# Selected topics in diffraction with protons and nuclei: past, present, and
future

L Frankfurt1,3, V Guzey2, A Stasto3, M Strikman3 1 Sackler School of Exact
Sciences, Tel Aviv University, Tel Aviv, 69978, Israel  2 National Research
Center “Kurchatov Institute”, Petersburg Nuclear Physics Institute (PNPI),
Gatchina, 188300, Russia  3 Department of Physics, Penn State University,
University Park, PA 16802, USA

###### Abstract

We review a broad range of phenomena in diffraction in the context of
hadron–hadron, hadron–nucleus collisions and deep inelastic
lepton–proton/nucleus scattering focusing on the interplay between the
perturbative QCD and non-perturbative models. We discuss inclusive diffraction
in DIS, phenomenology of dipole models, resummation and parton saturation at
low $x$, hard diffractive production of vector mesons, inelastic diffraction
in hadron–hadron scattering, formalism of color fluctuations, inclusive
coherent and incoherent diffraction as well as soft and hard diffraction
phenomena in hadron–hadron/nucleus and photon–nucleus collisions. For each
topic we review key results from the past and present experiments including
HERA and the LHC. Finally, we identify the remaining open questions, which
could be addressed in the continuing experiments, in particular in photon-
induced reactions at the LHC and the future Electron-Ion Collider (EIC) in the
US, Large Hadron electron Collider (LHeC) and Future Circular Collider (FCC)
at CERN.

††: Rep. Prog. Phys.

###### Contents

  1. 1 Introduction
  2. 2 Inclusive diffraction in DIS
    1. 2.1 Introduction
    2. 2.2 Collinear factorization in diffractive DIS
    3. 2.3 Diffractive fits to HERA data
    4. 2.4 Diffractive dijet and charm production
    5. 2.5 Predictions for EIC, LHeC, FCC-eh
  3. 3 Dipole model in diffraction
    1. 3.1 Introduction
    2. 3.2 Diffraction in dipole model
    3. 3.3 Models for the dipole cross section
    4. 3.4 Distribution of dipole sizes
    5. 3.5 Diffractive structure function from dipole model and higher twists
  4. 4 Low $x$, resummation and parton saturation
    1. 4.1 BFKL Hard Pomeron
    2. 4.2 Resummation at low $x$
    3. 4.3 Cross section taming
    4. 4.4 Color Glass Condensate and Balitsky–Kovchegov equation
  5. 5 Hard diffractive production of vector mesons
    1. 5.1 Space–time evolution and factorization of high energy processes
    2. 5.2 The role of GPDs in hard diffractive production of vector mesons
    3. 5.3 Modeling finite-$Q^{2}$ effects
    4. 5.4 Elastic photoproduction of $J/\psi$: from HERA to LHC
    5. 5.5 Lessons and open problems
    6. 5.6 Hard diffraction in DIS and rapidity gap processes
      1. 5.6.1 Probing fluctuations of the gluon field at $t\sim 0$.
      2. 5.6.2 Competition of gluon field fluctuations and break up mechanisms of diffraction at finite $-t\leq 0.5\;{\rm GeV}^{2}$.
      3. 5.6.3 Rapidity gaps at large $t$.
  6. 6 Inelastic diffraction in hadron–hadron scattering and formalism of cross section fluctuations
    1. 6.1 Inelastic diffraction and orthogonality of wave functions
    2. 6.2 Cross section fluctuations
    3. 6.3 Properties of distribution over interaction strength $P(\sigma)$
    4. 6.4 $P_{\rho}(\sigma)$ distribution for $\rho$ mesons
    5. 6.5 $P_{\gamma^{(\ast)}}(\sigma)$ distribution for real and virtual photons
  7. 7 Inclusive coherent and incoherent diffraction in $eA$ DIS
    1. 7.1 Connection of nuclear shadowing to diffraction
    2. 7.2 The model of leading twist nuclear shadowing
    3. 7.3 Nuclear diffractive structure functions and PDFs
  8. 8 Hard coherent diffraction in photoproduction of quarkonia and dijets on nuclei
    1. 8.1 Exclusive photoproduction of charmonia on nuclei in Pb-Pb UPCs
    2. 8.2 Diffractive dijet photoproduction in UPCs and at the EIC
  9. 9 Coherent and incoherent photoproduction of light vector mesons in heavy ion UPCs
    1. 9.1 Hadron structure of real photons and coherent and incoherent cross sections of photoproduction on nuclei
    2. 9.2 Predictions for the cross section of $\rho$ photoproduction in heavy ion UPCs and comparison to RHIC and LHC data
  10. 10 Soft diffraction in hadron–nucleus and photon–nucleus collisions and related phenomena
    1. 10.1 Coherent diffractive dissociation off nuclei
    2. 10.2 Distribution over the number of wounded nucleons
  11. 11 Hard diffraction in proton–proton scattering
    1. 11.1 Rapidity gap survival probability
    2. 11.2 Probing correlations of partons near the nucleon edge
  12. 12 Summary and outlook

##  1 Introduction

One first encounters the phenomenon of diffraction in optics, where it takes
place, for example, in the scattering of light off a completely absorptive
target or a hole in an absorptive screen. From the observed diffractive image,
one can extract information on the properties of the target. Similarly,
diffraction is also observed in quantum mechanics, for example, in the double
slit setup with the variation of the intensity of particles hitting the screen
determined by the distance between the slits and their width. In the case of
particle collisions at high energies, elastic scattering (or elastic
diffraction) provides information on the dependence of the strengths of
interaction on the impact parameter of the collision (the distance between the
projectile and the target in the transverse plane).

Another aspect of diffraction in hadron–proton scattering, which has no simple
analog in quantum mechanics, is the process of $h+p\to X+p$ referred to as
inelastic diffraction, when the minimum momentum transfer squared $t_{\rm
min}\to 0$. In such a process, a hadron $h$ scatters off a proton $p$ at rest
and transforms into a system of hadrons $X$ of a mass $M_{X}$, with $-t_{\rm
min}=m_{N}^{2}M_{X}^{4}/s^{2}$ ($m_{N}$ is the nucleon mass and $s$ is the
invariant energy squared of the collision). So, in the $s\to\infty$ limit, the
interacting proton remains at rest. The idea of how such a process can happen
has been put forward by Feinberg and Pomeranchuk [1, 2], who pointed out that
only if a hadron can exist in configurations interacting with the target with
different strengths, inelastic diffraction is possible. Trying to understand
this very difficult paper, Good and Walker111W.D. Walker, private
communication to M. Strikman (1993). suggested the cross section eigenstate
model [3], which is still widely used, though often beyond the range of its
applicability, namely, away from the $t_{\rm min}\sim 0$ limit, see the
discussion in Sec. 6.

To a large extent theoretical modeling of inelastic diffraction remained on
the sidelines of description of the strong interaction except for the
calculation of the inelastic shadowing phenomenon and a related fundamental
upgrade of the Glauber approximation for hadron–nucleus scattering [4].

With the advent of QCD, fluctuations of the interaction strength arise
naturally due to the asymptotic freedom: small-size configurations in hadrons
interact with significantly smaller strengths than the average one. This was
supported by theoretical and experimental studies of diffraction in the 90’s,
which to a large extent were stimulated by experiments at the electron–proton
collider HERA.

In this review we will discuss results of these studies and their spin-offs
for various phenomena involving interactions with nucleons and nuclei
emphasizing the interplay of non-perturbative and perturbative dynamics. A
special attention will be given to present and future programs to study
diffraction including ultraperipheral collisions (UPCs) at the LHC [5] and
lepton–ion scattering at the Electron–Ion Collider (EIC) in the US [6] and the
Large Hadron-electron Collider (LHeC) [7, 8] or Future Circular Collider (FCC)
[9] at CERN.

This review is structured as follows. In Sect. 2, we present the results of
theoretical and experimental studies of diffraction in electron–proton deep
inelastic scattering (DIS) and the most recent analyses of the HERA data,
which demonstrated that this process is dominated by the leading twist (LT)
mechanism in a wide $x$ and $Q^{2}$ range and has the energy dependence
consistent with expectations from models based on the soft Pomeron dynamics.
Predictions for the next lepton–proton colliders, EIC, LHeC and FCC-eh are
also given.

Section 3 reviews the phenomenological dipole models of diffraction and the
total small-$x$ DIS inclusive cross section, which enable one to include
higher-twist effects in the inclusive and diffractive DIS. These processes
allow one not only to constrain the dipole–nucleon cross section, but also to
determine the kinematics, where the interaction at small impact parameters may
approach the regime of complete absorption, the so-called black disk regime
(BDR), which is characterized by the breakdown of perturbative QCD
approximations.

This very high-energy asymptotic behaviour and related small-$x$ methods based
on resummation and leading to the notion of parton saturation are discussed in
Sec. 4.

In Sec. 5, we summarize results of theoretical studies of hard diffractive
electro- and photoproduction of vector mesons in the framework of the dipole
model, which in the limit of high $Q^{2}$ reproduces the leading twist
factorization theorem for exclusive processes. A special attention is payed to
the phenomenologically important case of elastic $J/\psi$ photoproduction in
UPCs in the LHC kinematics. The rapidity gap vector meson production is
discussed in various limits: at $t\sim 0$, where the gluon density
fluctuations are probed; in the intermediate $-t<0.5\;\rm GeV^{2}$ region,
which may be described as a break-up mechanism and, finally, in the large $t$
region, which offers unique opportunities to probe
Balitsky–Fadin–Kuraev–Lipatov (BFKL) Pomeron with minimal interference from
the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) kinematics.

A convenient formalism to treat inelastic diffraction in hadron–hadron
scattering is offered by the notion of cross section fluctuations introduced
above. In Sec. 6, we consider specific models of such fluctuations
parameterizing the composite hadronic structure of protons, pions, and
photons.

In Sec. 7, we review the deep connection between diffraction off nucleons and
the phenomenon of nuclear shadowing in scattering off nuclei. The application
of the factorization theorems for inclusive and diffractive DIS allows one to
predict nuclear inclusive and diffractive parton distribution functions (PDFs)
within the framework of the leading twist nuclear shadowing approximation.

A characteristic feature of this approach is the predictions of large nuclear
gluon shadowing. Since for the next decade ultraperipheral $pp$, $pA$, and
$AA$ collisions at the LHC would remain one of the best sources of information
on diffraction in photon–nucleon and photon–nucleus scattering, a comparison
to the UPC data presents an important testing ground for theory. A good
agreement of the predictions of the leading twist nuclear shadowing
approximation with the UPCs data on coherent $J/\psi$ photoproduction on heavy
nuclei is demonstrated in Sec. 8. We also make predictions for dijet
photoproduction, including direct and resolved photon contributions, in UPCs
at the LHC and in $ep$ scattering at the EIC.

A related subject is nuclear shadowing in coherent photoproduction of $\rho$
mesons on nuclei in heavy ion UPCs. In Sec. 9, we consider this process and
demonstrate that the shadowing effect is much stronger than in the approach
based on the vector meson dominance and Glauber models and significantly
stronger than in the $J/\psi$ case.

The importance of color (cross section) fluctuations in soft hadron–nucleus
scattering is further demonstrated in Sec. 10, where we present predictions
for the diffractive dissociation cross section and fluctuations in the
distribution over the number of wounded nucleons in $\gamma A$ scattering,
which could be measured at the LHC.

LHC experiments provide novel opportunities to study hard diffraction in
proton–proton ($pp$) scattering. In particular, in Sec. 11 we consider the
diffractive process $pp\to 2jets+X+p$ and estimate the corresponding gap
survival probability using information on elastic $pp$ scattering and
diffractive fragmentation function. We provide a numerical estimate of the
survival probability, which is consistent with the LHC results. A mechanism of
breakdown of soft factorization (the universal gap survival probability) is
suggested as well as several new reactions involving production of four jets.

Our conclusions and outlook are presented in Sec. 12.

##  2 Inclusive diffraction in DIS

###  2.1 Introduction

During the 90’s both H1 and ZEUS experiments performed a series of
observations of events which were characterized by the presence of large
rapidity gaps [10, 11, 12, 13]. In these events, which constituted a large
fraction of about $10\%$ of all DIS events, the proton was separated from the
rest of the particles by a large region in a detector which did not have any
activity - a rapidity gap. Two experimental scenarios were employed, the large
rapidity gap method (LRG) which relied on the observation of the large
rapidity gap, and the leading proton method (LP) where the elastically
scattered proton was directly measured in the forward instrumentation. There
are some differences in the both methods in that the LRG method does not
distinguish the case when the proton is scattered elastically from the cases
when the proton dissociates into the system with small mass $M_{Y}$. The LP
method also allows one to reconstruct the four-momentum squared $t$ at the
proton vertex.

Given the presence of the hard scale in the DIS process, the minus virtuality
of the photon $q^{2}=-Q^{2}$, it was only natural to ask if such processes are
tractable within the perturbative QCD. Veneziano and Trentadue in [14]
postulated that in DIS the semi-inclusive processes, where the hadron is
produced in the target fragmentation region, can be described within the
collinear approximation. For that purpose they introduced the notion of the
fracture functions which contain the information about the structure function
of a given target hadron once it has fragmented into another given final state
hadron. The diffractive processes, which can be classified as a special case
of the processes discussed in [14], were considered in Refs. [15, 16] where it
was demonstrated that they can be described within the collinear
approximation, in analogy to the standard non-diffractive processes in DIS.
The factorization proof, presented in [15], essentially followed that of the
inclusive case. Note that, diffractive factorization can also be applied to
other semi-inclusive processes in diffractive DIS like diffractive heavy quark
production or dijet production in the direct photon case (see discussion later
in this section). Also, factorization is valid for a more general case of
production of a hadron with a fixed momentum fraction $x_{F}$ and a transverse
momentum $p_{t}$ in the target fragmentation region.

The typical event with a rapidity gap in DIS is depicted in a diagram shown in
Fig. 1. An incoming electron or positron with four-momentum $k$ scatters off
the incoming proton with four-momentum $p$. The proton is scattered into the
final state $Y$ with four-momentum $p^{\prime}$. The proton may stay intact or
alternatively it can also dissociate into a low mass excitation with mass
$M_{Y}$. The process proceeds through the exchange of a single photon and
there is a rapidity gap between the final state $Y$ and the diffractive system
$X$, see the diagram in Fig. 1.

Figure 1: Diagram for diffractive DIS in the single photon approximation. The
diffractive mass $X$ is separated from the diffractive scattered proton (or
its excitation) $Y$ by a rapidity gap. See the text for the definition of the
variables. Figure from [17], https://doi.org/10.1103/PhysRevD.100.074022.

As any DIS process, the diffractive event is characterized by the standard set
of variables:

$q^{2}=-Q^{2}\;,\qquad x=\frac{Q^{2}}{2p\cdot q}\;,\qquad
W^{2}=(p+q)^{2}\;,\qquad y=\frac{p\cdot q}{p\cdot k}\;,$ (1)

being minus photon virtuality, Bjorken $x$, center-of-mass energy squared of
the photon-proton system and inelasticity, respectively. In addition to these
variables, there are also diffractive ones which are defined as follows

$t=(p-p^{\prime})^{2}\;,\qquad\xi=\frac{Q^{2}+M_{X}^{2}-t}{Q^{2}+W^{2}}\;,\qquad\beta=\frac{Q^{2}}{Q^{2}+M_{X}^{2}-t}\;,$
(2)

where $t$ is the momentum transfer squared at the proton vertex, $M_{X}^{2}$
is the mass squared of the diffractive system $X$, $\xi$ is the momentum
fraction carried by the diffractive exchange, and $\beta$ is the momentum
fraction carried by the struck parton with respect to the diffractive
exchange. Often $\xi$ is denoted by $x_{{I\\!\\!P}}$ in the literature. The
two momentum fractions satisfy the constraint $x=\xi\beta$. The variable $\xi$
can be related to the fraction $x_{L}$ of the longitudinal momentum of the
initial proton carried by the final proton, i.e. $\xi=1-x_{L}$. Thus typical
diffractive events are characterized by small $\xi$, or large $x_{L}$ meaning
that the final proton carries a large fraction of the initial momentum. The
double line in diagram in Fig. 1 depicts the diffractive exchange (often
referred to as the Pomeron) between the proton and the diffractive system $X$,
and is responsible for the presence of the rapidity gap.

The diffractive cross sections can be expressed by the two structure
functions. In the one-photon approximation

$\displaystyle\sigma_{\rm red}^{D(3)}$ $\displaystyle=$ $\displaystyle
F_{2}^{D(3)}(\beta,\xi,Q^{2})-\frac{y^{2}}{Y_{+}}F_{\mathrm{L}}^{D(3)}(\beta,\xi,Q^{2})\;,$
(3) $\displaystyle\sigma_{\rm red}^{D(4)}$ $\displaystyle=$ $\displaystyle
F_{2}^{D(4)}(\beta,\xi,Q^{2},t)-\frac{y^{2}}{Y_{+}}F_{\mathrm{L}}^{D(4)}(\beta,\xi,Q^{2},t)\;,$
(4)

where $Y_{+}=1+(1-y)^{2}$. In the above equations the reduced cross sections
are the rescaled differential cross sections

$\frac{d^{4}\sigma^{D(4)}}{d\xi d\beta dQ^{2}dt}\;=\;\frac{2\pi\alpha_{\rm
em}^{2}}{\beta Q^{4}}\,Y_{+}\,\sigma_{\rm red}^{D(4)}\,,$ (5)

or, upon the integration over $t$,

$\frac{d^{3}\sigma^{D(3)}}{d\xi d\beta dQ^{2}}\;=\;\frac{2\pi\alpha_{\rm
em}^{2}}{\beta Q^{4}}\,Y_{+}\,\sigma_{\rm red}^{D(3)}\,.$ (6)

The subscripts (3) and (4) in the above formulae denote the number of
variables that the diffractive cross sections or structure functions depend
on. Note that the structure functions $F_{2,L}^{D(4)}$ have dimension $\rm
GeV^{-2}$, whereas $F_{2,L}^{D(3)}$ are dimensionless. The contribution of the
longitudinal structure function to the reduced cross sections is rather small,
for the most part, except in the region of $y$ close to unity.

###  2.2 Collinear factorization in diffractive DIS

The standard perturbative QCD approach to diffractive cross sections is based
on the collinear factorization [15, 14, 16]. Similarly to the inclusive DIS
cross section, the diffractive cross section can be written in a factorized
form

$F_{2/L}^{D(4)}(\beta,\xi,Q^{2},t)\;=\;\sum_{i}\int_{\beta}^{1}\frac{dz}{z}\
C_{2/L,i}\left(\frac{\beta}{z},Q^{2}\right)\,f_{i}^{\rm D}(z,\xi,Q^{2},t)\;,$
(7)

where the sum is performed over all parton flavors (gluon, $d$-quark,
$u$-quark, etc.). In the case of the lowest order parton model process,
$z=\beta$. When higher order corrections are taken into account then
$z>\beta$. The coefficient functions $C_{2/L,i}$ can be computed
perturbatively in QCD and are the same as in inclusive deep inelastic
scattering case. The long distance part $f_{i}^{\rm D}$ corresponds to the
diffractive parton distribution functions (DPDF). Similarily to the inclusive
case one can provide operator definition for the diffractive parton densities
[16]. The quark diffractive distribution function is defined as

$\displaystyle f_{j}^{\rm
D}(z,\xi,\mu,t)\;=\;\frac{1}{4\pi}\frac{1}{2}\sum_{s}\int
dy^{-}e^{-izp^{+}y^{-}}\sum_{X,s^{\prime}}\langle
p,s|\tilde{\bar{\psi}}(0,y^{-},0_{T})|p^{\prime},s^{\prime};X\rangle$
$\displaystyle\times\gamma^{+}\langle
p^{\prime},s^{\prime};X|\tilde{\psi}(0)|p,s\rangle\,,$ (8)

and gluon diffractive distribution

$\displaystyle G^{\rm D}(z,\xi,\mu,t)\;=\;\frac{1}{2\pi
zp^{+}}\frac{1}{2}\sum_{s}\int
dy^{-}e^{-izp^{+}y^{-}}\sum_{X,s^{\prime}}\langle
p,s|\tilde{F}_{a}^{+\mu}(0,y^{-},0_{T})|p^{\prime},s^{\prime};X\rangle$
$\displaystyle\times\langle
p^{\prime},s^{\prime};X|\tilde{F}_{a\mu}^{+}(0)|p,s\rangle\,.$ (9)

In the above the quark field is defined as

$\tilde{\psi_{j}}(0,y^{-},0_{T})\;=\;{\cal
P}\,\exp\left(ig\int_{y^{-}}^{\infty}dx^{-}A_{c}^{+}(0,x^{-},0_{T})\,t_{c}\right)\psi_{j}(0,y^{-},0_{T})\,,$
(10)

whereas the gluon field

$\tilde{F_{a}}^{\mu\nu}(0,y^{-},0_{T})\;=\;{\cal
P}\,\exp\left(ig\int_{y^{-}}^{\infty}dx^{-}A_{c}^{+}(0,x^{-},0_{T})\,T_{c}\right)_{ab}F_{b}^{\mu\nu}(0,y^{-},0_{T})\,,$
(11)

where ${\cal P}$ denotes the path ordering in the exponential and
$t_{a},T_{a}$ are the generators of the $SU(3)$ group in the fundamental and
adjoint representations, respectively. The definitions (8) and (9) differ from
the standard definitions in the inclusive case by the fact that the final
state is the proton, and thus there is a sum over the spin of the proton and
over any other particles that form the diffractive state $X$. We note that the
above definitions and corresponding factorization theorem are valid for
general case, which includes the fragmentation kinematics, that is for any
fixed $\xi$ and $t$.

The arguments for the factorization were presented in [14, 16] and the proof
in [15]. The proof basically follows the one in the inclusive case [18]. The
leading regions in the Feynman graphs for the diffractive amplitude involve:
a) the beam jet which consists of partons collinear to proton $p$ and also
includes the diffractively scattered final state proton with momentum
$p^{\prime}$, b) one or more final state jets, which are not in the direction
of the initial proton, c) the hard interaction, which contains the lines with
virtualities of the order of $Q^{2}$ and connected to the virtual photon, d)
soft subgraph joined to final jets and beam jet by the soft gluons. Using
gauge invariance one can demonstrate that the soft gluons do not resolve the
final state jets. As a result, the gluon connections can be incorporated in
the gauge link which appears in the definition of the diffractive parton
densities, see Eqs. (10) and (11).

An alternative and intuitive way of stating the factorization may be also
illustrated schematically in Fig. 2. Upon changing the hard scale from
$Q_{0}^{2}$ to $Q^{2}$, an additional parton (or partons) is emitted in the
diffractive system $X$. The interaction of partons which would form $h$ is not
changed upon the variation of the scale since the overall interaction does not
resolve the quark-gluon system which is located at a distance $\ll 1/Q_{0}$.
Finally, we note that the collinear factorization does not require the concept
of the Pomeron and is distinct from the soft factorization at the proton
vertex discussed below.

Figure 2: Schematic illustration of the factorization theorem for diffractive
DIS. By changing the scale from $Q_{0}^{2}$ to $Q^{2}$, an additional parton
is emitted in the diffractive mass $X(M_{X})$. The blue oval indicates the
interaction of partons which form the final state hadron $h$ (or its low mass
excitation) with the the partons in the diffractive mass. This interaction
does not change since it cannot resolve the $qg$ system which is localized at
the transverse distance much smaller than $1/Q_{0}$.

As in the inclusive case the factorization applies to other processes, which
can include heavy quarks or jets. It also applies to photoproduction of jets
in the direct photon case, but is expected to fail in the resolved photon
case, since in effect the latter case is like the hadron-hadron scattering
where the factorization is known to be violated, for example, in the case of
the diffractive dijet production at Tevatron [19, 20, 21], see Sec. 11.

The diffractive parton densities can be interpreted as conditional
probabilities for finding partons in the proton, provided the proton is
scattered into the final state system $Y$ with a specified four-momentum
$p^{\prime}$. As mentioned above, since factorization holds for semi-inclusive
processes, the DPDFs should be universal and hence can be used from hard
process to hard process. They are evolved using the Dokshitzer-Gribov-Lipatov-
Altarelli-Parisi (DGLAP) evolution equations [22, 23, 24, 25] similarly to the
inclusive case.

Note that while the definitions in Eqs. (8) and (9) do not have the form of
parton number operators, it is still customary in the literature [16] to use
the term diffractive parton distributions with their ensuing interpretation as
conditional probabilities (see above).

###  2.3 Diffractive fits to HERA data

Both H1 [26] and ZEUS [27] experiments performed fits to the diffractive
structure functions based on the collinear factorization and DGLAP evolution.
We shall describe the details and results of the fits below.

Since the diffractive parton distribution depends a priori on 4 variables, a
large amount of physical information needs to be incorporated to correctly
describe their shape. It was empirically found that the description of the
experimental data was very good when the Ingelman-Schlein [28] proton vertex
factorization is assumed. This means that the DPDF is factorized into products
of two terms, one of which depends on $\xi$ and $t$ only and another one which
depends on $z$ and $Q^{2}$ [28]

$f_{i}^{D(4)}(z,\xi,Q^{2},t)=f^{p}_{{I\\!\\!P}}(\xi,t)\,f_{i}^{{I\\!\\!P}}(z,Q^{2})\;.$
(12)

A popular physical interpretation of this factorization is that the
diffractive exchange can be interpreted as a colorless object called a Pomeron
with a partonic structure given by the parton distributions
$f_{i}^{{I\\!\\!P}}(\beta,Q^{2})$, the variable $\beta$ corresponding to the
fraction of the Pomeron longitudinal momentum carried by the struck parton.

The Pomeron flux factor $f^{p}_{{I\\!\\!P}}(\xi,t)$ represents the probability
that a Pomeron with particular values of $\xi$ and $t$ couples to the proton.
It is worth to emphasize that in the diagrammatic language the Pomeron
exchange occurs over long space-time intervals, which is due to the fact that
the exchanged gluon quanta have small plus and minus momentum components. Also
let us note that the soft factorization is highly nontrivial as the structure
of the Pomeron could a priori depend on $Q^{2}$ \- like it happens in the case
of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) perturbative Pomeron [29, 30, 31].
In the latter case it is known, for example, that the Pomeron intercept will
depend on the value of the strong coupling which depends on the scales
involved in the process, such as $Q^{2}$ in the case of DIS, see Sec. 4.2.

There are number of diffractive fits in the literature, [26, 27, 32, 33, 34].
Both H1 [26] and ZEUS [27] collaborations performed DGLAP fits to their own
experimental data.

Analyses of the experimental data actually show a necessity to include another
component, that corresponds to the subleading term, (in a sense of the small
$\xi$ behavior) to which we refer as Reggeon and which is important at large
values of $\xi\geq 0.03$ and small $\beta$. Thus the parametrization employed
in the fits is a modification of (12) which results in a two-component model,
which is a sum of two exchange contributions, ${I\\!\\!P}$ and ${I\\!\\!R}$,
each satisfying the vertex factorization hypothesis:

$f_{i}^{D(4)}(z,\xi,Q^{2},t)=f^{p}_{{I\\!\\!P}}(\xi,t)\,f_{i}^{{I\\!\\!P}}(z,Q^{2})+f^{p}_{{I\\!\\!R}}(\xi,t)\,f_{i}^{{I\\!\\!R}}(z,Q^{2})\;.$
(13)

The fluxes $f^{p}_{{I\\!\\!R},{I\\!\\!P}}(\xi,t)$ are parametrized using the
form motivated by the Regge theory,

$f^{p}_{{I\\!\\!P},{I\\!\\!R}}(\xi,t)=A_{{I\\!\\!P},{I\\!\\!R}}\frac{e^{B_{{I\\!\\!P},{I\\!\\!R}}t}}{\xi^{2\alpha_{{I\\!\\!P},{I\\!\\!R}}(t)-1}}\;,$
(14)

where $A_{{I\\!\\!P},{I\\!\\!R}}$ are normalization constants for the Pomeron
and Reggeon, $B_{{I\\!\\!P},{I\\!\\!R}}$ are $t$ slopes, and
${\alpha_{{I\\!\\!P},{I\\!\\!R}}(t)=\alpha_{{I\\!\\!P},{I\\!\\!R}}(0)+\alpha_{{I\\!\\!P},{I\\!\\!R}}^{\prime}\,t}$
are linear trajectories .

We emphasize here that the notions of Pomeron and Reggeon in the context of
diffractive deep inelastic scattering differ from those familiar from the soft
$pp$ $(p\bar{p})$ interactions. In particular, the parameters of both
trajectories may be different.

Strictly speaking the parametrization in (13) is inspired by the Regge theory.
One does not assume anything here about the quantum numbers of the Reggeon.
Neutron production at large $z$ is strongly suppressed as compared to proton
production, see [35, 36]. Thus the Reggeon contribution in (13) most probably
cannot be interpreted as solely as a pion exchange. This is due to the fact
that in the case of the pion exchange, the ratio of the neutron to proton
production is equal to two as a consequence of the Clebsh–Gordan isospin
relations between the corresponding pion photoproduction amplitudes.

The diffractive parton distributions of the Pomeron at the initial scale
$\mu_{0}^{2}$ are modeled as a singlet quark distribution
$\Sigma=\sum_{i}(f_{i}^{I\\!\\!P}+\bar{f}_{i}^{I\\!\\!P})$ consisting of quark
and antiquark distributions and a gluon distribution $f_{g}^{{I\\!\\!P}}$. The
neutrality of the Pomeron implies
$f_{i}^{I\\!\\!P}\equiv\bar{f}_{i}^{I\\!\\!P}$. In both ZEUS and H1 fits it
was assumed that the light quark distributions are equal
$f_{u}^{{I\\!\\!P}}=f_{d}^{{I\\!\\!P}}=f_{s}^{{I\\!\\!P}}$ and that the charm
and beauty distributions are in the variable flavour number scheme (VFNS) and
are generated radiatively. Both quarks and gluons are parametrized using
similar functional forms to that used in the fits to the inclusive structure
function data. H1 and ZEUS used the following simple parametrization for the
distributions in the Pomeron

$zf_{i}^{I\\!\\!P}(z,\mu_{0}^{2})=A_{i}z^{B_{i}}(1-z)^{C_{i}}\;,$ (15)

where $i$ is a gluon or a light quark. The parameters $C_{q},C_{g}$ were
allowed to vary and in particular they were allowed to take both positive and
negative values. To ensure the vanishing of the distributions at $z=1$ for the
solutions to the DGLAP equation, an exponential regulating factor of
$\exp(-0.01/(1-z))$ has been included.

The parton distributions for the Reggeon component, $f_{i}^{{I\\!\\!R}}$ were
taken from a parametrization which was obtained from fits to the pion
structure function [37, 38].

The fits performed by H1 [26] and ZEUS [27] are very similar in general setup
but differ in the details of the choice and number of free parameters and the
selection of data sets. The fits performed in [26] were referred to as NLO H1
fit A and B. The two sets of parton densities differ mainly in the gluon
density at high fractional parton momentum. This region is very poorly
constrained by the inclusive diffractive scattering data at HERA kinematic
range. ZEUS collaboration also performed three separate fits, ZEUS C, S and SJ
fits [27]. Fits C and S differ in the form of gluon parametrization at large
$z$, whereas fit SJ is essentially based on fit S parametrization but in
addition to inclusive data, diffractive dijet data are included in the fit.

In Table 1 we show values of the selected parameters for ZEUS fits S, C and SJ
as well as H1 fits A and B. Note that the parametrizations for ZEUS were
specified at $\mu_{0}^{2}=1.8\,\rm GeV^{2}$ and for H1 for
$\mu_{0}^{2}=1.75\,\rm GeV^{2}$. We do not show the normalization parameters
as they cannot be compared directly due to different conventions for both
experiments.

Parameter | ZEUS S | ZEUS C | ZEUS SJ | H1 A | H1 B
---|---|---|---|---|---
$B_{q}$ | 1.34 $\pm$0.05 | 1.25 $\pm$ 0.03 | 1.23 $\pm$ 0.04 | 2.3$\pm$ 0.36 | 1.5 $\pm$ 0.12
$C_{q}$ | 0.34 $\pm$ 0.043 | 0.358 $\pm$ 0.043 | 0.332 $\pm$ 0.049 | 0.57 $\pm$ 0.15 | 0.45 $\pm$ 0.09
$B_{g}$ | -0.422 $\pm$ 0.066 | 0 | -0.161 $\pm$ 0.051 | 0 | 0
$C_{g}$ | -0.725 $\pm$ 0.082 | 0 | -0.232 $\pm$ 0.058 | -0.95 $\pm$ 0.20 | 0
$\alpha_{{I\\!\\!P}}(0)$ | 1.12 $\pm$ 0.02 | 1.11 $\pm$ 0.02 | 1.11 $\pm$ 0.02 | 1.118 $\pm$ 0.008 | 1.111 $\pm$ 0.007
$\alpha_{{I\\!\\!R}}(0)$ | 0.732 $\pm$ 0.031 | 0.668 $\pm$ 0.040 | 0.699 $\pm$ 0.043 | 0.5 | 0.5
$\alpha^{\prime}_{{I\\!\\!P}}$ | 0 |  0 ${\rm GeV^{-2}}$ |  0 ${\rm GeV^{-2}}$ |  0.06 ${\rm GeV^{-2}}$ |  0.06 ${\rm GeV^{-2}}$
$\alpha^{\prime}_{{I\\!\\!R}}$ |  0.9 ${\rm GeV^{-2}}$ |  0.9 ${\rm GeV^{-2}}$ |  0.9 ${\rm GeV^{-2}}$ |  0.3 ${\rm GeV^{-2}}$ |  0.3 ${\rm GeV^{-2}}$
$B_{{I\\!\\!P}}$ |  7 ${\rm GeV^{-2}}$ |  7 ${\rm GeV^{-2}}$ |  7 ${\rm GeV^{-2}}$ |  5.5 ${\rm GeV^{-2}}$ |  5.5 ${\rm GeV^{-2}}$
$B_{{I\\!\\!R}}$ |  2 ${\rm GeV^{-2}}$ |  2 ${\rm GeV^{-2}}$ |  2 ${\rm GeV^{-2}}$ |  1.6 ${\rm GeV^{-2}}$ |  1.6 ${\rm GeV^{-2}}$
Table 1: Values of the parameters for ZEUS S, C, SJ fits [27] and H1 A and B
fits [26]. Parameters in bold font have been fixed in the fit.

Figure 3:  The value of Pomeron intercept $\alpha_{I\\!\\!P}(0)$ obtained from
Regge fits in different $Q^{2}$ bins. The inner error bars represent the
statistical and systematic errors added in quadrature and the outer error bars
include model uncertainties. Figure from [39],
https://doi.org/10.1140/epjc/s10052-012-2074-2 .

The value of the Pomeron intercept $\alpha_{{I\\!\\!P}}(0)$ is about
$1.11-1.12$, which is of the same order as the value extracted from the $pp$
total cross section data. Both collaborations tested the proton vertex
factorization hypothesis, by investigating the dependence on $Q^{2}$. In the
analysis [39], full range in $Q^{2}$ is divided into six intervals, and for
each interval, a Pomeron intercept was introduced. In that way the vertex
factorization hypothesis was tested by allowing for a $Q^{2}$ dependence of
the Pomeron intercept in the fit procedure. The results of this analysis are
shown in Fig. 3. As is evident from this figure no significant $Q^{2}$
dependence of the Pomeron intercept was observed, which supports the proton
vertex factorization hypothesis. The average value for the Pomeron intercept
found in this analysis was $\alpha_{{I\\!\\!P}}(0)=1.113$, which is consistent
with the earlier fits.

The value of $\alpha_{{I\\!\\!P}}(0)$ in inclusive diffraction is
significantly smaller than the value extracted from the elastic diffractive
production of heavy vector mesons, for example, in the $J/\psi$
photoproduction [40, 41], see Sec. 5. The $Q^{2}$ independence of the
intercept also has to be contrasted with the increase of the effective Pomeron
intercept extracted from the inclusive $F_{2}(x,Q^{2})$ [42].

The value of $\alpha^{\prime}_{{I\\!\\!P}}$ from Table 1 is either fixed to
zero (ZEUS fits) or to very small value $0.06\;\rm GeV^{-2}$ in the case of H1
fits. This is much smaller than the value found from the fits to hadronic
cross sections $\alpha^{\prime}_{{I\\!\\!P}}=0.25\,\rm GeV^{-2}$, see for
example [43]. This parameter has been extracted by both collaborations in the
measurements of the $t$ differential cross section $\sigma^{D(4)}_{\rm red}$.
The H1 measurements using Forward Proton Spectrometer (FPS) gave value
$0.02-0.1\;\rm GeV^{-2}$ [44] depending on the range of $\xi$ and
$0.009-0.06\,\rm GeV^{-2}$ [45] depending on the range of $Q^{2}$. In the case
of ZEUS [46], the extracted value was $\alpha^{\prime}_{{I\\!\\!P}}=-0.01\;\rm
GeV^{-2}$ with large error bars.

In the same works, the $t$ slope parameter was extracted for the Pomeron with
the value of about $7.0\;\rm GeV^{-2}$ from ZEUS [46] and $5.17-5.78\;\rm
GeV^{-2}$ from H1 data [45]. This is somewhat larger slope than for exclusive
hard diffraction, $J/\psi$ production, which indicates that the size of the
inclusive diffractive vertex is not small. This is consistent with
$\alpha_{I\\!\\!P}(t=0)$ being smaller for inclusive diffraction than for hard
exclusive processes.

As mentioned above the biggest difference between ZEUS fits S and C and fits
H1 A and B is the behavior of the gluon at large $z$. Since both fits
described the inclusive diffractive HERA data very well, it was concluded that
the inclusive data are not sensitive enough to pin down the behavior of the
gluon density in this region. The data on diffractive production of dijets
turned out to be much more sensitive to the gluon distribution at large $z$
and favored ZEUS C and H1 B fits. The latter data on diffractive dijets were
included in addition to the inclusive data in ZEUS SJ fit. Examples of
diffractive PDFs from ZEUS and H1 are shown in Fig. 4.

Figure 4: Examples of diffractive parton distribution functions from ZEUS and
H1 analysis. Figures from [27],
https://doi.org/10.1016/j.nuclphysb.2010.01.014 and [26],
https://doi.org/10.1140/epjc/s10052-006-0035-3.

First diffractive fits at NNLO accuracy have also been performed recently
[34]. The fit used the combined HERA II data in addition to the HERA I data,
which allowed for the increased precision. In addition, similarly to previous
fits, the NNLO fit has been constrained by the diffractive dijet data. The
quark distribution is similar to the NLO extraction, whereas the gluon
density, which is more sensitive to higher order effects is reduced by about
$30\%$. Overall, the NNLO calculation fits the data very well, above
$Q^{2}>8.5\,\rm GeV^{2}$, where the effects from higher twists can be
neglected.

Figure 5: The $Q^{2}$ dependence of the reduced diffractive cross section,
multiplied by $x_{{I\\!\\!P}}$, at different fixed values of
$x_{{I\\!\\!P}}$=0.0003 (a), 0.001 (b), 0.003 (c) and 0.01 (d). The reduced
cross section values are multiplied by a scaling factor, $4^{l}$ for
$x_{I\\!\\!P}$=0.0003 and $3^{l}$ for $x_{{I\\!\\!P}}$=0.003, 0.001 and 0.01,
with l values as indicated in parentheses. Solid lines fit H1 B, dashed line
indicates an extrapolation. Figure from [39],
https://doi.org/10.1140/epjc/s10052-012-2074-2.

In Fig. 5, the calculations using the H1 Fit B [26] are compared with the more
recent analysis of the HERA data [39] on the reduced diffractive cross
section. The results are shown as a function of $Q^{2}$ in several bins of
$\beta$ and four bins bins of $\xi=x_{I\\!\\!P}$. One observes reasonable
description of the experimental data by the fits, with a certain deterioration
at low values of $Q^{2}$. Indeed, it was noted by both ZEUS and H1 that DGLAP
fits fail in the low $Q^{2}$ region, the breakdown point was about $8.5\;\rm
GeV^{2}$ as reported by H1, and $5\;\rm GeV^{2}$ for ZEUS. The solid line in
Fig. 5 indicates the fit H1 B which was only performed down to $8.5\;\rm
GeV^{2}$, while the dashed line is an extrapolation. From the figures we also
observe that the leading twist description deteriorates at small values of
$\beta$. The exact origin of this breakdown of DGLAP fits is unknown. It has
been though suggested [47, 48] that this may be an indication of the
importance of the higher twists in diffraction. Both analyses [47, 48]
demonstrate that inclusion of higher twists significantly improves the
description of the inclusive diffractive data. We briefly discuss the higher
twist contributions and their impact on the description of the experimental
data in Sec. 3.

###  2.4 Diffractive dijet and charm production

The power of the factorization lies in the observation that the parton
densities are universal, and once extracted from one process can be used to
make predictions for the rates of other hard processes. In the case of
diffraction, the diffractive parton densities can be used to calculate other
processes in rapidity gap DIS where the hard scale is present. At HERA, the
diffractive parton densities were used to calculate the diffractive charm
production and the dijet production both in DIS and in photoproduction.

The factorization formula for the diffractive dijet production in DIS takes
the following form

$d\sigma(e+p\rightarrow e+2{\rm jets}+X^{\prime}+Y)=\sum_{i}\int dt\int
d\xi\int dz\,d\hat{\sigma}(e+i\rightarrow{e+\rm
2jets})f_{i}^{D(4)}(z,\xi,\mu^{2},t)$ (16)

where the sum is over the contributing partons of type $i$, and
$d\hat{\sigma}$ is the partonic cross section. The scale $\mu^{2}$ is the
factorization scale, which is usually identified with the jet transverse
energy scale and $X^{\prime}$ denotes the part of the diffractive system $X$
which does not include the two jets.

Figure 6: Typical diagrams for diffractive dijet production in $ep$
scattering. Graphs $a$ and $b$ correspond to the direct and resolved photon
contributions, respectively.

In the case of dijet photoproduction, there are in general two ways the photon
can interact with the partons in the proton, namely the direct and the
resolved photon processes, see Fig. 6. In the former case, the photon acts as
a pointlike particle and interacts with the parton from the target. An example
is shown in graph ($a$) in Fig. 6, where the partonic interaction is the
photon–gluon fusion which results in the quark – antiquark pair in the final
state. In this process the collinear factorization is expected to hold.

In the second, resolved case, the photon may fluctuate into $q\bar{q}$ pairs
and rather low mass hadronic states. The low mass hadronic states are
dominated by vector mesons. The parton from the photon will then interact with
the parton from the proton, see graph ($b$) in Fig. 6. In this case the
interaction is more reminiscent of the hadron–hadron scattering. Therefore the
breaking of the collinear factorization is expected due to the fact that the
soft interactions between the target proton and the resolved photon component
are going to destroy the rapidity gap. It is usually accounted for by
introducing a model-dependent suppression factor (rapidity gap survival
probability) either globally or only for the resolved photon contribution [49,
50, 51, 52]. Note that the calculations at next-to-leading order (NLO) of QCD
[53] and beyond [54] demonstrate that the direct and resolved processes are
connected through the factorization of collinear initial-state singularities.
It is also possible that the resolved-photon suppression factor depends on the
parton flavor. Indeed, since the QCD factorization seems to hold for
diffractive photoproduction of open charm, the suppression factor should be
introduced only for light parton flavors and should be significantly smaller
compared to the situation, when all channels are suppressed equally. Moreover,
it can also depend on the momentum fraction $x_{\gamma}$ [52]. See also the
discussion in the end of the current subsection.

The cross section for the dijet photoproduction in diffraction, neglecting any
rapidity gap destruction effects, can be written as a convolution of the
diffractive parton distribution functions, photon parton distribution
functions and the hard partonic cross section [55, 56, 57, 51, 49]

$\displaystyle d\sigma(e+p\rightarrow e+2\,{\rm
jets}+X^{\prime}+Y)=\sum_{ij}\int\,dy\,f_{\gamma/e}(y)\int
dx_{\gamma}\,f_{j/\gamma}(x_{\gamma},\mu^{2})\times$ $\displaystyle\times\int
dt\int d\xi\int dz\,d\hat{\sigma}(i+j\rightarrow{\rm
2\,jets})\,f_{i}^{D(4)}(z,\xi,\mu^{2},t)\,,$ (17)

where $i$ and $j$ are parton flavors including the case when $i$ is the photon
for the photon direct contribution. Here, $x_{\gamma}$ is the fraction of the
longitudinal momentum of the photon carried by the parton entering the hard
process and $y$ is the longitudinal momentum fraction of electron carried by
the photon. The functions $f_{j/\gamma}(x_{\gamma},\mu^{2})$ are the parton
distributions in the photon and $f_{\gamma/e}(y)$ is the flux of equivalent
photons calculated in the Weizsäecker-Williams approximation [58, 59]. In the
case of the direct photon process, the photon PDF reduces to the delta
function $\delta(1-x_{\gamma})$. Finally, $d\hat{\sigma}$ is the partonic
cross section for scattering of partons $i$ and $j$ and producing two jets.

Thus the diffractive dijet photoproduction offers a unique bridge between the
diffractive DIS, for large $Q^{2}$, where the factorization should hold, and
the hadron–hadron scattering, where the breakdown of the factorization happens
due to the remnant–remnant interactions.

Both H1 and ZEUS performed measurements of diffractive dijets in DIS and
photoproduction [60, 61, 62, 63, 27, 64, 65, 66, 67]. In the case of the
dijets in DIS, both H1 and ZEUS data could be described very well when using
H1 B and ZEUS C and SJ DPDF sets, respectively. This was not the case with H1
A and ZEUS S set, where the main difference is the behavior of the gluon
density at large fractional momenta. Thus the dijet data are more sensitive to
this region than the inclusive data and can discriminate between different
forms of the DPDFs in this region. We note that the ZEUS SJ fit was performed
to the combination of the inclusive and diffractive data.

The H1 data on dijet photoproduction [63] show the suppression with respect to
the theoretical predictions of about $50\%$. The overall shapes of the single
differential cross sections are well reproduced when the calculations used H1
B set of DPDFs. However, the experiment did not observe the difference of the
suppression of the resolved enriched ($x_{\gamma}<0.75$) and the direct
enriched ($x_{\gamma}>0.75$) part of the cross section, contrary to the
theoretical expectations. For the ZEUS data [61] the predictions tend to
overestimate the data, but in general were consistent with no factorisation
breaking within the large uncertainties of NLO calculations.

In [68] the analysis of the published H1 and ZEUS data was performed to better
understand the difference between the different experiments. The conversion
between the phase space of two experiments was done. It was demonstrated that
the results were not very sensitive to the different photon structure
functions. The different hadronization corrections implemented in H1 and ZEUS
did not seem have an impact onto the results.

The factorization in diffractive DIS was also tested in the heavy quark
production. Measurements of $D^{*}$ and open charm production in
photoproduction and in DIS were performed by ZEUS and H1 [69, 70, 71]. The
experimental data were well described by the NLO calculations. Thus the
factorization is supported in charm production both in DIS and in $\gamma\,p$
interactions. Note here that the leading twist production of charm jets
contributes about $40\%$ of the diffractive cross section for transverse
momenta $Q^{2}\ll 4p_{t}^{2}$ and $p_{t}^{2}\gg m_{c}^{2}$. So the validity of
factorization for charm jets in photoproduction would require even larger
breakdown of factorization for production of light quark jets than the one
indicated by the inclusive diffractive dijet data.

###  2.5 Predictions for EIC, LHeC, FCC-eh

There are several proposed future DIS machines that could explore the
diffractive phenomena with much higher precision than at HERA as well as at
higher energies. The prospects of the measurements of the inclusive
diffraction and constraints on the diffractive parton distribution functions
have been studied in detail and presented in [17, 72, 73], and selected
results will be summarized below.

At lower energy end, of about $\sqrt{s}\sim 100\,\rm GeV$, the Electron-Ion
Collider (EIC) in the US [6] will be able to measure diffractive phenomena,
covering a smaller kinematic region than HERA in $ep$ but a completely novel
region in $e$A with respect to fixed target experiments where diffraction has
been barely studied. Thanks to very high luminosity $10^{34}\,{\rm
cm}^{-2}{\rm s}^{-1}$, and precise forward instrumentation EIC should be able
to potentially reach higher values of $\xi$ and possibly measure the
longitudinal diffractive structure function, see [74].

At higher energy end, there are two proposals for the electron-proton and
electron-ion collisions at CERN. The Large Hadron-electron Collider (LHeC) is
a project [75, 76, 77, 72] that would utilize the $7\,\rm TeV$ proton beam
from the High Lumi LHC and collide it with a $60(50)\,\rm GeV$ electron beam
accelerated by an energy recovery linac, thus reaching a centre-of-mass energy
$\sqrt{s}=1.3\,\rm TeV$. The instantaneous peak luminosity is projected [78,
79] to reach also $10^{34}\,{\rm cm}^{-2}{\rm s}^{-1}$, similar to EIC, and
about three orders of magnitude higher than HERA. The projected running of the
machine is over three periods. In the initial run period the total integrated
luminosity is estimated to be $50\;{\rm fb}^{-1}$. Throughout the entire
operation the LHeC is projected to reach $1\,{\rm ab}^{-1}$ integrated
luminosity. At yet higher energies, the next generation $ep$ collider would be
the Future Circular Collider in electron-hadron mode (FCC-eh), utilizing the
$50\,\rm TeV$ proton beam from the FCC [80, 81] which would probe DIS at
centre-of-mass energy of $\sqrt{s}=3.5\,\rm TeV$ with a total integrated
luminosity of several ${\rm ab}^{-1}$. Both machines would also have
capabilities to run in $e$A mode with center of mass energy $\sqrt{s}=812\,\rm
GeV$ for LHeC and $\sqrt{s}=2.2\,\rm TeV$ per nucleon for FCC-eh respectively.
The projected integrated luminosity for the $e$A collisions would be of the
order of $10\,{\rm fb}^{-1}$.

Figure 7: Kinematic phase space for inclusive diffraction in $(x,Q^{2})$ for
the EIC (magenta region), the LHeC (orange region) and the FCC-eh (dark blue
region) as compared with the HERA data (light blue region, ZEUS-LRG [46],
H1-LRG [39], HERA-FLPS [82]). The acceptance limit for the electron in the
detector design has been assumed to be $1^{\circ}$, and we take $\xi<0.4$.
Figure from [17], https://doi.org/10.1103/PhysRevD.100.074022.

In Figure 7 we show the accessible kinematic range in $(x,Q^{2})$ for the four
machines: HERA, EIC, LHeC and FCC-eh. The EIC region will extend the range of
HERA towards largest values of $x$ as well as lowest $Q^{2}$. For the LHeC
design the range in $x$ is increased by a factor $\sim 20$ over HERA and the
maximum available $Q^{2}$ by a factor $\sim 100$. The FCC-eh machine would
further increase this range with respect to LHeC by roughly one order of
magnitude in both $x$ and $Q^{2}$.

Simulations of the pseudodata were performed for these machines using
extrapolations of the reduced cross sections with the ZEUS-SJ DPDFs, see Eq.
(6). The pseudodata were generated with the errors given by the total error
consisting of uncorrelated $5\%$ systematic error and the statistical error
computed assuming luminosity of $2\,{\rm fb}^{-1}$.

In Fig. 8 we show an example of the pseudodata simulation for the LHeC case.
We show a subset of the simulated data for the diffractive reduced cross
section $\xi\sigma_{\rm red}$ as a function of $\beta$ in selected bins of
$\xi$ and $Q^{2}$. The errors are very small and are dominated by the
systematics.

Figure 8: Selected subset of the simulated data for the diffractive reduced
cross section as a function of $\beta$ in bins of $\xi$ and $Q^{2}$ for $ep$
collisions at the LHeC. The curves for $\xi=0.01,0.001,0.0001$ are shifted up
by 0.04, 0.08, 0.12, respectively. Figure from [17],
https://doi.org/10.1103/PhysRevD.100.074022.

Using these pseudodata one can estimate the experimental precision with which
the diffractive parton densities can be extracted. In [17] detailed studies
were performed, where the simulated data were used to predict possible
constraints on the DPDFs. This analysis demonstrated that the DPDFs
determination accuracy improves with respect to HERA by a factor of 5–7 for
the LHeC and 10–15 for the FCC-eh.

As an example, in Fig. 9 relative uncertainties on the diffractive PDFs are
shown for the LHeC and FCC-eh as a function of the longitudinal momentum
fraction $z$ and the scale $\mu^{2}$. In particular we show the variation of
the relative precision of DPDFs with the change of the minimal value of
$Q^{2}$ from $1.8\ {\rm GeV^{2}}$ (curves) to $5\,\rm GeV^{2}$ (bands). The
LHeC scenario is indicated in green and FCC-eh in red. There is a quite
substantial effect on the achieved precision depending on the minimal value of
$Q^{2}$, and it is clear from the figure that that both machines will be very
sensitive to the low $Q^{2}$ region and therefore potentially able to
constrain higher twists and/or saturation effects.

Figure 9: Relative uncertainties on the diffractive gluon PDF extraction for
four distinct scales $\mu^{2}=6,20,60,200\,\rm GeV^{2}$. The bands correspond
to the choice of the high cut-off on the data included in the fit $Q^{2}_{\rm
min}=5\,\rm GeV^{2}$ and the lines correspond to the lower choice $Q^{2}_{\rm
min}=1.8\,\rm GeV^{2}$. The green colour corresponds to the LHeC scenario and
red to the FCC-eh scenario. The cross-hatched areas show kinematically
excluded regions. The bands indicate only the experimental uncertainties, see
the text. Figure from [17], https://doi.org/10.1103/PhysRevD.100.074022.

Diffractive dijet photoproduction in $ep$ and $eA$ scattering in the EIC
kinematics using next-to-leading order (NLO) of QCD was considered in Ref.
[83]. The analysis established the kinematic reaches in most important
kinematic variables for various beam types, energies and kinematic cuts. It
also showed that the EIC will provide new information on DPDFs of the proton,
probe novel nuclear diffractive PDFs, and may illuminate the mechanism of
factorization breaking.

Dijets can also be measured in ultraperipheral collisions (UPC) at the LHC,
see discussion in Sec. 8.

##  3 Dipole model in diffraction

###  3.1 Introduction

The diffractive phenomena in DIS can also be described using the dipole model
of QCD [84]. In that approach, when viewed from the target rest frame, the
virtual photon fluctuates into a quark–antiquark pair, which has large
longitudinal distance to evolve, and then subsequently interacts with the
target [85, 86, 87]. On average, the distance over which this pair forms is
about $\sim 1/(x\,m_{N})$ [88]. The size of the configuration close to the
interaction point in impact parameter space of the dipole depends on the
polarization of the photon. For the case of the longitudinaly polarized
photon, it is of the order of $\sim 1/Q^{2}$ for $q\bar{q}$, where $Q^{2}$ is
the minus photon virtuality. In the case of the transversely polarized photon
this size becomes larger due to the so-called ‘aligned jet’ configurations
(see discussion later in this section). In the aligned jet configuration, the
photon converts into $q\bar{q}$ with small transverse momenta (e.g., 0.3
GeV/c) and over a distance $1/(x\,m_{N})$ it evolves into a hadronic size
configuration.

The $q\bar{q}$ pair size in impact parameter space can also become larger due
to the subsequent parton emissions. The lowest order correction to the
$q\bar{q}$ dipole stems from the single gluon emission and can be computed in
the perturbative QCD, provided the strong coupling is sufficiently small. Both
the quark–antiquark dipole and the quark–antiquark–gluon state can be
described using the lightcone wave functions. The gluon emission is higher
order in the strong coupling, but it is important for the description of the
diffractive states with higher masses $M_{X}^{2}$, or equivalently in the
region of small $\beta$. From the perspective of the DGLAP evolution the gluon
emissions are automatically resummed through the evolution equation in $\ln
Q^{2}$, and thus in such a case, at large $Q^{2}$, there is more than one
gluon emission taken into account.

The dipole picture has certain advantages and disadvantages over the standard
collinear approach. The advantage in that it is very convenient to include the
unitarization corrections. The latter ones are important, particularly at high
energy and low virtualities $Q^{2}$ and they are expected to play more
prominent role in the diffractive cross sections than in the inclusive ones
due to the fact that the diffractive cross sections are squared in the
amplitudes whereas the inclusive cross sections are proportional to the
imaginary part of the scattering amplitude. Thus the use of dipole model opens
up the possibility of convenient incorporation of unitarization and some of
the higher twist contributions into the description of diffraction.

The dipole picture has also some limitations, namely it is suitable when the
center-of-mass energy is high, i.e., Bjorken $x$ sufficiently small, and when
the ratio of $M_{X}^{2}/Q^{2}$ is not extremely large. Also, it parametrically
underestimates the strength of double (triple …) scattering of a small dipole
with 2 (3…) nucleons leading to diffraction for the longitudinal
$\sigma_{L}(x,Q^{2})$ cross section being a higher twist effect in difference
from the factorization theorem expectations.

###  3.2 Diffraction in dipole model

In the dipole approach the diffractive structure function $F_{2}^{D}$ can be
expressed as sum of the contributions from the transverse and longitudinally
polarized virtual photons. Usually the two-component model is considered with
$q\bar{q}$ and $q\bar{q}g$ contributions [89, 90, 91, 92]. The structure
function can thus be expressed as the sum of the following terms

$F_{2}^{D(3)}(\xi,\beta,Q^{2})\;=\;F_{T}^{q\bar{q}}+F_{L}^{q\bar{q}}+F_{T}^{q\bar{q}g}\;.$
(18)

The longitudinal contribution from $q\bar{q}g$ state is usually neglected
since it has no leading logarithm in $Q^{2}$ [93]. The $q\bar{q}$ components
for transversely and longitudinally polarized photons have the following
expressions [91]

$\displaystyle\xi
F_{T}^{q\bar{q}}(\xi,\beta,Q^{2})\;=\;\frac{3Q^{4}}{64\pi^{4}\beta
B_{d}}\sum_{f}e_{f}^{2}\int_{z_{f}}^{1}dzz(1-z)\times$
$\displaystyle\times\left\\{[z^{2}+(1-z)^{2}]Q_{f}^{2}\phi_{1}^{2}+m_{f}^{2}\phi_{0}^{2}\right\\}\;,$
(19)

and

$\xi F_{L}^{q\bar{q}}(\xi,\beta,Q^{2})\;=\;\frac{3Q^{6}}{16\pi^{4}\beta
B_{d}}\sum_{f}e_{f}^{2}\int_{z_{f}}^{1}dz\,z^{3}(1-z)^{3}\,\phi_{0}^{2}\;.$
(20)

In the above $f$ denotes quark flavors, $m_{f}$ is the quark mass, $B_{d}$ is
the diffractive slope stemming from the $t$ integration assuming an
exponential form, and $z$ is the photon momentum fraction carried by a quark
(antiquark) in the dipole. In addition, the variables used in the above are
defined as

$z_{f}=\frac{1}{2}\left(1-\sqrt{1-\frac{4m_{f}^{2}}{M_{X}^{2}}}\right),\;\;\;Q_{f}^{2}=Q^{2}z(1-z)+m_{f}^{2}\;,$
(21)

as well as the functions $\phi_{i}$ being defined as

$\phi_{i}=\int_{0}^{\infty}drr\,K_{i}(Q_{f}r)\,J_{i}(k_{f}r)\,\hat{\sigma}(\xi,r)\;\;\;,$
(22)

with $k_{f}=\sqrt{z(1-z)M_{X}^{2}-m_{f}^{2}}$, and $J_{i}$ and $K_{i}$ are the
Bessel functions. We recall that $\xi=(Q^{2}+M_{X}^{2})/(Q^{2}+W^{2})$ and
$\beta=Q^{2}/(Q^{2}+M_{X}^{2})$. The function $\hat{\sigma}(\xi,r)$ in Eq.
(22) is the dipole cross section which contains all the necessary information
about the interaction between the dipole and the proton. It depends on the
dipole size $r$ and it has energy dependence through the dependence on $\xi$.
It can be calculated from theory or modeled phenomenologically as we shall
discuss below.

The formulae (19) and (20) were derived in the case when the two gluons were
exchanged between the color dipole and the proton. All possible couplings were
added up in order to retain the gauge invariance property. The result was
obtained in the form of the $k_{T}$ factorization formula, with the
unintegrated gluon density describing the details of the target. Finally, the
formulae were rewritten in terms of the dipole cross section which can be
related to the unintegrated gluon density. The $q\bar{q}g$ component is given
by [89, 91]

$\displaystyle\xi
F_{T}^{q\bar{q}g}(\xi,\beta,Q^{2})=\frac{81\beta\alpha_{s}}{512\pi^{5}B_{d}}\sum_{f}e_{f}^{2}\int_{\beta}^{1}\frac{dz}{(1-z)^{3}}\left[\big{(}1-\frac{\beta}{z}\big{)}^{2}+\big{(}\frac{\beta}{z}\big{)}^{2}\right]$
$\displaystyle\times\,\int_{0}^{(1-z)Q^{2}}dk^{2}\,\log\bigg{(}\frac{(1-z)Q^{2}}{k^{2}}\bigg{)}\phi_{2}^{2}\;,$
(23)

where the function $\phi_{2}$ is defined as

$\phi_{2}=k^{2}\int_{0}^{\infty}drr\,K_{2}\left(\sqrt{\frac{z}{1-z}}kr\right)\,J_{2}(kr)\,\hat{\sigma}(\xi,r)\;,$
(24)

with $J_{2}$ and $K_{2}$ Bessel functions.

The contribution of Eq. (24) is of course higher order in $\alpha_{s}$. The
$q\bar{q}g$ term in Eq. (23) was computed in the approximation when the
transverse momenta are strongly ordered

$k_{Tq}\simeq k_{T\bar{q}}\gg k_{Tg}\;.$

In the large $N_{c}$ approximation, it can be effectively treated as the
gluonic color dipole. Thus a factor $C_{A}/C_{F}$ must be included in order to
rescale the interaction with respect to the $q\bar{q}$ dipole by changing the
scattering amplitude by the factor 9/4, for details see [94].

###  3.3 Models for the dipole cross section

The dipole cross section can be obtained from theory or it can be modeled
phenomenologically. One can broadly classify the dipole models into several
main categories: those based on DGLAP evolution, obtained from non-linear
evolution equations at small $x$, and phenomenological parametrizations.

In the limit of small dipole size, and in the leading logarithmic
approximation, accounting for the terms $\alpha_{s}\ln
Q^{2}/\Lambda^{2}_{QCD}$, there is an important relation between the dipole
cross section $\hat{\sigma}(x,r)$ and the collinear gluon density
$xg(x,\mu^{2})$ which was derived in [95, 96]

$\hat{\sigma}(x,r)\;=\;\frac{\pi^{2}}{3}r^{2}\left[\alpha_{s}(\mu^{2})xg(x,\mu^{2})\right]_{\mu^{2}=C/r^{2}}\;\;,$
(25)

where the coefficient $C$ can be estimated based on matching to $\sigma_{L}$
(cross section for longitudinally polarized photons) or phenomenologically to
match large and small size contributions smoothly, see [97, 98]. The above
formula has an important property that the dipole cross section vanishes for
small dipole sizes – the color transparency phenomenon. However, since the
cross section is proportional to the gluon density, the interaction can become
strong at sufficiently small $x$. Namely, the probability of interaction at a
fixed impact parameter can reach values close to unity.

The dipole cross section can also be derived in the limit of high energy
including the unitarization corrections in the form of parton saturation
effects. Parton saturation is a phenomenon expected to occur at very high
energy or equivalently at very small value of Bjorken $x$, when the parton
density is very high and gluon recombination is expected to occur [99, 100],
for a comprehensive review of low $x$ and saturation phenomena see Ref. [101].
This effect leads to the modification of the gluon evolution equations to
include the nonlinear terms. There are various approaches to saturation: Color
Glass Condensate (CGC) [102, 103, 104, 105, 106, 107, 108, 109, 110, 111],
Balitsky hierarchy [112, 113], and Balitsky–Kovchegov equation [114, 115]; we
shall describe more details of these approaches and the low $x$ physics in
Sec. 4.2.

The fundamental property of the parton saturation is the presence of the
dynamically generated saturation scale $Q_{s}(x,b)$ which depends on Bjorken
$x$, impact parameter as well as the target [99, 100, 102, 103]. To be precise
it can depend on the mass number $A$ of the nucleus. It divides the dilute and
dense partonic regimes, i.e., the regions where the parton density is low and
the evolution is linear, and the region where the parton density is high and
the evolution needs to be supplemented by the nonlinear terms which tame the
growth of the scattering amplitude. The schematic view of the dilute and dense
regions together with the saturation scale $Q_{s}$ is illustrated in Fig. 10
(in the case averaged over impact parameter).

Using the solution to the non-linear Balitsky–Kovchegov equation, numerous
fits to HERA data were performed. The fits were performed in the leading
logarithmic order approximation with running coupling [116], including
resummation in the form of kinematical constraints [117, 118] as well as very
recently by taking into account NLO corrections to the dipole wave function
[119] for light quarks.

Figure 10: Schematic representation of parton evolution $(x,Q^{2})$ plane. The
dots symbolize partons, with their number density increasing towards
decreasing $x$ and the resolution in transverse size, the latter one
decreasing when scale $Q^{2}$ is increased. Yellow diagonal line indicates the
saturation scale $Q_{s}^{2}(x)$ which divides the dilute regime where the
linear evolution is applicable (to the right of the line) and the dense regime
where the nonlienar evolution needs to be taken into account (to the left of
the line).

An alternative, phenomenological way to include the unitarization corrections
is to eikonalize the formula Eq. (25), see for example [120]

$\hat{\sigma}(x,r)\;=\;\sigma_{0}[1-\exp(\frac{\pi^{2}}{3\sigma_{0}}r^{2}\left[\alpha_{s}(\mu^{2})xg(x,\mu^{2})\right]_{\mu^{2}=C/r^{2}})]\;\;.$
(26)

In the limit of small dipole sizes and when $\alpha_{s}(\mu^{2})xg(x,\mu^{2})$
is not large the eikonalized formula can be expanded and it coincides with Eq.
(25). Otherwise in the limit of large dipole sizes and small $x$ it saturates
to a constant value set by the $\sigma_{0}$.

The above model can be thought of as an extension of the Golec-
Biernat–Wusthoff (GBW) model [121, 122] that includes the DGLAP evolution. The
GBW model posseses qualitative features that can be found when solving the
Balitsky–Kovchegov equation. It has been very successful in the description of
the inclusive HERA data on structure function $F_{2}$ as well as the
diffractive structure function. The consistency of the simultaneous
description of the non-diffractive and diffractive data is a very important
test for the dipole models. The GBW parametrization [121, 122] has the
following form for the $q\bar{q}$ dipole cross section

$\hat{\sigma}(\xi,r)\;=\;\sigma_{0}\big{[}1-\exp(-r^{2}Q_{s}^{2}(\xi)/4)\big{]}\;,$
(27)

where the normalisation constant was found to be $\sigma_{0}=29.12\;{\rm mb}$
from the fit. The saturation scale $Q_{s}$ is given by

$Q_{s}^{2}(\xi)=Q_{0}^{2}\,\bigg{(}\frac{\xi}{x_{0}}\bigg{)}^{-\lambda}\;,$
(28)

with $Q_{0}^{2}=1\,{\rm GeV^{-2}}$ $x_{0}=4\cdot 10^{-5}$ and $\lambda=0.277$.

By comparing Eq. (27) with Eq. (26) we observe that the saturation scale in
the GBW model approximates, roughly speaking, the product of the gluon density
and the coupling constant

$\alpha_{s}(\mu^{2})xg(x,\mu^{2})\rightarrow Q_{s}^{2}(x)\;.$ (29)

More recently, fits based on the DGLAP improved saturation model and Eq. (26)
were performed [123] to the HERA data. It was demonstrated, not surprisingly,
that good description of the HERA data at values of $Q^{2}>50\rm\;GeV^{2}$ can
only be achieved by including the DGLAP evolution in the form of the dipole
model (26).

An example of the model which has been successfully used to describe the
inclusive $F_{2}(x,Q^{2})$ is based on the analysis of the semi-analytical
solution to the nonlinear Balitsky-Kovchegov equation. The CGC parametrisation
of the dipole scattering amplitude (see discussion below) is given by the
following model [124, 125, 126]

$N(\xi,r,b)\;=\;S(b)\,{\cal N}(\xi,r)\;,$ (30)

here $S(b)$ is the impact parameter profile $S(b)=\exp(-b^{2}/(2B_{d}))$ with
the diffractive slope $B_{d}$ taken from the HERA data $B_{d}=6\,\rm GeV^{-2}$
[44] (see discussion later in this section about the impact parameter
dependence). Upon the integration of the scattering amplitude (30) over the
impact parameter $b$, the resulting dipole cross section reads

$\hat{\sigma}(\xi,r)\;=\;4\pi B_{d}\,{\cal N}(\xi,r)\;,$ (31)

With the value of the diffractive slope from HERA this gives $4\pi
B_{d}=29\,\rm mb$. The functional form of the CGC dipole cross section is
motivated by the analysis of the approximated solution to the BK equation.
Following [126] it reads

${\cal N}(\xi,r)=\left\\{\begin{array}[]{ll}{\cal
N}_{0}\bigg{(}\frac{rQ_{s}}{2}\bigg{)}^{2\gamma_{s}}\,\exp(\frac{2\ln^{2}(rQ_{s}/2)}{\kappa\lambda\ln(\xi)})&{\rm
for}\;rQ_{s}\leq 2\;,\\\ 1-\exp(-4\alpha\ln^{2}(\beta rQ_{s}))&{\rm
for}\;rQ_{s}>2\;.\end{array}\right.$ (32)

The fitted parameters in the saturation scale
$Q_{s}^{2}(\xi)=Q_{0}^{2}(x_{0}/\xi)^{\lambda}$ are $\lambda=0.22$,
$x_{0}=1.63\cdot 10^{-5}$ (with $Q_{0}^{2}=1\,{\rm GeV^{2}}$ being fixed). The
parameters $\alpha=0.615$, $\beta=1.006$ are chosen such that the amplitude
${\cal N}$ and its first derivative are continuous at point ${\cal
N}_{0}=0.7$. The parameter $\kappa=9.9$ was taken from the approximated
analytical solution to the non-linear BK equation. In the fit above with the
heavy quarks and parameter $\gamma_{c}=0.7376$ was also fitted, though other
fits presented in [126] were using the value which can be obtained from the
approximated analytical solution as well.

Figure 11: The dipole cross section in units of $\rm mb$ as a function of the
dipole size in $\rm fm$ for two different values of $x=10^{-2}$ and $10^{-4}$
(left and right panel respectively). Four models are shown: GBW, red solid,
[121]; GBS + DGLAP, magenta points, [123]; CGC, blue dashed, [126]; MFGS,
black points, [127].

In Fig. 11 several models have been collected for the dipole cross sections
and shown for comparison. The dipole cross sections are shown as a function of
the dipole size and for two different values of $x=10^{-2}$ and $x=10^{-4}$.
The models presented are GBW [121] Eq. (27), GBS with DGLAP [123] Eq. (26),
MFGS [127] and CGC [124, 125, 126]. We see that the dipole cross section grows
with increasing values of the dipole size and for large values of dipole sizes
the dipole cross section saturates to a constant value for GBW, GBS+DGLAP and
CGC. In the case of the MFGS model there is a residual increase at large
dipole sizes which is modeled by the soft Pomeron behavior $\sim x^{-0.08}$.
When $x$ is decreased the dipole cross section becomes larger overall.
Characteristically it becomes saturated for lower values of the dipole size
when $x$ is smaller. This is typical feature of the dipole models, and the
transition between the region where the dipole cross section is small and
large is given by the saturation scale, as discussed above. Note that in Fig.
11 the modelled dipole cross sections are similar in the perturbative domain,
whereas there are sizeable differences in the non-perturbative regime, which
could be attributed to different treatment of quark masses in the wave
function.

The dipole cross section depends on energy (through longitudinal momentum
fraction) and on the dipole size. It is related to the imaginary part of
dipole scattering amplitude through

$\hat{\sigma}(x,{\bf r})\;=\;2\int d^{2}{\bf b}\,N(x,{\bf r},{\bf b})\;,$ (33)

for the interaction of the dipole of size ${\bf r}$ interacting with the
target at impact parameter ${\bf b}$. In the above we put back explicitly
dependence on two-dimensional vectors ${\bf r}$ and ${\bf b}$. Note that, in
principle the amplitude depends on the relative angle between the two vectors.

It is the dipole scattering amplitude that should satisfy the unitarity
constraint

$N\leq 1\;.$ (34)

An interesting limit in the context of hadronic interactions at high energies
is the black disk limit. It is defined as the case when the probability of
inelastic interaction is equal to unity for all impact parameters less than
the size of the target $R$. This corresponds to the condition

$N(x,{\bf r},{\bf b})\;=\;\theta(R-b)\;.$ (35)

In this limit the inelastic cross section is equal to elastic one, and both of
them equal to half of the total cross section.

In general, one can define the saturation scale from the dipole scattering
amplitude for example through the condition

$N(x,r=1/Q_{s},{\bf b})={\rm const.}\;,$ (36)

where the $\rm const.$ is of the order of $1/2$. The above condition defines
$Q_{s}(x,b)$ as a function of $x$ and impact parameter. The solution to the
nonlinear Balitsky–Kovchegov equation [112, 113, 114, 115] satisfies the
unitarity condition (34), see Sec. 4.

Another form the dipole scattering amplitude that satisfies the unitarity
constraint in QCD, has been introduced by Mueller in [84]

$N(x,{\bf r},{\bf
b})=1-\exp\big{[}-\frac{\pi^{2}}{2N_{c}}r^{2}\alpha_{s}(\mu^{2})xg(x,\mu^{2})T({\bf
b})\big{]}\;,$ (37)

with the scale $\mu$ defined as in Eq. (25) and $T({}\bf b)$ is the profile
function for the target. This expression can be viewed as generalization of
Eq. (26) by including the dependence on the impact parameter ${\bf b}$.
Equation (37) is usually refered to as the Gribov–Glauber–Mueller model [84],
which takes into account the multiple scatterings of the dipole off the
target. The original idea goes back to Glauber [128, 129, 130] who treated it
in a quantum-mechanical model and to Gribov [4, 85] in the context of photon
and hadron–nucleus scattering. Gribov expressed amplitudes for hadron–nucleus
scattering through the diffractive (elastic plus inelastic) amplitudes of
interaction with one, two, three, $\dots$, nucleons in the nuclei, see Sec. 7.

Note that in the discussed eikonal approximation (37), the rescattering terms
corresponding to the interaction with $n$ nucleons are proportional to
$r^{2n}$. Hence they are higher twist terms. Thus, this approximation neglects
the contribution of multiparton configurations responsible for the leading
twist diffraction and leading twist nuclear shadowing, see Sec. 7 and Ref.
[131].

###  3.4 Distribution of dipole sizes

For a given value of the photon (minus) virtuality $Q^{2}$, the photon wave
function will have a distribution of the dipole configurations of different
sizes $r$. It is instructive to investigate this distribution in the case of
diffraction as compared to the inclusive cross section. The inclusive cross
section in the dipole model is given by

$\sigma_{T,L}^{\gamma^{*}p}(x,Q^{2})\;=\;\sum_{f}\int d^{2}{\bf
r}\,dz|\Psi_{T,L}^{f}({r},Q,z)|^{2}\,\hat{\sigma}(x,{r})\;,$ (38)

where the photon wave functions [132, 133, 134] have the following form for
the transverse case

$|\Psi_{T}^{f}({r},Q,z)|^{2}\;=\;\frac{3\alpha_{\rm
em}}{2\pi^{2}}e_{f}^{2}\left\\{[z^{2}+(1-z)^{2}]Q_{f}^{2}K_{1}^{2}(Q_{f}r)+m_{f}^{2}K_{0}^{2}(Q_{f}r)\right\\}\;,$
(39)

and for the longitudinal photon case

$|\Psi_{L}^{f}({r},Q,z)|^{2}\;=\;\frac{3\alpha_{\rm
em}}{2\pi^{2}}e_{f}^{2}\left\\{4Q^{2}z^{2}(1-z)^{2}K_{0}^{2}(Q_{f}r)\right\\}\;.$
(40)

In Fig. 12 we show the distribution of the dipole sizes for the case of the
dipole cross section from the GBW model for two different values of
$Q^{2}=2,10\rm\;GeV^{2}$ (left and right plot respectively). In particular we
plot the function $1/\sigma_{T,L}^{\gamma^{*}p}\;p_{T,L}(r,x,Q^{2})$ which is
defined as

$\sigma_{T,L}^{\gamma^{*}p}(x,Q^{2})\;=\;\int_{0}^{\infty}dr\,p_{T,L}(r,x,Q^{2})\;.$
(41)

Figure 12:  The normalized dipole size distribution for the inclusive (red)
and diffractive (blue) cross section using the GBW dipole cross section. Both
transverse (solid) and longitudinal (dashed) polarizations are shown. Two
values of photon virtuality are chosen : $Q^{2}=2\ {\rm GeV^{2}}$ (left plot)
and $Q^{2}=10\;{\rm GeV^{2}}$ (right plot). $x$ was fixed so that
$Q_{s}(x)=1\rm GeV$ and masses $m_{u}=m_{d}=m_{s}=0.14\;\rm GeV$.

Figure 13:  The median size distribution for inclusive and diffractive cross
section, based on the GBW model [121, 122], both longitudinal and transverse
photon polarization cases. Left plot: function of $x$, fixed $Q^{2}=10\;{\rm
GeV}^{2}$. Right plot: function of $Q$, fixed $x=x_{0}=0.4\times 10^{-5}$ such
that $Q_{s}(x_{0})=1\,\rm GeV$ in the GBW model.

Solid lines denote the dipole size distribution for the transverse photon and
the dashed lines for the longitudinal polarization. Two values of $Q^{2}$ are
chosen to illustrate the change in the distribution from larger to smaller
sizes. We observe that the distribution for transversely polarized photons has
a longer tail, extending to larger values of dipole sizes $r$ as compared to
the longitudinal one. This is due to the presence of the aligned jet
configurations in the $F_{T}(x,Q^{2})$ structure function originating from the
endpoint configurations $z=0$ and $z=1$.

This effect is also illustrated in Figs. 13, where the median of the
distribution is plotted as a function of $x$ and $Q$ (left and right plot
respectively). The median size $r_{\rm m}$, defined by the following relation

$\frac{\int_{0}^{r_{\rm
m}}dr\,p_{T,L}(r,x,Q^{2})}{\sigma_{T,L}^{\gamma^{\ast}p}}=\frac{1}{2}\,.$ (42)

One can see from the figure that $r_{\rm m}$ is a slowly increasing function
of $x$ due to the fact that the GBW dipole cross section becomes larger at
small $r$ when $x$ is decreased, and it always saturates to a constant at
large $r$. Thus the dipole size distribution shifts to smaller values of $r$
when $x$ decreases. In the case of the $Q$ dependence, the median dipole size
decreases with increasing $Q$. This is connected with the fact that the peak
of the integrand in the cross section moves to smaller values of $r$ with
increasing value of $Q$ due to the effect of the photon wave function. The
different dependence for the case of longitudinal and transverse photons is
evident. Note that in Fig. 13 we show the dipole sizes up to values of
$x\simeq 0.1$ which is most likely beyond the region of validity of the dipole
model. This region is shown for purely illustrative purposes to demonstrate
the behavior of the model extrapolation.

We also analyze this distribution for the case of the diffractive cross
section. In the approximation of high energy, when $\xi$ is small, such that
$\beta\simeq 1$, one can rewrite the formula for the cross section (19) and
(20) using the substitution of $\xi\simeq x$ inside the dipole cross section
and obtain the following form for the diffractive cross section, see for
example [135]

$\sigma_{T,L}^{\gamma^{*}p(D)}(x,Q^{2})\;=\;\frac{1}{16\pi B_{d}}\sum_{f}\int
d^{2}{\bf r}\,dz|\Psi_{T,L}^{f}({r},Q,z)|^{2}\,\hat{\sigma}^{2}(x,{r})\;.$
(43)

We note that only a $q\bar{q}$ component is included in this approximation.
The above formula can be interpreted as the realization of the Good–Walker
idea [3], which states that the diffraction occurs due to the different
absorption of eigenstates of the interaction operator (scattering matrix). In
the context of the dipole picture at small $x$, these are the $q\bar{q}$
dipoles with definite values of $r$ and $z$.

At this point, it is important to clarify the connection between the Good-
Walker formalism and the dipole model. As mentioned above, the quark–antiquark
wave functions corresponding to different dipole cross sections are
orthogonal, which explicitly realizes the assumption of the Good–Walker model
designed for soft diffractive processes, see discussion in Sec. 6. At the same
time, in contrast to this model, orthogonality of scattering eigenstates does
not hold for scattering at finite $t\neq 0$ since elastic scattering mixes
dipoles of different sizes. Thus, it is not clear whether one can build an
orthogonal set of $q\bar{q}$ and $q\bar{q}g$, etc. dipole states even for
$t=0$. As a result, while the effects of quantum evolution and the
quark–antiquark–gluon Fock states are important for the description of
diffraction [136], their connection to the formalism of Good–Walker
eigenstates in general cannot be established.

The distributions analogous to Eq. (41) for the case of diffraction (using Eq.
(43)) are shown in Figs. 12 and are indicated by the blue lines. As compared
with the inclusive case, the distributions are wider, with the peak shifted to
larger values of $r$. This is also evident in the median plots, Fig. 13 where
the median size shifts substantially to larger dipole sizes. Significant
difference between the cases of transverse and longitudinally polarized
photons is evident. This has physical consequence that the diffraction is
sensitive more to the soft physics. This qualitatively explains the smaller
$\alpha_{I\\!\\!P}(0)$ value for diffraction than the inclusive case. The
broader distribution for the transverse case is even more prominent in
diffractive case.

The fact that diffraction is dominated by the larger dipole sizes than the
inclusive process has important implications for the saturation. As showed in
[122] saturation of the dipole cross section is very important for
diffraction, with the saturation scale $Q_{s}(x)$ playing a role of a
regulator which reduces the contribution to the cross section from the
infrared regime. As a result, as shown in [122] the ratio of the diffractive
to inclusive cross section is approximately constant (modulo logarithmic
corrections) for the dipole model with saturation (when only $q\bar{q}$
contributions are considered).

Note, however, that the conclusion of Ref. [122] is sensitive to modeling of
the dipole cross section at large $|r|$, where it is taken to be energy
independent. In this approach, $N(x,r,{\bf b})$ first grows with a decrease of
$x$, but then starts to decrease so that the $N(x,r,{\bf b})=1$ limit is never
reached. On the other hand, in Ref. [127], where the dipole cross section at
large $|r|$ is motivated by soft Pomeron exchange, the ratio of the
diffractive to inclusive cross sections grows with an increase of energy.

It is worth mentioning that the general considerations about the distributions
are valid for large values of $Q^{2}$. When $Q^{2}$ is small, like in the case
of Fig.12 (left plot) and 13 (right plot, low $Q$ region), the scales related
to the non-zero quark masses have a non-negligible effect on the
distributions. Effectively, the larger masses tend to shift slightly the
dipole size distribution towards smaller values of dipole sizes and increase
the value of $\hat{\sigma}(x,r)$ to keep the same magnitude of
$F_{2}(x,Q^{2})$.

Figure 14: The diffractive structure function $F_{2}^{D(3)}$ from the dipole
model based on the GBW parametrisation of the dipole cross section. Dotted
lines indicate the components $q\bar{q}$, transverse and longitudinal and the
$q\bar{q}g$ transverse component. Solid line indicates the sum of all three
contributions. Figure from [94], https://doi.org/10.1103/PhysRevD.79.114010.

###  3.5 Diffractive structure function from dipole model and higher twists

In Fig. 14 the diffractive structure function $\xi F_{2}^{D(3)}$ is shown as a
function of $\beta$ for fixed value of $Q^{2}=8\;\rm{\rm GeV^{2}}$ and
$\xi=0.0042$. In addition to the total value, three components as given by Eq.
(18) are separately shown. We observe that they dominate the cross section in
different regions of diffractive masses, see [92, 94]. The $F_{T}^{q\bar{q}}$
dominates for $\beta\sim 1/2$ which corresponds to $M_{X}^{2}\sim Q^{2}$,
$F_{L}^{q\bar{q}}$ dominates for large $\beta\sim 1$, that is small
diffractive masses $M_{X}^{2}\ll Q^{2}$ , and the $F_{T}^{q\bar{q}g}$ is most
important for large diffractive masses $M_{X}^{2}\gg Q^{2}$, corresponding to
$\beta\ll 1$. By analyzing the formulae for $q\bar{q}$ contribution from
transverse and longitudinal cases one can demonstrate that in the small mass
limit $M^{2}\ll Q^{2}$, the longitudinal part dominates, even though it is
suppressed by an additional power of $1/Q^{2}$. This stems from the fact that
the transverse part of the cross section is dominated by the aligned jet
configurations which in turn involves large distances in the dipole size $r$
(see analysis above). As a result, in the small diffractive mass limit, this
component is suppressed. On the other hand the longitudinal cross section is
dominated by the symmetric configurations of $z\simeq 1/2$, and the cross
section is dominated by the small dipole sizes. Thus the diffractive cross
section is dominated by the contribution from the longitudinally polarized
photons in the small mass $M_{X}^{2}\ll Q^{2}$ or large $\beta\rightarrow 1$
limit. Note that the same holds for inclusive vector meson production, see
Sect. 5.

Figure 15:  Variation of the $\chi^{2}$ of the fit to the diffractive
structure function as a function of the cutoff $Q^{2}_{\rm min}$, where only
data for $Q^{2}>Q^{2}_{\rm min}$ were included in the fit. DGLAP fit - dhased
line, DGLAP supplemented with the higher twists from the dipole model MMS
[125, 137], solid line. Figure from [47],
https://doi.org/10.1103/PhysRevD.86.111501.

Fits to the diffractive data using the dipole model were performed in [94],
where good description of the data was obtained. The dipole model with
parametrization described above contains towers of higher twists effects which
go beyond the leading twist DGLAP approach. As mentioned in the previous
section, the DGLAP description of the diffractive inclusive data is inadequate
for low values of $Q^{2}$. To be precises the fits are inadequate for
$Q^{2}<5\;\rm GeV^{2}$ for ZEUS data and $Q^{2}<8.5\;\rm GeV^{2}$ for H1 data.
The fits deteriorate in a low $Q^{2}$ region where the $\chi^{2}$ deviates by
$100\%$. The problem is illustrated in Fig. 15 where the value of $\chi^{2}$
is shown as a function of $Q^{2}_{\rm min}$, the latter being defined as the
cutoff of the data used in DGLAP fits. To be precise, the fits were done with
subset of the ZEUS LRG data with $Q^{2}>Q_{\rm min}^{2}$. The NNLO order
analysis does not cure this problem, though it was limited to the same form of
the initial parametrizations for the diffractive structure function.

Figure 16: The experimental data from ZEUS [27] for $\xi\sigma^{D(3)}$ at low
values of $Q^{2}$. Green dashed line, DGLAP ZEUS-SJ fit, blue dashed line,
DGLAP+twist-4 corrections from the MMS dipole saturation model, red solid
line, DGLAP with twist-4 and 6 contributions from MMS dipole saturation model.
In yellow region the contribution due to the $q\bar{q}gg$ contribution may be
neglected. Figure from [47], https://doi.org/10.1103/PhysRevD.86.111501.

In [47] a twist analysis was performed, where the DGLAP fit was supplemented
by the higher twist expansion from the dipole model. In that approach, it was
shown that the dipole model formula can be systematically expanded in powers
of $1/Q^{2}$. Several calculations were performed and compared with each other
: DGLAP NLO, dipole approach with saturation models MMS [125, 137] and GBW
[121, 122], twist-2 truncation of the dipole approach DGLAP plus twist-4 part
of MMS dipole model, and DGLAP plus twist-4 and twist-6 parts for the MMS
dipole model. This study concluded that the DGLAP NLO description and twist-2
parts of dipole models were consistent in the description of the HERA data at
moderate and high $Q^{2}$ but both failed at low $Q^{2}$. The dipole models
gave better description of the data but were still below the data. The DGLAP
supplemented by the twist-4 part of the MMS saturation model describes the
data much better. The best description was given by the DGLAP supplemented by
the twist-4 and twist-6 parts of the dipole model.

The results are presented in Fig. 16, where we show the comparison of the
calculations to the HERA data. It is clear that the DGLAP supplemented by
twist-4 and twist-6 contributions provides best description of the data in the
low $Q^{2}$ region. We observe however that the curves are still lower than
the data in the region of low values of $\beta$. This is the region of high
diffractive masses, where more resolved photon fluctuations, $q\bar{q}gg$ may
become more relevant. In that analysis it was found that this correction may
be neglected for values of $\beta>0.035$.

The improvement of the description is demonstrated by the solid line in Fig.15
where the $\chi^{2}$ is shown for the case of the DGLAP supplemented by the
twist-4 and twist-6 components.

In summary, the dipole model has been successful in fitting the diffractive
data on structure function [94]. It provides a convenient way to parametrize
unitarization effects, either through the modeling or through the solution to
the nonlinear evolution equation. Models, where the additional twist
contributions were added to the DGLAP evolution, lead to the improved
description of the data at low values of $Q^{2}$. However, more studies need
to be done to pin down the origin of the slight discrepancy of the collinear
description based on the DGLAP evolution at low values of $Q^{2}$. For
example, studies with more flexible parametrizations of the initial conditions
for the DGLAP evolution, in particular relaxing the assumption of Regge
factorization, would need to be performed in order to find out how much of the
discrepancy can be accommodated in the initial conditions at very low $Q^{2}$.

##  4 Low $x$, resummation and parton saturation

###  4.1 BFKL Hard Pomeron

Perturbative QCD gives robust predictions for processes where hard scales are
involved. This is also true in the case of the diffractive processes in deep
inelastic scattering (DIS), where the collinear factorization of hard
scattering cross section and the diffractive parton distribution functions
(DPDFs) apply in the case of DIS with high values of $Q^{2}$, see Sec. 2. The
DPDFs are evolved with DGLAP evolution which resums powers of $\alpha_{s}\ln
Q^{2}/\Lambda^{2}$ up to the desired order of accuracy. The DGLAP framework is
applicable in the case of large $Q^{2}$ and fixed values of Bjorken $x$.
However, in the limit of high energies, or very low $x$, there are other types
of logarithms, $\ln 1/x$, which are potentially very large and thus need to be
resumed. This limit is usually referred to as the Regge limit, that is when
$s\gg|t|$ and $t={\rm const}$.

The resummation of the leading logarithms of $x$, that is powers
$(\alpha_{s}\ln 1/x)^{n}$, was performed in the seminal papers by Lipatov and
collaborators [30, 29, 31]. In these works, the evolution equation in variable
$\ln 1/x$ or $\ln s$ was derived for the gluon Green’s function, which is
known as the Balitsky–Fadin–Kuraev–Lipatov (BFKL) evolution equation. The BFKL
evolution equation can be written in the following form

$G(x;{\bf k},{\bf k}_{0})=\delta^{(2)}({\bf k}-{\bf
k}_{0})+\int_{x}^{1}\frac{dz}{z}\int d^{2}{\bf k}^{\prime}\,K(\alpha_{s},{\bf
k},{\bf k}^{\prime})\,G(\frac{x}{z};{\bf k}^{\prime},{\bf k}_{0})\;,$ (44)

where $G(x;{\bf k},{\bf k}_{0})$ is the gluon Green’s function which depends
on the transverse momenta of the gluons in the $t$-channel and $K$ is the BFKL
kernel which has an expansion in the strong coupling

$K(\alpha_{s},{\bf k},{\bf k}^{\prime})\;=\;\alpha_{s}K_{0}({\bf k},{\bf
k}^{\prime})\;+\;\alpha_{s}^{2}K_{1}({\bf k},{\bf k}^{\prime})\;+\dots\;$ (45)

A schematic representation of the gluon Green’s function in the context of
high energy scattering is shown in Fig. 17 which depicts the scattering of two
particles in the Regge limit $s\gg|t|$.

Figure 17: A schematic representation of the high energy scattering with the
exchange of the gluon Green’s function $G$. Functions $\Phi_{0}$ denote the
impact factors which describe the coupling of the $t$-channel gluons to the
incoming particles.

The solution to this equation can be obtained by performing the Mellin
transform of the variables, and one ends up in the following eigenvalue
equation

$\omega=\frac{\alpha_{s}N_{c}}{\pi}\,\chi(\gamma)\;,$ (46)

where $\omega$ is the Mellin variable conjugated to the longitudinal momentum
fraction $z$ and $\gamma$ is the conjugate variable to the transverse momentum
${\bf k}$. In the above, function $\chi(\gamma)$ is the BFKL eigenvalue [31]
which in the leading logarithmic approximation reads

$\chi_{0}(\gamma)=2\psi(1)-\psi(\gamma)-\psi(1-\gamma)\;,$ (47)

with

$\psi(\gamma)=\frac{1}{\Gamma(\gamma)}\frac{d\Gamma(\gamma)}{d\gamma}\,,$ (48)

the polygamma function, and the index ’0’ denotes the leading logarithmic
order in $\ln 1/x$, LLx. The solution to this equation exhibits powerlike
behavior in $x$, or Regge-type, $\sim x^{-\lambda}$, with
$\lambda=(N_{c}\alpha_{s}/\pi)\chi_{0}(\gamma=1/2)=(N_{c}\alpha_{s}/\pi)4\ln
2$ given by the saddle-point solution to Eq. (44). The important property of
QCD in high energy limit as demonstrated in [30, 29, 31] is the fact that the
gluon is reggeizzed, that is its propagator can be expressed as
$1/(k^{2})^{\omega(t)}$, with the trajectory $\omega(t)$ which can be
calculated in QCD. The other property of the BFKL evolution equation is the
diffusion in the transverse momenta along the ladder exchanged between the
scattering objects in the high energy limit. This leads to the effect, where
the momenta which are initially perturbative can diffuse into the infrared
region, [138]. The last property is related to the fact that the BFKL ladder
is not ordered in the transverse momenta. This can be seen in the Mellin
space, where the eigenvalue (47) can be approximated by the leading poles

$\chi_{0}(\gamma)\;\simeq\;\frac{1}{\gamma}\,+\,\frac{1}{1-\gamma}\;,$ (49)

where the pole $1/\gamma$ corresponds to the collinear limit $k^{2}\gg
k^{\prime 2}$ and the pole $1/(1-\gamma)$ to the anti-collinear limit
$k^{2}\ll k^{\prime 2}$.

The power-like solution, $x^{-\lambda}$, in the LLx approximation, turned out
to be too steep for the experimental data [139]. With moderate values of the
strong coupling $\alpha_{s}\simeq 0.2$, this gives the intercept
$\lambda\approx 0.5$, which is excluded by the HERA data on inclusive
structure function $F_{2}(x,Q^{2})$.

The next-to-leading logarithmic (NLLx) terms in $\ln 1/x$ to the BFKL
evolution were calculated in [140, 141]. The NLLx terms turn out to be very
large and negative, and also lead to some instabilities in the solution [142],
like the oscillating cross sections. The main source of the NLLx corrections
were identified to be the non-singular part of the DGLAP splitting function,
the choice of the energy scales and the running of the strong coupling, [142,
143] 222It is worth to mention, that the poor convergence of the series for
the total cross section in terms of powers of $\ln(s/\mu^{2})$, where $\mu$ is
electron mass, was first demonstrated in QED [144] by the direct calculation
of the lowest order diagrams for the $e^{+}e^{-}$ pair production in
electron–electron scattering. It was explained in [144] that a fast growth of
the coefficients in front of the powers of $\ln(s/\mu^{2})$ reflects the
highly restricted phase space for obtaining logarithmic contributions. .

The kernel eigenvalue at NLL in QCD has the following form [140, 141]

$\displaystyle\chi_{1}(\gamma)$ $\displaystyle=$
$\displaystyle-\frac{b}{2}[\chi^{2}_{0}(\gamma)+\chi^{\prime}_{0}(\gamma)]-\frac{1}{4}\chi_{0}^{\prime\prime}(\gamma)$
(50)
$\displaystyle-\frac{1}{4}\left(\frac{\pi}{\sin\pi\gamma}\right)^{2}\frac{\cos\pi\gamma}{3(1-2\gamma)}\left(11+\frac{\gamma(1-\gamma)}{(1+2\gamma)(3-2\gamma)}\right)$
$\displaystyle+\left(\frac{67}{36}-\frac{\pi^{2}}{12}\right)\chi_{0}(\gamma)+\frac{3}{2}\zeta(3)+\frac{\pi^{3}}{4\sin\pi\gamma}-\Phi(\gamma)\;,$

with

$\Phi(\gamma)=\sum_{n=0}^{\infty}(-1)^{n}\left[\frac{\psi(n+1+\gamma)-\psi(1)}{(n+\gamma)^{2}}+\frac{\psi(n+2-\gamma)-\psi(1)}{(n+1-\gamma)^{2}}\right]\;,$
(51)

where $b=(33-2N_{f})/(12\pi)$ with $N_{f}$ the number of (active) flavors.

The well known problem that arises at NLLx order in BFKL is due to the
presence of double and triple collinear poles. The double poles are arising
due to the running coupling and the non-singular (in $1/z$) part of the DGLAP
splitting function which appear at NLLx order. To be precise, keeping most
singular $\gamma\rightarrow 0$ and $\gamma\rightarrow 1$ contributions in Eq.
(50) gives for the corresponding terms

$-\frac{b}{2}[\chi^{2}_{0}(\gamma)+\chi^{\prime}_{0}(\gamma)]\;\;\;\longrightarrow\;\;\;-b\frac{1}{(1-\gamma)^{2}}\;,$
(52)

for the running coupling, and

$-\frac{1}{4}\left(\frac{\pi}{\sin\pi\gamma}\right)^{2}\frac{\cos\pi\gamma}{3(1-2\gamma)}\left(11+\frac{\gamma(1-\gamma)}{(1+2\gamma)(3-2\gamma)}\right)\;\;\;\longrightarrow\;\;\;-\frac{11}{12}\frac{1}{\gamma^{2}}\,-\,\frac{11}{12}\frac{1}{(1-\gamma)^{2}}\;,$
(53)

which is the DGLAP contribution. There are also triple collinear poles which
appear due to the kinematical constraint [145, 146]. Such constraint was
discussed in the BFKL context as originating from the improved kinematics, and
more precisely by the requirement that the exchanged momenta are dominated by
the transverse components [147, 148], for more recent work on kinematical
constraint see [149]. These contributions, when truncated at the NLLx order,
generate the double logarithms in transverse momenta in the kernel and in the
Mellin space they exhibit most singular behavior resulting in the triple
collinear poles. The corresponding term in the NLLx eigenvalue is

$-\frac{1}{4}\chi_{0}^{\prime\prime}(\gamma)\;\;\;\longrightarrow\;\;\;-\frac{1}{2}\frac{1}{\gamma^{3}}\,-\,\frac{1}{2}\frac{1}{(1-\gamma)^{3}}\;.$
(54)

As it has been demonstrated in [142, 143] this collinear approximation to the
NLLx eigenvalue

$\chi(\gamma)_{\rm
coll}=-b\frac{1}{(1-\gamma)^{2}}-\frac{11}{12}\frac{1}{\gamma^{2}}\,-\,\frac{11}{12}\frac{1}{(1-\gamma)^{2}}-\frac{1}{2}\frac{1}{\gamma^{3}}\,-\,\frac{1}{2}\frac{1}{(1-\gamma)^{3}}\;,$
(55)

accounts for the major part of the NLLx corrections given by $\chi_{1}$.

###  4.2 Resummation at low $x$

Resummation procedures were constructed in the early 2000’s to stabilize the
BFKL solution [150, 151, 152, 153, 154, 155, 145, 156, 146, 157, 158, 159,
160]. General setup for the CCSS [145, 156, 146] resummation scheme was based
on the analysis of poles in the Mellin space, but the final formulation and
the solution to the equation was given in the momentum space. A similar idea
for the resummation was formulated previously also in Ref. [161] by combining
the DGLAP and BFKL evolution with the kinematical constraint. In the CCSS
resummation one subtracts triple and double poles and incorporates the full
DGLAP splitting function and the kinematical constraint which both resum
double and triple poles respectively. In addition, more subtractions are
needed to ensure the conservation of the momentum sum rule.

In the original CCSS scheme [145, 146] one starts with the LLx+NLLx BFKL
kernel with LO DGLAP splitting function and puts in kinematical constraint
[148]. Imposing kinematical constraint means that the transverse momentum
integrals in Eq. (44) are limited by

$k^{\prime 2}\leq\frac{k^{2}}{z}\;.$ (56)

In the Mellin space this leads to the following modification of the LLx kernel
eigenvalue

$\chi_{0}(\gamma)\rightarrow\tilde{\chi}_{0}(\gamma,\omega)=2\psi(1)-\psi(\gamma)-\psi(1-\gamma+\omega)\;,$
(57)

with characteristic shift of the pole in $\gamma$ by $\omega$ (recall that
$\omega$ is the Mellin variable conjugated to longitudinal momentum fraction
$z$). Using the above modified kernel eigenvalue in the eigenvalue equation
(46) and expanding in $\omega$, one can see that it generates powers of
$1/(1-\gamma)^{2n+1}$ poles with $n=1,2,\dots$ at NLLx, NNLLx and higher
orders. It has been verified [149] that the kinematical constraint generates
correct poles at NNLLx level for the case of the $N=4$ SYM theory where the
calculation at this order is available [162]. The form of the kernel Eq. (57)
corresponds to the so-called asymmetric scale choice [141]. This is
appropriate for the case of the DIS where the scale is given by $Q^{2}$. On
the other hand, for the process like $\gamma^{*}\gamma^{*}$ scattering with
two similar virtualities, the scale choice would be given by $QQ_{0}$ where
$Q^{2}\simeq Q_{0}^{2}$ are the minus virtualities of both photons. For this
case one has to perform the scale change and this will result in the following
modified eigenvalue

$\tilde{\chi}_{0}^{\rm
sym}(\gamma,\omega)=2\psi(1)-\psi(\gamma+\omega/2)-\psi(1-\gamma+\omega/2)\;.$
(58)

Expanding this form will lead to the symmetric appearance of the triple
collinear poles exactly as in Eq. (54).

These triple poles need to be subtracted from the NLLx expression in order to
avoid the double counting. In addition, the non-singular DGLAP splitting
function in leading order is added and the strong coupling is running in front
of the LLx kernel. Thus the double poles also need to be subtracted from the
NLLx kernel since they are already incorporated by these modifications. The
expression in the Mellin space for the NLLx kernel with subtractions is

$\chi_{1}^{\rm
resum}(\gamma)\;=\;\chi_{1}(\gamma)+\frac{1}{2}\chi_{0}(\gamma)\frac{\pi^{2}}{\sin^{2}(\pi\gamma)}-\chi_{0}(\gamma)\frac{A_{1}(0)}{\gamma(1-\gamma)}+\frac{b}{2}(\chi_{0}^{\prime}+\chi_{0}^{2})\;,$
(59)

where $A_{1}(0)=-11/12$. The terms on the r.h.s of this equation are as
follows: the original NLLx eigenvalue; the subtraction due to the
$\omega$-shift (or kinematical constraint) giving the triple poles; the double
pole DGLAP terms; the double poles from running coupling with coefficient
proportional to the $\beta$ function in QCD.

The resummation removed the instability of the solutions, which remained
positive, and gave a reduced value of the intercept of the order of $0.2-0.3$,
more compatible with the experimental data. In Fig. 19 we show the value of
the BFKL intercept as a function of the strong coupling for LLx, NLLx and the
resummed case. We see that the resummed intercept value is in between the LLx
and NLLx values with the nonlinear dependence on the strong coupling.

In addition to the reduced value of the intercept, the resummation also leads
to the strong preasymptotic effects. As shown through detailed analysis in
[163] the onset of the BFKL growth may be delayed by several units of
rapidity, primarly due to the effects related to the kinematical constraint.
For example, for the gluon Green’s function $G(x;{\bf k},{\bf k}_{0})$ with
scales $k\simeq k_{0}$ of the order $3-100$ GeV the growth can be delayed by
$4-8$ units of rapidity. In the case of the $\gamma^{*}\gamma^{*}$ scattering
mediated by the BFKL Pomeron exchange, the resummed calculation leads even to
suppression of the cross section for energies $W\lesssim 100\,\rm GeV$ with
respect to the Born level, i.e., based on the two-gluon exchange, before the
BFKL growth overcomes the lowest order at higher energies, see Fig. 18.

Figure 18: The $\gamma^{*}\gamma^{*}$ as a function of energy W for two values
of $Q^{2}=2$ and $100\,\rm GeV$. The BFKL resummed calculation is compared to
the LLx BFKL calculation with the running coupling, and the Born case. Figure
from [163], https://doi.org/10.1140/epjc/s10052-019-7171-z.

Let us also note that, in practical applications, for example in DIS, the size
of the high energy logarithms is not given by $\alpha_{s}\ln 1/x$ but rather
by interval of rapidity available for the gluon emissions. In the context of
DIS, this is usually smaller than naive expectation of $\ln 1/x$ due to the
restrictions of the phase space in a process under consideration. For example,
for the case of structure function $F_{2}(x,Q^{2})$ evaluated using the
$k_{T}$ factorization the relevant large logarithm resummed by BFKL would be
given by $\ln 1/x_{g}$, where $x_{g}$ is the longitudinal momentum fraction of
the gluon entering the photon–gluon partonic subprocess. In other words the
phase space for BFKL gluon emissions is reduced by the energy needed to
produce $q\bar{q}$ pair.

Figure 19: The value of the BFKL intercept as a function of the coupling
constant, from the LL BFKL (yellow long dashed), NLL case (green dot- long
dashed), and three resummed schemes (black dashed, red dot-short dashed, blue
solid). Calculation done in the fixed coupling case. Figure from [146],
https://doi.org/10.1103/PhysRevD.68.114003.

Fits based on the resummed evolution were performed, see for example [164,
165, 153, 166, 167] and the overall description of the experimental data were
very good. In recent works [164, 168] it was shown that the fits with
resummation lead to a better description of the experimental data than the
fixed order calculation based on the DGLAP in the NNLO accuracy. The
resummation also suppresses strong diffusion into the infrared, characteristic
for the LLx evolution, although the diffusion is still present, see for
example [163].

###  4.3 Cross section taming

The other effect, which is expected to be present in the limit of high energy,
is the phenomenon of the parton saturation. The BFKL equation, even in the
resummed case predicts a power-like growth of the gluon density with
decreasing $x$. This leads to the untamed growth of the density and hence the
cross section, and ultimately will violate the unitarity of the S-matrix.
Thus, additional corrections are expected to become important and may be
related to the recombination of the gluons in the region, when the gluon
density is very large. This is known as the phenomenon of gluon saturation
[99, 100].

A heuristic argument for parton recombination can be formulated as follows.
The density of gluons in the proton per unit area is roughly proportional to
$\rho\sim xg(x,Q^{2})/\pi R_{gN}^{2}$, where $R_{gN}$ is the proton gluon
radius. The cross section for the gluon recombination can be estimated as
$\sigma\sim\alpha_{s}/Q^{2}$. Thus the gluon saturation is expected to occur
when $\rho\sigma\geq 1$ which results in $Q^{2}\leq Q_{s}^{2}(x)$, where the
saturation scale $Q_{s}$ is defined through the condition

$Q_{s}^{2}\sim\frac{\alpha_{s}(Q_{s}^{2})}{\pi
R_{gN}^{2}}\,xg(x,Q_{s}^{2})\;,$ (60)

which leads to the qualitative behavior $Q_{s}^{2}\sim x^{-\lambda}$ assuming
that $xg(x,Q_{s}^{2})\propto x^{-\lambda}$.

Another way to quantify the onset of proximity to the black disk regime, see
Sec. 3.3 and Eq. (35), is to consider the cross section of dipole–nucleon
scattering. Since the total cross section is proportional to the gluon
density, while the elastic cross section is proportional to the gluon density
squared, the ratio of the elastic and total cross sections grows and exceeds
the black disk limit value of 1/2. From the requirement that the interaction
at small impact parameters is completely absorptive, one obtains using the
optical theorem

${\hat{\sigma}\over 16\pi}\int dtF_{2g}^{2}(t)\leq\frac{1}{2}\;,$ (61)

where $\hat{\sigma}$ is the dipole–nucleon cross section333Note that the
actual inequality is significantly stronger since dipole cross section
$\hat{\sigma}$ corresponds to the case when elastic and diffractive
intermediate state are neglected. , see Sec. 3.3, and $F_{2g}(t)$ is the two-
gluon form factor which we take to have an exponential form

$F_{2g}^{2}(t)=\exp\big{(}B_{2g}(x)t\big{)}\;,$ (62)

where the slope is parametrized as [169]

$B_{2g}(x)=B_{2g}^{(0)}+2\alpha_{g}^{\prime}\ln(x_{0}/x)\;,$ (63)

with $x_{0}=0.0012$, $B_{2g}^{(0)}=4.1\,({}^{+0.3}_{-0.5})\,{\rm GeV}^{-2}$
and $\alpha_{g}^{\prime}=0.140\,({}^{+0.08}_{-0.08})\,{\rm GeV}^{-2}$. One can
rewrite Eq. (61) as

$\hat{\sigma}\leq 8\pi B_{2g}(x)\approx 40\,\mbox{mb}\;,$ (64)

for $x=10^{-3}$. We note that taking into account relation between the gluon
density and dipole cross section Eq. (25), the relation (64) is equivalent to
(60) (with fixed normalisation).

The above arguments can be extended to nuclei, in which case the saturation
scale obtains the modification due to the mass number $A$. It is coming from
the enhanced gluon density, which scales roughly like a volume, factor $A$
times reduction factors (a) the nuclear shadowing factor and (b) smaller
transverse density (nuclei are rather dilute objects) resulting in

$\frac{Q_{sA}^{2}}{Q_{sN}^{2}}=A\,\frac{R_{gN}^{2}}{R_{A}^{2}}\,\frac{g_{A}(x,Q^{2})}{Ag_{N}(x,Q^{2})}\;.$
(65)

Taking $R_{gN}^{2}(x=10^{-3})=0.6\,\mbox{fm}^{2}$ from analysis of the
$J/\psi$ elastic production, see Sec. 5,
$R_{A}^{2}=(1.1\,\mbox{fm}\,A^{1/3})^{2}$, and nuclear shadowing factor of 0.6
for $Q^{2}=3$ GeV2 and $x=10^{-3}$, we estimate the enhancement factor for
heavy nuclei ($A\sim 200$):

$\frac{Q_{sA}^{2}}{Q_{sN}^{2}}=0.3A^{1/3}\approx 1.75\;.$ (66)

A more accurate estimate avoiding edge effects can be done for the case of
scattering at small impact parameters. In this case we can estimate ratio
$Q_{sA}^{2}/Q_{sN}^{2}$ for small impact parameters by comparing the product
of the matter density at $b=0$,

$T_{A}(b=0)=\int_{-\infty}^{\infty}dz\rho_{A}(b=0,z)_{A=200}\approx\mbox{2
fm}^{-2}\;,$ (67)

times the shadowing factor $S_{A}(x)\sim 0.5$ with the transverse gluon
density in a nucleon:

${1\over\pi R^{2}_{gN\,tr}}={1\over\pi
R_{gN}^{2}(2/3)}\approx\frac{1}{2R_{gN}^{2}}\;.$ (68)

Using the same value of $R_{gN}^{2}$ as above we find the modification factor
for the saturation scale equal to

$Q_{sA}^{2}(b=0)/Q_{sN}^{2}=T_{A}(b=0)\cdot S_{A}(x,b=0)\cdot
2R_{gN}^{2}=1.2\;,$ (69)

for heavy nuclei. The difference is mainly due to neglect of the surface
effects in modeling the nuclear density.

In practice the black disk regime is difficult to reach experimentally,
nevertheless it is instructive to analyze the behavior of the cross sections
in this limit. It was first considered by Gribov [85] for the total cross
section for $\gamma^{*}$ \- heavy nucleus scattering. In this limit for
virtualities $Q^{2}<Q_{sA}^{2}$, where $Q_{sA}^{2}\gg\Lambda_{\rm QCD}^{2}$,
the cross section of dipole–nucleus scattering does not depend on the dipole
size for $1/r^{2}<Q_{sA}^{2}$ and is equal to $2\pi R_{A}^{2}$. As a result
Bjorken scaling is grossly violated: $\sigma_{tot}^{\gamma^{*}A}(x,Q^{2})$
does not drop with $Q^{2}$ and grows as $\ln(x_{0}/x)$ [85]. In this limit
$\sigma_{\rm diff}=\pi R_{A}^{2}$ and is dominated by the exclusive dijet
production [170]. Also, in this limit the absolute normalization of the vector
meson coherent production cross section is predicted. The cross section drops
with $Q^{2}$ by a factor $1/Q^{4}$ slower than in the LT limit since in the LT
cross section is proportional to the square of the dipole cross section. To be
more precise in this limit one expects

$\displaystyle\sigma_{L}(\gamma^{*}A\to V\,A)\;\propto\;1/Q^{2},$
$\displaystyle\sigma_{T}(\gamma^{*}A\to V\,A)\;\propto\;M_{V}^{2}/Q^{4}\;.$
(70)

For a detailed discussion of the prediction in the black disk limit, see [170,
171].

###  4.4 Color Glass Condensate and Balitsky–Kovchegov equation

The effective theory which describes parton saturation is the Color Glass
Condensate (CGC) [102, 103, 104], with the Jalilian-Marian - Iancu - McLerran
- Weigert - Leonidov - Kovner (JIMWLK) evolution equations [108, 109, 110,
111, 105, 106, 107] (see [172] for a review together with selected
phenomenological applications). In the CGC effective theory the relevant
degrees of freedom are the color sources $\rho({\bf x})$, which have large
values of $x$ and the gauge fields $A^{\mu}$ in the region of small $x$. Here
${\bf x}$ is the transverse spatial coordinate and thus $\rho({\bf x})$
describes the distribution of these color charges in the transverse coordinate
space. The color sources produce a current
$J^{\mu}=\delta^{\mu+}\delta(x^{-})\rho({\bf x})$ (for a target moving in the
positive $z$ direction, or $x^{+}$ direction). Due to the time dilation the
color sources are effectively frozen at the time scales relevant for the
strong interaction. The fast and slow degrees of freedom are then coupled
through the gauge field and the current, i.e. $A^{\mu}J_{\mu}$. The
distribution of the fast partons is a stochastic quantity which is different
in every collision and thus the central object in CGC is the statistical
distribution $W_{y}[\rho]$ of the color sources. From this distribution one
can calculate various operators through the averaging procedure

$\langle{\cal O}\rangle_{y}\;=\;\int[D\rho]\,W_{y}[\rho]\,{\cal O}[\rho]\;,$
(71)

where ${\cal O}[\rho]$ is the expectation value of the operator for the
particular configuration $\rho$ of the color sources. The statistical
distribution $W_{y}[\rho]$ encodes all the correlations of the color charge
density and it depends on the cutoff $y$, which divides the fast and slow
partons. Variable $y$ may be related to the rapidity, which in the leading
logarithmic approximation is given by $y=\ln 1/x$, where $x$ would be Bjorken
$x$ in the DIS case. The evolution of the distribution $W_{y}[\rho]$ of the
color sources is provided by renormalization group equation

$\frac{\partial W_{y}[\rho]}{\partial y}\;=\;{\cal
H}[\rho,\frac{\delta}{\delta\rho}]\;W_{y}[\rho]\;,$ (72)

with ${\cal H}$ being the JIMWLK Hamiltonian. This operator contains up to two
derivatives $\delta/\delta\rho$ and arbitrary powers of $\rho$. From Eq. (71)
and (72) one can derive evolution equations for the different operators.

More specifically, the scattering between the dipole and the fields generated
by the target can be described using the product of two Wilson lines, one for
the quark with transverse coordinate ${\bf x}$ and the antiquark with
transverse coordinate ${\bf y}$. The relevant operator is given by

$1-\frac{1}{N_{c}}{\rm Tr}(U^{\dagger}({\bf x})\,U({\bf y}))\;,$ (73)

where

$U^{\dagger}({\bf x})\;=\;P\,\exp\left[ig\,\int dx^{-}A_{a}^{+}(x^{+}\simeq
0,x^{-},{\bf x})t^{a}\right]\;,$ (74)

is the Wilson line operator, corresponding to the left moving quark. Here,
$t^{a}$ are the generators of $SU(N_{c})$ in the fundamental representation,
$P$ denotes the $x^{-}$ ordering of color matrices. This Wilson line describes
the quark propagating through the target, preserving the straight trajectory
in the eikonal approximation and undergoing color precession.

The physical dipole scattering amplitude is defined in this framework through
the average over all the color configurations of the target given by the
functional $W_{y}[\rho]$

$\displaystyle N_{y}({\bf x},{\bf y})$ $\displaystyle=$ $\displaystyle\langle
1-\frac{1}{N_{c}}{\rm Tr}(U^{\dagger}({\bf x})\,U({\bf y})))\rangle_{y}$ (75)
$\displaystyle=$
$\displaystyle\int[D\rho]\,W_{y}[\rho]\,\left[1-\frac{1}{N_{c}}{\rm
Tr}(U^{\dagger}({\bf x})\,U({\bf y}))\right]\;.$ (76)

By inserting the above definitions into the evolution equation for the weight
function Eq. (72) one can derive the following evolution equation

$\displaystyle\frac{\partial}{\partial y}\langle{\rm Tr}(U^{\dagger}({\bf
x})\,U({\bf y}))\rangle_{y}\;=\;$ $\displaystyle-\int d^{2}{\bf z}\;{\cal
K}({\bf x},{\bf y},{\bf z})\;\langle N_{c}{\rm Tr}((U^{\dagger}({\bf
x})\,U^{\dagger}({\bf y}))$ (77) $\displaystyle-{\rm Tr}(U^{\dagger}({\bf
x})\,U^{\dagger}({\bf z}))\,{\rm Tr}(U^{\dagger}({\bf z})\,U({\bf
y}))\rangle_{y}\;.$

Here ${\cal K}$ is the dipole kernel, which in the leading order case has the
form

${\cal K}({\bf x},{\bf y},{\bf z})\;=\;\frac{\alpha_{s}}{2\pi^{2}}\frac{({\bf
x}-{\bf y})^{2}}{({\bf x}-{\bf z})^{2}({\bf z}-{\bf y})^{2}}\;.$ (78)

The above equation for the S-matrix is not closed since it relates the
2-Wilson line correlator to the 4-Wilson line correlator. For the latter, one
can derive the evolution equation which in turn will involve 6-Wilson line
correlator. Thus Eq. (77) is just one of the set of equations which form an
infinite hierarchy. This hierarchy was derived in an alternative approach by
Balitsky [112, 113] using the operator product expansion for high energies.
One can make however some simplifications to Eq. (77), that is in the large
multicolor limit the line correlation function factorizes

$\langle{\rm Tr}(U({\bf x})\,U^{\dagger}({\bf z}))\,{\rm Tr}(U({\bf
z})\,U^{\dagger}({\bf y}))\rangle_{y}\rightarrow\langle{\rm Tr}(U({\bf
x})\,U^{\dagger}({\bf z}))\rangle_{y}\,\langle{\rm Tr}(U({\bf
z})\,U^{\dagger}({\bf y}))\rangle_{y}\;.$ (79)

As a result the first equation of the hierarchy decouples, and has the
following form

$\frac{\partial N_{y}({\bf x},{\bf y})}{\partial y}=\int d^{2}{\bf z}\;{\cal
K}({\bf x},{\bf y},{\bf z})\left[N_{y}({\bf x},{\bf z})+N_{y}({\bf z},{\bf
y})-N_{y}({\bf x},{\bf y})-N_{y}({\bf x},{\bf z})N_{y}({\bf z},{\bf
y})\right]\;,$ (80)

where we used definition Eq. (76) for the dipole scattering ampliutude.

The above equation is the Balitsky–Kovchegov (BK) equation for the dipole
scattering amplitude, derived independently by Kovchegov [114, 115] from the
description of the soft gluons in the dipole wave function at high energy
[173]. The linearlized version of this equation, without the $N_{y}({\bf
x},{\bf z})N_{y}({\bf z},{\bf y})$ term is equivalent to the famous BFKL
evolution equation [29, 30, 31] which is an equation allowing for the
resummation of the large logarithms in $\ln 1/x$. The Balitsky-Kovchegov
equation contains an extra negative and non-linear contribution which is
relevant when the parton density is large. It is evident that the solution
$N=1$ satisfies this equation, and one can demonstrate that this is the stable
point of the solution. The solution to this equation generates dynamically
saturation scale $Q_{s}^{2}(x)$ which has a power like behavior in $x$.

The BK equation is most often analyzed in the approximation where $N_{y}({\bf
x},{\bf y})=N_{y}(|{\bf x}-{\bf y}|)$, that is the amplitude depends on the
dipole size (absolute value) $r=|{\bf x}-{\bf y}|$. This simplifies greatly
the numerical solution to the BK equation. It however corresponds to the crude
approximation of the infinite nucleus. The full solution to the BK equation
with impact parameter was presented in [174, 175]. In that case the solution
for the amplitude as a function of the dipole size, for fixed values of the
impact parameter, drops down for very large dipole sizes. This is due to the
fact that very large dipoles will simply miss the target. Of course in
reality, there is confinement, which will cut off the large dipole sizes which
extend beyond the confinement scale. In [174, 175] it was also shown that even
after including the saturation effects, the BK solution will violate the
Froissart bound, i.e. the dipole cross section will increase as a power of
$1/x$, rather than $\ln^{2}1/x$. This is related with the fact that the BK
equation in the leading order approximation is scale invariant and the kernel
has power like tails, which need to be regulated by confinement.

The JIMWLK and Balitsky equations have been derived at leading and next-to-
leading logarithmic order [176, 177, 178]. Resummation schemes were also
applied to the non-linear evolution equations [179, 180, 181, 182], and a good
description of the experimental data on the structure function $F_{2}$ at HERA
was achieved based on the resummed nonlinear evolution [118]. It would be
interesting to investigate whether the energy-momentum conservation effects,
which are discussed for one-Pomeron exchange (see discussion in the previous
section) are amplified for the multi-Pomeron exchanges which are underpinning
the non-linear evolution equation Eq. (80).

A non-linear evolution equation in the small $x$ limit was also derived for
the case of the high-mass diffraction by Kovchegov and Levin [183], see also
[101]. The equation derived was for the $N^{D}({\bf x},{\bf y},Y,Y_{0})$ cross
section per impact parameter ${\bf b}$ for diffractive interaction with
rapidity gap greater than or equal to $Y_{0}$, and where $Y$ is the total
rapidity interval, so that $Y>Y_{0}$. The corresponding diffractive cross
section with rapidity gap greater than or equal to $Y_{0}$ in the
dipole–nucleus scattering is

$\hat{\sigma}_{\rm diff}=\int d^{2}{\bf b}\;N^{D}({\bf x},{\bf y},Y,Y_{0})\;.$
(81)

In order to find the diffraction cross section for a given fixed rapidity gap
$Y_{0}$ one needs to differentiate the $N^{D}$ with respect to $Y_{0}$

$M_{X}^{2}\frac{d\sigma_{\rm diff}}{dM_{X}^{2}}\;=\;-\int d^{2}{\bf
b}\;\frac{\partial N^{D}({\bf x},{\bf y},Y,Y_{0})}{\partial Y_{0}}\;,$ (82)

where the diffractive mass $M_{X}$ is related to the rapidity gap $Y_{0}$
through

$Y_{0}=\Delta Y_{\rm gap}\simeq\ln\frac{W^{2}}{M_{X}^{2}+Q^{2}}\;.$ (83)

The evolution equation for the diffractive amplitude in the LLx approximation
reads

$\displaystyle\frac{\partial}{\partial y}N^{D}({\bf x},{\bf y},Y,Y_{0})$
$\displaystyle=\int d^{2}{\bf z}\;{\cal K}({\bf x},{\bf y},{\bf z})$ (84)
$\displaystyle\times\left[N^{D}({\bf x},{\bf z},Y,Y_{0})+N^{D}({\bf z},{\bf
y},Y,Y_{0})-N^{D}({\bf x},{\bf y},Y,Y_{0})\right.$ $\displaystyle+N^{D}({\bf
x},{\bf z},Y,Y_{0})N^{D}({\bf z},{\bf y},Y,Y_{0})-2N({\bf x},{\bf
z},Y)N^{D}({\bf z},{\bf y},Y,Y_{0})$ $\displaystyle\left.-2N^{D}({\bf x},{\bf
z},Y,Y_{0})N({\bf z},{\bf y},Y)+2N({\bf x},{\bf z},Y)N({\bf z},{\bf
y},Y)\right]\;.$

Numerical solutions were studied in [184, 185], where it was shown that the
saturation scale has the same dependence on $x$ as the saturation scale for
the total cross section. It was also shown that the ratio of $\sigma_{\rm
diff}/\sigma_{\rm tot}$ has only mild dependence on the energy and
$M_{X}^{2}$.

We note that this equation was derived in LLx approximation, and similarly to
the inclusive case described previously, it is expected that the higher orders
will be very important (e.g., see Fig. 19). It will be interesting to see
whether the predictions from Eq. (84) reproduce the value of the
$\alpha_{{I\\!\\!P}}(t=0)\simeq 1.11$.

##  5 Hard diffractive production of vector mesons

Studies of vector meson production in hard diffractive processes off nucleons
and nuclear targets, $\gamma^{*}+T\to(V,V_{Q\bar{Q}},\gamma)+T$, where
$V=\rho,\omega,\phi$ or a heavy quarkonium $V_{Q\bar{Q}}=J/\psi,\Upsilon$, is
one of major areas of the theoretical and experimental studies of QCD. A
number of measurements has been performed using muon and electron beams with
fixed targets, electron–proton colliding beams at HERA, and more recently
using ultraperipheral heavy ion collisions at the LHC (see Sec. 8.1), for
reviews, see Refs. [186, 187].

Hard exclusive diffractive processes are the source of valuable information on
the origin of small $x$ processes, i.e., on the origin of the phenomenon known
as Pomeron. It enables one to probe the evolution of high energy processes
with a decrease of $x$ and to compare the data with the competing LLx and NLLx
$\ln(x/x_{0})$ approximations and with the resummation of these terms, see
Sec. 4. Also, studies of these processes allow one to access generalized
parton distributions (GPDs), which can be rigorously defined in the framework
of the QCD factorization theorem [188] and the DGLAP approximation. These
studies should allow one to establish spatial distributions of quarks and
gluons in the hadronic targets and to investigate the role of color in high
energy processes. We will begin our consideration from the formulation of
basic properties of hard diffractive production of vector mesons.

###  5.1 Space–time evolution and factorization of high energy processes

Vector meson production at small $x$ in the target rest frame can be described
as a three-stage process [189]:

(i) The virtual photon $\gamma^{\ast}$ with the large longitudinal momentum
$q$ converts into a $\bar{q}q$ pair, where the partons carry the longitudinal
momenta $zq$ and $(1-z)q$ and the transverse momenta ${\bf k}_{t}$ and $-{\bf
k}_{t}$, respectively; $z$ is the momentum fraction describing the
longitudinal momentum sharing between the quarks. The lifetime of such
fluctuations is also called the coherence length $l_{\rm coh}$ and is given by

$\tau_{i}=l_{\rm coh}/c={2q\over Q^{2}+{k_{\perp}^{2}+m_{q}^{2}\over
z(1-z)}}\approx{1\over m_{N}x}\,,$ (85)

which follows from the energy–time uncertainty principle. The values of
$l_{\rm coh}$, which could be reached at colliders, are $l_{\rm coh}\sim
10^{3}$ fm at HERA and $l_{\rm coh}\sim 10^{2}$ fm at the EIC.

As noted earlier in Sec. 3, in the case of longitudinally polarized photons,
the transverse size of the $q\bar{q}$ pair is $\propto 1/Q$, making it
possible to justify the applicability of perturbative QCD. In the case of
transversely polarized photons, the low $k_{\perp}$ contribution is not
parametrically suppressed leading to significant nonperturbative effects.

(ii) Then the $q\bar{q}$ pair scatters off the target with the dipole cross
section discussed in Sec. 3. Note that the $q\bar{q}$ dipole lives for the
time

$\tau_{f}=l_{f}/c={2q\over{k_{\perp}^{2}+m_{q}^{2}\over z(1-z)}}\,,$ (86)

so that $\tau_{f}\geq\tau_{i}$.

(iii) Finally, the vector meson in the final state is formed.

As a result, the amplitude of vector meson electroduction by longitudinal
photons ${\cal A}(\gamma^{\ast}_{L}+p\to V+p)$ can be written as a convolution
of the light-cone wave function of the photon, $\Psi_{L}^{f}({\bf r},Q,z)$,
the dipole cross section, and the wave function of the vector meson
$\Psi_{V}^{f}$ in a $q\bar{q}$ configuration,

${\cal A}(q_{t},Q)=\sum_{f}\int d^{2}{\bf b}\int d^{2}{\bf r}\int dz\,e^{i{\bf
q_{t}}\cdot{\bf b}}\,\Psi_{L}^{f}({\bf r},Q,z)\,N(x,{\bf r},{\bf
b})\,\Psi_{V}^{f}({\bf r},z)\,,$ (87)

where $N(x,{\bf r},{\bf b})$ is the dipole amplitude, ${\bf r}$ is the
transverse separation between $q$ and $\bar{q}$ (the dipole size); ${\bf b}$
is the impact parameter defined as the distance between the center of mass of
the dipole and the center of the target; ${\bf q_{t}}$ is the momentum
transfer to the target, which is assumed to be purely transverse here (the
effect of the non-zero longitudinal momentum transfer will be discussed
below); $\sum_{f}$ is a sum over quark flavors depending on the flavor
structure of the produced vector meson $V$. The corresponding graph is shown
in Fig. 20. The dipole amplitude $N(x,{\bf r},{\bf b})$ when integrated over
the impact parameter ${\bf b}$ is reduced to the dipole cross section
$\hat{\sigma}$, see Eqs. (33) and (25).

Figure 20: Electroproduction of vector mesons in the dipole model. The three
stages of the process involve the transition of the photon into a quark-
antiquark pair (dipole), its scattering off the target depicted as a two-gluon
exchange here, and formation of the final-state vector meson. The longitudinal
momentum fractions of the exchanged gluons are denoted $x_{1}$ and $x_{2}$.

Note that at large $Q^{2}$, QCD factorization into the three blocks in Eq.
(87) is valid for all $x$ and any two-body final states at fixed $t$ [188].
However, for small $x$ the space-time picture of the process is greatly
simplified. Note also that the same three-block description is valid for hard
diffractive production of heavy quarkonia, where the heavy quark mass
$m_{c,b}$ defines the hard scale of process [190, 191].

The $\gamma_{L}^{\ast}+p\to V+p$ differential cross section is given by the
usual expression

$\frac{d\sigma(\gamma_{L}^{\ast}+p\to V+p)}{dt}=\frac{1}{16\pi}|{\cal
A}(q_{t},Q)|^{2}\,,$ (88)

where $t\approx-{\bf q_{t}}^{2}$.

In the leading $\alpha_{s}\ln Q^{2}$ approximation and in the leading twist
limit, one can use the explicit relation of the dipole cross section to the
gluon density, see Eq. (25), to obtain the following expression for the cross
section of diffractive vector meson electroproduction [190, 98, 189, 192]

$\left.{d\sigma^{L}_{\gamma^{*}N\rightarrow VN}\over
dt}\right|_{t=0}={12\pi^{3}\Gamma_{V\rightarrow
e^{+}e^{-}}M_{V}\alpha_{s}^{2}(Q)\eta^{2}_{V}\left|\left(1+i{\pi\over
2}{d\over d\ln
x}\right)xg_{T}(x,Q^{2})\right|^{2}\over\alpha_{EM}Q^{6}N_{c}^{2}}\,,$ (89)

where $M_{V}$ is the vector meson mass; $\alpha_{s}(Q^{2})$ is the running
coupling constant; $xg_{T}(x,Q^{2})$ is the gluon density of the target;
$\alpha_{EM}$ is the fine-structure constant; $N_{c}=3$ is the number of
colors; $\Gamma_{V\rightarrow e^{+}e^{-}}$ is the $V\to e^{+}e^{-}$ decay
width. The parameter $\eta_{V}$ is defined as follows,

$\eta_{V}={1\over 2}\frac{\int\frac{dz}{z(1-z)}\,\Psi_{V}(r=0,z)}{\int
dz\,\Psi_{V}(r=0,z)}={1\over 2}{\int{dz\,d^{2}{\bf k_{t}}\over
z(1-z)}\,\Phi_{V}(k_{t},z)\over\int dz\,d^{2}{\bf
k_{t}}\,\Phi_{V}(k_{t},z)}\,,$ (90)

where $\Phi_{V}(k_{t},z)$ is the Fourier transform of the vector meson wave
function $\Psi_{V}(r,z)$. In perturbative QCD, $\Phi_{V}(k_{t},z)$ has the $Q$
dependence given by the ERBL equations and becomes proportional to $z(1-z)$ in
the $Q^{2}\to\infty$ limit. Hence, $\eta_{V}$ quantifies the deviation of the
$z$ dependence of the vector meson wave function $\Psi_{V}(r,z)$ from its