# Probing the structure of $X(3872)$ in photoproduction

E. Ya. Paryev
Institute for Nuclear Research of the Russian Academy of Sciences
Moscow, Russia

###### Abstract

We study the production of $X(3872)$ mesons in photon-induced nuclear
reactions near the threshold within the collision model based on the nuclear
spectral function. The model accounts for direct photon-nucleon $X(3872)$
production processes as well as five different scenarios for their internal
structure. We calculate the absolute and relative excitation functions for
$X(3872)$ production off 12C and 184W target nuclei at near-threshold incident
photon energies of 8–16 GeV, the absolute differential cross sections for
their production off these target nuclei at laboratory angles of 0∘–10∘ and
for incident photon energy of 13 GeV as well as the A dependences of the
relative (transparency ratios) cross sections for $X(3872)$ production from
${\gamma}A$ collisions at photon energies around 13 GeV within the adopted
scenarios for the $X(3872)$ meson internal structure. We show that the
absolute and relative observables considered reveal distinct sensitivity to
these scenarios. Therefore, the measurement of such observables in a dedicated
experiment at the CEBAF facility in the near-threshold energy range will allow
us to get valuable information on the $X(3872)$ inner structure.

## 1\. Introduction

The discovery of $X(3872)$ resonance (also known as $\chi_{c1}(3872)$) by the
Belle Collaboration in 2003 [1] as a narrow peak in the vicinity of the
$D^{0}{\bar{D}}^{*0}$ (${\bar{D}}^{0}{D}^{*0}$) mass threshold in the
${J/\psi}{\pi^{+}}\pi^{-}$ invariant mass distribution in exclusive
$B^{\pm}\to K^{\pm}X(3872)\to K^{\pm}({J/\psi}{\pi^{+}}\pi^{-})$ decays 111)It
was also confirmed in many other high-energy experiments. In particular, the
experimental results of studying the $X(3872)$ lineshape using the data
samples collected with the Belle and BESIII detectors have been reported in
very recent publications [2] and [3], respectively.) has opened a new era for
the study of exotic heavy hadrons, which exhibit properties incompatible with
the predictions of the traditional quark model for quark-antiquark mesons and
three-quark baryons. They are composed of four or five quarks (and antiquarks)
and usually are named as tetraquark and pentaquark states, respectively. Since
then, many unconventional charmonium- and bottomonium-like states, the so-
called $X,Y,Z$ mesons as well as the hidden-charm non-strange $P_{c}$ and
strange $P_{cs}$ pentaquark states, the doubly-charmed tetraquark
$T^{+}_{cc}(3875)$ state, the fully charmed tetraquark $X(6900)$ state and so
on have been observed in various experiments, as summarized in reviews [4–19].
The observation of such exotic hadronic states and studying their properties
are very important for uderstanding the nonperturbative aspects of QCD and for
extending our knowledge about the world of ”elementary” particles. Among these
exotic hadronic states, the $X(3872)$ attracted lots of attention because of
its rather enigmatic intrinsic structure. Since the experimental discovery of
the $X(3872)$ meson, this structure has been intensely debated. However, a
compelling understanding of the nature of this resonance is still lacking and
there is no consensus about its structure. Thus, in a molecular scenario, due
to the closeness of the observed $X(3872)$ mass ($m_{X(3872)}=3871.65\pm 0.06$
MeV [20]) to the $D^{0}{\bar{D}}^{*0}$ (${\bar{D}}^{0}{D}^{*0}$) mass
threshold ($m_{D^{0}}+m_{{\bar{D}}^{*0}}=3871.69\pm$0.07 MeV), the $X(3872)$
resonance can be naturally interpreted as an extremely narrow
$D^{0}{\bar{D}}^{*0}$+${\bar{D}}^{0}{D}^{*0}$ molecular state in a relative
S-wave with a width less than 1.2 MeV [20] and with a very small binding
energy (see, for instance, Refs. [21, 22] and those given below). However,
there exists another alternative explanations of the $X(3872)$. For example,
it is interpreted as a conventional $c{\bar{c}}$ charmonium $\chi_{c1}(2P)$
state [23], a diquark-antidiquark compact tetraquark state [24], a mixture of
a molecule and an excited charmonium state [25]. Such also plausible
interpretations of the $X(3872)$ exotic state were supported by many current
theoretical studies (see references herein below).

Most of the above interpretations are based on the comparison of the
calculated and experimental branching fractions for various two- and three-
body $X(3872)$ decays. However, the internal structure of $X(3872)$ could also
be investigated in high-energy proton–proton and heavy–ion collisions at
hadron colliders, since its production yield in these collisions could reflect
its structure. Thus, the rather large prompt (i.e. not from $B$ decays)
production rate of $X(3872)$ at hadron colliders is considered as an argument
against its interpretation as a weakly bound charm-meson molecule (cf. [26]).
There are many debates in the literature [26–32] on whether or not it is
compatible with such interpretation, leading to mutually conflicting
conclusions. Recently, the inclusive high transverse momentum distributions of
$X(3872)$, promptly produced in high-energy $pp$ [33, 34, 35] and PbPb [36]
collisions at rates commensurate with those of the $\psi(2S)$ charmonium, were
measured at the LHC, respectively, by the CMS, ATLAS, LHCb and CMS
Collaborations. A similarity of the $X(3872)$ production rate with the
$\psi(2S)$ production rate suggests the importance of a compact component in
the X(3872) wave function. Indeed, the measured in $pp$ collisions $X(3872)$
transverse momentum distributions are well described in Refs. [37, 38]
assuming a large charmonium $c{\bar{c}}$ component in this function.
Nevertheless, in spite of large efforts the full understanding the production
of $X(3872)$ and its structure in hadronic collisions remains a challenging
open problem [13, 16, 18, 19, 39]. Alternatively, the structure of $X(3872)$
mesons can also be studied in their photoproduction off nuclei at energies
close to the threshold for their production off a free nucleon. This has the
advantage compared to the hadronic collisions that the interpretation of data
from such experiments is less ambiguous owing to a negligible strength of
initial-state photon interaction and since the $X(3872)$ meson production
occurs through a few channels in a cleaner environment - in static cold
nuclear matter whose density is well known.

In this paper we present the detailed results for the absolute and relative
excitation functions for $X(3872)$ production off 12C and 184W target nuclei
as well as for the A dependences of the relative (transparency ratios) cross
sections for $X(3872)$ production from ${\gamma}A$ collisions in the threshold
energy region obtained in the framework of the first collision model, based on
the nuclear spectral function, within the different scenarios for the
$X(3872)$ meson internal structure. A comparison of these results with data,
which could be taken in the future JLab experiments at the CEBAF facility, can
provide deeper insights into the inner structure and properties of the
puzzling $X(3872)$ resonance.

## 2\. The model: direct $X(3872)$ photoproduction mechanism

Direct $X(3872)$ photoproduction on nuclear targets in the near-threshold
incident laboratory photon energy region $E_{\gamma}\leq 16$ GeV 222)Which
corresponds to the center-of-mass energies W of the photon-proton system
$W\leq 5.56$ GeV, or to the relatively ”low” excess energies
$\epsilon_{X(3872)p}$ above the $X(3872)p$ production threshold $W_{\rm
th}=m_{X(3872)}+m_{p}=$4.81 GeV ($m_{X(3872)}$ and $m_{p}$ are the $X(3872)$
meson and proton bare masses, respectively ) $0\leq\epsilon_{X(3872)p}\leq
0.75$ GeV and in which the exotic hadron $X(3872)$ can be observed in
${\gamma}p$ and ${\gamma}A$ reactions at the 12 GeV CEBAF facility and at the
proposed 24 GeV apgrade of this facility with novel magnet designs in the
existing recirculation arcs [40–44].) may proceed via the following elementary
processes with the lowest free production threshold ($\approx$ 11.86 GeV)
[45–47]:

${\gamma}+p\to X(3872)+p,$ (1) ${\gamma}+n\to X(3872)+n.$ (2)

The $X(3872)$ mesons and nucleons, produced in these processes, are
sufficiently energetic. Thus, for example, the kinematically allowed $X(3872)$
and final proton laboratory momenta in the direct process (1), taking place on
the free target proton at rest, vary within the momentum ranges of
8.481–11.775 GeV/c and 1.225–4.519 GeV/c, respectively, at initial photon beam
energy of $E_{\gamma}=13$ GeV. Since the neutron mass is approximately equal
to the proton mass, the kinematical characteristics of the $X(3872)$ mesons
and final neutrons, produced in the reaction (2), are close to those of final
particles ($X(3872)$ mesons and protons) in the process (1). Evidently, the
binding of target nucleons and their Fermi motion will distort the
distributions of the outgoing high-momentum $X(3872)$ mesons and nucleons as
well as lead to a wider accessible momentum intervals compared to those given
above. Since the medium effects are expected to be reduced for high momenta
[48–50], we will neglect the modification of the final high-momentum $X(3872)$
mesons and outgoing nucleons in the nuclear medium in the case when the
reactions (1), (2) proceed on a nucleons embedded in a nuclear target 333)It
should be pointed out that the behavior of $X(3872)$ in a nuclear environment
has been studied in the recent works [51, 52]. The authors of Ref. [51],
analyzing the $D{\bar{D}}^{*}$ scattering $T$–matrix within the mixed-
molecular scenario of $X(3872)$ have found that the width of the $X(3872)$
peak significantly grows when the nuclear density is increased, whereas its
position moves to higher energies as the molecular component is lowered. For
this component of the order of 60% for the $X(3872)$, this work predicts at
threshold widths for the resonance of around 30–40 MeV, and more modest
repulsive mass-shifts with a maximum of 10 MeV for the nuclear matter
saturation density $\rho_{0}$, which amounts approximately to 0.25% of its
free-space nominal mass. On the other hand, in Ref. [52] it was obtained using
QCD sum rule calculations based on a diquark-antidiquark picture for the
$X(3872)$ resonance that its mass is substantially influenced by the nuclear
medium. Indeed, in the approach of Ref. [52] the mass shift of the low-
momentum $X(3872)$ due to the nuclear matter is negative and is about 25% at
density $\rho_{0}$, which is questionable accounting for the quark content of
the $X(3872)$. But in any case in accordance with the above, one may expect
that the $X(3872)$-nucleus optical potential should vanish at high $X(3872)$
momenta of interest. The latter is also true for the nucleon–nucleus mean-
field potential at high nucleon momenta (cf. Refs. [53, 54]).) .

Then, ignoring the distortion of the incident photon in the energy range of
interest and describing the $X(3872)$ meson final-state absorption by the
absorption cross section $\sigma_{{X(3872)}N}$, we represent the total cross
section for the production of $X(3872)$ mesons on nuclei in the direct
photon–induced reaction channels (1), (2) as follows [55, 56]:

$\sigma_{{\gamma}A\to{X(3872)}X}^{({\rm
dir})}(E_{\gamma})=I_{V}[A,\sigma_{{X(3872)}N}]\left<\sigma_{{\gamma}p\to{X(3872)}p}(E_{\gamma})\right>_{A},$
(3)

where

$I_{V}[A,\sigma]=2{\pi}\int\limits_{0}^{R}r_{\bot}dr_{\bot}\int\limits_{-\sqrt{R^{2}-r_{\bot}^{2}}}^{\sqrt{R^{2}-r_{\bot}^{2}}}dz\rho(\sqrt{r_{\bot}^{2}+z^{2}})\exp{\left[-\sigma\int\limits_{z}^{\sqrt{R^{2}-r_{\bot}^{2}}}\rho(\sqrt{r_{\bot}^{2}+x^{2}})dx\right]},$
(4)
$\rho(r)=\rho_{p}(r)+\rho_{n}(r),\,\,\,r=\sqrt{r_{\bot}^{2}+z^{2}}\,\,\,{\rm
or}~{}r=\sqrt{r_{\bot}^{2}+x^{2}};$ (5)
$\left<\sigma_{{\gamma}p\to{X(3872)}p}(E_{\gamma})\right>_{A}=\int\int
P_{A}({\bf p}_{t},E)d{\bf
p}_{t}dE\sigma_{{\gamma}p\to{X(3872)}p}(\sqrt{s^{*}})$ (6)

and

$s^{*}=(E_{\gamma}+E_{t})^{2}-({\bf p}_{\gamma}+{\bf p}_{t})^{2},$ (7)
$E_{t}=M_{A}-\sqrt{(-{\bf p}_{t})^{2}+(M_{A}-m_{p}+E)^{2}}.$ (8)

Here, $\sigma_{{\gamma}p\to{X(3872)}p}(\sqrt{s^{*}})$ is the ”in-medium” total
cross section for the production of $X(3872)$ mesons in process (1) 444)In
equation (3) it is assumed that the $X(3872)$ meson production cross sections
in ${\gamma}p$ and ${\gamma}n$ interactions are the same.) at the ”in-medium”
${\gamma}p$ center-of-mass energy $\sqrt{s^{*}}$; $\rho_{p}(r)$, $\rho_{n}(r)$
($r$ is the distance from the nucleus center) and $P_{A}({\bf p}_{t},E)$ are
normalized to the numbers of protons $Z$, neutrons $N$ and to unity the local
proton, neutron densities and the spectral function of target nucleus with
mass number $A$ ($A=Z+N$), having mass $M_{A}$ and radius $R$ 555)The specific
information about the latter quantity, used in our calculations, is given in
Ref. [57]. The local nucleon densities, employed in the calculations, will be
defined below.); ${\bf p}_{t}$ and $E$ are the internal momentum and binding
energy of the struck target proton just before the collision; ${\bf
p}_{\gamma}$ and $E_{\gamma}$ are the momentum and energy of the incident
photon beam.

As in Ref. [57], we suggest that the ”in-medium” cross section
$\sigma_{{\gamma}p\to X(3872)p}({\sqrt{s^{*}}})$ for $X(3872)$ production in
process (1) is equivalent to the vacuum cross section
$\sigma_{{\gamma}p\to{X(3872)}p}({\sqrt{s(E_{\gamma})}})$, in which the vacuum
center-of-mass energy squared $s(E_{\gamma})$ for given photon energy
$E_{\gamma}$, presented by the formula

$s(E_{\gamma})=W^{2}=(E_{\gamma}+m_{p})^{2}-{\bf
p}_{\gamma}^{2}=m_{p}^{2}+2m_{p}E_{\gamma},$ (9)

is replaced by the in-medium expression (7). For the free total cross section
$\sigma_{{\gamma}p\to{X(3872)}p}({\sqrt{s(E_{\gamma})}})$ in the considered
photon energy range $W\leq$ 5.56 GeV we have used the following fit of the
results of the calculations of this cross section here within the approach
[47], in which the ${\gamma}p\to{X(3872)}p$ reaction is considered to be
proceeded by the $t$-channel vector mesons ($\rho$, $\omega$ and $\phi$)
exchange:

$\sigma_{{\gamma}p\to{X(3872)}p}({\sqrt{s(E_{\gamma})}})=125.0\left(1-\frac{s_{\rm
th}}{s(E_{\gamma})}\right)^{1.2}~{}[{\rm nb}],$ (10)

where

$s_{\rm th}=W_{\rm th}^{2}=(m_{X(3872)}+m_{p})^{2}.$ (11)

In this low-energy regime, the cross sections for the $X(3872)$ production in
${\gamma}p$ collisions are predicted to be of the order of 1–20 nanobarns
[47]. It should be pointed out that in the very recent work [58] the total
cross section of the ${\gamma}p\to X(3872)p$ reaction has been also estimated
assuming the coupled-channel production mechanism through the open-charm
meson-baryon intermediate states. For energies near the threshold, this cross
section is predicted to be of the order of tens of nanobarns, which are
generally larger than the model prediction [47] and which together with this
prediction are well experimentally measurable already within the capabilities
of the present CEBAF facility (cf. Refs. [59–61]). At the same time, the
authors of this work have underscored that their results provide merely an
order-of-magnitude estimate and should not be considered as a precise
quantitative prediction since no data are available so far near the threshold
to constrain the model parameters 666)It should be pointed out that virtual
exclusive photoproduction of the $X(3872)$ on a nucleon target with a muon
beam of 160 and 200 GeV/c momentum has been recently explored by the COMPASS
Collaboration [62]. A new charmonium-like structure, denoted by the
${\tilde{X}}(3872)$, has been observed in the ${J/\psi}\pi^{+}\pi^{-}$ mass
distribution with a 4.1 $\sigma$ statistical significance. Its mass and width
are consistent with those of the $X(3872)$. But the measured $\pi^{+}\pi^{-}$
mass distribution from the observed decay into the ${J/\psi}\pi^{+}\pi^{-}$
was found to be different from that measured in previous experiments for the
$X(3872)$. This suggests that this structure has the quantum numbers
$J^{PC}=1^{+-}$, while the quantum numbers previously determined for the
$X(3872)$ are $1^{++}$. The product of total cross section and branching ratio
of the decay of the observed ${\tilde{X}}(3872)$ state into the
${J/\psi}\pi^{+}\pi^{-}$ was determined to be 71$\pm$28(stat)$\pm$39(syst) pb.
Also, the COMPASS Collaboration has measured the upper bound for the $X(3872)$
photoproduction cross section at substantially higher energy – at an average
${\gamma}p$ c.m. energy of 13.7 GeV as $\sigma_{{\gamma}N\to
X(3872)N^{\prime}}$$\times$$Br[X(3872)\to{J/\psi}{\pi}\pi]<2.9$ pb (CL=90%).)
. Therefore, in our subsequent calculations we will use the empirical formulas
(10), (11) as a guideline for a reasonable estimation of the $X(3872)$ yield
from ${\gamma}A$ interactions.

We focus now on the local nucleon densities, used in our calculations of the
$X(3872)$ production on the considered in the present work target nuclei
${}^{12}_{6}$C, ${}^{27}_{13}$Al, ${}^{40}_{20}$Ca, ${}^{63}_{29}$Cu,
${}^{93}_{41}$Nb, ${}^{112}_{50}$Sn, ${}^{184}_{74}$W, ${}^{208}_{82}$Pb and
${}^{238}_{92}$U. For lightest nucleus ${}^{12}_{6}$C we use the same proton
and neutron density profiles of the harmonic oscillator model [63]. For nuclei
${}^{27}_{13}$Al, ${}^{40}_{20}$Ca and ${}^{63}_{29}$Cu for the proton and
neutron densities, $\rho_{p}(r)$ and $\rho_{n}(r)$, we have employed in our
present calculations the Woods-Saxon distributions with the same radial
parameters for protons and neutrons [63], viz.:

$\rho_{p}(r)=Z\rho(r),\,\,\,\rho_{n}(r)=N\rho(r),\,\,\,\rho(r)=\rho_{0}\left[1+\exp\left(\frac{r-R_{1/2}}{a}\right)\right]^{-1},$
(12)

where $\rho(r)$ represents the local nucleon density normalized to unity
($\int\rho(r)d^{3}r=1$) and parameters $R_{1/2}$ and $a$, determined from fits
to electron-scattering data, are taken from standard compilation [64]. They
are: $R_{1/2}=3.07$ fm, $a=0.519$ fm for ${}^{27}_{13}$Al; $R_{1/2}=3.51$ fm,
$a=0.563$ fm for ${}^{40}_{20}$Ca and $R_{1/2}=4.214$ fm, $a=0.586$ fm for
${}^{63}_{29}$Cu. For medium-weight ${}^{93}_{41}$Nb, ${}^{112}_{50}$Sn and
heavy ${}^{184}_{74}$W, ${}^{208}_{82}$Pb, ${}^{238}_{92}$U target nuclei, we
adopted for protons and neutrons the two-parameter Fermi density
distributions:

$\rho_{i}(r)=\rho_{0i}\left[1+\exp\left(\frac{r-c_{i}}{z_{i}}\right)\right]^{-1},\,\,\,i=p,n$
(13)

with proton density parameters: $c_{p}=4.87$ fm, $z_{p}=0.573$ fm for
${}^{93}_{41}$Nb; $c_{p}=5.375$ fm, $z_{p}=0.560$ fm for ${}^{112}_{50}$Sn and
$c_{p}=6.51$ fm, $z_{p}=0.535$ fm for ${}^{184}_{74}$W; $c_{p}=6.624$ fm,
$z_{p}=0.549$ fm for ${}^{208}_{82}$Pb; $c_{p}=6.805$ fm, $z_{p}=0.605$ fm for
${}^{238}_{92}$U, also inferred from nuclear charge distributions for these
target nuclei [64].

Figure 1: (Color online.) Proton and neutron densities for 93Nb. Neutron
density is calculated in the ’skin’ and ’halo’ forms (see text). Figure 2:
(Color online.) The same as in Fig. 1, but for the 208Pb target nucleus.
Figure 3: (Color online.) The same as in Fig. 1, but for the 238U target
nucleus.

To see the sensitivity of the results of calculations of the $X(3872)$ meson
total production cross sections on the medium-weight and heavy target nuclei
to the choice of the neutron density $\rho_{n}(r)$, which is not known to
sufficient accuracy for these nuclei, we employed for it the ’skin’ and ’halo’
forms of Ref. [65]. Their radial parameters $c_{n}$ and $z_{n}$ for each
nucleus were determined from the r.m.s. radius $r_{n}$ of $\rho_{n}(r)$, which
assumes in line with Ref. [65] larger value than that $r_{p}$ for proton
density distribution $\rho_{p}(r)$:

$r_{n}-r_{p}={\gamma}\frac{N-Z}{A}+\delta,$ (14)

using the relation [66]

$c_{n}=\sqrt{\frac{5}{3}r_{n}^{2}-\frac{7}{3}{\pi^{2}}z_{n}^{2}}.$ (15)

In Eq. (14) the parameters $\gamma$ and $\delta$ were chosen as $\gamma=1.05$
fm, $\delta=-0.035$ fm and the r.m.s. radius $r_{p}$ is considered to be known
and was set equal to that of the known nuclear charge density [64]. Then,
according to [65], for each value of $r_{n}$ found from Eq. (14) in the ’skin’
form the same diffuseness parameter for protons and neutrons, $z_{n}=z_{p}$,
is used and the radius parameter $c_{n}$ is determined from Eq. (15). In the
’halo’ form the same radius parameter, $c_{n}=c_{p}$, is assumed and the
diffuseness parameter $z_{n}$ is determined also from Eq. (15). Using the
radial parameters for the proton density distributions given above and their
r.m.s. radii: $r_{p}=4.310$ fm for ${}^{93}_{41}$Nb, $r_{p}=4.655$ fm for
${}^{112}_{50}$Sn, $r_{p}=5.420$ fm for ${}^{184}_{74}$W, $r_{p}=5.521$ fm for
${}^{208}_{82}$Pb, $r_{p}=5.8434$ fm for ${}^{238}_{92}$U [64] and going along
this way, we obtain the following values for the radius parameter $c_{n}$ and
for the diffuseness parameter $z_{n}$, respectively, for the ’skin’ and ’halo’
shapes of the neutron distribution for each value of its r.m.s. radius
$r_{n}$: $c_{n}=4.971$ fm for ${}^{93}_{41}$Nb, $c_{n}=5.487$ fm for
${}^{112}_{50}$Sn, $c_{n}=6.745$ fm for ${}^{184}_{74}$W, $c_{n}=6.882$ fm for
${}^{208}_{82}$Pb, $c_{n}=7.246$ fm for ${}^{238}_{92}$U and $z_{n}=0.6093$ fm
for ${}^{93}_{41}$Nb, $z_{n}=0.605$ fm for ${}^{112}_{50}$Sn, $z_{n}=0.649$ fm
for ${}^{184}_{74}$W, $z_{n}=0.673$ fm for ${}^{208}_{82}$Pb, $z_{n}=0.797$ fm
for ${}^{238}_{92}$U. The proton and neutron densities, for example, in 93Nb,
208Pb and 238U target nuclei, calculated in line with Eq. (13) using two
shapes of the neutron distribution mentioned above, are shown, respectively,
in Figs. 1, 2, 3. It is seen that the differences between two shapes become
the most pronounced for all nuclei at distances about 8–10 fm. This will
enable us to test their influence on the observables considered in the present
work (see below).

In order to proceed further, we have to specify the $X(3872)$–nucleon
absorption cross section $\sigma_{{X(3872)}N}$. The evaluation of this cross
section in different scenarios for the structure of $X(3872)$ proposed in the
literature is a crucial ingredient for its interpretation via, in particular,
the $X(3872)$ photoproduction off nuclei since there is a direct dependence of
the rate of this production on the absorption cross section
$\sigma_{{X(3872)}N}$ (cf. Eq.(3)), which is expected to be different for
different choices for the internal structure of $X(3872)$ (see below). In
other words, in our approach the sensitivity to this structure should be
encoded in the $X(3872)$ meson production rate. In this exploratory study we
consider four popular scenarios for the $X(3872)$ resonance: i) a conventional
2${}^{3}P_{1}$ $c{\bar{c}}$ state, the first radial excitation of $\chi_{c1}$,
i.e., the $\chi_{c1}(2P)$ state [37, 67–78] 777)It should be mentioned that
the bare charmonium state $\chi_{c1}(2P)$ would have the same quantum numbers
$J^{PC}=1^{++}$ as the $X(3872)$, it locates slightly above the
$D^{0}{\bar{D}}^{*0}$ (${\bar{D}}^{0}D^{*0}$) threshold and has not been found
yet [79].), ii) compact, $\sim$ 1 fm, diquark $[cq]$ and antidiquark
$[{\bar{c}}{\bar{q}}]$ tetraquark state with $q$ either $u$ or $d$ quark [52,
80–90], iii) a molecular state - a loosely bound, with large spatial size
characterized by the r.m.s. radius of the order of 5–10 fm or more, $S$-wave
state mainly in the $D^{0}{\bar{D}}^{*0}$+c.c. system consisting of a neutral
$D^{0}$ and ${\bar{D}}^{*0}$ or ${\bar{D}}^{0}$ and $D^{*0}$ mesons [26–28,
37, 72, 82, 84, 86, 89, 91–105] 888)In some works (see, for example, Refs.
[79, 99, 106–110]), it is assumed that in a molecular picture the $X(3872)$
flavor wave function contains also an additional component $D^{+}D^{*-}$+c.c.
composed of the charged $D$ and $D^{*}$ mesons. Because of the proximity of
the $X(3872)$ mass to the $D^{0}{\bar{D}}^{*0}$ (${\bar{D}}^{0}D^{*0}$)
threshold, the $X(3872)$ couples strongly only to the neutral
$D^{0}{\bar{D}}^{*0}$ and ${\bar{D}}^{0}D^{*0}$ channels (which is manifested
by the large branching ratios to the $D^{0}{\bar{D}}^{*0}$ and
$D^{0}{\bar{D}}^{0}\pi^{0}$ channels of about 37% and 40%, respectively,
[111]) and this ’charged’ component is of a minor importance in the $X(3872)$
wave function [106–109]. Thus, for instance, in the work [106] it was
predicted for the cutoff $\Lambda=0.5$ GeV that this function contains about
15% of the $c{\bar{c}}$ core state, 73% of the $D^{0}{\bar{D}}^{*0}$+c.c. and
12% of the $D^{+}D^{*-}$+c.c. molecular components. Therefore, within the
molecular hypothesis we will interpret the $X(3872)$ below as a
$D^{0}{\bar{D}}^{*0}$+c.c. molecule.), and iv) a hybrid state - a state in
which the $c{\bar{c}}$ charmonium core state $\chi_{c1}(2P)$ couples to the
neutral $D^{0}{\bar{D}}^{*0}$ and ${\bar{D}}^{0}D^{*0}$ molecular states [37,
73, 106–109, 112–115].

We begin here from a picture of the $X(3872)$ as a pure $c{\bar{c}}$ state.
Since the geometrical size of this state is expected to be similar to that of
the usual charmonium $J/\psi$, whose radius $\sim$ 0.2–0.3 fm (cf. [86, 87,
116] and [117, 118]), and they have the same quark content, it is natural to
assume for the $X(3872)$–nucleon absorption cross section
$\sigma_{{X(3872)}N}^{{\rm c{\bar{c}}}}$ in this picture for $X(3872)$ the
same absorption cross section as for the charmonium $J/\psi$, i.e.,
$\sigma_{{X(3872)}N}^{{\rm c{\bar{c}}}}=3.5$ mb [56].

In the second scenario, where the high-momentum (see above) $X(3872)$
resonance is treated as a compact tetraquark state with radius $r_{4q}$, its
breakup cross section $\sigma_{{X(3872)}N}^{{\rm 4{q}}}$ in this scenario can
be well approximated by a geometrical cross section $\sigma_{4q}^{\rm
geo}={\pi}r_{4q}^{2}$ [85, 86], i.e., $\sigma_{{X(3872)}N}^{{\rm
4{q}}}\approx{\pi}r_{4q}^{2}$. For $r_{4q}=0.65$ fm [85, 86], we obtain that
$\sigma_{{X(3872)}N}^{{\rm 4{q}}}=13.3$ mb (cf. [86]).

Within the ”pure” molecular interpretation of the $X(3872)$, whose particle
content in line with the above-mentioned is [27, 37, 73, 104, 108]

$|X(3872)>_{\rm
mol}=\frac{1}{\sqrt{2}}\left(|D^{0}{\bar{D}}^{*0}>+|{\bar{D}}^{0}D^{*0}>\right)$
(16)

and in which due to the closeness of the $X(3872)$ to the
$D^{0}{\bar{D}}^{*0}$ (${\bar{D}}^{0}D^{*0}$) threshold the constituents
(charmed mesons) are weakly bound and have large spatial separation in
coordinate space 999)The universal wave function for the $X(3872)$
$D^{0}{\bar{D}}^{*0}$ molecule in the coordinate representation
$\psi_{X(3872)}({\bf r})=\exp(-r/a)/(\sqrt{2{\pi}a}r)$ implies that the r.m.s.
size $r_{X}$ of this separation is $r_{X}=a/\sqrt{2}$, where the scattering
length $a=1/\sqrt{2\mu_{0}\delta_{X}}$ can be determined by knowing the
binding energy $\delta_{X}=m_{D^{0}}+m_{{\bar{D}}^{*0}}-m_{X(3872)}$ of the
molecule [27, 99, 104]. Here, $\mu_{0}$ is the $D^{0}{\bar{D}}^{*0}$ reduced
mass. Taking $\delta_{X}=0.04\pm 0.09$ MeV as input [111, 119], we find that
the charmed mesons in the $X(3872)$ have a surprisingly huge r.m.s.
separation: $r_{X}=15.9^{+\infty}_{-7.1}$ fm. The typical scale for the
relative momentum between the neutral charmed mesons [26, 27, 101],
$\sqrt{2\mu_{0}\delta_{X}}\simeq$10–16 MeV/c, is much smaller than their
average laboratory momenta $\sim$ 5 GeV/c (see above). With these, we expect
that the application (see below) of the quasi-free approximation to estimate
the $X(3872)$–nucleon absorption cross section in molecular scenario is
sufficiently well justified.), the $X(3872)$–nucleon absorption cross section
$\sigma_{{X(3872)}N}^{\rm mol}$ in this interpretation of the $X(3872)$ can be
naturally evaluated in the quasi-free approximation [120–122]. In this
approximation, the charmed mesons are taken to be on-shell, they are
considered to fly together with an average laboratory momentum of the order of
5 GeV/c each and their binding energy and mutual interactions are neglected.
In the quasi-free approximation the $X(3872)$ is absorbed when an intranuclear
nucleon (proton or neutron) interacts (elastically or inelastically) with the
$D^{0}$ or with the ${\bar{D}}^{*0}$ (with the ${\bar{D}}^{0}$ or with the
$D^{*0}$). In each of these interactions the other charmed meson is a
spectator. Then, assuming that the total cross sections of the free
${\bar{D}}^{*0}N$ and $D^{*0}N$ interactions are the same as those for the
${\bar{D}}^{0}N$ and $D^{0}N$ ones [123–126], we can evaluate the cross
section for $X(3872)$ absorption, $\sigma_{{X(3872)}N}^{\rm mol}$, as
[123–126]:

$\sigma_{{X(3872)}N}^{\rm mol}=\sigma_{{X(3872)}p}^{\rm
mol}=\sigma_{{X(3872)}n}^{\rm mol},$ (17)

where the $X(3872)p$ and $X(3872)n$ absorption cross sections,
$\sigma_{{X(3872)}p}^{\rm mol}$ and $\sigma_{{X(3872)}n}^{\rm mol}$, are given
by (cf. [127])

$\sigma_{{X(3872)}p(n)}^{\rm mol}\approx\sigma_{D^{0}p(n)}^{\rm
el}+\sigma_{D^{0}p(n)}^{\rm in}+\sigma_{{\bar{D}}^{0}p(n)}^{\rm
el}+\sigma_{{\bar{D}}^{0}p(n)}^{\rm in}+\sigma_{D^{0}p\to
D^{+}n}(\sigma_{{\bar{D}}^{0}n\to D^{-}p}).$ (18)

Here, $\sigma_{D^{0}p(n)}^{\rm el(in)}$ and $\sigma_{{\bar{D}}^{0}p(n)}^{\rm
el(in)}$ are the elastic (inelastic) cross sections of the free
$D^{0}p$($D^{0}n$) and ${\bar{D}}^{0}p$(${\bar{D}}^{0}n$) interactions,
respectively. And, $\sigma_{D^{0}p\to D^{+}n}$ and $\sigma_{{\bar{D}}^{0}n\to
D^{-}p}$ are the total cross sections of the free charge-exchange reactions
$D^{0}p\to D^{+}n$ and ${\bar{D}}^{0}n\to D^{-}p$, correspondingly. In our
calculations we adopt for them the following constants which are relevant to
the momentum regime above of 1 GeV/c of interest: $\sigma_{D^{0}p(n)}^{\rm
el}=\sigma_{{\bar{D}}^{0}p(n)}^{\rm el}=10$ mb, $\sigma_{D^{0}p(n)}^{\rm
in}=10$ mb, $\sigma_{{\bar{D}}^{0}p(n)}^{\rm in}=0$, $\sigma_{D^{0}p\to
D^{+}n}=\sigma_{{\bar{D}}^{0}n\to D^{-}p}=12$ mb [123–125]. Using these
values, we obtain that $\sigma_{{X(3872)}p}^{\rm mol}=\sigma_{{X(3872)}n}^{\rm
mol}=42$ mb and, in view of Eq. (17), $\sigma_{{X(3872)}N}^{\rm mol}=42$ mb.

In the hybrid approach, it is supposed that the $X(3872)$ wave function in the
center of the mass frame is a linear superposition of the charmonim
$c{\bar{c}}$ core state and the $D^{0}{\bar{D}}^{*0}$+c.c. hadronic molecular
state [37, 73, 106, 108, 113]:

$|X(3872)>_{\rm
hyb}=\alpha|c{\bar{c}}>+\frac{\beta}{\sqrt{2}}\left(|D^{0}{\bar{D}}^{*0}>+|{\bar{D}}^{0}D^{*0}>\right).$
(19)

Here, $\alpha^{2}$ and $\beta^{2}$ represent the probabilities to find a
charmonium and hadronic configurations, respectively, for the normalization

$\alpha^{2}+\beta^{2}=1.$ (20)

The limiting case of $\alpha^{2}=0$, $\beta^{2}=1$ corresponds to the pure
molecular interpretation of the $X(3872)$, while the case of $\alpha^{2}=1$,
$\beta^{2}=0$ refers to its pure charmonium treatment. In this paper,
following Ref. [37], we will assume that the $X(3872)$ wave function (19)
contains 50% of the genuine nonmolecular $c{\bar{c}}$ component and 50% of the
molecular $D^{0}{\bar{D}}^{*0}$+c.c. component ($\alpha^{2}=\beta^{2}=0.5$).
In addition, to extend the range of applicability of our model and to see the
sensitivity of the $X(3872)$ production cross sections from the direct
processes (1), (2) to the nonmolecular and molecular probabilities of the
$X(3872)$ in the $c{\bar{c}}$ and $D^{0}{\bar{D}}^{*0}$+c.c. channels we will
further yet adopt in calculations another additional representative option for
them, namely: 15% and 85% ($\alpha^{2}=0.15$ and $\beta^{2}=0.85$),
respectively, as was derived in a model [106]. These two options for the
nonmolecular and molecular probabilities of the $X(3872)$ cover the bulk of
theoretical and experimental information presently available in this field. It
should be pointed out that it is difficult at present to definitely conclude
what are the precise values of these probabilities. The available presently
data definitely rule out the possibility of a dominant nonmolecular component
[109]. The precise value of the molecular probability (which may be as big as
95% [109]) requires a more accurate determination of the scattering length and
effective range of the $D^{0}{\bar{D}}^{*0}$ channel, as well as the
measurement of these magnitudes for the $D^{+}D^{*-}$ channel which have not
been determined experimentally up to now [109]. Since the role of the
interference effects between the nonmolecular and molecular contributions to
the $X(3872)$–nucleon absorption cross section $\sigma_{{X(3872)}N}^{\rm hyb}$
in the hybrid picture (19) of the $X(3872)$ is expected to be insignificant,
we can represent it (to a good approximation) in the following incoherent
probability-weighted sum:

$\sigma_{{X(3872)}N}^{\rm hyb}=\alpha^{2}\sigma_{{X(3872)}N}^{\rm
c{\bar{c}}}+\beta^{2}\sigma_{{X(3872)}N}^{\rm mol}.$ (21)

In view of the above, we set $\sigma_{{X(3872)}N}^{\rm c{\bar{c}}}=3.5$ mb and
$\sigma_{{X(3872)}N}^{\rm mol}=42$ mb. With these values, the $X(3872)$
absorption cross section (21) is $\sigma_{{X(3872)}N}^{\rm hyb}=22.75$ as well
as 36.4 mb for the nonmolecular and molecular probabilities of the $X(3872)$
50% and 50% as well as 15% and 85%, respectively. We finalize here by
summarizing the results obtained above for the $X(3872)$–nucleon absorption
cross section $\sigma_{{X(3872)}N}$ in the considered scenarios for $X(3872)$:

$\sigma_{{X(3872)}N}=\left\\{\begin{array}[]{lll}3.5~{}{\rm mb}&\mbox{for pure
$c{\bar{c}}$ state},\\\ &\\\ 13.3~{}{\rm mb}&\mbox{for tetraquark (4q)
state},\\\ &\\\ 22.75~{}{\rm mb}&\mbox{for hybrid state (50\%,50\%)},\\\ &\\\
36.4~{}{\rm mb}&\mbox{for hybrid state (15\%,85\%)},\\\ &\\\ 42~{}{\rm
mb}&\mbox{for $D^{0}{\bar{D}}^{*0}$+{\it c.c.} molecule}.\end{array}\right.$
(22)

Consequently, the $X(3872)$ as a meson molecule has the largest absorption
cross section and, therefore, is expected to be more easily destroyed than its
other states in a hadronic medium. We will use these values for the quantity
$\sigma_{{X(3872)}N}$ throughout our calculations in the near-threshold energy
domain.

Ignoring the considered nuclear effects: the struck target nucleon binding and
Fermi motion, from Eq. (3) we get the following simple expression for the
total cross section $\sigma_{{\gamma}A\to{X(3872)}X}^{({\rm
dir})}(E_{\gamma})$:

$\sigma_{{\gamma}A\to{X(3872)}X}^{({\rm
dir})}(E_{\gamma})=I_{V}[A,\sigma_{{X(3872)}N}]\sigma_{{\gamma}p\to{X(3872)}p}(\sqrt{s(E_{\gamma})}),$
(23)

where the elementary cross section
$\sigma_{{\gamma}p\to{X(3872)}p}(\sqrt{s(E_{\gamma})})$ is given above by Eq.
(10). It is valid only at photon energies well above threshold 101010)The
difference between the expressions (3) and (23) is quite small (within several
percent) at incident photon energies around 13 GeV and it is $\sim$ 20% at
higher photon energies considered, as our calculations have shown. On the
other hand, it becomes substantial at far subthreshold beam energies (compare
magenta short-dashed and cyan short-dashed-dotted curves in figures 4–7 given
below).) and allows one to easily estimate here this cross section. As a
measure for the $X(3872)$ absorption cross section $\sigma_{{X(3872)}N}$ in
nuclei and, hence, for the $X(3872)$ internal structure we use the so-called
$X(3872)$ transparency ratio defined as [128–135]:

$S_{A}=\frac{\sigma_{{\gamma}A\to{X(3872)}X}(E_{\gamma})}{A~{}\sigma_{{\gamma}p\to{X(3872)}p}(\sqrt{s(E_{\gamma})})},$
(24)

i.e. the ratio of the inclusive nuclear $X(3872)$ photoproduction cross
section divided by $A$ times the same quantity on a free proton. It should be
noticed that this relative observable is more favorable compared to those
based on the absolute cross sections for the aim of getting the information on
the $X(3872)$ nuclear absorption, since it at photon energies well above
threshold is mainly sensitive to the $X(3872)N$ absorption cross section (cf.
[130]), whereas the theoretical uncertainties associated with the $X(3872)$
production mechanism substantially cancel out in it. The direct photon–induced
reaction channels (1), (2) is expected to be dominant in the $X(3872)$
production in ${\gamma}A$ reactions close to threshold 111111)Let’s say at
incident photon energies $\leq$ 13 GeV. Here, the elementary processes
${\gamma}N\to X(3872)N\pi$ and ${\gamma}N\to X(3872)N{\pi}\pi$ with one and
two pions in the final states are expected to be suppressed in $X(3872)$
production in ${\gamma}A$ reactions for kinematics of interest compared to
those of (1), (2) due to their larger production threshold energies
($\approx$12.56 and 13.28 GeV, respectively) in free ${\gamma}N$ interactions.
Moreover, since the main inelastic channel in ${\gamma}N$ collisions at these
beam energies is the multiplicity production of pions with comparatively low
energies the secondary ${\pi}N\to{X(3872)}X$ processes are energetically
suppressed at them as well. We expect that the full inclusive nuclear
$X(3872)$ photoproduction cross section
$\sigma_{{\gamma}A\to{X(3872)}X}(E_{\gamma})$, entering into Eq. (24), is
entirely exhausted by that of Eq. (3) from the direct $X(3872)$ production
channels (1), (2) not only at photon energies $E_{\gamma}\leq 13$ GeV, but
also for higher beam energies considered in the present work.). Then,
according to Eqs. (3) and (4), we have:

$S_{A}=\frac{\sigma_{{\gamma}A\to{X(3872)}X}^{({\rm
dir})}(E_{\gamma})}{A~{}\sigma_{{\gamma}p\to{X(3872)}p}(\sqrt{s(E_{\gamma})})}=\frac{I_{V}[A,\sigma_{{X(3872)}N}]}{A}\frac{\left<\sigma_{{\gamma}p\to{X(3872)}p}(E_{\gamma})\right>_{A}}{\sigma_{{\gamma}p\to{X(3872)}p}(\sqrt{s(E_{\gamma})})}.$
(25)

At photon energies around 13 GeV the difference between the cross sections
$\left<\sigma_{{\gamma}p\to{X(3872)}p}(E_{\gamma})\right>_{A}$ and
$\sigma_{{\gamma}p\to{X(3872)}p}(\sqrt{s(E_{\gamma})})$ can be ignored (cf.
footnote 10)). Then, from Eq. (25) we obtain (in a good approximation):

$S_{A}=\frac{1}{A}I_{V}[A,\sigma_{{X(3872)}N}].$ (26)

It is usually customary to normalize the transparency ratio of the meson to a
light nucleus like 12C [135–140]. With this, for $X(3872)$ we have:

$T_{A}=\frac{S_{A}}{S_{\rm
C}}=\frac{12~{}\sigma_{{\gamma}A\to{X(3872)}X}(E_{\gamma})}{A~{}\sigma_{{\gamma}{\rm
C}\to{X(3872)}X}(E_{\gamma})}.$ (27)

Here, $\sigma_{{\gamma}A\to{X(3872)}X}(E_{\gamma})$ and $\sigma_{{\gamma}{\rm
C}\to{X(3872)}X}(E_{\gamma})$ are the inclusive total cross sections for
$X(3872)$ production in ${\gamma}A$ and ${\gamma}{\rm C}$ collisions at
incident photon energy $E_{\gamma}$, respectively. In accordance with the
above, the expression (27) for the normalized to carbon transparency ratio
$T_{A}$ can be represented in the following form:

$T_{A}=\frac{12~{}\sigma_{{\gamma}A\to{X(3872)}X}^{({\rm
dir})}(E_{\gamma})}{A~{}\sigma_{{\gamma}{\rm C}\to{X(3872)}X}^{({\rm
dir})}(E_{\gamma})}=\frac{12~{}I_{V}[A,\sigma_{{X(3872)}N}]}{A~{}I_{V}[{\rm
C},\sigma_{{X(3872)}N}]}\frac{\left<\sigma_{{\gamma}p\to{X(3872)}p}(E_{\gamma})\right>_{A}}{\left<\sigma_{{\gamma}p\to{X(3872)}p}(E_{\gamma})\right>_{\rm
C}}.$ (28)

Ignoring the difference between the cross sections
$\left<\sigma_{{\gamma}p\to{X(3872)}p}(E_{\gamma})\right>_{A}$ and
$\left<\sigma_{{\gamma}p\to{X(3872)}p}(E_{\gamma})\right>_{\rm C}$, from (28)
we approximately get 121212)It is easily seen that, according to the equations
(23) and (28), the expression (29) describes the transparency ratio $T_{A}$
also in the case when direct ${\gamma}N\to X(3872)N$ processes proceed on a
free target nucleon being at rest.) :

$T_{A}\approx\frac{12~{}I_{V}[A,\sigma_{{X(3872)}N}]}{A~{}I_{V}[{\rm
C},\sigma_{{X(3872)}N}]}.$ (29)

As is easy to see, the integral (4) for the quantity
$I_{V}[A,\sigma_{{X(3872)}N}]$ can be transformed to a simpler expression:

$I_{V}[A,\sigma_{{X(3872)}N}]=\frac{\pi}{\sigma_{{X(3872)}N}}\int\limits_{0}^{R^{2}}dr_{\bot}^{2}\left(1-e^{-\sigma_{{X(3872)}N}\int\limits_{-\sqrt{R^{2}-r_{\bot}^{2}}}^{\sqrt{R^{2}-r_{\bot}^{2}}}\rho(\sqrt{r_{\bot}^{2}+x^{2}})dx}\right),$
(30)

which in the case of a uniform nucleon densities for a nucleus of a radius
$R=r_{0}A^{1/3}$ with a sharp boundary is reduced to even more simple form:

$I_{V}[A,\sigma_{{X(3872)}N}]=\frac{3A}{2a_{1}}\left\\{1-\frac{2}{a_{1}^{2}}[1-(1+a_{1})e^{-a_{1}}]\right\\},\,\,\,\,a_{1}=3A\sigma_{X(3872)N}/2{\pi}R^{2}.$
(31)

The simple formulas (23) and (31) allow one to easily estimate the total cross
section $\sigma_{{\gamma}A\to{X(3872)}X}^{({\rm dir})}(E_{\gamma})$ at above
threshold energies.

Before closing this subsection, we discuss now the momentum-dependent
differential cross section for $X(3872)$ production from the direct processes
(1) and (2) in ${\gamma}A$ reactions. At the incident photon energies of
interest the $X(3872)$ mesons are produced at very small laboratory polar
angles 131313)Thus, for example, the maximum angle of $X(3872)$ meson
production off a free proton at rest in reaction (1) is about 3.4∘ at photon
energy of 13 GeV.) . Therefore, we will calculate their momentum distribution
from considered target nuclei for the laboratory solid angle
${\Delta}{\bf\Omega}_{X(3872)}$ = $0^{\circ}\leq\theta_{X(3872)}\leq
10^{\circ}$, and $0\leq\varphi_{X(3872)}\leq 2{\pi}$. Then, accounting for the
results presented both in Ref. [141] and above by Eqs. (3)–(8), we can get the
following expression for this distribution:

$\frac{d\sigma_{{\gamma}A\to{X(3872)}X}^{({\rm
dir})}(p_{\gamma},p_{X(3872)})}{dp_{X(3872)}}=2{\pi}I_{V}[A,\sigma_{{X(3872)}N}]\int\limits_{\cos
10^{\circ}}^{1}d\cos{{\theta_{X(3872)}}}\left<\frac{d\sigma_{{\gamma}p\to{X(3872)}{p}}(p_{\gamma},p_{X(3872)},\theta_{X(3872)})}{dp_{X(3872)}d{\bf\Omega}_{X(3872)}}\right>_{A},$
(32)

where
$\left<\frac{d\sigma_{{\gamma}p\to{X(3872)}p}(p_{\gamma},p_{X(3872)},\theta_{X(3872)})}{dp_{X(3872)}d{\bf\Omega}_{X(3872)}}\right>_{A}$
is the off-shell inclusive differential cross section for the production of
$X(3872)$ mesons with momentum ${\bf p}_{X(3872)}$ in the channel
${\gamma}p\to{X(3872)}p$, averaged over the Fermi motion and binding energy of
the protons in the nucleus. It can be expressed by Eqs. (28), (31)–(39) from
Ref. [141], in which one needs to make the substitution: $\Upsilon(1S)\to
X(3872)$. For better readability of this paper, we do not give these
expressions here. In order to calculate the c.m. $X(3872)$ angular
distribution in process (1) (cf. Eq. (34) from Ref. [141]) one should know its
exponential $t$-slope parameter $b_{X(3872)}$ in the near-threshold energy
region. Since the $X(3872)$ angular distribution from the reaction
${\gamma}p\to{X(3872)}p$ is experimentally unknown and because of the
similarity of the $X(3872)$ production rate with the production rate of the
another radially excited $\psi(2S)$ meson in hadronic collisions (see above),
we will assume that the slope parameter $b_{X(3872)}$ is the same as an
exponential $t$-slope $b_{\psi(2S)}$ of the differential cross section of the
reaction ${\gamma}p\to{\psi(2S)}p$ in the c.m. system near the threshold
141414)The threshold photon energy for ${\psi(2S)}p$ production on a free
proton being at rest is 10.93 GeV.) . The slope for 2S-radially excited heavy
charmonium elastic photoproduction was found [142, 143] to have a smaller
value than for $J/\psi$:

$b_{\psi(2S)}(s(E_{\gamma}))=b_{J/\psi}(s(E_{\gamma}))-\Delta_{b}(s(E_{\gamma})).$
(33)

For the c.m. free space energy behavior of the diffraction slope
$b_{J/\psi}(s(E_{\gamma}))$ we use the standard Regge form [142, 143]:

$b_{J/\psi}(s(E_{\gamma}))=b_{0}+2{\alpha}^{\prime}(0){\rm
ln}\left(\frac{s(E_{\gamma})}{s_{0}}\right),$ (34)

where the parameters ${\alpha}^{\prime}(0)=0.171~{}{\rm GeV}^{-2}$ and
$b_{0}=1.54~{}{\rm GeV}^{-2}$ were fitted in [142] to high-energy data on
$J/\psi$ photoproduction with $s_{0}=1$ GeV2. It is worth noting that the
extrapolation of the simple fit (34) of the high-energy data to the
${J/\psi}p$ threshold energies (to the energies around 10 GeV) is also
compatible with available here data for an exponential $J/\psi$ $t$-slope
$b_{J/\psi}$ [141]. In line with Eq. (34), for $E_{\gamma}=13$ GeV we have
that $b_{J/\psi}\approx 2.7$ GeV-2. The factor $\Delta_{b}(s(E_{\gamma}))$,
entering into the Eq. (33), was parameterized in Ref. [142] as:

$\Delta_{b}^{T}(s(E_{\gamma}))=0.60-0.04{\rm
ln}\left(\frac{s(E_{\gamma})}{s_{0}}\right)~{}{\rm
GeV}^{-2},\,\,\,\,\,\,\Delta_{b}^{L}(s(E_{\gamma}))=1.53-0.12{\rm
ln}\left(\frac{s(E_{\gamma})}{s_{0}}\right)~{}{\rm GeV}^{-2}$ (35)

for photoproduction of $T$ and $L$ polarized $\psi(2S)$ mesons. Numerically,
$\Delta_{b}^{T/L}$ are obtained as:

$\Delta_{b}^{T}=0.47~{}{\rm
GeV}^{-2},\,\,\,\,\,\,\,\,\Delta_{b}^{L}=1.14~{}{\rm GeV}^{-2}$ (36)

for photon energy $E_{\gamma}=13$ GeV. A weighted average of the two values
(36) is about 0.7 GeV-2, i.e., the $t$-slope parameter $b_{\psi(2S)}\approx
2.0$ GeV-2 for incident photon beam energy of 13 GeV. And, thus, in line with
the aforementioned, the value of the $X(3872)$ slope parameter $b_{X(3872)}$
is $b_{X(3872)}\approx 2.0$ GeV-2 at this energy. We will employ it in our
subsequent differential cross-section calculations.

Figure 4: (Color online.) Excitation function for production of $X(3872)$
mesons off 12C from primary ${\gamma}N\to{X(3872)}N$ reactions proceeding on
an off-shell target nucleons and on a free ones being at rest. The curves are
calculations in the scenarios, in which the $X(3872)$ is treated as a purely
$c{\bar{c}}$ charmonium state, as a tetraquark state: a compact four quark
(4$q$) state, as a purely molecular state: a weakly coupled
$D^{0}{\bar{D}}^{*0}$ molecule, or as a hybrid state: a mixture of the
$c{\bar{c}}$ and $D^{0}{\bar{D}}^{*0}$ states in which there are,
respectively, 50% of the $c{\bar{c}}$ component and 50% molecular component as
well as 15% and 85% of the nonmolecular and molecular states. The arrow
indicates the threshold energy for $X(3872)$ photoproduction on a free
nucleon. Figure 5: (Color online.) The same as in Fig. 4, but for the 184W
target nucleus. Figure 6: (Color online.) Ratio between the $X(3872)$
production cross sections on 12C, shown in Fig. 4, and the cross section,
calculated in the molecular scenario of $X(3872)$ for an off-shell target
nucleons, as a function of photon energy. The arrow indicates the threshold
for the reaction ${\gamma}N\to{X(3872)}N$ proceeding on a free target nucleon.
Figure 7: (Color online.) Ratio between the $X(3872)$ production cross
sections on 184W, shown in Fig. 5, and the cross section, calculated in the
molecular scenario of $X(3872)$ for an off-shell target nucleons, as a
function of photon energy. The arrow indicates the threshold for the reaction
${\gamma}N\to{X(3872)}N$ proceeding on a free target nucleon. Figure 8: (Color
online.) Transparency ratio $S_{A}$ for the $X(3872)$ mesons from primary
${\gamma}N\to{X(3872)}N$ reactions proceeding on a free target nucleons being
at rest as a function of the nuclear mass number $A$ in the considered
scenarios for the $X(3872)$ internal structure. The lines are to guide the
eyes. Figure 9: (Color online.) Transparency ratio $T_{A}$ for the $X(3872)$
mesons from primary ${\gamma}N\to{X(3872)}N$ reactions proceeding on a free
target nucleons being at rest as a function of the nuclear mass number $A$ in
the considered scenarios for the $X(3872)$ internal structure. The lines are
to guide the eyes. Figure 10: (Color online.) Transparency ratio $T_{A}$ for
the $X(3872)$ mesons from primary ${\gamma}N\to{X(3872)}N$ reactions
proceeding on an off-shell and free target nucleons as a function of the
incident photon energy for combination 184W/12C in the considered scenarios
for the $X(3872)$ internal structure. The arrow indicates the threshold energy
for $X(3872)$ photoproduction on a free target nucleon at rest. Figure 11:
(Color online.) The direct momentum distribution of $X(3872)$ mesons, produced
in the reaction $\gamma$12C $\to X(3872)X$ in the laboratory polar angular
range of 0∘–10∘ and calculated in line with Eq. (32) at initial photon energy
of 13 GeV in the laboratory system. The curves are calculations in the
scenarios, in which the $X(3872)$ is treated as a purely $c{\bar{c}}$
charmonium state, as a tetraquark state: a compact four quark (4$q$) state, as
a purely molecular state: a weakly coupled $D^{0}{\bar{D}}^{*0}$ molecule, or
as a hybrid state: a mixture of the $c{\bar{c}}$ and $D^{0}{\bar{D}}^{*0}$
states in which there are, respectively, 50% of the $c{\bar{c}}$ component and
50% molecular component as well as 15% and 85% of the nonmolecular and
molecular states. Figure 12: (Color online.) The same as in Fig. 11, but for
the 184W target nucleus.

## 3\. Results and discussion

At first, we consider the excitation functions for production of $X(3872)$
mesons off 12C and 184W target nuclei. They were calculated on the basis of
equation (3) for five basic adopted options for the $X(3872)$ absorption cross
section $\sigma_{{X(3872)}N}$ in nuclei (cf. Eq. (22)) as well as in line with
formula (23) for the free target nucleon being at rest with the value of
$\sigma_{X(3872)N}=42$ mb corresponding to the molecular scenario for
$X(3872)$, and are given in figures 4 and 5, respectively. For heavy tungsten
nucleus for the neutron density $\rho_{n}(r)$ the ’skin’ form of Ref. [65]
(see also above) was used in the calculations 151515)These calculations show
that the replacement of the ’skin’ form of the neutron density in the case of
the 184W target nucleus by the ’halo’ one made little difference: the
calculated $X(3872)$ total production cross sections on this nucleus are
increased only by a small fractions about 1–2%. An analogous small fractions
are available in the cases of another medium-mass and heavy target nuclei
considered in the present work. Therefore, we will further report for 184W and
these nuclei the numerical results obtained only for the ’skin’ form of their
neutron densities.) . It is seen that the difference between calculations with
and without accounting for the target nucleon Fermi motion (between magenta
short-dashed and cyan short-dashed-dotted curves) is indeed sufficiently small
at above threshold photon energies $\sim$ 12.5–16 GeV and especially at
energies around 13 GeV, while at lower incident energies its influence on
$X(3872)$ yield is quite essential. It is seen yet from these figures that the
$X(3872)$ meson total cross sections reveal a certain sensitivity to the
$X(3872)$–nucleon absorption cross section $\sigma_{{X(3872)}N}$ and, hence,
to the $X(3872)$ inner structure, for both target nuclei and for photon
energies both below and above threshold energy of 11.86 GeV. The absolute
values of the total cross sections have at above threshold photon energies of
$\sim$ 12.5–16 GeV a well measurable strengths $\sim$ 30–200 and 200–3000 nb
for carbon and tungsten target nuclei, correspondingly. Here there are a well
separated and experimentally distinguishable differences (especially for the
heavy target nucleus 184W) between the considered choices:
$\sigma_{{X(3872)}N}=3.5$ mb and $\sigma_{{X(3872)}N}=13.3$ mb,
$\sigma_{{X(3872)}N}=13.3$ mb and $\sigma_{{X(3872)}N}=22.75$ mb,
$\sigma_{{X(3872)}N}=22.75$ mb and $\sigma_{{X(3872)}N}=36.4$ mb,
$\sigma_{{X(3872)}N}=36.4$ mb and $\sigma_{{X(3872)}N}=42$ mb or between the
considered scenarios: pure $c{\bar{c}}$ state and tetraquark state, tetraquark
state and hybrid state with the nonmolecular and molecular probabilities of
50% and 50%, hybrid state with the nonmolecular and molecular probabilities of
50% and 50% and hybrid state with the nonmolecular and molecular probabilities
of 15% and 85%, hybrid state with the nonmolecular and molecular probabilities
of 15% and 85% and pure molecular state for the $X(3872)$–nucleon absorption
cross section or for the $X(3872)$ internal structure. They are $\sim$ 21, 17,
22, 8% and 66, 42, 42, 12% in the cases of 12C and 184W target nuclei,
respectively. Therefore, one might expect to measure both these strengths and
these differences in the future high-precision experiments at the CEBAF
facility. To further motivate the conducting of such experiments at this
facility, it is highly desirable to evaluate the $X(3872)$ production rates in
the ${\gamma}^{12}$C and ${\gamma}^{184}$W reactions. For this purpose, we
translate the $X(3872)$ photoproduction total cross section predictions,
reported above, into the expected yields of the $X(3872)$ signals from the
reactions ${\gamma}^{12}{\rm C}(^{184}{\rm W})\to X(3872)X$,
$X(3872)\to{J/\psi}\pi^{+}\pi^{-}$, $J/\psi\to l^{+}l^{-}$ 161616)The lepton
pairs $l^{+}l^{-}$ denote both $\mu^{+}\mu^{-}$ and $e^{+}e^{-}$.) . We assume
that a conservative integrated luminosity of about 100 pb-1 could be reached
for a year of data taking 171717)Thus, for example, one year of running of the
GlueX (2023) experiment [60] devoted to the study of the
${\gamma}p\to{J/\psi}p$ reaction in the full near-threshold kinematic region
at the CEBAF facility delivered a total accumulated luminosity of about 320
pb-1.) . For the $X(3872)$ yield (for the total number of the $X(3872)$
events) estimates in a one-year run, one needs to multiply the above
luminosity by the $X(3872)$ production total cross sections of 30–200 and
200–3000 nb on the carbon and tungsten target nuclei, respectively, as well as
by the detection efficiency and by the appropriate branching ratios
$Br[X(3872)\to{J/\psi}\pi^{+}\pi^{-}]\approx$4.1% [90] and
$Br[{J/\psi}\to\l^{+}l^{-}]\approx$12%. Even with a relatively low (and
realistic) 10% detection efficiency, we estimate about of 1476–9840 and
9840–147600 events per year for the $X(3872)$ signal in the cases of the 12C
and 184W target nuclei, respectively. We see that a considerable amount of
$X(3872)$ could be observed 181818)Since the $X(3872)$ photoproduction total
cross sections of the elementary reaction ${\gamma}p\to X(3872)p$ have an
absolute values $\sim$ 10–20 nb at photon energies well above threshold of
interest [47, 58], the production rate of the $X(3872)$ in this reaction is
expected to be also large. Accounting for these $X(3872)$ photoproduction
total cross sections and the estimates of the $X(3872)$ yields given above, we
can evaluate the numbers of events expected in the measurements of the
$X(3872)$ mesons in the considered energy range for this reaction as about of
500–1000 with the one-year run. These numbers are commensurate with those of
the $J/\psi$ charmonium, measured in the reaction ${\gamma}p\to{J/\psi}p$ in
the GlueX(2023) experiment [60] at JLab. Thus, the values of the total cross
section of this reaction, measured in this experiment, are about of 0.1–1 nb
near the threshold. Using these values as well as values of the integrated
luminosity of 320 pb-1, of the detection efficiency of 10% and of the
branching ratio $Br[J/\psi\to e^{+}e^{-}]$ of 6%, we can estimate the measured
numbers of the $J/\psi$ events in this experiment as 180–1800 (cf. [60]).
Therefore, it will be possible to measure the $X(3872)$ state even through the
scan of its total photoproduction cross section on a proton target in the
near-threshold energy region, at least, in the future CEBAF experiments.) .
Hence, the measurements of the photonuclear $X(3872)$ meson production might
help to decipher its internal structure.

To see more clearly the sensitivity of the total cross sections, presented in
Figs. 4 and 5, to the $X(3872)$–nucleon absorption cross section, we show in
Figs. 6 and 7 the energy dependences of the ratios of these cross sections to
the cross section, calculated in the molecular scenario for $X(3872)$ for an
off-shell target nucleons, on a linear scale for 12C and 184W target nuclei,
respectively. It is nicely seen from these figures that there are indeed
experimentally distinguishable differences between the results corresponding
to the considered options for the $X(3872)$ meson–nucleon absorption cross
section or to the considered configurations for its inner structure
practically for both target nuclei and for both subthreshold and above
threshold incident photon energies. This means that the structure of $X(3872)$
mesons could be investigated at the CEBAF through the energy dependence of
their absolute (and relative) production cross sections in inclusive
${\gamma}A$ reactions in the near-threshold energy region. Figures 6 and 7
also show convincingly that the difference between the absolute $X(3872)$
production cross sections, calculated in the pure molecular scenario for
$X(3872)$ with allowance for the influence of the binding of target nucleons
and their Fermi motion on the direct processes (1), (2) and without it (cf.
Eqs. (3) and (23)) is quite inessential at initial photon energies around
energy of $E_{\gamma}=13$ GeV. This justifies the use of approximate formulas
(26) and (29) for calculation of the $X(3872)$ transparency ratios $S_{A}$ and
$T_{A}$, respectively, at least at this beam energy.

Figures 8 and 9 show the A–dependences of the transparency ratios $S_{A}$ and
$T_{A}$ from the primary ${\gamma}N\to{X(3872)}N$ reaction channels in
${\gamma}A$ ($A=$12C, 27Al, 40Ca, 63Cu, 93Nb, 112Sn, 184W, 208Pb, and 238U)
collisions, calculated for the incident photon energy of around 13 GeV on the
basis of equations (26) and (29), correspondingly, and for the same values of
the genuine $X(3872)$–nucleon absorption cross section $\sigma_{X(3872)N}$ as
those given above by Eq. (22). One can see that the results for the
transparency ratio $S_{A}$ depend more strongly on this cross section or on
the adopted configuration for $X(3872)$ than those for the quantity $T_{A}$.
Thus, for the first observable, we observe the experimentally separated
differences $\sim$ 35, 25, 30, 10% between the results, obtained assuming for
$X(3872)$ a pure charmonium and tetraquark scenarios, a tetraquark and
charmonium-molecule mixture with the charmonium and molecular probabilities of
50% and 50% pictures, a charmonium-molecule mixture with the charmonium and
molecular probabilities of 50% and 50% and a charmonium-molecule mixture with
the charmonium and molecular probabilities of 15% and 85% configurations, a
charmonium-molecule mixture with the charmonium and molecular probabilities of
15% and 85% and pure molecular pictures for comparatively ”light” nuclei
(27Al, 40Ca). For the medium-mass (93Nb, 112Sn) and heavy (184W, 238U) target
nuclei these differences are larger and are about 50, 35, 37, 11% and 60, 41,
42, 12%, respectively. In the case of the quantity $T_{A}$ the analogous
differences are somewhat smaller and they are about 10, 7, 6, 2%, 25, 15, 12,
3% and 40, 20, 17, 4%, respectively, in the cases of ”light”, medium-mass and
heavy target nuclei indicated above. We see yet that the highest sensitivity
of both considered observables $S_{A}$ and $T_{A}$ to the $X(3872)$ structure
is observed, as is expected, for heavy target nuclei. Therefore, one can
conclude that the observation, at least, of the A dependence of the
transparency ratio $S_{A}$, especially for large mass numbers $A$, in the
future CEBAF high-precision photoproduction experiments 191919)In which the
data should be collected with experimental accuracy better than 10%.) offers
the possibility to discriminate between all considered internal configurations
of the $X(3872)$ exotic state. On the other hand, the future precise $X(3872)$
photoproduction data on the A dependence of the transparency ratio $T_{A}$ in
the range of large A, obtained in such experiments, could also additionally
help to distinguish, at least, between a pure charmonium, tetraquark,
charmonium-molecule mixture with the charmonium and molecular probabilities
$\sim$ 50% and 50% and a pure molecular configurations of the $X(3872)$
resonance 202020)It should be noticed that such relative observable is
favorable for the aim of getting the information on the $X(3872)$ internal
structure from the experimental point of view, since it allows for a reduction
of systematic errors due to the cancelation of the efficiency corrections.) .

In addition, we have investigated the opportunity of discriminating between
the proposed configurations for the $X(3872)$ meson from the measurements of
the incident photon energy dependence of the transparency ratio $T_{A}$. Fig.
10 shows such dependence for the 184W/12C combination. It was calculated on
the basis of Eq. (28) for five employed options for the $X(3872)$ meson inner
structure and for the off-shell intranuclear nucleons as well as in line with
the simple formula (29) for its pure molecular picture and for the free target
nucleons being at rest. It can be seen from this figure that there is a
definite sensitivity of the transparency ratio $T_{A}$ to the considered
variations in the $X(3872)$ meson–nucleon absorption cross section or, which
is equivalent, to its proposed configurations at all studied photon energies.
This sensitivity is similar to that available in Fig. 9 for heavy target
nuclei. This means that this relative observable can also be useful to help
determine the $X(3872)$ inner structure. By looking at this figure, we see yet
that the simple expression (29) describes quite well the quantity $T_{A}$ at
above threshold photon energies, where it exhibits a practically flat
behavior. But in the far subthreshold region ($E_{\gamma}\sim$ 8–10 GeV) this
quantity increases as the photon-beam energy decreases. This can be explained
by the fact that the averaged over the Fermi motion and binding energy of the
nucleons in the nucleus $X(3872)$ production cross sections calculated for the
carbon nucleus decrease faster than the respective cross sections for the
tungsten nucleus as the photon energy becomes lower. This behavior of the
transparency ratio $T_{A}$ can also be used for discriminating between
possible configurations of the $X(3872)$ state.

Finally, the absolute $X(3872)$ meson momentum distributions from the direct
(1), (2) $X(3872)$ production processes in $\gamma$12C and $\gamma$184W
interactions, calculated on the basis of Eq. (32) for laboratory polar angles
of 0∘–10∘ and for incident photon energy of 13 GeV, are shown, respectively,
in Figs. 11 and 12. These momentum distributions were obtained for five
adopted values of the $X(3872)$–nucleon absorption cross section (cf. Eq.
(22)) in the considered scenarios for $X(3872)$. The absolute values of the
differential cross sections have a well measurable strength $\sim$ 10–103
nb/(GeV/c) in the high-momentum region of 11–12 GeV/c. They also show a rather
sizeable and experimentally distinguishable variations, especially for the
heavy target nucleus 184W 212121)Which are the same as those shown,
respectively, in Figs. 4 and 5.) , upon going over from the $X(3872)$–nucleon
absorption cross section value of 3.5 to 42.0 mb. This behavior of the
differential cross sections can also be used to unveil the $X(3872)$ structure
from comparison the present model calculations with future experimental data.

Taking into account the above considerations, we come to the conclusion that
such observables as the absolute total and differential cross sections for
production of $X(3872)$ mesons from ${\gamma}A$ reactions as well as their
relative (transparency ratios) yields can be useful at near-threshold photon
beam energies to help determine the genuine $X(3872)$ internal structure.

## 4\. Epilogue

In this paper we have investigated the production of $X(3872)$ mesons in
photon-induced nuclear reactions near the threshold on the basis of an
spectral function approach, which accounts for direct photon-nucleon $X(3872)$
production processes as well as five different scenarios for their internal
structure. We have calculated the absolute and relative excitation functions
for the production of $X(3872)$ mesons off 12C and 184W target nuclei at near-
threshold incident photon energies of 8–16 GeV, the absolute differential
cross sections for their production off these target nuclei at laboratory
angles of 0∘–10∘ and for incident photon energy of 13 GeV as well as the A
dependences of the relative (transparency ratios) cross sections for $X(3872)$
production from ${\gamma}A$ collisions at photon energies around 13 GeV within
the adopted scenarios for the $X(3872)$ meson internal structure. We have
shown that the absolute and relative observables considered reveal distinct
sensitivity to these scenarios, which implies that they may be an important
tool to get valuable information on its inner configuration. The measurements
of these observables could be performed in the current and forthcoming
experiments at the CEBAF facility [40–44], electron-ion colliders EIC [144]
and EicC [145].

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