# A comparison between the $P_{c}$ and $P_{cs}$ systems

Kan Chen1<EMAIL_ADDRESS>Zi-Yang Lin1<EMAIL_ADDRESS>Shi-Lin Zhu1
<EMAIL_ADDRESS>1School of Physics and Center of High Energy Physics, Peking
University, Beijing 100871, China

###### Abstract

We construct the effective potentials of the $P_{c}$ and $P_{cs}$ states based
on the SU(3)${}_{\text{f}}$ symmetry and heavy quark symmetry. Then we perform
the coupled-channel analysis of the lowest isospin $P_{c}$ and $P_{cs}$
systems. The coupled-channel effects play different roles in the $P_{c}$ and
$P_{cs}$ systems. In the $P_{c}$ systems, this effect gives minor corrections
to the masses of the $P_{c}$ states. In the $P_{cs}$ system, the
$\Lambda_{c}\bar{D}_{s}-\Xi_{c}\bar{D}$ coupling will shift the mass of the
$P_{cs}(4338)$ close to the $\Xi_{c}\bar{D}$ threshold. The
$\Lambda_{c}\bar{D}^{(*)}_{s}-\Xi_{c}\bar{D}^{(*)}$ coupling will also produce
extra $P_{cs}$ states. We discuss the correspondence between the $P_{c}$ and
$P_{cs}$ states. Our results prefer that the SU(3) partners of the observed
$P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ in the $P_{cs}$ system have
not been found yet.

##  I Introduction

Very recently, the LHCb Collaboration announced the observation of a
$P_{cs}(4338)$ signal from the $J/\Psi\Lambda$ mass spectrum in the
$B^{-}\rightarrow J/\Psi\Lambda\bar{p}$ process LHCb:2022jad . The mass and
width of this new pentaquark candidate were measured to be

$\displaystyle M_{P_{cs}}$ $\displaystyle=$ $\displaystyle 4338.2\pm 0.7\pm
0.4\quad\text{MeV},$ (1) $\displaystyle\Gamma_{P_{cs}}$ $\displaystyle=$
$\displaystyle 7.0\pm 1.2\pm 1.3\quad\text{MeV}.$ (2)

Meanwhile, the amplitude analysis prefers the $\frac{1}{2}^{-}$ spin-parity
quantum numbers. The central value of the mass of $P_{cs}(4338)$ is above the
$\Xi_{c}\bar{D}$ threshold. Thus, this state can not be directly assigned as
the $\Xi_{c}\bar{D}$ molecular state. However, the authors of Ref.
Meng:2022wgl  pointed out that the lineshape of this resonance could be
distorted from the conventional Breit-Wigner distribution if it lies very
close to and strongly couples to the threshold.

Besides the newly observed $P_{cs}(4338)$, the $P_{cs}(4459)$ was observed at
LHCb LHCb:2020jpq  as a candidate of a $\Xi_{c}\bar{D}^{*}$ molecular state,
which agrees well with the prediction from the chiral effective field theory
in Ref. Wang:2019nvm . The strange hidden-charm states were also discussed in
Refs. Peng:2020hql ; Chen:2020kco ; Liu:2020hcv ; Zhu:2021lhd ; Xiao:2021rgp ;
Wu:2010jy ; Santopinto:2016pkp ; Shen:2019evi ; Xiao:2019gjd  and reviewed
extensively in Refs. Chen:2016qju ; Guo:2017jvc ; Liu:2019zoy ; Meng:2022ozq ;
Chen:2022asf ; Dong:2021juy .

The mass of the $P_{cs}(4459)$ is about 19 MeV below the $\Xi_{c}\bar{D}$
threshold. In Ref. Burns:2022uha , the author argued that from heavy quark
symmetry, the $[\Xi_{c}\bar{D}]^{1/2^{-}}$, $[\Xi_{c}\bar{D}^{*}]^{1/2^{-}}$,
and $[\Xi_{c}\bar{D}^{*}]^{3/2^{-}}$ channels should share identical
potentials and have comparable binding energies. However, the heavy quark
symmetry is drastically violated in the charm system due to the rather small
charm quark mass. With the assignment of the $P_{cs}(4338)$ and $P_{cs}(4459)$
as the $[\Xi_{c}\bar{D}]^{1/2^{-}}$ and $[\Xi_{c}\bar{D}^{*}]^{1/2^{-}}$
($[\Xi_{c}\bar{D}^{*}]^{3/2^{-}}$) molecular states, the degeneracy of the
$[\Xi_{c}\bar{D}^{*}]^{1/2^{-}}$ and $[\Xi_{c}\bar{D}^{*}]^{3/2^{-}}$ channels
is removed by the couple-channel effects and recoil corrections.

Another novel phenomenon from the $M_{J/\Psi\Lambda}$ invariant spectrum
LHCb:2022jad  is that there seems to be a structure around $M=4254$ MeV. To
understand this signal, the LHCb checked the $m(J/\Psi\Lambda)$ distribution
close to the $\Lambda_{c}^{+}D_{s}^{-}$ threshold and found that this signal
is not statistically significant. Nevertheless, the authors in Ref.
Nakamura:2022jpd  investigated the $P_{cs}(4338)$ and $P_{cs}(4255)$ pole
positions from a unitary $\Xi_{c}\bar{D}-\Lambda_{c}\bar{D}_{s}$ coupled-
channel scattering amplitude. Besides, the $P_{cs}(4255)$ pole was also found
in a model with the coupling between the meson-baryon molecule and the compact
five-quark state Giachino:2022pws . The $P_{cs}(4255)$ state was also
suggested in an effective field theory framework Yan:2022wuz .

The analogy between the observed ($P_{c}(4312)$, $P_{c}(4440)$, $P_{c}(4457)$)
LHCb:2015yax ; LHCb:2019kea  and ($P_{cs}(4338)$, $P_{cs}$(4459)) states are
discussed in Ref. Karliner:2022erb ; Wang:2022mxy ; Wang:2022neq . However,
since the $\Sigma_{c}$ and $\Xi_{c}$ belong to different SU(3)${}_{\text{f}}$
multiplets, the relations between the discussed $P_{c}$ and $P_{cs}$ states
are not clear. Besides, the $P_{\Psi ss}^{N}$ pentaquark states as the
partners of the $P_{c}$ and $P_{cs}$ states are investigated in Ref.
Ortega:2022uyu .

If the $P_{cs}$ states and $P_{c}$ states can be related via
SU(3)${}_{\text{f}}$ symmetry, it is important to investigate the similarities
and differences between these two sets of molecular candidates. In Ref.
Chen:2021spf ; Chen:2021cfl , we discussed the symmetry properties of
different heavy flavor molecular systems via a quark level Lagrangian. We
proposed that the interactions of different heavy flavor molecules can be
related via a generalized flavor-spin symmetry Chen:2021spf . This framework
provides a suitable tool to discuss the similarities between the $P_{c}$ and
$P_{cs}$ states.

We also notice an important difference between the $P_{c}$ and $P_{cs}$
states. The minimal quark components of the $P_{c}$ and $P_{cs}$ states are
$c\bar{c}nnn$ and $c\bar{c}nns$ ($n=u,d$), respectively. For the
charmed/charmed-strange mesons and baryons, the SU(3)${}_{\text{f}}$ symmetry
breaking effects are reflected on their physical masses, and we need to
distinguish the $s$ quark from $u$, $d$ quarks when we study the $P_{cs}$
systems. Unlike the $P_{c}$ pentaquarks, the $P_{cs}$ states can couple to two
sets of channels, i.e., the $cns$-$\bar{c}n$ type and $cnn$-$\bar{c}s$ type
channels. In Table 1, we list the possible open-charm channels and their
thresholds for the $P_{c}$ and $P_{cs}$ systems.

Table 1: The thresholds of the meson-baryon channels associated with the $P_{c}$ and $P_{cs}$ systems, we adopt the isospin averaged masses for the ground charmed mesons and baryons ParticleDataGroup:2020ssz . All values are in units of MeV. $P_{c}$ | $P_{cs}$
---|---
$\Lambda_{c}\bar{D}$ | 4153.7 | $\Lambda_{c}\bar{D}_{s}$ | 4255.5 | $\Xi_{c}\bar{D}$ | 4336.7
$\Lambda_{c}\bar{D}^{*}$ | 4295.0 | $\Lambda_{c}\bar{D}_{s}^{*}$ | 4398.7 | $\Xi_{c}\bar{D}^{*}$ | 4478.0
$\Sigma_{c}\bar{D}$ | 4320.8 | $\Sigma_{c}\bar{D}_{s}$ | 4422.5 | $\Xi_{c}^{\prime}\bar{D}$ | 4446.0
$\Sigma_{c}^{*}\bar{D}$ | 4385.4 | $\Sigma_{c}^{*}\bar{D}_{s}$ | 4487.1 | $\Xi_{c}^{*}\bar{D}$ | 4513.2
$\Sigma_{c}\bar{D}^{*}$ | 4462.1 | $\Sigma_{c}\bar{D}_{s}^{*}$ | 4565.7 | $\Xi_{c}^{\prime}\bar{D}^{*}$ | 4587.4
$\Sigma_{c}^{*}\bar{D}^{*}$ | 4526.7 | $\Sigma_{c}^{*}\bar{D}_{s}^{*}$ | 4630.3 | $\Xi_{c}^{*}\bar{D}^{*}$ | 4654.5

In this work, we will take the $P_{cs}(4338)$ as a molecular candidate and
discuss the following three issues:

  * (1)

Can we understand the minor binding energy of the $P_{cs}(4338)$ (close to the
$\Xi_{c}\bar{D}$ threshold) through a
$\Xi_{c}\bar{D}$-$\Lambda_{c}\bar{D}_{s}$ coupled-channel effect?

  * (2)

Can we produce a $P_{cs}(4254)$ bound state by including the
$\Xi_{c}\bar{D}$-$\Lambda_{c}\bar{D}_{s}$ coupled-channel effect with the
potential constrained from SU(3)${}_{\text{f}}$ symmetry?

  * (3)

What is the correspondence between the $P_{c}$ and $P_{cs}$ states if the
interactions of the $P_{c}$ and $P_{cs}$ states obey a generalized flavor-spin
symmetry?

This paper is organized as follows. We present our theoretical framework in
Sec. II and the corresponding numerical results and discussions in Sec. III.
Sec. IV is the summary.

##  II Framework

In Ref. Chen:2021cfl , we proposed an isospin criterion and pointed out that
the $P_{c}$ and $P_{cs}$ states with the lowest isospin numbers are more
likely to form bound states. Based on the same Lagrangian, we only focus on
the $P_{c}$ and $P_{cs}$ states with isospin numbers $I=1/2$ and 0,
respectively. Thus, we will not include the
$\Sigma^{(*)}_{c}\bar{D}^{(*)}_{s}$ channels listed in Table 1 for the
$P_{cs}$ system.

For the $I=1/2$ $P_{c}$ states, we consider the following channels for the
$J=1/2$ and $3/2$ states

$\displaystyle J=\frac{1}{2}$ $\displaystyle:$
$\displaystyle\Lambda_{c}\bar{D},\Lambda_{c}\bar{D}^{*},\Sigma_{c}\bar{D},\Sigma_{c}\bar{D}^{*},\Sigma_{c}^{*}\bar{D}^{*},$
(3) $\displaystyle J=\frac{3}{2}$ $\displaystyle:$
$\displaystyle\Lambda_{c}\bar{D}^{*},\Sigma_{c}^{*}\bar{D},\Sigma_{c}\bar{D}^{*},\Sigma_{c}^{*}\bar{D}^{*}.$
(4)

Similarly, for the $I=0$ $P_{cs}$ states, we include the following channels
for the $J=1/2$ and $3/2$ states

$\displaystyle J=\frac{1}{2}$ $\displaystyle:$
$\displaystyle\Lambda_{c}\bar{D}_{s},\Lambda_{c}\bar{D}^{*}_{s},\Xi_{c}\bar{D},\Xi_{c}\bar{D}^{*},\Xi_{c}^{\prime}\bar{D},\Xi_{c}^{\prime}\bar{D}^{*},\Xi_{c}^{*}\bar{D}^{*},$
(5) $\displaystyle J=\frac{3}{2}$ $\displaystyle:$
$\displaystyle\Lambda_{c}\bar{D}_{s}^{*},\Xi_{c}\bar{D}^{*},\Xi_{c}^{*}\bar{D},\Xi_{c}^{\prime}\bar{D}^{*},\Xi_{c}^{*}\bar{D}^{*}.$
(6)

The result of the $P_{c}$ ($P_{cs}$) state with $J=5/2$ can be obtained from a
single-channel calculation and was predicted in Ref. Chen:2021cfl  in the same
framework. Thus, we will not discuss them further in this work.

###  II.1 Lagrangians for the baryon-meson systems

To describe the $S$-wave interactions between the ground charmed/charmed-
strange baryons and mesons, we introduce the following quark-level Lagrangian
Meng:2019nzy ; Wang:2019nvm ; Wang:2020dhf ; Chen:2021cfl

$\displaystyle\mathcal{L}=g_{s}\bar{q}\mathcal{S}q+g_{a}\bar{q}\gamma_{\mu}\gamma^{5}\mathcal{A}^{\mu}q.$
(7)

Here, $q=(u,d,s)$, $g_{s}$ and $g_{a}$ are two independent coupling constants
that describe the interactions from the exchanges of the scalar and axial-
vector meson currents. They encode the nonperturbative low energy dynamics of
the considered heavy flavor meson-baryon systems.

From this Lagrangian, the effective potential of the light quark-quark
interactions reads

$\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle\tilde{g}_{s}\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}+\tilde{g}_{a}\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}.$
(8)

Here,

$\displaystyle\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}=\lambda_{1}^{8}\lambda_{2}^{8}+\lambda_{1}^{i}\lambda_{2}^{i}+\lambda_{1}^{j}\lambda_{2}^{j},$
(9)

$i$ and $j$ sum from 1 to 3 and 4 to 7, respectively. The operators
$\lambda_{1}^{8}\lambda_{2}^{8}$
($\lambda_{1}^{8}\lambda_{2}^{8}(\bm{\sigma}_{1}\cdot\bm{\sigma}_{2})$),
$\lambda_{1}^{i}\lambda_{2}^{i}$
($\lambda_{1}^{i}\lambda_{2}^{i}(\bm{\sigma}_{1}\cdot\bm{\sigma}_{2})$), and
$\lambda_{1}^{j}\lambda_{2}^{j}$
($\lambda_{1}^{j}\lambda_{2}^{j}(\bm{\sigma}_{1}\cdot\bm{\sigma}_{2})$) arise
from the exchanges of the isospin singlet, triplet, and two doublets light
scalar (axial-vector) meson currents, respectively. The redefined coupling
constants are $\tilde{g}_{s}\equiv g_{s}^{2}/m^{2}_{\mathcal{S}}$ and
$\tilde{g}_{a}\equiv g_{a}^{2}/m^{2}_{\mathcal{A}}$.

The Lagrangian in Eq. (7) allows the exchanges of two types of scalar and
axial-vector mesons that have quantum numbers $I(J^{P})=0(0^{+})$, $1(0^{+})$,
$1/2(0^{+})$ and $I(J^{P})=0(1^{+})$, $1(1^{+})$, $1/2(1^{+})$, respectively.
At present, we can not specifically pin down the coupling parameter of each
exchanged meson in the above six meson currents. Alternatively, since the
mesons in each meson current have identical interacting Lorentz structure, we
use the coupling constant $\tilde{g}_{s}$($\tilde{g}_{a}$) to collectively
absorb the total dynamical effects from the exchange of each scalar (axial-
vector) meson current. In addition, the couplings $\tilde{g}_{s}$
($\tilde{g}_{a}$) for the scalar (axial-vector) meson currents with different
isospin numbers are the same in the SU(3) limit.

The effective potential between the $i$-th baryon-meson channel $B_{i}M_{i}$
and the $j$-th baryon-meson channel $B_{j}M_{j}$ with total isospin $I$ and
total angular momentum $J$ can be calculated as

$\displaystyle v_{ij}$ $\displaystyle=$
$\displaystyle\left\langle[B_{i}M_{i}]_{J}^{I}|\mathcal{V}|[B_{j}M_{j}]_{J}^{I}\right\rangle.$
(10)

Here, the $|[B_{i}M_{i}]_{J}^{I}\rangle$ is the quark-level flavor-spin wave
function of the considered $i$-th channel baryon-meson system

$\displaystyle|[B_{i}M_{i}]_{J}^{I}\rangle$ $\displaystyle=$
$\displaystyle\sum_{m_{I_{1}},m_{I_{2}}}C_{I_{1},m_{I_{1}};I_{2},m_{I_{2}}}^{I,I_{z}}\phi_{I_{1},m_{I_{1}}}^{B_{if}}\phi_{I_{2},m_{I_{2}}}^{M_{if}}$
$\displaystyle\otimes\sum_{m_{S_{1}},m_{S_{2}}}C^{J,J_{z}}_{S_{1},m_{S_{1}};S_{2},m_{S_{2}}}\phi_{S_{1},m_{S_{1}}}^{B_{is}}\phi_{S_{2},m_{S_{2}}}^{M_{is}}.$

In Eq. (LABEL:BMWF), the $\phi_{S_{1},m_{S_{1}}}^{B_{is}}$ and
$\phi_{S_{2},m_{S_{2}}}^{M_{is}}$ are the spin wave functions of the baryon
and meson, respectively. The total spin wave function can be obtained with the
help of SU(2) CG coefficient $C_{S_{1},m_{S_{1}};S_{2},m_{S_{2}}}^{J,J_{z}}$.
For the flavor wave functions of the considered baryons
($\phi_{I_{1},m_{I_{1}}}^{B_{if}}$) and mesons
($\phi_{I_{1},m_{I_{1}}}^{M_{if}}$), their explicit forms have been given in
Ref. Chen:2021spf . When constructing the total flavor wave functions of the
considered baryon-meson systems, we use the SU(2) CG coefficient and take the
$s$ quark as a flavor singlet.

The coupled-channel Lippmann-Schwinger equation (LSE) reads

$\displaystyle\mathbb{T}\left(E\right)$ $\displaystyle=$
$\displaystyle\mathbb{V}+\mathbb{V}\mathbb{G}\left(E\right)\mathbb{T}\left(E\right),$
(12)

with

$\displaystyle\mathbb{V}=\left(\begin{array}[]{ccccc}v_{11}&\cdots&v_{1i}&\cdots&v_{1n}\\\
\vdots&&\vdots&&\vdots\\\ v_{j1}&\cdots&v_{ji}&\cdots&v_{jn}\\\
\vdots&&\vdots&&\vdots\\\ v_{n1}&\cdots&v_{ni}&\cdots&v_{nn}\\\
\end{array}\right),$ (18)
$\displaystyle\mathbb{T}(E)=\left(\begin{array}[]{ccccc}t_{11}(E)&\cdots&t_{1i}(E)&\cdots&t_{1n}(E)\\\
\vdots&&\vdots&&\vdots\\\ t_{j1}(E)&\cdots&t_{ji}(E)&\cdots&t_{jn}(E)\\\
\vdots&&\vdots&&\vdots\\\ t_{n1}(E)&\cdots&t_{ni}(E)&\cdots&t_{nn}(E)\\\
\end{array}\right),$ (24)

and

$\displaystyle\mathbb{G}(E)=\text{diag}\left\\{G_{1}(E),\cdots,G_{i}(E),\cdots,G_{n}(E)\right\\}.$
(25)

Here,

$\displaystyle G_{i}=\frac{1}{2\pi^{2}}\int
dq\frac{q^{2}}{E-\sqrt{m_{i1}^{2}+q^{2}}-\sqrt{m_{i2}+q^{2}}}u^{2}(\Lambda).$

The $m_{i1}$ and $m_{i2}$ are the masses of the baryon and meson in the $i-$th
channel, respectively. In our previous work Chen:2021spf ; Chen:2021cfl , we
use a step function to exclude the contributions from higher momenta to
perform the single-channel calculation. In the coupled-channel case, we need
to further suppress the contributions from the channels that are far away from
the thresholds of the considered channels. Thus, we introduce a dipole form
factor $u(\Lambda)=(1+q^{2}/\Lambda^{2})^{-2}$ with regular parameter
$\Lambda=1.0$ GeV Nakamura:2022jpd ; Leinweber:2003dg ; Wang:2007iw .

The pole position of Eq. (12) satisfies $||\bm{1}-\mathbb{V}\mathbb{G}||=0$.
For the bound state below the lowest channel, we search the bound state
solution in the first Riemann sheet of the lowest channel. For the quasi-bound
state between the thresholds of the $i$-th and $j$-th channels, we adopt the
complex scaling method and replace the integration variable $q$ by
$q\rightarrow q\times\text{exp}(-i\theta)$ and maintain $0<\theta<\pi/2$ to
find the quasi-bound state solution in the first Riemann sheet of the higher
$j$-th channel and the second Riemann sheet of the lower $i$-th channel.
Liu:2016wxq .

##  III Numerical results

###  III.1 Determination of $\tilde{g}_{s}$ and $\tilde{g}_{a}$

We first determine the parameters $\tilde{g}_{s}$ and $\tilde{g}_{a}$ in our
model. We collect the matrix elements of
$\langle\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}\rangle$,
$\langle\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}\rangle$
for the $P_{c}$ and $P_{cs}$ states in Tables 2 and 3, respectively.

Table 2: The matrix elements of $[\langle\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}\rangle$, $\langle\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}\rangle]$ for the meson-baryon channels associated with the $J^{P}=1/2^{-}$ and $3/2^{-}$ $P_{c}$ systems. $\mathbb{V}_{1/2}^{P_{c}}$ | $\mathbb{V}_{3/2}^{P_{c}}$
---|---
Channel | $\Lambda_{c}\bar{D}$ | $\Lambda_{c}\bar{D}^{*}$ | $\Sigma_{c}\bar{D}$ | $\Sigma_{c}\bar{D}^{*}$ | $\Sigma_{c}^{*}\bar{D}^{*}$ | Channel | $\Lambda_{c}\bar{D}^{*}$ | $\Sigma_{c}^{*}\bar{D}$ | $\Sigma_{c}\bar{D}^{*}$ | $\Sigma_{c}^{*}\bar{D}^{*}$
$\Lambda_{c}\bar{D}$ | [$\frac{2}{3}$,0] | [0,0] | [0,0] | [0,$2\sqrt{3}$] | [0,$2\sqrt{6}$] | $\Lambda_{c}\bar{D}^{*}$ | [$\frac{2}{3}$,0] | [0,$-2\sqrt{3}$] | [0,2] | [0,$2\sqrt{5}$]
$\Lambda_{c}\bar{D}^{*}$ |  | [$\frac{2}{3}$,0] | [0,$2\sqrt{3}$] | [0,$-4$] | [0,$2\sqrt{2}$] | $\Sigma_{c}^{*}\bar{D}$ |  | [$-\frac{10}{3}$,0] | [0,$-\frac{10}{3\sqrt{3}}$] | [0,$-\frac{10\sqrt{\frac{5}{3}}}{3}$]
$\Sigma_{c}\bar{D}$ |  |  | [$-\frac{10}{3}$,0] | [0,$-\frac{20}{3\sqrt{3}}$] | [0,$\frac{10\sqrt{\frac{2}{3}}}{3}$] | $\Sigma_{c}\bar{D}^{*}$ |  |  | [$-\frac{10}{3}$,$-\frac{20}{9}$] | [0,$\frac{10\sqrt{5}}{9}$]
$\Sigma_{c}\bar{D}^{*}$ |  |  |  | [$-\frac{10}{3}$,$\frac{40}{9}$] | [0,$\frac{10\sqrt{2}}{9}$] | $\Sigma_{c}^{*}\bar{D}^{*}$ |  |  |  | [$-\frac{10}{3}$,$\frac{20}{9}$]
$\Sigma_{c}^{*}\bar{D}^{*}$ |  |  |  |  | [$-\frac{10}{3}$,$\frac{50}{9}$] |  |  |  |  |
Table 3: The matrix elements of [$\langle\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}\rangle$, $\langle\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}\rangle$] for the meson-baryon channels associated with the $J^{P}=1/2^{-}$ and $3/2^{-}$ $P_{cs}$ systems. $\mathbb{V}_{1/2}^{P_{cs}}$ | $\mathbb{V}_{3/2}^{P_{cs}}$
---|---
Channel | $\Lambda_{c}\bar{D}_{s}$ | $\Lambda_{c}\bar{D}_{s}^{*}$ | $\Xi_{c}\bar{D}$ | $\Xi_{c}\bar{D}^{*}$ | $\Xi_{c}^{\prime}\bar{D}$ | $\Xi_{c}^{\prime}\bar{D}^{*}$ | $\Xi_{c}^{*}\bar{D}^{*}$ | Channel | $\Lambda_{c}\bar{D}_{s}^{*}$ | $\Xi_{c}\bar{D}^{*}$ | $\Xi_{c}^{*}\bar{D}$ | $\Xi_{c}^{\prime}\bar{D}^{*}$ | $\Xi_{c}^{*}\bar{D}^{*}$
$\Lambda_{c}\bar{D}_{s}$ | $[-\frac{4}{3},0]$ | [0,0] | $[2\sqrt{2},0]$ | $[0,0]$ | $[0,0]$ | $[0,2\sqrt{2}]$ | $[0,4]$ | $\Lambda_{c}\bar{D}_{s}^{*}$ | $[-\frac{4}{3},0]$ | $[2\sqrt{2},0]$ | $[0,-2\sqrt{2}]$ | $[0,2\sqrt{\frac{2}{3}}]$ | $[0,2\sqrt{\frac{10}{3}}]$
$\Lambda_{c}\bar{D}_{s}^{*}$ |  | $[-\frac{4}{3},0]$ | $[0,0]$ | $[2\sqrt{2},0]$ | $[0,2\sqrt{2}]$ | $[0,-4\sqrt{\frac{2}{3}}]$ | $[0,\frac{4}{\sqrt{3}}]$ | $\Xi_{c}\bar{D}^{*}$ |  | $[-\frac{10}{3},0]$ | $[0,-2]$ | $[0,\frac{2}{\sqrt{3}}]$ | $[0,2\sqrt{\frac{5}{3}}]$
$\Xi_{c}\bar{D}$ |  |  | $[-\frac{10}{3},0]$ | $[0,0]$ | $[0,0]$ | $[0,2]$ | $[0,2\sqrt{2}]$ | $\Xi_{c}^{*}\bar{D}$ |  |  | $[-\frac{10}{3},0]$ | $[0,-\frac{10}{3\sqrt{3}}]$ | $[0,-\frac{10\sqrt{\frac{5}{3}}}{3}]$
$\Xi_{c}\bar{D}^{*}$ |  |  |  | $[-\frac{10}{3},0]$ | $[0,2]$ | $[0,-\frac{4}{\sqrt{3}}]$ | $[0,2\sqrt{\frac{2}{3}}]$ | $\Xi_{c}^{\prime}\bar{D}^{*}$ |  |  |  | $[-\frac{10}{3},-\frac{20}{9}]$ | $[0,\frac{10\sqrt{5}}{9}]$
$\Xi_{c}^{\prime}\bar{D}$ |  |  |  |  | $[-\frac{10}{3},0]$ | $[0,-\frac{20}{3\sqrt{3}}]$ | $[0,\frac{10\sqrt{\frac{2}{3}}}{3}]$ | $\Xi_{c}^{*}\bar{D}^{*}$ |  |  |  |  | $[-\frac{10}{3},\frac{20}{9}]$
$\Xi_{c}^{\prime}\bar{D}^{*}$ |  |  |  |  |  | $[-\frac{10}{3},\frac{40}{9}]$ | $[0,\frac{10\sqrt{2}}{9}]$ |  |  |  |  |  |
$\Xi_{c}^{*}\bar{D}^{*}$ |  |  |  |  |  |  | $[-\frac{10}{3},\frac{50}{9}]$ |  |  |  |  |  |

We can directly obtain the effective potentials associated with the $P_{c}$
and $P_{cs}$ states from Tables 2 and 3, respectively. For example, the
explicit form of the effective potential matrix for the $J=3/2$ $P_{c}$ states
is

$\displaystyle\mathbb{V}_{3/2}^{P_{c}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}\frac{2}{3}\tilde{g}_{s}&-2\sqrt{3}\tilde{g}_{a}&2\tilde{g}_{a}&2\sqrt{5}\tilde{g}_{a}\\\
-2\sqrt{3}\tilde{g}_{a}&-\frac{10}{3}\tilde{g}_{s}&-\frac{10}{3\sqrt{3}}\tilde{g}_{a}&-\frac{10\sqrt{\frac{5}{3}}}{3}\tilde{g}_{a}\\\
2\tilde{g}_{a}&-\frac{10}{3\sqrt{3}}\tilde{g}_{a}&-\frac{10}{3}\tilde{g}_{s}-\frac{20}{9}\tilde{g}_{a}&\frac{10\sqrt{5}}{9}\tilde{g}_{a}\\\
2\sqrt{5}\tilde{g}_{a}&-\frac{10\sqrt{\frac{5}{3}}}{3}\tilde{g}_{a}&\frac{10\sqrt{5}}{9}\tilde{g}_{a}&-\frac{10}{3}\tilde{g}_{s}+\frac{20}{9}\tilde{g}_{a}\\\
\end{array}\right).$ (31)

Similarly, the effective potential matrixes $\mathbb{V}_{1/2}^{P_{c}}$,
$\mathbb{V}_{1/2}^{P_{cs}}$, and $\mathbb{V}_{3/2}^{P_{cs}}$ can also be
obtained directly from Table 2 and 3.

We use the masses of the observed $P_{c}$ states as input to determine the
coupling constants $\tilde{g}_{s}$ and $\tilde{g}_{a}$. In our previous work,
we find that the Lagrangian in Eq. (7) can give a satisfactory description of
the observed $T_{cc}$ LHCb:2021auc ; LHCb:2021vvq , $P_{c}$, and $P_{cs}$
states if we assign the $P_{c}(4440)$ and $P_{c}(4457)$ as the
$I(J^{P})=1/2(1/2^{-})$ and $1/2(3/2^{-})$ states. For consistency, we still
adopt this set of assignments and use the masses of the $P_{c}(4440)$ and
$P_{c}(4457)$ as inputs. In the coupled-channel formalism, the bound/quasi-
bound states in the $J^{P}=1/2^{-}$ and $3/2^{-}$ $P_{c}$ systems satisfy the
following equations

$\displaystyle\text{Re}\left|\left|\bm{1}-\mathbb{V}_{1/2}^{P_{c}}\mathbb{G}_{1/2}^{P_{c}}\right|\right|=0,$
(33)
$\displaystyle\text{Im}\left|\left|\bm{1}-\mathbb{V}_{1/2}^{P_{c}}\mathbb{G}_{1/2}^{P_{c}}\right|\right|=0,$
(34)
$\displaystyle\text{Re}\left|\left|\bm{1}-\mathbb{V}_{3/2}^{P_{c}}\mathbb{G}_{3/2}^{P_{c}}\right|\right|=0,$
(35)
$\displaystyle\text{Im}\left|\left|\bm{1}-\mathbb{V}_{3/2}^{P_{c}}\mathbb{G}_{3/2}^{P_{c}}\right|\right|=0.$
(36)

These four equations can be solved numerically and we get

$\displaystyle\tilde{g}_{s}=8.28\quad\text{GeV}^{-2},\quad\tilde{g}_{a}=-1.46\quad\text{GeV}^{-2}.$
(37)

The imaginary part of the pole positions of the $P_{c}(4440)$ and
$P_{c}(4457)$ can also be obtained from Eqs. (33-36).

###  III.2 Flavor-spin symmetry of the $P_{c}$ and $P_{cs}$ systems in the
single-channel formalism

With the determined parameters $\tilde{g}_{s}$ and $\tilde{g}_{a}$, we first
present our single-channel results for the considered $P_{c}$ and $P_{cs}$
systems and demonstrate that we can relate the $P_{c}$ and $P_{cs}$ systems
from their interactions constrained by the SU(3) and heavy quark symmetries.

Although the $\Xi_{c}\bar{D}$ and $\Sigma_{c}\bar{D}$ belong to different
multiplets, in Ref. Chen:2021spf  we proposed that there exists a generalized
flavor-spin symmetry between two-body heavy-flavor systems. For two different
heavy-flavor meson-baryon systems, if they both possess the same flavor
($\langle
H^{f}_{1}H^{f}_{2}|\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}|H^{f}_{1}H^{f}_{2}\rangle$)
and spin ($\langle
H^{s}_{1}H^{s}_{2}|\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}|H^{s}_{1}H^{s}_{2}\rangle$)
matrix elements, they will still have identical effective potentials in the
SU(3) and heavy quark limits.

In the single-channel formalism, we present the masses and binding energies of
the $P_{c}$ and $P_{cs}$ states in Table 4. The theoretical uncertainties are
introduced by considering the experimental errors of the masses of the
$P_{c}(4440)$ and $P_{c}(4457)$. We collect the $P_{c}$ and $P_{cs}$ states
that share identical effective potentials in the same row. As listed in Table
4, the $P_{c}$ and $P_{cs}$ states have similar binding energies in the same
row and can be related via a flavor-spin symmetry.

Table 4: In the single-channel formalism, the binding energies of the $P_{c}$ and $P_{cs}$ states that share the same effective potentials in the SU(3) and heavy quark limits. All results are in units of MeV. $P_{c}$ | Mass | BE | $P_{cs}$ | Mass | BE | $V$ |
---|---|---|---|---|---|---|---
$[\Sigma_{c}\bar{D}]^{\frac{1}{2}}$ | $4312.7^{+4.1}_{-2.6}$ | $-8.1^{+4.1}_{-2.6}$ | $[\Xi_{c}\bar{D}]^{\frac{1}{2}}$ | $4328.5^{+4.1}_{-2.7}$ | $-8.2^{+4.1}_{-2.7}$ | $-\frac{10}{3}\tilde{g}_{s}$ |
$[\Xi_{c}\bar{D}^{*}]^{\frac{1}{2},\frac{3}{2}}$ | $4468.3^{+4.5}_{-2.9}$ | $-9.7^{+4.5}_{-2.9}$ |
$[\Sigma_{c}^{*}\bar{D}]^{\frac{3}{2}}$ | $4376.9^{+4.2}_{-2.7}$ | $-8.5^{+4.2}_{-2.7}$ | $[\Xi_{c}^{\prime}\bar{D}]^{\frac{1}{2}}$ | $4437.2^{+4.5}_{-2.8}$ | $-8.8^{+4.3}_{-2.8}$ |
$[\Xi_{c}^{*}\bar{D}]^{\frac{3}{2}}$ | $4503.9^{+4.4}_{-2.8}$ | $-9.3^{+4.4}_{-2.8}$ |
$[\Sigma_{c}\bar{D}^{*}]^{\frac{1}{2}}$ | $4438.9^{+4.9}_{-8.9}$ | $-23.2^{+4.9}_{-8.9}$ | $[\Xi_{c}^{\prime}\bar{D}^{*}]^{\frac{1}{2}}$ | $4562.9^{+2.8}_{-9.1}$ | $-24.5^{+2.8}_{-9.1}$ | $-\frac{10}{3}\tilde{g}_{s}+\frac{40}{9}\tilde{g}_{a}$ |
$[\Sigma_{c}\bar{D}^{*}]^{\frac{3}{2}}$ | $4457.5^{+3.7}_{-1.8}$ | $-4.6^{+3.7}_{-1.8}$ | $[\Xi_{c}^{\prime}\bar{D}^{*}]^{\frac{3}{2}}$ | $4582.2^{+4.0}_{-2.0}$ | $-5.2^{+4.0}_{-2.0}$ | $-\frac{10}{3}\tilde{g}_{s}-\frac{20}{9}\tilde{g}_{a}$ |
$[\Sigma_{c}^{*}\bar{D}^{*}]^{\frac{1}{2}}$ | $4498.8^{+6.6}_{-6.0}$ | $-27.9^{+6.6}_{-6.0}$ | $[\Xi_{c}^{*}\bar{D}^{*}]^{\frac{1}{2}}$ | $4625.3^{+6.8}_{-12.7}$ | $-29.2^{+6.8}_{-12.7}$ | $-\frac{10}{3}\tilde{g}_{s}+\frac{50}{9}\tilde{g}_{a}$ |
$[\Sigma_{c}^{*}\bar{D}^{*}]^{\frac{3}{2}}$ | $4510.3^{+4.1}_{-4.1}$ | $-16.4^{+4.1}_{-4.1}$ | $[\Xi_{c}^{*}\bar{D}^{*}]^{\frac{3}{2}}$ | $4637.9^{+4.3}_{-4.2}$ | $-16.6^{+4.3}_{-4.2}$ | $-\frac{10}{3}\tilde{g}_{s}+\frac{20}{9}\tilde{g}_{a}$ |

###  III.3 The masses of $P_{c}$ states in the multi-channel formalism

Then we explore how the coupled-channel effect influences the masses of the
$P_{c}$ states. As can be seen from Eq. (8), the effective potential consists
of two parts, i.e., the central term
($\tilde{g}_{s}\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}$) and the spin-spin
interaction
($\tilde{g}_{a}\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}$)
term. Since the determined $\tilde{g}_{s}$ is much larger than
$\tilde{g}_{a}$, the central term dominates the total effective potential, and
therefore determines whether the considered system can form a bound state.

As given in Table 2, for the matrix elements in the $P_{c}$ system, all the
diagonal matrix elements have central terms, and some of them have corrections
from the spin-spin interaction terms. The off-diagonal terms only consist of
the spin-spin interaction terms. Thus, before we perform a practical multi-
channel calculation of the $P_{c}$ system, we may anticipate that the coupled-
channel effect would have small corrections to the masses of the $P_{c}$
states.

As discussed in Sec. II, we include five and four channels to study the
$J=1/2$ and $J=3/2$ $P_{c}$ states, respectively. For the $J=1/2$ channels,
according to their thresholds, we consider five energy regions

$\displaystyle E$ $\displaystyle\leq$ $\displaystyle m_{\Lambda_{c}\bar{D}},$
(38) $\displaystyle m_{\Lambda_{c}\bar{D}}<$ $\displaystyle E$
$\displaystyle\leq m_{\Lambda_{c}\bar{D}^{*}},$ (39) $\displaystyle
m_{\Lambda_{c}\bar{D}^{*}}<$ $\displaystyle E$ $\displaystyle\leq
m_{\Sigma_{c}\bar{D}},$ (40) $\displaystyle m_{\Sigma_{c}\bar{D}}<$
$\displaystyle E$ $\displaystyle\leq m_{\Sigma_{c}\bar{D}^{*}},$ (41)
$\displaystyle m_{\Sigma_{c}\bar{D}^{*}}<$ $\displaystyle E$
$\displaystyle\leq m_{\Sigma_{c}^{*}\bar{D}^{*}}.$ (42)

We search the bound (quasi-bound) state solutions below the higher threshold
in each energy region on the first Riemann sheet. The bound (quasi-bound)
state solutions of the $J=3/2$ $P_{c}$, $J=1/2$ and $3/2$ $P_{cs}$ states can
be found by repeating the same procedure. We present the obtained $P_{c}$
states in Table 5. We do not find any bound states below the
$\Lambda_{c}\bar{D}$ threshold. Thus, all the obtained resonances ($E_{R}$)
listed in Table 5 should refer to quasi-bound states and have imaginary parts
($\text{Im}(E_{R})$). Since we only include the two body open-charm decay
channels, the estimated widths ($\Gamma$) in Table 5 are smaller than
experimental widths. By comparing the masses of $P_{c}$ states in Table 4 and
5, we find that the coupled-channel effect indeed have small influences to the
masses of the $P_{c}$ states.

Table 5: The results of $P_{c}$ states obtained in the coupled-channel formalism. Here, $\Gamma=-2\text{Im}(E_{R})$, all the results are in units of MeV. |  | Our | Exp
---|---|---|---
State | $J^{P}$ | Mass | $\Gamma$ | Mass | Width
$P_{c}(4312)$ | $\frac{1}{2}^{-}$ | $4308.2^{+2.6}_{-4.5}$ | $2.6^{+2.4}_{-1.7}$ | $4311.9_{-0.9}^{+7.0}$ | $10\pm 5$
$P_{c}(4440)$ | $\frac{1}{2}^{-}$ |  $4440.3^{+4.0}_{-5.0}$ (input) | $9.8^{+4.6}_{-5.8}$ | $4440.3^{+4.0}_{-5.0}$ | $21^{+10}_{-11}$
$P_{c}(4457)$ | $\frac{3}{2}^{-}$ |  $4457.7^{+4.0}_{-1.8}$ (input) | $2.0^{+1.4}_{-0.8}$ | $4457.3^{+4.0}_{-1.8}$ | $6.4^{+6.0}_{-2.8}$
$P_{c}(4380)$ | $\frac{3}{2}^{-}$ | $4373.3^{+3.4}_{-6.8}$ | $5.2^{+20.2}_{-3.5}$ | $-$ | $-$
$P_{c}(4500)$ | $\frac{1}{2}^{-}$ | $4501.4^{+5.0}_{-6.2}$ | $8.8^{+17.2}_{-5.4}$ | $-$ | $-$
$P_{c}(4510)$ | $\frac{3}{2}^{-}$ | $4513.4^{+5.8}_{-3.1}$ | $7.6^{+9.4}_{-0.0}$ | $-$ | $-$

###  III.4 A numerical experiment on the ($\Lambda_{c}\bar{D}^{(*)}_{s}$,
$\Xi_{c}\bar{D}^{(*)}$) coupled-channel systems

There exists an important difference between the effective potential matrices
in the $P_{c}$ and $P_{cs}$ systems. As presented in Tables 2 and 3, the
diagonal matrix elements in the $P_{cs}$ system are very similar to those of
the $P_{c}$ system. But for the off-diagonal matrix elements, the effective
potentials of the $\Lambda_{c}\bar{D}_{s}-\Xi_{c}\bar{D}$ and
$\Lambda_{c}\bar{D}_{s}^{*}-\Xi_{c}\bar{D}^{*}$ channels in the $P_{cs}$
system with $J=1/2$ or $3/2$ consist of central terms. These terms may give
considerable corrections to the spectrum of the $P_{cs}$ states.

For the $J=1/2$ and $J=3/2$ $P_{cs}$ systems, as given in Eq. (5-6) we need to
perform seven and five coupled-channel calculations. Before we perform such
complete calculations, we first perform a detailed discussion on the
($\Lambda_{c}\bar{D}_{s}$, $\Xi_{c}\bar{D}$) and
($\Lambda_{c}\bar{D}_{s}^{*}$, $\Xi_{c}\bar{D}^{*}$) coupled-channel systems.

The effective potential matrixes of the $J=1/2$ ($\Lambda_{c}\bar{D}_{s}$,
$\Xi_{c}\bar{D}$), ($\Lambda_{c}\bar{D}_{s}^{*}$, $\Xi_{c}\bar{D}^{*}$)
systems, and the $J=3/2$ ($\Lambda_{c}\bar{D}_{s}^{*}$, $\Xi_{c}\bar{D}^{*}$)
system share the same expressions in the heavy quark limit. From Table 3, we
obtain the corresponding effective potential matrix

$\displaystyle\mathbb{V}=\left(\begin{array}[]{cc}v_{11}&v_{12}\\\
v_{21}&v_{22}\\\ \end{array}\right)$ (45)

with

$\displaystyle v_{11}$ $\displaystyle=$
$\displaystyle-\frac{4}{3}\tilde{g}_{s},\quad
v_{22}=-\frac{10}{3}\tilde{g}_{s},$ (46) $\displaystyle v_{12}$
$\displaystyle=$ $\displaystyle v_{21}=2\sqrt{2}\tilde{g}_{s}g_{x}.$ (47)

Here, for the diagonal matrix elements listed in Table 2 and 3, their dominant
components are from the exchange of the non-strange light scalar meson
currents. Since the interactions of the off-diagonal channel
$\Lambda_{c}\bar{D}^{(*)}_{s}-\Xi_{c}\bar{D}^{(*)}$ are introduced via the
exchange of the strange scalar meson currents, we further introduce a factor
$g_{x}$ to estimate the SU(3) breaking effects. Compared with the exchange of
the non-strange light scalar meson currents, the off-diagonal matrix elements
should be suppressed by the mass of strange mesons. Thus, we assume $0\leq
gx\leq 1$. This factor also reflects the coupling strength of the
$\Lambda_{c}\bar{D}_{s}-\Xi_{c}\bar{D}$ channel. With $g_{x}=0$, the
$\Lambda_{c}\bar{D}^{(*)}_{s}$ does not couple to the $\Xi_{c}\bar{D}^{(*)}$
channel. With $g_{x}=1.0$, the $\Lambda_{c}\bar{D}^{(*)}_{s}$ couples to the
$\Xi_{c}\bar{D}^{(*)}$ channel and its coupling strength is set to be the
value in the SU(3) limit.

Figure 1: The variations of the masses for the possible bound states in the
($\Lambda_{c}\bar{D}^{(*)}_{s}$ $\Xi_{c}\bar{D}^{(*)}$) two-channel system as
the parameter $g_{x}$ increases. We use the blue-dotted lines to denote the
$\Lambda_{c}\bar{D}^{(*)}_{s}$ and $\Xi_{c}\bar{D}^{(*)}$ thresholds. The
masses of the $P_{cs}$ states are denoted with black lines. Figs. (a), (d),
Figs. (b), (e), and Figs. (c), (f) are obtained at
$v_{11}^{\delta}=1/5v_{11}$, 0, and $-1/5v_{11}$, respectively. The green
bands in Figs. (a), (d), (b), (e) denote that the bound states near the
$\Lambda_{c}\bar{D}_{s}$ ($\Lambda_{c}\bar{D}_{s}^{*}$) and $\Xi_{c}\bar{D}$
($\Xi_{c}\bar{D}^{*}$) thresholds can coexist in this $g_{x}$ region.

In Fig. 1 (b), we present the variation of the masses for the bound states
$P_{cs}(4338)$ and $P_{cs}(4255)$ as the parameter $g_{x}$ increases. The
masses of the $P_{cs}(4338)$ and $P_{cs}(4255)$ are denoted with black lines.
At $g_{x}=0$, the $\Lambda_{c}\bar{D}_{s}$ channel itself has a weak
attractive force $v_{11}=-4/3\tilde{g}_{s}$, and this force is too weak to
form a $\Lambda_{c}\bar{D}_{s}$ bound state. On the contrary, the
$\Xi_{c}\bar{D}$ channel can form a bound state and its mass is about $M=4329$
MeV, slightly smaller than the experimental value. As the $g_{x}$ increases,
the attractive force of the $P_{cs}(4338)$ decreases and its mass moves closer
to the $\Xi_{c}\bar{D}$ threshold. In a very narrow region $0.62\leq g_{x}\leq
0.64$, the attractive force is just enough to form a $P_{cs}(4338)$ bound
state at the $\Xi_{c}\bar{D}$ threshold and the weak attractive channel
$\Lambda_{c}\bar{D}_{s}$ starts to form a bound state due to the
$\Xi_{c}\bar{D}-\Lambda_{c}\bar{D}_{s}$ coupling. Only in this very narrow
region, the $P_{cs}(4338)$ and $P_{cs}(4255)$ can coexist as quasi-bound
states. At $g_{x}>0.64$, the $\Xi_{c}\bar{D}-\Lambda_{c}\bar{D}_{s}$ coupling
further weakens the attractive force of the $\Xi_{c}\bar{D}$ channel and the
$P_{cs}(4338)$ no longer exists as a quasi-bound state, while the attractive
force of the $\Lambda_{c}\bar{D}_{s}$ channel becomes stronger and its mass
will decrease. The observation of the $P_{cs}(4338)$ by LHCb seems to exclude
the parameter region $0.64<g_{x}<1.0$.

Here, we also check the pole position of the $P_{cs}(4338)$ at $g_{x}>0.64$ in
the energy region slightly above the $\Xi_{c}\bar{D}$ threshold. We find that
the pole of the $P_{cs}(4338)$ still exists in the first Riemann sheet. This
is mainly due to the fact that the $\Lambda_{c}\bar{D}_{s}-\Xi_{c}\bar{D}$
coupling leads the $P_{cs}(4338)$ to be a state that have a considerable
width, thus the central value of the $P_{cs}(4338)$ mass may cross the
$\Xi_{c}\bar{D}$ threshold. In this case, the $P_{cs}(4338)$ should be
interpreted as a quasi-bound state above the $\Xi_{c}\bar{D}$ threshold.
Nevertheless, in this work, we restrict our scope to the case that the masses
of the bound/quasi-bound states are below their corresponding thresholds.

To understand why the $g_{x}$ region that allows the $P_{cs}(4338)$ and
$P_{cs}(4255)$ states to coexist is so narrow, we further check the role of
the $\Lambda_{c}\bar{D}_{s}$ channel in our two-channel model. We allow the
effective potential of the $\Lambda_{c}\bar{D}_{s}$ channel to have a $20\%$
shift, i.e.,

$\displaystyle v^{\prime}_{11}=v_{11}+v^{\delta}_{11},\quad
v^{\delta}_{11}=0,\pm\frac{1}{5}v_{11},$ (48)

and further check how the masses of the $P_{cs}(4338)$ and $P_{cs}(4255)$
change as we increase the $g_{x}$.

The channel $\Lambda_{c}\bar{D}_{s}$ itself has a weak attractive force, as
presented in Fig. 1 (a), at $g_{x}=0$. After we increase this force by 20$\%$,
this single-channel still can not form a bound state. But the $g_{x}$ region
that allows these two $P_{cs}$ states to coexist becomes broader. On the
contrary, as illustrated in Fig. 1 (c), if we decrease the attractive force of
the $\Lambda_{c}\bar{D}_{s}$ channel by 20$\%$, the $P_{cs}(4338)$ and
$P_{cs}(4255)$ can not coexist no matter how we adjust the off-diagonal
$\Lambda_{c}\bar{D}_{s}-\Xi_{c}\bar{D}$ coupling. Thus, the narrow $g_{x}$
region that the $P_{cs}(4338)$ and $P_{cs}(4255)$ can coexist is due to the
fact that the $\Lambda_{c}\bar{D}_{s}$ channel has a small but non-negligible
attractive force.

The results for the $J=1/2$ and $3/2$
($\Lambda_{c}\bar{D}_{s}^{*}$,$\Xi_{c}\bar{D}^{*}$) coupled-channels are
presented in Fig. 1 (d-f). We find that the roles of the predicted
$P_{cs}(4477)$ and $P_{cs}(4398)$ with $J^{P}=1/2^{-}$ or $3/2^{-}$ are very
similar to those of the $P_{cs}(4338)$ and $P_{cs}(4255)$ with
$J^{P}=1/2^{-}$, respectively.

###  III.5 The results of $P_{cs}$ system in the coupled-channel formalism

Figure 2: The variations of the masses for the $P_{cs}$ states with
$J^{P}=1/2^{-}$ and $J^{P}=3/2^{-}$ as the $g_{x}$ increases. We use the blue-
dotted lines to denote the considered meson-baryon thresholds. The masses of
the obtain bound states are denoted with black lines. The green bands denote
that the bound states near the $\Lambda_{c}\bar{D}_{s}^{(*)}$ or
$\Xi_{c}\bar{D}^{(*)}$ thresholds can coexist in this $g_{x}$ region.

We present our complete multi-channel calculations on the $J=1/2$ and $J=3/2$
$P_{cs}$ systems in Fig. 2. We find that only the bound states close to the
$\Lambda_{c}\bar{D}_{s}^{(*)}$ and $\Xi_{c}\bar{D}^{(*)}$ channels have
significant dependence on the $g_{x}$, since these bound states can couple to
the $\Lambda_{c}\bar{D}_{s}^{(*)}$ and $\Xi_{c}\bar{D}^{(*)}$ channels through
non-negligible central terms, while the bound states that can only couple to
the $\Lambda_{c}\bar{D}_{s}^{(*)}$ and $\Xi_{c}\bar{D}^{(*)}$ channels via the
spin-spin interaction terms have very tiny dependence on the parameter
$g_{x}$.

To further present our numerical results, we fix the parameter $g_{x}$ at 0.5
and 0.62. We denote these two cases as case 1 and case 2. The case 1 and case
2 correspond to the results that the possible $P_{cs}(4255)$ signal does
not/does exist. The results of these two cases are listed in Table 6.

Table 6: The results of the $P_{cs}$ states calculated at $g_{x}=0.50$ and $g_{x}=0.62$ in the coupled-channel formalism. All results are in units of MeV. | $g_{x}=0.50$ | $g_{x}=0.62$
---|---|---
States | Mass | $\Gamma$ | BE | Mass | $\Gamma$ | BE
$[\Lambda_{c}\bar{D}_{s}]^{\frac{1}{2}}$ | $-$ | $-$ | $-$ | $4255.5_{-0.7}^{+0.0}$ | 0.0 | $-0.0_{-0.7}^{+0.0}$
$[\Lambda_{c}\bar{D}_{s}^{*}]^{\frac{1}{2}}$ | $-$ | $-$ | $-$ | $4398.1^{+0.2}_{-1.5}$ | 0.0 | $-0.6^{+0.2}_{-1.5}$
$[\Lambda_{c}\bar{D}_{s}^{*}]^{\frac{3}{2}}$ | $-$ | $-$ | $-$ | $4398.3^{+0.4}_{-1.3}$ | $0.0$ | $-0.4^{+0.4}_{-1.3}$
$[\Xi_{c}\bar{D}]^{\frac{1}{2}}$ | $4331.6_{-1.5}^{+2.6}$ | $17.8_{-7.6}^{+3.6}$ | $-5.0^{+2.6}_{-1.5}$ | $4335.9_{-2.1}^{+0.7}$ | $26.0^{+5.0}_{-10.2}$ | $-0.7_{-2.1}^{+0.7}$
$[\Xi_{c}\bar{D}^{*}]^{\frac{1}{2}}$ | $4472.1^{+2.5}_{-1.6}$ | $23.4^{+5.2}_{-6.0}$ | $-5.9^{+2.5}_{-1.6}$ | $4477.1^{+0.5}_{-0.8}$ | $33.0^{+6.0}_{-8.0}$ | $-0.9^{+0.5}_{-0.8}$
$[\Xi_{c}\bar{D}^{*}]^{\frac{3}{2}}$ | $4469.7^{+1.9}_{-6.6}$ | $14.6^{+4.4}_{-10.6}$ | $-8.3^{+1.9}_{-6.6}$ | $4473.7^{+2.7}_{-8.7}$ | $20.2^{+5.8}_{-13.6}$ | $-4.7^{+2.7}_{-8.7}$
$[\Xi_{c}^{\prime}\bar{D}]^{\frac{1}{2}}$ | $4433.8^{+3.4}_{-4.8}$ | $0.8_{-0.6}^{+5.0}$ | $-12.2^{+3.4}_{-4.8}$ | $4433.7_{-5.2}^{+3.5}$ | $0.4_{-0.0}^{+2.6}$ | $-12.3^{+3.5}_{-5.2}$
$[\Xi_{c}^{*}\bar{D}]^{\frac{3}{2}}$ | $4501.8^{+4.4}_{-3.4}$ | $7.7^{+22.3}_{-5.1}$ | $-11.4^{+4.4}_{-3.4}$ | $4501.2^{+4.2}_{-3.7}$ | $7.8^{+25.4}_{-5.3}$ | $-12.0^{+4.2}_{-3.7}$
$[\Xi_{c}^{\prime}\bar{D}^{*}]^{\frac{1}{2}}$ | $4564.4^{+3.4}_{-4.0}$ | $4.5^{+9.5}_{-2.7}$ | $-23.0^{+3.4}_{-4.0}$ | $4564.1_{-7.4}^{+3.3}$ | $4.8^{+9.2}_{-2.8}$ | $-23.3_{-7.4}^{+3.3}$
$[\Xi_{c}^{\prime}\bar{D}^{*}]^{\frac{3}{2}}$ | $4582.1^{+4.3}_{-2.0}$ | $1.9_{-1.1}^{+0.7}$ | $-5.3_{-2.0}^{+4.3}$ | $4582.1^{+4.2}_{-2.0}$ | $2.0^{+1.3}_{-1.2}$ | $-5.3^{+4.2}_{-2.0}$
$[\Xi_{c}^{*}\bar{D}^{*}]^{\frac{1}{2}}$ | $4628.2_{-7.1}^{+3.5}$ | $5.0_{-3.0}^{+9.0}$ | $-26.3_{-7.1}^{+3.5}$ | $4628.1^{+5.1}_{-7.3}$ | $5.2_{-3.2}^{+24.5}$ | $-26.4_{-7.3}^{+5.1}$
$[\Xi_{c}^{*}\bar{D}^{*}]^{\frac{3}{2}}$ | $4640.7^{+5.8}_{-3.2}$ | $7.0_{-0.8}^{+8.2}$ | $-13.8^{+5.8}_{-3.2}$ | $4640.6_{-3.2}^{+5.9}$ | $7.2^{+8.4}_{-4.4}$ | $-13.9^{+5.9}_{-3.2}$

Comparing the masses of the $P_{cs}$ states calculated in the single-channel
formalism (Table 4) with the results obtained in the coupled-channel formalism
(Table 6), we infer that the off-diagonal channels that only consist of the
spin-spin interaction terms have small influence on the masses of the $P_{cs}$
states, which is very similar to the $P_{c}$ system. From Table 6, we find
that there exist three extra $P_{cs}$ states below the
$\Lambda_{c}\bar{D}^{(*)}_{s}$ thresholds in the case 2.

For the $P_{cs}(4338)$ state, due to its strong coupling to the
$\Lambda_{c}\bar{D}_{s}$ channel, the width of this state is broader than the
result given by LHCb. Note that in our calculation, we only include the open-
charm two body meson-baryon channels. Thus, the width predicted by our model
should be regarded as the lower limit of the experimental width. Since the
$P_{cs}(4338)$ is reported in the $B\rightarrow J/\Psi\Lambda\bar{p}$ channel,
the narrow width of the $P_{cs}(4338)$ found by the LHCb may be due to the
small phase space of this $B$ meson decay process. Thus, confirming the
$P_{cs}(4338)$ in other decay processes is important to pin down its resonance
parameters.

Besides, we also find that the $P_{cs}$ states that are close to the
$\Xi_{c}\bar{D}^{(*)}$ states are broader than the other $P_{cs}$ states due
to its strong coupling to the $\Lambda_{c}\bar{D}_{s}^{(*)}$ channel. Thus,
our results suggest that there exist two $J^{P}=1/2^{-}$ and $J^{P}=3/2^{-}$
quasi-bound states near the $\Xi_{c}\bar{D}^{*}$ region. This region is close
to the reported $P_{cs}(4459)$, and the two-peak structure in this region has
been discussed in many literatures Xiao:2021rgp ; Wang:2019nvm ; Chen:2020uif
; Wang:2020eep ; Zhu:2021lhd . The results from our model provide a new
possibility, i.e., the two $P_{cs}$ structures in this region may have a
significant overlap in the $J/\Psi\Lambda$ invariant spectrum due to their
considerable widths. The decay behaviors of the $P_{cs}(4459)$ have been
discussed in Refs. Du:2021bgb ; Ortega:2022uyu ; Azizi:2021utt ; Chen:2021tip
; Shen:2019evi . The decay widths and decay patterns are valuable in
identifying the structure of the $P_{cs}(4459)$. Further investigations on the
total and partial decay widths will be crucial to accomplish a thorough
understanding on the $P_{c}$ and $P_{cs}$ states.

###  III.6 The correspondence between the $P_{c}$ and $P_{cs}$ systems

Figure 3: The mass spectra of the $P_{c}$ and $P_{cs}$ states in the multi-
channel formalism, we present the results of $P_{cs}$ system at $g_{x}=0.50$
and $g_{x}=0.62$. The meson-baryon thresholds $m_{\Lambda_{c}\bar{D}^{(*)}}$
$m_{\Lambda_{c}\bar{D}_{s}^{(*)}}$,$m_{\Sigma_{c}^{(*)}\bar{D}^{(*)}}$,
$m_{\Xi_{c}^{\prime(*)}\bar{D}^{(*)}}$ are illustrated with the blue-dotted
lines. The two extra thresholds $\Xi_{c}\bar{D}$ and $\Xi_{c}\bar{D}^{*}$ in
the $P_{cs}$ system are denoted with the green-dotted lines. We use the black
lines to denote the central values of the obtained $P_{c}$ and $P_{cs}$
states, their uncertainties are illustrated with the green and red rectangles.
The six green states in the $P_{c}$ system can directly correspond to the six
green states in the $P_{cs}$ system.

Finally, we compare the masses of the $P_{c}$ and $P_{cs}$ states obtained
from our multi-channel model. Since the mass of constituent $s$ quark is
heavier than that of the $u$, $d$ quarks by about 100 MeV. Thus, we shift the
mass plot of the $P_{cs}$ system by 100 MeV to check the similarities between
the $P_{c}$ and $P_{cs}$ states. We present the multi-channel results for the
$P_{c}$ system in Fig. 3 (a), and the multi-channel results for the $P_{cs}$
system calculated at $g_{x}=0.5$ and $g_{x}=0.62$ are given in Fig. 3 (b). As
can be seen from Fig. 3 (a) and (b), the meson-baryon thresholds in the
$P_{c}$ and $P_{cs}$ systems have the following analogies

$\displaystyle m_{\Lambda_{c}\bar{D}^{(*)}}$ $\displaystyle\leftrightarrow$
$\displaystyle m_{\Lambda_{c}\bar{D}_{s}^{(*)}},$ (49) $\displaystyle
m_{\Sigma_{c}^{(*)}\bar{D}^{(*)}}$ $\displaystyle\leftrightarrow$
$\displaystyle m_{\Xi_{c}^{\prime(*)}\bar{D}^{(*)}}.$ (50)

We denote these thresholds with the blue-dotted lines in Fig. 3. Besides,
there exist two extra meson-baryon thresholds $\Xi_{c}\bar{D}$ and
$\Xi_{c}\bar{D}^{*}$ in the $P_{cs}$ system. These two channels can not
directly correspond to the meson-baryon channels in the $P_{c}$ system. We
denote these two thresholds with the green-dotted lines.

As can be seen from Fig. 3, there exist six $P_{c}$ states with
$J^{P}=1/2^{-}$ or $3/2^{-}$. These six states can correspond to the six
states in the $P_{cs}$ system. We denote the masses of the central values of
these 12 states with black lines and their uncertainties are denoted with
green rectangles. According to Fig. 3, the experimentally observed $P_{c}$
states and the predicted $P_{cs}$ states should have the following analogies

$\displaystyle P_{c}(4312)$ $\displaystyle\leftrightarrow$ $\displaystyle
P_{cs}(4434),$ (51) $\displaystyle P_{c}(4440)$ $\displaystyle\leftrightarrow$
$\displaystyle P_{cs}(4564),$ (52) $\displaystyle P_{c}(4457)$
$\displaystyle\leftrightarrow$ $\displaystyle P_{cs}(4582).$ (53)

As indicated in Fig. 3, if we replace the $\Lambda_{c}\bar{D}_{s}^{*}$ and
$\Xi_{c}^{\prime}\bar{D}$ channels with the $\Xi_{c}\bar{D}$ and
$\Xi_{c}\bar{D}^{*}$ channels, respectively, we can reluctantly obtain the
following analogies

$\displaystyle P_{c}(4312)$ $\displaystyle\leftrightarrow$ $\displaystyle
P_{cs}(4472),$ (54) $\displaystyle P_{c}(4440)$ $\displaystyle\leftrightarrow$
$\displaystyle P_{cs}(4564),$ (55) $\displaystyle P_{c}(4457)$
$\displaystyle\leftrightarrow$ $\displaystyle P_{cs}(4582).$ (56)

The predicted $P_{cs}(4472)$ may correspond to the reported $P_{cs}(4459)$.
However, such an analogy indicates a considerable SU(3) breaking effect. In
both sets of analogies, the $P_{cs}(4338)$ can not directly correspond to the
lowest $P_{c}(4312)$ state.

There exist three and six extra $P_{cs}$ states that can not correspond to the
states in the $P_{c}$ system at $g_{x}=0.50$ and $g_{x}=0.62$, respectively.
We denote the masses of the central values of these states with the black
lines and their uncertainties are denoted with red rectangles. Further
experimental explorations on the $P_{cs}$ system may help us to distinguish
which case should be preferred.

##  IV Summary

Motivated by the recently discovered $P_{cs}(4338)$ from the LHCb
Collaboration, we have performed a multi-channel calculation of the $I=1/2$
$P_{c}$ and $I=0$ $P_{cs}$ systems and presented a comparison between the
interactions of the $P_{c}$ and $P_{cs}$ states in the SU(3)${}_{\text{f}}$
limit and heavy quark limit.

Unlike the $\bar{c}n-cnn$ ($n=u$, $d$) type meson-baryon channels in the
$P_{c}$ system, we need to consider two types of channels when we study the
$P_{cs}$ system, i.e., the $\bar{c}n-cns$ and $\bar{c}s-cnn$ meson-baryon
channels. This difference will lead to extra states in the $P_{cs}$ systems.

The effective potentials of the $P_{c}$ and $P_{cs}$ states are collectively
obtained via a quark-level Lagrangian, which allows us to construct the
correspondence between the $P_{c}$ and $P_{cs}$ systems.

We use the masses of the $P_{c}(4440)$ and $P_{c}(4457)$ as input to determine
the coupling parameters $\tilde{g}_{s}$ and $\tilde{g}_{a}$ in our model. We
first study the masses of the $P_{c}$ states in the single-channel and
coupled-channel formalisms. Since all the off-diagonal terms in the effective
potential matrices consist of the spin-spin interaction terms, the coupled-
channel effect provides very small corrections to the masses of the $P_{c}$
states.

There exists an important difference between the $P_{c}$ system and $P_{cs}$
system. In the $P_{cs}$ system, the off-diagonal terms
$\Lambda_{c}\bar{D}^{(*)}_{s}-\Xi_{c}\bar{D}^{(*)}$ in the effective potential
matrices consist of the central terms and will have considerable corrections
to the mass spectrum of the $P_{cs}$ states. To clarify the role of the
$\Lambda_{c}\bar{D}^{(*)}_{s}-\Xi_{c}\bar{D}^{(*)}$ coupling, we have
performed a numerical experiment on the
$(\Lambda_{c}\bar{D}^{(*)}_{s},\Xi_{c}\bar{D}^{(*)})$ coupled-channel system.
Our results suggest that the mass of the $P_{cs}(4338)$ may shift very close
to the $\Xi_{c}\bar{D}$ threshold by adjusting the coupling between the
$\Xi_{c}\bar{D}$ and $\Lambda_{c}\bar{D}_{s}$ channels. This coupling may also
lead to a $P_{cs}(4255)$ state in a reasonable $g_{x}$ region.

Then we present our complete multi-channel calculations of the $P_{cs}$
systems. Since the $P_{cs}(4255)$ is not confirmed by experiment, we present
our numerical results with $g_{x}=0.50/0.62$, corresponding to the case that
the $P_{cs}(4255)$ does not/does exist, respectively. Due to the strong
$\Lambda_{c}\bar{D}_{s}-\Xi_{c}\bar{D}$ couplings, our predicted width of
$P_{cs}(4338)$ is broader than the experimental value. The reported narrower
width may be due to the small phase space of the $B$ meson decay process.
Confirming the $P_{cs}(4338)$ state in other processes will be helpful to pin
down its resonance parameters. There exist two $P_{cs}$ states with
$J^{P}=1/2^{-}$ and $J^{P}=3/2^{-}$ below the $\Xi_{c}\bar{D}^{*}$ threshold.
The masses of these two states are close to the mass of the reported
$P_{cs}(4459)$. Due to the $\Lambda_{c}\bar{D}^{*}_{s}-\Xi_{c}\bar{D}^{*}$
coupling, these two states should have considerable widths and may have
significant overlap in the $J/\Psi\Lambda$ invariant spectrum. Further
experimental exploration would be important to test our predictions.

Finally, we present a complete correspondence between the $P_{c}$ and $P_{cs}$
states. The observed $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ do not
directly correspond to the observed $P_{cs}(4338)$ and $P_{cs}(4459)$. It is
particularly interesting to find the SU(3) $P_{cs}$ states that may correspond
to the observed $P_{c}$ states, and to investigate if such a correspondence
does exist. Further experimental researches on these topics will be helpful to
fulfill a complete picture on the spectra of the $P_{c}$ and $P_{cs}$ systems.

## Acknowledgments

This research is supported by the National Science Foundation of China under
Grants No. 11975033, No. 12070131001 and No. 12147168.

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# Social Network Extraction: Superficial Method and Information Retrieval

Mahyuddin K. M. Nasution Information Technology Department,
Fakultas Ilmu Komputer dan Teknologi Informasi (Fasilkom-TI)
Universitas Sumatera Utara, Padang Bulan, Medan 20155, Sumatera Utara,
Indonesia
<EMAIL_ADDRESS>

Knowledge Technology Research Group, Faculty of Information Science &
Technology
Universiti Kebangsaan Malaysia, Bangi 43600 UKM Selangor Malaysia
samn<EMAIL_ADDRESS>
Shahrul Azman Mohd. Noah Proceeding of International Conference on Informatics
for Development (ICID’11), c2-110 - c2-115 (2011).Information Technology
Department,
Fakultas Ilmu Komputer dan Teknologi Informasi (Fasilkom-TI)
Universitas Sumatera Utara, Padang Bulan, Medan 20155, Sumatera Utara,
Indonesia
<EMAIL_ADDRESS>

Knowledge Technology Research Group, Faculty of Information Science &
Technology
Universiti Kebangsaan Malaysia, Bangi 43600 UKM Selangor Malaysia
samn<EMAIL_ADDRESS>
Information Technology Department,
Fakultas Ilmu Komputer dan Teknologi Informasi (Fasilkom-TI)
Universitas Sumatera Utara, Padang Bulan, Medan 20155, Sumatera Utara,
Indonesia
<EMAIL_ADDRESS>

Knowledge Technology Research Group, Faculty of Information Science &
Technology
Universiti Kebangsaan Malaysia, Bangi 43600 UKM Selangor Malaysia
samn<EMAIL_ADDRESS>
Saidah Saad Information Technology Department,
Fakultas Ilmu Komputer dan Teknologi Informasi (Fasilkom-TI)
Universitas Sumatera Utara, Padang Bulan, Medan 20155, Sumatera Utara,
Indonesia
<EMAIL_ADDRESS>

Knowledge Technology Research Group, Faculty of Information Science &
Technology
Universiti Kebangsaan Malaysia, Bangi 43600 UKM Selangor Malaysia
samn<EMAIL_ADDRESS>

###### Abstract

Social network has become one of the themes of government issues, mainly
dealing with the chaos. The use of web is steadily gaining ground in these
issues. However, most of the web documents are unstructured and lack of
semantic. In this paper we proposed an Information Retrieval driven method for
dealing with heterogeneity of features in the web. The proposed solution is to
compare some approaches have shown the capacity to extract social relation:
strength relations and relations based on online academic database.

###### Keywords:

superficial, unsupervised, supervised, keyword, similarity, association rule,
semantic.

##  1 Introduction

Social networks is an approach for representing relations between individuals,
groups or organization [1]. The concept of social networks extraction is
designed to map the relationship of entities among them that can be observed,
to mark the patterns of ties between social actors, to measure the social
capital: the values obtained by the entities individually or in groups, to
present a variety of social structures according to the interests and its
implementation, based on different domains or information sources [2]. The
extracted social networks can further be processed in information systems,
i.e., in data mining systems that detect valuable trends in the social
networks, in cases when selecting content based on certain semantic classes,
or in expert systems that reason with the extracted knowledge in order to
support government decision making. For example, in counterterrorism and the
terrorism require networks for learning exchange of information and actions
[3]. In scientometric, so extraction of academic social networks aims to see
the performance of human resources for example in education [4].

Web has been chosen as main source to gain various digital information about
entities and their relations from all over the world. The web has been
becoming the largest text database contained information about social
networks, but most of data on the web are so unstructured that they can only
be understood by humans, the other way the amount of data is so huge that they
can only be processed efficiently by machines. The semantic representation of
documents currently forms the vision of semantic web [5], in the same manner
as the social network is a nucleus of semantic relation, i.e. possibilities of
attributes partnership between actors where domains generate the attributes
and the web documents provide the descriptions of entities and their
relations.

Extracting social network is more than the text mining process and has been
one of Artificial Intelligent (AI) research agendas, involving the task of
identifying entities and their relations for trusted social network. As the
manual semantic relation annotation of web documents is impractical and
unscalable, and so in the semantic web area there are two research streams for
extracting social networks from web documents: unsupervised method and
supervised method. The unsupervised methods utilize the Cartesian product for
clustering the nodes $A$ in the network. The concept of clustering is
$\gamma:A\times A\rightarrow R$ such that $\gamma(a,b)\in R$, $a,b\in A$ [6,
7, 8]. However, the clustering approach is difficult to generate the labels of
relations in a network. The supervised methods employ a function $\lambda$ for
classifying $Z$, i.e. $\lambda:Z\rightarrow C$ such that $\lambda(z)=c$, $z\in
Z$, and $c\in C$ is a class label, where $C=\\{c_{1},c_{2},\dots,c_{|C|}\\}$
is data set as special target attributes, $|C|\geq 2$ is the number of
classes, and $Z\cap C=\emptyset$ [9, 10, 11]. The classification approaches
however only concern with extraction of network based on predefined labels
only, and thus cannot be adapted to the other descriptions of relations.
Therefore, our goal is to enhance the superficial approach, an approach belong
to unsupervised method, for extracting social network from web by not only
relying on the co-occurrence but to consider other information.

##  2 Motivation and Related Works

An approach for extracting social network is by involving the superficial
strategy that depends heavily on the co-occurrence. An occurence, a singleton
event of $a$ is a for $a\in A$, $A$ is a set of social actors, ${\bf
a}\subset\Omega$, whereas a co-occurrence, the doubleton event of $a$ and $b$,
i.e., ${\bf a}\cap{\bf b}$, the subset of $\Omega$, and $a,b\in A$. $\Omega$
is a set of web pages indexed by a search engine, and
$P:\Omega\rightarrow[0,1]$ to be an uniform mass probability functions. To
make the semantic relation between $a$ and $b$ from Web, we can explore
singleton and doubleton, following a pioneer work [12] by implementing
similarity measure with conditions that $|{\bf a}|\geq|{\bf a}\cap{\bf b}|$
and $|{\bf b}|\geq|{\bf a}\cap{\bf b}|$ [13]. At the time of doing this
experiment, a Yahoo! search for ”Shahrul Azman Mohd Noah”, returned 1,200
hits, whereas for ”Abdullah Mohd Zin” Yahoo! Search Engine returned 3,870
hits. Searching the pages where both ”Shahrul Azman Mohd Noah” and ”Abdullah
Mohd Zin” = 13 hits, and showed that singletons is greater or equal to
doubleton, but we have found that the conditions is not met. Moreover,
singletons and doubleton always took along bias in relation, due to the
limitations of search engine which due to ambiguity of results.

###  2.1 Strength Relation and Similarity (SRS)

Relations differ in strength. Such strength can be operationalized in a number
of ways. With respect to event frequencies of the pairs of actors where they
may exchange large or small amounts of social capital: money, goods, papers,
or services. They may supply important or trivial information. Such aspects of
relationships measure different types of relational strength. Flink system has
been developed to extract, aggregate and visualize a social network [6]. In
POLYPHONET system has been created the procedures to expand superficial
strategy by providing keywords in query submitted to search engine [7].
However, these system took along special cases: Flink for building the social
network of a Semantic Web community by utilizing the friend-of-a-friend (FOAF)
semantic profile [6], and POLYPHONET developed to identify the relations in
the Japan AI conference [8].

The content of a relation refers to the resource that is exchanged that may
generate some labels of relation. In this case, each entity will be assigned
with multiple labels extracted from information sources related to that
entity. Suitable labels will be assigned to relations by using the Information
Retrieval technique, mainly the generative probabilistic model (PGM) [14]. The
parameters of GPM are used as modalities to get the knowledge from the corpus,
and generating strength relation based on labels of entities. For example, the
strength relation between entities based on participants in the same
conference or workshop [15].

Similarity measures the relation between two entities are a strategy to
generate strength relation. One of the used similarity measure widely is the
Jaccard coefficient, in singleton and doubleton we have

$sim_{jac}(a,b)=\frac{|{\bf a}\cap{\bf b}|}{(|{\bf a}|+|{\bf b}|-|{\bf
a}\cap{\bf b}|)}$ (1)

###  2.2 Underlying Strength Relation (USR)

Underlying strength relation is an approach to exploit URL (Uniform Resource
Locator) addresses and its organization since URL address is always available
in Web snippets, returned by any search engine [1]. Syntactically, an URL
represents a resource in Internet. A composition of URL contains a set of
tokens, $U=\\{s,d_{1},\dots,d_{m},p_{1},\dots,$ $p_{n-1}\\}$, satisfying a
structure: $s://d_{m}.\dots.d_{2}.d_{1}/p_{1}/p_{2}/$ $\dots/p_{n-1}$, i.e. a
string consists of scheme, authority, and path. The scheme is a token s, a
component contains a protocol that is used for communicating in Internet. For
example, http (Hypertext Transfer Protocol), https (http Security) and other
protocols. The authority is string of tokens as $dm.\dots.d_{2}.d_{1}$, i.e. a
component has three subcomponents: user information, host, and port:

  1. 1.

The user information may consist of a user name and, optionally, scheme-
specific information about how to gain authorization to access the resource.
Usually, it is followed by a commercial at-sign(”@”) that delimits it from the
host, if present like in an email address<EMAIL_ADDRESS>
  2. 2.

The host contains a location of a web server, where the location can be
describe as either a domain name system (DNS) or internet protocol (IP)
address.

  3. 3.

The port is a specific number. For instance, a default port number (80 for the
http protocol), i.e. $s://d_{m}.\dots.d_{2}.d_{1}:80/$. The colon symbol (”:”)
should be prefixed prior to the port number.

The last string of tokens $d_{1}/p_{1}/p_{2}/\dots/p_{n-1}$ is a path, i.e. a
component contains the directories including a web page and a file name of the
page, where a directory and a file are separated by the slash symbol (”/”).
The last token of path sometimes comes with two other components: query and
fragment. For example,
http://search.yahoo.com/search;_ylt=AjoEJrO9wuxK84pfA74_RvCbvZx4?vc=&fp_ip=my&p=Mahyuddin+K.+M.+Nasution&toggle=1&cop=mss&ei=UTF-8&fr=yfp-t-701.
The query is a component containing parameter names and values that may to be
supplied to web applications. The token of path and the query are separated by
the question symbol (”?”). The form of query is name=[value], where there is
equal symbol (”=”) between a parameter name and a parameter value. A pair of
name=[value] is separated each other by the ampersand symbol (”&”). The
fragment is a component for indicating a parameter part of a document. This
last component and the part previously mediated by a sharp symbol (”#”).
Consequently, be found same URL addresses, but presented in a different
threads. Therefore, necessary to the canonicalization of the URL.

URL address indicates the layered structure of a web site which can be
logically shown as a hierarchy. As such the URL of web pages which provides
and indicator of its logical position in the hierarchy structure that can be
considered as the underlying strength of the relationship, where site editors
usually tend to put similar or related web pages as close as possible
underlying relations among entities in the case that co-occurrence measures
unable to provide such relations.

For any web snippet produces as a results of the entity name query, there
exists a set of $k$ URL addresses. Therefore, there will be
$n_{1}+\dots+n_{k}$ URL addresses whereby $n_{i}$, $i=1,\dots,k$, is the
number of layers for each $i$. For these generated URLs, there is a
possibility of redundant URLs. Let u is the number of same URL address. For
each entity $a\in A$ we can derive a vector space $a=[a_{1},\dots,a_{K}]$
where $a_{j}=un_{i}$. We, therefore, can measure the distance between the two
entities based on the list of URL addresses from Web snippets [1].

###  2.3 Associaton Rule and Similarity (ARS)

Another implementation of co-occurrence is a formulation of basic data mining,
e.g. the association rule [16]. Assume $B=\\{b_{1},b_{2},\dots,b_{|B|}\\}$ is
a set of attribute literals, and a set of transaction $M_{i}$ are subsets of
attributes, or $M_{i}$ be subsets of $B$. Then, we define the implication,
$X\Rightarrow Y$ with two possible values $T=\textsc{TRUE}$ or
$F=\textsc{FALSE}$ as association rule if $X$ be a subset of $B$, $Y$ be a
subset $B$ and $X\cap Y\neq\emptyset$. Let $q$ = ”$a$ AND $x$” is a query,
where $x$ is a keyword and $a$ is a name of actor $a\in A$ as seeds, and
$Db_{i}$ is a collection of document containing names of actors $b_{i}\in A$,
then the transactions be $M_{i}=\\{q,b_{i}\\}$ or $\\{(q\Rightarrow
b_{i})\\}$, $q\in X$, $b_{i}\in Y$, $M_{i}\in M$. So, by making q always T,
(see Table 1), we obtain a conditional probability as follows.

Table 1: Transaction and implication Transaction | $~{}~{}q=(t_{a},t_{x})~{}~{}$ | $~{}~{}t_{b_{i}}~{}~{}$ | Implication
---|---|---|---
$M_{i}$ | $~{}~{}a~{}~{}$ | $t_{x}$ | $b_{i}$ | $a\Rightarrow t_{x}$ | $q\Rightarrow b_{i}$
$M_{1}$ | T | T | T | T | T
$M_{2}$ | T | T | $\dots$ | T | $\dots$
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
$M_{|M|}$ | T | T | $\vdots$ | T | $\vdots$

$p(b_{i}|a)=\frac{|{(q\Rightarrow b_{i})=T}|}{|M|}.$ (2)

The Jaccard coefficient in Eq. (1) is modified as follows.

$sim(a,b_{i})=\frac{|{(a\Rightarrow b_{i})=T}|}{(|M|+|Db_{i}|-|{(q\Rightarrow
b_{i})=T}|)}.$ (3)

We used the association rule for extracting social network from online
database such as DBLP and for enhancing the superficial method, but this
approach depend on the structure of DBLP web page. In previous research,
association rule can be generally defined as $b_{1},\dots,b_{|B|-1}\Rightarrow
b_{|B|}$, where $X=\\{b_{1},\dots,b_{|B|-1|}\\}$, and $Y=\\{a_{|B|}\\}$. If we
used Table 1 to generate labels of networks in a tree, then we can use TF.IDF
(term frequency - inverse document frequency) scheme for extracting label by
considering $\sigma=$ degree of node as the tree root, i.e.

$\begin{array}[]{rcl}TF.IDFw&=&tf(w)\cdot
idf(w)\cr&=&\Big{(}\sum_{j\in\\{1,\dots,N\\}}\sum_{i\in\\{1,\dots,m\\}}\frac{1}{n}\Big{)}\log\frac{N}{df(w)}\cr\end{array}$
(4)

where $n$ is the number of words in a document, $m$ is the number of word $w$
in document $j$, $N$ is the total number of documents, and $df(w)$ is the
number of documents containing the word $w$. The normalization of TF.IDF, is
defined as

$tfidf_{nor}=(TF.IDF)(N/\sigma)$ (5)

##  3 Extraction and Information Retrieval

First step for building any network is to determine nodes [17, 18].
Discretely, a node $v\in V$ in a graph $G$, is a representation of any object
in a network, i.e. $V=\\{v_{i}|i=1,\dots,n\\}$, $V\neq\emptyset$. The nodes in
a social network refer to actor names such as authors, recipients,
researchers, artists. Therefore, the first task of extracting social network,
$\xi$, is to identify the actors. This can be achieved by providing a list of
names as seeds in order to extract other names, recognize and disambiguate
them [19, 20]. The actors play some role in a social based on their
background, and they have some characteristics as attributes. Formally, such
attributes of actors we define as $Z=\\{z_{j}|j=1,\dots,m\\}$ which are
attributes / characteristics of entities whereby a pair of $\langle
A,Z\rangle$ is the instance of actors, where $Z_{i}$ are subsets of $Z$,
$Z_{i}$ are subsets of attributes of each entity $a_{i}$, i.e. $\langle
a_{i},Z_{i}\rangle$, $i=1,\dots,n$, or simply denotes a set of attributes of
entity a as $Z_{a}$.

We developed an approach to extracting keywords from web snippets for
disambiguating names in social networks. The methods for using keyword in
query are [22]:

noK | : | A name pair used as query without relation keywords.
---|---|---
K1 | : | A name pair and top-weighted relation keyword used as a query.
K2 | : | A name pair and second-weighted relation keyword used
|  | as a query.
K1+K2 | : | A name pair and top-weighted strongest relation keyword.

By considering a set of snippets $L$, each contains a reference to a person.
Let $P=\\{P_{1},\dots,P_{|P|}\\}$ be a partition of $\xi$ and $\zeta$ into
references to the same person, so for example
$P_{i}=\\{S_{1},S_{4},S_{5},S_{9}\\}$ might be a set of references to ”Abdul
Razak Hamdan” the information technology professor. We produce the current
context for each actor where keywords have two vector, $\nu$ and $\upsilon$:
first vector we define based on TF.IDF and second vector we generate from hit
count, and then we define delta $\delta$ as distance of two vector. We use
delta to select keyword from each classes of candidates that grouped by tree
of network semantic of words [1].

Let $C=\\{C_{1},\dots,C_{|C|}\\}$ be a collection of disjoint subset of $L$
created by algorithm and manually validated such that each $S$ has an
identifier, i.e. URL address. Then, we denote $L_{C}$ as references that be
the clusters based on collection. Based on measure were introduced, we define
an notation of recall $Rec()$ as follows

$Rec(S_{i})=\frac{|\\{S\in P(S_{i}):C(S)=C(S_{i})\\}|}{|\\{S\in P(S_{i})\\}|}$
(6)

and a notation of precision $Prec()$ as follows

$Prec(S_{i})=\frac{|\\{S\in C(S_{i}):P(S)=P(S_{i})\\}|}{|\\{S\in
C(S_{i})\\}|}$ (7)

where $P(S_{i})$ as a set $P_{i}$ containing reference $S_{i}$ and $C(S_{i})$
to be the set $C_{i}$ containing $S_{i}$. Thus, the precision of a reference
to ”Abdul Razak Hamdan” is the fraction of references in the same cluster that
are also to ”Abdul Razak Hamdan”. We obtain average of: recall (REC),
precision (PREC), and F-measure for the clustering C as follows:

$REC=\frac{\sum_{S\in L_{C}}Rec(S)}{|L_{C}|}$ (8) $PREC=\frac{\sum_{S\in
L_{C}}Prec(S)}{|L_{C}|}$ (9) $F=\frac{2\cdot REC\cdot PREC}{REC+PREC}$ (10)

Second step, for building any network is to determine an edge based on a
concept of graph theory [18]. In social network, the edges refer to
relationships between actors. Therefore, another main objective of extracting
social network is to identify relation among entities, i.e., second task of
social network extraction, $\zeta$. If there exists
$R=\\{r_{1},\dots,r_{k}\\}$ as a set of possible relations between social
actors, then we have $R=A\rightarrow A\times A=r(a,b)$, $\forall a,b\in A$,
that can be depicted as the overlap principle, i.e. the intersection their
attributes $r_{k}(a,b)=Z_{a}\cap Z_{b}$. This means that the relations among
actors are formed by sharing attributes, ideas, and concepts between them.

Social network extraction provides the technology to identify and describe
content. The technology is as an exertion to further acquire rich and trusted
social network. Such social network formally is $SN=\langle
V,E,A,R,Z,\xi,\zeta\rangle$, $V\neq\emptyset$, $A\neq\emptyset$ that satisfies
the following conditions:

  1. 1.

$\xi:A\stackrel{{\scriptstyle 1:1}}{{\rightarrow}}V$, $v=\xi(a)$, for every
$a\in A$ there is only one $v\in V$;

  2. 2.

$\zeta:R\rightarrow E$ so that $e_{j}=\zeta(r_{k}(a,b))=\zeta(Z_{a}\cap
Z_{b})$, $e_{j}\in E$, $r_{k}\in R$, $\forall a,b\in A$, where
$Z_{a},Z_{b},Z_{a}\cap Z_{b}\subset Z$.

The labeled social network is to present a variety of social structures
according to the interests and its implementation. The strength relation in
social network tends to ambiguity. One or more relations may connect a pair of
actors, if it applies, we have social network with many ties. Pairs may
maintain a tie based on one relation only, e.g., as members of the same
organization, or they may maintain a multiplex tie, based on many relations,
such as sharing information, giving financial support and attending conference
together. Semantically, the composition of a relation or a tie is derived from
the social attributes of both participants, for example is the tie between
different or same sex dyads, between a supervisor and an underling or between
two peers. Computer network tends to underplay the social cues of participants
than the connections, but once a computer network is social network. Once
again, we define the social network as $SN_{des}=\langle
V,E,A,R,Z,\xi,\zeta,\varphi_{1},\varphi_{2}\rangle$, $V\neq\emptyset$,
$A\neq\emptyset$, a labelled social network satisfying the following
conditions:

  1. 1.

There exists $\varphi_{1}:Z\rightarrow V$ so that $\forall z_{i}\in Z_{a}$,
$\varphi_{1}(z_{i})\in V$, and $\forall a\in A$ and $Z_{a}\subset Z$;

  2. 2.

There exists $\varphi_{2}:Z\rightarrow E$ so that $\forall z_{j}\in Z_{a}\cap
Z_{b}$, $\varphi_{2}(z_{j})\in E$, $\forall a,b\in A$, and
$Z_{a},Z_{b},Z_{a}\cap Z_{b}\subset Z$.

Extracting social network from web documents need technologies. Those
technologies will become an important component of any retrieval system or as
Information Retrieval (IR) where the representations of documents and query
are enriched with semantic information. From social network perspective, it
will give emersion that can adapted and advance the retrieval models, i.e. the
representations of query and document relations and by the function that
estimates the relevance of relations to a query. Therefore, we can consider
relationship extraction as an information retrieval task. Let $G_{1}=\langle
V_{1},E_{1}\rangle$ is a graph as the resulted social networks and
$G_{2}=\langle V_{2},E_{2}\rangle$ is a benchmark graph, we have Jaccard-
coefficient, or Eqs. (1) and (8)-(10) be [22]

$sim_{G}=\frac{|E_{1}\cap E_{2}|}{|E_{1}|+|E_{2}|-|E_{1}\cap E_{2}|},$
$Precision=\frac{|E_{1}\cap E_{2}|}{|E_{1}|},$ $Recall=\frac{|E_{1}\cap
E_{2}|}{|E2|}$

and

$F_{measure}=\frac{2(|E_{1}\cap E_{2}|)^{2}}{|E_{1}|\cdot|E_{1}\cap
E_{2}|+|E_{2}|\cdot|E_{1}\cap E_{2}|}.$

Information retrieval is concerned with answering information needs as
accurately as possible, where relations retrieval typically involves the
querying of structured data from unstructured information.

##  4 Result

###  4.1 Evaluation and Dataset

For evaluation of the approaches, we have gathered and labeled a dataset of
539 Web pages, Tabel 2, where the label we created based on URL address of
each web page in snippet.

Table 2: Statistics of our dataset Personal name — | Position | Number of pages
---|---|---
Abdul Razak Hamdan | Dean | 85
Abdullah Mohd Zin | Professor | 90
Shahrul Azman Mohd Noah | Professor | 134
Tengku Mohd Tengku Sembok | Professor | 189
Md Jan Nordin | Assoc. Prof. | 41
Total | 539

As banchmark of social network, we possess $|V_{2}|=67$ nodes and
$|E_{2}|=253$ edges. This graph we derived from online database DBLP to 67
academic persons, we extract by using association rule (ARS) and then
evaluated and corrected their relations based on author-coauthor relationship.

Figure 1: (TF.IDF, hit count, delta) vs (recall, precision, F-measure) of
candidate words for actor ”Abdul Razak Hamdan”.

###  4.2 Experiment

Let us consider information context of actors that includes all relevant
relationships as well as interaction history, where Yahoo! Search engine fall
short of utilizing any specific information, especially context information,
and just use full text index search in web snippets. In experiment, we use
maximum of 600 web snippets for search an actor a, and we consider words where
the TF.IDF value $>0.3\times$ highest value of TF.IDF, or maximum number is 30
words, see Fig. 1. We used Eqs. (6), (7) and (10) for computing recall,
precision and F-measure.

Table 3: Name disambiguation result Method | Recall | Precision | F-measure
---|---|---|---
Delta ($\delta$) | 45.8 | 29.5 | 35.9

Under recall, precision and F-measure: Eqs. (8)-(10), and results in Table 3,
this method shows something to consider, i.e. the number of words in the
cluster should be limited so that an average value of measurement is not
affected by the lower (see Fig. 1). Given the implementation of this method is
done with the Yahoo! search, this result is still reasonable.

We tes for 213 actor names and there are 22,683 potential relations, but at
the time of doing this experiment by using SRS there are only 12,621 (53%)
relations which satisfy threshold $\alpha=0.0001$. However, the method (USR)
that involves URL-computation able to identify 19,513 (86%) relations for
$\alpha=0.01$.

Table 4: Recall and precision of social network Method | Recall | Precision | F-measure
---|---|---|---
SRS | 120/253 (47%) | 120/12,621 (10%) | 27%
URS | 176/253 (70%) | 176/19,513 (9%) | 18%

Tabel 4 shows the result of correlation with the SRS and USR where based on
Jaccard coefficient $sim(SRS,ASR)=0.0094$ almost the same with
$sim(URS,ASR)=0.009$, but looking at those results USR shows good performance.

##  5 Conclusion and Future Work

The method used URL as object for extracting social network to be incorporated
into existing tracting method of social network. It shows how to uncover
underlying strength relations by exploiting Web snippets and URL structure.
This showed that a well-known paradigm of querying a document web more simple
for accessing by inputting keyword, but most simply by using URL address,
because on the word or keyword we have typical problems of synonymy and
ambiguity, but URL not exist. Our near future work is to further experiment he
proposed method and look into the possibility of enhancing IR performance by
using social network with developing banchmark manually.

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