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\begin{document}

\title{Meson model for $f_0(980)$ production in peripheral
  pion-nucleon reactions }

\author{F.P. Sassen}
\author{S. Krewald}
\author{J. Speth}
\affiliation{Institut f\"ur Kernphysik, \\
Forschungszentrum J\"ulich GmbH, 52425 J\"ulich,
Germany}

\date{03.12.2002}

\begin{abstract}
The J\"ulich  model for $\pi\pi$-scattering, based on an effective meson-meson Lagrangian is applied to the 
analysis of the $S$-wave production
amplitudes derived from the BNL E852 experiment $\pi^- p \rightarrow
\pi^0 \pi^0 n$ for a pion momentum of 18.3 GeV. The unexpected strong dependence
of the S-wave partial wave amplitude on the momentum transfer between
the proton and neutron in the vicinity of the $f_0(980)$ resonance
is explained in our analysis as interference effect between the correlated and uncorrelated $\pi^0\pi^0$ pairs.


\end{abstract}

\pacs{11.80.Gw,13.85.-t,14.40.Aq,14.40.Cs}

\keywords{ Pion-Nucleon reactions, scalar-isoscalar mesons,
    meson decay}

\maketitle

Meson spectroscopy in the scalar-isoscalar channel has received increasing 
interest motivated by the search for non-$q\bar{q}$ mesons, such as
glueballs\cite{Amsler:1998up}.
The large number of experimentally observed $0^{++}$resonances suggests that some of those
resonances
may  have a more complicated structure than the conventional 
$q\bar{q}$ structure \cite{Amsler:2002ey,Abele:2001pv}. 
The $f_0(980)$ has been a candidate for a non-$q\bar{q}$ meson for
more than two decades\cite{Jaffe:1977ig,Weinstein:1990gu,Lohse:1990ew,Barnes:1985cy,Pichowsky:2001qe,Oller:1998ng,Achasov:1998pu}.

Recently, the scalar-isoscalar $\pi\pi$ partial wave amplitudes have been
deduced from two pion interaction obtained via the
charge-exchange reaction $\pi^- p \rightarrow \pi^0 \pi^0 n$
measured for incident pion momentum of 18.3 GeV$/c$ by the
E852 collaboration at the Brookhaven National Laboratory\cite{Gunter:2000am}.
 In the
vicinity of the invariant two-pion mass $m_{\pi \pi}=980$ MeV, a
peculiar behavior of the $S$-wave amplitude has been observed.
 Such an effect has also previously been reported
by the GAMS collaboration for a beam momentum of 38 GeV/c\cite{Alde:1995jj}.
 While for
small momentum transfers between the proton and the neutron $( -t < 0.1\;\textrm{GeV}^2)$
the scalar amplitudes show a dip around 1 GeV, a sharp peak is seen
at the same energy for large momentum transfers $( -t > 0.4\;\textrm{GeV}^2 )$.

This observation has been interpreted as evidence for a hard component
in the $f_0(980)$ which would make the interpretation of this scalar
meson
as a $K\bar{K}$ molecule unconvincing
\cite{Klempt:2000ud,Kondashov:1998uh,Anisovich:1995jy,Anisovich:2002us}.
Here, we want to show that the strong dependence of the
$f_0(980)$-production on the momentum transfer between the proton and
the neutron is not in contradiction with a $K\bar{K}$
structure of the $f_0(980)$.
 Actually we will show in the following that this $t$-dependence is
 due to the interference between the resonance structure and the non-resonant background
 and does not depend on the detailed structure of the $f_0(980)$.

\begin{figure}
\resizebox{0.48\textwidth}{!}{%
\rotatebox{90}{
 \includegraphics{fig1.ps}
}}
\caption{\label{dsigmadt}The $\pi^0\pi^0$ production events as a
  function
of the square $t$ of the momentum transfer between proton and neutron.
 These data are used to determine the slope factors b.
Solid line: meson-exchange model including final state interactions
between the produced mesons ( see Feynman diagrams of the insert ).
Crosses: the BNL-E852 data\cite{Gunter:2000am}.}
\end{figure}

For ultrarelativistic beam momenta in the present kinematical regime the reaction
 $\pi^- p \rightarrow \pi^0\pi^0 n$
is a peripheral one. This implies a relatively simple reaction mechanism
which suppresses especially the excitation of nucleon resonances. The 
relevant 
Feynman diagrams are displayed in  Fig.\ref{dsigmadt}. In a peripheral
 reaction one assumes that the incoming pion interacts with the
meson cloud of the proton only once. On the other hand one fully considers the final state interaction between
the produced mesons.
 In  a peripheral charge-exchange reaction, only isovector
mesons have to be considered. The $\rho$-meson cannot contribute because of 
G-parity. This leaves the pion and the $a_1$-meson as
the only relevant mesons with parity $P=(-1)^{J+1}$ to be exchanged in the 
t-channel. e.g. the $a_2$ can not contribute in the reaction since it 
has quantum numbers $J^P=2^+$. However the
$a_1$-exchange is known to be important in peripheral $\pi 
N$-reactions\cite{Kaminski:1997gc,Achasov:1998pu}.


The final state interaction of the produced mesons is described by
an improved version of the J\"ulich meson-exchange
model\cite{Lohse:1990ew, Krehl:1997rk}. This means we use the 
Blankenbeclar Sugar scattering equation to generate our pion 
pion $T$-matrix. 
\begin{eqnarray}\nonumber 
&&T_{ij}(\vec k',\vec k;E)=V_{ij}(\vec k',\vec k;E)\\
&& \phantom{T_{ij}} \nonumber +\sum_l \int 
d^3\vec k''V_{il}(\vec k',\vec k'';E) G_l(\vec k'';E) %\\ 
%\nonumber && 
T_{lj}(\vec k'',\vec k;E)
\end{eqnarray}
Here $\vec k$ and $\vec k'$ are the momenta of the initial and final 
particles in the center of mass frame and $E$ is the total energy of the 
system. The propagator $G$ has been constructed in a way that ensures 
unitarity for the $S$-matrix and is given by:
\begin{eqnarray}
\nonumber G_l(\vec k;E)&=&\frac{\omega_1(\vec k)+\omega_2(\vec 
k)}{(2\pi)^3 
2\omega_1(\vec k)\omega_2(\vec k)} 
\frac{1}{E^2-(\omega_1(\vec k)+\omega_2(\vec k))^2} 
\end{eqnarray}
with $\omega_{1/2}(\vec k)=\sqrt{\vec k^2+m_{1/2}^2}$. Further more $V$ 
is calculated in the one boson exchange approximation including $s$- 
and $t$-channel graphs. The subscripts to 
the transition matrix $T$, the propagator $G$ and the potential $V$ 
indicate 
the coupled channels used in our analysis. They are the $\pi\pi$ and the 
$K\bar K$ channel as well as the newly added $\pi a_1$ reaction channels. 
When adding the later we used the 
Wess-Zumino Lagrangian\cite{Wess:1967jq} for the $a_1 \rho \pi$-coupling.

We also
investigated whether our $K\bar{K}$-molecule was artifically generated
by the independent choice of $s$- and $t$-channel form factors.
\begin{figure}
\resizebox{0.48\textwidth}{!}{%
 \includegraphics{fig2.ps}
}
\caption{\label{highlow}
The contribution of the $S$-wave to the total cross section is shown as a function of the
invariant two-pion mass $m_{\pi\pi }$. Solid line: the meson-exchange model;
dotted line: contribution generated by pion exchange at the
proton-neutron vertex;
dashed line: contribution generated by $a_1$ exchange at the
proton-neutron vertex.
In the upper part, the $S$-wave contributions to the cross section
  from \cite{Gunter:2000am} averaged for $0.01<-t<0.1\;\textrm{GeV}^2$
are shown as a function of the invariant two pion mass, while in the
lower part the corresponding data   averaged for $0.4<-t<1.5\;\textrm{GeV}^2$
are shown.
The data are scaled according to the limits given in \cite{Kaminski:2001hv}.}
\end{figure}
Correlating the form factors by
dispersion relations we found no hint in this direction.
In the original model, only one scalar meson $f_0(1400)$ was included.
Now we consider both the $f_0(1370)$ and the $f_0(1500)$ mesons
as $s$-channel diagrams. The couplings of  these mesons to the three reaction
channels considered were adjusted to reproduce the
two-pion decays of the resonances. We found $g_{f_0(1370)K\bar{K}} = 0.551$,
$g_{f_0(1370)\pi a_1} =0.268$,  and
 $g_{f_0(1500)\pi\pi }= g_{f_0(1500)K\bar{K}} =0.188$. These are effective couplings
which also simulate the influence of $4\pi$ decay channels.
This is a minimal extension of the original
J\"ulich model which allows  to discuss the structure of the $f_0(980)$, 
which is our main point of interest.
To analyze  the  decay structure of   the $f_0(1370)$ and the $f_0(1500)$ mesons,
 the inclusion of $4\pi$ decays would be required, however
 \cite{Abele:2001pv}.
 The $\pi\pi$ phase shifts obtained in the new
model are very similar to the  ones of Ref.  \cite{Lohse:1990ew}.

Given the large beam momentum, we describe the initial $\pi$- and $a_1$-meson exchanges
by the corresponding Regge trajectories.
In ultrarelativistic two-pion production reactions, the cross sections decrease
exponentially with the momentum transfer $t$. In the partial wave analysis of the
data, one therefore attaches a slope factor $e^{b_{\pi}(t-m_{\pi}^2)}$.
 The analysis
of the BNL data required the introduction of two different slope factors.
We interprete the two slope factors as effective  form factors of the $pn\pi$-
and the $pna_1$-vertices. Choosing $b_{\pi}=10.0\;\textrm{GeV}^{-2}$ and $b_{a_1}=5.0\;\textrm{GeV}^{-2}$,
the model can reproduce the experimental slope up to
$-t=2\;\textrm{GeV}^2$, see Fig.\ref{dsigmadt}. The full 
$t$-dependence 
is given by:
\begin{eqnarray} \nonumber
\frac{\partial^2\sigma}{\partial m_{\pi\pi}\partial 
t}&=&A_{\pi}\frac{-t}{(t-m_{\pi}^2)^2}e^{b_{\pi}(t-m_{\pi}^2)}\left| 
T_{\pi\pi\rightarrow\pi\pi}(m_{\pi\pi},t)\right|^2\\
\label{prodampli} &&+A_{a_1}(1+tC)^2e^{b_{a_1}t}
\left|T_{\pi a_1\rightarrow\pi\pi}(m_{\pi\pi},t)\right|^2
 \end{eqnarray}
Please note that $A_{\pi}$ and $A_{a_1}$ are not constant and that adding 
the absolute values squared is to account for the helicity structure 
as will be explained later. Further more $C$ should be considered a free 
parameter as explained in \cite{Achasov:1998pu} where our value of 
$C=-4.4\textrm{ GeV}^2$ was taken from. 

\begin{figure}
\resizebox{0.48\textwidth}{!}{%
 \includegraphics{fig3.ps}
}
\caption{\label{noKKbar}%
The contribution of the $S$-wave to the total $\pi\pi$ cross section is shown
as a function of the invariant two-pion mass $m_{\pi\pi}$.
The  transition potentials which couple to the
$K\bar{K}$-channel via meson-exchanges in the $t$-channel
are multiplied by a scaling factor,
long-dashed: $\lambda=0.0$,
  dotted:   $\lambda=0.75$,
 dash-dotted:   $\lambda=0.88$,
  solid:   $\lambda=1.0$.
 The data shown are taken from the BNL
  E852 Experiment \cite{Gunter:2000am}.
  The upper and lower part refer to small and large momentum transfers, as in
  Fig.\ref{highlow}.
  }
\end{figure}


In Fig.\ref{highlow}, the $S$-wave contribution to the total cross section
 is shown as a function of the invariant two-pion
mass. In the upper part, the data integrated over the momentum range
$0.01 < -t < 0.1\;\textrm{GeV}^2$ show a broad strength distribution from threshold to
about 1.5 GeV, interrupted by a dip near 980 MeV. 
Our microscopic meson-theoretical model
is able to reproduce this behavior nearly quantitatively. The model
 includes the $\pi\pi$, $K\bar{K}$, and $\pi a_1$ reaction channels, but no coupling to the $\rho\rho$-channel.  For the small momentum transfers displayed
in the upper part of Fig.\ref{highlow},
 the contribution due to the exchange of
a pion  in the initial $t$-channel is dominant.
For invariant masses $m_{\pi\pi}$ ranging from threshold to about 1 GeV,
the experimental $\pi\pi$ phase shifts in the $S$-wave rise
almost linearly  to about $100^{\circ}$. The corresponding partial wave amplitude therefore becomes negative
in the vicinity of  $m_{\pi\pi} = 980\;\textrm{MeV}$. This implies a destructive interference with the amplitude
which describes the $f_0(980)$ meson and generates the dip seen in the data.
At even higher energies the $f_0(1500)$ shows a similar behavior.
 At larger momentum transfers, the broad bump has disappeared in the data and one observes a narrow peak around 1 GeV.
  In that momentum regime (lower part of Fig.\ref{highlow})  the contribution due to the pion in the initial $t$-channel
is negligibly small within our meson exchange model and the $a_1$
 exchange gives the dominating contribution. (This can be traced back
 to the different slope factors.)
 Due to the spin structure interference effects between $a_1$- and
 $\pi$-exchange can be neglected since the $a_1$-emission mainly conserves
 the helicity of the nucleon whereas the $\pi$-emission dominantly
 flips the nucleon helicity. But since the resonant contribution is now
 in phase with the non-resonant background we observe the opposite
 behavior compared to the upper part: the $f_0(980)$ resonance shows
 as a peak.


In contrast to an empirical analysis which 
assumes a  smooth background to which parameterized resonances are added
\cite{Anisovich:1995jy,Achasov:1998pu},
the present approach derives the background in a consistent way within our model.
This is essential for interference effects.
 To illustrate this point, we performed a series of calculations
in which  the transition potentials connecting the $\pi\pi$ channel  and the
 $K\bar{K}$ channel via $t$-channel meson exchanges
 were multiplied by a scaling factor $\lambda$ which
we changed from 0 to 1. The $t$-channel meson exchanges within the $K\bar{K}$
channel were scaled by the same factor. For   $\lambda = 0$, the $\pi\pi$ and  $K\bar{K}$
channel can interact only via $s$-channel diagrams.
The corresponding contributions to the $S$-wave total cross sections are shown
in Fig.\ref{noKKbar}.

For small momentum transfers $t$ between the proton and the neutron ( upper part of  Fig.\ref {noKKbar}),
 one finds a broad strength distribution
extending up to 1500 MeV, if the  $t$-channel coupling to the $K\bar{K}$ channel is
switched off ($\lambda=0$).
 Allowing a small
coupling to the $K\bar{K}$ channel  ($\lambda=0.75$),
 the cross section decreases in the energy region
between $1000$ MeV and the onset of the $f_0(1370)$ resonance at about
 $1250$ MeV. As the scaling strength   $\lambda$  is further increased,
a dip develops near $m_{\pi\pi} = 980$ MeV.
For large momentum transfers $t$ between the proton and the neutron (lower part of  Fig.\ref {noKKbar}),
a bump in the vicinity of $m_{\pi\pi} = 980$ MeV appears when the coupling to the
$ K\bar{K}$ channel  is switched on.   

Near  $m_{\pi\pi} = 500$ MeV, a calculation without coupling to
the   $K\bar{K}$ channel overestimates the data. With increasing coupling strength  $\lambda$
the shape of the experimental strength distribution and the relative
size of the bumps centered at $m_{\pi\pi}=500$ MeV and
$m_{\pi\pi}=980$ MeV are reproduced. It is important to 
realize that this feature emerges in
a natural way from a model for the $\pi\pi$ phase shifts when
proper Regge production amplitudes are used. Here for example the 
$(1+tC)^2$ part 
of (\ref{prodampli}), which already appears in early analysis of 
$\pi N\rightarrow\pi\pi N$ scattering data e.g. \cite{Kimel:1977np}, is 
essential to the low energy part of the high momentum transfer case.

\begin{figure}
\resizebox{0.48\textwidth}{!}{%
 \includegraphics{fig4.ps}
}
\caption{\label{eventsk0k0}%
The contribution of the $S$-wave to the production $\pi^-p\rightarrow K^0\bar{K}^0n$ is shown
as a function of the invariant two-kaon mass $m_{K\bar{K}}$.
dashed: our model with overestimation due to missing $4\pi$-decays,
solid:  Model with additional $\sigma\sigma$-channel.
The data shown is taken from \cite{Etkin:1982sg} with the bin width of
$50$ MeV not shown.  
  }
\end{figure}

Finally, we compare the results of our model for the reaction $\pi^- p
\rightarrow K^0 \bar{K}^0 n$ with the published data\cite{Etkin:1982sg}.
The model works satisfactorily from threshold up to
about 1200 MeV. Beyond that energy, our model strongly overestimates
the production of neutral kaons (Dashed line in
Fig.\ref{eventsk0k0}). This is understood when comparing our effective
couplings to the decays of the $f_0(1370)$ and $f_0(1500)$ resonances
as listed in \cite{Hagiwara:2002pw}. We used a strong coupling to the $K\bar{K}$
to simulate decays which in reality go to $4\pi$-channels thus naturally
overestimating the kaon production. The solid line in
Fig.\ref{eventsk0k0} shows a version of the model where this shortcoming has been
removed by a mock $\sigma\sigma$-channel to
account for $4\pi$-decays. A good description of the data is obtained
even beyond $1200$ MeV where the partial wave amplitudes are strongly dominated by resonances.

We conclude that the J\"ulich model which describes the $f_0(980)$ as
a $K\bar{K } $ molecule can explain the strong dependence of the
$S$-wave production on the momentum transfer between the proton and
the neutron near $m_{\pi\pi}=980$ MeV by an interference mechanism.

\bibliography{sassen}
\end{document}

