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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 $900.03$. 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 $0.031\,000$ 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 $110150$ 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}