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\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.}  B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}
\def\EPJC{{\em Eur. Phys. J.} C}
\newcommand{\Pom}{{\hspace{ -0.1em}I\hspace{-0.2em}P}}

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\begin{document}

%\title{Exclusive Vector Meson and diffractive Jets at HERA }
\title{Exclusive Vector Meson at HERA }
\author{H. Kowalski}

\institute{Deutsches Elektronen-Synchrotron, DESY \\
Notkestr. 85, \\ 
22607 Hamburg, Germany\\ 
E-mail: Henri.Kowalski@desy.de}


\maketitle

\begin{abstract}
This talk describes the measurement of $F_2$ and inclusive and exclusive diffractive cross sections in the low-$x$ region by HERA experiments.  The abundance of diffractive reactions observed at HERA indicates the presence of perturbative multi-ladder exchanges. The exclusive diffractive vector-meson and diffractive dijet production  are discussed in terms of dipole models    which connect the measurement of $F_2$ with diffractive processes and in which multiple exchanges and saturation processes are natural. 
%The  diffractive dijets are also discussed within the diffractive parton density  approach. Good description of diffractive dijets in the dipole 
%picture and in the diffractive parton density approach indicates that these two seemingly different views on diffraction are not really distinct.
\end{abstract}


\section{$F_2$ and Diffraction at HERA}
\begin{wrapfigure}{r}{0.3\columnwidth}
%\begin{figure}[htp]
%\vspace{1.0cm}
%\hspace{0.0cm}
\epsfig{file=h1-event.eps,width=5.8cm,height=2.7cm,clip}
\epsfig{file=r31565e40151_zevis.eps,width=4.5cm,height=2.8cm,clip}
\caption{\sl Two examples of DIS events seen in the H1 (left) and ZEUS (right) detector. } 
\label{fig:h1event}
\vspace*{-0.5cm}
%\end{figure}
\end{wrapfigure}
The HERA machine is a large electron-proton collider, in which
electrons with energy of 27.5 GeV scatter on protons of 920 GeV.  The collision products are recorded by the two large, multipurpose experiments ZEUS and H1. The detectors consist of  inner tracking detectors surrounded by  large calorimeters measuring the spatial energy distribution, event by event. The calorimeters are in addition surrounded by  muon detector systems. Fig.~\ref{fig:h1event} shows, as an example, a picture of a high $Q^2$ DIS event measured by the H1 and ZEUS detectors. 
From the amount and positions of energy deposited by the scattered electron and the hadronic debris, the total $\gamma^* p$ CMS energy, $W$, and the virtuality of the exchanged photon, $Q^2$, are determined. Counting the events at given  $Q^2$ and $W^2$ allows the determination of the total cross section for the collisions of the virtual photon with the proton, $\sigma_{\gamma^{*} p}(W^2,Q^2)$, and in turn the structure function, 
$$F_2(x,Q^2)=\frac{Q^2}{4\pi^2 \alpha_{em}} \sigma_{\gamma^{*} p}(W^2,Q^2)$$
with  $x\approx Q^2/W^2$ when $Q^2\ll W^2$. 

Deep inelastic scattering and the structure function $F_2$ have a simple and intuitive interpretation  when viewed in the fast moving proton frame. The incoming electron scatters on the proton by emitting an intermediate photon with a virtuality $Q^2$. The incoming proton consists of a fluctuating cloud of quarks, antiquarks and gluons. Since the lifetime of the virtual photon is much shorter than the lifetime of the $q\bar q$-pair, the photon scans the ``frozen'' parton cloud and picks up quarks with longitudinal momentum $x$, see Fig.~\ref{fig:stdis}.  $F_2$ measures then the density of partons  with a size which is larger than the photon size, $1/Q$, at a given $x$. Fig.~\ref{fig:varf2} shows the structure function $F_2$ as measured by H1, ZEUS and fixed target experiments for selected $Q^2$ values~\cite{wolf}.   
\begin{wrapfigure}{r}{0.3\columnwidth}
%\begin{figure}[htp]
%\vspace{-0.5cm}
%\hspace{3.cm}
\epsfig{file=stdis.eps,width=4.5cm,height=2.5cm}
\vspace*{-1.2cm}
\caption{\sl Schematic view of deep inelastic scattering (DIS). } 
\label{fig:stdis}
%\end{figure}
\vspace*{-0.5cm}
\end{wrapfigure}

In the low-$x$ regime,  $F_2$ measured at HERA exhibits a striking behavior. At low $Q^2$ values, $Q^2<1$ GeV$^2$, where the photon is large, $F_2$ rises only moderately with diminishing $x$, whereas as $Q^2$ increases, i.e. the photon becomes smaller, the rise of $F_2$ accelerates quickly. The rise of $F_2$ at low $Q^2$ values, i.e. when the photon is of similar size as a hadron,
corresponds to the rise of the hadronic cross sections with energy. The fast rise at large $Q^2$ indicates the strong growth of the cloud of partons in the proton. The onset of the fast growth at $Q^2$ values larger than 1 GeV$^2$ indicates  that these partons are of perturbative origin. 
\begin{wrapfigure}{r}{0.45\columnwidth}
\vspace*{-0.8cm}
\hspace{0.0cm}
\epsfig{file=epsfig9.hera.4.eps,height=8cm,width=7cm}
\vspace*{-3.0cm} 
\caption{\sl The structure function $F_2$ as a function of $x$ as measured by H1, ZEUS and fixed target experiments for selected $Q^2$ between 0.1 and 150 GeV$^2$. } 
\label{fig:varf2}
\vspace*{-0.3cm}
\end{wrapfigure}

For sufficiently large $Q^2$ perturbative QCD provides a 
set of leading-twist linear evolution equations (DGLAP) which describe the 
variation of the cross section as a function of $Q^2$; 
see Fig.~\ref{fig:Ael2}. Moreover, a closer look at the $x$-dependence 
of the parton splitting functions has led to the prediction that 
the gluon density, at small $x$, should rise 
with $1/x$. This rise should translate into a growth with energy of the total $\gamma^* p$ cross section or, equivalently, of $F_2$ with diminishing $x$. The data show that the growth of $F_2$ starts in the low-$x$ regime which indicates that this is mainly due to the abundant gluon production.
 This is confirmed by all detailed theoretical investigations of HERA data. As an example Fig.~\ref{fig:sf-zeus} shows the results of the ZEUS and MRST analyses of parton densities. Both analyses show that in the low-$x$ region the gluon density dwarfs all quark densities with exception of the sea quarks. The sea quarks,  in  perturbative QCD, are generated from the gluon density.  

\begin{wrapfigure}{r}{0.4\columnwidth}
\vspace{-0.7cm}
\hspace{0.5cm}
\epsfig{file=agk1.eps,height=3.5cm,width=5cm}
\vspace{-1.3cm} 
\caption{\sl Illustration of the pQCD description of the total cross section 
$\sigma_{tot}^{\gamma^*p}$.
The gluon ladder represents the linear QCD evolution equations.} 
\label{fig:Ael2}
\vspace*{-0.3cm}
\end{wrapfigure}
One of the most important observations of the HERA experiments is that,   
in addition to the usual DIS events, in which the struck proton is transformed into a swarm of particles, there are also events in which the proton remains intact after collision. Whereas the usual DIS events are characterized by large energy depositions in the forward (proton) direction, see Fig.~\ref{fig:h1event}, the events with intact protons show no activity in this region; see Fig.~\ref{fig:diffev-zeus}. 


By analogy to the absorption of light waves on a black disk, the events of this type are called diffractive events and the process in which they are produced is called diffractive scattering. The intact forward proton corresponds in optics to the forward white spot observed in the center of the disc shadow.
The measurement of diffractive reactions requires the determination of two additional variables: the diffractive mass, $M_X$, and the square of the four-momentum transferred by the outgoing proton, $t$. The variable $M_X$,  which is equal to the invariant mass of all particles emitted in the  reaction with exception of the outgoing proton (or the proton dissociated system),
%\begin{figure}[htp]
\begin{wrapfigure}{r}{0.35\columnwidth}
\vspace*{-0.2cm}
%\hspace{0.5cm}
\epsfig{file=sf-zeus.eps,width=5.5cm,height=5.0cm}
\vspace*{-0.7cm}
\caption{\sl Quark and gluon densities at $Q^2=10$ GeV$^2$ as determined
from HERA data. Note that the gluon and sea-quark densities are displayed diminished by a factor 0.05.} 
\label{fig:sf-zeus}
\vspace*{-0.4cm}
\end{wrapfigure}

is determined from energy depositions recorded by the central detectors of the H1 and ZEUS experiments. The variable $t$ is determined by forward detectors, which measure the momentum of the outgoing diffractively scattered proton.  In  exclusive diffractive vector-meson production the $t$ variable can also be determined from the precise measurement  of the momenta of the vector-meson decay products measured in the tracking chamber systems of central detectors.

The analysis of the observed $\ln M_X^2$ distribution allows a separation of diffractive and non-diffractive events as indicated in Fig.~\ref{fig:mx-zeus}.
The plateau like structure, most notably seen  at higher $W$ values, is due to diffractive events since in diffraction  $d{ N}/d\ln M^2_X \approx $ const.
%\begin{figure}[htp]
\begin{wrapfigure}{r}{0.55\columnwidth}\vspace{0.0cm}
\vspace*{-0.3cm}
%\hspace{1.0cm}
\epsfig{file=r31558e2887_zevis.eps,width=4.5cm,height=2.5cm,clip}
\hspace*{-0.6cm}
\epsfig{file=ee2tr.eps,width=4.0cm,height=2.5cm,clip}
\vspace*{-0.7cm}
\caption{\sl Two examples of diffractive events seen in the ZEUS detector.} 
\label{fig:diffev-zeus}
\vspace*{-0.7cm}
\end{wrapfigure}
The high mass peaks in Fig.~\ref{fig:mx-zeus},  which are due to non-diffractive events,  have a steep exponential  fall-off, $d{ N}/d  \ln M_X^2 \propto \exp(\lambda \ln M_X^2)$, towards smaller $\ln M_X^2$ values.
This exponential fall-off is directly connected to the exponential suppression of large rapidity gaps in a single gluon ladder 
exchange diagram, Fig.~\ref{fig:Ael2}, which represents the dominant QCD contribution. 

In the ZEUS investigation~\cite{kow.zeus.2,zdiffFPC}  the diffractive contribution 
was therefore identified as the excess
of events at small $M_X$ above the exponential fall-off of the
non-diffractive contribution in $\ln M^2_X$. 
This selection procedure is called the $M_X$ method. 
%The diffractive measurements in which the forward proton was not directly detected cannot fully differentiate between the events with  intact or dissociated forward proton. In the $M_X$ method the ZEUS detector acceptance limits the mass of the dissociated forward proton, $N$, to $M_N<2.3$ GeV.  
In the H1 investigation~\cite{H1epf2d394}
the selection of diffractive events was performed
by the requirement of a large rapidity gap in the event. 
%This selection
%makes use of the information from the central and various forward detectors.
%The particles observed for each event are grouped into two systems, called
%Y and N,
%such that the maximum rapidity gap observed is between Y and N. The system
%N, which includes  particles produced in the direction of the incoming
%proton is required to have a mass $M_N < 1.6$ $GeV$.
%The presence of forward detectors allows H1
%to extend the diffractive measurement beyond the region
%accessible to the ZEUS $M_X$ measurement.     
The ZEUS $M_X$ and the H1  rapidity gap methods allow only  to 
measure the diffractive cross section integrated over the 
square of the four-momentum transfer~$t$. 

%The $t$ dependence of the diffractive cross section
%was measured by the ZEUS leading proton spectrometer (LPS)~\cite{Zeuslps} and will be measured by H1 very forward proton spectrometer (VFPS).
%The LPS  detector is  designed
%to measure small angle ($\sim$~1 mrad) scattered protons which would
%otherwise escape undetected inside the beam pipe. Isolation of diffractive
%events with the LPS is rather straight forward: 
%detection of a proton carrying a
%large fraction of the momentum of the incoming proton, 
%$x_L = p^{LPS}/p_{p_{beam}} > 0.97$, ensures a large rapidity gap
%between the outgoing proton and the system $X$. However, the event
%rate is considerably limited by the acceptance of the LPS. 

%Figure~\ref{fig:diff-zeus} shows the diffractive differential cross sections and the corresponding diffractive structure functions as measured with the $M_X$ method by ZEUS experiment and Figure~\ref{fig:diff-LPS-zeus}  
%shows  the measurement of the diffractive structure function 
%performed with the ZEUS LPS and the  central detector. Due to much larger acceptance the $M_X$ method covers a considerably larger kinematical region and has higher precision than LPS measurement. It cannot however fully differentiate between the diffractive processes in which a forward going proton stays intact or is dissociated. The comparison of two measurements allows the determination of the proton dissociation fraction within the $M_X$ sample as around 30\%.      
%\begin{figure}[htp]
%\vspace{-0.0cm}
%\hspace{-1.5cm}
%\epsfig{file=DESY-05-011_14.eps,width=9cm,height=11.cm}
%\epsfig{file=DESY-05-011_19.eps,width=9.5cm,height=11.0cm}
%\caption{\sl The diffractive differential cross section (left) and  the diffractive structure function (right) as measured with the  $M_X$ method by the ZEUS detector.} 
%\label{fig:diff-zeus}
%\end{figure}
%\begin{figure}[htp]
%\vspace{-0.0cm}
%\hspace{4.0cm}
%\epsfig{file=DESY-04-131_17.eps,width=8cm,height=9.cm}
%\caption{\sl  The diffractive structure function as measured by detection of the leading proton in the  ZEUS-LPS detector.} 
%\label{fig:diff-LPS-zeus}
%\end{figure}
%\begin{figure}[htp]
%\vspace{-0.0cm}
%\hspace{3.cm}
%\epsfig{file=H1prelim-02-012.fig2.eps,width=9cm,height=10.cm}
%\caption{\sl The diffractive structure function as measured with the rapidity gap method by the H1 detector.} 
%\label{fig:diff-h1}
%\end{figure}
The measured diffractive cross sections show a clear rise with increasing energy  $W$ in all
%The diffractive structure function measurements show a corresponding rise with decreasing $x_{IP}$ in all $\beta$ regions. The diffractive structure function measured by the H1 detector shows a similar behaviour, as seen in Fig.~\ref{fig:diff-h1}. 
 $M_X$ regions. It is interesting to note that the increase of the differential diffractive cross sections with $W$ is very similar to the increase of the total inclusive DIS cross sections, i.e. $\sigma_{diff}/\sigma^{tot}_{\gamma^{*}p}$ is approximately independent of energy in all $Q^2$ and $M_X$ regions as seen in Fig.~\ref{fig:rdiff}. The ratio of the diffractive to the total DIS cross section integrated over the whole accessible $M_X$ range, $M_X<35$ GeV, was evaluated at the highest energy of $W \approx 220$ GeV. 
 At $Q^2=4$ GeV$^2$, 
 \begin{wrapfigure}{r}{0.75\columnwidth}
%\begin{figure}[htp]
\vspace*{-0.4cm}
\hspace{-0.2cm}
\epsfig{file=nice_mx_fit3.eps,width=12cm,height=4.cm}
\vspace*{-0.8cm}
\caption{\sl Distribution of $M_X$ in terms of ln $M_X^2$. The straight lines give the non-diffractive contribution as obtained from the fits.
Note that the $\ln M_X^2$ distribution  can be viewed as a rapidity gap distribution since
$\Delta Y = \ln ( W^2 / M_X^2)$ for $M_X^2\gg Q^2$.
} 
\label{fig:mx-zeus}
\vspace*{-0.5cm}
\end{wrapfigure}
 $\sigma_{diff}/\sigma^{tot}_{\gamma^{*}p}$ reaches $\sim 16\%$. It decreases slowly with increasing $Q^2$, reaching $\sim 10\%$ at $Q^2=27$ GeV$^2$.   
\begin{wrapfigure}{r}{0.5\columnwidth}
%\begin{figure}[htp]
\vspace*{-0.9cm}
%\hspace{1.0cm}
\epsfig{file=rdiffmar1.eps,height=8cm,width=8cm}
\vspace*{-0.6cm} 
\caption{\sl The ratio of the inclusive diffractive and total DIS cross sections versus the $\gamma^* p $ energy $W$.} 
\label{fig:rdiff}
\vspace*{-0.5cm} 
\end{wrapfigure}


The observation of such a large fraction of diffractive events  was unexpected  since according to the intuitive interpretation of DIS  the incoming proton consists of a parton cloud and at least one of the partons is kicked out in the hard scattering process. In the language of QCD diagrams,    
at low-$x$ and not so small $Q^2$, the total cross section or $F_2$ is dominated by the abundant gluon emission as described by the  
single ladder exchange shown in Fig.~\ref{fig:Ael2}; the ladder 
structure also illustrates the linear DGLAP evolution equations that are 
used to describe the $F_2$ data. 
In the region of small $x$ gluonic ladders are expected to dominate over 
quark ladders. The cut line in Fig.~\ref{fig:Ael2} marks the final states produced in a DIS event: a cut parton (gluon) hadronizes and leads to jets or particles seen in the detector. It is generally expected that partons produced from 
a single chain  are unlikely to generate large rapidity 
gaps between them, since large gaps are exponentially suppressed as a function
of the gap size. This is a general property of QCD evolution equations of the DGLAP, BFKL or other types.

%\begin{figure}[htp]
In the single ladder contribution of Fig.~\ref{fig:Ael2}, 
diffractive final states can, therefore, only reside inside the blob at 
the lower end, i.e. lie below the initial scale $Q_0^2$ which separates the parton description from the non-perturbative strong interaction, as shown in Fig.~\ref{fig:agknpfan}.  The thick vertical wavy lines denote the non-perturbative Pomeron exchanges which generate the rapidity gap in DIS diffractive states \footnote{It is customary to call the exchange of a colourless system in scattering reactions a Pomeron. The simplest example of a (perturbative) Pomeron is given by the ladder diagram of Fig.~\ref{fig:Ael2}.}. 
The diagram of Fig.~\ref{fig:agknpfan} exemplifies therefore the ``Regge factorization" approach to diffractive parton densities as description of diffractive phenomena in DIS. 
In this approach the diffractive states are essentially of non-perturbative origin but they evolve according to the perturbative QCD evolution equations. 
Note, however, that the effective Pomeron intercept, $\alpha_\Pom $, extracted from diffractive DIS data lies significantly above the `soft' Pomeron intercept, indicating a substantial contribution to diffractive DIS from perturbative Pomeron exchange~\cite{zdiffFPC,h1dif}.
\begin{wrapfigure}{r}{0.3\columnwidth}
\vspace{-0.5cm}
\hspace{0.1cm}
\epsfig{file=agknp2fan.eps,height=3cm,width=5cm}
\vspace*{-1.0cm} 
\caption{\sl Diffractive final states as part of the initial condition to the evolution equation in $F_2$. The thick vertical wavy lines denote the non-perturbative Pomeron exchanges which generate the rapidity gap in DIS diffractive states.} 
\label{fig:agknpfan}
\vspace*{-0.4cm} 
\end{wrapfigure}


The properties of special diffractive reactions at HERA, like exclusive diffractive vector-meson and jets production, give clear indications that the diffractive processes could be hard and of perturbative origin. A significant contribution from perturbative multi-ladder exchanges should be present, in particular from the  double ladder exchange of Fig.~\ref{fig:Ael2d}. This diagram 
provides a potential source for the harder diffractive states: the cut blob at the upper end may contain $q\bar{q}$ and $q\bar{q} g$ states which hadronize into harder jets or particles. 
The evidence for the presence of multi-ladder contributions is emerging mostly from the interconnections between the various  DIS processes: 
inclusive $\gamma^* p$ reaction,  inclusive diffraction,  exclusive diffractive  vector-meson production and diffractive jet-jet production. These interconnections are naturally expressed in the    
 dipole saturation models, 
which have been shown to successfully describe
HERA $F_2$ data in the low-$x$ region.
These models are explicitly built on the idea of summing over multiple 
exchanges of single ladders. In the following we will discuss the exclusive and inclusive diffractive DIS processes and their connection with the total DIS cross section in terms of dipole models. 
\begin{wrapfigure}{r}{0.25\columnwidth}
%\begin{figure}[htp]
\vspace*{-1.cm}
%\hspace{2.5cm}
\epsfig{file=agk21.eps,height=3.5cm,width=4cm}
\vspace*{-1.8cm} 
\caption{\sl The double gluon ladder contribution to the 
inclusive diffractive $\gamma^*p$ cross section.  } 
\label{fig:Ael2d}
\vspace*{-0.1cm} 
\end{wrapfigure}

\section{Dipole Models}
In the dipole model,  deep inelastic scattering is viewed as interaction of a colour dipole, i.e. mostly a quark-antiquark pair, with the proton. The size of the pair is denoted by $r$ and a quark carries a fraction $z$ of the photon momentum. In the proton rest frame, the dipole life-time is much longer than the life-time of its interaction with the target proton. Therefore, the interaction is assumed to proceed in three stages: first the incoming virtual photon fluctuates into a quark-antiquark pair, then the $q\bar{q}$ pair elastically scatters on the proton, and finally the  $q\bar{q}$ pair recombines to form a virtual photon. 
%This is shown schematically in Fig.~\ref{fig:Dipole}.
The amplitude for the complete process is simply the product of these three processes. 

The amplitude of the incoming virtual photon to fluctuate into a quark-antiquark pair is given by the photon  wave function $\psi$, which is determined from light cone perturbation theory to leading order in the fermionic charge (for simplicity, the indices of the quark and antiquark helicities are suppressed). Similarly the amplitude for the $q \bar q$ to recombine to a virtual photon is $\psi^*$. 
The cross section for elastic scattering  of the $q\bar q$ pair with squared momentum transfer $\Delta^2 = -t$ is described by the elastic scattering amplitude, $A_{el}^{q \bar q}(x,r,\Delta)$, as
\begin{eqnarray}
 \frac{d\sigma_{q\bar q}}{dt} = \frac{1}{16\pi}|A_{el}^{q\bar q}(x,r,\Delta)|^2.
\label{eq:elsiggp}
\end{eqnarray}
To evaluate the connections between the total cross section and various diffractive reactions it is convenient to work in coordinate space and define the S-matrix element at a particular impact parameter $b$
\begin{eqnarray}
 S(b) = 1 + \frac{1}{2}\int d^2\Delta \exp(i\vec{b}\cdot\vec{\Delta}) A_{el}^{q\bar q}(x,r,\Delta).
\label{eq:smatrix}
\end{eqnarray}
This corresponds to the intuitive notion of impact parameter when the dipole size is small compared to the size of the proton. 
The Optical Theorem then connects  the total cross section of the $q\bar q$ pair to the imaginary part of $iA_{el}$ 
\begin{eqnarray}
 \sigma_{q\bar q}(x,r)=\Im i A_{el}^{q\bar q}(x,r,0)=\int d^2 b\, 2[1-\Re S(b)].
\label{eq:totsiggp}
\end{eqnarray}
The integration over the S-matrix element  motivates the definition of the elastic $q\bar q$ differential cross section as 
\begin{eqnarray}
 \frac{d\sigma_{q\bar q}}{d^2 b}= 2[1-\Re S(b)].
\label{eq:difsiggp}
\end{eqnarray}
The total cross section for $\gamma^* p$ scattering, or equivalently $F_2$, is obtained by averaging the dipole cross sections with the photon wave functions, $\psi(r,z)$: 
\begin{eqnarray}
\sigma^{\gamma^* p} = \int d^2r \int \frac{dz}{4\pi} \psi^*
\, \sigma_{q \bar q}(x,r)\, \psi.
\label{eq:siggp}
\end{eqnarray}
In the dipole picture the elastic vector-meson production appears in a similarly transparent way. The amplitude is given by
\begin{eqnarray}
A_{\gamma^* p\rightarrow pV}(\Delta) = \int d^2r \int \frac{dz}{4\pi} \int d^2b\, \psi^*_V\psi \exp(-i\vec{b}\cdot\vec{\Delta}) 2[1- S(b)].
\label{eq:amvecm}
\end{eqnarray}
We denote the wave function for a vector meson to fluctuate into a $q\bar q$ pair by $\psi_V$. Assuming that the S-matrix element is predominantly real, we may substitute $2[1-  S(b)]$ with $d \sigma_{q\bar q}/d^2b$. Then, the elastic diffractive cross section is 
\begin{eqnarray}
\frac{d\sigma^{\gamma^* p\rightarrow Vp}}{dt} = 
\frac{1}{16\pi}\left|\int d^2r \int \frac{dz}{4\pi} \int d^2b \, \psi^*_V\psi \exp(-i\vec{b}\cdot\vec{\Delta}) \frac{d\sigma_{q\bar q}}{d^2b}\right|^2.
\label{eq:xsvecm}
\end{eqnarray}
The equations (\ref{eq:siggp}) and (\ref{eq:xsvecm}) determine the inclusive and exclusive diffractive vector-meson production using the universal elastic differential cross section $d\sigma_{q\bar q}/d^2b$ which contains all the interaction dynamics.  

The inclusive diffractive cross section can be obtained from the eq. (\ref{eq:xsvecm}) summing over all (generalized) vector-meson states as
\begin{eqnarray}
 \left. \frac{d\sigma^{\gamma^* p}_{dif}}{dt}\right|_{t=0} = 
\frac{1}{16\pi}\int d^2r  \int \frac{dz}{4\pi} \psi^*\,  \sigma_{q\bar q}^2 \,\psi .
\label{eq:xsdif-incl}
\end{eqnarray}
Thus, properties of inclusive diffraction  are also  determined by the elastic cross section only and, contrary to vector-meson production,  are not dependent on the wave function of the outgoing diffractive state.    

\subsection{Dipole Cross Section and Saturation}

The dipole models became an important tool in investigations of deep inelastic scattering due to the initial observation of K. Golec-Biernat and M. W\" usthoff (GBW)~\cite{GBW} that a simple  ansatz for the dipole cross section integrated over the impact parameter $b$, $\sigma_{q\bar q}$, is able to describe simultaneously the total inclusive and diffractive DIS cross sections:
\begin{eqnarray} 
\sigma_{q\bar q}^{GBW} = \sigma_0 [1-\exp (-r^{2}/4R_{0}^{2})]
\label{eq:sigGBW}
\end{eqnarray}
where $\sigma_0 $ is a constant and $R_0$ denotes the $x$ dependent saturation radius $R_0^2 = (x/x_0)^{\lambda_{GBW}}\cdot(1/GeV^2)$. The parameters $\sigma_0=23$ mb, $\lambda_{GBW}$ and $x_0=3\cdot 10^{-4}$ were determined from a fit to the data. Although the dipole model is theoretically well justified for  small size dipoles only, the GBW model provides a good description of data from medium size $Q^2$ values ($\sim$30 GeV$^2$) down to low $Q^2$ ($\sim$0.1 GeV$^2$).
The inverse of the saturation radius $R_0$ is analogous to the gluon density. The exponent $\lambda_{GBW}$ determines therefore the growth of the total and diffractive cross sections with decreasing $x$. For dipole sizes which are large in comparison to $R_0$ the dipole cross section  saturates by approaching a constant value $\sigma_0$, which becomes independent of  $\lambda_{GBW}$.
It is a characteristic of the model that a good description of data is due to large saturation effects, i.e. the strong growth due to the factor  
$(1/x)^{\lambda_{GBW}}$ is, for large dipoles, significantly flattened by the  exponentiation in eq. (\ref{eq:sigGBW}). 

The assumption of dipole saturation provided an attractive theoretical background for investigation of the transition from the perturbative to non-perturbative regime in the HERA data. Despite the appealing simplicity and success of the GBW model  it suffers from clear shortcomings. In particular it does not include scaling violation, i.e. at large $Q^2$ it does not match with QCD evolution (DGLAP). Therefore,  Bartels,  Golec-Biernat and Kowalski (BGBK)~\cite{BGK} proposed  a modification of the original ansatz of eq.(~\ref{eq:sigGBW}) by replacing $1/R^2_0$ by a gluon density with explicit DGLAP evolution:
\begin{eqnarray} 
\sigma_{q\bar q}^{BGBK} = \sigma_0 [1-\exp (-\pi^2r^{2}\alpha_s(\mu^2)xg(x,\mu^2)/3\sigma_0)]
\label{eq:sigBGBK}
\end{eqnarray}
The scale of the gluon density, $\mu^2$, was assumed to be $\mu^2= C/r^2 +\mu_0^2$,  and the density was evolved according to DGLAP equations.  

The BGBK form of the dipole cross section led to significantly better fits to  the HERA $F_2$ data than the original GBW model, especially in the region of larger $Q^2$. The good agreement of the original model with the DIS diffractive HERA data was also preserved, as seen from  the comparison of the predictions of the model with data for the ratio of the diffractive to the total cross section,
 Fig.~\ref{fig:rdiff}.   

The BGBK analysis found, surprisingly, that there exist two distinct solutions  giving  very good description of HERA data, depending on the quark mass in the photon wave function. The first solution is obtained assuming $m_q =140$ MeV and leads to the initial gluon density distribution with the value of exponent $\lambda_g=0.28$, which is very similar to the $\lambda_{GBW}$. As in the original model, the good agreement with data is  due to substantial saturation effects.  In the second solution, $m_q \approx 0$, and the value of the  exponent is very different, $\lambda_g=-0.41$ . The initial gluon density no longer rises at small $x$, it is  valence-like, and QCD evolution plays a much more significant role than in the first solution. 

 \begin{wrapfigure}{r}{0.55\columnwidth}
%\begin{figure}[htp]
\vspace{0.cm}
\hspace{0.cm}
\epsfig{file=sigtotn.eps,height=4.3cm,width=4cm} 
\hspace*{-0.2cm}
\epsfig{file=dsn.eps,height=4.3cm,width=4cm}
\vspace*{-0.7cm} 
\caption{\sl LHS: The $\gamma^* p$ cross section as a function of $W^2$. 
RHS: The differential cross section for exclusive diffractive  $J/\Psi$ production as a function of the four-momentum transfer $t$. 
The solid line shows a fit by the IP saturation model (KT).}
\label{fig:sigtot}
\vspace*{-0.4cm} 
\end{wrapfigure}
The  DGLAP evolution, which is generally used in the analysis of HERA data, may not be appropiate when $x$ approaches the saturation region. Therefore, Iancu, Itakura and Munier (IIM)~\cite{IIM} proposed a new saturation model, the Colour Glass Condensate model, in which gluon saturation effects are incorporated via an approximate solution of the Balitsky-Kovchegov equation. Later, also Forshaw and Shaw (FS)~\cite{Forshaw} proposed a Regge type model with saturation effects. The IIM and FS models provide a description of  HERA  $F_2$ and diffractive data which is better than the original GBW model and comparable in quality to the BGBK analysis. Both models find strong saturation effects in HERA data comparable to the GBW model and the first solution of the BGBK model.  

All  approaches to  dipole saturation discussed so far ignored a possible impact parameter (IP) dependence of the dipole cross section. This dependence was introduced by  Kowalski and Teaney (KT)~\cite{KowTea},  who assumed that the dipole cross section is a function of the opacity $\Omega$:
\begin{eqnarray}
\frac{d\sigma_{qq}}{d^2b} = 2\, \left(1-\exp(-\frac{\Omega }{2})\right). 
\label{eq:xdip}
\end{eqnarray}
At small $x$ the opacity $\Omega$ can be directly related 
to the gluon density, $xg(x,\mu^2)$, and the transverse profile 
of the proton, $T(b)$:  
\begin{eqnarray}
\Omega = \frac{\pi^2}{N_C}\, r^2\, \alpha_s(\mu^2)\, xg(x,\mu^2)\, T(b).
\label{eq:omega}
\end{eqnarray}
The transverse profile is assumed to be of the form:
\begin{eqnarray}
T(b) = \frac{1}{2\pi B_G}\exp(-b^2/2B_G),
\label{eq:Tofb}
\end{eqnarray}
since  the Fourier transform of $T(b)$  has  the exponential form: 
\begin{eqnarray}
\frac{d\sigma_{VM}^{\gamma^{*}p}}{dt} = \exp(-B_G |t|)
\label{eq:DiVM}
\end{eqnarray}
The formula of eq.~(\ref{eq:xdip}) and (\ref{eq:omega}) is called the Glauber-Mueller dipole cross section. The diffractive cross section of this type was used around 50 years ago to study the diffractive dissociation of the deuterons by Glauber and reintroduced by A. Mueller~\cite{GaubMueller} to describe  dipole scattering in deep inelastic processes.  

The parameters of the gluon density  
are determined from the fit to the total inclusive DIS cross section,
as shown in Fig.~\ref{fig:sigtot}~\cite{KowTea}. The transverse profile was determined from the exclusive diffractive $J/\Psi$ cross sections shown in the same figure. In this approach the charm quark was explicitly taken into account with the mass $m_c = 1.25$ GeV. 

 For a small value of $\Omega$ the dipole cross section, eq. (\ref{eq:xdip}), is equal to $\Omega$ and therefore proportional to the gluon density. This allows one to identify the opacity with the single Pomeron exchange amplitude of Fig.~\ref{fig:Ael2}.


The KT model with parameters determined in this way  has predictive properties which go beyond the models discussed so far; it allows a description of the other measured reactions, e.g. the charm structure function~\cite{zeusf2ch} or elastic diffractive $J/\Psi$ production~\cite{zeusf0jpsi} shown in Fig.~\ref{fig:f2chjp}. 
\begin{wrapfigure}{r}{0.72\columnwidth}
%\begin{figure}[htp]
\vspace*{-0.8cm}
%\hspace*{0.1cm}
\epsfig{file=f2ch.eps,height=6cm,width=5.9cm} 
\hspace*{-0.4cm}
\epsfig{file=jpn.eps,height=6cm,width=4.9cm}
\vspace*{ -0.5cm} 
\caption{\sl LHS: Charm structure function,  $F_2^c$. 
RHS:  Total elastic $J/\Psi$  cross section. The solid line shows the result of the IP saturation model (KT).}
\label{fig:f2chjp}
\vspace*{ -0.5cm} 
\end{wrapfigure}

The initial gluon distribution determined in the model is valence-like, with $\lambda_g=-0.12$ and the fit pushes the quark mass to small values, $m_q \approx 50$ MeV. The resulting gluon distribution is therefore similar to the second solution of the BGBK model. The first solution of the BGBK model was disfavoured by the data. This behaviour is  presumably due to the assumption of the Gaussian-like proton shape, eq. (\ref{eq:Tofb}). In 
the tail of the Gaussian, the gluon density is low, but the relative contribution of the tail to the cross section  is large. The saturation effects cannot therefore be as large as in the GBW-like models (i.e. BGBK-1, IIM, FS).  In addition, as  noted in the KT paper and also in the Thorne analysis~\cite{Thorne}, the introduction of charm in the analysis of HERA data lowers the gluon density and therefore diminishes  the saturation effects.
Nevertheless, the KT analysis shows that in the center of the proton ($b\approx 0$) the saturation effects are similar to the ones in the GBW-like models in which charm is properly taken into account. This  can be seen from the evaluation of the saturation scale in the center of the proton in the KT paper and the comparison to the value of the saturation scale evaluated  with charm in the original GBW paper. 

\section{Exclusive Diffractive Vector-Meson Production}
The exclusive diffractive vector-meson production is very interesting because, in the low-$x$ region, it is driven by the square of the gluon density. It was, therefore, investigated by many authors~\cite{NN,MMS,CS,KowTea,DGKP,FS,Levy}.
In addition, the information contained in the $Q^2$, $W$ and $t$ dependence of the cross sections allows to determine vector-meson wave functions together with the proton shape. The analysis can also be  performed separately for the longitudinal and transverse photons.

The recent analysis of vector-meson production by Kowalski, Motyka and Watt (KMW)~\cite{KMW} shows that it is possible to describe the measured differential cross sections making simple assumptions about the vector-meson wave functions~\cite{NN,FS}. The analysis shows that using the gluon density determined from the total cross sections and the size of the interaction region determined from the $t$ 
distribution of the $J/\Psi$ meson at $Q^2=0$,  it is possible to simultaneously
  describe not only the shape of various differential cross sections as a function of $Q^2$, $W$ and $t$  but also their absolute magnitude.       In this analysis the assumption that vector-meson size should be much smaller than  proton size was relaxed. Following the work of Bartels, Golec-Biernat and Peters~\cite{BGBP} the Fourier transform of eq. (\ref{eq:xsvecm}) was modified to take into account the finite size of the vector meson: 
\begin{eqnarray}
\exp(-i\vec{b}\cdot\vec{\Delta})
 \rightarrow \exp(-i(\vec{b}+(1-z)\vec{r})\cdot\vec{\Delta}).
\label{eq:modfour}
\end{eqnarray}
In this way, the information about the size of the vector meson, contained in the wave function, is contributing to the size of the interaction region $B_D$, together with the size of the proton.   
 \begin{wrapfigure}{r}{0.8\columnwidth}
%\begin{figure}[htp]
\vspace*{-0.5cm}
\hspace*{0.cm}
\epsfig{file=crossq_jpsi.eps,height=4.5cm,width=3.0cm} 
\epsfig{file=crossq_phi.eps,height=4.5cm,width=3.0cm}
\epsfig{file=crossq_rho.eps,height=4.5cm,width=3.0cm}
\hspace*{0.65cm}
\epsfig{file=bd_jpsi.eps,height=4.5cm,width=3.3cm} 
\epsfig{file=bd_phi.eps,height=4.5cm,width=3.3cm}
\epsfig{file=bd_rho.eps,height=4.5cm,width=3.3cm}
\vspace*{ -0.4cm} 
\caption{\sl (Top) The exclusive diffractive  cross sections  for $J/\Psi$, $\phi$  and $\rho$ vector-meson production as a function of $Q^2+M_V^2$.
(Bottom) The interaction size $B_D$  defined by $d\sigma/dt \propto \exp(B_D t)$, extracted from $t$ distributions of $J/\Psi$, $\phi$  and $\rho$ vector meson as a function of $Q^2+M_V^2$. The solid line shows predictions of the KMW model. (Preliminary results)}
\label{fig:vmq2}
\vspace*{ -0.4cm} 
\end{wrapfigure}
As an example of results obtained in this analysis 
Fig.~\ref{fig:vmq2} shows the comparison of KMW model predictions for  the total exclusive diffractive vector-meson cross section and the size of the interaction region with  data. 
Here, the profile function is assumed to have a Gaussian form  (\ref{eq:Tofb}), with the parameter $B_G=4$ GeV$^{-2}$. The `boosted Gaussian' vector-meson wave functions~\cite{FS} are used. The light quark masses are $m_q = 140$ MeV, with $m_c=1.4$ GeV. 

%\section{Diffractive Dijets}
%The diffractive dijet production is analysed within QCD in two seemingly different ways: the diffractive parton density  approach and the dipole picture. The diffractive structure function is proportional to the diffractive parton density which, in the simplest case, factorizes into the Pomeron flux factor and the Pomeron parton density. The Pomeron flux factor gives the probability to find a Pomeron within a proton with the momentum fraction $\xPom$. The Pomeron parton density gives the probability to find a parton within a Pomeron  carrying a  Pomeron momentum fraction $z_\Pom$\cite{GWatt}. This parton interacts then with the incoming photon in a boson-gluon,  Fig.~\ref{fig:dfsgra}, or QCD-Compton process. 

%

%The diffractive parton distribution is extracted from the inclusive diffractive measurements  at an initial scale, $Q^2_0$, and
% then evolved according to the DGLAP evolution to the  scale  $Q^2$ of jet production. The events are  generated in the RAPGAP Monte Carlo in which the  $Q^2$, $x$, $\xPom$ and $z_\Pom$ variables are obtained from the diffractive structure function  and   the jet 
% variables, $E_t$, $\eta_{jets}$ and $\beta$ are obtained from the usual QCD matrix 
% \begin{wrapfigure}{r}{0.3\columnwidth}
%\vspace*{-0.6cm}
%%\hspace{3.5cm}
%\epsfig{file=295_1.ps,height=2.3cm,width=3.cm} 
%\vspace*{-0.3cm} 
%\caption{\sl   Diagram describing diffractive production of a $q\bar q g$ state in the diffractive structure function approach.}
%\label{fig:dfsgra}
%\vspace*{-0.5cm} 
%\end{wrapfigure}
% elements of the boson-gluon fusion and QCD-Compton processes. The results are compared to data in Fig.~\ref{fig:diffjets}~\cite{zdifjets}.  
% 
%In the dipole picture,   diffractive dijet production  can be directly determined from the unintegrated gluon density in the low-$x$ region, $f(x,l^2)$, where $\vec{l}$ is the transverse momentum of the gluon coupled to the quark pair. The unintegrated gluon 
%density is then obtained from the dipole cross section by~\cite{BGK}:
%\begin{eqnarray}
%\frac{\alpha_s f(x,l^2)}{l^4} = \frac{3}{8\pi^2 }\int_{0}^{\infty} dr\, r J_0(lr)[\sigma_{q\bar q}(x,\infty)- \sigma_{q\bar q}(x,r)].
%\label{eq:unintg}
%\end{eqnarray}
%The jet cross sections are then computed in  momentum space from diagrams shown in Fig.~\ref{fig:fgra}~\cite{GBW}. 
%%\begin{figure}[htp]
% In addition to the contributions of the $q\bar q$ states it is important  to include the contributions of the $q\bar q g$ final states~\cite{BEKW}.      

%The results from the evaluation of the diagrams of Fig.~\ref{fig:fgra}, obtained within the SATRAP Monte Carlo, are compared to data\cite{h1difjets,zdifjets} in Fig.~\ref{fig:diffjets}.
%The figure shows that the distributions of typical jet variables are well described in both the diffractive parton densities (RAPGAP) and dipole picture (SATRAP) approaches. It is remarkable that in addition to the shapes 
%also the absolute magnitudes of the diffractive jet cross sections are well described.   

%\begin{wrapfigure}{r}{0.35\columnwidth}
%%\begin{figure}[htp]
%\vspace*{-1.2cm}
%%\hspace{1.5cm}
%\epsfig{file=fgraqq.eps,height=3.0cm,width=2.5cm} 
%\epsfig{file=fgraqqg.eps,height=3.0cm,width=2.5cm} 
%\vspace*{ -0.5cm} 
%\caption{\sl   Diagrams describing the diffractive production of a $q\bar q$ and $q\bar q g$ system in the dipole picture approach.}
%\label{fig:fgra}
%\vspace*{ -0.2cm} 
%\end{wrapfigure}
%Finally, let us note for completeness that  
%the diffractive parton density approach uses as input  the inclusive diffractive data at low $Q^2$. It was observed that  small systematic differences of input data tend to be amplified by the fitting procedure~\cite{zdifjets}.  The dipole approach uses as input the dipole cross section determined from 
%the $F_2$ measurements, which have smaller systematic errors than the $F_2^D$ measurements.
%\begin{wrapfigure}{r}{0.5\columnwidth}
%%\begin{figure}[htp]
%\vspace*{-0.3cm}
%%\hspace{2.cm}
%\epsfig{file=295_3.eps,height=7.0cm,width=7.5cm} 
%\vspace*{-0.3cm} 
%\caption{\sl The differential diffractive jet cross sections shown as a function of jet variables. The solid lines show the predictions from the LO RAPGAP MC normalised by a factor 0.92. SATRAP predictions normalised by a factor 1.12 are shown as dashed lines.}
%\label{fig:diffjets}
%\end{wrapfigure}

\section{Conclusions}

One of the most important results of HERA measurements is the observation of the large amount of diffractive processes. Inclusive diffraction,  diffractive jet process and  exclusive diffractive vector-meson production are connected to  inclusive deep inelastic scattering and, in the dipole picture, can be successfully derived from the measured $F_2$. 
In the dipole  approach, the  Pomeron is essentially of the perturbative type, since the dipole models are explicitly built on the idea of summing over multiple exchanges of single ladders.  


Inclusive diffraction and diffractive dijet production are also well described
in the diffractive parton density approach, in which  the Pomeron could be of non-perturbative origin.  However,  the effective Pomeron intercept extracted from diffractive DIS data lies significantly above the soft Pomeron intercept\cite{zdiffFPC,h1dif}, indicating a substantial contribution to diffractive DIS from perturbative Pomeron exchange. In addition,  
the initial scale  chosen for the  analysis is relatively high, $Q^2_0=3$ GeV$^2$. At this scale $F_2$ exhibits a clear growth with diminishing $x$  indicating that the exchanged Pomeron should be of perturbative type.

The good agreement between the diffractive parton density and dipole model analysis in the description of diffractive dijets indicates that both approaches, although seemingly different,  are not  really distinct. An attempt to combine these two approaches is recently discussed in Ref.\cite{GWatt}.  

%\section*{Acknowledgments}
%I would like to thank Graeme Watt for discussions and help in preparing this manuscript.

% changed references:  comment 4 refs., remove ``.''  before Forshaw and reorder refs.  
\begin{footnotesize}
\providecommand{\etal}{et al.\xspace}
\providecommand{\href}[2]{#2}
\providecommand{\coll}{Coll.}
\catcode`\@=11
\def\@bibitem#1{%
\ifmc@bstsupport
  \mc@iftail{#1}%
    {;\newline\ignorespaces}%
    {\ifmc@first\else.\fi\orig@bibitem{#1}}
  \mc@firstfalse
\else
  \mc@iftail{#1}%
    {\ignorespaces}%
    {\orig@bibitem{#1}}%
\fi}%
\catcode`\@=12
\begin{mcbibliography}{1}
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\bibitem{kow.zeus.2}ZEUS Collab., J. Breitweg et al., {\EPJC} {6} {43} {1999}
\bibitem{zdiffFPC}ZEUS Collab., S. Chekanov at al., {\NPB} {713} {3} {2005}
\bibitem{H1epf2d394}H1 Collab., C. Adloff et al., {\ZPC} {76} {613} {1997}
%\bibitem{Zeuslps}ZEUS Collab., S. Chekanov at al., {\EPJC} {38} {43} {2004}.
\bibitem{h1dif}H1 Collab., paper 089, EPS03, Aachen.
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\bibitem{BGK}J. Bartels, K. Golec-Biernat and H. Kowalski {\PRD} {66} {014001} {2002}
\bibitem{IIM}E. Iancu, K. Itakura and S. Munier, {\PLB} {590} {199} {2004}
\bibitem{Forshaw}J. R. Forshaw and G. Shaw, {JHEP}{0412} {052} {2004}
\bibitem{KowTea}H. Kowalski and D. Teaney, {\PRD}{68} {114005} {2003}
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\bibitem{KMW} H. Kowalski, L. Motyka, G. Watt, in preparation
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\end{mcbibliography}

\end{footnotesize}
\end{document}




