
\def\xgo{x_\gamma^{\rm obs}}
\def\kt{{\rm k}_{\rm T}}
\def\met{\overline{E}_T}
\def\et2{\overline{E}_T^2}
\def\q2{Q^2}

\def\gp{\gamma^*p}
\def\gpjj{\gp\rightarrow {\rm jet}\ + \ {\rm jet}}
\def\g2{GeV$^2$}
\def\fxgo{\xgo = \frac{ \sum_{jets}E_{\rm T}^{\rm jet}e^{-\eta^{\rm jet}}}{2yE_e}}
\def\kt{k_T}
\def\etjetd{(E_{\rm T}^{\rm jet})^2}


\def\etjet{E_T^{\rm jet}}
\def\etjetd{(E_T^{\rm jet})^2}
\newcommand{\ft}{F_2}
\newcommand{\GeV}{\,\mathrm{Ge\kern-1pt V}}
\newcommand{\pb}{\,\mathrm{pb}^{-1}}
\newcommand{\pom}{{\rm I\! P}}
\newcommand{\reg}{{\rm I\! R}}
\newcommand{\mub}{\,\mu\mathrm{b}}

\newcommand{\plb}[1]{{\it Phys.\ Lett.\ }{\bf B #1}}
\newcommand{\npb}[1]{{\it Nucl.\ Phys.\ }{\bf B #1}}
\newcommand{\epjc}[1]{{\it Eur.\ Phys.\ J.\ }{\bf C #1}}
\newcommand{\prd}[1]{{\it Phys.\ Rev.\ }{\bf D #1}}
\newcommand{\cpc}[1]{Comp.\ Phys.\ Comm. {\bf #1}}

\documentclass{ws-procs9x6}

\begin{document}

\title{Structure Functions and the Transition Region from Photoproduction
to Deep Inelastic Scattering}

\author{DORIAN KCIRA}

\address{University of Wisconsin\\
ZEUS/DESY \\
Notkestrasse 85\\
22607 Hamburg\\
GERMANY\\
E-mail: dorian.kcira@desy.de}


\maketitle

\abstracts{
Precise inclusive and dijet measurements are presented that
extend the kinematic domain of HERA
and cover the transition region from deep inelastic scattering to photoproduction.
The rise of the structure function $F_2$ at low $x$ persists but is shallower at
lower $Q^2$. Dijet measurements can significantly constrain the
parton distribution functions of the virtual photon. 
}


\section{Introduction}

In deep inelastic $ep$ scattering (DIS) at HERA, a photon of negative squared
momentum (virtuality, $Q^2$) is exchanged. The highly virtual pointlike photon
is used as a probe and precise determinations of proton structure are
performed by inclusive measurements. Measurements of jet cross sections in DIS
provide further information, are sensitive to the proton structure
and allow precise tests of perturbative QCD (pQCD).

In photoproduction ($Q^2 \simeq 0$),
the requirement of events with high transverse energy jets provides for a hard
scale in the interaction and allows comparison to pQCD calculations.
In photoproduction the measurements are sensitive
to both the structure of the proton and the structure of the (quasi-real) photon.

Both the DIS and photoproduction regimes have been extensively studied at HERA
and pQCD predictions have been successful in describing the data. The physical
picture and the language used in the two regimes is not the same.
Therefore, data and predictions in the region between them are particularly interesting.

Until recently, not much data had been gathered at HERA in the
transition region between DIS and photoproduction mainly because of
the experimental difficulty of identifying the scattered lepton
at small scattering angles.
Recently, the H1 and ZEUS Collaborations have performed measurements
that ``scan'' this region using inclusive and dijet
events. It must be noted that these are conceptually two different
ways of crossing the transition region. In the case of inclusive
measurements the transition is from a hard regime (DIS,
with hard scale $Q^2$) to a soft regime (photoproduction).
In the case of dijet measurements the requirement of high transverse energy
jets allows application of pQCD throughout the transition from real ($Q^2\simeq 0$)
photons, which can have structure, to virtual pointlike photons.
In this second transition two hard scales play a role: $Q$ and
$E_T$, the transverse energy scale of the jets. Depending
on the interplay of these two hard scales, the photon may or may not display a structure.
Both these types of measurements are presented in this paper.

\section{Inclusive Measurements}

Measurements of the proton structure function $\ft$ in the
region of $Q^2 \gg 1$~GeV$^2$ have shown a good description of
the data by perturbative QCD (pQCD)~\cite{H1LOWQ2,H1QCD2003,ZEUSQCD2003}.
On the other hand the region
$Q^2 < 1$~GeV$^2$ is described by models
based on generalized vector meson dominance~\cite{GVDMTHEORY} and Regge
theory~\cite{REGGETHEORY}. Therefore, the region of $Q^2$ around $1$~GeV$^2$ deserves
particular attention.

In this section the latest HERA measurements
in the transition region from low to high $Q^2$ are presented
together with comparisons to perturbative and non-perturbative approaches.
These measurements use different experimental techniques for accessing
the low $Q^2$ data, which are reflected in the titles
of the subsections.

\subsection{Initial State Radiation Analysis}

Events with QED initial state radiation (ISR) have been
used to measure $\ft$.
The measurement is based on the data taken during 1996 with the ZEUS detector (3.78~pb$^{-1}$)~\cite{ZEUSISR}.
The emission of a hard photon from the initial-state positron leads to a reduction in the
center-of-mass energy of the hard scattering particles.
This means that lower values of $Q^2$ can be accessed.

\begin{figure}[ht]
\centerline{
\epsfxsize=2.1in\epsfbox{kcira.fig1.eps}
\epsfxsize=2.3in\epsfbox{kcira.fig2.eps}
}
\caption{
\underline{Left plot:} $\ft$ versus $x$ for fixed $Q^2$. The results of the ISR analysis are shown together with previously
published ZEUS results. The yellow band shows the predictions from the ZEUS NLO QCD fit while the
dashed curve those from ALLM97.
\underline{Right plot:} $\ft$ measurement from QEDC analysis (closed circles) compared with other HERA and fixed target
measurements. The solid line is the prediction from the ALLM97 parametrization while the dashed line
represents the fractal fit, plotted for $y > 0.0003$.
}\label{zeus_isr_1}
\end{figure}

The $\ft$ measurements are shown in Fig.~\ref{zeus_isr_1} (left plot) as a function
of $x$ at different values of $Q^2$ together with the most recently published
$\ft$ measurements~\cite{ZEUSF22001,ZEUSSVX95} and the BPC/BPT~\cite{BPCBPT}
results. The data sets are in good agreement in the region
of overlap, completing the coverage of the $x$-$Q^2$ plane at low $Q^2$.
The results are compared to the predictions of the ZEUS NLO QCD fit for $Q^2\geq1.3$~GeV$^2$
and to the predictions of ALLM97~\cite{ALLM97} over the whole $Q^2$ range. Both predictions give a good description
of the data.

\subsection{QED Compton Analysis}

The H1 Collaboration has performed measurements of $\ft$ using QED Compton scattering events\cite{H1QEDC}.
The measurements use $e^+p$ data with an integrated luminosity of 9.25~pb$^{-1}$ collected during 1997.

The QED Compton process corresponds to low $Q^2$ and a large
virtual electron mass. The final state topology is given by an
azimuthal back-to-back configuration of the outgoing positron and photon
detected under rather large scattering angles. Their transverse
momenta balance such that very low values of $Q^2$ are experimentally
accessible.

The $\ft$ values measured in QED Compton scattering are depicted in Fig.~\ref{zeus_isr_1} (right plot)
as a function of $x$ at fixed $Q^2$ and compared to other HERA and fixed target data. The kinematic
range is extended at very low $Q^2$ towards higher $x$ values, complementing the standard
inclusive and shifted vertex measurements. The results overlap with
the region measured in fixed target experiments. Good agreement is observed.

\subsection{Shifted Vertex Analysis}

A special run was performed in 2000 at HERA in which the interaction vertex was shifted
towards the proton beam direction by 70~cm. This increased the acceptance for positrons
with smaller scattering angles. This again allows lower values of $Q^2$ to
be accessed.
The data presented here\cite{H1SVTX} correspond to an integrated luminosity of 0.6~pb$^{-1}$,
a factor of four larger than
the previous shifted vertex data samples at H1\cite{H1SVX95} and ZEUS~\cite{ZEUSSVX95}.
The combination of larger statistics and the new Backward Silicon Tracker
(BST) allowed the H1 Collaboration to make measurements with higher precision than previously.

\begin{figure}[h]
\centerline{\epsfxsize=3.0in\epsfbox{kcira.fig3.eps}} 
\caption{
$\ft$ from H1 shifted vertex data (squares), H1 99 and 97 nominal vertex data (points and triangles),
larger $x$ data from BPC/BPT (triangles) and from NMC (stars).
}\label{h1_svtx_2}
\end{figure}


In Fig.~\ref{h1_svtx_2} the $\ft$ data are compared with phenomenological predictions obtained
from fits to previous data. The best description is given by the fractal model parametrization~\cite{FRACTMODEL}
(solid curve) the parameters of which were determined by fitting the ZEUS BPC/BPT data and the published low $Q^2$
H1 data~\cite{H1LOWQ2}. The ALLM97 parametrization (dashed-dotted curves) similarly used previous data
which leads to an acceptable description of the present $\ft$ data with a slightly stronger rise
with decreasing $x$ for $Q^2\geq 1$~GeV$^2$ than is observed in the H1 data.
The prediction from the H1 NLO QCD fit~\cite{H1LOWQ2} is also shown (dashed curves). This fit only
included data with $Q^2 \geq 3.5$~GeV$^2$. Its backward extrapolation undershoots
measurements at lower $Q^2$. At $Q^2\leq 1$~GeV$^2$ the strong coupling constant
$\alpha_S (Q^2)$ becomes large, making a comparison with NLO QCD meaningless.

In Fig.~\ref{h1_svtx_3} (left plot) the photon-proton cross section, $\sigma_{\gamma^* p}$, which is
proportional to $\ft / Q^2$, is plotted as a function of $Q^2$ at different values of $W^2 \simeq sy$.
$W$ is the invariant mass of the $\gamma^* p$ system.
The new data fill the gap between previous measurements at lower and higher $Q^2$ and reach similar accuracy.
The same three parametrizations are plotted as in Fig.~\ref{h1_svtx_2}. At $Q^2\simeq 1$~GeV$^2$ the ALLM97
parametrization is too large at high $W$ and the QCD fit fails, as expected. For $Q^2 \rightarrow 0$, the fractal model fit
is too steep to describe the total photoproduction cross section (not shown here). However, this could
be cured by considering mass effects in the fractal Ansatz~\cite{FRACTALMASS}.

\begin{figure}[ht]
\centerline{
\epsfxsize=2.3in\epsfbox{kcira.fig4.eps}
\epsfxsize=2.5in\epsfbox{kcira.fig5.eps}
} 
\caption{
\underline{Left plot}: measurements of the structure function $\ft$ represented as $\ft/Q^2$ compared
to the Fractal fit, H1 QCD fit and the ALLLM97 parametrization.
\underline{Right plot}: determination of the exponent $\lambda(x,Q^2)$ from fits to $\ft$ as measured
from H1 and ZEUS. The straight line represents a linear fit to the measurements for $Q^2\geq 3.5$~GeV$^2$.
}\label{h1_svtx_3}
\end{figure}

\subsection{The Rise of $\ft$ at low $x$}

To quantify the behaviour of the rise of $\ft$ towards low $x$ at fixed $Q^2$
the H1 and ZEUS Collaborations have measured the derivative:

\begin{equation}
\left( \displaystyle\frac{\partial \ln \ft(x,Q^2)}{\partial \ln x} \right) \equiv - \lambda(Q^2)
\nonumber
\end{equation}

Using the data presented in this paper, the measurement of $\lambda(Q^2)$ is
extended to the transition region as plotted in Fig.~\ref{h1_svtx_3} (right plot). For values
$Q^2 < 1$~GeV$^2$ the derivative $\lambda(Q^2)$ is seen to significantly deviate
from an extrapolation of the linear behaviour determined from the data in the deep
inelastic scattering region ($Q^2 \geq 3.5$~GeV$^2$).
From soft hadronic interaction models, it is expected
that for $Q^2 \rightarrow 0$, $\lambda \rightarrow 0.08$.
HERA data agree with these expectations.


\section{Dijet Measurements}

It has been established that the real photon ($Q^2\simeq 0$) has a partonic structure~\cite{PLUTO}.
In contrast, in hard collisions at high $Q^2$ the photon is considered to be pointlike.
In leading order (LO) QCD two types of processes contribute to jet photoproduction: direct, in which the photon couples
as a pointlike particle to quarks in the hard scatter; and resolved, in which the photon acts as
a source of partons. Both processes lead to two jets in the final state.
The $\xgo$ variable:
\begin{equation} \xgo = \frac{\sum_{\rm jets} ( E^{\rm jet}-p_z^{\rm jet} )}{\sum_{\rm hadrons} ( E - p_z ) }\nonumber\;\;,\end{equation}
which measures the
fraction of the photon momentum participating in the production of the dijet system, is used to separate the two
processes since resolved (direct) processes dominate at low (high) $\xgo$ values~\cite{OBSDIRPHP}.
In the above definition the upper sum runs over the two jets with highest
transverse energy and the lower sum runs over all final state hadrons in the laboratory frame.

Within fixed-order QCD, only pointlike photon interactions are expected to contribute to jet production in DIS. However,
two scales play a role in the interaction: $Q$ and the jet
transverse energy, $E_T^{\rm jet}$. QCD predicts that for high $Q^2$ ($Q^2\gg (E_{\rm T}^{\rm jet})^2$)
the photon will behave as a pointlike particle. For $Q^2\ll (E_{\rm T}^{\rm jet})^2$
the photon may present a partonic structure, even for relatively large values of $Q^2$,
which is resolved at a scale related to the transverse energy of the jets.
Therefore, resolved processes may contribute significantly to the total cross section.

Dijet measurements in photoproduction, DIS and the transition region
are presented in this section. Different LO and NLO pQCD calculations are compared
with the data.

The H1 Collaboration has measured dijet cross sections in the $\gamma^*p$ center-of-mass
frame using a sample of $16.3$~pb$^{-1}$ data collected in 1999\cite{H1DIJETS}. The phase space is defined
by: $2 < Q^2 < 80$~GeV$^2$, the inelasticity $0.1<y<0.85$, the transverse
energy of the two leading jets $E_{\rm T}^{\rm jet1,2} > 5$~GeV,
$\overline{E}_{\rm T} = (E_{\rm T}^{\rm jet1}+E_{\rm T}^{\rm jet2})/2 > 6$~GeV and
the pseudorapidity of the two leading jets $-2.5<\eta^{\rm jet1,2}<0$.

\begin{figure}[ht]
\centerline{
\epsfxsize=2.3in\epsfbox{kcira.fig6.eps}
\epsfxsize=2.3in\epsfbox{kcira.fig7.eps}
}
\caption{
Triple differential cross section as a function of $\xgo$, $\overline{E}^2_{\rm T}$ and $Q^2$. The data are compared
to HERWIG and RAPGAP MC predictions (left plot). Separately shown are the components of the cross section from
direct (dir) and resolved $\gamma^*$. Predictions of the HERWIG and CASCADE MC are also compared to the data (right plot).
}\label{h1virph_1}
\end{figure}

The triple differential dijet cross sections measured as a function of $Q^2$, $\overline{E}^2_{\rm T}$,
and $\xgo$ are shown in Fig.~\ref{h1virph_1}. The prediction of HERWIG~\cite{HERWIG} and RAPGAP~\cite{RAPGAP}
Monte Carlo (MC) simulations with the SaS~1D\cite{SAS} parametrization of the virtual photon parton distribution function (PDF),
as well as the pure direct contributions are compared to the data in Fig.~\ref{h1virph_1} (left plot).
In general HERWIG and RAPGAP tend to underestimate the measured cross sections.
The direct contributions reasonably describe the data in the highest $Q^2$ bin,
while a clear need for resolved processes is observed for $Q^2 \ll \overline{E}^2_{\rm T}$.
The measured cross sections are compared in Fig.~\ref{h1virph_1} (right plot) to the predictions of the CASCADE~\cite{CASCADE} MC program, which uses the CCFM evolution scheme. This theoretical approach is based on non-$k_T$-ordered parton
cascades and does not include the concept of virtual photon structure. CASCADE describes the data reasonably well
but not perfectly.

The ZEUS Collaboration has studied the dependence of dijet production on $Q^2$
by measuring dijet cross sections in the range 
$0 \lesssim Q^2 < 2000 {\;\rm GeV}^2$ using the 1996-1997 data sample\cite{ZEUSDIJETS},
which corresponds to an integrated luminosity of 38.6~pb$^{-1}$.
Inclusive dijet cross sections were measured for jets with
transverse energy $E_T^{1,2}>7.5,6.5$~GeV and pseudorapidities in
the photon-proton center-of-mass frame in the range $-3<\eta^{\rm jet}<0$.

Fig.~\ref{virph_1} presents the differential dijet cross section, $d\sigma/dQ^2$,
for the direct-enhanced region ($\xgo > 0.75$) and the resolved-enhanced region ($\xgo <0.75$)
together with the total dijet cross section.
The measurements are precise and cover a wide range of photon
virtualities, including the transition region from photoproduction to DIS.
As expected, the cross section for $\xgo< 0.75$ falls more rapidly than that
for $\xgo > 0.75$. Even though the dijet cross section is dominated by interactions with $\xgo > 0.75$ for $Q^2\gtrsim
10$~GeV$^2$, there is still a contribution of approximately $24\%$ from low-$\xgo$ events even for $Q^2$ as high as
$500$~GeV$^2$. The NLO QCD calculations of DISASTER++~\cite{DISASTER} with $\mu_R^2=\q2+\etjetd$ are
compared to the measurements.
The photoproduction measurement is compared with the NLO prediction from Frixione and Ridolfi
where $E_T^2$ is used as the scale.
The measured cross section for
$\xgo>0.75$ is reasonably well described by the calculation but the measured cross section for $\xgo < 0.75$
is significantly underestimated for high $Q^2$.

\begin{figure}[ht]
\centerline{\epsfxsize=2.5in\epsfbox{kcira.fig8.eps}}   
\caption{
a) Measured dijet cross-sections $d \sigma / d Q^2$ for $\xgo > 0.75$ (upwards triangles)
$d \sigma / d Q^2$ for $\xgo < 0.75$ (downwards triangles) and $d \sigma / d Q^2$ for the whole $\xgo$
region (black dots). The shaded band represents the uncertainty in the absolute energy scale of the jets.
The NLO QCD calculations of DISASTER++ ($\mu_R^2=\q2+\etjetd$) and Frixione ($\mu_R^2=\etjetd$)
are shown for each cross-section.
b) Relative difference of the measured cross-section to the DISASTER++
and Frixione calculations. The hatched band shows the theoretical uncertainty of the calculations.
}\label{virph_1}
\end{figure}

Fig.~\ref{virph_3} shows the ratio
$R=\frac{{\rm d}\sigma}{{\rm d} Q^2} (\xgo < 0.75) /\frac{{\rm d}\sigma}{{\rm d} Q^2} (\xgo > 0.75)$
as a function of $Q^2$ in three different regions of $\et2$.
The $Q^2$ dependence of the data is stronger at low $\et2$ than at higher $\et2$,
showing that the low $\xgo$ component is suppressed at low
$Q^2$ as $\et2$ increases and vice-versa.
Predictions from HERWIG assuming the SaS~1D and GRV\cite{GRV}  parametrizations of the photon
PDFs are compared to the data in Fig.~\ref{virph_3} (left plot).
The SaS~1D parametrization contains the suppression of the virtual photon structure
with increasing virtuality. The predictions fall with increasing $Q^2$ and reproduce
the fall of the data relatively well.
On the other hand the predictions using GRV which assumes no suppression of the virtual photon
structure, are relatively constant at the level of the photoproduction ratio of the data.
The NLO calculations are compared to the data in Figure~\ref{virph_3} (right plot).
For photoproduction, the calculations lie below the data for all $\et2$.
The NLO-prediction which used GRV for the photon PDFs is closer to the data than the one using AFG~\cite{AFG}.
However, they underestimate the relative cross section at low $\xgo$.
The DISASTER++ predictions show some suppression of the low $\xgo$ contribution with increasing $\q2$, however
they underestimate the proportion of low $\xgo$ relative to high $\xgo$.

\begin{figure}[ht]
\centerline{
\epsfxsize=2.3in\epsfbox{kcira.fig9.eps}
\epsfxsize=2.3in\epsfbox{kcira.fig10.eps}
}
\caption{
Measured ratio 
$R =\frac{{\rm d}\sigma}{{\rm d} Q^2} (\xgo < 0.75) /\frac{{\rm d}\sigma}{{\rm d} Q^2} (\xgo > 0.75)$ 
as a function of $Q^2$ in different regions of $\et2$ (black dots).
The LO calculations of HERWIG with using the SaS~1D and GRV photon pdfs (left plot) as well as the NLO
predictions from DISASTER++ and Frixione-Ridolfi (right plot) are also shown.
}\label{virph_3}
\end{figure}


\section{Summary and Conclusions}

New precise measurements of the structure function $\ft$
have been performed extending the kinematic domain of HERA
and covering the transition region from DIS to photoproduction.
The data are in agreement with the previous
HERA data at higher and lower $Q^2$ and those
from fixed targed experiments in the region of overlap.

Differential dijet cross sections have been measured in a large
range of photon virtualities, including the transition region
from photoproduction to DIS. The data are reasonably well described
by DGLAP LO models which introduce virtual photon structure, suppressed with
increasing $Q^2$. The non-$k_T$-ordered CCFM-based approach
also does a relatively good job in describing the measurements.
These data have the potential to significantly constrain virtual photon PDFs and should be taken
into account in future fits.
The currently available NLO QCD calculations have large uncertainties
at low $Q^2$, where the presence of the resolved photon contribution
is expected. Improved higher-order or resummed calculations are needed.
The NLO QCD predictions generally underestimate the cross section
at low $\xgo$ relative to that of high $\xgo$.

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\end{document}

