# The $X(3872)$ to $\psi(2S)$ yield ratio in heavy-ion collisions

L. M. Abreu1,2<EMAIL_ADDRESS>F. S. Navarra2<EMAIL_ADDRESS>H. P. L.
Vieira1<EMAIL_ADDRESS>1Instituto de Física, Universidade Federal da
Bahia, Campus Universitário de Ondina, 40170-115, Bahia, Brazil  2Instituto de
Física, Universidade de São Paulo, Rua do Matão, 1371, CEP 05508-090, São
Paulo, SP, Brazil

###### Abstract

In this work we evaluate the $X(3872)$ to $\psi(2S)$ yield ratio
($N_{X}/N_{\psi(2S)}$) in Pb Pb collisions, taking into account the
interactions of the $\psi(2S)$ and $X(3872)$ states with light mesons in the
hadron gas formed at the late stages of these collisions. We employ an
effective Lagrangian approach to estimate the thermally-averaged cross
sections for the production and absorption of the $\psi(2S)$ and use them in
the rate equation to determine the time evolution of $N_{\psi(2S)}$. The
multiplicity of these states at the end of mixed phase is obtained from the
coalescence model. The multiplicity of $X(3872)$, treated as a bound state of
$(D\bar{D}^{*}+c.c.)$ and also as a compact tetraquark, was already calculated
in previous works. Knowing these yields, we derive predictions for the ratio
($N_{X}/N_{\psi(2S)}$) as a function of the centrality, of the center-of-mass
energy and of the charged hadron multiplicity measured at midrapidity
$[dN_{ch}/d\eta\,(\eta<0.5)]$. Finally, we make predictions for this ratio in
Pb Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV to be measured by the ALICE
Collaboration in the Run 3.

##  I Introduction

Among the new hadrons observed in the last two decades Workman:2022ynf ,
several have properties incompatible with the quark model predictions and can
be classified as unconventional states. For recent reviews see yuan22 ; nora ;
zhu . Their properties remain subject of an intense debate and this makes the
exotic quarkonium spectroscopy a hot topic of research. The most relevant
question is: which is their structure? They can be weakly bound hadron
molecules, compact multiquark states, cusps generated from kinematical
singularities, excited conventional hadrons, glueballs, hybrids, etc., or even
a superposition of different configurations! So far there is no compelling
answer to this question.

One emblematic example is the first-observed and most famous exotic state, the
$X(3872)$ Workman:2022ynf ; Belle:2003nnu . Its intrinsic nature is still
matter of controversy. Two configurations are the most explored in the
literature yuan22 ; nora ; zhu : the shallow bound state of open charm mesons
$(D\bar{D}^{*}+c.c.)$ and the $c\bar{c}q\bar{q}$ compact tetraquark. The
ultimate goal of the present work is to contribute to the determination of the
$X(3872)$ structure.

A new era in the investigation of exotic charmonium states has started with
the first observation of $X(3872)$ in relativistic heavy ion collisions
reported recently by the CMS Collaboration CMS:2021znk . The data were
collected in lead-lead (Pb Pb) collisions at a center-of-mass energy
$\sqrt{s}=5.02$ TeV per nucleon pair, using the decay chain $X(3872)\to
J/\psi\pi^{+}\pi^{-}\to\mu^{+}\mu^{-}\pi^{+}\pi^{-}$. The rapidity and
transverse momentum intervals considered were $|y|<1.6$ and $15<p_{T}<50$ GeV.
The significance of the inclusive $X(3872)$ signal was $4.2\,\sigma$.
Interestingly, the prompt $X(3872)$ to $\psi(2S)$ yield ratio was found to be:

$\displaystyle\mathcal{R}$ $\displaystyle=$
$\displaystyle\frac{N_{X(3872)}}{N_{\psi(2S)}}$ (1) $\displaystyle=$
$\displaystyle 1.08\pm 0.49(\mathrm{stat})\pm 0.52(\mathrm{syst}).$

This central value is about one order of magnitude higher than the one
observed in $pp$ collisions lhcb1 , which is close to $0.09$. Additionally,
the LHCb Collaboration reported the observation of $X(3872)$ in $p$ Pb
collisions at both forward and backward rapidity and at $\sqrt{s}=8.16$ TeV
per nucleon pair LHCb:2022ixc . The decay chain was the same as the one
studied by the CMS Collaboration. The transverse momentum interval considered
was $p_{T}>5$ GeV, and the rapidity intervals were $1.5<y<4$ for $p$ Pb
(forward configuration) and $-5<y<-2$ for Pb $p$ (backward configuration). The
resulting ratios of the $X(3872)$ to $\psi(2S)$ multiplicities were $0.27\pm
0.08\pm 0.05$ in $p$ Pb and $0.36\pm 0.15\pm 0.11$ in Pb $p$. Both are bigger
than the one seen in $pp$ collisions but smaller than the one observed in Pb
Pb collisions.

At first sight, the comparison between the ratios measured in pp and Pb Pb
collisions shows a growth that we might be tempted to interpret as some medium
effect. However the $p_{T}$ values measured by the CMS are so large that it is
unlikely that the particles carrying these momenta have a thermal and/or
hydrodynamical origin. Rather, they are produced perturbatively at the very
beginning of the collision and do not experience any thermalization or
interactions with a medium.

On the theoretical side, some recent works have attempted to describe the
data. In Ref. Yun:2022evm  the coalescence model has been used to estimate
$N_{X}$ and the statistical hadronization model to obtain $N_{\Psi(2S)}$. The
obtained ratio has a central value of 0.806 for the $X(3872)$ in the molecular
configuration and 0.204 for the tetraquark configuration. In Ref. Yun:2022evm
the interaction of $X$ (and also of the $\psi(2S)$) with the light mesons in
the hadron gas was not considered. Here we try to estimate the effect of these
interactions on the ratio $\mathcal{R}$.

At the beginning of a heavy-ion collision quark-gluon plasma (QGP) is formed.
It expands, cools down and hadronizes into a hot hadron gas. The gas lives for
about $10$ fm and freezes out generating the observed particles. Conventional
and exotic hadrons formed in the end of the mixed phase can interact with the
(mostly light) particles in the hadron gas and their multiplicities may
experience modifications due to production and absorption processes, as was
pointed out in previous works Houng:2013 ; Torres:2014 ; abreu16 ;
Abreu:2018mnc ; Abreu:2021 ; Abreu:2022EPJC ; QCDSR:21 ; Abreu:2022jmi ;
Navarra:2022 ; Abreu:2023aqy ; Abreu:2023awj . The case of the $X(3872)$ has
been studied Houng:2013 ; Torres:2014 ; abreu16 ; QCDSR:21  and its final
multiplicity will depend on the interaction cross sections, which, in turn,
depend on the spatial configuration of the quarks. Meson molecules are larger,
and therefore have greater cross sections and stronger interaction with the
hadronic medium than compact tetraquarks. In order to have a more complete
description of the process these interactions should be taken into account.

In this work we evaluate the ratio $\mathcal{R}$ taking into account the
interactions of the $\psi(2S)$ and of the $X(3872)$ states with the hadron gas
formed in heavy ion collisions. We will make use of effective Lagrangians to
estimate the thermally averaged cross sections for $X(3872)$ and $\psi(2S)$
production and absorption and employ them in the rate equation to determine
the time evolution of the ratio $\mathcal{R}$. We will use the coalescence
model to compute the multiplicity of these states at the end of mixed phase.
The $X(3872)$ will be treated as a bound state of $(D\bar{D}^{*}+c.c.)$ and
also as a compact tetraquark. Finally, we will make predictions for
$\mathcal{R}$ to be measured by the ALICE Collaboration in the Run 3.

##  II Effective formalism

In what follows we describe the effective formalism used to evaluate the
interactions of the $X(3872)$ and $\psi(2S)$ states with the surrounding
hadronic medium. In particular, we consider the medium constituted of the
lightest and most abundant pseudoscalar and vector mesons, i.e. the pions and
$\rho$ mesons. The reactions involving the $X(3872)$ have already been studied
in previous works, and for the sake of conciseness we will not reproduce them
here. We refer the reader to Refs. Houng:2013 ; Torres:2014 ; abreu16 ;
QCDSR:21  for a detailed discussion.

$\psi(p_{1})$$\bar{D}(p_{3})$$\psi(p_{1})$$\bar{D}(p_{3})$$\pi(p_{2})$$D(p_{4})$$\pi(p_{2})$$D(p_{4})$$D^{*}$$D^{*}$$\psi(p_{1})$$\bar{D}(p_{3})$$\psi(p_{1})$$\bar{D}(p_{3})$$\pi(p_{2})$$D^{*}(p_{4})$$\pi(p_{2})$$D^{*}(p_{4})$$D$$D^{*}$$\psi(p_{1})$$\bar{D}(p_{3})$$\psi(p_{1})$$\bar{D}^{*}(p_{3})$$\pi(p_{2})$$D^{*}(p_{4})$$\pi(p_{2})$$D^{*}(p_{4})$$D^{*}$$D$$\psi(p_{1})$$\bar{D}^{*}(p_{3})$$\psi(p_{1})$$\bar{D}^{*}(p_{3})$$\pi(p_{2})$$D^{*}(p_{4})$$\pi(p_{2})$$D^{*}(p_{4})$$D$$D^{*}$$\psi(p_{1})$$\bar{D}^{*}(p_{3})$$\pi(p_{2})$$D^{*}(p_{4})$$D^{*}$

Figure 1: Born diagrams for the processes $\psi(2S)\pi\rightarrow\bar{D}D$
[(a) and (b)], $\psi(2S)\pi\rightarrow\bar{D}^{*}D$ [(c) - (e)], and
$\psi(2S)\pi\rightarrow\bar{D}^{*}D^{*}$ [(f) - (i)]. Here t The particle
charges are not specified. We denote $\psi(2S)\equiv\psi$.

$\psi(p_{1})$$\bar{D}(p_{3})$$\psi(p_{1})$$\bar{D}(p_{3})$$\rho(p_{2})$$D(p_{4})$$\rho(p_{2})$$D(p_{4})$$D$$D$$\psi(p_{1})$$\bar{D}(p_{3})$$\psi(p_{1})$$\bar{D}(p_{3})$$\rho(p_{2})$$D(p_{4})$$\rho(p_{2})$$D(p_{4})$$D^{*}$$D^{*}$$\psi(p_{1})$$\bar{D}(p_{3})$$\psi(p_{1})$$\bar{D}(p_{3})$$\rho(p_{2})$$D^{*}(p_{4})$$\rho(p_{2})$$D^{*}(p_{4})$$D$$D$$\psi(p_{1})$$\bar{D}(p_{3})$$\psi(p_{1})$$\bar{D}(p_{3})$$\rho(p_{2})$$D^{*}(p_{4})$$\rho(p_{2})$$D^{*}(p_{4})$$D^{*}$$D^{*}$$\psi(p_{1})$$\bar{D}^{*}(p_{3})$$\psi(p_{1})$$\bar{D}^{*}(p_{3})$$\rho(p_{2})$$D^{*}(p_{4})$$\rho(p_{2})$$D^{*}(p_{4})$$D$$D$$\psi(p_{1})$$\bar{D}^{*}(p_{3})$$\psi(p_{1})$$\bar{D}^{*}(p_{3})$$\rho(p_{2})$$D^{*}(p_{4})$$\rho(p_{2})$$D^{*}(p_{4})$$D^{*}$$D^{*}$

Figure 2: Born diagrams for the processes $\psi(2S)\rho\rightarrow\bar{D}D$
[(j) - (m)], $\psi(2S)\rho\rightarrow\bar{D}^{*}D$ [(n) - (q)] and
$\psi(2S)\rho\rightarrow\bar{D}^{*}D^{*}$ [(r) - (u)]. The particle charges
are not specified. We denote $\psi(2S)\equiv\psi$.

To the best of our knowledge, in contrast to the case of the $J/\psi$
Yongseok: 2001 ; Navarra:2005 ; Abreu:2018 , there is no effective theory for
the reactions involving the $\psi(2S)$. In the lack of works on this subject,
we adopt an effective approach, based on the $J/\psi$ studies, to describe the
reactions $\psi(2S)\pi\rightarrow\bar{D}^{(*)}D^{(*)}$ and
$\psi(2S)\rho\rightarrow\bar{D}^{(*)}D^{(*)}$, as well as the inverse
processes. In Figs. 1 and 2 we show the lowest-order Born diagrams
contributing to each process. To calculate their respective cross sections, we
will adapt the effective Lagrangians introduced in Yongseok: 2001 ;
Navarra:2005 ; Abreu:2018 . The couplings involving the $\psi(2S)$ and the
$D^{\ast}$ mesons are (we denote $\psi(2S)\equiv\psi$),

$\displaystyle\mathcal{L}_{\psi DD}$ $\displaystyle=$ $\displaystyle g_{\psi
DD}\psi_{\mu}(\partial^{\mu}D\bar{D}–D\partial^{\mu}\bar{D}),$
$\displaystyle\mathcal{L}_{\psi D^{*}D^{*}}$ $\displaystyle=$ $\displaystyle-
ig_{\psi
D^{*}D^{*}}[\psi^{\mu}(\partial_{\mu}D^{*\nu}\bar{D}_{\nu}^{*}–D^{*\nu}\partial_{\mu}\bar{D}_{\nu}^{*})$
$\displaystyle+(\partial_{\mu}\psi_{\nu}D^{*\nu}-\psi_{\mu}\partial_{\nu}D^{*\nu})\bar{D}^{*\mu}$
$\displaystyle+D^{*\mu}(\psi^{\nu}\partial_{\mu}\bar{D}_{\nu}^{*}-\partial_{\mu}\psi_{\nu}\bar{D}^{*\nu})],$
$\displaystyle\mathcal{L}_{\psi DD^{*}}$ $\displaystyle=$ $\displaystyle-
g_{\psi
DD^{*}}\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}\psi_{\nu}(\partial_{\alpha}D_{\beta}^{*}\bar{D}+D\partial_{\alpha}\bar{D}_{\beta}^{*});$

while the vertices involving the $D^{\ast}$ mesons and the pions and $\rho$
mesons are

$\displaystyle\mathcal{L}_{\pi DD^{*}}$ $\displaystyle=$ $\displaystyle
ig_{\pi
DD^{*}}D^{*\mu}\vec{\tau}\cdot(\bar{D}\partial_{\mu}\vec{\pi}-\partial_{\mu}\bar{D}\vec{\pi})+h.c.,$
$\displaystyle\mathcal{L}_{\rho DD}$ $\displaystyle=$ $\displaystyle ig_{\rho
DD}(D\vec{\tau}\partial_{\mu}\bar{D}-\partial_{\mu}D\vec{\tau}\bar{D})\cdot\vec{\rho}^{\mu},$
$\displaystyle\mathcal{L}_{\rho D^{*}D^{*}}$ $\displaystyle=$ $\displaystyle
ig_{\rho
D^{*}D^{*}}\left[(\partial_{\mu}D^{*\nu}\vec{\tau}\bar{D}^{*}_{\nu}-D^{*\nu}\vec{\tau}\partial_{\mu}\bar{D}^{*}_{\nu})\cdot\vec{\rho}^{\mu}\right.$
$\displaystyle+(D^{*\nu}\vec{\tau}\cdot\partial_{\mu}\vec{\rho}_{\nu}-\partial_{\mu}D^{*\nu}\vec{\tau}\cdot\vec{\rho}_{\nu})\bar{D}^{*\mu}$
$\displaystyle+\left.D^{*\mu}(\vec{\tau}\cdot\vec{\rho}^{\nu}\partial_{\mu}\bar{D}^{*}_{\nu}-\vec{\tau}\cdot\partial_{\mu}\vec{\rho}^{\nu}\bar{D}^{*}_{\nu})\right],$
$\displaystyle\mathcal{L}_{\pi D^{*}D^{*}}$ $\displaystyle=$ $\displaystyle-
g_{\pi
D^{*}D^{*}}\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}D^{8}_{\nu}\pi\partial_{\alpha}\bar{D}^{*}_{\beta},$
$\displaystyle\mathcal{L}_{\rho DD^{*}}$ $\displaystyle=$ $\displaystyle-
g_{\rho
DD^{*}}\epsilon^{\mu\nu\alpha\beta}(D\partial_{\mu}\rho_{\nu}\partial_{\alpha}\bar{D}^{*}_{\beta}+\partial_{\mu}D^{*}_{\nu}\partial_{\alpha}\rho_{\beta}\bar{D}).$

In Eqs. (LABEL:EffLagf1) and (LABEL:EffLagf2), $D^{(\ast)}$ and
$\bar{D}^{(*)}$ represent the isospin doublets for the pseudoscalar (vector)
charmed mesons; $\psi_{\nu}$ denotes the $\psi(2S)$ vector field; $\vec{\tau}$
represents the Pauli matrices in the isospin space; $\vec{\pi}$ and
$\vec{\rho}$ are the pion and $\rho$ isospin triplets, respectively. The
coupling constants $g_{\pi DD^{*}}$, $g_{\rho DD}$, $g_{\rho D^{*}D^{*}}$,
$g_{\pi D^{*}D^{*}}$ and $g_{\rho DD^{*}}$ will be discussed below.

These effective Lagrangians introduced above allow us to determine the
amplitudes of the $\psi(2S)$ absorption processes depicted in Figs. 1 and 2,

$\displaystyle\mathcal{M}_{\psi\pi\rightarrow\bar{D}D}$ $\displaystyle\equiv$
$\displaystyle\sum_{{i}=a,b}\mathcal{M}^{(\textbf{i})},$
$\displaystyle\mathcal{M}_{\psi\pi\rightarrow\bar{D}D^{*}}$
$\displaystyle\equiv$
$\displaystyle\sum_{\textbf{i}=c,d,e}\mathcal{M}^{(\textbf{i})},$
$\displaystyle\mathcal{M}_{\psi\pi\rightarrow\bar{D}^{*}D^{*}}$
$\displaystyle\equiv$
$\displaystyle\sum_{\textbf{i}=f,g,h,i}\mathcal{M}^{(\textbf{i})},$
$\displaystyle\mathcal{M}_{\psi\rho\rightarrow\bar{D}D}$ $\displaystyle\equiv$
$\displaystyle\sum_{\textbf{i}=j,k,l,m}\mathcal{M}^{(\textbf{i})},$
$\displaystyle\mathcal{M}_{\psi\rho\rightarrow\bar{D}D^{*}}$
$\displaystyle\equiv$
$\displaystyle\sum_{\textbf{i}=n,o,p,q}\mathcal{M}^{(\textbf{i})},$
$\displaystyle\mathcal{M}_{\psi\rho\rightarrow\bar{D}^{*}D^{*}}$
$\displaystyle\equiv$
$\displaystyle\sum_{\textbf{i}=r,s,t,u}\mathcal{M}^{(\textbf{i})},$ (4)

where $\mathcal{M}^{(\textbf{i})}$ denotes the contribution coming from the
specific reaction $(i)$; the expressions are explicitly summarized in Appendix
A.

As usual, we employ form factors to prevent the artificial growth of the
amplitudes with energy and to take into account the finite size of hadrons. To
the best of our knowledge there is so far no systematic study of the couplings
of Eqs. (LABEL:EffLagf1) and (LABEL:EffLagf2) involving the $\psi(2S)$. Due to
the similarities of these vertices with those involving the $J/\psi$, we
assume that the form factors for the $\psi(2S)$ are given by the same
parametrizations of those for the $J/\psi$, which have already been computed
with QCD sum rules in Refs. QCDSR:23 ; QCDSR:18 ; QCDSR:19 ; QCDSR:20 ;
QCDSR:21 ; Navarra:2005 . They are given by

$\displaystyle I)\,\,\,\,g_{M_{1}M_{2}M_{3}}$ $\displaystyle=$
$\displaystyle\frac{A}{Q^{2}+B},$ $\displaystyle
II)\,\,\,\,g_{M_{1}M_{2}M_{3}}$ $\displaystyle=$ $\displaystyle
A\,e^{-\left(\frac{B+Q^{2}}{C}\right)},$ (5)

where the values of the constants appearing above are given in Table 1. The
cross sections should depend on the size of the interacting hadron. From the
studies of charmonium spectroscopy, it is well known that the $\psi(2S)$ is
larger than the fundamental charmonium state by a factor of about 2 (see for
example the discussion in Godfrey:1985xj ). Therefore, based on geometrical
arguments we expect the $\psi(2S)$ to have bigger cross sections than the
$J/\psi$ by a factor of about 4. Hence, in the lack of reliable estimates of
the coupling constants and form factors for the $\psi(2S)$ vertices, we use
the parametrizations given in Eq. (5) and Table 1 but with the coupling
constants varying in the range $[g_{M_{1}M_{2}M_{3}},2g_{M_{1}M_{2}M_{3}}]$.
This will be the main source of uncertainties in our calculation.

Table 1: Form factors $g_{M_{1}M_{2}M_{3}}$ for the respective vertices $M_{1}M_{2}M_{3}$ presented in Eq. (5), computed with QCD sum rules in Refs. QCDSR:23 ; QCDSR:18 ; QCDSR:19 ; QCDSR:20 ; QCDSR:21 ; Navarra:2005  for the $J/\psi$. $M_{2}$ denotes the exchange particle. $M_{1}$ $M_{2}$ $M_{3}$ | Form | A | B | C
---|---|---|---|---
$\psi DD$ | II | 5.8 | 20 | 15.8
$\psi D^{*}D$ | II | 20 | 27 | 18.6
$\psi DD^{*}$ | II | 13 | 26 | 21.2
$\psi D^{*}D^{*}$ | II | 6.2 | 0 | 3.55
$\pi DD^{*}$ | I | 126 | 11.9 | -
$\pi D^{*}D$ | I | 126 | 11.9 | -
$\rho DD$ | I | 37.5 | 12.1 | -
$\rho D^{*}D^{*}$ | II | 4.9 | 0 | 13.3
$\pi D^{*}D^{*}$ | II | 4.8 | 0 | 6.8
$\rho DD^{*}$ | I | 234 | 44 | -
$\rho D^{*}D$ | I | 234 | 44 | -

##  III Cross sections for the processes involving $\psi(2S)$

###  III.1 Vaccum cross sections

The isospin-spin-averaged cross section in the center-of-mass (CM) frame for a
specific $ab\rightarrow cd$ process is given by

$\displaystyle\sigma_{ab\rightarrow
cd}=\frac{1}{64\pi^{2}sg_{a}g_{b}}\frac{|\vec{p}_{cd}|}{\vec{p}_{ab}|}\int
d\Omega\sum_{S,I}|\mathcal{M}_{ab\rightarrow cd}(s,\theta)|^{2}$ (6)

where $\sqrt{s}$ is the CM energy; $|\vec{p}_{ab}|$ and $|\vec{p}_{cd}|$ are
the magnitudes of three-momenta of initial and final particles in the CM
frame, respectively; $\sum_{S,I}$ means the sum over the spins and isospins of
the particles in the initial and final state, weighted by the isospin and spin
degeneracy factors $g_{a}=(2S_{a}+1)(2I_{a}+1)$ and
$g_{b}=(2S_{b}+1)(2I_{b}+1)$ of the two particles forming the initial state.
The cross sections for the respective inverse processes can be computed
through the use of the detailed balance relation,

$\displaystyle\sigma_{cd\rightarrow
ab}(s)=\frac{g_{a}g_{b}}{g_{c}g_{d}}\frac{|\vec{p}_{ab}|}{|\vec{p}_{cd}|}\sigma_{ab\rightarrow
cd}(s).$ (7)

The calculations are performed using the isospin-averaged masses of the light
and heavy mesons according to the values reported in the PDG Workman:2022ynf .
Since we use a range of values for the $\psi(2S)$ couplings (due to the
uncertainties) the results are shown as bands.

Figure 3:  Cross sections for the absorption processes
$\psi\pi\rightarrow\bar{D}^{(*)}D^{(*)}$ (2.a) and
$\psi\rho\rightarrow\bar{D}^{(*)}D^{(*)}$ (2.b), as functions of the relative
CM energy $\sqrt{s}-\sqrt{s_{0}}$. $\sqrt{s_{0}}$ is the threshold energy of
the respective channel.

Figure 4:  Cross sections for the production processes
$\bar{D}^{(*)}D^{(*)}\rightarrow\psi\pi$ (3.a) and
$\bar{D}^{(*)}D^{(*)}\rightarrow\psi\rho$ (3.b), as functions of the relative
CM energy $\sqrt{s}-\sqrt{s_{0}}$.

The cross sections for the $\psi(2S)$ absorption by comoving light mesons as
functions of the relative CM energy $\sqrt{s}-\sqrt{s_{0}}$ ($\sqrt{s_{0}}$
being the threshold energy of the respective channel) are plotted in Fig. 3.
Except for the channels $\psi\pi\rightarrow\bar{D}^{*}D$ and
$\psi\pi\rightarrow\bar{D}^{*}D^{*}$, all the cross sections are exothermic,
showing a substantial decrease right beyond the threshold. We remark that the
$\psi\pi\rightarrow\bar{D}D$ channel acquires large values very near the
threshold, not visible in the plot, and after that suffers a strong decrease.
In the region close to the threshold up to moderate energies, i.e.
$\sqrt{s}-\sqrt{s_{0}}\simeq 500\mbox{ MeV}$, the different channels present
magnitudes of the order of $10^{-4}-10^{1}\mbox{ mb}$, but with the reactions
with final state $\bar{D}D$ being suppressed with respect to the other ones.
In the case of the inverse processes, the cross sections are plotted in Fig.
4. As expected, very near the threshold they have an opposite behavior with
respect to the corresponding $\psi(2S)$ absorption reactions in Fig. 3. From
the region close to the threshold up to moderate energies, we observe that the
cross sections are of the order $10^{-4}-10^{0}\mbox{ mb}$, and in general
those for $\psi(2S)$ absorption reactions have smaller magnitudes than their
respective inverse processes. This is an important feature (since it appears
in the energy range relevant for heavy ion collisions) which can be attributed
to the differences in the phase space and the degeneracies encoded in Eq. (7)
Abreu:2022jmi .

###  III.2 Thermally-averaged cross sections

In a hadron gas the temperature determines the order of magnitude of the
collision energy. Hence it is more realistic to use the vacuum cross sections
weighted by the thermal momentum distributions of the colliding particles. For
processes with a two-particle initial state going into two final state
particles $ab\rightarrow cd$, it is given by Abreu:2022EPJC ; QCDSR:21 ;
Abreu:2022jmi ; Houng:2013 ; Koch:1986

$\displaystyle\langle\sigma_{ab\rightarrow cd}\upsilon_{ab}\rangle$
$\displaystyle=$ $\displaystyle\frac{\int
d^{3}\mathbf{p}_{a}d^{3}\mathbf{p}_{b}f_{a}(\mathbf{p}_{a})f_{b}(\mathbf{p}_{b})\sigma_{ab\rightarrow
cd}\,\,\upsilon_{ab}}{\int
d^{3}\mathbf{p}_{a}d^{3}\mathbf{p}_{b}f_{a}(\mathbf{p}_{a})f_{b}(\mathbf{p}_{b})}$
$\displaystyle=$
$\displaystyle\frac{1}{4\beta_{a}^{2}K_{2}(\beta_{a})\beta_{b}^{2}K_{2}(\beta_{b})}$
$\displaystyle\times\int_{z_{0}}^{\infty}dzK_{1}(z)\,\,\sigma_{ab\rightarrow
cd}(s=z^{2}T^{2})$
$\displaystyle\times\left[z^{2}-(\beta_{a}+\beta_{b})^{2}\right]\left[z^{2}-(\beta_{a}-\beta_{b})^{2}\right],$

where $\upsilon_{ab}$ denotes the relative velocity of the two initial
interacting particles; the function $f_{i}(\mathbf{p}_{i})$ is the Bose-
Einstein distribution; $\beta_{i}=m_{i}/T$, with $T$ being the temperature;
$z_{0}=max(\beta_{a}+\beta_{b},\beta_{c}+\beta_{d})$, and $K_{1}$ and $K_{2}$
are the modified Bessel functions of second kind. It can be seen from the
expression in the second line of Eq. (LABEL:Eq:AvCrossSection) that the
thermal average suppresses the configurations very close to the thresholds:
for sufficiently large $z_{0}$ the $K_{1}(z)$ function acquires very small
values.

Figure 5:  Thermally averaged cross sections for the absorption processes
$\psi\pi\rightarrow\bar{D}^{(*)}D^{(*)}$ (4.a) and
$\psi\rho\rightarrow\bar{D}^{(*)}D^{(*)}$ (4.b), as functions of the
temperature.

Figure 6:  Thermally averaged cross sections for the production processes
$\bar{D}^{(*)}D^{(*)}\rightarrow\psi\pi$ (5.a) and
$\bar{D}^{(*)}D^{(*)}\rightarrow\psi\rho$ (5.b), as functions of the
temperature.

In Fig. 5 we plot the thermally averaged cross sections as functions of the
temperature for the $\psi(2S)$ absorption by comoving light mesons using the
vacuum cross-section results obtained previously. In general, they have a weak
dependence on the temperature. The tendency seen in the vacuum at moderate
energies is reproduced here: reactions with $\bar{D}D$ final state are
suppressed with respect to the other ones. On the other hand, for the
$\psi(2S)$ production processes, shown in Fig 6, the reactions involving the
production of $\psi(2S)\rho$ present a strong dependence on the temperature.
Most importantly, the absorption cross sections are always larger than
production ones and the difference can reach two orders of magnitude,
depending on the temperature. According to previous studies, this feature may
have strong impact on the final yield of the $\psi(2S)$ in heavy ion
collisions. This issue will be addressed in the next section.

##  IV Multiplicities of $\psi(2S)$ and $X(3872)$

###  IV.1 Time Evolution

Here we will present the formalism used to determine the yields of $\psi(2S)$
and $X(3872)$ in a hadron gas formed in the final stage of heavy ion
collisions. We will use the thermally-averaged cross sections estimated in the
previous section for $\psi(2S)$. For $X(3872)$ we will use the results
reported in Ref. QCDSR:21  and include the contributions of the anomalous
vertex $XD^{*}\bar{D}^{*}$ presented in Refs. abreu16 ; Torres:2014 . These
quantities will be employed as entries into the time-evolution equations to
estimate the gain and loss terms. Explicitly, the rate equation is QCDSR:21 ;
abreu16 ; Nielsen:2011 ; EXHIC:2017 ; Abreu:2018 ; Houng:2013 ; Abreu:2023awj

$\displaystyle\frac{dN_{h}(\tau)}{d\tau}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}\bar{c}=\bar{D},\bar{D}^{*};\\\
c=D,D^{*};\\\
\varphi=\pi,\rho\end{subarray}}\left[\langle\sigma_{\bar{c}c\rightarrow\varphi
h}\upsilon_{\bar{c}c}\rangle n_{\bar{c}}(\tau)N_{c}(\tau)\right.$ (9)
$\displaystyle-\left.\langle\sigma_{\varphi
h\rightarrow\bar{c}c}\upsilon_{\psi\varphi}\rangle
n_{\varphi}(\tau)N_{h}(\tau)\right],$

where $N_{h}(\tau)$ represents the multiplicity of the state of type $h$
$(h=\psi(2S),X(3872))$; $n_{i}(\tau)$ and $N_{i}(\tau)$ denote the density and
the number of the meson of type $i$ at a given $\tau$. The pions, $\rho$ and
charmed mesons in the reactions discussed previously are assumed to be in
equilibrium, implying that with the Maxwell-Boltzmann approximation
$n_{i}(\tau)$ becomes

$\displaystyle n_{i}(\tau)$ $\displaystyle\approx$
$\displaystyle\frac{1}{2\pi^{2}}\gamma_{i}g_{i}m_{i}^{2}T(\tau)K_{2}\left(\frac{m_{i}}{T(\tau)}\right),$
(10)

where $\gamma_{i}$, $g_{i}$ and $m_{i}$ is the fugacity, degeneracy factor and
particle mass $i$, respectively. The multiplicity $N_{i}(\tau)$ is then
obtained by multiplying $n_{i}(\tau)$ by the volume $V(\tau)$. Following
preceding works, we model the hadron gas expansion by the boost invariant
Bjorken flow with an accelerated transverse expansion. The volume and
temperature profiles as a function of the proper time $\tau$ are as follows
abreu16 ; Nielsen:2011 ; EXHIC:2017 ; Abreu:2018 ; Houng:2013 :

$\displaystyle V(\tau)$ $\displaystyle=$
$\displaystyle\pi\left[R_{C}+\upsilon_{C}\left(\tau-\tau_{C}\right)+\frac{a_{C}}{2}\left(\tau-\tau_{C}\right)^{2}\right]^{2}\tau
c,$ $\displaystyle T(\tau)$ $\displaystyle=$ $\displaystyle
T_{C}-\left(T_{H}-T_{F}\right)\left(\frac{\tau-\tau_{H}}{\tau_{F}-\tau_{H}}\right)^{\frac{4}{5}},$
(11)

where $R_{C},\upsilon_{C},a_{C}$ and $T_{C}$ are the transverse size,
transverse velocity, transverse acceleration and temperature at the time
$\tau_{C}$, respectively; $T_{H}(T_{F})$ is the temperature at the
hadronization (kinetic freeze-out) time $\tau_{H}(\tau_{F})$. The parameters
in Eq. (11) are fixed according to Ref. EXHIC:2017  for a hadronic medium
formed in central Pb Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV; for
completeness they are given in Table 2.

Table 2: In the first three rows we list the set of parameters used in Eq. (11) for the hydrodynamic expansion and cooling of the hadronic medium formed in central PbPb collisions at $\sqrt{s_{NN}}=5.02$ TeV EXHIC:2017 . In the fourth row we list the multiplicity of the charm quark and light mesons used in the model. In the last two rows the quark masses and the frequency used in the coalescence model are listed QCDSR:21 .  $v_{C}$ (c) |  $a_{C}$ (c2/fm) |  $R_{C}$ (fm)
---|---|---
0.5 | 0.09 | 11
$\tau_{C}$ (fm/c) |  $\tau_{H}$ (fm/c) |  $\tau_{F}$ (fm/c)
7.1 | 10.2 | 21.5
$T_{C}(\mbox{ MeV})$ | $T_{H}(\mbox{ MeV})$ | $T_{F}(\mbox{ MeV})$
156 | 156 | 115
$N_{c}$ | $N_{\pi}(\tau_{F})$ | $N_{\rho}(\tau_{H})$
14 | 2410 | 184
$V_{C}$ (fm3) | $m_{q}(\mbox{ MeV})$ | $m_{c}(\mbox{ MeV})$
5380 | 350 | 1500
$\omega(\mbox{ MeV})$ |  |
220 |  |

Additionally, the multiplicities of the light mesons as well as of the charm
quarks in charmed mesons are also shown in Table 2. For the light mesons,
their fugacities in Eq. (10) appear as normalization parameters to adjust the
multiplicities given in Table 2. In the case of charm quarks, since they are
produced in the early stages of the collision, we assume that the total number
of charm quarks $(N_{c})$ in charmed hadrons remains approximately conserved
during the hydrodynamic expansion, which leads to the condition
$N_{c}=n_{c}(\tau)\times V(\tau)=const.$, engendering a time-dependent charm-
quark-fugacity factor $\gamma_{c}\equiv\gamma_{c}(\tau)$ in Eq. (10).

To fix the initial condition for the yield $N_{h}(\tau)$ appearing in the
integro-differential equation (9), i.e. the yield of the state $h$ at the end
of QGP, we employ the so-called coalescence model, which is characterized by
the convolution of the density matrix of the constituents of $h$ with its
Wigner function QCDSR:21 ; EXHIC:2017 ; Abreu:2021 . This model has the
advantage of carrying information on the intrinsic structure of the system,
such as angular momentum and the type and number of constituent quarks. More
concretely, according to this approach the yield of a hadronic state of type
$h$ can be written as

$\displaystyle N_{h}$ $\displaystyle\approx$ $\displaystyle
g_{h}\prod_{j=1}^{n}\frac{N_{j}}{g_{j}}\prod_{i=1}^{n-1}\frac{(4\pi\sigma_{i}^{2})^{\frac{3}{2}}}{V(\tau)(1+2\mu_{i}T(\tau)\sigma_{i}^{2})}$
(12)
$\displaystyle\times\left[\frac{4\mu_{i}T(\tau)\sigma_{i}^{2}}{3(1+2\mu_{i}T(\tau)\sigma_{i}^{2})}\right]^{l_{i}},$

where $g_{j}$ and $N_{j}$ are the degeneracy and number of the $j$-th
constituent of $h$ and $\sigma_{i}=(\mu_{i}\omega)^{-1/2}$; the quantity
$\omega$ is the oscillator frequency (taking an harmonic oscillator as a model
of the hadron internal structure) and $\mu$ the reduced mass, i.e.
$\mu^{-1}=m_{i+1}^{-1}+\left(\sum_{j=1}^{i}m_{j}\right)^{-1}$; $l_{i}$ is the
angular momentum of the system: it is 0 for an $S$ wave and 1 for a $P$ wave.
Table 2 summarizes the oscillation frequency for the charmed hadrons, the
charm quark number and masses used here.

From Eq. (12) we have the following multiplicities for $\psi(2S)$ and
$X(3872)$ at the end of the mixed phase:

$\displaystyle N_{\psi(2S)}(\tau_{H})$ $\displaystyle\approx$ $\displaystyle
1.8\times 10^{-4},$ $\displaystyle N_{X}^{(4q)}(\tau_{H})$
$\displaystyle\approx$ $\displaystyle 1.8\times 10^{-4},$ $\displaystyle
N_{X}^{(Mol.)}(\tau_{H})$ $\displaystyle\approx$ $\displaystyle 1.1\times
10^{-2},$ (13)

In the case of the molecular state $X(3872)^{(Mol.)}$, to calculate the
oscillation frequency, we have employed the expression $\omega=6B$, with $B$
being the binding energy of the $X(3872)$ considered as a
$(D^{0}\bar{D}^{*0}+c.c.)$ bound state. As it can be seen, the coalescence
mechanism generates initial conditions in which molecules are more abundant
than compact tetraquarks by a factor of about 60, reflecting the fact that
forming a loosely bound state is easier than a compact tetraquark.

Figure 7: Multiplicity of $\psi(2S)$ (upper panel), of $X(3872)$ (central
panel) and their ratio (lower panel) as a function of the proper time in
central Pb Pb collisions at $\sqrt{s_{NN}}=$ 5.02 TeV.

In Fig. 7 we show the time evolution of the $\psi(2S)$ and $X(3872)$
multiplicities as a function of the proper time. In the case of $\psi(2S)$,
the final yield increases by a factor of about one order of magnitude, which
means that the gain terms in the evolution equation (9) play a dominant role
at higher temperatures. Strictly speaking, at the beginning of the hadron gas
phase the densities and multiplicities of the charmed mesons are sufficient to
counterbalance the bigger magnitudes of the thermal cross sections for the
absorption processes multiplied by the densities of the light mesons and the
multiplicity of $\psi(2S)$. As the time evolves, the gain and loss terms
become almost of the same order (considering the uncertainties), and the
$\psi(2S)$ multiplicity suffers just a slight reduction.

For the $X(3872)$ abundance, we remark that its time evolution has already
been analyzed in Refs. Houng:2013 ; abreu16  for central Au Au collisions at
$\sqrt{s_{NN}}=200$ GeV. Very recently, in Ref. QCDSR:21  this analysis has
been redone for central Pb Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV, for the
purpose of comparison between the final yields of $T_{cc}^{+}(3985)$ and
$X(3872)$. Since we are interested in the $X(3872)$ to $\psi(2S)$ yield ratio,
for completeness we present again in Fig. 7 the $N_{X(3872)}(\tau)$. We
improved the calculation including the anomalous vertex $XD^{*}\bar{D}^{*}$ as
discussed in Torres:2014 . As can be seen from the figure, tetraquarks reach a
final yield which is much smaller (by a factor of about 50) than the one of
$(D\bar{D}^{*}+c.c.)$ molecules. Another feature is that in both cases the
$X(3872)$ abundance does not present a sizable change with time.

We also show in Fig. 7 the time evolution of the $X(3872)$ to $\psi(2S)$ yield
ratio. As expected from the discussion above, we observe a strong variation of
this ratio in the early times and after that a relative stabilization. In Ref.
Yun:2022evm  the coalescence model has been used to estimate the $N_{X}$ and
the statistical hadronization model to obtain $N_{\Psi(2S)}$. However, the
hadronic interactions have not been taken into account. This has resulted in a
ratio with central value of 0.806 for the $X(3872)$ in the molecular
configuration, argued to be consistent with the one observed by the CMS
Collaboration. As shown in Fig. 7 we find a bigger ratio for the molecular
configuration than that of Ref. Yun:2022evm .

###  IV.2 Source size dependence

The size of the source can be related to a measurable quantity, the charged-
particle pseudorapidity density at mid-rapidity,
$[dN_{ch}/d\eta\,(|\eta|<0.5)$, which, in turn, can be related to the freeze-
out temperature by means of the empirical formula LeRoux:2021adw ; QCDSR:21 ;
Abreu:2023aqy :

$\displaystyle T_{F}=T_{F0}\ e^{-b\mathcal{N}},$ (14)

where $T_{F0}=132.5\mbox{ MeV}$, $b=0.02$, and
$\mathcal{N}\equiv[dN_{ch}/d\eta\,(|\eta|<0.5)]^{1/3}$. Assuming that the
hadron gas undergoes a Bjorken cooling, i.e.
$T=T_{h}\left(\tau_{h}/\tau\right)^{1/3}$, then the freeze-out time $\tau_{F}$
can be written in terms of $\mathcal{N}$ as LeRoux:2021adw ; QCDSR:21 ;
Abreu:2023aqy :

$\displaystyle\tau_{F}=\tau_{H}\left(\frac{T_{H}}{T_{F0}}\right)^{3}e^{3b\mathcal{N}}.$
(15)

A larger source generates a bigger $\mathcal{N}$, which, from the equation
above, implies a bigger $\tau_{F}$, i.e. a longer hadron gas phase. As a
consequence, the use of Eq. (15) in (9) will give rise to a dependence of the
multiplicity $N_{h}$ on the size of the source.

As shown in Ref. Abreu:2023aqy , empirical formulas relating $\mathcal{N}$
with the volume of the system ($V$), charm quark number ($N_{c}$) and light
quark number ($N_{q}$) can also be obtained. They are

$\displaystyle V$ $\displaystyle=2.82\mathcal{N}^{3}$ $\displaystyle N_{c}$
$\displaystyle=7.9\times 10^{-5}\mathcal{N}^{4.8}$ $\displaystyle N_{u}$
$\displaystyle=N_{d}=0.37\mathcal{N}^{3}.$ (16)

We also assume that the charm quark number and the number of $D$ mesons
($N_{D}$) are proportional. Hence, the use of Eq. (16) in (13) to estimate the
dependence of the initial conditions with $\mathcal{N}$, together with (15) in
(9), generates a dependence of $N_{h}$ on $\mathcal{N}$.

Figure 8:  Ratio $\mathcal{R}$ as a function of $\mathcal{N}$.

In Fig. 8 we observe that the ratio $\mathcal{R}$ grows with $\mathcal{N}$.
This is in qualitative agreement with the existing data. As mentioned in
introduction, the LHCb Collaboration reported in Ref. LHCb:2022ixc  the growth
of the ratio $\mathcal{R}$ when we go from p p to p Pb and to Pb Pb
collisions. Our results give qualitative support to the conjecture that the
$X(3872)$ and $\psi(2S)$ experience a different dynamics in the hadronic
medium. Unfortunately a direct comparison is an involved task, as the data in
Ref. LHCb:2022ixc  for the p p, p Pb and Pb Pb (this last one from the CMS
Collaboration) collisions refer to different energies and ranges of rapidity
and transverse momentum. Interestingly, the curves of the molecular and
compact tetraquark configurations converge to each other at smaller sources,
but in the region of validity of this model (larger sources) the difference
between these configurations remains approximately of the same order.

###  IV.3 Centrality and energy dependence

Apart from its dependence on the mass number $A$, $\mathcal{N}$ also depends
on the centrality of the collision and on the center-of-mass collision energy
($\sqrt{s}$). For Pb Pb collisions at $5.02$ TeV the relation between
$\mathcal{N}$ and the centrality (denoted as $x$, in %) can be parametrized as
Abreu:2023aqy :

$\displaystyle\left.\frac{dN_{ch}}{d\eta}\right|_{|\eta|<0.5}$
$\displaystyle=$ $\displaystyle 2142.16-85.76x+1.89x^{2}-0.03x^{3}$ (17)
$\displaystyle+3.67\times 10^{-5}\ x^{4}-2.24\times 10^{-6}\ x^{5}$
$\displaystyle+5.25\times 10^{-9}\ x^{6},$

Similarly, the dependence of $\mathcal{N}$ on $\sqrt{s}$ can be parametrized
as:

$\displaystyle\left.\frac{dN_{ch}}{d\eta}\right|_{|\eta|<0.5}$
$\displaystyle=$ $\displaystyle-2332.12+491.69\,\log(220.06+\sqrt{s})$

In order to determine the initial $\psi(2S)$ and $X(3872)$ multiplicities with
the coalescence model in terms of the centrality and of $\sqrt{s}$, we insert
Eqs. (17) and (LABEL:relSN) into Eq. (16) and use these new equations in Eq.
(12). The final yields are then obtained solving Eq.(9) with these initial
conditions and up to a final time $\tau_{F}$ (which is also $\mathcal{N}$
dependent).

Figure 9: The ratio $\mathcal{R}$ as a function of centrality in Pb Pb
collisions at $\sqrt{s_{NN}}=$ 5.02 TeV.  Figure 10:  The ratio $\mathcal{R}$
as a function of energy $\sqrt{s}$ in central Pb Pb collisions.

In Fig. 9 we show the ratio $\mathcal{R}$ as a function of the centrality. The
$X(3872)$ final yield decreases faster than the one of $\psi(2S)$ as we move
from central to peripheral collisions, and therefore the ratio becomes
smaller. Also, the curves for the molecular and compact tetraquark
configurations converge to similar values for more peripheral collisions.

The ratio $\mathcal{R}$ as a function of the energy $\sqrt{s}$ in central Pb
Pb collisions is presented in Fig. 10. The $X(3872)$ final yield presents an
enhancement compared to that of $\psi(2S)$, resulting in a bigger ratio as
$\sqrt{s}$ increases; the difference between the curves of the molecular and
compact tetraquark configurations remain of the same order at different
energies.

Taking together the results in Figs. 7, 8, 9 and 10 we conclude that in
central Pb Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV the ratio $\mathcal{R}$
for minimum bias events is:

$\displaystyle\frac{N_{X}}{N_{\psi(2S)}}$ $\displaystyle\simeq$ $\displaystyle
5\hskip 35.56593pt\mbox{for molecules}$
$\displaystyle\frac{N_{X}}{N_{\psi(2S)}}$ $\displaystyle\simeq$ $\displaystyle
0.1\hskip 28.45274pt\mbox{for tetraquarks}$ (19)

This is the main result of our work and it is our prediction for future
measurements at the run 3 of the ALICE Collaboration, where it will be
possible to measure $X(3872)$ and $\psi(2S)$ at low transverse momentum, i.e.
$2<p_{T}<8$ GeV. With data taken in this $p_{T}$ range it will be possible,
for the first time, to investigate the medium effects on $X(3872)$, on
$\psi(2S)$ and on their ratio and check our predictions. Moreover, in view of
the numbers above we may have a good chance to discriminate between molecules
and tetraquarks. From the figures we also conclude that is easier to identify
the different configurations in larger systems and in more central collisions.
Changing the collision energy is not so relevant for this purpose.

We close this section mentioning that, apart from our work, there are other
papers Yun:2022evm ; Andronic:2019  where the $X(3872)$ and $\psi(2S)$ $p_{T}$
distributions have been calculated with hydrodynamical and statistical models.

##  V Concluding remarks

In this work we have studied the $X(3872)$ to $\psi(2S)$ yield ratio in heavy-
ion collisions, taking into account the interactions of the $\psi(2S)$ and
$X(3872)$ states with a hadron gas made of light mesons. To this end, the
thermally-averaged cross sections for the production and absorption of the
$\psi(2S)$ have been evaluated for the first time by using effective
Lagrangians. These cross sections, together with the thermally-averaged cross
sections for the $X(3872)$-production and absorption analyzed in previous
works, have been employed in the rate equation to determine the time evolution
of $N_{\psi(2S)}$, $N_{X}$ and of $N_{X}/N_{\psi(2S)}$. The coalescence model
has been used to compute the initial multiplicities, with the $X(3872)$ being
treated both as a molecular bound state and as a compact tetraquark. Our
results indicate that the ratio is strongly affected by the combined effects
of hadronic interactions and hydrodynamical expansion, and that the molecular
configuration has a final value bigger than one, while the compact tetraquark
gives a smaller ratio by a factor of about 50.

Other interesting finding is that the ratio $\mathcal{R}$ grows with the size
of the source, which is in qualitative agreement with the data from LHCb and
CMS Collaborations LHCb:2022ixc , although in our case this growth has a
different origin.

We believe that this study can be seen as a theoretical support to the idea
that the $X(3872)$ and the conventional charmonium $\psi(2S)$ have a different
dynamics in a hadronic medium.

###### Acknowledgements.

This work was supported by the Brazilian agencies CNPq, FAPESP and CNPq/FAPERJ
(Project INCT-Física Nuclear e Aplicações - Contract No. 464898/2014-5). The
authors are deeply grateful to R. Rapp, who encouraged us to study this
subject and with whom we had so many fruitful discussions. We would like to
thank Su-Houng Lee and Marcelo Munhoz for useful discussions.

##  Appendix A Amplitudes

Here the expressions for the contributions to the amplitudes in Eq. (4),
associated to the reactions depicted in Figs. 1 and. 2, are given:

$\displaystyle\mathcal{M}^{(a)}$ $\displaystyle=$ $\displaystyle g_{\psi
DD^{*}}g_{\pi
DD^{*}}\varepsilon_{1}^{\mu}\frac{1}{t–m_{D^{*}}^{2}}\epsilon_{\mu\beta\lambda\rho}$
(20)
$\displaystyle\times\left(-g^{\alpha\beta}+\frac{(p_{1}–p_{3})^{\alpha}(p_{1}–p_{3})^{\beta}}{m_{D^{*}}^{2}}\right)p_{2\alpha}p_{1}^{\lambda}p_{3}^{\rho},$

$\displaystyle\mathcal{M}^{(b)}$ $\displaystyle=$ $\displaystyle g_{\psi
DD^{*}}g_{\pi
DD^{*}}\varepsilon_{1}^{\mu}\frac{1}{u–m_{D^{*}}^{2}}\epsilon_{\mu\beta\lambda\rho}$
(21)
$\displaystyle\times\left(-g^{\alpha\beta}+\frac{(p_{1}–p_{4})^{\alpha}(p_{1}–p_{4})^{\beta}}{m_{D^{*}}^{2}}\right)p_{2\alpha}p_{1}^{\lambda}p_{4}^{\rho},$

$\displaystyle\mathcal{M}^{(c)}$ $\displaystyle=$ $\displaystyle 2g_{\psi
DD}g_{\pi
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{4}^{*\nu}\frac{1}{t–m_{D}^{2}}p_{2\nu}p_{3\mu},$

$\displaystyle\mathcal{M}^{(d)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{\psi D^{*}D^{*}}g_{\pi
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{4}^{*\nu}\frac{1}{u–m_{D^{*}}^{2}}$
(22)
$\displaystyle\times\left(-g^{\alpha\beta}+\frac{(p_{1}–p_{4})^{\alpha}(p_{1}–p_{4})^{\beta}}{m_{D^{*}}^{2}}\right)(p_{2}+p_{3})_{\alpha}$
$\displaystyle\times[2p_{4\mu}g_{\nu\beta}-(p_{1}+p_{4})_{\beta}g_{\mu\nu}+2p_{1\nu}g_{\mu\beta}],$

$\displaystyle\mathcal{M}^{(e)}$ $\displaystyle=$ $\displaystyle g_{\psi
DD^{*}}g_{\pi
D^{*}D^{*}}\varepsilon_{1}^{\mu}\varepsilon_{4}^{*\nu}\frac{1}{t–m_{D^{*}}^{2}}\epsilon_{\mu\gamma\delta\beta}\epsilon_{\nu\lambda\rho\alpha}$
(23)
$\displaystyle\times\left(-g^{\alpha\beta}+\frac{(p_{1}–p_{3})^{\alpha}(p_{1}–p_{3})^{\beta}}{m_{D^{*}}^{2}}\right)p^{\gamma}p_{2}^{\rho}p_{3}^{\delta}p_{4}^{\lambda},$

$\displaystyle\mathcal{M}^{(f)}$ $\displaystyle=$ $\displaystyle g_{\psi
DD^{*}}g_{\pi
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{3}^{*\lambda}\varepsilon_{4}^{*\nu}\frac{1}{t–m_{D}^{2}}\epsilon_{\mu\lambda\gamma\delta}p_{1}^{\gamma}p_{2\nu}p_{3}^{\delta},$

$\displaystyle\mathcal{M}^{(g)}$ $\displaystyle=$ $\displaystyle-g_{\psi
DD^{*}}g_{\pi
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{3}^{*\lambda}\varepsilon_{4}^{*\nu}\frac{1}{u–m_{D}^{2}}\epsilon_{\mu\nu\gamma\delta}p_{1}^{\gamma}p_{2\lambda}p_{4}^{\delta},$

$\displaystyle\mathcal{M}^{(h)}$ $\displaystyle=$ $\displaystyle g_{\psi
D^{*}D^{*}}g_{\pi
D^{*}D^{*}}\varepsilon_{1}^{\mu}\varepsilon_{3}^{*\lambda}\varepsilon_{4}^{*\nu}\frac{1}{t–m_{D^{*}}^{2}}\epsilon_{\nu\gamma\delta\alpha}$
(24)
$\displaystyle\times\left(-g^{\alpha\beta}+\frac{(p_{1}–p_{3})^{\alpha}(p_{1}–p_{3})^{\beta}}{m_{D^{*}}^{2}}\right)p_{4}^{\gamma}p_{2}^{\delta}$
$\displaystyle\times[2p_{3\mu}g_{\beta\lambda}–(p_{1}+p_{3})_{\beta}g_{\mu\lambda}+2p_{1\lambda}g_{\mu\beta}],$

$\displaystyle\mathcal{M}^{(i)}$ $\displaystyle=$ $\displaystyle g_{\psi
D^{*}D^{*}}g_{\pi
D^{*}D^{*}}\varepsilon_{1}^{\mu}\varepsilon_{3}^{*\lambda}\varepsilon_{4}^{*\nu}\frac{1}{u–m_{D^{*}}^{2}}\epsilon_{\lambda\gamma\delta\alpha}$
(25)
$\displaystyle\times\left(-g^{\alpha\beta}+\frac{(p_{1}–p_{4})^{\alpha}(p_{1}–p_{4})^{\beta}}{m_{D^{*}}^{2}}\right)p_{2}^{\gamma}p_{3}^{\delta}$
$\displaystyle\times[2p_{4\mu}g_{\beta\nu}–(p_{1}+p_{4})_{\beta}g_{\mu\nu}+2p_{1nu}g_{\mu\beta}],$

$\displaystyle\mathcal{M}^{(j)}$ $\displaystyle=$ $\displaystyle 4g_{\psi
DD}g_{\rho
DD}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\frac{1}{t–m_{D}^{2}}p_{3\mu}p_{4\nu},$

$\displaystyle\mathcal{M}^{(k)}$ $\displaystyle=$ $\displaystyle 4g_{\psi
DD}g_{\rho
DD}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\frac{1}{u–m_{D}^{2}}p_{3\nu}p_{4\mu},$

$\displaystyle\mathcal{M}^{(l)}$ $\displaystyle=$ $\displaystyle g_{\psi
DD^{*}}g_{\rho
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\frac{1}{t–m_{D^{*}}^{2}}\epsilon_{\nu\gamma\delta\alpha}\epsilon_{\mu\lambda\rho\beta}$
(26)
$\displaystyle\times\left(–g^{\alpha\beta}+\frac{(p_{1}–p_{3})^{\alpha}(p_{1}–p_{3})^{\beta}}{m_{D^{*}}^{2}}\right)p_{1}^{\lambda}p_{2}^{\gamma}p_{3}^{\rho}p_{4}^{\delta},$

$\displaystyle\mathcal{M}^{(m)}$ $\displaystyle=$ $\displaystyle-g_{\psi
DD^{*}}g_{\rho
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\frac{1}{u–m_{D^{*}}^{2}}\epsilon_{\nu\gamma\delta\alpha}\epsilon_{\mu\lambda\rho\beta}$
(27)
$\displaystyle\times\left(–g^{\alpha\beta}+\frac{(p_{1}–p_{4})^{\alpha}(p_{1}–p_{4})^{\beta}}{m_{D^{*}}^{2}}\right)p_{1}^{\lambda}p_{2}^{\delta}p_{3}^{\gamma}p_{4}^{\rho},$

$\displaystyle\mathcal{M}^{(n)}$ $\displaystyle=$ $\displaystyle-2g_{\psi
DD}g_{\rho
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\varepsilon_{4}^{*\lambda}\frac{1}{t–m_{D}^{2}}\epsilon_{\nu\lambda\gamma\delta}p_{2}^{\delta}p_{4}^{\gamma}p_{3\mu},$

$\displaystyle\mathcal{M}^{(o)}$ $\displaystyle=$ $\displaystyle 2g_{\psi
DD}g_{\rho
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\varepsilon_{4}^{*\lambda}\frac{1}{u–m_{D}^{2}}\epsilon_{\lambda\mu\gamma\delta}p_{1}^{\gamma}p_{4}^{\delta}p_{3\nu},$

$\displaystyle\mathcal{M}^{(p)}$ $\displaystyle=$ $\displaystyle g_{\psi
DD^{*}}g_{\rho
D^{*}D^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\varepsilon_{4}^{*\lambda}\frac{1}{t–m_{D^{*}}^{2}}\epsilon_{\mu\gamma\delta\beta}$
(28)
$\displaystyle\times\left(-g^{\alpha\beta}+\frac{(p_{1}-p_{3})^{\alpha}(p_{1}-{3})^{\beta}}{m_{D^{*}}^{2}}\right)p_{1}^{\gamma}p_{3}^{\delta}$
$\displaystyle\times[2p_{4\nu}g_{\lambda\alpha}–(p_{2}+p_{4})_{\alpha}g_{\nu\lambda}+2p_{2\lambda}g_{\nu\alpha}],$

$\displaystyle\mathcal{M}^{(q)}$ $\displaystyle=$ $\displaystyle g_{\psi
D^{*}D^{*}}g_{\rho
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\varepsilon_{4}^{*\lambda}\frac{1}{u–m_{D^{*}}^{2}}\epsilon_{\nu\gamma\delta\beta}$
(29)
$\displaystyle\times\left(–g^{\alpha\beta}+\frac{(p_{1}–p_{4})^{\alpha}(p_{1}–p_{4})^{\beta}}{m_{D^{*}}^{2}}\right)p_{2}^{\delta}p_{3}^{\gamma}$
$\displaystyle\times[2p_{4\mu}g_{\beta\lambda}–(p_{1}+p_{4})_{\beta}g_{\mu\lambda}+2p_{1\lambda}g_{\mu\beta}],$

$\displaystyle\mathcal{M}^{(r)}$ $\displaystyle=$ $\displaystyle-g_{\psi
DD^{*}}g_{\rho
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\varepsilon_{3}^{*\rho}\varepsilon_{4}^{*\lambda}\frac{1}{t–m_{D}^{2}}$
(30)
$\displaystyle\times\epsilon_{\nu\lambda\gamma\delta}\epsilon_{\mu\rho\alpha\beta}p_{1}^{\alpha}p_{2}^{\delta}p_{3}^{\delta}p_{4}^{\gamma},$

$\displaystyle\mathcal{M}^{(s)}$ $\displaystyle=$ $\displaystyle g_{\psi
DD^{*}}g_{\rho
DD^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\varepsilon_{3}^{*\rho}\varepsilon_{4}^{*\lambda}\frac{1}{u–m_{D}^{2}}$
(31)
$\displaystyle\times\epsilon_{\nu\rho\gamma\delta}\epsilon_{\mu\lambda\alpha\beta}p_{1}^{\alpha}p_{2}^{\gamma}p_{3}^{\delta}p_{4}^{\beta},$

$\displaystyle\mathcal{M}^{(t)}$ $\displaystyle=$ $\displaystyle g_{\psi
D^{*}D^{*}}g_{\rho
D^{*}D^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\varepsilon_{3}^{*\rho}\varepsilon_{4}^{*\lambda}\frac{1}{t–m_{D^{*}}^{2}}$
(32)
$\displaystyle\times\left(-g^{\alpha\beta}+\frac{(p_{1}–p_{3})^{\alpha}(p_{1}–p_{3})^{\beta}}{m_{D^{*}}^{2}}\right)$
$\displaystyle\times[2p_{4\nu}g_{\alpha\lambda}–(p_{2}+p_{4})_{\alpha}g_{\nu\lambda}+2p_{2\lambda}g_{\nu\alpha}]$
$\displaystyle\times[2p_{3\mu}g_{\beta\rho}–(p_{1}+p_{3})_{\beta}g_{\mu\rho}+2p_{1\rho}g_{\mu\beta}],$

$\displaystyle\mathcal{M}^{(u)}$ $\displaystyle=$ $\displaystyle g_{\psi
D^{*}D^{*}}g_{\rho
D^{*}D^{*}}\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\varepsilon_{3}^{*\rho}\varepsilon_{4}^{*\lambda}\frac{1}{u–m_{D^{*}}^{2}}$
(33)
$\displaystyle\times\left(-g^{\alpha\beta}+\frac{(p_{1}–p_{4})^{\alpha}(p_{1}–p_{4})^{\beta}}{m_{D^{*}}^{2}}\right)$
$\displaystyle\times[2p_{3\nu}g_{\alpha\rho}–(p_{2}+p_{3})_{\alpha}g_{\nu\rho}+2p_{2\rho}g_{\nu\alpha}]$
$\displaystyle\times[2p_{4\mu}g_{\beta\lambda}–(p_{1}+p_{4})_{\beta}g_{\mu\lambda}+2p_{1\lambda}g_{\mu\beta}],$

where $p_{1}$ and $p_{2}$ are the momenta of initial state particles, while
$p_{3}$ and $p_{4}$ are those of final state particles; $s,t,u$ are the
Mandelstam variables: $s=(p_{1}+p_{2})^{2},t=(p_{1}-p_{3})^{2},$ and
$u=(p_{1}-p_{4})^{2}$; and
$\epsilon_{\textbf{i}}^{(*)}\equiv\epsilon^{(*)}(p_{i})$ is the polarization
vector.

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