KLOE RESULTS ON $f_0$(980), $a_0$(980) SCALARS AND $\eta$ DECAYS
THE KLOE COLLABORATION
$\ $
% {%\scriptsize F.~AMBROSINO, A.~ANTONELLI, M.~ANTONELLI, C.~BACCI, P.~BELTRAME, G.~BENCIVENNI, S.~BERTOLUCCI, C.~BINI, C.~BLOISE, V.~BOCCI, F.~BOSSI, D.~BOWRING, P.~BRANCHINI, R.~CALOI, P.~CAMPANA, G.~CAPON, T.~CAPUSSELA, F.~CERADINI, S.~CHI, G.~CHIEFARI, P.~CIAMBRONE, S.~CONETTI, E.~DE~LUCIA, A.~DE~SANTIS, P.~DE~SIMONE, G.~DE~ZORZI, S.~DELL'AGNELLO, A.~DENIG, A.~DI~DOMENICO, C.~DI~DONATO, S.~DI~FALCO, B.~DI~MICCO, A.~DORIA, M.~DREUCCI, G.~FELICI, A.~FERRARI, M.~L.~FERRER, G.~FINOCCHIARO, C.~FORTI, P.~FRANZINI, C.~GATTI, P.~GAUZZI, S.~GIOVANNELLA, E.~GORINI, E.~GRAZIANI, M.~INCAGLI, W.~KLUGE, V.~KULIKOV, F.~LACAVA, G.~LANFRANCHI, J.~LEE-FRANZINI, D.~LEONE, M.~MARTINI, P.~MASSAROTTI, W.~MEI, S.~MEOLA, S.~MISCETTI, M.~MOULSON, S.~M\"ULLER, F.~MURTAS, M.~NAPOLITANO, F.~NGUYEN, M.~PALUTAN, E.~PASQUALUCCI, A.~PASSERI, V.~PATERA, F.~PERFETTO, L.~PONTECORVO, M.~PRIMAVERA, P.~SANTANGELO, E.~SANTOVETTI, G.~SARACINO, B.~SCIASCIA, A.~SCIUBBA, F.~SCURI, I.~SFILIGOI, T.~SPADARO, M.~TESTA, L.~TORTORA, P.~VALENTE, B.~VALERIANI, G.~VENANZONI, S.~VENEZIANO, A.~VENTURA, R.VERSACI, G.~XU}
% presented by S. GIOVANNELLA
The KLOE experiment running at the $\phi$-factory DA$\Phi$NE has collected $\sim 450\ pb^{-1}$ in the 2001--2002 data taking. We report preliminary results on light meson spectroscopy based on this data sample; particles are all produced through $\phi$ radiative decays. % The nature of $f_0$(980) and $a_0$(980) is investigated by studying the shape of the resulting mass spectra, which is sensitive to their structure. % A detailed study of the radiative decays involving scalar mesons and % the shape of the resulting mass spectra give the possibility of studying % the nature of these particles. % A detailed study of the $\eta\to\pi\pi\pi$ dynamics through a Dalitz plot analysis gives the possibility to extract information on the quark mass difference. % Finally, the branching ratio for the $\eta\to\pi^0\gamma\gamma$ decay is compared with previous measurements and with the expectations from Chiral Perturbation Theory.
kloe.tex
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\def\NIMA{{\em Nucl. Instrum. Methods} A}
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\def\PRL{\em Phys. Rev. Lett.}
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\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
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\def\ee{\end{equation}}
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\begin{document}
\vspace*{4cm}
\title{KLOE RESULTS ON $f_0$(980), $a_0$(980) SCALARS AND $\eta$ DECAYS}

\author{THE KLOE COLLABORATION\\ $\ $ \\
%
{%\scriptsize
  F.~AMBROSINO, A.~ANTONELLI, M.~ANTONELLI, C.~BACCI, P.~BELTRAME,
  G.~BENCIVENNI, S.~BERTOLUCCI, C.~BINI, C.~BLOISE, V.~BOCCI,
  F.~BOSSI, D.~BOWRING, P.~BRANCHINI, R.~CALOI, P.~CAMPANA, 
  G.~CAPON, T.~CAPUSSELA, F.~CERADINI, S.~CHI, G.~CHIEFARI,
  P.~CIAMBRONE, S.~CONETTI, E.~DE~LUCIA, A.~DE~SANTIS, P.~DE~SIMONE,
  G.~DE~ZORZI, S.~DELL'AGNELLO, A.~DENIG, A.~DI~DOMENICO, 
  C.~DI~DONATO, S.~DI~FALCO, B.~DI~MICCO, A.~DORIA, M.~DREUCCI, 
  G.~FELICI, A.~FERRARI, M.~L.~FERRER, G.~FINOCCHIARO, C.~FORTI, 
  P.~FRANZINI, C.~GATTI, P.~GAUZZI, S.~GIOVANNELLA, E.~GORINI, 
  E.~GRAZIANI, M.~INCAGLI, W.~KLUGE, V.~KULIKOV, F.~LACAVA, 
  G.~LANFRANCHI, J.~LEE-FRANZINI, D.~LEONE, M.~MARTINI, 
  P.~MASSAROTTI, W.~MEI, S.~MEOLA, S.~MISCETTI, M.~MOULSON, 
  S.~M\"ULLER, F.~MURTAS, M.~NAPOLITANO, F.~NGUYEN, M.~PALUTAN, 
  E.~PASQUALUCCI, A.~PASSERI, V.~PATERA, F.~PERFETTO, L.~PONTECORVO, 
  M.~PRIMAVERA, P.~SANTANGELO, E.~SANTOVETTI, G.~SARACINO, 
  B.~SCIASCIA, A.~SCIUBBA, F.~SCURI, I.~SFILIGOI, T.~SPADARO, 
  M.~TESTA, L.~TORTORA, P.~VALENTE, B.~VALERIANI, G.~VENANZONI, 
  S.~VENEZIANO, A.~VENTURA, R.VERSACI, G.~XU}\\ $\ $ \\
%
  presented by S. GIOVANNELLA}

\address{Laboratori Nazionali di Frascati dell'INFN, via Enrico Fermi 40, 
  00044 Roma, Italy}

\maketitle\abstracts{
  The KLOE experiment running at the $\phi$-factory DA$\Phi$NE has 
  collected $\rm \sim 450\ pb^{-1}$ in the 2001--2002 data taking.
  We report preliminary results on light meson spectroscopy based
  on this data sample; particles are all produced through $\phi$ 
  radiative decays.
%
  The nature of $f_0$(980) and $a_0$(980) is investigated by studying
  the shape of the resulting mass spectra, which is sensitive to their 
  structure.
%  A detailed study of the radiative decays involving scalar mesons and 
%  the shape of the resulting mass spectra give the possibility of studying 
%  the nature of these particles. 
%
  A detailed study of the $\eta\to\pi\pi\pi$ dynamics through
  a Dalitz plot analysis gives the possibility to extract information
  on the quark mass difference.
%
  Finally, the branching ratio for the $\eta\to\pi^0\gamma\gamma$ decay
  is compared with previous measurements and with the expectations from
  Chiral Perturbation Theory.
}

%==============================================================================
\section{Introduction}
%==============================================================================

The KLOE experiment \cite{KLOE} operates at DA${\Phi}$NE, \cite{DAFNE} 
the Frascati $e^+e^-$ collider, whose center of mass energy is equal
to the $\phi$ mass. Data collected in 2001-2002, corresponding to 
$\sim 450$ pb$^{-1}$, are used to study light scalar and pseudoscalar 
mesons produced through $\phi$ radiative decays. 
%Some preliminary results on $f_0$(980), $a_0$(980) and $\eta$ decays
%are reported.

%KLOE has been designed to study $CP$ violation in the $K^0\bar{K}^0$ 
%system. The apparatus, consisting of a 2 m radius fully stereo drift 
%chamber and a fine sampling lead/scintillating fibers electromagnetic 
%calorimeter, is also well suited for studying $\phi$ radiative decays.

%==============================================================================
\section{Light Scalar Mesons: $f_0$(980) and $a_0$(980)}
%==============================================================================

A complete study of the radiative decay of the $\phi$ to the scalar 
mesons $f_0(980)$ and $a_0(980)$ is in progress, involving the decays 
$f_0\to\pi^+\pi^-/\pi^0\pi^0$ and $a_0\to\eta\pi^0$, with 
$\eta\to\gamma\gamma$ and $\eta\to\pi^+\pi^-\pi^0$.
Since the mass spectra are sensitive to the nature of such mesons
\cite{AchIvanch} which are still puzzling,~\cite{qqbar,qqqq,kkbar} the 
data are compared with two different theoretical models. In the first one
the scalar amplitude is described by the kaon-loop model~\cite{SigmaKloop}
while in the second one a  point-like approach
is followed. In both cases, the interference with  
the background with the same final state is taken into
account in the fit procedure.

For the $\pi^+\pi^-\gamma$ final state there is a huge irreducible 
background of $e^+ e^- \to \pi^+ \pi^-$ with an additional photon
due to initial state (ISR) or final state radiation (FSR). However, 
requiring two tracks and a large angle photon a clean signal appears
in the $M_{\pi\pi}$ region above 850 MeV (see Fig.~\ref{Fig:ppg}.left).
Moreover, a forward-backward asymmetry 
$A=\frac{N^+(\theta > 90^\circ)-N^+(\theta < 90^\circ)}
{N^+(\theta > 90^\circ)+N^+(\theta < 90^\circ)}$
is expected due to the interference of FSR and ISR.~\cite{Asymm} 
In Fig.~\ref{Fig:ppg}.right we show this asymmetry as a function 
of $M_{\pi\pi}$, both for data and for theoretical predictions with 
ISR and FSR only. A clear discrepancy is observed in the $f_0$ region 
and in the mass range below 700 MeV, thus adding a further evidence
on the need of a scalar meson in the theoretical description.

\begin{figure}
\begin{center}
%\psfig{figure=ppg_minv.eps,height=6cm}
\begin{tabular}{cc}
\psfig{figure=ppg_minv.eps,width=0.5\textwidth,height=6cm} &
\psfig{figure=ppg_asymm.eps,width=0.45\textwidth,height=6cm} \\
\end{tabular}
\end{center}

\vspace{-0.2cm}
\caption{Left: two pion invariant mass for $\pi^+\pi^-\gamma$ events.
  The upper and lower curves are the result of the fit and the
  contribution due to FSR+ISR respectively.
  Right: forward-backward asymmetry as a function of $M_{\pi\pi}$.
  Experimental data are reported as dark triangles while light dots 
  represent the Monte Carlo expectations for FSR and ISR only.\hfill
  \label{Fig:ppg}}
\end{figure}

In the case of the $f_0\to\pi^0\pi^0$ decay we instead deal with a 
non-resonant  background with the same $\pi^0\pi^0\gamma$ signature,
produced  through $\omega\pi^0/\rho\pi^0$ intermediate states.
The intensity of this background is twice the signal. In order
to consider its interference with the scalar term we fit the Dalitz 
plot distribution. A smaller background contamination dominated by 
$\phi\to\eta\gamma$, with $\eta\to\pi^0\pi^0\pi^0$ and two lost
or merged photons, is estimated by Monte Carlo and subtracted from 
the Dalitz plot. When using the kaon-loop model we cannot describe data 
without introducing a scalar term due to a $\sigma(600)$ meson.

For the fully neutral search of $\phi \to  a_0 \gamma$,
the background with the same $\eta\pi^0\gamma$ final state is 
small and simplifies the fit procedure.
On the other hand, having a yield ten times smaller than the
$f_0\to\pi^0\pi^0$, it is contaminated by a large non-interfering 
background with a five photon signature.
%For the search of $\phi \to  a_0 \gamma$, the fully neutral final 
%state has a yield which is ten times smaller than $f_0\to\pi^0\pi^0$
%thus being contaminated by a large background with five photon
%signature which, however, do not interfere with the signal.
%%Having these two processes the same five photon signature, there is a 
%%much more severe background that do not interfere with the signal.
%On the contrary, the background with $\eta\pi^0\gamma$ final state 
%is smaller and simplify the fit procedure.
The $a_0$ decay chain with $\eta\to\pi^+\pi^-\pi^0$ has instead
a rate three times smaller than the neutral channel, but it is 
completely background free.
%
%Finally, in the case of $\eta\to\pi^+\pi^-\pi^0$ the background 
%is completely negligible, although it has a smaller rate
%(4180 events).
A combined fit of the two channels is in progress to extract 
the $a_0$ parameters.

%==============================================================================
\section{Dynamics of $\eta\to\pi\pi\pi$}
%\section{Dynamics of $\eta\to\pi\pi\pi$ and search for $\eta \to \pi^0\gamma \gamma$ decay}
%==============================================================================

The amplitude of $\eta\to\pi\pi\pi$ is related to the d-u
quark mass difference; a precise study of 
%the dynamic of 
this decay can lead to a very accurate measurement of 
$Q^2=(m_s^2-\hat{m}^2)/(m_d^2-m_u^2)$.
Using the 17 millions $\eta$ mesons produced in 2001/2002,
the dynamics of both $\pi^+\pi^-\pi^0$ and $\pi^0\pi^0\pi^0$ 
final states has been studied through a Dalitz plot analysis. 
The $\eta$ mesons are clearly tagged by detecting the 
monochromatic recoil photon of the $\phi\to\eta\gamma$ decay 
($E_{\rm recoil}=363$ MeV); the background is at the level of 
few per mill.

Concerning the $\pi^+\pi^-\pi^0$ final state, the conventional
Dalitz variables are $X\propto T_+ - T_-$ and $Y\propto T_0$, 
where $T$ is the kinetic energy of the pions. The measured 
distribution is parametrized as:
$|A(X,Y)|^2 = 1 + aY + bY^2 + cX + dX^2 + eXY + fY^3$.
As expected from $C$ parity conservation, the odd powers of $X$ 
are consistent with zero (see Tab.~\ref{Tab:FitDalitz}).
Using our fitted parameters, the value of $Q$ can be extracted.
For example, in Ref.~9 the value $Q=22.8\pm 0.4$ is obtained, the 
error being dominated by the $\eta\to\pi^+\pi^-\pi^0$ width.
This value is in agreement with Chiral Perturbation Theory
($\chi_{PT}$) predictions \cite{Qtheory} and with other evaluations
based on $\eta$ decays, \cite{Qeta1,Qeta2} which have larger errors.

\begin{table}[t]
\caption{Fitted parameters of the $\eta\to\pi^+\pi^-\pi^0$ Dalitz 
  plot.\label{Tab:FitDalitz}}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|}\hline
  $\rm N_{dof}$ & Prob($\chi^2$) (\%) & 
  \begin{tabular}{c} 
    $a$ \phantom{00000000000000000} $b$ \phantom{00000000000000000} $c$ \\
    $d$ \phantom{00000000000000000} $e$ \phantom{00000000000000000} $f$ \\ \end{tabular}\\ \hline
  147  &  60   & 
  \begin{tabular}{ccc} $-1.072\pm0.006^{+0.005^{\phantom{0}}}_{-0.007_{\phantom{0}}}$ &
                 $0.117\pm0.006^{+0.004}_{-0.006}$ & 
                 $0.0001\pm0.0029^{+0.0003}_{-0.0021}$ \\ 
                 $\phantom{-}0.047\pm0.006^{+0.004^{\phantom{0}}}_{-0.005_{\phantom{0}}}$ &
                 $0.006\pm0.008^{+0.013}_{-0.000}$ &
                 $0.13\phantom{00}\pm0.01\phantom{00}^{+0.02\phantom{00}}_{-0.01}$ \\ \end{tabular}\\ \hline 
  149  &  63   & 
  \begin{tabular}{ccc} $-1.072\pm0.005^{+0.005^{\phantom{0}}}_{-0.008_{\phantom{0}}}$ & 
                 $0.117\pm0.006^{+0.004}_{-0.006}$ & --- \\ 
                 $\phantom{-}0.047\pm0.006^{+0.004^{\phantom{0}}}_{-0.005_{\phantom{0}}}$ & --- & 
                 $0.13\phantom{00}\pm0.01\phantom{00}^{+0.02\phantom{00}}_{-0.01}$ \\ \end{tabular}\\ \hline 
\end{tabular}
\end{center}
\end{table}

For the $\eta\to\pi^0\pi^0\pi^0$ decay the Dalitz plot density is
described by a single parameter $\alpha$: $|A|^2\propto 1+2\,\alpha\, z$, 
where $z$ is related to the three pion energies in the $\eta$ 
rest frame. 
Photons are paired to $\pi^0$'s after kinematically constraining
the total 4-momentum to $M_\phi$, thus improving the energy resolution.
By fitting a sample with high purity on pairing (98.5\%), corresponding 
to an analysis efficiency of 4.5\%, we get:
%
\begin{equation}
{\rm \alpha = -0.013 \pm 0.005_{stat} \pm 0.004_{\,syst} }\,.
\label{Eq:Alpha}
\end{equation}

%==============================================================================
\section{The $\eta\to\pi^0\gamma\gamma$ Decay}
%==============================================================================

The $\eta\to\pi^0\gamma\gamma$ decay is an important test of $\chi_{PT}$
because of its sensitivity to $p^6$ on both the branching ratio (BR) and 
the $M_{\gamma\gamma}$ spectrum.~\cite{ChPT1,ChPT2}
The present experimental situation is not completely clear: the most 
accurate determination of the BR \cite{GAMS} is far from theoretical 
predictions while a more recent measurement, \cite{CrystalBall} with a 
larger relative error, gives a significantly lower value.
Moreover, all previous searches were done at hadron machines, using 
mainly $\pi^- p\to \eta\, n$. The value of the BR has decreased by three 
orders of magnitude in the last 40 years, due to the improved separation 
of the $\eta\to\pi^0\pi^0\pi^0$ background.

KLOE searches for this decay in a much cleaner environment, with 
different background topologies and experimental systematics. 
The two orders of magnitude higher background with the same five photon 
final state ($e^+e^-\to\omega\pi^0\to\pi^0\gamma\pi^0$, 
$\phi\to f_0\gamma\to\pi^0\pi^0\gamma$, 
$\phi\to a_0\gamma\to\eta\pi^0\gamma$ with $\eta\to\gamma\gamma$)
is reduced by vetoing the additional $\omega/\pi^0/\eta$ particles in
the event.
The remaining background is $\eta\to\gamma\gamma$ with additional
clusters from shower fragmentation or machine background and
$\eta\to\pi^0\pi^0\pi^0$ with merged/lost photons. 
We reject them with energy momentum conservation and a likelihood 
technique to identify merged clusters.
%while the $\eta\to\gamma\gamma/\pi^0\pi^0\pi^0$ are 
%rejected with energy momentum conservation and a likelihood technique
%to identify merged clusters.
The preliminary results obtained fitting the $\eta$ invariant mass 
spectrum (Fig.~\ref{Fig:Minv_pi0gg}) gives a BR in agreement with 
${\cal O}(p^6)$ $\chi_{PT}$ calculations, with a central value which 
is three times smaller than the previous measurement:
%
\begin{equation}
{\rm BR(\eta\to\pi^0\gamma\gamma) = 
  ( 8.4 \pm 2.7_{stat} \pm 1.4_{\,syst}) \times 10^{-5} }\, .
\label{Eq:BRpi0gg}
\end{equation}

\begin{figure}
\begin{center}
\psfig{figure=minv_pi0gg.eps,width=8cm,height=6cm}
\end{center}
\caption{Four photon invariant mass for $\eta\to\pi^0\gamma\gamma$ events.
  Data (crosses) are fitted with the signal and background contributions 
  evaluated from MC (solid histograms).\hfill
\label{Fig:Minv_pi0gg}}
\end{figure}

%==============================================================================
%\section*{Acknowledgments}
%==============================================================================

%We thank the DAFNE team for their efforts in maintaining low background 
%running conditions and their collaboration during all data-taking. 
%We want to thank our technical staff: 
%G.F.Fortugno for his dedicated work to ensure an efficient operation of 
%the KLOE Computing Center; 
%M.Anelli for his continuous support to the gas system and the safety of the
%detector; 
%A.Balla, M.Gatta, G.Corradi and G.Papalino for the maintenance of the
%electronics;
%M.Santoni, G.Paoluzzi and R.Rosellini for the general support to the
%detector; 
%C.Piscitelli for his help during major maintenance periods.
%This work was supported in part by DOE grant DE-FG-02-97ER41027; 
%by EURODAPHNE, contract FMRX-CT98-0169; 
%by the German Federal Ministry of Education and Research (BMBF) contract 06-KA-957; 
%by Graduiertenkolleg `H.E. Phys. and Part. Astrophys.' of Deutsche Forschungsgemeinschaft,
%Contract No. GK 742; 
%by INTAS, contracts 96-624, 99-37; 
%and by the EU Integrated Infrastructure
%Initiative HadronPhysics Project under contract number
%RII3-CT-2004-506078.

%==============================================================================
\section*{References}
%==============================================================================

\begin{thebibliography}{99}

\bibitem{KLOE}
    The KLOE Collaboration (A.~Aloisio et al.), 
    LNF-92/019 (IR) (1992) and LNF-93/002 (IR) (1993).

\bibitem{DAFNE}
    S.~Guiducci et al., Proc.~of the 2001 Particle Accelerator Conference 
    (Chicago, Illinois, USA), P.~Lucas S.~Webber Eds.~353 (2001).

\bibitem{AchIvanch}
    N.~N.~Achasov and V.~N.~Ivanchenko, \Journal{\NPB}{315}{465}{1989}.

\bibitem{qqbar}
    N.~A.~T\"{o}rnqvist, \Journal{\PRL}{49}{624}{1982}.

\bibitem{qqqq}
    R.~L.~Jaffe, \Journal{\PRD}{15}{267}{1997}.

\bibitem{kkbar}
    J.~Weinstein and N.~Isgur, \Journal{\PRL}{48}{659}{1982}.

\bibitem{SigmaKloop}
  N.~N.~Achasov and V.~V.~Gubin, \Journal{\PRD}{56}{4084}{1997}.

\bibitem{Asymm} 
  S.~Binner, J.~H.~Kuhn, K.~Melnikov, Phys.\ Lett.\ B 459 (1999) 279

\bibitem{QKLOE} 
  B.~V.~Martemyanov and V.~S.~Sopov, \Journal{\PRD}{71}{017501}{2005}.

\bibitem{Qtheory}
  J.~Donoghue and A.~Perez, \Journal{\PRD}{55}{7075}{1997};
  J.~Bijnens and J.~Prades, \Journal{\NPB}{490}{239}{1997}.

\bibitem{Qeta1}
  J.~Kambor, C.~Wiesendanger and D.~Whyler, \Journal{\NPB}{465}{215}{1996}.

\bibitem{Qeta2} 
  A.~V.~Anisovich and H.~Leutwyler, \Journal{\PLB}{375}{335}{1997}.

\bibitem{ChPT1}
  J.~Bijnens and J.~Gasser, Proc.~of Workshop on Production, Interaction 
  and Decay of the $\eta$ Meson (Uppsala, October 2001), 
  J.~Bijnens, G.~F\"aldt and B.~M.~K.~Nefkens Eds.,
  T99, 34 (Phys.~Scripta, Stockholm, 2002)

\bibitem{ChPT2}
  E.~Oset, J.~R.~Pelaez, L.~Roca, \Journal{\PRD}{67}{073013}{2003}.

\bibitem{GAMS}
  Alde et al., \Journal{\ZPC}{25}{225}{1984}.

\bibitem{CrystalBall}
  N.~Knecht et al., \Journal{\PLB}{589}{14}{2004}.

\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

hep-ex0505074.extracted_bib
\begin{thebibliography}{99}

\bibitem{KLOE}
    The KLOE Collaboration (A.~Aloisio et al.), 
    LNF-92/019 (IR) (1992) and LNF-93/002 (IR) (1993).

\bibitem{DAFNE}
    S.~Guiducci et al., Proc.~of the 2001 Particle Accelerator Conference 
    (Chicago, Illinois, USA), P.~Lucas S.~Webber Eds.~353 (2001).

\bibitem{AchIvanch}
    N.~N.~Achasov and V.~N.~Ivanchenko, \Journal{\NPB}{315}{465}{1989}.

\bibitem{qqbar}
    N.~A.~T\"{o}rnqvist, \Journal{\PRL}{49}{624}{1982}.

\bibitem{qqqq}
    R.~L.~Jaffe, \Journal{\PRD}{15}{267}{1997}.

\bibitem{kkbar}
    J.~Weinstein and N.~Isgur, \Journal{\PRL}{48}{659}{1982}.

\bibitem{SigmaKloop}
  N.~N.~Achasov and V.~V.~Gubin, \Journal{\PRD}{56}{4084}{1997}.

\bibitem{Asymm} 
  S.~Binner, J.~H.~Kuhn, K.~Melnikov, Phys.\ Lett.\ B 459 (1999) 279

\bibitem{QKLOE} 
  B.~V.~Martemyanov and V.~S.~Sopov, \Journal{\PRD}{71}{017501}{2005}.

\bibitem{Qtheory}
  J.~Donoghue and A.~Perez, \Journal{\PRD}{55}{7075}{1997};
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\end{thebibliography}