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\newcommand{\GeV}{\rm GeV}
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\newcommand{\rpv}{$R_P$-violating}
\newcommand{\grav}{$\tilde{G}$}
\newcommand{\sel}{$\tilde{e}$}


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\begin{document}

\pagestyle{empty}
\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
\begin{tabular}{llrr}
%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
                & & & \\
\multicolumn{4}{l}{{\bf
                International Europhysics Conference 
                on High Energy Physics, EPS03},
                July~17-23,~2003,~Aachen} \\
                (Abstract {\bf 676} & Parallel Session & {\bf 13}) & \\
                & & & \\
\multicolumn{4}{l}{{\bf
                XXI International Symposium on  
                Lepton and Photon Interactions, LP03},
                August~11-16,~2003,~Fermilab} \\ 
                & & & \\
                \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}

\begin{center}
  \Large
  {\bf 
    Search for Superlight Gravitinos at HERA}

  \vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent
A search for gravitinos produced in positron-proton scattering is presented
using the 
H1 detector at HERA. In models where the gravitino is the lightest
supersymmetric particle, an $R$-parity violating $t$-channel exchange of a
scalar electron can produce a neutralino, which subsequently decays into a
photon and a superlight gravitino. The resulting event signature involves
an isolated photon and missing transverse            
energy. No deviation from the Standard Model has been found. Limits on
$R$-parity violating Yukawa couplings have
been derived --- for the first time at HERA independently of squark masses
--- for different neutralino and selectron masses in the framework of a
Gauge Mediated Supersymmetry Breaking scenario.
\end{abstract}


\end{titlepage}

\pagestyle{plain}

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

At HERA, in previous analyses of $R$-parity ($R_P$) violating Supersymmetry
(SUSY) 
resonant squark production has been investigated \cite{rpvhera}. Limits
on various \rpv\ Yukawa couplings have been derived dependent on the
squark mass. In this analysis, neutralino ($\chi_1^0$) production has been
studied via 
\rpv\ t-channel selectron exchange in positron proton scattering. The
masses of squarks are assumed to be beyond the reach of HERA. 

In Gauge Mediated Supersymmetry Breaking (GMSB) scenarios the slepton masses are usually much lower than the squark
masses \cite{gmsbtheo}. Therefore, the above process is studied
within the framework of GMSB. Here the prompt
decay of the neutralino into a photon and a gravitino is studied, which
leads to prominent event signatures with a photon and large missing
transverse momentum. 
%
Thus, the theoretical model investigated in this analysis is a combination of
\rpv\ SUSY with a GMSB scenario. The gravitino is the lightest
supersymmetric particle (LSP), while the neutralino is assumed to be the
next-to-lightest supersymmetric particle (NLSP).

The data were taken from 1999 to 2000, when HERA was operating with a
positron beam and a center of mass energy of $\sqrt{s}=319\
\GeV$. They correspond to an integrated luminosity of $65.4\ \pb^{-1}$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theoretical Framework}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Neutralino Production}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$R$-parity is a discrete multiplicative symmetry. It can be written as
$R_P=(-1)^{3B+L+2S}$, where $B$ denotes the baryon number, $L$ the lepton
number and $S$ the spin of a particle. For all superfields of the
supersymmetric Standard Model, each Standard Model (SM) field has the quantum
number $R_P=+1$ and its superpartner $R_P=-1$. 

Given the particle content of the SM, the most general gauge and
supersymmetry invariant superpotential \cite{wein} contains an \rpv\ Yukawa
coupling between a left-handed selectron $\tilde{e}_L$, a left-handed
up-type quark $u_L^i$ and a right-handed down type anti-quark
$\bar{d}_R^j$, where $i$ and $j$ denote generation indices. The
corresponding part of the Lagrangian reads as:
%
\begin{equation}
  {\cal L}_{\rm RPV} = \lambda'_{1jk} (-\tilde{e}_L u_L^j \bar{d}_R^k) + h.c.
  \label{eq:rpv}
\end{equation}
%
At HERA, the presence of interactions (\ref{eq:rpv}) lead to neutralino
production 
via $t$-channel selectron exchange. The search presented here is performed under the simplifying
assumption that one of the couplings $\lambda'_{1j1}$ ($j=1,2$) dominates.
%This is the first investigation of such couplings performed independently
%of squark masses.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Neutralino Decay}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In GMSB models the gravitino mass is given by
%
\begin{equation}
  m_{\tilde{G}} = \frac{F}{\sqrt{3} M_P}
  \label{eq:gravmass}
\end{equation}
%
where $ M_P = 2.4 \cdot 10^{18}$ GeV is the reduced Planck mass and $F$ the
fundamental squared scale of Supersymmetry breaking. This leads to 
typically light ($<1$ GeV) gravitino masses. Therefore the gravitino is
the lightest SUSY particle. The couplings of SUSY particles to
the gravitino are small, so the 
gravitino only emerges in the decay of the NLSP (which is the neutralino in
our scenario), provided that the \rpv\ decay of the NLSP is not largely
dominant.  

The relevant part of the supersymmetric Lagrangian containing gravitino
interactions with gauginos is given by
%
\begin{equation} 
  {\cal L}_{\tilde{G}} = \frac{1}{8 M_P} \bar{\lambda}^A \gamma^\rho
  \sigma^{\mu\nu} \tilde{G}_\rho F^A_{\mu\nu} + h.c. \,\,,
\label{eq:lgr}
\end{equation}
%
where $\tilde{G}$ is the gravitino field, $\bar{\lambda}^A$ the gaugino
field and $F^A_{\mu\nu}$ the corresponding field strength. At the level
of an effective 
interaction, the spin-3/2 gravitino field can be well described by its
spin-1/2 goldstino component when it appears as an external state, i.e.
%
\begin{equation} 
  \tilde{G}_\mu = \sqrt{\frac{2}{3}} \frac{i}{m_{\tilde{G}}} \partial_{\mu}
  \tilde{G} \,\,.
  \label{eq:gold1/2}
\end{equation}
%
Using this limit (\ref{eq:gold1/2}) in (\ref{eq:lgr}), one derives the decay
width of the 
neutralino into a gravitino and a photon:
%
\begin{equation}
  \Gamma({\chi_i}^0 \rightarrow \gamma\tilde{G}) = \frac{\kappa_i
  m_{\chi^0_i}^5}{16 \pi \, F^2} \,\,.
\label{eq:neudecay}
\end{equation}
%
Here, $\kappa_i = |N_{i1} \cos\Theta_W + N_{i2} \sin\Theta_W|^2 $ is the mixing
parameter with $N_{ij}$ the $\chi_i^0$ ($i=1,2,3,4$) components in the
notation of~\cite{standardnotation}.
 
%The \rpv\ decays of the neutralino are not considered in the analysis.
The respective Feynman-Graph of the process analyzed in this paper is shown in
Fig.~\ref{fig:feyn}.

%\newpage
%%%%%%%%%%%%%%%%%%begin%%figure%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[H] 
\vspace{-2cm}
  \begin{center}
   \psfrag{a}[][][1.8][0]{$e^+$}
   \psfrag{b}[][][1.8][0]{$d$}
   \psfrag{c}[][][1.8][0]{$\chi^0$}
   \psfrag{d}[][][1.8][0]{$u^j$}
   \psfrag{k}[][][1.8][0]{\color{blue} $\lambda'_{1j1}$}
   \psfrag{s}[][][1.8][0]{\color{blue} $c$}
   \psfrag{t}[][][1.8][0]{\color{blue} $\tilde{e}$}
   \psfrag{gr}[][][1.8][0]{\color{red} $\tilde{G}$}
   \psfrag{ga}[][][1.8][0]{\color{red} $\gamma$}
   {\epsfig{file=H1prelim-03-064.fig1.eps,width=7cm,bbllx=0,bblly=190,bburx=180,bbury=0}}
  \end{center}
  \vspace{8cm}
  \caption{Neutralino production via selectron exchange, decaying into
    a gravitino and a photon.}
  \label{fig:feyn}
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%end%%figure%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The H1 Detector}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The detector components of the H1 experiment~\cite{Abt:1997hi} most
relevant for this analysis are shortly described in the following. 
The interaction region is surrounded by a system of drift and multi-wire proportional
chambers covering the polar angular range\footnote{The polar angle $\theta$ is measured with respect to the proton beam direction ($+z$).} $7^o < \theta < 176^o$. The tracking system is
placed inside a finely segmented liquid argon (LAr) calorimeter
covering the polar angular range 4$^o~<~\theta~<$~154$^o$~\cite{Andrieu:1993kh}.
Energy resolutions of
 $\sigma_E / E \simeq 12\% / \sqrt{E({\rm GeV})} \oplus 1\%$ for
 electrons and $ \sigma_E / E \simeq 50\% / \sqrt{E({\rm GeV})} \oplus 2\%$ for
 hadrons have been obtained in test beam measurements 
~\cite{Andrieu:1994yn,Andrieu}. The tracking system and calorimeters are surrounded by a
 superconducting solenoid and an iron yoke instrumented with streamer tubes.
Leakage of hadronic showers outside the calorimeter is measured by analogue 
charge sampling of the streamer tubes with a resolution~\cite{iron} of
$\sigma_E / E \simeq 100\% /~\sqrt{E({\rm GeV})}$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Monte Carlo Generation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% BACKGROUND SIMULATION:


For each possible SM background source, complete Monte Carlo
simulations of the H1 detector response are performed.
All of them correspond to a luminosity of more than 10 times 
that of the data.


% For the NC or CC DIS background estimates we use

For the simulation of the charged and neutral current (CC, NC) backgrounds, 
the DJANGO \cite{DJANGO} event generator is used, which
includes first order QED radiative corrections.
QCD radiation is treated following 
the approach of the Colour Dipole Model~\cite{CDM} 
and is implemented using
ARIADNE~\cite{ARIADNE}.
The hadronic final state is generated using the
string fragmentation model~\cite{JETSET74}.
%
The parton densities in the proton are taken
from the CTEQ5L~\cite{CTEQ} parameterization. 
%
For  direct and resolved photoproduction ($\gamma p$) 
of light and heavy flavors,
the PYTHIA event generator~\cite{PYTHIA} 
is used, which relies on first order QCD matrix elements
and uses leading-log parton showers
and string fragmentation~\cite{JETSET74}.
The GRV  (GRV-G) parton densities~\cite{SFGRVGLO} in the proton
(photon) are used. 
%???
The SM prediction for $ep \rightarrow e W^\pm X$ and $ep \rightarrow e Z X$
is calculated with EPVEC~\cite{EPVEC}. 
%???

% SIGNAL SIMULATION:
 
%
For the simulation of the signal, we use the SUSYGEN~\cite{SUSYGEN} 
event generator.
%
%, recently extended~\cite{SUSYGEN3} to allow 
%the generation of SUSY events in $ep$ collisions.
%Any gauge decay of the squark can be generated, and the
%cascade decays of the subsequent $\chi$'s or $\tilde{g}$ 
%are performed according to the corresponding matrix elements.
%
Initial and
final state parton showers are simulated following the
DGLAP~\cite{DGLAP} evolution equations, 
and string fragmentation~\cite{PYTHIA,JETSET74}
is used for the non-perturbative part of the hadronization.
%
The parton densities CTEQ5L~\cite{CTEQ} are evaluated at the
scale of the Mandelstam variable $-t$.
This scale is also chosen for the maximum virtuality of parton showers
initiated by the final state quark.
% ???
%
%
To allow a model independent interpretation of the results,
the signal topology was simulated for a wide range
of masses of the neutralino from
$50\ \GeV$ to $140\ \GeV$ in steps of typically $10\ \GeV$. The mass of the
selectron was varied from $m_{\chi^0_1} +1$ GeV to the theoretically allowed
limit in the given GMSB scenario. The mass spectrum of supersymmetric
particles has been calculated within GMSB using SUSPECT~\cite{SUSPECT}.
The events were passed through a complete simulation of the H1 detector.
%
These simulations allowed the determination of signal selection efficiencies
as a function of the 
masses of the neutralino and the selectron, since
the step sizes for the simulation were
small enough for a linear interpolation between them.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Event Selection}
\label{seq:selection}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To reject non-$ep$ background,
it is required
that the events are not rejected by a set of beam halo and cosmic muon 
filters~\cite{H1F2PAPER}, that
they satisfy constraints on their timing relative to the 
nominal time of the beam bunch crossings, and
that a primary interaction vertex is reconstructed within 
$\pm 35$ cm around the nominal vertex value. 


The decay $\chi^0 \rightarrow \gamma \tilde{G}$ is characterized 
by missing transverse energy and by an electromagnetic cluster in the
calorimeter. The main SM background is expected from CC~DIS.
Events are selected with a total vector sum of the transverse momentum in all
calorimeters \ptc\ greater than 25 GeV and an identified photon of
transverse momentum $P_{t,\gamma}$ greater than 15 GeV.

The photon is identified by a shower shape analysis of energy deposits in
the LAr calorimeter, selecting isolated and compact electromagnetic
clusters. 
To reject NC~DIS background in which the scattered electron
(sometimes misinterpreted as a photon) is dominantly scattered through
small angles, photon candidates are accepted in the forward region of the
detector only ($\theta_\gamma < 80^o$). The minimal polar angle of the photon is
$10^o$. For $\theta_\gamma > 20^o$ an electromagnetic cluster is only accepted
if it is {\em not} associated with a charged track in the central tracking
system fulfilling matching criteria of cluster energy and track momentum.
To reduce the influence of photons from collinear QED radiation from the
quark line, the photon must be isolated so that no other calorimeter
cluster with 
an energy larger than 500 MeV lies within a
pseudorapidity-azimuthal cone centered around the photon direction and
of opening $\sqrt{ (\Delta \eta)^2 + (\Delta \phi)^2 } = 0.5$,
where $\eta = -\ln \tan \frac{\theta}{2}$. 
%
Furthermore,
$\sum (E-p_z) > 15$ GeV is required\footnote{The summation runs over all
  energy deposits in the calorimeters.} to reduce CC-DIS background.

Having applied all selection cuts, 1~event is found for an expected background
of $ 2.55 \pm 1.30 $ events. The background is composed of 2.0~events from
CC~DIS, 0.35~events from NC~DIS and 0.20~events from $W$/$Z$ production with
negligible contributions from $\gamma p$. 
The selection efficiency for the signal ranges between 10\% and 30\%.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Systematic Uncertainties}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The error on
the expectation from Standard Model processes has been calculated
by taking into account the systematic errors described below. 
The experimental error sources considered are:
%
\begin{itemize}
 \item an uncertainty of $ 1.5 \%$ on the integrated luminosity; 
 \item an uncertainty of $ 2 \%$ on the track reconstruction efficiency;
 \item an uncertainty on the absolute calibration of the calorimeters
       for electromagnetic energies, ranging between $ 0.7\%$ in the
       central part to $ 3\%$ in the forward region of the LAr
       calorimeter;
 \item an uncertainty of $ 2 \%$ on the absolute hadronic energy
       scale. 
\end{itemize}
%
%For the electromagnetic and hadronic energy scale uncertainties, the analysis
%has been repeated shifting the central value by the given systematic
%errors.
%The overall systematic error on the SM expectations is then determined as
%the quadratic sum of these individual errors, the global errors for
%luminosity measurement, track efficiency and statistical uncertainty
%on the Monte Carlo simulation. 
% 
The following theoretical
uncertainties on the SM cross-sections are considered:
%
\begin{itemize}
 \item The lack of QED radiation from the quark line in the DJANGO
generator leads to an uncertainty of the CC DIS background expectation
which was conservatively estimated to be $40\%$ following~\cite{excneu}.
For NC DIS an uncertainty of $7 \%$ is attributed to the proton
structure. For $W$/$Z$ production an uncertainty of $30 \%$ is estimated
because only LO contributions are included. However, NLO contributions to W
production are small \cite{wnlo}.
\end{itemize}
%

As theoretical uncertainties on the signal cross-section, the following is
considered:
\begin{itemize}
 \item an uncertainty of $ 10\%$ from the parton densities; 
 \item an uncertainty of $ 20\%$ from the variation of the factorization
   scale;
 \item an uncertainty of $ 10 \%$ from the interpolation procedure used
   for the determination of signal detection efficiencies and signal cross
   sections, dependent on the neutralino and selectron masses;
 \item an uncertainty of $ 3\%$ due to limited statistics in the signal
   simulation.
\end{itemize}

%
%For each error source, the analysis has been repeated  shifting
%the central value  by $ 1$ standard deviation to estimate their
%individual contribution.
%The overall systematic error on SM expectations is then determined as
%the quadratic sum of these individual errors and the statistical uncertainty
%on the Monte Carlo simulation. \\
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{GMSB Model Dependent Interpretations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

No significant deviation from the SM has been observed (see Section
\ref{seq:selection}), so limits in a GMSB 
model are derived, using the search results 
including the effects of systematic errors on the signal detection
efficiencies and background expectation \cite{Junk:1999kv}.

While there is no unique GMSB model, there are typically
\cite{Dimopoulos:1996yq,Ambrosanio:1997rv,Giudice:1998bp} six new parameters
in addition to those of the SM:
\begin{equation}
  F,\; \Lambda,\; M,\; N,\; \tan\beta\; {\rm and}\; {\rm sign}(\mu) \,.
\end{equation}
The intrinsic SUSY breaking scale is $\sqrt{F}$, which also determines the 
\grav\ mass according to $m_{\tilde{G}} \simeq 2.5 \cdot F/(100\ {\rm TeV})^2
\, {\rm eV}$ corresponding to (\ref{eq:gravmass}). Since $\sqrt{F}$ affects
primarily the lifetime of the NLSP, we do not vary it for this paper, but
simply assume that this lifetime is short enough to have no effect on our
detection efficiencies. Furthermore, it is assumed that $\sqrt{F}$ is low
enough so that the \rpv\ decays of the neutralino are suppressed compared
to the decays $\chi^0_1 \rightarrow \gamma \tilde{G}$
(cf. Eq. (\ref{eq:neudecay})). Thus, the 
branching ratio ${\rm BR}(\chi^0_1 \rightarrow \gamma \tilde{G})$ is set to
$100\%$, which is fulfilled within an uncertainty of $3\%$ for all values of
$\lambda'_{1j1}$ presented in this paper, taking $\sqrt{F}$
above the present limits \cite{Acosta:2002eq}. The contributions due to
$\chi^0_i$ production ($i=2,3,4$) can be neglected.

The parameter $\Lambda$ sets the overall mass scale for SUSY particles, $M$
is the mass of the messenger particles, $N$ is the number of sets\footnote{
 $N$ is technically the Dynkin index of the gauge representation of the
 messenger fields. To preserve gauge coupling unification, the messengers
 are assumed to form a GUT representation. In the simplest form, each of
 the $N$ messenger particle sets has the quantum numbers of a $\bf 5 +
 \bar{5}$ of $\bf SU(5)$. The maximum number of messengers can be restricted
 by requiring the gauge interactions remain perturbative up the GUT scale,
 although this bound depends on $M$. For $M= 100$ TeV, $N \le 5$, while for
 $M = 10^{10}$ TeV, $N \le 10$.} of messenger particles, and
$\tan\beta$ is the usual ratio of the Higgs vacuum expectation values. The
final parameter sign($\mu$) is the sign of the Higgs sector mixing parameter,
$\mu$ (the magnitude of $\mu$ is
calculable from the other parameters in the minimal model by imposing
radiative electroweak symmetry breaking).

We work in a theoretical framework based on \cite{Dimopoulos:1996yq},
but assuming that the left-handed selectron mass is equal to
the right-handed one. The theoretical calculations are embedded in the
SUSYGEN \cite{SUSYGEN} generator. 
%
For the limits presented below, $\tan\beta=1.5$, $N=2$ and a negative $\mu$
is chosen. The remaining free GMSB parameters $\Lambda$ and $M$ have been
transformed into the parameters $m_{\tilde{e},R}$ ($ = m_{\tilde{e},L} =
m_{\tilde{e}}$) and $m_{\chi^0_1}$. So, in the following, the limits are
presented as a function of $m_{\tilde{e}}$ and $m_{\chi^0_1}$.
The parameter range considered in this analysis corresponds then to
a scan of {$17.3\le \Lambda \le 51.5 $ TeV} and {$450\le M \le 5.3 \cdot
  10^7$ TeV}. 
To determine whether a set of masses ($m_{\chi^0_1}$, $m_{\tilde{e}}$) is
excluded by this analysis, we use the numbers of observed and expected
events for which the invariant mass from the photon and the missing
particle lies 
within a mass interval of $\pm 15$ GeV centered on $m_{\chi^0_1}$.

In Figure~\ref{fig:mselmne}, excluded regions at the $95\%$ confidence
level are 
presented in the plane spanned by $m_{\tilde{e}}$ and $m_{\chi^0_1}$ for different
values of of the \rpv\ coupling.\footnote{The step from $m_{\chi^0_1}=98$ GeV to
$m_{\chi^0_1}=99$ GeV in the excluded region for $\lambda'_{1j1} =$ 1.0 comes
from the fact that the observed event has a reconstructed invariant mass
from the photon and the missing particle (which corresponds to the neutralino
mass) of 83.3 GeV. Due to the neutralino mass interval of $\pm 15$ GeV, for
$m_{\chi^0_1}=98$ GeV this event is considered in the limit calculation, but
for $m_{\chi^0_1}=99$ GeV it is not, which leads to an improved limit
and thus to a larger excluded region at $m_{\chi^0_1}=99$ GeV.}
For selectron masses very close to the neutralino mass and for
$\lambda'_{1j1} = 1.0$, neutralino masses up to 108 GeV can be excluded.

The excluded regions in Figure~\ref{fig:mselmne} can be compared to the
region excluded by OPAL \cite{Abbiendi:2000bd} analyzing $e^+
e^- \rightarrow \chi^0_1 \chi^0_1 \rightarrow \gamma \tilde{G} \gamma \tilde{G}$
in an $R$-parity-conserving SUSY scenario. OPAL are mostly sensitive to
right-handed 
selectrons (exchanged in the $t$-channel) and perform a scan over the GMSB
parameter space. Their result 
for $\tan\beta = 2$ excludes the region $m_{\chi^0_1} \lesssim 91$ GeV (almost
independently of the selectron mass) at the $95\%$
confidence level. Thus, for not too low values of $\lambda'_{1j1}$, H1
can extend the 
region excluded by OPAL.\footnote{The differences between our signal cross
  sections for $\tan\beta=1.5$ and $\tan\beta=2.0$ are only a few per cent.}

In Figure~\ref{fig:lambda}, lower limits at $95\%$ confidence level on
$\lambda'_{1j1}$ ($j=1,2$) are given as a function of $m_{\chi^0_1}$ for various
assumptions for the difference between selectron and neutralino
mass.\footnote{The two steps from $m_{\chi^0_1}=68$ GeV to
$m_{\chi^0_1}=69$ GeV and from $m_{\chi^0_1}=98$ GeV to
$m_{\chi^0_1}=99$ GeV in the limit curve come --- as discussed in the previous
footnote --- again from the fact that the observed event lies inside or
outside the considered neutralino mass interval leading to a weaker or
stronger limit, respectively.}
While $\lambda'_{111}$ is already tightly constrained by neutrinoless
double 
beta decay searches \cite{mohapatra}, these are the first limits derived on
$\lambda'_{121}$ which are independent of squark masses.

% ($\lambda'_{111} < 2.8 \cdot 10^{-9/4} m^2_{\tilde{f}}
%m^{\frac{1}{2}_{\chi^0}  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have searched for events containing a photon and large missing
transverse momentum, expected in \rpv\ models with gauge mediated
SUSY breaking, at a center-of-mass energy of $\sqrt{s}=319$ GeV with the H1
detector at HERA. No evidence for a deviation from the SM was found.
Limits which constrain minimal models of gauge mediated supersymmetry
breaking have been derived for different values of the \rpv\ coupling. For
the first time, limits on the \rpv\ coupling $\lambda'_{121}$ 
have been derived independently of squark masses.

%%%%%%%%%%%%%%%%%%begin%%figure%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[H] 
\vspace{1cm}
  \begin{center}
      \psfrag{x}[][][0.7][0]{$\bf\chi^0_1$}
      {\epsfig{file=H1prelim-03-064.fig2.eps,width=13cm,bbllx=25,bblly=660,bburx=575,bbury=130}}
  \end{center}
  \vspace{12cm}
  \caption{Excluded region at the $95\%$ confidence level in the plane spanned
    by selectron and neutralino mass for different values of $\lambda'_{1j1} =$ 0.5 (dark red), 1.0 (red), 1.5 (light
red) ($j=1,2$). Within the considered GMSB scenario, the region in the
upper left 
rectangle (dark grey) is theoretically not accessible. The region in the
lower right rectangle (light grey) is not analyzed because the neutralino
is required to be the next-to-lightest supersymmetric particle.}
  \label{fig:mselmne}
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%end%%figure%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%begin%%figure%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[H] 
%\vspace{2cm}
  \begin{center}
      \psfrag{x}[][][0.7][0]{$\bf\chi^0_1$}
   {\epsfig{file=H1prelim-03-064.fig3.eps,width=13cm,bbllx=25,bblly=660,bburx=575,bbury=130}}
  \end{center}
  \vspace{12cm}
  \caption{Upper limit at the $95\%$ confidence level on \rpv\ couplings
    $\lambda'_{1j1}$ ($j=1,2$) as a function
    of the neutralino mass for a difference
    between selectron and neutralino mass of 1 GeV (solid, dark), 5 GeV
    (dashed), 10 GeV (dashed-dotted) and 15 GeV (solid, light).}
  \label{fig:lambda}
\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%end%%figure%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   References for Gravitino paper
%
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%========================================================================
%
%%%%%%%> refs. from "Introduction" section
%
%========================================================================

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%
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%
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%

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%
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%
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%========================================================================
%
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%
%========================================================================


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%========================================================================
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%
%========================================================================



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%========================================================================
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%
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%================================================
%
%%%%%%%> refs. from "Event Selection" section
%
%================================================

\bibitem{H1F2PAPER}
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% Eur.\ Phys.\ J. C13 (2000) 609.
%\cite{Adloff:2000ah}
%\bibitem{Adloff:2000ah}
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%%CITATION = HEP-EX 9908059;%%
\vspace{-2mm}


%================================================
%
%%%%%%%> refs. from "Systematic Uncertainties" section
%
%================================================

%\cite{Adloff:2001me}
\bibitem{excneu}
C.~Adloff {\it et al.}  [H1 Collaboration],
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%%CITATION = HEP-EX 0110037;%%

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%%CITATION = HEP-PH 0203269;%%


%================================================
%
%%%%%%%> refs. from "GMSB Model Dependent Interpretations" section
%
%================================================


%\cite{Junk:1999kv}
\bibitem{Junk:1999kv}
T.~Junk,
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%%CITATION = HEP-EX 9902006;%%


%\cite{Dimopoulos:1996yq}
\bibitem{Dimopoulos:1996yq}
S.~Dimopoulos, S.~Thomas and J.~D.~Wells,
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Nucl.\ Phys.\ B {\bf 488} (1997) 39
[hep-ph/9609434].
%%CITATION = HEP-PH 9609434;%%


%\cite{Ambrosanio:1997rv}
\bibitem{Ambrosanio:1997rv}
S.~Ambrosanio, G.~D.~Kribs and S.~P.~Martin,
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Phys.\ Rev.\ D {\bf 56} (1997) 1761
[hep-ph/9703211].
%%CITATION = HEP-PH 9703211;%%


%\cite{Giudice:1998bp}
\bibitem{Giudice:1998bp}
G.~F.~Giudice and R.~Rattazzi,
%``Theories with gauge-mediated supersymmetry breaking,''
Phys.\ Rept.\  {\bf 322} (1999) 419
[hep-ph/9801271].
%%CITATION = HEP-PH 9801271;%%


%\cite{Acosta:2002eq}
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D.~Acosta {\it et al.}  [CDF Collaboration],
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%%CITATION = HEP-EX 0205057;%%


%\cite{Abbiendi:2000bd}
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%%CITATION = HEP-EX 0007014;%%

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\end{thebibliography}

%
%
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\end{document}




