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\begin{titlepage}

\noindent
\begin{center}
\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
                32nd International Conference 
                on High Energy Physics, ICHEP04},
                August~16,~2004,~Beijing} \\
                 & Abstract:        & {\bf 5-0161}    &\\
                 & 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 
    Determination of the longitudinal proton structure\\ 
function {\boldmath $F_L$} 
%    in {\boldmath $e q$} Contact Interactions at HERA}
    at low {\boldmath $Q^2$} at HERA }

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

\begin{abstract}

\noindent An extraction
of the longitudinal proton structure function $F_L(x,Q^2)$ from H1
data at low $Q^2$ $\sim$ 1 GeV$^2$ is reported. The analysis is based
on the data collected in a dedicated low $Q^2$ running period in 1999
and during shifted vertex runs in 2000.  Two methods of extracting
$F_L(x,Q^2)$ are discussed.  It is shown that results from both
methods are consistent.  Theoretical predictions are compared with the
measured $F_L(x,Q^2)$ points.
\end{abstract}


\end{titlepage}

\pagestyle{plain}

\section{Introduction}
Precise measurements of the inclusive scattering cross section at the
$ep$ collider HERA are important for the understanding of proton
substructure. In the one-photon exchange approximation, which is valid
in the kinematic domain explored here, the deep inelastic scattering
(DIS) cross section is given by the expression:
$$\frac{d^2 \sigma}{dxdQ^2} = \frac{2 \pi \alpha^2}{Q^4
  x}Y_{+}(F_2(x,Q^2)-y^2F_L(x,Q^2)), \label{xs} $$
where $
Y_{+}=1+(1-y)^2 $, $Q^2$ is the squared four-momentum transfer, $x$
denotes the Bjorken scaling variable, $y = Q^2/sx$ is the
inelasticity, with $s$ the $ep$ center of mass energy squared and
$\alpha $ is the fine structure constant. The structure functions
$F_2$ and $F_L$ are related to the cross sections $\sigma _T$ and
$\sigma _L$ for the interaction of transversely and longitudinally
polarized photons with protons:
\begin{equation} F_2(x,Q^2)=\frac{Q^2}{4\pi^2\alpha}(\sigma_T(x,Q^2)+\sigma_L(x,Q^2)), \end{equation}
\begin{equation} F_L(x,Q^2)=\frac{Q^2}{4\pi^2\alpha}(\sigma_L(x,Q^2)). \end{equation}
Due to the positivity of the cross sections, the structure functions
$F_2$ and $F_L$ obey the relation:
%
\begin{equation}
 0 \le F_L \le F_2. 
\end{equation}
%
The ``reduced'' cross section is defined as:
%
\begin{equation} 
\sigma_{r}= F_2(x,Q^2) - \frac{y^2}{Y_{+}} \cdot F_L(x,Q^2). 
\end{equation}
%
In the Quark Parton Model~\cite{qpm} the photon interacts with a spin
1/2 particle having only longitudinal momentum, which leads to the so
called Callan-Gross relation~\cite{gross}: $F_L(x) = 0$.  In QCD
quarks interact through gluons, which can split into quark anti-quark
or gluon pairs. This way, the quark struck by a virtual photon has
transverse momentum $\sim Q $, which leads to $F_L(x,Q^2) > 0$.  Due
to its origin $F_L$ is directly connected with the gluon distribution
in the proton and therefore can provide a sensitive test of
perturbative QCD.



In recent years HERA structure function analyses have focussed on the
measurement of $F_2$ which is the dominating contribution to the
inclusive cross section.  In these measurements an assumption was made
on $F_L$, and $F_2$ was obtained in a wide range of $x,Q^2$ and for
inelasticities $y$ from about 0.6 down to 0.002.  At high $y$, beyond
0.6, the $F_L$ contribution to the reduced cross section becomes
significant.  Therefore, the standard procedure of extracting
$F_2(x,Q^2)$ from the DIS cross section by subtracting the
theoretically computed $F_L$ contribution can be reversed: at high $y$
the $F_L(x,Q^2)$ contribution may be extracted from the measured cross
section by subtracting a calculation of $F_2$.

The H1 cross section measurements access $y$ values as high as 0.9
since the modified backward apparatus, a silicon strip detector and a
Spaghetti calorimeter with a drift chamber attached, enable scattered
electrons to be identified down to 3~GeV energy. This allowed the
longitudinal structure function to be accessed using different methods
developed to determine $F_2$. In this paper the new H1 low $Q^2$ data
\cite{xsec}, from dedicated runs in 1999 and with a vertex position
shifted in 2000, are used to obtain rather accurate new data on $F_L$
in the low $Q^2$ region, $Q^2 \sim 1$~GeV$^2$, where no data are
available at small Bjorken $x$ but the theoretical uncertainties
are particularly large \cite{mrst}.


The $F_L$ data are obtained using two methods, the derivative method
\cite{deriv,doris} introduced previously and a new ``shape method''
which employs the characteristic $y^2$ dependence in the cross section
eq. \ref{xs} in order to separate the $F_2$ and $F_L$ terms.
 


\section{Data and extraction methods}

The low $Q^2$ data used in this study were collected with the H1
detector at {\sc Hera} in a dedicated running period in 1999 and
during shifted vertex runs in 2000.  Details concerning data selection
and cross section analysis can be found elsewhere~\cite{xsec}. The
cross section measured in various $Q^2$ bins is shown in
Fig.~\ref{xsec}. For fixed $Q^2$, the cross section rises with
decreasing $x$. However, at very low $x$ (high $y$) a characteristic
bending of the cross section can be noticed. This occurs at all $Q^2$
values at fixed $y\sim 0.5$ and is attributed to the contribution from
the longitudinal structure function $F_L(x,Q^2)$.

\subsection{{\boldmath $F_L$} extraction with a derivative method}
For the $F_L(x,Q^2)$ extraction from the cross section the data are
rebinned in $Q^2$ taking into account systematic error correlations. A
method was used, called the derivative method, introduced
in~\cite{xsec,deriv}.  This method is based on the partial derivative
of the reduced cross section calculated at fixed $Q^2$,
\begin{equation}
\left(\frac{\partial\sigma_r}{\partial\ln y}\right)_{Q^2} = \left(\frac{\partial F_2}{\partial\ln y}\right)_{Q^2} - F_L\cdot y^2\cdot\frac{2-y}{{Y_+}^{2}} - \frac{\partial F_L}{\partial \ln y}\cdot \frac{y^2}{Y_+}. 
\label{eq_der} 
\end{equation} 
%
At large $y$ the $F_L$ contribution to the derivative is of similar
size as the $F_2$ contribution.  Thus $F_L$ can be extracted as the
difference between the cross section derivative and the contribution
of $\partial F_2/\partial \ln y$, which is estimated by a straight
line fit at low $y \leq 0.2$~\cite{deriv,doris}. $\partial
F_L/\partial \ln y$ is found to be smaller than the error of the
$\partial F_2/\partial \ln y$ term~\cite{doris}, and it is neglected.

In Fig.~\ref{deriv} the derivative of the reduced cross section for
the 2000 low $Q^2$ data is shown. For illustration also the line fit
to the low $y$ points and its extrapolation to the high $y$ region,
representing the subtracted $F_2$ derivation, is plotted.  The
$F_L(x,Q^2)$ data obtained from the 2000 shifted vertex data are shown
in Fig.~\ref{deriv00}.

The uncertainty of the line fit and its extrapolation, taking into
account the correlation of errors at low $y$ with those at high $y$,
are included into the systematic errors. Also the error due to
neglecting the $F_L$ derivative term is added considering its
calculated size as the resulting error.

The derivative method assumes a linear behaviour of $\partial
F_2/\partial \ln y$ with $\ln y$ and extrapolates the information
about $F_2(x,Q^2)$ from the low $y$ to the high $y$ region. It does
not make full use of the information provided by the cross section
measurement in the intermediate $y$ region, i.e. for the linear fit
the lowest $y$ points are used but the extraction is made only for the
points with highest $y$, thus at medium $y$ some cross section points
are not used. The result on $F_L$ consists in a few points close in
$y$ with sizeable errors. The precision of the measurement does not
allow to resolve the $x$ dependence of $F_L(x,Q^2)$ on this basis,
see Fig.~\ref{deriv00}.  Thus, a new, more precise method for the
$F_L(x,Q^2)$ extraction was developed.


\subsection{{\boldmath $F_L$} extraction with a shape method}

The new method employs the shape of the reduced cross section
distribution in a given $Q^2$ bin, which is governed by the $y^2$
dependence (Eq.~\ref{xs}), and thereforeit is called the shape method.
The shape of the reduced cross section at high $y$ (Fig.~\ref{xsec})
is driven by the kinematic factor $y^2/Y_+$, and to a lesser extent by
$F_L(x,Q^2)$ which is considered to be constant ($F_L=F_L(Q^2)$), for
each $Q^2$ bin in the narrow $x$ range, high $y$ range of sensitivity
to $F_L$ governed by the $y^2$ term. The method furthermore assumes,
in agreement with previous measurements~\cite{lambda}, that the
structure function $F_2(x,Q^2)$ behaves like $x^{-\lambda}$ at fixed
$Q^2$.  On this basis the reduced cross section distribution in each
$Q^2$ bin can be parametrised and fitted as:
%
\begin{equation}
                \sigma_{FIT}=c\cdot x^{-\lambda}-\frac{y^2}{1+(1-y)^2}F_L. 
\end{equation}
%
Fig.~\ref{shape} illustrates that this fit provides an excellent
description of the reduced cross section in the full kinematic range.
The $\lambda$ and $c$ values extracted from this fit turn out to be in
good agreement with previous measurements~\cite{lambda}.
  
For different $Q^2$ bins the $F_L(x,Q^2)$ points are thus determined from
the fit and a bin-centre procedure is applied to obtain the correct
$x$ value.  Statsictical uncertainties of the $F_L$ determination
include errors of the fit to the cross section points with their
statistical and uncorrelated systematic errors. The correlated
systematic errors are treated separately and added in quadrature to
the total error.  The $F_L(x,Q^2)$ points, as obtained with the shape
method, are compared with the values obtained with the derivative
method in Figs.~\ref{shape99} and~\ref{shape00}.  The results from
both approaches are consistent. However, the errors from the shape
method turn out to be significantly smaller than those from the
derivative method. Therefore, as the final result of this analysis,
only the $F_L$ points extracted with the shape method are used.

\section{Results}

The longitudinal structure function $F_L(x,Q^2)$ as determined from
the 1999 minimum bias and the 2000 shifted vertex H1 data is shown in
Fig.~\ref{fin1} together with predictions of different theoretical
models. The GBW dipole model~\cite{gbw} is a model based on the
concept of saturation for small $Q^2$ and small $x$, which includes
only three parameters to describe the DIS data. This model gives a
good description of the extracted $F_L$ points over the whole
kinematic region covered by this measurement.  The BKS (GRV off-shell)
model~\cite{bks} is based on the photon-gluon fusion mechanism
extrapolated to the region of low $Q^2$ and employing the soft pomeron
exchange mechanism for describing quarks of limited transverse
momentum.  This model gives a correct description of the data as well.

Fig.~\ref{fin2} shows the same H1 data, together with previously
published H1 results, compared here with higher-order QCD fits from H1
\cite{deriv}, ZEUS \cite{zeus}, MRST \cite{mrst} and
Alekhin~\cite{alekhin}.  The values of $F_L(x,Q^2)$ are consistent
with the previous determination of $F_L$ by the H1 Collaboration, but
are more precise and extend the kinematical region, in which
$F_L(x,Q^2)$ is determined, to the lower $Q^2$ region.  As can be seen
there is a significant uncertainty for the $F_L$ prediction in the NLO
QCD fits reflecting the uncertainty of the initial gluon distribution.
The H1 data clearly favour a positive, not small $F_L$ at low $Q^2$
and small $x$, as is preferred by the H1 and Alekhin's fits while the
MRST and ZEUS predictions are low \footnote{ Note that none of the QCD
  fits included low $x$ data for $Q^2$ around 1~GeV$^2$ which is the
  edge of the DIS region. The initial distributions usually are
  parametrised at larger $Q_0^2$. The minimum $Q^2$ of data included
  is (3.5, 2.5, 2.5, 2) GeV$^2$ for H1, ZEUS, Alekhin and MRST while
  the $Q_0^2$ is (4, 7, 9, 1) GeV$^2$, respectively.  Apart from the
  MRST fit, all QCD fits are thus backward extrapolated for most of the
  $F_L$ data points presented here.}.  A negative $F_L$ at small $x$
is experimentally ruled out, both from the extracted $F_L$ values,
fig.~\ref{fin2}, and basically by the measured turn-over of the
reduced cross section, fig.~\ref{shape}.  It becomes evident from
fig.~\ref{fin2} that the $x$ dependence of $F_L$ needs to be measured
which requires to operate HERA at reduced proton beam
energy~\cite{fl_beam}. The low $Q^2$ region, $Q^2 < 5$~GeV$^2$, will
be accessible only in a dedicated new phase of HERA as it requires to
rearrange the interaction region.
 

An overview of all current H1 data on $F_L(x,Q^2)$, from $Q^2 = 0.75$
GeV$^2$ to $Q^2 = 700$ GeV$^2$ and for fixed W=276 GeV, is given in
Fig.~\ref{fin_all}. It comprises the preliminary results of the low
$Q^2$ analysis described in this paper and the results based on data
collected in 96/97~\cite{deriv} and also the recently published high
$Q^2$ results from $e^+p$ and $e^-p$ data~\cite{highQ2}.  The
experimental points are in good agreement with the GBW dipole
model~\cite{gbw} in the whole $Q^2$ range. The BKS (GRV off-shell)
model~\cite{bks}, which evolves steeper at low and moderate $Q^2$, is
still able to describe the data. The H1 QCD fit agrees
with the data in the $Q^2$ region of its applicability beyond a few GeV$^2$
while its backward extrapolation exceeds the data for $Q^2 <$ 1 GeV$^2$. 

Fig.~\ref{fin_allq} presents the $Q^2$ dependence of $F_L$ in comparison
with the higher order QCD fits. All QCD fits describe the data at larger $Q^2$
while at lower $Q^2$ similar conclusions can be drawn as for the 
$x$ dependence discussed above.



\section{Summary}

Two methods of extraction of the longitudinal structure function
$F_L(x,Q^2)$ from H1 inclusive cross section data in the low $Q^2$
region are presented. The derivative method assumes that $\partial
F_2/\partial\ln y$ is linear in $\ln y$ and extrapolates the behaviour
of $F_2(x,Q^2)$ from the low $y$ to the high $y$ region. The newly
introduced shape method approximates the $F_2(x,Q^2)$ behaviour with
$c\cdot x^{-\lambda}$ at fixed $Q^2$ and assumes that $F_L(x,Q^2)$ can
be treated as constant, for a given $Q^2$ bin, in the narrow $x$ range
accesible to this measurement.  It is shown that both methods give
consistent results, however, the shape method proofs to be more
precise than the derivative one.

The measured data points are in agreement with previous results and
allow to extend the region in which $F_L$ is extracted into the very
low $Q^2$ region.  The strong $y$ dependence observed for the cross section at
high $y$ leads to positive results of the longitudinal structure
function down to lowest $Q^2 \sim $ 1 GeV$^2$. This requires the gluon
distribution to be positive at lowest $x$.  Collecting all H1 results,
$F_L(x,Q^2)$ data are presented in a wide $Q^2$ range, from 0.75 to
700 GeV$^2$.  A measurement of the $x$ dependence of $F_L(x,Q^2)$,
independent of assumptions on $F_2(x,Q^2)$ and more accurately, can be
performed  at
HERA with a variation of the proton beam energy.



%
%   References for F_L paper
%
\begin{thebibliography}{99}

% H1 CI paper
\bibitem{qpm} J.~D.~Bjorken, Phys. Rev. {\bf 179} (1969) 1547. 
\bibitem{gross} C.~G.~Callan and D.~J.~Gross, Phys. Rev. Lett. {\bf 22} (1969) 156.
\bibitem{xsec} H1 Collaboration,  
contributed paper 799 to EPS~2001, Budapest. \\
H1 Collaboration,contributed paper 979 to ICHEP~2002, Amsterdam. 
\bibitem{mrst}  A.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne, Eur. Phys. J. C {\bf 23} (2002) 73 [hep-ph/0110215]. \\
A.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne, Phys. Lett. B {\bf 531} (2002) 216 [hep-ph/0201127].
\bibitem{deriv} C.~Adloff {\em et al.} [H1 Collaboration],
                 Eur. Phys. J. {\bf C21} (2001) 33 [hep-ex/0012053]. 
\bibitem{doris} D.~Eckstein, Doctoral Thesis, Humboldt University of Berlin, 2002 [DESY-THESIS-2002-008].
\bibitem{lambda} C.~Adloff {\em et al.} [H1 Collaboration],
                    Phys. Lett. B {\bf 520} (2001) 183 [hep-ex/0108035].
\bibitem{gbw} K.~Golec-Biernat and M.~W\"{u}sthoff, Phys. Rev. D {\bf 59}  (1999) 014017 [hep-ph/9807513]. 
\bibitem{bks} B.~Bade\l ek, J.~Kwieci\'{n}ski and A.~Sta\'{s}to, Z. Phys. C {\bf 74} (1997) 297 [hep-ph/9603230].
%\bibitem{mrst}  A.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne, Phys. Lett. B {\bf 531} (2002) 216
\bibitem{zeus} S.~Chekanov {\em et al.} [ZEUS Collaboration],  
 Phys. Rev. D {\bf 67} (2003) 012007 [hep-ex/0208023]. 
\bibitem{alekhin} S.~I.~Alekhin Phys. Rev. D {\bf 68} (2003) 114002 [hep-ph/0211096].
\bibitem{highQ2} C.~Adloff {\em et al.} [H1 Collaboration],
Accepted by Eur. Phys. J.  C, [hep-ex/0304003]. 
\bibitem{fl_beam} L.~Bauerdick, A.~Glazov and M.~Klein, Proc.Workshop on Future Physics at HERA, eds. G.~Ingelmann, A.~De~Roeck and R.~Klanner, Hamburg DESY 77 (1996) [hep-ex/9609017].


\end{thebibliography}



\clearpage
% \input{figures}

\begin{figure}[h]
\mbox{
%\epsfig{file=sigma9900_col.eps,
\epsfig{file=H1prelim-03-043.fig1.eps,
height=17.cm,
width=1.0\linewidth,
%width=15.cm,
%bbllx=70pt,bblly=285pt,
%bburx=550pt,bbury=700pt,
clip=}
}
\caption{\label{xsec} 
  Measurements of the inclusive DIS cross section for the 1999 minimum
  bias and the 2000 shifted vertex data, compared to larger $x$ data
  from ZEUS (BPT 97) and NMC. The curves are phenomenological
  parametrisation of the cross section calculating $F_2(x,Q^2)$ within
  the fractal proton structure concept and with different assumptions
  on $F_L$.  }
\end{figure}

\newpage

\begin{figure}[h]
\mbox{
%\epsfig{file=xsec_r00sv_reb.ps,
\epsfig{file=H1prelim-03-043.fig2.ps,
%height=15.cm,
%width=1.0\linewidth,
width=15.cm,
bbllx=40pt,bblly=295pt,
bburx=540pt,bbury=720pt,
clip=}
}
\caption{ 
  Derivative of the reduced cross section $\sigma_r$ at fixed $Q^2$
  for shifted vertex 2000 data. The inner error bars represent the
  statistical errors. The full errors include the statistical,
  uncorrelated and correlated systematic errors added in quadrature. A
  linear fit to the low $y$ points and its extrapolation to the high y
  region is shown. In the derivative method, the longitudinal
  structure function $F_L(x,Q^2)$ is extracted from the deviation from
  the linear behaviour in the high $y$ region according to
  $(\partial\sigma_r/\partial\ln y)_{Q^2} = (\partial F_2/\partial\ln
  y)_{Q^2} - F_L\cdot y^2\cdot (2-y)/{Y_+}^{2}$, under the assumption
  that the linear behaviour of $\partial F_2/\partial\ln y$ persists
  in the high $y$ region. The dashed line represents the QCD fit to
  previous H1 cross section data extrapolated backwards to $Q^2$ below
  $Q^2_{min}$=3.5 GeV$^2$.}
\label{deriv}
\end{figure}

\newpage

\begin{figure}[h]
\mbox{
%\epsfig{file=fl00svder2.ps
%\epsfig{file=fl00svder2.ps,
\epsfig{file=H1prelim-03-043.fig3.ps,
%height=15.cm,
%width=1.0\linewidth,
width=15.cm,
bbllx=40pt,bblly=295pt,
bburx=540pt,bbury=720pt,
clip=}
}
\caption{\label{deriv00} 
  The longitudinal structure function $F_L(x,Q^2)$ (at fixed $Q^2$ )
  from 2000 shifted vertex data as extracted with the derivative
  method.  The inner error bars represent the statistical errors. The
  full errors include the statistical, uncorrelated and correlated
  systematic errors added in quadrature. The dashed line represents
  the QCD fit to previous H1 cross section data extrapolated backwards
  to $Q^2$ below $Q^2_{min}$=3.5 GeV$^2$.  }
\end{figure}

\newpage

\ \ \ \ \ \
\vspace{2cm}
\ \ \ \

\begin{figure}[h]
\mbox{
%\epsfig{file=xsect_qsumb_col.ps,
\epsfig{file=H1prelim-03-043.fig4.ps,
height=15.cm,
%width=1.0\linewidth,
width=15.cm,
bbllx=40pt,bblly=155pt,
bburx=540pt,bbury=570pt,
clip=}
}
\caption{\label{shape} 
  The reduced cross section as a function of $x$ for different $Q^2$
  bins. Data from 1999 minimum bias (squares) and 2000 shifted vertex
  (bullets) running periods are shown. The inner error bars represent
  the statistical errors. The full errors include the statistical and
  systematic errors added in quadrature. The dashed lines show a
  function of the form $\sigma_r = c\cdot x^{-\lambda}$ representing
  the $F_2$ contribution to the fitted cross section. The solid lines
  show fits of the form $\sigma_r = c\cdot x^{-\lambda} - y^2/Y_+ F_L$
  , from which $F_L$ is extracted in the shape method.  }
\end{figure}

\newpage
%\vspace{-18cm}
\begin{figure}[h]
\mbox{
%\epsfig{file=fl99_2met2b.ps ,
\epsfig{file=H1prelim-03-043.fig5.ps ,
%width=1.0\linewidth,
%height=13.cm,
width=15.cm,
bbllx=40pt,bblly=295pt,
bburx=540pt,bbury=720pt,
%angle=90,
clip=}
}
\caption{\label{shape99} 
  Comparison of $F_L(x,Q^2)$ results (for fixed $Q^2$) from 1999
  minimum bias data as extracted by the derivative (triangles) and
  shape (bullets) methods.  The inner error bars correspond to
  statistical errors. The full errors include the statistical,
  uncorrelated and correlated systematic errors added in quadrature.
  The solid line represents the QCD fit to previous H1 cross section
  data and the dashed line the QCD fit extrapolated backwards to $Q^2$
  below $Q^2_{min}$=3.5 GeV$^2$.  }
\end{figure}

\newpage


\begin{figure}[h]
\mbox{
%\epsfig{file=fl00svboth2b.ps,
\epsfig{file=H1prelim-03-043.fig6.ps,
%height=13.cm,
width=15.cm,
%width=1.0\linewidth,
bbllx=40pt,bblly=295pt,
bburx=540pt,bbury=720pt,
%angle=90,
clip=}
}
\caption{\label{shape00} 
  Comparison of $F_L(x,Q^2)$ results, for fixed $Q^2$, from 2000
  shifted vertex data as extracted by the derivative (triangles) and
  the shape (bullets) methods.  The inner error bars represent the
  statistical errors. The full errors include the statistical,
  uncorrelated and correlated systematic errors added in quadrature.
  The dashed line represents the QCD fit to previous H1 cross section
  data extrapolated backwards to $Q^2$ below $Q^2_{min}$=3.5 GeV$^2$.
  }
\end{figure}

\newpage

\begin{figure}[h]
\mbox{
%\epsfig{file=fl9900_mod.ps,
\epsfig{file=H1prelim-03-043.fig7.ps,
%height=13.cm,
width=15.cm,
%width=1.0\linewidth,
bbllx=40pt,bblly=295pt,
bburx=540pt,bbury=720pt,
clip=}
}
\caption{\label{fin1} 
  $F_L(x,Q^2)$, for fixed $Q^2$, from 1999 minimum bias (squares) and
  2000 shifted vertex  (bullets) data as extracted by the shape
  method.  The inner error bars correspond to statistical errors. The
  full errors include the statistical, uncorrelated and correlated
  systematic errors added in quadrature. The solid, black line
  represents the QCD fit to previous H1 cross section data and the
  dashed, black line the QCD fit extrapolated backwards to $Q^2$ below
  $Q^2_{min}$=3.5 GeV$^2$.  Other curves show predictions of 
  the GBW dipole model~\cite{gbw} (dashed-dotted
  line) and the BKS model~\cite{bks} (thin solid,green line).  }
\end{figure}

\newpage

\begin{figure}[h]
\mbox{
%\epsfig{file=fl9900_ps.ps,
\epsfig{file=H1prelim-03-043.fig8.ps,
%height=13.cm,
width=15.cm,
%width=1.0\linewidth,
bbllx=40pt,bblly=295pt,
bburx=540pt,bbury=720pt,
clip=}
}
\caption{\label{fin2} 
  $F_L(x,Q^2)$, for fixed $Q^2$, from 1999 minimum bias (squares) and
  2000 shifted vertex. (bullets) data as extracted by the shape
  method.  The inner error bars correspond to statistical errors. The
  full errors include the statistical, uncorrelated and correlated
  systematic errors added in quadrature. The solid, black line
  represents the QCD fit to previous H1 cross section data. The blue lines
  show results from Alekhin in NLO (solid) and NNLO (dashed).
  The green line is the result from the ZEUS fit. The red line is the prediction of the
  MRST 2001 fit in NLO.}
\end{figure}

\newpage

\begin{figure}[h]
\mbox{
%\epsfig{file=flq2_mod.ps,
\epsfig{file=H1prelim-03-043.fig9.ps,
%height=13.cm,
width=14.cm,
%width=1.0\linewidth,
bbllx=15pt,bblly=345pt,
bburx=540pt,bbury=760pt,
clip=}
}
\caption{\label{fin_all} 
  $Q^2$ dependence of $F_L(x,Q^2)$ (at fixed W=276 GeV), summarizing
  the data from the H1 experiment.  The inner error bars represent the
  statistical errors. The full errors include the statistical,
  uncorrelated and correlated systematic errors added in quadrature.
  The solid, black line represents the QCD fit to previous H1 cross
  section data and the dashed, black line the QCD fit extrapolated
  backwards to $Q^2$ below $Q^2_{min}$=3.5 GeV$^2$.  Other curves show
  predictions of the GBW dipole
  model~\cite{gbw} (dashed-dotted line) and the BKS model~\cite{bks}
  (thin solid, green line).  }
\end{figure}

\newpage

\begin{figure}[h]
\mbox{
%\epsfig{file=flq2th2_tst.ps,
\epsfig{file=H1prelim-03-043.fig10.ps,
%height=13.cm,
width=14.cm,
%width=1.0\linewidth,
bbllx=15pt,bblly=345pt,
bburx=540pt,bbury=760pt,
clip=}
}
\caption{\label{fin_allq} 
  $Q^2$ dependence of $F_L(x,Q^2)$ (at fixed W=276 GeV), summarizing
  the data from the H1 experiment.  The inner error bars represent the
  statistical errors. The full errors include the statistical,
  uncorrelated and correlated systematic errors added in quadrature.
  The solid, black line represents the QCD fit to previous H1 cross
  section data. The blue lines show results from Alekhin in NLO
  (solid) and NNLO (dashed).  The green line is the result from the
  ZEUS fit. The red line is the prediction of the MRST 2001 fit in
  NLO.}
\end{figure}

\end{document}

