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\hyphenation{author another created financial paper re-commend-ed}

% declarations for front matter
\title{On the scalar nonet lowest in mass}

\author{Peter Minkowski\address{Institute for Theoretical Physics,
     University of Bern,  CH-3012 Bern, Switzerland}
         \thanks{Work supported in part by Schweizerischer Nationalfonds}
         and
         Wolfgang Ochs\address{Max Planck Institut f\"ur Physik, Werner
      Heisenberg Institut,
            D-80805 Munich, Germany}}

\begin{document}

\begin{abstract}
The hypothesis that there exists a nonet of scalars mainly composed
of a valence quark-antiquark pair and mixed according to near 
singlet-octet separation : $f_{0} (980)$ singlet, 
\ $a_{0}^{+,0,-} (980) \ , \ K^{* \ +,0}_{0} (1430) \ , 
\ \overline{K}^{* \ -,0}_{0} (1430) \ f_{0} (1500)$ 
octet, is put to further tests from the three body decays 
$D^{\pm} , D_{s}^{\pm} \ \rightarrow \ PS^{\pm} \ \pi^{+} \pi^{-}$
with $PS^{\pm} \ = \ \pi^{\pm} \ , \ K^{\pm}$.
The analysis of decay phases supports the singlet nature of $f_{0} (980)$.
%The question of eventually significant mixing of the singlet $f_{0} (980)$
%with the nearby gluonic meson $gb_{0} (1000)$ is deferred to future analysis.
\end{abstract}

% typeset front matter (including abstract)
\maketitle

\section{Spectroscopy of $q \overline{q}$ p-waves}

Following the hypothesis given in the abstract we select the following
4 p-wave nonets : 
$J^{PC_{n}} \ = \ 0^{++} \ , \ 1^{++} \ , \ 2^{++} \ , \ 1^{+-}$. 
They are displayed together with the pseudoscalar nonet in figure \ref{fig1}.

\begin{figure}[htb]
\vspace*{-1.cm}
\begin{center}
\mbox{\epsfig{file=p-001071-Model.EPS,%
%bbllx=2.1cm,bblly=0.2cm,bburx=24.cm,bbury=20.5cm,
width=6.5cm,angle=90}}
\end{center}
\vspace*{-1.0cm}
\caption{$J^{PC_{n}} = 0^{++},0^{-+},1^{++},2^{++}, 1^{+-}$
$q \overline{q}$ nonets.}
\label{fig1}
\end{figure}
%\vspace*{-0.5cm}

As a consequence of the adopted selection criteria we 
exclude the following candidate states from the PDG listings \cite{PDG}:
$f_{0} (400-1200)$, $f_{0} (1370)$, $f_{1} (1410)$ and 
resonances with a mass exceeding $f_{2}^{'} (1525)$ in particular
$h_{1} (1595)$ and $f_{0} (1710)$, see also \cite{PMWO}.
\vspace*{-0.0cm}

We pose the following questions\footnote{For a rather different
description of low lying scalar mesons we refer e.g. to ref. \cite{CloTo}.}:

\begin{description}
\item - Is the flavor mixing pattern of the $0^{-+}$ and
$0^{++}$ nonets similar , i.e. near singlet - octet? 
The two nonets would be parity doublets due to chiral symmetry,
which is spontaneously and explicitely broken.      

\item - Does the scalar gluonic meson distort through
large mixing effects the scalar $q \overline{q}$ nonet 
beyond (spectroscopic) recognition?

In our analysis in \cite{PMWO} we started from the hypothesis,
that the answer to this question is no. Inspecting the
spectra of the p-wave nonets in Fig. 1 we observe no strong distortion
in the scalar sector indeed.
%The answer to this question is here no,
%{\em by hypothesis} and its spectroscopic verification.

\item - Can the last question be resolved through direct
observation of the scalar gluonic meson as a conventionally narrow
resonance and how reliable are the mass estimates from
purely gluonic lattice-QCD \cite{latQCD}

\begin{displaymath}
m_{gb} \ (0^{++}) \ = \ 1600 \ \mbox{MeV} 
\ \pm 10 \% \hspace*{0.3cm} \mbox{?}
\end{displaymath}

\item - Can QCD sum rules including 
local gluonic operators \cite{Nari} shed light on the above mass estimate?
The latter two questions are adressed in the accompanying paper
\cite{WO}.
\end{description}

{\bf The Gell-Mann - Okubo square mass formula} 

According to our analysis 
$f_{0} \ \rightarrow \ f_{<}$ (980) represents the SU3 singlet, whereas

$a_{0}$ (984.7) , $K_{0}$ (1412) , $f_{0} \ \rightarrow \ f_{>}$ (1507)

\noindent
form the associated octet.

\noindent
The Gell-Mann - Okubo (first order) mass square
relation then yields (in $\mbox{GeV}^{2}$ units)

\begin{equation}
\label{eq:2}
\begin{array}{l}
m^{2} (f_{>}) = 
\vspace*{0.2cm} \\
\hspace*{0.6cm} 
 m^{2} (a_{0}) + \frac{4}{3} 
 \left (  m^{2} (K_{0}) -  m^{2} (a_{0}) \right )
\vspace*{0.2cm} \\
2.271 \ = \ 0.970 \ + \ 1.365 \ = \ 2.335
\end{array}
\end{equation}

\noindent
the deviation amounts to $0.064/2.271 \ = \ 2.8 \ \%$
\footnote{Eq. \ref{eq:2} is numerically a
refinement with respect to ref. \cite{PMWO}.}

\noindent
There is no sign - yet - of any major distortion.

\noindent
The degeneracy in mass of $f_{<}$ and $a_{0}$, while
not offending any basic principles, indicates further
dynamic simplicity to be explained.

\section{Further evidence for (near) octet-singlet flavor 
phase structure} 

The aim is to study and eventually confirm
the nonstrange versus strange $q \overline{q}$ flavor
structure of the two singlet-octet assigned isoscalar scalars :

\begin{displaymath}
\begin{array}{lll ll}
f_{<} \ (980) & = & \cos \ \vartheta
\ \left | \ {\bf 0} \ \right \rangle
& - & \sin \ \vartheta 
\ \left | \ {\bf 8} \ \right \rangle
\vspace*{0.2cm} \\
f_{>} \ (1500) & = & \sin \ \vartheta
\ \left | \ {\bf 0} \ \right \rangle
& + & \cos \ \vartheta 
\ \left | \ {\bf 8} \ \right \rangle
\end{array}
\end{displaymath}

\noindent
The following phase convention shall be chosen in
the flavor basis

\begin{equation}
\label{eq:3}
\begin{array}{l}
f \ = \ \sum_{q} \ c_{q} 
\ \left | \ {\bf q \overline{q}} \ \right \rangle
\hspace*{0.1cm} , \hspace*{0.1cm} 
q \ = \ u,d,s
\vspace*{0.2cm} \\
c_{u}  =  c_{d}  =  \frac{1}{\sqrt{2}} \ c_{ns}
\hspace*{0.0cm} ; \hspace*{0.0cm} 
c_{ns}  =  \sin \varphi , c_{s}  =  \cos \varphi
\vspace*{0.2cm} \\
 \left | \ {\bf 0} \ \right \rangle
 =  \frac{1}{\sqrt{3}} \left (  1 , 1 , 1  \right )
\hspace*{0.1cm} ; \hspace*{0.1cm} 
 \left | \ {\bf 8} \ \right \rangle
 =  \frac{1}{\sqrt{6}} \left (  1 , 1 , -2  \right )
\vspace*{0.2cm} \\
 \left | \ {\bf ns} \ \right \rangle
\ = \ \frac{1}{\sqrt{2}} \left (  1 , 1 , 0  \right )
\hspace*{0.1cm} ; \hspace*{0.1cm} 
 \left | \ {\bf s} \ \right \rangle
 =  \left (  0 , 0 , 1  \right )
\end{array}
\end{equation}

\noindent
and in the ns-s basis we have

\begin{equation}
\label{eq:4}
\begin{array}{l}
 \left | \ {\bf 0} \ \right \rangle
\ = \ \sin \ \varphi_{*} 
 \left | \ {\bf ns} \ \right \rangle
\ + \ \cos \ \varphi_{*} 
\ \left | \ {\bf s} \ \right \rangle
\vspace*{0.2cm} \\
 \left | \ {\bf 8} \ \right \rangle
\ = \ \cos \ \varphi_{*} 
\ \left | \ {\bf ns} \ \right \rangle
\ - \ \sin \ \varphi_{*}
\ \left | \ {\bf s} \ \right \rangle
\vspace*{0.2cm} \\
\varphi_{*} \ = \ \mbox{arccot} \ \frac{1}{\sqrt{2}} \ = \ 54.74^{o}
\end{array}
\end{equation}

\noindent
so in the flavor basis we have

\begin{equation}
\label{eq:5}
\begin{array}{l}
f_{<} \ (980) = \sin \varphi
\ \left |  {\bf ns}  \right \rangle
 +  \cos \varphi 
\ \left | {\bf s} \right \rangle
\vspace*{0.2cm} \\
f_{>} \ (1500)  =  \cos \varphi
\ \left | {\bf ns} \right \rangle
 -  \sin \varphi 
\ \left | {\bf s}  \right \rangle
\vspace*{0.2cm} \\
\vartheta \ = \ \varphi_{*} \ - \ \varphi 
\end{array}
\end{equation}

\noindent
As conjectured range of the
singlet-octet angle $\vartheta$ we consider

\begin{equation}
\label{eq:6}
\begin{array}{l}
0 \ \leq \ \vartheta \ \leq \ \vartheta_{m}
\ , \ \vartheta_{m}  =  \mbox{arcsin} \frac{1}{3} \ = \ 19.47^{o}
\end{array}
\end{equation}

\noindent
The corresponding range for $\varphi$ becomes

\begin{equation}
\label{eq:7}
\begin{array}{l}
35.26^{o} \ \leq \ \varphi \ \leq \ \varphi_{*}
\ = \ 54.74^{o}
\end{array}
\end{equation}

\noindent
Previously \cite{PMWO} we have analysed the decays involving $f_{0} (980)$ :
$J/\Psi \rightarrow \phi f_{0}, \omega f_{0}$, the
radiative decays $f_{0},a_{0} \rightarrow \gamma \gamma$
and $f_{0} \rightarrow K \overline{K}, \pi \pi$ and concluded
on a large flavor mixing similar to $\eta$-$\eta^{'}$ in the pseudoscalar
nonet. Now we extend our analysis to D decays.

\noindent
Our analysis of the mixing pattern shall focus
on the two ratios of strange to nonstrange components

\begin{equation}
\label{eq:8}
\begin{array}{l}
R_{<} \ = \ R \ ( f_{0} (980) ) \ = \ \mbox{cot} \ \varphi
\vspace*{0.2cm} \\
R_{>} \ = \ R \ ( f_{0} (1500) ) \ = \ - \ \mbox{tg} \ \varphi
\end{array}
\end{equation}

\noindent
If the premise of mainly singlet $f_{0}$ (980) is correct
we infer the ranges $R_{<} \ > \ 0$ and
$R_{>} \ = \ - \ 1 / R_{<} < 0$ . The relation
$R_{<} \ R_{>} \ = \ -1$ follows from orthogonality.

\noindent
The restricted ranges are

\begin{equation}
\label{eq:9}
\begin{array}{rll lr}
\frac{1}{\sqrt{2}} & \leq  & R_{<} & \leq & \sqrt{2}
\vspace*{0.2cm} \\
- \ \sqrt{2} & \leq & R_{>} & \leq & - \ \frac{1}{\sqrt{2}}
\end{array}
\end{equation}

\noindent
We determine $\varphi$ considering the three three body
decays of the charmed mesons $D$ and $D_{s}$ \cite{E791}:

%\begin{description}
%\item Charmed meson decays \cite{E791}

\begin{equation}
\label{eq:10}
\begin{array}{l}
A) \ D^{+}_{s} \ \rightarrow \ f_{0} \ (980) \ \pi^{+} 
\vspace*{0.2cm} \\
B) \ D^{+} \ \rightarrow \ f_{0} \ (980) \ \pi^{+} 
\vspace*{0.2cm} \\
C) \ D^{+} \ \rightarrow \ K_{0}^{*} \ (1430) \ \pi^{+} 
\end{array}
\end{equation}

%\vspace*{1.0cm}

 \begin{figure}[htb]
\begin{center}
\mbox{\epsfig{angle=90,file=p-001072-Model.EPS,width=6.cm}}
\mbox{\epsfig{angle=90,file=p-001073-Model.EPS,width=6.cm}}
\end{center}
\caption{The two color suppressed quark flavor flow diagrams ($\propto
\epsilon a$) in reaction B.}
\label{fig4}
 \end{figure}
\vspace*{0.0cm}

%\begin{center}
%\vspace*{-0.5cm}
% \begin{figure}[htb]

%\epsfig{file=T2.eps,width=6.cm}
%\caption{Quark flavor flow in reaction B) (2)}
% \end{figure}
%\end{center}

%\end{description}

\noindent
We consider the color favored amplitudes ($\propto$ a)
which contribute to all processes in eq. \ref{eq:10},
and color suppressed amplitudes 
($\propto$ $\epsilon a$) obtained by 
$\overline{d}_{1} \ \leftrightarrow \ \overline{d}_{2}$ 
(see Fig. \ref{fig4}) and obtain

%The diagrams in Fig. 2 
%and $\overline{d}_{1} \ \leftrightarrow \ \overline{d}_{2}$ yield,
%extended to the three reactions in eq. \ref{eq:10}


\begin{equation}
\label{eq:11}
\begin{array}{l}
A \ = \ \cos\varphi \ V_{ud}V_{cs}^* \ a 
\vspace*{0.2cm} \\
B \ = \ \frac{\sin\varphi}{\sqrt{2}}
           V_{ud}V_{cd}^* 
 (1+\epsilon)  a  +  \cos\varphi  V_{us}V_{cs}^*
 \epsilon \ a 
\vspace*{0.2cm} \\
C \ =  \ V_{ud}V_{cs}^*  \  (1+\epsilon) \ a 
  \label{damp}
\end{array}
\end{equation}

\noindent
We consider the two ratios of partial decay widths $A / B$
and $A / C$ :

\begin{equation}
\label{eq:12}
\begin{array}{l}
R_{A/B}  =  \frac{\Gamma(D_s^+\to f_0(980) \pi^+)} % f_{<} \pi^+)}
{\Gamma(D^+\to f_0(980) \pi^+)}% {<} \pi^+)}
\vspace*{0.2cm} \\
R_{A/C}  =  \frac{\Gamma(D_s^+\to f_0(980) \pi^+)} %f_{<} \pi^+)}
{\Gamma(D^+\to K_0(1430) \pi^+)}
\end{array}
\end{equation}

\noindent
In the approximation $V_{ud}=V_{cs}=\cos\vartheta_c$,
$V_{us}=-V_{cd}=\sin\vartheta_c$ with Cabibbo angle $\vartheta_{c}$,
we find

\begin{equation}
\label{eq:13}
\begin{array}{l}
R_{A/B}  =  2 \frac{\Phi_1}{\Phi_2} \cot^2\vartheta_c\cot^2\varphi
     \frac{1}{|1-(\sqrt{2}\cot\varphi -1)\epsilon |^2}
\vspace*{0.2cm} \\
R_{A/C}  =   \frac{\Phi_1}{\Phi_3} \cos^2\varphi
     \ \frac{1}{|1+\epsilon |^2}
\label{r4r5}
\end{array}
\end{equation}

\noindent
where $\Phi_{1,2,3} = (p_{\pi^+})_{1,2,3}$ denote the phase space in
s-wave decays, proportional to the $\pi^+$ momentum in the decay resonance
rest frame.
Using the branching fractions established by the E791 Collaboration
and the PDG results we find numerically

\begin{equation}
\label{eq:14}
\begin{array}{l}
\cot^2\varphi  /  |  1-(\sqrt{2}\cot\varphi -1)\epsilon  |^2 
 = 
\vspace*{0.2cm} \\
\hspace*{1.0cm} = 1.26 \ (1.0 \pm 0.4 )
\vspace*{0.2cm} \\
\cos^2 \varphi \ / \ | \ 1+\epsilon  \ |^2  
 =  0.52 \ ( 1.0 \pm \ 0.3 )
\label{angle1}
\end{array}
\end{equation}

\noindent
Then we obtain as solutions two bands (a and b)
due to the quadratic nature of the relations in eq. \ref{eq:14}

\begin{equation}
\label{eq:15}
\begin{array}{l}
 \cot \varphi_{a}  =  1.11^{\ + 0.33}_{\ - 0.20}
\hspace*{0.1cm} , \hspace*{0.1cm} 
\varphi_{a} = \left . 42.14^{\ + 5.8}_{\ - 7.3} \right .^\circ
\vspace*{0.2cm} \\
 \epsilon_{a}  =  ( 2.85 \ \pm \ 5.35 ) \ 10^{-2} 
\vspace*{0.2cm} \\
 \tan \varphi_{b}  =  - 0.34^{\ + 0.46}_{\ - 0.37}
\hspace*{0.1cm} , \hspace*{0.1cm} 
\varphi_{b} = \left . 161.2^{\ + 25.7}_{\ - 16.5} \right .^\circ
\vspace*{0.2cm} \\
 \epsilon_{b}  =   0.31^{\ + 0.004}_{\ - 0.13} 
\label{phieps}
\end{array}
\end{equation}

The angle $\varphi$ is defined modulo $180^{\circ}$.
The two solutions in eq. \ref{eq:15} can be distinguished
through the sign of the quantity $R_{<}$ (or $R_{>}$) defined
in eq. \ref{eq:8}.

%\noindent
The phase (+ --) structure of $f{<}$ , $f_{>}$
is determined from interference with other resonances in 
$D^{+}$ and $D^{+}_{s}$ decays
into $3 \pi$ and $\pi K \overline{K}$ \cite{E791,E687}.
%
The amplitudes A , B , C in eq. \ref{eq:11} exhibit the  
+ -- phase structure shown in table 1.

\begin{table*} 
%[ht]
\caption{Decay phases (in degrees) for resonances in $D$ and $D_s$ decays
as measured by experiments E687 
and E791 . In each line an overall phase
 has been fixed arbitrarily. The states marked by ($*$) are not directly
evident in the plots.
Comparison with theoretical expectations for the mixing angle $\varphi$
and predictions for $f_0(980)$ near flavour
singlet and $f_0(1500)$ as octet partner
($0^\circ<\varphi<90^\circ$) with arbitrary angle $\alpha$,
standard choice is $\alpha=0$. }
\vspace*{0.3cm}

$
 \begin{array}{lc@{\hspace*{3.2cm}}c@{\hspace*{3.2cm}}c}
 \hline
 D\to 3\pi   &d \bar d \to  & &
  \\
%\hline
             &  \rho(770)        &  f_0(980)         & f_2(1270)^*
  \\
\mbox{E687}  &  27\pm 14 \pm 11  & 197 \pm 28 \pm 24 & 207 \pm 17 \pm 4 \\
\mbox{E791}  & 0 (\mbox{fixed})  & 151.8 \pm 16.0      & 102.6 \pm 16.0  \\
\mbox{Theory}& -d\bar d          & d\bar d \sin(\varphi) & d\bar d \\
             & \alpha+180^\circ  &  \alpha           & \alpha \\
\hline
 D_s\to 3\pi  &  s \bar s \to & &   \\
             &  f_0(980)         & f_2(1270)^*         &f_0(1500)
  \\
\mbox{E687}  & 0(\mbox{fixed})   & 83\pm 16          & 210 \pm 10   \\
\mbox{E791}  & 0(\mbox{fixed})   & 133 \pm 13\pm 28  & 198 \pm 19 \pm 27 \\
\mbox{Theory}& s\bar s \cos(\varphi)  & \varepsilon s\bar s &
                   - s\bar s \sin(\varphi)\\
             & \alpha           &  \alpha          & \alpha+180^\circ\\
\hline
D_s\to \pi K\bar K \qquad  &  s \bar s \to & &   \\
             &  f_0(980)^*         & \phi(1020)  &
  \\
\mbox{E687}  &  159 \pm 22 \pm 16 & 178 \pm 20\pm 24 &    \\
\mbox{Theory}& s\bar s \cos(\varphi)  & s\bar s    &    \\
             &   \alpha           &   \alpha         &    \\
\hline
  \end{array} 
$
\label{tab:phases}
\end{table*}

\section{Conclusions and outlook}

It becomes clear from the results in table 1 and the assumed
form of the amplitudes in eq. \ref{eq:11}, that only the solution
in band a) in eq. \ref{eq:15} is compatible with the data.
This implies 
$\varphi = \left . 42.14^{\ + 5.8}_{\ - 7.3} \right .^\circ$
confirming the near singlet quark flavor mixing of $f_{0} (980)$,
the ideal singlet corresponds to
$\varphi_{*} = 54.74^\circ$.

The present analysis leaves the mixing with the 
scalar gluonic meson(s)
completely open. Here we refer to our present results
on glueballs in ref. \cite{WO}. Future work will hopefully
establish the full structure of the 
scalar nonet lowest in mass including the scalar glueball $gb (0^{++}$).

\section*{Acknowledgement}

It is a pleasure to thank all organizers and participants
of the QCD2002 conference, in particular Stephan Narison,
for enabling the meeting to evolve in a
lively and fruitful atmosphere. 

%begin{table*}
% space before first and after last column: 1.5pc
% space between columns: 3.0pc (twice the above)
%\setlength{\tabcolsep}{1.5pc}
% -----------------------------------------------------
% adapted from TeX book, p. 241
%\newlength{\digitwidth} \settowidth{\digitwidth}{\rm 0}
%\catcode`?=\active \def?{\kern\digitwidth}
% -----------------------------------------------------
%\caption{Biologically treated effluents (mg/l)}
%\label{tab:effluents}
%\begin{tabular*}{\textwidth}{@{}l@{\extracolsep{\fill}}rrrr}
%\hline
%                  & \multicolumn{2}{l}{Pilot plant}
%                  & \multicolumn{2}{l}{Full scale plant} \\
%\cline{2-3} \cline{4-5}
%                  & \multicolumn{1}{r}{Influent}
%                  & \multicolumn{1}{r}{Effluent}
%                  & \multicolumn{1}{r}{Influent}
%                  & \multicolumn{1}{r}{Effluent}         \\
%\hline
%Total cyanide    & $ 6.5$ & $0.35$ & $  2.0$ & $  0.30$ \\
%Method-C cyanide & $ 4.1$ & $0.05$ &         & $  0.02$ \\
%Thiocyanide      & $60.0$ & $1.0?$ & $ 50.0$ & $ <0.10$ \\
%Ammonia          & $ 6.0$ & $0.50$ &         & $  0.10$ \\
%Copper           & $ 1.0$ & $0.04$ & $  1.0$ & $  0.05$ \\
%Suspended solids &        &        &         & $<10.0?$ \\
%\hline
%\multicolumn{5}{@{}p{120mm}}{Reprinted from: G.M. Ritcey,
%                              Tailings Management,
%                              Elsevier, Amsterdam, 1989, p. 635.}
%\end{tabular*}
%\end{table*}



%\begin{figure}[hbt]
%\vspace{9pt}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
%\caption{Good sharp prints should be used and not (distorted) photocopies.}
%\label{fig:largenenough}
%\end{figure}
%
%\begin{figure}[htb]
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
%\caption{Remember to keep details clear and large enough.}
%\label{fig:toosmall}
%\end{figure}


\begin{thebibliography}{9}
\bibitem{PDG} K. Hagiwara et al., Phys. Rev. D66 (2002) 010001.
\bibitem{PMWO} P. Minkowski and W. Ochs, EPJ C9 (1999) 283.
\bibitem{CloTo} F. Close and N. Tornqvist, .
\bibitem{latQCD} P. R\"{u}fenacht,  

                 G.S. Bali, .
\bibitem{Nari} S.Narison,these proceedings,

               H.G. Dosch and S. Narison, .

\bibitem{WO} W. Ochs and P. Minkowski, these proceedings.

\bibitem{E791} E.M. Aitala et al., E791 collaboration, 
               Phys. Rev. Lett. 86, (2001) 770, 
%D+ ---> PI- PI+ PI+ DECAY.

               and Phys. Rev. Lett. 86, (2001) 765, .
% D+(S) ---> PI- PI+  PI+  f0 mass width

\bibitem{E687} P.L. Frabetti et al., E687 collaboration, 
               Phys. Lett. B407 (1997) 79,
%D+ ---> PI- PI+ PI+ DECAY.
% D+(S) ---> PI- PI+  PI+ decay 

               and Phys. Lett. B351 (1995) 591.
%D+ ---> K- K+ PI+ DECAY.
%D+(S) ---> K- K+ PI+ DECAY.

\end{thebibliography}
\end{document}
\clearpage
\newpage

\section*{$f_{0} (980)$ from phase shift analysis}

Digressing into the search for the gluonic $0^{++}$ state
we show the original result by the CERN-Munich collaboration \cite{Hyams}
in figure \ref{fig2}.

\begin{center}
 \begin{figure}[htb]
\label{fig2}
\vspace*{-0.0cm}
\epsfig{file=fzeros2.eps,width=6.cm}
\vspace*{-1.2cm}
\caption{I=J=0 elastic $\pi \pi$ phase shift.}
 \end{figure}
\vspace*{-0.0cm}
\end{center}

If the $f_{0}$ (980) phase variation, i.e. its width,
would be much smaller and also much smaller
than the energy resolution, this situation is equivalent
to 'operating out' the narrow resonance. 
This is shown in figure 3. 

\begin{center}
\begin{figure}[htb]
\label{afig3} 
\vspace*{-0.2cm}
\epsfig{file=fzeros3.eps,width=6.cm}
\vspace*{-1.2cm}
\caption{The wide resonance below $f_{0} (980)$.}
 \end{figure}
\vspace*{-0.3cm}
\end{center}

The width of the wide resonance is then given by the
derivative of the phase with respect to $\sqrt{s}$

\begin{equation}
\label{eq:1}
\begin{array}{c}
\left .
d \ \delta \ / \ d \ \sqrt{s} \ \right |_{res} \ = 
\ \left ( \ \Gamma \ / \ 2 \ \right )^{\ -1}
\vspace*{0.2cm} \\
\Gamma \ \sim \ ( 0.8 - 1 )\ \mbox{GeV}
\end{array}
\end{equation}

\begin{thebibliography}{9}

\bibitem{Hyams} B. Hyams et al., Nucl. Phys. B64 (1973) 134.

\end{thebibliography}

\end{document}

