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\title{Demystifying generalized parton distributions}
\author{A.~Freund}
\institute{Institut f{\"u}r Theoretische Physik, Universit{\"a}t Regensburg, D-93040 Regensburg, Germany}

\date{\today}
%\medskip
%\noindent Keywords: Generalized parton distributions, deeply virtual 
%Compton scattering


\abstract{In this paper, I will explain in as simple and intuitive
  physical terms as possible what generalized parton distributions
  are, what new information about the structure of hadrons they convey
  and therefore what picture of the hadron will emerge. To develop
  this picture, I will use the example of deeply virtual Compton
  scattering (DVCS) and exclusive  meson electroproduction processes.
  Based on this picture, I will make some general predictions for DVCS
  and hard, exclusive meson production.
%\end{abstract}
\PACS{{11.10.Hi} {11.30.Ly} {12.38.Bx}}}

\maketitle

\section{Introduction}

Scientists have striven for centuries to unravel the dynamics and the
structures involved in the physical systems they have been
investigating, from large scale structures in our universe over
biological systems down to the smallest scales achievable in todays
high energy experiments. At these smallest scales the questions one is
trying to answer are ``What are the substructures of hadrons, what are
the dynamics of these substructures and what three dimensional picture
of hadrons is emerging ?''.

In the theory of strong or color interactions (QCD) parton
distribution functions (PDFs) encode the long distance or bound state
i.e.  non-perturbative information about hadrons. These PDFs are
precisely what we need in order to construct a dynamical as well as
geometrical picture of these objects.  Unfortunately, most high energy
experiments analyzing hadronic substructure study inclusive reactions
such as deep inelastic scattering (DIS) $e+p\to e+X$; in other words the
object they would like to study is destroyed in the reaction. Although
PDFs can be extracted from inclusive data, these functions are only
single particle distributions precisely because the target is
destroyed.  Hence, vital information about the three dimensional
distribution of substructure is lost and they can only give a one
dimensional picture of a hadron, as I will explain below.  In order to
gain insight into its three dimensional structure one has to measure
particle correlation functions which encode additional information on
how the object as a whole reacts to an outside probe.  Correlations in
hadrons refer to the dynamical influence during the reaction one or
more particles or partons found in a particular configuration in the
hadron have on one or more other partons found in a different
configuration inside the same hadron.  These particle correlation
functions can only be measured if the hadron stays intact in the
reaction since otherwise the dynamical relationship between the
different partons is destroyed. This can only be achieved if there are
no large color forces, responsible for a break-up, occurring during
the reaction.  This requirement simply means that such a reaction has
to be mediated by color neutral objects such as color singlets or, at
the very least, that color is locally saturated.  The experimental
signature of such a process is a so-called rapidity gap meaning that
the produced particles which are well localized in the detector, are
clearly separated from the intact final state hadron with no detector
activity in between the two.  There are many reactions of this kind
such as hard diffraction $e+p\to e+p+X$ or, in particular, deeply
virtual Compton scattering (DVCS) $e+p\to e+p+\gamma$
\cite{mrgdh,ji,rad1,jcaf,diehl97} which is the most exclusive example
of hard diffraction. Hard is meant here in the sense of the presence
of a large scale in the reaction such as a large momentum transfer
from probe to target.  In the perturbative QCD description of fully
exclusive hard reactions such as DVCS, we finally encounter the
objects we have been looking for: particle correlation functions. They
appear in the collinear factorization theorems of these reactions
\cite{jcaf,jfs} where collinear refers to the physics being dominated
by what is happening on the light cone neglecting internal transverse
momenta.  Factorization theorems state that, within QCD, one can
factorize the leading term in the cross section or scattering
amplitude of a particular hard reaction to all orders in perturbation
theory into a convolution of a finite or infra-red safe, hard
scattering function and an infrared sensitive, non-perturbative
function, a PDF. The other terms in the cross section or amplitude are
suppressed in the large scale of the reaction and can be disregarded,
at least in the limit of large scales. The hard scattering function is
particular to each reaction but computable to all orders in
perturbation theory.  The PDFs which are universal objects and can be
used in other hard, exclusive reactions, cannot be computed within
perturbative QCD save for their momentum scale dependence induced by
the renormalization of the theory. They are given, in a quantum field
theoretic language, as a Fourier transformation of a matrix element of
non-local, renormalized, operators. The key thing, in this context,
are the in and out states of these matrix elements. In inclusive
reactions such as DIS, the in and out states are the same since the
scattering amplitude can be directly related through the optical
theorem to a reaction which has the same in and out state. In hard,
exclusive reactions the in and out state differ, at least, in their
momenta. This is due to a finite momentum transfer in the $t$-channel
of the reaction onto the outgoing hadron, most commonly a nucleon.
These PDFs depend on more variables, namely those characterizing the
momentum difference of the in and out state, than the PDFs in
inclusive reactions which only depend one one momentum variable, apart
from the momentum scale dependence and therefore carry only one
dimensional information on the hadron. The behavior of these PDFs
called generalized parton distributions (GPDs)
\cite{mrgdh,ji,ji2,rad1,poly} under a change of their variables
encodes the response of the entire hadron, i.e. its substructure, to
the outside probe.  Therefore, these GPDs are particle correlation
functions and a complete mapping in all their variables through
experiments would give us for the first time a full three dimensional
picture of hadrons. Please note here that GPDs are by no means the
only particle correlation functions encountered in high energy
reactions. For example, so called, higher twist matrix elements in
DIS, which contain more than just two elementary operators, are
correlation functions since the momenta of the third, fourth etc.
operator in the matrix element depend on the momenta of the other
operators involved.  Furthermore, generalized distribution amplitudes
\cite{afex,polyrev,diehl2} encountered in exclusive $\gamma\gamma^*$ reactions
or transition GPDs in, for example, $e+p\to e+n+\pi^+$ are also
correlation functions (for a review see \cite{polyrev} and references
therein). Since the aim of this paper is not completeness but rather
an intuitive understanding of at least some of the physics involved,
we will only discuss afore mentioned GPDs and their physical
implications.

\begin{figure}
\centering
\mbox{\epsfig{file=fig.eps,width=8.5cm,height=4.5cm}}
\vskip+0.2in
\caption{a) DVCS graph, b) Bethe-Heitler with photon from final state lepton
and  c) with photon from initial state lepton.}
\label{dvcsfig}
\end{figure}

In order to directly extract GPDs from experiment one has to access
scattering amplitudes. Unfortunately, the cross section for exclusive
processes is the amplitude times its complex conjugate, $|A|^2$,
compared to inclusive processes where the cross section is just given
by the imaginary part of the amplitude. Though we are accessing both
the real and imaginary part of the amplitude in exclusive processes,
their phase structure i.e. each part individually, cannot be cleanly
separated unless there is a ``phase filter''.  A "phase filter" would
be a well understood process with which the exclusive reaction
interferes. Fortunately, there is such a process in the case of DVCS,
the QED Compton or Bethe-Heitler (BH) process (see Fig.\ 
\ref{dvcsfig}), first discussed in \cite{bordclose}. The interference
term between the two processes allows one to directly access both the
imaginary and real part of the DVCS scattering amplitude which
contain, {\it simultaneously}, four distinct structures, namely ${\cal
  H}$, an unpolarized amplitude with no hadron spin-flip, ${\cal
  \tilde H}$, a polarized amplitude with no hadron spin-flip, ${\cal
  E}$, an unpolarized amplitude with hadron spin-flip, ${\cal \tilde
  E}$, a polarized amplitude with hadron spin-flip.  The imaginary
part is accessible through the measurement of the beam polarization
asymmetry (longitudinal polarization in and opposite to the beam
direction) also called single spin asymmetry (SSA) and the real part
through the beam charge asymmetry (reversal of the lepton charge) or
simply called charge asymmetry (CA) \cite{diehl97,afmmlong,bemu4}.
This ``filtering'' has been aptly named ``nucleon holography'' by the
authors of \cite{bemunew}, since it employs the same principle of
interference as regular holography. Note that the nucleon spin-flip is
only made possible because of a finite momentum transfer $t$ onto the
final state nucleon as compared to DIS where $t=0$ and thus
there is no spin flip. This last statement means that ${\cal E}$ and ${\cal
  \tilde E}$ have no inclusive analog and hence contain unique
information on the nucleon only accessible in exclusive reactions!
This alone justifies an in depth study of these objects.

Note, furthermore, that whereas on the amplitude level we have
``nucleon holography'', on the deep structure level of the GPDs we
will, as I will explain in a later section, have ``nucleon
tomography'' \cite{fps} (see also \cite{ral1}), since for each value
of $x_{bj}$ and each value of $t$ we are studying the dynamics of a
slice of a nucleon and so, when we put all the slices together we
obtain a three dimensional image of a nucleon, as one obtains a three
dimensional image of a person when putting enough MRI pictures
together.

In Sec.\ \ref{picture}, I will define GPDS and then develop a
picture of what they mean in an intuitive way based on the example of
DVCS and exclusive meson production. In Sec.\ \ref{prediction}, I will
make general predictions about DVCS in particular and hard exclusive
reactions in general at facilities such as the planned EIC at BNL, the
proposed HERA III or a dedicated fixed target experiment. I will then
conclude in Sec.\ \ref{conc}.





\section{What is the physical picture GPDs convey?}
\label{picture}

\subsection{GPD Definition}

Whenever I will talk about GPDs in the following, I mean GPDs in a
nucleon, since I will mainly concern myself with hard electro
production reactions involving protons. However, the statements below
are much more general in nature and apply to any hadron target. For
brevity and ease of presentation, I will restrict myself to nucleons.

GPDs, first introduced in \cite{mrgdh} and later rediscovered in
\cite{ji,rad1}, are generally defined through the Fourier transform of
matrix elements of renormalized, non-local twist-two operators.
Twist-two operators are composite operators containing only two
elementary fields of the theory. These are situated at different
positions on a light ray making them non-local and are sandwiched
between {\it unequal} momentum nucleon states. The essential feature
of such two parton correlation functions, where the difference in the
in and out state is responsible for the correlations, is the presence
of a finite momentum transfer, $\Delta = p-p'$, in the $t$-channel ($p, p'$
are the initial and final state nucleon momenta). Hence, the partonic
structure of the nucleon is tested at {\it distinct} momentum
fractions.

There are many representations of GPDs (see
\cite{mrgdh,ji,rad1,gb1,ffgs}).  In this paper I will use the
off-diagonal PDFs, ${\cal F}^i (X,\zeta)$, defined by Golec-Biernat and
Martin \cite{gb1} and used in the numerical solution of the
renormalization group or evolution equations in \cite{afmmgpd} (for
other treatments see \cite{gb1,ffgs,rad3,bemuevolve}).
This representation will allow us a very intuitive insight into GPDs
as I will explain now.
 
The GPDs in this representation depend on the momentum fraction $X \in
[0,1]$ of the {\it incoming} proton's momentum, $p$, and the
skewedness variable $\zeta = \Delta^+/p^+$ (so that $\zeta = x_{bj}$ for DVCS and
meson production). This is analogous to the case of forward
PDFs where $x_{bj}$ is also defined with respect to the incoming
proton's momentum.

\begin{figure} 
\centering 
\mbox{\epsfig{file=HF.eps,width=7.5cm,height=5.5cm}} 
\vskip+0.2in
\caption{The relationship between ${\cal F}^q (X,\zeta)$, ${\cal F}^{\bar q} (X,\zeta) $ and $H^q (v,\xi)$ with $v \in [-1,1]$ and $X \in [0,1]$.} 
\label{figfdd} 
\end{figure}

For the quark case, the relationship of the quark and anti-quark
distributions, ${\cal F}^q (X,\zeta), {\cal F}^{\bar q} (X,\zeta) $, to the more
widely used $H^q (v,\xi)$ \cite{ji} where the GPDs are defined with
respect to the average of $p$ and $p'$ ($v \in [-1,1]$ and $\xi = \zeta/(2-\zeta) \in
[0,1]$) is shown in Fig.~\ref{figfdd}.  More explicitly, for $v \in
[-\xi,1]$:
\begin{equation}
{\cal F}^{q,a} \left(X = \frac{v+\xi}{1 + \xi},\zeta\right) = \frac{H^{q,a} 
(v,\xi)}{1-\zeta/2} \, ,
\label{curlyq}
\end{equation}
\noindent and for $v \in [-1,\xi]$
\begin{equation}
{\cal F}^{{\bar q},a} \left(X = \frac{\xi -v}{1 + \xi},\zeta\right) = 
-\frac{H^{q,a} (v,\xi)}{1-\zeta/2} \, .
\label{curlyqbar}
\end{equation}

The two distinct transformations between $v$ and $X$ for the quark and
anti-quark cases are shown explicitly on the left hand side of
eqs.(\ref{curlyq},\ref{curlyqbar}). There are two distinct regions:
the DGLAP region, $X > \zeta$ ($|v| > \xi$), in which the GPDs behave like
regular parton distributions and obey a generalized form of the so
called DGLAP equations for PDFs, and the so called ERBL region, $X<\zeta$
($|v| < \xi$), where the GPDs behave like distributional
amplitudes/meson wavefunctions and obey a generalized form of the ERBL
equations for distributional amplitudes (see
\cite{gb1,ffgs,afmmgpd,rad3,bemuevolve}). In the ERBL region, due to
the fermion symmetry, ${\cal F}^q$ and ${\cal F}^{\bar q}$ are not
independent anymore. In fact ${\cal F}^{q} (X,\zeta) = -{\cal F}^{\bar q}
(\zeta-X,\zeta)$, which leads to an anti-symmetry of the unpolarized quark
singlet distributions (summed over flavor $a$), ${\cal F}^S = \sum_a
{\cal F}^{q,a} + {\cal F}^{{\bar q},a}$, about the point $\zeta/2$ (the
non-singlet and the gluon, ${\cal F}^g$, which is built from $v
H^g_{Ji} (v,\xi)$, are symmetric about this point). For a detailed
review on the mathemtical properties see for example \cite{ji2}.

The operator definition of the ${\cal F}$'s is analogous to the one
for the $H$'s:
\begin{align}
&{\cal F}^q(X,\zeta) =\nonumber\\
&\int \frac{dz^-}{4\pi}~e^{-i(X-\zeta)p^+z^-}\langle 
p|\psi\left(-\frac{z^-}{2}\right){\cal P}\gamma^+{\bar 
\psi}\left(\frac{z^-}{2}\right)|p'\rangle\nonumber\\
&{\cal F}^g(X,\zeta) =\nonumber\\
&\int \frac{dz^-}{2\pi Xp^+}~e^{-i(X-\zeta)p^+z^-}\langle 
p|G_{+\nu}\left(-\frac{z^-}{2}\right){\cal 
P}G^{\nu}_+\left(\frac{z^-}{2}\right)|p'\rangle
\label{gpddef}
\end{align}
except that the Fourier conjugate momentum fraction and the momenta of
the in and out states are different compared to the symmetric
approach. Note that one could have, more conventionally, chosen $X$ to
be the Fourier conjugate momentum to $z^-$.  Since the crucial points
$X=0$ and $X=\zeta$ are related via the above symmetry arguments, it does
not matter whether one chooses one or the other. Nonetheless, the
variable $X-\zeta$ will prove convenient later on since it will be zero
for $X=\zeta$ which is a special point and signals that large, strictly
speaking infinite, light-like separations of the operators will play a
very important role in the GPD. As we will see below, this point in the
GPD is of paramount importance in hard exclusive reactions like DVCS
and meson production. In the symmetric representation \cite{ji},
the uniqueness of this point in terms of separation of operators on the
light ray is not as obvious and thus I prefer a representation here
where the uniqueness of this point is directly obvious. This does not
mean that one representation is better than another but rather that
sometimes one representation is more convenient to use than another.

\subsection{Why does DVCS help us understand GPDs better?}
\label{dvcsfirst}

The first question one has to answer is: Why is it that DVCS (see
Fig.\ \ref{dvcsfig}) is the cleanest process within which to measure
GPDs in a nucleon ? The reason for this is quite simple. With a real
photon, one has an elementary, point-like particle in the final state
rather than a bound state like a meson or an even more complicated
state like several mesons/hadrons or jets adding other unknown,
non-perturbative functions.  Note that the contribution of the
non-point-like part of the real photon wave function which would be
similar to a meson wavefunction, is power suppressed in DVCS
\cite{rad1,jcaf}. The factorization theorem for the DVCS scattering
amplitude \cite{rad1,jcaf} is merely a simple convolution of a hard
scattering function with only {\it one} GPD rather than with a GPD
plus another non-perturbative function as in meson production.

\begin{figure}
\centering
\mbox{\epsfig{file=hand.eps,height=4.5cm,width=8.5cm}}
\vskip+0.2in
\caption{LO handbag diagram for DVCS. Here $x_1=X$ and $x_2=X-\zeta$.}
\label{hand}
\end{figure}

The DVCS amplitude is ${\cal T}\simeq Im{\cal T}\propto {\cal F}(\zeta,\zeta,Q^2)$ in
leading order (LO) of perturbation theory (see for example
\cite{mrgdh,ji,rad1}).  This is true up to a $\zeta=x_{bj}\simeq 0.2-0.3$ even
when taking next-to-leading order (NLO) effects into account
\cite{afmmlong,bemu4,bemu2,afmmamp}. Hence, DVCS is dominated, at
least in a very broad region of phase space, by the crossover point
between the DGLAP and ERBL region. At this particular point in phase
space, $X=\zeta$, the parton line carrying momentum fraction $x_2$ in
Fig.\ \ref{hand} is becoming ``soft'' and all the momentum is carried
by the incoming quark with fraction $x_1=X=\zeta$.  Also note that the
quark connecting the two photon vertices, which is usually hard, is on
or almost on mass shell and carries only a large - momentum (see
again Fig.\ \ref{hand}).  Factorization for DVCS still holds in this
situation \cite{jcaf}, with the hard interaction now being the
photon-quark vertex, however, the point $X=\zeta$ in the GPD is indeed
rather peculiar. One should recall that the GPD is defined by a
Fourier transform of a non-local matrix element on a light ray and
that the Fourier conjugate variables are the light-ray separation
$z^-$ between operators and a momentum fraction variable.  Here this
is either $X$ or $X-\zeta$ (see eq.\ (\ref{gpddef})). This means then that
for $X-\zeta\to 0$ , $z^-\to\infty$ and therefore the operators have an infinite
separation on the light ray or more physically speaking that there is
bad resolution of the probed object in the - space-time direction.
This situation is analogous to inclusive DIS in the limit of
$x_{bj}\to0$.  Thus inclusive scattering at small $x_{bj}$ and DVCS up
to a large $x_{bj}$ in the valence region is dominated by the same
type of particle configurations with the only difference being that
the configurations in DVCS remain correlated since the proton stays
intact! What does the last statement mean from a physical point of
view?

\subsection{The physical picture of DVCS and its connection to GPDs}
\label{dvcspic}

The answer to the last question in Sec.\ \ref{dvcsfirst} is simply:
The particle configurations dominating the DVCS cross section are much
bigger, in their extension on the light ray, than the probed object
itself. Since the produced final state is a point-like particle these
particle configurations which one would normally call ``end-point''
contributions are not suppressed as in, for example, a meson wave
function describing an object of ``finite'' size \cite{meson}! Even
deep into the valence region, one is not probing the actual quark
structure or lowest Fock states of the proton but rather the QCD
vacuum as influenced by and interacting with the three valence quarks.
I will elaborate more on this later on. Note a caveat here, though:
The operators are not literally separated by an infinite light-like
distance, this would only be true in the limit $Q^2\to\infty$, but rather by
a distance which is inversely proportional to, at most, $X-\zeta =
\zeta\frac{\Lambda^2_{QCD}}{Q^2}$ (see \cite{afmmshort} for a derivation) which
acts as a lower bound and is motivated by considering possible
intermediate states in the reaction.  To be definite compare this to
DIS at $x_{bj} = 0.2$ and an initial, non-perturbative, scale
$Q^2_0=1~\mbox{GeV}^2$. $X-\zeta=X-x_{bj}$ would be then bounded by
$0.2\cdot(0.2)^2/1=0.08$, which is not too small but still $2.5$ times
smaller, and at $Q^2=5~\mbox{GeV}^2$, $12.5$ times smaller, than the
respective distances encountered in DIS. The basic claim is: {\it DVCS
  probes a larger, light-like distance than DIS for the same
  $x_{bj}$.}

\begin{figure}
\centering
\mbox{\epsfig{file=proton.eps,width=8.5cm,height=8cm}}
\caption{}
\label{figproton}
\end{figure}

\begin{figure}
\centering
\mbox{\epsfig{file=proton1.eps,width=8.5cm,height=8cm}}
\caption{}
\label{figproton1}
\end{figure}

There is a very intuitive picture of why the above interpretation is
indeed true and one is not really probing the valence structure of the
proton within DVCS.  Consider the following situation (Fig.\ 
\ref{figproton}): In the infinite momentum frame, the proton is moving
along the $+$ direction of the light cone i.e. in the positive $3$ or
+ $z$-direction with each valence quark carrying about a third of the
total momentum, $X\simeq1/3$.  If a valence quark were to be struck by a
virtual photon which has large + ($-x_{bj}P_+\simeq -P_+/3$) and -
($Q^2/2x_{bj}P_+\simeq 3Q^2/2P_+$) components with $P_+ \simeq O(Q)$, it would
then only have a large - component but a quasi zero + component since
$X\simeq x_{bj}$.  This means that the struck valence quark would have a
large momentum in the $-z$-direction, opposite to that of the other
two quarks, then radiate a real photon which moves in the
$-z$-direction, creating a rapidity gap necessary for this type of
reaction. After radiating the photon the quark then becomes ``soft''
i.e.  has no large momentum components. The transition matrix element
i.e.  the overlap integral, between an initial bound state with three
collinear or ``fast'' quarks to a final bound state with two collinear
quarks and one soft or ``slow'' quark is suppressed.  This is due to
the probability of two collinear and one ``soft'' quark forming a
proton in the final state being exponentially suppressed with the
relative light-like separation or in momentum space with the momentum
fraction of the ``slow'' quark.  However, DVCS is observed at large
$x_{bj}$ \cite{hermclas} and, therefore, the only alternative picture
(Fig.\ \ref{figproton1}) is the one where the virtual photon is not
scattering on a valence quark but rather on a $q$ or $\bar q$ from the
sea i.e. the QCD vacuum where either the $q$ or the $\bar q$ has a
large + momentum fraction matching the one from the virtual photon.
In other words, either the struck $q$ or the struck $\bar q$ starts to
move in the -z direction and then annihilates with a ``soft''
($X-\zeta\simeq0$) $\bar q$ or $q$ from the sea into a real photon. This photon
then has large - momentum i.e. moves along the -z direction as it
should.  None of the valence quarks are involved in the reaction and
therefore, it is not very difficult for the proton to stay intact.
Note, however, that the sea configurations at large $x_{bj}$ probed in
DVCS cannot directly be probed in inclusive DIS, since this would
involve too large a light-light separation as compared to the one
allowed in an inclusive reaction at large $x_{bj}$. One might argue
that this situation could violate the momentum sum rule, since one
would have more than $3$ quarks with large momentum fractions. This is
not true, though, since one is summing over {\it all} possible final
states in constructing regular PDFs and therefore, one allows for a
proper rearrangement of momentum between all the constituents in the
hadronization. To put it simply, {\it the questions one asks of the
  proton in DIS and DVCS are different}. In DIS in the valence region,
one wants to know if there are quarks with large momentum fractions in
the proton, in DVCS, on the other hand, one asks a much more specific
question of how the constituents in the proton must conspire to make
DVCS happen and therefore obtains a much more specific answer. Thus,
one of the first conclusions is that {\it there exist QCD vacuum
  configurations, the exact nature of which can only be probed in hard
  exclusive reactions like DVCS, which are encoded in a GPD in the
  region around $X\simeq\zeta$}.

\subsection{The origin of the asymmetric parton configurations}
\label{origin} 

The exact origin of these asymmetric $q\bar q$ pairs at high $Q^2$ for
any $x_{bj}$ is clear.  Perturbative evolution i.e. the change of the
GPD under a change in the renormalization or momentum scale, strongly
enhances the $X\simeq\zeta=x_{bj}$ region in the quark singlet GPD as compared
to the evolution effect in forward PDFs at the same $x_{bj}$ (see for
example \cite{ffgs,afmmgpd} for a detailed analysis of this
phenomenon). This enhancement effect is driven by the gluon GPD which
itself, however, is not as strongly enhanced as the quark GPD. This is
similar to the forward case where the gluon PDF drives the rise of the
quark sea, however not as strong as in the GPD case at $X\simeq\zeta$. Note
that the gluons responsible for the enhancement at higher $Q^2$
originate themselves from quarks at higher values of $X$ and lower
values of $Q^2$ i.e. are collinearly radiated from the valence quarks
in the proton which are found {\it not} in asymmetric configurations
but rather symmetric ones as encountered in DIS as discussed in
Sec.~\ref{dvcspic}. Thus evolution creates higher and higher Fock
states inside the proton and the valence quarks at low $Q^2$ become
more and more ``dressed'' at higher $Q^2$ (see Fig.\ 
\ref{figproton1}). One can say then that at low $Q^2$ and large $X\neq\zeta$
the quark GPD and the inclusive quark PDF should be the same since
they are both dominated by the same type of symmetric configurations
at low $Q^2$.  However, since the evolution is different for the GPD
and PDF the two functions will be different at higher $Q^2$.

The question of the origin of the asymmetric $q\bar q$ pairs at low
$Q^2$ where perturbative evolution is either not valid anymore or its
use is questionable, is more difficult to answer. They should be a
non-perturbative feature of the QCD vacuum rather than the valence
structure which is found in symmetric configurations. However, one
might expect that non-perturbative asymmetric $q\bar q$ configurations
would be suppressed since they would look like end-point
configurations in a meson. At large $x_{bj}$, the valence quarks of
the proton dominate and thus the expectation is that there are no such
$q\bar q$ pairs except the ones in the non-perturbative QCD vacuum
(see Fig.  \ref{figproton}). However, at very small $x_{bj}$ when one
enters the high gluon density or non-linear regime, one might still be
able to answer this question from a perturbative point of view. This
is true as long as the natural scale of the problem is the so-called
saturation scale $Q_s\sim1/x_{bj}^{2\lambda}$ with $\lambda\sim0.15-0.2$. Saturation means
that gluon recombination and other non-linear effects (see for example
\cite{weigert} and references therein) slow down the rapid increase of
the number of gluons in the nucleon as $x_{bj}$ decreases. This does
not mean that the photon has virtuality $Q_s^2$ but rather that the
internal scale of the gluon couplings in the system is $\alpha_s(Q^2_s)$,
which is small at sufficiently small $x_{bj}$ rather than $\alpha_s(Q^2)$
which at $Q^2 \leq 1~\mbox{GeV}^2$ is large.  This smallness of $\alpha_s$
allows a perturbative treatment of the gluonic degrees of freedom and
their evolution in $x_{bj}$. In this regime, one can therefore say
that the asymmetric $q\bar q$ configurations again originate from
perturbatively treatable gluon configurations, as at large $Q^2$,
though these configurations come from completely different regions of
phase space. Let me add a note of caution here as far as the
identification of high density gluons with a gluon GPD is concerned.
The non-linear, small $x_{bj}$ evolution does not rely on a twist
expansion but rather includes {\it all} twists. In fact higher twist
contributions provide the essential non-linearities in the evolution
equations.

Summarizing one can say that {\it the main source of the asymmetric
  $q\bar q$ configurations are gluons originating themselves either
  from symmetric valence configurations or are part of the
  non-perturbative QCD vacuum at small $x_{bj}$}

\subsection{Meson production and GPDs}

If one were to consider other reactions like meson production, the
situation, previously discussed, obviously changes since one does not
want to produce an elementary particle which is predominantly
point-like and therefore easily allows particle configuration of
``infinite'' extent in its creation but rather a bound state with a
``finite'' size.  As I will explain below, only some details are
adjusted, the overall picture, however, remains unaltered.

As in DVCS in LO of perturbation theory, the imaginary part of the
scattering amplitude in meson production is proportional to ${\cal
  F}(\zeta,\zeta,Q^2)$. Depending on the produced meson i.e. its
quantum numbers, a particular combination or particular types of GPDs
are probed in contrast to DVCS where only the quark singlet is
directly probed (at least in LO). Therefore, the mesons act as a ``GPD
filter''. For example $\pi^0$ production, being a pseudo scalar, singles
out the polarized quark GPD in an unpolarized reaction \cite{jfs}!

Consider the following picture of meson production, again in the
infinite momentum frame: The proton moves along the + direction of the
light cone and is struck by a highly virtual, longitudinally
polarized, so as to maintain factorization, photon again having large
+ and - light cone momenta. In order to produce a meson there has to
be the exchange of at least one gluon or equivalently the splitting of
a gluon into a $q\bar q$ pair. These can be, in keeping with the
factorization theorem for meson production \cite{jfs}, either hard or
collinear to the proton i.e. the $+$ direction, or collinear to the
produced meson i.e.  the $-$ direction.  In order not to be power
suppressed the collinear regions have to be included either in the GPD
or the meson wave function as collinear splitting i.e. evolution. We
are now particularly interested in the situation when the struck
collinear quark in the proton (valence or not) carries the initial
momentum fraction $X\simeq x_{bj}$ with another accompanying
quark/anti-quark being ``soft'' i.e. $X-\zeta\simeq0$ as in DVCS.  This
situation can only be achieved through the exchange of at least one
hard gluon.  I will elaborate on this in detail below. Also, one can
probe the gluon GPD directly, as in for example $J/\psi$ production
\cite{jfs}. This corresponds to the case when a collinear gluon
carrying momentum fraction $X\simeq x_{bj}$, splits into a $q\bar q$ pair,
one of which interacts with the virtual photon and the other one with
a second, ``soft'', gluon. They then go on to form the meson in the
final state.  In both instances, one directly probes the point
$X=\zeta=x_{bj}$ in the GPD associated with a large light-like separation
of operators as in DVCS. Let me discuss the quark case in more detail
first and then speak about the gluon case.

When the collinear quark, which will eventually interact with the
virtual photon, radiates a hard gluon, the quark itself becomes hard.
There are several momentum configurations for the quark and the gluon.
For example, the gluon could have large - and + components i.e.  the
hard quark also has large + and - momentum components, decaying into a
$q\bar q$ pair where the $\bar q$ has a large - component and the $q$
a large + component. The $\bar q$ will then form the meson with the
quark after the interaction with the virtual photon, since this will
then have only a large - component, exactly as in DVCS.  The $q$ with
large + momentum replaces the struck one in the proton, keeping it
intact.  Since the gluon is hard, the $q\bar q$ originates from the
same space-time point where the virtual photon strikes the collinear
quark, the hard quark is also in the same space-time point as the
gluon and hence it seems to an observer, as if nothing really happened
to the struck collinear quark.  However, this is not the situation we
are interested in. We need the situation where the + component of the
hard gluon is small i.e. it is on or almost on mass shell, in exact
analogy to the quark connecting the two photon vertices in DVCS in
Fig.\ \ref{hand} for the situation $X\simeq\zeta$ as explained in
Sec.~\ref{dvcspic}.  The quark remains hard and at the photon-quark
vertex, the struck quark starts to move along the $-$ direction, since
the + components of the photon and quark cancel. The gluon now splits
either into a $q\bar q$ pair with the anti-quark carrying large $-$
momentum or it hits a ``soft'' anti-quark in the proton transferring
its large $-$ momentum. In both instances the soft quark will be
associated with the proton. In order to keep the proton intact, the
struck collinear quark could not have been a valence quark since there
would be no other collinear i.e.  ``fast'' quark to replace it, only a
``soft'' i.e.  slow one. Thus it must have come from the sea. In this
way, the situation is analogous to the DVCS case.  Hence, the
interpretation of the exact particle configurations probed in the GPD
in meson production compared to DVCS for $X\simeq x_{bj}$ does not change
for the case of quark scattering. What happens when we have a
collinear gluon as mentioned above? The situation is quite similar to
the quark case. The collinear gluon splits into a $q\bar q$ pair with
both of them being hard.  Either of them interacts with the virtual
photon and, with the + momentum components canceling once more, ending
up with only large - momentum. The other hard $q$/$\bar q$ radiates a
gluon with large + momentum and ends up also with only large -
momentum. This situation is analogous to the just discussed quark case
and once more not the one we are looking for. To produce an asymmetric
configuration, again either the hard $q$ or $\bar q$ has to go on or
near mass shell only carrying large - momentum, implying that the
initial, collinear gluon has a + momentum fraction $X\simeq\zeta$, and then
radiating a ``soft'' gluon required to match the color of the initial
gluon. Again we have the same situation as in the quark case and
therefore the same interpretation except that we have now the gluon
GPD rather than the quark GPD at $X=\zeta$ as already stated above. The
origin of these asymmetric gluon configurations is the same as the one
for the asymmetric $q\bar q$ configurations explained in
Sec.~\ref{origin}.  In fact the asymmetric gluons can be the source of
asymmetric $q\bar q$ through collinear splitting. Also it should be
noted that the gluon configurations entering DVCS are more symmetric
than in meson production. This is so because gluons enter DVCS mainly
through evolution, at least in LO, in contrast to the meson case where
they enter directly and, therefore, the configurations from which
asymmetric configurations arise, are themselves more
symmetric as explained in Sec.~\ref{origin}. The only instance when
the gluon GPD directly enters DVCS is suppressed by an extra power of
$\alpha_s$ compared to the quark contribution i.e. in NLO and arises when
the collinear gluon with $X\simeq\zeta$ splits as, in the meson case, into a
hard $q\bar q$ pair interacting with the virtual photon and
immediately i.e. out of the same space-time point, radiating a real
photon and a ``soft'' gluon $X-\zeta\simeq0$ necessary for color matching. 

The fact that the interpretation about dominant particle
configurations encoded in the GPD does not change in going from DVCS
to meson production means that GPDs are indeed universal objects as
proven to all orders in the factorization theorems \cite{jcaf,jfs}.
Again, I would like to stress that this picture of dominant particle
configurations in meson production is only valid if the
imaginary part of the scattering amplitude is larger than the
real part. In fact, for the real part where the regions $X>>x_{bj}$
and $X<<x_{bj}$ are very important, valence quarks and symmetric gluon
configurations do play an important role. This is due to the fact that
the region of phase space, where the exchanged gluon or $q$ and $\bar
q$ is hard, becomes large. It is also clear that, as the mass of the
produced vector meson or $Q^2$ increases, it starts to act in a
similar fashion to a point particle i.e directly emerges from the hard
scattering space-time point.

In summary on can say that {\it GPDs and the information about
  particle configurations encode in them are universal since the same
  particle configurations are probed in DVCS and meson production.}

\subsection{The $t$ and $Q^2$ dependence of GPDs}

Up until now, I have neither talked about the role of the
$t$-dependence nor of the precise meaning of $Q^2$ or more precisely
the renormalization scale $\mu^2$. In \cite{diehl1} a beautiful
exposition of the physical meaning of these two variables for GPDs has
been given which I will only briefly reiterate: The scale $\mu^2$
defines from what scale, or, in space-time, from what resolution in
the transverse plane onwards one can speak of several or just one
parton.  In other words, the better the resolution $\sim1/\mu$ of the
probe, the more partons or substructure of one parton one can observe
(see Fig.\ \ref{figproton1}). As $\mu$ defines the resolution of the
probe in the transverse plane, the $t$-dependence gives the relative
transverse position of the probed parton in the DGLAP region or the
probed $q\bar q$ pair in the ERBL region with respect to the proton (see
Fig.\ \ref{figproton}). If $\mu^2\simeq-t=$ several $\mbox{GeV}^2$, the exact
meaning between resolution and position becomes lost, including the
above simple picture of DVCS and meson production, since the
hierarchy of scales necessary for a factorized approach to these
processes is lost and thus its simple physical picture.

In contrast, in the case of $\mu^2=Q^2>>-t$, one has a very interesting
picture emerging (see Fig.\ \ref{figproton1}): since $t$ is up to
corrections of $O(M_N^2\zeta^2)$, which are very small, equal to
$-(p_{\perp}-p'_{\perp})^2$, the relative transverse momentum difference
between initial and final state, small $t$ corresponds to a large
distance in the transverse plane from the proton ``center'' and large
$t$ to a small distance. Here ``center'' is meant with respect to the
relative transverse positional difference between initial and final
state. This implies that the asymmetric $q\bar q$ configurations I
have been talking about are clearly separated from the ``center'' of
the proton, if they are non-perturbative in nature since they will be
situated in the ``pion cloud'', for lack of a better word, at
the ``edge'' of the proton. However, the perturbative configurations
having a resolved size of $O(1/Q) $ in the transverse plane, will be
closer to the ``center'' since they are associated with the valence
quarks through evolution and those are situated well within the proton
radius, $r_p\sim1$ Fermi. In fact, the helicity non-flip valence quark
density should peak near the center, since the $s$-wave component
should dominate this quantity. Similar to the charge distribution in a
hydrogen atom which in the s-wave state is spheric and largest at the
center. Higher states like p- and d-waves correspond to high angular
momentum excitations which in the proton can come only through
momentum transfer in the $t$ channel. Thus $E$ and $\tilde E$ which
probably correspond to such p- and d-wave states, have no inclusive
analog since such states can simply not be excited at $t=0$.

The emerging three dimensional picture of the asymmetric $q\bar q$ or
gluon configurations as well as their symmetric ``parents'' can be
stated as follows: These configurations are basically located
``inside'' of the proton with the non-perturbative configurations
towards the edge and the perturbative configurations more towards the
``center'' but very spread out on the light cone.  For example, at the
average $t$ of HERMES of about $-0.2~\mbox{GeV}^2$, these
configurations are located only about $0.4$ Fermi away from the
``center'', clearly ``inside'' the proton radius $r_p$ with respect to
the transverse position (only for a $t < -0.04~\mbox{GeV}^2$ would
they be located ``outside'' of the proton).  Since we restrict our
considerations to the region of $-t\leq 1~\mbox{GeV}^2$, the relative
distance to the ``center'' is never closer than about $0.2$ Fermi.

\section{Going beyond the nucleon: Qualitative predictions from the above 
picture}
\label{prediction}

The above considerations are not limited to a nucleon target but are
also valid for example for a nuclear target. There are some
interesting consequences emerging from the above considerations: The
fact that the same large light-like distances are involved in
conventional PDFs for $x_{bj}\to0$ and in GPDs for $X\simeq\zeta$ together with
the observed enhancement of this region through perturbative
evolution, suggests that for the same $x_{bj}$ of the process, {\it
  GPDs probe the configuration content of the proton and its effect on
  the QCD vacuum at relatively smaller momentum fractions than PDFs}.
This is borne out by the analysis carried out in \cite{fmsnew} which
shows that a good GPD input capable of describing all available DVCS
data \cite{hermclas,h1,zeus2} in a NLO QCD analysis is obtained by
using conventional forward PDFs at a momentum fraction $X$ shifted to
a smaller value by an amount of $O(\zeta)$.  This in turn implies
\begin{itemize}
\item Earlier onset of saturation effects in DVCS observables
  dominated by the imaginary part of the scattering amplitude compared
  to inclusive observables. This is particularly true for nuclear
  targets since saturation is a strongly $x_{bj}$ dependent phenomenon
  \cite{sgb}! A concrete prediction would be the presence of geometric
  scaling in the $\gamma^*p$ DVCS cross section in either $ep$ or $eA$
  scattering up to an $x_{bj}$ where it normally would break down in
  $F_2^{p,A}$.
\item Nuclear shadowing corrections for DVCS should set in at larger
  values of $x_{bj}$ as compared to the inclusive case. Moreover, at
  comparable values of $x_{bj}$, the nuclear shadowing corrections
  should be stronger in DVCS compared to DIS. Since nuclear shadowing
  is only a weak function of $x_{bj}$ except for the transition region
  between $0.01<x_{bj}<0.1$ (see for example \cite{fgms}), the
  enhancement effect should only be visible in this region.
\item Since varying $t$ changes the relative transverse position at
  which the target is probed, it will allow one to scan through the
  ``grey'', where non-linear perturbative QCD is still applicable, and
  the ``black'' area or total absorption region of the target. In
  these two regions, the target behavior will be qualitatively
  different and this difference should be reflected in different
  geometric scaling curves for different values of $t$. I do not claim
  here that DVCS in the black disc limit is very different from DIS in
  this limit, quite on the contrary \cite{fgms1}. However, the $t$
  dependence allows one to discern between two regions of different
  target behavior.
\end{itemize}



These predictions could be verified at the future EIC with its high
luminosity both for $ep$ and $eA$ scattering, as well as at HERA III
with nuclei in the HERA ring or a dedicated, high luminosity, fixed
target experiment.

Furthermore, the fact that one cannot really probe the valence
distributions at $X=\zeta$ and leave the proton intact, leads one to
conclude that as $X\to\zeta$ the actual unpolarized valence quark GPD should
become very small or tend to zero at the input scale. Evolution will
change this and the valence GPD will start to grow also at $X=\zeta$ since
higher and higher Fock states will be present in the valence GPD at
higher $Q^2$ as previously discussed in Sec.\ \ref{origin}. This
prediction is supported by two facts.  First, calculations both in the
chiral-quark-soliton model \cite{wgp} and in the constituent quark
model \cite{scopetta} show that the valence GPD becomes either small
or vanishes at $X=\zeta$.  Secondly, since $X=\zeta$ corresponds to large
light-like separations as in the inclusive case for $x_{bj}\to0$, one
might expect that the valence quark GPD vanishes at $X=\zeta$ as the
forward valence quark PDF vanishes for $x_{bj}\to0$. Experimentally,
this could be verified through a flavor separation in $\nu p$ DVCS at
the COMPASS experiment.

One can also make some interesting statement about the $t$-dependence.
The slope of it in DVCS at low $Q^2\sim$ a few $\mbox{GeV}^2$ should be
larger than the one for light meson production for the same
kinematics. Whereas at large $Q^2$, the two slopes should be the same
as stated in factorization theorems \cite{jcaf,jfs}. The reason for
this is quite simple in the region of $x_{bj}$ and $Q^2$ where the
imaginary part of the amplitude dominates. Consider the following (see
Fig.\ \ref{figproton1}): For DVCS at low $Q^2$ the main origin of the
required $q\bar q$ configurations will not yet be perturbative
collinear gluon splitting into $q\bar q$ pairs as at large $Q^2$ but
rather some non-perturbative property of the QCD vacuum. In the
transverse plane, the perturbatively created asymmetric configurations
have to be rather ``close'' to their ``parents'', the valence quarks.
The ``parent'' statement simply means that in the convolution of the
evolution kernel with the PDF, necessary for solving the evolution
equations, the main contribution to the integral originates from large
values of $X$ in the valence region.  ``Close'' in this context refers
to the gluon as well as the $q\bar q$ pair being part of the valence
quarks substructure. This substructure is only resolved in the
transverse plane once the probing scale $1/Q$ is smaller than the
scale $1/\mu_0$, where $1/\mu_0$ is the scale at which one can still speak
of the ``parent'' as a single valence quark.  For example , if
$\mu_0=1~\mbox{GeV}$ and $Q\simeq 1.4~\mbox{GeV}$ then one would say that up
to a transverse resolution of about $0.2$ Fermi one can speak of just
one valence quark but that at the experimental $Q$ above one already
resolves the valence quarks substructures up to a relative distance of
about $0.14$ Fermi. Having said that, the non-perturbative
configurations on the other hand should be more or less situated in
the ``pion cloud'' i.e. more towards the edge of the proton in the
transverse plane. This means that, at low $Q^2$, as one approaches the
``center'' of the proton i.e.  as $t$ increases, the number of
asymmetric configurations suitable to facilitate DVCS should drop
since the non-perturbative configurations sit at the edge rather than
in the ``center'', while at the same time the perturbative
configurations as part of the substructure of the valence quarks, are
not as well resolved yet and hence less than at large $Q^2$.
Therefore, the number of asymmetric $q\bar q$ configurations is less
at larger $t$ than at smaller $t$, and as a consequence, the cross
section drops faster with the increase in $t$ at low $Q^2$ than at
large $Q^2$.  The above means that the slope of the $t$-dependence
should be $Q^2$ dependent and decrease as $Q^2$ increases. Also, the
slope should only weakly depend on $x_{bj}$ as long as $x_{bj}$ is in
the region where the imaginary part of the amplitude dominates, since
the above observation does not strongly depend on $x_{bj}$. These two
statements are borne out both by the observations made in
\cite{fmsnew} where a $Q^2$ dependent but $x_{bj}$ independent slope
gives very good agreement, within the experimental errors, between the
DVCS data and NLO QCD calculations and by experimental measurements
(see for example \cite{brho} and references therein). Why should the
slopes for DVCS and meson production at low $Q^2$ be different
though?

In fact, it is almost not so, except that meson production, as pointed
out above, is sensitive to both asymmetric $q\bar q$ and gluon
configurations. Since one now has a larger number with which to
facilitate the production, one is not as dependent as in DVCS on the
number of asymmetric $q\bar q$ configurations in the proton. In
consequence, the slope of the $t$-dependence should be smaller for the
meson case as compared to the DVCS case at low $Q^2$. At large $Q^2$,
where there are now a lot of suitable $q\bar q$ configurations, even
close to the ``center'', originating almost exclusively from gluons,
this additional possibility becomes less important.  The conclusion
for low $Q^2$ is supported by the findings in \cite{fmsnew} where a
larger slope for DVCS at relatively low $Q^2\sim 2-4~\mbox{GeV}^2$ was required to
obtain a good agreement between data and theory than in the case of,
for example, $\rho^0$ production \cite{brho}.


\section{Conclusions}
\label{conc}

To summarize once more, I have presented a concise, simple and
intuitive picture of what GPDs mean in the sense of carrying new
information about the three dimensional structure of nucleons compared
to regular parton distributions or form factors. To achieve this I
have developed a simple picture through which type of particle
configurations encoded in the GPDs, DVCS and meson production proceed
and that these configurations can only be correctly identified in
exclusive reactions. However, they originate mainly from symmetric
valence configurations through perturbative evolution. Furthermore,
based on this picture, I conclude that the unpolarized valence quark
GPD at a non-perturbative scale should be either small or vanish near
the crossover point between ERBL and DGLAP region. I have also made
verifiable, qualitative predictions for DVCS and meson
production in $ep$ and $eA$ collisions such as an early onset of
saturation, different geometric scaling curves for different $t$
values, determining the sizes of the ``grey'' and ``black'' areas of
the target, stronger nuclear shadowing corrections in the transition
region $0.01<x_{bj}<0.1$ and a difference in the slope of the
$t$-dependence at low $Q^2$ between the two processes, using the above
picture. These predictions/conclusions are already partially supported
by both experimental as well as theoretical observations.

I would like to thank Nikolai Kivel, Mark Strikman and Christian
Weiss, for useful discussions as well as Moskov Amarian, Vladimir
Braun, Einan Gardi, Mark Strikman, and Heribert Weigert for a careful
reading of the manuscript.  This work was supported by the
Emmi-Noether grant of the DFG.

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\end{thebibliography}

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