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% event classes
\newcommand{\ee}{\mbox{$e$-$e$}}
\newcommand{\emu}{\mbox{$e$-$\mu$}}
\newcommand{\ej}{\mbox{$e$-$j$}}
\newcommand{\enp}{\mbox{$e$-$\nu$}}
\newcommand{\epho}{\mbox{$e$-$\gamma$}}
\newcommand{\mumu}{\mbox{$\mu$-$\mu$}}
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\newcommand{\eee}{\mbox{$e$-$e$-$e$}}
\newcommand{\eej}{\mbox{$e$-$e$-$j$}}
\newcommand{\eenp}{\mbox{$e$-$e$-$\nu$}}
\newcommand{\eepho}{\mbox{$e$-$e$-$\gamma$}}
\newcommand{\eemu}{\mbox{$e$-$e$-$\mu$}}
\newcommand{\emumu}{\mbox{$e$-$\mu$-$\mu$}}
\newcommand{\emunu}{\mbox{$e$-$\mu$-$\nu$}}
\newcommand{\jmumu}{\mbox{$j$-$\mu$-$\mu$}}
\newcommand{\mumunp}{\mbox{$\mu$-$\mu$-$\nu$}}
\newcommand{\mumupho}{\mbox{$\mu$-$\mu$-$\gamma$}}
\newcommand{\emuj}{\mbox{$e$-$\mu$-$j$}}
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\newcommand{\jjpho}{\mbox{$j$-$j$-$\gamma$}}
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\newcommand{\ejjjj}{\mbox{$e$-$j$-$j$-$j$-$j$}}
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\begin{titlepage}

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%H1prelim-04-063\\
%July 28, 2004
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\multicolumn{4}{l}{{\bf
                33rd International Conference
                on High Energy Physics, ICHEP06},
                July.~26-Aug. 2,~2006,~Moscow} \\
                 & Abstract:        &     &\\
                 & Parallel Session: & Beyond the Standard Model   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
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\vspace{2cm}

\begin{center}
\begin{Large}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf \boldmath A general search for new phenomena in $e^-p$ scattering\\ at HERA}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{2cm}

H1 Collaboration

\end{Large}
\end{center}

\vspace{1cm}

\begin{abstract}
\noindent
  A model-independent search for
  deviations from the Standard Model prediction is performed
  in $e^- p$ collisions at HERA~II using H1 data recorded during the years   $2005$--$2006$ and
  corresponding to an integrated luminosity of $159$ $\mbox{pb}^{-1}$.
  All event topologies 
  involving isolated electrons, photons, muons, neutrinos and jets with
  high transverse momenta are investigated in a single analysis.
  Events are assigned to exclusive classes according to their
  final state.
  A statistical algorithm is used to search for
  deviations from the Standard Model in the distributions of the scalar sum of
  transverse momenta or invariant mass of final state particles and to quantify their significance.
  A good agreement with the Standard Model prediction is observed in most
  of the event classes.
  No significant deviation is observed in the phase-space and in the event topologies covered by this analysis.


\end{abstract}


\vspace{1.5cm}

\end{titlepage}

\newpage

\pagestyle{plain}

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

At HERA electrons\footnote{
  In this paper ``electrons'' refers to both electrons and positrons, if
  not otherwise stated.}
and protons collide at a centre-of-mass energy of up to $319$~GeV. 
These high-energy electron-proton interactions provide a 
testing ground for the Standard Model (SM) complementary to $e^+e^-$ and $p\overline{p}$ scattering. 
% It is widely believed that the SM is incomplete and 
% that new physics signals may appear below energies of $1$ TeV. 
% Many extensions to the SM have been constructed during
% the last decades predicting various phenomena which may be visible at high energies 
% or large transverse momenta ($P_T$). HERA data have been used to test some 
% of these models of new processes by analysing their anticipated experimental signatures
% and limits on their parameters have been 
% derived~\cite{Kuze:2002vb}.

The approach described in this paper consists of a comprehensive
and generic search for deviations from the SM prediction at large 
transverse momenta. The present analysis follows closely the strategy of our previous publication \cite{Aktas:2004pz}.
All high $P_T$ final state configurations involving
electrons ($e$), muons ($\mu$), jets ($j$), photons ($\gamma$) or neutrinos ($\nu$) are systematically 
investigated. The analysis covers phase space regions where the SM 
prediction is sufficiently precise to detect anomalies and does not 
rely on assumptions concerning the characteristics of any SM extension. 
% Such a model-independent approach might discover 
% unexpected manifestations of new physics. Therefore it addresses the important question of whether evidence for new physics  
% might still be hidden in the data recorded at collider experiments.
% A similar strategy for a model-independent search was previously presented in~\cite{Abbott:2001ke}. 

All final states containing at least two objects ($e$, $\mu$, $j$, $\gamma$, $\nu$) with 
$P_T >$~$20$~GeV in the polar angle\footnote{ 
  The origin of the H1 
  coordinate system is the nominal $ep$ interaction point, with 
  the direction of the proton beam defining the positive 
  $z$-axis (forward region). The transverse momenta are measured 
  in the $xy$ plane. 
  The 
  pseudorapidity $\eta$ is related to the polar 
  angle $\theta$ by $\eta = -\ln \, \tan (\theta/2)$.}
range  $10^\circ < \theta < 140^\circ$ are investigated. 
The complete HERA II $e^-p$ data sample ($2005$--$2006$) is used, corresponding 
to an integrated luminosity of $159$~pb$^{-1}$.
All selected events are classified into exclusive event classes 
according to the number and types of objects detected in the final state 
(e.g.  \ej, \mujnp, \jjjjj). 
These exclusive event classes ensure a clear
separation of final states and allow an 
unambiguous statistical interpretation of deviations. 
All experimentally accessible combinations of objects have been studied and 
data events are found in $21$ of them.

In a first analysis step the global event yields of the event
classes are compared with the SM expectation.
The distributions of the invariant mass $M_{all}$ and of the scalar sum of transverse momenta
$\sum P_T$ of high $P_T$ final state objects are presented.
New physics may be visible as an excess or a deficit 
in one of these distributions. 
Therefore, in a second step these distributions are systematically investigated using
a dedicated algorithm which locates 
the region with the largest deviation of the data from the SM
prediction. 
The probability of occurrence of such a deviation
is derived, both for each event class individually and globally for all classes combined.
                    
% This paper is organised as follows. Section $2$ describes the Standard Model 
% processes at HERA and their Monte Carlo simulation. The 
% H1 detector, the event selection and measurement procedure
% are described in section $3$.
% The event yields and distributions for each event class are presented in
% section $4$. The search strategy and results are explained in section $5$. 
% Section $6$ summarises the paper.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Standard Model processes and Monte Carlo generation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Several Monte Carlo event 
generators are combined to simulate events for all dominant
SM processes, avoiding double-counting. 
All processes 
are generated with an integrated luminosity significantly higher than
that of the data sample and events are passed through a full detector
simulation~\cite{Brun:1987ma}.
At high transverse 
momenta the dominant SM processes are the photoproduction of two jets and 
neutral current (NC) deep-inelastic scattering (DIS).
In the following the abbreviation $X$ represents the reaction products not belonging to the high $P_T$ objects considered.

\paragraph{Photoproduction of jets and photons}
To simulate the direct and resolved photoproduction of jets 
$ep \rightarrow jj X$, prompt photon production $ep \rightarrow \gamma  j X$ 
and the resolved photoproduction of photon pairs 
$ep \rightarrow \gamma \gamma X$, the PYTHIA $6.1$ event 
generator~\cite{Sjostrand:2000wi} is used. Light and heavy flavoured jets 
are generated. The simulation contains the Born level hard scattering
matrix elements and radiative QED corrections. 

\paragraph{Neutral current deep-inelastic scattering}
The Born, QCD Compton and Boson Gluon Fusion matrix elements are used in 
the RAPGAP~\cite{Jung:1993gf} event generator to model NC DIS events. 
The QED radiative effects arising from real photon emission from both the 
incoming and outgoing electrons are simulated using the 
HERACLES~\cite{Kwiatkowski:1990es} generator. Hence the NC DIS prediction 
contains the processes $ep \rightarrow e j X$, $ ep \rightarrow e j j X$ and 
$ep \rightarrow e \gamma j X$. If the electron has $P_T<20$~GeV,
NC DIS may contribute to the $ep \rightarrow j j X$ and $ep \rightarrow \gamma j X$ processes.

\paragraph{Charged current deep-inelastic scattering}
Charged current (CC) DIS events are also simulated
using the RAPGAP~\cite{Jung:1993gf} program.
%Parton cascades are generated using the 
%colour-dipole model in ARIADNE~\cite{Lonnblad:1992tz}. 
This prediction contributes to the final states $ep \rightarrow \nu j X$,
$ep \rightarrow \nu j j X$  and to final states with an additional radiated photon.

\paragraph{QED Compton scattering}
Elastic and quasi-elastic Compton processes $ep \rightarrow e \gamma X$
are simulated with the WABGEN~\cite{Berger:kp} generator. The inelastic 
contribution is already included in the NC DIS RAPGAP sample.

\paragraph{Electroweak production of lepton pairs}
Multi-lepton events ($ee$, $\mu\mu$, $\tau\tau$) are generated with the GRAPE ~\cite{Abe:2000cv}
program, which 
includes all electroweak matrix elements at tree level. 
Multi-lepton production via $\gamma \gamma$, $\gamma Z$, $ZZ$ collisions, 
internal photon conversion and the decay of virtual or real $Z$ bosons 
is considered. Initial and final state QED radiation is included.
The complete hadronic final state is obtained via interfaces to 
PYTHIA and SOPHIA~\cite{Mucke:1999yb} for the inelastic and quasi-elastic regimes, respectively.
Consequently, GRAPE predicts $ep \rightarrow \mu \mu X$ and
$ep \rightarrow e e X$, as well as $ep \rightarrow e \mu \mu X$ and
$ep \rightarrow e e e X$ if the scattered electron is detected. Processes with an additional radiated photon are also modelled.

\paragraph{W production}
The production of $W$ bosons $ep \rightarrow W X$ and $ep \rightarrow W j X$
is modelled using EPVEC~\cite{Baur:1991pp}. Next-to-leading order QCD 
corrections ~\cite{Diener:2002if} are taken into account by reweighting the 
events as a function of the transverse momentum and rapidity of the $W$ 
boson~\cite{Diener:2003df}.

\paragraph{}
Processes with the production of three or more jets, e.g. $ep \rightarrow jjjX$ or 
$ep \rightarrow jjjjX$, are accounted for using leading
logarithmic parton showers as a representation of higher order QCD radiation.
Hadronisation is modelled using Lund string fragmentation~\cite{Sjostrand:2000wi}.
The prediction of processes with two or more high transverse momentum jets, e.g. $ep \rightarrow jjX$, $ep \rightarrow ejjX$ is scaled by a factor of $1.2$ to normalise the leading order Monte Carlos to next-to-leading 
order QCD calculations~\cite{Adloff:2002au}. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experimental technique}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The H1 detector}
The H1 detector \cite{Abt:1996xv} components relevant to the 
present analysis are briefly described here. Jets, 
photons and electrons are measured with the Liquid
Argon (LAr) calorimeter~\cite{Andrieu:1993kh}, which covers the polar angle range 
$4^\circ < \theta < 154^\circ$ with full azimuthal acceptance. 
Electromagnetic shower energies are measured with a precision of 
$\sigma (E)/E = 12\%/ \sqrt{E/\mbox{GeV}} \oplus 1\%$ and hadronic energies 
with $\sigma (E)/E = 50\%/\sqrt{E/\mbox{GeV}} \oplus 2\%$, as measured in test beams. The central and forward tracking detectors are used to
measure charged particle trajectories, to reconstruct the interaction 
vertex and to supplement the measurement of the hadronic energy. The LAr and inner tracking detectors are enclosed in a super-conducting magnetic coil with a strength of $1.15$~T. 
The return yoke of the coil is the outermost part of the detector and is 
equipped with streamer tubes forming the central muon detector 
($4^\circ < \theta < 171^\circ$). It is also used to supplement the 
measurement of hadrons. In the forward region of the detector ($3^\circ < \theta < 17^\circ$) a set of drift chamber layers (the forward muon system) detects muons and, together with an iron toroidal magnet, allows a momentum measurement. The luminosity measurement is based on the Bethe-Heitler process  $ep \rightarrow ep \gamma$, 
where the photon is detected in a calorimeter located
downstream of the interaction point.

The main trigger for events with high transverse momentum is provided 
by the LAr calorimeter. The trigger efficiency is close 
to $100\%$ for events having an electromagnetic deposit in the LAr
(electron or photon) with transverse momentum greater than 
$20$~GeV~\cite{Adloff:2003uh}. Events triggered only by jets have a trigger 
efficiency close to $100\%$ for $P_T^{jet}>20$~GeV.
For events with missing transverse momentum above $20$ GeV, determined from an 
imbalance in the transverse momentum measured in the calorimeter, the trigger efficiency is 
$\sim$ $90\%$.
The muon trigger is based on single muon signatures from the central muon detector, which are combined with signals from the central tracking detector.
The trigger efficiency for di-muon events is
about $70\%$.

\subsection{Event selection}
At HERA electrons or positrons with an energy 
of $27.6$ GeV collide with protons at an energy of $920$ GeV resulting in a
centre-of-mass energy of $\sqrt{s}$ = $319$~GeV.
The event sample studied consists of the 
full $2005$--$2006$ HERA~II data set. It corresponds to an integrated luminosity of  $159$ $\mbox{pb}^{-1}$ recorded in electron--proton collisions only.  

The data selection requires at least one isolated electromagnetic cluster, 
jet or muon  to be found in the detector acceptance. 
Energy deposits in the calorimeters and tracks in the inner tracking system 
are combined to reconstruct the hadronic energy. 
To reduce background it is demanded that the event 
vertex be reconstructed within $35$~cm in $z$ of the nominal position
and that $\sum_i \left(E_i-P_{z,i}\right)< 75$~GeV, where $E_i$ is the particle's energy and 
$P_{z,i}$ is the $z$ component of the particle momentum.
Here, the index $i$ runs over all hadronic energy deposits, electromagnetic clusters
and muons found in the event. Due to energy-momentum conservation events are expected to have a value of \mbox{$\sum_i \left(E_i-P_{z,i}\right)=55.2$}~GeV, twice the electron beam energy,
if only longitudinal
momentum along the proton beam direction is unmeasured. 
Events with topologies 
typical of cosmic ray and beam-induced background are rejected.
Moreover, the timing of the event is required to coincide with that of the $ep$ bunch crossing.

The identification criteria for each type of particle are similar to those applied in our previous general search~\cite{Aktas:2004pz}. 
The following paragraphs describe the identification criteria for the different objects and give the identification efficiencies for the kinematic region considered in the analysis.
 
\paragraph{Electron identification}
The electron identification is based on the measurement 
of a compact and isolated electromagnetic shower in the LAr calorimeter. 
The hadronic energy within a distance in the pseudorapidity-azimuth
($\eta-\phi$) plane $R=\sqrt{(\Delta \eta)^2+(\Delta \phi)^2}<0.75$ around the 
electron is required to be below $2.5\%$ of the electron energy.
This calorimetric electron identification is complemented by 
tracking conditions. A high quality 
track is required to geometrically match the electromagnetic cluster within a 
distance of closest approach to the cluster centre-of-gravity of $12$~cm. 
No other good track is allowed within $R<0.5$ around the electron direction. 
In the central region ($20^\circ < \theta <  140^\circ$) the distance between the first 
measured point in the central drift chambers and the beam axis is required to 
be below $30$~cm in order to reject photons that convert late in the central 
tracker material. In addition, the transverse 
momentum measured from the associated track $P_T^{e_{tk}}$ is required 
to match the calorimetric measurement $P_T^e$ with 
$1/P_T^{e_{tk}} - 1/P_T^e < 0.04$~GeV$^{-1}$. 
In the region not fully covered by the central drift chambers 
($10^\circ < \theta <  37^\circ$) a wider isolation 
cone of $R=1$ is required to reduce the contribution of fake electrons from hadrons. 
The resulting electron finding efficiency 
is $\sim 90$\% in the central region and $70\%$ in the forward region.

\paragraph{Photon identification}
The photon identification relies on the measurement of an 
electromagnetic shower and on the same calorimetric isolation criteria against 
hadrons as for the electron identification.
In addition, photons are required to be separated from jets with $P_T^{jet} > 5$~GeV by a distance of $R>1$ to the jet axis. 
Vetoes on any charged track pointing to the electromagnetic cluster 
are applied. No track with a distance of closest approach to the cluster below $24$~cm should be present.
The resulting photon identification efficiency as derived using 
elastic QED Compton events is $\sim 85\%$.

\paragraph{Muon identification}
The muon identification is based on a track in the forward muon
system or in the inner tracking systems associated with
a track segment or an energy deposit in the central muon detector. The muon momentum is measured from the track 
curvature in the toroidal or solenoidal magnetic fields. A muon candidate 
should have no more than $5$~GeV deposited in a
cylinder of radius $25$ cm and $50$ cm in the electromagnetic and hadronic
sections of the LAr calorimeter, respectively, centred on the muon track direction. In di-muon events, the requirement of an opening angle between the two muons smaller than $165^\circ$ discards events with 
muons coming from cosmic rays. Beam halo muons are rejected by requiring 
that the muons originate from the event vertex. 
Finally, misidentified  hadrons 
are almost completely suppressed by requiring that the  muon candidate is 
separated from the closest jet with $P_T>3$~GeV by $R>1$. The efficiency to 
identify muons is greater than $90\%$.

\paragraph{Jet identification}
Jets are defined using the 
inclusive $k_{\bot}$ algorithm \cite{Ellis:1993tq,Catani:1993hr}. 
The algorithm is applied in the laboratory frame with a separation parameter
of $1$ and using a $P_T$ weighted recombination scheme \cite{Ellis:1993tq} in which the jets are
treated as massless. 
The jet algorithm is run on all combined cluster-track objects not previously 
identified as electron or  photon candidates. The scattered electron may fake a jet.
This effect is important for multi-jet events, especially 
at high transverse momenta. To reject these fake jets, the first radial moment of the jet transverse energy \cite{Giele:1997hd,frising} is required to be greater than $0.02$ and the quantity
$M^{obj}/P_T^{jet}$ greater than $0.1$~\cite{Adloff:2002au,frising}. 
The invariant mass $M^{obj}$ is obtained using the four-vector of all objects belonging to the jet. If the fraction of the jet energy contained in the 
electromagnetic part of the LAr calorimeter is greater than $0.9$, the above
criteria are tightened to $0.04$ and $0.15$, respectively. The jet selection efficiency is $97\%$.

\paragraph{Neutrino identification}
A neutrino candidate is defined in events with missing transverse momentum
above $20$~GeV. The missing momentum is derived from all identified particles and
energy deposits in the event. Fake missing transverse momentum may also arise from the
mismeasurement of an identified object. This effect is reduced
by requiring that the neutrino\footnote{
  The four-vector of the neutrino is calculated under 
  the assumption $\sum_i \left(E_i-P_{z,i}\right) + \left(E_\nu -P_{z,\nu}\right) =55.2$~GeV.} be isolated from 
all identified objects with a transverse momentum above $20$~GeV.
Requiring $\sum_i \left(E_i-P_{z,i}\right)<48$~GeV discards neutrino 
candidates from NC processes where the missing transverse momentum is caused by energy leakage in the 
forward region. 
% An additional criterion is applied to reduce NC and lepton pair
% background events where one particle's energy is mismeasured. These events 
% typically have values of $\Delta \phi(l-X_{tot}) \approx 180^\circ$, where 
% $\Delta \phi(l-X_{tot})$ is the azimuthal angle difference between the 
% charged lepton and the direction of the system $X_{tot}$ built of all energies 
% measured in the calorimeters. 
If exactly one electron or muon object is found, a neutrino object is only assigned to an event 
if $\Delta \phi(l-X_{tot})<170^\circ$, where $\Delta \phi$ is the difference in azimuthal angle between the lepton $l$ and the direction of the system $X_{tot}$ built of all hadronic energies. 
 
\paragraph{Event classification}
The common phase space for electrons, photons, muons and jets is defined 
by $10^\circ<\theta<140^\circ$ and $P_T > 20$~GeV. The  neutrino phase space 
is defined by missing transverse momentum above $20$~GeV and  
$\sum_i \left(E_i-P_{z,i}\right)<48$~GeV. These values are chosen to retain a high 
selection and trigger efficiency. All particles with $P_T > 20$~GeV, 
including the neutrino defined by its reconstructed four-vector, are required 
to be isolated compared with each other by a minimum distance $R$ of one unit in the 
$\eta-\phi$ plane. The events are 
classified, depending on the number and types of objects, into
exclusive event classes. Events with an isolated calorimetric 
object in the considered phase space which is not identified as a photon, 
electron or jet are discarded from the analysis in order to minimise wrong 
classifications.  

Based on these identification criteria purities can be 
derived for each event class with a sizeable SM expectation. Purity is defined as the ratio of SM events reconstructed 
in the event class in which they are generated to the total number of 
reconstructed events in this class. 
Most purities are found to be above $60\%$ and are close to $100\%$ for the 
\jj, \ej, \jnp~and \mumu~event classes. 



\subsection{Systematic uncertainties}
This section describes the sources of experimental and theoretical systematic 
uncertainties considered. Experimental systematic uncertainties arising from 
the measurement of the objects are listed in table~\ref{tab:objectunc}.
  
\begin{itemize}
\item  
  The electromagnetic energy scale uncertainty varies between $1\%$ and $3\%$ depending on the particle's impact point on the LAr calorimeter surface. 
  The polar angular measurement uncertainty is $3$~mrad.
  The identification of electron and photon candidates 
  depends on the tracking efficiency, which is known with a precision ranging from $3\%$ for polar 
  angles above $37^\circ$ to $5\%$ in the forward region. 

\item
  The hadronic energy scale of the LAr calorimeter is known to $2\%$. 
  The uncertainty on the jet polar angle determination is $5$ mrad 
  for $\theta<30^\circ$ and $10$ mrad for $\theta>30^\circ$.
\item
  The uncertainty on the transverse momentum measurement for muons is taken to be $5\%$. 
  The uncertainty on the polar angle is $3$~mrad. The muon identification efficiency is known with a precision of 5\%.
\item
  The trigger uncertainties for each class are determined by 
  the object with the highest trigger efficiency. The uncertainty on the
  trigger efficiency is estimated to be $3\%$ if the event is triggered
  by a jet or missing transverse momentum and $10\%$ if it is triggered by a muon. For electrons and photons the 
  uncertainty on the trigger efficiency is negligible.
\item
  The uncertainty in the integrated luminosity results in an
  overall normalisation error of $2.5\%$.
\end{itemize}


Depending on the dominant production process different theoretical uncertainties are used as listed in table~\ref{tab:modelunc}. The errors attributed to 
the predictions for $ep\rightarrow jj X$, $ep \rightarrow j\gamma X$,
$ep\rightarrow j\nu X$, $ep \rightarrow jeX$, 
$ep\rightarrow jj\nu X$, $ep \rightarrow jjeX$ 
and $W$ production include uncertainties in the parton distribution functions and those
due to missing higher order corrections~\cite{Adloff:2002au,Andreev:2003pm,frising,martin}. 
The error attributed to $ep \rightarrow \mu\mu X$ and $ep \rightarrow ee X$ results 
mainly from uncertainties in the structure functions~\cite{Aktas:2003sz,Aktas:2003jg}. 
The error on the 
QED Compton cross section is estimated to be $5$\% for elastic
and 10\% for inelastic production. An additional theoretical error of 
$20\%$ is applied for each jet produced by parton shower processes 
(e.g. 20\% for the \jjj~event class).
An uncertainty of $50\%$ is added to the prediction for NC DIS 
events with missing transverse 
momentum above $20$~GeV and a high $P_T$ electron. This uncertainty is 
estimated by a comparison of the missing transverse momentum distribution 
between NC DIS events with a low $P_T$ electron ($P_T<20$~GeV) with the SM prediction.

All systematic errors are added in quadrature and are assigned to the SM predictions.
%  For example, the resulting total uncertainties for \ej~events are $10\%$ and $35\%$ at low and high invariant mass $M_{all}$, respectively. In the \jj~event 
% class the errors are typically $20\%$ and reach $40\%-50\%$ for $M_{all}$ and
% $\sum P_T$ values around $250$~GeV.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Event yields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
All possible event classes with at least two objects are investigated\footnote{The \munp~ event class is discarded from the present analysis.
It is dominated by events in which a poorly reconstructed muon
gives rise to missing transverse momentum, which fakes the
neutrino signature.}. 
The event yields subdivided 
into event classes are presented for the data and SM expectation
in figure~\ref{fig:summaryplot}. All event classes with observed data events or with
a SM expectation greater than $1$ event are shown. 
In each class, a good description of the number of observed data events by the SM prediction is seen.

No data events are observed in the event classes \mujnp~and \ejnp. Those classes correspond mainly to high $P_T$ $W$ production with a subsequent leptonic decay and the total SM expectation amounts to $1.2 \pm 0.2$ and $2.5 \pm 0.8$ in \mujnp~and \ejnp~classes, respectively. The probability $p$ (see section $4.1$) to observe such a downwards fluctuation in the data given the SM expectation is $0.35$ and $0.12$ in the \mujnp~and \ejnp~classes, respectively.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Search for deviations from the Standard Model}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Search algorithm and strategy}
In order to quantify the level of agreement between 
the data and the SM expectation and to identify regions of possible 
deviations, the same search algorithm as developed for our previous publication~\cite{Aktas:2004pz} is used.
It locates the region of largest deviation of the data from the SM in these 
distributions. 

\paragraph{Definition of regions}
A region in the $\sum P_T$ and $M_{all}$ distributions is defined as a set of connected histogram bins\footnote{
  In order to minimise binning effects, a bin size
smaller than the resolution of the researched quantity is 
used.
All values presented in this study are derived using a 5 GeV bin 
  size for all distributions. Further reduction of the bin size has a negligible
  effect on the results.} with a size of at least twice the resolution.
All possible regions of any width and at any position in the histograms are considered. 
The number of data events ($N_{obs}$), the SM 
expectation ($N_{SM}$) and its total systematic uncertainty ($\delta N_{SM}$) are calculated for each region.

\paragraph{Probability of a fluctuation of the data in a region}
A statistical estimator $p$ is defined to judge which region is of 
most interest. This estimator is derived from the convolution of the
Poisson probability density function (pdf) to account for statistical 
errors with a Gaussian pdf, $G(b;N_{SM},\delta N_{SM})$, with mean $N_{SM}$ and width $\delta N_{SM}$, to include the effect of 
non negligible systematic uncertainties. The estimator
is defined via 
\begin{equation*}
  p  = \left\{ \begin{array}{ll}
      A \int\limits_0^{\infty} db  \, G(b;N_{SM},\delta N_{SM}) \, \sum\limits_{i=N_{obs}}^{\infty} \frac{e^{-b}
        b^i} {i!} & \textrm{ if } N_{obs} \ge N_{SM} \\
      A \int\limits_0^{\infty} db  \, G(b;N_{SM},\delta N_{SM}) \,\,\,\, \sum\limits_{i=0}^{N_{obs}} \frac{e^{-b}
        b^i} {i!} & \textrm{ if } N_{obs} < N_{SM} 
    \end{array} \right.
\end{equation*}
\begin{equation*}
    \textrm{ with } \,\,\,\, A= 1 / \left[ \int\limits_0^{\infty} db  \, G(b;N_{SM},\delta
  N_{SM}) \, \sum\limits_{i=0}^{\infty} \frac{e^{-b} b^i} {i!}  \right].
\end{equation*}
The factor $A$ ensures normalisation to unity. If $G$ is replaced 
by a Dirac delta function $\delta(b-N_{SM})$ the estimator $p$ becomes the usual 
Poisson probability. The value of $p$ gives an estimate of the probability of a fluctuation of the SM expectation upwards (downwards) to at least (at most) the observed number of data events in the region considered.

\paragraph{Determination of the most interesting region}
A possible sign of new physics is found if the expectation significantly
disagrees with the data. A disagreement is quantified by the estimator $p$.
The region of greatest interest (of greatest deviation) is the region having the smallest $p$-value, $\pmin$. 




\paragraph{Significance per event class}
The possibility that a fluctuation with a value $\pmin$ occurs
anywhere in the distribution is estimated
using the following method. 
Many independent hypothetical data histograms are made by filling each bin
with an event number diced
according to the pdfs of the SM expectation
(again a convolution of Poisson and Gaussian pdfs). 
For each of 
those hypothetical data histograms the algorithm is run to find the region of 
greatest deviation and the corresponding $\pmin^{SM}$ is calculated. 
The probability $\hat{P}$ is then defined as the 
fraction of hypothetical data histograms with a $\pmin^{SM}$ equal to or smaller 
than the $\pmin$ value obtained from the real data.
The $\hat{P}$ values of event classes with no data event and a SM expectation
$\lsim 1$ are $1$.
$\hat{P}$ is a measure of the statistical significance of the deviation observed in the data.
If the event classes are exclusive, $\hat{P}$ can be used 
to compare results of different event classes. Consequently the event class of 
most interest for a search is the one with the smallest $\hat{P}$ value.
Depending on the final state, a $\pmin$-value of $5.7 \cdot 10^{-7}$ (``$5\sigma$'') corresponds 
to a value of $- \log_{10}{\hat{P}}$, the negative decade logarithm of $\hat{P}$, between $5$ and $6$. 


\paragraph{Global significance}
The overall degree of agreement with the SM can further be quantified by 
taking into account the large number of event 
classes studied in this analysis.
Among all studied classes there is some chance that small 
$\hat{P}$ values occur. 
This probability can be calculated with 
MC experiments. 
A MC experiment is defined as
a set of hypothetical data histograms
(either in $M_{all}$ or in
$\sum P_T$) following 
the SM expectation with an integrated luminosity of 159 pb$^{-1}$.

The complete search algorithm and statistical analysis are
applied to the MC experiments like for data. This procedure is repeated many times.
The expectation for the $\hat{P}$ values 
observed in the data is then given by the 
distribution of $\hat{P}^{SM}$ values obtained from all MC experiments.
The probability to find in MC experiments a $\hat{P}$ value smaller than in the data can be calculated and gives us the global significance of the observed deviation.



\subsection{Search results}
The $\hat{P}$ values observed in the real data in all event classes are compared in figure~\ref{fig:scan} to
the distribution of $\hat{P}^{SM}$ obtained from the large set of MC experiments, normalised to one experiment. 
The comparison is presented for the scans of the invariant mass distributions and $\sum
P_T$ distributions.
All $\hat{P}$ values range from $0.01$ to $0.99$, corresponding 
to event classes
where no significant discrepancy between data and the SM expectation 
is observed. 
These results are in agreement with the expectation from 
MC experiments.

The regions selected by the algorithm in $\sum P_T$ and $\Mall$ distributions of each class
are presented in figures \ref{fig:1} and \ref{fig:2}, respectively. 


Due to the uncertainties of the SM prediction in the \jjjj~and \jjjjnp~event classes at highest $M_{all}$ and $\sum P_T$ (see~\cite{Aktas:2004pz}), where data events are observed, no reliable $\hat{P}$ values can be calculated for these classes. These event classes are not considered to search for deviations from the SM in this extreme kinematic domain. 
Consequently, these event classes are not taken into account to determine the overall degree of agreement between the data and the SM.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data collected with the H1 experiment during the years $2005$--$2006$ (HERA~II) 
have been investigated for deviations from the SM prediction at high transverse 
momentum. All event topologies involving isolated electrons, photons, 
muons, neutrinos and jets are investigated in a single analysis.
This is the first general search performed on a large set of data from electron--proton collisions.
A good agreement between data and SM expectation is found 
in most event classes.
In each event class the invariant mass and sum of transverse momenta 
distributions of particles have been 
systematically searched for 
deviations using a statistical algorithm. 
No significant deviation is observed in the phase-space and in the event topologies covered by this analysis.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\cite{Aktas:2003jg}
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A.~Aktas {\it et al.}  [H1 Collaboration],
%``Multi-electron production at high transverse momenta in e p collisions at HERA,''
Eur.\ Phys.\ J.\ C {\bf 31} (2003) 17
[hep-ex/0307015].
%%CITATION = HEP-EX 0307015;%%

% %\cite{Aktas:2003yd}
% \bibitem{Aktas:2003yd}
% A.~Aktas {\it et al.}  [H1 Collaboration],
% %``Search for single top quark production in e p collisions at HERA,''
% Eur.\ Phys.\ J.\ C {\bf 33}, 9 (2004)
% [hep-ex/0310032].
% %%CITATION = HEP-EX 0310032;%%


% %\cite{Kon:1997bz}
% \bibitem{Kon:1997bz}
% T.~Kon, T.~Matsushita and T.~Kobayashi,
% %``Possible excess in charged current events with high-Q**2 at HERA from  stop and sbottom production,''
% Mod.\ Phys.\ Lett.\ A {\bf 12} (1997) 3143
% [hep-ph/9707355].
% %%CITATION = HEP-PH 9707355;%%

% %%%%% ref for H1 e+p LQ search 94-97
% \cite{Adloff:1999tp}
% \bibitem{Adloff:1999tp}
% C.~Adloff {\it et al.}  [H1 Collaboration],
% %%``A search for leptoquark bosons and lepton flavor violation in e+ p
% %%collisions at HERA,''
% Eur.\ Phys.\ J.\ C {\bf 11} (1999) 447
% [Erratum-ibid.\ C {\bf 14} (2000) 553]
% [arXiv:hep-ex/9907002].
% %%CITATION = HEP-EX 9907002;%%


% %%%%% ref for H1 e-p LQ search
% %\cite{Adloff:2001cp}
% \bibitem{Adloff:2001cp}
% C.~Adloff {\it et al.}  [H1 Collaboration],
% %``A search for leptoquark bosons in e- p collisions at HERA,''
% Phys.\ Lett.\ B {\bf 523} (2001) 234
% [hep-ex/0107038].
% %%CITATION = HEP-EX 0107038;%%
% 

% %%%%%---- ref from presentation in conf of prelim H1 results ...
% %\cite{Sztuk:vh}
% \bibitem{Sztuk:vh}
% J.~Sztuk  [H1 and ZEUS Collaborations],
% %``Leptoquark Searches At Hera,''
% Acta Phys.\ Polon.\ B {\bf 33} (2002) 3893.
% %%CITATION = APPOA,B33,3893;%%

% %\cite{Chekanov:2003af}
% \bibitem{Chekanov:2003af}
% S.~Chekanov {\it et al.}  [ZEUS Collaboration],
% %``A search for resonance decays to lepton + jet at HERA and limits on
% %leptoquarks,''
% Phys.\ Rev.\ D {\bf 68} (2003) 052004
% [arXiv:hep-ex/0304008].
% %%CITATION = HEP-EX 0304008;%%



\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newpage


%%%%%%%%% Paper tables %%%%%%%%%%%%%%
\clearpage

\begin{table}[b]
  \begin{center}
    \begin{tabular}{|l|c|c|c|}
      \hline
      Object & Energy Scale & $\theta$ &  Identification \\
      &          &    (mrad)    &     efficiency \\ 
      \hline                                        
      Jet &  2\%             &  5--10   &  -- \\ 
      Electron & 1--3\%    &  1--3    &  3--5\%   \\    
      Photon & 1--3\%      &  1--3    &  5--15\% \\ 
      Muon &  5\%            &  3       &  5\% \\ 
      \hline
    \end{tabular}
    \caption{Systematic uncertainties attributed to the measurement of energies, polar angles and to the identification efficiencies of particles.}
    \label{tab:objectunc}
  \end{center}  
\end{table}

\begin{table}[b]
  \begin{center}
    \begin{tabular}{|l|c|}
      \hline
      Process & Uncertainty \\
      \hline
      $ep\rightarrow jj X$ and $ep \rightarrow j\gamma X$ &15\% \\
      $ep\rightarrow j\nu X$ and $ep \rightarrow jeX$  & 10\% \\
      $ep \rightarrow jj\nu X$ and $ep \rightarrow jjeX$  & 15\% \\
      $ep \rightarrow \mu\mu X$ and $ep \rightarrow ee X$ & 3\%\\
      $ep \rightarrow W X$ and $ep \rightarrow WjX$ & 15\%\\
      $ep \rightarrow e\gamma X$ and $ep \rightarrow e\gamma j$& 10\% \\
      $ep \rightarrow e\gamma p$ & 5\% \\
      \hline
    \end{tabular}
    \caption{Theoretical uncertainties attributed to the simulation of different SM processes.}
    \label{tab:modelunc}
  \end{center}
\end{table}

\clearpage


\begin{figure}[p]
  \center
  \includegraphics[width=\textwidth]{H1prelim-06-161.fig1.eps}
  \caption{The data and the SM expectation for all event classes 
    with observed data events or a SM expectation greater than $1$ events.
    The analysed data sample corresponds to an integrated luminosity of 159~pb$^{-1}$. 
    The error bands on the predictions include model uncertainties and 
    experimental systematic errors added in quadrature.
  }
  \label{fig:summaryplot}
\end{figure}
\clearpage


\begin{figure}[p]
  \includegraphics[width=\textwidth]{H1prelim-06-161.fig2.eps}
  \caption{The number of data events and the SM expectation as a 
    function of $\sum P_T$ for classes with at least one event. 
    The shaded areas show the regions of greatest deviation 
    chosen by the search algorithm. No search is performed for the \jjjj~and \jjjjnp~ classes.}
  \label{fig:1}
\end{figure}
\clearpage

\begin{figure}[p]
  \includegraphics[width=\textwidth]{H1prelim-06-161.fig3.eps}
  \caption{The number of data events and the SM expectation as a
    function of $\Mall$ for event classes with at least one event. 
    The shaded areas show the regions of greatest deviation 
    chosen by the search algorithm. No search is performed for the \jjjj~and \jjjjnp~ classes.}
  \label{fig:2}
\end{figure}


\begin{figure}[p]
  \center
  \includegraphics[width=0.75\textwidth]{H1prelim-06-161.fig4.eps}\\[0.5cm]
  \includegraphics[width=0.75\textwidth]{H1prelim-06-161.fig5.eps}  
  \caption{The $-\log{\hat{P}}$ values for the data event classes and the 
    expected distribution from MC experiments as derived by investigating
    the $\sum P_T$ distributions (top) and $\Mall$ distributions
    (bottom) with the search algorithm. 
    } 
  \label{fig:scan}
\end{figure}
\clearpage


\end{document}

