\subsection{Generalised parton distributions and hard exclusive reactions}
\label{subsec:physics.gpd.intro}

Generalised Parton Distributions~\cite{Mueller:1998fv,Ji:1996ek,
Ji:1996nm,Radyushkin:1996ru,Radyushkin:1997ki} provide a
novel and comprehensive description of the nucleon's partonic structure
and contain a wealth of new information. In particular, they embody both
nucleon electromagnetic form factors, \ie\ ratios of the observed
elastic electron scattering to that predicted for a point-like nucleon,
and Parton Distribution Functions (PDFs) measured in DIS, \ie\
parton number and helicity densities. Very importantly, GPDs provide a
novel description of the nucleon as an extended object, referred sometimes
to as 3-dimensional ``nucleon tomography''~\cite{Burkardt:2000za,
Burkardt:2002hr,Burkardt:2004bv}, which correlates
(transverse) spatial and (longitudinal) momentum degrees of freedom of
quarks and gluons. Moreover, the evaluation of GPDs may for the first time
provide an insight into orbital momenta of quarks and gluons, another
fundamental property of the nucleon \cite{Ji:1996ek,Ji:1996nm}.

GPDs, just like ordinary PDFs, describe the structure of the nucleon
independently of the specific reaction by which the nucleon is probed,
\ie\ they are expected to be universal quantities. The mapping of
nucleon GPDs, which very recently became one of the key objectives of
high-energy nuclear physics, requires a comprehensive programme of measuring 
various hard exclusive processes in a broad kinematic range, in particular
Deeply Virtual Compton Scattering (DVCS). In addition, Deeply Virtual Meson
Production (DVMP) is expected to add independent and complementary information.

The kinematic variables on which GPDs depend can be illustrated using the
``handbag'' diagram shown in Fig.~\ref{fig:physics.gpd.handbag_diagram} which
describes the DVCS process at leading twist in the Bjorken limit ($Q^2 
\rightarrow \infty$ for fixed $x_B$ and $t$, \ie\ $|t|/Q^2$ small).
%
\begin{figure}[h]
\begin{center}
\includegraphics*[width=6cm]{gpd_intro_handbag}
\end{center}
\caption{Handbag diagram for the DVCS process at leading twist.}
\label{fig:physics.gpd.handbag_diagram}
\end{figure}
%
GPDs depend on the photon virtuality $Q^2=-q^2$, the total four-momentum 
squared $t=(p-p')^2=(q-q')^2$ transferred between initial and final nucleon 
states, and on $x$ and $\xi$. The
latter two variables represent respectively average and half the difference 
between the initial and final longitudinal momentum fractions of the nucleon, 
carried by the parton throughout the process. While in Deeply Inelastic 
Scattering (DIS), the momentum fraction $x$ carried by the struck parton is
identified with the Bjorken variable $x_B=\frac{Q^2}{2 \, p \cdot q}$, 
in the hard exclusive DVCS 
and DVMP processes $x$ is an internal variable that is integrated over 
in a convolution of the given GPD with a kernel describing the hard 
virtual-photon quark interaction (see, \eg\ Eq.~(\ref{eq:reH_LO})). As such, 
$x$ must not be identified with $x_B$, which in these kinematics is related 
to the skewness as $\xi \simeq x_B/(2-x_B)$ in the Bjorken limit. Hence the 
skewness coverage of a given leptoproduction measurement is approximately 
given by the range it covers in $x_B$.

Based on the factorisation theorem~\cite{Collins:1998be}, the short-distance 
information specific to the virtual-photon quark interaction can be separated 
from the long-distance information about nucleon structure contained in the 
GPDs. In the handbag approximation of DVCS, GPDs can be understood as 
describing the quantum mechanical amplitude for ``kicking out'' a parton of
the fast moving nucleon by the virtual photon and ``putting it back'' with a 
different momentum after it has radiated the real photon.

The DVCS final state is identical to that of the competing Bethe--Heitler (BH) 
(bremsstrahlung) process and hence both processes interfere on the level of 
amplitudes. The resulting interference term permits access to certain 
linear combinations of GPDs, which makes hard exclusive single-photon 
leptoproduction a powerful tool to study GPDs. Complementarily, DVMP will 
allow independent access to different bilinear combinations of GPDs.

The GPDs $H^f$ and $\widetilde{H}^f$ $(f=u,d,s,g)$, which describe the case of 
nucleon-helicity conservation in the scattering process, include as limiting 
cases the well-measured parton density and helicity distributions $q^f$ and 
$\Delta q^f$, respectively. The case of nucleon-helicity flip is described by 
the GPDs $E^f$ and $\widetilde{E}^f$, for which there are no such limiting 
cases. Gluon GPDs enter in DVCS only beyond leading order in $\alpha_s$ 
(LO), analogous to DIS.

GPDs attracted much attention after it was shown that the total angular 
momentum of a given parton species $f$ is related to the 2$^{\mathrm{nd}}$ 
moment of the sum of the two GPDs $H^f$ and $E^f$ via the Ji 
relation~\cite{Ji:1996ek}
%
\be
J^f(Q^2)=\; \frac{1}{2} \; \; \lim_{t \rightarrow 0} \int_{-1}^1 \; \dx \; x \;
      \left[H^f(x,\xi,t,Q^2)+E^f(x,\xi,t,Q^2)\right],
\label{Ji-relation}
\ee
%
which holds for any value of $\xi$.
This finding in 1997 was one of two major reasons that strongly boosted
experimental and theoretical activities towards determinations of GPDs as
it opened the way to constrain the contributions $J^q$ from total quark
angular momenta to the nucleon's spin budget
%
\be
\frac{1}{2}=\sum_{q=u,d,s}J^q(Q^2) + J^g(Q^2),
\ee
where $J^g$ is the total gluon contribution to the nucleon spin. The Ji
relation Eq.~(\ref{Ji-relation}) is the only known---though not 
straightforward---way 
to constrain the quark total angular momentum contributions to the 
nucleon spin budget. Progress in its evaluation, on the one hand, will have to 
rely on global analyses of experimental data for various exclusive processes in
the broadest possible kinematic range. Here, sufficient knowledge about the
kinematic dependencies of GPDs in the full kinematic range is essential
as \eg\ unknown nodes in an unmeasured region would render reliable GPD 
fits impossible. On the other hand, QCD simulations on an Euclidean lattice 
have been used to determine the second moments of the 
GPDs $H$ and $E$ that enter the Ji relation. Using chiral perturbation theory
to extrapolate to the physical pion mass, the results for $u$ and $d$ quarks
are $J^u=0.236\pm0.006$ and $J^d=0.002\pm0.004$. The uncertainties are only 
statistical, as some of the systematic uncertainties are very hard to quantify
~\cite{Hagler:2007xi,Brommel:2007sb,Bratt:2010jn}. No practical way is known 
to evaluate the Ji relation for partons other than $u$ and $d$ quarks.

A particularly simple physical interpretation for GPDs as probability density
exists in the limiting case $\xi=0$ where the parton carries the same 
longitudinal momentum fraction $x$ in initial and final state and hence the 
momentum transfer $t \equiv -\Delta^2 = -\Delta_L^2 - \Delta_\perp^2$ is purely transverse,
$t = -\Delta_\perp^2$. In this case, in analogy to the case of form factors, 
the Fourier transform of the $-\Delta_\perp^2$ dependence of the GPD 
$H^f(x,0,-\Delta_\perp^2)$ for fixed
$x$ describes the spatial distribution of partons of species $f$ carrying the
longitudinal momentum fraction $x$, with respect to their transverse distance
$\bT$ from the centre of momentum of the nucleon (impact-parameter
representation)~\cite{Burkardt:2002hr}
%
\be
q^f(x, \bT) = \int \frac{\di^2\DT}{(2\pi)^2 }
e^{-i\DT\cdot \bT} H^f(x,0, - \DT^2).
\label{IPD}
\ee
%
The ``three-dimensional'' impact-parameter-dependent parton distribution
$q^f(x, \bT)$ can be interpreted as providing a set of ``tomographic images'' of
the nucleon, as illustrated by the cartoon shown in
Fig.~\ref{fig:physics.gpd.Jlab_tomography}. Nucleon tomography continues 
attracting great attention, more than 300 publications on both experimental and
theoretical aspects have appeared over the last 10~years.

\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=12cm]{gpd_intro_Jlab_tomography}
\end{center}
\caption{
Nucleon tomography: (a) The Fourier transform of the $-\Delta_\perp^2$ 
dependence of the GPD $H^f(x,0,-\Delta_\perp^2)$ for fixed $x$ describes the
distribution of the transverse distance $b\equiv|\bT|$ of partons carrying the
fraction $x$ of the nucleon's longitudinal momentum $P$, from the centre of
momentum of the nucleon. (b) Sketch of tomographic views of the transverse 
spatial parton distribution in the nucleon at certain parton longitudinal 
momentum fractions $x$. Figure adapted from Ref.~\cite{JLab12GeV}.
}
\label{fig:physics.gpd.Jlab_tomography}
\end{figure}
In the study of the transverse structure of the nucleon, there are two quantities 
of particular importance. The first one is the GPD at $\xi=0$ which does have a 
probabilistic interpretation. It is related via Fourier transform to the 
distribution in the impact-parameter $b\equiv|\bT|$, which represents the 
transverse distance between struck quark and centre of momentum of the whole
nucleon~\cite{Burkardt:2002hr}.
Lattice calculations can determine the expectation value of $b$, averaged over 
$x$ with different weights $x^n$~\cite{:2003is}. The transverse distance between 
the struck parton and the centre of momentum of the spectator system is given by 
$r_{\perp} = b/(1-x)$ and provides an estimate of the overall transverse size of 
the nucleon. One expects that its expectation value remains finite due to 
confinement, which implies that the expectation value of $b$ must tend to zero 
for $x \to 1$.

The second important quantity is the GPD at  $x=\xi$. It has no 
probabilistic interpretation but nevertheless its Fourier transform is connected 
to the distance $r_{\perp}$ between struck parton and spectator
system~\cite{Diehl:2002he,Burkardt:2007sc}.
At leading order in $\alpha_s$ (LO), the corresponding average 
$\langle r_\perp^2 (x) \rangle$  can be directly obtained from the imaginary 
part of amplitudes of exclusive processes. At small $x_B$, where amplitudes are
predominantly imaginary, one has the relation  
$\langle r_\perp^2 (x_B) \rangle \approx 2 \cdot B(x_B)$ if the exclusive 
cross section is parametrised as 
$\frac{\di\sigma}{\di t} \propto \exp(-B(x_B)|t|)$.

Results on the transverse size of the nucleon derived from measurements of the 
experimental electromagnetic form factor correspond to an average over the 
longitudinal momentum fractions $x$ carried by the struck parton. The measured
transverse r.m.s.\ charge radius of the nucleon is about 0.7~fm. The present understanding 
of the $x$ dependence of the ``partonic'' transverse size of the nucleon is 
sketched in Fig.~\ref{fig:physics.gpd.Jlab_tomography}. At very large 
$x$, for $x \rightarrow 1$, the width of the distribution $q^f(x, \bT)$ in 
$b\equiv|\bT|$ is expected to vanish since the active parton becomes the centre 
of momentum of the entire nucleon. In the valence quark region, denoted by 
$x \sim 0.3$ in Fig.~\ref{fig:physics.gpd.Jlab_tomography}, we expect to mainly 
``see'' the core of valence quarks. 
Lattice calculations find average values of $\langle { b}  (x) \rangle $ 
decreasing from about 0.5~fm to about 0.25~fm for typical values of the 
longitudinal momentum fraction $x$ ranging from about 0.2 to about
0.4~\cite{Negele:2004iu,Hagler:2007xi,Bratt:2010jn}. The low-$x$ range, denoted 
by ``$x \sim 0.003$'' in Fig.~\ref{fig:physics.gpd.Jlab_tomography}, is dominated 
by sea quarks and gluons. This 
domain has been investigated by the \hera\ collider experiments 
($x_B = 10^{-2} \ldots 10^{-3}$). An average transverse proton radius 
$\langle r_\perp (x_B) \rangle $ of 0.65$\pm$0.02~fm was determined~\cite{:2007cz}
from the $t$-slope $B(x_B)$ of the DVCS cross section.  Although this is a
different quantity than the average $b$ value determined on the lattice, an
increase of about 20--30\% in transverse nucleon size compared to the valence 
region is expected in a chiral-dynamics approach that is applicable for 
$x \ll M_\pi/M_p \approx 0.15$, with $M_\pi$ ($M_p$) the pion (proton) 
mass~\cite{Strikman:2003gz,Strikman:2009bd}. This model predicts a corresponding
increase in the transverse size of the nucleon due to the ``pion cloud'' which is 
expected to enhance the gluon density with decreasing $x$. The variation in 
transverse nucleon size vs. $x$ was also investigated using model-dependent GPD 
fits based on most recent \hera\ DVCS measurements, confirming its growth with 
decreasing $x$~\cite{Kumericki:2009uq}. However, absolute numbers from these fits
still vary in the range 20--30\%  due to model uncertainties.  We emphasise 
that there exists no direct measurement of 
$\langle r_\perp (x_B) \rangle $ in the $x_B$ range above $10^{-2}$.

The transverse distribution of partons with $x > 0.01$ also plays
an important role in the theory and phenomenology of high-energy 
$pp/\bar p p$ collisions with hard processes at the Tevatron and LHC.
Because events with hard processes require binary parton--parton 
collisions, their centrality dependence and underlying event structure
are generally very different from those of minimum-bias 
events \cite{Frankfurt:2003td}. 
Knowing the transverse distribution of the partons in the colliding protons
from independent measurements through exclusive $ep/\mu p$ scattering,
one can calculate the centrality dependence and model the spectator
interactions in much more detail. This is particularly important
for estimating the rate of multijet production in $pp$ collisions 
at LHC, which is expected to be high and represents an important
background in $pp$ events with new physics processes. Precise
information about the transverse distribution of quarks and gluons with 
$x > 0.01$ will be highly valuable for developing next-generation Monte Carlo 
generators for multijet events (for a recent summary see Ref.~\cite{Jung:2009eq}), 
and for revealing dynamical multiparton correlations which enhance
multiple hard processes beyond their geometric probability
\cite{Frankfurt:2004kn,Frankfurt:2008vi}. The transverse distribution of
partons also determines the rapidity gap survival probability 
\cite{Frankfurt:2006jp} in central exclusive diffraction 
$pp \rightarrow p + ({\rm gap}) + H + ({\rm gap}) + p$, which is being 
considered as a clean channel for Higgs boson production at the 
LHC~\cite{Albrow:2008pn} (for a recent summary see Ref.~\cite{Deile:2010mv}. 
The parton momentum fraction probed in production of a 
Standard Model Higgs boson of $M_H \sim 10^{2}~\GeV$ at the LHC at central rapidity are 
$x_{1, 2} \sim M_H/\sqrt{s} \sim 10^{-2}$, which is exactly the region
where their transverse distribution will be measured in exclusive
processes at \compass.

The transverse spatial distribution of partons measured in exclusive 
$ep/\mu p$ scattering is an essential ingredient in studies of the 
regime of high parton densities in high-energy QCD (``saturation'') 
at \rhic, LHC and a future $ep/eA$ collider \cite{Frankfurt:2005mc}.
The transverse distribution of quarks and gluons at $x > 0.01$ defines 
the initial conditions for the non-linear QCD evolution equations
\cite{Balitsky:1995ub,Kovchegov:1996ty,JalilianMarian:1
996xn} describing the approach to the saturation 
regime~\cite{McLerran:1993ka,McLerran:1993ni}. We note that the transverse 
profile of partons in the nucleon also figures in the estimates of the 
nuclear enhancement of the saturation scale \cite{Kowalski:2007rw}, 
which determines the region
of applicability of these concepts in $AA$ collisions at \rhic\ and LHC.
The dynamically generated saturation scale $Q_{\rm sat}$ is proportional 
to the density of small-$x$ gluons in transverse area, and thus directly 
sensitive to the spatial distribution of colour charges in the 
initial condition at larger $x$. 
Present studies of saturation in the QCD dipole model 
\cite{Kowalski:2003hm,Rogers:2003vi} rely on the 
limited data from exclusive $\jpsi$ production at HERA 
\cite{Aktas:2005xu,Chekanov:2004mw} for 
information on the transverse distribution of partons at $x < 0.01$. 
Measurements of exclusive processes with \compass\  would map
the spatial distribution of partons in the unknown region $x > 0.01$,
and separate the distributions of gluons and singlet quarks
by comparing $\jpsi$ and $\gamma$ production (DVCS), and thus
provide new information about the initial conditions
for small-$x$ evolution.