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\begin{document}  
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\begin{titlepage}

\noindent
%Submitted to the 30th International Conference on 
%High-Energy Physics ICHEP2000, \\ 
%Osaka, Japan, July 2000

%{\bf 
%Draft Version : 0.1 \\
%Date :  \today  \\
%Editors : C.Diaconu, J.Dingfelder, F.Keil, E.Perez \\
%Referee : M.Erdmann }\\
%\vspace*{3.0cm}

\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=H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                International Europhysics
                Conference on High Energy Physics},
                July~12,~2001,~Budapest} \\
 {\bf EPS 2001:} 
                 & Abstract:        & {\bf 824}    &\\
                 & Parallel Session & {\bf 6}   &\\[.7em]
%                 & Plenary Session  & {\bf S11}   &\\[.7em]
\multicolumn{4}{l}{{\bf
               XX International Symposium on Lepton and Photon Interactions}, 
               July~23,~2001,~Rome} \\ 
{\bf LP 2001:}  
                 & Abstract:        & {\bf 516} &\\
                 & Plenary Session  & {\bf S11}   &\\ \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}


%\vspace{2cm}

\begin{center}
\begin{Large}

{\bf  Search for Single Top Production  in e$^\pm$p collisions at HERA }

\vspace{2cm}

H1 Collaboration

\end{Large}
\end{center}

\vspace{1cm}

\begin{abstract}
%%%%%%%%%%%%%%%%%%% ABSTRACT%%%%%%%%%%%%%%%%%%%%%
 A search for anomalous top quark production 
 mediated by a flavour changing neutral current via a $\gamma$-$u$-$t$ coupling is performed in $e^{\pm}p$ collisions at HERA.
 The search exploits data collected  from 1994 to 2000 and is motivated by the previous observation of outstanding events with isolated leptons,
 missing transverse momentum and a jet with large transverse momentum.
 The top decay into a $b$-jet and a $W$-boson is considered in both 
 the leptonic and hadronic decay modes of the $W$.
 In leptonic decay modes, five events are found to be compatible with the hypothesis
 of anomalous top quark production while $1.8$ events are expected from Standard Model.
 No excess above the Standard Model expectation 
 is found in the analysis of the hadronic decay channel.
% A limit on the effective photon-$u$-$t$ coupling of 
% $\kappa_\gamma=0.23$ is determined.
 An upper limit on the anomalous $\gamma$-$u$-$t$ coupling is established in the framework of  recent NLO calculations. The sensitivity of HERA to flavour changing neutral currents is found to be competitive
 with  that at other existing high energy colliders.


\end{abstract}

%\vfill
%\begin{flushleft}
%  {\bf Abstract: 961 } \\
%  {\bf Parallel session: 11} \\
%  {\bf Plenary talk: 7~b } 
%\end{flushleft}



\end{titlepage}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% body
\section{Introduction}
 The H1 collaboration has first reported~\cite{h1isol} the observation of events 
 with high-$P_T$ isolated leptons and missing transverse momentum in 
 the positron-proton data collected in 1994-1997. 
% Within 36.5~$pb^{-1}$ of $e^+p$ data, H1 has observed 1 electron event 
% for  $2.4\pm0.5$ expected and 5 muon events for  $0.8\pm0.2$ expected. 
% Three muon events were found in a region of large transverse momentum 
% of the hadronic system.
%
 In  more recent data collected from 1998 to 2000, new events with 
 the same topology have been
 observed~\cite{h1isolbudapest}. 
 Some of the events were further characterized by a large transverse momentum of the hadronic system, which
% This characteristics 
turned out to be an atypical kinematic property when
 compared to the Standard Model  (SM)
 calculations which are dominated by
 direct production of $W$-bosons.
% In the more recent data, new events with 
%% this topology have been 
% the same topology have been
% observed~\cite{h1isolbudapest}. 
 \par
% The lepton and the missing transverse momentum are likely to be associated
% with a $W$-boson decaying 
% into the lepton and a neutrino.
% leptonically.
% The large transverse momentum of the hadronic system, however, 
This may indicate
 presence of a further production mechanism involving beyond Standard Model.
% $W$ production.
 A possible origin of these events is production of top quarks which
 predominantly decay into a $b$-quark jet and a $W$-boson.
The lepton and the missing transverse momentum would then be associated with a $W$-boson decaying leptonically.
 Within the Standard Model, top quark production is negligible in $e^{\pm}p$ collisions.
% In the framework of physics beyond the Standard Model, however, top
% production is predicted to occur via the flavour changing neutral
% current
% (FCNC) mechanism.
 However, in several extensions of the SM, the top quark
 is predicted to have significantly large flavour changing neutral
 current (FCNC) interactions, which could lead to a sizeable
 single top production cross-section at HERA.
%
 \par
 In this paper we present 
 a search 
% the first search
 for top quark production using 
 the H1 detector at HERA.
% With respect to \cite{h1toposaka} the luminosity in the leptonic decay channel
% has been updated giving a total integrated luminosity of $115.2$ pb$^{-1}$. 
The analysis for the leptonic decay channel makes uses of all data collected from 1994 to 2000 and corresponding to an integrated luminosity of $115.2$ pb$^{-1}$.
 In addition to the leptonic decay channel, 
 hadronic decays of the top quark into $3$ or more jets are exploited using a data sample corresponding to an integrated luminosity of $36.5$~pb$^{-1}$.
 The large jet transverse momenta involved ensure that a potential signal
 can be found above the jet production expected from QCD processes.



%============================
\section{Phenomenology}
%============================

In the Standard Model  neutral current interactions 
preserve the involved flavours in the leading order.
Therefore flavour changing neutral current processes
only arise via higher order radiative corrections and are
highly suppressed. Their low SM rates make FCNC processes
an ideal place to look for new physics at high energy colliders.
Due to the large top quark mass close to the electroweak symmetry 
breaking scale, deviations from the SM might be observed first in
the top sector. FCNC interactions involving the
top quark have thus received considerable interest since the discovery
of the top at the Fermilab Tevatron.
Such interactions are present in many extensions of the SM like
models with an extended Higgs sector~\cite{ATWOOD},
supersymmetry~\cite{SUSY},  dynamical  
breaking of the electroweak symmetry~\cite{PECCEI}
or of an additional symmetry~\cite{FRITZSCH}.

At HERA, FCNC interactions 
coupling a top quark to a light quark ($u$ or $c$)
and a gauge boson would lead to single top production as
illustrated in Fig.~\ref{fig:diagtop}.
The crossed process $e^+ e^- \rightarrow t \bar{q} (\bar{t} q)$
is actively searched for at the LEP collider since its 
centre-of-mass energy is above the top production threshold.

%-----------------------------------------------------------

We consider here the most general effective Lagrangian proposed 
in~\cite{JOANNE} to describe FCNC top interactions involving
electroweak bosons. It reads as~:
$$ {\cal{L}}_{eff} =  \sum_{U = u,c}
 i \frac{e e_U}{\Lambda} \bar{t} \sigma_{\mu \nu} q^{\nu} 
 \kappa_{\gamma,U} U A^{\mu} + 
 \frac{g}{2 \cos \theta_W} \bar{t}
 \left[ \gamma_{\mu} (v_{Z,U} - a_{Z,U} \gamma^5) +
 i \frac{1}{\Lambda} \sigma_{\mu \nu} q^{\nu} K_{Z,U} \right] 
 U Z^{\mu} \quad +  {\mbox {h.c.}}  ,$$
where $\sigma_{\mu \nu} = (i/2) \left[ \gamma^{\mu}, \gamma^{\nu} \right]$,
$\theta_W$ is the Weinberg angle, $q$ the four-momentum of
the exchanged boson, 
$e$ and $g$ denote the gauge couplings relative to $U(1)$ and
$SU(2)$ symmetries respectively, 
$e_U$ denotes the electric charge of up-type quarks, 
$A^{\mu}$ and $Z^{\mu}$ the fields of the photon and $Z$ boson,
and $\Lambda$ denotes the scale up
to which the effective theory is assumed to hold.
By convention we set $\Lambda = m_t$ in the following.
Only magnetic
operators allow FCNC $t q \gamma$ couplings denoted by
$\kappa_{\gamma, q}$, while $q-t$ transitions
involving the $Z$ boson may also occur via vector interactions due
to the non-vanishing $Z$ mass. 
The single top production in $ep$
collision is however largely dominated by the $t$-channel exchange of
a $\gamma$ 
% with very low virtuality $Q^2$ 
and therefore we
concentrate on the $\kappa_{\gamma,q}$ couplings only\footnote{
A different convention for the $\kappa_\gamma$ coupling is used here with respect to~\cite{h1toposaka}( $\kappa^0_\gamma$). The difference is that the quark electric charge introduced here in the normalization of the anomalous coupling
$\kappa_\gamma=\kappa^0_\gamma/e_u =3/2*\kappa_\gamma^0 $. With this notation the limit on $\kappa^0_\gamma$ obtained in ~\cite{h1toposaka} $\kappa^0_\gamma<0.25$ reads  $\kappa_\gamma<0.375$.}  

The possibility of anomalous single top production at HERA was first 
investigated
in~\cite{FRITZSCH} where a model was built in which 
$\kappa_{\gamma, q} \propto m^2_{q}$, and therefore only the $tc \gamma$ 
coupling was relevant. We do not rely here on any assumption concerning
the underlying theory and thus allow both couplings $\kappa_{\gamma,c}$
and $\kappa_{\gamma,u}$ to be present and not necessarily
related to each other.
The sensivity of HERA is naturally much higher for the coupling
$\kappa_{\gamma,u}$ than for $\kappa_{\gamma,c}$ due to the
more favourable parton density.
The most stringent existing bound on $\kappa_{\gamma, u}$ comes from 
upper limits
on radiative top decays set by the CDF Collaboration~\cite{CDF}~:
$BR ( t \rightarrow u \gamma) < 3.2 \%$ at $95 \%$ confidence level (CL),
leading to $\kappa_{\gamma, u} < 0.42$~\cite{JOANNE}.
When $\kappa_{\gamma, u}$ is saturated to its upper bound, the single
top cross-section at HERA has been calculated
to be $\sim 1$ pb, sufficiently large to allow
several single top events to be produced with the current integrated luminosity. 

Cross section calculations have recently been improved for next-to-leading order terms~\cite{nlotop}. The correction increases the leading order value by roughly 20\%, and is taken into account in the results derived in section 7. 


%======================================
\section{Monte-Carlo Event Generators}
%======================================

A complete Monte Carlo simulation of the H1 detector response is
performed for each possible SM background source and
for the single top signal.

For the neutral current (NC) or charged current (CC)  deep inelastic scattering (DIS) background estimates we use
the DJANGO~\cite{DJANGO} event generator, which
includes first order QED radiative corrections.
Simulation of real bremsstrahlung photons, based
on HERACLES~\cite{HERACLES}, is also included. 
%
QCD radiation at all orders is treated following
the approach of the Color Dipole Model~\cite{CDM} and are implemented using
ARIADNE~\cite{ARIADNE}.
The hadronic final state is generated using the
string fragmentation model~\cite{JETSET74}.
The parton densities in the proton used to estimate DIS
expectations are taken
from the MRST~\cite{MRST} parametrization.
The RAPGAP~\cite{rapgap} generator has also been used for the study of multijet production in NC DIS.

For the direct and resolved photoproduction of light and heavy flavours,
we use the PYTHIA MC event generator~\cite{PYTHIA}
which relies on first order QCD matrix element
% corrections to first order in $\alpha_s$,
and uses leading-log parton showers
and string fragmentation~\cite{JETSET74}.
The GRV LO (GRV-G LO) parton densities~\cite{SFGRVGLO} in the proton
(photon) are used. The photoproduction expectation has also been calculated using JetVip~\cite{jvp} program.

The production of electroweak vector bosons $Z^0$ and
$W^{\pm}$ was modelled using the EPVEC~\cite{EPVEC} event generator.
Contributions from two-photon processes where one $\gamma$ originates
from the proton were calculated using the
LPAIR~\cite{LPAIR} event generator.

The simulation of the single top signal relies on a specific
event generator ANOTOP using the matrix elements of the complete
$e + q \rightarrow e + t \rightarrow e + b + W \rightarrow e + b + f + \bar{f'}$
process as obtained from the CompHEP~\cite{COMPHEP} program.
This allows a proper description of angular distributions.
The BASES/SPRING~\cite{BASES} package is used to perform the numerical
integration of the amplitudes and to generate events according to
the differential cross-section. 
The parton showers approach~\cite{JETSET74} relying
on the DGLAP~\cite{DGLAP} evolution
equations is used
to simulate QCD corrections in the initial and final states.
The parton densities used~\cite{MRST} are evaluated at the scale of the
top mass, in analogy with leptoproduction of heavy quarks.


\section{Experimental Conditions}

This analysis is based on $e^{\pm}p$ collisions recorded by the H1 experiment in the 1994-2000 period. 
It is restricted to periods  where the  central drift chambers, the liquid argon calorimeter and the luminosity system are  active and fully operational. 
\par

A detailed description of the H1 detector can be found in \cite{H1detector}.
Only components  essential  for this analysis are shortly described 
here. A tracking system of central and forward drift chambers is used to measure the
charged particle trajectories and to determine the interaction vertex. Particle transverse momenta are determined from the curvature of
the trajectories in a magnetic field of 1.15 Tesla.
%yields  a measurement of the particle momenta with a nominal 
%resolution  $\sigma_{P}/P^2=3\times10^{-3}$ GeV$^{-1}$.

Hadronic and electromagnetic final state particles 
are absorbed in a highly segmented calorimeter 
\cite{h1cal}, which is   5 to 8 interaction 
lengths deep depending on the polar angle of the particle.
%Electromagnetic shower energies are measured with a precision of 
%$\sigma (E)/E = 12 \% / \sqrt{E/\mathrm{GeV}} \oplus 1\%$ and
% hadronic shower energies with 
%$\sigma (E)/E = 50 \% / \sqrt{E/\mathrm{GeV}} \oplus 2 \%$. 
%The electro-magnetic energy scale is known to the level of 0.7-3\%
%depending on the polar angle. The hadronic energy scale is understood at the 2\% 
%level when comparing to the Monte Carlo expectations.
The calorimeter is surrounded by a superconducting coil 
and an iron yoke instrumented with streamer 
tubes. Leakage of hadronic showers outside the  calorimeter is measured 
 from an analogue  charge sampling of the streamer tubes (Tail Catcher).
% with a resolution of $\sigma (E)/E = 100 \% / \sqrt{E/\mathrm{GeV}}$ . 
Tracks of 
penetrating charged particles, such as muons, escaping the calorimeter
 are reconstructed from 
their hit pattern in the streamer tubes with an efficiency higher than 90\%. 

The trigger condition for high transverse energy interactions is based on calorimeter signals.
 The trigger efficiency is higher than 95\% for events 
with a scattered electron with an energy
above 10 GeV, and higher than 85\% for events with 
a calorimetric missing transverse momentum  greater than 25 GeV. For multi-jet events the trigger efficiency is 100\%.
 

\section{Search for Single Top Production in the Leptonic Channel}
The search of top quark decays in the leptonic channel is based on the selection of events with isolated leptons and missing transverse energy performed in~\cite{h1isolbudapest}.
The basic criteria are the presence of a lepton at high transverse momentum $P_T^\ell >10$~GeV  with missing $P_T$ in the event above 12 GeV.
This selection (W search criteria) is further optimized to separate top production from Standard Model W production:
\begin{itemize}
\item {\bf Cut on the transverse momentum of the hadronic system}\\
In the top decays, the hadronic system is expected to have a high transverse momentum. The events are accepted if the transverse momentum of the hadronic system $P_T^X$ is higher than 25~GeV.
The expected $P_T$ distributions of the highest $P_T$ jet in two angular domains are presented in the figure~\ref{ptthcut}. 
The highest $P_T$ hadronic jet is required to have $P_T^{jet}
>25$ GeV. This threshold is set to 35 GeV in the case where the highest $P_T$ jet has a polar angle with respect to the proton direction of $\theta_{jet}<35^\circ$.

\item {\bf Transverse mass of lepton-neutrino system}\\
This quantity is defined as the invariant mass of the two massless vectors obtained by projecting the lepton and neutrino momenta in the transverse plane, where the transverse neutrino momentum is calculated from the missing transverse momentum $P_T^{miss}$ in the event:
\begin{equation}
 M_T^{\ell \nu} = \sqrt{\left(|\vec{P}_T^\ell|+|\vec{P}_T^{miss}| \right)^2 - \left(\vec{P}_T^\ell+\vec{P}_T^{miss} \right)^2} .
\label{eq:mt}
\end{equation}

For W leptonic decays, the transverse mass of the lepton-neutrino system is centered around the W mass and has a tail towards low masses. In the standard model W production, the production of the charged lepton and neutrino can also proceed through photon-W scattering, while only real W decays produce  leptonic final states in top induced events.
In order to reduce the contribution from
off-shell processes, a cut is applied on the transverse mass $M_T^{\ell\nu}>10$~GeV.

\item {\bf Charge of the observed lepton}\\ The production of top quarks in $\gamma-u$ fusion yields t-quarks with charge $+2/3$  (the $\bar{\mathrm t}$ production via fusion with sea quarks $\gamma-\bar{u}$ is supressed by a factor of ~80). The subsequent decay $t \rightarrow bW^+ (\rightarrow \ell^+\bar{\nu})$ produces only positively charged isolated leptons. In order
% to reduce the standard $W^\pm$ production contribution 
to reduce contributions other than top
to the final sample, leptons for which the charge measurement yields the wrong sign with a precision better than $2\sigma$ are rejected.

The reliability of the charge measurement at high $P_T$ has been tested using a sample of neutral current events collected in e$^+$p collisons. The scattered electrons are required to have transverse momentum measured from calorimeter deposit above 20 GeV 
and corresponding track transverse momentum above 10 GeV. The track corresponding to the scattered electron has to be isolated and must have a momentum measurement better than two standard deviations. The wrong sign assignment rate is below 1\% in this sample. In a 
subsample of forward electrons with the polar angle 
$\theta <50^\circ$, where most of the searched signal is expected, the wrong charge assignment has been measured in the data to be less than 2\%.

\end{itemize}

\par
The overall results are presented in the tables~\ref{tableback1} and 
\ref{tableback2} for an analysed data sample corresponding to 101.6~pb$^{-1}$ in e$^+$p collisons and 13.6~pb$^{-1}$ in e$^-$p collisions. 
In $e^+p$ data 4 events containing electrons and 6 events containing muons have been observed at $P_T^X>25 $ GeV~\cite{h1isolbudapest}. 
After the full top selection described above, 3 electron events and 2 muon events remain as top candidates for an expectation of  $0.75\pm0.18$ ($e^+p$) plus $0.13\pm0.04$ ($e^-p$) for the electron channel and $0.77\pm0.21$ ($e^+p$) plus $0.12\pm0.03$ ($e^-p$) for the muon channel. 
%No candidate is found in  the $e^-p$ data sample.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[hhh]
\begin{center}
\begin{tabular}{|c|c|c|c|c||} \hline
 & \multicolumn{2}{|c|}{W search cuts and $P_T^X>25$~GeV} &  \multicolumn{2}{c|}{Top selection}  \\ \hline 
 &  Electron Channel &  Muon Channel
 &  Electron Channel &  Muon Channel \\
 \hline \hline
Data      &       4           &   6           &     3              &   
2            \\ \hline \hline
Total SM  &     $1.29\pm 0.33$ & $1.54\pm 0.41$ &  $0.75\pm 0.18$  & 
$0.77\pm 0.21$ \\ \hline
W only    &     $1.05\pm 0.32$ & $1.29\pm 0.39$ &  $0.51\pm 0.16$  & 
$0.68\pm 0.19$ \\ \hline
Top efficiency & 43\%   & 51\%  & 37\% & 45\% \\ \hline
\end{tabular}
\caption{Observed and predicted number of events with isolated leptons and missing transverse momentum for $e^+p$ data in $101.6$ pb$^{-1}$}
\label{tableback1}
\end{center}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[hhh]
\begin{center}
\begin{tabular}{|c|c|c|c|c||} \hline
 & \multicolumn{2}{|c|}{W search cuts and $P_T^X>25$~GeV} &  \multicolumn{2}{c|}{Top selection}  \\ \hline 
 &  Electron Channel &  Muon Channel
 &  Electron Channel &  Muon Channel \\
 \hline \hline
Data           & 0               &0    &  0   &  0  \\ \hline \hline
Total SM       & $0.26  \pm 0.06 $  & $ 0.20 \pm 0.06 $ & $0.13\pm0.04$&
$0.12\pm0.03$ \\ \hline
$W$ only & $0.16  \pm 0.05 $  & $ 0.18 \pm 0.07$  & $0.07\pm0.02$&
$0.10\pm0.03$\\ \hline
Top efficiency & 43\%   & 51\%  & 37\% & 45\% \\ \hline
\end{tabular}
\caption{Observed and predicted number of events with isolated leptons and missing transverse momentum for $e^-p$ data in $13.6$ pb$^{-1}$ }
\label{tableback2}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par
 In order to calculate the invariant masses of the lepton-neutrino and lepton-neutrino-hadrons systems, the event kinematics has to be reconstructed. 
The following variables are used to quantify the event kinematics:
\begin{itemize}
\item $\vec{P}_T^\ell$: Transverse momentum vector of the isolated lepton.
The calculation of the transverse momentum is performed using calorimetric
information for electrons and central tracker information for muons. 
%\item $\theta^\ell$ : Polar angle of the isolated lepton. The polar angle
%is measured relative to the $z$-axis which points in the direction of the
%incoming proton.
\item $E^\ell$: Energy of the isolated lepton.       
\item $\vec{P}_T^X$: Transverse momentum vector of the hadronic system.
It is calculated as the vector sum of all energy deposits in the liquid
argon calorimeter and the tail catcher calorimeter not including energy
deposited by any additional leptons.
%(i.e. the positron in event MUON-3 and ELECTRON-3).
%\item $\theta^X$ : Polar angle of the hadronic system. 
\item $E^X$: Energy of the hadronic system.
\item $\vec{P}_T^{miss}$: Total missing transverse momentum vector of the event. It is defined as the vector sum of transverse momenta of all final state particles: $\vec{P}_T^{miss} =-(\vec{P}_T^{lepton(s)} + \vec{P}_T^{X})$. The missing transverse momentum is attributed to the hypothetical neutrino.
\item $\delta^{vis}$: Total visible event balance in longitudinal momentum, defined as the scalar sum of $\left( E-P_z \right)$ over all final state particles:  $\delta^{vis} = \left( E-P_z \right)^{lepton(s)} + \left( E-P_z \right)^X$
%\item $M_T^{\ell \nu} = \sqrt{\left(|\vec{P}_T^\ell|+|\vec{P}_T^{miss}| \right)^2 - \left(\vec{P}_T^\ell+\vec{P}_T^{miss} \right)^2}$ : Transverse mass of the lepton-neutrino system. The missing transverse momentum is attributed to the hypothetical neutrino.
\item  Transverse mass of the lepton-neutrino system defined in equation~\ref{eq:mt}.


\end{itemize}

\par

In order to compute the complete event kinematics the neutrino four-vector
must be reconstructed.
The transverse momentum of the neutrino corresponds to the total
missing transverse momentum $\vec{P}_T^{miss}$ of the event. 
Regarding the longitudinal momentum of the neutrino two cases have
to be treated separately:
\begin{itemize}
\item {\it Tagged events:} In this case the scattered electron
is detected in  one of the calorimeters. In the muon channel this means that 
the event contains an identified electron  and in the electron channel 
the event contains an additional electron of lower transverse momentum. 
For tagged events, the invariant mass of the lepton-neutrino-hadron system is computed using the neutrino kinematics deduced from longitudinal energy conservation. We  obtain 
$\left( E-P_z \right)^{\nu} = 55$ GeV $- \delta^{vis}$
using the event balance in longitudinal momentum as mentioned above. 
 For two events out of the ten  with $P_T^X>25$~GeV which are quoted in table~\ref{tableback1} this method can be applied.

\item {\it Untagged events}
The scattered electron is not detected and
therefore its longitudinal momentum unknown. A constraint on the invariant 
mass of the lepton and neutrino from the $W$ decay has to be applied:
\begin{equation}
 M_{\ell\nu} = \sqrt{P_\ell^2 + P_\nu^2 + 2P_eP_\nu}\ \approx \sqrt{2P_eP_\nu} = M_W = \ 80.2 \ \mbox{GeV}
\end{equation}
 with $P_l, P_\nu$ denoting the four-vectors of the lepton and neutrino
 respectively.
This constraint on the $W$ mass yields two possible solutions for
$\left( E-p_z \right)^{\nu}$, one corresponding to a backward neutrino (solution 1), 
the other to a forward neutrino (solution 2) relative to the lepton direction.
Using the two possible solutions for the neutrino, the invariant mass of the lepton-neutrino-hadrons system $M(\ell\nu X)$ can be calculated. All solutions that do not meet one of the following requirements are
rejected as being unphysical:

\begin{itemize}
\item $M(\ell\nu X)<\sqrt{s}$ where $\sqrt{s}$ is the electron-proton cms energy (300 GeV or 320 GeV depending on data taking period).
\item $0$ GeV $<\left( E-P_z \right)^{\nu}<55$ GeV
\item $\left( E-P_z \right)^{\nu} + \delta^{vis} <75$ GeV
\item $E^\nu+E^X+E^{lepton(s)} < 1000$ GeV 
\end{itemize}
The latter two requirements ensuring energy and $E-p_z$ conservation were 
optimized on a sample of generated top events in order to leave some scope for 
fluctuations in the energy measurement. \\
In case no such solution exists, the measurement errors on the energy of the hadronic final state $E^X$
(electron channel) or the transverse momentum of the muon $P_T^\mu$
(muon channel) are taken into account being the dominant errors in 
the particular channel.  New masses are computed using values of $E^X$ 
or $P_T^\mu$ varied by $\pm 1\sigma$. If a solution is still not found then a  $\pm 2\sigma$ variation is performed. 
The solution yielding the lowest mass is chosen as lower $1\sigma$ or  $2\sigma$ limit. 
\end{itemize}
This mass reconstruction procedure has 80\% reconstruction efficiency in the electron channel and 65\% in the muon channel for Monte Carlo top events. 
In figure \ref{topmasres} distributions of the reconstructed
top mass are shown for ANOTOP Monte Carlo events in both electron and 
muon channel. The widths of the mass distributions are determined to 
$\sigma (M_{top})= 20$ GeV for electrons and $\sigma (M_{top})= 22$ GeV 
for muons.\\
The mass solutions thus obtained for the isolated  lepton events with $P_T^X > 25$~GeV can be found in table \ref{tab_topmasses}. Two events (one electron and one muon) contain the scattered electron and therefore the ``tagged'' method can be applied. For the two tagged events, masses reconstructed with the ``tagged'' and one of the solutions of the ``untagged'' method are compatible.
In the final top selection as defined above, the first three of the four observed electron events and the two last muon events in table~\ref{tab_topmasses} are selected as top candidates.

\begin{table}[htb]
\begin{tabular}{|l|l|cc|cc|c|} \hline
        & PT(X) & \multicolumn{2}{|c|}{Untagged Solution 1} &  \multicolumn{2}{|c|}{Untagged Solution 2} & Tagged Solution\\
        &       & Mass & $E-P_z$  & Mass & $E-P_z$   & Mass\\
\hline
$e^+$  top cand. &   41& $ 159\pm  7$ & 61& $ 136\pm  6$ & 21 & -- \\ 
$e^+$  top cand.  &   43&    --            & --  & $ 166\pm  8$ & 54 & $ 165\pm  6$ \\
$e^+$   top cand. &   39& $ 178\pm  5$ & 39& $ 156\pm  7$ & 21 & -- \\
$e^+$  &   27& $ 234 \pm 15 $ & 65 & $ 199 \pm 14 $ & 38 & -- \\
 \hline
$\mu^+$ &   42&    --            & --  & $ 146\pm  7$ & 19 & --\\
$\mu^-$ &   27& $ 152\pm 12$ & 53& $ 129\pm  6$& 34& $ 156\pm 10$\\
$\mu^-$$ (1\sigma)$ &   59& $ 180 $ & 67& $ 188 $ & 37 & --\\
$\mu  $ $(2\sigma)$ &   30& $ 157$ & 53& $ 139$ & 22 & --\\
$\mu^+$ top cand. &   67&    --            & --  & $ 181\pm  10$& 19 & --\\
$\mu^+$ top cand. &   50&   --             & --  & $ 174\pm 12$ & 39 & -- \\
\hline
\end{tabular}
\caption{$M_{\ell\nu X}$ calculation for the ten events with $P_T^X > 25$~GeV. ``Untagged'' solutions (mass and total $E-P_z$)  using a W constraint are presented for all the events.
In two events the scattered electron is detected and the ``tagged'' mass solution using an $E-P_z$ constraint can be obtained. The events selected as top candidates are indicated.}
\label{tab_topmasses}
\end{table}

The total lepton-neutrino-hadrons mass distributions of the final top candidates are compared to the Standard Model prediction in  figure~\ref{mtmlnux}.



\section{Search for Single Top Production in the Hadronic Channel}
\label{sec:hadrons}

The top hadronic channel has been searched in a smaller data sample taken in the period 1994-1997 in $e^+p$ mode at $\sqrt{s}=300$~GeV, corresponding to an integrated  luminosity of 36.5~pb$^{-1}$.
The decay cascade $t\rightarrow bW(\rightarrow q\bar q)$ yields events with at least three jets with
high transverse momentum $P_T$. The main Standard Model QCD background is the electro- or photoproduction 
of high $P_T$ jets.

The search for those events is performed using a $K_T$ algorithm \cite{kt} based on calorimetric energy
deposits. Events with at least three jets fulfilling $P_T^{Jet1}>25$ GeV, $P_T^{Jet2}>15$ GeV and
$P_T^{Jet3}>10$ GeV are selected. In order to reduce the neutral current (NC DIS) contribution, events with an identified electron are removed from the sample. NC DIS events where the electron is being misidentified as a hadronic jet 
are supressed by requiring either the electromagnetic energy fraction of the reconstructed jets to be less 
than 90\% or the jet size defined in the $\eta -\phi$ plane to be larger than 0.1. In figure \ref{jet:control}
the number of selected events is shown as a function of the transverse momentum of the first jet 
$P_T^{Jet1}$ and the invariant mass $M_{ij}$ of the three dijet systems together with the prediction of
the Monte Carlo generator PYTHIA \cite{PYTHIA}. Using an overall normalization factor of 1.2 determined to
fit the measured cross sections, which could account for corrections to PYTHIA (leading order with parton showers) calculation, the data are well described by the MC prediction. The remaining
NC background estimated with the Monte Carlo model RAPGAP \cite{rapgap} is less than 7\% on average.

Top induced events are expected to have high total transverse energy and 2- and 3-jet invariant masses
close to the $W$ and top mass, respectively. To select top candidates in the data sample defined above the 
invariant dijet mass of at least one of the three dijet systems has to fulfill $70<M_{ij}<90$ GeV, the total
invariant mass has to be reconstructed in a  window around the top mass $150<M_{tot}<198$ GeV, 
and the total transverse energy $E_{T,tot}$ has to be larger than 120 GeV. $E_{T,tot}$ and $M_{tot}$ are
calculated from all jets found in a given event. Their distribution is also shown in figure \ref{jet:control}
and found to be in good agreement with the PYTHIA prediction. The top selection efficiency was estimated 
with the Monte Carlo model ANOTOP to be 31\%. 
%Using $M_{tot}$ and not the invariant mass of the 
%three highest $P_T$ jets increases the selection efficiency by 10\%.
With these requirements 10 event candidates are selected in good agreement with $8.3^{+4.2}_{-1.9}({\it exp.})\pm 4.2({\it theory})$ events expected from the standard processes dominated by the photoproduction as seen in table \ref{jet:res}.

%In the data sample corresponding to 36.5 pb$^{-1}$ collected between 1994 and 1997, 10 candidates are selected with these requirements. This is 
%in good agreement with the Standard Model expectation of 8.3 events, dominated by the photoproduction processes as shown in the  table \ref{jet:res}. 

The main experimental systematic error in this
analysis of $\approx 40$\% is due to the uncertainty in the absolute hadronic energy calibration of the 
calorimeter. Systematic effects due to the uncertainty in the luminosity measurement and trigger 
inefficiencies are negligible. 
Figure \ref{jet:mass} shows the invariant mass $M_{tot}$ of the top candidates together with the Standard model prediction. 


\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|}
\hline
& Events\\
\hline
Data & 10\\
\hline
Standard Model & 8.3$^{+ 4.2}_{- 1.9}(exp.)\pm4.2(theory)$\\
\hline\hline
Photoproduction & 7.8$^{+ 4.1}_{- 1.8}$\\
\hline
Neutral Current & 0.5 $\pm$ 0.5\\
\hline\hline
Top efficiency & 31\%\\
\hline
\end{tabular}
\end{center}
\caption{Three jet selection of 1994-1997 data (L=36.5~pb$^{-1}$).}
\label{jet:res}
\end{table}

\par

The Standard Model expectation has also been determined using the program JetVip \cite{jvp}. 
JetVip includes hard scattering matrix elements up to $O(\alpha_s^3)$ und thus allows 
the calculation of three parton final states in leading order QCD. The calculated number of
$4.7^{+2.2}_{-1.4}$ events is in agreement with the PYTHIA prediction. The quoted error results
from a variation of the renormalization scale $\mu_r^2$, chosen to be the highest jet transverse 
momentum of a given event, by factors of 1/4 and 4. Based on this estimation, 
the theoretical error considered is 50\%. 

%=======================
\section{Results}
%=======================

No deviation from the SM prediction has been observed in the
three high $P_T$ jets channel as seen in section~\ref{sec:hadrons}.
Assuming Poisson distributions for the SM background and for the top signal,
an upper limit at $95 \%$ confidence level (CL) 
on the number of events coming from singly produced tops
decaying hadronically
is obtained using a standard Bayesian prescription.
This limit number is then converted into an upper limit
on the top production cross-section at $\sqrt{s} = 300$ GeV
by folding in the corresponding efficiency and branching ratio.
The branching $t \rightarrow b W$ is assumed to be $100 \%$, which
is a safe approximation as seen from the CDF upper bound on radiative top
decays
into $q \gamma$.
This results in an upper bound 
$\sigma_{300} = \sigma (e p \rightarrow e t X, \sqrt{s} = 300$ GeV) $< 1.17$
pb.
In turn, this leads to upper limits on the number of top induced events
one could observe in the leptonic channel in the $e^+ p$ datasets.
At $95 \%$ CL, we obtain that less than 3.2 (9.7) events coming
from single top production should
be observed with our selection criteria
in the leptonic channels ($e + \mu$) in the $e^+ p$ data
taken at $\sqrt{s} = 300$ GeV ($\sqrt{s} = 320$ GeV), while 1 (4)
candidate(s) is (are) observed.
The upper limit derived from the hadronic channel therefore does not
rule out an interpretation of the observed events in the leptonic
channel as anomalous top production.
%It is thus legitimate to assume that the full observation
%contains at most two components, an unknown top signal and a known
%expectation from the SM.
The number of observed and expected events in all the
different channels analysed for the  $e^+ p$ and $e^- p$ data  can be combined to set an upper bound on
$\sigma (e p \rightarrow e t X)$ at a given centre of mass
energy, which is then translated into
constraints on the anomalous coupling $\kappa_{\gamma}$.
%
Each channel contributes in the limits derivation via its branching ratio,
the numbers of observed and expected events satisfying the selection cuts,
and the corresponding selection efficiencies. 
The three considered datasets are treated as independent channels,
weighted by their relative luminosities and the ratios
of single top cross-section for the corresponding running conditions
to $\sigma_{320} = \sigma(e p \rightarrow e t X)$ at $\sqrt{s} = 320$ GeV.
The resulting upper limit $N_{\lim}$ on the number of signal events simply
translates
into an upper bound on the cross-section $\sigma_{320}$ via
% $N_{\lim} = ({\cal{L}}_{1} + {\cal{L}}_{2} + {\cal{L}}_{3}) \times \sigma_{320}$.
$N_{\lim} = {\cal{L}}_{tot} \times \sigma_{320}$.
Both systematic and statistical errors have been folded in channel
by channel as described in~\cite{H1LQ94}.
\par
The resulting upper bound on the single top cross-section
at $\sqrt{s} = 320 \GeV$ is~: 
$$ \sigma(e p \rightarrow e + t + X, \sqrt{s} = 320 \GeV)
   < 0.87 \;{\mbox{pb}} \qquad {\mbox{ at }} 95 \% {\mbox{ CL}} \quad .$$ 
%
This results into an upper limit on the anomalous
$tu \gamma$ magnetic coupling~:
$$ \kappa_{\gamma, u} < 0.305 \quad, $$
%
%which is close to the CDF upper bound coming from radiative top decays.
which improves the CDF upper bound coming from radiative top decays.

Single top searches at LEP2 have also been interpreted in terms
of bounds on the anomalous couplings.
%~\cite{ALEPH}.
Since $e^+ e^-$ collisions also provide
sensitivity to anomalous couplings $tq Z$, the couplings
$\kappa_{\gamma}$ and $v_Z$ have been considered
simultaneously.
% The obtained limit on $\kappa_{\gamma}$ is competitive with the CDF upper bound
% but however it does not yet supersede it.
%Under the conservative assumption that only $u$
%quarks participate to the considered FCNC processes, the 
%current status on the constraints on $\kappa_{\gamma, u}$ and
%$v_{Z,u}$ is represented in Fig.~\ref{fig:limits}.
%
The preliminary results obtained by OPAL ~\cite{OPAL}, using data up to $209$~GeV, are shown in Fig.~\ref{fig:limits} which represents the current status of the constraints on $\kappa_\gamma$ and $v_Z$. 
The sensitivity of the HERA and TeVatron colliders on such FCNC 
processes will significantly increase~\cite{SASHA} with the large
luminosities expected in the near future.


\section{Summary}

 We observe an increasing number of events containing an isolated lepton 
 at large transverse momentum, missing transverse momentum and a jet
 at large transverse momentum.
 The signature of the lepton and missing transverse momentum are likely 
 to result from $W$ decay.
 The number of events exceeding the Standard Model expectations 
 together with the large transverse momenta found in these events
 motivated the search for top quark production performed in this
 contribution.
% (full dot after "contribution")
 \par
Five isolated lepton events have been selected by this dedicated search in data samples corresponding to 101.6~pb$^{-1}$ $e^+p$ and 13.6~pb$^{-1}$  $e^-p$ .
% Five isolated lepton events are 
% compatible with having a positively charged lepton. 
% observed with a positively charged lepton.
% They are compatible with anomalous top production in the neutral current 
% channel which is expected to dominantly result from the $u$-valence 
% quarks of the proton.
 The observed number of events is to be compared with $1.8$ events 
 expected from Standard Model calculations.
\par
The analysis of $n$-jet production ($n\ge 3$) shows no excess above the
 Standard Model expectation within the large uncertainty resulting from
 the treatment of higher order corrections and the dependence on the
 renormalization scale.
 However the limit on top production derived from the hadronic channel
 alone does not rule out the possible single top interpretation
 for the candidates observed in the leptonic channels. 
 
%The limit on top production derived from hadronic channel alone is compatible 
%with the number of events observed in the data in the leptonic channels.

% Testing compatibility of the observed lepton and $n$-jet
% events within the top quark interpretation, 
% the $n$-jet events have been used to calculate upper limits for the 
% number of events which are allowed to result from the anomalous top
% production in the leptonic decay channel:
% good compatibility is found with the hypothesis of anomalous top
% production causing the $5$ isolated lepton events.
 \par
%
% Assuming that the observed data contain only Standard Model processes
% and an
% anomalous top production mechanism, the lepton and $n$-jet events are
% used 
% to derive an upper limit for anomalous top production from $u$-quarks:
% $\kappa_\gamma=0.23$.
% This value improves the limits found at the Tevatron and LEP colliders.
%
 Assuming that the observed data contain only Standard Model processes
 and possibly an anomalous top production mechanism,
 the lepton and $n$-jet events are used to derive an upper limit 
 on the $t \leftrightarrow u$ transition mediated by a coupling
 to the photon.
 This bound is slightly better than the previously most stringent
 limit obtained from constraints on radiative top decays
 set at the Tevatron collider.




\section*{Acknowledgements}

We are grateful to the HERA machine group whose outstanding
efforts have made and continue to make this experiment possible. 
We thank
the engineers and technicians for their work in constructing and now
maintaining the H1 detector, our funding agencies for 
financial support, the
DESY technical staff for continual assistance, 
and the DESY directorate for the
hospitality which they extend to the non-DESY 
members of the collaboration. 
We thank A.S. Belyaev for useful discussions and fruitful collaboration.


%============================
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\vspace*{3cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% figures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centering
\epsfig{file=  H1prelim-01-163.fig1.eps,width=10.0cm} 
\caption{Example diagram for anomalous single top production 
            via flavour changing neutral current interaction  
            at HERA.
}
\label{fig:diagtop}
\end{figure}


%-----------------------------------------------------------
%  FIGURE : DIAGRAM FOR SINGLE TOP PRODUCTION
%-----------------------------------------------------------
% \begin{figure}[htb]
% \begin{center}
% \begin{tabular}{p{0.58\textwidth}p{0.37\textwidth}}
%     \raisebox{-120pt}{
%     \mbox{\epsfxsize=0.5\textwidth
%         \epsffile{H1prelim-01-163.fig1.eps}}}
% &
%         \caption
%         {\small \label{fig:diagtop}
%            Example diagram for anomalous single top production 
%            via flavour changing neutral current interaction  
%            at HERA. }
% \end{tabular}
% \end{center}
%\end{figure}


\begin{figure}[htb]
\centering
\epsfig{file=H1prelim-01-163.fig2.eps,width=14.0cm} 
\caption{    Expected distribution for hadronic jet transverse momenta for
         W- production (shaded) and top-production (open) for two
         jet angular regions. The cut value applied in the 
         analysis is indicated.}

\label{ptthcut}
\end{figure}


\begin{figure}[htb]
  \begin{center}
  \epsfig{file= H1prelim-01-163.fig3.eps,width=12.0cm}
\caption{ Reconstructed top mass as invariant mass of lepton, neutrino and hadrons in the electron channel (left) and muon channel (right).} 
\label{topmasres}
  \end{center}
\end{figure}

\begin{figure}[htb]
\centering 
\epsfig{file=H1prelim-01-163.fig4.eps,width=14.0cm}
\caption{
Distribution of the lepton-neutrino-hadrons mass for the top candidates (symbols) and the Standard Model expectation (white histogram). Shaded histogram represents the SM W contribution. The dashed histogram  represents the ANOTOP prediction for a rate corresponding to one event. All the solutions obtained by the procedure described in the text are displayed.  
In the data there are 3 electron candidates with 5 possible solutions and  2 muon candidates with 2 solutions.  The precision of individual mass measurement is better than 9 GeV for electron candidates and better that 13 GeV for the muon candidates.}
\label{mtmlnux} 
\end{figure}

\begin{figure}[htb]
\centering 
\epsfig{file= H1prelim-01-163.fig5.eps,width=14.0cm}
\caption{Control plots for three jet sample in 1994-1997 data (36.5~pb$^-1$).}
\label{jet:control} 
\end{figure}

\begin{figure}[htb]
\centering 
\epsfig{file= H1prelim-01-163.fig6.eps,width=14.0cm}
\caption{Reconstructed mass for candidate three jet events in 1994-1997 data (36.5~pb$^-1$) and Standard Model expectation. The top Monte Carlo is also displayed with an arbitrary normalization.}


\label{jet:mass} 
\end{figure}


\begin{figure}[htb]
\centering
\epsfig{file=  H1prelim-01-163.fig7.eps,width=14.0cm} 
\caption{
% Limits on $k_\gamma$ and $k_Z$ obtained from Tevatron (CDF), LEP (ALEPH) and HERA (H1)
 Limits on the anomalous $t q \gamma$ (magnetic)
 coupling $\kappa_\gamma$ and on the
 anomalous $t q Z$ (vector) coupling
 $v_Z$ obtained at the TeVatron 
 (CDF), LEP (OPAL) and HERA (H1) colliders.
 The H1 limit applies to the coupling $\kappa_{\gamma,u}$ only.
}
\label{fig:limits}
\end{figure}



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\end{document}
