octet mechanism
of baryons production via a single gluon emission.
For the \lala and \alal pairs the depletion of the S~=~1 component was
found below Q~=~1.8~GeV, implying the dominance of the S~=~0 component.
Assuming that the observed depletion is due to the Pauli exclusion
principle, the dimension of the
\lala emitter may be extracted. The dependence of the
the $\epsilon$ parameter on $Q$ was parametrised using the formula
proposed by G. Goldhaber
which was originally aimed to describe
the Bose-Einstein correlations:
\begin{equation}
\epsilon(Q) = 0.75\cdot (1 - e^{- R^2\cdot Q^2})
\label{GOLDHABER}
\end{equation}
where the parameter $R$ measures the size of the emitter.
The determined $\epsilon(Q)$ has been fitted event-by-event using the maximum
likelihood method.
The result of the fit is shown in Fig~\ref{EPSPLOT}.
The systematic uncertainty of $R$ has been evaluated in the same way as for the
parameter $\epsilon$.
The fitted value of the emitter size is:
$R = 0.11^{+0.05}_{-0.03}(stat)\pm 0.01(syst)$~fm, or expressed as the upper limit :
$R<0.2$ fm at 95\% c.l.
The result is in agreement with the OPAL measurement~\cite{OPAL} performed
in the region $Q>0.5 GeV$ and the preliminary result of
the ALEPH~\cite{ALEPH}.
\section*{Acknowlegments}
\vskip 3mm
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. We also thank
A.Bialas, J.Turnau and K.Zalewski for many useful discussions on the subject.
This work was supported in part by the KBN Grant 621/E78/SPUB/P03/113/97.
\newpage
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