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\pagestyle{empty}
\begin{titlepage}

\noindent
% {\bf H-UM \& JS: version of \today} \\[.3em] 
Submitted to the 30th International Conference on 
High-Energy Physics ICHEP2000, \\ 
Osaka, Japan, July 2000


\vspace*{3cm}

\begin{center}
  \Large
{\bf 
Measurement of Neutral and Charged Current \\
 Cross Sections in Electron-Proton Collisions \\
at High {\boldmath{$Q^2$}} at HERA }

\vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}

\begin{abstract}
\noindent
  The inclusive $e^- p$~single and double differential cross sections
  for neutral and charged current processes are measured with the H1
  detector at HERA. The cross sections are measured in the range of
  four-momentum transfer squared $Q^2$ between $200$ and $30\,000$
  GeV$^2$, and Bjorken $x$ between $0.0032$ and $0.65$. The data were
  taken with the H1 detector at HERA in 1998 and 1999 with a
  centre-of-mass energy of 320 GeV, and correspond to an integrated
  luminosity of $15.3 \ {\rm pb}^{-1}$. The data are compared with
  recent measurements of the inclusive neutral and charged current
  $e^+ p$ cross sections. Clear evidence is observed for neutral
  current parity violating $Z^0$ exchange and the structure function
  \Fz~ is extracted. The data are found to be in good agreement with
  Standard Model predictions.
\end{abstract}
\vfill
\begin{flushleft}
  {\bf Abstract: 971 } \\
  {\bf Parallel session: 11 } \\
  {\bf Plenary talk: 7~b } 
\end{flushleft}

\end{titlepage}

\pagestyle{plain}


\newpage

\section{Introduction}

\noindent
Inclusive Deep Inelastic Scattering (DIS) cross sections have long
been used as sensitive probes of proton structure and QCD dynamics.
Since 1992 the experiments H1 and ZEUS have used the colliding
lepton--proton beams of the HERA accelerator to further extend the
phase space of such measurements into new kinematic regions of large
four-momentum transfer squared, $Q^2$, and small $x$, where $x$ is the
Bjorken scaling variable. The large integrated luminosity collected by
the experiments has allowed measurements to be made in the very high
$Q^2$ range up to $30\,000$ GeV$^2$. In this region where $Q^2 \simeq
M_Z^2$ or $M_W^2$, the $Z^0$ and $W^{\pm}$ boson masses squared, the effects of the
electroweak sector of the Standard Model can be tested in DIS. In addition
signals of new physics beyond the Standard Model may be expected to
arise at the highest $Q^2$ where the smallest distance scales of proton
structure are probed.

Both contributions to DIS, neutral current (NC) interactions,
\mbox{$ep \rightarrow eX$}, and charged current (CC) interactions,
\mbox{$ep \rightarrow \nu X$} can be measured at HERA and give
complementary information on the QCD and electroweak parts of the
Standard Model. The cross sections are defined in terms of two of the
kinematic variables $Q^2$, $x$, and $y$, where $y$ quantifies the
inelasticity of the interaction.
%\begin{equation}
%Q^2 = -q^2\equiv(k-k')^2
%\hspace*{1.1cm}
%x  = \frac{Q^2} {2 p \cdot q}
%\hspace*{1.1cm}
%y = \frac{p \cdot q} {p \cdot k},
%\hspace*{1.1cm}
%s \equiv (p+k)^2 = Q^2/xy
%\end{equation}
%where $k (k^{'})$ and $p$ are the four-momenta of the incident
%(scattered) lepton and proton, and where the proton and lepton rest
%masses are neglected. 
%The quantity $-Q^2$ is the four-momentum
%transfer squared, $x$ is the Bjorken scaling variable, $y$ specifies
%the inelasticity, and $\sqrt{s}$ is the $ep$ centre-of-mass energy of
%the $ep$ interaction. 

%Measurements by the fixed target experiments
%span the $x$ range $0.01$ to $0.9$ and up to a maximum $Q^2$ of $250$
%\Gevv.  Previous measurements by the HERA experiments, H1 and ZEUS,
%extend to lower values of $x \approx\ 10^{-5}$ at low $Q^2$, and at
%higher $x$ to $Q^2$ of $5000$ \Gevv~\cite{H194,ZEUS94}. 

%The measurements presented in this paper extend the $Q^2$ range to
%$30\,000$ \Gevv.
%high four-momentum transfer
%squared, $Q^2$, and low Bjorken $x$. 

Preliminary measurements of the $Q^2$ dependence of the NC and CC
cross sections from $e^-p$ scattering have already been
presented\cite{wlad}. These results were based on an integrated
luminosity of $5.2 \ {\rm pb}^{-1}$. In this paper we report on the NC
and CC cross section measurements using the full $e^-p$ integrated
luminosity taken during the 1998/1999 running period. The data sample
consists of $15.3 \ {\rm pb}^{-1}$ obtained when HERA operated with
electron beams at 27.6 GeV colliding with proton beams at an increased
energy of 920 GeV, compared with 820 GeV in previous years, yielding a
centre-of-mass energy $\sqrt{s} \approx 320$ GeV. The data are
compared with NC and CC measurements~\cite{h1hiq2} from H1 of $e^+p$
scattering taken at a centre-of-mass energy $\sqrt{s} \approx 300$
GeV. Recently measurements of the NC and CC cross sections at high
\qsq have been reported by the ZEUS experiment~\cite{zeushiq2}.

%This paper is divided into 5 sections. In section 2 the data and
%event selection are described. The calibration procedure
%is given in section 3, and the cross section measurements and
%comparisons are presented in section 4. In section 5 the paper is
%summarised.



\section{Neutral and Charged Current Cross Sections}

The NC cross section for the process $e^{\pm}p\rightarrow e^{\pm}X$
with unpolarized beams is given by
%
\begin{eqnarray}
\label{Snc1}
\frac{{\rm d}^2\sigma_{NC}^{\pm}}{{\rm d}x\;{\rm d}\QQ}
& = & \frac{2\pi \alpha^2}{xQ^4}    
\left[Y_+ \Ftwo \mp Y_{-}x\Fz -y^2 \FL \right]\;\;\;, 
\end{eqnarray}
%where
%\begin{eqnarray}
%\label{Snc}
%\phi_{NC}^{\pm}(x,Q^2) & = & 
%Y_+ \Ftwo (x,\QQ ) \pm Y_{-}x\Fz (x,\QQ )-y^2 \FL (x,\QQ ) .
%\end{eqnarray}
%
where $\alpha$ is the fine structure constant taken to be $\alpha
\equiv \alpha(Q^2=0)$ and the helicity dependences of the electroweak
interactions are contained in the functions \ $Y_{\pm} \equiv 1 \pm
(1-y)^2$. The dominant contribution to the cross section comes from
the structure function \Ftwo. The \FL contribution is largest at high
$y$ and is expected to diminish with increasing $Q^2$, whereas the
structure function $x$\Fz\ contributes in the high $Q^2$ regime of
$Z^0$ exchange.  Note that for unpolarized beams, \Ftwo\ is the same
for electron and for positron scattering, while the $x$\Fz\ 
contribution changes sign as can be seen in eq.~\ref{Snc1}.

It is convenient to derive from the measured
${{\rm d}^2\sigma}/{{\rm d}x{\rm d}Q^2}$ the NC ``reduced cross section'' in which
part of the $Q^2$ dependence of ${{\rm d}^2 \sigma}/{{\rm d}x{\rm d}Q^2}$ due to
the photon propagator is removed
\begin{equation}
\label{Rnc}
\tilde{\sigma}_{NC}(x,Q^2) \equiv  \frac{1}{Y_+} \ 
\frac{ Q^4 \ x  }{2 \pi \alpha^2}
\          \frac{{\rm d}^2 \sigma_{NC}}{{\rm d}x{\rm d}Q^2}\;\;\;.
\end{equation}

The leading order double differential CC cross section 
for $e^-p \rightarrow \nu X$ can be written as
\begin{equation}
\frac{{\rm d}^2\sigma_{CC}}{{\rm d}x\;{\rm d}\QQ} =
 \frac{G_F^2}{2\pi x } \left(\frac{M_W^2}{M_W^2+Q^2} \right)^2 
x \left [ (u+c)+(1-y)^2(\bar{d}+\bar{s}) \right ]\;\;\;,
\label{Scc}
\end{equation}
where $G_F$ is the Fermi coupling constant, and $u$, $c$, $\bar{d}$,
$\bar{s}$ are the quark distributions. The description of $e^+p
\rightarrow \bar{\nu} X$ is obtained by changing all quarks with
anti-quarks (and vice-versa) in eq.~\ref{Scc}. The CC reduced 
cross section is defined as
\begin{equation}
\label{Rcc}
\tilde{\sigma}_{CC}(x,Q^2) \equiv  
\frac{2 \pi  x}{ G_F^2}
\left( \frac {M_W^2+Q^2} {M_W^2} \right)^2
          \frac{{\rm d}^2 \sigma_{CC}}{{\rm d}x{\rm d}Q^2}\;\;\;.
\end{equation}

\section{Experimental Technique}

%\subsection{Detector and Simulation}
\subsection{Detector and Kinematic Reconstruction}
The H1 detector is described in detail elsewhere~\cite{h1detector}.
The co-ordinate system of H1 defines the positive $z$ axis to be in
the direction of the incident proton beam. The polar angle $\theta$ is
then defined with respect to the positive $z$ axis. The forward
direction is the region of increasing $z$. 

In order to determine acceptance corrections and background
contributions for the DIS cross section measurements, the detector
response to events produced by various Monte Carlo generation programs
is simulated in detail, and is described in ~\cite{h1hiq2}.
% using a program based on GEANT~\cite{GEANT}.
%These simulated events are then subjected to the same reconstruction
%and analysis chain as the real data.

%DIS processes are generated using the DJANGO~\cite{django} program
%which is based on HERACLES \cite{heracles} for the electroweak
%interaction and on LEPTO~\cite{lepto}, using the colour dipole model
%as implemented in ARIADNE \cite{cdm} to generate the QCD dynamics.
%The JETSET program is used for the hadron fragmentation~\cite{jetset}.
%The simulated events are produced with the MRSH \cite{mrsh} parton
%distributions, reweighted to the H1 $e^+p$ QCD fit described in
%~\cite{h1hiq2} which then gives a good description of the data.

%The dominant background contribution to NC and CC processes is due to
%photoproduction events.  These are simulated using the
%PYTHIA~\cite{pythia} generator with GRV leading order parton
%distribution functions for the proton and photon~\cite{ggrv}.

The present analysis closely follows that of ref.~\cite{h1hiq2}, and
is briefly described below. The NC event kinematics are reconstructed
using the energy and polar angle of the scattered electron,
$E^{\prime}_e$ and $\theta_e$ respectively.  The total $E-p_z$ of the
event is then obtained from the relation $E{\rm-}p_z = \Sigma
+E^{\prime}_e(1-\cos{\theta_e})$. Here $\Sigma=\sum_i{(E_i-p_{z,i})}$,
where the summation is performed over all hadronic final state
particles.  These quantities are used to determine the kinematic
variables $x$, \qsq, and $y$ in the $e\Sigma$ method~\cite{esigma}:
\begin{equation}
  y_{e\Sigma} = 2 E_e \frac{\Sigma}{(E{\rm-}p_z)^2} 
  \hspace*{1.5cm}
  Q^2_{e\Sigma} = \frac{P_{T,e}^2}{ 1-y_e} 
  \hspace*{1.5cm}
  x_{e\Sigma} = \frac{P_{T,e}^2}{s \ y_{\Sigma} (1-y_{\Sigma})}\;\;\;,
  \label{kinematics1}
\end{equation}
where $E_e$ is the incident electron beam energy, and
\begin{equation}
  y_{\Sigma} = \frac{\Sigma} {E{\rm-}p_z}
  \hspace*{1cm}
  y_e =1- \frac {E_e^{'} (1-\cos{\theta_e})} {2 \ E_e}
  \hspace*{1cm}
  P_{T,e} = E^{\prime}_e \sin{\theta_e}\;\;\;.
\end{equation}

The resolution of the reconstruction method is controlled by requiring
the purity and stability of any ($x$, \qsq) bin to be larger than $30\%$.
The stability (purity) is defined as the fraction of events which
originate from a bin and which are reconstructed in it, divided by the
number of generated (reconstructed) events in that bin.  
%In addition
%the acceptance is required to be more than $20\%$ in any bin, where
%the acceptance is the ratio of events reconstructed in a bin to the
%number of events generated in that bin.

The CC event kinematics can only be determined with the hadron method
($h$ method)~\cite{jbmethod}. 
The $h$ method kinematic variables are reconstructed using the
relations
\begin{equation}
   y_{h} = \frac{\Sigma}{ 2 \ E_e }
   \hspace*{2cm}
   Q^2_{h} = \frac{P_{T,h}^2}{ 1-y_{h}}
   \hspace*{2cm}
    x_h=\frac{Q^2_h} {s \ y_h}\;\;\;,
\end{equation}
where $P_{T,h}=\sqrt{(\sum_i{p_{x,i}})^2+(\sum_i{p_{y,i}})^2}$ and is
summed over all particles of the hadronic final state. This method is
influenced by particle losses in the beam pipe and fluctuations of the
detector response to hadronic final state particles, and therefore has
moderate precision.

%\subsection{Detector Calibration}
%The electromagnetic calibration of the detector performed on the 1997
%high statistics data in~\cite{h1hiq2} is applied to the current data
%set. The same calibration procedure is then repeated for the
%individual calorimeter modules where there are sufficient statistics.
%The new calibration constants are found to be within $1\%$ of those
%determined in 1997. The systematic uncertainty of the absolute
%electromagnetic energy scale varies from 1\% in the backward region to
%3\% in the forward region of the calorimeter.

%The hadronic response of the detector is calibrated in a two step
%procedure after first applying the calibration of~\cite{h1hiq2}. In
%the first step the simulation is used to determine a $z$ dependant
%energy correction to high $E_t$ jets. The corrections show a marked
%$z$ dependence following the 

% Then jets with $P_T\geq 5$ GeV are used to determine
%$z$ dependant energy correction factors with the calibrated electron
%energy as the reference scale. Finally the hadronic $P_T$.....

\subsection{Measurement Procedure}

The electromagnetic and hadronic response of the detector is
calibrated using the analysis described in~\cite{h1hiq2}. The
procedure is found to give good agreement between data and simulation.
The systematic uncertainty of the absolute electromagnetic energy
scale varies from 1\% in the backward region to 3\% in the forward
region of the calorimeter. For the hadronic energy scale a
conservative uncertainty of 3\% is assigned.

High \qsq NC events are selected by requiring that the event has a
compact electromagnetic cluster, taken to be the scattered electron,
in addition to a vertex position within $\pm 35$ cm of its nominal
position. The cluster is validated by requiring that an extrapolated
track have a distance of closest approach to the cluster of less than
$12$ cm. This is only applied for $\theta_e\geq 40^{\circ}$, where
$\theta_e$ is the polar angle of the scattered electron. In this
analysis the polar angle is determined using the position of the
electromagnetic cluster and assigned a systematic uncertainty of $3$
mrad.  The energy of the cluster, $E_e^{\prime}$, is required to be
larger than $11$ GeV where the trigger is very efficient. The total
$E-P_z$ summed over all particles is required to be larger than $35$
GeV to reduce the photo-production background, and the influence of
QED radiative corrections to the measured cross sections.  Fiducial
cuts are also made to remove local regions where the electromagnetic
shower of the scattered electron is not fully contained in the
calorimeter, and where the trigger is not fully efficient. Since the
signal to background ratio is reduced in the region of high $y$, a
further kinematic cut is made requiring $y_e \leq 0.9$. The remaining
photo-production contribution to the NC cross sections is never more
than 5\% at the highest $y$ and negligible elsewhere. The double
differential NC reduced cross section data have a statistical
precision of $\approx$ 4\% at low \qsq compared with a total systematic
uncertainty of $\approx$ 7\% rising to 15\% at high $x$. The total
uncertainty is dominated by the statistical error for $Q^2>1\,000$
GeV$^2$. The final sample of selected data consists of about $40\,000$
events. The comparison of the data and the simulation is shown in
fig.\ref{nc_cont} for the scattered electron energy spectrum, and the
polar angle $\theta_e$.  Both distributions are well described by the
simulation.

The selection of CC events is based on the expectation that such
events have a large missing transverse momentum, $P_{T,h}$, assumed to
be carried by an unseen neutrino. Therefore a requirement that
$P_{T,h}\geq 12$~GeV is made. In addition the event must have a
reconstructed vertex within $\pm 35$ cm of its nominal position. The
non-$ep$ and $ep$ background is rejected using the same method as
detailed in~\cite{h1hiq2}. The remaining $ep$ background is dominantly
due to photo-production events and is negligible for most of the
measured kinematic domain, though it may reach up to 3\% at
$Q^2=500\,{\rm GeV}^2$.  The contribution is subtracted statistically
from the CC data sample with a systematic uncertainty of 30\% of the
subtracted events. In order to restrict the measurement to a region
where the kinematic reconstruction is optimal the events are required
to have $y_h<0.85$. The CC trigger efficiency is determined using NC
events in which all information associated to the scattered electron
is suppressed. This method gives a precise measure of
the efficiency which is found to be $78\%$ at $Q^2=500\,{\rm GeV}^2$
and reaches 99\% at $Q^2=5000\,{\rm GeV}^2$.  The measurement is
restricted to the region where the trigger efficiency is acceptable by
demanding $y_h>0.03$. The systematic uncertainty assigned to the CC
trigger efficiency varies from 2\% at low $Q^2=500$ GeV$^2$ to 1\% at
high $Q^2$. Of the systematic sources of uncertainty studied the most
significant contribution to the measured CC cross sections comes from
the 3\% uncertainty on the hadronic energy scale. This results
in an uncertainty of 4\% at low \qsq and 15\% at high \qsq compared with
a total systematic error of 6.5\% at low \qsq, and 16\% at high \qsq .
This is comparable to the statistical precision which varies from 11\%
to 19\% over the same \qsq range. After all selection criteria are
applied, the final CC data sample contains about 700 events. The data
and simulation are compared in fig.\ref{cc_cont} for the $P_{T,h}$ and
$y_h$ spectra. In both cases the simulation gives a good description
of the data.

The selected events are corrected for detector acceptance, and
migrations using the simulation and converted to bin centered
cross sections using the prediction from the H1 $e^+p$ QCD
fit~\cite{h1hiq2}. The cross sections are corrected for the effects of
QED radiation.

\section{Results}

The single differential cross sections ${\rm d}\sigma/\rm{ d}Q^2$ for NC and CC
$e^-p$ data are shown in fig.\ref{dsdq2} for $y\leq 0.9$.
\footnote{These measured NC cross sections are corrected for the small
  effect of the cut $E_e^{\prime}>11$ GeV. Similarly the CC
  cross sections are corrected for the cuts $0.03<y<0.85$} The data
are compared with the H1 $e^+p$ measurements and the NLO QCD fit
performed on low $Q^2$ fixed target data from NMC and BCDMS and H1
$e^+p$ data called ``H1 $e^+p$ QCD fit'' in the following.  The new
$e^-p$ NC and CC measurements presented here were not included in the
QCD fit. The lower figures show the ratio of the preliminary
measurements to the Standard Model expectation. The global 3\%
luminosity uncertainty of the new data is not included in the error
bars.  The Standard Model uncertainty represents the uncertainty of
the expectation due to assumptions made in the H1 $e^+p$ QCD fit, as
well as the uncertainties of the experimental data entering the fit,
and is detailed in~\cite{h1hiq2}. 

The data are found to be in good agreement with the expectation. For
$Q^2>3\,000$ GeV$^2$ the NC $e^-p$ cross section is observed to be
systematically larger than the $e^+p$ cross section. The influence of
the increased centre-of-mass energy is indicated by the dashed line
and predicts an increased cross section of $\approx 7\%$ for
$Q^2<1\,000$ GeV$^2$ rising to 50\% at $Q^2=30\,000$ GeV$^2$. However,
at high $Q^2$ this is approximately an order of magnitude smaller than
the increase expected from the different lepton polarity. The
cross sections are consistent with the Standard Model expectation of the
parity violating contribution of $Z^0$ exchange.

The CC cross section $Q^2$ dependence is shown in fig.\ref{dsdq2}b and
compared with the $e^+p$ measurements.  The $e^-p$ data are found to have a
larger cross section everywhere, by up to a factor of ten at
$Q^2=15\,000$GeV$^2$. The effect of the increased centre-of-mass
energy is expected to be relatively small, and is shown as the dashed
curve in fig.\ref{dsdq2}b. The CC electron scattering cross section is
in good agreement with the Standard Model expectation based on the H1
$e^+p$ QCD fit. The ratio of data to expectation is shown in
fig.\ref{dsdq2}d together with the Standard Model Uncertainty.

The integrated $x$ dependence of the NC cross sections are shown in
fig.\ref{nc_dsdx} for both $e^-p$ and $e^+p$ scattering. The data are
shown for $Q^2>1\,000$ GeV$^2$ and $y<0.9$ in fig.\ref{nc_dsdx}a,b and are in
agreement with the expectation from the Standard Model. The larger
centre-of-mass energy for the $e^-p$ scattering data yields a larger
cross section by $\simeq 25\%$ at low $x$.  The cross sections are
also measured for $Q^2>10\,000$ GeV$^2$, $y<0.9$ (fig.\ref{nc_dsdx}c,d) and
are approximately a factor of four larger than for $e^+p$
scattering at $x \simeq 0.2$ of which 12\% is due to the different
centre-of-mass energies. The dashed curves show the Standard Model
expectation under the assumption of pure photon exchange. For electron
scattering the $Z^0$ exchange contribution enhances the cross section,
whilst in $e^+p$ scattering it reduces the cross section.

The integrated CC cross section ${\rm d}\sigma/{\rm d}x$ is measured for
$Q^2>1\,000$ GeV$^2$, and $y<0.9$ and is shown in fig.\ref{cc_dsdx}
compared with the same measurement in $e^+p$ scattering. The new data
are found to be in good agreement with the expectation. The
cross sections are larger than for positron scattering by a factor of
two at low $x$ and a factor of four at high $x$. The influence of the
centre-of-mass energy accounts for a small part of the increased
cross section, the remainder being due to the different quarks
which couple to the $W^{\pm}$ in $e^{\pm}p$ scattering.

The double differential NC reduced cross section is shown in
fig.\ref{nc_stamp} over the full $x$ and $Q^2$ range of the
measurement which reaches $x=0.65$ and $Q^2=30\,000$ GeV$^2$. The data
are compared with the H1 $e^+p$ QCD fit, and are found to give a good
prediction of the $x$, $Q^2$ behaviour of the data. In
fig.\ref{nc_hix} the data at high $x$ are compared with the measurements
of $e^+p$ scattering as a function of $Q^2$. They are compared with the
cross sections determined from the H1 $e^+p$ QCD fit. The data are
found to be in agreement with the $e^+p$ measurements for $Q^2<2\,000$
GeV$^2$. 


At large $Q^2$, and at lower $x$, the two data-sets differ as
expected from $x$\Fz  contributions to the NC cross sections.
This structure function is extracted using the equation
\begin{eqnarray}
\tilde{\sigma}^{-}_{NC} - \tilde{\sigma}^{+}_{NC} & = &
x\Fz \left[ 
\frac{ Y_{-~920}} {Y_{+~920}} + \frac{Y_{-~820}} {Y_{+~820}} 
\right]
- \FL \left[ \frac{y^2_{920}}{Y_{+~920}} + \frac{y^2_{820}}{Y_{+~820}} 
\right]\;\;\;, 
\end{eqnarray}

where the functions $Y_{\pm~920}$ and $Y_{\pm~820}$ are the usual
electroweak helicity functions evaluated for the given proton beam
energy. Since the cross section measurements are made for fixed $x$
and $Q^2$, any change in centre-of-mass energy translates into a
change in $y$ and hence affects the helicity functions. For the
determination of $x$\Fz~the small contribution of \FL~is taken from
the QCD fit. In order to optimise the sensitivity to $x$\Fz, both the
$e^+p$ and the $e^-p$ reduced cross sections are rebinned into two $Q^2$ bins 
 with centres $Q^2=2\,500$ GeV$^2$ and $Q^2=10\,000$ GeV$^2$. The
reduced cross sections  and the resulting 
measurement of $x$\Fz~ are shown in fig.\ref{xf3} 
compared with the H1 $e^+p$ QCD fit. The systematic errors for each reduced 
cross section are evaluated for these bins. 
The data are in good agreement with the
expectation. The data are observed to peak at $x\simeq 0.1$ reflecting
the valence structure of the constituent quarks.

The double differential CC reduced cross section is shown in
fig.\ref{cc_stamp} compared with the measurements from $e^+p$
scattering. The data are well described by the H1 $e^+p$ QCD fit. The
cross sections measured in $e^-p$ scattering are found to be
consistently larger than for $e^+p$ scattering everywhere. The
difference in the cross sections is found to increase with $Q^2$. This is
expected within the Standard Model and is due to the different quark
densities probed in $e^+p$ and $e^-p$ scattering.
  
\section{Summary}
The NC and CC cross sections have been measured for $e^-p$ scattering
at a centre-of-mass energy of $\sqrt{s} \approx 320$ GeV. Standard
Model expectations based on the H1 $e^+p$ QCD fit to NMC, BCDMS, and
H1 $e^+p$ data~\cite{h1hiq2} are able to provide a good description of
all the measured cross sections.

Comparisons of these cross sections with the H1 measurements of
NC and CC cross sections in $e^+p$ scattering are made.
The NC $e^-p$ measurement of ${\rm d}\sigma/{\rm d}Q^2$ shows a clear
increase with respect to the positron scattering cross sections at
high $Q^2$, consistent with the expectation of the contribution of
$Z^0$ exchange.  The CC cross section is observed to be larger for
electron scattering than for positron scattering by up to a factor of
ten at high \qsq .  The influence of larger centre-of-mass energy is
responsible for only a small part of the cross section increase.  The
NC and CC cross sections ${\rm d}\sigma/{\rm d}x$ are presented for
\qsq $>1\,000$ GeV$^2$ and $y<0.9$. In addition the NC cross section
is also measured for \qsq $>10\,000$ GeV$^2$ and found at low $x$ to
be approximately a factor of four larger than for $e^+p$ scattering.

The double differential NC reduced cross sections are measured in the
\qsq range $200 \leq Q^2 \leq 30\,000$ GeV$^2$, and $0.0032 \leq x
\leq 0.65$. The data agree well with measurements for $e^+p$
scattering for $Q^2<2\,000$ GeV$^2$. At higher $Q^2$ the expected
enhancement of the cross section due to $Z^0$ exchange is observed and
the structure function $x$\Fz~ is measured in the range $0.032<x<0.65$
and $2\,500 \leq Q^2 \leq 10\,000$ GeV$^2$. The double differential CC reduced
cross section is presented for the range $500 \leq Q^2 \leq 15\,000$
GeV$^2$, and $0.013 \leq x \leq 0.4$. The data are found to be larger
than for positron scattering as predicted by the Standard Model.


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

\begin{thebibliography}{99}
\bibitem{wlad} 
V. Chekelian, Rencontres de Moriond on Electroweak
  Interactions and Unified Theories, March 1999, Les Arcs, France.


\bibitem{h1hiq2} H1 Collab., `Measurement of Neutral and Charged
  Current Cross Sections in Positron-Proton Collisions at Large
  Momentum Transfer at HERA',
 paper 157ai contributed to 
International Europhysics Conference on High Energy Physics, 15-21 July
1999, Tampere, Finland HEP99

%H1 Collab., C. Adloff et al., Contri
%To be submitted to European Physical Journal
%\Journal{\PLB}{428}{1998}{206}

\bibitem{zeushiq2} 
ZEUS Collab., J.Breitweg et al. , DESY 99-056 (1999), {\em accepted by Eur. Phys. J.}\\
ZEUS Collab., J.Breitweg et al. , DESY 99-057 (1999), {\em submitted to Eur. Phys. J.}

\bibitem{h1detector}
H1 Collab., I.~Abt et al., \Journal{\NIMA}{386}{1997}{310 and 348}.

\bibitem{esigma} 
  U. Bassler and G. Bernardi, Nucl. Instr. Meth. {\bf A361} (1995) 197; \\
  U. Bassler and G. Bernardi,  Nucl. Instr. Meth. {\bf A426} (1999) 583.

\bibitem{jbmethod}
A.~Blondel and F.~Jacquet, Proceedings of the Study of an $ep$ Facility for
Europe, ed. U.~Amaldi, DESY 79/48 (1979) 391.

\end{thebibliography}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]   
\begin{center} 
\begin{picture}(160,70)(0,0)
\setlength{\unitlength}{1 mm}
\put( 0, 30){\epsfig{file=H1prelim-00-151.fig1a.eps,width=15cm}}
\put(48,-20){\epsfig{file=H1prelim-00-151.fig1b.eps,width=9cm}}
\put(20,60){\bf (a)}
\put(90,60){\bf (b)}
\end{picture}
\end{center}    
\caption{Distributions of (a) $E_e^{\prime}$ and (b) $\theta_e$ for NC data 
  (solid points) and simulation (solid line). The lower histograms
  show the photo-production contribution. The inset in (b) shows an
  enlargement of the low $\theta_e$ distribution.}
\label{nc_cont}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]   
\begin{center} 
\begin{picture}(160,70)(0,0)
\setlength{\unitlength}{1 mm}
\put(0,-10){\epsfig{file=H1prelim-00-151.fig2.eps,width=15cm}}
\put( 62,62){\bf (a)}
\put(132,62){\bf (b)}
\end{picture}
\end{center}    
\caption{Distributions of (a) $P_{T,h}$ and (b) $y_h$ for CC data 
  (solid points) and simulation (solid line). The lower histograms
  show the photo-production contribution.}
\label{cc_cont}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h*]
\setlength{\unitlength}{1 mm}
\begin{center}
\begin{picture}(160,120)(0,0)

\put(-10, 50){\epsfig{file=H1prelim-00-151.fig3a.eps,width=9.5cm}}
%bbllx=30pt,bblly=200pt,bburx=550pt,bbury=630pt}}
\put(-10, -5){\epsfig{file=H1prelim-00-151.fig3b.eps,width=10.0cm}}
\put(7.,22){\epsfig{file=H1prelim-00-151.fig3c1.eps,width=6.0cm}}
\put(40,38){$\frac{\delta {\cal L}}{{\cal L}}=3 \%$}

\put(80,50){\epsfig{file=H1prelim-00-151.fig3c2.eps,width=9.5cm}}
\put(80, -5){\epsfig{file=H1prelim-00-151.fig3d.eps,width=10.0cm}}
\put(135,38){$\frac{\delta {\cal L}}{{\cal L}}=3 \%$}
\put( 10,75){\bf (a)}
\put(100,75){\bf (b)}
\put( 10,15){\bf (c)}
\put(100,15){\bf (d)}
\end{picture}
\end{center}
\caption{The $Q^2$ dependence of the NC (a) and CC (b) cross sections $d\sigma/dQ^2$ are shown for the preliminary $e^-p$ (solid points) and $e^+p$ (open points) measurements. The data are compared with the Standard Model expectation determined from the H1 $e^+p$ QCD fit including the H1 $e^+p$ data. The influence of the increased centre-of-mass energy is shown as the dashed curve. The ratio of the $e^-p$ data to the Standard Model expectation is shown in figures (c) and (d) for NC and CC respectively. The Standard Model Uncertainty is shown as the shaded band. The 3\% luminosity uncertainy is not included in the error bars.}
\label{dsdq2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
\setlength{\unitlength}{1 mm}
\begin{center} 
\begin{picture}(160,120)(0,0)
\put( 10, 0){\epsfig{file=H1prelim-00-151.fig4.eps,width=15cm}}
\put( 35,85){\bf (a)}
\put( 85,85){\bf (b)}
\put( 35,50){\bf (c)}
\put( 85,50){\bf (d)}
\end{picture}
\end{center}
\caption{The NC cross sections $d\sigma/dx$ for the preliminary $e^-p$ data are shown in (a) for $Q^2>1\,000$ GeV$^2$ and in (c) for $Q^2>10\,000$ GeV$^2$. The H1 $e^+p$ cross sections are shown in (b) and (d) for $Q^2>1\,000$ GeV$^2$ and $10\,000$ GeV$^2$ respectively. The solid curves show the Standard Model expectation based on the H1 $e^+p$ QCD fit. The dashed curves show the contribution of photon exchange only. All cross sections are shown for $y<0.9$.}
\label{nc_dsdx} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
  \center \epsfig{file=H1prelim-00-151.fig5.eps,width=15cm}
\caption{The CC cross section $d\sigma/dx$ for $Q^2>1\,000$ GeV$^2$ and $y<0.9$ is shown for the preliminary H1 $e^-p$ data (solid points) and the H1 $e^+p$ data (open points). The solid curves show the Standard Model expectation based on the H1 $e^+p$ QCD fit. The dashed curve shows the influence of the increased centre-of-mass energy.}
\label{cc_dsdx} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
\center
\epsfig{file=H1prelim-00-151.fig6.eps,width=15cm}
\caption{The NC $e^-p$ reduced cross section $\tilde{\sigma}_{NC}(x,Q^2)$ is compared with the H1 $e^+p$ QCD fit. The inner error bars represent the statistical error, and the outer error bars show the total error.}
\label{nc_stamp} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
\center \epsfig{file=H1prelim-00-151.fig7.eps,width=15cm}
\caption{The NC reduced cross section $\tilde{\sigma}_{NC}(x,Q^2)$ is shown at high $x$ compared with the H1 $e^+p$ QCD fit. The preliminary $e^-p$ data with $\sqrt{s} \approx 320$ GeV (solid points) are compared with the H1 $e^+p$ data at $\sqrt{s} \approx 300$ GeV (open points). The solid curves represent the Standard Model expectation based on the H1 $e^+p$ QCD fit. The inner error bars represent the statistical error, and the outer error bars show the total error.}
\label{nc_hix} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
\center \epsfig{file=H1prelim-00-151.fig8.eps,width=15cm}
\caption{The NC reduced cross section $\tilde{\sigma}_{NC}(x,Q^2)$ is shown  
compared with the H1 $e^+p$ QCD fit at (a) $Q^2=2\,500$ GeV$^2$ and (b) 
$Q^2=10\,000$ GeV$^2$. The preliminary 
$e^-p$ data with $\sqrt{s} \approx 320$ GeV 
(solid points) are compared with the H1 $e^+p$ data at $\sqrt{s} \approx 300$ GeV (open points). 
The solid curves represent the Standard Model expectation based on the H1 $e^+p$ QCD fit. 
The parity violating structure function $x$\Fz~ is compared with
the H1 $e^+p$ QCD fit at (c) $Q^2=2\,500$ GeV$^2$ and 
(d) $Q^2=10\,000$ GeV$^2$. 
In all figures the inner error bars represent the statistical
error, and the outer error bars show the total error.}
\label{xf3}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
\center \epsfig{file=H1prelim-00-151.fig9.eps,width=15cm}
\caption{The CC reduced cross section $\tilde{\sigma}_{CC}(x,Q^2)$
  is shown for $e^-p$ scattering at $\sqrt{s} \approx 300$ GeV (solid
  squares) and $e^+p$ scattering at with $\sqrt{s} \approx 320$ GeV
  (open squares). The data are compared with the H1 $e^+p$ QCD fit. The
  inner error bars represent the statistical error, and the outer
  error bars show the total error. }  \label{cc_stamp}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

