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\begin{document}  
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\newcommand{\ptmiss}{$P_{T}^{\rm miss}$}
\newcommand{\epz} {$E{\rm-}p_z$}
\newcommand{\vap} {  $V_{ap}/V_p$}
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% Journal macro
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\def\NCA{\em Nuovo Cimento}
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\def\CPC{\em Comp. Phys. Commun.}

% jobby  check the parameters used in the fit - is there a E_du param used?
% jobby  included the text that states that there should be more paprams for
% jobby  ubar than for dbar as well as for u rather than d
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\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
\begin{tabular}{llrr}
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi
,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                31st International Conference 
                on High Energy Physics, ICHEP02},
                July~24,~2002,~Amsterdam} \\
                 & Abstract:        & {\bf 978}    &\\
                 & Parallel Session & {\bf 5}   &\\ \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 
    Measurement and QCD Analysis of\\
    Inclusive Deep Inelastic Scattering
    at High {\boldmath $Q^2$} and Large {\boldmath $x$}}

  \vspace*{2cm}
    {\Large H1 Collaboration} 
\end{center}

\vspace{2cm}

\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 $\sqrt{s} \simeq 320$ GeV and correspond to
an integrated luminosity of $65.2 \ {\rm pb}^{-1}$.  The cross
sections are measured in the range of four-momentum transfer squared,
$Q^2$, between $100$ and $30\,000$ GeV$^2$, and Bjorken $x$ between
$0.0013$ and $0.65$. The new measurements are found to be consistent
with previous data and are well described by next-to-leading
order QCD fits in the framework of the Standard Model.
Flavour separated parton densities are extracted.
The neutral current analysis is extended to small energies of the
scattered electron and therefore to higher values of inelasticity $y$
allowing a determination of the longitudinal structure function $F_L$
at high $Q^2$ ($110-700$~GeV$^2$). 
\end{abstract}

\vspace{1.5cm}

\end{titlepage}

%
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% from /h1/iww/ipublications/h1auts.tex 

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\newpage
%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%
\label{intro}

%\begin{itemize}
%\item $sigma_cc^{tot}$ theory error; quantify no. of sigmas meas is from theory
%\end{itemize}

In 1992 the HERA accelerator began operation colliding lepton--proton
beams within the H1 and ZEUS experiments. Both experiments have since
analysed over $100~{\rm pb}^{-1}$ of luminosity per experiment.
The phase space covered by HERA inclusive deep
inelastic scattering (DIS) cross section measurements ranges from
small Bjorken $x$, to the electroweak (EW) regime of the Standard
Model where the four-momentum transfer squared, $Q^2 \sim
10\,000$~GeV$^2$ is similar to the squared masses of the
EW boson propagators. At these high $Q^2$ values the measurements
provide an insight into the partonic structure of matter and the
dynamics of strong interactions, as described by quantum chromodynamics
(QCD), at the smallest distance scales. QCD, formulated in the context
of perturbative evolution, provides an excellent predictive tool with
which to compare the data. Any deviations from this may be due to
exotic phenomena beyond the Standard Model.

Both neutral current (NC) interactions, \mbox{$ep \rightarrow eX$} via
$\gamma$ or $Z^0$ exchange, and charged current (CC) interactions,
\mbox{$ep \rightarrow \nu X$} via $W$ exchange, can be observed at
HERA yielding complementary information on the QCD and EW
parts of the Standard Model. The cross sections are defined in terms
of three kinematic variables $Q^2$, $x$, and $y$, where $y$ quantifies
the inelasticity of the interaction. The kinematic variables are
related via $Q^2=sxy$, where $\sqrt{s}$ is the $ep$ centre-of-mass
energy.

Measurements of the NC and CC cross sections in $e^+p$ scattering have
been made by H1 and ZEUS based on $\simeq 40$ pb$^{-1}$ data sets between
$1994$ to $1997$ \cite{h1hiq2,zeushiq2} when the $ep$ centre-of-mass
energy $\sqrt{s}$ was about $300$ GeV. 
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 with increased precision using a luminosity of $65.2$ pb$^{-1}$.
The data provide the most accurate neutral and charged
current cross sections measured by H1 at high $Q^2$ based on the first
phase of HERA operation.

The accuracy and kinematic coverage of the H1 neutral and charged
current cross section data enable dedicated QCD analyses which both
test the prediction of logarithmic scaling violations over four orders
of magnitude in $Q^2$ and unfold parton distributions. 
The inclusive cross sections are principally sensitive to the up and down
quark and anti-quark distributions. This is used in a novel,
next-to-leading order (NLO) QCD analysis of the H1 data which is also
extended to include the accurate proton and deuteron data from the
BCDMS muon scattering experiment~\cite{bcdms}. The fit parameter space is carefully
narrowed using theoretical constraints adapted to the new ansatz and
the experimental and phenomenological uncertainties are systematically
approached. This leads to a description of the complete set of NC
and CC data as well as to new determinations of the parton
distribution functions (PDFs) and their uncertainties.

Finally the NC analysis is extended to high $y$ for $800 > Q^2 >
100$~GeV$^2$. This extension of the kinematic range allows a
determination of the longitudinal structure function, $F_L(x,Q^2)$, to
be made at high $Q^2$. This analysis is performed
on both the $99-00$ $e^+p$ data and the $e^-p$ data set taken in
$1998$ and $1999$ with a luminosity of $16.4$ pb$^{-1}$. The
extended $e^{-}p$ analysis and $F_L$ extraction 
was not been reported on in~\cite{h1elec}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Neutral and Charged Current Cross Sections}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:theory}

After correction for QED radiative effects, the NC cross section for the process $e^{\pm}p\rightarrow e^{\pm}X$
with unpolarised beams and corrected
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]
(1+\Delta^{weak}_{NC})\,\,\,,
\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 $\alpha \equiv \alpha(Q^2=0)$ is the fine structure constant. The $\Delta^{weak}_{NC}$ corrections are
defined in~\cite{workshop} with the Fermi coupling constant, $G_F$,
and the $Z$ boson mass, $M_Z$, as the other main electroweak inputs.  The weak corrections,
are typically less than 1\% %jobby
and have not been applied to
the measured cross sections.  The helicity dependences of the electroweak
interactions are contained in \mbox{$Y_{\pm} \equiv 1 \pm (1-y)^2$}.
The generalised structure functions $\Ftwo$ and $x\Fz$ can be
decomposed as~\cite{klein}
\begin{eqnarray}
\label{f2p}
\Ftwo  \equiv & \Fem & - \ v_e  \ \frac{\kappa_w  \QQ}{(\QQ + M_Z^2)}
 \hspace*{0.2cm} \Fint  + (v_e^2+a_e^2)  
\left(\frac{\kappa_w  Q^2}{\QQ + M_Z^2}\right)^2 \Fwk \\
\label{f3p}
x\Fz    \equiv &      & - \ a_e  \ \frac{\kappa_w  \QQ}{(\QQ + M_Z^2)} 
x\Fzint + \hspace*{0.3cm} (2 v_e a_e) \hspace*{0.3cm}
\left(\frac{\kappa_w  Q^2}{\QQ + M_Z^2}\right)^2  x\Fzwk,
\end{eqnarray} 

where $\kappa_w^{-1}=4\frac{M_W^2}{M_Z^2}(1-\frac{M_W^2}{M_Z^2})$ in
the on-mass-shell scheme~\cite{pdg} and $M_W$ is defined in terms of the
electroweak inputs.  The quantities $v_e$ and $a_e$ are the vector and
axial couplings of the electron to the $Z^{0}$~\cite{pdg}.  The
electromagnetic structure function $\Fem$ originates from photon
exchange only. The functions $\Fwk$ and $x \Fzwk$ are the
contributions to $\Ftwo$ and $x\Fz$ from $Z^0$ exchange and the
functions $\Fint$ and $x\Fzint$ are the contributions from $\gamma Z$
interference. The longitudinal structure function $\FL$ may be
decomposed in a manner similar to $\Ftwo$. Its contribution is
significant only at high $y$.

Over most of the kinematic domain at HERA the dominant contribution to
the cross section comes from the electromagnetic structure function
$F_2$. Only at large values of $Q^2$ do the contributions due to the $Z^0$
boson become important.  For longitudinally unpolarised lepton
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}. 

In the quark parton model (QPM) the structure functions $F_2$,
$F_2^{\gamma Z} $ and $F_2^Z$ are related to the sum of the quark and
anti-quark densities
\begin{equation}
\label{eq:f2}
[F_2,F_2^{\gamma Z},F_2^{Z}] = x \sum_q 
[e_q^2, 2 e_q v_q, v_q^2+a_q^2] 
\{q+\bar{q}\} 
\end{equation}
and the structure functions $xF_3^{\gamma Z} $ and $xF_3^Z$ to their
difference which determines the valence quark distributions, $u_v$ and  $d_v$
\begin{equation}
\label{eq:xf3}
[ x F_3^{\gamma Z},x F_3^{Z} ] = x \sum_q 
[2 e_q a_q, 2 v_q a_q]
\{q -\bar{q} \} = x \sum_q [2 e_q a_q, 2 v_q a_q] q_v\,\,\,.
\end{equation}
The functions $q(x,Q^2)$ and $\bar{q}(x,Q^2)$ are the 
PDFs for quarks and anti-quarks, $e_q$ is the charge of quark $q$ in
units of the electron charge and $v_q$ and $a_q$ are the vector and
axial-vector weak coupling constants of the quarks.

For CC interactions the cross section corrected for QED radiative
effects may  be expressed as
\begin{eqnarray}
\label{eq:cccross}
\frac{{\rm d} ^2 \sigma_{\rm CC}^{\pm}}{{\rm d} x\; {\rm d} Q^2} & = &
\frac{G_F^2 }{2 \pi x} \left[ \frac{M_W^2}{(Q^2+M_W^2)^2} \right] 
\;\phi_{\rm CC} ^\pm \; (1+\Delta^{weak}_{CC})\\
\mbox{with } \hspace{1cm} \phi_{\rm CC}^\pm & =& 
\frac{1}{2}(Y_+ \wtwogen^\pm   \mp Y_- \xwthreegen^\pm - y^2 \wlgen^\pm),  
\end{eqnarray}
where $\Delta^{weak}_{CC}$ represents the CC weak radiative corrections.  The
CC structure functions $\wlgen^\pm$, $\wtwogen^\pm$, and
$\xwthreegen^\pm$ are defined in a similar manner as the NC structure
functions. In the QPM (where $\wlgen^\pm \equiv 0$) they may be interpreted 
as sums and differences of quark and anti-quark densities and are
given by
\begin{eqnarray}
\label{ccstf}
   \wtwogen^{+}  =  2x (\bar{U}+D)\hspace{0.1cm}\mbox{,}\hspace{0.3cm}
\xwthreegen^{+}  =  2x (\bar{U}-D)\hspace{0.1cm}\mbox{,}\hspace{0.3cm}
   \wtwogen^{-}  =  2x (U+\bar{D})\hspace{0.1cm}\mbox{,}\hspace{0.3cm}
\xwthreegen^{-}  =  2x (U-\bar{D})\,\,\,.
\end{eqnarray}
Here $U$ represents the sum of up-type, and $D$ the sum of down-type
quark densities and includes the electroweak couplings (cf. section~\ref{sec:QCDAnsatz}).

For the presentation of the subsequent measurements it is convenient
to define the NC and CC ``reduced cross sections'' as
\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},\;\;\;
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{expt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{H1 Apparatus and Trigger}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{det}

The co-ordinate system of H1 is defined such that the positive $z$
axis is in the direction of the outgoing proton beam (forward
direction). The polar angle $\theta$ is then defined with respect to
this axis. The detector components most relevant to this analysis are
the Liquid Argon (LAr) calorimeter, which measures the positions and
energies of particles over the range $4^\circ<\theta<154^\circ$, a
lead-fibre calorimeter (SPACAL) covering the range
$153^\circ<\theta<177^\circ$, the PLUG calorimeter covering the range
$0.7^\circ<\theta<3.3^\circ$, and the inner tracking detectors which
measure the angles and momenta of charged particles over the range
$7^\circ<\theta<165^\circ$.
In the central region, $25^{\circ}\gapprox\theta\gapprox165^{\circ}$, 
the central jet chamber (CJC) is
supplemented by two $z$ drift chambers to improve the $\theta$
measurement of the tracks in the CJC. The forward tracking detector, $\theta\lapprox30^{\circ}$, is
used to determine the vertex of events with no tracks in the CJC.

The $ep$ luminosity is determined by
measuring the QED bremsstrahlung event rate $ep \rightarrow ep\gamma$
in the photon tagger located at $z=-103$m.  An electron tagger is
placed at $z=-33$m adjacent to the beam-pipe. It is used to check the
luminosity measurement and to provide information on $ep\rightarrow
eX$ events at very low $Q^2$ (photoproduction) where the electron
scatters through a small angle ($\pi - \theta < 5$~mrad). A full
description of the H1 detector can be found in~\cite{h1detector}
and~\cite{spacal}.

NC events are triggered mainly using information
from the LAr calorimeter. The trigger requires an electromagnetic
trigger ``tower'' pointing to the vertex. For electrons with energy
above $11$~GeV this is $100\%$ efficient.  At lower energies the
triggers based on LAr information are supplemented by using additional
information from the tracking detectors.  In $1998$ the LAr
calorimeter electronics were upgraded in order to
trigger on scattered leptons as low as $6$~GeV, the minimum value
considered in this analysis. This gives access to the
high $y$ kinematic region. For electron energies of $6$~GeV$6$~GeV, the combined trigger
efficiency is $96\%$ for the $e^+p$ data set, and $90\%$ for the
earlier $e^-p$ data set.

The characteristic feature of CC events is a large missing transverse
momentum, $P_T^{miss}$, which is identified at the trigger level using the LAr calorimeter
vector sum of trigger towers. At low $P_T^{miss}$ the efficiency is
enhanced by use of an additional trigger requiring hadronic energy
pointing to the event vertex in combination with track information from the
inner tracking chambers. For the minimum $P_T^{miss}$ considered in the analysis of $12$~GeV the efficiency
is $60\%$, rising to $90\%$ for $P_T^{miss}$ of $25$~GeV.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simulation Programs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{mc}

In order to determine acceptance corrections DIS processes are
generated using the DJANGO~\cite{django} Monte Carlo (MC) simulation
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 to simulate the hadronisation process~\cite{jetset}.
The simulated events are produced with PDFs from an NLO QCD fit (H1 97
PDF Fit) which
includes previous H1 $e^+p$ data and is detailed in~\cite{h1hiq2}.

The detector response to events produced by the various generation
programs is simulated in detail 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.


The dominant $ep$ background contribution to deep inelastic scattering
processes is due to photoproduction ($\gamma p$) processes.  These are
simulated using the PYTHIA~\cite{pythia} generator with leading order
parton distribution functions for the proton and photon taken
from~\cite{ggrv}. Further background from QED-Compton processes,  lepton pair production via two photon interactions, prompt photon production, and heavy
gauge boson ($W^{\pm},Z^0$) production are included in the background
simulation. Non-$ep$ backgrounds are negligible. Further details are given in~\cite{h1hiq2}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Neutral Current Measurement Procedure}
\subsection{Cross Section Measurement Procedure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{ncmeas}

Inelastic $ep$ interactions are required to have a vertex within $\pm
35$~cm of its nominal $z$ position. High \qsq neutral current events
are selected by requiring that the event has a compact and isolated
energy deposition (cluster) in the electromagnetic part of the LAr
calorimeter.
The scattered electron is identified as the cluster of highest
transverse momentum. In the central detector region,
$\theta>40^{\circ}$, the cluster has to be associated with a track
measured in the inner tracking chambers.  Energy-momentum conservation
requires the variable $E-P_z$ summed over all final state particles to
be approximately equal to twice the initial electron beam energy.
Restricting $E-P_z$ to be greater than $35$~GeV considerably reduces
the photoproduction background and the radiative corrections in which
the scattered electron or bremsstrahlung photon escape undetected in
the electron beam direction. In order to further suppress the
photoproduction background, which increases with $y$, cuts are applied on  $y_e=1-\frac{E_e^\prime}{E_e} \sin^2{\theta_e/2}$. The restriction is $y_e < 0.63$ for
$Q^2<890$~GeV$^2$  and $y_e < 0.90$ for
$Q^2>890$~GeV$^2$, which defines  the kinematic region\footnote{At $Q^2=90$~GeV$^2$ $y=0.63$ corresponds to $E_e^{\prime}=11$~GeV.} of the
{\em nominal analysis}.

Based on the recent upgrade of LAr electronics the neutral current
analysis is extended to lower energies of the scattered electron,
$E_e^{\prime}>6$~GeV, giving access to the high $y$ region  $0.63<y_e < 0.90$
for $90<Q^2<890$~GeV$^2$. This defines the kinematic region
of the {\em high-$y$ analysis}. At low energies of the scattered
electron the photoproduction background, where $\pi^0\rightarrow \gamma \gamma$ decays give rise to fake electron candidates,  plays an increasingly
important role . Part of this background is suppressed by
requiring a well measured track linked to the calorimeter cluster.
The track is further required to have the same charge as the beam
lepton.  Finally, the remaining background is determined directly from
the data by identifying fake leptons with the wrong charge.  This
background is
statistically subtracted from the correctly charged sample. The charge
symmetry between fake lepton candidates is
determined using the $e^+p$ and $e^-p$ data sets and is measured to be
$0.99\pm 0.07$. Further details are given in~\cite{burkard,dubak}.

The NC kinematics in the nominal analysis are reconstructed using the
$e\Sigma$ method~\cite{esigma} which has the best resolution and least
sensitivity to QED radiative corrections over the accessible phase
space. In the {\em high-$y$ analysis} the electron reconstruction method is
used to reconstruct the event kinematics using only the measured
energy and angle of the scattered electron. This is the most precise
method at high $y$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Charged Current Measurement Procedure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\label{ccmeas}

The selection of charged current events requires a large missing
transverse momentum, $P_T^{miss}\geq 12$~GeV, assumed to be carried by
an unseen neutrino. In addition the event must have a reconstructed
vertex within $\pm 35$ cm of its nominal position. The kinematic variables $y_h$ and $Q^2_h$  are determined
using the hadron kinematic reconstruction
method~\cite{jbmethod}.
In order to restrict the measurement to a region with good
kinematic resolution the events are required to have $y_h<0.85$ and $P_{T,h}>12$~GeV.
The measurement is restricted to the region where the trigger
efficiency is acceptable by demanding $y_h>0.03$.  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 $79\%$ at
$Q^2=300\,{\rm GeV}^2$ and reaches 98\% at $Q^2=3000\,{\rm GeV}^2$.
The $ep$ background is dominantly due to photoproduction events and is
suppressed as discussed in ~\cite{h1elec}.
The residual $ep$
background is negligible for most of the measured kinematic domain,
though it reaches 15\% at the lowest $Q^2$ and the highest
$y$. This contribution is subtracted statistically from the CC data
sample with a systematic uncertainty of 30\% of the subtracted events.
The non-$ep$ background is rejected using the same method as 
in~\cite{h1hiq2}. For further details see~\cite{juergen,zhang}.
 

For both the NC and CC analyses 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 DIS cross sections $\sigma_{NC(CC)}$ for NC and CC processes in
% $e^{\pm}p$ collisions may be factorised as $\sigma_{NC(CC)} =
% \sigma_{NC(CC)}^{Born}(1+\Delta^{qed}_{NC(CC)})(1+\Delta^{weak}_{NC(CC)})$
% where $\sigma_{NC(CC)}^{Born}$ is the Born cross section and
% $\Delta^{qed}_{NC(CC)}$ and $\Delta^{weak}_{NC(CC)}$ are the QED and
% weak radiative corrections respectively.  The measured cross sections
% presented in section~\ref{results}, in which the effects of QED
% radiation have been corrected for, correspond to the differential
% cross sections ${\rm d}^2 \sigma_{NC(CC)}/{\rm d}x{\rm d}Q^2$ defined
% in eq.~\ref{Snc1} and~\ref{eq:cccross}.  

The ``QED radiative corrections'' are defined in~\cite{h1hiq2}
and were calculated using the program HERACLES~\cite{heracles} as
implemented in DJANGO~\cite{django} and verified with the numerical analysis
program HECTOR~\cite{hector}.  

%>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%>\subsection{Systematic Uncertainties}
%>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%>\label{syserr}

The uncertainties on the measurement lead to systematic errors on the
cross sections, which can be split into bin--to--bin correlated and
uncorrelated parts.  All the correlated systematic errors are found to
be symmetric to a good approximation.
Full details about the origin of the correlated and the uncorrelated
systematic errors of the NC and CC cross section measurements
can be found in~\cite{h1hiq2,dubak,beate,zhang,burkard}.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Combination with Previous Measurements at Different {\boldmath $\sqrt{s}$}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\label{comb}

The measurements of high $Q^2$ $e^+p$ cross sections from different running periods are combined to
provide a convenient summary of the first stage of HERA
operation. This combination improves the statistical precision which
is the dominant error source over the full $Q^2$ range for the CC
cross sections and for $Q^2 \gapprox 1000$~GeV$^2$ for the NC cross
sections.
For the purpose of combination the systematic errors are assumed to be fully correlated.
The combined data correspond to a luminosity of $100.8$ pb$^{-1}$.

\section{QCD Analysis}
\label{qcdana}
%
The cross section data presented here cover a huge range in momentum
transfer squared, $Q^2$, and Bjorken $x$. Combined with the low $Q^2$
neutral current data~\cite{h1lowq2}, recently published by the H1
Collaboration, $e^{\pm}p$ data on structure functions of much
increased accuracy are now available which allow the predictions of
perturbative QCD to be tested over four orders of magnitude in $Q^2$,
from about $1$~\gv to above $10^4$~\gv, and $x$, from below $10^{-4}$ to
$0.65$. The joint measurement of neutral and charged current scattering
cross sections provides sensitivity to different quark flavour
distributions and the gluon distribution, $xg(x,Q^2)$.
This is used to determine the sum of up, anti-up, down and anti-down
quark distributions  ($xU~,x\bU~,xD~,x\bD$) and to unfold the valence-quark
distributions, $u_v$ and $d_v$.

With the current beam energies the HERA collider data do not access the
medium $Q^2$, large $x$ region of deep inelastic scattering.
Complementary information on quark distributions is provided by
lepton-deuteron scattering data which are not yet available from HERA.
Therefore in the subsequent analysis the H1 data are also combined
with the precise BCDMS muon-proton and muon-deuteron scattering data,
and the results are compared with the analysis of the H1 data alone.
%
\subsection{Ansatz}
%
\label{sec:QCDAnsatz}
Traditionally, QCD analyses of inclusive deep inelastic cross-section
data use parameterisations of the valence-quark distributions and of a
sea-quark distribution imposing additional assumptions on the flavour
decomposition of the sea. The neutral and charged current cross
section data presented here, however, are  sensitive to four
combinations of up-type and down-type quark distributions
which for $Q^2$ less than the bottom quark production threshold are written as:
\begin{eqnarray}\label{ud}
 U &=& u + c  \nonumber\\
\bU&=& \bu + \bc \nonumber\\
 D &=& d + s \nonumber\\
\bD&=& \bd + \bs. 
\end{eqnarray}
% 
This ansatz weakens the influence of necessary assumptions on the flavour
decomposition of the sea in the fit. The
valence-quark distributions are obtained according to
 %
\begin{eqnarray}\label{equvdv}
   u_v=U-\bU,  \quad d_v=D-\bD
\end{eqnarray}
%
and are not fitted directly.

The charged current cross sections, eq.~\ref{eq:cccross}, are
superpositions of the distributions given in eq.\ref{ud} according to
%
\begin{eqnarray}\label{Scc}
 \phi^+_{CC} = x\bU +(1-y)^2 xD, \quad \phi^-_{CC} = xU +(1-y)^2 x\bD
\end{eqnarray}
%
with the helicity determined $y$ dependence.  The neutral current
cross section is dominated by the structure function \Fc which can be
written as
%
\begin{eqnarray}\label{f2ud}
 F_2 =  \frac{4}{9} \cdot (xU +x\bU) + \frac{1}{9} \cdot (xD +x\bD).
\end{eqnarray}
% 
In the high $Q^2$ neutral current data complementary sensitivity is
obtained from the interference structure function $xF_3^{\gamma Z}
\propto x [2(U-\bU) + (D-\bD)]$ but still higher luminosity is
required to exploit the $xF_3$ function for a dedicated determination
of the valence quarks. With an extra assumption on the $\bu$ and $\bd$
quark distributions, the $xU,~x\bU,~xD,~x\bD$ and $xg$ distributions
may be simultaneously determined using H1 NC and CC data alone. This
assumption may be relaxed if deuteron data are additionally used in
the fit.

The isoscalar nucleon structure function $F_2^N$  is determined by
the singlet combination of parton distributions and a  small
contribution from the difference of strange and charm quark
distributions, i.e.
%
\begin{eqnarray}\label{f2d}
 F_2^N =  \frac{5}{18} \cdot x(U +\bU  + D +\bD) + \frac{1}{6} \cdot x(c+\bc -s-\bs).
\end{eqnarray}
% 
The  nucleon  data are obtained from the  muon-deuteron cross 
sections measured by the BCDMS collaboration, for $x \geq 0.01$, and nuclear
corrections are applied following~\cite{nucor}.

Experimental results are used to determine the fractions of charm
\cite{h1f2c,zeusf2c} and strange \cite{ccfr} quarks. The present QCD analysis is
rather insensitive to their actual values since in the fit to the H1
and BCDMS data $c$ and $s$ explicitly occur only as small corrections
to $F_2^N$.

The analysis is performed in the $\overline{MS}$ renormalisation
scheme using the QCD evolution equations~\cite{dglap} in
NLO~\cite{furmanski}. The structure function formulae given here are
thus replaced by integral convolutions of coefficient functions and PDFs.
An approach is used whereby all quarks are taken to be massless including
the charm  and bottom  quarks which provides an adequate
description of the parton densities, in the high $Q^2$ kinematic
range of the data presented here. The bottom quark density, $xb$, is
evolved assuming that $xb(x,Q^2)=0$ for $Q^2<m_b^2$~GeV$^2$ where
$m_b$ is the bottom quark mass.

Fits are performed to the measured cross sections calculating the
longitudinal structure functions to order $\alpha_s^2$ and assuming
the strong coupling constant to be equal to \amz$=0.1185$~\cite{pdg}.
The analysis uses an $x$ space program developed inside the H1
collaboration~\cite{qcdfit}, with cross checks performed using an
independent program~\cite{qcdnum}.  In the fit procedure, a $\chi^2$
function is minimised which is defined in~\cite{h1lowq2}. The
minimisation takes into account correlations of data points caused by
systematic uncertainties allowing the error parameters, including the
relative normalisation of the various data sets, to be determined by
the fit. 

\subsection{Parameterisations \label{param}}
%
%Following previous analyses, \cite{MRST,h1lowq2}, 
The initial parton distributions, i.e.  $xP=xg,~xU,~xD,~x\bU,~x\bD$,
are parameterised at $Q^2 = Q^2_0$ in the following general form
%
\begin{equation}
xP(x) = A_qx^{B_q}(1-x)^{C_q}
      [1+E_{q}x + D_{q}\sqrt{x} +F_{q}x^2].
\label{eqpara}
\end{equation}
%
A QCD analysis requires a choice to be made for $Q^2_0$, the initial
scale, and $Q^2_{min}$, the minimum $Q^2$ of the data considered in
the analysis. Variations of both $Q^2_0$ and $Q^2_{min}$ are studied.
As in~\cite{h1lowq2} the $Q^2_0$ is chosen to be
$4$~GeV$^2$, and data are included for $Q^2\geq 3.5$~GeV$^2$.
Reasonable variations of these choices, see table~\ref{tabmodunc}, are
considered as part of the model uncertainties of the parton
distributions determined.

The general ansatz, eq.~\ref{eqpara}, represents an
overparameterisation of the description of the data considered here.
This is clear both from the saturation behaviour of the $\chi^2$
function and also from fit instabilities occasionally observed.
Moreover, the choice of parameters depends on the data sets included
in the fit. The H1 data can be fitted with less parameters than the
combined H1 and BCDMS data due to the precise BCDMS
proton and deuteron data in the large $x$ region where the
cross section variations with $x$ are particularly strong.

Careful consideration of the parameter space  leads to the following set of 
parameterisations in the fit to the H1 and BCDMS data

\begin{eqnarray}
xg(x)  &=& A_Gx^{B_G}(1-x)^{C_G}             \cdot [1+E_{G}x+D_{G}\sqrt{x} ] \nonumber\\
xU(x)  &=& A_Ux^{B_U}(1-x)^{C_U}             \cdot [1+E_{U}x+D_{U}\sqrt{x}+F_{U}x^2 ] \nonumber\\
xD(x)  &=& A_Dx^{B_D}(1-x)^{C_D}             \cdot [1+E_{D}x+D_{D}\sqrt{x} ] \nonumber\\
x\bU(x)&=& A_{\bU}x^{B_{\bU}}(1-x)^{C_{\bU}} \cdot [1+E_{\bU}x ] \nonumber\\
x\bD(x) &=& A_{\bD}x^{B_{\bD}}(1-x)^{C_{\bD}} \,\,\,.
\label{eqparh1b}
\end{eqnarray}
% \begin{eqnarray}
% xg(x)  &=& A_Gx^{B_G}(1-x)^{C_G}             \cdot [1+E_{G}x+a_{G}\sqrt{x} ] \nonumber\\
% xU(x)  &=& A_Ux^{B_U}(1-x)^{C_U}             \cdot [1+E_{U}x+a_{U}\sqrt{x}+c_{U}x^2 ] \nonumber\\
% xD(x)  &=& A_Dx^{B_D}(1-x)^{C_D}             \cdot [1+E_{D}x+a_{D}\sqrt{x} ] \nonumber\\
% x\bU(x)&=& A_{\bU}x^{B_{\bU}}(1-x)^{C_{\bU}} \cdot \nonumber\\
% \bD(x) &=& A_{\bD}x^{B_{\bD}}(1-x)^{C_{\bD}} \cdot \,\,\,.
% \label{eqparh1b}
% \end{eqnarray}
%
% \begin{table}[h]
% \begin{center}
% \begin{tabular}{ccccccccccc}
% $xg(x)  $&=&$ A_Gx^{B_G}(1-x)^{C_G} $&$\cdot $&$[~1~+~$&$E_{G}x$&$+$&$D_{G}\sqrt{x}$&$ $&$ $&$]$\\
% $xU(x)  $&=&$ A_Ux^{B_U}(1-x)^{C_U} $&$\cdot $&$[~1~+~$&$E_{U}x$&$+$&$D_{U}\sqrt{x}$&$+$&$F_{U}x^2 $&$]$\\
% $xD(x)  $&=&$ A_Dx^{B_D}(1-x)^{C_D} $&$\cdot $&$[~1~+~$&$E_{D}x$&$+$&$D_{D}\sqrt{x}$&$ $&$ $&$]$\\
% $x\bU(x)$&=&$ A_{\bU}x^{B_{\bU}}(1-x)^{C_{\bU}} $&$\cdot $&$[~1~+~$&$ $&$ $&$ $&$ $&$ $&$]$\\
% $\bD(x) $&=&$A_{\bD}x^{B_{\bD}}(1-x)^{C_{\bD}}  $&$\cdot $&$[~1~+~$&$ $&$ $&$ $&$ $&$ $&$]$\\
% \label{eqparh1b}
% \end{tabular}                                   
% \end{center}
% \end{table}
The variations in the number of parameters for the different parton distributions
are broadly in accord with expectation. The gluon distribution
is not sensitive to a high $x$ parameter, $F_Gx^2$, since at large $x$
the scaling violations are due to gluon bremsstrahlung, i.e.
independent of the gluon distribution. The $U$ and $D$ distributions
require more parameters than the anti-quark distributions $\bU$ and
$\bD$ because the former are a superposition of valence and sea
quarks, in contrast to the latter. Due to the different electric charges,
$e_u^2=4 e_d^2$, and the $y$ dependence of the charged current cross
section, the data are much more sensitive to the up quark than to the
down quark distributions. Thus less parameters are needed for $D$ than
for $U$.  The specific choice of these parameterisations is obtained
from saturation of the $\chi^2$: an additional parameter $D,E$, or $F$ is
considered only when its introduction improves the $\chi^2$ by more
than one unit. Furthermore solutions are excluded which yield
unphysical behaviour, e.g. negative cross sections.

A number of relations between parameters can be introduced naturally
in the ansatz.  At low $x$ the valence quark distributions are
expected to vanish. Thus the low $x$ parameters $A_q$ and $B_q$ are
required to be the same for $U,~\bU$ and for $D,~\bD$ as the sea-quark
and the anti-quark densities can be assumed to be equal. In the
absence of deuteron data from HERA there is no distinction possible of
the rise towards low $x$ between $U$ and $D$.  Thus the corresponding
parameters $B$ and $D$ are required to be equal, i.e.  $B_U=B_D=B_q$
and $D_U=D_D=D_q$.  Further contraints are the conventional momentum
sum rule and the valence quark counting rules. The total number of
free parameters of the five parton distributions is thus equal to 13
in the fit to the H1 and BCDMS data. In the fit to the H1 data alone
inclusion of the parameter $D_q$ does not improve the $\chi^2$ and it
is omitted from the fit.

The present fit differs from traditional QCD analyses which directly
parameterise either the difference $(\bd-\bu)$~\cite{mrst2001} or
the ratio $\bd/\bu$ \cite{cteq6}.  The functional forms in these
ansatzes are chosen to require $(\bd-\bu) \rightarrow 0$ when
$x\rightarrow 0$.  This behaviour can be imposed to the current
fit assuming that $x\bc$ and $x\bs$, at the starting scale of
$Q^2_0=4~$GeV$^2$, can be expressed as fractions $f_s$ and $f_c$ of
the $\bU$ and $\bD$ distributions, respectively. 
In the low $x$ limit the ratio $\bd/\bu$ can 
then be expressed in terms of the
parton distribution parameters as
\begin{equation}
\bd/\bu=\frac{1-f_s}{1-f_c} \frac{A_{\bD}}{A_{\bU}} x^{(B_{\bD}-B_{\bU})}
\label{eqlowxud}
\end{equation}
The condition $\bd/\bu\rightarrow 1$ as $x \rightarrow 0$ is imposed
with the constraint \mbox{$A_{\bU}=A_{\bD}\cdot(1-f_s)/(1-f_c)$}, 
as well as requiring $B_{\bD}=B_{\bU}$.
%
\subsection{Fit Results \label{h1fit}}
%
The fit to the H1 and BCDMS data, termed ``Preliminary H1 2002 PDF Fit'', uses the
data sets as specified in table~\ref{dataset}. 
%The results of the fit
%are presented in table~\ref{bcdmsfit} in which the $\chi^2$ for each
%data set as well as the optimised relative normalisation as determined
%from the fit are given. 
The total $\chi^2$ per degree of freedom (ndf) 
in this fit is $917/(1014-13)=0.92$. 
The initial parton distributions
are shown in fig.~\ref{figpdfhb}. The inner error band describes the
experimental uncertainty while the outer band represents the
experimental and the model uncertainties added in quadrature. The
solid line is the result of the fit to the H1 data alone, which gives
a $\chi^2$/ndf$=548/(621-11)=0.90$.
%
 \begin{table}[htb]
 \scriptsize
 \begin{center}
 \begin{tabular}{|c|l|l|r|r|c|r|c|}
 \hline
 experiment          &  $x_{min}$         & $x_{max}$ & $Q^2_{min}$ &$Q^2_{max}$ & 
$\delta \mathcal{L}$ &    ref.            & comment   \\ 
                     &                    &           &  (GeV$^2$)  & (GeV$^2$)  &
  $(\%)$             &                    &           \\
 \hline
H1 min bias  $96-97$&  $0.00008$ &  $0.02$        &  $3.5$   & $12$      
              &  $1.8$     &~\cite{h1lowq2} &  $e^+p$ $\sqrt{s}\simeq 300$~GeV\\
H1 low $Q^2$ $96-97$&  $0.000161$ &  $0.20$        &  $12$   & $150$      
              &  $1.8$     &~\cite{h1lowq2} &  $e^+p$ $\sqrt{s}\simeq 300$~GeV\\
H1 NC  $94-97$&  $0.0032 $ &  $0.65$        &  $150$   & $30\,000$
              &  $1.5$     &~\cite{h1hiq2}  &  $e^+p$ $\sqrt{s}\simeq 300$~GeV\\
H1 CC  $94-97$&  $0.013  $ &  $0.40$        &  $300$   & $15\,000$  
              &  $1.5$     &~\cite{h1hiq2}  &  $e^+p$ $\sqrt{s}\simeq 300$~GeV\\
H1 NC  $98-99$&  $0.0032$  &  $0.65$        &  $150$   & $30\,000$ 
              &  $1.8$     &~\cite{h1elec}  &  $e^-p$ $\sqrt{s}\simeq 320$~GeV\\
H1 CC  $98-99$&  $0.013  $ &  $0.40$        &  $300$   & $15\,000$
              &  $1.8$     &~\cite{h1elec}  &  $e^-p$ $\sqrt{s}\simeq 320$~GeV\\
H1 NC  $99-00$&  $0.0032$  &  $0.65$        &  $150$   & $30\,000$
              &  $1.5$     & this rep.      &  $e^+p$
 $\sqrt{s}\simeq 320$~GeV; incl. {\em high-$y$ anal.}\\
H1 CC  $99-00$&  $0.013  $ &  $0.40$        &  $300$   & $15\,000$ 
              &  $1.5$     & this rep.      &  $e^+p$ $\sqrt{s}\simeq 320$~GeV\\
H1 NC  $98-99$&  $0.00131$ &  $0.0105$      &  $100$   & $800$ 
              &  $1.5$     & this rep.      &  $e^+p$
 $\sqrt{s}\simeq 320$~GeV; {\em high-$y$ analysis}\\
BCDMS-p       &  $0.07   $ &  $0.65$        &  $7.5$   & $230$ 
              &  $3.0$     & ~\cite{bcdms}  &  $\mu p$; require
 $y_{\mu}>0.3$ \& $x<0.7$ \\
BCDMS-D       &  $0.07   $ &  $0.65$        &  $7.5$   & $230$   
      &  $3.0$     & ~\cite{bcdms}  &  $\mu D$; require $y_{\mu}>0.3$ \& $x<0.7$ \\
 \hline
 \end{tabular}
 \end{center}
 \caption[RESULT]
 {\sl \label{dataset} Table of data sets used in the QCD fit to the 
H1 $ep$ and the BCDMS $\mu $-proton and $\mu$-Deuteron scattering data.
 As for the previous H1 QCD analysis~\cite{h1lowq2},
 the original BCDMS data are used 
 imposing the constraint  $y_{\mu}>0.3$. The normalisation
 uncertainties of each data set ($\delta \mathcal{L}$) are given as
 well as the kinematic ranges in $x$ and $Q^2$.}
 \end{table}

The experimental accuracy of the initial distributions is of the order
of a few percent at small $x$. The gluon distribution is consistent
with the distribution obtained previously if the effect of the
different heavy flavour treatment is taken into account, see
\cite{h1lowq2}. 

The parameters of this analysis are specified in table~\ref{tabmodunc}.
The model uncertainties are determined in a similar manner to \cite{h1lowq2},
and the sources of uncertainty are specified in table~\ref{tabmodunc}.
Within the considered functional form, see
eq.~\ref{eqpara}, the parameterisation given in eq.~\ref{eqparh1b} 
is found uniquely. Possible
variations within the $\Delta\chi^2 \simeq 1$ region of the
parameter space do not lead to noticeably different
distributions. Thus no account is made for a parameterisation
uncertainty in this analysis. A completely different ansatz, however,
may well lead to different initial distributions as seen, for example,
in the complicated shape of $xg$ chosen in \cite{mrst2001}. The DGLAP
evolution mechanism equalises such possible initial differences to a
remarkable extent in the high $Q^2$ region to which the data presented
here belong.

%
\begin{table}[h]
  \begin{center}
    \begin{tabular}{|l|c|c|}
\hline
   source of  uncertainty &  central value &  variation \\
\hline
    $ Q^2_{min}$,\gv             &    $ 3.5   $  &  $2.0\:-\:5.0$  \\
    $ Q^2_{0} $,\gv              &    $ 4.0   $  &  $2.0\:-\:6.0$  \\
%    $y_{\mu} >$                 &    $ 0.3   $  &  $0.3\:-\:0.4$ \\
    $\alpha_s(M_Z^2)$            &    $ 0.1185$  &  $0.1165\:-\:0.1205$ \\
$f_s$,strange fraction of $\bD$  &    $ 0.33  $  &  $0.25\:-\:0.40$  \\ 
$f_c$,charm  fraction of $\bU$   &    $ 0.15  $  &  $0.10\:-\:0.20$  \\ 
    $ m_c            $,\,GeV     &    $ 1.4   $  &  $1.2\:-\:1.6$    \\ 
    $ m_b            $,\,GeV     &    $ 4.5   $  &  $4.0\:-\:5.0$ \\
\hline
    \end{tabular}
    \caption{ \sl Model uncertainties considered 
     in the QCD analysis.}
    \label{tabmodunc}
  \end{center}
\end{table}
%

%%%%%%%%%%%%%%%%%
\section{Results}
%%%%%%%%%%%%%%%%%
\label{results}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{NC and CC Cross Sections 
{\boldmath ${\rm d}\sigma/\rm{d}Q^2$}
% and {\boldmath $\sigma_{CC}^{tot}$}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{integ}

The $e^+p$ neutral current cross section ${\rm d}\sigma/\rm{ d}Q^2$ from the 99-00 data
is shown in fig.\ref{dsdq2nc}(a). The data are
compared to previous H1 $e^+p$ measurements
made at $\sqrt{s}\simeq 300$~GeV.
The new cross sections are found to be higher than the measurement
from $94-97$ as expected due to the increase in centre-of-mass energy.
Both cross sections are well described by the Preliminary H1 2002 PDF Fit.

Fig.\ref{dsdq2nc} (b) shows the ratios of the measurements to the
corresponding Standard Model expectation.  The Standard Model
uncertainty represents the effect of the assumptions and the
experimental uncertainty of the QCD analysis and is detailed in
section~\ref{param}. The new data are observed to agree well with the
previously published data. There
is a tendency observed for the data to be larger than the expectation
by a few percent which is consistent with the normalisation shift
imposed by the QCD fit.

The $Q^2$ dependence of the charged current cross section from the 99-00 data is shown in
fig.\ref{dsdq2cc}(a) after correction for the kinematic cuts
$0.03<y<0.85$ and $P_{T,h}>12$~GeV using the Preliminary H1 2002 PDF Fit. The data
are compared to the previous measurement taken at lower centre-of-mass
energy. 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. At high $Q^2$ the data
tend to be larger than the fit result which is essentially driven by the BCDMS data in this region.

Fig.\ref{fig:dsdq2nccc} shows the $Q^2$ dependence of the NC and CC
cross sections representing the total $e^+p$ and $e^-p$ data sets
taken in the first phase of HERA operation. At low $Q^2$ the NC cross
section exceeds the CC cross section by more than two orders of magnitude. The sharp
increase of the NC cross section with decreasing $Q^2$ is due to the
dominating photon exchange cross section $\propto 1/Q^4$.
In contrast, the CC cross section $\sim
\left[M_W^2/(Q^2+M_W^2)\right]^2$ approaches a constant at low
$Q^2$. The CC and NC cross sections are of comparable size at $Q^2\sim
10^4$~GeV$^2$ where the photon and $Z^0$ exchange contributions to
the NC process are also of similar size. These measurements thus
demonstrate unification of the EW interaction in deep inelastic
scattering. The difference of CC cross sections between $e^+p$ and
$e^-p$ scattering is due to the different up and down quark densities
and the less favourable helicity factor in $e^+p$ interactions.

%---------------------------------------------
\subsection{NC and CC Double Differential Cross Sections}
%---------------------------------------------

The double differential NC reduced cross
section,~$\tilde{\sigma}_{NC}$ (defined in eq.~\ref{Rnc}), is shown in
fig. \ref{nc_stamp} for the $99-00$ $e^+p$ data for both the nominal
and high-$y$ analyses (see section~\ref{ncmeas}). In addition the
high-$y$ analysis results of the $98-99$ $e^-p$ data are presented.
The data agree well with the predictions of the Preliminary H1 2002 PDF Fit which
is also shown\footnote{Note, the normalisation as determined by the
  QCD fit is not applied to the data shown in
  the figure.}. The rise of the Standard Model cross section towards
low $x$ departs from a monotonic behaviour at the lowest $x$ due to the contribution of
the longitudinal structure function that enters approximately $\propto y^2$.

In fig~\ref{fig:nc_hixc} the new $e^+p$ NC cross section data at
$\sqrt{s}\simeq 320$~GeV are compared with the data obtained
previously~\cite{h1hiq2} at $\sqrt{s}\simeq 300$~GeV.  The two data sets are
found to be in agreement with each other and with the Preliminary H1 2002 PDF Fit.
Fig.\ref{fig:nc_hixc} also shows the data from recent H1 measurement at lower
$Q^2$ \cite{h1lowq2} and the fixed-target data from
BCDMS~\cite{bcdms}.  At the highest $Q^2$ a decrease of the cross
section is expected due to the negative $\gamma Z$ interference in
$e^+p$ scattering which needs still higher luminosity to be studied in
detail.

In fig.~\ref{cc_stamp} the reduced CC cross section,
~$\tilde{\sigma}_{CC}$ (defined in eq.~\ref{Rnc}), is shown for the
new data and the data taken in $94-97$. These data are found to be consistent.
The combined result is thus shown in fig. \ref{cc_stampcomb} and is well
described by the Preliminary H1 2002 PDF Fit.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Extraction of the Proton Structure Functions 
{\boldmath $F_2$} and {\boldmath $F_L$} }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{f2fl}
The measured NC double differential cross section is dominated by the
structure function $F_2$ in most of the kinematic range. In order to
extract $F_2$ the reduced cross section must be corrected for the contribution
from the other structure functions and for $Z^0$ exchange according to
the equation
\begin{equation}
\label{f2corr}
\tilde{\sigma}_{NC}=F_2(1+\Delta_{F_2}+\Delta_{F_3}+\Delta_{F_L})
(1+\Delta^{weak}_{NC})=F_2(1+\Delta_{all})\;\;\;.
\end{equation}
Here the correction terms $\Delta_{F_2}$ and $\Delta_{F_3}$ account
for the relative contribution of pure $Z^0$ exchange and $\gamma Z^0$
interference to $\Ftwo$ and $x\Fz$, whilst $\Delta_{F_L}$ originates from
the longitudinal structure function $\FL$. 
$\Delta_{all}$ represents the combination of these corrections with the weak radiative corrections.
At high $y$ and low $Q^2
(\lapprox 1000$~GeV$^2)$ $\Delta_{F_L}$ is sizeable thus allowing the
cross section measurement to be used to determine $F_L$. The
extraction of $F_2$ in this $Q^2$ region is restricted to the kinematic
range $y<0.6$. It is extended to higher $y$ at larger $Q^2$ in order
to study the scaling violations.

Fig.\ref{f2_plot} shows the extracted $F_2$ data using the combined
$94-97$ and $99-00$ high $Q^2$ $e^+p$ data sets.
% as described in section~\ref{comb}.
In addition the figure also shows the recent H1
$F_2$ measurements at lower $Q^2$. The H1 data cover a range of five
orders of magnitude in $x$ and $Q^2$. The Preliminary H1 2002 PDF fit is able to
provide a good description of the data over the full region, and in
particular of the scaling violations observed in the measurements.

As stated above the cross sections at high $y>0.63$ and $Q^2 \lapprox
1000$~GeV$^2$ are used here to determine $F_L$. In order to optimise the
sensitivity to $F_L$, the measured cross sections in two neighbouring
$Q^2$ bins are combined. Statistical errors are added in quadrature
and systematic uncertainties are assumed to be fully correlated between
the two bins. $F_L$ is then determined from the measured reduced cross
sections after combination using the formula:
%\begin{equation}
%F_L = \frac{1}{y^2}\left[ Y_+\Ftwo - Y_-x\tilde{F}_3
%      -\tilde{\sigma}_{NC}\right],
%\end{equation}
\begin{equation}
\tilde{F}_L = \frac{Y_+}{y^2}\left[ \Ftwo
        -\frac{Y_-}{Y_+}x\tilde{F}_3
        -\tilde{\sigma}_{NC} \right]\,\,.
\label{eqfl:ew}
\end{equation}
The extraction of $F_L$ using eq.~\ref{eqfl:ew} relies upon the
extrapolation of $F_2$ into the high $y$ region. In the kinematic
range considered the EW contributions to eq.~\ref{eqfl:ew} are small
and are corrected for. This equation
reduces to 
\begin{equation}
F_L = \frac{Y_+}{y^2} \left[F_2 -\frac{\tilde{\sigma}_{NC}}{1+\Delta^{weak}_{NC}}\right]\,\,,
\label{eqfl}
\end{equation}
where $F_2$ is determined from a dedicated QCD fit to H1 data only,
restricted to $y<0.35$~\cite{flpaper}. This fit follows exactly the
same procedure as described in section~\ref{qcdana}. It results in a
$\chi^2/$ndf$=424/(455-15)=0.96$ and agrees with the H1 only fit over
the full $y$ range. 

In the extraction of the longitudinal structure function the
experimental cross sections are renormalised using the results of the
fit. In addition small shifts of the data arising from the optimised
correlated systematic uncertainties are applied. This is necessary in
order to have a consistent description of the data at low $y<0.35$ and
at high $y$. The uncertainties on $F_L$ due to the correlation 
of systematic error
sources common to the region $y<0.35$ and high $y$ are taken into
account. The additional systematic uncertainty due to the background
charge asymmetry is added in quadrature. 

In fig.\ref{fl_plot} the determinations of $F_L$ at high $Q^2$
are shown at fixed $y=0.75$ for both the $e^+p$ and the $e^-p$ data
sets. The results from both data sets are mutually consistent and are in agreement with the
Preliminary H1 2002 PDF Fit. The extreme values allowed for $F_L$ ($F_L=0$ and
$F_L=$\Ftwo) are clearly excluded by the data.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Quark Distributions {\boldmath $xu$} and {\boldmath
    $xd$} at Large {\boldmath $x$}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{quarks}

The flavour composition of the proton at high $x$ may be disentangled
by exploiting the NC and CC cross section measurements. The $e^+p$ CC
cross section at large $x$ is dominated by the $d$ quark contribution
as may be inferred from fig.~\ref{cc_stampcomb}. Similarly the $u$ density
dominates the $e^-p$ CC cross section. In addition the $e^+p$ and  $e^-p$ NC cross
sections are dominated by the $u$ density at high enough $x$ for the sea to be neglected. Requiring the $u$ or $d$ contribution to provide a
minimum $70\%$ of the cross section, as inferred from the Preliminary H1 2002 PDF Fit,  allows the up and down quarks to
be determined locally.
The extraction relies on weighting the differential cross section measurement with the expected local flavour contribution and is illustrated in fig.~\ref{fig:xuxd}. The method of
extraction relies directly on the {\em local} cross section
measurements and is complementary to performing a QCD fit since the
extraction is insensitive to QCD assumptions.

The data are compared to the results of the Preliminary H1 2002 PDF Fit and to
other global parameterisations from the MRST~\cite{mrst2001} and
CTEQ~\cite{cteq6} collaborations. This procedure is an indirect
determination of the $u$ and $d$ distributions in
the valence quark region. Note, however, that the separation between
quarks and anti-quarks, and thus the extraction of the valence quark
distributions, is entirely dependent to the helicity structure of the CC
cross sections and the $Z^0$ exchange parts of the NC cross section. A
measurement of the valence quark densities from H1 data alone must
be deferred to the high luminosity phase of HERA.

%%%%%%%%%%%%%%%%%%
\section{Summary}
%%%%%%%%%%%%%%%%%%
\label{summary}

New data are presented on inclusive deep inelastic neutral and charged
current scattering cross sections at high momentum transfers $Q^2 >
100$\,GeV$^2$ from recent ($1999-2000$) positron-proton runs at HERA
with a luminosity of $65.2$~pb$^{-1}$. These data taken together with
previous analyses of the $1994-1997$ $e^+p$ data ($35.6$~pb$^{-1}$) and
$1998-1999$ $e^-p$ runs ($16.4$~pb$^{-1}$) represent the now complete
analysis of the high $Q^2$ data obtained during the first phase of
HERA operation over nearly a decade of data taking.

The accuracy of the NC measurements presented here has reached the
level of a few \% in the mid $Q^2$ range of $Q^2<3000$\,GeV$^2$, 
a precision which is largely due to a better understanding of the  LAr calorimeter performance and
calibration of the electromagnetic and the hadronic energy scales. Both the CC and the very high $Q^2$ ($Q^2>M_Z^2$) NC
data are still limited by their statistical accuracy to about
$10-20$\% uncertainties which are expected to be diminished
in the future high luminosity phase of HERA.

For both positron and electron scattering data the region of very
large inelasticity is explored which allows a determination of the
longitudinal structure function $F_L(x,Q^2)$ in the
large momentum transfer range, $100< Q^2<700$~GeV$^2$.

The NC and CC cross sections at high $Q^2$ measure linear combinations
of the sums of up and down quark and antiquark momentum densities,
$U,D,\bar{U}$, and $\bar{D}$.  Introducing these four distributions
into the minimisation procedure, a novel NLO QCD analysis is performed
aiming at a determination of the partonic nucleon structure from
inclusive DIS scattering data alone. In this Preliminary H1 2002 PDF fit the low
$Q^2$ region is constrained by recent H1 $ep$ scattering data at
low $x$ and by muon-nucleon data of the BCDMS collaboration at high
$x$.

The QCD analysis leads to a good description of all the NC and CC
cross section data and of the derived structure functions over the large
range of nearly five orders of magnitude in $x$ and $Q^2$.  The
analysis presented here leads to the first simultaneous unfolding of
four quark distributions and the gluon distribution with consistent results from
the analysis of H1 data alone with the Preliminary H1 2002 PDF fit.

% The NC and CC cross sections have been measured for $e^+p$ scattering
% at a centre-of-mass energy of $\sqrt{s} \simeq 320$ GeV.  Standard Model
% expectations based on the H1 2002 PDF Fit to all H1 NC and CC data 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.

% The data are compared to a previous measurement of H1 NC and CC cross
% sections with $\sqrt{s} \approx 300$ GeV. 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. 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.


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\bibitem{qcdfit} C. Pascaud and F. Zomer, LAL preprint, LAL/95-05 (1995)

\bibitem{qcdnum} M. Botje, QCDNUM15 program, {\em write-up in
preparation}; J. Bl\"umlein et al., Proceedings of the Workshop
``Future Physics at HERA'', vol. 1, eds. G. Ingelman, A. De Roeck,
R. Klanner, DESY (1996) 23.

\bibitem{mrst2001}
%\cite{Martin:2001es}
A.~D.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne,
%``MRST2001: Partons and alpha(s) from precise deep inelastic scattering  and Tevatron jet data,''
Eur.\ Phys.\ J.\ C {\bf 23} (2002) 73 [hep-ph/0110215].
%%CITATION = HEP-PH 0110215;%%

\bibitem{cteq6} 
%\cite{Pumplin:2002vw}
J.~Pumplin, D.~R.~Stump, J.~Huston, H.~L.~Lai, P.~Nadolsky and W.~K.~Tung,
%``New generation of parton distributions with uncertainties from global  QCD analysis,''
JHEP {\bf 0207} (2002) 012 [hep-ph/0201195].
%%CITATION = HEP-PH 0201195;%%

\bibitem{flpaper} 
%\cite{Adloff:1996yz}
C.~Adloff {\it et al.}  [H1 Collaboration],
%``Determination of the longitudinal proton structure function  F(L)(x,Q**2) at low x,''
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%%CITATION = HEP-EX 9611017;%%

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%\bibitem{minuit} F. James, CERN Program Library, D506.

%
%
\end{thebibliography}
\newpage

%-----------------------------------------------------------
% pdfs H1+BCDMS Fig1
\begin{figure}[p] 
\setlength{\unitlength}{1 mm}
\begin{center}
\begin{picture}(160,150)(0,0)
\put(0,0) {\epsfig{file=H1prelim-02-053.fig1.eps,width=16cm}}
\put( 40,137){\bf (a)}
\put(120,137){\bf (b)}
\put( 40, 87){\bf (c)}
\put(120, 87){\bf (d)}
\put( 40, 37){\bf (e)}
\end{picture}
\end{center}
\caption{Parton distributions (a) $xU$, (b) $x\bar{U}$,(c) $xD$, (d)
  $x\bar{D}$, and (e) $xg$ as determined from the Preliminary H1 2002 PDF Fit to
  H1 and BCDMS data. The distributions are shown at $Q^2=4$~GeV$^2$
  with experimental and model uncertainties. The valence quark
  densities $xu_v$ (a) and $xd_v$ (c) are also shown. 
  determined with H1 and BCDMS. For comparison, the central values from the fits to H1 data alone are also shown.} 
\label{figpdfhb}
\end{figure} 

%-----------------------------------------------------------
\begin{figure}[htbp]
\setlength{\unitlength}{1 mm}
\begin{center}
\begin{picture}(160,185)(0,0)
\put(5,52) {\epsfig{file=H1prelim-02-053.fig2a.eps,width=0.9\textwidth}}
\put(5,-12) {\epsfig{file=H1prelim-02-053.fig2b.eps,width=0.9\textwidth}}
\put(35,183){\bf (a)}
\put(35, 57){\bf (b)}
\end{picture}
\end{center}
\caption{\sl 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 Preliminary H1 2002 PDF Fit. The
  full (dashed) line corresponds to a center-of-mass energy of
  $\sqrt{s}\simeq 300$ GeV ($\sqrt{s}\simeq 320$ GeV).  The ratio of the $94-97$ and
  $99-00$ data to their respective Standard Model expectations is shown in
  (b). 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}[htbp]
\setlength{\unitlength}{1 mm}
\begin{center}

\begin{picture}(160,185)(0,0)
\put(5,52) {\epsfig{file=H1prelim-02-053.fig3a.eps,width=0.9\textwidth}}
\put(5,-12) {\epsfig{file=H1prelim-02-053.fig3b.eps,width=0.9\textwidth}}
\put(35,183){\bf (a)}
\put(35, 57){\bf (b)}
\end{picture}
\end{center}
\caption{\sl 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 Preliminary H1 2002 PDF Fit. The
  full (dashed) line corresponds to a center-of-mass energy of
  $\sqrt{s}\simeq 300$ GeV ($\sqrt{s}\simeq 320$ GeV). The ratio of the $94-97$ and
  $99-00$ data to their respective Standard Model expectations is shown in
  (b). 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}[htbp]
\setlength{\unitlength}{1 mm}
\begin{center}
\begin{picture}(160,160)(0,0)
%\put(0,0){\epsfig{file=nccc_dsdq2_3.eps,width=\textwidth}}
%AAA\put(0,0){\epsfig{file=h1-02-pdf/nccc_dsdq2.eps,width=\textwidth}}
\put(0,0){\epsfig{file=H1prelim-02-053.fig4.eps,width=\textwidth}}
\end{picture}
\end{center}
\caption{\sl The $Q^2$ dependences of the NC (circles) and CC (squares)
cross sections $d\sigma/dQ^2$ are shown for the combined $94-00$ $e^+p$
(solid points) and $e^-p$ (open points) measurements. The data are
compared to the Standard Model expectations determined from the Preliminary H1 2002
PDF Fit.   The 1.5\% luminosity uncertainty is not
  included in the error bars.}
\label{fig:dsdq2nccc}
\end{figure}
%-----------------------------------------------------------
\begin{figure}[htbp]
\center
%\epsfig{file=f2.ps,bbllx=20,bblly=45,bburx=570,bbury=805,width=14cm}
%AAA\epsfi1g{file=h1-02-pdf/f2_errband_bw.ps,bbllx=20,bblly=45,bburx=570,bbury=805,width=14cm}
%\epsfig{file=plots/f2_9900pub_0.eps,bbllx=20,bblly=45,bburx=570,bbury=805,width=14cm}
\epsfig{file=H1prelim-02-053.fig5.eps,width=15cm}
\caption{\sl The NC reduced cross section $\tilde{\sigma}_{NC}$ is
compared to the Preliminary H1 2002 PDF Fit. The $99-00$ $e^+p$ data and the {\em high-$y$
analysis} $e^-p$ data of $98-99$ data are shown. The inner error bars
represent the statistical error, and the outer error bars show the
total error.  The 1.5\% luminosity uncertainty is not
  included in the error bars.}
\label{nc_stamp} 
\end{figure}
%-----------------------------------------------------------
\begin{figure}[htbp]
\setlength{\unitlength}{1 mm}
\begin{center}
%\begin{picture}(160,160)(0,0)
%\put(-3,-12){
%\epsfig{file=fig_redx_hix_1_3.eps,width=15cm}}
%AAA\epsfig{file=h1-02-pdf/fig_redx_hix_bw_errband.eps,width=15cm}}
\epsfig{file=H1prelim-02-053.fig6.eps,width=15cm}
%}
%\end{picture}
\end{center}
\caption{\sl The NC reduced cross sections $\tilde{\sigma}_{NC}$ from the
$94-97$ and $99-00$ $e^+p$ data are shown at high $x$ compared to the Preliminary H1 2002 PDF
Fit. Also shown are data from H1 measured at lower $Q^2$, as well
as from the fixed-target experiment BCDMS. The dashed (solid) curve
represents the Standard Model expectation based on the Preliminary H1 2002 PDF Fit
for $\sqrt{s}\simeq 320$ GeV ($\sqrt{s} \simeq 300$ GeV). The inner error bars
represent the statistical error, and the outer error bars show the
total error.  The 1.5\% luminosity uncertainty is not
  included in the error bars.}
\label{fig:nc_hixc} 
\end{figure}
%-----------------------------------------------------------
\begin{figure}[htbp]
\center 
\epsfig{file=H1prelim-02-053.fig7.eps,width=15cm}
\caption{\sl The CC reduced cross section $\tilde{\sigma}_{CC}(x,Q^2)$ is
shown for $e^+p$ scattering for the new data from $99-00$ at
$\sqrt{s} \simeq 320$ GeV (solid circles) and the data from $94-97$
at $\sqrt{s} \simeq 300$ GeV (open circles). The data are compared to
the Preliminary H1 2002 PDF Fit with the full and dashed curves showing respectively
predictions for $\sqrt{s} \simeq 320$ GeV and $\sqrt{s} \simeq 300$ GeV. 
The inner error bars represent the
statistical error, and the outer error bars show the total error.  The 1.5\% luminosity uncertainty is not
  included in the error bars.}
\label{cc_stamp}
\end{figure}
%-----------------------------------------------------------
\begin{figure}[htbp]
\center 
%\epsfig{file=ccplot_3.eps,width=15cm}
%AAA\epsfig{file=h1-02-pdf/ccplot_3_errband_bw.eps,width=15cm}
\epsfig{file=H1prelim-02-053.fig8.eps,width=15cm}
\caption{\sl 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
$\sqrt{s} \simeq 320$ GeV. The full line shows the Preliminary H1 2002 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.  The 1.5\% luminosity uncertainty is not
  included in the error bars.}
\label{cc_stampcomb}
\end{figure}
%-----------------------------------------------------------
\begin{figure}[htbp]
\center 
\epsfig{file=H1prelim-02-053.fig9.eps,width=\textwidth}
\caption{\sl The proton structure function $F_2$ from H1 data compared
to the Preliminary H1 2002 PDF Fit. Also shown are the $F_2$ $\mu p$ scattering
measurements from BCDMS.}
\label{f2_plot}
\end{figure}
%-----------------------------------------------------------
\begin{figure}[htbp]
\center 
\epsfig{file=H1prelim-02-053.fig10.eps,width=\textwidth}
\caption{\sl 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 inner error bar represents the
statistical error, the intermediate error bar shows the systematic
uncertainty contribution, and the outer error bar includes the uncertainty
arising from the extrapolation of $F_2$. The lower shaded band shows
the expectation of $F_L$, and its uncertainty, from the Preliminary H1 2002 PDF
fit. The upper shaded band shows the $F_2$ extrapolation at
$y=0.75$. This is theoretically the maximum allowed value for $F_L$. }
\label{fl_plot}
\end{figure}
%-----------------------------------------------------------
\begin{figure}[htbp]
\begin{center}
\begin{picture}(50,160)
\put(-60,0){\epsfig{file=H1prelim-02-053.fig11.eps,
width=\textwidth}}
\end{picture}
\end{center}
\caption{\sl The quark distributions $xu$ and $xd$ determined 
both with the Preliminary H1 2002 PDF 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 
parameterisations which included data from a variety of other scattering processes.}
\label{fig:xuxd}
\end{figure}
%-----------------------------------------------------------
% \begin{figure}[htb]
% \setlength{\unitlength}{1 mm}
% \begin{center}
% \begin{picture}(50,160)
% \put(-60,0){\epsfig{file=plots/paper_pdf_h1.eps,width=\textwidth}}
% \end{picture}
% \end{center}
% \caption{\sl The PDFs (a) $xu$, (b) $xd$, (c) $xg$, and (d) $xSea$
%   from the Preliminary H1 2002 PDF Fit at $Q^2=5$GeV$^2$. The inner shaded band
%   represents the uncertainty arising from the experimental
%   measurements. The outer band includes the theoretical uncertainty of
%   the fit. Also shown are comparisons with global parameterisations
%   MRST (dashed) and CTEQ (dotted).}
% \label{h1pdf}
% \end{figure}
%-----------------------------------------------------------

\end{document}




