# Production of $P_{cs}(4459)$ from $\Xi_{b}$ decay

Qi Wu1 Dian-Yong Chen1111Corresponding author<EMAIL_ADDRESS>Ran Ji2 1
School of Physics, Southeast University, Nanjing 210094, China
2 Research School of Physics, Australian National University, Canberra 2601,
Australia

###### Abstract

Inspired by the $P_{cs}(4459)$ reported by the LHCb collaboration recently, we
investigate the $P_{cs}(4459)$ production from $\Xi_{b}$ decay in a molecular
scenario using an effective Lagrangian approach. With different $J^{P}$
assignments to $P_{cs}(4459)$, the magnitude of branching fractions of
$\Xi_{b}\to P_{cs}(4459)K$ is estimated, which is of the order of $10^{-4}$.
Together with the decay properties of $P_{cs}(4459)$, the present estimations
could be further testified by precise measurements and contribute to a better
understanding of the molecular interpretations and the exploration of $J^{P}$
quantum numbers of $P_{cs}(4459)$.

###### pacs:

13.87.Ce, 13.30.Eg, 14.20.Pt,

##  I Introduction

The notion of pentaquark can date back to the birth of the quark model. The
searches of pentaquark states become one of the most crucial inspections on
the quark model. A lot of efforts have been made from both experimental and
theoretical aspects and progresses have been achieved in recent decades (see
recent reviews Liu:2019zoy ; Chen:2016qju ; Guo:2017jvc  for more details).
Among these achievements, the observations of $P_{c}$ states are undoubtedly
one of the most important experimental breakthroughs, which was initially
reported by the LHCb collaboration in 2015. The two pentaquark candidates,
$P_{c}(4380)$ and $P_{c}(4450)$, were observed in the $J/\psi p$ invariant
mass distribution of $\Lambda_{b}\to KJ/\psi p$ process Aaij:2015tga . And
later in 2019, with more data samples, the LHCb collaboration updated their
analysis of the $J/\psi p$ invariant mass distribution of $\Lambda_{b}\to
KJ/\psi p$ and discovered the split of structures $P_{c}(4380)$ and
$P_{c}(4450)$ into three structures, namely $P_{c}(4312)$, $P_{c}(4440)$ and
$P_{c}(4457)$ Aaij:2019vzc .

Since these $P_{c}$ states were observed in the $J/\psi p$ invariant mass
distribution, the most possible quark components are $c\bar{c}uud$, which
indicates that these $P_{c}$ states can be good candidates of pentaquark
states. In the pentaquark scenario, the spectroscopy and decay properties have
been investigated using the QCD sum rule Chen:2015moa ; Wang:2015epa ;
Wang:2019got  and the constituent quark model Ortega:2016syt ; Park:2017jbn ;
Weng:2019ynv ; Zhu:2019iwm . In the vicinity of $P_{c}$ states, there are
abundant thresholds of a baryon and a meson, such as
$\Sigma_{c}^{(\ast)}\bar{D}^{(\ast)}$, $\Lambda_{c}\bar{D}^{\ast}$,
$\chi_{c1}p$, $\psi(2S)p$. In particular, the mass of $P_{c}(4312)$ is very
close to the threshold of $\Sigma_{c}\bar{D}$, while $P_{c}(4440)$ and
$P_{c}(4457)$ are close to the threshold of $\Sigma_{c}\bar{D}^{\ast}$.
Considering the $S$ wave interaction, the possible $J^{P}$ quantum numbers of
$\Sigma_{c}\bar{D}$ are $1/2^{-}$, while for $\Sigma_{c}\bar{D}^{\ast}$
system, the possible $J^{P}$ quantum numbers are $1/2^{-}$ and $3/2^{-}$,
which illustrates that $P_{c}(4440)$ and $P_{c}(4457)$ could be assigned as a
$\Sigma_{c}\bar{D}^{\ast}$ molecular state with different $J^{P}$ quantum
numbers. Along this way, the mass spectrum Chen:2015loa ; He:2015cea ;
Chen:2019bip ; Liu:2019tjn ; Azizi:2016dhy ; Huang:2015uda ; Chen:2019asm ;
He:2019ify ; He:2019rva ; Zhang:2019xtu , decay properties Wang:2015qlf ;
Lu:2016nnt ; Shen:2016tzq ; Lin:2017mtz ; Guo:2019fdo ; Xiao:2019mvs ;
Wang:2019hyc ; Xu:2019zme ; Lin:2019qiv ; Dong:2020nwk  and production
behaviors Wang:2019krd ; Wu:2019rog ; Wang:2019dsi  of these $P_{c}$ states
have been investigated in various methods, such as the QCD sum rule
Azizi:2016dhy ; Chen:2019bip ; Wang:2019hyc ; Xu:2019zme ; Zhang:2019xtu , the
potential model Chen:2015loa ; He:2015cea ; Huang:2015uda ; Chen:2019asm ;
Liu:2019tjn ; He:2019ify ; He:2019rva  and the effective Lagrangian approach
Lu:2016nnt ; Xiao:2019mvs ; Wang:2019krd ; Wu:2019rog ; Wang:2019dsi ;
Lin:2019qiv .

Inspired by the $P_{c}$ states, the authors in Refs. Hofmann:2005sw ;
Wu:2010vk ; Cheng:2015cca ; Anisovich:2015zqa ; Wang:2015wsa ; Feijoo:2015kts
; Chen:2015sxa ; Chen:2016ryt ; Lu:2016roh ; Xiao:2019gjd ; Wang:2019nvm ;
Park:2018oib ; Shen:2020gpw ; Zhang:2020cdi ; Dong:2021juy  investigated the
existence of the hidden-charm pentaquark states with strangeness, named as
$P_{cs}$ states. Based on the SU(3) flavor symmetry, the authors in Ref.
Cheng:2015cca  advised to search the $P_{cs}$ states in $\Xi_{b}\rightarrow
P_{\Sigma}K$ and $\Omega_{b}\rightarrow P_{\Xi}K$ which may consist rates
comparable to $\Lambda_{b}\rightarrow P_{p}K$. In Refs. Anisovich:2015zqa ;
Wang:2015wsa , hidden-charm pentaquark states with strangeness were studied in
the diquark-diquark-antiquark picture. The $J/\psi\Lambda$ invariant mass
distribution was studied in reactions of $\Lambda_{b}\rightarrow
J/\psi\eta\Lambda$ Feijoo:2015kts , $\Lambda_{b}\rightarrow J/\psi
K^{0}\Lambda$ Lu:2016roh  and $\Xi^{-}_{b}\rightarrow K^{-}J/\psi\Lambda$
Chen:2015sxa ; Shen:2020gpw  to examine the existence of the hidden-charm
pentaquark states with strangeness. In the molecular scenario, the prediction
of hidden-charm pentaquark states with strangeness was provided in Refs.
Chen:2016ryt ; Xiao:2019gjd ; Zhang:2020cdi .

Experimentally, one predicted $P_{cs}$ state was reported very recently by the
LHCb collaboration in the $J/\psi\Lambda$ invariant mass spectrum of the
$\Xi^{-}_{b}\rightarrow J/\psi\Lambda K^{-}$ decays for the first time
Aaij:2020gdg . The measured mass and width are, respectively,

$\displaystyle m_{P_{cs}}$ $\displaystyle=$ $\displaystyle 4458.8\pm
2.9^{+4.7}_{-1.1}\ \mathrm{MeV},$ (1) $\displaystyle\Gamma_{P_{cs}}$
$\displaystyle=$ $\displaystyle 17.3\pm 6.5^{+8.0}_{-.57}\ \mathrm{MeV}.$ (2)

However, the $J^{P}$ quantum numbers of $P_{cs}(4459)^{0}$ were not
determined.

It should be noticed that the mass of $P_{cs}(4459)^{0}$ is about 19 MeV below
the threshold of $\Xi_{c}\bar{D}^{\ast}$, which naturally assigns
$P_{cs}(4459)^{0}$ to a hadronic molecule state composed of
$\Xi_{c}\bar{D}^{\ast}$ with possible $J^{P}$ quantum numbers $1/2^{-}$ or
$3/2^{-}$. As a strange partner of $P_{c}$, the properties of $P_{cs}(4459)$
was investigated in molecular Chen:2020uif ; Peng:2020hql ; Chen:2020opr ;
Chen:2020kco ; Liu:2020hcv  and pentaquark Wang:2020eep  scenarios after the
observation. Along the way of $\Xi_{c}\bar{D}^{\ast}$ molecular scenario, we
investigate the production mechanism of $\Xi^{-}_{b}\rightarrow
P_{cs}(4459)^{0}K^{-}$ using an effective Lagrangian approach. In such an
estimation, we regard $P_{cs}(4459)^{0}$ as a $\Xi_{c}\bar{D}^{\ast}$ molecule
state, while the probable conditions $J^{P}=1/2^{-}$ and $3/2^{-}$ are both
considered. Similar to the case of $P_{c}$ state Wu:2019rog ; Xiao:2019mvs ,
the present estimation helps to test the molecular interpretations and verify
the quantum number of $P_{cs}$ state in the molecule ansatz.

The present work is organized as the following. After the introduction, the
formulae of the productions of $\Xi_{b}\to P_{cs}(4459)K$ are shown, including
the relevant effective Lagrangians and production amplitudes. Also, we exhibit
our numerical estimation and some discussions on our results. Final, we give a
short summary.

##  II The productions of $\Xi_{b}\to P_{cs}K$

|  |
---|---|---
$(a)$ | $(b)$ | $(c)$
Figure 1: Diagrams contributing to $\Xi_{b}\rightarrow P_{cs}K$ at the quark
level (diagram(a)) and hadron level (diagrams(b)-(c)).

The quark components of $\Xi_{b}$ are $bsd$, while the quarks in the final
states are $s\bar{u}(K)csd(\Xi_{c})\bar{c}u(\bar{D}^{\ast})$. At the quark
level, the process $\Xi_{b}\to P_{cs}K$ (Here and after, $P_{cs}$ refers to
$P_{cs}(4459)$.) occurs via the following subprocess from the phenomenological
point of view: (1) The bottom quark transits to charm quark by emitting a
$W^{-}$ boson which couples to $\bar{c}s$; (2) the $\bar{c}s$ and the
$u\bar{u}$ pair created from vacuum transits to $K^{-}$ and $\bar{D}^{\ast}$,
then $\bar{D}^{\ast}$ and $\Xi_{c}$ form a bound state, i.e., $P_{cs}$. For
simplicity, $H_{W}$ and $H_{T}$ are used to denote the first and the second
subprocess respectively. The production of $P_{cs}$ can be written as $\langle
KP_{cs}|H|\Xi_{b}\rangle=\langle KP_{cs}|H_{W}H_{T}|\Xi_{b}\rangle$. The
direct estimations of this production process at quark level are much more
difficult. In the present work, we qualitatively estimate this process at
hadron level by inserting a complete basis formed by a meson and a baryon
between $H_{W}$ and $H_{T}$, then the production process of $P_{cs}$ becomes,

$\displaystyle\langle KP_{cs}|H|\Xi_{b}\rangle=\sum_{B,M}\langle
KP_{cs}|H_{T}|BM\rangle\langle BM|H_{W}|\Xi_{b}\rangle.$ (3)

In principle, all the possible basis that can connect the initial $\Xi_{b}$
and $KP_{cs}$ should be considered. In the molecular scenario, the molecule
should couple to its constituent hadrons more strongly than to other hadrons.
Moreover, the loops composed by the ground states of the baryons and mesons
are regarded as the major contributions as indicated in Refs. Colangelo:2002mj
; Colangelo:2003sa ; Xiao:2016hoa ; Chen:2016byt ; Xiao:2018kfx ; Chen:2015igx
. With such an assumption, the dominant diagrams contributing to $\Xi_{b}\to
P_{cs}K$ at hadron level are presented in diagrams (b) and (c) in Fig. 1.

###  II.1 Effective Lagrangian

In the present work, the diagrams at the hadron level are estimated by an
effective Lagrangian approach. In the molecular scenario, the $P_{cs}$ is
considered to be molecular state composed of $\bar{D}^{\ast}\Xi_{c}$ with
$J^{P}=1/2^{-}$ or $3/2^{-}$ and the simplest coupling of $P_{cs}$ with its
components are Zou:2002yy

$\displaystyle\mathcal{L}^{1/2^{-}}_{p_{cs}\Xi_{c}\bar{D}^{\ast}}$
$\displaystyle=$ $\displaystyle g_{p_{cs}}^{1/2^{-}}\
\bar{\Xi}_{c}\gamma_{5}\left(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m^{2}}\right)\gamma^{\nu}P_{cs}\bar{D}^{\ast\mu},$
$\displaystyle\mathcal{L}^{3/2^{-}}_{p_{cs}\Xi_{c}\bar{D}^{\ast}}$
$\displaystyle=$ $\displaystyle g_{p_{cs}}^{3/2^{-}}\
\bar{\Xi}_{c}P_{cs\mu}\bar{D}^{\ast\mu},$

where $\mathcal{L}^{1/2^{-}}_{p_{cs}\Xi_{c}\bar{D}^{\ast}}$ and
$\mathcal{L}^{3/2^{-}}_{p_{cs}\Xi_{c}\bar{D}^{\ast}}$ denote the Lagrangians
for $J^{P}=1/2^{-}$ and $3/2^{-}$ respectively; $p$ and $m$ are the momentum
and mass of $P_{cs}$, respectively.

The effective Lagrangians relevant to
$\Xi_{b}\rightarrow\Xi_{c}D^{(\ast)}_{s}$ are Cheng:1996cs

$\displaystyle\mathcal{L}_{\Xi_{b}\Xi_{c}D_{s}}$ $\displaystyle=$
$\displaystyle i\bar{\Xi}_{c}\left(A+B\gamma_{5}\right)\Xi_{b}D_{s},$
$\displaystyle\mathcal{L}_{\Xi_{b}\Xi_{c}D^{\ast}_{s}}$ $\displaystyle=$
$\displaystyle
D^{\ast\mu}_{s}\bar{\Xi}_{c}\left(A_{1}\gamma_{\mu}\gamma_{5}+A_{2}p_{2\mu}\gamma_{5}+B_{1}\gamma_{\mu}+B_{2}p_{2\mu}\right)\Xi_{b},$

where $p_{2}$ is the momentum of $\Xi_{c}$. $A,\ B$, $A_{1},\ A_{2},\ B_{1}$
and $B_{2}$ are the combinations of the form factors $f_{1}$, $f_{2}$, $g_{1}$
and $g_{2}$, which are Cheng:1996cs

$\displaystyle A$ $\displaystyle=$ $\displaystyle\lambda
a_{1}f_{D_{s}}\left(m-m_{2}\right)f_{1},$ $\displaystyle B$ $\displaystyle=$
$\displaystyle\lambda a_{1}f_{D_{s}}\left(m+m_{2}\right)g_{1},$ $\displaystyle
A_{1}$ $\displaystyle=$ $\displaystyle-\lambda
a_{1}f_{D^{\ast}_{s}}m_{1}\left[g_{1}+g_{2}\left(m-m_{2}\right)\right],$
$\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle-2\lambda
a_{1}f_{D^{\ast}_{s}}m_{1}g_{2},$ $\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\lambda
a_{1}f_{D^{\ast}_{s}}m_{1}\left[f_{1}-f_{2}\left(m+m_{2}\right)\right],$
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle 2\lambda
a_{1}f_{D^{\ast}_{s}}m_{1}f_{2},$ (6)

with $\lambda=\frac{G_{F}}{\sqrt{2}}V_{cb}V^{\ast}_{cs}$, and $a_{1}=1.07$
Li:2012cfa . Here $m$, $m_{1}$ and $m_{2}$ are the masses of $\Xi_{b}$,
$D^{(\ast)}_{s}$ and $\Xi_{c}$ respectively; $f^{(\ast)}_{D_{s}}$ is the decay
constant of a charmed strange meson, which is estimated by a twisted-mass
Lattice QCD Carrasco:2014poa . The form factors $f_{i}$ and $g_{i}$ (i=1,2)
will be discussed in the following.

Following Refs. Lin:1999ad ; Oh:2000qr  and starting with SU(4) Lagrangian for
pseudoscalar and vector meson, we can construct the effective Lagrangian
relevant for $D_{s}^{(\ast)}D^{\ast}K$,

$\displaystyle\mathcal{L}_{KD_{s}D^{\ast}}$ $\displaystyle=$ $\displaystyle
ig_{KD_{s}D^{\ast}}D^{\ast\mu}\left[\bar{D}_{s}\partial_{\mu}K-\left(\partial_{\mu}\bar{D}_{s}\right)K\right]+H.c.,$
$\displaystyle\mathcal{L}_{KD^{\ast}_{s}D^{\ast}}$ $\displaystyle=$
$\displaystyle-
g_{KD^{\ast}_{s}D^{\ast}}\varepsilon^{\mu\nu\alpha\beta}\left(\partial_{\mu}\bar{D}^{\ast}_{\nu}\partial_{\alpha}D^{\ast}_{s\beta}\bar{K}+\partial_{\mu}D^{\ast}_{\nu}\partial_{\alpha}\bar{D}^{\ast}_{s\beta}K\right),$

where the coupling constants are $g_{KD_{s}D^{\ast}}=5.0$ and
$g_{KD^{\ast}_{s}D^{\ast}}=7.0\,\mathrm{GeV}^{-1}$ Lin:1999ad ; Oh:2000qr .

###  II.2 Decay Amplitude

With the above effective Lagrangians, we can obtain the amplitudes
corresponding to the diagrams (b) and (c) in Fig. 1. With the assumption that
the $J^{P}$ quantum numbers of $P_{cs}(4459)$ are $\frac{1}{2}^{-}$, the decay
amplitudes could be derived as follows:

$\displaystyle\mathcal{M}_{b}^{1/2}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\Big{[}g_{p_{cs}}^{1/2^{-}}\bar{u}(p_{4})\gamma^{\nu}\gamma_{5}\left(g_{\mu\nu}-\frac{p_{4\mu}p_{4\nu}}{m^{2}_{4}}\right)\Big{]}$
$\displaystyle\times\Big{(}p_{2}\\!\\!\\!\\!\\!/+m_{2}\Big{)}\Big{[}i(A+B\gamma_{5})u(p)\Big{]}\Big{[}ig_{KD^{\ast}D_{s}}(ip_{3\alpha}+ip_{1\alpha})\Big{]}$
$\displaystyle\times\frac{1}{p^{2}_{1}-m^{2}_{1}}\frac{1}{p^{2}_{2}-m^{2}_{2}}\frac{-g^{\mu\alpha}+q^{\mu}q^{\alpha}/m^{2}_{E}}{q^{2}-m^{2}_{E}}\mathcal{F}(q^{2},m_{E}^{2}).$
$\displaystyle\mathcal{M}_{c}^{1/2}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\Big{[}g_{p_{cs}}^{1/2^{-}}\bar{u}(p_{4})\gamma^{\nu}\gamma_{5}(g_{\mu\nu}-\frac{p_{4\mu}p_{4\nu}}{m^{2}_{4}})\Big{]}$
$\displaystyle\times\Big{(}p_{2}\\!\\!\\!\\!\\!/+m_{2}\Big{)}\Big{[}\Big{(}A_{1}\gamma_{\alpha}\gamma_{5}+A_{2}p_{2\alpha}\gamma_{5}+B_{1}\gamma_{\alpha}+B_{2}p_{2\alpha}\Big{)}$
$\displaystyle\times
u(p)\Big{]}\Big{[}-g_{KD^{\ast}D^{\ast}_{s}}\varepsilon_{\rho\sigma\tau\lambda}iq^{\rho}(-i)p^{\tau}_{1}\Big{]}\frac{-g^{\alpha\lambda}+p_{1}^{\alpha}p_{1}^{\lambda}/m^{2}_{1}}{p^{2}_{1}-m^{2}_{1}}$
$\displaystyle\times\frac{1}{p^{2}_{2}-m^{2}_{2}}\frac{-g^{\mu\sigma}+q^{\mu}q^{\sigma}/m^{2}_{E}}{q^{2}-m^{2}_{E}}\mathcal{F}(q^{2},m_{E}^{2}).$

In the same way, with the assumption that the $J^{P}$ quantum numbers of
$P_{cs}$ are $3/2^{-}$, the amplitudes corresponding to diagrams (b) and (c)
in Fig. 1 could be derived as follows:

$\displaystyle\mathcal{M}_{b}^{3/2}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\Big{[}g_{p_{cs}}^{3/2^{-}}\bar{u}_{\mu}(p_{4})\Big{]}\Big{(}p_{2}\\!\\!\\!\\!\\!/+m_{2}\Big{)}\Big{[}i(A+B\gamma_{5})u(p)\Big{]}$
$\displaystyle\times\Big{[}ig_{KD^{\ast}D_{s}}(ip_{3\nu}+ip_{1\nu})\Big{]}\frac{1}{p^{2}_{1}-m^{2}_{1}}\frac{1}{p^{2}_{2}-m^{2}_{2}}$
$\displaystyle\times\frac{-g^{\mu\nu}+q^{\mu}q^{\nu}/m^{2}_{E}}{q^{2}-m^{2}_{E}}\mathcal{F}(q^{2},m_{E}^{2}).$
$\displaystyle\mathcal{M}_{c}^{3/2}$ $\displaystyle=$ $\displaystyle
i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\Big{[}g_{p_{cs}}^{3/2^{-}}\bar{u}_{\mu}(p_{4})\Big{]}\Big{(}p_{2}\\!\\!\\!\\!\\!/+m_{2}\Big{)}$
$\displaystyle\times\Big{[}(A_{1}\gamma_{\nu}\gamma_{5}+A_{2}p_{2\nu}\gamma_{5}+B_{1}\gamma_{\nu}+B_{2}p_{2\nu})u(p)\Big{]}$
$\displaystyle\times\Big{[}-g_{KD^{\ast}D^{\ast}_{s}}\varepsilon_{\rho\sigma\tau\lambda}iq^{\rho}(-i)p^{\tau}_{1}\Big{]}\frac{-g^{\nu\lambda}+p_{1}^{\nu}p_{1}^{\lambda}/m^{2}_{1}}{p^{2}_{1}-m^{2}_{1}}$
$\displaystyle\times\frac{1}{p^{2}_{2}-m^{2}_{2}}\frac{-g^{\mu\sigma}+q^{\mu}q^{\sigma}/m^{2}_{E}}{q^{2}-m^{2}_{E}}\mathcal{F}(q^{2},m_{E}^{2}).$

In order to depict the structure and off shell effects, a form factor in
monopole form is introduced Cheng:2004ru ; Tornqvist:1993vu ; Tornqvist:1993ng
; Locher:1993cc ; Li:1996yn ,

$\displaystyle\mathcal{F}(q^{2},m^{2})=\frac{m^{2}-\Lambda_{E}^{2}}{q^{2}-\Lambda_{E}^{2}},$
(10)

where the parameter $\Lambda_{E}$ is reparameterized as
$\Lambda_{E}=m_{D^{\ast}}+\alpha\Lambda_{QCD}$ with $\Lambda_{QCD}=220$ MeV;
and $m_{D^{\ast}}$ is the mass of the exchanged $D^{\ast}$ meson. The model
parameter $\alpha$ should be of the order of unity Tornqvist:1993vu ;
Tornqvist:1993ng ; Locher:1993cc ; Li:1996yn . However, its precise value
cannot be estimated by the first principle. In practice, the value of $\alpha$
is usually determined by comparing theoretical estimations with the
corresponding experimental measurements.

With the above amplitudes, the partial width of $\Xi_{b}\rightarrow P_{cs}K$
could be estimated by

$\displaystyle\Gamma_{\Xi_{b}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{1}{8\pi}\frac{|\vec{p}|}{m^{2}}\overline{\big{|}\mathcal{M}\big{|}^{2}},$
(11)

where the factor $1/2$ results from the average of $\Xi_{b}$ spin and
$\vec{p}$ is the momentum of $P_{cs}$ or $K$ in the rest frame of $\Xi_{b}$.
The overline indicates the sum over the spins of final states.

##  III Numerical Results and discussion

###  III.1 Coupling Constants and Transition Form Factors

The coupling constants relevant to the molecular states can be estimated by
the compositeness condition with the assumption that the observed $P_{cs}$ is
in a molecule state which gives Xiao:2020Pre

$\displaystyle g^{1/2^{-}}_{P_{cs}}=1.61^{+0.13}_{-0.08},\ \ \
\,g^{3/2^{-}}_{P_{cs}}=2.82^{+0.20}_{-0.14},$ (12)

where the center values are estimated with the model parameter $\Lambda=1.0\
\mathrm{GeV}$, while the uncertainties of the coupling constants are obtained
by varying the parameter $\Lambda$ from $0.8\ \mathrm{GeV}$ to $1.2\
\mathrm{GeV}$. The parameter $\Lambda$ in Ref.Xiao:2020Pre  comes from an
exponent form of form factor, i.e. $\mathrm{exp}(p^{2}/\Lambda^{2})$. It
depicts the componential distribution in the molecular states, and it also
plays the role of removing the ultraviolet divergences Xiao:2019mvs ;
Chen:2015igx .

As an $S$-wave shallow bound state, the coupling constants of $P_{cs}$ and its
components $\Xi_{c}\bar{D}^{\ast}$ could be estimated under non-relativistic
conditions Weinberg:1965zz ; Baru:2003qq

$\displaystyle
g^{2}=\frac{4\pi}{4m_{0}m_{2}}\frac{(m_{1}+m_{2})^{5/2}}{(m_{1}m_{2})^{1/2}}\sqrt{32E_{b}},$
(13)

where $m_{0}$, $m_{1}$ and $m_{2}$ are the masses of $P_{cs}$, $D^{\ast}$ and
$\Xi_{c}$ respectively, and $E_{b}=m_{1}+m_{2}-m_{0}$ is the binding energy of
the $S-$wave shallow bound state. With the measured mass of $P_{cs}$, the
coupling constant is estimated to be $g=2.00$, which is very close to the
estimation under compositeness conditions. Moreover, the coupling constants
are also similar to those of $g_{P_{c}(4440)\bar{D}^{\ast}\Sigma_{c}}$ and
$g_{P_{c}(4457)\bar{D}^{\ast}\Sigma_{c}}$ Xiao:2019mvs ; Wu:2019rog , which is
consistent with the expectation of $\mathrm{SU}(3)$ symmetry.

The transition form factors of $\Xi_{b}\to\Xi_{c}$ could be parameterized in
the form of Cheng:1996cs

$\displaystyle f(Q^{2})=\frac{f(0)}{(1-Q^{2}/m^{2}_{V})^{2}},\ \ \
g(Q^{2})=\frac{g(0)}{(1-Q^{2}/m^{2}_{A})^{2}},$ (14)

where $m_{V}(m_{A})$ is the pole mass of the vector (axial-vector) meson. For
the $b\rightarrow c$ transition, the pole mass are $m_{V}=6.34$ GeV and
$m_{A}=6.73$ GeV Cheng:1996cs . In Table. 1, the parameters related to the
transition form factors of $\Xi_{b}\to\Xi_{c}$ Cheng:1996cs  are collected. It
should be noticed that considering the form factor in Eq. (10) and the
transition form factors, the loop integrals contain no ultraviolet
divergences.

Table 1: The values of the parameters $f_{i}(0)$ and $g_{i}(0)$ in the form factors of $\Xi_{b}\to\Xi_{c}$ transitionCheng:1996cs . Parameter |  Value |  Parameter |  Value
---|---|---|---
$f_{1}(0)$ | $f_{2}(0)m_{\Xi_{b}}$ | $g_{1}(0)$ | $g_{2}(0)m_{\Xi_{b}}$
0.533 | -0.124 | 0.580 | -0.019

###  III.2 Partial Width of $\Xi_{b}\to P_{cs}K$

Figure 2: The branching fractions of $\Xi_{b}\rightarrow P_{cs}K$ with
different $J^{P}$ assignments (left panel) and the ratio of the branching
fractions (right panel) depending on the parameter $\alpha$.

To date, the $J^{P}$ quantum numbers of $P_{cs}$ have not been determined by
LHCb collaboration. In the $S$-wave $\bar{D}^{\ast}\Xi_{c}$ molecular
scenario, the possible assignments can be $J^{P}=1/2^{-}$ and $3/2^{-}$. In
the present work, the branching ratios of $\Xi_{b}\to P_{cs}K$ with two
different $J^{P}$ assignments are estimated. The model parameter $\alpha$
varies from 0.8 to 1.2, which is the same as the one of $P_{c}$ production
Wu:2019rog . In this $\alpha$ range, the branching ratio is estimated to be
$(0.84\sim 1.31)\times 10^{-4}$ for $J^{P}=1/2^{-}$ and $(8.85\sim 14.1)\times
10^{-4}$ for $J^{P}=3/2^{-}$. It indicates that the branching ratio of
$\Xi_{b}\to P_{cs}K$ should be of the order of $10^{-4}$. Moreover, the
$\alpha$ dependences for two different assignments are found to be very
similar, the ratio of the branching fractions with different $J^{P}$
assignments is presented in the right panel of Fig. 2. The branching ratio
with $J^{P}=3/2^{-}$ is about one order of magnitudes larger than that with
$J^{P}=1/2^{-}$. Our estimation indicates that the ratio is around 10.6, and
its dependence on the model parameter is weak. Thus, a molecular state of
$\bar{D}^{\ast}\Xi_{c}$ with $J^{P}=3/2^{-}$ is more simply produced from
$\Xi_{b}$ decay.

In Ref. Wu:2019rog , the authors argued that the production of $P_{c}$ states
occurs via $\Lambda_{b}\to\Sigma_{c}D_{s}^{(\ast)}\to P_{c}K$. However, the
weak transition of $\Lambda_{b}\to\Sigma_{c}$ should be suppressed with a
suppression factor $R=0.09$ due to isospin violation and spin flip of the
light di-quark. In Refs. Liu:2015fea ; Du:2021fmf , the authors indicated that
there can be important contribution from other color suppressed diagrams. The
present estimation indicate that the branching fraction of $\Xi_{b}\to
P_{cs}K$ occur via a very similar process but without isospin violation and
spin flip of the light di-quark. The branching ratio is of order of $10^{-4}$,
which is about two orders larger than that of $\Lambda_{b}\to P_{c}K$.

Moreover, the experimental analysis of LHCb collaboration indicated that the
fit fraction of $P_{cs}$ is $(2.7^{+1.9+0.7}_{-0.6-1.3})\%$, which indicates
Aaij:2020gdg

$\displaystyle\frac{\mathcal{B}\left(\Xi_{b}\to P_{cs}K\to J/\psi\Lambda
K\right)}{\mathcal{B}\left(\Xi_{b}\to J/\psi\Lambda
K\right)}=\left(2.7^{+1.9+0.7}_{-0.6-1.3}\right)\%,$ (15)

while the branching fraction of $\Xi_{b}\to J/\psi\Lambda K$ is measured to be
Aaij:2017bef

$\displaystyle\frac{f_{\Xi_{b}}}{f_{\Lambda_{b}}}\frac{\mathcal{B}(\Xi_{b}\to
J/\psi\Lambda K)}{\mathcal{B}(\Lambda_{b}\to J/\psi\Lambda)}=\left(4.19\pm
0.29\pm 0.15\right)\times 10^{-2},$ (16)

with $f_{\Xi_{b}}$ and $f_{\Lambda_{b}}$ being the fragmentation fractions of
$b\to\Xi_{b}$ and $b\to\Lambda_{b}$ transitions respectively. The branching
fraction of $\Lambda_{b}\to J/\psi\Lambda$ has been measured by CDF
collaboration, which is Abe:1996tr

$\displaystyle\mathcal{B}(\Lambda_{b}\to J/\psi\Lambda)=(3.7\pm 1.7\pm
0.7)\times 10^{-4}.$ (17)

This measured branching fraction is consistent with the theoretical estimation
in Ref. Cheng:1995fe , which is $2.1\times 10^{-4}$. As for the fragmentation
fractions, the theoretical estimations are model dependent Cheng:2015ckx ;
Hsiao:2015txa ; Voloshin:2015xxa ; Jiang:2018iqa ; Wang:2019dls . Using the
data sample at $\sqrt{s}=7,\ 8$TeV, the recent experimental analysis from the
LHCb collaboration shows that the ratio of the fragmentation fractions is
Aaij:2019ezy ,

$\displaystyle\frac{f_{\Xi_{b}}}{f_{\Lambda_{b}}}=(6.7\pm 0.5\pm 0.5\pm
2.0)\times 10^{-2},$ (18)

which is very consistent with the estimations in Refs. Jiang:2018iqa ;
Wang:2019dls . With the above experimental results, the branching ratio of
$\Xi_{b}\to P_{cs}K\to J/\psi\Lambda K$ could be obtained as follows:

$\displaystyle\mathcal{B}(\Xi_{b}\to P_{cs}K\to J/\psi\Lambda
K)=\left(6.25^{+5.98}_{-4.98}\right)\times 10^{-6}.$ (19)

Considering the small width of $P_{cs}$, an approximation of
$\mathcal{B}(\Xi_{b}\to P_{cs}K)\times\mathcal{B}(P_{cs}\to
J/\psi\Lambda)\simeq\mathcal{B}(\Xi_{b}\to P_{cs}K\to J/\psi\Lambda K)$ could
be obtained. The present estimations indicate the branching ratio of
$\Xi_{b}\to P_{cs}K$ is of the order of $10^{-4}$, thus the expected branching
ratio of $P_{cs}\to J/\psi\Lambda$ should be of the order of $10^{-2}\sim
10^{-1}$. In Ref. Shen:2019evi , bound states of $\bar{D}^{\ast}\Xi_{c}$ are
found and the branching fraction of $J/\psi\Lambda$ mode is estimated to be of
the order of $10^{-2}$ for $J^{P}=1/2^{-}$ and $10^{-1}$ for $J^{P}=3/2^{-}$,
which is consistent with the above analysis. In addition, in the estimations
of Ref. Shen:2019evi , the dominant decay modes of $P_{cs}$ state are
$\rho\Sigma$ and $K^{\ast}\Xi$.

Before the end of this work, it is worth mentioning that the two resonances
have been fitted by the LHCb collaboration Aaij:2020gdg . However, the
analysis indicate that the two-peak hypothesis cannot be confirmed due to
limit of the current data. In addition, in the two resonance fit scheme, the
masses of the resonances are fitted to be $4454.9\pm 2.7$ MeV and $4467.8\pm
3.7$ MeV, respectively, which are consistent with that of $P_{cs}(4459)$
within uncertainty even when the systematically uncertainties are excluded in
the two-resonance scenario. Experimentally, more precise data is expected in
future, which would provide more information on the charmed-strange pentaquark
states.

##  IV Summary

In summary, we have presented an investigation on the production of the newly
observed $P_{cs}$ state from $\Xi_{b}$ decay in a $\bar{D}^{\ast}\Xi_{c}$
molecular scenario. By analyzing the $P_{cs}$ production at quark level, the
dominant contributions to $P_{cs}$ productions are found to be the charmed-
strange hadron loops, where the initial $\Xi_{b}$ couples to
$\Xi_{c}\bar{D}^{(\ast)}_{s}$ by weak interaction, and then the
$\bar{D}^{(\ast)}_{s}$ transits into $\bar{D}^{\ast}$ via kaon emission, and
the recoil $\bar{D}^{\ast}$ and $\Xi_{c}$ form the molecular state, i.e.,
$P_{cs}$.

The production process of $\Xi_{b}\rightarrow P_{cs}K$ is estimated at hadron
level with an effective Lagrangian approach. Our estimation indicates that the
branching fraction of $\Xi_{b}\to P_{cs}K$ is of the order of $10^{-4}$ for
both $J^{P}=1/2^{-}$ and $3/2^{-}$. Moreover, we find the ratio of the
branching fractions with $J^{P}=3/2^{-}$ and $1/2^{-}$ is about 10.6 and
almost independent of the model parameter. Our analysis indicates that the
estimated productions fraction in the present work are consistent with the
experimental measurements and the theoretical estimations in the literature
available.

##  V ACKNOWLEDGMENTS

D. Y. Chen would like to thank Professor Wei Wang and Professor Fu-Sheng Yu
for useful discussions. This work is supported by the National Natural Science
Foundation of China under the Grant No. 11775050. Qi Wu was also supported by
the Scientific Research Foundation of Graduate School of Southeast University
(Grants no. YBPY2028).

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