\documentstyle [12pt,a4,epsfig]{article}
\oddsidemargin=-.5cm
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\newcommand{\pbar}{\overline{p}}
\newcommand{\pbarp}{\overline{p}p}
\newcommand{\op}{\omega\pi^0}
\newcommand{\ctp}{\pi^0\pi^0\pi^0}
\newcommand{\cte}{\pi^0\pi^0\eta}
\newcommand{\ptp}{\overline{p}p\rightarrow\pi^0\pi^0\pi^0}
\newcommand{\ppom}{\overline{p}p\rightarrow\pi^0\omega}
\newcommand{\pkk}{\overline{p}p\rightarrow K_S K_L}
\newcommand{\ptpe}{\overline{p}p\rightarrow\pi^0\pi^0\eta}
\newcommand{\pgx}{\pi^0\rightarrow\gamma X}              
\newcommand{\egx}{\eta\rightarrow\gamma X}              

\begin{document}
%\begin{center}
%EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH \\
%\end{center}
%\begin{flushright}
%CERN-PPE/94-xx \\
%16.03.1994 \hspace{14mm} \\
%\end{flushright}
\begin{center}
\begin{Large}
        {\bf SEARCH FOR A NEW LIGHT GAUGE BOSON IN DECAYS OF  
$\pi^0$ AND $\eta$ }  \\
\end{Large}
\bigskip

C.~Amsler$^{14}$, D.~S.~Armstrong$^{1}$, I.~Augustin$^{7,a}$,
C.A.~Baker$^{4}$, B.M.~Barnett$^{10}$, C.J.~Batty$^{4}$,
K.~Beuchert$^{2}$, P.~Birien$^{1}$, P.~Bl\"um$^{7}$,
R.~Bossingham$^{1}$, K.~Braune$^{11}$, J.~Brose$^{10}$,
D.V.~Bugg$^{8}$, M.~Burchell$^{5,b}$, T.~Case$^{1}$, 
A.~Cooper$^{8}$, K.M.~Crowe$^{1}$, T.~Degener$^{2}$,
H.P.~Dietz$^{11}$, S.~von~Dombrowski$^{14}$, M.~Doser$^{5}$,
W.~D\"unnweber$^{11}$, D.~Engelhardt$^{7}$, M.~Englert$^{11}$,
M.A.~Faessler$^{11}$, C.~Felix$^{11}$, R.~Hackmann$^{10}$,
R.P.~Haddock$^{9}$, F.H.~Heinsius$^{1}$, N.P.~Hessey$^{5}$,
P.~Hidas$^{3}$, P.~Illinger$^{11}$, D.~Jamnik$^{11,c}$,
Z.~J\'avorfi$^{3}$, H.~Kalinowsky$^{10}$, B.~K\"ammle$^{6}$,
T.~Kiel$^{7}$, J.~Kisiel$^{11,d}$, E.~Klempt$^{10,e}$,
M.~Kobel$^{5,f}$, H.~Koch$^{2}$, C.~Kolo$^{11}$,
K.~K\"onigsmann$^{11,g}$, M.~Kunze$^{2}$, R.~Landua$^{5}$,
F.~Loser$^{6}$, J.~L\"udemann$^{2}$, H.~Matth\"ay$^{2}$,
M.~Merkel$^{10,h}$, J.P.~Merlo$^{10}$, C.A.~Meyer$^{12}$,
L.~Montanet$^{5}$, A.~Noble$^{14}$, F.~Ould-Saada$^{14}$,
K.~Peters$^{2}$, C.N.~Pinder$^{4}$, G.~Pinter$^{3}$,
S.~Ravndal$^{2}$, J.~Salk$^{2}$, A.H.~Sanjari$^{8,i}$,
E.~Sch\"afer$^{10}$, B.~Schmid$^{14,j}$, P.~Schmidt$^{6}$,
S.~Spanier$^{10}$,
C.~Stra\ss burger$^{10,e}$, U.~Strohbusch$^{6}$, M.~Suffert$^{13}$,
D.~Urner$^{14}$, C.~V\"olcker$^{11}$, F.~Walter$^{10}$,
D.~Walther$^{2}$, U.~Wiedner$^{6}$, N.~Winter$^{7}$,
B.~Zou$^{8}$, J.~Zoll$^{5}$, \v{C}.~Zupan\v{c}i\v{c}$^{11}$ \\

\vspace*{0.8cm}
{\bf Crystal Barrel Collaboration}   \\
\vspace*{3mm}
\bigskip
                             %
$^{1}$   University of California, LBL, Berkeley, CA 94720, USA \\
$^{2}$   Universit\"at Bochum, D-44780 Bochum, Germany \\
$^{3}$   Academy of Science, H-1525 Budapest, Hungary \\
$^{4}$   Rutherford Appleton Laboratory, Chilton, Didcot OX11, 0QX, UK \\
$^{5}$   CERN, CH-1211 Gen\`eve, Switzerland \\
$^{6}$   Universit\"at Hamburg, D-22761 Hamburg, Germany \\
$^{7}$   Universit\"at Karlsruhe, D-76344 Karlsruhe, Germany  \\
$^{8}$   Queen Mary and Westfield College, London E1\,4NS, UK \\
$^{9}$   University of California, Los Angeles, CA 90024, USA \\
$^{10}$   Universit\"at Mainz, D-55099 Mainz, Germany \\
$^{11}$   Universit\"at M\"unchen, D-85748 M\"unchen, Germany \\
$^{12}$   Carnegie Mellon University, Pittsburgh, PA, USA \\
$^{13}$   Centre de Recherches Nucl\'eaires,
F-67037 Strasbourg, France \\
$^{14}$   Universit\"at Z\"urich, CH-8001 Z\"urich, Switzerland \\
%
\bigskip
\vspace*{3mm}
Submitted to Physics Letters B \\
\end{center}
%
\begin{flushleft}
\vfill
\noindent \rule[0.5ex]{13cm}{0.6pt} \\
\begin{small}
$a)$ Now at University of Siegen, Siegen, Germany \\
$b)$ Now at University of Kent, Canterbury, UK \\
%$c)$ On leave of absence from BNL, Upton, NY \\
$c)$ On leave of absence from the University of Ljubljana,
Ljubljana, Slovenia  \\
$d)$ On leave of absence from the University of Silesia,
Katowice, Poland \\
$e)$ Now at University of Bonn, D-55127 Bonn, Germany \\
$f)$ Now at University of Freiburg, D 79104 Freiburg i. Br., Germany \\
$g)$ Now at Max Planck Institut, Heidelberg, Germany \\
$h)$ Now at CERN, Gen\`eve, Switzerland \\
$i)$ Now at State University of New York, Stony Brook, NY, USA \\
$j)$ Now at University of Carlifornia, Irvine, CA, USA \\
\end{small}
\end{flushleft}
\pagebreak
\begin{center}
{\bf Abstract} \\
%\vspace*{1cm}
\end{center}
\noindent
We have searched for new light gauge bosons produced in $\pi^0$
and $\eta$
decays by studying the kinematically well constrained reactions
$\pbarp \rightarrow \pi^0\pi^0\pi^0$ and 
$\pbarp \rightarrow \pi^0\pi^0\eta$, where one $\pi^0$ 
or the $\eta$ decay through
the emission of a single photon recoiling against a missing-state $X$.
%The data were collected with
%stopped antiprotons by the Crystal Barrel detector at LEAR.
%$15$ millions 0-prong triggered events have been analysed.
No signal has been observed and branching ratio upper-limits of
$5 \times 10^{-5}$ at $90~\% \ C.L.$ have been obtained.
% for masses of the gauge
%boson between $\sim 60~MeV$ and $130~MeV$ ($\pi^0$ decay), and 
%$5 \times 10^{-5}$ at $90~\% \ C.L.$, for $X$ masses 
%between $\sim 150~MeV$ and $540~MeV$ ($\eta$ decay).
The $\pi^0$ limit represents a factor of ten improvement when compared to the
existing limit, whereas the $\eta$ decay limit has been measured for the 
first time. Current models which predict $\pi^0$ and $\eta$
decay branching ratios into $\gamma X$ at the level of $10^{-3}$ -
$10^{-4}$ can be ruled out. \\

\vspace{1.0cm}

There exist various extensions of the Standard Model 
($SU(3) \times SU(2) \times U(1)$) of
strong and electroweak interactions. Some of them 
require an extra gauge group $U'(1)$ and hence, a new gauge boson $X$. 
This possibility is realised in grand unified theories \cite{guts},
in supersymmetric theories \cite{susy}, in superstring inspired
models \cite{sustring}
and in models including a new long-range interaction, the fifth force
\cite{fifthf}.
Some $X$ bosons could be light enough to be detected in rare or (SM) forbidden
decays of light mesons ($\pi$, $K$, $\eta$, ...) \cite{pigx1,pigx2}.	
To look for these
new light particles, with masses of the order of the pion mass and with
small couplings, high rates and precision are needed.
Table 1 summarises the theoretical expectations concerning the 
branching ratios for the decay of a 
pseudoscalar $P\rightarrow\gamma X$, $P$ being a $\pi^0$, an $\eta$ or an
$\eta'$ \cite{pigx1}. Two possiblities are considered 
depending on the interaction of the $X$ boson:
(I) $X$ interacts weakly with both leptons and quarks, in which case it 
decays in our detector ($P\rightarrow \gamma e^+e^-$ or                         
$P\rightarrow\gamma\nu\overline{\nu}$).
The decay $X\rightarrow\nu\overline{\nu}$ is experimentally more favourable to
study since $X\rightarrow e^+ e^-$ is overwhelmed by the 
Dalitz decay $\pi^0\rightarrow\gamma e^+e^-$;
(II) $X$ interacts weakly only with quarks
and hence has a long lifetime and does not decay in our detector 
\footnote{
The advantage of pseudoscalar decays into $\gamma X$ over 
$K^+\rightarrow\pi^+ X$ is 
the unambiguous theoretical interpretation: if there is a signal
it can only be a spin $1$ gauge boson \cite{pigx1}.}.
In both cases the analysis consists in looking for a photon and a missing mass.

\vspace{0.4cm}

Consider the decay $ \pi^0\rightarrow\gamma \gamma $.
In the rest frame of the decaying $\pi^0$, the photon energy is given by
$ E_{\gamma}^{\ast} = \frac{m_{\pi^0}}{2} $.         
Because of detector inefficiencies, one of the two photons can be lost,
and hence undetected.  
This leads to a peak in the energy distribution with a width determined by 
the experimental resolution.
If one replaces the missing photon by a massive missing gauge boson $X$
the photon energy distribution peaks at
\begin{equation}
E_{\gamma}^{\ast} = \frac{m_{\pi^0}}{2} \ (1 \ - \ \frac{m_X^2}{m_{\pi^0}^2} )
\label{eq-egps}
\end{equation}
For $\eta$ decay one replaces $m_{\pi^0}$ by $m_{\eta}$.

\vspace{0.4cm}

To look for hypothetical gauge bosons $X$ in the decays
$\pi^0\rightarrow\gamma X$  or $\eta\rightarrow\gamma X$,        
we exploit the fact that the annihilation of antiprotons
with protons is a rich source of mesons($\pi^0 , \eta , \omega $)
and that the Crystal Barrel detector is ideally designed  
for measurements of final states with many photons.
We use the well constrained reactions 
$\ptp$ and $\ptpe$ because of the relatively high branching ratios
$BR(\pbarp \rightarrow\pi^0\pi^0\pi^0) = (0.76\pm 0.23 \% )$  
\cite{devons}
and $BR(\pbarp \rightarrow\pi^0\pi^0\eta) = (0.65\pm 0.07 \% )$ 
\cite{curtis}.
Before describing in more detail the data selection and analysis,
we first summarise the capabilities of our detector.

\vspace{0.4cm}

The data have been collected using the Crystal Barrel Detector on beam line
C2 at the Low Energy Antiproton Ring (LEAR) at CERN. 
Antiprotons with a momentum of $200~MeV/c$ are stopped in a $4$ cm long liquid
hydrogen target. A matrix of silicon counters in front of the target defines
and monitors the incident $\pbar$ beam (typically $10^4~\pbar /s$).
The target is surrounded by two cylindrical proportional wire chambers (PWC),
a jet drift chamber (JDC) to detect charged particles and a barrel-shaped calorimeter
consisting of $1380$ CsI(Tl) crystals, $16$ radiation lengths deep, with 
photodiode readout. 
The CsI calorimeter covers the polar angles between $12^{\circ} $ and 
$168^{\circ} $ with full coverage in azimuth. 
The useful acceptance for shower detection is $0.95\times 4\pi $ sr.
The energy calibration is provided by $\pi^0 \rightarrow 2\gamma $ decays
by imposing the mass of the $\pi^0$. 
The energy resolution, which varies slightly from run to run, is given by
$\sigma_E / E_{\gamma} = 2.5\%/ \sqrt[4]{E_{\gamma}/GeV}$.  
The angular resolution (polar and azimuthal) $\sigma_{\theta,\phi}$
varies with the photon energy from $20$~mrad to $35$~mrad
for isolated showers and is $\sim \ 35$~mrad 
for two or more showers in the same cluster of crystals.
The whole assembly is located in a solenoid magnet providing a homogeneous 
field of $1.5~T$ parallel to the incident $\pbar $ beam.
A more detailed description of the detector,
the trigger conditions and the reconstruction procedure
is given in \cite{cbdet}.

\vspace{0.4cm}

The analysis is based on $14.6$ million triggered all-neutral events
taken in 1990 and 1991. 
The triggering is performed using the PWC and 
the inner layers of the JDC to veto charged tracks.
Events with residual charged tracks are removed  in the offline data
selection. $1'019'634$ 
events containing exactly $5$ electromagnetic showers (PEDs) with energy deposits
exceeding the threshold energy $E_{cut}=20~MeV$ are selected.
%Events due to various sources of background are identified and ignored: 
Events for which at least one PED is located around 
the beam-pipe are removed to suppress energy leakage.
Pseudo `photons', due to shower fluctuations, the so-called
split-offs, are characterised by their small energy and their proximity
to a higher energy PED. Such split-off events are suppressed by requiring
the relative energy deposition of two neighbouring showers, the centroids
of which are separated by less than $14^{\circ}$, to be larger than  $0.18$.  
The next step is to reconstruct two $\pi^0$ s out of the $5\gamma$ sample
($818'655$ events). 
All combinations of the invariant masses of two photons
are calculated and events containing two $\pi^0$ s and one single $\gamma$ 
are kept. 
When more than one combination is found, only the best configuration 
with a confidence level $C.L. \ >2.7\%$ is chosen. 
The remaining data sample consists of $187'145$ events.
Merged $\pi^0$ s whose photon
showers are not isolated (below $1~\%$ effect) are discarded.
Once the $\pi^0\pi^0\gamma$ events are selected, 
a kinematic fit with $3$ constraints
(3C-fit) ($\pi^0\pi^0(\pi^0)_{miss}$ or $\pi^0\pi^0(\eta)_{miss}$) is performed
ignoring the remaining single photon ($\gamma_5$).
A confidence level greater than $15\%$ is required.
We are left with $95'392$ $\pi^0\pi^0(\pi^0)_{miss}$ and $17'943$
$\pi^0\pi^0\eta_{miss}$ fit events.

The method used to search for an $X$ signal now consists in plotting the energy
$E_{\gamma}^{\ast}$
of the single $\gamma$ in the rest frame of the missing $\pi^0$ (or $\eta$).
To calculate $E_{\gamma}^{\ast}$ the non-fit values of the fifth photon
(which is ignored by the fitting procedure) and the fit values of $\pi^0_{miss}$ 
or $\eta_{miss}$ are used.

\vspace{0.4cm}

The background contribution can be split into two categories:
$5\gamma$ final states without and with missing energy and momentum $E/P$.
The first source of background
comes from real $\pi^0\pi^0\gamma$ events, due to the finite resolution of 
the detector. Here it is mainly the channel 
$\pbarp \rightarrow \omega\pi^0$ ($\omega\rightarrow\pi^0\gamma$)
which feeds into the fit $\pi^0\pi^0\pi^0_{miss}$ and to a lesser extent into
$\pi^0\pi^0\eta_{miss}$. To reduce this background the selected
$5\gamma$ data events are kinematically fit to the following final states: 
$\pbarp \rightarrow \pi^0\omega $ 
($\omega \rightarrow \pi^0 \gamma $) (7C-fit),
$\pbarp \rightarrow \eta \omega $ 
($\omega \rightarrow \pi^0 \gamma $ and 
$\eta \rightarrow \gamma \gamma $) (7C-fit),
$\pbarp \rightarrow \pi^0 \eta \gamma $ (6C-fit) and
$\pbarp \rightarrow \eta \eta \gamma $ (6C-fit). 
If one of the $5\gamma$ combinations fits one of the above final states
with a $C.L.>1\%$, the event is rejected.
About $ 25\% $ of events are rejected for $\pi^0\pi^0\pi^0_{miss}$,
while the proportion of rejected events is negligible 
in the case of $\pi^0\pi^0\eta_{miss}$. 
The second source of background consists of the following final states:
$\pbarp \rightarrow \pi^0\pi^0\pi^0 $, where one photon 
from a $\pi^0$ is undetected and
$\pbarp \rightarrow \pi^0\pi^0\eta $, where one photon from
the $\eta$ is undetected.
There are two main reasons for a photon to escape detection.
Either the $\gamma$ is lost in the beam pipe ($ \theta  \geq 
168^{\circ}$ or $ \theta \leq 12^{\circ}$) 
or between the two halves of the barrel ($\theta \sim 90^{\circ}$)
or its energy is below the detection threshold of $20~MeV$.
In order to reduce the missing photon contribution, an event is discarded 
if $\mid cos ( \theta_{\gamma_{miss}}) \mid \ \geq \ 0.95$ (missing photon in 
beam pipe) or
$\mid cos (\theta_ {\gamma_{miss}}) \mid \ \leq \ 0.02$ (missing photon in dead
region between the two halves of the Crystal Barrel detector). 
We are left with 
$48'158$ $\pi^0\pi^0(\pi^0)_{miss}$ and $10'827$
$\pi^0\pi^0\eta_{miss}$ events.

For the $\eta$ decay we also take into account
the channel $\pbarp \rightarrow K_S K_L$, where 
$K_S \rightarrow \pi^0 \pi^0 $ and $K_L$ interacts in the barrel and
produces one PED. This background is relevant 
since the mass of $K_L$ ($497.67$) is close to the $\eta$ mass.
The $K_L$ and $K_S$ fly back to back and
therefore events are rejected for which the invariant mass of the two $\pi^0$s
is in the region $455 \leq M_{\pi^0\pi^0} \leq 506~MeV/c^2 $ 
and the angle between the the two $\pi^0$s and the fifth photon is
less than $13^{\circ}$.
About $50\%$ of the events far from the $\eta\rightarrow\gamma\gamma$ 
peak are rejected. This reduces the $\pi^0\pi^0\eta_{miss}$ sample to
$8'961$ events.  

To further reduce badly fit events
the ratio $r_E = E_{\gamma}^{lab}/E_{\gamma}^{\ast}$ is used. 
The data show an accumulation of events at high values of the ratio $r_E$.  
The following cuts are therefore applied:
$ E_{\gamma}^{lab} / E_{\gamma}^{\ast} \ \leq \ 2.6$ for $\eta$ decay and
$ E_{\gamma}^{lab} / E_{\gamma}^{\ast} \ \leq \ 10.0 $ for $\pi^0$ decay.
Almost $50\%$ of bad events ($\pi^0$ decay) 
and $\sim 30 \% $ ($\eta$) are rejected by these cuts.  
The final sample consists of $24'503$ $\pi^0\pi^0\pi^0_{miss}$ and 
$5'121$ $\pi^0\pi^0\eta_{miss}$ events. 

\vspace{0.4cm}

The next step is to check the method and 
determine the acceptance and energy resolution. A Monte Carlo program, 
based on the GEANT \cite{geant} package modified to reproduce 
our detector, is used to generate and simulate the various $\pbarp$
final states (signal and background events).
The simulated events are then passed through the same reconstruction and 
analysis as the real data.
In the generation of signal events it is assumed that 
the particle $X$ does not interact in the detector.
Its mass is varied between $0$ and $m_{\pi^0}$
(or $m_{\eta}$). In the $\pi^0\pi^0\pi^0$ generated sample, one $\pi^0$
is forced to decay into $\gamma X$.
For masses smaller than $\sim 60~MeV/c^2$ (which correspond to high 
$\gamma$ energies)
a possible $X$ signal would be overwhelmed by the missing photon background 
($E_{\gamma}^\ast \sim m_{\pi^0}/2$). 
The higher mass range (smaller $\gamma$ energy) is limited by the detection
threshold of the photon ($20~MeV$). 
The detector is therefore more sensitive to $X$ masses between $\sim 60$ 
and $\sim 130~MeV/c^2$ for $\pi^0$ decay.
With the $\eta$ decay we can investigate higher $X$ masses between 
$\sim 150$ and $\sim 540~MeV/c^2$.

The photon energy resolutions vary between
$\sigma = (4.0 - 5.0)~MeV $ and $\sigma = (3.5 - 4.5)~MeV$, 
depending on the $X$ mass, for $\pi^0$ and $\eta$ decay. 
On the $\pi^0\rightarrow\gamma\gamma$ peak the resolution is about $10~MeV$.
The lower and upper-limits correspond to the above upper and lower
limits for the $X$ mass.

Important background sources are also simulated:
(i) $\pbarp \rightarrow \pi^0\omega $, 
($\omega \rightarrow \pi^0 \gamma $), 
(ii) $\pbarp \rightarrow \pi^0\pi^0\pi^0 $  
($\pi^0 \rightarrow \gamma \gamma $), and
(iii) $\pbarp \rightarrow \pi^0\pi^0\eta $ 
($\pi^0,\eta \rightarrow \gamma \gamma $).
We find that the $5\gamma$ final state $\pi^0\omega$ and 
the $6\gamma$ final state $\pi^0\pi^0\pi^0$ (where one photon from a $\pi^0$ 
is not reconstructed) reproduce the shape of the data.
The normalisation is reproduced if we take into account the known branching 
ratios: $BR(\pbarp \rightarrow\pi^0\pi^0\pi^0) = (0.76\pm 0.23) \% $,  
$BR(\pbarp \rightarrow\pi^0\omega) = (0.57\pm 0.05) \% $ 
(\cite{cbomega}) and $BR(\omega\rightarrow\pi^0\gamma) = (8.5\pm 0.5 \% )$.  
Other final states, like $\pi^0\eta\gamma$, $\eta\eta\gamma$,
$\omega\pi^0\pi^0$ (where two photons are not reconstructed), and 
$\omega\omega$, $\omega\rightarrow\pi^0\gamma$ (where one photon is not 
reconstructed), give a negligible contribution.

%Before showing the data and presenting the results we need the overall efficiencies. 
The signal events (with $\pi^0$ or $\egx$) are used to
calculate the ratio between the 
number of simulated and accepted events and the number of generated events. 
The following efficiencies are obtained:
$\epsilon^{\pi^0} = (11\pm 1)\% $ for the $\pi^0$ decay and
$\epsilon^{\eta} = (16\pm 1)\% $
for the $\eta$ decay. The errors include the dependence of the efficiency
on the mass of $X$.
The number of expected events for a given branching 
ratio $BR(X) = BR(\pi^0,\eta\rightarrow\gamma X)$ is
$N_X = N_{\pbar} \times       
BR(\pbarp \rightarrow\pi^0\pi^0\pi^0 (\eta)) \times BR(X) \times \epsilon $.
For the $\pi^0$ decay a multiplicative factor $3$ is taken into account,
corresponding to the $3 \pi^0$ final state.
The $14.6 \times 10^6$ analysed $0$-prong events correspond to
$N_{\pbar} = 356 \times 10^6$ annihilations at rest if we assume a 
$BR(\pbarp \rightarrow 0-prong)$ of $(4.1^{+0.2}_{-0.6})\%$ \cite{Ghesq}.
We then expect $N_X^{\pi^0} = 97$ and $N_X^{\eta} = 35$ 
signal events for a branching ratio $BR(X)$ of $10^{-4}$. 

\vspace{0.4cm}
 
After having discussed the background reduction and the  
Monte Carlo simulation we now present the data. 
First we discuss the $\pi^0$ decay.  

The energy of the single $\gamma$ in the $\pi^0$ rest frame is shown 
in fig. \ref{fig-xpi0f2}.
A ten-parameter fit to the data includes the sum of three gaussians and a 
constant background term. 
The three gaussians have maxima around $m_{\pi^0}/2$ (at $69.2$, 
$69.7$ and $72.0~MeV$). 
The resolutions ($4.2$, $10.3$ and $26.0~MeV$) are in agreement with what 
we expect for $\pi^0\omega$ events (no missing photon),
$\pi^0\pi^0\pi^0$ events (where one photon is missing), and
other smaller background sources like $\omega\pi^0\pi^0$ (where $2$ photons are
missing). The third gaussian is, as expected, wider than the first two and 
its amplitude much smaller ($32$ compared to $187$ for the first and $263$ for
the second gaussian).
The backgound term is due to events where the fifth photon found
does not belong to the missing $\pi^0$. 
The global fit shows that the spectrum is well understood. 
To look for a signal
a gaussian with two free parameters (position and number of events) is fit 
to the data in the region between $E_{\gamma}^{\ast}=10$ and $67.5~MeV$.
The width is fixed to the experimental resolution.
The result is shown in fig. \ref{fig-xpi02.res}. 
%The ``signal'' found around $17.5~MeV$ contains $\sim 10$ events 
%(corresponding to $\sim 10^{-5}$ BR)
%but the error is very large ($\pm 8$), so that the significance is poor. 
No significant signal is found. A signal corresponding
to $BR(\pi^0\rightarrow\gamma X) = 5 \times 10^{-4}$, the present existing 
limit \cite{xbnl}, is indicated (dashed curve).
An upper-limit of 
$BR(\pi^0\rightarrow\gamma X) \ < \ 5 \times 10^{-5}$ 
is obtained at $90\% \ C.L.$ for masses of the $X$ boson varying between 
$60~MeV/c^2$ and $130~MeV/c^2$.
The upper-limits as a function of mass are shown in fig. \ref{fig-brxvsmx2}(a).

\vspace{0.4cm}

Next we discuss the $\eta$ decay.
The energy of the single $\gamma$ in the $\eta$ rest frame shows
a flatter and lower background than for $\pi^0$ (fig. \ref{fig-xeta2.res}).
The main reason is the reduced contribution from $6\gamma$ events ($\pi^0\pi^0\eta$,
where one photon from the $\eta$ is lost).
This is due to the smaller BR for $\eta\rightarrow\gamma\gamma$, the higher
$\gamma$ efficiency for the $\eta$ decay and the lower final state multiplicity
(one for $\pi^0\pi^0\eta$ vs three for $\pi^0\pi^0\pi^0$). 
In fig. \ref{fig-xeta2.res}
a six-parameter fit to the data is performed
which includes one gaussian and a background term (second order polynomial).
No significant signal is found on top of the background curve.
An upper-limit of 
$BR(\eta\rightarrow\gamma X) \ < \ 5 \times 10^{-5}$ 
is obtained at $90\% \ C.L.$ for masses of the $X$ boson between $150~MeV/c^2$
and $540~MeV/c^2$. 
A signal corresponding to $BR(\eta\rightarrow\gamma X) = 5 \times 10^{-4}$ 
is also indicated (dashed curve). 
The upper-limits 
as a function of mass are given in fig. \ref{fig-brxvsmx2}(b).

\vspace{0.4cm}

To summarise,
we have searched for new light gauge bosons produced in $\pi^0$
and $\eta$ decays by studying the kinematically well constrained reactions
$\pbarp \rightarrow \pi^0\pi^0\pi^0$ and 
$\pbarp \rightarrow \pi^0\pi^0\eta$, where one $\pi^0$ (in one case)
or the $\eta$ (in the other case) decay through
the emission of a single photon recoiling against a missing state $X$.
The different sources of background are well understood.
We have analysed $14.6$ million 0-prong triggered events.
No signal has been observed and branching ratio upper-limits of
$5 \times 10^{-5}$ at $90~\% \ C.L.$ have been obtained for masses of the gauge
boson between $60~MeV$ and $130~MeV$ ($\pi^0$ decay), and 
of $5 \times 10^{-5}$ at $90~\% \ C.L.$, for $X$ masses 
between $150~MeV$ and $540~MeV$ ($\eta$ decay).
The $\pi^0$ limit represents a factor of ten improvement when compared to the
existing limit \cite{xbnl}, whereas the $\eta$ upper limit has been measured for the first 
time. Current models which predict $\pi^0$ and $\eta$
decay branching ratios into $\gamma X$ at the level of $10^{-3}$ -
$10^{-4}$ \cite{pigx1} can be ruled out.
%It is possible to extend the search for $X$ to the $\eta'$
%an hence the decay $\pbarp \rightarrow \pi^0\pi^0\eta'$ 
%($\eta'\rightarrow\gamma X$) is being studied. 

%
\vspace{0.5cm}
%
We would like to thank the technical staff of the LEAR machine
group and of all the participating institutions for their
invaluable contributions to the success of the experiment.
We acknowledge financial support from the German
Bundesministerium f\"ur Forschung und Technologie, the
Schweizerischer Nationalfonds, the British Science and Engineering
Research Council and the U.S.~Department of Energy (contract
No.~DE-FG03-87ER40323 and DE-AC03-76SF00098).
We thank Dr. S. Gninenko who first brought this research subject to our 
attention.

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\vspace{1.2cm}

\begin{table}[hbpt]\centering
\label{tab-pred}
\begin{tabular}[t]{|l|c|c|}
\hline
    Decay                  &{\bf $I$ }              & {\bf $II$ } \\ \hline 
$\pi^0\rightarrow\gamma X$ & $ 6.2 \times 10^{-6} $      &  $2.7 \times 10^{-3} $  \\
$\eta \rightarrow\gamma X$ & $ 1.6-3.8 \times 10^{-4} $  &  $2.0-4.7 \times 10^{-4} $ \\
$\eta'\rightarrow\gamma X$ & $ 0.7-5.7 \times 10^{-6} $  &  $0.3-2.3 \times 10^{-6} $ \\
                                                              \hline
\end{tabular}
\caption[]{\it Theoretical predictions (upper-limits) 
         for the decay of a pseudoscalar into a photon
         and a gauge boson $X$. Two cases are considered: (I) $X$ interacts
         with leptons and quarks, (II) $X$ interacts only with quarks
         \cite{pigx1}.} 
\end{table}
%
\newpage
\begin{figure}[htbp]
\vspace{18.0cm}
\caption[]{\it Energy of the single $\gamma$ in the $\pi^0$ rest frame 
           for the $\pi^0\pi^0\pi^0_{miss}$ fit sample after background
           reduction.
           The different contributions to the ten-parameter fit
           (full line) described in the text are shown separately:
           a narrow gaussian corresponding to $\pi^0\omega$ (dashed line),
           a broader gaussian corresponding to $\pi^0\pi^0\pi^0$ (dotted
           line), a much broader gaussian corresponding to other 
           final states (like $\pi^0\pi^0\omega$), where 2 photons are lost
           (dashed-dotted line)
           on top of a constant background.
                   }
\label{fig-xpi0f2}
\end{figure} 
%
\begin{figure}[htbp]
\vspace{18.0cm}
\caption[]{\it Energy of the single $\gamma$ in the $\pi^0$ rest frame 
           for the $\pi^0\pi^0\pi^0_{miss}$ fit sample after background
           reduction. The curve corresponds to a fit to the data
           (1 gaussian on top of the curve of fig. \ref{fig-xpi0f2}).
           The width of the gaussian is $4.5~MeV$, the position and amplitude
           being the free parameters. 
           A signal corresponding to a branching ratio 
           for $\pgx$ with $m_X = 120~MeV/c^2$ of 
           $5 \times 10^{-4}$ is also drawn (dashed line). The ``signal'' 
           found around $E_{\gamma}^{\ast}=18~MeV$ (full line) is not 
           significant.     
                   }
\label{fig-xpi02.res}
\end{figure} 
\begin{figure}[htbp]
\vspace{18.0cm}
\caption[]{\it Energy of the single $\gamma$ in the $\eta$ rest frame 
           for the $\pi^0\pi^0\eta_{miss}$ fit sample after background
           reduction. 
           The full curve corresponds to a six-parameter fit to the data
           (gaussian + $2^{nd}$ order polynomial).
           A signal corresponding to a branching ratio 
           for $\egx$ with $m_X = 500~MeV/c^2$ 
           of $5 \times 10^{-4}$ is also indicated (dashed line).   
                   }
\label{fig-xeta2.res}
\end{figure} 
%
\begin{figure}[htbp]
\vspace{20.0cm}
\caption[]{\it Branching ratio upper-limit for the decays 
          $\pi^0\rightarrow\gamma X$ (a) and
          $\eta\rightarrow\gamma X$ (b) 
          as function of $E_{\gamma}^{\ast}$ (lower $x$-axis) and $X$ mass
          (upper $x$-axis).
                   }
\label{fig-brxvsmx2}
\end{figure} 
%
\end{document}

