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\begin{document}
\vspace*{4cm}
\title{MODEL-DEPENDENT SEARCHES FOR NEW PHYSICS AT HERA}
\author{G. BARBAGLI}
\address{on behalf of the H1 and ZEUS Collaborations\\
INFN Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino (FI),
Italy}
\maketitle\abstracts{
Some of the searches for phenomena and particles beyond the Standard Model
performed at HERA relying on specific theoretical models
and using almost all the collected
luminosity are discussed here. They particularly concern
leptoquarks, lepton
flavour violation, excited fermions, the anomalous top coupling and
contact interactions, with improved limits on the quark radius.}
\section{Introduction}
The HERA collider delivered luminosity for 15 years colliding
$e^+$ ($e^-$) with $p$ at a centre-of-mass energy
$\sqrt{s} \simeq$ 300-320 GeV. The data taking started in 1992 and
continued until 2000
(HERA I phase). Then a
substantial upgrade program involved both the machine and the experiments and
the data taking
was resumed in 2003 and continued until 2007 (HERA II phase),
during which longitudinally
polarised $e^\pm$ were available in most of data.
The two general purpose experiments H1 and ZEUS ended
data taking in summer 2007, after collecting a total integrated
luminosity of about 1 fb$^{-1}$.
At HERA an extensive program of searches for new particles
and phenomena beyond the Standard Model (SM) has been carried out
in a unique $ep$ environment.
The focus in this paper will be on recent results of searches inspired or
driven by specific theoretical models.
\begin{figure}[h]
\vskip 2.5cm
\psfig{figure=h1lqnc.eps,height=2.5in}
\psfig{figure=h1lqcc.eps,height=2.5in}
\caption{Mass spectra of HERA I $+$ II $e^\pm p$ data for Neutral Current
(left) and Charged Current (right) events in the H1 leptoquark search.
Data points are compared to Standard Model (SM) expectations.
\label{fig:h1lq}}
\end{figure}
\section{Leptoquarks}
Starting from the symmetry between the quark and the lepton sectors
many extensions of the SM predict bosons with fractional
electromagnetic charge and both lepton and
baryon numbers. A widely used model for leptoquarks is the
phenomenological model of Buchm\"uller-R\"uckl-Wyler (BRW)
\, \cite{BRW} which assumes
invariance under $SU(3)_C \times SU(2)_L \times U(1)_Y$,
conservation of the lepton number L and the baryon number B and a set of
7 scalar and 7 vector leptoquarks (4 decaying into both $eq$ and $\nu
q$) classified according to the fermion number $F=3B+L=0, 2$ and
coupling to either left handed or right handed leptons, but not to both,
with fixed branching ratio into $e \nu$ (1, 1/2), $\nu q$ (0, 1/2).
At HERA, leptoquarks can be resonantly produced in the $s$ channel or
exchanged in the $u$ channel between the incoming lepton and the quark
from the proton. The resonant production shows up as a peak in
the mass spectrum or an enhancement in $x$ distribution at the
value corresponding to the mass $M$ of the leptoquark: $x=M^2/s$.
As a consequence of quark densities in the proton, $e^-p$ and $e^+p$
collisions offer respectively best sensitivities to
$F=2$ and $F=0$ leptoquarks.\\
The availability of polarisation of both signs within the HERA II sample
has the advantage of
enhancing the sensitivity to individual leptoquarks species.
H1 searched for leptoquarks studying the inclusive Neutral Current and
Charged Current Deep Inelastic Scattering
high $Q^2$ $e^\pm p$ samples from HERA I and HERA II
and using an integrated luminosity of 482 pb$^{-1}$.
\begin{figure}[h]
\vskip 2.5cm
\psfig{figure=h1lq2b.eps,height=2.5in}
\psfig{figure=h1lfv3.eps,height=2.5in}
\caption{H1 exclusion limits at 95 \% C.L. on the coupling $\lambda$
as a function of the mass for the scalar leptoquark coupling to the first
generation $S^0_L$ (left).
H1 exclusion limits at 95 \% C.L. on the coupling $\lambda_{\mu q}=\lambda_{eq}$
as a function of the mass for the scalar leptoquark mediating lepton flavour violation $S^0_L$ (right).
\label{fig:h1lqlfv}}
\end{figure}
No excess was seen in the $e-jet$, $\nu-jet$ mass spectra
(fig. \ref{fig:h1lq})
and limits were set on the couplings and masses of the different
leptoquark types\, \cite{h1lq} (fig. \ref{fig:h1lqlfv}).\\
\section{Lepton Flavour Violation}
Leptoquarks can couple to different fermion generations and mediate lepton
flavour violation processes in family non diagonal models.\\
H1 searched for $F=2$ leptoquarks coupling to $eq$ and $\mu q$
using $e^- p$ HERA II data and an integrated luminosity of 158 pb$^{-1}$.
No evidence for leptoquarks mediating lepton flavour
violation was obtained and limits were set on couplings
and masses of leptoquarks coupling to 1st and 2nd generation fermions
(fig. \ref{fig:h1lqlfv}).
For an electromagnetic type coupling masses below 291-433 GeV can be
excluded depending on the leptoquark type \, \cite{h1lfv}.
\section{Excited leptons}
To try to explain the hierarchy problem, models of compositeness
introduce substructures to SM fermions, implying the existence of
fermion excited states. Couplings between excited fermions and SM
fermions can be described with phenomenological gauge mediated models
\, \cite{lstartheory1, lstartheory2, lstartheory3}.
Excited fermion states have spin and isospin 1/2 with both
left-handed ($F_L^*$) and right-handed ($F_R^*$) components in weak
iso-doublets. They can decay into fermions and gauge bosons. Magnetic
type transitions between SM fermions $F$ and excited states $F^*$ can take
place.
Weight factors $f$, $f'$ and $f_s$ are used to set the coupling strength to
the three gauge groups (U(1), SU(2) and SU(3)) .
The branching ratios of excited lepton decays can be fixed by assuming a
specific relation between $f$ and $f'$ and then the production cross
section
depends only on $f/\Lambda$ where $\Lambda$ is the compositeness scale.
H1 searched for $e^* \rightarrow e \gamma$,
$e^* \rightarrow e Z$ with $Z \rightarrow q \bar{q}$,
and $e^* \rightarrow \nu W$ with $W \rightarrow q q'$
using $e^\pm p$ data and an integrated luminosity of 475
pb$^{-1}$. No evidence for $e^*$ production was observed.
Improved limits
with respect to LEP and Tevatron were set \, \cite{h1estar}.
Due to the helicity structure of electroweak interactions and the valence
quark densities in the proton, signals for excited neutrinos
are expected to be stronger in $e^- p$ rather than in $e^+ p$ data.
H1 searched for $\nu^* \rightarrow \nu \gamma$,
$\nu^* \rightarrow \nu Z$ with $Z \rightarrow q \bar{q}$,
and $\nu^* \rightarrow e W$ with $W \rightarrow q q'$
using $e^- p$ data and an integrated luminosity of 184 pb$^{-1}$.
No evidence was found and new limits were set \, \cite{h1nustar}(fig.
\ref{fig:h1nustar}).
Masses were excluded in the range up to 213 GeV ($f=-f'$) and 196 GeV
($f=f'$). The H1 analysis has entered regions of masses
not previously explored.
\begin{figure}[h]
\vskip 2.5cm
\psfig{figure=h1estarnew.eps,height=2.5in}
\psfig{figure=h1nustar2a.eps,height=2.5in}
\caption{
H1 exclusion limits at 95 \% C.L. on the coupling $f/\Lambda$
as a function of the mass of the $e^*$ for gauge mediated interactions,
with the assumption $f=+f'$ (left).
H1 exclusion limits at 95 \% C.L. on the coupling $f/\Lambda$ as
a function of the mass of the $\nu^*$ assuming $f=-f'$ (right).
\label{fig:h1enustar}}
\end{figure}
\section{Anomalous top coupling}
At HERA top quarks can only be singly produced.
SM single-top production proceeds via the Charged Current
reaction $ep\rightarrow \nu t \bar b {\rm X}$.
As the SM cross section at HERA is less than $1$~fb
any observed
single-top event must come from physics beyond the SM.
In a Flavour Changing Neutral Current reaction the incoming lepton
exchanges a $\gamma$ or $Z$ with an up-type quark in the proton, yielding
a top quark in the final state most sensitive to a coupling of the type
$tq\gamma$. The $u$-quark dominates at large $x$ and therefore the
production
of single top quark is related to the coupling $tu\gamma$.
H1 searched for single top events in a sample of isolated leptons with high
$p_t$
using $e^\pm p$ data and an integrated luminosity of 482 pb$^{-1}$.
The analysis searched for anomalous production of $t$ decaying into
$b$ and
$W$ with subsequent decay of $W$ into an electron or a muon.
A multivariate
discrimination, based on a phase space density estimator with a range
searching algorithm
was used to separate the signal from the SM background
(mostly real $W$ production).
The upper limit on the cross section set by H1 \, \cite{h1anomtop}
is $\sigma_{ep \rightarrow
etX}<$ 0.16 pb, leading to the most stringent
limit to date on $k_{tu\gamma}<$0.14 at 95 \% C.L.
(fig.\ref{fig:h1anomtop}).
\begin{figure}[h]
\vskip 3cm
\psfig{figure=h1tugamma.eps,height=2.9in}
\caption{Exclusion limits at 95 \% C.L. on the anomalous top coupling
$k_{tu\gamma}$ from H1 and ZEUS compared to limits from LEP and
Tevatron (anomalous couplings to charm are neglected, the top mass
is set to 175 GeV).
\label{fig:h1anomtop}}
\end{figure}
\begin{figure}[h]
\vskip 2.5cm
\psfig{figure=cirq.eps,height=2.2in}
\psfig{figure=h1qr.eps,height=2.5in}
\caption{Ratio of inclusive neutral current deep inelastic scattering data
obtained by ZEUS in $e^\pm$ (left) and single differential cross sections
obtained by H1 in $e^+p$ (top right) $e^-p$ (bottom right) to SM
expectations as a function of $Q^2$, compared with 95 \%
C.L. limits on the effective mean square radius of the electroweak charge
of the quark.
\label{fig:zeush1qr}}
\end{figure}
\section{Contact interactions and quark radius}
Four-fermion contact interactions describe effects from processes
at much higher scales, which could alter the SM distributions
at high $Q^2$ and interfere with the predictions at
intermediate $Q^2$.
These effects modify the tree level amplitude $e q \rightarrow e q$.
Let us focus on vector terms (as scalar and tensor terms are already
costrained by previous searches). The Lagrangian can be written as:
\begin{equation}
L_{CI}=\sum_{\alpha,\beta=L,R}^{q=u,d}\eta_{\alpha \beta}^{q}
(\bar{e}_{\alpha}\gamma^{\mu}e_{\alpha})(\bar{q}_{\beta}\gamma_{\mu}q_{\beta})
\end{equation}
The equation:
\begin{equation}
\eta_{\alpha \beta}=\epsilon \frac{g_{CI}^2}{\Lambda^2}
\end{equation}
where $g_{CI}=4 \pi$ $\epsilon=\pm$ 1
defines the structure of the model.
Contact interaction effects could come from the exchange of extra gauge
bosons (Z'), the production or exchange of leptoquarks or squarks,
compositeness, gravitational effects (extra-dimensions) or from a finite
quark
radius.\\
ZEUS analysed inclusive Neutral Current Deep Inelastic
$e^\pm p$
data from HERA I and HERA II corresponding to
an integrated luminosity of 330 pb$^{-1}$,
comparing the data to SM predictions and
performing a QCD fit where experimental and theoretical
uncertainties are taken into account\, \cite{zeusci}.
Besides general model independent limits on contact interactions
(values of the scale $\Lambda_{eeqq}$ in the range 2.0-8.0 TeV)
depending on the chiral structure,
limits were also set on the heavy leptoquark
(beyond the available CM energy) couplings to the first generation
($M_{LQ}/\lambda$ in the range 0.29-2.08
TeV).\\
In some $4+n$ dimensional string theories
\, \cite{extradim1,extradim2,extradim3}
compactified
extra dimensions have size $R \simeq $ 1 mm. The effective Planck scale
$M_S$ related to the Planck scale $M_P \simeq 10^{19}$ GeV:
$M_P^2=M_S^{2+n}R^n$ can be as small as 1 TeV.
Graviton can propagate into the extra dimension, visible in
the ordinary 4 dimensions as a Kaluza-Klein tower of excited states
with spacing $\Delta m= \frac{1}{R}$. Such states can be summed
up to $M_S$, give sizeable effects, equivalent to
a contact interaction term $\eta^G \simeq\frac{\pm \lambda}{M_S^4}$
where $\lambda \simeq 1$ \, \cite{extradim4}.
The interference with the SM can be constructive or
destructive.\\
Constraints were derived by ZEUS for such extra dimension scales:
$M_S>$ 0.9 TeV for $\lambda=-$1 and
$M_S>$ 0.88 TeV for $\lambda=+$1.\\
As far as the finite size of the quark is concerned in a classical
approach to the quark
substructure a
charge distribution of radius $R_q$ in the quark
can be described using a form factor:\\
\begin{equation}
\frac{d \sigma}{d Q^2}=\frac{d \sigma^{SM}}{d Q^2}
\cdot (1-\frac{R_q^2}{6} \cdot Q^2)^2)
\end{equation}
This effect leads to a decrease of cross sections at high $Q^2$.
An upper limit on quark radius was extracted from the ZEUS analysis:
$R_q < 0.62 \cdot 10^{-16}$ cm.
A study of high $Q^2$ Neutral Currents single differential cross section
by H1 using the complete HERA I and HERA II data and
an integrated luminosity of 270 pb$^{-1}$
($e^+ p$) and 165 pb$^{-1}$ ($e^- p$) \, \cite{h1qr}
led to a limit: $R_q < 0.74 \cdot 10^{-16}$ cm at 95 \% C.L.
(fig. \ref{fig:zeush1qr}).
\section{Conclusions}
The complete statistics of 15 years of data taking is being exploited
by H1 and ZEUS to improve the sensitivity of the searches for new physics
in the unique HERA environment. H1 and ZEUS at HERA have performed a number of
model dependent searches finding no evidence for leptoquarks or lepton
flavor violation, for excited electrons or excited neutrinos.
Looking for single top production new limits on the anomalous top coupling
are set.
Limits on the contact interaction scales
and quark radius have been updated fitting the Deep Inelastic Scattering
differential cross sections
at high $Q^2$. For some of these searches the two collaborations are going
to provide a combination of H1 and ZEUS data.
\section*{References}
\begin{thebibliography}{99}
\bibitem{BRW}
W. Buchmuller, R. Ruckl and D. Wyler \Journal{\PLB}{191}{1987}{442}.\\
W. Buchmuller, R. Ruckl and D. Wyler \Journal{\PLB} {448}{1987}{320}.
\bibitem{h1lq}H1 Collaboration,
Contributed paper to DIS08 H1prelim-07-164
and references therein.
\bibitem{h1lfv}H1 Collaboration,
Contributed paper to DIS08 H1prelim-07-167
and references therein.
\bibitem{lstartheory1}
K.Hagiwara, S. Komamiya and D.Zeppenfeld,
\Journal{\ZPC}{29}{1985}{115}.
\bibitem{lstartheory2}
U.Baur, M.Spiras and P.M. Zerwas,
\Journal{\PRD}{42}{1990}{815}.
\bibitem{lstartheory3}
F. Boudjema, A. Djouadi and J.L. Kneur,
\Journal{\ZPC}{57}{1993}{425}.
\bibitem{h1estar}F.D. Aron {\em et al.} [H1 Collaboration],
DESY 08-052 arxiv:0805.4530 Submitted to Phys. Lett. B.
and references therein.
\bibitem{h1nustar}F.D. Aron {\em et al.} [H1 Collaboration],
DESY 08-009 arxiv:0802.1858
\Journal{\PLB}{663}{2008}{382}.
and references therein.
\bibitem{h1anomtop}H1 Collaboration,
Contributed paper to EPS07 H1prelim-07-163
and references therein.
\bibitem{zeusci}ZEUS Collaboration,
Contributed paper to LP07 ZEUS-prel-07-28
and references therein.
\bibitem{extradim1}
N. Arkani-Hamed, S. Dimopoulos and G.Dvali,
\Journal{\PLB}{429}{1998}{263}.
\bibitem{extradim2}
I. Antioniadis et al.,
\Journal{\PLB}{436}{1998}{257}.
\bibitem{extradim3}
N. Arkani-Hamed, S. Dimopoulos and G.Dvali,
\Journal{\PRD}{59}{1999}{086004}.
\bibitem{extradim4}
G. F. Giudice, R. Rattazzi and J. D. Wells,
\Journal{\NPB}{544}{1999}{3}.
\bibitem{h1qr}H1 Collaboration,
Contributed paper to LP07 H1prelim-07-141
and references therein.
\end{thebibliography}
\end{document}
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