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% A.F.Zarnecki       Warsaw University/ZEUS
%
% `Contact Interactions: Results from ZEUS and a Global Analysis'
%
% contribution to DIS'99 proceedings:  W 1 - 24
%                                     ==========
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% May 31, 1999
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\documentstyle[twoside,fleqn,npb,epsfig]{article}


\def\pbi{\,{\rm pb}^{-1}}
\def\tev{\,{\rm TeV}}
\def\gev{\,{\rm GeV}}
\def\epslam{\epsilon / \Lambda^2}



\title{Contact Interactions: Results from ZEUS and a Global Analysis}

\author{A.F.\.Zarnecki\address{Institute of Experimental Physics, 
        Warsaw University, Ho\.za 69, 00-681 Warszawa, Poland}
        (zarnecki@fuw.edu.pl) }

\begin{document}

\begin{abstract}
In a search for signatures of physics processes beyond the Standard Model,
various $eeqq$ vector contact interaction hypotheses have been tested
in the ZEUS experiment.
%
No significant evidence of a contact interaction signal has been found.  
%
The analysis is based on NC $e^+p$ DIS data corresponding
to an integrated luminosity of $47.7\pbi$ and results in 95\% CL limits on the
effective mass scales $\Lambda$ ranging from 1.7 to $5\tev$ 
for the different one-parameter contact interaction scenarios considered.

Within the global analysis, including data from other experiments as well,
any contact interactions with mass scale below $2.1\tev$ are  
excluded at 95\% CL.
%
Combined mass scale limits for specific one-parameter scenarios range
from 5.1 to $18\tev$.
%
Upper limits on 
possible effects to be observed in future HERA, LEP and Tevatron running
are estimated. 
%
The total hadronic cross-section at LEP and $e^{-}p$ scattering
cross-section at HERA are strongly constrained by existing data,
whereas large cross-section deviations are still possible for 
Drell-Yan lepton pair production at the Tevatron.
\end{abstract}

% typeset front matter (including abstract)
\maketitle

% ----------------------------------------------------------------------------
%       Introduction
% ----------------------------------------------------------------------------
\section{INTRODUCTION}
\label{sec-int}

The HERA $ep$ collider has extended the kinematic range available for the study
of deep--inelastic scattering (DIS) by two orders of magnitude to values of
$Q^2$ up to about $50000\gev^2$.
%
 Measurements in this
domain allow new searches for physics processes beyond the Standard Model (SM)
at characteristic mass scales in the $\tev$ range. 
% 
The recent analyses were stimulated
in part by an excess of events over the SM expectation for $Q^2 > 20000\gev^2$
reported in 1997 by the ZEUS \cite{zeus97} and H1 \cite{h1_97}
collaborations, for which electron-quark contact interactions (CI)
have been suggested as possible explanations.

%----------------------------------------------------------------------------
%  Scenarios and mass scale limit table
%----------------------------------------------------------------------------

\begin{table*}[t]
%
\caption{Coupling structure and the 95\% CL limits on the effective 
mass scales for different one-parameter contact interaction models.
Results from the ZEUS analysis based on high-$Q^{2}$ $e^{+}p$ NC DIS data
are compared to results based on the combined electron/muon NC data
(global analysis not assuming SU(2) invariance) and those corresponding to 
all available data (global analysis with SU(2) universality).
}
\label{tab-limits}
%
\setlength{\tabcolsep}{1mm}
\begin{tabular*}{\textwidth}%
    {@{}l@{\hspace{5mm}}cccc@{\extracolsep{\fill}}rrrrrr}
\hline
% & & & & & \multicolumn{6}{c}{95\% CL mass scale limits  [TeV]}\\ \cline{6-11}
  & & & & & \multicolumn{2}{c}{\hspace*{-1cm}ZEUS analysis\hspace*{-1cm}}  
          & \multicolumn{4}{c}{Global analysis}  \\ \cline{8-11}
  Model   & \multicolumn{4}{c}{Couplings}  
          & \multicolumn{2}{c}{ \hspace*{-1cm}
                               $e^{+}p$ NC DIS data \hspace*{-1cm}}  
          & \multicolumn{2}{c}{$e/\mu$ NC data}
          & \multicolumn{2}{c}{All data} \\ 
            \cline{2-5} \cline{6-7} \cline{8-9} \cline{10-11}
          & $\eta^{eq}_{LL}$    & $\eta^{eq}_{LR}$
          & $\eta^{eq}_{RL}$    & $\eta^{eq}_{RR}$ 
          & $\Lambda_{min}^{-}$ & [TeV]~~ $\Lambda_{min}^{+}$
          & $\Lambda_{min}^{-}$ & [TeV]~~ $\Lambda_{min}^{+}$ 
          & $\Lambda_{min}^{-}$ & [TeV]~~ $\Lambda_{min}^{+}$  \\ \hline

 %  & \multicolumn{4}{c}{$q \; = \; u, d, s, c, b$} & & & & & & \\ \cline{2-5}
 %
 VV & + & + & + & + & 5.0 & 4.7 &  9.8 & 10.7 &  9.6 & 11.4 \\
 AA & + & --& --& + & 2.6 & 2.5 & 10.5 & 10.1 &  9.9 & 11.1 \\
 VA & + & --& + & --& 3.7 & 2.6 &  6.6 &  6.2 &  6.3 &  8.0 \\
 % \hline
 %
 X1 & + & --&   &   & 2.8 & 1.8 &  8.7 &  8.1 &  8.1 &  9.5 \\
 X2 & + &   & + &   & 3.1 & 3.4 &  8.2 &  8.4 &  7.8 &  9.6 \\
 X3 & + &   &   & + & 2.8 & 2.9 &  9.9 & 10.2 &  9.5 & 11.1 \\
 X4 &   & + & + &   & 4.3 & 4.0 &  5.7 &  5.2 &  6.0 &  5.4 \\
 X5 &   & + &   & + & 3.3 & 3.5 &  5.9 &  6.4 &  6.2 &  6.4 \\
 X6 &   &   & + & --& 1.7 & 2.8 &  6.2 &  5.8 &  6.2 &  5.8 \\
 \hline
 %  & \multicolumn{4}{c}{$q \; = \; u, c$} & & & & & & \\ \cline{2-5}
 %
 U1 & + & --&   &   & 2.6 & 2.0 &  6.4 &  7.7 &  7.9 & 17.0 \\
 U2 & + &   & + &   & 3.9 & 4.0 &  6.9 &  9.1 &      &      \\
 U3 & + &   &   & + & 3.5 & 3.7 &  8.5 & 11.7 &  8.6 & 18.2 \\
 U4 &   & + & + &   & 4.8 & 4.4 &  5.1 &  5.5 &      &      \\
 U5 &   & + &   & + & 4.2 & 4.0 &  6.4 &  8.8 &  7.1 &  8.8 \\
 U6 &   &   & + & --& 1.8 & 2.4 &  7.0 &  5.6 &      &       \\ 
 \hline
 %
\end{tabular*}
\end{table*}


% ----------------------------------------------------------------------------
%       Contact Interactions
% ----------------------------------------------------------------------------
\section{CONTACT INTERACTIONS}
\label{sec-ci}

Four-fermion contact interactions are an effective theory, which 
allows us to describe, in the most general way, possible low energy 
effects  coming from ``new physics'' at much higher energy scales. This 
includes the possible existence of second-generation heavy weak bosons, 
leptoquarks as well as electron and quark compositeness \cite{cidef,cihera}.
%
As strong limits beyond the HERA
sensitivity have already been placed on the scalar and 
tensor terms \cite{cihera},
only the vector $eeqq$ contact interactions
are considered in this study.
%
They can be represented 
as additional term in the Standard Model Lagrangian \cite{cihera}:
%
\begin{eqnarray}
L_{CI} & = & \epsilon \frac{g^2}{\Lambda^2}
           \sum_{i,j=L,R} \eta^{eq}_{ij} (\bar{e}_{i} \gamma^{\mu} e_{i} )
              (\bar{q}_{j} \gamma_{\mu} q_{j}) 
\end{eqnarray}
where the sum runs over electron and quark helicities,
$\epsilon$ is the overall sign of the CI Lagrangian,
$g$ is the coupling, and $\Lambda$ is the effective mass scale.
%
Helicity and flavour structure of contact interactions is described by 
set of parameters $\eta^{eq}_{ij}$.
%
Since $g$ and $\Lambda$ always enter in the
combination $g^2/\Lambda^2$, we fix the coupling by adopting
the convention $g^2=4\pi$.
%
In the ZEUS analysis 30 specific CI scenarios are considered.
%
Assumed relations between different couplings are
listed in Table \ref{tab-limits}.
%
Each line in the table represents 
two scenarios, one for $\epsilon=+1$ and one for $\epsilon=-1$.
%
For the models VV, AA, VA and X1 to X6 all quark flavours are assumed 
to have the same contact interaction couplings and
each of the $\eta_{ij}^{eq}$ is either zero or $\pm 1$.
%
For the U1 to U6 models only couplings of up-type quarks ($u$ and $c$)
are considered.

The global analysis combining data from different experiments 
(see sections \ref{sec-global} and \ref{sec-pred})  also 
takes into account three less constrained models, 
in which different couplings can vary independently.
%
    The {\bf General Model} assumes that contact interactions couple only 
    electrons to $u$ and $d$ quarks (8 independent couplings). 
    All other couplings (for $s, c, b, t, \mu , \tau$) are 
    assumed to vanish.
%
    The model with {\bf Family Universality} assumes lepton universality
    ($e$=$\mu$) and quark family universality ($u$=$c$ and 
    $d$=$s$=$b$). There are also 8 independent couplings.
%
    In a model assuming ${\bf SU(2)_{L} \times U(1)_{Y}}$ gauge invariance,
    the number of free model parameters is reduced from 8 to 7
    ($\eta^{eu}_{RL}$=$\eta^{ed}_{RL}$). In this model the $eeqq$ 
    contact interaction couplings can be also related to 
    $\nu \nu qq$  and $e \nu q q'$ couplings \cite{ci_su2}.


% ----------------------------------------------------------------------------
%       ZEUS analysis
% ----------------------------------------------------------------------------
\section{ZEUS ANALYSIS}
\label{sec-zeus}

This analysis \cite{zeus_ci} is based on  $47.7\pbi$ of
NC $e^+p$ DIS data collected by the ZEUS experiment in 1994-97.
Monte Carlo simulation, event selection, kinematic reconstruction, 
and assessment of systematic effects is that of
the NC DIS analysis described in \cite{zeus_nc}. 
%
The event sample used in the CI analysis is limited to 
$0.04\!<\!x\!<\!0.95$, $0.04\!<\!y\!<\!0.95$ and $Q^{2}>500\gev^{2}$.


A cross-section increase at highest $Q^{2}$, corresponding to 
the direct ``new physics'' contribution, is expected for 
most CI scenarios, as shown in Figure \ref{fig-model}.
%
At intermediate $Q^{2}$ a moderate increase or decrease 
due to CI-SM interference terms is possible.
%
As the helicity structure of new interactions can be different 
from that of the Standard Model, also the differential cross-section
$d\sigma / dx$ (for fixed $Q^{2}$) is modified.
%
Sensitivity to many CI scenarios
is significantly improved by considering
the two-dimensional event distribution.


The ZEUS CI analysis compares the distributions of the measured kinematic 
variables with the corresponding distributions from a MC simulation 
of $e^+p\to e^+X$ events reweighted to simulate the CI scenarios.
%
An unbinned log--likelihood technique is used to calculate
$L(\epslam)$ from the individual kinematic event coordinates $(x_i,y_i)$:
%
\begin{equation}
  L(\epslam)=-\sum_{i\in{\rm data}}\log{p(x_i,y_i;\epslam)} \; ,
\end{equation}
%
where the sum runs over all events in the selected data sample and 
$p(x_i,y_i;\epslam)$ is the probability that an event $i$
observed at  $(x_i , y_i)$ results from the model described by coupling 
$\epslam$. $L$ tests the shape of the $(x,y)$--distribution but
is independent of its absolute normalisation.




The best estimates, $\Lambda^{\pm}_\circ$, for the different CI scenarios 
are given by the positions of the respective minima of $L(\epslam)$ for
$\epsilon$=$\pm 1$.
%
All results are consistent with the Standard Model,
the probability that the observed values of $\Lambda^{\pm}_{\circ}$
result from the Standard Model does not fall below 16\%.
%
The $95\%$ C.L. lower limits $\Lambda_{min}$ on the effective
mass scale $\Lambda$ are defined as the mass
scales for which MC experiments have a 95\% chance to result in 
$\Lambda_{\circ}$ values smaller than that observed in data.
%
The lower limits on $\Lambda$ ($\Lambda^\pm_{min}$ for $\epsilon$=$\pm 1$)
are summarized in Table \ref{tab-limits}.
The $\Lambda$ limits range from 1.7 to $5\tev$. 

\begin{figure}[t]
\epsfig{figure=w1_24_fig1.eps,width=7.5cm,clip=} 
\vspace{-1.0cm}
\caption{Examples of the relative influence of a CI on the $e^{+}p$ NC DIS 
cross-section $d\sigma/dQ^{2}$.}
\label{fig-model}
\end{figure}

% ----------------------------------------------------------------------------
%       Global analysis
% ----------------------------------------------------------------------------
\section{GLOBAL ANALYSIS}
\label{sec-global}

The global analysis \cite{my_ci} of $eeqq$ contact interactions
combines relevant data from different experiments:
%
ZEUS and H1 high-$Q^{2}$ NC DIS results;
%
Tevatron data on high-mass Drell-Yan lepton pair production;
%
LEP2 results on the hadronic cross-section 
            $\sigma ( e^{+}e^{-} \rightarrow q \bar{q} (\gamma) )$,
 the heavy quark production ratios $R_{b}$ and $R_{c}$, and the
 forward-backward asymmetries  $A^{b}_{FB}$, $A^{c}_{FB}$, $A^{uds}_{FB}$;
%
data from low-energy $eN$ and $\mu N$ scattering
and from atomic parity violation (APV) measurements.


For models assuming $SU(2)_{L} \times U(1)_{Y}$ universality,
additional constraints come from 
HERA $e^{+}p$  CC DIS results,
data on $\nu N$ scattering,
unitarity  of the CKM matrix and  electron-muon universality.


The combined data are consistent with the Standard Model predictions.
%
The mass scale limits  $\Lambda_{min}^{-}$ and $\Lambda_{min}^{+}$
obtained from fitting one-parameter models to all available data
are summarized in Table \ref{tab-limits}.
%
For models not assuming $SU(2)_{L} \times U(1)_{Y}$  universality
(only $e$/$\mu$ NC data used) the mass limits range
from  5.1 to   $11.7\tev$.
%
With $SU(2)_{L} \times U(1)_{Y}$  universality (using also $\nu N$ and CC data)
the limits extend up to about $18\tev$.

Limits for single couplings derived in multi-parameter models
(of Section \ref{sec-ci})
are weaker than in the case of one-parameter models, as no relation
between separate couplings is assumed.
%
The mass limits obtained for the general model range 
from 2.1 to $5.1\tev$.
%
All limits improve significantly  
and reach  3.5 to $7.8\tev$ 
for the SU(2) model with family universality.

Taking into account possible correlations between couplings,
any contact interaction with a mass scale below $2.1\tev$
($3.1\tev$ when SU(2) universality is assumed) is excluded at 95\% CL. 


% ----------------------------------------------------------------------------
%       Predictions
% ----------------------------------------------------------------------------
\section{PREDICTIONS}
\label{sec-pred}

Likelihood function for the possible cross-section deviations
from the Standard Model predictions is calculated as the weighted
average over all contributing CI coupling combinations \cite{my_ci}.
%
The results for HERA, in terms of the 95\% C.L. limit bands on the
ratio of predicted and the Standard Model cross-sections as a function
of $Q^{2}$, are shown  in Figure \ref{fig-predict},
for the general model and the SU(2) model with family universality.


The allowed increase in the integrated $e^{+}p$ NC DIS cross-section 
for $Q^{2} \; >$ 15,000 GeV$^{2}$
is about 40\% for the general model and about 30\% for the SU(2) model.
%
In order to reach the level of statistical precision
which would allow to confirm a possible discrepancy of this 
size, the HERA experiments would have to collect $e^{+}p$
integrated luminosities of the order of 100-200$\pbi$ 
(depending on the model). 
%
This will be possible after the HERA upgrade planned for year 2000.

\begin{figure}[t]
\epsfig{figure=w1_24_fig2.eps,width=7.5cm,clip=} 
\vspace{-1.0cm}
\caption{The 95\% CL limit band on the ratio of predicted
to the Standard Model cross-section for  $e^{+}p$ and $e^{-}p$ NC DIS
scattering at HERA.}
\label{fig-predict}
\end{figure}
%
Constraints on possible deviations from the Standard Model 
predictions are much stronger in the case of $e^{-}p$ NC DIS.
%
For the general model
deviations larger than about 20\% are excluded for $Q^{2} >$15,000 GeV$^{2}$,
whereas for the SU(2) model with family universality the limit goes down
to about 7\%. 
%
In such a case it would be very hard to detect contact interactions 
in future HERA $e^{-}p$ running.
%
However, for scattering with 60\% longitudinal $e^{-}_{R}$ polarisation, 
the maximum allowed deviations increase to 28\% and 19\%, respectively,
and significant effects could be observed already for 
integrated luminosities of the order of 120$\pbi$.

For the hadronic cross-section at LEP, for 
\mbox{$\sqrt{s}\!\sim\!200 \gev$},
the possible deviations from the Standard Model are
only about 8\%. 
%
However, significant deviations
are still possible for the heavy quark production ratios 
$R_{c}$ and $R_{b}$, and for the forward-backward asymmetries 
$A^{c}_{FB}$ and $A^{b}_{FB}$.
%
Significant cross-section deviations will be possible in 
the Next Linear Collider (NLC), for  $\sqrt{s}\! > \! 300 \gev$.
%
The largest cross-section deviations from the Standard Model predictions are 
still allowed at the Tevatron. For Drell-Yan lepton pair production,
deviations of the cross-section at  $M_{ll}$=500 GeV up to a factor of 5 
are still not excluded. 

\begin{figure}[t]
\epsfig{figure=w1_24_fig3.eps,width=7.5cm,clip=} 
\vspace{-1.0cm}
\caption{The 68\% and 95\% CL contours for the possible deviation from
the Standard Model predictions, for $e^{+}p$ NC DIS
cross-section at HERA, at $Q^{2}$ = 30,000 GeV$^{2}$, total
$e^{+}e^{-} \rightarrow q\bar{q}$ cross-section at $\sqrt{s}$ = 400 GeV
and Drell-Yan lepton pair production cross-section at
the Tevatron, at $M_{ll}$ = 500 GeV, as indicated on the plot. 
The limits are calculated using the SU(2) contact interaction
model with family universality.}
\label{fig-rel}
\end{figure}
%
In Figure \ref{fig-rel} relations between possible cross-section deviations
at HERA, NLC and the Tevatron are presented. There are no clear correlations
between different experiments.
%
All experiments should continue to analyse their data in terms of
possible new electron-quark interactions, as constraints resulting from
different processes are, to large extent, complementary.


\begin{thebibliography}{9}

\bibitem{zeus97}    % ZEUS high Q2 results from 94-96 data
The ZEUS Collaboration, J.Breitweg et al., \\ Z. Phys. C74 (1997) 207.

\bibitem{h1_97}     % H1 high Q2 results from 94-96 data
The H1 Collaboration, C.Adloff et al., \\ Z. Phys. C74 (1997) 191.

\bibitem{cidef}     % Applications of CI approach
R.R\"uckl, Phys. Lett. 129 B (1983) 363. \\
E.Eichten, K.Lane, M.Peskin, Phys. Rev. Lett. 50 (1983) 811.

\bibitem{cihera}     % Lagrangian for CI
P.Haberl, F.Schrempp, H.U.Martyn, \\
in {\it Proc. Workshop Physics at HERA},  \\
ed. W. Buchm\"uller and G.Ingelman, \\
Hamburg 1991, Vol. 2, pg. 1133.

\bibitem{ci_su2}    % SU(2)xU(1) in CI 
V. Barger, K. Cheung, K. Hagiwara, D. Zeppenfeld Phys. Rev. D57 (1998) 391.

\bibitem{zeus_ci}    % ZEUS final CI results 1994-97
The ZEUS Collaboration, J.Breitweg et al., \\ DESY 99-058,
% {\it Search for Contact Interactions in Deep-Inelastic 
% $e^{+}p \rightarrow e^{+}X$ Scattering at HERA,}
submitted to  Euro.Phys.J.~C.

\bibitem{zeus_nc}    % ZEUS final high-Q2 NC DIS results 1994-97
The ZEUS Collaboration, J.Breitweg et al., \\ DESY 99-056,
% {\it Measurement of High-$Q^{2}$ Neutral-Current $e^{+}p$ 
% Deep Inelastic Scattering Cross Sections at HERA,}
submitted to Euro.Phys.J.~C.
\\ R.Pawlak, 
{\it ZEUS High $Q^{2}$ $e^{+}p$ and $e^{-}p$ Cross Sections,} 
these proceedings.

\bibitem{my_ci}    % My CI results
A.F.\.Zarnecki, IFD-02/99,   \\
submitted to Euro.Phys.J.~C.

\end{thebibliography}

\end{document}


