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%
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\begin{document}

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\begin{titlepage}

\noindent
March 2010 \\
H1prelim-10-043 \\
ZEUSprelim-10-001 \\


\vspace*{4cm}

\begin{center}
\begin{Large}

{\bfseries Combined Measurement of the Inclusive $\mathbold{e^+p}$
Scattering
 Cross Sections  at HERA for Reduced Proton Beam Energy Runs and Determination of Structure Function $\mathbold{F_L}$}

\vspace*{2cm}

H1 and ZEUS Collaborations

\end{Large}
\end{center}

\vspace*{2cm}

\begin{abstract} \noindent
A combination of the inclusive deep inelastic cross
sections measured by the H1 and ZEUS Collaborations
%in neutral and charged current
for $ep$ scattering with nominal and reduced proton-beam energies, $E_p=920$~GeV, $E_p=460$~GeV and $575$~GeV, is presented.
%The data span six orders of magnitude in
%negative four-momentum-transfer squared, $Q^2$, and in Bjorken $x$.
%and were taken in the first period of operating the
%collider.
The combination method used takes the correlations of
systematic uncertainties into account,
% and represents
% a cross calibration of the consistent H1 and ZEUS measurements
resulting in improved accuracy. From the combined data the proton structure
function, $F_L$, is extracted in the region of $2.5$ $\le$ $Q^{2}$ $\le$ $800$ GeV$^{2}$. \newline

\end{abstract}

\vspace*{1.5cm}

\begin{center}
%{\slshape Submitted to }
\end{center}

\end{titlepage}

\section{Introduction}
At low absolute four momentum transfer squared, $Q^2$, the neutral
current deep-inelastic
scattering (DIS) cross section is defined by the two structure functions, $F_2$
and $F_L$. In its reduced form the cross section can be written as
\begin{equation}
\sigma_r(x,Q^2) = F_2(x,Q^2) - f(y)F_L(x,Q^2),~~~~~~ f(y)= \frac{y}{1+(1-y)^2}.
\end{equation}
Here $x$ is the Bjorken-$x$ variable and $y$ is the inelasticity of the process.
The structure function $F_2$ has the dominant contribution to the cross section,
while $F_L$ is only visible at large values of $y$. Measurements of the reduced cross sections at
fixed $(x,Q^{2})$ and different $y$ would allow to extract $F_{2}$ and $F_{L}$ simultaneously.
%To measure both
%structure functions it is necessary to measure the reduced cross section
%for the same values of $x$ and $Q^2$ at different $y$.
For this purpose, the HERA collider undertook reduced proton-beam energy
runs with $E_p=460$~GeV and $E_p=575$~GeV at the end of its operation.
These data were used by the H1 and ZEUS collaborations to perform the
first measurements of $F_L$~\cite{h1fl,Chekanov:2009na}.

In this note a combined measurement of the DIS cross sections at the reduced
proton-beam energies is presented.
%The combination is based on the published %ZEUS~\cite{Chekanov:2009na} and preliminary H1 results~\cite{h1prelimSpa,h1prelimLar}.
The combination is based on the method developed
in~\cite{glazov,Collaboration:2009bp} and used
previously for the combination of the H1 and ZEUS data collected at $E_p=820$~GeV and $E_p=920$~GeV~\cite{h1zeus}. The combined cross sections at reduced and nominal $E_p$ are used to extract $F_L$.
%and to perform an Next-to-Leading Order (NLO) QCD analysis.

\section{Data Sets}
Input data for the combination are the ZEUS measured cross sections at $E_p=460$, $575$ and $920$~\cite{Chekanov:2009na},
preliminary H1 measurements at reduced $E_p$ using SpaCal~\cite{h1prelimSpa} and LAr
calorimeter~\cite{h1prelimLar} as well as published H1 measurement at
$E_p=920$~GeV~\cite{Collaboration:2009bp,Aaron:2009kv}. The data cover
wide range in $2.5\le Q^2 \le 800$~GeV$^2$ and $0.85\le y \le 0.1$.

Both H1 and ZEUS collaborations use electron method for kinematic reconstruction
which has optimal resolution at high $y$. The systematic errors arise from
electromagnetic and hadronic energy scales, detector alignment, background
subtraction and electron reconstruction efficiency.

\section{Procedure}

%The method for the combination is adopted from earlier combination of H1 and ZEUS data ~\cite{Collaboration:2009bp}.
For the combination a $\chi^2$ minimisation method is used.
%The $\chi^2$ function takes into account the correlated systematic uncertainties for the H1 and ZEUS cross-section measurements.
The combination procedure allows to average H1 and ZEUS data taking into account
correlations due to systematic uncertainties.
Before the combination, data are corrected to a common $x,Q^2$ grid.
%For $Q^2\le 15$~GeV$^2$
%and $Q^2\ge 150$~GeV$^2$, the grid is chosen based on H1 data since
%there are no new ZEUS results reported in this kinematic domain.
%For intermediate $Q^2$ region, $y$ values are taken from the H1
%grid for the measurement at $E_p=920$~GeV, and $Q^2$ values from the ZEUS
%grid, $x$ values are calculated
%as $x=Q^2/sy$.
%The $E_p=920$~GeV $y$ binning is chosen since it is the finest in $y$.
Swimming corrections are based on HERAPDF1.0 parameterisation
for the structure functions $F_2$ and $F_L$.
%The swimming is performed to the bin closest in $Q^2$ and in $y$
%(for HERAPDF1.0 average, bin closest in $Q^2$ and $x$ was chosen).

% TEXT on SENSITIVITY of x-sections, FL to the swimming corrections.
%Sensitivity of the result to the assumptions made for $F_2$ and $F_L$
%are tested by using H1PDF2000 parameterisation for $F_2$ and various
%assumptions used for $F_L$. Assuming $F_L = R/(R+1)F_2$ with $R=0.25$ and $R=0.5$
%changes the average by at most $1.1\%$ at
%highest $y=0.85$ which is negligible
%compared to the experimental uncertainty in this kinematic domain
%(about $5\%$). For modification of $F_2$, the observed variations are also found negligible.
%compared to experimental uncertainties  .



%TEXT on SWIMMING to same X,Q2 within EXPERIMENT\\
%ZEUS binning for measurement of the cross sections at reduced $E_p$ is finer then H1 binning. If more then one measured cross section for ZEUS correspond to a single  grid point, these measurements are first averaged using
%statistical uncertainties  and then added to the combination with H1 data.

%\begin{table}
%\caption{\label{tab:tab1}$\chi^2/\dof$ for separate combinations of the H1 and ZEUS at different $E_p$.}
%\begin{center}
%\begin{tabular}{ccc}
%\hline
%   $E_e=460$~GeV      & $E_p=575$~GeV   &  $E_p = 920$   \\
%%\hline
%\end{tabular}
%\end{center}
%\end{table}

\section{Combination Results}

To extract $F_L$, cross sections measured at two or more available proton beam energies
%$E_p=920$~GeV
in the same $x,Q^2$ binning are required.

At $y<0.35$ the sensitivity of the cross section to the structure
function $F_L$ is small, therefore the data sets for all
$E_p$ are averaged in this kinematic domain. This allows to reduce relative normalization uncertainty among the data sets.

The cross-section average for the low energy combination is shown in \Fig~\ref{fig:figxs} for all $Q^2$ bins and in \Fig~\ref{fig:figxszoom} for the
$Q^2$ bins were both H1 and ZEUS measurements contribute to the average.


\section{Extraction of $\boldsymbol{F_L}$}

Extraction of the structure function $F_L$ is performed using the offset method. The central values are
obtained fitting cross sections with statistical uncertainties only. The total uncertainty on $F_L$ is calculated performing the
fit with each of the systematic errors separately, and adding the difference to the total error, together with the uncorrelated part.


The structure function $F_L$ is measured as a slope of a linear fit of $\sigma_r(x,Q^2,f(y))$
versus $f(y)$. For illustration, these fits are shown for $Q^2=32$~GeV$^2$ bin in \Fig~\ref{fig:rosen}.
%At low $y$, the difference in $f(y)$ for the measurements at different $E_p$ is small. This region
%is used to perform cross normalisation of the data at reduced $E_p$, which is corrected to $E_p=920$~GeV.
%The resulting correction factors are  $+X.X\pm 0.X\%$ for $E_p=460$ and $-Y.Y\pm 0.Y\%$ for $E_p=575$~GeV
%data.
The measured structure function $F_L$ is shown in \Fig~\ref{flcomb}.
% It is, somehow, positive.
The measurements are compared to the prediction of the HERAPDF1.0 fit.


The range in $x$ covered by
the measurements for each $Q^2$ value is limited and for this range variation of $F_L$
is expected to be small.
The measurements are therefore averaged for each $Q^2$ bin to obtain
more compact representation of the data.
This average is performed using total uncertainties: the systematic errors have strong $y$ depencence and
hence neighbouring in $x$ $F_L$ points are to a large extend uncorrelated.
The averaged structure function $F_L$
is shown in \Fig~\ref{fig:flave} and it is compared to HERAPDF1.0 prediction.
The region were both H1 and ZEUS measurements are present is zoomed in
\Fig~\ref{fig:flzoom}.


\section{Conclusions \label{sec:sum}}

An averaging of the reduced proton beam energy DIS cross-section measurements by the H1 and ZEUS collaborations
is reported. The average allows to represent the results by a single dataset with reduced total
uncertainty. The averaged cross sections are used to extract the structure function $F_L$
%which is found
%to be consistent with HERAPDF1.0 predictions for $Q^2\ge 12$~GeV$^2$, for lower $Q^2$ the data are above the
%prediction.

\section*{Acknowledgements}
\refstepcounter{pdfadd} \pdfbookmark[0]{Acknowledgements}{s:acknowledge}
%\begin{theacknowledgments}
%
We are grateful to the HERA machine group whose outstanding
efforts have made this experiment possible.
We thank the engineers and technicians for their work in constructing
and maintaining the H1 and ZEUS detectors, our funding agencies for
financial support, the DESY technical staff for continual assistance
and the DESY directorate for support and for the hospitality
which they extend to the non-DESY members of the collaborations.
%\end{theacknowledgments}

%\begin{flushleft}
%\input{h1zeus-auth-oct2009-V1}
%\end{flushleft}

%\newpage
%\tableofcontents

%\newpage
\clearpage
\bibliography{H1prelim-10-043}

\clearpage
%\input{tables}
% \input{figures}

\clearpage

\begin{figure}
\centerline{\epsfig{file=H1prelim-10-043.fig1.eps,width=\linewidth}}
\caption{\label{fig:figxs}Reduced average H1 and ZEUS cross section
for $2.5\le Q^2 \le 800$~GeV$^2$ taken at different proton beam energies $E_p$
of $920$~GeV (full boxes), $575$~GeV (stars) and $460$~GeV (full circles).
The error bars show total experimental uncertainties. Theoretical
predictions of reduced cross section are shown as solid lines
for $920$~GeV (blue), $575$~GeV (black) and $460$~GeV (magenta).
Theoretical expectations are derived from HERAPDF1.0 fit.}
\end{figure}

\begin{figure}
\centerline{\epsfig{file=H1prelim-10-043.fig2.eps,width=\linewidth}}
\caption{\label{fig:figxszoom}Reduced average H1 and ZEUS
cross section for bins were both H1 and ZEUS measurements are present
 taken at different proton beam energies $E_p$
of $920$~GeV (full boxes), $575$~GeV (stars) and $460$~GeV (full circles).
The error bars show total experimental uncertainties. Theoretical
predictions of reduced cross section are shown as solid lines
for $920$~GeV (blue), $575$~GeV (black) and $460$~GeV (magenta).
Theoretical expectations are derived from HERAPDF1.0 fit.
}
\end{figure}

%\begin{figure}
%\centerline{\epsfig{file=figs/pull.eps,width=0.8\linewidth}}
%\caption{\label{fig:pulls}Pulls to the average for $E_p=460$ and $E_p=575$~GeV
%datasets.}
%\end{figure}

\begin{figure}
\centerline{\epsfig{file=H1prelim-10-043.fig3.eps,width=0.8\linewidth}}
\caption{
The reduced cross section
$\sigma(x,Q^2,f(y))$ measured at $Q^2=32$~GeV$^2$ and different $x$ valies
as a function of $f(y)$. The full boxes represent data at $E_p=920$~GeV,
stars represnt data at $E_p=575$~GeV and full circles represent data at $E_p=460$~GeV. The inner error bars are the statistical uncertainties and the full
error bars are the total uncertainties. The lines show the linear fits used
to determine $F_L$.
\label{fig:rosen}}
\end{figure}

\begin{figure}
\centerline{\epsfig{file=H1prelim-10-043.fig4.eps,width=0.8\linewidth}}
\caption{\label{flcomb}The structure function $F_L$ measured a s function
of $x$ at fixed values of $Q^2$ obtained from the
combined H1 and ZEUS cross-section data. The inner error bars are the statistical uncertainties and the full
error bars are the total uncertainties. The line represents a QCD prediction
based on HERAPDF1.0 fit with parameterization (green) model (yellow) and experimental (red) uncertainties. }
\end{figure}

\begin{figure}
\centerline{\epsfig{file=H1prelim-10-043.fig5.eps,width=0.8\linewidth}}
\caption{\label{fig:flave}The structure function $F_L$ averaged in $x$ at given
valie of $Q^2$ using combined H1 and ZEUS cross-section data. 
The resulting $x$ values of the averaged $F_L$ measurements are given in the
figure for each point in $Q^2$. The inner error bars are the statistical uncertainties and the full
error bars are the total uncertainties. The band represents a QCD prediction
based on HERAPDF1.0 fit with parameterization (green) model (yellow) and experimental (red) uncertainties.}
\end{figure}

\begin{figure}
\centerline{\epsfig{file=H1prelim-10-043.fig6.eps,width=0.8\linewidth}}
\caption{\label{fig:flzoom}H1 and ZEUS average structure function $F_L$ 
(full circles) compared
to the individual measurements of the H1 (open circles) and ZEUS (open boxes) 
collaborations. 
The inner error bars are the statistical uncertainties and the full
error bars are the total uncertainties. The band represents a QCD prediction
based on HERAPDF1.0 fit.
}
\end{figure}


%\begin{figure}
%\centerline{\epsfig{file=figs/rosen_24.eps,width=0.8\linewidth}}
%\caption{
%The reduced cross section
%$\sigma(x,Q^2,f(y))$ measured at $Q^2=24$~GeV$^2$ and different $x$ valies
%as a function of $f(y)$. The full boxes represent data at $E_p=920$~GeV,
%stars represnt data at $E_p=575$~GeV and full circles represent data at $E_p=460$~GeV. The inner error bars are the statistical %uncertainties and the full
%error bars are the total uncertainties. The lines show the linear fits used
%to determine $F_L$.
%\label{fig:rosen24}}
%\end{figure}

%\begin{figure}
%\centerline{\epsfig{file=figs/rosen_45.eps,width=0.8\linewidth}}
%\caption{
%The reduced cross section
%$\sigma(x,Q^2,f(y))$ measured at $Q^2=45$~GeV$^2$ and different $x$ valies
%as a function of $f(y)$. The full boxes represent data at $E_p=920$~GeV,
%stars represnt data at $E_p=575$~GeV and full circles represent data at $E_p=460$~GeV. The inner error bars are the statistical uncertainties and the full
%error bars are the total uncertainties. The lines show the linear fits used
%to determine $F_L$.
%\label{fig:rosen45}}
%\end{figure}

%\begin{figure}
%\centerline{\epsfig{file=figs/rosen_60.eps,width=0.8\linewidth}}
%\caption{
%The reduced cross section
%\sigma(x,Q^2,f(y))$ measured at $Q^2=60$~GeV$^2$ and different $x$ valies
%as a function of $f(y)$. The full boxes represent data at $E_p=920$~GeV,
%stars represnt data at $E_p=575$~GeV and full circles represent data at $E_p=460$~GeV. The inner error bars are the statistical %uncertainties and the full
%error bars are the total uncertainties. The lines show the linear fits used
%to determine $F_L$.
%\label{fig:rosen60}}
%\end{figure}

%\begin{figure}
%\centerline{\epsfig{file=figs/rosen_80.eps,width=0.8\linewidth}}
%\caption{
%The reduced cross section
%$\sigma(x,Q^2,f(y))$ measured at $Q^2=80$~GeV$^2$ and different $x$ valies
%as a function of $f(y)$. The full boxes represent data at $E_p=920$~GeV,
%stars represnt data at $E_p=575$~GeV and full circles represent data at $E_p=460$~GeV. The inner error bars are the statistical uncertainties and the full
%error bars are the total uncertainties. The lines show the linear fits used
%to determine $F_L$.
%\label{fig:rosen80}}
%\end{figure}

%\begin{figure}
%\centerline{\epsfig{file=figs/rosen_110.eps,width=0.8\linewidth}}
%\caption{
%The reduced cross section
%$\sigma(x,Q^2,f(y))$ measured at $Q^2=110$~GeV$^2$ and different $x$ valies
%as a function of $f(y)$. The full boxes represent data at $E_p=920$~GeV,
%stars represnt data at $E_p=575$~GeV and full circles represent data at $E_p=460$~GeV. The inner error bars are the statistical %uncertainties and the full
%error bars are the total uncertainties. The lines show the linear fits used
%to determine $F_L$.
%\label{fig:rosen110}}
%\end{figure}


%\begin{figure}
%\centerline{\epsfig{file=figs/flaveh1.eps,width=0.8\linewidth}}
%\caption{\label{fig:flaveh1}H1 and ZEUS average structure function $F_L$
%(solid circles) compared to H1 preliminary result (open circles).
%The  $x$ values of the averaged H1 and ZEUS $F_L$ measurements are given in the
%figure for each point in $Q^2$. The inner error bars are the statistical uncertainties and the full
%error bars are the total uncertainties. The band represents a QCD prediction
%based on HERAPDF1.0 fit.
%}
%\end{figure}



\end{document}

