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\markboth{\small \em \quad A.B.~Kaidalov\hfill}{\hfill\small \em 
Difractive processes and QCD\quad}

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\begin{document}
\sloppy

\title{DIFFRACTIVE PROCESSES AND QCD}

\author{A. B. KAIDALOV}

\address{Institute of Theoretical and Experimental Physics, B.Cheremushkinskaya
25, \\ Moscow 117259, RUSSIA\\E-mail: kaidalov@vitep5.itep.ru}

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\maketitle \abstracts{  Diffractive processes are considered from
both s--channel and t--channel points of view. Soft and hard
diffractive processes are discussed. An importance of unitarity
(shadowing) effects for a relation between hard diffraction at
HERA and Tevatron is emphasized.
 A model, which takes into account unitarity effects is developed
 for to predict interaction of high--energy virtual photons with nucleons. It is shown that
 this model gives a good description of HERA data on both total $\gamma^*p$
 total cross section and diffractive dissociation of virtual photons in a
 broad region of $Q^2$. Diffractive production of jets in hadronic interactions
 is investigated and an important role of unitarization effects is emphasized.
 It is shown how to describe the CDF data on diffractive jet production at
 Tevatron using an information on distribution of partons in the pomeron from
 HERA experiments. Double--pomeron production of jets at Tevatron shows a clear
 violation of factorization and is in a good agreement with our prediction.}

\vspace*{5mm}

\tableofcontents

\newpage
\section{Introduction}

 This review paper on diffractive processes is devoted to a memory of Misha
 Marinov. I has met Misha 40 years ago when I started to visit ITEP as a student
 of I.Ya.Pomeranchuk. I was always impressed by his very serious and deep
 style of thinking. He tried to understand all subtleties in the field of
 high--energy physics. His deep knowledge of many complicated problems in our
 field was well known and highly appreciated by physicists of ITEP. Misha had a great
 pedagogical talent which unfortunately was not used to full strength. Only once we
 asked him to give lectures on path integral methods. These lectures became a
 basis for his famous review on this subject. In 60-ies many people in ITEP including Misha
 were interested in high--energy hadronic interactions. The theoretical
 approach based on Regge theory was very popular at that time.
 Misha Marinov has investigated scattering of particles with spins at high energies.
I learned a lot on this subject from his papers and discussions with him.
At present there is a revival of
interest to Regge approach mostly due to successful applications of this method
to deep--inelastic scattering in the small--x region, studied experimentally at HERA
and theoretical investigation of the role of Regge singularities in QCD.

Investigation of diffractive scattering of hadrons gives an
important information on structure of hadrons and mechanisms of their interactions. There are two complementary points of view to these processes.
\subsection*{The s--channel view of diffraction}
\addcontentsline{toc}{subsection}{\numberline{}The s--channel view of diffraction}
 It is well known that an absorption of an initial wave due to many inelastic
 channels leads by unitarity to diffractive elastic scattering. At high
 energies lifetimes of hadronic fluctuations are large $\tau\sim E/m^2$ and
 hadronic constituents can undergo elastic scattering . This process leads
 to diffraction dissociation of hadrons. An elegant
 interpretation of diffraction dissociation in terms of eigenstates of
 diffractive part of S-matrix has been given by Good and Walker.\cite{Good}
 Consider the states $\phi_k$ which diagonalize the diffractive part
of the $T$ matrix.  Such eigenstates of diffraction only undergo
elastic
 scattering.  Let us denote the orthogonal matrix which diagonalizes Im~$T$
 by $C$, so that
\be \label{eq:a3}
{\rm Im}~T \; = \; CFC^T \quad\quad {\rm with}
\quad\quad \langle \phi_k |F| \phi_j \rangle \; = \; F_j \:
\delta_{jk}.
\ee
Now consider the diffractive dissociation of an
arbitrary incoming state
\be
\label{eq:a4} | i \rangle \; = \;
\sum_k \: C_{ik} \: | \phi_k \rangle.
\ee
The elastic scattering amplitude for this state satisfies
\be \label{eq:a5}
\langle i
|{\rm Im}~T| i \rangle \; = \; \sum_k \: |C_{ik}|^2 \: F_k \; = \;
\langle F \rangle,
\ee
where $F_k \equiv \langle \phi_k |F| \phi_k
\rangle$ and where the brackets of $\langle F \rangle$ mean the
average of $F$ over the initial probability distribution of
diffractive eigenstates.  After the diffractive scattering
described by $T_{fi}$, the final state $| f \rangle$ will, in
general, be a different superposition of eigenstates than those of
$| i \rangle$ shown in (\ref{eq:a4}). At high energies one can
neglect by the real parts of the diffractive amplitudes, then for
cross sections at a given impact parameter $b$ we have
\bea
\label{eq:a6}
\frac{d \sigma_{\rm tot}}{d^2 b} & = & 2 \: {\rm Im}
\langle i |T| i \rangle \; = \; 2 \: \sum_k
\: |C_{ik}|^2 \: F_k \; = \; 2 \langle F \rangle \nonumber \\
& & \nonumber \\
\frac{d \sigma_{\rm el}}{d^2 b} & = & \left | \langle i |T| i
\rangle \right |^2 \; = \; \left (
\sum_k \: |C_{ik}|^2 \: F_k \right )^2 \; = \; \langle F \rangle^2 \\
& & \nonumber \\
\frac{d \sigma_{\rm el \: + \: SD}}{d^2 b} & = & \sum_k \: \left |
\langle \phi_k |T| i \rangle \right |^2 \; = \; \sum_k \:
|C_{ik}|^2 \: F_k^2 \; = \; \langle F^2 \rangle. \nonumber
\eea
It follows that the cross section for the single diffractive
dissociation of a proton,
\be \label{eq:a7}
\frac{d \sigma_{\rm
SD}}{d^2 b} \; = \; \langle F^2 \rangle \: - \: \langle F
\rangle^2,
\ee
is given by the statistical dispersion in the
absorption probabilities of the diffractive eigenstates.

Note that if all the components $\phi_k$ of the incoming
diffractive state $| i \rangle$ were absorbed equally then the
diffracted superposition would be proportional to the incident one
and  the inelastic diffraction would be zero.  Thus if, at
very high energies, the amplitudes $F_k$ at small impact
parameters are equal to the black disk limit, $F_k = 1$, then
diffractive production will be equal to zero in this impact
parameter domain and so will only occur in the peripheral $b$
region.  This behaviour takes place in $pp$ (and
$p\bar{p}$) interactions at Tevatron energies.

Let us consider as an example just two diffractive channels\,\cite{BTM,GLM,KMR} (say, $p, N^*$), and assume, for simplicity,
that the elastic scattering amplitudes for these two channels are
equal.  Then the $T$ matrix has the form
\be \label{eq:a8}
{\rm Im}~T \; = \; 1 \: - \: e^{- \Omega/2},
\ee
where the eikonal matrix $\Omega$ has elements
\be \label{eq:a9}
\Omega_{f^\prime
i^\prime}^{fi} \; = \; \Omega_0 \: \omega^{fi} \: \omega_{f^\prime
i^\prime}.
\ee
The individual $\omega$ matrices, which correspond
to transitions from the two incoming hadrons, each have the form
\be \label{eq:b9}
\omega \; = \; \left ( \begin{array}{cc} 1 &
\gamma \\[1mm] \gamma & 1 \end{array} \right ).
\ee
The parameter
$\gamma (s, b)$ determines the ratio of the inelastic to elastic
transitions.  The overall coupling $\Omega_0$ is also a function
of the energy $\sqrt{s}$ and the impact parameter $b$.

With the above form of $\omega$, the diffractive eigenstates are
\be \label{eq:c9}
| \phi_1 \rangle \; = \; \frac{1}{\sqrt{2}}
\left (| p \rangle + | N^* \rangle \right ), \quad\quad | \phi_2
\rangle \; = \; \frac{1}{\sqrt{2}} \left (| p \rangle - | N^*
\rangle \right ).
\ee
In this basis, the eikonal has the diagonal form
\be \label{eq:d9}
\Omega_{m^\prime n^\prime}^{mn} \; = \;
\Omega_0 \: r^{mn} \: r_{m^\prime n^\prime},
\ee
where $m, n =
\phi_1, \phi_2$ and \vspace*{-2mm}
\be \label{eq:e9} 
r \; = \; \left (
\begin{array}
{cc} 1 + \gamma & 0 \\[1mm] 0 & 1 - \gamma
\end{array}
\right ).
\ee
In the case where $\gamma$ is close to unity,
$\gamma = 1 - \varepsilon$, one of the eigenvalues is small.

 \subsection*{The t--channel point of view on diffraction}
\addcontentsline{toc}{subsection}{\numberline{}The t--channel point of view on diffraction}
 The t--channel approach is based on the Regge model for diffractive processes.
  Regge poles have been introduced in particle physics in the beginning of
 60-ies\,\cite{Gribov,Chew} and up to present time are widely used for
 description of high-energy interactions of hadrons and nuclei. Regge
 approach establishes an important connection between high energy scattering
 and spectrum of particles and resonances.
 In this approach diffractive processes are mediated by an exchange by the
 pomeron ($P$) -- the leading Regge pole with vacuum quantum numbers. The
 pomeron plays a role of an exchanged "particle" and gives factorizable
 contributions to scattering amplitudes. In impact parameter space Regge
 amplitude has asymptotically gaussian form. This contradicts to a peripheral
 form for inelastic diffraction, which follows from unitarity in the s-channel
 picture. However in the Regge theory it is necessary to take into account
 not only Regge poles but also Regge cuts, which corresspond to exchange by
 several Regge poles in the t-channel. These contributions restore unitarity
 of the theory and for inelastic diffraction they lead to a peripheral form
 of impact parameter distributions and allow to understand an energy dependence
 of the corresponding cross sections.

For all diffractive processes (Fig.~1) there is a large rapidity
gap between groups of
 produced particles. For example for single diffraction dissociation there is
a gap between the particle $1^{\prime}$ and the rest system of hadrons.
 This rapidity gap $\Delta y\approx ln (1/1-x),$ where $x$ the $x_F$ for
 hadron $1^{\prime}$ . A mass of diffractively excited state at
 large $s$ can be large. The only condition for diffraction dissociation is
$s_i\ll s$.
\begin{figure}
\vskip0.05in \centering
\includegraphics[width=1.5in]{fig7.eps}
%\vskip 0.1in
\caption{ Diagrams for diffractive production of hadrons in the
Regge pole model. \label{fig:diffr}}
\end{figure}

The cross section for inclusive single diffraction dissociation in
the Regge pole model can be written in the following form
\be
\frac{d^2\sigma }{d\xi _2dt}=\frac{(g_{11}(t))^2}{16\pi}
|G_p(\xi^{\prime},t)|^2\sigma_{P2}^{(tot)}(\xi_2,t)
\label{diffcross}
\ee
where $\xi_2\equiv ln(s_2/s_0), \xi^{\prime}= ln(s/s_2)$ and
 $G_p(\xi^{\prime},t)=\eta (\alpha _p(t))exp[(\alpha _p(t)-1)\xi^{\prime}]$
 is the pomeron Green function. The quantity $\sigma _{P2}^{(tot)}(\xi _2,t)$
 can be considered as the pomeron--particle total interaction cross section.\cite{Kai}
 Note that this quantity is not directly observable one and it is defined 
by its
 relation to the diffraction production cross section, Eq.\,(\ref{diffcross}).
 This definition is useful however because at large $s_2$ this cross section
 has the same Regge behavior as usual cross sections
\be 
\sigma _{P2}^{(tot)}(s_2, t)=\sum_k g^k_{22}(0)r^{\alpha
_k}_{PP}(t)\left(\frac{s_2}{s_0}\right)^{\alpha _k(0)-1} 
\ee 
where the
$r^{\alpha _k}_{PP}(t)$ is the triple--reggeon vertex ,
 which describes coupling of two pomerons to reggeon $\alpha _k$.

In this kinematical region $s\gg s_2\gg m^2$ the inclusive
diffractive cross
 section is described by the triple--Regge diagrams (Fig.~2)and has the form
\be
f^1=\sum_k G_k(t)(1-x)^{\alpha _k(0)-2\alpha_P(t)}
(\frac{s}{s_0})^{\alpha _k(0)-1}
\ee
\begin{figure}
\vskip0.05in
\centering
\includegraphics[width=1.5in]{fig8.eps}
%\vskip 0.1in
\caption{ Triple--Regge diagram. \label{fig:triple}}
\end{figure}
The pomeron--proton total
cross section and triple--Regge vertices $r^P_{PP},
 r^f_{PP}$ have been determined from analysis of experimental data on
 diffractive production of particles in hadronic collisions
 (see review\,\cite{Kaidiff}).


\section {Small-x physics}

An interesting class of diffractive processes is studied in deep
inelastic scattering (DIS) at small values of the Bjorken variable
x .
 They became especially actual due to recent experimental
investigation of the small-x region at HERA accelerator.

 Experiments at HERA have found two extremely important properties of small-$x$
 physics~: a fast increase of parton densities as $x$ decreases\,\cite{H1F2,ZEUSF2}
 and the existence of substantial diffractive production in deep inelastic scattering
 (DIS).\cite{ZEUSd,H1d}

 In the paper\,\cite{9r} it was suggested that the increase of the effective
 intercept of the pomeron, $\alpha_{eff} = 1 + \Delta_{eff}$, as $Q^2$
 increases from zero to several GeV$^2$ is mostly due to a decrease of
 shadowing effects with increasing $Q^2$. A parametrization of the $Q^2$
 dependence of $\Delta_{eff}$ such that
 $\Delta_{eff} \approx 0.1$ for $Q^2 \approx 0$ (as in soft hadronic
 interactions) and $\Delta_{eff} \approx 0.2$ (bare pomeron intercept) for
 $Q^2$ of the order of a few GeV$^2$, gives a good description\,\cite{9r,10r} of all
 existing data on $\gamma^*p$ total cross-sections in 
the region of
 $Q^2 \leq  5 \div 10$~GeV$^2$. At larger $Q^2$ effects
 due to QCD evolution become important. Using the above parametrization as
 the initial condition in the QCD evolution equation, it is possible to
 describe the data in the whole region of $Q^2$ studied at HERA.\cite{9r,11r}

 In the reggeon approach there are good reasons to believe that the fast
 increase of the $\sigma^{(tot)}_{\gamma^*p}$ with energy in the HERA energy range
 will change to a softer increase at much higher energies. This
 is due to multi--pomeron effects, which are related to shadowing in highly dense
 systems of partons - with eventual ``saturation'' of densities.
This problem was investigated recently in our paper,\cite{CFKS}
where reggeon approach was applied to the processes of diffractive
$\gamma^* p$ interaction.

 In the reggeon calculus\,\cite{regtech}
  the amount of rescatterings is closely related to diffractive
 production. AGK-cutting rules\,\cite{AGK} allow to calculate the cross-section
 of inelastic diffraction if contributions of
 multi--pomeron exchanges to the elastic scattering amplitude are known.
 Thus, it is very important for self-consistency of theoretical models to
 describe not only total cross sections, but, simultaneously, inelastic
 diffraction. In particular in the reggeon calculus the variation of $\Delta_{eff}$
 with $Q^2$ is related to the corresponding variation of the ratio
 of diffractive to total cross sections.

In the paper\,\cite{CFKS} an explicit model for the contribution of
 rescatterings was constructed which leads to the pattern of energy behavior
 of $\sigma_{\gamma^*p}^{(tot)} (W^2,Q^2)$ for different
 $Q^2$ described above. Moreover, it allows to describe simultaneously
 diffraction production by real and virtual photons. In this model it is
possible to study quantitatively a regime of "saturation" of
parton densities.

Let us discuss briefly the qualitative picture of diffractive
dissociation of a highly virtual photon at high energies. It is
convenient to discuss this process in the lab. frame, where the
quark--gluon fluctuations of a photon live a long time $\sim 1/x$
(Ioffe time\,\cite{Ioffe}). A virtual photon fluctuates first to
$q\bar q$ pair. There are two different types of configurations of
such pair, depending on transverse distance between
quarks (or $k_{\perp})$.\\
a) Small size configurations with $k^2_{\perp}\sim Q^2$. These
small dipoles ($ r\sim 1/k_{\perp}\sim 1/Q$) have a small ($\sim
r^2$) total interaction
cross section with the proton.\\
b) Large size configurations with $r\sim 1/ \Lambda_{QCD}$ and
 $ k_{\perp}\sim \Lambda_{QCD}\ll Q$. They have a large total interaction
cross section, but contribute with a small phase space at large
$Q^2$, because
 these configurations are kinematically possible only if the fraction of
 longitudinal momentum carried by one of the quarks is very small
 $x_1\sim k_{\perp}^2/Q^2\ll 1$. This configuration corresponds to the "aligned jet",
introduced by Bjorken and Kogut.\cite{BJ}

Both configurations lead to the same behaviour of
 $\sigma_{\gamma^*p}^{(tot)}\sim 1/Q^2$,
but they behave differently in the process of the diffraction
dissociation of a virtual photon.\cite{Frank,Nik} The cross
section of such a process is proportional to a square of modulus
of the corresponding diffractive amplitude and for a small size
configuration it is small ($\sim 1/Q^4$). For large size
configurations a smallness is only due to the phase space and the
inclusive cross section for diffractive dissociation of a virtual
photon decreases as $1/Q^2$, i.e. in the same way as the total
cross section. This is true only for the total inclusive
diffractive cross section, where characteristic masses of produced
states are $M^2\sim Q^2$. For exclusive channels with fixed mass
(for example production of vector mesons) situation is different
and these cross sections decrease faster than $1/Q^2$ at large
$Q^2$.

Inclusive diffractive production of very large masses ($M^2\gg
Q^2$) can be described in the first approximation by triple--Regge
diagrams.\cite{15r} From the point of view of the quark-gluon
fluctuation of the fast photon triple--pomeron contribution
corresponds to diffractive scattering of very slow (presumably
gluonic) parton, which has a small virtuality.

The model\,\cite{CFKS} uses the picture of diffraction dissociation
of a virtual photon outlined above and is a natural
 generalization of models used for the description of high-energy hadronic
 interactions. The interaction of the small size component in the wave function
of a virtual photon is calculated using QCD perturbation theory.

The main parameter of the model -- intercept of the pomeron was fixed from
 a phenomenological study of hadronic interactions\,\cite{Ponomar}
 ($\Delta_P = 0.2$) and was found to give a good description of
 $\gamma^*p$-interactions in a broad range of $Q^2$ ($0\le Q^2< 10~GeV^2$).
 Another important parameter of the theory, the triple--pomeron vertex, obtained from
  a fit to the data ($r_{PPP}^{(0)}/g_{pp}^P(0) \approx 0.1$) is also in a reasonable
 agreement with the analysis of soft hadronic interactions.\cite{Ponomar,15r}
 The model describes experimental data on the structure
 function $F_2$ and diffractive structure function $F_{2D}^{(3)}$
 quite well (Figs.~3,~4). It can be used to predict
structure functions and partonic distributions at higher energies
or smaller $x$.
\begin{figure}
%\label{fig:sigmat}
%\vskip0.05in
\centering
\includegraphics[width=5.0in]{fig16.ps}
%\vskip 0.1in 
%\centerline{\epsfig{file=fig16.eps,width=5in}}
\caption{Structure function $F_2$ as a function of
$x$ for different values of
 $Q^2$ compared with experimental data. Dashed lines denote small distance
 contributions and dotted lines -- large distance ones.\label{fig:sigmat}}
\end{figure}
\begin{figure}
%\vskip0.05in 
\centering
\includegraphics[width=5in]{fig18.eps}
%\vskip 0.1in
%\centerline{\epsfig{file=fig18.eps,width=5in}}
\caption{ The diffractive structure function $x_P F_2^{D(3)}$ as a
function of $x_P=x/\beta$ for fixed values of $Q^2$ and
$\beta=Q^2/(Q^2+M^2)$. \label{fig:diff2}}
\end{figure}

   Investigation of the $Q^2$-dependence of diffraction dissociation of a
 highly virtual photon gives a possibility to determine a distribution of
 gluons in the pomeron.\cite{H1d,15r,Stirg} It turns out that contrary to
 the proton case both distributions of quarks and gluons inside the pomeron
 are rather hard. This indicates to an important contrubution of "valence"
 gluons in the pomeron. It should be noted however that the form of the
 distribution of gluons in the pomeron is not well known at present. This
 is partly due to systematic differences in $Q^2$ dependence of diffractive
 structure functions of H1 and ZEUS (for recent discussion of these points
 see\,\cite{Bartels}).

\subsection*{Shadowing effects for nuclear structure functions}
\addcontentsline{toc}{subsection}{\numberline{}Shadowing effects for nuclear structure functions}

 A study of the shadowing effects for structure functions of nuclei in the small x
 region provides a stringent test of the reggeon approach to small--x problem.
The shadowing effects are enhanced for nuclei ($\sim A^{1/3}$)
and lead to deviations from $A^1$ behaviour for structure
functions of nuclei. Glauber--Gribov\,\cite{Glauber,Gribovn}
approach to interactions of particles with nuclei gives a
possibility to calculate rescattering corrections for interaction
of a virtual photon with a nucleus in terms of diffractive
interaction of a photon with a nucleon, which was discussed above.

A contribution of a double rescattering term to the
$\sigma_{\gamma^{\ast}A}$ is directly expressed in terms of the
differential cross section for a diffraction dissociation of a
virtual photon in $\gamma^{\ast}N$-interactions
\bel{double} 
\sigma^{(2)}=\! -4\pi\!\int\! d^2b\, T_A^2(b)\!\int
\!
dM^2\,\frac{d\sigma^{DD}_{\gamma^{\ast}N}(t=0)}{dM^2dt}\,F_A(t_{min})
\ee 
where
\mbox{$F_A(t_{min})=
\exp(R^2_At_{min}/3),~t_{min}\approx-m_N^2x_P^2,$} and $T_A(b)$
is the nuclear profile function $(\int d^2b T_A(b)=A)$.

Higher order rescatterings are model dependent and in the
generalized Schwimmer model\,\cite{paris13} we obtain in the
region of small $x$
\bel{F2A} F_{2A}/F_{2N}=\int
d^2b\,\frac{T_A(b)}{1+F(x,Q^2)T_A(b)}
\ee
with
$$F(x,Q^2)=4\pi \int dM^2\,
\frac{d\sigma^{DD}_{\gamma^{\ast}N}(t=0)}{dM^2 dt}\,\frac{F_A(t_{min})}{\sigma_{\gamma^{\ast}N}(x,Q^2)}\,.$$

Theoretical predictions,\cite{nuclshad} based on
Eq.\,(\ref{F2A}) and the model for diffraction dissociation of
Ref.\,\protect\citebk{15r} are in a very good agreement with NMC-data on
nuclear structure functions at very small $x$.\cite{paris14} I
believe that this appraoch gives reliable predictions for nuclear
shadowing effects in the region of smaller $x$ not yet studied
experimentally. This region will play an important role in
dynamics of heavy ions collisions at super--high (LHC) energies.

\section{Hard diffraction in hadronic interactions}

  Diffractive production of large mass states, which include hard subprocesses,
has been studied recently in high--energy hadronic interactions.
Most of the results have been obtained at Tevatron where
diffractive production of jets, W-bosons, heavy quarks and heavy
quarkonia was observed.\cite{CDFj,CDFW,Goulex}

The most interesting results were obtained for diffractive dijet
production. These observations, coupled with the diffractive
measurements by H1\,\cite{H1d} and ZEUS\,\cite{ZEUSd} at HERA, offer
the opportunity to explore the relation between hard diffraction
at HERA and Tevatron. The processes are shown schematically in
Fig.~5, in the absence of rescattering corrections.
\begin{figure}[htb]
\begin{center}
\epsfig{figure=fig2n.ps,height=3in}
\caption{Schematic diagrams for diffractive dijet production at
the Tevatron and for diffractive deep icelastic scattering at
HERA.
\label{fig:fig2}}
\end{center}
\end{figure}
Without rescattering corrections the cross section for diffractive
dijet production , integrated over $t$, may be written as
\be
\label{eq:a12}
\sigma =  \sum_{i,j} \int F_P (\xi)  f_i^P (\beta,
E_T^2)  f_j^p (x_1, E_T^2) \hat{\sigma} d \beta dx_1,
\ee
where $\hat{\sigma}$ is the cross section to produce dijets from
partons carrying longitudinal momentum fractions $x_1$ and
$\beta$ of the proton and pomeron respectively. It was
found\,\cite{CDFj,Goulex} that calculation of the cross section,
based on this factorized formula with diffractive structure
functions obtained from HERA data, leads to a large discrepancy with
the CDF measurements both in the normalization and in the shape of the
observed distribution. Even taking into account uncertainties in gluonic
distribution of the pomeron
 discussed above the calculation lies about a factor of 10 above the data. A
 fast increase of the cross section for $\beta\to 0$ observed at Tevatron
 differs strongly from predictions of  models based on HERA data. The last
 result is difficult to understand in the models which take into account an
 extra suppression in hadronic interactions ("survival probability"\,\cite{BJs})
 due to rescatterings in the eikonal approximation.\cite{Levin,Block}

 This problem was considered in Ref.\,\protect\citebk{KKMR}. It was emphasized that a
 large cross section for single diffraction dissociation in hadronic collisions
 indicate to a large dispersion of eigen cross sections for diagonal states.
 It was assumed that the sea quarks and gluons mainly occur in large size
 configurations of the incident proton and thus have large cross sections,
 while the valence quarks occupy predominantly small--size configurations and
 are absorbed with substantially smaller cross sections.

  In such a two-channel model the survival probability of the
gaps can be written as follows
\be \label{eq:a16} | S |^2  =
\frac{\int  d^2 b \left ( |  M_ v|^2  e^{- \Omega_ v (s,b)}  +
 | M_{sea}|^2  e^{-\Omega_{sea} (s,b)} \right )}{\int  d^2 b \left ( |  M_ v|^2
  +  | M_{sea}|^2 \right )},
\ee
where $ M_{v, sea}$ are the probability amplitudes (in impact
parameter space) of the hard diffractive process corresponding to
the valence quark and to the sea quarks and gluons respectively.
The functions $\Omega_i$ can be parametrized in the form
\be
\label{eq:a17} \Omega_i  =  K_i  \frac{(g_{pp}^P)^2
(s/s_0)^\Delta}{4 \pi B}  e^{-b^2/4B},
\ee
with $i = v, sea $, and
where the slope of the pomeron amplitude is
\be \label{eq:a18}
B =  \textstyle{\frac{1}{2}} B_0  +  \alpha^\prime  \ln (s/s_0),
\ee
with $s_0 = 1~{\rm GeV}^2$.  We take $K_{\rm v} = 1 - \gamma$ and
$K_{\rm sea} = 1 + \gamma$, consistent with the simple physical
model introduced above.  The value of the parameters $g_{pp}^P,
B_0, \Delta$ and $\alpha^\prime$ were determined in a two-channel
global description of the total, elastic differential and soft
diffraction cross sections,\cite{KMR} in which the parameter
$\gamma$ was fixed to be 0.4.

In this model the soft rescattering effects $(\Omega_i \neq 0)$ of
the model based on (\ref{eq:a16}) modify the $\beta$ distribution
of the dijet process in a characteristic way. First note that the
CDF measurements cover a narrow $\xi$ interval, $0.035 \leq \xi
\leq 0.095$, and hence that the invariant mass squared of the
diffractively produced state, $M^2 = \xi s$, remains close to the
average value $2 \times 10^5~{\rm GeV}^2$.  Also the mass squared
of the produced dijet system
\be \label{eq:a19}
M_{jj}^2 \; = \; x_1 \beta M^2,
\ee
 does not change much compared to its average value of $1.2 \times 10^3~{\rm
GeV}^2$.  Thus $x_1 \beta \simeq 0.006$ and so for $\beta \geq
0.3$ we have $x_1 \leq 0.02$, whereas for $\beta \sim 0.03$ we
have $x_1 \sim 0.2$.  Therefore for large $\beta$ (small $x_1$)
sea quarks and gluons will give the dominant contribution, while
for small $\beta$ the valence quarks play an important role. Hence
the survival probability should increase as $x_1$ increases and
$\beta$ decreases. Results of calculations based on this model
(Fig.~6) are in a good agreement with CDF data both in magnitude
and form of $\beta$ distribution.\cite{KKMR} In particular an
anomalously strong increase of this distribution as $\beta\to 0$
is reasonably described.
\begin{figure}[h]
\begin{center}
\epsfig{figure=diffig4.ps,height=3.5in}
\caption{The predictions for diffractive dijet production at the
Tevatron, obtained from two alternative sets of \lq HERA\rq
diffractive parton distributions , compared with the CDF data.$^{32}$
The upper two curves correspond to the neglect of
rescattering corrections, whereas the lower four curves show the
effect of including these corrections in two models~ A(continuous
curves) and model~B (dashed curves) for the diffractive
eigenstates (see Ref.\,{38}).
\label{fig:fig6}}
\end{center}
\end{figure}
This calculation of diffractive dijet production illustrates a
crucial ingredient necessary in the description of rapidity gap
processes.  Namely that the survival probability of a gap can
depend on $x_1$ of the partons of the proton .  This leads to many
experimental consequences for processes with rapidity gaps.\cite{KKMR}

\subsection*{Double--pomeron jet production}
\addcontentsline{toc}{subsection}{\numberline{}Double--pomeron jet production}

The process of particle production by double--pomeron exchange
process (DPE) gives an important  new information on mechanism of
diffraction at high energies. Production of dijets with two large
rapidity gaps gaps ($x_p$ and $x_{\bar p}$ are  close to 1) has
been observed at Tevatron.\cite{Goulex} Comparison of cross
section of this process $(\sigma_{DP}^{jj})$ with cross section of
jet production in the process of single diffraction dissociation
$(\sigma_{SD}^{jj})$ and in ordinary inelastic events
$(\sigma_{in}^{jj})$ allows to study factorization properties of
inelastic diffractive processes. The ratio
$\sigma_{SD}^{jj}/\sigma_{in}^{jj}$ can be written as:
\begin{equation}
R_1\equiv\frac{\sigma_{SD}^{jj}}{\sigma_{in}^{jj}}=\frac{F_P(\xi)f_g^P
(\beta)}{f_g^p (x)}S_1^2, \label{a}
\end{equation}
where $S_1^2$ is the "survival probability" for single diffraction
dissociation. For simplicity I consider only contribution of
gluons. Account of quarks is straightforward.

The ratio of jet production in double--pomeron process to those for
single one has the form
\begin{equation}
R_2\equiv\frac{\sigma_{DP}^{jj}}{\sigma_{SD}^{jj}}=\frac{F_P(\xi_1)f_g^P
(\beta_1)} {f_g^p(x_1)}\frac{S_2^2}{S_1^2}, \label{b}
\end{equation}
where $S_2^2$ is the "survival" probability for DPE.

So the ratio $R_1/R_2$ for the situation when $\xi=\xi_1$,
$\beta=\beta_1$ ($x=x_1$) is equal to
\begin{equation}
R\equiv\frac{R_1}{R_2}=\frac{(S_1^2)^2}{S_2^2} \label{c}
\end{equation}

For a single Regge pole exchange ($S_i^2=1$) $R=1$ and thus
deviations of R from unity signals a breakdown of the
factorization. With account of absorption $S_1^2=0.1$,
$S_2^2=0.05$  at Tevatron energies.\cite{KMR} So from Eq.\,(8) it
follows that we can expect $R\approx 0.2$ at these energies.
Recent results of CDF show that $R=0.19\pm 0.07$  in a good
agreement with theoretical expectation.\cite{KKMR1} This result
shows that factorization  is strongly violated in inelastic
diffractive processes.

\section{Conclusions}
  A study of diffractive processes  provides an important
 information on interplay of soft and hard mechanisms in high energy interactions
 of hadrons. A new field of hard diffractive processes give possibility to study
 not only partonic content of the pomeron but also cross sections for absorption
 of different partonic configurations of the proton. The theory of supercritical
 pomeron with the intercept $\alpha_P(0)\approx 1.2$ , which takes into account
 multi--pomeron exchanges gives a unified description of diffractive processes
 both for small--x DIS and hadronic interactions at high energies.

\section*{Acknowledgments}
\addcontentsline{toc}{section}{\numberline{}Acknowledgments}

I would like to thank K.Boreskov, A.Capella, E.G. Ferreiro, O.V. Kancheli,
 V.A. Khoze, A. Martin, C. Merino, C.A. Salgado, K.A. Ter-Martirosyan  and
 J.Tran Thanh Van for useful discussions and collaboration.
  This work is supported in part by the grants: INTAS 00-00366, NATO PSTCLG-
977275, RFBR 00-15-96786, 01-02-17383.

%\vspace{1cm}
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 \end{document}
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