# A survey of heavy-antiheavy hadronic molecules

Xiang-Kun Dong1,2<EMAIL_ADDRESS>Feng-Kun Guo1,2<EMAIL_ADDRESS>Bing-
Song Zou1,2,3<EMAIL_ADDRESS>1CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
2School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
3School of Physics, Central South University, Changsha 410083, China

###### Abstract

Many efforts have been made to reveal the nature of the overabundant resonant
structures observed by the worldwide experiments in the last two decades.
Hadronic molecules attract special attention because many of these seemingly
unconventional resonances are located close to the threshold of a pair of
hadrons. To give an overall feature of the spectrum of hadronic molecules
composed of a pair of heavy-antiheavy hadrons, namely, which pairs are
possible to form molecular states, we take charmed hadrons for example to
investigate the interaction between them and search for poles by solving the
Bethe-Salpeter equation. We consider all possible combinations of hadron pairs
of the $S$-wave singly-charmed mesons and baryons as well as the narrow
$P$-wave charmed mesons. The interactions, which are assumed to be meson-
exchange saturated, are described by constant contact terms which are resummed
to generate poles. It turns out that if a system is attractive near threshold
by the light meson exchange, there is a pole close to threshold corresponding
to a bound state or a virtual state, depending on the strength of interaction
and the cutoff. In total, 229 molecular states are predicted. The observed
near-threshold structures with hidden-charm, like the famous $X(3872)$ and
$P_{c}$ states, fit into the spectrum we obtain. We also highlight a
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state that has a pole consistent with the
cross section of the $e^{+}e^{-}\to\Lambda_{c}\bar{\Lambda}_{c}$ precisely
measured by the BESIII Collaboration.

###### Contents

  1. I Introduction
  2. II Lagrangian from heavy quark spin symmetry
    1. II.1 Heavy mesons
    2. II.2 Heavy baryons
  3. III Potentials
    1. III.1 Conventions
    2. III.2 Potentials from light vector exchange
    3. III.3 Potentials from vector charmonia exchange
  4. IV Molecular states from constant interactions
    1. IV.1 Poles
    2. IV.2 Discussions of selected systems
      1. IV.2.1 $D^{(*)}\bar{D}^{(*)}$: $X(3872)$, $Z_{c}(3900)$ and their partners
      2. IV.2.2 $D_{s}^{(*)}\bar{D}_{s}^{(*)}$ virtual states
      3. IV.2.3 $D^{(*)}\bar{D}_{s}^{(*)}$: $Z_{cs}$ as virtual states
      4. IV.2.4 $D^{(*)}\bar{D}_{1,2}$: $Y(4260)$ and related states
      5. IV.2.5 $\Lambda_{c}\bar{\Lambda}_{c}$: analysis of the BESIII data and more baryon-antibaryon bound states
      6. IV.2.6 $\bar{D}^{(*)}\Sigma_{c}^{(*)}$: $P_{c}$ states
      7. IV.2.7 $\bar{D}^{(*)}\Xi_{c}^{(\prime)}$: $P_{cs}$ and related states
  5. V Summary and discussion
  6. A Vertex factors for direct processes
  7. B List of the potential factor $F$
  8. C Amplitude calculation for cross processes

##  I Introduction

The formation of hadrons from quarks and gluons is governed by quantum
chromodynamics (QCD), which at low energy is nonperturbative. Therefore, it is
difficult to tackle with the hadron spectrum model-independently. The
traditional quark model Gell-Mann (1964); Zweig (1964), where hadrons are
classified as mesons, composed of $q\bar{q}$ and baryons, composed of $qqq$,
provides quite satisfactory description of the hadrons observed in last
century. The last two decades witnessed the emergence of plenty of states or
resonant structures in experiments, many of which do not fit the hadron
spectrum predicted by the naive quark model and thus are candidates of the so-
called exotic states. Many efforts have been devoted to understand the nature
of these states but most of them still remain controversial (see Refs. Chen
_et al._ (2016a); Hosaka  _et al._ (2016); Richard (2016); Lebed  _et al._
(2017); Esposito  _et al._ (2017); Guo  _et al._ (2018); Ali  _et al._ (2017);
Olsen  _et al._ (2018); Altmannshofer  _et al._ (2019); Kalashnikova and
Nefediev (2019); Cerri  _et al._ (2019); Liu  _et al._ (2019a); Brambilla  _et
al._ (2020); Guo  _et al._ (2020); Yang  _et al._ (2020a); Ortega and Entem
(2020) for recent reviews).

The observation that most of these structures are located near the threshold
of a pair of heavy-antiheavy hadrons may shed light on identifying the nature
of these structures. To name a few famous examples, let us mention the
$X(3872)$, aka $\chi_{c1}(3872)$ Choi  _et al._ (2003) and $Z_{c}(3900)^{\pm}$
Ablikim  _et al._ (2013a); Liu  _et al._ (2013); Ablikim  _et al._ (2014a)
around the $D\bar{D}^{*}$ threshold, the $Z_{c}(4020)^{\pm}$ Ablikim  _et al._
(2014b, 2013b) near the $D^{*}\bar{D}^{*}$ threshold, the $Z_{b}(10610)^{\pm}$
and $Z_{b}(10650)^{\pm}$ Bondar  _et al._ (2012); Garmash  _et al._ (2016)
near the $B\bar{B}^{*}$ and $B^{*}\bar{B}^{*}$ thresholds, the
$Z_{cs}(3985)^{-}$ Ablikim  _et al._ (2020a) near the $\bar{D}_{s}D^{*}$ and
$\bar{D}_{s}^{*}D$ thresholds, and the $P_{c}$ states Aaij  _et al._ (2019)
near the $\bar{D}^{(*)}\Sigma_{c}$ thresholds, respectively. These resonant
structures were widely investigated in many models, including assigning them
to be the molecular states of the corresponding systems. We refer to Ref. Guo
_et al._ (2018) for a comprehensive review of hadronic molecules.

Although these near-threshold resonant structures have been explored by many
works using various methods, our understanding of these structures will
greatly benefit from a whole and systematic spectrum of heavy-antiheavy
hadronic molecules based on one single model. In this paper, we provide such a
spectrum of hadronic molecules composed of a pair of heavy-antiheavy hadrons
including $S,P$-wave heavy mesons and $S$-wave heavy baryons. The interactions
between these hadron pairs are assumed to be contact terms saturated by meson
exchanges as in, e.g., Refs. Ecker  _et al._ (1989); Epelbaum  _et al._
(2002); Peng  _et al._ (2020a). In order to have a unified treatment in all
systems, we will not consider coupled-channel effects. In Ref. Peng  _et al._
(2020a) a similar study, focusing on the possible bound states composed of
$\bar{D}^{(*)}$ and $\Sigma_{c}^{(*)}$, are performed. Also note that such
interactions have been obtained in many other works, e.g. Refs. Liu  _et al._
(2008); Ding (2009); Wu  _et al._ (2010, 2011); Yang  _et al._ (2012); Pavon
Valderrama (2020), to look for possible bound states associated with the near-
threshold resonant structures. These works used different models and
conventions and in some cases the results are inconsistent with each other.

In this work we consider the resonance saturation of the contact interaction
due to the one-boson-exchange (light pseudoscalar and vector mesons) together
with heavy quark spin symmetry (HQSS), chiral symmetry and SU(3) flavor
symmetry, to estimate the potentials of the heavy-antiheavy systems, and
special attention is paid to the signs of coupling constants to get self-
consistent results. Heavy quark flavor symmetry (HQFS) implies that the
potentials between bottomed hadron pairs are the same as those between charmed
ones and thus we take charmed hadrons for example. The obtained potentials are
used to solve the Bethe-Salpeter equation in the on-shell form to determine
the pole positions of these heavy-antiheavy hadron systems and a systematic
spectrum of these hadronic molecules is obtained.

Besides the possible molecular states, the interaction between a pair of
heavy-antiheavy hadrons at threshold is crucial to understand the line shape
of the invariant mass distribution of related channels, as discussed in Ref.
Dong  _et al._ (2020a). It is known that unitarity of the $S$ matrix requires
the threshold to be a branch point of the scattering amplitude. Therefore, the
square of amplitude modulus always shows a cusp at the threshold and a
nontrivial structure may appear in the line shape of the invariant mass
distribution. The detailed structure of the cusp, a peak, a dip or buried in
the background, depends on the interaction of the two relevant particles near
the threshold. More specifically, the cusp at the threshold of a heavy-
antiheavy hadron pair will generally show up as a peak in the invariant mass
distribution of a heavy quarkonium and light hadron(s), which couple to the
heavy-antiheavy hadron pair, if the interaction is attractive at threshold and
a nearby virtual state comes into existence (for subtlety and more detailed
discussions, we refer to Ref. Dong  _et al._ (2020a)). If the attraction is
strong and a bound state is formed, the peak will be located at the mass of
the bound state below threshold. Therefore, the potentials and the pole
positions obtained in this work are of great significance for the study of
near-threshold line shapes.

Note that the leading order interaction, which is just a constant, between a
heavy-antiheavy hadron pair, when resummed, works well for both purposes
mentioned above if we only focus on the near-threshold pole and line shape.
Therefore, we only keep the leading order interaction, which is saturated by
the vector meson exchange as discussed in the following, and its resummation.

This paper is organized as follows. In Section II, the Lagrangians of heavy
hadron and light meson coupling are presented. In Section III, the potentials
of different systems are obtained. We calculate the pole positions of
different systems and compare them with experimental and other theoretical
results in Section IV. Section V is devoted to a brief summary. Some details
regarding the potentials are relegated to Appendices A, B and C.

##  II Lagrangian from heavy quark spin symmetry

For hadrons containing one or more heavy quarks, additional symmetries emerge
in the low energy effective field theory for QCD due to the large heavy quark
mass Isgur and Wise (1990, 1989). On the one hand, the strong suppression of
the chromomagnetic interaction, which is proportional to
${\bm{\sigma}\cdot\bm{B}}/{m_{Q}}\sim{\Lambda_{\rm QCD}}/{m_{Q}}$ with
$\Lambda_{\rm QCD}\sim 200-300$ MeV the typical QCD scale, implies that the
spin of the heavy quark $s_{Q}$ is decoupled with the angular momentum
$s_{\ell}$ of light quarks in the limit of $m_{Q}\to\infty$. Therefore, HQSS
emerges, which means that the interaction is invariant under a transformation
of $s_{Q}$. On the other hand, the change of the velocity of the heavy quark
in a singly-heavy hadron during the interaction, $\Delta v={\Delta
p}/{m_{Q}}\sim{\Lambda_{\rm QCD}}/{m_{Q}}$, vanishes in the limit of
$m_{Q}\to\infty$, and the heavy quark behaves like a static color source,
independent of the quark flavor. Therefore, it is expected that the potentials
between bottomed hadron pairs are the same as those of the charmed ones, and
in turn it is sufficient to focus on the charm sector.

###  II.1 Heavy mesons

To construct a Lagrangian that is invariant under the heavy quark spin
transformation and chiral transformation, it is convenient to represent the
ground states of charmed mesons as the following superfield Wise (1992);
Casalbuoni  _et al._ (1992, 1997)

$\displaystyle
H_{a}^{(Q)}=\frac{1+\not{v}}{2}\left[P_{a}^{*(Q)\mu}\gamma_{\mu}-P_{a}^{(Q)}\gamma_{5}\right],$
(1)

where $a$ is the SU(3) flavor index,

$\displaystyle P^{(Q)}=(D^{0},D^{+},D_{s}^{+}),\quad
P^{*(Q)}_{\mu}=(D^{*0}_{\mu},D^{*+}_{\mu},D^{*+}_{s\mu}),$ (2)

and $v^{\mu}=p^{\mu}/M$ is the four-velocity of the heavy meson satisfying
$v\cdot v=1$. The heavy field operators contain a factor $\sqrt{M_{H}}$ and
have dimension 3/2. The superfield that creates heavy mesons is constructed as

$\bar{H}_{a}^{(Q)}=\gamma_{0}H_{a}^{(Q)\dagger}\gamma_{0}.$ (3)

The superfields that annihilate or create mesons containing an antiheavy quark
are not $\bar{H}_{a}^{(Q)}$ or $H_{a}^{(Q)}$ but the following ones Grinstein
_et al._ (1992):

$\displaystyle H_{a}^{(\bar{Q})}$
$\displaystyle=C\left(\mathcal{C}H_{a}^{(Q)}\mathcal{C}^{-1}\right)^{T}C^{-1}$
$\displaystyle=\left[P_{a\mu}^{*(\bar{Q})}\gamma^{\mu}-P_{a}^{(\bar{Q})}\gamma_{5}\right]\frac{1-\not{v}}{2},$
(4) $\displaystyle\bar{H}_{a}^{(\bar{Q})}$
$\displaystyle=\gamma_{0}H_{a}^{(\bar{Q})\dagger}\gamma_{0},$ (5)

with

$\displaystyle P^{(\bar{Q})}=(\bar{D}^{0},D^{-},D_{s}^{-}),\quad
P_{\mu}^{*(\bar{Q})}=(\bar{D}_{\mu}^{*0},D_{\mu}^{*-},D^{*-}_{s\mu}).$ (6)

$\mathcal{C}$ is the charge conjugation operator and $C=i\gamma^{2}\gamma^{0}$
is the charge conjugation matrix, where we have taken the phase convention for
charge conjugation as
$\mathcal{C}P_{a}^{(Q)}\mathcal{C}^{-1}=P_{a}^{(\bar{Q})}$ and
$\mathcal{C}P_{a}^{*(Q)}\mathcal{C}^{-1}=-P_{a}^{*(\bar{Q})}$.

The $P$-wave heavy mesons have two spin multiplets, one with $s_{\ell}=1/2$
represented by $S$ while the other with $s_{\ell}=3/2$ represented by $T$ Falk
(1992); Falk and Luke (1992),

$\displaystyle S_{a}^{(Q)}=$
$\displaystyle\,\frac{1+\not{v}}{2}\left[P_{1a}^{\prime(Q)\mu}\gamma_{\mu}\gamma_{5}-P_{0a}^{*(Q)}\right],$
(7) $\displaystyle T_{a}^{(Q)\mu}=$
$\displaystyle\,\frac{1+\not{v}}{2}\bigg{[}P_{2a}^{*(Q)\mu\nu}\gamma_{\nu}$
$\displaystyle-\sqrt{\frac{3}{2}}P_{1a\nu}^{(Q)}\gamma_{5}\left(g^{\mu\nu}-\frac{1}{3}\gamma^{\nu}\left(\gamma^{\mu}-v^{\mu}\right)\right)\bigg{]}.$
(8)

Analogous with Eqs. (3,4,5), we have

$\displaystyle\bar{S}_{a}^{(Q)}=$
$\displaystyle\,\gamma_{0}S_{a}^{(Q)\dagger}\gamma_{0},$ (9)
$\displaystyle\bar{T}_{a}^{(Q)\mu}=$
$\displaystyle\,\gamma_{0}T_{a}^{(Q)\mu\dagger}\gamma_{0},$ (10)
$\displaystyle S_{a}^{(\bar{Q})}=$
$\displaystyle\,\left[P_{1a}^{\prime(\bar{Q})\mu}\gamma_{\mu}\gamma_{5}-P_{0a}^{*(\bar{Q})}\right]\frac{1-\not{v}}{2},$
(11) $\displaystyle T_{a}^{(\bar{Q})\mu}=$
$\displaystyle\,\left[P_{2a}^{(\bar{Q})\mu\nu}\gamma_{\nu}-\sqrt{\frac{3}{2}}P_{1a\nu}^{(\bar{Q})}\right.$
$\displaystyle\left.\times\gamma_{5}\left(g^{\mu\nu}-\frac{1}{3}\left(\gamma^{\mu}-v^{\mu}\right)\gamma^{\nu}\right)\right]\frac{1-\not{v}}{2},$
(12) $\displaystyle\bar{S}_{a}^{(\bar{Q})}=$
$\displaystyle\,\gamma_{0}S_{a}^{(\bar{Q})\dagger}\gamma_{0},$ (13)
$\displaystyle\bar{T}_{a\mu}^{(\bar{Q})}=$
$\displaystyle\,\gamma_{0}T_{a\mu}^{(\bar{Q})\dagger}\gamma_{0}.$ (14)

The mesons in the $T$ multiplet are

$\displaystyle P_{1}^{(Q)}$
$\displaystyle=(D_{1}(2420)^{0},D_{1}(2420)^{+},D_{s1}(2536)^{+}),$
$\displaystyle P_{2}^{(Q)}$
$\displaystyle=(D_{2}(2460)^{0},D_{2}(2460)^{+},D_{s2}(2573)^{+}),$ (15)

which can couple to $D^{*}\pi/K$ only in $D$-wave in the heavy quark limit.
While the $P$-wave charmed mesons with $s_{\ell}=1/2$ can couple to
$D^{*}\pi/K$ in $S$-wave without violating HQSS, there are issues in
identifying them. The $D_{0}^{*}(2300)$ and $D_{1}(2430)$ listed in the Review
of Particle Physics (RPP) Zyla  _et al._ (2020) could be candidates for the
charm-nonstrange ones. However, on the one hand, they have rather large widths
such that they would have decayed before they can be bound together with
another heavy hadron Filin  _et al._ (2010); Guo and Meißner (2011); on the
other hand, they were extracted using the Breit-Wigner parameterization which
has deficiencies in the current case Du  _et al._ (2019) and has been
demonstrated Du  _et al._ (2020a) to lead to resonance parameters for the
$D_{0}^{*}(2300)$ in conflict with the precise LHCb data of the $B^{-}\to
D^{+}\pi^{-}\pi^{-}$ process Aaij  _et al._ (2016). For the ones with
strangeness, the lowest positive-parity $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$
are widely considered as molecular states of $DK$ and $D^{*}K$ Barnes  _et
al._ (2003); van Beveren and Rupp (2003); Kolomeitsev and Lutz (2004); Chen
and Li (2004); Guo  _et al._ (2006, 2007), see Ref. Guo (2019) for a recent
review collecting evidence for such an interpretation. This multiplet $S$,
therefore, will not be considered in the rest of this work. For studies of
three-body hadronic molecular states involving the $D_{s0}^{*}(2317)$ as a
$DK$ subsystem, we refer to Refs. Ma  _et al._ (2019); Martínez Torres  _et
al._ (2019); Wu  _et al._ (2019, 2020).

The light pseudoscalar meson octet can be introduced using the nonlinear
realization of the spontaneous chiral symmetry breaking of QCD as
$\Sigma=\xi^{2}$ and $\xi=e^{i\Pi/(\sqrt{2}F_{\pi})}$ with $F_{\pi}=92$ MeV
the pion decay constant and

$\displaystyle\Pi=\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{array}\right).$ (19)

The effective Lagrangian for the coupling of heavy mesons and light
pseudoscalar mesons is constructed by imposing invariance under both heavy
quark spin transformation and chiral transformation Wise (1992); Falk and Luke
(1992); Grinstein  _et al._ (1992),

$\displaystyle\mathcal{L}_{PP\Pi}$ $\displaystyle=\,ig\left\langle
H_{b}^{(Q)}\not{\mathcal{A}}_{ba}\gamma_{5}\bar{H}_{a}^{(Q)}\right\rangle+ik\left\langle
T_{b}^{(Q)\mu}\not{\mathcal{A}}_{ba}\gamma_{5}\bar{T}_{a\mu}^{(Q)}\right\rangle+i\tilde{k}\left\langle
S_{b}^{(Q)}\not{\mathcal{A}}_{ba}\gamma_{5}\bar{S}_{a}^{(Q)}\right\rangle+\left[ih\left\langle
S_{b}^{(Q)}\not{\mathcal{A}}_{ba}\gamma_{5}\bar{H}_{a}^{(Q)}\right\rangle\right.$
$\displaystyle\left.+\,i\tilde{h}\left\langle T_{b}^{(Q)\mu}\mathcal{A}_{\mu
ba}\gamma_{5}\bar{S}_{a}^{(Q)}\right\rangle+i\frac{h_{1}}{\Lambda_{\chi}}\left\langle
T_{b}^{(Q)\mu}\left(D_{\mu}\not{\mathcal{A}}\right)_{ba}\gamma_{5}\bar{H}_{a}^{(Q)}\right\rangle+i\frac{h_{2}}{\Lambda_{\chi}}\left\langle
T_{b}^{(Q)\mu}\left(\not{D}\mathcal{A}_{\mu}\right)_{ba}\gamma_{5}\bar{H}_{a}^{(Q)}\right\rangle+h.c.\right]$
$\displaystyle+ig\left\langle\bar{H}_{a}^{(\bar{Q})}\not{\mathcal{A}}_{ab}\gamma_{5}H_{b}^{(\bar{Q})}\right\rangle+ik\left\langle\bar{T}_{a}^{(\bar{Q})\mu}\not{\mathcal{A}}_{ab}\gamma_{5}T_{b\mu}^{(\bar{Q})}\right\rangle+i\tilde{k}\left\langle\bar{S}_{a}^{(\bar{Q})}\not{\mathcal{A}}_{ab}\gamma_{5}S_{b}^{(\bar{Q})}\right\rangle+{\bigg{[}}ih\left\langle\bar{H}_{a}^{(\bar{Q})}\not{\mathcal{A}}_{ab}\gamma_{5}S_{b}^{(\bar{Q})}\right\rangle$
$\displaystyle+i\tilde{h}\left\langle\bar{S}_{a}^{(\bar{Q})}\mathcal{A}_{\mu
ab}\gamma_{5}T_{b}^{(\bar{Q})\mu}\right\rangle+i\frac{h_{1}}{\Lambda_{\chi}}\left\langle\bar{H}_{a}^{(\bar{Q})}\Big{(}\not{\mathcal{A}}\stackrel{{\scriptstyle\leftarrow}}{{D_{\mu}^{\prime}}}\Big{)}_{ab}\gamma_{5}T_{b}^{(\bar{Q})\mu}\right\rangle+i\frac{h_{2}}{\Lambda_{\chi}}\left\langle\bar{H}_{a}^{(\bar{Q})}\Big{(}\mathcal{A}_{\mu}\stackrel{{\scriptstyle\leftarrow}}{{\not{D}^{\prime}}}\Big{)}_{ab}\gamma_{5}T_{b}^{(\bar{Q})\mu}\right\rangle+\text{h.c.}{\bigg{]}},$
(20)

where $D_{\mu}=\partial_{\mu}+\mathcal{V}_{\mu}$,
$D^{\prime}_{\mu}=\partial_{\mu}-\mathcal{V}_{\mu}$, $\langle\cdots\rangle$
denotes tracing over the Dirac $\gamma$ matrices, $\Lambda_{\chi}\simeq 4\pi
F_{\pi}$ is the chiral symmetry breaking scale, and

$\displaystyle\mathcal{V}_{\mu}$
$\displaystyle=\frac{1}{2}\left(\xi^{\dagger}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{\dagger}\right),$
(21) $\displaystyle\mathcal{A}_{\mu}$
$\displaystyle=\frac{1}{2}\left(\xi^{\dagger}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{\dagger}\right)$
(22)

are the vector and axial currents which contain an even and odd number of
pseudoscalar mesons, respectively.

The coupling of heavy mesons and light vector mesons can be introduced by
using the hidden local symmetry approach Bando  _et al._ (1985, 1988); Meißner
(1988), and the Lagrangian reads Casalbuoni  _et al._ (1992, 1993, 1997)

$\displaystyle\mathcal{L}_{PPV}=$ $\displaystyle\,i\beta\left\langle
H_{b}^{(Q)}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ba}\bar{H}_{a}^{(Q)}\right\rangle+i\lambda\left\langle
H_{b}^{(Q)}\sigma^{\mu\nu}F_{\mu\nu}(\rho)_{ba}\bar{H}_{a}^{(Q)}\right\rangle+i\beta_{1}\left\langle
S_{b}^{(Q)}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ba}\bar{S}_{a}^{(Q)}\right\rangle$
$\displaystyle+i\lambda_{1}\left\langle
S_{b}^{(Q)}\sigma^{\mu\nu}F_{\mu\nu}(\rho)_{ba}\bar{S}_{a}^{(Q)}\right\rangle+i\beta_{2}\left\langle
T_{b}^{(Q)\lambda}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ba}\bar{T}_{a\lambda}^{(Q)}\right\rangle+i\lambda_{2}\left\langle
T_{b}^{(Q)\lambda}\sigma^{\mu\nu}F_{\mu\nu}(\rho)_{ba}\bar{T}_{a\lambda}^{(Q)}\right\rangle$
$\displaystyle+\left[i\zeta\left\langle
H_{b}^{(Q)}\gamma^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ba}\bar{S}_{a}^{(Q)}\right\rangle+i\mu\left\langle
H_{b}^{(Q)}\sigma^{\lambda\nu}F_{\lambda\nu}(\rho)_{ba}\bar{S}_{a}^{(Q)}\right\rangle+i\zeta_{1}\left\langle
T_{b}^{(Q)\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ba}\bar{H}_{a}^{(Q)}\right\rangle\right.$
$\displaystyle\left.+\mu_{1}\left\langle
T_{b}^{(Q)\mu}\gamma^{\nu}F_{\mu\nu}(\rho)_{ba}\bar{H}_{a}^{(Q)}\right\rangle+h.c.\right]$
$\displaystyle-i\beta\left\langle\bar{H}_{a}^{(\bar{Q})}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ab}H_{b}^{(\bar{Q})}\right\rangle+i\lambda\left\langle\bar{H}_{a}^{(\bar{Q})}\sigma^{\mu\nu}F_{\mu\nu}(\rho)_{ab}H_{b}^{(\bar{Q})}\right\rangle-i\beta_{1}\left\langle\bar{S}_{a}^{(\bar{Q})}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ab}S_{b}^{(\bar{Q})}\right\rangle$
$\displaystyle+i\lambda_{1}\left\langle\bar{S}_{a}^{(\bar{Q})}\sigma^{\mu\nu}F_{\mu\nu}(\rho)_{ab}S_{b}^{(\bar{Q})}\right\rangle-i\beta_{2}\left\langle\bar{T}_{a\lambda}^{(\bar{Q})}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ab}T_{b}^{(\bar{Q})\lambda}\right\rangle+i\lambda_{2}\left\langle\bar{T}_{a\lambda}^{(\bar{Q})}\sigma^{\mu\nu}F_{\mu\nu}(\rho)_{ab}T_{b}^{(\bar{Q})\lambda}\right\rangle$
$\displaystyle+\left[i\zeta\left\langle\bar{S}_{a}^{(\bar{Q})}\gamma^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ab}H_{a}^{(\bar{Q})}\right\rangle+i\mu\left\langle\bar{S}_{a}^{(\bar{Q})}\sigma^{\lambda\nu}F_{\lambda\nu}(\rho)_{ab}H_{b}^{(\bar{Q})}\right\rangle-i\zeta_{1}\left\langle\bar{H}_{a}^{(\bar{Q})}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)_{ab}T_{b}^{(\bar{Q})\mu}\right\rangle\right.$
$\displaystyle\left.+\mu_{1}\left\langle\bar{H}_{a}^{(\bar{Q})}\gamma^{\nu}F_{\mu\nu}(\rho)_{ab}T_{b}^{(\bar{Q})\mu}\right\rangle+\text{h.c.}\right],$
(23)

with
$F_{\mu\nu}=\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}+\left[\rho_{\mu},\rho_{\nu}\right]$,
and

$\displaystyle\rho=i\frac{g_{V}}{\sqrt{2}}V=i\frac{g_{V}}{\sqrt{2}}\left(\begin{array}[]{ccc}\frac{\omega}{\sqrt{2}}+\frac{\rho^{0}}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&\frac{\omega}{\sqrt{2}}-\frac{\rho^{0}}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right),$ (27)

which satisfies $\mathcal{C}V\mathcal{C}^{-1}=-V^{T}$.

Remind that in the following we are only interested in the potential near
threshold and will not consider coupled channels. Therefore, the Lagrangian
that results in potentials proportional to the transferred momentum
$\bm{q}^{2}$ will have little contributions. At the leading order of the
chiral expansion, the light pseudoscalar mesons as Goldstone bosons only
couple in derivatives, as demonstrated in Eq. (20), so all pseudoscalar
exchanges have subleading contributions near threshold in comparison with the
constant contact term that can generate a near-threshold pole after
resummation. Moreover, coupled channels are not taken into account here, and
we do not consider the $s_{\ell}=1/2$ mesons. To this end we can just keep the
$\beta$, $\beta_{2}$ and $\zeta_{1}$ terms in Eq. (23). Expanding these terms
we obtain

$\displaystyle\mathcal{L}_{PPV}=-\sqrt{2}\beta
g_{V}\left(P_{a}^{(Q)}P_{b}^{(Q)\dagger}-P_{b}^{(\bar{Q})}P_{a}^{(\bar{Q})\dagger}\right)v_{\mu}V^{\mu}_{ab}$
$\displaystyle+\sqrt{2}\beta
g_{V}\left(P_{a}^{*(Q)\nu}P_{b\nu}^{*(Q)\dagger}-P_{b}^{*(\bar{Q})\nu}P_{a\nu}^{*(\bar{Q})\dagger}\right)v_{\mu}V^{\mu}_{ab}$
$\displaystyle-\sqrt{2}\beta_{2}g_{V}\left(P_{1a}^{(Q)\nu}P_{1b\nu}^{(Q)\dagger}-P_{1b}^{(\bar{Q})\nu}P_{1a\nu}^{(\bar{Q})\dagger}\right)v_{\mu}V^{\mu}_{ab}$
$\displaystyle+\sqrt{2}\beta_{2}g_{V}\left(P_{2a}^{(Q)\alpha\beta}P_{2b\alpha\beta}^{(Q)\dagger}-P_{2b}^{(\bar{Q})\alpha\beta}P_{2a\alpha\beta}^{(\bar{Q})\dagger}\right)v_{\mu}V^{\mu}_{ab}$
$\displaystyle+\Big{[}\sqrt{2}\zeta_{1}g_{V}\left(P_{2a}^{(Q)\mu\nu}P_{b\nu}^{(Q)*\dagger}+P_{2b}^{(\bar{Q})\mu\nu}P_{a\nu}^{(\bar{Q})*\dagger}\right)V_{ab\mu}$
$\displaystyle-\frac{i\zeta_{1}g_{V}}{\sqrt{3}}\epsilon_{\alpha\beta\gamma\delta}\left(P_{1a}^{(Q)\alpha}P_{b}^{(Q)*\dagger\beta}+P_{1b}^{(\bar{Q})\alpha}P_{a}^{(\bar{Q})*\dagger\beta}\right)v^{\gamma}V_{ab}^{\delta}$
$\displaystyle-\frac{2\zeta_{1}g_{V}}{\sqrt{3}}\left(P_{1a\mu}^{(Q)}P_{b}^{(Q)\dagger}-P_{1b\mu}^{(\bar{Q})}P_{a}^{(\bar{Q})\dagger}\right)V_{ab}^{\mu}+{\rm
h.c.}\Big{]}.$ (28)

Assuming vector meson dominance, the coupling constants $g_{V}$ and $\beta$
were estimated to be $5.8$ Bando  _et al._ (1988) and $0.9$ Isola  _et al._
(2003), respectively. In Ref. Dong  _et al._ (2020b), it is estimated that
$\beta_{2}\approx-\beta=-0.9$ under the assumption that the coupling of
$D_{1}D_{1}V$ is the same as that of $DDV$ and $\zeta_{1}\approx 0.16$ from
the decay of $K_{1}\to K\rho$.

###  II.2 Heavy baryons

In the heavy quark limit, the ground states of heavy baryons $Qqq$ form an
SU(3) antitriplet with $J^{P}=\frac{1}{2}^{+}$ denoted by $B^{(Q)}_{\bar{3}}$
and two degenerate sextets with $J^{P}=(\frac{1}{2},\frac{3}{2})^{+}$ denoted
by $(B^{(Q)}_{6},B^{(Q)*}_{6})$ Yan  _et al._ (1992),

$\displaystyle B^{(Q)}_{\bar{3}}$
$\displaystyle=\left(\begin{array}[]{ccc}0&\Lambda_{c}^{+}&\Xi_{c}^{+}\\\
-\Lambda_{c}^{+}&0&\Xi_{c}^{0}\\\
-\Xi_{c}^{+}&-\Xi_{c}^{0}&0\end{array}\right),$ (32) $\displaystyle
B^{(Q)}_{6}$
$\displaystyle=\left(\begin{array}[]{ccc}\Sigma_{c}^{++}&\frac{1}{\sqrt{2}}\Sigma_{c}^{+}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime+}\\\
\frac{1}{\sqrt{2}}\Sigma_{c}^{+}&\Sigma_{c}^{0}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime
0}\\\ \frac{1}{\sqrt{2}}\Xi_{c}^{\prime+}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime
0}&\Omega_{c}^{0}\end{array}\right),$ (36) $\displaystyle B_{6}^{(Q)*}$
$\displaystyle=\left(\begin{array}[]{ccc}\Sigma_{c}^{*++}&\frac{1}{\sqrt{2}}\Sigma_{c}^{*+}&\frac{1}{\sqrt{2}}\Xi_{c}^{*+}\\\
\frac{1}{\sqrt{2}}\Sigma_{c}^{*+}&\Sigma_{c}^{*0}&\frac{1}{\sqrt{2}}\Xi_{c}^{*0}\\\
\frac{1}{\sqrt{2}}\Xi_{c}^{*+}&\frac{1}{\sqrt{2}}\Xi_{c}^{*0}&\Omega_{c}^{*0}\end{array}\right).$
(40)

Here we do not consider the $P$-wave heavy baryons since they are not well
established experimentally. The two sextets are collected into the superfield
$S_{\mu}$,

$\displaystyle S^{(Q)}_{\mu}$
$\displaystyle=B_{6\mu}^{(Q)*}-\frac{1}{\sqrt{3}}\left(\gamma_{\mu}+v_{\mu}\right)\gamma^{5}B^{(Q)}_{6},$
(41) $\displaystyle\bar{S}^{(Q)}_{\mu}$
$\displaystyle=\bar{B}_{6\mu}^{(Q)*}+\frac{1}{\sqrt{3}}\bar{B}^{(Q)}_{6}\gamma^{5}\left(\gamma_{\mu}+v_{\mu}\right),$
(42)

where $B_{6\mu}$ is the Rarita-Schwinger vector-spinor field Rarita and
Schwinger (1941). The fields that annihilate anti-baryons are obtained by
taking the charge conjugation of $B^{(Q)}_{\bar{3}}$, $B^{(Q)}_{6}$ and
$B_{6}^{(Q)*}$,

$\displaystyle B^{(\bar{Q})}_{{3}}$
$\displaystyle=\left(\begin{array}[]{ccc}0&\Lambda_{c}^{-}&\Xi_{c}^{-}\\\
-\Lambda_{c}^{-}&0&\bar{\Xi}_{c}^{0}\\\
-\Xi_{c}^{-}&-\bar{\Xi}_{c}^{0}&0\end{array}\right),$ (46) $\displaystyle
B^{(\bar{Q})}_{\bar{6}}$
$\displaystyle=\left(\begin{array}[]{ccc}\Sigma_{c}^{--}&\frac{1}{\sqrt{2}}\Sigma_{c}^{-}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime-}\\\
\frac{1}{\sqrt{2}}\Sigma_{c}^{-}&\bar{\Sigma}_{c}^{0}&\frac{1}{\sqrt{2}}\bar{\Xi}_{c}^{\prime
0}\\\
\frac{1}{\sqrt{2}}\Xi_{c}^{\prime-}&\frac{1}{\sqrt{2}}\bar{\Xi}_{c}^{\prime
0}&\bar{\Omega}_{c}^{0}\end{array}\right),$ (50) $\displaystyle
B_{\bar{6}}^{(\bar{Q})*}$
$\displaystyle=\left(\begin{array}[]{ccc}\Sigma_{c}^{*--}&\frac{1}{\sqrt{2}}\Sigma_{c}^{*-}&\frac{1}{\sqrt{2}}\Xi_{c}^{*-}\\\
\frac{1}{\sqrt{2}}\Sigma_{c}^{*-}&\bar{\Sigma}_{c}^{*0}&\frac{1}{\sqrt{2}}\bar{\Xi}_{c}^{*0}\\\
\frac{1}{\sqrt{2}}\Xi_{c}^{*-}&\frac{1}{\sqrt{2}}\bar{\Xi}_{c}^{*0}&\bar{\Omega}_{c}^{*0}\end{array}\right),$
(54)

where we have used the phase conventions such that
$\mathcal{C}B^{(Q)}\mathcal{C}^{-1}=B^{(\bar{Q})}$. The corresponding
superfields now read

$\displaystyle S^{(\bar{Q})}_{\mu}$
$\displaystyle=B_{\bar{6}\mu}^{(\bar{Q})*}-\frac{1}{\sqrt{3}}\left(\gamma_{\mu}+v_{\mu}\right)\gamma^{5}B^{(\bar{Q})}_{\bar{6}},$
(55) $\displaystyle\bar{S}^{(\bar{Q})}_{\mu}$
$\displaystyle=\bar{B}_{\bar{6}\mu}^{(\bar{Q})*}+\frac{1}{\sqrt{3}}\bar{B}^{(\bar{Q})}_{\bar{6}}\gamma^{5}\left(\gamma_{\mu}+v_{\mu}\right).$
(56)

The Lagrangian for the coupling of heavy baryons and light mesons is
constructed as Liu and Oka (2012)

$\displaystyle\mathcal{L}_{\mathcal{B}}=$
$\displaystyle\,\mathcal{L}_{B_{3}}+\mathcal{L}_{S}+\mathcal{L}_{\text{int}},$
(57) $\displaystyle\mathcal{L}_{B_{{\bar{3}}}}=$
$\displaystyle\,\frac{1}{2}\operatorname{tr}\left[\bar{B}^{(Q)}_{\bar{3}}(iv\cdot
D)B^{(Q)}_{\bar{3}}\right]$
$\displaystyle+i\beta_{B}\operatorname{tr}\left[\bar{B}^{(Q)}_{\bar{3}}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)B^{(Q)}_{\bar{3}}\right],$
(58) $\displaystyle\mathcal{L}_{S}=$
$\displaystyle\,-\operatorname{tr}\left[\bar{S}^{(Q)\alpha}\left(iv\cdot
D-\Delta_{B}\right)S^{(Q)}_{\alpha}\right]$
$\displaystyle+\frac{3}{2}g_{1}\left(iv_{\kappa}\right)\epsilon^{\mu\nu\lambda\kappa}\operatorname{tr}\left[\bar{S}^{(Q)}_{\mu}\mathcal{A}_{\nu}S^{(Q)}_{\lambda}\right]$
$\displaystyle+i\beta_{S}\operatorname{tr}\left[\bar{S}^{(Q)}_{\mu}v_{\alpha}\left(\mathcal{V}^{\alpha}-\rho^{\alpha}\right)S^{(Q)\mu}\right]$
$\displaystyle+\lambda_{S}\operatorname{tr}\left[\bar{S}^{(Q)}_{\mu}F^{\mu\nu}S^{(Q)}_{\nu}\right],$
(59) $\displaystyle\mathcal{L}_{\rm int}=$
$\displaystyle\,g_{4}\operatorname{tr}\left[\bar{S}^{(Q)\mu}\mathcal{A}_{\mu}B^{(Q)}_{\bar{3}}\right]$
$\displaystyle+i\lambda_{I}\epsilon^{\mu\nu\lambda\kappa}v_{\mu}\operatorname{tr}\left[\bar{S}^{(Q)}_{\nu}F_{\lambda\kappa}B^{(Q)}_{\bar{3}}\right]+{\rm
h.c.},$ (60)

where $D_{\mu}B=\partial_{\mu}B+\mathcal{V}_{\mu}B+B\mathcal{V}_{\mu}^{T}$,
and $\Delta_{B}=m_{6}-m_{\bar{3}}$ is the mass difference between the anti-
triplet and sextet baryons. The coupling constants $\beta_{B}$ and $\beta_{S}$
are estimated in Ref. Liu and Oka (2012) where $\beta_{S}=-2\beta_{B}=1.44$ or
$2.06$ from quark model and $\beta_{S}=-2\beta_{B}=1.74$ from the vector meson
dominance assumption. However, there is a sign ambiguity (only the absolute
value of the $\rho NN$ coupling was determined from the $NN$ scattering
Machleidt  _et al._ (1987)). It turns out that the sign choice in Ref. Liu and
Oka (2012) yields potentials of the anti-charmed meson and charmed baryon
systems with an opposite sign compared to the ones obtained by SU(4) relations
Wu  _et al._ (2011). It also conflicts with the famous $P_{c}$ states Aaij
_et al._ (2019), which are believed to be molecular states of
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ with isospin $1/2$—with a positive
$\beta_{S}$, these systems will be repulsive (see below). These issues can be
fixed by choosing the signs of $\beta_{B}$ and $\beta_{S}$ opposite to those
taken in Ref. Liu and Oka (2012), just like what Ref. Chen  _et al._ (2019a)
did.

For the coupling of antiheavy baryons and light mesons, by taking the charge
conjugation transformation of the above ones, we have

$\displaystyle\mathcal{L}^{\prime}_{\mathcal{B}}=$
$\displaystyle\,\mathcal{L}^{\prime}_{B_{3}}+\mathcal{L}^{\prime}_{S}+\mathcal{L}^{\prime}_{\text{int}},$
(61) $\displaystyle\mathcal{L}^{\prime}_{B_{{\bar{3}}}}=$
$\displaystyle\,\frac{1}{2}\operatorname{tr}\left[\bar{B}^{(\bar{Q})}_{{3}}(iv\cdot
D)^{T}B^{(\bar{Q})}_{{3}}\right]$
$\displaystyle-i\beta_{B}\operatorname{tr}\left[\bar{B}^{(\bar{Q})}_{{3}}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)^{T}B^{(\bar{Q})}_{{3}}\right],$
(62) $\displaystyle\mathcal{L}^{\prime}_{S}=$
$\displaystyle\,-\operatorname{tr}\left[\bar{S}^{(\bar{Q})\alpha}\left(iv\cdot
D-\Delta_{B}\right)^{T}S^{(\bar{Q})}_{\alpha}\right]$
$\displaystyle+\frac{3}{2}g_{1}\left(iv_{\kappa}\right)\epsilon^{\mu\nu\lambda\kappa}\operatorname{tr}\left[\bar{S}^{(\bar{Q})}_{\mu}\mathcal{A}_{\nu}^{T}S^{(\bar{Q})}_{\lambda}\right]$
$\displaystyle-i\beta_{S}\operatorname{tr}\left[\bar{S}^{(\bar{Q})}_{\mu}v_{\alpha}\left(\mathcal{V}^{\alpha}-\rho^{\alpha}\right)^{T}S^{(\bar{Q})\mu}\right]$
$\displaystyle+\lambda_{S}\operatorname{tr}\left[\bar{S}^{(\bar{Q})}_{\mu}(F^{\mu\nu})^{T}S^{(\bar{Q})}_{\nu}\right],$
(63) $\displaystyle\mathcal{L}^{\prime}_{\rm int}=$
$\displaystyle\,g_{4}\operatorname{tr}\left[\bar{S}^{(\bar{Q})\mu}\mathcal{A}^{T}_{\mu}B^{(\bar{Q})}_{{3}}\right]$
$\displaystyle+i\lambda_{I}\epsilon^{\mu\nu\lambda\kappa}v_{\mu}\operatorname{tr}\left[\bar{S}^{T}_{\nu}(F_{\lambda\kappa})^{T}B^{(\bar{Q})}_{{3}}\right]+{\rm
h.c.},$ (64)

with the transpose acting on the SU(3) flavor matrix. Notice that the spinor
for an antibaryon is $u$ instead of $v$ since here the fields of heavy baryons
and heavy antibaryons are treated independently.

Similar with the Lagrangian for heavy mesons, we can focus on the vector
exchange contributions, and only the following terms are relevant,

$\displaystyle\mathcal{L}_{BBV}$
$\displaystyle=\,i\beta_{B}\mathrm{tr}\left[\bar{B}^{(Q)}_{\bar{3}}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)B^{(Q)}_{\bar{3}}\right]$
$\displaystyle-i\beta_{B}\mathrm{tr}\left[\bar{B}^{(\bar{Q})}_{{3}}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)^{T}B^{(\bar{Q})}_{{3}}\right]$
$\displaystyle+i\beta_{S}\operatorname{tr}\left[\bar{S}_{\nu}^{(Q)}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)S^{(Q)\nu}\right]$
$\displaystyle-i\beta_{S}\operatorname{tr}\left[\bar{S}_{\nu}^{(\bar{Q})}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)^{T}S^{(\bar{Q})\nu}\right].$
(65)

##  III Potentials

###  III.1 Conventions

In this paper, we take the following charge conjugation conventions:

$\displaystyle\mathcal{C}\left|D\right\rangle$
$\displaystyle=\left|\bar{D}\right\rangle,$
$\displaystyle\mathcal{C}\left|D^{*}\right\rangle$
$\displaystyle=-\left|\bar{D}^{*}\right\rangle,$
$\displaystyle\mathcal{C}\left|D_{1}\right\rangle$
$\displaystyle=\left|\bar{D}_{1}\right\rangle,$
$\displaystyle\mathcal{C}\left|D_{2}\right\rangle$
$\displaystyle=-\left|\bar{D}_{2}\right\rangle,$
$\displaystyle\mathcal{C}\left|B_{\bar{3}}\right\rangle$
$\displaystyle=\left|\bar{B}_{\bar{3}}\right\rangle,$
$\displaystyle\mathcal{C}\left|B^{(*)}_{6}\right\rangle$
$\displaystyle=\left|\bar{B}^{(*)}_{6}\right\rangle,$ (66)

which are consistent with the Lagrangians in Section II. Within these
conventions, the flavor wave functions of the flavor-neutral systems that are
charge conjugation eigenstates, including $\left|DD^{*}\right\rangle_{c}$,
$\left|DD_{1}\right\rangle_{c}$, $\left|D^{*}D_{1}\right\rangle_{c}$,
$\left|DD_{2}\right\rangle_{c}$, $\left|D^{*}D_{2}\right\rangle_{c}$,
$\left|D_{1}D_{2}\right\rangle_{c}$ and
$\left|B_{6}B_{6}^{*}\right\rangle_{c}$, can be expressed as

$\displaystyle\left|A_{1}A_{2}\right\rangle_{c}=\frac{1}{\sqrt{2}}\left(\left|A_{1}\bar{A}_{2}\right\rangle\pm(-1)^{J-J_{1}-J_{2}}c\,c_{1}c_{2}\left|A_{2}\bar{A}_{1}\right\rangle\right),$
(67)

where $J_{i}$ is the spin of $\left|A_{i}\right\rangle$, $J$ is the total spin
of the system $\left|A_{1}A_{2}\right\rangle_{c}$, $c_{i}$ is defined by
$\mathcal{C}\left|A_{i}\right\rangle=c_{i}\left|\bar{A}_{i}\right\rangle$, and
the plus and minus between the two terms are for boson-boson and fermion-
fermion systems, respectively. These systems satisfy
$\mathcal{C}\left|A_{1}A_{2}\right\rangle_{c}=c\left|A_{1}A_{2}\right\rangle_{c}$
with $c=\pm 1$.

For the isospin conventions, we use the following ones:

$\displaystyle\left|u\right\rangle$
$\displaystyle=\left|\frac{1}{2},\frac{1}{2}\right\rangle,\quad~{}~{}\,\left|d\right\rangle=\left|\frac{1}{2},-\frac{1}{2}\right\rangle,$
$\displaystyle\left|\bar{d}\right\rangle$
$\displaystyle=\left|\frac{1}{2},\frac{1}{2}\right\rangle,\quad\left|\bar{u}\right\rangle=-\left|\frac{1}{2},-\frac{1}{2}\right\rangle.$
(68)

Consequently, we have

$\displaystyle\left|D^{+}\right\rangle$
$\displaystyle=\left|\frac{1}{2},\frac{1}{2}\right\rangle,$
$\displaystyle\left|D^{-}\right\rangle$
$\displaystyle=\left|\frac{1}{2},-\frac{1}{2}\right\rangle,$
$\displaystyle\left|D^{0}\right\rangle$
$\displaystyle=-\left|\frac{1}{2},-\frac{1}{2}\right\rangle,$
$\displaystyle\left|\bar{D}^{0}\right\rangle$
$\displaystyle=\left|\frac{1}{2},\frac{1}{2}\right\rangle,$
$\displaystyle\left|D_{s}^{+}\right\rangle$
$\displaystyle=\left|0,0\right\rangle,$
$\displaystyle\left|D_{s}^{-}\right\rangle$
$\displaystyle=\left|0,0\right\rangle,$
$\displaystyle\left|\Lambda_{c}^{+}\right\rangle$
$\displaystyle=\left|0,0\right\rangle,$
$\displaystyle\left|\Lambda_{c}^{-}\right\rangle$
$\displaystyle=-\left|0,0\right\rangle,$
$\displaystyle\left|\Xi_{c}^{(^{\prime}*)+}\right\rangle$
$\displaystyle=\left|\frac{1}{2},\frac{1}{2}\right\rangle,$
$\displaystyle\left|\Xi_{c}^{(^{\prime}*)-}\right\rangle$
$\displaystyle=\left|\frac{1}{2},-\frac{1}{2}\right\rangle,$
$\displaystyle\left|\Xi_{c}^{(^{\prime}*)0}\right\rangle$
$\displaystyle=\left|\frac{1}{2},-\frac{1}{2}\right\rangle,$
$\displaystyle\left|\bar{\Xi}_{c}^{(^{\prime}*)0}\right\rangle$
$\displaystyle=-\left|\frac{1}{2},\frac{1}{2}\right\rangle,$
$\displaystyle\left|\Sigma_{c}^{(*)++}\right\rangle$
$\displaystyle=\left|1,1\right\rangle,$
$\displaystyle\left|\Sigma_{c}^{(*)--}\right\rangle$
$\displaystyle=\left|1,-1\right\rangle,$
$\displaystyle\left|\Sigma_{c}^{(*)+}\right\rangle$
$\displaystyle=\left|1,0\right\rangle,$
$\displaystyle\left|\Sigma_{c}^{(*)-}\right\rangle$
$\displaystyle=-\left|1,0\right\rangle,$
$\displaystyle\left|\Sigma_{c}^{(*)0}\right\rangle$
$\displaystyle=\left|1,-1\right\rangle,$
$\displaystyle\left|\bar{\Sigma}_{c}^{(*)0}\right\rangle$
$\displaystyle=\left|1,1\right\rangle,$
$\displaystyle\left|\Omega_{c}^{(*)0}\right\rangle$
$\displaystyle=\left|0,0\right\rangle,$
$\displaystyle\left|\bar{\Omega}_{c}^{(*)0}\right\rangle$
$\displaystyle=\left|0,0\right\rangle.$ (69)

The isospin states of $D^{*}$, $D_{1}$ and $D_{2}^{*}$ are the same as those
of $D$. The flavor wave functions of the systems considered below with certain
isospins can be easily computed using Clebsch-Gordan coefficients with these
conventions.

The potential we calculate is $V=-\mathcal{M}$ with $\mathcal{M}$ the $2\to 2$
invariant scattering amplitude so that a negative $V$ means an attraction
interaction. This convention is the same as the widely used one in the on-
shell Bethe-Salpeter equation $T=V+VGT$ Oller and Oset (1997), and it is also
the same as the nonrelativistic potential in the Schrödinger equation up to a
mass factor.

###  III.2 Potentials from light vector exchange

With the Lagrangian and conventions presented above we are ready to calculate
the potentials of different systems. We will use the resonance saturation
model to get the approximate potentials as constant contact terms. Note that
the resonance saturation has been known to be able to well approximate the
low-energy constants (LECs) in the higher order Lagrangians of chiral
perturbation theory Ecker  _et al._ (1989); Donoghue  _et al._ (1989), and it
turns out therein that whenever vector mesons contribute they dominate the
numerical values of the LECs at the scale around the $\rho$-meson mass, which
is called the modern version of vector meson dominance.

The general form of $\mathcal{M}$ for a process
$A_{1}(p_{1})\bar{A}_{2}(p_{2})\to A_{1}(k_{1})\bar{A}_{2}(k_{2})$ by the
vector meson exchange reads

$\displaystyle
i\mathcal{M}=ig_{1}v_{\mu}\frac{-i(g^{\mu\nu}-{q^{\mu}q^{\nu}}/{m_{\rm
ex}^{2}})}{q^{2}-m_{\rm
ex}^{2}+i\epsilon}v_{\nu}ig_{2}\approx-i\frac{g_{1}g_{2}}{m_{\rm ex}^{2}},$
(70)

where $g_{1}$ and $g_{2}$ account for the vertex information for $A_{1}$ and
$\bar{A}_{2}$, respectively, $q=p_{1}-k_{1}$, and we have neglected terms
suppressed by $\mathcal{O}(\vec{q}\,^{2}/m_{\text{ex}}^{2})$. For different
particles $g_{1}$ and $g_{2}$ are collected in Appendix A. It is worth
mentioning that the spin information of the component particles is irrelevant
here since the exchanged vectors only carry the momentum information, see Eqs.
(28,65). Hence for a given system with different total spins, the potentials
at threshold are the same. With the vertex factors evaluated, the potentials
of different systems have a uniform expression,

$V\approx-F\tilde{\beta}_{1}\tilde{\beta}_{2}g_{V}^{2}\frac{2m_{1}m_{2}}{m_{\rm
ex}^{2}},$ (71)

where $m_{1},m_{2}$ and $m_{\rm ex}$ are the masses of the two heavy hadrons
and the exchanged particle, respectively. $\tilde{\beta}_{1}$ and
$\tilde{\beta}_{2}$ are the coupling constants for the two heavy hadrons with
the vector mesons, and, given explicitly, $\tilde{\beta}_{i}=\beta$ for the
$S$-wave charmed mesons, $\tilde{\beta}_{i}=-\beta$ for the $P$-wave charmed
mesons, $\tilde{\beta}_{i}=\beta_{B}$ for the anti-triplet baryons, and
$\tilde{\beta}_{i}=-\beta_{S}/2$ for the sextet baryons. $F$ is a group theory
factor accounting for the light-flavor SU(3) information, and in our
convention a positive $F$ means an attractive interaction. The values of $F$
are listed in Tables 8 and 9 in Appendix B for all combinations of heavy-
antiheavy hadron pairs.

For a system which can have different $C$-parities, like those in Eqs. (67),
the potential is expressed as

$V=V_{d}\pm(-1)^{J-J_{1}-J_{2}}c\,c_{1}c_{2}V_{c}$ (72)

with $V_{d}$ the potential from the _direct_ process, e.g., $D\bar{D}^{*}\to
D\bar{D}^{*}$ and $V_{c}$ from the _cross_ one, e.g., $D\bar{D}^{*}\to
D^{*}\bar{D}$. $V_{d}$’s for these systems are covered by Eq. (71), while for
the cross processes, it turns out that $V_{c}$’s for
$\left|DD^{*}\right\rangle_{c}$, $\left|DD_{2}\right\rangle_{c}$,
$\left|D_{1}D_{2}\right\rangle_{c}$ and
$\left|B_{6}B_{6}^{*}\right\rangle_{c}$ systems vanish at threshold. Explicit
calculation shows that for the other three systems,
$\left|DD_{1}\right\rangle_{c}$, $\left|D^{*}D_{1}\right\rangle_{c}$, and
$\left|D^{*}D_{2}\right\rangle_{c}$,

$\displaystyle V_{c}\approx
F_{c}F\zeta_{1}^{2}g_{V}^{2}\frac{m_{1}m_{2}}{m_{\rm ex}^{2}-\Delta m^{2}},$
(73)

where $F$ is the same as in Eq. (71), $\Delta m^{2}=(m_{1}-m_{2})^{2}$, and
the additional factor $F_{c}$ accounts for the spin information, which are
shown in Appendix C. However, $V_{c}$ is much smaller than $V_{d}$ and has
little influence on the pole positions compared with the cutoff dependence
(see below).

Table 1: Values of the coupling parameters used in the calculations. $g_{V}$ | $\beta$ | $\beta_{2}$ | $\zeta_{1}$ | $\beta_{B}$ | $\beta_{S}$
---|---|---|---|---|---
5.8 | 0.9 | $-0.9$ | 0.16 | 0.87 | $-1.74$
Bando  _et al._ (1988) | Isola  _et al._ (2003) | Dong  _et al._ (2020b) | Dong  _et al._ (2020b) | Liu and Oka (2012); Chen  _et al._ (2019a) | Liu and Oka (2012); Chen  _et al._ (2019a)

In Table 1, we collect the numerical values of the coupling parameters used in
our calculations.

###  III.3 Potentials from vector charmonia exchange

In principle, the $J/\psi$, as well as the excited $\psi$ vector charmonia,
can also be exchanged between charmed and anti-charmed hadrons. Being vector
mesons, their couplings to the charmed mesons have the same spin-momentum
behavior as that of the light vectors. According to Eq. (71), such
contributions should be suppressed due to the much larger masses of the $\psi$
states than those of the light vectors by a factor of
${m_{\rho}^{2}}/{m_{\psi}^{2}}\sim 0.1$, up to the difference of coupling
constants. Therefore, the exchange of light mesons, if not vanishing,
dominates the potentials at threshold. While for the systems where
contributions from light vectors vanish (or the $\rho$ and $\omega$ exchanges
cancel each other), the vector charmonia exchange, as the sub-leading term,
will play an important role in the near-threshold potentials.

To be more precise, let us take the $J/\psi$ exchange for example, for which
the Lagrangian reads

$\displaystyle\mathcal{L}_{{DD}J/\psi}=ig_{DDJ/\psi}\psi^{\mu}\left(\partial_{\mu}D^{\dagger}D-D^{\dagger}\partial_{\mu}D\right),$
(74)

with $g_{DDJ/\psi}\approx 7.64$ Lin and Ko (2000). The resulting potential in
the nonrelativistic limit is

$\displaystyle V\sim-g_{DDJ/\psi}^{2}\frac{4m_{1}m_{2}}{m_{\rm ex}^{2}},$ (75)

which is about 40% of the potential from the $\phi$ exchange between $D_{s}$
and $\bar{D}_{s}$. The contributions from the other vector charmonia will be
similar since their masses are of the same order. Notice that for all charmed
and anti-charmed hadron systems, the vector charmonia exchange yields
attractive interactions. Unfortunately, it is not easy to quantitatively
estimate their contributions because the masses of these charmonia are much
larger than the energy scale of interest, and there is no hierarchy among them
to help selecting the dominant ones. Nevertheless, it could be possible to
use, e.g., the $Z_{c}(3900)$, as a benchmark to estimate the overall
contribution of the charmonia exchange. Given the controversy regarding its
pole position Ablikim  _et al._ (2013a, 2014a); Albaladejo  _et al._ (2016a);
Pilloni  _et al._ (2017); Gong  _et al._ (2018), we refrain from doing so here
(for further discussion, see Section IV.2.1).

##  IV Molecular states from constant interactions

###  IV.1 Poles

Now that we have obtained the constant interactions between a pair of heavy-
antiheavy hadrons, we can give a rough picture of the spectrum of possible
molecular states. We search for poles of the scattering amplitude by solving
the single channel Bethe-Salpeter equation which factorizes into an algebraic
equation for a constant potential,

$\displaystyle T=\frac{V}{1-VG},$ (76)

where $G$ is the one loop two-body propagator. Here we adopt the dimensional
regularization (DR) to regularize the loop integral Veltman (2012),

$\displaystyle G(E)=$
$\displaystyle\frac{1}{16\pi^{2}}\bigg{\\{}a(\mu)+\log\frac{m_{1}^{2}}{\mu^{2}}+\frac{m_{2}^{2}-m_{1}^{2}+s}{2s}\log\frac{m_{2}^{2}}{m_{1}^{2}}$
$\displaystyle+\frac{k}{E}\Big{[}\log\left(2kE+s+\Delta\right)+\log\left(2kE+s-\Delta\right)$
$\displaystyle-\log\left(2kE-s+\Delta\right)-\log\left(2kE-s-\Delta\right)\Big{]}\bigg{\\}},$
(77)

where $s=E^{2}$, $m_{1}$ and $m_{2}$ are the particle masses, and

$\displaystyle k=\frac{1}{2E}\lambda^{1/2}(E^{2},m_{1}^{2},m_{2}^{2})$ (78)

is the corresponding three-momentum with
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2xz$ for the Källén triangle
function. Here $\mu$, chosen to be 1 GeV, denotes the DR scale, and $a(\mu)$
is a subtraction constant. The branch cut of $k$ from the threshold to
infinity along the positive real $E$ axis splits the whole complex energy
plane into two Riemann sheets (RSs) defined as Im$(k)>0$ on the first RS while
Im$(k)<0$ on the second RS. Another way to regularize the loop integral is
inserting a Gaussian form factor, namely,

$\displaystyle G(E)=$
$\displaystyle\,\int\frac{l^{2}dl}{4\pi^{2}}\frac{\omega_{1}+\omega_{2}}{\omega_{1}\omega_{2}}\frac{e^{-2l^{2}/\Lambda^{2}}}{E^{2}-(\omega_{1}+\omega_{2})^{2}+i\epsilon},$
(79)

with $\omega_{i}=\sqrt{m_{i}^{2}+l^{2}}$. The cutoff $\Lambda$ is usually in
the range of $0.5\sim 1.0$ GeV. The subtraction constant $a(\mu)$ in DR is
determined by matching the values of $G$ from these two methods at threshold.
We will use the DR loop with the so-determined subtraction constant for
numerical calculations.

For a single channel, if the interaction is attractive and strong enough to
form a bound state, the pole will be located below threshold on the first RS.
If it is not strong enough, the pole will move onto the second RS as a virtual
state, still below threshold. In Tables 2, 3, 4, 5, 6 and 7, we list all the
pole positions of the heavy-antiheavy hadron systems which have attractive
interactions, corresponding to the masses of hadronic molecules. For better
illustration, these states, together with some hadronic molecule candidates
observed in experiments, are also shown in Figs. 1, 2, 3, 4, 5 and 6. In
total, we obtain a spectrum of 229 hadronic molecules considering constant
contact interactions, saturated by the light vector mesons, with the coupled-
channel effects neglected.

Figure 1: The spectrum of hadronic molecules consisting of a pair of charmed-
anticharmed hadrons with $(I,S)=(0,0)$. $0^{--}$, $1^{-+}$ and $3^{-+}$ are
exotic quantum numbers. The colored rectangle, green for a bound state and
orange for a virtual state, covers the range of the pole position for a given
system with cutoff $\Lambda$ varies in the range of $[0.5,1.0]$ GeV.
Thresholds are marked by dotted horizontal lines. The rectangle closest to,
but below, the threshold corresponds to the hadronic molecule in that system.
In some cases where the pole positions of two systems overlap, small
rectangles are used with the left (right) one for the system with the higher
(lower) threshold. The blue line (band) represents the center value (error) of
the mass of the experimental candidate of the corresponding molecule. The
averaged central value and error of the $\psi(4230)$ mass are taken from RPP
Zyla  _et al._ (2020). Figure 2: The spectrum of hadronic molecules consisting
of a pair of charmed-anticharmed hadrons with $(I,S)=(0,0)$. $2^{+-}$ are
exotic quantum numbers. The parameters of the $X(3872)$ and $\tilde{X}(3872)$
are taken from RPP Zyla  _et al._ (2020) and Ref. Aghasyan  _et al._ (2018),
respectively. See the caption for Fig.1.

Figure 3: The spectrum of hadronic molecules consisting of a pair of charmed-
anticharmed hadrons with $(I,S)=(\frac{1}{2},0)$ and unit baryon number. The
left orange band and right green band for each pole represent that the pole
moves from a virtual state on the second RS to a bound state on the first RS
when the cutoff $\Lambda$ changes from 0.5 to 1.0 GeV. The parameters of these
three $P_{c}$ states are taken from Ref. Aaij  _et al._ (2019), whose $J^{P}$
have not been determined experimentally. See the caption for Fig. 1.

Figure 4: The spectrum of hadronic molecules consisting of a pair of charmed-
anticharmed hadrons with $(I,S)=(0,1)$ and unit baryon number. The parameters
of $P_{cs}(4459)$ are taken from Ref. Aaij  _et al._ (2020a), whose $J^{P}$
have not been determined experimentally. See the caption for Fig. 3. Figure 5:
The spectrum of hadronic molecules consisting of a pair of charmed-anticharmed
hadrons with $(I,S)=(1,0)$. See the caption for Fig. 1.

Figure 6: The spectrum of hadronic molecules consisting of a pair of charmed-anticharmed hadrons with $(I,S)=(\frac{1}{2},1)$. See the captions for Figs. 1 and 3. Table 2: Pole positions of heavy-antiheavy hadron systems with $(I,S)=(0,0)$. $E_{\rm th}$ in the second column is the threshold in MeV. The number 0.5 (1.0) in the third (fourth) column means that the cutoff $\Lambda=0.5$ ($1.0$) GeV for Eq. (79) is used to determine the subtraction constant $a(\mu)$ in Eq. (77). In the last two columns, the first number in the parenthesis refers to the RS where the pole is located while the second number means the distance between the pole position and the corresponding threshold, namely, $E_{\rm th}-E_{\rm pole}$, in MeV. System | $E_{\rm th}$ | $J^{PC}$ | Pole (0.5) | Pole (1.0)
---|---|---|---|---
$D\bar{D}$ | 3734 | $0^{++}$ | (1, 1.31) | (1, 35.8)
$D\bar{D}^{*}$ | 3876 | $1^{+\pm}$ | (1, 1.56) | (1, 36.2)
$D_{s}\bar{D}_{s}$ | 3937 | $0^{++}$ | (2, 35.5) | (2, 4.72)
$D^{*}\bar{D}^{*}$ | 4017 | $(0,2)^{++},1^{+-}$ | (1, 1.82) | (1, 36.6)
$D_{s}\bar{D}_{s}^{*}$ | 4081 | $1^{+\pm}$ | (2, 31.0) | (2, 3.15)
$D_{s}^{*}\bar{D}_{s}^{*}$ | 4224 | $(0,2)^{++},1^{+-}$ | (2, 26.7) | (2, 1.92)
$D\bar{D}_{2}$ | 4330 | $2^{-\pm}$ | (1, 2.2) | (1, 36.7)
$D_{s}\bar{D}_{s2}$ | 4537 | $2^{-\pm}$ | (2, 21.3) | (2, 0.713)
$D_{1}\bar{D}_{1}$ | 4844 | $(0,2)^{++},1^{+-}$ | (1, 3.01) | (1, 36.7)
$D_{1}\bar{D}_{2}$ | 4885 | $(1,2,3)^{+\pm}$ | (1, 3.06) | (1, 36.6)
$D_{2}\bar{D}_{2}$ | 4926 | $(0,2,4)^{++},(1,3)^{+-}$ | (1, 3.1) | (1, 36.6)
$D_{s1}\bar{D}_{s1}$ | 5070 | $(0,2)^{++},1^{+-}$ | (2, 11.7) | (1, 0.074)
$D_{s1}\bar{D}_{s2}$ | 5104 | $(1,2,3)^{+\pm}$ | (2, 11.3) | (1, 0.104)
$D_{s2}\bar{D}_{s2}$ | 5138 | $(0,2,4)^{++},(1,3)^{+-}$ | (2, 10.9) | (1, 0.139)
$\Lambda_{c}\bar{\Lambda}_{c}$ | 4573 | $0^{-+},1^{--}$ | (1, 1.98) | (1, 33.8)
$\Sigma_{c}\bar{\Sigma}_{c}$ | 4907 | $0^{-+},1^{--}$ | (1, 11.1) | (1, 60.8)
$\Xi_{c}\bar{\Xi}_{c}$ | 4939 | $0^{-+},1^{--}$ | (1, 4.72) | (1, 42.2)
$\Sigma_{c}^{*}\bar{\Sigma}_{c}$ | 4972 | $1^{-\pm},2^{-\pm}$ | (1, 11.0) | (1, 60.1)
$\Sigma_{c}^{*}\bar{\Sigma}_{c}^{*}$ | 5036 | $(0,2)^{-+},(1,3)^{--}$ | (1, 10.9) | (1, 59.5)
$\Xi_{c}\bar{\Xi}_{c}^{\prime}$ | 5048 | $0^{-\pm},1^{-\pm}$ | (1, 4.79) | (1, 41.9)
$\Xi_{c}\bar{\Xi}_{c}^{*}$ | 5115 | $1^{-\pm},2^{-\pm}$ | (1, 4.84) | (1, 41.6)
$\Xi_{c}^{\prime}\bar{\Xi}_{c}^{\prime}$ | 5158 | $0^{-+},1^{--}$ | (1, 4.87) | (1, 41.5)
$\Xi_{c}^{*}\bar{\Xi}_{c}^{\prime}$ | 5225 | $1^{-\pm},2^{-\pm}$ | (1, 4.91) | (1, 41.3)
$\Xi_{c}^{*}\bar{\Xi}_{c}^{*}$ | 5292 | $(0,2)^{-+},(1,3)^{--}$ | (1, 4.95) | (1, 41.0)
$\Omega_{c}\bar{\Omega}_{c}$ | 5390 | $0^{-+},1^{--}$ | (1, 4.17) | (1, 38.0)
$\Omega_{c}^{*}\bar{\Omega}_{c}$ | 5461 | $1^{-\pm},2^{-\pm}$ | (1, 4.22) | (1, 37.8)
$\Omega_{c}^{*}\bar{\Omega}_{c}^{*}$ | 5532 | $(0,2)^{-+},(1,3)^{--}$ | (1, 4.26) | (1, 37.6)
Table 3: Pole positions of heavy-antiheavy hadron systems with $(I,S)=(0,0)$. See the caption for Table 2. In these systems different total spins and $C$-parities yield slightly different pole positions. System | $E_{\rm th}$ | $J^{PC}$ | Pole (0.5) | Pole (1.0)
---|---|---|---|---
$D\bar{D}_{1}$ | 4289 | $1^{-+}$ | (1, 1.78) | (1, 34.9)
|  | $1^{--}$ | (1, 2.53) | (1, 38.4)
$D^{*}\bar{D}_{1}$ | 4431 | $0^{-+}$ | (1, 2.55) | (1, 37.4)
|  | $0^{--}$ | (1, 2.29) | (1, 36.3)
|  | $1^{-+}$ | (1, 2.36) | (1, 36.6)
|  | $1^{--}$ | (1, 2.49) | (1, 37.1)
|  | $2^{-+}$ | (1, 2.36) | (1, 36.6)
|  | $2^{--}$ | (1, 2.49) | (1, 37.1)
$D^{*}\bar{D}_{2}$ | 4472 | $1^{-+}$ | (1, 2.54) | (1, 37.1)
|  | $1^{--}$ | (1, 2.4) | (1, 36.5)
|  | $2^{-+}$ | (1, 2.68) | (1, 37.7)
|  | $2^{--}$ | (1, 2.26) | (1, 35.9)
|  | $3^{-+}$ | (1, 2.89) | (1, 38.6)
|  | $3^{--}$ | (1, 2.05) | (1, 34.9)
$D_{s}\bar{D}_{s1}$ | 4503 | $1^{-+}$ | (2, 24.2) | (2, 1.4)
|  | $1^{--}$ | (2, 19.6) | (2, 0.402)
$D_{s}^{*}\bar{D}_{s1}$ | 4647 | $0^{-+}$ | (2, 17.5) | (2, 0.179)
|  | $0^{--}$ | (2, 19.2) | (2, 0.402)
|  | $1^{-+}$ | (2, 18.7) | (2, 0.402)
|  | $1^{--}$ | (2, 17.9) | (2, 0.227)
|  | $2^{-+}$ | (2, 18.7) | (2, 0.342)
|  | $2^{--}$ | (2, 17.9) | (2, 0.402)
$D_{s}^{*}\bar{D}_{s2}$ | 4681 | $1^{-+}$ | (2, 17.4) | (2, 0.177)
|  | $1^{--}$ | (2, 18.3) | (2, 0.402)
|  | $2^{-+}$ | (2, 16.6) | (2, 0.402)
|  | $2^{--}$ | (2, 19.2) | (2, 0.418)
|  | $3^{-+}$ | (2, 15.4) | (2, 0.023)
|  | $3^{--}$ | (2, 20.6) | (2, 0.402)
Table 4: Pole positions of heavy-antiheavy hadron systems with $(I,S)=(1/2,0)$ and unit baryon number. See the caption for Table 2. System | $E_{\rm th}$ | $J^{P}$ | Pole (0.5) | Pole (1.0)
---|---|---|---|---
$\bar{D}\Sigma_{c}$ | 4321 | $\frac{1}{2}^{-}$ | (2, 2.04) | (1, 7.79)
$\bar{D}\Sigma_{c}^{*}$ | 4385 | $\frac{3}{2}^{-}$ | (2, 1.84) | (1, 8.1)
$\bar{D}^{*}\Sigma_{c}$ | 4462 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (2, 1.39) | (1, 8.95)
$\bar{D}^{*}\Sigma_{c}^{*}$ | 4527 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{-}$ | (2, 1.23) | (1, 9.26)
$\bar{D}_{1}\Sigma_{c}$ | 4876 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (2, 0.417) | (1, 11.5)
$\bar{D}_{2}\Sigma_{c}$ | 4917 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (2, 0.366) | (1, 11.7)
$\bar{D}_{1}\Sigma_{c}^{*}$ | 4940 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{+}$ | (2, 0.34) | (1, 11.8)
$\bar{D}_{2}\Sigma_{c}^{*}$ | 4981 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2})^{+}$ | (2, 0.294) | (1, 12.0)
Table 5: Pole positions of heavy-antiheavy hadron systems with $(I,S)=(0,1)$ and unit baryon number. See the caption for Table 2. System | $E_{\rm th}$ | $J^{P}$ | Pole (0.5) | Pole (1.0)
---|---|---|---|---
$\bar{D}\Xi_{c}$ | 4337 | $\frac{1}{2}^{-}$ | (2, 2.14) | (1, 7.53)
$\bar{D}\Xi_{c}^{\prime}$ | 4446 | $\frac{1}{2}^{-}$ | (2, 1.82) | (1, 8.05)
$\bar{D}^{*}\Xi_{c}$ | 4478 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (2, 1.47) | (1, 8.69)
$\bar{D}\Xi_{c}^{*}$ | 4513 | $\frac{3}{2}^{-}$ | (2, 1.65) | (1, 8.34)
$\bar{D}^{*}\Xi_{c}^{\prime}$ | 4587 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (2, 1.21) | (1, 9.21)
$\bar{D}^{*}\Xi_{c}^{*}$ | 4655 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{-}$ | (2, 1.08) | (1, 9.51)
$\bar{D}_{1}\Xi_{c}$ | 4891 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (2, 0.455) | (1, 11.3)
$\bar{D}_{2}\Xi_{c}$ | 4932 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (2, 0.4) | (1, 11.5)
$\bar{D}_{1}\Xi_{c}^{\prime}$ | 5001 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (2, 0.326) | (1, 11.8)
$\bar{D}_{2}\Xi_{c}^{\prime}$ | 5042 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (2, 0.28) | (1, 12.0)
$\bar{D}_{1}\Xi_{c}^{*}$ | 5068 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{+}$ | (2, 0.262) | (1, 12.1)
$\bar{D}_{2}\Xi_{c}^{*}$ | 5109 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2})^{+}$ | (2, 0.222) | (1, 12.3)
Table 6: Pole positions of heavy-antiheavy hadron systems with $(I,S)=(1/2,1)$. See the caption for Table 2. System | $E_{\rm th}$ | $J^{P}$ | Pole (0.5) | Pole (1.0)
---|---|---|---|---
$\Lambda_{c}\bar{\Xi}_{c}$ | 4756 | $(0,1)^{-}$ | (2, 1.29) | (1, 8.42)
$\Lambda_{c}\bar{\Xi}_{c}^{\prime}$ | 4865 | $(0,1)^{-}$ | (2, 1.05) | (1, 8.93)
$\Xi_{c}\bar{\Sigma}_{c}$ | 4923 | $(0,1)^{-}$ | (1, 5.98) | (1, 46.4)
$\Lambda_{c}\bar{\Xi}_{c}^{*}$ | 4932 | $(1,2)^{-}$ | (2, 0.92) | (1, 9.23)
$\Xi_{c}\bar{\Sigma}_{c}^{*}$ | 4988 | $(1,2)^{-}$ | (1, 6.01) | (1, 46.1)
$\Sigma_{c}\bar{\Xi}_{c}^{\prime}$ | 5032 | $(0,1)^{-}$ | (1, 6.03) | (1, 45.9)
$\Sigma_{c}^{*}\bar{\Xi}_{c}^{\prime}$ | 5097 | $(1,2)^{-}$ | (1, 6.06) | (1, 45.6)
$\Sigma_{c}\bar{\Xi}_{c}^{*}$ | 5100 | $(1,2)^{-}$ | (1, 6.05) | (1, 45.6)
$\Sigma_{c}^{*}\bar{\Xi}_{c}^{*}$ | 5164 | $(0,1,2,3)^{-}$ | (1, 6.08) | (1, 45.2)
$\Xi_{c}\bar{\Omega}_{c}$ | 5165 | $(0,1)^{-}$ | (2, 8e-5) | (1, 15.9)
$\Xi_{c}\bar{\Omega}_{c}^{*}$ | 5235 | $(1,2)^{-}$ | (1, 0.002) | (1, 16.2)
$\Xi_{c}^{\prime}\bar{\Omega}_{c}$ | 5274 | $(0,1)^{-}$ | (1, 0.006) | (1, 16.4)
$\Xi_{c}^{*}\bar{\Omega}_{c}$ | 5341 | $(1,2)^{-}$ | (1, 0.016) | (1, 16.6)
$\Xi_{c}^{\prime}\bar{\Omega}_{c}^{*}$ | 5345 | $(1,2)^{-}$ | (1, 0.016) | (1, 16.6)
$\Xi_{c}^{*}\bar{\Omega}_{c}^{*}$ | 5412 | $(0,1,2,3)^{-}$ | (1, 0.030) | (1, 16.8)
Table 7: Pole positions of heavy-antiheavy hadron systems with $(I,S)=(1,0)$. See the caption for Table 2. System | $E_{\rm th}$ | $J^{PC}$ | Pole (0.5) | Pole (1.0)
---|---|---|---|---
$\Lambda_{c}\bar{\Sigma}_{c}$ | 4740 | $(0,1)^{-\pm}$ | (1, 2.19) | (1, 33.9)
$\Lambda_{c}\bar{\Sigma}_{c}^{*}$ | 4805 | $(1,2)^{-\pm}$ | (1, 2.27) | (1, 33.9)
$\Sigma_{c}\bar{\Sigma}_{c}$ | 4907 | $0^{-+},1^{--}$ | (1, 8.28) | (1, 53.3)
$\Xi_{c}\bar{\Xi}_{c}$ | 4939 | $0^{-+},1^{--}$ | (2, 18.2) | (2, 0.39)
$\Sigma_{c}^{*}\bar{\Sigma}_{c}$ | 4972 | $(1,2)^{-\pm}$ | (1, 8.27) | (1, 52.8)
$\Sigma_{c}^{*}\bar{\Sigma}_{c}^{*}$ | 5036 | $(0,2)^{-+},(1,3)^{--}$ | (1, 8.25) | (1, 52.3)
$\Xi_{c}\bar{\Xi}_{c}^{\prime}$ | 5048 | $(0,1)^{-\pm}$ | (2, 16.5) | (2, 0.19)
$\Xi_{c}\bar{\Xi}_{c}^{*}$ | 5115 | $(1,2)^{-\pm}$ | (2, 15.6) | (2, 0.11)
$\Xi_{c}^{\prime}\bar{\Xi}_{c}^{\prime}$ | 5158 | $0^{-+},1^{--}$ | (2, 14.9) | (2, 0.061)
$\Xi_{c}^{*}\bar{\Xi}_{c}^{\prime}$ | 5225 | $(1,2)^{-\pm}$ | (2, 14.0) | (2, 0.020)
$\Xi_{c}^{*}\bar{\Xi}_{c}^{*}$ | 5292 | $(0,2)^{-+},(1,3)^{--}$ | (2, 13.2) | (2, 0.002)

###  IV.2 Discussions of selected systems

It is worthwhile to notice that the overwhelming majority of the predicted
spectrum is located in the energy region that has not been experimentally
explored in detail. Searching for these states at BESIII, Belle-II, LHCb and
other planned experiments will be important to establish a clear pattern of
the hidden-charm states and to understand how QCD organizes the hadron
spectrum.

In the following, we discuss a few interesting systems that have experimental
candidates.

####  IV.2.1 $D^{(*)}\bar{D}^{(*)}$: $X(3872)$, $Z_{c}(3900)$ and their
partners

Within the mechanism considered here, the interactions of $D\bar{D}$,
$D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$ are the same. For the meson pairs to be
isospin scalars, the attractions are strong enough to form bound states with
similar binding energies, see Table 2 and Fig. 2, while for the isovector
pairs, the contributions from the $\rho$ and $\omega$ exchanges cancel each
other, see Table 8 in Appendix B.

The $X(3872)$ observed by the Belle Collaboration Choi  _et al._ (2003) is
widely suggested as an isoscalar $D\bar{D}^{*}$ molecule with $J^{PC}=1^{++}$
Törnqvist (2003); Wong (2004); Swanson (2004); Törnqvist (2004) (for reviews,
see, e.g., Refs. Chen  _et al._ (2016a); Guo  _et al._ (2018); Kalashnikova
and Nefediev (2019); Brambilla  _et al._ (2020)). Actually such a hadronic
molecule was predicted 10 years before the discovery by Törnvist considering
the one-pion exchange Törnqvist (1994). Our results show that the light vector
exchange leads to a near-threshold isoscalar bound state that can be
identified with the $X(3872)$ as well, together with a negative $C$-parity
partner of $X(3872)$ with the same binding energy (see also Refs. Gamermann
_et al._ (2007); Gamermann and Oset (2007)). There is experimental evidence of
such a negative $C$-parity state, named as $\tilde{X}(3872)$,111It should be
called $h_{c}(3872)$ according to the RPP nomenclature. reported by the
COMPASS Collaboration Aghasyan  _et al._ (2018). A recent study of the
$D_{(s)}^{(*)}\bar{D}_{(s)}^{(*)}$ molecular states using the method of QCD
sum rules also finds both $1^{++}$ and $1^{+-}$ $D\bar{D}^{*}$ states Wang
(2020a).

The potential predicting the $X(3872)$ as an isoscalar $D\bar{D}^{*}$ bound
state, also predicts the existence of isoscalar $D\bar{D}$ and
$D^{*}\bar{D}^{*}$ bound states. By imposing only HQSS, there are two
independent contact terms in the $D^{(*)}\bar{D}^{(*)}$ interactions for each
isospin AlFiky  _et al._ (2006); Nieves and Valderrama (2012), which can be
defined as Guo  _et al._ (2018)

$\displaystyle C_{H\bar{H},0}$
$\displaystyle=\left\langle\frac{1}{2},\frac{1}{2},0\right|\mathcal{H}_{I}\left|\frac{1}{2},\frac{1}{2},0\right\rangle,$
$\displaystyle C_{H\bar{H},1}$
$\displaystyle=\left\langle\frac{1}{2},\frac{1}{2},1\right|\mathcal{H}_{I}\left|\frac{1}{2},\frac{1}{2},1\right\rangle,$
(80)

where $\mathcal{H}_{I}$ is the interaction Hamiltonian, and $\left|s_{\ell
1},s_{\ell 2},s_{\ell}\right\rangle$ denotes the charmed meson pair with
$s_{\ell}$ being the total angular momentum of the light degrees of freedom in
the two-meson system and $s_{\ell 1,\ell 2}$ for the individual mesons. Such
an analysis leads to the prediction of a $2^{++}$ $D^{*}\bar{D}^{*}$ tensor
state as the HQSS partner of the $X(3872)$ considering the physical charmed
meson masses Nieves and Valderrama (2012); Hidalgo-Duque  _et al._ (2013a);
Guo  _et al._ (2013) and three partners with $0^{++},1^{+-}$ and $2^{++}$ in
the strict heavy quark limit Hidalgo-Duque  _et al._ (2013b); Baru  _et al._
(2016) that depend on the same contact term as the $X(3872)$. The resonance
saturation by the light vector mesons in fact leads to a relation
$C_{H\bar{H},0}=C_{H\bar{H},1}$, and consequently 6 $S$-wave
$D^{(*)}\bar{D}^{(*)}$ bound states.

The existence of an isoscalar $D\bar{D}$ bound state has been predicted by
various phenomenology models Zhang  _et al._ (2006); Gamermann  _et al._
(2007); Liu  _et al._ (2009); Wong (2004); Nieves and Valderrama (2012);
Hidalgo-Duque  _et al._ (2013a), and more recently by lattice QCD calculations
Prelovsek  _et al._ (2020). Despite attempts Gamermann and Oset (2008); Dai
_et al._ (2020); Wang  _et al._ (2020a) to dig out hints for such a state from
the available experimental data Uehara  _et al._ (2006); Pakhlov  _et al._
(2008); Aubert  _et al._ (2010), no clear evidence has yet been found.
However, this could be because its mass is below the $D\bar{D}$ threshold so
that no easily detectable decay modes are available.

As for the isoscalar $2^{++}$ $D^{*}\bar{D}^{*}$ bound state, it can decay
into $D\bar{D}$ in $D-$wave, and the width was predicted to be in the range
from a few to dozens of MeV Albaladejo  _et al._ (2015); Baru  _et al._
(2016). No evidence has been found so far. One possible reason is that the
coupling to ordinary charmonia could either move the $2^{++}$ pole deep into
the complex energy plane and thus invisible Cincioglu  _et al._ (2016)222For a
discussion of the intricate interplay between a meson-meson channel with
multiple quark model states, see Ref. Hammer  _et al._ (2016). or make the
$D^{*}\bar{D}^{*}$ interaction in the $2^{++}$ sector unbound Ortega  _et al._
(2018); Ortega and Entem (2020).333Mixing of two energy levels will push them
further apart. Thus, mixing of the $D^{*}\bar{D}^{*}$ state with a lower-mass
$\chi_{c2}(2P)$ can effectively provide a repulsive contribution to the
$D^{*}\bar{D}^{*}$ interaction. For more discussions regarding the mixing of
charmonia with meson-meson channels, we refer to Refs. Kalashnikova (2005);
Zhou and Xiao (2017); Cincioglu  _et al._ (2020).

The $Z_{c}(3900)$ Ablikim  _et al._ (2013a); Liu  _et al._ (2013); Ablikim
_et al._ (2014a) was also suggested to be an isovector $D\bar{D}^{*}$ molecule
with quantum numbers $J^{PC}=1^{+-}$ Wang  _et al._ (2013); Guo  _et al._
(2013) even though the light vector exchange vanishes in this case. Recall
that the vector charmonia exchange will also yield an attractive interaction,
as discussed in Section III.3, which can possibly lead to a virtual state
below threshold. In fact, it has been suggested that the $J/\psi$-exchange is
essential in the formation of the $Z_{c}(3900)$ in Ref. Aceti  _et al._
(2014). It was shown in Refs. Albaladejo  _et al._ (2016a); He and Chen
(2018); Ortega  _et al._ (2019) that a virtual state assignment for the
$Z_{c}(3900)$ is consistent with the experimental data.444Notice that the
coupling of the $D\bar{D}^{*}$ to a lower channel, which is $J/\psi\pi$ in
this case, induces a finite width for the virtual state pole so that it
behaves like a resonance, i.e. a pole in the complex plane off the real axis.
The same is true for all other poles generated here. Furthermore, it was shown
in Ref. Albaladejo  _et al._ (2016b) that the finite volume energy levels are
also consistent with the lattice QCD results which did not report an
additional state Prelovsek  _et al._ (2015). Similarly, the
$Z_{c}(4020)^{\pm}$ Ablikim  _et al._ (2013b, 2014b) with isospin-1 near the
$D^{*}\bar{D}^{*}$ threshold can be a virtual state as well. It was recently
argued that a near-threshold virtual state needs to be understood as a
hadronic molecule Matuschek  _et al._ (2020). Analysis of the Belle data on
the $Z_{b}$ states Bondar  _et al._ (2012); Garmash  _et al._ (2016) using
constant contact terms to construct the unitary $T$-matrix also supports the
$Z_{b}$ states as hadronic molecules Cleven  _et al._ (2011); Hanhart  _et
al._ (2015); Guo  _et al._ (2016); Wang  _et al._ (2018a); Baru  _et al._
(2020). The molecular explanation of the $Z_{c}$ and $Z_{b}$ states is further
supported by their decay patterns studied using a quark exchange model Wang
_et al._ (2019); Xiao  _et al._ (2020). Without a quantitative calculation, as
commented in Section III.3, we postulate that there can be 6 isovector
hadronic molecules (with the same $J^{PC}$ as the isoscalar ones) as virtual
states of $D^{(*)}\bar{D}^{(*)}$, which will show up as prominent threshold
cusps (see Ref. Dong  _et al._ (2020a) for a general discussion of the line
shape behavior in the near-threshold region).

####  IV.2.2 $D_{s}^{(*)}\bar{D}_{s}^{(*)}$ virtual states

Here we find that the potential from the $\phi$ exchange is probably not
enough to form bound states of $D_{s}^{(*)}\bar{D}_{s}^{(*)}$. Instead,
virtual states are obtained, see Table 2 and Fig. 2.

On the contrary, based on two prerequisites that

  * 1)

$X(3872)$ is a marginally bound state of $D^{0}\bar{D}^{*0}$ with binding
energy $0\sim 1$ MeV, and

  * 2)

$D_{s}\bar{D}_{s}$ can form a bound states with a binding energy of $2.4\sim
12.9$ MeV from the lattice result Prelovsek  _et al._ (2020) and the
$\chi_{c0}(3930)$ mass determined by LHCb Aaij  _et al._ (2020b, c),

Ref. Meng  _et al._ (2020a) obtained $D_{s}^{(*)}\bar{D}_{s}^{(*)}$ bound
systems with binding energies up to 80 MeV.

Figure 7: Thresholds of charm-strange meson pairs in the energy range relevant
for the $B^{+}\to J/\psi\phi K^{+}$. Here, $D_{s0}^{*}$ denotes
$D_{s0}^{*}(2317)$, $D_{s1}$ and $D_{s1}^{\prime}$ denote $D_{s1}(2536)$ and
$D_{s1}(2460)$, respectively, and $D_{s2}$ denotes $D_{s2}(2573)$. The data
are taken from Ref. Aaij  _et al._ (2017a).

The $X(4140)$ first observed by the CDF Collaboration Aaltonen  _et al._
(2009) was considered as a molecule of $D^{*}_{s}\bar{D}_{s}^{*}$ with
$J^{PC}=0^{++}$ or $2^{++}$ in Refs. Liu and Zhu (2009); Branz  _et al._
(2009); Albuquerque  _et al._ (2009); Ding (2009); Zhang and Huang (2010);
Chen  _et al._ (2015); Karliner and Rosner (2016), which is, however,
disfavored by the results of LHCb Aaij  _et al._ (2017a, b) where the $J^{PC}$
of $X(4140)$ were suggested to be $1^{++}$ (and thus the $X(4140)$ was named
as $\chi_{c1}(4140)$ in the latest version of RPP). Actually in our
calculation, it is not likely for the $D^{*}_{s}\bar{D}_{s}^{*}$ to form such
a deeply bound state, noticing that the $X(4140)$ is about 80 MeV below the
threshold of $D^{*}_{s}\bar{D}_{s}^{*}$. Instead, it is interesting to notice
that just at the $D^{*}_{s}\bar{D}_{s}^{*}$ threshold there is evidence for a
peak in the invariant mass distribution of $J/\psi\phi$, see Fig. 7.555This
structure around the $D^{*}_{s}\bar{D}^{*}_{s}$ threshold drew the attention
of Ref. Wang  _et al._ (2018b) where two resonances, a narrow $X(4140)$ and a
broad $X(4160)$, were introduced to fit the $J/\psi\phi$ invariant mass
distribution from the threshold to about 4250 MeV. There the broad $X(4160)$
was considered as a $D^{*}_{s}\bar{D}^{*}_{s}$ molecule. Following the
analysis in Ref. Dong  _et al._ (2020a), if the interaction of the
$D^{*}_{s}\bar{D}_{s}^{*}$ is attractive but not strong enough to form a bound
state, a peak will appear just at the $D^{*}_{s}\bar{D}_{s}^{*}$ threshold in
the invariant mass distribution of $J/\psi\phi$, and the peak is narrow if
there is a nearby virtual state pole. A detailed study of this threshold
structure can tell us whether the attraction between a pair of charm-strange
mesons is strong enough to form a bound state or not.

The difference between the $J/\psi\phi$ and $D_{s}\bar{D}_{s}^{*}$ thresholds
is merely 36 MeV. Thus, the shallow $D_{s}\bar{D}_{s}^{*}$ virtual state with
$1^{++}$ could be responsible for the quick rise of the $J/\psi\phi$ invariant
mass distribution just above threshold observed in the LHCb data Aaij  _et
al._ (2017a).

####  IV.2.3 $D^{(*)}\bar{D}_{s}^{(*)}$: $Z_{cs}$ as virtual states

No light vector can be exchanged here and the attractive interaction from
vector charmonia exchange is crucial, similar to the isovector
$D\bar{D}^{(*)}$ systems. A virtual state pole could exist below threshold. In
particular, if the $Z_{c}(3900)$ exists as a virtual state, the same
interaction would induce $D^{(*)}\bar{D}_{s}^{(*)}$ virtual states.

Recently, a near-threshold enhancement in the invariant mass distribution of
$D_{s}^{-}D^{*0}+D_{s}^{*-}D^{0}$ was reported by the BESIII Collaboration
Ablikim  _et al._ (2020a) and an exotic state $Z_{cs}(3985)^{-}$ was claimed.
This state has been widely investigated Wang  _et al._ (2020b); Wan and Qiao
(2020); Wang  _et al._ (2020c); Meng  _et al._ (2020b); Yang  _et al._
(2020b); Chen and Huang (2020); Du  _et al._ (2020b); Cao  _et al._ (2020);
Sun and Xiao (2020); Wang  _et al._ (2020d, e); Wang (2020b); Azizi and Er
(2020); Jin  _et al._ (2020); Simonov (2020); Süngü  _et al._ (2020); Ikeno
_et al._ (2020); Xu  _et al._ (2020), some of which regard it as a molecule of
$D_{s}^{-}D^{*0}+D_{s}^{*-}D^{0}$ while some others object such an
explanation. In Ref. Yang  _et al._ (2020b), it was found that a virtual or
resonant pole together with a triangle singularity can well reproduce the line
shape of the BESIII data, consistent with the analysis here.

####  IV.2.4 $D^{(*)}\bar{D}_{1,2}$: $Y(4260)$ and related states

It is possible for the isoscalar $D\bar{D}_{1}$ pair to form a bound state
with a binding energy from a few MeV to dozens of MeV. Note that this system
can have $J^{PC}=1^{-\pm}$ and the $1^{--}$ state is slightly more deeply
bound than the $1^{-+}$ one which has exotic quantum numbers.

The $Y(4260)$ was discovered by the BABAR Collaboration Aubert  _et al._
(2005) with a mass of $(4259\pm 8^{+2}_{-6})$ MeV and a width of 50 $\sim$ 90
MeV and later confirmed by other experiments He  _et al._ (2006); Yuan  _et
al._ (2007). Now it is called $\psi(4230)$ due to a lower mass from the more
precise BESIII data and a combined analysis in four channels,
$e^{+}e^{-}\rightarrow\omega\chi_{c0}$ Ablikim  _et al._ (2016),
$\pi^{+}\pi^{-}h_{c}$ Ablikim  _et al._ (2017a), $\pi^{+}\pi^{-}J/\psi$
Ablikim  _et al._ (2017b) and $D^{0}D^{*-}\pi^{+}+c.c.$ Ablikim  _et al._
(2019), yielding a mass of $(4219.6\pm 3.3\pm 5.1)$ MeV and a width of
$(56.0\pm 3.6\pm 6.9)$ MeV Gao  _et al._ (2017). This state is a good
candidate of exotic states (see reviews, e.g., Refs. Chen  _et al._ (2016a);
Guo  _et al._ (2018); Brambilla  _et al._ (2020)). It was argued that the
isoscalar $D\bar{D}_{1}$ plays important roles in the structure of $Y(4260)$
in, e.g., Refs. Wang  _et al._ (2013); Qin  _et al._ (2016); Chen  _et al._
(2019b). The binding energy of the isoscalar $D\bar{D}_{1}$ system with
$J^{PC}=1^{-\pm}$ via the vector meson exchange was calculated by solving the
Schrödinger equation in spatial space in Ref. Dong  _et al._ (2020b), and the
results are consistent with this work except that the sign of $V_{c}$, which
has a minor impact, is not correct there. Note that the mass of the isoscalar
$D\bar{D}_{1}$ bound state obtained here is larger than the nominally mass of
the $\psi(4230)$, see Fig. 1, but the mixing of the $D\bar{D}_{1}$ molecule
with a $D$-wave vector charmonium Lu  _et al._ (2017) may solve this
discrepancy.

From the results in Table 2 and Fig. 1, the isoscalar $D\bar{D}_{1}$ bound
state has quite some partners, either of HQSS or of SU(3) flavor. In
particular, several of them have vector quantum numbers, including a
$D^{*}\bar{D}_{1}$ bound state with a mass about 4.39 $\sim$ 4.43 GeV, a
$D^{*}\bar{D}_{2}$ bound state with a mass about 4.43 $\sim$ 4.47 GeV, and
three virtual states of $D_{s}\bar{D}_{s1}$, $D^{*}_{s}\bar{D}_{s1}$ and
$D^{*}_{s}\bar{D}_{s2}$.

The current status of the vector charmonium spectrum around 4.4 GeV is not
clear, and the peak structures in exclusive and inclusive $R$-value
measurements are different (for a compilation of the relevant data, see Ref.
Yuan ). Thus, it is unclear which structure(s) can be identified as the
candidate(s) of the $D^{*}\bar{D}_{1(2)}$ bound states. Nevertheless, the
$Y(4360)$, aka $\psi(4360)$, and $\psi(4415)$ have been suggested to
correspond to the $D_{1}\bar{D}^{*}$ and $D_{2}\bar{D}^{*}$ states,
respectively Wang  _et al._ (2014); Ma  _et al._ (2015); Cleven  _et al._
(2015); Hanhart and Klempt (2020). A determination of the poles around 4.4 GeV
would require a thorough analysis of the full data sets including these open-
charm channels, and the first steps have been done in Refs. Cleven  _et al._
(2014); Olschewsky (2018).

As for the virtual states with hidden-strangeness, they are expected to show
up as narrow threshold cusps in final states like $J/\psi f_{0}(980)$ and
$\psi(2S)f_{0}(980)$. They could play an important role in generating the
$Y(4660)$, aka $\psi(4660)$, peak observed in the
$\psi(2S)f_{0}(980)\to\psi(2S)\pi^{+}\pi^{-}$ invariant mass distribution Lees
_et al._ (2014); Wang  _et al._ (2015).666The $Y(4660)$ was suggested to be a
$\psi(2S)f_{0}(980)$ bound state in Ref. Guo  _et al._ (2008) to explain why
it was seen only the $\psi(2S)\pi^{+}\pi^{-}$ final state with the pion pair
coming from the $f_{0}(980)$. Although it was proposed Cotugno  _et al._
(2010); Guo  _et al._ (2010) that the $Y(4630)$ structure observed in the
$\Lambda_{c}\bar{\Lambda}_{c}$ spectrum Pakhlova  _et al._ (2008) could be the
same state as the $Y(4660)$ one, the much more precise BESIII data Ablikim
_et al._ (2018), however, show a different behavior up to 4.6 GeV in the
$\Lambda_{c}\bar{\Lambda}_{c}$ invariant mass distribution (see below).
Further complications come from the $1^{--}$ structures around 4.63 GeV
reported in the $D_{s}\bar{D}_{s1}+c.c.$ Jia  _et al._ (2019) and
$D_{s}\bar{D}_{s2}+c.c.$ Jia  _et al._ (2020) distributions, the former of
which has been proposed to be due to a molecular state from the
$D_{s}^{*}\bar{D}_{s1}$-$D_{s}\bar{D}_{s1}$ interaction He  _et al._ (2020).
Suffice it to say that the situation of the $Y(4630)$ is not unambiguous. With
more precise data that will be collected at BESIII and Belle-II, we suggest to
search for line shape irregularities (either peaks or dips) at the
$D_{s}^{*}\bar{D}_{s1}$ and $D_{s}^{*}\bar{D}_{s2}$ thresholds in open-charm-
strangeness final states such as $D_{s}^{(*)}\bar{D}_{s}^{(*)}$ and
$D_{s}\bar{D}_{s1(s2)}$.

There are also hints in data for positive $C$-parity $D_{s}\bar{D}_{s1}$ and
$D_{s}^{*}\bar{D}_{s1}$ virtual states (see Table 3 and Fig. 1), whose
thresholds are at 4503 MeV and 4647 MeV, respectively. As can be seen from
Fig. 7, there is a peak around 4.51 GeV and a dip around 4.65 GeV in the
$J/\psi\phi$ energy distribution measured by the LHCb Collaboration Aaij  _et
al._ (2017a), and the energy difference between the dip and peak approximately
equals to the mass splitting between the $D_{s}^{*}$ and $D_{s}$. We also
notice that the highest peak in the same data appears at the
$D_{s0}^{*}(2317)\bar{D}_{s}$ threshold.777The coincidence of the peak
position with the threshold and the highly asymmetric line shape suggests a
$D_{s0}^{*}(2317)\bar{D}_{s}$ virtual state. Such systems will be studied in a
future work. All these channels, together with the $D_{s}\bar{D}_{s}^{*}$ and
$D_{s}^{*}\bar{D}_{s}^{*}$ discussed in Section IV.2.2, need to be considered
in a reliable analysis of the $B^{+}\to J/\psi\phi K^{+}$ data, which is
however beyond the scope of this paper.

Notice that because the $D_{1(2)}$ and $D_{s1(s2)}$ have finite widths, the
molecular states containing one of them can decay easily through the decays of
$D_{1(2)}$ or $D_{s1(s2)}$. The structures at the thresholds of
$D_{s}^{(*)}\bar{D}_{s1(s2)}$, for which virtual states are predicted, will
get smeared by the widths of $D_{s1(s2)}$. Thus, the $D_{s}^{(*)}\bar{D}_{s2}$
threshold structures should be broader and smoother than the
$D_{s}^{(*)}\bar{D}_{s1}$ ones since the width of the $D_{s2}$, $(16.9\pm
0.7)$ MeV Zyla  _et al._ (2020), is much larger than that of the $D_{s1}$,
$(0.92\pm 0.05)$ MeV Zyla  _et al._ (2020).

####  IV.2.5 $\Lambda_{c}\bar{\Lambda}_{c}$: analysis of the BESIII data and
more baryon-antibaryon bound states

From Table 2 and Fig. 1, in the spectrum of the isoscalar $1^{--}$ states, in
addition to those made of a pair of charmed mesons, we predict more than 10
baryon-antibaryon molecules. The lowest one is the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state, and the others are above 4.85 GeV.
While those above 4.85 GeV are beyond the current reach of BESIII (there is a
BESIII data-taking plan in the energy region above 4.6 GeV Ablikim  _et al._
(2020b)), there is strong evidence for the existence of a
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state in the BESIII data Ablikim  _et
al._ (2018).

The $\Lambda_{c}\bar{\Lambda}_{c}$ system can form a bound state with a
binding energy in the range from a few MeV to dozens of MeV, depending on the
cutoff. Therefore, we predict that there is a pole below the
$\Lambda_{c}\bar{\Lambda}_{c}$ threshold and the pole position can be
extracted from the line shape of the $\Lambda_{c}\bar{\Lambda}_{c}$ invariant
mass distribution near threshold.

The cross section of $e^{+}e^{-}\to\Lambda_{c}\bar{\Lambda}_{c}$ was first
measured using the initial state radiation by Belle Pakhlova  _et al._ (2008)
and a vector charmonium-like structure $Y(4630)$ was observed. The BESIII
Collaboration measured such cross sections at four energy points just above
threshold much more precisely Ablikim  _et al._ (2018). The energy dependence
of the cross sections at these four points has a weird behavior: it is almost
flat. This can be understood as the consequence of the Sommerfeld factor,
which makes the distribution nonvanishing even exactly at threshold, and the
existence of a near-threshold pole, which counteracts the increasing trend of
the phase space multiplied by the Sommerfeld factor to result in an almost
flat distribution. Here we fit BESIII data to estimate where the pole is
located.

The Sommerfeld factor Sommerfeld (1931) accounting for the multi-photon
exchange between the $\Lambda_{c}^{+}$ and $\bar{\Lambda}_{c}^{-}$ reads,

$\displaystyle S_{0}(E)=\frac{2\pi x}{1-e^{-2\pi x}},$ (81)

where $x=\alpha\mu/k$ with $\alpha\approx 1/137$, $\mu=m_{\Lambda_{c}}/2$, and
$k$ is defined in Eq. (78). The cross section of
$e^{+}e^{-}\to\Lambda_{c}\bar{\Lambda}_{c}$ is now parameterized as

$\displaystyle\sigma(E)=N\cdot
S_{0}(E)\cdot|f(E)|^{2}\cdot\frac{\rho(E)}{E^{2}},$ (82)

with $N$ a normalization constant and $\rho(E)=k/(8\pi E)$ the phase space.
Here $f(E)$ denotes the nonrelativistic scattering amplitude, and the $S$-wave
one,

$\displaystyle
f_{0}(E)=\left(\frac{1}{a_{0}}-i\sqrt{2\mu(E-2m_{\Lambda_{c}})}\right)^{-1},$
(83)

is sufficient in the immediate vicinity of the threshold. Note that we take
the scattering length $a_{0}$ complex to take into account the couplings
between the $\Lambda_{c}\bar{\Lambda}_{c}$ and lower channels Dong  _et al._
(2020a). Finally, we have 3 parameters, Re$(1/a_{0})$, Im$(1/a_{0})$ and $N$,
to fit four experimental data.

Figure 8: Top: pole positions of Eq. (83) on the first RS with different
scattering length ($a_{0}$) values and the color represents the $\chi^{2}$ (in
a logarithmic form for better illustration) of the fit to BESIII data Ablikim
_et al._ (2018). The pole on the second RS is at the same position if we
change the sign of Re$(1/a_{0})$, which does not change the fit. Bottom:
examples of some fits, which yield poles below threshold at $4456-19i$ MeV
(red), $4468-14i$ MeV (blue dash-dotted) and $4566-6i$ MeV (green dashed).

The fitted results are shown in Fig. 8. We can see that the best fit leads to
a pole located close to the real axis but above threshold. A pole below
threshold, as we predicted, is also possible, see the bottom one in Fig. 8.
These fits, though with larger $\chi^{2}$, are reasonable since we have only
four points. The obtained $a_{0}$ values from these fits yield poles several
MeV below threshold with an imaginary part of dozens of MeV. Such poles are
located on the first RS corresponding to a bound state, which moves from the
real axis onto the complex plane due to the coupling to lower channels. There
is another pole located at the symmetric position on the second RS,
corresponding to a virtual state. Actually, with the scattering length
approximation in Eq. (83), we cannot determine on which RS the pole is located
since the poles on different RSs below threshold have the same behavior above
threshold. More data are needed to pin down the exact pole position
corresponding to our predicted $\Lambda_{c}\bar{\Lambda}_{c}$ bound state,
which should be different from the $Y(4630)$ or $Y(4660)$. In Ref. Dai  _et
al._ (2017), the BESIII Ablikim  _et al._ (2018) and Belle Pakhlova  _et al._
(2008) data are fitted together using an amplitude with a pole around 4.65
GeV. While the Belle data of the $Y(4630)$ peak can be well described, the
much more precise BESIII data points in the near-threshold region cannot. We
conclude that the data from $\Lambda_{c}\bar{\Lambda}_{c}$ threshold up to 4.7
GeV should contain signals of at least two states: the
$\Lambda_{c}\bar{\Lambda}_{c}$ molecule and another one with a mass around
4.65 GeV.

As for the isoscalar vector states above 4.85 GeV, the structures could be
more easily identified from data than those around 4.3 GeV. This is because
the charmonium states in that mass region should be very broad while these
hadronic molecules are narrower due to the small binding energies,
corresponding to large spatial extensions. We expect the
$\Sigma_{c}\bar{\Sigma}_{c}$, $\Xi_{c}\bar{\Xi}_{c}$ and
$\Sigma_{c}\bar{\Sigma}_{c}^{*}$ below 5 GeV to be seen in the forthcoming
BESIII measurements, and the ones higher than 5 GeV can be searched for in
future super tau-charm facilities Barniakov (2019); Peng  _et al._ (2020b).

There are isovector $1^{--}$ baryon-antibaryon molecular states above 4.7 GeV,
see Table 7 and Fig. 5. It is more difficult to observe these states than the
isoscalar ones in $e^{+}e^{-}$ collisions since the main production mechanism
of vector states should be driven by a vector $\bar{c}\gamma_{\mu}c$ current,
which is an isoscalar, coupled to the virtual photon. However, they could be
produced together with a pion, and thus can be searched for in future super
tau-charm facilities with center-of-mass energies above 5 GeV.

####  IV.2.6 $\bar{D}^{(*)}\Sigma_{c}^{(*)}$: $P_{c}$ states

The $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ systems with isospin-$1/2$ are attractive,
and a near-threshold pole can be found for each combination. This pole is a
virtual state or (mostly) a bound state depending on the cutoff, see Table 4
and Fig. 3.

Such systems have drawn lots of attention Wu  _et al._ (2010, 2011, 2012a);
Wang  _et al._ (2011); Yang  _et al._ (2012); Wu  _et al._ (2012b); Xiao  _et
al._ (2013); Karliner and Rosner (2015) especially after the pentaquark
states, $P_{c}(4450)$ and $P_{c}(4380)$, were observed by LHCb Aaij  _et al._
(2015). In the updated measurement Aaij  _et al._ (2019), the $P_{c}(4450)$
signal splits into two narrower peaks, $P_{c}(4440)$ and $P_{c}(4457)$. There
is no clear evidence for the previous broad $P_{c}(4380)$, and meanwhile a new
narrow resonance $P_{c}(4312)$ shows up. Several models have been applied by
tremendous works to understand the structures of these states, and the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular explanation stands out as it can
explain the three states simultaneously, see e.g. Refs. Liu  _et al._ (2019b);
Xiao  _et al._ (2019a); Du  _et al._ (2020c). Particularly in Ref. Du  _et
al._ (2020c), the LHCb data are described quite well by the interaction
constructed with heavy quark spin symmetry and actually four $P_{c}$ states,
instead of three, show up, corresponding to $\bar{D}\Sigma_{c}$,
$\bar{D}\Sigma_{c}^{*}$ and $\bar{D}^{*}\Sigma_{c}$ molecules. A hint of a
narrow $P_{c}(4380)$ was reported in the analysis of Ref. Du  _et al._
(2020c). The rest three $P_{c}$ states related to $\bar{D}^{*}\Sigma_{c}^{*}$
predicted there have no signals up to now.

In the vector meson saturation model considered here, the two contact terms
constructed considering only HQSS Liu  _et al._ (2018, 2019b); Sakai  _et al._
(2019); Du  _et al._ (2020c), corresponding to the total angular momentum of
the light degrees of freedom to be $1/2$ and $3/2$, are the same, similar to
the $H\bar{H}$ interaction discussed in Section IV.2.1. As a result, 7
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states Xiao  _et al._ (2013) with
similar binding energies are obtained, and the two $\bar{D}^{*}\Sigma_{c}$
states with different total spins, corresponding to the $P_{c}(4440)$ and
$P_{c}(4457)$, degenerate. The degeneracy will be lifted by considering the
exchange of pion and other mesons and keeping the momentum-dependent terms of
the light vector exchange.

####  IV.2.7 $\bar{D}^{(*)}\Xi_{c}^{(\prime)}$: $P_{cs}$ and related states

It is natural for the isoscalar $\bar{D}^{*}\Xi_{c}$ to form bound states if
the above $P_{c}$ states are considered as the isospin-$1/2$
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecules since the interactions from the
light vector exchange are the same in these two cases, see Table 8. Note that
such states have been predicted by various works Hofmann and Lutz (2005); Chen
_et al._ (2017); Anisovich  _et al._ (2015); Wang (2016); Feijoo  _et al._
(2016); Lu  _et al._ (2016); Xiao  _et al._ (2019b); Chen  _et al._ (2016b);
Wang  _et al._ (2020f); Zhang  _et al._ (2020).

Recently, Ref. Aaij  _et al._ (2020a) reported an exotic state named
$P_{cs}(4459)$ in the invariant mass distribution of $J/\psi\Lambda$ in
$\Xi_{b}^{-}\to J/\psi K^{-}\Lambda$. Even though the significance is only
3.1$\sigma$, several works Liu  _et al._ (2020); Chen (2020); Wang (2020c);
Peng  _et al._ (2020c); Chen  _et al._ (2020) have explored the possibility of
$P_{cs}(4459)$ being a molecule of $\bar{D}^{*}\Xi_{c}$, and the finding here
supports such an explanation that the structure could be caused by two
isoscalar $\bar{D}^{*}\Xi_{c}$ molecules.

Furthermore, Ref. Wang  _et al._ (2020g) moved forward to the double
strangeness systems and claimed that $\bar{D}_{s}^{*}\Xi_{c}^{\prime}$ and
$\bar{D}_{s}^{*}\Xi_{c}^{*}$ may form bound states with $J^{P}=3/2^{-}$ and
$5/2^{-}$, respectively. The $\phi$ exchange for such systems yields repulsive
interaction at leading order, see Table 8, and the bound states obtained in
Ref. Wang  _et al._ (2020g) result from other contributions, including the
exchange of pseudoscalar and scalar mesons, the subleading momentum dependence
from the $\phi$ exchange and coupled-channel effects.

##  V Summary and discussion

The whole spectrum of hadronic molecules of a pair of charmed and anticharmed
hadrons, considering all the $S$-wave singly-charmed mesons and baryons as
well as the $s_{\ell}=3/2$ $P$-wave charmed mesons, is systematically obtained
using $S$-wave constant contact potentials saturated by the exchange of vector
mesons. The coupling of charmed heavy hadrons and light mesons are constructed
by implementing HQSS, chiral symmetry and SU(3) flavor symmetry.

The spectrum predicted here should be regarded as the leading approximation of
the spectrum for heavy-antiheavy molecular states, and gives only a general
overall feature of the heavy-antiheavy hadronic molecular spectrum. Specific
systems may differ from the predictions here due to the limitations of our
treatment. We considered neither the effects of coupled channels, nor the
spin-dependent interactions, which arises from momentum-dependent terms that
are of higher order in the very near-threshold region, nor the contribution
from the exchange of pseudoscalar and scalar mesons, nor the mixing with
charmonia. Nevertheless, the spectrum shows a different pattern than that
considering only the one-pion exchange (see, e.g., Ref. Karliner and Rosner
(2015)), which does not allow the molecular states in systems such as
$D\bar{D}$ and $\Sigma_{c}\bar{D}$, where the one-pion exchange is forbidden
without coupled channels, to exist.

In total 229 hidden-charm hadronic molecules (bound or virtual) are predicted,
many of which deserve attentions:

  * 1)

The pole positions of the isoscalar $D\bar{D}^{*}$ with positive and negative
$C$-parity are consistent with the molecular explanation of $X(3872)$ and
$\tilde{X}(3872)$, respectively. There is a shallow bound state in the
isoscalar $D\bar{D}$ system, consistent with the recent lattice QCD result
Prelovsek  _et al._ (2020).

  * 2)

The spectrum of the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ systems is consistent with
the molecular explanations of famous $P_{c}$ states: the $P_{c}(4312)$ as an
isospin-$1/2$ $\bar{D}\Sigma_{c}$ molecule, and $P_{c}(4440)$ and
$P_{c}(4457)$ as isospin-$1/2$ $\bar{D}^{*}\Sigma_{c}$ molecules. With the
resonance saturation from the vector mesons, the two $\bar{D}^{*}\Sigma_{c}$
molecules are degenerated. In addition, there is an isospin-$1/2$
$\bar{D}\Sigma_{c}^{*}$ molecule, consistent with the narrow $P_{c}(4380)$
advocated in Ref. Du  _et al._ (2020c), and three isospin-$1/2$
$\bar{D}^{*}\Sigma_{c}^{*}$ molecules, consistent with the results in the
literature.

  * 3)

There are two isoscalar $\bar{D}^{*}\Xi_{c}$ molecules, which may be related
to the recently announced $P_{cs}(4459)$. In addition, more negative-parity
isoscalar $P_{cs}$-type molecules are predicted: one in $\bar{D}\Xi_{c}$, one
in $\bar{D}\Xi_{c}^{\prime}$, one in $\bar{D}\Xi_{c}^{*}$, two in
$\bar{D}^{*}\Xi_{c}^{\prime}$, and three in $\bar{D}^{*}\Xi_{c}^{*}$.

  * 4)

Instead of associating the $X(4140)$ with a $D_{s}^{*}\bar{D}_{s}^{*}$
molecule like some other works did, our results prefer the
$D_{s}^{*}\bar{D}_{s}^{*}$ to form a virtual state. The peak in the invariant
mass distribution of $J/\psi\phi$ measured by LHCb just at the
$D_{s}^{*}\bar{D}_{s}^{*}$ threshold is consistent with this scenario,
according to the discussion in Ref. Dong  _et al._ (2020a).

  * 5)

The isoscalar $D^{(*)}\bar{D}_{1(2)}$ can form negative-parity bound states
with both positive and negative $C$ parities. The $D\bar{D}_{1}$ bound state
is the lowest one in this family, and the $1^{--}$ one is consistent with the
sizeable $D\bar{D}_{1}$ molecular component in the $\psi(4230)$.

  * 6)

$\Lambda_{c}\bar{\Lambda}_{c}$ bound states with $J^{PC}=0^{-+}$ and $1^{--}$
are predicted. The vector one should be responsible to the almost flat line
shape of the $e^{+}e^{-}\to\Lambda_{c}\bar{\Lambda}_{c}$ cross section in the
near-threshold region observed by BESIII Ablikim  _et al._ (2018).

  * 7)

Light vector meson exchanges either vanish due to the cancellation between
$\rho$ and $\omega$ or are not allowed in the isovector $D^{(*)}\bar{D}^{(*)}$
systems and $D^{(*)}\bar{D}_{s}^{(*)}$ systems. However, the vector charmonia
exchanges may play an important role as pointed out in Ref. Aceti  _et al._
(2014), and the $Z_{c}(3900,4020)$ and $Z_{cs}(3985)$ could well be the
$D^{(*)}\bar{D}^{*}$ and $D^{*}\bar{D}_{s}$ – $D\bar{D}_{s}^{*}$ virtual
states.

When the light vector meson exchange is allowed, the results reported here are
generally consistent with the results from a more complete treatment of the
one-boson exchange model (e.g., by solving the Schrödinger equation). For
example, the binding energy of the isoscalar $D\bar{D}$ Zhang  _et al._ (2006)
and $D\bar{D}_{1}$ Dong  _et al._ (2020b) bound states from the $\rho$ and
$\omega$ exchanges fit the spectrum well; a similar pattern of molecular
states related to the $X(3872)$ was obtained in Ref. Liu  _et al._ (2009) and
the light vector exchange was found necessary to bind $D\bar{D}^{*}$ together;
the $\bar{D}^{*}\Sigma_{c}$ bound states corresponding to the $P_{c}(4440)$
and $P_{c}(4457)$ were obtained via one boson exchange in Ref. Liu  _et al._
(2019c), and the degeneracy of the two states with $J=1/2$ and $3/2$ is lifted
by the pion exchange and higher order contributions from the $\rho$ and
$\omega$ exchange. We should also notice that there can be systems whose
contact terms receive important contributions from the scalar-meson exchanges.

We expect that there should be structures in the near-threshold region for all
the heavy-antiheavy hadron pairs that have attractive interactions at
threshold. The structure can be either exactly at threshold, if the attraction
is not strong enough to form a bound state, or below threshold, if a bound
state is formed. Moreover, the structures are not necessarily peaks, and they
can be dips in invariant mass distributions, depending on the pertinent
production mechanism as discussed in our recent work Dong  _et al._ (2020a).

When the predicted states have ordinary quantum numbers as those of charmonia,
the molecular states must mix with charmonia, and the mixing can have an
important impact on the spectrum. Yet, in the energy region higher than 4.8
GeV, where plenty of states are predicted as shown in Figs. 1 and 2, normal
charmonia should be very broad due to the huge phase space while the molecular
states should be relatively narrow due to the large distance between the
consistent hadrons. Thus, narrow structures to be discovered in this energy
region should be mainly due to the molecular structures, being either bound or
virtual states.

Among the 229 structures predicted here, only a minority is in the energy
region that has been studied in detail. The largest data sets from the current
experiments have the following energy restrictions: direct production of the
$1^{--}$ sector in $e^{+}e^{-}$ collisions goes up to 4.6 GeV at BESIII; the
hidden-charm $XYZ$ states produced through the weak process $b\to c\bar{c}s$
in $B\to K$ decays should be below 4.8 GeV; the hidden-charm $P_{c}$
pentaquarks produced in $\Lambda_{b}\to K$ decays should be below 5.1 GeV. To
find more states in the predicted spectrum, we need to have both data in these
processes with higher statistics and data at other experiments such as the
prompt production at hadron colliders, PANDA, electron-ion collisions and
$e^{+}e^{-}$ collisions above 5 GeV at super tau-charm facilities.

The potentials in the bottom sector are the same as those in the charm sector,
if using the nonrelativistic field normalization, due to the HQFS, and we
expect the same number of molecular states in the analogous systems therein.
Because of the much heavier reduced masses of hidden-bottom systems, the
virtual states in the charm sector will move closer to the thresholds or even
become bound states in the bottom sector, and the bound states in the charm
sector will be more deeply bound in the bottom sector. There may even be
excited states for some deeply bound systems. For these deeply bound systems,
the constant contact term approximation considered here will not be
sufficient. However, due to the large masses, such states are more difficult
to be produced than those in the charm sector.

###### Acknowledgements.

We would like to thank Chang-Zheng Yuan for useful discussions, and thank Fu-
Lai Wang for a communication regarding Ref. Wang  _et al._ (2020g). This work
is supported in part by the Chinese Academy of Sciences (CAS) under Grant No.
XDB34030000 and No. QYZDB-SSW-SYS013, by the National Natural Science
Foundation of China (NSFC) under Grant No. 11835015, No. 12047503 and No.
11961141012, by the NSFC and the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through the funds provided to the Sino-German
Collaborative Research Center TRR110 “Symmetries and the Emergence of
Structure in QCD” (NSFC Grant No. 12070131001, DFG Project-ID 196253076), and
by the CAS Center for Excellence in Particle Physics (CCEPP).

##  Appendix A Vertex factors for direct processes

The vertex factors in Eq. (70) for different particles are calculated in the
following:

  * •

$D$

$\displaystyle g_{1}$ $\displaystyle=-\sqrt{2}\beta
g_{V}m_{D}(P_{a}^{(Q)}V_{ab}P^{(Q)T}_{b}),$ (84) $\displaystyle g_{2}$
$\displaystyle=\sqrt{2}\beta
g_{V}m_{D}(P_{a}^{(\bar{Q})}V_{ab}P^{(\bar{Q})T}_{b}).$ (85)

  * •

$D^{*}$

$\displaystyle g_{1}$ $\displaystyle=\sqrt{2}\beta
g_{V}m_{D^{*}}\epsilon\cdot\epsilon^{*}(P_{a}^{*(Q)}V_{ab}P^{*(Q)T}_{b})$
$\displaystyle\approx-\sqrt{2}\beta
g_{V}m_{D^{*}}(P_{a}^{*(Q)}V_{ab}P^{*(Q)T}_{b}),$ (86) $\displaystyle g_{2}$
$\displaystyle=-\sqrt{2}\beta
g_{V}m_{D^{*}}\epsilon\cdot\epsilon^{*}(P_{a}^{*(\bar{Q})}V_{ab}P^{*(\bar{Q})T}_{b})$
$\displaystyle\approx\sqrt{2}\beta
g_{V}m_{D^{*}}(P_{a}^{*(\bar{Q})}V_{ab}P^{*(\bar{Q})T}_{b}).$ (87)

  * •

$D_{1}$

$\displaystyle g_{1}$
$\displaystyle=-\sqrt{2}\beta_{2}g_{V}m_{D_{1}}\epsilon\cdot\epsilon^{*}(P^{(Q)}_{1a}V_{ab}P^{(Q)T}_{1b}),$
$\displaystyle\approx\sqrt{2}\beta_{2}g_{V}m_{D_{1}}(P^{(Q)}_{1a}V_{ab}P^{(Q)T}_{1b}),$
(88) $\displaystyle g_{2}$
$\displaystyle=\sqrt{2}\beta_{2}g_{V}m_{D_{1}}\epsilon\cdot\epsilon^{*}(P^{(\bar{Q})}_{1a}V_{ab}P^{(\bar{Q})T}_{1b}),$
$\displaystyle\approx-\sqrt{2}\beta_{2}g_{V}m_{D_{1}}(P^{(\bar{Q})}_{1a}V_{ab}P^{(\bar{Q})T}_{1b}).$
(89)

  * •

$D_{2}^{*}$

$\displaystyle g_{1}$
$\displaystyle=\sqrt{2}\beta_{2}g_{V}m_{D_{2}^{*}}\epsilon^{\mu\nu}\epsilon_{\mu\nu}^{*}(P^{*(Q)}_{2a}V_{ab}P^{*(Q)T}_{2b})$
$\displaystyle\approx\sqrt{2}\beta_{2}g_{V}m_{D_{2}^{*}}(P^{*(Q)}_{2a}V_{ab}P^{*(Q)T}_{2b}),$
(90) $\displaystyle g_{2}$ $\displaystyle=-\sqrt{2}\beta
g_{V}m_{D_{2}^{*}}\epsilon^{\mu\nu}\epsilon_{\mu\nu}^{*}(P^{*(\bar{Q})}_{2a}V_{ab}P^{*(\bar{Q})T}_{2b})$
$\displaystyle\approx\sqrt{2}\beta
g_{V}m_{D_{2}^{*}}(P^{*(\bar{Q})}_{2a}V_{ab}P^{*(\bar{Q})T}_{2b}).$ (91)

  * •

$B_{\bar{3}}$

$\displaystyle g_{1}$
$\displaystyle=\frac{1}{\sqrt{2}}\beta_{B}g_{V}\bar{u}(k_{1})u(p_{1}){\rm
tr}\left[(B_{\bar{3}}^{(Q)T}VB_{\bar{3}}^{(Q)}\right]$
$\displaystyle\approx\sqrt{2}\beta_{B}g_{V}m_{B_{\bar{3}}}{\rm
tr}\left[(B_{\bar{3}}^{(Q)T}VB_{\bar{3}}^{(Q)}\right],$ (92) $\displaystyle
g_{2}$
$\displaystyle=-\frac{1}{\sqrt{2}}\beta_{B}g_{V}\bar{u}(k_{2})u(p_{2}){\rm
tr}\left[B_{3}^{(\bar{Q})T}V^{T}B_{3}^{(\bar{Q})}\right]$
$\displaystyle\approx-\sqrt{2}\beta_{B}g_{V}m_{B_{\bar{3}}}{\rm
tr}\left[B_{3}^{(\bar{Q})T}V^{T}B_{3}^{(\bar{Q})}\right].$ (93)

  * •

$B_{6}$

$\displaystyle g_{1}$
$\displaystyle=-\frac{1}{3}\frac{\beta_{S}g_{V}}{\sqrt{2}}\bar{u}(k_{1})\gamma^{5}(\gamma^{\mu}+v^{\mu})^{2}\gamma^{5}u(p_{1})$
$\displaystyle\quad\times{\rm tr}\left[B_{6}^{(Q)T}VB^{(Q)}_{6}\right]$
$\displaystyle\approx-\sqrt{2}m_{B_{6}}\beta_{S}g_{V}{\rm
tr}\left[B_{6}^{(Q)T}VB^{(Q)}_{6}\right],$ (94) $\displaystyle g_{2}$
$\displaystyle=\frac{1}{3}\frac{\beta_{S}g_{V}}{\sqrt{2}}\bar{u}(k_{2})\gamma^{5}(\gamma^{\mu}+v^{\mu})^{2}\gamma^{5}u(p_{2})$
$\displaystyle\quad\times{\rm
tr}\left[B_{6}^{(\bar{Q})T}V^{T}B^{(\bar{Q})}_{6}\right]$
$\displaystyle\approx\sqrt{2}m_{B_{6}}\beta_{S}g_{V}{\rm
tr}\left[B_{6}^{(\bar{Q})T}V^{T}B^{(\bar{Q})}_{6}\right].$ (95)

  * •

$B_{6}^{*}$

$\displaystyle g_{1}$
$\displaystyle=\frac{\beta_{S}g_{V}}{\sqrt{2}}\bar{u}_{\mu}^{*}(k_{1})u^{*\mu}(p_{1}){\rm
tr}\left[B_{6}^{*(Q)T}VB_{6}^{*(Q)}\right]$
$\displaystyle\approx-\sqrt{2}m_{B_{6}^{*}}\beta_{S}g_{V}{\rm
tr}\left[B_{6}^{*(Q)T}VB_{6}^{*(Q)}\right],$ (96) $\displaystyle g_{2}$
$\displaystyle=-\frac{\beta_{S}g_{V}}{\sqrt{2}}\bar{u}_{\mu}^{*}(k_{2})u^{*\mu}(p_{2}){\rm
tr}\left[B_{6}^{*(\bar{Q})T}V^{T}B_{6}^{*(\bar{Q})}\right]$
$\displaystyle\approx\sqrt{2}m_{B_{6}^{*}}\beta_{S}g_{V}{\rm
tr}\left[B_{6}^{*(\bar{Q})T}V^{T}B_{6}^{*(\bar{Q})}\right].$ (97)

In the above deductions we have used $\epsilon\cdot\epsilon^{*}=-1$,
$\epsilon^{\mu\nu}\cdot\epsilon_{\mu\nu}^{*}=1$, $\bar{u}(k_{1})u(p_{1})=2m$
and $\bar{u}_{6\mu}^{*}u_{6}^{*\mu}=-2m$ at threshold. Note that the factors
such as $P_{a}^{(Q)}V_{ab}P^{(Q)T}_{b}$, ${\rm
tr}\left[B_{\bar{3}}^{(Q)T}VB_{\bar{3}}^{(Q)}\right]$ in the above expressions
contain only the SU(3) flavor information and the properties of the
corresponding fields have already been extracted.

##  Appendix B List of the potential factor $F$

The details of interactions between all combinations of heavy-antiheavy hadron
pairs are listed in Tables 8 and 9.

Table 8: The group theory factor $F$, defined in Eq. (71), for the interaction of charmed-anticharmed hadron pairs with only the light vector-meson exchanges. Here both charmed hadrons are the $S$-wave ground states. $I$ is the isospin and $S$ is the strangeness. Positive $F$ means attractive. For the systems with $F=0$, the sub-leading exchanges of vector-charmonia also lead to an attractive potential at threshold.  System | $(I,S)$ | Thresholds (MeV) | Exchanged particles | $F$
---|---|---|---|---
$D^{(*)}\bar{D}^{(*)}$ | (0,0) | $(3734,3876,4017)$ | $\rho,\omega$ | $\frac{3}{2},\frac{1}{2}$
| (1,0) |  | $\rho,\omega$ | $-\frac{1}{2},\frac{1}{2}$
$D_{s}^{(*)}\bar{D}^{(*)}$ | $(\frac{1}{2},1)$ | $(3836,3977,3979,4121)$ | $-$ | $0$
$D^{(*)}_{s}\bar{D}^{(*)}_{s}$ | (0,0) | $(3937,4081,4224)$ | $\phi$ | $1$
$\bar{D}^{(*)}\Lambda_{c}$ | $(\frac{1}{2},0)$ | $(4154,4295)$ | $\omega$ | $-1$
$\bar{D}_{s}^{(*)}\Lambda_{c}$ | $(0,-1)$ | $(4255,4399)$ | $-$ | $0$
$\bar{D}^{(*)}\Xi_{c}$ | $(1,-1)$ | $(4337,4478)$ | $\rho,\omega$ | $-\frac{1}{2},-\frac{1}{2}$
| $(0,-1)$ |  | $\rho,\omega$ | $\frac{3}{2},-\frac{1}{2}$
$\bar{D}_{s}^{(*)}\Xi_{c}$ | $(\frac{1}{2},-2)$ | $(4438,4582)$ | $\phi$ | $-1$
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ | $(\frac{3}{2},0)$ | $(4321,4385,4462,4527)$ | $\rho,\omega$ | $-1,-1$
| $(\frac{1}{2},0)$ |  | $\rho,\omega$ | $2,-1$
$\bar{D}_{s}^{(*)}\Sigma_{c}^{(*)}$ | $(1,-1)$ | $(4422,4486,4566,4630)$ | $-$ | $0$
$\bar{D}^{(*)}\Xi_{c}^{{}^{\prime}(*)}$ | $(1,-1)$ | $(4446,4513,4587,4655)$ | $\rho,\omega$ | $-\frac{1}{2},-\frac{1}{2}$
| $(0,-1)$ |  | $\rho,\omega$ | $\frac{3}{2},-\frac{1}{2}$
$\bar{D}_{s}^{(*)}\Xi_{c}^{{}^{\prime}(*)}$ | $(\frac{1}{2},-2)$ | $(4547,4614,4691,4758)$ | $\phi$ | $-1$
$\bar{D}^{(*)}\Omega_{c}^{(*)}$ | $(\frac{1}{2},-2)$ | $(4562,4633,4704,4774)$ | $-$ | $0$
$\bar{D}_{s}^{(*)}\Omega_{c}^{(*)}$ | $(0,-3)$ | $(4664,4734,4807,4878)$ | $\phi$ | $-2$
$\Lambda_{c}\bar{\Lambda}_{c}$ | $(0,0)$ | $(4573)$ | $\omega$ | $2$
$\Lambda_{c}\bar{\Xi}_{c}$ | $(\frac{1}{2},1)$ | $(4756)$ | $\omega$ | $1$
$\Xi_{c}\bar{\Xi}_{c}$ | $(1,0)$ | $(4939)$ | $\rho,\omega,\phi$ | $-\frac{1}{2},\frac{1}{2},1$
| $(0,0)$ |  | $\rho,\omega,\phi$ | $\frac{3}{2},\frac{1}{2},1$
$\Lambda_{c}\bar{\Sigma}_{c}^{(*)}$ | $(1,0)$ | $(4740,4805)$ | $\omega$ | $2$
$\Lambda_{c}\bar{\Xi}_{c}^{{}^{\prime}(*)}$ | $(\frac{1}{2},1)$ | $(4865,4932)$ | $\omega$ | $1$
$\Lambda_{c}\bar{\Omega}_{c}^{(*)}$ | $(0,2)$ | $(4982,5052)$ | $-$ | $0$
$\Xi_{c}\bar{\Sigma}_{c}^{(*)}$ | $(\frac{3}{2},-1)$ | $(4923,4988)$ | $\rho,\omega$ | $-1,1$
| $(\frac{1}{2},-1)$ |  | $\rho,\omega$ | $2,1$
$\Xi_{c}\bar{\Xi}_{c}^{{}^{\prime}(*)}$ | $(1,0)$ | $(5048,5115)$ | $\rho,\omega,\phi$ | $-\frac{1}{2},\frac{1}{2},1$
| $(0,0)$ |  | $\rho,\omega,\phi$ | $\frac{3}{2},\frac{1}{2},1$
$\Xi_{c}\bar{\Omega}_{c}^{(*)}$ | $(\frac{1}{2},1)$ | $(5165,5235)$ | $\phi$ | $2$
$\Sigma_{c}^{(*)}\bar{\Sigma}_{c}^{(*)}$ | $(2,0)$ | $(4907,4972,5036)$ | $\rho,\omega$ | $-2,2$
| $(1,0)$ |  | $\rho,\omega$ | $2,2$
| $(0,0)$ |  | $\rho,\omega$ | $4,2$
$\Sigma_{c}^{(*)}\bar{\Xi}^{{}^{\prime}(*)}_{c}$ | $(\frac{3}{2},1)$ | $(5032,5097,5100,5164)$ | $\rho,\omega$ | $-1,1$
| $(\frac{1}{2},1)$ |  | $\rho,\omega$ | $2,1$
$\Sigma_{c}^{(*)}\bar{\Omega}^{(*)}_{c}$ | $(0,2)$ | $(5149,5213,5219,5284)$ | $-$ | $0$
$\Xi_{c}^{{}^{\prime}(*)}\bar{\Xi}_{c}^{{}^{\prime}(*)}$ | $(1,0)$ | $(5158,5225,5292)$ | $\rho,\omega,\phi$ | $-\frac{1}{2},\frac{1}{2},1$
| $(0,0)$ |  | $\rho,\omega,\phi$ | $\frac{3}{2},\frac{1}{2},1$
$\Xi^{{}^{\prime}(*)}_{c}\bar{\Omega}_{c}^{(*)}$ | $(\frac{1}{2},1)$ | $(5272,5341,5345,5412)$ | $\phi$ | $2$
$\Omega_{c}^{(*)}\bar{\Omega}_{c}^{(*)}$ | $(0,0)$ | $(5390,5461,5532)$ | $\phi$ | $4$
Table 9: The group theory factor $F$, defined in Eq. (71), for the interaction of charmed-anticharmed hadron pairs with only the light vector-meson exchanges. Here one of the charmed hadrons is an $s_{\ell}=3/2$ charmed meson. See the caption of Table 8.  System | $(I,S)$ | Thresholds (MeV) | Exchanged particles | $F$
---|---|---|---|---
$D^{(*)}\bar{D}_{1,2}$ | (0,0) | $(4289,4330,4431,4472)$ | $\rho,\omega$ | $\frac{3}{2},\frac{1}{2}$
| (1,0) |  | $\rho,\omega$ | $-\frac{1}{2},\frac{1}{2}$
$D^{(*)}\bar{D}_{s1,s2}$ | $(\frac{1}{2},-1)$ | $(4390,4431,4534,4575)$ | $-$ | $0$
$D_{s}^{(*)}\bar{D}_{1,2}$ | $(\frac{1}{2},1)$ | $(4402,4436,4544,4578)$ | $-$ | $0$
$D^{(*)}_{s}\bar{D}_{s1,s2}$ | (0,0) | $(4503,4537,4647,4681)$ | $\phi$ | $1$
$D_{1,2}\bar{D}_{1,2}$ | (0,0) | $(4844,4885,4926)$ | $\rho,\omega$ | $\frac{3}{2},\frac{1}{2}$
| (1,0) |  | $\rho,\omega$ | $-\frac{1}{2},\frac{1}{2}$
$D_{s1,s2}\bar{D}_{1,2}$ | $(\frac{1}{2},1)$ | $(4957,4991,4998,5032)$ | $-$ | $0$
$D_{s1,s2}\bar{D}_{s1,s2}$ | (0,0) | $(5070,5104,5138)$ | $\phi$ | $1$
$\Lambda_{c}\bar{D}_{1,2}$ | $(\frac{1}{2},0)$ | $(4708,4750)$ | $\omega$ | $-1$
$\Lambda_{c}\bar{D}_{s1,s2}$ | $(0,-1)$ | $(4822,4856)$ | $-$ | $0$
$\Xi_{c}\bar{D}_{1,2}$ | $(1,-1)$ | $(4891,4932)$ | $\rho,\omega$ | $-\frac{1}{2},-\frac{1}{2}$
| $(0,-1)$ |  | $\rho,\omega$ | $\frac{3}{2},-\frac{1}{2}$
$\Xi_{c}\bar{D}_{s1,s2}$ | $(\frac{1}{2},-2)$ | $(5005,5039)$ | $\phi$ | $-1$
$\Sigma_{c}^{(*)}\bar{D}_{1,2}$ | $(\frac{3}{2},0)$ | $(4876,4917,4940,4981)$ | $\rho,\omega$ | $-1,-1$
| $(\frac{1}{2},0)$ |  | $\rho,\omega$ | $2,-1$
$\Sigma_{c}^{(*)}\bar{D}_{s1,s2}$ | $(1,-1)$ | $(4989,5023,5053,5087)$ | $-$ | $0$
$\Xi_{c}^{{}^{\prime}(*)}\bar{D}_{1,2}$ | $(1,-1)$ | $(5001,5042,5068,5109)$ | $\rho,\omega$ | $-\frac{1}{2},-\frac{1}{2}$
| $(0,-1)$ |  | $\rho,\omega$ | $\frac{3}{2},-\frac{1}{2}$
$\Xi_{c}^{{}^{\prime}(*)}\bar{D}_{s1,s2}$ | $(\frac{1}{2},-2)$ | $(5114,5148,5181,5215)$ | $\phi$ | $-1$
$\Omega_{c}^{(*)}\bar{D}_{1,2}$ | $(\frac{1}{2},-2)$ | $(5117,5158,5188,5229)$ | $-$ | $0$
$\Omega_{c}^{(*)}\bar{D}_{s1,s2}$ | $(0,-3)$ | $(5230,5264,5301,5335)$ | $\phi$ | $-2$

##  Appendix C Amplitude calculation for cross processes

In the following we show the deduction of Eq. (73).

  * •

$D\bar{D}_{1}$ and $D_{s}\bar{D}_{s1}$

$\displaystyle V$
$\displaystyle=i\left(i\frac{-2\zeta_{1}g_{V}}{\sqrt{3}}\right)\left(i\frac{2\zeta_{1}g_{V}}{\sqrt{3}}\right)m_{D}m_{D_{1}}$
$\displaystyle\times\epsilon_{1}^{*\mu}\frac{-i(g_{\mu\nu}-q_{\mu}q_{\nu}/m_{\rm
ex}^{2})}{q^{2}-m_{\rm ex}^{2}+i\epsilon}\epsilon_{2}^{\nu}F$
$\displaystyle\approx FF_{c}\zeta_{1}^{2}g_{V}^{2}\frac{m_{D}m_{D_{1}}}{m_{\rm
ex}^{2}-\Delta{m}^{2}}$ (98)

with $F_{c}=4/3$.

  * •

$D^{*}\bar{D}_{1}$ and $D_{s}^{*}\bar{D}_{s1}$

$\displaystyle V$
$\displaystyle=i\left(i\frac{i\zeta_{1}g_{V}}{\sqrt{3}}\right)\left(i\frac{-i\zeta_{1}g_{V}}{\sqrt{3}}m_{D^{*}}m_{D_{1}}\right)\epsilon_{\alpha\beta\gamma\delta}\epsilon_{\alpha_{1}\beta_{1}\gamma_{1}\delta_{1}}$
$\displaystyle\times\epsilon_{1}^{\beta}\epsilon_{2}^{*\alpha}v^{\gamma}\frac{-i(g^{\delta\delta_{1}}-q^{\delta}q^{\delta_{1}}/m_{\rm
ex}^{2})}{q^{2}-m_{\rm
ex}^{2}+i\epsilon}\epsilon_{3}^{\alpha_{1}}\epsilon_{4}^{*\beta_{1}}v^{\gamma_{1}}F$
$\displaystyle\approx\frac{1}{3}\zeta_{1}^{2}g_{V}^{2}\frac{m_{D^{*}}m_{D_{1}}}{m_{\rm
ex}^{2}-\Delta{m}^{2}}(\bm{\epsilon}_{1}\times\bm{\epsilon}_{2}^{*})\cdot(\bm{\epsilon}_{3}\times\bm{\epsilon}_{4}^{*})F$
$\displaystyle=-\frac{1}{3}F\zeta_{1}^{2}g_{V}^{2}\frac{m_{D^{*}}m_{D_{1}}}{m_{\rm
ex}^{2}-\Delta{m}^{2}}\bm{S}_{1}\cdot\bm{S}_{2}$
$\displaystyle=FF_{c}\zeta_{1}^{2}g_{V}^{2}\frac{m_{D^{*}}m_{D_{1}}}{m_{\rm
ex}^{2}-\Delta{m}^{2}},$ (99)

where $\bm{S}_{i}$ is the spin-1 operator. Explicitly,
$\bm{S}_{1}\cdot\bm{S}_{2}=-2,-1$ and $1$ for total spin $J=0,1$ and $2$,
respectively and in turn we obtain $F_{c}=2/3,1/3$ and $-1/3$.

  * •

$D^{*}\bar{D}_{2}$ and $D_{s}^{*}\bar{D}_{s2}$

$\displaystyle V$
$\displaystyle=i\left(i\sqrt{2}\zeta_{1}g_{V}\right)\left(i\sqrt{2}\zeta_{1}g_{V}\right)m_{D^{*}}m_{D_{2}}$
$\displaystyle\times\epsilon_{1\mu}\epsilon_{2}^{*\mu\nu}\frac{-i(g_{\nu\alpha}-q_{\nu}q_{\alpha})/m_{\rm
ex}^{2}}{q^{2}-m_{\rm
ex}^{2}+i\epsilon}\epsilon_{3}^{\alpha\beta}\epsilon_{4\beta}F$
$\displaystyle\approx{2\zeta_{1}^{2}g_{V}^{2}}\frac{m_{D^{*}}m_{D_{2}}}{m_{\rm
ex}^{2}-\Delta{m}^{2}}\epsilon_{1\mu}\epsilon_{2}^{*\mu\nu}\epsilon_{3\nu}^{\beta}\epsilon_{4\beta}F$
$\displaystyle=FF_{c}\zeta_{1}^{2}g_{V}^{2}\frac{m_{D^{*}}m_{D_{2}}}{m_{\rm
ex}^{2}-\Delta{m}^{2}},$ (100)

where $F_{c}=-1/3,1$ and $-2$ for total spin $J=1,2$ and $3$, respectively.

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