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\title*{Measurements of $\mathbf{F_2}$ and the 
total photon-proton cross sections at low $\mathbf{x}$ and low $\mathbf{Q^2}$}
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\toctitle{Measurements of $\mathbf{F_2}$ and the Total Photon--Proton 
Cross Sections at Low $\mathbf{x}$ and Low $\mathbf{Q^2}$}
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\titlerunning{Measurements of $F_{2}$ and $\sigma_{tot}^{\gamma^{*}p}$ 
at Low $x$ and Low $Q^{2}$}
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\author{Vincenzo Monaco}
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\authorrunning{V.Monaco}
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\institute{Universit\`a di Torino, INFN,
           Via Pietro Giuria 1, 10125 Torino, Italy}


\maketitle              % typesets the title of the contribution

\begin{center}
 On behalf of the H1 and ZEUS Collaborations
\end{center}

\begin{abstract} 

The HERA measurements of the proton structure function $F_{2}$
at low $Q^{2}$ and very low values of the Bjorken variable $x$ are presented.
These measurements cover the transition region between the 
photoproduction limit ($Q^{2}=0$) and the DIS regime at 
$Q^{2}>1$ $\mathrm{GeV}^{2}$. 
Whereas the behaviour of $F_{2}$ at moderate and high $Q^{2}$ is
described by DGLAP evolution,
in the low $Q^{2}$ region, the $F_{2}$ behaviour can be 
described by 
phenomenological models inspired by Regge theory and the 
Vector Dominance Model.

\end{abstract}



\section{Introduction}
 
One of the early results from HERA was the discovery that
the proton structure function 
$F_{2}$ is characterized by a steep rise with decreasing values 
of the Bjorken variable $x$~\cite{h1.1,zeus.1}. This behaviour persists
at low $Q^{2}$ values of the order of 
$Q^{2}\sim 1.5 $ $\mathrm{GeV}^{2}$~\cite{h1.2,zeus.2}
and can be interpreted as a purely perturbative phenomenon in terms
of an increase of the gluon ($xg$) and sea quark ($x\bar{q}$) densities 
when smaller and smaller values of $x$ are probed.
Gl\"{u}ck, Reya and Vogt (GRV)~\cite{grv.1}
predicted this behaviour well before the first HERA measurements; in their
model the parton distributions are assumed to be non-singular (valence-like) 
at a very low $Q^{2}$ starting scale 
($Q^{2}_{0}\simeq 0.34$ $\mathrm{{GeV}^{2}}$). The rise of $xg$
and $x\bar{q}$ with decreasing $x$ at higher $Q^{2}$ is generated dynamically 
via the DGLAP~\cite{dglap1} evolution. 

The proton structure function $F_{2}$ is related to the total 
virtual photon-proton cross section 
$\sigma_{tot}^{\gamma^{*}p}$ through the following relation, valid at low $x$ 
and low $Q^{2}$
\begin{equation}
  \sigma_{tot}^{\gamma^{*}p}(W,Q^{2}) \simeq \frac{4\pi^{2}\alpha}{Q^{2}} 
 F_{2}(x,Q^{2}) \; , \label{eq1.1}   
\end{equation}
\noindent 
where $W^{2} \simeq Q^{2}/x$ is the squared center of mass energy of the 
$\gamma^{*}p$ system.

From (\ref{eq1.1}) it 
follows that a measurement of $F_{2}(x,Q^{2})$
is equivalent to a measurement of the total cross section 
$\sigma_{tot}^{\gamma^{*}p}(W,Q^{2})$,
and that the rise of the structure function $F_{2}$ with decreasing $x$ 
at moderate and high $Q^{2}$ and
at low $x$ corresponds to a rise of the total cross section 
with $W$ at high values of $W$. 

The $W$ dependence of $\sigma_{tot}^{\gamma^{*}p}$  is 
shown in Fig.~\ref{fig1}, where a partial compilation of measured 
cross sections is plotted as a function of $W^{2}$ for different $Q^{2}$ bins.
The steep $W$ dependence of the cross section 
at high and moderate $Q^{2}$ is well described 
by global next-to-leading order (NLO) perturbative QCD (pQCD) fits 
(the prediction of the 
MRS(R1)~\cite{mrsr1} parameterization is shown in Fig.~\ref{fig1}).
The photoproduction cross section at $Q^{2}=0$,
shown in the same figure, exhibits
a much softer $W$ dependence.
\begin{figure}[th]
\begin{center}
\includegraphics[width=.7\textwidth, height=.67\textwidth]{monaco.1}
\end{center}
\caption[]{Total $\gamma^{*}p$ cross section as a function of the $\gamma^{*}p$
center-of-mass energy in different bins of $Q^{2}$.
The curves represent the ALLM97~\cite{allm97}  
(\emph{full line}) and the 
MRS(R1)~\cite{mrsr1} (\emph{dotted line}) parameterizations. The points 
are data from H1, ZEUS and fixed target experiments}
\label{fig1}
\end{figure}

As shown by Donnachie and Landshoff (DL)~\cite{DL.1} 
the $W$ dependence of the photoproduction cross section  
is similar to the 
$W$ dependence of the cross section of 
hadronic processes, described by a two-component 
Regge parameterization
\begin{equation}
 \sigma_{tot}^{\gamma p}(W)=A_{I\!R}{(W^{2})}^{\alpha_{I\!R}-1}+
           A_{I\!P}{(W^{2})}^{\alpha_{I\!P}-1} \; , \label{eq1.2}
\end{equation}
\noindent
where $I\!R$ and $I\!P$ denote the Reggeon and Pomeron contributions,
with intercepts $\alpha_{I\!R}\simeq 0.5$ and $\alpha_{I\!P}\simeq 1.08$
respectively. 
At high $W$, where the Reggeon term is negligible,
 the energy dependence of the total cross section can be parameterized as 
$\sigma_{tot}(W)\propto {W}^{2\lambda}$, with $\lambda=1-\alpha_{I\!P}$
small and close to 0 in the photoproduction limit.
%In contrast to the soft behaviour of the photoproduction cross section,
Conversely,
the behaviour of the DIS data with $Q^{2}> 1$ $\mathrm{GeV}^{2}$  at 
high $W$ 
corresponds to a much higher effective power $\lambda$ (between 0.2 and 0.4).

A transition between the different $W$ dependences of the
cross section in the two regimes is expected. 
One of the reasons for expecting new phenomenology in this region 
comes from the observation that the total photoproduction cross section 
is finite; this, because of (\ref{eq1.1}), 
implies that $F_{2}\rightarrow 0$
as $Q^{2}\rightarrow 0$, which entails strong scaling violations 
at low $Q^{2}$.
Therefore,
the description of the structure functions based on pQCD,
 which still works 
at a $Q^{2}$ scale of a few $\mathrm{GeV}^{2}$, should fail at lower $Q^{2}$.
It is interesting to see down to which low $Q^{2}$ value the
NLO DGLAP formalism can be extended;
at very low $Q^{2}$, however, pQCD predictions are not reliable at all 
and non-perturbative 
models have to be used.

At HERA, low $Q^{2}$ values correspond to very low values of $x$, where
the average longitudinal distance travelled by the 
hadronic fluctuation of the photon in the proton rest frame 
($l\sim 1/(2m_{N}x)$) is much longer than the target size. Then 
$ep$ scattering can be viewed as the interaction of a color dipole $q\bar{q}$
with the proton, and the soft Regge-like behaviour of 
the photoproduction cross section is expected
to hold also at 
finite but low values of $Q^{2}$. However, the Regge formalism 
cannot be pushed to too high $Q^{2}$ values without introducing hard 
contributions.

The question is at which $Q^{2}$ and $x$ values the behaviour of the structure 
function changes from being described by perturbative QCD 
to being dominated by non-perturbative contributions. 
Different models and parameterizations exist to interpolate between 
the soft and hard behaviour of the cross 
section~\cite{ckmt,bk,aby,sc.1,allm97,dl98}. 

\subsection*{Experimental methods}

The structure function $F_{2}$ and the cross section 
$\sigma_{tot}^{\gamma^{*}p}$ have been measured in the transition region
using $e^{+}p$ collisions from the 1995-1997 HERA runs.

The 4-momentum transfer squared, 
$Q^{2}$, is related to the energy $E_{e}'$ and the 
angle $\vartheta_{e}'$ of the scattered positron by
\begin{equation}
 Q^{2}=4E_{e}E_{e}'\cdot \sin^{2} \frac{\vartheta_{e}'}{2} \; , \label{eq1.3}
\end{equation}
where $\vartheta_{e}'$ is measured with respect to the direction of the 
positron beam and $E_{e}$ is the positron beam energy.
The limited angular acceptance of the main calorimeters of H1 and ZEUS
fixes the lowest $Q^{2}$ to $\sim 1.5$ $\mathrm{GeV}^{2}$.
In order to measure $F_{2}$ at lower 
values of $Q^{2}$, the final state positron
must be identified at very small scattering angles $\vartheta_{e}'$
or the beam energy $E_{e}$ must be reduced.

``Shifted Vertex'' 
runs have been provided by HERA in 1995; in these runs 
the nominal  interaction vertex was shifted in the proton beam direction by 
+70 cm, thereby increasing the angular acceptance of the detectors to smaller 
positron scattering angles.
Both the H1 and ZEUS experiments have measured $F_{2}$
using data collected in these runs, extending the coverage of the 
kinematic plane down to $Q^{2}$ values of
$\sim 0.35$ $\mathrm{GeV}^{2}$ (ZEUS)~\cite{svtx_zeus} and 
$\sim 0.6$ $\mathrm{GeV}^{2}$ (H1)~\cite{svtx_h1}.
 
In 1995 ZEUS extended the kinematic acceptance to low angles by
installing two small beam pipe calorimeter modules
(BPC) on two sides of the beam pipe
at $\sim 3$ m
from the interaction point in the rear direction. 
Each module is a tungsten-scintillator sampling calorimeter,
with an energy resolution of  
$\Delta E/E= 17\%/\sqrt{E}$. 
%Thin exit windows allows the positron to leave the beam pipe;
%the projection of the exit windows onto the BPC face defines
%the fiducial area of the detector, whose acceptance is limited to
The BPC acceptance is
$18\le \vartheta_{e}' \le 32$ mrad.
Data collected in 1995 with the BPC were used for the
measurement of $F_{2}$ ~\cite{zeus_bpc} in the kinematic range 
$0.11\le Q^{2}\le 0.65$ $\mathrm{GeV}^{2}$ and 
$2\times 10^{-6}\le x\le 6\times 10^{-5}$. 

Another method to extend the measurement of $F_{2}$ to lower $Q^{2}$ 
makes use of $ep$ events with photons radiated collinear to the 
incident positron, resulting in an effective beam energy $E_{e}$ 
reduced with 
respect to ``non-radiative events''. 
This method allows extension of the detector acceptance to
low values of $Q^{2}$ in a complementary
higher $x$ range with respect to the BPC and shifted vertex
measurements, but it
suffers from large statistical and systematic errors.

The typical systematic error of the Shifted Vertex and BPC 
analyses is between 6 
and 12 \%, depending on the bin. 
The statistical errors are much lower.


\section{Phenomenological analysis of the results}

The results of the $F_{2}$ measurements based on the 1995 data 
are shown in 
 Fig.~\ref{fig2} in $Q^{2}$ bins between $Q^{2}=0.11$ and $6.0$
$\mathrm{GeV}^{2}$. The results from the Shifted Vertex runs are labeled as 
H1 SVX95~\cite{svtx_h1}, ZEUS SVX95~\cite{svtx_zeus} 
those from BPC as ZEUS BPC95~\cite{zeus_bpc}; 
some points from previous ZEUS analyses and 
from the E665 experiment~\cite{e665} are also shown.
The curves are the parameterizations from two fits of the ZEUS data
based on non-perturbative models (ZEUSREGGE fit) and on NLO pQCD (ZEUSQCD fit).
The data indicate that the rise of $F_{2}$ with decreasing 
$x$ is prominent for values of $Q^{2}>1$ $\mathrm{GeV}^{2}$, but becomes less 
steep for smaller $Q^{2}$ values.
\begin{figure}[hb]
\begin{center}
\includegraphics[width=.76\textwidth, height=.68\textwidth]{monaco.2}
\end{center}
\caption[]{$F_{2}(x,Q^{2})$ at low $Q^{2}$ from ``Shifted Vertex'' runs
(H1 SVX95 and ZEUS SVX95), ZEUS BPC and 
E665 data. The curves are described in the text} 
\label{fig2}
\end{figure}

Figure~\ref{fig3} shows the cross section as a function of 
$W^{2}$ derived from the measured $F_{2}$ using (\ref{eq1.1}). 
The two curves shown are the predictions of Donnachie and Landshoff 
(DL)~\cite{DL.1} and of GRV94~\cite{grv.1}. 
The DL model
describes the $W$ dependence of the 
data at very low values of $Q^{2}$, but above $Q^{2}=0.65$ 
$\mathrm{GeV}^{2}$ it predicts a shallower rise of the cross section than the 
data exhibit. The increasing rise of $\sigma_{tot}^{\gamma^{*}p}$ with $W^{2}$
for $Q^{2}$ values above 1 $\mathrm{GeV}^{2}$ is described by the GRV94 curves.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.66\textwidth, height=.66\textwidth]{monaco.3}
\end{center}
\caption[]{The total $\gamma^{*}p$ cross section, $\sigma_{tot}^{\gamma^{*}p}$,
as a function of $W^{2}$ for different $Q^{2}$ bins. 
Low $Q^{2}$ DIS measurements 
and photoproduction cross sections~\cite{fixed_t} 
are compared with the predictions of DL 
({\it dotted line}) and GRV94 ({\it dashed line})}
\label{fig3}
\end{figure}

In~\cite{svtx_zeus}, a 
simplified Generalized Vector Dominance Model (GVDM)~\cite{gvdm}
approach was used to parameterize the $Q^{2}$ dependence 
of the cross section in the BPC range;
in the fit described in~\cite{svtx_zeus} the longitudinal $\gamma^{*}p$ 
cross section is neglected, while the transverse cross section 
$\sigma_{T}(W^{2},Q^{2})$ is related to the corresponding cross section at 
$Q^{2}=0$, $\sigma_{0}(W^{2})$, by
\begin{equation}
  \sigma_{T}(W^{2},Q^{2})=\frac{M_{0}^{2}}{M_{0}^{2}+Q^{2}}\sigma_{0}(W^{2})
  \; ,
  \label{eq1.4}
\end{equation}
\noindent
where only the contribution from continuum vector states from a 
cut-off-mass $M_{0}^{2}$ has been kept. 
A fit of the BPC $F_{2}$ points to (\ref{eq1.4}) gives
$M_{0}^{2}=0.53$ $\mathrm{GeV}^{2}$.

The $W$ dependence of $\sigma_{0}(W^{2})$ is parameterized by Regge theory
through (\ref{eq1.2}). 
The $Q^{2}$  
and $W$ dependence of the cross section $\sigma_{tot}^{\gamma^{*}p}$
used in the ZEUSREGGE fit is obtained
by combining (\ref{eq1.2}) and (\ref{eq1.4})
\begin{equation}
  \sigma_{tot}^{\gamma^{*}p}(W^{2},Q^{2})=
        \left( \frac{M_{0}^{2}}{M_{0}^{2}+Q^{2}} \right)
        \left( A_{I\!R}{(W^{2})}^{\alpha_{I\!R}-1}+
               A_{I\!P}{(W^{2})}^{\alpha_{I\!P}-1} 
        \right) \; .
   \label{eq1.5}
\end{equation}

The ZEUS BPC data and low $W$ photoproduction data from fixed target 
experiments~\cite{fixed_t} were fitted to
(\ref{eq1.5}), with $A_{I\!R}$, $A_{I\!P}$ and $\alpha_{I\!P}$ 
left as free parameters. 
The intercept of the Pomeron 
is determined to be $\alpha_{I\!P}=1.097\pm 0.002$,
compatible with the value of the DL fit. 
%From Fig.~\ref{fig2} it results that 
The fit result is shown in Fig.~\ref{fig2};
the ZEUSREGGE fit 
gives a good description of the low $Q^{2}$ data in the BPC region
$Q^{2}\le 0.65$ $\mathrm{GeV}^{2}$,
but at larger $Q^{2}$ the curves 
fall below the data.

The continuous curve in Fig.~\ref{fig2} (ZEUSQCD) is the result of a NLO QCD
fit to the ZEUS data with $Q^{2}>1$ $\mathrm{GeV}^{2}$;
data from the NMC~\cite{nmc} and the BCDMS~\cite{bcdms} experiments with 
$W^{2}>10$ $\mathrm{GeV}^{2}$, not shown in Fig.~\ref{fig2}, are also 
included to constrain the fit at high $x$.
The details of the fit are described in~\cite{svtx_zeus}.
The NLO DGLAP evolution equations are solved in the 
$\overline{\mathrm{MS}}$ scheme.
The input scale is chosen to be $Q_{0}^{2}=7$ $\mathrm{GeV}^{2}$,
and backward evolution is performed to fit the data with $Q^{2}<Q_{0}^{2}$.
The quality of the fit is good (the $\chi^{2}$ is 1474 for 1120 data points
and 11 free parameters) 
as can be seen in Fig.~\ref{fig2}, where the resulting $F_{2}$ is shown 
together with the measured values.

Figure~\ref{fig4} shows the gluon ($xg$) and singlet ($x\Sigma$) distributions 
obtained from the fit as a function of $x$ for $Q^{2}$ at 1, 7 and 20 
$\mathrm{GeV}^{2}$.
At the smallest $Q^{2}$ the gluon distribution is almost flat and 
compatible with zero, while the quark singlet distribution is still rising 
at small $x$. 
\begin{figure}[hb]
\begin{center}
\includegraphics[width=.58\textwidth, height=.54\textwidth]{monaco.4}
\end{center}
\caption[]{The quark singlet momentum distribution $x\Sigma$ (\emph{shaded})
and the gluon momentum distribution $xg$ (\emph{hatched}) as a function of $x$
at $Q^{2}= 1$, 7 and 20 $\mathrm{GeV}^{2}$ from the ZEUSQCD fit.
The left-hand plots show the results of the fit including data with $Q^{2}>1$
$\mathrm{GeV}^{2}$; the results in the right-hand plots include 
data with $Q^{2}>4$ $\mathrm{GeV}^{2}$. The error bands correspond to the quadratic sum of all the error sources}
\label{fig4}
\end{figure}

The quark singlet distribution
is dominated at low $x$ by the contribution from the 
$q\bar{q}$ sea distribution $xS$, and its evolution in $Q^{2}$ is 
coupled to that of the gluons. It seems rather unphysical that the gluon
does not drive the sea evolution at low $Q^{2}$; this behaviour 
(also observed, for instance, in the result of the fit by
 Martin et al. (MRST)~\cite{mrst1}) 
is incompatible 
with the hypothesis that the rapid rise in $F_{2}$ is driven by the rapid 
increase of the gluon density at small $x$ induced by parton splitting.
However, no breakdown of the technical validity 
of the NLO QCD fit has been found in the kinematic range explored; 
the experimental data are well described down to $Q^{2}$ 
$\sim 1$ $\mathrm{GeV}^{2}$.

The low $x$ behaviour of $F_{2}$ as a function of $x$ and $Q^{2}$ 
can be quantified through the slopes $d\ln{F_{2}}/d\ln{(1/x)}$
at fixed $Q^{2}$ and  $dF_{2}/d\ln{Q^{2}}$ at fixed $x$.

The slope $\lambda_{eff}=d\ln{F_{2}}/d\ln{(1/x)}$, shown in Fig.~\ref{fig5},
 is extracted by fitting 
the data at fixed $Q^{2}$ to the form 
$F_{2}=Ax^{-\lambda_{eff}}$,
equivalent to the parameterization 
$\sigma_{tot}^{\gamma^{*}p}(Q^{2},W^{2})=A'\cdot {W}^{2\lambda_{eff}}$
for the total cross section. 
The slope $\lambda_{eff}$ at $Q^{2}<1$ $\mathrm{GeV}^{2}$ is compatible 
with the $Q^{2}$-independent Regge prediction 
$\lambda= 1-\alpha_{I\!P} \simeq 0.1$, while at higher $Q^{2}$ it gradually 
increases with $Q^{2}$. This tendency at high $Q^{2}$ is not followed
by the dominant Pomeron term of soft Regge phenomenology, but is qualitatively 
described by pQCD fits. As an example, the prediction of the GRV94 
parameterization and the result of the ZEUSQCD fit
for $Q^{2}>1$ $\mathrm{GeV}^{2}$, obtained using the same 
$x$ range of the data, are shown in Fig.~\ref{fig5}.
\begin{figure}[bh]
\begin{center}
 \mbox{
  \leavevmode
  \includegraphics[width=.46\textwidth, height=.47\textwidth]{monaco.5}
  \hspace{.3cm}
  \leavevmode
  \includegraphics[width=.47\textwidth]{monaco.6}
  \hspace{.4cm}
 }
\end{center}
\caption[]{The slope $\lambda_{eff}=d\ln{F_{2}}/d\ln{(1/x)}$ 
as a function of $Q^{2}$ as determined by a fit to H1 ({\bf left}) and ZEUS 
({\bf right}) data}
\label{fig5}
\end{figure}

The logarithmic slope $dF_{2}/d\ln{Q^{2}}$ is derived from the data by 
fitting $F_{2}=a+b\ln{Q^{2}}$ in bins of fixed $x$. 
The slope $dF_{2}/d\ln{Q^{2}}$ at low $x$ is dominated  by the convolution 
of the splitting function $P_{qg}$ and the gluon density 
($dF_{2}/d\ln{Q^{2}}\propto \alpha_{S}P_{qg}\otimes xg$), and can be 
considered as  a quantitative evaluation of the scaling violations caused 
by gluon bremsstrahlung and quark pair creation. 
\begin{figure}[th]
\begin{center}
  \includegraphics[width=.52\textwidth]{monaco.7}
\end{center}
\caption[]{The slope $d{F_{2}}/d\ln{Q^{2}}$ as a function of $x$  
as determined by the fit of ZEUS data}
\label{fig6}
\end{figure}

The dependence of the logarithmic $Q^{2}$ slope on $x$, as determined from 
the fit of the ZEUS data, is shown in 
Fig.~\ref{fig6}, where for each bin the mean value of $Q^{2}$ is also shown.
For values of $x$ down to $3\times 10^{-4}$ the slope increases as $x$ 
decreases. In this $x$ range, where the mean values of 
$Q^{2}$ are high enough that pQCD fits are expected to be reliable,
the ZEUSQCD fit and the GRV94 parameterization are in good agreement with 
the data.
The trend of the slope in this region reflects the rapid increase of the 
gluon density with decreasing values of $x$ obtained by the same ZEUSQCD 
fits.

At lower values of $x$ and $Q^{2}$ the slope has the opposite behaviour, it 
decreases for decreasing $x$, 
with a ``turn-over'' at $x\sim 3\times 10^{-4}$ and at a mean 
$Q^{2}$ value of a few $\mathrm{GeV}^{2}$.
It should be noticed that the exact position of the ``turn-over'' could 
depend on the correlations between the $x$ and $Q^{2}$
values of the data points included in the fit.
In the same $Q^{2}$ range the slope determined from fixed-target data 
continues to increase with decreasing values of $x$, but these data are 
at larger values of $x$.

The parameterization of DL and the ZEUSREGGE fit describe the $x$ 
dependence of the slope only at very low values of $x$.
As shown by the result of the ZEUSQCD fit, 
the standard NLO DGLAP equations are able to describe the behaviour of the 
slope down to $Q^{2}\sim 1$ $\mathrm{GeV}^{2}$.

It is not surprising that the GRV94 parameterization, obtained by a fit 
not constrained by data at low $Q^{2}$, does not follow the 
turn-over of the slope between $Q^{2}\sim 1$ and 
$Q^{2}\sim 10$ $\mathrm{GeV}^{2}$; in this region
the evolution of the sea and gluon distributions predicted by the DGLAP 
equations depends strongly on the shape 
of the parton distributions at the starting scale and on the value of the 
strong coupling constant $\alpha_{s}$.
The ZEUSQCD fit, tuned to the same data from which the slopes have been 
determined, follows the $x$ dependence of $dF_{2}/d\ln{Q^{2}}$ 
down to lower values of $Q^{2}$. The flattening of the slope at $Q^{2}\sim 1$
$\mathrm{GeV}^{2}$ reflects the flat $x$-dependence of the gluon 
distributions determined from the ZEUSQCD fit at this $Q^{2}$ scale.

While in the pQCD regime the logarithmic $Q^{2}$ slope is sensitive to 
the gluon distribution,
the $x$ dependence of the structure function $F_{2}$ is related  to the 
sea quark distribution. The fact that at $Q^{2}\sim 1$ $\mathrm{GeV}^{2}$ 
$F_{2}$ continues to rise with decreasing values of $x$ is consistent with the
$x$ dependence of the 
singlet quark distribution obtained at these $Q^{2}$ values from the ZEUSQCD 
fit.

\section{New experimental results}

In order to reduce the background and the systematic error of the $F_{2}$ 
measurement in the transition region, ZEUS installed 
a tracker (BPT), consisting of 
two silicon microstrip detectors, in front of the 
Beam Pipe Calorimeter in 1997. 

The BPT complements the information of the BPC for the reconstruction of the 
scattered positron and the measurement of the scattering angle.
There are several advantages to using a tracker in front of the Beam 
Pipe Calorimeter:
\begin{itemize}
 \item The tracker can be used to reduce the main source of background in 
       the BPC analysis, due to neutral particles in the final state
       from photoproduction events misidentified as positrons. The reduction 
       of the background allows extension of the kinematic coverage 
       to higher values of $y$ with respect to the 1995 analysis.
 \item 
       With the BPT the scattering angle can be measured more precisely and 
       independently of the knowledge of the interaction vertex; the $F_{2}$ 
       measurement is thus less dependent on the 
       efficiency of the vertex reconstruction and on the simulation of the 
       hadronic final state in the Monte Carlo.  
 \item The use of the BPT in the reconstruction of the track coordinates 
       gives a better control of the fiducial volume cuts and 
       of the position dependent corrections in the BPC energy 
       reconstruction. The energy calibration, performed on 1997 data using 
       kinematic peak events, determines the energy scale of the calorimeter 
       to within $\sim 0.3\%$; the energy scale is uniform within $\pm 0.3\%$
       over the entire active area of the detector. 
\end{itemize}

The data taken with the BPT during six weeks at the end 
of the 1997 running period, corresponding to an integrated luminosity of 3.9 
$\mathrm{pb}^{-1}$, have been used to measure $F_{2}$.
The kinematic region covered by this analysis is
$0.045<Q^{2}<0.65$ $\mathrm{GeV}^{2}$ and $6\times 10^{-7}<x<1\times 10^{-3}$,
corresponding to $25<W<270$ GeV and $0.007<y<0.8$. 
The kinematic coverage has been extended
to lower and higher $y$ with respect to the 1995 BPC analysis.
The lowest $y$ bins 
overlap with the measurements of the E665 experiment.

For $y>0.08$, the event kinematics are reconstructed with the electron method.
At lower $y$, the $y$ resolution is improved by combining positron and 
hadronic final state variables with the $e\Sigma$ method~\cite{rec_kin}.

The simulation of $ep$ events in this analysis 
makes use of two Monte Carlo generators: 
non-diffractive events are generated with DJANGO~\cite{django}, 
while diffractive 
-- low mass and low multiplicity -- events are generated
with RAPGAP~\cite{rapgap3}. 
The two Monte Carlo samples are mixed in a proportion determined 
by fitting distributions of hadronic variables in data with a weighted 
sum of DJANGO and RAPGAP events.

The value of $R=F_{L}/(F_{2}-F_{L})$ is taken from the BKS model~\cite{bks1}.
Due to the low $Q^{2}$ of this analysis, 
the effect of $F_{L}$ is small, reaching at most $3\%$ in few bins at 
high $y$ and moderate $Q^{2}$.

The data are shown in Fig.~\ref{fig7}, together with two fits 
(ALLM97~\cite{allm97}, DL98~\cite{dl98}) which include the
1995 measurements, as well as with the original DL parameterization.
\begin{figure}[h]
\begin{center}
 \mbox{
  \leavevmode
  \includegraphics[width=.46\textwidth]{monaco.8}
  \hspace{.25cm}
  \leavevmode
  \includegraphics[width=.46\textwidth]{monaco.9}
 }
\end{center}
\caption[]{$F_{2}$ as a function of $x$ in bins of $Q^{2}$ from the BPT 
   analysis compared 
   with previous measurement from ZEUS (BPC 95) and E665 results}
\label{fig7}
\end{figure}

The average statistical error, including the statistical error of the 
Monte Carlo simulation, is 2.6\%. The typical systematic error of 3.3\% is 
similar to the statistical one and has been reduced by a factor 2 to 3 
compared to the previous measurement.

The BPT $F_{2}$ data are converted to total $\gamma^{*}p$ cross sections using 
(\ref{eq1.1}) and are shown in Fig.~\ref{fig8} as a function of $Q^{2}$ 
for different $W$ bins. 
The measured cross sections for fixed  $W$ have been fitted 
to (\ref{eq1.4}), with the values of the cross sections $\sigma_{0}(W^{2})$
at $Q^{2}=0$ left as free parameters. 
The fit has a reasonable $\chi^{2}/dof$ of 1.3, taking into account
statistical errors only;
this indicates that the simplified GVDM model of (\ref{eq1.4}) 
describes well the $Q^{2}$ dependence of the cross section.

The extrapolated cross sections $\sigma_{0}$ are shown in Fig.~\ref{fig9} 
as a function of the center of mass energy $W$; in the same figure the 
direct measurements from H1~\cite{h1.php} and ZEUS~\cite{zeus.php} 
and data from other experiments 
at low $W$~\cite{fixed_t} 
are also shown, and compared with the DL98~\cite{dl98} and 
ALLM97~\cite{allm97} parameterizations.
The magnitude of the model dependence, estimated 
by extrapolating $\sigma_{tot}^{\gamma^{*}p}$ with the $Q^{2}$-dependence 
of DL, DL98 and ALLM97 parameterizations, is shown as a shaded band in Fig.~\ref{fig9}.

A fit of the cross sections $\sigma_{0}(W)$ extrapolated from the BPT data,
and of the low $W$ measurements with the Regge-type form of (\ref{eq1.2}) 
fixing $\alpha_{I\!R}=0.5$, gives a value of the Pomeron intercept 
$\alpha_{I\!P}=1.105\pm 0.001(stat)\pm 0.007(sys)$.
\begin{figure}
\begin{center}
\begin{minipage}[b]{.49\textwidth}
 \includegraphics[width=.95\textwidth]{monaco.10}
 \caption[]{$\sigma_{tot}^{\gamma^{*}p}$ versus $Q^{2}$ in bins of $W$ from 
 the BPT $F_{2}$ analysis and other measurements from ZEUS and H1}
 \label{fig8}
\end{minipage}
\hfill
\begin{minipage}[b]{.47\textwidth}
 \includegraphics[width=\textwidth]{monaco.11}
 \caption[]{Extrapolated and measured $\sigma_{tot}^{\gamma p}$ versus $W^{2}$.
The curves are the ALLM97 and DL98 parameterizations and the result of the Regge fit described in the text}
 \label{fig9}
\end{minipage}
\end{center}
\end{figure}


\section{Conclusions}

The HERA measurements of $F_{2}$ 
and $\sigma_{tot}^{\gamma^{*}p}$ at low $Q^{2}$ and $x$, 
in the transition region between DIS and 
the photoproduction limit,
have been presented.
The data with $Q^{2}<0.65$ $\mathrm{GeV}^{2}$ can be described 
by a simple GVDM and Regge-inspired parameterization, which however
 fails at higher $Q^{2}$.
The rapid fall of $F_{2}$ at small $x$ for $Q^{2}>1$ $\mathrm{GeV}^{2}$
is described by NLO QCD fits.

The slopes $d\ln F_{2}/d\ln{(1/x)}$ at fixed $Q^{2}$ and 
$dF_{2}/d\ln{Q^{2}}$ at fixed $x$ have been detemined from the HERA data. 
In the regime where pQCD is reliable, the behaviour of the slopes is
compatible with the results of a NLO QCD fit.



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