%================================================================
% LaTeX file with preferred layout for the contributed papers to
% the ICHEP Conference 98 in Vancouver
% process with:  latex hep98.tex
%                dvips -D600 hep98
%================================================================
\documentclass[12pt]{article}
\usepackage{epsfig}
\usepackage{amsmath}
\usepackage{hhline}
\usepackage{amssymb}
\usepackage{times}

\renewcommand{\topfraction}{1.0}
\renewcommand{\bottomfraction}{1.0}
\renewcommand{\textfraction}{0.0}
\newlength{\dinwidth}
\newlength{\dinmargin}
\setlength{\dinwidth}{21.0cm}
\textheight23.5cm \textwidth16.0cm
\setlength{\dinmargin}{\dinwidth}
\setlength{\unitlength}{1mm}
\addtolength{\dinmargin}{-\textwidth}
\setlength{\dinmargin}{0.5\dinmargin}
\oddsidemargin -1.0in
\addtolength{\oddsidemargin}{\dinmargin}
\setlength{\evensidemargin}{\oddsidemargin}
\setlength{\marginparwidth}{0.9\dinmargin}
\setlength{\parindent}{0.0mm}
\marginparsep 8pt \marginparpush 5pt
\topmargin -42pt
\headheight 12pt
\headsep 30pt \footskip 24pt
\parskip 3mm plus 2mm minus 2mm

%===============================title page=============================

% Some useful tex commands
%
\newcommand{\GeV}{\rm GeV}
\newcommand{\TeV}{\rm TeV}
\newcommand{\pb}{\rm pb}
\newcommand{\cm}{\rm cm}
\newcommand{\hdick}{\noalign{\hrule height1.4pt}}
% The rest
\newcommand{\pom}{{I\!\!P}}
\newcommand{\reg}{{I\!\!R}}
\newcommand{\slowpi}{\pi_{\mathit{slow}}}
%\newcommand{\gevsq}{\mathrm{GeV}^2}
\newcommand{\fiidiii}{F_2^{D(3)}}
\newcommand{\fiidiiiarg}{\fiidiii\,(\beta,\,Q^2,\,x)}
\newcommand{\n}{1.19\pm 0.06 (stat.) \pm0.07 (syst.)}
\newcommand{\nz}{1.30\pm 0.08 (stat.)^{+0.08}_{-0.14} (syst.)}
\newcommand{\fiidiiiful}{F_2^{D(4)}\,(\beta,\,Q^2,\,x,\,t)}
\newcommand{\fiipom}{\tilde F_2^D}
\newcommand{\ALPHA}{1.10\pm0.03 (stat.) \pm0.04 (syst.)}
\newcommand{\ALPHAZ}{1.15\pm0.04 (stat.)^{+0.04}_{-0.07} (syst.)}
\newcommand{\fiipomarg}{\fiipom\,(\beta,\,Q^2)}
\newcommand{\pomflux}{f_{\pom / p}}
\newcommand{\nxpom}{1.19\pm 0.06 (stat.) \pm0.07 (syst.)}
\newcommand {\gapprox}
   {\raisebox{-0.7ex}{$\stackrel {\textstyle>}{\sim}$}}
\newcommand {\lapprox}
   {\raisebox{-0.7ex}{$\stackrel {\textstyle<}{\sim}$}}
\def\gsim{\,\lower.25ex\hbox{$\scriptstyle\sim$}\kern-1.30ex%
\raise 0.55ex\hbox{$\scriptstyle >$}\,}
\def\lsim{\,\lower.25ex\hbox{$\scriptstyle\sim$}\kern-1.30ex%
\raise 0.55ex\hbox{$\scriptstyle <$}\,}
\newcommand{\pomfluxarg}{f_{\pom / p}\,(x_\pom)}
\newcommand{\dsf}{\mbox{$F_2^{D(3)}$}}
\newcommand{\dsfva}{\mbox{$F_2^{D(3)}(\beta,Q^2,x_{I\!\!P})$}}
\newcommand{\dsfvb}{\mbox{$F_2^{D(3)}(\beta,Q^2,x)$}}
\newcommand{\dsfpom}{$F_2^{I\!\!P}$}
\newcommand{\gap}{\stackrel{>}{\sim}}
\newcommand{\lap}{\stackrel{<}{\sim}}
\newcommand{\fem}{$F_2^{em}$}
\newcommand{\tsnmp}{$\tilde{\sigma}_{NC}(e^{\mp})$}
\newcommand{\tsnm}{$\tilde{\sigma}_{NC}(e^-)$}
\newcommand{\tsnp}{$\tilde{\sigma}_{NC}(e^+)$}
\newcommand{\st}{$\star$}
\newcommand{\sst}{$\star \star$}
\newcommand{\ssst}{$\star \star \star$}
\newcommand{\sssst}{$\star \star \star \star$}
\newcommand{\tw}{\theta_W}
\newcommand{\sw}{\sin{\theta_W}}
\newcommand{\cw}{\cos{\theta_W}}
\newcommand{\sww}{\sin^2{\theta_W}}
\newcommand{\cww}{\cos^2{\theta_W}}
\newcommand{\trm}{m_{\perp}}
\newcommand{\trp}{p_{\perp}}
\newcommand{\trmm}{m_{\perp}^2}
\newcommand{\trpp}{p_{\perp}^2}
\newcommand{\alp}{\alpha_s}

\newcommand{\alps}{\alpha_s}
\newcommand{\sqrts}{$\sqrt{s}$}
\newcommand{\LO}{$O(\alpha_s^0)$}
\newcommand{\Oa}{$O(\alpha_s)$}
\newcommand{\Oaa}{$O(\alpha_s^2)$}
\newcommand{\PT}{p_{\perp}}
\newcommand{\JPSI}{J/\psi}
\newcommand{\sh}{\hat{s}}
%\newcommand{\th}{\hat{t}}
\newcommand{\uh}{\hat{u}}
\newcommand{\MP}{m_{J/\psi}}
%\newcommand{\PO}{\mbox{l}\!\mbox{P}}
\newcommand{\PO}{I\!\!P}
\newcommand{\xbj}{x}
\newcommand{\xpom}{x_{\PO}}
\newcommand{\ttbs}{\char'134}
\newcommand{\xpomlo}{3\times10^{-4}}  
\newcommand{\xpomup}{0.05}  
\newcommand{\dgr}{^\circ}
\newcommand{\pbarnt}{\,\mbox{{\rm pb$^{-1}$}}}
\newcommand{\gev}{\,\mbox{GeV}}
\newcommand{\WBoson}{\mbox{$W$}}
\newcommand{\fbarn}{\,\mbox{{\rm fb}}}
\newcommand{\fbarnt}{\,\mbox{{\rm fb$^{-1}$}}}


\newcommand{\vtab}{\rule[-1mm]{0mm}{4mm}}
\newcommand{\htab}{\rule[-1mm]{0mm}{6mm}}
\newcommand{\photoproduction}{$\gamma p$}
\newcommand{\ptmiss}{$P_{T}^{\rm miss}$}
\newcommand{\epz} {$E{\rm-}p_z$}
\newcommand{\vap} {  $V_{ap}/V_p$}
\newcommand{\Zero}   {\mbox{$Z^{\circ}$}}
\newcommand{\Ftwo}   {\mbox{$\tilde{F}_2$}}
\newcommand{\Ftwoz}   {\mbox{$\tilde{F}_{2,3}$}}
\newcommand{\Fz}   {\mbox{$\tilde{F}_3$}}
\newcommand{\FL}   {\mbox{$\tilde{F}_{_{L}}$}}
\newcommand{\Wtwo}   {\mbox{$W_2$}}
\newcommand{\Wz}   {\mbox{$W_3$}}
\newcommand{\WL}   {\mbox{$W_{_{L}}$}}
\newcommand{\Fem}  {\mbox{$F_2$}}
\newcommand{\Fgam}  {\mbox{$F_2^{\gamma}$}}
\newcommand{\Fint} {\mbox{$F_2^{\gamma Z}$}}
\newcommand{\Fwk}  {\mbox{$F_2^{Z}$}}
\newcommand{\Ftwos} {\mbox{$F_2^{\gamma Z, Z}$}}
\newcommand{\Fzz} {\mbox{$F_3^{\gamma Z, Z}$}}
\newcommand{\Fintz} {\mbox{$F_{2,3}^{\gamma Z}$}}
\newcommand{\Fwkz}  {\mbox{$F_{2,3}^{Z}$}}
\newcommand{\Fzint} {\mbox{$F_3^{\gamma Z}$}}
\newcommand{\Fzwk}  {\mbox{$F_3^{Z}$}}
\newcommand{\Gev}  {\mbox{${\rm GeV}$}}
\newcommand{\Gevv}{\mbox{${\rm GeV}^2$}}
\newcommand{\QQ}  {\mbox{${Q^2}$}}
%\newcommand{\lapprox}{\stackrel{<}{_{\sim}}}
%\newcommand{\gapprox}{\stackrel{>}{_{\sim}}}


%
% Some useful tex commands
%
\newcommand{\qsq}{\ensuremath{Q^2} }
\newcommand{\gevsq}{\ensuremath{\mathrm{GeV}^2} }
\newcommand{\et}{\ensuremath{E_t^*} }
\newcommand{\rap}{\ensuremath{\eta^*} }
\newcommand{\gp}{\ensuremath{\gamma^*}p }
\newcommand{\dsiget}{\ensuremath{{\rm d}\sigma_{ep}/{\rm d}E_t^*} }
\newcommand{\dsigrap}{\ensuremath{{\rm d}\sigma_{ep}/{\rm d}\eta^*} }
% Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2} (#3) #4}
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} {\bf A}}
\def\NPB{{\em Nucl. Phys.}   {\bf B}}
\def\PLB{{\em Phys. Lett.}   {\bf B}}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.}    {\bf D}}
\def\ZPC{{\em Z. Phys.}      {\bf C}}
\def\EJC{{\em Eur. Phys. J.} {\bf C}}
\def\CPC{\em Comp. Phys. Commun.}

\begin{document}

\pagestyle{empty}
\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
\begin{tabular}{llrr}
%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                International Europhysics
                Conference on High Energy Physics},
                July~12,~2001,~Budapest} \\
 {\bf EPS 2001:} 
                 & Abstract:        & {\bf 787,803}    &\\
                 & Parallel Session & {\bf Hard High Energy QCD and Structure Functions}   &\\
%                 & Plenary Session  & {\bf --}   &\\
[.7em]
\multicolumn{4}{l}{{\bf
               XX International Symposium on Lepton and Photon Interactions}, 
               July~23,~2001,~Rome} \\ 
{\bf LP 2001:}  
                 & Abstract:        & {\bf 481,496} &\\
                 & Parallel Session & {\bf 8}   &\\
%                 & Plenary Session  & {\bf --}   &\\ 
\hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}

\begin{center}
  \Large
  {\bf 
    Inclusive Measurement of Deep Inelastic Scattering at high
    {\boldmath $Q^2$} in {\boldmath $ep$} Collisions 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 data were taken in 1999 and 2000 at a
  centre-of-mass energy of $\sqrt{s}=320$ GeV and correspond to an
  integrated luminosity of $45.9 \ {\rm pb}^{-1}$.  The cross-sections
  are measured in the range of four-momentum transfer squared $Q^2$
  between $150$ and $30\,000$ GeV$^2$, and Bjorken $x$ between
  $0.0032$ and $0.65$.  These are compared to previous H1 results
  based on $35.6 \pb^{-1}$ of data taken in 1994 to 1997 at $\sqrt{s}
  = 300$ GeV.  The new measurements are found to be fully consistent
  with previous ones and well described by next-to-leading order QCD
  fits in the framework of the Standard Model.  In addition the
  neutral current analysis is
  extended to small energies of the scattered electron and therefore
  to values of Bjorken $y$ upto 0.9 allowing a determination of the
  longitudinal structure function $F_L$ for the first time at high
  $Q^2$. The analysis is performed on both the $e^+p$ data, and 
  $16.4~{\rm pb}^{-1}$ of $e^-p$ taken in 1998 and 1999 and $F_L$ is found 
  to be independent of lepton beam.
\end{abstract}
\end{titlepage}

\pagestyle{plain}

\section{Introduction}

Inclusive Deep Inelastic Scattering (DIS) cross-sections are sensitive
to the proton structure and QCD dynamics.  Since 1992 the experiments
H1 and ZEUS have used the colliding lepton--proton beams of the HERA
accelerator to extend considerably 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 $x$. 

Measurements of the NC and CC cross-sections in $e^+p$ scattering have
been made by H1 and ZEUS based on the data taken in 1994 to 1997
\cite{h1hiq2,zeushiq2} when the $ep$ centre-of-mass energy $\sqrt{s}$
was about $300$ GeV. In 1998 and 1999 HERA operated with an electron
beam, and the proton beam energy was increased from $820$~GeV to
$920$~GeV ($\sqrt{s} \simeq 320$GeV). H1 collected 16pb$^{-1}$ of
$e^-p$ data which was analysed and used together with the $e^+p$ data
to extract the structure function $x$\Fz~\cite{h1elec}.  Here, new
$e^+p$ NC and CC cross-section measurements based on data taken at a
centre-of-mass energy of $320$ GeV in 1999 and 2000 are presented. The
luminosity of this data set is $45.9$ pb$^{-1}$.

The new data are compared to those from 1994-1997. In particular the
region of high $Q^2$ is interesting since a slight excess of NC events
over the Standard Model expectation was observed in 1994 to 1996 for
$Q^2 > 15\,000$ GeV$^2$ \cite{h1exc,zeusexc}. The NC and CC
cross-sections in 1999-2000 are combined with the data from 1994-1997
taking into account the difference in centre-of-mass energy. The
measurements are compared to the Standard Model expectations.

Additionally the NC analysis is extended to higher $y$ for $Q^2 <
890$~GeV$^2$. This extension of the kinematic range allows a
determination of the longitudinal structure function, $F_L$(x,\QQ ),
to be made at high $Q^2$ in a region where the influence of $x$\Fz~is
expected to be negligible. This analysis is performed on both the
1999/2000 $e^+p$ data and the 1998/1999 $e^-p$ data set. The later
data have not been reported on in~\cite{h1elec}.


%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 $e^+p\rightarrow e^+X$
with unpolarised beams is given by
%
\begin{eqnarray}
\label{Snc1}
%\left(
\frac{{\rm d}^2\sigma_{NC}}{{\rm d}x\;{\rm d}\QQ}
%\right)_{Born}
& = & \frac{2\pi \alpha^2}{xQ^4}    
%\left(\frac{1}{\QQ}\right)^2  \hspace*{0.2cm} 
%\phi_{NC}^{\pm}(x,Q^2) 
%\left[Y_+ \Ftwo (x,\QQ ) \pm Y_{-}x\Fz (x,\QQ )-y^2 \FL (x,\QQ )\right] 
\left[Y_+ \Ftwo - Y_{-}x\Fz -y^2 F_L \right]\;\;\;, 
%(1+\delta_{NC}^{qed})
%+\delta_{NC}^{weak}).
\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 dependencies 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 generalised structure function \Ftwo. The $F_L$ 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. 

It is convenient to make use of the NC ``reduced 
cross-section'' in which the $Q^2$  
dependence 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}

%Note that \FL\ and $x$\Fz\ can be neglected over most of the kinematic range
%of this measurement such that the reduced cross-section is given
%essentially by
%\Ftwo. 
Only at high $y$ and high $Q^2$ do the contributions from $F_L$ and
$x$\Fz\ become important. In these regions the reduced cross-section differs
for different centre-of-mass energies while in most of the kinematic
range it is independent of the beam energies. 

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 [ (\bar{u}+\bar{c})+(1-y)^2(d+s) \right ]\;\;\;, 
\label{Scc}
\end{equation}
where $G_F$ is the Fermi coupling constant, and $\bar{u}$, $\bar{c}$,
$d$, $s$ are the quark distributions. 
%zhang The description of $e^+p
%zhang \rightarrow \bar{\nu} X$ is obtained by changing all quarks with
%zhang 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 this 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 ref.~\cite{h1hiq2}.  The
present analysis closely follows that of ref.~\cite{h1hiq2}.  The NC
event kinematics in the standard analysis 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}

At high $y$ the kinematic variables are best determined using
information from the scattered lepton alone. Therefore, in the
{\em extended analysis} the event kinematics are reconstructed using the
electron method ($e$-method) where
\begin{equation}
  Q^2_{e} = 4 E_e^{'} E_e \cos^2{\frac{\theta_e}{2}}
  \hspace*{1cm}
  x_e = \frac{Q^2_{e}} {s y_e}
\end{equation}
and $\sqrt{s}$ is the centre-of-mass energy.

The resolution of both reconstruction methods 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}

%\subsection{Selection of NC events}
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 loose cluster-track matching requirement 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 efficiency is greater than $99.5\%$. 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 $\sim$ 2\% at low \qsq compared to a total systematic
uncertainty of $\sim$ 4\%. At the highest $x$ the statistical error
increases to about 20\% and the systematic errors to about 10\%. 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 $130\,000$ 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.

In 1998 the LAr calorimeter electronics were upgraded allowing the
possibility to trigger on scattered leptons as low as $6$~GeV and
therefore giving access to the high $y$ kinematic region.  An NC {\em extended
analysis} is performed in the region $0.9>y>0.63$ and
$E_e^{\prime}>6$~GeV for $90<Q^2<890$GeV$^2$. In the lower $Q^2$ range
$y=0.63$ corresponds to $E_e^{\prime}=11$~GeV. At low energies of the
scattered lepton the photo-production background plays an increasingly
important role. In order to keep this background under control it is
suppressed by requiring a well measured track to validate the
scattered lepton cluster. The track is furthermore required to have
the same charge as the beam lepton. Finally, the remaining background
is statistically subtracted using fake leptons with the wrong charge
as described in~\cite{burkard}. The charge symmetry between right and
wrong charge lepton candidates in the background was found to be unity
within 10\%, and was taken into account in the systematic uncertainties.
%This was determined using events where the scattered lepton was
%detected in the low angle electron tagger of the luminosity system.

In total $24\,000$ $e^+p$ events and $5\,000$ $e^-p$ events are
selected in the {\em extended analysis}. Fig. \ref{lowe_cont} shows the
scattered lepton energy spectrum, the polar angle distribution, and
the $E-P_z$ spectrum for the positron data set. The simulation
provides a good description of both the $e^+p$ and the $e^-p$
data. $F_L$ is determined from the measured reduced cross-sections
using the formula:

\begin{equation}
F_L = \frac{Y_+}{y^2}\left[ \Ftwo -
      A \tilde{\sigma}_{NC}(x,Q^2)\right],
\end{equation}
%The dominant part of the cross-section comes from the contribution of
%\Ftwo\ and is taken from the H1 97 PDF Fit.  The fit does not include
%either of the data sets used to determine $F_L$ and therefore careful
%account is taken of the possible normalisation differences between the
%measured cross-sections presented here and \Ftwo\ from the QCD fit.
where $\Ftwo$~is taken from the H1 97 PDF Fit.
The factor $A$ in the formula accounts for the relative normalisation of the
data with respect to the H1 97 PDF Fit (which does not include this
data). This factor was obtained by normalising the measured
cross-sections in the region $y<0.4$ with respect to the prediction of
the QCD fit. The factors $A$ are found to be $0.99$ and $1.00$ for the
$e^-p$ and $e^+p$ data respectively. The positron data set have a
statistical precision on $F_L$ of between $7$\% and $40$\% and a
systematic uncertainty which ranges between $25$\% and $50$\%.

%\subsection{Selection of CC events}
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
$y_h>0.03$. After all selection criteria are applied, the final CC
data sample contains about 1000 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 electromagnetic and hadronic response of the detector is
calibrated using NC events similar to 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.

The selected events are corrected for detector acceptance and
migrations using the simulation and converted to bin centred
cross-sections using the prediction from the H1 97 PDF
Fit~\cite{h1hiq2}. The cross-sections are corrected for the effects of
QED radiation.

%\subsection{Systematic Uncertainties}
%The following systematic uncertainties were considered:
%\item An error of $1-3$ \% on the electromagnetic energy scale;
%\item an error of $3$ \% on the hadronic energy scale;
%\item an error of $25$ \% on the subtracted noise contribution;
%\item an error of $3$ mrad on the electron scattering angle; 
%\item an error of $30$ \% on the normalisation of the photoproduction
%  background; 
%\item an error of $0.5$ ($2-7$) \% on the trigger efficiency in the NC
%  (CC) analysis;
%\item an error of $1$ \% on the track-link efficiency in the NC analysis;
%\item an error of $2-5$ \% on the vertex efficiency only in the CC
%  analysis; 
%\item an error on the $V_

\subsection{Combination with previous measurements}
The new data are combined with the previously published 94-97 $e^+p$
data in order to improve the statistical precision which is the
dominant error source for $Q^2>1\,000$ GeV$^2$ in the NC cross-section
and at all $Q^2$ in the CC cross-section.

For this combination the 94-97 data are scaled to the new
centre-of-mass energy using the prediction from the H1 97 PDF Fit
\cite{h1hiq2}.  The combined measurement of any measured cross-section
$\sigma_i$ is given by

\begin{equation}
\sigma_{i}=\frac{\sigma^{meas}_{i,820} \cdot {\cal L}^{820}+
\sigma^{meas}_{i,920} \cdot {\cal L}^{920}} {{\cal
L}^{820}\cdot(\sigma^{th}_{i,820}/\sigma^{th}_{i,920})+{\cal
L}^{920}}
\end{equation}

where $\sigma^{meas}_{i,E_p}$ and $\sigma^{th}_{i,E_p}$ are the
measured and theoretical cross-sections at a proton energy $E_p$
respectively, and ${\cal L}^{820}$ and ${\cal L}^{920}$ are the
luminosities of the respective data sets.  The statistical error is
determined correspondingly and the systematic error is assumed to be
$100 \%$ correlated between the two data sets and conservatively taken
from the 99-00 data set. These combined data correspond to a
luminosity of $81.5$ pb$^{-1}$.

For the NC reduced cross-sections the correction factor for the beam
energy is relatively small. For $Q^2<5000$ GeV$^2$ it is only required
for the lowest $x$ bins and is always smaller than $2$ \%. The largest
correction of about $10$ \% is required at the highest $Q^2$ and low
$x$. For the CC reduced cross-section it varies between $5$ \% at low
$Q^2$ and $20$ \% at high $Q^2$.  For the cross-sections $\rm d \sigma
/\rm d Q^2$ the correction factor is about $5\%$ at low $Q^2$ and $50
\%$ at the highest $Q^2$.

\section{Results}

\subsection{Cross-section measurement}

The reduced $e^+p$ NC cross-section,~$\tilde{\sigma}_{NC}$, is shown in
fig. \ref{nc_stamp} where the results from the new measurement at
$\sqrt{s}=320$ GeV are compared to those obtained at $\sqrt{s}=300$
GeV.  Both sets of measurements are seen to provide comparable
precision and are found to be fully compatible.  Both data sets also
agree well with the predictions of the H1 97 PDF Fit which is also
shown. This NLO QCD fit was performed on low $Q^2$ fixed target data
from NMC \cite{nmc} and BCDMS \cite{bcdms} and the previous H1 high
$Q^2$ $e^+p$ data from 1994-1997.  It is shown for the two different
centre-of-mass energies of the two data sets.

%The reduced NC cross-sections measured at $\sqrt{s}=300$ GeV and
%$\sqrt{s}=320$ GeV are shown in
%fig.\ref{fig:nc_hix4}a,b as a function
%of $Q^2$ for the two highest $x$ values. The two data sets are
%found to be in agreement with each other and with the expectation from
%the NLO QCD fit. 
%In fig.\ref{fig:nc_hix4}c,d the combined result of all 1994 to 2000 $e^+p$ data
%is shown. The H1 data are also
%compared to the precise data from fixed-target experiments BCDMS, NMC and SLAC
%\cite{slac}.
%The strong fall of the cross-section with
%$Q^2$ due to the scaling violations is observed over more than four
%orders in magnitude.  
%At $Q^2=20\,000$ GeV$^2$ and $x=0.4$ the new data are below the
%expectation from the NLO QCD fit while the 94-97 measurement exceeded
%the expectation. The combined result from all $e^+p$ data shows no
%significant deviation from the expectation except for a slight deficit
%at $Q^2=12\,000$ GeV$^2$ and a slight excess above expectation at
%$Q^2=30\,000$ GeV$^2$.

%The $Q^2$ dependence of the reduced NC cross-section is also shown in
%fig. \ref{fig:nc_hix} for $x \geq 0.08$ for the 94-00 data. Also shown are
%the measurements at lower $Q^2$ by the fixed-target experiments
%BCDMS and NMC and from H1 at lower $Q^2$ \cite{h1lowq2}. 
%It is seen that the QCD Fit provides a good description
%of the $Q^2$ dependence for all data sets in the valence quark
%region. At $x=0.4$ overlap between the H1 data and the fixed-target data
%is achieved. At the highest $Q^2$ a decrease of the cross-section is
%expected due to the negative $\gamma Z$ interference in $e^+p$
%scattering. This decrease is clearly seen in the data at $x=0.18$ and
%$x=0.25$. 

The reduced NC cross-sections measured at $\sqrt{s}=300$ GeV and
$\sqrt{s}=320$ GeV are shown in fig.~\ref{fig:nc_hixc} as a function of
$Q^2$ for the high $x$ region. The two data sets are found to be in
agreement with each other and with the expectation from the H1 97 PDF
Fit. Fig.\ref{fig:nc_hixc} also shows the recent H1 measurement at lower $Q^2$
\cite{h1lowq2} and the fixed-target data from BCDMS and NMC.
%The strong fall of the cross-section with
%$Q^2$ due to the scaling violations is observed over more than four
%orders in magnitude.
The strong scaling violation is observed over more than four orders of
magnitude in $Q^2$.  At the highest $Q^2$ a decrease of the
cross-section is expected due to the negative $\gamma Z$ interference
in $e^+p$ scattering. This decrease is clearly seen in the data at
$x=0.18$ and $x=0.25$.  At $Q^2=20\,000$ GeV$^2$ and $x=0.4$ the new
data undershoot the expectation from the H1 97 PDF Fit while the
measurement in 1994-1997 exceeded the expectation. In
fig. \ref{fig:nc_hix} the combined results of the data from 1994 to
2000 are shown, where the $300$~GeV data have been corrected
individually for the change of centre-of-mass energy.
The combined result from all $e^+p$ data shows no
significant deviation from the expectation.
%zhang except for a slight deficit
%zhang at $Q^2=12\,000$ GeV$^2$ and a slight excess above expectation at
%zhang $Q^2=30\,000$ GeV$^2$. 

In fig. \ref{cc_stamp} the reduced CC cross-section is shown for the
new data and the data taken in 1994-1997. These data are consistent
within the statistical errors. The kinematic range for the
cross-section measurement was slightly extended at high and low $x$
mainly due to an improved trigger efficiency and hadronic
calibration. The combined result is shown in fig. \ref{cc_stampcomb}
and is found to be in good agreement with the theoretical prediction
from the H1 97 PDF Fit. The contribution to the CC reduced
cross-section from the $d$-quark density is also shown as given by the
fit. This illustrates that the CC cross-section at high $x$ is mainly
sensitive to the $d$ quark (see eq. 3). The precision of the CC
cross-section at $x=0.4$ from all data 1994-2000 is about $20$ \%.

The single differential NC cross-section ${\rm d}\sigma/\rm{ d}Q^2$
for $e^+p$ data is shown\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
$0.03<y<0.85$.}  in fig.\ref{dsdq2nc}(a) for $y\leq 0.9$.  The new
data are compared to the H1 $e^+p$ measurements based on data in
1994-1997 and the H1 97 PDF Fit.  The new preliminary cross-sections
are higher than the measurement from 1994-1997 due to the increase in
centre-of-mass energy.

Figures \ref{dsdq2nc} (b) show the ratios of the measurements to their
corresponding Standard Model expectation.  The global 1.5 \%
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 97 PDF Fit, as well as
the uncertainties of the experimental data entering the fit, and is
detailed in~\cite{h1hiq2}. The new data are observed to agree
well with the published data.
%Only at $Q^2=20\,000$ GeV$^2$ where
%the 94-97 data slightly exceed the expectation a deficit is seen in the
%new data. 
In fig. \ref{dsdq2nc} (c) the ratio of the combined $e^+p$
cross-section from 1994-2000 to the Standard Model are shown and the data
exhibit a similar feature as seen in fig. \ref{fig:nc_hix}.
%with a slight deficit at
%$Q^2=12\,000$ GeV$^2$ and a slight excess at $Q^2=30\,000$ GeV$^2$. 

The $Q^2$ dependence of the CC cross-section is shown in
fig.\ref{dsdq2cc}(a) and compared to the previous measurements.  Again
the effect of the increased centre-of-mass energy is seen resulting in
a higher cross-section for the new data set. The ratio of data to
expectation is shown in fig.\ref{dsdq2cc}(b) together with the
Standard Model uncertainty. The two data sets agree well with each
other and with the expectation from the Standard Model. The ratio of
the combined 94-00 measurement is also shown in fig. \ref{dsdq2cc}(c).

% The CC data are also sensitive to the propagator mass $M_W$ as is
% evident from eq.~\ref{Scc}. Determining this mass in the space-like
% regime provides an important check of the predictions of the Standard
% Model. A fit to the data is performed using the Standard Model
% expectation in which only $M_W$ is allowed to vary. The PDFs used are
% taken from the ``Low $Q^2$ QCD Fit''~\cite{h1hiq2} in which no data
% above $Q^2=150$~GeV$^2$ was included. The 94/97 $e^+p$ data were
% fitted independently of the $e^-p$ data, and both yield a consistent
% value for the propagator mass:
% $$e^+p:~~~M_{W}~=~80.9 \pm 3.3 (\rm stat.) \pm 1.7 (\rm syst.)  \pm 3.7 (\rm
% theo.)~\rm~GeV,$$
% $$e^-p:~~~M_{W}~=~79.9 \pm 2.2 (\rm stat.) \pm 0.9 (\rm syst.)  \pm 2.1 (\rm
% theo.)~\rm~GeV.$$
% The Standard Model uncertainty (theory) is evaluated by varying the
% assumptions for the Low $Q^2$ fit and is detailed in~\cite{h1hiq2}.
% Despite the smaller luminosity, the $e^-p$ data set has smaller
% uncertainties than the $e^+p$ data set due to the larger cross
% section, and its dominant contribution from the relatively well known
% $xu_v(x,Q^2)$ PDF.

% The total CC cross section has been measued for $e^-p$ in the region
% $Q^2 < 1\,000$~GeV$^2$ and $y < 0.9$:
% $$\sigma^{tot}_{CC}(e^-p)~=~43.08 \pm 1.84 (\rm stat.) \pm 1.74(\rm
% syst.) \rm ~pb,$$
% where the $1.8\%$ normalisation uncertainty is included in the
% systematic error. This is consistent with the expectation from the H1 97
% PDF Fit where \mbox{$\sigma^{tot}_{CC}(e^-p)~=~42.70~\pm~1.65~\rm~pb.$}

The $Q^2$ dependence of the NC and CC cross-sections are compared in
fig. \ref{fig:dsdq2nccc} for the combined 94-00 data.  At low $Q^2$
the NC cross-section is about 1000 times larger than the CC
cross-section since the CC cross-section is suppressed due to the
propagator term in eq. \ref{Scc}. At the highest values of $Q^2$ they
are of similar size as expected from the Standard Model, although in
$e^+p$ data the NC cross section remains larger.

%The integrated $x$ dependence of the NC and CC cross-sections are shown in
%fig.\ref{nc_dsdx}. 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 and with the
%94-97 data. The combined cross-sections from 94-97 data and 99/00 data
%are shown in fig.\ref{nc_dsdx}c,d. 

In fig.\ref{fl_plot} the measurements of $F_L$ are shown at fixed $y=0.75$
for both the $e^+p$ and the $e^-p$ data sets. Both data sets are
mutually consistent and in agreement with the QCD fit. The extreme
values allowed for $F_L$ ($F_L=0$ and $F_L=$\Ftwo) are clearly excluded
by the data.

\subsection{Valence quark distributions at high $Q^2$ and high $x$}

The new combined $e^+p$ double differential NC and CC cross-sections
together with the $e^-p$ NC and CC cross-sections measured previously
based on the data taken in years from 1998 to 1999~\cite{h1e-9899} are
used in a new NLO QCD fit to determine the dominant valence quark
distributions $xu_v$ and $xu_d$ at high $Q^2$ and high $x$.  The fit
is performed using the NLO DGLAP evolution equations~\cite{dglap} in
the $\overline{\rm MS}$ factorization scheme and treating the heavy
flavours as massless quarks. In addition to the two valence quarks,
gluon, up-type and down-type sea quarks are parameterised at
$Q^2_0=15\,{\rm GeV}^2$ in an MRS like form
$A_qx^{B_q}(1+x)^{C_q}(1+a_qx^{b_q})$~\cite{mrst} with $b_q=1/2$ for
the valence quarks and $b_q=1$ for the rest of quark components.  The
usual sum rules are imposed.  In order to have a reliable constraint
on the gluon and sea quarks, the new H1 low $Q^2$ data~\cite{h1lowq2}
are also included. The minimum $Q^2$-cut on the data is $Q^2_{\rm
min}=20$ ${\rm GeV}^2$.

The resulting valence quark densities of this fit are shown in
fig. \ref{fig:xuxd} labelled ``NLO QCD Fit: H1 only'' with the
estimated uncertainty. The result is compared to other
parameterisations MRST~\cite{mrst}, CTEQ5~\cite{cteq5}, and the
previous H1 fit~\cite{h1hiq2} all of which used the fixed target data of
BCDMS~\cite{bcdms} and NMC~\cite{nmc}.  The uncertainty on the parton
densities was estimated from the experimental errors on the data
following the prescription given in~\cite{zomer}. The relative
precision on the $u$ valence varies between 6\% for $x=0.25$ and 10\%
for $x=0.65$. The $d$ valence is essentially constrained by the $e^+p$
CC data only and has a precision of about 20\%.  The fit agrees with
the other parameterisations within the errors at all values of $x$
shown for both $xu_v$ and $xd_v$ except for $xu_v$ at $x=0.65$ where
it is $\sim 17\%$ lower than the other parameterisations with little
dependence in the range of $Q^2$ shown.  The difference remains within
about two standard deviations. The uncertainty on the valence quark
densities comes solely from the high $Q^2$ data as no difference was
found in the central value or the uncertainty when raising the
$Q^2_{min}$ cut to $200$ GeV$^2$

Also shown in the figure are the valence quark densities determined
with the local extraction method which was introduced in the previous
publication~\cite{h1hiq2}:
\begin{equation}
xq_v(x,Q^2)=\sigma_{\rm meas}(x,Q^2)\left(\frac{xq_v(x,Q^2)}
{\sigma(x,Q^2)}\right)_{\rm para}\,,
\end{equation}
where $\sigma_{\rm meas}(x,Q^2)$ is the measured NC or CC double
differential cross-sections, and the second factor on the
right-hand-side of the equation is the theoretical expectation from
the previous H1 fit~\cite{h1hiq2}. Only those points where the $xq_v$
contribution is greater than 70\% of the total cross section are
selected. The extracted parton densities are thus rather independent
of the theoretical input as the uncertainty on the dominant valence
quark contribution and that of the corresponding cross-section largely
cancel in the ratio.  The extracted valence quark densities combining
$\sigma^{e^\pm p}_{\rm NC}$ and $\sigma^{e^-p}_{\rm CC}$ represent an
improved statistical precision of typically 50\% and up to 100\% at
high $Q^2$ compared to using the 1994 to 1997 $e^+p$ data only.  The
local measurements are in good agreement with the global fit.

\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 97 PDF Fit to NMC, BCDMS, and H1
94-97 $e^+p$ data~\cite{h1hiq2} are able to provide a good description
of all the measured cross-sections. The NC analysis has been extended
to higher $y$ allowing, for the first time, $F_L$ to be determined
in the region $110<Q^2<700$~GeV$^2$ at $y=0.75$ for $e^+p$ and $e^-p$ data. $F_L$ is
observed to be consistent for $e^+p$ and $e^-p$ scattering as expected
in the Standard Model, which is in agreement with the data.

Comparisons of these cross-sections with the H1 measurements of NC and
CC cross-sections in $e^+p$ scattering from 94-97 data which were
taken at a centre-of-mass energy of $\sqrt{s} \approx 300$ GeV are
made.  The influence of the different centre-of-mass energy is seen in
the $\rm d\sigma / \rm d Q^2$ cross-section which is about $5$ \% higher
at low $Q^2$ and $50$ \% at the highest $Q^2$. This difference is in
agreement with the Standard Model expectation.

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 new data agree well with the measurements from 94-97.

The double differential CC reduced cross-sections are measured in the
\qsq range $300 \leq Q^2 \leq 15\,000$ GeV$^2$, and $0.008 \leq x \leq
0.4$. The new data agree well with the theoretical prediction and with
the 94-97 measurements.  Since the dominating error is statistical for
the CC cross-section the precision is improved from about 30 \% to 20
\% for each data point when combining the 99/00 data with those from
94-97.

A new NLO QCD fit was performed using all available cross-sections
measured by H1, but did not include the measurements of $F_L$.  It can
be seen for the first time that the valence quark distributions $xu_v$
and $xd_v$ can be separately constrained from the HERA high $Q^2$ data
alone with an experimental precision of about 10\% and 20\%
respectively for $xu_v$ and $xd_v$ at $x=0.65$ and $x=0.4$. The $u$
valence quark density at high $x$~is found to be about $17$ \% lower than
other parameterisations using the fixed target data.  The parton
densities determined with the local extraction method are in good
agreement with the global QCD fit.

\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{h1hiq2} 
H1 Collab., C. Adloff et al., Eur. Phys. J. {\bf C13} (2000) 609-639.
\bibitem{zeushiq2} 
ZEUS Collab., J.Breitweg et al. , Eur. Phys. J. {\bf C11} (1999) 427-445.\\
ZEUS Collab., J.Breitweg et al. , Eur. Phys. J. {\bf C12} (2000) 411-428.
\bibitem{h1elec} 
H1 Collab., C. Adloff et al., Eur. Phys. J. {\bf C19} (2001) 269-288 

\bibitem{h1exc}   
  H1 Collab., C. Adloff et al., Z. Phys. {\bf C74} (1997) 191.
\bibitem{zeusexc}
  ZEUS Collab., J. Breitweg et al., Z. Phys. {\bf C74} (1997) 207.
\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.

\bibitem{burkard} 
B. Reisert, Ph.D. Thesis, Max-Planck-Institut f\"ur Physik,Munich, MPI-PhE/2000-26.\\
Also available at 
http://www-h1.desy.de/publications/theses\underline{ }list.html
\bibitem{nmc} NMC Collab., M. Arneodo et al.,
Phys. Lett. {\bf B364} (1995) 107.
\bibitem{bcdms} BCDMS Collab., A.C. Benvenuti et al., 
Phys. Lett. {\bf B223} (1989) 485.
%\bibitem{slac} E. D. Bloom et al., Phys. Rev. Lett. {\bf 23} (1969) 930;\\
%M. Breitenbach et al., Phys. Rev. Lett. {\bf 23} (1969) 935.
\bibitem{h1lowq2} H1 Collab., Conf. Paper 944, 30th Intern. Conf. on
High-Energy Physics, Osaka, Japan (2000), July 2000

\bibitem{h1e-9899} H1 Collab., Conf.\ Paper 971, 30th Intern.\ Conf.\ on
High-Energy Physics, Osaka, Japan (2000), July 2000 

\bibitem{dglap}L.~N.~Lipatov, {\em Sov.\ J.\ Nucl.\ Phys.}\ {\bf 20} (1975)
 95; V.~N.~ Gribov and L.~N.~Lipatov, {\em Sov.\ J.\ Nucl.\ Phys.}\ {\bf 15}
 (1972) 438; G.~Altarelli and G.~Parisi, {\em Nucl.\ Phys.}\ {\bf B126}
 (1977) 298; Yu.~L.~Dokshitzer {\em Sov.\ Phys.\ JETP} {\bf 46} (1977) 641.
\bibitem{zomer} C. Pascaud and F. Zomer, LAL preprint, LAL/95-05 (1995)
\bibitem{mrst}A.~D.~Martin {\it et al}., {\em Eur.\ Phys.\ J.}\ {\bf C4} 
 (1998) 463; hep-ph/9803455.

\bibitem{cteq5}See e.g. CTEQ Collab., H.~L.~Lai {\it et al}., 
{\em Phys.\ Rev.}\ {\bf D}55 (1997) 1280; hep-ph/9701256, 
hep-ph/9903282.

\end{thebibliography}
\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]   
\begin{center} 
\begin{picture}(140,122.5)(0,0)
\setlength{\unitlength}{1 mm}
\put( 0, -10){\epsfig{file=H1prelim-01-053.fig1.eps,width=14cm}}
\put(25,97){\bf (a)}
\put(25,35){\bf (b)}
\put(83,35){\bf (c)}
\end{picture}
\end{center}    
\caption{Distributions of (a) $\theta_e$ and $E_e^{\prime}$ for (b)
  $Q^2>150$ GeV$^2$ and (c) $Q^2>5000$ GeV$^2$
  (solid points) and simulation (solid line) for $e^+p$ data. The lower histograms
  show the photo-production contribution.}
\label{nc_cont}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]   
\begin{center} 
\begin{picture}(140,122.5)(0,0)
\setlength{\unitlength}{1 mm}
\put( 0, -10){\epsfig{file=H1prelim-01-053.fig2.eps,width=14cm}}
\put(15,85){\bf (a)}
\put(85,85){\bf (b)}
\put(15,35){\bf (c)}
\end{picture}
\end{center}    
\caption{Distributions of (a) $E_e^{\prime}$, (b) $\theta_e$ and (c)
$E-P_z$ for $e^+p$ data (solid points) and simulation (solid line) in
the {\em extended analysis}.}
\label{lowe_cont}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]   
\begin{center} 
\begin{picture}(140,55)(0,0)
\setlength{\unitlength}{1 mm}
\put(10,-10){\epsfig{file=H1prelim-01-053.fig3.eps,bbllx=40,bblly=260,bburx=520,bbury=520,width=12cm}}
\put(55,25){\bf (a)}
\put(113,25){\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}[hhh]
\center
%\epsfig{file=fig/f2.eps,width=15cm}
\epsfig{file=H1prelim-01-053.fig4.eps,bbllx=48,bblly=30,bburx=550,bbury=780,width=14cm}
\caption{The NC $e^+p$ reduced cross-section
  $\tilde{\sigma}_{NC}$ is compared to the H1 97 PDF Fit. 
  The 99-00 and the 94-97 measurements are shown with the H1 97 PDF Fit predictions
  for the different centre-of-mass energies. 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]
%\setlength{\unitlength}{1 mm}
%\begin{center}
%\begin{picture}(160,160)(0,0)
%\epsfig{file=H1prelim-00-152.fig4.eps,width=15cm}
%\put(-130,110){\bf (a)}
%\put(-55,110){\bf (b)}
%\put(-130,30){\bf (c)}
%\put(-55,30){\bf (d)}
%\end{picture}
%\end{center}
%\caption{The NC reduced cross-section $\tilde{\sigma}_{NC}$ is
%  shown at $x=0.4$ and $x=0.65$ compared to the NLO QCD fit. Fig. (a) and (b)
%  show $\tilde{\sigma}_{NC}$ both the new 99-00 data and the
%  94-97 data. Fig. c) and d) show the result from all $e^+p$ data
%  (94-00). The inner error bars represent the statistical error, and the
%  outer error bars show the total error.}
%\label{fig:nc_hix4} 
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
\setlength{\unitlength}{1 mm}
\begin{center}
\begin{picture}(160,160)(0,0)
\put(-3,-12){
\epsfig{file=H1prelim-01-053.fig5.ps,bbllx=40,bblly=70,bburx=530,bbury=730,,width=15cm}}
\end{picture}
\end{center}
\caption{The NC reduced cross-sections $\tilde{\sigma}_{NC}$ from the
94-97 and 99-00 data are shown at high $x$ compared to the H1 97 PDF
Fit. Also shown are data from the H1 measured at lower $Q^2$, as well
as fixed-target experiments BCDMS and NMC. The solid (dashed) curve
represents the Standard Model expectation based on the H1 97 PDF Fit
for $\sqrt{s}=320$ GeV ($\sqrt{s}=300$ GeV). The inner error bars
represent the statistical error, and the outer error bars show the
total error.}
\label{fig:nc_hixc} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
\epsfig{file=H1prelim-01-053.fig6.ps,bbllx=40,bblly=70,bburx=530,bbury=730,width=15cm}
\caption{The NC reduced cross-section $\tilde{\sigma}_{NC}$ from the
  combined 94-00 data is shown at
  high $x$ compared to the H1 97 PDF Fit. Also shown are data from
  the fixed-target experiments BCDMS and NMC. The solid curves represent
  the Standard Model expectation based on the H1 97 PDF Fit. The
  inner error bars represent the statistical error, 
and the outer error bars show the total error.}
\label{fig:nc_hix} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
\center \epsfig{file=H1prelim-01-053.fig7.ps,bbllx=40,bblly=140,bburx=560,bbury=660,width=15cm}
\caption{The CC reduced cross-section $\tilde{\sigma}_{CC}(x,Q^2)$ is
shown for $e^+p$ scattering for the new data from 1999-2000 at
$\sqrt{s} \approx 320$ GeV (solid squares) and the data from 1994-1997
at $\sqrt{s} \approx 300$ GeV (open squares). The data are compared to
the H1 97 PDF Fit with the full and dashed curves showing respectively
for $\sqrt{s} \approx 320$ GeV and $\sqrt{s} \approx 300$ GeV. The
full line refers to the H1 97 PDF Fit and the dashed curve indicates
the $d$ contribution in that model. The inner error bars represent the
statistical error, and the outer error bars show the total error. }
\label{cc_stamp}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hhh]
\center \epsfig{file=H1prelim-01-053.fig8.ps,bbllx=40,bblly=140,bburx=560,bbury=660,width=15cm}
\caption{The CC reduced cross-section $\tilde{\sigma}_{CC}(x,Q^2)$
  for $e^+p$ scattering is shown for the combined data from 94-00 at
  $\sqrt{s} \approx 320$ GeV. The data are compared to the H1 97 PDF Fit. The
  inner error bars represent the statistical error, and the outer
  error bars show the total error. }  \label{cc_stampcomb}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h*]
\setlength{\unitlength}{1 mm}
\begin{center}
\begin{picture}(160,185)(0,0)

\put(15,-10){\epsfig{file=H1prelim-01-053.fig9.ps,bbllx=94,bblly=73,bburx=500,bbury=720,height=19cm}}
\put(34,128){\bf (a)}
\put(34,66){\bf (b)}
\put(34,20){\bf (c)}
\end{picture}
\end{center}
\caption{The $Q^2$ dependence of NC cross-sections $d\sigma/dQ^2$ is
  shown for the new preliminary $e^+p$ (solid points) and published
  94-97 $e^+p$ (open points) measurements. The data are compared to the
  Standard Model expectation determined from the H1 97 PDF Fit. The dashed (full)
  line corresponds to a center-of-mass energy of $\sqrt{s}=300$ GeV 
  ($\sqrt{s}=320$ GeV).
  The ratio of the
  94-97 and 99-00 data to their respective Standard Model expectation is shown 
  in figure (b). In fig. (c) the ratio of the combined 94-00 NC
  cross-section to the Standard Model is shown. The Standard Model
  Uncertainty is shown as the shaded band. The 1.5\% luminosity uncertainty
  is not included in the error bars.} 
\label{dsdq2nc}

\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h*]
\setlength{\unitlength}{1 mm}
\begin{center}
\begin{picture}(160,185)(0,0)

\put(15,-10){\epsfig{file=H1prelim-01-053.fig10.ps,bbllx=94,bblly=73,bburx=500,bbury=720,height=19cm}}
\put(34,128){\bf (a)}
\put(34,70){\bf (b)}
\put(34,26){\bf (c)}
\end{picture}
\end{center}
\caption{The $Q^2$ dependence of CC cross-section $d\sigma/dQ^2$ is
shown for the new preliminary $e^+p$ (solid points) and published
94-97 $e^+p$ (open points) measurements. The data are compared to the
Standard Model expectation determined from the H1 97 PDF Fit. The
dashed (full) line corresponds to a center-of-mass energy of
$\sqrt{s}=300$ GeV ($\sqrt{s}=320$ GeV). The ratio of the 94-97 and
99-00 data to their respective Standard Model expectation is shown in
figures (b). In fig. (c) the ratio of the combined 94-00 CC
cross-section to the Standard Model is shown. The Standard Model
Uncertainty is shown as the shaded band. The 1.5\% luminosity
uncertainty is not included in the error bars.}
\label{dsdq2cc}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h*]
\setlength{\unitlength}{1 mm}
\begin{center}
\begin{picture}(160,160)(0,0)


\put(0,0){\epsfig{file=H1prelim-01-053.fig11.ps,bbllx=40,bblly=150,bburx=550,bbury=660,height=16cm}}
\end{picture}
\end{center}
\caption{The $Q^2$ dependence of the NC (circles) and CC (squares) 
  cross-sections $d\sigma/dQ^2$ is shown for the combined 94-00 
  $e^+p$ measurements. The data are compared to the
  Standard Model expectations determined from the H1 97 PDF Fit
  including the 94-97 H1 $e^+p$ data. The 1.5\% luminosity uncertainty
  is not included in the error bars.} 
\label{fig:dsdq2nccc}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]   
\begin{center} 
\begin{picture}(140,122.5)(0,0)
\setlength{\unitlength}{1 mm}
\put( 0, -10){\epsfig{file=H1prelim-01-053.fig12.eps,width=14cm}}
\end{picture}
\end{center}    
\caption{Determination of $F_L$ shown for $e^-p$ and $e^+p$ data at
fixed $y=0.75$ as a function of $Q^2$ (lower scale), or equivalently
$x$ (upper scale). The shaded band shows the expectation of $F_L$, and
its uncertainty, from the QCD fit.}
\label{fl_plot}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{picture}(50,160)
\put(-65,-50){\epsfig{file=H1prelim-01-053.fig13.ps,bbllx=0pt,bblly=0pt,
bburx=594pt,bbury=842pt,width=190mm}}
\end{picture}
\end{center}
\caption{The valence quarks distributions $xu_v$ and $xd_v$ determined 
both with the NLO QCD fit (shaded error bands) using all cross-section
measurements from H1 only and with the local 
extraction method (data points with the inner and full error bars showing 
respectively the statistical and total errors) in comparison with other 
parameterizations which used fixed target data at low $Q^2$.}
\label{fig:xuxd}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[hhh]
%\setlength{\unitlength}{1 mm}
%\begin{center} 
%\begin{picture}(160,120)(0,0)
%\put( 10, 0){\epsfig{file=fig/dsigdxnc.hqn.both.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 NLO 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=fig/dsigdxcc.hqn.both.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 NLO QCD fit. The dashed curve shows the influence of the increased centre-of-mass energy.}
%\label{cc_dsdx} 
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{document}

