# Insights into the inner structures of the fully charmed tetraquark state
$X(6900)$

Zhi-Hui Guoa,b111<EMAIL_ADDRESS>J. A. Ollerc222<EMAIL_ADDRESS>
a School of Physics, Southeast University, Nanjing 211189, China
b Department of Physics and Hebei Advanced Thin Films Laboratory,
Hebei Normal University, Shijiazhuang 050024, China
c Departamento de Física. Universidad de Murcia. E-30071 Murcia. Spain

###### Abstract

The recently discovered fully charmed tetraquark candidate $X(6900)$ is
analyzed within the frameworks of effective-range expansion, compositeness
relation and width saturation, and a coupled multichannel dynamical study. By
taking into account constraints from heavy-quark spin symmetry, the coupled-
channel amplitude including the $J/\psi J/\psi,~{}\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ is constructed to fit the experimental di-$J/\psi$ event
distributions around the energy region near $6.9$ GeV. Another dynamical two-
coupled-channel amplitude with the $J/\psi J/\psi$ and $\psi(3770)J/\psi$ is
also considered to describe the same datasets. The three different theoretical
approaches lead to similar conclusions that the two-meson components do not
play dominant roles in the $X(6900)$. Our determinations of the resonance
poles in the complex energy plane from the refined coupled-channel study are
found to be consistent with the experimental analyses. The coupled-channel
amplitudes also have another pole corresponding to a narrow resonance
$X(6825)$ that we predict sitting below the $\chi_{c0}\chi_{c0}$ threshold and
of molecular origin. We give predictions to the line shapes of the
$\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ channels, which could provide a
useful guide for future experimental measurements.

##  1 Introduction

The first fully heavy-flavor tetraquark meson candidate $X(6900)$ was recently
observed in the di-$J/\psi$ spectra by the LHCb Collaboration [1]. This
intriguing observation has sparked fruitful theoretical discussions [2]. Among
the many interpretations, diquark-antidiquark cluster mechanism in the
valence-quark picture is currently the most popular theoretical model to
explain the narrow state observed around 6.9 GeV [2]. Although different
theoretical methods, including the quark models, QCD sum rules and effective
Lagrangians, could account for the right mass of the tetraquark candidate
$X(6900)$ without significant difficulty, its $J^{PC}$ quantum numbers and
decay patterns are still under vivid debate. Our study aims at pushing forward
the clarification of the nature of the $X(6900)$, its possible decay patterns,
and the extraction of its resonance pole position from the experimental event
distributions by employing a sophisticated coupled-channel framework based on
general principles of $S$-matrix theory.

In a series of recent works [3, 4, 5, 6, 7, 8, 9, 10], we developed a
theoretical framework that is especially useful for bringing insights into the
inner structures of the resonance states near some underlying two-hadron
thresholds. It is based on the effective-range expansion (ERE) and the
compositeness relation [11, 12, 13, 14, 15, 16, 3, 10] and has been widely and
successfully used to study many possible exotic hadrons, such as the charmed
baryon $\Lambda_{c}(2545)$ [4], the hidden bottom tetraquark candidates
$Z_{b}(10610)/Z_{b}(10650)$ [5], the narrow state $X(3872)$ [6], the
pentaquark candidates $P_{c}(4312)$, $P_{c}(4440)$, $P_{c}(4457)$ [8] and the
hidden charm mesons $Z_{c}(3900)$, $X(4020)$, $\chi_{c1}(4140)$, $\psi(4260)$,
$\psi(4660)$ [7]. This formalism is especially powerful for describing the
elastic single-channel scattering. Just with the inputs of the mass and width
of the resonance, we can estimate the scattering length, effective range, and
compositeness coefficient [3], whic is the probability of finding the two-
hadron component inside the resonance. In this work we will first tentatively
apply this formalism to the elastic scattering processes of
$\chi_{c0}\chi_{c0}$, $\chi_{c1}\chi_{c1}$ and $\psi(3770)J/\psi$ to explore
the newly observed $X(6900)$.

Clearly a realistic study needs to include the $J/\psi J/\psi$ channel, in
which the $X(6900)$ is observed by the experiments [1]. A similar situation
also happens for the pentaquark candidates
$P_{c}(4312),P_{c}(4440),P_{c}(4457)$, which requires one to simultaneously
include at least the $J/\psi p$ and $\Sigma_{c}D^{(*)}$ channels [8].
Generally speaking, when the mass of the resonance is below or rather close to
the underlying two-hadron threshold, other lighter hadronic degrees of freedom
(d.o.f) usually need to be introduced to account for the decay width of the
observed resonances. Then a coupled-channel formalism should instead be used
in this situation. A simple but efficient approach based on the simultaneous
requirements of the decay widths and the compositeness coefficients was
developed in Refs. [9, 5, 8]. The most important merit of this approach is
that we can predict the partial compositeness coefficients (namely, the
probabilities of finding the different components inside the considered
resonance) and the partial decay widths of the resonance with only the minimal
inputs, i.e. the total compositeness of the considered channels, apart from
the mass and width of the resonance from the experimental determinations.
Within the aforementioned formalism we will perform a three-coupled-channel
($J/\psi J/\psi,\chi_{c0}\chi_{c0},\chi_{c1}\chi_{c1}$) study and another
tentative two-coupled-channel [$J/\psi J/\psi$, $\psi(3770)J/\psi$] one for
the $X(6900)$. We point out that the quantum numbers $J^{PC}=0^{++}$ are
assumed for the $X(6900)$ throughout, which is also one of the possibilities
for the ground states in the quark model picture [2].

The total compositeness coefficient entering into the above approach should be
provided beforehand, and it is usually taken as a free parameter in practice.
To reach more definite conclusions on the nature of the $X(6900)$, we further
construct the scattering amplitudes involving the $J/\psi J/\psi$,
$\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ channels by imposing the heavy-
quark symmetry to reduce the number of free parameters. We use a general
coupled-channel near-threshold (of the $\chi_{c}\chi_{c}$ states)
parametrization driven by the presence of a Castillejo-Dalitz-Dyson (CDD) pole
[17] without assuming a specific dynamical model, which is indeed more general
than the ERE [6] or a Flatté parametrization [6, 18]. The resulting coupled-
channel scattering amplitudes are used to fit the experimental di-$J/\psi$
event distributions by taking into account the rescattering due to the two-
hadron systems out of a $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ $S$-wave
source. The resonance pole positions, their couplings to the $J/\psi
J/\psi,\chi_{c0}\chi_{c0},\chi_{c1}\chi_{c1}$ channels and the compositeness
coefficients can then be obtained. We also give further predictions for the
event distributions of the $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$.
Subsequently, similar studies involving the $J/\psi J/\psi$ and
$\psi(3770)J/\psi$ channels are also carried out. Our study also predicts the
presence of a resonance just below the $\chi_{c0}\chi_{c0}$ threshold, named
$X(6825)$, which couples very strongly with the $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ channels and weakly to $J/\psi J/\psi$. As a result, it
manifests as a narrow peak at the $\chi_{c0}\chi_{c0}$ threshold, being mostly
a virtual state of $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$.

This paper is organized as follows. First the tentative study of the elastic
scattering of $\chi_{c0}\chi_{c0}$, $\chi_{c1}\chi_{c1}$ and
$\psi(3770)J/\psi$ based on the ERE is explored to address the emergence of
the $X(6900)$ at its experimental pole position. Then we combine the decay
widths and compositeness relations to perform a three-coupled-channel study by
assuming several different values for the total compositeness coefficients.
Next the fits to the experimental $J/\psi J/\psi$ event distributions with the
$J/\psi J/\psi,$ $\chi_{c0}\chi_{c0},$ and $\chi_{c1}\chi_{c1}$ $S$-wave
coupled-channel scattering amplitudes are carried out. As a result, the pole
positions of the $X(6900)$, their coupling strengths, total and partial
widths, and compositeness coefficients are determined, and compared to those
in the previous approaches. We then make the predictions for the line shapes
of the $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$. Another analysis is also
provided by including the coupled $J/\psi J/\psi$ and $\psi(3770)J/\psi$
channels, which, however, is disadvantaged relative to the previous case.
Finally we give a short summary and conclusions.

##  2 Effective range expansion for the elastic scattering

The standard ERE formalism for the elastic $S$-wave two-body scattering is
given by

$\displaystyle T(E)=\frac{1}{-\frac{1}{a}+\frac{1}{2}r\,k^{2}-i\,k}\,,$ (1)

where $k$ represents the three-momentum in the center of mass (CM) frame, and
$a$ and $r$ in this case correspond to the scattering length and the effective
range, respectively. The CM three-momentum $k$ in the nonrelativistic limit is
related to the CM energy $E$ via

$\displaystyle k=\sqrt{2\mu_{m}(E-m_{\rm th})}\,,$ (2)

where the reduced mass $\mu_{m}$ and the threshold $m_{\rm th}$ are given by

$\displaystyle\mu_{m}=\frac{m_{1}m_{2}}{m_{1}+m_{2}}\,,\quad m_{\rm
th}=m_{1}+m_{2}\,,$ (3)

with $m_{1}$ and $m_{2}$ the masses of the two scattering particles. It is
easy to verify that the ERE amplitude in Eq. (1) fulfills the unitarity
relation

${\rm Im}\,T(E)^{-1}=-k\,,\quad(E>m_{\rm th})\,.$ (4)

Loosely speaking, the ERE formula in Eq. (1) works well in the energy region
sufficiently near the two-particle threshold. Three main facts may hinder the
applicability of the ERE formalism to the energy region where the resonance
lies: the left-hand cuts, the preexisting bare poles [19, 20, 18, 6] (which
can also be introduced as CDD poles [17]), and other important nearby
thresholds. Since the exchanges of the color-singlet light hadrons between the
two-charmonium states are highly suppressed, the contributions from the left-
hand cuts can be safely neglected in the two-charmonium scattering. The
situation for the CDD pole is more subtle. If the CDD pole is distant from the
two-particle threshold, this is a favorable situation in which to apply the
ERE formalism in Eq. (1). While, in the special circumstance in which the CDD
pole is close to the threshold, the ERE formula in Eq. (1) becomes inaccurate
for the description of the near-threshold dynamics [4]. As a consequence, one
should explicitly introduce the CDD pole terms into the scattering amplitudes.
However, it is usually impracticable to predict whether or not the CDD poles
near the thresholds exist. Alternatively, Refs. [4, 5] provide an indirect but
practical way to discern the validity of the ERE formula. It is obtained in
these references that when the mass of the CDD pole $M_{CDD}$ approaches the
threshold $m_{\rm th}$ the resulting $a$ and $r$ will be linearly and
quadratically inversely proportional to $M_{CDD}-m_{\rm th}$ respectively,
i.e.

$\displaystyle a\propto M_{CDD}-m_{\rm th}\,,\qquad
r\propto\frac{1}{(M_{CDD}-m_{\rm th})^{2}}\,.$ (5)

Since other sources could also contribute to the scattering length,333The
quantum mechanical example of a square well is analyzed in Refs. [5, 21],
where one can observe explicitly that the size of $|a|$ could be very
different from the radius $R$ of the square-well potential due to small
changes in its depth. it may not be reliable to infer the CDD pole information
from the value of $a$. In contrast, making a comparison with the standard
strong interaction scale around 1 fm, a large value of the magnitude for the
effective range $r$, would strongly hint at the existence of the near-
threshold CDD pole. In other words, the large effective range $r$ clearly
provides an intuitive and practicable criterion for the existence of the CDD
pole near threshold, which also indicates that the ERE formalism in Eq. (1)
probably is invalid for describing the dynamics around the threshold. On the
contrary, if the resulting magnitude of the effective range $r$ is around 1
fm, it is unlikely that one would necessarily need to introduce necessarily a
CDD pole around the threshold energy region, and the ERE formula in Eq. (1)
would be sufficient to describe the underlying physics.

Through the analytical continuation one can extrapolate the amplitudes into
the second Riemann sheet (RS), where the resonance poles lie. The scattering
amplitude in the second RS, $T_{II}(E)$, takes the form

$\displaystyle T_{II}(E)=\frac{1}{-\frac{1}{a}+\frac{1}{2}r\,k^{2}\,+i\,k}\,.$
(6)

The imaginary part of the three-momentum $k$ should be taken to be positive,
i.e. ${\rm Im}k>0$, in Eqs. (1) and (6). Alternatively, one can still use Eq.
(1) for $T(E)$, but for calculating $k=\sqrt{2\mu_{m}(E-m_{\rm th})}$ in the
first RS the argument of the radicand is taken between $[0,2\pi)$, while in
the second RS it is between $[2\pi,4\pi)$. The resonance pole
$E_{R}=M_{R}-i\Gamma_{R}/2$, with $M_{R}$ the resonance mass and $\Gamma_{R}$
the width, corresponds to the solution of $T_{II}(E_{R})^{-1}=0$, that is,

$\displaystyle-\frac{1}{a}+\frac{1}{2}r\,k_{R}^{2}+i\,k_{R}=0\,.$ (7)

$k_{R}$ stands for the three-momentum at the pole $E_{R}$, i.e.
$k_{R}=\sqrt{2\mu_{m}(E_{R}-m_{\rm th})}$. For simplicity, we introduce
$k_{r}$ and $k_{i}$ to denote the real and imaginary parts of $k_{R}$,
respectively,

$\displaystyle k_{r}={\rm Re}\,k_{R}\,,\quad k_{i}={\rm Im}\,k_{R}\,,$ (8)

where $k_{i}>0$ is taken, consistently with the convention for $k$ in Eqs. (1)
and (6). It is straightforward to solve Eq. (7) to obtain $a$ and $r$ in terms
of $k_{r}$ and $k_{i}$. The solutions of $a$ and $r$ were worked out in Ref.
[4],

$\displaystyle a=-\frac{2k_{i}}{|k_{R}|^{2}}\,,\quad r=-\frac{1}{k_{i}}\,.$
(9)

By combining Eqs. (9) and (6), the Laurent expansion of the $S$-wave
scattering amplitude in the second RS reads [5]

$\displaystyle T_{II}(k)=\frac{-k_{i}/k_{r}}{k-k_{R}}+\ldots\,,$ (10)

where one can easily identify $-k_{i}/k_{r}$ as the residue of the partial-
wave amplitude (PWA) at the pole position in the variable $k$. In our previous
study [3, 5], the compositeness coefficient $X$, corresponding to the weight
of the two-particle component inside the resonance, is shown to be equal to
this residue,

$\displaystyle X=-\frac{k_{i}}{k_{r}}\,.$ (11)

It is also proved in Ref. [5] that, when the mass of the resonance pole lies
above the considered threshold [3], $X$ in the previous equation is bounded
within the range $[0,1]$, and hence it meets the requirement for a
probabilistic interpretation.

According to Eqs. (9) and (11), once the mass and width of the resonance are
known, the scattering length, effective range, and the compositeness
coefficient can be correspondingly predicted under an assumption of single-
channel scattering. This assumption in the present case of interest on the
$X(6900)$ can be considered simplistic since at least three a priori relevant
channels near the nominal mass of the resonance, the $\chi_{c0}\chi_{c0}$,
$\chi_{c1}\chi_{c1}$, and $J/\psi\,\psi(3770)$, can be identified. Their
thresholds lie at $6829.4$, $7021.3$ and $6870.6$ MeV, respectively, and have
$\bar{c}\bar{c}cc$ as a valence-quark composition, so that their mutual
interactions are Okubo-Zweig-Iizuka (OZI) allowed.

Resonance | Mass | Width | Threshold | $a$ | $r$ | $X$
---|---|---|---|---|---|---
| (MeV) | (MeV) | (MeV) | (fm) | (fm) |
$X(6900)$-I | $6905\pm 13$ | $80\pm 38$ |  $\chi_{c0}\chi_{c0}$ (6829.4) | $-0.18\pm 0.07$ | $-1.52\pm 0.69$ | $0.25\pm 0.11$
$X(6900)$-II | $6886\pm 16$ | $168\pm 77$ |  $\chi_{c0}\chi_{c0}$ (6829.4) | $-0.32\pm 0.06$ | $-0.72\pm 0.26$ | $0.53\pm 0.16$
Table 1:  Scattering length $a$, effective range $r$ and compositeness
coefficient $X$ from the ERE study. The two different sets of mass and width
values for the $X(6900)$ are taken from the LHCb determinations [1].

We give the numerical results in Table 1 for the $\chi_{c0}\chi_{c0}$
uncoupled scattering, where the two different sets of masses and widths of the
$X(6900)$ from the models I and II of LHCb [1] are separately analyzed. The
values of the mass and width of the $X(6900)$ resonance from Ref. [1] are

$\displaystyle\text{Model I:}~{}M$ $\displaystyle=6905\pm 11\pm
7~{}\text{MeV}\,,~{}\Gamma=80\pm 19\pm 33~{}\text{MeV}\,,$ (12)
$\displaystyle\text{Model II:}~{}M$ $\displaystyle=6886\pm 11\pm
11~{}\text{MeV}\,,~{}\Gamma=168\pm 33\pm 69~{}\text{MeV}\,,$

where the distinction is based on the treatment of the nonresonant [with
respect to the $X(6900)$] background. For both sets, the masses of the
$X(6900)$ are below the $\chi_{c1}\chi_{c1}$ threshold, which is $7021.3$ MeV,
and hence we cannot interpret the $X$ defined in Eq. (11) as the probability
[3]. The scattering length and effective range resulting in the elastic
$\chi_{c1}\chi_{c1}$ channel are found to be

$\displaystyle a=-0.59\pm 0.04\,,\quad r=-0.31\pm 0.02\,,~{}({\rm case-I})\,,$
$\displaystyle a=-0.51\pm 0.05\,,\quad r=-0.28\pm 0.02\,,~{}({\rm case-
II})\,,$ (13)

given in units of femtometers.

The small value of the $X$ obtained for $\chi_{c0}\chi_{c0}$ scattering
indicates that there is an active extra source of dynamics beyond the
explicitly included channel, for instance because of the nearby presence of a
CDD pole or due to other channels not explicitly included or both effects
simultaneously. This could hamper the applicability of the elastic ERE to
reproduce the pole position of the $X(6900)$, and further study is required.
On the other hand a value of $X\simeq 1$ would perfectly confirm our onset
assumption on the dominance of this channel in order to justify the single-
channel treatment. Therefore, this result clearly indicates that a coupled-
channel analysis is required for the $X(6900)$. Similar conclusions can be
also made for case II, although the value of $X$ is larger in this case, as it
is compatible with the previous one at the level of 1 standard deviation. The
mild values for the effective range $r$ imply that the scenario with a near-
threshold CDD pole is disfavored in the uncoupled case. It should be pointed
out that the total width of the $X(6900)$ state is implicitly assumed to be
saturated by the $\chi_{c0}\chi_{c0}$ or $\chi_{c1}\chi_{c1}$ in this
simplified framework based on assuming the dominance of only one channel
within the elastic ERE. This assumption could be unrealistic, since the
partial decay width to the $J/\psi J/\psi$ channel is likely non-negligible.
Therefore, a more realistic study requires the information of the partial
decay width to $\chi_{c0}\chi_{c0}$, which will be worked out in the next
section within a coupled-channel analysis, after performing fits to the
experimental event distributions.

An analogous discussion is in order, when we apply the ERE study to the
$\psi(3700)J/\psi$ channel, for which the scattering length $a$, effective
range $r$ (given in units of ${\rm fm}$), and compositeness coefficient $X$
are determined to be

$\displaystyle a=-0.39\pm 0.12\,,\quad r=-1.12\pm 0.46\,,\quad X=0.46\pm
0.18\,,~{}({\rm case-I})\,,$ $\displaystyle a=-0.47\pm 0.10\,,\quad r=-0.57\pm
0.16\,,\quad X=0.83\pm 0.17\,,~{}({\rm case-II})\,.$ (14)

The width of the $\psi(3770)$ is $27.2\pm 1.0$ MeV [22] and we take it into
account by employing a complex mass, following the approach of Refs. [4, 7].
The resulting values found are now

$\displaystyle a=-0.33\pm 0.18\,,\quad r=-1.60\pm 1.06\,,\quad X=0.34\pm
0.22\,,~{}({\rm case-I})\,,$ $\displaystyle a=-0.50\pm 0.12\,,\quad r=-0.64\pm
0.22\,\quad X=0.80\pm 0.20\,,~{}({\rm case-II})\,.$ (15)

Comparing the two sets of values in Eqs. (2) and (2) it is obvious that the
finite-width effects from the $\psi(3770)$ are small. The resulting values of
$a$ and $r$ are similar to those of the $\chi_{c0}\chi_{c0}$ case, although
the compositeness coefficients become larger because the threshold of the
$\psi(3770)J/\psi$ is closer to the resonance mass. However, the fact that
several important thresholds are involved in this energy region, as indicated
above, warns us that the single-channel ERE is probably unreliable, which also
agrees with the findings in Sec 3.1 as explained therein.

##  3 Coupled-channel studies

In the previous discussion, one notices that the coupled-channel formalism is
needed to describe the $X(6900)$. We implement two approaches to consider the
couplings between several channels, as we detail next.

###  3.1 Three-coupled-channel case

We consider the $J/\psi J/\psi$ (1), $\chi_{c0}\chi_{c0}$ (2) and
$\chi_{c1}\chi_{c1}$ (3) three-coupled channels, with the label for every
channel given in parentheses. The contribution from the $\eta_{c}\eta_{c}$
channel, whose threshold is rather distant from the energy region of interest
around 6.9 GeV, is effectively reabsorbed into the $J/\psi J/\psi$ channel. In
the infinite quark mass limit, the spin symmetry of the heavy quarks becomes
exact, which in turn predicts special patterns for the heavy-hadron spectra.
For a heavy-quarkonium system with charm quarks, heavy-quark symmetry predicts
that the $J/\psi$ and $\eta_{c}$ will form a spin doublet, and the
$\chi_{c0},\chi_{c1},\chi_{c2}$, and $h_{c}$ will form another spin multiplet
[23]. These predictions are in reasonable agreement with the experimental
measurements [22]. To reduce the number of free parameters, we impose heavy-
quark symmetry on the couplings between the charmonia and the $X(6900)$ state.
To be more specific, the coupling of the $J/\psi$ pair with $X(6900)$ is
denoted as $g_{a}$. The couplings of the $\chi_{c0}\chi_{c0}$,
$\chi_{c1}\chi_{c1}$ and the $X(6900)$ are $g_{b}$ and $g_{b}/\sqrt{3}$,
respectively, since they are the same aside from a Clebsch-Gordan factor. To
properly account for the threshold effects, we take the masses of the
charmonia according to their physical values [22].

To fix the two couplings $g_{a}$ and $g_{b}$, we solve the two equations of
the decay width and the compositeness relation. The partial compositeness
coefficient $X_{j}$, i.e. the fraction of the two-particle state of the $j$th
channel in the resonance, is given by [3]

$\displaystyle X_{j}$ $\displaystyle=$
$\displaystyle|g_{j}|^{2}\left|\frac{dG_{j}^{({\rm
II})}(s_{R})}{ds}\right|\,,$ (16)

where $g_{j}$ denotes the coupling strength between the two particles in the
$j$th channel and the resonance. Depending on whether the pole lies
below(above) the $j$th-channel threshold, one should take the two-point one-
loop function $G_{j}(s)$[$G_{j}^{\rm II}(s)$] in the first(second) RS. The
explicit expression of $G_{j}(s)$ from dimensional regularization by replacing
the divergent term with a subtraction constant takes the form

$\displaystyle G_{j}(s)$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}\left[a(\mu^{2})+\log\frac{m_{2}^{2}}{\mu^{2}}-x_{+}\log\frac{x_{+}-1}{x_{+}}-x_{-}\log\frac{x_{-}-1}{x_{-}}\right]\,,$
$\displaystyle x_{\pm}$ $\displaystyle=$
$\displaystyle\frac{s+m_{1}^{2}-m_{2}^{2}}{2s}\pm\frac{q_{j}(s)}{\sqrt{s}}\,.$
(17)

The expression given in Eq. (3.1) represents $G_{j}(s)$ in the first or
physical RS, and its corresponding formula in the second RS reads [24]

$\displaystyle G^{\rm II}_{j}(s)=G_{j}(s)-i\frac{q_{j}(s)}{4\pi\sqrt{s}}\,.$
(18)

The imaginary part of $G^{\rm II}_{j}(s)$ has the opposite sign of that of
$G_{j}(s)$ above threshold. In Eq (3.1) one has to fix $m_{1}$ and $m_{2}$ to
the masses of the two particles in this channel. The CM three-momentum in the
channel $j$ is $q_{j}(s)$, given by the standard kinematical expression

$\displaystyle
q_{j}(s)=\frac{\sqrt{[s-(m_{j1}+m_{j2})^{2}][s-(m_{j1}-m_{j2})^{2}]}}{2\sqrt{s}}\,.$
(19)

The function $G_{j}(s)$ is independent of the regularization scale $\mu$ due
to the mutual cancellation of the $\mu$ dependence between the first two terms
in Eq. (3.1), and to be more specific we set $\mu=770$ MeV throughout this
work. Notice that the derivative of the $G_{j}^{(II)}(s)$ function is
independent of the subtraction constant term $a(\mu)$. We note that there are
several proposals in the literature about the extension of Weinberg’s
compositeness relation [11] from the bound state to the resonance situation
[12, 13, 14, 15, 16, 3]. We refer to the reader Ref. [7] for further
comparisons between different approaches.

When the mass of the resonance clearly lies above the threshold of the $j$th
channel, its partial decay width is calculated by the standard formula [22]

$\displaystyle\Gamma_{j}=|g_{j}|^{2}\frac{q_{j}(M_{R}^{2})}{8\pi
M_{R}^{2}}\,,$ (20)

with $q_{j}(M_{R}^{2})$ being the relativistic CM three momentum at the
resonance mass. In the situation when the mass of the resonance lies close to
or even below the $j$th threshold, the above formula is no longer applicable
and we introduce the Lorentzian distribution to account for the finite-width
effect [9, 5],

$\displaystyle\Gamma_{j}=|g_{j}|^{2}\int_{m_{\rm
th}}^{M_{R}+2\Gamma_{R}}dw\,\frac{q_{j}(w^{2})}{16\pi^{2}\,w^{2}}\frac{\Gamma_{R}}{(M_{R}-w)^{2}+\Gamma_{R}^{2}/4}\,,$
(21)

which naturally recovers the standard decay-width formula (20) in the narrow-
width limit for $M_{R}>m_{\rm th}$. The equations to fix $g_{a}$ and $g_{b}$
then read

$\displaystyle X=X_{1}+X_{2}+X_{3}$
$\displaystyle=|g_{a}|^{2}\left|\frac{dG_{1}^{\rm
II}(s_{R})}{ds}\right|+|g_{b}|^{2}\left|\frac{dG_{2}^{\rm
II}(s_{R})}{ds}\right|+\frac{|g_{b}|^{2}}{3}\left|\frac{dG_{3}(s_{R})}{ds}\right|\,,$
(22)

and

$\displaystyle\Gamma=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}$
$\displaystyle=|g_{a}|^{2}\frac{q_{1}(M_{R}^{2})}{8\pi
M_{R}^{2}}+|g_{b}|^{2}\int_{m_{\rm
th,2}}^{M_{R}+2\Gamma_{R}}dw\,\frac{q_{2}(w^{2})}{16\pi^{2}\,w^{2}}\frac{\Gamma_{R}}{(M_{R}-w)^{2}+\Gamma_{R}^{2}/4}$
$\displaystyle+\frac{|g_{b}|^{2}}{3}\int_{m_{\rm
th,3}}^{M_{R}+2\Gamma_{R}}dw\,\frac{q_{3}(w^{2})}{16\pi^{2}\,w^{2}}\frac{\Gamma_{R}}{(M_{R}-w)^{2}+\Gamma_{R}^{2}/4}\,,$
(23)

where the $1/3$ factors in $X_{3}$ and $\Gamma_{3}$ correspond to the Clebsch-
Gordan coefficient from the angular-momentum superposition. Equations (22) and
(3.1) can uniquely determine the coupling strengths $|g_{a}|$ and $|g_{b}|$ as
a function of the total compositeness $X$. In terms of these couplings we can
then calculate the partial decay widths and compositeness coefficients for the
$J/\psi J/\psi$, $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ channels. The
results are summarized in Table 2, where $X$ in Eq. (22) should be taken as an
external input (which we also fix below by implementing a coupled-channel
dynamical study). It is found that there is a maximum value of $X$, which is
0.4 for the case I and 0.9 for the case II, that allows Eqs. (22) and (3.1) to
have solutions. We have tried several different values for the $X$. Larger
values of $X$ lead to a smaller magnitude of $|g_{a}|$ and a bigger one for
$|g_{b}|$. As a result, the partial decay width of the $J/\psi J/\psi$ channel
becomes smaller and the decay width of $\chi_{c0}\chi_{c0}$ tends to increase.
Since the threshold of the $\chi_{c1}\chi_{c1}$ is clearly higher than the
resonance mass, its partial decay width is always very small. With increasing
values of $X$, the partial compositeness coefficient of the
$\chi_{c0}\chi_{c0}$ also becomes larger, while the compositeness values for
$J/\psi J/\psi$ and $\chi_{c1}\chi_{c1}$ always remain small.444Despite the
threshold for $\chi_{c1}\chi_{c1}$ being clearly larger than $M_{R}$, we have
still calculated the compositeness for this channel because its resulting
values are clearly meaningful. This is because they are driven by
$|g_{b}|^{2}/3$, with $g_{b}$ being the same as for $\chi_{c0}\chi_{c0}$,
times the modulus squared of the derivative of the $\chi_{c1}\chi_{c1}$
$G_{3}(s)$ function. The latter is also necessarily smaller than for
$\chi_{c0}\chi_{c0}$ because its threshold is farther from the resonance pole
position. An analogous analysis by taking the channels $J/\psi J/\psi$ and
$\psi(3770)J/\psi$ is discussed in Sec. 3.2.

Channel | $|g_{a}|$ | $|g_{b}|$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $X_{1}$ | $X_{2}$ | $X_{3}$ |
---|---|---|---|---|---|---|---|---|---
| (GeV) | (GeV) | (MeV) | (MeV) | (MeV) |  |  |  |
$X(6900)$-I |  |  |  |  |  |  |  |  |
$X=0.1$ | $7.1_{-2.1}^{+2.2}$ | $6.8_{-0.8}^{+0.6}$ | $64.7_{-33.4}^{+42.5}$ | $15.2_{-4.6}^{+5.3}$ | $0.1_{-0.1}^{+0.3}$ | $0.02_{-0.01}^{+0.02}$ | $0.06_{-0.01}^{+0.01}$ | $0.01_{-0.00}^{+0.00}$ |
$X=0.4$ | $0.6_{-0.5}^{+5.7}$ | $15.4_{-0.5}^{+0.5}$ | $0.4_{-0.4}^{+49.7}$ | $79.2_{-12.5}^{+11.1}$ | $0.4_{-0.4}^{+1.8}$ | $0.00_{-0.01}^{+0.02}$ | $0.33_{-0.01}^{+0.01}$ | $0.07_{-0.01}^{+0.01}$ |
$X(6900)$-II |  |  |  |  |  |  |  |  |
$X=0.1$ | $11.3_{-3.5}^{+2.7}$ | $5.0_{-3.1}^{+1.6}$ | $160.7_{-83.6}^{+83.3}$ | $7.0_{-6.1}^{+6.8}$ | $0.3_{-0.3}^{+0.1}$ | $0.06_{-0.03}^{+0.03}$ | $0.03_{-0.03}^{+0.03}$ | $0.01_{-0.01}^{+0.01}$ |
$X=0.4$ | $8.9_{-5.5}^{+3.0}$ | $15.0_{-1.0}^{+0.5}$ | $100.9_{-86.2}^{+79.0}$ | $64.3_{-16.9}^{+13.7}$ | $2.8_{-2.6}^{+3.7}$ | $0.04_{-0.03}^{+0.03}$ | $0.31_{-0.03}^{+0.03}$ | $0.06_{-0.01}^{+0.01}$ |
$X=0.9$ | $1.0_{-1.0}^{+6.6}$ | $23.7_{-1.4}^{+1.5}$ | $1.3_{-1.3}^{+71.9}$ | $159.9_{-38.9}^{+44.4}$ | $6.8_{-4.8}^{+10.1}$ | $0.00_{-0.00}^{+0.03}$ | $0.76_{-0.02}^{+0.02}$ | $0.14_{-0.02}^{+0.01}$ |
Table 2:  The solutions of Eqs. (22) and (3.1). The total compositeness
coefficient $X$ should be provided as an external input. The maximum values of
$X$ for Eqs. (22) and (3.1) admitting solutions are 0.4 for case I and 0.9 for
case II, respectively.

To reach a more definite conclusion using the above approach, it is necessary
to pin down the value of the total compositeness coefficient $X$, which is,
however, not known in advance. Another way to proceed is to further constrain
the coupling strengths $g_{a}$ and $g_{b}$, which will in turn give more
definite values for the $X$. In the following, we perform fits to the $J/\psi
J/\psi$ invariant-mass distributions from the LHCb Collaboration [1], in order
to obtain more definite values for the couplings and the resonance pole
position.

Assuming the quantum numbers $J^{PC}=0^{++}$ for the $X(6900)$, the three
channels $J/\psi J/\psi$ (1), $\chi_{c0}\chi_{c0}$ (2) and
$\chi_{c1}\chi_{c1}$ (3), are all in the $S$ wave. The coupled-channel
scattering amplitudes take the form [25]

$\displaystyle\mathcal{T}(s)=[1-\mathcal{V}(s)\cdot
G(s)]^{-1}\cdot\mathcal{V}(s)\,,$ (24)

where the right-hand cut is generated through the diagonal matrix $G(s)$,
whose diagonal matrix elements are the functions $G_{j}(s)$ given in Eq.
(3.1). As a result, Eq. (24) satisfies unitarity [25, 26]. The interacting
kernel $\mathcal{V}(s)$ is given by

$\displaystyle\mathcal{V}(s)=\left(\begin{matrix}0&b_{12}&b_{13}\\\
b_{12}&\frac{b_{22}}{M_{J/\psi}^{2}}(s-M_{CDD}^{2})&\frac{b_{23}}{M_{J/\psi}^{2}}(s-M_{CDD}^{2})\\\
b_{13}&\frac{b_{23}}{M_{J/\psi}^{2}}(s-M_{CDD}^{2})&\frac{b_{33}}{M_{J/\psi}^{2}}(s-M_{CDD}^{2})\\\
\end{matrix}\right)\,,$ (25)

where the $b_{ij}$ parameters are dimensionless. There is a normalization
difference between the unitarized phenomenological amplitude $\mathcal{T}$ in
Eq. (24) and the ERE amplitude $T$ in Eq. (1). They are related by
$\mathcal{T}=8\pi\sqrt{s}\,T$. To reduce the free parameters, we impose the
heavy-quark symmetry to constrain the parameters in the perturbative
amplitudes

$\displaystyle b_{13}=\frac{b_{12}}{\sqrt{3}}\,,\quad
b_{23}=\frac{b_{22}}{\sqrt{3}}\,,\quad b_{33}=\frac{b_{22}}{3}\,,$ (26)

where the Clebsch-Gordan coefficients of the angular momenta for
$\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ have been taken into account. We
point out that general parametrizations of the perturbative amplitudes
$\mathcal{V}(s)$ have been exploited by introducing polynomial terms, and it
turns out that the successful fits generally prefer the form of
$s-M_{CDD}^{2}$ in the $\mathcal{V}(s)$ for the $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ scattering amplitudes, which corresponds to a single CDD
pole associated with the unique resonance around it. For the mixing amplitudes
$J/\psi J/\psi\to\chi_{c0}\chi_{c0}\,,\chi_{c1}\chi_{c1}$ we find that it is
enough to take the constant terms to obtain reasonable fits, as shown in Eq.
(25). This fact indicates a smooth direct interaction kernel involving the
$J/\psi J/\psi$ channel with the other channels, as expected for a far-
threshold channel relevant only in providing the total decay width of the
resonance.

The formula to describe the experimental $J/\psi J/\psi$ event distributions
reads

$\displaystyle\frac{d\mathcal{N}(s)}{d\sqrt{s}}=|B_{1}(s)|^{2}\frac{q_{{J/\psi}{J/\psi}}(s)}{M_{J/\psi}^{2}}\,,$
(27)

where the production amplitudes are given by the general parametrization [26]

$\displaystyle B(s)=[1-\mathcal{V}(s)\cdot G(s)]^{-1}\cdot\mathcal{P}\,.$ (28)

In this equation $\mathcal{P}$ is the vector of production vertexes, which is
taken as the constant array

$\displaystyle\mathcal{P}=\left(\begin{matrix}d_{1}\\\ d_{2}\\\
d_{3}\end{matrix}\right)\,,$ (29)

subsequently modulated by the final-state interactions driven by the
$\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ strong rescattering [27, 28].
The production parameters $d_{j}$ are dimensionless, due to the introduction
of the $M_{J/\psi}^{2}$ in Eq. (27).

Again we impose the heavy-quark symmetry to further constrain the vertexes, so
that $d_{3}=d_{2}/\sqrt{3}$. To be consistent with the assumption in Eq. (25)
that the $J/\psi J/\psi$ channel is weakly coupled to the $X(6900)$ state, we
set the production vertex $d_{1}$ to zero and allow $d_{2}$ to vary only in
the fits. It is pointed out that when releasing $d_{1}$ the fits indeed prefer
very large ratios of $d_{2}/d_{1}$, but typically give extremely large
uncertainties, showing strong correlation between the two parameters.
Therefore our treatment to fix $d_{1}=0$ not only is motivated by the
scattering amplitude (25) but also helps to stabilize the various fits.

For the subtraction constant, its natural value can be estimated by matching
the functions $G_{j}(s)$ calculated in dimensional regularization and with a
three-momentum cutoff $q_{\rm max}$ at threshold, as explained in Refs. [26,
29, 30]. This leads to

$\displaystyle a(\mu)=-2\log\left(1+\sqrt{1+\frac{m^{2}}{q_{\rm
max}^{2}}}\right)+\cdots\simeq-3.0\,,$ (30)

by taking $\mu=q_{\rm max}=1.0$ GeV and $m=m_{\chi_{c0}}$. We will take a
universal value for the subtraction constants in the three channels (since
masses are rather similar) and fix it to the one given in Eq. (30). It is
further verified that other natural values ranging from $-3$ to $-2$ lead to
quite similar results, as explicitly shown later.

Regarding the parameter $M_{CDD}$, we have scanned its values around the range
of 6.9 GeV so that the other free parameters are fitted for every value of
$M_{CDD}$ fixed, and then there is a clear minimum for the resulting
$\chi^{2}$. To obtain the stable fits, we fix $M_{CDD}$ at the values
providing the minimum of $\chi^{2}$ from the scanning process.

We focus on the experimental data in the energy region around 6.9 GeV [1],
which amount to 12 data points, as shown in Fig. 1. We take the background
contributions from the experimental analyses [1], called there models I and
II, to distinguish the two different types of fits that we then perform and
which are denoted by Fits I and II, respectively.555 Here “background” denotes
all the other contributions in the experimental analysis by the LHCb
Collaboration [1] that do not correspond to the resonance signal of the
$X(6900)$. The latter is analyzed here within a more sophisticated coupled-
channel framework. The main difference between models I and II in Ref. [1] is
that the latter reproduces a dip in the data at around 6.75 GeV. In the fits
the free parameters within our approach are finally $b_{12}$, $b_{22}$ in
$\mathcal{V}(s)$ (25) and $d_{2}$ in $\mathcal{P}(s)$ (29), with $M_{CDD}$
determined as explained.

| $\chi^{2}/{\rm d.o.f}$ | $a(\mu)$ | $M_{CDD}$ | $b_{22}$ | $b_{12}$ | $d_{2}$
---|---|---|---|---|---|---
Fit-I | $1.6/(12-3)$ | $-3.0^{*}$ | $6910^{*}$ | $10817_{-2096}^{+8378}$ | $151_{-99}^{+153}$ | $2213_{-316}^{+2106}$
Fit-II | $4.9/(12-3)$ | $-3.0^{*}$ | $6885^{*}$ | $21085_{-7359}^{+15141}$ | $484_{-112}^{+239}$ | $3646_{-714}^{+1325}$
Table 3:  Fit-I and -II obtained with background contributions taken from
models I and II in Ref. [1], respectively. The entries marked with asterisks
are fixed during the fits, as explained in the text, cf. Eq. (30) and the full
paragraph following it. $M_{CDD}$ is given in units of MeV, and the parameters
$b_{22}$, $b_{12}$, and $d_{2}$ are dimensionless.

We give the outputs of the central fits in Table 3, labeled as Fit-I and Fit-
II. For each fit performed $M_{CDD}$ is fixed to some value and, as indicated
above, after this scanning there is a clear minimum in the $\chi^{2}$ for the
values $M_{CDD}=6910$ and 6885 MeV in Fit-I and Fit-II, respectively. With the
fitted parameters, we then calculate the resonance pole positions, their
residues, and the compositeness coefficients. The resonance poles lie in the
complex energy plane of an unphysical RS, which can be accessed via the
analytical extrapolation of the $G_{j}(s)$ functions. Different unphysical RSs
of the coupled-channel scattering amplitudes in Eq. (24) can be accessed by
properly taking $G_{j}(s)$ or $G^{\rm II}_{j}(s)$ for different channels. The
second RS is labeled as $(-,+,+)$, where the plus(minus) sign in the $j$th
entry indicates taking $G_{j}(s)$($G^{\rm II}_{j}(s))$ in the $j$th channel.
In this convention, the first, third, fourth and fifth sheets are labeled as
$(+,+,+)$, $(-,-,+)$, $(+,-,+)$ and $(-,-,-)$, respectively. The most relevant
resonance poles are found to lie in the third RS (which connects continuously
with the physical RS between the $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$
thresholds). The matrix elements of the scattering matrix in the unphysical RS
around the resonance pole region can be written as

$\displaystyle\mathcal{T}_{kj}(s)=\frac{\gamma_{k}\gamma_{j}}{s-M_{\rm
pole}^{2}}+\cdots\,,$ (31)

where $\gamma_{k,j=1,2,3}$ are the couplings of the resonance to the
corresponding channels and can then be obtained by working out the residue of
the PWAs at the resonance pole $M_{\rm pole}$. The omitted terms in Eq. (31)
are the regular parts in the $s-M_{\rm pole}^{2}$ Laurent expansion. The pole
positions and the resonance couplings $|\gamma_{i=1,2,3}|$ to the different
channels are summarized in Table 4. The partial compositeness coefficients
$X_{i=1,2,3}$ can be calculated via Eq. (16), and the results are also given
in Table 4. The total compositeness $X$ is given simply by the sum of
$X_{i=1,2,3}$. The masses and widths of the resonances from Fit-I and Fit-II
are very compatible with the experimental determinations [1] given in Eq.
(12). As a result, when taking the masses and widths from the pole positions
of the resonances, the ERE parameters in Table 1 change only very slightly.

Let us stress the small value obtained for the total compositeness with
$X<0.2$ for the two fits. This fact clearly indicates the dominance of a bare
component for the $X(6900)$ and the small weight of the two-hadronic
components in its nature. This conclusion has been reached without assuming
any specific dynamical model, instead relying on a general $S$-matrix
parametrization, Eqs. (24) and (25). The smallness of $X$ is a reflection of
the value of $M_{CDD}$ lying so close to the resonance mass [6, 20]. This is
the basic point stressed in the Morgan’s counting-pole criterion on the nature
of a resonance [20, 19], because it drives to the proliferation of similar
pole positions in different RSs [20]. We have checked that this is the case
here too, and poles are found in different RSs associated with the inelastic
channel $\chi_{c1}\chi_{c1}$ with little variation in their positions, as
required by this criterion.

Figure 1:  The $J/\psi J/\psi$ event distribution with the data points taken from Ref. [1] is shown. The blue and red dotted lines represent the background contributions extracted from models I and II in Ref. [1], respectively. The shaded areas correspond to the error bands at the 1 standard deviation by using the parameters shown in Table 3. The histogram given by the green line is obtained by averaging the blue solid line over the bin width for the Fit-I. The left and right vertical lines correspond to the thresholds of $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$, respectively.  | Mass | Width/2 | $|\gamma_{1}|$ | $|\gamma_{2}|$ | $|\gamma_{3}|$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X=\sum_{i=1}^{3}X_{i}$
---|---|---|---|---|---|---|---|---|---
| (MeV) | (MeV) | (GeV) | (GeV) | (GeV) |  |  |  |
Fit-I | $6907_{-3}^{+5}$ | $33_{-10}^{+14}$ | $4.6_{-2.8}^{+2.5}$ | $9.7_{-2.6}^{+1.4}$ | $5.6_{-1.5}^{+0.8}$ | $0.01_{-0.01}^{+0.01}$ | $0.13_{-0.06}^{+0.04}$ | $0.03_{-0.01}^{+0.01}$ | $0.17_{-0.07}^{+0.04}$
Fit-II | $6892_{-2}^{+2}$ | $80_{-17}^{+24}$ | $10.3_{-1.4}^{+1.8}$ | $6.9_{-1.9}^{+1.4}$ | $4.0_{-1.1}^{+0.8}$ | $0.05_{-0.01}^{+0.02}$ | $0.06_{-0.03}^{+0.03}$ | $0.01_{-0.01}^{+0.01}$ | $0.13_{-0.03}^{+0.03}$
Table 4:  The resonance poles of the $X(6900)$ in the third Riemann sheet from
the three-coupled-channel fits are given. $|\gamma_{i=1,2,3}|$ represent the
resonance couplings to the $J/\psi J/\psi,\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ channels, respectively. $X_{i=1,2,3}$ denote the partial
compositeness coefficients, i.e., the probabilities of finding the $J/\psi
J/\psi$, $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ components in the
$X(6900)$ state, respectively.

The resulting fits are plotted in Fig. 1, where the error bands at the 1
standard deviation for the Fit-I (blue solid line) and Fit-II (red dashed
line), are provided. The blue and red dotted lines in Fig. 1 represent the
background contributions of models I and II, respectively, extracted from Ref.
[1]. A cusp effect is clearly seen at the threshold of the
$\chi_{c0}\chi_{c0}$ indicated by the vertical left line, while the right one
corresponds to the $\chi_{c1}\chi_{c1}$ threshold. However, this narrow peak
washes out when performing the averages over the width around 27 MeV of the
experimental energy bins. This is explicitly shown by the histogram in Fig. 1,
which results were obtained by performing the average inside each bin width.
So as not to overload the plot, only the histogram obtained in this way for
Fit-I is explicitly shown. The histogram from Fit-II shows quite a similar
trend.

Indeed, we have checked to see that this remarkable cusp effect in the $J/\psi
J/\psi$ event distribution obtained with our results is unveiling the presence
of a pole quite close to the $\chi_{c0}\chi_{c0}$ threshold which enhances its
contributions. Of course, future experimental measurements with higher
statistics and better energy resolution in the narrow energy region around the
$\chi_{c0}\chi_{c0}$ threshold will then be crucial in order to definitely
discriminate this possible cusp effect and associated resonance. The existence
of this new fully charmed tetraquark resonance, which we name $X(6825)$, is a
neat prediction of our dynamical approach that should be considered in future
experimental searches.

The pole of the $X(6825)$ lies in fourth RS, $(+,-,+)$, and its positions and
residues obtained from the different fits are (primes are affixed to all the
symbols referring to this resonance):666The fourth RS $(+,-,+)$ connects
continuously with the second RS $(-,+,+)$ by crossing the physical $s$ axis
above the $\chi_{c0}\chi_{c0}$ threshold. In turn the second RS connects
continuously with the physical one below the same threshold.

Values for Fit-I: $\displaystyle
E^{\prime}_{R}=6827.0_{-4.8}^{+1.6}-i1.1_{-1.0}^{+1.3}\,,\,\,\,|\gamma^{\prime}_{1}|=1.4_{-0.9}^{+0.6}\,,\,\,\,|\gamma^{\prime}_{2}|=11.9_{-3.1}^{+3.2}\,,\,\,\,|\gamma^{\prime}_{3}|=6.8_{-1.8}^{+1.8}\,,$
(32) Values for Fit-II: $\displaystyle
E^{\prime}_{R}=6820.6_{-2.7}^{+3.0}-i4.0_{-1.6}^{+1.7}\,,\,\,\,|\gamma^{\prime}_{1}|=2.5_{-0.6}^{+0.5}\,,\,\,\,|\gamma^{\prime}_{2}|=15.8_{-0.6}^{+0.7}\,,\,\,\,|\gamma^{\prime}_{3}|=9.1_{-0.4}^{+0.4}\,,$

where $E^{\prime}_{R}$ is given in MeV and the residues are given in GeV.
Comparing them to the residues of the $X(6900)$ in Table 4, we see that the
$X(6825)$ couples even more strongly to the $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ channels. The couplings to these channels follow the rule
that $|\gamma^{\prime}_{3}|\approx|\gamma^{\prime}_{2}|/\sqrt{3}$, according
to the heavy-quark symmetry. However, the coupling to $J/\psi J/\psi$ is much
weaker and this fact explains the much smaller width of the $X(6825)$,
identified as minus twice the imaginary part of its pole position, as compared
with the $X(6900)$. One can semiquantitatively understand the magnitude of the
ratio $|\gamma_{1}/\gamma^{\prime}_{1}|$, by assuming dominance of the $J/\psi
J/\psi$ decay width for the $X(6825)$ and $X(6900)$. In this way, according to
Eq. (20) one has

$\displaystyle\left|\frac{\gamma_{1}}{\gamma^{\prime}_{1}}\right|\approx\sqrt{\frac{q(M^{\prime}_{R})\Gamma}{q(M_{R})\Gamma^{\prime}}}~{},$
(33)

whose central value is equal to 5.3 and 4.3 for Fit-I and -II, respectively.
These figures are pretty much compatible with the numbers of
$|\gamma_{1}/\gamma^{\prime}_{1}|$ given in Table 4 and Eq. (32). The facts
that the pole couples strongly to the $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ channels and that it lies so close to the
$\chi_{c0}\chi_{c0}$ threshold in the $(+,-,+)$ sheet, are clear indications
that the $X(6825)$ is a virtual state made up predominantly of these channels
(loosely speaking, this is usually referred as “molecular origin”). Regarding
the application of Morgan’s counting-pole criterion, the pole position of this
resonance lies only in the sheet indicated, which is also in agreement with
this interpretation. Indeed, in the limit of $b_{12}\to 0$, while keeping the
values for $b_{22}$, $M_{CDD}$ and $a(\mu)$ from the fits, the coupling to
$J/\psi J/\psi$ is zero and the pole becomes a pure virtual state, around 6827
and 6825 MeV for Fit-I and Fit-II respectively. Another interesting limit is
to take the mass of the $\chi_{c1}$ equal to the physical mass of the
$\chi_{c0}$. Then the pole moves to the first RS and it is a bound state. From
this point if $b_{12}$ is taken back to its fit value in Table 3 then this
bound state becomes a resonance in the second RS due to the coupling to
$J/\psi J/\psi$ with finite width. By continuously increasing the mass of the
$\chi_{c1}$, the jump of the pole to the fourth RS is seen slightly above the
$\chi_{c0}\chi_{c0}$ threshold and then, after a small increase in
$m_{\chi_{c1}}$, the pole turns left and finally evolves up to its final
position in Eq. (32), below the two $\chi_{c0}$ threshold for the physical
mass of the $\chi_{c1}$.

Figure 2:  Our predictions for the scattering amplitudes. The top left and
right panels show the amplitudes of $J/\psi J/\psi\to J/\psi J/\psi$ and
$J/\psi J/\psi\to\chi_{c0}\chi_{c0}$. The bottom left and right panels give
the predictions for the amplitudes of $J/\psi J/\psi\to\chi_{c1}\chi_{c1}$ and
$\chi_{c0}\chi_{c0}\to\chi_{c0}\chi_{c0}$, respectively. Since the heavy-quark
symmetry is imposed, the shapes of the amplitudes involving $\chi_{c1}$ look
rather similar to those with $\chi_{c0}$ and we do not explicitly show other
amplitudes.

Due to the large couplings of the $X(6825)$ to $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ one would expect a prominent peak at the
$\chi_{c0}\chi_{c0}$ threshold in the PWAs involving these channels. However,
its form is asymmetric because of the $\chi_{c0}\chi_{c0}$ threshold. The
predictions for the scattering amplitudes from different fits are shown
explicitly in Fig. 2, where the shaded areas correspond to the error bands at
the 1$\sigma$ level from the Fit-I and Fit-II in Table 3. Strong cusp effects
around the $\chi_{c0}\chi_{c0}$ threshold are clearly seen in all the
scattering amplitudes, though, as expected, this peak is much less prominent
for $T_{11}$. It is much more relevant for the $T_{12}$ and $T_{13}$, and huge
in the diagonal $T_{22}$ and $T_{33}$. Of course, this hierarchy is not more
than a reflection of the sizes of the $|\gamma^{\prime}_{i}|$ given in Eq.
(33). It is also visible in Fig. 2 that the peak at the cusp is broader for
Fit-II than for Fit-I. Regarding the $X(6900)$ in Fig. 2, for Fit-I the
resonant enhancement obviously shows up in the amplitudes $J/\psi J/\psi\to
J/\psi J/\psi,\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$, while in the
transition amplitudes $\chi_{0,1}\chi_{0,1}\to\chi_{0,1}\chi_{0,1}$ the
resonance manifests as a dip due to the destructive interference with the
$X(6825)$. For the Fit-II case, except for $J/\psi J/\psi\to J/\psi J/\psi$,
the $X(6900)$ barely shows any structure in the scattering amplitudes, which
seems to be consistent with the rather large widths from Fit-II [we can say
that the wide $X(6900)$ for Fit-II is eaten up by the $X(6825)$ strong signal
in the amplitudes involving the channels $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$].

Other fits by taking different values of the subtraction constants $a(\mu)$
are also obtained. To make a clear comparison of the fits in Table 3, the
values of $M_{CDD}$ of the Fit-I and Fit-II will be fixed at the same values
as shown in the previous table. For the Fit-IA by fixing $a(\mu)=-2.5$, the
fitted parameters, $\chi^{2}$, resonance pole position $E_{R}$ and $X$ are (in
order): $b_{22}=5234.6$, $b_{12}=155.0$ and $d_{2}=1229.7$, $\chi^{2}=3.5$,
$E_{R}=6925-i\,41$ MeV and $X=0.11$. In the same way for Fit-IIA we have:
$b_{22}=3442.7$, $b_{12}=200.0$ and $d_{2}=1549.6$, $\chi^{2}=7.7$,
$E_{R}=6909-i\,99$ MeV and $X=0.23$. We now take $a(\mu)=-3.5$ which is below
the nominal value $-3.$ Fit-IB and its results are $b_{22}=45382.3$,
$b_{12}=538.0$ and $d_{2}=3353.1$, $\chi^{2}=3.5$, $E_{R}=6923-i\,42$ MeV and
$X=0.04$. For Fit-IIB one has $b_{22}=46782.2$, $b_{12}=871.3$ and
$d_{2}=5734.4$, $\chi^{2}=10.1$, $E_{R}=6879.5-i\,104$ MeV and $X=0.09$. Here
we show only the most relevant resonance poles found in the third RS for the
$X(6900)$. It is noticeable that in all cases the total compositeness
coefficients $X$ clearly remain small, the pole positions vary only moderately
and the $\chi^{2}$ gets worse by around a factor of 2. The resulting curves
obtained in the scenarios of the Fit-IA/B and Fit-IIA/B are not explicitly
shown, since they look quite similar as those from the Fit-I and Fit-II,
respectively. In this way we verify that different fits by taking different
values for the subtraction constants $a(\mu)$ within the same type of
background contributions lead to rather similar results for the di-$J/\psi$
event distributions, while the two different types of fits by using different
models for the background, as provided by the LHCb analyses [1], give
obviously different results.

Since now the total compositeness coefficient $X$ is known from the fits, it
is interesting to redo the analyses by combining Eqs. (22) and (3.1). For case
I when fixing $X=0.17$, the solutions of Eqs. (22) and (3.1) read

$\displaystyle|g_{a}|=6.2~{}{\rm GeV}\,,\quad|g_{b}|=9.5~{}{\rm
GeV}\,,\quad\Gamma_{1}=49.7~{}{\rm MeV}\,,\quad\Gamma_{2}=30.1~{}{\rm
MeV}\,,\quad\Gamma_{3}=0.2~{}{\rm MeV}\,,$ $\displaystyle X_{1}=0.018\,,\quad
X_{2}=0.126\,,\quad X_{3}=0.026\,,$ (34)

which agree well with the Fit-I results from the sophisticated coupled-channel
study in Table 4. For case II when fixing $X=0.13$, the solutions of Eqs. (22)
and (3.1) are

$\displaystyle|g_{a}|=11.1~{}{\rm GeV}\,,\quad|g_{b}|=6.7~{}{\rm
GeV}\,,\quad\Gamma_{1}=154.7~{}{\rm MeV}\,,\quad\Gamma_{2}=12.8~{}{\rm
MeV}\,,\quad\Gamma_{3}=0.5~{}{\rm MeV}\,,$ $\displaystyle X_{1}=0.06\,,\quad
X_{2}=0.06\,,\quad X_{3}=0.01\,,$ (35)

which are also in good accord with the Fit-II results in Table 4. When
obtaining the values in Eqs. (3.1) and (3.1), we take the masses and widths of
the $X(6900)$ from the experimental analyses in Ref. [1]. These equations also
provide the partial decay widths of the $X(6900)$ to the different channels.
It is worth pointing out that when using the masses and widths of the
resonances from our fits in Table 4, the agreement of the residues and partial
compositeness coefficients turn out to be excellent. Therefore, we provide
here a solid demonstration that the coupled-channel methods by utilizing the
compositeness relations and the decay widths, namely, Eqs. (22) and (3.1),
indeed offer a very convenient and reliable approach to studying the resonance
dynamics.

When we consider the results in Table 1 from the single-channel ERE study, it
seems that the compositeness coefficients predicted by the elastic ERE are
larger and in contradiction to the results in Table 4 from the coupled-channel
fits. However, it should be stressed that in the elastic ERE study the total
width of the $X(6900)$ is assumed to be saturated by the $\chi_{c0}\chi_{c0}$
channel, which does not seem to be consistent with the partial decay width
predicted by the coupling/residue in Table 4. To be more specific, we
calculate next the partial decay width to $\chi_{c0}\chi_{c0}$, $\Gamma_{2}$,
by removing from the central values of the total width in Table 4 the easily
calculable $J/\psi J/\psi$ decay widths in terms of the central values of
$|\gamma_{1}|$, cf. Eq. (20) with $|g_{1}|$ substituted for $|\gamma_{1}|$.
The relative uncertainty for the resulting $\Gamma_{2}$ is estimated as twice
the relative error for $|\gamma_{2}|$ (because it depends quadratically on the
coupling). We then obtain the values

$\displaystyle\Gamma_{2}=40^{+11}_{-20}~{}{\rm MeV}\,\,({\rm
Fit-I})\,,\quad\Gamma_{2}=26^{+11}_{-14}~{}{\rm MeV}\,\,({\rm Fit-II})\,.$
(36)

By taking these realistic partial decay widths, the central values for the
single-channel ERE parameters turn out to be

$\displaystyle a=-0.10~{}{\rm fm}\,,\quad r=-3.0~{}{\rm fm}\,,\quad
X=0.13\,\,(\rm Fit-I)\,,$ (37)

and

$\displaystyle a=-0.09~{}{\rm fm}\,,\quad r=-4.1~{}{\rm fm}\,,\quad
X=0.10\,\,(\rm Fit-II)\,.$ (38)

It is clear that the compositeness coefficients from the ERE are now much
smaller than in Table 1, as they should be, and indeed they are fairly
compatible with the results for $X_{2}$ in Table 4 from the fits.

Therefore, after taking a more sophisticated coupled-channel analysis, our
results confirm the findings of Ref. [1] for the $X(6900)$. The resonance pole
positions from Fit-I and Fit-II are very compatible with the masses and widths
determined in models I and II in Ref. [1], respectively. As a novelty, our
coupled-channel study provides new information on the couplings of the
$\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$, which in turn allows us to
calculate the partial compositeness coefficients; cf. Eqs. (3.1) and (3.1).
Furthermore, the coupled-channel analyses also enable us to predict the line
shapes of the distributions of the $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$, as shown in Fig. 3, which could provide useful
guidelines for the experimental study in the next step. Future measurements on
the distributions of the $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ will
definitely be helpful for discriminating among different scenarios proposed
here and for reaching a more definite conclusion for the properties of the
$X(6900)$ state, since the $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ event
distributions look very different for Fit-I and Fit-II, particularly for the
former channel.

Figure 3:  Our predictions for the distributions of (left panel)
$\chi_{c0}\chi_{c0}$ and (right panel) $\chi_{c1}\chi_{c1}$.

###  3.2 Coupled-channel systems with $\psi(3770)J/\psi$

Other channels that one could include in the coupled-channel study are the
$\psi(2S)J/\psi$ and $\psi(3770)J/\psi$ [2], since their mutual interactions
and those with the channels $J/\psi J/\psi$, $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ are OZI allowed. As a complementary study to the
discussions in the previous section, coupled-channel analyses including the
$\psi(3770)J/\psi$ channel (whose threshold is 6870.6 MeV [22]) will be
further explored. We cannot afford to study simultaneously all the four
channels $J/\psi J/\psi$, $\chi_{c0}\chi_{c0}$, $\chi_{c1}\chi_{c1}$ and
$\psi(3770)J/\psi$ due to the proliferation of free parameters.777At least
four more free parameters would be needed when following a similar scheme as
before, namely, $b_{12}$, $b_{14}$, $b_{22}$, $b_{24}$, $b_{44}$, $d_{2}$ and
$d_{4}$, with the subscript 4 referring to the $\psi(3770)J/\psi$ channel..
Because of this, and also due to the fact that the threshold of
$\psi(2S)J/\psi$ is at 6783.0 MeV [22] below the energy region for the
resonance signal at the lower end of Fig. 1, we do not further consider this
channel.

We again consider the quantum numbers $J^{PC}=0^{++}$ for the $X(6900)$ and
study the $S$-wave scattering with the two coupled channels $J/\psi J/\psi$
(1) and $\psi(3770)J/\psi$ (2). We follow the same formalism as in the
previous section and write the perturbative kernel as

$\displaystyle\hat{\mathcal{V}}(s)=\left(\begin{matrix}0&\hat{b}_{12}\\\
\hat{b}_{12}&\frac{\hat{b}_{22}}{M_{J/\psi}^{2}}(s-\hat{M}_{CDD}^{2})\end{matrix}\right)\,,$
(39)

where the hat symbol is introduced over the parameters in order to distinguish
them from those in Eq. (25) that are used for the three-coupled-channel case.
We use this same convention to distinguish everything between these two
scenarios. Needless to say, the unitarized scattering amplitudes and the
production amplitudes will be constructed according to Eqs. (24) and (28),
respectively, with the obvious changes of the corresponding thresholds and
relevant parameters. The subtraction constant $\hat{a}(\mu)$ is fixed by
taking $q_{\rm max}=1$ GeV and $m=m_{\psi(3770)}$, which leads to
$\hat{a}(\mu)=-3.2$. Taking exactly the same background terms and fit
strategies as the previous three-channel study, we give the parameters from
the two-channel fits in Table 5. This fit is not well fixed, as one can
clearly infer because of the large uncertainties in the free parameters
$\hat{b}_{12}$, $\hat{b}_{22}$, and $\hat{d}_{2}$ written in Table 5, which
are essentially undetermined due to their large mutual correlations.
Furthermore, the fit is also unstable around the threshold of the
$\psi(3770)J/\psi$, giving rise to quite different $J/\psi J/\psi$ event
distributions in this energy region that somewhat wash out when averaging
along the width of the experimental bin. This fact potentially disfavors this
type of fits in relation to the three-coupled-channel ones.

Nonetheless, we give for reference the resulting resonance poles, residues,
and the compositeness coefficients for the two-coupled-channel fits in Table 6
so that some trends are common to all the explored fits of this kind. One can
observe that the CDD locations for Fit-I and -II are similar in Tables 4 and
6, and that they lie almost on top of the resonance mass. Indeed, the pole
positions given are also close to each other, with the biggest difference
being the discrepancy between the masses of the resonance in the Fit-II case,
although it is much smaller than the rather large width of the resonance, so
this fact is not really relevant. It is also the case in the two-coupled
channel analysis that virtual poles are present close to the
$\psi(3770)J/\psi$ threshold at around 6855 for Fit-I and Fit-II, similarly to
the $X(6825)$ in the three-coupled-channel case. The conclusion that the
$X(6900)$ resonance originated mainly from the bare CDD pole in our study is
quite robust, regardless of the dynamical channels included near the energy
region around 6.9 GeV. As a matter of fact, the value of $\hat{X}$ in the two-
coupled-channel study is even smaller than in the three-coupled-channel case.
For the two-coupled-channel case $\hat{X}_{1}$ is the largest, with a much
smaller $\hat{X}_{2}$, while for the three-coupled-channel scattering the
opposite situation occurs.

Another interesting fact that points toward a preference for the the three-
coupled-channel case over the two-coupled-channel one is the magnitude of the
couplings $|\gamma_{2}|$ and $|\hat{\gamma_{2}}|$. If one compares their
central values given in Tables 4 and 6, the result is that the values for
$|\gamma_{2}|$ are around a factor of 5 (Fit-I) and 3.5 (Fit-II) larger than
for $|\hat{\gamma}_{2}|$. This is quite a difference and it indicates that the
$X(6900)$ seems to couple much more strongly to $\chi_{c0}\chi_{c0}$, and also
to $\chi_{c1}\chi_{c1}$, than to $\psi(3770)J/\psi$. This result would justify
considering the reduction of possible channels more realistic in the three-
coupled-channel scenario than to the two-coupled-channel one. The smallness of
the residue $|\hat{\gamma}_{2}|$ is caused because of the large value of the
parameter $\hat{b}_{22}$, which tends to be more than an order of magnitude
larger than $b_{22}$.

It is also interesting to redo the analyses of the saturation of width and
compositeness in Eqs. (22) and (3.1) in terms of the channels $J/\psi J/\psi$
and $\psi(3770)J/\psi$. By imposing the total compositeness coefficients from
the fits in Table 5, it is straightforward to solve the equations that follow
from the saturation of the width and compositeness. The solutions are

$\hat{X}$ from Fit-IA: $\displaystyle|\hat{g}_{1}|$ $\displaystyle=7.9~{}{\rm
GeV},\,\,|\hat{g}_{2}|=2.5~{}{\rm GeV},\,\,\hat{\Gamma}_{1}=78.6~{}{\rm
MeV},\,\,\hat{\Gamma}_{2}=1.4~{}{\rm
MeV},\,\,\hat{X}_{1}=0.03,\,\,\hat{X}_{2}=0.01\,,$ (40) $\hat{X}$ from Fit-
IIA: $\displaystyle|\hat{g}_{1}|$ $\displaystyle=11.5~{}{\rm
GeV},\,\,|\hat{g}_{2}|=2.2~{}{\rm GeV},\,\,\hat{\Gamma}_{1}=167.0~{}{\rm
MeV},\,\,\hat{\Gamma}_{2}=1.0~{}{\rm
MeV},\,\,\hat{X}_{1}=0.06,\,\,\hat{X}_{2}=0.01\,,$ (41)

which are consistent with the results for Fit-$\hat{\rm I}$ and Fit-$\hat{\rm
II}$ given in Table 6. To obtain these values the experimental masses and
widths of the $X(6900)$ from Ref. [1] are used. We further verify that the
results are barely affected when the masses and widths are taken from the
resonance poles given in Table 6.

As mentioned above a more sophisticated coupled-channel analysis would be to
simultaneously include the four dynamical channels $J/\psi J/\psi$ (1),
$\chi_{c0}\chi_{c0}$ (2), $\chi_{c1}\chi_{c1}$ (3) and $\psi(3770)J/\psi$ (4).
The biggest challenge in this case is the large number of free parameters. To
perform a check of stability of our previous results with the three-coupled
channels, we treat the $\psi(3770)J/\psi$ channel in a perturbative manner
similar to the $J/\psi J/\psi$, so that $b_{14}=b_{44}=0$ and $b_{12}$,
$b_{24}$, and $d_{2}$ are allowed to float, with $b_{34}=b_{24}/\sqrt{3}$,
$b_{13}=b_{12}/\sqrt{3}$ and $d_{3}=d_{2}/\sqrt{3}$, because of the heavy-
quark symmetry. In this way, only one additional free parameter is needed, see
the discussions in Sec. 3.1. The resulting parameters and the results from
such four-channel fits turn out to be almost identical to the values in Tables
3 and 4, respectively, which points favorably toward the stability of the
results from the three-coupled-channel study.

Alternatively, we could switch the roles of the
$\chi_{c0}\chi_{c0},\chi_{c1}\chi_{c1}$ channels and the $\psi(3770)J/\psi$
one and include the two former channels perturbatively in the two-coupled-
channel study. However, the fit does not become more stable than the previous
one with only the $J/\psi J/\psi$ and $\psi(3770)J/\psi$ channels, and the fit
results are essentially the same as in Table 6 (despite having one more free
parameter). Therefore, we do not dwell further on these results and refer to
the discussion above.

| $\chi^{2}/{\rm d.o.f}$ | $\hat{a}(\mu)$ | $\hat{M}_{CDD}$ | $\hat{b}_{22}$ | $\hat{b}_{12}$ | $\hat{d}_{2}$
---|---|---|---|---|---|---
Fit-$\hat{\rm I}$ | $2.8/(12-3)$ | $-3.2^{*}$ | $6900^{*}$ | ${(2.4_{-1.7}^{+4.6})}\cdot 10^{5}$ | $1303_{-597}^{+1243}$ | $7825_{-3495}^{+6318}$
Fit-$\hat{\rm II}$ | $2.4/(12-3)$ | $-3.2^{*}$ | $6880^{*}$ | ${(1.5_{-0.6}^{+1.8})}\cdot 10^{5}$ | $1356_{-305}^{+741}$ | $9675_{-2043}^{+4674}$
Table 5:  Results from the two-channel fits including the $J/\psi J/\psi$ and $\psi(3770)J/\psi$.  | Mass | Width/2 | $|\hat{\gamma}_{1}|$ | $|\hat{\gamma}_{2}|$ | $\hat{X}_{1}$ | $\hat{X}_{2}$ | $\hat{X}=\sum_{i=1}^{2}\hat{X}_{i}$
---|---|---|---|---|---|---|---
| (MeV) | (MeV) | (GeV) | (GeV) |  |  |
Fit-$\hat{\rm I}$ | $6900_{-1}^{+2}$ | $44_{-16}^{+20}$ | $8.2_{-1.6}^{+1.7}$ | $1.9_{-0.9}^{+1.5}$ | $0.03_{-0.01}^{+0.01}$ | $0.01_{-0.00}^{+0.01}$ | $0.04_{-0.01}^{+0.02}$
Fit-$\hat{\rm II}$ | $6877_{-2}^{+1}$ | $78_{-14}^{+21}$ | $11.1_{-1.1}^{+1.4}$ | $2.0_{-0.7}^{+0.6}$ | $0.06_{-0.01}^{+0.02}$ | $0.01_{-0.00}^{+0.00}$ | $0.07_{-0.01}^{+0.02}$
Table 6:  Resonance poles of the $X(6900)$ in the third Riemann sheet from the
dynamical study including the $J/\psi J/\psi$ and $\psi(3770)J/\psi$ channels
are shown. $|\hat{\gamma}_{i=1,2}|$ stand for the resonance couplings to the
$J/\psi J/\psi$ and $\psi(3770)J/\psi$ channels, respectively.
$\hat{X}_{i=1,2}$ are the partial compositeness coefficients of the $J/\psi
J/\psi$ and $\psi(3770)J/\psi$, respectively.

##  4 Conclusions

In this work we focus on the narrow peak named $X(6900)$ around 6.9 GeV
observed in the $J/\psi J/\psi$ event distributions from the LHCb measurements
[1], which is the first discovered fully heavy-flavored tetraquark candidate.
Several different theoretical approaches, including the effective-range
expansion, a combination of the compositeness relations, and saturation of the
decay width, as well as unitarized phenomenological amplitudes, are used to
investigate the $X(6900)$ state. It is remarkable that different theoretical
methods lead to similar conclusions for the $X(6900)$: the $J/\psi J/\psi$,
$\chi_{c0}\chi_{c0}$, $\chi_{c1}\chi_{c1}$ and $\psi(3770)J/\psi$ components
do not play dominant roles in the $X(6900)$. The most important component
should be a bare or elementary one, e.g., a compact four-charm-quark state or
another microscopic degrees of freedom. This is due to the presence of a CDD
pole lying almost on top of the resonance mass, which makes the compositeness
$X$ very small and to the applicability of the Morgan’s counting-pole
criterion with similar poles lying in different Riemann sheets. We have also
provided the resulting pole positions, couplings to the different channels and
partial as well as total compositeness coefficients. A prediction for the
coupled-channel dynamics between the $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ channels is the emergence of a virtual state just below
the $\chi_{c0}\chi_{c0}$ threshold that couples strongly to those channels and
weakly to the $J/\psi J/\psi$ channel, making its width is very small. We name
this state $X(6825)$. Its presence could be ascertained experimentally by
improving the energy resolution and the statistics of the $J/\psi J/\psi$
event distribution in future experiments.

Our sophisticated coupled-channel study confirms that two different types of
resonance poles can be obtained for the $X(6900)$, by taking two estimates of
the background contributions from the LHCb [1]. Different line shapes for the
$\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$ distributions are predicted.
Future experimental measurements on the $\chi_{c0}\chi_{c0}$ and
$\chi_{c1}\chi_{c1}$ will definitely be helpful to further pinning down the
properties of the $X(6900)$.

## Acknowledgements

We thank Feng-Kun Guo for useful discussions. This work is funded in part by
the Natural Science Foundation of China under Grants No. 11975090 and No.
11575052, the Natural Science Foundation of Hebei Province under Contract No.
A2015205205, and MINECO (Spain) and FEDER (EU) Grants No. FPA2016-77313-P and
MICINN (Spain) No. PID2019-106080GB-C22.

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