# Strong decays of the isovector-scalar $D^{\ast}\bar{D}^{\ast}$ hadronic
molecule

Jin-Cheng Deng  College of Physics Science and Technology, Hebei University,
Baoding 071002, China  Bo Wang  0000-0003-0985-2958<EMAIL_ADDRESS>College
of Physics Science and Technology, Hebei University, Baoding 071002, China
Hebei Key Laboratory of High-precision Computation and Application of Quantum
Field Theory, Baoding, 071002, China  Hebei Research Center of the Basic
Discipline for Computational Physics, Baoding, 071002, China

###### Abstract

We adopt the effective Lagrangian approach to study the strong decays of the
$1^{-}(0^{++})$ $D^{\ast}\bar{D}^{\ast}$ molecular state [denoted as $T_{\psi
0}^{a}(4010)$ according to the LHCb naming convention] through triangle
diagrams. The decay channels include the open-charm $D\bar{D}$, and the
hidden-charm $\eta_{c}\pi$, $J/\psi\rho$, and $\chi_{c1}\pi$. The coupling
between the $T_{\psi 0}^{a}(4010)$ and its constituents
$D^{\ast}\bar{D}^{\ast}$ is obtained by solving for the residue of the
scattering T-matrix at the pole. Our calculation yields a total width of
$(12.0$$-$$35.4)$ MeV for the $T_{\psi 0}^{a}(4010)$ state, with its main
decay channels being $\eta_{c}\pi$ and $\chi_{c1}\pi$. The $X(4100)$ and
$X(4050)$ have similar masses and widths, with both masses being close to the
$D^{\ast}\bar{D}^{\ast}$ threshold. Additionally, their decay final states are
consistent with those of the $T_{\psi 0}^{a}(4010)$. Therefore, it is likely
that they represent the same state and both potentially correspond to the
$T_{\psi 0}^{a}(4010)$. We suggest that future experiments focus on searching
for the $T_{\psi 0}^{a}(4010)$ signal in the final states $\eta_{c}\pi^{-}$,
$\chi_{c1}\pi^{-}$ and $D^{0}D^{-}$ of the $B^{0}\to\eta_{c}\pi^{-}K^{+}$,
$\chi_{c1}\pi^{-}K^{+}$ and $D^{0}D^{-}K^{+}$ processes, respectively, as well
as further investigating its resonance parameters with Flatté-like formula.

##  I Introduction

In 2018, the LHCb Collaboration reported a resonance state, denoted as
$X(4100)^{-}$ [1], in the invariant mass spectrum of $\eta_{c}\pi^{-}$ through
the process $B^{0}\to\eta_{c}\pi^{-}K^{+}$ with a significance exceeding three
standard deviations [2]. The measured mass and width of the resonance are
$m=4096\pm 20^{+18}_{-22}$ MeV and $\Gamma=152\pm 58^{+60}_{-35}$ MeV,
respectively. However, the current data sample is insufficient to determine
whether its spin-parity quantum numbers $J^{P}$ are $0^{+}$ or $1^{-}$. It can
be inferred that this charged structure is an isovector state, with quark
composition $c\bar{c}d\bar{u}$. Like the well-established states such as
$Z_{c}(3900)$ [3] and $Z_{c}(4020)$ [4], the $X(4100)$ is also evidently an
exotic charmoniumlike state.

Several studies have explored the properties of the $X(4100)$ resonance. For
instance, Wang et al. employed the QCD sum rule calculations, supporting the
interpretation that $X(4100)$ is a scalar tetraquark [5]. Wu et al. also
favored the assignment of $X(4100)$ as a $0^{++}$ tetraquark based on
calculations using the chromomagnetic interaction model [6]. In contrast,
Voloshin proposed that $X(4100)$ is a hadrocharmonium—a structure formed by a
compact charmonium $\eta_{c}$ coupled to light quark excitations with pion
quantum number through a QCD analogue of van der Waals force [7]. In Ref. [8],
Zhao proposed two possible interpretations for the $X(4100)$: (i) arising from
the rescattering effect of the S-wave $D^{\ast}\bar{D}^{\ast}$, (ii) a genuine
P-wave resonance of the $D^{\ast}\bar{D}^{\ast}$ system. Cao et al. [9], on
the other hand, proposed that the $X(4100)^{-}$ is the charge conjugate of the
$X(4050)^{+}$ [with mass and width $m=4051\pm 14^{+20}_{-41}$ MeV and
$\Gamma=82^{+21+47}_{-17-22}$ MeV] previously observed by the Belle
Collaboration in the invariant mass spectrum of $\chi_{c1}\pi^{+}$ via the
decay $\bar{B}^{0}\to\chi_{c1}\pi^{+}K^{-}$ [10]. For the other related works,
see Refs. [11, 12].

The proximity of the $X(4100)$ resonance to the $D^{\ast}\bar{D}^{\ast}$
threshold naturally raises question about its potential connections with other
resonances near the $D\bar{D}^{\ast}$ and $D^{\ast}\bar{D}^{\ast}$ thresholds.
These resonances include the well-established $X(3872)$ and $Z_{c}(3900)$ near
the $D\bar{D}^{\ast}$ threshold, and the $Z_{c}(4020)$ near the
$D^{\ast}\bar{D}^{\ast}$ threshold. Despite ongoing debates regarding their
exact nature, with interpretations ranging from hadronic molecules to compact
tetraquarks and kinematic effects, the molecular picture has emerged as the
most prevalent framework for understanding these states [13]. Within the
molecular paradigm, their existence is tightly linked to the interactions of
the $D^{\ast}\bar{D}^{(\ast)}$ systems with different quantum numbers.

Since the discovery of $X(3872)$ [14], a series of studies have been conducted
to investigate the interactions of the $D^{\ast}\bar{D}^{(\ast)}$ systems and
explore the possible molecular states [15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26]. Inspired by the recent observation of the $T_{cs0}(2900)$ [27] and
$T_{c\bar{s}0}^{a}(2900)$ [28], we proposed to connect different hadronic
molecules from the quark-level perspective in our recent works [29, 30]. If
$X(3872)$ and $Z_{c}(3900)$ are the bound and virtual states of the isoscalar
and isovector $D\bar{D}^{\ast}$ system, respectively, we can obtain a virtual
state in the $1(0^{+})$ $D^{\ast}K^{\ast}$ system [29], which is very likely
to correspond to the newly observed $T_{c\bar{s}0}^{a}(2900)$ by the LHCb
[28]. At the same time, we found that the interaction of the $1^{-}(0^{++})$
$D^{\ast}\bar{D}^{\ast}$ system is equal to that of the $1(0^{+})$
$D^{\ast}K^{\ast}$ system, and its interaction strength is about twice that of
the $1^{+}(1^{+-})$ $D^{\ast}\bar{D}^{(\ast)}$ system (see Table IV of Ref.
[29], in which the interaction is dominated by the $\tilde{c}_{a}$ term). That
is to say, if $Z_{c}(3900)/Z_{c}(4020)$ are molecular states of the
$1^{+}(1^{+-})$ $D\bar{D}^{\ast}/D^{\ast}\bar{D}^{\ast}$ systems, then there
must also exist a $1^{-}(0^{++})$ $D^{\ast}\bar{D}^{\ast}$ molecular state.
Since its interaction is stronger, it corresponds to a bound state rather than
a virtual state. The mass range of the $1^{-}(0^{++})$
$D^{\ast}\bar{D}^{\ast}$ state we obtain is $(4007.2$$-$$4016.7)$ MeV [29].

The $D^{\ast}\bar{D}^{\ast}$ molecule with $1^{-}(0^{++})$ will decay into the
open-charm $D\bar{D}$, and hidden-charm $\eta_{c}\pi$, $J/\psi\rho$ and
$\chi_{c1}\pi$ channels. Therefore, further experimental measurements of the
mass and width of $X(4100)/X(4050)$ with higher precision, as well as its
partial decay widths to the $D\bar{D}$, $\eta_{c}\pi$, $J/\psi\rho$ and
$\chi_{c1}\pi$ channels, will be very helpful to distinguish whether $X(4100)$
[2] and $X(4050)$ [10] are the same state, and whether they are both the
$1^{-}(0^{++})$ $D^{\ast}\bar{D}^{\ast}$ molecule. In this work, we employ the
triangle diagram approach to calculate its strong decays, providing useful
information such as total width, partial widths, and branching
fractions/ratios. The coupling constant to its constituents
$D^{\ast}\bar{D}^{\ast}$ will be determined from the residue of the scattering
T-matrix at the pole [31]. The triangle diagram approach, also known as the
hadron loop mechanism, has been widely applied to deal with the strong
transitions of highly excited charmonium (bottomonium) states via the dipion
($\eta$, $\phi$, $\omega$) emissions [32, 33, 34, 35, 36, 37], as well as the
decays [38, 39] and productions [40, 41] of hadronic molecules.

The structure of this article is arranged as follows: In Sec. II, we will
present the effective Lagrangians, coupling constants, and decay amplitudes.
In Sec. III, we will provide our results and related discussions. In Sec. IV,
we will summarize this work.

##  II Effective Lagrangians, coupling constants, and decay amplitudes

###  II.1 Effective Lagrangians

Following the recent LHCb naming convention [42], the $D^{\ast}\bar{D}^{\ast}$
molecule with $1^{-}(0^{++})$ quantum numbers should be named $T_{\psi
0}^{a}(4010)$, with the mass in parentheses coming from our prediction [29].
Throughout the subsequent sections, we will refer to it simply as $T_{\psi
0}^{a}$. The decay processes of this state to $D\bar{D}$, $\eta_{c}\pi$,
$J/\psi\rho$ and $\chi_{c1}\pi$ are shown in Fig. 1, from which it can be seen
that we need the following effective Lagrangians.

Figure 1: The triangle loop diagrams for the strong decays of the $T_{\psi
0}^{a}(4010)$ state. The double line, single line, and dashed line denote the
$T_{\psi 0}^{a}(4010)$, the $D/D^{\ast}$, and the light mesons $\pi/\rho$,
respectively, while the charmonia $\eta_{c}$, $J/\psi$ and $\chi_{c1}$ are
denoted by the thick lines. The contributions arising from the G-parity
transformation are implied in figures (b), (c) and (d).

  * (i)

The Lagrangians for $T_{\psi 0}^{a}D^{\ast}\bar{D}^{\ast}$ coupling.—The
$T_{\psi 0}^{a}$ couples to $D^{\ast}\bar{D}^{\ast}$ via the S-wave
interaction, thus their Lagrangian can be written as

$\displaystyle\mathcal{L}_{0}$ $\displaystyle=$ $\displaystyle
g_{0}\tilde{P}^{\ast\dagger}_{\mu}T_{\psi
0}^{a}P^{\ast\dagger\mu}+\mathrm{H.c.},$ (1)

where $\tilde{P}^{\ast\dagger}_{\mu}=(\bar{D}^{\ast
0\dagger},D^{\ast-\dagger})_{\mu}$, $P^{\ast\dagger}_{\mu}=(D^{\ast
0\dagger},D^{\ast+\dagger})_{\mu}^{T}$. The $T_{\psi 0}^{a}$ here denotes its
isospin triplet,

$\displaystyle T_{\psi 0}^{a}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\frac{1}{\sqrt{2}}T_{\psi
0}^{a0}&T_{\psi 0}^{a+}\\\ T_{\psi 0}^{a-}&-\frac{1}{\sqrt{2}}T_{\psi
0}^{a0}\end{array}\right].$ (4)

The extraction of the coupling constant $g_{0}$ from the residual of
$D^{\ast}\bar{D}^{\ast}$ scattering T-matrix will be demonstrated in Sec.
II.2.

  * (ii)

The Lagrangians for $D^{\ast}D^{(\ast)}\pi$ and $D^{\ast}D^{(\ast)}\rho$
couplings.—The corresponding Lagrangians within the superfield representations
are given by [13, 43]

$\displaystyle\mathcal{L}_{\pi}$ $\displaystyle=$ $\displaystyle
g_{b}\langle\mathcal{H}\gamma^{\mu}\gamma^{5}u_{\mu}\bar{\mathcal{H}}\rangle+g_{b}\langle\bar{\tilde{\mathcal{H}}}\gamma^{\mu}\gamma^{5}u_{\mu}\tilde{\mathcal{H}}\rangle,$
(5) $\displaystyle\mathcal{L}_{\rho}$ $\displaystyle=$ $\displaystyle
i\beta\left\langle\mathcal{H}v^{\mu}(\Gamma_{\mu}-\rho_{\mu})\bar{\mathcal{H}}\right\rangle+i\lambda\left\langle\mathcal{H}\sigma^{\mu\nu}F_{\mu\nu}\bar{\mathcal{H}}\right\rangle$
(6)
$\displaystyle+i\beta\left\langle\bar{\tilde{\mathcal{H}}}v^{\mu}(\Gamma_{\mu}-\rho_{\mu})\tilde{\mathcal{H}}\right\rangle+i\lambda\left\langle\bar{\tilde{\mathcal{H}}}\sigma^{\mu\nu}F_{\mu\nu}\tilde{\mathcal{H}}\right\rangle,$

where the notation $\langle x\rangle$ denotes the trace of $x$ in spinor
space, and

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{1+\not{v}}{2}\left(P_{\mu}^{\ast}\gamma^{\mu}+iP\gamma^{5}\right),$
(7) $\displaystyle\tilde{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\left(\tilde{P}_{\mu}^{\ast}\gamma^{\mu}+i\tilde{P}\gamma^{5}\right)\frac{1-\not{v}}{2},$
(8)

with $\bar{\mathcal{H}}=\gamma^{0}\mathcal{H}^{\dagger}\gamma^{0}$,
$\bar{\tilde{\mathcal{H}}}=\gamma^{0}\tilde{\mathcal{H}}^{\dagger}\gamma^{0}$.
The $P^{(\ast)}$ and $\tilde{P}^{(\ast)}$ are respectively given by
$P^{(\ast)}=(D^{(\ast)0},D^{(\ast)+})$ and
$\tilde{P}^{(\ast)}=(\bar{D}^{(\ast)0},D^{(\ast)-})^{T}$. $v^{\mu}$ denotes
the four velocity of heavy mesons. The coupling constant $|g_{b}|=0.59$ is
extracted from the partial decay width of the $D^{\ast}$ meson, e.g., the
process $D^{\ast+}\to D^{0}\pi^{+}$ [1]. Meanwhile, the axial-vector current
$u_{\mu}$ and chiral connection $\Gamma_{\mu}$ are defined as

$\displaystyle
u_{\mu}=\frac{i}{2}\left\\{\xi^{\dagger},\partial_{\mu}\xi\right\\},\qquad\Gamma_{\mu}=\frac{1}{2}\left[\xi^{\dagger},\partial_{\mu}\xi\right],$
(9)

with

$\displaystyle\xi^{2}=U=\exp\left(\frac{i\varphi}{f_{\pi}}\right),\varphi$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\pi^{0}&\sqrt{2}\pi^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}\end{array}\right],$ (12)

and the pion decay constant $f_{\pi}=92.4$ MeV. The antisymmetric tensors
$\sigma^{\mu\nu}$ and $F_{\mu\nu}$ are given by

$\displaystyle\sigma^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left[\gamma^{\mu},\gamma^{\nu}\right],$ (13)
$\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}+\left[\rho_{\mu},\rho_{\nu}\right],$
(14)

with

$\displaystyle\rho_{\mu}=i\frac{g_{V}}{\sqrt{2}}V_{\mu},\qquad V_{\mu}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\frac{\omega+\rho^{0}}{\sqrt{2}}&\rho^{+}\\\
\rho^{-}&\frac{\omega-\rho^{0}}{\sqrt{2}}\end{array}\right]_{\mu},$ (17)

and the constants $g_{V}=m_{\rho}/(\sqrt{2}f_{\pi})=5.9$, $\beta=0.9$, and
$\lambda=0.56$ GeV-1 [43].

  * (iii)

The Lagrangians for $\eta_{c}D^{(\ast)}\bar{D}^{\ast}$, $J/\psi
D^{(\ast)}\bar{D}^{\ast}$, and $\chi_{c1}D\bar{D}^{\ast}$ couplings.—For heavy
quarkonium, such as charmonium, the heavy quark flavor symmetry is badly
broken, while the heavy quark spin symmetry still holds. Therefore, charmonia
with the same orbital angular momentum $L$ but different spins $S$ form
multiplets. For example, the S-wave ($L=0$) spin doublet $\mathcal{J}$
containing the $J/\psi$ and $\eta_{c}$ are given by

$\displaystyle\mathcal{J}=\frac{1+\not{v}}{2}\left(\psi^{\mu}\gamma_{\mu}+i\eta_{c}\gamma_{5}\right)\frac{1-\not{v}}{2},$
(18)

while the P-wave quartet $\mathcal{J}^{\prime\mu}$ including the $\chi_{c2}$,
$\chi_{c1}$, $\chi_{c0}$ and $h_{c}$ are

$\displaystyle\mathcal{J}^{\prime\mu}$ $\displaystyle=$
$\displaystyle\frac{1+\not{v}}{2}\bigg{(}\chi_{c2}^{\mu\alpha}\gamma_{\alpha}+\frac{1}{\sqrt{2}}\epsilon^{\mu\alpha\beta\delta}v_{\alpha}\gamma_{\beta}\chi_{c1\delta}$
(19)
$\displaystyle+\frac{1}{\sqrt{3}}(\gamma^{\mu}-v^{\mu})\chi_{c0}+h_{c}^{\mu}\gamma_{5}\bigg{)}\frac{1-\not{v}}{2}.$

Then the corresponding Lagrangians are given by [44]

$\displaystyle\mathcal{L}_{1}$ $\displaystyle=$ $\displaystyle
g_{1}\langle\mathcal{J}^{\prime\mu}\bar{\tilde{\mathcal{H}}}\gamma_{\mu}\bar{\mathcal{H}}\rangle+\mathrm{H.c.},$
(20) $\displaystyle\mathcal{L}_{2}$ $\displaystyle=$ $\displaystyle
g_{2}\langle\mathcal{J}\bar{\tilde{\mathcal{H}}}\overleftrightarrow{\not{\partial}}\bar{\mathcal{H}}\rangle+\mathrm{H.c.},$
(21)

in which the coupling constants $g_{1}$ and $g_{2}$ are respectively related
to the decay constants of $\chi_{c0}$ and $J/\psi$ with the vector meson
dominance model [44].

$\displaystyle
g_{1}=-\sqrt{\frac{m_{\chi_{c0}}}{3}}\frac{1}{f_{\chi_{c0}}},\qquad
g_{2}=\frac{\sqrt{m_{\psi}}}{2m_{D}f_{\psi}},$ (22)

with $f_{\chi_{c0}}=510\pm 40$ MeV being estimated from the QCD sum rule [45],
while the $f_{\psi}$ can be determined through the electromagnetic decay of
$J/\psi$, i.e., with the decay width of $J/\psi\to e^{+}e^{-}$ [1]

$\displaystyle\Gamma[J/\psi\to
e^{+}e^{-}]=C_{\psi}\frac{4\pi}{3}\frac{\alpha^{2}}{m_{\psi}}f_{\psi}^{2},$
(23)

where $C_{\psi}=\frac{4}{9}$ is the charge square of charm quark,
$\alpha=\frac{1}{137}$ is the fine-structure constant. One can obtain that
$f_{\psi}=415$ MeV.

###  II.2 Determination of the coupling constant $g_{0}$

In this section, we will demonstrate how to determine the coupling constant
$g_{0}$ in Eq. (1) using the residue of the scattering T-matrix at the pole.
To begin, we consider the scattering of particles $A$ and $B$ (with masses
$m_{1}$ and $m_{2}$, respectively) within the framework of field theory.
Assuming that their interaction is strong enough, allowing for the formation
of bound states, we must treat their scattering in a nonperturbative manner,
which means using the Bethe-Salpeter (BS) equation,

$\displaystyle
T^{\mathrm{FT}}=V^{\mathrm{FT}}+V^{\mathrm{FT}}G^{\mathrm{FT}}T^{\mathrm{FT}},$
(24)

where the superscript “$\mathrm{FT}$” means that we are working in a
relativistic field theoretical approach. The two-body relativistic propagator
$G^{\mathrm{FT}}$ reads

$\displaystyle G^{\mathrm{FT}}$ $\displaystyle=$ $\displaystyle
i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}-m_{1}^{2}+i\epsilon}\frac{1}{(P-q)^{2}-m_{2}^{2}+i\epsilon}$
(25) $\displaystyle=$
$\displaystyle\int\frac{q^{2}dq}{(2\pi)^{2}}\frac{\omega_{1}+\omega_{2}}{\omega_{1}\omega_{2}}\frac{1}{P_{0}^{2}-(\omega_{1}+\omega_{2})^{2}+i\epsilon},$

where $P=p_{1}+p_{2}$, with $p_{1}$ and $p_{2}$ the momentum of $A$ and $B$
particles, respectively, and $P_{0}$ denotes the center of mass energy of $AB$
system. The result in the second line of Eq. (25) is obtained with the residue
theorem. The $\omega_{1}$ and $\omega_{2}$ are given by

$\displaystyle\omega_{1}^{2}=\bm{q}^{2}+m_{1}^{2},\qquad\omega_{2}^{2}=\bm{q}^{2}+m_{2}^{2}.$
(26)

Then, we perform a nonrelativistic reduction

$\displaystyle\omega_{i}$ $\displaystyle=$ $\displaystyle
m_{i}+\frac{\bm{q}^{2}}{2m_{i}}+\dots,\quad(i=1,2),$ (27)
$\displaystyle\frac{\omega_{1}+\omega_{2}}{\omega_{1}\omega_{2}}$
$\displaystyle\approx$
$\displaystyle\frac{1}{m_{1}}+\frac{1}{m_{2}}=\frac{1}{\mu},$ (28)

and

$\displaystyle\frac{1}{P_{0}^{2}-(\omega_{1}+\omega_{2})^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{P_{0}-(\omega_{1}+\omega_{2})}\frac{1}{P_{0}+(\omega_{1}+\omega_{2})}$
(29) $\displaystyle\simeq$
$\displaystyle\frac{1}{P_{0}-m_{1}-m_{2}-\frac{\bm{q}^{2}}{2\mu}}\frac{1}{2P_{0}}.$

With the nonrelativistic reduction, the $G^{\mathrm{FT}}$ becomes

$\displaystyle G^{\mathrm{FT}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\mu\sqrt{s}}G,$ (30)

where

$\displaystyle G$ $\displaystyle=$
$\displaystyle\int\frac{q^{2}dq}{(2\pi)^{3}}\frac{1}{\sqrt{s}-m_{1}-m_{2}-\frac{\bm{q}^{2}}{2\mu}}$
(31)

represents the nonrelativistic two-body propagator. Note that we have replaced
the $P_{0}$ with $\sqrt{s}$, where $s=(p_{1}+p_{2})^{2}$ is the invariant mass
of the $AB$ system, and it equals to the $P_{0}^{2}$ in the center of mass
frame.

For nonrelativistic scattering, such as the interaction of the $AB$ system
near the threshold energy, we can use the Lippmann-Schwinger (LS) equation to
handle it, i.e,

$\displaystyle t=v+vGt,$ (32)

where $G$ is given in Eq. (31).

The integral equation in Eqs. (24) and (32) can be transformed into algebraic
equations with, e.g., the on-shell factorization approach [46],

$\displaystyle T^{\mathrm{FT}}$ $\displaystyle=$
$\displaystyle(1-V^{\mathrm{FT}}G^{\mathrm{FT}})^{-1}V^{\mathrm{FT}},$ (33)
$\displaystyle t$ $\displaystyle=$ $\displaystyle(1-vG)^{-1}v.$ (34)

It can be seen that the position of the pole in the T-matrix is determined by
the product of $V^{\mathrm{FT}}$ ($v$) and $G^{\mathrm{FT}}$ ($G$). The
following condition ensures that the pole position remains unchanged after the
reduction,

$\displaystyle V^{\mathrm{FT}}G^{\mathrm{FT}}=vG.$ (35)

With the relation in Eq. (30), one can obtain that
$V^{\mathrm{FT}}=\frac{\mu\sqrt{s}}{\pi}V$. Similarly, the $T^{\mathrm{FT}}$
and $t$ will satisfy the same relationship, i.e.,

$\displaystyle T^{\mathrm{FT}}=\frac{\mu\sqrt{s}}{\pi}t.$ (36)

Now, we assume that there exists a pole with the squared mass $s_{R}$ in the
T-matrix, e.g., corresponding to a bound state. Then, in the vicinity of this
pole, the T-matrix can be approximately written as

$\displaystyle T^{\mathrm{FT}}=\frac{(g^{\mathrm{FT}})^{2}}{s-s_{R}},$ (37)

where $g^{\mathrm{FT}}$ represents the coupling constant of the bound state
$R$ to its component $AB$. The relation in Eq. (37) allows us to extract the
coupling constant from the residue of T-matrix,

$\displaystyle(g^{\mathrm{FT}})^{2}$ $\displaystyle=$ $\displaystyle\lim_{s\to
s_{R}}(s-s_{R})T^{\mathrm{FT}}$ (38) $\displaystyle=$ $\displaystyle\lim_{s\to
s_{R}}(\sqrt{s}+\sqrt{s_{R}})\frac{\mu\sqrt{s}}{\pi}(\sqrt{s}-\sqrt{s_{R}})t$
$\displaystyle=$ $\displaystyle\frac{2\mu s_{R}}{\pi}\lim_{s\to
s_{R}}(\sqrt{s}-\sqrt{s_{R}})t,$

and

$\displaystyle\lim_{s\to s_{R}}(\sqrt{s}-\sqrt{s_{R}})t=\lim_{s\to
s_{R}}\left[\frac{d}{d\sqrt{s}}t^{-1}(\sqrt{s})\right]^{-1},$ (39)

where we have used the l’Hôpital rule.

We adopt sharp cutoff to regularize the integral in Eq. (31). For the bound
state case, i.e., $\sqrt{s}<m_{1}+m_{2}$, its expression reads

$\displaystyle
G_{\Lambda}(\sqrt{s})=\frac{\mu}{4\pi^{3}}\left[\gamma_{b}\arctan\left(\frac{\Lambda}{\gamma_{b}}\right)-\Lambda\right],$
(40)

where $\gamma_{b}=\sqrt{2\mu(m_{1}+m_{2}-\sqrt{s})}$ denotes the binding
momentum, and $\Lambda$ is the cutoff parameter.

If the effective potential $v$ does not depend on energy, as in the case given
in Ref. [29]. Then the Eq. (39) will equal to

$\displaystyle g^{\prime 2}$ $\displaystyle=$ $\displaystyle\lim_{s\to
s_{R}}\left[-\frac{d}{d\sqrt{s}}G_{\Lambda}(\sqrt{s})\right]^{-1}$ (41)
$\displaystyle=$
$\displaystyle\frac{\gamma_{b}}{\frac{\mu^{2}}{4\pi^{3}}\left[\arctan\left(\frac{\Lambda}{\gamma_{b}}\right)-\frac{\gamma_{b}\Lambda}{\gamma_{b}^{2}+\Lambda^{2}}\right]}.$

The $g^{\mathrm{FT}}$ is obtained by combining the results in Eqs. (38)-(41).
For the small binding case $\gamma_{b}\to 0$ and using $\Lambda\to\infty$, one
can get $g^{\mathrm{FT}}=4\sqrt{\pi\gamma_{b}s_{R}/\mu}$. This is consistent
with the relation used in Ref. [31].

To obtain the $g_{0}$ in Lagrangian (1), we still need to perform one final
step: We constructed the Lagrangian (1) to describe the coupling between the
bound state $T_{\psi 0}^{a}$ and its components $D^{\ast}\bar{D}^{\ast}$.
Using this Lagrangian, we can describe the elastic scattering process shown in
Fig. 2 (replacing $A$, $B$, and $R$ with $D^{\ast}$, $\bar{D}^{\ast}$, and
$T_{\psi 0}^{a}$, respectively). The form of the scattering amplitude after
nonrelativistic reduction is

$\displaystyle
i\mathcal{M}=-i\frac{g_{0}^{2}}{s-s_{R}}(\bm{\varepsilon}_{1}\cdot\bm{\varepsilon}_{2})(\bm{\varepsilon}_{1}^{\dagger}\cdot\bm{\varepsilon}_{2}^{\dagger}),$
(42)

where the $s_{R}$ here denotes the squared mass of $T_{\psi 0}^{a}$, and
$\bm{\varepsilon}_{1}$ ($\bm{\varepsilon}_{1}^{\dagger}$) and
$\bm{\varepsilon}_{2}$ ($\bm{\varepsilon}_{2}^{\dagger}$) represent the
polarization vectors of the initial (final) state $D^{\ast}$ and
$\bar{D}^{\ast}$, respectively. Eq. (37) is also equivalent to an elastic
scattering process shown in Fig. 2. The scattering T-matrix is obtained in the
partial wave basis. To match Eq. (42) with Eq. (37) equivalently, the
scattering amplitude in Eq. (42) should also be projected onto the partial
wave basis (S-wave). With the spin transition operators [47], the
$(\bm{\varepsilon}_{1}\cdot\bm{\varepsilon}_{2})(\bm{\varepsilon}_{1}^{\dagger}\cdot\bm{\varepsilon}_{2}^{\dagger})$
equals to $(\bm{S}_{1}\cdot\bm{S}_{2})^{2}-1$, with $\bm{S}_{1}$
($\bm{S}_{2}$) the spin operator of the vector meson $D^{\ast}$
($\bar{D}^{\ast}$). Then the Eq. (42) in the partial wave basis for the
${}^{1}S_{0}$ case is given by

$\displaystyle i\mathcal{M}_{{}^{1}S_{0}}=-i\frac{3g_{0}^{2}}{s-s_{R}}.$ (43)

Finally, the matching condition

$\displaystyle\mathcal{M}_{{}^{1}S_{0}}=-\frac{(g^{\mathrm{FT}})^{2}}{s-s_{R}}$
(44)

yields

$\displaystyle g_{0}^{2}=\frac{2\mu_{D^{\ast}\bar{D}^{\ast}}m_{T_{\psi
0}^{a}}^{2}}{3\pi}g^{\prime 2},$ (45)

where $\mu_{D^{\ast}\bar{D}^{\ast}}$ and $m_{T_{\psi 0}^{a}}$ denote the
reduced mass of $D^{\ast}\bar{D}^{\ast}$ and the mass of $T_{\psi 0}^{a}$,
respectively. The expression for $g^{\prime 2}$ is given in Eq. (41). The
range of $g_{0}$ is

$\displaystyle g_{0}\in[2.9,8.0]~{}\mathrm{GeV},$ (46)

where we have used $\sqrt{s_{R}}=m_{T_{\psi 0}^{a}}\in[4007.2,4016.7]$ MeV,
and $\Lambda=0.4$ GeV [29].

Figure 2: The elastic scattering of $A$ and $B$ particles, where the
$s$-channel is saturated with their bound state $R$.

###  II.3 Decay amplitudes

By expanding the Lagrangians in Eqs. (1), (5), (6), (20) and (21), we can
obtain the vertices required for each Feynmann diagram shown in Fig. 1.
Considering the decay of $T_{\psi 0}^{a}$ to the final states $f_{2}f_{1}$
through the process

$T_{\psi 0}^{a}\to D^{\ast}(p_{1}+q)\bar{D}^{\ast}(p_{2}-q)[M(q)]\to
f_{2}(p_{2})f_{1}(p1),$

where the quantity in parenthesis represents the momentum of each particle,
and the notation $[M]$ means that the $D^{\ast}\bar{D}^{\ast}$ transitions to
$f_{2}f_{1}$ via exchanging the meson $M$. The amplitudes for each process
from the diagrams in Fig. 1 are

$\displaystyle i\mathcal{M}_{\textbf{(a)}}^{[\pi]}$ $\displaystyle=$
$\displaystyle
C_{\textbf{(a)}}^{[\pi]}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{g_{\mu}^{\alpha}-(p_{1}+q)_{\mu}(p_{1}+q)^{\alpha}/m_{D^{\ast}}^{2}}{(p_{1}+q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{g^{\mu\beta}-(p_{2}-q)^{\mu}(p_{2}-q)^{\beta}/m_{D^{\ast}}^{2}}{(p_{2}-q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{q_{\alpha}q_{\beta}}{q^{2}-m_{\pi}^{2}+i\epsilon},$
(47) $\displaystyle i\mathcal{M}_{\textbf{(b)}}^{[D]}$ $\displaystyle=$
$\displaystyle
iC_{\textbf{(b)}}^{[D]}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{g_{\alpha}^{\nu}-(p_{1}+q)_{\alpha}(p_{1}+q)^{\nu}/m_{D^{\ast}}^{2}}{(p_{1}+q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{g^{\alpha\mu}-(p_{2}-q)^{\alpha}(p_{2}-q)^{\mu}/m_{D^{\ast}}^{2}}{(p_{2}-q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{(p_{2}-2q)_{\mu}p_{1\nu}}{q^{2}-m_{D}^{2}+i\epsilon},$
(48) $\displaystyle i\mathcal{M}_{\textbf{(b)}}^{[D^{\ast}]}$ $\displaystyle=$
$\displaystyle
iC_{\textbf{(b)}}^{[D^{\ast}]}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{g_{\delta\omega}-(p_{1}+q)_{\delta}(p_{1}+q)_{\omega}/m_{D^{\ast}}^{2}}{(p_{1}+q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{g_{\alpha}^{\delta}-(p_{2}-q)^{\delta}(p_{2}-q)_{\alpha}/m_{D^{\ast}}^{2}}{(p_{2}-q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{g_{\gamma\nu}-q_{\gamma}q_{\nu}/m_{D^{\ast}}^{2}}{q^{2}-m_{D^{\ast}}^{2}+i\epsilon}$
(49)
$\displaystyle\times\left[\epsilon^{\mu\nu\alpha\beta}\epsilon^{\rho\omega\gamma\lambda}q_{\lambda}(p_{2}-2q)_{\mu}p_{1\rho}p_{2\beta}\right],$
$\displaystyle i\mathcal{M}_{\textbf{(c)}}^{[D]}$ $\displaystyle=$
$\displaystyle
iC_{\textbf{(c)}}^{[D]}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{g_{\gamma\delta}-(p_{1}+q)_{\gamma}(p_{1}+q)_{\delta}/m_{D^{\ast}}^{2}}{(p_{1}+q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{g_{\alpha}^{\delta}-(p_{2}-q)^{\delta}(p_{2}-q)_{\alpha}/m_{D^{\ast}}^{2}}{(p_{2}-q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{1}{q^{2}-m_{D}^{2}+i\epsilon}$
(50)
$\displaystyle\times\left[\epsilon^{\mu\nu\alpha\beta}\epsilon^{\kappa\omega\gamma\lambda}q_{\lambda}(2q-p_{2})_{\mu}p_{1\kappa}p_{2\beta}\varepsilon_{\bm{\psi}\nu}^{\dagger}\varepsilon_{\bm{\rho}\omega}^{\dagger}\right],$
$\displaystyle i\mathcal{M}_{\textbf{(c)}}^{[D^{\ast}]}$ $\displaystyle=$
$\displaystyle
iC_{\textbf{(c)}}^{[D^{\ast}]}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{(p_{2}-2q)_{\mu}}{\left[(p_{1}+q)^{2}-m_{D^{\ast}}^{2}+i\epsilon\right]\left[(p_{2}-q)^{2}-m_{D^{\ast}}^{2}+i\epsilon\right]\left(q^{2}-m_{D^{\ast}}^{2}+i\epsilon\right)}$
(51)
$\displaystyle\times\sum_{\lambda_{1},\lambda_{2},\lambda_{3}}\varepsilon_{1\delta}^{\dagger}(\lambda_{1})\varepsilon_{2}^{\dagger\delta}(\lambda_{2})\left[\varepsilon_{2\alpha}(\lambda_{2})\varepsilon_{3}^{\dagger\mu}(\lambda_{3})\varepsilon_{\bm{\psi}}^{\dagger\alpha}+\varepsilon_{2}^{\mu}(\lambda_{2})\varepsilon_{3\alpha}^{\dagger}(\lambda_{3})\varepsilon_{\bm{\psi}}^{\dagger\alpha}-\varepsilon_{2\alpha}(\lambda_{2})\varepsilon_{3}^{\dagger\alpha}(\lambda_{3})\varepsilon_{\bm{\psi}}^{\dagger\mu}\right]$
$\displaystyle\times\left\\{\beta
q_{\kappa}\varepsilon_{1\omega}(\lambda_{1})\varepsilon_{3}^{\omega}(\lambda_{3})\varepsilon_{\bm{\rho}}^{\dagger\kappa}+2\lambda
m_{D^{\ast}}p_{1\kappa}\varepsilon_{\bm{\rho}}^{\dagger\omega}\left[\varepsilon_{1}^{\kappa}(\lambda_{1})\varepsilon_{3\omega}(\lambda_{3})-\varepsilon_{1\omega}(\lambda_{1})\varepsilon_{3}^{\kappa}(\lambda_{3})\right]\right\\},$
$\displaystyle i\mathcal{M}_{\textbf{(d)}}^{[D]}$ $\displaystyle=$
$\displaystyle
C_{\textbf{(d)}}^{[D]}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{g^{\alpha\mu}-(p_{1}+q)^{\alpha}(p_{1}+q)^{\mu}/m_{D^{\ast}}^{2}}{(p_{1}+q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{g_{\alpha\nu}-(p_{2}-q)_{\alpha}(p_{2}-q)_{\nu}/m_{D^{\ast}}^{2}}{(p_{2}-q)^{2}-m_{D^{\ast}}^{2}+i\epsilon}\frac{p_{1\mu}\varepsilon_{\bm{\chi_{c1}}}^{\dagger\nu}}{q^{2}-m_{D}^{2}+i\epsilon},$
(52)

in which we used the notations, such as $\mathcal{M}_{\textbf{(a)}}^{[\pi]}$,
where the subscript represents the label of diagram in Fig. 1, and the
superscript means the exchanged particle is pion. The
$\varepsilon_{\bm{\psi}}$, $\varepsilon_{\bm{\rho}}$ and
$\varepsilon_{\bm{\chi_{c1}}}$ denote the polarization vectors of $J/\psi$,
$\rho$ and $\chi_{c1}$ in order. The corresponding coupling constants from
three vertices are packed into the coefficients, such as
$C_{\textbf{(a)}}^{[\pi]}$, and their expressions are given in the following,

$\displaystyle C_{\textbf{(a)}}^{[\pi]}$
$\displaystyle=\frac{g_{0}g_{b}^{2}}{f_{\pi}^{2}}m_{D}m_{D^{\ast}},$
$\displaystyle C_{\textbf{(b)}}^{[D]}$
$\displaystyle=\frac{2\sqrt{2}g_{0}g_{2}g_{b}}{f_{\pi}}m_{D}m_{D^{\ast}}\sqrt{m_{\eta_{c}}},$
$\displaystyle C_{\textbf{(b)}}^{[D^{\ast}]}$
$\displaystyle=\frac{2\sqrt{2}g_{0}g_{2}g_{b}}{f_{\pi}}\frac{m_{D^{\ast}}}{\sqrt{m_{\eta_{c}}}},$
$\displaystyle C_{\textbf{(c)}}^{[D]}$
$\displaystyle=4\sqrt{2}g_{0}g_{2}\lambda g_{V}\frac{m_{D}}{\sqrt{m_{\psi}}},$
$\displaystyle C_{\textbf{(c)}}^{[D^{\ast}]}$
$\displaystyle=2\sqrt{2}g_{0}g_{2}g_{V}m_{D^{\ast}}\sqrt{m_{\psi}},$
$\displaystyle C_{\textbf{(d)}}^{[D]}$
$\displaystyle=-\frac{4g_{0}g_{1}g_{b}}{f_{\pi}}m_{D}m_{D^{\ast}}\sqrt{m_{\chi_{c1}}}.$
(53)

The expression for $\mathcal{M}_{\textbf{(c)}}^{[D^{\ast}]}$ is lengthy, so we
have written it in the form of Eq. (51), where its specific form can be
obtained by expanding the numerator and then summing over the polarization
vectors in the following manner,

$\displaystyle\sum_{\lambda_{1}=-1,0,1}\varepsilon_{1\mu}^{\dagger}(p_{1}+q,\lambda_{1})\varepsilon_{1\nu}(p_{1}+q,\lambda_{1})$
$\displaystyle=$ $\displaystyle-
g_{\mu\nu}+\frac{(p_{1}+q)_{\mu}(p_{1}+q)_{\nu}}{m_{D^{\ast}}^{2}},$ (54)
$\displaystyle\sum_{\lambda_{2}=-1,0,1}\varepsilon_{2\mu}^{\dagger}(p_{2}-q,\lambda_{2})\varepsilon_{2\nu}(p_{2}-q,\lambda_{2})$
$\displaystyle=$ $\displaystyle-
g_{\mu\nu}+\frac{(p_{2}-q)_{\mu}(p_{2}-q)_{\nu}}{m_{\bar{D}^{\ast}}^{2}},$
(55)
$\displaystyle\sum_{\lambda_{3}=-1,0,1}\varepsilon_{3\mu}^{\dagger}(q,\lambda_{3})\varepsilon_{3\nu}(q,\lambda_{3})$
$\displaystyle=$ $\displaystyle-
g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{D^{\ast}}^{2}}.$ (56)

The decay amplitudes for $T_{\psi 0}^{a}\to D\bar{D}$, $\eta_{c}\pi$,
$J/\psi\rho$ and $\chi_{c1}\pi$ are respectively given by

$\displaystyle\mathcal{M}_{D\bar{D}}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{\textbf{(a)}}^{[\pi]},$ (57)
$\displaystyle\mathcal{M}_{\eta_{c}\pi}$ $\displaystyle=$ $\displaystyle
2\left(\mathcal{M}_{\textbf{(b)}}^{[D]}+\mathcal{M}_{\textbf{(b)}}^{[D^{\ast}]}\right),$
(58) $\displaystyle\mathcal{M}_{J/\psi\rho}$ $\displaystyle=$ $\displaystyle
2\left(\mathcal{M}_{\textbf{(c)}}^{[D]}+\mathcal{M}_{\textbf{(c)}}^{[D^{\ast}]}\right),$
(59) $\displaystyle\mathcal{M}_{\chi_{c1}\pi}$ $\displaystyle=$ $\displaystyle
2\mathcal{M}_{\textbf{(d)}}^{[D]}.$ (60)

With the amplitudes, the partial decay width of $T_{\psi 0}^{a}$ can be
expressed as

$\displaystyle\Gamma_{f_{2}f_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\frac{|\bm{p}_{1}|}{m_{T_{\psi
0}^{a}}^{2}}\overline{|\mathcal{M}_{f_{2}f_{1}}|^{2}},$ (61)

where

$\displaystyle|\bm{p}_{1}|$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\mathcal{K}(m_{T_{\psi
0}^{a}}^{2},m_{f_{1}}^{2},m_{f_{2}}^{2})}}{2m_{T_{\psi 0}^{a}}},$ (62)
$\displaystyle\mathcal{K}(\alpha,\beta,\gamma)$ $\displaystyle=$
$\displaystyle\alpha^{2}+\beta^{2}+\gamma^{2}-2\alpha\beta-2\alpha\gamma-2\beta\gamma,$
(63)

and the overline represents a sum over the polarization(s) of the $\chi_{c1}$
($J/\psi\rho$) in the final states.

The loop integrals in Eqs. (47)-(52) are calculated with the cutoff
regularization. Since we are dealing with the decays of a bound state, the
constituent particles $D^{\ast}$ and $\bar{D}^{\ast}$ are off-shell, meaning
that there are no right-hand cuts in the loop integrals. We take the following
scalar integral as an example to illustrate our calculation,

$\displaystyle\mathcal{I}_{0}^{\Lambda^{\prime}}$ $\displaystyle=$
$\displaystyle\int^{\Lambda^{\prime}}\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{(p_{1}+q)^{2}-m_{D^{\ast}}^{2}}\frac{1}{(p_{2}-q)^{2}-m_{\bar{D}^{\ast}}^{2}}$
(64) $\displaystyle\times\frac{1}{q^{2}-m_{M}^{2}}$ $\displaystyle=$
$\displaystyle-\frac{3i}{16\pi^{2}}\int_{0}^{1}dx\int_{0}^{1-x}dy\int_{0}^{\Lambda^{\prime}}d\ell\frac{\bm{\ell}^{2}}{\left(\sqrt{\bm{\ell}^{2}+\Delta}\right)^{5}}$
$\displaystyle=$
$\displaystyle-\frac{i}{16\pi^{2}}\int_{0}^{1}dx\int_{0}^{1-x}dy\frac{\Lambda^{\prime
3}}{\Delta\left(\Delta+\Lambda^{\prime 2}\right)^{3/2}},$

where $\ell=q+xp_{1}-yp_{2}$, and
$\Delta=xm_{D^{\ast}}^{2}+y(m_{\bar{D}^{\ast}}^{2}-m_{f_{2}}^{2})+xy(m_{f_{1}}^{2}+m_{f_{2}}^{2}-m_{T_{\psi
0}^{a}}^{2})+x^{2}m_{f_{1}}^{2}-xm_{f_{1}}^{2}+y^{2}m_{f_{2}}^{2}-(x+y-1)m_{M}^{2}$
(where $m_{M}$, $m_{f_{1}}$ and $m_{f_{2}}$ denote the masses of the exchanged
particles, the final states $f_{1}$ and $f_{2}$, respectively). We have used
the Feynman parameterization to combine the denominators of the propagators,
and employed the residue theorem to integrate out the $\ell_{0}$ component. It
can be easily proven that when $\Lambda^{\prime}$ tends to infinity, the
result of $\mathcal{I}_{0}^{\Lambda^{\prime}}$ is equivalent to that obtained
within the dimensional regularization.

##  III Numerical results and discussions

With the aforementioned preparations, we can now proceed to discuss the
dependence of the (partial) decay width of the $T_{\psi 0}^{a}(4010)$ state on
the cutoff parameter $\Lambda^{\prime}$ in the loop integrals, as well as
compare the total width with the current experimental data. In Ref. [29], the
mass of the $D^{\ast}\bar{D}^{\ast}$ molecular state with quantum numbers
$1^{-}(0^{++})$ was obtained using $\Lambda=0.4$ GeV. In order to maintain
consistency with Ref. [29], we also use $\Lambda=0.4$ GeV in Eq. (41) to
calculate the coupling constant $g_{0}$. The cutoff parameter $\Lambda$
typically reflects the interaction radius $R$ ($R\sim 1/\Lambda$) of a loosely
bound molecular system. Therefore, in principle, the value of
$\Lambda^{\prime}$ used in the loop integrals should be consistent with the
one used in calculating the mass spectrum. Here, we will vary the value of
$\Lambda^{\prime}$ within the range of $0.4$ to $0.7$ GeV to study the
dependence of the width on the cutoff.

In Fig. 3, we present the dependence of the total width on $\Lambda^{\prime}$
and also plot the widths of the $X(4100)$ reported by LHCb [2] and the
$X(4050)$ observed by Belle [10]. These two states have similar masses and
widths, and the observed decay channels are precisely the decay modes of
$T_{\psi 0}^{a}(4010)$. Therefore, it is possible that they are the same
state, serving as candidate for the $1^{-}(0^{++})$ $D^{\ast}\bar{D}^{\ast}$
molecular state. From Fig. 3, it can be observed that the total width
increases with increasing $\Lambda^{\prime}$. When $\Lambda^{\prime}$ reaches
around $0.6$ GeV, the theoretically calculated width overlaps with the
experimental widths obtained by LHCb and Belle. However, it is worth noting
that the widths measured by both experiments are obtained using the Breit-
Wigner parameterization, which is known to be insufficient in describing the
near-threshold states. In such cases, a Flatté-like formula is needed to fit
the line shape and extract the pole width. A typical example is the case of
$Z_{c}(3900)$ [3] and $T_{cc}(3875)$ [48], where the width obtained using the
Flatté-like formula is much smaller than the values obtained using the Breit-
Wigner parameterization. Therefore, we suggest that after accumulating more
data, the experimental widths be remeasured using the Flatté-like formula in
the $\eta_{c}\pi$ and $\chi_{c1}\pi$ decay channels. Our theoretical
estimation for the total width of $T_{\psi 0}^{a}(4010)$ will be provided at
$\Lambda^{\prime}=0.4$ GeV:

$\displaystyle\Gamma_{T_{\psi 0}^{a}}$ $\displaystyle=$
$\displaystyle(12.0\mathrm{-}35.4)~{}\mathrm{MeV}.$ (65)

One can see that the width is of similar size with those of its partners,
$Z_{c}(3900)$ and $Z_{c}(4020)$ [1].

Figure 3: The dependence of the total decay width of $T_{\psi 0}^{a}(4010)$ on
the cutoff parameter $\Lambda^{\prime}$. The red shaded area represents our
results, and the gray and blue shaded areas represent the widths of $X(4100)$
[2] and $X(4050)$ [10], respectively.

In Fig. 4, we present the dependence of each partial width on
$\Lambda^{\prime}$, and it can be seen that the partial widths also increase
with increasing $\Lambda^{\prime}$. Similarly, when $\Lambda^{\prime}$ is set
to $0.4$ GeV, the range of each partial width is as follows:

$\displaystyle\Gamma_{D\bar{D}}$
$\displaystyle=(2.0\mathrm{-}6.6)~{}\mathrm{MeV},$
$\displaystyle\Gamma_{\eta_{c}\pi}$
$\displaystyle=(5.8\mathrm{-}17.2)~{}\mathrm{MeV},$ (66)
$\displaystyle\Gamma_{J/\psi\rho}$
$\displaystyle=(1.2\mathrm{-}3.4)~{}\mathrm{MeV},$
$\displaystyle\Gamma_{\chi_{c1}\pi}$
$\displaystyle=(3.0\mathrm{-}8.2)~{}\mathrm{MeV},$ (67)

Figure 4: The dependence of the partial decay widths of $T_{\psi 0}^{a}(4010)$
on the cutoff parameter $\Lambda^{\prime}$.

The dependence of the branching fractions $\mathcal{B}_{i}$ and branching
ratios $\mathcal{R}_{i}$ on $\Lambda^{\prime}$ are shown in Fig. 5 and Fig. 6,
respectively. From these plots, it can be read that although each partial
width is sensitive to the variation of $\Lambda^{\prime}$, the dependence of
each $\mathcal{B}_{i}$ and $\mathcal{R}_{i}$ on $\Lambda^{\prime}$ is very
weak. Therefore, in addition to the total width, measuring the
$\mathcal{B}_{i}$ and $\mathcal{R}_{i}$ experimentally will be very helpful in
determining whether $X(4100)$ and $X(4050)$ are the same state and whether
they correspond to the $D^{\ast}\bar{D}^{\ast}$ molecular state in reality.
The branching fractions are predicted as:

$\displaystyle\mathcal{B}_{1}$
$\displaystyle\equiv\frac{\Gamma_{D\bar{D}}}{\Gamma_{T_{\psi 0}^{a}}}\simeq
0.18,$ $\displaystyle\mathcal{B}_{2}$
$\displaystyle\equiv\frac{\Gamma_{\eta_{c}\pi}}{\Gamma_{T_{\psi 0}^{a}}}\simeq
0.49,$ (68) $\displaystyle\mathcal{B}_{3}$
$\displaystyle\equiv\frac{\Gamma_{J/\psi\rho}}{\Gamma_{T_{\psi 0}^{a}}}\simeq
0.09,$ $\displaystyle\mathcal{B}_{4}$
$\displaystyle\equiv\frac{\Gamma_{\chi_{c1}\pi}}{\Gamma_{T_{\psi
0}^{a}}}\simeq 0.24,$ (69)

The branching ratios are predicted as:

$\displaystyle\mathcal{R}_{1}$
$\displaystyle\equiv\frac{\Gamma_{\eta_{c}\pi}}{\Gamma_{D\bar{D}}}\simeq 2.8,$
$\displaystyle\mathcal{R}_{2}$
$\displaystyle\equiv\frac{\Gamma_{\eta_{c}\pi}}{\Gamma_{J/\psi\rho}}\simeq
5.2,$ (70) $\displaystyle\mathcal{R}_{3}$
$\displaystyle\equiv\frac{\Gamma_{\eta_{c}\pi}}{\Gamma_{\chi_{c1}\pi}}\simeq
2.0,$ $\displaystyle\mathcal{R}_{4}$
$\displaystyle\equiv\frac{\Gamma_{D\bar{D}}}{\Gamma_{J/\psi\rho}}\simeq 1.9,$
(71) $\displaystyle\mathcal{R}_{5}$
$\displaystyle\equiv\frac{\Gamma_{D\bar{D}}}{\Gamma_{\chi_{c1}\pi}}\simeq
0.7,$ $\displaystyle\mathcal{R}_{6}$
$\displaystyle\equiv\frac{\Gamma_{J/\psi\rho}}{\Gamma_{\chi_{c1}\pi}}\simeq
0.4.$ (72)

Figure 5: The dependence of the branching fractions $\mathcal{B}_{i}$ on the
cutoff parameter $\Lambda^{\prime}$.

Figure 6: The dependence of the branching ratios $\mathcal{R}_{i}$ on the
cutoff parameter $\Lambda^{\prime}$.

It is evident that the $\eta_{c}\pi$ channel is the dominant decay mode of the
$T_{\psi 0}^{a}$ state, with a decay width approximately twice that of the
$\chi_{c1}\pi$ channel. However, the width of $\chi_{c1}\pi$ channel is still
larger than the remaining two channels, namely $D\bar{D}$ and $J/\psi\rho$.
Therefore, $\eta_{c}\pi$ and $\chi_{c1}\pi$ are considered the golden channels
for detecting the $T_{\psi 0}^{a}$ state. Furthermore, these two channels
respectively correspond to the final states in which the $X(4100)$ and
$X(4050)$ resonances were observed, which may suggest, to some extent, that
$X(4100)$ and $X(4050)$ are molecular state of the $1^{-}(0^{++})$
$D^{\ast}\bar{D}^{\ast}$ system.

In addition, future experiments can also explore the charged $T_{\psi 0}^{a}$
state by studying the invariant mass spectrum of
$D^{+}\bar{D}^{0}+\mathrm{c.c}$. For example, with the weak decay processes
$B^{0}\to D^{-}D^{0}K^{+}$, and $\bar{B}^{0}\to D^{+}\bar{D}^{0}K^{-}$ [49].
The angular distribution of $D^{+}\bar{D}^{0}+\mathrm{c.c.}$ will exhibit the
characteristic of a flat S-wave distribution. The
$D^{+}\bar{D}^{0}+\mathrm{c.c.}$ channel can effectively exclude contributions
from conventional charmonia, such as the $\psi(4040)$. Therefore, the signal
observed in the $D^{+}\bar{D}^{0}+\mathrm{c.c.}$ channel may be cleaner and
provide a clearer indication of the presence of the $T_{\psi 0}^{a}$ state.
This makes the $D^{+}\bar{D}^{0}+\mathrm{c.c.}$ channel a promising avenue for
studying the $T_{\psi 0}^{a}$ state and distinguishing it from other
resonances.

##  IV Summary

Recently, we investigated the interactions of the $D^{(\ast)}\bar{D}^{(\ast)}$
systems based on a quark-level potential model [29]. We found that if
$X(3872)$ and $Z_{c}(3900)$ are the isoscalar and isovector molecular states
of the $D\bar{D}^{\ast}$ system, respectively, then there must exist a bound
state in the $1^{-}(0^{++})$ $D^{\ast}\bar{D}^{\ast}$ system, denoted as
$T_{\psi 0}^{a}(4010)$. This state would decay into the $D\bar{D}$,
$\eta_{c}\pi$, $J/\psi\rho$ and $\chi_{c1}\pi$ channels. It is noteworthy that
the LHCb and Belle Collaborations have observed the $X(4100)$ [2] and
$X(4050)$ [10] in the final states of $\eta_{c}\pi$ and $\chi_{c1}\pi$,
respectively. The masses and widths of these two states are of similar size
within the experimental uncertainties, and their masses are close to the
$D^{\ast}\bar{D}^{\ast}$ threshold. Furthermore, the final states of their
decays are consistent with the decay channels of $T_{\psi 0}^{a}(4010)$.
Therefore, it is significant to investigate whether these two states are the
same one and whether they correspond to the molecular state of
$D^{\ast}\bar{D}^{\ast}$.

In this work, we used the effective Lagrangian approach to investigate the
strong decays of the $T_{\psi 0}^{a}(4010)$ state through triangle loop
diagrams. We found that its main decay channels are $\eta_{c}\pi$ and
$\chi_{c1}\pi$, which may explain why the LHCb and Belle Collaborations
reported the signals of $X(4100)$ and $X(4050)$ in these two channels,
respectively. This also suggests that $X(4100)$ and $X(4050)$ might be
identified as the $T_{\psi 0}^{a}(4010)$ state. We also investigated the
dependence of the total (partial) width(s) and branching fractions (ratios) on
the cutoff parameter $\Lambda^{\prime}$ in the loop integrals. We noticed that
the width shows a strong dependence on the cutoff, while the branching
fractions (ratios) exhibit a very weak dependence. Therefore, experimental
measurements of the branching fractions (ratios) would help to identify the
properties of $X(4100)$ and $X(4050)$ and their relationship with the $T_{\psi
0}^{a}(4010)$. Our calculation predicts a total width of
$(12.0$$\mathrm{-}$$35.4)$ MeV for the $T_{\psi 0}^{a}(4010)$, which is
consistent with the experimentally measured widths of its partners
$Z_{c}(3900)$ and $Z_{c}(4020)$.

We propose that future experiments focus on the study of the resonance
parameters of the $T_{\psi 0}^{a}(4010)$ state in the $\eta_{c}\pi^{-}$,
$\chi_{c1}\pi^{-}$ and $D^{0}D^{-}$ invariant mass spectra of the
$B^{0}\to\eta_{c}\pi^{-}K^{+}$, $\chi_{c1}\pi^{-}K^{+}$ and $D^{0}D^{-}K^{+}$
processes (or the charge conjugate channels). This would be crucial for
constructing the mass spectrum of the hadronic molecules in the
$D^{(\ast)}\bar{D}^{(\ast)}$ systems and, in turn, understanding the
properties of the exotic hadrons such as $X(3872)$ and $Z_{c}(3900)$.

## Acknowledgement

B. Wang is very grateful to Dr. Lu Meng for helpful discussions. This work is
supported by the National Natural Science Foundation of China under Grants No.
12105072. B. Wang is also supported by the Start-up Funds for Young Talents of
Hebei University (No. 521100221021).

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# Doubly-Charged $T_{cc}^{++}$ States in the Dynamical Diquark Model

Halil Mutuk<EMAIL_ADDRESS>Department of Physics, Faculty of Sciences,
Ondokuz Mayis University, 55200, Samsun, Türkiye

###### Abstract

One of the celebrated tools in explaining the Hydrogen atom is Born-
Oppenheimer approximation. The resemblance of $QQ\bar{q}\bar{q}$ tetraquarks
to Hydrogen atom within Quantum chromodynamics (QCD) implies usage of Born-
Oppenheimer approximation for these multiquark states. In this work, we use
dynamical diquark model to calculate mass spectra and sizes of doubly charmed
and charged tetraquark states denoted as $T_{cc}^{++}$. Our results for mass
spectra indicate some bound state candidates with respect to corresponding
two-meson thresholds. Calculation of expectation values of $\sqrt{\langle
r^{2}\rangle}$ reflects that doubly charmed and charged tetraquark states are
compact.

diquark, doubly-charmed hadrons, $T_{cc}^{++}$ states

##  I Prologue

It is well known that the existence of exotic states composed of more than
three quarks or having unusual quantum numbers is not ruled out in QCD.
Conventional hadrons can be grouped into mesons ($q\bar{q}$ states) and
baryons ($qqq$ states). Exotic states in this perspective cannot fit into the
conventional hadron configuration. Exotic particles have been observed in many
experiments throughout the previous few decades. It is worth to mention that
these exotic states were conjectured almost a half century ago
phenomenologically [1, 2, 3, 4, 5].

With the accumulation of experimental information about these exotic hadrons,
many different models have been proposed to enlighten internal structure of
these states [6, 7, 8, 9, 10, 11, 12, 13, 14]. Such endeavors are very
valuable and important to understand perturbative and nonperturbative aspects
of QCD.

Recently, the LHCb Collaboration announced the first observation of doubly-
charmed tetraquark state $T_{cc}^{+}$ in the $D^{0}D^{0}\pi^{+}$ invariant
mass spectrum [15, 16]. This state has a mass $M=3875.1+\delta m_{exp}$, where
$\delta m_{exp}=-273\pm 61\pm 5^{+11}_{-14}~{}\mathrm{keV}$ and width
$\Gamma=410\pm 165\pm 43_{-38}^{+18}~{}\mathrm{keV}$, and has a quark content
$cc\bar{u}\bar{d}$. It is below two-meson $D^{0}D(2010)^{\ast+}$ threshold and
it is the longest-living exotic meson detected to date because of its very
small width. The spin-parity quantum number of $T_{cc}^{+}$ state is estimated
as $J^{P}=1^{+}$ by the experiment. Due to the its interesting properties, the
observation of $T_{cc}^{+}$ triggered many studies such as hadronic molecule
generated by the $DD^{*}$ interaction [17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34], compact tetraquark [35, 36, 32], or triangle
singularity [37].

Heavy quark spin symmetry (HQSS) [38] suggests that when the heavy quark
masses are taken to infinity $m_{Q}\to\infty$, the spin of the quark decouples
from the dynamics which refers the strong interactions in the system are
independent of the heavy quark spin. HQSS predicts the presence of partners
for the particles involving heavy quarks. For example using HQSS, if
$\chi_{c1}(3872)$ (formerly known as $X(3872)$) is a bound state of $D$ and
$D^{\ast}$ mesons, then it should have partners in heavy quark limit [39, 40].
In the same manner, locations of possible $T_{cc}^{++}$ and $T_{cc}^{0}$
states were determined in Ref. [41]. In the same work, $I(J^{P})=0(1^{+})$ and
$I(J^{P})=1(2^{+})$ HQSS partners below $D^{\ast+}D^{\ast 0}$ thresholds were
predicted. In Ref. [42], mass and coupling of the $T_{cc}$ states with light
diquark content $\bar{s}\bar{s}$ and $\bar{d}\bar{s}$ were studied in QCD sum
rule method. Scalar, axialvector and tensor doubly-charmed molecular four-
quark states with isospin $I=0,1/2$ and $I=1$ have been studied in QCD sum
rule framework [43]. Alongside $T_{cc}^{+}$ state, masses of $T_{cc}$ states
with single and double strangeness have been examined using one-meson exchange
potential model [44]. In Ref. [45], magnetic moments of hypothetical
$T_{cc}^{++}$ states were calculated in molecular picture using light-cone QCD
sum rule method. Ref. [46] studied $QQ\bar{q}\bar{q}$ system in the Born-
Oppenheimer approximation within string theory.

It is well known that doubly charged four-quark states are pure exotic states
since they cannot be explained as traditional mesons. The most recent example
is the observation of a doubly charged open-charm tetraquark state with
possible quark content $(c\bar{s}u\bar{d})$ by LHCb Collaboration in 2022
[47]. Doubly charmed four-quark states can be interpreted as compact
tetraquark states or molecular states. Furthermore if this state has double
electric charge, it could not be interpreted as loosely bound molecule even if
it was experimentally observed close to any open charm threshold, since an
extended object is subject to range of strong interactions where Coulomb
repulsion would cause a fall apart decay.

To the best of our knowledge, there is no study about doubly charmed and
charged states conducted in dynamical diquark model. In this paper, we present
mass spectra and sizes of doubly charmed and charged tetraquark using
dynamical diquark model. In Section II, we introduce dynamical diquark model
and in Section III we give brief information about Born-Oppenheimer
approximation and potential. Section IV presents results of numerical
calculations and Section V summarizes our results.

##  II Dynamical Diquark Model

The resolution of a compact tetraquark can be written as a diquark+antidiquark
state in which diquark (antidiquark) is treated as a degrees-of-freedom. A
diquark is a colored bound quark-quark pair $(qq)$ and antidiquark is a
colored bound antiquark-antiquark pair $(\bar{q}\bar{q})$. These colored pairs
can have colorless combinations. A consideration of the structure and
spectroscopy of multiquark states may even make diquark degrees-of-freedom
important. For more discussion about diquarks see reviews [48, 49].

The tetraquark interpretation of a four-quark state can be done by using
diquarks and antidiquarks as $[(qq)(\bar{q}\bar{q})]$ combination. In this
configuration, the quark $q$ is in the fundamental color representation 3 and
the antiquark $\bar{q}$ is in the fundamental color representation $\bar{3}$.
Combining two quarks as $|qq\rangle$ gives $3\otimes 3=6\oplus\bar{3}$,
whereas combining two antiquarks $|\bar{q}\bar{q}\rangle$ gives
$\bar{3}\otimes\bar{3}=\bar{6}\oplus 3$. Accordingly, color singlet tetraquark
$|qq\bar{q}\bar{q}$ state may be produced from $\bar{3}\otimes 3$ or
$\bar{6}\times 6$.

The dynamical diquark model proposed in Ref. [50] suggests that confinement
mechanism is responsible for binding in the exotic states. The diquark-
antidiquark pair forms instantly at the production point and rapidly moves
from each other due to the kinematic effects of the production process. Due to
the fact that diquark and antidiquark are colored objects, they cannot
separate far apart; instead they elongate creating a color flux tube or string
between them. This model was developed and upgraded in Refs. [51, 52] where
diquarks $\delta$ and antidiquarks $\bar{\delta}$ are treated as quasi-bound
hadronic subcomponents.

The states for the $\delta-\bar{\delta}$ system can be written

$\displaystyle J^{PC}=0^{++}:$ $\displaystyle
X_{0}\equiv|0_{\delta},0_{\bar{\delta}}\rangle_{0}\,,\ \
X_{0}^{\prime}\equiv|1_{\delta},1_{\bar{\delta}}\rangle_{0},$ $\displaystyle
J^{PC}=1^{++}:$ $\displaystyle
X_{1}\equiv\frac{1}{\sqrt{2}}\left(|1_{\delta},0_{\bar{\delta}}\rangle_{1}+|0_{\delta},1_{\bar{\delta}}\rangle_{1}\right),$
$\displaystyle J^{PC}=1^{+-}:$ $\displaystyle Z\
\equiv\frac{1}{\sqrt{2}}\left(|1_{\delta},0_{\bar{\delta}}\rangle_{1}-|0_{\delta},1_{\bar{\delta}}\rangle_{1}\right),$
$\displaystyle Z^{\prime}\,\equiv|1_{\delta},1_{\bar{\delta}}\rangle_{1}\,,$
$\displaystyle J^{PC}=2^{++}:$ $\displaystyle
X_{2}\equiv|1_{\delta},1_{\bar{\delta}}\rangle_{2}\ ,$ (1)

where the basis of good diquark-spin eigenvalues with labels such as
$1_{\delta}$ are used. Here outer subscripts on the kets indicate total quark
spin $J$. Since we study $S$-wave states, it equals $J=S$. These six states
constitute the lowest multiplet $\Sigma^{+}_{g}(1S)$ within the Born-
Oppenheimer approximation. The core states $X_{0}$, $X_{0}^{\prime}$ and $Z$,
$Z^{\prime}$ have the same $J^{PC}$ quantum numbers and can mix. Instead
equivalent eigenstates [53] can be defined as $X_{1}$, $X_{2}$, and

$\displaystyle{\tilde{X}}_{0}$ $\displaystyle\equiv$
$\displaystyle|0_{Q\bar{Q}},0_{q^{\prime}\bar{q}^{\prime}}\rangle_{0}=+\frac{1}{2}X_{0}+\frac{\sqrt{3}}{2}X_{0}^{\prime}\,,$
$\displaystyle{\tilde{X}}_{0}^{\prime}$ $\displaystyle\equiv$
$\displaystyle|1_{Q\bar{Q}},1_{q^{\prime}\bar{q}^{\prime}}\rangle_{0}=+\frac{\sqrt{3}}{2}X_{0}-\frac{1}{2}X_{0}^{\prime}\,,$
$\displaystyle{\tilde{Z}}$ $\displaystyle\equiv$
$\displaystyle|1_{Q\bar{Q}},0_{q^{\prime}\bar{q}^{\prime}}\rangle_{1}=\frac{1}{\sqrt{2}}\left(Z^{\prime}\\!+Z\right)\,,$
$\displaystyle{\tilde{Z}}^{\prime}$ $\displaystyle\equiv$
$\displaystyle|0_{Q\bar{Q}},1_{q^{\prime}\bar{q}^{\prime}}\rangle_{1}=\frac{1}{\sqrt{2}}\left(Z^{\prime}\\!-Z\right)\,.$
(2)

If the quarks inside of $\delta$ or $\bar{\delta}$ pairs are identical, then
the space-spin wave function of the related diquark must be symmetric. As a
result, the total wave function together with color part of the corresponding
diquark obeys the Fermi statistic. The dynamical diquark model assumes no
orbital excitation within the diquarks. Therefore spatial wave functions and
spin wave functions of diquarks should be symmetric which leaves only
$1_{\delta}$ and $1_{\bar{\delta}}$ survive [54]. This is the case for $cc$,
$\bar{d}\bar{d}$, and $\bar{s}\bar{s}$ diquarks but not in the
$\bar{d}\bar{s}$ which can combine together to have $0_{\bar{\delta}}$. This
elaboration requires more involved analysis which is beyond the assumption of
this work. It is assumed that the tetraquark states in this present study are
composed of colour antitriple ($\bar{3}_{c}$) diquark and colour triplet
($3_{c}$) antidiquark. Therefore $X_{0}^{\prime}$, $Z^{\prime}$ and $X_{2}$
are the eigenstates that remain and no mixing occurs. Furthermore, the states
that are considered is this work do not have a specified $C$\- parity. As a
result for these states, it is not a ”good” quantum number.

##  III Born-Oppenheimer Approximation and Potential

The original Born-Oppenheimer (BO) approximation was proposed to study quantum
states of molecules made of nuclei and electrons. In this approximation, it is
possible to separate the motion of the nuclei and the motion of the electrons,
i.e., the separation of nuclear and electronic coordinates facilitates to
solve Schrödinger equation for multibody systems. The nuclei is sufficiently
heavy and its motion is approximately frozen where the electrons around the
nuclei move very much faster. The nuclei constitute a big attractive force
such that electrons cannot “disappear” from the atom.

From the four-quark point of view, a minimum of two heavy sources and degrees-
of-freedom for light quarks are needed for the Born-Oppenheimer approximation.
In the doubly-heavy $QQ\bar{q}\bar{q}$ tetraquarks heavy quark mass is much
bigger than the light quark mass, $M_{Q}>>m_{q}$. In this limit, fast motion
of light quarks in the presence of the heavy color sources produces an
effective potential. This potential regulates the slow motion of heavy quarks.
The slow motion of heavy quarks falls within the so-called Born-Oppenheimer
potential. [55].

In this work, mass spectrum of the $T_{cc}^{++}$ states is computed by a
specific BO potential, $\Gamma$ ($\Sigma_{g}^{+}$, $\Sigma_{u}^{+}$, _etc_.)
which gives rise to a multiplet of states. These potentials are obtained by
several groups on lattice [56, 57, 58, 59, 60]. Having masses of a diquark
$m_{\delta}$ and antidiquark $m_{\bar{\delta}}$, one can solve numerically the
corresponding Schrödinger equation.

The Born-Oppenheimer approximation in QCD has a numerically accessible
potential obtained by lattice QCD (see for example Refs. [57, 61]) which
describe successfully heavy quark-antiquark systems. Refs. [62, 63, 64, 65,
66, 67] used Born-Oppenheimer approximation using lattice QCD potentials. Due
to this feature, Born-Oppenheimer approximation prevails being one of the
promising tools for studying both conventional and unconventional heavy quark-
antiquark states [68, 69, 70].

The radial Schrödinger equations for the uncoupled BO potentials
$V_{\Gamma}(r)$ can be written as, in natural units for reduced mass $\mu$,

$\left[-\frac{1}{2\mu
r^{2}}\partial_{r}r^{2}\partial_{r}+\frac{\ell\left(\ell+1\right)}{2\mu
r^{2}}+V_{\Gamma}(r)\right]\psi_{\Gamma}^{(n)}(r)=E_{n}\psi_{\Gamma}^{(n)},$
(3)

and the coupled BO potentials read:

$\displaystyle\left[-\frac{1}{2\mu
r^{2}}\partial_{r}r^{2}\partial_{r}+\frac{1}{2\mu
r^{2}}\begin{pmatrix}\ell\left(\ell+1\right)+2&2\sqrt{\ell\left(\ell+1\right)}\\\
2\sqrt{\ell\left(\ell+1\right)}&\ell\left(\ell+1\right)\end{pmatrix}\right.$
$\displaystyle\left.\begin{pmatrix}V_{\Sigma_{u}^{-}}&0\\\
0&V_{\Pi_{u}^{+}}\end{pmatrix}\right]\begin{pmatrix}\psi_{\Sigma_{u}^{-}}^{(n)}(r)\\\
\psi_{\Pi_{u}^{+}}^{(n)}(r)\end{pmatrix}=E_{n}\begin{pmatrix}\psi_{\Sigma_{u}^{-}}^{(n)}(r)\\\
\psi_{\Pi_{u}^{+}}^{(n)}(r)\end{pmatrix}.$ (4)

We use the functional form of $V(r)$ given by lattice simulations [58, 59].
The $S$-wave Hamiltonian can be written as

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{0}+2\left[\kappa_{QQ}\left(\mathbb{s}_{Q}\cdot\mathbb{s}_{Q}\right)+\kappa_{\bar{q}\bar{q}}\left(\mathbb{s}_{\bar{q}}\cdot\mathbb{s}_{\bar{q}}\right)\right].$
(5)

The eigenvalues of $H$ are computed as

$M=M_{0}+\Delta M_{\kappa_{QQ}}+\Delta M_{\kappa_{\bar{q}\bar{q}}},$ (6)

where

$\Delta
M_{\kappa_{QQ}}=\frac{1}{2}\kappa_{QQ}\left[2s_{\delta}\left(s_{\delta}+1\right)-3\right],$
(7)

and

$\Delta
M_{\kappa_{\bar{q}\bar{q}}}=\frac{1}{2}\kappa_{\bar{q}\bar{q}}\left[2s_{\bar{\delta}}\left(s_{\bar{\delta}}+1\right)-3\right].$
(8)

In the above expressions, $s_{\delta}$ represents the spin of the heavy
diquark and $s_{\bar{\delta}}$ represents the spin of the light antidiquark.

The Hamiltonian in Eq. (5) first appeared in Ref. [71] and also included
orbital and spin-orbit terms. Ref. [52] used S-wave Hamiltonian to study
$c\bar{c}q\bar{q}^{\prime}$ states. The same Hamiltonian emerged in Ref. [53]
for hidden-bottom and hidden-charm-strange states and in Ref. [54] for all-
heavy tetraquark states. The authors obtained reliable results compared to
literature.

##  IV Mass Spectrum

The possible quark content of doubly charmed and charged tetraquark states are
$cc\bar{d}\bar{d}$, $cc\bar{d}\bar{s}$, and $cc\bar{s}\bar{s}$. We assume that
diquarks are pointlike objects. Diquark and antidiquark masses
$(m_{\delta},m_{\bar{\delta}})$ are need to be adjusted to the model as input.
In order to make predictions that do not heavily depend on nonrelativistic
quark models [72, 73], we use diquark mass values that are extracted from QCD
sum rule $M_{ud}(1^{+})=0.81\pm 0.06~{}\text{GeV}$ and $M_{qs}(1^{+})=0.92\pm
0.04~{}\text{GeV}$ from Ref. [74], and $M_{cc}(1^{+})=3.51\pm
0.35~{}\text{GeV}$ from Ref. [75]. A recent study conducted in QCD sum rule
for $J^{P}=1^{+}$ $ss$ diquark is failed to stabilize masses [76]. Therefore
we use doubly-strange diquark mass from Ref. [77] as $M=1203~{}\text{MeV}$.
The coupling constants are taken as $\kappa_{qq}=\frac{1}{2}\kappa_{q\bar{q}}$
(which is a general consideration for the diquark potential is the half of the
corresponding quark-antiquark potential) where
$\kappa_{q\bar{q}}=315~{}\text{MeV}$, $\kappa_{s\bar{q}}=195~{}\text{MeV}$,
and $\kappa_{s\bar{s}}=121~{}\text{MeV}$ [78].

###  IV.1 $cc\bar{d}\bar{d}$ State

The mass spectrum and expectation values for $\sqrt{\langle r^{2}\rangle}$ of
$cc\bar{d}\bar{d}$ state are given in Table 1. As can be seen from the Table,
mass splitting of $cc\bar{d}\bar{d}$ multiplet is 97 MeV. We observe from
$\sqrt{\langle r^{2}\rangle}$ values that the predicted states are compact.

Table 1:  Predicted mass spectra and $\sqrt{\langle r^{2}\rangle}$ for the $cc\bar{d}\bar{d}$ state. The superscripts denote the spin of the diquark/antidiquark whereas the subscripts denote the color structure. $\\{~{}\\}$ denotes the symmetric flavor wave functions of the two quarks and antiquarks subsystems. Mass results are in MeV and $\sqrt{\langle r^{2}\rangle}$ values are in fm units. $J^{P}$ | Configuration | Mass | $\sqrt{\langle r^{2}\rangle}$
---|---|---|---
$0^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{d}\\}^{1}_{\mathbf{3}_{c}}\right]^{0}_{\mathbf{1}_{c}}$ | 3904 | 0.64
$1^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{d}\\}^{1}_{\mathbf{3}_{c}}\right]^{1}_{\mathbf{1}_{c}}$ | 3952 | 0.65
$2^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{d}\\}^{1}_{\mathbf{3}_{c}}\right]^{2}_{\mathbf{1}_{c}}$ | 4001 | 0.65

We also show values of the difference of the doubly-heavy tetraquark and
threshold masses, $\Delta=M-T$ in Table 2. A negative $\Delta$ means the state
is below corresponding two-meson threshold and could be observed as a bound
state. Positive but small values of $\Delta$ correspond to resonance states
which can be observed as resonances. Remaining values of $\Delta$ reflect that
states should be broad and are difficult to observe experimentally.

Table 2: Corresponding $\Delta$ values for thresholds of $cc\bar{d}\bar{d}$ state. The superscripts denote the spin of the diquark/antidiquark whereas the subscripts denote the color structure. $\\{~{}\\}$ denotes the symmetric flavor wave functions of the two quarks and antiquarks subsystems. All results are in MeV. $J^{P}$ | Configuration | Mass | Threshold | $\Delta$
---|---|---|---|---
$0^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{d}\\}^{1}_{\mathbf{3}_{c}}\right]^{0}_{\mathbf{1}_{c}}$ | 3904 | $DD$ | 170
$1^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{d}\\}^{1}_{\mathbf{3}_{c}}\right]^{1}_{\mathbf{1}_{c}}$ | 3952 | $DD^{\ast}$ | 76
$2^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{d}\\}^{1}_{\mathbf{3}_{c}}\right]^{2}_{\mathbf{1}_{c}}$ | 4001 | $D^{\ast}D^{\ast}$ | -16

It can be seen from Table 2 that the masses of $J^{P}=0^{+}$ and $J^{P}=1^{+}$
$cc\bar{d}\bar{d}$ states are above corresponding thresholds. They could be
resonance states. The mass of $J^{P}=2^{+}$ $cc\bar{d}\bar{d}$ state is 16 MeV
below corresponding threshold. This state could be a bound state candidate.

###  IV.2 $cc\bar{d}\bar{s}$ State

The mass spectrum and expectation values for $\sqrt{\langle r^{2}\rangle}$ of
$cc\bar{d}\bar{s}$ state are given in Table 3. As can be seen from the table,
mass splitting of $cc\bar{d}\bar{s}$ multiplet is 87 MeV. We observe from
$\sqrt{\langle r^{2}\rangle}$ values that the predicted states are compact.

Table 3: Same as Table 1 but for $cc\bar{d}\bar{s}$ state. $J^{P}$ | Configuration | Mass | $\sqrt{\langle r^{2}\rangle}$
---|---|---|---
$0^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{0}_{\mathbf{1}_{c}}$ | 3982 | 0.64
$1^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{1}_{\mathbf{1}_{c}}$ | 4031 | 0.65
$2^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{2}_{\mathbf{1}_{c}}$ | 4069 | 0.65

We also show values of the $\Delta=M-T$ for the $cc\bar{d}\bar{s}$ tetraquarks
in Table 4. As can be seen from the Table, masses of $J^{P}=0^{+}$ and
$J^{P}=1^{+}$ $cc\bar{d}\bar{s}$ states are above corresponding thresholds.
They could be resonance states. The mass of $J^{P}=2^{+}$ $cc\bar{d}\bar{s}$
state is 52 MeV below corresponding threshold. This state could be a bound
state candidate.

Table 4: Same as Table 2 but for $cc\bar{d}\bar{s}$ state. $J^{P}$ | Configuration | Mass | Threshold | $\Delta$
---|---|---|---|---
$0^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{0}_{\mathbf{1}_{c}}$ | 3982 | $DD_{s}$ | 146
$1^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{1}_{\mathbf{1}_{c}}$ | 4031 | $D^{\ast}D_{s}/D_{s}^{\ast}D$ | 54/52
$2^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{d}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{2}_{\mathbf{1}_{c}}$ | 4069 | $D^{\ast}D_{s}^{\ast}$ | -52

###  IV.3 $cc\bar{s}\bar{s}$ State

The mass spectrum and expectation values for $\sqrt{\langle r^{2}\rangle}$ of
$cc\bar{s}\bar{s}$ state are given in Table 5. As can be seen from the table,
mass splitting of $cc\bar{d}\bar{s}$ multiplet is 57 MeV. We observe from
$\sqrt{\langle r^{2}\rangle}$ values that the predicted states are compact.

Table 5: Same as Table 1 but for $cc\bar{s}\bar{s}$ state. $J^{P}$ | Configuration | Mass | $\sqrt{\langle r^{2}\rangle}$
---|---|---|---
$0^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{s}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{0}_{\mathbf{1}_{c}}$ | 4121 | 0.64
$1^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{s}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{1}_{\mathbf{1}_{c}}$ | 4148 | 0.65
$2^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{s}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{2}_{\mathbf{1}_{c}}$ | 4178 | 0.65

$\Delta=M-T$ values for the $cc\bar{s}\bar{s}$ tetraquarks can be seen in
Table 6.

Table 6: Same as Table 2 but for $cc\bar{s}\bar{s}$ state. $J^{P}$ | Configuration | Mass | Threshold | $\Delta$
---|---|---|---|---
$0^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{s}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{0}_{\mathbf{1}_{c}}$ | 4121 | $D_{s}D_{s}$ | 184
$1^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{s}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{1}_{\mathbf{1}_{c}}$ | 4148 | $D_{s}D_{s}^{\ast}$ | 67
$2^{+}$ | $\left[\\{cc\\}^{1}_{\bar{\mathbf{3}}_{c}}\\{\bar{s}\bar{s}\\}^{1}_{\mathbf{3}_{c}}\right]^{2}_{\mathbf{1}_{c}}$ | 4178 | $D^{\ast}D_{s}^{\ast}$ | -46

As can be seen from the Table 6, masses of $J^{P}=0^{+}$ and $J^{P}=1^{+}$
$cc\bar{s}\bar{s}$ states are above corresponding thresholds. They could be
resonance states. The mass of $J^{P}=2^{+}$ $cc\bar{s}\bar{s}$ state is 46 MeV
below corresponding threshold. This state could be a bound state candidate.

##  V Discussion and Final Remarks

We obtain mass spectra and expectation values of $\sqrt{\langle r^{2}\rangle}$
for the $cc\bar{d}\bar{d}$, $cc\bar{d}\bar{s}$, and $cc\bar{s}\bar{s}$
tetraquark states using dynamical diquark model. The calculated $\sqrt{\langle
r^{2}\rangle}$ values reflect the distance between heavy diquark and light
antidiquark. As can be seen from Tables 2, 4, and 6 we obtain some bound state
candidates. A calculation of strong decays for the $cc\bar{d}\bar{d}$,
$cc\bar{d}\bar{s}$, and $cc\bar{s}\bar{s}$ tetraquark states can give
information about existence of bound states. From the Tables 1, 3, and 5 it
can be seen that the $cc\bar{d}\bar{d}$, $cc\bar{d}\bar{s}$, and
$cc\bar{s}\bar{s}$ tetraquark states are compact. The size of the tetraquarks
is less than $<1~{}\text{fm}$ which is called confinement radius.

The observation of $T_{cc}^{+}$ state is as important as the observation of
$\chi_{c1}(3872)$ (formerly known as X(3872)). The possible observation of
$T_{cc}^{++}$ states in future experimental studies would constitute a new
establishment for multiquark states. We hope that our results will help for
this endeavour.

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