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\begin{document} \title{Gap-Survival Probability and Rescattering 
in Diffraction at the LHC } 
\author{Michele Arneodo$^1$\protect\footnote{\ \ speaker}}
\institute{$^1$Universit\`a del Piemonte Orientale, I-28100 Novara, and 
INFN-Torino, I-10125 Torino, Italy,\\ 
on behalf of the CMS Collaboration}
\maketitle
\begin{abstract}
The feasibility is discussed of rediscovering hard diffraction at the LHC
with the first 10-100~pb$^{-1}$ collected by the CMS detector. Studies are
presented of single-diffractive di-jet production in $pp$ 
collisions at $\sqrt{s}=14$~TeV, single-diffractive $W$ boson production, 
and exclusive $Y$ photoproduction. The prospects of assessing the 
rapidity-gap survival probability are discussed.
\end{abstract}

\section{Introduction}

A substantial fraction of the total proton-proton cross section is due to
diffractive reactions of the type $pp \to XY$, where $X$, $Y$ are either
protons or low-mass states which may be a resonance or a continuum state.
In all cases, the energy of the outgoing protons or the states $X$, $Y$ is
approximately equal to that of the incoming beam particles, to within a
few per cent.  The two (groups of) final-state particles are well
separated in phase space and have a large gap in rapidity between them
(``large rapidity gap", LRG). Diffractive hadron-hadron scattering can be
described within Regge theory (see e.g.~\cite{Collins:1977jy}).  In this
framework, diffraction is characterised by the exchange of a specific
trajectory, the ``Pomeron'', which has the quantum numbers of the vacuum
and notably no colour (hence the LRG).

The effort to understand diffraction in QCD has received a great boost 
from the seminal studies of diffractive $p\bar{p}$ collisions with the UA8 
experiment at CERN~\cite{Brandt:1992zu} and more recently from studies of 
diffractive events in $ep$ collisions at HERA and $p\bar{p}$ collisions at 
Fermilab (see 
e.g.~\cite{BaronePredazzi,Alekhin:2005dy,Albrow:2006xt,Arneodo:2005kd, 
solano,gallinaro,Jung:2009eq} and references therein).  A key to this 
success are factorisation theorems for $ep$ diffractive scattering, which 
allow to express the cross section in terms of diffractive parton 
distribution functions and generalised parton distributions. These 
functions can be extracted from measurements and contain information about 
small-$x$ partons that can only be obtained in diffractive processes. To 
describe hard diffractive hadron-hadron collisions is more challenging 
since factorisation is broken by rescattering between spectator partons.  
These rescattering effects, often quantified in terms of the so-called 
``rapidity-gap survival probability"~\cite{Bj, Frankfurt}, are of interest 
in their own right because of their relation with multiple parton 
scattering.

This paper summarises some recent feasibility studies carried out by the 
CMS Collaboration, aiming at ``rediscovering" hard-diffraction with the 
early LHC data and at quantifying the rapidity-gap survival probability at 
LHC energies by means of the single-diffractive (SD) reaction $pp \to Xp$, 
in which $X$ includes either a $W$ boson or a di-jet system. This 
reaction is sensitive to the diffractive structure function (dPDF) of the 
proton, specifically its gluon component (see e.g.~\cite{BaronePredazzi}). 
It is also sensitive to the rapidity-gap survival probability, $\langle 
|S^2|\rangle$; to first approximation, the cross section is directly 
proportional to $\langle |S^2|\rangle$, independent of kinematics. This 
process has been studied at the Tevatron, where the ratio of the yields 
for SD and inclusive di-jet production has been measured to be 
approximately 1\%~\cite{Abe:1997rg,Abbott:1999km,gallinaro}. Theoretical 
expectations for LHC are at the level of a fraction of a per 
cent~\cite{Cox:2000jt,Khoze:2008cx,Frankfurt,Gotsman:2005rt,Khoze:2006gg,Miller:2006bi}. 
There are, however, significant uncertainties in the predictions, notably 
due to the uncertainty of $\langle |S^2|\rangle$. While there is some 
consensus that $\langle |S^2|\rangle \simeq 
0.05$~\cite{Gotsman:2005rt,Khoze:2006gg} for hard diffractive processes at 
LHC energies, values of $\langle |S^2|\rangle$ as low as 0.004 and as high 
as 0.23 have been proposed~\cite{Miller:2006bi}. Exclusive photoproduction 
of $\Upsilon$ mesons, $pp \to p \Upsilon p$ is also briefly discussed. 
This reaction is sensitive to the structure of the proton, notably the 
generalised (or skewed) gluon density, but the rapidity-gap survival 
probability should in this case be close to unity~\cite{ryskin}.

The CMS apparatus is described in detail elsewhere~\cite{:2008zzk}. Two
experimental scenarios are considered here. In the first, no forward
detectors beyond the CMS forward calorimeter HF are assumed. In this case
the pseudo-rapidity coverage is limited to $|\eta|<5$. In the second,
additional coverage at $-6.6<\eta<-5.2$ is assumed by means of the CASTOR
calorimeter. HF and CASTOR are briefly discussed in the next section. 

For more details on the analyses presented here, the reader is referred 
to~\cite{antonio,felipe,jonathan}.

\section{The HF and CASTOR calorimeters}

The forward part of the hadron calorimeter, HF, is located 11.2~m from the
interaction point. It consists of steel absorbers and embedded radiation
hard quartz fibers, which provide a fast collection of Cherenkov light.
Each HF module is constructed of 18 wedges in a nonprojective geometry
with the quartz fibers running parallel to the beam axis along the length
of the iron absorbers. Long (1.65~m) and short (1.43~m) quartz fibers are
placed alternately with a separation of 5~mm. These fibers are bundled at
the back of the detector and are read out separately with phototubes.

CASTOR is a sampling calorimeter located at $\simeq  14$~m from the 
interaction point, with tungsten plates as absorbers and fused silica
quartz plates as active medium. The plates are inclined by $45^{\circ}$
with respect to the beam axis. The calorimeter has the shape of an
octagonal cylinder.  Particles passing through the quartz emit Cherenkov
photons which are transmitted to photomultiplier tubes through air-core
light-guides. The electromagnetic section is 22 radiation length deep with
2 tungsten-quartz sandwiches, and the hadronic section consists of 12   
tungsten-quartz sandwiches. The total depth is 10.3 interaction lengths.
The calorimeter read-out has azimuthal and longitudinal segmentation (16
and 14 segments, respectively). There is no segmentation in $\eta$.


\section{SD \boldmath{$W$} and di-jet production}

The analyses described here are planned for the first LHC data, and can be 
carried out on data samples with integrated luminosities of 
10-100~pb$^{-1}$ and with negligible pile-up. A centre-of-mass energy of 
14~TeV is used. No near-beam proton tagger is assumed, and the selection 
of diffractive events has therefore to rely on the observation of a 
rapidity gap.

The single diffractive signals were simulated with the {\sc pomwig} Monte 
Carlo generator~\cite{Cox:2000jt}. Non-diffractive events were simulated 
with {\sc pythia}~\cite{Sjostrand:2006za} or {\sc 
madgraph}~\cite{Alwall:2007st}.

\subsection{Event selection}

\subsubsection{$W\to \mu \nu$ production}

The selection of the events with a candidate $W$ decaying to $\mu \nu$ is 
the same as that used for inclusive $W \to \mu \nu$ 
production~\cite{inclusive}.  Events with a candidate muon in the 
pseudo-rapidity range $|\eta|>2.0$ and transverse momentum $p_T<25$~GeV 
were rejected, as were events with at least two muons with $p_T>20$~GeV. 
Muon isolation was imposed by requiring $\sum{p_T}<3$~GeV in a cone with 
$\Delta R<0.3$.  The transverse mass was required to be $M_T>50$~GeV. The 
contribution from top events containing muons was reduced by rejecting 
events with more than 3 jets with $E_T>40$~GeV (selected with a cone 
algorithm with radius of 0.5) and events with acoplanarity ($\zeta=\pi- 
\Delta{\phi}$) between the muon and the direction associated to $E_T^{\rm 
miss}$ greater than 1 rad.

\subsubsection{Di-jet production}

At the trigger level, events were selected by requiring at least 2 jets
with average uncorrected transverse energy greater then 30~GeV.  Offline,
jets were reconstructed with the SiSCone5~\cite{Salam:2007xv} algorithm  
and jet-energy scale (JES) corrections were applied. At least two jets  
with $E_T~>~55$~GeV were required. All plots shown in this paper are for 
energy-corrected jets.

\subsubsection{Diffractive selection}

The left panel of Fig.~\ref{fig-etagen} shows the generated 
energy-weighted $\eta$ distribution for stable particles in 
single-diffractive and non-diffractive $W$ production events; only 
diffractive events with the scattered proton at positive rapidities (the 
peak at $\eta \gtap 10$) are included in the plot. Diffractive events 
have, on average, lower multiplicity both in the central region and in the 
hemisphere that contains the scattered proton, the so-called ``gap side", 
than non-diffractive events. The right panel of Fig.~\ref{fig-etagen} 
shows the multiplicity distribution in the central tracker for $|\eta|<2$ 
after the di-jet selection cuts. Diffractive events have a multiplicity 
distribution that peaks at low values, unlike that of non-diffractive 
events. Diffractive event candidates were therefore selected on the basis 
of the multiplicity distribution in the central tracker, in the HF as well 
as in CASTOR.

\begin{figure}[bth]
\begin{center}
\includegraphics[width=0.49\textwidth]{genEnergyvsEta_wLabels_new}
\includegraphics[width=0.50\textwidth]{trackMult_new}
\caption{
Left panel:
Generated energy-weighted $\eta$ distribution for stable particles
(excluding neutrinos) in diffractive ({\sc pomwig}, continuous line) and
non-diffractive ({\sc pythia}, dashed line) $W$ production events. 
The HF coverage and that of the CASTOR calorimeter are also shown. The
diffractive events were
generated with the gap side in the positive $\eta$ hemisphere.
The peak at $\eta \protect\gtap 10$ is due to the scattered proton.
The area under the histograms is normalised to unity.
Right panel:
Track multiplicity distribution in the central tracker after the $W$
selection cuts for diffractive ({\sc pomwig}, continuous line) and  
non-diffractive ({\sc pythia}, dashed line) events. The track
corresponding to
the $\mu$ candidate is excluded.
The area under the histograms is normalised to unity.
}
\label{fig-etagen}
\end{center}
\end{figure}

The gap side was selected as that with lower energy sum in the HF. This
selection was made for all events though the concept is relevant only for
diffractive events. 

In addition, for the di-jet analysis, the two leading jets were required 
to 
be between $-4<\eta<1$ for events with the gap side at positive rapidities 
and $-1<\eta<4$ for events with the gap side at negative rapidities.  
When CASTOR is used, only events with the gap on the negative side are 
considered, since CASTOR will be installed on that side first. The 
rapidity separation between the two leading jets was required to be 
$\Delta \eta < 3$.

Finally, a cut was applied on the track multiplicity in the central
tracker. The plots shown in this paper were obtained with maximum
multiplicity for $|\eta|<2$, $N^{\max}_{\rm track}$, of 1, 5 and no cut
at all. For the events passing this cut, multiplicity distributions in
the HF and CASTOR calorimeters were studied, from which a diffractive
sample can be extracted.

\section{Results}

\subsection{SD \boldmath{$W\to \mu \nu$} production}

Figure~\ref{HF-HF-NT5} shows the HF tower multiplicity for the low-$\eta$ 
(``central slice", $2.9 <\eta<4.0$) and high-$\eta$ HF (``forward slice", 
$4.0<\eta<5.2$) regions for events with central tracker multiplicity 
$N_{\rm track} \le 5$.  In the figure, the top left and top right plots 
show the distributions expected for the diffractive $W$ events with 
generated gap in the positive and negative $Z$ direction, 
respectively\footnote{The $Z$ axis is along the beam direction.}; they 
exhibit a clear peak at zero multiplicity. Conversely, the non-diffractive 
$W$ events have on average higher multiplicities, as shown in the bottom 
left plot; this distribution is interesting in its own right as it is 
sensitive to the underlying event in non-diffractive interactions.  
Finally, the bottom right plot shows the sum of the {\sc pomwig} and {\sc 
pythia} distributions -- this is the type of distribution expected from 
the data. The diffractive signal at low multiplicities is visible. The 
significance is highest when the $N_{\rm track}$ cut is most strict (see 
\cite{antonio}).

\begin{figure}[bth]
\begin{center}
\resizebox{!}{7cm}{
\includegraphics{nHFLowEtaSlicevsnHFFwdSlice_nTrkMax_5_100pb_new} }
\caption{
Low-$\eta$ (``central slice") vs high-$\eta$ (``forward
slice") HF tower multiplicity distributions for events with
track multiplicity in the central tracker $N_{\rm track} \le 5$.
Top left: {\sc pomwig} events with gap generated in the positive $Z$
direction.
Top right: {\sc pomwig} events with gap generated in the negative $Z$
direction.
Bottom left: {\sc pythia} events.
Bottom right: Sum of the {\sc pythia} and {\sc pomwig} distributions.
}
\label{HF-HF-NT5}  
\end{center}  
\end{figure}

The HF tower multiplicity vs CASTOR $\phi$ sector multiplicity was also 
studied for the gap side.  Since CASTOR will be installed at first on the 
negative side of the interaction point, only events with the gap on that 
side (as determined with the procedure discussed above)  were considered. 
The CMS software chain available for this study did not include 
simulation/reconstruction code for CASTOR; therefore, the multiplicity of 
generated hadrons with energy above a 10~GeV threshold in each of the 
CASTOR azimuthal sectors was used. Figure~\ref{HF-CASTOR-NT5} shows plots 
analogous to those of Fig.~\ref{HF-HF-NT5} for the combination of HF and 
CASTOR. The top plots show the {\sc pomwig} distributions; the few events 
in the top left plot are those for which the gap-side determination was 
incorrect.  The signal to background ratio improves greatly with respect 
to the HF only case since a wider $\eta$ coverage suppresses 
non-diffractive events, where the gap is due to statistical fluctuations 
in the rapidity distribution of the hadronic final-state. Here as well, 
the significance is highest for small central tracker multiplicity cuts 
but still acceptable even when no cut is applied (see~\cite{felipe}). The 
plots also indicate that if only the CASTOR multiplicity is used, the 
diffractive signal is further enhanced. The accepted events with zero 
multiplicity in both the HF and CASTOR, i.e. the events with a candidate 
rapidity gap extending over HF and CASTOR and $N_{\rm track} \le 5$, 
typically have $\xi \ltap 0.01$, and thus populate the region where 
Pomeron exchange is expected to dominate over sub-leading exchanges. Here 
$\xi$ indicates the fractional momentum loss of the proton. The $\xi$ 
coverage for different $N_{\rm track}$ cuts is similar and so is that of 
the HF only case.

A sample of diffractive events can be obtained by using the 
zero-multiplicity bins, where the diffractive events cluster and the 
non-diffractive background is small. As an example, when an integrated 
effective luminosity for single interactions of 100~pb$^{-1}$ becomes 
available, SD $W$ production can then be observed with $\mathcal{O}(100)$ 
signal events if CASTOR is used.


\begin{figure}[bth]
\begin{center}
\resizebox{!}{7cm}{
\includegraphics{nHFvsnCASTOROnlyMinus_nTrkMax_5_100pb_new} }
\caption{HF tower multiplicity
vs CASTOR sector multiplicity distribution for events with
track multiplicity in the central tracker $N_{\rm track} \le 5$.
Top left: {\sc pomwig}  events with gap generated in the positive $Z$
direction
(opposite
side to CASTOR).
Top right: {\sc pomwig}  events with gap generated in the negative $Z$
direction
(same
side as CASTOR).
Bottom left: {\sc pythia} events.
Bottom right: Sum of the {\sc pythia}  and {\sc pomwig}  distributions.
}
\label{HF-CASTOR-NT5}
\end{center}
\end{figure}

\subsection{SD di-jet production}

Figure~\ref{evidence-track} shows the HF-only and HF vs CASTOR gap-side 
multiplicity distributions for different cuts on the central tracker; 
these plots are the equivalent of the bottom right ones of 
Figs.~\ref{HF-HF-NT5} and~\ref{HF-CASTOR-NT5}. The size of the enhancement 
in the zero-multiplicity bins relative to the rest of the distribution 
increases monotonically when the $N^{\max}_{\rm track}$ cut is tightened 
-- the opposite of what would happen if the enhancement were a statistical 
fluctuation. The relative size of the enhancement also increases when 
going from the HF-only coverage to the HF plus CASTOR coverage: again, a 
wider $\eta$ coverage suppresses non-diffractive events, where the gap is 
due to statistical fluctuations in the rapidity distribution of the 
hadronic final-state. Plots of this type, along with others presented 
in~\cite{felipe}, can be used to demonstrate the existence of a SD di-jet 
signal in a data-driven, model-independent way.

Once the existence of the signal is established, here again, a sample of 
diffractive events can be obtained by using the zero-multiplicity bins, 
where the diffractive events cluster and the non-diffractive background is 
small. For example, when an integrated effective luminosity for single 
interactions of 10~pb$^{-1}$ becomes available, SD di-jet production can 
then be observed with $\mathcal{O}(300)$ signal events.

\begin{figure}[bth]
\begin{center}
\includegraphics[angle=90,width=0.32\textwidth]{inf_nHFSlices}
\includegraphics[angle=90,width=0.32\textwidth]{five_nHFSlices}
\includegraphics[angle=90,width=0.32\textwidth]{zero_nHFSlices}
\includegraphics[angle=90,width=0.32\textwidth]{inf_nHF_nCASTOR}
\includegraphics[angle=90,width=0.32\textwidth]{five_nHF_nCASTOR}
\includegraphics[angle=90,width=0.32\textwidth]{zero_nHF_nCASTOR}
\caption{HF-only (top row) and HF vs CASTOR (bottom row) multiplicity
distributions
for signal plus background events with no cut on the track multiplicity in
the central tracker (left column), $N^{\max}_{\rm track}=5$ (central
column) and $N^{\max}_{\rm track}=1$ (right column).   
}
\label{evidence-track}
\end{center}
\end{figure}

\subsubsection{Sensitivity to the value of the rapidity-gap survival
probability}
\label{gapsurv}

Table~\protect\ref{2d-0-0-bins} gives the expected SD di-jet signal and 
background yields in the zero-multiplicity bins also for values of the 
rapidity-gap survival probability $\langle |S|^2\rangle=0.004$ and 
$\langle |S|^2\rangle=0.23$. In the former case, the observable signal 
becomes marginal, even with the widest possible $\eta$ coverage 
(HF+CASTOR).  Conversely, $\langle |S|^2\rangle=0.23$ gives rise to a very 
prominent signal, also in the HF-only case.

In order to assess the significance of these yields, a preliminary, 
conservative estimate of the systematic uncertainties was obtained by 
summing in quadrature the contributions due to the sensitivity to the HF 
threshold ($\pm 15\%$), the jet-energy scale ($\pm 30\%$), the use of 
different jet algorithms ($\pm 20\%$) and a $+ 30\%$ contribution due to 
proton dissociation (see~\cite{felipe}), yielding a $^{+50}_{-40}$\% 
systematic uncertainty.

Observation of an event yield of $ 236 \pm 15 (\rm stat.) ^{+120}_{-90} 
({\rm syst.})$ (cf. Table~\ref{2d-0-0-bins}, $N^{\max}_{\rm track}=1$ and
HF+CASTOR) or $409 \pm 20 (\rm stat.) ^{+200}_{-160} ({\rm syst.})$ (cf.
Table~\ref{2d-0-0-bins}, $N^{\max}_{\rm track}=5$ and HF+CASTOR) would
exclude $\langle |S|^2\rangle=0.004$, for which no signal is visible. 

\begin{table}[htbp]
\caption{Diffractive and non-diffractive di-jet event yields
expected with
(1) zero HF multiplicity,
(2) zero HF and CASTOR multiplicity, as a function of  $N^{\max}_{\rm 
track}$.
The signal yields are given for $\langle |S|^2\rangle=0.05$ (nominal) as
well as  $\langle |S|^2\rangle=0.004$ and $\langle |S|^2\rangle=0.23$.
The uncertainties are computed as $\sqrt{N}$.
}
\begin{center}
\setlength{\tabcolsep}{0.8ex}
\begin{tabular}{|l|r|r|r|r|r|}
\hline
\multicolumn{1}{|r|}{$N_{HF}$ = 0} & $N^{\max}_{\rm track}$ &
$N_{\rm diff}$&
$N_{\rm diff}$&
$N_{\rm diff}$& $N_{\rm non-diff}$  \\
                                  &  &
$\langle |S|^2\rangle=0.05$&
$\langle |S|^2\rangle=0.004$&
$\langle |S|^2\rangle=0.23$ &  \\
 \hline
 & no cut & $1047 \pm  32$ &$84 \pm 9$  & $4816 \pm 69$ & $1719 \pm 41$  
\\\hline
 & 5      & $803  \pm 28 $ &$64 \pm 8$  & $3694 \pm 61$ & $943  \pm 31$   
\\ \hline
 & 1      & $362  \pm 19 $ &$29 \pm 5$  & $1665 \pm 41$ & $276  \pm 16$  
\\ \hline
\multicolumn{1}{|r|}{$N_{\rm HF} = 0$, $N_{\rm CASTOR} = 0$} & & & &  &   
\\
\hline
 & no cut & $504 \pm 22$ &$40 \pm 6$ & $2318 \pm 48$  & $67 \pm 8$
\\\hline
 & 5      & $409 \pm 20$ &$33 \pm 4$ & $1881 \pm 43$  & $31 \pm 6$
\\\hline
 & 1      & $236 \pm 15$ &$19 \pm 4$ & $1086 \pm 33$  & $8  \pm 3$
\\\hline
\end{tabular}
\end{center}
\label{2d-0-0-bins}
\end{table}

\section{$\Upsilon$ photoproduction}

An important term of comparison for the early determination of the 
rapidity-gap survival probability is exclusive $\Upsilon$ photoproduction, 
$pp \to p \Upsilon p$, in which one of the protons radiates a quasi-real 
photon which interacts, via colour-singlet exchange, with the other 
proton. This reaction has been studied at HERA, and can be investigated at 
CMS with the early LHC data \cite{jonathan}. A few hundred events events 
are expected in 100 pb$^{-1}$. This process is interesting in its own 
right as a window on the generalised parton distribution functions of the 
proton. In addition, the rapidity-gap survival probability in this case is 
expected to be close to unity~\cite{ryskin}. The yield of exclusive 
$\Upsilon$ photoproduction should thus be essentially unsuppressed -- and 
can be used to further constrain the understanding of the rapidity-gap 
survival probability.

\section{A look at the future: near-beam proton taggers}

CMS (and ATLAS~\cite{campanelli}) will be able to carry out a forward and 
diffractive physics program also at the highest LHC instantaneous 
luminosities if the FP420 program~\cite{Albrow:2008pn} is approved. FP420 
at CMS aims at instrumenting the $\pm 420$ m region. This addition will 
allow measuring forward protons with values of the fractional momentum 
loss of the proton $0.002 \ltap \xi \ltap 0.02$.

An articulate joint CMS-TOTEM research program is also 
foreseen~\cite{Albrow:2006xt, ferro}, with coverage in the region $0.02 
\ltap \xi \ltap 0.2$, complementary to that of FP420.

\section{Summary and outlook}

In summary, CMS has detailed, quantitative plans to re-discover hard 
diffraction with the early data by means of the rapidity-gap signature. 
The simple measurement of event yields may give early information on the 
rapidity-gap survival probability. Also, the shape of the background is 
sensitive to the underlying event in non-diffractive interactions. Once a 
hard-diffractive signal is established, the plan is to move on to the 
measurement of the ratio of diffractive to inclusive yields \`a la CDF and 
D0. Significant improvements are expected as soon as forward proton 
coverage becomes available via TOTEM and FP420.


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  %%CITATION = HEP-PH/0511060;%%

\bibitem{Khoze:2006gg}
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  %%CITATION = EPHJA,C56,39;%%

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\bibitem{inclusive}  CMS Collaboration, CMS PAS EWK-07-002 (2007).

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\bibitem{ferro} F. Ferro, these Proceedings.

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