Analysis of the $\Lambda p$ Final State Interaction\\ in the Reaction $p+p\to K^+(\Lambda p)$.
F. Hinterberger\inst{1}
A. Sibirtsev\inst{2,3}
The missing mass spectrum measured in high resolution studies of the reaction $pp\to K^+X$ is analyzed with respect to the strong final state interaction near the $\Lambda{p}$ production threshold. The observed spectrum can be described by factorizing the reaction amplitude in terms of a production amplitude and a final state scattering amplitude. Parametrizing the $\Lambda{p}$ final state interaction in terms of the inverse Jost function allows a direct extraction of the low-energy phase-equivalent potential parameters. Constraints on the singlet and triplet scattering lengths and effective ranges are deduced in a simultaneous fit of the $\Lambda{p}$ invariant mass spectrum and the total cross section data of the free $\Lambda{p}$ scattering using the effective range approximation. % \PACS{ {13.75.Ev} {Hyperon-nucleon interactions} \and {21.30.Fe} {Forces in hadronic systems and effective interactions} \and {24.10.-i} {Nuclear reaction models and methods} \and {25.40.-h} {Nucleon-induced reactions} \and {25.80.Pw} {Hyperon-induced reactions} } % end of PACS codes
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%
%\documentclass[epj,referee]{svjour}
\documentclass[epj]{svjour}
% Remove option referee for final version
\usepackage{psfig}
\begin{document}
%
\title{Analysis of the $\Lambda p$ Final State Interaction\\ 
in the 
Reaction $p+p\to K^+(\Lambda p)$.}
\author{F. Hinterberger\inst{1} \and A. Sibirtsev\inst{2,3}}   

\institute{Helmholtz-Institut f\"ur Strahlen- und Kernphysik, 
Universit\"at Bonn, Nu{\ss}allee 14-16, D-53115, Bonn, Germany
\and Institut f\"ur Kernphysik,   Forschungszentrum J\"ulich, 
D-52425 J\"ulich, Germany \and
Special Research Center for the Subatomic Structure of Matter (CSSM) 
and Department of Physics and Mathematical Physics, 
University of Adelaide, SA 5005, Australia  
}
%
\date{Received: date / Revised version: date}
% The correct dates will be entered by Springer
%
\abstract{ 
The missing mass spectrum measured in high resolution studies of
the reaction $pp\to K^+X$ is analyzed with respect to the
strong final state interaction near the $\Lambda{p}$ production threshold. The
observed spectrum can be described by factorizing the reaction amplitude in
terms of a production amplitude and a final state scattering amplitude.
Parametrizing the $\Lambda{p}$ final state interaction in terms of
the inverse Jost function allows a direct extraction of the low-energy
phase-equivalent potential parameters. Constraints on the singlet and triplet
scattering lengths and effective ranges are deduced in a simultaneous fit of
the $\Lambda{p}$ invariant mass spectrum and the total
cross section data of the
free $\Lambda{p}$ scattering using the effective range approximation.
%
\PACS{ {13.75.Ev} {Hyperon-nucleon interactions} \and
       {21.30.Fe} {Forces in hadronic systems and effective interactions} \and
       {24.10.-i} {Nuclear reaction models and methods} \and
       {25.40.-h} {Nucleon-induced reactions} \and
       {25.80.Pw} {Hyperon-induced reactions}   
     } % end of PACS codes
} %end of abstract
%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction.}
Experimental information on the $\Lambda{p}$ interaction has been derived from
the analysis of hypernuclei, $\Lambda{p}$ scattering experiments and studies
of the $\Lambda{p}$ final state interaction (FSI) in strangeness transfer reactions.
The binding energy of light hypernuclei shows that the low-energy $\Lambda{p}$
interaction is attractive. In addition, the $\Lambda{p}$ interaction is
spin-dependent and the singlet interaction is stronger than the triplet
one~\cite{fhint:dal65,fhint:dal81,fhint:dov84}.  The free $\Lambda{p}$ scattering was studied in
bubble chamber measurements~\cite{fhint:ale68,fhint:sec68,fhint:ale69}.  In
the low momentum region the elastic cross sections were analyzed in terms of
the $S$-wave singlet and triplet scattering lengths and effective
ranges. However these determinations are characterized by large variances and
covariances since the data only support the determination of a spin-averaged
scattering length and effective range~\cite{fhint:ale69}.

The strong effect due to the $\Lambda{p}$ FSI was observed
in strangeness transfer reactions~\cite{fhint:tai69,fhint:pig77,fhint:pie88}
and and in associated strangeness production
reactions~\cite{fhint:mel65,fhint:ree68,fhint:hog68,fhint:sie94,fhint:bal98,fhint:bil98,fhint:mag01}. The
$\Lambda{p}$ production in the $K^-d{\to}\pi^-\Lambda{p}$,
$\pi^+d{\to}K^+\Lambda{p}$, $\gamma{d}{\to}K^0\Lambda{p}$ and
$pp{\to}K^+\Lambda{p}$ reactions can provide substantial improvement in an
evaluation of the low energy $\Lambda{p}$ scattering.

Intensive theoretical studies of the hyperon-nucleon interaction with the
Nijmegen~\cite{fhint:nag77,fhint:nag79,fhint:mae89,fhint:sto99} and
J\"ulich~\cite{fhint:hol89,fhint:reu94,Haidenbauer} potential models predict
the hyperon-nucleon potentials and phase shift parameters.  These models also
predict the sing\-let and triplet scattering length and effective range
parameters of the free $S$-wave $\Lambda{p}$ interaction.

The present paper refers to the associated strangeness reaction
$pp{\to}K^+(\Lambda{p})$ which is characterized by a strong FSI 
near the $\Lambda{p}$ production threshold.  In most
experiments the $\Lambda{p}$ system was measured inclusively.  
Exclusive measurements of the $pp{\to}K^+\Lambda{p}$ reaction were performed
at COSY~\cite{fhint:bal98,fhint:bil98} and
SATURNE~\cite{fhint:mag01}. Theoretical analyses of the reactions were
done~\cite{fhint:del89,fhint:lag91,Sibirtsev1,fhint:fae97,fhint:sib98,Sibirtsev2,Tsushima1,Sibirtsev3,Shyam1,Gasparian1,fhint:kel00,Shyam2,Gasparian2,Gasparian3}
by applying the meson exchange model and including the $\Lambda{p}$
FSI generally modelled by Nijmegen or J\"ulich potentials.

The aim of the present paper is to perform an analysis of the $\Lambda{p}$ FSI
in the reaction $pp{\to}K^+(\Lambda{p})$ and an evaluation of the $\Lambda{p}$
low energy interaction parameters. Using the Watson-Migdal
approximation~\cite{wat52,mig55,gol64,Gillespie} the reaction amplitude is factorized in
terms of a production matrix element and a FSI enhancement factor, which can be
represented by the inverse Jost function~\cite{jos47,jos52,bar49a,bar49b}. We
analyze experimental results from SATURNE collected by Siebert et
al.~\cite{fhint:sie94}, which are characterized by a high statistical accuracy
and a high invariant mass resolution. Furthermore, in the fitting procedure
the missing mass resolution is taken into account by folding the theoretical
expressions with the experimental resolution function.

The analysis shows that the shape of the sharply rising invariant mass
spectrum depends strongly on the singlet and triplet scattering length and
effective range parameters. But only two parameters, the spin-averaged
scattering length and effective range parameters, can be deduced within an
acceptable confidence level by fitting the $\Lambda{p}$ missing mass
spectrum.  Additional information can be obtained by taking the total cross
section data for the free $\Lambda{p}$ scattering into account and including
these data in an overall fit.  At low energies the total cross section can be
described in a model-independent way using the effective range
approximation~\cite{fhint:sch47,fhint:bet49,fhint:bla54}.  Thus, by fitting
simultaneously the $\Lambda{p}$ invariant mass spectrum and the available total cross
section data of the free $\Lambda{p}$ scattering severe constraints
on the singlet and triplet scattering length and effective range parameters
can be deduced.  This method allows also to test theoretical model
predictions.

\section{The formalism.}
\subsection{Phase space distribution.}
The $pp{\to}K^+\Lambda{p}$ double differential cross section is given as
\begin{equation}
\frac{d^2\sigma}{d\Omega_{K}dM_{\Lambda{p}}}= |{\tilde{\cal M}|^2} \, \Phi_3 ,
\end{equation}
where ${\tilde {\cal M}}$ is the Lorentz-invariant reaction amplitude and the 
three-body phase space distribution function is 
\begin{equation}
\Phi_3{=} \frac{\pi}{16(2\pi)^5} \frac{p_{K}^2 q}{p_p m_p
[(E_{p}+m_p)p_{K}-E_{K}p_{p}\cos\theta_{K}]},
\label{phase}
\end{equation}
where $q$ is the momentum of $\Lambda$ in the Gottfried-Jackson rest-frame of the
produced two-particle subsystem $X{=}\Lambda{+}p$,  $M_{\Lambda{p}}$ is the
corresponding invariant mass and $p_p$, $E_p$, $p_K$, $E_K$, $\theta_K$,
$\Omega_K$ are defined in the laboratory system. Obviously, in inclusive
measurements the invariant mass $M_{\Lambda{p}}$ is equal to the missing mass
$M_X$ below the $\Sigma$-hyperon production threshold.  Eq.(\ref{phase}) is
consistent with the kinematical definitions of
Refs.~\cite{fhint:byc73,fhint:pdg98}.

\subsection{Final state interaction.}
In the Watson-Migdal approximation~\cite{wat52,mig55,gol64} the FSI is taken
into account by introducing a FSI enhancement factor $|C_{FSI}|^2$,
\begin{equation}
\frac{d^2\sigma}{d\Omega_{K} dM_{\Lambda{p}}}= |{\cal M}|^2 \,
|C_{FSI}|^2 \, \Phi_3,
\label{watson}
\end{equation}
where now ${\cal M}$ is pure production matrix element and 
the FSI amplitude $C_{FSI}$ depends on the internal momentum $q$ of the
$\Lambda{p}$ subsystem. It converges to 1 for $q{\to}\infty$ where the
$S$-wave FSI enhancement vanishes.

Applying the factorization we assume that the production
operator ${\cal M}$ is constant, i.e. does not depend 
on the internal kinetic energy of the $\Lambda{p}$ subsystem.
%any variable defined
%under the phase space integration. 
In case of the $pp{\to}K^+\Lambda{p}$
reaction this assumption is supported by the kinematics which provides
a focus onto the $\Lambda{p}$ FSI. The internal kinetic energy
of the $\Lambda{p}$ subsystem is almost zero 
near the $\Lambda{p}$ threshold whereas
the $K^+ \Lambda$- and $K^+{p}$-subsystems have
large internal kinetic energies.
Even if the $pp{\to}K^+\Lambda{p}$ reaction is dominated
\cite{Sibirtsev2,Tsushima1,Sibirtsev3,Shyam1} by 
intermediate baryonic resonances coupled to
the $K^+\Lambda$ system a small variation of
the invariant $\Lambda{p}$ mass does practically not affect
the production amplitude.




%In case of the $pp{\to}K^+\Lambda{p}$
%reaction it might be incorrect, since reaction can be 
%dominated~\cite{Sibirtsev2,Tsushima1,Sibirtsev3,Shyam1} by
%intermediate baryonic resonances coupled to the $K^+\Lambda$ system. In that
%case the projection of the Dalitz plot on the invariant mass of the
%$\Lambda{p}$ system would explicitly reflect the  $K^+\Lambda$  resonance
%structures and introduce partial distortion of the {\it pure}   $\Lambda{p}$
%distribution. It is not obvious whether other dependencies will produce
%kinematical reflections and distort the $\Lambda{p}$ invariant mass spectrum.

The methods for studying the FSI between the particles have
been developed in different areas of physics, ranging
from atomic physics to high energy particle physics \cite{Gillespie}. 
Taking the inverse Jost function~\cite{jos47,jos52} the correction due to the
FSI is given as 
\begin{equation}
C_{FSI}=\frac{q-i\beta}{q+i\alpha},\; \; \;
|C_{FSI}|^2=\frac{q^2+\beta^2}{q^2+\alpha^2}.
\end{equation} 
The potential parameters $\alpha$ and $\beta$ can be used to establish
phase-equivalent Bargmann potentials~\cite{bar49a,bar49b}.  They are related
to the scattering lengths $a$, and effective ranges $r$ of the low-energy
$S$-wave scattering
\begin{equation}
\alpha=\frac{1}{r}\left(1-\sqrt{1-2\frac{r}{a}}\right),\; \; \; \beta
=\frac{1}{r}\left(1+\sqrt{1-2\frac{r}{a}}\right).
\end{equation}

The $\Lambda{p}$ system can couple to singlet $^1S_0$ and triplet $^3S_1$
states. Near production threshold the singlet-triplet transitions due to the
final state interaction cannot occur. Therefore, the contributions of the
spin-singlet and spin-triplet final states can be added incoherently.  Taking
the spin-statistical weights into account the unpolarized double differential
cross section may be written as
\begin{eqnarray}
\frac{d^2\sigma}{d\Omega_{K}dM_{\Lambda{p}}}= \Phi_3 \left[\, 0.25 \, |{\cal
M}_s|^2 \, \frac{q^2+\beta_s^2}{q^2+\alpha_s^2} \right. \nonumber \\
{+}\left.\, 0.75 \, |{\cal M}_t|^2 \,
\frac{q^2+\beta_t^2}{q^2+\alpha_t^2}\right].
\label{miss}
\end{eqnarray}
This equation leaves six free parameters, the singlet and triplet potential
parameters $\alpha_s$, $\beta_s$, $\alpha_t$, $\beta_t$ and the production
matrix elements $|{\cal M}_s|$ and $|{\cal M}_t|$.  Instead of the parameters
$\alpha_s$, $\beta_s$, $\alpha_t$ and $\beta_t$ one can equally well use the
singlet and triplet scattering length and effective range parameters $a_s$,
$r_s$, $a_t$ and $r_t$.  The functional dependence on the invariant mass
$M_{\Lambda{p}}$ can be evaluated by inserting the corresponding expression for the
internal momentum $q$ of the $\Lambda{p}$ system,
\begin{equation}
q=\frac{\sqrt{M_{\Lambda{p}}^2-(m_\Lambda+m_p)^2}
\sqrt{M_{\Lambda{p}}^2-(m_\Lambda-m_p)^2}} {2M_{\Lambda{p}}}.
\end{equation}

\subsection{Missing mass resolution.}
The theoretical missing mass spectrum of Eq.(\ref{miss}) has to be folded with
the missing mass resolution function before comparing with the
data. Introducing the missing mass resolution function $g({\tilde
M}_{\Lambda{p}}{-}M_{\Lambda{p}})$ and denoting the r.h.s. of Eq.(\ref{miss})
by $f_0(M_{\Lambda{p}})$ yields the distribution function $f(M_{\Lambda{p}})$
which has to be compared with the data,
\begin{equation}
f(M_{\Lambda{p}})=\int \!\!f_0({\tilde M}_{\Lambda{p}}) \, g({\tilde
M}_{\Lambda{p}}{-}M_{\Lambda{p}}) \, d{\tilde M}_{\Lambda{p}}
\label{res}
\end{equation}
where the resolution function is given as
\begin{equation}
g({\tilde M}_{\Lambda{p}}{-}M_{\Lambda{p}}){=} \frac{1}{\sqrt
{2\pi}}\frac{1}{\sigma_M} \exp-\frac{({\tilde
M}_{\Lambda{p}}{-}M_{\Lambda{p}}{+} \Delta M_{\Lambda{p}})^2}{2\sigma_{M}^2}.
\end{equation}
Here, $\sigma_{M}$ denotes the one standard deviation width and the shift
$\Delta M_{\Lambda{p}}$ takes a systematic calibration error of the invariant
mass scale into account.  For the comparison with the SATURNE data the
values $\Delta M_{\Lambda{p}}{=}1.7$~MeV and 
$\sigma_M{=}2.0$~MeV \cite{fhint:sie94} are used.

\subsection{The  $\Lambda{p}$ cross section.} 
In the effective range
approximation~\cite{fhint:sch47,fhint:bet49,fhint:bla54} 
the total
cross section of the free $\Lambda{p}$ elastic scattering can be expressed in
terms of the singlet and triplet scattering length and effective range
parameters $a_s$, $a_t$, $r_s$ and $r_t$,
\begin{equation}
\sigma_{\Lambda{p}}{=} \frac{\pi}{q^2{+}\left(-\frac{1}{a_s}{+}\frac{r_s
q^2}{2}\right)^2} {+} \frac{3\pi}{q^2{+}\left(-\frac{1}{a_t}{+}\frac{r_t
q^2}{2}\right)^2}.
\label{cross}
\end{equation}
Here, $q$ is the cm-momentum of the $\Lambda{p}$ scattering.  This equation
can be applied at low energies where $S$-wave contributions dominate.

\section{The results of the fits.}
The program Minuit of the CERN program library~\cite{fhint:cer94} was used in
order to perform nonlinear least-square fits.  
We fit the experimental results ~\cite{fhint:sie94} for missing mass spectrum from
$pp{\to}K^+X$ reaction measured at proton beam energy $T_p{=}2.3$~GeV and kaon
emission angle of $\theta_K{=}10.3^\circ$ using
Eqs.(\ref{miss}) and (\ref{res}), respectively. 
%The $\Lambda$ and $\Sigma$-hyperon production thresholds corresponding to
%$M_X{\simeq}$2054 and 2128~MeV can be clearly identified in the missing mass
%spectrum shown in Fig.\ref{spectrum}.
We fit the strong FSI enhancement near the $\Lambda{p}$
threshold assuming that the $S$-wave FSI is dominant.
Therefore, we take only data corresponding
to very low c.m. momenta  $q$ in the $\Lambda{p}$ system.
The fits include 29 experimental points of the missing
mass spectrum from the $\Lambda{p}$ threshold at 2054~MeV up to the invariant
mass of the $\Lambda{p}$ system of 2095~MeV.  This $M_{\Lambda{p}}$
range corresponds to c.m.  momenta $q$ in the $\Lambda{p}$ system between 0
and 200 MeV/c. 


Low energy total $\Lambda{p}$
cross section data from bubble chamber
measurements~\cite{fhint:ale68,fhint:sec68,fhint:ale69} are fitted  
simultaneously using
Eq.(\ref{cross}). The fits include 12 total
$\Lambda{p}$ cross section data points covering c.m. momentum range
60${\le}q{\le}$135~MeV/c. 

%\begin{figure}[b]\vspace*{-7mm}
%\psfig{file{=}sigtot.ps,width=9.2cm,height=9.2cm}\vspace*{-3mm}
%\caption{Total $\Lambda{p}$ cross 
%section\cite{fhint:ale68,fhint:sec68,fhint:ale69} as a function of the
%c.m. momentum $q$. The solid line shows the result of the three-parameter
%fit with an averaged  parameters given by Eq.(\ref{fit1}).
%The dashed and dotted lines indicate singlet and triplet contributions,
%respectively.}
%\label{sigtot}
%\end{figure}


%It is worthwhile to mention that the number of experimental points available
%from the $pp{\to}K^+X$ reaction at small $q$ is substantially larger than
%that from the  $\Lambda{p}{\to}\Lambda{p}$ measurements. Also the
%statistical accuracy of the missing mass spectrum is
%substantially higher.
%Therefore the  data on  $\Lambda{p}$ invariant mass spectrum seem very
%attractive to evaluate the  low-energy phase-equivalent 
%potential parameters as far as the evaluation  remains model
%independent. 
It is worthwhile to mention that the  $pp{\to}K^+X$ reaction
provides high quality data even at very small $q$ near the
$\Lambda p$ threshold whereas $\Lambda{p}{\to}\Lambda{p}$ data
are available only for $q{\geq}60$~MeV/c. 
Also the number of data and the accuracy of the $pp{\to}K^+X$
measurement is substantially higher.
%Therefore the $pp{\to}K^+X$ data seem very
%attractive to evaluate the  low-energy phase-equivalent 
%potential parameters as far as the evaluation  remains model
%independent. 
Therefore it is very attractive to include the $pp{\to}K^+X$ data 
in the evaluation of the  low-energy phase-equivalent 
potential parameters as far as the evaluation  remains model
independent. 

The quality of the least-square fits is given by total $\chi^2$ and reduced
$\chi^2/ndf$ where $ndf$ is the number of degrees of freedom given by the
number of data minus the number of the fit parameter. In calculating the
parameter errors nonlinearities and parameter correlations were taken into
account. The error for a given parameter is defined as the change of that
parameter which causes $\chi^2$ to increase by one while re-fitting all other
free parameters.  Due to the nonlinearities the resulting error intervals are
in general asymmetric.



\subsection{Three-parameter fit.}


In a three-parameter fit we determine  spin-averaged parameters 
by applying the 
constraints
\[
|{\cal M}_s|^2{=}|{\cal M}_t|^2{=}|\bar{{\cal M}}|^2,\; \; 
a_s{=}a_t{=}\bar{a},\; \; r_s{=}r_t{=}\bar{r}.
\]
The three-parameter fit yields the production matrix element
squared $|\bar{\cal M}|^2$ and the spin-averaged scattering length $\bar{a}$
and effective range $\bar{r}$.

In a first step we fit only the missing mass spectrum
without taking the $\Lambda p$  total cross section data into account.
The three-parameter fit yields an excellent description
of the missing mass spectrum 
but fails completely to reproduce the total
cross section data ($\chi^2/ndf{=}18.0$, see Fig.~\ref{mm3cross0}).
The fit parameters of the missing mass spectrum are
\begin{eqnarray}
|\bar{\cal M}|^2&=&15.4_{-1.6}^{+1.5}\;{\rm b/sr},\; \chi^2/ndf=0.98 \nonumber \\
\bar{a}&=&-2.57_{-0.23}^{+0.20}\; {\rm fm},\; 
\bar{r}=2.47_{-0.24}^{+0.23}\; {\rm fm}.  
\label{fitmm3cross0}
\end{eqnarray}
\begin{figure}[h!]\vspace*{-3mm}
\begin{center}
\psfig{file=spectrum11.ps,width=9cm,height=8cm}\vspace*{-10mm}
\psfig{file=sigtot00.ps,width=9cm,height=8cm}\vspace*{-3mm}
\end{center}
\caption{Top: Missing mass spectrum  of the reaction $pp{\to}K^+X$
 measured at $T_p{=}2.3~GeV$
and  $\theta_K{=}10.3^\circ$ \cite{fhint:sie94}. 
The upper axis indicates
the c.m.  momentum $q$ of the $\Lambda{p}$ system. 
Bottom: Total $\Lambda{p}$ cross 
section\cite{fhint:ale68,fhint:sec68,fhint:ale69} as a function of the
c.m. momentum $q$. 
Solid lines: Fit curves with parameters given by
Eq.(\ref{fitmm3cross0}) from a 
three-parameter fit of the missing mass spectrum alone, 
dashed line: phase space distribution,
dotted lines: singlet contributions,
dashed-dotted lines: triplet contributions.}
\label{mm3cross0}
\end{figure}
The dotted and dashed-dotted lines in Fig. \ref{mm3cross0}
and in the following Figs. \ref{mm1cross2} - \ref{mm5cross2}
show the corresponding singlet and triplet contributions.
The dashed lines show the phase space distributions
without the FSI enhancement factor, i.e.
$(0.25|{\cal M}_s|^2 + 0.75|{\cal M}_t|^2)\Phi_3$.



Vice versa, we take only
the total cross section data into account and
determine $\bar{a}$ and $\bar{r}$ in a two-parameter fit.
The resulting fit parameters are
\begin{equation}
\bar{a}=-1.81_{-0.21}^{+0.18}\; {\rm fm},\; 
\bar{r}=3.24_{-0.48}^{+0.48}\; {\rm fm},\; 
\chi^2/ndf=0.39.
\label{cross2}
\end{equation}
Taking those parameters fixed and fitting only $|\bar{\cal M}|^2$
yields for the missing mass spectrum
\begin{equation}
|\bar{\cal M}|^2=19.5_{-0.2}^{+0.2}\;{\rm b/sr},\; 
\chi^2/ndf=3.7.
\label{fitmm1cross2}
\end{equation}
This procedure yields an excellent fit
of the total cross section data but fails completely to describe the
missing mass spectrum (see Fig.~\ref{mm1cross2}).
\begin{figure}[h!]\vspace*{-3mm}
\begin{center}
\psfig{file=spectrum00.ps,width=9cm,height=8cm}\vspace*{-10mm}
\psfig{file=sigtot11.ps,width=9cm,height=8cm}\vspace*{-3mm}
\end{center}
\caption{
Same as Fig.~\ref{mm3cross0}.
Solid lines: Fit curves with $\bar{a}$ and $\bar{r}$ given by
Eq.~(\ref{cross2}) from a 
two-parameter fit of the total cross section data alone
and  $|\bar{\cal M}|^2$ given by Eq.~(\ref{fitmm1cross2}), 
dashed line: phase space distribution,
dotted lines: singlet contributions,
dashed-dotted lines: triplet contributions.}
\label{mm1cross2}
\end{figure}

In a next step, we determine spin-averaged parameters
in a combined fit, i.e. by fitting simultaneously
the missing mass spectrum and the total cross section data.
The resulting parameters are
\begin{eqnarray}
|\bar{\cal M}|^2&=&16.9_{-1.2}^{+1.2}\;{\rm b/sr},\; 
\chi^2/ndf=2.2, \nonumber \\
\bar{a}&=&-1.91_{-0.11}^{+0.10}\; {\rm fm},\; 
\bar{r}=2.74_{-0.20}^{+0.20}\; {\rm fm}.
\label{fitmm3cross2}
\end{eqnarray}
This procedure fails to  describe both,
the missing mass spectrum and 
the total cross section data (see Fig.~\ref{mm3cross2}).
This failure is a direct indication that the spin dependence
of the $\Lambda{p}$ interaction must be taken into account.

\begin{figure}[h!]\vspace*{-3mm}
\begin{center}
\psfig{file=spectrum22.ps,width=9cm,height=8cm}\vspace*{-10mm}
\psfig{file=sigtot22.ps,width=9cm,height=8cm}\vspace*{-3mm}
\end{center}
\vspace*{-5mm}
\caption{
Same as Fig.~\ref{mm3cross0}.
Solid lines: Fit curves with parameters given by
Eq.(\ref{fitmm3cross2}) from a combined  
three-parameter fit of the missing mass spectrum and the 
total cross section data, 
dashed line: phase space distribution,
dotted lines: singlet contributions,
dashed-dotted lines: triplet contributions.}
\label{mm3cross2}
\end{figure}

\begin{figure}[h!]
\vspace*{-3mm}
\begin{center}
\psfig{file=spectrum33.ps,width=9cm,height=8cm}\vspace*{-10mm}
\psfig{file=sigtot33.ps,width=9cm,height=8cm}\vspace*{-3mm}
\end{center}
\vspace*{-5mm}
\caption{
Same as Fig.~\ref{mm3cross0}.
Solid lines: Fit curves with parameters given by
Eq.(\ref{fitmm5cross2}) from a combined  
five-parameter fit of the missing mass spectrum and the 
total cross section data, 
dashed line: phase space distribution,
dotted lines: singlet contributions,
dashed-dotted lines: triplet contributions.}
\label{mm5cross2}
\end{figure}




\subsection{Five-parameter fit.}

Now the data on total  $\Lambda{p}$ cross section and $pp{\to}K^+X$
missing mass spectrum are fitted in a combined fit
with the singlet and triplet scattering lengths and effective ran\-ges
$a_s$, $r_s$, $a_t$, $r_t$ 
as separate free parameters.
Taking the unknown quantities  
$|{\cal M}_s|^2$ and $|{\cal M}_t|^2$
into account
a six-parameter fit should be performed.
However, it turned out that the $\chi^2$-criterion cannot be
used to determine simultaneously $|{\cal M}_s|^2$ and $|{\cal M}_t|^2$.
This is due to the fact that the resulting $\chi^2$ 
depends only weakly on the ratio
$|{\cal M}_t|^2{/}|{\cal M}_s|^2$ as is indicated in
Table~\ref{tab1}. 
Therefore, five-parameter fits were performed
taking  $|{\cal M}_s|^2$ as  free parameter and 
the ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$
as  fixed parameter.  
By this method valuable
constraints on the
singlet and triplet scattering lengths
and effective ranges can be deduced from the data.





It should be mentioned that the spin-statistical weights 0.25 and 0.75 of the
singlet and triplet contributions have already been taken
into account in the theoretical ansatz (\ref{miss}).
Therefore the five-parameter search was started with the constraint
$|{\cal M}_t|^2/|{\cal M}_s|^2{=}1$ yielding
a solution with $\chi^2/ndf{=}0.82$.
The resulting parameters are
\begin{eqnarray}
|{\cal M}_s|^2&=&15.0_{-1.6}^{+1.4}\;{\rm b/sr},\; 
\chi^2/ndf=0.82, \nonumber \\
a_s&=&-3.2_{-0.6}^{+0.4}\; {\rm fm},\; 
r_s=1.25_{-0.15}^{+0.13}\; {\rm fm}, \nonumber \\
a_t&=&-1.3_{-0.5}^{+0.4}\; {\rm fm},\; 
r_t=5.4_{-1.6}^{+1.6}\; {\rm fm}.
\label{fitmm5cross2}
\end{eqnarray}
Both,  the missing mass spectrum and
the total cross section data (see Fig.\ref{mm5cross2})
are perfectly reproduced.
Then, the ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$
was varied over a wide range  of values
and best fit solutions of similar quality with 
$\chi^2/ndf$ varying between 0.80 and 0.83 were found.
The resulting  fit parameters are listed in 
Table~\ref{tab1}
for $0{\leq} |{\cal M}_t|^2/|{\cal M}_s|^2{\leq} 8$.




A characteristic feature of the best fit solutions
is the fact that the singlet FSI enhancement factor at $q{=}0$
is much larger than the triplet one,
e.g. $\beta_s^2/\alpha_s^2{=}49.1$ and  $\beta_t^2/\alpha_t^2{=}3.9$
for $|{\cal M}_t|^2/|{\cal M}_s|^2{=}1$.
In spite of the statistical weight 0.25 the 
singlet contribution dominates the
${\rm p}{\rm p}{\to}{\rm K}^+ \Lambda {\rm p}$ cross section 
at $q{=}0$,
e.g. $0.25|{\cal M}_s|^2\beta_s^2/\alpha_s^2{=}183$ b/sr
and $0.75|{\cal M}_t|^2\beta_t^2/\alpha_t^2{=}43.5$~b/sr
for $|{\cal M}_t|^2/|{\cal M}_s|^2{=}1$.
Solutions where  the triplet contribution is
larger than the singlet contribution  
do not fit the FSI enhancement of the
missing mass spectrum near $q=0$
and can be excluded on the
basis of the $\chi^2$ criterion. 
This holds true even if one varies the ratio
$|{\cal M}_t|^2/|{\cal M}_s|^2$ in a wide range.


The resulting best fit parameters are shown in Fig.~\ref{correl3}
as a function of the ratio  $|{\cal M}_t|^2/|{\cal M}_s|^2$.
\begin{figure}[b]\vspace*{-7mm}
\hspace*{-1mm}\psfig{file=correl33.ps,width=9.2cm,height=9.2cm}\vspace*{-3mm}
\caption{The singlet (circles) and triplet (squares) scattering lengths and
effective ranges as a function of the ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$
resulting from the overall five-parameter fit to the total $\Lambda{p}$ cross
section and missing mass spectrum measured in $pp{\to}K^+X$ reaction.}
\label{correl3}
\end{figure}
The
variation of the ratio  $|{\cal M}_t|^2/|{\cal M}_s|^2$
causes rather small 
\begin{table*}[t]
\begin{center}
\caption{Five-parameter fit results for $a_s$, $r_s$,
$a_t$, $r_t$ and $|{\cal M}_s|^2$
%and for  an overall production matrix element squared $|{\cal M}_s|^2$  and 
and different ratios $|{\cal M}_t|^2/|{\cal M}_s|^2$ of the triplet and
singlet production matrix elements.}
\label{tab1}
\begin{tabular}{cccccccc}
\hline\noalign{\smallskip}
$|{\cal M}_t|^2/|{\cal M}_s|^2$&$ |{\cal M}_s|^2$ (b/sr) &
$a_s$   (fm) & $r_s$ (fm) &
$a_t$ (fm) & $r_t$ (fm) & $\chi^2$
& $\chi^2/ndf$\\
\noalign{\smallskip}\hline\noalign{\smallskip}
0.00 & $61.4_{-6.3}^{+5.9}$ & 
$-2.6_{-0.2}^{+0.2}$ & $2.47_{-0.24}^{+0.23}$ & 
$-1.5_{-0.3}^{+0.2}$ & $3.6_{-0.9}^{+0.9}$ &
29.0  & 0.805\\
0.10 & $46.7_{-4.5}^{+4.2}$ & 
$-2.7_{-0.4}^{+0.3}$ & $2.25_{-0.24}^{+0.23}$ & 
$-1.5_{-0.3}^{+0.3}$ & $3.8_{-1.0}^{+1.0}$ &
29.0 & 0.807\\
0.25 & $34.4_{-3.2}^{+3.0}$ & 
$-2.9_{-0.5}^{+0.3}$ & $1.97_{-0.23}^{+0.22}$ & 
$-1.4_{-0.4}^{+0.3}$ & $4.0_{-1.2}^{+1.1}$ &
29.1 & 0.809\\
0.50 & $24.0_{-2.3}^{+2.1}$ & 
$-3.1_{-0.6}^{+0.4}$ & $1.63_{-0.19}^{+0.18}$ & 
$-1.3_{-0.4}^{+0.4}$ & $4.5_{-1.3}^{+1.3}$ &
29.2 & 0.812\\
%0.75 & $18.5_{-2.2}^{+1.8}$ &
%$-3.2_{-0.7}^{+0.5}$ & $1.41_{-0.19}^{+0.17}$ &
%$-1.29_{-0.49}^{+0.40}$ & $4.9_{-1.6}^{+1.5}$ &
%1.14\\
1.00 & $15.0_{-1.6}^{+1.4}$ &
$-3.2_{-0.6}^{+0.4} $ & $1.25_{-0.15}^{+0.13}$ &
$-1.3_{-0.5}^{+0.4}$ & $5.4_{-1.}^{+1.6}$ &
29.4 & 0.816\\
%1.25 & $12.7_{-1.6}^{+1.3}$ &
%$-3.3_{-0.7}^{+0.5}$ & $1.13_{-0.15}^{+0.14}$ &
%$-1.24_{-0.56}^{+0.44}$ & $5.7_{-1.8}^{+1.8}$ &
%1.14\\
%1.50 & $11.0_{-1.4}^{+1.1}$ & 
%$-3.3_{-0.7}^{+0.5}$ & $1.04_{-0.14}^{+0.12}$ &
%$-1.23_{-0.59}^{+0.45}$ & $6.0_{-2.0}^{+1.9}$ &
%1.15\\
2.00 & $8.7_{-1.0}^{+0.8}$ & 
$-3.3_{-0.6}^{+0.4}$ & $0.90_{-0.10}^{+0.09}$ &
$-1.2_{-0.6}^{+0.5}$ & $6.5_{-2.1}^{+2.0}$ &
29.6 & 0.821\\
4.00 & $4.7_{-0.5}^{+0.4}$ & 
$-3.3_{-0.6}^{+0.4}$ & $0.63_{-0.07}^{+0.06}$ &
$-1.2_{-0.8}^{+0.5}$ & $7.8_{-2.7}^{+2.7}$ &
29.8 & 0.827\\
8.00 & $2.4_{-0.2}^{+0.2}$ & 
$-3.4_{-0.6}^{+0.4}$ & $0.44_{-0.05}^{+0.04}$ &
$-1.2_{-1.0}^{+0.6}$ & $9.2_{-3.5}^{+3.5}$ &
29.9 & 0.832\\
\noalign{\smallskip}\hline
\end{tabular}
\end{center}
\end{table*}
variations 
of the parameters 
$a_s$, $a_t$  
and rather large variations of the parameters
$r_s$, $r_t$,
respectively.
Therefore one can deduce important constraints
on the singlet and triplet scattering lengths
and effective ranges of the low energy $\Lambda{p}$ interaction:
\begin{eqnarray}
-4.1\; {\rm fm}&<&a_s < -2.3\; {\rm fm},\; \; \; 
r_s < 2.7\; {\rm fm}, \nonumber \\
-1.8\; {\rm fm} &<&a_t < -0.6\; {\rm fm},\; \; \; 
r_t > 2.7\; {\rm fm}. 
\label{constraint}
\end{eqnarray}






\section{Tests of potential model results.}
Available  meson-exchange potential model predictions
of the S-wave $\Lambda{p}$
singlet and triplet scattering length
and effective range parameters are listed in Table~\ref{tab2}. Here the 
results from Nijmegen model are denoted as Nijm~D~\cite{fhint:nag77}, 
Nijm~F~\cite{fhint:nag79} and Nijm~a-e~\cite{fhint:sto99}.  The  NSC 
parameters are taken from  Ref.~\cite{fhint:mae89}. The parameters 
given by J\"ulich model are denoted as J\"ul~A,B~\cite{fhint:hol89} and
J\"ul~$\tilde{\rm A}$,$\tilde{\rm B}$~\cite{fhint:reu94}. The most 
recent  J\"ulich parameters are indicated as J\"ul~03~\cite{Haidenbauer}. 


In Fig.~\ref{correl2} the model predictions of 
$a_s$, $r_s$, $a_t$ and $r_t$ are compared with
the five-parameter fit results of Sect.~3.
The experimentally allowed regions deduced from 
five-parameter fits are indicated by hatched rectangles.
All model predictions of ($a_s$, $r_s$) and most predictions
of ($a_t$, $r_t$)
lie outside the experimentally allowed regions. 
%((intervals))((reasonable ranges))
In most cases  the singlet scattering lengths $a_s$ are too small
and the triplet scattering lengths $a_t$ are too large.
\begin{figure}[t]\vspace*{-3mm}
\hspace*{-3mm}\psfig{file=correl22.ps,width=9.4cm,height=9.2cm}\vspace*{-3mm}
\caption{
Experimentally allowed regions (hatched rectangles) for
$\Lambda{p}$ singlet and triplet scattering lengths
and effective ranges, $a_s$, $r_s$ and $a_t$, $r_t$
deduced from five-parameter fits of the $pp\to K^+ X$
missing mass spectrum and the total $\Lambda{p}$ cross
section data. 
The symbols indicate
theoretical results for singlet and triplet parameters from different models:
circles - Nijmegen\cite{fhint:nag77,fhint:nag79}, inverse triangles -
J\"ulich~\cite{fhint:reu94,fhint:hol89} and triangle -
NSC~\cite{fhint:mae89}. The square shows most recent J\"ulich
result~\cite{Haidenbauer}.}
\label{correl2}
\end{figure}



%the results of the analysis in Sect. 3
%the experimental limits
%the experimentally allowed parameter ranges 
%the constraints (allowed values) 
%allowed regions
%ranges deduced from 
%five-parameter fits (see Sect.~3).

\begin{table*}[t]
\begin{center}
\caption{Test of potential model results performed with two-parameter fit.}
\label{tab2}
\begin{tabular}{lccccccccc}
\hline\noalign{\smallskip}
Model   & 
$ a_s$ \, (fm)  & $r_s$ \, (fm) &
$a_t$ \, (fm)  & $r_t$ \, (fm) & $|{\cal M}_s|^2$ \, (b/sr) &
 $|{\cal M}_t|^2$ \, (b/sr) & $\chi^2$ & $\chi^2/ndf$\\
\noalign{\smallskip}\hline\noalign{\smallskip}
Nijm a & -0.71 & 5.86 & -2.18 & 2.76&$0.\pm 0.1$&$22.8\pm 0.2$ & 47.6 & 1.22 \\
Nijm b & -0.90 & 4.92 & -2.13 & 2.84&$0.\pm 0.1$ &$23.4\pm 0.2$ & 53.9 & 1.38 \\
Nijm c & -1.20 & 4.11 & -2.08 & 2.92 &$0.\pm 0.1$&$23.9\pm 0.2$ & 62.2 & 1.60\\ 
Nijm d & -1.71 & 3.46 & -1.95 & 3.08 &$0.\pm 0.1$ &$25.0\pm 0.2$ & 83.3 & 2.14\\
Nijm e & -2.10 & 3.19 & -1.86 & 3.19 &$77.5\pm 0.7$ &$0.\pm 0.1$& 76.6 & 1.97\\
Nijm f & -2.51 & 3.03 & -1.75 & 3.32 &$74.7\pm 0.7$&$0.\pm 0.1$ & 44.8 & 1.15\\ 
J\"ul $\tilde{\rm A}$ & -2.04 & 0.64 & -1.33 & 3.91 
& $7.3\pm 0.5$& $7.8\pm 1.6$ & 55.9 & 1.43\\
J\"ul $\tilde{\rm B}$ & -0.40 & 12.28 & -2.12 & 2.57 
& $ 0. \pm 0.1$& $ 21.3\pm 0.2$ & 43.6 & 1.12\\
J\"ul A & -1.56 & 1.43 & -1.59 &3.16 &$33.8\pm 0.3$ &$0.\pm 0.1$ & 80.2&2.06\\ 
J\"ul B & -0.56 & 7.77 & -1.91 & 2.43 
& $0.\pm 0.1$ & $20.1\pm 0.2$ & 55.5 &1.42\\ 
NSC & -2.78 & 2.88 & -1.41 & 3.11 & $71.6\pm 0.7$  & $0.\pm 0.1$ & 32.8 &0.84\\ 
Nijm D & -1.90 & 3.72 & -1.96 & 3.24 
&  $ 0.\pm 0.2$  &  $ 26.1\pm 0.2$ & 91.6&2.35\\
Nijm F & -2.29 & 3.17 & -1.88 & 3.36 & $77.4\pm 0.7$ & $0.\pm 0.1$ & 62.6&1.61\\
J\"ul 03 & -1.02 & 4.49 & -1.89 & 2.57 
& $0.\pm 0.1$ & $21.3\pm 0.2$ & 62.4&1.60 \\ 
\noalign{\smallskip}\hline
\end{tabular}
\end{center}
\end{table*}




A more direct test of the model
predictions can be performed using
Eqs.(\ref{miss}) and (\ref{cross}). 
Keeping the predicted parameters fixed and fitting only the
two free parameters $|M_s|^2$ and $|M_t|^2$
the calculated cross sections are compared
with the ${\rm p}{\rm p}{\to}{\rm K}^+{\rm X}$
missing mass spectrum as well as the total $\Lambda{p}$ cross section data
and the $\chi^2$  are deduced  (see Table~\ref{tab2}).
Excepting the model J\"ul~${\tilde{\rm A}}$ this procedure yields
either $|M_s|^2{=}0$ or $|M_t|^2{=}0$.
This is due to the fact that the models predict
either a dominating singlet or triplet FSI enhancement.
%The resulting values of $\chi^2$ with
%$ndf{=}39$ can be used as a measure of the significance level. 
The resulting values of $\chi^2$ should be compared
with the best-fit value $\chi^2{=}29.0$ obtained in direct five-parameter
fits (see Table~\ref{tab1}).
Excepting the result of the NSC model all $\chi^2$ values 
are  essentially larger than 29.0.
%1.6 which corresponds to a significance level
%of less than 1~\%. 
%0.81 which was the best-value
%((is the limit of a 99~\% confidence level.)) 
They indicate that those model predictions fail to
reproduce simultaneously  
the missing mass spectrum and the
total cross section data.
%As an example the results of the recent J\"ulich model J\"ul03
%are shown in Fig.~\ref{??}.
%as can be seen in Figs~\ref{??}-\ref{??}. 
%It should be noted that the
%best-fit value $\chi^2/ndf{=}0.82$ was obtained in direct five-parameter
%fits of the data (see Table~\ref{tab1}). 
%They should be compared
%with the best-fit value $\chi^2/ndf{=}0.82$ obtained in direct five-parameter
%fits (see Table~\ref{tab1}).

 







\section{Summary and conclusions.}

We analyzed the high resolution $pp{\to}K^+X$ 
data of Siebert et al. \cite{fhint:sie94}
with respect to the strong FSI 
near the $\Lambda{p}$ production
threshold.
% and used for evaluation of the low-energy phase-equivalent
%$\Lambda{p}$ potential parameters. 
The observed missing mass spectrum was described by
factorizing the reaction amplitude in terms of a production amplitude and 
FSI amplitude which was
parametrized in terms of the inverse Jost function.
It was found that a three-parameter fit with 
spin-averaged scattering length and effective range 
parameters $\bar{a}$ and $\bar{r}$ can reproduce the 
missing mass spectrum but fails to describe simultaneously
the total $\Lambda{p}$ cross section data.
Vice versa deducing $\bar{a}$ and $\bar{r}$
from a fit to the total cross section
data fails to describe the missing mass spectrum. 
Also a combined   three-parameter fit of
the mis\-sing mass spectrum and
the total $\Lambda{p}$ cross section data
fails to reproduce simultaneously both data sets 
with spin-averaged parameters $\bar{a}$ and $\bar{r}$.

Therefore the singlet and triplet scattering lengths
and effective ranges $a_s$, $r_s$, $a_t$, $r_t$
are fitted as separate free parameters.
Taking $|{\cal M}_s|^2$ as free parameter and 
the ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$ 
as fixed parameter 
both the missing mass spectrum and the total
cross scetion data are perfectly reproduced
in five-parameter fits.
The ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$
was varied over a wide range of values
and important  constraints on
the parameters $a_s$, $r_s$, $a_t$, $r_t$
were deduced.
These are indicated in Fig.~\ref{correl2} 
as experimentally allowed regions.


A characteristic feature of the best-fit 
solutions 
is the fact that
the $\Lambda{p}$ FSI in the reaction
$pp{\to}K^+\Lambda{p}$
is dominated by the
singlet contribution. 
Another important result 
follows from a comparison of the singlet and triplet
scattering lengths.
The fact that $-a_s > -a_t$ means that the $\Lambda{p}$ interaction
is more attractive in the singlet state than in the triplet
state. This result is in accordance with the
expectation deduced from an analysis of the
binding energies of light hypernuclei \cite{fhint:dal65}.





%The $\chi^2$ values become slightly larger with 
%increasing $|{\cal M}_t|^2/|{\cal M}_s|^2$ so that
%ratios  larger than 8 can be excluded on the basis
%of the $\chi^2$ criterion. 

Though the $\chi^2$ values are only weakly dependent on
$|{\cal M}_t|^2/|{\cal M}_s|^2$ 
ratios with $|{\cal M}_t|^2/|{\cal M}_s|^2{>}8$
can be excluded on the basis of the $\chi^2$ criterion.
In this context it is interesting to note that the 
fit yields for $|{\cal M}_t|^2/|{\cal M}_s|^2{>}1$
abnormally small singlet and large triplet effective
ranges, $r_s{<}1$~fm and $r_t{>}6$~fm.
Taking the definition of the effective range  
as interaction range 
such values are questionable.
However, this problem can only  be decided by experiments
which allow to determine separately the absolute
values of the singlet and triplet
production matrix elements.  




Most previous experiments provide only informations
on the spin-averaged parameters $\bar{a}$ and $\bar{r}$
and the deduced values are given without errors estimates.
But since the concept of spin-averaged parameters fails to describe
simultaneously the $pp{\to}K^+ X$ FSI enhancement  and the 
$\Lambda {p}$ total cross sections we do not compare our spin-averaged
parameters with previous determinations of $\bar{a}$ and $\bar{r}$.
In this context we mention the general
problem of defining spin-averaged parameters if singlet and triplet
parameters are different.
%In addition
%there is the general problem of 
%defining spin-averaged parameters if singlet and triplet
%parameters are different.
%relating singlet and triplet
%parameters with spin-averaged parameters.



%Inspecting the world data set 
%there is one experiment which provides direct information on the
%triplet parameters $a_t$ and $r_t$.
%The data on the $\Lambda p$ production in the reaction
%$K^-d\to \pi^-\Lambda p$ \cite{fhint:tai69}
%show a marked FSI enhancement near the $\Lambda p$ threshold.
%The $K^- d$ capture at rest occurs mainly from 
%S-wave orbital admixtures.
%Spin and parity considerations imply that
%the final $\Lambda p$ state has spin one as the deuteron.
%Near threshold the amplitude is dominated by the $^3{\rm S}_1$-state.
%Under those assumptions, the triplet scattering length and effective range
%parameter $a_t$ and $r_t$ can directly be deduced
%from those data. The deduced values $a_t=-2.0\pm 0.5$~fm
%and $r_t=3.0\pm 1.0$~fm \cite{fhint:tai69}
%agree within one standard deviation with
%the experimentally allowed region
%of ($a_t, r_t$). They favor  our five-parameter
%solutions for small ratios $|{\cal M}_t|^2/|{\cal M}_s|^2$.

Inspecting the world data set 
there is one experiment which provides direct information on the
triplet parameters $a_t$ and $r_t$.
The data on the $\Lambda{p}$ production in the reaction
$K^-d{\to}\pi^-\Lambda{p}$ \cite{fhint:tai69}
show a marked FSI enhancement near the $\Lambda{p}$ threshold.
Tai Ho Tan deduced $a_t$ and $r_t$ assuming that
the $K^- d$ capture at rest occurs mainly from 
S-wave orbital admixtures.
Spin and parity considerations imply that
the final $\Lambda{p}$ state has spin one as the deuteron.
%Neglecting a possible contribution from a $^3{\rm D}_1$
Assuming that the FSI amplitude near threshold  is dominated by the 
S-wave contribution, i.e. the $^3{\rm S}_1$-state,
the triplet scattering length and effective range
parameter $a_t$ and $r_t$ can directly be extracted
from those data. The deduced values $a_t{=}-2.0\pm 0.5$~fm
and $r_t{=}3.0\pm 1.0$~fm \cite{fhint:tai69}
agree within one standard deviation with
the experimentally allowed region
of ($a_t, r_t$). They favor  our five-parameter
solutions for small ratios $|{\cal M}_t|^2/|{\cal M}_s|^2$.
It should be mentioned that a reanalysis of the
$K^-d{\to}\pi^-\Lambda{p}$ data is highly wanted
in view of the importance and the high quality of those data.







%(((The triplet parameters $a_t=-2.0\pm 0.5$~fm
%and $r_t=3.0\pm 1.0$~fm deduced from
%a study of the $K^-d\to \pi^-\Lambda p$ reaction
%\cite{fhint:tai69} also  favor  our five-parameter
%solutions for small ratios $|{\cal M}_t|^2/|{\cal M}_s|^2$.
%They agree within one standard deviation with
%the experimentally allowed region
%of ($a_t, r_t$).)))






The meson theoretical model predictions 
of the scattering lengths and effective ranges were compared with 
the experimentally allowed regions of ($a_s$, $r_s$)
and ($a_t$, $r_t$) deduced from five-parameter fits of the data.
In most cases the predicted singlet scattering lengths are too small and
the triplet scattering lengths are too large.
Only one model prediction is in agreement with the
constraints deduced from the data.
This finding was confirmed by a more direct test of the
model predictions. 
The test was performed 
by keeping  the predicted parameters in the fit routine fixed
and fitting only $|M_s|^2$ and $|M_t|^2$. 
With one exception the model predictions fail to
reproduce simultaneously the
missing mass spectrum near the $\Lambda{p}$ threshold
and the total $\Lambda{p}$  cross section data. 



It should be emphasized that the shape of the FSI enhancement
near the $\Lambda{p}$ threshold is the essential experimental
information in order to evaluate the FSI parameters
from the reaction $pp{\to}K^+ \Lambda{p}$.
Since the  most important part of the FSI enhancement is
located in the sharply rising part of the missing mass spectrum
a high missing mass resolution and sufficient statistical
accuracy are needed. The effective missing mass resolution
of the SATURNE experiment \cite{fhint:sie94} corresponded
to a $1\sigma$ width of 2~MeV. 
Improving this value by an order of magnitude would allow to
study the $\Lambda{p}$ FSI  near $q{=}0$ with higher accuracy.
But not only the improvement
of precision but also a systematic check
of the method are important. An essential assumption of
the factorization ansatz (Eq.~\ref{watson}) is that
the production matrix element is 
%sufficiently 
constant, i.e. does not depend  on the internal energy
of the $\Lambda{p}$ final state. A direct check is to
measure the reaction $pp{\to}K^+ \Lambda{p}$
at different energies and angles. 
%Such a measurement should yield the same FSI parameters.
%Such experiments are planned at the cooler synchrotron COSY in J\"ulich
%\cite{fhint:mai97}.

%Another important improvement of the experimental input
%would be a program to measure the total $\Lambda p$ cross sections
%at lower momenta and with higher accuracy. 

 
Ultimately, measurements with
polarized beams and polarized targets are needed in order to
disentangle the spin singlet and triplet contributions
in the $pp{\to}K^+\Lambda{p}$ reaction.
Such measurements would allow to determine separately
the singlet and triplet scattering lengths and effective ranges.
The feasibility depends not only on  the
polarization of beam and target but also on the
effective missing mass resolution and luminosity.




We appreciate discussions with J. Haidenbauer, C. Hanhart and H. Rohdje\ss{}.
During the preparation of the paper we were informed
about a  
theoretical study \cite{Gasparian3} of
the $pp{\to}K^+\Lambda{p}$ reaction
using dispersion relations.
 



%During the preparation of this paper we found that Gasparian et al.
%\cite{fhint:gas03} deduced spin-averaged parameters from the 
%SATURNE data applying a different method.



%They are also planned at the cooler synchrotron COSY.

%They can be performed at an accelerator like
%the cooler synchrotron COSY where polarized
%proton beams and polarized proton targets are
%available.

%The feasibility depends not only on  the
%polarization of beam and target but also on the
%effective missing mass resolution and luminosity.
%The cooler synchrotron COSY is ideally
%suited for this kind of measurements.
%They can be performed provided
%a polarized proton beam
%and a polarized proton target with sufficient
%luminosity are available.
%using the polarized beam of e.g.    
%within the polarization program at COSY. 



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\begin{figure}[t]\vspace*{3mm}
\psfig{file=spectrum.ps,width=9.1cm,height=9.2cm}\vspace*{-3mm}
\caption{Missing mass spectrum  from $pp{\to}K^+X$
reaction measured~\cite{fhint:sie94} at proton beam energy of $T_p{=}2.3~GeV$
and kaon emission angle of $\theta_K{=}10.3^\circ$. The upper axis indicates
the c.m.  momentum $q$ of the $\Lambda{p}$ system. The arrow shows
$\Sigma$-hyperon production threshold .  The solid line shows the result of
spin averaged three-parameter fit. The dashed line illustrates the phase
space distribution.}
\label{spectrum}
\end{figure}


\begin{eqnarray}
|\bar{\cal M}|^2&=&17.2_{-1.3}^{+1.3}\;{\rm nb/sr}, \nonumber \\
\bar{a}&=&-1.88_{-0.11}^{+0.10}\; {\rm fm}, \nonumber \\
\bar{r}&=&2.79_{-0.23}^{+0.23}\; {\rm fm}, 
\label{fit1}
\end{eqnarray}
with the reduced $\chi^2/ndf$=2.2.

\begin{figure}[t]\vspace*{3mm}
\psfig{file=spectrum.ps,width=9.1cm,height=9.2cm}\vspace*{-3mm}
\caption{Missing mass spectrum  from $pp{\to}K^+X$
reaction measured~\cite{fhint:sie94} at proton beam energy of $T_p{=}2.3~GeV$
and kaon emission angle of $\theta_K{=}10.3^\circ$. The upper axis indicates
the c.m.  momentum $q$ of the $\Lambda{p}$ system. The arrow shows
$\Sigma$-hyperon production threshold .  The solid line shows the result of
spin averaged three-parameter fit. The dashed line illustrates the phase
space distribution.}
\label{spectrum}
\end{figure}



\begin{figure}[b]\vspace*{-7mm}
\hspace*{1mm}\psfig{file=spectrum1.ps,width=9.2cm,height=9.2cm}\vspace*{-3mm}
\caption{Missing mass spectrum  from $pp{\to}K^+X$
reaction measured~\cite{fhint:sie94} at proton beam energy of $T_p{=}2.3~GeV$
and kaon emission angle of $\theta_K{=}10.3^\circ$.  The solid line shows the
result of five-parameter fit. The dashed line illustrates the phase space
distribution.}
\label{spectrum1}
\end{figure}




\begin{table*}[b]
\begin{center}
\caption{The results of the five-parameter fit for singlet $a_s$, $r_s$ and
triplet  $a_t$, $r_t$ parameters and for  an overall production matrix element
squared $|{\cal M}_s|^2$  obtained for different ratios of the triplet and
singlet matrix elements $|{\cal M}_t|^2/|{\cal M}_s|^2$.}
\label{tab1}
\begin{tabular}{ccccccc}
\hline\noalign{\smallskip}
$|{\cal M}_t|^2/|{\cal M}_s|^2$&$ |{\cal M}_s|^2$ (b/sr) &
$a_s$   (fm) & $r_s$ (fm) &
$a_t$ (fm) & $r_t$ (fm) &
$\chi^2/ndf$\\
\noalign{\smallskip}\hline\noalign{\smallskip}
0.00 & $61.6_{-7.2}^{+6.7}$ & 
$-2.6_{-0.3}^{+0.2}$ & $2.50_{-0.27}^{+0.26}$ & 
$-1.54_{-0.30}^{+0.23}$ & $3.6_{-0.9}^{+0.9}$ &
1.13\\
0.10 & $46.9_{-5.2}^{+4.7}$ & 
$-2.8_{-0.4}^{+0.3}$ & $2.26_{-0.28}^{+0.27}$ & 
$-1.47_{-0.33}^{+0.26}$ & $3.7_{-1.0}^{+1.0}$ &
1.14\\
0.25 & $34.5_{-3.7}^{+3.4}$ & 
$-2.9_{-0.6}^{+0.4}$ & $1.98_{-0.26}^{+0.25}$ & 
$-1.40_{-0.38}^{+0.33}$ & $4.0_{-1.2}^{+1.2}$ &
1.14\\
0.50 & $24.0_{-2.6}^{+2.4}$ & 
$-3.1_{-0.7}^{+0.5}$ & $1.64_{-0.22}^{+0.20}$ & 
$-1.33_{-0.44}^{+0.37}$ & $4.5_{-1.4}^{+1.3}$ &
1.14\\
%0.75 & $18.5_{-2.2}^{+1.8}$ &
%$-3.2_{-0.7}^{+0.5}$ & $1.41_{-0.19}^{+0.17}$ &
%$-1.29_{-0.49}^{+0.40}$ & $4.9_{-1.6}^{+1.5}$ &
%1.14\\
1.00 & $15.0_{-1.8}^{+1.5}$ &
$-3.2_{-0.7}^{+0.5} $ & $1.25_{-0.17}^{+0.15}$ &
$-1.26_{-0.53}^{+0.42}$ & $5.3_{-1.7}^{+1.6}$ &
1.14\\
%1.25 & $12.7_{-1.6}^{+1.3}$ &
%$-3.3_{-0.7}^{+0.5}$ & $1.13_{-0.15}^{+0.14}$ &
%$-1.24_{-0.56}^{+0.44}$ & $5.7_{-1.8}^{+1.8}$ &
%1.14\\
%1.50 & $11.0_{-1.4}^{+1.1}$ & 
%$-3.3_{-0.7}^{+0.5}$ & $1.04_{-0.14}^{+0.12}$ &
%$-1.23_{-0.59}^{+0.45}$ & $6.0_{-2.0}^{+1.9}$ &
%1.15\\
2.00 & $8.7_{-1.1}^{+0.9}$ & 
$-3.3_{-0.7}^{+0.5}$ & $0.90_{-0.12}^{+0.11}$ &
$-1.21_{-0.64}^{+0.48}$ & $6.5_{-2.2}^{+2.1}$ &
1.15\\
4.00 & $4.7_{-0.5}^{+0.5}$ & 
$-3.3_{-0.7}^{+0.5}$ & $0.63_{-0.08}^{+0.07}$ &
$-1.18_{-0.80}^{+0.54}$ & $7.8_{-2.9}^{+2.8}$ &
1.15\\
8.00 & $2.5_{-0.3}^{+0.3}$ & 
$-3.4_{-0.7}^{+0.4}$ & $0.44_{-0.06}^{+0.05}$ &
$-1.16_{-1.0}^{+0.61}$ & $9.2_{-3.8}^{+3.7}$ &
1.16\\
\noalign{\smallskip}\hline
\end{tabular}
\end{center}
\end{table*}


is not necesarily fulfilled.

Taking the singlet and triplet parameters
$|{\cal M}_s|^2$, $a_s$, $r_s$ and $|{\cal M}_t|^2$,
$a_t$, $r_t$ as independent free parameters a
six-parameter fit must be performed.


oder

one must take into account that 

oder


oder

In principle, six-parameter fits must be performed
when taking the singlet and triplet parameters
$|{\cal M}_s|^2$, $a_s$, $r_s$ and $|{\cal M}_t|^2$,
$a_t$, $r_t$ as independent free parameters.
However, it turned out that the $\chi^2$ criterion
cannot be used to fit simultaneously
$|{\cal M}_s|^2$ and $|{\cal M}_t|^2$.
(((This is due to the fact that the resulting $\chi^2/ndf$ does
practically not depend on the ratio
$|{\cal M}_t|^2{/}|{\cal M}_s|^2$
(see Table\ref{tab1}).)))
Therefore, five-parameter fits were performed
taking the ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$
as a fixed parameter.
By this method valuable
constraints on the
singlet and triplet scattering lengths
and effective ranges can be deduced from the data.


Starting with $|{\cal M}_t|^2/|{\cal M}_s|^2=1$ a
best fit solution with $\chi^2/ndf=1$ was found.
Both  the missing mass spectrum as well
as the total cross section data (see Fig.\ref{spectrum})
are perfectly reproduced.
Then, the ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$
was varied over a wide range  
and best fit solutions of similar quality with practically
constant $\chi^2/ndf=1$ were found.
For $0\le |{\cal M}_t|^2/|{\cal M}_s|^2\le 8$
the resulting  fit parameters are listed in 
Tables \ref{table1} and \ref{table2}.





Now the data on total  $\Lambda{p}$ cross section and $pp{\to}K^+X$
missing mass spectrum are fitted in a combined fit
with the singlet and triplet scattering lengths and effective ranges
$a_s$, $r_s$, $a_t$, $r_t$ 
as separate free parameters.
Taking the unknown quantities  
$|{\cal M}_s|^2$ and $|{\cal M}_t|^2$
into account
a six-parameter fit should be performed.
However, it turned out that the $\chi^2$-criterion cannot be
used to determine simultaneously $|{\cal M}_s|^2$ and $|{\cal M}_t|^2$.
This is due to the fact that the resulting $\chi^2/ndf$ does
practically not depend on the ratio
$|{\cal M}_t|^2{/}|{\cal M}_s|^2$
(see Table~\ref{tab1}). Therefore,
we performed five-parameter fits 
for different ratios $|{\cal M}_t|^2{/}|{\cal M}_s|^2$.

Since the production matrix elements are also unknown


In order to deduce the singlet and triplet parameters
$a_s$, $r_s$, $a_t$, $r_t$ 
as well as the production matrix elements squared
$|{\cal M}_s|^2$ and $|{\cal M}_t|^2$
a six-parameter fit should be performed.
However, it turned out that the $\chi^2$-criterion cannot be
used to determine simultaneously $|{\cal M}_s|^2$ and $|{\cal M}_t|^2$.
This is due to the fact that the resulting $\chi^2/ndf$ does
practically not depend on the ratio
$|{\cal M}_t|^2{/}|{\cal M}_s|^2$
(see Table~\ref{tab1}). Therefore,
we performed five-parameter fits 


Now the data on total  $\Lambda{p}$ cross section and $pp{\to}K^+X$
missing mass spectrum are fitted in a combined fit
 with 
$a_s$, $r_s$, $a_t$, $r_t$ 
and $|{\cal M}_s|^2$ as free parameters
assuming  different ratios of the
triplet and singlet  matrix element squared 
$|{\cal M}_t|^2{/}|{\cal M}_s|^2$.

assuming the constraint $|{\cal M}_t|^2{/}|{\cal M}_s|^2=1$.


The singlet production matrix element sqared, $|{\cal M}_s|^2$
is taken as free fit parameter 
assuming the constraint of a common production amplitude,
i.e. $|{\cal M}_t|^2{/}|{\cal M}_s|^2=1$
 and overall
production matrix element squared $|{\cal M}_s|^2$, but for different ratios
of the triplet and singlet  matrix element squared 
$|{\cal M}_t|^2{/}|{\cal M}_s|^2$. The results of the five-parameter fit are
listed in Table~\ref{tab1}.

Now both total $\Lambda{p}$ cross section and the missing mass spectrum are
perfectly reproduced. As an illustration in Fig.\ref{mm5cross2} we show only
the missing mass spectrum calculated for the ratio $|{\cal M}_t|^2/|{\cal
M}_s|^2$=1. It is clear from the results given in Table~\ref{tab1} that the
agreement between data and the calculations with different ratios can be
considered at same confidence level.





Moreover, we found that the reduced $\chi^2/ndf$ does not practically depend
on the ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$. Although the final reduced
$\chi^2/ndf$ corresponds to high confidence level of analysis, one can
not extract from the data the constraint on singlet and triplet contribution
to $pp{\to}K^+\Lambda{p}$ reaction. This finding clearly illustrates that a
separation of singlet and triplet contribution can not be done in a model
independent way.

Furthermore, we also note, the six-parameter fit with $|{\cal M}_s|^2$, 
$|{\cal M}_t|^2$, $a_s$, $r_s$, $a_t$, $a_t$ as free parameters did not 
result in a stable solution. This is due to the fact that the
$\chi^2$-criterion cannot be used to fit simultaneously $|{\cal M}_s|^2$ and
$|{\cal M}_t|^2$ since the resulting $\chi^2/ndf$ is almost stable with
respect to the ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$. This fact actually
motivates us to use a five-parameter fit.





The results shown in Table~\ref{tab1} illustrate that the variation of the
ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$ causes rather small modifications of the
parameters $a_s$ and $a_t$, respectively. In contrast to that, the parameters
$r_s$ and $r_t$ depend on the ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$.
Fig.\ref{correl3} shows the singlet and triplet parameters as a function of
ratio $|{\cal M}_t|^2/|{\cal M}_s|^2$. The solid lines indicate the mean
values for the parameters obtained by averaging the Minuit results over the
studied range of the ratio $0.1{\le}|{\cal M}_t|^2/|{\cal M}_s|^2{\le}4$,
where the reduced $\chi^2/ndf$ was found to be around
${\simeq}1.14{\div}1.15$.





The mean singlet and triplet values averaged over
six results shown at Table~\ref{tab1} and total $\chi^2$ finally are given as
\begin{eqnarray}
a_s &=& -2.99 \pm  0.21 \, {\rm fm}, \,\,\, \chi^2 = 1.41 \nonumber  \\
r_s &=& 0.97 \pm  0.05  \,\,\,\,\,\, {\rm fm}, \,\, \, \chi^2 = 86.3 
\nonumber  \\
a_t &=&  -1.36 \pm 0.17  \, {\rm fm}, \,\,\,\, \chi^2 = 0.35 \nonumber  \\
r_t &=&  4.49 \pm  0.58  \,\,\,\,\, {\rm fm},  \,\,\,\,\, \chi^2 = 3.26
\label{mean}
\end{eqnarray}

Furthermore, the singlet and triplet parameters listed in the Table~\ref{tab1}
as well as an mean values given by Eq.(\ref{mean}) do not match theoretical
results explicitly shown in Table~\ref{tab2}. Therefore we conclude that the
results from five-parameter fit of the total $\Lambda{p}$ cross section and
simultaneously the missing mass spectrum measured in  $pp{\to}K^+X$ reaction
are in substantial disagreement with the results from potential models listed
in Table~\ref{tab2}.

At this stage we must make a comment. When variating the ratio of the triplet
to singlet squared matrix elements within the large range, the fit to the
missing mass spectrum alone can not anymore to distinguish between the singlet
and triplet scattering parameters, as is clearly illustrated by
Eq.(\ref{miss}). At this case the identification of the singlet and triplet
parameters comes entirely from the fit to the total $\Lambda{p}$ cross
section, where the ratio of the triplet to singlet contribution is fixed
explicitly. For instance considering the parameters shown in
Fig.\ref{correl3} one may naively switch them around and to readjust the
calculations of the  total $\Lambda{p}$ cross section. 

However exchanging the triplet parameters for singlet and vice versa indeed
can not improve the agreement with potential model calculations, since one of
the scattering length, called as $a_s$ in Table~\ref{tab1} and
Fig.\ref{correl3} is far away from any theoretical results listed in
Table~\ref{tab2}. Furthermore, to identify the results of the fit as a singlet
and triplet parameters one needs more constraints. At present stage this can
not be done in a model independent way. A different approach to resolve the
dilemma is discussed in next section.


\section{Discussion}






The solid line in Fig.\ref{sigtot} shows the result of the three-parameter
fit in comparison to data on the total $\Lambda{p}$ cross 
section~\cite{fhint:ale68,fhint:sec68,fhint:ale69}. The dashed and dotted
lines  indicate the singlet and triplet contributions to the total cross  
section respectively. The solid line in Fig.\ref{spectrum}  shows the 
experimental missing mass  spectrum~\cite{fhint:sie94} together with the 
result of the fit. The dashed line is the pure phase space distribution 
calculated with $|C_{FSI}|^2{=}1$ and with squared matrix element 
$|\bar{\cal M}|^2$ from Eq.(\ref{fit1}).

\begin{figure}[b]\vspace*{-7mm}
\hspace*{-1mm}\psfig{file=correl1.ps,width=9.4cm,height=9.2cm}\vspace*{-3mm}
\caption{The correlation between averaged scattering length and effective
range resulting from the three-parameters fit. The contour lines show
the levels with $\chi^2{+}1$, $\chi^2{+}2$, $\chi^2{+}3$ and $\chi^2{+}4$
respectively, with minimal total $\chi^2/ndf{=}88.5$. The symbols indicate
theoretical results from different models: circles -
Nijmegen\cite{fhint:nag77,fhint:nag79}, inverse triangles -
J\"ulich~\cite{fhint:reu94,fhint:hol89} and triangles
- NSC~\cite{fhint:mae89}. The square shows most recent J\"ulich 
result~\cite{Haidenbauer}.}
\label{correl1}
\end{figure}

We consider the quality of the fit with $\chi^2/ndf{=}2.2$ as a rather bad.
Especially around the $\Lambda$-hyperon threshold region of the missing mass
spectrum the steeply rising enhancement is not reproduced by the fit.  This
might indicate that the enhancement factor at $q{=}0$ which is given by the
ratio $\bar{\beta}^2/\bar{\alpha}^2$ is not large enough.  In addition, the
simultaneous fit cannot reproduce the total $\Lambda{p}$ cross section data
with sufficient accuracy. The total cross section data are generally below the
fit curve.  It should be noted that substantially better fit with reduced
$\chi^2{\simeq}1$ can be obtained when fitting {\it separately} the missing
mass spectrum and the total cross section data. But then the resulting
spin-averaged parameters $\bar{\alpha}$ and $\bar{\beta}$ are different.

Fig.\ref{correl1} shows the correlation between the averaged scattering
length and effective range. The contour lines correspond to the confidence
levels with $\chi^2{+}1$, $\chi^2{+}2$, $\chi^2{+}3$ and $\chi^2{+}4$
respectively. The symbols in Fig.\ref{correl1} show the results from different
theoretical models\cite{fhint:nag77,fhint:nag79,fhint:reu94,fhint:hol89,fhint:mae89,Haidenbauer}.
The models provide separately the singlet and triplet scattering lengths and
effective ranges and to compare them with results of the fit we average model
parameters as
\begin{equation}
\bar{a}= \frac{a_s+3a_t}{4}, \,\,\, 
\bar{r}= \frac{r_s+3r_t}{4}.
\label{para}
\end{equation}
Fig.\ref{correl1} illustrates that none of the model calculation can be
compatible with the result of the overall fit. However at this stage we can
not conclude whether this discrepancy comes from an ambiguities of the
hyperon-nucleon potential models or results from an inadequate comparison with
an average parameters of Eq.(\ref{para}). 

Indeed the singlet and triplet contributions to the $pp\to K^+\Lambda{p}$
reaction depends on the production mechanism. Now Fig.\ref{correl1} shows the
comparison between the results of three-parameter fit with the singlet and
triplet scattering lengths and effective ranges separately. It is clear
that the triplet parameters from different models are not compatible with the
results of the fit even at very low confidence level. At the same time some
solutions for the singlet parameters are in reasonable agreement with the fit.

\begin{figure}[t]\vspace*{-3mm}
\hspace*{-3mm}\psfig{file=correl2.ps,width=9.4cm,height=9.2cm}\vspace*{-3mm}
\caption{The correlation between averaged scattering length and effective
range resulting from the three-parameters fit. The contour lines show the
levels with $\chi^2{+}1$, $\chi^2{+}2$, $\chi^2{+}3$ and $\chi^2{+}4$
respectively, with minimal reduced $\chi^2/ndf{=}2.2$. The symbols indicate
theoretical results for singlet and triplet parameters from different models:
circles - Nijmegen\cite{fhint:nag77,fhint:nag79}, inverse triangles -
J\"ulich~\cite{fhint:reu94,fhint:hol89} and triangles -
NSC~\cite{fhint:mae89}. The square shows most recent J\"ulich
result~\cite{Haidenbauer}.}
\label{correl2}
\end{figure}

From the comparison given by Fig.\ref{correl2} one may conclude that the
$pp{\to}K^+\Lambda{p}$ reaction is dominated by the triplet $\Lambda{p}$ final
state. However as we will show in the following that conclusion is absolutely
incorrect. Although the averaged parameters can be extracted from simultaneous
fit of the data on total $\Lambda{p}$ cross section and the missing mass
spectrum, these parameters can not be interpreted unambiguously.







The model predictions of 
$a-s$, $r_s$, $a_t$ and $r_t$ can be tested
using Eqs.(\ref{miss})
and (\ref{cross}). 
comparing the resulting curves
with the ${\rm p}{\rm p}{\to}{\rm K}^+{\rm X}$
missing mass spectrum and the total $\Lambda{p}$ cross section data.

Keeping the predicted parameters fixed and fitting only the
two free parameters $|M_s|^2$ and $|M_t|^2$
the resulting calculated values are compared
with the ${\rm p}{\rm p}{\to}{\rm K}^+{\rm X}$
missing mass spectrum and the total $\Lambda{p}$ cross section data.




The results are shown in Table~\ref{tab2}. 
The values of $\chi^2/ndf$ with
$ndf{=}43$ can be used as a measure of the fit quality. 
They should be compared
with the best-fit values $\chi^2/ndf{=}1.13$ obtained in direct five-parameter
fits and listed in Table~\ref{tab1}.





by inserting those parameters into Eqs.(\ref{miss})
and (\ref{cross}) and by 
comparing the resulting curves
with the ${\rm p}{\rm p}{\to}{\rm K}^+{\rm X}$
missing mass spectrum and the total $\Lambda{p}$ cross section data.
The two free parameters 
$|M_s|^2$ and $|M_t|^2$ are taken as free parameters
simultaneously the experimental results for
missing mass spectrum and the total $\Lambda{p}$ cross section data.  Now we
use $|M_s|^2$ and $|M_t|^2$ as two free parameters. The model parameters were
fitted to the same set of data on total  $\Lambda{p}$ cross section and in
principle one can perform fit only to the missing mass spectrum. However we
continue with analysis of both sets of data in order to obtain an overall
$\chi^2$ as a quantitative control of our analysis.

The results are shown in Table~\ref{tab2}. The values of $\chi^2/ndf$ with
$ndf{=}43$ can be used as a measure of the fit quality. They should be compared
with the best-fit values $\chi^2/ndf{=}1.13$ obtained in direct five-parameter
fits and listed in Table~\ref{tab1}.

We found that almost all fits result in omission either the singlet or
triplet contribution to the missing mass spectrum from $pp{\to}K^+X$
reaction. These results actually indicate that conclusion about the
contribution from singlet and triplet amplitudes to the reaction strongly
depends on the model applied. Taking the best $\chi^2/ndf{=}1.12$ obtained
with the parameters from  NSC model~\cite{fhint:mae89} one may decide that the
$pp{\to}K^+\Lambda{p}$ reaction at beam energy of $T_p{=}2.3$~GeV is dominated
by the singlet production amplitude ${\cal M}_s$. In contrast, the fit with
the parameters from J\"ulich~$\tilde{\rm B}$~\cite{fhint:reu94} model shows the
dominance of the triplet state, although in that case the reduced
$\chi^2/ndf{=}1.32$. 







Including the experimental information on the
triplet scattering length and effective range
deduced from the $K^- d \to pi^- \Lambda p$
experiment \cite{fhint:tai69} 
six-parameter fits were performed.
By this method the singlet and
triplet parameters $a_s$, $r_s$, $a_t$, $r_t$
were deduced in a unique manner.
The method relies on the fact that the ratio of 
spin-singlet and spin-triplet contributions
is well known for the total $\Lambda p$ cross section (0.25/0.75)
and the  $K^- d \to pi^- \Lambda p$
reaction (0/1). This allows to determine the 
singlet and triplet contributions in
the missing mass spectrum by fitting 
$|{\cal M}_s|^2$ and $|{\cal M}_t|^2$ in a combined
six-parameter fit.



%It should be mentioned that 
%we also tried to deduce the
%four parameters $a_s$, $r_s$, $a_t$, $r_t$
%by fitting only the
%$\Lambda p$ cross section data with $a_s$ and $r_s$
%as free parameters and taking $a_t=-2.0\pm 0.5$~fm 
%and $r_t=3.0\pm 1.0$~fm
%from the $K^- d \to pi^- \Lambda p$ experiment.
%This procedure yields
%\begin{eqnarray}
%a_s&=&-1.2_{-1.2}^{+0.5}\; {\rm fm},\; 
%r_s=4.4_{-3.8}^{+4.4}\; {\rm fm}, \; \chi^2/ndf=0.36, \nonumber \\
%a_t&=&-2.0_{-0.5}^{+0.5}\; {\rm fm},\; 
%r_t=3.0_{-1.0}^{+1.0}\; {\rm fm}.
%\label{fitcross2}
%\end{eqnarray}
%Though the errors are very large
%the resulting parameters are consistent with the
%results quoted in (\ref{fitmm6cross2}).


********************************************************************************
\subsection{Six-parameter fit}


The simultaneous five-parameter fit of the $pp\to K^+X$ missing mass
spectrum and the total cross section data
has shown that additional experimental information is needed
in order to determine the singlet and triplet
scattering lengths and effective ranges $a_s$, $r_s$, $a_t$ 
and $r_t$ in a unique manner.
The data on the $\Lambda p$ production in the reaction
$K^-d\to \pi^-\Lambda p$ \cite{fhint:tai69} can provide this information.
This reaction was studied 
in a large bubble chamber with $K^-$ stopping in liquid deuterium.
The data show a marked FSI enhancement in the $\Lambda p$ 
invariant mass spectrum near the $\Lambda p$ threshold.
The $K^- d$ capture at rest occurs mainly
from S-wave orbital admixtures.
Spin and parity considerations imply that the final
$\Lambda p$ state has spin one as the deuteron. 
Near threshold the amplitude is dominated by the 
$^3{\rm S}_1$-state.
Therefore, the triplet scattering length and effective range parameter
$a_t$ and $r_t$ can be deduced from those data.


Fortunately, the 
$K^-d\to \pi^-\Lambda p$ data have already been analyzed
with respect to the FSI enhancement.
Using the FSI formalism developed by Karplus and  Rodberg
\cite{fhint:kar59} and Kotani and Ross \cite{fhint:kot59}
Tan deduced the following values,
\begin{equation}
a_t=-2.0\pm 0.5\; {\rm fm},\; \; \; r_t=3.0\pm 1.0\; {\rm fm}
\label{tan}
\end{equation}
These values of the triplet parameters favor our
five-parameter fit solutions for small ratios 
$|{\cal M}_t|^2/|{\cal M}_s|^2$ (see Table~\ref{tab1}).

The two values in Eq.~\ref{tan} represent the complete result of the experiment
with respect to the $\Lambda p$ FSI enhancement.
Introducing these values and the error bars 
as two additional experimental data points 
into the fit routine yields an important experimental
constraint in the determination
of the full parameter set $a_s$, $r_s$, $a_t$, $r_t$. 
With this additional information it was possible to perform
a six-parameter fit and to find a unique solution.
The resulting parameters are
\begin{eqnarray}
|{\cal M}_s|^2&=&61.5_{-24.0}^{+5.8}\;{\rm b/sr},\; 
|{\cal M}_t|^2=0.0_{-0.0}^{+7.5}\;{\rm b/sr},\; 
 \nonumber \\
a_s&=&-2.6_{-0.4}^{+0.2}\; {\rm fm},\; 
r_s=2.5_{-0.4}^{+0.2}\; {\rm fm}, \; \chi^2/ndf=0.81, \nonumber \\
a_t&=&-1.6_{-0.2}^{+0.2}\; {\rm fm},\; 
r_t=3.5_{-0.6}^{+0.6}\; {\rm fm}.
\label{fitmm6cross2}
\end{eqnarray}
Numerically, the fit prefers $|{\cal M}_t|^2=0$ yielding $\chi^2=30.13$
and $\chi^2/ndf=0.814$ for 43 data points and $ndf=37$.
But this does not mean that $|{\cal M}_t|^2$
is exactly zero. 
Within one standard deviation 
($\chi^2\leq 31.13$) solutions
with $|{\cal M}_t|^2\leq 7.5$~b/sr and  $|{\cal M}_s|^2\geq 37.5$~b/sr
corresponding to ratios $|{\cal M}_t|^2/|{\cal M}_s|^2\leq 0.2$
can be found.
The resulting values for $a_s$, $r_s$, $a_t$, $r_t$
are within one standard deviation 
in agreement with the values quoted in (\ref{fitmm6cross2}). 
For instance a solution with $|{\cal M}_t|^2/|{\cal M}_s|^2 = 0.1$
yields $\chi^2=30.62$ and best-fit parameters like in the second row
of Table~\ref{tab1}. 
**************************************************************************  





































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