# Productions of $D^{*}_{s0}(2317)$ and $D_{s1}(2460)$ in $B_{(s)}$ and
$\Lambda_{b}(\Xi_{b})$ decays

Ming-Zhu Liu  Frontiers Science Center for Rare isotopes, Lanzhou University,
Lanzhou 730000, China  School of Nuclear Science and Technology, Lanzhou
University, Lanzhou 730000, China  Xi-Zhe Ling  Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  Li-Sheng Geng
<EMAIL_ADDRESS>School of Physics, Beihang University, Beijing
102206, China  Beijing Key Laboratory of Advanced Nuclear Materials and
Physics, Beihang University, Beijing 102206, China  Peng Huanwu Collaborative
Center for Research and Education, Beihang University, Beijing 100191, China
Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou 516000, China

###### Abstract

Recent studies show that $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ contain
large molecular components. In this work, we employ the naive factorization
approach to calculate the production rates of $D_{s0}^{\ast}(2317)$ and
$D_{s1}(2460)$ as hadronic molecules in $B_{(s)}$ and $\Lambda_{b}(\Xi_{b})$
decays, where their decay constants are estimated in the effective Lagrangian
approach. With the so-obtained decay constants $f_{D_{s0}^{\ast}(2317)}$ and
$f_{D_{s1}(2460)}$, we calculate the branching fractions of the $b$-meson
decays $B_{(s)}\to\bar{D}_{(s)}^{(*)}D_{s0}^{*}$ and
$B_{(s)}\to\bar{D}_{(s)}^{(*)}D_{s1}$ and the $b$-baryon decays
$\Lambda_{b}(\Xi_{b})\to\Lambda_{c}(\Xi_{c})D_{s0}^{*}$ and
$\Lambda_{b}(\Xi_{b})\to\Lambda_{c}(\Xi_{c})D_{s1}$. Our results show that the
production rates of $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ in the $B_{s}$,
$\Lambda_{b}$ and $\Xi_{b}$ decays are rather large that future experiments
could observe them. In particular, we demonstrate that one can extract the
decay constants of hadronic molecules via the triangle mechanism because of
the equivalence of the triangle mechanism to the tree diagram established in
calculating the decays $B\to\bar{D}^{(*)}D_{s0}^{\ast}(2317)$ and
$B\to\bar{D}^{(*)}D_{s1}(2460)$.

##  I Introduction

In 2003, the BarBar Collaboration discovered $D_{s0}^{*}(2317)$ in the
$D_{s}^{+}\pi^{0}$ mass distribution in the $e^{+}e^{-}$ annihilation process
Aubert  _et al._ (2003), which was later confirmed by the CLEO Collaboration
Besson  _et al._ (2003) and Belle Collaboration Mikami  _et al._ (2004) in the
same process. Moreover, the BESIII Collaboration observed the
$D_{s0}^{*}(2317)$ in the process of $e^{+}e^{-}\to
D_{s}^{*+}D_{s0}^{*-}(2317)$ Ablikim  _et al._ (2018). In addition to the
above inclusive processes, $D_{s0}^{*}(2317)$ was also observed in the
exclusive process of the $B$ decay by the Belle Collaboration Krokovny  _et
al._ (2003) and BarBar Collaboration Aubert  _et al._ (2004a). $D_{s1}(2460)$
as the heavy quark spin symmetry (HQSS) partner of $D_{s0}^{*}(2317)$ was
first discovered in the $D_{s}^{\ast+}\pi^{0}$ mass distribution by the CLEO
Collaboration Besson  _et al._ (2003), and then confirmed by several other
experiments Mikami  _et al._ (2004); Krokovny  _et al._ (2003); Aubert  _et
al._ (2004b). Treating $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ as conventional
$P$-wave $c\bar{s}$ mesons, the masses obtained in the Goldfrey-Isgur (GI)
model are larger than the experimental ones by 140 MeV and 100 MeV Godfrey and
Isgur (1985), which have motivated extensive discussions on their nature.

By analyzing their masses, several interpretations were proposed for the
internal structure of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$. In Ref. Song  _et
al._ (2015), the authors found that the masses of $D_{s0}^{*}(2317)$ and
$D_{s1}(2460)$ still deviate from the experimental data, even adding the
screen potential to the conventional quark model. However, as the $DK$ and
$D^{*}K$ components were embodied into the conventional quark model, the mass
puzzle of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ is resolved Albaladejo  _et
al._ (2018); Yang  _et al._ (2022); Luo  _et al._ (2021), which indicates that
the $D^{(*)}K$ components play an important role in forming $D_{s0}^{*}(2317)$
and $D_{s1}(2460)$. Therefore, $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ are
proposed to be hadronic molecules of $DK$ and $D^{*}K$ to explain their
masses, especially their mass splitting Barnes  _et al._ (2003); Gamermann
_et al._ (2007); Guo  _et al._ (2006); Xie  _et al._ (2010); Cleven  _et al._
(2011); Guo  _et al._ (2015); Wu  _et al._ (2019). It should be noted that in
the lattice QCD simulation of the $DK$ interaction, a bound state below the
$DK$ mass threshold was identified Liu  _et al._ (2013); Mohler  _et al._
(2013); Lang  _et al._ (2014); Bali  _et al._ (2017); Alexandrou  _et al._
(2020). Furthermore, with the $D^{(*)}K$ potentials supplemented by the
$c\bar{s}$ core couplings to the $D^{(*)}K$ components, $D_{s0}^{*}(2317)$ and
$D_{s1}(2460)$ can be dynamically generated Weinberg (1965); Martínez Torres
_et al._ (2015); Albaladejo  _et al._ (2016); Song  _et al._ (2022),
indicating that the $D^{(*)}K$ molecular components account for a large
proportion of their wave functions in terms of the Weinberg compositeness rule
$1-Z$ Weinberg (1965). Studying the masses of $D_{s0}^{*}(2317)$ and
$D_{s1}(2460)$, one can conclude that they contain both $D^{(*)}K$ molecular
components and a $c\bar{s}$ core. The natural next step forward is to study
their decays.

According to the review of particle physics (RPP) Workman  _et al._ (2022),
the $D_{s0}^{*}(2317)^{+}$ dominantly decays to $D_{s}^{+}\pi^{0}$, which
means that $D_{s0}^{*}(2317)^{+}$ must be narrow since the decay of
$D_{s0}^{*}(2317)^{+}\to D_{s}^{+}\pi^{0}$ breaks isospin. The dominant decay
of $D_{s1}(2460)$ into $D_{s}^{*+}\pi^{0}$ and $D_{s}^{+}\gamma$ is
responsible for its narrow width. The narrow widths of $D_{s0}^{*}(2317)$ and
$D_{s1}(2460)$ are quite different from the widths of their SU(3)-flavor
partners $D^{*}_{0}(2300)$ and $D_{1}(2430)$, which reflects the exotic
properties of these excited charmed mesons. In Refs. Fayyazuddin and Riazuddin
(2004); Ishida  _et al._ (2004); Wei  _et al._ (2006); Nielsen (2006); Song
_et al._ (2015), the authors proposed that the decays of $D_{s0}^{*}(2317)$
and $D_{s1}(2460)$ as the $c\bar{s}$ excited states into $D_{s}\pi$ and
$D_{s}^{*}\pi$ proceed via the $\pi-\eta$ mixing, resulting in widths of tens
of keV. Treating $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ as hadronic molecules,
their widths are of the order of $100$ keV Faessler  _et al._ (2007a, b); Fu
_et al._ (2022). Up to now, there are no precise experimental measurements of
the widths of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$, but only their upper
limits of $3.8$ MeV. From the perspective of their widths, one can obtain the
same conclusion as from the studies of their masses regarding the nature of
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$. It is worth noting that a model-
independent method has been proposed to verify the molecular nature of
$D_{s0}^{*}(2317)$ by experimental searches for its three-body counterparts
$DDK$ and $D\bar{D}K$ Wu  _et al._ (2019); Huang  _et al._ (2020); Wu  _et
al._ (2022); Wu and Geng (2023).

The discoveries of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ in the inclusive and
exclusive processes in $e^{+}e^{-}$ collisions triggered a series of
theoretical works to investigate their production mechanism. Assuming
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ as $D^{(*)}K$ hadronic molecules and
$c\bar{s}$ excited states, Wu et al. estimated that their production rates in
$e^{+}e^{-}$ collisions are of the order of $10^{-3}$ Wu and Geng (2023),
consistent with the experimental data Aubert  _et al._ (2006). As for the
exclusive process, Faessler et al. calculated the decays $B\to
D_{s0}^{*}(2317)\bar{D}^{(*)}$ and $B\to D_{s1}(2460)\bar{D}^{(*)}$ assuming
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ as $D^{(*)}K$ molecules Faessler  _et
al._ (2007c). The results are a bit smaller than the experimental data.
Assuming $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ as $c\bar{s}$ excited states,
the decays $B\to D_{s0}^{*}(2317)\bar{D}^{(*)}$ and $B\to
D_{s1}(2460)\bar{D}^{(*)}$ were investigated as well, but the results suffer
from large uncertainties Colangelo  _et al._ (1991); Veseli and Dunietz
(1996); Cheng  _et al._ (2004); Colangelo  _et al._ (2005); Cheng and Chua
(2006); Segovia  _et al._ (2012). Moreover, the productions of
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ in the $\Lambda_{b}$ decays have been
explored Datta  _et al._ (2004), finding that their production rates in the
$\Lambda_{b}$ decays are larger than those in the corresponding $B$ decays.
Recently, the $DK$ femtoscopic correlation function was investigated to
elucidate the nature of $D_{s0}^{*}(2317)$ Liu  _et al._ (2023); Ikeno  _et
al._ (2023), which can be accessed in high energy nucleon-nucleon collisions
in the future Adamczyk  _et al._ (2015); Collaboration  _et al._ (2020).

Until now, the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ have only been observed
in the exclusive process via $B$ decays. In this work, we systematically
explore the productions of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ in $B_{(s)}$
and $\Lambda_{b}(\Xi_{b})$ decays with the factorization ansatz Chau (1983);
Chau and Cheng (1987). Following Ref. Faessler  _et al._ (2007c), we employ
the effective Lagrangian approach to estimate the decay constants of
${D_{s0}^{*}(2317)}$ and ${D_{s1}(2460)}$, which are dynamically generated via
the $DK-D_{s}\eta$ and $D^{*}K-D_{s}^{*}\eta$ coupled-channel potentials
described by the contact-range effective field theory (EFT), and then
calculate the production rates of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ in
$B_{(s)}$ and $\Lambda_{b}(\Xi_{b})$ decays. Another motivation of this work
is to test the universality of the approach that we proposed to calculate the
decay constant of a hadronic molecule via the triangle mechanism Wu  _et al._
(2023). Based on our previous study of the decays $B\to
D_{s0}^{*}(2317)\bar{D}^{(*)}$ and $B\to D_{s1}(2460)\bar{D}^{(*)}$ via the
triangle mechanism, the decay constants of $D_{s0}^{*}(2317)$ and
$D_{s1}(2460)$ can be extracted Liu  _et al._ (2022). The effective Lagrangian
approach in this work can further check the validity of our approachWu  _et
al._ (2023).

This paper is organized as follows. In Sec. II, we introduce the effective
Lagrangian approach to calculate the productions of $D_{s0}^{*}(2317)$ and
$D_{s1}(2460)$ in $B_{(s)}$ and $\Lambda_{b}(\Xi_{b})$ decays and the decay
constants of $f_{D_{s0}^{*}}$ and $f_{D_{s1}}$. Results and discussions are
given in Sec. III, followed by a summary in Sec. IV.

##  II  Theoretical formalism

###  II.1  Effective Lagrangians for nonleptonic Weak decays

In this work, we focus on the productions of $D_{s0}^{*}(2317)$ and
$D_{s1}(2460)$ in $B_{(s)}$ and $\Lambda_{b}(\Xi_{b})$ decays. At quark level,
the Cabibbo-favored decays $B^{+}\to\bar{D}^{(\ast)0}D_{s}^{(\ast)+}$ and
$\Lambda_{b}\to\Lambda_{c}D_{s}^{(\ast)+}$ mainly proceed via the external
$W$-emission mechanism shown in Fig. 1 according to the topological
classification of weak decays Chau and Cheng (1987); Ali  _et al._ (1998,
2007); Li  _et al._ (2012), which the naive factorization approach can well
describe. With the factorization ansatz Bauer  _et al._ (1987), the amplitudes
of the weak decays $B^{+}\to\bar{D}^{(\ast)0}D_{s}^{(\ast)+}$ and
$\Lambda_{b}\to\Lambda_{c}D_{s}^{(\ast)+}$ can be expressed as products of two
current matrix elements

$\displaystyle\mathcal{A}\left(B^{+}\to
D_{s}^{(\ast)+}\bar{D}^{(\ast)0}\right)$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}\left\langle
D_{s}^{(\ast)+}|(s\bar{c})|0\right\rangle\left\langle\bar{D}^{(\ast)0}|(c\bar{b})|B^{+}\right\rangle,$
(1) $\displaystyle\mathcal{A}\left(\Lambda_{b}\to
D_{s}^{(\ast)+}\Lambda_{c}\right)$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}\left\langle
D_{s}^{(\ast)+}|(s\bar{c})|0\right\rangle\left\langle\Lambda_{c}|(c\bar{b})|\Lambda_{b}\right\rangle,$
(2)

where $G_{F}$ is the Fermi constant, $V_{cb}$ and $V_{cs}$ are the Cabibbo-
Kobayashi-Maskawa (CKM) matrix elements, and $a_{1}$ is the effective Wilson
coefficient. In principle, $a_{1}$ is expressed by the Wilson Coefficients of
the QCD-corrected effective weak Hamiltonian, which depends on the
renormalization scale Chau and Cheng (1987); Cheng and Tseng (1993); Cheng and
Chiang (2010). In this work, we parameterize the non-factorization
contributions with the effective Wilson coefficient $a_{1}$, which can be
determined by reproducing relevant experimental data.

Figure 1:  External $W$-emission mechanism for
$B^{+}\to\bar{D}^{(\ast)0}D_{s}^{(\ast)+}$ (a) and
$\Lambda_{b}\to\Lambda_{c}D_{s}^{(\ast)+}$ (b).

The current matrix elements of
$\left\langle\bar{D}^{0}|(c\bar{b})|B^{+}\right\rangle$ and
$\left\langle\bar{D}^{\ast 0}|(c\bar{b})|B^{+}\right\rangle$ describing the
hadronic transitions are parameterized by six form factors Cheng and Chua
(2006)

$\displaystyle\left\langle\bar{D}^{\ast
0}|(c\bar{b})|B^{+}\right\rangle=\epsilon_{\alpha}^{*}\left\\{-g^{\mu\alpha}(m_{\bar{D}^{\ast
0}}+m_{B^{+}})A_{1}\left(q^{2}\right)+P^{\mu}P^{\alpha}\frac{A_{2}\left(q^{2}\right)}{m_{\bar{D}^{\ast
0}}+m_{B^{+}}}\right.+i\varepsilon^{\mu\alpha\beta\gamma}P_{\beta}q_{\gamma}$
(3) $\displaystyle\left.\frac{V\left(q^{2}\right)}{m_{\bar{D}^{\ast
0}}+m_{B^{+}}}+q^{\mu}P^{\alpha}\left[\frac{m_{\bar{D}^{\ast
0}}+m_{B^{+}}}{q^{2}}A_{1}\left(q^{2}\right)-\frac{m_{B^{+}}-m_{\bar{D}^{\ast
0}}}{q^{2}}A_{2}\left(q^{2}\right)-\frac{2m_{\bar{D}^{\ast
0}}}{q^{2}}A_{0}\left(q^{2}\right)\right]\right\\},$
$\displaystyle\left\langle\bar{D}^{0}|(c\bar{b})|B^{+}\right\rangle=\left[P^{\mu}-\frac{m_{B^{+}}^{2}-m_{\bar{D}^{0}}^{2}}{q^{2}}q_{\mu}\right]F_{1}(q^{2})+\frac{m_{B^{+}}^{2}-m_{\bar{D}^{0}}^{2}}{q^{2}}q_{\mu}F_{0}(q^{2}),$
(4)

where the momenta $q=p_{B^{+}}-p_{\bar{D}^{(*)0}}$ and
$P=p_{B^{+}}-p_{\bar{D}^{(*)0}}$, and $F_{1}(q^{2})$, $F_{2}(q^{2})$,
$A_{0}(q^{2})$, $A_{1}(q^{2})$, $A_{2}(q^{2})$, and $V(q^{2})$ are form
factors. The current matrix element
$\langle\Lambda_{c}(p^{\prime})|c\bar{b}|\Lambda_{b}(p)\rangle$ is given by
Gutsche  _et al._ (2018)

$\displaystyle\langle\Lambda_{c}(p^{\prime})|c\bar{b}|\Lambda_{b}(p)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})[f_{1}^{V}(q^{2})\gamma_{\mu}-f_{2}^{V}(q^{2})\frac{i\sigma_{\mu\nu}q^{\nu}}{m}+f_{3}^{V}(q^{2})\frac{q^{\mu}}{m}$
$\displaystyle-(f_{1}^{A}(q^{2})\gamma_{\mu}-f_{2}^{A}(q^{2})\frac{i\sigma_{\mu\nu}q^{\nu}}{m}+f_{3}^{A}(q^{2})\frac{q^{\mu}}{m})\gamma^{5}]u(p),$

where
$\sigma^{\mu\nu}=\frac{i}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$
and $q=p-p^{\prime}$, and $f_{i}^{V/A}(q^{2})$ are form factors. In general,
the form factors are parameterized in the following form:

$\frac{F_{i}(0)}{1-a\,\frac{q^{2}}{m^{2}}+b\,(\frac{q^{2}}{m^{2}})^{2}},$ (6)

where $F_{i}(0)$, $a$, and $b$ are parameters determined in phenomenological
models. In this work, we take these parameters determined in the quark model
Verma (2012); Gutsche  _et al._ (2015); Faustov and Galkin (2018).

The current matrix element $\left\langle D_{s}^{+}|(s\bar{c})|0\right\rangle$
describes the process of creating a $D_{s}^{+}$ meson from the vacuum via the
axial current, which is parameterized by the decay constant $f_{D_{s}^{+}}$
and the momentum of the $D_{s}^{+}$ meson. Following Ref. Cheng  _et al._
(2004), the current matrix elements for the $D_{s}$, $D_{s}^{*}$,
$D_{s0}^{*}(2317)$, and $D_{s1}(2460)$ mesons created from the vacuum are

$\displaystyle\left\langle D_{s}^{+}|(s\bar{c})|0\right\rangle$
$\displaystyle=$ $\displaystyle
if_{D_{s}^{+}}p^{\mu}_{D_{s}^{+}},~{}~{}~{}~{}~{}~{}\left\langle
D_{s}^{\ast+}|(s\bar{c})|0\right\rangle=m_{D_{s}^{\ast+}}f_{D_{s}^{\ast+}}\epsilon_{\mu}^{*},$
(7) $\displaystyle~{}~{}\left\langle D_{s0}^{*+}|(s\bar{c})|0\right\rangle$
$\displaystyle=$ $\displaystyle
f_{D_{s0}^{*+}}p^{\mu}_{D_{s0}^{*+}},~{}~{}~{}~{}~{}\left\langle
D_{s1}^{+}|(s\bar{c})|0\right\rangle=m_{D_{s1}^{+}}f_{D_{s1}^{+}}\epsilon_{\mu}^{*}.$

The decay constants of $D_{s}$ and $D_{s}^{*}$ as $c\bar{s}$ ground states are
determined to be $f_{D_{s}}=250$ MeV and $f_{D_{s}^{*}}=272$ MeV Verma (2012).
Due to the exotic properties of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$, the
estimations of the decay constants $f_{D_{s0}^{*}}$ and $f_{D_{s1}}$ are quite
uncertain. In this work, we estimate the values of $f_{D_{s0}^{*}}$ and
$f_{D_{s1}}$ in the molecular picture. In addition, assuming SU(3)-flavor
symmetry, the $B\to\bar{D}^{(*)}$ and $\Lambda_{b}\to\Lambda_{c}$ transitions
can be related with the $B_{s}\to\bar{D}_{s}^{(*)}$ and $\Xi_{b}\to\Xi_{c}$
transitions, and the production mechanism of $D_{s0}^{*}(2317)$ and
$D_{s1}(2460)$ in the $B_{s}$ and $\Xi_{b}$ decays are similar to those in the
$B$ and $\Lambda_{b}$ decays as illustrated in Fig. 1. In the following, we
only present the amplitudes for the decays
$B\to\bar{D}^{(\ast)}D_{s}^{(\ast)}$ and
$\Lambda_{b}\rightarrow\Lambda_{c}\bar{D}_{s}^{(*)}$, and the amplitudes for
the other decays have similar expressions.

With the above effective Lagrangian, we obtain the amplitudes for the decays
$B\to\bar{D}^{(\ast)}D_{s}^{(\ast)}$:

$\displaystyle\mathcal{A}(B\to D_{s}\bar{D}^{\ast})$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}f_{D_{s}}\\{-q_{1}\cdot\varepsilon(q_{2})(m_{\bar{D}^{\ast}}+m_{B})A_{1}\left(q_{1}^{2}\right)$
(8)
$\displaystyle+(k_{0}+q_{2})\cdot\varepsilon(q_{2})q_{1}\cdot(k_{0}+q_{2})\frac{A_{2}\left(q_{1}^{2}\right)}{m_{\bar{D}^{\ast}}+m_{B}}+(k_{0}+q_{2})\cdot\varepsilon(q_{2})$
$\displaystyle[(m_{\bar{D}^{\ast}}+m_{B})A_{1}(q_{1}^{2})-(m_{B}-m_{\bar{D}^{\ast}})A_{2}(q_{1}^{2})-2m_{\bar{D}^{\ast}}A_{0}(q_{1}^{2})]\\},$
$\displaystyle\mathcal{A}(B\to D_{s}\bar{D})$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}^{\prime}f_{D_{s}}(m_{B}^{2}-m_{D}^{2})F_{0}(q_{1}^{2}),$
$\displaystyle\mathcal{A}(B\to D_{s}^{\ast}\bar{D})$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}^{*}m_{D_{s}^{\ast}}f_{D_{s}^{\ast}}(k_{0}+q_{2})\cdot\varepsilon(q_{1})F_{1}(q_{1}^{2}),$
$\displaystyle\mathcal{A}(B\to D_{s}^{\ast}\bar{D}^{\ast})$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}^{*\prime}m_{D_{s}^{\ast}}f_{D_{s}^{\ast}}\varepsilon_{\mu}(q_{1})\left[(-g^{\mu\alpha}(m_{\bar{D}^{\ast}}+m_{B})A_{1}\left(q_{1}^{2}\right)\right.$
$\displaystyle+\left.(k_{0}+q_{2})^{\mu}(k_{0}+q_{2})^{\alpha}\frac{A_{2}\left(q_{1}^{2}\right)}{m_{\bar{D}^{\ast}}+m_{B}}+i\varepsilon^{\mu\alpha\beta\gamma}(k_{0}+q_{2})_{\beta}q_{1\gamma}\frac{V\left(q_{1}^{2}\right)}{m_{\bar{D}^{\ast}}+m_{B}}\right]\varepsilon_{\alpha}(q_{2}).$

The weak decays $\Lambda_{b}\rightarrow\Lambda_{c}\bar{D}_{s}^{(*)}$ can be
characterised by the following Lagrangian Cheng (1997):

$\begin{split}\mathcal{L}_{\Lambda_{b}\Lambda_{c}D_{s}}&=i\bar{\Lambda}_{c}(A+B\gamma_{5})\Lambda_{b}D_{s},\\\
\mathcal{L}_{\Lambda_{b}\Lambda_{c}D_{s}}&=\bar{\Lambda}_{c}(A_{1}\gamma_{\mu}\gamma_{5}+A_{2}\frac{p_{2\mu}}{m}\gamma_{5}+B_{1}\gamma_{\mu}+B_{2}\frac{p_{2\mu}}{m})\Lambda_{b}D_{s}^{*\mu},\end{split}$
(9)

where $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $A$, and $B$ represent the
transition form factors of $\Lambda_{b}$ to $\Lambda_{c}$:

$\begin{split}A&=\lambda
f_{D_{s}}[(m-m_{2})f_{1}^{V}+\frac{m_{1}^{2}}{m}f_{3}^{V}],\quad B=\lambda
f_{D_{s}}[(m+m_{2})f_{1}^{A}-\frac{m_{1}^{2}}{m}f_{3}^{A}],\\\ A_{1}&=-\lambda
f_{D_{s}^{*}}m_{1}[f_{1}^{A}-f_{2}^{A}\frac{m-m_{2}}{m}],\quad\quad
B_{1}=\lambda f_{D_{s}^{*}}m_{1}[f_{1}^{V}+f_{2}^{V}\frac{m+m_{2}}{m}],\\\
A_{2}&=2\lambda
f_{D_{s}^{*}}m_{1}f_{2}^{A},\quad\quad\quad\quad\quad\quad\quad\quad
B_{2}=-2\lambda f_{D_{s}^{*}}m_{1}f_{2}^{V},\end{split}$ (10)

with $\lambda=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}$ and $m,m_{1},m_{2}$
referring to the masses of $\Lambda_{b}$, $\bar{D}_{s}^{(*)}$, and
$\Lambda_{c}$.

With the amplitudes for the weak decays given above, one can compute the
corresponding partial decay widths

$\displaystyle\Gamma=\frac{1}{2J+1}\frac{1}{8\pi}\frac{|\vec{p}|}{m^{2}}{|\overline{M}|}^{2},$
(11)

where $J$ is the total angular momentum of the initial state and $|\vec{p}|$
is the momentum of either final state in the rest frame of the initial state.

###  II.2  Decay Constants

Figure 2:  Feynman diagrams for the $W$ boson transiting to $D_{s0}^{*}(2317)$
in the $DK$ and $D_{s}\eta$ molecular picture.

The decay constants $f_{D_{s0}^{*}}$ and $f_{D_{s1}}$ are defined in Eq.(7).
To obtain the values of $f_{D_{s0}^{*}}$ and $f_{D_{s1}}$, one usually
constructs the amplitudes for the $D_{s0}^{*}$ and $D_{s1}$ created from the
vacuum and then extracts the coefficients of $p^{\mu}_{D_{s0}^{*+}}$ and
$m_{D_{s1}^{+}}\epsilon_{\mu}^{*}$ Colangelo  _et al._ (1991); Cheng  _et al._
(2004); Thomas (2006); Colangelo  _et al._ (2005). Following the same
principle, we calculate the $f_{D_{s0}^{*}}$ and $f_{D_{s1}}$ decay constants
in the molecular picture. Assuming that $D_{s0}^{*}$ is dynamically generated
by the $DK$ and $D_{s}\eta$ coupled-channel interactions, the current matrix
element $\left\langle D_{s0}^{*+}|(s\bar{c})|0\right\rangle$ is illustrated in
Fig. 2. Considering HQSS, we replace the above $D$ and $D_{s}$ mesons with the
$D^{*}$ and $D_{s}^{*}$ mesons, dynamically generating the $D_{s1}$. In the
following, we introduce the effective Lagrangian approach to calculate the
decay constants of molecules.

The effective Lagrangians for the $D^{(\ast)}(D_{s}^{(\ast)})$ mesons
transiting to the $K(\eta)$ mesons and $W$ boson are given by

$\displaystyle\mathcal{L}_{VDK}$ $\displaystyle=$ $\displaystyle
f_{1}^{DK}(0)V^{\mu}(D\partial_{\mu}K-\partial_{\mu}DK),~{}~{}~{}\mathcal{L}_{VD_{s}\eta}=f_{1}^{D_{s}\eta}(0)V^{\mu}(D_{s}\partial_{\mu}\eta-\partial_{\mu}D_{s}\eta)$
(12) $\displaystyle\mathcal{L}_{AD^{*}K}$ $\displaystyle=$
$\displaystyle(m_{D}+m_{K^{*}})A_{1}^{DK^{*}}(0)A^{\mu}D^{*}_{\mu}K,~{}~{}~{}\mathcal{L}_{AD_{s}^{*}\eta}=(m_{D_{s}}+m_{\phi})A_{1}^{D_{s}\phi}(0)A^{\mu}D^{*}_{s\mu}\eta$

where $f_{1}^{DK}(0)[f_{1}^{D_{s}\eta}(0)]$ and
$A_{1}^{DK^{*}}(0)[A_{1}^{D_{s}\phi}]$ are the form factors at $q^{2}=0$. Such
parameters can be determined by fitting the corresponding semileptonic
branching fractions. In this work, we take the following values:
$f_{0}^{DK}=0.74$ Cheng and Chiang (2010); Cheng  _et al._ (2016); Zhang  _et
al._ (2018), $A_{1}^{DK^{*}}=0.78$, $A_{1}^{D_{s}\phi}=0.77$ Chang  _et al._
(2019), $F_{+}^{DK}=0.77$, $F_{+}^{D_{s}\eta}=0.49$, $A_{0}^{DK^{*}}=2.08$,
$A_{0}^{D_{s}\phi}=2.13$ Ivanov  _et al._ (2019)

The effective Lagrangians describing the couplings of the hadronic molecules
to their constituents are written as

$\displaystyle\mathcal{L}_{D_{s0}^{*}DK}$ $\displaystyle=$ $\displaystyle
g_{D_{s0}^{*}DK}D_{s0}^{*}DK,~{}~{}~{}~{}\mathcal{L}_{D_{s0}^{*}D_{s}\eta}=g_{D_{s0}^{*}D_{s}\eta}D_{s0}^{*}D_{s}\eta$
(13) $\displaystyle\mathcal{L}_{D_{s1}D^{*}K}$ $\displaystyle=$ $\displaystyle
g_{D_{s1}D^{*}K}D_{s1}^{\mu}D^{*}_{\mu}K,~{}~{}\mathcal{L}_{D_{s1}D_{s}^{*}\eta}=g_{D_{s1}D_{s}^{*}\eta}D_{s1}^{\mu}D_{s\mu}^{*}K,$

where $g_{D_{s0}^{*}DK}$, $g_{D_{s0}^{*}D_{s}\eta}$, $g_{D_{s1}D^{*}K}$, and
$g_{D_{s1}D_{s}^{*}\eta}$ are the coupling constants between
$D_{s0}^{*}(D_{s1})$ and their constituents. In this work, we employ the
contact-range EFT to dynamically generate the $D_{s0}^{*}$ and $D_{s1}$ and
further determine the couplings between the molecular states and their
constituents from the residues of the corresponding poles, which are widely
applied to study hadronic molecules Xiao  _et al._ (2019); Du  _et al._
(2020); Pan  _et al._ (2023).

With the above preparations, we can write the amplitude of Fig. 2 as

$\displaystyle\mathcal{A}_{a}$ $\displaystyle=$ $\displaystyle
g_{D_{s0}^{*}DK}f_{1}^{DK}(0)\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{k_{1}^{2}-m_{1}^{2}}\frac{1}{k_{2}^{2}-m_{2}^{2}}(k_{1}^{\mu}-k_{2}^{\mu})\varepsilon_{\mu}(V),$
(14) $\displaystyle\mathcal{A}_{b}$ $\displaystyle=$ $\displaystyle
g_{D_{s0}^{*}D_{s}\eta}f_{1}^{D_{s}\eta}(0)\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{k_{1}^{2}-m_{1}^{2}}\frac{1}{k_{2}^{2}-m_{2}^{2}}(k_{1}^{\mu}-k_{2}^{\mu})\varepsilon_{\mu}(V),$

where the subscripts 1 and 2 denote the $D$ and $K$ mesons in amplitude
$\mathcal{A}_{a}$, and the $D_{s}$ and $\eta$ mesons in amplitude
$\mathcal{A}_{b}$. Similarly, we obtain the amplitudes describing the $D_{s1}$
created from the vacuum as

$\displaystyle\mathcal{A}_{a}$ $\displaystyle=$ $\displaystyle
g_{D_{s1}D^{*}K}(m_{D}+m_{K^{*}})A_{1}^{DK^{*}}(0)\int\frac{d^{4}q}{(2\pi)^{4}}\varepsilon_{\mu}(k_{0})\frac{-g^{\mu\nu}+\frac{k_{1}^{\mu}k_{1}^{\nu}}{m_{1}^{2}}}{k_{1}^{2}-m_{1}^{2}}\frac{1}{k_{2}^{2}-m_{2}^{2}}\varepsilon_{\nu}(A),$
(15) $\displaystyle\mathcal{A}_{b}$ $\displaystyle=$ $\displaystyle
g_{D_{s1}D_{s}^{*}\eta}(m_{D_{s}}+m_{\phi})A_{1}^{D_{s}\phi}(0)\int\frac{d^{4}q}{(2\pi)^{4}}\varepsilon_{\mu}(k_{0})\frac{-g^{\mu\nu}+\frac{k_{1}^{\mu}k_{1}^{\nu}}{m_{1}^{2}}}{k_{1}^{2}-m_{1}^{2}}\frac{1}{k_{2}^{2}-m_{2}^{2}}\varepsilon_{\nu}(A).$

Once the amplitudes of Fig. 2 are obtained, the decay constants
$f_{D_{s0}^{*}}$ and $f_{D_{s1}}$ can be easily extracted considering their
definitions. In the following, we show how to calculate the relevant loop
functions in the dimensional regularisation scheme.

With the Feynman parameter approach, we obtain the following integrals

$\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{1}{(k_{1}^{2}-m_{1}^{2})}$
$\displaystyle=$
$\displaystyle\frac{m_{1}^{2}}{16\pi^{2}}(ln\frac{m_{1}^{2}}{\mu^{2}}-1),$
(16)
$\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{1}{(k_{1}^{2}-m_{1}^{2})[(p-k_{1})^{2}-m_{2}^{2}]}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\int_{0}^{1}dxln\frac{\Delta^{2}}{\mu^{2}},$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}}{(k_{1}^{2}-m_{1}^{2})[(p-k_{1})^{2}-m_{2}^{2}]}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\int_{0}^{1}dx[\Delta^{2}(2(ln\frac{\Delta^{2}}{\mu^{2}}-1)+1)+p^{2}x^{2}ln\frac{\Delta^{2}}{\mu^{2}}],$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{1}^{\mu}k_{1}^{\nu}}{(k_{1}^{2}-m_{1}^{2})[(p-k_{1})^{2}-m_{2}^{2}]}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\int_{0}^{1}dx[g^{\mu\nu}\frac{\Delta^{2}}{2}\,(ln\frac{\Delta^{2}}{\mu^{2}}-1)+p^{\mu}p^{\nu}x^{2}\,ln\frac{\Delta^{2}}{\mu^{2}}],$

where $\Delta^{2}=p^{2}x^{2}-m_{1}^{2}(x-1)-x(p^{2}-m_{2}^{2})$,
$p=k_{1}+k_{2}$, and the renormalization scale $\mu$ depends on the specific
physical process under consideration. To extract the decay constants of
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$, the loop functions of Eq. (14) and Eq.
(15) are converted into the following form

$\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{k_{1}^{\mu}-k_{2}^{\mu}}{(k_{1}^{2}-m_{1}^{2})[(p-k_{1})^{2}-m_{2}^{2}]}=\frac{p^{\mu}}{16\pi^{2}}\int_{0}^{1}dx(2x-1)ln\frac{\Delta^{2}}{\mu^{2}},$
(17)
$\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{-g^{\mu\nu}+\frac{k_{1}^{\mu}k_{1}^{\nu}}{m_{1}^{2}}}{(k_{1}^{2}-m_{1}^{2})[(p-k_{1})^{2}-m_{2}^{2}]}=-\frac{1}{16\pi^{2}}\int_{0}^{1}dx\\{g^{\mu\nu}\left[ln\frac{\Delta^{2}}{\mu^{2}}-\frac{1}{2m_{1}^{2}}\Delta^{2}(ln\frac{\Delta^{2}}{\mu^{2}}-1)\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{1}{m_{1}^{2}}p^{\mu}p^{\nu}x^{2}ln\frac{\Delta^{2}}{\mu^{2}}\\}.$

Finally, we obtain the analytic form of the decay constants of
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$

$\displaystyle f_{D_{s0}^{*}}^{m_{1}m_{2}}$ $\displaystyle=$ $\displaystyle
g_{D_{s0}^{*}m_{1}m_{2}}f_{1}^{m_{1}m_{2}}(0)\frac{1}{16\pi^{2}}\int_{0}^{1}dx(2x-1)ln\frac{\Delta^{2}}{\mu^{2}},$
(18) $\displaystyle f_{D_{s1}}^{m_{1}m_{2}}$ $\displaystyle=$
$\displaystyle\frac{g_{D_{s1}m_{1}m_{2}}(m_{1}+m_{2})A_{1}^{m_{1}m_{2}}(0)}{m_{D_{s1}}}\frac{1}{16\pi^{2}}\int_{0}^{1}dx~{}\left[\frac{1}{2m_{1}^{2}}\Delta^{2}(ln\frac{\Delta^{2}}{\mu^{2}}-1)-ln\frac{\Delta^{2}}{\mu^{2}}\right],$
(19)

where $m_{1}$ and $m_{2}$ refer to $D(D_{s})$ and $K(\eta)$ for
$D_{s0}^{*}(2317)$ and $D^{*}(D_{s}^{*})$ and $K(\eta)$ for $D_{s1}(2460)$.
The decay constants of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ are calculated as
the sum of $f_{D_{s0}^{*}}^{DK}$ and $f_{D_{s0}^{*}}^{D_{s}\eta}$ and the sum
of $f_{D_{s1}}^{D^{*}K}$ and $f_{D_{s1}}^{D_{s}^{*}\eta}$.

###  II.3 Contact-range effective field theory approach

In the following, we briefly introduce the contact-range effective field
theory (EFT) approach. The scattering amplitude $T$ is responsible for the
dynamical generations of molecules, which is obtained by solving the following
Lippmann-Schwinger equation

$\displaystyle T(\sqrt{s})=(1-VG(\sqrt{s}))^{-1}V,$ (20)

where $V$ is the coupled-channel potential determined by the contact-range EFT
approach, and $G(\sqrt{s})$ is the loop function of the two-body propagator.

The coupled-channel potential $V$ in matrix form reads

$V_{DK-D_{s}\eta}^{J^{P}=0^{+}}=\begin{pmatrix}-2C_{a}&\sqrt{3}C_{a}\\\
\sqrt{3}C_{a}&0\end{pmatrix},\quad
V_{D^{*}K-D_{s}^{*}\eta}^{J^{P}=1^{+}}=\varepsilon(k_{1})\cdot\varepsilon(k_{1}^{\prime})\begin{pmatrix}-2C_{a}&\sqrt{3}C_{a}\\\
\sqrt{3}C_{a}&0\end{pmatrix},$ (21)

where the coefficient $C_{a}$ needs to be determined by fitting the
$D_{s0}^{*}$ and $D_{s1}$ masses. The loop functions of $D_{s0}^{*}$ and
$D_{s1}$ are

$\displaystyle G(\sqrt{s})^{D_{s0}^{*}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\int_{0}^{1}dxln\frac{\Delta^{2}}{\mu^{2}},$
(22) $\displaystyle G(\sqrt{s})^{D_{s1}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\int_{0}^{1}dx~{}\left[ln\frac{\Delta^{2}}{\mu^{2}}-\frac{1}{2m_{1}^{2}}\Delta^{2}(ln\frac{\Delta^{2}}{\mu^{2}}-1)\right]$

with $\Delta^{2}=sx^{2}-m_{1}^{2}(x-1)-x(s-m_{2}^{2})$. We note that the loop
function of $D_{s1}$ contains an additional term, which is induced by the
$\frac{k_{1}^{\mu}k_{1}^{\nu}}{m_{1}^{2}}$ term in the loop integral. One can
see that the loop integrals depend on the renormalization scale $\mu$.

With the potentials obtained above, we can search for poles generated by the
coupled-channel interactions and determine the couplings between the molecular
states and their constituents from the residues of the corresponding poles,

$\displaystyle
g_{i}g_{j}=\lim_{\sqrt{s}\to\sqrt{s_{0}}}\left(\sqrt{s}-\sqrt{s_{0}}\right)T_{ij}(\sqrt{s}),$
(23)

where $g_{i}$ denotes the coupling of channel $i$ to the dynamically generated
state and $\sqrt{s_{0}}$ is the pole position.

##  III Results and Discussions

Table 1: Masses and quantum numbers of relevant hadrons needed in this work Workman  _et al._ (2022).  Hadron | $I(J^{P})$ | M (MeV) | Hadron | $I(J^{P})$ | M (MeV) | Hadron | $I(J^{P})$ | M (MeV)
---|---|---|---|---|---|---|---|---
$K^{\pm}$ | $1/2(0^{-})$ | $493.677$ | $K^{0}$ | $1/2(0^{-})$ | $497.611$ | $\Lambda_{b}$ | $0(1/2^{+})$ | $5619.60$
$\bar{D}^{0}$ | $1/2(0^{-})$ | $1864.84$ | $D^{-}$ | $1/2(0^{-})$ | $1869.66$ | $\Lambda_{c}^{+}$ | $0(1/2^{+})$ | $2286.46$
$\bar{D}^{\ast 0}$ | $1/2(1^{-})$ | $2006.85$ | $D^{\ast-}$ | $1/2(1^{-})$ | $2010.26$ | $B^{+}$ | $1/2(0^{-})$ | $5279.34$
$J/\psi$ | $0(1^{-})$ | $3096.90$ | $\eta_{c}$ | $0(0^{-})$ | $2983.90$ | $B^{0}$ | $1/2(0^{-})$ | $5279.65$
$D_{s}^{\pm}$ | $0(0^{-})$ | $1968.35$ | $D_{s}^{\ast\pm}$ | $0(1^{-})$ | $2112.2$ | $B_{s}^{0}$ | $0(0^{-})$ | $5366.91$

In Table 1, we tabulate the masses and quantum numbers of relevant particles.
One can see that there exists an unknown parameter $\mu$ (renormalization
scale) in both Eq. (17) and Eq. (22), for which a consistent value is adopted
in this work. First, we employ the contact-range EFT approach to dynamically
generate the poles corresponding to $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ by
varying $\mu$ and then obtain the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$
couplings to their constituents as well as their decay constants. With the so-
obtained decay constants $f_{D_{s0^{*}}(2317)}$ and $f_{D_{s1}(2460)}$ we
further study the productions of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ in the
$B_{(s)}$ and $\Lambda_{b}(\Xi_{b})$ decays, where the naive factorization
approach works well as mentioned above. In this work, assuming that the decay
mechanisms of $H_{b}\to H_{c}D_{s}^{(\ast)}$($H_{b}$ and $H_{c}$ denote bottom
and charm hadrons of interest) and $H_{b}\to H_{c}D_{s0}^{*}(D_{s1})$ are the
same, we parameterize the unknown non-factorization contributions with the
effective Wilson coefficients. In other words, we determine $a_{1}$ by
reproducing the experimental branching fractions of $H_{b}\to
H_{c}D_{s}^{(\ast)}$ decays, and then calculate the branching fractions of the
corresponding decays $H_{b}\to H_{c}D_{s0}^{*}(D_{s1})$ using the so-obtained
$a_{1}$.

Table 2: $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ couplings to their constituents (in units of MeV).  Couplings | $\mu=1000$ | $\mu=1500$ | $\mu=2000$ | Fu et al. Fu  _et al._ (2022)
---|---|---|---|---
$g_{D_{s0}^{*}DK}$ | 11.75 | 11.92 | 11.95 | 9.4
$g_{D_{s0}^{*}D_{s}\eta}$ | 8.13 | 7.47 | 7.32 | 7.4
$g_{D_{s1}D^{*}K}$ | 12.06 | 12.16 | 12.15 | 10.1
$g_{D_{s1}D_{s}^{*}\eta}$ | 8.78 | 7.76 | 7.53 | 7.9

In Refs. Gamermann  _et al._ (2007); Molina  _et al._ (2010); Chen  _et al._
(2023), the loop function is regularised in the dimensional regularization
scheme, which shows that $\mu$ is around $1.5$ GeV in the charm sector. To
quantify the uncertainty of the renormalization scale, we vary $\mu$ from $1$
GeV to $2$ GeV in this work. For $\mu$ of $1.0$ GeV, $1.5$ GeV, and $2.0$ GeV,
the values of $C_{a}$ for $DK-D_{s}\eta$ ($D^{*}K-D_{s}^{*}\eta$) contact-
range potentials are determined as $74.78$, $34.53$, and $25.04$ ($98.60$,
$42.45$, and $30.34$). With the so-obtained scattering amplitude $T$, we
obtain the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ couplings to their
constituents shown in Table 2, a bit different from the estimations of Ref. Fu
_et al._ (2022)111 One should note that an additional parameter, e.g.,
subtraction constant, is introduced in Ref. Fu  _et al._ (2022). . Finally,
the decays constants of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ are determined
as shown in Table 3. One can see that $f_{D_{s0}^{*}(2317)}$ is almost
independent of the renormalization scale, as can also seen from Eq. (18). The
slight variation of $f_{D_{s0}^{*}(2317)}$ stems from the weak dependence of
the couplings $g_{D_{s0}^{*}DK}$ and $g_{D_{s0}^{*}D_{s}\eta}$ on the
renormalization scale $\mu$. However, the decay constant $f_{D_{s1}(2460)}$ is
dependent on $\mu$ as shown in Table 3. In the following calculations, we
adopt the values of $f_{D_{s0^{*}}(2317)}$ and $f_{D_{s1}(2460)}$ at $\mu=1.5$
GeV, e.g., $f_{D_{s0}^{*}}=58.74$ GeV and $f_{D_{s1}}=133.76$ GeV, which are
consistent with the results of Ref. Faessler  _et al._ (2007c), but smaller
than the results of lattice QCD Bali  _et al._ (2017).

Table 3: Decay constants of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ as hadronic molecules (in units of MeV).  Decay Constants | $\mu=1000$ | $\mu=1500$ | $\mu=2000$ | Faessler et al. Faessler  _et al._ (2007c)
---|---|---|---|---
$f_{D_{s0}^{*}(2317)}$ | 59.36 | 58.74 | 58.59 | 67.1
$f_{D_{s1}(2460)}$ | 56.10 | 133.76 | 187.48 | 144.5
Table 4: Values of $F(0)^{B\to D^{(*)}}$, $a^{B\to D^{(*)}}$, $b^{B\to D^{(*)}}$ in the $B\rightarrow D^{(*)}$ transition form factors and $F(0)^{B_{s}\to D_{s}^{(*)}}$, $a^{B_{s}\to D_{s}^{(*)}}$, $b^{B_{s}\to D_{s}^{(*)}}$ in the ${B_{s}\to D_{s}^{(*)}}$ transition form factors Verma (2012).  | $F_{0}$ | $F_{1}$ | $V$ | $A_{0}$ | $A_{1}$ | $A_{2}$ |  | $F_{0}$ | $F_{1}$ | $V$ | $A_{0}$ | $A_{1}$ | $A_{2}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$F(0)^{B\to D^{(*)}}$ | 0.67 | 0.67 | 0.77 | 0.68 | 0.65 | 0.61 | $F(0)^{B_{s}\to D_{s}^{(*)}}$ | 0.67 | 0.67 | 0.75 | 0.66 | 0.62 | 0.57
$a^{B\to D^{(*)}}$ | 0.63 | 1.22 | 1.25 | 1.21 | 0.60 | 1.12 | $a^{B_{s}\to D_{s}^{(*)}}$ | 0.69 | 1.28 | 1.37 | 1.33 | 0.76 | 1.25
$b^{B\to D^{(*)}}$ | -0.01 | 0.36 | 0.38 | 0.36 | 0.00 | 0.31 | $b^{B_{s}\to D_{s}^{(*)}}$ | 0.07 | 0.52 | 0.67 | 0.63 | 0.13 | 0.56

Up to now, $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ have only been observed in
$B$ decays. Therefore, we first focus on the decays of
$B\to\bar{D}^{(\ast)}D_{s}^{(\ast)}$ and
$B\to\bar{D}^{(\ast)}D_{s0}^{*}(D_{s1})$. In Table 4, we present the
parameters of the form factors in the $B\to D$ and $B\to D^{*}$ transitions,
which are taken from Ref. Verma (2012). By reproducing the experimental
branching fractions of the decays $B\to\bar{D}^{(\ast)}D_{s}^{(\ast)}$, we
determine the effective Wilson coefficient $a_{1}$. Using $a_{1}$ and
$f_{D_{s0}^{*}}(f_{D_{s}})$ obtained above, we calculate the branching
fractions of the decays $B\to\bar{D}^{(\ast)}D_{s0}(D_{s1})$ , which are shown
in Table 5. Our results are a bit smaller than those of Ref. Faessler  _et
al._ (2007c) because of the smaller values for the decay constants and the
effective Wilson coefficient. Interestingly, our results are consistent with
our previous calculations using the triangle mechanism for the decay
$B^{+}\to\bar{D}^{\ast 0}D_{s0}^{*+}$ Liu  _et al._ (2022) 222We note that in
the triangle diagram, the relative phase between the $D^{(*)}$ exchange and
the $\eta$ exchange is fixed in such a way that they add constructively, which
produces results in better agreement with data. This may not hold in the decay
of $B^{+}\to\bar{D}^{\ast 0}D_{s0}^{*+}$. , which indicates that the triangle
diagram and tree diagram accounting for the decays
$B\to\bar{D}^{(\ast)}D_{s0}(D_{s1})$ are equivalent. In principle, one can
replace the triangle diagram with one vertex, resulting in an effective
description for the weak decay $B^{+}\to\bar{D}^{0}D_{s0}^{*+}(2317)$ at tree
level as shown in Fig. 3, which indicates that it is reasonable to extract the
decay constants of hadronic molecules using the triangle mechanism. In Ref. Wu
_et al._ (2023), with this approach, we extract the decay constants of
$X(3872)$ as a $\bar{D}D^{*}$ molecule. Here, we note that the relative phase
among various amplitudes may lead to uncertainties in extracting the decay
constants. As a result, it is better to select relevant amplitudes with no or
small relative phases.

Table 5: Branching fractions of ($10^{-3}$) of $B\to\bar{D}^{(\ast)}D_{s0}(D_{s1})$.  Decay modes |  Exp Workman  _et al._ (2022) |  $a_{1}$ | Decay modes |  $\mathrm{Ours}$ |  Triangle Liu  _et al._ (2022) |  Exp Workman  _et al._ (2022)
---|---|---|---|---|---|---
$B^{+}\to\bar{D}^{0}D_{s}^{+}$ |  $9.0\pm 0.9$ |  0.80 | $B^{+}\to\bar{D}^{0}D_{s0}^{*+}(2317)$ |  $0.48$ |  $0.68$ |  $0.80^{+0.16}_{-0.13}$
$B^{+}\to\bar{D}^{\ast 0}D_{s}^{+}$ |  $8.2\pm 1.7$ |  0.93 | $B^{+}\to\bar{D}^{\ast 0}D_{s0}(2317)$ |  $0.39$ |  $1.21$ |  $0.90^{+0.70}_{-0.70}$
$B^{+}\to\bar{D}^{0}D_{s}^{\ast+}$ |  $7.6\pm 1.6$ |  0.81 | $B^{+}\to\bar{D}^{0}D_{s1}(2460)$ |  $1.39$ |  $1.26$ |  $3.1^{+1.0}_{-0.9}$
$B^{+}\to\bar{D}^{\ast 0}D_{s}^{\ast+}$ |  $17.1\pm 2.4$ |  0.83 | $B^{+}\to\bar{D}^{\ast 0}D_{s1}(2460)$ |  $4.36$ |  $3.07$ |  $12.0\pm 3.0$
Figure 3:  Equivalence of the triangle diagram and the tree diagram depict the
decay of $B^{+}\to\bar{D}^{0}D_{s0}^{*+}(2317)$.

Along this line, we investigate the decays of
$B_{s}\to\bar{D}_{s}^{(*)}D_{s}^{(*)}$ and
$B_{s}\to\bar{D}_{s0}^{*}(\bar{D}_{s1})D_{s}^{(*)}$, which are related to the
decays of $B\to\bar{D}^{(\ast)}D_{s}^{(\ast)}$ and
$B\to\bar{D}^{(\ast)}D_{s0}^{*}(D_{s1})$ via SU(3)-flavor symmetry as shown in
Fig. 1. The amplitudes for the decays of
$B_{s}\to\bar{D}_{s}^{(*)}D_{s}^{(*)}$ and
$B_{s}\to\bar{D}_{s0}^{*}(\bar{D}_{s1})D_{s}^{(*)}$ are the same as those of
their SU(3) symmetric partners. The unknown parameters in the form factors of
the $B_{s}\to D_{s}^{(*)}$ transitions are taken from Ref. Verma (2012),
tabulated in Table 4. Following the same strategy, we calculate the branching
fractions of the decays $B_{s}\to\bar{D}_{s}^{(\ast)}D_{s0}^{*}$ and
$B_{s}\to\bar{D}_{s}^{(\ast)}D_{s1}$. The results are shown in Table 6. One
can see that the branching fractions of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$
in the $B_{s}$ decays are similar to those in the $B$ decays, following the
SU(3)-flavor symmetry. Such large production rates means that the
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ are likely to be detected in future
experiments.

Table 6: Branching ratios ($10^{-3}$) of $B\to\bar{D}_{s}^{(\ast)}D_{s0}(D_{s1})$.  decay modes |  Exp Workman  _et al._ (2022) |  $a_{1}$ | decay modes |  $\mathrm{Ours}$
---|---|---|---|---
$B_{s}^{0}\to D_{s}^{+}D_{s}^{-}$ |  $4.4\pm 0.5$ |  0.87 | $B_{s}^{0}\to D_{s}D_{s0}^{\ast}(2317)$ |  $0.47$
$B_{s}^{0}\to D_{s}^{\ast+}D_{s}^{-}+D_{s}^{+}D_{s}^{\ast-}$ |  $13.9\pm 1.7$ |  0.83 | $B_{s}^{0}\to D_{s}^{*}D_{s0}(2317)$ |  $0.27$
$B_{s}^{0}\to D_{s}^{\ast+}D_{s}^{-}+D_{s}^{+}D_{s}^{\ast-}$ |  $13.9\pm 1.7$ |  0.77 | $B_{s}^{0}\to{D}_{s}D_{s1}(2460)$ |  $1.18$
$B_{s}^{0}\to D_{s}^{\ast+}D_{s}^{\ast-}$ |  $14.4\pm 2.1$ |  0.84 | $B_{s}^{0}\to D_{s}^{\ast}D_{s1}(2460)$ |  $4.11$

In addition to the productions of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ in the
$B_{(s)}$ decays, it is interesting to investigate their productions in the
$\Lambda_{b}$($\Xi_{b}$) decays. As indicated in Fig. 1, the decays of
$\Lambda_{b}\to D_{s}^{(\ast)}\Lambda_{c}$ and $\Lambda_{b}\to
D_{s0}^{*}(D_{s1})\Lambda_{c}$ share the same mechanism as those of $B\to
D_{s}^{(\ast)}\bar{D}^{(*)}$ and $B\to D_{s0}^{*}(D_{s1})\bar{D}^{(*)}$ at
quark level, which proceeds via the decays $b\to c\bar{c}s$. In terms of
SU(3)-flavor symmetry, we also investigate the decays of $\Xi_{b}\to
D_{s}^{(\ast)}\Xi_{c}$ and $\Xi_{b}\to D_{s0}^{*}(D_{s1})\Xi_{c}$. With the
naive factorization approach, the amplitudes for these decays are given by the
effective Lagrangian shown in Eq. (9), where the parameters in the form
factors of the $\Lambda_{b}\to\Lambda_{c}$ and $\Xi_{b}\to\Xi_{c}$ transitions
are obtained in the quark model Gutsche  _et al._ (2015); Faustov and Galkin
(2018); Lu  _et al._ (2021), tabulated in Table 7. The decay constants of the
charmed-strange mesons $D_{s}^{(*)}$ and $D_{s0}^{*}(D_{s1})$ are calculated
in the same way as explained above.

Table 7: Values of $F(0)$, $a$, $b$ in the $\Lambda_{b}\rightarrow\Lambda_{c}$ and $\Xi_{b}\rightarrow\Xi_{c}$ transition form factors Gutsche  _et al._ (2015); Faustov and Galkin (2018).  | $F_{1}^{V}$ | $F_{2}^{V}$ | $F_{3}^{V}$ | $F_{1}^{A}$ | $F_{2}^{A}$ | $F_{3}^{A}$
---|---|---|---|---|---|---
$F(0)^{\Lambda_{b}\to\Lambda_{c}}$ | 0.549 | 0.110 | $-0.023$ | 0.542 | 0.018 | $-0.123$
$a^{\Lambda_{b}\to\Lambda_{c}}$ | 1.459 | 1.680 | 1.181 | 1.443 | 0.921 | 1.714
$b^{\Lambda_{b}\to\Lambda_{c}}$ | 0.571 | 0.794 | 0.276 | 0.559 | 0.255 | 0.828
$F(0)^{\Xi_{b}\to\Xi_{c}}$ | 0.467 | 0.145 | $0.086$ | 0.447 | $-0.035$ | $-0.278$
$a^{\Xi_{b}\to\Xi_{c}}$ | 1.702 | 2.530 | 1.742 | 1.759 | 2.675 | 2.270
$b^{\Xi_{b}\to\Xi_{c}}$ | 0.531 | 1.581 | 0.758 | 0.356 | 1.789 | 1.072
Table 8: Branching fractions ($10^{-3}$) of $\Lambda_{b}\to\Lambda_{c}D_{s0}^{*}(D_{s1})$ and $\Xi_{b}\to\Xi_{c}D_{s0}^{*}(D_{s1})$.  decay modes |  Exp Workman  _et al._ (2022); Aaij  _et al._ (2023) |  $a_{1}$ | decay modes |  Ours
---|---|---|---|---
$\Lambda_{b}\to\Lambda_{c}D_{s}^{-}$ |  $11\pm 1.0$ |  $0.88$ |  $\Lambda_{b}\to\Lambda_{c}D_{s0}^{*}(2317)$ |  $0.70$
$\Lambda_{b}\to\Lambda_{c}D_{s}^{\ast-}$ |  $18.568\pm 1.102$ |  $0.76$ |  $\Lambda_{b}\to\Lambda_{c}D_{s1}(2460)$ |  $4.34$
decay modes |  Ours |  $a_{1}$ | decay modes |  Ours
$\Xi_{b}\to\Xi_{c}D_{s}^{-}$ |  $8.52$ |  $0.88$ |  $\Xi_{b}\to\Xi_{c}D_{s0}^{*}(2317)$ |  $0.58$
$\Xi_{b}\to\Xi_{c}D_{s}^{\ast-}$ |  $16.30$ |  $0.76$ |  $\Xi_{b}\to\Xi_{c}D_{s1}(2460)$ |  $4.29$

One should note that only the branching fraction of the decay $\Lambda_{b}\to
D_{s}\Lambda_{c}$ is available in the RPP. Very recently the ratio of
$\mathcal{B}(\Lambda_{b}\to D_{s}^{*}\Lambda_{c})/\mathcal{B}(\Lambda_{b}\to
D_{s}\Lambda_{c})=1.688\pm 0.022^{+0.061}_{-0.055}$ is reported by the LHCb
Collaboration Aaij  _et al._ (2023), and one can obtain the branching fraction
of the decay $\Lambda_{b}\to D_{s}^{*}\Lambda_{c}$. With the branching
fractions of $\mathcal{B}(\Lambda_{b}\to D_{s}\Lambda_{c})$ and
$\mathcal{B}(\Lambda_{b}\to D_{s}^{*}\Lambda_{c})$ as inputs, we determine
$a_{1}$, and then predict the branching fractions of the decays of
$\Lambda_{b}\to D_{s0}^{*}\Lambda_{c}$ and $\Lambda_{b}\to D_{s1}\Lambda_{c}$,
which are shown in Table 8. We can see that the production rates of
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ in the $\Lambda_{b}$ decay are of the
order of $10^{-3}$, which are large enough to be detected in future
experiments. Since the effective Wilson coefficients in the $B$ decays and
$B_{s}$ decays are similar, as shown in Table 5 and Table 6, we can take the
same values for $a_{1}$ in the $\Lambda_{b}$ decays and the $\Xi_{b}$ decays.
Similarly, we predict the branching fractions of the decays $\Xi_{b}\to
D_{s}^{(\ast)}\Xi_{c}$ and $\Xi_{b}\to D_{s0}^{*}(D_{s1})\Xi_{c}$ in Table 8.
The production rates of ground-state mesons $D_{s}^{(*)}$ and excited mesons
$D_{s0}^{*}(D_{s1})$ in the $\Xi_{b}$ decays are of the order of $10^{-2}$ and
$10^{-3}$,, which are likely to be detected in future experiments. In Ref.
Datta  _et al._ (2004), the authors estimated the ratios
$\mathcal{B}(\Lambda_{b}\to\Lambda_{c}M)/\mathcal{B}(B\to DM)$, where $M$
represents the ground-state $D_{s}^{(*)}$ and excited $D_{s0}^{*}(D_{s1})$
mesons. In Table 9, we show the ratios
$R_{u}^{M}=\mathcal{B}(\Lambda_{b}\to\Lambda_{c}M)/\mathcal{B}(B\to DM)$ and
$R_{s}^{M}=\mathcal{B}(\Xi_{b}\to\Xi_{c}M)/\mathcal{B}(B_{s}\to D_{s}M)$
obtained in this work, which are consistent with Ref. Datta  _et al._ (2004).
We can see that the production rates of ground-state mesons $D_{s}^{(*)}$ and
excited mesons $D_{s0}^{*}(D_{s1})$ in the $\Lambda_{b}(\Xi_{b})$ decays are
larger than those in the $B_{(s)}$ decays because the
$\Lambda_{b}\to\Lambda_{c}$($\Xi_{b}\to\Xi_{c}$) form factors are larger than
the corresponding $B_{(s)}\to D_{(s)}$ form factor as shown in Ref. Datta  _et
al._ (2004).

Table 9:  Ratios of $R_{u}^{M}=\mathcal{B}(\Lambda_{b}\to\Lambda_{c}M)/\mathcal{B}(B\to DM)$ and $R_{s}^{M}=\mathcal{B}(\Xi_{b}\to\Xi_{c}M)/\mathcal{B}(B_{s}\to D_{s}M)$.  $R_{u}^{M}$ |  Ours |  Ref. Datta  _et al._ (2004) | $R_{s}^{M}$ |  Ours
---|---|---|---|---
$R_{u}^{D_{s}}$ |  $1.22$ |  $1.75$ | $R_{s}^{D_{s}}$ |  1.94
$R_{u}^{D_{s}^{\ast}}$ |  $2.44$ |  $3.47$ | $R_{s}^{D_{s}^{\ast}}$ |  $2.35$
$R_{u}^{D_{s0}^{\ast}}$ |  $1.46$ |  $1.58$ | $R_{s}^{D_{s0}^{\ast}}$ |  $1.23$
$R_{u}^{D_{s1}}$ |  $3.12$ |  $4.76$ | $R_{s}^{D_{s1}}$ |  $3.64$

##  IV Summary and outlook

In this work, we utilized the effective Lagrangian approach to compute the
decay constants of $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ as hadronic
molecules dynamically generated by the $DK-D_{s}\eta$ and
$D^{*}K-D_{s}^{*}\eta$ contact-range potentials, and then with the naive
factorization approach systematically investigated the productions of
$D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ in the $B_{(s)}$ and
$\Lambda_{b}(\Xi_{b})$ decays, which proceed via the decay $b\to c\bar{c}s$ at
quark level. The decay constants of $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$
are estimated to be $58.74$ MeV and $133.76$ MeV. In particular, the decay
constant of $D_{s0}^{\ast}(2317)$ is almost independent of the renormalization
scale $\mu$ in the loop functions.

As for the branching fractions of the decays
$B\to\bar{D}^{(*)}D_{s0}^{\ast}(2317)$ and $B\to\bar{D}^{(*)}D_{s1}(2460)$,
our results are smaller than the experimental data, but are consistent with
our previous results obtained in the triangle mechanism, which indicates that
$D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ may contain components other than
hadronic molecules, such as a $c\bar{s}$ core. The values of
$\mathcal{B}[B_{s}\to{D}_{s}^{(*)}D_{s0}^{\ast}(2317)]$ and
$\mathcal{B}[B_{s}\to{D}_{s}^{(*)}D_{s1}(2460)]$ are similar to those of
$\mathcal{B}[B\to\bar{D}^{(*)}D_{s0}^{\ast}(2317)]$ and
$\mathcal{B}[B\to\bar{D}^{(*)}D_{s1}(2460)]$, which reflects the underlying
SU(3)-flavor symmetry. In addition, we predicted the branching fractions of
the decays $\Lambda_{b}\to\Lambda_{c}D_{s0}^{\ast}(2317)$ and
$\Lambda_{b}\to\Lambda_{c}D_{s1}(2460)$ as well as
$\Xi_{b}\to\Xi_{c}D_{s0}^{\ast}(2317)$ and $\Xi_{b}\to\Xi_{c}D_{s1}(2460)$,
which are much larger than the corresponding ones in the $B$ and $B_{s}$
decays and indicates that the productions of $D_{s0}^{\ast}(2317)$ and
$D_{s1}(2460)$ in the decays of bottom baryons are likely to be detected in
future experiments.

Our study shows that, because of the equivalence of the triangle mechanism to
the tree diagram established in calculating the branching fractions of the
decays $B\to\bar{D}^{(*)}D_{s0}^{\ast}(2317)$ and
$B\to\bar{D}^{(*)}D_{s1}(2460)$, one can extract the decay constants of
$D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ as hadronic molecules via the
triangle mechanism. This provides an effective approach to calculating the
decay constants of hadronic molecules, which can then be used in studies of
these hadronic molecules in other related processes. We hope that our present
work can stimulate more studies along this line.

##  V Acknowledgments

We are grateful to Prof. Fu-Sheng Yu for stimulating discussions. This work is
supported in part by the National Natural Science Foundation of China under
Grants No.11975041 and No.11961141004. M.Z.L acknowledges support from the
National Natural Science Foundation of China under Grant No.12105007. X.Z.L
acknowledges support from the National Natural Science Foundation of China
under Grant No. 12247159 and China Postdoctoral Science Foundation under Grant
No. 2022M723149.

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