# Systematics of the heavy flavor hadronic molecules

Kan Chen1 Rui Chen2 Lu Meng3 Bo Wang4,5<EMAIL_ADDRESS>Shi-Lin Zhu1
<EMAIL_ADDRESS>1School of Physics and Center of High Energy Physics, Peking
University, Beijing 100871, China
2Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of
Ministry of Education, Department of Physics and Synergetic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha
410081, China
3Institut für Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum,
Germany
4School of Physical Science and Technology, Hebei University, Baoding 071002,
China
5Key Laboratory of High-precision Computation and Application of Quantum Field
Theory of Hebei Province, Baoding 071002, China

###### Abstract

With a quark level interaction, we give a unified description of the loosely
bound molecular systems composed of the heavy flavor hadrons
$(\bar{D},\bar{D}^{*})$, $(\Lambda_{c},\Sigma_{c},\Sigma_{c}^{*})$, and
$(\Xi_{c},\Xi_{c}^{\prime},\Xi_{c}^{*})$. Using the $P_{c}$ states as inputs
to fix the interaction strength of light quark-quark pairs, we reproduce the
observed $P_{cs}$ and $T_{cc}^{+}$ states and predict another narrow
$T_{cc}^{\prime+}$ state with quantum numbers $[D^{*}D^{*}]_{J=1}^{I=0}$. If
we require a satisfactory description of the $T_{cc}^{+}$ and $P_{c}$ states
simultaneously, our framework prefers the assignments of the $P_{c}(4440)$ and
$P_{c}(4457)$ as the $[\Sigma_{c}\bar{D}^{*}]_{J=1/2}^{I=1/2}$ and
$[\Sigma_{c}\bar{D}^{*}]_{J=3/2}^{I=1/2}$ states, respectively. We propose the
isospin criterion to explain naturally why the experimentally observed
$T_{cc}$, $P_{c}$, and $P_{cs}$ molecular candidates prefer the lowest isospin
numbers. We also predict the loosely bound states for the bottom di-hadrons.

##  I Introduction

The conventional mesons ($q\bar{q}$) and baryons ($qqq$) have been extensively
discussed at the birth of quark model GellMann:1964nj ; Zweig:19xx ;
Jaffe:1976ig , and have become the main part of hadron spectrum nowadays
ParticleDataGroup:2020ssz . Besides conventional mesons and baryons, quantum
chromodynamics (QCD) also allows the existence of hadrons with more
complicated configurations, such as $qq\bar{q}\bar{q}$, $\bar{q}qqqq$,
$qqqqqq$ ($qqq\bar{q}\bar{q}\bar{q}$), etc.. In the past decades, many $XYZ$
states have been observed Chen:2016qju ; Liu:2019zoy ; Liu:2013waa ;
Guo:2017jvc ; Hosaka:2016pey ; Lebed:2016hpi ; Esposito:2016noz ;
Brambilla:2019esw  since the discovery of $X(3872)$ in 2003 Belle:2003nnu .
Some of these states are below the thresholds of di-hadrons from several to
several tens MeVs. The molecular explanations were widely proposed to
understand their underlying structures.

In 2015, the LHCb Collaboration reported two structures $P_{c}(4380)$ and
$P_{c}(4450)$ in the $J/\psi p$ mass spectrum LHCb:2015yax . Their masses are
consistent with the predictions of the hidden-charm pentaquarks Wu:2010jy ;
Yang:2011wz ; Wang:2011rga . In 2019, the LHCb Collaboration updated their
analyses with larger data samples and found that the $P_{c}(4450)$ consists of
two $P_{c}$ states, i.e., $P_{c}(4440)$ and $P_{c}(4457)$ LHCb:2019kea .
Besides, they further reported another near threshold hidden-charm pentaquark
$P_{c}(4312)$. These important results from LHCb provide strong evidences for
the existence of the hidden-charm molecular pentaquarks Chen:2019asm ;
Liu:2019tjn ; He:2019ify ; Xiao:2019aya ; Meng:2019ilv ; Yamaguchi:2019seo ;
PavonValderrama:2019nbk ; Chen:2019bip ; Burns:2019iih ; Du:2019pij ;
Wang:2019ato . Thus, it is desirable to see whether there exist the hadronic
molecules in other heavy flavor di-hadron systems.

If enlarging the flavor symmetry group to SU(3), one may expect that the
$\Xi_{c}\bar{D}^{(*)}$, $\Xi^{\prime}_{c}\bar{D}^{(*)}$, and
$\Xi_{c}^{*}\bar{D}^{(*)}$ systems may also form molecular bound states. The
$P_{cs}$ pentaquarks were investigated in Refs. Wu:2010jy ; Chen:2016ryt ;
Santopinto:2016pkp ; Shen:2019evi ; Xiao:2019gjd ; Wang:2019nvm  and the most
promising production channel $\Xi_{b}^{-}\rightarrow J/\psi\Lambda K$ was
suggested in Refs. Santopinto:2016pkp ; Chen:2015sxa . Later, the LHCb
reported the evidence of $P_{cs}(4459)$ in the $J/\psi\Lambda$ invariant mass
spectrum LHCb:2020jpq , which agrees very well with the prediction from the
chiral effective field theory in Ref. Wang:2019nvm . However, this state still
needs further confirmation due to limited data samples at present LHCb:2020jpq
. Very probably, the $P_{c}$ and $P_{cs}$ pentaquarks share a very similar
binding mechanism.

Very recently, the LHCb reported a very narrow structure $T_{cc}^{+}(3875)$ in
the $D^{0}D^{0}\pi^{+}$ spectrum LHCb:2021auc ; LHCb:2021vvq . Its mass is
slightly below the $D^{*+}D^{0}$ threshold about $300$ keV. This signal tends
to confirm the predictions of the $DD^{*}$ molecular state with quantum
numbers $I(J^{P})=0(1^{+})$ Li:2012ss . The doubly heavy tetraquark states
have been extensively studied and debated in the literatures Carlson:1987hh ;
Silvestre-Brac:1993zem ; Semay:1994ht ; Pepin:1996id ; Gelman:2002wf ;
Vijande:2003ki ; Janc:2004qn ; Cui:2006mp ; Navarra:2007yw ; Vijande:2007rf ;
Ebert:2007rn ; Lee:2009rt ; Yang:2009zzp ; Du:2012wp ; Feng:2013kea ;
Ikeda:2013vwa ; Molina:2010tx ; Carames:2011zz ; Sakai:2017avl ; Luo:2017eub ;
Karliner:2017qjm ; Eichten:2017ffp ; Wang:2017uld ; Cheung:2017tnt ;
Park:2018wjk ; Li:2012ss ; Xu:2017tsr . One can refer to Ref. Liu:2019zoy  for
a review of the $QQ\bar{q}\bar{q}$ system. This inspiring discovery also
stimulated a series of theoretical studies Li:2021zbw ; Weng:2021hje ;
Agaev:2021vur ; Chen:2021vhg ; Dai:2021wxi ; Feijoo:2021ppq ; Ling:2021bir ;
Meng:2021jnw ; Wu:2021kbu ; Yan:2021wdl ; Hu:2021gdg ; Jin:2021cxj ;
Yang:2021zhe ; Ren:2021dsi ; Chen:2021tnn ; Azizi:2021aib ; Fleming:2021wmk ;
Dong:2021bvy .

The minimal valance quark components for the $P_{c}$, $P_{cs}$, and
$T_{cc}^{+}$ states are $c\bar{c}uud$, $c\bar{c}uds$, and $cc\bar{u}\bar{d}$,
respectively. Such states can not be accommodated within the conventional
quark model and thus give us a golden platform to study the structures and
dynamics of the multiquark states.

We propose the following picture to understand the above heavy flavor di-
hadron systems. The heavy quark ($c$ or $b$) behaves like a static color
triplet source in each hadron. In the heavy quark limit, its velocity $v$ does
not change with time. The non-relativistic property of heavy quarks is
stimulative to stabilize a molecular system Lee:2011rka ; Li:2012bt ;
Li:2014gra . On the other hand, if the particular combinations of the light
quark components within the two heavy hadrons can coincidentally provide the
attractive force, then the residual strong interaction that mainly comes from
their light degrees of freedom (d.o.f) may render this system to be bound.
This picture is similar to a hydrogen molecule in QED, where the two electrons
are shared by the two protons and the residual electromagnetic force binds
this system. Therefore, the question of which heavy flavor di-hadron can form
a bound state becomes which kind of light quark combinations in the di-hadron
system can provide the enough attractive force.

In this work, we adopt the quark level interaction to address the above
question. We relate the heavy flavor di-hadron effective potentials to their
matrix elements in the flavor and spin spaces of their light d.o.f, so that we
can qualitatively determine which heavy flavor di-hadron is possible to form a
bound state.

This paper is organized as follows. In Sec. II, we introduce our theoretical
framework. Then we present our numerical results and discussions in Sec. III.
In Sec. IV, we conclude this work with a short summary.

##  II Theoretical framework

We consider the two-body systems that are the combinations of the ground-state
hadrons $(\bar{D},\bar{D}^{*})$, $(\Lambda_{c},\Sigma_{c},\Sigma_{c}^{*})$,
and $(\Xi_{c},\Xi_{c}^{\prime},\Xi_{c}^{*})$. We list the physical allowed
heavy flavor meson-meson, meson-baryon, and baryon-baryon systems
$[H_{1}H_{2}]_{J}^{I}$ in Table 1, where $H_{1}$ and $H_{2}$ denote the
considered heavy flavor hadrons, while $J$ and $I$ are the total angular
momentum and total isospin of the di-hadron system, respectively. If $H_{1}$
and $H_{2}$ are the general identical particles (e.g., $\bar{D}\bar{D}$ and
$\bar{D}^{*}\bar{D}^{*}$), then the quantum numbers of the
$[H_{1}H_{2}]_{J}^{I}$ (fermion or boson) systems must satisfy the following
selection rule

$\displaystyle L+S_{\text{tot}}+I_{\text{tot}}+2i=\text{Even number},$ (1)

where $L$ is the orbital angular momentum between $H_{1}$ and $H_{2}$ ($L=0$
for the $s$-wave case in our calculations), while $S_{\text{tot}}$ and
$I_{\text{tot}}$ are the total spin and total isospin of the general identical
di-hadron system, respectively. $i$ denotes the isospin of $H_{1}$ or $H_{2}$,
e.g., $i=1/2$ for the $D$ meson, and $i=1$, $\frac{1}{2}$ for the $\Sigma_{c}$
and $\Xi_{c}$ baryons, respectively. The $2s$ ($s$ is the spin of $H_{1}$ or
$H_{2}$) is omitted from Eq. (1) since it must be an odd or even number for
the two-body systems of the identical fermions or bosons, respectively.

Table 1: The allowed heavy flavor di-hadron systems ($[H_{1}H_{2}]^{I}_{J}$) that are considered in this work, where the quantum numbers of the general identical systems are constrained by Eq. (1). Meson-meson | $[\bar{D}\bar{D}]^{1}_{0}$ | $[\bar{D}\bar{D}^{*}]^{0,1}_{1}$ | $[\bar{D}^{*}\bar{D}^{*}]^{1}_{0,2}$
---|---|---|---
$[\bar{D}^{*}\bar{D}^{*}]^{0}_{1}$ |  |
Baryon-meson | $[\Lambda_{c}\bar{D}]^{\frac{1}{2}}_{\frac{1}{2}}$ | $[\Lambda_{c}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2}}^{\frac{1}{2}}$ | $[\Sigma_{c}\bar{D}]^{\frac{1}{2},\frac{3}{2}}_{\frac{1}{2}}$
$[\Sigma_{c}D^{*}]_{\frac{1}{2},\frac{3}{2}}^{\frac{1}{2},\frac{3}{2}}$ | $[\Sigma_{c}^{*}\bar{D}]_{\frac{3}{2}}^{\frac{1}{2},\frac{3}{2}}$ | $[\Sigma^{*}_{c}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2},\frac{5}{2}}^{\frac{1}{2},\frac{3}{2}}$
$[\Xi_{c}\bar{D}]_{\frac{1}{2}}^{0,1}$ | $[\Xi_{c}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2}}^{0,1}$ | $[\Xi_{c}^{\prime}\bar{D}]_{\frac{1}{2}}^{0,1}$
$[\Xi_{c}^{\prime}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2}}^{0,1}$ | $[\Xi_{c}^{*}\bar{D}]_{\frac{3}{2}}^{0,1}$ | $[\Xi_{c}^{*}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2},\frac{5}{2}}^{0,1}$
Baryon-baryon | $[\Lambda_{c}\Lambda_{c}]^{0}_{0}$ | $[\Lambda_{c}\Sigma_{c}]^{1}_{0,1}$ | $[\Sigma_{c}\Sigma_{c}]^{0,2}_{0}$
$[\Sigma_{c}\Sigma_{c}]^{1}_{1}$ | $[\Lambda_{c}\Sigma_{c}^{*}]^{1}_{1,2}$ | $[\Sigma_{c}\Sigma_{c}^{*}]_{1,2}^{0,1,2}$
$[\Sigma_{c}^{*}\Sigma_{c}^{*}]_{1,3}^{1}$ | $[\Sigma_{c}^{*}\Sigma_{c}^{*}]_{0,2}^{0,2}$ | $[\Xi_{c}\Xi_{c}]_{0}^{1}$
$[\Xi_{c}\Xi_{c}]_{1}^{0}$ | $[\Xi_{c}\Xi_{c}^{\prime}]_{0,1}^{0,1}$ | $[\Xi_{c}\Xi_{c}^{*}]_{1,2}^{0,1}$
$[\Xi_{c}^{\prime}\Xi_{c}^{\prime}]_{0}^{1}$ | $[\Xi_{c}^{\prime}\Xi_{c}^{\prime}]_{1}^{0}$ | $[\Xi_{c}^{\prime}\Xi_{c}^{*}]_{1,2}^{0,1}$
$[\Xi_{c}^{*}\Xi_{c}^{*}]^{0}_{1,3}$ | $[\Xi_{c}^{*}\Xi_{c}^{*}]^{1}_{0,2}$ |

As discussed in the introduction, we assume that the interactions in a heavy
flavor di-hadron systems are mainly from the interactions of their light quark
components. We neglect the corrections from the heavy d.o.f and study the
residual strong interactions induced from their light d.o.f. Then the heavy
flavor meson-meson, meson-baryon, and baryon-baryon can be studied
simultaneously in the same formalism by checking the interactions with all
possible quantum numbers.

We focus on the two-body $s$-wave interactions among the considered ground
heavy flavor mesons/baryons. One can formulate the corresponding quark level
Lagrangians as Meng:2019nzy ; Wang:2019nvm ; Wang:2020dhf

$\displaystyle\mathcal{L}=g_{s}\bar{q}\mathcal{S}q+g_{a}\bar{q}\gamma_{\mu}\gamma^{5}\mathcal{A}^{\mu}q,$
(2)

where $q=(u,d,s)$, $g_{s}$ and $g_{a}$ are two independent coupling constants.
They encode the nonperturbative dynamics between light quarks of two color
singlet hadrons and can be determined from the experimental data.

From Eq. (2), we can see that the systems listed in Table 1 can only couple to
the isospin triplet and isospin singlet fields. The systems that can couple to
the strange isospin doublet fields are not considered in the present work.
Then the fictitious scalar field $\mathcal{S}$ and axial-vector field
$\mathcal{A}^{\mu}$ reduce to the form

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\mathcal{S}_{3}\lambda^{i}+\mathcal{S}_{1}\lambda^{8},$ (3)
$\displaystyle\mathcal{A}^{\mu}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{3}^{\mu}\lambda^{i}+\mathcal{A}^{\mu}_{1}\lambda^{8},$
(4)

where $\lambda^{i}$ ($i=1,2,3$) and $\lambda^{8}$ are the generators of SU(3)
group. $\mathcal{S}_{3}$ ($\mathcal{A}_{3}^{\mu}$) and $\mathcal{S}_{1}$
($\mathcal{A}_{1}^{\mu}$) denote the isospin triplet and isospin singlet
fields, respectively.

The effective potential of light quark-quark interactions can be deduced from
Eq. (2), and we have

$\displaystyle V_{qq}$ $\displaystyle=$
$\displaystyle\tilde{g}_{s}\left(\lambda_{1}^{8}\lambda_{2}^{8}+\lambda_{1}^{i}\lambda_{2}^{i}\right)+\tilde{g}_{a}\left(\lambda_{1}^{8}\lambda_{2}^{8}+\lambda_{1}^{i}\lambda_{2}^{i}\right)\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}.$

where the effective potential $V_{qq}$ is reduced to the local form when we
integrate out the exchanged spurions (which is analogous to the resonance
saturation model Epelbaum:2001fm ). The redefined coupling constants are
$\tilde{g}_{s}=g_{s}^{2}/m_{\mathcal{S}}^{2}$ and
$\tilde{g}_{a}=g_{a}^{2}/m_{\mathcal{A}}^{2}$. Then the heavy flavor di-hadron
effective potential from the interactions of their light quark components can
be written as

$\displaystyle
V_{[H_{1}H_{2}]_{J}^{I}}=\left\langle[H_{1}H_{2}]_{J}^{I}\left|V_{qq}\right|[H_{1}H_{2}]_{J}^{I}\right\rangle,$
(6)

where $|[H_{1}H_{2}]_{J}^{I}\rangle$ denotes the quark-level spin-flavor wave
function of $H_{1}H_{2}$ system with total isospin $I$ and total angular
momentum $J$, which is the direct product of spin and flavor wave functions

$\displaystyle\left|[H_{1}H_{2}]_{J}^{I}\right\rangle$ $\displaystyle=$
$\displaystyle\sum_{m_{I_{1}},m_{I_{2}}}C_{I_{1},m_{I_{1}};I_{2},m_{I_{2}}}^{I,I_{z}}\phi^{H_{1f}}_{I_{1},m_{I_{1}}}\phi^{H_{2f}}_{I_{2},m_{I_{2}}}$
(7)
$\displaystyle\otimes\sum_{m_{l_{1}},m_{l_{2}}}C_{l_{1},m_{l_{1}};l_{2},m_{l_{2}}}^{l,l_{z}}\phi^{H_{1s}}_{l_{1},m_{l_{1}}}\phi^{H_{2s}}_{l_{2},m_{l_{2}}}.$

The $C_{I_{1},m_{I_{1}};I_{2},m_{I_{2}}}^{I,I_{z}}$ and
$C_{l_{1},m_{l_{1}};l_{2},m_{l_{2}}}^{l,l_{z}}$ are the Clebsch-Gordan (CG)
coefficients. ($\phi^{H_{1f}}_{I_{1},m_{I_{1}}}$,
$\phi^{H_{1s}}_{l_{1},m_{l_{1}}}$), ($\phi^{H_{2f}}_{I_{2},m_{I_{2}}}$,
$\phi^{H_{2s}}_{l_{2},m_{l_{2}}}$) are the quark-level (flavor, spin) wave
functions for the $H_{1}$ and $H_{2}$ hadrons, respectively. $l$ denotes the
total light spin of this two-body system.

From Eqs. (LABEL:Vqq) and (6), we can see that the residual strong interaction
of a specific $[H_{1}H_{2}]_{J}^{I}$ system can be divided into four parts,
i.e., the scalar type ($\lambda^{8}\lambda^{8}$), the isospin related type
($\lambda^{i}\lambda^{i}$), the spin related type
($\lambda^{8}\lambda^{8}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}$) and the isospin-
spin related type
$(\lambda^{i}\lambda^{i}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2})$ interactions. In
Table 2, we present the matrix elements of these four types of operators for
the considered di-hadron systems in Table 1.

Table 2: The matrix elements of the operators $\mathcal{O}_{1}$ ($\lambda^{8}_{1}\lambda^{8}_{2}$), $\mathcal{O}_{2}$ ($\lambda^{i}_{1}\lambda^{i}_{2}$), $\mathcal{O}_{3}$ ($\lambda^{8}_{1}\lambda^{8}_{2}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}$), and $\mathcal{O}_{4}$ ($\lambda^{i}_{1}\lambda^{i}_{2}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}$) for the considered heavy flavor hadron-hadron systems ($[H_{1}H_{2}]_{J}^{I}$) listed in Table 1.  System | $\mathcal{O}_{1}$ | $\mathcal{O}_{2}$ | $\mathcal{O}_{3}$ | $\mathcal{O}_{4}$ | System | $\mathcal{O}_{1}$ | $\mathcal{O}_{2}$ | $\mathcal{O}_{3}$ | $\mathcal{O}_{4}$
---|---|---|---|---|---|---|---|---|---
$[\bar{D}\bar{D}]_{0}^{1}$ | $\frac{1}{3}$ | $1$ | $0$ | $0$ | $[\bar{D}\bar{D}^{*}]^{0,1}_{1}$ | $\frac{1}{3},\frac{1}{3}$ | $-3,1$ | $0,0$ | $0,0$
$[\bar{D}^{*}\bar{D}^{*}]_{0,2}^{1}$ | $\frac{1}{3},\frac{1}{3}$ | $1,1$ | $-\frac{2}{3},\frac{1}{3}$ | $-2,1$ | $[\bar{D}^{*}\bar{D}^{*}]_{1}^{0}$ | $\frac{1}{3}$ | $-3$ | $-\frac{1}{3}$ | $3$
$[\Lambda_{c}\bar{D}]^{\frac{1}{2}}_{\frac{1}{2}}$ | $\frac{2}{3}$ | $0$ | $0$ | $0$ | $[\Lambda_{c}\bar{D}^{*}]^{\frac{1}{2}}_{\frac{1}{2},\frac{3}{2}}$ | $\frac{2}{3},\frac{2}{3}$ | $0,0$ | $0,0$ | $0,0$
$[\Sigma_{c}\bar{D}]_{\frac{1}{2}}^{\frac{1}{2},\frac{3}{2}}$ | $\frac{2}{3},\frac{2}{3}$ | $-4,2$ | $0,0$ | $0,0$ | $[\Sigma_{c}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2}}^{\frac{1}{2}}$ | $\frac{2}{3},\frac{2}{3}$ | $-4,-4$ | $-\frac{8}{9},\frac{4}{9}$ | $\frac{16}{3},-\frac{8}{3}$
$[\Sigma_{c}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2}}^{\frac{3}{2}}$ | $\frac{2}{3},\frac{2}{3}$ | $2,2$ | $-\frac{8}{9},\frac{4}{9}$ | $-\frac{8}{3},\frac{4}{3}$ | $[\Sigma_{c}^{*}\bar{D}]_{\frac{3}{2}}^{\frac{1}{2},\frac{3}{2}}$ | $\frac{2}{3},\frac{2}{3}$ | $-4,2$ | $0,0$ | $0,0$
$[\Sigma_{c}^{*}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2},\frac{5}{2}}^{\frac{1}{2}}$ | $\frac{2}{3},\frac{2}{3},\frac{2}{3}$ | $-4,-4,-4$ | $-\frac{10}{9},-\frac{4}{9},\frac{2}{3}$ | $\frac{20}{3},\frac{8}{3},-4$ | $[\Sigma_{c}^{*}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2},\frac{5}{2}}^{\frac{3}{2}}$ | $\frac{2}{3},\frac{2}{3},\frac{2}{3}$ | $2,2,2$ | $-\frac{10}{9},-\frac{4}{9},\frac{2}{3}$ | $-\frac{10}{3},-\frac{4}{3},2$
$[\Xi_{c}\bar{D}]_{\frac{1}{2}}^{0,1}$ | $-\frac{1}{3},-\frac{1}{3}$ | $-3,1$ | $0,0$ | $0,0$ | $[\Xi_{c}\bar{D}^{*}]^{0}_{\frac{1}{2},\frac{3}{2}}$ | $-\frac{1}{3},-\frac{1}{3}$ | $-3,-3$ | $0,0$ | $0,0$
$[\Xi_{c}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2}}^{1}$ | $-\frac{1}{3},-\frac{1}{3}$ | $1,1$ | $0,0$ | $0,0$ | $[\Xi_{c}^{\prime}\bar{D}]_{\frac{1}{2}}^{0,1}$ | $-\frac{1}{3},-\frac{1}{3}$ | $-3,1$ | $0,0$ | $0,0$
$[\Xi_{c}^{\prime}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2}}^{0}$ | $-\frac{1}{3},-\frac{1}{3}$ | $-3,-3$ | $\frac{4}{9},-\frac{2}{9}$ | $4,-2$ | $[\Xi_{c}^{\prime}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2}}^{1}$ | $-\frac{1}{3},-\frac{1}{3}$ | $1,1$ | $\frac{4}{9},-\frac{2}{9}$ |  -$\frac{4}{3},\frac{2}{3}$
$[\Xi_{c}^{*}\bar{D}]_{\frac{3}{2}}^{0,1}$ | $-\frac{1}{3},-\frac{1}{3}$ | $-3,1$ | $0,0$ | $0,0$ | $[\Xi_{c}^{*}\bar{D}^{*}]^{0}_{\frac{1}{2},\frac{3}{2},\frac{5}{2}}$ | $-\frac{1}{3},-\frac{1}{3},-\frac{1}{3}$ | $-3,-3,-3$ | $\frac{5}{9},\frac{2}{9},-\frac{1}{3}$ | $5,2,-3$
$[\Xi_{c}^{*}\bar{D}^{*}]_{\frac{1}{2},\frac{3}{2},\frac{5}{2}}^{1}$ | $-\frac{1}{3},-\frac{1}{3},-\frac{1}{3}$ | $1,1,1$ | $\frac{5}{9},\frac{2}{9},-\frac{1}{3}$ | $-\frac{5}{3},-\frac{2}{3},1$ |  |  |  |  |
$[\Lambda_{c}\Lambda_{c}]_{0}^{0}$ | $\frac{4}{3}$ | $0$ | $0$ | $0$ | $[\Lambda_{c}\Sigma_{c}]_{0,1}^{1}$ | $\frac{4}{3},\frac{4}{3}$ | $0,0$ | $0,0$ | $0,0$
$[\Sigma_{c}\Sigma_{c}]_{0}^{0,2}$ | $\frac{4}{3},\frac{4}{3}$ | $-8,4$ | $-\frac{16}{9},-\frac{16}{9}$ | $\frac{32}{3},-\frac{16}{3}$ | $[\Sigma_{c}\Sigma_{c}]_{1}^{1}$ | $\frac{4}{3}$ | $-4$ | $\frac{16}{27}$ | $-\frac{16}{9}$
$[\Lambda_{c}\Sigma_{c}^{*}]_{1,2}^{1}$ | $\frac{4}{3},\frac{4}{3}$ | $0,0$ | $0,0$ | $0,0$ | $[\Sigma_{c}\Sigma_{c}^{*}]^{0}_{1,2}$ | $\frac{4}{3},\frac{4}{3}$ | $-8,-8$ | $-\frac{40}{27},\frac{8}{9}$ | $\frac{80}{9},-\frac{16}{3}$
$[\Sigma_{c}\Sigma_{c}^{*}]^{1}_{1,2}$ | $\frac{4}{3},\frac{4}{3}$ | $-4,-4$ | $-\frac{40}{27},\frac{8}{9}$ | $\frac{40}{9},-\frac{8}{3}$ | $[\Sigma_{c}\Sigma_{c}^{*}]^{2}_{1,2}$ | $\frac{4}{3},\frac{4}{3}$ | $4,4$ | $-\frac{40}{27},\frac{8}{9}$ | $-\frac{40}{9},\frac{8}{3}$
$[\Sigma_{c}^{*}\Sigma_{c}^{*}]^{1}_{1,3}$ | $\frac{4}{3},\frac{4}{3}$ | $-4,-4$ | $-\frac{44}{27},\frac{4}{3}$ | $\frac{44}{9},-4$ | $[\Sigma_{c}^{*}\Sigma_{c}^{*}]^{0}_{0,2}$ | $\frac{4}{3},\frac{4}{3}$ | $-8,-8$ | $-\frac{20}{9},-\frac{4}{9}$ | $\frac{40}{3},\frac{8}{3}$
$[\Sigma_{c}^{*}\Sigma_{c}^{*}]^{2}_{0,2}$ | $\frac{4}{3},\frac{4}{3}$ | $4,4$ | $-\frac{20}{9},-\frac{4}{9}$ | $-\frac{20}{3},-\frac{4}{3}$ |  |  |  |  |
$[\Xi_{c}\Xi_{c}]_{0}^{1}$ | $\frac{1}{3}$ | $1$ | $0$ | $0$ | $[\Xi_{c}\Xi_{c}]_{1}^{0}$ | $\frac{1}{3}$ | $-3$ | $0$ | $0$
$[\Xi_{c}\Xi_{c}^{\prime}]_{0,1}^{0}$ | $\frac{1}{3},\frac{1}{3}$ | $-3,-3$ | $0,0$ | $0,0$ | $[\Xi_{c}\Xi_{c}^{\prime}]_{0,1}^{1}$ | $\frac{1}{3},\frac{1}{3}$ | $1,1$ | $0,0$ | $0,0$
$[\Xi_{c}\Xi_{c}^{*}]_{1,2}^{0}$ | $\frac{1}{3},\frac{1}{3}$ | $-3,-3$ | $0,0$ | $0,0$ | $[\Xi_{c}\Xi_{c}^{*}]_{1,2}^{1}$ | $\frac{1}{3},\frac{1}{3}$ | $1,1$ | $0,0$ | $0,0$
$[\Xi_{c}^{\prime}\Xi_{c}^{\prime}]_{0}^{1}$ | $\frac{1}{3}$ | $1$ | $-\frac{4}{9}$ | $-\frac{4}{3}$ | $[\Xi_{c}^{\prime}\Xi_{c}^{\prime}]_{1}^{0}$ | $\frac{1}{3}$ | $-3$ | $\frac{4}{27}$ | $-\frac{4}{3}$
$[\Xi_{c}^{\prime}\Xi_{c}^{*}]_{1,2}^{0}$ | $\frac{1}{3},\frac{1}{3}$ | $-3,-3$ | $-\frac{10}{27},\frac{2}{9}$ | $\frac{10}{3},-2$ | $[\Xi_{c}^{\prime}\Xi_{c}^{*}]_{1,2}^{1}$ | $\frac{1}{3},\frac{1}{3}$ | $1,1$ | $-\frac{10}{27},\frac{2}{9}$ | $-\frac{10}{9},\frac{2}{3}$
$[\Xi_{c}^{*}\Xi_{c}^{*}]_{1,3}^{0}$ | $\frac{1}{3},\frac{1}{3}$ | $-3,-3$ | $-\frac{11}{27},\frac{1}{3}$ | $\frac{11}{3},-3$ | $[\Xi_{c}^{*}\Xi_{c}^{*}]_{0,2}^{1}$ | $\frac{1}{3},\frac{1}{3}$ | $1,1$ | $-\frac{5}{9},-\frac{1}{9}$ | $-\frac{5}{3},-\frac{1}{3}$

After we obtain the effective potential of the $[H_{1}H_{2}]_{J}^{I}$ system,
we need to check whether this system can form a bound state. This can be
achieved by solving the following Lippmann-Schwinger equation (LSE),

$\displaystyle
T(p^{\prime},p)=V(p^{\prime},p)+\int\frac{d^{3}q}{\left(2\pi\right)^{3}}\frac{V\left(p^{\prime},q\right)T\left(q,p\right)}{E-\frac{q^{2}}{2m_{\mu}}+i\epsilon},$
(8)

where $m_{\mu}$ is the reduced mass of the $H_{1}$ and $H_{2}$. $p$ and
$p^{\prime}$ represent the momentum of the initial and final states in the
center of mass frame, respectively.

Here, we introduce a hard regulator to exclude the contributions from higher
momenta Meng:2021jnw ; Meng:2021kmi

$\displaystyle
V(p,p^{\prime})=V_{[H_{1}H_{2}]_{J}^{I}}\Theta\left(\Lambda-p\right)\Theta\left(\Lambda-p^{\prime}\right),$
(9)

where $\Theta$ is the step function. The amplitude $T(p^{\prime},p)$ is a
function of $p^{\prime}$, $p$, and binding energy $E$ with a separable form

$\displaystyle
T\left(p^{\prime},p\right)=\beta(E)\Theta(\Lambda-p^{\prime})\Theta(\Lambda-p).$
(10)

Then the LSE can be reduced to an algebraic equation

$\displaystyle\beta(E)=\frac{V_{[H_{1}H_{2}]_{J}^{I}}}{1-V_{[H_{1}H_{2}]_{J}^{I}}G},$
(11)

with

$\displaystyle
G=\frac{m_{\mu}}{\pi^{2}}\left[-\Lambda+k\tan^{-1}\left(\frac{\Lambda}{k}\right)\right],k=\sqrt{-2m_{\mu}E}.$

We can search for the pole position of Eq. (11) to obtain the binding energy
of the $[H_{1}H_{2}]_{J}^{I}$ system.

##  III Numerical results

###  III.1 The results of the $P_{c}$, $P_{cs}$, and $T_{cc}$ states

We first use the masses of the $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$
in Ref. LHCb:2019kea  to fix the parameters in our model. In our previous work
Wang:2019ato , we suggested that the $P_{c}(4312)$, $P_{c}(4440)$, and
$P_{c}(4457)$ have the assignments $[\Sigma_{c}\bar{D}]_{1/2}^{1/2}$,
$[\Sigma_{c}\bar{D}^{*}]_{1/2}^{1/2}$, and
$[\Sigma_{c}\bar{D}^{*}]_{3/2}^{1/2}$, respectively. With the matrix elements
listed in Table 2, we can easily read out the effective potentials for these
three $P_{c}$ states,

$\displaystyle V_{P_{c}(4312)}$ $\displaystyle=$
$\displaystyle-\frac{10}{3}\tilde{g}_{s},$ (13) $\displaystyle
V_{P_{c}(4440)}$ $\displaystyle=$
$\displaystyle-\frac{10}{3}\tilde{g}_{s}+\frac{40\tilde{g}_{a}}{9},$ (14)
$\displaystyle V_{P_{c}(4457)}$ $\displaystyle=$
$\displaystyle-\frac{10}{3}\tilde{g}_{s}-\frac{20\tilde{g}_{a}}{9}.$ (15)

There exist three undetermined parameters in Eq. (11), the light quark-quark
coupling constants $\tilde{g}_{s}$, $\tilde{g}_{a}$, and the momentum cutoff
$\Lambda$. We use the experimental mass of the $P_{c}$ states LHCb:2019kea  to
precisely extract these three parameters. The solutions are
$\tilde{g}_{s}=11.739$ GeV-2, $\tilde{g}_{a}=-2.860$ GeV-2, and
$\Lambda=0.409$ GeV. In our convention, a positive (negative)
$V_{[H_{1}H_{2}]_{J}^{I}}$ means a(n) repulsive (attractive) interaction. Once
we determine the signs of the $\tilde{g}_{s}$ and $\tilde{g}_{a}$, we can
directly find out whether the considered systems have repulsive or attractive
forces from the values in Table 2.

From the point of view of the potential model, the spin-spin interaction is
suppressed by a factor of $1/(m_{\Sigma_{c}^{(*)}}m_{\bar{D}^{(*)}})$, which
is roughly consistent with our obtained ratio
$|\tilde{g}_{a}|/|\tilde{g}_{s}|\approx 0.24$.

The cutoff $\Lambda$ is smaller than the masses of the ground scalar or axial-
vector mesons, which are regarded as the hard scales and integrated out in the
effective field theory. In principle, there may exist contributions from the
pion-exchange interactions. Although the two ground heavy meson/baryons can
easily couple to the pion field via the $p$-wave interactions, our
calculations based on the chiral effective field theory Chen:2021htr ;
Meng:2019ilv ; Wang:2019ato ; Wang:2019nvm ; Wang:2020dhf  showed that the
magnitude of the one-pion-exchange (OPE) interaction is comparable to that of
the next-to-leading order two-pion-exchange (TPE) interaction. The OPE and TPE
have considerable corrections to the binding energies of the bound states, but
they are not the main driving force of the formation of the bound states. In
this work, we do not include the pion exchange dynamics.

In Table 3, we list the masses of the experimentally observed molecular
candidates and the results from our model. The center values of the
$P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ are used as inputs to
determine the values of $\tilde{g}_{s}$, $\tilde{g}_{a}$, and $\Lambda$. We
also predict a $[\Sigma_{c}^{*}\bar{D}]^{1/2}_{3/2}$ molecular state with the
mass 4376.2 MeV, which may correspond to the observed $P_{c}(4380)$ state
LHCb:2015yax . The bound state $[\Sigma_{c}^{*}\bar{D}]^{1/2}_{3/2}$ is also
obtained in different models Chen:2019asm ; Liu:2019tjn ; Xiao:2019aya ;
Yamaguchi:2019seo ; PavonValderrama:2019nbk ; Chen:2019bip ; Du:2019pij ;
Wang:2019ato .

Table 3: The experimental data LHCb:2015yax ; LHCb:2019kea ; LHCb:2020jpq ; LHCb:2021auc ; LHCb:2021vvq  and our results of the masses and binding energies (BE) for the $T_{cc}^{+}$, $P_{c}(4312)$, $P_{c}(4380)$, $P_{c}(4440)$, $P_{c}(4457)$, and $P_{cs}(4459)$. We adopt the isospin averaged masses for the single-charm mesons and baryons ParticleDataGroup:2020ssz . The listed values are all in units of MeV.  | Mass (Expt.) | BE (Expt.) | Mass (Our) | BE (Our)
---|---|---|---|---
$T_{cc}(3875)^{+}$ | $3874.8$ | $-1.0$ | $3874.5$ | $-1.8$
$P_{c}(4312)^{+}$ | $4311.9\pm 0.7^{+6.8}_{-0.6}$ |  $-8.9$ (input) | $4311.9$ | $-8.9$
$P_{c}(4380)^{+}$ | $4380\pm 8\pm 29$ | $-6.2$ | $4376.2$ | $-9.1$
$P_{c}(4440)^{+}$ | $4440.3\pm 1.3^{+4.1}_{-4.7}$ |  $-21.8$ (input) | $4440.2$ | $-21.8$
$P_{c}(4457)^{+}$ | $4457.3\pm 0.6^{+4.1}_{-1.7}$ |  $-4.8$ (input) | $4457.3$ | $-4.8$
$P_{cs}(4459)^{0}$ | $4458.8\pm 2.9^{+4.7}_{-1.1}$ | $-19.7$ | $4468.1$ | $-10.0$

We further adopt our picture to study the recently observed $T_{cc}^{+}$
state. We assign the $T_{cc}^{+}$ as the $[DD^{*}]_{1}^{0}$ molecular state
and calculate its mass with the same parameters extracted from the $P_{c}$
states. As shown in Table 3, our approach gives a rather good description of
the mass of $T_{cc}^{+}$. This nice agreement indicates that neglecting the
corrections from the heavy degrees of freedom is a fairly good approximation
in this case. The heavy flavor meson-meson and baryon-baryon systems share the
same binding mechanism that is dominated by their light degrees of freedom.

The $P_{cs}(4459)$ is close to the threshold of $\Xi_{c}\bar{D}^{*}$, which
can be assigned as a $[\Xi_{c}\bar{D}^{*}]^{0}_{1/2}$ or
$[\Xi_{c}\bar{D}^{*}]_{3/2}^{0}$ molecular state Wang:2019nvm . The spin of
the light diquark in the $\Xi_{c}$ baryon is 0, so the $\Xi_{c}\bar{D}^{*}$
system has the vanishing spin-spin interaction from its light d.o.f. The
$[\Xi_{c}\bar{D}^{*}]^{0}_{1/2}$ and $[\Xi_{c}\bar{D}^{*}]_{3/2}^{0}$ states
are degenerate in our formalism, as can be seen from Table 2. The inclusion of
the spin-spin interaction from heavy degrees of freedom or pion-exchange shall
distinguish these two states, which is beyond the scope of the present work.
However, from a serious calculation within the framework of chiral effective
field theory Wang:2019nvm , the mass gaps induced from the spin-spin
interactions are within several MeVs. In this sense, our prediction is
consistent with the observed $P_{cs}(4459)$. Indeed, the LHCb collaboration
also fitted the data using two resonances with masses $4454.9\pm 2.7$ MeV and
$4467.8\pm 3.7$ MeV LHCb:2020jpq . However, the limited data samples cannot
confirm or refute the two-peak hypothesis. An updated analysis with more data
samples is desired to clarify this issue.

We further swap the assignments of the $P_{c}(4440)$ and $P_{c}(4457)$, and
regard them as the $[\Sigma_{c}\bar{D}^{*}]_{3/2}^{1/2}$ and
$[\Sigma_{c}\bar{D}^{*}]_{1/2}^{1/2}$ molecular states, respectively. We can
also find a set of solutions that can reproduce the masses of the three
$P_{c}$ states. However, the cutoff $\Lambda$ is at 1.763 GeV, which is far
away from the scale of light scalar mesons. Moreover, we can not reproduce the
$T_{cc}^{+}$ state in this case. Thus, we rule out this set of assignments for
the $P_{c}(4440)$ and $P_{c}(4457)$. In our framework, we can identify the
quantum numbers of the $P_{c}(4440)$ and $P_{c}(4457)$ states if we require a
satisfactory description of the $T_{cc}^{+}$ and $P_{c}$ states
simultaneously.

###  III.2 $T_{cc}^{\prime}$ state and other heavy flavor molecular states

In the previous section, we have shown that our framework gives a nice
description of the observed $T_{cc}$, $P_{c}$, and $P_{cs}$ states. In the
following, we further adopt the fitted parameters $\tilde{g}_{s}$,
$\tilde{g}_{a}$, and $\Lambda$ to calculate the other heavy flavor di-hadron
systems listed in Table 1. The effective potentials for these systems can be
easily read from Table 2. Their calculated masses and binding energies are
listed in Table 4.

Table 4: The predicted masses and binding energies (BE) for the charmed di-hadrons ($[H_{1}H_{2}]_{J}^{I}$) in Table 1. We adopt the isospin averaged masses for the single-charm hadrons ParticleDataGroup:2020ssz . The values are all in units of MeV.  System | $[\bar{D}^{*}\bar{D}^{*}]_{1}^{0}$ | $[\Sigma_{c}^{*}\bar{D}^{*}]_{\frac{1}{2}}^{\frac{1}{2}}$ | $[\Sigma_{c}^{*}\bar{D}^{*}]_{\frac{3}{2}}^{\frac{1}{2}}$ | $[\Sigma_{c}^{*}\bar{D}^{*}]_{\frac{5}{2}}^{\frac{1}{2}}$ | $[\Xi_{c}\bar{D}]_{\frac{1}{2}}^{0}$ | $[\Xi_{c}\bar{D}^{*}]_{\frac{3}{2}}^{0}$ | $[\Xi_{c}^{\prime}\bar{D}]_{\frac{1}{2}}^{0}$ | $[\Xi_{c}^{\prime}\bar{D}^{*}]_{\frac{1}{2}}^{0}$ | $[\Xi_{c}^{\prime}\bar{D}^{*}]^{0}_{\frac{3}{2}}$ | $[\Xi_{c}^{*}\bar{D}]_{\frac{3}{2}}^{0}$ | $[\Xi_{c}^{*}\bar{D}^{*}]_{\frac{1}{2}}^{0}$
---|---|---|---|---|---|---|---|---|---|---|---
Mass | $4009.7$ | $4501.3$ | $4510.1$ | $4523.8$ | $4327.7$ | $4468.1$ | $4436.7$ | $4564.9$ | $4582.1$ | $4503.6$ | $4628.5$
BE | $-7.4$ | $-25.4$ | $-15.9$ | $-2.9$ | $-8.9$ | $-10.0$ | $-9.4$ | $-22.5$ | $-5.2$ | $-9.6$ | $-26.0$
System | $[\Xi_{c}^{*}\bar{D}^{*}]_{\frac{3}{2}}^{0}$ | $[\Xi_{c}^{*}\bar{D}^{*}]_{\frac{5}{2}}^{0}$ | $[\Sigma_{c}\Sigma_{c}]_{0}^{0}$ | $[\Sigma_{c}\Sigma_{c}]^{1}_{1}$ | $[\Sigma_{c}\Sigma_{c}^{*}]_{1}^{0}$ | $[\Sigma_{c}\Sigma_{c}^{*}]_{2}^{0}$ | $[\Sigma_{c}\Sigma_{c}^{*}]^{1}_{1}$ | $[\Sigma_{c}\Sigma_{c}^{*}]^{1}_{2}$ | $[\Sigma_{c}^{*}\Sigma_{c}^{*}]_{1}^{1}$ | $[\Sigma_{c}^{*}\Sigma_{c}^{*}]_{3}^{1}$ | $[\Sigma_{c}^{*}\Sigma_{c}^{*}]_{0}^{0}$
Mass | $4638.0$ | $4651.3$ | $4825.4$ | $4903.9$ | $4894.3$ | $4931.9$ | $4958.4$ | $4969.3$ | $5021.9$ | $5035.1$ | $4946.5$
BE | $-16.5$ | $-3.2$ | $-81.7$ | $-3.2$ | $-77.4$ | $-39.8$ | $-13.3$ | $-2.4$ | $-14.4$ | $-1.2$ | $-89.8$
System | $[\Sigma_{c}^{*}\Sigma_{c}^{*}]_{2}^{0}$ | $[\Xi_{c}\Xi_{c}]^{0}_{1}$ | $[\Xi_{c}\Xi_{c}^{\prime}]_{0,1}^{0}$ | $[\Xi_{c}\Xi_{c}^{*}]^{0}_{1,2}$ | $[\Xi_{c}^{\prime}\Xi_{c}^{\prime}]_{1}^{0}$ | $[\Xi_{c}^{\prime}\Xi_{c}^{*}]_{1}^{0}$ | $[\Xi_{c}^{\prime}\Xi_{c}^{*}]_{2}^{0}$ | $[\Xi_{c}^{*}\Xi_{c}^{*}]_{1}^{0}$ | $[\Xi_{c}^{*}\Xi_{c}^{*}]_{3}^{0}$ |  |
Mass | $4996.1$ | $4933.1$ | $5042.1$ | $5109.0$ | $5153.6$ | $5210.3$ | $5221.7$ | $5276.4$ | $5290.3$ |  |
BE | $-40.1$ | $-5.8$ | $-6.2$ | $-6.4$ | $-4.0$ | $-14.4$ | $-3.0$ | $-15.6$ | $-1.7$ |  |

We check all the physically allowed $D^{(*)}D^{(*)}$ systems and find that
there exists another $T_{cc}^{\prime+}$ state with $[D^{*}D^{*}]_{1}^{0}$
assignment. The $T_{cc}^{\prime+}$ lies about 7 MeV below the $D^{*}D^{*}$
threshold. This state has no hidden-charm strong decay channels due to its
$cc\bar{u}\bar{d}$ valance quark component, thus should decay into
$D^{0}D^{0}\pi^{0}\pi^{+}$ or $D^{0}D^{+}\pi^{0}\pi^{0}$ final states. In
addition, because of the small phase space for $D^{*+}\rightarrow
D^{+}\pi^{0}$ ($D^{0}\pi^{+}$) and $D^{*0}\rightarrow D^{0}\pi^{0}$, if the
$T_{cc}^{\prime+}$ does exist, similar to the $T_{cc}^{+}$, the
$T_{cc}^{\prime+}$ should also be a narrow state in a loosely bound molecular
picture. We suggest the LHCb Collaboration to look for this state in the
future.

The results for the charmed baryon-meson (baryon) systems are also presented
in Table 4. The determined $\tilde{g}_{s}$ is a positive value and about $3$
times larger than the $\tilde{g}_{a}$. From Eq. (LABEL:Vqq) we can see that
for the lowest isospin di-hadron systems, the isospin-isospin matrix elements
are negative and dominate the whole effective potentials of the di-hadron
systems. Thus, if a heavy flavor two-body system has a large negative
$\lambda^{i}_{1}\lambda_{2}^{i}$ eigenvalue, this two-body system will have an
attractive force and may form a bound state. As shown in Table 2 and 4, this
feature is universal for all the studied heavy flavor meson-meson, meson-
baryon, and baryon-baryon systems. The $\lambda_{1}^{i}\lambda_{2}^{i}$
reduces to $\bm{\tau}_{1}\cdot\bm{\tau}_{2}$ in the SU(2) case, and can be
calculated with

$\displaystyle\bm{\tau}_{1}\cdot\bm{\tau}_{2}=2\left[I(I+1)-I_{1}(I_{1}+1)-I_{2}(I_{2}+1)\right].$
(16)

As shown in Eq. (16), the lowest total isospin generally leads to a negative
eigenvalue and corresponds to an attractive force. Our formalism gives a very
practical criterion to understand why the currently observed $T_{cc}^{+}$,
$P_{c}$, and $P_{cs}$ states all prefer the lowest isospins.

###  III.3 Implications for the bottom hadron-hadron systems

In our calculations, we neglect the corrections from the heavy quarks in the
charmed two-body systems and obtain a good description of the observed
$T_{cc}^{+}$, $P_{c}$, and $P_{cs}$ states. If we adopt the same approximation
for the bottom di-hadrons, then the ($T_{cc}^{+}$, $P_{c}$, $P_{cs}$) and
($T_{bb}^{-}$, $P_{b}$, $P_{bs}$) molecular states share the identical
effective potentials from their light d.o.f.

In Fig. 1, we present the variation of binding energies for some typical
molecular states as their corresponding reduced masses gradually increase. In
each system, there exists a critical reduced mass at $E_{\text{BE}}=0$, from
which the system starts to form a bound state. Then the absolute values of
binding energies increase as their reduced masses increase. The increased rate
depends on the different types of light quark combinations in the two-body
heavy flavor systems. We mark the $T_{cc}^{+}$, $T_{cc}^{\prime+}$,
$P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$, as well as their bottom
partners in Fig. 1. For the rest of the considered bottom meson-meson, meson-
baryon and baryon-baryon systems, we list our predictions in Table 5. As shown
in Fig. 1, due to the large reduced masses of the bottom di-hadron systems, if
there exist bound states in the charm di-hadrons, there should also exist the
bottom partners with deeper binding energies as well.

Figure 1: The variation of binding energies for the $T_{cc}^{+}$, $T_{cc}^{\prime+}$, $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ states as their reduced masses increase. At $m_{Q}=m_{b}$, we have their bottom partners $T_{bb}^{-}$, $T_{bb}^{\prime-}$, $[\Sigma_{b}B]_{1/2}^{1/2}$, $[\Sigma_{b}B^{*}]_{1/2}^{1/2}$, and $[\Sigma_{b}B^{*}]_{3/2}^{1/2}$, respectively. Table 5: The predicted masses and binding energies (BE) for the bottom di-hadrons ($[H_{1}H_{2}]_{J}^{I}$) in Table 1. We adopt the isospin averaged masses for the single-bottom hadrons ParticleDataGroup:2020ssz . The values are all in units of MeV.  System | $[BB^{*}]_{1}^{0}$ | $[B^{*}B^{*}]_{1}^{0}$ | $[\Sigma_{b}B]_{\frac{1}{2}}^{\frac{1}{2}}$ | $[\Sigma_{b}B^{*}]_{\frac{1}{2}}^{\frac{1}{2}}$ | $[\Sigma_{b}B^{*}]_{\frac{3}{2}}^{\frac{1}{2}}$ | $[\Sigma_{b}^{*}B]_{\frac{3}{2}}^{\frac{1}{2}}$ | $[\Sigma_{b}^{*}B^{*}]_{\frac{1}{2}}^{\frac{1}{2}}$ | $[\Sigma_{b}^{*}B^{*}]_{\frac{3}{2}}^{\frac{1}{2}}$ | $[\Sigma_{b}^{*}B^{*}]_{\frac{5}{2}}^{\frac{1}{2}}$ | $[\Xi_{b}B]_{\frac{1}{2}}^{0}$ | $[\Xi_{b}B^{*}]^{0}_{\frac{1}{2},\frac{3}{2}}$ | $[\Xi_{b}^{\prime}B]_{\frac{1}{2}}^{0}$
---|---|---|---|---|---|---|---|---|---|---|---|---
Mass | $10584.9$ | $10621.7$ | $11064.1$ | $11094.9$ | $11116.2$ | $11083.5$ | $11110.7$ | $11121.5$ | $11139.1$ | $11048.0$ | $11093.1$ | $11181.2$
BE | $-19.3$ | $-27.7$ | $-28.5$ | $-42.9$ | $-21.6$ | $-28.6$ | $-46.6$ | $-35.8$ | $-18.1$ | $-28.5$ | $-28.6$ | $-33.3$
System | $[\Xi_{b}^{\prime}B^{*}]^{0}_{\frac{1}{2}}$ | $[\Xi_{b}^{\prime}B^{*}]_{\frac{3}{2}}^{0}$ | $[\Xi_{b}^{*}B]_{\frac{3}{2}}^{0}$ | $[\Xi_{b}^{*}B^{*}]^{0}_{\frac{1}{2}}$ | $[\Xi_{b}^{*}B^{*}]_{\frac{3}{2}}^{0}$ | $[\Xi_{b}^{*}B^{*}]_{\frac{5}{2}}^{0}$ | $[\Sigma_{b}\Sigma_{b}]_{0}^{0}$ | $[\Sigma_{b}\Sigma_{b}]_{1}^{1}$ | $[\Sigma_{b}\Sigma_{b}^{*}]_{1}^{0}$ | $[\Sigma_{b}\Sigma_{b}^{*}]_{2}^{0}$ | $[\Sigma_{b}\Sigma_{b}^{*}]_{1}^{1}$ | $[\Sigma_{b}\Sigma_{b}^{*}]_{2}^{1}$
Mass | $11216.6$ | $11238.0$ | $11206.1$ | $11233.3$ | $11244.1$ | $11261.8$ | $11523.2$ | $11609.3$ | $11547.5$ | $11586.4$ | $11615.6$ | $11630.5$
BE | $-43.1$ | $-21.7$ | $-28.7$ | $-46.7$ | $-35.9$ | $-18.3$ | $-103.0$ | $-16.9$ | $-98.2$ | $-59.3$ | $-30.0$ | $-15.1$
System | $[\Sigma_{b}^{*}\Sigma_{b}^{*}]_{1}^{1}$ | $[\Sigma_{b}^{*}\Sigma_{b}^{*}]_{3}^{1}$ | $[\Sigma_{b}^{*}\Sigma_{b}^{*}]_{0}^{0}$ | $[\Sigma_{b}^{*}\Sigma_{b}^{*}]_{2}^{0}$ | $[\Xi_{b}\Xi_{b}]_{1}^{0}$ | $[\Xi_{b}\Xi_{b}^{\prime}]^{0}_{0,1}$ | $[\Xi_{b}\Xi_{b}^{*}]_{1,2}^{0}$ | $[\Xi_{b}^{\prime}\Xi_{b}^{\prime}]_{1}^{0}$ | $[\Xi_{b}^{\prime}\Xi_{b}^{*}]_{1}^{0}$ | $[\Xi_{c}^{\prime}\Xi_{c}^{*}]_{2}^{0}$ | $[\Xi_{b}^{*}\Xi_{b}^{*}]^{0}_{1}$ | $[\Xi_{b}^{*}\Xi_{b}^{*}]_{3}^{0}$
Mass | $11634.1$ | $11652.6$ | $11554.7$ | $11605.8$ | $11573.4$ | $11711.3$ | $11731.6$ | $11852.9$ | $11860.0$ | $11875.0$ | $11879.4$ | $11898.0$
BE | $-31.0$ | $-12.4$ | $-110.4$ | $-59.3$ | $-20.6$ | $-20.7$ | $-20.7$ | $-17.2$ | $-30.3$ | $-15.4$ | $-31.3$ | $-12.7$

##  IV Summary

In this work, we use a quark level effective potential to give a universal
description of the heavy flavor hadronic molecules that are composed of the
ground $(\bar{D},\bar{D}^{*})$, $(\Lambda_{c},\Sigma_{c},\Sigma_{c}^{*})$, and
$(\Xi_{c},\Xi_{c}^{\prime},\Xi_{c}^{*})$ hadrons. Based on this quark-level
effective Lagrangian, we neglect the contributions from heavy quarks and
relate the effective potentials of di-hadrons to their flavor and spin
interaction operators of light degrees of freedom.

In our approach, we only introduce three parameters $\tilde{g}_{s}$,
$\tilde{g}_{a}$, and $\Lambda$. They can be well extracted from the observed
$P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$. We exclude the assignments of
$P_{c}(4440)$ and $P_{c}(4457)$ as the $[\Sigma_{c}\bar{D}^{*}]_{3/2}^{1/2}$
and $[\Sigma_{c}\bar{D}^{*}]_{1/2}^{1/2}$ states, respectively, due to the
poor description of $T_{cc}^{+}$ in this case. Our results strongly indicate a
very similar binding mechanism between the heavy flavor meson-meson and meson-
baryon systems, i.e., they are bound dominantly by the interactions of their
light degrees of freedom. We further generalize this similarity to the heavy
flavor baryon-baryon systems.

We predict another $T_{cc}^{\prime+}$ state with the assignment
$[D^{\ast}D^{\ast}]_{1}^{0}$. From our calculations, the $T_{cc}^{+}$ and
$T_{cc}^{\prime+}$ are the only two molecular states in the $D^{(*)}D^{(*)}$
systems. We suggest the LHCb to look for this state in the future. We also
predict other possible heavy flavor hadronic molecules in the charmed and
bottom sectors (e.g., see Tables 4 and 5).

## Acknowledgments

K. Chen thanks Zi-Yang Lin for helpful discussions. B. Wang is supported by
the Youth Funds of Hebei Province (No. 042000521062) and the Start-up Funds
for Young Talents of Hebei University (No. 521100221021). This project is also
supported by the National Natural Science Foundation of China under Grants
11975033 and 12070131001.

## References

  * (1) M. Gell-Mann, Phys. Lett. 8, 214 (1964).
  * (2) G. Zweig, An SU(3) model for strong interaction symmetry and its breaking. Version 1. CERN-TH-401.
  * (3) R. L. Jaffe, Phys. Rev. D 15, 267 (1977).
  * (4) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020)
  * (5) H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Phys. Rept. 639, 1-121 (2016)
  * (6) Y. R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Prog. Part. Nucl. Phys. 107, 237-320 (2019)
  * (7) X. Liu, Chin. Sci. Bull. 59, 3815-3830 (2014)
  * (8) F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, no.1, 015004 (2018)
  * (9) A. Hosaka, T. Iijima, K. Miyabayashi, Y. Sakai and S. Yasui, PTEP 2016, no.6, 062C01 (2016)
  * (10) R. F. Lebed, R. E. Mitchell and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143-194 (2017)
  * (11) A. Esposito, A. Pilloni and A. D. Polosa, Phys. Rept. 668, 1-97 (2017)
  * (12) N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. P. Shen, C. E. Thomas, A. Vairo and C. Z. Yuan, Phys. Rept. 873, 1-154 (2020)
  * (13) S. K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003).
  * (14) R. Aaij et al. [LHCb], Phys. Rev. Lett. 115, 072001 (2015)
  * (15) J. J. Wu, R. Molina, E. Oset and B. S. Zou, Phys. Rev. Lett. 105, 232001 (2010)
  * (16) Z. C. Yang, Z. F. Sun, J. He, X. Liu and S. L. Zhu, Chin. Phys. C 36, 6-13 (2012)
  * (17) W. L. Wang, F. Huang, Z. Y. Zhang and B. S. Zou, Phys. Rev. C 84, 015203 (2011)
  * (18) R. Aaij et al. [LHCb], Phys. Rev. Lett. 122, no.22, 222001 (2019)
  * (19) R. Chen, Z. F. Sun, X. Liu and S. L. Zhu, Phys. Rev. D 100, no.1, 011502 (2019)
  * (20) M. Z. Liu, Y. W. Pan, F. Z. Peng, M. Sánchez Sánchez, L. S. Geng, A. Hosaka and M. Pavon Valderrama, Phys. Rev. Lett. 122, no.24, 242001 (2019)
  * (21) J. He, Eur. Phys. J. C 79, no.5, 393 (2019)
  * (22) C. W. Xiao, J. Nieves and E. Oset, Phys. Rev. D 100, no.1, 014021 (2019)
  * (23) L. Meng, B. Wang, G. J. Wang and S. L. Zhu, Phys. Rev. D 100, no.1, 014031 (2019)
  * (24) Y. Yamaguchi, H. García-Tecocoatzi, A. Giachino, A. Hosaka, E. Santopinto, S. Takeuchi and M. Takizawa, Phys. Rev. D 101, no.9, 091502 (2020)
  * (25) M. Pavon Valderrama, Phys. Rev. D 100, no.9, 094028 (2019)
  * (26) H. X. Chen, W. Chen and S. L. Zhu, Phys. Rev. D 100, no.5, 051501 (2019)
  * (27) T. J. Burns and E. S. Swanson, Phys. Rev. D 100, no.11, 114033 (2019)
  * (28) M. L. Du, V. Baru, F. K. Guo, C. Hanhart, U. G. Meißner, J. A. Oller and Q. Wang, Phys. Rev. Lett. 124, no.7, 072001 (2020)
  * (29) B. Wang, L. Meng and S. L. Zhu, JHEP 11, 108 (2019)
  * (30) R. Chen, J. He and X. Liu, Chin. Phys. C 41, no.10, 103105 (2017)
  * (31) E. Santopinto and A. Giachino, Phys. Rev. D 96, no.1, 014014 (2017)
  * (32) C. W. Shen, J. J. Wu and B. S. Zou, Phys. Rev. D 100, no.5, 056006 (2019)
  * (33) C. W. Xiao, J. Nieves and E. Oset, Phys. Lett. B 799, 135051 (2019)
  * (34) B. Wang, L. Meng and S. L. Zhu, Phys. Rev. D 101, no.3, 034018 (2020)
  * (35) H. X. Chen, L. S. Geng, W. H. Liang, E. Oset, E. Wang and J. J. Xie, Phys. Rev. C 93, no.6, 065203 (2016)
  * (36) R. Aaij et al. [LHCb], Sci. Bull. 66, 1278-1287 (2021)
  * (37) R. Aaij et al. [LHCb], [arXiv:2109.01056 [hep-ex]].
  * (38) R. Aaij et al. [LHCb], [arXiv:2109.01038 [hep-ex]].
  * (39) N. Li, Z. F. Sun, X. Liu and S. L. Zhu, Phys. Rev. D 88, no.11, 114008 (2013)
  * (40) H. Xu, B. Wang, Z. W. Liu and X. Liu, Phys. Rev. D 99, no.1, 014027 (2019).
  * (41) J. Carlson, L. Heller and J. A. Tjon, Phys. Rev. D 37, 744 (1988)
  * (42) B. Silvestre-Brac and C. Semay, Z. Phys. C 57, 273-282 (1993)
  * (43) C. Semay and B. Silvestre-Brac, Z. Phys. C 61, 271-275 (1994)
  * (44) S. Pepin, F. Stancu, M. Genovese and J. M. Richard, Phys. Lett. B 393, 119-123 (1997)
  * (45) B. A. Gelman and S. Nussinov, Phys. Lett. B 551, 296-304 (2003)
  * (46) J. Vijande, F. Fernandez, A. Valcarce and B. Silvestre-Brac, Eur. Phys. J. A 19, 383 (2004)
  * (47) D. Janc and M. Rosina, Few Body Syst. 35, 175-196 (2004)
  * (48) Y. Cui, X. L. Chen, W. Z. Deng and S. L. Zhu, HEPNP 31, 7-13 (2007)
  * (49) F. S. Navarra, M. Nielsen and S. H. Lee, Phys. Lett. B 649, 166-172 (2007) doi:10.1016/j.physletb.2007.04.010
  * (50) J. Vijande, E. Weissman, A. Valcarce and N. Barnea, Phys. Rev. D 76, 094027 (2007)
  * (51) D. Ebert, R. N. Faustov, V. O. Galkin and W. Lucha, Phys. Rev. D 76, 114015 (2007)
  * (52) S. H. Lee and S. Yasui, Eur. Phys. J. C 64, 283-295 (2009)
  * (53) Y. Yang, C. Deng, J. Ping and T. Goldman, Phys. Rev. D 80, 114023 (2009)
  * (54) M. L. Du, W. Chen, X. L. Chen and S. L. Zhu, Phys. Rev. D 87, no.1, 014003 (2013)
  * (55) G. Q. Feng, X. H. Guo and B. S. Zou, [arXiv:1309.7813 [hep-ph]]
  * (56) Y. Ikeda, B. Charron, S. Aoki, T. Doi, T. Hatsuda, T. Inoue, N. Ishii, K. Murano, H. Nemura and K. Sasaki, Phys. Lett. B 729, 85-90 (2014)
  * (57) R. Molina, T. Branz and E. Oset, Phys. Rev. D 82, 014010 (2010)
  * (58) T. F. Carames, A. Valcarce and J. Vijande, Phys. Lett. B 699, 291-295 (2011)
  * (59) S. Sakai, L. Roca and E. Oset, Phys. Rev. D 96, no.5, 054023 (2017)
  * (60) S. Q. Luo, K. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Eur. Phys. J. C 77, no.10, 709 (2017)
  * (61) M. Karliner and J. L. Rosner, Phys. Rev. Lett. 119, no.20, 202001 (2017)
  * (62) E. J. Eichten and C. Quigg, Phys. Rev. Lett. 119, no.20, 202002 (2017)
  * (63) Z. G. Wang, Acta Phys. Polon. B 49, 1781 (2018)
  * (64) G. K. C. Cheung et al. [Hadron Spectrum], JHEP 11, 033 (2017)
  * (65) W. Park, S. Noh and S. H. Lee, Nucl. Phys. A 983, 1-19 (2019)
  * (66) N. Li, Z. F. Sun, X. Liu and S. L. Zhu, Chin. Phys. Lett. 38, 092001 (2021)
  * (67) X. Z. Weng, W. Z. Deng and S. L. Zhu, [arXiv:2108.07242 [hep-ph]].
  * (68) S. S. Agaev, K. Azizi and H. Sundu, [arXiv:2108.00188 [hep-ph]].
  * (69) R. Chen, Q. Huang, X. Liu and S. L. Zhu, [arXiv:2108.01911 [hep-ph]].
  * (70) L. Y. Dai, X. Sun, X. W. Kang, A. P. Szczepaniak and J. S. Yu, [arXiv:2108.06002 [hep-ph]].
  * (71) A. Feijoo, W. H. Liang and E. Oset, [arXiv:2108.02730 [hep-ph]].
  * (72) X. Z. Ling, M. Z. Liu, L. S. Geng, E. Wang and J. J. Xie, [arXiv:2108.00947 [hep-ph]].
  * (73) L. Meng, G. J. Wang, B. Wang and S. L. Zhu, [arXiv:2107.14784 [hep-ph]].
  * (74) T. W. Wu, Y. W. Pan, M. Z. Liu, S. Q. Luo, X. Liu and L. S. Geng, [arXiv:2108.00923 [hep-ph]].
  * (75) M. J. Yan and M. P. Valderrama, [arXiv:2108.04785 [hep-ph]].
  * (76) Y. Hu, J. Liao, E. Wang, Q. Wang, H. Xing and H. Zhang, [arXiv:2109.07733 [hep-ph]].
  * (77) Y. Jin, S. Y. Li, Y. R. Liu, Q. Qin, Z. G. Si and F. S. Yu, [arXiv:2109.05678 [hep-ph]].
  * (78) G. Yang, J. Ping and J. Segovia, [arXiv:2109.04311 [hep-ph]].
  * (79) H. Ren, F. Wu and R. Zhu, [arXiv:2109.02531 [hep-ph]].
  * (80) X. Chen, [arXiv:2109.02828 [hep-ph]].
  * (81) K. Azizi and U. Özdem, [arXiv:2109.02390 [hep-ph]].
  * (82) S. Fleming, R. Hodges and T. Mehen, [arXiv:2109.02188 [hep-ph]].
  * (83) X. K. Dong, F. K. Guo and B. S. Zou, [arXiv:2108.02673 [hep-ph]].
  * (84) N. Lee, Z. G. Luo, X. L. Chen and S. L. Zhu, Phys. Rev. D 84, 014031 (2011)
  * (85) N. Li and S. L. Zhu, Phys. Rev. D 86, 014020 (2012)
  * (86) X. Q. Li and X. Liu, Eur. Phys. J. C 74, no.12, 3198 (2014)
  * (87) L. Meng, B. Wang and S. L. Zhu, Phys. Rev. C 101, no.6, 064002 (2020)
  * (88) B. Wang, L. Meng and S. L. Zhu, Phys. Rev. D 101, no.9, 094035 (2020)
  * (89) E. Epelbaum, U. G. Meissner, W. Gloeckle and C. Elster, Phys. Rev. C 65, 044001 (2002).
  * (90) L. Meng, G. J. Wang, B. Wang and S. L. Zhu, [arXiv:2109.01333 [hep-ph]].
  * (91) K. Chen, B. Wang and S. L. Zhu, Phys. Rev. D 103, no.11, 116017 (2021)