asymptotic form, which leads to $\eta_{V}\to 3$ at high $Q^{2}$.

For the first time the expression for the cross section of $J/\psi$
electroproduction and photoproduction neglecting the Fermi motion of quarks
and the real part of the scattering amplitude was derived in Ref. [190].

Note that in the leading twist approximation, the integration over quark
transverse momenta in the graph in Fig. 20 extends up to $k_{t}\sim Q$. As a
result, the expression for the cross section of vector meson electroproduction
involves $\Psi_{V}$ at $|{\bf r}|\propto 1/Q$, i.e., the vector meson wave
function at the origin (the vanishing separation between the quark and
antiquark), see Eqs. (89) and (90).

On the phenomenology side, the rapid onset of the leading twist behavior for
the $\sigma(e^{+}e^{-}\to hadrons)$ electron–positron annihilation cross
section suggests that for $\rho$ and $\phi$ mesons, $\Phi_{V}(z,k_{t})$ and
hence $\eta$ should already be close to the asymptotic value at $Q^{2}\sim$ a
few GeV2.

###  5.2 The role of GPDs in hard diffractive production of vector mesons

In the framework of collinear factorization for exclusive processes [188],
cross sections of hard exclusive electroproduction of vector mesons are
expressed in terms generalized parton distributions (GPDs) of the target [193,
194, 195, 196, 197, 198, 199]. GPDs are defined as matrix elements of quark
and gluon QCD operators between states with non-equal momenta. For instance,
the gluon GPD of the proton target reads in the symmetric notation [195]

$F^{g}(x,\xi,t)=\frac{1}{(\bar{p}\cdot
n)}\int\frac{d\lambda}{2\pi}e^{ix(\bar{p}\cdot z)}n_{\rho}n_{\sigma}\langle
p^{\prime}|F^{\rho\mu}\left(-\frac{z}{2}\right)F^{\
\sigma}_{\mu}\left(\frac{z}{2}\right)|p\rangle|_{z^{+}=z_{\perp}=0,z=\lambda
n}\,,$ (91)

where $F_{\mu\nu}$ is the gluon field strength tensor; $n$ is a light-like
vector in the direction of the initial photon momentum; $x_{1}=x+\xi$ and
$x_{2}=x-\xi$ are the light-cone momentum fractions of the gluons along the
$\bar{p}=(p+p^{\prime})/2$ direction, see Fig. 21. The averaging and summation
over colors is assumed.

Figure 21: Momentum fractions in the definition of GPDs in the symmetric
notation.

The momentum fraction $\xi$ is associated with the longitudinal momentum
transfer and determined by the kinematics of the process. For electroduction
of vector mesons,

$\xi=\frac{x}{2}=\frac{1}{2}\frac{Q^{2}+M_{V}^{2}}{W^{2}}\,.$ (92)

In the forward limit, i.e., in the $\xi=0$ and $t=0$ limit, one obtains from
Eq. (91):

$\displaystyle F^{g}(x,\xi=0,t=0)$ $\displaystyle=$
$\displaystyle\frac{1}{(p\cdot n)}\int\frac{d\lambda}{2\pi}e^{ix(p\cdot
z)}n_{\rho}n_{\sigma}\langle p|F^{\rho\mu}\left(-\frac{z}{2}\right)F^{\
\sigma}_{\mu}\left(\frac{z}{2}\right)|p\rangle|_{z^{+}=z_{\perp}=0,z=\lambda
n}$ (93) $\displaystyle=$ $\displaystyle xg(x)\,,$

where $g(x)$ is the usual gluon distribution (we suppress the scale dependence
for brevity), see, e.g. Ref. [200].

Note that in Eq. (89) the difference between the light-cone fractions of the
gluons attached to the $q\bar{q}$ pair, $x_{1}-x_{2}=2\xi$, see Fig. 20, was
neglected, and the cross section was expressed in terms of the usual gluon
distribution of the target. One can correct for this effect phenomenologically
by examining the $Q^{2}$ evolution of GPDs [201]. In particular, making a
natural assumption that at the input evolution scale the difference between
$x_{1}$ and $x_{2}$ can be neglected, the effect of $x_{1}\neq x_{2}$ at
higher $Q^{2}$ scales is generated by the evolution, see also Refs. [202,
203]. The natural qualitative feature of this method is that the effect of
$x_{1}\neq x_{2}$ increases with an increase of $Q^{2}$ and the mass of the
produced vector meson.

In the phenomenologically important case of $J/\psi$ production, the gluon GPD
of the target can be approximated well by the usual gluon density evaluated at
$x_{\rm eff}=(x_{1}+x_{2})/2=\xi=x/2$ multiplied by the factor of $R_{g}$
[203, 204, 205]

$R_{g}=\frac{2^{3+2\lambda}}{\sqrt{\pi}}\frac{\Gamma(\frac{5}{2}+\lambda)}{\Gamma(4+\lambda)}\,,$
(94)

where the parameter $\lambda$ parametrizes the small-$x$ behavior of the gluon
density, $xg(x)\sim 1/x^{\lambda}$. For the realistic value of $\lambda=0.2$,
$R_{g}=1.2$. Note however that the result of Eq. (94) was derived in the limit
when $x_{1}\gg x_{2}$, which may not be the case for $J/\psi$ production in
the dipole model [192]. This topic thus deserves further investigation. In the
case of $\Upsilon$ production, the discussed affect is more pronounced [206].

###  5.3 Modeling finite-$Q^{2}$ effects

In general, the $Q^{2}$ dependence of the differential cross section of vector
meson electroduction is contained in the amplitude in Eq. (87). It comes from
the intrinsic $Q^{2}$ dependence of the photon wave functuon
$\Psi_{L}^{f}(r,Q,z)$ and the action of the scattering operator on the energy
denominator of $\Psi_{L}^{f}({\bf r},Q,z)$ in momentum space, see discussion
in Refs. [98, 192]. This leads to the following overall $Q^{2}$ dependence of
the cross section

${Q^{2}\over\left(Q^{2}+{k_{\perp}^{2}+m^{2}\over
z(1-z)}\right)^{4}}\to{1\over Q^{6}}\,,$ (95)

where $M^{2}_{q{\bar{q}}}=(k_{\perp}^{2}+m_{q}^{2})/[z(1-z)]$ is the invariant
mass squared of the $q{\bar{q}}$ dipole. The asymptotic behavior of $1/Q^{6}$
corresponds to the leading twist approximation in the $Q^{2}\to\infty$ limit.
Alternatively, one can obtain this result by working in coordinate space and
noticing that in the leading twist approximation, the integral over the dipole
size in Eq. (87) is dominated by $|{\bf r}|\propto 1/Q$.

To obtain a quantitative picture of the dipole sizes characteristic for
different processes, it is instructive to introduce the median dipole size
$r(\rm med)$, which is defined as the value of the upper limit in the
integration in Eq. (87) corresponding to half of the full answer. Figure 22
shows $r(\rm med)$ as a function of the photon virtuality $Q^{2}$ for
electroproduction of light ($\rho$) and heavy ($J/\psi$) vector mesons. For
comparison, we also give $r(\rm med)$ for the total photoabsorption cross
section $\sigma_{L}(x,Q^{2})$. The calculations were carried out using the
MFGS dipole model [127] and the Gauss-LC vector meson wave function [207].

Figure 22: The dependence of the median dipole size $r(\rm med)$ on the photon
virtuality $Q^{2}$ for electroproduction of light and vector mesons and also
the total photoabsorption cross secion $\sigma_{L}(x,Q^{2})$.

One can see from the figure that, as expected, the median dipole size for
$J/\psi$ is smaller than that for $\rho$ and $\sigma_{L}(x,Q^{2})$ at low
$Q^{2}$. As $Q^{2}$ is increased, the median sizes for light and heavy vector
mesons converge since the details of their wave functions become unimportant.

The results presented in Fig. 22 agree with those of Refs. [98, 192].

A related effect is that at pre-asymptotic energies, one cannot substitute
$\Psi_{V}(r,z)$ by $\Psi_{V}(r=0,z)$ as has been done in Eq. (89). The higher-
twist correction associated with this substitution can be quantified by the
following suppression factor

$T(Q^{2})={\left|\int d^{2}{\bf
r}\,dz\,\Psi_{L}^{f}(r,Q,z)\hat{\sigma}(x,r)\phi_{V}(r,z)\right|^{2}\over\left|\int
d^{2}{\bf
r}\,dz\,\Psi_{L}^{f}(r,Q,z)\hat{\sigma}(x,r)\phi_{V}(r=0,z)\right|^{2}}\,.$
(96)

This suppression factor is closely related to the term
$M^{2}_{q\bar{q}}=(k_{\perp}^{2}+m^{2})/[z(1-z)]$ in Eq. (95). Indeed, even if
one took the minimal value of this term for light vector mesons $\sim 1$ GeV2,
one would still find a reduction of the cross section by 30% at $Q^{2}=10$
GeV2. In fact, the transverse momentum distribution appears to be rather broad
due to a singular structure of the $q\bar{q}$ component of the meson wave
function leading to an even larger value of $M^{2}_{q\bar{q}}$ and, hence, to
a larger suppression. The suppression factor gradually disappears with an
increase of $Q^{2}$ leading to a slower decrease of the cross section with an
increase of $Q^{2}$ than in the leading twist approximation. Note that the
suppression effect is stronger for electroproduction of heavy vector mesons
than for light ones.

The suppression factor of $T(Q^{2})$ as a function of $Q^{2}$ and the trends
of its behavior discussed above are presented in Fig. 23. It also allows one
to explain the $Q^{2}$ dependence of $J/\psi$ electroproduction [192], which
is too steep in the model of [190], where the Fermi motion of quarks is
neglected.

Figure 23: The suppression factor of $T(Q^{2})$, see Eq. (96), for
electroproducion of light and heavy vector mesons.

The finite-$Q^{2}$ effects considered in this section are nothing but higher-
twist effects. As we mentioned above, they can be effectively modeled and
estimated in the dipole formalism, for a review, see [207]. This is also
intertwined with possible gluon saturation effects in exclusive vector meson
production at HERA [207, 208], see also the discussion below.

###  5.4 Elastic photoproduction of $J/\psi$: from HERA to LHC

The phenomenologically important case of vector meson production is elastic
photoproduction of $J/\psi$, where the hard scale is provided by the mass of
$J/\psi$ (mass of the charm quark). The $\gamma+p\to J/\psi+p$ differential
cross section reads [190, 192, 191] [compare to Eq. (89)]

$\frac{d\sigma^{\gamma p\to J/\psi p}(t=0)}{dt}=\frac{12\pi^{3}}{\alpha_{\rm
e.m.}}\frac{\Gamma_{V}M_{V}^{3}}{(4m_{c}^{2})^{4}}\left[\alpha_{s}(Q^{2}_{\rm
eff})xg(x,Q^{2}_{\rm eff})\right]^{2}C(Q^{2}=0)\,,$ (97)

where $Q_{\rm eff}$ is the effective hard scale of the process (see the
discussion below). The factor of $C(Q^{2}=0)$ depends on the details of the
vector meson wave function and takes into account the intrinsic motion
(transverse momentum) of charm quarks in the diagram in Fig. 20. Hence,
$C(Q^{2}=0)$ describes the effect of higher-twist effects in the $\gamma+p\to
J/\psi+p$ cross section. It is given by the following expression,

$C(Q^{2}=0)=\left(\frac{\eta_{V}}{3}m_{c}^{4}\right)^{2}T(0)R(0)\,,$ (98)

where

$T(0)R(0)=\frac{1}{M_{V}^{4}}\left[\frac{\int\frac{dz}{z(1-z)}\int
drr^{3}m_{c,r}^{2}\phi_{V}(z,r)\phi_{\gamma}(z,r)}{\int\frac{dz}{z(1-z)}\phi_{V}(z,r=0)}\right]^{2}\,.$
(99)

Here $\phi_{\gamma}(z,r)=K_{0}(m_{c}r)$; $m_{c,r}$ is the running mass of the
charm quark.

To compare predictions of Eq. (97) to those available in the literature, it is
convenient to cast it in the following form

$\frac{d\sigma^{\gamma p\to J/\psi p}(t=0)}{dt}=\frac{\pi^{3}}{48\alpha_{\rm
e.m.}}\frac{\Gamma_{V}M_{V}^{3}}{Q_{\rm eff}^{8}}\left[\alpha_{s}(Q^{2}_{\rm
eff})xg(x,Q^{2}_{\rm eff})\right]^{2}\tilde{C}(Q^{2}=0)\,,$ (100)

where

$\tilde{C}(Q^{2}=0)=\frac{9}{4}\left(\frac{\eta_{V}}{3}\right)^{2}\frac{Q_{\rm
eff}^{8}}{\left(\frac{M_{V}}{2}\right)^{2}m_{c}^{4}}\left[\frac{\int\frac{dz}{z(1-z)}\int
drr^{3}(m_{c,r}m_{c})^{2}\phi_{V}(z,r)\phi_{\gamma}(z,r)}{\int\frac{dz}{z(1-z)}\phi_{V}(z,r=0)}\right]^{2}\,.$
(101)

Using the LC-Gauss wave function of $J/\psi$ [207], one can explicitly
calculate the overlap of the photon and vector meson wave functions and obtain

$\tilde{C}(Q^{2}=0)\approx\frac{1}{2}\frac{Q_{\rm
eff}^{8}}{\left(\frac{M_{V}}{2}\right)^{2}m_{c}^{4}}\,.$ (102)

As one can see from Eq. (102), the coefficient $\tilde{C}(Q^{2}=0)$ very
strongly depends on the effective scale $Q_{\rm eff}$ and the value of
$m_{c}$, which affects the normalization of the $\gamma+p\to J/\psi+p$.

Using as an example the MFGS dipole model with the LC-Gauss wave function of
$J/\psi$, one can estimate that $Q_{\rm eff}^{2}=\lambda/d_{t}^{2}(\rm
med)=2.8$ GeV2, where $d_{t}(\rm med)$ is the corresponding median dipole
size, see the discussion in the previous section. This value should be
compared to $Q_{\rm eff}^{2}=M_{V}^{2}/4=2.4$ GeV2 obtained in Ref. [190] and
$Q_{\rm eff}^{2}=3$ GeV2, which was determined phenomenologically [209] by
requiring that Eq. (100) correctly reproduces the measurements of $W$
dependence of the $\gamma+p\to J/\psi+p$ cross section at high energies. For
the latter, one usually assumes the exponential $t$ dependence with the slope
$B_{J/\psi}(W)$ giving

$\sigma^{\gamma p\to J/\psi p}=\frac{1}{B_{J/\psi}(W)}\frac{d\sigma^{\gamma
p\to J/\psi p}(t=0)}{dt}\,.$ (103)

The slope $B_{J/\psi}(W)$ can be parameterized in the form
$B_{J/\psi}(W)=4.5+0.4\ln(W/90\ {\rm GeV})$, which is consistent with the HERA
measurements [210, 211, 212].

Predictions of Eq. (100) for the $\gamma+p\to J/\psi+p$ cross section as a
function of $W$ and comparison to the available H1 [213, 211], ZEUS [212],
LHCb [214, 215], and ALICE [216] data are shown in Fig. 24. For the
calculation, we used $Q_{\rm eff}^{2}=3$ GeV2, the CTEQ6L gluon density [217],
and $\tilde{C}(Q^{2}=0)=0.7$, see details in [209]. One can see from the
figure that Eq. (100) provides a good description of the $W$ dependence and
normalization of the cross section of elastic $J/\psi$ photoproduction on the
proton covering a very wide range of energies extending into a TeV-range.

Figure 24: The cross section of elastic $J/\psi$ photoproduction on the proton
as a function of $W$: predictions of Eq. (100) vs. the H1 [213, 211], ZEUS
[212], LHCb [214, 215], and ALICE [216] data.

The data on $J/\Psi$ exclusive photoproduction can be described using
collinear approach with a modeling of some of the NLO effects [204, 218, 219].
The resulting gluon density from the fit has the $x$ dependence of the form
$\sim x^{-\lambda}$ with $\lambda\simeq 0.13$ which is not far from
$\alpha_{{I\\!\\!P}}$ in the soft regime.

Alternatively, a good description of these data can also be achieved within
the color dipole framework employing different models of the dipole cross
sections [207] or the unintegrated gluon density [220, 221]. In work [207] it
was shown that the saturation based models provide very good description of
the experimental data. In [220, 221] it was found that while the data can be
described by both linear and non-linear low-$x$ QCD evolution, the former
requires unnaturally large perturbative corrections. The authors consider it
as a strong hint of the gluon saturation in exclusive $J/\psi$
photoproduction. Finally, we mention that similar conclusions were reached in
[208] and [222] by extracting the S-matrix from the $t$ dependence of this
process. For example, for $Q^{2}=2\,\rm GeV^{2}$ and $x\simeq 10^{-4}$ the
interaction of the dipole at small impact parameters becomes predominantly
absorptive, see Ref. [222].

At the same time, one should mention that this range of impact parameters
gives a rather small contribution to the cross section integrated over the
impact parameter $b$.

Further, combing the dipole model with fluctuations of the gluon density,
which is taken into account in the spirit of the Good-Walker formalism applied
to the proton or nuclear targets, one can provide simultaneous description of
coherent and incoherent vector meson production [223, 224, 225, 226].

###  5.5 Lessons and open problems

The HERA data, for reviews, see [186, 187], and the LHC data on production of
$J/\psi$ in exclusive photoproduction in ultraperipheral collisions (UPCs),
see Sec. 8, have confirmed the following basic predictions of perturbative
QCD.

(i) The rapid increase with energy of cross sections of vector meson
production, which are proportional to $(xG_{N}(x,Q^{2}_{\rm eff}))^{2}\propto
W^{0.8}$ for $Q^{2}_{\rm eff}\sim 4$ GeV2 in the case of $\rho$ production for
$Q^{2}=10-20$ GeV2 and in the case of $J/\psi$ production for $Q^{2}\leq 10$
GeV2. For $\Upsilon$ production, $Q^{2}_{\rm eff}\approx 40$ GeV2, which leads
to $\sigma(W)\propto W^{1.7}$. This prediction can be tested in the
ultraperipheral collisions at the LHC, but so far the statistics is not high
enough.

(ii) The data on elastic photoproduction of $J/\psi$ can be described rather
well within the collinear approximation, see the discussion in the previous
section. This can be interpreted as an example of the DGLAP dominance. At the
same time, the strong discrepancy between the LLx approximation and the data
can be accommodated within the resummation approach, see Sec. 4. The
restriction on the region of applicability of the leading $\log(x_{0}/x)$
approximation follows from the necessity to take into account energy–momentum
conservation. Indeed, in multi-Regge kinematics, the interval in rapidity
between adjacent radiations within the ladder is $\Delta y\gg 2$. This number
is comparable with the interval in rapidities achieved (to be achieved) in
DIS:

$\Delta y=\ln(1/x)+2\ln(Q/m_{N})\,.$ (104)

For the edge of the kinematics achieved at HERA, $\Delta y\approx 10$. Since
four units of rapidity are occupied by the two fragmentation regions, only
two-to-three gluons are allowed to be radiated in this kinematics.

The energy dependence of the $\rho$ production at small $-t$ is consistent
with expectations of soft dynamics, $\alpha(t)\sim 1.1$, however for large
negative $t\sim 1.5$ GeV2 the trajectory appears to be flattening out at
$\alpha(t)<1$, [227]. In the perturbative regime we would expect these values
to be above $1$, see Fig.19. In this region we expect weak dependence on $t$,
see discussion in Sec. 5.6.

(iii) The absolute values of the cross sections of vector meson production are
well reproduced, provided that the factor $T(Q^{2})$ (Eq. (96)) is taken into
account. In the case of $\Upsilon$ photoproduction, the skewedness effects due
to a large difference between $x_{1}$ and $x_{2}$ as well as the large value
of the real part of the amplitude are also important. Together they increase
the predicted cross section by a factor of about four [206, 165]. Note that
the decrease of the $\sigma^{L}_{\gamma^{\ast}p\to Vp}$ cross section with an
increase of $Q^{2}$ is slower than ${1/Q^{6}}$ because of the
$\left(\alpha_{S}G_{N}\right)^{2}$ and $T(Q^{2})$ factors. The cross section
ratio $\sigma_{L}/\sigma_{T}\gg 1$ for $Q^{2}\gg m_{V}^{2}$.

(iv) There is a universal $t$ dependence for large $Q^{2}$ originating solely
from the two-gluon form factor of the nucleon. The model of Ref. [192], which
takes into account transverse squeezing of $\gamma_{L}^{\ast}$ (decrease of
the average $r$) with $Q^{2}$, provides a reasonable description of the
convergence of the $t$-slopes of light mesons and $J/\psi$ production and
makes the observation that the slope of $J/\psi$ production is practically
$Q^{2}$ independent (Fig. 25). The observed dependence of the $t$-slopes on
$Q^{2}$ indicates that in the case of light vector mesons, the LT dominance
(universality regime) is reached only at $Q^{2}\geq 15\;\mbox{GeV}^{2}$.

A confirmation of a late onset of the regime of the universality follows also
from the measurements of the $\phi$ and $\rho$ cross sections [227] up to
$Q^{2}\simeq\mbox{30 GeV}^{2}$, where the measured cross section ratio is
$\sigma_{\phi}/\sigma_{\rho}=0.191\pm 0.007({\rm
stat.})^{+0.008}_{-0.006}({\rm syst.})\pm 0.008({\rm norm.})$. The
measurements are close to the value expected from quark charge counting $2:9$,
but they tend to be slightly lower. On the other hand the LT prediction of Eq.
(89) for $\sigma_{\phi}/\sigma_{\rho}\times 9/2$ is expected to be $1.06\pm
0.02$ (here we use the values of $\Gamma_{V}$ and $M_{V}$ from [228]) and the
theoretical expectation is that at this value of $Q^{2}$, the wave functions
of $\phi$ and $\rho$ should be very close.

(v) The observed $t$ dependence of the slopes of hard diffractive processes in
the LT regime provides unique information on the distribution of quarks and
gluons within hadronic targets. The interior of the nucleon of the radius
$\approx$ 0.5 fm is filled with quarks and gluons, but not mesons (hadrons) as
was popular before the onset of QCD and currently in the low energy nuclear
physics. A small number of antiquarks in the nucleon (nucleus) indicates that
the observation of hard diffractive processes off nuclear target will allow
one to demonstrate the absence of short-range meson currents in nuclei.

Figure 25: The convergence of the $t$-slopes, $B$, of $\rho$ and $J/\psi$
electroproduction off a nucleon at high $Q^{2}$. The data are from [229, 230];
the curves are the predictions of [98]. The figure is from section
“Diffractive phenomena in high energy processes”, L. Frankfurt and M.
Strikman, part of book “100 Years of Subatomic Physics”, Eds. E.M. Henley and
S.D. Ellis, 2013, World Scientific.

(vi) In the case of light vector mesons, one can estimate the effective size
of a $q\bar{q}$ dipole as

${B(Q^{2})-B_{2g}\over B(Q^{2}=0)-B_{2g}}\sim{R^{2}({\rm dipole})\over
R^{2}_{\rho}}\,,$ (105)

where $B_{2g}$ is the slope of the square of the two-gluon form factor. Based
on the HERA data [230], we conclude that

${R^{2}({\rm dipole})(Q^{2}\geq 3\ {\rm GeV}^{2})/R^{2}_{\rho}}\leq{1/2-1/3}$
(106)

for collider energies. Accordingly, it appears that the soft energy dependence
of the cross section persists over a significant range of the dipole sizes.
This is consistent with the observed similarity of the energy dependence of
$\rho$ and $\phi$ photoproduction.

(vii) In the pQCD regime, the $t$-slope the dipole–nucleon amplitude should be
a weak function of $s=W^{2}$, $B(s)=B(s_{0})+2\alpha^{\prime}_{\rm
eff}\ln(s/s_{0})$, since the Gribov diffusion in the hard regime is small (see
the discussion in Sect. 4.2. Hence, a significant contribution to
$\alpha^{\prime}$ comes from the variation of the $t$ dependence of the gluon
GPD with a decrease in $x$ at $Q_{0}^{2}$. There are indications that
$\alpha_{\rm eff}(t)$ for $\rho$-meson exclusive electroproduction drops below
unity at $t\lesssim-1$ GeV2 though in the limit $-t\sim Q^{2}$, one expects
$\alpha_{\rm eff}>1$. This may indicate that the soft physics dominates in
GPDs at such $t$.

(viii) The contribution of soft QCD physics in the overlapping integral
between the wave functions of the virtual photon and the transversely
polarized vector meson is suppressed by the Sudakov form factor (see, e.g.,
the discussion in [231]), which is absent in the case of the processes
initiated by longitudinally polarized photons. This is relevant for the
understanding of similar dependence of $\sigma_{L,T}$ on $x$ and on $t$ that
were observed at HERA. Note, however, the indication of a slightly larger $t$
slope for $\sigma_{T}$ for $Q^{2}=8.6\,\rm GeV^{2}$ as reported by the H1
collaboration [227].

(ix) The account of NLO effects was so far concerned with NLO effects in the
scattering amplitude, while the contribution of the $q\bar{q}g$ components was
not analyzed. For the charmonium case, such effects appear to be important,
see the discussion below.

(x) There exists no formal proof of factorization for photo/electroproduction
of quarkonia in the $Q^{2}\lesssim M_{V}^{2}$ limit. However, such
factorization seems natural due to small sizes of the lowest mass quarkonium
states. Still there are substantial difference between low $Q^{2}$ limit and
the limit, where the factorization theorem was derived. Below we list only
several of the important effects, for an extensive discussion, see [192, 127]
.

(a) In the case of $J/\psi$ photoproduction, the $c\bar{c}g$ component of the
wave function is comparable to the $c\bar{c}$ component since the $\alpha_{s}$
factor is compensated by a large numerical coefficient.

(b) The nonrelativisitic approximation is not valid for $J/\psi$ production
because in the overlap integrals, the region $k\sim m_{c}$ gives a significant
contribution. Overall the quark Fermi motion results in the strong suppression
of the cross section as compared to the approximation, when the Fermi motion
is neglected [190], see also the discussion above. The relativistic
corrections to the charmonoium wave function and their effect on the cross
section of elastic $J/\psi$ photoproduction were recently considered in Refs.
[232, 233].

(c) The predicted cross section strongly depends on $m_{c}$ so one in
principle needs to take into account the $Q^{2}$ dependence of $m_{c}$ in such
a way that the condition that the intermediate mass
$M^{2}_{c\bar{c}}>m^{2}_{J/\psi}$ is satisfied.

(d) In the case of $\psi^{\prime}$ production, the application of the dipole
model appears problematic due to the proximity of the $D\bar{D}$ threshold.
Hence, one may expect that the $\psi^{\prime}$ wave function at large
distances $r$ effectively describes $D\bar{D}$ configurations, which have a
small overlapping integral with the photon wave function.

Since production of $J/\psi$ serves as one of probes of the $x$ dependence of
gluon density at moderate $Q^{2}$ and small $x$, it is important to
investigate to what accuracy the effects mentioned above could be factorized
so that they do not result in significant corrections to the ratio of gluon
densities of nuclei and proton, see discussion in Sec. 8.1.

###  5.6 Hard diffraction in DIS and rapidity gap processes

Another process of interest is the production of heavy vector mesons with a
rapidity gap, which constitutes a rich source of information on high energy
QCD dynamics, in particular, on the gluonic structure of the nucleon and the
BFKL dynamics in small dipole–parton scattering.

The exclusive process $\gamma^{\ast}p\rightarrow Vp$, which is discussed in
the previous section, dominates at low values of the momentum transfer, i.e.,
$|t|<1\;{\rm GeV^{2}}$. The requirement of the heavy vector meson in the final
state allows for the selection of the small size $q\bar{q}$ dipole pair, which
scatters elastically of a nucleon. As discussed previously, this process has a
steep dependence on the momentum transfer $t$. In the diffractive vector meson
production with a rapidity gap, see Fig. 26,

$\gamma^{\ast}+p\to V+{\rm rapidity~{}gap}+X\;,$ (107)

one can regulate hardness of the interaction by changing the virtuality of the
photon, quark content of the meson, and considering a large momentum transfer
$t=(p_{\gamma}-p_{V})^{2}$.

Figure 26: A typical QCD diagram for the rapidity gap process (107).

Of particular interest is the momentum transfer dependence of this process.
Below we shall discuss separately different regimes of $t$ and the underlying
physical mechanisms.

####  5.6.1 Probing fluctuations of the gluon field at $t\sim 0$.

Let us first consider the reaction given by Eq. (107) in the limit $t\to 0$
and large $Q^{2}$ or heavy onium production. In this limit, the two-gluon
ladder couples only to one parton in the target in the leading twist
approximation. If the strength of the coupling to all configurations in the
nucleon containing a parton with a given $x$ was the same, it would be
impossible to produce an inelastic final system $X$ at $t=0$. As a result, the
discussed process measures the quantum fluctuations (variance) of the gluon
field. It is given by the ratio of the diffraction dissociation and elastic
cross sections for vector meson production at $t=0$, see [234],

$\omega_{g}={d\sigma^{\gamma^{\ast}+p\to V+{\rm rapidity\
gap}+X}(x,Q^{2})/dt\over d\sigma^{\gamma^{\ast}+p\to
V+p}(x,Q^{2})/dt}\bigg{|}_{t=0}\,.$ (108)

To see that this ratio is proportional to the fluctuations of the gluon
density, we shall follow the arguments presented in [234]. Let us assume that
the initial proton state can be expanded in a set of partonic states
characterized by the number of partons and their transverse positions. The
states can be schematically labeled as $|n\rangle$, and the expansion can thus
be written as

$|p\rangle=\sum_{n}a_{n}|n\rangle\;.$ (109)

Each configuration given by $n$ has a definite gluon density $G(x,Q^{2}|n)$,
given by the expectation value of the twist–2 gluon operator in the state
$|n\rangle$. The average gluon density in the proton is

$G(x,Q^{2})\;\;=\;\;{\textstyle\sum_{n}}\,|a_{n}|^{2}\,G(x,Q^{2}|n)\;\equiv\;\langle
G\rangle.$ (110)

The assumption of factorization implies that the the partonic states appear
“frozen” on the typical timescale of the hard scattering process. Therefore
the QCD factorization allows one to calculate the amplitude for vector meson
production configuration by configuration. It is (up to small calculable
corrections) proportional to the gluon density in that particular
configuration [189]. An essential point is now that in the leading-twist
approximation the hard scattering process attaches to a single parton, and,
moreover, does not transfer momentum to that parton (in the low $t$ limit).
Therefore the partonic state $|n\rangle$ will not change. Making use of the
completeness of partonic states, we find that the elastic ($X=p$) and total
diffractive ($X$ is arbitrary) cross sections are proportional to

$\displaystyle\frac{d\sigma_{\rm el}}{dt}\bigg{|}_{t=0}$
$\displaystyle\propto$
$\displaystyle\left[{\textstyle\sum_{n}}|a_{n}|^{2}G(x,Q^{2}|n)\right]^{2}\equiv\;\langle
G\rangle^{2}\;,$ (111) $\displaystyle\frac{d\sigma_{\rm
diff}}{dt}\bigg{|}_{t=0}$ $\displaystyle\propto$
$\displaystyle{\textstyle\sum_{n}}|a_{n}|^{2}\left[G(x,Q^{2}|n)\right]^{2}\equiv\;\langle
G^{2}\rangle\;.$ (112)

Therefore for the cross section of the diffraction with a rapidity gap, we
have

${\sigma^{\gamma^{*}p\rightarrow V+{\rm rapgap}+X}}={\sigma_{\rm
diff}}-{\sigma_{{\rm el}}}\;,$ (113)

and thus obtain

$\omega_{g}\;\;\equiv\;\;{d\sigma^{\gamma^{\ast}+p\to V+{\rm
rap\;gap}+X}/dt\over d\sigma^{\gamma^{\ast}+p\to
V+p}/dt}\bigg{|}_{t=0}=\frac{\langle G^{2}\rangle-\langle
G\rangle^{2}}{\langle G\rangle^{2}}\;.$ (114)

The $x$ and $Q^{2}$ evolution of $\omega_{g}$ originates from the DGLAP
evolution of the gluon distribution [234] and from multiparton correlations
induced by the parton splitting (the effect similar to the one discussed for
the double parton scattering, see review in [235]).

The value of $\omega_{g}\approx 0.15$ for electroproduction of $\rho$-mesons
in reaction (107) was reported in [236]. A very similar value of $\omega_{g}$
for production of $J/\psi$ mesons was derived using the data [211]. The
$J/\psi$ data also indicate that $\omega_{g}$ tends to decrease with increase
of $W$. Indeed, the $J/\psi$ data were also fitted in the Regge phenomenology
spirit using an effective Pomeron trajectory for both elastic and rapidity gap
channels with $\alpha_{el}=1.2\pm 0.01$ and $\alpha_{pd}=1.09\pm 0.02$ leading
to $\omega_{g}$ decreasing with an increase of $W$ as

$\omega_{g}\propto(W/W_{0})^{\Delta_{R}}\,,\quad\Delta_{R}=-0.44\pm 0.08\,,$
(115)

where $\Delta_{R}=4(\alpha(0)-\alpha_{el}(0))$.

In Sec. 6 the fluctuations of soft interactions are discussed. There are
indications that fluctuations of the strength of soft interactions given by
$\omega_{\sigma}$ may be related to the fluctuation of the overall size of
configurations in the nucleon. Since the gluon field in small configurations
should be screened and hence reduced, this picture leads to fluctuations of
the strength of the gluon field in nucleons. A model [234] based on this
picture allows one to reproduce the magnitude of $\omega_{g}$. However, it
does not include the effect of the increase of fluctuations with $Q^{2}$ due
to parton splitting and resulting pQCD parton–parton correlations (these
issues require further theoretical studies).

####  5.6.2 Competition of gluon field fluctuations and break up mechanisms of
diffraction at finite $-t\leq 0.5\;{\rm GeV}^{2}$.

It was argued above that for $t=0$, inelastic hard diffraction is possible
only due to quantum fluctuations of the gluon field. However, for finite $t$,
a two-gluon ladder, which is attached to one parton of the nucleon, transfers
to the parton the transverse momentum $\sqrt{-t}$, which would most likely
lead to inelastic diffractive states, if $\sqrt{-t}\geq\left<k_{t}\right>$
($\left<k_{t}\right>$ is the parton average transverse momentum). At the same
time, at moderate $t$ the final state could remain intact leading to a
contribution to the elastic scattering.

The fraction of the elastic and inelastic channels can be estimated as
follows,

$\sigma_{el}\propto F_{N}^{2}(t),\;\;\;\sigma_{break}\propto
1-F_{N}^{2}(t)\,,$ (116)

where $F_{N}(t)$ is the two-gluon nucleon form factor, which has approximately
the same $t$ dependence as the elastic process $\gamma^{*}+p\to V+p$. Note
here that $\sigma_{break}$ goes to zero at $t=0$ in line with the general
arguments presented above since it does not include the gluon field
fluctuations.

In this intermediate $t$ regime, the cross section can be expressed as a sum
of the fluctuation and break up contributions

$\frac{d\sigma(\gamma^{*}+p\to
V+X)}{dt}\;=\;\omega_{g}\;\frac{d\sigma(\gamma^{*}+p\to
J/\psi+p)}{dt}\bigg{|}_{t=0}\exp(\tilde{B}t)+\bigg{(}1-F_{N}^{2}(t)\bigg{)}\,\phi(t)\;,$
(117)

where $\phi(t)$ contains the residual $t$ dependence due to the coupling of
two gluons to the parton of the nucleon. Here we used the observation that
fluctuations, which contribute to the cross section, are close to the average
size since $\omega_{g}$ is rather small, but nondiagonal transitions should
lead to a somewhat slower $t$ dependence of this term than for for elastic
$J/\psi$ production (similarly to the case of hadron–nucleon diffractive
scattering).

Equation (117) gives a qualitative explanation of the trend observed in [211],
where the fit to the diffractive cross section systematically exceeds the data
at small $t$.

For the function $\phi(t)$ in the second term in Eq. (117) in the $t$-range
$-t<m_{V}^{2}$, we expect that the main contribution to the $t$ dependence
originates from the cross section of gluon ladder–parton elastic scattering.
(The other sources of the $t$-dependence involve the interaction at the
$M_{V}$ or $Q$ scales.) Therefore the function $\phi(t)$ should have the
$t$-dependence similar to that of $\gamma^{\ast}$–quark (or gluon) scattering,
i.e.,

$\phi(t)\propto 1/(1-t/m_{0}^{2})\,,$ (118)

where $m_{0}^{2}\sim 0.5-0.8$ GeV2 is the scale describing the soft-hard
transition.

We find that indeed Eq. (118) with $m_{0}^{2}=0.7$ GeV2 gives a good
description of the diffractive data at $-t\leq 0.8$ GeV2 providing a strong
support to the two-component picture of diffraction described above.

Note that in the model based on the notion of “hot spots” in the nucleon [224,
223], it is assumed that the dominant contribution originated from multiple
rescatterings of small dipoles and the smallness of the rescattering amplitude
is compensated by introducing strong positive short-range correlations. It is
not clear how to reconcile such an assumption with the observed strength of
double parton interactions, see the discussion in Sec. 11. Also, phenomenology
based on the dominance of a single gluon ladder in the $t$-channel (Fig. 26)
provides a good description of the data.

####  5.6.3 Rapidity gaps at large $t$.

Finally, let us consider the limit of large $-t$. In this case, on the
microscopic level this process is elastic scattering of a small dipole off a
parton – a quark or a gluon coming from the proton. The proton dissociates,
but its fragmentation remnants are still well separated from the vector meson
by the large rapidity gap. Thanks to the presence of hard scales, $Q^{2}$
and/or $m_{J/\psi}$ and momentum transfer $t$, the DGLAP evolution is strongly
suppressed as well as the diffusion in the transverse momenta in the gluon
ladder which is exchanged between the parton and the vector meson, see Fig.
26. Hence, in the high energy limit, this process can be suitably described in
terms of the exchange of the perturbative BFKL Pomeron [237]. The latter one
can be obtained from the solution to the BFKL equation in the non-forward
case, that is at $t\neq 0$ [31]. Due to the presence of large scales, that is
a large momentum transfer scale $-t$, a heavy vector meson scale and the fact
that the dissociated proton has a larger mass than the proton mass, the
Pomeron ladder is ‘squeezed’ at both ends. To be more precise, this means that
the typical scales at both ends of the Pomeron are perturbative and large.
This results in the suppression of the diffusion of the transverse momenta
along the Pomeron ladder to the infrared regime. In addition, the multi-
Pomeron exchanges which can fill the rapidity gap, are suppressed in this
kinematics. The presence of the two hard scales allows for the study of the
energy dependence of the Pomeron, and also to separate the effects stemming
from the summation of different types of large logarithms. As discussed in
Sec. 4.2, the $\ln 1/x$ logarithms are relevant at low $x$, and need to be
resummed. This kinematics allows one to separate $\ln 1/x$ effects from $\ln
Q^{2}$ logarithms. In this way, by studying the dependence of the hard
diffractive vector meson cross section as a function of the rapidity gap, one
can learn about the energy dependence of BFKL Pomeron.

The process discussed here is schematically represented in Fig. 26. The
electron–proton cross section may be factorized into a universal photon flux
and the $\gamma p$ subprocess $\gamma p\rightarrow VY$. Since we are
considering the photoproduction, the photon is quasi-real. It interacts with
the incoming proton, and produces state $Y$ as well as the vector meson
$V(p)$, separated by the rapidity gap of size $\Delta y$. At large $-t$, such
that $|t|\gg\Lambda_{QCD}^{2}$ the process is tractable within the
perturbative QCD by means of the perturbative Pomeron given by the non-forward
BFKL equation [29, 30, 31]. The cross section can be written in the factorized
form, which is the convolution of the partonic cross section describing the
interaction of the photon with the quark or a gluon from the proton and the
collinear parton distribution function

$d\sigma\;=\;\sum_{i=g,q,\bar{q}}\,\int
dx\,f_{i}(x,\mu)\,d\hat{\sigma}_{\gamma i}(\hat{s},t,\mu)\;.$ (119)

Here, $\hat{s}=xs$ is the photon–proton invariant mass squared and $f_{i}$ is
the collinear parton distribution for the parton of type $i$, i.e., gluon,
quark or anti-quark. Also, $x$ is the fraction of the longitudinal momentum of
the proton carried by the parton, a quark or a gluon that participates in the
sub-process, and $\mu$ is the factorization scale that is given by the
transverse momentum in the process.

The partonic cross section can be written as

$d\hat{\sigma}_{\gamma i}\;=\;C_{\gamma i}\,\frac{1}{16\pi\hat{s}^{2}}\,|{\cal
A}(\hat{s},t)|^{2}\frac{d^{2}{\bf p}}{\pi}\;,$ (120)

where $C_{\gamma i}$ is the color factor for $i=g,q$ gluon or a quark and
${\cal A}$ is the amplitude to produce a vector meson through the Pomeron
exchange which can be written as

${\rm Im}\,{\cal A}(\hat{s},t=-p^{2})\;=\;\hat{s}\int\frac{d^{2}{\bf
k}}{2\pi}\frac{\Phi_{V}({\bf k},{\bf p})\,\Phi_{q}(\Delta y,{\bf k},{\bf
p})}{({\bf k}^{2}+s_{0})[({\bf p}-{\bf k})^{2}+s_{0}]}\;,$ (121)

The diagrams which are taken into account in the partonic subprocess are shown
in Figs. 27. The ${\bf k}$ and ${\bf k}-{\bf p}$ are the transverse momenta of
the exchanged gluons in the $t$ channel and ${\bf p}$ is the transverse
momentum of the produced vector meson. In this kinematical approximation,
$t=-{\bf p}^{2}$. Also, $\Phi_{V}$ and $\Phi_{q}$ are the impact factors of
the vector meson and quark, respectively. Parameter $s_{0}$ is the infrared
cutoff, also set in the solution of the BFKL equation.

Figure 27: Diagrams for the partonic sub-process of the diffractive $J/\psi$
production in scattering of the photon off the parton (gluon and quark).
Vertical dashed line indicates the diffractive cut, the upper blob denotes the
photon-meson impact factor and the lower blob indicates the gluon Green’s
function of the BFKL hard Pomeron. Figure from [163],
https://doi.org/10.1103/PhysRevD.103.014022.

The expression for the vector meson impact factor was calculated in Ref. [238]
using a non-relativistic model for the wave function, which differs
significantly from the light-cone $q\bar{q}$ wave function discussed in the
beginning of Sec. 5.

The parton impact factor depends on the rapidity interval $\Delta y$, and in
the calculation can be obtained from the BFKL equation in the following form

$\Phi_{q}(\Delta y,{\bf k},{\bf p})\;=\;\int d^{2}{\bf
k}^{\prime}\,\Phi_{q}({\bf k}^{\prime},{\bf p})\,{\cal G}(\Delta y,{\bf
k},{\bf k}^{\prime},{\bf p})\;,$ (122)

where the function ${\cal G}$ is the gluon Green’s function for the non-
forward BFKL equation, compare Eq. (44) with the Dirac delta
$\delta^{(2)}({\bf k}-{\bf k}^{\prime})$ as the initial condition. The un-
evolved impact factor in the lowest order is just $\Phi_{q,0}=\alpha_{s}$.

The setup described above was used to compute the cross section and describe
the data for diffractive production of $J/\psi$ at HERA at large $-t$. Both
ZEUS [239] and H1 collaborations [240] have performed the measurements of this
process, which are consistent with each other. The ZEUS [239] data were
performed in the $\gamma p$ energy range of $80-120\;\rm GeV$ and H1 [240] at
$50-200\;\rm GeV$. The range of the momentum transfer was $|t|<12\;\rm
GeV^{2}$ and $2<|t|<30\;\rm GeV^{2}$ for ZEUS and H1 respectively.

Figure 28: The differential cross section $d\sigma/dt$ for the diffractive
$J/\psi$ production vs H1 data [240]. The calculation was done with LL BFKL
using the coupling constant as a fit parameter equal to
$\bar{\alpha}_{s}=0.14$. Figure from [241],
https://doi.org/10.1007/JHEP07(2019)129.

In Fig. 28 a comparison of the calculation following [241] with the
experimental data from H1 [240] is shown. We note that, the calculation
presented here is applicable for large values of $t$. There are data from H1
[211] which extend to $|t|<5.7\;{\rm GeV^{2}}$, in the region where both sets
of data have common range of $t$, they are consistent with each other. We show
in Fig. 28 only data from [240]. The mass of the diffractively produced state
is restricted to be $M_{Y}^{2}<|t|/x$ with $M_{Y}=30\;\rm GeV$. For this
calculation the rapidity gap interval was defined as

$\Delta y=\log\left(\frac{\hat{s}}{E_{T}^{2}}\right)\;,$

with $E_{T}=\sqrt{M^{2}_{V}+p_{T}^{2}}$. The more exact definition of the
rapidity gap is [242]

$\Delta y=\log\left(\frac{\hat{s}}{\sqrt{-t(-t+M_{V}^{2})}}\right)\;,$

which coincides with the one above when $|t|\gg M_{V}^{2}$. The choice of the
scale $\mu$ appearing in Eq. (119) both in PDFs and in the coupling constant
was taken to be $\mu^{2}=E_{T}^{2}=p_{T}^{2}+M_{V}^{2}$, with CT14nlo PDF set.
The parameter $s_{0}$ was set to be equal to $0.5\;\rm GeV^{2}$. It is
important to note that, the setup is finite for $s_{0}\;\rightarrow 0$,
including the BFKL equation which is infrared finite. However this parameter
was introduced, following [243], to mimic the effects of the confinement.

As evident from Fig. 28 the calculation describes the experimental data very
well, which was also found in earlier calculations based on the BFKL equation
[244]. However, the HERA data and in particular the experimental setup of the
detectors at HERA did not allow for the precise measurement of the energy
dependence on the rapidity gap interval. The energy dependence of the cross
section in Eq. (119) stems from the two sources: the energy dependence of the
BFKL Pomeron and the $x$ dependence of the collinear distribution function. In
order to be able to extract the Pomeron energy dependence it is important to
perform a measurement in which the $x$ of the parton is fixed, or in other
words to measure for fixed rapidity gap intervals, and not integrate over the
entire range.

In [163] analysis was performed of the different experimental scenarios for
the measurement of the rapidity gaps in this process at the Electron Ion
Collider (EIC). Two cases for detection were considered:

  1. 1.

Request the detection of the $J/\psi$ vector meson, a rapidity gap and an
activity in the detector in the direction of the proton beam. The latter,
formed by the fragments of the dissociated target, is separated from the
vector meson by the rapidity gap.

  2. 2.

Request detection of the $J/\psi$ meson and a rapidity gap. No other activity
in the central detector is registered.

In the second scenario, given the limitations in the coverage of the central
part of the detector, one does not have a knowledge of the exact size of the
rapidity gap $\Delta y$. Therefore, one needs to integrate over the
longitudinal momentum fraction $x$ of the parton in the range of rapidity not
accessible by the detector. Since the maximum rapidity covered is related to
the minimum polar angle, in the calculation the cut was applied on the polar
angle of the last particle at the edge of the rapidity gap, i.e.
$\theta_{j}>\theta_{\rm min}$.

In Fig. 29 the structure of the phase space of vector meson production in
diffraction is shown. The two-dimensional plots indicate the number of events
differential in the energy $W$ as a function of $W$ and the $\Delta Y_{\rm
min}$, which is the minimum rapidity gap. The multiplicity is also integrated
over $t$ and $x$ over the bins. The smaller the value of the momentum transfer
$|t|$ and the larger the $x$, the larger accessible rapidity gap. For example,
we see that for $|t|\in(1,2)$ and $x\in(0.1,0.3)$ the range in $\Delta y$ is
well above $4$ and even up to $6$ for largest energies, left top panel in Fig.
29. On the other hand for larger $|t|\in(4,8)$ and smaller $x\in(0.01,0.06)$
the rapidity range is at most $4$ for largest energies at the EIC, left lower
panel in Fig. 29. In the right panels, the calculations is shown when the
experimental cut on the angle is applied. We observed that for certain bins in
$x$ and $t$ this cut acts as a veto on the vector meson production in this
process. This is particularly striking in the upper right panel of Fig. 29
since large $x$ and small $t$ mean lower angle at which the vector meson is
produced.

Thus it is seen that the EIC, with the proper detector setup will be able to
measure range in the rapidity sizes and extract some limited range of the
dependence from this measurement. Obviously, the higher energy machine, like
the LHeC would have a much wider range of the rapidity, thus allowing for the
longer lever arm and possibility of the more precise extraction of the energy
behavior of the Pomeron. Finally, studies of rapidity gap in the $J/\psi$ and
$\psi^{\prime}$ process are feasible also in ultraperipheral $pA$ and $AA$
collisions at the LHC [245, 5].

Figure 29: Differential number of events in $W$ in bins in $t$ and $x$ as a
two-dimensional function of $W$ and $\Delta Y_{\rm min}$ for the EIC
kinematics. Left column: no cuts on angle, right column: restriction on angles
$4^{\circ}$. Upper row: bin in $|t|\in(1,2){\rm GeV^{2}}$ and $x\in(0.1,0.3)$,
lower row: bin in $|t|\in(4,8){\rm GeV^{2}}$ and $x\in(0.01,0.05)$. Integrated
luminosity ${\cal L}=10\;\rm fb^{-1}$. Figure from [163],
https://doi.org/10.1103/PhysRevD.103.014022.

##  6 Inelastic diffraction in hadron–hadron scattering and formalism of cross
section fluctuations

###  6.1 Inelastic diffraction and orthogonality of wave functions

Inelastic diffraction in hadron–hadron scattering at high energies has been
attracting attention of theorists since the 50’s. In a sense it appears as a
counterintuitive process: a high energy hadron $h$ can scatter off a
stationary nucleon and transform into a hadronic system of the mass $M$
leaving the target nucleon practically at rest with the longitudinal momentum
$p_{L}=(M^{2}-m_{h}^{2})/(2p_{h})\to 0$, when the projectile momentum
$p_{h}\to\infty$. The corresponding minimum invariant momentum transfer
squared is

$-t_{\rm min}=p_{L}^{2}=\frac{(M^{2}-m_{h}^{2})^{2}}{(2p_{h})^{2}}\approx
0\,.$ (123)

To see why the existence of such processes is non-trivial, it is instructive
to consider an example of scattering of a bound state of two nucleons (e.g. a
deuteron) off a stationary hadron in the impulse approximation. Assuming for
simplicity that the cross sections of proton–hadron and neutron–hadron
interactions are equal, the cross section of deuteron–hadron inelastic
diffraction can be written in the following form

${d\sigma(D+h\to M+h)\over dt}=2{d\sigma(N+h\to M+h)\over
dt}\left|\langle\psi_{D}\left(k,-k\right)|\psi_{M}\left(k+\frac{q}{2},-k-\frac{q}{2}\right)\rangle\right|^{2},$
(124)

where $\psi_{D}$ and $\psi_{M}$ are the wave functions of the incoming
deuteron and the produced state $M$, respectively, which depend on the momenta
in the two-nucleon center of mass system. In particular, $k$ is the initial
proton momentum and $k+q/2$ is the proton momentum after the scattering, where
$q$ is the momentum transfer.

If $M$ is a two-nucleon system, the condition of orthogonality of the wave
functions of the continuum and bound states leads to

${d\sigma(D+h\to"pn"+h)\over dt}\bigg{|}_{t=0}\,=\,0\,,$ (125)

i.e., to the absence of inelastic diffraction at $t=0$.

In a general case of $t\neq 0$, summing over the states $M$ and using their
completeness in Eq. (124), one can write

$\displaystyle{d\sigma(D+h\to"pn"+h)\over dt}\bigg{|}_{\rm
incoh}\,=\,2{d\sigma(N+h\to N+h)\over dt}\big{(}1-F^{2}_{D}(4t)\big{)}\,,$
$\displaystyle{d\sigma(D+h\to D+h)\over dt}\,=\,4{d\sigma(N+h\to N+h)\over
dt}F_{D}^{2}(t)\,,$ (126)

where $F_{D}(t)$ is the deuteron elastic form factor. Equation (126) indicates
that inelastic diffraction is impossible at $t=0$, if $h$ interacts with just
one nucleon. At the same time at $-t>0$, inelastic diffraction is present and
dominates for $t$ satisfying the condition $-t\geq 1/R^{2}_{D}$ ($R_{D}$ is
the effective deuteron radius).

Note that no cancellation of inelastic diffraction at $t=0$ takes place, if
$h$ can interact (at the amplitude level) with both nucleons of the deuteron.
This is the case of the nuclear shadowing correction to the total
pion–deuteron (hadron–nucleus) cross section discussed in Sec. 7.

###  6.2 Cross section fluctuations

The idea that inelastic diffraction can take place at small $t$ due to the
presence of configurations in the nucleon, which can interact with different
strengths, was first suggested in the paper of Feinberg and Pomeranchuk [1,
2]. In this work, as an example of fluctuation of the interaction strength,
the authors considered fluctuations of the nucleon into the nucleon and the
pion, where the latter could originate from the pion field of the nucleon.

A model illustrating this idea was suggested by Good and Walker [3]. They
assumed that the projectile can interact with different interaction strengths
in contributing configurations, which do not change while the wave packet
passes through the target. The corresponding coherence length (time) $l_{c}$
denotes the distance, over which the incoming hadron remains in the state with
the mass $M^{\ast}$,

$l_{c}=\frac{1}{\Delta E}=\left(\sqrt{M^{\ast\,2}+p_{\rm
lab}^{2}}-\sqrt{m_{h}^{2}+p_{\rm lab}^{2}}\right)^{-1}\simeq\frac{2p_{\rm
lab}}{M^{\ast 2}-m_{h}^{2}}\ \gg R_{\rm target}\,,$ (127)

where $R_{\rm target}$ is the radius of the target. From Eq. (127), one can
immediately see that the coherence length linearly grows with an increase of
the energy of the incoming hadron. Therefore, the range of masses $M^{\ast}$,
which contribute to the fluctuations and which can be considered “frozen”,
increases.

Good and Walker [3] assumed that the state of an energetic incident hadron
$|\Psi\rangle$ can be represented as a coherent superposition of eigenstates
$|\Psi_{k}\rangle$ of the scattering matrix

$|\Psi\rangle=\sum_{k}c_{k}|\Psi_{k}\rangle\ ,$ (128)

where

$\displaystyle{\rm Im}T|\Psi_{k}\rangle$ $\displaystyle=$ $\displaystyle
t_{k}|\Psi_{k}\rangle\ ,$ $\displaystyle\sum_{k}|c_{k}|^{2}$ $\displaystyle=$
$\displaystyle 1\ .$ (129)

Here, $T$ is the scattering operator, and $t_{k}$ is the imaginary part of the
eigenvalue corresponding to the eigenstate $|\Psi_{k}\rangle$.

Various states $|\Psi_{k}\rangle$ interact with the target with different
cross sections $\sigma_{k}$. By the optical theorem, $\sigma_{k}$ is related
to the imaginary part of the scattering amplitude $t_{k}$,

$\sigma_{k}=t_{k}\,.$ (130)

Thus, the coherent superposition of the eigenstates, which form the final
state emerging after the scattering, could be different from the initial
state. Note that introduction of states, which interact with different cross
sections, is natural in QCD, see the discussion in Sec. 3, where (at least in
the perturbative regime) the strength of interaction is related to the area
occupied by color.

Thus, the formalism of eigenstates of the scattering matrix is natural for
describing diffractive dissociation of hadrons. However, the model is valid
only for small $t$, which is the kinematics considered in the original paper
[3], since the authors discussed diffractive dissociation for scattering off
nuclei. Thus, they effectively assumed that $t$ is very small, $-t\leq
2/R_{A}^{2}$, where $R_{A}$ is the effective nucleus size. Later on in a
number of papers it was assumed that Eq. (129) can be applied in a wide range
of the momentum transfer, which seems problematic. Indeed, elastic scattering
of one of the constituents of the diffracting hadron can break it at finite
$t$ even in the absence of fluctuations, see the case of the deuteron–proton
scattering considered above and the rapidity gap process discussed in Sec.
5.6. The assumption that cross section eigenstates are orthogonal at $t\neq 0$
is in contradiction with calculations in the dipole model, where one obtains
for two states with different transverse sizes $r_{t}$ and $r_{t}^{\prime}$

$\langle\psi(r_{t})|T(t\neq 0)|\psi(r_{t}^{\prime})\rangle\neq 0\;.$ (131)

Also, additional evidence comes from the analysis of soft diffraction in $pp$
scattering [246], which shows that spin-flip amplitudes become important and
dominate at large $-t\geq 0.2-0.3$ GeV2.

Thus, the formalism of eigenstates of the scattering matrix is suitable for
describing diffractive dissociation of hadrons for $t\sim 0$.

Using Eqs. (128)–(130), diffractive dissociation can be presented as follows.
Diffractive scattering occurs when the final state carries the same quantum
numbers as the initial state, i.e., whenever the initial state overlaps with
any $|\Psi_{k}\rangle$. Then, the total diffractive differential cross section
at $t=0$ can be presented as

$\Big{(}\frac{d\sigma}{dt}\Big{)}^{\rm
diff}_{t=0}=\frac{1}{16\pi}\sum_{k}|\langle\Psi_{k}|{\rm
Im}T|\Psi\rangle|^{2}=\frac{1}{16\pi}\sum_{k}|c_{k}|^{2}t_{k}^{2}\equiv\frac{1}{16\pi}\langle\sigma^{2}\rangle\
.$ (132)

In Eq. (132), we have used the completeness of the set of states
$|\Psi_{k}\rangle$ and the optical theorem (130). Similarly, the elastic
differential cross section at $t=0$ reads

$\Big{(}\frac{d\sigma}{dt}\Big{)}^{\rm
el}_{t=0}=\frac{1}{16\pi}|\langle\Psi|{\rm
Im}T|\Psi\rangle|^{2}=\frac{1}{16\pi}\Big{(}\sum_{k}|c_{k}|^{2}t_{k}\Big{)}^{2}\equiv\frac{1}{16\pi}\langle\sigma\rangle^{2}\,.$
(133)

In these equations, we introduced the first and second moments of the
distribution over cross sections

$\displaystyle\langle\sigma\rangle$ $\displaystyle=$
$\displaystyle\sum_{k}|c_{k}|^{2}t_{k}\,,$
$\displaystyle\langle\sigma^{2}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k}|c_{k}|^{2}t_{k}^{2}\,.$ (134)

Subtracting the elastic cross section from the total diffractive cross
section, one obtains the diffractive dissociation cross section (inelastic
diffractive cross section)

$\Big{(}\frac{d\sigma}{dt}\Big{)}^{\rm
diss}_{t=0}=\Big{(}\frac{d\sigma}{dt}\Big{)}^{\rm
diff}_{t=0}-\Big{(}\frac{d\sigma}{dt}\Big{)}^{\rm
el}_{t=0}=\frac{1}{16\pi}\Big{(}\langle\sigma^{2}\rangle-\langle\sigma\rangle^{2}\Big{)}\,.$
(135)

Equation (135) was first derived in [247] to describe diffractive dissociation
within the framework of cross section fluctuations. It explicitly demonstrates
that diffractive dissociation occurs only if different components
$|\Psi_{k}\rangle$ of the incident hadron interact with the target with
different strengths $t_{k}$, i.e., when cross section fluctuations take place
in the wave function of the incoming hadron.

###  6.3 Properties of distribution over interaction strength $P(\sigma)$

Within the eigenstate scattering formalism it is useful to introduce the
distribution over cross sections $P(\sigma)$, which gives the probability to
find in an energetic projectile (proton, pion, real or virtual photon) a
hadronic (quark–gluon) configuration interacting with the target with the
cross section $\sigma$. This distribution is a continuum version of the
descrete formalism discussed in the previous section,

$\displaystyle\sum_{k}|c_{k}|^{2}$ $\displaystyle\to$ $\displaystyle\int
d\sigma P(\sigma)\,,$ $\displaystyle\langle\sigma\rangle$ $\displaystyle=$
$\displaystyle\int d\sigma P(\sigma)\sigma\,,$
$\displaystyle\langle\sigma^{2}\rangle$ $\displaystyle=$ $\displaystyle\int
d\sigma P(\sigma)\sigma^{2}\,.$ (136)

Note that the $P(\sigma)$ distribution was introduced by Miettinen and Pumplin
[247] in the context of a parton model, where the strength of interaction is
proportional to the number of slow partons. It was revived in Ref. [95] within
the color screening framework.

The distribution of $P(\sigma)$ satisfies the normalization and the average
cross section sum rules

$\displaystyle\int d\sigma P(\sigma)$ $\displaystyle=$ $\displaystyle 1\,,$
$\displaystyle\int d\sigma P(\sigma)\sigma$ $\displaystyle=$
$\displaystyle\sigma_{\rm tot}\,,$ (137)

where $\sigma_{\rm tot}$ is the total projectile–target cross section.

The dispersion of $P(\sigma)$ around $\langle\sigma\rangle$ is naturally
related to the cross section of diffractive dissociation, see Eq. (135),

$\int d\sigma
P(\sigma)(\sigma^{2}/\sigma_{tot}^{2}-1)\,=\,\Big{(}\frac{d\sigma}{dt}\Big{)}^{\rm
diss}_{t=0}\Big{/}\Big{(}\frac{d\sigma}{dt}\Big{)}^{\rm
el}_{t=0}\,\equiv\,\omega_{\sigma}\,.$ (138)

The shape of the distribution $P(\sigma)$ can be modeled using Eqs. (137) and
(138) and making a natural assumption that fluctuations around
$\langle\sigma\rangle$ should be approximately Gaussian. In addition one can
use the analysis [248] of the coherent diffraction off the deuteron, which
indicates that $\left<(1-\sigma/\sigma_{\rm tot})^{3}\right>\approx 0$.

The behavior of $P(\sigma\to 0)$ is determined by the interplay of the
probability for a hadron to be in a small-size configuration and the cross
section for such a configuration leading to

$P_{h}(\sigma)\propto\sigma^{n_{q}-2}\ ,$ (139)

where $n_{q}$ is the number of valence quarks in the hadron. Thus, for protons
and pions, when $\sigma$ is much smaller than the average value of $\sigma$,
i.e., when $\sigma\ll\langle\sigma\rangle$, one obtains that

$\displaystyle P_{p}(\sigma)$ $\displaystyle\sim$ $\displaystyle\sigma\,,$
$\displaystyle P_{\pi}(\sigma)$ $\displaystyle\sim$ $\displaystyle{\rm
const}\,.$ (140)

The parameterizations of the proton and pion distribution functions
$P(\sigma)$ suggested in the literature [95, 248] describe correctly the
behaviour at small and large $\sigma$ discussed above,

$\displaystyle P_{p}(\sigma)$ $\displaystyle=$ $\displaystyle
N_{p}\,\frac{\sigma/\sigma_{0}}{\sigma/\sigma_{0}+1}e^{-(\sigma-\sigma_{0})^{2}/(\Omega\sigma_{0})^{2}}\
,$ $\displaystyle P_{\pi}(\sigma)$ $\displaystyle=$ $\displaystyle
N_{\pi}\,e^{-(\sigma-\sigma_{0})^{2}/(\Omega\sigma_{0})^{2}}\,,$ (141)

where $\sigma_{0}\sim\sigma_{\rm tot}$ and $\Omega$ determine the peak and the
width of $P(\sigma)$, respectively. These parameters are found using the sum
rules of Eqs. (137) and (138), and, therefore, they depend on energy.

In the pion case one can go further and use pQCD to calculate the probability
of small-size configurations, i.e., the constant entering Eq. (140). Indeed,
using the QCD factorization theorem [96, 95] for the interaction of a small-
size dipole with hadrons, one finds for $P_{\pi}(\sigma)$ at
$\sigma\ll\langle\sigma\rangle$,

$P_{\pi}(\sigma\ll\langle\sigma\rangle)\,=\,\frac{6f^{2}_{\pi}}{5\alpha_{s}(4k^{2}_{t})\bar{x}G_{N}(\bar{x},4k^{2}_{t})}\;.$
(142)

In this equation, $\alpha_{s}(4k^{2}_{t})$ is the QCD running coupling
constant; $G_{N}(\bar{x},4k^{2}_{t})$ is the gluon distribution in the
nucleon; $\bar{x}=4k^{2}_{t}/s_{\pi N}$, where $s_{\pi N}$ is the center of
mass energy squared; $k^{2}_{t}\propto 1/d_{t}^{2}$, where $d_{t}$ is the
transverse size of the $q\bar{q}$ pair in the pion wave function; $f_{\pi}$ is
the constant of the $\pi\to\mu\nu$ decay.

This estimate gives a correct magnitude of $P_{\pi}(\sigma\ll\sigma_{tot}(\pi
N)$, see Fig. 2 of Ref. [249].

To present explicit forms of $P_{p}(\sigma)$ and $P_{\pi}(\sigma)$, we used
the COMPETE parametrization of the total proton–proton and pion–proton cross
sections [228].

The variance of the $P_{p}(\sigma)$ distribution for protons,
$\omega_{\sigma}$ in Eq. (138), can be extracted from the data at fixed-target
and collider energies. It was found [248, 250] that $\omega_{\sigma}$
initially grows with energy reaching a broad maximum value
$\omega_{\sigma}\sim 0.30\pm 0.05$ around $\sqrt{s}\sim\mbox{60 GeV}$ and then
starts to decrease for $\sqrt{s}>200$ GeV toward the Tevatron and LHC energies
reaching the value of $\omega_{\sigma}\sim 0.1$ at $\sqrt{s}\sim\mbox{8 TeV}$.
Thus, one can parameterize the energy dependence of $\omega_{\sigma}$ for the
proton in the following simple form

$\omega_{\sigma}(s)=\left\\{\begin{array}[]{ll}\beta\,\sqrt{s}/(24{\rm\,GeV})\,,&\sqrt{s}<24\
{\rm GeV}\,,\\\ \beta\,,&24<\sqrt{s}<200\ {\rm GeV}\,,\\\
\beta-0.056\ln(\sqrt{s}/200{\rm\,GeV})\,,&\sqrt{s}>200\ {\rm
GeV}\,,\end{array}\right.$ (143)

where $\beta=0.30\pm 0.05$.

The resulting $P_{p}(\sigma)$ as a function of $\sigma$ for three typical
values of energies ($\sqrt{s}=200$ GeV, $\sqrt{s}=1.8$ TeV, and $\sqrt{s}=13$
TeV) is presented in Fig. 30. One can see from the figure that the
distribution $P_{p}(\sigma)$ remains rather broad for all studied energies
since a decrease of $\omega_{\sigma}$ is compensated by an increase of
$\sigma_{\rm tot}$ [228].

Figure 30: The distribution $P_{p}(\sigma)$ as a function of $\sigma$ for
$\sqrt{s}=200$ GeV (solid red), $\sqrt{s}=1.8$ TeV (blue short-dashed), and
$\sqrt{s}=13$ TeV (green dot-dashed).

For the pion projectile, one can use the constituent quark counting rule for
the ratio of the nucleon–nucleon and the pion–nucleon total cross sections
[251] to obtain the following simple estimate for $\omega_{\sigma}$ for pions

$\omega_{\sigma}^{\pi}=\frac{3}{2}\omega_{\sigma}\,.$ (144)

The resulting $P_{\pi}(\sigma)$ distribution for pions as a function of
$\sigma$ for $\sqrt{s}=46$ GeV and $\sqrt{s}=62$ GeV is shown in Fig. 31.
These values correspond to the invariant photon–nucleon energies accessed in
photoproduction of $\rho$ mesons in heavy-ion ultraperipheral collisions at
central rapidities at the LHC, see next section.

Figure 31: The distribution $P_{\pi}(\sigma)$ as a function of $\sigma$ for
$\sqrt{s}=46$ GeV (solid red) and $\sqrt{s}=62$ GeV (blue dashed).

###  6.4 $P_{\rho}(\sigma)$ distribution for $\rho$ mesons

Color fluctuation phenomena can also be studied in $\rho$ meson
photoproduction, which was explored at HERA and in heavy-ion ultraperipheral
collisions at the LHC. We note that $P_{\rho}(\sigma)$ corresponds actually to
the $\gamma\to\rho$ transition (we shall use $P_{\rho}$ as a shorthand
notation).

Based on the constituent quark counting rule, it is generally expected that
the $P_{\rho}(\sigma)$ distribution for $\rho$ mesons should be similar to
that for pions. However, this does not seem to be supported by the HERA data
on $\rho$ photoproduction. Indeed, the model based on the combination of the
assumption that $\sigma_{\rho N}=\sigma_{\pi N}$ ($\sigma_{\rho N}$ and
$\sigma_{\pi N}$ are the total $\rho$ meson–nucleon and pion–nucleon total
cross sections, respectively) with the vector meson meson dominance (VMD)
model somewhat overestimates the HERA data on the $\sigma_{\gamma p\to\rho p}$
cross section of elastic $\rho$ photoproduction on the proton [252, 253, 254,
255]. This calls for modifications of $P_{\rho}(\sigma)$ compared to
$P_{\pi}(\sigma)$. First, a natural mechanism of reduction of the
$\sigma_{\gamma p\to\rho p}$ cross section is offered by the color dipole
model, where due to the point-like coupling of the photon to quarks, the
overlap between the real photon and $\rho$ meson light-cone wave functions
selects on average dipoles with a smaller transverse sizes than those
characteristic for the pion ($\rho$ meson) wave function. In the language of
$P_{\rho}(\sigma)$, it leads to an enhanced contribution of small $\sigma$,
which can be modeled in the following form [256]

$P_{\rho}(\sigma)=N_{\rho}\frac{1}{(\sigma/\sigma_{0})^{2}+1}e^{-(\sigma-\sigma_{0})^{2}/(\Omega\sigma_{0})^{2}}\,.$
(145)

Second, small-size quark–antiquark dipoles are characterized by the large
relative transverse momentum and the large invariant mass. To take this into
account, one should model the variance of the $P_{\rho}(\sigma)$ distribution,
$\omega_{\sigma}^{\rho}$, using information on photon diffractive dissociation
on the proton. This can be done as follows [256]. Using the formalism of cross
section fluctuations, the cross section of photon diffractive dissociation on
the proton can be written in the following form [compare to Eq. (138)]

$\frac{d\sigma_{\gamma p\to
Xp}(t=0)}{dt}=\frac{1}{16\pi}\left(\frac{e}{f_{\rho}}\right)^{2}\int d\sigma
P_{\rho}(\sigma)(\sigma^{2}-\sigma_{\rho
N}^{2})=\frac{1}{16\pi}\left(\frac{e}{f_{\rho}}\right)^{2}\omega_{\sigma}^{\rho}\sigma_{\rho
N}^{2}\,,$ (146)

where $f_{\rho}$ is the $\gamma-\rho$ coupling constant fixed by the
$\Gamma(\rho\to e^{+}e^{-})$ width of the $\rho\to e^{+}e^{-}$ decay,
$f_{\rho}^{2}/(4\pi)=2.01\pm 0.1$. In Eq. (146), $\sigma_{\rho N}$ is the
total $\rho$–nucleon cross section, which is determined by fitting the
available fixed-target and HERA experimental data on the elasic
$d\sigma_{\gamma p\to\rho p}(t=0)/dt$ cross section,

$\sigma_{\rho N}=\int d\sigma
P_{\rho}(\sigma)\sigma=\frac{f_{\rho}}{e}\sqrt{16\pi\frac{d\sigma_{\gamma
p\to\rho p}(t=0)}{dt}}\,.$ (147)

To proceed with the determination of $\omega_{\sigma}^{\rho}$, one invokes the
result of the analysis in Ref. [257], which demonstrated that the cross
sections of photon and pion diffractive dissociation can be related as follows

$\frac{d\sigma_{\gamma p\to Xp}(t=0)/dt}{\sigma_{\gamma
p}}\approx\frac{d\sigma_{\pi p\to Xp}(t=0)/dt}{\sigma_{\pi
p}}=\frac{\omega_{\sigma}^{\pi}}{16\pi}\sigma_{\pi p}\,,$ (148)

where $\sigma_{\gamma p}$ is the total photoabsorption cross section. Combing
Eqs. (146) and (148), one obtains

$\omega_{\sigma}^{\rho}=\left(\frac{f_{\rho}}{e}\right)^{2}\frac{\sigma_{\gamma
p}\sigma_{\pi p}}{\sigma_{\rho N}^{2}}\,\omega_{\sigma}^{\pi}\,.$ (149)

Figure 32 shows the distribution $P_{\rho}(\sigma)$ for $\rho$ mesons as a
function of $\sigma$ at $\sqrt{s}=46$ GeV and $\sqrt{s}=62$ GeV. For
comparison with the pion case, we also give the corresponding
$P_{\pi}(\sigma)$ by thin curves.

Figure 32: The distribution $P_{\rho}(\sigma)$ for $\rho$ mesons as a function
of $\sigma$ for $\sqrt{s}=46$ GeV (solid red) and $\sqrt{s}=62$ GeV (blue
dashed). For comparison, the thin curves show $P_{\pi}(\sigma)$ for pions.

###  6.5 $P_{\gamma^{(\ast)}}(\sigma)$ distribution for real and virtual
photons

It is well known that real and virtual photons also reveal their hadron-like
nature in strong interactions. For example, in the vector dominance (VMD)
model, approximately 70% of the total photoabsorption cross section comes from
the contribution of $\rho$, $\omega$ and $\phi$ mesons [258].

In QCD, it is instructive to discuss the hadronic structure of real and
virtual photons in the language of the color dipole model, see Sec. 3. In
general, the photon at high energies can be viewed as superposition of the
following two types of components. First, the photon can fluctuate into
aligned quark–antiquark pairs with the invariant mass $M$, where the quarks
share asymmetrically the photon longitudinal momentum and have small
transverse momenta $p_{t}$. Such configurations are characterized by large
cross sections of the order of $\sigma_{\rho N}$ and small probabilities of
the order of $\mu^{2}/M^{2}$, where $\mu$ is a soft QCD scale. The latter is
required to comply with the approximate Bjorken scaling of the total virtual
photon–nucleon cross section [259]. Second, in addition to the aligned pairs,
there are also configurations with large $p_{t}$, which are characterized by
small cross sections of the order of $\alpha_{s}(p_{t}^{2})/p_{t}^{2}$ and
large probabilities to find in the photon wave function [87]. The relative
importance of these two types of contributions depends on the photon
virtuality $Q^{2}$, the longitudinal momentum, and the invariant mass of the
produced diffractive state.

As in the cases of protons, pions, and $\rho$ mesons considered above, it is
convenient to quantify the hadronic structure of photons in terms of the
distribution $P_{\gamma^{(\ast)}}(\sigma)$. Starting with the real photon
case, the corresponding distribution $P_{\gamma}(\sigma)$ satisfies the
following constraints,

$\displaystyle\int d\sigma P_{\gamma}(\sigma)\sigma$ $\displaystyle=$
$\displaystyle\sigma_{\gamma p}(W)\,,$ $\displaystyle\int d\sigma
P_{\gamma}(\sigma)\sigma^{2}$ $\displaystyle=$ $\displaystyle
16\pi\frac{d\sigma_{\gamma p\to Xp}(t=0)}{dt}\,,$ (150)

where $\sigma_{\gamma p}$ is the total photon–nucleon cross section;
$d\sigma_{\gamma p\to Xp}(t=0)/dt$ is the cross section of photon diffractive
dissociation on the proton including the $\rho$ meson peak. Note that, the
distribution $P_{\gamma}(\sigma)$ is not normalizable, i.e., the integral
$\int d\sigma P_{\gamma}(\sigma)$ is divergent at the lower integration limit
due to the infinite renormalization of the photon Green’s function (the vacuum
polarization).

A model for $P_{\gamma}(\sigma)$ should interpolate between the regimes of
small and large $\sigma$; the following presentation in based on Ref. [260].
For small $\sigma\ll\sigma_{\pi N}$, we use the color dipole model, which
allows one to readily present $P_{\gamma}(\sigma)$ in the following form, (see
[260] for details)

$P_{\gamma}^{\rm dipole}(\sigma)=\sum_{q}e_{q}^{2}\left|\frac{\pi d{\bf
r}^{2}}{d\sigma_{q\bar{q}}(r,m_{q})}\right|\int
dz|\Psi_{\gamma}(z,r(\sigma_{q\bar{q}}),m_{q})|^{2}_{\
\big{|}\sigma_{q\bar{q}}(r,m_{q})=\sigma}\;,$ (151)

where $\Psi_{\gamma}(z,r(\sigma_{q\bar{q}}),m_{q})$ is the photon wave
function, which depends on the the quark longitudinal momentum fraction $z$,
the dipole transverse size $r$, and the quark mass $m_{q}$. In Eq. (151),
$r(\sigma_{q\bar{q}})$ is related to the dipole cross section using the
following implementation (see Sec. 3) and Eq. (25) of the dipole formalism
[127]

$\sigma_{q\bar{q}}(r,m_{q})=\frac{\pi^{2}}{3}r^{2}\alpha_{s}(Q_{\rm
eff}^{2})x_{\rm eff}g(x_{\rm eff},Q_{\rm eff}^{2})\,,$ (152)

where $Q_{\rm eff}^{2}=\lambda/r^{2}$ for light quarks and $Q_{\rm
eff}^{2}=m_{q}^{2}+\lambda/r^{2}$ for heavy quarks (note that heavy quarks
give a negligible contribution to the discussed quantities); $x_{\rm
eff}=4m_{q}^{2}/W^{2}+0.75\lambda/(W^{2}r^{2})$; $m_{q}=300$ MeV for light
$u$, $d$ and $s$ quarks and $m_{c}=1.5$ GeV. This choice of the quark masses
ensures that the average transverse size of $q\bar{q}$ configurations in the
photon wave function is close to that of the pion, $d_{\pi}=0.65$ fm, and also
leads to a smoother interpolation between small and large $\sigma$ regimes.
The parameter $\lambda=4$ is chosen to best reproduce the HERA data on
diffractive $J/\psi$ photoproduction [191].

For large $\sigma\geq\sigma_{\rho N}$, the distribution $P_{\gamma}(\sigma)$
can be approximated by $P_{\rho}(\sigma)$ for $\rho$ mesons. Taking the sum of
the $\rho$, $\omega$ and $\phi$ meson contributions, the resulting
distribution reads:

$P_{(\rho+\omega+\phi)/\gamma}(\sigma)=\frac{11}{9}\left(\frac{e}{f_{\rho}}\right)^{2}P_{\rho}(\sigma)\,,$
(153)

where the coefficient of $11/9$ takes into account the $\omega$ and $\phi$
contributions in the flavor SU(3) approximation.

Interpolating between the regimes of small and large $\sigma$, one arrives at
the following hybrid model for $P_{\gamma}(\sigma)$

$P_{\gamma}(\sigma,W)=\left\\{\begin{array}[]{ll}P_{\gamma}^{\rm
dipole}(\sigma,W)\,,&\sigma\leq 10\ {\rm mb}\,,\\\ P_{\rm
int}(\sigma,W)\,,&10\ {\rm mb}\leq\sigma\leq 20\ {\rm mb}\,,\\\
P_{(\rho+\omega+\phi)/\gamma}(\sigma,W)\,,&\quad\sigma\geq 20\ {\rm
mb}\,.\end{array}\right.$ (154)

where $P_{\rm int}(\sigma,W)$ is a smooth interpolating function.

The resulting model for $P_{\gamma}(\sigma)$ satisfies the constraints of Eq.
(150) and gives a good description of the total and diffraction dissociation
photon–proton cross sections. Indeed, at the typical $W=100$ GeV, it gives
$\sigma_{\gamma p}=\int d\sigma\sigma P_{\gamma}(\sigma)=135$ $\mu$b in
agreement with the PDG value of $\sigma_{\gamma p}=146$ $\mu$b [261] and
$d\sigma_{\gamma p\to Xp}(t=0)/dt=\int
d\sigma\sigma^{2}P_{\gamma}(\sigma)/(16\pi)=240$ $\mu$b/GeV2 in agreement with
the estimate of $d\sigma_{\gamma p\to Xp}(t=0)/dt\approx 220$ $\mu$b/GeV2,
which is obtained by integrating the data of [257] over the produced
diffractive masses and extrapolating the resulting cross section to the
desired $W=100$ GeV.

In the virtual photon case, the distribution $P_{\gamma^{\ast}}(\sigma)$ can
be constructed similarly to the model outlined above. Namely, for the
small-$\sigma$ part one uses Eq. (151), where the real photon wave function is
substituted by the one of the virtual photon. In addition, the large-$\sigma$
contribution due to vector mesons receives a $Q^{2}$-dependent suppression
factor dictated by the vector meson dominance model,

$P_{(\rho+\omega+\phi)/\gamma^{\ast}}(\sigma)=\frac{11}{9}\left(\frac{e}{f_{\rho}}\right)^{2}\frac{m_{\rho}^{2}}{Q^{2}+m_{\rho}^{2}}P_{\rho}(\sigma)\,.$
(155)

In the intermediate-$\sigma$ region, we use smooth interpolation as in Eq.
(154).

The resulting distributions $P_{\gamma}(\sigma)$ for the real photon and
$P_{\gamma^{\ast}}(\sigma)$ for the virtual photon at $Q^{2}=4$ GeV2 as a
function of $\sigma$ at $W=100$ GeV are shown in Fig. 33. Note that for small
$\sigma$, the sensitivity to the choice of the quark mass $m_{q}$ is very
weak.

Figure 33: The distributions $P_{\gamma}(\sigma)$ and
$P_{\gamma^{\ast}}(\sigma)$ for the real and virtual photons at $W=100$ GeV.

Note that the distribution $P_{\gamma}(\sigma)$ can be probed in measurements
of the total photon–nucleus cross sections at high energies, which can done
using, e.g., ultraperipheral heavy ion collisions in next run (Run 3) of the
LHC operations. Indeed, in this formalism, the photon–nucleus cross section
can be written as a series in terms of moments of the distribution
$\langle\sigma^{k}\rangle=\int d\sigma P_{\gamma}(\sigma)\sigma^{k}$, where
each term corresponds to the simultaneous interaction with $1\leq k\leq A$
nucleons of the nuclear target; the terms with $k\geq 2$ contribute to the
shadowing correction modifying the impulse approximation given by the $k=1$
term, see Sec. 7. Moreover, the effective number of $\langle\sigma^{k}\rangle$
terms, which one needs to take into account, increases with the nucleus mass
number $A$. Thus, the photon-nucleus cross section allows one to probe higher
moments of the distribution $P_{\gamma}(\sigma)$ giving additional constraints
on it compared to the free proton case.

It is expected that the effect of nuclear shadowing in this cross section is
stronger than in the structure function $F_{2A}(x,Q^{2})$ measured in
lepton–nucleus DIS and has a significant $W$ dependence extending into a TeV
range.

Another application of the distribution $P_{\gamma}(\sigma)$ is for the
wounded nucleon model in the fixed-target case, see Sec. 10.

##  7 Inclusive coherent and incoherent diffraction in $eA$ DIS

###  7.1 Connection of nuclear shadowing to diffraction

It is well known that at high energies, cross sections of hadron–nucleus
scattering are smaller than the sum of individual hadron–nucleon cross
sections. This phenomenon originates from destructive interference among the
amplitudes for the interaction with one and two, tree, etc. nucleons of the
nuclear target. In the literature it is called nuclear shadowing (NS) because
it can be interpreted as a decrease of the effective number of nucleons of the
nuclear target due to their overlap or mutual geometric shadowing in the
transverse plane [128, 130, 258, 4]; the resulting theoretical approach is
called the Gribov–Glauber model of nuclear shadowing.

Note here that Glauber considered a quantum–mechanical potential model, which
is different from the high-energy scattering situation, where the amplitude is
predominantly imaginary. Moreover for this amplitude the diagrams considered
by Glauber are cancelled out in the high energy limit. The reason for this
disappearance is that the projectile does not have time to merge back to its
initial state between the first and second interactions. Gribov found the
essential diagrams for hadron–nucleus scattering and, in the case of the
interaction with two nucleons, expressed the correction to the total nuclear
cross section through the diffractive cross section on the nucleon, see Eq.
(156). At the same time, for multiple scatterings, one needs modelling, which
can be done by introducing the notion of cross section fluctuations, see
discussion in Sec. 6.

A classic example of nuclear shadowing is the total pion–deuteron cross
section $\sigma^{\pi D}_{\rm tot}(s)$. The corresponding forward pion–deuteron
scattering amplitude is shown in Fig. 34, where graphs $a$ and $b$ correspond
to the interaction with one and two nucleons of the target, respectively. An
evaluation of these graphs gives [4]

$\sigma^{\pi D}_{\rm tot}(s)\,=\,2\sigma^{\pi N}_{\rm tot}(s)-2\int
dt\,\rho_{D}(4t)\,\frac{d\sigma^{\pi N}_{\rm diff}(s,t)}{dt}\,,$ (156)

where the first term coming from graph $a$ is the impulse approximation given
by twice the total pion–nucleon cross section $\sigma^{\pi N}_{\rm tot}(s)$
(we do not distinguish between protons and neutrons here) and the second term
coming from graph $b$ is the negative NS correction, which is expressed in
terms of the pion–nucleon diffractive cross section $d\sigma^{\pi N}_{\rm
diff}(s,t)/dt$. Note that $d\sigma^{\pi N}_{\rm diff}(s,t)/dt$ includes both
the elastic and inelastic diffractive contributions; the former corresponds to
the elastic intermediate state, which is implicitly included in graph $b$. The
distribution of nucleons is given by the deuteron form factor $\rho_{D}(4t)$
evaluated at double the momentum transfer, which takes into account the effect
of nuclear recoil; the integration runs over the invariant momentum transfer
squared $t$. Finally, all involved cross sections depend on the invariant
collision energy squared $s$.

Figure 34: The forward pion–deuteron scattering amplitude: graphs $a$ and $b$
correspond to the interaction with one and two nucleons of the nucleus and
lead to the impulse approximation and the nuclear shadowing correction to
$\sigma^{\pi D}_{\rm tot}(s)$, respectively.

The diagrams corresponding to the sequential interaction of a projectile with
target nucleons via elastic intermediate states, which lead to the Glauber
shadowing correction, vanish in the high energy limit. At the same time, since
the characteristic longitudinal distance over which the projectile–nucleus
interaction develops (coherence length) increases with an increase of energy
[88, 262], the diagrams corresponding to the simultaneous interaction with all
target nucleons contribute to this process; their contribution can be
formulated in terms of diffractive scattering on a single nucleon.

Note that the original derivation of Eq. (156) used the Pomeron trajectory
with the intercept equal to unity, which resulted in the vanishing real part
of the scattering amplitude. The effect of the real part can be readily
included using the Gribov–Migdal relation [263], see Eq. (157) below.

This connection between NS and diffraction reflects the dominance of large
longitudinal distances at high energies. It also holds in the case of the
interaction of real and virtual photons with deuterons and can be further
generalized to heavier nuclei. In particular, the
$\sigma^{\gamma^{(\ast)}D}(W)$ total photon–deuteron cross section at high
energies can be written in the following form

$\sigma^{\gamma^{(\ast)}D}(W)=2\sigma^{\gamma^{(\ast)}N}(W)-2\frac{1-\eta^{2}}{1+\eta^{2}}\int_{x_{{I\\!\\!P},{\rm
min}}}^{0.1}dx_{{I\\!\\!P}}d|\vec{k}_{t}|^{2}\rho_{D}\left(4t\right)\frac{d\sigma^{\gamma^{(\ast)}N}_{\rm
diff}(W,t,x_{{I\\!\\!P}})}{dx_{{I\\!\\!P}}d|\vec{k}_{t}|^{2}}\,,$ (157)

where $\sigma^{\gamma^{(\ast)}N}(W)$ is the total photon–nucleon cross
section; $d\sigma_{\gamma^{(\ast)}N}^{\rm
diff}(W,t,x_{{I\\!\\!P}})/(dx_{{I\\!\\!P}}d|\vec{k}_{t}|^{2})$ is the cross
section for the $\gamma^{(\ast)}+N\to X+N$ inclusive diffractive scattering on
the nucleon; $x_{{I\\!\\!P}}=\xi$ is the longitudinal momentum fraction of the
diffractive exchange, see Eq. (2); $\vec{k}_{t}$ is the transverse momentum
transfer to the target nucleons so that
$-t=(x_{{I\\!\\!P}}m_{N})^{2}+|\vec{k}_{t}|^{2}$.

The lower $x_{{I\\!\\!P}}$ integration limit is defined by the lowest possible
diffractively produced mass corresponding to the $\rho$ vector meson,

$x_{{I\\!\\!P},{\rm min}}=\frac{M_{\rho}^{2}+Q^{2}}{W^{2}+Q^{2}}\,.$ (158)

In the case of DIS with $Q^{2}={\cal O}(\rm few\ GeV^{2}$),
$x_{{I\\!\\!P},{\rm min}}=Q^{2}/(2p\cdot q)=x$. The upper integration limit is
defined by the usual condition on diffraction $M_{X}^{2}/W^{2}\leq 0.1$. In
fact the cutoff is stronger due to the presence of the nuclear form factor.
Note that, as we discussed in Sec. 2, the diffractive exchange with the vacuum
quantum numbers dies out for $x_{{I\\!\\!P}}>0.03$ and non-vacuum sub-leading
contributions give a dominant contribution.

In Eq. (157), $\eta$ is the ratio of the real and imaginary parts of the
$\gamma^{(\ast)}+N\to X+N$ amplitude, which can be estimated using the
asymptotic energy behavior of the corresponding cross section [263]

$\eta=\frac{\Re e{\cal A}_{\gamma^{(\ast)}+N\to X+N}}{\Im m{\cal
A}_{\gamma^{(\ast)}+N\to X+N}}\approx\frac{\pi}{2}\frac{\partial\ln\Im m{\cal
A}_{\gamma^{(\ast)}+N\to X+N}}{\partial\ln
s}=\frac{\pi}{2}\left(\alpha_{{I\\!\\!P}}(0)-1\right)\,,$ (159)

where $\alpha_{{I\\!\\!P}}(0)$ determines the energy dependence of the
imaginary part of the amplitude. It corresponds to the intercept of the
Pomeron exchange, which gives the dominant contribution at high $W$ and
$M_{X}^{2}/W^{2}\leq 0.03$.

Equation (157) presents a general connection between the nuclear shadowing
effect originating from the interaction with two nucleons of a nuclear target
and diffraction on the nucleon. It is based on such general properties of
scattering amplitudes as unitarity, which can also be formulated in form of
the Abramovsky–Gribov–Kancheli (AGK) cutting rules [264], and analyticity. It
is important to emphasize that it does not require the decomposition over
twists, i.e., it is valid for both leading-twist and higher-twist
contributions and in the entire range of the photon virtualities including the
$Q^{2}=0$ photoproduction limit.

Equation (157) can be readily cast into the expression for the deuteron
structure function (SF) $F_{2D}(x,Q^{2})$ probed in inclusive lepton–deuteron
deep inelastic scattering (DIS) at small $x$,

$F_{2D}(x,Q^{2})=2F_{2N}(x,Q^{2})-2\frac{1-\eta^{2}}{1+\eta^{2}}\int_{x}^{0.1}dx_{{I\\!\\!P}}\,d|\vec{k}_{t}|^{2}\rho_{D}\left(4t\right)F_{2N}^{D(4)}(\beta,Q^{2},x_{{I\\!\\!P}},t)\;,$
(160)

where $F_{2N}(x,Q^{2})$ and $F_{2N}^{D(4)}$ are the usual and diffractive
nucleon structure functions (SFs) in inclusive $l+N\to l^{\prime}+X$ and
diffractive $l+N\to l^{\prime}+X+N^{\prime}$ lepton–nucleon DIS, respectively
(see Sec. 2). Various approaches using Eq. (160), which describe the available
data on the deuteron SFs in a wide range of photon virtualities $Q^{2}$
including the $Q^{2}=0$ photoproduction limit, differ mainly in modeling and
parametrization of the nucleon diffractive SF. In particular, the
phenomenological parameterizations included the scaling, quark–antiquark
parton model contribution to the Pomeron part of the diffractive structure
function and the contribution of $\rho$, $\omega$, and $\phi$ vector mesons to
low-mass diffraction [265, 266]; the Pomeron, vector meson, and meson exchange
contributions [267, 268]; the vector meson and quark–antiquark continuum
contributions to the nuclear structure function [269]; the aligned-jet model
[87]; the leading-twist contribution based on next-to-leading order QCD
analysis of hard diffraction in $ep$ DIS at HERA [270]. In all the cases, the
effect of NS is a few percent correction to the total lepton–deuteron cross
section.

The effect of NS is much larger for heavy nuclei and increases with an
increase of the atomic mass number $A$. However, a generalization of Eqs.
(157) and (160) to heavier nuclear targets requires modeling of interactions
with three and more nucleons of the target. This can be done using the eikonal
or quasi-eikonal approximations [131], which assume that the intermediate
diffractive state elastically scatters on $N\geq 3$ nucleons of the nuclear
target with the effective cross section $\sigma_{XN}$

$\sigma_{XN}=\frac{1}{16\pi}\frac{1}{(1+\eta^{2})\sigma^{\gamma^{(\ast)}N}(W)}\int_{x}^{0.1}dx_{{I\\!\\!P}}\frac{d\sigma^{\gamma^{(\ast)}N}_{\rm
diff}(W,t,x_{{I\\!\\!P}})}{dx_{{I\\!\\!P}}d|\vec{k}_{t}|^{2}}\bigg{|}_{t=0}\;.$
(161)

Figure 35 shows the forward photon–nucleus scattering amplitude: graphs $a$,
$b$, and $c$ present first three terms of the multiple scattering series
corresponding to the photon interaction with one, two, and three nucleons of
the nucleus target; the graphs for the interaction with $A\geq N>3$ nucleons
are not explicitly shown, but included in the final answer. In graphs $b$ and
$c$, $X$ denotes the intermediate diffractively produced state $X$; in graph
$c$, the middle darker vertex denotes the interaction with the cross section
$\sigma_{XN}$.

Figure 35: The multiple scattering series for the forward (virtual) photon-
nucleus scattering amplitude: graphs $a$, $b$, $c$ correspond to the
interaction with one, two, and three nucleons of the target; the interaction
with $A\geq N>3$ nucleons is not shown, but implied.

An evaluation and summation of these graphs gives the following expression for
the (virtual) photon–nucleus cross section [87, 269, 271, 272, 273],

$\displaystyle\sigma^{\gamma^{(\ast)}A}(W)=A\sigma^{\gamma^{(\ast)}N}(s)-8\pi
A(A-1)\Re
e\frac{(1-i\eta)^{2}}{1+\eta^{2}}\int_{x}^{0.1}dx_{{I\\!\\!P}}\frac{d\sigma^{\gamma^{(\ast)}N}_{\rm
diff}(W,t,x_{{I\\!\\!P}})}{dx_{{I\\!\\!P}}d|\vec{k}_{t}|^{2}}\bigg{|}_{t=0}$
$\displaystyle\times\int
d^{2}\vec{b}\int^{\infty}_{-\infty}dz_{1}\int^{\infty}_{z_{1}}dz_{2}\rho_{A}(\vec{b},z_{1})\rho_{A}(\vec{b},z_{2})e^{i(z_{1}-z_{2})x_{{I\\!\\!P}}m_{N}}e^{-\frac{A}{2}(1-i\eta)\sigma_{XN}\int_{z_{1}}^{z_{2}}dz^{\prime}\rho_{A}(\vec{b},z^{\prime})}\,,$
(162)

where $\vec{b}$ and $z$ refer to the transverse and longitudinal coordinates
of the involved nucleons, respectively. The derivation of Eq. (162) assumes
independent nucleons in the nuclear ground state, where $\rho_{A}(\vec{b},z)$
is the nuclear density calculated using the Woods–Saxon (two-parameter Fermi
model) parametrization [274], and neglects the slope of the
$\gamma^{(\ast)}+N\to X+N^{\prime}$ diffractive cross section compared to that
of the nuclear form factor. The latter means that the diffractive cross
section on the nucleon is evaluated at $t=0$ and all involved nucleons are
positioned at the same transverse distance (impact parameter) $\vec{b}$.

The main difference between various approaches based on the Gribov–Glauber
model lies in different parametrizations of the diffractive cross section on
the nucleon (see the discussion above) and different implementations of the
interaction with $N\geq 3$ nucleons of the nuclear target. For instance, these
interactions can be written in the form of summed fan diagrams [275, 276, 277,
278], which leads to a different form of the expression for
$\sigma^{\gamma^{(\ast)}A}(W)$.

The approaches based on the connection of nuclear shadowing to diffraction
provide a good description and interpretation of data on inclusive cross
section in lepton–nucleus and photon–nucleus scattering at high energies. The
useful feature of Eq. (162) is that it is applicable in a wide range of photon
virtualities including the $Q^{2}=0$ limit, which allows one to study the
interplay of different mechanisms of NS and also higher-twist effects in NS.

In the following we will consider hard scattering off nuclei and an approach
allowing one to generalize Eq. (162) to nuclear parton distribution functions
(nPDFs). Since in this kinematics the involved structure functions can be
expressed in terms of leading twist parton distributions, this approach is
called the leading twist nuclear shadowing model [279, 280, 281].

###  7.2 The model of leading twist nuclear shadowing

The model of leading twist nuclear shadowing is based on the combination of
the following key elements:

  1. 1.

The Gribov–Glauber model of nuclear shadowing allowing one to evaluate the
effect of NS in hadron–nucleus scattering;

  2. 2.

The QCD collinear factorization theorems for inclusive [200] and diffractive
DIS [15] and QCD analyses of inclusive diffraction in lepton–proton DIS at
HERA [26, 44, 27], see Sec. 2, allowing one to make predictions for parton
distribution functions (PDFs) of different flavors $i$ in nuclei;

  3. 3.

A model for the hadronic structure of a hard probe to calculate the
interaction with $N\geq 3$ nucleons of the target, see Sec. 6.

Note that this approach employs the equivalence of pictures of hard scattering
in the nuclear rest frame and in the frame, where the nucleus is fast.

The multiple scattering series for nuclear PDFs $f_{i/A}(x,Q^{2})$ is
presented in Fig. 36 (compare to Fig. 35), where graphs $a$, $b$, $c$
correspond to the interaction with one, two, and three nucleons of the nuclear
target. The graphs for the interaction with $N>3$ nucleons are implied and
included in the resulting expression for $xf_{i/A}(x,Q_{0}^{2})$, see Eq.
(163). The dashed vertical lines denote the imaginary part of the
corresponding graphs and correspond to the total parton–nucleon cross section
in graph $a$ and to the so-called diffractive cut in graphs $b$ and $c$, where
the outer vertices correspond to the parton–nucleon diffractive scattering.
The middle darker vertex in graph $c$ stands for the interaction with $N>3$
nucleons, which is modeled using the $\sigma_{\rm soft}^{i}(x,Q_{0}^{2})$
effective cross section, see Eq. (166).

Figure 36: The multiple scattering series for nPDFs $f_{i/A}(x,\mu^{2})$:
graphs $a$, $b$, $c$ correspond to the interaction with one, two, and three
nucleons of the target; the interaction with $A\geq N>3$ nucleons is not
shown, but implied. The dashed vertical lines stand for the imaginary part;
$X$ denotes the diffractively produced intermediate state.

Applying the steps listed above to Eq. (162) and evaluating the graphs in Fig.
36, nPDFs at small $x$ can be presented in the following form [281] [compare
to Eq. (162)]

$\displaystyle xf_{i/A}(x,Q_{0}^{2})=Axf_{i/N}(x,Q_{0}^{2})-8\pi A(A-1)\Re
e\frac{(1-i\eta)^{2}}{1+\eta^{2}}\int_{x}^{0.1}dx_{{I\\!\\!P}}\beta
f_{i}^{D(4)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}},t=0)$ $\displaystyle\times\int
d^{2}\vec{b}\int^{\infty}_{-\infty}dz_{1}\int^{\infty}_{z_{1}}dz_{2}\rho_{A}(\vec{b},z_{1})\rho_{A}(\vec{b},z_{2})e^{i(z_{1}-z_{2})x_{{I\\!\\!P}}m_{N}}\frac{\langle\sigma^{2}e^{-\frac{A}{2}(1-i\eta)\sigma\int_{z_{1}}^{z_{2}}dz^{\prime}\rho_{A}(\vec{b},z^{\prime})}\rangle_{i}}{\langle\sigma^{2}\rangle_{i}}\;,$
(163)

where $i$ is the parton flavor; $f_{i}^{D(4)}$ are diffractive PDFs of the
proton with $\beta=x/x_{{I\\!\\!P}}$; $Q_{0}$ is the input, factorization
scale chosen such that higher-twist effects in the proton diffractive PDFs are
small. Note that the exponential factor
$e^{i(z_{1}-z_{2})x_{{I\\!\\!P}}m_{N}}$ automatically suppresses the shadowing
term at large $x$.

In our analysis, $Q_{0}=2$ GeV. At smallest $x$ there is an indication that
the higher twists effects in inclusive diffraction in $ep$ DIS at HERA may not
be negligible up to $Q_{0}=3\,\rm GeV$, see Sect. 2 and 3.5. This effect is
masked to some extent by the parametrization of proton diffractive PDFs
$f_{i}^{D(4)}$ extracted from the fits to the diffractive data.

The $\langle\dots\rangle_{i}$ term in Eq. (163) sums the contributions
corresponding to the interaction with $N\geq 3$ nucleons of the nuclear target
and is expressed in terms of moments of the distribution $P_{i}(\sigma)$,
which takes into account the composite hadronic structure of a hard probe, see
Sec. 6,

$\langle\sigma^{k}\rangle_{i}\equiv\int_{0}^{\infty}d\sigma
P_{i}(\sigma)\sigma^{k}\,,$ (164)

where we explicitly indicated the dependence on parton flavor $i$. It is
important to note that the $\langle\dots\rangle_{i}$ term and the entire
expression in Eq. (163) does not depend on the type of hard probe and is valid
for virtual photons, gauge bosons, neutrinos, etc. Hence, the resulting nPDFs
obey the QCD collinear factorization.

While the interaction with $N>3$ target nucleons needs to be modeled in Eq.
(163), the interaction with two target nucleons is unambiguously expressed in
terms of usual and diffractive PDFs of the protons [compare to Eq. (161)]

$\sigma_{2,i}(x)=\frac{\langle\sigma^{2}\rangle_{i}}{\langle\sigma\rangle_{i}}=\frac{1}{16\pi}\frac{\int_{x}^{0.1}dx_{{I\\!\\!P}}f_{i}^{D(4)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}},t=0)}{(1+\eta^{2})f_{i}(x,Q_{0}^{2})}\,,$
(165)

where $f_{i}(x,Q_{0}^{2})$ are usual PDFs of the proton.

In the following, for definiteness we assume that the nuclear PDFs are probed
by virtual photons and, hence, $P_{i}(\sigma)$ describes the hadronic
structure of the virtual photon. While in general hadronic fluctuations of
small and large transverse sizes (small and large $\sigma$) are present in
virtual photons, the structure of Eq. (163) indicates that the
$\langle\dots\rangle_{i}$ term is dominated by large-$\sigma$ fluctuations.
Hence, it can be evaluated in terms of a single cross section

$\sigma_{\rm
soft}^{i}(x,Q_{0}^{2})=\frac{\langle\sigma^{3}\rangle_{i}}{\langle\sigma^{2}\rangle_{i}}\,,$
(166)

which can be modeled using soft, non-perturbative part of the $P_{i}(\sigma)$
distribution. In the present approach, this is done by assuming that
$P_{i}(\sigma)$ can be either calculated using a particular version of the
color dipole model [127], i.e., $P_{i}(\sigma)=P_{\gamma^{\ast}}^{\rm
dipole}(\sigma)$, or can be identified with that of the pion [248], i.e.
$P_{i}(\sigma)=P_{\pi}(\sigma)$. For details, see Sec. 6. Further, since the
dispersion of $P_{i}(\sigma)$ for the calculation of the
$\langle\dots\rangle_{i}$ term in Eq. (163) can be safely neglected, i.e.,

$\frac{\langle\sigma^{k}\rangle_{i}}{\langle\sigma^{2}\rangle_{i}}=[\sigma_{\rm
soft}^{i}(x,Q_{0}^{2})]^{k-2}\,,$ (167)

one obtains the following final expression for the nPDF
$xf_{i/A}(x,Q_{0}^{2})$

$\displaystyle xf_{i/A}(x,Q_{0}^{2})=Axf_{i/N}(x,Q_{0}^{2})-8\pi A(A-1)\Re
e\frac{(1-i\eta)^{2}}{1+\eta^{2}}\int_{x}^{0.1}dx_{{I\\!\\!P}}\beta
f_{i}^{D(4)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}},t=0)$ $\displaystyle\times\int
d^{2}\vec{b}\int^{\infty}_{-\infty}dz_{1}\int^{\infty}_{z_{1}}dz_{2}\rho_{A}(\vec{b},z_{1})\rho_{A}(\vec{b},z_{2})e^{i(z_{1}-z_{2})x_{{I\\!\\!P}}m_{N}}e^{-\frac{A}{2}(1-i\eta)\sigma_{\rm
soft}^{i}(x,Q_{0}^{2})\int_{z_{1}}^{z_{2}}dz^{\prime}\rho_{A}(\vec{b},z^{\prime})}\,.$
(168)

The use of two plausible models for $\sigma_{\rm soft}^{i}(x,Q_{0}^{2})$
results in a spread of predictions for $xf_{i/A}(x,Q_{0}^{2})$, which
quantifies the theoretical uncertainty of the present approach. This is
illustrated in Fig. 37 showing predictions for the
$f_{i/A}(x,Q_{0}^{2})/[Af_{i/N}(x,Q_{0}^{2})]$ ratios for $\bar{u}$ quarks and
gluons as a function of $x$. The upper row of panels corresponds to the
nucleus of Ca-40, while the lower panels are for Pb-208. In the calculations,
we used the 2006 H1 (Fit B) diffractive PDFs [26]. Note that this set of
diffractive PDFs is considered to be standard in the literature since it not
only provides a good description of the cross sections of inclusive
diffraction in DIS measured at HERA, but also gives an adequate description of
diffractive dijet production, for reviews, see [187, 282]. Other analyses of
diffractive PDFs, notably, the 2009 ZEUS Fit SJ [283] and that based on the
xFitter framework [32], give comparable results for the quark and gluon
diffractive PDFs on the proton. Note that the 2006 H1 Fit A is characterized
by a very large gluon density at large momentum fractions leading to a poor
agreement with the HERA diffractive jet data. Thus, the uncertainties
associated with diffractive PDFs are rather modest and are smaller than the
theoretical uncertainties of the leading twist nuclear shadowing model
associated with modeling of the interaction with $N\geq 3$ nucleons, see Fig.
37.

Note that Eq. (168) is applied to calculate the effect of nuclear shadowing
for sea quark and gluon nuclear PDFs for $x\leq 0.1$. For larger $x$, it is
assumed that the sea quark distributions in nuclei are not modified at the
input scale. For the gluon distribution, one additionally adds the effect of
nuclear antishadowing in the interval $0.03\leq x\leq 0.2$ by requiring that
in the momentum sum rule for nPDFs, it compensates the suppression of nPDFs
due to nuclear shadowing, see Ref. [281]. A similar dynamical model of the
gluon antishadowing was proposed in [284], where the mechanism of
antishadowing was related to merging of two parton ladders attached to two
different nucleons of the nuclear target. Alternatively, the enhancement of
quark nuclear PDFs around $x=0.1$ was proposed to arise because of the real
part the quark–nucleon scattering amplitude driven by the Reggeon exchange
[285]. However, this model contradicts the experimental observation that there
is no enhancement of sea quarks in nuclei in this interval of $x$.

Figure 37: The leading twist nuclear shadowing model predictions for the
$f_{i/A}(x,Q_{0}^{2})/[Af_{i/N}(x,Q_{0}^{2})]$ ratios of the nuclear and
nucleon PDFs for $\bar{u}$ quarks and gluons as a function of $x$ at the input
scale $Q_{0}=2$ GeV. The upper and lower panels correspond to Ca-40 and
Pb-208, respectively.

Figure 37 presents the important feature of the leading twist nuclear
shadowing model that the magnitude of NS is significant and nuclear shadowing
for nuclear gluons is larger than that for quarks. This is a direct
consequence of the used connection between NS and diffraction on the proton
and QCD factorization for inclusive diffraction in lepton–proton DIS. In
particular, since the gluon diffractive PDF is found to be large [26, 44, 27],
this naturally leads to a large suppression of
$g_{A}(x,Q_{0}^{2})/[Ag_{N}(x,Q_{0}^{2})]$ for small $x$.

In addition, it is predicted that the effect of NS increases with a decrease
of $x$, see Fig. 37. This is different from predictions of global fits of
nuclear PDFs [286, 287, 288, 289], which assume that the
$f_{i/A}(x,Q_{0}^{2})/[Af_{j/N}(x,Q_{0}^{2})]$ ratios become $x$-independent
at small $x$.

One can see from Fig. 37 that the sensitivity to modeling of the interaction
with $N\geq 3$ nucleons of the nuclear target by the effective cross section
$\sigma_{\rm soft}^{i}(x,Q_{0}^{2})$ becomes reduced with an increase of $x$.
In particular, at $x=10^{-3}$, the uncertainties in the predictions for
$xf_{i/A}(x,Q_{0}^{2})$ are at the level of 10% for sea quarks, 15% for
gluons, and smaller for larger $x$.

Note that unlike the quark channel, where the magnitude of nuclear shadowing
for quark distributions can be inferred from the behavior of the ratio
$F_{2A}(x,Q^{2})/[AF_{2N}(x,Q^{2})]$ of the structure functions, the nuclear
gluon distribution at small $x$ can be predicted only using the collinear
factorization for inclusive and diffractive DIS.

A good indication of the magnitude of the gluon nuclear shadowing is given by
the ratio of the longitudinal nuclear and nucleon structure functions,
$F_{LA}(x,Q^{2})/[AF_{LN}(x,Q^{2})]$. Thus, future measurements of this
quantity at EIC will rather directly constrain $xg_{A}(x,Q^{2})$ at small $x$
and test predictions of the model of the leading twist nuclear shadowing.

The leading twist model of nuclear shadowing naturally makes predictions for
the impact parameter $\vec{b}$ dependence of nuclear PDFs,

$\displaystyle xf_{i/A}(x,b,Q_{0}^{2})\;=$ $\displaystyle
Axf_{i/N}(x,Q_{0}^{2})T_{A}(\vec{b})-8\pi A(A-1)\Re
e\frac{(1-i\eta)^{2}}{1+\eta^{2}}\int_{x}^{0.1}dx_{{I\\!\\!P}}\beta
f_{i}^{D(4)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}},t=0)$
$\displaystyle\times\int^{\infty}_{-\infty}dz_{1}\int^{\infty}_{z_{1}}dz_{2}\rho_{A}(\vec{b},z_{1})\rho_{A}(\vec{b},z_{2})e^{i(z_{1}-z_{2})x_{{I\\!\\!P}}m_{N}}e^{-\frac{A}{2}(1-i\eta)\sigma_{\rm
soft}^{i}(x,Q_{0}^{2})\int_{z_{1}}^{z_{2}}dz^{\prime}\rho_{A}(\vec{b},z^{\prime})}\,,$
(169)

where $T_{A}(\vec{b})=\int dz\rho_{A}(\vec{b},z)$ is the so-called nuclear
optical density. Equation (169) encodes the dynamical picture of NS resulting
from an overlap of target nucleons in the transverse plane and naturally leads
to an increase of NS with a decrease of $|\vec{b}|$, where the nuclear density
is larger. This leads to correlations of $x$ and $\vec{b}$ in the impact-
parameter dependent nPDFs $f_{i/A}(x,b,Q_{0}^{2})$.

The upper panels of Fig. 38 show the
$f_{i/A}(x,b,Q_{0}^{2})/[AT_{A}(b)f_{i/N}(x,Q_{0}^{2})]$ ratios for $\bar{u}$
quarks (left) and gluons (right) in Pb-208 as a function of $x$ at $b=0$ and
$Q_{0}^{2}=4$ GeV2. The lower panels compare the predictions for
$f_{i/A}(x,b,Q_{0}^{2})$ to those of the $\vec{b}$-integrated
$f_{i/A}(x,Q_{0}^{2})$ and give the ratios

$\frac{f_{i/A}(x,b,Q_{0}^{2})}{AT_{A}(b)f_{i/N}(x,Q_{0}^{2})}\Big{/}\frac{f_{i/A}(x,Q_{0}^{2})}{Af_{i/N}(x,Q_{0}^{2})}=\frac{f_{i/A}(x,b,Q_{0}^{2})}{T_{A}(b)f_{i/A}(x,Q_{0}^{2})}$
(170)

for $\bar{u}$ quarks (left) and gluons (right) as a function of $x$. The
suppression of these ratios indicates that the effect of NS is larger at $b=0$
than in the $\vec{b}$-integrated case.

Figure 38: The leading twist nuclear shadowing model predictions for the
$\vec{b}$-dependence of nuclear PDFs. Upper panels: The
$f_{i/A}(x,b,Q_{0}^{2})/[AT_{A}(b)f_{j/N}(x,Q_{0}^{2})]$ ratios for $\bar{u}$
quarks (left) and gluons (right) in Pb-208 as a function of $x$ at $b=0$ and
$Q_{0}^{2}=4$ GeV2. Lower panels: The
$f_{i/A}(x,b,Q_{0}^{2})/[{T_{A}(b)f_{i/A}(x,Q_{0}^{2})}]$ ratios, see Eq.
(170), for $\bar{u}$ quarks (left) and gluons (right) in Pb-208 as a function
of $x$.

The global fits of nuclear PDFs can also in principle model the dependence of
nPDFs on the impact parameter $b$, see, e.g. [290]. However, it depends on the
assumed parametrization of the $b$ dependence and requires data with several
nuclei to fix the additional free parameters.

###  7.3 Nuclear diffractive structure functions and PDFs

Using the framework of the leading twist nuclear shadowing model, one can
readily and without additional assumptions make predictions for nuclear
diffractive structure functions and nuclear diffractive PDFs. These quantities
can be probed, for instance, in $e+A\to e^{\prime}+X+A$ inclusive coherent
diffraction in DIS with nuclei. As in the case of usual nPDFs, in the leading
twist nuclear shadowing model nuclear diffractive PDFs $f_{i/A}^{D}$ at small
$x$ are obtained by summing the multiple scattering series for hard probe
interaction with one, two, $\cdots$, $A$ nucleons of the nuclear target; first
three terms of this series are shown in Fig. 39.

Figure 39: The multiple scattering series for nuclear diffractive PDFs
$f_{i/A}^{D}(\beta,\mu^{2},x_{{I\\!\\!P}})$ at the amplitude level: graphs
$a$, $b$, $c$ correspond to the interaction with one, two, and three nucleons
of the target; the interaction with $A\geq N>3$ nucleons is not shown, but
implied. The diffractively produced final state and the intermediate states
are denoted by $X$.

Summing these graphs, one obtains the following expression for nuclear
diffractive PDFs $f_{i/A}^{D(3)}(\beta,Q^{2},x_{{I\\!\\!P}})$

$\displaystyle\beta f_{i/A}^{D(3)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}})$
$\displaystyle=$ $\displaystyle 4\pi A^{2}\beta
f_{i/N}^{D(4)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}},t_{\rm min})$ (171)
$\displaystyle\times$ $\displaystyle\int
d^{2}\vec{b}\left|\int^{\infty}_{-\infty}dz\rho_{A}(\vec{b},z)e^{izx_{{I\\!\\!P}}m_{N}}e^{-\frac{A}{2}(1-i\eta)\sigma_{\rm
soft}^{i}(x,Q_{0}^{2})\int_{z}^{\infty}dz^{\prime}\rho_{A}(\vec{b},z^{\prime})}\right|^{2}\,.$

The structure of final states in diffraction with nuclei is richer than that
in the proton case and, in addition to coherent diffraction, there is a class
of diffractive events corresponding to disintegration of the target nucleus,
which is called incoherent diffraction (without meson production). Using
completeness of final nuclear states $A^{\prime}\neq A$, one obtains for the
nuclear structure function characterizing $e+A\to e^{\prime}+X+A^{\prime}$
incoherent diffraction in DIS with nuclei

$\displaystyle F_{2A,{\rm incoh}}^{D(3)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}})$
$\displaystyle=$ $\displaystyle
AF_{2N}^{D(3)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}})\int
d^{2}\vec{b}dz\rho_{A}(b,z)e^{-A\sigma_{\rm soft}^{\rm
inel}(x,Q_{0}^{2})\int_{z}^{\infty}dz^{\prime}\rho_{A}(\vec{b},z^{\prime})}$
(172) $\displaystyle-$ $\displaystyle 4\pi A^{2}\frac{\sigma_{\rm soft}}{8\pi
B_{X}}F_{2N}^{D(4)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}},t_{\rm min})\int
d^{2}\vec{b}\Big{|}\int^{\infty}_{-\infty}dz\rho_{A}(\vec{b},z)$
$\displaystyle\times$ $\displaystyle
e^{izx_{{I\\!\\!P}}m_{N}}e^{-\frac{A}{2}(1-i\eta)\sigma_{\rm
soft}(x,Q_{0}^{2})\int_{z}^{\infty}dz^{\prime}\rho_{A}(\vec{b},z^{\prime})}\Big{|}^{2}\,,$

where $\sigma_{\rm soft}(x,Q_{0}^{2})$ is identified with $\sigma_{\rm
soft}^{i}(x,Q_{0}^{2})$ for quarks, and

$\sigma_{\rm soft}^{\rm inel}(x,Q_{0}^{2})=\sigma_{\rm
soft}(x,Q_{0}^{2})-(1+\eta^{2})[\sigma_{\rm soft}(x,Q_{0}^{2})]^{2}/(16\pi
B_{X})\;,$ (173)

with $B_{X}=6-7$ GeV-2.

We would like to emphasize that as in the inclusive case, in Eqs. (171) and
(172), the contributions due to the interaction with two target nucleons are
model-independent. For the $N\geq 3$ terms, we made a simplifying assumption
that all configurations contributing to $\sigma_{\rm soft}^{i}(x,Q_{0}^{2})$
have similar cross sections. This is natural since $\sigma_{\rm
soft}^{i}(x,Q_{0}^{2})$ is already large.

Multiple interactions with target nucleons encoded in Eqs. (171) and (172)
lead to a significant suppression of the predicted nuclear diffractive PDFs
and structure functions at small $x$ due to the leading twist nuclear
shadowing. A convenient way to illustrate its magnitude is to introduce the
nuclear suppression factor of $R_{i}$ with respect to the impulse
approximation (IA):

$\displaystyle f_{i/A}^{D(3)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}})$
$\displaystyle=$ $\displaystyle R_{i}(\beta,Q_{0}^{2},x_{{I\\!\\!P}})4\pi
A^{2}\beta f_{i/N}^{D(4)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}},t_{\rm min})\int
d^{2}b\,(T_{A}(b))^{2}$ (174) $\displaystyle=$ $\displaystyle
R_{i}(\beta,Q_{0}^{2},x_{{I\\!\\!P}})A^{2}\beta
f_{i/N}^{D(4)}(\beta,Q_{0}^{2},x_{{I\\!\\!P}},t_{\rm min})\int
dt|F_{A}(t)|^{2}\,,$

where $F_{A}(t)$ is the nuclear form factor. Figure 40 shows predictions for
$R_{i}(\beta,Q_{0}^{2},x_{{I\\!\\!P}})$ for Pb-208 as a function of
$x_{{I\\!\\!P}}$ at $\beta=0.1$ and $Q_{0}=4$ GeV2. The left and right panels
correspond to $\bar{u}$ quarks and gluons, respectively. One can see from the
figure that the predicted nuclear suppression factor depends weakly on the
parton flavor and $x_{{I\\!\\!P}}$ in the studied kinematic range.

Figure 40: The suppression factor of $R_{i}(\beta,Q_{0}^{2},x_{{I\\!\\!P}})$
for nuclear diffractive PDFs as a function of $x_{{I\\!\\!P}}$ at $\beta=0.1$
and $Q_{0}=4$ GeV2 for Pb-208. The left and right panels correspond to
$\bar{u}$ quarks and gluons, respectively.

One should note that the uncertainty bands for the predicted nuclear
diffractive PDFs are much larger than those for the inclusive nPDFs, compare
Figs. 37 and 40. Mostly this is a consequence of the fact that in the case of
coherent diffraction, the nuclear diffractive PDFs involve squares of the
corresponding scattering amplitudes, which essentially doubles the resulting
uncertainties.

Predictions of the leading twist nuclear shadowing model for nuclear structure
functions and nuclear diffractive structure functions can be best tested at
the future Electron-Ion Collider [291, 6]. Morevover, in the EIC kinematics
with $Q^{2}\geq 2$ GeV2 and $x\geq 10^{-3}$, predictions of the presented
approach have small uncertainties and cover a wide range of nuclei from
deuteron to lead.

Note that the strong nuclear shadowing of nuclear diffractive PDFs and
structure functions leads to the ratio of the diffractive to total cross
sections on nuclei, which is similar to that on the proton [281]. Thus, the LT
nuclear shadowing makes nuclei more transparent to high-energy probes and,
hence, delays the onset of the black disk limit.

##  8 Hard coherent diffraction in photoproduction of quarkonia and dijets on
nuclei

###  8.1 Exclusive photoproduction of charmonia on nuclei in Pb-Pb UPCs

Further progress in constraining nPDFs at small $x$ relies on studies of high
energy hard processes with nuclei at collider energies, notably, in
proton–nucleus ($pA$) scattering at the LHC [292] and lepton–nucleus ($eA$)
scattering at the future EIC [6] and LHeC [76]. However, the QCD analyses of
the data on various hard processes in $pA$ scattering at the LHC during Runs 1
and 2 [293, 294, 295, 296, 297, 298, 299, 300] showed that while they provide
additional restrictions on nPDFs, the remaining uncertainties at small $x$ are
still significant. Therefore, it is important to explore the potential of
complementary probes of small-$x$ nPDFs at the LHC.

It has been realized that collisions of ultrarelativistic ions at large impact
parameters, when the strong interaction is suppressed and the ions interact
electromagnetically via the emission of quasi-real photons in the so-called
ultraperipheral collisions (UPCs), give an opportunity to study photon–photon,
photon–proton and photon–nucleus scattering at unprecedentedly high energies
[5].

This program was realized during Runs 1 and 2 at the LHC by measuring
exclusive photoproduction of charmonia ($J/\psi$ and $\psi(2S)$ vector mesons)
in Pb-Pb UPCs at $\sqrt{s_{NN}}=2.76$ TeV [301, 302, 303, 304] and
$\sqrt{s_{NN}}=5.02$ TeV [305, 306, 307]. Further high statistics studies are
planned during Runs 3 and 4 at the LHC.

A diagram for coherent $J/\psi$ photoproduction on nuclei in UPCs of ions $A$
is shown in Fig. 41.

Figure 41: Coherent $J/\psi$ photoproduction on nuclei in UPCs of ions $A$.
The cross section is given by a sum of two terms, where each ion serves either
as a source of equivalent photons or as a target.

Using the method of equivalent photons [58, 59], the cross section of this
process in nucleus–nucleus UPCs can be written in the following form [5]:

$\frac{d\sigma_{AA\to AAJ/\psi}(y)}{dy}=N_{\gamma/A}(y)\sigma_{\gamma A\to
AJ/\psi}(y)+N_{\gamma/A}(-y)\sigma_{\gamma A\to AJ/\psi}(-y)\,,$ (175)

where $N_{\gamma/A}(y)\equiv\omega dN_{\gamma/A}(\omega)/d\omega$ is the
photon flux; $y=\ln(2\omega/M_{J/\psi})=\ln(W^{2}_{\gamma
p}/(2\gamma_{L}m_{N}M_{J/\psi}))$ is the $J/\psi$ rapidity, where $\omega$ is
the photon energy in the laboratory frame, $W_{\gamma p}$ is the invariant
photon–proton center-of-mass energy, $M_{J/\psi}$ is the mass of $J/\psi$ and
$m_{N}$ is the nucleon mass. Note that interference between the two terms in
Eq. (175) is sizable only at very small values of the $J/\psi$ transverse
momentum [308, 309] and at central rapidities and, hence, can be safely
neglected in the $t$-integrated UPC cross section. The presence of two terms
in Eq. (175) is a reflection of the fact that in UPCs, both colliding nuclei
serve as a source of quasi-real photons and a target. For non-central
rapidities $y\neq 0$, this leads to ambiguity in the value of probed
$W_{\gamma p}$,

$W_{\gamma p}^{\pm}=\sqrt{2E_{N}M_{J/\psi}}\,e^{\pm y/2}\,,$ (176)

where $E_{N}$ is the nuclear beam energy per nucleon.

The photon flux is calculated as convolution of the flux of quasireal photons
emitted by an ultrarelativistic charged ion $N_{\gamma/A}(\omega,\vec{b})$
[58, 59] with the probability not to have inelastic strong ion–ion
interactions $\Gamma_{AA}(\vec{b})=\exp(-\sigma_{NN}\int
d^{2}\vec{b}_{1}T_{A}(\vec{b}_{1})T_{A}(\vec{b}-\vec{b}_{1}))$,

$N_{\gamma/A}(W_{\gamma p})=\int
d^{2}\vec{b}\,N_{\gamma/A}(\omega,\vec{b})\Gamma_{AA}(\vec{b})\,,$ (177)

where $\sigma_{NN}$ is the total nucleon–nucleon cross section and
$T_{A}(\vec{b})$ is the nuclear optical density.

As we discussed in Sec. 3, coherent photoproduction of $J/\psi$ probes the
gluon density of the target, see Eqs. (97) and (100). Thus, in the case of
photoproduction on nuclei, this process allows one to access the
$xg_{A}(x,\mu^{2})$ nuclear gluon distribution at small $x$ [190],

$\frac{d\sigma_{\gamma A\to J/\psi A}(W_{\gamma
p},t=0)}{dt}\,\propto\,[xg_{A}(x,\mu^{2})]^{2}\,.$ (178)

In Sec. 3, we discussed and quantified within the MFGS dipole model
corrections to Eqs. (97) and (100), hence, to the relation in Eq. (178). One
should also mention additional corrections, which include the effects of the
transverse momentum in the gluon loop and the charmonium wave function [310],
relativistic corrections to the charmonium wave function [311, 98, 192, 312,
233], and next-to-leading order (NLO) QCD radiative corrections in the
framework of collinear factorization for hard exclusive processes and
generalized parton distribution functions (GPDs) [313, 218].

Keeping in mind all these corrections and using Eq. (178), one can express the
cross section of coherent $J/\psi$ photoproduction on nuclei integrated over
$t$ in the following form [314, 209, 315]

$\displaystyle\sigma_{\gamma A\to J/\psi A}(W_{\gamma p})$
$\displaystyle=\kappa_{A/N}^{2}\frac{d\sigma_{\gamma p\to J/\psi p}(W_{\gamma
p},t=0)}{dt}\int_{|t_{\rm
min}|}^{\infty}dt\left[\frac{xg_{A}(x,t,\mu^{2})}{Axg_{N}(x,\mu^{2})}\right]^{2}$
(179) $\displaystyle\approx\kappa_{A/N}^{2}\frac{d\sigma_{\gamma p\to J/\psi
p}(W_{\gamma
p},t=0)}{dt}\left[\frac{xg_{A}(x,\mu^{2})}{Axg_{N}(x,\mu^{2})}\right]^{2}\int_{|t_{\rm
min}|}^{\infty}dt|F_{A}(t)|^{2}\,,$

where $|t_{\rm min}|=x^{2}m_{N}^{2}$ is the minimal momentum transfer squared.
In Eq. (179), $xg_{A}(x,t,\mu^{2})$ is the nuclear gluon GPD in the limit,
when both gluon lines carry the same light-momentum fraction $x$, i.e., it is
a two-gluon form factor of the target. Fourier transform of this quantity
relates it to the $b$-dependent nuclear gluon distribution, which we discussed
in Sect. 7.2,

$g_{A}(x,t,\mu^{2})=\int
d^{2}\vec{b}\,e^{i\vec{q}_{\perp}\cdot\vec{b}}g_{A}(x,b,\mu^{2})\,,$ (180)

where $q_{\perp}$ is the transverse component of the momentum transfer,
$t\approx-q_{\perp}^{2}$. In the second line of Eq (179), we used the commonly
used factorized form $g_{A}(x,t,\mu^{2})=F_{A}(t)g_{A}(x,\mu^{2})$, where
$F_{A}(t)$ is the nuclear form factor. This approximation works sufficiently
well for the $t$-integrated cross section. At the same time, for the
differential cross section, the impact parameter dependence of the leading
twist gluon nuclear shadowing modifies the shape of the $t$ distribution
[315], see also the discussion below.

The factor of
$\kappa_{A/N}^{2}=(1+\eta_{A}^{2})R_{g,A}^{2}/[(1+\eta_{p}^{2})R_{g,p}^{2}]$
takes into account the slightly different $x$ dependence of the nuclear and
proton gluon distributions, where $\eta$ is the ratio of the real to the
imaginary parts of the $\gamma T\to J/\psi T$ amplitude, and $R_{g}$ is a
phenomenological enhancement factor relating the usual gluon density to the
gluon generalized parton distribution, see Eq. (94). In the case, when the
gluon distributions in a nucleus and the proton have a similar small-$x$
behavior, i.e., $R_{g}(x,\mu^{2})=xg_{A}(x,\mu^{2})/[Axg_{N}(x,\mu^{2})]$ is a
slow function of $x$, $\kappa_{A/N}\approx 1$ with a good precision.

To quantify nuclear modifications of the $J/\psi$ photoproduction cross
section and minimize the effects affecting Eqs. (178) and (179), which we
mentioned above, it is useful to introduce the nuclear cross section in the
impulse approximation (IA)

$\sigma_{\gamma A\to J/\psi A}^{\rm IA}(W_{\gamma p})=\frac{d\sigma_{\gamma
N\to J/\psi N}(W_{\gamma p},t=0)}{dt}\int_{|t_{\rm
min}|}^{\infty}dt|F_{A}(t)|^{2}\,.$ (181)

It is important to point out that the theoretically-defined $\sigma_{\gamma
A\to J/\psi A}^{\rm IA}(W_{\gamma p})$ is practically model-independent since
its estimate is based on the experimental data.

Then, taking the square root of the ratio of the cross sections in Eqs. (179)
and (181), one introduces the nuclear suppression factor of $S_{Pb}(x)$ [314,
209, 316]

$S_{Pb}(x)=\sqrt{\frac{\sigma_{\gamma A\to J/\psi A}(W_{\gamma
p})}{\sigma_{\gamma A\to J/\psi A}^{\rm IA}(W_{\gamma
p})}}=\kappa_{A/N}\frac{xg_{A}(x,\mu^{2})}{Axg_{N}(x,\mu^{2})}\equiv\kappa_{A/N}R_{g}(x,\mu^{2})\,.$
(182)

It is expected that almost all kinematic factors and mentioned corrections
cancel in the ratio of the nuclear and IA( proton) cross sections. Thus, Eq.
(182) establishes a direct correspondence between the suppression factor of
$S_{Pb}(x)$ and the ratio of the nuclear and nucleon gluon distributions
$R_{g}(x,\mu^{2})$. Further, since at central rapidities $|y|\approx 0$, the
$d\sigma_{AA\to AAJ/\psi}(y)/dy$ cross section is unambiguously related to the
$\sigma_{\gamma A\to J/\psi A}(W_{\gamma p})$ photoproduction cross section at
the definite value of $W_{\gamma p}=\sqrt{2E_{N}M_{J/\psi}}$, Eq. (182) gives
a one-to-one correspondence between the measured UPC cross section at central
rapidities and $R_{g}(x,\mu^{2})$ at $x=M_{J/\psi}/(2E_{N})$.

Figure 42: The nuclear suppression factor of $S_{Pb}(x)$ as a function of the
gluon momentum fraction of $x$: the values extracted from the Run 1 [301, 302,
304] and the central rapidity Run 2 [307] UPC data on coherent $J/\psi$
photoproduction in Pb-Pb UPCs vs. predictions of the leading twist model of
nuclear shadowing and global fits of nPDFs. The bands indicate the
uncertainties for the LTA model (yellow) and EPS09 parametrization (blue).

Figure 42 shows a comparison of the values of $S_{Pb}(x)$ extracted from the
Run 1 [301, 302, 304] and the central rapidity Run 2 [307] UPC data on
coherent $J/\psi$ photoproduction in Pb-Pb UPCs with $R_{g}(x,\mu^{2})$
predicted in the leading twist model of nuclear shadowing and global QCD fits
of nPDFs. Note that following the analysis of Ref. [209], we take advantage of
the ambiguity in the exact values of the scale $\mu$ and take $\mu^{2}=3$ GeV2
to best reproduce the available HERA and LHCb data on the $W_{\gamma p}$
dependence of the cross section of exclusive $J/\psi$ photoproduction on the
proton. The good agreement with the predictions of the leading twist nuclear
shadowing model and the EPS09 nPDFs, which however have much larger
uncertainties, gives direct and weakly model-dependent evidence of large
nuclear gluon shadowing at small $x$,

$R_{g}(x=6\times 10^{-4}-10^{-3},\mu^{2}=3\ {\rm GeV}^{2})\approx 0.6\,.$
(183)

Note that the analysis of Ref. [316] extracted the nuclear suppression factor
of $S_{Pb}(x)$ in a wide range of $x$, $10^{-5}\leq x\leq 0.04$ using all
available Run 1 and 2 data on coherent $J/\psi$ photoproduction in Pb-Pb UPCs.
However, due to the ambiguity of the two terms in Eq. (175), such a procedure
is in general model dependent and leads to significant uncertainties in
$S_{Pb}(x)$ for $x<6\times 10^{-4}$ and $x>0.01$. In this respect one should
also mention the analysis of [317], where $S_{Pb}(x)$ was extracted from
measurements of coherent $J/\psi$ photoproduction in ultraperipheral and
peripheral Pb-Pb collisions at the LHC at 2.76 TeV. The results of that
analysis broadly agree with the trend of the nuclear suppression presented in
Fig. 42.

The significant leading twist gluon nuclear shadowing also affects the
differential cross section of coherent $J/\psi$ photoproduction on nuclei,

$\frac{d\sigma_{\gamma A\to J/\psi A}(W_{\gamma
p},t)}{dt}=\kappa_{A/N}^{2}\frac{d\sigma_{\gamma p\to J/\psi p}(W_{\gamma
p},t=0)}{dt}\left[\frac{xg_{A}(x,t,\mu^{2})}{Axg_{N}(x,\mu^{2})}\right]^{2}\,.$
(184)

Figure 43 shows the $d\sigma_{\gamma A\to J/\psi A}(W_{\gamma p},t)/dt$ cross
section normalized to its value at $|t|=t_{\rm min}$ as a function of $t$ at
$W=124$ GeV. This value corresponds to Pb-Pb UPCs during Run 2 at the LHC with
$\sqrt{s_{NN}}=5.02$ TeV and the central rapidity $y=0$. The red solid curve
is the prediction of Eq. (184), where for $xg_{A}(x,t,\mu^{2})$ and
$xg_{A}(x,b,\mu^{2})$, see Eq. (180), we used predictions of the leading twist
nuclear shadowing model for the impact parameter dependent nuclear PDFs, see
Sec. 7. The blue dot-dashed curve gives the $t$ dependence of the nuclear form
factor squared $[F_{A}(t)/A]^{2}$. One can see from the figure that the impact
parameter dependence of the leading twist nuclear shadowing, i.e., the
correlation between $b$ and $x$ in $xg_{A}(x,b,\mu^{2})$, noticeably shifts
the minimum of the $t$ distribution toward lower values of $t$. This can be
interpreted as broadening in impact parameter space of the small-$x$ gluon
distribution in nuclei as a consequence of the fact that nuclear shadowing
increases with a decrease of $b$ (increase of the nuclear density).

Figure 43: The $d\sigma_{\gamma A\to J/\psi A}(W_{\gamma p},t)/dt$ cross
section normalized to its value at $|t|=t_{\rm min}$ as a function of $t$ at
$W=124$ GeV: predictions of the leading twist model of nuclear shadowing (red
solid curve) vs. the factorized approximation (blue dot-dashed curve). The
figure is from [315], https://doi.org/10.1103/PhysRevC.95.025204.

The predictions for the shift of the $t$ dependence of the $d\sigma_{\gamma
A\to J/\psi A}(W_{\gamma p},t)/dt$ cross section shown in Fig. 43 have been
nicely confirmed by the recent ALICE measurements [318].

The program of UPC measurements will continue with Runs 3 and 4 at the LHC
with the increased collision energy and luminosity [319], which will enable
one to probe and constrain the nuclear gluon distribution down to $x\approx
10^{-4}$. An additional opportunity to access small-$x$ nuclear gluon
distributions is offered by $J/\psi$ photoproduction accompanied by neutron
emission due to electromagnetic excitation of one or both colliding nuclei,
which allows one in principle to separate the contributions of two terms in
Eq. (175), see [320].

###  8.2 Diffractive dijet photoproduction in UPCs and at the EIC

The program of UPC measurements also includes photoproduction of jets on the
proton and nuclei, which provides information on the proton and nucleus
structure in QCD, which is complementary to that obtained in DIS. As discussed
in Sec. 2.4, measurements of dijets in the presence of a rapidity gap allow
one to access diffractive parton distributions of the target, test their
universality, and get a new handle of factorization breaking in this process.

Figure 44 shows typical Feynman graphs for (quark) dijet photoproduction in
the UPCs of ions $A$ and $B$, where graphs $a$ and $b$ correspond to the
direct and the resolved photon contributions, respectively. They are
completely analogous to graphs in Fig. 6 with the only difference that the
electron and the proton are replaced by nuclei.

Figure 44: Typical Feynman graphs for diffractive dijet photoproduction in
UPCs. Graphs $(a)$ and $(b)$ correspond to the direct and resolved photon
contributions, respectively.

The corresponding cross section for symmetric heavy ion UPCs reads

$\displaystyle d\sigma(AA\to A+2\,{\rm jets}+X^{\prime}+A)$ $\displaystyle=$
$\displaystyle d\sigma(AA\to A+2\,{\rm jets}+X^{\prime}+A)^{(+)}$ (185)
$\displaystyle+$ $\displaystyle d\sigma(AA\to A+2\,{\rm
jets}+X^{\prime}+A)^{(-)}\,,$

where $X^{\prime}$ stands for the produced diffractive final state $X$ after
removing two jets. The two terms in Eq. (185) have the same meaning as in Eq.
(175) and correspond to two possible directions of the diffracting nucleus.
They are related to each other by inverting the sign of the $\eta_{1}$ and
$\eta_{2}$ jet rapidities:

$d\sigma(AA\to A+2\,{\rm jets}+X^{\prime}+A)^{(-)}=d\sigma(AA\to A+2\,{\rm
jets}+X^{\prime}+A)^{(+)}_{|\eta_{1}\to-\eta_{1},\,\eta_{2}\to-\eta_{2}}\,.$
(186)

The cross section of dijet photoproduction in $AA$ UPCs is given by the
following convolution in analogy to the photoproduction in DIS collisions, see
Eq. (17) in Sec. 2.4,

$\displaystyle d\sigma(AA\to A$ $\displaystyle+$ $\displaystyle 2\,{\rm
jets}+X^{\prime}+A)^{(+)}=\sum_{a,b}\int dt\int
dx_{{I\\!\\!P}}\int_{0}^{1}dz_{{I\\!\\!P}}\int dy\int dx_{\gamma}$ (187)
$\displaystyle\times$ $\displaystyle
f_{\gamma/A}(y)f_{a/\gamma}(x_{\gamma},\mu^{2})f^{D(4)}_{b/A}(x_{{I\\!\\!P}},z_{{I\\!\\!P}},t,\mu^{2})d\hat{\sigma}_{ab\to{\rm
jets}}^{(n)}\,,$

where $a$ and $b$ are parton flavors including the case, when $a$ stands for
the photon for the direct photon contribution; $y$, $x_{\gamma}$,
$x_{{I\\!\\!P}}$, and $z_{{I\\!\\!P}}$ are longitudinal momentum fractions
carried by photons, partons in the photon, the diffractive exchange in the
target nucleus, and partons in the diffractive exchange, respectively;
$f_{\gamma/A}(y)$ is the photon flux calculated in the equivalent photon
approximation; $f_{a/\gamma}(x_{\gamma},\mu^{2})$ and
$f^{D(4)}_{b/A}(x_{{I\\!\\!P}},z_{{I\\!\\!P}},t,\mu^{2})$ are PDFs of the
photon in the resolved photon case and the diffractive PDFs of the target
nucleus, respectively; $d\hat{\sigma}_{ab\to{\rm jets}}^{(n)}$ is the cross
section for production of an $n$-parton final state from two initial partons
$a$ and $b$.

As we mentioned above, Eq. (187) should be in principle complemented by the
effect of factorization breaking in diffractive dijet photoproduction. In
connection to this, one of the best observables to distinguish various schemes
of the factorization breaking is provided by the observed photon momentum
fraction of $x_{\gamma}^{\rm jets}$,

$x_{\gamma}^{\rm jets}=\frac{m_{\rm jets}}{y\sqrt{s_{NN}}}e^{y_{\rm jets}}\,,$
(188)

where

$m_{\rm
jets}=\left[(E_{1}+E_{2})^{2}-(\vec{p}_{1}+\vec{p}_{2})^{2}\right]^{1/2}\,,\quad
y_{\rm
jets}=\frac{1}{2}\ln\frac{E_{1}+E_{2}+p_{1z}+p_{2z}}{E_{1}+E_{2}-p_{1z}-p_{2z}}\,.$
(189)

In Eq. (189), $E_{1,2}$, $\vec{p}_{1,2}$ and $p_{1,2z}$ are the energies,
momentum vectors, and the longitudinal component of the momentum of two most
energetic jets.

Using the framework presented above, one can make predictions for various
distributions in diffractive dijet photoproduction in UPC in the LHC
kinematics [321]. This analysis used the $k_{T}$-cluster algorithm with the
jet radius $R=1$ and assumed that $E_{1}\geq 20$ GeV for the leading jet and
$E_{2}\geq 18$ GeV for the subleading one. An example is presented in Fig. 45,
where the cross section of dijet photoproduction in Pb-Pb UPCs at
$\sqrt{s_{NN}}=5.02$ TeV is shown as a function $x_{\gamma}^{\rm jets}$. The
results are shown for two schemes of diffractive QCD factorization breaking.
Namely, the red solid curve corresponds to the global suppression factor of
$R_{\rm glob}=0.1$, while the blue dot-dashed curve assumes the suppression
only for the resolved photon contribution by the factor of $R_{\rm res}=0.04$.

Thus, studies of this process on nuclei may shed some light on the mechanism