# A survey of heavy-heavy 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

The spectrum of hadronic molecules composed of heavy-antiheavy charmed hadrons
has been obtained in our previous work. The potentials are constants at the
leading order, which are estimated from resonance saturation. The experimental
candidates of hadronic molecules, say $X(3872)$, $Y(4260)$, three $P_{c}$
states and $P_{cs}(4459)$, fit the spectrum well. The success in describing
the pattern of heavy-antiheavy hadronic molecules stimulates us to give more
predictions for the heavy-heavy cases, which are less discussed in literature
than the heavy-antiheavy ones. Given that the heavy-antiheavy hadronic
molecules, several of which have strong experimental evidence, emerge from the
dominant constant interaction from resonance saturation, we find that the
existence of many heavy-heavy hadronic molecules is natural. Among these
predicted heavy-heavy states we highlight the $DD^{*}$ molecule and the
$D^{(*)}\Sigma_{c}^{(*)}$ molecules, which are the partners of the famous
$X(3872)$ and $P_{c}$ states. Quite recently, LHCb collaboration reported a
doubly charmed tetraquark state, $T_{cc}$, which is in line with our results
for the $DD^{*}$ molecule. With the first experimental signal of this new kind
of exotic states, the upcoming update of the LHCb experiment as well as other
experiments will provide more chances of observing the heavy-heavy hadronic
molecules.

##  I Introduction

The fact that quantum chromodynamics (QCD) is nonperturbative at low energy
makes the calculation of the whole hadron spectrum from first-principle too
difficult at the present stage. The quark model proposed in Refs. Gell-Mann
(1964); Zweig (1964) successfully classify plenty of hadrons and its later
developments after the birth of QCD (see, e.g., Refs. Godfrey and Isgur
(1985); Capstick and Isgur (1985)) provide a remarkable description of the
hadron spectrum of $q\bar{q}$ mesons and $qqq$ baryons. However, there was
little clear evidence for the multiquark states predicted in Ref. Gell-Mann
(1964) until the discovery of the $X(3872)$ Choi  _et al._ (2003), also known
as $\chi_{c1}(3872)$ in the Review of Particle Physics (RPP) Zyla  _et al._
(2020). Since then many near-threshold structures, e.g. the
$Z_{c}(3900)^{\pm}$ Ablikim  _et al._ (2013a); Liu  _et al._ (2013); Ablikim
_et al._ (2014a), the $Z_{c}(4020)^{\pm}$ Ablikim  _et al._ (2014b, 2013b),
the $Z_{b}(10610/10650)^{\pm}$ Bondar  _et al._ (2012); Garmash  _et al._
(2016), the $Z_{cs}(3985)^{-}$ Ablikim  _et al._ (2021) and the $P_{c}$ states
Aaij  _et al._ (2019), have been observed in the worldwide high energy
experiments. These so-called exotic states are clearly outside the scope of
the traditional quark model consisting of $q\bar{q}$ mesons and $qqq$ baryons
but their inner structures are still under debate (see Refs. Chen  _et al._
(2016); 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 (2021);
Dong  _et al._ (2021a) for recent reviews of the multiquark states).

One peculiar and thus important property of these exotic states mentioned
above is that they are all located quite close to the thresholds of a pair of
hadrons that they can couple to. Therefore it is natural to consider them as
hadronic molecules111Note that the concept of hadronic molecules have been
extended from bound states Weinberg (1963a, b, 1965) to near-threshold
resonances Baru  _et al._ (2004) and virtual states Matuschek  _et al._
(2021).  composed of the corresponding hadron pairs and such an idea was
widely explored in the literature, see Ref. Guo  _et al._ (2018) for a review
of hadronic molecules and Ref. Dong  _et al._ (2021b) for a discussion of the
general behaviors of near-threshold structures. Although many works have used
different methods to explain certain properties of these near-threshold
states, a whole and systematic spectrum of hadronic molecules will deepen our
understanding of these exotic states. In Ref. Dong  _et al._ (2021a), we have
provided such a spectrum for heavy-antiheavy (taking charmed hadrons for
example) hadronic molecules by solving the single channel Bethe-Salpeter (BS)
equation with constant interactions, which are assumed to be vector meson-
exchange saturated. In this work we extend the work in Ref. Dong  _et al._
(2021a) to the heavy-heavy systems to make the spectrum more complete.

Note that all the experimentally established exotic states mentioned above are
hidden-charm or hidden-bottom ones. It is much more difficult to produce the
states with doubly heavy quarks than those with heavy-antiheavy quarks since
that require at least two heavy quark-antiquark pairs to be produced and two
heavy (anti)quarks to move in a particular phase space region. It is only
until very recently that the first experimental evidence for a double-charm
tetraquark state was reported Aaij  _et al._ (2021a, b), which immediately
stimulated a series of theoretical studies Li  _et al._ (2021); Agaev  _et
al._ (2021a); Ling  _et al._ (2021); Meng  _et al._ (2021a); Chen  _et al._
(2021a); Feijoo  _et al._ (2021); Yan and Valderrama (2021); Wang and Liu
(2021); Xin and Wang (2021); Fleming  _et al._ (2021); Azizi and Özdem (2021);
Chen (2021); Ren  _et al._ (2021); Jin  _et al._ (2021). Despite that, many
attempts have been made to investigate the possible states with doubly heavy
quarks, see the discussions in Sec. IV. On the one hand, the heavy
antiquark–diquark symmetry for antiheavy baryons may lead to the corresponding
doubly-heavy tetraquark states. On the other hand, like the heavy-antiheavy
systems, the heavy-heavy hadronic molecules are also expected to exist.

This work is organized as follows. In Sec. II, we give a brief summary of the
interactions between heavy hadrons following the heavy quark spin symmetry. In
Sec. III, the potentials resulting from these interactions are presented and
the molecular states obtained by solving the single channel BS equation are
listed. Some selected states are discussed in Sec. IV, together with an
incomplete review on the status of such systems in the literature. We then
close this paper with a brief summary in Sec. V.

##  II Lagrangian from heavy quark spin symmetry

The interactions between hadrons consisting of one or more heavy quarks, say
$c$ and $b$ due to their much larger masses than the typical QCD scale, can be
constructed systematically under the guidance of heavy quark symmetries222For
the spirits of heavy quark symmetries, we refer to the pioneer works Voloshin
and Shifman (1987); Politzer and Wise (1988a, b); Isgur and Wise (1990, 1989)
and reviews Neubert (1994); Manohar and Wise (2000) and such Lagrangians can
be found in, e.g. Refs. Wise (1992); Casalbuoni  _et al._ (1992, 1997);
Grinstein  _et al._ (1992); Falk (1992); Falk and Luke (1992); Yan  _et al._
(1992); Casalbuoni  _et al._ (1993, 1997); Liu and Oka (2012). The relevant
Lagrangians and some details have been collected in our previous work Dong
_et al._ (2021a) and here we will just give the final results that are needed.
Note that in the following we are interested in the potential near threshold
and will not consider coupled channels. Therefore, we only consider the
coupling between heavy hadrons and light vector mesons and omit the Lagrangian
that results in potentials proportional to the square of transferred momentum,
$\bm{q}^{2}$.

###  II.1 Coupling of light vector mesons and heavy mesons

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:1987ge and the leading order Lagrangian is expressed as Casalbuoni
_et al._ (1993, 1997); Dong  _et al._ (2021a)

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

where $a,b$ are the SU(3) flavor indices and

$\displaystyle
V=\left(\begin{array}[]{ccc}\frac{\omega}{\sqrt{2}}+\frac{\rho^{0}}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&\frac{\omega}{\sqrt{2}}-\frac{\rho^{0}}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right),$ (5)

which satisfies $\mathcal{C}V\mathcal{C}^{-1}=-V^{T}$. The fields for heavy
mesons are collected into

$\displaystyle P^{(c)}$ $\displaystyle=(D^{0},D^{+},D_{s}^{+}),$
$\displaystyle P^{*(c)}$ $\displaystyle=(D^{*0},D^{*+},D^{*+}_{s}),$
$\displaystyle P_{1}^{(c)}$
$\displaystyle=(D_{1}(2420)^{0},D_{1}(2420)^{+},D_{s1}(2536)^{+}),$
$\displaystyle P_{2}^{(c)}$
$\displaystyle=(D_{2}(2460)^{0},D_{2}(2460)^{+},D_{s2}(2573)^{+}),$
$\displaystyle P^{(b)}$ $\displaystyle=(B^{+},B^{0},B_{s}^{0}),$
$\displaystyle P^{*(b)}$ $\displaystyle=(B^{*+},B^{*0},B^{*0}),$
$\displaystyle P_{1}^{(b)}$
$\displaystyle=(B_{1}(5721)^{+},B_{1}(5721)^{0},B_{s1}(5830)^{+}),$
$\displaystyle P_{2}^{(b)}$
$\displaystyle=(B_{2}(5747)^{+},B_{2}(5747)^{0},D_{s2}(5840)^{+}).$ (6)

Note that $P_{1}$ and $P_{2}$ are $P$-wave mesons where the total angular
momentum of light degrees of freedom is $s_{l}=3/2$. The $P$-wave mesons with
$s_{l}=1/2$ will not be considered here because the charmed ones are too wide
Zyla  _et al._ (2020) to form bound states Filin  _et al._ (2010); Guo and
Meißner (2011) or considered as molecular states Guo (2019) and the bottomed
ones have not been found experimentally.

###  II.2 Coupling of light vector mesons and 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),

$\displaystyle B^{(c)}_{\bar{3}}$
$\displaystyle=\left(\begin{array}[]{ccc}0&\Lambda_{c}^{+}&\Xi_{c}^{+}\\\
-\Lambda_{c}^{+}&0&\Xi_{c}^{0}\\\
-\Xi_{c}^{+}&-\Xi_{c}^{0}&0\end{array}\right),$ (10) $\displaystyle
B^{(c)}_{6}$
$\displaystyle=\left(\begin{array}[]{ccc}\Sigma_{c}^{++}&\frac{1}{\sqrt{2}}\Sigma_{c}^{+}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime+}\\\
\frac{1}{\sqrt{2}}\Sigma_{c}^{+}&\Sigma_{c}^{0}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime
0}\\\ \frac{1}{\sqrt{2}}\Xi_{c}^{\prime+}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime
0}&\Omega_{c}^{0}\end{array}\right),$ (14) $\displaystyle B_{6}^{(c)*}$
$\displaystyle=\left(\begin{array}[]{ccc}\Sigma_{c}^{*++}&\frac{1}{\sqrt{2}}\Sigma_{c}^{*+}&\frac{1}{\sqrt{2}}\Xi_{c}^{*+}\\\
\frac{1}{\sqrt{2}}\Sigma_{c}^{*+}&\Sigma_{c}^{*0}&\frac{1}{\sqrt{2}}\Xi_{c}^{*0}\\\
\frac{1}{\sqrt{2}}\Xi_{c}^{*+}&\frac{1}{\sqrt{2}}\Xi_{c}^{*0}&\Omega_{c}^{*0}\end{array}\right),$
(18) $\displaystyle B^{(b)}_{\bar{3}}$
$\displaystyle=\left(\begin{array}[]{ccc}0&\Lambda_{b}^{0}&\Xi_{b}^{0}\\\
-\Lambda_{b}^{0}&0&\Xi_{b}^{-}\\\
-\Xi_{b}^{0}&-\Xi_{b}^{-}&0\end{array}\right),$ (22) $\displaystyle
B^{(b)}_{6}$
$\displaystyle=\left(\begin{array}[]{ccc}\Sigma_{b}^{+}&\frac{1}{\sqrt{2}}\Sigma_{b}^{0}&\frac{1}{\sqrt{2}}\Xi_{b}^{\prime
0}\\\
\frac{1}{\sqrt{2}}\Sigma_{b}^{0}&\Sigma_{b}^{-}&\frac{1}{\sqrt{2}}\Xi_{b}^{\prime-}\\\
\frac{1}{\sqrt{2}}\Xi_{b}^{\prime
0}&\frac{1}{\sqrt{2}}\Xi_{b}^{\prime-}&\Omega_{b}^{-}\end{array}\right),$ (26)
$\displaystyle B^{(b)*}_{6}$
$\displaystyle=\left(\begin{array}[]{ccc}\Sigma_{b}^{*+}&\frac{1}{\sqrt{2}}\Sigma_{b}^{*0}&\frac{1}{\sqrt{2}}\Xi_{b}^{*0}\\\
\frac{1}{\sqrt{2}}\Sigma_{b}^{*0}&\Sigma_{b}^{*-}&\frac{1}{\sqrt{2}}\Xi_{b}^{*-}\\\
\frac{1}{\sqrt{2}}\Xi_{b}^{*0}&\frac{1}{\sqrt{2}}\Xi_{b}^{*-}&\Omega_{b}^{*-}\end{array}\right).$
(30)

Note that the heavy baryon multiplets have not been completely established
experimentally and the experimental candidates in RPP Zyla  _et al._ (2020) of
the above baryons predicted by quark model are listed in Table 1. The isospin
averaged masses will be used in this work and no isospin breaking will be
considered. The $\Sigma_{b}^{(*)0}$, $\Xi_{b}^{\prime 0}$ and
$\Omega_{b}^{*0}$ have no experimental candidates, and thus we use
$m_{\Sigma_{b}^{(*)}}=\left(m_{\Sigma_{b}^{(*)+}}+m_{\Sigma_{b}^{(*)-}}\right)/2$
and $m_{\Xi_{b}^{\prime}}=m_{\Xi_{b}^{\prime-}}$.

Table 1: The experimental candidates of heavy baryons predicted by quark model. The notations of experimental states are taken from RPP Zyla  _et al._ (2020). The $\Sigma_{b}^{(*)0}$, $\Xi_{b}^{\prime 0}$ and $\Omega_{b}^{*0}$ do not have experimental candidates yet. model |  experimental | model |  experimental
---|---|---|---
$\Lambda_{c}^{+}$ | $\Lambda_{c}^{+}$ | $\Lambda_{b}^{0}$ | $\Lambda_{b}^{0}$
$\Xi_{c}^{+}$ | $\Xi_{c}^{+}$ | $\Xi_{b}^{0}$ | $\Xi_{b}^{0}$
$\Xi_{c}^{0}$ | $\Xi_{c}^{0}$ | $\Xi_{b}^{-}$ | $\Xi_{b}^{-}$
$\Sigma_{c}$ | $\Sigma_{c}(2455)$ | $\Sigma_{b}$ | $\Sigma_{b}$
$\Xi_{c}^{\prime+}$ | $\Xi_{c}^{\prime+}$ | $\Xi_{b}^{\prime 0}$ | $-$
$\Xi_{c}^{\prime 0}$ | $\Xi_{c}^{\prime 0}$ | $\Xi_{b}^{\prime-}$ | $\Xi_{b}^{\prime}(5935)^{-}$
$\Omega_{c}^{0}$ | $\Omega_{c}^{0}$ | $\Omega_{b}^{-}$ | $\Omega_{b}^{-}$
$\Sigma_{c}^{*}$ | $\Sigma_{c}(2520)$ | $\Sigma_{b}^{*}$ | $\Sigma_{b}^{*}$
$\Xi_{c}^{*+}$ | $\Xi_{c}(2645)^{+}$ | $\Xi_{b}^{*0}$ | $\Xi_{b}(5945)^{0}$
$\Xi_{c}^{*0}$ | $\Xi_{c}(2645)^{0}$ | $\Xi_{b}^{*-}$ | $\Xi_{b}(5955)^{-}$
$\Omega_{c}^{*0}$ | $\Omega_{c}(2770)^{0}$ | $\Omega_{b}^{*0}$ | $-$

The Lagrangian for the coupling of heavy baryons and light mesons is
constructed as Liu and Oka (2012); Dong  _et al._ (2021a)

$\displaystyle\mathcal{L}_{BBV}$
$\displaystyle=\,i\beta_{B}\mathrm{tr}\left[\bar{B}^{(Q)}_{\bar{3}}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)B^{(Q)}_{\bar{3}}\right]$
$\displaystyle-i\beta_{B}\mathrm{tr}\left[\bar{B}^{(\bar{Q})}_{{3}}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)^{T}B^{(\bar{Q})}_{{3}}\right]$
$\displaystyle+i\beta_{S}\operatorname{tr}\left[\bar{S}_{\nu}^{(Q)}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)S^{(Q)\nu}\right]$
$\displaystyle-i\beta_{S}\operatorname{tr}\left[\bar{S}_{\nu}^{(\bar{Q})}v^{\mu}\left(\mathcal{V}_{\mu}-\rho_{\mu}\right)^{T}S^{(\bar{Q})\nu}\right],$
(31)

where the heavy baryons are expressed as bispinors,

$\displaystyle S^{(Q)}_{\mu}$
$\displaystyle=B_{6\mu}^{(Q)*}-\frac{1}{\sqrt{3}}\left(\gamma_{\mu}+v_{\mu}\right)\gamma^{5}B^{(Q)}_{6},$
(32) $\displaystyle\bar{S}^{(Q)}_{\mu}$
$\displaystyle=\bar{B}_{6\mu}^{(Q)*}+\frac{1}{\sqrt{3}}\bar{B}^{(Q)}_{6}\gamma^{5}\left(\gamma_{\mu}+v_{\mu}\right),$
(33)

$v_{\mu}$ is the four-velocity of the heavy field, and tr denotes the traces
over both the spinor and light flavor spaces.

##  III Molecular states from resonance-saturated constant interactions

In the following we will solve the Bethe-Salpeter equation $T=V+VGT$ Oller and
Oset (1997) to search for poles of the scattering amplitude $T$. The
interaction kernel (potential) $V$ is defined as $V=-\mathcal{M}$ with
$\mathcal{M}$ the $2\to 2$ invariant scattering amplitude so that a negative
$V$ means an attraction interaction. Such a potential is also the same as the
nonrelativistic potential in the Schrödinger equation up to a mass factor.

Table 2: 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._ (2020) | Dong  _et al._ (2020) | Liu and Oka (2012); Chen  _et al._ (2019) | Liu and Oka (2012); Chen  _et al._ (2019)

In Table 2, we list the numerical values of the coupling constants used in
this work with the corresponding references, which have been used in our
previous work Dong  _et al._ (2021a). Note that the signs of $\beta_{B}$ and
$\beta_{S}$ adapted in this work are different from those in Ref. Liu and Oka
(2012), the choice of which is in conflict with the molecular interpretation
of the famous $P_{c}$ states as well as those obtained by flavor SU(4)
relations Wu  _et al._ (2011).

###  III.1 Potentials from light vector meson exchange

The constant potentials of different systems assuming the saturation of the
light vector meson exchange can be expressed uniformly as333 For the systems
composed of two identical particles, such as $DD$, the symmetry factor
$\frac{1}{2}$ cancels the additional $u-$channel contribution and finally this
equation holds valid.

$V\approx-F\tilde{\beta}_{1}\tilde{\beta}_{2}g_{V}^{2}\frac{2m_{1}m_{2}}{m_{\rm
ex}^{2}},$ (34)

with $m_{1},m_{2}$ and $m_{\rm ex}$ 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 are explicitly given as terms of the couplings in Eqs.
(1) and (31) as

  * •

$\tilde{\beta}_{i}=\beta$ 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 charmed baryons,

  * •

and $\tilde{\beta}_{i}=-\beta_{S}/2$ for the sextet charmed 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 $F$ for charmed-(anti)charmed and
bottomed-(anti)bottomed systems are listed in Tables 7, 8, 9 and 10 in
Appendix A for all combinations of heavy-(anti)heavy hadron pairs (the ones
for the heavy-antiheavy systems have been given in our previous work Dong  _et
al._ (2021a)). Notice that we use the usual relativistic normalization for all
the involved fields, i.e., a factor of $m_{1}m_{2}$ has been multiplied to the
amplitude derived from the Lagrangian in Eqs. (1, 31) so as to get $V$ in Eq.
(34).

It is important to note that the isoscalar meson exchange yields potentials
with opposite signs for heavy-heavy and heavy-antiheavy systems while the
isovector exchange leads to potentials with the same sign. Such observations
are confirmed formally in Appendix A.

Table 3: Pole positions of double-charm-hadron systems with $I=0$ and $P=+$. $E_{\rm th}$ in the second column is the threshold in MeV. The results as given in the last columns corresponds to using the cutoff $\Lambda=0.5$ ($1.0$) GeV for Eq. (38) used to determine the subtraction constant $a(\mu)$ in Eq. (36), respectively. In the last two columns, the first number in the parenthesis refers to the Riemann sheet (RS) where the pole is located while the second number means the distance between the pole position and the corresponding threshold, namely, $E_{B}\equiv E_{\rm th}-E_{\rm pole}$, in units of MeV. System |  $E_{\rm th}$ [MeV] | $J^{P}$ | (RS, $E_{B}$ [MeV])
---|---|---|---
|  |  |  $0.5$ GeV | 1.0 GeV
$DD^{*}$ | 3876 | $1^{+}$ | (2, 3.58) | (1, 5.96)
$D^{*}D^{*}$ | 4017 | $1^{+}$ | (2, 2.68) | (1, 7.07)
$D_{1}D_{1}$ | 4844 | $1^{+}$ | (2, 0.321) | (1, 12.2)
$D_{1}D_{2}$ | 4885 | $(1,2,3)^{+}$ | (2, 0.277) | (1, 12.4)
$D_{2}D_{2}$ | 4926 | $(1,3)^{+}$ | (2, 0.237) | (1, 12.6)
$\Sigma_{c}{\Sigma}_{c}$ | 4907 | $0^{+}$ | (1, 2.72) | (1, 35.2)
$\Xi_{c}{\Xi}_{c}$ | 4939 | $1^{+}$ | (2, 43.4) | (2, 10.1)
$\Sigma_{c}{\Sigma}_{c}^{*}$ | 4972 | $(1,2)^{+}$ | (1, 2.79) | (1, 35.1)
$\Sigma_{c}^{*}{\Sigma}_{c}^{*}$ | 5036 | $(0,2)^{+}$ | (1, 2.86) | (1, 35.1)
$\Xi_{c}{\Xi}_{c}^{\prime}$ | 5048 | $(0,1)^{+}$ | (2, 40.1) | (2, 8.55)
$\Xi_{c}{\Xi}_{c}^{*}$ | 5115 | $(1,2)^{+}$ | (2, 38.3) | (2, 7.73)
$\Xi_{c}^{\prime}{\Xi}_{c}^{\prime}$ | 5158 | $1^{+}$ | (2, 36.9) | (2, 7.14)
$\Xi_{c}^{*}{\Xi}_{c}^{\prime}$ | 5225 | $(1,2)^{+}$ | (2, 35.2) | (2, 6.4)
$\Xi_{c}^{*}{\Xi}_{c}^{*}$ | 5292 | $(1,3)^{+}$ | (2, 33.4) | (2, 5.7)
${D}_{1}\Xi_{c}$ | 4891 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (1, 2.78) | (1, 35.5)
${D}_{2}\Xi_{c}$ | 4932 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (1, 2.83) | (1, 35.5)
${D}_{1}\Xi_{c}^{\prime}$ | 5001 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (1, 2.89) | (1, 35.4)
${D}_{2}\Xi_{c}^{\prime}$ | 5042 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (1, 2.94) | (1, 35.4)
${D}_{1}\Xi_{c}^{*}$ | 5068 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{+}$ | (1, 2.96) | (1, 35.3)
${D}_{2}\Xi_{c}^{*}$ | 5109 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2})^{+}$ | (1, 3.0) | (1, 35.3)
$D_{s1}{\Omega}_{c}$ | 5230 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (1, 0.0298) | (1, 17.4)
$D_{s2}{\Omega}_{c}$ | 5264 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (1, 0.039) | (1, 17.5)
$D_{s1}{\Omega}_{c}^{*}$ | 5301 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{+}$ | (1, 0.0474) | (1, 17.6)
$D_{s2}{\Omega}_{c}^{*}$ | 5335 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2})^{+}$ | (1, 0.0588) | (1, 17.7)
Table 4: Pole positions of double-charm-hadron systems with $I=0$ and $P=-$. See the caption for Table 3. System |  $E_{\rm th}$ [MeV] | $J^{P}$ | (RS, $E_{B}$ [MeV])
---|---|---|---
|  |  |  $0.5$ GeV | 1.0 GeV
$D\ {D}_{1}$ | 4289 | $1^{-}$ | (2, 2.48) | (1, 6.94)
$D\ {D}_{2}$ | 4330 | $2^{-}$ | (2, 1.65) | (1, 8.69)
$D^{*}{D}_{1}$ | 4431 | $(0,1,2)^{-}$ | (2, 1.35) | (1, 9.12)
$D^{*}{D}_{2}$ | 4472 | $(1,2,3)^{-}$ | (2, 1.0) | (1, 10.1)
${D}\ \Xi_{c}$ | 4337 | $\frac{1}{2}^{-}$ | (1, 1.92) | (1, 35.3)
${D}\ \Xi_{c}^{\prime}$ | 4446 | $\frac{1}{2}^{-}$ | (1, 2.04) | (1, 35.4)
${D}^{*}\Xi_{c}$ | 4478 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (1, 2.19) | (1, 35.5)
${D}\ \Xi_{c}^{*}$ | 4513 | $\frac{3}{2}^{-}$ | (1, 2.11) | (1, 35.4)
${D}^{*}\Xi_{c}^{\prime}$ | 4587 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (1, 2.31) | (1, 35.5)
${D}^{*}\Xi_{c}^{*}$ | 4655 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{-}$ | (1, 2.38) | (1, 35.5)
$D_{s}{\Omega}_{c}$ | 4664 | $\frac{1}{2}^{-}$ | (2, 0.168) | (1, 14.3)
$D_{s}{\Omega}_{c}^{*}$ | 4734 | $\frac{3}{2}^{-}$ | (2, 0.129) | (1, 14.6)
$D_{s}^{*}{\Omega}_{c}^{\prime}$ | 4807 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (2, 0.0507) | (1, 15.3)
$D_{s}^{*}{\Omega}_{c}^{*}$ | 4878 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{-}$ | (2, 0.0308) | (1, 15.6)
Table 5: Pole positions of double-charm-hadron systems with $I=1/2$ and $P=-$. See the caption for Table 3. System |  $E_{\rm th}$ [MeV] | $J^{P}$ | (RS, $E_{B}$ [MeV])
---|---|---|---
|  |  |  $0.5$ GeV | 1.0 GeV
${D}\ \Lambda_{c}$ | 4154 | $\frac{1}{2}^{-}$ | (2, 3.44) | (1, 5.62)
${D}^{*}\Lambda_{c}$ | 4295 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (2, 2.53) | (1, 6.73)
${D}\ \Sigma_{c}$ | 4321 | $\frac{1}{2}^{-}$ | (1, 5.81) | (1, 50.5)
${D}\ \Sigma_{c}^{*}$ | 4385 | $\frac{3}{2}^{-}$ | (1, 5.85) | (1, 50.2)
${D}^{*}\Sigma_{c}$ | 4462 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (1, 5.97) | (1, 49.7)
${D}^{*}\Sigma_{c}^{*}$ | 4527 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{-}$ | (1, 6.01) | (1, 49.5)
$D_{s}{\Xi}_{c}$ | 4438 | $\frac{1}{2}^{-}$ | (2, 25.7) | (2, 1.76)
$D_{s}{\Xi}_{c}^{\prime}$ | 4547 | $\frac{1}{2}^{-}$ | (2, 23.7) | (2, 1.29)
$D_{s}^{*}{\Xi}_{c}$ | 4582 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (2, 21.8) | (2, 0.882)
$D_{s}{\Xi}_{c}^{*}$ | 4614 | $\frac{3}{2}^{-}$ | (2, 22.6) | (2, 1.05)
$D_{s}^{*}{\Xi}_{c}^{\prime}$ | 4691 | $(\frac{1}{2},\frac{3}{2})^{-}$ | (2, 20.0) | (2, 0.564)
$D_{s}^{*}{\Xi}_{c}^{*}$ | 4758 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{-}$ | (2, 19.0) | (2, 0.416)
Table 6: Pole positions of double-charm-hadron systems with $I=1/2$ and $P=+$. See the caption for Table 3. System |  $E_{\rm th}$ [MeV] | $J^{P}$ | (RS, $E_{B}$ [MeV])
---|---|---|---
|  |  |  $0.5$ GeV | 1.0 GeV
${D}_{1}\Lambda_{c}$ | 4708 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (2, 1.04) | (1, 9.31)
${D}_{2}\Lambda_{c}$ | 4750 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (2, 0.95) | (1, 9.51)
${D}_{1}\Sigma_{c}$ | 4876 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (1, 6.25) | (1, 47.5)
${D}_{2}\Sigma_{c}$ | 4917 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (1, 6.27) | (1, 47.3)
${D}_{1}\Sigma_{c}^{*}$ | 4940 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{+}$ | (1, 6.28) | (1, 47.2)
${D}_{2}\Sigma_{c}^{*}$ | 4981 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2})^{+}$ | (1, 6.29) | (1, 47.0)
$D_{s1}{\Xi}_{c}$ | 5005 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (2, 14.2) | (2, 0.00911)
$D_{s2}{\Xi}_{c}$ | 5039 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (2, 13.8) | (2, 0.00176)
$D_{s1}{\Xi}_{c}^{\prime}$ | 5114 | $(\frac{1}{2},\frac{3}{2})^{+}$ | (2, 12.8) | (1, 0.00636)
$D_{s2}{\Xi}_{c}^{\prime}$ | 5148 | $(\frac{3}{2},\frac{5}{2})^{+}$ | (2, 12.4) | (1, 0.0175)
$D_{s1}{\Xi}_{c}^{*}$ | 5181 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2})^{+}$ | (2, 12.0) | (1, 0.0319)
$D_{s2}{\Xi}_{c}^{*}$ | 5215 | $(\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2})^{+}$ | (2, 11.6) | (1, 0.0532)

###  III.2 Poles of molecular states

Given the constant interactions between a pair of heavy-heavy hadrons, we
solve the single channel BS equation that is factorized into an algebraic
equation,

$\displaystyle T=\frac{V}{1-VG},$ (35)

to search for poles of the scattering amplitude and in turn to give a rough
but overall picture of the spectrum of possible molecular states. Here $G$ is
the one loop two-body propagator that after the dimensional regularization
(DR) Veltman (2012) reads444There are typos in the expression of $G(E)$ in the
published version of the previous work Dong  _et al._ (2021a).

$\displaystyle G(E)=$
$\displaystyle\frac{1}{16\pi^{2}}\bigg{\\{}a(\mu)+\log\frac{m_{1}^{2}}{\mu^{2}}+\frac{m_{2}^{2}-m_{1}^{2}+s}{2s}\log\frac{m_{2}^{2}}{m_{1}^{2}}$
$\displaystyle+\frac{k}{E}\Big{[}\log\left(2kE+s+\Delta\right)+\log\left(2kE+s-\Delta\right)$
$\displaystyle-\log\left(2kE-s+\Delta\right)-\log\left(2kE-s-\Delta\right)\Big{]}\bigg{\\}},$
(36)

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

$\displaystyle k=\frac{1}{2E}\lambda^{1/2}(E^{2},m_{1}^{2},m_{2}^{2})$ (37)

is the corresponding three-momentum magnitude with
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2xz$ for the Källén triangle
function. The DR scale $\mu$ is chosen to be 1 GeV and its variation can be
absorbed into the subtraction constant $a(\mu)$. For the single channel case
considered in our paper, there are two Riemann sheets (RSs) that are 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 to introduce a Gaussian form
factor, namely555 Note that unlike the dimensional regularization, this
Gaussian form factor will violate unitarity, i.e. ${\rm
Im}\left[T^{-1}(E)\right]=-i\rho(E)$ with $\rho(E)$ the two body phase space
factor, if the same constant contact term is used. However, if we focus on the
near threshold bound or virtual states, such a difference between Eq. (36) and
Eq. (38) is negligible.,

$\displaystyle G(E)=$
$\displaystyle\,\int\frac{l^{2}dl}{4\pi^{2}}\frac{\omega_{1}+\omega_{2}}{\omega_{1}\omega_{2}}\frac{e^{-2l^{2}/\Lambda^{2}}}{E^{2}-(\omega_{1}+\omega_{2})^{2}+i\epsilon},$
(38)

with $\omega_{i}=\sqrt{m_{i}^{2}+l^{2}}$. The cutoff $\Lambda$ is chosen in
the range of $0.5\sim 1.0$ GeV, which is believed phenomenologically adequate
Epelbaum  _et al._ (2009); Nieves and Valderrama (2012); Guo  _et al._ (2013),
and then the subtraction constant $a(\mu)$ in DR is determined by matching the
values of $G$ from these two methods at threshold. In the following we will
use the DR loop with the so-determined subtraction constant for numerical
calculations.

In Tables 3, 4, 5 and 6, we list all the pole positions of the double-charm-
hadron systems which have attractive interactions, corresponding to the masses
of hadronic molecules (bound states on 1st RS or virtual states on 2nd RS).
For better illustration, these states are also shown in Figs. 1, 2, 3, 4, 5
and 6 together with the corresponding thresholds. Considering the constant
contact interactions saturated by the light vector meson exchange with the
coupled-channel effects neglected, we obtain a spectrum of 124 hadronic
molecules in total. At least the same number of molecules are expected to
exist for each of the charm-bottom and bottom-bottom systems since it is
easier to form a bound state with the same attraction strength due to the
heavier reduced masses; there could be even more as if the ground state is
deeply bound excited states might exist as well as illustrated in the Jülich
meson-exchange model for hidden-bottom pentaquark-like hadronic molecules in
Ref. Shen  _et al._ (2018).

Figure 1: The spectrum of hadronic molecules consisting of a pair of charmed
mesons or baryons with $I=0$ and $P=+$. 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 the cutoff $\Lambda$ varying 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, e.g., $DD^{*}$, there are two
rectangles for one system, with the upper edges exactly at the threshold. This
corresponds to the situation that the pole moves from the second RS (left
orange) to the first RS (right green) when $\Lambda$ increases in the
considered range. In some other 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.  Figure 2: The spectrum of hadronic
molecules consisting of a pair of charmed mesons with $I=0$ and $P=-$. See the
caption for Fig. 1.

Figure 3: The spectrum of hadronic molecules consisting of a pair of charmed
meson and charmed baryon with $I=1/2$ and $P=-$. See the caption for Fig. 1.
The right part of the dashed line for $D^{*}\Lambda_{c}$ marks the real
threshold while the left part is deformed to avoid being covered by the
rectangle of $D\Sigma_{c}$ system.

Figure 4: The spectrum of hadronic molecules consisting of a pair of charmed
meson and charmed baryon with $I=0$ and $P=-$. See the caption for Fig. 1.

Figure 5: The spectrum of hadronic molecules consisting of a pair of charmed
meson and charmed baryon with $I=1/2$ and $P=+$. See the caption for Fig. 1.

Figure 6: The spectrum of hadronic molecules consisting of a pair of charmed
meson and charmed baryon with $I=0$ and $P=+$. See the caption for Fig. 1.

Besides the predicted molecular states mentioned above, where only
$\omega,\rho$ and $\phi$ exchanges are considered, we need pay some extra
attentions to the $K^{*}$ exchange, which is absent in the heavy-antiheavy
systems due to the sizeable symmetry breaking between $s$ quark and $u/d$
quark. In most cases, $K^{*}$ exchange, if allowed, will contribute a
repulsive potential, see Table 7, unless two channels are close enough and can
be linearly combined to an isospin-like (called $U/V-$spin in Ref. Meng  _et
al._ (2020)) eigenstates. Explicitly, the thresholds of $DD^{*}_{s}$ and
$D^{*}D_{s}$ are close enough and they may form two $U/V-$spin eigenstates,

$\displaystyle U/V=\pm:DD_{s}^{*}\pm D^{*}D_{s}$ (39)

The potentials of $K^{*}$ exchange are repulsive and attractive, respectively,
the strength of which can be also described by Eq. (34) with $|F|=1$ and
$m_{\rm ex}=m_{K^{*}}$. Therefore, we expect a near threshold virtual or bound
states in the $U/V=-1$ systems of $DD_{s}^{*}$. Some other similar systems,
such as $\Sigma_{c}\Xi^{*}_{c}$ and $\Sigma_{c}^{*}\Xi_{c}$, the thresholds
are not that close and more efforts should be paid to the coupled channel
efforts, which is beyond the scope of this work.

##  IV Discussions of selected systems

###  IV.1 Heavy meson-meson molecules v.s. doubly heavy tetraquarks

From Table 7, one sees that the attractions strength for the isoscalar
$D^{(*)}D^{*}$ is half of that for the isoscalar $C=+$ $D^{(*)}\bar{D}^{*}$
pairs.666For the isoscalar systems,
$V_{D^{(*)}D^{*}}=\frac{3}{2}V_{\rho}-\frac{1}{2}V_{\omega}$, which is about
half of $V_{D^{(*)}\bar{D}^{*}}=\frac{3}{2}V_{\rho}+\frac{1}{2}V_{\omega}$
considering $V_{\rho}\approx V_{\omega}$, where $V_{\rho}$ and $V_{\omega}$
are the potentials due to the exchange of $\rho$ and $\omega$, respectively.
Thus, for the $X(3872)$ as a $D\bar{D}^{*}$ bound state with
$J^{PC}=1^{++}$,777Notice that the narrow structure $X(4014)$ reported
recently by Belle Wang  _et al._ (2021a) is an excellent candidate for the
$2^{++}$ $D^{*}\bar{D}^{*}$ molecule Tornqvist (1994); Molina and Oset (2009);
Nieves and Valderrama (2012); Guo  _et al._ (2013); Albaladejo  _et al._
(2015); Baru  _et al._ (2016); Dong  _et al._ (2021a), and its mass
$(4014.4\pm 4.1\pm 0.5)$ MeV and width $(6\pm 16\pm 6)$ MeV nicely agree with
the predictions in Ref. Albaladejo  _et al._ (2015). The $D\bar{D}$ bound
state obtained in the same way Dong  _et al._ (2021a) (for early predictions,
see Refs. Zhang  _et al._ (2006); Gamermann  _et al._ (2007); Liu  _et al._
(2009); Wong (2004); Nieves and Valderrama (2012); Hidalgo-Duque  _et al._
(2013)) also receives support from a recent lattice QCD calculation Prelovsek
_et al._ (2021).  the $D^{(*)}D^{*}$ system would be less. From Fig. 2 we see
indeed that the isoscalar $D^{(*)}D^{*}$ systems are at the edge of forming
bound states. While for the isovector ones, the potential from $\rho$ and
$\omega$ exchange is repulsive and thus no molecules are expected. The
situations for $D^{(*)}D_{(1,2)}$ and $D_{(1,2)}D_{(1,2)}$ systems are all
similar.

The $T_{cc}^{+}$ state was recently observed in the invariant mass
distribution of $D^{0}D^{0}\pi^{+}$ by LHCb Aaij  _et al._ (2021a, b). The
pole from an analysis using a unitarized Breit-Wigner parameterization,
considering the momentum-dependent width of the $D^{*}$ from its decays, is
located in the complex energy plane at

$\displaystyle-360\pm 40^{+4}_{-0}-i\,(24\pm 2^{+0}_{-7})~{}{\rm keV},$ (40)

with the real part defined as the energy relative to the $D^{0}D^{*+}$
threshold Aaij  _et al._ (2021b). 888A simple Breit-Wigner parametrization of
the peak convolved with energy resolution leads to a width of $(410\pm 165)$
keV Aaij  _et al._ (2021a). Note that the $T_{cc}^{+}$ is very close to the
$D^{0}D^{*+}$ threshold, and thus the Flatté parameterization is more proper
than the Breit-Wigner one; the former leads to a asymmetric line shape
automatically while the latter does not. It is also worthwhile to notice that
the width of a Flatté line shape can be much smaller than that of the Breit-
Wigner one when convolved with the energy resolution, as nicely illustrated in
the careful analysis of the $X(3872)$ line shape by the LHCb Collaboration
Aaij  _et al._ (2020).  This new state is a good candidate of the isoscalar
$DD^{*}$ molecule, in agreement with our result. Before this experiment
signal, lots of predictions of such doubly charmed tetraquark states, being
either hadronic molecules Janc and Rosina (2004); Ohkoda  _et al._ (2012); Li
_et al._ (2013); Liu  _et al._ (2019b, 2020); Yang  _et al._ (2009) or compact
tetraquark states Carlson  _et al._ (1988); Silvestre-Brac and Semay (1993a,
b); Semay and Silvestre-Brac (1994); Gelman and Nussinov (2003); Vijande  _et
al._ (2004); Navarra  _et al._ (2007); Ebert  _et al._ (2007); Vijande  _et
al._ (2007); Lee and Yasui (2009); Abud  _et al._ (2010); Yang  _et al._
(2009); Karliner and Nussinov (2013); Feng  _et al._ (2013); Luo  _et al._
(2017); Karliner and Rosner (2017); Eichten and Quigg (2017); Wang (2018a);
Hyodo  _et al._ (2017); Cheung  _et al._ (2017); Park  _et al._ (2019a);
Junnarkar  _et al._ (2019); Deng  _et al._ (2020); Yang  _et al._ (2020b); Tan
_et al._ (2020); Lü  _et al._ (2020); Braaten  _et al._ (2021); Gao  _et al._
(2020); Cheng  _et al._ (2021); Noh  _et al._ (2021); Faustov  _et al._
(2021), have been made with various methods.

####  IV.1.1 Heavy meson-meson molecules

The doubly heavy tetraquark states in molecular configurations were widely
investigated in the literature Carlson  _et al._ (1988); Richards  _et al._
(1990); Manohar and Wise (1993); Tornqvist (1994); Mihaly  _et al._ (1997);
Stewart and Koniuk (1998); Barnes  _et al._ (1999); Janc and Rosina (2004);
Yang  _et al._ (2009); Molina  _et al._ (2010); Carames  _et al._ (2011);
Ohkoda  _et al._ (2012); Li  _et al._ (2013); Sun  _et al._ (2012, 2012); Xu
_et al._ (2019); Caramés  _et al._ (2019); Michael and Pennanen (1999);
Pennanen  _et al._ (2000); Wagner (2011); Guerrieri  _et al._ (2015); Ikeda
_et al._ (2014); Detmold  _et al._ (2007); Bali and Hetzenegger (2011); Brown
and Orginos (2012); Bicudo and Wagner (2013); Bicudo  _et al._ (2015, 2016);
Bicudo and Wagner (2013); Brown and Orginos (2012); Bicudo  _et al._ (2015,
2016, 2017a); Karliner and Rosner (2015); Sanchez Sanchez  _et al._ (2018);
Sakai  _et al._ (2017); Wang  _et al._ (2019); Yu  _et al._ (2020); Liu  _et
al._ (2019b); Ding  _et al._ (2020); Liu  _et al._ (2020); Meng  _et al._
(2021b); Ding  _et al._ (2021) where predictions of many doubly heavy
tetraquark states including the $T_{cc}$ mentioned above as well as other
systems were made.

The interaction between a pair of heavy mesons was estimated using the Born-
Oppenheimer approximation in the MIT bag model in Ref. Carlson  _et al._
(1988). The $I(J^{P})=0(1^{+})$ di-meson $T(bb\bar{q}\bar{q})$ was found to be
a bound state about 70 MeV below the $BB^{*}$ threshold while the situations
of $T(bc\bar{q}\bar{q})$ and $T(cc\bar{q}\bar{q})$ were uncertain.

By solving the double-charm tetraquark system with two realistic potential
models, it was found in Ref. Janc and Rosina (2004) that the ground state
tetraquark state has a configuration of $DD^{*}$ molecule. In the constituent
quark model, it was also found that the di-meson configurations of
$QQ\bar{q}\bar{q}$ can be bound Yang  _et al._ (2009).

In Ref. Yu  _et al._ (2020), the $B^{(*)}B^{(*)}$ scattering amplitude was
explored in the constituent interchange model and the $I(J^{P})=0(1^{+})$
$BB^{*}$ and $B^{*}B^{*}$ bound states together with virtual states with some
other quantum numbers were found.

In Ref. Manohar and Wise (1993), it was found that the long-range potential
from one-pion exchange may be attractive enough to bind the $BB^{*}$ system
but insufficient for the $DD^{*}$ system to bind, which has a smaller reduced
mass. The existence of $DD^{*}$ was also disfavored by the one pion exchange
in Ref. Tornqvist (1994). While the one-boson exchange model predicts that the
isoscalar $DD^{*}$ system can form an $S$-wave bound state with a binding
energy of $62.3$ MeV (depending on the cutoff) with the $\pi,\rho$ and
$\omega$ exchanges Ohkoda  _et al._ (2012), $3\sim 40$ MeV with the $\pi,\rho$
and $\omega$ exchanges Liu  _et al._ (2020), $0.47\sim 43$ MeV (depending on
the cutoff) after including the $\pi,\sigma,\rho$ and $\omega$ exchanges and
coupled channel effects Li  _et al._ (2013) or $3^{+15}_{-~{}4}$ MeV without
coupled channels but with the cutoff fixed by producing the correct binding
energy of $X(3872)$ Liu  _et al._ (2019b). Besides, some other doubly heavy
molecules including $D_{(s)}^{(*)}D_{(s)}^{(*)}$,
$\bar{B}_{(s)}^{*}\bar{B}_{(s)}^{(*)}$ and $D^{(*)}\bar{B}^{(*)}$ with
different quantum numbers are predicted in these works as well as in Ref. Ding
_et al._ (2021). In Ref. Xu  _et al._ (2019), the potential between $DD^{*}$
from the one-pion exchange supplemented by the contact term and the exchange
of two pions was investigated in a chiral effective field theory, and an
isovector state was found to be bound while the scalar one was not.

With the potential from a chiral constituent quark model, the
Lippmann–Schwinger equation was solved for the $DD$-$DD^{*}$-$D^{*}D^{*}$
coupled channels in Ref. Carames  _et al._ (2011) and a stable doubly charmed
meson with $I(J^{P})=0(1^{+})$ was predicted. In Ref. Caramés  _et al._
(2019), the similar strategy yielded isoscalar $0^{+}$ and $1^{+}$
$bc\bar{q}\bar{q}$ bound states, which are stable against strong interaction,
but the isovector systems were found unbound.

We can notice that the results from these works, as well as those obtained
here, do not all agree with each other. There are at least two possible
reasons: the form of the potential, in particular the treatment of the short-
distance part, is different; the parameter values are different. In the spirit
of effective field theory, different treatments of the short-distance
potential correspond to taking different values for the contact terms. In any
case, most of the literature tends to agree that it is easier for the
isoscalar $DD^{*}/BB^{*}$ to bind than the isovector combinations, as is the
conclusion in our paper as well. Our model assumes the short-distance contact
terms are saturated by the light-vector-meson exchange, and the long- or mid-
range attractive potential from the $\pi$ or $\sigma$ exchanges and the
coupled channel effects may change the hadronic molecule spectrum
quantitatively to some extent.

There are also lattice calculations of the potential between a pair of heavy
mesons Richards  _et al._ (1990); Mihaly  _et al._ (1997); Stewart and Koniuk
(1998); Wagner (2011); Guerrieri  _et al._ (2015). The $DD^{(*)}$ interactions
were calculated on lattice Ikeda  _et al._ (2014) and it was found that the
potentials of the isovector systems are repulsive while those of the isoscalar
systems are attractive, qualitatively in line with the contact interactions
from the vector meson exchange reported here (see Table 7). The isoscalar
$B^{(*)}B^{(*)}$ interaction from lattice calculations Pennanen  _et al._
(2000); Detmold  _et al._ (2007); Bali and Hetzenegger (2011); Brown and
Orginos (2012); Bicudo and Wagner (2013); Bicudo  _et al._ (2015, 2016) in the
static $b$ quark limit is attractive in the short range while for the
isovector one, the attraction is weaker. It was found plausible for
$\bar{b}\bar{b}qq$ di-mesons to be stable under strong interactions Michael
and Pennanen (1999). In Refs. Bicudo and Wagner (2013); Brown and Orginos
(2012); Bicudo  _et al._ (2015, 2016, 2017a), the Schrödinger equation with
the obtained potential yields results that the $I(J^{P})=0(1^{+})$
$\bar{b}\bar{b}qq$ system has an attractive potential between two $B^{(*)}$
mesons strong enough to form bound states but not strong enough in the
isovector case. A recent analysis on lattice Bicudo  _et al._ (2021) shows
that the meson-meson component in the $I(J^{P})=0(1^{+})$ $\bar{b}\bar{b}ud$
state has a fraction around 60%.

####  IV.1.2 Compact tetraquark states

Besides the molecular assignment, many works have investigated the compact
doubly heavy tetraquarks via various methods, including quark potential models
Ader  _et al._ (1982); Ballot and Richard (1983); Zouzou  _et al._ (1986);
Heller and Tjon (1987); Brink and Stancu (1998); Silvestre-Brac and Semay
(1993b, a); Semay and Silvestre-Brac (1994); Schaffner-Bielich and Vischer
(1998); Czarnecki  _et al._ (2018); Vijande  _et al._ (2009); Hyodo  _et al._
(2013); Xing and Zhu (2018); Cui  _et al._ (2007); Silvestre-Brac (1992);
Hyodo  _et al._ (2017); Meng  _et al._ (2021c), quark models with heavy quark
symmetries Gelman and Nussinov (2003); Eichten and Quigg (2017); Karliner and
Rosner (2017), QCD sum rules Navarra  _et al._ (2007); Wang  _et al._ (2011);
Dias  _et al._ (2011); Chen  _et al._ (2014); Wang (2018a); Wang and Yan
(2018); Wang and Chen (2020); Agaev  _et al._ (2019); Tang  _et al._ (2020);
Agaev  _et al._ (2020a, b, 2021b); Wang and Chen (2020) and lattice QCD Green
and Pennanen (1998); Francis  _et al._ (2017); Cheung  _et al._ (2017);
Junnarkar  _et al._ (2019); Leskovec  _et al._ (2019); Hudspith  _et al._
(2020); Francis  _et al._ (2019); Mohanta and Basak (2020).

In the quark potential models, it was found that the $\bar{Q}\bar{Q}qq$ is
possible to be bound below the two-meson threshold for certain potentials Ader
_et al._ (1982); Ballot and Richard (1983); Zouzou  _et al._ (1986); Heller
and Tjon (1987) but no bound states can be found if all quarks have the same
mass, which was confirmed recently in a four-body calculation Czarnecki  _et
al._ (2018). It was also found Vijande  _et al._ (2009) that although the
possible configurations of tetraquarks proliferate, only five of the
candidates are stable, namely, $cc\bar{q}\bar{q}$ with
$J^{P}(L,S,I)=1^{+}(0,1,0)$ and $bb\bar{q}\bar{q}$ with
$J^{P}(L,S,I)=1^{+}(0,1,0),\ 3^{-}(1,2,1),\ 0^{+}(0,0,0)$ and $1^{-}(1,0,0)$,
where $L$ and $S$ refer to the orbital angular momentum and total spin of the
two mesons that couple to the tetraquark, among which the last one has a
molecular nature. The doubly charmed system was also confirmed to be bound in
Ref. Valcarce  _et al._ (2011) using different potential models while in Ref.
Pepin  _et al._ (1997) only one doubly-bottomed system with
$I(J^{P})=0(1^{+})$ was found to be located below the threshold of the
corresponding meson pair. With an adequate treatment of the four-body dynamics
in the quark model picture of tetraquark states, it was found in Ref. Richard
_et al._ (2018) that the $I(J^{P})=0(1^{+})$ $cc\bar{u}\bar{d}$ system is at
the edge of binding while the doubly-bottomed system is easier to be bound. On
the other hand, the analysis within the chiral SU(3) quark model Zhang  _et
al._ (2008) or the relativistic quark model Ebert  _et al._ (2007) found that
the $I(J^{P})=0(1^{+})$ $cc\bar{u}\bar{d}$ state is not bound but above the
thresholds for decays into open charm mesons. In Ref. Meng  _et al._ (2021b),
two bound states of $I(J^{P})=0(1^{+})$ $bb\bar{u}\bar{d}$ were found in a
constituent quark model, one deeply bound compact tetraquark and one $BB^{*}$
molecule.

In Ref. Lipkin (1986), the masses of tetraquark states were obtained model-
independently with known hadron masses as input and it was found that four-
quark states containing two identical heavy quarks have a good probability of
being stable against strong decay. In Ref. Karliner and Rosner (2017), the
ground states of hadrons are described by the quark model and therein the mass
of the predicted $J^{P}=1^{+}$ $cc\bar{u}\bar{d}$ state reads $(3882\pm 12)$
MeV using the measured $\Xi_{cc}^{++}$ mass Aaij  _et al._ (2017) as input,
which nicely covers the LHCb result. Ref. Karliner and Rosner (2017) also
predicts a $1^{+}$ $bb\bar{u}\bar{d}$ state to be at $(10389\pm 12)$ MeV,
which is well below the $BB^{*}$ threshold and thus stable under strong and
electromagnetic interactions.

By implementing the heavy antiquark-diquark symmetry Savage and Wise (1990),
the mass of doubly-heavy tetraquark states may be predicted using the relation

$\displaystyle m(QQ\bar{q}\bar{q})-m(QQq)\approx m(Qqq)-m(Q\bar{q}),$ (41)

the essence of which can be traced back to Refs. Eichten (1988); Lepage and
Thacker (1988). For instance, after the double-charm baryon $\Xi_{cc}^{++}$
was observed Aaij  _et al._ (2017), the lightest double-bottom tetraquark
state was predicted in Ref. Eichten and Quigg (2017) to be at 10482 MeV, which
is below the $BB^{*}$ threshold and stable under strong and electromagnetic
interactions, with $J^{P}=1^{+}$, using the doubly-bottom baryon masses in
Ref. Karliner and Rosner (2014) as input. The mass of $cc\bar{u}\bar{d}$ was
determined in this way to be 3845 or 3905 MeV Gelman and Nussinov (2003), 3978
MeV Eichten and Quigg (2017), $(3947\pm 11)$ MeV Braaten  _et al._ (2021),
3900 MeV Karliner and Nussinov (2013) and 3929 MeV Cheng  _et al._ (2021), see
also the discussion in Ref. Mehen (2017). The spectrum of some other doubly-
heavy tetraquark states were also obtained in these works, including stable
$bb\bar{q}\bar{q}$ tetraquark states. Note that different from the case of
doubly-heavy baryons, where the heavy diquark must be in a color anti-triplet,
the two heavy quarks inside a doubly-heavy tetraquark can be in either a color
anti-triplet or a color sextet. The approximate heavy antiquark-diquark
symmetry is applicable only when the color sextet component of the diquark can
be neglected and when the two heavy quarks are close to each other so that
they acts as a pointlike color-antitriplet source just like a heavy antiquark.
Thus, it cannot be applied to relate the heavy-heavy molecular systems to
singly-heavy baryons. Tetraquarks with the two charm quarks as a compact
diquark may exist in addition to the molecular states. The mixing of these two
configurations will make the spectrum more complicated.

The spectrum of $QQ\bar{q}\bar{q}$ tetraquark states are explored via the
method of QCD sum rules in Refs. Navarra  _et al._ (2007); Wang  _et al._
(2011); Dias  _et al._ (2011); Chen  _et al._ (2014); Wang (2018a); Wang and
Yan (2018); Wang and Chen (2020); Agaev  _et al._ (2019); Tang  _et al._
(2020); Agaev  _et al._ (2020a, b, 2021b); Wang and Chen (2020). In Ref. Dias
_et al._ (2011) it was argued that the molecular current of $DD^{*}$ yields a
similar mass with that of the $D\bar{D}^{*}+c.c.$ molecule and thus the result
will perfectly match $T_{cc}^{+}$ mass measured by LHCb if $X(3872)$ is a
$D\bar{D}^{*}+c.c.$ molecule. The mass of the $cc\bar{u}\bar{d}$ ground state
with $J^{P}=1^{+}$ was determined to be $(3.90\pm 0.09)$ GeV in Refs. Wang
(2018a); Wang and Yan (2018), which is also consistent with LHCb result. While
in other works, it was found that usually the $J^{P}=1^{+}$ $bb\bar{u}\bar{d}$
state lies below the threshold of $BB^{*}$ Navarra  _et al._ (2007); Agaev
_et al._ (2019); Tang  _et al._ (2020) while the charmed one is above the
corresponding $DD^{*}$ threshold Navarra  _et al._ (2007); Wang  _et al._
(2011); Tang  _et al._ (2020).

The explorations of doubly-heavy compact tetraquark states on lattice can be
seen in, e.g., Refs. Green and Pennanen (1998); Francis  _et al._ (2017);
Cheung  _et al._ (2017); Junnarkar  _et al._ (2019); Francis  _et al._ (2019);
Leskovec  _et al._ (2019); Hudspith  _et al._ (2020); Mohanta and Basak
(2020); Bicudo  _et al._ (2017b). Deeply bound tetraquark states,
$ud\bar{b}\bar{b}$, $qs\bar{b}\bar{b}$ and $ud\bar{b}\bar{c}$ with
$J^{P}=1^{+}$ are predicted in Refs. Francis  _et al._ (2017, 2019), which are
$189\pm 10$, $98\pm 7$ and $15\sim 61$ MeV below the corresponding free two-
meson thresholds, respectively. Ref. Junnarkar  _et al._ (2019) obtained
similar results for the above systems and predicted another three bound states
of $uc\bar{b}\bar{b}$, $ud\bar{c}\bar{c}$ and $us\bar{c}\bar{c}$ located just
below the corresponding free two-meson thresholds, and the masses of the
double-charm states are in agreement with those obtained in Ref. Cheung  _et
al._ (2017). In Ref. Mohanta and Basak (2020), the tetraquark operators for
$bb\bar{u}\bar{d}$ are constructed in both diquark-antidiquark
($[bb][\bar{u}\bar{d}]$) and molecular ($[b\bar{u}][b\bar{d}]$) configurations
and two states were found on lattice, a compact tetraquark located at $189\pm
18$ MeV below $BB^{*}$ threshold and a molecular one $17\pm 14$ MeV above the
same threshold.

###  IV.2 Heavy meson-baryon molecules v.s. doubly heavy baryons

Different from the meson-meson case, from Table 7 we can see that double-charm
meson-baryon systems are more attractive or less repulsive than the hidden-
charm ones. Specifically, the $D^{(*)}\Sigma_{c}^{(*)}$ systems with
isospin-$1/2$ are more attractive than the isospin-$1/2$
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ systems while the latter are widely believed
to be able to form bound states with experimental candidates, namely, the
famous $P_{c}$ states Aaij  _et al._ (2019). Therefore, it is natural that
more deeply bound states of $D^{(*)}\Sigma_{c}^{(*)}$ exist, see Fig. 3.
Similar conclusions can be drawn for $D_{1,2}\Sigma_{c}^{(*)}$ systems, as
shown in Fig. 5, since they have the same form of leading order interactions.
Other systems including $D^{(*)}\Xi_{c}^{(\prime*)}$,
$D_{1,2}\Xi_{c}^{(\prime*)}$, $D_{s}^{(*)}\Omega_{c}^{(*)}$ and
$D_{s1,s2}\Omega_{c}^{(*)}$ are also predicted to be bound easily, the spectra
of which are shown in Figs. 4 and 6.

Similarly with the coupled channel analysis used in the pioneering works Wu
_et al._ (2010, 2011) of pentaquark states where the $P_{c}$ states were
successfully predicted, Ref. Dias  _et al._ (2018) extended such a study to
the double-charm systems and some deeply bound states of
$D^{(*)}\Sigma_{c}^{(*)}$ with binding energies of $\mathcal{O}$(100 MeV) were
found. It was extended further to the charm-bottom system and double-bottom
systems in Refs. Yu  _et al._ (2019); Dias  _et al._ (2020), and more deeply
bound states of $D^{(*)}\Sigma_{b}^{(*)}$ and $\bar{B}^{(*)}\Sigma_{c}^{(*)}$
with binding energies of $\mathcal{O}$(300 MeV) and
$\bar{B}^{(*)}\Sigma_{b}^{(*)}$ with binding energies of $\mathcal{O}$(400
MeV) were obtained. There it was also found that there are poles located about
100 MeV below the $\Lambda_{b}D$, $\Lambda_{c}\bar{B}$ and
$\Lambda_{b}\bar{B}$ thresholds, respectively. Such conclusions are
qualitatively consistent with our results that $D^{(*)}\Sigma_{c}^{(*)}$ and
$D^{(*)}\Xi_{c}^{(*)}$ are attractive and the former is stronger. The meson-
baryon transitions between the coupled channels $J/\psi
N$-$\Lambda_{c}\bar{D}^{(*)}$-$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ were explored in
Ref. Shimizu and Harada (2017) and it was found that a doubly-charmed state,
$\Xi_{cc}^{*}(4380)$, exists with almost the same mass as $P_{c}(4380)$. An
S-wave scattering of ground state doubly-charmed baryons with the light
pseudoscalar mesons were first investigated in Ref. Guo and Meißner (2011) and
then in Ref. Guo (2017) by means of unitarized chiral effective theory and
several doubly charmed baryon resonances were predicted. The spectrum was
modified by including the effects of the $P$-wave excitation inside the charm
diquark in Ref. Yan  _et al._ (2018).

The interaction between $D\Lambda_{c}$ or $\bar{B}\Lambda_{b}$ from two-pion
exchange are investigated in Ref. Xu  _et al._ (2012) and it was claimed that
a $\bar{B}\Lambda_{b}$ bound state from such an interaction is possible. In
Ref. Chen  _et al._ (2017), the $D\Lambda_{c/b}$, $\bar{B}\Lambda_{c/b}$
systems were found possible to be bound by the $\sigma/\omega$ exchange
interaction. Systematic studies on the $\Sigma_{c}^{(*)}D^{(*)}$ interactions
within the framework of chiral effective field theory Chen  _et al._ (2021b)
or one boson exchange Liu  _et al._ (2020) were performed, and the $I=1/2$
systems may form bound states with binding energies of about several or dozens
MeV, consistent with the results obtained here, see Table 5. Therein, deeper
bound states of the $\Sigma_{c}^{(*)}\bar{B}^{(*)},\Sigma_{b}^{(*)}D^{(*)}$
and $\Sigma_{b}^{(*)}\bar{B}^{(*)}$ systems were also predicted to exist due
to the larger reduced masses, as expected.

The doubly-heavy pentaquark states with compact configurations were explored
within the color-magnetic interaction model Zhou  _et al._ (2018), non-
relativistic constituent quark model Zhu  _et al._ (2019) and QCD sum rules
Xing and Niu (2021); Wang (2018b) where some narrow exotic pentaquark states
were predicted, constituent quark model Park  _et al._ (2019b) where one
strong-interaction stable state, $ccud\bar{s}$ was predicted.

From Table 5 and Fig. 3, one sees that the lightest double-charm meson-baryon
molecule is the one in the $D\Lambda_{c}$ system. The state has a mass large
enough for it to decay into $\Xi_{cc}\pi$ and could be broad. So are the other
similar states.

###  IV.3 Heavy di-baryons

For heavy di-baryons, the leading order interaction from the vector meson
exchange leads to evident binding only for the
$\Sigma_{c}^{(*)}\Sigma_{c}^{(*)}$ systems, see Table 3 and Fig. 1. The
attractions for the $\Xi_{c}\Xi_{c}$ related systems are too weak and only
remote virtual poles are found, which are not robust somehow and can get
sizeably modified by the omitted momentum-dependent terms.

In our simple model the leading order interaction for the
$\Lambda_{c}\Lambda_{c}$ system is repulsive and thus it cannot be bound.
Within various models, it was found that the $\Lambda_{c}\Lambda_{c}$ system
can not be bound by itself Lee  _et al._ (2011); Meguro  _et al._ (2011);
Huang  _et al._ (2014); Oka (2013); Carames and Valcarce (2015); Garcilazo and
Valcarce (2020) but Refs. Meguro  _et al._ (2011); Huang  _et al._ (2014); Oka
(2013); Li and Zhu (2012) showed that the coupling to the strongly attractive
$\Sigma_{c}^{(*)}\Sigma_{c}^{(*)}$ system may lead to a states below
$\Lambda_{c}\Lambda_{c}$ threshold . In an analysis with QCD sum rules Wang
_et al._ (2021b), the states with the same quark components as
$\Lambda_{c}\Lambda_{c}$ were found all above the $\Lambda_{c}\Lambda_{c}$
threshold. In Refs. Gerasyuta and Matskevich (2012); Lu  _et al._ (2019); Chen
_et al._ (2017), on the contrary, the single channel $\Lambda_{c}\Lambda_{c}$
can be bound by itself; however, Ref. Lu  _et al._ (2019) warns that a more
thorough theoretical exploration is needed to determine whether the
$\Lambda_{c}\Lambda_{c}$ system really binds.

Some other doubly-heavy di-baryon systems were also explored. Several
realistic phenomenological nucleon-nucleon interaction models are employed in
Ref. Froemel  _et al._ (2005). It was found there that the
$\Xi^{(\prime)}_{c}\Xi^{(\prime)}_{c}$ and $\Sigma_{c}\Sigma_{c}$ systems can
be bound in some models. From one-boson exchange with coupled channel effects
included, Refs. Lee  _et al._ (2011); Yang  _et al._ (2019) found that
$\Xi_{c}^{(\prime*)}\Xi_{c}^{(\prime*)}$ and
$\Omega_{c}^{(*)}\Omega_{c}^{(*)}$ may be loosely bound while the isoscalar
$\Sigma_{c}^{(*)}\Sigma_{c}^{(*)}$ systems can be deeply bound. Similar
conclusion was obtained in Ref. Huang  _et al._ (2014) within the framework of
quark delocalization color screening model that the
$\Sigma_{c}^{(*)}\Sigma_{c}^{(*)}$ single-channel system can be deeply bound.
In Ref. Garcilazo and Valcarce (2020), the potential of $\Sigma_{c}\Sigma_{c}$
was derived from a constituent quark model and a bound state with
$I(J^{P})=0(0^{+})$ was obtained with a binding energy about 6.2 MeV. The
long-range pion exchange force is strong enough to form molecules of
$\left[\Sigma_{Q}\Xi_{Q}^{\prime}\right]_{J=1}^{I=1/2}$,
$\left[\Sigma_{Q}\Lambda_{Q}\right]_{J=1}^{I=1}(Q=b,c)$,
$\left[\Sigma_{b}\Xi_{b}^{\prime}\right]_{J=1}^{I=3/2}$ and
$\left[\Xi_{b}\Xi_{b}^{\prime}\right]_{J=1}^{I=0}$ where the $S$-$D$ mixing is
necessary Li and Zhu (2012). In Ref. Lu  _et al._ (2019), the
$\Sigma_{Q}^{(*)}\Sigma_{Q}^{(*)}$ were claimed to be good candidates of bound
states from the one-pion and vector meson exchanges while it concludes that a
more thorough analysis is necessary to determine whether there is a binding
for the $\Lambda_{Q}\Sigma_{Q}^{(*)}$.

The di-baryon systems with two heavy quarks were investigated in Ref. Vijande
_et al._ (2016) with a simple quark model but no bound or metastable state was
found.

##  V Summary

In this work we have obtained an overall spectrum of hadronic molecules
composed of a pair of charmed hadrons, including all the $S$-wave singly-
charmed mesons and baryons as well as the $s_{\ell}=1/2$ $P$-wave charmed
mesons. The interaction is assumed to be dominated by the light vector meson
exchange and approximated by constants at leading order, which are derived
systematically from the couplings that satisfy HQSS and SU(3) flavor symmetry.

One should keep in mind that the spectrum predicted here should be regarded as
the leading approximation of the spectrum for heavy-heavy molecular states,
which gives only a general overall feature of the heavy-heavy hadronic
molecular spectrum. The numerical results can receive large quantitative
corrections due to the limitations of our treatments, which we discuss
qualitatively in the following.

  * •

We have only considered the leading interactions described by constant contact
terms. The momentum dependent terms (including both spin-dependent and spin-
independent contributions) may change the spectrum we obtained visibly,
especially for the systems where the poles are far away from the corresponding
thresholds. The spin-dependent terms will also lift the degeneracy of the same
system with different total spins.

  * •

The coupled channel effects have been neglected. In some cases the coupled-
channel effects may play an important role in the formation of near threshold
states. However, it is common and natural that the near-threshold pole found
in a coupled-channel system dominantly couples to a single channel, see, e.g.,
the $D_{s0}^{*}(2317)$, which is dynamically generated in the $DK$ and
$D_{s}\eta$ system but couples dominantly to $DK$ Kolomeitsev and Lutz (2004);
Guo  _et al._ (2006); Gamermann  _et al._ (2007), and the $\Xi_{cc}(4083)$
state with $J^{P}=1/2^{-}$, which is dynamically generated in the
$\Sigma_{c}D$ and $\Xi_{c}^{\prime}D_{s}$ system but couples dominantly to
$\Sigma_{c}D$ Dias  _et al._ (2018).999 The $\Sigma_{c}D$ bound state obtained
in this work has a mass around 4.3 GeV, much closer to the threshold; see
Table 5 and Fig. 3.

  * •

The hadronic molecules shown in Figs. 3-6 can couple to normal double-charm
baryons as well as channels with a double-charm baryon and a light meson. It
is expected that each of these two types of systems also forms a spectrum. The
physical spectrum of double-charm baryons should incorporates the mixing among
all the three spectra. Coupled channels including both the charm-baryon–charm-
meson channels and light-meson–double-charm-baryon channels have been
considered in, e.g., Ref. Dias  _et al._ (2018) for the $\Xi_{cc}$ type
molecular states. Mixing of light-meson–double-charm-baryon molecular states
with the normal double-charm baryons with a $P$-wave excitation inside the
charm diquark has been considered in Ref. Yan  _et al._ (2018). Yet, a model
considering all the three kinds of channels does not exist so far.

  * •

The exchange of other particles, such as the light scalar mesons, charmed
mesons and charmonia, are not considered, the effect of which can be partly
covered by varying the cutoff. In addition, the interactions considered here
are of leading order in the $1/N_{c}$ expansion, where $N_{c}$ is the number
of colors, i.e., the Okubo-Zweig-Iizuka violating interactions have been
neglected. Such contributions will also lift the degeneracy of the same system
with different total spins.

Therefore, although we expect the spectrum given here should present an
overall pattern of the hadronic molecules formed by a pair of charmed mesons
and/or baryons, specific systems may quantitatively differ from the
predictions here due to the limitations of our treatments.

In total we obtained 124 double-charm hadronic molecules and we summarize the
main feature of this spectrum in the following.

  1. 1.

Unlike the isovector $D^{*}D^{(*)}$ systems that are repulsive, the isoscalar
ones have attractive interaction from the light vector meson exchange and the
total potential makes the systems at the edge of forming near-threshold
molecules. With a reasonable cutoff regularizing the loop integral, the
binding energy of the $I(J^{P})=0(1^{+})$ $DD^{*}$ system is consistent with
the double charm tetraquark $T_{cc}^{+}$ in LHCb observation. If the hadronic
molecule structure of the $T_{cc}^{+}$ is confirmed, which is rather natural
given the closeness to the $DD^{*}$ threshold, many other similar states with
$I=0$ including $D^{*}D^{*}$, $D^{(*)}D_{1,2}$, $D_{1,2}D_{1,2}$ can also
exist.

  2. 2.

Given that the famous $P_{c}$ states are hadronic molecules of
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$, it is natural to expect the existence of
double-charm $D^{(*)}\Sigma_{c}^{(*)}$ and $D_{1,2}\Sigma_{c}^{(*)}$ states
with $I=1/2$ because the attraction from the light vector meson exchange for
the latter is stronger than that for the former. Similar conclusions can be
made for the $D^{(*)}\Xi_{c}^{(\prime*)}$ and $D_{1,2}\Xi_{c}^{(\prime*)}$
channels, especially when the $P_{cs}$ states are established experimentally.
The hadronic molecules in other systems including $D^{(*)}\Lambda_{c}$,
$D_{1,2}\Lambda_{c}$, $D_{s}^{(*)}\Xi_{c}^{(\prime*)}$,
$D_{s1,2}\Xi_{c}^{(\prime*)}$ $D_{s}^{(*)}\Omega_{c}^{(*)}$ and
$D_{s1,2}\Omega_{c}^{(*)}$ are also predicted.

  3. 3.

Within our simple model, in the double-charm di-baryon sector, only isoscalar
$\Sigma_{c}^{(*)}\Sigma_{c}^{(*)}$ systems are expected to be good candidates
of bound di-baryon states. As discussed in the literature, the inclusion of
other contributions to the interaction as well as the coupling to other
channels may make additional di-baryon bound states possible.

Due to the heavy quark flavor symmetry, the potentials in the bottom sector
are the same as those in the charm sector, if using the nonrelativistic field
normalization, and we expect at least the same number of molecular states in
the analogous systems therein. Because of the much heavier reduced masses of
hidden-bottom system, it will be easier to form bound states than the charmed
systems, and there may even be excited states if the ground states are deeply
bound, the treatment of which, however, requires momentum-dependent
interactions and is beyond the scope of this paper.

###### Acknowledgements.

We would like to thank Marek Karliner and Eulogio Oset for useful comments and
Ming-Zhu Liu for pointing out a mistake about the quantum numbers of a pair of
identical fermions. This work is supported in part by the Chinese Academy of
Sciences (CAS) under Grant No. XDPB15, 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, and by the NSFC and the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through
funds provided to the Sino-German Collaborative Research Center “Symmetries
and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
Project-ID 196253076 – TRR110).

##  Appendix A The flavor factor $F$

In this section we list the factor $F$ that accounts for the flavor
information for the exchange of different mesons in different systems. The
essential point is that the vector mesons, $\rho,\omega$ and $\phi$, only
couple to the light quark in heavy or antiheavy hadron. The conclusion is that
the isoscalar vector, $\omega$ and $\phi$, exchange have opposite signs for
$qq$ and $q\bar{q}$ interactions while the isovector meson, $\rho$, exchange
has the same sign in these two cases. Note that the negative $C$-parity of
vector mesons has been taken into account. The conclusion for isoscalar meson
exchange is apparent and in the following we give a brief deduction for the
case of isovector exchange.

For simplicity, we consider the hadrons that contain only one $u$ or $d$
quark. The Lagrangian for the isospin structure, taking the $\rho$ meson
exchange for example, reads

$\mathcal{L}=\psi^{\dagger}{\bm{\rho\cdot\tau}}\psi+\rm c.c.$ (42)

with $\psi=(u,d)^{T}$, $\psi^{\dagger}=(u^{\dagger},d^{\dagger})$, $\bm{\tau}$
the Pauli matrices, where $T$ means transpose. Note that here we are only
interested in the flavor structure and hence we denote the field creating a
$u$ quark by $u^{\dagger}$ instead of $\bar{u}$ to avoid confusion with the
field for antiquark below. The charge conjugation term, which needs special
attentions, is

$c\bar{\psi}^{{}^{\prime}\dagger}(\bm{\rho\cdot\tau})^{T}\bar{\psi}^{\prime}$
(43)

with $\bar{\psi}^{\prime}=(\bar{u},\bar{d})^{T}$,
$\bar{\psi}^{{}^{\prime}\dagger}=(\bar{u}^{\dagger},\bar{d}^{\dagger})$ and
$c=-1$ the charge conjugation factor for the $\rho$ meson. Using the common
convention for the isospin eigenstates,

$\left|u\right\rangle=\left|\uparrow\right\rangle,\left|d\right\rangle=\left|\downarrow\right\rangle,\left|\bar{d}\right\rangle=\left|\uparrow\right\rangle,\left|\bar{u}\right\rangle=-\left|\downarrow\right\rangle,$
(44)

we rearrange the charge conjugation term as

$c\bar{\psi}^{\dagger}\tau_{2}(\bm{\rho\cdot\tau})^{T}\tau_{2}\bar{\psi}=-c\bar{\psi}^{\dagger}(\bm{\rho\cdot\tau})\bar{\psi}$
(45)

where $\bar{\psi}=(\bar{d},-\bar{u})^{T}=i\tau_{2}\bar{\psi}^{\prime}$. Now we
are ready to calculate the isospin factors for $qq$ and $q\bar{q}$
interactions via the isovector meson exchange. Explicitly, they are given by

$\displaystyle V_{qq}$
$\displaystyle\propto\bm{\tau_{1}\cdot\tau_{2}}=\left\\{\begin{array}[]{ll}1&\text{for
}I=1\\\ -3&\text{for }I=0\end{array}\right.,$ (48) $\displaystyle
V_{q\bar{q}}$ $\displaystyle=-cV_{qq}.$ (49)

Table 7: The group theory factor $F$, defined in Eq. (34), for the interaction of charm-anticharm/charm-charm hadron pairs with only the light vector-meson exchanges. Here both charm hadrons are the $S$-wave ground states. $I$ is the isospin and $S$ is the strangeness. Note that we have collect the pairs with the heavy hadrons in the same spin multiples (such as $DD$, $DD^{*}$, etc.) in one row, and the several numbers in the column of “Thresholds” represent the thresholds of these different pairs in an increasing order. Positive $F$ means that the interaction attractive. The values in the column of “$F$” correspond to those for the exchanged particles in the column of “Exchanged particles” in order. System | $I$ | $S$ | Thresholds [MeV] | Exchanged particles | $F$
---|---|---|---|---|---
$D^{(*)}\bar{D}^{(*)}/D^{(*)}D^{(*)}$ | 1 | 0/0 | $(3734,3876,4017)$ | $\rho,\omega$ | $-\frac{1}{2},\frac{1}{2}/-\frac{1}{2},-\frac{1}{2}$
| 0 |  |  |  |  $\frac{3}{2},\frac{1}{2}$/$\frac{3}{2},-\frac{1}{2}$
$D_{s}^{(*)}\bar{D}^{(*)}$/$D_{s}^{(*)}D^{(*)}$ | $\frac{1}{2}$ | 1/1 | $(3836,3977,3979,4121)$ | ${K^{*}}$ |  $0$/$-1$
$D^{(*)}_{s}\bar{D}^{(*)}_{s}$/$D^{(*)}_{s}D^{(*)}_{s}$ | 0 | 0/2 | $(3937,4081,4224)$ | $\phi$ |  $1$/$-1$
$\bar{D}^{(*)}\Lambda_{c}$/$D^{(*)}\Lambda_{c}$ | $\frac{1}{2}$ | 0/0 | $(4154,4295)$ | $\omega$ |  $-1$/$1$
$\bar{D}_{s}^{(*)}\Lambda_{c}$/$D_{s}^{(*)}\Lambda_{c}$ | 0 | $-1/1$ | $(4255,4399)$ | $-$ |  $0$/$0$
$\bar{D}^{(*)}\Xi_{c}$/$D^{(*)}\Xi_{c}$ | 1 | $-1/-1$ | $(4337,4478)$ | $\rho,\omega$ |  $-\frac{1}{2},-\frac{1}{2}$/$-\frac{1}{2},\frac{1}{2}$
| 0 |  |  |  | $\frac{3}{2},-\frac{1}{2}/\frac{3}{2},\frac{1}{2}$
$\bar{D}_{s}^{(*)}\Xi_{c}$/$D_{s}^{(*)}\Xi_{c}$ | $\frac{1}{2}$ | $-2/0$ | $(4438,4582)$ | $\phi$ |  $-1$/$1$
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$/$D^{(*)}\Sigma_{c}^{(*)}$ | $\frac{3}{2}$ | 0/0 | $(4321,4385,4462,4527)$ | $\rho,\omega$ |  $-1,-1$/$-1,1$
| $\frac{1}{2}$ |  |  |  |  $2,-1$/2,1
$\bar{D}_{s}^{(*)}\Sigma_{c}^{(*)}$/$D_{s}^{(*)}\Sigma_{c}^{(*)}$ | 1 | $-1/1$ | $(4422,4486,4566,4630)$ | $-$ | $0/0$
$\bar{D}^{(*)}\Xi_{c}^{{}^{\prime}(*)}$/$D^{(*)}\Xi_{c}^{{}^{\prime}(*)}$ | 1 | $-1/-1$ | $(4446,4513,4587,4655)$ | $\rho,\omega$ |  $-\frac{1}{2},-\frac{1}{2}$/$-\frac{1}{2},\frac{1}{2}$
| 0 |  |  |  |  $\frac{3}{2},-\frac{1}{2}$/$\frac{3}{2},\frac{1}{2}$
$\bar{D}_{s}^{(*)}\Xi_{c}^{{}^{\prime}(*)}$/$D_{s}^{(*)}\Xi_{c}^{{}^{\prime}(*)}$ | $\frac{1}{2}$ | $-2/0$ | $(4547,4614,4691,4758)$ | $\phi$ | $-1/1$
$\bar{D}^{(*)}\Omega_{c}^{(*)}$/$D^{(*)}\Omega_{c}^{(*)}$ | $\frac{1}{2}$ | $-2/0$ | $(4562,4633,4704,4774)$ | $-$ | $0/0$
$\bar{D}_{s}^{(*)}\Omega_{c}^{(*)}$/$D_{s}^{(*)}\Omega_{c}^{(*)}$ | 0 | $-3/-1$ | $(4664,4734,4807,4878)$ | $\phi$ | $-2/2$
$\Lambda_{c}\bar{\Lambda}_{c}$/$\Lambda_{c}\Lambda_{c}$ | 0 | 0/0 | $(4573)$ | $\omega$ |  $2$/$-2$
$\Lambda_{c}\bar{\Xi}_{c}$/$\Lambda_{c}\Xi_{c}$ | $\frac{1}{2}$ | $1/-1$ | $(4756)$ | $\omega{/K^{*}}$ |  $1{,0}$/$-1{,-1}$
$\Xi_{c}\bar{\Xi}_{c}$/$\Xi_{c}\Xi_{c}$ | 1 | $0/-2$ | $(4939)$ | $\rho,\omega,\phi$ |  $-\frac{1}{2},\frac{1}{2},1$/$-\frac{1}{2},-\frac{1}{2},-1$
| 0 |  |  |  |  $\frac{3}{2},\frac{1}{2},1$/$\frac{3}{2},-\frac{1}{2},-1$
$\Lambda_{c}\bar{\Sigma}_{c}^{(*)}$/$\Lambda_{c}\Sigma_{c}^{(*)}$ | 1 | 0/0 | $(4740,4805)$ | $\omega{/K^{*}}$ |  $1{,0}$/$-1{,-1}$
$\Lambda_{c}\bar{\Xi}_{c}^{{}^{\prime}(*)}$/$\Lambda_{c}\Xi_{c}^{{}^{\prime}(*)}$ | $\frac{1}{2}$ | $1/-1$ | $(4865,4932)$ | $\omega$ |  $1$/$-1$
$\Lambda_{c}\bar{\Omega}_{c}^{(*)}$/$\Lambda_{c}\Omega_{c}^{(*)}$ | 0 | $2/-2$ | $(4982,5052)$ | $-$ |  $0$/$0$
$\Sigma_{c}^{(*)}\bar{\Xi}_{c}$/$\Sigma_{c}^{(*)}\Xi_{c}$ | $\frac{3}{2}$ | $1/-1$ | $(4923,4988)$ | $\rho,\omega,K^{*}$ | $-1,1,0/-1,-1,-2$
| $\frac{1}{2}$ |  |  |  |  $2,1,0$/$2,-1,-2$
$\Xi_{c}\bar{\Xi}_{c}^{{}^{\prime}(*)}$/$\Xi_{c}\Xi_{c}^{{}^{\prime}(*)}$ | 1 | $0/-2$ | $(5048,5115)$ | $\rho,\omega,\phi$ |  $-\frac{1}{2},\frac{1}{2},1$/$-\frac{1}{2},-\frac{1}{2},-1$
| 0 |  |  |  |  $\frac{3}{2},\frac{1}{2},1$/$\frac{3}{2},-\frac{1}{2},-1$
$\Xi_{c}\bar{\Omega}_{c}^{(*)}$/$\Xi_{c}\Omega_{c}^{(*)}$ | $\frac{1}{2}$ | $1/-3$ | $(5165,5235)$ | $\phi,K^{*}$ |  $2,0$/$-2,-2$
$\Sigma_{c}^{(*)}\bar{\Sigma}_{c}^{(*)}$/$\Sigma_{c}^{(*)}\Sigma_{c}^{(*)}$ | 2 | 0/0 | $(4907,4972,5036)$ | $\rho,\omega$ |  $-2,2$/$-2,-2$
| 1 |  |  |  |  $2,2$/$2,-2$
| 0 |  |  |  |  $4,2$/$4,-2$
$\Sigma_{c}^{(*)}\bar{\Xi}^{{}^{\prime}(*)}_{c}$/$\Sigma_{c}^{(*)}\Xi^{{}^{\prime}(*)}_{c}$ | $\frac{3}{2}$ | $1/-1$ | $(5032,5097,5100,5164)$ | $\rho,\omega,K^{*}$ |  $-1,1,0$/$-1,-1-2$
| $\frac{1}{2}$ |  |  |  |  $2,1,0$/$2,-1,-2$
$\Sigma_{c}^{(*)}\bar{\Omega}^{(*)}_{c}$/$\Sigma_{c}^{(*)}\Omega^{(*)}_{c}$ | 0 | $2/-2$ | $(5149,5213,5219,5284)$ | $-$ |  $0$/$0$
$\Xi_{c}^{{}^{\prime}(*)}\bar{\Xi}_{c}^{{}^{\prime}(*)}$/$\Xi_{c}^{{}^{\prime}(*)}\Xi_{c}^{{}^{\prime}(*)}$ | 1 | $0/-2$ | $(5158,5225,5292)$ | $\rho,\omega,\phi$ |  $-\frac{1}{2},\frac{1}{2},1$/$-\frac{1}{2},-\frac{1}{2},-1$
| 0 |  |  |  |  $\frac{3}{2},\frac{1}{2},1$/$\frac{3}{2},-\frac{1}{2},-1$
$\Xi^{{}^{\prime}(*)}_{c}\bar{\Omega}_{c}^{(*)}$/$\Xi^{{}^{\prime}(*)}_{c}\Omega_{c}^{(*)}$ | $\frac{1}{2}$ | $1/-3$ | $(5272,5341,5345,5412)$ | $\phi,K^{*}$ |  $2,0$/$-2,-2$
$\Omega_{c}^{(*)}\bar{\Omega}_{c}^{(*)}$/$\Omega_{c}^{(*)}\Omega_{c}^{(*)}$ | 0 | $0/-4$ | $(5390,5461,5532)$ | $\phi$ |  $4$/$-4$
Table 8: The group theory factor $F$, defined in Eq. (34), for the interaction of charm-anticharm/charm-charm hadron pairs with only the light vector-meson exchanges. Here one of the charm hadrons is an $s_{\ell}=3/2$ charm meson. See the caption of Table 7.  System | $I$ | $S$ | Thresholds [MeV] | Exchanged particles | $F$
---|---|---|---|---|---
$D^{(*)}\bar{D}_{1,2}$/$D^{(*)}D_{1,2}$ | 0 | 0/0 | $(4289,4330,4431,4472)$ | $\rho,\omega$ |  $\frac{3}{2},\frac{1}{2}$/$\frac{3}{2},-\frac{1}{2}$
| 1 | 0/0 |  |  |  $-\frac{1}{2},\frac{1}{2}$/ $-\frac{1}{2},-\frac{1}{2}$
$D^{(*)}\bar{D}_{s1,s2}$/$D^{(*)}D_{s1,s2}$ | $\frac{1}{2}$ | $1/-1$ | $(4390,4431,4534,4575)$ | $-$ |  $0$/$0$
$D_{s}^{(*)}\bar{D}_{1,2}$/$D_{s}^{(*)}D_{1,2}$ | $\frac{1}{2}$ | $-1/1$ | $(4402,4436,4544,4578)$ | $-$ |  $0$/$0$
$D^{(*)}_{s}\bar{D}_{s1,s2}$/$D^{(*)}_{s}D_{s1,s2}$ | 0 | 0/$-2$ | $(4503,4537,4647,4681)$ | $\phi$ |  $1$/$-1$
$D_{1,2}\bar{D}_{1,2}$/$D_{1,2}D_{1,2}$ | 0 | 0/0 | $(4844,4885,4926)$ | $\rho,\omega$ |  $\frac{3}{2},\frac{1}{2}$/$\frac{3}{2},-\frac{1}{2}$
| 1 |  |  |  |  $-\frac{1}{2},\frac{1}{2}$/$-\frac{1}{2},-\frac{1}{2}$
$D_{s1,s2}\bar{D}_{1,2}$/$D_{s1,s2}D_{1,2}$ | $\frac{1}{2}$ | 1/1 | $(4957,4991,4998,5032)$ |  |  $0$/0
$D_{s1,s2}\bar{D}_{s1,s2}$/$D_{s1,s2}D_{s1,s2}$ | 0 | $0/-2$ | $(5070,5104,5138)$ | $\phi$ |  $1$/1
$\Lambda_{c}\bar{D}_{1,2}$/$\Lambda_{c}D_{1,2}$ | $\frac{1}{2}$ | 0/0 | $(4708,4750)$ | $\omega$ |  $-1$/1
$\Lambda_{c}\bar{D}_{s1,s2}$/$\Lambda_{c}D_{s1,s2}$ | 0 | $-1/1$ | $(4822,4856)$ | $-$ |  $0$/0
$\Xi_{c}\bar{D}_{1,2}$/$\Xi_{c}D_{1,2}$ | 1 | $-1/-1$ | $(4891,4932)$ | $\rho,\omega$ |  $-\frac{1}{2},-\frac{1}{2}$/$-\frac{1}{2},\frac{1}{2}$
| 0 |  |  |  |  $\frac{3}{2},-\frac{1}{2}$/$\frac{3}{2},\frac{1}{2}$
$\Xi_{c}\bar{D}_{s1,s2}$/$\Xi_{c}D_{s1,s2}$ | $\frac{1}{2}$ | $-2/0$ | $(5005,5039)$ | $\phi$ |  $-1$/1
$\Sigma_{c}^{(*)}\bar{D}_{1,2}$/$\Sigma_{c}^{(*)}D_{1,2}$ | $\frac{3}{2}$ | 0/0 | $(4876,4917,4940,4981)$ | $\rho,\omega$ |  $-1,-1$/$-1,1$
| $\frac{1}{2}$ |  |  |  |  $2,-1$/2,1
$\Sigma_{c}^{(*)}\bar{D}_{s1,s2}$/$\Sigma_{c}^{(*)}D_{s1,s2}$ | 1 | $1/-1$ | $(4989,5023,5053,5087)$ | $-$ |  $0$/0
$\Xi_{c}^{{}^{\prime}(*)}\bar{D}_{1,2}$/$\Xi_{c}^{{}^{\prime}(*)}D_{1,2}$ | 1 | $-1/-1$ | $(5001,5042,5068,5109)$ | $\rho,\omega$ |  $-\frac{1}{2},-\frac{1}{2}$/$-\frac{1}{2},\frac{1}{2}$
| 0 |  |  |  |  $\frac{3}{2},-\frac{1}{2}$/$\frac{3}{2},\frac{1}{2}$
$\Xi_{c}^{{}^{\prime}(*)}\bar{D}_{s1,s2}$/$\Xi_{c}^{{}^{\prime}(*)}D_{s1,s2}$ | $\frac{1}{2}$ | $-2/0$ | $(5114,5148,5181,5215)$ | $\phi$ |  $-1$/1
$\Omega_{c}^{(*)}\bar{D}_{1,2}$/$\Omega_{c}^{(*)}D_{1,2}$ | $\frac{1}{2}$ | $-2/-2$ | $(5117,5158,5188,5229)$ | $-$ |  $0$/0
$\Omega_{c}^{(*)}\bar{D}_{s1,s2}$/$\Omega_{c}^{(*)}D_{s1,s2}$ | 0 | $-3/-1$ | $(5230,5264,5301,5335)$ | $\phi$ |  $-2$/2
Table 9: The group theory factor $F$, defined in Eq. (34), for the interaction of bottom-antibottom/bottom-bottom hadron pairs with only the light vector-meson exchanges. Here both bottom hadrons are the $S$-wave ground states. See the caption of Table 7.  System | $I$ | $S$ | Thresholds [MeV] | Exchanged particles | $F$
---|---|---|---|---|---
$B^{(*)}\bar{B}^{(*)}/B^{(*)}B^{(*)}$ | 1 | 0/0 | $(10559,10604,10649)$ | $\rho,\omega$ | $-\frac{1}{2},\frac{1}{2}/-\frac{1}{2},-\frac{1}{2}$
| 0 |  |  |  |  $\frac{3}{2},\frac{1}{2}$/$\frac{3}{2},-\frac{1}{2}$
$B_{s}^{(*)}\bar{B}^{(*)}$/$B_{s}^{(*)}B^{(*)}$ | $\frac{1}{2}$ | 1/1 | $(10646,10695,10692,10740)$ | ${K^{*}}$ |  $0$/${-1}$
$B^{(*)}_{s}\bar{B}^{(*)}_{s}$/$B^{(*)}_{s}B^{(*)}_{s}$ | 0 | 0/2 | $(10734,10782,10831)$ | $\phi$ |  $1$/$-1$
$\bar{B}^{(*)}\Lambda_{b}$/$B^{(*)}\Lambda_{b}$ | $\frac{1}{2}$ | 0/0 | $(10899,10944)$ | $\omega$ |  $-1$/$1$
$\bar{B}_{s}^{(*)}\Lambda_{b}$/$B_{s}^{(*)}\Lambda_{b}$ | 0 | $-1/1$ | $(10986,11035)$ | $-$ |  $0$/$0$
$\bar{B}^{(*)}\Xi_{b}$/$B^{(*)}\Xi_{b}$ | 1 | $-1/-1$ | $(11074,11119)$ | $\rho,\omega$ |  $-\frac{1}{2},-\frac{1}{2}$/$-\frac{1}{2},\frac{1}{2}$
| 0 |  |  |  | $\frac{3}{2},-\frac{1}{2}/\frac{3}{2},\frac{1}{2}$
$\bar{B}_{s}^{(*)}\Xi_{b}$/$B_{s}^{(*)}\Xi_{b}$ | $\frac{1}{2}$ | $-2/0$ | $(11161,11210)$ | $\phi$ |  $-1$/$1$
$\bar{B}^{(*)}\Sigma_{b}^{(*)}$/$B^{(*)}\Sigma_{b}^{(*)}$ | $\frac{3}{2}$ | 0/0 | $(11093,11138,11112,11157)$ | $\rho,\omega$ |  $-1,-1$/$-1,1$
| $\frac{1}{2}$ |  |  |  |  $2,-1$/2,1
$\bar{B}_{s}^{(*)}\Sigma_{b}^{(*)}$/$B_{s}^{(*)}\Sigma_{b}^{(*)}$ | 1 | $-1/1$ | $(11180,11228,11199,11248)$ | $-$ | $0/0$
$\bar{B}^{(*)}\Xi_{b}^{{}^{\prime}(*)}$/$B^{(*)}\Xi_{b}^{{}^{\prime}(*)}$ | 1 | $-1/-1$ | $(11215,11260,11233,11279)$ | $\rho,\omega$ |  $-\frac{1}{2},-\frac{1}{2}$/$-\frac{1}{2},\frac{1}{2}$
| 0 |  |  |  |  $\frac{3}{2},-\frac{1}{2}$/$\frac{3}{2},\frac{1}{2}$
$\bar{B}_{s}^{(*)}\Xi_{b}^{{}^{\prime}(*)}$/$B_{s}^{(*)}\Xi_{b}^{{}^{\prime}(*)}$ | $\frac{1}{2}$ | $-2/0$ | $(11302,11350,11321,11369)$ | $\phi$ | $-1/1$
$\bar{B}^{(*)}\Omega_{b}^{(*)}$/$B^{(*)}\Omega_{b}^{(*)}$ | $\frac{1}{2}$ | $-2/0$ | $(11326,11371,11349,11395)$ | $-$ | $0/0$
$\bar{B}_{s}^{(*)}\Omega_{b}^{(*)}$/$B_{s}^{(*)}\Omega_{b}^{(*)}$ | 0 | $-3/-1$ | $(11413,11462,11437,11485)$ | $\phi$ | $-2/2$
$\Lambda_{b}\bar{\Lambda}_{b}$/$\Lambda_{b}\Lambda_{b}$ | 0 | 0/0 | $(11239)$ | $\omega$ |  $2$/$-2$
$\Lambda_{b}\bar{\Xi}_{b}$/$\Lambda_{b}\Xi_{b}$ | $\frac{1}{2}$ | $1/-1$ | $(11414)$ |  $\omega$, $K^{*}$ |  $1{,0}$/$-1{,-1}$
$\Xi_{b}\bar{\Xi}_{b}$/$\Xi_{b}\Xi_{b}$ | 1 | $0/-2$ | $(11589)$ | $\rho,\omega,\phi$ |  $-\frac{1}{2},\frac{1}{2},1$/$-\frac{1}{2},-\frac{1}{2},-1$
| 0 |  |  |  |  $\frac{3}{2},\frac{1}{2},1$/$\frac{3}{2},-\frac{1}{2},-1$
$\Lambda_{b}\bar{\Sigma}_{b}^{(*)}$/$\Lambda_{b}\Sigma_{b}^{(*)}$ | 1 | 0/0 | $(11433,11452)$ | $\omega$ |  $2$/$-2$
$\Lambda_{b}\bar{\Xi}_{b}^{{}^{\prime}(*)}$/$\Lambda_{b}\Xi_{b}^{{}^{\prime}(*)}$ | $\frac{1}{2}$ | $1/-1$ | $(11555,11573)$ |  $\omega$, $K^{*}$ |  $1{,0}$/$-1{,-1}$
$\Lambda_{b}\bar{\Omega}_{b}^{(*)}$/$\Lambda_{b}\Omega_{b}^{(*)}$ | 0 | $2/-2$ | $(11666,11690)$ | $-$ |  $0$/$0$
$\Xi_{b}\bar{\Sigma}_{b}^{(*)}$/$\Xi_{b}\Sigma_{b}^{(*)}$ | $\frac{3}{2}$ | $-1/-1$ | $(11608,11627)$ | $\rho,\omega,K^{*}$ | $-1,1,0/-1,-1,-2$
| $\frac{1}{2}$ |  |  |  |  $2,1,0$/$2,-1,-2$
$\Xi_{b}\bar{\Xi}_{b}^{{}^{\prime}(*)}$/$\Xi_{b}\Xi_{b}^{{}^{\prime}(*)}$ | 1 | $0/-2$ | $(11729,11748)$ | $\rho,\omega,\phi$ |  $-\frac{1}{2},\frac{1}{2},1$/$-\frac{1}{2},-\frac{1}{2},-1$
| 0 |  |  |  |  $\frac{3}{2},\frac{1}{2},1$/$\frac{3}{2},-\frac{1}{2},-1$
$\Xi_{b}\bar{\Omega}_{b}^{(*)}$/$\Xi_{b}\Omega_{b}^{(*)}$ | $\frac{1}{2}$ | $1/-3$ | $(11841,11864)$ | $\phi,K^{*}$ |  $2,0$/$-2,-2$
$\Sigma_{b}^{(*)}\bar{\Sigma}_{b}^{(*)}$/$\Sigma_{b}^{(*)}\Sigma_{b}^{(*)}$ | 2 | 0/0 | $(11626,11646,11665)$ | $\rho,\omega$ |  $-2,2$/$-2,-2$
| 1 |  |  |  |  $2,2$/$2,-2$
| 0 |  |  |  |  $4,2$/$4,-2$
$\Sigma_{b}^{(*)}\bar{\Xi}^{{}^{\prime}(*)}_{b}$/$\Sigma_{b}^{(*)}\Xi^{{}^{\prime}(*)}_{b}$ | $\frac{3}{2}$ | $1/-1$ | $(11748,11768,11767,11786)$ | $\rho,\omega,K^{*}$ |  $-1,1,0$/$-1,-1,-2$
| $\frac{1}{2}$ |  |  |  |  $2,1,0$/$2,-1,-2$
$\Sigma_{b}^{(*)}\bar{\Omega}^{(*)}_{b}$/$\Sigma_{b}^{(*)}\Omega^{(*)}_{b}$ | 0 | $2/-2$ | $(11859,11879,11883,11903)$ | $K^{*}$ |  $0$/$-4$
$\Xi_{b}^{{}^{\prime}(*)}\bar{\Xi}_{b}^{{}^{\prime}(*)}$/$\Xi_{b}^{{}^{\prime}(*)}\Xi_{b}^{{}^{\prime}(*)}$ | 1 | $0/-2$ | $(11870,11889,11908)$ | $\rho,\omega,\phi$ |  $-\frac{1}{2},\frac{1}{2},1$/$-\frac{1}{2},-\frac{1}{2},-1$
| 0 |  |  |  |  $\frac{3}{2},\frac{1}{2},1$/$\frac{3}{2},-\frac{1}{2},-1$
$\Xi^{{}^{\prime}(*)}_{b}\bar{\Omega}_{b}^{(*)}$/$\Xi^{{}^{\prime}(*)}_{b}\Omega_{b}^{(*)}$ | $\frac{1}{2}$ | $1/-3$ | $(11981,12000,12005,12024)$ | $\phi,K^{*}$ |  $2,0$/$-2,-2$
$\Omega_{b}^{(*)}\bar{\Omega}_{b}^{(*)}$/$\Omega_{b}^{(*)}\Omega_{b}^{(*)}$ | 0 | $0/-4$ | $(12092,12116,12140)$ | $\phi$ |  $4$/$-4$
Table 10: The group theory factor $F$, defined in Eq. (34), for the interaction of bottom-antibottom/bottom-bottom hadron pairs with only the light vector-meson exchanges. Here one of the bottom hadrons is an $s_{\ell}=3/2$ bottom meson. See the caption of Table 7.  System | $I$ | $S$ | Thresholds [MeV] | Exchanged particles | $F$
---|---|---|---|---|---
$B^{(*)}\bar{B}_{1,2}$/$B^{(*)}B_{1,2}$ | 0 | 0/0 | $(11005,11051,11018,11063)$ | $\rho,\omega$ |  $\frac{3}{2},\frac{1}{2}$/$\frac{3}{2},-\frac{1}{2}$
| 1 | 0/0 |  |  |  $-\frac{1}{2},\frac{1}{2}$/ $-\frac{1}{2},-\frac{1}{2}$
$B^{(*)}\bar{B}_{s1,s2}$/$B^{(*)}B_{s1,s2}$ | $\frac{1}{2}$ | $1/-1$ | $(11093,11141,11105,11154)$ | $-$ |  $0$/$0$
$B_{s}^{(*)}\bar{B}_{1,2}$/$B_{s}^{(*)}B_{1,2}$ | $\frac{1}{2}$ | $-1/1$ | $(11108,11153,11119,11165)$ | $-$ |  $0$/$0$
$B^{(*)}_{s}\bar{B}_{s1,s2}$/$B^{(*)}_{s}B_{s1,s2}$ | 0 | 0/$-2$ | $(11196,11207,11244,11255)$ | $\phi$ |  $1$/$-1$
$B_{1,2}\bar{B}_{1,2}$/$B_{1,2}B_{1,2}$ | 0 | 0/0 | $(11452,11464,11477)$ | $\rho,\omega$ |  $\frac{3}{2},\frac{1}{2}$/$\frac{3}{2},-\frac{1}{2}$
| 1 |  |  |  |  $-\frac{1}{2},\frac{1}{2}$/$-\frac{1}{2},-\frac{1}{2}$
$B_{s1,s2}\bar{B}_{1,2}$/$B_{s1,s2}B_{1,2}$ | $\frac{1}{2}$ | 1/1 | $(11555,11566,11567,11578)$ |  |  $0$/0
$B_{s1,s2}\bar{B}_{s1,s2}$/$B_{s1,s2}B_{s1,s2}$ | 0 | $0/-2$ | $(11657,11669,11680)$ | $\phi$ |  $1$/1
$\Lambda_{b}\bar{B}_{1,2}$/$\Lambda_{b}B_{1,2}$ | $\frac{1}{2}$ | 0/0 | $(11346,11358)$ | $\omega$ |  $-1$/1
$\Lambda_{b}\bar{B}_{s1,s2}$/$\Lambda_{b}B_{s1,s2}$ | 0 | $-1/1$ | $(11448,11459)$ | $-$ |  $0$/0
$\Xi_{b}\bar{B}_{1,2}$/$\Xi_{b}B_{1,2}$ | 1 | $-1/-1$ | $(11520,11533)$ | $\rho,\omega$ |  $-\frac{1}{2},-\frac{1}{2}$/$-\frac{1}{2},\frac{1}{2}$
| 0 |  |  |  |  $\frac{3}{2},-\frac{1}{2}$/$\frac{3}{2},\frac{1}{2}$
$\Xi_{b}\bar{B}_{s1,s2}$/$\Xi_{b}B_{s1,s2}$ | $\frac{1}{2}$ | $-2/0$ | $(11623,11634)$ | $\phi$ |  $-1$/1
$\Sigma_{b}^{(*)}\bar{B}_{1,2}$/$\Sigma_{b}^{(*)}B_{1,2}$ | $\frac{3}{2}$ | 0/0 | $(11539,11551,11559,11571)$ | $\rho,\omega$ |  $-1,-1$/$-1,1$
| $\frac{1}{2}$ |  |  |  |  $2,-1$/2,1
$\Sigma_{b}^{(*)}\bar{B}_{s1,s2}$/$\Sigma_{b}^{(*)}B_{s1,s2}$ | 1 | $1/-1$ | $(11642,11653,11661,11672)$ | $-$ |  $0$/0
$\Xi_{b}^{{}^{\prime}(*)}\bar{B}_{1,2}$/$\Xi_{b}^{{}^{\prime}(*)}B_{1,2}$ | 1 | $-1/-1$ | $(11661,11673,11680,11692)$ | $\rho,\omega$ |  $-\frac{1}{2},-\frac{1}{2}$/$-\frac{1}{2},\frac{1}{2}$
| 0 |  |  |  |  $\frac{3}{2},-\frac{1}{2}$/$\frac{3}{2},\frac{1}{2}$
$\Xi_{b}^{{}^{\prime}(*)}\bar{B}_{s1,s2}$/$\Xi_{b}^{{}^{\prime}(*)}B_{s1,s2}$ | $\frac{1}{2}$ | $-2/0$ | $(11764,11775,11783,11794)$ | $\phi$ |  $-1$/1
$\Omega_{b}^{(*)}\bar{B}_{1,2}$/$\Omega_{b}^{(*)}B_{1,2}$ | $\frac{1}{2}$ | $-2/-2$ | $(11772,11784,11796,11808)$ | $-$ |  $0$/0
$\Omega_{b}^{(*)}\bar{B}_{s1,s2}$/$\Omega_{b}^{(*)}B_{s1,s2}$ | 0 | $-3/-1$ | $(11875,11886,11899,11910)$ | $\phi$ |  $-2$/2

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