# Spectrum of the doubly charmed molecular pentaquarks in chiral effective
field theory

Kan Chen1,2<EMAIL_ADDRESS>Bo Wang1,2<EMAIL_ADDRESS>Shi-Lin Zhu1,2
<EMAIL_ADDRESS>1Center of High Energy Physics, Peking University, Beijing
100871, China
2School of Physics and State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing 100871, China

###### Abstract

We perform a systematic study on the interactions of the
$\Sigma_{c}^{(*)}D^{(*)}$ systems within the framework of chiral effective
field theory. We introduce the contact term, one-pion-exchange and two-pion-
exchange contributions to describe the short-, long-, and intermediate-range
interactions. The low energy constants of the $\Sigma_{c}^{(*)}D^{(*)}$
systems are estimated from the $N\bar{N}$ scattering data by introducing a
quark level Lagrangian. With three solutions of LECs, all the
$\Sigma_{c}^{(*)}D^{(*)}$ systems with isospin $I=1/2$ can form bound states,
in which different inputs of LECs may lead to distinguishable mass spectra. In
addition, we also investigate the interactions of the charmed-bottom
$\Sigma_{c}^{(*)}\bar{B}^{(*)}$, $\Sigma_{b}^{(*)}D^{(*)}$, and
$\Sigma_{b}^{(*)}\bar{B}^{(*)}$ systems. Among the obtained bound states, the
bindings become deeper when the reduced masses of the corresponding systems
are heavier.

##  I Introduction

The existence of the $qqqq\bar{q}$ pentaquarks are proposed by Gell-Mann and
Zweig GellMann:1964nj ; Zweig:1981pd ; Zweig:1964jf  at the birth of the quark
model in 1964. In 2003, the LEPS group Nakano:2003qx  reported a narrow
resonance signal at 1540 MeV with $S=+1$, called $\Theta^{+}(1540)$, whose
quark component should be $uudd\bar{s}$. Although further experiments did not
confirm this state MartinezTorres:2010zzb , it triggered extensive theoretical
and experimental studies on possible pentaquark states Zhu:2004xa ;
Liu:2014yva .

In 2015, the LHCb Collaboration Aaij:2015tga ; Aaij:2016phn  measured the
$\Lambda_{b}^{0}\rightarrow J/\psi K^{-}p$ decay process and reported two
hidden-charm pentaquark-like states $P_{c}(4380)$ and $P_{c}(4450)$ in the
$J/\psi p$ channel, indicating that these two states have a minimal quark
content of $uudc\bar{c}$. In 2019, the LHCb Collaboration announced
Aaij:2019vzc  the observation of three narrow peaks in the $J/\psi p$
invariant mass spectrum. They found that the $P_{c}(4450)$ is actually
composed of two substructures, the $P_{c}(4440)$ and $P_{c}(4457)$ with
5.4$\sigma$ significance. Moreover, they also reported a new state below the
$\Sigma_{c}\bar{D}$ threshold, namely the $P_{c}(4312)$ with 7.3$\sigma$
significance. Before the discovery of the LHCb Collaboration in 2015, several
groups had predicted Wu:2010jy ; Yang:2011wz ; Wang:2011rga  the existence of
molecular pentaquarks.

The LHCb experiments keep giving us surprise. Very recently, they reported the
first evidence of a charmonium pentaquark candidate with strangeness in the
$\Xi_{b}^{-}\rightarrow J/\psi\Lambda K^{-}$ decay process Aaij:2020gdg . Its
mass and width are determined to be $4458.8\pm 2.9^{+4.7}_{-1.1}$ MeV and
$17.3\pm 6.5_{-5.7}^{+8.0}$ MeV, respectively. However, its significance just
exceeds 3$\sigma$ after considering all systematic uncertainties. Further
studies on the $P_{cs}^{0}$ pentaquark are still needed. As the strange
partner of the $P_{c}$ pentaquark states, it has been predicted in Refs.
Wu:2010jy ; Chen:2016ryt ; Santopinto:2016pkp ; Shen:2019evi ; Xiao:2019gjd ;
Wang:2019nvm ; Chen:2015sxa . Especially, the mass predicted from chiral
effective field theory agrees very well with the experimental data
Wang:2019nvm .

Besides the pentaquarks with hidden-charm quark components, the existence of
the double-charm pentaquarks is also an interesting topic (see Refs.
Chen:2016qju ; Guo:2017jvc ; Liu:2019zoy ; Lebed:2016hpi ; Esposito:2016noz ;
Brambilla:2019esw ; Hosaka:2016pey  for reviews of the exotic hadrons). For
the double-charm pentaquarks, two straightforward configurations are the
compact $ccqq\bar{q}$ pentaquarks and $(cqq)$-$(c\bar{q})$ baryon-meson
molecular states. Based on the compact pentaquark configuration, the mass
spectra of the pentaquarks with $QQqq\bar{q}$ ($Q=b$, $c$, and $q=u$, $d$,
$s$) quark components were estimated systematically in the framework of the
color-magnetic interaction model Zhou:2018bkn . The authors of Ref.
Park:2018oib  used similar approach to estimate possible stable pentaquark
states. In addition, the chiral quark model Yang:2020twg  and QCD sum rule
Wang:2018lhz  were exploited to analyze the doubly charmed pentaquark states.
For the case of the latter configuration, some theoretical calculations were
performed in the meson exchange models Xu:2010fc ; Chen:2017vai ;
Shimizu:2017xrg . We can qualitatively capture some features of the double-
charm pentaquarks from above works, while a systematic study of the
$\Sigma_{c}^{(*)}D^{(*)}$ systems is still absent.

The chiral effective field theory has achieved great success in describing the
interactions of the $NN$ systems Bernard:1995dp ; Epelbaum:2008ga ;
Machleidt:2011zz ; Meissner:2015wva ; Hammer:2019poc ; Machleidt:2020vzm . It
is also a very useful tool to study the interactions of the two-body hadron
systems with heavy flavors Liu:2012vd ; Xu:2017tsr ; Wang:2018atz ;
Meng:2019ilv ; Meng:2019nzy ; Wang:2019ato ; Wang:2019nvm ; Wang:2020dhf ;
Wang:2020dko ; Wang:2020htx . In the framework of heavy hadron chiral
effective theory, we consider the one-pion-exchange, two-pion-exchange, and
contact contributions to account for the long-, intermediate-, and short-range
interactions of the $\Sigma_{c}^{(*)}D^{(*)}$ systems, respectively. Among
them, the one-pion-exchange diagrams can be easily calculated with the
standard procedure. For the two-pion-exchange box diagrams, Weinberg
Weinberg:1990rz ; Weinberg:1991um  suggested that we should only consider the
contributions from two-particle-irreducible (2PI) graphs, since the two-
particle-reducible (2PR) part can be recovered by inserting the one-pion-
exchange potentials into the nonperturbative iterative equations. This
treatment can be done with the help of the principle-value integral method.
For the low energy constants (LECs) associated with the contact terms,
generally, they should be fixed from the experimental scattering data or
lattice QCD simulations. In Refs. Wang:2020dhf ; Wang:2019nvm ; Meng:2019nzy ,
we proposed an approach which can relate the contact effective potentials
derived at the hadron level to those derived at the quark level, so that the
LECs can be determined from the quark model. For example, to estimate the
contributions from the contact terms in the $\Sigma_{c}^{(*)}D^{(*)}$ systems,
we can derive the contact effective potentials of the
$\Sigma_{c}^{(*)}D^{(*)}$ systems at the quark level, the coupling constants
in the contact terms can be determined from the $N\bar{N}$ scattering data.
Thus, the contributions from the unknown contact terms can also be estimated.
We have a complete framework to study the interactions of the
$\Sigma_{c}^{(*)}D^{(*)}$ systems, and which is also used to investigate the
interactions of the charmed-bottom $\Sigma_{c}^{(*)}\bar{B}^{(*)}$,
$\Sigma_{b}^{(*)}D^{(*)}$, and $\Sigma_{b}^{(*)}\bar{B}^{(*)}$ systems.

This paper is organized as follows. In Sec. II, we present the effective
chiral Lagrangians and the effective potentials. In Sec. III, we present our
numerical results and discussions. In Sec. IV, we conclude this work with a
short summary. Some supplemental materials for loop diagrams and the results
for charmed-bottom systems are given in the Appendices A and B, respectively.

##  II Effective chiral Lagrangians and analytical effective potentials

We consider the leading order contact and one-pion-exchange interactions, and
the next-to-leading order two-pion-exchange contributions to describe the
scattering amplitudes of the $\Sigma_{c}D$, $\Sigma_{c}^{*}D$,
$\Sigma_{c}D^{*}$, and $\Sigma_{c}^{*}D^{*}$ systems. We first briefly
introduce the effective Lagrangians for the pionic and contact interactions.

###  II.1 Effective chiral Lagrangians

In the heavy baryon reduction formalism Scherer:2002tk , the leading order
nonrelativistic chiral Lagrangians describing the interactions between the
charmed baryons and pion can be constructed as

$\displaystyle\mathcal{L}_{B\phi}$ $\displaystyle=$
$\displaystyle\text{Tr}\left[\mathcal{\bar{B}}_{3}\left(iv\cdot
D-\delta_{c}\right)\mathcal{B}_{3}\right]+2g_{5}\text{Tr}\left(\bar{\mathcal{B}}_{3^{*}}^{\mu}\mathcal{S}\cdot
u\mathcal{B}_{3^{*}\mu}\right)$
$\displaystyle-\text{Tr}\left[\bar{\mathcal{B}}^{\mu}_{3^{*}}\left(iv\cdot
D-\delta_{d}\right)\mathcal{B}_{3^{*}\mu}\right]+2g_{1}\text{Tr}\left(\bar{\mathcal{B}}_{3}\mathcal{S}\cdot
u\mathcal{B}_{3}\right)$
$\displaystyle+2g_{2}\text{Tr}\left(\bar{\mathcal{B}}_{3}\mathcal{S}\cdot
u\mathcal{B}_{1}+\text{H.c.}\right)+\frac{1}{2}\text{Tr}\left[\bar{\mathcal{B}}_{1}\left(iv\cdot
D\right)\mathcal{B}_{1}\right]$
$\displaystyle+g_{3}\text{Tr}\left(\mathcal{\bar{B}}_{3^{*}}^{\mu}u_{\mu}\mathcal{B}_{3}+\text{H.c.}\right)+g_{4}\text{Tr}\left(\bar{\mathcal{B}}_{3^{*}}^{\mu}u_{\mu}\mathcal{B}_{1}+\text{H.c.}\right),$

where $\mathcal{S}^{\mu}=\frac{i}{2}\gamma_{5}\sigma^{\mu\nu}v_{\nu}$ is the
operator for spin-$\frac{1}{2}$ baryon. The covariant derivative is defined as
$D_{\mu}\psi=\partial_{\mu}\psi+\Gamma_{\mu}\psi+\psi\Gamma_{\mu}^{T}$, where
$\Gamma_{\mu}^{T}$ is the transposition of $\Gamma_{\mu}$. The chiral
connection $\Gamma_{\mu}$ and axial current $u_{\mu}$ are defined as

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

with

$\displaystyle\xi^{2}=U=\text{exp}\left(\frac{i\phi}{f_{\pi}}\right),\quad\phi=\left(\begin{array}[]{cc}\pi^{0}&\sqrt{2}\pi+\\\
\sqrt{2}\pi^{-}&-\pi^{0}\\\ \end{array}\right).$ (5)

Here, $f_{\pi}=92.4$ MeV is the pion decay constant.

The charmed baryons $\Lambda_{c}$ and $\Sigma_{c}^{(*)}$ form the SU(2)
isosinglet and isotriplets, respectively. The spin-$\frac{1}{2}$ isosinglet is

$\displaystyle\psi_{1}=\left(\begin{array}[]{cc}0&\Lambda_{c}^{+}\\\
-\Lambda_{c}^{+}&0\\\ \end{array}\right),\quad$ (8)

the isotriplet with spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ are labeled as
$\psi_{3}$ and $\psi_{3^{*}}^{\mu}$, respectively. They have the matrix form

$\displaystyle\psi_{3}=\left(\begin{array}[]{cc}\Sigma_{c}^{++}&\frac{\Sigma_{c}^{+}}{\sqrt{2}}\\\
\frac{\Sigma_{c}^{+}}{\sqrt{2}}&\Sigma_{c}^{0}\\\
\end{array}\right),\quad\psi^{\mu}_{3^{*}}=\left(\begin{array}[]{cc}\Sigma_{c}^{*++}&\frac{\Sigma_{c}^{*+}}{\sqrt{2}}\\\
\frac{\Sigma_{c}^{*+}}{\sqrt{2}}&\Sigma_{c}^{*0}\\\ \end{array}\right)^{\mu}.$
(13)

The heavy baryon field can be decomposed into the light and heavy components
$\mathcal{B}_{i}$ and $\mathcal{H}_{i}$, which read

$\displaystyle\mathcal{B}_{i}=e^{iM_{i}v\cdot
x}\frac{1+\not{v}}{2}\psi_{i},\quad\mathcal{H}_{i}=e^{iM_{i}v\cdot
x}\frac{1-\not{v}}{2}\psi_{i},$ (14)

where $\psi_{i}$ denote the heavy baryon fields $\psi_{1}$, $\psi_{3}$, and
$\psi_{3^{*}}$. $v_{\mu}=\left(1,\textbf{0}\right)$ is the four-velocity of
heavy baryon. The $\mathcal{B}_{i}$ fields contribute at the leading order,
whereas the $\mathcal{H}_{i}$ are supressed by power of $1/m_{Q}$. $M_{i}$ are
the masses of the heavy baryons. In this work, we adopt the following mass
splittings Zyla:2020zbs

$\displaystyle\delta_{a}$ $\displaystyle=$ $\displaystyle
M_{3^{*}}-M_{3}\simeq 65\;\rm{MeV},$ $\displaystyle\delta_{c}$
$\displaystyle=$ $\displaystyle M_{3}-M_{1}\simeq 168.5\;\rm{MeV},$
$\displaystyle\delta_{d}$ $\displaystyle=$ $\displaystyle
M_{3^{*}}-M_{1}\simeq 233.5\;\rm{MeV}.$ (15)

In Eq. (II.1), the couplings $g_{2}=-0.60$ and $g_{4}=1.04$ can be calculated
from the partial decay widths of the $\Sigma_{c}\rightarrow\Lambda_{c}\pi$ and
$\Sigma_{c}^{*}\rightarrow\Lambda_{c}\pi$ processes Zyla:2020zbs ,
respectively. The $g_{1}$, $g_{3}$, and $g_{5}$ can be related to $g_{2}$ via
the quark model Meguro:2011nr ; Liu:2011xc ; Meng:2018gan , which read

$\displaystyle g_{1}=0.98,\quad g_{3}=0.85,\quad g_{5}=-1.47.$ (16)

The leading order chiral Lagrangians for the interactions between the charmed
mesons and pion are Manohar:2000

$\displaystyle\mathcal{L}_{H\phi}$ $\displaystyle=$
$\displaystyle-\left\langle\left(iv\cdot\partial\mathcal{H}\right)\bar{\mathcal{H}}\right\rangle+\left\langle\mathcal{H}v\cdot\Gamma\bar{\mathcal{H}}\right\rangle+g\left\langle\mathcal{H}\not{u}\gamma_{5}\bar{\mathcal{H}}\right\rangle$
(17)
$\displaystyle-\frac{1}{8}\delta_{b}\left\langle\mathcal{H}\sigma^{\mu\nu}\bar{\mathcal{H}}\sigma_{\mu\nu}\right\rangle,$

where $\delta_{b}=m_{D^{*}}-m_{D}=142.0$ MeV Zyla:2020zbs . $g=-0.59$
represents the axial coupling constant, its value is calculated from the
partial decay width of $D^{*+}\rightarrow D^{0}\pi^{+}$ process Zyla:2020zbs
and its sign is determined from the quark model.

In the above Lagrangian, the $\mathcal{H}$ denotes the super-field of the
$\left(D,D^{*}\right)$ doublet in the heavy quark limit,

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{1+\not{v}}{2}\left(P^{*}_{\mu}\gamma^{\mu}+iP\gamma_{5}\right),$
$\displaystyle\bar{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\gamma^{0}H^{\dagger}\gamma^{0}=\left(P^{*\dagger}_{\mu}\gamma^{\mu}+iP^{\dagger}\gamma_{5}\right)\frac{1+\not{v}}{2},$
$\displaystyle P$ $\displaystyle=$
$\displaystyle\left(D^{0},D^{+}\right),\qquad
P^{*}_{\mu}=\left(D^{*0},D^{*+}\right)_{\mu}.$ (18)

Accordingly, the mass splittings for the bottom baryons and mesons are
Zyla:2020zbs

$\displaystyle\delta_{a}$ $\displaystyle=$ $\displaystyle
m_{\Sigma_{b}^{*}}-m_{\Sigma_{b}}\simeq 20\;\rm{MeV},$
$\displaystyle\delta_{b}$ $\displaystyle=$ $\displaystyle
m_{B^{*}}-m_{B}\simeq 45\;\rm{MeV},$ $\displaystyle\delta_{c}$
$\displaystyle=$ $\displaystyle m_{\Sigma_{b}}-m_{\Lambda_{b}}\simeq
191\;\rm{MeV},$ $\displaystyle\delta_{d}$ $\displaystyle=$ $\displaystyle
m_{\Sigma_{b}^{*}}-m_{\Lambda_{b}}\simeq 211\;\rm{MeV}.$ (19)

In the bottom sector, the axial coupling $g=-0.52$ is taken from the Lattice
QCD calculations Ohki:2008py ; Detmold:2012ge . $g_{2}=-0.51$ and $g_{4}=0.91$
are obtained from the partial decay widths of the
$\Sigma_{b}\rightarrow\Lambda_{b}\pi$ and
$\Sigma_{b}^{*}\rightarrow\Lambda_{b}\pi$ Zyla:2020zbs . Similarly, $g_{1}$,
$g_{3}$, and $g_{5}$ are determined from the quark model Meguro:2011nr ;
Liu:2011xc ; Meng:2018gan ,

$\displaystyle g_{1}=0.83,\quad g_{3}=0.72,\quad g_{5}=-1.25.$ (20)

In order to describe the contact interactions of $\Sigma_{c}^{(*)}$ and
$D^{(*)}$, we construct the following Lagrangians,

$\displaystyle\mathcal{L}_{HB}$ $\displaystyle=$ $\displaystyle
D_{a}\left\langle\mathcal{H}\bar{\mathcal{H}}\right\rangle\text{Tr}\left(\bar{\psi}^{\mu}\psi_{\mu}\right)$
(21)
$\displaystyle+iD_{b}\epsilon_{\sigma\mu\nu\rho}v^{\sigma}\left\langle\mathcal{H}\gamma^{\rho}\gamma_{5}\bar{\mathcal{H}}\right\rangle\text{Tr}\left(\bar{\psi}^{\mu}\psi^{\nu}\right)$
$\displaystyle+E_{a}\left\langle\mathcal{H}\tau^{i}\bar{\mathcal{H}}\right\rangle\text{Tr}\left(\bar{\psi}^{\mu}\tau_{i}\psi_{\mu}\right)$
$\displaystyle+iE_{b}\epsilon_{\sigma\mu\nu\rho}v^{\sigma}\left\langle\mathcal{H}\gamma^{\rho}\gamma_{5}\tau^{i}\bar{\mathcal{H}}\right\rangle\text{Tr}\left(\bar{\psi}^{\mu}\tau_{i}\psi^{\nu}\right),$

where

$\displaystyle\psi^{\mu}$ $\displaystyle=$
$\displaystyle\mathcal{B}_{3^{*}}^{\mu}-\frac{1}{\sqrt{3}}\left(\gamma^{\mu}+v^{\mu}\right)\gamma^{5}\mathcal{B}_{3},$
$\displaystyle\bar{\psi}^{\mu}$ $\displaystyle=$
$\displaystyle\bar{\mathcal{B}}_{3^{*}}^{\mu}+\frac{1}{3}\bar{\mathcal{B}}_{3}\gamma^{5}\left(\gamma^{\mu}+v^{\mu}\right)$
(22)

denote the super-fields of $(\mathcal{B}_{3},\mathcal{B}_{3^{*}})$ doublet
Cho:1992cf ; Cho:1992gg . The $D_{a}$, $D_{b}$, $E_{a}$, and $E_{b}$ are the
low energy constants that account for the central potential, spin-spin
interaction, isospin-isospin interaction, and isospin related spin-spin
interaction, respectively. In Sec. III, we will use the LECs fitted from the
$N\bar{N}$ scattering data Kang:2013uia  to estimate the contributions of the
leading order contact terms in the $\Sigma_{c}^{(*)}D^{(*)}$ systems.

###  II.2 Effective potentials

To obtain the effective potentials in the momentum space, we first calculate
the scattering amplitude $\mathcal{M}$. The scattering amplitude $\mathcal{M}$
is related to the effective potential $\mathcal{V}(\bm{q})$ by the following
relation

$\displaystyle\mathcal{V}\left(\bm{q}\right)=-\frac{\mathcal{M}}{\sqrt{\prod_{i=1}^{4}2M_{i}}},$
(23)

where $M_{i}$ are the masses of the scattering particles. We can obtain the
effective potential $\mathcal{V}(r)$ in the coordinate space via the following
Fourier transformation,

$\displaystyle\mathcal{V}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}\bm{q}}{\left(2\pi\right)^{3}}e^{-i\bm{q}\cdot\bm{r}}\mathcal{V}\left(\bm{q}\right)\mathcal{F}\left(\bm{q}\right).$
(24)

where a Gaussian regulator $\mathcal{F}\left(\bm{q}\right)={\rm
exp}\left(-\bm{q}^{2n}/\Lambda^{2n}\right)$ is introduced to regularize the
divergence in this integral. This type of regulator has been widely used in
the $NN$ and $N\bar{N}$ systems Machleidt:2011zz ; Epelbaum:2014efa ;
Epelbaum:2003xx ; Kang:2013uia ; Entem:2003ft . In this work, we use the LECs
fitted from the $N\bar{N}$ scattering Kang:2013uia  to estimate the LECs of
the $\Sigma_{c}^{(*)}D^{(*)}$ systems, thus we use $n=3$ as adopted in Ref.
Kang:2013uia  for consistency, and take a typical cutoff $\Lambda=0.4$ GeV to
suppress the contributions from higher momenta Wang:2020dhf .

The contact and one-pion-exchange interactions contribute to the leading order
effective potentials. The corresponding Feynman diagrams are collected in Fig.
1, where the $\Sigma_{c}D$ and $\Sigma_{c}^{*}D$ systems do not have the one-
pion-exchange diagrams due to the forbidden $DD\pi$ vertex.

Figure 1: The leading order Feynman diagrams for the $\Sigma_{c}D$
($X_{1.1}$), $\Sigma_{c}^{*}D$ ($X_{2.1}$), $\Sigma_{c}D^{*}$ ($X_{3.1}$,
$H_{3.1}$), and $\Sigma_{c}^{*}D^{*}$ ($X_{4.1}$, $H_{4.1}$) systems. We use
the thick line, heavy-thick line, thin line, double-thin line, and dashed line
to denote the $\Sigma_{c}$, $\Sigma_{c}^{*}$, $D$, $D^{*}$, and $\pi$,
respectively.

The explicit expressions of the contact potentials for the $\Sigma_{c}D$,
$\Sigma_{c}^{*}D$, $\Sigma_{c}D^{*}$, and $\Sigma_{c}^{*}D^{*}$ systems are

$\displaystyle\mathcal{V}_{\Sigma_{c}D}^{X_{1.1}}$ $\displaystyle=$
$\displaystyle-D_{a}+2\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)E_{a},$
(25) $\displaystyle\mathcal{V}_{\Sigma_{c}^{*}D}^{X_{2.1}}$ $\displaystyle=$
$\displaystyle-D_{a}+2\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)E_{a},$
(26) $\displaystyle\mathcal{V}^{X_{3.1}}_{\Sigma_{c}D^{*}}$ $\displaystyle=$
$\displaystyle-D_{a}+2\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)E_{a}$
(27)
$\displaystyle+\frac{2}{3}\left[-D_{b}+2E_{b}\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\right]\bm{\sigma}\cdot\bm{T},$
$\displaystyle\mathcal{V}^{X_{4.1}}_{\Sigma_{c}^{*}D^{*}}$ $\displaystyle=$
$\displaystyle-D_{a}+2\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)E_{a}$
(28)
$\displaystyle+\left[-D_{b}+2E_{b}\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\right]\bm{\sigma}_{rs}\cdot\bm{T}.$

where $\mathbf{I}_{1}$ and $\mathbf{I}_{2}$ are the isospin operators of the
$\Sigma_{c}^{(*)}$ and $D^{(*)}$, respectively. The matrix elements of
$\mathbf{I}_{1}\cdot\mathbf{I}_{2}$ can be obtained via

$\displaystyle\left\langle\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right\rangle=\frac{1}{2}\left[I(I+1)-I_{1}(I_{1}+1)-I_{2}(I_{2}+1)\right],$
(29)

where $I$ is the total isospin of the $\Sigma_{c}^{(*)}D^{(*)}$ systems. The
cross product of the final and initial polarization vectors for $D^{*}$ mesons
($\bm{\varepsilon}^{\dagger}$ and $\bm{\varepsilon}$, respectively) is given
in terms of $\bm{T}$ operator

$\displaystyle-i\bm{T}=\bm{\varepsilon}^{\dagger}\times\bm{\varepsilon}.$ (30)

where the spin operator $\bm{S}_{v}$ of the $D^{*}$ meson can be related to
the $\bm{T}$ operator via

$\displaystyle\bm{S}_{v}=-\bm{T}.$ (31)

The spin operators $\bm{S}_{s}$ of $\Sigma_{c}$ and $\bm{S}_{rs}$ of
$\Sigma_{c}^{\ast}$ are related to the Pauli matrix $\bm{\sigma}$ and
$\bm{\sigma}_{rs}$ via

$\displaystyle\bm{S}_{s}=\frac{1}{2}\bm{\sigma},\qquad\bm{S}_{rs}=\frac{3}{2}\bm{\sigma}_{rs}.$
(32)

Then the matrix elements of the $\bm{\sigma}\cdot\bm{T}$ and
$\bm{\sigma}_{rs}\cdot\bm{T}$ can be obtained from Eqs. (31)-(32) by

$\displaystyle\bm{\sigma}\cdot\bm{T}$ $\displaystyle=$
$\displaystyle-2\bm{S}_{1}\cdot\bm{S}_{2}$ $\displaystyle=$
$\displaystyle-\left[S(S+1)-S_{1}(S_{1}+1)-S_{2}(S_{2}+1)\right],$
$\displaystyle\bm{\sigma}_{rs}\cdot\bm{T}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\bm{S}_{1}\cdot\bm{S}_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\left[S(S+1)-S_{1}(S_{1}+1)-S_{2}(S_{2}+1)\right].$

where $\bm{S}_{1}\equiv\bm{S}_{s}(\bm{S}_{rs})$ and
$\bm{S}_{2}\equiv\bm{S}_{v}$ denote the spin operators of the
$\Sigma^{(*)}_{c}$ baryon and $D^{*}$ meson, respectively.

The one-pion-exchange diagrams for the $\Sigma_{c}D^{*}$ and
$\Sigma_{c}^{*}D^{*}$ systems are depicted in graphs ($H_{3.1}$) and
($H_{4.1}$) of Fig. 1. The corresponding effective potentials read

$\displaystyle\mathcal{V}_{\Sigma_{c}D^{*}}^{H_{3.1}}$ $\displaystyle=$
$\displaystyle\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\frac{gg_{1}}{2f_{\pi}^{2}}\frac{\left(\bm{q}\cdot\bm{\sigma}\right)\left(\bm{q}\cdot\bm{T}\right)}{\bm{q}^{2}+m_{\pi}^{2}},$
(34) $\displaystyle\mathcal{V}_{\Sigma_{c}^{*}D^{*}}^{H_{4.1}}$
$\displaystyle=$
$\displaystyle-\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\frac{gg_{5}}{2f_{\pi}^{2}}\frac{\left(\bm{q}\cdot\bm{\sigma}_{rs}\right)\left(\bm{q}\cdot\bm{T}\right)}{\bm{q}^{2}+m_{\pi}^{2}}.$
(35)

One can notice that there is a minus sign between the one-pion-exchange
amplitudes of the $\Sigma_{c}^{(*)}D^{*}$ and $\Sigma_{c}^{(*)}\bar{D}^{*}$
systems Wang:2019ato . This minus sign comes from the $G$-parity
transformation between the ($\bar{D}^{(*)0}$, $D^{(*)-}$) and ($D^{(*)+}$,
$D^{(*)0}$) doublets.

Figure 2: Two-pion-exchange diagrams that account for the effective potentials
of the $\Sigma_{c}D$ system at next-to-leading order. These diagrams include
the football diagram ($F_{1.1}$), triangle diagrams ($T_{1.i}/\bar{T}_{1.i}$),
box diagrams ($B_{1.i}/\bar{B}_{1.i}$), and crossed box diagrams
($R_{1.i}/\bar{R}_{1.i}$). The $\bar{T}_{1.3}$, $\bar{B}_{1.1}$, and
$\bar{R}_{1.1}$ denote the diagrams with $\Lambda_{c}$ as the intermediate
state. The notations are the same as those in Fig. 1.
Figure 3: Two-pion-exchange diagrams for the $\Sigma^{*}_{c}D$ system. The
notations are the same as those in Fig. 2.
Figure 4: Two-pion-exchange diagrams for the $\Sigma_{c}D^{*}$ system. The
notations are the same as those in Fig. 2.
Figure 5: Two-pion-exchange diagrams for the $\Sigma^{*}_{c}D^{\ast}$ system.
The notations are the same as those in Fig. 2.

The two-pion-exchange diagrams for the $\Sigma_{c}D$, $\Sigma_{c}^{*}D$,
$\Sigma_{c}D^{*}$, and $\Sigma_{c}^{*}D^{*}$ systems are illustrated in Figs.
2, 3, 4, and 5, respectively. The analytical results for the football diagrams
($F_{i.j}$), triangle diagrams ($T_{i.j}$), box diagrams ($B_{i.j}$), and
crossed box diagrams ($R_{i.j}$) generally have the following forms,

$\displaystyle\mathcal{V}_{\rm{sys}}^{F_{i.j}}$ $\displaystyle=$
$\displaystyle\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\frac{1}{f_{\pi}^{4}}J_{22}^{F}(m_{\pi},q),$
(36) $\displaystyle\mathcal{V}_{\rm{sys}}^{T_{i,j}}$ $\displaystyle=$
$\displaystyle\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\frac{\mathcal{C}_{\rm{sys}}^{T_{i.j}}}{f_{\pi}^{4}}\left[-\bm{q}^{2}\mathcal{C}_{1}^{T_{i.j}}\left(J_{24}^{T}+J_{33}^{T}\right)\right.$
(37)
$\displaystyle\left.+\mathcal{C}_{2}^{T_{i.j}}J_{34}^{T}\right]\left(m_{\pi},\mathcal{E}^{T_{i.j}},q\right),$
$\displaystyle\mathcal{V}_{\rm{sys}}^{B_{i.j}}$ $\displaystyle=$
$\displaystyle\left(1-\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\frac{\mathcal{C}_{\rm{sys}}^{B_{i.j}}}{f_{\pi}^{4}}\left[-\bm{q}^{2}\mathcal{C}_{1}^{B_{i.j}}J_{21}^{B}\right.$
(38)
$\displaystyle+\bm{q}^{4}\mathcal{C}_{2}^{B_{i.j}}(J_{22}^{B}+2J_{32}^{B}+J_{43}^{B})-\bm{q}^{2}\mathcal{C}_{3}^{B_{i.j}}(J_{31}^{B}$
$\displaystyle+J_{42}^{B})\left.+\mathcal{C}_{4}^{B_{i.j}}J_{41}^{B}\right]\left(m_{\pi},\mathcal{E}_{1}^{B_{i.j}},\mathcal{E}^{B_{i.j}}_{2},q\right),$
$\displaystyle\mathcal{V}_{\rm{sys}}^{R_{i.j}}$ $\displaystyle=$
$\displaystyle\left(1+\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\frac{\mathcal{C}_{\rm{sys}}^{R_{i.j}}}{f_{\pi}^{4}}\left[-\bm{q}^{2}\mathcal{C}_{1}^{R_{i.j}}J_{21}^{R}\right.$
(39)
$\displaystyle+\bm{q}^{4}\mathcal{C}_{2}^{R_{i.j}}\left(J_{22}^{R}+2J_{32}^{R}+J_{43}^{R}\right)-\bm{q}^{2}\mathcal{C}_{3}^{R_{i.j}}(J_{31}^{R}$
$\displaystyle+J_{42}^{R})\left.+\mathcal{C}_{4}^{R_{i.j}}J_{41}^{R}\right]\left(m_{\pi},\mathcal{E}_{1}^{R_{i.j}},\mathcal{E}^{R_{i.j}}_{2},q\right),$

where the subscript “sys” denotes the corresponding $\Sigma_{c}^{(*)}D^{(*)}$
system. The superscripts $F_{i.j}$, $T_{i.j}$, $B_{i.j}$, and $R_{i.j}$ are
the labels of Feynman diagrams illstrated in Figs. 2-5. The $J_{x}^{T}$,
$J_{x}^{B}$, $J_{x}^{R}$ are scalar loop functions defined in Appendix A of
Ref. Wang:2019ato . $\mathcal{E}^{T_{i.j}}$, $\mathcal{E}_{1(2)}^{B_{i.j}}$,
$\mathcal{E}^{R_{i.j}}_{1(2)}$ are the residual energies.

In Ref. Wang:2019ato , we found the contributions of $\Lambda_{c}$ in the
loops of two-pion-exchange diagrams have considerable corrections to the
effective potentials. Thus, in this work, we also consider the contributions
from intermediate $\Lambda_{c}$ state. The general expressions for the
corresponding triangle diagrams ($\bar{T}_{i.j}$), box diagrams
($\bar{B}_{i.j}$), and crossed box diagrams ($\bar{R}_{i.j}$) read

$\displaystyle\mathcal{V}^{\bar{T}_{i.j}}_{\rm{sys}}$ $\displaystyle=$
$\displaystyle\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\frac{C_{\rm{sys}}^{\bar{T}_{i.j}}}{f_{\pi}^{4}}\left[-\bm{q}^{2}\mathcal{C}_{1}^{\bar{T}_{i.j}}\left(J_{24}^{T}+J_{33}^{T}\right)\right.$
(40)
$\displaystyle\left.+\mathcal{C}_{2}^{\bar{T}_{i.j}}\right]\left(m_{\pi},\mathcal{E}^{\bar{T}_{i.j}},q\right),$
$\displaystyle\mathcal{V}_{\rm{sys}}^{\bar{B}_{i.j}}$ $\displaystyle=$
$\displaystyle\left(1-2\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\frac{\mathcal{C}_{\rm{sys}}^{\bar{B}_{i.j}}}{f_{\pi}^{4}}\left[-\bm{q}^{2}\mathcal{C}_{1}^{\bar{B}_{i.j}}J_{21}^{B}\right.$
(41)
$\displaystyle+\bm{q}^{4}\mathcal{C}_{2}^{\bar{B}_{i.j}}\left(J_{22}^{B}+2J_{32}^{B}+J_{43}^{B}\right)-\bm{q}^{2}\mathcal{C}_{3}^{\bar{B}_{i.j}}(J^{B}_{31}$
$\displaystyle+J_{42}^{B})\left.+\mathcal{C}_{4}^{\bar{B}_{i.j}}J_{41}^{B}\right]\left(m_{\pi},\mathcal{E}_{1}^{\bar{B}_{i.j}},\mathcal{E}_{2}^{\bar{B}_{i.j}},q\right),$
$\displaystyle\mathcal{V}_{\rm{sys}}^{\bar{R}_{i.j}}$ $\displaystyle=$
$\displaystyle\left(1+2\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\frac{\mathcal{C}_{\rm{sys}}^{\bar{R}_{i.j}}}{f_{\pi}^{4}}\left[-\bm{q}^{2}\mathcal{C}_{1}^{\bar{R}_{i.j}}J_{21}^{R}\right.$
(42)
$\displaystyle+\bm{q}^{4}\mathcal{C}_{2}^{\bar{R}_{i.j}}\left(J_{22}^{R}+2J_{32}^{R}+J_{43}^{R}\right)-\bm{q}^{2}\mathcal{C}_{3}^{\bar{R}_{i.j}}(J^{R}_{31}$
$\displaystyle+J_{42}^{R})\left.+\mathcal{C}_{4}^{\bar{R}_{i.j}}J_{41}^{R}\right]\left(m_{\pi},\mathcal{E}_{1}^{\bar{R}_{i.j}},\mathcal{E}_{2}^{\bar{R}_{i.j}},q\right),$

One can see Appendix A for the explicit values of the coefficients defined in
Eqs. (37)-(42).

We notice the expressions of the two-pion-exchange diagrams for the
$\Sigma_{c}^{(*)}D^{(*)}$ systems are identical to those of the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ systems Wang:2019ato . This interesting
results can be easily understood as follows: the differences between the two-
pion-exchange amplitudes of the $\Sigma_{c}^{(*)}D^{(*)}$ and
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ systems are completely caused by the pionic
coupling of the charmed and anti-charmed mesons. As mentioned before, the one-
pion vertices [from the $u_{\mu}$ in Eq. (2)] between the charmed and anti-
charmed mesons have a minus sign difference, but they appear in pairs in the
two-pion-exchange diagrams. Besides, the two-pion vertices [from the
$\Gamma_{\mu}$ in Eq. (2)] is invariant under the $G$-parity transformation.

We have subtracted the 2PR contributions of the box diagrams in our
calculations. This can be achieved by the principal-value integral method
proposed in Ref. Wang:2019ato , in which a detailed derivation is presented in
the Appendix B.

##  III Numerical results and discussions

To get the numerical results, we need to determine the four LECs defined in
Eq. (21). At present, there are no experimental data or lattice QCD
simulations for the possible $P_{cc}$ states. In Refs. Wang:2020dhf ;
Wang:2019nvm ; Meng:2019nzy , we proposed to bridge the LECs determined from
the $NN$ ($N\bar{N}$) scattering data to the unknown LECs of the di-hadron
systems via a quark level contact Lagrangian. In this work, we apply this
approach to estimate the contributions of the contact terms for the
$\Sigma_{c}^{(*)}D^{(*)}$ systems, likewise. Then we search for binding
solutions via solving the Schrödinger equation and discuss the numerical
results.

###  III.1 Determining the LECs of the $\Sigma_{c}^{(*)}D^{(*)}$ systems

It is assumed that the contact terms are mimicked by exchanging heavy mesons
through the $S$-wave interaction Wang:2020dhf ; Wang:2019nvm ; Meng:2019nzy ,
in which a general quark-level Lagrangian is constructed as

$\displaystyle\mathcal{L}=g_{s}\bar{q}\mathcal{S}q+g_{a}\bar{q}\gamma_{\mu}\gamma^{5}\mathcal{A}^{\mu}q,$
(43)

where $q=(u,d)$, $c_{s}$ and $c_{t}$ are two independent coupling constants.
The fictitious scalar ($\mathcal{S}$) and axial-vector ($\mathcal{A}^{\mu}$)
fields with positive parity are introduced to account for the central
potential and spin-spin interaction, respectively. From Eq. (43), the
$q\bar{q}$ contact potential is obtained as

$\displaystyle
V_{q\bar{q}}=c_{s}\left(1-3\bm{\tau}_{1}\cdot\bm{\tau}_{2}\right)+c_{t}\left(1-3\bm{\tau}_{1}\cdot\bm{\tau}_{2}\right)\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}.$
(44)

In Table 1, we present the quark-level matrix elements of the operators
related to the contact potentials.

Table 1: The quark-level matrix elements of two-body interaction operators $\mathcal{O}_{ij}$ for the $N\bar{N}$ and $\Sigma_{c}D^{*}$ systems. $\mathcal{O}_{ij}$ | $\bm{1}_{ij}$ | $\bm{\tau}_{i}\cdot\bm{\tau}_{j}$ | $\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$ | $\left(\bm{\tau}_{i}\cdot\bm{\tau}_{j}\right)\left(\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}\right)$
---|---|---|---|---
$\left[N\bar{N}\right]^{I=1}_{J=1}$ | 9 | 1 | 1 | $\frac{25}{9}$
$\left[N\bar{N}\right]^{I=1}_{J=0}$ | 9 | 1 | -3 | -$\frac{25}{3}$
$\left[N\bar{N}\right]^{I=0}_{J=1}$ | 9 | -3 | 1 | -$\frac{25}{3}$
$\left[N\bar{N}\right]^{I=0}_{J=0}$ | 9 | -3 | -3 | 25
$\left[\Sigma_{c}D^{*}\right]_{J=\frac{3}{2}}^{I=\frac{3}{2}}$ | 2 | 2 | $\frac{4}{3}$ | $\frac{4}{3}$

Based on the Lagrangian in Eq. (43) and the matrix elements in Table 1, the
authors of Ref. Wang:2020dhf  derived the contact potential of the $N\bar{N}$
system with quantum numbers $I=1$ and ${}^{2S+1}L_{J}=^{3}S_{1}$,

$\displaystyle V_{N\bar{N}}^{{}^{3}S_{1}}$ $\displaystyle=$
$\displaystyle\left\langle
N\bar{N}\left|V_{q\bar{q}}\right|N\bar{N}\right\rangle=6c_{s}-\frac{22}{3}c_{t}.$
(45)

One can as well as obtain the contact potentials of the
$[N\bar{N}]_{J=0}^{I=1}$, $[N\bar{N}]_{J=1}^{I=0}$, and
$[N\bar{N}]_{J=0}^{I=0}$ systems, accordingly.

Similarly, the $\Sigma_{c}D^{*}$ contact potential can be obtained from Eq.
(44) and Table 1 as

$\displaystyle V_{\Sigma_{c}D^{*}}$ $\displaystyle=$ $\displaystyle
2c_{s}-12c_{s}\mathbf{I}_{1}\cdot\mathbf{I}_{2}-\frac{4}{3}c_{t}\bm{\sigma}\cdot\bm{T}$
(46)
$\displaystyle+8c_{t}\left(\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right)\left(\bm{\sigma}\cdot\bm{T}\right).$

Comparing Eq. (27) with Eq. (46) we get

$\displaystyle D_{a}=-2c_{s},\quad E_{a}=-6c_{s},\quad D_{b}=2c_{t},\quad
E_{b}=6c_{t}.$

In Ref. Kang:2013uia , based on the $N\bar{N}$ scattering data, the LECs for
the $I=0$ ($J=0,1$) and $I=1$ ($J=0,1$) $N\bar{N}$ systems are fitted. With
the LECs of these four $N\bar{N}$ systems, we obtain six sets of solutions for
the $c_{s}$ and $c_{t}$. Among them, four sets of $c_{s}$ and $c_{t}$ are
consistent with each other in sizes and signs:

  * Set 1: $c_{s}=-5.84$ GeV-2, $c_{t}=2.50$ GeV-2;

  * Set 2: $c_{s}=-8.10$ GeV-2, $c_{t}=0.65$ GeV-2;

  * Set 3: $c_{s}=-8.25$ GeV-2, $c_{t}=0.52$ GeV-2;

  * Set 4: $c_{s}=-7.71$ GeV-2, $c_{t}=0.38$ GeV-2.

The remaining two sets of solutions either have the different signs or are too
large, leading to unstable numerical results in our calculations.

When checking the above four sets of LECs, we notice that the $c_{s}$ value in
Set 1 is smaller than those of other sets, the input with $c_{s}$ in Set 1
will lead to relatively small central potentials. On the contrary, the value
of $c_{t}$ in Set 1 is larger than those of other sets, the spin-spin
corrections would be important with this set of LECs. This is the first case
we want to discuss, we label this set of LECs solution as Case 1. The LECs in
Set 2 have been successfully applied to study the interactions of the
$D^{(*)}N$ systems Wang:2020dhf , the small value of $c_{t}$ shows that with
this set of solution, the spin-spin interaction serves as the perturbation to
the $D^{(*)}N$ multiplets. The LECs in Sets 3 and 4 are very close to that of
Set 2, and they give very similar results. Thus, we will use the LECs in Set 2
as our Case 2. In addition, we also use the least square method to fit a best
solution from these four sets of LECs, the solution are obtained as

$\displaystyle c_{s}=-7.46\rm{GeV}^{-2},\qquad c_{t}=1.02\rm{GeV}^{-2}.$ (48)

we label this set of LECs as the Case 3.

###  III.2 Numerical results of the effective potentials

Figure 6: The effective potentials for the $[\Sigma^{(*)}_{c}D]^{I=1/2}_{J}$
systems. Their $I(J^{P})$ numbers are illustrated in each subfigure. The red
dashed line and blue dot-dashed line denote the effective potentials from the
contact term and two-pion-exchange, respectively. The black solid line denote
the total effective potential for each system.

We use the LECs in Case 3 to present the effective potentials of all the
$\Sigma_{c}^{(*)}D^{(*)}$ systems. In Fig. 6, we plot the effective potentials
of the $\Sigma_{c}D$ and $\Sigma_{c}^{*}D$ systems. The contact terms of the
$\Sigma_{c}D$ and $\Sigma_{c}^{*}D$ are the same in the heavy quark limit,
which can be checked from the line shapes of the contact effective potentials
in Fig. 6.

For the $[\Sigma_{c}D]^{I=1/2}_{J=1/2}$ system, the two-pion-exchange
interaction provides a weakly repulsive force. The contact interaction
provides a strong attractive force, which is also true for the
$[\Sigma^{*}_{c}D]^{I=1/2}_{J=3/2}$ system. In the
$[\Sigma^{*}_{c}D]^{I=1/2}_{J=3/2}$ system, the two-pion-exchange interaction
provides a weakly attractive potential and forms a deeply bound
$[\Sigma^{*}_{c}D]^{I=1/2}_{J=3/2}$ state together with the strong attractive
contact term.

From the right panel of Fig. 6, we can see that the two-pion-exchange
interactions provide considerable attractive force in the
$[\Sigma^{(*)}_{c}D]^{I=3/2}_{J}$ systems. However, for the $I=3/2$ case, the
contact terms provide strong repulsive forces and the total effective
potentials are repulsive, i.e., we can not find any bound states. This is also
true for the $[\Sigma^{(*)}_{c}D^{*}]^{I=3/2}_{J}$ systems. Thus, in the
following, we only discuss the $\Sigma_{c}^{(*)}D^{(*)}$ systems with $I=1/2$.

Figure 7: The effective potentials for the
$[\Sigma^{(*)}_{c}D^{*}]^{I=1/2}_{J}$ systems. Their $I(J^{P})$ numbers are
illustrated in each subfigure. The red dashed line, green dashed line and blue
dot-dashed line denote the effective potentials from the contact term, one-
pion- and two-pion-exchange interactions, respectively. The black solid line
denote the total effective potential for each system.

In Fig. 7, we present the effective potentials of the $\Sigma_{c}^{(*)}D^{*}$
systems. From Fig. 7, we can see that in the $[\Sigma_{c}D^{*}]_{J}^{I=1/2}$
and $[\Sigma_{c}^{*}D^{*}]_{J}^{I=1/2}$ systems, the contact terms provide
strong attractive force. The one-pion- and two-pion-exchange interactions
supply very weak repulsive forces in the $[\Sigma_{c}D^{*}]_{J=1/2}^{I=1/2}$,
$[\Sigma^{*}_{c}D^{*}]_{J=1/2(3/2)}^{I=1/2}$ systems. For the
$[\Sigma_{c}D^{*}]_{J=3/2}^{I=1/2}$ and
$[\Sigma_{c}^{*}D^{*}]_{J=5/2}^{I=1/2}$ systems, the one-pion- and two-pion-
exchange potentials supply comparable attractive and repulsive forces,
respectively. Thus, the total effective potentials for these two systems are
nearly equivalent to their contact potentials.

Our calculation shows that the contact terms are important to the
$[\Sigma_{c}^{(*)}D^{(*)}]^{I=1/2}_{J}$ systems. In our framework, the
fictitious scalar mesons account for the mainly attractive interactions in the
$[\Sigma_{c}^{(*)}D^{(*)}]^{I=1/2}_{J}$ systems, and the exchange of axial-
vector mesons result in mass splittings in spin multiplets. In Eq. (43), the
$c_{s}$ and $c_{t}$ are related to the strength of the scalar-exchange and
axial-vector-exchange forces, respectively. From Table 2, we notice that the
axial-vector-exchange interactions which are related to the $c_{t}$ provide
small corrections to the final effective potentials in Cases 2 and 3. However,
in Case 1, we have $|c_{t}/c_{s}|=0.43$, this can be regarded as our upper
limit of the $|c_{t}/c_{s}|$. The relatively small value of $c_{t}$ can be
traced to the large masses of the axial-vector particles since their masses
exceed $1$ GeV Zyla:2020zbs .

###  III.3 The binding energies of the $\Sigma_{c}^{(*)}D^{(*)}$ systems

Table 2: The binding energies, masses and root-mean-square radii for all the $[\Sigma_{c}^{(*)}D^{(*)}]^{I=1/2}_{J}$ systems. The subscript denotes the total angular momentum of this system. The adopted LECs in Cases 1, 2, and 3 are ($c_{s}=-5.84$, $c_{t}=2.50$) GeV-2, ($c_{s}=-8.10$, $c_{t}=0.65$ ) GeV-2, and ($c_{s}=-7.46$, $c_{t}=1.02$) GeV-2, respectively. |  | $\left[\Sigma_{c}D\right]_{\frac{1}{2}}$ | $\left[\Sigma^{*}_{c}D\right]_{\frac{3}{2}}$ | $\left[\Sigma_{c}D^{*}\right]_{\frac{1}{2}}$ | $\left[\Sigma_{c}D^{*}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{c}^{*}D^{*}\right]_{\frac{1}{2}}$ | $\left[\Sigma_{c}^{*}D^{*}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{c}^{*}D^{*}\right]_{\frac{5}{2}}$
---|---|---|---|---|---|---|---|---
Case 1 | BE (MeV) | -15.4 | -25.0 | -31.8 | -8.0 | -32.8 | -18.2 | -3.5
$R_{rms}$ (fm) | 1.45 | 1.25 | 1.20 | 1.65 | 1.20 | 1.38 | 1.91
Case 2 | BE (MeV) | -31.3 | -42.9 | -30.3 | -31.7 | -26.6 | -25.4 | -29.7
$R_{rms}$ (fm) | 1.23 | 1.11 | 1.22 | 1.20 | 1.26 | 1.27 | 1.22
Case 3 | BE (MeV) | -26.5 | -37.7 | -29.1 | -25.0 | -26.4 | -22.6 | -22.2
$R_{rms}$ (fm) | 1.27 | 1.14 | 1.23 | 1.27 | 1.26 | 1.31 | 1.30

The binding energies, masses, and the root-mean-square radii in the above
three cases are presented in Table 2. We find bound state solutions only for
the $I=1/2$ channels. The $R_{rms}$ are about 1$-$2 fm for all the considered
$\Sigma_{c}^{(*)}D^{(*)}$ systems, which are the typical sizes of the hadronic
molecules. From Table 2, we can see that the binding of the
$[\Sigma_{c}^{*}D]_{J=3/2}^{I=1/2}$ system is deeper than that of the
$[\Sigma_{c}D]_{J=1/2}^{I=1/2}$ system. In the heavy quark limit, the
$[\Sigma_{c}D]_{J=1/2}^{I=1/2}$ and $[\Sigma_{c}^{*}D]_{J=3/2}^{I=1/2}$
systems share the same contact term. Thus, the difference of the binding
energy is from the contributions of the two-pion-exchange interactions.
Besides, in Case 1, the $\Sigma_{c}^{(*)}D^{*}$ systems with lower total
angular momentum $J$ are more compact. This situation is very similar to the
$\Sigma_{c}^{(*)}\bar{D}^{*}$ systems Wang:2019ato . However, the results in
Cases 2 and 3 show that the binding energies for the different
$\Sigma_{c}^{(*)}D^{(*)}$ systems are comparable to each other, and they have
very similar spatial sizes.

The parameters $c_{s}$ and $c_{t}$ are related to the central potentials and
spin-spin interactions, respectively. In Case 1, $|c_{t}/c_{s}|=0.43$, the
spin-spin corrections in contact terms have considerable contributions in the
$\Sigma_{c}^{(*)}D^{*}$ systems [note that the spin-spin corrections do not
contribute to the $\Sigma_{c}^{(*)}D$ systems, e.g., see Eqs. (25)-(26)]. The
spin-spin corrections are much larger than the contributions from the one-
pion-exchange and two-pion-exchange interactions. Thus, in Case 1, the mass
splittings among different $[\Sigma_{c}^{(*)}D^{*}]_{J}^{I=1/2}$ systems are
mainly caused by the corrections of the spin-spin interactions. The results
obtained from the Cases 2 and 3 are close to each other. In contrast to Case
1, $|c_{t}/c_{s}|\approx 0.1$ in these two cases, i.e., the central potentials
are dominant and the spin-spin potentials are small. The contributions from
the spin-spin interactions are comparable to those of the one-pion-exchange
and two-pion-exchange interactions. As shown in Table 2, in these two cases,
the $\Sigma_{c}^{(*)}D^{*}$ systems with higher total angular momenta have
deeper binding energies.

The results for the $\Sigma_{c}^{(*)}\bar{B}^{(*)}$,
$\Sigma_{b}^{(*)}D^{(*)}$, and $\Sigma_{b}^{(*)}\bar{B}^{(*)}$ systems are
given in Appendix B.

###  III.4 Possible decay patterns of the $\Sigma_{c}^{(*)}D^{(*)}$ molecules

The $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ are produced in the
$\Lambda_{b}^{0}\rightarrow J/\psi K^{-}p$ decay process and reconstructed in
the $J/\psi p$ channel Aaij:2019vzc . Similarly, the $P_{cs}(4459)$ is
produced in the $\Xi_{b}^{-}\rightarrow J/\psi K^{-}\Lambda$ process and
observed in the $J/\psi\Lambda$ invariant mass spectrum Aaij:2020gdg . In this
subsection, we discuss the possible decay patterns of the $P_{cc}$ states.
They may be considered as the reconstructive channels from the $pp$ collisions
at LHCb.

Figure 8: The mass spectrum of the $P_{cc}$ pentaquarks. The results are
obtained using the LECs in Case 3. The black solid lines denote the $P_{cc}$
pentaquarks. The blue and red dotted lines denote the thresholds of the
$\Sigma^{(*)}_{c}D^{(*)}$ and possible decay channels, respectively.

In Fig. 8, we present the mass spectrum of the $P_{cc}$ pentaquarks based on
the inputs in Case 3, and some relevant thresholds. Due to the $cc$ pairs in
the $P_{cc}$ pentaquarks, the decay behaviors of the $P_{cc}$ states are
different from that of the hidden-charm pentaquarks. There exist two types of
decay modes for the $P_{cc}$ states, i.e., the $(cqq)$-$(c\bar{q})$ and
$(ccq)$-$(q\bar{q})$ modes. Note that the $\Lambda_{c}\pi$ and $D\pi$ are the
dominant decay channels for the $\Sigma_{c}^{(*)}$ baryons and $D^{*}$ meson,
respectively. For simplicity, we only consider the ground $\Lambda_{c}$, $D$,
and $\pi$ as our decay final states in the first mode. From Fig. 8, we can see
that the $P_{cc}(4296)$ state with $J^{P}=1/2^{-}$ is near the threshold of
the $\Lambda_{c}D\pi$. Thus, it is very difficult for this state to decay into
this three-body final state due to the small phase space. But the
$P_{cc}(4296)$ can easily decay into $\Lambda_{c}D$ two-body final states.
Further study on the branching ratio of this decay process is still needed.
The $P_{cc}(4350)$ with $J^{P}=3/2^{-}$ can decay into the $\Lambda_{c}D\pi$
via the $S$-wave, while decaying into the $\Lambda_{c}D$ is $D$-wave
suppressed. One can perform similar analyses for the other five $P_{cc}$
states.

Now we discuss the $(ccq)$-$(q\bar{q})$ decay mode, i.e., the $P_{cc}$ states
decay into the ground $\Xi_{cc}$ baryon and a pseudoscalar or a vector meson.
The threshold of $\Xi_{cc}\pi$ channel is about 3760 MeV, which is much lower
than the $P_{cc}$ states and is not presented in Fig. 8. The predicted
$P_{cc}$ states with $J^{P}=1/2^{-}$ can decay into this channel through
$S$-wave, thus, this should be an important strong decay channels for the
$J^{P}=1/2^{-}$ states due to the large phase spaces. The states that are
composed of the $\Sigma^{(*)}_{c}$ and $D^{*}$ can also decay into the
$\Xi_{cc}\omega$ and $\Xi_{cc}\rho$ final states. One can also extract the
decay properties for the other $P_{cc}$ pentaquarks in Fig. 8.

##  IV Summary

Inspired by the recently observed $P_{c}$ Aaij:2019vzc  and $P_{cs}$
Aaij:2020gdg  pentaquarks, we perform a systematic study on the interactions
of the $\Sigma_{c}^{(*)}D^{(*)}$ systems to explore the possible $P_{cc}$
states. We include the contact term, one-pion-exchange, and two-pion-exchange
interactions within the framework of chiral effective field theory.

Due to $G$-parity transformation law, the expressions of the one-pion-exchange
and two-pion-exchange effective potentials of the $\Sigma_{c}^{(*)}D^{(*)}$
systems are opposite and identical to those of the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ systems Wang:2019ato , respectively. With the
LECs fitted from the $N\bar{N}$ scattering data, we obtain four sets of
($c_{s}$, $c_{t}$) parameters describing the contributions of the contact
terms. We present three cases to study the binding energies of the
$\Sigma_{c}^{(*)}D^{(*)}$ systems.

The mass spectrum of the $[\Sigma_{c}^{(*)}D^{(*)}]^{I=1/2}_{J}$ molecules
depend on the values of the LECs. In Case 1, a relatively small central
potential and a large spin-spin interaction are introduced. The obtained
$P_{cc}$ mass spectrum is very similar to that of the
$\Sigma_{c}^{(*)}\bar{D}^{(*)}$ systems. However, the mass spectra obtained in
Cases 2 and 3 are different from that of the Case 1. Further experimental
studies may help us to clarify the predicted mass spectra of the
$\Sigma_{c}^{(*)}D^{(*)}$ systems in different cases.

We briefly discuss the strong decay behaviors of the $P_{cc}$ pentaquarks. The
$(cqq)$-$(c\bar{q})$ and $(ccq)$-$(q\bar{q})$ are the two types of decay
modes. Correspondingly, the $\Lambda_{c}D$, $\Lambda_{c}D\pi$, and
$\Xi_{cc}\pi$ are expected to be important channels to search for these
$[\Sigma_{c}^{(*)}D^{(*)}]^{I=1/2}_{J}$ molecules.

We also study the interactions of the $\Sigma_{c}^{(*)}\bar{B}^{(*)}$,
$\Sigma_{b}^{(*)}D^{(*)}$, and $\Sigma_{b}^{(*)}\bar{B}^{(*)}$ to search for
possible $P_{cb}$, $P_{bc}$, and $P_{bb}$ pentaquarks. The corresponding
systems with $I=1/2$ can also form molecular states. In addition, among the
studied systems, the binding becomes deeper when the reduced masses of the
systems are heavier.

## Acknowledgments

This project is supported by the National Natural Science Foundation of China
under Grants 11975033 and 12070131001.

##  Appendix A Supplements for the two-pion-exchange expressions

In Sec. II.2, we present the general expressions for the football diagrams
($F_{i.j}$), triangle diagrams ($T_{i.j}$), ($\bar{T}_{i.j}$), box diagrams
($B_{i.j}$), ($\bar{B}_{i.j}$) and crossed box diagrams $(R_{i.j})$,
($\bar{R}_{i.j}$) in Eqs. (36)-(42). In this appendix, we give their explicit
coefficients defined in Eqs. (36)-(42).

Specifically, we collect the coefficients $\mathcal{C}_{\rm{sys}}^{T_{i.j}}$
($\mathcal{C}_{\rm{sys}}^{\bar{T}_{i.j}}$),
$\mathcal{C}_{\rm{sys}}^{(B/R)_{i.j}}$
($\mathcal{C}_{\rm{sys}}^{(\bar{B}/\bar{R})_{i.j}}$) in Table 3, the
coefficients of triangle diagrams $\mathcal{C}_{m}^{T_{i.j}}$
($\mathcal{C}_{m}^{\bar{T}_{i.j}}$) and $\mathcal{E}^{T_{i.j}}$
($\mathcal{E}^{\bar{T}_{i.j}}$) in Table 4, the coefficients of box and
crossed box diagrams $\mathcal{C}_{m}^{B_{i.j}}$, $\mathcal{C}_{m}^{R_{i.j}}$
and $\mathcal{E}_{n}^{(B/R)_{i.j}}$ in Table 5, and the coefficients of box
and crossed box diagrams $\mathcal{C}_{m}^{\bar{B}_{i.j}}$,
$\mathcal{C}_{m}^{\bar{R}_{i.j}}$ and
$\mathcal{E}_{n}^{(\bar{B}/\bar{R})_{i.j}}$ in Table 6.

Table 3: The coefficients $\mathcal{C}_{\rm{sys}}^{T_{i.j}}$ ($\mathcal{C}_{\rm{sys}}^{\bar{T}_{i.j}}$), $\mathcal{C}_{\rm{sys}}^{(B/R)_{i.j}}$ ($\mathcal{C}_{\rm{sys}}^{(\bar{B}/\bar{R})_{i.j}}$) defined in Eqs. (37)-(42). The superscripts denote the corresponding diagrams illustrated in Figs. 2, 3, 4, and 5, respectively.  | $\mathcal{C}_{\rm{sys}}^{T_{i.1}}$ | $\mathcal{C}_{\rm{sys}}^{T_{i.2}}$ | $\mathcal{C}_{\rm{sys}}^{T_{i.3}}$ | $\mathcal{C}_{\rm{sys}}^{T_{i.4}}$ | $\mathcal{C}_{\rm{sys}}^{(B/R)_{i.1}}$ | $\mathcal{C}_{\rm{sys}}^{(B/R)_{i.2}}$ | $\mathcal{C}_{\rm{sys}}^{(B/R)_{i.3}}$ | $\mathcal{C}_{\rm{sys}}^{(B/R)_{i.4}}$ | $\mathcal{C}_{\rm{sys}}^{\bar{T}_{i.3}}$ | $\mathcal{C}_{\rm{sys}}^{(\bar{B}/\bar{R})_{i.1}}$ | $\mathcal{C}_{\rm{sys}}^{(\bar{B}/\bar{R})_{i.2}}$ | $\mathcal{C}_{\rm{sys}}^{(\bar{B}/\bar{R})_{i.3}}$ | $\mathcal{C}_{\rm{sys}}^{(\bar{B}/\bar{R})_{i.4}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\Sigma_{c}D$ | $g^{2}$ | $\frac{g_{3}^{2}}{4}$ | $\frac{g_{1}^{2}}{4}$ | - | $\frac{g^{2}g_{1}^{2}}{8}$ | $\frac{g^{2}g_{3}^{2}}{8}$ | - | - | $\frac{g_{2}^{2}}{2}$ | $\frac{g^{2}g_{2}^{2}}{8}$ | - | - | -
$\Sigma_{c}^{*}D$ | $g^{2}$ | $\frac{5g_{5}^{2}}{36}$ | $\frac{g_{3}^{2}}{4}$ | - | $\frac{5g^{2}g_{5}^{2}}{72}$ | $\frac{g^{2}g_{3}^{2}}{8}$ | - | - | $\frac{g_{4}^{2}}{2}$ | - | $\frac{g^{2}g_{4}^{2}}{8}$ | - | -
$\Sigma_{c}D^{*}$ | $g^{2}$ | $g^{2}$ | $\frac{g_{1}^{2}}{4}$ | $\frac{g_{3}^{2}}{4}$ | $\frac{g^{2}g_{1}^{2}}{8}$ | $\frac{g^{2}g_{1}^{2}}{8}$ | $\frac{g^{2}g_{3}^{2}}{8}$ | $\frac{g^{2}g_{3}^{2}}{24}$ | $\frac{g_{2}^{2}}{2}$ | $\frac{g^{2}g_{2}^{2}}{8}$ | $\frac{g^{2}g_{2}^{2}}{8}$ | - | -
$\Sigma_{c}^{*}D^{*}$ | $g^{2}$ | $g^{2}$ | $\frac{g_{3}^{2}}{4}$ | $\frac{5g_{5}^{2}}{36}$ | $\frac{g^{2}g_{5}^{2}}{24}$ | $\frac{g^{2}g_{5}^{2}}{24}$ | $\frac{g^{2}g_{3}^{2}}{32}$ | $\frac{g^{2}g_{3}^{2}}{32}$ | $\frac{g_{4}^{2}}{2}$ | - | - | $\frac{g^{2}g_{4}^{2}}{32}$ | $\frac{g^{2}g_{4}^{2}}{32}$
Table 4: The coefficients $\mathcal{C}_{m}^{T_{i.j}}$ ($\mathcal{C}_{m}^{\bar{T}_{i.j}}$) and $\mathcal{E}^{T_{i.j}}$ ($\mathcal{E}^{\bar{T}_{i.j}}$) defined in Eqs. (37) and (40). The superscript denotes the corresponding diagrams illustrated in Figs. 2, 3, 4, and 5, respectively.  | $\mathcal{C}_{1}^{T_{i.1}}$ | $\mathcal{C}_{2}^{T_{i.1}}$ | $\mathcal{E}^{T_{i.1}}$ | $\mathcal{C}_{1}^{T_{i.2}}$ | $\mathcal{C}_{2}^{T_{i.2}}$ | $\mathcal{E}^{T_{i.2}}$ | $\mathcal{C}_{1}^{T_{i.3}}$ | $\mathcal{C}_{2}^{T_{i.3}}$ | $\mathcal{E}^{T_{i.3}}$ | $\mathcal{C}_{1}^{T_{i.4}}$ | $\mathcal{C}_{2}^{T_{i.4}}$ | $\mathcal{E}^{T_{i.4}}$ | $\mathcal{C}_{1}^{\bar{T}_{i.3}}$ | $\mathcal{C}_{2}^{\bar{T}_{i.3}}$ | $\mathcal{E}^{\bar{T}_{i.3}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\Sigma_{c}D$ | 1 | 3 | $\mathcal{E}-\delta_{b}$ | $\frac{2}{3}$ | 2 | $\mathcal{E}-\delta_{a}$ | 1 | 3 | $\mathcal{E}$ | - | - | - | 1 | 3 | $\mathcal{E}+\delta_{c}$
$\Sigma_{c}^{*}D$ | 1 | 3 | $\mathcal{E}-\delta_{b}$ | 1 | 3 | $\mathcal{E}$ | $\frac{1}{3}$ | 1 | $\mathcal{E}+\delta_{a}$ | - | - | - | $\frac{1}{3}$ | 1 | $\mathcal{E}+\delta_{d}$
$\Sigma_{c}D^{*}$ | $\frac{2}{3}$ | 2 | $\mathcal{E}$ | $\frac{1}{3}$ | 1 | $\mathcal{E}+\delta_{b}$ | 1 | 3 | $\mathcal{E}$ | $\frac{2}{3}$ | 2 | $\mathcal{E}-\delta_{a}$ | 1 | 3 | $\mathcal{E}+\delta_{c}$
$\Sigma_{c}^{*}D^{*}$ | $\frac{2}{3}$ | 2 | $\mathcal{E}$ | $\frac{1}{3}$ | 1 | $\mathcal{E}+\delta_{b}$ | $\frac{1}{3}$ | 1 | $\mathcal{E}+\delta_{a}$ | 1 | 3 | $\mathcal{E}$ | $\frac{1}{3}$ | 1 | $\mathcal{E}+\delta_{d}$
Table 5: The coefficients $\mathcal{C}_{m}^{B_{i.j}}$, $\mathcal{C}_{m}^{R_{i.j}}$ and $\mathcal{E}_{n}^{(B/R)_{i.j}}$ defined in Eqs. (38) and (39). The superscripts denote the corresponding diagrams illustrated in Figs. 2, 3, 4, and 5, respectively.  | $\mathcal{C}_{1}^{B_{i.1}}$ | $\mathcal{C}_{1}^{R_{i.1}}$ | $\mathcal{C}_{2}^{(B/R)_{i.1}}$ | $\mathcal{C}_{3}^{(B/R)_{i.1}}$ | $\mathcal{C}_{4}^{(B/R)_{i.1}}$ | $\mathcal{E}_{1}^{(B/R)_{i.1}}$ | $\mathcal{E}_{2}^{(B/R)_{i.1}}$
---|---|---|---|---|---|---|---
$\Sigma_{c}D$ | 1 | 1 | 1 | 10 | 15 | $\mathcal{E}$ | $\mathcal{E}-\delta_{b}$
$\Sigma_{c}^{*}D$ | 1 | 1 | 1 | 10 | 15 | $\mathcal{E}$ | $\mathcal{E}-\delta_{b}$
$\Sigma_{c}D^{*}$ | $\frac{2+A}{3}$ | $\frac{2-A}{3}$ | $\frac{2}{3}$ | $\frac{20}{3}$ | 10 | $\mathcal{E}$ | $\mathcal{E}$
$\Sigma^{*}_{c}D^{*}$ | $B^{2}$ | $\frac{3B^{2}-2B}{3}$ | $\frac{10}{9}$ | $\frac{20+12B^{2}-4B}{3}$ | $10+6B^{2}-2B$ | $\mathcal{E}$ | $\mathcal{E}$
| $\mathcal{C}_{1}^{B_{i.2}}$ | $\mathcal{C}_{1}^{R_{i.2}}$ | $\mathcal{C}_{2}^{(B/R)_{i.2}}$ | $\mathcal{C}_{3}^{(B/R)_{i.2}}$ | $\mathcal{C}_{4}^{(B/R)_{i.2}}$ | $\mathcal{E}_{1}^{(B/R)_{i.2}}$ | $\mathcal{E}_{1}^{(B/R)_{i.2}}$
$\Sigma_{c}D$ | $\frac{2}{3}$ | $\frac{2}{3}$ | $\frac{2}{3}$ | $\frac{20}{3}$ | 10 | $\mathcal{E}-\delta_{a}$ | $\mathcal{E}-\delta_{b}$
$\Sigma_{c}^{*}D$ | $\frac{1}{3}$ | $\frac{1}{3}$ | $\frac{1}{3}$ | $\frac{10}{3}$ | 5 | $\mathcal{E}+\delta_{a}$ | $\mathcal{E}-\delta_{b}$
$\Sigma_{c}D^{*}$ | $\frac{1+A}{3}$ | $\frac{1-A}{3}$ | $\frac{1}{3}$ | $\frac{10}{3}$ | 5 | $\mathcal{E}$ | $\mathcal{E}+\delta_{b}$
$\Sigma^{*}_{c}D^{*}$ | $\frac{5-3B^{2}+2B}{3}$ | $\frac{5-3B^{2}}{3}$ | $\frac{5}{9}$ | $\frac{15-6B^{2}+2B}{3}$ | $15-6B^{2}+2B$ | $\mathcal{E}$ | $\mathcal{E}+\delta_{b}$
| $\mathcal{C}_{1}^{B_{i.3}}$ | $\mathcal{C}_{1}^{R_{i.3}}$ | $\mathcal{C}_{2}^{(B/R)_{i.3}}$ | $\mathcal{C}_{3}^{(B/R)_{i.3}}$ | $\mathcal{C}_{4}^{(B/R)_{i.3}}$ | $\mathcal{E}_{1}^{(B/R)_{i.3}}$ | $\mathcal{E}_{1}^{(B/R)_{i.3}}$
$\Sigma_{c}D^{*}$ | $\frac{4-A}{9}$ | $\frac{4+A}{9}$ | $\frac{4}{9}$ | $\frac{40}{9}$ | $\frac{20}{3}$ | $\mathcal{E}-\delta_{a}$ | $\mathcal{E}$
$\Sigma^{*}_{c}D^{*}$ | $2-B^{2}+B$ | $2-B^{2}-\frac{B}{3}$ | $\frac{8}{9}$ | $\frac{40-12B^{2}+4B}{3}$ | $20-6B^{2}+2B$ | $\mathcal{E}+\delta_{a}$ | $\mathcal{E}$
| $\mathcal{C}_{1}^{B_{i.4}}$ | $\mathcal{C}_{1}^{R_{i.4}}$ | $\mathcal{C}_{2}^{(B/R)_{i.4}}$ | $\mathcal{C}_{3}^{(B/R)_{i.4}}$ | $\mathcal{C}_{4}^{(B/R)_{i.4}}$ | $\mathcal{E}_{1}^{(B/R)_{i.4}}$ | $\mathcal{E}_{1}^{(B/R)_{i.4}}$
$\Sigma_{c}D^{*}$ | $\frac{2-A}{3}$ | $\frac{2+A}{3}$ | $\frac{2}{3}$ | $\frac{20}{3}$ | 10 | $\mathcal{E}-\delta_{a}$ | $\mathcal{E}+\delta_{b}$
$\Sigma^{*}_{c}D^{*}$ | $\frac{3B^{2}+B-2}{3}$ | $\frac{3B^{2}-3B-2}{3}$ | $\frac{4}{9}$ | $\frac{12B^{2}-4B}{3}$ | $6B^{2}-2B$ | $\mathcal{E}+\delta_{a}$ | $\mathcal{E}+\delta_{b}$
Table 6: The coefficients $\mathcal{C}_{m}^{\bar{B}_{i.j}}$, $\mathcal{C}_{m}^{\bar{R}_{i.j}}$ and $\mathcal{E}_{n}^{(\bar{B}/\bar{R})_{i.j}}$ defined in Eqs. (41) and (42). The superscripts denote the corresponding diagrams illustrated in Figs. 2, 3, 4, and 5, respectively.  | $\mathcal{C}_{1}^{\bar{B}_{1.1}}$ | $\mathcal{C}_{1}^{\bar{R}_{1.1}}$ | $\mathcal{C}_{2}^{(\bar{B}/\bar{R})_{1.1}}$ | $\mathcal{C}_{3}^{(\bar{B}/\bar{R})_{1.1}}$ | $\mathcal{C}_{4}^{(\bar{B}/\bar{R})_{1.1}}$ | $\mathcal{E}_{1}^{(\bar{B}/\bar{R})_{1.1}}$ | $\mathcal{E}_{2}^{(\bar{B}/\bar{R})_{1.1}}$
---|---|---|---|---|---|---|---
$\Sigma_{c}D$ | 1 | 1 | 1 | 10 | 15 | $\mathcal{E}+\delta_{c}$ | $\mathcal{E}-\delta_{b}$
| $\mathcal{C}_{1}^{\bar{B}_{2.2}}$ | $\mathcal{C}_{1}^{\bar{R}_{2.2}}$ | $\mathcal{C}_{2}^{(\bar{B}/\bar{R})_{2.2}}$ | $\mathcal{C}_{3}^{(\bar{B}/\bar{R})_{2.2}}$ | $\mathcal{C}_{4}^{(\bar{B}/\bar{R})_{2.2}}$ | $\mathcal{E}_{1}^{(\bar{B}/\bar{R})_{2.2}}$ | $\mathcal{E}_{2}^{(\bar{B}/\bar{R})_{2.2}}$
$\Sigma_{c}^{*}D$ | $\frac{1}{3}$ | $\frac{1}{3}$ | $\frac{1}{3}$ | $\frac{10}{3}$ | 5 | $\mathcal{E}+\delta_{d}$ | $\mathcal{E}-\delta_{b}$
| $\mathcal{C}_{1}^{\bar{B}_{3.1}}$ | $\mathcal{C}_{1}^{\bar{R}_{3.1}}$ | $\mathcal{C}_{2}^{(\bar{B}/\bar{R})_{3.1}}$ | $\mathcal{C}_{3}^{(\bar{B}/\bar{R})_{3.1}}$ | $\mathcal{C}_{4}^{(\bar{B}/\bar{R})_{3.1}}$ | $\mathcal{E}_{1}^{(\bar{B}/\bar{R})_{3.1}}$ | $\mathcal{E}_{2}^{(\bar{B}/\bar{R})_{3.1}}$
$\Sigma_{c}D^{*}$ | $\frac{2+A}{3}$ | $\frac{2-A}{3}$ | $\frac{2}{3}$ | $\frac{20}{3}$ | 10 | $\mathcal{E}+\delta_{c}$ | $\mathcal{E}$
| $\mathcal{C}_{1}^{\bar{B}_{3.2}}$ | $\mathcal{C}_{1}^{\bar{R}_{3.2}}$ | $\mathcal{C}_{2}^{(\bar{B}/\bar{R})_{3.2}}$ | $\mathcal{C}_{3}^{(\bar{B}/\bar{R})_{3.2}}$ | $\mathcal{C}_{4}^{(\bar{B}/\bar{R})_{3.2}}$ | $\mathcal{E}_{1}^{(\bar{B}/\bar{R})_{3.2}}$ | $\mathcal{E}_{2}^{(\bar{B}/\bar{R})_{3.2}}$
$\Sigma_{c}D^{*}$ | $\frac{1+A}{3}$ | $\frac{1-A}{3}$ | $\frac{1}{3}$ | $\frac{10}{3}$ | 5 | $\mathcal{E}+\delta_{c}$ | $\mathcal{E}+\delta_{b}$
| $\mathcal{C}_{1}^{\bar{B}_{4.3}}$ | $\mathcal{C}_{1}^{\bar{R}_{4.3}}$ | $\mathcal{C}_{2}^{(\bar{B}/\bar{R})_{4.3}}$ | $\mathcal{C}_{3}^{(\bar{B}/\bar{R})_{4.3}}$ | $\mathcal{C}_{4}^{(\bar{B}/\bar{R})_{4.3}}$ | $\mathcal{E}_{1}^{(\bar{B}/\bar{R})_{4.3}}$ | $\mathcal{E}_{2}^{(\bar{B}/\bar{R})_{4.3}}$
$\Sigma^{*}_{c}D^{*}$ | $2-B^{2}+B$ | $2-B^{2}-\frac{B}{3}$ | $\frac{8}{9}$ | $\frac{40-12B^{2}+4B}{3}$ | $20-6B^{2}+2B$ | $\mathcal{E}+\delta_{d}$ | $\mathcal{E}$
| $\mathcal{C}_{1}^{\bar{B}_{4.4}}$ | $\mathcal{C}_{1}^{\bar{R}_{4.4}}$ | $\mathcal{C}_{2}^{(\bar{B}/\bar{R})_{4.4}}$ | $\mathcal{C}_{3}^{(\bar{B}/\bar{R})_{4.4}}$ | $\mathcal{C}_{4}^{(\bar{B}/\bar{R})_{4.4}}$ | $\mathcal{E}_{1}^{(\bar{B}/\bar{R})_{4.4}}$ | $\mathcal{E}_{2}^{(\bar{B}/\bar{R})_{4.4}}$
$\Sigma_{c}^{*}D^{*}$ | $\frac{3B^{2}+B-2}{3}$ | $\frac{3B^{2}-3B-2}{3}$ | $\frac{4}{9}$ | $\frac{3B^{2}-B}{3}$ | $6B^{2}-2B$ | $\mathcal{E}+\delta_{d}$ | $\mathcal{E}+\delta_{b}$

##  Appendix B The binding energies of the $\Sigma_{c}^{(*)}\bar{B}^{(*)}$,
$\Sigma_{b}^{(*)}D^{(*)}$, and $\Sigma_{b}^{(*)}\bar{B}^{(*)}$ systems

We also study the interactions of the $\Sigma_{c}^{(*)}\bar{B}^{(*)}$,
$\Sigma_{b}^{(*)}D^{(*)}$, and $\Sigma_{b}^{(*)}\bar{B}^{(*)}$ systems with
the three cases of LECs. The results for the possible $P_{cb}$
($\Sigma_{c}^{(*)}\bar{B}^{(*)}$), $P_{bc}$ ($\Sigma_{b}^{(*)}D^{(*)}$), and
$P_{bb}$ ($\Sigma_{b}^{(*)}\bar{B}^{(*)}$) pentaquarks are collected in Tables
7, 8, and 9, respectively.

In our calculation, we have already adopted the approach developed in Ref.
Wang:2019ato  to keep the effects from the mass splittings in the
($\Sigma_{b}^{*}$, $\Sigma_{b}$, $\Lambda_{b}$) baryons and ($\bar{B}^{*}$,
$B$) mesons.

We find binding solutions for all the
$[\Sigma_{c}^{(*)}\bar{B}^{(*)}]^{I=1/2}_{J}$,
$[\Sigma_{b}^{(*)}D^{(*)}]^{I=1/2}_{J}$, and
$[\Sigma_{b}^{(*)}\bar{B}^{(*)}]^{I=1/2}_{J}$ systems. The mass spectra of the
possible $P_{cb}$, $P_{bc}$, and $P_{bb}$ pentaquarks are similar to those of
the $\Sigma_{c}^{(*)}D^{(*)}$ systems.

We also notice that among the $[\Sigma_{c}^{(*)}D^{(*)}]^{I=1/2}_{J}$,
$[\Sigma_{c}^{(*)}\bar{B}^{(*)}]^{I=1/2}_{J}$,
$[\Sigma_{b}^{(*)}D^{(*)}]^{I=1/2}_{J}$, and
$[\Sigma_{b}^{(*)}\bar{B}^{(*)}]^{I=1/2}_{J}$ systems, the absolute values of
the binding energies generally have the following relation,

$\displaystyle|E_{P_{cc}}|<|E_{P_{cb}}|\approx|E_{P_{bc}}|<|E_{P_{bb}}|.$ (49)

Table 7: The binding energies and root-mean-equare radii for all the $[\Sigma_{c}^{(*)}\bar{B}^{(*)}]^{I=1/2}_{J}$ systems. The adopted LECs in Cases 1, 2, and 3 are ($c_{s}=-5.84$, $c_{t}=2.50$) GeV-2, ($c_{s}=-8.10$, $c_{t}=0.65$ ) GeV-2, and ($c_{s}=-7.46$, $c_{t}=1.02$) GeV-2, respectively. |  | $\left[\Sigma_{c}\bar{B}\right]_{\frac{1}{2}}$ | $\left[\Sigma^{*}_{c}\bar{B}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{c}\bar{B}^{*}\right]_{\frac{1}{2}}$ | $\left[\Sigma_{c}\bar{B}^{*}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{c}^{*}\bar{B}^{*}\right]_{\frac{1}{2}}$ | $\left[\Sigma_{c}^{*}\bar{B}^{*}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{c}^{*}\bar{B}^{*}\right]_{\frac{5}{2}}$
---|---|---|---|---|---|---|---|---
Case 1 | BE (MeV) | -24.3 | -24.8 | -39.8 | -13.0 | -47.6 | -32.7 | -11.0
$R_{rms}$ (fm) | 1.16 | 1.15 | 1.04 | 1.35 | 1.00 | 1.08 | 1.38
Case 2 | BE (MeV) | -43.5 | -44.0 | -38.0 | -40.9 | -40.3 | -41.5 | -44.1
$R_{rms}$ (fm) | 1.01 | 1.01 | 1.06 | 1.03 | 1.04 | 1.03 | 1.01
Case 3 | BE (MeV) | -37.9 | -44.9 | -36.8 | -33.3 | -40.1 | -38.1 | -35.3
$R_{rms}$ (fm) | 1.05 | 1.00 | 1.06 | 1.08 | 1.04 | 1.05 | 1.06
Table 8: The binding energies and root-mean-equare radii for all the $[\Sigma_{b}^{(*)}D^{(*)}]_{J}^{I=1/2}$ systems. The adopted LECs in Cases 1, 2, and 3 are ($c_{s}=-5.84$, $c_{t}=2.50$) GeV-2, ($c_{s}=-8.10$, $c_{t}=0.65$ ) GeV-2, and ($c_{s}=-7.46$, $c_{t}=1.02$) GeV-2, respectively. |  | $\left[\Sigma_{b}D\right]_{\frac{1}{2}}$ | $\left[\Sigma^{*}_{b}D\right]_{\frac{3}{2}}$ | $\left[\Sigma_{b}D^{*}\right]_{\frac{1}{2}}$ | $\left[\Sigma_{b}D^{*}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{b}^{*}D^{*}\right]_{\frac{1}{2}}$ | $\left[\Sigma_{b}^{*}D^{*}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{b}^{*}D^{*}\right]_{\frac{5}{2}}$
---|---|---|---|---|---|---|---|---
Case 1 | BE (MeV) | -23.2 | -21.2 | -37.3 | -11.5 | -43.0 | -28.0 | -7.8
$R_{rms}$ (fm) | 1.21 | 1.25 | 1.09 | 1.42 | 1.05 | 1.16 | 1.55
Case 2 | BE (MeV) | -41.7 | -39.3 | -35.6 | -38.4 | -36.0 | -36.5 | -38.6
$R_{rms}$ (fm) | 1.06 | 1.08 | 1.10 | 1.07 | 1.10 | 1.10 | 1.07
Case 3 | BE (MeV) | -36.3 | -34.0 | -34.4 | -31.0 | -35.8 | -33.2 | -30.2
$R_{rms}$ (fm) | 1.10 | 1.12 | 1.11 | 1.13 | 1.10 | 1.12 | 1.13
Table 9: The binding energies and root-mean-equare radii for all the $[\Sigma_{b}^{(*)}\bar{B}^{(*)}]_{J}^{I=1/2}$ systems. The adopted LECs in Cases 1, 2, and 3 are ($c_{s}=-5.84$, $c_{t}=2.50$) GeV-2, ($c_{s}=-8.10$, $c_{t}=0.65$ ) GeV-2, and ($c_{s}=-7.46$, $c_{t}=1.02$) GeV-2, respectively. |  | $\left[\Sigma_{b}\bar{B}\right]_{\frac{1}{2}}$ | $\left[\Sigma^{*}_{b}\bar{B}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{b}\bar{B}^{*}\right]_{\frac{1}{2}}$ | $\left[\Sigma_{b}\bar{B}^{*}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{b}^{*}\bar{B}^{*}\right]_{\frac{1}{2}}$ | $\left[\Sigma_{b}^{*}\bar{B}^{*}\right]_{\frac{3}{2}}$ | $\left[\Sigma_{b}^{*}\bar{B}^{*}\right]_{\frac{5}{2}}$
---|---|---|---|---|---|---|---|---
Case 1 | BE (MeV) | -33.2 | -30.5 | -53.0 | -20.0 | -60.0 | -41.8 | -15.1
$R_{rms}$ (fm) | 0.97 | 0.99 | 0.87 | 1.09 | 0.85 | 0.92 | 1.16
Case 2 | BE (MeV) | -54.8 | -53.5 | -51.0 | -52.2 | -51.5 | -51.7 | -52.5
$R_{rms}$ (fm) | 0.86 | 0.87 | 0.88 | 0.87 | 0.88 | 0.87 | 0.87
Case 3 | BE (MeV) | -48.5 | -46.7 | -49.5 | -43.6 | -51.3 | -47.9 | -42.7
$R_{rms}$ (fm) | 0.88 | 0.89 | 0.89 | 0.91 | 0.88 | 0.89 | 0.91

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the spectrum. It would be interesting to see if this connection between the
Loschmidt echo and spectral form factor can help reveal the true meaning and
significance of various DQPTs we encountered in this work.

###### Acknowledgements.

Research at the University of Maryland was supported by US-ARO Contract
No.W911NF1310172, NSF DMR-2037158 (V.G. and C.R.). The collaborative work
between the University of Colorado and the University of Maryland was
supported by the Simons Collaboration on Ultra-Quantum Matter, which is a
grant from the Simons Foundation (651440, V.G. and V.G.). A. Z. is partially
supported by Grant No. 2018058 from the United States-Israel Binational
Science Foundation (BSF)

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