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%\documentclass[draft]{ws-procs9x6}  
\documentclass{ws-procs9x6}

\def\FPom{F_2^{\Pom}}
\def\F3D{F_2^{D(3)}}
\def\LPF2{F_2^{D(4)}}
\def\Pom{I\!\!P}
\def\Reg{I\!\!R}
\def\xP{x_{I\!\!P}}
\def\aP{\alpha_{\Pom}}
\def\aR{\alpha_{\Reg}}
\def\aPP{\alpha_{\Pom}^{\prime}}
\def\as{\alpha_s}

\begin{document}

\title{Diffractive Structure Functions and QCD Fits}

\author{S. LEVONIAN}

\address{DESY, \\ Notkestrasse 85, \\ 22607 Hamburg, Germany \\ 
E-mail: levonian@mail.desy.de}


\maketitle

\abstracts{
Recent progress is reviewed in our understanding of inclusive diffraction
in the deeply inelastic regime at HERA. New precision measurements are now available 
from both H1 and ZEUS collaborations. A new set of diffractive parton
distribution functions, determined from the H1 data, is presented. 
It can be used to test factorization properties of diffractive scattering
in different reactions.} 


\section{Introduction}
%=====================
Significant progress has been achieved over the last decade in understanding
the nature of {\it diffractive phenomena} at high energies.
This is to a large extent due to the electron-proton collider HERA, which is 
an ideal place to study hard diffraction in deep-inelastic scattering (DIS).
Already the term `deep-inelastic diffraction' itself sounds as a paradox
and goes against our intuition, based upon the experience of hadronic interactions
in which diffraction was known to be a predominantly soft peripheral process.
HERA for the first time offers the possibility for the partonic structure
of colour singlet exchange to be probed.

At the hadronic level diffraction is best described in the framework of
Regge formalism\cite{Regge,Gribov} as a $t$-channel exchange of a leading trajectory
with the vacuum quantum numbers, named {\it Pomeron}. The specific interest
in diffraction as  `physics of the Pomeron' is related to the fact that Pomeron
exchange asymptotically dominates over all other contributions to the scattering
amplitude, and thus represents the essence of strong interactions in the high energy
limit. Translated into  modern partonic language this reveals that 
colourless exchange is important in the low-$x$ regime (which is the high energy limit
of QCD) where gluons are expected to dominate.
Since the diffractive cross section, being proportional to the gluon density
squared, rises with energy faster than the total cross section,
unitarity correction effects, e.g. in the form of gluon saturation, are expected
to be first seen in diffraction. Hence an interesting question is:
can this be observed already at HERA?

Another aspect in which diffraction may play a key r\^{o}le is in the  
interplay between soft and hard processes. Since both short distance and
long distance physics contribute to diffractive DIS, HERA has a high potential 
to provide new insight into non-perturbative QCD phenomena, as well as to bridge
the soft (Regge) and hard (pQCD) domains.  


\subsection{Kinematics of Diffractive DIS at HERA}
%-------------------------------------------------
Figure~\ref{FD} sketches the generic diffractive process in electron-proton scattering.
In addition to the usual DIS kinematic variables: photon virtuality, $Q^2=-q^2$,
Bjorken-x, $x=-q^2/2P\cdot q$ and inelasticity $y=Q^2/sx$ this process is characterised
by the 4-momentum squared transferred at the proton vertex, $t=(P-p_Y)^2$ and the
longitudinal momentum fraction of the colour singlet exchange relative to the incoming 
proton, $\xP = q\cdot (P-p_Y)/q\cdot P$. 
Here $P$ and $q$ are the 4-momenta of the interacting
proton and photon respectively, and $s$ is the $ep$ centre of mass energy squared. 
The ratio of Bjorken-x to $\xP$ defines another diffractive variable, $\beta=x/\xP$, which
can be interpreted as the longitudinal momentum fraction of the exchange that is carried
by the struck quark.
%
\begin{figure}[htb]
\centerline{\epsfxsize=2.5in\epsfbox{Levonian_fig1.eps}}   
\caption{The generic diffractive process $ep \to eXY$ at HERA,
in which the interaction is mediated by colour singlet exchange between
the photon and proton vertecies, leading to a large rapidity interval 
separating final state systems $X$ and $Y$. 
\label{FD}}
\end{figure}

\noindent
After integrating over $M_Y$, which is not well measured at HERA, 
a fourfold-differential diffractive {\it reduced cross section}, $\sigma_r^{D(4)}$,
can be defined through
\begin{equation}
\frac{d^4\sigma^D}{d\xP~dt~d\beta~dQ^2} = \frac{4\pi\alpha^2}{\beta Q^4}
      \left( 1-y+\frac{y^2}{2}\right)\sigma_r^{D(4)}(\xP, t, \beta, Q^2),
\label{eq:sD4}
\end{equation}
which is related to the diffractive structure functions $F_2^D$ and $F_L^D$ by
\begin{equation}
   \sigma_r^D = F_2^D - \frac{y^2}{1+(1-y)^2} F_L^D. 
\label{eq:DSF}
\end{equation} 
To a very good approximation $\sigma_r^{D}=F_2^D$ everywhere
except possibly at the highest values of $y$. Furthermore, if the outgoing proton
is not detected, the measurements integrate over $t$: 
$\sigma_r^{D(3)} = \int \sigma_r^{D(4)} dt$.

\subsection{Understanding Diffraction in DIS}\label{subsec:models} 
%-----------------------------------------------------------------
The variety of  phenomenological models for diffractive DIS can be classified
into three main categories.
%\begin{itemize}

%\noindent
1. Regge motivated models with a factorizable Pomeron\cite{RPM}. The internal partonic 
   structure of such an object can be resolved in DIS. Regge factorization
   means that diffractive cross sections can be factorized into a Pomeron flux and 
   its structure function. More generally, sub-leading reggeon contributions,
   which are essential at high values of $\xP$, should also be taken into account:
        \begin{equation}
            F_2^{D(4)}(\xP,t,\beta,Q^2) = f_{\Pom}(\xP,t) F_2^{\Pom}(\beta,Q^2) +
                                          f_{\Reg}(\xP,t) F_2^{\Reg}(\beta,Q^2)
        \label{eq:RF}  
        \end{equation}
   with flux factors defined as
        \begin{equation}
            f_{i}(\xP,t) = \xP^{1-2\alpha_i(t)} e^{bt} ,  
        \label{eq:flux}
        \end{equation}  
   where $\alpha_i(t) = \alpha_i(0)+\alpha_i^{\prime} t ~~ (i=\Pom,\Reg)$ 
   are Pomeron and reggeon trajectories respectively with intercept $\alpha_i(0)$
   and slope $\alpha_i^{\prime}$.

%\noindent 
2. Non-Pomeron mechanisms generating large rapidity gap events. For example,
   in the soft colour interaction model\cite{SCI} diffraction occurs through the
   normal boson-gluon fusion with additional emission of soft gluons, neutralizing
   the colour flow. As a consequence the model predicts leading gluon behaviour
   in which one gluon takes most of the momentum of the colour singlet exchange.

%\noindent 
3. Colour dipole approach\cite{CDA} to DIS in which the interaction is viewed 
   in the proton rest frame. This is especially convenient at low $x$, in which case
   the virtual photon chooses a specific Fock state to fluctuate into long before 
   arriving at the proton target ($c\tau \approx 1/xM_p \simeq 1000$ fm at HERA).
   Hence the resulting cross section can be represented as a simple convolution
   of the photon wave function $|\gamma\rangle = \alpha_1|q\bar{q}\rangle +
   \alpha_2|q\bar{q}g\rangle + ...$
   with the colour dipole cross section ($\sigma_{(q\bar{q})p}$,
   or $\sigma_{(q\bar{q}g)p}$). The models of this class\cite{CDM} differ in
   the way they parameterise dipole cross sections. Their common strong feature
   is related to the fact, that once fitted to the inclusive DIS data, the same
   dipole cross sections with only one additional free parameter, related
   to the $t$-dependence, should also describe diffractive phenomena in DIS. 
%\end{itemize}

Alternatively, one can employ a model independent approach using the concept
of {\it diffractive parton distributions, $f_i^D$}. It is based on the rigorous 
proof\cite{QCD_fact} that QCD hard scattering factorization is valid for 
diffractive DIS, and hence the diffractive cross section~(\ref{eq:sD4}) 
at fixed values of $\xP$ and $t$ can be expressed 
as a convolution of universal partonic cross sections 
$\hat{\sigma}^{\gamma^*i}$ with $f_i^D$:
\begin{equation}
   \sigma_r^D(4) \propto \sum_i \hat{\sigma}^{\gamma^*i}(x,Q^2) \otimes
   f_i^D(x,Q^2; \xP,t).
\label{eq:collins}  
\end{equation} 
The partonic cross sections $\hat{\sigma}^{\gamma^*i}$ are the same as in inclusive DIS,
and $f_i^D$, which are not known from first principles, should obey the DGLAP evolution 
equations. Hence the standard framework of NLO QCD can be applied to diffractive DIS 
in a similar manner as to inclusive DIS.  
% The question to be addressed is whether or not experimentally observed
% Regge factorization still holds, and if not - what is a mechanism of the
% factorization breaking. 

\section{Experimental Techniques and data Samples}
%=================================================
Several experimental techniques are used to measure diffractive processes at HERA.
They have different advantages and drawbacks and are complementary to each other.

The clearest way is to detect the scattered proton in the forward
direction using so called {\it Roman pot} technique - several detectors inserted
in the beam pipe which together with machine magnets form a proton spectrometer.
Such a method was exploited by both H1 and ZEUS experiments. 
It provides a diffractive data sample free of proton dissociative admixture and
in addition enables to measure $t$, which in other methods is not directly measured
and has to be integrated over. The method however suffers from low statistics,
due to limited acceptance and the dependence on beam conditions, 
which do not allow full luminosity to be collected.

To obtain high statistics sample one can use characteristic properties of the
hadronic final state, as shown in Fig.~\ref{FD}. H1 and ZEUS follow slightly
different approaches trying  to fully utilize their detector capabilities.

The method iof H1 is based on the requirement of a large rapidity gap separating 
the leading baryon system $Y$ from the photon dissociation system $X$.
The rapidity gap is positively identified by the absence of activity in the
pseudorapidity interval $3.2 < \eta < 7.5$, covered by various detector components.
In this approach non-diffractive background is small and can be estimated
using Monte Carlo simulation. 

The extraction of the diffractive contribution in ZEUS is performed using 
the so called $M_X$ method, which is based on the fact that diffractive
and non-diffractive final states have very different $\ln M_X^2$
distributions.  The non-diffractive contribution,
exhibiting an exponential fall-off towards lower $M_X$ values, is extrapolated
underneath the diffractive plateau and statistically
subtracted from the total $M_X$ distribution. 

% New Forward Plug Calorimeter\cite{zeus_plug}, covering the pseudorapidity 
% range $4 < \eta < 5$, was installed in 1998 and significantly improved capability
% of ZEUS detector for diffractive physics.

Diffractive samples selected with the latter two methods are dominated by 
the
single dissociative process $ep \to eXp$ with a small admixture of double dissociation
in which the low mass proton dissociation system $M_Y$ escapes in the beam hole undetected.
Hence the measured cross sections are corrected to the region $M_Y<1.6$ GeV and
$|t|<1$ GeV$^2$ in case of H1 and to $M_Y<2.3$ GeV in case of ZEUS.

Both collaborations have released recently several new measurements with a
 2- to
5-fold increase in statistics and a significant extension in the covered phase space
as compared to previously published data\cite{HERA_round1}. 

New measurements of the diffractive reduced cross section
in the range $1.5 < Q^2 < 120$ GeV$^2$  by the H1 collaboration is presented in 
Fig.~\ref{dsdxP}. Two data samples\cite{h1_lowq_data,h1_medq_data} 
based on the rapidity gap technique are in a good agreement 
with each other and with a third measurement\cite{h1_FPS_data}, covering
$2 < Q^2 < 50$ GeV$^2$, in which the leading proton is detected in the forward 
spectrometer.

\begin{figure}[htb] 
\centerline{\epsfxsize=3.8in\epsfbox{Levonian_fig2.eps}}   
\caption{The diffractive reduced cross section as measured in three different
samples and with different techniques is shown as a function
of $\xP$ in bins of $\beta$ and $Q^2$.
\label{dsdxP}}
\end{figure}

\begin{figure}[htb] 
\centerline{\epsfxsize=3.2in\epsfbox{Levonian_fig3.eps}}   
\caption{The differential cross section $d\sigma^{\rm diff}_{\gamma^*p\to XN}
/dM_X,~M_N<2.3$ GeV, as a function of $W$ in bins of $M_X$ and $Q^2$.
\label{dsdmx}}
\end{figure}
\begin{figure}[htb]
\centerline{\epsfxsize=2.9in\epsfbox{Levonian_fig4.eps}}
\caption{$Q^2$ dependence of the diffractive cross section $d\sigma^{D}_{\gamma^*p}/dM_X$ 
at different values of $W$ and $M_X$.The solid (dashed) lines represent results of a
specific dipole model fit$^{16}$ (and its extrapolation). The $q\bar{q}g$
contribution is shown by the dotted curves.
\label{LPS}}
\end{figure}  

Inclusive diffractive cross sections as measured by the ZEUS collaboration\cite{ZEUS_mx_data}
are  shown in Fig.~\ref{dsdmx}. With the exception of the lowest $M_X$ bin a strong rise
with energy, $W$, is observed. The data\cite{ZEUS_LPS} in the transition region between 
DIS and quasi-real photon-proton interactions (see Fig.~\ref{LPS}) exhibit a behaviour 
very similar to that of the total photon-proton cross section\cite{ZEUS_stot}.
Namely, the diffractive cross section falls rapidly with $Q^2$ at high virtualities,
while for $Q^2 \to 0$ the cross section dependence on $Q^2$ flattens off. 
This behaviour is broadly described by the colour dipole model\cite{Bartels} in which 
the virtual photon fluctuates into $q\bar{q}$ and $q\bar{q}g$ states.

\section{General Properties of Inclusive Diffraction at HERA}
%================================================================

To compare the dynamics of diffractive DIS with those of inclusive DIS
the ratio of the diffractive to the total inclusive cross sections is studied as
a function of energy and $Q^2$. Fig.~\ref{ratios} represents just two of several
such measurements. The ratio is observed to be everywhere remarkably flat, both in $Q^2$ 
and in $W$, except at high $\beta$ (low $M_X$). 
The latter can be readily explained by simple kinematics
(suppression of available phase space for gluon radiation close to the kinematic
limit), or by the onset of higher twist effects. 
%
It is fair to note that such behaviour is naturally expected in models 
of the 2-nd class (see Sec.~\ref{subsec:models}), 
in which rapidity gap formation is due to purely probabilistic mechanisms. 
%
\begin{figure}[htb] 
\centerline{\epsfxsize=4.2in\epsfbox{Levonian_fig5.eps}}   
\caption{Ratio of diffractive to inclusive DIS cross sections 
as a function of $W$ (left) and $Q^2$ (right).
\label{ratios}}
\end{figure}

Approximately flat ratio as a function of $W$ implies the same energy
dependence for diffractive and inclusive $\gamma^*p$ cross section, which is
not expected neither in the Regge picture, nor in a simple two-gluon exchange
QCD model for diffractive DIS.

To quantify the difference between the energy dependence of $\sigma^D$ and $\sigma^{tot}$
one can compare the values of the Pomeron intercept, $\aP(0)$, extracted using 
Regge motivated fits from diffractive and inclusive DIS data.
In the inclusive case, the proton structure function can be parameterised
at low $x < 0.01$ as $F_2(x,Q^2) = cx^{-\lambda(Q^2)}$ with 
$\lambda=\aP(0)-1$.
In diffractive DIS $\aP(0)$ can be obtained, after correcting for the finite
values of $t$, from the $W$-, or equivalently from the $\xP$-dependence of the diffractive 
cross section. 

The results are summarized in Fig.~\ref{aPom}.  $\aP(0)$ is observed to rise 
with $Q^2$, although the error bars are still large in case of diffraction.  
In contrast to photoproduction, where the same universal `soft' Pomeron trajectory
describes both inclusive and diffractive data, in the DIS regime the Pomeron intercept
extracted in diffraction is approximately half of that obtained in inclusive data.
This is consistent with the above conclusion that the energy dependence is 
very similar in diffractive and inclusive DIS. 
This surprising result is not so easy to explain, unless some sort of 
unitarity corrections are assumed to play a significant r\^{o}le
in diffraction at HERA. Alternatively, it could point to the possibility
that different proportions of soft and hard physics contribute to diffractive
processes as compared to fully inclusive ones, suggesting that $Q^2$ is not 
completely equivalent or the only relevant scale in those two cases. 

\begin{figure}[ht] 
\centerline{\epsfxsize=4.6in\epsfbox{Levonian_fig6.eps}}   
\caption{The intercept of the Pomeron trajectory, $\aP(0)$, as a function of
$Q^2$ in diffractive and inclusive DIS, as determined by ZEUZ(left) and H1 (right).
The cross hatched band represents `half' of the $W$-rise of the total $\gamma^*p$ 
cross section, $\alpha_{\Pom/2} = 1+(\alpha_{\Pom}^{\rm tot}(0)-1)/2$. 
\label{aPom}}
\end{figure}

\section{NLO QCD Fit to H1 Data and Diffractive PDFs}
%====================================================

H1 has performed\cite{h1_qcd_fit} a new NLO DGLAP QCD fit to
the medium $Q^2$ data sample\cite{h1_medq_data} from which
diffractive parton densities are obtained. To simplify the fit
Regge factorization~(\ref{eq:RF}) was assumed, implying that the shape 
of pdf's is independent on $\xP$. At the present level of precision this 
assumption is justified by the data, as can be seen in Fig.~\ref{ratio},
in which the $\beta$ and $Q^2$ dependences are compared to the fit results.

\begin{figure}[htb] 
\centerline{\epsfxsize=4.9in\epsfbox{Levonian_fig7.eps}}   
\caption{$\beta$ (left) and $Q^2$ (right) dependence of the reduced
cross section scaled at each $\xP$ by the values assumed for the $t$-integrated
Pomeron flux in the QCD fits.
\label{ratio}}
\end{figure}

Both $\Pom$ and $\Reg$ contributions are included in the fit.
The Pomeron structure is parameterised by a light flavour singlet, $\Sigma$,
and a gluon, $g$, distribution, while for the reggeon term the
{\it pion} pdf\cite{Owens} is used. The latter choice does not significantly
influence the results, as the sub-leading contribution is negligible 
at $\xP < 0.01$.  
Diffractive parton densities ({\it dpdf's}) are parameterised at a starting scale 
$Q_0^2 = 3$ GeV$^2$ and evolved to higher $Q^2$ using NLO DGLAP equations. 
313 data points from the region $Q^2 >6.5$ GeV$^2$ and $M_X > 2$ GeV are used 
to constrain 7 free parameters (3 for each of the distributions, 
i.e. for the singlet and the gluon, and one for the 
normalisation of the sub-leading exchange at high $\xP$).
It should be emphasised that for the first time in diffraction the experimental
and model uncertainties are fully propagated to obtain error bands for resulting
dpdf's. 

The result of the fit is shown in Fig.~\ref{H1_pdf}.
Whereas the singlet part is well constrained, there is a substantial
uncertainty in the gluon distribution at large fractional momenta $z$ (or $\beta$),
mainly due to model assumptions. 
The diffractive pdf's remain large up to large $z$ and are dominated by gluons.
In total $75\pm 15\%$ of the exchanged momentum is carried by gluons.
Fig.~\ref{pdf_comp} presents the  $\beta$ and $Q^2$ dependences of the
diffractive reduced
cross section at two selected values of $\xP$. Even extrapolated beyond the region
used in the fit, the new pdf's provide a very good description of new 
H1 data\cite{h1_higq_data} at high $Q^2$. One can also see that at high $\xP$
the reggeon contribution becomes important. 
It is worth mentioning that new precise
low $Q^2$ data\cite{h1_lowq_data} have a large potential to further constrain
dpdf's.

\enlargethispage{1cm}
Another possible test involves comparisons with diffractive final states\cite{Vlasov}
at HERA and TEVATRON.
\clearpage
\begin{figure}[htb] 
\centerline{\epsfxsize=3.2in\epsfbox{Levonian_fig8.eps}}   
\caption{Diffractive parton densities obtained from the QCD fits
and normalized such that the `Pomeron flux' is unity at $\xP=0.003$.
\label{H1_pdf}}
%\end{figure}
%
%\begin{figure}[hb] 
\centerline{\epsfxsize=4.2in\epsfbox{Levonian_fig9.eps}}   
\caption{Diffractive reduced cross section at fixed $\xP$ as a function
of $\beta$ in bins of $Q^2$(left) and as a function of $Q^2$ in bins
of $\beta$ (right). Also shown is the prediction for $\xP\sigma_{\rm r}^{D(3)}$
at $\sqrt{s}=319$ GeV from the NLO QCD fit performed to the medium $Q^2$ data. 
\label{pdf_comp}}
\end{figure}
%
\clearpage
\section{Summary}
%----------------
New precision measurements of inclusive diffractive DIS are now
available from HERA. These data are extremely valuable to further constrain
QCD motivated phenomenological models and to distinguish between different
approaches to diffractive phenomena in the DIS regime. 
%In general colour dipole
%models in a consistent picture give fair description of the the data. 
A new generation of diffractive parton distributions together with their
uncertainties have been determined from H1 data in NLO QCD framework.
They can be used to test factorization properties of diffraction in 
different reactions.
The energy dependence of the diffractive cross section has no simple 
explanation at the moment. It may indicate that the onset of unitarity corrections 
(or saturation effects) is already seen in diffraction at HERA. Alternatively,
it could be a manifestation of a complicated interplay between soft and
hard phenomena in diffractive DIS.
In spite of the recent progress a complete understanding of the nature 
of colour singlet exchange remains a major challenge in QCD.

\enlargethispage{1cm}
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\end{document}
