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\begin{document} 

\title{Rescattering and gap survival probability at HERA} 
\author{Ada Solano, on behalf of the H1 and ZEUS Collaborations}
\institute{Univ. of Torino and INFN}
\maketitle

\begin{abstract}
Diffractive dijet photoproduction and leading neutron data measured with 
the H1 and ZEUS detectors at HERA are presented. These data allow to study
rescattering and gap survival probability in $ep$ interactions.

\end{abstract}

\section{Introduction}
The role of rescattering and gap survival probability in $ep$ interactions 
at HERA has been studied by the H1 and ZEUS Collaborations looking at 
diffractive dijet photoproduction and leading neutron production.

Diffractive $ep$ events, $ep\rightarrow eXp$, are characterized by the presence 
in the final state of a fast forward proton, scattered at a very small angle 
having lost only a small fraction of the incoming proton energy, and a large 
rapidity gap (LRG) with no particle flow between the scattered proton and 
the hadronic system X from the dissociated photon. 
This event topology is ascribed to the absence of colour flow between the 
proton and the system X, due to the exchange of an object with vacuum 
quantum numbers, historically called pomeron.
Both characteristics have been used at HERA to select diffractive events, 
either by measuring the fast scattered proton with detectors placed along the 
proton beamline at distances between 20 and 90 m from the interaction point, 
or by searching for LRG in the central detectors. The diffractive samples 
for the dijet photoproduction analyses presented here were selected by both 
Collaborations using the LRG method. 

Leading neutron events, $ep\rightarrow eXn$, are characterized by the 
presence in the final state 
of a fast forward neutron carrying a relevant fraction of the incoming proton 
beam energy. This neutron escapes along the beamline and is detected by both 
Collaborations by means of forward neutron calorimeters placed at about 
100 m from the interaction point.

\section{QCD factorization in diffraction}
According to the quantum chromodynamics (QCD) factorization 
theorem~\cite{Collins}, the cross section for diffractive processes 
in deep inelastic scattering (DIS) can be expressed 
as a convolution of partonic hard scattering cross sections, which are 
calculable in perturbative QCD (pQCD), and universal diffractive parton 
density functions (DPDFs) of the proton, which are analogous to the 
usual proton PDFs under the condition that the proton stays intact in the 
interaction. 

At HERA, various sets of DPDFs~\cite{DPDFs} have been determined from QCD 
fits to inclusive diffractive cross section measurements in DIS. It was found 
that most of the momentum of the diffractive exchange is carried by gluons.

The DPDFs extracted from inclusive data have been used for calculating 
next-to-leading order (NLO) 
predictions of semi-inclusive DIS diffractive final states, in particular 
dijet and open charm production, for which the presence of hard scales 
ensures that the partonic cross sections are perturbative calculable.
Both H1 and ZEUS data on the DIS diffractive production of open 
charm~\cite{DISopencharm} and dijets~\cite{DISdijets,H1jetDPDFs} agree with 
NLO predictions within the uncertainties, which represents an experimental 
proof of the validity of QCD factorization in diffractive DIS. 
This also allowed to include dijet data in the QCD fits to better constrain 
the DPDFs, in particular the gluon one~\cite{H1jetDPDFs}.

QCD factorization is not expected to hold in diffractive hadron-hadron 
interactions. Actually, QCD calculations with HERA DPDFs as input 
overestimate the cross section for single diffractive dijet production 
in $p\bar{p}$ collisions at the Tevatron by approximately a factor 
10~\cite{Tevatron}. This violation of factorization has been understood 
in terms of secondary interactions and rescattering between spectator 
partons, which may fill the rapidity gap, leading to a breakdown of 
hard-scattering factorization and causing a suppression of the diffractive
cross section. Models including rescattering corrections via multi-pomeron
exchanges are able to describe the suppression observed~\cite{GSP}, 
which is often quantified by a 'rapidity gap survival probability'. 
This is also of great interest for the forthcoming LHC data analyses.

The increasing role of rescattering in the transition from DIS to 
hadron-hadron interactions can be studied at HERA by comparing processes 
in DIS and in photoproduction (PHP), since in photoproduction the quasi-real 
photon, with virtuality $Q^2\sim 0$, can develop a hadronic structure. 

At leading order (LO) two types of processes contribute to PHP events 
(see Fig.~\ref{diagrams}), 
direct- and resolved-photon processes. When the photon participates 
directly in the hard scattering as a point-like probe the processes 
are expected to be similar to the DIS ones and diffractive QCD factorization 
is expected to hold as in DIS. 
In contrast, processes in which the photon is first resolved into
partons which then engage in the hard scattering resemble hadron-hadron
interactions. In this latter case, the additional photon remnant opens up 
the possibility of secondary remnant-remnant interactions and diffractive 
QCD factorization is not expected to hold. 

\begin{figure}
  \hspace{1cm}
  \includegraphics[height=.25\textheight]{diagram_a}
  \includegraphics[height=.25\textheight]{diagram_b}
  \caption{Left panel: Direct-photon diagram for diffractive dijet photoproduction.
           Right panel: Resolved-photon diagram for the same process.}
  \label{diagrams}
\end{figure}

\section{Diffractive dijets in photoproduction: gap survival probability and 
its $E_T$ dependence}

Diffractive photoproduction of dijets has been studied by the H1 and ZEUS 
Collaborations as an interesting process to test the QCD factorization 
hypothesis and measure a possible rapidity gap survival probability 
in $ep$ interactions. A reasonably high transverse energy, $E_T$, of the 
jets provides the hard scale, ensuring the applicability of pQCD at the 
small photon virtualities considered. The variable $x_\gamma$, which is 
the fraction of the photon momentum entering in the hard scattering, is used
to separate direct- and resolved-photon events, where the latter have $x_\gamma<1$. 

A first sample of H1 diffractive data~\cite{H1-1} has been analyzed in the 
kinematic region $Q^2<0.01$~GeV$^2$, $x_{\pom}<0.03$, where $x_{\pom}$ is the 
fraction of the proton momentum carried by the pomeron, $E_T^{jet1}>5$~GeV and 
$E_T^{jet2}>4$~GeV. Since the data were selected with the LRG method, 
where the diffractive proton is not measured, the sample includes events 
in which the proton dissociates into low mass states, up to $M_Y<1.6$~GeV,
that escape detection going into the beampipe.
Figure~\ref{H1p8} shows a few differential distributions measured 
with this sample. The H1 data, corrected to the hadron level, are compared 
with NLO calculations obtained assuming factorization with a program by 
Frixione et al.~\cite{Frixione}. H1 2006 Fit B DPDFs have been used as input
and one can see that the NLO predictions, also corrected to the hadron level, 
agree with the data if scaled by a factor 0.5. 
Two conclusions can be drawn: NLO calculations overestimate
the measured cross sections by a factor $\sim$2 both in the direct and in the
resolved region, in contrast to the expectation the only resolved-photon
processes should be suppressed; as expected the suppression in $ep$
events is much smaller than in $p\bar{p}$ interactions.

\begin{figure}
  \begin{center}
  \includegraphics[height=.40\textheight]{H1p8}
  \caption{Differential cross sections for the diffractive photoproduction 
           of dijets. H1 data are compared to NLO calculations by Frixione et al.}
  \label{H1p8}
  \end{center}
\end{figure}

In Fig.~\ref{ZEUSp9H1DR}, left panel, the ZEUS measurement~\cite{ZEUS-PH} 
of the differential cross section in $x_\gamma$ and the ratio of data 
to NLO calculation are shown.
NLO predictions have been obtained assuming factorization with a program 
by Klasen and Kramer~\cite{KK}. The ZEUS data were selected in the 
kinematic region $Q^2<1$~GeV$^2$, $x_{\pom}<0.025$, $E_T^{jet1}>7.5$~GeV 
and $E_T^{jet2}>6.5$~GeV. Cross sections were corrected to the hadron level 
and the contribution due to proton dissociative events ($16\pm4$\%) 
was subtracted. A correction for the proton dissociative contribution was 
also applied when using the H1 DPDFs, since these are extracted from 
inclusive diffractive samples including proton dissociation with $M_Y<1.6$~GeV.
As in the H1 analysis presented above, data do not show any difference 
between the resolved and the direct photon region. However, the ZEUS data 
show a very weak, if any, suppression, which mainly originates from the 
lower $E_T^{jet1}$ region. NLO calculations tend to 
overestimate the measured cross sections but within the large theoretical 
uncertainties the data are still compatible with QCD factorization.

\begin{figure}
  \begin{center}
  \includegraphics[height=.40\textheight]{ZEUSp9}
  \hspace{1cm}
  \includegraphics[height=.27\textheight]{H1DR}
  \caption{Left panel: a) Differential cross section in $x_\gamma$ for the 
           diffractive photoproduction of dijets; b) ratio of data to NLO prediction. 
           ZEUS data are compared to NLO calculations by Klasen and Kramer.
           Right panel: Cross section double ratio of H1 data to NLO predictions
           for PHP and DIS as function of $x_\gamma$.}
  \label{ZEUSp9H1DR}
  \end{center}
\end{figure}

The discrepancy between H1 and ZEUS has been attributed to the different 
$E_T$ regions of the two analyses. Indeed, both H1 and ZEUS data have a 
harder $E_T$ distribution than in NLO. The possible $E_T$ dependence of the
suppression can be better seen in the double ratio shown in Fig.~\ref{ZEUSp9H1DR}, 
right panel, obtained by dividing the ratio of measured to predicted cross 
section in photoproduction by the corresponding ratio in DIS. In this double
ratio many experimental errors and also theoretical scale errors cancel to 
a large extend. The plot gives a clear signal that the rapidity gap survival
probability might increase with $E_T$.

To better study the $E_T$ dependence, a more recent H1 analysis~\cite{H1-2} 
has been performed, based on a three times higher integrated luminosity with 
respect to the previous one. This allowed selecting two samples with different 
$E_T$ cuts: for the first sample (Low $E_T$ one) all the cuts were the same
as in the previous H1 analysis, in particular $E_T^{jet1}>5$~GeV and 
$E_T^{jet2}>4$~GeV, to be able to cross check the results; instead, the 
second sample (High $E_T$ one) covered a kinematical region similar to that
of the ZEUS analysis, with $E_T^{jet1}>7.5$~GeV and $E_T^{jet2}>6.5$~GeV. 
Two independent NLO calculations have been compared to the measurements, 
that by Frixione et al. and that by Klasen and Kramer, using three sets of 
DPDFs, H1 2006 Fit A and Fit B and H1 2007 Fit Jets. Figure~\ref{H1-2}, 
left panel, shows the $x_\gamma$ distribution and the ratio of data to 
theory expectation for the 'Low $E_T$' sample, while Fig.~\ref{H1-2}, 
right panel, shows the same plots for the 'High $E_T$' sample.

\begin{figure}
  \begin{center}
  \includegraphics[height=.45\textheight]{H1lowEt}
  \hspace{1cm}
  \includegraphics[height=.45\textheight]{H1highEt}
  \caption{Differential cross section in $x_\gamma$ for the diffractive 
           photoproduction of dijets and ratio of H1 data to NLO predictions. 
           Left panel: 'Low $E_T$' sample. Right panel: 'High $E_T$' sample.}
  \label{H1-2}
  \end{center}
\end{figure}

In both cases, data confirm that there is no sign of a dependence in $x_\gamma$ 
of the rapidity gap survival probability, as already observed in the previous 
H1 and ZEUS analyses. The survival probabilities measured with the 
'Low $E_T$' sample are in the range 0.43-0.65, depending on the DPDFs but 
always compatible within uncertainties, and also compatible with the one 
of the previous H1 analysis. The survival probabilities measured with the 
'High $E_T$' sample are in the range 0.44-0.79, 
that is slightly higher than in the 'Low $E_T$' case and closer to the 
ZEUS results, confirming a possible $E_T$ dependence of the suppression.

H1 data have also been compared to NLO calculations assuming factorization 
breaking and suppression of the resolved component only. The result is a 
much worse agreement in the $x_\gamma$ distribution. Awaiting for more
theoretical work, the experimental data seem to prefer an unexpected 
global suppression. 

\section{Leading neutron production: rescattering and absorption}

The measurement of leading neutron (LN) production at HERA is particularly 
interesting for studying rescattering effects in $ep$ collisions.
Although the production mechanism of leading neutrons is not 
completely understood, exchange models give a reasonable description
of the data. In this picture, the incoming proton emits a virtual particle 
which scatters on the photon emitted from the beam electron. 
In particular, one-pion exchange is a significant contributor to LN 
production for large values of $x_L$~\cite{early-data}, where $x_L$ is 
the fraction of the beam proton ener\-gy carried by the leading neutron. 
In exchange models, neutron absorption can occur through 
rescattering~[15-18], 
%rescattering~\cite{NikZak-KMR},  
which can thus be studied measuring neutron yields and distributions.

Figure~\ref{ZEUS-LN}, left panel, shows the measurement with the ZEUS 
data~\cite{ZEUS-LN} of the ratio of the normalized cross section for 
LN photoproduction as a function of $x_L$ to the same distribution in DIS. 
The ratio is below 1 at low $x_L$ values and rises with increasing $x_L$. 
As shown by the comparison with the theoretical curves, data are consistent
with a $\pi$-exchange model by D'Alesio and Pirner, which includes 
absorption via a geometrical picture~\cite{dAlesio}. In this picture, 
if the size of the $n-\pi$ system is small compared to the size of the 
photon, besides the $\pi$ also the neutron can scatter on the photon, 
escaping then detection, 
which can be seen as neutron absorption. Since the size of the virtual 
photon is inversely related to $Q^2$, more absorption is expected in 
photoproduction than in DIS. Moreover, since parametrizations of the 
pion flux in general show that the mean value of the $n-\pi$ separation 
increases with $x_L$, less absorption is expected at high $x_L$ than at 
low $x_L$. Both behaviours are confirmed by the data. Figure~\ref{ZEUS-LN} 
also shows that the data are reasonably consistent with a Regge-based 
model with multi-pomeron exchanges~\cite{NikZak}.

\begin{figure}
  \begin{center}
  \includegraphics[height=.30\textheight]{ZEUS-LN1}
  \hspace{1cm}
  \includegraphics[height=.40\textheight]{ZEUS-LN2}
  \caption{Left panel: Ratio of the normalized $x_L$ distributions for PHP and DIS.
           Right panel: a) Exponential slopes $b$ for DIS; b) difference 
           of the exponential slopes $b$ for PHP and DIS.}
  \label{ZEUS-LN}
  \end{center}
\end{figure}

The presence of a forward neutron tracker, a scintillator hodoscope installed 
in the calorimeter at a depth of one interaction length, allowed the 
measurement of neutron transverse momenta in the range $p_T \le 0.69~x_L$~GeV.
The $p_T^2$ distributions in the different $x_L$ bins are all compatible 
with a single exponential distribution. In Fig.~\ref{ZEUS-LN}a, right panel, 
is shown the measurement of the exponential slopes $b$ in DIS, while in 
Fig.~\ref{ZEUS-LN}b is presented the difference of the exponential 
slopes for photoproduction and DIS. Data are compared to a $\pi$-exchange 
model with enhanced neutron absorption based on multi-pomeron exchanges, 
which also accounts for the migration of neutrons in ($x_L, p_T^2$) after 
rescattering~\cite{KMR}. Including secondary exchanges ($\rho, a_2$) allows
the model to give a good description of the $b$ slopes. Finally, since the
size of the $n-\pi$ system is inversely proportional to the neutron $p_T$, 
rescattering removes neutrons with large $p_T$. Thus rescattering results
in a depletion of high $p_T$ neutrons in photoproduction relative to DIS.

A possible suppression has also been looked for by H1 in a sample of 
photoproduction dijet events with a leading neutron~\cite{H1-LN-JJ}. 
Jets were selected with transverse energies $E_T^{jet1}>7$~GeV and 
$E_T^{jet2}>6$~GeV.
No suppression has been observed since NLO calculations by Klasen and 
Kramer~\cite{KK-LN1}, which assume factorization, agree with the data 
if corrections to the hadron level are introduced. 
A more recent analysis by Klasen and Kramer~\cite{KK-LN2} concludes 
instead for the observation of factorization breaking.

\section{Summary and conclusions}

Diffractive dijet photoproduction has been studied at HERA to test possible 
QCD factorization breaking, expected for resolved-photon processes only, as 
in $p\bar{p}$ collisions at the Tevatron. Ra\-pi\-di\-ty gap survival probabilities 
have been measured in the range 0.4-0.9, higher than in $p\bar{p}$. Both H1 and 
ZEUS data, in contrast to the expectation, prefer a global suppression 
for direct and resolved components of the photon, with a possible 
$E_T$ dependence of the suppression factor.

Leading neutron data show the effects of rescattering through the neutron 
absorption observed at low $x_L$ and high $p_T$ in photoproduction  
with respect to DIS. $\pi$-exchange models with enhanced absorptive 
corrections, including migration and secondary exchanges, are able 
to describe the data. 
Absorptive effects may equally be described in terms of gap survival probability. 
It is worth to note that the HERA data can be used to get reliable 
predictions for the gap survival probability in $pp$ interactions~\cite{KMR2}, 
which is a crucial input to calculations of diffractive processes at the LHC.

%------------------------------------------------------------------------------
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\end{document}
