\documentclass{ws-p8-50x6-00}

\begin{document}
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\title{New Tests of Chiral Perturbation Theory in
    $\eta$ Decays using the Crystal Ball}
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\author{B.M.K. Nefkens and S. Prakhov}
\address{for the Crystal Ball Collaboration
\footnote{
The Crystal Ball Collaboration:
E.~Berger, M.~Clajus, A.~Maru\u{s}i\'c, S.~McDonald,
B.M.K.~Nefkens, N.~Phaisangittisakul, S.~Prakhov, M.~Pulver,
A.~Starostin and W.B.~Tippens, {\em UCLA},
D.~Isenhower and M.~Sadler, {\em  ACU},
C.~Allgower and H.~Spinka, {\em ANL},
J.~Comfort, K.~Craig and T. Ramirez, {\em ASU},
T.~Kycia, {\em BNL},
J.~Peterson, {\em UCo},
W.~Briscoe and A.~Shafi, {\em GWU},
H.M.~Staudenmaier, {\em UKa},
D.M.~Manley and J.~Olmsted, {\em KSU},
D.~Peaslee, {\em UMd},
V.~Bekrenev, A.~Koulbardis, N.~Kozlenko, S.~Kruglov and I.~Lopatin, {\em PNPI},
G.M.~Huber, N.~Knecht, G.J.~Lolos and Z.~Papandreou, {\em UReg},
I.~Slaus and I.~Supek, {\em RBI},
D.Grosnick, D.~Koetke, R.~Manweiler and S.~Stanislaus, {\em ValU}.}
\footnote{Supported in part by US DOE, NSF, NSERC, RMS and VS.}
}
\address{UCLA, Los Angeles, CA 90095, USA}
\maketitle

The G-parity violating $\eta\to 3\pi^0$ decay occurs primarily as a 
consequence of the up-down quark mass difference, $m_u-m_d$.
In the standard notation~\cite{RPP}
 $A(\eta\to 3\pi^0) \sim (m_{u}-m_{d})
 (1+\alpha z)$ and
 $\Gamma(\eta \to 3\pi^0) \sim (m_u-m_d)^2 (1+2\alpha z)$,
 where $\alpha$ is the quadratic slope parameter,
 $z = (\rho /\rho_{\mathrm{max}})^2 \sim \sum(E^{\pi^o}_i-m_{\eta}/3)^2$~,
  $\rho$ is the  distance from the center of the Dalitz plot.
% $E_i$ is the energy of the $i^{th}$ pion in the $\eta$ rest frame. 
 Thus far $\chi$PTh has not
succeeded in accounting fully for the experimental $\eta\to3\pi$ 
decay rate.  The expanded ${\cal O} ({p^4})$ calculation which
includes pion-pion rescattering to all orders~\cite{Kambor} is still 
too low by $(34 \pm10)\%.$  It has been suggested~\cite{Holstein}
 that this might be due to dynamical effects with 
terms ${\cal O}({p^6})$ such as
$\mathcal{L}^6 \sim F^2_\pi trm D^{\mu} U D_\mu U^{\dag} tr 
 D^\nu U D_\nu U^{\dag}.$
 This possibility may be
explored by a precise measurement of the quadratic slope parameter 
$\alpha$ in $\eta\to 3\pi^0$ decay.  The
existing data are inconclusive.  GAMS-2000~\cite{Alde} obtained
$\alpha= -(22\pm23)\times 10^{-3}$.
The Crystal Barrel \cite{Abele} has reported
 $\alpha=-(52\pm 15\pm 10)\times 10^{-3}$.
 The theoretical prediction~\cite{Kambor} is 
$\alpha = -(14~\mathrm{to}~7)\times 10^{-3}$.

 In our recent experiment at the AGS we have produced a 
 sample of $6\times10^6$ $\eta\to 3\pi^0$ in $\pi^-p\to\eta n$ near
 threshold using the Crystal Ball, CB~\cite{Prakhov}.
 We have selected a very pure subsample of
$1\times10^6$ $\eta\to3\pi^0$ decays with a background $<1\%$ using 
a kinematic fit to the process
$\pi^-p\to\eta n\to3\pi^0n\to 6\gamma n.$
%  This includes testing the 15 possible permutations of 6
%clusters to give uniquely the $6\gamma\to3\pi^0\to\eta$ event.
  Our data are compared in great detail to a
full scale Monte Carlo simulation.
% which accurately represents our experimental particulars such as our
% $\eta$ source, the resolution of the CB and the trigger conditions.
A straight line fit of the ratio of the z-distributions of our data 
and the MC gives the slope parameter, see Fig.\ref{fig:eta1}a~:
 $\alpha=-(33\pm 3_{\mathrm{stat}}\pm 6_{\mathrm{syst}})\times 10^{-3}$.
 We agree with theory~\cite{Kambor} in the
sign but not in the value of $\alpha$.
%
\begin{figure}[htb]
%\figurebox{20pc}{15pc}{} % to have a box alone
\epsfxsize=27pc % will enlarge or reduce the postscript figures based on the xsize
\epsfbox{nefkens.eta.fig.eps} % postscript image file name
\caption{
 ~a) - Ratio of the $z$-distributions of our data and MC;
 normalizing $A_0\to 1.0$ we have $\alpha =\frac{1}{2} A_1$.
 ~b) - The invariant mass distribution of a subsample
 of $\pi^0\gamma\gamma$ candidates before (solid line) and
 after subtraction of the background from $\eta \to 3\pi^0$
 (shown by crosses).  The expected contribution
 from $\eta \to \pi^0\gamma\gamma$ events occuring at the decay rate of
 GAMS-2000 is shown by the hatched histogram.  
  \label{fig:eta1}}
\end{figure}
%

 From $\alpha_{\mathrm{dyn}}=\alpha_{\mathrm{exp}}-\alpha_{\mathrm{th}}$,
 see Ref~\cite{Holstein}, we obtain
 $\alpha^{000}_{\mathrm{dyn}}=-(26\pm 2\pm 6)\times 10^{-3}$, using the
 value $\alpha^{000}_{\mathrm{th}}=-7\times 10^{-3}$ from Ref~\cite{Kambor}. 
 Isospin invariance implies
that $\alpha^{000}_{\mathrm{dyn}}=\alpha^{+-0}_{\mathrm{dyn}}$.
 The meager experimental data~\cite{Gormley} supports this within errors,
$\alpha^{+-0}_{\mathrm{dyn}}=-(23\pm15)\times 10^{-3}.$  The WASA 
collaboration is planning a precise measurement of
$\alpha^{+-0}$. 
%$\alpha^{+-0}$~\cite{Kullander}. 
 It is interesting to compare our 
$\eta\to 3\pi^0$ slope with the one for
$K^{0}_{L}\to 3\pi^0$ reported in Ref.~\cite{Somalwar}.  The kaon data 
has the same sign as the $\eta$ but the
value is much smaller:
 $\alpha^{000}_{K}=-(1.6\pm 0.6\pm 0.4)\times 10^{-3}$.

 The rare decay $\eta\to \pi^0\gamma\gamma$ has drawn much 
attention over the years, especially
in connection with testing $\chi$PTh as summarized in
 Table~\ref{tab:thcalc}~. 
 The only known measurement~\cite{Alde} yielded 
$\Gamma(\eta\to\pi^0\gamma\gamma)=0.84\pm0.19$~eV. This
value is greater than every prediction based on $\chi$PTh. 
 Table~\ref{tab:thcalc} shows that the decay $\eta\to\pi^0\gamma\gamma$
 is a unique test of the ${\cal O}(p^6)$ of $\chi$PTh.  We have 
used the CB $\eta$ run which yielded $18.9\times 10^6~\eta$`s
 to search for $\eta\to\pi^0\gamma\gamma$ decays.  
  After applying a series of carefully crafted cuts
 to the initial sample of $7\times 10^6$ four cluster events
 and making a kinematic fit to the
 $\pi^-p \to \pi^0\gamma\gamma n \to 4\gamma n$ 
 hypothesis we are left with a set of
 events, their invariant mass distribution shows a peak at 
the $\eta$ mass atop a background remaining from reaction
 $\pi^-p \to \pi^0\pi^0 n$~.
 This peak is mostly produced
 by $\eta \to 3 \pi^0$ decays with $2 \times 2$
overlapping clusters. We have made detailed investigations 
of this decay with the CB. 
 So we feel confident of subtracting it from the
% data (see the result in Fig.\ref{fig:eta1}b). Thus
 data, the result is shown in Fig.\ref{fig:eta1}b. Thus
 far we have not identified a clear signal.  At 
present we prefer to state only the upper limit
 $\Gamma(\eta\to\pi^0\gamma\gamma)\ < 0.3$~eV
 which is consistent with $\chi$PTh, ${\cal O}({p^6})$.
%  This limit is supported by a totally independent
% analysis being carried out by the Regina branch of the CB collaboration.
% Our upper limit on $\eta\to\pi^0\gamma\gamma$ is consistent with
% $\chi$PTh, ${\cal O}({p^6})$.

 An independent demonstration that the GAMS result on 
$\Gamma(\eta\to\pi^0\gamma\gamma)$ is too large
comes from the CB experiment on $\eta\to 3\gamma$ which is a test of 
$C$-invariance.  We have achieved
$BR(\eta\to 3\gamma)<5\times 10^{-5}$ which is a factor 10 better 
than the published limit.  We have found
$(16\pm33)$ $\eta\to 3\gamma$ events.  The largest background in this 
search is from $\eta\to\pi^0\gamma\gamma.$  If the GAMS result
were correct we expect to have 78 events in our $\eta\to3\gamma$
sample due to $\eta\to\pi^0\gamma\gamma$.
% decays. 
%
\begin{table}[t]
\caption{Theoretical prediction for the $\eta \to \pi^0\gamma\gamma$
 decay rate. \label{tab:thcalc}}
\begin{center}
\footnotesize
\begin{tabular}{|l|c|l|}
\hline
Theory  &
 \raisebox{0pt}[13pt][7pt]{$\Gamma (\eta \to \pi^0 \gamma \gamma)[eV]$}
 & Reference  \\
\hline
$\chi$PTh, ${\cal O}(p^2)$ &  0 & Y. Nemoto,
 {\it PR} {\bf D54} (1996) 6777 \\

$\chi$PTh, ${\cal O}(p^4)$ & 0.004 & Ll. Amettler..,
 {\it PL} {\bf B276} (1992) 185 \\

$\chi$PTh, ${\cal O}(p^6) +. . $  & 0.42 $\pm$ 0.20 & Ll. Amettler..,
 {\it PL} {\bf B276} (1992) 185 \\

$\chi$PTh, ${\cal O}(p^6)+. . $ & 0.47 & P.Ko,
 {\it PL} {\bf B349} (1995) 555 \\

$\chi$PTh, ENJL &  0.58 $\pm$ 0.03 & S. Belluci..,
 {\it NP} {\bf B452} (1995) 626 \\

$\chi$PTh, ENJL & 0.27 $\pm$ 0.18 &J. Bijnens..,
 {\it PL} {\bf B379} (1996) 209 \\

VMD & 0.30 $\pm$ 0.15 &J. Ng ..,
 {\it PR} {\bf D46} (1992) 5034 \\

Q box & 0.70 & J. Ng.., {\it PR} {\bf D47} (1993) 4937 \\

VMD & 0.4 & W. Allen.., {\it NC} {\bf A45} (1966) 272 \\
\hline
\end{tabular}
\end{center}
\end{table}

For an interesting excercise in analysis we are looking for the rare
 decay $\eta\to \pi^0\pi^0\gamma\gamma$.
  This is particularly challenging in view of the
monumental background from $\eta\to3\pi^0$.  Our preliminary result is
$BR(\eta\to\pi^0\pi^0\gamma\gamma)<3.1\times10^{-3}$.
   A theoretical prediction~\cite{Knoechlein} based on
$\chi$PTh ~${\cal O}(p^4)$ is about $10^{-6}$. 
%depending on the $2\gamma$ invariant mass cut off.
%
\begin{thebibliography}{99}
\bibitem{RPP} Review of Particle Physics,
 \Journal{\PRD}{50}{1454}{1994}.
\bibitem{Kambor}J. Kambor {\it et al},
 \Journal{\NPB}{465}{215}{1996}.
\bibitem{Holstein} B. Holstein, private communication.
\bibitem{Alde} The GAMS-2000 Collaboration, D. Alde {\it et al},
  \Journal{\ZPC}{25}{225}{1984}.  
\bibitem{Abele}The Crystal Barrel Col., A. Abele {\it et al},
 \Journal{\PLB}{417}{193}{1998}.  
\bibitem{Prakhov}The Crystal Ball Col., S.Prakhov {\it et al},
 \Journal{\PRL}{84}{4802}{2000}.  
\bibitem{Gormley}M. Gormley {\it et al}, 
  \Journal{\PRD}{2}{501}{1970}.  
%\bibitem{Kullander}S. Kullander (The WASA Collaboration),
% private communication.
\bibitem{Somalwar}S. Somalwar {\it et al},
  \Journal{\PRL}{68}{2580}{1992}.  
\bibitem{Knoechlein}G. Knoechlein {\it et al},
  \Journal{\PRD}{53}{3634}{1996}.  
\end{thebibliography}
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\end{document}
