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\begin{document}

\title{Search for beyond the Standard Model physics at HERA}

\author{E. GALLO}

\address{INFN Firenze, \\
Via Sansone 1, \\ 
50019 Sesto Fiorentino, Italy,\\ 
E-mail: gallo@fi.infn.it}



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\maketitle

\abstracts{
The H1 and ZEUS Collaborations have almost completed a rich program of
searches for physics beyond the Standard Model using the data
collected in the HERA I period. 
The main highlights, such as leptoquarks and signatures with
isolated leptons at high energy, are reviewed.
}


\section{Introduction}

The H1 and ZEUS Collaborations have each collected of the order
of $110 \div 130~pb^{-1}$ of $e^+p$ and $15~pb^{-1}$ of $e^-p$ 
collisions data in the HERA I running period. These data have been used for
 a wide spectrum of searches for physics beyond the
Standard Model (SM). In these proceedings some of the highlights
are presented, together with some of the prospects for the HERA II running.


\section{Leptoquarks}

Leptoquarks (LQ) are scalar or vector triplet bosons, carrying both
lepton ($L$) and baryon ($B$) number, and are foreseen in many
extensions of the SM. 
HERA is the ideal accelerator to look for leptoquarks, as, 
for large masses, they are produced 
from the fusion of the electron/positron 
beam with a valence $u,d$-quark in the proton. 
The Buchm\"uller-R\"uckl-Wyler model
classifies them in 14 different types, according {\em i.e.} to
their quark flavour content (up or down), their fermion number (7 with $F=3B+L$=0 and 7 with $F=2$), their spin (7 scalar and 7 vector), and their decay ($eq$ or $\nu q$). 
For masses $M_{LQ}< \sqrt{s}=300,318~\mathrm{GeV}$, leptoquarks are produced mainly in the 
$s$-channel. The cross-section, in  the narrow-widthapproximation, is proportional to $\lambda^2 q(x, Q^2$), where
$\lambda$ is the Yukawa coupling of the leptoquark to the lepton and quark and
the quark density  $q(x,Q^2)$ is evaluated  at 
Bjorken $x={M^2_{LQ}/s}$ and photon virtuality $Q^2=M^2_{LQ}$. The signature is therefore a narrow resonance decaying into electron+jet or neutrino+jet, with mass peaking at $M_{LQ}\simeq \sqrt{xs}$.
The final state is in principle not distinguishable from  the SM DIS processes $ep \rightarrow eX$ (NC) or $ep \rightarrow \nu X$ (CC), however
the inelasticity $y$ distribution, which is related to the scattering
angle in the $eq$($\nu q$) rest frame, is different for LQs ($d \sigma/dy \sim y$ for scalar, $\sim (1-y)^2$ for vectors) and for DIS ($\sim 1/y^2$). For $M_{LQ} \gg \sqrt{s}$,
the effect of the leptoquark exchange can be treated as a contact-interaction term.

\begin{figure}[h]
\mbox{
\epsfxsize=6cm   %width of figure - will enlarge/reduce the figures
\epsfbox{gallo.fig1a.eps}
\hspace{-1cm}
\epsfxsize=6cm   %width of figure - will enlarge/reduce the figures
\epsfbox{gallo.fig1b.eps}}
\caption{Example of limits on leptoquark production from the HERA I data,
compared also to LEP and Tevatron results.
The left plot shows the limit of the Yukawa coupling $\lambda$ as a function of the LQ mass for the $S_{1/2}^L$ type leptoquark. The right plot shows
the limit on the branching ratio $\beta_e$ versus the LQ mass.\label{gallolqfigure}}
\end{figure}

 
Both H1 and ZEUS have performed sophisticated fits of the
SM+LQ+interference cross-section to their high $Q^2$ and high$-x$ data, using the two-dimensional
information in the $x-y$ plane\cite{lqpapers}. No evidence of a leptoquark signal
has been found in the data. H1 has used the neutral current data at high $Q^2$, while ZEUS has also done a fit to the charged current data, providing strong limits for the 4 types of leptoquark decaying both to $eq$ and
to $\nu q$. An example of these limits is shown for the $S_{1/2}^L$ leptoquark (produced in $e^+ p$, {\em i.e.} $F=0$) in Fig.~\ref{gallolqfigure} (left), where
the limits of the coupling $\lambda$ versus the LQ mass is presented
for H1 and ZEUS, in comparison to L3 and the Tevatron combined analysis.

In the mass region between 240 and 300 GeV, HERA set the most stringent limits,
excluding couplings in the range of $10^{-2}\div 10^{-1}$. 
At LEP, the strongest limits come from indirect searches from the reaction
$e^+ e^- \rightarrow q \bar q$ and are competitive only in the contact-interaction region for few types of leptoquarks. At Tevatron, leptoquarks are
produced in pairs and the cross-section is independent from
the Yukawa coupling $\lambda$, as shown in the figure. However their limits 
depend on the branching ratio $\beta_e$ to electron and quark in the final state, as illustrated in Figure~\ref{gallolqfigure} (right). HERA has better 
sensitivity than D0 for small branching ratios
and there is still a window open for discovery for small values
of $\lambda$ and $\beta_e$. The search for leptoquarks remain therefore
a {\em must} also for the HERA II searches, which will complement
the Tevatron II results in this field.    



\section{Lepton Flavour Violation}

As there is now clear evidence for oscillations in the neutrino sector, it is natural
to look for flavour violation interactions in the charged leptonic sector (LFV). HERA is basically
background-free for reactions of the type $e q \rightarrow \mu X$ or $eq \rightarrow \tau X$,
which are predicted by many extensions of the SM, like leptoquarks coupling also to the second and third generation, here taken
as a possible model.

ZEUS has recently\cite{lfvpaper} presented a new analysis in the $\tau$ channel,
which makes use of a new $\tau$ identification technique for the hadronic decay\cite{isoltau}. 
In this method, variables which are chosen to discriminate
between QCD and $\tau$-jets, making use of the fact that taus are low-multiplicity
and collimated jets, are combined with a multi-observable discrimination technique. The
signal efficiency is therefore improved with respect to previous analyses. The LFV signature
is a $\tau$, either decaying to electron, muon or hadrons, aligned with the missing 
transverse momentum in the event. 

\begin{figure}[h]
\mbox{
\epsfxsize=6cm   %width of figure - will enlarge/reduce the figures
\epsfbox{gallo.fig2a.eps}
\hspace{-1cm}
\epsfxsize=6cm   %width of figure - will enlarge/reduce the figures
\epsfbox{gallo.fig2b.eps}}
\caption{Example of limits on lepton flavour violation production from the ZEUS data,
compared also to limits extracted from rare decays. 
\label{gallolfvfigure}}
\end{figure}


No event with such a signature was selected by ZEUS in the
99-00 $e^+p$ data. The result is used to set limits on the production of leptoquarks
which decay to $e \tau$. An example, for the two leptoquark types $\tilde{S}_{1/2}^L$ and
$V_0^R$, is shown in Figure~\ref{gallolfvfigure}, where the limit of the Yukawa coupling $\lambda_{eq}$, assuming
that it is equal to the coupling $\lambda_{\tau q}$, is presented as a function of the LQ mass.
For couplings of the order of the electromagnetic interaction, $\lambda=0.3$, leptoquarks
with LFV are excluded in the range $279 \div 299~\mathrm{GeV}$. The limits are also shown in comparison
to results obtained indirectly from rare decays of $B,K$ mesons and taus. ZEUS sets the
strongest limits in a wide region, especially when a heavy quark in the final
 state is
involved.   




\section{$R-$parity violating supersymmetry}

The Minimal Supersymmetric Model (MSSM) is widely accepted as a possible
extension of the SM.
In SUSY models, every particle has its own supersymmetric partner
and their spin $S$ differ by 1/2. A new quantity, the $R$-parity,
defined as $R_p=(-1)^{3B+L+2S}$, is $+1$ for
particle and $-1$ for sparticle. While HERA is not competitive with
LEP and Tevatron for reactions which require the conservation of $R_p$,
it can instead look for $R_p$-violating reactions. In this case the Lagrangian
has 3 additional terms:
\noindent
\begin{equation}
{L} =
{\lambda_{ijk}} L_i L_j \tilde{E}_k +
{\lambda^\prime_{ijk}} L_i Q_j \tilde{D}_k +
{\lambda^{\prime \prime}_{ijk}} U_i D_j \tilde{D}_k. 
\label{eq:gallorpviolation}
\end{equation}
While LEP is more sensitive to the first term and Tevatron to the last, HERA
can search for reactions with the term $\lambda^\prime$, involving the coupling
of squarks to an electron and a quark.

\begin{figure}[h]
\mbox{
\epsfxsize=5.5cm   %width of figure - will enlarge/reduce the figures
\epsfbox{gallo.fig3a.eps}
\epsfxsize=5.5cm   %width of figure - will enlarge/reduce the figures
\epsfbox{gallo.fig3b.eps}}
%\epsfxsize=12cm   %width of figure - will enlarge/reduce the figures
\caption{Limits on $R_p$-violating squark production for $\tilde{u},\tilde{c}$ squarks
in the unconstrained MSSM model (left) and on the stop production in the mSugra
model (right).
\label{gallosusyfigure}}
\end{figure}


An example in $e^+ p$ interaction is the process $e^+ d \rightarrow \tilde{u}_L^j$,
where the squark ${u}_L^j$ can decay to: $e^+ d$, giving a signature identical to
the one expected by a leptoquark; or via a gauge decay to $ q\chi^0_\alpha$, $q \tilde{g}$ or
$q \chi^+_\beta$. The neutralinos $\chi^0_{1,2,3,4}$, the charginos $\chi^+_{1,2}$ and
the gluino $\tilde{g}$ are unstable and can decay via $R-$parity violating coupling. For instance
the decay $\chi^0 \rightarrow e^\pm q \bar q$, gives in the final state a total signature
of 3 jets and an electron or positron, half of the time of opposite sign compared to
the initial beam lepton, which is a background-free process. 

The H1 Collaboration has analyzed the 98-00 data and searched for squarks in seven
different topologies, according to all possible decays\cite{susy}.
No evidence was found for a signal, therefore limits on squark production were determined.
The limits were derived scanning a wide range of the SUSY parameters $\mu,M_2$ and for two
values of the $\tan \beta$ parameters (2,6). However the limits do not depend 
strongly on these variations, as shown by the grey band in Figure~\ref{gallosusyfigure} (left).       
In general $\tilde{u}_L,\tilde{c}_L$ squarks with masses below $240~\mathrm{GeV}$
 are ruled out for
$\lambda^\prime_{111},\lambda^\prime_{121}=0.3$, respectively.

H1 has also calculated limits in the framework of the Minimal Supergravity Model (mSUGRA).
An example is shown for the stop quark in Figure~\ref{gallosusyfigure} (right), where the exclusion region
in the two mSUGRA parameters $m_0,m_{1/2}$ is shown for $\lambda^\prime_{131}=0.3$.
The excluded area corresponds approximately to a value of the $\tilde{t}_L$-mass of
$270~\mathrm{GeV}$, as shown by the dashed line in the figure. The present Tevatron limits are around 
$130~\mathrm{GeV}$ and the sensitivity of Tevatron Run II is around $250~\mathrm{GeV}$, showing the 
importance of HERA II for stop searches. 




\section{Isolated leptons and missing $p_T$ events at HERA}


The H1 collaboration has observed regularly, during the HERA I running,
an excess of events characterized by an isolated lepton $e,\mu$ at
high transverse energy, missing transverse momentum $p_T$ and high
transverse momentum of the remaining hadronic system $p_T^X$.
Both H1 and ZEUS have published their final results on this study,
which are  summarized in Table~\ref{gallotab1}\cite{singletop}.
 


\begin{table}[h]
\tbl{Summary of isolated lepton search at H1 and ZEUS.\vspace*{1pt}}
{\footnotesize
\tabcolsep7pt
\begin{tabular}{|c|r|r|r|}
\hline
{}                             & Electron & Muon      & Tau      \\[-1.5ex]
{}                             & obs./exp & obs./exp. & obs./exp. \\[1ex]
\hline
{} &{} &{} &{} \\[-1.5ex]
H1 $p_T^X~>~25~\mathrm{GeV}$   & {4 / 1.49}           & {6 / 1.44}  & - \\\hline
H1 $p_T^X~>~40~\mathrm{GeV}$   & {3 / 0.54}           & {3 / 0.55}  & - \\ \hline
\hline
ZEUS $p_T^X~>~25~\mathrm{GeV}$ &  2 / 2.90   & 5 / 2.75  & {2 / 0.20} \\ \hline
ZEUS $p_T^X~>~40~\mathrm{GeV}$ &  0 / 0.94   & 0 / 0.95  & {1 / 0.07} \\ \hline
\hline
\end{tabular}\label{gallotab1} }
\vspace*{-13pt}
\end{table}

The main SM expectation in the table comes from the production 
of a $W$ radiated from a quark line, and then
decaying into a lepton and a neutrino, giving therefore the observed topology. However, while
for the SM expectation the $p_T^X$ distribution peaks at low values, for H1 there is a clear
excess of events at $p_T^X>40~\mathrm{GeV}$, where 6 events are observed in the $\mu+e$-channels,
while less than 1 event is expected. The ZEUS experiment does not confirm
the result and observes no events for $p_T^X>40~\mathrm{GeV}$. 

A possible process beyond the SM which could give this signature
is the flavour-changing neutral-current (FCNC) reaction $eu \rightarrow (e)t \rightarrow (e)bW$
(where the scattered electron is seen in the detector only for high $Q^2$).
Top decays could be produced in the SM via CC processes like  $eg \rightarrow \nu t b$,
however the cross-section is less than $1~fb$ and not measurable with the present
integrated luminosity. Any excess of this kind of events could then be due to FCNC 
reactions. Both ZEUS and H1 have performed a search optimized for single-top
production, both in the leptonic and hadronic decay of the $W$. In the leptonic channel,
H1 observes 5 events compatible with a single-top signal, while $1.8$ are expected
from the SM contribution; no deviation is observed in the hadronic channel,
where however the contribution from the SM QCD background is high. ZEUS observes no
deviation in both channels, setting therefore strong limits on the FCNC couplings.


\begin{figure}[h]
\mbox{
\epsfxsize=3cm   %width of figure - will enlarge/reduce the figures
\epsfbox{gallo.fig4a.eps}
\epsfxsize=6cm   %width of figure - will enlarge/reduce the figures
\epsfbox{gallo.fig4b.eps}
}
\caption{Diagram for FCNC top production at HERA (left) and limit (right)
on the FCNC couplings obtained by H1, ZEUS, CDF
and the combined LEP preliminary result. The dashed lines show also the
prospects on the limits at HERA II and Tevatron II, assuming that no deviation 
from the SM is observed.
\label{gallofcncfigure}}
\end{figure}

ZEUS has calculated the limit taking into account both the $t u \gamma$ and 
the $t u Z$ contribution at leading order, as shown in Figure~\ref{gallofcncfigure} (right). 
H1 has calculated the limit using the NLO order cross-section and neglecting the 
$Z$ contribution: the excess in the leptonic channel is cause of the slightly weaker   
limit in $k_{tu \gamma}$ compared to ZEUS. 
The limits are compared to the ones derived by the LEP experiments,
looking for the decay $e^+ e^- \rightarrow t u$\footnote{The charm contribution has been 
put to zero in this comparison, as in the HERA high$-x$ regime of interest to this process
the  charm density is negligible compared to the $u$-density.
The LEP and CDF results have therefore been rescaled by a factor $\sqrt{2}$. 
Another factor of $\sqrt{2}$, compensating the previous
one, derives from the different Lagrangian convention used by HERA and LEP/Tevatron.}, and to the CDF
contraints derived from the branching ratios $t \rightarrow q \gamma$, $t \rightarrow q Z$. 
The strongest limit on the $\gamma$ coupling, $k_{t u \gamma}< 0.174$, comes from ZEUS. 

At HERA II, the search for this kind of events will
be one of the high priority studies. Even hoping to see still an excess,
an absence of this signal will provide very strong limits on the
FCNC couplings, as shown in Figure~\ref{gallofcncfigure} for an expected integrated luminosity
of $1~fb^{-1}$. 



The ZEUS experiment has extended the previous search to the hadronic 
$\tau$-decays\footnote{The decays $\tau \rightarrow e \nu, \tau \rightarrow
\mu \nu$ are included in the previous search.}. The same technique described
in Section 3 has been used to separate the taus from the QCD jets. 
The results\cite{isoltau} are summarized in Table~\ref{gallotab1}, where, with some surprise, 2 outstanding
events with $p_T^X>25~\mathrm{GeV}$ remain after all cuts, with a SM expectation of
$0.20 \pm 0.05$\footnote{The results here reported are an update of what was presented at the
Workshop, with the recently published results.}. 
%One of these two events is shown in Figure 5.
Note that, due to the low efficiency of selecting $\tau$-hadronic decays, 
the cross-section corresponding to these 2 events is far higher than the single-top
cross-section already excluded by the $e,\mu$ and hadronic-channels.
A possible explanation of these $\tau$ events beyond the SM requires 
that there is no $W$-production involved. It is possible, in SUSY models
with large $\tan\beta$, that the stop decays with preference to a $\tau$, {\em i.e.}
$\tilde{t} \rightarrow \tau \tilde{\nu}_\tau b, \tilde{\tau} \nu_\tau b$, 
with subsequent decays, giving the observed topology.



\section{Multi-lepton events at HERA}

Also the study of events with two or more leptons ($e$ or $\mu$) at high energy,
can provide information on new possible processes beyond the SM. H1 has recently
published the final results on this search, 
ZEUS has presented preliminary results, both use the full HERA I data sets\cite{multilepton}.

For the multi-electron search, both experiments require two well-reconstructed 
{\em central} electrons\footnote{The term electron here refers to both electrons and positrons,
unless specified.}, 
well inside the acceptance of the tracking chambers ($20^\circ < \theta_e < 150^\circ$
for H1 and $17^\circ < \theta_e < 164^\circ$ for ZEUS), and with transverse energy 
$p_T^e>10~\mathrm{GeV}$.
At high-$Q^2$ values, the scattered electron can be observed in the detector, giving a topology
of 3-$e$ in the final states: the third electron is allowed
in a wider angular region ($5^\circ < \theta_e < 175^\circ$).
In general both H1 and ZEUS observe, in all distributions, a good agreement 
with the SM prediction, which
is dominated by the Bethe-Heitler reaction $\gamma \gamma \rightarrow l^+ l^-$. 

\begin{table}[h]
\tbl{Summary of multi-electron search at H1 and ZEUS for $M_{12}>100~\mathrm{GeV}$.\vspace*{1pt}}
{\footnotesize
\tabcolsep7pt
\begin{tabular}{|c|c|c||c|c|}
\hline
{}                             & {} & {}  & {} & {}\\[-1.5ex]
{}                             & DATA H1 & SM H1 & DATA ZEUS & SM ZEUS \\[1ex]
\hline
{}   &{}   &{}                  & {}  & {}                \\[-1.5ex]
2$e$ & {3} & {$0.30 \pm 0.04$}  & {2} & {$0.77 \pm 0.08$}  \\\hline
3$e$ & {3} & {$0.23 \pm 0.04$}  & {0} & {$0.37 \pm 0.04$}  \\\hline
\hline
\end{tabular}\label{gallotab2} }
\vspace*{-13pt}
\end{table}


H1 observes
in total 108 $2e$ events and 17 $3e$ events, in good agreement with the SM predictions
of $117.1 \pm 8.8$ and $20.3 \pm 2.1$, respectively. However, the invariant mass distribution,
$M_{12}$, of the two highest-$p_T^e$ electrons,  shows 6 outstanding events at mass greater
than $100~\mathrm{GeV}$, 3 with 2 electrons and 3 with 3 electrons, while $0.53$ events in total
are expected.  The $M_{12}$ distribution is shown in Figure~\ref{gallodielfigure} and the predicted number of events
is shown
in Table~\ref{gallotab2}. The figure also shows that the events at high mass have a different
topology in the $2e$- and in the $3e$-case: while in the $2e$-case the two electrons
with high invariant mass have also high transverse momentum, this is not the
case for the $3e$-events, suggesting that, if it is new physics, it could be of
different origin. 
  



\begin{figure}[h]
\begin{center}
\mbox{
\epsfxsize=8cm   %width of figure - will enlarge/reduce the figures
\epsfbox{gallo.fig5.eps}
}
\end{center}
\caption{
Invariant mass distribution of the two highest-$p_T^e$ electrons in the H1 data,
for the $2e$- and $3e$- samples separately, compared to the sum of the SM expectations. 
The two plots on the left show the correlation
between the mass and the sum of the transverse momentum of the two highest-$p_T^e$
electrons.
\label{gallodielfigure}}
\end{figure}

The ZEUS collaboration has looked for a similar topology in the HERA I data. In
total 191 $2e$-events and 26 $3e$-events are selected, in broad agreement
with the SM prediction of $213.9 \pm 3.9$ and $34.7 \pm 0.5$, respectively.
However no deviation is observed at high mass, where two $2e$-events 
have $M_{12}>100~\mathrm{GeV}$, while $1.2$ are expected in total (see Table 2). In addition, no
deviation from the SM is observed in the muon channel by both experiments.

A possible explanation of the 6 outstanding events observed by H1 is the
production of doubly-charged Higgs. A $H^{++}(H^{--})$ can be produced at
HERA, radiated from the  positron(electron) line. It decays then
in two leptons of the same positive (negative) sign. Although there are already very strong
limits from OPAL, a dedicated search was done by H1, optimizing
the selection cuts for this signal. Only one of the 6 events survive these
criteria, making unlikely that these events can be explained by
doubly-charged Higgs production. The analysis is instead used to set limits
on this process, excluding $H^{++}/H^{--}$ masses of $131~\mathrm{GeV}$ for a value of
the Yukawa coupling of the Higgs to the electrons of 0.3. 
Also this search will be an important chapter of the HERA II program, with
expected sensitivities that compete with the present limits from LEP.

%\begin{figure}[h]
%\mbox{
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{gallo.fig6.eps}
%}
%\caption{
%Result of the general search performed by H1.
%\label{gallogensefigure}}
%\end{figure}



\section{Summary}

The H1 and ZEUS Collaborations have looked for signature beyond
the SM model in many processes, ranging from $s$-channel resonances
like leptoquarks, squarks and excited fermions, to contact interactions and
to special signatures in the final state like those with high-transverse-momentum
leptons. It is natural to ask if anything was forgotten: for this
purpose H1 has performed a model-independent search\cite{gensearch}, selecting
all events with at least two objects (electrons, photons, muons, jets, and 
neutrinos {\em i.e.} missing
$p_T$) at high transverse energy ($E_T>20~\mathrm{GeV}$). The number of selected events
has been compared to the expectation, calculated from the sum of the 
Monte Carlo prediction of various models. 
% Figure 6 shows the number of
% events (dots), compared to the SM (histogram) for the various final states.
The largest deviation is observed in the $\mu-jet-\nu$ channel, corresponding to
the events with isolated muons, high $p_T^X$ and high missing $p_T$, as described
in Section 5. All other observations are compatible with all previous analyses
and no other anomaly is found. With the higher integrated luminosity promised
for HERA II, it will be exciting to confirm or solve the remaining puzzles
from the 1994-2000 running period.



\begin{thebibliography}{0}
\bibitem{lqpapers} ZEUS Coll., {\it Phys. Rev.} {\bf D68} 052004 (2003), and references
therein; H1 Coll., paper 105 contributed to the EPS03 conference, Amsterdam. 

\bibitem{lfvpaper} ZEUS Coll., EPS03 paper 498.

\bibitem{isoltau} ZEUS Coll., DESY-03-182.

\bibitem{susy} H1 Coll., EPS03 paper 101.

\bibitem{singletop} H1 Coll., {\it Phys. Lett.} {\bf B561} 241 (2003)
                              and EPS03 paper 100; ZEUS Coll.,   
{\it Phys. Lett.} {\bf B559} 153 (2003) and EPS03 paper 495. 


\bibitem{multilepton} H1 Coll., hep-ex/0307015 and EPS03 paper 104; ZEUS Coll., ICHEP02 paper 910.

\bibitem{gensearch} H1 Coll., EPS03 paper 118.

\end{thebibliography}

\end{document}





